Transport Processes and Separation Technologies [1st ed.] 9783030478551, 9783030478568

Table of contents :
Front Matter ....Pages i-vii
Clay Ceramic Materials: From Fundamentals and Manufacturing to Drying Process Predictions (A. G. Barbosa de Lima, J. M. P. Q. Delgado, L. P. C. Nascimento, E. S. de Lima, V. A. B. de Oliveira, A. M. V. Silva et al.)....Pages 1-29
Vegetable Fiber Drying: Theory, Advanced Modeling and Application (J. F. Brito Diniz, A. R. C. de Lima, I. R. de Oliveira, R. P. de Farias, F. A. Batista, A. G. Barbosa de Lima et al.)....Pages 31-60
Foam-Mat Drying Process: Theory and Applications (E. R. Mangueira, A. G. Barbosa de Lima, J. de Assis Cavalcante, N. A. Costa, C. C. de Souza, A. K. F. de Abreu et al.)....Pages 61-87
Drying Process of Jackfruit Seeds (T. M. Q. de Oliveira, R. A. de Medeiros, V. S. O. Farias, W. P. da Silva, C. M. R. Franco, A. F. da Silva Júnior)....Pages 89-103
Spouted Bed Drying of Fruit Pulps: A Case Study on Drying of Graviola (Annona muricata) Pulp (F. G. M. de Medeiros, I. P. Machado, T. N. P. Dantas, S. C. M. Dantas, O. L. S. de Alsina, M. F. D. de Medeiros)....Pages 105-150
Osmo-convective Dehydration of Fresh Foods: Theory and Applications to Cassava Cubes (T. R. Bezerra Pessoa, A. G. Barbosa de Lima, P. C. Martins, V. C. Pereira, T. C. O. Alves, E. S. da Silva et al.)....Pages 151-183
Heat Transfer in a Packed-Bed Elliptic Cylindrical Reactor: Theory, Heterogeneous Transient Modeling, and Applications (A. S. Pereira, R. M. da Silva, R. S. Santos, A. G. Barbosa de Lima, R. O. de Andrade, W. M. P. B. de Lima et al.)....Pages 185-214

Citation preview

Advanced Structured Materials

J. M. P. Q. Delgado A. G. Barbosa de Lima   Editors

Transport Processes and Separation Technologies

Advanced Structured Materials Volume 133

Series Editors Andreas Öchsner, Faculty of Mechanical Engineering, Esslingen University of Applied Sciences, Esslingen, Germany Lucas F. M. da Silva, Department of Mechanical Engineering, Faculty of Engineering, University of Porto, Porto, Portugal Holm Altenbach , Faculty of Mechanical Engineering, Otto von Guericke University Magdeburg, Magdeburg, Sachsen-Anhalt, Germany

Common engineering materials reach in many applications their limits and new developments are required to fulfil increasing demands on engineering materials. The performance of materials can be increased by combining different materials to achieve better properties than a single constituent or by shaping the material or constituents in a specific structure. The interaction between material and structure may arise on different length scales, such as micro-, meso- or macroscale, and offers possible applications in quite diverse fields. This book series addresses the fundamental relationship between materials and their structure on the overall properties (e.g. mechanical, thermal, chemical or magnetic etc.) and applications. The topics of Advanced Structured Materials include but are not limited to • classical fibre-reinforced composites (e.g. glass, carbon or Aramid reinforced plastics) • metal matrix composites (MMCs) • micro porous composites • micro channel materials • multilayered materials • cellular materials (e.g., metallic or polymer foams, sponges, hollow sphere structures) • porous materials • truss structures • nanocomposite materials • biomaterials • nanoporous metals • concrete • coated materials • smart materials Advanced Structured Materials is indexed in Google Scholar and Scopus.

More information about this series at http://www.springer.com/series/8611

J. M. P. Q. Delgado A. G. Barbosa de Lima •

Editors

Transport Processes and Separation Technologies

123

Editors J. M. P. Q. Delgado CONSTRUCT-LFC, Department of Civil Engineering University of Porto Porto, Portugal

A. G. Barbosa de Lima Department of Mechanical Engineering Federal University of Campina Grande Campina Grande, Paraíba, Brazil

ISSN 1869-8433 ISSN 1869-8441 (electronic) Advanced Structured Materials ISBN 978-3-030-47855-1 ISBN 978-3-030-47856-8 (eBook) https://doi.org/10.1007/978-3-030-47856-8 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Contents

1 Clay Ceramic Materials: From Fundamentals and Manufacturing to Drying Process Predictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. G. Barbosa de Lima, J. M. P. Q. Delgado, L. P. C. Nascimento, E. S. de Lima, V. A. B. de Oliveira, A. M. V. Silva, and J. V. Silva 1.1 Ceramic Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 Fundamental Concepts . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.3 The Ceramic Industry and Clay Products . . . . . . . . . . . 1.1.4 Red Ceramic Product Manufacturing Process . . . . . . . . 1.2 The Drying Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 General Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 The Mathematical Modeling of the Drying Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Lumped Model Application: Drying of Clay Ceramic Brick . . . 1.3.1 The Experimental Data . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 Phenomenological Mathematical Modeling . . . . . . . . . . 1.3.3 Results Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Vegetable Fiber Drying: Theory, Advanced Modeling and Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . J. F. Brito Diniz, A. R. C. de Lima, I. R. de Oliveira, R. P. de Farias, F. A. Batista, A. G. Barbosa de Lima, and R. O. de Andrade 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Drying of Sisal Fibers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Experimental Study . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Theoretical Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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3 Foam-Mat Drying Process: Theory and Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E. R. Mangueira, A. G. Barbosa de Lima, J. de Assis Cavalcante, N. A. Costa, C. C. de Souza, A. K. F. de Abreu, and A. P. T. Rocha 3.1 Drying Theory of Porous Materials . . . . . . . . . . . . . . . . . . . . . 3.1.1 Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 Mathematical Modeling in Drying . . . . . . . . . . . . . . . . 3.2 Foam-Mat Drying . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 General Foundations . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Different Methods for Foam Formation . . . . . . . . . . . . . 3.2.3 Foaming Agents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Applications: Drying of Egg White and Yolk of Duck Egg . . . . 3.3.1 Material Preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Experimental Planning . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.3 Experiment of Foam-Mat Drying . . . . . . . . . . . . . . . . . 3.3.4 Analysis of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Final Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Drying Process of Jackfruit Seeds . . . . . . . . . . . . . . . . . . T. M. Q. de Oliveira, R. A. de Medeiros, V. S. O. Farias, W. P. da Silva, C. M. R. Franco, and A. F. da Silva Júnior 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Mathematical Modeling . . . . . . . . . . . . . . . . . 4.3 Results Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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5 Spouted Bed Drying of Fruit Pulps: A Case Study on Drying of Graviola (Annona muricata) Pulp . . . . . . . . . . . . . . . . . . . . . . . . F. G. M. de Medeiros, I. P. Machado, T. N. P. Dantas, S. C. M. Dantas, O. L. S. de Alsina, and M. F. D. de Medeiros 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Fundamentals of Spouted Bed Drying . . . . . . . . . . . . . . . . . . . . 5.3 Spouted Bed Drying of Fruit Pulps . . . . . . . . . . . . . . . . . . . . . . 5.4 Phytochemicals on Spouted Bed Dried Fruit Powders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 Impact of Spouted Bed Drying on the Phytochemicals Content . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.2 Use of Drying Carriers . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Spouted Bed Drying of Graviola (Annona muricata) Pulp: A Case Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.1 Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.2 Materials and Methods . . . . . . . . . . . . . . . . . . . . . . . . . .

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5.5.3 Results and Discussion 5.5.4 Final Comments . . . . . 5.6 Conclusions . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . .

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6 Osmo-convective Dehydration of Fresh Foods: Theory and Applications to Cassava Cubes . . . . . . . . . . . . . . . . . . . . T. R. Bezerra Pessoa, A. G. Barbosa de Lima, P. C. Martins, V. C. Pereira, T. C. O. Alves, E. S. da Silva, and E. S. de Lima 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.1 Drying Fundamentals . . . . . . . . . . . . . . . . . . . . . . . 6.1.2 The Focus of This Work . . . . . . . . . . . . . . . . . . . . 6.2 Application: Hybrid Drying of Cassava Cubes . . . . . . . . . . 6.2.1 The Raw Material . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.2 Osmotic Dehydration Tests . . . . . . . . . . . . . . . . . . 6.2.3 Convective Drying Tests . . . . . . . . . . . . . . . . . . . . 6.2.4 Results and Analysis . . . . . . . . . . . . . . . . . . . . . . . 6.3 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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7 Heat Transfer in a Packed-Bed Elliptic Cylindrical Reactor: Theory, Heterogeneous Transient Modeling, and Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. S. Pereira, R. M. da Silva, R. S. Santos, A. G. Barbosa de Lima, R. O. de Andrade, W. M. P. B. de Lima, and G. S. de Lima 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Porous Media and Packed-Bed Reactors . . . . . . . . . . . . . . . . . 7.2.1 Porous Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.2 Chemical Reactors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Heat Transfer in Fixed-Bed Elliptical Reactor via Two-Phase Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.1 Physical Problem and Geometry . . . . . . . . . . . . . . . . . . 7.3.2 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.3 Numerical Treatment of Heat Transport Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Application: Heat Transfer in an Elliptic Cylindrical Reactor Filled with Spheroidal Particles . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Chapter 1

Clay Ceramic Materials: From Fundamentals and Manufacturing to Drying Process Predictions A. G. Barbosa de Lima, J. M. P. Q. Delgado, L. P. C. Nascimento, E. S. de Lima, V. A. B. de Oliveira, A. M. V. Silva, and J. V. Silva Abstract This chapter is devoted to study heat and mass transfer and dimension variations of arbitrary-shaped porous materials. The focus is on the drying process of clay ceramic materials. Here, different topics related to history, manufacturing, drying process, phenomenological lumped modeling, and parameters estimation are present and discussed. Emphasis is given to industrial clay bricks, with theoretical and experimental approaches. Keywords Drying · Brick · Experimental · Simulation · Lumped model

A. G. B. de Lima (B) · L. P. C. Nascimento · E. S. de Lima · A. M. V. Silva · J. V. Silva Department of Mechanical Engineering, Federal University of Campina Grande, Av. Aprígio Veloso, 882, Bodocongó, Campina Grande, PB 58429-900, Brazil e-mail: [email protected] L. P. C. Nascimento e-mail: [email protected] E. S. de Lima e-mail: [email protected] A. M. V. Silva e-mail: [email protected] J. V. Silva e-mail: [email protected] J. M. P. Q. Delgado CONSTRUCT-LFC, Civil Engineering Department, Faculty of Engineering, University of Porto, Porto, Portugal e-mail: [email protected] V. A. B. de Oliveira State University of Paraiba, Rodovia PB 075, S/N, km 1, Guarabira, PB 58200-000, Brazil e-mail: [email protected] © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 J. M. P. Q. Delgado and A. G. Barbosa de Lima (eds.), Transport Processes and Separation Technologies, Advanced Structured Materials 133, https://doi.org/10.1007/978-3-030-47856-8_1

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1.1 Ceramic Materials 1.1.1 History The art of pottery is one of the oldest in the world due mainly to the abundance of clay and the ease of extraction and fabrication. There is evidence of activity of this art in almost all peoples of antiquity and to improve their quality of life, man has always been seeking to perfect the various uses of ceramic materials. Pottery was invented in the Neolithic (polished stone age) in 25000 BC and during this period prehistoric man-made wicker baskets with clay, that is, the first objects were intended to store grain and liquids and were just simple objects. Later, the plasticity of clays was discovered, where it was noted that by adding water the clay could be molded, dried in the sun, and hardened when exposed to high temperatures. Following, ceramics were widely used for various purposes, such as pieces with nozzles and handles made with relief images, or with living paintings that were considered decorative objects (Cavalcanti 2010). Each civilization and each culture have developed its own forms and characteristics in the use of clay, so that pottery is one of the greatest auxiliaries in historical research. One of the greatest ancient peoples who have strong ties to ceramics is the Greeks, who for a long time produced the finest pieces in the Mediterranean world. It was common at that time to sell these products at fairs and there was a continuous export of generally ovoid and handled vessels (Phoenician amphora), which could often be used to serve water, wine, and olive oil (Silva 2009). In addition to the Greeks and Romans, other ancient peoples such as the Byzantines and Arabs were responsible for transmitting their practices throughout Europe, which consequently have varied styles of construction in their territories. It was precisely with the growth of civil construction that the manufacture of ceramic pieces evolved from a more artisanal activity to an industrial one. Initially, around 1850, the first bricks were made on animal-powered molding machines, only later that the manufacturing would go through a major leap. Production systems were stagnant until the nineteenth century, i.e., drying was still done in the sun, burning in trapezoidal ovens and production was still mostly by hand. Only with the emergence of the first steam-powered machines, it was possible to increase production as raw material extraction, preparation, and forming operations became mechanized. Thus, in the modern era countries like Spain, France and Germany stood out in the market as producers of red ceramics and as equipment manufacturers. It is important to highlight that Italy was one of the great pioneers in the production of bricks in series with good quality (Silva 2016). Later, in the mid-twentieth century, the technological development of the ceramic industry boosted the manufacture of high strength and low weight cast structural blocks, a major evolution compared to previously manufactured solid bricks. At this time, including Brazil, there was a resurgence of structural masonry with ceramic products, competing economically with conventional reinforced concrete structures in medium-sized buildings (up to about 8–10 floors) (Silva 2009).

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Following, the ceramic industry underwent major developments, now based on research, technology, and studies by specialized laboratories. Along with the study of ceramics, the study of ovens, better glazing, molding apparatus, dry molding, high strength porcelain was developed and it was possible that the field of use of ceramics grew a lot, enabling aerospace and technology applications, such as space shuttle thermal shielding, nanofilm production, sensors to detect toxic gases, and among others. With regard to Brazil, construction ceramics currently occupies a prominent place in the national economy. Great growth came in the industry when the Government’s Growth Acceleration Programs (PAC) and My House My Life (MCMV) were implemented. Therefore, because it represents a sector of great importance in job creation and income distribution, it has received the attention of government sectors, research institutes, universities, and various entities (Rodrigues Neto and Mota 2016).

1.1.2 Fundamental Concepts Ceramic or ceramic material can be defined as any non-metallic and inorganic material whose structure, after heat treatment at high temperatures, is wholly or partially crystallized. They are composed of total or predominantly ionic interatomic bonds, but having some covalent character. Ceramics are known to have different raw materials in their composition, but the main one is clay, which can be defined as an earthy, thin, and natural material that, by adding water, acquires a certain plasticity and can be easily molded (Callister 2007; Callister and Rethwisch 2008). Ceramic materials have a wide range of structural arrangement types. The existence of several ceramic phases makes possible the combinations of metallic and non-metallic atoms (which form many structural arrangements) making them widely applicable in various sectors besides construction. It is noteworthy that the structure of the ceramic material defines its properties (Silva 2009; Callister 2007; Callister and Rethwisch 2008).

1.1.3 The Ceramic Industry and Clay Products The ceramic industry sector plays a very important role in Brazil’s economy, with a share of approximately 1% of GDP. Gaining prominence, the evolution of Brazilian companies has been very fast, mainly due to the abundance of natural raw material, alternative sources of energy, and the availability of practical technologies. Among the regions of the country, the ones that stand out and have a large concentration of industries are the Southeast and the South; this is because they have higher demographic density, greater industrial and agricultural activity, better infrastructure, and better income distribution. It is noteworthy that the other regions of the country have shown a certain degree of development, especially in the northeast due to the large

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occurrence of mineral resources, abundance of natural gas, expanding market, and great export potential (Silva 2009). Despite the greatness of the Brazilian ceramics industry and its great potential, it is quite heterogeneous. In addition to the red ceramics industries, several mining companies, ceramic tiles, sanitary ware, thermal ceramics, enamels, and others have already been installed or are in phase of deployment. However, within the current scenario of globalization, it can be said that the segments that are best adapted and structured are the covering, refractory, and sanitary ware. In the other segments, there are some modern companies that stand out from the others, but this contingent is not so expressive. One of the most important areas in the industry is related to red ceramic products. According to data from SEBRAE/Brazil (SEBRAE 2019), there is a range of 8500– 11,000 companies in the country, generating around 300,000 direct jobs and 1.5 million indirect jobs. Despite having good numbers, the production activity of the sector has a great technological backwardness, since most of the companies are of small or medium size and family order. It is also worth noting that this large number of jobs that the sector generates is caused by the low level of knowledge and investment required to start activities. In the northeast region, and especially in the State of Paraíba (Brazil), there is a marked industrial activity in this area. There are around 60 active red ceramic factories throughout the state, distributed in at least 30 towns, offering about 3000 direct jobs. Research carried out in the state of Paraíba shows, as to geographic regions and watersheds, that studies in the area are concentrated in some specific regions of the state, especially in the coast and in the Agreste. On the other hand, in the Sertão and Cariri regions there is a large concentration of potentially usable deposits, but, to date, there is no systematic study regarding their exploitation and use. Red ceramics encompass various products such as blocks, tiles, solid bricks, plumbing pipes, slabs, castings, and also expanded clay, which are often used in construction. It is also present in household items such as filters, decorative vases, and clay pots. This type of ceramics has the nomenclature “red” due to the presence of ferrous compounds that develop reddish coloration. The basic raw material of structural ceramics is common clay, which is used in a single dough to shape products, unlike other segments of the ceramics industry that mix clay with other substances such as talc, kaolin, and others (Callister 2007; Callister and Rethwisch 2008; Cabral et al. 2008). Natural clay seeks an ideal composition of plasticity and fusibility so that it provides good workability and mechanical resistance during firing. Red ceramic products are classified according to the manufacturing process used and can be pressed or extruded. In summary, red or structural ceramics can be grouped generically according to Table 1.1, as follows. The clay used for the production of red ceramics is composed of a large amount of amorphous material, in other words, those that do not have long-distance spatial ordering; however, the crystalline material predominates, which is grouped in welldefined mineralogical species. From the physicochemical point of view, clays can be

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Table 1.1 Types and characterization of red ceramics (Silva 2009) Types

Description

Porous

Solid bricks, pressed solid bricks, laminated bricks, hollow bricks, prefabricated panels, tiles, components for slabs, tiles, conductors for electrical cables, and others (sills, cladding plates, etc.)

Glazed

Tiles, glazed glazes, laminated bricks, pipes, glazed internally, glazed internally and externally, and unglazed

Expanded clay Obtained from thermo-expansion of some types of clays (illite). In the production process, mineral oil is added to the ceramic mass. They are launched in an inclined rotary kiln with a burner at the bottom

considered as dispersed mineral systems in which particles below 2 µm in diameter predominate (Silva 2016). A great advantage of the clay used in red ceramics is that it has great plasticity while wet, allowing the manufacture of pieces of various shapes using simple equipment. Another important point is that when cooked at more than 800 °C it has a good mechanical resistance, making the final product suitable for various applications (Brito 2016). To produce bricks and tiles the clay used is generally quaternary and sometimes tertiary. One of the main characteristics is to present, in large quantities, iron, and alkalis in their composition. They are fine-grained and have a considerable organic matter content, factors that are responsible for their high plasticity. It is recommended that the clays used have easy molding, flexural strength before and after sintering, have a reddish color after sintering, with a minimum of cracks and warping. It is noteworthy that high levels of bivalent iron and alkaline elements may reduce the range of vitrification and cause undesirable coloration (Silva 2009).

1.1.4 Red Ceramic Product Manufacturing Process Red ceramics can generate a wide variety of products and for this it goes through a specific production process, which is sometimes still poorly evolved compared to other segments of the ceramic industry. However, due to the increasing emergence of technological innovations in some companies, we can find good quality production processes with high production rates. Most of these technological advances are related to equipment automation and, consequently, the reduction of labor costs. The production process, exemplified in Fig. 1.1, is common to all red ceramic companies in general, with slight variations depending on the particular characteristics of each raw material or end product. For example, some companies use rudimentary equipment and others have more modern equipment, or some have a much higher degree of production, among other differences. The production process of pieces with red ceramic comprises several steps that can be divided into four major stages, namely, extraction and preparation of raw

6

A. G. B. de Lima et al.

Fig. 1.1 Manufacturing flowchart of red ceramic pieces

materials, mechanical forming, thermal processing, and shipping. The following best describes these steps:

1.1.4.1

Extraction and Preparation of Raw Materials

The manufacturing process begins with the extraction of clay, which is removed from the deposits with the aid of backhoes and then transported to storage sheds, which may be owned by companies or third parties. At this stage, the material goes through a “rest”, thus undergoing chemical changes and being unpacked. Shed storage also ensures continued production in rainy seasons. After this phase, we have the dosage, in which the clays are proportionally dosed in a feeder coffin obeying their ceramic characteristics. Following the manufacturing process is disintegration, which is the step responsible for bringing the hardest and most compacted clays to a disintegrator that will crush the larger clumps of clay to facilitate subsequent operations. Then, the raw material goes to the mixer, where it will be homogenized, thus allowing the addition of water in the mixture to obtain adequate moisture and plasticity for extrusion.

1 Clay Ceramic Materials: From Fundamentals …

7

The last step of this first major stage is lamination, which is responsible for a thickening of the mixture, eliminating air bubbles or clumps that may have remained so far. With the end of this stage, the raw material already prepared can be directed to the extruders, which may even have a rolling mill attached to them.

1.1.4.2

Mechanical Forming

The mechanical conformation stage is responsible for transforming the clay plastic mass into products with different shapes and sizes. Thus, according to the type of product to be obtained and also depending on the plasticity characteristics of the available raw material, it will be possible to choose the appropriate forming system. The main systems of this stage are extrusion and cutting. Firstly, the clay mass will take the desired shape upon entering the extruder, which contains a steel plate perforated in a vacuum chamber. Then, through the manual or automatic cutter, the extruded block is cut to standard sizes, thus obtaining products such as bricks, tiles, ceramic tubes, and among others (Oliveira and Bernils 2006).

1.1.4.3

Thermal Processing

This stage consists of the drying and burning steps of the already formed parts. This is where the composition and structure transformations will occur, generating the final properties of the product, such as color, gloss, porosity, flexural strength, high temperatures, and among others (Silva 2009). During drying, a large amount of thermal energy is used to slowly and evenly evaporate the water added during the molding process. This step usually takes place inside drying chambers and aims to reduce the moisture content of the products from 20–25% to 3–10% after the process. An important property of any clay is that it has water in the constitution of its crystal lattice. Thus, during the drying process, the water that has been added is easily removed, with the temperature starting from room temperature and reaching approximately 110 °C. However, water that is in the clay crystal lattice will only be removed at temperatures above 400 °C and may vary to even higher values depending on the type of clay. During the drying process, the clay may contract as the spaces that were occupied by water inside the material become empty after evaporation. This shrinkage is proportional to the degree of moisture removed. Thus, it is important to control the process well, as a possible consequence of this shrinkage is that it can cause deformation or cracking in the material. Following is the firing step, in which the product is taken to a kiln and, as well as drying, will receive an even greater amount of thermal energy. Once these steps are completed, the product will have lower porosity and greater mechanical strength and will also be ready for commercialization and use.

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1.1.4.4

A. G. B. de Lima et al.

Expedition

Shipment is the final stage of the production process, where finished product is inspected to identify excessively cracked, broken, chipped, or burned products. Then, the parts are stored in a covered area until they leave for delivery to the customer. In Brazil, transportation of the parts is usually made by trucks on the highways of the country. The thermal processing stage must be performed correctly, otherwise the parts could present a series of defects and thus, the products will not be able to perform their respective functions. Given this, the most common defects are as follows (Silva 2009; Silva et al. 2011): (a) Commitments—This defect is a deformation of the part usually caused by residual shrinkage stresses, which arise when one side of the material dries faster than the other, i.e., it is important that the drying is done evenly. Commitments may also arise due to poor positioning of the product on the drying support. (b) Cracks—It is important that during the drying process the air velocity and temperature are controlled, because when we have a very fast drying, it is common the appearance of cracks, which are nothing more than small fissures that start at the edges and spread until the center of the piece. Cracks may also appear in the firing step, which may be by heating or cooling. The heating ones are characterized by being open, little winding, and with jagged edges, while the cooling ones are characterized by being closed and very thin, usually S-shaped edges. It is important to point out that all drying starts must be done with the plastic-covered part, to prevent a very fast outflow of water that is closer to the surface, causing a localized shrinkage that can cause cracks. (c) Black heart—This type of defect is black or gray spots that can be seen along the cross section of the part and appear after the firing process. The existence of the “black heart” is associated with the presence of carbon-containing compounds, which are formed due to the small amount of oxygen, preventing the complete oxidation of carbon compounds and organic matter. (d) Efflorescence—Efflorescence occurs on the outer surface of the product and is a salt deposit accumulated in some regions, which may cause undesirable stains and colors. This defect appears as the water interferes with salts. If the piece, after burning, absorbs moisture, the salts will be dissolved; however, if the external environment becomes dry, the opposite process occurs, the surface water is evaporated and the crystallization of the salts occurs. (e) Defects related to steps before or after drying—It is common for small cracks to occur when the clay paste is improperly mixed in the mixing step. This defect is most pronounced in areas with higher moisture content and is quite common in manual manufacturing processes. Finally, it is worth mentioning the problem of moisture absorption. Depending on the type of clay, if the time elapsed from the clay leaves the dryer to when it is introduced into the kiln is large and the ambient absolute humidity is very high, a rehydration (reabsorption) process may occur, which may cause breakage and/or explosion when material enters the kiln.

1 Clay Ceramic Materials: From Fundamentals …

9

1.2 The Drying Process 1.2.1 General Principles Drying can be explained as a thermodynamic process responsible for the partial removal of a liquid, usually water, from the porous material by providing energy to it and providing water loss by evaporation. In this process, there is a simultaneous heat and mass transfer, and the transport of moisture from the interior to the surface of the material may occur in the form of liquid and/or vapor, depending on the percentage of moisture present and the type of product (Brooker et al. 1992; Strumillo and Kudra 1986). The drying process has become, among many other uses, one of the most important steps in the manufacture of ceramic parts. In the case of red ceramic, this step is of relevant importance, since if the moisture is not removed properly, severe stresses occur inside the part, causing deformations, cracks, and reducing the quality of the product post-drying process. Thus, it is noteworthy that the in-depth study related to drying of ceramic materials increases the overall efficiency of the ceramic sector by reducing losses and increasing material quality and provides an environment conducive to progress and sustainable development. There are three ways to classify drying: natural, artificial, or mixed. Whatever the type of drying, it has to fulfill four basic functions: the transport of the heat necessary for water evaporation, the removal of the produced water vapor, the reduction of the saturated vapor layer formed on the product surface and the movement of liquid, and/or vapor inside the part. The process time depends on the special conditions of the drying environment, such as temperature, relative humidity, and air velocity, and may reach periods of up to six weeks (natural drying). Artificial drying is carried out in drying chambers or dryers, usually taking advantage of the residual heat of the kiln, which significantly reduces the drying time. In addition, the artificial drying period also depends on the characteristics of the raw material, the shape of the parts, and the type of dryer. Convective drying technique differs from other separation techniques such as osmotic dehydration, evaporation, and decantation by the way water is removed from the solid. In convective drying, there is a difference between the partial pressure of the water vapor at the surface of the product and the surrounding air, which allows the migration of the liquid from the inside and consequently the removal of the water molecules from it. In osmotic dehydration, for example, this moisture removal may occur due to a pressure difference between the product and a hypertonic solution, due to a density difference, or due to temperature increase (Silva 2016; Brooker et al. 1992; Strumillo and Kudra 1986). In order, to perform a thermodynamic analysis of drying it is necessary to understand the influence of velocity, relative humidity, and temperature of the drying air on the process. Relative humidity can be defined as the ratio of the vapor partial pressure in the air to the vapor saturation pressure, which varies with temperature. The ability of air to absorb water vapor will be higher when the saturation pressure of water

10

A. G. B. de Lima et al.

vapor is greater than the partial pressure of water vapor. Therefore, the ability of air to absorb water vapor increases with temperature, so that the higher the air temperature, the greater its drying capacity, in fixed conditions of the air relative humidity. In addition, if the air is warmer, the volume of air needed for drying decreases and, as a result, the powers of the hoods and air circulators are reduced, reducing drying costs. The speed with which the product is dried can be affected by many factors, such as moisture movement mechanism, product shape, external environment conditions, and green product porosity. Thus, it is of great importance to verify the influence of the shape and volume of the pores in the part, because it is inside them that is the moisture, that even under favorable conditions can be retained inside these pores. This occurs when the surface of the part is dried very quickly, as the pores, being very narrow, reduce moisture migration for a rate less than the evaporation rate. Another important point is that with a higher drying air temperature and lower relative humidity there will be an increase in drying rate. The drying process is generally divided into four distinct phases: adaptation, colloidal water outlet, void formation, and interstitial moisture expulsion. In the first phase occurs the adaptation of the product to environmental conditions (temperature, relative humidity, and pressure), in which drying will be performed. In the second phase, there is evaporation of the colloidal water, and sensible variations in the dimensions of the part occur due to the approximation of the particles of its microstructure. Even at this stage water continually migrates to the surface of the part, constantly forming an evaporating saturated wet film. In the third phase occurs the disappearance of the water film on the surface of the piece, which provokes changes in color. The last drying phase, which is not always reached in the dryers and is often performed in the kilns, is the expulsion of the last amounts of moisture from interstitial origin, in which the moisture removal rate decreases to near zero (Silva 2009). Given the importance and complexity of the drying process, a large number of researchers have been working intensively on its analysis. Some focus on external air conditions, such as temperature, relative humidity, and velocity, correlated with the product’s drying rate, while others consider the internal conditions of the product, with emphasis on the mechanisms of moisture movement and their effects on it. In this regard, several drying theories have been proposed to describe heat and mass transport in capillary porous media, namely, (a) (b) (c) (d) (e) (f) (g) (h)

Liquid diffusion theory; Vaporization–condensation theory; Cappilary theory; Kricher’s theory; Luikov’s theory; Philip and De Vrie’s theory; Berger and Pei’s theory; Fortes and Okos theory.

1 Clay Ceramic Materials: From Fundamentals …

11

A more detailed discussion of drying theories can be found in the literature (Brooker et al. 1992; Strumillo and Kudra 1986; Lima et al. 2014). According to the drying theories listed before, the following mechanisms of moisture transport in porous material have been cited in the literature: (a) Transport of liquid by diffusion due to moisture concentration gradients; (b) Transport of vapor by diffusion due to moisture concentration gradients and vapor partial pressure (caused by temperature gradients); (c) Transport by effusion (Knudsen flow) that occurs when the average free path of vapor molecules is of the same order as the pore diameter. It is important for high vacuum conditions such as freeze drying; (d) Transport of vapor by thermofusion due to temperature gradients; (e) Transport of liquids by capillary forces due to capillarity phenomena; (f) Transport of fluid by osmotic pressure due to osmotic force; (g) Transport of liquid due to gravity; (h) Transport of liquid and vapor due to total pressure difference caused by external pressure, shrinkage, high temperature, and capillarity; (i) Transport of liquid and vapor by surface diffusion due to the migration of these phases through the pores of the product surface. Then, based on the drying theories and moisture migration mechanisms, several drying models have been reported in the literature. This topic will be discussed following.

1.2.2 The Mathematical Modeling of the Drying Process The main objective of an appropriated drying modeling is to mathematically describe the physical phenomena, so that it is possible to choose appropriate operating conditions, the most appropriate method of drying and also to control and know the process deeply. Thus, we can optimize the steps of drying and eliminate or minimize existing irregularities. The development of mathematical models to describe the drying process is increasingly recurrent and has been studied for several decades. This is because the process has great importance in the production of different products and also involves complex phenomena of heat and mass transfer, linear momentum, and dimension variations of the product. The principle of modeling is based on having a system of mathematical equations that completely characterizes the system to be modeled. In particular, the solution of these equations makes it possible to predict process parameters as a function of drying time based only on initial and boundary conditions, and some simplifications. The starting point in mathematical modeling is the definition of the process to be modeled, in particular the description of the input data that influence the process, as well as the variables that depend on the process behavior.

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The complexity of the drying process depends on the geometric and thermophysical parameters of the material and thickness of the material layer in study. They can then be classified in thin-layer drying models (particle level models) and thicklayer models (dryer models). The dryer mathematical models (thick-layer model) most used by the researchers take into account the thermophysical properties, drying kinetics, and mass and energy balance in the device. Some researchers have applied dryer model to predict drying process of clay ceramic materials with particular reference to industrial clay bricks (Almeida et al. 2013; Tavares et al. 2014; Almeida et al. 2016; Silva 2018). From a practical point of view, thin-layer drying is very limited. But to have a good understanding of the thick-layer drying process it is necessary to have thin-layer equations for the drying kinetics of a particular material under certain predetermined operating conditions (Macedo 2016). Several thin-layer mathematical models have been proposed to describe the rate of moisture loss during drying and can be divided into two large groups: lumped and distributed models. Distributed models express heat and mass transfer rates as a function of position within the part and drying time, taking into account external and internal resistances. Lumped models, on the other hand, express the same rates only as a function of process time and ignoring the existing internal resistance for heat end mass transfer. The following general balance equation (distributed model) has been applied to predict drying process (by diffusion only) of irregularly-shaped porous body: d(λΦ) = ∇ · (Γ Φ ∇Φ) + Φ  dt

(1.1)

where λ and Γ Φ are transport properties. Φ is the unknow, Φ  is the source term, and t is the time. Distributed models based on the liquid diffusion theory have been applied to predict drying of ceramic porous materials. For example, clay plates (Silva et al. 2009), clay pipes (Santos 2018), roof tiles (Farias et al. 2013; Silva et al. 2012; Farias et al. 2012), and bricks (Araújo et al. 2019a, b, 2017; Brito et al. 2017; Araújo et al. 2017; Silva et al. 2011; Lima et al. 2015; Santos et al. 2020). This chapter addresses the use of the lumped model to describe the drying process. The equations of the lumped model can be classified as empirical, semi-empirical, and theoretical. It is noteworthy that in this analysis the effects of temperature and moisture variation inside the material are neglected during the process. When it comes to empirical equations, they have a direct link between moisture content and drying time, while semi-empirical ones are analogous to Newton’s law of cooling, assuming that the drying rate is proportional to the difference between moisture content of the product and its equilibrium moisture content for the specified drying conditions. Theoretical equations generally use heat and mass balances between the product and air surrounding it, taking account different physical phenomena during the process. Some researchers have applied lumped models to describe drying process of clay porous materials. For example, clay pipes (Silva et al. 2016), bricks (Silva 2009; Almeida et al. 2013; Tavares et al. 2014; Silva 2018;

1 Clay Ceramic Materials: From Fundamentals …

13

Fig. 1.2 Representative scheme of the drying process of an arbitrarily-shaped solid based on a lumped analysis

Silva et al. 2011; Lima et al. 2015), and others geometries (Silva et al. 2016; Lima et al. 2018; Lima 2017). For a better understanding of the lumped analysis method (theoretical model), consider the solid with arbitrary geometry, illustrated in Fig. 1.2. In this scheme, the arbitrary solid will receive on its surface a flux per unit area of the potential of interest Φ and has uniformly distributed internal generation per unit volume. According to what has already been mentioned, when applying the lumped analysis method, the effects of the potential variation within the material are neglected. Thus, all flux of Φ received and generated will diffuse instantly through the solid. In order for this condition to be physically possible and well approximated, the flux resistance within the solid must be much lower than the flux resistance between the solid and its vicinity. Thus, the balance of Φ (potential of interest) can be obtained as follows:   d λΦ 

V

dt

= Φ  S + Φ  V

(1.2)

in which Φ  and Φ  are flux of Φ per unit area and source term, respectively. Further, λ includes transport parameters and S and V are the surface area and volume of the porous material, respectively.



1.3 Lumped Model Application: Drying of Clay Ceramic Brick As an application, in this topic will be developing new research methods and techniques, particularly process modeling and simulation involving heat and mass transport in solid–liquid systems, with particular reference to drying of clayey ceramic materials, via lumped models.

14

A. G. B. de Lima et al.

The focus is to develop a phenomenological mathematical modeling and its analytical solution via method of separation of variables to predict heat and mass transfer in clayey, cast, and arbitrary-shaped ceramic materials (industrial ceramic bricks).

1.3.1 The Experimental Data The materials used for drying in oven were parallelepiped-shaped ceramic bricks with 8 rectangular holes (industrial ceramic bricks). Figure 1.3 illustrates the test body model used, as well as the positions where the measurements of length (R1 ), width (R2 ), height (R3 ), and dimensions that characterize the brick holes, a1 , a2 , a3 , and a4 , were obtained. Initially, dimensions were measured with a digital caliper, mass with a digital scale, brick temperature (vertex) with infrared thermometer, and room temperature and relative humidity with thermohygrometer. Then, the samples were taken inside the forced-air oven where drying was performed. In this process, the internal temperature of the oven was set as desired with the temperature controller. At predefined intervals, the brick was taken from the oven and measured its temperature, mass, and dimensions. Table 1.2 summarizes, for each experimental condition, the product, and air data. Table 1.3 presents, for each operating condition, the dimensions, volume, and surface area of the sample before the drying process begins. During the process, measurements were taken every 10 min until the mass had minimal variation. Then, the measurements were changed every 30 min, and the next measurements were taken every 60 min until it reached constant mass. Soon after, the

Fig. 1.3 Hollow brick with dimensions

1 Clay Ceramic Materials: From Fundamentals …

15

Table 1.2 Experimental air and brick parameters for each drying test (Silva 2009) T (°C)

Air

Brick

Time, t (h)

UR (%)

V(m/s)

M o (db)

M f (db)

M e (db)

θ o (°C)

θ f (°C)

50

80

0.05

0.13969

0.0

0.00011

20.6

41.0

18.5

60

79

0.06

0.14795

0.0

0.00268

20.5

50.2

13.7

70

69

0.07

0.15414

0.0

0.00076

26.0

64.5

17.8

80

66

0.08

0.15248

0.0

0.00039

21.4

69.2

15.0

90

68

0.09

0.15921

0.0

0.00151

21.0

78.5

11.5

100

52

0.10

0.16903

0.0

0.00038

26.1

93.2

12.3

sample was dried for 24 h at the same drying temperature to obtain the equilibrium mass and then, for another 24 h at 105 °C to obtain the mass of the dried product. All experiment was performed by Silva (2009). This author also performed an adjustment of experimental data related to mass transfer (moisture content) during the process and proposed an exponential equation with two terms and four parameters. The equation has the form: M = A1 exp(k1 t) + A2 exp(k2 t)

(1.3)

where t is given in minutes. The A1 , A2 , k 1 , and k 2 parameters were estimated using the Statistica® Software, the Rosembrock and Quasi-Newton numerical method, and a convergence criterion of 0.001. After fitting, Silva (2009) presented the parameters reported in Table 1.4. The experimental data of the brick vertex temperature was fitted to an equation with four parameters. The equation has the form:   θ = B1 + B2 log10 t K 1 + B3

(1.4)

where t is given in minutes. Parameters B1 , B2 , K 1 , and B3 were estimated using Statistica® Software, the Quasi-Newton numerical method, and with a convergence criterion of 0.0001. Table 1.5 summarizes the coefficients of Eq. 1.4 obtained after fitting to the experimental data.

1.3.2 Phenomenological Mathematical Modeling To predict the drying process was developed an advanced and phenomenological mathematic model. It is based on the following hypotheses:

R1 (mm)

93.36

92.75

93.16

92.76

93.10

92.80

T (°C)

50

60

70

80

90

100

198.00

197.00

197.00

197.00

195.00

197.00

R2 (mm)

202.00

201.00

201.00

203.00

200.00

200.00

R3 (mm)

1.70

8.88

8.16

8.54

8.34

9.04

a1 (mm)

9.41

7.95

7.20

9.87

7.32

7.10

a2 (mm)

Table 1.3 Brick dimensions before the drying process begins (Silva 2009)

8.74

6.57

7.84

7.99

7.11

7.88

a3 (mm)

8.00

6.78

6.66

6.96

6.45

6.30

a4 (mm)

1,734,026.10

1428,426.08

1,408,074.95

1,621,580.85

1,367,269.30

141,5643.80

V o (mm3 )

36,116.49

37,233.87

37,214.46

162,158.85

369,020.69

371,100.44

S o (mm2 )

16 A. G. B. de Lima et al.

1 Clay Ceramic Materials: From Fundamentals …

17

Table 1.4 Parameters of Eq. 1.3 obtained after fitting to experimental data of average moisture content T (°C)

Parameter A1 (–)

k 1 (mm−1 )

A2 (–)

R (–)

Explained variance (%)

k 2 (mm−1 )

50

0.576178

−0.004711

0.482232

−0.004711

0.997676745

0.995358888

60

0.547740

−0.005945

0.513349

−0.005945

0.997968284

0.995940696

70

0.000000

−0.006781

1.045050

−0.0070948

0.999112861

0.998226509

80

0.535201

−0.009190

0.527668

−0.009190

0.998502641

0.997007523

−0.014298

−9.613313

−0.015018

0.998876724

0.997754709

−0.008383

−3.827964

−0.007881

0.998297496

0.996597890

90 100

10.63554 4.875507

Table 1.5 Parameters of Eq. 1.4 obtained after fitting to experimental data of the vertex temperature T (°C)

Parameter

R (–)

Explained variance (%)

0.960840804

0.923215051

B1 (°C)

B2 (°C/min)

k 1 (–)

B3 (min)

50

−546.0430

283.1605

0.42554

101.18296

60

−48.7454

39.22594

0.86804

66.362934

0.981233190

0.962818573

70

−18.4408

40.19277

0.698315

11.943974

0.953411275

0.908993060

80

−21.3661

37.35810

0.871389

14.410538

0.970896765

0.942640529

90

−30.7995

33.11958

1.222654

47.338410

0.981074082

0.962506354

15.41788

2.234665

0.984632771

0.969501694

100

−2.86969

118.38213

(a) Brick is composed of liquid water and solid matter; (b) Water migrates from the interior of the brick in liquid form and evaporates on the surface; (c) On the solid surface there is thermal convection, evaporation, and heating of produced vapor; (d) Dimensional variations were considered during drying process; (e) Heat and mass generation were neglected; (f) Constant mechanical and thermophysical properties. 1.3.2.1

Geometric and Dimensional Analysis

From the various measurements of the brick dimensions, made during the drying process, mathematical equations were proposed to calculate the volume and surface area of the brick (Fig. 1.3). The brick volume at any time t was calculated as follows: Vf = aV aH R3 (brick holes volume)

(1.5)

V = (R1 R2 R3 ) − (8Vf )(brick volume)

(1.6)

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A. G. B. de Lima et al.

The brick surface area at any time t was determined by using the following equation: S = (2R1 R3 ) + (2R2 R3 ) + 2[(R1 R2 ) − (8aH aV )] + 8[(2aH R3 ) + (2aV R3 )] (1.7) where in Eqs. 1.5, 1.6, and 1.7: aV = (R2 − 2a1 − 3a3 )/4

(height of a hole)

(1.8)

aH = (R1 − 2a2 − a4 )/2

(width of a hole)

(1.9)

After determination of the volume and surface area at different moments of drying, it was possible to adjust them to mathematical models that describe the volumetric variation and surface area of the brick during the drying process. This procedure was realized by using Statistica® software (Simplex numerical method and convergence criterion of 0.00001). For this, a third-degree polynomial model was proposed for both volume and surface area, as follows:

1.3.2.2

V (t) = C1 t 3 + C2 t 2 + C3 t + C4

(1.10)

S(t) = D1 t 3 + D2 t 2 + D3 t + D4

(1.11)

Mass Transfer Analysis

The complexity of the drying process depends, among other parameters, on the analysis taken into account. Distributed models express heat and mass transfer rates as a function of position within the part and drying time, taking into account external and internal resistances. Already the lumped models express the same rates only as a function of the process time and ignoring the existing internal resistance for this transfer. This study makes use of the lumped model analysis to describe the drying process of ceramic brick. Thus, from Eq. 1.2, we have the following mass balance: V

dM = −h m S(M − Me ) + V M˙ dt

(1.12)

where S and V represent the surface area and volume of the solid at any time t, hm is the convective mass transfer coefficient, M is the average moisture content, M e is the equilibrium moisture content of the brick, and t is the time. Considering M  = M − M e , it is valid dM  = dM. Therefore, it is possible to write:

1 Clay Ceramic Materials: From Fundamentals …

V

19

dM  = −h m S M  + V M˙ dt

(1.13)

Separating the variables and rearranging the terms, Eq. 1.13 results in: dM  hm S   =− dt V M˙ V  (M ) − h m S

(1.14)

Since that M = M 0 at t = 0, and that there are no reactions that can generate water ˙ = 0. So, Eq. 1.14 can be integrated from the inside the product, it was considered M initial condition. Thus, it is possible to write: M−M e

M0 −Me

dM  =− (M  )

t

 hm S dt V

(1.15)

0

Putting Eqs. 1.10 and 11.11 into Eq. 1.15 and integrating it, we obtain the following equation, which defines the mass transfer, considering dimensional variations during the process: ⎧ ⎧ ⎡    ⎫⎤⎫  ⎨ a 1 t + a 2 arctan a 3 a 4 + 2t + a 5 log a 6 − t ⎬ ⎬ ⎨   ⎦ + Me M = (M0 − Me ) exp⎣−h m ⎭ ⎭ ⎩ + a 7 log a 8 + a 9 t + t 2 − a 10 ⎩ 



















(1.16)



where the coefficients a k are specified according to drying conditions.

