The Use of Surge Chambers in the Limiting of Gas Pulsations Produced by Reciprocating Compressors

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TÎIS USE OF SURGE CHAMBERS IN THE LIMITING OF Ga s

pulsations

produced

BY

reciprocating

compressors

THESIS

Submitted in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE IN APPLIED MECHANICS at the POLYTECHNIC INSTITUTE OP BROOKLYN

David Seltzer June 1950

Approved:

666 Thesis Adviser

Head of Department

ProQuest Number: 27591584

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uest ProQuest 27591584 Published by ProQuest LLO (2019). C opyright of the Dissertation is held by the Author. All rights reserved. This work is protected against unauthorized copying under Title 17, United States C o d e M icroform Edition © ProQuest LLO. ProQuest LLO. 789 East Eisenhower Parkway P.Q. Box 1346 Ann Arbor, Ml 4 8 1 0 6 - 1346

v it a

The author was horn in New York City on April 5, 1922.

He received his B.M.S. at C.C.N.Y.

in September, 1942. Upon graduation he was employed by the Lummus Company, contractors for the design and erection of petroleum refineries and chemical plants.

He was em­

ployed in the Pump and Compressor Section of the Engineering Department for a year and one-half when he was Inducted into the Army. After about two years as Aerial Phototopographer in the Army, he returned to his position in the Lummus Company where he has since been employed.

David Seltzer

ABSTRACT

On the basis of non-steady one-dimensional frictionless flow, the gas pressure variation induced in the suction and discharge piping to a reciprocating compressor is dependent on compressor speed, gas condi­ tions and the geometry of the system.

These character­

istics can produce a resonant build-up of pressure variation. Using the limitation of the maximum pressure varia­ tion to 5^ of the mean pressure as an empirically satis­ factory criterion, permits the checking of the adequacy of the suction or discharge piping volume.

If the piping

volume is inadequate, surge chambers may be used to pro­ vide the additional required volume. Providing adequate volume for one set of operating conditions may not be sufficient since a change in these conditions may produce excessive pressure variation. This makes arbitrary surge chamber sizing particularly susceptible to error.

TABLa' OF CONTENTS

Acknowledgments.......................................... 1 List of Symbols.......................................... 2 Introduct ion............................................. 5 Procedure......................................

9

Analysis.................................................10 Basic One-Dimensional Equations................... 11 Application of lave Equation...................... 15 Gas Velocity in 5urge Chamber..................... 18 Gas Pressure in Surge Chamber.................... .19 Effect of Pressure Variation on Chamber Sizing....20 Application............................................. 21 Introduction....................................... 22 Methods of Application.............................24 Discussion of Application......................... 27 Suggestions for Further Study..................... 29 Conclusions............................................. 31 Appendix.................................................34 Bibliography............................................ 37 Illustrations & Curves.................................. 40

ILLUSTRATIONS & CURVES Pa£e FIGURES 1....Density & Pressure Vaves, for Adiabatic Process...41 2....Forces on Element, for Equations of Motion........41 3....Flow thru Element, for Equations of Continuity....41 4....Gas Tube, for Application of Wave Equation........42 5... .Compressor Cylinder & Surge Chambers.............. 42

CURVES 1 ... .Variation of |#p| with II... .Variation

of Rwith

........................ 43 ............................ 44

III... .Variation

of R with N ............................. 45

IV. .. .Variation

of R with T ............................. 45

V. .. .Variation

of R with

............................. 46

-1-

ACKN0WLGDGMENT8

The author is grateful for the cooperation of the representatives of the companies listed below, in making available articles and data pertinent to this study. Burgea8-Manning Company Clark Brothers Company Cooper-Bessemer Corporation Fluor Corporation Inger3oil-Rand Company Worthington Pump & Machinery Corporation Appreciation is due to the Lummus Company for encouraging this study.

-2-

LIST OF SYMBOLS

English Alphabet

a

Velocity of sound at gas conditions.

A

Arbitrary constant.

B

Arbitrary constant.

c

Clearance, percentage of swept cylinder volume

d

Diameter of surge chamber.

\D\

Determinant,

e

Constant, 2.781...

E

Cross-sectional area of element of volume,

f

Factor for single- or double-acting cylinder,

g

Gravitational acceleration.

