Heatmetry: The Science and Practice of Heat Flux Measurement [1 ed.] 3030408531, 9783030408534

Table of contents :
Preface
Acknowledgements
Contents
Acronyms
1 Heat Flux Measurement and Heat Flux Sensor
1.1 Heat Flux Measurement
1.2 Surface-Mounted Heat Flux Sensors
1.3 Modern Heat Flux Sensors
References
2 Gradient Heat Flux Sensors
2.1 Theory and Design
2.2 GHFS Materials and Constructions
2.3 Calibration
2.4 Digital Signal Processing
References
3 Transient Heat Flux Measurements
3.1 GHFS Response Time
3.2 Thermal Model of GHFS
References
4 Multifunctional Performance of Gradient Heat Flux Sensors
4.1 Thermometry
4.2 Determination of GHFS Thermal Characteristics
4.3 Emissivity Factor Determination
4.4 Wall-Shear Stress Measurements
4.5 Fluid Flow Rate Measurement and Recording
4.6 Measurement of Electric Circuit Parameters
References
5 Validation and Science Experiment
5.1 Validation
5.1.1 Free Convection Near Vertical Plate
5.1.2 Cross Flow Around Circular Cylinder
5.2 Science Experiments
5.2.1 Cross and Non-cross Flow Around Cylinder with Turbulisators
5.2.2 Surfaces with Intensifiers
5.2.3 Heatmetry in Shock Tubes
5.3 Heat Transfer During Phase Changes
5.3.1 Condensation
5.3.2 Heat Transfer During Boiling
5.4 Radiative Heat Transfer
References
6 Industrial Experiments
6.1 Heatmetry in the Diesel Engine
6.2 Heat and Mass Transfer in a Vertical Tube (Nuclear Fuel Storage Facility)
6.3 Electric Machines
6.4 Furnace of a Steam Boiler
References

Citation preview

Heat and Mass Transfer

Sergey Z. Sapozhnikov Vladimir Yu. Mityakov Andrey V. Mityakov

Heatmetry The Science and Practice of Heat Flux Measurement

Heat and Mass Transfer Series Editors Dieter Mewes, Universität Hannover, Hannover, Germany Franz Mayinger, München, Germany

This book series publishes monographs and professional books in all fields of heat and mass transfer, presenting the interrelationships between scientific foundations, experimental techniques, model-based analysis of results and their transfer to technological applications. The authors are all leading experts in their fields. Heat and Mass Transfer addresses professionals and researchers, students and teachers alike. It aims to provide both basic knowledge and practical solutions, while also fostering discussion and drawing attention to the synergies that are essential to start new research projects.

More information about this series at http://www.springer.com/series/4247

Sergey Z. Sapozhnikov Vladimir Yu. Mityakov Andrey V. Mityakov •

Heatmetry The Science and Practice of Heat Flux Measurement

123

•

Sergey Z. Sapozhnikov Science Education Centre “Energy Thermophysics” Institute of Energy Peter the Great St. Petersburg Polytechnic University St. Petersburg, Russia

Vladimir Yu. Mityakov Saint-Petersburg, Russia

Andrey V. Mityakov Saint-Petersburg, Russia

ISSN 1860-4846 ISSN 1860-4854 (electronic) Heat and Mass Transfer ISBN 978-3-030-40853-4 ISBN 978-3-030-40854-1 (eBook) https://doi.org/10.1007/978-3-030-40854-1 © Springer Nature Switzerland AG 2020 This work is subject to copyright. All rights are reserved 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

Preface

Heat flux measurements (HEATMETRY) as an important part of heat engineering experiments have not yet been adequately developed, primarily due to the small range and low level of modern heat flux sensors. Most of the sensors developed and available on the market include measuring the temperature difference on the auxiliary wall using differential thermocouples and Peltier elements. These sensors have insufficient response time and thermostability, as well as low-tech. Gradient Heat Flux Sensors (GHFS), proposed as a means of heatmetry, realize the transverse Seebeck effect: their thermo-EMF is proportional to the temperature gradient linearly related to the heat flux density. In the modern scientific and industrial experiments, the digital technologies play great role: In the data recording and analyzing. With digital technologies, scientists can make experiments much quicker with better quality. At the same time, scientists use same sensors for years. The performance gap between the sensors and the signal converting path is constantly growing. The following obstacles remain on the way to the development of digital technologies: 1. Many parameters (heat transfer coefficient, velocity, heat flux, etc.) are compelled to be obtained by calculation, since there are no sensors whose signal is proportional to a value of a desired quantity. 2. One sensor rarely allows you to measure several values (speed, temperature, heat flux, etc.). 3. The number of transmission channels is limited (especially in transport, in space technology, in energy). Therefore, a bottleneck arises between the sensors and equipment, which cannot be expanded. Heat flux—medium or local—is measured tens and hundreds of times less often than temperature. A typical heat engineering laboratory resembles an electric laboratory, where one hundred voltmeters account for a single ammeter. This is primarily due to the lack of cheap, and therefore common and attractive to the experimenter heat flux sensors.

v

vi

Preface

At the first stage of our research, we used heat flux sensors proposed and constructed for demonstrational physical experiments by N. P. Divin. In addition to heat flux measurement problems, these sensors (called Gradient Heat Flux Sensors) were successfully used to measure temperature, thermophysical characteristics of materials, surface emissivity, fluid flow rate, wall shear stress, and electric circuit parameters. In 2007, we created a new generation of layered composite-based sensors (called Heterogeneous Gradient Heat Flux Sensors), whose advantages are high technology and thermal stability. The use of Gradient Heat Flux Sensors in laboratory and industrial conditions confirmed their reliability, showed high information content, and allowed to obtain a number of priority results. The results of our research were published in monographs in 2003, 2007, and 2012 in Russia. Since then the abovementioned new Heterogeneous Gradient Heat Flux Sensors have appeared and then, as a result, new data; we have looked in a different way at some of our previous results. This and our desire to introduce a wide circle of specialists to gradient heat flux measurement triggered the appearance of this book. Saint-Petersburg, Russia December 2019

Sergey Z. Sapozhnikov Vladimir Yu. Mityakov Andrey V. Mityakov

Acknowledgements

We are deeply grateful to Associate Professor N. P. Divin, Ph.D., whose bright brain and skillful fingers created a bismuth-based gradient heat flux sensor and whose good soul was supporting our research for many years. We wish to thank Academician of the Russian Academy of Sciences (RAS) A. I. Leontiev for his multi-faceted support of our work. The talks about heat flux measurement with Member of the Royal Society, Foreign Member of the RAS, Prof. D. B. Spalding became pleasant and useful. Many ideas appeared were evaluated or, vice versa, rejected (sometimes more valuable) in the conversations with the co-workers of the Kutateladze Institute of Thermophysics, Siberian Branch of the RAS: Academician RAS S. V. Alkseenko, Academician RAS D. M. Markovich, Prof. V. I. Terekhov et al. Great attention and comprehensive assistance in the work were provided by the senior scientist Yu. A. Zeygarnik with employees of the Joint Institute for High Temperatures of the RAS. It became important and successful for us to cooperate with Prof. S. V. Bobashev, Head of the Laboratory of Physical Gas dynamics at the A. I. Ioffe Institute of RAS. Studies and discussions with Prof. S. A. Isaev, Prof. Yu. S. Chumakov, Ph.D., senior researcher N. P. Mende, and researcher V. A. Sakharov were very fruitful. We got a lot of progress in our work as a result of the discussions with researcher A. S. Guzeev. As for modern approaches and metrology potentialities, we were patiently assisted by the co-workers of the D. I. Mendeleyev Institute for Metrology, Prof. A. I. Pokhodun, senior researcher N. A. Sokolov, and researcher E. N. Korchagina. Director of the Open Joint-Stock Company “Etalon” (Omsk) V. A. Nikonenko, Ph.D. and engineer V. A. Svirkov made a major contribution to the development of equipment for high-temperature calibration of sensors. The cooperation with senior researcher A. A. Snarsky from Kiev Polytechnic Institute and researcher M. I. Zhenirovsky from Clemson University (USA) became beneficial. Invaluable assistance in the diffusion welding of workpieces was provided to the authors by employees of K. E. Tsiolkovsky Moscow Aviation Technology Institute, Prof. O. A. Barabanova and researcher S. V. Nabatchikov. vii

viii

Acknowledgements

We would like to thank our colleagues of Peter the Great St. Petersburg Polytechnic University, whose participation and discussion of results were always friendly and professional. We would like to express our sincere appreciation to all of them, as well as to many other colleagues, to whom we are very grateful for their assistance in our work and for their tolerance of our weaknesses, professional and human. At last, our special thanks to those who had a full opportunity to put spokes in the wheels at various work stages and directions, but for different reasons did not take this exciting opportunity. Saint-Petersburg, Russia December 2019

Sergey Z. Sapozhnikov Vladimir Yu. Mityakov Andrey V. Mityakov

Contents

1 Heat Flux Measurement and Heat Flux Sensor 1.1 Heat Flux Measurement . . . . . . . . . . . . . . . 1.2 Surface-Mounted Heat Flux Sensors . . . . . . 1.3 Modern Heat Flux Sensors . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

1 1 3 11 15

2 Gradient Heat Flux Sensors . . . . . . . . 2.1 Theory and Design . . . . . . . . . . . . 2.2 GHFS Materials and Constructions 2.3 Calibration . . . . . . . . . . . . . . . . . . 2.4 Digital Signal Processing . . . . . . . References . . . . . . . . . . . . . . . . . . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

19 19 27 30 34 37

3 Transient Heat Flux Measurements 3.1 GHFS Response Time . . . . . . . 3.2 Thermal Model of GHFS . . . . . References . . . . . . . . . . . . . . . . . . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

39 39 49 58

4 Multifunctional Performance of Gradient Heat Flux Sensors 4.1 Thermometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Determination of GHFS Thermal Characteristics . . . . . . . 4.3 Emissivity Factor Determination . . . . . . . . . . . . . . . . . . . 4.4 Wall-Shear Stress Measurements . . . . . . . . . . . . . . . . . . . 4.5 Fluid Flow Rate Measurement and Recording . . . . . . . . . 4.6 Measurement of Electric Circuit Parameters . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

59 59 63 66 68 75 80 85

5 Validation and Science Experiment . . . . . . . . . . 5.1 Validation . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 Free Convection Near Vertical Plate . . 5.1.2 Cross Flow Around Circular Cylinder .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

87 87 87 92

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

ix

x

Contents

5.2 Science Experiments . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Cross and Non-cross Flow Around Cylinder with Turbulisators . . . . . . . . . . . . . . . . . . . 5.2.2 Surfaces with Intensifiers . . . . . . . . . . . . . . 5.2.3 Heatmetry in Shock Tubes . . . . . . . . . . . . . 5.3 Heat Transfer During Phase Changes . . . . . . . . . . . 5.3.1 Condensation . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Heat Transfer During Boiling . . . . . . . . . . . 5.4 Radiative Heat Transfer . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

........... . . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

6 Industrial Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Heatmetry in the Diesel Engine . . . . . . . . . . . . . . . . . . . . 6.2 Heat and Mass Transfer in a Vertical Tube (Nuclear Fuel Storage Facility) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Electric Machines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Furnace of a Steam Boiler . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

99 99 108 124 140 140 154 159 170

. . . . . . 175 . . . . . . 175 . . . .

. . . .

. . . .

. . . .

. . . .

. . . .

185 193 199 208

Acronyms

ADC AFC ALTP ATE BDC CC GHFS HFS HGHFS HTHFS ICE RG ST TCR TDC WWS a A Amin An b B B B0 c C C0 C1 , C2 , C3 Cf

Analog-to-digital converter Amplitude-frequency characteristic Atomic layer thermopile Anisotropic thermoelement Bottom dead center Combustion chamber Gradient heat flux sensor Heat flux sensor Heterogeneous gradient heat flux sensor High-temperature heat flux sensor Internal combustion engine Radiation gauge Shock tube Temperature coefficient of resistance Top dead center Wall shear stress Thermal diffusivity, m2 /s Area, m2 Minimal area of GHFS, m2 Amplitude function Width, m Width, m Magnetic induction vector Magnetic induction, T Specific mass heat capacity, J/(kg
K) Constant coefficient, 1/K Absorption coefficient of absolutely black body (5:67
108 W=ðm2
K4 )) Crystallographic axes in ATE Resistance coefficient

xi

xii

Acronyms

d D Dss E e1 , e2 E0 Emax Eq eX EX EZ 0

0

Ex ; Ez E GHFS E meas E eR Ek E? !

E f g h H h0 h hw hu hu ðsÞ i I I Tm Iq j k ke ks 0 ks 0 Ka

Diameter, m Dispersion of heat flux density, W2 /(m4 ) Flow rate of superheated steam, t/h Electro-motive force, V Thermo-EMF of thermocouples, V Electric signal, V Maximum measuring value of output signal E, V Spectral power density of heat flux fluctuations, s Transverse thermo-EMF of ATE, V Thermoelectric field strength in the direction of x axis, V Thermoelectric field strength in the direction of z axis, V Vector components of electric field strength in the directions of x and z axes, respectively, V Signal of GHFS, V Measured signal of GHES, V Thermal noise level, V Total thermo-EMF, V Longitudinal thermo-EMF, V Transverse thermo-EMF, V Electric field strength vector, V Frequency, Hz Acceleration of gravity, m/s2 Thickness, m Manometer indications, mm Thickness of hot layer, m Heat transfer coefficient, W/(m2
K) Heat transfer coefficient at flat wall, W/(m2
K) Time-average heat transfer coefficient at azimuth angle, W/(m2
K) Time-average heat transfer coefficient at azimuth angle, W/(m2
K) Instantaneous current strength, ÐŘ Current strength, ÐŘ Intensity of temperature fluctuations Intensity of heat flux density fluctuations Flux of charge carriers ADC resolution Sensitivity coefficient of thermocouple, V/K “Coefficient of HFS”, W
V/(m2
K2 ) Sensitivity coefficient of battery HFS, W/(m2
V) Coefficient of transfer of operational amplifier

Acronyms

k k kf k11 , k22 , k33 kf L l l0 m M n N ns nj p; pa P p Pdrum q Q ! q q q0 qv qmax QZ !

Q Z qZ Qloss qd qw qu 0 qu

xiii

Thermal conductivity, W/(m
K) Tensor of thermal conductivity Thermal conductivity of fluid, W/(m
K) Tensor components of thermal conductivity in ATE along C1 , C2 , and C3 axes, respectively, W/(m
K) Thermal conductivity of air at temperature Tf , W/(m
K) Distance, m Length, m Characteristic dimension, m Safety factor cutting off thermal noise Mach number Number of elements Power, W Safety factor determined by requirement for precision of measurements Number of thermocouple junctions Wall pressure and atmospheric pressure, Pa Heater power, W Instantaneous power, W Drum pressure, kgs/cm2 Heat flux density (heat flux per unit area), W/m2 Heat flux, W Heat flux vector, W/m2 Mean-arithmetic measured heat flux, W/m2 Constant heat flux, W/m2 Volumetric heat flux density, W/m3 Maximum heat flux, W/m2 External heat flux, W Resultant vector of heat flux in ATE

qu ðsÞ qffiffiffiffiffiffi qu0 2

External heat flux, W/m2 Heat loss, W Heat flux in dimple, W/m2 Heat flux at flat surface, W/m2 Average heat flux at azimuth angle u, Characteristic fluctuation amplitude of W/m2 Local heat flux at fixed azimuth angle Local heat flux at fixed azimuth angle

r R r0

Current coordinate, m Resistance, Ohm Radius, m

W/m2 heat flux, u, W/m2 u, W/m2

xiv

R0 RE1;E2 Rqs RT Rsh S S0 S0max S0 Sl Sq ST Ss T T0 T1 , T2 Tmax Tf Tw Ts Tt Tss Tf :w: TsA TsB U V V_ w W x; y; z Q b c Db DEmeas Dq DT DT Dx

DTm

Acronyms

Resistance of a GHFS at zero temperature, Ohm Correlation function Coefficient of autocorrelation function of heat flux density fluctuations Resistance of a GHES at temperature T, Ohm Shunt resistance, Ohm Tube pitch, m Volt-watt sensitivity, V/W Maximum sensitivity, V/W Volt-watt sensitivity of sensitive element, V/W Linear volt-watt sensitivity, (V
m)/W Operating coefficient of GHFS, (V
m2 )/W Volt-degree sensitivity of GHFS, V/K Operating coefficient of SS sensor Temperature,  C, K Initial temperature, K Constant and different temperatures kept at ATE faces, K Maximum operating temperature,  C Fluid temperature, K Wall temperature, K Screen temperature, K Tube temperature, K Superheated steam temperature,  C Feed water temperature,  C Steam temperature ahead of the screens in line A (right),  C Steam temperature ahead of the screens in line B (right),  C Voltage, V Volume, m3 Volume flow rate, m3 /s Average incoming flow velocity, m/s Velocity, m/s Spatial coordinates Q-factor Volumetric gas expansion factor, 1/K Rotation angle of ICE crankshaft,  Insulation thickness, m Absolute measuring error of ADC signal, V Probable relative measuring error of radiant heat flux, Temperature drop, K Finite-difference analog of temperature gradient, K/m Temperature difference on material layer, K

Acronyms

DTHFS ¼ T1HFS  T2HFS DT ¼ Tw  Tf Dx DS Dy d d dHFS e ered e1 , e2 e e11 , e22 , e33 g gR = Rr gx = hx h hR hopt hðsÞ 0 H ¼ TT T0 lc ln m n q qb qair qlav r rc rq rT R0 s smin scalc min sw u v w

xv

Temperature difference T1HFS and T2HFS at opposite HFS surfaces Temperature difference at convective heat transfer, K Thickness of a material layer, m Total standard uncertainty in determining sensitivity Total standard uncertainty in determining the value Thickness, m Momentum thickness, m Thickness of a HFS, m Emissivity Reduced emissivity Components of thermo-EMF of layers of a HGHFS Tensor of differential thermo-EMF Tensor components of differential thermo-EMF in ATE along C1 , C2 , and C3 axes, respectively, V/K Intensity of heat flux fluctuations Dimensionless radius Dimensionless thickness Inclination angle of a trigonal plane in an ATE,  Dimensionless temperature Optimal value of an angle h Transient temperature, K Dimensionless excess temperature Dynamic viscosity, Pa
s Characteristic equation roots Kinematic viscosity, m2 /s Drag coefficient Density, kg/m3 Bismuth density, kg/m3 Air density, kg/m3 Lavsan density, kg/m3 Root-mean-square deviation Specific electric conductivity, 1/Ohm Intensity of heat flux fluctuations Intensity of temperature fluctuations Dimensionless volt-watt sensitivity Time, s Time constant or response time, s Minimal possible (calculated) value of response time, s Wall shear stress, Pa Angular coordinate (azimuth angle) at the cylinder surface,  Thermal resistance coefficient, 1/K Optimal angle of installing intensifiers, 

xvi

Acronyms

Coefficient of temperature field nonuniformity Angular velocity, s1 Biot number Fourier number for semi-infinite body Fourier number for semi-sphere Grashof number

wT Bi ¼ ah k Fo ¼ as x2 Fo ¼ Ras2 Grx ¼ gbDTx m2 c 1 q1 h 1 K ¼ c2 q h 2

3

Dimensionless storage capacity ratio of composite plate (GHFS) upper part to lower one (substrate) Modified Kirpichev number (dimensionless depth) Nusselt number at frontal point (u = 0), where a0 = au Nusselt number for plate Nusselt number at fixed azimuth angle u

2

q0 x Ki ¼ kT 0 Nu0 ¼ ak0fx Nux = ax kf au d kf R 2p 1 2p 0

Nuu =

Nu = Prf , Prw

Cylinder circumference-averaged Nusselt number

Nuu du

Pradtl number at fluid and wall temperature, respectively Reynolds number for cylinder Reynolds number for plate Stewart number

Re ¼ wd m Rex ¼ wx m B20 l20 rc qw ¼ fdw

S¼

Sh Tu rTz

!

rT ¼ @T @x i þ

@T @y

!

j þ

@T @z

!

k

Strouhal number Turbulence degree, % Projection of temperature gradient onto z axis Temperature gradient vector

Chapter 1

Heat Flux Measurement and Heat Flux Sensor

1.1 Heat Flux Measurement Nowadays, many methods exist for measuring the temperature and only few for measuring heat and heat flux, while the concepts of heat, heat flux and temperature are well understood. Why? Let us look at the US Patent and Trade Mark Office database [1]. Searching the US Patent Collection we found 39 patents with following keywords (heat AND (flow OR flux) AND (sensor OR gauge), compared with hundreds of thousands of patents for temperature measurement. Heat flux is the energy transferred through a given surface. The unit for heat flux is Joule per second, or Watt: Q = d Q τ /dτ,

(1.1)

where Q, W, is the heat flux; Q τ , J, is the energy transferred through the surface (heat); τ , s, is the time. The temperature field is one of the basic definitions in heat transfer: T (x, y, z, τ ),

(1.2)

where x, y, z are three Cartesian’s. The heat flux varies in space and time, and we can also describe the heat flux field: Q(x, y, z, τ ).

(1.3)

The heat flux per unit area is q = d Q/d A, where A, m2 , is the surface area. From now, we shall refer to the heat flux density (or heat flux per unit area) as the heat flux. The difference between the temperature and the heat flux is clear. The temperature is scalar and the heat flux is a vector. This gives us the direction of energy flow. For example, if the temperature inside an electronic control unit is increasing, there are two general reasons for this: either the inside electronics might be overheated because of high current or the cooling system (fan) might broken. Any temperature sensors © Springer Nature Switzerland AG 2020 S. Z. Sapozhnikov et al., Heatmetry, Heat and Mass Transfer, https://doi.org/10.1007/978-3-030-40854-1_1

1

2

1 Heat Flux Measurement and Heat Flux Sensor

installed inside the unit would show that temperature is increasing but they would not explain why it happened. If we were able to measure the heat flux somehow, we could obtain the heat flux direction and understand the reason why the temperature increased. The heat Qτ is used in different fields, including scientific research, industrial experiments, energy saving, environmental thermal pollution control, and other applications; this quantity is determined as: τ Qτ =

Q(τ )dτ.

(1.4)

0

The Q(τ ) and their integration with respect to the time [2] are to be measured. However, a different method is often used for this purpose. The temperature field is determined experimentally (and discretely, since experimental opportunities are limited) by formula (1.2), function (1.3) is calculated next, and only then is (1.4) integrated. Heat leakage is estimated just from the temperature of the heat transfer surface, for example, using infrared imaging systems, heat-reflecting paint, etc. However, these advanced and well-advertised technologies mainly provide qualitative results. Recognition of colored thermograms is otherwise difficult. Finally, we are able to better control thermal processes: thermometers, which are widely used as sensors only to detect thermal action results, do not allow for automation systems (or operators in case of manual control) to predict how the system will change in the future, knowing the Q(τ ). Let us give an example. The main disadvantage of feedback control systems is that disturbance starts forcing at the object before the actuating element can neutralize it. Feed Forward Control, e.g., control according to variation of the surrounding temperature [3], is an idea that has emerged recently. For example, if it gets cold outdoors, the heat in the room should be increased beforehand depending on the value received from the outdoor thermometer, but not when the room temperature has decreased. In other words, the corrective action is ahead of the disturbance and does not allow the disturbance to affect the controlled parameter. Heat flux sensors have proved their competitive potential in Forward Control Systems and have substantial advantages over temperature sensors in a number of cases. There are no uniform classification has been created for methods and sensors used to determine heat flux: local, area-averaged or time-averaged, etc. The classification proposed by Gerashchenko [4] in the 1960s can be considered one of the most successful. Without a detailed description of different heat measurement’s type, it should be noted that it uses: – heat measurement of phase changes; – measurement of fluid or gas enthalpy drop; – measurement of the Joule heat flux;

1.1 Heat Flux Measurement

3

– “gradient methods” (in Fyodorov’s terminology [5]): measurement of the temperature difference across a layer of certain thickness with known thermal conductivity); – method of electro-thermal analogy; – calorimetric methods (using a bomb calorimeter); – optical and resistometric correlation methods; – thermometric methods (radial heat flux measurement); – surface-mounted heat flux measurement method. Heat flux measurement methods or HEATMETRY [6] review are beyond the scope of the present work that entirely fits into the framework of the last mentioned method, surface-mounted heat flux measurement. Such a choice is determined by the fact that this method: – relatively weakly distorts the temperature field in the measuring zone; – allows to determine local (averaged just by the area of the sensor) heat flux; – uses a well-known method to measure DC voltage.

1.2 Surface-Mounted Heat Flux Sensors Let us be specific and narrow down the subject discussed: 1. There is a foreign body in the area measured, namely, a heat flux sensor (HFS). 2. We discuss measurement of local (averaged over the plan area of the HFS) heat flux. When the surface-mounted HFSs are used in experiments, assumptions are valid (Fig. 1.1). A sensor is an element of a shell (cylindrical, spherical or of any other shape). It oriented with its smaller size normal to isothermal surfaces, is placed either at the surface of a body (1) or inside it (2). In the both cases, heat fluxes q1 and q2 are assumed to be constant over HFS areas A1 and A2 (q1 = const, q2 = const). An HFS generates a signal (usually electric), in terms of which the heat flux q1 or q2 are estimated. Surface-mounted HFSs installed at the surface of the body or Fig. 1.1 Heatmetry with surface-mounted sensors

q1 T1=const A1 q2

1

A2 2

T2=const

4

1 Heat Flux Measurement and Heat Flux Sensor

(a)

q

E

(b)

q E

Fig. 1.2 Heat flux sensors: a thermocouple- and b anisotropic-type

inside it inevitably disturb the temperature field in this body. The uncertainty was estimated, for example, by Van der Graaf [7, 8]. It’s quantity is the smaller, the less is the thermal resistance of the HFS. Most of the surface-mounted HFSs are made as plates, with differential thermocouple junctions mounted (Fig. 1.2a) at the surfaces. Peltier elements sometimes serve as HFSs. In [8], sensors based on thermocouples are called thermocouple-type (or longitudinal-type) HFSs. The thermocouple-type HFS’s theory was described by Kutz [9]. According to the temperature difference, the voltage output E of such sensors is E = nST (T1 − T2 ),

(1.5)

where n is the number of thermocouples, ST is the Seebeck coefficient of the materials (volts per degree centigrade). The corresponding sensitivity of the heat flux sensor is: nST δ E = . (1.6) S= q k In experiments the signal using thermocouple-type HFSs is proportional to the temperature difference over the HFS cross-section. In steady-state mode, the temperature difference (and HFS sensitivity) linearly decreases as the plate thickness and electrical resistance decrease. In transient state ∂T − → q = −k , ∂x

(1.7)

→ where x is the coordinate in the direction collinear to the heat flux vector − q . In T HFS ∂T this case, ∂ x = , the relationship between sensitivity and thickness becomes δHFS nonlinear. This problem will be considered in more detail below. Now, let us confine ourselves to noting that the discrepancy between sensitivity and response time of longitudinal-type HFS is fundamental and cannot be eliminated. The simplest longitudinal-type HFSs are single sensors designed in the 1960s at the Institute of Engineering Thermophysics of the Academy of Sciences of Ukraine

1.2 Surface-Mounted Heat Flux Sensors

5

3 1

4

2

5

Fig. 1.3 Single thermocouple-type HFS made by Gerashchenko [4]: 1—nickel coverage; 2— copper thermoelectrodes; 3—intermediate thermoelectrode; 4—current-collecting wires; 5— insulating bushes

(a)

1

(b)

2

q

3

4

2 1

3 (c)

1

(d)

Fig. 1.4 Longitudinal-type HFS based on multijunction differential thermocouples: a schematic diagram; b thermal battery section; c disc-shaped thermal battery; d scheme of Gerashchenko’s sensor design [4]. Figures correspond to: 1—substrate; 2—hot junction; 3—cold junction; 4— surface with HFS mounted

under the guidance of Gerashchenko [4]. A single sensor (Fig. 1.3) is a flattened differential thermocouple which intermediate thermoelectrode serves as a supporting wall. A significant disadvantage of a single HFS is low volt-watt sensitivity S0 = SHFS ≈ 2 × 10−6 mV/W (here A is the plan area of the HFS). A Differential multi-junction thermocouple sensors developed during the 1960s– 80s usually consist of material (rubber, plastic) layer 1, on which thermal battery 2, 3 (Fig. 1.4) is mounted. There are several methods to produce such thermal batteries. One of the first methods was the technology of “semi-coated” wire [7] (Fig. 1.4b). Constantan wire is coiled at plastic base 1. Parts of the wire turns on one side of the HFS are coated (with a galvanic method) with copper or silver, thus forming junctions of battery 2

6

1 Heat Flux Measurement and Heat Flux Sensor

and 3. Plastic tape with the thermocouples can be either folded in a zigzag shape or rolled into a spiral (Fig. 1.4c). This is providing compactness (disc diameter is from 10 to 300 mm, thickness from 2 to 5 mm) and high level of signal (the number of differential thermocouple junctions is from 100 to 2000 or more). However, fundamentally different HFSs, for example, the so-called supporting wall-type sensors (Fig. 1.2b) are now available. They are made of anisotropic materials with different thermal conductivity, electrical resistance and thermopower. Their operation is based on Seebeck’s transverse effect [10]: the transverse temperature difference appears in HFS cross-sections normal to the external heat flux vector. Thermopower E⊥ is generated proportional to this difference. In [11], sensors made of anisotropic materials are referred to as transverse-type HFSs. Since, based on the Fourier law, the heat flux is proportional to the temperature gradient, the term “gradient heat flux sensor” (GHFS) is proposed in our works [11, 12]. L. Geiling was the first who proposed to use a synthetic HFS in 1947. Studying a system made of alternating layers of two materials with different thermoelectric coefficients and thermal conductivities, situated at an angle to the heat flux vector (Fig. 1.9). He found that heat flux in this system induced thermopower normal to the heat flux vector [13]. The author did not discuss practical application of such HFSs. The idea of a transverse-type (synthetic) HFS has attracted the attention of researchers again in the last decade, perhaps for the first time after Geiling’s publications. FORTECH HTS GmbH (Germany) is advertising its ALTP (Atomic Layer Thermo Pile) sensor [14, 15]. it’s operation is based on Seebeck’s transverse effect (Fig. 1.5a, b) in an anisotropic element. Initiation of thermopower in a hightemperature superconducting film was mentioned for the first time in the study published in 1990 [16]. The sensor is laser-irradiated for tens of nanoseconds, exhibiting high fast response and output signal sufficient for recording. The ALTP sensor’s scheme and the general view are shown in Fig. 1.5a, b. A film with 1 µm in thickness is made by sputtering alternating layers of YBa2 Cu3 O7−d and CuO2 . The film is deposited on a 2 mm thick SrTiO3 base. The entire assembly is shaped as a cylinder of 6 mm in diameter and 20 mm in length. It is proposed to use ALTP sensors in studies on supersonic flow. As shown in Fig. 1.5b, such a sensor is a fairly complex structure, whose thickness is not small from an experimental perspective. Its application beyond the above-mentioned field of research might prove a challenging task. According to the above-mentioned company’s data, the time response of these HFSs is τmin = 10−6 s, which corresponds to the heat flux frequency of 300 kHz. The developers propose to estimate the sensor signal E by the relation E=

sin2θ l εT , A 2

(1.8)

where l is the active length (1. . . 3 mm), δ F is the film thickness (500...700 nm), ε is the thermoelectric coefficients difference in a crystal (normal and along the C axis), θ is the inclination angle of the C axis, and T = (TFFS − TFBS ) is the temperature difference between front and back surfaces of the film. The ALTP sensors’s volt-

1.2 Surface-Mounted Heat Flux Sensors

7

Fig. 1.5 ALTP sensor by Fortech HTS GmbH (Germany) [17]: a principal structure of ALTP, b terminology of some temperatures and general view

(a)

S ,K

(b)

z x

S ,K Constantan Copper Fig. 1.6 HFS with copper and constantan layers manufactured and tested by Zahner et al. [19]

watt sensitivity is S0 = 0.77 mV/W, θ = 10◦ . It is assumed that T . The process is δl also close to steady-state at the time τmin due to small thickness of δl (and Fourier number). The ALTP sensors’s heat stability does not exceed 350 K. The sensors examined by Oparichev [18] have the same disadvantages. Using Geiling’s idea, Zahner et al. [19] manufactured and tested an HFS with copper and constantan layers alternating over the cross-section. The thermal anisotropy was provided by inclination of layers of a composite (Fig. 1.6) produced by sintering a foil stack. The 0.1 mm thick copper and constantan plates alternated. Upon sintering, the stack was cut at different angles θ (apparently, to search for the optimum); the cutting pitch was 1 mm. While low thermal stability considerably limits wide application of HFSs in practice, it can be eliminated for HFSs of both of these types. By selecting metal materials and coupling compounds, we managed to attain the level of 500...700 K. Since only

8

1 Heat Flux Measurement and Heat Flux Sensor

(a)

qz

(b)

Steel Brass Electrical insulator Solder

V Fig. 1.7 Scheme (a) and general view (b) of high-temperature heat flux sensor prototype

the thermal stability of the compound is critical, attempts have been made to manufacture HFSs only of metals and alloys. Raphael-Mabel [20], a post-graduate student of the Virginia Polytechnic Institute and State University (USA) who was guided by Dr. Thomas Diller (who renowned specialist in heat flux measurement and Vice-President of the Vatell Company) manufactured an synthetic sensor composed of 46 alternating steel and brass layers at the inclination angle θ = 45◦ . The tests revealed low sensitivity of the sensor (5 µV at a heat flux of 15 kW/m2 ), precluding its further application. To measure heat flux at high temperatures, a “straight-layered” high-temperature heat flux sensor (HTHFS) with overall dimensions 1.7 × 1.1 × 0.1 cm was fabricated from five steel layers and from six brass layers (Fig. 1.7). Evidently, a longitudinal-type HFS was produced as result but it proved to be worse than the above-described equivalents in many ways. The HTHFS prototype was calibrated at a special setup against a standard industrial HFS by Vatell; its sensitivity amounted to 20.4 µV/(W/cm2 ), which corresponded to the volt-watt sensitivity of 10.96 µV/W. The mean sensitivity of the HTHFS prototype did not exceed 20% of the sensitivity of Vatell’s serial HFS. The author argued that the proposed technology would allow to produce HFSs with thousand junctions and to initiate a large output signal in the future. Clearly he was not aware that such battery sensors were designed in Kiev more than 40 years ago by Gerashchenko [4]. Derryberry and Mann’s attempts to create bismuth and titanium compositions, as well as to use bismuth telluride also did not allow to design a thermally stable HFS with acceptable characteristics. It is interesting that Gerashchenko [4] had already tried to calculate the sensitivity of the synthetic HFS in the 1960s. The meaning of his considerations is illustrated in Fig. 1.8. The author stated that the optimal value of the angle θ was approximately 23◦ for different compositions, and the maximum of the volt-watt sensitivity was rather flat. The following notations are used for calculation of an synthetic sensor fabricated from two materials with the thicknesses δ1 and δ2 , respectively:

1.2 Surface-Mounted Heat Flux Sensors

9

Fig. 1.8 Diagram of synthetic HFS considered in [4]

– – – – – – – – – –

x, y are the coordinates related to the “plate after cutting”; − → q is the heat flux vector directed along the y axis; l is the calculated length of the working sensor; e is the calculated thermopower initiated due to the effect of the heat flux q; 1, 2 are the subscripts referring to the materials with the layer thicknesses δ1 and δ2 , respectively; ξ , η are the coordinates related to layers 1 and 2; S1 , S2 are Seebeck’s coefficients, µV/K;  δ = δδ21 is the layer thickness ratio;  k = kk21 is the thermal conductivity ratio;  σ = σσ21 is the electrical conductivity ratio.

The quantity  δ is of the greatest practical interest. This quantity defines the ratio of δ1 and δ2 in the initial stack. If layers are made by plastic yield,  δ can be varied over wide ranges. Otherwise, the available foil layers thickness is substituted. The specific (per unit length l) thermopower is  δ ( k − σ) S1 − S2 e = q  + l 2k (δ σ )( δ + k)



( δ + k) 1 . 1 + δ · k k

(1.9)

The available references did not allow to establish the origin of this dependence, and its validity is not experimentally confirmed. Another way is associated with designing a GHFS based on homogeneous anisotropic materials. The number of natural materials available for designing GHFSs is small. The radiation detector developed by Ordin [21] is particularly noteworthy. This detector contains a sensitive single-crystal higher element based on manganese silicide MnSi1.7 . It is made in the shape of a plate directed at an angle θ = 45◦ to the crystallographic C1 axis. The author states that the time response of such an ATE is of the order of 10−11 . . . 10−13 s.

10

1 Heat Flux Measurement and Heat Flux Sensor

The radiation detector in [22, 23] is equipped with a cadmium antimonide (CdSb)based anisotropic thermoelement (ATE) equivalent to that of a battery composed of several hundreds of copper-constantan thermocouples. A detecting area (14 × 14 mm) is made from 0.02 mm thick copper foil and is coated with camphor. Each detection element contains three sections 14 mm in length, 1.2 mm in width, and 0.3 mm in thickness. The electric resistance of the detector is 2. . . 3 k, its sensitivity is 150 mV/W [23, 24]. Both HFSs have been made only in laboratory and have not found wide use. In spite of their sufficient sensitivity and high response time, they possess two drawbacks. Firstly, their very brittle elements are badly connected. Secondly, volt-watt sensitivity is temperature-dependent and a maximum point, which makes the measurement result near this point unambiguous. A better idea is to use single-crystal bismuth in GHFS designs, since it possesses pronounced anisotropy of physical properties. First GHFSs based on 0.9999 pure bismuth were designed by Divin [25]. Bismuth-based GHFSs have the volt-watt sensitivity S0 = 5 . . . 65 mV/W, the response time τmin = 10−9 . . . 10−8 s, and the operating temperature range of 20. . . 544 K. Figure 1.9 illustrates the battery GHFS comprising bismuth strips 1. The “mirror” alternation of trigonal planes in two adjacent strips 1 allows to sum transverse thermopowers appearing in them. A battery is usually installed on mica sheet 2. Strips 1 are separated one from another by thin (5 µm) spacers (for example, made of mica) 5 and are kept on substrate 2 with glue. Pure bismuth connectors 3 join strips 1 into a series circuit. Outside elements are equipped with output wires 4. Battery GHFS have plan-area sizes from 1 × 1 mm to 10 × 10 mm or more. Now their minimum thickness is reduced to 0.1 mm. Some of the most important and fundamentally unavoidable bismuth-based GHFSs disadvantages are: – low thermal stability (up to 544 K which is the melting point of bismuth); – considerable complexity and low availability of the technology for industrial; – technological limitation of the thickness to 0.1 mm, which hinders investigation of unsteady processes. Our studies, started in 1996, consisted of two stages. In the beginning, the abovedescribed bismuth-based sensors served as an investigation’s object. Their response time was successfully determined and used in industrial and laboratory experiments. GHFSs based on composites (metal + metal, metal + alloy, semiconductor + semiconductor, semiconductor + metal) with thermal stability up to 1300 K or more were subsequently designed. Since such sensors operated as macroheterogeneous structures, they were called heterogeneous GHFSs (HGHFS). The sensitivity of HGHFSs is within a very wide range, achieving in some cases the level of 60– 70 mV/W. Their manufacturability is substantially better than that of bismuth sensors. The both types of the GHFS are outlined in Chap. 2.

1.3 Modern Heat Flux Sensors

11

(a)

(b) 4 1

3

q

E

5

2

Fig. 1.9 Scheme (a) and general view (b) of Divin’s GHFS design (scale in mm). Figures denote: 1—anisotropic bismuth strips; 2—silica substrate; 3—pure bismuth junctions; 4—current outputs; 5—lavsan spacers

1.3 Modern Heat Flux Sensors Although thermocouple- and transverse-type HFSs are used for more than half a century, we have been unable to uncover any systematic information on their improvement let alone the modern state of scientific developments and market in the literature available to us. The basic parameters of HFSs and, as a result, their attractiveness for users include: – – – –

volt-watt sensitivity; response time; working temperature range; HFS sizes in plan-area.

Moreover, the technical level of HFSs is affected by their stability against aggressive media, mechanical strength, possibility of extensive manufacturing, etc. Figure 1.10 contains the data from the Internet, provided by companies. While these data are promotional in nature, it is generally clear that, despite the possibilities offered by modern technologies, the existing HFSs differ little from the prototypes designed more than 30 years ago [4, 8, 22, 32] and others. Figure 1.11, a does not demonstrate some part of the abscissa scale 10−11 ...10−13 s, which should be provided with the information on MnSi1.7 -based sensors [21] (see Sect. 1.2.2). We did it, because the cited works are informative only, but our experience shows that it is very difficult to correctly estimate the response time of HFS even over the range 10−8 . . . 10−9 s [11]. As follows from Fig. 1.11, a, the bismuth-based GHFS volt-watt sensitivity is much downscale (by 3. . . 4 decades) only to the HFS prototype designed at the Physical Electronics Laboratory (Switzerland). However the Swiss analog has hardassigned sizes of the plate (35 × 35 mm), which noticeably narrows its potential uses. The sensors from the IET of NAS of Ukraine are more sensitive than GHFSs by an order of magnitude; however, application of modern ADCs (see Sect. 2.4) makes this advantage insignificant.

12

1 Heat Flux Measurement and Heat Flux Sensor

Fig. 1.10 Modern longitudinal type HFS: a Vatell (USA) [26]; b TNO (Netherlands) [27]; c WUNTRONIC GmbH (Germany) [17]; d Captec Entreprise (France) [28]; e Hukseflux (Netherlands) [27]; f Laboratory of Physical Electronics (Switzerland) [29]; Institute of Engineering Thermophysics of NAS of Ukraine; [30]; RdF Corporation (USA) [31]

1.3 Modern Heat Flux Sensors

(a) S0, mV/W

≈

105

be

7

tte

r

wo

rs

103

e

2

9

1

5

101

4 1

3

10-7

10-9

6

8

10-1

≈

10-1

τ, s

101

(b) T, °C 600 400

HGFHS

Fig. 1.11 Comparison of modern heat flux sensors by sensitivity S0 and response time τmin (a) and temperature limit (b): 1—GHFS based on bismuth (a) and HGHFS (b); 2—Academy of Science, Ukraine; 3—Vatell (USA); 4—Wuntronic (Germany); 5—Captec (France); 6—Hukseflux (Netherlands); 7—Physical Electronics Laboratory (Switzerland); 8—Newport (USA); 9—TNO (Netherlands); 10—ALTP from FORTECH HTS GmbH (Germany) (no data on a working temperature)

13

0

GFHS

200

-273 1

2

3

4

5

6

7

8

9

All other analysed sensors fall behind the GHFSs by 3 orders or more, which confirms the advantages of the GHFSs used for studying unsteady processes. It is demonstrable from Fig. 1.11b that, only the sensor from Captec (France) is capable of working at cryogenic temperatures. At the same time, the upper temperature level of the sensor from Vatell (USA) considerably exceeds that of the bismuth-based GHFS. The limitation of the single-crystal bismuth-based GHFS temperature imposed by the bismuth melting point (544 K). It is the fundamental disadvantage of this sensor. Heterogeneous gradient heat flux sensors (HGHFSs) designed at our laboratory in 2007 are described in detail below. Now their thermal stability is 1300 K and can be increased by using more refractory materials (tungsten, titanium, niobium, etc.). It follows from Table 1.1 that the area size of modern HFSs is not less than 4 mm2 . This value can be reduced by a factor of 4. . . 5 for GHFSs, while the maximum area is limited only by the sensor’s complexity and cost (Tables 1.1 and 1.2).

