Gerlach, C. R.

Parker, J. D.

English

NASA Technical Reports Server

Fluid Power Controls Laboratory 
School of Mechanical Engineering 
OKLAHOMA STATE UNIVERSITY 
Stillwater, Oklahoma 
https://ntrs.nasa.gov/search.jsp?R=19650011752 2020-03-16T23:49:57+00:00Z
I 
PROGRESS REPORT NO. 64-5 
STUDY I N  FLUID TRANSIENTS I N  
CLOSED CONDUITS 
Contract: NAS 8 11302 
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PROGRESS REPORT NO: 64-5 
Per iod  of Report: 1 November 1964 through 30 November 1964 
Cont rac tor :  Oklahoma S t a t e  Univers i ty ,  S t i l l w a t e r ,  Oklahoma 
Segment Generating Report: School of Mechanical Engineering 
F lu id  Power and Con t ro l s  Laboratory 
STUDY OF FLUID TRANSIENTS I N  CLOSED CONDUITS 
Cont rac t :  NAS 8 11302 
Cont ro l  Number: DCN-1-4-50-Oll53-Ol(IF) 
CPB 02- 1269-64 
Date: 7 December 1964 
Prepared f o r :  George C.  Marshall Space F l i g h t  Center,  Hun t sv i l l e ,  Alabama 
Prepared by: D r .  J. D. Parker,  P r o j e c t  Go-Director 
C .  R. Gerlach, P ro jec t  Leader 
Approved : 
/ E.  C.  F i t c h  
CONTENTS 
Section Page 
I. Authorization and Program Schedule. . , . , . . . 1 
11. Information Report. . . . . . . . . . . . . . . . 3 
111. Monthly Progress. . . . . . . . . . . . . . . . . 9 
I V .  Anticipated Progress ... . . . . . . . . . . . . . 19 
PROGRESS REPORT ON 
FLUID TRANSIENTS STUDY 
I. Author iza t ion  
Th i s  r e p o r t  covers  t h e  work performed dur ing  t h e  pe r iod  1 
November 1964 through 30 November 1964 and t h e  work t h a t  w i l l  
be pursued during t h e  per iod  1 December 1964 through 31 December 
1964, a s  au tho r i zed  by: 
Contract:  NAS 8 11302 
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11. Information Report 
A .  Dis t r ibu ted -  and Lumped-Parameter Conduit Systems 
I n  a condui t  system t h e  e f f e c t  of t h e  mass, r e s i s t a n c e ,  e t c .  
are d i s t r i b u t e d  wi th  r e spec t  t o  space and t i m e .  I n  o the r  words 
a condui t  i s  a d i s t r ibu ted-parameter  system and hence must be 
descr ibed  by p a r t i a l  d i f f e r e n t i a l  equat ions.  There are some 
circumstances under which we  may reduce t h e  desc r ib ing  equat ions  
t o  o rd ina ry  d i f f e r e n t i a l  equat ions  by cons ider ing  t h e  e f f e c t s  of 
t h e s e  d i s t r i b u t e d  parameters t o  be concent ra ted  a t  one loca t ion .  
Th i s  i s  c a l l e d  lumping and t h e  r e s u l t i n g  equat ions then  desc r ibe  
a lumped-parameter system. This  method has been found t o  be 
v a l i d  i f  t h e  f requencies  involved are not  g r e a t e r  than  about one- 
e i g h t h  of t h e  f i rs t  c r i t i ca l  frequency of t h e  system. The lumped- 
parameter approach g ives  a f i r s t  approximation i n  t h e  s o l u t i o n  of 
a problem. To compare both types  of systems le t  us  consider  t h e  
s i m p l e  case  of a one-dimensional condui t  wi th  no f r i c t i o n  lo s ses .  