G

Gross-sectional area of surge chamber.

i

Imaginary unity, ))-l .

k

Cofficient of cubic elasticity.

J

Length of surge chamber,

ra.w.

Molecular Weight,

n

Variable,

N

Compressor rotative speed,

p

Pressure,

r

Compression ratio.

R

Surge volume ratio.

3

Condensation,

t

Time.

-3-

T

Absolute temperature,

u

Velocity in direction of flow,

V

Velocity perpendicular to flow,

V

Volume.

Vq

Volume at cylinder midstrpke.

Vj

Volume of swept

cylinder.

Vjj

Volume of surge

chamber.

X

Displacement in direction of flow,

y

Displacement perpendicular to flow.

V

Resistance to flow.

Z

Impedance.

Zqq

Driving-point impedance.

Greek Alphabet o(

Attenuation factor,

/3

Compression cycle constant,

y

Adiabatic process exponent.

S

Variation in condition. Operator.

4

Velocity potential.

7T

Constant, 3.141...

p

Density.

ip

Forcing function.

uj

Angular velocity of compressor crank.

-4-

General Subscripts Pl,

Pg,

etc.

Refer to conditions

of cylinder cycle

Pq , U q , Tq , etc.

Refer to conditions

at x

p^ , u^,

Refer to conditions

at x =

, etc.

u^, T^, etc. Tg,

etc.

s 0.

Refer to mean values. Refer to suction and discharge.

-5-

INTRODUCTION

THE.USE OP SURGE CHAMBERS IN THE LIMITING OF GAS p u l s a t io n s .PRODUCED BY RECIPROCATING CQMPrÆSSORS

The established usage of reciprocating compressors has led to many studies of the various vibrations and pulsations resulting from such compression.

This thesis

will concern itself only with those pulsations of the gas itself which are induced in the piping to and from the compressor by the very nature of the reciprocating compression cycle.

Particular attention will be given

to the possible limiting of these pulsations by the use of large capacity "surge chambers" adjacent to the compressor suction and discharge connections. For this purpose it is assumed that adequate study for a given installation has been made of the compressor running balance, foundations, piping sizing and supports. For even when such proper design is attained, surge chambers are often used to limit gas pulsation. This limitation is commonly based on keeping the gas pressure variation within b% of the mean gas pressure (references 15 and 19).

This empirical criterion is pre­

dicated not only on restricting pressure variation in the piping system, but also on maintaining proper compressor performance without starving or supercharging the cylinder

—7 —

and without attendant compressor speed fluctuations and horsepower losses. Although orifice restrictions and "pulsation dampeners" {combining surge and restriction character­ istics) are also used for pressure variation limitation (references 9, 16, 17), this thesis will only consider surge chambers, the type which the writer has found to be the most common. (In passing, it is interesting to note that by electrical analogy, the surge chamber may be replaced by capacitance and the orifice by inductance.

Many

complex compressor installations have been studied by setting up and analysing equivalent electrical circuits (reference 10).) For practical purposes, the sizing of the surge chambers considered is of the utmost importance.

Indeed

it was the inadequacy of usual empirical sizing methods which aroused the writer’s interest in this study. Within the writer’s experience, surge chamber vol­ umes are usually set by rule-of-thumb to be five to ten times the total compressor cylinder swept volume.

Even

when pulsation is decreased, this arbitrary ratio may

-8-

lead to costly oversizing of surge chambers.

When

troubles arise, a suspected contributing cause is that the surge chamber volume causes a resonant condition. Therefore, in studying the limitation of gas pulsation by the use of surge chambers, consideration will be given to the variation of the ratio of chamberto-cylinder volume as affected by such criteria as compressor speed, gas temperature and compression ratio.

-9-

PRQCSDURS

In order to determine the effect of surge chambers in limiting gas pressure variations which are produced by reciprocating compressors, the following procedure will be adopted; A.

Consideration of basic one-dimensional acou­

stical theory. B.

Consideration of one-dimensional wave equation

as applied to harmonic piston acting in a tube. C.

Consideration of velocity of gas in surge cham­

ber as created by harmonic piston in adjacent cylinder. D.

Combination of B & C to find an equation for

pressure variation in the surge chamber. E.

Consideration of sizing of surge chamber to

limit pressure variation. F.

Consideration of effect of compressor operating

conditions on surge volume ratio. G.