14

1 Heat Flux Measurement and Heat Flux Sensor

Table 1.1 Modern thermocouple HFS Manufactures Mark Vatell Corporation

Wuntronic GmbH

HFM-6D/H, HFM-7E/H, HFM-8E/L, HFM-8E/H, HFM-7E/L With water cooling TG-1000 TG-2000 TG-9000 FR-75D

High temperature Heat flux sensor FCR

Captec Entreprise

Heat flux sensor

Ultrathin flexible radiant flux sensor

ITIC

HT-50

Hukseflux

NF01, NF02

HF05

RHF with air and oil cooling

Parameters T = 620 . . . 1070 K; τ = 17 µs; S0 = 105 . . . 1.5 × 106 ; d = 6.3 mm T = 470 . . . 870 K; q = 50 . . . 10,000 kW/m2 ; S0 = 0.0033 . . . 0.2 µV m2 /W; d = 4.7 . . . 25.4 mm T = 240 . . . 420 K; q = 10 kW/m2 ; S0 = 14.5 µV m2 /W; d = 19 mm T = 820 K; q = 15.8 kW/m2 S0 = 1.8 µV m2 /W; d = 19 mm; σ = 3.6 mm T = 90 . . . 470 K; q = 500 kW/m2 ; S0 = 0.3 . . . 180 µV m2 /W; τ = 0.3 s; σ = 0.17 . . . 0.4 mm; F = 1 . . . 900 cm2 T = 90 . . . 470 K; q = 500 kW/m2 ; S0 = 0.1 . . . 90 µV m2 /W; τ = 0.05 s; σ = 0.4 mm; F = 1 . . . 900 cm2 T = 490 . . . 1255 K q = 3155 kW/m2 ; S0 = 3 . . . 21 µV m2 /W; τ = 0.1 s; σ = 2.54 mm; d = 8 . . . 25.4 mm T = 490 . . . 1255 K; q = 3155 kW/m2 ; d = 4 mm T = 440 K; q = 0 . . . 600 kW/m2 ; S0 = 15 µV m2 /W; σ = 5 mm; d = 40 mm T = 1070 K; S0 = 0.004 µV m2 /W; d = 30 mm

1.3 Modern Heat Flux Sensors

15

Table 1.2 Modern thermocouple HFS Thermoflux

Omega

RdF Corporation

RHF Range

NPO Etalon

T = 220 . . . 500 K; S0 = 0.15 . . . 60 µV m2 /W; σ = 0.4 . . . 2.5 mm; F = 1 . . . 100 cm2 TFX-191, TFX-178 T = 520 K; q = 0.01 . . . 100 kW/m2 ; S0 = 10,000 µV m2 /W; d = 4.5 mm HFS-3, HFS-4 T = 70 . . . 420 K; S0 = 1 . . . 2.1 µV m2 /W; τmin = 0.6 s; ζ = 0.00178 K m2 /W; σ = 0.18 mm; F = 28.5 × 35.1 mm Micro-foil in-depth thermopile T = 90 . . . 530 K; PA-HFS S0 = 0.8 × 106 µV m2 /W; τmin = 0.02 . . . 0.5 s; σ = 1.2 . . . 2.0 mm; q = 94.6 kW/m2 RHF Range T = 90 . . . 470 K; S0 = 0.1 . . . 10 µV m2 /W; τmin = 0.05 s; q = 500 kW/m2 ; σ = 0.4 mm; F = 1 . . . 100 cm2 SHF Range T = 90 . . . 470 K; S0 = 1 . . . 2 µV m2 /W; τmin = 0.3 s; ζ = 0.00015 K×m2 /W; q = 500 kW/m2 ; σ = 0.4 mm; F = 1 . . . 25 cm2 T = 280 . . . 420 K; S0 = 27 µV m2 /W; q = 10 . . . 2000 kW/m2 ; d = 20 . . . 300 mm; F = 4 . . . 441 cm2 TFX-161

References 1. US Patent and Trade Mark Office database. http://www.uspto.org/. 2. Cherepanov, V. Ya. (2005). Measurement of heat transfer parameters. World of Measurements, 9, 4–15. 3. Lartz, D., Cudney, H., & Diller, T. (1994). Heat flux measurement used for feedforward temperature control. 261–266. https://doi.org/10.1615/IHTC10.3100. 4. Gerashchenko, O. A. (1971). Osnovy teplometrii (The basics of heat metering). Kiev: Naukova Dumka.

16

1 Heat Flux Measurement and Heat Flux Sensor

5. Fedorov, V. G. (1974). Teplometriya v pishevoy promishlennosti (Heatmetry in the food industry) (p. 176). Moscow: Pishevaya promishlennost. 6. Gusakov, A., Kosolapov, A., Markovich, D., et al. (2014). Simultaneous PIV and gradient heat flux measurement of a circular cylinder in cross-flow. Applied Mechanics and Materials, 629, 444–449. 7. Van der Graaf, F. (1990). Heat flux sensors. In: Gopel (Ed.), “Thermal sensors” of the multivolume work “Sensors, a comprehensive series” (Chap. 8, Vol. 4). 8. Dekusha, L. V., Grishenko, T. G., & Mendeleeva, T. V. (2001). Teoreticheskiye osnovy metrologii (Theoretical foundation of heat metrology in heatburner). Promislennaya teplotehnika, 23(4–5), 175–180. 9. Kutz, M. (2013). Handbook of measurement in science and engineering (Vol. 1). Hoboken, NJ: Willey. 10. Anatichuk, L. I., & Bulat, L. P. (2001). Poluprovodniki v ekstremalnikh temperaturnykh usloviyakh (Semiconductors in extreme temperature conditions). Saint-Petersburg: Nauka. 11. Mityakov, A. V. (2000). Gradientnyye datchiki teplovogo potoka v nestatsionarnoy teplometrii (Heat flax sensors in non-stationary heat metering). Dissertation, Saint-Petersburg State Technical University. 12. Sapozhnikov, S. Z., Mityakov, V. Y., & Mityakov, A. V. (1998). Heat flux sensor for heat transfer investigation. In 11th International Heat Transfer Conference, Kyongju, Korea (Vol. 4, pp. 77–79). 13. Geling, L. (1951). Das Thermoelement als Strahlungsmesser. Zeitschrift Fur Angewandte Physik, 3(12), 467–477. 14. Knauss, H., Gaisbauer, U., Wagner, S., et al. (2002). Calibration experiments of a new active fast response heat flux sensor to measure total temperature fluctuations. Introduction to the problem. In International Conference on the Methods of Aerophysics, Novosibirsk (pp. 632–643). 15. Knauss, H., Gaisbauer, U., Wagner, S., et al. (2006). Novel sensor for fast heat flux measurements. AIAA, 2006–3637, 32. 16. Chang, C. L., Kleinhammes, A., Moulton, W. G., et al. (1990). Symmetry-forbidden laserinduced voltages in YBA2 Cu3 O7 . Physical Review B, 41(16), 11564–11567. 17. https://www.wuntronic.de/en/. 18. Oparichev, A. B. (2006). Issledovaniye naklonnokondensirovannykh plonochykh materialov dlya termoelektricheskikh preobrazovateley lazernogo izlucheniya (Research of inclinedcondensed film materials for thermoelectric converters of laser radiation). Dissertation, Moscow State University of Fine Chemical Technologies named after M.V. Lomonosov. 19. Zahner, Th., Forg, R., & Lengfellner, H. (1998). Transverse thermoelectric response of a tilted metallic multilayer structure. Applied Physics Letters, 73(10), 1364–1366. 20. Raphael-Mabel, S. (2005). Design and calibration of novel high temperature heat flux sensor. Dissertation, Virginia Polytechnic Institute and State University. 21. Ordin, S. V., & Shelykh, A. I. (2002). Elektronnoye otrazheniye i zonnaya struktura vysshego silitsida margantsa (Electron reflection and band structure of higher manganese silicide). In Thermoelectrics and their application: Tr. FTI them (p. 34). A. F. Ioffe RAS. 22. Anatachuk, L. I. (1979). Termoelementy i termoelektricheskiye ustroystva: Spravochnik (Thermoelements and thermoelectric devices: Reference book). Kiev: Nauka dumka. 23. Pilate, I. M., Vetoshnikov, V. S., & Khokhlachev, K. I. (1974). Termoelektricheskiy priyemnik izlucheniya na anizotropnykh elementakh (Thermoelectric radiation detector on anisotropic elements) (pp. 2–7). Leningrad: Teplovyye priyemniki izlucheniya. 24. Pilate, I. M., Samoilovich, A. G., & Anatychuk, L. I. (1969). Thermocouple. USSR Patent 230915, 13. 25. Divin, N. P. (1998). Heat flux sensor. Russian Patent 9959, 16 May 1999. 26. http://www.vatell.com/. 27. https://www.hukseflux.com/. 28. https://www.captec.fr/. 29. Hagleitner, C., et al. (2001, November 15). Smart single-chip gas sensor microsystem. In C. Hagleitner, A. Hierlemann, D. Lange, A. Kummer, N. Kerness, O. Brand, et al. (Eds.). Nature, 414, 293–296.

References 30. http://ittf.kiev.ua/. 31. http://www.rdfcorp.com/. 32. TNO Industrial Research. (1992). Heat flux sensors catalog.

17

Chapter 2

Gradient Heat Flux Sensors

2.1 Theory and Design As mentioned in Sect. 1.2, the gradient heat flux sensors (GHFSs) method is based on generation of the transverse component of electric field in a medium with anisotropic thermal conductivity, electric conductivity, and thermoelectric coefficient when a heat flux vector passes in the direction opposite to the principal axes of the anisotropic medium [1, 2]. Such a medium is considered to be continuum, and anisotropic single crystals are the most similar to this model. A rougher approximation is represented by layered anisotropic media. The thinner the layers, the closer is the structure to continuum. Let us consider, as an object of investigation, an element of finite sizes and simplest shape (for example, a parallelepiped); at A. G. Samoylovich’s suggestion, it is called an anisotropic thermoelement (ATE) [3]. The ATE shown in Fig. 2.1 was cut from an anisotropic single crystal so that its x, y, and z axes do not coincide with the principal crystallographic C1 and C3 axes (anisotropy of properties in the direction of the C2 axis is neglected). ATE’s elementary theory is constructed in the framework of the stationary heat conduction problem and reduces to the following [1, 2]. Let an external heat flux Qz (Fig. 2.2) be supplied to the ATE. In accordance with the Fourier law, the projection of the temperature gradient ∇Tz will be directed along − → the z axis in the direction opposite to the direction of the vector Q z . Since the heat flux vector in all sections of the ATE, except for the z = 0 plane, will deviate from the original direction, the temperature difference will arise not only in the direction of the z axis, but also in the direction of the x axis. Let the thermal conductivity tensor components for anisotropic bismuth ⎞ k11 0 0  k = ⎝ 0 k22 0 ⎠ . 0 0 k33 ⎛

© Springer Nature Switzerland AG 2020 S. Z. Sapozhnikov et al., Heatmetry, Heat and Mass Transfer, https://doi.org/10.1007/978-3-030-40854-1_2

19

20

2 Gradient Heat Flux Sensors

Fig. 2.1 Anisotropic thermoelement: dimensions l × b × h; crystallographic axes C1 , C2 , C3

z

C1

90°

C3

θ

0 C2 y

x h

l b

Fig. 2.2 Heat flux vector deviation in anisotropic material layer

z Tz

C1 A

u t

C3 B x

O

m

s r

h Qz*

n

Qz

in the directions of the axes C1 and C3 are equal k11 and k33 , respectively. The Fourier equation for an anisotropic medium has the form − → Q = − k∇T F, − → − → where Q is resulting heat flux vector, ∇T = ∂∂Tx i + gradient vector, F = l × b is ATE’s plan-area. ∇Tz =

(2.1) ∂T ∂y

− → j +

∂T − → k. ∂z

∂T ∂z

− → k is temperature

(2.2)

Here ∇T is vector component directed along z axis. We decompose the vector along the axes C1 and C3 O A = ∇Tz cosθ, O B = ∇Tz sinθ.

(2.3)

The components of the heat flux vector qz in the projections on the same axis are equal to

2.1 Theory and Design

21

on = Q 1z = −k11lb∇Tz cosθ, om = Q 3z = −k33lb∇Tz sinθ.

(2.4)

Now project the components of the vectors Q 1z and Q 3z on z and x axes; we get or + os = Q∗z = Q 3z sinθ + Q 1z cosθ = −(k33 sin2 θ + k11 cos2 θ )lb∇Tz , ou + ot = Q∗x = −Q 3z sinθ + Q 1z cosθ = −(k11 − k33 )sinθ cosθlb∇Tz .

(2.5)

The resulting heat flux vector in the ATE is rotated relative to the external heat flux vector by an angle ϕ determined by the relation tgθ =

Q∗x (k11 − k33 )sinθ cosθ . ∗ = Qz k33 sin2 θ + k11 cos2 θ

(2.6)

Under the influence of an external heat flux Q z , an electric field of intensity E arises in the ATE (Fig. 2.2). This is possible due to the anisotropy of thermal conductivity and the coefficient of differential thermoEMF does not coincide in direction with either the external heat flux Q z and the heat flux Q ∗z . The equation for the electric field strength E arising in the ATE is outwardly similar to Eq. (2.1) − → E = − e∇T. (2.7) 



We calculate the components of the electric field vector E z and Ex . Then, we write the components of the electric field vector caused by the contribution of the components e33 and e11 in the projections on the C1 and C3 axes, and then project the vector components at the z and x axes ⎛

⎞ e11 0 0  e = ⎝ 0 e22 0 ⎠ . 0 0 e33 ⎧ E 3z = −e33 ∇Tz sinθ ⎪ ⎪ ⎪ ⎨ E = −e ∇T cosθ 1z 11 z ∗ ⎪ Ez = E 3z sinθ + E 1z cosθ = −(e33 sin2 θ + e11 cos2 θ )∇Tz ⎪ ⎪ ⎩ ∗ Ex = −E 3z sinθ + E 1z cosθ = −(e33 − e11 )sinθ cosθ ∇Tz .

(2.8)

− → − → The vectors E and Q z are rotated one relative to the other at an angle ψ, determined from the relation (Fig. 2.3) tgψ =

E∗x (e33 − e11 )sinθ cosθ . = E∗z e33 sin2 θ + e11 cos2 θ

(2.9)

T2 − T1 , h

(2.10)

We can assume that ∇Tz 

22

2 Gradient Heat Flux Sensors

Fig. 2.3 The heat flux vectors directional pattern and electric field strength in the ATE

z C1

p Ex

Tz

O

v

C3

x

w d

h

E

Ez b Qz*

Qz

for an ATE with a sufficiently small thickness h. Here T1 < T2 are constant and different temperatures maintained on the faces of the ATE z = h and z = 0, respectively. Seebeck’s longitudinal effect causes thermopower E z = −E z∗ × h  (e33 sin2 θ + e11 cos2 θ )∇Tz × h = = (e33 sin2 θ + e11 cos2 θ ) × (T2 − T1 ).

(2.11)

As we see, the E z does not depend on the sizes l, b, h and is determined only by the angle θ . The transverse Seebeck effect is manifested in the fact that thermoEMF acts in the direction of the x axis. E E x = −E x∗ × l = (e33 − e11 )sinθ cosθ ∇Tz × l   (e33 − e11 )sinθ cosθ × hl × (T2 − T1 ).

(2.12)

It follows from Eq. (2.12) that the modulus of E x is not proportional to the temperature difference (T1 − T2 ). But according to the Fourier law, E x is proportional to the gradient modulus, the heat flux modulus Q ∗z . On the basis of the first law of thermodynamics, Q ∗z  Q z (the approximation reminds that a small part of the heat is spent on the production of thermoEMF).   If we recall that the quantities Q z and Q x are the projections of the vector on the z and x axes, then we can state that  2 2 Q z = −klb∇Tz = − k33 sin2 θ + k11 cos2 θlb∇Tz , where from ∇Tz = 

Qz 2 k33 sin2 θ

2 + k11 cos2 θlb

.

(2.13)

(2.14)

2.1 Theory and Design

23

After substituting the ∇Tz from (2.12) we obtain (e33 − e11 )sinθ cosθl Q z (e33 − e11 )sinθ cosθ Fqz =  , Ex =  2 2 2 2 2 k33 sin θ + k11 cos2 θlb k33 sin2 θ + k11 cos2 θ b

(2.15)

Qz the average density of the external heat flux at the site is A = l × b. where qz = l×b Thus, thermoEMF E x is linearly related to the heat flux per unit area. ATE voltwatt sensitivity Ex (e33 − e11 )sinθ cosθ . (2.16) S0 = =  Qz A b k 2 sin2 θ + k 2 cos2 θ 33

11

Since for each ATE the values e11 , e33 , k11 , e33 , θ are constant and set, the S0 depends on the linear size b and angle θ . Looking ahead, we note that both of these parameters are very important for the manufacture of GHFS. From the formula (2.16) it follows that the ATE’s width b must be minimized to technologically limits. d S0 (2.16) on the extremum. It follows from the condiWe examine the function dθ opt tion that the optimal angle θ . θopt = ±arctg

k11 . k33

(2.17)

For anisotropic single-crystal bismuth with a purity of 0.9999 at a temperature of about 300 K, k11 = 10 W/(m K), k33 = 5.5 W/m K), e11 = 50 µV/K, e33 = 100 µV/K, which corresponds to θopt = 53 24. The curve S0 (q) constructed for bismuth according to formula (2.16) is shown in Fig. 2.8. The number of natural materials that are suitable for the creation of GHFS extremely small. Note the detector cell which was developed at the Physicotechnical Institute named after A. F. Ioffe RAS S. V. Ordin [4, 5]. The cell’s contains a sensitive element based on monocrystalline higher manganese silicide (MnSi1.7 ). The sensitive element is made as a plate oriented at an angle θ = 45◦ to the crystallographic axis C1 . The development’s author claims that the ATE response time is of the order of 10−11 . . . 10−13 s. In a detector cell [4, 6], the use of antimicrobium oxide from CdSb is equivalent to the action of a battery of several hundred copper-constantan thermocouples. The receiving area (14 × 14 mm) is made of 0.02 mm thick copper foil and coated with camphored black. Each receiving element contains three ATEs with a length of 14 mm, a width of 1.2 mm and a thickness of 0.3 mm. The electrical resistance of the receiver is 2. . . 3 k, the sensitivity is 0.15 V/W [3]. A signal initiated by passing the heat flux through the ATE is often too weak for reliable recording, so a GHFS is usually a cascade battery of ATEs (Fig. 1.9). This is equivalent to a proportional increase in the size l at b = idem.

24

2 Gradient Heat Flux Sensors

Such a battery has a number of advantages, including, in addition to a high signal level. Moreover, bifilar location of conductors that weaken the influence of external electric and magnetic interferences. The width of the GHFS’s battery is B = b × n + b × (n − 1),

(2.18)

where b is the width of gaskets and n is the number of ATEs that form the GHFS. The thermopower of such a sensor is E =e×n =e

B + b , b + b

(2.19)

and the volt-watt sensitivity is E=

E e B + b = × . qz Bl qz Bl b + b

(2.20)

It was mentioned in Sect. 1.2. that the number of natural anisotropic metals is small and their effective kinetic coefficients cannot be affected. In addition, the restrictions imposed by low thermal stability of such materials and, in some cases, by the nonmonotonic temperature dependence of the functions S0 and Sl cannot be eliminated in principle. Therefore, the interest of researchers [7–9] in synthetic anisotropic media is clear. The sensors based on such substances are called heterogeneous gradient heat flux sensors (HGHFS) in Sect. 1.1. The HGHFS is schematically sketched in Fig. 2.4. ∼ ∼ The quasi-crystallographic x and z axes of a layered medium (equivalents of the C1 and C2 axes in a single crystal) are rotated by the angle θ relative to the “laboratory”

Fig. 2.4 Synthetic composite of HGHFS (microphotography)

x ~

z 1 2

2

,k

2

,S

1

qz

2

1

,k 1

x

,S

~

z

2.1 Theory and Design

25

x and y axes. Synthetic ATE is assumed to be oriented so that the external heat flux vector is directed along the z axis. Let’s introduce the following notations: k1 , k2 are the thermal conductivities of the layers; σ1 , σ2 are the electrical conductivities of layers; S1 , S2 are the thermoelectric (Seebeck) coefficients of layers; δ1 , δ2 are the thicknesses of layers. Let us also use dimensionless values ∼

k=

k2 ∼ σ2 ∼ S2 ∼ δ2 ; σ = ; S= ; δ = . k1 σ1 S1 δ1

(2.21)

The tensors of the effective kinetic coefficients are expressed in terms of the local kinetic coefficients and thickness of layers in the crystallographic axes: For thermal conductivity ⎛ k x∼x =

∼

⎞

δ1 + δ2 ⎜ 1+ δ ⎟ = k1 ⎝ ∼ ⎠, δ1 δ2 + k1 k2 1 + ∼δ

(2.22)

k

k zz∼ =

1 δ1 + δ2

  σ1 σ2 (S1 − S2 )δ1 δ2 δ1 k1 + δ2 k2 + ×T ≈ δ1 σ1 + δ2 σ2 ⎞ ⎛ ∼ ∼∼ 1+ δ1 k1 + δ2 k2 k1 + δ k2 k δ ⎠. ≈ = k1 ⎝ = ∼ ∼ δ1 + δ2 1+ δ 1+ δ

(2.23)

Here T is the mean thermodynamic temperature of the given medium. The quantity Z T = σ1 σ2 (S1 − S2 )δ1 δ2 /(δ1 σ1 + δ2 σ2 )T is the Q-factor of a medium. It is negligibly small for metal materials and this allows to simplify E q (2.37). For thermopower coefficient Sx∼x =

S1 δ1 σ1 δ1 σ1

+ +

S2 δ2 σ2 δ2 σ2

⎛ = S1 ⎝1 +

∼

S

∼

σ

⎞ ∼

δ⎠ ,

⎞ ⎛ ∼ ∼∼ σδ 1+ δ1 σ1 S1 + δ2 σ2 S2 S ⎠. Szz∼ = = S1 ⎝ ∼∼ δ1 σ1 + δ2 σ2 1+ σ δ

(2.24)

(2.25)

Assuming that k x∼x ≡ k11 , k zz∼ ≡ k33 , Sx∼x ≡ S11 , Szz∼ ≡ S33 in (2.22)–(2.25) allow to determine the values of the HGHFS, as well as the maximum volt-watt sensitivity, where, according to equalities (2.17), (2.24) and (2.25)

26

2 Gradient Heat Flux Sensors ∼

1+ δ .  ∼ ∼∼ 1 + ∼δ (1+ δ k )

θopt = arctg 

(2.26)

k

Production of composite structures is based on the technological specifics of their diffusion bonding (selected pairs of materials should be mutually soluble, not yielding eutectics, refractory oxides, etc.). Moreover, it is important to provide thermal stability of composite, ease commutation and mounting of the HGHFS, etc. The ∼

parameter δ = δδ21 is the most easily attainable, since modern rolling technologies allow to produce billets with a thickness of several micrometers and high dimensional stability. ∼ ∼

∼

Let us optimize a HGHFS through volt-watt sensitivity at defined k , σ and S . The volt-watt sensitivity of layered material will be characterized by the dimensionless equivalent of the function S0 

∼ ∼∼

1+ S σ δ

S0 bk1

0 = = S1

∼∼

1+σ δ ∼∼

1+k δ ∼

1+ δ

∼∼

−

S 1+ ∼ δ σ ∼

sinθ cosθ

1+ ∼δ σ

sin θ + 2



∼

1+ δ

∼ 1+ ∼δ k

cos2 θ

.

(2.27)

where the value θ ≡ θopt is chosen by formula (2.26); further, by default, the module of the function | 0 | is considered. The function 0 has an extremum with respect to the parameter (Fig. 2.5) and is ∼ ∼

∼

monotonic with respect to the parameters S , k and σ . Curves 1. . . 8 correspond to the values of these parameters for the materials used in practice (see below). ∼

∼

Figure 2.6 plots θopt as a function of δ and k . It can be seen that the optimal angle ∼

∼

is within the 30◦ –40◦ range for practically any ratios of δ and k . The quantity S0 is very sensitive to the accuracy of all assigned arguments (except for θopt : here the maximum appears to be rather sloping for most materials). The scatter of the data on material properties given in literature usually exceeds 10–20%; many of the characteristics are simply not available in the technologically important temperature range. So, the formula of form (2.27) must be used carefully at different temperatures, first, to estimate the unknown quantity in terms of magnitude and then, preferably, to get a preliminary idea whether the curve S0 (T ) is monotonic or has an extremum. Figure 2.7 illustrates the results for two compositions. Both of the above-mentioned problems were particularly pronounced for the nickel + stainless steel composition, since the data on the thermoelectric properties of these materials is very contradictory. Thus, creating synthetic layered media as initial billets for HGHFSs is an extremely complex and difficult task, and only the first steps have been taken in

2.1 Theory and Design

27

0

1

0.03

2

0.02

3

6

4 5

0.01 8 7

0

2

4

6

8 ∼

Fig. 2.5 Dimensionless volt-watt sensitivity of synthetic ATE. Numbers correspond to: 1—σ = 1.5, ∼

∼

∼

∼

∼

∼

∼

∼

∼

S = 0.43, k = 6.067; 2—σ = 1.23, S = 1.1, k = 6.067; 3—σ = 1.23, S = 0.43, k = 5.0; 4—σ = 1.23, ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ S = 0.43, k = 6.067; 5—σ = 1.23, S = 0.43, k = 7.0; 6—σ = 1.23, S = 0.43, k = 7.0; 7—σ = 1.23, ∼ ∼ ∼ ∼ ∼ S = 0.43, k = 10.0; 8—σ = 1.1, S = 6.067, k = 10.0 ∼

opt

0.6 0.4 0.2 20 60 40

100

~

∼

~

200 k

∼

Fig. 2.6 Values of function θopt at different δ and k

dealing with it. It is necessary, in particular, to have reliable data on thermal, electrophysical and thermoelectric characteristics or to confine the consideration to crude estimates and to optimize construction by experimentally founding points in the curves S0 (T ).

2.2 GHFS Materials and Constructions As discussed in Sect. 1.2, let us first focus in detail on a GHFS based on anisotropic bismuth single crystals.

28

2 Gradient Heat Flux Sensors

(a)

(b)

S0 , mkV/W

S0 , mkV/W

1 2

20

100

10

0

1 2

50 200

T, oC

400

0

200

T, oC

400

Fig. 2.7 Preliminary estimates of volt-watt sensitivity of sensors manufactured from: a chromel + alumel; b nickel + stainless steel. Numbers correspond to: 1—calculated and 2—experimental curves

S0, mV/W

b=0,05 mm (present conditions)

60

40 b=0,1mm

20 b=0,2 mm

20

40

60 opt=53°24’

80

,°

Fig. 2.8 Volt-watt sensitivity of bismuth-based ATE

GHFSs manufactured from 99.99% pure bismuth have the volt-watt sensitivity S0 = 5 . . . 65 mV/W, the response time τmin ≈ 10−9 …0−8 s and the working temperature range from 20 to 544 K. Figure 1.9 shows the GHFS incorporating a bismuth-based ATE, which described in detail in Sect. 2.1. For anisotropic 99.99% pure bismuth at a temperature about 300 K: k x x = 10 W/(m×K); k zz = 5.5 W/(m K); Sx x = 50 mV/K; Szz = 100 mV/K, which corresponds to θopt = 53◦ 24 . The curve S0 (q) plotted by formula (2.16) for bismuth is shown in Fig. 2.8.

2.2 GHFS Materials and Constructions

29

Fig. 2.9 Microstructure of GHFS manufactured from stainless steel (mesh) + nickel (scale in mm)

In view of the unavoidable disadvantages of the bismuth-based GHFS mentioned in Chap. 1, we have paid attention to the technique and technology of layered composites. HGHFSs were created for the first time in 2007 at Peter the Great St. Petersburg Polytechnic University. As a result of the first steps, sensors based on compositions of nickel + stainless steel, chromium steel + nickel, chromel + alumel, and iron + constantan were designed. These HGHFSs could operate up to a temperature of 1300 K or more. In 2010, HGHFSs fabricated from tungsten + nickel compositions were developed, with their thermal stability enhanced to 2300 K; a very difficult task of calibrating the sensors at high temperatures was posed simultaneously that remains as yet unsolved. In 2011, there was an important work on the sensors to prepare silicon + aluminum and silicon + nickel compositions exceeding bismuthbased GHFSs in volt-watt sensitivity. It became possible to use not only solid, but also permeable materials as layers: fibrous, mesh, with regular perforation, etc. Figure 2.9 illustrates the structure of the stainless steel (mesh) + nickel composition. Intersections of mesh threads are welded in contact zones. Upon cutting, a sensor becomes penetrable for fluid or gas flows, drag does not exceed tens of Pa/mm (it can be regulated by selecting a mesh, slope angle θ , and plate thickness). Interpretation of a signal from such an HGHFS is a different task, but applying it to research of injection systems, gas screens, etc., gives distinct unique advantages. All billets for HGHFSs are produced by the diffusion welding method. The quality of bonding at the microlevel was controlled by the metallographic method (Fig. 2.10 demonstrates the microstructure of one of the compositions), as well as by the microanalyser “Camebax-Microbeam” (Cameca, France). Observations were made under an optic microscope (×400) and by secondary electrons by an electron probe (×100...20000) in scanning mode. The chemical composition was determined by the method of quantitative X-ray microanalysis using wave dispersion spectrometers. Analysis was carried out with the following conditions. The electron beam energy of 10 keV; current of10 nA; Ni–Fe system of elements; NiKa , xFeKa analytical lines. The references used for preparing the composite were metal nickel and steel.

30

2 Gradient Heat Flux Sensors

(a)

(b) Diffusion zones

100 µm Fig. 2.10 Microstructure of stainless steel + nickel compositions made of 0.1 mm thick layers: a prior to etching of layers; b after etching of layers (light diffusion zone is marked by dashed lines) Table 2.1 HGHES’s volt-watt sensitivity at approximately 300 K Composition Average sensitivity (mV/W) Working temperature (◦ C) Nickel + stainless steel Chromel + alumel Titan + molybdenum Nichrome + steel Copper + constantan Chromel + alumel Silicon + aluminium p-silicon + n-silicon

0.40...0.8 0.35 0.02 0.5 0.2 0.1 1.0 1.5

1400 1200 1660 1100 1000 1450 700 800

The accuracy of analysis was 5% relative, the transverse spatial resolution (diameter of the analyzed region) was 1.5. . . 2.5 µm. The longitudinal spatial resolution (in depth) was 0.7 µm. Analysis has revealed that the width of the diffusion zone is about 5 µm, which provides mechanical strength in the joint and does not touch the source materials for 90–95% of their thickness. To make “working” plates, metal billets were cut by an electric spark cutting device. Silicon-containing billets were cut on a special test bench using a steel jeweler’s saw. Some data on the volt-watt sensitivity of the HGHFS are presented in Table 2.1.

2.3 Calibration Usually, a GHFS is calibrated by the absolute method. Heat flux in the steady-state thermal regime is kept invariable in time and identical over the entire sensor plane. The bismuth-based GHFS was calibrated for the first time by Divin [1, 2]; the schematic diagram of his calibration setup is shown in Fig. 2.11a. Calibrated GHFS 1 is mounted at base 2. Electric heater 3 serves as a heat flux source; its power is determined through current strength and voltage difference, with

2.3 Calibration

31

(a)

(b) qz, kW/m2

mV2

5 4 3 1

0...2 V

mV1

800 600 400

2

200

V 0...2 V

1 2

A

0

50

100 150

Ex, mV

Fig. 2.11 Scheme of calibration setup [1, 2] (a) and calibration results of typical bismuth GHES (b). Points stand for data obtained by alternative method [10, 11]. Numbers indicate: 1—calibrated GHFS; 2—base; 3—heater; 4—zero indicator GHFS; 5—“protective” heater

(a) (b)

2

6 3 1 mV1

2

V 1

3

6 0...2 V

A

Fig. 2.12 GHES calibration made in [10, 11]: a disposition of elements on setup; b scheme of setup. Numbers indicate: 1—calibrated GHFS; 2—base; 3—heater; 4—thermal insulation

a zero value of heat flux maintained by “protective” heater 5 through additional GHFS 4 serving as a zero indicator. The “lateral” leakage of heat flux does not exceed 0.5 W in this calibration setup. Calibration is possible up to the bismuth melting point. The linearity of the calibrated characteristic (Fig. 2.11b) is preserved at an external pressure up to 30 MPa. The GHFS sensitivity changes depending on temperature do not exceed 3%. Such a calibration scheme is not the only one possible. In our experiments [10, 11], GHFS 1 with the 4 × 7 mm dimensions in plan was installed on massive silumin base 2 (head of a cylinder block of a diesel engine described in Sect. 3.4) (Fig. 2.12a). Heater 3 was installed at the top of GHFS 1, mounted at a fiber-glass plastic plate 4. It was a 4 × 7 × 0.04 mm nichrome plate, 2 mm2 two copper wires were drawn through the holes drilled in fiber-glass plastic plate 4. The calibrated module was covered with a layer of thermal insulation [10]. Heater 3 was powered by direct current through the rheostat from an accumulator (12 V, 66 A h). Power input was varied by the rheostat and controlled by a voltmeter

32

2 Gradient Heat Flux Sensors

and an ammeter. The maximum power output on heater 3 was 30 W, which corresponded to a heat flux of 1.07 × 106 W/m2 in terms of the GHES area. A GHFS signal was recorded by a V7-42 mV. The calibration curve obtained on Divin’s setup and our experimental points are compared in Fig. 2.11b. The scatter in the experimental data from the averaged linear characteristic over the temperature range 293. . . 544 K lies within 0.2. . . 1.0% on the greater part of the curve and attains 6. . . 8% in the upper corner of the figure. The temperature of the entire assembly does not exceed 200 ◦ C in this region, so the obviously crude experiment has reduced accuracy due to procedural error. However, it is clear that calibration by both schemes yields fairly good agreement of results. In all other cases, the bismuth-based GHFS was calibrated by Divin’s scheme, as it provides acceptable accuracy and is essentially temperature-independent in the measuring assembly. The calibration of HGHFS by the above methods at high temperatures is impossible. Contribution of convective heat transfer and radiation loss increases, thermal stability of the material is insufficient. To solve this task, we designed the setup (Fig. 2.12) and developed the method described below. The setup has steel cylindrical body 3, at the axis of which nickel foil tube 2 with a tungsten heater inside is positioned. Tube is fixed in special holders 5. The surfaces of tube 2 are equipped with calibrated HGHFSs 1 and thermocouples 6. Body 3 is closed with covers 4 and sealed with silicon joint sealant. One of covers 4 is provided with a sealed connector for power supply to the heater and for signal removal of HGHFS 1 and with thermocouples 6. The setup cavity is treated under vacuum through vacuum connector 7. This allows to eliminate convective heat transfer. Heat flux from the heater is transferred by radiation in the radial direction. Thermocouple 6 serves for determining the reference temperature. As shown in Chap. 1, the HGHFS signal is E = S0 × q × A.

(2.28)

The power output of the heater is P = U × I,

(2.29)

where U and I are the current strength and the heater circuit current, respectively. The heat flux per unit area at the surface of tube 2 is (Fig. 2.13) q=

U×I P = , π ×d ×l π ×d ×l

(2.30)

where d and l are the diameter and the length of tube 2 (d x = 15 mm, l = 220 mm). Thus, the volt-watt sensitivity of the HGHFS is S0 =

E ×π ×d ×l . U×I×A

(2.31)

2.3 Calibration

33

(a)

5

4

1

2

6

3

(b)

Fig. 2.13 Setup scheme (a) and general view (b) of high-temperature calibration of GHFS. Numbers correspond to: 1—calibrated GHFSs; 2—tube; 3—body; 4—cover; 5—holder; 6— thermocouple

The heater is powered by 220 V alternating current through an autotransformer. The power output is controlled by a voltmeter and an ammeter. The maximum power output of the heater is 190 W in our experiments, which corresponds to 22.5 kW/(m2 ) heat flux per unit area. HGHFS signals are supplied to the panel of a PCLD-789D switchboard, and from it to an ADC PCL-818HG and are processed in Genie (monitoring procedure is outlined in Sect. 2.4). Low thermal resistance of nickel foil, from which tube 2 and HGHFSs 1 are fabricated, has made it possible to conduct calibration in dynamic mode: signals of calibrated sensors 1 and thermocouple 6 have been fixed simultaneously. Axial symmetry of the scheme enables simultaneous calibration of several HGHFSs 1. Below, the calibration results of HGHFSs manufactured from stainless steel + nickel and chromel + alumel (Table 2.2, Fig. 2.14) are presented as examples. It can be seen from the graph (Fig. 2.14) that the characteristic of the HGHFS with the chromel + alumel composition is monotonic over the temperature range 250. . . 400 ◦ C, which is of the greatest interest for heat flux measurement in the boiler furnace; however the sensitivity of the sensor based on the stainless steel +

34

2 Gradient Heat Flux Sensors

Table 2.2 HGHFS calibration results Reference Heat flux Sensitivity Sensitivity temperature (W/m2 ) steel + nickel chromel + (◦ C) (mV/W) alumel (mV/W)

Experiment time (s)

Number of processed points (–)

3170 572 200 297 281

6340 1140 400 594 562

104 206 256 336 380

2600 5690 9670 15,200 22,500

Fig. 2.14 HGHFS’s calibration curves: 1—stainless steel + nickel; 2—chromel + alumel. Results are presented after averaging and removal of noise and harmonics induced by network of frequency (50 Hz)

1.120 1.620 1.400 1.180 0.844

0.599 0.562 0.461 0.443 0.422

S 0 , mV/W 1,2 1,0

1 2

0,8 0,6 0,4 0,2 0

100

200

300

400

T, oC

nickel composition is higher by almost an order of magnitude, making this HGHFS the best candidate for industrial experiments (see below).

2.4 Digital Signal Processing The following tasks have to be solved for measurement, recording, presentation, and use of GHFS signals: – measure the average level and fluctuations of the weak DC signal; – statistically handle a variable-frequency signal with selection of the average level, dispersion, and distribution in different experimental conditions; – estimate the relationships between area and sensitivity of the HFS, as well as between the expected level of heat flux and the capabilities of the data conversion system; – visualize measurements in a rather common form.

2.4 Digital Signal Processing

35

A system designed for operation with the GHFS specifically for the given experiment or in experiments close in signal properties typically solves these and many other tasks. Such a system can be enhanced with modern digital technologies, making it sufficiently flexible, but it is important to take into account the specifics of GHFSs themselves, to understand how the system comprising the heat flux, the sensor’s and the equipment functions. It follows from relations (2.28) that the GHFS signal is proportional to A, which is the sensor’s plan-area, for the given qz and b. It would seem evident the area A of the GHFS should be increased for measuring lower heat fluxes qz , and vice versa. However larger sensors are much more difficult and expensive to manufacture; moreover, a larger sensor cannot measure heat flux over small areas. In the modern experiments, sensor signals are usually converted into digital code using analog-to-digital converters (ADC). Hundreds of such devices are offered on the market but only few of them allow to measure microvolt signals reliably [12, 13]. PCL 818 HG, a 12-bit analog-to-digital converter produced by Advantech Co., Ltd. (USA), is an example of such a device. A built-in low-noise operational amplifier permits to amplify input signal by 1000 times before it is converted to digital. One of the main attributes of the ADC is the number of bits, defines the minimal difference in signal values that is measured by a system. So, a 12-bit ADC can form 212 = 4096 different values of output signal (code). If the measuring limit at the output is 5 V, the resolution is 5/4096 = 1.221 × 10−3 V. When the scheme comprises an operational amplifier with the transfer coefficient K y = 1000, the minimal level of input signal is E 0 = 1.221 × 10−6 V. A 24-bit ADC, AD7714 by Analog Devices (USA), allows to measure the signal E 0 = 3/224 = 0.179 × 10−6 V at a full scale of 3 V (the additional amplification cascade is not taken into account in this case). In 2003, 32-bit ADCs first appeared, which proved to be redundant: E 0 ≈ 7 × 10−10 V for these ADCs, less than the thermal noise level. On the other hand, the E 0 must essentially exceed thermal noise in metals E ∗ that approaches 10−9 V at a temperature up to 500 K [14, 15]. Developers and manufacturers of equipment usually require for the signal-to-noise ratio to be no less than 6 dB per each ADC bit., i.e., the condition EE0∗ = 100.6k must be satisfied. Thus, the resolution of the scheme should be in the interval E max  E 0  100.6k × E ∗ , K y × 2k

(2.32)

where E max is the upper limit of the ADC output signal E 0 . In view of the above, let us consider non-strict equality (2.24) in a “truncated” form E max . (2.33) E0 = K y 2k The GHFS signal E GHFS = S0 q Akmin should exceed E 0 . So E GHFS = S0 q Akmin = 10n 3 E 0 ,

(2.34)

36

2 Gradient Heat Flux Sensors

where n 3 is the reserve coefficient determined by the requirement for measuring accuracy (in practical terms, n 3 = 2…3). It follows from formulae (2.33) and (2.34) that the minimal GHFS area is found from the equation (2.35) E GHFS = S0 q Akmin = 10n 3 E 0 , and the required number of ADC bits is   Akmin K y S0 q . k  3.321 × n 3 − lg E max

(2.36)

Let’s consider an example where the expected heat flux density q = 10 W/m2 , the measuring channel uses a PCL 818 HG-type ADC and an operational amplifier (k = 12; E max = 5 V; K y = 103 ), while the GHFS’s volt-watt sensitivity is S0 = 10−2 V/W. Moreover, let us assume that n 3 = 2. Calculation by formula (2.36) yields 2 A12 min = 1222 mm . If such a scheme uses an AD7714-type ADC (E max = 3 V; k = 24), it is found that 2 A24 min = 0.18 mm . It is impossible yet to produce a GHFS of such small dimensions, but the obtained result means that at k = 24 the heat flux q = 10 W/m2 can be reliably measured by practically any of the existing GHFSs whose dimensions are more than 1 × 1 mm. Judging by the level of modern technology, we can assume that Amin = 10−6 m2 , which implies using (see formula (2.36)) an ADC with the following number of digits.   10−6 × 103 × 10−2 × 10 = 21.5 ≈ 22, k = 3.321 × 2 − lg 3

(2.37)

(here it was assumed that E max = 3 V, although the type of ADC was not known before; calculation can be refined by the step-by-step approach, but the result essentially does not vary). Using digital techniques and software of last generations, in particular, TEKTRONIX oscilloscopes allows, for example, to visualize the heat flux field by the readings from a small number of GHFSs as initial data (see Sect. 5.2, describing the solution of a problem on local heat flux in a spherical dimple). Such methods apply in all similar cases, and their results are particularly important. Recall that a GHFS: – generates “smooth” thermopower E x ; for a bismuth-based GHFS it is practically linearly related to the heat flux Q z over the sensor cross-section, and the slope of the straight line E x (Q z ) does not depend on the temperature within the greater part of the working range; – has a “small” (by our estimates) electric resistance of the order of 0.1. . . 10  depending on the design and dimensions of the GHFS. (Exceptions are siliconbased HGHFSs, but the for them resistance does not exceed 1. . . 2 k).

