ML20091M871

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Rept of Independent Assessment of Calculation of TSP Forces
ML20091M871
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Site: Byron, Braidwood  Constellation icon.png
Issue date: 08/25/1995
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MPR ASSOCIATES, INC.
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NUDOCS 9508300324
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{{#Wiki_filter:i i Report of an Independent Assessment of the Calculation of TSP Forces d 4 9508300324 950825 PDR ADDCK 05000454 P PDR

f_ 1 Ls. Introduction The' investigation reported here was directed at answering the following two questions. bantheforcesonanorificeplatoresulting'fromthe

1.. transient, blo'wdown-type flow of a compressible fluid

.in a pipe be calculated from a form-loss? R 2. Using a method other than finite differences, how can the pressures and velocities needed to determine such forces be calculated? a We have concluded that the answer to the first question is' "yes". In what follows three independent analyses are presented in answer to the second question. However, before getting into these details the following more general comments should be made. At the outset of the investigation a' literature search was made to determine whether or not the characterization of an ori-fice be a form loss was appropriato under transient conditions. The consensus of several papers is that it is. Reference (1) is typical of these and includes experimental verification. A copy of this paper is included in this report. At the same timerthe appropriateness of using the same method of analysis in both com-pressible and incompressible flows was also considered. Referenco (2) is submitted in support of the conclusion that it can. The general characteristics of the three analyses of unste'ady pipe flows should also be described. The first of these is by far. the simplest, most direct and easiest to~ apply. It is, in fact, -identical to the standard analysis of water hammer (see (1)) except that-the sound speed and other physical properties of the fluid arc those of a gas. It's drawback is that it is no' doubt limited to L some extent to ucch waves in which the fluid can be considered quasi-incompressible.

i 2 1 The second analysis is the magt genorcl and alco the most In it the fluid is treated as completely compressible. complex. I h applies the theory of characteristics in the flow .This approac away from the orifice and the open end. These regions are then treated by assuming steady flow and applying the basic conserva-tion laws. Treatment of the orifice in this manner does not re-quire supplemental empirical information such as the orifice coefficient K. In fact I can be deduced from the results. This l analysis is better suited to strong waves and also provides a check on the more direct approach. It is also better suited to a numer-ical'. calculation of the long-time conditions in the pipe, The third analysis is a compromise betweed the other two. i CompressibilityisincludedbuttheorificeKI-factormustbeknown. All three of these analyses show that the pressure drop across l the orifice is always given by an expression of the form hy = K { $ All three analyses agree with regar'd to the calculated values the l fluid velocity and its pressure for the low Mach number case con-sidered. The K -factor determined from the results of the second.> i analysis verifies the value used in the other two. It is recommended that caution be used in applying these findings'to the results of finite difference calculations'. The I i velocity through the orifice is critically dependent on the I I reflection and transmission of the wave at the orifice. In our experience finite difference calculations are not reliable in this respect. Thus we strongly recommend -that you run our two test casos on your program and comparc the results with those A7 presented in Figurc 7 of the text. The " reservoir" caso can be

'3 + I s P: ~ < simulated by using:an upstream pipa diamatar equal to:100Ltimeo 'i theidownstream.value. t 4 e 4 d( t 9 E / e e P 4 e I 1 k '9 I = ? :. i-j '2' 9 f j-tl, 4 k (- G e 5 4 e 4 ..'.t .-,Y,.- w 9 4 h 4 I I g e _,m_m.

4 A. Gunsi-Incomornssible Analycic ) Density changes in forced ' gas. flows in which the velocity does not excced thirty percent of the local sound speed are j not large (3). It is therefore expected that an analysis that. treats the gas as incompressible, but with a finite sound speed will be valid over a significant range.- The followirig analysis assumes these condition {. Two different cases are considered. In the first the orifice is assumed to be located somewhere in the middle of a ng pipe. Thus flow on both sides l J is unsteady.. In the second case the orifice is considered to be located at the entrance to the pipe from a reservoir sufficiently large so that the upstream conditions remain constant. l Pipe Problem Consider a wave travelling to the right as shown below: f g I 4'/ hg,! I I I i i i x N A):. i 1 (Le .f [ g i ' m 4 rot volme. I x E Al l pre. Applying a momentum balance to the control volume indicated: FA C

