1. E.

FUNK The Transient Response of Orifices and

Professor o f Mechanical Engineering.
Mem. ASME
Very Short Lines1
D. J. WOOD
Professor o f Civil Engineering. It is generally assumed that orifices and valves follow closely their steady-state character-
Mem. ASME istics during transient operation. However, this assumption of quasisteady behavior
•may lead to errors in predicting transient flow conditions under certain circumstances.
S. P. CHAO In order to evaluate the transient behavior of an orifice, a differential equation relating
the flow through and the pressure drop across an orifice was derived. An extension was
G r a d u a t e Research Assistant.
made to include an axial dimension for the orifice. The solution of this equation for
University of Kentucky, Lexington, Ky. transient flow through an orifice subjected to a step change in pressure drop across the
orifice is significantly different than that obtained using the steady state relationship.
An experiment was designed to evaluate the theoretical results in which an orifice on
the end of a line was subjected to a sudden pressure change and the resulting transient
pressures were observed. It was found that a significant short term transient occurs
before the orifice flow reaches the new steady state condition. The observed short term
transient agrees well with that predicted by the theory. It is concluded that the be-
havior of an orifice can deviate considerably from that predicted by steady-state equa-
tions during periods of rapid pressure or flow changes. The dynamic description of
orifice flow may be combined with a larger system analysis (e.g., using the method of
characteristics) to more accurately predict the overall transient performance of flow
systems.

Introduction duce significant errors. There is a need for the development of

relationships describing the transient behavior of various flow
L
I ECHNIQTJES for carrying out transient flow analyses system components which can be used in situations where the
in piping systems have been well developed. With the use of use of quasisteady approximations is unacceptable.
digital computers it is possible to analyze flow systems with rela- This investigation deals with the transient behavior of orifices
tively complex geometry, containing components such as pumps, which may have a significant axial dimension or very short lines.
valves, and orifices, and including viscous effects. However, The orifice is a common component utilized in the analysis of
these analyses are sometimes limited by a lack of knowledge of fluid flow system. Certain valves can be adequately repre-
the transient behavior of various components in the piping sys- sented by an orifice. In transient flow analyses viscous effects
tem. It is generally assumed that the pressure-flow character- are often simulated by a series of orifices distributed throughout
istics of a component at any instant are identical to the steady- the flow system [1, 2]. 2 It is commonly assumed that steady-
state characteristics which correspond to the flow conditions state characteristics properly describe an orifice during both
at that instant. This quasisteady approximation has been used transient and steady flow [1, 2, 3, 4, 5]. The validity of this
extensively to describe the behavior of various flow system com- assumption has not be demonstrated. It is likely that situa-
ponents and to describe viscous effects when making transient tions exist where short term transient effects are quite important.
flow calculations. It is apparent that some errors will be intro- For example, the response of an engine to the injection of fuel
duced by assuming quasisteady behavior and it is important to would be very much dependent on the transient behavior of the
be able to evaluate when this approximation is likely to intro- injection line and orifice.
Orifices are widely used as meters for the measurement of
fluid flowrates. These devices are sometimes used during
1
Symposium Paper: State-of-the-Art: Fluid Transients. transient as well as steady flow. The operation of the orifice
Contributed by the Fluids Engineering Division and presented meter, however, is normally described by carefully measured
at the Winter Annual Meeting, Washington, D. C, November 28- steady-state characteristics. The steady-state behavior of
December 2, 1971, of THE AMERICAN SOCIETY OF MECHANICAL
ENGINEERS. Manuscript received at ASME Headquarters, July
23, 1971. Paper No. 71-WA/FE-14. Numbers in brackets designate References at end of paper.