2.4 Digital Signal Processing

37

It is important that processing of GHFS signals does not differ in principle from that of thermocouple signals. The existing equipment does not require readjustment, additional amplification channels, etc. In 2000–2002, we constructed a scheme of digital processing of GHFS signals in a large-scale experiment on free convective heat transfer in vertical tubes (see Sect. 6.2). The improved version of the described measuring system was implemented in industrial experiment at the Elektrosila manufacturing plant (St. Petersburg) in late 2002 (Sect. 6.3). Thus, neither is measurement of GHFS signals a problem now, nor will it be in the future. The possibilities offered by digital technologies allow to change the approach to experiment, to start monitoring heat transfer. The amount of information clearly exceeds the level “established beforehand” and is limited by the capacity of the digital channel. It is important from a methodological standpoint that the entire set of measurements is taken in the course of one experiment, and the results can answer those important (as is usually the case) questions that arise only in analyzing results, formulating similarity equations, etc. [16].

References 1. Divin, N., Kirillov, A., & Sapozhnikov, S. (1996). Gradientenartige Messgeber fur die Messung des Warmestromes. Messtechnik zur Undersuchung von Vorgangen in thermischen Energieanlagen. XXVIII. Kraftwerkstechnisches Kolloquium und 6 (pp. 155–160). Dresden: Kolloquium Messtechnik fur Energieanlagen. 2. Divin, N., & Sapozhnikov, S. (1995). Gradiyentnyye datchiki teplovogo potoka: prilozheniye dlya issledovaniy tepla (Gradient heat-flux transducers: Application for heat investigations). In Proceeding of International Symposium in Power Machinery, Moscow (p. 79). 3. Pilat, I. M., Samoylovich, A. G., & Anatychuk, L. I. (1969). Termoelement. USSR Patent 230915, 13 Sept 1969. 4. Ordin, S. V., & Shelykh, A. I. (2002). Elektronnoye otrazheniye i zonnaya struktura vysshego silitsida margantsa (Electron reflection and band structure of higher manganese silicide). In Thermoelectrics and their application: Tr. FTI them (p. 34). A. F. Ioffe RAS. 5. Ordin, S. V. (1996). Radiation receiver. Application No. 93036965/25 of 03/20/1996. // S. V. Ordin, Russian Agency for Patents and Trademarks. 6. Anatychuk, L. I. (1969). Poluprovodniki v ekstremal’nykh temperaturnykh usloviyakh (Thermoelements and thermoelectric devices) (p. 768). Dumka, Kiev: Science. 7. Anatychuk, L. I., & Bulat, L. P. (2001). Poluprovodniki v ekstremal’nykh temperaturnykh usloviyakh (Semiconductors in extreme temperature conditions) (p. 224). St. Petersburg: Nauka. 8. Oparichev, A. B. Issledovaniye naklonnokondensirovannykh plonochnykh materialov dlya termoelektricheskikh preobrazovateley lazernogo izlucheniya (Research of inclined-condensed film materials for thermoelectric converters of laser radiation). Dissertation, National Research University “Moscow Power Engineering Institute”. 9. Pshenai-Severin, D. A., Ravich, Yu. I., & Vedernikov, M. V. (2000). Iskusstvenno anizotropnyy termoelektricheskiy material s poluprovodnikovymi i sverkhprovodyashchimi sloyami (Artificially anisotropic thermoelectric material with semiconductor and superconducting layers). Physics and Technology of Semiconductors, 34(10), 1265–1269.

38

2 Gradient Heat Flux Sensors

10. Mityakov, A. V. (2000). Gradientnyye datchiki teplovogo potoka v nestatsionarnoy teplometrii (Heat flax sensors in non-stationary heat metering) (p. 134). Dissertation, Saint-Petersburg State Technical Universit. 11. Sapozhnikov, S. Z., Mityakov, V. Y., & Mityakov, A. V. (1997). Teplometriya v tsilindre dvigatelya vnutrennego sgoraniya s ispol’zovaniyem gradiyentnykh datchikov teplovogo potoka (Thermometry in the cylinder of an internal combustion engine using gradient heat flow sensors). News of higher educational institutions and energy CIS. Energy, 9–10, 53–57. 12. Brindley, K. (19991). Izmeritel’nyye preobrazovateli (Measuring transducers) (p. 144). Moscow: Energoatomizdat. 13. Goll, P. (1999). Kak prevratit’ personal’nyy komp’yuter v izmeritel’nyy kompleks (How to turn a personal computer into a measuring complex) (p. 144). Moscow: DMK. 14. Van der Zil, A. (1979). Shumy pri izmereniyakh (Noises during measurements) (p. 294). Moscow: Mir. 15. Mac-Donald, D. (1964). Vvedeniye v fiziku shumov i fluktuatsiy (Introduction to the physics of noise and fluctuations) (p. 158). Moscow: Mir. 16. Vinogradova, N. A., Gaiduchenko, V. V., Karjakin, A. I., Sviridova, V. G., et al. (Eds.). (2004). Osnovy postroyeniya informatsionno-izmeritel’nykh sistem: posobiye po sistemnoy integratsii (Fundamentals of building information-measuring systems: A manual on system integration) (p. 268). Moscow: MPEI.

Chapter 3

Transient Heat Flux Measurements

3.1 GHFS Response Time Heatmetry in transient processes is of primary interest because heat flux is subjected to fluctuations caused by turbulent convective heat transfer, environmental temperature instability, and other factors even in processes assumed to be steady-state [1–3]. Therefore, we can assess how accurately experimental data reflect the actual heat flux variation by taking into account thermal inertia of the HFS. For most of sensors, response time is determined in “artificial” thermal regimes. The main requirement in this case is to be able to record external disturbances. The response time τmin numerically equal to the sensor signal attains (ex p1)−1 ≈ 0.632 the stationary value time (here the leading edge of a disturbing signal is considered to be vertical). The response time of a thermocouple-type sensor [4, 5] is estimated analytically by heat conduction theory. Thermal models include the sensor’s installation specifics, as well as the convective heat transfer at its “free” surface. In this case “free” surface is the back surface of the HFS opposite to the disturbing action of the heat flux (Fig. 3.1a, c), or the back surface of the massive plate (Fig. 3.1b) where the sensor is mounted. It is established in monograph [5] that the response time under such thermal boundary conditions is determined by the relations – for the Fig. 3.1a

2 + Bi 1 ; 2Bi 1

(3.1)

η1 (1 + χ1 η1 ) + Bi 2 [η1 + 21 χ1 (1 + η1 2 )] ; Bi 2 × χ1

(3.2)

Fo1 min = – for the Fig. 3.1b Fo1 min = – for the Fig. 3.1c

© Springer Nature Switzerland AG 2020 S. Z. Sapozhnikov et al., Heatmetry, Heat and Mass Transfer, https://doi.org/10.1007/978-3-030-40854-1_3

39

40

3 Transient Heat Flux Measurements

Fig. 3.1 Thermal models for response time calculation in thermocouple-type HFS [5]: a mounting without substrate; b mounting on plate; c mounting with protective coating. Numbers correspond to: 1—HFS and 2—plate

(a)

q

k1,

1

1

k 2,

, c1

1

(b)

, c2

2

2

q

1

k1,

1

1

, c1

2

(c)

2 k2,

2

, c2

2 k2,

2

q , c2

1

2

Fo1 min =

1

k1,

1

, c1

η1 (1 + χ1 η1 ) + Bi 2 [η1 + 21 χ1 (1 + η1 2 )] . Bi 2 × η1 2 χ1

(3.3)

The following notations are used in formulae (3.1)–(3.3). 1 2 Here Fo1 min = a1δτ2min ; Fo2 min = a2δτ2min ; Bi 1 = hδ ; Bi 2 = hδ and h is the heat k1 k2 1 2   k ρ c δ2 a1 transfer coefficient at the “free” (cold) surface; η1 = δ1 a2 ; χ1 = k21 ρ21 c12 . The system is clear from Fig. 3.1. In the physical realm, the dimensionless quantities is common. Thus, the response time τmin grows as the thickness of the HFS (δ1 or δ2 ) increases. For the modern thermocouple-type HFSs, τmin is about seconds or tens of seconds (Fig. 1.10). In literature transverse-type HFSs response time descriptions are few and not consistent. It is argued, that signal of a GHFS does not depend on its thickness, that’s why it is possible to infinitely reduce the sensor response time by decreasing its thickness,

3.1 GHFS Response Time

41

1 2

8

5

3

4

6

7

Fig. 3.2 Experimental setup scheme for studying response time of GHFS. Numbers correspond to: 1—laser, 2—laser beam, 3—mirror, 4—GHFS, 5—laser detector, 6—photodiode, 7—aluminium substrate layer, 8—oscilloscope

without reducing its sensitivity [5]. This assessment is undoubtedly valid for steadystate conditions but it should be proved for the initial stage of the process. In [6], the inertia in HFS is estimated when the heat flux at the sensor surface varies harmonically. The authors select an active layer of thickness (2k1 ρ1 c1 ω)0.5 (here ω is the frequency of heat flux fluctuations), pointing out that the “remaining” thickness of the HFS “creates a parasitic load on the active layer which decreases the resulting thermal EMF” [6]. It is established that the HFS signal coincides with the disturbing action in frequency but delays in phase. The reason is that the relations for the longitudinal-type HFS remain valid in case of step changes in heat flux; the thickness of the HFS is the characteristic dimension in the Fourier and Biot numbers. In our opinion, a valid model for the thermal EMF generated by the GHFS can be constructed using the methods of solid state physics, which is beyond the scope of this work. So, we describe the experiments, that estimate the order of for the GHFS based on different compositions. Also there were used heat conduction theory to interpret the results obtained. At the first stage of our studies (in 1996) we assumed that the response time of GHFSs depends on their thickness. Therefore, GHFSs of different thickness (from 0.2 to 4.0 mm) were used in experiments [7]. This assumption was not confirmed (see below), but the general idea of the experiment was chosen correctly. The experimental setup for investigating the behavior of GHFS is shown in Fig. 3.2. Delta-201 laser 1 operated in Q-switching mode with the following parameters [8]. The wavelength of 1.06 μm; pulse time of 0.15 ms; power of 1250 kW/m2 ; pulse interval of 60 ms.

42 Fig. 3.3 GHFS signal with OGM-20 laser (ruby laser with 720 J power) operating in single-pulse mode

3 Transient Heat Flux Measurements E, mV

E, mV 200 100

200

0 2,5 2,6 2,7 2,8

, µs

190 ns 100

0

2

4

6

8

, µS

Beam 2 with an initial diameter of 0.1 mm passed from laser 1 through optical system and was deflected by mirror 3; following that, beam 2 was set to the diameter of 15 mm and traveled to photodiode 6, GHFS 4 and laser radiation detector 5. All these elements were fixed on base 8 with thermal grease and connected to storage oscilloscope 8 of model C1-94. Notice that photodiode 6 with a response time of approximately 1 μm was used in the scheme for starting up oscilloscope 8 virtually simultaneously with the start of irradiation. At first, the experiments involved a 4 × 7 × 0.2 mm GHFS with a volt-watt sensitivity of 9.3 mV/W. The sensor surface was coated with a special composition with an emissivity of 0.98. Nest, six other GHFSs with the thicknesses of 0.5, 1.0, 1.5, 2.0, 2.5, and 4.0 mm, with the same plan dimensions (4 × 7 mm), and the same surface emissivity were considered one after the other. The experimental results are identical for all sensors. Thus, the initial assumption was not confirmed: the response time did not depend on the GHFS’s thickness. In our early works [9], we assumed that the response time τmin ≈ 0.05 ms to be reliably established, and there was no doubt that the heat flux changed its value abruptly. However, photodiode 6 only served for starting up oscilloscope 8 and the condition qmax ≈ const could only be estimated from the laser’s 1 operation description. We subsequently managed to confirm that the quantity τmin = 0.05 ms was overestimated by several orders of magnitude. This was associated with slow response of oscilloscope 8. A new series of experiments1 led to this conclusion. The experimental setup differed from the one in Fig. 3.3 only by the absence of mirror 4: GHFS 6 on a special holder and photodiode 6 were positioned on the optical axis of laser 1. In experiments, an OGM-20 ruby laser with a radiation wavelength of 694.3 nm and an FD-2 photodiode were used. At the first stage, GHFS and photodiode signals were recorded by a C8-17 oscilloscope. The pulse time in single-phase mode (Fig. 3.3) was equal to about 30 ns at a radiation power of about 107 W (a semitransparent lavsan film screen was placed at a distance of 10 mm from the receivers. That why they would not be damaged by radiation). As seen from Fig. 3.3, the duration of 1 Prof. S. V. Bobashev, Dr. N. P. Mende, and senior researcher V. A. Sakharov from the Ioffe Institute of the Russian Academy of Sciences participated in the studies.

3.1 GHFS Response Time Fig. 3.4 GHFS signal with OGM-20 laser operating in free generation mode: 1—corresponds to GHFS, 2—to FD-2 photodiode

43

E, mV 1 20

2

15 10 5

0

100

200

300

400

, µS

the both stages, heating and cooling, does not exceed 100 ns. The leading edge is so steep that it is impossible to estimate the time constant in a “classical” manner in terms of exponential approximation. But it is not important, since the magnitude itself is valuable: characteristic frequencies for the heat flux can reach the level of 105 …106 Hz with the response time of 10−8 s, which previously could only be achieved in ALTP sensors (see Sect. 1.2). Figure 3.4 shows the results of one of the experiments from the second series when the laser pulse was 400 μm in free oscillation mode. The FD-2 photodiode was not subjected to special calibration and operated as a detector; the GHFS had the volt-watt sensitivity of 9.3 mV/W, its surface was not calibrated for emissivity. The experiment was not aimed at heatmetry. Only the dynamics of signals was estimated. The ripples appear almost simultaneously on both curves, which submits the early estimating for the GHFS response time. The GHFS signal is slightly delayed on the ascending branch of the curves and considerably delayed on the descending one, where it becomes smoother and approaches an exponent in shape. An explanation for this cooling pattern is that GHFS’s inertia is more substantial than the photodiode. At the next stage, we managed to use a modern high-speed “TEKTRONIX” oscilloscope, which substantially increased the quality of the data obtained in the experiment. Figure 3.5 shows heat flux in both generation modes (on different scales along the time axis). It is seen that the GHFS is much more precise than the photodiode and shows how the power varies even over the nanosecond range. Measurements with a maximum time resolution showed (Fig. 3.6) that the GHFS describes the process specifics for times of 10−7 s, which was by 1–2 orders less than the characteristic times for most heat transfer processes. It can be seen that GHFS is much more accurate than a photodiode, since it shows a power change in the nanosecond range. Maximum temporal resolution measurement showed that GHFS allows us to detail the process at time about 10−7 s. This is 1. . . 2 orders of magnitude less than the time characteristic of most heat transfer processes. So, bismuth-based GHFS time constant has a level of 10−8 . . . 10−9 s. This makes GHFS on a practical level a non-inertia equipment for thermal engineering research. The HGHF’s response time were found in a similar manner. High thermal conductivity of these sensors required a higher level of the disturbing heat flux. We

44 Fig. 3.5 Example of screenshot (photographed) from TEKTRONIX oscilloscope with synchronized signal recording: a GHFS and FD-2 photodiode, nanosecond pulse; b GHFS and photomultiplier, free oscillation mode

3 Transient Heat Flux Measurements

(a) GHFS

2V

1 µs

Photodiode FD-2

(b) 200 mV

GHFS

50 µs

Photomultiplier

GHFS

used a double-pulse Nb:YAG laser with a wavelength of 635 nm, a pulse energy of 50–120 mJ and a pulse repetition frequency of 1–10 Hz as the source of disturbing heat flux. Stainless steel + nickel and silicon + aluminum HGHFSs served as objects of investigation. Typical oscillograms (Fig. 3.7) demonstrate that the response time for these sensors has the above-mentioned order of magnitude, and the response of silicon-based HGHFSs is faster than that of bismuth-based ones by about an order of magnitude. The reliability of the results is confirmed by comparison with the other researchers data, among other things. Experiments by Zahner et al. on calibrating a HFS with synthetic anisotropy [10] showed the linear behavior of the characteristics up to 1 MHz. The HFS model had plan dimensions of 8 × 6 mm and was cut out at an angle θ = 15…35◦ . The sensor with a layer slope angle of 35◦ and a thickness of 10 μm was irradiated with a pulse laser (wavelength of 1064 nm, pulse time of 15 ns). Signal amplitude at an energy per unit area of 0.5 mJ/cm2 amounted to 1 mV and attenuation occurred within several μm (Fig. 3.8).

3.1 GHFS Response Time

45

Fig. 3.6 Heatmetry results during nanosecond pulse (screen-shot from “TEKTRONIX” oscilloscope photographed)

(a)

(b)

Fig. 3.7 Oscillograms of GHFS response time: a for bismuth-based sensors (100 mV/div, 10 ns/div); b for n-silicon + p-silicon compositions (50 mV/div, 100 ns/div)

Characteristics of ALTP sensors were determined for Y BC O films on a SrTiO3 substrate (see Fig. 1.5) [11]. Film thickness varied from 35 to 400 nm, while the slope angle of the layers was about 10◦ . Irradiation was carried out at a wavelength of 308 nm with a pulse time of 30 ns. The signal of the model silicon detector is shown at the bottom of Fig. 3.9. The signals of the ALTP sensor with different thickness films are given at the top (with a shift, but on the same scale along the ordinate axis). Here the dashed lines stand for the calculated curves; as we can see, the agreement is fairly good except for the 400 nm thick film where the theory predicts a more steep decrease of the signal. The authors attribute this to the presence of the substrate that was neglected in the calculation: such a divergence is absent in a more thermally conductive MgO substrate even for 500 nm thick films.

46 Fig. 3.8 Signal for 10 μm thick HFS irradiated with pulse laser [10]

3 Transient Heat Flux Measurements E, mV 0.6 0.4 0.2 0.0 0

2

1

Fig. 3.9 Pulse laser signals for several YBCO films on SrTiO3 substrate [11]

, µs

Film 400 nm

Output voltage

Film 250 nm

Film 80 nm

Film 35 nm

Laser signal

0

50

100

150

, ns

The authors of this study participated in experiments on comparative calibration of dynamic characteristics of bismuth-based HGFS and ALTP sensors [12] in a hypersonic wind tunnel IT-302 at the Khristianovich Institute of Theoretical and Applied Mechanics of the Siberian Branch of Russian Academy of Sciences (Novosibirsk). Both sensors were subjected to preliminary stationary calibration at the Institute of Aerodynamics and Gas Dynamics (Aerospace Engineering and Geodesy, University of Stuttgart, Stuttgart, Germany). Sensors were mounted on a plate (Fig. 3.10) in supersonic flow at a zero angle of impact at a distance of 183 mm downstream from the sharp edge. The distance between the sensors was 16 mm in the transverse direction, implying that the unsteady flow conditions were identical in the boundary layer near both sensors.

3.1 GHFS Response Time Fig. 3.10 Location of HFSs in experiments [12]

47

183 mm w

Fig. 3.11 Signals of GHFSs (curve 1) and ALTP sensors (curve 2) obtained during response time calibration

16 mm

ALTP GHFS

q, W/cm2 1 2

80 60 40 20 0

Fig. 3.12 Results for GHFS response time calibration obtained at Kutateladze Institute of Thermophysics, Siberian Branch of the RAS [12]

1

2

, ms

3

GHFS Laser

E, V 0,04 0,00 -0,04 -0,08 -0,12 0,00

0,01

0,02

0,03

0,04

,s

The ALTP sensor’s thickness was equal to 0.5 × 10−6 m; that of the bismuth-based GHFS to 2 × 10−4 m; both dimensions were close to the technologically attainable minimum. Figure 3.11 illustrates the time dependences of heat flux calculated by the signals from the ALTP sensor and the GHFS and by the results of stationary calibration. The heat flux calculated by the GHFS signal (curve 1) has a substantially smaller amplitude and a less sharp leading edge compared with that calculated by the ALTP sensor signal (curve 2). Since the GHFS has higher thermal inertia due to its thickness, the heat flux spectrum has no high-frequency fluctuations that are clearly observed in the signal from the ALTP sensor. Another methodical approach was implemented in experiments on bismuth-based GHFSs at the Institute of Thermophysics of SB RAS [12] (Fig. 3.12). A CO2 laser with a continuous power of 25 mW, generating rectangular pulses was used. The GHFS’s response to a rectangular disturbing pulse is shown in Fig. 3.13 on a larger scale. The GHFS’s frequency response is illustrated in Fig. 3.14. The

48 Fig. 3.13 GHFS responses time to rectangular disturbing pulses: a at frequency of 0.1 Hz; b at frequency of 20 Hz

3 Transient Heat Flux Measurements

(a) Laser GHFS

E, V 0,25 0,20 0,15 0,10 0,05 0,00 -0,05

0

2

4

6

8

(b)

,s

10 Laser GHFS

E, V 0,05 0,00 -0,05 -0,10 0,00

Fig. 3.14 Frequency response of bismuth-based GHFS [12]

0,01

0,02

0,03

0,04

,s

1 0,9 0,8

A/A0

0,7 0,6 0,5 0,4 10

100

1000

f, Hz

comparison bismuth-based GHFSs and ALTP sensors frequency response is given in Fig. 3.15. The comparison of the curves indicates that the GHFS has a narrower frequency range (∼1 kHz) compared to the ALTP sensor (∼1 MHz). Presumably, this gap can be reduced or overcome by decreasing the GHFS’a thickness up to 10−6 m. While such processing is fundamentally impossible for bismuth-based GHFSs, it is quite feasible for HGHFSs.

3.1 GHFS Response Time

49

Fig. 3.15 Comparison of frequency response of GHFS and ALTP sensor made in [12]

1 0,9

GHFS ALTP

0,8

A/A 0

0,7 0,6 0,5 10

1000

10000

f, Hz

z T1 T2 T3 ...

Fig. 3.16 Isotherms T1 , T2 , T3 . . . Ti in ATE [13] h and l are thickness and length of element, respectively; T1 > T2 > T3 > · · · > Ti are temperatures over ATE cross-section

100

Ti x l

Thus, the volt-watt sensitivity of composite-based GHFSs is not inferior to the characteristics of sensors made of natural anisotropic materials and even exceeds those of semiconductor-based ones. The absence of a relationship between the response time of the GHFSs and their thickness is anomalous from a traditional standpoint and cannot be fully explained within the framework of theory of heat conduction. However, we can still use some of the possibilities offered by this theory.

3.2 Thermal Model of GHFS It is essential to establish the degree to which it would be valid to ignore anisotropic heat conduction of a medium in further considerations. It was first assumed in [13] that the slope of isotherms over the cross-section of an anisotropic single crystal was extremely small (Fig. 3.16). From a physical standpoint, it was clear that it was the slope of the lines T 1 , T 2 , T 3 , etc., that allowed for the crystal to generate thermoEMF along the x axis, but this slope was inessential for the entire ATE with the l × h cross-section. Numerical simulation [9], where the finite-element method was adopted, showed similar locations of the calculated isotherms (Fig. 3.17). It can be seen from these figures that heat conduction anisotropy noticeably curves the isotherms only at the edges of the ATE, while the temperature field in the middle

50

3 Transient Heat Flux Measurements

Fig. 3.17 Numerical simulation of temperature field in anisotropic plate by finite-element method. Left: temperature scale (in K)

part of the crystal is close to the one in the isotropic medium. Thus, the boundary conditions Fourier problem can be the basis of the model. Bearing in mind that radiation flow is absorbed in a very thin (several micrometers) surface layer of the GHFS, let us consider the second boundary conditions Fourier problem for an infinite plate or a semi-infinite body. The GHFS is seldom heated by more than several degrees in unsteady-state experiments (see above), so it is reasonable to confine the consideration to the linear approximation. Following [5, 13] dealing with heat conduction processes in the thermocoupletype HFS, let us consider several models of single- and two-layer systems and further determine which one is better for analysis. Let us examine three models: a semi-infinite body, an infinite homogeneous plate and an identical composite plate whose lower layer has considerably larger thermal conductivity than the upper one [5, 14, 15]. They correspond to the cases when the GHFS does not “detect” the plate (Fig. 3.18a), when the lower surface of the GHFS is at constant temperature (Fig. 3.18b) and when the GHFS with the thickness δ1 is mounted on a plate with the thickness δ2 , whose thermal conductivity greatly exceeds that of the GHFS itself (Fig. 3.18c). The heat equation ∂T ∂2T =a 2, ∂τ ∂x

(3.4)

is complemented with one of the three sets of boundary conditions [9, 14, 16, 17]

3.2 Thermal Model of GHFS Fig. 3.18 Models of heat conduction in GHFS: a semi-infinite body, b plate, c plate with bottom layer

51

(a)

q T 0

x

T(x,0) = T0 k1,

1

k1,

1

, c1

T = T0

q

(b) 0

T(x,0) = T0

1

, c1

T(x= 1) = T0 x

(c)

x1

1

q

T(x,0) = T0 0

2

x2

– for a semi-infinite body

1 k1,

1

, c1

2 k2,

, c2 2

T

T = T0

⎧ T (x, 0) = T0 ⎪ ⎪ ⎪ ∂ T (0,τ ) ⎨ k ∂x + q = 0 ∂ T (∞,τ ) ⎪ =0 ⎪ ⎪ ⎩ ∂x T (∞, τ ) = T0 .

(3.5)

– for an infinite homogeneous plate ⎧ ⎪ ⎨T (x, 0) = T0 ) k ∂ T∂(0,τ +q =0 x ⎪ ⎩ T (δ1 , τ ) = T0 .

(3.6)

52

3 Transient Heat Flux Measurements

– for an infinite composite plate, provided that k2 >> k1 ⎧ ⎪ ⎨T1 (x, 0) = T2 (x, 0) = T0 ) k ∂ T∂(0,τ +q =0 x ⎪ ⎩ T2 (δ2 , τ ) = T0 .

(3.7)

Solutions of problems (3.4)–(3.6) [18, 19] take the form, respectively  



∞  x q0 x x 2 a1 τ exp −μn 2 − An sin μn 1 − T = T0 + 1− , k1 δ1 n=1 δ1 δ1

(3.8)





 ¯ ∞  x a1 τ q0 x x 2 exp −μn 2 − An sin μn 1 − 1− T = T0 + . k1 δ1 n=1 δ1 δ1

(3.9)

    



   δ12 a1 τ x 2 x 2 q 0 δ1 δ1 2 1− 1+ T = T0 + exp − + exp − − × 2 k π δ1 2 4a1 τ δ1 4a1 τ δ1

 

 



2c2 ρ2 δ2 x x δ1 δ1 x x + − 1− erfc √ − 1+ erfc √ + 1− 1+ δ1 2 a1 τ δ1 δ1 c1 ρ1 δ 1 2 a1 τ δ1 

 

x c1 ρ1 δ 1 2c2 ρ2 δ2 c 1 ρ1 δ 1 2 a 1 τ × exp 1 + + × + c1 ρ1 δ 1 δ 1 c2 ρ2 δ 2 c2 ρ2 δ 2 δ12  

√ c1 ρ1 δ 1 a 1 τ δ1 x × erfc √ + 1+ 2 a1 τ δ1 c2 ρ2 δ 2 δ 1

where T0 is the initial temperature; c1 , ρ1 , δ1 are the heat capacity, the density, the thickness of the GHFS (and the upper part of the composite plate), respectively; c2 , ρ2 , δ2 are the heat capacity, the density, the thickness of the lower part of the composite plate, respectively; An = (−1)n+1 μ22 is the amplitude function; μn = n μ . (2n − 1) π2 are the roots of the characteristic equation ctgμ = Bi In a dimensionless form, the relations (3.6) have the form  

 1 1 Fo exp − − erfc √ θ = Ki 2 , (3.10) π 4Fo 2 Fo  θ = Ki 1 − ηx −

∞ 

 2 2  An sin [μn (1 − ηx )] exp −μn ηx Fo .

(3.11)

n=1 0 where θ = T −T is the dimensionless excess temperature; K i = kq10Tx0 is the modified T0 Kirpichev number (dimensionless depth); Fo = ax12τ is the Fourier number (dimensionless time); ηx = δx1 is the dimensionless thickness; χ 2 = cc12ρρ12 δδ21 is the dimen-

3.2 Thermal Model of GHFS

53

sionless heat storage capacity ratio of the upper and lower parts of the composite plate. We choose an aluminum plate (c2 = 900 J/(kg K); ρ2 = 2700 kg/m3 ; δ2 = 5 × −3 10 m) as a fairly typical substrate for calculating the dimensionless temperature by formula (3.7). In other cases, we take the effective thermophysical properties of the GHFS calculated in our study [9] and set its thickness to δ1 = 0.2 × 10−3 m. Calculation by all three formulae (3.7) yields practically the same results for the Fourier number ranging from 5 × 10−3 to 2 × 10−1 , where the maximum value is known to overlap the time of heating (Fig. 3.19). This means that the plate has little effect on temperature field formation in the surface (“hot”) layer of the GHFS, the same as the sensor thickness h1 that substantially exceeds the thickness of this layer. In view of this, let us examine the simplest boundary conditions (3.4)–(3.5) and its solution (3.7). The thickness of the layer was estimated, where the GHFS signal is formed, using the integral heat balance method [20]. According to this method, heat does not propagate beyond the so-called thermally heated layer (this is equivalent to the assumption that the heat propagation velocity is finite, although the question about the magnitude of this velocity is remains unanswered).

Fig. 3.19 Temperature distribution in GHFS: a θ(K i) curve for different Fourier numbers; b θ(Fo) curve for different Kirpichev numbers

(a)

0.06

Fo=0.02 Fo=0.015 Fo=0.01 Fo=0.005

0.04 0.02

0

0.1

0.2

0.3

0.4

Ki

(b) Ki=0.1

0.06 0.04

Ki=0.2 0.02

0

Ki=0.3 Ki=0.4 .4 0.005

0.010

0.015

Fo

54

3 Transient Heat Flux Measurements

Within the framework of problem (3.4)–(3.5), let us impose the condition that its solution should satisfy not the initial heat conduction equation but the averaged one. Integrating Eq. (3.6) by the heated layer thickness δ0 , we obtain the heat balance integral  δ0  δ0 2 ∂T ∂ T d x. (3.12) dx = a ∂τ ∂x2 0 0 Leibniz’s formula was applied to it d dτ



β(τ )

α(τ )

 f (x, τ )d x =

β(τ )

α(τ )

dβ dα ∂f d x + f [β(τ ), τ ] − f [α(τ ), τ ] . (3.13) ∂τ dτ dτ

Relation (3.8) takes the form 

 δ0  d d  ∗ ∂T dδ0 dx = = θ − T (x, 0) × δ0 , T (x, τ )d x − T (δ0 , τ ) ∂τ dτ 0 dτ dτ 0 (3.14)  δ where θ ∗ = 0 0 T (x, τ )d x. The right side of equality (3.8) is represented in the form

  δ0 ∂ T (δ0 , τ ) ∂ T (0, τ ) ∂ ∂T dx = a − . (3.15) a ∂x ∂x ∂x 0 ∂x δ0

After substituting integrals (3.10) and (3.11) into the initial equality (3.8), we obtain  d  ∗ ∂ T (δ0 , τ ) ∂ T (0, τ ) θ − T (x, 0) × δ0 = a − . (3.16) dτ ∂x ∂x The initial boundary conditions is complemented with those on the boundary x = δ0  T (δ0 , τ ) = T (x, 0) (3.17) ∂ T (δ0 ,τ ) = 0. ∂x Solution to problem (3.12)–(3.13) is sought in the form of a quadratic polynomial T (x, τ ) = b0 (τ ) + b1 (τ )x + b2 (τ )x 2 ,

(3.18)

where coefficients b0 , b1 and b2 depend on τ and are yet unknown. They are with using the boundary conditions

3.2 Thermal Model of GHFS

55

⎧ 2 ⎪ ⎪ T (δ0 , τ ) = T (0, τ ) = b0 + b1 δ0 + b2 δ0 ⎨ dT (δ0 ,τ )  = 0 = b1 + 2b2  dx x=δ0 ⎪ ⎪ ⎩ −kb = q. 1

therefore, b1 = − qk ; b2 =

b1 2δ0

q = − 2kδ ; 0

b0 = T (δ0 , τ ) − b1 δ0 − b2 δ02 = T (δ0 , τ ) +

qδ0 k

−

qδ02 2kδ0

= T (δ0 , τ ) +

qδ0 . 2k

The unknown polynomial (3.15) assume the form T (x, τ ) = T (δ0 , τ ) +

 qx qx2 qδ0 q  2 − + δ0 − 2δ0 x + x 2 = = T (δ0 , τ ) + 2k k 2kδ0 2kδ0 q = T (δ0 , τ ) + (3.19) (δ0 − x)2 . 2kδ0

Variable δ0 that is part of equality (3.15) is determined from the heat balance integral: 

 δ0 q T (x, τ )d x = T (δ0 , τ ) × δ0 − θ = (δ0 − x)2 d x = 2kδ0 0 0  δ0 q = T (h 0 , τ ) × δ0 − (δ0 − x)2 d (δ0 − x) = 2kδ0 0 q qh 0 = T (h 0 , τ ) × δ0 − (δ0 − x)|δ00 = T (h 0 , τ ) × h 0 + 2kδ0 × 3 6k ∗

δ0

(3.20)

Taking into account the substitutions made, heat balance integral (3.15) assume the form  ⎧  qδ02 d ⎪ T (δ = a qk , τ ) × δ − − T (δ , τ ) × δ 0 0 0 0 ⎪ dτ 6k ⎨   qδ0 d = aq dτ 6k k ⎪ ⎪   ⎩1 d 2 qδ = aq. 0 6 dτ Since h0 = 0 at the initial moment (τ = 0), we obtain  q(τ )δ02 = 6a

τ 0

q(τ )dτ.

56

3 Transient Heat Flux Measurements

Table 3.1 Characteristics of sensors Sensor type

δ0 (m)

a (m2 /s) 10−6

6.0 × 8.16 × 10−6 4.7 × 10−6 1.92 × 10−6

Battery GHFS based on bismuth single crystals HGHFS based on nickel + stainless steel HGHFS based on chromel + alumel HGHFS based on silicon + aluminum

6.0 × 10−7 7.0 × 10−7 5.3 × 10−7 3.4 × 10−7

The thickness of the heated layer  δ0 =

6a q(τ )



τ

q(τ )dτ

21

,

(3.21)

0

with q = const, the equality (3.16) has the form δ0 =

√ 6aτ .

(3.22)

Assuming from the results of Sect. 3.1 that τ ≈ 10−8 s, we obtain the results summarized in Table 3.1 for different types of sensors. As we can see, all values of δ0 exceed the level of (1…2) × 10−7 m, where thermoelectric phenomena are possible [21]. Now let us compare the capabilities of longitudinal and transverse-type HFSs in steady and unsteady heat flux measurements. Conditions of measurement and calibration, and other features of HFSs of both types are close in steady-state regime, but there are still differences between them. The larger the signal E | , the higher is the temperature difference T| across the HFS thickness δ. This means that the sensor must be “thermally thick”, which increases the temperature field distortions and, consequently, the methodical error in determining the heat flux. At the same time, the transverse-type HFS can be made as thin as technologically possible: the temperature gradient in steady-state regime does not depend on the thickness of the sensor. To transient regime both types of HFSs are a surface layer of a homogeneous semi-infinite body and constant heat flux of q (boundary condition (3.4)–(3.5) is generated on the surface of this body constant. The fundamental difference between the two types of sensors is that the GHFS forms the signal E ⊥ in the layer δ0 δ, while the longitudinal-type HFS forms signal E | proportional to the temperature difference T| across the entire thickness δ 

1 , (3.23) E ⊥ = S0⊥ × A⊥ × q × erfc √ 2 Fo0 E = S 0 × A × [T (0, τ ) − T (δ, τ )] = S 0 × A × T ,

(3.24)

3.2 Thermal Model of GHFS

57

Fig. 3.20 Functions ϕ (Foδ ) and erfc 2√1Fo

(Fo ), erfc

0

1 2 Fo0 0,8 0,6

(Fo ) 0,4

erfc 0,2 0 0,001 0,01

0,1

1

10

1 2 Fo0

100 1000 Foh,Fo0

where the subscripts || and ⊥ stand for the corresponding types of HFS, and Fo0 = aδ2τ . 0 The problem’s solution (3.4)–(3.5) is well known [19]: 2q × δ 

T = × Foδ × k



1 1 √ − ierfc √ π 2 Foδ

 =q×

δ × ϕ (Foδ )), (3.25) k

  √ ϕ (Foδ ) = 2 Foδ × √1π − ierfc 2√1Fo . δ   1 √ The functions ϕ (Foδ ) and erfc 2 Fo are shown in Fig. 3.19. Evidently, the function ϕ (Foδ ) attains the level 0.98…0.99 corresponding to the measurement error of 1…2% only at Foδ > 100 ϕ (Foh ) coincides with ϕ (Fo0 ) up to the symbols). Curves (3.19) and (3.20) were compared. For comparison, we use a modern thermocouple-type sensor, the heat flux converter 1 B.11.2.1.11.P.00.1.16.00.0 (Russian standard GOST 30619-98, Ukrainian standard 3756-98) with the thickness h = 2 × 103 m. Values of its effective thermal diffusivity a = 27 × 106 m2 /s are taken from monograph [5]. Calculation shows (Fig. 3.20) that the functions ϕ (Foδ ) and erfc 2√1Fo reach the 0 level of 0.99 with the Fourier numbers of the order of 102 . This means that the response time of the thermocouple-type HFS is where Foδ =

a,τ ; δ2

τ|| =

102 ×0.0022 27×10−6

= 14.8 s,

which is close to the value τ|| ≥ 15 s given in the sensor’s manual. The calculation of the quantity τ⊥ for the Fourier number is equal to 102 and for the thickness δ0 taken from Table 3.1 yields the values close to 6 × 10−7 s for all compositions. Thus, the response time of the transverse-type HFS is approximately 2.5 × 107 times less than the thermocouple-type HFS.

58

3 Transient Heat Flux Measurements

References 1. Koshkin, V. K., Kalinin, E. K., Dreytser, G. A., et al. (1973). Nestatsionarnyy teploobmen (Unsteady heat transfer). Moscow: Mashinostroyeniye. 2. Loytsyanskiy, L. G. (1987). Mekhanika zhidkosti i gaza (Mechanics of fluid and gas). Moscow: Nauka. 3. Shlikhting, G. (1969). Teoriya pogranichnogo sloya (Theory of the boundary layer). Moscow: Nauka. 4. Anatychuk, L. I., & Bulat, L. P. (2001). Poluprovodniki v ekstremal’nykh temperaturnykh usloviyakh (Semiconductors in extreme temperature conditions). Saint-Petersburg: Nauka. 5. Gerashchenko, O. A. (1971). Osnovy teplometrii (The basics of heat metering). Kiyev: Naukova dumka. 6. Grabov, V. M., Divin, N. P., & Komarov, V. A. (2002). Bystrodeystviye anizotropnogo elementa (The performance of an anisotropic element). In Termoelektriki i ikh primeneniye (Thermoelectrics and their application) (pp. 85–88). Saint-Petersburg: Institute of Physics and Technology named after A. F. Ioffe Russian Academy of Sciences. 7. Mitiakov, V., Sapoznikov, S., & Mitiakov, A. (2000). Transient phenomena in gradient heat flux sensor. Paper presented at the 3rd European Thermal Sciences Conference, Heidelberg, Germany. 8. Laser Technologies Center—Laser equipment, technology, material, technical support, service. http://www.ltc.ru/about/history-en.shtml. 9. Mityakov, A. V. (2000). Gradiyentnyye datchiki teplovogo potoka v nestatsionarnoy teplometrii (Gradient heat flux sensors in non-stationary heat flux measurement: dis.). Dissertation, SaintPetersburg State Technical University. 10. Zakhner, T., Forg, R., & Lengfelner, G. (1998). Transverse thermoelectric response of a tilted metallic multilayer structure. Applied Physics Letters, 73(10), 1364–1366. 11. Zeuner, S., Lengfellner, H., & Prettl, W. (1995). Thermal boundary resistance and diffusivity for YBA2 Cu3 O7 . Physical Review B, 51(17), 11903–11908. 12. Sapozhnikov, S. Z., Terekhov, V. I., & Mityakov, V. Y., et al. (2008). Testing and using of gradient heat flux sensors. Paper presented at the Heat Transfer Research. 13. Anatachuk, L. I. (1979). Termoelementy i termoelektricheskiye ustroystva: Spravochnik (Thermoelements and thermoelectric devices: Reference book). Kiev: Nauka dumka. 14. Grigoryev, B. A. (1974). Impul’snyy nagrev izlucheniyami. Nestatsionarnyye temperaturnyye polya pri impul’snom luchistom nagreve chast 2 (Pulse heating by radiation. Non-stationary temperature fields during pulsed radiant heating, part 2). Moscow: Nauka. 15. Grigoryev, B. A. (1974). Impul’snyy nagrev izlucheniyami: Kharakteristiki impulsnogo oblucheniya i luchistogo nagreva chast 1 (Pulse heating by radiation: Characteristics of pulsed irradiation and radiant heating, part 1). Moscow: Nauka. 16. Pekhovich, A. I., & Zhidkikh, V. M. (1976). Raschety teplovogo rezhima tverdykh tel (Calculation of the thermal regime of solids). Leningrad: Energiya. 17. Solodov, F. F., & Tsvetkov, A. P. (1986). Praktikum po teploperedache (Workshop on heat transfer: Textbook for universities). Moscow: Energoatomizdat. 18. Karslou, G., & Yeger, D. (1964). Teploprovodnost’ tverdykh tel (Thermal conductivity of solids). Moscow: Nauka. 19. Kartashov, E. M. (1979). Analiticheskiye metody v teploprovodnosti tverdykh tel (Analytical methods in the thermal conductivity of solids: Textbook for universities). Moscow: Vysshaya shkola. 20. Belyayev, N. M., & Ryadno, A. A. (1978). Metody nestatsionarnoy teploprovodnosti (Methods of unsteady heat conduction: Textbook for universities). Moscow: Vysshaya Shkola. 21. Blatt, F. Dzh., Shreder, P. A., & Belashchenko, D. K. (Eds.). (1980). Termoelektrodvizhushchaya sila metallov (Thermoelectromotive force of metals). Moscow: Metallurgiya.