== ' rate of change of momentum. a

5 l Eut-because the wave is moving to the right with 'wava-veloci a, ) g$t. mass of fluid is accel'erated from' U.g. to ut - in. time ~ d t.. - T h u s, S hk ~ N " d g. Irateofchangeofmomentum = -[% L St Therefore t' k*) G lA i.- lag 4-ug. =. c p - Q - UI i Pu-hg.=3.(a-u,& 3 o c g CL (M - hp. (l - asu 8} (A1) ~ ~ (. >g, = L .q

Now if the wave is moving to the left, a mass of fluid goes from I

I up.to M.. The only -effect is to reverse the sign of the rate t t F of change of, momentum term: i p / at + ku h i (~I b 0- " '- NA j (A2) O ' ;3r *j 1 c Now consider an initially uniform and stationary fluid. 1 5 e

l F

Po 4' i UM W yn. D r e 55tA.YC. e x. 1A. I I A Veloc.k x L. Rg uc. A 2, j l ~ i

+ l 6 ..e At some time the fluid at the right is accelerated to IA and a i ' decompression wave starts'down' the tube: 1 0 brukjn.ssim j e tu m v ' 13 4= lef-i-n. I Fluid buels b Flu:1 dabo.cg. k.w $ht b'" *- m ~ i o nRtc. K F,'y us A~b Using. equation ( A2), 'with ?de ?, ha. = hi, Urr.= Mi and-Lli = R. = o fo - P, = (~ U ' 1 (A3) ~ If Do << E, the /a. term in the brackets is negligible and we drop it in what follows. So c f o t - 6 LA, (A4) g I t Figure A3 applies until the left-travelling wave hits the orifice; we have ~ 3 I l Po e g i4 I t I X R. 1 Fluid movikg b tiry kb 4 I ~ , & c. n n) ave h4 -

7. 'cnd;just after, ws hava i p h l I T % 3* *.J l kl - si m u m 4r m is P' i-es reae. +a.4 -s e g' lg wave. h.av. h % ris pt. l-i W g n n d m ovi g b dyb ~ us. u - l. u

l__,

s s L <-l l on9cc =. X T tg u.v e A s-i Note that the pressure is' discontinuous because of the orifice pressure drop, but the velocity is continuous. We now have 2 waves, one transmitted left wave, one reflected'right wave. 1 Ass'uming as before U <t O., we get: For reflected wave: ^ i l g. (4-u fN G n (AS) ~g - ?, For transmitted wave: L 0~U2-I fa-f.2. (A6) 1 =. b & Lk 2 t 'd The pressure drop across f.he orifice is 'l 9 at 1 L 3 (A7) 3 ~ { ~ g i Adding (A5) and (AG), we got 33 - { h. ).1. ( Ia - %.) ~2 & N.- R Me ,y r

8 Substituting,for' Ma.'.from'(A6) and' tai -from (A4), we obtain

2. (?e pi) A{ P. 3) f(h-M
3 Y./

. AD. 6 or: i 7 h.- ho * (P.;-k)4k(f3-h2.)/ (^S) l '. J which says that:the original pressure' disturbance is equal to the 4 Ltransmitted disturbance plus on,e half the orifice drop. In terms ' of ' velocities,. (' A9) becomes l k g u.2. i

  • f. o. a - - o a i 4-2.

.-.[ T 4 'd r

r or k4 O. R. -

(A u g = 0' 2 I 40. 2. I f, d. E ,f T (A10) i f' lAt -- { 2 Obviously, we want the positive root. k l b 6 c6 K .p. - 1 K K gt i r K ui ~ g- ( 2 a_, { () {t .(A11) l , lA g,