1.3.2.3

Heat Transfer Analysis

Similarly, to mass transfer, for heat transfer analysis, considering constant the heat flux per area unit, the following energy balance is given: ρV Cp

dθ = [h c S(θ∞ − θ )] + q˙ V dt

(1.17)

where ρ and C p represent the density and specific heat of the brick, respectively, hc is the convective heat transfer coefficient, θ and θ ∞ represent, respectively, the average product temperature at any time t and the equilibrium temperature (which is equal to the drying air temperature). Considering T  = θ∞ − θ , it turns out that dT  = −dθ. Then, putting this result into Eq. 1.17, separating the variables, this equation can be rewritten as follows: 

dT  T

+

q˙ V (h c S)

 =−

hc S dt ρV Cp

(1.18)

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A. G. B. de Lima et al.

Since that θ = θ 0 at t = 0, and that there are no chemical reactions that can generate heat inside the product, it is possible to consider q˙ = 0. So, Eq. 1.18 can be integrated from the initial condition. Thus, we have that: θ ∞ −θ

θ∞ −θo

dT  =− [T  ]

t 0

hc S dt ρV Cp

(1.19)

Now, putting Eqs. 1.10 and 1.11 into Eq. 1.19 and integrating it, we obtain as results the following equation, which defines the heat transfer, considering dimensional variations during the process: ⎧ ⎧ ⎡     ⎫⎤⎫ ⎪ ⎪ ⎨ b1 t + b2 arc tan b3 b4 + 2t + b5 log b6 − t ⎪ ⎨ ⎬ ⎪ ⎬ ⎢ hc ⎥ θ = θ∞ − (θ∞ − θo ) exp⎣−   ⎦ ⎪ ⎪ ρCp ⎪ ⎩ + b7 log b8 + b9 t + t 2 − b10 ⎩ ⎭ ⎪ ⎭ 



















(1.20) 

where the coefficients bk are specified according to drying conditions. Equations 1.16 and 1.20 were fitted to the experimental data of the average moisture content (Eq. 1.3) and surface temperature (Eq. 1.4) of the ceramic brick using the Statistica® software (Quasi-Newton numerical method and convergence criterion of 0.0001). From the non-linear regression, it was possible to estimate the convective mass transfer (hm) and heat transfer (hc) coefficients.

1.3.3 Results Analysis 1.3.3.1

Dimensional Variations

Tables 1.6 and 1.7 summarize the parameters obtained for Eqs. 1.10 and 1.11, respectively. Table 1.6 Parameters of Eq. 1.10 that describe the volumetric behavior of the brick during drying process T (°C) Parameter C1

(m3 /min3 )

R (–) C2

(m3 /min2 )

C3

(m3 /min)

C4

(m3 )

Explained variance (%)

50

−0.000862

1.76499

−1099.66

1,381,400 0.98227288 96.486

60

−0.001538

2.46177

−1213.49

1,356,500 0.98812668 97.639

70

−0.000678

1.36975

−911.113

1,546,700 0.94004504 88.368

80

−0.001273

2.02475

−977.457

1,374,700 0.94563064 89.422

90

−0.003695

4.40681

−1596.91

1,392,300 0.9535488

100

−0.003480

5.278373

−2066.011

1,551,571 0.81039933 65.675

90.926

1 Clay Ceramic Materials: From Fundamentals …

21

Table 1.7 Parameters of Eq. 1.11 that describe the surface area behavior of the brick during drying process T (°C)

Parameter

R (–)

Explained variance (%)

371,912

0.99130038

98.268

368,692

0.98169467

96.372

−192.89

366,801

0.93426766

87.286

−190.31

370,685

0.9335808

87.157

0.761024

−271.251

373,572

0.95696019

91.577

0.322700

−134.0546

355,682

0.82398847

67.896

D1 (m2 /min3 )

D2 (m2 /min2 )

D3 (m2 /min)

D4 (m2 )

50

−0.000126

0.259821

−163.671

60

−0.000315

0.45662

−202.289

70

−0.000200

0.366684

80

−0.000267

0.41566

90

−0.000628

100

−0.000226

Statistica® software also provides graphs of the estimated functions compared to the collected experimental points. Thus, Figs. 1.4 and 1.5 show the transient volume and surface area variations under operating conditions from 50 to 100 °C. After analysis of Figs. 1.4 and 1.5, it is possible to notice that the volume and surface area have a decreasing behavior over time. This is because the water inside the brick is being evaporated during drying (shrinkage) and it is being heated during the process (volumetric expansion). Since that, drying at higher temperatures provokes increases in the drying and heating rates, these phenomena are more intensive. It is also possible to see that the experimental data found at 100 °C have a less accurate adjustment for both volume and surface area variation. This is due to possible measurement errors with brick drying and large temperature variations due to the fact that the brick was removed from the oven so that measurements could be made. It can be verified that at 50 °C, the brick volume decreased by 17.88% and at 100 °C a reduction of 20.57% occurred. At 100 °C, the amount of evaporated water is greater. With regard to surface area at temperature 50 °C the brick was reduced in surface area by 8.46% and at 100 °C by 5.83%. Thus, we can see that the area reduction was much smaller than the volume, which shows that the drying was done properly and the brick did not suffer large deformations, maintaining its original shape, but in a smaller size.

1.3.3.2

Drying Process

Table 1.8 summarizes the coefficients of Eqs. 1.16 and 1.20. With this, it was possible to adjust these equations to the experimental data of moisture content (Eq. 1.3) and surface temperature (Eq. 1.4) and to estimate the convective heat transfer and mass transfer coefficients. Figures 1.6 and 1.7 illustrate a comparison between the predicted and experimental brick average moisture content as a function of time for drying at 50 and 100 °C, respectively.

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A. G. B. de Lima et al.

Fig. 1.4 Predicted (---) and experimental (ooo) volume variations of the brick during drying. a 50 °C and b 100 °C

1 Clay Ceramic Materials: From Fundamentals …

23

Fig. 1.5 Predicted (---) and experimental (ooo) surface area variations of the brick during drying. a 50 °C and b 100 °C

24

A. G. B. de Lima et al.

Table 1.8 Parameters of Eqs. 1.16 and 1.20 that describe mass transfer and heat transfer of the brick during drying process Parameter 



a 4 = b4 



a 5 = b5 



a 9 = b9 

100

0.15

0.20

0.29

0.21

0.17

0.06

95.11

48.35

47.31

39.10

44.57

88.20

0.0005

0.0006

0.0005

0.0006

0.0008

0.0009

−218.45

−128.91

−85.22

−44.78

−83.38

−194.00

−51.82

−14.11

−25.59

−21.74

−27.83

−34.53

1829.10

1471.72

1935.06

1545.75

1109.26

1322.77

24.85

22.52

14.76

14.41

12.28

20.15

876141.16

599290.09

1.18 × 106

698617.29

339691.29

337060.33

−218.45

−128.91

−85.22

−44.78

−83.38

−194.00

−60.3635

192.68

401.945

33.22

−41.86

−6.50



a 8 = b8 

90



a 7 = b7 

80



a 6 = b6 

70



a 3 = b3 

60



a 2 = b2 

50 

a 1 = b1 

T (°C)



a 10 = b10

Fig. 1.6 Predicted and experimental average moisture content of ceramic brick as a function of drying time (T = 50 °C)

From the analyzes of the figures, we can state that the drying of brick occurred in the falling drying rate period, for two reasons: the drying rate isn’t constant and the brick temperature arises during the drying process. Further, the drying rate increases with increasing drying temperature. For all drying temperatures, it can be considered that after the first 1000 min of process, the average moisture content varies slightly. Thus, we can state that the equilibrium moisture content was reached at this time.

1 Clay Ceramic Materials: From Fundamentals …

25

Fig. 1.7 Predicted and experimental average moisture content of ceramic brick as a function of drying time (T = 100 °C)

Analyzing the graphs, it is noted that there is an agreement between the experimental values and the predicted values by the model used, confirming that the modeling used to find the average moisture content, as a function of drying time, is effective.

1.3.3.3

Heating Process

Similar to the procedure adopted for the moisture content, to find the convective heat transfer coefficients, at the different drying temperatures, a comparison between the predicted and experimental brick vertex temperature (Eq. 1.4) was made, until it reached a minimum error. Figures 1.8 and 1.9 illustrate the brick temperature adjustment curves as a function of time for drying at 50 and 100 °C, respectively. Through analysis, it can be proved that, for drying at higher temperatures, the brick reaches its equilibrium temperature in a shorter process time. When evaluating the graphs, it is noted that there was a good agreement between the experimental values and the values determined through the proposed model. The adjustment efficiency was not as good as in mass transfer, but this was due to the fact that the analytical model developed took into account only the heat transfer, disregarding the simultaneous mass transfer effect and the energy to be used in the phase change of water. Therefore, even with a not so refined agreement between the experimental and predicted brick temperatures, it can be said that the modeling used to estimate the process parameters is effective.

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A. G. B. de Lima et al.

Fig. 1.8 Predicted and experimental brick surface temperature as a function of drying time (T = 50 °C)

Fig. 1.9 Predicted and experimental brick surface temperature as a function of drying time (T = 100 °C)

1.3.3.4

Estimation of Transport Parameters

After gravimetric and thermal analysis, the process parameters were found for the different drying conditions. After fitting of Eqs. 1.16 and 1.20 to experimental data of average moisture content and temperature, the heat and mass transfer coefficients were estimated. Table 1.9 summarizes the estimated values of convective heat and mass transfer coefficients, with the respective errors obtained with Eqs. 1.21 and 1.22. In these equations, the number of experimental points is n = 110.

1 Clay Ceramic Materials: From Fundamentals …

27

Table 1.9 Convective heat and mass transfer coefficients estimated from fitting of Eqs. 1.16 and 1.20 to experimental data ERM (kg/kg)2

hc (W/m2 °C)

ERT (–)

50

2.6500 ×

10−7

0.000293

0.36985

0.823922

60

3.3266 × 10−7

0.000307

0.50538

0.375189

70

4.7300 × 10−7

0.000176

0.75000

0.525849

80

5.2783 ×

10−7

0.000276

0.88250

0.792176

90

5.7216 × 10−7

0.000462

1.00000

0.539377

100

6.9816 × 10−7

0.000263

1.33333

0.689403

T (°C)

hm (m/s)

ERM =

n  

Mpred − Mexp

2

(1.21)

i=1

ERT =

 n   θpred − θexp 2 i=1

θpred

(1.22)

Analyzing the convective heat and mass transfer coefficients as a function of drying temperatures, it is noted that there is an increase in the value of these coefficients with increasing drying temperature. Increasing the temperature implies an increase in the drying and heating rates of the brick, which makes it possible to reach its thermal and hygroscopic equilibrium conditions faster. The small values of the heat transfer coefficient are equivalent to the free convection heat transfer condition.

1.4 Concluding Remarks From the studies performed, it can be concluded that: (a) The drying process at high temperatures takes place in a shorter process time. (b) Parameters for third-degree polynomial functions describe well the behavior of volume and surface area during the drying process. (c) At high temperatures, the volumetric variation was 20.57% while the surface area varied by 5.83%. This point explains that the drying process was done correctly as the integrity of the brick shape was maintained. It is important to evaluate the behavior of the brick shape during drying, as this can make process control and therefore avoid possible defects in the parts. (d) The mathematical modeling developed taking into account the dimensional variations during the drying process was considered satisfactory. (e) The convective heat and mass transfer coefficients increased with the evolution of the drying temperature.

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Acknowledgments The authors thank CNPq, CAPES, FINEP (Brazilian Research Agencies), and PIBIC/CNPq-UFCG scientific initiation undergraduate program for the financial support, and the researchers cited in the text, who helped in the improvement of the investigation made.

References Almeida, G.S., Silva, J.B., Silva, C.J., Swarnakar, R., Neves, G.A., Lima, A.G.B.: Heat and mass transport in an industrial tunnel dryer: modeling and simulation applied to hollow bricks. Appl. Thermal Eng. 55, 78–86 (2013) Almeida, G.S., Tavares, F.V.S., Lima, W.M.P.B., Lima, A.G.B.: Energetic and exergetic analysis of the clay bricks drying in an industrial tunnel dryer. Def. Diff. Forum 369, 104–109 (2016) Araújo, M.V., Pereira, A.S., Oliveira, J.L., Brandão, V.A.A., Brasileiro Filho, F.A., Silva, R.M., Lima, A.G.B.: Industrial ceramic brick drying in oven by CFD. Mater 12(10), 1612–1634 (2019a) Araújo, M.V., Santos, R.S., Silva, R.M., Nascimento, J.B.S., Santos, W.R.G., Lima, A.G.B.: Drying of industrial hollow ceramic brick: a numerical analysis using CFD. Def. Diff. Forum 391, 48–53 (2019b) Araújo, M.V., Santos, R.S., Silva, R.M., Lima, A.G.B.: Drying of industrial hollow ceramic brick: analysis of the moisture content and temperature parameters. Def. Diff. Forum 380, 72–78 (2017a) Araújo, M.V., Delgado, J.M.P.Q., Lima, A.G.B.: On the use of CFD in thermal analysis of industrial hollow ceramic brick. Diff. Found. 10, 70–82 (2017b) Brito, M.K.T.: Theoretical study of heat and mass transfer in the drying of ceramic bricks with parallelepiped shape. Master’s dissertation in Mechanical Engineering, Federal University of Campina Grande, Campina Grande, Brazil (2016). (In Portuguese) Brito, M.K.T., Almeida, D.B.T., Lima, A.G.L., Rocha, L.A., Lima, E.S., Oliveira, V.A.B.: Heat and mass transfer during drying of clay ceramic materials: a three-dimensional analytical study. Diff. Found. 10, 93–106 (2017) Brooker, D.B., Bakker-Arkema, F.W., Hall, C.W.: Drying and Storage of Grains and Oilseeds. AVI Book, New York (1992) Cabral Jr., M., Motta, J.F.M., Almeida, A.S., Tanno, L.C.: Clay for red ceramics. Industrial rocks and minerals. CETEM 2(1), 747–770 (2008). (In Portuguese) Callister Jr., W.D.: Materials Science and Engineering: An Introduction, 7th edn. Wiley, USA (2007) Callister Jr., W.D., Rethwisch, D.G.: Fundamentals of Materials Science and Engineering: An Integrated Approach, 3rd edn. Wiley, USA (2008) Cavalcanti, M.S.L.: Development of ceramic masses for sanitary stoneware using flat glass residue as a flux in partial replacement to feldspar, Doctoral Thesis in Process Engineering, Federal University of Campina Grande, Campina Grande, Brazil (2010). (In Portuguese) Farias, V.S.O., Silva, W.P., Silva, C.M.D.P.S., Delgado, J.M.P.Q., Farias Neto, S.R., Lima, A.G.B.: Transient diffusion in arbitrary shape porous bodies: numerical analysis using boundary-fitted coordinates. In: Delgado, J.M.P.Q., Barbosa de Lima, A.G., Silva, M.V. (eds.) Numerical Analysis of Heat and Mass Transfer in Porous Media, vol. 27, pp. 85–119. Springer, Heidelberg, Germany (2012) Farias, V.S.O., Silva, W.P., Silva, C.M.D.P.S., Rocha, V.P.T., Lima, A.G.B.: Drying of solids with irregular geometry: numerical study and application using a three-dimensional model. Heat Mass Transfer 49(5), 695–709 (2013) Lima, A.G.B., Delgado, J.M.P.Q., Santos, I.B., Santos, J.P.S., Barbosa, E.S., Silva, C.J.: GBI method: A powerful technique to study drying of complex shape solids. In: Delgado, J.M.P.Q., Barbosa de Lima, A.G. (eds.) Transport Phenomena and Drying of Solids and Particulate Materials, vol. 48, pp. 25–431, Springer International Publishing, Heidelberg, Germany (2014)

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Lima, A.G.B., Silva, J.B., Almeida, G.S., Nascimento, J.J.S., Tavares, F.V.S., Silva, V.S.: Clay products convective drying: foundations, modeling and applications. In: Drying and Energy Technologies, vol. 63, pp. 43–70. Springer, Heidelberg, Germany (2015) Lima, E.S., Lima, W. M.P.B., Lima, A.G.B., Farias Neto, S.R., Silva, E.G., Oliveira, V.A.B.: Advanced study to heat and mass transfer in arbitrary shape porous materials: Foundations, phenomenological lumped modeling and applications In: Delgado, J.M.P.Q., Lima, A.G.B. (eds.) Transport Phenomena in Multiphase Systems, 93, pp. 181–217. Springer International Publishing, Heidelberg, Germany (2018) Lima, W.M.P.B.: Heat and mass transfer in porous solids with complex geometry via concentrated analysis: modeling and simulation. Master,s dissertation in Mechanical Engineering, Federal University of Campina Grande, Campina Grande, Brazil (2017) Macedo, R.F.: Continuous drying of bentonite clay in an industrial rotary dryer: modeling, simulation and experimentation. Master’s dissertation in Mechanical Engineering, Federal University of Campina Grande, Campina Grande, Brazil (2016) Oliveira, M.C., Bernils, M.F.: Environmental technical guide for the white ceramics and coatings industry. In: CETESB—Companhia de Tecnologia de Saneamento Ambiental, São Paulo (2006). (In Portuguese) Rodrigues Neto, A., Mota, J.A.: Local productive arrangements in the red ceramic industry: a case study in Brazil Northeastern. Rev. Econ. NE. 47(1), 127–142 (2016). (In Portuguese) Santos, J.P.S.: Drying of ceramic materials with complex shape: a theoretical study via CFX. Doctoral Thesis in Process Engineering, Federal University of Campina Grande, Campina Grande, Brazil (2018) Santos, R.S., Farias Neto, S.R., Lima, A.G.B., Silva Jr., J.B., Silva, A.M.V.: Drying of ceramic bricks: thermal and mass analysis via CFD. Diff. Found. 25, 133–153 (2020) SEBRAE (Serviço Brasileiro de Apoio às Micro e Pequenas Empresas). Red Ceramics: Overview of the market in Brazil. http://www.bibliotecas.sebrae.com.br/chronus/ARQUIVOS_CHRONUS/ bds/bds.nsf/b877f9b38e787b32594c8b6e5c39b244/$File/5846.pdf. Accessed 13 Jan 2019 Silva, A.A., Nascimento, J.J.S., Lima, A.G.B: Analytical study of ceramic tiles drying using the Galerkin-based integral method and Dirichlet boundary condition Rev. Eletrônica Mater. Process. (UFCG) 4(2), 48–55 (2009). (In Portuguese) Silva, A.M.V.: Drying of industrial ceramic blocks: Modeling, simulation and experimentation. Doctoral thesis in Process Engineering, Federal University of Campina Grande, Campina Grande, Brazil (2018) Silva, J.B.: Simulation and experimentation of the drying of holed ceramic bricks. Doctoral Thesis in Process Engineering, Federal University of Campina Grande, Campina Grande, Brazil (2009). (In Portuguese) Silva, V.S.: Heat and mass transfer in complex shaped materials via the lumped analysis method. Case study: drying of ceramic materials. Doctoral Thesis in Process Engineering, Federal University of Campina Grande, Campina Grande, Brazil (2016). (In Portuguese) Silva, J.B., Almeida, G.S., Lima, W.C.P.B., Neves, G.A., Lima, A.G.B.: Heat and mass diffusion including shrinkage and hygrothermal stress during drying of holed ceramics bricks. Def. Diff. Forum 312–315, 971–976 (2011) Silva, W.P., Farias, V.S.O., Neves, G.A., Lima, A.G.B.: Modeling of water transport in roof tiles by removal of moisture at isothermal conditions. Heat Mass Transfer 48(5), 809–821 (2012) Silva, V.S., Delgado, J.M.P.Q., Lima, W.M.P.B., Lima, A.G.B.: Heat and mass transfer in holed ceramic material using lumped model. Diff. Found. 7, 30–52 (2016) Strumillo, C., Kudra, T.: Drying: Principles, Science and Design. Gordon and Breach Science Publishers, New York (1986) Tavares, F.V.S., Farias Neto, S.R., Barbosa, E.S., Lima, A.G.B., Silva, C.J.: Drying of ceramic hollow bricks in an industrial tunnel dryer: a finite volume analysis. Int. J. Multiphys. 8(3), 297–312 (2014)

Chapter 2

Vegetable Fiber Drying: Theory, Advanced Modeling and Application J. F. Brito Diniz, A. R. C. de Lima, I. R. de Oliveira, R. P. de Farias, F. A. Batista, A. G. Barbosa de Lima, and R. O. de Andrade

Abstract This chapter aims to study the drying of sisal fibers. The interest in this type of material is related to its high mechanical performance. Several important topics such as theory, experiments, lumped and distributed mathematical modeling, and technological applications of fibers are presented and discussed. Emphasis is given to advanced distributed modeling that describes the heat and mass transfer in a wet fiber bed during drying. The model includes different effects such as bed porosity, fiber and bed moisture, coupling between heat and mass transport, and conduction, convection, and evaporation heat transfer. Results of fiber drying and heating kinetics, temperature distribution, and water vapor concentration in the fibrous bed are presented, compared with experimental data and analyzed.

J. F. Brito Diniz Department of Mathematics, Federal University of Campina Grande, Av. Aprígio Veloso 882, Bodocongó, 58429-900 Campina Grande, PB, Brazil e-mail: [email protected] A. R. C. de Lima · I. R. de Oliveira · A. G. B. de Lima (B) · R. O. de Andrade Department of Mechanical Engineering, Federal University of Campina Grande, Av. Aprígio Veloso 882, Bodocongó, 58429-900 Campina Grande, PB, Brazil e-mail: [email protected] I. R. de Oliveira e-mail: [email protected] R. O. de Andrade e-mail: [email protected] R. P. de Farias Department of Agriculture Science, State University of Paraiba, Catolé do Rocha, PB 58884-000, Brazil e-mail: [email protected] F. A. Batista Department of Physics, State University of Paraiba, R. das Baraúnas, 351, Campina Grande, PB 58429-500, Brazil e-mail: [email protected] © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 J. M. P. Q. Delgado and A. G. Barbosa de Lima (eds.), Transport Processes and Separation Technologies, Advanced Structured Materials 133, https://doi.org/10.1007/978-3-030-47856-8_2

31

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Keywords Drying · Sisal fiber · Experimental · Simulation · Finite-Volume

2.1 Introduction Vegetable fibers are structural tissues of plants. It is a body consisting of fibers (solid) and pores (filled with fluid) as illustrated in Fig. 2.1. The main chemical components of vegetable fibers are polar substances such as cellulose, hemicellulose (or polyoses), and lignin, with lower percentages of other components such as pectin, proteins, wax, inorganic salts, and other water-soluble substances. Its chemical composition varies slightly according to cultivation region, soil type, and climatic conditions (Silva 2003). Vegetable fiber is made up of several elemental fibers strongly bonded together by a resinous material consisting primarily of lignin. Each elemental fiber is essentially a composite in which rigid cellulose microfibrils are encased in an amorphous matrix of lignin and hemicellulose. Lignin acts as a resinous material, uniting microfibrils, Fig. 2.1 Longitudinal micrographs of untreated sisal fiber obtained by Scanning Electron Microscope (SEM): a 100× magnification and b 200× magnification

2 Vegetable Fiber Drying: Theory, …

33

while hemicellulose acts as an interface between cellulose microfibril and lignin (Silva 2003). The structure of an elemental plant fiber consists of a thick wall formed by several microfibril spirals along the fiber axis, having a lumen in the center. There are several types of vegetable fibers such as caroá, curauá, pineapple leaf, juta, and sisal. Sisal is a plant of the cactaceae family with the scientific name Agave Sisalana Perrine, being cultivated in semi-arid regions, being resistant to aridity and intense sun. The main and best-known product of sisal is biodegradable yarn. The cycle of transformation of sisal into natural yarn begins at 3 years of plant life or when its leaves reach about 150 cm in length. Plant growth depends, among other factors, on water availability: The plant stores water in the rainy season (winter) to consume it in the dry season (summer). Its useful life is 6–7 years, and the leaves are cut every 6 months. Sisal can produce between 200 and 250 leaves before flowering, each leaf measuring 6–10 cm wide, 150–200 cm long, and containing approximately 700–1400 bundles of fiber ranging in length from 0.5 to 1.0 m. The leaf of sisal consists of a structure composed of approximately 4% fiber, 1% film (cuticle), 8% dry matter, and 87% water. Except fiber, these materials are considered processing waste, being used as organic fertilizer, animal feed, and by the pharmaceutical industry (Martin et al. 2009; Wei and Meyer 2014). The sisal fiber extraction process, which consists in the elimination of the pulp from the fibers, can be done manually, by maceration or by a mechanical process called decorticating (Silva 2008). In the decorticating process, the sisal leaves are crushed by passing between two blunt-bladed wheels (defibrillator), so that only the fibers remain (Silva 2008). In the Brazilian northeastern, the defibration is performed through a machine called “agave engine”. Benefited or industrialized, the sisal fiber generates more than half a million direct or indirect jobs through its service chain, which begins with the activities of crop maintenance, harvesting, leaf cutting, fiber shredding and processing, and finally with industrialization and varied use (Martin et al. 2009). It has been used in handicrafts, baling fodder, and ropes of various uses. Sisal is also used in the production of upholstery, paste for the cellulose industry, the production of tequila, decorative rugs, medicines, biofertilizers, animal feed, organic fertilizer, and sacks. Fibers can also be used in the automotive industry, replacing fiberglass. In recent years, with increasing awareness of environmental preservation and pollution control, interest in the use of natural fibers in polymer composite materials has increased significantly. In this context, the use of vegetable fibers as reinforcement in polymer composites, with the objective of totally or partially replacing synthetic fibers, has received attention from the researchers. This is because vegetable fibers have important advantages such as low cost, low density, high specific strength and stiffness, and low abrasiveness to process equipment are biodegradable, nontoxic, non-polluting, which reduces environmental problems. They also come from renewable sources and are available worldwide (Cruz et al. 2011; Melo Filho et al. 2013; Nóbrega et al. 2010; Zhou et al. 2014). However, plant fibers are very susceptible to humidity and temperature, which strongly affect their mechanical properties, especially when used as reinforcement in polymeric materials.

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J. F. Brito Diniz et al.

Because it is a lignocellulosic fiber, light and non-toxic, with high modulus and specific resistance, costing about ten times lower than fiberglass (inorganic fiber), causing less abrasion damage to equipment and molds, sisal can have its commercial value multiplied if used as reinforcement in polymer composites (Angrizani et al. 2006; Barreto et al. 2011). The use of sisal fiber in high performance composites requires study of the mechanical behavior of the fibers. There is a large discrepancy between the values reported in the literature for tensile strength and elastic modulus of sisal fiber. The variability in the properties of these fibers can be attributed to three main factors: test parameters and conditions, plant characteristics, and the cross-sectional measurement method. Among the parameters or test conditions that may influence the mechanical properties of the fibers, we can mention the precision of the instruments, the fiber length, the test speed, the types of claws used, and the sensitivity of the equipment itself. The characteristics of the plants themselves include the origin of the plant, age, type of processing (extraction process) as well as its microstructure. The measurement of the cross-section may also cause variation in the measurement of mechanical properties due to the indefinite shape of the section and the variation itself along the fiber (Silva 2008).

2.2 Drying of Sisal Fibers Sisal fibers when extracted from plants are moist, which requires drying them for later use in different applications. Drying is a thermodynamic process whereby a body’s moisture is reduced and its temperature increased by supplying energy. It involves complex phenomena of heat and mass transfer, momentum, and dimensional variations (Lima et al. 2016). The control of the drying process is of fundamental importance to determine the ideal drying conditions, minimizing product losses, and energy consumption. According to Ferreira et al. (2012), the cellulose polysaccharide chains are more tightly arranged with the removal of water during drying and thus the microfibrils come together in the dry state as a result of increased packaging. Fiber voids are progressively closed with drying and cannot be completely reopened with rehumidification. A direct consequence of reduced absorption is the decrease in fiber dimensional variation between the dry and saturated state. Thus, we have a greater dimensional stability of the fiber. Moisture transport from the interior to the material surface may occur as a liquid and/or vapor, depending on the type of product and the percentage of moisture present. The diffusion process should occur in a controlled way, avoiding high moisture and temperature gradients within the material that may affect material properties. Thus, understanding the mechanisms involved in diffusion is a fundamental requirement in the study of solutions that minimize such problems.

2 Vegetable Fiber Drying: Theory, …

35

2.2.1 Experimental Study In this work, the phenomena of heat and mass transport in fibrous bodies (sisal fibers) were investigated. Agave sisalana variety sisal fibers with average moisture content of 11.2% (db) were used. The fibers were submitted to oven drying with forced air circulation at 70 °C. Figure 2.2 illustrates the sisal fibers used in the experiment. Table 2.1 contains the fiber and drying air information. During the drying process, moisture loss was measured by taking the sample from the oven and periodically weighing (predefined intervals) using a 0.1 g precision digital electronic device, and the surface temperature was measured using an infrared thermometer. In the experiment, measurements were taken every 5 min until the mass had minimal variation (about 30 min), after 10, 15, 20, 25, and 30 min. Then, the measurements were changed every 60 min until the constant mass was reached. Fig. 2.2 Sisal fibers used in the experiments

T (°C)

70

RH (%)

6.89

Air

0.07

v (m/s)

0.1

2R1 (m) 0.05

2R1 (m)

Table 2.1 Experimental parameters of air and fibrous medium

0.1

2R1 (m) 0.11148

M o (db) 0.02015

M eq (db)

Fibrous medium 31.5

T o (o C)

67.3

T f (o C)

Process time 5.7

t (h)

36 J. F. Brito Diniz et al.

2 Vegetable Fiber Drying: Theory, …

37

Following, the sample was dried for 24 h to obtain the equilibrium mass and then for a further 24 h at 105 °C to obtain the dry mass.

2.2.2 Theoretical Study To describe (theoretically) the moisture and heat transfer within a fibrous medium and to analyze the effects of certain parameters on their mechanical properties, it is necessary that these transport phenomena within the fibrous medium be well represented by a mathematical model. Therefore, it is important to fully insert all effects within mathematical models so that the physical phenomenon can be described with great realism, and to increase the reliability of the results obtained. Analytical and numerical solutions to the transient diffusion problem for various geometries have been reported in the literature; however, there are few studies related to transient and three-dimensional problems, particularly those related to plant fibers and incorporating the effect of porosity inside the fibrous medium.

2.2.2.1

Lumped Modeling

To describe the drying behavior of the fibrous media (sisal fibers) and to predict it under different operating conditions, it is necessary to model the drying process. For this, mathematical models are used in an attempt to predict the drying and heating kinetics, which, as a rule, predominantly follow a falling rate period. Several empirical, semi-empirical, and theoretical models have been proposed to describe the drying process. Theoretical models take into account only internal resistance, while semi-empirical and empirical models (thin-layer drying models) take into account only external resistance to heat and moisture transfer between material and air (internal resistance is insignificant). Currently, there are few models that represent the drying of fibrous bodies, for this, the following models have been proposed: (a) Drying model

M = c1 exp(−k1 t) + c2 exp(−k2 t) + c3 exp(−k3 t)

(2.1)

(b) Heating model

T = c1 log2 [(k1 t + k2 )c2 · (k3 t + k4 )c3 ]

(2.2)

38

J. F. Brito Diniz et al.

Fig. 2.3 Geometric configuration of the physical problem

where M is the average moisture content on dry basis, T is the temperature, and t corresponds to the process time. The ci and ki parameters of the proposed thinlayer models were obtained by nonlinear regression analysis by the Quasi-Newton method using the STATISTICA 7.0 software. In addition, some statistical parameters (correlation coefficient and variance) were determined for each proposed model.

2.2.2.2

Advanced Distributed Modeling

Governing Equations This section presents the mathematical models required for the development of heat and moisture (vapor) transport simulations in porous solids, with particular reference to plant fibers. The entire mathematical formulation was developed in a fibrous region in the form of a parallelepiped, according to Figs. 2.2 and 2.3. (a) Vapor diffusion equation The porous media consists of a series of rigid and inert fibers. Water vapor is free to diffuse between the voids of the fiber bed (interfiber) and to be sorbed or desorbed by the fiber (intrafiber). In addition, changes in fiber bed volume due to moisture absorption/desorption may be neglected. To describe the water vapor concentration in the fiber bed throughout the process, it was considered: (a) that the vapor diffusion through the voids and across the fibers is proportional to the concentration gradient in the usual way. The diffusion through the pores will in many cases be greater than through the fibers, but even so, both processes can be represented by assuming that the vapor in the fiber is always in equilibrium with the surrounding air vapor and that the absorption isotherm has a linear behavior, given by Eq. (2.5); (b) that moisture absorption or desorption through the voids inside the fiber. The mass and heat conservation equations will be expressed in terms of a unit volume of the air-fiber mixture. Water vapor accumulates in the volume element in both voids (interfiber) and fibers (intrafiber). Thus, the differential equation that describes the phenomenon of vapor diffusion is as follows:   ∂M ∂C + (1 − ε)ρs ∇ · ε D  ∇C = ε ∂t ∂t

(2.3)

2 Vegetable Fiber Drying: Theory, …

39

where ε is the porosity of the fibrous medium, D  is the diffusion coefficient, M is the amount of moisture absorbed per unit mass of fiber, ρs is the fiber density, C is the water vapor concentration in the voids (inter fibers), and t is the time. (b) Heat conduction equation Changes in heat flux in the element arise from various processes: heat conduction into or out of the element, water phase change (sorption or desorption), changes in temperature of the solid and gaseous phases. The contribution due to the last cause is small and will be neglected. To determine the change in the temperature of the fiber volume element, it was considered: (a) heat conduction through air and fibers; (b) the heat released when moisture is absorbed by the fibers. Thus, the differential equation that describes the heat diffusion phenomenon is as follows: ρ · cP

∂T ∂M = ∇ · (K ∇T ) + h s ρ ∂t ∂t

(2.4)

where cp is the specific heat, K is the thermal conductivity of the material, h s is the enthalpy, and T is the temperature. In Eq. (2.4), the assumption that air energy is negligible compared to fiber is used. An important point to note is that both the vapor diffusion Eq. (2.3) and the energy Eq. (2.4) involve M (fiber moisture). This shows that the two processes, moisture transfer and heat transfer, are coupled, so in general, one process cannot be considered without considering the other simultaneously. (c) Equilibrium equation According to Crank (1975), one can always consider that the fiber comes into balance with its immediate surroundings. In addition, it is possible to assume linear dependence on temperature and moisture content and write: M = α + σ C − βT

(2.5)

where M is the amount of moisture absorbed per unit mass of fiber in kgvapor /kgdry fiber , C is the concentration of water vapor in the voids (interfibers) expressed in kg/m3 , T is the temperature in °C, and α, σ and β are constants. It is just a reasonable approximation over small ranges of moisture and temperature. The average value of the quantity of interest Φ can be obtained by a weighted average using the volume of each control volume as follows: Φ=

1 ∫ ΦdV VV

(2.6)

where Φ = C, T, or even M (Eq. 2.5) and V is the volume of the fibrous medium.

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J. F. Brito Diniz et al.

Considering a three-dimensional solid with vapor distribution C = C (x, y, z), temperature T = T (x, y, z), and moisture M = M (x, y, z), the transient equations of vapor diffusion (2.3) and heat conduction (2.4) in Cartesian coordinates are given by:        ∂ ∂C ∂ ∂C ∂ ∂C ∂C = D + D + D + SC [ε + (1 − ε)ρs σ ] ∂t ∂x ∂x ∂y ∂y ∂z ∂z (2.7) and ρ(cP + h s β)

       ∂ ∂T ∂ ∂T ∂ ∂T ∂T = K + K + K + ST ∂t ∂x ∂x ∂y ∂y ∂z ∂z

(2.8)

where ∂M ∂C ∂T =σ −β ∂t ∂t ∂t

(2.9)

D = ε D

(2.10)

S C = (1 − ε)ρs β

∂T ∂t

(2.11)

and S T = h s ρσ

∂C ∂t

(2.12)

In the models, the following initial and boundary conditions were used: • Initial condition:

C(x, y, z, t = 0) = Co ; (Mass transfer)

(2.13)

T (x, y, z, t = 0) = To . (Heat transfer)

(2.14)

• Boundary conditions for mass transfer:    ∂C  −D = h m C − Ceq , ∀(x = R1 , y, z, t > 0); ∂ x x=R1

(2.15)

2 Vegetable Fiber Drying: Theory, …

   ∂C  −D = h m C − Ceq , ∀(x, y = R2 , z, t > 0); ∂ y  y=R2    ∂C  −D = h m C − Ceq , ∀(x, y, z = R3 , t > 0).  ∂z z=R3

41

(2.16) (2.17)

• Boundary conditions for heat transfer:    ∂ T  = h c T − Teq , ∀(x = R1 , y, z, t > 0);  ∂ x x=R1    ∂ T  −K = h c T − Teq , ∀(x, y = R2 , z, t > 0); ∂y  −K

(2.18) (2.19)

y=R2

   ∂ T  −K = h c T − Teq , ∀(x, y, z = R3 , t > 0).  ∂z z=R3

(2.20)

Numerical Procedure Often a diffusive problem is so complex and contains intense nonlinear equations that it cannot be solved by analytical solutions. However, it can be solved by numerical methods. The numerical method consists of replacing a continuous domain by a discrete domain and the partial differential equation is replaced by several algebraic equations, one for each control volume of the discrete domain. One of the main advantages of this method is the possibility of finding numerical solutions for the diffusion equation for the most diverse situations, such as non-homogeneous and non-isotropic medium, variable volume and diffusivity, and any geometry and shape of the body (Maliska 2004). There are several numerical methods reported in the literature. The Finite-Volume Method (FVM) does not present problems of instability or convergence, ensuring that in each discretized volume, the property under study obeys the conservation law, giving a conservative characteristic. It is one of the most used in the discretization of partial differential equations. This method works with control volumes, thus preserving the finite-volume level. Therefore, FVM is widely used in solving problems involving heat and/or mass transfer, and fluid flow (Maliska 2004). In the finite-volume method, any continuous quantity can be approximated by a discrete model composed of a set of continuous step or linear functions, defined under a finite number of subdomains. Subdomains are called the control volumes and the nodal points are well known as centroid of the control volume. In this method, the partial differential equation that governs the phenomenon is numerically discretized by integrating it into elementary volumes and time, thus obtaining

42

J. F. Brito Diniz et al.

Fig. 2.4 Scheme used to exploit the inherent symmetry condition of the parallelepiped: a domain, b highlight to 1/8 of the parallelepiped, and c new domain

a system of algebraic equations, which must be solved by specific mathematical techniques. In order to obtain the numerical solution of the diffusion equation for a parallelepiped-shaped porous material, the diffusion process takes into account the following assumptions: • • • • • •

The thermophysical and mechanical properties are constant; The diffusion coefficient is constant throughout the diffusion process; The solid is homogeneous and isotropic; There is symmetry in each central plane of the solid; The only mechanism of water transport within the solid is diffusion; Convective boundary conditions at the solid surface, with moisture content and temperature depending on position and time; • The field of moisture content and temperature inside the body is uniform at the beginning of the process; • Convective mass and heat transfer coefficients are constant for all faces of the solid. Because the geometric shape of the parallelepiped is regular, one can use the symmetry condition and numerically solve the diffusive problem for only one symmetrical part of the domain, such as studying only the region illustrated in Fig. 2.4c. Figure 2.5 illustrates the discretized domain used for discretization of the governing equation. The numerical solution of the diffusion equation for a parallelepiped was obtained considering a fully implicit formulation. This formulation was chosen because it presents no instability or convergence problems.

2 Vegetable Fiber Drying: Theory, …

43

Fig. 2.5 Discretized three-dimensional domain with 27 types of control volumes

Considering Fig. 2.5, and integrating Eq. (2.7) into space and time, the following result is obtained for a control volume P: [ε + (1 − ε)ρs σ ]

CP − CP0 t

        ∂C  ∂C  ∂C  C ∂C  y z + DnC x z − Dw − DsC x y z = DeC    ∂x e ∂x w ∂y n ∂ y s       ∂C  ∂C  + DfC x y + S C x y z − DbC (2.21) ∂z f ∂z b

where superscript zero means that the term must be evaluated at time t prior to the time of interest, whereas terms without superscript are evaluated at the time of interest. The subscripts “e”, “w”, “n”, “s”, “f”, and “b” mean the east, west, south, north, front, and back boundaries, respectively, of a considered control volume, P is the nodal point, centered on this control volume, and N, S, E, W, F, and B refer to the neighbors to the north, south, east, west, front, and back, respectively (Figs. 2.5 and 2.6). From the analysis of Figs. 2.5 and 2.6, the following derivatives for Eq. (2.21) are obtained:  CE − CP ∂C  = (2.22)  ∂x e δxe  ∂C  CP − CW = (2.23) ∂ x w δxw

44

J. F. Brito Diniz et al.

Fig. 2.6 Internal control volume of nodal point P and its neighbors

 ∂C  ∂ y n  ∂C  ∂ y s  ∂C  ∂z b  ∂C  ∂z f

=

CN − CP δyn

(2.24)

=

CP − CS δys

(2.25)

=

CP − CB δz b

(2.26)

=

CF − CP δz f

(2.27)

Substituting Eqs. (2.22)–(2.27) in Eq. (2.21) and arranging terms, we have for the internal control volumes the following algebraic equation: Ap C P = Ae C E + Aw C W + An C N + As C S + Af C F + Ab C B + B

(2.28)

where y z y z x z x y z + DeC + DwC + DnC t δxe δxw δyn x z x y x y + DsC + DfC + DbC (2.29) δys δz f δz b

Ap = [ε + (1 − ε)ρs σ ]

Ae = DeC

y z δxe

(2.30)

Aw = DwC

y z δxw

(2.31)

An = DnC

x z δyn

(2.32)

As = DsC

x z δys

(2.33)

2 Vegetable Fiber Drying: Theory, …

45

Fig. 2.7 Scheme used to identify the symmetry condition in the parallelepiped (new domain)

Af = DfC

x y δz f

(2.34)

Ab = DbC

x y δz b

(2.35)

B = [ε + (1 − ε)ρs σ ]

x y z 0 CP + S C x y z t

(2.36)

Equation (2.28) has important physical significance. The coefficients Ae , Aw , An , As , Af , and Ab represent the conductance between point P and its neighbors. The term CP0 represents the influence of the value of variable C at the previous time on its value at the present time. In this equation, V = x y z is the volume of the infinitesimal element considered in Fig. 2.6. Equation (2.28) is applied to all points within the computational domain except boundary points, where boundary conditions must be incorporated into the formulation. In this case, volumes adjacent to the body surface, called boundary control volumes, are used. For such volumes, the integration of the conservation equation is preceded, as described above, considering the existing boundary conditions. For example, in the new domain under study (Fig. 2.4c), considering the symmetry condition, it is found that the mass flux in the west, back, and south boundaries is zero, in    other words, Cw = 0, Cb = 0, and Cs = 0 (Fig. 2.7). Consequently, for the proposed discretization, with boundary condition of the third type, it is sufficient to impose h mw = h mb = h ms = 0, (where h mw = h mb = h ms = h m ). Similarly, to what was obtained for the vapor diffusion equation as applied to the internal control volumes, the following algebraic equation for the heat transfer is obtained: Ap TP = Ae TE + Aw TW + An TN + As TS + Af TF + Ab TB + B where

(2.37)

46

J. F. Brito Diniz et al.

Fig. 2.8 Two control volumes with transport coefficients DPC e DEC

Ap =

y z y z x z x z ρ(cP + h s β) x y z + K wT + K nT + K bT + K eT t δxe δxw δyn δys x y x y + K fT + K bT (2.38) δz f δz b

B=

Ae = K eT

y z δxe

(2.39)

Aw = K wT

y z δxw

(2.40)

An = K nT

x z δyn

(2.41)

As = K sT

x z δys

(2.42)

Af = K fT

x y δz f

(2.43)

Ab = K bT

x y δz b

(2.44)

ρ(cP + h s β) x y z 0 TP + S T x y z t

(2.45)

The discretization of the diffusion equation requires knowledge of the D C values not only at the nodal point, but at the east, west, north, south, front and back faces of each control volume, as illustrated in Fig. 2.8. The value of DeC is the value of the property at the interface of the nodal points. So, it is the value of D C on the common face between P and E. Such value is given by:

2 Vegetable Fiber Drying: Theory, …

DeC =

47

DEC DPC (1 − f d )DPC + f d DEC

(2.46)

dP dP + dE

(2.47)

where fd =

where dP e dE are the distances from the interface “e” to the nodal points P and E, respectively. Considering a uniform mesh, we have that f d = 1/2, since in this case dP = dE . Thus, Eq. (2.46) results in: DeC =

2DEC DPC DEC + DPC

(2.48)

The average value of C can be obtained by a weighted average using each control volume as follows (Eq. 2.6): C=

npx−1 npy−1 npz−1 1     Ci jk Vi jk V i=2 j=2 k=2

(2.49)

with   

npx−1 npy−1 npz−1

V =

i=2

j=2



Vi jk

(2.50)

k=2

where V is the volume of the solid, i, j, and k define the position nodal point, in the  control volume considered, Vi jk is the volume value of this elemental volume, npx − 2, npy − 2, and npz − 2 define the number of control volumes along the x, y, and z directions, respectively. In the case of a uniform mesh, it’s given: one-dimensional approach is described as: 

Vi jk = x y z

(2.51)

To solve the systems of algebraic equations generated by Eqs. (2.7) and (2.8), a computer code using Mathematica® software was developed. In it, the systems of linear equations are iteratively solved using the Gauss–Seidel method. It was assumed that the numerical solution converged when, starting from an initial condition, the following criterion was met at each nodal point in the computational domain at a certain time:   n+1 Φ − Φ n  ≤ 10−8

(2.52)

48

J. F. Brito Diniz et al.

where Φ can be C or T, and n represents the nth iteration at each time point. This criterion from the physical and numerical point of view is sufficiently precise to guarantee the physical realism of the obtained results. To obtain the results it was considered a numerical mesh of 20 × 20 × 20 nodal points and a t = 20 s. These parameters were obtained after a mesh and time step refining was performed. A program operation flowchart in block diagram form is shown in Fig. 2.9.