Chapter 4

Multifunctional Performance of Gradient Heat Flux Sensors

4.1 Thermometry Electrical resistance of all gradient heat flux sensors (GHFSs) is temperaturedependent, which makes it possible to regard them as resistance temperature detectors. The resistance temperature coefficient of metal- and/or alloy-based GHFSs is practically temperature-independent, but only bismuth-based GHFSs can be used as resistant thermometers since the resistance of metal-based HGHFSs is negligibly small. Conversely, HGHFSs made of n-silicon + p-silicon compositions have the resistance about tens of k, making them impossible to use in thermometry. HGHFSs based on silicon + aluminum and silicon + nickel are quite promising. Their ohmic resistance does not exceed several k. The temperature dependence of the TCR (Temperature Coefficient of Resistance) of semiconductor-based GHFSs is exponential. This fact must be taken of during signal processing. GHFSs can be used as resistant thermometers in the traditional temperature measurement designs, “immersing” them fully into the tested medium to exclude the effect of transverse thermopower. In this case, the GHFS’s instrument lag shaped . We can as a plate with thickness h is determined by the Fourier number Fo = aτ h2 describe this type of thermometry as “active” since external current should be passed through the resistance thermometer. However, as the GHFS can form a signal in the surface layer in thickness h 0  h, a different “passive” temperature measurement circuit can be proposed. Here the sensor serves as a source of thermopower [1]. It is enough to measure electric current I and thermopower E in the experiment and subsequently calculate temperature. For metal-based GHFSs I =

E , Rsh + R0 (1 + χ T )

© Springer Nature Switzerland AG 2020 S. Z. Sapozhnikov et al., Heatmetry, Heat and Mass Transfer, https://doi.org/10.1007/978-3-030-40854-1_4

(4.1)

59

60

4 Multifunctional Performance of Gradient Heat Flux Sensors

(b) (a)

R0

q, kW/m2

T 50 40

600

30

400

E

T = T(E) q = q(E)

20

Rsh

switch

200 0

0

0.5

1.0

E, mV

Fig. 4.1 Electric circuit (a) and calibration results (b) of “passive” thermometer with access to one output channel (denoted by the frame)

where Rsh is the loading (shunt) resistance, R0 is the GHFS resistance at the onset temperature (for example, at 273 K), and χ is the GHFS’s material TCR. For bismuth, χ ’s magnitude is close to the ideal gas expansion factor (1/273). In our experiments, calibration was done in the thermostat at temperatures from 20 to 200 ◦ C. The χ was within the range 0.003…0.004 1/K. It follows from formula (4.1) that 1 T = R0 χ



 E − (Rsh + R0 ) , I

(4.2)

therefore, current and thermopower should be measured separately using two output channels. If the GHFS with the resistance RT = R H + R0 (1 + χ T ) is switched on according to the circuit in Fig. 4.1. It is calibrated by: – for the heat flux (with the switch open); – for the temperature (with the switch closed), then two linear dependences T (E) and q(E) are obtained. The one of the experiments results are plotted in Fig. 4.1b. The relationship between the HGHFS’s temperature and the HGHFS’s electrical resistance is determined by Steinhart-Hert’s equation [2] 1 = a + b ln R + c ln3 R, T

(4.3)

where the quantities a, b, and c (Steinhart-Hert’s parameters) should be determined as empirical constants. In most cases, we can assume that c = 0. Equality (4.3) assumes the form

4.1 Thermometry

61

Fig. 4.2 HGHFS’s electrical resistance based on silicon + aluminum as function of temperature

R, experiment approximation

4

3

2 1 0 300

340

1 1 1 = + ln T T0 B



R R0

380

420

T, K

 ,

(4.4)

where B is the constant, T 0 is the temperature (in Kelvin scale) at which the resistance R0 is determined, and the resistance at the temperature T is    1 1 . − R = R0 exp B T T0

(4.5)

The desired temperature is 

T = ln

B

R R0

 .  exp − TB0

(4.6)

The dependence R(T ) for a typical HGHFS made of the silicon + aluminum is shown in Fig. 4.2. The coefficient B is equal to the asymptotic slope of curve R(1/T ). Thus, it is sufficient to install the GHFS at the desired location to lay wires to the recording zone, and then to determine temperature and heat flux using a single channel. In our opinion, the GHFS can also be used for measuring unsteady temperatures. It is generally accepted in unsteady thermometry [3, 4] that the true unsteady temperature value in the medium with a thermometer is θ = θ (τ ) = T (τ ) + τ˜min ×

dT , dτ

(4.7)

where T (τ ) are the readings of a thermometer and τ˜min is the thermometer’s response time.

62

4 Multifunctional Performance of Gradient Heat Flux Sensors

Fig. 4.3 Setup’s scheme for measuring unsteady temperature using GHFS

D1

D2

Solid mV

Bismuth-based GHFSs D1 and D2 be mounted at a solid substrate as shown in Fig. 4.3. The temperature T (τ ) is calculated by formula (4.2) and τ˜min is determined experimentally. Schmidt’s finite-difference scheme is used to determine the derivative dT dτ a dT ≈ dτ x



T x

 +

 −

T x

  −

,

(4.8)

where a is the GHFS’s thermal diffusivity, x is the thickness of one of the n layers the thickness of the infinite plate is conveniently divided, and

T  into 

Twhich , are the finite-difference equivalents of the derivatives dT at the right x + x − dx and the left of the middle plane of the nth layer.



 and T are related to the D1 and D2 signals linearly Let us assume that T x + x − equal to E 1 and E 2 , respectively. In this case E2 − E1 dT E2 − E1 ≈a = , dτ δx S0 Aks ρs cs AδS0

(4.9)

where k s , ρs , cs are the thermal conductivity, density, and heat capacity of the GHFS, respectively; δ ≡ x is the GHFS’s thickness and A is the GHFS’s plan-form area. The temperature θ (τ ) is calculated by formulae (4.5)–(4.7) 1 θ (τ ) = R0 χ



 E E2 − E1 − (Rsh + R0 ) + τ˜min · . I ρs cs AδS0

(4.10)

4.1 Thermometry

63

Thus, GHFSs allow to control not only the heat flux but also the temperature over some surface’s section, as well as the temperature of a fluid or a gas that is rapidly varying with time.

4.2 Determination of GHFS Thermal Characteristics The possibilities offered by heatmetry are rarely used in this area of research. However Gerashchenko [5] created the concept of such experiments as early as in 1971. There are no new methods of “heat flux measurement” have been developed recently, except for the classical determining thermal conductivity of materials in steady-state regime [6]. We are going to show, at least, at the level of “imaginary experiments” and the simplest experiments what can be done using GHFSs. As the first example, we consider the heat conduction problem for a semi-infinite rod under Derichlet’s boundary conditions. The experiment scheme is shown in Fig. 4.4. A semi-infinite rod with the temperature T0 is set in contact with a medium with the temperature T f > T0 at the moment τ . Uniformly distributed heat sources qv , W/m3 , are getting in the rod. Adiabatic conditions are imposed at the side surface of the rod. The thermal conductivity k, density ρ and heat capacity c of the rod material are to be determined experimentally. At the time τ , the heat flux in the free end of the rod is [7]

q = T f − T0





ρck − 2qv πτ



aτ . π

(4.11)

At the moment τ = τ0 , when q = 0,

T f − T0





1 ρ Ak · √ = 2qv π τ0



a √ · τ0 , π

Fig. 4.4 Semi-infinite rod model

Tf>T0 T0

qv k, ρ, c

64

4 Multifunctional Performance of Gradient Heat Flux Sensors

from where τ0 =

 ρA

T f − T0 . 2qv

ρA =

2τ 0 qv . T f − T0

(4.12)

Thus, the volumetric heat capacity ρc can be determined from the τ0 at which the GHFS at the free end of the rod signal changes its sign, even without measuring heat flux. We are going to estimate the order of τ0 . For example, for copper (ρ = 8900 kg/m3 , c = 400 J/(kg K)) with qv = 106 W/m3 , T f − T0 = 0.1 K, we obtain τ0 = 0.18 s, which is quite acceptable for GHFSs measurements [8, 9]. To determine thermal conductivity, we represent equality (4.9) in the form

q = T f − T0 from where





ρc √ k − 2qv πτ

q 2π k=

 Z T f − T0 −



2qv Z

τ √ k, ρcπ

2 ,

(4.13)

2qv where Z = · τ0 is the quantity determined from τ0 and random τ = τ0 (T f −T0 ) τ during the experiment. We determine the thermal diffusivity a = k/(ρc) as the parameter after calculations by formulae (4.11) and (4.12). Another possible experiment can be carried out by placing the GHFS at a fixed distance from the sample surface. Such experiment can be proposed for investigation of granular materials (Fig. 4.5). We use an isothermal surface at Tw0 = const (for example, evaporator or boiler wall) and kept at the temperature T0 < Tw0 in the experiment. The GHFS is installed at a distance δ from the sample surface. After the sample has contacted with the isothermal surface, the heat flux at the depth δ > 0 nonmonotonically changes (Fig. 4.5b); with [7, 10] qmax × δ aτmax = 0.484; = 0.5, k (Tw0 − T0 ) δ2 where qmax and τmax are the maximum heat flux and the time when it’s maximum achieved, respectively. It follows from the equality that a=

δ2 , 2τmax

(4.14)

4.2 Determination of GHFS Thermal Characteristics

(a)

65

(b) q Tw0=const qmax

GHFS

T0 Tw (T f is the fluid temperature) and “falls” with T f < Tw . In any case, the heat flux q and the thermopower E that are generated by the GHFS change, which is recorded by an actuating system (for example, a computer). Unlike the existing indicators, such an indicator has no moving parts, does not require special orientation in space and operates over a wide range of temperatures and pressures. Its possible applications include systems of fuel supply to internal combustion engines, systems for cooling linear electric engines, compressors, etc. The results presented here show that using GHFSs in problems of fluid flow rate measurement and fluid flow indication is a fairly competitive method in comparison with the available temperature devices. It is planned to design devices for specific engineering applications in the future.

80 Fig. 4.23 Distribution of wall temperature and heat flux in fluid flow indicator

4 Multifunctional Performance of Gradient Heat Flux Sensors

GHFS

(a)

Tw Hot fluid

Without moving

Cold fluid

(b) q Hot fluid

Without moving

Cold fluid

4.6 Measurement of Electric Circuit Parameters Measurement of electric energy consumption (and electric energy consumption in industrial and living conditions monitoring) becomes increasingly important. The modern market of active energy meters is very competitive. The one of the most unfilled niches is the fight against unauthorized use of electric energy. The performance capabilities of HGFS in this area are established. The principle of measurement is the current in the electric line in terms of Joule heat flux at the surface of the conductor. While most of the HFSs listed in Table 1.1 are applicable for such measurements, it is the GHFS with response allows us to control several electrical network parameters at a time. Electric current is supplied to a house (Fig. 4.24) (flat, or any living area where energy consumption must be taken into account) according to the two-wire system. Phase wire 1 on the “measuring section” is covered with or surrounded by HFSs that fix time-variable values of the Joule heat flux (for subsequent processing in an individual active energy meter, archiving, payment, etc.). It is extremely difficult to “cheat” such a system. Any current in a conductor causes a GHFS signal. In this case, heat transfer conditions, air temperature, etc., have no effect on the level of these signals. It should be possible to control energy consumption by choosing GHFS dimensions and converter type (see Sect. 2.4).

4.6 Measurement of Electric Circuit Parameters

81

Fig. 4.24 Measuring circuit of electric power and energy consumption in dwelling house

GHFS 0

Electric meter

Conductor with a current I includes an a × b × L area (with b  a). Both sides (total area is A = 2a L) are equipped with GHFSs with the voltwatt sensitivity S 0 . The thermopower induced by the Joule heat flux through seriesconnected GHFSs is equal to E. Assume that the rated voltage U in the circuit is constant, load is active, and output power on the “measuring element” can vary from Pmin to Pmax . Heat flux is removed to the environmental medium at the constant heat transfer coefficient h. The electric resistance R of the conductor does not depend on temperature, the electric conductivity of the conductor on the “measuring section” is  equal to ρ. The system parameters provided that the measuring error of E min E max = ε should not exceed a fixed level determined in advance. Maximum Joule heat flux through both GHFSs  2 R= Q max = Imax

Pmax U

2

 R=

Pmax U

2 ρ

L , ab

(4.29)

where Imax is the current, U is the voltage drop on the “measuring element”, induces the thermopower   Pmax 2 L (4.30) E max = S0 Q max = S0 ρ . U ab Superheating of the GHFS surface with respect to the environmental medium T is defined from the heat balance equation T =

Q max = Fh



Pmax U

2

1 ρL × = ab 2a Lh



Pmax U

2

ρ . 2h × a 2 b

(4.31)

82

4 Multifunctional Performance of Gradient Heat Flux Sensors

The thermopower corresponding to the power Pmin is, similar to the previous relation   Pmin 2 L E min = S0 ρ , (4.32) U ab and the measuring error is E min = ε= E max



Pmin Pmax

2 .

(4.33)

Physical and economic constraints in such a circuit. All quantities taken as limiting are bracketed. The cost of a GHFS is almost linearly related to its area, therefore, [A] is considered to be an economic restriction. The bismuth melting point (544 K) yields a temperature limit [T ]. A metrological restriction [ε] was mentioned in the statement of the problem. Design and economic considerations restrict the limiting dissipated power [Qmax ]. A minimum level of signal [E min ] is dictated by the capabilities of the equipment. Ideally, it should be imposed that the following inequalities be satisfied simultaneously under any conditions: A ≤ [A] , T ≤ [T ] , ε ≤ [ε] , Q max ≤ [Q max ] , E min ≤ [E min ] .

(4.34)

Generally speaking, the set of non-strict equalities 4.34 is not a system (in the mathematical sense). A relationship between limiting parameters is established. For obvious reasons, the sign of the inequality is neglected. It follows from formula 4.29, 4.31 and 4.32 that 1 [Q] × [ε] = . S0 [E min ]

(4.35)

Equalities 4.31 and 4.32 together with the obvious relation [A] = a L ⎧

2 ρ P 2 ⎪ ⎪ Umax 2h[T ] = a b ⎪ ⎨ Pmin 2 ρ S0 U = ab L [E min ] ⎪ ⎪ ⎪ ⎩ [A] = a L .

(4.36)

Making transformations, we obtain 

Pmax U

2

ρ = S0 2h [T ]



Pmin U

2

[E min ] = 2h S0 , [ε] [A] [T ]

ρ [A] , [E min ] (4.37)

4.6 Measurement of Electric Circuit Parameters

83

which coincides in sense with equality 4.37 obtained earlier. It is also possible to write a combined formula [E min ] [E min ] = = S0 . [ε] [Q max ] [ε] [A] [T ]

(4.38)

All equalities (4.35, 4.36 and 4.38) are equivalent. Since h, [T ], S 0 and [E min ] are usually known beforehand, it is more correct to calculate [ε] =

1 [E min ] × , 2S0 h [T ] [A]

(4.39)

As expected, the error [ε] is hyperbolically related to the sensor area [A]. The problem of measuring the active power in the circuit with the maximum current Imax = 20 A is considered. The accuracy class of the sensor is equal to 0.5. Assume that the GHFS has the volt-watt sensitivity S 0 = 10 mV/W, the limiting temperature increase is [T] = 250 K, and the heat transfer coefficient at the surface is h = 5 W/(m2 K). The heating element is made of copper (ρ = 10−8  m), its thickness is b = −4 10 m. An AD7714 device by ANALOG DEVICES (E 0 = 0.179 × 10−3 mV) is chosen as an ADC. The heat flux at maximum load is Q max =

2 ρL Imax = h [T ] × 2a L ab

from here a = Imax

 ρ 10−8 = 20 = 4 × 10−3 m. 2h [T ] b 2 × 5 × 250 × 10−4

At minimum load I min = 0.005I max = 0.005 × 20 = 0.1 A. Therefore Q min =

E0 0.179 × 10−3 = 0.18 × 10−4 W. = S0 10

The heating element length L is found from the relation L=

Q min ab 0.18 × 10−4 × 4 × 10−3 × 10−4 = = 0.072 m. 2 0.12 × 10−8 Imin ρ

Thus, a heater with the L × a × b = 72 × 4 × 0.1 mm dimensions should be made, the GHFS area is 72 × 4 × 2 = 576 mm2 = 5.76 cm2 . It is possible to manufacture a sensor of such dimensions, even thoughit is somewhat too “large”. Imagine  that the ADC has 32 bits. In this case, E 0 = E max 232 ≈ 3 232 = 6.98 × 10−10 V =

84

4 Multifunctional Performance of Gradient Heat Flux Sensors

0.7 nV = is the level that should be surpassed at least by 2 orders of magnitude in measurements. Assume E 0 = 70 nV, then Q min = L=

0.07 × 10−3 = 0.07 × 10−4 W, 10

Q min ab 0.07 × 10−4 × 4 × 10−3 × 10−4 = = 0.028 m. 2 0.12 × 10−8 Imin ρ

The GHFS area is 28 × 4 × 2 = 224 mm2 = 2.24 cm2 . The response time of the GHFSs established in Chap. 3 allows to use them to control the a.c. frequency. Let sinusoidal current have the frequency f = ω 2π , with amplitude values of voltage and current equal to U m and I m , respectively. Instantaneous values of voltage and current are u = Um sin ωτ , i = Im sin (ωτ − ϕ) , where ϕ is the phase shift angle. Since P = U I cos ϕ, the power coefficient cosϕ is immediately found from the “heatmetry” value of P and the “usual” product UI. The instantaneous power is

 p = ui = Um Im cos ϕ sin2 ωτ − sin ϕ sin ωτ cos ωτ = = Um2Im (cos ϕ − cos ϕ cos 2ωτ − sin ϕ sin 2ωτ ) = = Um2Im [cos ϕ − cos (2ωτ − ϕ)] .

(4.40)

It is possible to record the power p(τ ) through the GHFS signal, since the sensor both for the industrial frequency (50 Hz) and for the majority response time τmin  2π ω of the increased frequencies used. The average power UI can be easily measured by standard devices. Here U, I are the average-integral values of voltage and current, respectively. Then, 2p p (τ ) = cos ϕ − cos (2ωτ − ϕ) , = Um I m UI but cos ϕ =

p¯ UI

. Here p¯ is the average power. Therefore

from where ω =

1 2τ

p (τ ) − p¯ = − cos (2ωτ − ϕ) , UI    ¯ p(τ ) arccos p− + ϕ , UI

4.6 Measurement of Electric Circuit Parameters

85

Fig. 4.25 Measuring section (scheme)

Q

a

L

E b Q

I Fig. 4.26 Calculation of the AC frequency

p, p p

0

Fig. 4.27 Structure diagram of frequency measurement

U I q

Averaging out q( )

Calculations f p Computer

1 f = 4π τ

    p¯ − p (τ ) arccos +ϕ . UI

(4.41)

Thus the frequency f for a thermally thin conductor is uniquely associated with the value of the power p(τ ) measured by the GHFS (Fig. 4.25). If τ = τ0 is fixed (Fig. 4.26) with p¯ = p (τ0 ), then arccos 0 = π2 , f =

 1 π +ϕ , 4π τ0 2

(4.42)

where ϕ corresponds to the phase angle at the time τ0 . The structure diagram for frequency control can be of the type shown in Fig. 4.27.

References 1. Mityakov, V. Y. (2005). Vozmozhnosti gradiyentnykh datchikov teplovogo potoka na osnove vismuta v teplotekhnicheskom eksperimente (Possibilities of gradient bismuth-based heat flux sensors in a thermotechnical experiment). Dissertation, Saint-Petersburg State Polytechnical University.

86

4 Multifunctional Performance of Gradient Heat Flux Sensors

2. Tauts, Y. A. (1962). Foto- i termoelektricheskiye yavleniya v poluprovodnikakh (Photo- and thermoelectric phenomena in semiconductors) (M. P. Mikhaylovoy, trans., T. B. Kolomiytsa, ed.). Moscow: Izdatel’stvovo inostrannoy literatury. 3. Sergeyev, O. A. (1972). Metrologicheskiye osnovy teplofizicheskikh izmereniy (Metrological foundations of thermophysical measurements). Moscow: Izdatel’stvo standartov. 4. Yaryshev, N. A. (1967). Teoreticheskiye osnovy izmereniya nestatsionarnykh temperature (Theoretical foundations of measuring non-stationary temperatures). Leningrad: Energiya. 5. Gerashenko, O. A. (1971). Osnovy teplometrii (Basics of heat metering). Kiev: Nauka dumka. 6. Lykov, A. V., & Smolsky, B. M. (1966). Issledovaniye nestatsionarnogo teplo- i massoobmena (The study of unsteady heat and mass transfer). Nauka i tekhnika, 252. 7. Carslow, G., & Jaeger, D. (1964). Teploprovodnost’ tverdykh tel (Thermal conductivity of solids). Moscow: Nauka. 8. Mitiakov, V. Y., Sapoznikov, S. Z., & Mitiakov, A. V. (2000). Transient phenomena in gradient heat flux sensor. In 3rd European Thermal Sciences Conference, Heidelberg, Germany (Vol. 2, pp. 687–690). 9. Mitiakov, V. Y., Sapozhnikov, S. Z., Chumakov, Y. S., & Mitiakov, A. V. (2001). Experimental investigation of the convective heat transfer using gradient heat flux sensors. In 5th World Conference on Experimental Heat Transfer, Fluid Mechanics and Thermodynamics, Thessaloniki, Greece (pp. 111–116). 10. Kartashov, E. M. (1979). Analiticheskiye metody v teploprovodnosti tverdykh tel (Analytical methods in the thermal conductivity of solids. A Textbook for universities). Moscow: Vysshaya shkola. 11. Mityakov, V. Yu., Mityakov, A. V., & Sapozhnikov, S. Z. (2002). Opredeleniye radiatsionnykh i teplofizicheskikh kharakteristik materialov metodami gradiyentnoy teplometrii (Determination of radiation and thermophysical characteristics of materials by gradient heatmetry methods). In The Siberian Thermophysical Seminar, Institute of Thermophysics Siberian Branch of the Russian Academy of Sciences, Novosibirsk. 12. Kornilov, V. I., & Litvinenko, Yu. A. (2001). Sravnitel’nyy analiz metodov izmereniy poverkhnostnogo treniya v neszhimayemom gradiyentnom turbulentnom pogranichnom sloye (A comparative analysis of methods for measuring surface friction in an incompressible gradient turbulent boundary layer). Preprint of the Institute of Theoretical and Applied Mechanics Siberian Branch of the Russian Academy of Sciences, 1, 44. 13. Sapozhnikov, S. Z., Mityakov, V. Yu., & Mityakov, A. V. (2003). Gradiyentnyye datchiki teplovogo potoka (Gradient heat flux sensors). Saint-Petersburg: Izdatel’stvo SPbGPU. 14. Sapozhnikov, S. Z., Mitiakov, V. Y., & Mitiakov, A. V. (2003). Capabilities of gradient sensors in the measurement of the heat fluxes, temperatures, tangential stresses, and thermophysical characteristics of materials. Journal of Engineering Thermophysics, 12(1), 49–71. 15. Preobrazhensky, V. P. (1978). Teplotekhnicheskiye izmereniya i pribory (Hermotechnical measurements and devices. Textbook for universities). Moscow: Energiya. 16. Kutateladze, S. S. (1990). Teploperedacha i gidrodinamicheskoye soprotivleniye: Spravochnoye posobiye (Heat transfer and hydrodynamic resistance. A reference guide). Moscow: Energoatomizdat. 17. Il’inskiy, M. (1970). Beskontaktnoye izmereniye raskhodov (Noncontacting rate measurement). Energiya, 112. 18. Korotkov, P. A., Belyaev, D. V., & Azimov, R. K. (1969). Teplovyye raskhodomery (Thermal flow meters) (p. 176). Leningrad: Mechanical Engineering. 19. Kakhanovich, V. S. (1970). Izmereniye raskhoda veshchestva i tepla pri peremennykh parametrakh (Measurement of the flow of matter and heat with variable parameters). Moscow: Energiya. 20. Kremlevskiy, P. P., & Shornikova, Ye. A. (Eds.). (2004). Raskhodomery i schetchiki kolichestva: Spravochnik (Flowmeters and counters of quantity). Saint-Petersburg: Politekhnika. 21. Levin, V. M. (1972). Raskhodomery malykh raskhodov dlya skhem promyshlennoy avtomatik (Low flow meters for industrial automation circuits). Moscow: Energiya.

Chapter 5

Validation and Science Experiment

The reliability heatmetry as a new approach to investigating heat transfer can be verified only by testing of the phenomena already well-studied by other experimental methods, supported by analytical descriptions, etc. At the same time, the features of gradient heat flux sensors (GHFS), in particular, their response time allow us to obtain substantially new results.

5.1 Validation 5.1.1 Free Convection Near Vertical Plate We considered the classical problem of heat transfer at a vertical plate heated by electricity. For comparison, we used reliable data from hot-wire anemometry measurements obtained by Chumakov et al.1 [1]. The hot-wire anemometry has substantial unavoidable disadvantages, first of all, the distortions introduced by a hot-wire anemometer present in the boundary layer. In our experiments, bismuth single crystal-based GHFSs were mounted at the surface of the plate with heat-conducting compound and were moved from point to point many times. Their coordinates were defined with an accuracy of 1 mm. The GHFSs had the plan-area dimensions of 4 × 7 mm, the thickness up to 0.2 mm, and the volt-watt sensitivity of 9.3 mV/W. Vertical plate 1 is made of aluminum; it is 4950 mm in height, 900 mm in width, and 20 mm in thickness [2] (Fig. 5.1). The back side of plate 1 is equipped with 25 heaters 2 which power is regulated by means of a special electronic system. It maintains the heat regime for 6–8 h and allows 1 Experiments

were carried out on the setup of the Fluid Dynamics Department of Peter the Great St. Petersburg Polytechnic University, Professor Yu. S. Chumakov, Doctor of Physical and Mathematical Sciences, took part in experiments. © Springer Nature Switzerland AG 2020 S. Z. Sapozhnikov et al., Heatmetry, Heat and Mass Transfer, https://doi.org/10.1007/978-3-030-40854-1_5

87

88 Fig. 5.1 Setup for studying free convective heat transfer: 1—plate, 2—heater, 3—GHFS, 4—heat probe, 5—heat insulation, 6—measuring unit electric driver

5 Validation and Science Experiment

2 1 5

3

4

6

to model different methods of heating throughout the plate height (in particular, a constant temperature regime of +70 ◦ C at the entire surface). Such a regime was chosen for two reasons: – hot-wire anemometry data were obtained in [3] for the temperature range of +70…+80 ◦ C; – it was found in [4] that the contribution of radiative heat transfer is close to minimum over this temperature range. The temperature at the plate surface is measured by thermocouples installed flush with the surface. The temperature and the air velocity in the boundary layer are determined from the resistance thermometer and a hot-wire anemometer installed at heat probe 4 (tungsten wires 5 m in dia and 3 mm long serve as sensors). Each heat probe 4 has two such wires: a “hot” wire measures velocity and a “cold” one measures temperature. Movements of heat probe 4 are provided by electric driver 6 (the accuracy of movement is 10 mm, whereas that of movement across the boundary layer is about 1 m). Movements of heat probe 4 are remotely controlled to exclude noticeable disturbances in free convective air flow. Setup was previously used to calculate heat flux from temperature measurement data. Thermocouples were mounted at the plate surface, a time-averaged value of heat flux was obtained after processing the experimental data (taking into account the power of heaters 2). The methods of heatmetry by GHFS were adopted to investigate three flow in the boundary layer: laminar, transient, and turbulent up to the Grashof number Gr x = g β T x 3 = 3.5 × 1011 (here g is the gravity, β is the air expansion factor, x is the ν2 distance from the plate to the measurement point).

5.1 Validation Fig. 5.2 Nusselt numbers versus Grashof number in free convective boundary layer at vertical plate

89 Nu x , Nu x 10 3

Nu x Nu x

10 2

10 1 10 5

10 7

10 9

10 11

Grx

The local Nusselt number in experiments on temperature measurement [5, 6] was determined by the relation qw × x , (5.1) N ux = k f × T where qw = −k f (∂ T /∂z) is the heat flux at the surface. It was determined as the average temperature gradient in the boundary layer, k f is thermal conductivity of air at the temperature T f = 295…298 K, and T is the difference between the plate surface temperature (Tw = 341…344 K) and the temperature T f . The local Nusselt number in case of heatmetry is u x = N

q×x , k f × T

(5.2)

where q is the heat flux following to the GHFS signal. In our experiments, Nusselt numbers 5.2 were determined at the same x that enter into relation 5.1. The results of both experiments (“old” and “new”) are shown in Fig. 5.2. The both groups of points are sufficiently similar and can be describe approximated by logarithmic curves with close coefficients. However, divergence reaches 30…50%. In our opinion, the GHFS’s emissivity (close to 0.7) in heatmetry is substantially different from that of polished aluminum (from 0.04 to 0.1). In subsequent experiments, the GHFS’s external surface was covered with a 0.02 mm thick aluu x and N u x curves merged together (within the experimental minium layer and the N error). Using GHFS allowed to obtain new data on non-stationary distribution of instantaneous heat flux at the heated vertical surface and to estimate the fluctuation component of flux for the flow conditions. √ q¯2 Figure 5.3 shows the heat flux fluctuation intensity distribution Iq = q˜w (q¯ is the heat flux fluctuation component at the current coordinate x and q˜w is the average

90

5 Validation and Science Experiment

Fig. 5.3 Heat flux fluctuation and temperature intensity

I

Iq ITm 0.2

0.1

0.0

10 6

10 7

10 8

10 9

10 10

10 11

Gr x

heat flux at the same value of x) over the plate height. For comparison, this figure plots the air temperature fluctuationsintensity’s maximum over the current cross¯2

T section of the boundary layer IT m = Tf as a function of coordinate [1]. Here T¯ f is the temperature fluctuation component at a constant temperature difference T ). A small delay (in the Grashof number Gr x ) can be seen in the increase of the Iq in comparison with the IT m . This seems logical. As the Grashof number increases, the motion of air fluctuations intensity grows first (which favors a sharp increase in mass inflow of cold air into the boundary layer). With cold air reaching the surface only after that. As a consequence, the fraction of heat to be transferred by conduction increases against the background of increasing heat flux fluctuations. We used correlation and spectral analysis of the heat flux fluctuation to gain a more detailed understanding of heat transfer at the surface. Along this surface the boundary layer develops and flow regimes change (from laminar to developed turbulent regime). Figure 5.4 illustrates the auto-variance coefficient for fluctuations of heat flux dis)·q(τ +τ ) . tribution in the zone of transition from laminar to turbulent regime Rqτ = q(τ√ 2 q (τ )

Here τ is the current time and τ is the time interval between measurements. It is also possible to assess the heat flux periodic oscillations in the transition zone using the frequency spectrum. In Fig. 5.5 the spectral power density of heat flux fluctuations are shown  ∞ Rqτ (τ ) × cos(2π f τ )d(τ ) Eq = 0

as a function of current frequency f . The intense oscillation process with the dominant frequency over the range of 3.5…4.0 Hz almost immediately occurs at the start of the transition zone (x = 800 mm). As x increases, the spectral density in the low-frequency spectral region (0.1…0.5 Hz) noticeably grows. Moreover, one more harmonic with a dominant frequency of about 7.0 Hz appears at x = 1000 mm, while the frequency spectrum at x = 1100 mm already corresponds to the developed turbulent regime.

5.1 Validation Fig. 5.4 Auto-variance coefficient for fluctuations of heat flux in laminar transition zone

91 R qx

x, mm

0,8

800 900 1000

0,4

1100

0,0

-0,4 0,0

Fig. 5.5 Power density spectrum of heat flux fluctuations in transition zone

0,2

0,4

0,6

0,8

E q, s x, mm 800 900 1000 1100

1.0 10 -3

5.0 10 -4

0.0

0

2

4

6

8

10

f, Hz

The coefficient Rqτ indicates that periodic oscillations with a dominant frequency of about 3.5 Hz are present in the heat flux fluctuations. This frequency coincides with the fundamental frequency of air temperature fluctuations [3]. However, if the length of the periodic oscillation for temperature fluctuations is about 600 mm (starting with the longitudinal coordinate x = 600 mm [7]), a similar region for heat flux fluctuations of is much narrower (about 200 mm). Also its start shifts downstream to x = 800 mm. Both the onset and the decay of heat flux fluctuation’s periodic oscillations of occur rather sharply. It can be seen from Fig. 5.5 that the periodic oscillation is still pronounced at x = 1000 mm, but the oscillations already disappear at x = 1100 mm. The auto-variance coefficients and the frequency spectra corresponding to welldeveloped turbulent flow are shown in Figs. 5.6 and 5.7, respectively. Both quantities depend weakly on the Grashof number. A small disturbance in the monotonic behavior of the curves (it is especially noticeable in the auto-variance coefficient’s distribution at x = 1100 mm) may be associated with the large vortex structures forming that periodically destroy the boundary layer almost entirely.

92 Fig. 5.6 Auto-variance coefficient in turbulent flow regime

5 Validation and Science Experiment Rq

x, mm 1100

0,8

2200 3000 0,4

0,0

0,0

Fig. 5.7 Frequency spectrum of heat flux fluctuations in turbulent flow regime

0,2

0,4

0,6

0,8

Eq, s 10 -3

x, mm

10 -4

0,01

1100 2200 3000

0,1

1

10

f, Hz

It follows from the results shown in Figs. 5.2, 5.3, 5.4 and 5.5 that the parameters of free convective heat transfer (Fig. 5.7) determined by both methods are rather close. However, the characteristics of heat flux fluctuations could be obtained with the GHFS, which made the experiments more informative. Gradient heatmetry made the experiment easier. There was no need to use a electric driver (which would distort the flow) and signal processing was simplified. Moreover, it was more correct from a methodological standpoint to determine the surface heat flux experimentally rather than calculate it from temperature measurements in the boundary layer.

5.1.2 Cross Flow Around Circular Cylinder This section deals with another classical problems: cross flow around a cylinder. We have additionally considered some related problems allowing to understand the features and advantages of gradient heatmetry.

5.1 Validation

93

(a)

(b)

2

7 1

3 4

5

6

Fig. 5.8 Cylinder used in experiments: scheme (a) and general view (b). Numbers indicate: 1— GHFS, 2—cylinder, 3—table with measuring circle of angle meter, 4—steam supply; 5—condensate drain, 6—U-shaped manometer, 7—hot-wire anemometer sensor

This problem was discussed in great detail in the available literature [8–17]. The local and the average heat transfer coefficient over the cylinder perimeter were determined over a Reynolds numbers. The fluid nature’s influence, flow turbulence, cylinder surface roughness, etc., on heat transfer were studied. Heat transfer was examined both at constant heat flux and at constant wall temperature. However, it was at constant heat flux that the majority of experiments were carried out [10], with average heat transfer coefficient determined by means of electric calorimeters, and wall and flow temperatures determined by means of thermocouples and resistance thermometers. One of the GHFSs advantages is that they allow immediately determining the local (within the sensor area A) heat transfer coefficient h=

Ex q = .  Tw − T f S0 A Tw − T f

The working section of the wind tunnel (Tu ≤ 0.8%, w ≤ 25 m/s) was equipped with round smooth-wall cylinders 2 (Fig. 5.8a, b) 25 and 66 mm in dia, 166 and 500 mm in length. The cylinders wall thickness was equal to 0.1 mm. As experiments showed, such a small thickness practically did not reduce the rigidity of the cylinders in air flow. Saturated steam was supplied from a steam generator to cylinder 2 via connecting leg 4. Steam pressure was controlled by manometer 6. Steam condensate was poured out into a condensate tank via connecting leg 5. The heat transfer surface had a temperature close to the steam saturation in this case. Cylinder 2 was mounted at

94

5 Validation and Science Experiment

Fig. 5.9 Fluctuations of: 1—heat flux and 2—static pressure at cylinder surface (ϕ = 90◦ , Re = 5 × 104 )

q, kW/m 2

p, Pa

2,0

200

1,0

100

0,0

0

1 2

-1,0

-100 -200

-2,0 0,1

0,2

0,3

0,4

rotary table 3 with an angle meter whose measuring circle had the division of 1◦ . During the experiment, the cylinder equipped with the GHFS rotated around its axis by a azimuth angle ϕ. The frontal angle (ϕ = 0◦ ) was taken as the reference. GHFSs with the 4 × 7 × 0.2 mm, 5 × 5 × 0.2 mm and 15 × 2 × 0.2 mm dimensions were placed flush with the cylinders surface. The GHFS’s volt-watt sensitivity was within 9.8…20.0 mV/W. The azimuth angle covered by GHFSs was 18.5◦ for 25 mm dia cylinders, 3.5…7◦ for 66 mm dia cylinders, and 2.8◦ for 66 mm dia cylinders. Moreover, static pressure fluctuations at the cylinder wall were recorded in a separate series of experiments. For this purpose, a static pressure receiver connected with an electric pressure sensor of membrane type with a frequency about 30 Hz was placed at one generating line with the GHFS. Typical synchronous pressure and heat flux curves recorded with a light-beam oscilloscope N-145 are shown in Fig. 5.9. The curves shape corresponds to the traditional understanding of the Reynolds analogy. Local heat transfer coefficients were calculated by the formula h ϕ (τ ) =

qϕ (τ ) , Tw − T f

(5.3)

where qϕ (τ ) are the local heat flux measured at a fixed ϕ. The experimental results were processed as the relations  N uϕ 1 N uϕ (ϕ), N u = N u ϕ  Re=const (ϕ), √ (ϕ), N u0 2π Re



2π

N u ϕ dϕ, 0

where N u 0 (ϕ) is the Nusselt number at the frontal point (ϕ = 0), and the Reynolds was determined in terms of the average flow velocity w and the number Re = wd ν kinematic viscosity of air ν taken at the temperature T f . The variation of local heat transfer coefficients along the cylinder within the different Reynolds numbers is shown in Fig. 5.10a, b. The data obtained in these experiments are fairly close to the well-known results [10, 18]. Interestingly, our data coincide with the results of [17] where the authors used an HFS made by Vatell.

5.1 Validation Fig. 5.10 Local heat transfer coefficient variation at isothermal surface of cylinder in cross flow

95

(a) Nu

Re = 25·104 Re = 15·104 Re = 9·104

600

Re = 5·104 Re = 3·104

400

200

30

0

Nu Re0.5

(b)

60

90

120

150

120

150

Re = 25·104 Re = 15·104 Re = 9·104

1.2

Re = 5·104 Re = 3·104 0.9

0.6

0.3 0

30

60

90

Good coincidence is observed in all cases, including minimum N u near ϕ = 85◦ . This supports the validity of the estimate for time-averaged and “fluctuating” Nusselt numbers in our experiment. In particular, an area in front of a repeatedly attaching region of the flow is present in N u distributions over the range ϕ = 110…130◦ . Such heat transfer variation was also obtained in [17]. However, maintaining the temperature Tw = const causes the heat transfer coefficients average over cylinder to deviate from the values obtained at q = const. For example, in case of subcritical flow around a cylinder (103 < Re < 2 × 105 ), to calculate the average heat transfer coefficient over cylinder, it is recommended to use the relation [17] N u = 0.22Re0.6 .

96

5 Validation and Science Experiment

The processing of our experimental data has yielded the similarity equation N u = 0.29Re0.55 . Comparison shows that comparable temperatures and Reynolds numbers, the average heat transfer coefficient is lower by 10…20% at the isothermal cylinder’s surface than at the surface of the cylinder with constant (on average over the surface) heat flux. This is consistent, in particular, with the conclusions of the known studies [19]. There is an increasingly high demand for computations’s verification in modern numerical experiments [20–22]. It is especially important to obtain experimental data on the velocity characteristics, temperature and heat flux in turbulent flows. Experience of numerical experiments shows that it is impossible to use modern turbulence models without verification at Re > 103 , and the confidence for numerical simulation data is significantly reduced at smaller Re values. Methods of hot-wire anemometry [10] have found wide use as a verification instrument and are successful for recording temperature fluctuations near the surface. These data are considered to be the most reliable, since the wire diameter of a hot-wire anemometer is about 10−6 m and the response time is equal to 10−3 s. To examine the correlation between heat transfer coefficient and velocity fluctuations, a hot-wire anemometer was placed at a distance of 1.5 mm from the GHFS and at the same angle ϕ. Its signals were recorded simultaneously with those for heat flux fluctuations. Although the signal from the hot-wire anemometer was affected by temperature fluctuations of air, this effect appeared to be insignificant because of high temperature of the wire. At first, it is interesting to compare the fluctuation characteristics measured by the hot-wire anemometer and the GHFS. Secondly, it is important to specifically use the heat flux fluctuations for verification of numerical experiments (since the GHFS’s response time is less than that of the best modern hot-wire anemometers by several magnitude). The equipment described in Sect. 2.4 allows to convert analog GHFS signal to digital one with a frequency up to 30 kHz. All measurements were taken 6 × 104 times with a frequency of 2 kHz. Experimental data were processed using the criterion for the heat flux fluctuation’s intensity [23, 24]  qϕ 2 η= × 100%, q¯ϕ  where qϕ 2 is the RMS of heat flux fluctuations at fixed angle ϕ and q¯ϕ is the average heat flux at the same angle ϕ. Statistical data processing by standard programs made it possible to obtain in each experiment a set of n of the average heat flux qi that can be assumed to be a discrete

5.1 Validation

97

and randomly varying quantity. The required function in this case is represented by the heat flux dispersion [25] n D=σ = 2

 2 qi − q¯ , n−1

i=1

where σ is the standard oscillation and q¯ is the arithmetic mean value of the measured heat flux. The spectral heat flux fluctuation at fixed angle ϕ was computed by fast Fourier transform [26]. The signals from the hot-wire anemometer were processed in a similar manner. Typical heat flux oscillograms are plotted in Fig. 5.11. It can be seen that the flow in the boundary layer is laminar at ϕ = 0 . . . 80◦ and individual fluctuation bursts and flow separation are observed starting with ϕ ≈ 90◦ . The fluctuation’s character is preserved up to the rear stagnation point (ϕ = 180◦ ). Notably, similar character of heat flux fluctuations was also observed in [17, 19], where a film sensor of a hot-wire anemometer and a heat flux sensor by Vatell were positioned at the cylinder surface. Figure 5.12 shows typical velocity fluctuation’s oscillograms at different Reynolds numbers for the angle ϕ = 150◦ . Large-scale fluctuation spikes in the curves correspond to the flow. Figure 5.13 shows the power spectral density’s distributions for heat flux and flow velocity fluctuations at ϕ = 150◦ for different Reynolds numbers. As the Reynolds number increases, the amplitude maximum of fluctuations is shifted towards high frequencies. The amplitude is almost linearly related to the frequency. This spectral energy region has peaks corresponding to the vortex separation’s frequency. These properties of the energy spectrum in the repeatedly attaching flow region also correspond to the results obtained in [17]. The pronounced maxima at the frequencies of 33.0 Hz and 58.0 Hz reach 10 (W/m2 )/Hz at ϕ = 150◦ , which coincides up to 0.1 Hz with the maxima in the energy spectra of velocity. The vortex separation’s frequency also coincides with the frequencies calculated in terms of Strouhal numbers taking into account the flow blockage degree (Sh = 0.316) [10]. The fluctuation intensity η and dispersion D of heat flux as functions of angle ϕ are shown in Fig. 5.14a, b. The curves shape for η (ϕ) and D (ϕ) (Fig. 5.14a, b) agrees with the results of [10, 17, 19] both qualitatively and quantitatively. The heat flux fluctuation intensity and dispersion preserve minimal and almost constant within laminar flow in the boundary layer (about 4% for ϕ ≤ 60◦ ). The growth of both characteristics is observed immediately behind the separation point in the region where the local heat transfer coefficient are the minimal. This growth reaches 15…20% near the rear stagnation point (ϕ = 180◦ ). Local minimum and maximum heat transfer coefficients are seen at ϕ = 110…130◦ , which indicates that the near-wall separated flow in this region has a very complex character.