  • 7

} t -1 4 4 l i I ) f a q

9 4 The key. equations'are-thus: M I (A12-a) o L r (A12-b) 1 { t D' .-l l_ "- N g 2,,,, g g e %.- h. + E. (LAi- (4 ) (A12-c) i 23 - v* o o L =.. P CA. LA n (A12-d) f ~ k-o - The ab'ovelresults are now applied to the following numerical ' example (the numbers'are' chosen to facilitato a later comparison j - with other results): I-0 =- 2033 ft/sec. K=sw, ? u, = 2o.3 ft/sec p/. =. .o47 slug s /ft* [ p, =- i44,ss i psf. i . The so-called acoustic impedance is therefore l 6 I S ct. = 97.72. 9 From equation (A12-a) 4.. - R N. g 363S = -is g 63; g, -, h, s - (3'l. 3 2. (20. = - *Theflosses inLorifice-flow como primarily from the sudden expan- ~ 'sion on:the downstream side.. This:value of K was therefore ,obtained from the analytical expression ~ 1 l{ - -. (l-l - for a sudden ' expansion: in incompressible flow (10). 2 . ~.. ,,~.t

E m-mwe ,A -am -h4 4-4.da aa AAJ. a& 3 4--+4 -rabirab b--4a Es4 -+ed-s.c J A..p-.# 4 et_K.m. s ..de. m ,wJ F 6 Edm-+m-6 10 .. -l [.' ~ - ~ ~. 1 Using'(A12-b) f {2) (2oM ) ],[+(gg)h.o.$).'I t - (.2.os5) j a 2. gg-s- q =. ll.3 ff.[sec. l i I 8 i -i ~ Using (A12-c) l N T S. 8. t-D. 8 3. '(\\\\ 3

  • 2 O
  • k

- p, s 6 -19. f 5 2.3 l,/ 2. as - = y t r i Using (A12-d) - f i l1Q9.N ks[ f h1,St.)(ll.% fr h c: =. t o 6 ]~ 11 (s v - o =. ~ Reservoir Problem The same analysis is now applied to the cases in which the upstream side of the orifice is a reservoir. Up to the time t;he wave hits the orifice, the situation is the same as'above. After i the wave hits the orifice, we have: O . lPi I l Ps j 1 and 11-6Eskicc g. g 4 - l - l us 3 lm x. s v e $ 6. ~ Y

31 Thus,-using (A1) LAs Ui) (A13) of g

  • Q and the orifice equation 2

+ ' k.- b. (A14)- o D 1 j. L p(y, (A15) - buti = and from. (A12-a) ' J3 r g 0 U-i IA16) 9, - ? = i i 4 Substituting, g. ki) 2 -N 'Y f 0 M (A17) G. R i 2. i { $5 i-Oc (A3 - 20. Lt thus =. o g 2. - or ' 41 M t N 4.1A{M =- 0 (A18) 3 3 g j withisolution l b . (A i -I (A19) LA3 s t

.. ~ 12 j

Thua ths key equatiens for the reservoir'$s::o era i

+: 4 1 [ CL lA t ; (A20-a) - 3..; - g Mge' h eik ~ '(A20-b)" ~l

K'(1:.

-a e and-i: i \\ (A20-c) I $ - p, y (' p, - o, c, (us-u. ) y 1 i i ' Numerical' Example 4 4 Using the.same. data as before we obtain the following results.' l ? a .-l9Es,8 gst ~ z-po = i _. i s, g : osi = ) zo-d +- (4 )cs x x 2 3) g s =_

],

-I l cu, 4 (2cas)- j ? [.. .uf =4 15a 2f frt/sec.. - c lg -so - - 19 9 S.7 + (.n 82. ) (I sci 8 - 2.o. 3 ) = c-f 4.- -- ~ 2( 23. T hs[ 2 ~' ,2,1 /o i., =. s -s. \\ N i The schematic below' summarizes both cases for comparison, j p j r k e

A. e9 4 4 A 1a L w w a. O e b s-A.s.n-- I. g b~ j ' q. 6 s m -.).- t m 19.1 iL3-Y a. - l 4 tt"l 1/~ p e ' 4 1/ [f 2 i I e b

9. h * '

p do is.r lS.2. 1 l's l i h r -l g K v s 4-d54WVer -m g-t L 1 I -e _.c Fr p AT N ~ . j ~% j 4 l 1 l

  • * 'l
.W.