Journal of Basic Engineering Copyright © 1972 by ASME JUNE 1 9 7 2 / 483

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 orifice meters has been thoroughly studied and abundant informa-
tion pertaining to design and operation is available [6, 7]. I t
has been pointed out t h a t errors can occur when using the orifice
meter for pulsating flow measurements. Stearns, et al., [7],
indicated t h a t while it was not possible to predict the error
incurred during pulsating flow, observed errors of 20 to 30 per-
cent were common while errors of over 100 percent could occur.
Mosely [8] carried out a study of errors which occur in orifice + rco -roo
meter measurement of pulsating water flow. He concluded
Po
that significant errors can occur at high frequencies and t h a t
these errors always indicate a higher flowrate than actually
exists. Sparks [9] studied the effects of pulsating compressible
flow and concluded t h a t significant measurement errors can
occur, b u t to precisely define these errors would require a large
amount of additional research.
The purpose of the study summarized in this paper was to
Fig. 1 Flow conditions in vicinity of orifice
develop an analytical description of the dynamic behavior of
orifices and very short lines (or orifices with extended axial di-
mensions) and to experimentally evaluate this description.
i)(uA)
Consideration of an orifice with a significant axial dimension is 0 (2)
a?-
included in the study so the results can be applied to orifice type
hydraulic components with additional inertia and resistance in where A is the hemispherical flow area and is given by
the vicinity of the component (such as most valves), in addition
to the direct application to very short lines. The analytical A = 2irr2 (3)
development applies to any transient flow condition. The
The velocity in terms of the volumetric flowrate, q, and the
analytical results are experimentally evaluated by subjecting
radial position, r, is given by
the flow in a finite line terminated by an orifice to a step change
and observing the pressure response. The observed pressure
g
transient is compared to t h a t predicted using the dynamic orifice (4)
"2-n-r2
relationship developed herein.
The momentum and continuity equations are combined with
(4) to give
Theoretical Development
(A) Basic Orifice Equation. Fig. 1 shows the assumed flow field dp dq PT
(S)
in and around an orifice. Potential flow is assumed with flow &7 2-wrt At 27r2r6
from the left into a sink and out of a source to the right. I t is
Equation (5) can be integrated with respect to r and evaluated
expected t h a t this flow field will reasonably describe conditions
from + r r a to ra to give the total pressure change for the acceler-
near the orifice provided the orifice is sufficiently small relative
ating fluid between those points. The result is:
to the line diameter. The momentum equation for an inviscid,
incompressible fluid expressed in angular independent spherical PT
coordinates is Pi ~ Pi (6)
27rr, 2(2Trra*y
du 1 dp du I t has been assumed t h a t r„ ^> ra. A similar analysis for the
(1)
p dr br decelerating fluid between ra and — r„ gives the pressure change

where p is the fluid density and u and p denote the fluid velocity _P_ dq PT
and static pressure, respectively which are functions of the radial Pe - P2 (7)
2irra _dt J 2(2xr„ 2 ) 2
position, r, and time, t.
The continuity equation is given by The irreversible pressure change across the orifice itself at

^Nomenclature-

A = surface area of hemisphere, Pi, Pi Pi pressure upstream and down- Si, St, S3 coefficients of dimensionless
V- stream from orifice, FL~* differential equation, n.d.
AL = line area, L 2 Pi Pe pressure at entrance and exit t time, T
Ao = orifice area, L2 of orifice, FZ,~ 2 V dimensionless time, n.d.
Ar = ratio of orifice to line area, 1 volumetric flowrate, LST~X Ti, T2 coefficients of differential
n.d. a' dimensionless flowrate, n.d. equation
<?••> Uivy initial flow in line, valve and velocity as function of radial
a = orifice radius, L <lio

C = wave velocity, LT~X orifice, VT~l position, LT~l

u aj flowrate in orifice before and Wo = initial line velocity, LT~l
CD = discharge coefficient, n.d.
after wave, L3T_1 p = mass density, ML~
3
, CB = coefficients for closed form
1v, 5/ flowrate in valve before and Ap< = change in pressure in line,
solution, n.d.
after wave, L3T_1 FL->
/ = friction factor, n.d. flowrate in line before and
iu ?l' Ap = change in pressure across
K = constant for closed form solu- after wave, LZT~X short tube, FL'*
tion, n.d. radial dimension, L Aq change in flowrate, L'T'1
L = axial length for orifice, L radius at minimum flow a, fi, y dimensionless coefficients,
p = pressure, FL~2 area for orifice, L n.d.

484 / JUNE 1972 Transactions of the ASME

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 \
tzztzzzzzz
<zzzzzzzA
i t/;;;/////;/;//////;/;;;;/;//S77

po
//////,

ZEZ
Fig. 3 Experiment for measuring orifice dynamics
y77777777

The pressure drop across the tube can be added to the pressure
Fig. 2 Orifice with a x i a l dimension change across the orifice given by equation (10) to yield the
following differential equation for flow through an orifice with
a significant axial dimension.
any time is assumed to be described by steady-state energy pL dq P pfL
considerations Pi - p2 =
2CVAo! + 4o4 2