Estimation of Process Parameters (a) Estimation of transport coefficients With Eqs. (2.1) and (2.2) adjusted, “data capture” moments were established throughout the process in which the average moisture content and temperature could be determined so that the distribution of these points was approximately uniform. Subsequently, these equations were used in the computer code to adjust the diffusive and convective transport coefficients. Transport coefficients were obtained by varying their values to minimize the sum of quadratic deviations between predicted and experimental results. Deviations between experimental and calculated values and variance were obtained as follows: ERMQ =

n   2 Φi,Num − Φi,Exp

(2.53)

i=1

ERMQ  S¯ 2 =  n−n

(2.54)





where n is the number of experimental points and n number of adjusted parameters (number of degrees of freedom). The initial value of the convective heat transfer coefficient  used  during the adjust¯ , Reynolds (Re), and ment was obtained using the correlations for the Nusselt Nu Prandtl numbers (Incropera and De Witt 2002), applied to a plane plate as follows: 

h c, j =

¯ Nuk Rj

(2.55)

where R j can be R1 or R2 or R3 (Fig. 2.2), and 1/ 3 ¯ j = 0.664Re1j / 2 Pr Nu

(2.56)

2 Vegetable Fiber Drying: Theory, …

Fig. 2.9 Computational algorithm diagram

49

50

J. F. Brito Diniz et al.

being ρv R j μ

Re j =

(2.57)

with validity range (5 × 105 < Re ≤ 1 × 108 ). The equation used to obtain the initial convective mass transfer coefficient   calculated for air was obtained using the correlations for Sherwood numbers Sh and Schmidt (Sc), as follows (Incropera and De Witt 2002): 

h m, j =

ShDAB Rj

(2.58)

where DAB is the diffusivity of water vapor in the air, and the Sherwood number given by: Sh j = 0.664Re j / Sc1/ 3 1 2

(2.59)

To obtain the apparent density of the samples, we used the following equation: ρsample =

m fiber Vsample

(2.60)

The equations used to obtain the thermal conductivity and the specific heat calculated for the sample are given as follows: ksample = (1 − ε)k(fiber) + εk(air)

(2.61)

cp,sample = (1 − ε)cp,(fiber) + εcp,(air)

(2.62)

and

(b) Estimation of equilibrium equation parameters Considering the equilibrium equation, Eq. (2.5) can also be written as follows: M = a1 + a2 T

(2.63)

where a1 = α + σ C e a2 = −β. With the values of T eq and M eq for each experimental condition, a linear fit of Eq. (2.63) to the experimental data was performed using the Quasi-Newton numerical method using the Statistica® Software, with a convergence criterion of 0.000099, from which we obtained the values of a1 and a2 parameters.

2 Vegetable Fiber Drying: Theory, …

51

Table 2.2 Thermophysical and process parameters of the sample and fiber used in the simulation for T = 70 °C C o (kgvapor /m3 )

C eq (kgvapor /m3 )

ρ(sample) (kg/m3 )

ρ(fiber) (kg/m3 )

0.04125

0.01381

81.956

1450.00

cp,(sample) (J/kg K)

cp,(fiber) (J/kg K)

k(sample) (W/m K)

k(fiber) (W/m K)

961.617968

149.65

0.03141034

0.067

On the other hand, the equation of state for an ideal gas is given by: P = ρair · R · T

(2.64)

where P is the atmospheric pressure, ρ is the density, R is the particular gas constant (atmospheric air), and T is the drying temperature. Thus, air density within the fibrous medium can be determined as follows: ρair =

P R·T

(2.65)

To calculate the equilibrium water vapor concentration in the fibrous medium (between the fibers), consider the following formula: Ceq = ρair · UA

(2.66)

where UA is the air absolute humidity in voids, obtained from the temperature and relative humidity of the drying air. So, using the data from C eq and M eq and the value of β obtained from the adjustment (Eq. 2.63), there is a system of equations with two unknowns, which allows obtaining the α and σ parameters.

Thermophysical Properties of Materials Table 2.2 presents some thermophysical properties and process parameters of the fibrous medium for the drying experiment. These data were used in the simulation. Table 2.3 presents some thermophysical properties and process parameters of the drying air at atmospheric pressure for the experiment. These data were used to obtain the parameters of Table 2.4 and the transport coefficients reported in Table 2.7. Table 2.4 presents some calculated transport parameters. These data were used to determine the transport coefficients reported in Table 2.7.

52

J. F. Brito Diniz et al.

Table 2.3 Thermophysical and process parameters of drying air at atmospheric pressure for T = 70 °C ρair (kg/m3 )

k (W/m K)

μ (N s/m2 )

1.020411   DAB m2 /s

0.028260

31.1693 × 10−6

UA (kgvapor/ kgdry air )

UAsat (kgvapor/ kgdry air )

19.96557 × 10−6

0.01353746

0.2765

cp (kJ/kg K)

hs (kJ/kg)

Pr

Sc

1.033492

2333.26

0.73016

0.627739

Table 2.4 Transport parameters calculated for the experimental test in T = 70 °C

Dimensionless parameters Rex

Re y

Rez

127.7713

63.8856

127.7713

Nux

Nu y

Nuz

6.75860

4.77905

6.75860

Shx

Sh y

Shz

6.42655

4.54425

6.42655

2.2.3 Results 2.2.3.1

Lumped Analysis

From the analysis of the experimental data obtained, it was observed that the moisture content decreases with time. In general, the drying rate increases with higher drying air temperature and lower air relative humidity. Therefore, increasing the drying air temperature resulted in a shorter processing time. The surface temperature of the fibrous medium varied during the drying process and reached thermal equilibrium faster at the highest drying temperature. Since the surface temperature of the fiber rises during drying, the process occurs at a falling drying rate, i.e., the migration rate of water from the fiber to its surface is less than the water removal rate from the surface by the heated air. The fiber temperature increases as the drying rate decreases. Similar results have been reported for different researchers (Zhou et al. 2014; Santos et al. 2017). It is noticed that at the end of the process, the drying rate tends to zero when the moisture content approaches the hygroscopic equilibrium condition and the fiber temperature stabilizes, that is, the fiber approaches its thermal equilibrium. All details of the results and statistical parameters obtained with the model adjustments to the experimental data are presented in Tables 2.5 and 2.6. Figure 2.10 shows the fitted curves for the average moisture content of the samples versus drying time.

2 Vegetable Fiber Drying: Theory, …

53

Table 2.5 Parameters of Eq. (2.1) obtained after adjustment to experimental data of sisal fiber moisture content T (°C) 70

Parameters c1

k 1 (min−1 )

c2

k 2 (min−1 )

c3

k 3 (min−1 )

0.779910

0.031757

0.022230

0.000215

0.011298

0.243766

R (kg/kg)2

Proportion of variance (kg/kg)2

Loss function (kg/kg)2

0.99995

0.99989

0.000001574

Table 2.6 Parameters of Eq. (2.2) obtained after adjustment to experimental data of sisal fiber temperature T (°C) Parameters c1 (°C) 70

k 1 (min−1 )

0.536510 0.000406

k 2 (–)

−5.85983

0.006903 8.947182 13.39565

R (°C)2

Proportion of Loss variance function (°C)2 (°C)2

0.99737

0.99474

c3 (–)

k 3 (min−1 ) k 4 (–)

c2 (–)

3.234171

6.887782641

__ T = 70o C Model: M = c1 exp(-k1 t) + c2 exp(-k2 t) + c3 exp(-k3 t) __ M = (0.077991) exp[-(0.031757) t]+(0.02223) exp[-(0.000215) t]+(0.011298) exp[-(0.243766) t] 0.12 0.10

M (kg/kg)

0.08 0.06 0.04 0.02 0.00

0

100

200

300

400

500

t (min)

Fig. 2.10 Average moisture content of the sample during drying at 70 °C. Experimental (ooo) and predicted (__)

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T s = 70oC Model: T = c1 log2[(k1 t + k2)c2 (k3 t + k4 )c3] T = (0.53651) log2{[(0.000406) t + (0.006903)] (-5.8598) [(13.3956) t + (3.23417)](8.94718)}

70 60

T (ºC)

50 40 30 20 10 0

0

100

200

300

400

500

t (min)

Fig. 2.11 Sample surface temperature during drying at 70 °C. Experimental (ooo) and predicted (__)

Figure 2.11 shows the curves of the fiber surface temperature versus drying time. From the analysis of Tables 2.5 and 2.6 and Figs. 2.10 and 2.11, it can be seen that a good fit was obtained, with correlation coefficient R above 0.988, in all cases.

2.2.3.2

Equation Equilibrium Analysis

Based on the methodology presented before, we obtained the following equilibrium equation: M = 0.044702 + 2.217337C − 0.000784T

(2.67)

Equation (2.67) describes the linear dependence of temperature (T ), water vapor concentration (C), and moisture content (M), fundamental for the coupling between the vapor diffusion equation and the heat conduction equation. A correlation coefficient of 0.99 was obtained in this regression.

2.2.3.3

Distributed Analysis

Drying and Heating Kinetics In this topic will be presented the results of drying and heating kinetics, and distribution of moisture content and temperature inside the fibrous medium, for drying air

2 Vegetable Fiber Drying: Theory, …

55

Table 2.7 Initial values of the transport coefficients used during the nonlinear regression (T = 70 °C) Parameters 





h mx (m/s)

h my (m/s)

h mz (m/s)

4.00622 × 10−3    h cx W/m2 K

5.66564 × 10−3    h cy W/m2 K

4.00622 × 10−3    h cz W/m2 K

3.81996

5.40224

3.81996

Table 2.8 Parameters Estimated transport coefficients for the drying air temperature T = 70 °C Parameters h mx (m/s) 5.0622 × 10−4

h my (m/s)

21.6564 × 10−4   D  (m2 /s) α m2 /s

1.612 × 10−6

h mz (m/s)

    h cx W/m2 K h cy W/m2 K

  h cz W/m2 K

5.0622 × 10−4

5.538942

7.833248

5.538942

ERMQM (kg/kg)2

S M (kg/kg)2

ERMQT (°C)2

S T (°C)2

7.52571 × 10−7

967.74872

9.976791

3.9855592 × 0.0000722468 10−7

2

2

Fig. 2.12 Comparison between numerical and experimental values of the average moisture content of the fibrous medium as a function of time (T = 70 °C)

temperature of 70 °C. Later, on Tables 2.7 and 2.8 will be presented and discussed the values of the diffusive and convective transport coefficients obtained after comparing the predicted data with the experimental data for the experiment performed. Figure 2.12 presents the comparison of the numerical and experimental values of the average moisture content (on dry basis) of the fibrous medium as a function of

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Fig. 2.13 Comparison between numerical and experimental values of the fibrous medium surface temperature as a function of time (T = 70 °C)

time for the air temperature of 70 °C. From the analysis of this figure, we can see that data of there is a good agreement between the numerical and experimental the average moisture contents throughout the drying process. The temperature behavior of sisal fibers during drying was described by Eq. (2.4). The properties of the fibrous medium, which are thermal conductivity (k) and specific heat (cp ) were obtained using Eqs. (2.61) and (2.62), respectively. On the surface of the fibrous medium occurs both convective heat transport and heat transfer associated with moisture evaporation. Taking into account only the convective heat transport on the surface of the fibrous medium, the boundary condition of third kind was considered, in which the heat flux on the surface of the fibrous medium is proportional to the difference between the surface temperature of the fibrous medium and the drying air temperature (equilibrium temperature). The proportionality constant hc (convective heat transfer coefficient) was obtained applying the best square error technique between the numerical and experimental data of fiber temperature during drying at 70 °C. The best value of hc corresponds to the lowest value of S¯ 2 . Figure 2.13 shows the comparison of numerical and experimental values of the fibrous medium surface temperature as a function of time for the temperature of 70 °C. From the analysis of this figure, we can note that there is good agreement between numerical and experimental surface temperature values in the first 4000 s of process. Since the surface temperature of the fiber rises during drying, the process occurs at a falling drying rate. The fiber temperature increases as the rate of drying decreases. It can be seen in Fig. 2.13 that at the end of the process, the drying rate tends to

2 Vegetable Fiber Drying: Theory, …

a) t = 200 s

57

b) t = 700 s

c) t = 8000 s

Fig. 2.14 Water vapor concentration distribution (kgvapor /m3 ) in the plane yz at x = 0.025 m (R1 /2) to a drying air temperature of 70 °C

zero when the moisture approaches the equilibrium moisture content and the fiber temperature stabilizes, that is, the fiber approaches its equilibrium thermal. Analyzing the result, it can be stated that the model used to describe the heating kinetics of the fibrous medium, considering constant volume and transport parameters can be considered satisfactory, even if a discrepancy between the simulated and experimental data is perceived from t = 4000 s. This discrepancy may have been caused by errors in the temperature measurement process during the experiments or even by the measuring device itself. These differences observed between the simulated results and the experimental data of the temperature after the process 4000 s may also indicate that the convective heat coefficient hc from this time should be lower than in the initial moments. In this case, the hypothesis of hc variable throughout the process would be a more appropriate choice.

Water Vapor Concentration and Temperature Distributions Figures 2.14 and 2.15 show the distribution of water vapor concentration and temperature inside the fibrous medium, analyzed in the planes x = 0.025 m (R1 /2), for three times 200 s, 700 s, and 8000 s, respectively. It is important to remember that all results are plotted to 1/8 of the fibrous body volume, due to the symmetry that exists in the physical problem and in the geometry of the sample. By analyzing these figures, we can see that water vapor concentration presented the highest results in the central regions of the body at any time. The decrease in vapor concentration over time at any position was also noted, tending toward its equilibrium value for sufficiently long drying times (Fig. 2.12).

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a) t = 200 s

b) t = 700 s

c) t = 8000 s

Fig. 2.15 Temperature distribution (°C) in the plane yz at x = 0.025 m (R1 /2) to a drying temperature of 70 °C

Temperature has the lowest results in the central regions of the body at any time. The temperature also increases over time in any position, tending to its equilibrium value for sufficiently long drying times. This shows that heat flux occurs from the surface to the center of the material, in contrast to the moisture flux that occurs from the center to the surface of the material.

Estimation of Transport Coefficients (D, hm and hc ) The mass diffusion coefficient measures the tendency of water molecules to migrate from one region of high concentration to another of lower concentration. The higher the diffusion coefficient, the faster the diffusion of one species relative to another. This coefficient is directly related to the temperature and moisture content.   The initial convective transport coefficients h c and h m were obtained using Eqs. (2.55) and (2.58), respectively. From these initial values, an optimization process was performed to obtain the ideal values of the convective coefficients, as described earlier. Table 2.7 gives the obtained values of these parameters. Estimation of the transport coefficients h c , h m , and D was made by minimizing the sum of the squares of the residues, as mentioned earlier. Table 2.8 summarizes the obtained values of these coefficients as well as relative error and variance for the experimental test. In general, the numerical results showed a good agreement with the experimental data of moisture content and temperature of sisal fibers submitted to drying. The small errors and variances indicate that the methodology used to estimate the transport coefficients is satisfactory.

2 Vegetable Fiber Drying: Theory, …

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Specifically, with respect to the mass diffusion coefficient (D), it can be said that it is a multiplication of the vapor diffusion coefficient inside the fibrous medium by the bed porosity that is, D = ε D  . Since the porosity of the medium ε > 0.91, it can be noted that the drying of the fibrous medium resembles the drying of individual fibers. This in fact is almost true and can be proved by the value of the D  which is smaller than the DAB (Table 2.3) for the experimental test. This indicates that the vapor flux inside the fibrous medium is more difficult than in the air outside the fibrous medium, as expected. Comparison between the mass diffusivities of porous materials reported in the literature becomes difficult due to the lack of specific studies of these materials. In general, it is important to note that differences in mass diffusion coefficient can be attributed to the different factors, such as geometric considerations, calculation method, initial and equilibrium moisture contents, physical structure of the material used, and porosity of the material and boundary conditions.

2.3 Conclusions In this chapter, the physical problem of simultaneous heat and mass transfer in a parallelepiped porous bed has been studied. Due to the great importance of plant fibers, emphasis is given to drying of sisal fibers. A transient three-dimensional mathematical modeling, written in Cartesian coordinates was proposed, and its numerical solution based on the finite-volume method is presented and discussed. Results of the drying and heating kinetics and the vapor concentration and temperature distributions inside the fiber bed at different times of the process are presented, compared with experimental data and discussed. From the obtained results it can be concluded that: (a) The proposed lumped models can satisfactorily be used to describe the drying process of sisal fibers, considering that they presented good agreement with the experimental data, with correlation coefficient greater than 0.988; (b) the finite-volume method proved to be adequate to predict the phenomenon of heat and mass transfer within the fibrous medium; (c) drying of sisal fibers occurred at a falling drying rate; (d) there is a difference in water vapor concentration and temperature between the central region and the surface of the fibrous medium; (e) the largest water vapor concentration and temperature gradients are located in the regions near the vertices of the fiber porous bed. Since these regions are in more intense contact with the drying air, fibers located in these regions are more susceptible to deformation and thermal effects. Acknowledgments The authors thank CNPq, FINEP and CAPES (Brazilian Research Agencies), and the Federal University of Campina Grande (Brazil) for financial support.

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References Angrizani, C.A., Vieira, C. A.B., Zattera, A.J., Freire, E., Santana, R.M.C., Amico, S.C.: Influence of sisal fiber length and its chemical treatment on the properties of polyester composites. In: 17º CBECIMat—Brazilian Congress of Materials Science and Engineering, Foz do Iguaçu, PR, Brazil (2006). (In Portuguese) Barreto, A.C.H., Rosa, D.S., Fechine, P.B.A., Mazetto, S.E.: Properties of sisal fibers treated by alkali solution and their application into cardanol-based biocomposites. Compos. Part A: Appl. Sci. Manufac. 42(5), 492–500 (2011) Crank, J.: The Mathematics of Diffusion. Oxford University Press, London (1975) Cruz, V.C.A., Nóbrega, M.M.S., Silva, W.P., Carvalho, L.H., Lima, A.G.: B: An experimental study of water absorption in polyester composites reinforced with macambira natural fiber. Materialwiss. Werkstofftech. 42(11), 979–984 (2011) Ferreira, S.R., Lima, P.R.L., Silva, F.A., Toledo Filho, R.D.: Effect of sisal fiber humidification on the adhesion with portland cement matrices. Rev. Matéria 17(2), 1024–1034 (2012). (In Portuguese) Incropera, F.P., De Witt, D.P.: Fundamentals of Heat and Mass Transfer. Wiley, New York, USA (2002) Lima, A.G.B., Silva, J.B., Almeida, G.S., Nascimento, J.J.S., Tavares, F.V.S., Silva, V.S.: Clay products convective drying: Foundations, modeling and applications. In: Delgado, J.M.P.Q., Barbosa de Lima, A.G. (eds.) Drying and Energy Technologies. Series: Advanced Structured Materials, vol. 63, 63edn, pp. 43–70. Springer, Heidelberg (Germany) (2016 Maliska, C.R.: Computational Heat Transfer and Fluid Mechanics, p. 453. LTC, Rio de Janeiro, Brazil (2004) Martin, A.R., Martins, M.A., Mattoso, L.H.C., Silva, O.R.R.F.: Chemical and structural characterization of sisal fibers from Agave Sisalana variety. Polímeros: Ciência e Tecnologia 19(1), 40–46 (2009). (In Portuguese) Melo Filho, J.A., Silva, F.A., Toledo Filho, R.D.: Degradation kinetics and aging mechanisms on sisal fiber cement composite systems. Cem. Concr. Compos. 40, 30–39 (2013) Nóbrega, M.M.S., Cavalcanti, W.S., Carvalho, L.H., Lima, A.G.B.: Water absorption in unsaturated polyester composites reinforced with caroá fiber fabrics: modeling and simulation. Materialwiss. Werkstofftech 41(5), 300–305 (2010) Santos, D.G., Lima, A.G.B., Costa, P.S.: The effect of the drying temperature on the Moisture removal and mechanical properties of sisal fibers. Def. Diff. Forum 380, 66–71 (2017) Silva, J. S.: Drying and Storage of Agricultural Products. Aprenda Fácil, Viçosa,) 560 p. (2008). (In Portuguese) Silva, R.V.: Polyurethane Resin Composite Derived from Castor Oil and Vegetable Fibers. Doctoral Thesis in Science and Materials Engineering. University of São Paulo, São Carlos, SP, Brazil (2003) (In Portuguese) Wei, J., Meyer, C: Improving degradation resistance of sisal fiber in concrete through fiber surface treatment. Appl. Surf. Sci. 289, 511–523 (2014) Zhou, F., Cheng, G., Jiang, B.: Effect of silane treatment on microstructure of sisal fibers. Appl. Surf. Sci. 292, 806–812 (2014)

Chapter 3

Foam-Mat Drying Process: Theory and Applications E. R. Mangueira, A. G. Barbosa de Lima, J. de Assis Cavalcante, N. A. Costa, C. C. de Souza, A. K. F. de Abreu, and A. P. T. Rocha

Abstract The duck egg is an ideal product to increase human nutrition because it has a large amount of protein and vitamins. This chapter focuses on the foammat drying technique applied to duck egg white and yolk. The aim is to obtain powder of these materials after drying. Herein, different topics related to foundations, experiments, and lumped underling are presented and discussed. Drying experiments with and without emulsifiers to obtain stable foam were performed based on the complete factorial experimental design. The idea is to assist researchers, engineers, and academics in the understanding of this important topic related to food preservations. E. R. Mangueira · J. de Assis Cavalcante · N. A. Costa · C. C. de Souza Department of Chemical Engineering, Federal University of Paraiba (UFPB), 58051-900 João Pessoa, PB, Brazil e-mail: [email protected] J. de Assis Cavalcante e-mail: [email protected] N. A. Costa e-mail: [email protected] C. C. de Souza e-mail: [email protected] A. G. B. de Lima (B) Department of Mechanical Engineering, Federal University of Campina Grande, Av. Aprígio Veloso, 882, Bodocongó, Campina Grande 58429-900, PB, Brazil e-mail: [email protected] A. K. F. de Abreu Department of Technology and Development, Federal University of Campina Grande (UFCG), Sumé, PB 58540-000, Brazil e-mail: [email protected] A. P. T. Rocha Department of Food Engineering, Federal University of Campina Grande (UFCG), Campina Grande 58429-900, PB, Brazil e-mail: [email protected] © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 J. M. P. Q. Delgado and A. G. Barbosa de Lima (eds.), Transport Processes and Separation Technologies, Advanced Structured Materials 133, https://doi.org/10.1007/978-3-030-47856-8_3

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Keywords Foam-mat drying · Egg white · Egg yolk · Experimental · Theoretical

3.1 Drying Theory of Porous Materials 3.1.1 Fundamentals Drying is one of the oldest processes used by humans in food preservation. In general, it is a process in which water is removed from the product, involving simultaneous heat and mass transfer, and phase change of the water present in the food. In this process, a large amount of water is eliminated, consequently reducing weight of the product and the water activity that affects microbial growth, enzymatic reactions, and other reactions of chemical and physical reactions (Gava 2008). Two of the main factors governing drying are the removal of moisture from the product surface and the migration of moisture inside the product. The rate of moisture removal from the surface of the product is a function of both the surface area of the product exposed to air and the ability of the air to remove water of the surface (drying potential). The larger the surface area of the product, the larger the heat and mass exchange area with the airflow, facilitating the removal of water. In this step, the driving force related to water removal is directly related to the difference between the water vapor pressure at the material surface, the water vapor pressure in the air passing through the dryer. When the moisture is removed from the concentration gradient is created internally in the product, causing a migratory process of moisture from the center to the product surface (Fioreze 2004). The migration of moisture from the interior to the surface of the product depends on the particle size, its internal structure, and the driving force for this migration (concentration gradient). The larger the particle, the greater the distance to be traveled by heat from surface to center and by moisture from center to surface of the product to be evaporated. Different products have different internal structures, facilitating or hindering moisture migration, according to their porosity and the positive and negative charges of the carbon chains of the product. Increased driving force for water migration can be observed by increasing temperature and/or decreasing relative humidity of the drying air. This increases drying rate and the differences in concentrations between the inner and surface of the product (Fioreze 2004). In convective drying process, the water vapor present at the surface of the wet porous material is removed by air flow, either in natural or forced convection. For successful drying of biological products, it is well known that several variables must be taken into account during the process. The main variables are: air relative humidity; air temperature; air velocity; initial moisture of the product; final moisture of the product; product shape; and product type (Fioreze 2004). Some products, when dried, keep their physical and nutritional characteristics intact and return to their natural appearance or undergo few changes when rehumidified. This characteristic makes the drying process a viable way of preserving food for human consumption (Cornejo et al. 2003; Mayor and Sereno 2004).

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Fig. 3.1 Typical drying curve of a wet porous material

The drying process has evolved from the use of solar energy to current techniques, including but not limited to tray drying, tunnel drying, spray drying, rotary drying, freeze drying, osmotic dehydration, extrusion, fluidized bed drying, and microwave and radiofrequency use (Vega-Mercado et al. 2001). Therefore, there are several types of dryers; however, the choice of an appropriated method depends on several factors, among which stand out: product type, dryer availability, drying cost, and purpose of the dehydrated product (Sagar and Kumar 2010). When a wet porous material is subjected to the drying process, it can lose water at a constant velocity throughout the process. As drying progresses under fixed conditions, the rate of water removal decreases (Meloni 2003). This can be seen in Fig. 3.1 which shows the relationship of product moisture content with time in a typical drying curve. The importance of studying the drying curves of a wet porous product is that they indicate the rate of water removal at any time measured from the beginning of the process. A higher or lower slope of the curve indicates the ease or difficulty of removing water during the drying process (Meloni 2003). Drying curves can help in choosing a desirable drying time, with the aim of obtaining the product with the required moisture, and thus increasing a good quality product. The A–B section of the curve represents the initial stage of drying when the solid is heated or cooled and goes from the initial temperature T0 to the wet-bulb temperature T bu . This stretch is called the stabilization period, in which the surface conditions of the solid balance with those of the drying air. In general, this stage is described by a short period and, in general, is negligible of the total drying cycle (Ordonéz 2005). During this time the drying rate may increase or decrease under the effect of drying temperature. In B–C section, the drying rate and product temperature become constant. From this point, the temperature increases and the drying rate drops rapidly. At this stage, liquid water evaporates at the surface of the product in the same rate as the liquid water moves inside it. The surface of the solid remains wet and at a temperature close to the wet-bulb temperature of the drying air. This stage is known as the constant drying rate period and continues until it reaches the critical moisture that corresponds to the point C in Fig. 3.1 (Fellows 2000).

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Fig. 3.2 Typical drying rate curves

The C–D section is called the first falling drying rate period. It occurs when the velocity of water migration from the interior of the product to the surface is reduced, and therefore the partial pressure of water vapor at the surface of the product decreases progressively, and it begins to dry. In this period, the drying rate is limited mainly by the velocity of moisture movement within the solid, reducing the effects of external factors, especially air velocity. At this stage, water is entrapped in the structure of the moisture, and its movement through the dehydrated product is very slow. Therefore, for the drying rate to be significant, it is necessary to increase the temperature of the product to provide sufficient desorption heat and to raise the water vapor pressure inside of the material (Ordonéz 2005). The D–E stretch is well known as the second-rate period, where the product dries to reach the equilibrium moisture content (hygroscopic equilibrium condition). Variations in product moisture content overtime during the drying process give rise to the drying rate curve (Fig. 3.2). The drying curve is linked to heat and mass transfer phenomena (Strumillo and Kudra 1986). In the initial drying period, the solid and its surface are covered by a liquid layer. The constant drying rate period considers the mass transfer resistance and the limiting factors of the drying rate are the external conditions and the gaseous boundary layer at the surface of the product. In the falling drying rate period, the amount of moisture that reaches the surface of the material decreases gradually. As a result, the water vapor partial pressure on the material surface also decreases and the drying rate is controlled by the moisture transport that depends on the moisture concentration gradient inside the material (Mangueira 2017). In general, the falling drying rate period is almost always the only one observed for the drying of agricultural and food products.

3.1.2 Mathematical Modeling in Drying The study of drying kinetics aims at understanding the behavior of the product during the process and the prediction of drying time. On the one hand, drying experiments are

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of great importance, on the other hand, process modeling plays important role in the development and optimization of dryers, as well as enabling process standardization and the reduction of exhaustive drying tests, and predicting the drying behavior of various materials quickly. Predicting the falling drying rate is more complex than the constant drying rate and encompasses not only external heat and mass transfer mechanisms but also internal mechanisms in the product. The complexity of phenomena during drying leads researchers to propose numerous theories and multiple empirical formulas to predict the drying rate adequately. These theories can be summarized as being derived from two other theories: diffusional theory and capillary (Park et al. 2007). Diffusional theory is based on Fick’s law, which expresses mass flow rate per unit area as proportional to the water concentration gradient within the product. It is a model used for the falling drying rate period. However, the analytical solution of Fick’s diffusional model requires that boundary conditions be known and that the effective means diffusivity be specified. These limitations, in addition to the requirement of knowing the material geometry, often lead searchers to use empirical or semi-empirical models. Semi-theoretical models are generally derived from the simplification of a solution in series of Fick’s second law (Doymaz 2005). The empirical method is used for drying analysis using experimental data, which can be determined in laboratory and in the use of dimensionless analysis (Gouveia et al. 2002). This method is generally based on drying external conditions such as temperature, relative humidity, and drying-air velocity (Carlesso et al. 2007), without providing information on energy or mass transport within the product. From the drying data, it is possible to study the drying kinetics, with the aid of the drying characteristic curve of the product. In fact, to adequately product the drying, it is necessary to know the initial moisture content of the product to be drier, the relationship of water with the solid structure, and the mechanism of water migration from inside the material to its surface (Park et al. 2001). Some empirical and semi-empirical mathematical models are summarized in Table 3.1. These models can be fitted to experimental data by nonlinear regression using appropriately statistical software (Doymaz 2005; Biazus 2006; Marques Table 3.1 More common empirical and semi-empirical models used to describe the drying process

Name

Model

References

Henderson and Pabis

RU = a exp(−Kt)

Park et al. (2002)

Midilli et al.

RU = a exp(−Kt n ) + Midilli et al. (2002) bt

Page

RU = exp(−Kt n )

Page (modified)

RU = a exp(−Kt n )

Mangueira (2017)

Newton

RU = exp(−Kt)

Liu et al. (1997)

Zhang and Litchfield (1991)

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2009). In Table 3.1, RU = X/X o is the moisture content ratio of the product; t is the time; K is the drying coefficient; and a, b, and n are constants.

3.2 Foam-Mat Drying 3.2.1 General Foundations Foam-mat drying is a technique used to obtain powdered food products. In this technique, liquid or semi-liquid foods are transformed into stable foams by the addition or not of foaming agents (which is intended to keep the foam stable during the process) and incorporation of air, nitrogen, or other gases in blenders or other foam generation equipment (Brennan 1994). Foam is a colloidal dispersion in which gas is dispersed in a continuous liquid phase. The dispersed phase is referred to as the internal phase and the continuous phase is called the external phase (Baniel et al. 1997). Based on the ratio of dispersed phase to continuous phase, the foams can be classified into polyhedral foam and diluted bubbling foam. In polyhedral foams, the proportion is large, resulting in a large number of bubbles. As the number of bubbles increases, they push themselves to form a honeycomb structure. Egg white foam and beer foam are good examples of polyhedral foam. In diluted foams, the proportion is small; therefore, individual bubbles retain their spherical shape. Chocolate mousse is a good example of diluted bubbling foams (Prins 1988). The gas phase (usually air) is incorporated as evenly distributed small particles. The idea is to the texture and appearance of the product (Narchi et al. 2009). The foams have thin, flat, and liquid films or lamella between bubbles. The coverslips meet at a point called the plateau border (Fig. 3.3). As described in Fig. 3.3, gas bubbles are confined to the structures formed by the foam and plateau border coverslips. Other foam characteristics depend on the air interface that determines foam

Fig. 3.3 Schematic representation of a foam structure

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stability. The use of viscous liquids for foaming results in stable foaming. The foam texture is also influenced by proteins and surfactants as they help in the formation of stable foam (Vernon-Carter et al. 2001). The mechanical strength of the lamella determines the stability of the foam along with its air/water interface properties. If viscous liquids are used for foam production, they would produce more stable foams due to increased lamella elasticity (Dickinson 1998). Foam increases drying efficiency because it increases surface area and the heat and mass transfer. In addition, capillarity through the foam pores facilitates moisture loss. This makes drying in a foam mat approximately three times faster than drying in a similar liquid layer (Muthukumaran et al. 2008; Rajkumar et al. 2007). Foam thickness directly influences drying time, but this effect is greater for foods that are dehydrated in solid form (Kadam et al. 2010). After production, the foam is spread as a thin sheet or mat and exposed to hot airflow to be dried until the desired moisture content. Drying is performed at relatively low temperatures to form a thin porous layer which is disintegrated to produce a freeflowing powder. The larger surface area exposed to the drying air is the main cause of accelerated moisture removal (Brygidyr et al. 1977). During drying, moisture can be reduced to a level ranging from 1 to 5%, which prevents microbial and enzymatic deterioration. In addition to a substantial reduction in weight and volume, it minimizes packaging, storage, and transportation costs (Falade and Solademi 2010). Foam-mat drying is used in heat-sensitive foods as it requires lower dehydration temperatures and drying time due to a larger surface area exposed to air, and consequently, a faster drying rate, thus accelerating the water removal process and obtaining an easily rehydrated porous product (Karim and Chee-Wai 1999; Rajkumar et al. 2007). This technique has some advantages over other liquid drying techniques, such as simpler technique, lower operating cost and allows the use of lower temperatures, which better preserves the taste and nutritional value. According to Franco et al. (Franco 2015), the powder produced by this method is easily reconstituted, presenting characteristics of texture, color, taste, and nutritional composition very similar to the original material. This consequently increases the commercial possibilities of obtaining dehydrated products by the method, especially for heat-sensitive foods. The main disadvantages of this technique are related to the need for a large drying surface area to meet high production rates, which increases the investment cost (Francis 2000). Furthermore, additives can modify the taste, aroma, and color characteristics of the food. In addition, a lack of foam stability may occur during heating or drying processes (Karim 1999), thus some variables such as the chemical nature of the raw materials, soluble solids, type, and concentration of foaming agent have influenced the foam stability (Hart et al. 1963).

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3.2.2 Different Methods for Foam Formation The quantity and quality of the foam produced is determined by the different foaming techniques. The most common foaming methods are whipping, shaking, and bubbling (Dehghannya et al. 2018).

3.2.2.1

Whipping or Beating

Whipping is a process that involves adding a large amount of air to a known amount of liquid to generate foam (Lomakina and Mikova 2006). This can be accomplished using various devices such as mixers and homogenizers. These devices can mix a variety of food materials including fruits, vegetables, and liquids. In this process, the stirring air gets entrapped in a liquid. The amount of air entrapped in a liquid increases with increasing agitation. Incorporation of air into the liquid initially results in large bubbles, which upon further agitation are reduced to a smaller size, providing a homogeneous foamy structure to the food material. The final size of the air bubble depends on the stirring speed, equipment design, and rheological properties of the liquid. This technique is widely applied in the food processing sector and also for the basic study of foam (Dehghannya et al. 2018).

3.2.2.2

Shaking

In this method, the foam is generated by vigorous stirring of the liquid. The volume of foam produced by this method depends on the magnitude and frequency of agitation, and the shape and size of the container, temperature, type, and concentration of foaming agent (Arzhavitina and Steckel 2010). This method is slower compared to the whipping or bubbling method, as under similar agitation conditions they produce less foam volume (Lomakina and Mikova 2006).

3.2.2.3

Bubbling or Sparging

In this method, gas is injected into a known amount of liquid through small openings (Arzhavitina and Steckel 2010), where bubbles of uniform size are produced. The size of the bubbles can be controlled by adjusting the size of the opening through which air is injected. The volume of foam produced depends on the amount of liquid and foaming agent. In this method, the liquid can be completely foamed if a large amount of air is injected (Mounir 2017).

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3.2.3 Foaming Agents Food foams are composed of air and liquid and sometimes it is necessary to use an active agent on the surface (Kinsella 1981). A foaming agent is a surfactant that facilitates foaming when present in small amounts. The foaming agent reduces the surface tension between two liquid materials or between a solid and liquid material that results in foam generation. A good foaming agent must be able to stabilize the foam adequately and rapidly at low concentrations; effectively function over a wide range of pH values and ability to perform effectively in the presence of foam inhibitors such as fat, flavoring substances, and alcohol (Zayas 1997). The most commonly used foaming agents are egg albumin, milk protein, soy protein, and gelatin.

3.2.3.1

Egg Albumin

Egg albumin is a natural protein found in eggs with good foaming properties (Sangamithra et al. 2015). By rapidly beating the egg albumin, the air/liquid interface is denatured and interact with each other to form a stable film (Lomakina and Mikova 2006). Therefore, the whipping time required by egg albumin is comparatively shorter compared to other foaming agents. This implies that egg white proteins could be adsorbed faster at the air–liquid interface and more rapidly denatured than other proteins (Townsend and Nakai 1983).

3.2.3.2

Whey Protein

Whey protein is obtained from the dairy industry as a byproduct during cheese production (Tariq et al. 2003). This is one of the main sources of protein in industrial foods because of its properties, which include its use as an emulsifier, its nutritional content, and its ability to form gel and stable foam (Broch et al. 2014). Due to the higher solubility of whey protein in water and on its surface, whey protein has the ability to form a high-quality foam (Sangamithra et al. 2015). Whey protein has the ability to retard oxidation reactions in dry food materials. Furthermore, whey protein has a greater ability to bind flavored compounds, making it highly conducive to working with vegetables with sensitive and volatile constituents (Zhang et al. 2017).

3.2.3.3

Soy Protein

Soy protein can be obtained by removing soy oil at a lower temperature. Isolated soy protein is a highly purified form of protein, with a minimum protein content of 90% (Nishinari et al. 2014). Isolated soy protein has many functional properties such as

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solubility, water and fat absorption, water retention capacity, viscosity, foaming or whipping, emulsification, and gelling (Liu et al. 1958).

3.3 Applications: Drying of Egg White and Yolk of Duck Egg 3.3.1 Material Preparation Duck eggs were purchased from the popular commerce of the city of João Pessoa, Paraíba, Brazil. Following, they were cleaned and sanitized with chlorinated water (50 ppm) according to the industrial and sanitary inspection of products regulation of animal origin (Mapa 2005). After washing, the duck eggs were broken, and the egg white and yolk were separated manually. Then, they were weighed and beaten in a food mixer to obtain the foam.

3.3.2 Experimental Planning Experimental planning is used in basic and technological research, where many factors can be varied at the same time, and the analysis of variance (ANOVA) is used to determine which factors are statistically significant. The analysis of the significance of the parameters can also be performed using the F-test values, which can be obtained by the ratio between the quadratic means associated with the regression and the residuals. The use of the central point allows to add a third level for each factor, thus enabling the factorial study using the response surface methodology, as well as quantifying the significance of possible curvature and errors associated with individual effects and interactions between them. For the accomplishment of the experiments, a complete factorial experimental design was carried out 23 + 3 central points, with the input variables for the egg white one: temperature (50, 60, and 70 °C), agitation rate (6, 7, and 8 levels), and stirring time (4, 5, and 6 min), and for the yolk, the input variables were temperature (50, 60, and 70 °C), emulsifier concentration (Emustab®) (7, 10, and 13%),. and stirring time (4, 5, and 6 min). The following output variables were considered: final moisture of the product and drying time (min) for both. Tables 3.2 and 3.3 present the coded and actual values of the independent variables and the matrix of the complete factorial experimental design for egg white, respectively. Tables 3.4 and 3.5 present the coded and actual values of the independent variables and the matrix of the complete factorial experimental design for the yolk, respectively. The complete factorial experimental design methodology was used aiming the proposition of statistical models able to adequately predict the characteristics of the

3 Foam-Mat Drying Process: Theory and Applications Table 3.2 Coded and real values of the independent variables for the egg white

71

Independent variables

Levels −1

vag (level)

0

6

t ag (min) T (°C)

1

7

8

4

5

6

50

60

70

vag —stirring rate; t ag —stirring time; T —temperature

Table 3.3 Matrix of complete factorial experimental design 23 + 3 central points for the egg white

Table 3.4 Coded and actual values of independent variables for the yolk

Experiment

Independent variables vag (level)

t ag (min)

T (°C)

1

(−1) 6

(−1) 4

(−1) 50

2

(1) 8

(−1) 4

(−1) 50

3

(−1) 6

(1) 6

(−1) 50

4

(1) 8

(1) 6

(−1) 50

5

(−1) 6

(−1) 4

(1) 70

6

(1) 8

(-1) 4

(1) 70

7

(−1) 6

(1) 6

(1) 70

8

(1) 8

(1) 6

(1) 70

9

(0) 7

(0) 5

(0) 60

10

(0) 7

(0) 5

(0) 60

11

(0) 7

(0) 5

(0) 60

Independent

Levels

variables

−1

t ag (min)

4

5

6

C (%)

7

10

13

T (°C)

50

60

70

0

1

t ag —stirring time; T —temperature; C—emulsifier concentration

powder obtained after the foam-mat drying of the egg white and the yolk. Statistical analysis was performed using the Statistica® 12 software, where the obtained data were interpreted by analysis of variance (ANOVA), for the comparison of arithmetic means, calculating the main effects and interactions of the variables on the obtained responses.

72 Table 3.5 Matrix of complete factorial experimental design 23 + 3 central points for the yolk

E. R. Mangueira et al. Experiment

Independent variables

1

C (%)

t ag (min)

T (°C)

(−1) 7

(−1) 4

(−1) 50

2

(1) 13

(−1) 4

(−1) 50

3

(−1) 7

(1) 6

(−1) 50

4

(1) 13

(1) 6

(−1) 50

5

(−1) 7

(−1) 4

(1) 70

6

(1) 13

(−1) 4

(1) 70

7

(−1) 7

(1) 6

(1) 70

8

(1) 13

(1) 6

(1) 70

9

(0) 10

(0) 5

(0) 60

10

(0) 10

(0) 5

(0) 60

11

(0) 10

(0) 5

(0) 60

3.3.3 Experiment of Foam-Mat Drying The foam obtained after beating using the Arno Deluxe Planetary Mixer SX80 was placed in aluminum trays (Fig. 3.4) and set was placed in the oven at a constant temperature of 50, 60, and 70 °C. At regular time intervals, the trays were weighed Fig. 3.4 Foam of a egg white and b yolk of the duck egg arranged in a tray

a)

b)

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on a semi-analytical digital scale, accurate to ±0.01 g until they reached constant weight. The dried material was removed from the tray with the aid of spatulas, packed in polyethylene bags, and closed. In this stage, we studied the drying kinetics of the egg white and yolk, where the experiments were performed according to the complete factorial design 23 + 3 central points as described in Table 3.1. After completion of drying, some empirical models were fitted to the experimental moisture content data.

3.3.4 Analysis of Results 3.3.4.1

Duck Egg White

Duck egg white has good air incorporation capacity where the addition of a foaming and/or stabilizing agent is not required to produce a stable foam. This is due to the action of its proteins, which move through the aqueous phase and are spontaneously absorbed by the liquid–gas interface where the viscoelastic film is subsequently formed. The result of protein adsorption is related to reduction in surface tension, which improves the foaming ability, as well as the ability to encapsulate and retain incorporated air (Davis and Foegeding 2007). Table 3.6 shows the results of moisture content (X f ) and drying time (t f ) for the egg white of the duck at the end of foam-mat drying process. Figures 3.5, 3.6, and 3.7 show the drying kinetics for the duck egg white foam at temperatures of 50, 60, and 70 °C, respectively. Figures 3.8, 3.9, and 3.10 illustrate Table 3.6 Moisture content values on dry basis and drying time at the end of drying of the duck egg white Experiment

Independent variables

Dependent variables

vag (level)

t ag (min)

T (°C)

X f (db)

t f (min)

1

(−1) 6

(−1) 4

(−1) 50

0.1475

250

2

(+1) 8

(−1) 4

(−1) 50

0.4559

250

3

(−1) 6

(+1) 6

(−1) 50

0.0596

250

4

(+1) 8

(+1) 6

(−1) 50

0.0689

250

5

(−1) 6

(−1) 4

(+1) 70

0.1182

150

6

(+1) 8

(−1) 4

(+1) 70

0.4158

110

7

(-1) 6

(+1) 6

(+1) 70

0.0623

110

8

(+1) 8

(+1) 6

(+1) 70

0.0461

180

9

(0) 7

(0) 5

(0) 60

0.0224

180

10

(0) 7

(0) 5

(0) 60

0.1810

210

11

(0) 7

(0) 5

(0) 60

0.1896

130

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Fig. 3.5 Drying curves of duck egg white foam at a temperature of 50 °C

Fig. 3.6 Drying curves of duck egg white foam at a temperature of 60 °C

the specific water mass flowrate (m ˙  ) as a function of drying time, obtained for each experiment performed. After analysis of these figures, it can be observed that the drying curves presented both constant (approximately) and falling drying rate period. It can be observed that there was a variation in the drying time for the different temperatures, being the drying at 50 °C the longest, not exceeding 250 min, and the fastest, using the

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Fig. 3.7 Drying curves of duck egg white foam at a temperature of 70 °C

Fig. 3.8 Specific drying rate of duck egg white during foam-mat drying at temperature of 50 °C

temperature of 70 °C, drying approximately 180 min. This is due to the higher heat and mass transfer between air and foam layer, which proves that temperature positively influences the drying of egg white and that drying occurs with moderate velocity even at low temperatures. In addition, it was found that both levels and stirring time affect the moisture reduction of the product, especially at low temperatures. This implies that the dominant variable in the low-temperature foam-mat drying process is the heat and mass exchange area available in the product. However, at high temperatures, it is the

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Fig. 3.9 Specific drying rate of duck egg white during foam-mat drying at temperature of 60 °C

Fig. 3.10 Specific drying rate of duck egg white during foam-mat drying at temperature of 70 °C

temperature of the drying air that determines the phenomenon behavior. Obviously, in both cases, the drying-air potential is directly related to the relative humidity and the drying-air velocity (in the constant drying rate period). Pereira (Pereira 2015), in his studies on chicken egg white drying, reports values for the drying time, of approximately 360 min at temperatures of 50, 60, and 70 °C. From some mathematical models reported in Table 3.1, nonlinear regressions were made to the experimental data. For each drying condition, Table 3.7 shows the results of the statistical parameters obtained with this procedure. The best-fit model

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Table 3.7 Statistic parameters of the modified Page’s model after fitting to the experimental data for the drying of the duck egg white at 50, 60, and 70 °C Experiment

Parameters X 0 (db)

K (min−1 )

a (–)

n (–)

R2 (–)

S (–)

1

6.1017

0.0010

0.9878

1.4886

0.9988

0.0051

2

9.7615

0.0138

1.0394

1.0477

0.9945

0.0219

3

5.6573

0.0009

0.9599

1.4450

0.9965

0.0119

4

5.7980

0.0008

0.9678

1.5687

0.9984

0.0072

5

6.0155

0.0027

0.9766

1.5177

0.9988

0.0044

6

8.9434

0.0055

0.9872

1.3774

0.9994

0.0016

7

4.4871

0.0027

0.9681

1.5458

0.9981

0.0056

8

3.9617

0.0028

0.9684

1.5619

0.9983

0.0070

9

2.9781

0.0043

0.9822

1.3707

0.9976

0.0095

10

2.9775

0.0023

0.9774

1.4369

0.9969

0.0126

11

3.1212

0.0070

0.9955

1.2698

0.9979

0.0060

(modified model Page’s) was chosen taking into consideration the statistical analysis (coefficient of determination above 0.99 and estimation errors less than 0.01). The graph showing the result predicted by the Modified Page’s model and the experimental dimensionless moisture content at 50, 60, and 70 °C are illustrated in Figs. 3.11, 3.12, and 3.13, respectively.