The d is t r ibu ted-parameter  model f o r  t h i s  ca se  i s  represented  by 
t h e  equat ion  of motion 
and by t h e  c o n t i n u i t y  r e l a t i o n  f o r  a l i q u i d  
3 
Solu t ion  of t h e s e  equat ions l eads  t o  t h e  t r a n s f e r  equat ions  
between s e c t i o n  0 and L of a condui t  
and 
Going back now and assuming t h e  compress ib i l i t y  t o  be lumped a t  
z=O and a l s o  lumping the  mass w e  may rewrite Equat ions 1) and 
2)  i n  transformed form as  (See F igure  1) 
'0  
and 
Figure  1 Lumped F r i c t i o n l e s s  Conduit 
4 
. 
From 5) and 
7) 
and 
8) 
Notice t h a t  
by assuming 
6 )  t h e  t r a n s f e r  equat ions  a r e  obta ined  as 
Equations 7) and 8) may be obta ined  from 3) and 4 )  
(%) t o  b e  small  and then  w r i t i n g  t h e  appropr i a t e  
forms f o r  c o s h L  and sinh& f o r  small  agrument. 
c* c, 
There are many ways of lumping condui t  systems, depending on 
t h e  s p a t i a l  placement of t h e  lumped parameters.  For example, we 
might have put one-half  of t h e  compress ib i l i t y  e f f e c t  a t  each end 
in s t ead  of a t  only one end. There are  many p o s s i b i l i t i e s .  
One should not  be misled i n  t h e  use  of a lumped-parameter model, 
It i s  important t o  remember t h a t  i t  i s  good only  f o r  r e l a t i v e l y  l o w  
frequency (less than  one-eighth t h e  f i r s t  c r i t i c a l  frequency).  
Because of t h i s  r e s t r i c t i o n ,  t h e  lumped model i s  only adequate f o r  
a frequency a n a l y s i s  o r  for  a time domain a n a l y s i s  i n  which t h e r e  a r e  
no sharp inpu t s  which e x c i t e  high f requencies .  There seem t o  be 
t h r e e  main reasons  f o r  using a lumped-parameter model: 
a )  Allows use of e l e c t r i c a l  ana log ie s  
b)  To g e t  f i r s t  approximation answers 
c )  Easy t o  solve a complex system f o r  time 
domain response on analog computer. 
5 
d) Permits  the use  of d i f f e r e n t  proper ty  va lues ,  such a s  
and G, , f o r  each lump, approximating t h e  v a r i a b l e  F r o p e r t y  case.  This  would be p a r t i c u l a r l y  important 
where the re  a r e  l o c a l i z e d  changes of phase s i n c e  
p rope r ty  v a r i a t i o n s  are o f t e n  q u i t e  la rge .  
B. Analog Methods i n  Conduit: Analysis  
Most of our e f f o r t s  t hus  f a r  have been d i r e c t e d  toward t h e  
development of t r a n s f e r  equat ions f o r  condui t s  which y i e l d  good 
information wi th  r e spec t  t o  frequency a n a l y s i s  bu t  a r e  r a t h e r  
d i f f i c u l t  t o  use wi th  respec t  t o  t i m e  domain ana lys i s .  One method 
of a n a l y s i s  which has been found u s e f u l  by previous i n v e s t i g a t o r s  
wi th  regard  t o  the  t i m e  domain i s  t h e  analog method. So lu t ions  by 
t h e  analog method genera l ly  f a l l  under one of two groups: 
A)  Simulat ion of t h e  Model 
B) So lu t ion  of Model Equations.  
Simulat ion involves  bu i ld ing  a r e p r e s e n t a t i o n  of t h e  model 
us ing  computer elements ( capac i to r s ,  r e s i s t o r s ,  m u l t i p l i e r s ,  d i v i d e r s ,  
func t ion  gene ra to r s ,  e t c . ) .  The s imula t ion  of a condui t  system 
on an analog computer i s  b e s t  s u i t e d  t o  a lumped-parameter model of 
t h e  system. Using a pressure-vol tage  analogy t h e  e l e c t r i c  equiva len t  
of t h e  lumped f r i c t i o n l e s s  model represented  by 5) and 6 )  i s  shown 
i n  F igure  2. 