98 Fig. 5.11 Oscillograms for heat flux per fluctuations at different angles ϕ with Re = 5 × 104

5 Validation and Science Experiment q, kW/m 2

q, kW/m 2

(a) 4

= 0˚

(f)

2

2

0

q,

(b)

0.4

0

0.8

kW/m 2

q,

= 30˚

4

0

0.4

0

0.8

= 60˚

4

(h) 4

0.4

0.8

0

= 70˚

4

0.4

0.8

= 150˚

0.4

0.8

q, kW/m 2 = 70˚

4

(j)

= 180˚

4 2

0.4

0.8

w, m/s

0

0.4

0.8

w, m/s

(a) 16

(b) 16

12

12

8

8

4

4

0

0.4

4

0

0.8

2

Fig. 5.12 Oscillograms of velocity fluctuations at angle ϕ = 150◦ with Re = 3 × 104 (a), Re = 5.3 × 104 (b)

= 120˚

2

q, kW/m 2

0

0.8

q, kW/m 2

(i)

2

(e)

0.4

2

q, kW/m 2

0

= 100˚

q, kW/m 2

2

(d)

0.8

2

q, kW/m 2

0

0.4

kW/m 2

(g) 4

2

(c)

= 90˚

4

1.0

2.0

0

1.0

2.0

5.2 Science Experiments Fig. 5.13 Distributions of power spectral density for fluctuations of velocity (a, c) and heat flux (b, d) at angle ϕ = 150◦ and with different Reynolds numbers

99 w/f, (m/s)/Hz

(a)

w/f, (m/s)/Hz

(c)

0,4

0,4

0,3

0,3

0,2

0,2

0,1

0,1

0

80 f, Hz

40

50

q/f, (W/m 2 )/Hz

q/f, (W/m 2 )/Hz

(b)

(d) 80

15

60

f, Hz

60

f, Hz

60

10

40 5

20 0

Fig. 5.14 Heat flux: a fluctuation intensity and b dispersion

80 f, Hz

40

η, %

(a)

50

D 10 -5

(b) 4

20

Re=15 104 Re=9 104 Re=5 104

2

10 Re=15 10 4 Re=9 10 4 Re=5 10 4

0

60

120

0

60

120

5.2 Science Experiments 5.2.1 Cross and Non-cross Flow Around Cylinder with Turbulisators The next stage of the study was to examine heat transfer at the cylinder’s surface in a cross flow with artificial intensifiers to enhance convective heat transfer. It is known from Prandtl’s experiments [27] that installing a thin wire ring at the sphere surface ahead of the equatorial line changes the flow in the boundary layer. Separation delay decreases the drag coefficient. The experiments of Fage and Warsap [28, 29] show that mounting wires along the generating lines of a cylinder in cross flow at yawed angles ψ = ±65◦ causes a sharp reduction in drag. Using GHFSs allowed to find the local heat transfer coefficients at a circular cylinder with intensifiers that were wires with the diameter of 1.5…2.0 mm (Fig. 5.15) placed along the generating lines. The experimental procedure and processing of the results are similar to those outlined above. Figure 5.16 shows how local heat transfer coefficients vary around the cylinder circumference.

100

5 Validation and Science Experiment

2 7 1 w

3 4

5

6

Fig. 5.15 Layout of intensifiers mounted at cylinder surface. Numbers indicate: 1—GHFS, 2— cylinder, 3—rotary table, 4—steam supply, 5—condensate drain, 6—U-shaped manometer, 7— intensifiers Nu Re 0.5

D 10 -4

(a)

Re = 9 10 4

1 2

2.0

(b)

1.5

30

1.0

20

0.5

10

0

30

60

90 120 150

1 2

40

0

30

60

90 120 150

Fig. 5.16 Local dimensionless heat transfer coefficients at the surface Local dimensionless heat transfer coefficients at surface (a) and heat flux dispersion (b) in flow around smooth cylinder (1) and around cylinder with turbulisaitors installed at angles ψ = ±55◦ to incoming flow (2)

The optimal angle for installing intensifiers was chosen for the experiments. With ψ = ±55◦ the local heat transfer coefficient at the angle ϕ = 70◦ increases by a factor of 4 in comparison with a smooth cylinder. The average heat transfer coefficient over cylinder increases by a factor of 1.25. It can be seen from Fig. 5.16 how the heat transfer coefficient increases immediately behind intensifiers and their influence appears to be pronounced closer to the flow separation point. The curves are similar to those corresponding to a smooth cylinder but the local heat transfer coefficients prove to be larger by a factor of 2…3. There is practically no influence of intensifiers over the ϕ ≈ 100…120◦ .

5.2 Science Experiments

101

The dispersion of heat flux fluctuations (Fig. 5.16b) for the cylinder with intensifiers is also substantially different from that for the smooth cylinder. Here the turbulence affects the heat flux in a different manner. The dispersion slightly increases up to the ϕ = 70…90◦ but a sharp burst is observed over the 90…120◦ . The flow “remembers” additional disturbance, fluctuations in the boundary layer increase far beyond the separation point, but their influence on heat transfer is reduced to zero long before the dispersion approaches the level achieved on the smooth cylinder. It is interesting to note that heat transfer “optimization” in terms of the angle ψ takes less than an hour during the experiment. If GHFS are mounted around the circumference of the cylinder with a step of 10…15◦ , the complexity of optimization could be additionally reduced. Numerical experiment would doubtless make such “optimization” longer and more expensive. Full-scale experiments were carried out in 2014–2019. The relationship between the flow and heat transfer suggests that combined studies of both processes in real time yield sufficiently complete and reliable information on the nature of both processes. We tried to combine gradient heatmetry with the PIV (Particle Image velocimetry) as our following step. We used a GHFS based on single-crystal bismuth and the POLIS system produced by the Institute of Thermophysics, SB RAS (Novosibirsk) for PIV. The experiments were performed in a wind tunnel at the Peter the Great St. Petersburg Polytechnic University. The system consists of a double-pulse Quantel BSL laser, a 4 Mpix crosscorrelation camera, a synchronizing device, a fog-machine and image-processing software. A solid-state double-pulse Nd:YAG (yttrium aluminum garnet doped with neodymium ions serves as its active medium) laser was used. The lasing wavelength was 532 nm, pulse energy was 2 × 220 MJ, beam diameter was 3 mm, pulse duration was 7 ns. The cross-correlation camera has a spatial resolution of 2048 × 2048 (4 Mpix) and a time resolution up to 10 µs. The model was made of 0.1 mm steel sheet and heated by saturated steam. Its surface temperature was close to 100◦ . The cylinder was electrically driven to rotate around its axis by ±180◦ . The yawed angle β varied within 90. . . 45◦ . A GHFS mounted flush with the surface can record local heat flux at different angles ϕ (Fig. 5.17). The experiments covered a Reynolds numbers range from 104 to 8 × 105 . Threedimensional flow around the cylinder was visualized using the stereo PIV whose scheme [30]. It is shown in Fig. 5.18. The experimental setup structure diagram is shown in Fig. 5.19 Triggered by the synchronizing unit, the laser emits a double pulse, cameras 1 and 2 record the tracers, and camera 3 records the GHFS signal measured by an upgraded N-145 light-beam oscilloscope. Data from the cameras are transmitted to a computer for processing. The standard laser used in the POLIS complex appeared to generate EMI, so it was impossible to convert low signals from the GHFS even with an advanced ADC converter (National Instruments, USA). For this reason, the N-145 light beam oscilloscope with the mercury lamp replaced by a laser pointer was used to measure

102

5 Validation and Science Experiment

Fig. 5.17 Circular cylinder model

800

66 = 45˚...90˚ w Steam

Condensate

GHFS (4×7×0.2 mm, S0=8.4 mV/W)

Fig. 5.18 Stereo PIV scheme for flow around a cylinder: Stereo PIV scheme for flow around a cylinder: 1—GHFS, 2—cylinder, 3—camera No. 1, 4—camera No. 2, 5—laser

5

2 1

4

3 wy w z wx

14 4

3

5

6

1 2

12 11

13 7

q, kW/m2 1400 1300

10

8 9

1200 0

40

80

Fig. 5.19 Diagram of experimental setup: 1—GHFS, 2—cylinder, 3—laser beam, 4—mirror, 5— laser, 6—camera, 7—synchronization device, 8—camera No. 2, 9—scale, 10—voltmeter N-145, 11—light beam, 12—computer, 13—heat flux graph, 14—velocity field

5.2 Science Experiments Nu

103 qmax

90 qmin

80 90

95

100

200

211

222 w, m/s 4 2 0

Fig. 5.20 Angular heat flux graph and corresponding velocity fields

GHFS signal. Pointer’s beam was deflected by a galvanometer mirror and recorded by a digital photo camera. The signal was fed to a computer for processing. The magnetic field surrounding the oscilloscope galvanometer helped eliminate all EMI in the experiment. The experiments were conducted for a cylinder in cross-flow (β = 90◦ ) and a yawed cylinder (with the flow directed at an acute angle to its axis, 90◦ > β > 45◦ ). Separate experiments were run for flow around a cylinder heated to 100◦ and a cylinder without heating whose temperature was equal to the free-stream airflow temperature. Models of cylinders 66 mm in dia and 600 mm long (L/d = 9.1), and 18 mm in dia and 900 mm long (L/d = 50) were used to simulate end effects. The parameters of the flow and heat transfer were analyzed in the experiments. The wake behind the cylinder was considered as well. The model of the cylinder was rotated around its axis at a constant rate in each experiment, making a half-turn from ϕ = 0◦ to ϕ = 180◦ during the observation period. Flow and heat transfer parameters (for the heated cylinder) were recorded simultaneously and continuously. A single experiment yielded 1000 instantaneous velocity fields and local heat flux values. Accordingly, each measurement of the instantaneous velocity field corresponded to a reading from the GHFS with its position known. Heat flux depending on time (“time-dependent heat flux graph”) is replaced with the term “angular heat flux graph” for a model rotated at low speed (see Fig. 5.20). Heat transfer studies at different yawed angles β indicate that heat flux can be both increased and decreased by varying the angle. Sparrow and Yanez Moreno [31] established that minimum heat flux corresponds to β = 65…70◦ . A significant increase in the Nusselt number is observed for β = 45…50◦ , reaching higher than in case of cross-flow (β = 90◦ ). These results are compared with our data in Fig. 5.21.

104

5 Validation and Science Experiment Nu Re 9 103 13 103 26 103 42 103 70 103

200 150

Our data

100

9 103 Sparrow and 42 103 Yanez Moreno (1987) 70 103 50 50

60

70

80

Fig. 5.21 Comparison of our data with Sparrow and Moreno [31]

Figure 5.21 shows the surface-averaged distribution of the Nusselt number for different yawed angles. The strongest differences are observed near the rear face due to nature of the flow and to vortex separation in that zone. Notably, the separation point shifts downstream with decreasing angle β. Figure 5.22 shows the 3D averaged velocity fields in the cylinder wake. Vectors indicate the direction of the flow in the cross-section of the laser sheet. Color shows the magnitude of the third velocity component, Wz . It changes direction after yawed angles above β = 65◦ , correlating with the minimum in the average Nusselt number (Fig. 5.21). Assuming that this change of direction is responsible for the increase in the Nusselt number, we introduced two additional models to check that assumption: a cylinder 41 mm in dia, 3 m long (to eliminate the end effects) and the initial model 66 mm in dia, now mounted vertically. The study was concluded by expanding the concept of the rotating cylinder, attaching wires to it to rod-turbulizers. They were d = 0.86. . .2.0 mm in diameter, spaced symmetrically by an angle with respect to the front stagnation point. Rods were attached both flush with the cylinder generatrix and with a gap δ. We managed to experimentally select the best ψ, d and δ rather quickly. In particular, we confirmed that a stagnation region in front of the rod “vanishes”, and heat follows the trend shown in Fig. 5.23. We also established the yawed angle β influence. The experiments proved that the local heat transfer coefficient grows by 2.5 times near the turbulizers, while the average heat transfer coefficient grows by 1.14 times. The final series of experiments combining gradient heatmetry with PIV was dedicated to flow and heat transfer near circular fins [32–34]. The velocity and temperature fields around fin elements is interesting from both scientific and applied standpoints. In particular, this problem could provide a test dataset for flow and heat transfer numerical simulation. However, unlike problems on flow around a single cylinder, this problem still lacks a commonly accepted reasonable solution. Non-cross flow around finned tubes and bundles is also of interest. Deviation of the tube axis from the normal to the free-stream flow is as well as theoretical as it is encountered in manufacturing and installing heat exchangers [35].

5.2 Science Experiments

105

=85°

=75°

=65°

=60°

=55°

=45°

-1.2

-0.8

-0.4

0.0

0.4

Wz, m/s

Fig. 5.22 3D average velocity fields for cylinder wake

Previous experiments combining measurement techniques were conducted with models of hollow structures internally heated with saturated steam. That essentially made it possible to measure the heat transfer coefficient along with heat flux, as the temperatures of the ambient fluid and model surface hardly changed during the experiments. Flow and heat transfer around finned tubes is a more complex problem because the thermal parameters are measured at a surface with a variable temperature. A third technology to be used together with flow visualization and heatmetry is temperature measurement (either contact or non-contact).

106 Fig. 5.23 Local heat transfer coefficients for cylinder with rods-turbulizers

5 Validation and Science Experiment Nu

(a) Re

1.2

0.8

0.4

0

30

60

90

120

150

Without turbulizers With turbulizers Nu

(b) Re 1.2

0.8

0.4

0

30

60

90

120

150

This stage consisted of flow and heat transfer studies in both cross and noncross flow around a tube with circular fins. We used five single-crystal bismuth GHFSs. Three of them had the dimensions of 2 × 2 mm, 4 × 7 mm and the 5 × 5 mm. All GHFSs were 0.2 mm thick. The GHFSs volt-watt sensitivity was found using calibration by the Joule-Lenz heat flux and was about 10 mV/W. Temperature measurements were conducted using thermal imaging diagnostics (a non-contact method) and semi-artificial thermocouples (contact method). A FLIR P640 (Forward-Looking Infrared) camera was used to measure temperature at fin surface. Bodies whose temperature is different from absolute zero emit thermal electromagnetic radiation. The spectral power density of such radiation has a maximum whose wavelength is temperature-dependent. The position of the maximum shifts to shorter wavelengths with increasing temperature. Bodies heated to 40. . . 100 ◦ C have a maximum in the mid-infrared region. The camera was equipped with software that could simultaneously measure temperature at several points at the

5.2 Science Experiments

107

Fig. 5.24 Thermal images with an isothermal and non-isothermal fin

=0...15˚

d = 66 mm D = 106 mm Fig. 5.25 Cylinder with a single hollow fin

fin surface with almost no time delay and with an accuracy of 1 K. Figure 5.24 shows two examples of thermal images. A hollow cylinder heated with saturated steam was used as a prototype for all models used in the study. The following models were considered: – a cylinder with a hollow (“ideal”) circular fin; – a cylinder with a circular fin made of VT 22 alloy; – a cylinder with a circular fin made of VT 22 alloy and simulating fins of plexiglass. The supporting tube had the same size in all experiments, while fin height and spacing could vary. The model of a cylinder with a hollow fin is shown in Fig. 5.25. The sizes and materials were chosen for the fins in view of the experimental goals. Thermal conductivity of the VT22 alloy (9 W/m K) ensured that a non-isothermal temperature field could be controlled experimentally. Fin thickness significantly

108

5 Validation and Science Experiment

exceeded that typically used in practice, making it possible to evaluate heat transfer coefficient and effectiveness in a fin with an impractical design. This confirmed that the new approach can be applied to non-isothermal heat transfer surfaces of any configuration. Plexiglass fins served to generate the flow in the spaces between the fins and played practically no role in heat transfer. We used semi-artificial thermocouples made of VT22 and copper for thermometric measurements in the spaces between the fins. The model of a finned tube included hot junctions positioned at the surface of a fin symmetrically with the GHFS to monitor the temperature Tw1,2,3 . A fine needle of VT22 alloy was mounted on the fin, with the needle’s tip protruding into the flow and forming a cold junction with the copper wire. The local heat transfer coefficients are calculated by finding the temperature difference between the free-stream flow and the fin surface; the cold junction should be located in oncoming airflow with the temperature T f . The flow patterns obtained for a fin with H = 20 mm correlate well with the flow scheme described in [36]. Figure 5.26 shows average velocities above the fin and before the front stagnation point. Velocity vectors indicate the direction of the flow in the cross-section of the laser sheet. PIV studies of a 60 mm high fin revealed the presence of a vortex, shown in Fig. 5.27. Maximum heat transfer coefficient is achieved at h = 20 mm, due to a reverse vortex forming during flow separation. Figure 5.28 shows a three-component distribution of the local heat transfer coefficient for fins 20 and 60 mm high, respectively. It correlates with the results given in [36] (Fig. 5.29).

5.2.2 Surfaces with Intensifiers The next stage was to study local heat flux at heat transfer surfaces equipped with intensifiers (dimples, cavities, trenches, etc.) widespread for heat transfer enhancement. In a number of cases, an increase in local heat transfer coefficients is followed by a reduction in drag. Searching for the optimal size ratio of the intensifiers themselves and the distance between them, for best mutual arrangement of intensifiers at a smooth surface, etc., is a multiparameter problem. It has been solved satisfactorily only for some particular cases [37–43]. Data on the efficiency of intensifiers are often contradictory [44, 45]. Traditionally, most of the experiments are performed for a heat transfer surface heated by electric current. The reason for this is that HFS are either unavailable or imperfect. At the same time, it is known that in practice, the surface temperature of any local intensifier varies very little and it is therefore more reasonable to maintain constant temperature in experiments, measuring local heat flux. This approach was adopted in [45, 46], where the integral characteristics of surfaces equipped with intensifiers

5.2 Science Experiments

109

Re=1.8 104

steam

steam

Re=0.4 104

shadow

0

shadow

0.5

1

0

1.5

W, m/s

Re=3.2 104

steam

steam

Re=2.1 104

shadow

0

3

W, m/s

2.5

shadow

5

0

3.5

W, m/s

W, m/s

steam

Re=4.1 104

shadow

0

4.5

9

W, m/s Fig. 5.26 Average velocity fields near the 20-mm-fin for different Reynolds number

7

110

5 Validation and Science Experiment

W, m/s

Re=0.4 104

0.9

fin

steam

0.6

0.3

shadow 0

W, m/s

Re=2.1 104

5.1

steam

3.4

1.7

fin shadow

0

W, m/s

Re=4.1 104

steam

9

fin

6

3

shadow 0 Fig. 5.27 Average velocity fields near the 60-mm-fin for different Reynolds number

5.2 Science Experiments

111

(a)

(b)

h, W/(m2 K) 150 100 50

(c)

0

(d)

Fig. 5.28 3D distribution of local heat transfer coefficients for: a hollow fin 20 m high; b solid fin 20 mm high; c hollow fin 60 mm high; d solid fin 60 mm high

were found. Using GHFS offers a more comprehensive solution for the problem, finding local heat transfer coefficients during full-scale experiments. Simulating the conditions T = const is no more difficult than q = const in numerical experiments, therefore, verifying computational schemes does not cause any difficulties. Scarce data on heatmetry are a more likely problem. We investigated heat transfer at the surface of spherical and elongated dimples, a transverse cylindrical trench and a trench with a trapezoid cross-section. Constant temperature (different from that for flow stagnation) was maintained at heat transfer surfaces by steam heating of the models. Characteristic sizes of these models are shown in Fig. 5.29. In our works [39, 40, 48], the results were compared with the data of numerical simulation2 and experiments.3 During the experiments, GHFSs were installed in different zones of intensifiers, as well as at the smooth surface of the plate. In front of an intensifier (for comparison) and behind it (to estimate the influence of the intensifier on a “loop” change in heat transfer coefficient). Box-type flat panels 1 equipped with intensifiers were installed at the lower wall of open-type wind tunnel 2 (Fig. 5.30). Air was pumped through the setup via fan 3 connected to the outlet connection of wind tunnel 2. The entrance of wind tunnel 2 was equipped with Witoszynski nozzle 2 Professor S. A. Isaev (Doctor of Physical and Mathematical Sciences) participated in this stage of

the study. results of V. I. Terehov.

3 The

112

5 Validation and Science Experiment

(a)

d h

w

Condensate Steam

D

(b) h

w

d

Condensate Steam

(c) w h d Condensate Steam

(d) w h d Condensate Steam

Fig. 5.29 Spherical dimple (a), elongated dimple (b), cylindrical trench (c), trench with trapezoid cross-section (d)

5.2 Science Experiments

113

(b)

(a) 1

3

2

4 4 3

2

6 7 5 Fig. 5.30 Scheme (a) and general view (b) of experimental setup: 1—steam-heated model, 2— wind tunnel, 3—fan (conventionally rotated), 4—contraction 4 at entrance (Witoszynski profile), 5—steam generator, 6—condensate drain, 7—measuring/computing complex (MCC)

4 that provided a uniform velocity field over the channel cross-section. Air flow rate through the wind tunnel was regulated by a shutter mounted at the outlet of fan 3. Air velocity was determined by the pressure drop in nozzle 4 and varied from 5 to 15 m/s in different experiments. Preliminary experiments showed that the turbulence degree was Tu ≈ 1% in these conditions. The thickness of the dynamic boundary layer in front of the intensifier was 3…5 mm and the thermal one was 3.3…5.4 mm. Air temperature was determined from the readings of the room thermometer. Model 1 was a box-type construction with the 150 × 450 × 10 mm dimensions (for a dimple and a cylindrical trench) and with the 150 × 250 × 100 mm dimensions (for a trapezoid cavity). This construction was manufactured from a 0.1 mm thick steel sheet and had connecting leg 5 for steam supply from the steam generator and connecting leg 6 for condensate drain. Saturated steam temperature was determined by the steam generator pressure, and it was close to 100 ◦ C at atmospheric pressure. GHFS signals were recorded and processed using MCC 7 based on a personal computer (Sect. 2.2). The first series of experiments were performed on heat transfer in a spherical dimple 65 mm in dia and 9 mm deep, located at a distance of 400 mm from the inlet nozzle (Fig. 5.31). GHFSs with the 4 × 7 × 0.2 mm dimensions and the average volt-watt sensitivity of 9.7 mV/W were positioned in different zones of the dimple. Identical GHFS was placed on the model in front of the dimple (Fig. 5.31) to determine the heat transfer coefficient at the smooth (roughness height was less than 0.001 mm) surface. The results were compared with the known similarity equations [18] N u x = 0.332Rex0.5 Pr 0.33 at 103 < Rex < 5 × 105 N u x = 0.0288Rex0.8 Pr 0.4 at 104 < Rex < 106 .

114

5 Validation and Science Experiment

66 mm 30 mm w 60 mm

400 mm

Fig. 5.31 Spherical dimple on isothermal plate (view from above). Black squares correspond to GHFSs locations Re=6.4 10 4 Re=4.2 10 4 Re=2.5 10 4

hs /h w

(a)

hs /hw

(b)

1.6

1.2

1.2 1.0 0.8

1.0

0.4

0.6

0.8

40

20

0

d/2, mm

20

40

40

20

0

20

40

d/2, mm

Fig. 5.32 Relative local heat transfer coefficients over longitudinal (a) and transverse (b) central cross-sections of dimple. Dashed lines correspond to numerical simulation data obtained by Isaev

Data processing revealed that the local heat transfer coefficient at the smooth plate was reliable and the measuring error did not exceed 1%. Local heat transfer coefficients determined at the surface of the dimple were correlated with those at the smooth surface in front of the dimple. Measurements were processed in the form of the relations hhws (Re), qqws (Re), Here the subscript “s” corresponds to the quantities to be measured at the spherical surface of the dimple where and the subscript “w” to the quantities to be measured on the plane, Re = wd ν “d” is the diameter of the dimple. Experimental results are plotted in Fig. 5.32. The heat transfer coefficient over the longitudinal cross-section first decreases (with respect to h w ) and then increases by a factor of 1.8…1.95 closer to the back edge (Fig. 5.32a). Heat transfer over the cross-section is the least pronounced near the edges of the dimple for all the Reynolds numbers. The heat transfer coefficient increases towards the dimple’s center and reaches its maximum either at a radius of 16 mm (Re = 2.5 × 104 ), or at the center of the dimple (Re = 4.2 × 104 and Re = 6.4 × 104 ) (Fig. 5.32b).

5.2 Science Experiments

115

Fig. 5.33 Jet-vortex structures identification in spherical dimple by labeled liquid particle method. Re = 2.5 × 104 . Data obtained by Isaev et al.: a view from above, b cross-section of dimple

It seems interesting to compare the experimental results with the numerical simulation data. Figure 5.33 illustrates the flow in the spherical dimple obtained by method of labeled particles in flow-forming structural elements, self-generating in the dimple. This indicates the existence of tornado-like swirled jet flows. They evolve in the foci and transport fluid locations from the dimple’s periphery to its central zone, followed by fluid outflow along the trajectories adjacent to the dimple’s walls. These jets are surrounded by vortex braids with fluid particle nutation. In the direction of fluid motion in jets at small radii and in the opposite direction at large radii. The absence of vortices in the equatorial zone (in the flow direction) correlates qualitatively with the “dip” in both experimental and predicted curves. At the same time, it can be seen that quantitative agreement of numerical simulation and full-

116

5 Validation and Science Experiment hs /hw

1.1

1.0

0.9

2

4

6

Re 10 -4

Fig. 5.34 Relative heat transfer coefficients (average over dimple surface): solid line corresponds to physical experiment, dashed line to computation according to numerical simulation data [42]

2.5d 2.0d 1.5d 1.25d d

w

1 5

9

2

3

4

6

7

8

d Fig. 5.35 Arrangement of GHFSs for heatmetry at plane behind spherical dimple

scale experiments cannot be considered satisfactory, especially in case of heat transfer variations along the flow. In particular, the dips in the experimental curves (Fig. 5.33a) are not reflected in the computations, the maxima in the curves at Re = 2.5 × 104 are shifted, and the results at Re = 6.4 × 104 (Fig. 5.33b) are differ by almost two times. Thus, heatmetry makes it possible to improve numerical experiments. The flow pattern also corresponds to the water tunnel experiment’s data, which we carried out together with senior researcher Guzeev [47] at the Krylov Shipbuilding Research Institute (Fig. 5.37). Dimensionless heat transfer coefficients h¯ s averaged over the dimple’s surface were found as well (Fig. 5.34). Evidently, the heat transfer in the dimple does not intensify with Re < 2.6 × 104 , since the flow is not turbulent enough. As Re 4 increases up to 6.5 × 10 , intensification reaches 19%. The dimple with the dimenh sion ratio d = 0.139 is chosen arbitrarily and optimization was not performed for this parameter.

5.2 Science Experiments

117

(a) h s /h w

Re=2,5·10 4 Re=3,5·10 4

1,3

Re=4,5·10 4 Re=5,5·10 4

1,2

Re=6,5·10 4 1,1

1,0

0,6

0,9

1,2

1,5

1,8

d

(b) h s /h w

Re=2,5·10 4 Re=3,5·10 4

1,0

Re=4,5·10 4 0,8

Re=5,5·10 4 Re=6,5·10 4

0,6 0,4 0,2

0,6

0,9

1,2

1,5

1,8

d

Fig. 5.36 Local relative heat transfer coefficients at plate behind spherical dimple. Squares indicate locations where GHFSs are mounted, corresponding to those in Fig. 5.32

The next stage of experiments was to determine local heat transfer coefficients behind the dimple. The GHFSs location is shown in Fig. 5.35. The evolution of the flow is such that no “damping” of the flow occurs at a distance up to 2d behind the dimple (Fig. 5.36). This means that the next dimple placed at this distance will be in disturbed flow. The experiments revealed that the GHFSs were extremely difficult to install and switching was complex. Moreover, even thin (0.2 mm) GHFSs mounted in large quantities distort the flow in the wall layer and generate procedural error which is hard to estimate (Fig. 5.37). To reduce these disadvantages, new methods have been proposed to investigate many axis-symmetric protrusions and valleys on the steam-heated model. Let us consider how these methods are implemented for the spherical dimple.4 The idea is based on the model of a spherical dimple with the rotary part mounted flush with the plate (Fig. 5.38). The dimensions of the model and the arrangement of GHFSs are shown in Fig. 5.39. 4 Post-graduate

student S. A. Mozhaisky participated in this stage of the study.

118

5 Validation and Science Experiment

Fig. 5.37 Vortex formation in dimple with relative depth 0.2 at different moments (a)–(d). Data by A. S. Guzeev

Measurements were taken for a dimple 72 mm in dia and with h = 15 mm depth (h/d = 0.2). Readings of the bismuth-based GHFS with the 5 × 5 mm2 dimensions were recorded with averaging over a time interval of 20 s. The GHFS located at the plate (4) was used to obtain “reference” values corresponding to surface heat transfer without intensification. The plate was rotated through 180◦ with a step of 10◦ (Fig. 5.38). The Surfer 5.0 software was used to process experimental data and to construct three-dimensional distributions of heat flux depending on the polar coordinates with its center at the dimple axis. This software allowed us to be represented in the form of a system of isolines. The results of some experiments are plotted in Fig. 5.40. GHFS sensitivity and coordinates are listed in Table 5.1. r/r s is the relative radius, where r is the radius corresponding to the GHFS center’s position and r s is the dimple radius. This example illustrates that a small number of GHFSs is sufficient to construct an informative and illustrative picture of heat transfer. It is especially important that time-averaged heat transfer coefficients were obtained as a experiment’s result, as it

5.2 Science Experiments

119

(a)

w

Rotaitio

n GHFS

Steam Condensate

(b)

Fig. 5.38 Schematic sketch (a) and general view (b) of model of spherical dimple with rotary part

Ø72

151

Ø50

S2 S3 w

S1

155

Fig. 5.39 Model with rotary part, view from above

121

120

5 Validation and Science Experiment

(a) Re = 3 10 3

Re = 7 10 4

Re = 1 10 5

(b)

Fig. 5.40 Heat flux field in spherical dimple with relative depth h/d = 0.2: a three-dimensional visualization of field, b isolines at dimple surface (relative coordinate are plotted on the axis, the lines correspond to heat flux) Table 5.1 Characteristics and arrangement Notation S 0 (mV/W) GHFS1 GHFS2 GHFS3 GHFS4

14 14 16 19

r/r s 0.91 0.63 0.086 1.77

is these quantities that are significant for optimization of heat exchange elements in power plants. In addition, the method allows us to compare heatmetry data with the fundamentally different studies results, for example, with the data on flows obtained by the PIV method. Numerical simulation of flow and heat transfer in an axis-symmetric dimple entailed a series of experiments that had to be conducted for verification. A◦ dimh l ple (Fig. 5.41a) with the dimension ratio d = 0.13, d = 0.5, and ϕ = 45 was considered in Isaev’s model and in our experiments; the dimple diameter remained the same (d = 66 mm). Making the dimple asymmetric alters the flow around it. The two-cell vortex structure in the dimple is replaced by a mono-vortex tornado-like one (Fig. 5.41b). This causes not only local (Fig. 5.42a) but also integral heat transfer to change.

5.2 Science Experiments

121

d/2

l

(a)

w

(b) w

w

Fig. 5.41 Shape of the asymmetric dimple shape of asymmetric dimple (a) and conditions of flow around it, compared to round dimple (b)

It can be seen (Fig. 5.42a, b) that the elongated dimple behaves similarly over both cross-sections. It has no symmetry in the heat transfer coefficient distribution relative to the longitudinal axis. This is also supported by the data numerical simulation (Fig. 5.41b). To make analysis of the curves more informative, we performed experiments on the trench (Fig. 5.43). The experimental results are shown in Fig. 5.42. It can be seen that the heat transfer coefficient at the entrance to the trench falls almost as in the dimple. Next, approaching the trench’s back edge, it increases up to the exceeding the heat transfer level at the plate by a factor of two. The deviation of the from the initial one at a distance of 3d behind the trench is 10…25%. As the curves show, the “loop” of enhanced heat transfer runs a distance of (5…10) d or more (Fig. 5.43). The comparison of average heat transfer coefficients for spherical and asymmetric dimples, as well as for the trench is shown in Fig. 5.44. The asymmetric dimples are clearly inferior not only to the spherical dimple, but also to the trench. The result is not surprising since the shape and the orientation of the asymmetric dimple are selected randomly. What is interesting here (and demands further research) is the non-monotonic character of the curve for the cylindrical trench [40].

122

5 Validation and Science Experiment

Elongated dimple

(a)

Round dimple

hs /h w

Re = 2.5 104 Re = 3.5 104

4

Re = 2.5 10 Re = 4.5 104 Re = 6.5 104

2.0

Re = 4.5 104 Re = 5.5 104 Re = 6.5 104

1.5

A

A

w

1.0 0.5 0.0

40

20

20

0

40

x, mm

Round dimple Elongated dimple

(b) hs /hw

Re = 2.5 104

Re = 2.5 104 Re = 4.5 104 Re = 6.5 104

B-B

Re = 3.5 104 Re = 4.5 104

1.5

Re = 5.5 104 Re = 6.5 104

1.0

B

0.5

B w

0.0

40

20

0

20

40

y, mm

Fig. 5.42 Local heat transfer over transverse (a) and longitudinal (b) cross-sections (relative to incoming flow) for elongated dimple

w

GHFS Fig. 5.43 GHFS’s locations

Experiments were also performed in a transverse trench with trapezoid crosssection at the wall inclination angle of 45◦ (Fig. 5.47). Measurements were taken at air velocities from 5 to 15 m/s, corresponding to Reynolds numbers (2.2 × 104 …5.7 × 104 ). The trench depth h = 60 mm (Fig. 5.45) was assumed to be the determining dimension.

5.2 Science Experiments Fig. 5.44 Relative heat transfer coefficients in trench and in wake behind it

123 hs /hw

10 4 10 4 10 4 10 4 10 4

Re=2.5 Re=3.5 Re=4.5 Re=5.5 Re=6.5

1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2

Fig. 5.45 Average relative heat transfer coefficients for spherical (1) and asymmetric (2) dimples, for cylindrical trench (3) and Isaev’s numerical simulation data for spherical dimple (4) [48]

100

50

0

l, mm

150

hs /hw

1 2 3 4

1.1

1.0

3 10 4

4 10 4

5 10 4

6 10 4

Re

The local boundary layer’s thickness is varied from 0.39 to 0.63 mm. The construction of the working section and the experimental conditions were chosen close to those in the studies by Terekhov et al. [49, 50], where the walls were heated by electric current. We chose the dimensions, cavity shape and heating zone to exactly match the data of [49]. In our experiments, heat transfer in the trench was compared with air flow had no “thermal history” in front of the cavity. Only the working section was steam-heated. Experimental results are shown in Fig. 5.46b. Figure 5.47 shows the comparison of the data obtained at Re = 4.0 × 104 with the results of [49], obtained under the similar conditions. It can be seen that heat transfer coefficients are close at the bottom of the trench, but differ over the side cross-sections. Additional analysis is required in this case as well.

124

(a)

w

45˚

60 mm

Fig. 5.46 Scheme of working section of model (a) and results of investigation (b) of heat transfer over trench with trapezoid cross-section

5 Validation and Science Experiment

60 mm

(b)

h, W/(m2 K) 70 60

Re=2.5 10 4 Re=3.5 10 4 Re=4.5 10 4 Re=5.5 10 4 Re=6.5 10 4

50 40 30 20 10 0

Fig. 5.47 Heat transfer coefficient distribution over generating line of cavity: T = const (1), q = const (2) (according to data [49])

50

100

150

200

l, mm

h, W/(m2 K) 80 70 60

1 2

50 40 30 20 10 0

50

100

150

200

x, mm

5.2.3 Heatmetry in Shock Tubes Thermal processes in the shock tube (ST) have a characteristic time of the order of 10−3 s and less [51], while heat transfer in the gas–wall system is complicated and has not been studied sufficiently. In experiments, the pressure is measured traditionally in the ST cavity and the temperature at the cylindrical and end walls [52]. To the best

5.2 Science Experiments

125

of our knowledge, heat flux has not been measured on wall surfaces this far because there is no HFS with an acceptable time constant (less than 10−5 s). As for the cases when heat transfer in the ST occurs with electric and magnetic fields present, even calculated estimates have not been obtained for the heat flux until recently. It is possible to estimate the noise stability of GHFSs for flows of ionized gas and low-temperature plasma in the electromagnetic field and, as a consequence, make forecasts for using the sensors in similar industrial conditions (for example, in studies of heat transfer in electric machines and equipment). Quick-response film thermocouples and thermometers successfully used for ST measurements have limited thermal stability (usually no more than 473 K) [53]. This does not hinder heat flux measurement even in cases when the gas temperature reaches (7…8)103 K—because the experiments are short. Estimates obtained using heat conduction theory [17, 54] show that the temperature of the inner surface of the ST wall increases only by 100…200 K during the entire experiment, so bismuthbased GHFSs are quite suitable for operation in such conditions. The measurements were taken at the Laboratory of Gasdynamics of the Ioffe Institute [52] and in the Fluid Dynamics and Thermodynamics and Heat Transfer Department of Peter the Great St. Petersburg Polytechnic University [55]. 5 GHFSs with the area of 4 × 7 mm2 area and the thickness of 0.2 mm placed on Plexiglas or Teflon were used in all experiments. First shock tube experiments at the Ioffe Institute (Fig. 5.48) were carried out in an atmosphere of xenon heated by a shock wave with the Mach number M = 6…7. GHFSs were installed flush with the inner cylindrical wall of the ST over the crosssections at distances of l = 100 and 425 mm from the closed end wall and were oriented with their smaller side along the ST axis. Piezoelectric pressure sensors with a sensitive element 4 mm in diameter were mounted over these cross-sections. Thus, heat flux and pressure were recorded first behind the incident shock wave and then behind the shock wave reflected from the closed end wall of the ST in each experiment. Figure 5.49 shows the measurements taken over the cross-section at a distance of 100 mm from the end wall of the ST. The pressure sensor’s signal increases sharply at the moment when the incident (τ = 0.35 ms) and reflected (τ = 0.75 ms) shock waves are passing through the measuring cross-section. The heat flux time variation correlates with the pressure sensor’s signal and the heat flux behind the reflected shock wave (the gas temperature is T f ≈ 7000 K) reaches 1.4 MW/m2 . Both sensors were calibrated before the start of experiments. The measuring error is about 2% for the heat flux and no more than 10% for the pressure. The shock tube at the Peter the Great St. Petersburg Polytechnic University had a closed end. The experiments were carried out for air gas with the Mach number M = 1.9. GHFSs were located at the inner cylindrical ST’s wall over the cross5 Prof.

S. V. Bobashev, N. P. Mende, Ph.D. and research fellow V. A. Sakharav from the Ioffe Institute, R. L. Petrov, Ph.D., research fellow V. V. Grigoriev from the Peter the Great St. Petersburg Polytechnic University participated in the experiments.

126

5 Validation and Science Experiment

8

(a)

4

6 9

3

5

7

10

2

1

(b)

Fig. 5.48 Shock tube at Ioffe Physical-Technical Institute of the Russian Academy of Sciences: a ST; b vacuum chamber with plasma. Numbers correspond to: 1—high-pressure chambers; 2— vacuum pump; 3—model with GHFS; 4—optical metrology system; 5—vacuum chamber; 7— pressure-shock front; 8—low-pressure chamber; 9—high-pressure chamber; 10—pressure sensors

Fig. 5.49 Pressure and heat flux in xenon medium at ST’s wall at Ioffe Institute (St. Petersburg, Russia) over cross-section at distance of 100 mm from end wall

p, MPa

q, MW/m 2 1,6

0,8

1,2

0,6

p q

0,8 0,4

0,4 0,2 0

0

0,5

1,0

1,5

5.2 Science Experiments

127

Fig. 5.50 Working end part of ST at Peter the Great St. Petersburg Polytechnic University (a), ST at Department of Thermodynamics and Heat Transfer (b), and GHFSs mounted on shock tube wall (c) and on end wall (d)

sections at distances of 3 m, 98 mm and 40 mm from the end wall and at the end plug’s centre of the channel (Fig. 5.50). Figures 5.51 and 5.52 illustrate the heatmetry results. As in the previous experiments, the start of pulses coincides with the arrival of the incident shock wave to the GHFS [56]. The GHFS at the side wall at a distance of 3 m from the end wall of the ST records a heat flux increase at the moment when the incident shock wave passes (the temperature behind it is T f ≈ 460 K) and its subsequent decrease (at τ > 2.5 ms) after cold gas arrives to the GHFS behind the contact surface (Fig. 5.51a). It was found for the GHFS located at distances of 98 and 40 mm from the ST’s end wall that the signal amplitude decreases as the distance from the end wall of the ST

128 Fig. 5.51 Heat flux (a) and pressure (b) fluctuations behind shock wave in air at side wall of ST. Distance from end wall is 3 m

5 Validation and Science Experiment q, kW/m 2

(a)

300

200

100

0

1

2

3

4

p 10 -6, MPa

(b) 1.5

1.0

0.5

0

1

2

3

4

decreases (Fig. 5.51a). At the moment when the shock wave arrives and is reflected, the GHFS at the ST’s end wall (Fig. 5.52b) is immediately surrounded with a region of immobile gas with a higher temperature (T f ≈ 900 K). The heat flux reaches a practically invariable level approximately in τ = 1.5 ms after the shock wave reflection. It is preserved up to τ ≈ 7 ms, when the ST’s end wall is approached by the contact surface that separates gas particles in high and low pressure chambers. The maximum heat flux at the cylindrical wall of the channel is twice as high as at the end wall. In our opinion, this is evidence of the difference in heat transfer mechanisms. Heat transfer at the ST’s end wall occurs in the same manner as between two semi-infinite rods [54], while convective heat transfer prevails at the cylindrical wall. This assumption is supported by measurements of heat flux behind the reflected wave in shock tube experiments carried out at the Ioffe Institute. The GHFS in these experiments was located within the boundary layer formed by co-flow behind the incident shock wave, and the reflected wave moving towards the co-flow interacted with this layer. Unlike the gas near the ST’s end wall was not at rest, making convective heat transfer possible.