Correlation of theoretical surge volume ratios

with those of actual practice.

-1 0 -

ANALYSIS

-1

ANALYSIS

A.

Basic One-Dlmenalonal Equations In Fluid Plow

1.

Adiabatic Process Assuming perfect gas laws and referring

to figure 1, define condensation, s, as follows: 8 :

(la)

P =

(1+3)

(lb)

Since pyO is constant for adiabatic flow, ■p" "

n

+ 8)"^

(2)

Assuming that s is small enough so that 8^, s^, etc may be neglécteé, (1 4- a )^ = 1 -V '5 s

(3a)

Combining this with equation (2), £- ,

^

a 1 4 b S

(3b)

Sp = Pjjï s s ks

(3c)

p = p^ + ks

(3d)

where k is defined as the coefficient of cubic elasticity, (reference 2).

—12-

2.

Equation of Motion From the forces on the element shown in

figure 2, the unbalanced force = [p - (p 4 dp)] E. -E dp » E

dx

» E yO^ u

Combining with -k ds =

du

(4)

dp - k ds (from 3d),

dx

(5a)

(5b)

3.

Equation of Continuity. Non-Steady Flow From figure 3, note that the mass flow

into the element ~ mass flow u K =

4

out ♦ element change; (u 4 du) E 4 E dx II

(6a)

or neglecting (c^ du), ^

. 0

(6b,

Combining with

Aa

H

ds (from lb),

♦ An «S i * Ai i f ' 0

or neglecting second order terms,

“ s|-ft=fÿ

(7c)

4.

Wave Equation, Steady Flow For

* 0, equation (6b) becomes: « 0

(60

^ = 0 'Yn

(6d)

From equation (4b): ^

%

du = 0

(4c)

Combining equations (6d) & (4c), -An du . Ug^ d/: = ^

(8a)

u2 r |E

(8b)

From dp s k ds & d/> *

ds,

ul ■ ^

(9a)

Letting sonic velocity be a,^

Z

=^

^

(9b)

Combining equations (5b), (7c), (9b), (10a) This equation was developed for steady rather than non-steady flow since the velocities to be dealt with are small compared to the sonic velocity.

Equation (10a) could have been derived by using the concept of the velocity potential (refer­ ence 5).

However, the above development is presented

as the writer’s cruder first study, while the velocity potential method is left for the two-dimensional case considered in the appendix hereto.

5.

Excess Pressure Equation Changing the notation of equation (lOa): Vi” s Ü

(au* 4 Û )

(lOb)

(Au’ - Û) * 0 (11)

Combining this with (7b),

I f = - It = -è^

(12a)

a = i

(12b)

I w

Combining equations (3c), (9b), (12b): Sp =

ks = ±

gp s t Pjjj a u

Pm 4% JL u (13)

—I L —

B.

Appl&cation of One-Dimensional Wave Equation Consider a gas tube of length S- as shown in

figure 4.

The piston ât x » 0 is displaced by a harmonie

forcing function, ^

cos Wt (or real part of

).

The free piston at x =

is driven by the excess pressure

of the sound waves transmitted to and reflected by its surface.

(Reference 2.) Since the general solution of equation (10a) is

tt S A Pj(t ♦ f ) ♦ * Pg(t - f.)

(14)

consider the particular solution for the gas tube wave as u = (A .1“*

B «"1“*)

where n » ^

(16)

This can be thought of as the sum of the generated velocity wave B

e^^^ to the right and the reflected

velocity wave A e ^ ^ e^^^ to the left.

Consistently, from

equation (13), Sp - /O^u Is to the right, Sp =. - p^^au

is

to the left. Thus, combining equations (15) and (13), Sp -

a e ^

(-A e ^ ^ 4^ B e“^“

)

(17)

-1C'

Note that in equations (15) and (17) frictional dissipation has been assumed to be negligible.

If resis­

tance, Y, were present, the exponentials would take the form

and

lux)

factor is defined as o< «

the attenuation

(reference 2).

2/54

(18)

To solve for A and B in equation (15) and (17), note the following boundary conditions: @ x r O & t » 0 Ug = A + B

(19a)

SPo =(B - A ) p^a @ X

(19b)

& t= 0 = A Sp^ » (-A

Define Impedance,

♦ B

(20a)

+ B

(20b)

Z

- Le. u

(21)

Then ® x =

0 :

%o *o

@ X =

j :

Zj Uj : Sp^

Letting

(22a) (8 2 b)

= L, & combining (19a) thru (22b):

to =

Zo (A ♦ B) -

d

Z

=

to = 0

to " SPg,

( ^ . BL) *p^Z^)(L®+l)

(23b)

.