Fig. 3.11 Comparison between predicted (modified Page’s model) and experimental results of the moisture content ratio during the foam-mat drying process of the duck egg white at 50 °C

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Fig. 3.12 Comparison between predicted (modified Page’s model) and experimental results of the moisture content ratio during the foam-mat drying process of the duck egg white at 60 °C

Fig. 3.13 Comparison between predicted (modified Page’s model) and experimental results of the moisture content ratio during the foam-mat drying process of the duck egg white at 70 °C

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3.3.4.2

79

Duck Egg Yolk

For the production of duck egg yolk foam, the Emustab® emulsifier (Sousa 2017) was required. In studies realized by Negreiros (Negreiros 2016) related to foam-mat drying chicken egg yolk, the egg white was used as emulsifier. In the duck egg yolk, it was not possible to produce the foam using the egg white because, despite the emulsifying characteristic of the duck egg yolk, this product has a high-fat content, which makes the formation of foam difficult. Table 3.8 shows the results of moisture content (X f ) and drying time (t f ), for the duck egg yolk at the end of drying process. Figures 3.14, 3.15, and 3.16 show the drying kinetics for the duck egg yolk foam at temperatures of 50, 60, and 70 °C, respectively. Figures 3.16, 3.17, and 3.18 illustrate the specific water mass flowrate (m ˙  ) as a function of time, obtained for each experiment performed. After analysis of these figures, it can be observed in Figs. 3.14, 3.15, and 3.16 that the drying curves present approximately constant and falling drying rate periods. It can also be observed that there was a variation in the drying time for the different temperatures, being the drying at 50 °C the longest, occurring in approximately 450 min, and the fastest, using the temperature of 70 °C, in a drying time approximately 250 min, almost half of drying realized at 50 ° C. With this, it was possible to prove the influence of temperature on foam-mat drying process. Approximate drying time values were found by Negreiros (Negreiros 2016) in their studies related to foam-mat drying of chicken egg yolk at the same temperatures. The experiments show a good reproducibility in the behavior of the drying curve to the center point cited in Table 3.5 (Experiments 9–11). Table 3.8 Moisture content values on dry basis and drying time at the end of drying for duck egg yolk Experiment

Independent variables C (%)

Dependent Variables

t ag (min)

T (°C)

X f (db)

t f (min)

1

(−1) 7

(−1) 4

(−1) 50

0.0144

450

2

(+1) 13

(−1) 4

(−1) 50

0.1406

450

3

(−1) 7

(+1) 6

(−1) 50

0.0058

450

4

(+1) 13

(+1) 6

(−1) 50

0.0024

390

5

(−1) 7

(−1) 4

(+1) 70

0.0008

210

6

(+1) 13

(−1) 4

(+1) 70

0.1400

180

7

(−1) 7

(+1) 6

(+1) 70

0.0124

250

8

(+1) 13

(+1) 6

(+1) 70

0.0065

210

9

(0) 10

(0) 5

(0) 60

0.0842

290

10

(0) 10

(0) 5

(0) 60

0.0223

290

11

(0) 10

(0) 5

(0) 60

0.0866

250

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Fig. 3.14 Drying curves of duck egg yolk foam at a temperature of 50 °C

Fig. 3.15 Drying curves of duck egg yolk foam at a temperature of 60 °C

Figures 3.17, 3.18, and 3.19 show the specific drying rates for the duck egg yolk at temperatures of 50, 60, and 70 °C, respectively. From the analysis of these figures, we state the existence of both constant and falling drying rate periods. Now, as reported for duck egg white, for some mathematical models reported in Table 3.1, nonlinear regression was made to the experimental data. The results

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Fig. 3.16 Drying curves of duck egg yolk foam at a temperature of 70 °C

Fig. 3.17 Specific drying rate of duck egg yolk during foam-mat drying at temperature of 50 °C

presented in Table 3.9 are only of the model with statistically significant fitting (modified Page’s model). The graphs showing the result predicted by the Modified Page’s model and the experimental dimensionless content at 50, 60, and 70 °C are shown in Figs. 3.20, 3.21, and 3.22, respectively.

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Fig. 3.18 Specific drying rate of duck egg yolk during foam-mat drying at temperature of 60 °C

Fig. 3.19 Specific drying rate of duck egg yolk during foam-mat drying at temperature of 70 °C

3.4 Final Considerations In this chapter, the physical problem of foam-mat drying has been addressed. Special attention is given to egg white and yolk, which are protein-rich foods. Here, the theoretical (via grouped models) and experimental (observed in experimental design) approaches are made with the aim of obtaining powder product.

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Table 3.9 Statistic parameters of the modified Page’s model after fitting the experimental data for the drying of the duck egg yolk at temperature of 50, 60, and 70 °C Experiment

Parameters X 0 (db)

K (min−1 )

a (–)

n (–)

R2 (–)

S (–)

1

0.8338

0.0026

0.9877

1.1945

0.9992

0.0041

2

1.1588

0.0042

1.0002

1.0556

0.9985

0.0061

3

0.8095

0.0027

0.9634

1.2139

0.9975

0.0131

4

0.8370

0.0013

0.9686

1.3740

0.9985

0.0078

5

0.8616

0.0040

0.9678

1.3419

0.9979

0.0086

6

1.1551

0.0072

0.9951

1.1194

0.9992

0.0024

7

0.7836

0.0061

0.9832

1.2222

0.9992

0.0036

8

0.8348

0.0036

0.9848

1.3750

0.9994

0.0027

9

1.0756

0.0047

0.9844

1.1253

0.9994

0.0023

10

0.9976

0.0042

0.9765

1.1962

0.9989

0.0044

11

1.0133

0.0081

0.9987

1.0961

0.9983

0.0063

Fig. 3.20 Comparison between predicted (modified Page’s model) and experimental results of the moisture content ratio during the foam-mat drying process of duck egg yolk at 50 °C

From the obtained results, it can be concluded that: (a) For the duck egg white, the drying-air temperature, the stirring rate, and the stirring time influenced the final moisture content of the obtained powder. For the duck egg yolk, the drying-air temperature, Emustab® concentration and

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Fig. 3.21 Comparison between predicted (modified Page’s model) and experimental results of the moisture content ratio during the foam-mat drying process of duck egg yolk at 60 °C

Fig. 3.22 Comparison between predicted (modified Page’s model) and experimental results of the moisture content ratio during the foam-mat drying process of duck egg yolk at 70 °C

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stirring time influenced the final moisture content. Verifying that the higher the temperature the shorter the drying time. (b) The drying curves for both duck egg white and yolk showed approximately constant and falling drying rate. The modified Page’s model was the empirical model with best fit to the experimental data. (c) Duck egg white foams dried at 50 and 70 °C were considered to be egg white powder only for a stirring time greater than 5 min (moisture content, X < 8%). For the duck egg yolk, yolk powder was obtained in all experimental conditions. (d) The production of the duck egg powder obtained with the foam-mat drying process proved to be a viable alternative. A batch averaged time at 3 h for the egg white and 5 h for the egg yolk were obtained. Acknowledgments The authors are grateful for the financial support provided by CNPq, CAPES, and FINEP (Brazilian Research Agencies). We also acknowledge scientific support from the authors mentioned in this chapter.

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Fioreze, R.: Principles of Drying Biological Products, vol. 229. University Publisher/UFPB, João Pessoa (2004) Francis, F.J.: Encyclopedia of Food Science and Technology, vol. 12907, 2nd edn. Wiley, New York (2000) Franco, T.S.: Dehydration of yacon juice by the foam mat method. Doctoral Thesis in Food Engineering. Federal University of Paraná, Curitiba, Brazil (2015). (In Portuguese) Gava, A.J.: Food Technology: Principles and Applications. Nobel, São Paulo (2008) Gouveia, J.P.G., Moura, R.S.F., Almeida, F.A.C., Oliveira, A.M.V., Silva, M.M.: Evaluation of cashew drying kinetics through experimental design. Braz. J. Agri. Environ. Eng. 6(3), 471–474 (2002) Hart, M.R., Graham, R.P., Ginnette, L.F., Morgan Jr., A.I.: Foam-mat drying requires stiff, stable foams. Food Technol. 17(10), 90–92 (1963) Kadam, D.M., Wilson, R.A., Kaur, S.: Determination of biochemical properties of foam-mat dried mango powder. Int. J. Food Sci. Technol. 45(8), 1626–1632 (2010) Karim, A.A., Chee-Wai, C.: Characteristic of foam prepared from starfruit (Averrhoa carambola L.) puree by using methyl cellulose. Food Hydrocolloids 13(2), 203–210 (1999) Kinsella, J.: Functional properties of proteins: possible relationships between structure and function in foams. Food Chem. 7, 273–288 (1981) Liu, K., Johnson, L. A., Knapp, H.: Soybeans as Functional Foods and Ingredients. In: Soyfoods, L., KeShun (eds.) Chapter 1: Soybeans as a Powerhouse of Nutrients and Phytochemicals, pp. 1958– 2053. Champaign: AOCS Press (2004) Liu, Q., Martins, M.D., Bakker-Arkema, F.W.: Stochastic modelling of grain drying: Part 1: experimental investigation. J. Agri. Eng. Res. 66(4), 267–273 (1997) Lomakina, K., Mikova, K.: A study of the factors affecting the foaming properties of egg white—a review. Czech J. Food Sci. 24(3), 110–118 (2006) Mangueira, E.R.: Study of duck egg (foam mat drying). Master´s dissertation, in Chemical Engineer. Federal University of Paraiba, João Pessoa (2017). (In Portuguese) Mapa, M.D.: Regulation of industrial and sanitary inspection of animal origin product (2005) Marques, G.M.R.: Drying of sugarcane juice in foam mat and sensory evaluation of the product. Dissertation (Master in Food Engineering)—State University of the Southwest of Bahia, Itapetinga, Brazil (2009). (In Portuguese) Mayor, L., Sereno, A.M.: Modelling shrinkage during convective drying of food materials: a review. J. Food Eng. 61(3), 373–386 (2004) Meloni, P.L.S.: Dehydration of Fruits and Vegetables. Frutal Institute (2003). http://www.eteavare. com.br/arquivos/20_1959.pdf. Accessed October (2019) Meloni, P.L.S.: Dehydration of Fruits and Vegetables, p. 87. Frutal Institute, Fortaleza (2003) Midilli, A., Kucuk, H., Yapar, Z.A.: New model for single-layer drying. Dry. Technol. 20(7), 1503– 1513 (2002) Mounir, S.: Foam Mat Drying. In: Prabhat, K., Nema, B.P.K., Mujumdar, A.S. (eds.) Drying Technologies for Foods-Fundamentals and Applications. CRC Press, New York (2017) Muthukumaran, A., Ratti, C.; Raghavan, V.G.S.: Foam-mat freeze drying of egg white and mathematical modeling. Part I: optimization of egg white foam stability. Dry. Technol. 26(4), 508–512 (2008) Narchi, I., Vial, C., Djelveh, G.: Effect of protein–polysaccharide mixtures on the continuous manufacturing of foamed food products. Food Hydrocolloids 23, 188–201 (2009) Negreiros, J.K.S.: Study of the kinetics of drying of egg in a foam mat (foam-mat drying) and the characterization of the obtained product. 63 p. Final course report (Undergraduate in Chemical Engineering), Federal University of Paraíba, João Pessoa, Brazil (2016). (In Portuguese) Nishinari, K., Fang, Y., Guo, S., Phillips, G.O.: Soy proteins: a review on composition, aggregation and emulsification. Food Hydrocolloids 39, 301–318 (2014) Ordonéz, J.A.: Food Technology, vol. I. Artmed, Porto Alegre (2005) Park, K.J., Yado, M.K.M., Brod, F.P.R.: Drying study of sliced Bartlett pear (Pyrus sp.). Food Sci. Technol. 21(3), 288–292 (2001)

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Park, S.K., Shanbhag, S.R., Dubin, A.E., De Bruyne, M., Wang, Q., Yu, P., Shimoni, N., D’Mello, S., Carlson, J.R., Harris, G.L., Steinbrecht, R.A., Pikielny, C.W.: Inactivation of olfactory sensilla of a single morphological type differentially affects the response of Drosophila to odors. J. Neurobiology. 51(3), 248–260 (2002) Park, K.J., Antonio, G.C., Oliveira, R.A., Park, K.J.B.: Process concepts and drying equipment. Campinas, 121 p. (2007). http://www.feagri.unicamp.br/ctea/manuais/concproceqsec_07. pdf. Accessed October (2007). (In Portuguese) Pereira, T.S.: Foam-mat drying study of white egg. Master´s dissertation in Agri-industrials Systems, Federal University of Campina Grande, Pombal, Brazil (2015). (In Portuguese) Prins, A.: Principles of foam stability. In: Dickinson, E., Stainsby, G. (eds.) Advances in Food Emulsions and Foams, pp. 91–122. Elsevier, London (1988) Rajkumar, P., Kailappan, R., Viswanathan, R., Raghavan, G.S.V., Ratti, C.: Foam mat drying of alphonso mango pulp. Dry. Technol. 25(2), 357–365 (2007a) Rajkumar, P., Kailappan, R., Viswanathan, R., Raghavan, G.S.V.: Drying characteristics of foamed alphonso mango pulp in a continuous type foam mat dryer. J. Food Eng. 79, 1452–1459 (2007b) Sagar, V.R., Kumar, S.P.: Recent advances in drying and dehydration of fruits and vegetables: a review. J. Food Sci. Technol. 47(1), 15–26 (2010) Sangamithra, A., Venkatachalam, S., John, S.G., Kuppuswamy, K.: Foam mat drying of food materials: a review. J. Food Process. Preserv. 39(6), 3165–3174 (2015).https://doi.org/10.1111/jfpp. 12421 Sousa, C.C.: Definition of parameters for foam-mat drying of anas platyrhynchos domesticus egg white and yolk. João Pessoa, 54 p. Final course report (Undergraduate in Chemical Engineering), Federal University of Paraíba, João Pessoa, Brazil (2017). (In Portuguese) Strumillo, C., Kudra, T.: Drying: Principles. Applications and Design. Gordon and Breach Science Publishers, New York (1986) Tariq, M.R., Sameen, A., Khan, M.I., Huma, N., Yasmin, A.: Nutritional and therapeutic properties of whey. Ann. Food Sci. Technol. 14(1), 19–26 (2003) Townsend, A.A., Nakai, S.: Relationships between hydrophobicity and foaming characteristics of food proteins. J. Food Sci. 48(2), 588–594 (1983) Vega-Mercado, H., Gongora-Nieto, M.M., Barbosa-Canovas, G.V.: Advances in dehydration of foods. J. Food Eng. 49(4), 271–289 (2001) Vernon-Carter, E.J., Espinosa-Paredes, G., Beristain, C.I., Romero-Tehuitzil, H.: Effect of foaming agents on the stability, rheological properties, drying kinetics and flavour retention of tamarind foam-mats. Food Res. Int. 34(4), 587–598 (2001) Zayas, J.F.: Functionality of Proteins in Food. Chapter 5: Foaming Properties of Proteins, pp. 122– 134. Springer, Heidelberg (1997) Zhang, Q., Litchfield, J.B.: An optimization of intermittent corn drying in a laboratory scale thin layer dryer. Dry. Technol. 9, 383–395 (1991) Zhang, M., Bhandari, B., Fang, Z.: Handbook of Drying of Vegetables and Vegetable Products. CRC Press, Boca Raton (2017)

Chapter 4

Drying Process of Jackfruit Seeds T. M. Q. de Oliveira, R. A. de Medeiros, V. S. O. Farias, W. P. da Silva, C. M. R. Franco, and A. F. da Silva Júnior

Abstract This chapter presents the application of an analytical solution for the diffusion equation in cylindrical coordinates, considering a boundary condition of the third kind. This diffusive model was used to verify the influence of the presence of the seed coat in jackfruit seeds on the mass transfer at the product surface and on the drying time. For this, jackfruit seeds were dried with and without the seed coat at 60 and 70 °C. In order to obtain the optimal values of effective mass diffusivity and Biot number, the analytical solution was coupled to an optimizer developed from an inverse method. A program was developed in Fortran language to execute the optimizer coupled to the analytical solution. The results showed that the seed coat had a strong influence on the drying time and on the Biot number, indicating that the boundary condition of the third kind is the most suitable for the drying simulation of this type of product.

T. M. Q. de Oliveira · R. A. de Medeiros Postgraduate Program in Natural Sciences and Biotechnology, Federal University of Campina Grande, Olho D’Água da Bica, S/N, Cuité, PB 58175-000, Brazil e-mail: [email protected] R. A. de Medeiros e-mail: [email protected] V. S. O. Farias · C. M. R. Franco · A. F. da Silva Júnior (B) Physics and Mathematics Department, Federal University of Campina Grande, Olho D’Água da Bica S/N, Cuité, PB 58175-000, Brazil e-mail: [email protected] V. S. O. Farias e-mail: [email protected] C. M. R. Franco e-mail: [email protected] W. P. da Silva Physics Department, Federal University of Campina Grande, Av. Aprígio Veloso 882 Bodocongó, Campina Grande, PB 58429-900, Brazil e-mail: [email protected]; [email protected] © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 J. M. P. Q. Delgado and A. G. Barbosa de Lima (eds.), Transport Processes and Separation Technologies, Advanced Structured Materials 133, https://doi.org/10.1007/978-3-030-47856-8_4

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Keywords Diffusion equation · Analytical solution · Mass transfer · Seed coat · Optimization

4.1 Introduction Jackfruit (Artocarpus heterophyllus Lam.) is a species widely cultivated in the Asian continent and in tropical climate areas, including Brazil. Its fruits reach an average of 3.5 kg to a maximum of 25 kg and are composed of arils with yellowish and sweet pulp and brown seeds wrapped in a hard shell (Swami et al. 2012). Jackfruit pulp has high nutritional value and is rich in sugars, mainly sucrose, fructose, and glucose, as well as minerals, dietary fiber, carboxylic acids, and vitamins. Due to the great versatility of its use as food, it can be eaten fresh or processed by adding sugar or another component into products such as jams and cakes (Anaya-Esparza et al. 2018). Another very important component is the seeds, which represent about 10–15% of the total weight of the fruit and stand out for having high amounts of protein, fiber, minerals, and fatty acids (Pacheco et al. 2015; Tulyathan et al. 2002). Its great technological value directly influences its economic potential, which is linked to the wide possibility of use in biotechnology, especially as a food source. It can be eaten cooked, candied, in flour form, and used as preparation and/or meal enrichment ingredients or as a substitute for people with dietary restrictions (Anaya-Esparza et al. 2018). The seed when kept fresh has favorable conditions for rapid deterioration, causing loss and damage. Therefore, the drying of these seeds to obtain the flour has been widely used as an alternative to increase the useful life and expansion of technological applications, especially in the food industry. Due to its nutritional content consisting of 78% carbohydrates, 11.2% protein, and 0.99% lipids, it is being implemented as an enrichment strategy in the development of cappuccinos, breads, and meatballs (Tulyathan et al. 2002; Landim et al. 2015; Santos 2012; Spada et al. 2018). Although drying is one of the most commonly used preservation techniques, it is known that its use requires a large expenditure of energy, in addition to modifying the nutritional and sensory properties of the product, resulting in financial losses for industries. With a view to reducing costs and obtaining a good quality product, the description of drying kinetics through mathematical simulation is a strategy that can be used to predict variables such as time, temperature, and dryer types best suited for use in the processing of the studied product (Gan and Poh 2014). In addition to the variables often studied in the drying process such as temperature, pressure, and air velocity, product constituents such as shells can influence important process parameters. However, most studies available in the literature do not analyze the influence of such elements on the thermo-physical parameters of drying. Doymaz and Pala (Doymaz and Pala 2003) performed the drying of corn grains with and without ethyl oleate pretreatment at 55, 65, and 75 °C. One of the models used to predict drying kinetics and to determine effective water diffusivity was a

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simplification of the solution of the diffusion equation in spherical coordinates (only the first term was considered). In addition, a boundary condition of the first kind was imposed. Although these simplified models fit the experimental data well, the influence of product constituents (such as the seed coat) on the surface resistance to mass transfer should be evaluated. However, this is only possible by assuming a boundary condition of the third kind. A similar study was performed by Leite et al. (Leite et al. 2019) with germinated seeds of jackfruit. In this research, germinated seeds of jackfruit were dried at 55, 65, and 75 °C with air velocities of 1.0 and 1.3 m s−1 . For the description of drying kinetics, 12 empirical models were tested. To determine the effective diffusivity of water, the same simplification of the diffusive model used by Doymaz and Pala (Doymaz and Pala 2003) was adopted. Since the boundary condition was again of the first kind, it was not possible to evaluate the influence of the seed coat on mass transfer at the product surface. In order to analyze the effect of seed coat presence on the drying process parameters, four treatments were carried out (drying at 60 and 70 °C of seeds with and without seed coat). An analytical solution for the diffusion equation in cylindrical coordinates for the two-dimensional case, assuming a boundary condition of the third kind, was considered to describe the processes.

4.2 Methodology 4.2.1 Experiments The study was carried out using “soft” and “hard” jackfruit varieties obtained at the local market of the city of Cuité, Paraiba, Brazil, pre-selected according to physiological integrity and absence of mechanical damage to the fruit. The technological processes of the experiment were carried out at the Food Technology Laboratory of the Federal University of Campina Grande—UFCG, Campus of Cuité, Brazil. The seeds used in the drying process were obtained through fruit disinfection, pulping, characterized by pulp and seed separation, and washing of the seed in drinking water. In addition, to remove the shell (seed coat), the jackfruit seeds were immersed in boiling water for about 1 min to facilitate the process. Then these seeds were placed on a stainless steel sieve to remove surface water, and the shells were removed with the help of a knife, as shown in Fig. 4.1. Immediately after, the seeds with and without seed coat were placed separately in a sealed plastic container and subjected to the drying process. An oven previously stabilized at 60 and 70 °C was used for drying. The samples were divided into four treatments reproduced in triplicate: T1—Jackfruit seeds with seed coat submitted to 60 °C, T2—Jackfruit seeds without seed coat submitted to 60 °C, T3—Jackfruit seeds with seed coat submitted to 70 °C, and T4—Jackfruit seeds without seed coat submitted to 70 °C. In each treatment, the samples were

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Fig. 4.1 Seed shell removal with a stainless steel knife

Fig. 4.2 Jackfruit seeds without the seed coat arranged in baskets for the drying process

placed in baskets with mass previously measured on a semi-analytical scale, as shown in Fig. 4.2. The masses of the samples were measured before drying began (time t = 0), and then at intervals of 2, 5, 10, 20, and 30 min and 1 hour (h), 2 h, and 3 h until they reached equilibrium. Then, the samples were dried in an oven previously stabilized at 105 °C for 24 h to obtain the dry mass. The moisture content of the samples ranged from 1.069 to 1.321 (dry basis, d.b.) for the seeds with seed coat and from 0.977 to 1.35 (d.b.) for the seeds without seed coat.

4.2.2 Mathematical Modeling In the present study, the liquid diffusion model for water migration in a product with finite cylinder geometry was considered adequate to describe thin-layer drying of jackfruit seed. This is a widely accepted model in the literature (Pacheco-Aguirre et al. 2014; Silva Júnior et al. 2018; Arunsandeep and Chandramohan 2018; Chayjan and Kaveh 2014). In addition, as the objective of this study was to analyze the influence of the presence of the seed coat on the mass transfer at the product surface, a boundary condition of the third kind was assumed.

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4.2.2.1

93

The Model

The analytical solution of the diffusion equation will be presented for the finite cylinder, where the diffusive process is subject to the following hypotheses: (1) the cylinder must be considered homogeneous and isotropic; (2) the distribution of moisture content within the cylinder must have radial symmetry and must be initially uniform; (3) the conditions of the drying medium remain the same throughout the process; (4) the only water transport mechanism inside the cylinder is liquid diffusion; (5) dimensions of the cylinder do not vary during diffusion; (6) the effective diffusivity does not vary during the process; and (7) the boundary condition is of the third kind. For the previously established hypotheses, the diffusion equation has analytical solution for several simple geometries, among which is the finite cylinder geometry. It is noteworthy that a finite cylinder can be obtained by intersecting two even simpler solids: the infinite cylinder and the infinite wall, as shown in Fig. 4.3a. In order to present the analytical solution of the diffusion equation for the geometry of a finite cylinder of radius R and length L, such geometry is outlined as shown in Fig. 4.3b. The three-dimensional diffusion equation in cylindrical coordinates (r, y, ) is given by:       ∂X 1 ∂ ∂X ∂ ∂X 1 ∂ ∂X rD + 2 D + D = ∂t r ∂r ∂r r ∂θ ∂θ ∂y ∂y

(4.1a)

For a symmetrical diffusion with respect to the r- and y-axes, only the radial and axial flows were considered and, therefore, the flow in the angular direction was neglected. Thus, for the cylindrical geometry shown in Fig. 4.3b, the two-dimensional diffusion equation can be written as follows:

(a)

(b)

Fig. 4.3 a Intersection of an infinite cylinder and an infinite wall; b Finite cylinder of radius R and length L

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    1 ∂ ∂X ∂ ∂X ∂X = rD + D ∂t r ∂r ∂r ∂y ∂y

(4.1b)

In Eq. (4.1b), r is defined relative to the central axis of the cylinder and, together with the y-coordinate, defines the position of a point (r, y) within the solid to be studied. Also, in this equation, D is the effective mass diffusivity, X is the moisture content on dry basis, and t is the time. The boundary condition is of the third kind, which is expressed by imposing equality between the internal (diffusive) flow on the surface of the finite cylinder and the external (convective) flow in the vicinity of this surface:    ∂ X (r, y, t)  = h X (r, y, t)r =R − X eq r =R ∂r

(4.2)

   ∂ X (r, y, t)  = h X (r, y, t) y=±L/2 − X eq y=±L/2 ∂y

(4.3)

−D and −D

4.2.2.2

Exact Solution

For a homogeneous and isotropic cylinder of radius R and length L with uniformly distributed initial moisture content X i and equilibrium moisture content X eq , the solution X(r, y, t) of Eq. (4.1b) for the boundary conditions defined by Eqs. (4.2) and (4.3) can be obtained by separating the variables (Luikov 1968; Crank 1975) and results in:    r y cos μm,2 An,1 Am,2 J0 μn,1 R L/2 n=1 m=1 μ2m,2 + Dt (4.4) (L/2)2

X (r, y, t) = X eq + (X i − X eq )

 × exp −

μ2n,1 R2

∞  ∞ 

As mentioned earlier, this solution considers the idea that a finite cylinder can be obtained by intersecting two even simpler solids: an infinite cylinder of radius R and an infinite wall of thickness L. Returning to Eq. (4.4), it should be noted that X(r, y, t) is the moisture content on dry basis at a cylinder position (r, y) at time t, and D is the effective mass diffusivity. Also, with respect to Eq. (4.4), the coefficients An,1 and Am,2 are defined as follows: An,1 =

2Bi1 J0 (μn,1 )(Bi21 + μ2n,1 )

(4.5)

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and Am,2 = (−1)m+1

2Bi2 (Bi22 + μ2m,2 )1/2 μm,2 (Bi22 + Bi2 + μ2m,2 )

(4.6)

In Eqs. (4.5) and (4.6), variables referring to the terms of the second member will be defined later. On the other hand, the expression for the average moisture content at time t is given as follows: X (t) =

1 V

 X (r, y, t)d V

(4.7)

The solution of the diffusion equation for the mean value in a finite cylinder at time t is obtained by substituting Eq. (4.4) in Eq. (4.7), which results in:



μ2n,1 μ2m,2 X (t) = X eq + (X i − X eq ) Bn,1 Bm,2 exp − + Dt (4.8) R2 (L/2)2 n=1 m=1 ∞  ∞ 

where X¯ (t) is the average moisture content on dry basis at time t. The coefficient Bn,1 is defined as follows: Bn,1 =

4Bi21 μ2n,1 (Bi21 + μ2n,1 )

(4.9)

where Bi1 is the Biot number for the infinite cylinder and is given by Bi1 =

hR D

(4.10)

The coefficient Bm,2 is defined as follows: Bm,2 =

μ2m,2 (Bi22

2Bi22 + Bi2 + μ2m,2 )

(4.11)

and in this expression Bi2 is the Biot number referring to the infinite wall, given by the expression Bi2 =

h(L/2) D

(4.12)

In Eqs. (4.2), (4.3), (4.10), and (4.12), h is the convective mass transfer coefficient, and, in the presented solution, the same value of h was imposed to all external surfaces of the cylinder.

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In Eqs. (4.5), (4.8), and (4.9), μn,1 are the roots of the characteristic equation for the infinite cylinder and are calculated by the following transcendental equation: J0 (μn,1 ) μn,1 = J1 (μn,1 ) Bi 1

(4.13)

where J 0 and J 1 are Bessel functions of the first-order type 0 and 1, respectively. In Eqs. (4.6), (4.8), and (4.11), μm,2 are the roots of the characteristic equation for the infinite wall and are calculated by the following transcendental equation: cot μm,2 =

μm,2 Bi 2

(4.14)

From the foregoing, Eqs. (4.4) and (4.8) can be used to determine X(r, y, t) and X¯ (t) for any Biot numbers of interest. In the present study, aiming at the computational implementation of Eq. (4.8), the first 20 roots of Eqs. (4.13) and (4.14) were calculated. The choice of this number of roots is based on the study by Silva et al. (2012), which points out the relationship between the number of terms in the series (Eq. 4.8) and the Biot number. Note that for each term in this series, a root for transcendental Eqs. (4.13) and (4.14) is required. A detailed description of the obtaining of these roots will be presented in the section related to computational code development. It is worth noting that the Biot number is a widely used parameter in the literature to determine if the internal resistance to mass flow is relevant (Incropera et al. 2012). Such resistance may be considered negligible if Bi < 0.1. In this case, the moisture distribution inside the product becomes uniform during the process. Equation (4.8) can be rearranged to express the moisture ratio, which is defined as follows: ∗

X =

X (t) − X eq X i − X eq

(4.15)

In this study, the experimental data obtained for drying kinetics were used in the dimensionless form using Eq. (4.15).

4.2.2.3

Optimization Procedure

A program in the Fortran language was developed on the Windows platform. Equation (4.8), which presents the average moisture content for any time instant, was implemented in this program. However, to obtain the average moisture content through Eq. (4.8), it is necessary to find the roots of the transcendental equations, which in turn depend on Bessel functions and Biot numbers. The input data of the computational code are the experimental data, initial moisture content, equilibrium moisture content, and the length and radius of the cylinder. Once

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input data are provided, the developed optimizer begins the process of determining the Biot numbers Bi1 and Bi2 for the infinite cylinder and the infinite wall, respectively, and the effective water diffusivity. For each value of the Biot numbers supplied by the optimizer, it is necessary to calculate the roots μn,1 and μm,2 . For the calculation of μn,1 , the code proceeds as follows: the Bessel functions of order 0 and 1 are calculated for each n, considering 80 terms of the factorial in the expression. Then their values are replaced in the transcendental equation and, after that, the Secant method with an accuracy of 10−12 is used to calculate these roots. Thus, the value of μn,1 is obtained and, consequently, by Eq. (4.13), the value of Bi1 is obtained for each n. For the calculation of μm,2 , the Newton Method with an accuracy of 10−16 is applied to Eq. (4.14). Thus, for each m, the respective value of μm,2 is obtained. The optimizer used in the present work was developed in order to obtain the optimal values of parameters D, Bi1 , and Bi2 . This optimizer was developed through an inverse method, which initially requires the user to provide initial values for the three parameters. These values are then corrected to minimize an objective function, which in this case is chi-square, defined as: χ = 2

Np   i=1

exp

Xi

− X¯ isim (D, Bi1 , Bi2 )

2 1 , σi2

(4.16)

exp where X i is the i-th experimental point; X¯ isim (D, Bi1 , Bi2 ) is the average value of exp X obtained by the analytical solution at the same time as X i ; σi is the standard deviation of the experimental average moisture content at point i; D is the effective mass diffusivity; and Np is the number of experimental points. In the present work, all σi were considered equal to 1. Once all the necessary elements for the determination of the value of χ 2 are obtained, the next step is to adjust the initial values of the parameters Bi1 , Bi2 , and D, by minimizing the chi-square. For this, the optimizer follows the following steps: First, with the initial user-supplied parameter data, the optimizer calculates the first value for chi-square. Second, the optimizer starts looking for ranges where the optimal values for the parameters Bi1 , Bi2 , and D are found. Initially, with the Bi2 and D parameter values fixed, the algorithm corrects the initial Bi1 parameter value by adding 0.1% to the current value (if the optimal value is higher) or subtracting 0.1% from the current value (if the optimal value is lower). At each correction of parameter Bi1 , a new value for χ 2 is calculated and compared with the previous one. This process is repeated it is considered that the until the χ 2 value is higher than the previous   one. Finally, k , where the indices k and k − , Bi optimal value of Bi1 is within the interval Bik−1 1 1 1 represent the iterations where the chi-square increases and the previous iteration, respectively. Third, once the range containing the optimal value for the parameter is found, the midpoint is calculated, which will divide the range into two new ranges. Then, the chi-square at the midpoints of the two new intervals obtained is calculated, and the

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choice of the best midpoint is made by decreasing the chi-square. This process is repeated until the chi-square can no longer be minimized. In each iteration, the second and third steps should be applied to each parameter. At the end of each iteration, the optimizer checks the relative error of each parameter. The optimization process is terminated when the relative error of each parameter is less than 10−16 . This error is calculated by the following formula: E rel

   Pcurrent − Pprevious  = , Pprevious

(4.17)

where E rel denotes the relative error, Pcurrent is the parameter value in the current iteration, and Pprevious is the parameter value in the previous iteration. In the optimizations performed, no significant differences were observed between Bi1 and Bi2 . Thus, the values of these parameters were considered equal to a single value Bi.

4.3 Results Analysis The physical parameter optimization processes were performed using the analytical solution presented in Sect. 4.2. The results of these processes are presented in Table 4.1. It is possible to notice the influence of the presence of the seed coat also on the drying time by observing the graphs shown in Fig. 4.4. From the observed drying times, it can be concluded that the drying of the samples with the presence of seed coat at temperatures of 60 and 70 °C lasted about 66% longer than the drying of the samples without seed coat. Thus, the energetic cost for seed processing with seed coat is high, being justified only if these seeds have relevant nutritional indicators. Figure 4.5 shows the drying kinetics for the four treatments performed. By analyzing the kinetics for seed with seed coat, one can observe the influence of temperature. On the other hand, this influence is not observed on the kinetics for seeds without seed coat. However, further studies are necessary considering other Table 4.1 Values obtained for the parameters of the drying process of jackfruit seeds with and without the presence of the seed coat Temperature/Presence of seed coat

Dw (m2 m−1 )

h (m m−1 )

Bi

R2

χ2

60 °C/present

6.46 × 10−7

6.21 × 10−6

8.84 × 10−2

0.991

3.64 × 10−2

60 °C/absent

6.21 × 10−8

1.32 × 10−4

19.01

0.998

4.15 × 10−3

70 °C/present

7.99 × 10−7

7.99 × 10−6

9.19 × 10−2

0.998

6.92 × 10−3

70 °C/absent

6.43 × 10−8

1.28 × 10−4

18.85

0.999

2.04 × 10−3

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(a)

(b)

(c)

(d)

Fig. 4.4 Fits obtained for samples with seed coat at temperatures of a 60 °C and b 70 °C and for samples without seed coat at temperatures of c 60 °C and d 70 °C

Fig. 4.5 Simulation of drying kinetics for the four treatments

temperatures in order to analyze whether this influence is dominated by the presence of the seed coat. Figure 4.6 presents simulations of the moisture distribution inside the seeds for the four treatments. As can be seen, the simulations presented in Fig. 4.6 correspond to the half of the cylinder that represents the seeds. It is also possible to notice a very small moisture gradient. This phenomenon may be related to the high resistance observed through

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(a)

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Fig. 4.6 Simulation of moisture distribution for treatment at 60 °C with seed coat at times: a t = 30.72 min, b t = 120 min, and c 330.20 min

the Biot number presented in Table 4.1. This resistance causes a kind of “brake” to the water flow, resulting in the uniformity of moisture inside the product. By comparing Figs. 4.6 and 4.7, it can be observed that the seed coat is what determines the existence of moisture gradients inside the product, since the gradients (which were small in Fig. 4.6) increase in Fig. 4.7. This phenomenon can also be observed when comparing the simulations for 70 °C, presented in Figs. 4.8 and 4.9. By comparing the simulations of 60 and 70 °C, one can note an influence of temperature on the distribution of moisture inside the product. Moreover, in simulations for the time near 120 min, gradient is observed only in the drying at 60 °C. In the simulation for drying at 70 °C, a uniform distribution of moisture inside the product is noted. The influence of the seed coat on the moisture distribution inside the product observed in the simulations for drying at 60 °C is also seen in Figs. 4.8 and 4.9. Moreover, when comparing the simulations for drying at 60 and 70 °C of the samples

(a)

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Fig. 4.7 Simulation of moisture distribution for treatment at 60 °C without seed coat at times: a t = 30.36 min, b t = 120.1 min, and c 330 min

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Fig. 4.8 Simulation of moisture distribution for treatment at 70 °C with seed coat at times: a t = 30.40 min, b t = 120.70 min, and c 330.60 min

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Fig. 4.9 Simulation of moisture distribution for treatment at 70 °C without seed coat at times: a t = 30.24 min, b t = 120.20 min, and c 330.50 min

without seed coat, there are slight differences between the moisture distributions. This can also be observed through the kinetics presented in Fig. 4.5.

4.4 Concluding Remarks In this chapter, the effect of the presence of the seed coat in jackfruit seeds on the process parameters was studied. It was concluded that the presence of this element contributed to the increase in drying time and influenced the resistance of the product surface to mass transfer. The proposed model adequately described the drying kinetics of seeds with and without seed coat. Moreover, from the values obtained for the Biot number, the most appropriate boundary condition for describing the drying of this

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type of seed is that of the third kind. Finally, the methodology used in this chapter can be applied to describe the drying of other seeds in order to verify the influence of their seed coat on the process parameters. Acknowledgments Prof. Wilton would like to thank CNPq (Conselho Nacional de Desenvolvimento Científico e Tecnológico) for his research grant (Process Number 301708/2019-3; PQ-1A).

References Anaya-Esparza, L.M., González-Aguilar, G.A., Domínguez-Ávila, J.A., Olmos-Cornejo, J.E., Pérez-Larios, A., Montalvo-González, E.: Effects of minimal processing technologies on jackfruit (Artocarpus heterophyllus Lam.) Qual. Paramet. Food Biopro. Technol. 11, 1761–1774 (2018) Arunsandeep, G., Chandramohan, V.P.: Numerical solution for determining the temperature and moisture distributions of rectangular, cylindrical, and spherical objects during drying. J. Eng. Phys. Thermophys. 91(4), 895–906 (2018) Chayjan, R.A., Kaveh, M.: Physical parameters and kinetic modeling of fix and fluid bed drying of terebinth seeds. J. Food Process. Preserv. 38, 1307–1320 (2014) Crank, J.: The Mathematics of Diffusion, 414 p. Clarendon Press, Oxford, UK (1975) Doymaz, I., Pala, M.: The thin-layer drying characteristics of corn. J. Food Eng. 60, 125–130 (2003) Gan, P.L., Poh, P.E.: Investigation on the effect of shapes on the drying kinetics and sensory evaluation study of dried jackfruit. Int. J. Sci. Eng. 7, 193–198 (2014) Incropera, F.P., De Witt, D.P., Bergman, T.L., Lavine, A.S.: Fundamentals of Heat and Mass Transfer. LTC, Rio de Janeiro (2012). (In Portuguese) Landim, L.B., Bonomo, R.C.F., Reis, R.C., Silva, N.M.C., Veloso, C.M., Fontan, R.C.I.: Kibbeh formulation with jackfruit flour. J. Health Sci. 14, 87–93 (2015). (In Portuguese) Leite, D.D.F., Queiroz, A.J.M., Figueirêdo, R.M.F., Lima, L.S.L.: Mathematical drying kinetics modeling of jackfruit seeds (Artocarpus heterophyllus Lam.), vol. 50, pp. 361–369. Agronomic Science Magazine (2019). In Portuguese Luikov, A.V.: Analytical Heat Diffusion Theory, 685 p. Academic Press, Inc. Ltd, London (1968) Pacheco, C.S.V., Ferreira, A.N., Rocha, T.J.O., Tavares, I.M.C., Franco, M.: Use of the jackfruit seed for obtaining endoglucanase from aspergillus niger by solid state fermentation. J. Health Sci. 14, 25–29 (2015). (In Portuguese) Pacheco-Aguirre, F.M., Ladrón-González, A., Ruiz-Espinosa, H., García-Alvarado, M.A., RuizLópez, I.I.: A method to estimate anisotropic diffusion coefficients for cylindrical solids: application to the drying of carrot. J. Food Eng. 125, 24–33 (2014) Santos, D. B. French bread development with the addition of jackfruit flour (Artocarpos integrifólia L.). Biosph. Encycl. 8, 597–602 (2012). (In Portuguese) Silva Júnior, A.F., Silva, W.P., Farias, V.S.O., Silva, C.M.D.P.S., Lima, A.G.B.: Description of osmotic dehydration of banana slices dipped in solution of water and sucrose followed by complementary drying using hot air. In: Transport Phenomena in Multiphase Systems: Advanced Structured Materials, 1 edn, pp. 273–304. Springer International Publishing (2018) Silva, W.P., Farias, V.S.O., Neves, G.A., Lima, A.G.B.: Modeling of water transport in roof tiles by removal of moisture at isothermal conditions. Heat Mass Transf. 48, 809–821 (2012) Spada, F.P., Silva, P.P.M., Mandro, G.F., Margiotta, G.B., Spoto, M.H.F., Canniatti-Brazaca, S.G.: Physicochemical characteristics and high sensory acceptability in cappuccinos made with jackfruit seeds replacing cocoa powder. PLoS One 13, 1–12 (2018)

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Swami, S.B., Thakor, N.J., Haldankar, P.M., Kalse, S.B.: Jackfruit and its many functional components as related to human health: a review. Compr. Rev. Food Sci. Food Saf. 11, 565–576 (2012) Tulyathan, V., Tananuwong, K., Songjinda, P., Jaiboon, N.: Some physicochemical properties of jackfruit (Artocarpus heterophyllus Lam) seed flour and starch. Sci. Asia 28, 37–41 (2002)

Chapter 5

Spouted Bed Drying of Fruit Pulps: A Case Study on Drying of Graviola (Annona muricata) Pulp F. G. M. de Medeiros, I. P. Machado, T. N. P. Dantas, S. C. M. Dantas, O. L. S. de Alsina, and M. F. D. de Medeiros Abstract The spouted bed dryer with inert particles has been researched as an alternative for the drying fruit pulps in order to obtain powdered products. Depending on the composition and physical properties of the pulps, the dryer is subject to agglomeration and accumulation problems that can be minimized by the addition of drying adjuvants, especially carbohydrates, such as maltodextrin; sources of proteins, such as whey protein and milk itself. In this work, the advantages of the spouted bed dryer, regarding its mixing capacity, temperature uniformity, high heat and mass transfer rates, and reduced processing time are emphasized. A review of the fundamentals of the spouted bed is presented in this chapter, as well as the relevant and recent works related to drying fruit pulps in the spouted bed, addressing the use of adjuvants and the impact of the process on phytochemicals present in fruits and other vegetables. The chapter is concluded with the presentation of a case study on the drying of graviola fruit pulp with the addition of milk in the spouted bed dryer, where the results related F. G. M. de Medeiros (B) · I. P. Machado · S. C. M. Dantas · M. F. D. de Medeiros (B) Department of Chemical Engineering, Federal University of Rio Grande do Norte, Av. Senador Salgado Filho, 3000, Natal, RN 59078-970, Brazil e-mail: [email protected] M. F. D. de Medeiros e-mail: [email protected]; [email protected] I. P. Machado e-mail: [email protected] S. C. M. Dantas e-mail: [email protected] T. N. P. Dantas Federal Institute of Education, Science and Technology of Rio Grande do Norte, Campus Currais Novos, R. Manoel Lopes Filho, 773, Currais Novos, RN 59380-000, Brazil e-mail: [email protected] O. L. S. de Alsina Department of Chemical Engineering (retired), Federal University of Campina Grande, R. Aprigio Veloso, 743, Campina Grande, PB 58429-140, Brazil e-mail: [email protected] © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 J. M. P. Q. Delgado and A. G. Barbosa de Lima (eds.), Transport Processes and Separation Technologies, Advanced Structured Materials 133, https://doi.org/10.1007/978-3-030-47856-8_5

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to production, thermal efficiency, product characteristics, and process modeling are presented with intermittent pulp feeding. Keywords Spouted bed dryer · Graviola pulp · Phytochemicals · Process modeling

5.1 Introduction Brazil is the third-largest fruit producer in the world and ranks among the leading exporters of several tropical such as pineapple, papaya, mango, oranges, melons, and others (Altendorf 2017). However, due to the perishable condition of fruits, the development of efficient post-harvest processing strategies is a major concern of the food industry, in order to avoid food waste. According to FAO (FAO 2019), 20–30% of world’s fruits and vegetables’ production goes to waste between the post-harvest and retail levels. For many years, drying has been regarded as an efficient conservation method and a versatile post-harvest processing alternative for fruits and other highly perishable materials (Zhang et al. 2020; Souza da Silva et al. 2019). The lowering of moisture content on dried fruit is a key parameter for extending products’ shelf life and maintaining a stable fruit products’ supply chain, since low water activity on dried products is related to reduced microorganism growth and delayed enzymatic process, which leads to higher storage stability (Rocha et al. 2011; Karam et al. 2016). The growing market demand for natural-based products has been pushing the food industry toward innovative product-oriented technologies in order to take full advantage of the dietary and phytochemical values of fruits (Belwal et al. 2018; Demirkol and Tarakci 2018). In this sense, the versatility of dehydration as a processing technique is highlighted on a new product-development point of view. From ready-to-eat and ready-to-drink products (Dantas et al. 2019; Cappato et al. 2018) to food ingredients (Correia et al. 2017; Moraes et al. 2017), dried fruit products have been attracting attention and space on the food industry. The choice of drying method is, therefore, of utmost importance, since the impact of the heat and mass transfers during the dehydration process are directly related to the final composition and quality aspects of the dried product (Demirkol and Tarakci 2018). In addition to the industrially popular spray dryer and freeze dryer, the spouted bed dryer with inert particles is an alternative drying technique that allows the production of powdered matrices from liquid and paste-like foods (Dantas et al. 2018). Combining operation flexibility with lower costs, when compared to the spray dryer, several studies have demonstrated the efficiency of the spouted bed drying for the production of high-quality powdered products (Rocha et al. 2011; Medeiros et al. 2002; Lucas et al. 2018).