5.2 Science Experiments Fig. 5.52 Gradient heatmetry data: 1—obtained from GHFSs mounted at distances of 98 mm and 2—40 mm (a); obtained from GHFSs mounted on ST’s end wall (b)

129 q, kW/m 2

(a) 200

1 2 100

0

5

10

15

0.5

1.0

1.5

q, kW/m 2

(b) 120

80

40

0

Of special interest are experiments where pressure’s variations, heat flux and temperature (measured by a platinum resistance thermometer) are recorded simultaneously at a distance of 3 m from the ST’s end wall during multiple shock wave reflections (Fig. 5.53). The heat flux and temperature variations corresponds qualitatively to the developed physical concepts, however, the start of the process (up to τ ≈ 5. . .10 ms) deserves more attention. It is commonly believed that the boundary layer does not develop at the ST’s end wall and the surface temperature changes abruptly with the arrival of the shock wave. Figure 5.54 plots the temperature and the heat flux calculated for the conditions of the experiment performed. Abrupt variation in temperature at the initial time was set in the calculations, therefore, according to the heat conduction theory [54], the initial heat flux must be infinitely large. Previous attempts to obtain the leading edge of the HFS signal close to the vertical one were unsuccessful. This was attributed to

130 Fig. 5.53 Variations of pressure sensor signals (curve 1), heat flux (curve 2), and temperature (curve 3) in ST during multiple shock wave reflections

5 Validation and Science Experiment E1, V

E2, V E3, V

1 2 3

2.0

1.0

0 -1.0 0

Fig. 5.54 Calculated temperature’s (curve 3) and heat flux (curve 1) variations at ST’s end wall compared to experiment (curves 2 and 4, respectively)

10

40

30

20

1.0

0.1

0.5

0.05

0

0

-0.5

-0.05

, ms

q, kW/m2

T, K 1 2 3 4

750

305

500

300

250

295

0

1

2

3

4

, ms

290

insufficiently fast response time of the sensor. As shown in Sect. 3.1, the response time of the bismuth-based GHFS does not exceed 10−8 s, so these sensors certainly would have detected the “spike” in the heat flux, had it occurred. The difference between the response time and the process time in the ST is more than 5 orders of magnitude. The experiments show that temperature and heat flux have completely different variations: heat flux is almost constant (except for the initial time equal to 1 ms), whereas the temperature smoothly increases by 1.9 K without an extremum (within 5 ms). In our opinion, the “electrothermal analogy” is appropriate here. In electrical engineering, the law of commutation states that the capacity voltage cannot change abruptly. The temperature on the surface of a solid body also cannot change abruptly, although for more complex physical reasons. Otherwise, heat flux at the start of the process would be infinite, which contradicts both the physical concepts and the experimental results. It is the experimentally found shape of the curves shown in Fig. 5.54 that should be used for numerical simulation of heat transfer in the ST, for example, to calculate the time-variable heat transfer coefficient on the ST’send wall.

5.2 Science Experiments

131

(b)

(a) 4

6

2 1

3

5 8 3

7

6

Fig. 5.55 Scheme (a) and general view (b) of ST’s end wall equipped with nozzle at Ioffe Institute with external magnetic field. Numbers correspond to: 1—ST; 2—pressure sensor; 3—probes with GHFS; 4—nozzle; 5—wedge-shaped plate; 6—electric magnets; 7—gas motion; 8—reflected shock wave

The next series of experiments was carried out in the ST at the Ioffe Institute (Fig. 5.55) [5], dedicated to investigating the effect of a very strong external magnetic field in supersonic plasma flow in a diverging nozzle. GHFSs 3 were installed on the wedge-shaped plate (Fig. 5.56) that is a continuation of wall 4 of the supersonic nozzle (Fig. 5.57). The chamber with the supersonic nozzle was additionally equipped with electromagnets 6 to initiate a pulse magnetic field at the moment when gas was passing through the nozzle. Supersonic flow measurements were taken on the experimental setup for studying the interaction between low-ionized gas flow and the external magnetic field [5]. A working rectangular section was made of dielectric material (Teflon) for this purpose. The section was positioned at the end of the ST channel and was separated from it with a thin plastic diaphragm (to maintain the initial pressure not higher than 1 Pa in the section). Wedge-shaped nozzle 1 (Fig. 5.57) with the width of 75 mm and the exit nozzle throat height of 9 mm was located inside this section. Plate 2 was adjacent to the outlet edge of the supersonic nozzle. Consequently, the supersonic − → flow with the velocity vector W was deflected the plate by an angle of 15◦ . A pair of electrodes E (with the 6 × 4 mm plan-area dimensions) was mounted near the dihedral angle vertex formed by the plate and the nozzle wall, with the external voltage supply connected to these electrodes to initiate the current I in plasma flow. Two co-axial electromagnetic induction coils were placed at the working section top and the bottom and initiated a pulse magnetic field with magnetic induction B up to 1.5 T for 4.5 ms. Two 4 × 7 mm GHFSs D1 and D2 were installed flush with the surface. One was mounted at the center of the plate and the other was mounted at its edge. The working section’s vertical side walls had glass windows for gas flow optical diagnostics using a shaded display oriented to view the cross-section of the outlet nozzle. Analysis of the shadow patterns allows to assess the shock waves structure in the flow.

132

5 Validation and Science Experiment

Fig. 5.56 GHFS at Teflon nozzle element

Fig. 5.57 Sketch of supersonic nozzle: 1—nozzle; 2—deflector plate; electrodes E; GHFSs D1, D2; I, B, W are vectors of current, magnetic induction, and plasma flow velocity, respectively

D2 B

w I

w

D1 1

E

2

As plasma flow reaches the electrodes, current is initiated in plasma. The duration of this process depends on the time of steady plasma outflow and equals about 1 ms. Magneto-hydrodynamic (MHD) interaction changing the flow pattern takes place during this period. Shadow patterns and GHFS signals with different intensity of the MHD interaction, i.e., with varying current through the plasma and of the magnetic field induction were recorded in the experiments. Considering the shadow patterns, we can draw conclusions about the shape of shock wave surface above the plate when the MHD intensity affects the supersonic

5.2 Science Experiments

133

Fig. 5.58 General view of shock wave surface under MHD interaction according to data from [5]

D1 D2 E

flow. The hypothetic shape of the shock wave surface [5] and the location of electrodes and GHFSs are shown in Fig. 5.58. Experiments have shown that the central part of the shock wave surface located above sensor D1 practically did not change the slope to the plate. While the peripheral sections were substantially sloped, it moving away from the surface with increasing MHD intensity. Both GHFSs, D1 and D2, were tested for noise stability. Their signals were compared for this purpose. The signals were recorded either with only external magnetic field applied, or only with the current (up to 500 A) passing through the plasma, or in the absence of these two disturbing factors. It has been found that the effect of all these factors on the readings of the GHFS is within the accuracy of measurement and experimental reproducibility. Notably, the effect of MHD interaction on the flow was not observed in all of the given cases. The MHD interaction considerably affects the signal from sensor D2 and practically does not affect the signal from D1. Figure 5.59 illustrates the GHFS signals D2 at different intensities of MHD interaction (the curves for this quantity are characB2l σ terized by the dimensionless Stewart number S = 0ρw0 c , where B0 is the characteristic of magnetic induction, w is the medium velocity, l0 is the plates characteristic dimension, σc and ρ are the specific electrical conductivity and the medium density, respectively). The plots demonstrate that the heat flux under the MHD interaction (S > 0) is substantially higher than in the case S = 0, and this difference grows with increasing S. The GHFS records rather rapid changes in heat flux and yields quantitative information about the process in view of the above-found noise stability. The heat flux increases sharply in 500 µs for the curve corresponding to S = 0.42 and in 1.1 ms for the curve corresponding to S = 0.32. We believe that this is associated with the separated flow initiated by MGD interaction (the curve at S = 0) in the nozzle in front of the plate. Recall that steady outflow continues for about 1 ms, so further description of the flow pattern is only hypothetical. It was demonstrated in the above-described experiments that bismuth single crystal-based GHFSs were capable of working in pulsing high-temperature gas flow. The experiment showed that these sensors could operate in the conditions of high external electromagnetic fields without special signal amplifiers and could be used in studies of processes with characteristic frequencies up to 1 MHz.

134 Fig. 5.59 Heat flux at nozzle wall as function of presence of magnetic field and its strength

5 Validation and Science Experiment q, kW/m2 S = 0.32

1200 1000 S = 0.42 800 600 S=0 400 200

0

0.5

1.0

1.5

2.0

, ms

The next step was to conduct experiments allowing to combine all elements of the electromagnetic system in a test object streamlined by supersonic gas flow or by plasma flow. The electromagnetic device is sketched in Fig. 5.60. A test model that is a rotation body fabricated from dielectric material is shaped as a cylinder 26 mm in diameter and 40 mm long, coupled with a cone with the vertex angle of 60◦ . Solenoid 1 comprising 20 turns of 1 mm dia copper wire is located co-axially with the model on the cylindrical part. Ring electrode 2 is located flush with the surface at the coupling point of the cone and the cylinder. One of the solenoid coils is connected to the electrode and the other to the voltage supply. A lumped parameter line composed of 14 LC-cells, charged up to the voltage of 300…700 V prior to experiment, is used as a pulse voltage supply. The other pole of the voltage supply is connected via secondary winding of a highvoltage transformer to central electrode 3 located along the model axis. Electrode 3 is a 6 mm dia brass rod with a conical nose. Pulse with a voltage of 12 kV and a duration of 1 µm is supplied to the transformer’s primary winding to start up the device. Pulse of the secondary winding with the amplitude up to 30 kV causes electric breakdown in the discharge gap between central electrode 3 and ring electrode 2. As a result, the supply is discharged for 1.5 ms and the pulse current of about 100 A is initiated in the circuit consisting of the plasma gap and the solenoid. Magnetic field induced as the current passes through the solenoid interacts with the current in plasma 4. Consequently, plasma starts rotating around the body in the azimuth direction. Experiments were carried out on the experimental setup based on a large shock tube at the Ioffe Institute [52]. The rectangular section with a supersonic nozzle was located at the end wall of the low-pressure channel. Nitrogen supersonic flow over the outlet cross-section of the nozzle had the following parameters: pressure 5 kPa, density of 0.04 kg/m3 , temperature of 440 K, velocity of 1600 m/s, Mach number 4. Steady outflow of nitrogen lasted for 1.5 ms.

5.2 Science Experiments

135

1

2 3

1 ms

Tr

4 Fig. 5.60 Sketch of electromagnetic device and its electric circuit [152] Fig. 5.61 Model (streamlined body) with electrodes and GHFSs at surface

GHFS 1

O-ring electrode

End electrode GHFS 2

The body of rotation was positioned inside the working section behind the nozzle outlet cross-section. The general view of the body with GHFSs at its surface is shown in Fig. 5.61. It was found in the experiments [6] that the plasma rotation frequency substantially depended on the external voltage supply connection polarity. The average rotation frequency was 30 kHz with the annular anode serving as the cathode, and 15 kHz with the annular electrode serving as the anode. Heat flux at the surface of the rotation body was measured for both types of electrode connection. GHFSs with the 2 × 2 mm plan-area dimensions were mounted on the rotation body cylindrical part at a distance of 15 mm from the annular electrode (Fig. 5.63). The volt-watt sensitivity of the GHFSs was 5.3 mV/W.

136 Fig. 5.62 Heat flux variation at cylindrical surface in static air for different variants of connecting the voltage supply when the annular electrode serves as: 1—cathode, 2—anode

5 Validation and Science Experiment q, MW/cm 2 2,0 1,5

1 2

1,0 0,5 0,0

0

200

400

600

800

1000

1200

, µs

In the first series of experiments, the heat flux was measured in static air at a pressure of 50 kPa for two different variants of connecting the external voltage supply. The supply voltage was 500 V, while the plasma current reached 700 A. Measurement results are shown in Fig. 5.62. It can be seen from the curves that the quasi-stationary heat flux is reached approximately 300 µs after the electromagnetic device is started up. Stationary heat flux is larger for the case when the annular electrode serves as the cathode (curve 1) by almost an order of magnitude, compared to the case of opposite polarity. In the second series of experiments (Fig. 5.63) on nitrogen supersonic flow (M = 4) around the rotation body. The plasma current did not vary and remained the same as in experiments without flow for both types of electrode connection. However, the same as in experiments without flow, the average heat flux in the case when the annular electrode serves as the cathode (curve 1) is much larger than that in the case of opposite polarity (curve 2). This is might be attributed to non-uniform distribution of the potential in gas discharge. Since the potential gradient near the cathode is larger than that near the anode, heat release in gas discharge should be larger in the cathode vicinity. When the annular electrode serves as the anode (curve 2), the pulsation frequency of 15 kHz corresponds to the rotation frequency of the plasma measured by a photographic recorder. When the annular electrode serves as the cathode, the plasma rotation frequency is twice as much. In this case, the type of the GHFS signal does not allow detecting the characteristic pulsation frequency (curve 1). However, in 300 µs from the start of the process, the average heat flux becomes practically invariable in time and the pulsation amplitude is small with respect to the GHFS’s signal. Since the energy spent for exciting gas discharge plasma is compared with the kinetic energy of the supersonic flow in the given experiments, heat flux to the body surface appears to be larger during operation of the electromagnetic device than when the device is switched off (Fig. 5.63, curve 3). Measurements have shown that it is possible to achieve the regime of gas discharge plasma flow around a body when heat flux at the surface is close to stationary. The

5.2 Science Experiments

137

q, 2

MW/cm

7 6 5 4

1

3

2 3

2 1 0

0

200

400

600

800

, µs

Fig. 5.63 Heat flux variation at the model’s surface in nitrogen supersonic flow for different connections of voltage supply with annular electrode serving as: 1—cathode, 2—anode; 3—curve corresponds to heat flux with MHD device switched off

scheme can then be used for simulating high-temperature supersonic gas flow around bodies, for studying the results of MHD interaction upon general flow near bodies and the heat flux to the body surface. Another series of experiments was devoted to the measurement of thermal characteristic during a supersonic flow around a model without an MHD effect. Reliable surface distribution data for models heat flow. For this, a special model was created from plexiglass (Fig. 5.64). At the conical and cylindrical surfaces of the model along 6 GHFS were installed. All GHFS had the same geometric dimensions in the plan 2.2 × 2.2 mm and the thickness of 0.2 mm. The first series of experiments was performed using a model without device MHD. Numbering of GHFS at conical and cylindrical the surface of the model is carried out sequentially, starting from the top of the cone (Fig. 5.65). In data experiments, changing the parameters of the gas in the nozzle’s outlet section of the was carried out by varying the pressure P in shock wave’s front in the low-pressure chamber. The results of the first two experiments, allowed a comparative analysis of heat fluxes at the model with and without exposure. Heat flux obtained in the third experiment were used when comparing with numerical simulation of supersonic flow. In Fig. 5.66 shows the GHFS signals on a conical surface models registered in experiment No. 1 (a) and in experiment No. 2 (b). It can be seen that in both figures, the signals all three sensors have a similar shape and are quite close. The maximum signal in experiment No. 1 is approximately 2 times less than in experiment No. 2, which is practically due to twofold difference in gas density in a supersonic flow. Also, in the experiment No. 1 there are minor signal oscillations. This is apparently due to a greater degree boundary layer turbulence due to lower gas density in this flow mode.

138

5 Validation and Science Experiment

GHFS

Fig. 5.64 Model with 6 GHFS Fig. 5.65 GHFS location and numbering at the surface of the model

4

5

6

3 2 1

In Fig. 5.67 the GHFS signals on a cylindrical surface. In this case, a similar character is also observed the behavior of all the curves. Comparing the numerical values of these signals, it can be seen that they are much smaller than the value of the sensor signals on the conical surface. Note that the difference in sensor signals at a cylindrical surface in experiment No. 1 and No. 2 less the corresponding difference of the sensor signals on the conical surface. Since the signals of the sensors in each group are quite close in its value for calculating the heat flux in each experiment it makes sense to choose the smoothest curve. In Fig. 5.68 shown heat fluxes to the conical surface of the model, calculated at the signal of the State Customs Service No. 2. The black curve shows the heat flux, corresponding to experiment No. 1, and the blue curve corresponding experiment No. 2.

5.2 Science Experiments

139

(a)

(b)

U,mV

U,mV

15

30

10

20

5

10

0

0 0.0

0.5

1.0

1.5

0.0

,s

0.5

1.5

1.0

,s

Fig. 5.66 GHFS signals on the conical surface of the model obtained in experiment No. 1 (a) and experiment No. 2 (b)

(a)

(b)

U, mV

U, mV 4

3

3 2 2 1

0

GHFS

4

GHFS

5

GHFS

0.00

0.25

0.50

0.75

6

1 0

1.00

0.00

0.25

0.50

GHFS

4

GHFS

5

GHFS

6

0.75

1.00

, ms

, ms

Fig. 5.67 GHFS signals on the cylindrical surface of the model obtained in experiment No. 1 (a) and experiment No. 2 (b) Fig. 5.68 Heat flux to the conical surface of the model, calculated by the signal GHFS 2

q0, kW2 m 600 500 400 300 200 100 2 0 0.0

0.5

1.0

1.5

, ms

140 Fig. 5.69 Heat flux to the cylindrical surface of the model, calculated by the signal GHFS 5

5 Validation and Science Experiment q0, kW2 m 80

60

40

20 2

0 0.0

0.2

0.4

0.6

0.8

, ms

In Fig. 5.69 shows the heat flux to a cylindrical surface. The black curve is shown heat flux corresponding to experiment No. 1, and the blue curve corresponding to experiment No. 2. The results show that the heat fluxes to the surface of the model in experiment No. 2 is smaller than in experiment No. 1, which is also caused different gas densities.

5.3 Heat Transfer During Phase Changes 5.3.1 Condensation Study of heat transfer during condensation is important for improving the technical and economical properties of thermal power plants, ensuring the safety of nuclear power plants, increasing the effectiveness of condensers in power-generating and cooling facilities, introducing new heat equipment, etc. An extensive database has been accumulated from experimental studies of heat transfer during condensation, mostly involving thermometry [57–60]. However, these experiments do not yield sufficiently reliable results since thermometric measurements allow to calculate only time-averaged heat fluxes and heat transfer coefficients. Unsteady heat transfer, waves in condensate films, etc., cannot be described within this approach. The GHFS has a small response time, making it a nearly inertia-free measuring device that can track the oscillatory processes during condensation. Our experiments involved heterogeneous gradient heat flux sensors (HGHFS) based on the 12H18N9T steel + nickel composition with a sensitivity of 0.008 mV/W. A circular pipe was chosen as an experimental object to assess the applicability of the new technique, since heat transfer during condensation at the surface of such pipes is well understood both theoretically and experimentally. Heat transfer during condensation was considered both on the outer and inner surface of a single pipe.

5.3 Heat Transfer During Phase Changes

141

Fig. 5.70 HGHFS with commutating wires (a) and its location at pipe surface (b, c)

5.3.1.1

Condensation at Outer Surface of the Pipe

We used HGHFSs for our experiments since their design seemed more reliable for operation in a mixture of steam and liquid. HGHFSs made of the 12H18N9T steel + nickel had the plan-area dimensions of 10 × 10 (Fig. 5.70). The HGHFS’s sensitivity depends on its temperature, so its signals can be interpreted by using thermometry in the measurement zone. Thermometry is also necessary for calculating the heat transfer coefficient. That is why the HGHFS was connected using a three-wire scheme: the sensor signal was read from two chromel wires, and a copel wire was connected to one of them to form a thermocouple to measure temperature. The experimental zone consisted of two coaxially arranged pipes: an internal pipe of 12H18N9T stainless steel and an external pipe made of reinforced rubber hose. Steam from a steam generator was injected into the annular gap between the pipe and its housing, with the generator power varied within 2–12 kW [61]. Condensation heat was removed by the cooling water fed into the metal pipe. The inlet water temperature was controlled within 15–55 ◦ C. The water flow rate was adjusted using a valve and a flow meter. Condensate was removed to a condensate drum with a measurement scale. A thermocouple was mounted in the condensate drain line to measure the condensate temperature (Fig. 5.71). The experimental setup allowed to incline the pipe setup within 0◦ –90◦ (Fig. 5.72a) with a step of 10◦ using separating discs, one of which was mounted on the stationary part of the setup and the other on the movable part. The pipe could be rotated so as to minimize the number of GHFSs and thermocouples needed and simplify recording and processing data. This way we could measure the local heat flux and calculate the local heat transfer coefficient at all azimuth angles using a single set of primary converters. GHFSs and thermocouples were mounted on the same generatrix of the pipe in all measurement zones.

142

5 Validation and Science Experiment

5

Water

Steam

T’1 E1 T1

4

T’2 E2 T2

2

T’3 E3 T3

6

T’4 E4 T4

1

Water Condensate q, kW/m2

7 3 ,s

Fig. 5.71 Simplified scheme of pilot setup. 1—Steam generator, 2—experimental part, 3— condensate drain, 4—voltmeter, 5—videocamera, 6—computer, 7—heat flux graph

The setup is constructed to rotate the experimental part within 0◦ –180◦ at a step of 15 . The rotation device is made as a butt hinge (Fig. 5.72b) with a latch mechanism. Steam was fed into the annular space between the pipe and its housing from above and tap water could be injected either counter- or co-directional with the steam flow. Residual steam and condensate were removed to the condensate collector. Signals were taken using an upgraded recording N339 ammeter/voltmeter, fitted with a DCA. The automatic plotter was replaced with a mirror and a laser (Fig. 5.73). Removing the plotter helped lower the response time of the device and increase its sensitivity. The laser beam was reflected to the scale, while the signal was recorded by a video camera. The video recordings were digitized and used to plot time-dependent heat flux graphs. The applicability of gradient heatmetry to condensation studies was experimentally verified using a setup with a vertical pipe. Steam and cooling water were running in counter-current. Steam was injected into the annular space from above, and the cooling water from the bottom. The average temperature along the length of the pipe wall detected by the thermocouples was 78 ◦ C. Parameters required for the calculations were borrowed from the tables in [62]. The average heat flux determined experimentally corresponds to the values found using Labuntsov’s formula [63]. Condensate film flow was lam◦

5.3 Heat Transfer During Phase Changes

143

(a)

(b) Fig. 5.72 Structure of inclination (a) and rotation (b) devices

f ilm

inar, the Reynolds number Re = 659 < Recr < 1600. The average heat transfer coefficient along the length of the vertical pipe was h calcul = 6.10 kW/(m2 K). The experiments with the pilot setup revealed that HGHFS No. 1 was not mounted flush with the pipe wall, but was actually located deeper in it, creating a cavity; this meant that the signal it provided was not sufficiently reliable. Figure 5.74 shows a time-dependent heat flux graph plotted using the readings from the other three HGHFSs [64]. While the flow in the vertical pipe is practically axisymmetric, such symmetry is impossible in the horizontal pipe because of gravity. The measurements in our experiments could be taken at any azimuth angle ϕ = 0. . .360◦ , measured from the upper (frontal) generatrix corresponding to ϕ = 0◦ . The step of the angle ϕ in our experiments was 15◦ . The main results shown in Fig. 5.75.

144

5 Validation and Science Experiment

Fig. 5.73 Upgraded device H339 Fig. 5.74 Time-dependent heat flux graph for vertically installed pipe

q, kW/m2 200 175 150 125 100 75 50 0

5 HGHFS No

10 1

15 3

20

,s

4

pipe lenght averaged value

The results were generalized (see the central area in the figure) using the dimensionless local heat flux q(0) (5.4) q= q(ϕ)

5.3 Heat Transfer During Phase Changes

145

Fig. 5.75 Distribution of local heat flux along surface of horizontal pipe

~ 0 q 30

330 1.0

300

60

0.5 90

270

240

120 150

210 180

where q(ϕ) is the local heat flux near the pipe generatrix with the azimuth angle ϕ; q(0) is the local heat flux near the upper (frontal) generatrix with the azimuth angle ϕ = 0◦ . hori zontal The heat flux averaged along the perimeter of the horizontal pipe was h ex p = 5.79 kW/(m2 K) which differs from the above calculated by 4.3%. Thus, our method gives good results for horizontal pipes as well Calculation by the Nusselt formula

hor h cal

= 0.728 ×

4

k 3f gρr ν f dt

= 6.05 kW/(m2 K)

(5.5)

If the pipe deviates from the vertical, condensate flow is no longer axisymmetric. It is rearranged and splits into the main and near-bottom regions. Studies of heat transfer in an inclined pipe involve finding the heat flux distribution along the pipe perimeter. The pipe’s inclination angle ψ varied from 0◦ to 90◦ in our experiments. The angular heat flux graphs are were plotted for the cross-sections spaced 300, 700 and 800 mm away from the upper cross of the pipe. Same as in the previous section, the angular distribution of heat flux is represented in its dimensionless form q = q(ϕ) (Fig. 5.76). The pipe inclination angle ψ was taken equal to 0◦ for the vertical pipe and 90◦ for the horizontal one. Angular heat flux graph, plotted using the readings from the HGHFS, provide data on the evolution of condensate film flow. As the inclination angle ψ increases, the flow becomes more complex and completely different from that in the vertical pipe, with diverging effects on the local heat flux. Figure 5.77 shows the variation in the average dimensionless heat flux depending on the pipe inclination angle ψ and Fig. 5.78 shows the average heat transfer coefficient as a function of the same angle ψ.

146

5 Validation and Science Experiment

(a) q 2.0

(b)

(c)

~ q

0

2.0

30

60

1.0

0

~ q

0

60

1.0

90

2.0

30

0

0 30 60

1.0

0

90

90

120

120 150

120

150 HGHFS No

150 3

1

4

Fig. 5.76 Angular heat flux graphs for pipe inclined at angle ψ equal to: 30◦ (a), 60◦ (b), 80◦ (c) Fig. 5.77 Variation of average relative heat flux depending on pipe inclination angle ψ

~ q

~ q

1.1 12.6%

~ q

1.0 19.1%

~ q 0.9

0

20

40

60

80

Figure 5.78 shows the dependence of the averaged heat transfer coefficient on the pipe inclination angle during condensation. The highest surface-averaged heat transfer coefficient is found for the pipe inclination of 20◦ , h = 6.94 kW/(m2 K), which exceeds the vertical pipe by 14.9%. The difference with heat transfer coefficient for the horizontal pipe is 25.3%. Comparing the heat flux and heat transfer coefficient, both local and averaged, obtained using our heatmetry technique with the values calculated using the Nusselt model for vertical and horizontal pipes, we can confirm that our new approach is reliable.

5.3 Heat Transfer During Phase Changes

147

h, kW/(m2

6.5

14.9% 25.3%

6.0

5.5

5.0

0

20

40

60

80

Fig. 5.78 Variation of average heat transfer coefficient as function of pipe inclination angle ψ Nu 1.6

W. Nusselt P. L. Kapitsa S. S. Kutateladze D. A. Labuntsov W. M. Nozhat W. Roshenow

1.2 0.8 0.4 0.0

1

10

100

Re

Fig. 5.79 Comparison of different calculations of Nusselt number

GHFS-based heatmetry allows to plot distributions of local heat flux and local heat transfer coefficient along the pipe perimeter and length for any pipe orientation, from vertical to horizontal. The technique allows us to find the optimal inclination angle for the pipe, increasing the average heat transfer coefficient corresponding to the pipe’s vertical or horizontal orientation by 14. . . 25%.

5.3.1.2

Condensation at Inner Surface

Even though condensation in pipes, particularly if steam is injected from the bottom, is very complex, most calculations are based on the model offered by Nusselt in 1916 [65]. Results obtained by various authors differ by more than two times, which means that local heat transfer coefficients inside the pipe require experimental adjustment (Fig. 5.79).

148

5 Validation and Science Experiment

Water

3

Tin

Steam

E1

T’1

T1 E2

T’2

T2 E3

T’3

T3 1

E4

T’4

T4 Tin

Water

Condensate 2

Fig. 5.80 Scheme of experimental setup: 1—steam generator, 2—condensate collector, 3— experimental part

We used HGHFSs made of 12H18N9T stainless steel (18% Cr, 9% Ni, 2% Mn, 0.8% Ti) + nickel again (Fig. 5.70) for our experiments. These HGHFSs can measure temperature in a range limited only by the melting temperature ( 1400 ◦ C), and have a uniq response time (τmin 10−8 . . .10−9 s). Thermal resistance of the HGHFS is close to that of stainless steel used to produce the experimental section, so installing the sensor only slightly distorts the temperature field. HGHFSs are simpler to manufacture than single-crystal bismuth GHFSs, but they have a lower volt-watt sensitivity [66]. The experimental setup’s structure diagram is shown in Fig. 5.80. Steam was supplied to the pipe by a steam generator. Three tubular electric heaters with the overall power of 12 kW were installed at the bottom of a 150 l stainless steel tank, which allowed to change its output power in steps.

5.3 Heat Transfer During Phase Changes

149

(b)

HGHFS-3

HGHFS-4

0.2 m 0.2 m

HGHFS-2

0.2 m

HGHFS-1

0.2 m

0.2 m

(a)

Fig. 5.81 Pipe without segment (a), GHFS position (b)

A T-joint with a hermetically sealed condensate collector was assembled to inject steam into the pipe from the bottom. The flow rate of cooling water was controlled within 0.025. . . 0.05 kg/s. The experimental setup includes a line for supplying air to steam. A T joint with a shutoff valve preventing steam from entering the air flow meter was connected to the steam duct for that purpose. Air flow rate was controlled within 0. . . 0.35 g/s. It is important to maintain a smooth surface inside the pipe during the experiment in order to avoid distortions in the film flow. A DK7725 EDM machine tool was used to cut 4 segments in the pipe. The first was 300 mm from the pipe’s upper crosssection, and all the following ones were spaced 200 mm from each other (Fig. 5.81). The cuts were only 0.2 mm wide, making it possible to solder the segments almost without distorting the inner surface of the pipe. The scheme of the measuring segment is shown in Fig. 5.82. “Cuttings out” for HGHFS were milled in the segments and holes were drilled for HGHFS and thermocouple wires (Fig. 5.83). The “cuttings out” were depth-sized using a boring machine to install the HGHFSs flush with the surface. HGHFSs were mounted on insulating mica substrates 0.05 mm thick and then glued with epoxy resin. Excess resin was removed and the surface was polished after the sensors were

150

5 Validation and Science Experiment

Fig. 5.82 Scheme of measuring segment: 1—pipe segment, 2—HGHFS, 3—mica, 4—wires, 5—thermocouple, 6—ceramic tube

1 3

4

6 2 5

installed. The thermocouple junction was brought flush with the surface through a ceramic tube and soldered. After the HGHFS and the thermocouple were installed, each segment was soldered to its cut-out spot. The applicability of the HGHFS in the first experiments was tested using thermocouples installed on the outer surface of the pipe. After all segments were installed, the pipe was placed into plastic housing and fixed using two rubber plugs, which allowed to quickly assemble and disassemble the setup. All wires were brought out through pinholes in the upper plug, so that the setup remained watertight (Fig. 5.84). A measurement system by National Instruments was used in the study. A flowchart illustrating the processing and archiving of signals is given in Fig. 5.85. HGHFS signals were transferred to the TB-2706 terminal board and forwarded from it to the measuring board. The signals were digitized and transferred to a computer with LabVIEW software via the PXI-1050 chassis. The signals were then processed and recorded to file. The scheme for measuring HGHFS signals deserves special attention. A PXI-6289 measuring board was used to measure low signals from the HGHFS. According to specification, the board sensitivity within the lower measurement range (±0.1 V) is 0.8 µV. However, it was observed during measurements that the alien crosstalk generates far greater noise. A 0. . . 45 Hz low-frequency filter removed the 50 Hz crosstalk from electric power circuits. According to studies on condensate films oscillations [67–69], the oscillation frequency is within 5. . . 25 Hz, so it can be assumed that the true signal is not removed by such filtration. Experiments with traditional injection of steam from above were conducted to verify whether the HGHFSs are applicable to studying condensation inside pipes.

5.3 Heat Transfer During Phase Changes

151

Fig. 5.83 Step by step manufacturing of measuring segment

Reducing the output power of a steam generator from 12 to 7 kW only insignificantly changes the heat flux and the total heat balance in the experimental section. Full condensation of water steam was observed at an output power of 5 kW, which is confirmed by measurements of the condensate flow rate and the signal from HGHFS No. 4. Figure 5.86 shows time-dependent heat flux graphs observed during the experiment with 12 kW output power of the steam generator. Fluctuations of heat flux were detected by HGHFSs No. 3 and 4, corresponding to wavy flow of condensate film. Heat flux detected by HGHFSs No. 1 and 2 decrease because air is present in the steam, while the heat flux detected by HGHFS No. 3 increases due to a decrease in

152

5 Validation and Science Experiment

Fig. 5.84 Measuring segment soldered into pipe (a), pipe section with wires brought out through upper rubber plug (b) and assembled setup (c)

the condensate film thickness. Air does not affect the readings from HGHFS No. 4, but the effect of condensate film thickness is more pronounced (Fig. 5.87). Based on the experimental results, we plotted the spectral power density for HGHFSs No. 3 and 4 at different power outputs of the steam generator and different flow rates of cooling water. Figure 5.87a shows the spectral power density function for HGHFS No. 3 at different power outputs of the steam generator and the cooling water flow rate of 0.05 kg/s. Peak frequencies are highlighted in the graphs. The graphs obtained for the cooling water flow rate of 0.033 kg/s are qualitatively similar to the previous ones. The fluctuation’s characters and peak frequencies are the same, pointing to a

5.3 Heat Transfer During Phase Changes 4 channels (signals from HGHFS)

TB-2706

153

PXI-6289 Computer with LabVIEW software PXI-1050

10 channels (signals from thermocouples)

SCXI-1303

SCXI-1102C

Fig. 5.85 Flow-chart for recording experimental data

non-random nature of such fluctuations. Peak frequencies are found within 2–17 Hz, corresponding to the traditional understanding of their nature [67–69]. The condensate flow rate does not change if steam is injected from above with the output power of the steam generator varying within 12...7 kW, which means that the Reynolds numbers also remain unchanged. Therefore, it is safe to assume that the oscillation frequency is determined by the steam velocity and does not depend on the Reynolds number for condensate film. Steam injection from the bottom is rare for traditional power industry but is widely used in geothermal power generation. Understanding condensation in vertical pipes is important for designing wells and pipelines for geothermal power. Steam may also be injected from the bottom during emergency operation of nuclear power plants equipped with steam generators with U-pipes (Figs. 5.88 and 5.89). As the steam generator produces an output power of 12 or 10 kW, the condensate “sticks” to the pipe wall. Some liquid flows down, while most of it is driven up the pipe by the steam flow. This regime is the most effective in terms of heat removal. Two alternating regimes are established with the steam generator power reduced to 7 kW: reflux condensation and plug mode. The transition from plug mode to reflux condensation is accompanied by plug oscillations formed by the condensate. The plug velocity reached 2 m/s. With the power further reduced to 5 kW, the condensate plug dropped down, as seen from. The heat flux at HGHFS No. 1 remained equal to zero during the entire experiment. Neither steam nor condensate film reached that sensor. Fluctuations of heat flux at HGHFSs Nos. 3 and 4 were found to be periodic: the plug kept oscillating only at that height, its frequency close to 0.5 Hz (Fig. 5.90).

154

5 Validation and Science Experiment

q, kW/m2 100

Steam

50

Water 0

5

10

15

20

,s

5

10

15

20

,s

5

10

15

20

,s

5

10

15

20

,s

2

q, kW/m

100 50

0

q, kW/m2 100 50

0

Water

Condensate q, kW/m2 HGHFS 100 TK 50

0

Fig. 5.86 Heat flux graph with steam generator output of 12 kW

5.3.2 Heat Transfer During Boiling Heat transfer at high-temperature surfaces during water boiling has been studied since the first half of the 20th century. Nukiyama, Bromley, Labuntsov, Zenkevich, Petukhov [70–72] and other Russian and foreign scientists [73] made great contributions to that line of research. Experimental studies of that subject are still relevant,

5.3 Heat Transfer During Phase Changes

155

2.5

(a) SPD, V2

3.9

10-13

14.6

5.5

10-14 10-15 10-16 10-17 0

10

Gsteam, g/s

20

30 40

5.3

f, Hz

4.4 3.1 2.1

2.0

(b) SPD, V2

3.7

10-13

16.0

4.0

10-14 10-15 10-16 10-17 0

10

20

30 40

f, Hz

Fig. 5.87 Spectral power density for HGHFS No. 3 with different power outputs of steam generator: a cooling water flow rate is 0.05 kg/s; b cooling water flow rate is 0.033 kg/s

especially concerning crucial matters such as NPP safety. We used HGHFSs made of 12H18N9T stainless steel + nickel composition for our study. Grey surface of the HGHFS allows to determine the integral heat flux with the radiation component included. The thermal conductivity of the HGHFS is 15 W/(m K), so mounting the sensor on materials with similar thermal conductivity (e.g., VT22) would insignificantly distort the temperature field. To study unsteady heat transfer during water boiling at high temperature surfaces, we assembled an experimental setup mounted on a metal frame (Fig. 5.91). The basic elements of the setup included a pusher-type muffle furnace, a water tank, a lever

156

5 Validation and Science Experiment

q, kW/m2 300 200 100

Condensate Water

0 -100 0

5

10

15

20

,s

10

15

20

,s

10

15

20

,s

q, kW/m2 300 200 100

0

5

2

q, kW/m

300 200 100 0 -100 0

Water

5

Steam q, kW/m2

Condensate 100 50

0

5

10

Fig. 5.88 Heat flux graphs for steam generator power of 12 kW

15

20

,s

5.3 Heat Transfer During Phase Changes Fig. 5.89 Thermal load under variable conditions

157 Reflux condensation

Plug regime

q, kW 6 5 4 3 2 1

0

100

200

300

400

,s

with guiding rails to lift and lower the sample, an Evercam 1000-4-C high-speed camera and a system with a PXI-1303 board by National Instruments. The furnace consists of a ceramic tube with 30 mm inner dia, 1 mm nichrome wire, ceramic insulating beads, thermal insulation of the muffle, frame, 3 kW power controller and a thermocouple to monitor temperature. The following samples were used: an L68 brass cylinder, L68 brass and titanium spheres (balls). All samples had a threaded connection to a stainless-steel holding rod on the side of the sample opposite to the HGHFS. The L68 brass cylinder of 25 mm diameter, 60 mm long was machined on a 16K20 lathe machine, and holes were drilled in it on a drilling machine (see Fig. 5.92). After achieving acceptable surface roughness, 1.7 mm through holes were made in the sample for chromium-aluminum thermocouples and HGHFS wires, plus blind 7 mm deep holes with M3 thread for holding rods of stainless steel. Test samples were made on a milling machine using low-diameter milling tools (Fig. 5.93). HGHFSs were sealed with the Donny Deal high-temperature compound. Since chromium-aluminum thermocouples were used only to monitor the sample temperature, the junctions were put inside the titanium sample (Fig. 5.92a). We additionally planned to obtain the distribution of local heat flux over the surface of a titanium ball. The holder rod was modified for that purpose and fitted with a rotating device (Fig. 5.92) that was a locknut graded every 10◦ , fixing the HGHFS position. The ball had a 0◦ notch to set the angle of its rotation [74]. All samples were heated in the vertical pusher-type muffle furnace to 450. . . 500 ◦ C and put into water of 25 ◦ C temperature. The geometry of the bodies (cylinder and ball) was selected to cover both the applied and theoretical aspects of the problem. Using a ball shape in a laboratory

158

5 Validation and Science Experiment q, kW/m2 500 250 0

Water

5

10

15

20

,s

0

5

10

15

20

,s

0

5

10

15

20

,s

0

5

10

15

20

,s

0

q, kW/m2 500 250 0

q, kW/m2 500 250 0 -250

q, kW/m2

Water Steam Condensate

500 250 0 -250

Fig. 5.90 Heat flux graphs for steam generator power of 5 kW

experiment reduces the end effects, which is necessary for developing a physical model. The experimental results are shown in Figs. 5.94 and 5.95. Points on the curve in Fig. 5.94 correspond to the frames of high-speed video recording. Negative values of heat flux found in the time-dependent thermogram (Fig. 5.95) for the brass ball may be explained by separation of oxide film from the surface. It prevented further experiments, as cavities formed on the surface of the ball.

5.3 Heat Transfer During Phase Changes

159

2

1

3

4

5 Fig. 5.91 Experimental setup: 1—muffle furnace; 2—PC + National Instruments; 3—model; 4— high-speed camera; 5—water tank

The titanium ball was selected to study heat flux and heat transfer coefficient distribution over the surface in detail. Mounting one HGHFS on the ball and using the rotating device (see above), we obtained the following dependences (see Fig. 5.96). End positions of the ball (0◦ and 180◦ ) were recorded first, and the remaining points followed at a 45◦ increment. The experiment was repeated 4 times for each point to test data integrity. Distributions in Fig. 5.97 were obtained averaging the 4 experiments. Heat flux distribution over the ball surface indicates that the first critical value of heat flux for saturated water was exceeded for the rotation angle of the ball ϕ > 50◦ . Maximum heat flux was about 4.5 MW/m2 at ϕ = 135◦ . The following time dependences were plotted for that case (Fig. 5.98), complemented with still frames from high-speed video recording. Boiling regimes changed almost instantly (in 0.2 s). Temperature and time dependences of heat flux and heat transfer coefficient were obtained for different models. The relative measurement uncertainty of local heat flux was 8%. It was confirmed that gradient heatmetry was applicable and informative for studying boiling of subcooled water in different regimes and at different surfaces.

5.4 Radiative Heat Transfer Radiative heat transfer studies are associated with designing a radiation gauge (RG) for indicating the radiation of an object by a photons powerful beam. Since the radiation spectrum was not given in advance, GHFS as “grey” body prove to be preferable to their selective equivalents.