Pi - P.
PT
2(2irrJY
(8) 4 CDA0ir A0 dt 0

(12)
This is the same relationship which is normally used to describe (C) Application to Transient Flow. Fig. 3 shows the apparatus
steady-state orifice flow. Again it is assumed t h a t r„ is large used to test the validity of equation (12). Flow in the line
compared to ra and the kinetic energy of the fluid approaching exits through two orifices, one at A and one a t B. With the
the orifice is neglected when obtaining equation (8). Even system flowing steadily the valve at A is closed very rapidly.
when the ratio r„/ra is relatively small the effect of neglecting This causes a rapid change in line flow and an associated rapid
the upstream kinetic energy is not serious. For example, when change in line pressure. Under these conditions the flow through
}•„/?•„ = 2.0 the coefficient relating the orifice discharge and the orifice B then adjusts to a new steady-state value. Changes in
net pressure drop across the orifice would only change by about line pressure and flowrate can be related through the basic fluid
3.2 percent if the upstream kinetic energy were considered. line impedance relationship
I t is noted t h a t the flow area will undergo some sort of con-
traction a t the orifice such t h a t the orifice area is not the same (13)
Ap = — Aq
as the minimum flow area. A discharge coefficient, CD, is de-
fined such t h a t
where AL is the cross-sectional area of the line.
CDA0 = 27IT„ (9)
1 Instantaneous Valve Closure. A closed form solution of equa-
where Aa is the cross-sectional area of the orifice. Combining tion (12) can be obtained for the case where the valve at A is
equations (6), (7), (8), and (9) gives the following relationship closed completely at time t = 0. At any instant the line pressure
for the dynamic characteristic of the orifice. is related to the initial pressure and discharge, po and qit and the

Pi - Pi

V CDAair m + 2(C Ao7

PT
D
(10)
discharge through the orifice, q, by

Pi
pC pC
(14)

For a steady-state flow this reduces to the usual steady-state T h e exit pressure, p%, is taken as zero in this case. If equation
characteristic equation for an orifice. The time dependent (14) is substituted into equation (12.) and then rearranged by
term represents the effect of decelerating and accelerating the using the following dimensionless parameters
fluid into and out of the orifice. In equation (10) the discharge
coefficient CD is known to vary with flow conditions. However, t' =
tc (15)
in the subsequent analysis this term will be treated as constant a Ah
for two reasons. First of all, the change in CD in steady flow
is slight over small changes in Reynolds number. Secondly, the the basic nondimensional differential equation becomes
variation in Co for transient flow is unknown. Therefore in the
1
analyses the value of CD was taken as constant and equal to its —+ 1 T
' <lALCAr' |_CV «J
initial steady state value.
(B) Extension to Include Axial Dimension. If the orifice has a
significant axial dimension as shown in Fig. 2 both the inertia
a Ar
L# a

(16)
and the frictional resistance in the short section will affect the where
total pressure change across the orifice. Since the axial dimen-
sion is short, the fluid in t h a t region can be assumed to act as pCqi
a = —-—
transient plug flow in a circular tube which implies t h a t fully PoAL
developed flow conditions exist in this region. The pressure
drop in the tube is given by The following additional dimensionless parameters are defined

pL fdql L
pfW Si =
Ap, (11)
A0 \_dtj + 4aA 0
2
CD a

. #
where a is the radius of the orifice and L is the axial length. The
friction factor, / , is also known to depend on the flow conditions.
In the following analyses it is assumed to be constant and the
(*- + £) (17)
4:ALCAr*
reasons for this assumption are the same as previously stated re-
garding the discharge coefficient.

Journal of Basic Engineering JUNE 1 9 72 / 485

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 dq go - go
+1 dt At
(22)

Combining (16) and (17) yields The pressure change is related to the flowrate change by the
impedance equation
(18)
pC
Pt' = pi + Ap = pi + —-(qx - g / ) (23)
subjected to the initial condition
Continuity relationships give
g'(0) = ^ (19)
gi gi = go + g„ (24)
where g« is the initial discharge through the orifice. and
T h e solution of equations (18) and (19) is
gi go' + g„ (25)
a, ztanh +<7 1
"i[ [(s *)f3- . (20) Equations (21)-(25) are solved simultaneously to yield the fol-
lowing quadratic in qi'
where