F igure  2 E l e c t r i c a l  Analogy f o r  Lumped Loss l e s s  Conduit 
6 
I f  t h e  condui t  i s  not  l o s s l e s s  but  has  f r i c t i o n  then  one e l e c t r i c a l  
r e p r e s e n t a t i o n  i s  shown i n  F igure  3 where R = a  f o r  s teady  
flow v e l o c i t y  v. I f  the flow i s  laminar then  R i s  a cons tan t  f o r  
v 
a given condui t  and f l u i d ,  but  i f  t h e  flow i s  tu rbu len t  then  R i s  
some func t ion  of v. 
F igure  3 Electr ical  Analogy f o r  Lumped Conduit wi th  F r i c t i o n  
One method which has  been used t o  improve t h e  accuracy of t h e s e  
lumped models i s  t o  represent  segments of a l i n e  by an. analogy 
such as above ganged toge ther  i n  series. Suppose, f o r  example, 
we r ep resen t  t h e  condui t  of F igure  3 by segments i n s t ead  
of a s i n g l e  segment. The new analogy i s  shown i n  F igure  4 .  
Figure  4 Analogy f o r  n-Segmented Lumped Conduit wi th  F r i c t i o n  
7 
There a r e  a g r e a t  variety of lumped models a v a i l a b l e  f o r  use i n  
analog s imulat ion;  however, i t  must be remembered t h a t  t h e i r  use 
must be l imi t ed  t o  those  a p p l i c a t i o n s  involving r e l a t i v e l y  l o w  
f requencies .  When t h e  f requencies  involved a r e  such t h a t  t h e  
lumped-parameter model i s  not  v a l i d ,  then  t h e  d is t r ibu ted-parameter  
model must be used. 
Severa l  i n v e s t i g a t o r s  have r ep resen ted  a l o s s l e s s  d i s t r i b u t e d  
parameter condui t  on an  analog computer by w r i t i n g  t h e  t r a n s f e r  
equat ions  i n  terms of t i m e  de lays ,  t hus  
and 
where Z i s  the  c h a r a c t e r i s t i c  impedance f o r  t h e  l i n e .  The block 
diagram r e p r e s e n t a t i o n  of such a model i s  shown i n  F igure  5 ( a ) .  
C 
There appears  t o  be much room f o r  improvement i n  t h e  develop- 
ment of analog models fo r  d i s t r i b u t e d  parameter systems. It 
appears  t o  t h e  wr i te r ,  however, t h a t  i t  w i l l  be d i f f i c u l t  t o  
a t t a i n  t h e  degree of exactness  i n  analog condui t  models t h a t  i s  
a t t a i n e d  i n  t h e  t r a n s f e r  equat ion models develaped thus  f a r .  
8 
(a) Loss l e s s  D i s t r ibu ted  Parameter Conduit Block Diagram 
A h  
Summation Block 
&El B = A * C  
M u l t i p l i c a t i o n  Block 
(b) Block Notation 
Figure 5 
111. Monthly P rogres s  
Experimental I n v e s t i g a t i o n  
The major e f f o r t  during t h e  r e p o r t  per iod  on t h e  p a r t  of 
t h e  one-phase group w a s  t h e  completion of t h e  f i r s t  t e s t  se tup .  
As i n d i c a t e d  i n  previous p rogres s  r e p o r t s ,  t h i s  f i r s t  experimental  
appara tus  c o n s i s t s  of a conduit  wi th  a cons tan t  p r e s s u r e  r e s e r v o i r  
a t  one end and an o s c i l l a t i n g  p i s t o n  a t  t h e  o the r  end as shown 
schemat ica l ly  i n  F igure  6 .  See a l s o  F igures  7-9. 