160

5 Validation and Science Experiment

Fig. 5.92 Experimental models: a, b brass cylinder; c, d titanium balls

It is known that all solid-state radiation receivers are either meant for specific irradiation (i.e., exposure) times or have a forced cooling system [75]. In this case, the RG falls under the following restrictions: – maximum exposure is determined by the surface temperature of the RG. By the irradiation end, it should not exceed the thermal stability limits for sensors, compounds, insulation and other structural elements and materials; – maximum dimensions of the RG depend on the sensor’s sensitivity and response time (this relationship can be obtained by solving the direct heat conduction problem). An irradiation indicator with a flux to 0.5 × 106 W/m2 , a mass no more than 50 g, and an output signal no less than 0.3 V has to be designed for our purposes. The exposure time could be as long as 30 s, and the initial temperature range of the RG

5.4 Radiative Heat Transfer

161

(a)

(b)

(c)

(d)

cutter

HGHFS

K-type TC

Fig. 5.93 Brass ball: a in the metal turning lathe; b in the continuous milling machine; c with a technical holes; d with a transducers

was 123…393 K. The uncertainty in irradiation angle characteristics led us to the idea of constructing an RG with a receiving surface in the shape of a hemisphere. Using bismuth-based GHFS limited the upper temperature range for surface heating to a level of 523 K (in view of the thermal stability of other GHFS materials).

162

5 Validation and Science Experiment

q, MW/m2

T, K

3

Tw Tf q

3 4

1

2

600

0 ms

5

4 qcr1

1 6

130 ms

5

6

200

qcr2

1 0

80 ms

400

2

2

3

0.5

1.0

100 ms

,s

140 ms

410 ms

Fig. 5.94 Thermogram and heat flux graph with high-speed video recording during boiling at the cylinder

q, MWt m2

T, ºC

3 300 2 200 1 100 0

0

0,25

0,50

0,75

,s

Fig. 5.95 Thermogram and heat flux graph during boiling at the brass ball

The RG6 constructed is hemisphere 1 (Fig. 5.99), with “terraces” made on its surface and covered with ribbon GHFSs 2 (the total length of the ribbon of the ATE with the 0.25 × 0.25 mm cross-section, placed on the 20 mm dia hemisphere, exceeded 14 m). GHFSs 2 are connected in series and their current leads 3 are passed through the channels in hemisphere 1 and are then connected to the compact socket (not shown in the figure). The RG was mounted on the object in a manner that allowed to neglect heat transfer from the lower hemisphere plane. As a model, we therefore used the problem on unsteady heat conduction of a sphere with constant heat flux at the surface. GHFS 2 placed on the “terraces” of hemisphere 1 made its surface rather smooth:

6 Associate

professor N. P. Divin participated in construction of the RG.

5.4 Radiative Heat Transfer

(a)

163

(b)

HGHFS TK

Ceramic tubes Fig. 5.96 Titanium ball: a 3D model; b holder

(a) 135

90

hcr1

hmax

(b) q, h, kW/(m2 K) MW/m2 20 4 3 15 10 2 1 5 0 0 180° 135°

45°

90 qmax qcr1

90° 45

135

0°

45

Fig. 5.97 Heat transfer coefficient distribution (a) and heat flux graph (b) at the titanium ball surface

⎧ ∂ T (r,τ ) ⎪ =a ⎪ ⎨ ∂τ  ∂ 2 T (r,τ ) 2 ∂ T (r,τ ) + 2 ∂r r ∂r ⎪ ⎪ ⎩(τ > 0; 0 < r < R) , here R is the sphere radius and T0 is its initial temperature. The solution to boundary problem:

(5.6)

164

5 Validation and Science Experiment q, MW m2

h, kW m2 K q

Zone A

10

4

5

2

0 0

0.1

0.2

0.3

,s

0.4

MW Zone A q, m2

4.4

4.2

4.0 0.020

0.025

0.030

0.035

,s

Fig. 5.98 Heat flux graph with high-speed video recording during boiling at the ball

∞    An λ (T − T0 ) η2 = 3Fo R + r − 0.3 − sin (μn ηr ) exp −μ2n Fo R , qR 2 η n=1 r (5.7) where Fo = aRτ2 is the Fourier number, ηr = Rr is the dimensionless radius (r is the 2 current coordinate), An = μ2 cos is the amplitude function, μn is the nth root of the μn n characteristic equation μ = tg μ. It is known [54] that at Fo > 0.5:

θR =

θ R ≈ 3Fo R +

ηr2 − 0.3. 2

(5.8)

5.4 Radiative Heat Transfer Fig. 5.99 Design of RG. Numbers indicate: 1—hemisphere, 2—GHFS, 3—current leads, 4—shield, 5—supporting screws

165

4 2

1

3

5

Therefore, θ1 = 3Fo R + 0.2 at the sphere’s surface (where η = 1) and θ0 = 3Fo R − 0.3 at its center (where η = 0). The temperature difference between the sphere’s surface and its center is 5.9 T = T |ηr =1 − T |ηr =0 = (θ1 − θ0 ) ×

qR qR = . k 2k

(5.9)

Selecting a material for the solid frame of the RG, we settled on aluminum (k = 224 W/(m K), a = 8.33 × 10−5 m2 /s, ρ = 2700 kg/m3 ). It satisfies the requirements for thermal stability, it is substantially lighter than copper and its alloys and is not far behind in heat conduction. Limitations on mass (M = 50 g) set the RG radius to  13   13  −3 R = 3M 2πρ = 3 · 50 · 10 2π × 2700 ≈ 0.02 < 0.02 m. According to equalities (5.8) and (5.9), the maximum surface temperature of the RG is   q 3a τ + 0.2R + T0 . (5.10) T |ηr =1 = k R In τ = 30 s, the heat flux at area q = 0.5 × 106 W/m2 brings the temperature from the initial T0 = 393 K to T |ηr =1 =1239 K 523 K. Giving T |ηr =1 = 523 K, we find from equality (5.10) that qmax = 7.7 × 104 W/m2 is the maximum permissible heat flux. This level of q should be provided to shield the RG. Half the surface of the hemisphere is subjected to irradiation if the photon beam is oriented in the “least beneficial” (lateral) direction. The output signal of the RG for the GHFS with the volt-watt sensitivity S0 ≈ 5 mV/W is E = S0 qmax π R 2 = 5 × 7.7 × 104 × 3.14 × 0.022 ≈ 480 = 0.48 V which is more than enough to satisfy the given requirements. So, analysis for the solid frame of the RG has revealed that the aluminum hemisphere with a radius of 20 mm should be shielded, providing heat flux of about

166

(a)

5 Validation and Science Experiment

(b) q 2

,c2

2 1

,c1

1

1

2

R2 R1

q=0

Fig. 5.100 Sensor with radiation shield: a general view (shield and hemisphere equipped with GHFS), b sketch explaining heat model

0.8 × 105 W/m2 at the surface of the RG. The construction and the heat circuit of the RG with a hemispherical radiation shield are shown in Fig. 5.100. Let us consider the physical model of the shielded RG [76]. The following notations are used for constructing the model: (a) geometric parameters: δ1 is the shield thickness; SR1 is the shield radius (calculated at the axial surface of the shield due to small thickness δ1 ); R2 is the hemisphere radius; V1 = 2π · R12 δ1 is the shield volume; 2π×R 3 V2 = 3 2 is the hemisphere volume; A1 = 2π × R12 is the shield surface (outer surface A1e or inner surface A1i . It can be assumed that A1e ≈ A1i = A1 ); A2 = 2π × R22 is the hemisphere surface covered with the ribbon GHFS;  2 A2 R2 = is the ratio of areas of the hemisphere and the shield; A1 R1 (b) thermophysical parameters: c1 , c2 are the specific mass capacities of shield and hemisphere materials, respectively; ρ1 , ρ2 are the densities of shield and hemisphere materials, respectively; ε1e is the integral emissivity at the outer surface of the shield; εk is the spectral emissivity at the outer surface of the shield; ε1i is the integral emissivity at the inner surface of the shield; 2 is the integral emissivity at the surface of the hemisphere;   2  −1 R2 1 1 ε1−2 = ε2 + R1 −1 is the reduced emissivity of the “shield– ε1i hemisphere” system;

5.4 Radiative Heat Transfer

167

(c) parameters related to temperature and heat flux: T1 is the absolute (cross-section-average) temperature of the shield; T2 is the absolute (calculated) temperature of the hemisphere surface; T1 T2 U1 = 100 , U2 = 100 are the temperature factors; qk is the radiation flux incident on the RG; C0 = 5.67 m2WK4 is the absorption coefficient of an absolutely black body. We used the following assumptions in the calculations: The shield is thermally thin: the temperature is the same at all of its points and depends only on the current time τ . The gap between the shield and the hemisphere is isothermal, heat conduction and convective heat transfer in the gap (according to the use of the RG) is negligible. The heating of the hemisphere is a regular process (second-kind regular regime). 0 of temperature field non-uniformity, where T is the The coefficient ψT = TT1−T −T0 mass-average temperature of the hemisphere and T0 is the initial temperature of the system, is estimated using the relation T − T0 = 3Fo R (valid for Fo R > 0.5 [77]) and equality (5.8):   0.2 −1 T¯ − T0 3Fo R = 1+ ψT = = . T1 − T0 3Fo R + 0.2 3Fo R −5

×30 With τ = 30 s, a = 8.33 × 10−5 m2 /s; Fo R = 8.33×10 = 6.25 0.5, and 0.022 ψT = 0.989 → 1.0. It can therefore be assumed that the RG temperature is practically equalized at the end of exposure. The heat balance equation for the shield has the form

A1 ρ1 V1 · 100

 F2  4 dU1 = qλ · ελ · F1 − C0 ε1e F1 U14 − C0 ε1−2 U1 − U24 , dτ F1

from where dU1 = dτ

  2   2  R2 4 qλ ελ − C0 ε1e + ε1−2 R1 U1 − ε1−2 RR21 U24 100 A1 ρ1 δ1 dU2 (U 4 − U24 ) × 3C0 ε1−2 = 1 . dτ 100c2 ρ2 R2

.

(5.11)

(5.12)

Equalities (5.11) and (5.12) form a system of ordinary differential equations with respect to variables U 1 and U 2 , which is solved in a standard manner, for example, in MathCad 2000. Analyzing the solution, we assume the quantities: R2 = 0.02 m; R1 = 0.035 m; ε2 = 0.98; qk = 0.5 × 106 W/m2 ; c2 = 995.5 J/(kg K); ρ2 = 2700 kg/m3 to be invariable. In addition, the material and the shield thickness should be chosen in advance.

168

5 Validation and Science Experiment

=0.98, =0.98, 1i=0.98 =0.1, =0.1, 1i=0.98 10 =0.1, =0.1, 1i=0.1 10 10

(a)

(b)

T,

E, V

1600

Shield

4

1200

3 2

800

Hemisphere 400 273

0

5

10

15

20

25

,s

1

0

5

10

15

20

25

,s

Fig. 5.101 Sensitive element with radiation shield: a general view (shield and hemisphere equipped with GHFS), b sketch explaining heat model

Let us assume that ε1−2 ≈ 2 . If the surface temperature of the hemisphere does not exceed 523 K, then it follows from the Stefan-Boltzmann equation that the shield temperature is equal to 1089 K. This means that the shield should be manufactured from corrosion-resistant steel (for example, stainless steel of grade, for which ρ1 = 8000 kg/m3 , c1 = 500 J/(kg K), and δ1 = 0.002 m. Calculation results are shown in Fig. 5.101. The curves were obtained at different emissivities at shield surfaces. It can be seen (Fig. 5.101a) that the RG signal is maximum at ε10 = εk = ε1i = 0.98 and is 4.6…4.7 V, while the time constant is equal to 2 s. However, in this case, the “lifetime” of the RG does not exceed 10 s. The signal E > 1 V can be reached at ε10 = 0.1 while providing the required lifetime for the RG. While constructing a test model for the RG appeared to be fairly complex, no fundamental obstacles were encountered. It proved to be more difficult to calibrate such a model because: – the direction of the radiation flux is not known in advance and the sensitivity of individual ATEs is not the same “in different directions”; – the area of the ATE (total) noticeably differs both from the area of the hemisphere with the radius R2 and from the area of the surface where “terraces” are made to place the ATE. Due to this, we also paid close attention to the methods and the setup for calibrating the RG (Fig. 5.102). The fully assembled RG (with a blackened surface but without a shield) was mounted under semi-spherical cavity 2, whose dimensions and emissivity were identical to the shield parameters (R2 , ε1i ). Cavity 2 was a part of the bottom of vessel 4. Heater 6 provided water boiling in the vessel at a temperature of about 373 K (or oil heating to a higher temperature) and mixer 3 provided uniform

5.4 Radiative Heat Transfer Fig. 5.102 Scheme of setup for calibrating RG. Numbers indicate: 1—RG, 2—blackened surface of radiated cavity, 3—mixer, 4—liquid vessel; 5—equipment block, 6—heater, 7—thermocouple, 8—control GHFSs, 9—millivoltmeter

169 6 5

3 7 4

2 1

8

heat supply to the streamlined “back” surface of cavity 2. Thus, the condition 1 = const corresponding to the one included in model (5.11)–(5.12) and reproducible in subsequent calibrations was imitated. The controlling and measuring system of the setup comprised thermocouple 7, pre-calibrated GHFSs 8, millivoltmeter 9, and equipment block 5 (Fig. 5.102). The setup allows us to record the shield temperature, heat flux at the RG surface and the output signal of the RG. These parameters are sufficient for determining the calculated volt-watt sensitivity of the RG S0∗ =

E 2π R22 q

(5.13)

where E is the output signal of the ATE battery and q is the heat flux (average) measured at the surface of the hemisphere. Calibration has shown that the volt-watt sensitivity of the S0∗ = 1.7. . .1.8 mV/W. It is substantially lower than for most other GHFSs (Sect. 1.2). The reason for this is that ATE ribbons could only be placed on the “terraces” of the hemisphere with a comparatively low density. The setup for laboratory tests of the RG is schematically sketched in Fig. 5.103. Electric bulb 1 (220 V, 500 W) which is used as a heat flux source is placed in the focus of parabolic reflector 2. A galvanometer of a light-beam oscilloscope N-145 4 (model M017-300) provides recording of the characteristic q(τ ). The distance from radiator 1 to RG 3 is regulated by means of optical bench elements over the range 100. . . 300 mm. Comparison of experiment with computation shows (Fig. 5.104) that the “flow saturation” level is very similar in both cases but the initial section is less steep in the experiment compared with the computation.

170

5 Validation and Science Experiment

Fig. 5.103 Scheme of setup for tests of RG. Numbers indicate: 1—electric bulb, 2—parabolic reflector, 3—RG, 4—light-beam oscilloscope equipped with galvanometer

1 Fig. 5.104 Comparison of experimental and computational characteristics of RG: 1—numerical simulation, 2—analytical solution, 3—experiment

2

3

4

E, V 0.8

0.6

1 2

0.4

3

0.2

0

30

60

90

120

,s

This is apparently connected with the schematic assumptions made in model (5.11)–(5.12). At the same time, numerical simulation confirmed the sensitivity found experimentally and allowed to optimize the construction of the RG during numerical simulation, but not during full-scale experiment.

References 1. Nikolskaya, S. B., & Chumakov, Yu. S. (2000). Eksperimental’noye issledovaniye pul’satsionnogo dvizheniya v svobodno-konvektivnom pogranichnom sloye (An experimental study of pulsation motion in a free-convective boundary layer). In Hight temperature (Vol. 38, no. 2, pp. 249–256). Moscow. 2. Mityakov, V. Yu., Mityakov, A. V., Sapozhnikov, S. Z., et al. (2002). Ispol’zovaniye poperechnogo effekta Zeyebeka dlya izmereniya mgnovennogo znacheniya teplovogo potoka na vertikal’noy nagretoy poverkhnosti v usloviyakh svobodno-konvektivnogo teploobmena (Using the transverse Seebeck effect to measure the instantaneous value of the heat flux on a vertical heated surface under conditions of free convective heat transfer). In Hight temperature (Vol. 40, no. 4, pp. 647–669). Moscow. 3. Kuzmitsky, O. A., Nikolskaya, S. B., & Chumakov, Yu. S. (1996). Spektral’nyye i korrelyatsionnyye kharakteristiki pul’satsiy skorosti i temperatury v svobodno-konvektivnom pogranichnom sloye (Spectral and correlation characteristics of velocity and temperature pulsations in a freely convective boundary layer). In 3rd Minsk International Forum on Heat and Mass Transfer, Minsk 1996 (Vol. 1, no. 2, pp. 117–122).

References

171

4. Kuzmitsky, O. A., & Chumakov, Yu. S. (1990). Struktura temperaturnogo polya v svobodnokonvektivnom pogranichnom sloye okolo vertikal’noy izotermicheskoy poverkhnosti (The structure of the temperature field in a free convective boundary layer near a vertical isothermal surface). In Hight temperature, (Vol. 28, no. 6, pp. 1142–1148). Moscow. 5. Bobashev, S. V., Mende, N. P., Sakharov, V. A., et al. (2004). Vozmozhnosti gradiyentnykh datchikov pri issledovanii protsessov v udarnykh trubakh (Possibilities of gradient sensors in the study of processes in shock tubes). In VII Siberian Thermophysical Workshop, Novosibirsk, 2004. 6. Bobashev, S. V., Mende, N. P., & Sakharov, V. A. (2005). Primeneniye datchika gradiyentnogo teplovogo potoka v eksperimentakh s udarnymi trubami (Application of gradient heat flux sensor in shock tube experiments). In 43rd AIAA Aerospace Sciences Meeting and Exhibit, Reno, Nevada, 10–13 January 2005. 7. Chumakov, Y. S., & Nikolskaja, S. B. (1998). Features of dynamic and heat structure of free convection boundary layer near vertical heated surface. In Turbulent heat transfer-2 (Vol. 2, pp. 9–19). Manchester. 8. Akylbaev, Zh. S., Isataev, S. I., & Polzik, V. V. (1972). Sryv vikhrey s poverkhnosti plokhoobtekayemykh tel i yego vliyaniye na teploobmen (Disruption of vortices from the surface of poorly streamlined bodies and its effect on heat transfer). Teplo - i massoperenos, 1(1), 291–295. 9. Dyban, E. P., Epic, E. Ya., & Kozlova, L. G. (1975). Sovmestnoye vliyaniye stepeni prodol’nogo masshtaba turbulentnosti i uskorennosti vozdushnogo potoka na teploobmen krugovogo tsilindra (The joint influence of the degree of the longitudinal scale of turbulence and the acceleration of air flow on the heat transfer of a circular cylinder). In Teploobmen 1974. Sovetskiye issledovaniya (pp. 110–116). Moscow: Nauka. 10. Zhukauskas, A. A. (1982). Konvektivnyy perenos v teploobmennikakh (Convective transfer in heat exchangers). Moscow: Nauka. 11. Oka, S. (1974). Teploobmen odinochnogo tsilindra pri razlichnykh usloviyakh obtekaniya (Heat transfer of a single cylinder under various flow conditions). In Problemy teplofiziki i fizicheskoy gidrodinamiki, Novosibirsk, 1974. Collection of scientific papers (pp. 47–71). Nauka. 12. Chzhen, P. (1973). Otryvnyye techeniya (Separate currents) (pp. 1–3). Moscow: Mir. 13. Chzhen, P. (1979). Upravleniye otryvom potoka (Management of flow separation). Moscow: Mir. 14. Achenbach, E. (1966). Distribution of local pressure and skin friction around a circular cylinder in cross-flow up to Re = 5 × 106 . Journal of Fluid Mechanics, 34(625), 639. 15. Achenbach, E. (1975). Total and local heat transfer from a smooth circular cylinder in cross-flow at high Reynolds number. Heat and Mass Transfer, 18(11), 1387–1396. 16. Chang, B. H., & Mills, A. F. (2004). Effect of aspect ratio on forced convection heat transfer from cylinders. Heat Mass Transfer, 47, 1289–1296. 17. Nakamura, H., & Igarashi, T. (2002) Unsteady heat transfer in separated flow behind a circular cylinder. In 12th International Heat Transfer Conference, Grenoble, France, 2002 (Vol. 2, pp. 729–734). 18. Kutateladze, S. S. (1990). Teploperedacha i gidrodinamicheskoye soprotivleniye: Spravochnoye posobiye (Heat transfer and hydrodynamic resistance. A reference guide). Moscow: Energoatomizdat. 19. Maher, J., & David, C. (1974). Dynamics of heat transfer from cylinders in a turbulent air stream. Heat and Mass Transfer, 17, 767–783. 20. Sergeyev, O. A. (1972). Metrologicheskiye osnovy teplofizicheskikh izmereniy (Metrological foundations of thermophysical measurements). Moscow: Izdatel’stvo standartov. 21. Sagittarius, M. H. (2001). Metod modelirovaniya otsoyedinennykh vikhrey i yego primeneniye dlya rascheta otryvnykh turbulentnykh techeniy (The method of modeling detached eddies and its application for calculating separated turbulent flows). In School-Seminar for Young Scientists and Specialists, Moscow Power Engineering Institute, Moscow. 22. Strelets, M. (2001). Detached eddy simulation of massively separated flows. In AIAA (18 pp.).

172

5 Validation and Science Experiment

23. Boulos, M. J., & Pei, D. C. T. (1974). Dynamics of heat transfer from cylinders in a turbulent air stream. Heat Mass Transfer, 17, 767–783. 24. Liu, C., Tai, Y. C., Huang, J. B., & Ho, C. M. (1994). Surface micromachined thermal shear stress sensor. In Application of Microfabrication to Fluid Mechanics 1994 presented at 1994 ASME International Mechanical Engineering Congress and Exposition, Chicago, USA. 25. Agapiev, B. D., Belov, V. N., & Kesamanly, F. P. (1999). Obrabotka eksperimental’nykh dannykh (Processing of experimental data: Textbook allowance). Saint-Petersburg: SPbGTU. 26. Dyakonov, V. P. (1987). Spravochnik po algoritmam i programmam na yazyke Beysik dlya personal’nykh EVM (Reference on algorithms and programs in the BASIC language for personal computers). Moscow: Nauka. 27. Rowe, D. M. (Ed.). (1995). CRC handbook of thermoelectrics. London, New York, Washington: Boca Raton. 28. Goldstein, S. (Ed.). (1948). Sovremennoye sostoyaniye gidrodinamiki vyazkoy zhidkosti (Current state of viscous fluid hydrodynamics). 29. Fage, A., & Falkner, V. M. (1930). An experimental determination of the intensity of friction on the surface of an airfoil. London: Proceedings of the Royal Society. 30. Babich, A. Y., Zainullina, E. R., Sapozhnikov, S. Z., et al. (2017). Hydrodynamics and heat transfer of yawed circular cylinder. Heat and Mass Transfer, 115, 333–339. 31. Sparrow, E. M., & Moreno, A. A. Y. (1987). Effect of yaw on forced convection heat transfer from a circular cylinder. Heat and Mass Transfer, 30(3), 427–435. 32. Babich, A. Y., Mityakov, A. V., Zainullina, E. R., et al. (2017). Investigating heat transfer augmentation using gradient heat flux measurement and PIV method. In 33rd Siberian Thermophysical Seminar, STS 2017, Kutateladze Institute of Thermophysics SB RAS Novosibirsk, Russian Federation, 6–8 June 2017. 33. Seroshtanov, V. V., Sapozhnikov, S. Z., Pavlov, A. V., et al. (2017). Comprehensive study of flow and heat transfer at the surface of circular cooling fin. In International Conference on Problems of Thermal Physics and Power Engineering 2017, PTPPE 2017, National Research University Moscow Power Engineering Institute (NRU MPEI) Moscow, Russian Federation, 9–11 October 2017. 34. Seroshtanov, V. V., Sapozhnikov, S. Z., Mityakov, V. Y., et al. (2018). Investigation of flow and heat transfer at the surface of a single circular cooling fin. Engineering and Technology, 7(4), 33–36. 35. Seroshtanov, V. V., Gusakov, A. A., Mityakov, V. Y., et al. (2018). Investigation of flow and heat transfer at the circular fins. In International Scientific Conference on Energy, Environmental and Construction Engineering, EECE 2018, Congress Center of Peter the Great St. Petersburg Polytechnic University, 19–20 November 2018. 36. Pismennyy, E. N. (2004). Teploobmen i aerodinamika paketov poperechno-orebrennykh trub (Heat exchange and aerodynamics of packages of cross-finned tubes). Kiev: Alterpres. 37. Alekseev, V. V., Gachechiladze, I. A., Kiknadze, G. I., & Oleinikov, V. G. (1998). Smerchevoy energoobmen na trekhmernykh vognutykh rel’yefakh struktura samoorganizuyushchikhsya techeniy, ikh vizualizatsiya i mekhanizmy obtekaniya poverkhnostey (Tornado energy exchange on three-dimensional concave reliefs—The structure of self-organizing flows, their visualization and mechanisms of flow around surfaces). In The Second Russian National Conference on Heat Transfer, 1998 (Vol. 6, pp. 33–42). 38. Afanasyev, V. N., Veselkin, V. Yu., & Leontiev, A. I. (1991). Gidrodinamika i teploobmen pri obtekanii odinochnykh uglubleniy na iskhodno gladkoy poverkhnosti (Hydrodynamics and heat transfer during flow around single recesses on an initially smooth surface). Moscow: Moskovskiy Gosudarstvennyy tekhnicheskiy institut im. N. E. Baumana. 39. Isaev, S. A., Leontiev, A. I., Pyshny, I. A., et al. (2003). Intensifikatsiya smerchevogo turbulentnogo teploobmena v asimmetrichnykh lunkakh na ploskoy stenke (Intensification of tornado turbulent heat transfer in asymmetric holes on a flat wall). Inzhenerno-fizicheskiy, 76(2), 31–34. 40. Isaev, S. A., Leontiev, A. I., Mityakov, A. V., et al. (2002). Mestnyye koeffitsiyenty teplootdachi na poverkhnosti vytyanutoy lunki (Local heat transfer coefficients on the surface of an elongated hole). In 3th Russian National Conference on Heat Transfer. Intensification of Heat Transfer.

References

41.

42.

43.

44.

45.

46.

47.

48.

49.

50. 51.

52.

53.

54.

173

Radiation and Complex Heat Transfer, Moscow Power Engineering Institute, Moscow, 2002 (Vol. 6, pp. 214–218). Kiknadze, G. I., & Oleinikov, V. G. (1990). Samoorganizatsiya smercheobraznykh vikhrevykh struktur v potokakh gazov i zhidkostey i intensifikatsiya teplo- i massoobmena (Selforganization of tornado-like vortex structures in gas and liquid flows and intensification of heat and mass transfer). Novosibirsk: Institut teplofiziki Sibirskogo Otdeleniya akdemii nauk SSSR. Gortyshev, Yu. F., Popov, I. A., Olimpiev, V. V., Schelchkov, A. V., et al. (2009). Teplogidravlicheskaya effektivnost’ perspektivnykh sposobov intensifikatsii teplootdachi v kanalakh teploobmennogo oborudovaniya (Thermohydraulic efficiency of promising methods of heat transfer intensification in the channels of heat exchange equipment). Kazan: Kazan National Research Technical University A. N. Tupolev. Sokolov, N. P., Polishchuk, V. G., Andreev, K. D., et al. (2012). Teploobmen i gidravlika v kanalakh s oblunennymi poverkhnostyami (Heat transfer and hydraulics in channels with exposed surfaces). Saint-Petersburg: Izdatel’stvo Politekhnicheskogo universiteta. Belenky, M. Ya., Lebedev, M. E., & Fokin, B. S. (1996). Konvektivnyy teploobmen pri obtekanii poverkhnosti so sfericheskimi lunkami (Convective heat transfer when flowing around a surface with spherical dimples: Textbook). Saint-Petersburg: Izdatel’stvo Politekhnicheskogo universiteta. Baranov, P. A., Isaev, S. A., Leontiev, A. I., et al. (2002). Fizicheskoye i chislennoye modelirovaniye vikhrevogo teploobmena pri turbulentnom obtekanii sfericheskoy lunki na ploskosti (Physical and numerical simulation of vortex heat transfer during turbulent flow around a spherical hole on a plane). Teplofizika i aeromekhanika, 9(4), 521–532. Ermishina, A. V., & Isaeva, S. A. (2001). Upravleniye obtekaniyem tel s vikhrevymi yacheykami v prilozhenii k letatelnym apparatam integralnoy komponovki (chislennoye i fizicheskoye modelirovaniye) (Control of the flow around bodies with vortex cells as applied to aircraft of the integrated layout (numerical and physical modeling)). Moscow: Saint-Petersburg. Guzeev, A. S., Lebedev, A. O., Mityakov, A. V., et al. (2009). O zadymlyayemosti transportnykh sudov (On the smokiness of transport ships). In Optical flow research methods, Moscow, 2009, June 23–26. Isaev, S. A., Mityakov, A. V. (2009). Chislennoye modelirovaniye konvektivnogo teploobmena v nizkoskorostnykh otryvnykh techeniyakh neodnorodnykh sred (Numerical modeling of convective heat transfer in low-speed separated flows of inhomogeneous media). In ShkolaSeminar, Zhukovsky, 2009. Terekhov, V. I., Yarygin, N. I., & Dyachenko, A. Yu. (2002). Intensifikatsiya teplootdachi pri perestroyke techeniya v poperechnoy naklonnoy kaverne (The intensification of heat transfer during the restructuring of the flow in a transverse inclined cavity). In VI Siberian Thermophysical Seminar, Institute of Thermophysics Siberian Branch of the Russian Academy of Sciences, Novosibirsk, 2002. Terekhov, V. I., Mshvidobadze, Yu. M., & Kalinina, S. V. Heat transfer coefficient and aerodynamic resistance on surface with single dimple. Enhancement Heat Transfer, 4(2), 131–145. Akatnov, N. I., et al. (1982). Issledovaniye na udarnoy trube s soplom sverkhzvukovykh MGD kanalov na neravnovesnoy plazme inertnogo gaza (Research on a shock tube with a nozzle of supersonic MHD channels on a nonequilibrium inert gas plasma). Zhurnal tekhnicheskoy fiziki, 52(5), 884–892. Maslennikov, V. G., & Sakharov, V. A. (1997). Dvukhdiafragmennaya udarnaya truba Fizikotekhnicheskogo instituta (Double-diaphragm shock tube of the Physicotechnical Institute). Zhurnal tekhnicheskoy fiziki, 67(11), 88–95. Zhilin, Yu. V. (1976). Metodika izmereniya statsionarnykh teplovykh potokov s pomoshch’yu plenochnykh datchikov soprotivleniya (Method for measuring stationary heat fluxes using film resistance sensors). In Preprint No. 2-005, Joint Institute for High Temperatures (JIHT), RAS, Moscow, 1976. Lykov, A. V. (1967). Teoriya teploprovodnosti (Theory of thermal conductivity). Moscow: Vysshaya shkola.

174

5 Validation and Science Experiment

55. Babinsky, M. G., et al. (1976). Nekotoryye aerodinamicheskiye issledovaniya v giperzvukovoy udarnoy trube Leningradskogo politekhnicheskogo instituta imeni M.I. Kalinina (Some aerodynamic studies in a hypersonic shock tube of Leningrad Polytechnic Institute named after M.I. Kalinina). In Mechanics and Mechanical Engineering, Leningrad Polytechnic Institute named after M.I. Kalinina, Leningrad, 1976. Proceedings of the Leningrad Polytechnic Institute named after M.I. Kalinina (Vol. 352, pp. 100–104). 56. Sapozhnikov, S. Z., Mityakov, V. Yu., Mityakov, A. V., et al. (2004). Izmereniye teplovogo potoka na vnutrennikh stenkakh kanala udarnoy truby (Measurement of heat flux on the inner walls of the channel of the shock tube). Journal of Technical Physics, 30(2), 76–80. 57. Kim, S. J., & No, H. Ch. (2000). International Journal of Heat and Mass Transfer, 43, 4031. 58. Lee, K.-W., No, H. Ch., Chu, I.-Ch., et al. (2006). International Journal of Heat and Mass Transfer, 49, 1813. 59. Fan, G., Tong, P., Sun, Z., et al. (2018). Annals of Nuclear Energy, 113, 139. 60. Hu, H. W., Tang, G. H., & Niu, D. (2016). Applied Thermal Engineering, 100, 699. 61. Babich, A. Y., Zainullina, E. R., Sapozhnikov, S. Z., et al. (2017). Gradient heat flux measurement while researching of saturated water steam condensation. In International Conference on Problems of Thermal Physics and Power Engineering 2017, PTPPE 2017, National Research University Moscow Power Engineering Institute (NRU MPEI) Moscow, Russian Federation, 9–11 October 2017. 62. Rivkin, S. L., & Aleksandrov, A. A. (1980). Teplofizicheskiye svoystva vody i vodyanogo para (Thermophysical properties of water and water vapor). Moscow: Energiya. 63. Labuntsov, D. A. (1957). O vliyanii na teplootdachu pri plenochnoy kondensatsii zavisimosti fizicheskikh parametrov ot temperatury (On the effect on the heat transfer during film condensation of the dependence of physical parameters on temperature). Teploenergetika. 64. Babich, A. Y., Zainullina, E. R., Sapozhnikov, S. Z. et al. (2018). Gradient heat flux measurement in condensation study at inner and outer surfaces of the pipe. In 34th Siberian Thermophysical Seminar Dedicated to the 85th Anniversary of Academician A. K. Rebrov, STS 2018, Kutateladze Institute of Thermophysics of Siberian Branch of Russian Academy of Sciences Novosibirsk, Russian Federation, 27–30 August 2018. 65. Nusselt, W. (1916). Die Oberflachenkondensation des Wasserdampfes. Zeitchrift des VDI, 60(27), 541–546, 568–575. 66. Babich, A. Y., Zainullina, E. R., Sapozhnikov, S. Z., et al. (2019). The study of heat flux measurement for heat transfer during condensation at pipe surfaces. Technical Physics Letters, 45(4), 321–323. 67. Gross, U., Storch, Th., Philipp, Ch., et al. (2009). Wave frequency of falling liquid films and the effect on reflux condensation in vertical tubes. Multiphase Flow, 35, 398–409. 68. Salazar, R. P., & Marschall, E. (1978). Statistical properties of the thickness of a falling liquid film. Acta Mechanica, 29, 239–255. 69. Nikoglou, A. A., Hinis, E. P., & Simopoulos, S. E. (2015). Statistical characteristics of free falling water film. In NURETH-16, Chicago, IL, August 30–September 4, 2015. 70. Labuntsov, D. A., & Gomelauri, A. V. (1976). Tr. MEI, 310, 50. 71. Nukiyama, S. (1984). Heat and Mass Transfer, 27(7), 956–970. 72. Petukhov, B. S. (1952). Opytnoye izucheniye protsessov teploperedachi: uchebnoye posobiye (Experimental study of heat transfer processes: A training manual). Leningrad: Gosenergoizdat. 73. Yagov, V. V., Zabirov, A. R., Kanin, P. K., et al. (2017). Inzh.-Fiz. Zh., 90, 287. 74. Subbotina, V. V., Sapozhnikov, S. Z., Mityakov, V. Y., et al. (2019). An experimental investigation of the film boiling of subcooled water by gradient heat flux measurement. Technical Physics Letters, 45(3), 253–255. 75. Kruse, P. W., McGlauchlin, L. D., McQuistan, R. B., et al (1962). Elements of infrared technology. N.Y. 76. Sapozhnikov, S. Z., Mityakov, V. Yu., & Mityakov, A. V. (2003). Gradiyentnyye datchiki teplovogo potoka (Gradient heat flux sensors). Saint-Petersburg: Izdatel’stvo SPbGPU. 77. Carslow, G., & Jaeger, D. (1964). Teploprovodnost tverdykh tel (Thermal conductivity of solids). Moscow: Nauka.

Chapter 6

Industrial Experiments

6.1 Heatmetry in the Diesel Engine According to common view, the processes occurring in the cylinder of an internal combustion engine (ICE) are an example of complex and unsteady heat transfer. Meticulous investigations [1] based on experiment data and numerical simulation do not fully describe these phenomena, do not allow modulating them, and, what is more important in practical terms, to control the engine parameters associated with the processes inside the cylinder. ICEs are primarily used for speeding up the operating process, which increases heat loads on parts of the cylinder-piston group. Heat flux reaches 106 W/m2 or more during fuel combustion. Its value varies significantly during the cycle and is not uniform on heat transfer surfaces [2, 3]. For estimating the thermal state of the ICE parts, it is necessary not only to have surface-average but also local heat fluxes [1] as a function of the crankshaft rotation angle. The experiments conducted in 1996 were pilot tests and had substantial regime limitations, since bismuth-based GHFS were used. The invention of heat-resistant heterogeneous gradient heat flux sensors (HGHFS) provides significantly new opportunities. Studies should be resumed, because as far as we know, heatmetry have not been performed in ICE cylinders and combustion chambers so far. To summarize, the experiments described below have yielded tentative results and require further consideration. Using GHFSs allowed to investigate unsteady heat transfer in the combustion chamber of Indenor XL4D, a four-stroke diesel engine manufactured by PSA Peugeot Citroen. Basic engine characteristics include: a turbulence chamber, with a compression degree of 23, maximum power of 35 kW at 5000 rev/min and maximum torque of 84.3 Nm at 2500 rev/min. The engine installed on the frame (Fig. 6.1) was connected to the power, start, cooling, exhaust of waste gases, signal measurement and recording systems. The locations of the mounted GHFSs are shown in Fig. 6.2. © Springer Nature Switzerland AG 2020 S. Z. Sapozhnikov et al., Heatmetry, Heat and Mass Transfer, https://doi.org/10.1007/978-3-030-40854-1_6

175

176

6 Industrial Experiments

Fig. 6.1 Indenor XL4D on testing installation Fig. 6.2 Numbers indicate: 1—GHFS, 2—cylinder head, 3—pre chamber, 4—intake valve, 5—exhaust valve, 6—engine cooling system

(a)

(b)

2

A-A

6 4

2

1

3 1 3 A

A

5

GHFS 1 with the 4 × 7 × 0.2 mm dimensions and a sensitivity of 8.4 mV/W was used to measure local heat flux at the flame deck of a block cover group of cylinders 2 (Fig. 6.3). The sensor signal was measured by a light-beam oscilloscope H-145. Additionally, time marks (50 Hz) and top dead center (TDC) marks for a test cylinder were recorded by an oscilloscope. GHFS 1 was mounted on the head of the block of cylinders and calibrated according to the specially developed methods (see Sect. 2.3) along with the light-beam oscilloscope. In the first series of experiments, the crankshaft was rotated without fuel supply (at the frequencies of 250 and 870 r.p.m) and in the second series, fuel was supplied

6.1 Heatmetry in the Diesel Engine

177

Fig. 6.3 General view of cylinder head with GHFS mounted (before calibration)

in a staggered manner and frequencies were equal to 900 and 1320 r.p.m. Figure 6.4 shows variations in the oscilloscope records in local heat flux per unit area. It can be also seen that maximum heat flux coincides with the TDC. As follows from thermodynamic analysis of the ICE cycle, it is most efficiency to burn fuel as close to the TDC as possible. However, practically all studies dealing with heat flux measurement in ICE show that heat flux depends on the crankshaft rotation angle, where maximum heat flux lies behind the TDC. This is mainly due to a great response time of conventional heat receivers, as well as to the fact that the variation of combustion chamber wall temperature (and consequently of thermocouples) usually falls behind that of gas temperature, resulting in a phase error in the heat flux calculated from heatmetry data. In our experiments, a difference is also observed in the oscillation amplitude of the heat flux in the neighboring cycles (Fig. 6.4a–d), which has already attracted the attention of other experimental scientists. This may be caused both by irregular charge motion [4] and by fuel supply in cycles (for example, about 10% for a Bosch fuel pump [5]). The heat flux decrease at the moment when the exhaust valve opens (Fig. 6.4a, b) is seen in the curve. The crankshaft is rotating in the regime without fuel supply (Fig. 6.4a, b) the local heat flux increases from zero, reaching a maximum at the TDC, and then decreases, which agrees with data [1, 4].

178 Fig. 6.4 Heat flux graphs: a without fuel supply, 250 rpm, b without fuel supply, 870 rpm, c combustion, 900 rpm, d combustion, 1320 rpm

6 Industrial Experiments

(a) q, kW/m2 100

without fuel supply n = 250 rev/min

50

TDC

TDC

(b) q, kW/m2 100

n = 870 rev/min

without fuel supply

20 ms

50

TDC

TDC

TDC

(c) q, kW/m 2

n = 870 rev/min

100

conbustion 50 20 ms TDC

TDC

TDC

(d) q, kW/m2

conbustion

100

n = 1320 rev/min 50

20 ms TDC

TDC

TDC

TDC

As shown in Fig. 6.4c, d, two heat flux maxima during fuel combustion are observed. This is characteristic for engines with divided combustion chambers. The first maximum is reached in the TDC while the second one—at 70. . . 80◦ from the TDC. The theory and the physical concepts of such a process are covered in literature [6, 7]. However, the “double maximum” has not been experimentally confirmed prior to these experiments because a sufficiently fast HFS was unavailable. The oscilloscope record also demonstrates the moments of exhaust valve opening and fuel injection. When valves close (near the TDC), heat flux varies by approximately 2 kW/m2 . This is probably associated with heat transfer during cylinder scavenging.

6.1 Heatmetry in the Diesel Engine

179

Fig. 6.5 Pictures of a HGHFS and b HGHFS mounted on probe

It is point to note that heat flux does not change its direction in all curves. This indicates that the flame deck in the measuring zone does not cool down in the suction stroke and heat flux fluctuates, with its sign preserved, which is confirmed by the data obtained by Japanese researchers [3]. According to studies [4], the sign-variable regime is also possible in other sections of the combustion chamber and the cylinder. The series of heatmetry experiments continued from 2015 till 2019. New measurements were performed using HGHFS of nickel + 12H18N9T steel composition (Fig. 6.5). Thermal stability of such sensors (around 1500 ◦ C) means that they can be used in harsh conditions of the combustion chamber (CC) for a long time. Sensors were sized 3.0 × 3.0 × 0.2 mm. The sensor’s plane-area was As = 9 × 10−6 m2 , and the flame deck area was A p = 4.8 × 10−3 m2 . The ratio of HGHFS area to flame deck area, As /A p = 530, makes it possible to regard the measurement as local. There are certain problems connected to measuring the gas temperature in the CC. The temperature is typically reconstructed using the indexing results, data yielded by different optical methods, etc. To correctly measure the gas temperature with a thermocouple, its junction should be placed outside the thermal boundary layer, in the flow core. The gap between the piston in the top dead center and the flame deck is 0.8 mm. HGHFSs 1 and 2 were mounted above the flat part of the piston, and HGHFSs 3 and 4 were above the chamber in the piston, keeping the thermocouples away from the firing surface (Fig. 6.6). The first experiments were conducted with a single HGHFS mounted flush with the surface of the flame deck (Fig. 6.7). A process hole usually closed with a plug was provided for indexing in the cylinder head (above the first cylinder).