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5.2 Fundamentals of Spouted Bed Drying The spouted bed technique was reported for the first time by Mathur and Gishler (1955), when the authors investigated the performance of this unconventional technique for the drying of wheat grains. In that original work, the authors reported that the vigorous air and particles circulation through the system facilitated the removal of water from the wheat grains, when compared to traditional fluidized bed dryers. In the late 1960s, the Leningrad Institute of Technology successfully applied the spouted bed dryer with inert particles for the drying of solutions. The authors reported that the spouted bed drying of organic dyes, salt and sugar solutions, and chemical reagents resulted in quality fine powders (Mathur and Epstein 1974). Several designs have been proposed for processing suspensions and pastes in a spouted bed dryer (Costa et al. 2006; Passos et al. 1997; Pallai et al. 2007), but the conventional dryer is composed of a cylindrical vessel (drying column or drying chamber) with a conical base with an inlet orifice for air injection (Fig. 5.1). Inert particles are used as drying support for the feed suspension. A thin layer of material is accumulated around the particles, covering them, and, after drying, the powders are separated from the particles due to the friction of the particles’ bed and carried by the inlet air (Araújo et al. 2015). The spouted bed drying system is initiated with inlet air injection at the conical base of the dryer’s chamber. When air flow is sufficient to pneumatically move the inert particles, an upward movement is noticed and the inert particles are carried to

Fig. 5.1 Schematic representation of a conventional spouted bed dryer. (1) Air blower; (2) Air valve; (3) Heater; (4) Temperature sensor; (5) High-density polyethylene bed; (6) Control panel; (7) Cylindrical column; (8) Lapple cyclone; (9) Outlet air temperature and humidity sensors; (10) Digital thermo-hygrometer and anemometer; (11) Powder collector

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levels above the bed, forming a high-porosity spouting region. A low-porosity region is formed in the annular section of the drying chamber by the downward movement of the inert particles. The particles then return to the conical base and are carried again into the spouting region, in a cyclic movement (Delgado and Lima 2014; Nascimento et al. 2015). The feed pastes and suspensions may be drip-fed or sprayed through a nozzle onto the moving and spouting particles bed, which allows for an increased contact surface area. The inlet feed is usually performed on the top of the drying chamber, since studies have shown that this leads to higher processing stability and lower material accumulation inside the system (Costa et al. 2006; Freire et al. Freire et al. 2011). The study of the fluid dynamic behavior of the spouted bed dryer with inert particles, taking into account the presence of the feed suspension in the bed is extremely relevant, although neglected by many authors. The fluid dynamic behavior influences the process operation conditions, which can be optimized for greater production efficiency, absence of instability, more economical process, and better product quality. Among the factors that influence the fluid dynamic behavior of inert particles in the spouted bed, the minimum spouting air velocity, the stable spouting air velocity, and the maximum pressure drop are worth to highlight (Vieira et al. 2004). In addition to the parameters of the inlet drying air, the geometric configuration of the dryer, the physical characteristics and composition of the suspensions, and properties of the inert material are also responsible for influencing the process stability (Nascimento et al. 2015; Moreira da Silva et al. 2019). The heat transfer in the spouted bed is carried out by conduction in the inert particles and convection of the hot drying air. The heat and mass transfers promote the drying of the material accumulated in the thin layer around the inert particles which progressively become fragile and friable. These characteristics of the dried material allow it to be removed, in the powder form, by the successive collisions to which the inert particles are subjected. The powder is then carried by the inlet air and collected in separation cyclone. In a spouted bed drying system, removal rates should be high enough to avoid agglomeration and mass accumulation inside the drying chamber (Freire et al. 2011; Sousa et al. 2019). The inert particles’ collision energy is also affected by a number of variables, including the solid circulation rates, the ratio between paste/suspension inlet feed and the inert particles load in the bed and the drying rate. The solids circulation rate determines the time required for a complete drying cycle: from particle coating to dried film removal. The increasing drying rate favors increased dried film friability and positively affects the process. The paste/suspension inlet feed rate should be carefully controlled to prevent bed collapse caused by fluid dynamic instability, particle growth, or agglomeration. The material inlet feeding rate must be low or moderate, resulting in low production of dry products, compared to the spray dryer, and is therefore an important limitation of this drying technique (Freire et al. 2012; Santos et al. 2015; Benelli et al. 2013b). In addition, the high pressure drop through the inert particles bed, the high air flow necessary for maintaining spouting stability, the difficult process scaling up, the

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strong adhesion of the dry powder to the equipment walls and the particle agglomeration, which is related to the composition of the paste/suspension feed, are among the main limitations involved with the spouted bed drying process (Pallai et al. 2007; Bacelos et al. 2007). Despite said limitations, the spouted bed drying technique has several advantages, which are responsible for attracting interest from research and development sectors, such as promoting good mixture between the particles and the fed suspension, minimal friction, temperature uniformity throughout the bed and elevated heat and mass transfer coefficients. In addition, the spouted bed drying process presents high drying rates due to the large surface contact area between the inert particles and the drying gas, that results in reduced processing times, which is indicated for heatsensitive products, such as phytochemicals (Pallai et al. 2007; Pablos et al. 2018; Niksiar and Nasernejad 2017; Niksiar et al. 2013).

5.3 Spouted Bed Drying of Fruit Pulps Drying is one of the most commonly used techniques for handling foods with high moisture content, such as fruits and vegetables (Zhang et al. 2020; Kumar et al. 2014). The drying of fruits and fruit pulps on a spouted bed dryer is a simpler and cost-reduced alternative technology, when compared to traditional freeze or spray drying (Rocha et al. 2011; Medeiros et al. 2002). Over the last three decades, several studies have described the drying of fruit pieces and fruit pulps using a spouted bed dryer with inert particles. Such studies have mainly focused on the influence of feed composition (Rocha et al. 2011; Medeiros et al. 2002; Larrosa et al. 2015; Braga and Rocha 2013), drying parameters (Fujita et al. 2013; Nascimento et al. 2019; Sales et al. 2019), powder production and quality parameters (Lucas et al. 2018; Benelli et al. 2013a, b; Braga and Rocha 2013; Braga and Rocha 2015), and use of drying carriers (Dantas et al. 2019; Fujita et al. 2013; Butzge et al. 2015, 2016; Rocha et al. 2018). Medeiros et al. (2002) investigated the influence of the composition of fruit pulps, using mango pulp as a model formulation, on the fluid dynamics and powder production efficiency for spouted bed drying. In this study, the composition of the fruit pulps was adjusted in relation to the contents of reducing sugars, starch, pectin, lipids, fibers, water, and total acidity. The authors reported that the fiber content of modified fruit pulps did not influence the fluid dynamic behavior of the drying system. Other carbohydrates, on the other hand, were related to instabilities on the spouted bed dryer: reducing sugars promoted instability of the bed, while starch and pectin accounted for higher spouting stability. In addition, the authors reported that lipids, starch, and pectin concentrations positively influence the powder production efficiency. These results were further confirmed by Rocha et al. (2011) have also reported that while high reducing sugars concentrations were related to spouting stability, fruit pulps with higher concentrations of starch and lipids promoted a more stable fluid dynamic regime of the spouting bed. In addition, the authors also verified that,

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despite a significant sharp decrease on pressure drop just after the pulp feeding into the drying chamber, pressure drop increased as the drying process reached a stable regime. Moreover, authors also highlighted that the powder retention inside the drying chamber, due to inadequate powder removal rates, may also influence the fluid dynamics of the spouting bed. In a recent study, Nascimento et al. (2019) reported the obtention of dried bacaba fruit powder by spouted bed drying. The authors reported the optimization on processing conditions in order to obtain a high yielding and quality dried product. Authors investigated the influence of temperature, maltodextrin concentration (as a drying carrier) and drying air velocity on the drying yield, moisture content, anthocyanins, and phenolic compounds retention. In this study, the authors reported findings that agree with Costa et al. (2015), who investigated the spouted bed drying of açaí fruit pulp: increased temperature is related to lower moisture content on the dried product and maltodextrin concentrations around 20% increased process production and phytochemicals retention. In addition, both studies have suggested that there must be a balance between drying temperature (>70 °C) and drying air velocity in order to avoid further degradation on nutritional and bioactive components of fruit pulps. Braga and Rocha (2015) evaluated the spouted bed drying of pure blackberry pulp and milk-added blackberry pulp. In addition, they also evaluated the impact of maltodextrin, casein, and palm oil as composition-modulators on the performance of the spouted bed drying of blackberry pulp. The authors reported that despite the modifications on protein and lipid content of the pulp composition, the drying of pure blackberry pulp was not possible on the spouted bed dryer, since the pulp feeding was responsible for significant instability that resulted in bed collapse due. The use of milk as a drying carrier, on the other hand, was reported successful in this study. The authors described that the addition of 25% (v/v) of whole milk in the paste formulation yielded a high-quality (around 3% moisture, 77 mg/100 g anthocyanins, and 19% protein) product and increased fluid dynamic stability. In a recent study by our research group, Dantas et al. (2019) reported the drying of acerola pulp (Malpighia emarginata DC) using milk and milk whey protein as drying carriers. The authors analyzed the influence of these diary adjuvants on both the fluid dynamics and the final product quality of dried acerola-based powders. It was reported that the addition of 1% whey protein to the acerola pulp jeopardize spouted bed drying due to high pressure drop on the spouting bed that led to instability and collapse. On the other hand, confirming the results previously reported by Braga and Rocha (2013, 2015), the use of milk as a drying carrier for fruit pulps was successful. Dantas et al. (2019) used a model formulation in which the drying carrier corresponded to 50% of the total solids on the final composition, and reported that the addition of milk powder increased the production yield, prevented spouting instability and allowed the most efficient processing in terms of the equipment thermal requirements. On product quality parameters, the authors also reported that the use of milk as a drying adjuvant increased calcium content on the final product and permitted a high ascorbic acid retention (around 70%) after thermal processing.

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5.4 Phytochemicals on Spouted Bed Dried Fruit Powders Over the last decade, the growing awareness for the benefits of health-promoting diets has led the food industry toward research and development strategies in order to take full advantage of the functional potential of fruits and vegetables (Cappato et al. 2018; Willett et al. 2019). Studies have shown that fruits play a major role in the balanced diets due to their vast phytochemical and fiber contents, which have protective properties against several diseases (Siriamornpun et al. 2012; Habauzit and Morand 2012). Phytochemicals are naturally occurring extra-nutritional plant metabolites, such as vitamins, phenolic acids, flavonoids, that can be related to biological activities in the human organism (Chang et al. 2016; Fang and Bhandari 2017). In fact, the bioactive performance of several groups of phytochemicals has already been documented. Anti-inflammatory activity of blueberries (Grace et al. 2019), wound-healing properties of strawberries and blackberries (Van de Velde et al. 2019), neuroprotective effects of camu–camu (Myrciaria dubia HBK McVaug) (Azevêdo et al. 2016) and antioxidant potential of acerola (Malpighia emarginata DC) (Cruz et al. 2019) have been consistently reported. Despite the several health benefits related to these bioactive components, the potential of such phytochemicals is still in the verge of exploitation. In addition to their occurrence in small quantities in fruits (Kris-Etherton et al. 2002), their availability for the final consumer is limited due to the seasonal aspect of fruit production, as well as the low storage stability, which is usually associated with these compounds (Karam et al. 2016). Studies have shown that drying of fruits can, in addition to extending products’ shelf life, help stabilizing the phytochemical contents of the dried powders (Correia et al. 2017; Moraes et al. 2017). Naturally sensitive, these bioactive compounds may be affected by a number of processing, storage, and delivery conditions, such as pH variations, temperature, presence of light, and oxygen (Patras et al. 2010), which may alter their molecular structure and compromise the biological functionality. In this sense, the use of the spouted bed drier has been regarded as an effective alternative for dealing heat-sensitive materials (Lucas et al. 2018; Alves et al. 2016), due to the possibility of milder operation conditions and the use of drying adjuvants that may act as a protection for the labile components (Dantas et al. 2019; Costa et al. 2015). Some aspects of the impact of spouted bed drying on the phytochemicals found on fruit pulps will be discussed as follows, which focus on total phenolics, anthocyanins, and ascorbic acid.

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5.4.1 Impact of Spouted Bed Drying on the Phytochemicals Content 5.4.1.1

Total Phenolic Compounds

The known potential for promoting health benefits has drawn much attention for increasing the presence of phenolic compounds on the human diet. Phenolic compounds are plant secondary metabolites structured around benzene rings with hydroxyl substituents (Lin et al. 2016). The anti-inflammatory, cardio- and neuroprotective properties, antioxidant and chemo-preventive activities of phenolic compounds can be related to several biological mechanisms, some of them still not fully described on the literature, that involve radical scavenging and inactivation, metal-ion chelation and single oxygen quenching (Oliveira et al. 2016; Sauceda et al. 2018). Studies have shown that the biological activities of polyphenols are concentrationdependent (Heleno et al. 2015), which highlights the drying strategy as an effective way of producing polyphenol-rich products that increase these phytochemicals’ concentration and stability while maintaining functionality (Correia et al. 2017; Hoskin et al. 2019). The total phenolic content was evaluated on spouted bed dried pepper and aromatic extracts (Benelli et al. 2013a, b), camu–camu (Fujita et al. 2013), bacaba fruit (Nascimento et al. 2019), cubiu fruit (Sales et al. 2019) and vegetable pastes (Larrosa et al. 2015). Camu–camu (Myrciaria dubia HBK McVaugh) is a small, round Amazonian fruit known for its high nutritional and nutraceutical value. It is known for its high contents of phytochemicals (phenolic compounds, β-carotene, vitamin C) and micronutrients (potassium, iron, phosphorus, amino acids) (Akter et al. 2011). Fujita et al. (2013, 2015) conducted studies on the impact of spouted bed drying on the physico- and phytochemical characteristics of camu–camu pulp. In addition, the research group also evaluated the impact of other drying techniques (spray drying and freeze drying) on the biological activities of camu–camu phytochemicals. The drying of camu–camu pulp was carried in a classical conical base spouted bed dryer, using high-density polyethylene (HDPE) for the inert particles bed. In this study, different temperatures (60, 80, 95, and 110 °C) were evaluated in order to assess the final impact on the products’ quality. Freeze drying of camu–camu pulp was performed for comparison reasons (Fujita et al. 2013). Authors reported a high concentration of phenolic compounds on the fresh camu–camu pulp (81.6 ± 6.5 milligrams of equivalent gallic acid per gram of dried sample [mg GAE/g DW]) and the average impact of the drying process on the phenolic content of camu–camu pulp was around 33–42%, but the increasing temperature did not statistically affect the phenolic results. In fact, although temperature-sensitive, the increase in temperature was followed by a reduction in processing time, which may have compensated the losses (Vega-Gálvez et al. 2012). In a follow-up study, the authors investigated the biological activities of dried camu–camu (Fujita et al. 2015). Although the drying technique reported in this

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study was spray drying, the authors indicated that the presence of several phenolic compounds such as gallic acid, syringic acid, ellagic acid, quercetin, and myricetin on the camu–camu pulp and dried powders were linked to anti-diabetic and antimicrobial activities, and cellular regeneration properties on planaria models. Cubiu (Solanum sessiliflorum Dunal) is an Amazonian fruit of the Solanaceae family, which is known by the indigenous communities in the Amazonia forest for its health-promoting benefits (Andrade Júnior et al. 2012). The biological activities associated with the phytochemicals from the cubiu fruit range from hypoglycemic and hypocholesterolemic control, anti-genotoxic, and antioxidant activities (Hernandes et al. 2014). Sales et al. described the drying of the cubiu pulp on a lab-scale spouted bed dryer in order to evaluate the influence of the drying temperature on the degradation of the phenolic compounds (Sales et al. 2019). The authors investigated the influence of two drying temperatures (50 and 70 °C) in the total phenolic content of cubiu pulp dried without the addition of drying carriers. The reduction in the total phenolic content from the dried cubiu pulp was temperature-dependent. In this case, the degradation of the phenolic compounds increased from 33.54 to 59.12% following the increase in temperature. Studies have shown that the use of drying carriers may help to reduce the degradation of phenolic compounds, as well as other phytochemicals, during spouted bed drying of fruit pulps. The use of such drying adjuvants will be discussed in the following section.

5.4.1.2

Anthocyanins

Anthocyanins are water-soluble flavonoid pigments found in tubers, flowers, and fruits. Chemically, the anthocyanins are known for their particular structure of an aromatic ring bonded to an oxygen-containing heterocyclic ring and linked to a third aromatic structure. As pigments, this group of bioactive compounds is responsible for the red, blue, and purple colors, while as nutraceuticals, the anthocyanins are linked to several health benefits (Frank et al. 2003). The low bioavailability of anthocyanins is related to their low stability in food systems (Khoo et al. 2017). Several factors such as temperature, pH variations, oxygen, and light exposure, type of solvent and co-pigment, contact with degrading enzymes, metal ions, and some antioxidant agents are responsible for affecting the molecular stability of these compounds and jeopardizing their functionality (Laleh et al. 2006; Castañeda-Ovando et al. 2009). On the other hand, recent studies have shown that drying is an effective way of increasing anthocyanins stability (Correia et al. 2017; Moraes et al. 2017; Roopchand et al. 2013). In addition, the in vivo and in vitro biological activities related to the anthocyanins content on dried powders have also been documented, and studies report the presence of hypoglycemic (Roopchand et al. 2012; Grace et al. 2009), anti-inflammatory (Esposito et al. 2014), cardio-protective (Bell and Gochenaur 2006), and anticarcinogenic (Wang et al. 2009) activities, among others. The anthocyanins content was evaluated on spouted bed dried grapes (Butzge et al. 2015), bacaba fruit (Nascimento

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et al. 2019), açai fruit (Lucas et al. 2018; Costa et al. 2015), blueberries (Feng et al. 1999), blackberries (Braga and Rocha 2015) and purple flesh sweet potato (Liu et al. 2015). Lucas et al. (2018) compared the impact of three drying processes (spouted bed drying, freeze drying, and spray drying) on the production of açai powder without the addition of drying adjuvants. On a physicochemical point of view, the authors reported that the moisture content of spouted bed dried açai powder (4.75 ± 0.21%) did not differ statistically from the moisture of the spray dried samples (4.75 ± 0.22%). Medeiros et al. (2002) have previously indicated that, with the proper tuning on operation conditions, the spouted bed obtained products can meet the overall quality parameters of spray dried powders, with lower production costs. When it comes to the phytochemical content, the anthocyanins content was considerably affected according to the chosen drying method. The authors reported that the total anthocyanins content on spouted bed dried açai powders (1.36 ± 0.02 mg/g) represented a 30% lower degradation of these phytochemicals, when compared to the powders obtained by spray drying (0.53 ± 0.04 mg/g) (Lucas et al. 2018). While the constant agitation of the spouted bed may provide a higher contact with oxygen (Oliveira et al. 2016), the temperature required for the spouted bed process was lower, when compared to spray drying (90 °C and 210 °C, respectively), which may explain the higher retention on the spouted bed dried samples (Braga and Rocha 2015). Regarding the pigments’ color stability, the authors (Lucas et al. 2018) also reported that the spray dried samples presented the highest total color difference (E; 16.28 ± 0.30), while the E for the spouted bed dried samples did not differ from the freeze dried samples (11.79 ± 0.47 and 12.22 ± 0.26, respectively).

5.4.1.3

Ascorbic Acid

Ascorbic acid, or vitamin C, is a water-soluble vitamin found in fruits and vegetables. Essential micronutrient, the bioactive capacity of vitamin C is well-documented in the literature, being related to strengthening of the immune system, antioxidant activities, skin health promotion, scurvy, cancer, and chronical diseases prevention (ManelaAzulay et al. 2003; Pullar et al. 2017). Vitamin C is a particularly heat-sensitive compound, and evaluating the impact of the drying processes on the degradation of this phytochemical is a key parameter for assessing the overall quality and efficiency of the process (Kamiloglu et al. 2016; Santos and Silva 2008). Fujita et al. (2013) investigated the impact of the spouted bed drying process on the vitamin C content on the camu–camu fruit pulp. In that occasion, the authors determined that the ascorbic acid retention on the dried powders was temperaturedependent. Despite the lower drying period associated with the higher temperatures, the sensitivity of the vitamin C to the increasing temperature was more significant and the retention varied from 55 to 36%, when temperature increased from 60 to 110 °C. The spouted bed drying of Tommy-variety mango pulp was described by Cunha et al. (2006) and the authors also evaluated the impact of the drying process on the

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ascorbic acid content. The authors described that the degradation of vitamin C was both temperature- and time-dependent. In the continuous process described in that study, authors reported that by the increasing of processing time, the accumulated mass on the spouted bed dryer was submitted to longer periods of exposure to temperature and oxygen, which are degrading factors for the ascorbic acid. However, authors highlighted that a freeze-drying process was performed for comparison purposes and the degradation of vitamin C on freeze dried samples were around 50%, while the degradation after spouted bed reached 65%. In this sense, the hypothesis was that in addition to temperature, other deleterious factors also contributed to the degradation of ascorbic acid.

5.4.2 Use of Drying Carriers Feed composition plays a major role on the efficiency of the spouted bed drying process. Studies have shown that the chemical composition is directly related to the glass transition temperature of the inlet feed, which may be responsible for altering the heat and mass transfer phenomena on the drying process. The presence of high concentrations of reducing sugars (glucose, fructose, lactose), for example, shows a negative effect on both powder production and the products’ quality parameters (Medeiros et al. 2002; Braga and Rocha 2015; Souza et al. 2009). Drying carriers have been used in the spouted bed drying process in order to alter the feed composition and improve both the process efficiency and the final products’ quality (Souza and Oliveira 2012). Maltodextrins, cyclodextrins, protein sources (whey protein, collagen, plant proteins), modified starch, and milk are some examples that have been mentioned in the literature as effective drying carriers for increasing powder production, solubility, and storage stability (Dantas et al. 2019; Butzge et al. 2016, 2015; Rocha et al. 2018; Costa et al. 2015). The combination of the adequate drying carriers and drying processing conditions may result in the encapsulation of the phytochemicals found on the fruit pulps. For encapsulation, the drying carrier is used in order to build a protective coat around the targeted compounds, in this case, the phytochemicals, which will be less exposed to deleterious agents (pH, moisture, temperature, light) and, hence, increasing storage stability (Correia et al. 2017; Fang and Bhandari 2017; Fang and Bhandari 2010). Studies have shown that the spouted bed dryer is an alternative for producing encapsulated products, which are usually core-shell type large microcapsules formed by coating mechanisms favored by the fluid dynamics of the spouted bed process (Jono et al. 2000; Baracat et al. 2008). Milk has been used as an alternative drying carrier in order to produce ready-toconsume dried powders. The use of milk powder and reconstituted milk in formulations for spouted bed drying has been reported for acerola fruit (Dantas et al. 2019) and blackberries (Braga and Rocha 2013). In both cases, the use of the drying carriers helped in increasing the phytochemicals retention on the dried powders by protecting these compounds from further degradation due to the thermal processing.

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Dantas et al. (2019) reported that the use of milk powder yielded a 72.9% of ascorbic acid retention on acerola fruit formulations, while Braga and Rocha (2013) described a 14% degradation on the anthocyanins content on the blackberry dried powders. Maltodextrin is a versatile form of hydrolyzed starch and it is frequently used as a drying carrier in order to modulate both the feed composition, process efficiency, and final powder characteristics. Souza and Oliveira (2012). Several are the reported benefits of maltodextrin as a drying carrier, such as high solubility, mild residual flavor, high glass transition temperature, and low hygroscopicity. In addition, authors have reported increased encapsulation efficiency and production under storage when maltodextrin is used (Zhang et al. 2018; Ballesteros et al. 2017; Vidovi´c et al. 2014). Costa et al. (2015) reported that maximum anthocyanins retention on the spouted bed drying of açai fruit pulp was achieved when 20% maltodextrin was used as drying carrier, 65 °C of air temperature, and air velocity corresponding to 1.25-times the minimum spouting air velocity. The same operation conditions were used by Nascimento et al. (2019) to produce bacaba fruit powder. The authors reported that using 20% maltodextrin resulted in anthocyanins and phenolic compounds retention around 90% and 80%, respectively.

5.5 Spouted Bed Drying of Graviola (Annona muricata) Pulp: A Case Study The studies regarding the spouted bed drying of fruit pulps carried out by the research groups at the Federal University of Rio Grande do Norte (UFRN) and the Federal University of Campina Grande (UFCG), in partnership with the State University of Campinas (UNICAMP) and the University of Tiradentes (UNIT) have successfully evolved in the last 20 years (Conrado et al. 2019). Based on the studies at UNICAMP (Braga and Rocha 2013, 2015), the UFRN research group developed work on the spouted bed drying of graviola fruit (Annona muricata) using milk and albumin as drying carriers, which presented promising results for processing efficiency and production yields (Machado et al. 2015; Machado 2015; Dantas and Machado 2015). In a follow-up study, Dantas et al. (2018; Dantas 2018) investigated the influence of the physical properties of graviola fruit mixtures and the operational conditions (drying carriers concentration and temperature) on the modeling of mathematical equations that described the powder production and the drying behavior on the spouted bed dryer, taking into account the intermittent feeding strategy. From the mixtures’ physical properties and the material accumulation data, several mathematical models were proposed in order to describe the powder production and the outlet air temperature behaviors. Such models were validated with the previous set of experimental data from Machado (2015), with satisfactory adjusts to the predicted variables. This case study is comprised of the relevant data gathered during the

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development of a master (Machado 2015) and a doctoral (Dantas 2018) thesis, which describes the drying of graviola fruit pulp and mixtures with milk in a spouted bed dryer.

5.5.1 Fundamentals Graviola (Annona muricata L.) is a popular fruit grown in all tropical regions, native from the Caribbean, Central and South America, and valued for its pleasant characteristics: moderate, aromatic acidity, juicy pulp, and distinct flavor (Quek et al. 2013). The fruits of the graviola are oval, large, and wide, about 10–30 cm long, and can weigh up to 4.5 kg (Nwokocha and Williams 2009). The edible pulp of the fruit corresponds to about 67.5% of the total fruit mass (Badrie and Schauss 2010). When it reaches physiological maturity, the graviola completes ripening within six days, which makes the graviola fruit extremely perishable (Oliveira et al. 2019). Among the enzymes found in the graviola fruit, pectinesterase stands out, a more heat resistant enzyme, which can lead to geleification and precipitation of pectin in pulps and graviola juices and polyphenol oxidase. This enzyme is responsible for darkening the fruit pulp (Badrie and Schauss 2010). Studies on the presence of compounds with phytochemical and pharmacological properties are being carried out and reveal the existence of new acetogenins in all parts of the graviola fruit. Among the pharmacological properties, the following stand out: cytotoxicity and anti-leishmanicide activity of the fruit pericarp; antiviral ability against the herpes-causing HSV-1 virus; anticarcinogenic and genotoxic effects (existence of acetogenins with antitumor property); activity for healing injuries and antimicrobial capacity (Coria-Téllez et al. 2018). Antibacterial activities against Staphylococcus aureus, Stapylococcus epidermidis, Propionibacterium acne, and Pseudomonas aeruginosa were also found positive (Pai 2016). Due to the fruit’s fragility and ease of suffering injuries, graviola is usually indicated for processing, being used in the manufacture of juices, nectars, syrups, shakes, sweets, jams, ice cream, powders, and flakes (Quek et al. 2013; Gratão et al. 2007). In parallel with the growth of fruit production, there has been an increase in consumption of fruit-based beverages in recent years. In these preparations fruits are usually associated with dairy compounds such as milk and whey protein. The development of these flavored dairy foods becomes an alternative that adds functional and more attractive value to the product (Moura et al. 2015). Considering the characteristics of graviola fruit and the importance of the study of drying this fruit minimizing the use of additives and incorporating nutritional ingredients that act as agents that enable the good performance of the process, the drying of the graviola pulp with addition of milk in a spouted bed dryer with inert particles was investigated (Machado 2015). The effects of milk concentration on the mixture, drying air temperature, intermittent feeding time and air flow on yield, production rate, powder moisture, drying rate, and thermal efficiency, as well as the impact of the process on the physicochemical characteristics and physical properties

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of mixtures reconstituted by post rehydration were assessed. This study was expanded (Dantas et al. 2018; Dantas 2018) investigating the influence of the physical properties of graviola pulp + milk mixtures and temperature in order to generate mathematical equations and models for the prediction of recovered powder and the spouted bed dryer behavior, based on mass and energy balances, using intermittent feeding.

5.5.2 Materials and Methods 5.5.2.1

Materials

Graviola pulp (GP) was obtained by manual depulping of ripe fresh graviola fruits acquired in the local market (Natal, Brazil). The fruit pulp was processed with a domestic blender and sieved through a nylon cloth (0.5 mm) in order to further remove any parts of peels and seeds. The GP was then frozen to −20 °C until use. Pasteurized whole milk was obtained from the local market. High-density polyethylene (HDPE) particles were used for the inert bed. The particles were characterized through the mean diameter (as a sphere of equal volume) and density by liquid-phase picnometry. The inert bed apparent density was calculated as the ratio between the inert load mass (2.5 kg) and the apparent volume of the bed. The porosity of the static bed (ε) was estimated according to Eq. 5.1. All measures were taken in triplicate. ρap = (1 − ε)ρinert

(5.1)

where ρap is the apparent density of the inert bed, ε is the porosity of the static bed, and ρinert is the density of the inert particles.

5.5.2.2

Drying Apparatus

The spouted bed dryer used in this work is similar to the apparatus shown in Fig. 5.1. The dryer was composed of a stainless steel cylindrical column (72 cm height, 18 cm diameter) with a conical base (60° angle, 13 cm height, 3 cm inlet air diameter) and a Lapple-type cyclone (40 cm height, 10 cm diameter, 5 cm overflow, 2.5 cm underflow). The inlet air was supplied by a 7 hp centrifugal blower (model CR-6, IBRAMWeq, Brazil) and heated by a set of 2 kW electrical resistances. The mixtures inlet feed was atomized using a twin-fluid nozzle coupled with a peristaltic pump. A low power compressor supplied the atomization air. A digital thermo-hygrometer, digital anemometer, digital K-type thermocouple thermometers, and a U-type manometer were used to measure the air temperature and relative humidity, air flow, air temperature in the cyclone outlet and in the drying chamber walls, and the bed pressure drop, respectively.

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Table 5.1 Study variables of the full 24 factorial experimental design Value

X L (%)

Tge (°C)

tinter (min)

v ∗ /vjm

−1

30

70

10

1.2

0

40

80

12

1.35

+1

50

90

14

1.5

Legend: X L —milk concentration; Tge —inlet air temperature; tinter —intermittent feeding time; v∗ /vjm —ratio between inlet air velocity and minimum spouting air velocity

5.5.2.3

Experimental Design and Drying Conditions

A full 24 factorial design with three repetitions on the central point (total 19 experiments) was used to investigate the influence of milk concentration (X L ; %), intermittent feed time (tinter ; min), drying temperature (Tge ; °C), and ratio between inlet air velocity and minimum spouting air velocity (v∗ /vjm ) on the drying yield (Y; %), powder moisture (Upo ; %), powder production rate (Wpo ; g/min), drying rate (K; g/s), and process thermal efficiency (EFF; %). The levels for each one of the four study variables are shown in Table 5.1. After the operation conditions were set, the feeding was initiated at a rate of 7.0 ± 0.8 mL/min using a twin-fluid atomizer nozzle. Six feeding cycles of 6 min were performed, while the intermittent feeding time was defined as the interval between the stop after 6 min feeding and the start of a new feeding cycle. Throughout the drying experiments, outlet air temperature and humidity were measured every 2 min. After the six feeding cycles, the inert particles were weighted in order to assess the amount of fruit powder not entrained and accumulated within the bed.

5.5.2.4

Samples Characterization

GP samples and GP + milk mixtures were characterized for soluble solids, density and viscosity. Soluble solids (SST; °Brix) measures were performed by direct read in a digital refractometer (model Smart-1, Atago, USA). The density was assessed by picnometry at 25 °C, according to AOAC method 952.22 (AOAC 2006). For viscosity, the rheological data of all samples were taken by a digital viscosimeter (model DV-II + Pro, Brookfield Engineering, USA), for 300 mL samples. Reads were performed in 5 min. Total acidity (ATT) was determined by titration and results were expressed as citric acid equivalents (g/100 g) (AOAC 2006).

5.5.2.5

Drying Yields

Drying yield (Y; %) was calculated as total solids recovery. It was defined as the ratio between the total solids content on the powdered samples (recovered at the cyclone outlet) and the total solids content on the initial sample formulations (GP +

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milk), according to Daza et al. (2016). For the spouted bed drying experiments, the powder production data was used to adjust a linear model and determine the powder production rate (Wpo ; g/min).

5.5.2.6

Powder Characterization

For the dried powders, samples were assessed for pH, moisture content, water activity, total titratable acidity, solubility, and reconstitution time. Samples pH was determined using a digital potentiometer (model Tec-5, Tecnal, Brazil), water activity was determined using a digital dew point hygrometer Aqualab® (model series 3 TE, Decagon Devices, USA) at room temperature (23 °C). Moisture was assessed by the gravimetric method at 70 °C (Tontul et al. 2018). Solubility was assessed according to Rocha et al. (2011). Samples (1 g) were mixed with 100 mL of distilled water, stirred for 5 min, and centrifuged (3000 rpm, 5 min). Supernatant aliquots (20 mL) were transferred to Petri dishes and dried at 70 °C to constant weight. Solubility was calculated as percentage of material soluble in the supernatant (%). Reconstitution time was evaluated by dissolving the powdered samples in distilled water under constant stirring until the initial soluble solids content of the non-processed mixture was achieved and no agglomerated material was observed.

5.5.2.7

Thermal Efficiency

The thermal efficiency of the spouted bed drying process was evaluated based on energy and mass balances and calculations according to Passos et al. (2004) and Saldarriaga et al. (2015). The overall thermal efficiency (EFF; %) was defined as the ratio between the energy (heat) used in the drying process for water evaporation (Q evap ) and the energy (heat) provided by the inlet drying air flow (Q inlet ), calculated according to Eqs. 5.2–5.4. Q evap (kJ/s) = Wevap × Λ

(5.2)

  Q inlet (kJ/s) = Winlet × Cp × Tge − Tea

(5.3)

EFF (%) = Q evap /Q inlet × 100

(5.4)

where Wevap is the evaporated water molar flow (kmol/s), Λ is the enthalpy of vaporization of water (kJ/kmol), Winlet is the inlet air molar flow (kmol/s), Tge is the inlet air temperature (°C), and Tea is the room temperature (°C).

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121

Mathematical Modeling

For constructing a mathematical model of the spouted bed drying of graviola pulp + milk mixtures, further drying experiments were performed, as described in Sect. 5.2.2. The sample mixtures of graviola pulp + 30% milk (GP-30M), graviola pulp + 40% milk (GP-40M), and graviola pulp + 50% milk (GP-50M) were dried at three different temperatures (60, 70 and 80 °C) using two different inert particles for the dryer’s bed, polypropylene (PP) and high-density polyethylene (HDPE). The intermittent feeding period defined for the experiments was 4 min. The physicochemical composition of the sample mixtures can be found in Table 5.2. The physical properties of mixtures and inert particles, as a function of the composition of materials and the temperature of the drying process, considered for the definition of the equation describing the production of graviola-based powders in the spouted bed dryer were viscosity, surface tension, density, contact angle, adhesion work, fat concentration, and reducing sugars concentration. The Vaschy–Buckingham theorem was applied in order to mathematically represent the inlet flow of the mixtures. The energy balance of the process was constructed based on the outlet air temperature, the mathematical model for powder production (powder removal from the dryer), the operational conditions of the drying process, the correlations for drying rates, and the dryer wall temperature. The mathematical model for the energy balance, based on the first law of Thermodynamics, considering the powder accumulation inside the dryer, can be written as Eq. 5.5. Wge cpge Tge + Wpe cppe Tpe − Wgs cpgs Tgs − cppo Tgs Wpo − kH v − Q˙ dTgs = dt m g cpgs + m pi cppi + m j cp j Tgs + Mcppo (5.5) where Tgs is the temperature of the outlet air flow (°C), Wge is inlet air mass flow (g/s), cpge is the specific heat of the inlet air flow (J/g K), Tge is the temperature of the inlet air flow (°C), Wpe is the mixture inlet mass flow (g/s), cppe is the specific heat of the mixture inlet flow (J/g K), Tpe is the temperature of the mixture inlet flow (°C), Wgs is the outlet air mass flow (g/s), cpgs is the specific heat of the outlet air flow (J/g K), cppo is the specific heat of the product powder (J/g K), Wpo is the Table 5.2 Physicochemical composition of the graviola pulp + milk sample mixtures

Parameter Reducing sugars (%) Fat (%) Water content (%)

GP-30M 3.45 ± 0.21

GP-40M 2.75 ± 0.17

GP-50M 2.07 ± 0.13

0.94 ± 0.08

1.25 ± 0.10

1.56 ± 0.11

12.42 ± 0.98

11.94 ± 0.95

11.63 ± 0.87

Legend: GP-30M—graviola pulp + 30% milk; GP-4M—graviola pulp + 40% milk; GP-50M—graviola pulp + 50% milk

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Table 5.3 Inert particles characterization results

Properties

Results (g/cm3 )

0.875 ± 0.468

Mean diameter (cm)

0.320 ± 0.050

Real density

Apparent density (g/cm3 )

0.537

Static bed porosity

0.386

powder production rate (g/s), k is the drying rate (g/s), H v is the enthalpy of water vaporization (J/g), Q˙ is the heat lost to the spouted bed dryer surroundings (J/s), m g is the outlet air flow mass (g), m pi is the inert particles mass load (g), cppi is the specific heat of the inert particles (J/g K), and t is time (s). The procedure for estimating the unknown parameter of the model, the heat lost to the dryer surroundings (Q), was based on minimizing the objective function of the least square technique using the heuristic method of optimization PSO (Particle Swarm Optimization), an algorithm developed by Kennedy and Eberhart (1995) and also known as a particle swarm method (Prata et al. 2009).

5.5.3 Results and Discussion 5.5.3.1

Inert Material Characterization

The results for the inert HDPE particles are presented in Table 5.3. From the characteristic curve of the inert particle bed, which represents the pressure drop in the bed due to the air surface velocity inside the column, the minimum spouting air velocity 0.8 m/s and stable spouting pressure drop of 319.37 Pa were determined. In order to work under stable fluid dynamic conditions, we chose to work in an air velocity range between 20 and 50% above the minimum spouting air velocity found for the bed without adding the mixtures of GP and milk.

5.5.3.2

Characterization of Graviola Pulp and Sample Formulations

Table 5.4 shows the physicochemical characterization of graviola pulp and the sample formulation with the addition of 30% (GP-30M), 40% (GP-40M), and 50% (GP50M) of pasteurized whole milk, prior to the spouted drying. According to Brazilian legislation (BRASIL 2000), the graviola pulp should present minimum values for total soluble solids of 9 °Brix, pH 3.5, total acidity of 0.6 g/100 g, and total solids of 12%. The results found in this study, where the graviola pulp was obtained without water addition, meet the legislation parameters. All found values are above the minimum required. Canuto et al. (2010) analyzed the GP produced in the Brazilian state of Pará and found values of 88.1%, 12 °Brix, and 3.7 for moisture, total soluble solids, and pH,

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Table 5.4 Physicochemical characterization of graviola pulp and graviola pulp + milk sample formulations Parameter

GP

Milk

GP-30M

GP-40M

GP-50M

Moisture (%)

82.4 ± 0.9

87.60 ± 0.35

85.33 ± 0.49

85.55 ± 0.14

85.73 ± 0.64

SST (°Brix)

15.7 ± 0.34

12.99 ± 0.06

12.89 ± 0.07

11.76 ± 0.30

9.69 ± 0.16

pH

4.13 ± 0.02

6.57 ± 0.02

4.17 ± 0.08

4.22 ± 0.08

4.47 ± 0.02

ATT

0.77 ± 0.04

0.53 ± 0.04

0.48 ± 0.02

0.46 ± 0.04

ND

Legend: GP—graviola pulp; GP-30M—graviola pulp + 30% milk; GP-40M—graviola pulp + 40% milk; GP-50M—graviola pulp + 50% milk; SST—total soluble solids; ATT—total titratable acidity; ND—not detected

respectively. Marcellini and Cordeiro (2003) presented results of analysis of graviola fruits produced and marketed in the Brazilian state of Sergipe: 88.3% moisture, 12.21 °Brix for soluble solids, pH 4.36 and 0.578 g/100 g of total acidity. The GP characterized in the present study contains less water, higher concentration of soluble solids and acidity, and pH among the values cited by the authors. It is important to mention that the in natura pulps present variations in their physicochemical characteristics due to variations in the physiological conditions of the fruit. The addition of milk with lower concentration of soluble solids and higher moisture content results in wetter mixtures with lower soluble solids content than natural GP. Since milk presents a basic character, its addition elevates the pH of the GP + milk mixtures compared to that of fruit pulp, and consequently lowers the acidity (Table 5.4).

5.5.3.3

Drying Process Performance

The matrix for the full 24 factorial experimental design and the results observed in the spouted bed drying of GP + milk formulations are presented in Table 5.5. In all drying experiments, the fluid dynamic conditions of the spouted bed dryer remained stable, without instabilities or bed collapse. However, in most experiments, low powder production was observed, which can be related to the accumulation of dried material in the dryer walls and the retention of powder in the inert bed (20.7 ± 5.7 g, on average) For all experiments, the powder production data as a function of time were linearized and the correlation coefficients presented good adjusts (>0.90). The angular coefficient for all the adjusted linear models represents the powder production rates. Figure 5.2 illustrates the linear adjusted models representing the cumulative powder production for experiments carried out with intermittent feeding time of 10 min. The other experiments presented similar behavior. These results are compatible with those reported by previous authors (Dantas et al. 2019; Braga and Rocha 2015; Souza Júnior 2012).