The o s c i l l a t o r  d r ive r  i s  powered by a hydrau l i c  motor. By 
c o n t r o l l i n g  t h e  flow r a t e  of t h e  d r iv ing  f l u i d  t o  t h e  hydrau l i c  
motor, i t s  speeds may be v a r i e d  from near ze ro  t o  4000 rpm. The 
p ipe  l eng th  and diameter may be v a r i e d  up t o  80 f e e t  and 3 inches  
9 
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r e s p e c t i v e l y .  
t h e  second c r i t i c a l  frequency wi th  some f l u i d s .  For water t h e  second 
c r i t i c a l  frequency w i l l  be less than  t h e  peak rpm of t h e  hydrau l i c  
motor and most of t h e  f l u i d s  which w e  w i l l  be concerned wi th  have 
sound v e l o c i t i e s  which a r e  smaller than  t h a t  of water.  
With an  80 f o o t  p i p e  it should be  p o s s i b l e  t o  r each  
The reason  t h i s  p a r t i c u l a r  experimental  model was chosen i s  
because it  al lows good c o n t r o l  of t h e  f l u i d  v e l o c i t y .  Unsteady 
f l u i d  v e l o c i t i e s  a r e  very d i f f i c u l t  t o  measure a c c u r a t e l y  bu t  by 
knowing t h e  o s c i l l a t o r  d r ive r  amplitude and speed w e  have good c o n t r o l  
of t he  f l u i d  v e l o c i t y .  P re s su res ,  on t h e  o the r  hand, are  not  s o  
d i f f i c u l t  t o  measure with a good p res su re  t ransducer ,  so w e  w i l l  
l e t  p re s su re  be our va r i ab le .  By p lac ing  p res su re  t r ansduce r s  
a long t h e  p ipe ,  w e  can experimental ly  determine t h e  amplitude 
r a t i o  and phase ang le  between p res su re  and p i s t o n  d r i v i n g  v e l o c i t y .  
Th i s  da t a  can then  by used t o  p a r t i a l l y  check t h e  a n a l y t i c a l  mQdel. 
By varying t h e  p ipe  length,  diameter ,  th ickness ,  and material and 
a l s o  t h e  f l u i d  w e  should be a b l e  t o  g e t  a good i n d i c a t i o n  of t h e  
accuracy of t h e  a n a l y t i c a l  model. It should be mentioned t h a t  t h i s  
b a s i c  u n i t  might be va r i ed  i n  a mul t i tude  of ways t o  r ep resen t  
o the r  models, such a s  p u t t i n g  r e s t r i c t i o n s  i n  t h e  l i n e ,  having 
abrupt  changes i n  c ros s - sec t ion ,  e t c .  
There a r e  o ther  important concepts  which might be eva lua ted  
wi th  t h e  present  experimental  setup.  For example, t h e  gene ra t ion  
and a t t e n u a t i o n  of t h e  var ious  v e l o c i t y  modes could be checked t o  
see i f  i t  complies wi th  t h e  t h e o r e t i c a l  p red ic t ions .  The o s c i l l a t i n g  
p i s t o n  should genera te  many of t h e  higher  modes and a t  t h e  
14 
r e l a t i v e l y  low frequencies  t h e  experiments w i l l  be performed, t h e s e  
higher  modes w i l l  a l l  a t t e n u a t e  wi th in  a s h o r t  d i s t ance  from t h e  
p i s ton .  
near t h e  p i s ton .  
Th i s  means the re  should be a cons iderably  hea t ing  e f f e c t  
B. Approximate E f f e c t  of Tube Longi tudina l  Def lec t ions  
Some condui t  systems of p r a c t i c a l  i n t e r e s t  may involve t i m e  
varying long i tud ina l  d e f l e c t i o n s  which may not  be neglected.  An 
e x a c t  a n a l y s i s  f o r  such a case  would involve  solving t h e  f l u i d  
equat ions  and tube  equat ions and s e t t i n g  t h e i r  v e l o c i t i e s  equal  a t  
t h e  tube  w a l l .  An approximate approach i s  presented  he re  which i s  
much less involved and should g ive  good f i r s t  approximation answers. 