s; ... gg __.__m, 4m- ~ =

I 14 i ~ B. General Compressible Analysis _ Use of the analysis in Section A in problems involving strong waves would no doubt lead to significant error. To provide for ~ l I such cases the following analysis -of unsteady compressible flow in a pipe or duct is also included. The bulk of the material presented is taken from the book on the subject by Rudinger \\ (reference 4). i General Approach' In. general,. problems involving the propagation and inter-l action of finite amplitude uaves in a compressible fluid are l nada too complicated to be treated analytically. Fortunately, i the mathematical properties -of the equations involved are part-icularly useful in constructing a finito difference scheme and this is "the approach that has'been chosen in this case. The basis for the technique.is what is known as the theory of j ~ characteristics. \\ In the case of two independent variables the characteristics ? of.the class of partial differential equations that frequently 4 arise in fluid mechanics are curves in the plane of these variables. l They can be identified by any of a number of properties, but from i an operational point of view the most important one is that the dependent. variables of the problem are related along each of - thesc curves by an ordinary differential equation. The form of these j reintions and the e:<pression~s for the characteristic curves can be i obt'ained from the governing equations. Asaresultgoftheseh) rop-i erties thc_ basic scheme can be siuply c::plained in terms of the t follouing co:amplc. In a one'-dinensional, unsteady, isentropic [ flow problem the~ depend.:nt variables are the velocity LA and one thermodynamic variabic. It in frequently very convenient-i

15

  • e-J' to u23 the sound 1spcod. Om as.this second variable.

Through each 'J 4

point in the sit ~ plane.would pass two -characteristic curves of different slope.- Along each of these curves u.and o.would be

~ related through a known, ordinary > differential equation. Now considerla point 3 a small-b' 3 dis'tance'away from a non- ~ ~ haracteristiejeurve along c which the solution is known. This curve can either be 1. .a. boundary curve.or'one on which the values of n and 1 '0Ehave been determined by X r . previous calculation. The l two characteristics are seen Figure B-1 to provide a set of simultaneous equations for. U-3 and o in j g terms of b,u.,d and d. If the distances involved are i s i 2 sufficiently small the curves'can be taken as straight lines and the equations can be written in simple difference form and solved j algebraically. In what follows the above approach will be applied to the l problem of unsteady flow in a duct or pipe containing an orifice. ] The above: general picture will thus be considerably filled in in i the process. No attempt will be made to present the aspects'of-the. theory of. characteristics which lead to the steps taken. Such j ~ i "a complete development would be icngthy and is.already available 1 jelsewhere. -You are referred to any of-references (5) through (7), with'the first two boing,vory similar.and recommended. Xerox i 4 ~copics of these' arc appended to this report. 4

J: 16' V. - Basic Eauntions cnd Problem Formulation I"* The problem considered is the flow of gas Ln a duct of uniform 4 i cross-section. At some point inside the duct here is an orifice. ~ Initially the gas is at rest in the state determined by isentropically j compressing it to a given pressure from atomospheric pressure (15.0 l l. psia) and temperature '(60 degrees F). At time zero the right end t of the pipe is partially opened to the atmosphere. The force on J i j tho' orifice due to the resulting expansion is to be determined. I i The gas will be assumed to be perfect. In addition the diame.ter i j of the pipe is assumed to,be sufficiently small in relation to its 1 3 length that t e flow can be treatedJas one dimensional. The flow l h i is taken to be adiabatic and it is assumed that the losses in the i orifice are sufficiently larger than frictional offects that the l latter can be neglected. Under these circumstances the equations i t of conservation,of, mass and,m_gne_ntum can be uritten as l i

  • O (B'1) l and-

\\- QW I eg = _ _l S_e 1 m 9t. e u. @ f 9K (B2) j LA-is f are the pressure and density and: respectively. O and j. tho' fluid velocity. The thermodynamic variables are related by the equation of state r [ t ./ 4 4 _ _. _, ~