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Table 5.5 Full 24 factorial design and responses for the spouted bed drying of graviola pulp + milk formulations Run X L (%) Tge (°C) tinter (min) v ∗ /vjm

Y (%) Upo (%) Wpo (g/min) K (g/s) EFF (%)

1

−1 (30) −1 (70) −1 (10)

−1 (1.2)

14.28 6.39

0.078

0.083

38.49

2

1 (50) −1 (70) −1 (10)

−1 (1.2)

24.54 5.48

0.181

0.142

60.38

3

−1 (30)

1 (90) −1 (10)

−1 (1.2)

3.90 6.98

0.018

0.056

20.22

4

1 (50)

1 (90) −1 (10)

−1 (1.2)

17.32 6.17

0.099

0.091

27.86

0.024

0.092

44.65

5

−1 (30) −1 (70)

1 (14)

−1 (1.2)

4.70 9.99

1 (50) −1 (70)

1 (14)

−1 (1.2)

34.17 6.72

0.142

0.100

45.43

7

−1 (30)

1 (90)

1 (14)

−1 (1.2)

5.16 6.00

0.021

0.084

29.41

8

1 (50)

1 (90)

1 (14)

−1 (1.2)

10.98 5.39

0.036

0.081

33.62

6

9 10 11 12

−1 (30) −1 (70) −1 (10)

1 (1.5)

16.97 7.49

0.105

0.094

41.80

1 (50) −1 (70) −1 (10)

1 (1.5)

42.65 6.24

0.303

0.102

55.07

−1 (30)

1 (90) −1 (10)

1 (1.5)

10.95 5.43

0.069

0.116

22.40

1 (50)

1 (90) −1 (10)

19.87

1 (1.5)

33.96 4.40

0.203

0.070

13

−1 (30) −1 (70)

1 (14)

1 (1.5)

32.47 7.11

0.143

0.045

16.86

14

1 (50) −1 (70)

1 (14)

1 (1.5)

30.44 5.74

0.155

0.098

49.83

15

−1 (30)

1 (90)

1 (14)

1 (1.5)

10.06 5.26

0.055

0.067

16.82

16

1 (50)

1 (90)

1 (14)

1 (1.5)

36.28 4.18

0.151

0.066

21.99

17

0 (40)

0 (80)

0 (12)

0 (1.35) 20.72 6.32

0.087

0.083

33.39

18

0 (40)

0 (80)

0 (12)

0 (1.35) 15.97 7.27

0.067

0.104

29.33

19

0 (40)

0 (80)

0 (12)

0 (1.35) 17.44 5.85

0.089

0.081

28.71

Legend: X L —milk concentration; Tge —inlet air temperature; tinter —intermittent feeding time; v ∗ /vjm —ratio between inlet air velocity and minimum spouting air velocity; Y —powder production yield; Upo —powder moisture; Wpo —powder production rate; K—drying rate; EFF—thermal efficiency

As presented in Table 5.5, the moisture content of the powdered samples varied from 4.18 to 9.99%. Lower moisture values are noted for tests performed at the highest temperature. These results corroborate the values found by Souza (2009) for the spouted bed drying mixtures of mango, umbu, and seriguela pulps (4.4–7.5%). The powder production rates ranged from a minimum of 0.018 g/min in Run 3 to a maximum of 0.303 g/min in Run 10 with yields of 3.9% and 42.5%, respectively. By analyzing the responses for the full 24 factorial experimental design, the influence of the independent variables (study factors) on the moisture content, powder production rate and yields can be verified in the Pareto diagrams (Fig. 5.3), at a confidence level of 95%. As shown in Fig. 5.3a, none of the study variables, alone or combined with the other variables, presented significant effects on the moisture content, although it is noticeable in the Pareto diagram the trend for higher temperatures to produce powders

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Fig. 5.2 Representative behavior of powder production for the spouted bed drying of graviola pulp + milk formulations, at intermittent feeding time of 10 min

with lower moisture contents. These results can be justified by the low feeding flow of the mixture, kept virtually constant. In Fig. 5.3b, c are illustrated the Pareto diagrams for the powder production rate and drying yield, respectively. For the powder production rate, all study factors independently have significant effects on the response. The effects of air flow and solid concentration are positive and of greater intensity, while inlet air temperature and intermittent feeding time negatively influence the powder production. These results are compatible with phenomenological observations of the process and with the literature. The positive effect of milk concentration is justified by its fat content (Medeiros et al. 2002) and high glass transition temperature. The negative effect of the inlet air temperature on the powder production rate is cited by several authors. With the inert bed heated above the glass transition temperature of the dehydrated fruit pulp, the film adhered to the surface of the inert material. It behaves like a rubber-like material with hygroscopic characteristics, and its compromised detachment only occurs due to the action of shock and friction of inert material (Collares et al. 2004; Hofsetz et al. 2007). When the mixture feeding is suspended for a longer period and the air temperature is high, the bed becomes more heated and the glass transition temperature of the dehydrated pulp is achieved resulting in powder adherence and compromising powder production (Souza 2009). Higher air flow rates promote higher solid circulation rate and frequency of shocks between the inert particles, facilitating detachment of adhered film. Figure 5.3c shows that only the air flow showed a significant and positive effect on drying yield. The effects of milk concentration and inlet air temperature were almost significant, positive and negative, respectively, as expected since yield is a function of powder production. In the calculation of yield, the moistures of the fed mixture and the powder produced are included. The moisture of the mixtures has undergone small variations that depended on the characteristics of the processed graviola pulp. On

126 Fig. 5.3 Pareto diagrams for the influences of the study variables on the drying responses: a moisture content, b powder production rate, c drying yields

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the other hand, the moisture of the inlet air flow (not controlled in the experiments) and the temperature interfere in the moisture of the recovered powder. In a combined way, these variables ended up interfering in the yield and nullifying the statistical significance of the other variables. These interactions justify why the other operating variables presented statistically less significant and lower intensity effects on yield, although with the same trend observed in the powder production rate. Since none of the independent variables presented significant effects on the powder moisture content, statistical models were adjusted to the experimental data of the powder production rate and drying yield, and they are represented in Eqs. 5.6–5.7. For the powder production rate, effects that did not present statistical significance were eliminated from the models. For yield, the “almost significant” effects of inlet air temperature (Tge ) and v∗ /vjm ratio was considered in the models. Wpo = 0.109 + 0.044 × X L − 0.033 × Tge − 0.025 × tinter + 0.040 v∗ /vjm (5.6) Y = 20.155 + 8.24 × X L − 4.476 × Tge + 6.171 v∗ /v jm

(5.7)

where X L —milk concentration; Tge —inlet air temperature; tinter —intermittent feeding time; v∗ /vjm —ratio between inlet air velocity and minimum spouting air velocity; Y—powder production yield; Wpo —powder production rate. The adjusted models for both the powder production rate and yield have a reasonable quality of agreement with the experimental data. The determination coefficients indicate a satisfactory adjustment between the values observed and predicted by the correlations (R2 > 0.80). However, F tests for both regression models and lack of fit tests indicate that both models are statistically significant and that there is a satisfactory adjustment of the first-order model to experimental observations. Figure 5.4 illustrates the satisfactory adjust between the experimental data and the values predicted by the adjusted regression models.

5.5.3.4

Drying Rates and Thermal Efficiency

Before analyzing the behavior of drying rates and thermal efficiency of the spouted bed drying in the processing of GP + milk formulations, it is necessary to evaluate the temperature and humidity conditions of the outlet air flow, as these variables were used in the mass and energy balances and calculation of the drying and heat exchange rates and, consequently, in determining the thermal efficiency of the dryer. Figure 5.5 illustrates the experimental data of air humidity (Fig. 5.5a) and temperature (Fig. 5.5b) for the dryer outlet air flow, for a representative set of experimental conditions. This set of conditions includes the behavior observed in all experiments. In all curves, the same behavior reported by Dantas (2013) is observed, temperature oscillations due to discontinuation of the feeding flow (intermittent feed). These oscillations become more evident in the tests with longer intermittent feeding times and greater flow of air, represented by the v∗ /vjm ratio.

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Fig. 5.4 Experimental data versus adjusted mathematical models for powder production rate (a) and yield (b)

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Fig. 5.5 Representative experimental behavior of air humidity a and temperature b of the spouted bed dryer’s outlet air flow

As shown in Fig. 5.5, the behavior of air humidity is the same observed for temperature, with alternating oscillations as a function of the intermittent feeding time of GP + milk formulations. It is also observed that oscillations are attenuated when the intermittent feeding time is shorter. There is also a small influence of the inlet air temperature on the air humidity at the dryer’s outlet air flow. The air leaves the dryer with lower humidity content when drying occurs at higher temperatures. However, an important effect of the initial air humidity is observed, which corresponds to the humidity of the air fed to the dryer. Coincidentally, in experiments carried out at 90 °C, the fed air was much drier. This variable may have interfered more significantly in the air humidity curve along the drying process. The drying rates are subjected to variations of the initial air humidity (not controlled in this work), since they were calculated from the mass and energy balances for the inlet and outlet air flows. On the other hand, the GP + milk formulations feed flow also suffered mild variations (7.0 ± 0.8 mL/min) due to the physical properties

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(density and viscosity) of the sample formulations. Figure 5.6 illustrates, at the same scale, the feeding and drying rates of a representative set of experiments. The drying rate results presented in Table 5.5 represent an average of the drying rates observed during the feeding periods, since the drying rate was almost null during the periods when feeding was suspended. The results regarding the drying rates are consistent with those found by other authors (Souza Júnior 2012; Dantas 2013) and can predict the condition of constant drying rate with continuous pulp/mixture feeding, as cited by Moraes Filho (2013). The drying rates are subject to variations in the initial air humidity (not controlled). The heat exchange rates used for water evaporation during the drying process and the heat exchange rates lost to the dryer surroundings were calculated and are shown in Fig. 5.7. Figure 5.7 shows the same oscillatory behavior verified for the other variables previously analyzed, resulting from the intermittent feeding of the GP + milk formulations. The heat exchange rates used for water evaporation follow the same behavior as drying rates, and present lower values than the energy lost to the dryer’s surroundings (Fig. 5.7a, b). It is important to highlight the gap in the peaks of each curve and the most pronounced heat losses in the period in which the feed is suspended, when the heat spent on evaporation is minimal. As a result, the heat exchange rates lost fluctuate less over the drying process. Considering that the mixture feeding flow was constant in all experiments, an analysis of the results demonstrates that the dryer operates above the thermal conditions necessary to enable the evaporation rates required in the drying process. In addition to the temperature, the air flow required to maintain the stability conditions of the spouting bed with high circulation rate of solids and sufficient friction for the breakup of the adhered dried film extrapolates the thermal demands of the drying process. In the conditions illustrated in Fig. 5.7c, the behavior is different from that observed in Fig. 5.7a, b. It is observed that the peaks in heat exchange rates spent on evaporation coincide with the heat losses. Oscillatory behavior is maintained in these conditions, being less evident in the heat losses data, with a tendency to become constant. Heat losses to the dryer surroundings are in a range lower than those observed in Fig. 5.7a, b, which can be justified by the concomitant conditions of lower temperature and inlet air flow to the dryer. The drying rates ranged from a minimum of 0.046 g/s in Run 13 to a maximum of 0.142 g/s in Run 2 (Table 5.5). For Run 2 the high evaporation rate may be due to the high mixture feeding flow. Regarding the dryer’s thermal efficiency, the values observed in Table 5.5 are low and, for the most part, below 50% as expected. The Runs 2, 13, and 15 presented the highest and lowest thermal efficiencies, 60.80%, 16.86%, and 16.82%, respectively. Figure 5.8 shows the Pareto diagrams for the responses drying rate (Fig. 5.8a) and thermal efficiency (Fig. 5.8b) in relation to the study factors. From the analysis of Fig. 5.8, it is verified that none of the study factors, alone or combined, had statistically significant effects on the drying rates. This result was expected and agrees with other results found in the literature (Dantas 2013;

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Fig. 5.6 Representative experimental behavior of feeding and drying rates for a Run 3, b Run 6, and c Run 7

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Fig. 5.7 Representative experimental behavior of the heat exchange rates used on water evaporation during the drying process (Q evap ) and the heat exchange rates lost to the dryer’s surroundings (Q p ) for a Run 3, b Run 6, and c Run 7

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Fig. 5.8 Pareto diagrams for the influences of the study variables on the drying responses: a drying rate, b thermal efficiency

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Bacelos 2005) which shows the strong dependence on drying rates with the mixture feeding flow. As in this study, except for the variations due to the control difficulties, the mixture feeding flow was virtually constant, and the effects of the other study variables were not statistically significant. Regarding the dryer thermal efficiency, only the variable intermittent feeding time showed no significant effect. The effects of temperature and inlet air flow are negative, with the temperature presenting the highest intensity, which was expected and agrees with the previous discussion on extrapolation of these operating variables. The concentration of milk also interfered significantly and positively on dryer thermal efficiency. The influence of this study factor may be attributed to the higher inlet water flow to be evaporated since the higher concentration of milk means a lower percentage of solids in the feeding mixture. As previously mentioned, milk contains a lower solids content when compared to graviola pulp. In addition, the higher fat content in the inlet feeding mixture facilitates the particles’ flowability and bed stability, which can also imply better use of thermal potential for water evaporation. Some interactions between the variables also had significant effects on thermal efficiency, as observed in the Pareto diagram (Fig. 5.8b). Since none of the study factors presented significant effects on the drying rate, a mathematical model was adjusted only to the experimental data of thermal efficiency (Eq. 5.8). Non-significant effects from study factors and variables interactions were not considered in the model, which presented good adjust to the experimental data (R2 = 0.892). EFF = 33.50 + 5.18 × X L − 10.05 × Tge − 3.50 × v∗ /vjm − 3.37 × X L × Tge + 3.19 × Tge × tinter + 3.22 × X L × tinter × v∗ /vjm

(5.8)

Figure 5.9 shows the relation between the experimental data for the dryer thermal efficiency and the predicted values by the adjusted mathematical model, representing the satisfactory agreement between them.

5.5.3.5

Powder Characterization

For the physicochemical characterization of dried powder, the experimental conditions that presented the highest yields were chosen. Therefore, Run 10 (X L = 50%, Tge = 70 ◦ C, tinter = 10 min, v∗ /vjm = 1.5, Y = 42.65%), Run 13 (X L = 30%, Tge = 70 ◦ C, tinter = 14 min, v∗ /vjm = 1.5, Y = 32.47%), and Run 16 (X L = 50%, Tge = 90 ◦ C, tinter = 14 min, v∗ /vjm = 1.5, Y = 36.28%) were selected. The results for moisture content, water activity (aw ), total titratable acidity, and solubility of the dried powders are presented on Table 5.6. Moisture content and water activity are important parameters for food conservation. The moisture content represents the total water present in the food matrix, while water activity represents the free or available water content in the food matrix to be used on microbial, enzymatic, and biochemical reactions (Nóbrega et al. 2015;

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Fig. 5.9 Experimental data versus adjusted mathematical model for the dryer thermal efficiency

Table 5.6 Physicochemical characterization of the dried graviola pulp + milk on selected high-yielding conditions

Parameter Moisture content (%)

Run 10

Run 13

Run 16

5.68 ± 0.61

7.17 ± 0.09

4.41 ± 0.32

0.331 ± 0.001

0.375 ± 0.003

0.274 ± 0.007

ATT (g/100 g)

3.22 ± 0.13

4.46 ± 0.02

3.81 ± 0.02

Solubility (%)

64.18 ± 0.19

70.20 ± 1.38

66.52 ± 0.21

aw

Legend: aw —Water activity

Casciatori et al. 2015). The range established for dried and stable foods, from a microbiological point of view, is aw < 0.6 and moisture content below 25% (Dantas et al. 2019; Moraes et al. 2017). The powders obtained here are within the desirable ranges in relation to the two parameters. Dantas et al. (Dantas et al. 2019) reported similar values for moisture content (5.28–7.04%) and water activity (0.362–0.374) for spouted bed dried acerola pulp using milk powder and concentrated whey protein as drying carriers. Regarding the total acidity (expressed in g citric acid/100 g), comparing the results of the powders (Table 5.6) with those of the natural mixtures (Table 5.4), it is observed that the powders are more acidic, which was expected due to water evaporation. There

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Fig. 5.10 Dried powders and water reconstituted mixtures for a Run 13 and b Run 16

is also the highest acidity of the powder produced from the drying of the mixtures with 30% milk, due to the higher concentration of the fruit with acid characteristic. Solubility is an important physical property that can be defined as the ability of the solute (dried powders) to remain in homogeneous mixture with water (Borges et al. 2016). The powders presented high solubility, with values close to those found by Souza (Souza 2009) for the mixtures of siriguela, umbu, and mango with the addition of starch, pectin, and palm fat, 60.15 and 67.82%. The solubility results are comparable to those reported by Dantas et al. (2019) for spouted bed dried acerola pulp (61.5–75.4%), and higher than those reported by Correia et al. (2017) for spray dried blueberry extract using vegetal proteins as drying carriers (28.1–52.4%) Figure 5.10a, b display the images of the powders obtained in Runs 13 e 16, respectively, and the reconstituted mixtures obtained by rehydration with water. As observed, the reconstituted mixtures are homogeneous and very similar to natural mixtures. Table 5.7 shows the results for the physicochemical attributes of GP + milk mixtures prior to drying and the reconstituted mixtures from the rehydration of the powders obtained in Runs 10, 13, and 16. The pH of the reconstituted mixture remained close to that of the fresh mixtures, and in the acid foods classification range. No relevant changes were verified between the physicochemical attributes of initial GP + milk mixtures and the water reconstituted powders. The minor variations can be attributed to the eventual losses on volatile and other heat-sensitive components due to the drying process (Souza 2009; Borges et al. 2016), which may not be noticeable, from a sensory point of view, in face of the physicochemical results presented here.

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Table 5.7 Physicochemical attributes of graviola pulp + milk mixtures and reconstituted powders Parameter

In natura GP-30M

Reconstituted GP-50M

Run 10

Run 13

Run 16

SST (°Brix)

12.89 ± 0.07 11.76 ± 0.30 12.59 ± 0.33

12.99 ± 0.07

10.91 ± 0.16

pH

4.17 ± 0.08

4.22 ± 0.08

4.48 ± 0.02

4.04 ± 0.06

4.43 ± 0.106

ATT

0.53 ± 0.04

0.48 ± 0.02

0.496 ± 0.000 0.775 ± 0.131 0.543 ± 0.022

–

80

Reconstitution – time (s)

120

90

Legend: GP-3M—graviola pulp + 30% milk; GP-50M—graviola pulp + 50% milk; SST—total soluble solids; ATT—total titratable acidity

Powders with 50% milk were reconstituted faster than powder with 30% milk. It can be noted from Fig. 5.10 that the powder with 50% milk presents a looser aspect, different from the powder with 30% milk, which apparently presents more agglomerate.

5.5.3.6

Density and Rheological Attributes

For a more detailed evaluation of the effect of milk addition, the study of the rheological behavior of the in natura GP and GP + milk mixtures at room temperature was performed. Also, the rheogram of the reconstituted mixture from Run 10 was obtained. Figure 5.11 shows viscosity curves as a function of viscosimeter rotation. Both the GP and GP + milk mixtures in different concentrations, and the reconstituted mixture (Run 10) presented the same rheological behavior. They behave as non-Newtonian fluids (n < 1) with pseudoplastic characteristics, since viscosity decreases with increasing the rotation speed of the viscosimeter.

Fig. 5.11 Rheograms of samples of graviola pulp (GP), graviola pulp + 30% milk (GP-30M), graviola pulp + 40% milk (GP-40M), graviola pulp + 50% milk (GP-50M) and water reconstituted Run 10 powder

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Table 5.8 Rheological attributes of graviola pulp + milk mixtures Sample

n

μ (cP) 30 rpm

μ (cP) 60 rpm

ρ (g/cm3 )

GP

0.483

1080

963.9

1.015

GP-30M

0.359

535.9

395.9

1.019

GP-40M

0.395

539.9

427.9

1.023

GP-50M

0.374

567.9

264.4

1.029

Reconstituted Run 10

0.356

304.9

205

1.048

Milk

1.000

1.65

1.65

1.032

Legend: GP—graviola pulp; GP-30M—graviola pulp + 30% milk; GP-40M—graviola pulp + 40% milk; GP-50M—graviola pulp + 50% milk; μ—apparent viscosity; ρ—density; n—fluid flow index; cP—centipoise

Table 5.8 shows the values of the apparent viscosity of the unprocessed (GP, GP30M, GP-40M, and GP-50M) and water reconstituted mixtures (Run 10) for 30 and 60 rpm, as well as fluid flow index and density of samples. The values of the apparent viscosity for the rotations of 30 and 60 rpm are compatible with those reported by Medeiros et al. (Medeiros et al. 2002) for tropic fruit pulps. The apparent viscosities of GP-50M to 30 rpm is higher than that of the Run 10 reconstituted mixture; however, as the viscosimeter rotation increases (60 rpm) the values become closer. The specific mass (density) of the Run 10 reconstituted mixture is higher which can be attributed to possible changes in its composition, or changes in the physicochemical characteristics caused by the heating process. The density of mixtures increases with the concentration of milk which is justified by the higher density of milk.

5.5.3.7

Mathematical Modeling of the Spouted Bed Drying of Graviola Pulp with Intermittent Feeding

Based on experimental results and using the correlations for predicting physical properties as a function of temperature and milk concentration, the relationship between the powder production rate (Wpo ) and the reducing sugar and fat contents is represented by Eq. 5.9 (Dantas 2018).  Wpo =3.04 × 10

−6

×

× (m fat × σ )0.5

−2.44

η 1/6

ρ 1/3 × σ 1/2 × m fat

 ×

m rs m fat

2.29

 ×

Wad σ

−6.13

(5.9)

where Wpo is the powder production rate (g/s), η is the apparent viscosity (Pa s), ρ is density (g/cm3 ), m fat is the fat content (g), m rs is the reducing sugars content (g), σ is the superficial tension (mN/cm), Wad is the adhesion work (mN/cm).

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Evaluating the relationship between powder production and apparent viscosity, it was noticed that the influence on this response due to the change in the inert type is much more evident in conditions in which this property has lower values, resulting from the increase in temperature and the highest percentage of milk in the mixture. In the condition of lower concentration of milk and, consequently, higher viscosity, the type of polymeric material did not interfere so intensely. The relationship between powder recovery and density is centered on the variation of milk concentration, in which mixtures with a higher percentage of this ingredient resulted in larger densities and consequently higher powder production. There was little influence of process temperature on the variation of this physical property. The relationship between powder recovery and surface tension indicates that tensions with lower values implicated higher yields, which is related to increased temperature and percentage of milk in the mixture. There was an important increase in powder production with the increase in the variables of milk concentration and drying temperature, when drying was performed with HDPE, in these cases there was a reduction in the values of the adhesion work, which indicates that this inert was best suited for drying these materials. For graviola pulp mixtures with lower percentage of milk, the large amount of reducing sugars resulted in lower powder production during the drying process in the spouted bed dryer. However, in mixtures with a greater amount of milk, due to the increase in the fat concentration, there was a reduction in adhesion forces and the significant increase in product recovery (Araújo et al. 2015; Benelli et al. 2013b; Braga and Rocha 2013). By evaluating the adjust quality of Eq. 5.9 to the experimental data, it was found that 95.23% of Wpo variation is explained by the regression model, which indicates that it satisfactorily represents the experimental data. The adhesion work and surface tension have the greatest influence on the estimation of the powder production (Eq. 5.9), a fact observed by the higher values of exponents for these properties, when compared to the other terms. This reflects the importance of the interaction between fluid and the inert particle used in the spouted bed dryer. For the validation of powder production adjusted model (Eq. 5.9), data from Machado (2015) (Table 5.5) on the production of graviola pulp + milk powders were used. The mixtures of graviola pulp + milk (30–50%) were dried at 70–90 °C using HDPE. The experimental data and curves adjusted by the model (Eq. 5.9) are illustrated in Fig. 5.12. In order to evaluate the application of the adjusted model based on the mass and energy balances given by Eq. 5.5, the value of heat lost to the dryer surroundings (Q) was considered constant, determining an average value for each experiment. The determined results for heat lost to the dryer surroundings were equal to 203.71 ± 4.69 W, 232.88 ± 3.72 W, and 281.63 ± 5.85 W, for experiments carried out at 60 °C, 70 °C, and 80 °C, respectively. These values agree with those reported by Machado (2015). In this work, the authors found that 267 and 222 W corresponded to the heat losses for the processing of graviola + milk mixtures with 30% milk and 50% processed at 70 °C, respectively. Using these experiments for validation of Eq. 5.5 and Q = 232.88 W, the adjusted

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Fig. 5.12 Experimental data for graviola + milk powder production (Machado 2015) versus adjusted model (Eq. 5.9)

Fig. 5.13 Experimental data for graviola + milk (Machado 2015) versus adjusted model for heat loss (Eq. 5.5)

models can be found on Fig. 5.13. The deviations observed for these simulations were 2.35% and 4.82% for Figs. 5.13a and 5.13b, respectively. As observed in the curves adjusted to the experimental data, the model was able to predict the behavior of air temperature at the outlet of the dryer in these experiments. The most evident deviations observed in Fig. 5.13b are due to the difficulty faced in controlling the feeding flow of the mixture in this experiment, in which in some periods of intermittent feeding, the flow of the paste distanced itself from the predicted mean.

5.5.4 Final Comments The spouted bed dryer presented stable fluid dynamic behavior with uniform production rates although some tests presented low yield and large amount of material retained in bed and adhered to the walls of the equipment.

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All process variables presented significant effects on the powder production rate, with positive of milk concentration and air flow, and negative effects of temperature and intermittent feeding time. For yield, the important positive effect was air flow. None of the process variables had a significant effect on powder moisture; however, drier powders were obtained at higher air temperature. The first-order statistical models adjusted to the data of the production rate and yield were shown to be significant and useful for predictive purposes. The analytical resolution of the equations obtained by Vaschy–Buckingham theorem generated an empirical equation capable of predicting the behavior of the graviola pulp + milk powder production rate, considering process variables and physical properties of the mixtures. This model was used in the energy balance, in determining the air temperature at the outlet of the dryer, considering the accumulation of material in the drying system. The model described by the differential equations adapted to the intermittent feeding condition considering the accumulation of mass in the dryer bed was able to describe the phenomena of heat and mass transfer occurred during the spouted bed drying of graviola pulp + milk under stable operation. The heat exchange analysis demonstrated that the dryer operated under conditions that extrapolated the drying thermal requirements. The first-order statistical model adjusted to thermal efficiency data was significant and useful for predictive purposes. The obtained graviola + milk powders presented moisture and water activity in the indicated range for dry foods, short reconstitution time, and high solubility compatible with the literature. The reconstituted mixtures presented physicochemical and rheological characteristics close to those of the unprocessed mixture, which demonstrates the low impact of the process on the dehydrated product.

5.6 Conclusions Considering the most recent data reported on the drying of fruit pulps, it is perceived the current importance of research that seeks to enable the production of fruit powders through low impact technologies on the product. The current search for products rich in bioactive compounds and the consumption of healthier foods is an incentive to investigate drying methods that promote the production of high added value food components. The spouted bed dryer used in the drying of fruit pulps has presented promising results mainly with regard to the use of adjuvants such as milk that promotes the enrichment of the product without compromising sensory characteristics, and the rehydration capacity in water. Low performance problems are related to powder production or thermal efficiency need to be reevaluated by reusing the thermal potential of exhaust gas and techniques that will favor powder recovery. Studies on process modeling with intermittent feeding correlated with the composition and physical properties of fruit pulp are important and need to be expanded and validated with data from other fruits.

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Acknowledgments Fábio Medeiros was supported by Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES, Brazil), grant number 88882.375732/2019-01. The authors would like to thank the Federal University of Rio Grande do Norte (UFRN) for the technical support. We also acknowledge scientific support from the authors mentioned along this chapter.

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Chapter 6

Osmo-convective Dehydration of Fresh Foods: Theory and Applications to Cassava Cubes T. R. Bezerra Pessoa, A. G. Barbosa de Lima, P. C. Martins, V. C. Pereira, T. C. O. Alves, E. S. da Silva, and E. S. de Lima Abstract This chapter focuses on the hybrid process of osmotic dehydration and convective air drying of foods. Emphasis has been done to cassava cubes (Manihot esculenta Crantz.). The fresh cassava cubes had a water activity content of about 0.954, 60.45% moisture on a wet basis, 2.27% sucrose, and 0.13% sodium chloride on a wet basis. Herein, the kinetics of osmotic dehydration of cassava cubes were studied under an optimized operating condition of the convective drying kinetics of fresh and osmotically dehydrated cassava cubes which were evaluated at different drying conditions. Under mathematical point of view, different lumped approaches are used to estimation of average effective mass diffusivities (moisture and solids). Transient results of moisture loss, solids gain, sodium chloride, and sucrose incorporation, and moisture content are presented, compared with experimental data, and discussed. T. R. Bezerra Pessoa (B) · P. C. Martins · V. C. Pereira · T. C. O. Alves Department of Food Engineering, Federal University of Paraiba, João Pessoa, PB 58051-900, Brazil e-mail: [email protected] P. C. Martins e-mail: [email protected] V. C. Pereira e-mail: [email protected] T. C. O. Alves e-mail: [email protected] A. G. B. de Lima · E. S. de Lima Department of Mechanical Engineering, Federal University of Campina Grande, Av. Aprígio Veloso, 882, Bodocongó, Campina Grande, PB 58429-900, Brazil e-mail: [email protected] E. S. de Lima e-mail: [email protected] E. S. da Silva Department of Civil and Environmental Engineering, Federal University of Paraíba, João Pessoa, PB 58051-900, Brazil e-mail: [email protected] © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 J. M. P. Q. Delgado and A. G. Barbosa de Lima (eds.), Transport Processes and Separation Technologies, Advanced Structured Materials 133, https://doi.org/10.1007/978-3-030-47856-8_6

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The hybrid process has generated a dry product with 11% moisture, 16.89% sucrose, and 5.94% sodium chloride on a dry basis at better operation conditions. The product obtained from the cassava cubes hybrid process can be used in the food production at cassava base. Keywords Osmotic dehydration · Cassava · Drying · Hybrid drying process

6.1 Introduction 6.1.1 Drying Fundamentals Water is a major constituent of fresh food. It is intimately related to most of the physicochemical changes taking place in food. This constituent is vital in maintaining the quality and extending the shelf life of the food materials (Li et al. 1998). Since water activity is the thermodynamic measure of chemical potential, it can be used to determine the state of water in a solution or in a solid (Lewicki 2004). The water content is considered one of the most critical quality parameters, which influences the microbial growth and sensory attributes (texture, appearance, and flavor) of fresh foods (Hills et al. 1990). In this context, reduction in water content improves food materials stability and extends their shelf life. Drying is one of the most frequently used preservation methods of fresh foods, particularly vegetables, fruits, and meats that contain high water concentration. Drying provokes increases in the concentration of solids in food by removing water either by evaporation or sublimation (Kudra 2004; Koyuncu et al. 2007). Different drying methods have been applied in the industry to remove water from foods, such as spray drying, fluidized bed drying, foam-mat drying, microwave drying, osmotic dehydration convective drying, etc. The drying kinetics measurement and modeling can provide better insights into physical and chemical processes that take place when food materials undergo drying. Such insights provide reliable determination of drying time, help meet quality specifications, and lead to better energy conservation (Feng et al. 2014). Osmotic dehydration produces foods with an intermediate moisture content, whose shelf life is relatively short, and it can improve sensory characteristics (color, taste, texture, and others) of them; this drying technique has been used for many researches (Assis et al. 2018; Pessoa et al. 2017; Mierzwa and Kowalski 2016; Corrêa et al. 2017; Kumar et al. 2017). Convective drying is a very simple water removal operation applied to wet porous solids that has better moisture removal results and products with longer storage and consumption times under ambient conditions (Mercali et al. 2010; Isquierdo et al. 2013; Yadav and Singh 2014). Hybrid drying technologies are being developed to better preserve the natural food quality, better control of the residual moisture content, and to enhance drying efficiency (Feng et al. 2014). Despite the importance of the single drying process, the hybrid process plays an important role in this area. The use of combined processes aims to use the advantages of each technique of a particular preservation method

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transformation of raw materials into differentiated products that can improve their use in formulations and consumption. Thus, osmotic dehydration associated with convective drying of fresh foods represents a combined process of moisture reduction techniques that can provide products with better color and texture stability and longer shelf life. This new product should exhibit differentiated sensory characteristics from only convection dried material due to incorporation or solute loss during osmotic dehydration and increased storage time (24–48 months) at ambient conditions promoted by convective drying, which may increase its acceptance and expand its commerce and consumer market (Castro et al. 2018; Pessoa et al. 2011).

6.1.2 The Focus of This Work Brazil is one of the world’s largest cassava producers (Manihot esculenta Crantz), occupying the third position in the world ranking (Chicherchio 2014). This vegetable is a shrub, perennial plant, from Euforbiaceas family and dicots class. It has tuberous roots, which has high starch content in its composition. It is the only species of its genus commercially cultivated, aiming at the production of tuberous roots rich in starch (Fialho and Vieira 2011). There are several products that can be obtained from the cassava root. In the human food, it is consumed cooked, fried, and in other forms (Chicherchio 2014).

6.2 Application: Hybrid Drying of Cassava Cubes 6.2.1 The Raw Material 6.2.1.1

Samples Preparation

Experimental tests were performed with cassava (Manihot esculenta Crantz) purchased from local commerce in João Pessoa, Paraíba state, Brazil. The cassava was peeled by hand until the peel was completely removed and the root was cut with a 2.5 × 2.5 × 2.5 cm3 cube slicer. For osmotic dehydration, solutions were prepared with 56% w/w solute, 46% w/w sucrose, and 10% w/w sodium chloride. Concentrations of the solutions were verified through a benchtop refractometer. The previously weighed cassava cubes were placed in screw-capped 250 mL glass jars along with the osmotic solution. The material: solution ratio of 1:15 was used to ensure that the solution concentration remained constant throughout the process. Vials containing the samples were brought to a refrigerated digital shaker bench incubator (model LS4900-TZH, Alpax, Brazil) for 190 min at 52 °C and 180 rpm. Figure 6.1 shows the raw material and devices used in the experiments.

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Fig. 6.1 Fresh material, aluminum cutter, and incubator

6.2.1.2

Samples Physical and Chemical Characterization

The characterization experiments (moisture content, water activity, sodium chloride content, and non-reducing sucrose sugars) of the cassava cubes in natura and dehydrated osmotically were performed in triplicate. Moisture content was determined by the gravimetric method using heat, which is based on the weight loss of the material when heated to 70 °C in a vacuum oven until it reaches constant weight, according to the AOAC methodology (AOAC 1995). For water activity, the LabMaster equipment (NOVASINA, Switzerland) was used at 25 °C. Sodium chloride content was obtained by the Mohr method, based on titration with silver nitrate, using potassium chromate as an indicator (Jeffery et al. 1989). Non-reducing sucrose sugars were determined by the Lane-Eynon method, consisting of the complete reduction of the cupric ions of Fehling’s reagent reducing to cuprous ions under the action of heat in alkaline medium (Lane and Eynon 1923).

6.2.2 Osmotic Dehydration Tests 6.2.2.1

Experimental Procedure

The samples were taken from the incubator at 15, 30, 45, 60, 75, 90, 110, 130, 150, 180, 240, 300, 360, 420, and 480 min that correspond to 8 h of operation. In the test was observed the behavior of the mass transfer process between the solid and the solution. After removal, they were drained on stainless steel mesh to remove excess dehydrating solution and weighed. Thereafter, triplicate determinations of moisture content, sucrose, and sodium chloride were performed. The osmotic dehydration kinetics were studied by monitoring moisture loss (PU), solids gain (GS), and incorporations of sodium chloride (INaCl) and sucrose (ISac). Figure 6.2 illustrates the samples before osmotic dehydration process.

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Fig. 6.2 Fresh material and fresh cassava cubes inside osmotic solutions

6.2.2.2

Theoretical Procedure

The osmotic dehydration kinetics was theoretically studied using the Azuara’s model (1992). The authors developed an empirical model based on a mass balance that can be used for different geometries (without geometric constraint) including cube. The processes can be performed at short time intervals, as the model does not require that the equilibrium point has actually reached for the mass transport parameter prediction. Starting from a mass balance in the material that undergoes dehydration, the following equations are obtained for moisture loss, total solids gain, and sodium chloride and sucrose incorporations: MLt =

t(ML∞ ) S1 t(ML∞ ) = 1 1 + S1 t +t S1

(6.1)

SGt =

t(SG∞ ) S2 t(SG∞ ) = 1 1 + S2 t +t S2

(6.2)

NaClt = Suct =

t(NaCl∞ ) S3 t(NaCl∞ ) = 1 1 + S3 t +t S3 t(Suc∞ ) S4 t(Suc∞ ) = 1 1 + S4 t +t S4

(6.3)

(6.4)

where MLt is the moisture loss fraction at any time; SGt is the solid gain fraction at any time; NaClt is the sodium chloride incorporation at any time; Suct is the sucrose incorporation any time; ML∞ is the moisture loss fraction at equilibrium condition; SG∞ is the solid gain fraction at equilibrium condition; NaCl∞ is the sodium chloride incorporation fraction at equilibrium condition; Suc∞ is the sucrose incorporation fraction at equilibrium condition; S1 is a constant related to the water diffusion rate out from product; S2 is a constant related to the rate of solids diffusion in the product; S3 is a constant related to the rate of sodium chloride incorporation in the product; and S4 is a constant related to the rate of sucrose incorporation in the

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product. Rewriting Eqs. 6.1, 6.2, 6.3, and 6.4, in the linear form we obtain t 1 t = + MLt S1 ML∞ ML∞

(6.5)

1 t t = + SGt S2 SG∞ SG∞

(6.6)

t 1 t = + NaClt S3 NaCl∞ NaCl∞

(6.7)

1 t t = + Suct S4 Suc∞ Suc∞

(6.8)

Crank (1975) obtained a simplified equation for the Fick’s physics–mathematical model, considering a small process time, transient regime, diffusion in a semi-infinite medium, constant osmotic solution concentration, and external resistance to mass transfer negligible. This model is represented by the following equations:   DeffML t 1/2 MLt =2 ML∞ π L2   SGt DeffSG t 1/2 =2 SG∞ π L2   NaClt DeffNaCl t 1/2 =2 NaCl∞ π L2   Suct DeffSuc t 1/2 =2 Suc∞ π L2

(6.9)

(6.10)

(6.11)

(6.12)

where DeffML is the effective moisture loss diffusivity; DeffSG is the effective solids gain diffusivity; DeffNaCl is the effective sodium chloride incorporation diffusivity; DeffSuc is the effective sucrose incorporation diffusivity, and L is the characteristic dimension. Replacing Eqs. 6.9, 6.10, 6.11, and 6.12 in the Eqs. 6.5, 6.6, 6.7, and 6.8, respectively, four simple expressions are obtained to calculate the effective diffusivity of moisture loss, solids gain, sodium chloride incorporation, and sucrose incorporation at different times, as follows:   mod 2 ML∞ · exp t ML∞    2   S2 L SG∞mod πt · DeffSG t = exp 4 1 + SSG t SG∞



DeffML



=

πt 4



S1 L 1 + SML t

(6.13)

(6.14)

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2   NaCl∞mod · exp t NaCl∞ 2      S4 L Suc∞mod πt DeffSuc t = · exp 4 1 + SSuc t Suc∞

DeffNaCl





πt 4

=

S3 L 1 + SNaCl t

(6.15)

(6.16)

where ML∞mod is the moisture loss value at equilibrium condition obtained by Eq. 6.1; mod SGmod ∞ is the solids gain value at equilibrium condition obtained by Eq. 6.2; NaCl∞ is the sodium chloride incorporation value at equilibrium condition obtained by Eq. 6.3; Sucmod ∞ is the sucrose incorporation value at equilibrium condition obtained exp by Eq. 6.4; ML∞ is the moisture loss value at equilibrium condition obtained experiexp mentally; SG∞ is the solids gain value at equilibrium condition obtained experimenexp tally; NaCl∞ is the sodium chloride incorporation value at equilibrium condition exp obtained experimentally, and Suc∞ is the sucrose incorporation value at equilibrium condition obtained experimentally. Applying Eqs. 6.9, 6.10, 6.11, and 6.12 for a cubic geometry, considering the characteristic dimension L as the edge of the cube, and replacing them in Eqs. 6.1, 6.2, 6.3, and 6.4, respectively, we obtain the following equations: DeffML = DeffSG = DeffNaCl =



π 1

4t 3 

π 1

4t 3 

π 1

4t 3

DeffSuc =

1

4t 3

2   mod  3 ML∞ · exp ML∞

S2 L 3 1 + S2 t

2   mod  3 SG∞ · exp SG∞

S3 L 3 1 + S3 t



π

S1 L 3 1 + S1 t

2   mod  3 NaCl∞ · exp NaCl∞

S4 L 3 1 + S4 t

2   mod  3 Suc∞ · exp Suc∞

(6.17)

(6.18)

(6.19)

(6.20)

Thus, the general average effective diffusivity obtained from the transport of materials (water outflow, total solids input, sodium chloride, and sucrose) during of cassava cube can be calculated by using Eq. 6.21, as follows: N 

Deff =

(Deff )i

i=1

N

(6.21)

where Deff is the average effective diffusivity in time; Deff is the effective diffusivities for each time and N is the number of data used. The mean relative error (E) was calculated using Eq. 6.22 to evaluate whether the model was or not predictive (E < 10%).

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E=

N

100

Vexp − Vpre



N i=1

Vexp

(6.22)

where Vexp is the experimental value, Vpre is the predicted value, and N is the number of experimental points.

6.2.3 Convective Drying Tests 6.2.3.1

Experimental Procedure

Drying tests were performed at different temperatures with fresh cassava cubes osmotically dehydrated. The process was carried out in a convective fixed bed dryer with perpendicular air flow. Figure 6.3 shows the convective drying equipment. The drying air temperatures chosen were 50, 60, and 70 °C using constant velocity (1.35 m/s) and absolute humidity (0.060 kg water/kg solid) conditions of the drying air. Prior screening of the air velocity profile was performed using a hot wire anemometer (model AK833, China) and the temperature and relative humidity in the drying chamber using a portable thermo-hygrometer with probe (model AK625, brand AKSO). Figure 6.4 illustrates these devices. Convective drying kinetics was obtained by experimental measurements of the mass of the cassava cubes as a function of the drying time, with 60 min intervals until Fig. 6.3 Convective dryer used in the experiments

6 Osmo-convective Dehydration of Fresh …

159

Fig. 6.4 Anemometer and thermo-hygrometer

the final drying period. Drying air conditions were monitored during the experiments and the tests were performed in triplicate. The drying analysis was performed through the experimental curves of the moisture content as a function of the time and the drying rate as a function of the moisture content. Drying curves were expressed with moisture content on a dry basis, according to literature (Moyers et al. 1999). The average final moisture content data of the cassava cubes (fresh dried and osmotically dehydrated) at three drying air temperatures were subjected to variance analysis at a significance level of 5% and Tukey test mean comparison by using STATISTICA® software (STATSOFT 1997). According to Montgomery (Montgomery 2008), variance analysis, also known as ANOVA, is an approach used to compare various interest groups. It seeks to assess if there are considerable differences between the investigated groups. The Tukey test is based on an amplitude distribution of the T function. It is used to calculate the minimum significant difference and comparing it with the means difference obtained for each treatment, assuming a statistical significance level (Nogueira 2007).

6.2.3.2

Transport Parameter Estimation

The average effective moisture diffusivity of cassava cubes exposed to convective drying was obtained through the Fick’s model simplified by Crank (1975), considering uniform initial moisture distribution, absence of thermal effect in mass transfer, and applied to an infinite flat plate and long drying times. For a cube, all sides were

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considered of equal dimension; thus, the solution of the Fick’s model applied to a cube will be given as follows: 

M − Me M0 − Me



 =

8 π2



 3 π 2 Defft exp − ∴ (2L)2

L=

L sample 2

(6.23)

where Defft represent the effective diffusivity, M is the average moisture content, Me is the equilibrium moisture content, M0 is the initial moisture content, t is the time, L corresponds to the characteristic length (half the thickness on each side of the cube).

6.2.4 Results and Analysis 6.2.4.1

Physicochemical Characterization of Cassava

The physicochemical characterization of Manihot esculenta crants in natura was carried out according to the referenced methods in Sect. 2.1.2. Table 6.1 presents the results obtained. The cassava roots consist basically of water and sucrose, according to results in Table 6.1. The chemical composition of this material consisted of 60.45% moisture, 2.27% sucrose, and 0.13% sodium chloride. The water activity value found in the cassava root is above the minimum water activity values for the development of pathogenic microorganisms; thus, this raw material is not microbiologically stable (Chirife and Favetto 1992). Luna et al. (2013) presented results for the moisture (35.31%) of raw cassava roots well different from those obtained in this work. These differences may be related to variations in soil moisture. Maieves (2010) evaluated the moisture content of cassava tubers at various harvest times and concluded that the roots collected in February showed higher moisture than those collected in the month of May. He says that roots grown in sandy soils have an average moisture of 60% (wet basis). The values of non-reducing sugars in sucrose obtained in this research differ from those found by Aguiar et al. (2014), which obtained higher values, about 3.94% in the root of the frozen Mandiocaba variety. Differences in sugar content vary with the Table 6.1 Physicochemical characterization of the fresh cassava cubes

Fresh cassava cubes average valuesa

Analysis

0.954 ± 0.00

Water activity Moisture (w.b.

%)b

60.45 ± 0.03

Sucrose (%)

2.27 ± 0.19

Sodium chloride (%)

0.13 ± 0.02

value ± standard deviation basis

a Mean b Wet

6 Osmo-convective Dehydration of Fresh …

161

Fig. 6.5 Moisture loss and total solids gain kinetics in the osmotic dehydration of cassava cubes in ternary solution

harvest period. Couto (2013) observed that there was a decrease in the total sugar content in cassava roots harvested later. The plant’s development cycle consists of five physiological phases and in the first phase it needs more energy, producing a higher sugar concentration. Rinaldi et al. (2015) studied the effect of different freezing forms on cassava roots and concluded that the use of this conservation process causes less changes in the structure of the raw material, prolonging the conservation of its initial nutritional characteristics. Valduga et al. (2011) found significant differences in the sodium levels of five cassava cultivars (BRS Rosada, Casca Roxa, BRS Dourada, BRS Egg Yolk, and Saracura) harvested at 8 months after planting. The cultivar Casca Roxa showed a significant content of sodium chloride 0.19% (dry basis); this value was close to what was obtained in this work (0.13%). The authors concluded that the chemical composition of cassava is specific not only for each cultivar, but also depends, mainly, on associated genetic factors.