K = V l + 4S2S3
C A = 2&g'(0) + 1.0 pC
+ T*qv'* - Pl (26)
and

This relationship can be solved for the change in flowrate in the

C B = - log
l_i - cyxj line knowing the flowrate out the closing valve (g„') and the
conditions in the line at the beginning of the time interval.
2 Quick Valve Closure. Significant dynamic effects occur over From this the flowrate out the orifice (go') can be computed
a very short period of time and it is not physically possible to using equation (25).
instantaneously close the valve which generates the transient In this manner a stepwise plane wave solution is generated
condition. I t is therefore necessary to modify the analysis to describing flow conditions for the orifice. Since the valve
consider the finite time taken to generate the disturbance. I t is closure occurs over a period of time which is much less than the
not possible to obtain a closed form solution for this situation so wave travel and return time (2L/C) for the system, no reflected
a plane wave analysis, based on the generation of pressure waves waves from the reservoir need be considered in this analysis.
due to the disturbance occurring over very short time intervals,
is carried out. Fig. 4 shows conditions which exist in the line
a short time interval, At, apart. T h e flow variation out of the Analytical Results
valve, g„, is taken to be a known time dependent function. An inspection of the basic differential equation, (16), shows
The characteristic equation relating the pressure inside the that the response of the orifice is a function of geometry, hy-
orifice and the flowrate after the step change is given by equation draulic characteristics, and initial conditions.
(12) as Geometric parameters include the ratio of the orifice area
to line area, A„ and the ratio of the axial dimension to the
dq
_
~
T Tl
dt TV2 (21) radius, — i for the orifice. The discharge coefficient, CD, and
a
where the friction factor, /, are important hydraulic characteristics.
The parameters y = pCV0/po and 13 = V0/C depend on the initial
line velocity, Vo, and pressure p 0 , as does g'(0) which is the
CDA0TT
+^ A0 fraction of the initial line flow which passes through the fixed
orifice. Table 1 summarizes these parameters

Table 1 List of important parameters

D 24o
!
+' 4aA 0
2
J symbol type
The derivative is given in finite difference form as L/a geometric
AR geometric
CD hydraulic
f hydraulic
I .1 . initial condition
initial condition
|q 0 i g'(0) initial condition
P,,q,
•qyi 1 Parameters representing geometry and initial conditions can
vary over a wide range. However, the value of the hydraulic
parameters (discharge coefficient and friction factor) are limited
to rather small variations. For example, the discharge coeffi-
t q02 I cient for the orifice would be expected to vary between 0.6 and
p ,q 0.8. Therefore it is neither necessary nor practical to carry out
P,.q, i' 'i _ P 1t+At parametric studies of these variables. I t is expected t h a t the
%z I geometric parameters will have the greatest influence on the
response of the orifice and t h a t the initial flow condition will
affect the response to a lesser degree. Parameter studies on
Fig. 4 Conditions before and after small step change for plane wave
analysis these variables were carried out to determine these effects.

486 / JUNE 1972 Transactions of the AS ME

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 Response Using Equations (14) and (20)
• Response Using Steady State
Characteristics

O.OOI 0.002 0.003

Time Time (sec)
Fig. 5 Typical pressure response Fig. 6 Analytical results w i t h L/a variable and Ar = 1 / 1 6 (equations
(14) and (20))

(A) Typical Response. The pressure and flow variations at the

orifice were determined for the condition under which the initial
flow out the valve is extinguished instantaneously. The sche-
matic of the flow system is shown in Fig. 3. At the instant the

valve is closed a pressure increase equal to Ap = — qiv is gener-

•A-L

ated which subsequently decays to a new steady-state value.

A typical pressure response is shown in Fig. 5 along with the
response which would be predicted using the steady-state orifice
equation. This response is a step change from the initial line
pressure, p<>, to a new value, pf and is considerably different
than the predicted transient response.
O.OOI 0.002 0.003
For a particular situation the magnitude of the initial pressure
increase, Ap, is determined solely by the initial flowrate through Time (sec)
the valve as just given. The final pressure, pf, is a function of Fig. 7 Analytical results with Ar variable and L/a = 52 (equations (14)
all the parameters. The decay time is primarily a function and (20))
of the geometric parameters, Ar and L/a and is of particular im-
portance in determining whether it is necessary to consider
transient orifice dynamics in a larger system analysis. stant. For these calculations Ar = Vi6 and L/a = 52. The
(B) Numerical Results. In order to further illustrate the tran- initial orifice flow and line pressure were varied over a wide range.
sient response, calculations were carried out for a system similar The initial flow out the valve was kept constant for each calcula-
to the one which was used to obtain the experimental data. tion. The results are shown in Fig. 8, and indicate t h a t the
The parameters used in the calculations are shown in Table 2. initial flow conditions do not have a large effect on the response.