L e t  u s  consider  now t h e  c a s e  of a condui t  as shown bel.ow 
The length  of t h e  conduit  i s  assumed t o  be changing wi th  t i m e  
(the condui t  diameter remains cons t an t ) .  Assuming end (1) t o  
remain s t a t i o n a r y ,  w e  may w r i t e  t h e  t r a n s f e r  equat ions a s  
and 
15 
C .  
I f  L(s )  does not  vary  g r e a t l y  from L 
by s e t t i n g  L(s)  = L i n  the hyperbol ic  func t ions .  
then  w e  may f u r t h e r  s impl i fy  
0’ 
0 
A Homogeneous Model f o r  Two-Phase Flow 
Work has begun on the development of a t r a n s i e n t ,  two-phase 
f l u i d  condui t  model w i th  e f f o r t  a t  f i r s t  being placed on a simple 
homogeneous model. By homogeneous is  meant t h a t  t h e  concept of a 
f l u i d  continuum holds  and thus  t h e  governing equat ions  of motion 
a r e  v a l i d  f o r  t h e  homogeneous f l u i d  mixture.  I n  t h i s  a n a l y s i s  w e  
assume t h e  l i q u i d  flow t o  con ta in  a uniform d i s t r i b u t i o n  of vapor 
o r  gaseous p a r t i c l e s  which a r e  moving along wi th  t h e  l i q u i d ,  w i th  
no s l i p  between t h e  phases. 
The one-dimensional equat ion of motion i s  solved f o r  each 
phase alone, t hus  obta in ing  a s o l u t i o n  f o r  t h e  r e l a t i o n s h i p  between 
t h e  v e l o c i t y  and p res su re  f o r  each case .  It i s  then  assumed t h a t  
t h e  v e l o c i t y  f o r  t h e  mixture may be r ep resen ted  i n  t h e  Laplace 
domain by 
where F i s  some func t ion  of t h e  mass r a t i o  of gas  t o  l i q u i d  and 
a l s o  of S, t h e  Laplace va r i ab le .  In t roduc t ion  of 13) i n t o  t h e  
equat ion of motion of t h e  mixture g ives  an equat ion  dependent on 
F,  So lu t ion  of t h i s  equation y i e l d s  t h e  va lue  f o r  F. 
This  two-phase model r ep resen t s  t h e  f i r s t  s t e p  i n  t h e  develop- 
ment of two-phase condui t  t r a n s f e r  equat ions .  
16 
D. E f fec t  of Phase Chanqe on t h e  Ve loc i ty  of Sound 
The e f f e c t  of phase change, such as  t h e  formation of vapor 
bubbles,  on t h e  dens i ty  and bulk modulus of t h e  f l u i d  has  been 
discussed previous ly  i n  Monthly Progress  Report 6 4 - 3 ,  Sect ion  111. 
As would be expected, the  e f f e c t  of t h e  vapor formation i s  t o  make 
t h e  f l u i d  more compressible.  Since t h e  v e l o c i t y  of sound i s  
dependent upon t h e  f l u i d  compress ib i l i t y ,  t h e r e  i s  a s i g n i f i c a n t  
e f f e c t  upon t h e  sound v e l o c i t y  when a phase change occurs.  
A r ecen t  survey of t he  sub jec t  has  been g iven  i n  r e fe rence  (1). 
The material p e r t i n e n t  t o  t h e  present  s tudy i s  summarized below. 
The v e l o c i t y  of sound i n  a l i q u i d  conta in ing  small  
amounts of vapor uniformly d ispersed  w i l l  be  q u i t e  low, 
much lower than  t h e  v e l o c i t y  of sound i n  t h e  l i q u i d  phase, 
and lower than  t h e  v e l o c i t y  of sound i n  t h e  vapor phase. 
This i s  based upon the assumption of no s l i p ,  and thermo- 
dynamic equi l ibr ium between t h e  phases. The p res su re  
i s  assumed t o  be  low re la t ive  t o  t h e  c r i t i c a l  p ressure .  