6.2.4.2

Osmotic Dehydration Process

Drying Kinetics in the Optimal Condition After different experimental tests of osmotic dehydration of cassava cubes were chosen the best drying condition for the study. Optimum conditions obtained in ternary solution of sucrose, sodium chloride, and water were 52 °C, 56% w/w solute, 10% w/w sodium chloride, 190 min and 180 rpm. The kinetic parameters result from the osmotic dehydration of cassava cubes are presented through the moisture loss curves and solids gains shown in Fig. 6.5. From the analysis of Fig. 6.5, in the early stages of operation (first 15 min) the highest gradients of moisture loss and solids gain occurred. After this period, the solids gain almost reached equilibrium after 30 min of operation. Moisture loss continued to increase until reaching a maximum value of 24.89% and stabilized after 180 min of operation. An unexpected event occurred after 360 min of osmotic dehydration of the cassava cubes. There was a small decline in moisture loss and a

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sudden increase in the solids gain of samples, whose parameters were re-established in values equal to 22.7 and 16.45% for moisture loss and solids gain, respectively. The solids gain stabilized before the moisture loss probably due to lower internal mass transfer resistance of the solutes (sucrose and salt) as compared to moisture. The behavior of the mass transfer rates of moisture and solutes between the cassava cubes and the hypertonic osmotic solution may be associated with the chemical and physical characteristics of these solutes. Junqueira et al. (2017) have reported that high molecular weight solutes such as sucrose are responsible for increasing moisture loss in the dehydrated product and promote low solids gain. The sucrose promotes an increase in the viscosity of the osmotic solution, which may lead to the formation of a barrier on the vegetable surface, limiting its impregnation in the product (Pereira et al. 2006). However, low molecular weight solutes provide a higher solid gain when compared to high molecular weight solutes because they are able to penetrate into the food cell, increasing their concentration in the product (Junqueira et al. 2017; Pereira et al. 2006; Ruiz-López et al. 2011; Sritongtae et al. 2011). Considering the above reports, sucrose was probably responsible for the greater moisture loss, and sodium chloride by the solids gain in the dehydration process of cassava cubes in ternary solutions. High moisture loss and solids gain rates in the early hours of the osmotic dehydration process of guava were reported by Castro et al. (2018). Similar results were found by Silva et al. (2012), when studying the osmotic dehydration of acerola. According to Ramya et al. (2014), the early stages of osmotic dehydration play an important role because the transport phenomena are faster, causing greater impact on the process progress. Results of moisture loss and solids gain similar to those found in this research were found by Alam et al. (2013) when using binary solution of sodium chloride and water. They found a final value of 28.42% of moisture loss and 11.24% for the solids gain in osmotic dehydration of onion under the conditions 25 °C, 6 h of process and concentration salt of 25 °Brix in the solution. However, when using a ternary solution with 55 °Brix sucrose and 15 °Brix of salt, under these same conditions of temperature and time, the moisture loss increased to 48.08% and the solids gain increased slightly to 13.53%. According to these authors, an increase in moisture loss and solids gain occurs as the concentration of the solution increases. Figure 6.6 shows the diffusion profile over time of sucrose and sodium chloride incorporations for osmotic dehydration of cassava cubes in ternary solution of sucrose, sodium chloride, and water. The analysis of the results shown in Fig. 6.6 indicated that the kinetics of sucrose and sodium chloride incorporation presented the same behavior. Sucrose and salt gain occurred more intensely in the first 15 min of the process when it had already incorporated approximately 3% and 1%, respectively. Then, we noticed a slower gain of these two response variables in a time interval of up to 180 min. From 180 to 300 min of process, there was again an increase in sucrose and sodium chloride incorporations, rising to 6% and 3%, respectively. At 300 min, stability in mass transfer rates was observed for all incorporations. At the end of the experiment, the incorporations were approximately 9% for sucrose and 4% for sodium chloride.

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Fig. 6.6 a Sucrose and b Sodium chloride incorporations kinetics in the osmotic dehydration of cassava cubes in ternary solution

In the study of osmotic dehydration kinetics of cassava slices in ternary solution, Carmo et al. (2017) found that in the early hours of the process, the presence of a high concentration of sugars provided a rapid mass transfer, leading to greater gain of soluble solids. The sugars balance condition was reached from 200 min with a final value of 10%. Similar results were found by Aires et al. (2016) and Castro et al. (2014) for guava osmotic dehydration, and by Borsato et al. (2010) for apple osmotic dehydration. Araújo et al. (2014) studying the process of osmotic dehydration of carrots in temperature of 50 °C, 240 min and solution concentration of 50 °Brix, found solids incorporation values using sucrose solution, similar to this research, of 22.13%. However, Hadipernata and Ogawa (2016) studying potato osmotic dehydration at 20 °C in 8 h with 20% solution concentration, obtained in their research a smaller sucrose gain of 1.75%, only. Mercali (2009), when studying mass transfer kinetics in banana fruit, reported a high rate of sodium chloride incorporation at the beginning of the dehydration process followed by lower rates in the final stages of the process. Vázquez-Vila et al. (2009) also observed that solids gain using sodium chloride solution at 17.22 and 26% w/w at 25.35 and 45 °C increased at the beginning of the carrot osmotic dehydration process. Silva Júnior et al. (2015) observing that the incorporation of solids in dehydrated green beans with sodium chloride solution increased at the beginning of the process. Further, the authors found that the effective diffusivity of the solid increased with the increase of sodium chloride concentration in the solution from 20 to 26.5%. Regarding the stabilization time of the transfer rate of soluble solids, sucrose, and sodium chloride, Mayor et al. (2006), similarly to the information obtained in this research, found that the balance occurred around 5 h in pumpkin osmotic dehydration in sodium chloride solutions between 5 and 25% w/w, at 12.25 and 38 °C with times from 0 to 9 h. Kumar et al. (2017) found a solid incorporation content of 4.20% by employing sodium chloride solution in osmotically dehydrated chayote cubes. The conditions used were temperature 35 °C, process time 180 min, 10% sodium

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Fig. 6.7 Osmotically dehydrated cassava cube

chloride concentration, and 1:6 ratio between fruit and solution. Figure 6.7 illustrates the cassava cube osmotically dehydrated on the specific condition.

Moisture and Total Solids Effective Diffusivities Estimation The determination of the moisture and solutes mass transport parameter in the osmotic dehydration of cassava cubes was carried out using the empirical model proposed by Azuara et al. (1992). Initially, a linear regression related to moisture loss, solids gain, and sucrose and sodium chloride incorporations as a function of time, represented by t/ML, t/SG, t/NaCl, and t/Suc was performed. Figure 6.8 presents the linearized experimental results of these parameters for the determination of parameters of the (Eqs. 6.5–6.8). From the analysis of the analyses of Fig. 6.8, we can see that the model used satisfactorily represented the experimental data for the four studied variables (in these linear regressions the correlation coefficients (R2 ) were varied from 0.943 to 0.999). From the results of the fitted parameters, it was possible to determine diffusivities of moisture, solids gain, sucrose, and sodium chloride for osmotic dehydration of cassava cubes in the desired optimum condition. The values of these parameters are presented in Table 6.2. By analyzing Table 6.2, the Azuara et al. (1992) model presented relative mean error (E) ranging from 3.4 to 22.2%, and determination coefficients (R2 ) close to 1.00 for all parameters indicating a good fit to the experimental data. According to Table 6.1, the constants of sodium chloride (S3 ) and sucrose (S4 ) incorporations and the effective diffusion coefficient showed similar values, indicating that these two solutes were incorporated with the same velocity into the dehydrated cassava cubes. This fact probably occurred due to the material structure as well as the competition between the fluxes of the two solutes during the process. Since the molecular weight of the salt is lower than the sucrose, it spontaneously enters the plant cell, resulting in a reduction in the sucrose mass transfer coefficient. Further, the total solid gain velocity constant (S2 ) was higher than the moisture loss constant

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165

Fig. 6.8 Comparison between the experimental and theoretical (Azuara et al. 1992) results of the moisture loss, solids gain, and sodium chloride and sucrose incorporations as a function of time

(S1 ). The absorption rate of total solids in the osmotically dehydrated cassava cubes was about 63% higher than the water outlet velocity, resulting in the higher average effective diffusivity of the total solids in relation to moisture effective diffusivity. This behavior is probably related to the amount, type, and concentration of solutes used in the osmotic solution and the operating temperature. Comparison between the effective diffusivity data cited in the literature is a complicated task due to the different estimation methods and various models employed, associated with changes in the chemical composition of the food and its physical structure. In addition, effective diffusivity varies throughout the process and the osmotic moisture gradient is not constant (Azoubel and Da Silva 2008; Mercali et al. 2011). The effective diffusivities values for moisture and solids transport obtained in this research were of the order of 10−8 m2 /s, in which the average value of these transport parameters for osmotic dehydration of cassava cubes was 1.88 × 10−8 m2 /s. Values from the same order of magnitude were obtained for several osmotically dehydrated vegetables such as carrot (Sutar and Prasad 2011), mango (Arias et al. 2017) nectarine (Rodríguez et al. 2013), and banana (Góis et al. 2010).

166 Table 6.2 Fitted parameters of the Eqs. 6.5–6.8 in the best-operating conditions

T.R. Bezerra Pessoa et al. Variables

Parameters

Value

Moisture loss

T (min)

360

MLmod ∞ (%)

27.967

S1

Solids gain

8.265 × 10−4

R2

0.995

E (%)

8.407

Deff (m2 /s)

1.99 × 10−8

t (min)

360

SGmod ∞ (%)

12.923

S2

22.521 × 10−4

R2

0.999

E (%)

3.389

Deff (m2 /s)

2.77 × 10−8

Sodium chloride incorporation t (min)

480

NaClmod ∞ (%) 4.489 S3

Sucrose incorporation

1.883 × 10−4

R2

0.966

E (%)

15.678

Deff (m2 /s)

1.36 × 10−8

t (min)

480

Sucmod ∞ (%)

9.573

S4

1.802 × 10−4

R2

0.943

E (%)

22.224

Deff (m2 /s)

1.40 × 10−8

MLmod ∞ = value of the moisture loss at equilibrium obtained by the model; SGmod ∞ = value of the solids gain at equilibrium obtained by the model; INaClmod ∞ = value of the sodium chloride incorporation at equilibrium obtained by the model; ISucmod = value of the ∞ sucrose incorporation at equilibrium obtained by the model

Arias et al. (2017) and Rodríguez et al. (2013) associated the high diffusion coefficients with the temperature used in the osmotic dehydration process. Escobar et al. (2007) explained that high values of diffusivities are due to the effect of measurements that cause cell death of plant tissue before osmotic pretreatment, facilitating mass transfer. The authors cite as an example bleaching. Maldonado et al. (2008) justified that high diffusion coefficients are linked to high moisture loss values. Góis et al. (2010) reported that the differences in the order of magnitude of diffusivities probably derive from the type of geometry used in the material. Allali et al.

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167

(2008) and Corzo et al. (2008) stated that the magnitude of effective moisture diffusivity for food materials varies in the range of 10−8 to 10−12 m2 /s. This variation is explained by the type of experimental analysis, composition and physiology of the food and the method of data treatment. Azarpazhooh and Ramaswamy (2012) observed that the effective moisture and solid diffusivities in osmotic dehydration of apple cylinders reached maximum values at temperatures of 50 °C and 50 °Brix solution concentration, close to the results presented in this work for osmotic dehydration of cassava cubes. Assis et al. (2017) found that the higher temperature (60 °C) promoted higher diffusion of water and solute in the osmotic dehydration process of apple cubes. This behavior was also observed by Abbasi Souraki et al. (2012) in osmotically dehydrated green beans ranging from 30 to 50 °C. Lower diffusivity results (order 10−9 m2 /s) were obtained by Rodríguez et al. (2017) in plum osmotic dehydration. Assis et al. (2017) in osmotically dehydrated apple cubes, Singh et al. (2008) for pretreated carrot cubes in sodium chloride solution, Hamedi et al. (2018) on osmotic dehydration of ultrasound-assisted agar gel cylinders, and Azarpazhooh and Ramaswamy (2012) analyzing osmotically dehydrated apple cylinders. Cassava cubes were dehydrated in ternary solution of sucrose, salt, and water at high concentration (56% w/w) combined with high temperature (52 °C). High temperatures increase the solubility of solutes and cause changes in plant cell structure, increasing the diffusion rate of sodium chloride and sucrose into the cassava cube compared to water diffusion. Several authors have observed solid diffusivity values greater than water diffusivity in osmotic dehydration of vegetables (Arias et al. 2017; Azarpazhooh and Ramaswamy 2012; Assis et al. 2017; Hamedi et al. 2018; Zúñiga and Pedreschi 2012). Assis et al. (2017) found lower values of water diffusivity in relation to total solids in osmotic dehydration of apple cubes in sucrose and sorbitol solution, at temperatures ranging from 25 to 60 °C. They report that the sucrose solution has high viscosity due to the high molecular weight of the solute, making it difficult to transport water between the product and the solution. Arias et al. (2017) found lower effective moisture diffusivity values in sleeves at 50 °C by increasing the sucrose solution concentration from 45 to 60 °Brix. They explained that this behavior is due to the fact that the more dilute the solution can penetrate better into tissues, as opposed to concentrated solutions that are more viscous and can form a sucrose surface layer, making it difficult for water to escape from the material. Similarly, Assis et al. (2017) found lower values of water diffusivity compared to solids in the osmotic dehydration of apple cubes in sucrose and sorbitol solution, at temperatures ranging from 25 to 60 °C. They proved that the type of osmotic agent used, as well as its molecular weight, influenced the diffusion of water, obtaining smaller diffusivities when the osmotic dehydration was performed with sucrose solution. The authors explained that sucrose solution is more viscous than sorbitol and that sucrose has higher molecular weight, making it difficult to transport water between the product and the solution. Hamedi et al. (2018) observed higher values of solid diffusivity in relation to water, in the osmotic dehydration of an agar gel cylinder, in sucrose solution in an ultrasonic

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T.R. Bezerra Pessoa et al.

Fig. 6.9 Dimensions of moisture as a function of time for drying process of in natura and osmotically dehydrated cassava cubes at various process temperatures

bath (100% power). The authors explained that increasing the concentration of the osmotic solution resulted in greater absorption of solids in the sample due to the increased osmotic pressure gradient between the solution and the dehydrated product. Finally, we state that migration rates of total solids and moisture content, as shown in Fig. 6.6, and demonstrate that total solid gain reaches equilibrium more rapidly in relation to moisture loss in cassava cubes. This indicates a greater difficulty in migrating water from the solid matrix to the surrounding osmotic solution as compared to its transfer of solutes (sucrose and salt) into the solid. Other considerations, such as the higher amount of moisture in migration, the synergistic effect of salt for the penetration of sucrose in the vegetal membrane, and others, can clarify this difference of values between the average effective diffusivities of total solids and humidity.

6.2.4.3

Convective Drying Process

Drying Kinetics for Cassava Cubes Dried in Natura and Osmotically Dehydrated Figure 6.9 shows the dimensionless moisture content as a function of time for two different drying physical situations: dried in natura and osmotically dehydrated cassava cubes at the various drying air temperatures. By comparing the curves for the two sample types it is evident that the shortest drying time was detected for the dried samples without pretreatment. The in natura cassava cube dried at 70 °C presented the shortest drying time (≈12 h), and the osmotically dehydrated cassava cubes at the three temperatures had equal drying times, totaling 24 h of process, twice from the in natura sample time dried at 70 °C. Castiglioni et al. (2013) reported that the shortest convective drying time of cassava fibrous mass was reached at the highest temperature studied at 67 °C.

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169

Analyzing Fig. 6.9 for drying kinetics of fresh material dried at 50 and 60 °C, a similar trend is observed. However, by checking the drying curves separately for the fresh sample at 50 and 70 °C and 60 and 70 °C, another type of behavior was observed, as the temperature increased, the curve became sharper. This, the time required to reduce the moisture content to an almost equilibrium condition is reduced too, as stated above. On the other hand, more than 50% of the moisture of fresh cassava cubes was removed during the first 5 h of drying at 70 °C, and during the first 8 h for drying at 50 and 60 °C. The remaining water content was eliminated over a comparatively longer period. This probably occurred due to the shrinkage of the cube with the corresponding pore reduction that increased the resistance of water transport during the drying process (Pereira et al. 2009). Osmotically dehydrated samples took longer drying times when compared to fresh samples at three temperatures. It was observed that the moisture of the osmotically dehydrated material decreased rapidly at higher temperatures, generating more pronounced curves by increasing this parameter. More than 50% of moisture of the osmotically pretreated materials at the three temperatures analyzed were removed within the initial 10 h of the drying process. Pavkov et al. (2011) observed a similar result when studying the drying kinetics of osmotically dehydrated nectarine seeds at 60 °C. Kowalski and Łechta´nska (2015) also reported similar behavior when comparing the drying time for fresh beet and osmotically dehydrated in 25% NaCl solution at 65 °C. The authors reported that the longer drying time in the dehydrated material was due to the crystallization of sucrose contained in the beet, and the impregnation of the solutes on the food surface. The substances acted as a barrier to moisture output from the product. However, Kowalski and Mierzwa (2011) stated that vegetables submitted to the osmotic dehydration process should dry out faster than fresh vegetables due to their lower moisture content. Osidacz and Ambrosio-Ugri (2013) found that the osmotically dehydrated eggplant in 10% NaCl solution achieved a shorter drying time at 70 °C compared to in nature eggplant dried at the same temperature. The drying rates of in natura and osmotically dehydrated cassava cubes as a function of time for all drying condition are presented in Fig. 6.10. This figure reveals that there is no constant drying rate period for both types of materials studied, and that the entire drying process occurs in the decreasing rate period, being represented by the first and second phase of the falling drying rate with internal migration control of moisture. Several authors have also observed this behavior for various types of vegetables, such as Akoy (2014), Cabrera et al. (2016), Kek et al. (2013), and Rayaguru and Routray (2012). In the period of decreasing rate, it gradually decreases due to decreased evaporation of water inside the material; the process is controlled by the moisture migration mechanism inside the material, so that a dry layer begins to form on the product surface. After the removal of bound moisture from the food, the phase change of the various types of intrinsic moisture in the food occurs, which are bound to the components by physical and chemical forces (Pavkov et al. 2011). The influence of selected drying air temperature levels and sample type on drying kinetics is clearly visible in the images shown in Fig. 6.6. By comparing the drying

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T.R. Bezerra Pessoa et al.

Fig. 6.10 Drying rate as a function of moisture content of in natura and osmotically dehydrated cassava cubes at various process temperatures

rate for both sample types at all temperatures analyzed. It should be noted that the highest drying rates were achieved in fresh materials due to their higher initial moisture content compared to osmotically dehydrated samples. Kaya et al. (2016) researching the drying of fresh and osmotically pretreated Kiwui found a higher drying rate in fresh samples at higher temperatures. They justified this behavior, as due to the high moisture value of the fresh samples and explained that the increase in the temperature difference between the drying air and the product accelerates the water migration. The larger temperature difference between drying air and fresh cubes increased the convective heat transfer coefficient, influencing the heat and mass transfer rates. Several authors have reported similar results for the drying of fresh and osmotically dehydrated fruits such as figs (Babalis and Belessiotis 2004; Gupta and Patil 2014; Sacilik and Elicin 2006; Togrul 2005). The impregnation of sucrose and sodium chloride in dehydrated material during osmotic pretreatment of cassava cubes was probably responsible for the decrease in the drying rate. Similar results were verified by several authors in their studies with the drying of osmotically pretreated vegetables, such as Pavkov et al. (2011) for nectarine seeds, Osidacz and Ambrosio-Ugri (2013) for eggplant, Dionello et al. (2009) for pineapple, Kaya et al. (2016) for carrots, Andrés et al. (2007) for mango

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171

and Sanjinez Argandoña et al. (2005) for guava, Azoubel et al. (2009) for cashew nuts, Guiné (2006) and Park et al. (2002) for pear, Korsilabut et al. (2010) for melon, Riva et al. (2005) for apricot, Sankat et al. (1996) for bananas, and Singh and Gupta (2007) for carrots. Dionello et al. (2009) working with drying fresh pineapple and dehydrated in solutions of sucrose and invert sugar explained that the low humidity rates found in the osmotically dehydrated samples occurred due to the concentration of the osmotic solution and the type of solute used in it. Lower molecular weight solutes such as glucose and fructose found in invert sugar penetrate more easily than sucrose into the tissues of the top layer of the vegetable, entering more intensely to make it difficult to draw water inside the food, thus reducing the drying rate. Andrés et al. (2007) and Sanjinez Argandoña et al. (2005) found that the presence of sucrose molecules in pretreated mango and guava tissue, respectively, increased the internal resistance to water diffusion. Pavkov et al. (2011), studying drying of nectarine seeds in natura and osmotically dehydrated, explained that the difference between the partial pressures of water molecules on the surface of the fresh material and in moist air is greater. Such behavior promotes a high convective mass transfer coefficient at the surface of the material, generating faster evaporation and increased drying rate. The reduction in moisture of the osmotically pretreated sample generates a decrease between the partial pressures of water molecules on the material surface and in the moist air during the drying process. As the solute concentration in the plant tissue increases, the effective moisture diffusivity decreases, this increasing the drying time of the osmotically dehydrated sample. The parameters that characterize the convective drying of the cassava cubes in natura and osmotically dehydrated are presented in Table 6.3. The humidity of both types of samples was obtained dynamically, weighing the cubes until reaching a constant weight in the convective drying process. The parameter studied was the drying air temperature, which ranged from 50 to 70 °C. As stated earlier, there was no constant drying rate period for both types of materials. It can be seen from Table 6.3, through the transition moisture content (M tr ) values, that there were two periods of decreasing drying rate with internal moisture migration control, before that osmotically dehydrated and fresh cassava cubes reach the final moisture content (M f ). The times for the 1st stage and the 2nd stage correspond to the periods of the first and second phase of the decreasing rate, respectively. Table 6.3 shows that the in natura samples reached equilibrium moisture content in a shorter time than the osmotically dehydrated samples, as shown in Fig. 6.10. According to this table, the moisture contents found for the fresh and osmotically dehydrated cassava cubes at all temperatures analyzed were below 13% on a wet basis. These values are within the values established by Standard Resolution RDC No. 23 of December 14, 2005 (Brazil 2005). This regulation establishes tolerance limits for starch products derived from cassava root, which require moisture values of less than 14% and 15% on a wet basis for starch and tapioca, respectively.

0.76

0.98

70

0.49

0.43

0.43

0.65

0.63

0.58

0.60

0.57

0.57

1.00

1.09

1.11

0.37

0.36

0.36

0.50

0.52

0.53

(w.t.)

0.09

0.11

0.15

0.05

0.06

0.07

(d.b.)

M af

0.081

0.102

0.128

0.051

0.060

0.065

(w.t.)

2

1

1

2

2

1

t tr 1st stage (h)

22

23

23

10

18

20

t 2nd stage (h)

T = Temperature; M 0 = Initial moisture content; M tr = Transition moisture content; M f = Final moisture content; t = Time a Total average value in triplicate in the dry and wet basis, tr = transition time

0.74

60

1.88

70

50

1.67

60

Osmotically dehydrated

1.38

50

In natura

(d.b).

(d.b.)

(w.t.)

M tr

T (°C)

Sample

M a0

Table 6.3 Characterization of convective drying of cassava cubes in natura and osmotically dehydrated

24

24

24

12

20

21

T otal time (h)

172 T.R. Bezerra Pessoa et al.

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173

Table 6.4 Fitted parameters of the Fick’s model at different convective drying condition Temperature (°C) Parameter (Deff × 1010 m2 /s) in natura cube Osmotically dehydrated cube 50

5.45

2.75

60

5.83

3.61

70

9.85

4.82

From Table 6.3, we can see that the equilibrium moisture content of the fresh dried cubes was almost 50% lower than cubes dried by hybrid process. This fact occurred due to the high concentration of solids incorporated in the material during the osmotic dehydration step, forming an obstacle for water outlets in the convective drying step. The in natura cube dried at 70 °C reached the lowest equilibrium moisture content (5.11% on wet basis). A similar result was obtained by Pornpraipech et al. (2017) during drying process of cassava rectangular slices at 80 °C. However, Lugo et al. (2018) found 12% final moisture content for dry cassava at 70 °C in a hybrid drying system. Correa et al. (2017) observed lower equilibrium moisture content values in fresh dried samples and higher samples in osmotically dehydrated samples in sucrose solution when drying pineapple at 70 °C assisted by ultrasound. They justified that drying at high temperatures probably caused the caramelization of the incorporated sugars, establishing additional barriers to the outflow of water from the solid pineapple matrix. As in this paper, other authors studying the drying of fresh and osmotically pretreated products also found lower final moisture content values in fresh products (Osidacz and Ambrosio-Ugri 2013; Kaya et al. 2016; Singh and Singh Hathan 2016). In general, it is observed in Table 6.3 that the drying times of the in natura dried cubes were also shorter than the drying times of the osmotically pretreated cubes at the same temperatures, decreasing with increasing temperature. However, for osmotically dehydrated cassava cubes, the total time was equal at the three temperatures. These results agree with those obtained by Kaya et al. (2016), who observed longer drying times in osmotically dehydrated carrot slices in sucrose and sodium chloride solution at 35 °C, while studying the drying of fresh and osmotically pretreated carrots. Garcia-Noguera et al. (2010) found drying times of 612 and 891 min for strawberries dried in natura and osmotically dehydrated previously (50% sucrose solution), respectively.

Moisture Effective Diffusivity Estimation In this research, the simplified diffusional model (Fick’s 2nd Law) was fitted to experimental data of convective drying of fresh and osmotically dehydrated cassava cubes for long drying times, without considering shrinkage. The aim was to estimate the moisture diffusion coefficient.

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Table 6.5 Variance analysis for final moisture content of in natura cassava cubes Final moisture content Source

SS

DF

MS

p-Value*

Fc

F tab

Samples

1.0860

2

0.5430

1.1460

1.1460

6.94

Temperatures

2.8152

2

1.4076

2.9708

2.9708

6.94

Error

1.8952

4

0.4738

Total

5.7964

8

SS = sum of squares; DF = degrees of freedom; MS = mean square; F c = Calculated F; F tab = Tabulated F; *Statistically significant at p < 0.05 level

Table 6.4 presents the values of the effective moisture diffusivity obtained in this non-linear regression as applied to convective drying at temperatures 50, 60, and 70 °C. According to Table 6.4, an increase of the effective moisture diffusivity with increased drying temperature for the two samples analyzed was observed. Fresh samples showed higher effective diffusivity values than osmotically dehydrated samples due to the higher initial moisture content. The results obtained are in agreement with those reported in the literature (Singh and Gupta 2007; Aires et al. 2018; Ruiz-López et al. 2010). The osmotically dehydrated cassava cubes presented lower diffusivity values due to their lower initial moisture content and high concentration of incorporated solids during the drying process. Osmotic pretreatment resulted in a reduction of free water in these samples, contributing to the reduction of the mass transfer rate in convective drying. Aires et al. (2018) and Zuñiga and Pedreschi (2012) reported that solids gain can cause the formation of a barrier, making it difficult to mass transfer within the product during convective drying. Dehghannya et al. (2018) observed that potato cubes pretreated in solutions with higher sucrose concentrations (50 and 70%) showed lower effective moisture diffusivity values than samples pretreated in solutions with lower sucrose concentrations (10 and 30%), after convective drying; they explained that the use of this high solute concentration in osmotic solutions caused changes in potato texture, which in turn made it difficult to remove moisture from the product. According to Dehghannya et al. (2015), osmotic solutions in high concentration can degrade the texture of the vegetable and stop the moisture diffusion during drying.

Determination of the Best Drying Condition The determination of the best drying condition was based on the moisture content of the product and the operating time for both types of samples, according to the data presented in Table 6.5.

6 Osmo-convective Dehydration of Fresh … Table 6.6 Tukey test for comparison of averages final moisture content of cassava cubes and dried by hybrid process

175

Temperatures (°C)

Average final moisture content1

50

12.79(a)

60

10.22(ab)

70

8.05(b)

1 Means followed by the same lowercase letter in the same column

do not differ from each other by the Tukey test at 5% probability

According to Table 6.4, it is observed that for the fresh material, the drying time at 70 °C was shorter compared to the other drying times of fresh samples dried at 50 and 60 °C. However, in terms of final moisture content, there is a proximity of the results obtained for fresh samples at the three temperatures analyzed. Thus, an analysis of variance (ANOVA) was performed at a significance level of 5%, to verify if there was a significant difference between the values of these parameters at the three temperatures. According to the analysis of variance for final moisture content of the fresh samples shown in Table 6.5, it can be seen that there was no significant difference between samples and temperatures (p > 0.05), with calculated F values lower than the tabulated F. Thus, the best drying condition for fresh cassava cubes was at 70 °C, as it presented a shorter drying time (12 h). The Tukey test was used to verify the differences between the average final moisture content of the subjected to hybrid process (osmo-convective drying) cassava cubes at the three temperatures analyzed, as shown in Table 6.6. By Tukey test, the average final moisture content of the products at temperatures between 50 and 60 °C and 60 and 70 °C showed no significant difference. However, between 50 and 70 °C, there was a significant difference, so the temperature of 60 °C was chosen for convective drying of the osmotically dehydrated cassava cubes. As a final comment, despite the importance of the statistical methodology applied here in aiming to obtain the best convective drying condition of in natura and osmotically dehydrated cassava cubes, this is a food product. In this case, new statistical tests taking account nutritional aspects of the dried cassava cubes are strongly recommended, in order to obtain the best drying conditions based on the final moisture content, energy consumed nutritional and sensory parameters.

6.2.4.4

Evaluation of the Product Obtained by the Hybrid Process

To verify some changes that the hybrid process (osmotic dehydration and convective drying) can provoke on the final product, the physical–chemical characterization of this material was carried out, comparing the results with the physical–chemical information obtained for the cubes only dried at 60 °C and in natura cubes. Table 6.7 presents the chemical composition results of the cassava cubes in natura and osmotically dehydrated and dried at 60 °C.

176 Table 6.7 Composition of cassava cubes in natura and osmotically dehydrated and dried at 60 °C

T.R. Bezerra Pessoa et al. Parameter

In natura Cassava cubes

Cassava cubes dried at 60 °C

Water activity

0.954 ± 0.00

0.232 ± 0.00

Sucrose (g/100 g m s)

5.745 ± 0.60

16.894 ± 0.42

Sodium chloride (g/100 g m s)

0.340 ± 0.05

5.946 ± 0.06

Mean value ± standard deviation

The hybrid moisture reduction process (combination of osmotic dehydration and convective drying) produced a material with desirable shelf life, as it reduced the water activity value for values below 0.6, which prevents the pathogenic microorganisms’ growth that is responsible to deteriorate the food (Chirife and Favetto 1992). The values of water activity close to those presented by the osmotically dehydrated and dried cubes were found by Silva et al. (2013). These authors found water activity value of 0.180 in tapioca flour. Vieira et al. (2010) found a water activity value of 0.310 for sweet biscuits prepared with 15% cassava starch. The water activity reduction is more pronounced during convective drying, because the free water in food is evaporated, reducing the water vapor pressure, which acts on the food (Moreno et al. 2010). During the osmotic dehydration process, the reduction in water activity occurs slowly, because diffusion phenomena occur in an aqueous medium. Sodium is an alkali metal that constitutes approximately 40% of salt, i.e., in mean, 1 g salt contains 0.4 g of sodium (He and Macgregor 2010). The sodium value (2,136 mg) found in 100 g of the osmotically dehydrated and dried cassava cube at 60 °C is close to the daily nutrient reference values established in Standard RDC nº 360, on December 23, 2003. This standard reports the nutritional values of packaged foods and establishes 2,400 mg sodium as the daily nutrient reference value (Brazil 2003). The 2,136 mg value represents 89% sodium in a food portion. The maximum sodium limit recommended by the World Health Organization (WHO) is 2 g per day (Sarno et al. 2013). Then, the sodium content found in 100 g of the osmotically dried cassava cube at 60 °C exceeded the maximum sodium limit recommended by WHO. According to the data in Table 6.7, sucrose in the osmotically dehydrated cube, and dried at 60 °C showed a value three times greater than its initial value. However, sodium chloride showed a higher incorporation power than sucrose, showing a 17fold increase in its initial value. Solutes with a higher molecular weight such as sucrose have lower mass diffusivities due to its high molecule size, allowing less mobility in food materials through existing pores and free spaces of plant tissues. Sugars are generally hydrophilic, uncharged molecules that exhibit slow diffusion rates (Ruiz-López et al. 2011; Agnieszka and Andrzej 2010; Udomkun et al. 2015). Udomkun et al. (2015) observed high concentrations of glucose, fructose, and lower sucrose contraction value in osmotically pretreated and dried papaya cubes. They explained that this fact may be related to the solute lower molecular weight of glucose

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and fructose that penetrate more easily in the vegetable’s superficial layer. Brandão et al. (2003) found higher sucrose concentrations in osmotically dehydrated and sun-dried mangabas when compared to the mangaba in natura.

6.3 Concluding Remarks In this chapter, the drying process of food has been studied. Emphasis is given to the hybrid drying process (combination of osmotic dehydration and convective drying) as applied to cassava cubes. The study encompasses two analyses: experimental and theoretical (used to process parameters estimation). Herein transient results of the moisture loss solid gain and sodium chloride and sucrose incorporation along the process are presented. From the obtained results it can be concluded that (a) The fresh cassava cubes showed 60.45% moisture (wet basis), 2.27% sucrose, and 0.13% sodium chloride. (b) The study of osmotic dehydration kinetics in the optimized condition showed that the moisture loss reached equilibrium in 180 min of process, the total solids gain in 30 min, and the incorporation of sodium chloride and sucrose in 300 min. (c) The highest values of the osmotic dehydration kinetics study parameters were 24.90% moisture loss, 16.46% solids gain, 8.87% sucrose incorporation, and 4.07% sodium chloride. (d) The average effective moisture diffusivities, solid gain, incorporation of sodium chloride and sucrose presented results equal to 1.99 × 10−8 m2 /s, 2.77 × 10−8 m2 /s, 1.36 × 10−8 m2 /s, and 1.40 × 10−8 m2 /s, respectively. (e) Convective drying of fresh cassava cubes presented shorter drying times than previously osmotically dehydrated cassava cubes due to the presence of sucrose on the outer surface of the cubes making it difficult to remove moisture from the inside of the material. (f) The entire convective drying process of the fresh and osmotically dehydrated cubes at 50, 60, and 70 °C occurred during the falling moisture migration rate. (g) The fresh samples in convective drying showed higher drying rates than the osmotically pretreated material in sucrose solution and sodium chloride. (h) The convective drying temperatures that produced a material with lower final moisture content was 70 °C for fresh cassava cubes and 60 °C for osmotically dehydrated cassava cubes. (i) The average effective moisture diffusivities obtained through the simplified Fick model for the convective drying of cassava cubes varied from2.75 × 10−10 m2 /s (Osmotically dehydrated cube at 50 °C) to 9.85 × 10−10 m2 /s (in natura cube at 70 °C). (j) Cassava cubes dehydrated osmotically and dried at 60 °C showed 11% humidity, 16.89% sucrose, and 5.94% sodium chloride on a dry basis

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(k) The good results obtained with the cassava cubes after submitted to the hybrid process of osmotic dehydration and convective drying proved that it can be used in food production as raw material for the development of new cassava-based products. Acknowledgments The authors thanks to CNPQ, CAPES and FINEP (Brazilian Research Agencies) for their financial support.

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Kudra, T.: Energy aspects in drying. Dry. Technol. 22(5), 917–932 (2004) Kumar, A., Islam, S., Dash, K.K., Sarkar, S.: Optimization of process parameters for osmotic dehydration of chayote cubes by response surface methodology. Int. J. Agr. Environ. Biotechnol. 10(6), 725–737 (2017) Lane, H., Eynon, L.: Determination of reducing sugar by means of Fehling’s solution with methylene blue as internal indicatior. J. Soc. Chem. Ind., London 42, 32T–37T (1923) Lewicki, P.P.: Water as the determinant of food engineering properties: a review. J. Food Eng. 61(4), 483–495 (2004) Li, S., Dickinson, L.C., Chinachoti, P.: Mobility of “unfreezable” and “freezable” water in waxy corn starch by 2H and 1H NMR. J. Agr. Food Chemistry 46(1), 62–71 (1998) Lugo, P.J.M., Padilla, K.A.M., Gallo, R.T., Fandiño, J.M.M., Vasquez, R.D.G.: Experimental evaluation of venezuelan cassava drying by using a hybrid air heating system. Rev. Chilena de Ingeniería 26(2), 329–338 (2018) Luna, A.T., Rodrigues, F.F.G., Costa, J.G.M., Pereira, A.O.B.: Physical-chemical study, microbial and bromatological of Manihot esculenta Crantz (mandioca). Rev. Inter.: Saúde, Humanas e Tecnolo 1(3), 1–11(2013). (In Portuguese) Maieves, H.A.: Physical, physical chemical and technological potential of new Cassava cultivars. Master’s dissertation in Food Engineering, Food Engineering College, Federal University of Santa Catarina, Florianópolis, Brazil (2010). (In Portuguese) Maldonado, S., Santapaola, J.E., Singh, J., Torrez, M., Garay, A.: Cinética de la transferencia de masa durante la deshidratación osmótica de yacón (Smallanthus sonchifolius). Food Sci. Technol. 28(1), 251–256 (2008) Mayor, L., Moreira, R., Chenlo, F., Sereno, A.M.: Kinectcs of osmotic dehydration of pumpkin with sodium chloride solutions. J. Food Eng. 74(2), 253–262 (2006) Mercali, G.D.: Study of mass transfer in banana osmotic dehydration (Musa sapientum, shum.). Master’s dissertation in Chemical Engineering, Federal University of the Rio Grande do Sul, Porto Alegre, Brazil (2009). (In Portuguese) Mercali, G.D., Kechinski, C.P., Coelho, J.A., Tessaro, I.C., Marczak, L.D.F.: Study of mass transfer during the osmotic dehydration of blueberry. Braz. J. Food Tecnol. 13(2), 91–97 (2010). (In Portuguese) Mercali, G.D., Kechinski, C.P., Coelho, J.A., Tessaro, I.C., Marczak, L.D.F.: Study of mass transfer during the osmotic dehydration of blueberry. Braz. J. Food Technol. 13(2), 91–97 (2011). (In Portuguese) Mierzwa, D., Kowalski, S.J.: Ultrasound-assisted osmotic dehydration and convective drying of apples: process kinetics and quality issues. Chem. Process Eng. 37(3), 383–391 (2016) Montgomery, D.E.: Introduction to Statistical Quality Control, Sixt edn. Wiley, New York (2008) Moreno, A., León, D., Giraldo, G., Rios, E.: Estudio de la cinética fisicoquímica del mango (Mangiferia indica L. Var. tommy Atkins) tratado por métodos combinados de secado, DYNA, 77(162), 75–84 (2010) Moyers, C.G., Baldwin, G.W.: Psychrometry, evaporative cooling and solids drying. In: Perry, R.H., Green, D.W. (eds.) Perry’s Chemical Engineers’ Handbook, 7 th edn, pp. 12-1–12-90, Mac Graw-Hill, Nova York (1999). (CD-ROM) Nogueira, M.C.S.: Agronomic Experimentation I-Concepts, Planning and Statistical Analysis. Esalq, Piracicaba (2007). (In Portuguese) Osidacz, R.C., Ambrosio-Ugri, M.C.B.: Physicochemical quality of eggplant dehydrated with varied pretreatments. Acta Scient. Technol. 35(1), 175–179 (2013) Park, K.J., Bin, A., Pedro, R.B.F.: Drying of pear dAnjou with and without osmotic dehydration. J. Food Eng. 56(1), 97–103 (2002) Pavkov, I., Babi´c, L., Babi´c, M., Radojˇcin, M., Stojanovi´c, C.: Effects of osmotic pre-treatment on convective drying kinetics of nectarines halves (Pyrus persica L.). J. Process. Energy Agr. 15(4), 217–222 (2011)

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Pereira, L.M., Ferrari, C.C., Mastrantonio, S.D.S., Rodrigues, A.C.C., Hubinger, M.D.: Kinetic aspects, texture, and color evaluation of some tropical fruits during osmotic dehydration. Dry. Technol. 24(4), 475–484 (2006) Pereira, L.M., Carmello-Guerreiro, S.M., Hubinger, M.D.: Microscopic features, mechanical and thermal properties of osmotically dehydrated guavas. LWT Food Sci. Technol. 42(1), 378–384 (2009) Pessoa, T., Amaral, D.S., Duarte, M.E.M., Cavalcanti Mata, M.E.R.M., Gurjão, F.F.: Sensory assessment of passed guavas obtained by combined techniques of osmotic dehydration and drying. Holos 4(27), 137–147 (2011). (In Portuguese) Pessoa, T., Silva, D.R.S., Duarte, M.E.M., Cavalcanti Mata, M.E.M.R., Gurjão, F.F., Miranda, D.S.A.: Physical and physicochemical yam sticks of submitted to osmotic dehydration in saline. Holos 7(33), 30–38 (2017) Pornpraipech, P., Khusakul, M., Singklin, R., Sarabhorn, P., Areeprasert, C.: Effect of temperature and shape on drying performance of cassava chips. Agr. Nat. Resour. 51(5), 402–409 (2017) Ramya, H.G., Kumar, S., Kumar, M.: Mass exchange evaluation during optimization of osmotic dehydration for oyster mushrooms (Pleurotus sajor-caju) in salt-sugar solution. J. Appl. Nat. Sci. 6(1), 110–116 (2014) Rayaguru, K., Routray, W.: Mathematical modeling of thin layer drying kinetics of stone apple slices. Int. Food Res. J. 19(4), 1503–1510 (2012) Rinaldi, M.M., Vieira, E.A., Fialho, J.F., Malaquias, J.V.: Effect of different freezing forms on cassava roots. Braz. J. Food Thechnol. 18(2), 93–101 (2015). (In Portuguese) Riva, M., Campolongo, S., Leva, A.A., Maestrelli, A., Torreggiani, D.: Structure-property relationships in osmo-air dehydrated apricot cubes. Food Res. Int. 38(5), 533–542 (2005) Rodríguez, M.M., Arballo, J.R., Campañone, L.A., Cocconi, M.B., Pagano, A.M., Mascheroni, R.H.: Osmotic dehydration of nectarines: influence of the operating conditions and determination of the effective diffusion coefficients. Food Bioproc. Technol. 6(10), 2708–2720 (2013) Rodríguez, M.M., Arballo, J.R., Campañone, L.A., Mascheroni, R.H.: Analysis of operating conditions on osmotic dehydration of plums (Prunus Domestica L.) and 3D-numerical determination of effective diffusion coefficientsAnalysis of operating conditions on osmotic dehydration of plums (Prunus Domestica L.) and 3D-numerical determination of effective diffusion coefficients. Int. J. Food Eng. 13(11), 1–13 (2017) Ruiz-López, I.I., Huerta-Mora, I.R., Vivar-Vera, M.A., Martínez-Sánchez, C.E., Herman-Lara, E.: Effect of osmotic dehydration on air-drying characteristics of chayote. Dry. Technol. 28(10), 1201–1212 (2010) Ruiz-López, I.I., Ruiz-Espinosa, H., Herman-Lara, E., Zárate-Castillo, G.: Modeling of kinetics, equilibrium and distribution data of osmotically dehydration carambola (Averrhoa carambola L.) in sugar solutions. J. Food Eng. 104(2), 218–226 (2011) Sacilik, K., Elicin, A.K.: The thin layer drying characteristics of organic apple slices. J. Food Eng. 73(3), 281–289 (2006) Sanjinez-Argandoña, E.J., Cunha, R.L., Menegalli, F.C., Hubinger, M.D.: Evaluation of total carotenoids and ascorbic acid in osmotic pretreated guavas during convective drying. Italian J. Food Sci. 17(3), 305–314 (2005) Sankat, C.K., Castaigne, F., Maharaj, R.: The air drying behaviour of fresh and osmotically dehydrated banana slices. Int. J. Food Sci. Technol. 3(2), 123–135 (1996) Sarno, R.M., Levy, R.B., Bandoni, D.H., Monteiro, C.A.: Estimated sodium intake for the Brazilian population, 2008-2009. Rev. Saúde Pública 47(3), 571–578 (2013) Silva Júnior, A.F., Aires, J.E.F., Cleide, K.LC.A.F.A., Silva, M.D.P.S., Farias, V.S.O.: Effects of salt concentration on osmotic dehydration of green bean. J. Agr. Stud. 3(1), 60–78 (2015) Silva, M.A.C., Silva, Z.E., Mariani, V.C., Darche, S.: Mass transfer during the osmotic dehydration of West Indian cherry. LWT, Food Sci. Technol. 45(2), 246–252(2012) Silva, P.A., Cunha, R.L., Lopes, A.S., Pena, R.S.: Characterization of tapioca flour obtained in Pará state, Brazil. Ciencia Rural 43(1), 185–191 (2013). (In Portuguese)

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Singh, B., Gupta, A.K.: Mass transfer kinetics and determination of effective diffusivity during convective dehydration of pre-osmosed carrot cubes. J. Food Eng. 79(2), 459–470 (2007) Singh, B., Singh Hathan, B.: Convective dehydration kinetics and quality evaluation of osmoconvective dried beetroot candy. Italian J. Food Sci. 28(4), 669–682 (2016) Singh, B., Panesar, P.S., Nanda, V.: Osmotic dehydration kinetics of carrot cubes in sodium chloride solution. Int. J. Food Sci. Technol. 43(8), 1361–1370 (2008) Sritongtae, B., Mahawanich, T., Duangmal, K.: Drying of osmosed cantaloupe: effect of polyols on drying and water mobility. Dry. Technol. 29(5), 527–535 (2011) STATSOFT. Statistica for Windows, Tulsa, USA (1997) Sutar, P.P., Prasad, S.: Modeling mass transfer kinetics and mass diffusivity during osmotic dehydration of blanched carrots. Int. J. Food Eng. 7(4), 1–20 (2011) Togrul, H.: Simple modeling of infrared drying of fresh apple slices. J. Food Eng. 71, 311–323 (2005) Udomkun, P., Argyropoulos, D., Nagle, M., Mahayothee, B., Müller, J.: Sorption behaviour of papayas as affected by compositional and structural alterations from osmotic pretreatment and drying. J. Food Eng. 157, 14–23 (2015) Valduga, E., Tomicki, L., Witschinski, F., Colet, R., Peruzzolo, M., Ceni, G.C.: Evaluation of acceptability and mineral components of different cassava cultivars (Manihot esculenta Crantz) after cooking. Alimentos e Nutrição Araraquara 22(2), 205–210 (2011). (In Portuguese) Vázquez-Vila, M.J., Chenlo-Romero, F., Moreira-Martínez, R., Pacios-Penelas, B.: Dehydration kinetics of carrots (Daucus carota L.) in osmotic and air convective drying processes. Span. J. Agr. Res. 7(4), 869–875 (2009) Vieira, J.C., Montenegro, F.M., Lopes, A.S., Pena, R.S.: Physical and sensorial quality of sweet cookies with cassava starch. Ciência Rural 40(12), 2574–2579 (2010) Yadav, A.K., Singh, S.V.: Osmotic dehydration of fruits and vegetables: a review. J. Food Sci. Technol. 51(9), 1654–1673 (2014) Zúñiga, R.N., Pedreschi, F.: Study of the pseudo-equilibrium during osmotic dehydration of apples and its effect on the estimation of water and sucrose effective diffusivity coefficients. Food Bioproc. Technol. 5(7), 2717–2727 (2012)

Chapter 7

Heat Transfer in a Packed-Bed Elliptic Cylindrical Reactor: Theory, Heterogeneous Transient Modeling, and Applications A. S. Pereira, R. M. da Silva, R. S. Santos, A. G. Barbosa de Lima, R. O. de Andrade, W. M. P. B. de Lima, and G. S. de Lima Abstract This chapter focuses on the study of heat transfer in packed-bed elliptic cylindrical reactor. Based on the local thermal non-equilibrium, a heterogeneous mathematical model was developed. The transient model is composed for one solid phase and another fluid phase, in which the balance equation for each constituent, written in elliptic cylindrical coordinates, is applied separately, and the proposed model includes different phenomena such as geometry of the particles and reactor, bed porosity, fluid velocity, conduction, and convection heat transfer between the solid particles and fluid flowing inside the bed, and heat generations in the involved A. S. Pereira (B) Federal Institute of Education, Science and Technology Baiano - IFBaiano, Highway BR 420, Rural Zone, Santa Inês, BA Zip Code: 45320-000, Brazil e-mail: [email protected] R. M. da Silva Federal Institute of Education, Science and Technology of Paraíba, IFPB, R. Tranqüilino Coelho Lemos, 671, Dinamérica, Campina Grande, PB Zip Code: 58432-300, Brazil e-mail: [email protected] R. S. Santos Rural Federal University of the Semi-Arid, Av. Francisco Mota, 572, Mossoró, RN Zip Code: 59625-900, Brazil e-mail: [email protected] A. G. B. de Lima (B) · R. O. de Andrade · W. M. P. B. de Lima · G. S. de Lima Department of Mechanical Engineering, Federal University of Campina Grande, Av. Aprígio Veloso, 882, Bodocongó, Campina Grande, PB Zip Code: 58429-900, Brazil e-mail: [email protected] R. O. de Andrade e-mail: [email protected] W. M. P. B. de Lima e-mail: [email protected] G. S. de Lima e-mail: [email protected] © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 J. M. P. Q. Delgado and A. G. Barbosa de Lima (eds.), Transport Processes and Separation Technologies, Advanced Structured Materials 133, https://doi.org/10.1007/978-3-030-47856-8_7

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phases. Application has been done to a specific geometry of the reactor with aspect ratio 2. Keywords Heat transfer · Fixed-bed reactor · Two-phase model · Simulation

7.1 Introduction The study of heat transfer in porous media (structure composed by connected and for unconnected voids and solid material) has been an important topic of researches around the world. This morphology is present both in nature, as in most of the Chemical Engineering unitary operations, such as filtration, distillation, absorption and adsorption, drying and catalytic reactions in fixed and fluidized beds (Freire 2004). On catalytic reactions, fixed-bed tubular reactors are often used in industry to promote such highly exothermic or endothermic heterogeneous gas-solid reactions. However, in order to have a realistic and safety design of such equipment, theoretical and experimental studies, and the development of accurate mathematical models, based on cold (or heated) flow experiments, must be performed. The models of porous media heat transfer can be divided into two groups: (a) pseudo-homogeneous—where there is no distinction between the phases, and the solid–fluid mixture heat transfer occurs in the same temperature at each location of the bed (local thermal equilibrium) and (b) heterogeneous model—where each phase has its own heat transfer dynamics, and there is a parameter that coupling the heat transfer between the phases (local thermal non-equilibrium). These models can be further subdivided into 1D, 2D, or 3D models according to the analyzed geometry. Each model may present variations due to the considered simplifying assumptions; however, the heterogeneous model is the most accurate and realistic under the physical point of view. Heat transfer studies in fixed-bed reactors are still limited to simple cylindrical geometries, pseudo-homogeneous model, and variations in some system thermophysical properties of the involved materials. Therefore, this chapter aims to realize a study of the heat transfer in a fixed-bed reactor with elliptic cylindrical geometry and applying a two-phase mathematical model (heterogeneous model).