Table 2 Values for system parameters

Experimental Results
fluid density, p = 1.94 slugs/ft 3
wave velocity, C = 4256 fps An experiment corresponding to the arrangement shown in
Fig. 3 was constructed to evaluate the theoretical results. The
initial line velocity, -Jp- = 10.27 fps system was constructed of a twenty ft section of xl% in. copper
initial line pressure, p0 = 6120 psf tube with a y 8 in. I.D. short exit tube located near the quick
discharge coefficient, Cn = 0.625 closing valve. This valve was rigidly attached to the floor to
friction factor, / , = 0.05 prevent motion. Three tests were run. For the first, the length
orifice diameter, a, = 0.0625 in.
initial orifice discharge, q'(Q), = 0.1275 (dimensionless) of the exit tube was 3 y 4 in. and, for the second, 3 /s of an in.
For the third test the Vs in. tube was removed leaving a 1/i in.
opening with a thickness equal to the wall thickness of the copper
With these values the L/a ratio was varied while the area
pipe. The valve, which generates the disturbance, could be
ratio, AR, was held constant at Vi6- The transient static pres-
closed in about 1.5 ms. Pressure was directly measured using
sures computed using equation (20) are shown in Fig. 6. The
a piezoelectric pressure transducer, a charge amplifier, and an
transient pressure is normalized by dividing by the initial surge
oscilloscope, and the results photographed. The pressure re-
magnitude (pCqiv/AL). For this example the initial pressure
sponse for three cases is shown in Fig. 9. The pertinent data,
surge magnitude is about 530 psi. As shown in Fig. 6, this
which were measured prior to each test, are as follows:
parameter has some effect on the final value of the pressure.
However, the primary effect is on the decay time. For the
Case I
limiting case of L/a = 0 the pressure decays to the new steady-
state value in a fraction of a millisecond.
po = 1310 psf
Calculations were also made for the case where L/a was 16
held constant and Ar was varied. For this example L = 3.25 in.
L = 3.25 in. CD = 0.704
and L/a = 52. The results are shown in Fig. 7. The area
ratio has a large effect on the final value of the pressure b u t
does not seem to greatly affect the decay time. ^=52 / = 0.050
a
Calculations were made to determine the effect of the initial
flow on the response if the geometric parameters were held con- V0 = 9.6 fps g'(0) = 0.128

Journal of Basic Engineering JUNE 1 97 2 / 487

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 1.0
p-p
__0
q' (0) Po
~P
0.5 0.16 1714
0.08 357
0.008 3

o L..-_---J_ _ ~ _ ___L ___l~

Time
o 0.001 0.002 0.003 a. Case I
Time (sec)
Fig,8 Analytical results with initial canditions variable and A r = 1/16,
L/a = 52 (equations (14) and (20»

Time (ms)

b. Cose D

Fig.9 Experimental results

Fig, 10 Closed form predictians corresponding to test conditions (equa-

tic:ms (14) and ( 2 0 » · .

Prediction Using Eqs. (23-26)

o Experimental Data
300

Case I

Case :II

Case m
0.001 0.002 0.003
Time 1 (sec)
Fig, 11 Comparison of plane wave predictions and experimental results