-
The v e l o c i t y  of sound i n  a two-phase mixture  can be c a l c u l a t e d  
us ing  t h e  assumptions j u s t  s t a t e d  and s t a r t i n g  wi th  t h e  equat ion  f o r  
a s i n g l e  phase. 
14) C Z  ’ (ap), =-v ‘(@f)J where v = s p e c i f i c  volume 
s = s p e c i f i c  entropy 
or  (f2 =y2 (si (@$l (51 See r e fe rence  ( 2 )  pp. 342 & 343 
I n  t h e  two-phase region 
thus  
@JT and (@= 
@a I 
17 
Using t h e  d e f i n i t i o n  of q u a l i t y ,  X ,  and assuming t h a t  t h e  s p e c i f i c  
volume of t h e  l i q u i d  i s  much less than  t h a t  of t h e  vapor 
Thus, i f  temperature-entropy-volume d a t a  i s  a v a i l a b l e ,  t h e  
v e l o c i t y  of sound can be  c a l c u l a t e d  f o r  s p e c i f i e d  q u a l i t i e s .  The 
v e l o c i t y  of sound i n  t h e  two r eg ions  was c a l c u l a t e d  by Karplus ( 3 )  
and i s  shown i n  F igure  10 below 
0.1 J 
.00001 Dool xK)I .01 .I I 
OUALITY OF MIXTURE,X, Lay VAPOWLBy MIXTURE 
Figure  10 
Ve loc i ty  of Sound i n  Two-Phase Region f o r  Water 
It i s  p a r t i c u l a r l y  important t o  n o t i c e  t h e  low v e l o c i t y  of 
sound t h a t  occurs  a t  low q u a l i t i e s  such as i n  bubble flow. 
I f  t h e  vapor phase i s  p re sen t  but not d i spe r sed  t h e  v e l o c i t y  
of sound w i l l  be lower than f o r  t h e  l i q u i d  phase a lone ,  but not 
n e a r l y  so low as  ind ica t ed  by equat ion  15) .  
18 
I V .  An t i c ipa t ed  Progress  
A. Experimental I n v e s t i g a t i o n  
The f i r s t  s e t  of tests w i l l  being on t h e  newly completed 
test se tup  shown i n  F igure  1. Other experimental  equipment 
is  being designed t o  f u r n i s h  information on t h e  two-phase 
a s p e c t s  of t h e  problem. Two types  of equipment a r e  being 
considered and w i l l  be proposed i n  d e t a i l  i n  t h e  next  monthly 
r e p o r t .  One proposed p i e c e  of equipment i s  a hydrodynamic 
tunne l  s u i t a b l e  f o r  s tudying c a v i t a t i n g  flows of cryogenic  
and o ther  f l u i d s  through components such a s  nozz les ,  o r i f i c e s ,  
bends and valves .  The o ther  proposed equipment i s  an  observat ion 
chamber f o r  t h e  study of bubble growth and co l l apse .  
B. Analog Models 
Work w i l l  cont inue on t h e  development of analog condui t  
models. 
C. Mathematical Models 
Work w i l l  cont inue on t h e  development of t r a n s i e n t  one- 
phase and homogeneous two-phase condui t  models. 
19 
8 I 
REFERENCES 
1. S .  W .  Gouse, J r . ,  and George A.  Brown, “A Survey of the 
Veloc i ty  of Sound i n  Two-Phase Mixtures,” A.S .M.E.  Paper 
64-WA/FE-35. 
2 .  J. H. Keenan, ‘Thermodynamics,” John Wiley, 1941. 
3 .  H. B .  Karplus, “Propagation of Pressure Waves i n  a Mixture 
of Water and Steam,” ARF 4132-12, I l l i n o i s  I n s t .  of  Tech,,  
January, 1961. 
20 


Study of fluid transients in closed conduits progress report no. 64-5, 1-30 nov. 1964

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