7.2 Porous Media and Packed-Bed Reactors 7.2.1 Porous Media For a material to be considered as a porous medium it must be checked whether it contains relatively small voids, generally called pores, within the solid or semi-solid matrix. The pores usually are filled by a fluid, such as air, water, or a fluid mixture, moreover, they must be permeable, i.e., these fluids may penetrate the porous medium

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Fig. 7.1 Particle filling state, initial (left) and final (right), after compression sintering and /or heating

through one face and emerge on another face, or migrate from the volume interior to surface due to the action of some external agent, such as heat pressure or concentration gradients. Porous media can be classified according to structure as granular or fibrous. Granules are usually formed by a set of particles or grains, spherical or not, arranged regularly or randomly, and represent the vast majority of porous media. Fibers are formed by a set of very long inclusions, called fibers, that can be natural or synthetic, straight or curved, being randomly arranged or in regular distributions. Examples of porous media are particle beds, porous rocks, fibrous cluster such as tissues and filters, and extremely small microspore-containing catalytic particles (Mendes 1997). Figure 7.1 illustrates a porous medium consisting of particles with different shapes and dimensions, uncompressed, and compressed by pressure and/or heating. Except for metal structures, dense rocks, and some plastics, on a microscopic scale, solids can be considered porous media materials (Dullien 1992). Having adequate models to predict the behavior of the momentum, mass, and heat transport phenomena inside of porous media can be fundamental in the scientific, technological, and industrial areas, such as the fixed-bed reactors. The ability to model the porous medium behavior under different conditions allows to accelerate the development and to improve the efficiency of processes involving porous media and thus, to allow economic and environmental gains. It is difficult to precisely describe the porous media geometry due to its complex structure. In theory, this material can be specified by analytical equations that define the shape and pores dimensions. For practical purposes, it is impossible to fully describe these equations and, therefore, some geometric approximations are considered, obtaining characteristics very close to the real ones. As described in Eq. (7.1), the porosity is the ratio of void volume (V v ), which are the zones not occupied by the solid material and the total volume of the porous medium (V T ) which is the sum of the void volume and the solid particle volumes (Dullien 1992). ϕ=

VV VT

(7.1)

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Fig. 7.2 Distribution of particles in a porous bed

Thus, porosity indicates the void volume percentage in relation to the total volume; Fig. 7.2 can assist in the understanding of these parameters. The Eq. (7.2) shows a relationship between porosity and density of the bed (bulk density) that contain particle and fluid: ρbed =

m total Vtotal

(7.2)

where the total mass is the sum of solid and fluid mass, as follows: m total = (1 − ϕ)(SH)ρs + ϕ(SH)ρf

(7.3)

where ρ s and ρ f are, respectively, the particle and fluid densities, and Vtotal = (SH)Vtotal = (SH)

(7.4)

Inserting Eqs. (7.3) and (7.4) in Eq. (7.2), the available volume to flow is defined as ϕ=

ρbed − ρs ρf − ρs

(7.5)

Another way to determine the porous medium is by using the packing factor (PF), which represents the actual fraction occupied by the particles. The mathematical expression for this purpose is

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 ϕ = 1 − (PF) = 1 −

N · VP Vtotal

 (7.6)

where N is the effective particle number, V p is the particle volume, and V total is the unit cell volume (porous media). The particle geometric characteristics: shape, size, and distribution influence the behavior of the unconsolidated porous media. The bed porosity increases as the particle shape is very different from a spherical shape, thus, the sphericity is less than 1. Given this, many works report the use of an equivalent diameter for the particles. The equivalent diameter (d p ) is defined as the sphere diameter with the same particle volume (McCabe et al. 1985). It can be determined by the ratio d p = 6/Ap , if the specific surface area (Ap ), particle surface area divided by particle volume, is known. The Ap represents the contact area between the solid and the fluid phase, which is a very important parameter in some processes. An equivalent nominal diameter may also be obtained by sieving if an equivalent diameter measurement is not available. This mathematical expression for the diameter is most commonly used for particles with shape closer to the spherical. For particles with regular or irregular geometry, a sphericity factor β is defined by  β=

VP Vsc

1/3 (7.7)

where the values of this factor differ for particles of different shapes. In Eq. (7.7), V p and V sc , respectively, represent the particle and circumscribed sphere volumes, as shown in Fig. 7.3 (Curray and Griffiths 1955). In many applications, the bed particles do not have a uniform diameter, but a size distribution. Thus, it is common to determine an average value for the particle diameter (equivalent particle diameter). The consequence of this approach is that a Fig. 7.3 Relationship between particle volume and circumscribed sphere volume

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Fig. 7.4 Scheme of an unconventional fixed-bed reactor

real bed with various particle sizes distribution will be represented by a bed formed by spherical particles with the same equivalent diameter.

7.2.2 Chemical Reactors 7.2.2.1

Reactor Fundamentals

Chemical reactors are equipment in which chemical reactions occur. They can be found in two basic types, tanks or tubes, aiming to maximize the generation of desired products with higher added value; produce the highest yield at the lowest cost; generate intermediates chemical products for new processes, and to generate profits, operating within pre-established safety (controlled), according to environmental legislation (Fábrega 2012). The ideal reactors (for which a specific mathematical model is developed from pre-established conditions and predict properly the physical phenomena behavior when realistic conditions are applied) and non-ideal reactors (for which treatment and specific mathematical function due to reaction and/or reactor peculiarities are required) have been reported in the literature. Batch, tubular, and mixing are the three main types of ideal reactors. Figure 7.4 illustrates a schematic of a fixed-bed reactor with fluid inlet and outlet, the particle packing, and the fluid flow direction in the bed.

7.2.2.2

Reactors Modeling

Due to the development of computers and the availability of industrial simulators, it is not recommended to expand the reactor scale without first developing any modeling as simple as it may be, to have a better knowledge of the equipment and process behavior. Modeling can vary from the fundamental, which can use simpler differential

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equations by separately evaluating each mechanism that influences the process, to later add more information from the same mechanisms to the model, providing a more complex reactor simulation, or it can be based on pilot experimentation to adjust the model effective parameters (Froment and Hofmann 1987). The reactor to be adequately controlled must be very well known and, therefore, a model that describes its behavior as operating with great accuracy is necessary. On the other hand, applications involving a dynamic model usually require a large computational effort with direct impact on the processing time. In addition, to the development of models with high prediction potential, it is necessary to consider the difficulty in their solution, both in computational terms and in the availability of the necessary (Bunnell et al. 1949). The simulation of chemical reactors had large use in recent years. These simulations have served several purposes: reactor design, reactor start and stop strategies, determination of desired and hazardous operating conditions for process control, and optimization, using sometimes detailed and heterogeneous models. The heat transfer related catalytic bed models can be divided into heterogeneous model (solid phase + fluid phase) also known as two-phase model and single model (pseudo-homogeneous). Each model may also present variations due to the used assumptions. For example, in a tubular reactor, one-dimensional models do not consider radial gradients of temperature or concentration, grouping all resistance to heat transport on the wall. The two-dimensional consider the existence of a non-planar radial profile of temperatures and concentrations, which presupposes the knowledge of the effective radial thermal conductivity, and the film coefficient on the wall (Giordano 1991). Performance analyses of different two-dimensional models (pseudohomogeneous and heterogeneous) show that the pseudo-homogeneous model is consistent having a simpler computational solution and programming than the other models. In the two-phase model (solid + fluid), each phase has its own dynamics of heat transfer, which is physically more realistic. Nevertheless, few scientists have studied this model for inherent modeling reasons, which are considerably more complicated solution of the energy equations for the solid and fluid phases, experimental difficulty in determining the solid–fluid heat transfer coefficient required for heterogeneous model, and the difficulty in spot temperature measurement for each phase.

7.3 Heat Transfer in Fixed-Bed Elliptical Reactor via Two-Phase Model 7.3.1 Physical Problem and Geometry The packed-bed elliptical reactor studied in this research is illustrated in Fig. 7.5. The reactor bed is percolated by a heated fluid (fluid 1), which exchanges heat with

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Fig. 7.5 Scheme of a packed-bed elliptical cylindrical reactor

the particles, and the system is cooled at the wall by fluid 2 which has a temperature less than the inlet fluid temperature. Fluid 1 flows in the laminar regime. This physical problem with the use of pseudo-homogeneous model has been studied by Oliveira et al. (2004), Silva Filho et al. (2013), Silva et al. (2017, 2018); however, based on the heterogeneous model no works have been reported in the literature. The system geometric shape proposed in this chapter suggests the use of a particular coordinate system that best fits this geometry and, consequently, will lead to greater efficiency and confidence of the results (Lima 1999). In this case, the elliptic cylindrical coordinate system is most appropriate. Thus, a change of variables is a natural requirement. The general relations between the Cartesian coordinate system (x, y, z) and the elliptical cylindric coordinate system (τ, φ, z) are given as follows (Magnus et al. 1966):

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x = L cosh τ cos φ

(7.8)

y = L senh τ sen φ

(7.9)

z=z

(7.10)

whereL is the ellipse focal length, mathematically calculated by the expression L = L 22 − L 21 , where L 1 and L 2 are the ellipse minor and major axes, respectively (Fig. 7.5). To obtain the desired transformation, consider the following variables: ξ = cosh τ

(7.11)

η = cos φ

(7.12)

Thus, the substitution of these variables ξ and η in Eqs. (7.8)–(7.9) provides the direct relationships between the two coordinate systems. Thus, the following relationships are obtained for x, y, and z, in terms of ξ and η: x = Lξ η y=L

   1 − η2 ξ 2 − 1 z=z

(7.13) (7.14) (7.15)

The domain of validity of variables ξ, η, and z in the elliptic cylindrical system are 1 ≤ ξ ≤ L 2 /L, 0 ≤ η ≤ 1, and 0 ≤ z ≤ H . The elimination of variable φ, in the Eqs. (7.8) and (7.9), generating the xy plan curves (Fig. 7.6), characterized by the parameters ξ = ξ0 (constant). The surfaces Fig. 7.6 Representative scheme of the elliptic cylindrical coordinate system in ξ η and xy planes

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ξ0 > 1 are revolution ellipsoids with center in origin. The generated ellipses have the same focus. The two-ellipse focus are located along the x-axis at the points (x = ±L, y = 0). The surface ξ = 1 it is a straight line that joins the origin (z = 0) and the focal point (z = +L). According to Figs. 7.5 and 7.6, when L 2 → L 1 , the elliptic cylinder tends to a circular cylinder with diameter of 2L 1 . Thus, at the limit when the interfocal distance tends to zero, the elliptical coordinate system is reduced to the cylindrical: Lξ → r and η → cos θ, when ξ → ∞, where r and θ are the cylindrical coordinates.

7.3.2 Governing Equations The transient energy equations (in terms of temperature) for the fluid and solid phases in the Cartesian coordinate system are, respectively (Nield and Bejan 1992): • Fluid phase  ∂ Tf    DP + ρcp f v · ∇Tf = + ϕ∇ · (K f ∇Tf ) + μψ + ϕ q˙f + h(Ts − Tf ) ϕ ρcp f ∂t Dt

(7.16)

• Solid phase

(1 − ϕ)(ρc)s

∂ Ts = (1 − ϕ)∇ · (K s ∇Ts ) + (1 − ϕ)q˙s + h(Tf − Ts ) ∂t

(7.17)

where f and s refer to the fluid and solid phases, respectively, T is the temperature, t is the time, v is the fluid average velocity, ϕ is the media porosity, ρ is the density, cp is the specific heat at constant pressure, K is the thermal conductivity, μψ is the viscous dissipation term, P is the pressure, q˙ internal energy generation per volume unit, and h is the specific heat transfer coefficient by convection, between the solid and fluid phases. The first term of Eq. (7.16) represents the transient transfer of energy; and the second term, the convective heat transport, and where the relation of Dupuit–Forchheimer v = ϕV, V the fluid velocity was used. The third term represents the substantive derivative of pressure, the fourth term is the conduction heat transport, the fifth term corresponds to viscous dissipation, the sixth term is the internal energy generation, and the seventh term represents the convection heat exchange between the solid and fluid phases. The first term of Eq. (7.17) represents the transient transfer of energy and the second term represents the conductive heat flux through the solid. The third term represents the heat generation inside the solid, and the fourth term refers to the heat exchange between the solid and fluid phases, by convection. Some values of h can be obtained indirectly (Polyaev et al. 1996). One of the correlations for a porous particle bed is given by Dixon and Cresswell (1979):

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h = asf h sf

(7.18)

where hsf is the convective heat transfer coefficient to both phases and afs is the specific surface area, given by the relationships between the total area to heat transfer and the particle bed volume. For example, to a spherical particle bed this parameter is given as follows: asf = 6(1 − ϕ)/dP

(7.19)

where d P is the particle diameter and the parameter hsf is given as follows: 1 dP dP = + h sf Nusf K f β Ks

(7.20)

where β is the sphericity of the porous bed particles obtained by (see Eq. 7.7):  β=

a2 b2

1/3 (7.21)

being a and b positive real numbers that determine the particle dimensions and shape (see Fig. 7.3). The Nusselt number Nusf can be obtained by the following expression (Handley and Heggs 1968): 1

1

Nusf = (0.225/ϕ)Pr 3 Rep3

(7.22)

valid for particle Reynolds numbers ReP > 100. In this equation, Pr is the Prandtl number. Another correlation for Nusselt number is given as follows (Wakao and Kaguei 1982): 1

Nusf = 2.0 + 1.1Pr 3 Re0.6 p

(7.23)

Equation (7.23) is valid for low values of ReP . In this case, predictions of Nusf vary between 0.1 and 12.4 (Miyauchi et al. 1981; Wakao et al. 1976; Wakao and Kaguei 1979). Several other authors, such as Alazami and Vafai (2000), Grangeot et al. (1994), Saito and Lemos (2005), Quintard et al. (1997), Quintard and Whitaken (2000) and Nield (2002) have reported alternative expressions for determining the parameters hsf and asf . Considering viscous dissipation and the substantive derivative of pressure negligible, the general conservation equation applied for the fluid and solid phases for a generic variable Φ and writing in any coordinate system (Maliska 1995) are given by         ∂ ∂ ∂ Φ Φ Φ Φ ∂ ϕ(ρcP )f + + + = (ρcP )f U (ρcP )f V (ρcP )f W ∂t J ∂ξ J ∂η J ∂z J

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  ∂ ∂Φ ∂Φ ∂Φ α11 ϕ J Γ Φ + α12 ϕ J Γ Φ + α13 ϕ J Γ Φ ∂ξ ∂ξ ∂η ∂z   ∂Φ ∂Φ ∂Φ ∂ α21 ϕ J Γ Φ + α22 ϕ J Γ Φ + α23 ϕ J Γ Φ + ∂η ∂ξ ∂η ∂z   qf ∂Φ ∂Φ ∂Φ ∂ α31 ϕ J Γ Φ + α32 ϕ J Γ Φ + α33 ϕ J Γ Φ + ϕ + h(Φs − Φf ) + ∂z ∂ξ ∂η ∂z J (7.24) and   ∂ Φ ∂Φ Φ ∂Φ Φ ∂Φ α31 (1 − ϕ)J Γ + α32 (1 − ϕ)J Γ + α33 (1 − ϕ)J Γ ∂z ∂ξ ∂η ∂z h qs +(1 − ϕ) + (Φf − Φs ) (7.25) J J where J represents the Jacobian, determined mathematically as follows:

J −1

∂x ∂ξ ∂y = ∂ξ ∂z

∂x ∂η ∂y ∂η ∂z ∂ξ ∂η

∂x ∂z ∂y ∂z ∂z ∂z



(7.26)

The coefficients α ij , on the Eqs. (7.24) and (7.25) are determined by the following relations: α11 =

a J2

α12 = α21 = α22 =

(7.28) (7.29)

e J2

c J2

α23 = α32 = where

d J2

b J2

α13 = α31 = α33 =

(7.27)

(7.30) (7.31)

f J2

(7.32)

7 Heat Transfer in a Packed-Bed Elliptic …





a = 



b = 

∂ξ ∂x ∂η ∂x

2

197

 +

2

 + 

∂η ∂y

2

 +

2

 +

2

∂ξ ∂z

(7.33)

∂η ∂z

2 (7.34)



 ∂z 2 ∂z       ∂ξ ∂η ∂ξ ∂η ∂ξ ∂η  + + d = ∂x ∂x ∂y ∂y ∂z ∂z       ∂z ∂ξ ∂z ∂ξ ∂z ∂ξ e = + + ∂x ∂x ∂y ∂y ∂z ∂z       ∂z ∂η ∂z ∂η ∂z ∂η f = + + ∂x ∂x ∂y ∂y ∂z ∂z c =

∂z ∂x

2

∂ξ ∂y

+

∂z ∂y

2

+

(7.35) (7.36) (7.37) (7.38)

For the elliptic cylindrical coordinate system, the determination of the Jacobian inverse provides as a result: J

−1

  L 2 ξ 2 − η2  =−    ξ 2 − 1 1 − η2

(7.39)

In the general equation (Eqs. 7.24 and 7.25), the terms containing α ij with i = j are the diffusive terms referring to the non-orthogonality of the mesh. Thus, checking the orthogonality of the adopted coordinate system becomes an important requirement. In this case, the necessary and sufficient conditions for a coordinate system to be orthogonal are (McRobert 1967):   

∂x ∂x ∂ξ ∂η ∂x ∂x ∂η ∂z ∂x ∂x ∂z ∂ξ



 +



 +



 +

∂y ∂y ∂ξ ∂η ∂y ∂y ∂η ∂z ∂y ∂y ∂z ∂ξ



 +



 +



 +

∂z ∂z ∂ξ ∂η ∂z ∂z ∂η ∂z ∂z ∂z ∂z ∂ξ

 =0

(7.40)

=0

(7.41)

=0

(7.42)

 

It can be verified that all these conditions are satisfactory for the elliptic cylindrical coordinate system. Then, the coefficients d  , e and f  are null, and Eq. (7.24) will be reduced as follows:         ∂ Φ Φ Φ Φ ∂ ∂ ∂ ϕ(ρcP )f + + + = (ρcP )f u ξ (ρcP )f u η (ρcP )f u z ∂t J ∂ξ J ∂η J ∂z J

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      ∂ ∂Φ ∂Φ ∂Φ ∂ ∂ qf h α11 ϕ J Γ Φ + α22 ϕ J Γ Φ + α33 ϕ J Γ Φ + ϕ + (Φs − Φf ) ∂ξ ∂ξ ∂η ∂η ∂z ∂z J J

(7.43)

Substituting Eqs. (7.27), (7.29), (7.31), and (7.39) in Eq. (7.24) and rearranging the terms, where Φ = T and Γ Φ = K f , we obtain for the fluid phase, the following equation:

⎤

⎤ ⎤ ⎡  ⎡ ⎡  2 2  2 2  2 2 2 2 2 ∂ ⎣ ϕ ρcp f L ξ − η Tf ⎦ ∂ ⎣ ρcp f L ξ − η u ξ Tf ⎦ ∂ ⎣ ρcp f L ξ − η u η Tf ⎦    + +         ∂t ∂ξ ∂η ξ 2 − 1 1 − η2 ξ 2 − 1 1 − η2 ξ 2 − 1 1 − η2

⎤ ⎡  2 2         2 ∂T f ∂T f ξ2 − 1 1 − η2 ∂ ⎣ ρcp f L ξ − η u z Tf ⎦ ∂ ∂   ϕ Kf   ϕ Kf  + = +   ∂z ∂ξ ∂ξ ∂η ∂η 1 − η2 ξ2 − 1 ξ 2 − 1 1 − η2 +





⎡ ⎤ 2 2 2 L 2 ξ 2 − η2   ∂ Tf ⎦ ∂ ⎣ L ξ −η   ϕ K + f      ϕqf + h(Ts − Tf ) ∂z ∂z ξ 2 − 1 1 − η2 ξ 2 − 1 1 − η2

(7.44)

Considering u z  u ξ and u z  u η , the Eq. (7.44) assumes the way: ⎤ ⎡    ⎡    ⎤  2 2 2 2 2 2 L − η L − η T ξ T ξ u ρc ϕ ρc ∂⎣ p f z f ⎦ + ∂ ⎣ p f ⎦=  f      ∂t ∂z ξ 2 − 1 1 − η2 ξ 2 − 1 1 − η2         ξ2 − 1 1 − η2 ∂ ∂ Tf ∂ Tf ∂   ϕ Kf   ϕ Kf + ∂ξ ∂ξ ∂η ∂η 1 − η2 ξ2 − 1 ⎡ ⎤     L 2 ξ 2 − η2 L 2 ξ 2 − η2 ∂ Tf ⎦ ∂ ⎣  +    ϕ K f ∂z +   [ϕqf + h(Ts − Tf )] ∂z ξ 2 − 1 1 − η2 ξ 2 − 1 1 − η2 (7.45) Similarly, for the energy equation applied to the solid phase, and substituting Φ = T and Γ φ = K s , we obtain ⎡     ⎤     ξ2 − 1 ∂ ∂ Ts ∂ ⎣ (1 − ϕ) ρcp s L 2 ξ 2 − η2 Ts ⎦   (1 − ϕ)K s  =   ∂t ∂ξ ∂ξ 1 − η2 ξ 2 − 1 1 − η2 ⎡ ⎤       2 2 2 2 1 − η ξ L − η ∂ ∂ Ts ∂ Ts ⎦ ∂   (1 − ϕ)K s + + ⎣   (1 − ϕ)K s ∂z  ∂η ∂η ∂z ξ2 − 1 2 2 ξ −1 1−η   L 2 ξ 2 − η2 +   [(1 − ϕ)qs + h(Tf − Ts )]  ξ 2 − 1 1 − η2 (7.46)

7 Heat Transfer in a Packed-Bed Elliptic …

199

Fig. 7.7 Scheme showing the heat exchanges considered in the boundary for the solid and fluid phases

Since Eqs. (7.45) and (7.46) are second order in space and first order in time differential equations, their solution requires at least two boundary conditions for each direction and one initial condition. Moreover, it is an elliptical equation, making it necessary to define these boundary conditions along the entire boundary of the considered domain. Thus, considering Fig. 7.7, the following initial and boundary conditions are given: (a) Prescribed temperature condition at the reactor inlet: T (ξ, η, z = 0, t) = T0

(7.47)

(b) Parabolic condition at reactor output: ∂T (ξ, η, z = H, t) = 0 ∂z

(7.48)

∂T (ξ = 1, η, z, t) = 0 ∂ξ

(7.49)

(c) Symmetry conditions:

(d) Reactor wall conditions

∂T (ξ, η = 0, z, t) = 0 ∂η

(7.50)

∂T (ξ, η = 1, z, t) = 0 ∂η

(7.51)

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It is considered that in the reactor wall heat diffusive and convective fluxes (only for the fluid phase) occur on the inner side of the wall that equals the diffusive heat flux at the reactor wall, which is equal to the convective heat flux, at the external side of the reactor wall. Thus, can be written as   ξ 2 − 1 ∂ T   + h win (TP − Twin ) = h wext (Twext − Tm ) = ξ 2 − η2 ∂ξ (ξ =ξin )   2 ξ − 1 ∂ T kwr   (7.52) − L ξ 2 − η2 ∂ξ

k q =− L 

(ξ =ξext )

where ξ in = L 2 /L on the surface (Fig. 7.5) and the subscript wr means the reactor wall location.

7.3.3 Numerical Treatment of Heat Transport Equations It is well known by the scientific community that physical problems related to transport phenomena (energy, linear momentum, and mass) have a high complexity and inevitably are governing by partial differential equations. To obtain an analytical solution to problems of this insignificance, when it is achieved, will require laborious and rigorous mathematical treatment, with severe considerations. Thus, the use of numerical techniques emerges as an alternative for the interpretation and solution of the physical problem with great consistence and realism. The advance in the study of physical problems involving computer simulation, as well as the speed of computer processing today, has significantly increased the search for the analysis of such problems through numerical solutions. There are several numerical methods that are being used by the scientific community. In this chapter, we will use the finite-volume method, which has as its basic principle the transformation of partial differential equations into elementary algebraic equations. The fundamental concept of the finite-volume method is that any continuous quantity can be approximated by a discrete model, consisting of a set of continuous functions, defined in a finite number of subdomains, so-called control-volumes, with nodal points located in the centroid of them (Patankar 1980; Maliska 1995). Figure 7.8 illustrates the computational domain used to represent the physical domain in the fixed-bed elliptic cylindrical reactor, where there is symmetry in the four quadrants (see Figs. 7.5 and 7.6). In Fig. 7.8, the control-volume is associated with the nodal point P and the lines ξ and η constants defining the contour. The points F, T, N, S, E, and W are the neighboring nodal points (top, bottom, north, south, east, and west, respectively). Considering the control-volume outlined in Fig. 7.8, it is possible to make the discretization of Eqs. (7.45) (fluid phase) and (7.46) (solid phase) by integrating each term in volume and time. Assuming a fully implicit formulation and the WUDS

7 Heat Transfer in a Packed-Bed Elliptic …

201

Fig. 7.8 Numerical mesh control-volume in the elliptical cylindric coordinate system

scheme as a spatial interpolation function for the convective and diffusive fluxes over the control-volume, after rigorous mathematical treatment we obtain the energy equations for both phases in the discretized form, as follows. • Fluid phase

APf TPf = AEf TEf + AWf TWf + AN f TNf + ASf TSf + AFf TFf + ATf TTf + A0Pf TP0f + Bf (7.53) where     L 2 ξf2 − ηf2 ϕ K ff βff − (0.5 − αf )ρff cPff u z ξ η A Ff =     δz f ξf2 − 1 1 − ηf2     L 2 ξt2 − ηt2 ϕ K tf βtf ATf =  ξ η + + α c u (0.5 )ρ t tf P z tf   δz t ξ 2 − 1 1 − η2 t

t

(7.54)

(7.55)

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  1 − η2 ϕ K ef βef AEf =  2 e  ξ z δηe ξe − 1   1 − ηw2 ϕ K wf βwf  AWf =  2 ξ z δηw ξw − 1 ⎧ ⎨ (ξn2 −1) ϕ K nf βnf ηz, Internal points. (1−ηn2 ) δηn A Nf = ⎩ 0, Boundary points.   ξs2 − 1 ϕ K sf βsf  A Sf =  ηz δηs 1 − ηs2   ϕ 0 ρf0 cP0f ξ ηz L 2 ξ 2 − η2 A0Pf =    t 1 − η2 ξ 2 − 1

(7.56)

(7.57)

(7.58)

(7.59)

(7.60)

APf = AEf + Awf + ANf + ASf     L 2 ξf2 − ηf2 ϕ K ff βff +  + + α c u (0.5 )ρ f ff Pff z ξ η   δz f ξf2 − 1 1 − ηf2     L 2 ξt2 − ηt2 ϕ K tf βtf +  − (0.5 − αt )ρtf cPtf u z ξ η + SMf   δz t ξt2 − 1 1 − ηt2     ϕρ c L 2 ξ 2 − η2 f Pf + a (7.61) +  h sf sf ξ ηz   t 1 − η2 ξ 2 − 1 ⎧  ⎪ (ξn2 −ηP2 ) ⎪ ⎪ Lηz ⎪ (1−ηP2 ) ⎪ ⎪ ⎪ ⎞ ⎤ , Boundary points. ⎛ ⎪ ⎡ ⎪ ⎪ ⎨ ⎝1+ Kf + hwinf + Kf U + hwinf δξn ⎠ K wr h wextf h wextf δξn ⎥ SMf = ⎢ K U ⎥ ⎢ ⎛ ⎞ wr ⎪ ⎪ ⎦ ⎣ ⎪ KfU ⎪ ⎝ ⎠ ⎪ h winf + δξn ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 0, Internal points 





(7.62)

7 Heat Transfer in a Packed-Bed Elliptic …

Bf =



⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

L ⎡⎛ ⎢ ⎢ ⎣

203

(ξn2 −ηP2 ) ηz (1−ηP2 )

⎞⎤,



⎝1+ K f + h winf + K f U + h winf δξn ⎠ K wr h h δξn 

wextf

⎛

wextf



⎞

K wr

U

Boundary points.

⎥ ⎥ ⎦

⎝h winf + K f U ⎠ ⎪ ⎪ δξn ⎪ ⎪ ⎪ ⎪   ⎪ ⎪ ⎪ L 2 ξ 2 − η2 ⎪ ⎪  [ϕ q˙f + asf h sf TS )]ξ ηz, Internal points. ⎪ ⎪ ⎩ 1 − η2 ξ 2 − 1

(7.63)

• Solid phase

APs TPs = AEs TEs + Aws Tws + ANs TNs + ASs TSs + AFs TFs + ATs TTs + A0Ps TP0s + Bs (7.64) where   1 − ηe2 (1 − ϕ)K es βes  ξ z AEs =  2 δηe ξe − 1   1 − ηw2 (1 − ϕ)K ws βws  A ws =  2 ξ z δηw ξw − 1 ⎧ 0, Internal points. ⎨ 2 A Ns = ξn −1) (1−ϕ)K ns βns ⎩ (1−η ηz, Boundary points. δηn ( n2 )   ξs2 − 1 (1 − ϕ)K ss βss  A Ss =  ηz δξs 1 − ηs2   L 2 ξf2 − ηf2 (1 − ϕ)K fs βfs AFs =  ξ η   δz f ξ 2 − 1 1 − η2 f

(7.65)

(7.66)

(7.67)

(7.68)

(7.69)

f

  L 2 ξt2 − ηt2 (1 − ϕ)K ts βts ATs =  ξ η   δz t ξt2 − 1 1 − ηt2     1 − ϕ 0 ρs0 cP0s ξ ηz L 2 ξ 2 − η2 A0Ps =    t 1 − η2 ξ 2 − 1



  L 2 ξ 2 − η2 (1 − ϕ)ρs cPs APs = AEs + Aws + ANs + ASs + AFs + ATs +  + asf h sf + SMs   t 1 − η2 ξ 2 − 1

(7.70)

(7.71)

(7.72)

204

A. S. Pereira et al.  ⎧ (ξn2 −ηP2 ) ⎪ ⎪ Lηz ⎪ (1−ηP2 )⎞ ⎪ ⎪ ⎡⎛ ⎤ , Boundary points. ⎪ ⎪ ⎨ ⎜ K KU ⎟ 

SMs =

Bs =

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

s + s ⎢ ⎝1+ K wr h wexts δξn ⎠ ⎥ ⎢ ⎥ ⎢ ⎥ ⎛ ⎞ ⎢ ⎥ ⎣ ⎦ ⎜ KsU ⎟

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

(7.73)



⎝ δξ n

⎠

0, Internal points.  ξn2 −ηP2 ) L ( 1−η ηz ( P2 ) & ⎤ , Boundary points. ⎡% s + KsU 1+ KKwr h wexts δξn ⎥ ⎢ % & ⎦ ⎣ KsU 



⎪ δξn ⎪ ⎪ ⎪ ⎪   ⎪ ⎪ L 2 ξ 2 − η2 ⎪ ⎪ ⎪  [(1 − ϕ)q˙s + asf h sf Tf )]ξ ηz, Internal points. ⎪ ⎪ ⎩ 1 − η2 ξ 2 − 1

(7.74) The coefficients AK , K = P, represent the components of diffusive and convective heat transfer from the neighboring points toward the P point. The effects of the variable T in on previous time its value in the present time are computed in the coefficient A0P . These effects are gradually reduced as the process tends to steady-state condition. The applications of Eqs. (7.53) and (7.64) are restricted to the computational domain. Nodal points on the domain contour are not into the set of linear algebra equations to solve. After the system of equations has been solved, the estimation of the variable T is performed in these nodal points. For example, for symmetry axes, it is assumed that the conduction heat flux leaving the point adjacent to the symmetry point is equal to the conduction heat flux arriving at these points, as illustrated in Fig. 7.9. These points can be mathematically expressed for the fluid phase, for example, as follows.   % & & %   1 − η2 ∂ T 1 − η2 ∂ T Kf Kf     (7.75) − = − L L ξ 2 − η2 ∂η ξ 2 − η2 ∂η e

w

Thus, by discretizing Eq. (7.75) and rearranging the common terms, the temperature at the nodal points in η = 0, is given by ⎞⎤ ⎞ ⎛  (1−ηw2 ) (1−ηw2 ) K fw 2 2 2 2 ⎢ ⎜ δηw (ξP −ηw ) ⎟ ⎜ (ξP −ηw ) ⎟⎥ ⎟⎥ ⎟ ⎜ ⎜ TEf = ⎢ ⎣1 + ⎝  1−η2 ⎠⎦TPf − ⎝  1−η2 ⎠TWf ( e) ( e) K fP K fP δηe δηe (ξP2 −ηe2 ) (ξP2 −ηe2 ) ⎡



⎛

K fw δηw

(7.76)

7 Heat Transfer in a Packed-Bed Elliptic …

205

Fig. 7.9 Schematic representation of a symmetry point on face η = 0

The representative equation for determining the fluid phase temperature in the inner wall of the porous bed (see Fig. 7.7) is given by  Twinf =



KfU h wext δξn

h win δξn



Tm + TPf + + + K wr U   h Kf h win KfU win δξn 1 + K wr + h wext + h wext δξn + K wr U Kf K wr

h win h wext





(7.77)



7.4 Application: Heat Transfer in an Elliptic Cylindrical Reactor Filled with Spheroidal Particles In this chapter, as an application will be considered the heated air flow percolating a bed of prolate spheroidal particles in the local thermal equilibrium condition and in steady state. The reactor wall is cooled by water at room temperature T m . The thermophysical and geometrical parameters used in the simulation can be seen in Table 7.1. In the software Mathematica® a computational code was developed to solve the set of discretized equations, using Gauss–Seidel solution method, considering a convergence criterion of 10−9 . All results were obtained using a mesh with 20 × 20 ×20 control-volumes obtained after some refinements.

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Table 7.1 Thermophysical and geometrical parameters used in the simulation Reactor

Air (fluid phase)

L 1 (m) L 2 (m)

k f (W/m hwext (W/m2 K) ρ f (kg/m3 ) k s (W/mK) b (m) K)

Particles (solid phase) a (m)

0.05

0.10

2.47 × 10−3

2.0 × 10−3

H (m) 0.20

2.5 × 10−3

1.09488

5.64

K wr (W/mK) uz (m/s) cpf (J/kg K)

μf (kg/m s)

ρ s (kg/m3 ) cps (J/kg ϕ K)

401

2.025 × 10−5

487

0.10

2.69

1000

T m (°C) T 0 (°C)

β

30

0.85

70

5500

0.44

The results presented here take into account the dimensionless temperature profiles (T − T m )/(T 0 − T m ) in different xy planes (z/H = 0.0833, 0.5277 and 0. 9722) and on the yz(x/L = 0.0) and xz(y/L = 0.0) planes. Percolating fluid has the convection heat transfer coefficient (h wint ) calculated by correlations involving the particle Reynolds number (Beek 1975), as follows: Rep =

2ρu z ϕdp 3μ(1 − ϕ)

(7.78)

The Colburn factor (JH ) and the Stanton number (St ) both for heat transfer are given as follows (Incropera and De Witt 1992): JH = 2.06Re−0.575 p

(7.79)

and St =

JH Pr 2/3

(7.80)

Since that the Stanton number is defined by St =

h ρV cP

(7.81)

we can write h wint = St ρu z ϕcp

(7.82)

Figures 7.10, 7.11, 7.12 illustrate the dimensionless temperature distribution in the xy plane at different axial positions for the solid and fluid phases. Figures 7.13,

7 Heat Transfer in a Packed-Bed Elliptic …

207

(a) Fluid phase, z/H= 0.0833.

(b) Solid phase z/H = 0.0833.

(c) Fluid phase, z/H = 0.5277.

(d) Solid phase, z/H = 0.5277.

(e) Fluid phase, z/H = 0.9722.

(f) Solid phase, z/H = 0.9722.

Fig. 7.10 Dimensionless temperature distribution within a fixed-bed elliptic cylindrical reactor filled with prolate spheroidal particles (hwint = 1.03 × 10−4 W/m2 K)

7.14, 7.15 illustrate the dimensionless temperature distribution in the xz and yz planes for the solid and fluid phases. Considering the inner wall of the reactor, it can be observed that an increase in the heat transfer coefficient on the inner wall of the reactor (hwint ) from 1.03 × 10−4 W/m2 K to 40 W/m2 K does not cause significant changes in the temperature field for both phases inside the reactor. Major changes can be seen in the region near the outlet (z/H = 0.9722) on the surface (x = L 2 ). Radial dimensionless temperature gradients are high at the reactor inlet due to the thermal inlet effects. As the axial position increases along the reactor, these radial gradients decrease. It can be seen at any axial position along the bed that the radial dimensionless temperature gradients are slightly most affected by larger for the fluid phase than the solid phase, and that the region near the reactor surface is more affected by the cold wall.

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(a) Fluid phase, z/H = 0.0833.

(b) Solid phase z/H = 0.0833.

(c) Fluid phase, z/H = 0.5277.

(d) Solid phase, z/H = 0.5277.

(e) Fluid phase, z/H = 0.9722

(f) Solid phase, z/H = 0.9722.

Fig. 7.11 Dimensionless temperature distribution within fixed-bed elliptical cylindric reactor filled with prolate spheroidal particles (hwint = 2.0 W/m2 K)

7.5 Concluding Remarks In this chapter, the physical problem of heat transfer and fluid flow in a particle-filled fixed-bed has been studied. Emphasis is given to elliptical cross-sectional reactors. A transient mathematical modeling written in elliptic cylindrical coordinates applied to the particulate and fluid phases was proposed, and its numerical solution, which is based on the finite-volume method, is presented. Results of temperature distribution within the reactor at different planes have been presented and discussed. From the results obtained, the following conclusions are given: (a) Under the physical point of view the heterogeneous mathematical model proved to be satisfactory for the study of heat transfer in fixed-bed reactor with an elliptic cylindrical geometry; (b) The temperature distribution of the phases indicates that heat flux occurs from the center toward the reactor wall (angular and radial directions) and in the axial direction from the inlet to the outlet region;

7 Heat Transfer in a Packed-Bed Elliptic …

209

(a) Fluid phase, z/H = 0.0833.

(b) Solid phase z/H = 0.0833.

(c) Fluid phase, z/H = 0.5277.

(d) Solid phase, z/H = 0.5277.

(e) Fluid phase, z/H = 0.9722.

(f) Solid phase, z/H = 0.9722.

Fig. 7.12 Dimensionless temperature distribution within fixed-bed elliptic cylindrical reactor filled with prolate spheroidal particles (hwint = 40 W/m2 K)

(c) Axial temperature gradients are most relevant in the region near the reactor inlet; (d) Radial temperature gradients are larger near the reactor wall; (e) The temperature distribution within the reactor was higher for solid phase as compared to the fluid phase;

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a) Fluid phase (y/L=0.0).

c) Solid phase (y/L=0.0).

b) Fluid phase (x/L=0.0).

d) Solid phase (x/L=0.0).

Fig. 7.13 Dimensionless temperature distribution of the fluid and solid phases inside the elliptic cylindrical reactor (hwint = 1.03 × 10−4 W/m2 K)

7 Heat Transfer in a Packed-Bed Elliptic …

a) Fluid phase (y/L=0.0).

c) Solid phase (y/L=0.0).

211

b) Fluid phase (x/L=0.0).

d) Solid phase (x/L=0.0).

Fig. 7.14 Dimensionless temperature distribution of the fluid and solid phases inside the elliptic cylindrical reactor (hwint = 2.0 W/m2 K)

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a) Fluid phase (y/L=0.0).

b) Fluid phase (x/L=0.0).

c) Solid phase (y/L=0.0).

d) Solid phase (x/L=0.0).

Fig. 7.15 Dimensionless temperature distribution of the fluid and solid phases inside the elliptic cylindrical reactor (hwint = 40 W/m2 K)

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Acknowledgments The authors thank to FINEP, CAPES and CNPq (Brazilian Research Agencies) for financial support to this research, and also to the researchers for their referenced studies which helped in improving the quality of this work.

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