(^a)

= -

(24b)

|d 1 B FI Substituting (24a) & (24b) in (15) & (17),

U = [ (Zjj-Z'^a.)!.® Sp

-

(Z^.^a)e-^’“ j

« [-(Zj,-Pn^4)L®

(£5 a) (25b)

Define Z^ q as driving-point impedance: Zoo =

@ X = 0

(26)

l^o) m a x . Assume end resistances are negligible so Zq =Z^«0, From (22a),

SPq =

From (25a), (23a) © x K\max =

^ Am ^ V - A m

(27a)

= 0, j J t = ^ :

>^01

(27b )

Combining equations (26), (27a), (27b):

=

tanh (in^) =

tan (nC)

(28a)

-18-

C9

Q&3 Velocity In Surge Chamber Referring to figure 5, consider one single-acting

compressor cylinder with surge chambers on suction and discharge sides.

Assume simple harmonic piston motion,

instantaneous valve action without throttling, frictionless direct connections between cylinder and chambers. On thé compreaaor p-V diagram, relations (29), (30), (31) below are known quantities for a particular compressor. Swept cylinder volumet Vj s Vg clearance, c a y — 2

4

Compression ratio,

r «

Then define midstroke volume,

V. - Ts ^ ^4 c 2

Let S s ^2^ " Combining this with (29) Be (30),

rhss Combining (29), (32), (33b), V i ; 2AVe Rttm In general, where iu= — — & K is rpm, V = Vg -

cos uL>t

where t s 0 at the crank end dead center.

(36)

-19-

Asauming that the velocities in the surge cham­ bers near the compressor valves are the same as the pis­ ton velocity, then when the valves are open, the particle displacement may be derived from equation (36) to be;

O _ dy Ug -

~ G

♦ lïs cos wt G

(37)

“ "y """ sin wt

(38a)

The general equation for u^ can be set up in a Fourier series to account for valve timing (reference 8). However, this analysis is concerned only with the absolute maximum value of Ug.

This occurs at wt = ^

during suction , and at w t =

(♦’2TT, 44*11,...)

(42TT,...) during discharge.

'"oLax - “T T "

D.

(58b)

Gas Pressure in Surge Chambers Assume that the surge chamber boundary conditions

are similar to those of the gas tube in section B.

Thus

for one dimensional flow, with negligible chamber end im­ pedances, the relations of Sections B & C may be combined by using equations (26), (28a), (38b) to lead to: SPq =

ten ri

(39)

-2 0 -

E.

Effect of Pressure Variation on Chamber Sizing The most widely accepted criterion for surge

chamber sizing is the selection of that ratio, R, of surge chamber volume to swept cylinder volume,

,

which will limit the maximum gas pressure variation in the chamber to some percentage of the mean gas pressure. As an empirically satisfactory limit (references 14, 16, 19), consider:

= ,06

(40)

Rearranging equation (39) where only absolute values are considered, and noting that n z

)SPo|

= ^

■tP5£m n ^

(41a) |tan

(41b)

Combining (41b) & (9b), whence cf rm

" R = T&Ô

if I

(42=)

This was derived for a single-acting cylinder but may be extended to double-acting by introducing a factor, f, where f = 1 for single smd f = 2 for double-acting.

« = rfe

1

Equation (42b) is plotted as Curve II.

(42*)

—2 1 —

APPLICATION —

METHODS. DISCUSSION. SUGGESTIONS

••22—

APFLiofmcm

A,

wmcm,

dimmsim^ m m m T x c m

Igtroductlcm 1. Purpoee The coiaslder&tioîi of typical installations to ill­

ustrate the application of the amalyeia developed will serve four pur­ poses - a. To check actual euz^e chamber @lal%. h.

To investigate the effect of varying operating

conditions oa the surge volume ratio. c.

To permit the correlation of this study with

present practice. d. To izidicate what analysis revisions can best serve this aj^licaticm.

2. Additional %l#tipns In order to use the data below, as It Is usually avail­ able for ft ç