488 / J UNE 1 972 Transactions of the ASME

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 Case II Conclusions
A time dependent description of the dynamic response of
po = 430.6 psf orifices and short lines is formulated. This expression gives
' 16
results which compare well with experimental data. The good
L = 0.375 in. CD = 0.63 agreement between the predicted and observed results indicates
that the various assumptions made in the analysis are accept-
L able. The study shows t h a t significant transient effects can
/ = 0.050
- = 6 occur due primarily to the inertia of the fluid in the vicinity of,
a
and within the orifice. However, these effects are short term
V0 = 9.0. fps g'(0) = 0.128
and decay rapidly. As the axial dimension of the orifice de-
Case III creases the decay time becomes very small. I t is, therefore,
likely t h a t in many practical engineering situations the use of
p0 = 265.7 psf steady-state characteristics to describe the transient behavior
of the orifice is acceptable. However, in systems with rapidly
Occurring transients or those requiring a rapid response^ it may
L = 0.049 CJD = 0.617
be necessary to consider the dynamic behavior of orifices or other
L hydraulic components in the system.
- = 0.39 / = 0.050
a
V0 = 12.0 g'(0) = 0.140 Acknowledgment
This work was carried out under the sponsorship of the Army
The zero points for pressure and time are noted in Fig. 9. Research Office of Durham, N . C.
I t should be pointed out t h a t the zero point could hot be preset
since the oscilloscope was set to 'trigger the sweep internally oh
a certain vertical deflection of the beam. As less total deflec- References
tion occurred it took a greater time to reach the minimum input 1 Bergeron, L., Water Hammer in Hydraulics and Wave Surges in
to trigger the scope. However, the traces could be extrapolated Electricity, Wiley, New York, 1961.
back until they intersected the zero pressure line and this is how 2 Wood, D. J., Dorsch, R., and Lightner, C , "Wave Plan
the zero point was determined. In each case the zero point was Analysis of Unsteady Flow in Conduits," Journal of the Hydraulics
Division, ASCB, Vol. 92, No. HY5, Mar. 1966, pp. 83-110.
about 1.5 ms from the peak pressure which occurs at the instant 3 Parmakian, J., Water Hammer Analysis, Prentice-Hall, Bngle-
the valve is completely closed. This agrees with the known 1.5 wood Cliffs, N. J., 1955.
millisec valve closing time. 4 Rich, G. R., Hydraulic Transients, Engineering Society Mono-
graphs, McGraw-Hill, New York, 1961.
5 Streeter, V. L., and Wylie, C. B., Hydraulic Transients, Mc-
Graw-Hill, New York, 1967.
Comparison of Experimental and Analytical Results 6 Flowmeter Computational Handbook, American Society of Me-
chanical Engineers, New York, 1961.
The three cases corresponding to experimental conditions 7 Stearns, R, F., Jackson, R. M., Johnson, R. R., and Larson,
were solved using the closed form solution of equation 16 for C. A., Flow Measurement With Orifice Meters, D. Van Nostrand
instantaneous valve closure. The results for the pressure re- Company, 1951.
sponse are shown in Fig. 10. There is a considerable discrepancy 8 Moseley, D. C , "Measurement Error in the Orifice Meter on
Pulsating Water Flow," Flow Measurement Symposium, ASME,
between these and the experimental data shown in Fig. 9. How- 1966.
ever, since the decay times are on the order of the closing time, 9 Sparks, C. R., "A Study of Pulsation Effects on Orifice Meter-
such a discrepancy is to be expected. To more closely approach ing of Compressible Flow," Flow Measurement Symposium, ASME,
experimental conditions, it was necessary to use the modified 1966.
plane wave analysis which considers the valve closing over a
specified time interval. This analysis assumes t h a t the flow
through the valve was linearly decreased from its initial value DISCUSSION
to zero in 1.5 millisec. The analytical results for the three cases
are shown in Fig. 11. The experimental data presented in C. F. Holt 3 and J. M. Robertson"
Fig. 9 are also shown on these curves. As the authors note, in certain cases the transient response of
I t can be seen t h a t the analytical and the experimental results an orifice can differ greatly from the generally assumed instan-
are in good agreement. The maximum pressures are compared taneous steady state. We have been analyzing the transient
in Table 3. compressible flow occurrences in positive displacement pumps of
The primary difference between the predicted and observed re- fluid power systems and find t h a t the assumptions made for such
sults is t h a t the calculated pressure peak is very sharp while a transient response can markedly affect the results. The
that observed is quite roimded. There are several reasons for authors' paper presents a welcome discussion of this phenomena.
this. One is t h a t the valve closure function is not completely Since our transient analysis differs somewhat from the authors'
known and would actually be somewhat different than the linear approach, these comments are based on our familiarity with the
one assumed. Some of the discrepancy could be partly due to flow problem. Specifically we note some contradiction in the
nonlinear compressibility of the liquid. Also some errors may use of a discharge coefficient for an outflow assumed to be a
be introduced because / and CD were assumed constant for vary- source. A discharge coefficient of the order of the usual value
ing flow conditions. In general, it is felt t h a t the comparison (about 0.6) implies a separated jet flow downstream which is not
is satisfactory and t h a t equation (12) is a reasonably accurate permitted in the model chosen. An alternative approach permits
description of the dynamics of short lines and orifices. one to separate the two concepts.
In considering the transient response of ah orifice one may
Table 3 Maximum predicted and observed pressure surges apply t h e unsteady Bernoulli equation from a point for upstream

Observed Predicted
Case surge (psi) surge (psi) 3
Research Associate in Theoretical and. Applied Mechanics,
I 320 328 University of Illinois at Urbana T Champaign, Urbana, 111.
4
II 140 146 Professor of Theoretical and Applied Mechanics, University of
III 45 47 Illinois at Urbana-Champaign, Urbana, 111. Mem. ASME.

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