180

6 Industrial Experiments

4

4´

3

3´

2 1´

1

2´

Fig. 6.6 Compatible points at flame deck (left) and piston (right)

Ø78

HGHFS

Fig. 6.7 Installation scheme for single HGHFS

The measuring probe was made to follow the probe shape (Fig. 6.8). Its tip was interchangeable, allowing to use different types of sensors. After the tip was installed in the probe, its lead out wires were put in a metal jacket that served as an EMI screen. The probe opening was sealed to prevent oil from entering it, and the metal jacket was soldered to the probe. At the second stage, four HGHFSs were mounted flush with the surface of the flame deck (Fig. 6.9). Two K-type thermocouples were installed near each sensor. Four grooves were milled on the cylinder head surface for the HGHFS sized 5 × 5 × 0.5 mm. The positions of the HGHFSs were selected so that they would be located on approximately the same radius at different azimuth angles. HGHFSs on mica substrate 0.25 mm thick were attached to the cylinder head with epoxy glue (thermally stable up to 150 ◦ C). Glue layer thickness was about 0.05 mm. Grooves 2 mm wide and 1.5 mm deep were cut in the head surface to lay wires. After laying the wires, the rest of the groove was filled with epoxy to level the surface and protect the wires from heat. Commutation wires were brought outside through the process hole.

6.1 Heatmetry in the Diesel Engine

181

Fig. 6.8 Probe with HGHFS HGHFS-3

12

5°

R 29

HGHFS-4

Commutation wires

57

A-A

°

R 31

HGHFS-2 Ø28

A -62 °

HGHFS-1 R 28

A

R 27

8°

-9

Fig. 6.9 HGHFS group mounted at the flame deck’s surface

HGHFS

182 Fig. 6.10 Effect of rotation speed on heat flux profile

6 Industrial Experiments q, kW/m 2 500

400

300

800 1100 1400

200

900 1200 1500

1000 1300 1600

100 -360

-180

TDC

180

We used a measurement system by National Instruments (USA) to record all signals from the sensors. The pressure sensor was connected to the NI PXI-4461 board (24-bit capacity, maximum sampling rate of 204.8 kHz, measuring range of +1…+12 V) [8]. Signals from HGHFSs and thermocouples, as well as signals from the rotation rate sensor were recorded by the NI PXI-6289 board (18-bit capacity, maximum sampling rate of 500 kHz, measuring range of −10…+10 V) [9]. Signal wires were connected to the NI TB-2706 terminal that was connected to the board. All sensors were connected using differential circuitry. The effect of the speed rate on the heat flux is shown in Fig. 6.10. Dual combustion was observed in all modes, with the second maximum higher than the first. Dual combustion, typical for swirl-chamber diesel engines occurs in the entire CC. It is manifested differently in different zones of the CC. The average q for n = 800 rpm is significantly less than in any other mode. Heatmetry allow us to analyze the effect of mixture composition on the heat flux graphs. The lower flammability limit against the air/fuel ratio is 1.05 for diesel fuel. However, the air/fuel ratio in diesel ICEs is higher, ranging from 2 to 5. Enriching the fuel-air mixture reduces the heat flux. With the throttle closed almost completely (10%), the maximum heat flux is reduced by 20% at n = 1300 rpm (Fig. 6.11a) and by 40% at n = 1500 rpm (Fig. 6.11b). The parameters of the working medium are taken to be constant for the entire volume of the CC in calculations of ICE operation. This is not true in reality: gas parameters differ across the CC volume, and that difference is especially great at the boundary between the fuel-air mixture and spent gases. We discussed heat flux variation during probe diagnostics, i.e. at the only point accessible without disassembling the engine, in the previous section. Our task was to confirm whether it was possible

6.1 Heatmetry in the Diesel Engine

183

(a)

(b)

q, kW/m2

q, kW/m2

140

140

120

120

100

100

80

80

60

60

40 -360

-180

TDC

180

The degree of throttle opening, %

40 -360

100

-180

TDC

66

33

10

Fig. 6.11 Effect of mixture composition on heat flux at: a n = 1300 rpm and b n = 1500 rpm

to assess the variation in the heat flux throughout the entire CC by measuring it at a single point on the wall of the CC. Heat transfer coefficient was calculated as the ratio of heat flux to the difference between the readings of the thermocouples. The results indicate that heat is nonuniformly distributed over the surface of the flame deck (Fig. 6.12). The zero line and the variation in heat flux are the most pronounced at the location of HGHFS 2. The variation in heat flux across the engine cycle is barely noticeable for sensor 4 compared to other HGHFSs, even though the heat flux remains positive there during the whole cycle (heat is transferred to the CC wall). The average heat flux for HGHFS 4 is about 20 kW/m2 . Heat flux graphs for HGHFSs 1 and 2 are similar, while the heat flux graph for HGHFS 3 has a unique feature. Heat flux changes its sign within the range of ±60◦ . The intensity of heat transfer is depends on both the difference between the gas and wall temperatures, and gas flow. The calculated heat transfer coefficients (Fig. 6.13b) vary throughout the engine cycle similar to the heat flux graphs. The calculated heat transfer coefficient for HGHFS 3 changes insignificantly, with a slight decrease, remaining at the level of 200 W/(m2 K). HGHFS 1 comes third in terms of heat flux, but heat transfer coefficient reaches its maximum at that point. In general, the results indicate that the heat flux and heat transfer coefficient are vastly divergent, which means that multi-zone models of CC should be used instead of single-zone models. Beside measuring the gas temperature with thermocouples, we applied an alternative technique; it consists in reconstructing the gas temperature by the indexing results: the CC is regarded as an isolated thermodynamic system from the start of compression stroke to the end of operating stroke, and its medium is treated as ideal gas. Despite these oversimplified assumptions, they allow to obtain fairly accurate gas temperatures in the CC. Wall temperature was measured using the thermocouple, same as in other experiments.

184 Fig. 6.12 Angular heat flux graphs (a) and heat transfer coefficient (b) for compression mode

6 Industrial Experiments

(a) q, kW/m2 80 60 40 20 0 -360

(b) h, kW/(m2

-180

TDC

180 HGHFS-3

HGHFS-1 HGHFS-2

HGHFS-4

K)

600 500 400 300 200 100 0 -360

-180

TDC

180

The heat transfer coefficients calculated using the reconstructed gas temperature are shown in Fig. 6.13. In general, the curves of heat transfer coefficient still follow the thermogram profiles. As fuel is fed to the CC, the average heat flux almost doubles (Fig. 6.14a) Heat flux graphs for HGHFSs 2 and 3 have two peaks each, the first maximum appearing later than in compression mode: fuel needs some time to mix and combust. All HGHFSs still have different heat flux, since the combustion of the fuel-air mix intensifies heat transfer, while the flow pattern near the flame deck remains similar to that in compression mode. Curves of the heat transfer coefficient are similar to heat flux graphs, the same as in compression mode (Fig. 6.14b). Recall the difference observed in compression mode that calculations of heat transfer coefficient demonstrated. Peaks in the heat transfer coefficient sharpen (almost doubling in case of HGHFS No. 2), but they do not appear simultaneously. The distance between heat transfer coefficient peaks reaches 30◦ of the full rotation of the crankshaft. We believe that the effect is due to injection of a hot stream of fuel

6.1 Heatmetry in the Diesel Engine

185

(a)

(b)

(c)

(d)

Fig. 6.13 Heat flux and heat transfer coefficients, n = 1500 rpm, no fuel supply: a HGHFS 1; b HGHFS 2; c HGHFS 3; d HGHFS 4

(this mixture formation is typical for turbulence-chamber engines), which changes the pattern of the flow around the flame deck in the locations where the HGHFSs are installed. The maximum heat transfer coefficient for the gas temperatures reconstructed by the indexing data is reached simultaneously across the entire CC. The values of heat transfer coefficient are lower and the differences between them are smaller in the locations where the HGHFSs are installed (Fig. 6.15). Using the gas temperature that is average over the CC yields a conservative value of the heat transfer coefficient, since it is calculated similarly to the single-zone model.

6.2 Heat and Mass Transfer in a Vertical Tube (Nuclear Fuel Storage Facility) This section describes only some of the results obtained in experiments at I. I. Polzunov Scientific and Development Association on the Research and Design of Power Equipment, formerly, Central Boiler and Turbine Institute (CBTI) (St. Petersburg, Russia).1 The scheme for heatmetry using GHFSs was implemented during a large-scale and multifactor experiment for the first time ever. The extensive experimental material became the subject of analysis going far beyond heat flux measurements. Therefore, the results provided below are only illustrative. 1 Prof.

Yu. S. Chumakov, Dr. E. M. Lebedev, and others participated in the experiments.

186

6 Industrial Experiments

Fig. 6.14 Heat heat flux graphs (a) and heat transfer coefficients (b) with fuel supply

(a) q, kW/m2 140 120 100 80 60 40 20 0 -360

(b)

h, kW/(m2 K)

-180

TDC

HGHFS-1 HGHFS-2

180 HGHFS-3 HGHFS-4

1400 1200 1000 800 600 400 200 0 -360

-180

TDC

180

The experimental object was a vertical steel tube with metric converter Product ID98 mm 98 mm inner diameter that was 6000 mm high (Fig. 6.16). Windows 6 were made in the channel of tube 1 for laser flow diagnostics. Boundary conditions on the tube surface from the air side were different. Direct current heating allowed to achieve the regime q = const on individual sections and throughout the tube height. In this case, power supply cables with a circuit compensating for heat removal were brought out to conducting clamps 2. Water heaters 3 permitted to provide the regime T w = const (in different variants). The tube’s outer surface was covered with heat insulation layer 4 and entrance section 7 was fixed on supporting plate 8. To conduct experiments, GHFSs were fitted at inner (11 pieces) and outer (4 pieces) tube surfaces (in the latter case, they were used for control of heat losses through the insulation). The arrangement of GHFSs on the tube’s outer surface is shown in Fig. 6.17 and at the inner one in Fig. 6.18. The GHFS dimensions (5 × 20 mm) were chosen in order to ensure reliable signal recording by the measurement system. It was subsequently taken into account that

6.2 Heat and Mass Transfer in a Vertical Tube (Nuclear Fuel Storage Facility)

(a)

(b)

(c)

(d)

187

Fig. 6.15 Heat flux graphs and heat transfer coefficients, n = 1500 rpm, fuel supply: a HGHFS 1; b HGHFS 2; c HGHFS 3; d HGHFS 4

the sensor batch average volt-watt sensitivity was at a level of 10 mV/W. GHFSs were kept at the surface of a cooling channel by means of a thin layer of KPT-08 paste. Mounting GHFSs on the tube’s inner surface presents certain difficulties. To facilitate the mounting, 5 operational 22 mm dia holes located in the direct vicinity of the calculated location of GHFSs fastening were made in the tube wall. These holes were later covered with lids. Remaining 5 GHFSs were placed near the working holes for a laser Doppler anemometer, making it easier to mount them (see Fig. 6.18). After the working section was machined at the factory, it was subjected to finishing: burrs were removed from edges, and beds for mounting sensors were sanded. Additional 1.5 mm dia holes at the mounting places served to connect wires. The marks of the position (distances from the working section entrance) where GHFSs were mounted are listed in Table 6.1. 0.1 mm dia and 6.2 m long nichrome wire ohmic heater was placed at the tube axis to calibrate GHFS after they were installed and connected to the measuring equipment. Electric power in the calibration heater circuit was controlled by a voltmeter and an ammeter via an autotransformer connected to the AC mains (220 V, 50 Hz). The working section was blanked off from both ends with heat-insulating plugs in these experiments. Therefore, the heater heat flux differed little from the total heat flux recorded by a GHFS in steady-state conditions. Figure 6.8 illustrates the relationship between GHFS readings and the electric power. The deviation of GHFS readings from the input power value does not exceed 8% over the entire measurement range.

188

6 Industrial Experiments

(a)

(b) 10 2

9 5

9 2

1 9

6m

5

3 2 5

5

9

9

2 7 8

4

Fig. 6.16 General view and scheme of working section: 1—cooling channel, 2—conducting clamp, 3—coil-type water heater, 4—heat insulation, 5—probes measuring air temperature, 6—LDA windows, 7—entrance section, 8—plate imitating floor, 9—GHFS

Standard thermocouples were used along with GHFSs in the experiments. After junctions and thermal electrodes were mounted and connected to commutators, thermocouple calibration was performed with reference to the K-type thermocouple. Heat fluxes and temperatures were measured in the experiments at an operator’s command, with all of the GHFSs signals recorded. As shown in Fig. 6.19 (left part of the working section sketch), the experiment protocol contains the air velocity profiles in the working section that are typically measured over four cross-sections throughout the height (these cross-sections are “attached” to the working section by footnotes). A section corresponding to the scale used for plotting all profiles is shown below. An average air velocity value over a given cross-section obtained by integrating the corresponding velocity profile is shown above the profile. It should be noted that air velocities were measured in terms

6.2 Heat and Mass Transfer in a Vertical Tube (Nuclear Fuel Storage Facility)

189

GHFS

Fig. 6.17 Heatmetry with surface-mounted sensors

Fig. 6.18 Window of LDA system with studs for installing fasteners. GHFS mounted at inner tube surface is shown (a and b)

of diameters in the drawing plane, whereas air temperatures were measured in terms of diameters perpendicular to this plane. The height distribution of heat flux over the inner surface of the working section is shown on the left of the velocity profiles. The experiment results contains the air temperature profiles in the tube measured by a resistance thermometer located to the right of the working section. The wall

190

6 Industrial Experiments

Table 6.1 GHFS positions At inner surface HFS No. Entrance distance (m) 1 2 3 4 5 6 7 8 9 10 11

0.642 1.172 1.822 2.435 2.976 3.538 4.195 4.636 5.228 5.785 6.287

Fig. 6.19 Heat balance of facility

At outer surface HFS No.

Entrance distance (m)

1 2 3 4 – – – – – – –

1.814 2.440 4.210 5.763 – – – – – – –

QGHFS/Qel 1,8 1,6 1,4 1,2 1,0 0,8 0,6 0,4 0,2 0,0 40

60

80

100

120

140

Qel,W

temperature profiles measured throughout the tube height are shown on the right of the air temperature profiles. All the diagrams shown in the protocol are based on the averaged measurement results recorded. After the experiment was completed, the records are send to an ADC. Heat flux fluctuations were observed on the inner surface of the working section for all regimes during the experiments. Notably, the measuring circuit was not intended for registering a real oscilloscope record: the GHFS sampling frequency was about 1 Hz. Therefore, the fluctuation diagrams (Fig. 6.20) are the recording of random heat fluxes with a time internal of 1 s. They do not allow assessing the frequency characteristics. However, they do point to the fluctuation amplitude value (Fig. 6.21). As heat flux fluctuations is an integral property inherent to the given physical process, their characteristics vary with parameters of the thermal process. This conclusion may have new consequences, since as far as we know, vertical tube exper-

6.2 Heat and Mass Transfer in a Vertical Tube (Nuclear Fuel Storage Facility) Fig. 6.20 Typical experiment results

(a) w=0.662 m/s

191

t=31.62 °C h, m 6

w=0.681 m/s 5

t=31.82 °C

4

w=0.683 m/s 3

2 t=25.84 °C w=0.649 m/s 1

= 55 m/s

= 25 °C

50 tw, °C

50 q, W/m2

(b)

0

w=0.752 m/s

t=27.9 °C

w=0.776 m/s

t=29.3 °C w=0.752 m/s

t=18.18 °C w=0.745 m/s

= 55 m/s 100 q, W/m2

= 25 °C 50 tw, °C

192 Fig. 6.21 Heat flux fluctuations (regime T2.5N12)

6 Industrial Experiments q, W/m2 GHFS-2

200

GHFS-4 GHFS-5 GHFS-1 GHFS-8

100 0 -100

without honeycomb 0

Fig. 6.22 Heat flux fluctuations (regime T2.5N6)

1

2

with honeycomb 3

4

5

6

, min

q, W/m2 50

GHFS-2 GHFS-4 GHFS-5 GHFS-1 GHFS-8

0 -50

0

1

2

3

4

, min

iments combined with direct local heat flux measurement have not been conducted so far. As an example, the nature of heat flux fluctuations for the T2.5N6 regime is considered (the notations here are: heating metric converter Product ID2.5 m 2.5 m long at T w = const, N is natural convection, F is forced convection, Q is tube heating at q = const). Two series of recordings are obtained. The working section entrance in the second series (at a dia of 98 mm) was equipped with a 100 mm high honeycomb. Figure (Fig. 6.22) presents the recordings of GHFS signals for both series (the position of the respective sensors is shown on the heat flux per unit area curve). It can be seen that air flow “smoothing” at the entrance significantly reduces the amplitude of fluctuations recorded by the sensors positioned near the entrance (sensors 2 and 4). The amplitude of the fluctuations recorded by the sensors farther away from the entrance practically did not change. Thus, the change in flow regime alters the heat flux fluctuation characteristics. Note that a relative range of fluctuations at the heated section entrance is about 60% of the average value and increases to approximately 75% in the air flow direction. An even larger range of fluctuations is seen within the T2.5N12 regime which is close to the nominal regime in heat power (Fig. 6.22). If it is about 65% at the entrance, then it increases to values larger than 100% downstream. Heat flux fluctuations are the result of both velocity fluctuations arising at the tube entrance and the fluctuating character of the processes occurring in the heat supply zone and associated with formation, development and decay of large-scale structures.

6.2 Heat and Mass Transfer in a Vertical Tube (Nuclear Fuel Storage Facility) Fig. 6.23 Heat transfer coefficient (free convection, heating q = const)

193

Nu 8 7 6 5

Nu=0.083Ra q0.293

4 3

2

1 10 5

10 6

Ra q 107

The role of the first mechanism is reduced downstream, while that of the second one grows, which is confirmed by the explanation of the increase in relative amplitude in this direction. Similarity equations were obtained for heat transfer calculation for each series of experiments. Nusselt numbers (N u) as a function of Rayleigh numbers (Ra) were found for free convection conditions, and of Rayleigh and Reynolds numbers for mixed convection conditions. An example of data processing is shown in Fig. 6.23.

6.3 Electric Machines An electric machine can undergo unsteady thermal actions that result heating during tests and operation. A higher temperature damages the winding insulation, bearings, commutator-brush units, etc. For this reason, there are increasingly stringent requirements imposed on systems and methods for cooling electric machines. Using GHFS for heatmetry of electric machines helps to obtain information about the thermal state. The first series of experiments was test of turbogenerator TZFG-160-2MUZ with a power of 160 MW at Elektrosila OJSC (Fig. 6.24). The efforts of the factory’s specialists were coordinated by Prof. E. I. Gurevich (Head of Department) and by Yu. V. Pafomov (Head of Electric Machine Cooling Laboratory). GHFSs were positioned at the channel surfaces of the cooling system and in the gap between a rotor and a stator. Thermocouples were installed in near the heat flux sensors. Tests were conducted in the following operation modes: idle mode at rated voltage and at 10% excess of rated voltage, short circuit mode, and operation mode without excitation. 10 series of measurements confirmed that GHFSs were capable of operating in the high-power electromagnetic field zone, yielding fundamentally new information

194

6 Industrial Experiments

Fig. 6.24 Turbogenerator in assembly workshop of Elektrosila plant Fig. 6.25 Sensors location: 1—stator teeth, 2—stator spacer, 3—stator wedge, 4—foam with thermocouple, 5—glass textolite with thermocouple

4

5

3

1

2 Thermocouple GHFS

(earlier concepts of heat flux provided only heat flux measurement results comparable with the calculated heat transfer coefficients). GHFS’s and thermocouples locations at setup are shown in Fig. 6.25, the measured heat flux shown in Fig. 6.26. Another experiment with the use of GHFS in electric machines was the heatmetry in an electric machine with permanent magnets.2 Nowadays, permanent magnet electric machines are becoming increasingly popular. However, Present-day neodymium2 The

experiments were conducted in conjunction with the Lappeenranta University of Technology (Finland). Prof. U. Purhonen and Dr. H. Yussila took part in the work.

6.3 Electric Machines

195

Table 6.2 Machine main parameters Output power (kW) Speed (min−1 ) Line-to-line terminal voltage in star connection (V) Roted torque (Nm) Roted current (A) Thickness of PM (mm) PM remanent flux 20 ◦ C PM remanent flux 80 ◦ C Mass of magnets (N d FeB) (kg)

Fig. 6.26 Heat flux graph in short circuit mode

37 2400 400 147 60 16 1.1 1.03 3.9

q, W/m2 -0.5 -1.0 -1.5 -2.0 -2.5 -3.0 0

20

40

60

80

100

120

, min

iron-boron permanent magnets are sensitive to higher temperatures, which requires accurate determination of the parameters of the motor cooling system [10]. In our tests [11] one GHFS was installed on the surface of the stator of an electric motor (Fig. 6.27). The main parameters of the motor are given in Table 6.2 [11] (Fig. 6.26). Heat flux sensor 10 × 10 mm2 , installed using thermal paste on the stator slot wedge (Fig. 6.27b). The choice of the sensor installation location was due to the fact that the maximum gap between the stator and the rotor does not exceed 2 mm, and with a sensor thickness of about 0.2 mm, taking into account the thickness of the layer of heat-conducting paste, the possibility of damage to the sensor while rotating the rotor remained. Therefore, the sensor was installed in a recess on the surface of the winding. In any case, the sensor is installed inside an electric machine where a strong electromagnetic field is present. The no-load tests were performed in the generator mode using the DC machine drive as a prime mover. Figure 6.28 illustrates the test setup. The no-load test was performed to evaluate the heat flux in one local point, the induced back-EMF of the machine, stator iron losses, the Joule losses of the permanent magnets, and the mechanical loss in no-load conditions. The main idea is to use

196

6 Industrial Experiments

Fig. 6.27 One stator equipped with windings (a); heat flux sensor (b) Fig. 6.28 No-load test arrangement for the concentrated winding axial flux PM

Torque riding unit

Frequency unit

DC machine

Portotype machine

Heat flux Temperature reading unit reading unit

Power analyzer

PC

sensor for on-line heatmetry inside permanent magnet machines and similar electric drives. The stator phase winding temperatures at no load at different rotational speeds (Fig. 6.29). Pt100 temperature sensors were used in the phase windings. The temperatures increased during the tests (30 min) because of the heat capacity of the machine and did not reach the steady-state condition. The rotor maximum temperature was measured from top of the magnet using temperature labels, which stick on the rotor. The marker color changed from white to gray when the temperature maximum for this label was exceeded (Fig. 6.30). The local heat fluxes in the air gap at no load at different rotational speeds varying from 1200 to 2400 min−1 are shown in (Fig. 6.31). Test results were processed in a dimensionless form. Nusselt number was calculated as hr (6.1) Nu = k where h is the heat transfer coefficient (W/(m2 K)), r the radius where the sensor is installed (m), and k the air thermal conductivity (0.025 W/(m K)). The Reynolds number is calculated as in [10]:

6.3 Electric Machines

197

Fig. 6.29 Temperatures of windings as a function of time at different rotational speeds

Fig. 6.30 Temperature has reached 65 ◦ C on the top of a magnet during the measurements

Reω =

ωr 2 ρωr 2 = μ ν

(6.2)

where ρ (kg/m3 ) is the density of the fluid, μ [kg/(ms)] dynamic viscosity of the fluid, ν is the kinematic air viscosity (1.5 × 10−5 m2 /s), and ω (s−1 ) is the rotor speed on the radius r (m).

198

6 Industrial Experiments

Fig. 6.31 Heat flux graphs in the air gap as a function of time at different rotational speeds. The direction of the heat flux is from the rotor towards the stator slot

Fig. 6.32 Local N u as a function of Re: 1—measured data, 2—laminar flow, 3—turbulent free rotor, 4—laminar free rotor, 5—air gap ratio δ/r = 0.0106, 6—δ/r = 0.0212, 7—δ/r = 0.0297

Local heat transfer coefficient is calculated as follows: h=

q T2 − T1

(6.3)

where q is the heat flux, measured by the sensor (W/m2 ), T2 and T1 are the temperatures of the stator, measured by the thermopile, and air, far from the motor (20 ◦ C), respectively. In addition, Fig. 6.32 shows the local measured Nusselt number from the investigation made by Howey et al. [12] and data for laminar flow from [13]. As presented in [13] for the air gap ratio δ/r  0.01 and 2 × 104 < Re < 150 × 104 , the local N u does not depend on Re (curve 2 in Fig. 6.32). For a disc with δ/r < 0.05, there is a

6.3 Electric Machines

199

Couette flow between the rotor and the stator. For higher Reynolds number, the N u depends on Re with different ratios. Our measured data for 6000 < Re < 20,000 are lower than the predicted N u. This may be explained by the fact that the sensor was installed in a cavity on the fiber insulation plates. For 7000 < Re < 20,000, the local Nu increased from 15 to 150. For 20,000 < Re < 40,000, the local N u decreased from 150 to 70, which is practically the same as curve 4 in Fig. 6.32. In our further research, we will increase the number of sensors and record heat fluxes pulsations.

6.4 Furnace of a Steam Boiler Analysis of the processes occurring in furnaces of steam boiler is difficult because there are little data of heat flux distribution at heat-receiving surfaces. Local heat flux can vary by order of magnitude or more both with the thermal conditions and with nonuniform heating of different zones of the furnace surface. This quantity characterizes the processes of heat transfer and fluid dynamics in the furnace to a considerable extent, affecting the tube’s temperature conditions and, ultimately, the efficiency, reliability, and safety of a boiler as a whole. Using traditional heat flux and temperature sensors (thermal inserts, heat metercalorimeters, thermal probes) proves to be labor-consuming. The experiments yield scant data, and their results are not always reliable. However, thermally resistant HGHFSs also have not found industrial use up to now, but for another reason: they only appeared in 2007. There have been extremely few studies and applications of the sensors. For this reason, a series of laboratory and pilot experiments have been conducted prior to industrial experiments. A hot water boiler VTG-80 with a power of 80 kW operating on light diesel fuel (GOST 305-82) or on natural gas (GOST 5542-87) served as an object of industrial tests; its schematic sketch is shown in Fig. 6.33. The boiler is meant for heating water

Fig. 6.33 Scheme of fire-tube boiler VTG-80. Numbers indicate: 1—front cover, 2—boiler furnace, 3—smoke tubes, 4—tube boards, 5—chimney part of boiler, 6—hatch in boiler chimney part, 7—burner facility

5

4

2

3

1

7 6 3

4

200

6 Industrial Experiments

Fig. 6.34 Arrangement of HGHFSs and thermocouples: sensors 1…4, thermocouples T 1 , T 2

1 T1 2

3

T2

4

R230 R400 R315

to 95 ◦ C at a pressure of 4 kg/cm2 and consist a body, a flue, and a bundle of smoke tubes. The goal of the experiments was to estimate the “viability” of HGHFSs and pipelines, to check the efficiency of the system, and to assess the potential amount of data it provides as a rough approximation.3 HGHFSs based on a stainless steel + nickel composition were used in our study. The layer thickness was 0.1 mm, the angle of section 45 ± 5◦ to the blank plane, the sensor thickness 0.2 mm, dimensions in plane were 7 × 10 mm. Current-collecting conductors were connected at the extreme points of the plates. The arrangement of HGHFSs at the inner wall of the front cooled cover is shown in Fig. 6.34 and Fig. 6.35. Control chromel-alumel thermocouples are also located here. HGHFS wires were brought out through the connecting pipes located on the cover sides. HGHFS signals were transferred to a PCLD-789D amplifier and multiplexer board and from it to a PCL-818HG ADC and were processed using Genie software. Tests were performed at different loads over the power range from 25 to 45 kW. Heatmetry results (without processing) are shown in Fig. 6.36a. Figure 6.36a demonstrates time-averaged heat flux, Fig. 6.36b the same quantities processed using Butterworth dispersed filters: high-frequency noise is removed, harmonics whose amplitude is less than 10% of that of a reference frequency are suppressed. All manipulations are done in Matlab.

3 Students

O. V. Klyuchka and I. A. Sivakov from the Saint Petersburg State Institute of Fine Mechanics and Optics took part in the experiments.

6.4 Furnace of a Steam Boiler

201

Fig. 6.35 Sensors (a) and their mounting (b) on boiler cover Fig. 6.36 Typical results of heatmetry in boiler VTG-80: a without processing, b after averaging and processing by Butterworth filters. Notations are the same as in Fig. 6.18

(a) q, kW/m2

1 3

2 4

100

150

4 3 2 1

0

50

(b) q, kW/m2

1 3

200

2 4

4 3 2 1

0

50

100

150

200

Experiments with the boiler VTG-80 confirmed the efficiency of GHFSs. The heat transfer surface is a set of tubes and fins forming a closed loop. It was planned to use standard CKTI inserts both for control and, upon modification, for location of GHFSs. As for the planned locations of paired GHFSs on the measuring insert (frontal and lateral), two standard K-type thermocouples were used as wires. The chromel wires

202

6 Industrial Experiments

(a)

(b) 5

4

1 2

3

Fig. 6.37 Measuring insert and calibrated cell: a general view of insert equipped with GHFSs, b calibration manifold of sensor-equipped insert (b): 1—HGHFSs, 2—burner, 3—radiator serving as heating source, 4—tube, 5—electric heater

were connected to one of the GHFSs, the alumel ones to the other. Thermal stability of the assembly was subsequently maintained and the contribution of electrode thermopower was excluded. Measuring inserts equipped with HGHFSs were calibrated in laboratory conditions on a specially designed test setup (Fig. 6.36a), whose distinctive feature is modeling the conditions of heat impact on the tube wall, provided that absolute (Joule) calibration of sensors is possible. Calibration procedure is shown in Fig. 6.37b. An electric heater is coaxial with the tube insert. Its power regulated by an autotransformer maintains a “reference” wall temperature over the range 370…630 K. Thus, the real thermal state of a wall cooled by a water-steam mixture is successfully modeled. Prior to calibration, the insert surface was covered with soot to the emissivity of 0.98…0.99. The surface was heated from the outside by a gas burner. The heat flux was calculated by the Stefan-Boltzmann equation where temperatures were determined through the readings of the thermocouples positioned on the heating source and at the tube surface. HGHFS signals were recorded simultaneously with the readings of test thermocouples. It was found that the volt-watt sensitivity of different HGHFSs was 110…190 µV/W. Since calibration was done in the conditions close to operating ones, individual characteristics of each insert were also used for data processing in the industrial experiment. As mentioned above, modified CKTI inserts were used to install HGHFSs. One sensor was located on the frontal part of an insert and the other with a 45◦ displacement. Both HGHFSs were fastened on the high-temperature compound-milled 8 × 10 mm areas covered with a mica layer and a stainless steel protective shield spot-welded to the loop. The industrial experiment part was carried out in the furnace of the boiler BKZ 210-140F mounted at TEP-4 in the Kirov city.4 4 Employees

of the Reactor and Steam Generator Engineering Department of Peter the Great St. Petersburg Polytechnic University, Dr. K. A. Grigoriev and Dr. V. Ye. Skuditsky, and student P. G. Anoshin participated in the experiments.

6.4 Furnace of a Steam Boiler

203

Fig. 6.38 Longitudinal section of boiler BKZ-210-140F

An updated steam boiler BKZ-210 (with a boiler rating of 58.3 kg/s, with superheated steam parameters: pressure of 13.8 MPa, temperature of 540 ◦ C, heat power of 143 MW) started operating on eddy burning of coal, peat, and natural gas. The surface of the gasproof furnace was provided with updated and calibrated CKTI inserts with HGHFSs in the eddy zone of active burning (shown in Fig. 6.38).

204 Fig. 6.39 Heatmetry, 1—TK No. 1 CKTI, 2—TK No. 2 CKTI, 3—average TK No. 1 and No. 2 CKTI, 4—frontal HGHFS, 5—side HGHFS, 6—average 4 and 5

6 Industrial Experiments

q, kW/m2 300

1 2 3 4 5 6

200

100

0

Kindling

18

19

20

21

22

23

Stationary gas load (D=32 kg/s)

Local heat flux obtained by an HGHFS and a standard (with thermocouples) CKTI insert were compared during a long experiment (lasting over 4 months). At this time, the boiler was operating at loads of 0.5…1.2 of the rating and in different operating conditions. The heat stress of the furnace space was varied over the range of 75…180 kW/m2 and the average absorbed heat flux per unit area was 72…155 kW/m2 . The flame temperature in the eddy zone, measured by an optical pyrometer depending on boiler steam load and burnt fuel, varied over the range of 1270…1670 K. As a whole, the furnace’s eddy zone appeared to be sufficiently isothermal (the difference between maximum and average flame temperatures did not exceed 100 K in individual operating conditions of the boiler). The data on the local heat flux variations over time, obtained using CKTI inserts and HGHFSs are plotted in Fig. 6.39. It can be seen that the character of the dependences is qualitatively close (with the exception of thermocouple No. 1) and correlates with the variations in the boiler’s operating condition (periods of kindling and stationary load). As for qualitative correlation, however, the heat flux was calculated by the thermocouple methods and by HGHFS signals are different by 200 kW/m2 . The estimates obtained for the processed results of the balance experiment indicate that the values of local heat flux per unit area calculated by thermocouple methods exceed average furnace values by a factor of 2.4…3.6, while those found through the HGHFS signals exceed the average value by a factor of 1.2. The later agrees well with the data given in literature and the physical concepts of heat transfer patterns in chamber furnaces. The foulings of the furnace waterwalls had on HGHFS readings was found during the boiler’s operation on solid fuel. For example, it was discovered in adjustment and alignment tests that increasing the fraction of secondary air together with fine grinding of fuel elevated a temperature maximum in the active combustion zone and enhanced the slagging of furnace waterwalls, which follows from a sharp decrease in HGHFS signal (typical curves are shown in Fig. 6.40). Reduced secondary air fraction and especially coarse grinding of fuel favored a temperature decrease in the

6.4 Furnace of a Steam Boiler Solid fuel (D=56 kg/s)

300

4 5

200

Granular grind

q, kW/m2

Slagging

Fig. 6.40 Heatmetry during slagging of thermal inserts (example). Notation of the curves see Fig. 6.39

205

6 100

0

5

10

15

20

Gas fuel (D=48 kg/s)

combustion core and self-cleaning of the furnace waterwalls from foulings, which was evidenced by an increase in HGHFS signal and confirmed visually. The above facts prove that using HGHFSs for determining absorbed heat flux is justified. This method correlates qualitatively with the methods developed at CKTI. However the results are more accurate and valid. Concerning stability, no qualitative variations in the readings were detected during operation of the sensors. This is not a shortcoming inherent to the CKTI methods: the heat flux was calculated through the thermocouple’s readings took implausible values in seemingly good conditions. The relative error of heatmetry in the furnace of the steam-generating unit amounted to 17%. Thus, the efficiency of HGHFSs at the furnace fire surface and their high information capability were confirmed during industrial experiments. Readings of the sensors correspond to boiler regimes and operating conditions. An HGHFS can serve as a tool for slagging diagnostics. Knowing furnace waterwall parameters such as absorbed heat flux and surface temperature, it is possible to determine the slagging (fouling) degree of a furnace or a fraction of cumulative damage of furnace waterwalls. Another practically important use of HGHFSs is identifying the flame position in the furnace volume: different flame positions are consistent with different values of incident and absorbed heat fluxes at the furnace’s enclosure walls. Thus, it is possible to indirectly identify the flame position by obtaining the furnace width and depth distribution of absorbed heat flux. Even though industrial experiments have yielded positive outcomes, it was found that mounting HGHFSs on standard CKTI inserts was incredibly complex and timeconsuming. Structurally and technologically, it is more common to mount a sensor at the surface of a fin connecting the tubes: this is easier for commutation and thermal shielding of HGHFSs, there is no need to treat a tube surface area for mounting a

206

6 Industrial Experiments

Fig. 6.41 Calculation of angular coefficients in “flame-finned tube” system: a heating scheme, b equivalent model

sensor, etc. Since the fin is welded to the tube with a continuous seam, it can be assumed to have a rectangular cross-section with an effectiveness of at least 0.95 [14]. To pass from the local heat flux measured at the middle of the fin to the surfaceaverage value in the furnace, let us consider the scheme shown in Fig. 6.41 illustrating the conditions of flame heating of a “representative element” at the furnace surface and the substitution model [15] where the role of the flame is played by a flat wall closing a system of surfaces in the line ABCDBA. Calculating angular coefficients by the crossed-string method proposed by Polyak in 1935, we obtain the relations. φ12

  l3 l5 1 (l5 + l6 ) − (l3 + l4 ) , 1+ − = , φ13 = 2l1 2 l1 l1

where l1 = S, l2 = S −  l5 = l6 =



d 2 − δ 2 ≈S − d, l3 = l4 ≈

1 2 S− d − δ2 2

2 +

πd π d − 2δ ≈ , 4 4

2  1 d d − d 2 − δ2 ≈ S − , 4 2

the approximation takes into account that δ  d. Since the temperature is almost the same for all irradiated surfaces, mutual irradiation of surfaces AC, CD and DB is excluded and the closing condition assumes the form φ12 + 2φ13 = 1. The average heat flux at the surfaces of tubes and fins is

6.4 Furnace of a Steam Boiler

207

Fig. 6.42 Plot of q/qmax versus S/d for finned furnace waterwall tubes

q/q max 0,85 0,80 0,75 0,70 0,65 0,60 1

q¯ =

Q 12 + 2Q 13 = F2 + 2F3

ε pr 0



TM 100

4

−



T 100

4

2



F1 = ε0

F2 + F3

TM 100

S/d

3



4 −

TM 100

4 

l1 , l2 + 2l3

(6.4) and an HGHFS mounted at the middle of the fin takes the heat flux

     TM 4 T 4 − qmax = ε pr 0 100 100

(6.5)

the desired calculated relation then takes the form q¯ qmax

=

l1 S = l2 + 2l3 S−d +

πd 2

=

1 1+

π/2−1 S/d

≈

1 1+

0.571 S/d

;

(6.6)

this correction (6.6) is determined only through a dimensionless step S/d, does not depend on operating parameters, and should be factored into the calculations of a heat transfer surface in general (Fig. 6.42). Two variants of mounting HGHFSs at the fin are possible: an HGHFS can be “pasted” on a cleaned fin surface and closed with a stainless steel protective shield, as shown in Fig. 6.41. It is not difficult at the stage when the boiler is installed; however, either construction staging or steeplejack services are required for mounting an HGHFS on operating equipment, since a sensor cannot be repaired. It is more effective to use a measuring cell shown in Fig. 6.43. Measuring cell 3 is screwed-in into the throughfeed in fin 2 so that the cell end would be located flush with a “hot” surface of the fin. HGHFS 4 covered with a thermally resistant insulation layer (or lined with mica) is located at the cell bottom; the distance from the “hot” side of the HFS to the fire surface does not exceed 0.2…0.5 mm. Steel insert 5 with slots for wires is pressed so that its “cold” end finds itself in the “cold” plane of the fin. Thermally resistant paste 6 is kept due to adhesion and is additionally fixed by circular boring. Conductors 7 are brought out beyond the setting loop and are connected to measuring equipment. Some part of cell 3 projecting beyond the fin has faces for screwing-in into fin 2 from the side of the setting opening.

208 Fig. 6.43 Method of mounting HGHFSs at the fin. 1—Waterwall tubes, 2—fin, 3—measuring cell, 4—HGHFS under electric insulation layer, 5—steel insert with slots for wires, 6—thermoresistant paste, 7—commutation wires

6 Industrial Experiments 2

3

1

4 5 6

The main advantage of such a construction is that a cell with the HGHFS can be mounted or replaced from the external areas of the boiler. An outer heat-insulating layer of the furnace waterwall has to be removed to access the mounting zone of the cell. GHFS-based heatmetry in steam-generating units is still in its infancy stage, and its development is one the directions of our further research.

References 1. Kavtaradze, R. Z. (2007). Lokal’nyy teploobmen v porshnevykh dvigatelyakh (Local heat transfer in piston engines: Textbook for universities). Moscow: MGTU imeni N.E. Baumana. 2. Petrichenko, R. M. (1983). Fizicheskiye osnovy vnutritsilindrovykh protsessov v dvigatelyakh vnutrennego sgoraniya (The physical foundations of in-cylinder processes in internal combustion engines: Textbook for universities). Leningrad. 3. Isii, A., Nagano, K., Adachi, K., et al. (2000). Measurement of instantaneous heat flux flowing into metallic and ceramic combustion chamber walls. SAE Technical Papers, 01–1815, 1–24. 4. Rozenblit, G. B. (1977). Teploperedacha v dizelyakh (Heat transfer in diesel engines). Moscow: Mashinostroyeniye. 5. (1981). Toplivoraspredelitel’nyy nasos tipa VE: Tekhnicheskoye opisaniye (VE type fuel injection pump: Technical description). BOSCH GmbH. 6. Kostin, A. K., Pugachev, B. P., & Kochinev, Y. U. Y. U. (1989). Rabota dizeley v usloviyakh ekspluatatsii (The operation of diesel engines in operating conditions: Reference). Leningrad: Mashinostroyeniye. 7. Semenov, B. N., Pavlov, Ye. P., & Koptsev, V. P. (1990). Rabochiy protsess vysokooborotnykh dizeley maloy moshchnosti (The workflow of high-speed diesel engines of low power). Leningrad: Mashinostroyeniye. 8. Vintsarevich, A. V., Sapozhnikov, S. Z., Pavlov, A. V., et al. (2017). Gradient heat flux measurement as monitoring method for the diesel engine. In International Conference on Problems of Thermal Physics and Power Engineering 2017, PTPPE 2017, National Research University Moscow Power Engineering Institute (NRU MPEI) Moscow, Russian Federation, 9–11 October 2017. 9. Vintsarevich, A. V., Mityacov, A. V., Gerasimov, D., et al. (2018). Gradient heat flux measurement as monitoring tool for the diesel engine. In International Scientific Conference on

References

10. 11.

12.

13. 14. 15.

209

Energy, Environmental and Construction Engineering, EECE 2018, Congress Center of Peter the Great St. Petersburg Polytechnic University, 19–20 November 2018. Howey, D. A., Childs, P. R. N., & Holmes, A. S. (2012). Air-gap convection in rotating electrical machines. IEEE Transactions on Industrial Electronics, 59(3), 1367–1375. Jussila, H. (2009). Concentrated winding multiphase permanent magnetmachine design and electromagnetic properties-Case axial flux machine. Ph.D. dissertation, Acta Universitatis Lappeenrantaensis 374, Lappeenranta Univ. Technol., Lappeenranta, Finland. Howey, D. A., Holmes, A. S., & Pullen, K. R. (2009). Measurement of stator heat transfer in aircooled axial flux permanent magnet machines. In Proceedings of the IEEE Annual Conference on 35th Industrial Electronics, Porto, Portugal, 3–5 November 2009 (pp. 1197–1202). Harmand, S., Watel, B., & Desmet, B. (2000). Local convective heat exchanges from a rotor facing a stator. International Journal of Thermal Sciences, 39(3), 404–413. Lykov, A. V. (1967). Teoriya teploprovodnostim (Theory of thermal conductivity). Moscow: Vysshaya shkola. Blokh, A. G., Zhuravlev, Y. A., & Ryzhkov, L. N. (1991). Teploobmen izlucheniyem (Radiation heat transfer: Reference book). Moscow: Energoatomizdat.