17 where ~ k is the gas constant and I the temperature. L Since the sound, speed will be used as one of our independent thermodynamic variables we will also have cause to use an alter-f native form of (B3-a). Sound speed is defined as { Q. =~ 3 5 Since if 5 is codtant at::st hM{ where 5 is the entropy. are_related by cowst d Y = i CP/c we have . here Y is the ratio of specific heats w y i [ = } d

w. 'If hT p

(B3-b) The fourth equation is required to close this set and the most i convenient choice involves the entropy. For the portion of the 1 flow away from the orifice this will simply be D=0 1-y I where the standard operator notation L D_ b ~ Dt bh 4. t.A. d b has been used. As explained below the orifice is to be treated in a manner different from the main flow. Although equation (B4) applies everywhere else the necessity to patch the two analysis .together-requires that the entropy be kept* explicit in the formu-lation. The most convenient set of variables for the problem turn out ) to be, S 0-and-S' Equations (B1) and (B2) are therefore nou Put in that form. The results are then manipulated to obtain the 2 i-form appropriate to analysis tising characteristics. While the bachground has not been presented it is cicar that if-thece equation: i.* i 'are to'bc trasnformod into ordin;try differential equntions valid b ^ - - - - e-y .+4. m

skmaw., Lee,unll ksuc b cew414 % h hw b',%e s clong certain curves the derivativos in the direction of thess g It is this end that motivatos the following operations. curves. F.irst.the pressure and density can be climinated from (B1) and (B2) 'in f avor of G-and 5 by use of the following thermo-e j dynamic relations valid for a perfect gas with constant specific d5 = g ck kT -

d. f

? yd. TvSk [5 C Using the relation P-- Cp-C = y l and the equation of state in the form dc YET These equations can be written'as d w-R 5 f s. c g (B5) cl& -=- f; cl L a-c\\ s G Similarly equations (B1) and (B2) can be conveniently changed to 4 LL -{- =. O 3t 9x 9A (BG) 9.S -- _ N D ~> 3_b 4a 9t 9x r. 9x Thus, in terms of G,n. and s, the continuity and momentum equations-become -2 9ct a 1 d:.t UI DS pi 3 g 4 gi.";g + o ps c y 5E. (n?) tt -= 4 4 e-r- e v

- 19 .:4 - and Bg:.t-u. 2.!h + T-l-o_ J._a_ .=. c) 2. 6%

2..

9X 9K T P 9 X. (as) 9t 4 The desired combination of derivatives mentioned above can - now be obtained simply by alternatively adding and subtracting equation (B8) and equation (B7). The result is ( rI, 0 6 a) AER 4- ( R'1 2 y )_S \\ G. Y [ DsDt.:k OL (B9) l j Y 9Y j = ?y The left hand side of this equation represents the derivatives 1 of the variables G. i U. in the dir.cationsgiven by dX ~ as =

u. s a It is convenient to define P= g,cu u-Q=y,n.-u.

and also to introduce the operator notation }. ((A+0.h St Dt 2x i p (u. - 0 ) 2 x se 3i

2O' Thus (B9) became2 - 59 ST / Ds

0. D_.! h 4

G= j Q Df 5 'b h ) S_Q = o g T / D_ s & b._s 3 g y_ ( De r 2x), i r Since j 'D D fx _ h+ 4 St Dt 6t D t-the notation can be made consistent. Thus g4? G &..s + (r_c ) c qT Ds Q ~ o h1E St Pf2. Dt (B10) i 4.h_t)(,L (B11) St:- = & rR se ra Dt It should be einphasized that equation (B10) is valid along the curve whose slope at every point is d% =. (.L +Q_ ' dt

while (D'11) holds along curves specified by i

fy = (L - G_ l -db We nou non-dimensionalizo all velocities by dividing by the, spcod of sound at atmospheric conditions. Similarly the entropy l divided by N[g7 is our non-dimensional entropy. Mon-dimon-L sional dictance and timo, defined by 9 e,

21 t =. d a. D b I L ta o where 'lka.:is the atmospheric sound speed and b a convenient ) length, are also introduced. For.the present problem b is taken to be the distance from the'end of the duct to the orifice.' I . Equations -(B10) ' and (B11) thus become t._ t' O. h-I ~ s e. =. G. se Dt (B12) Ds-s_cp g_ s + O b-h Q CL (B13) st st i confleled The system of equations is formally c=:pm:Ed by adding the i f generalization of equation (B4). This could be written as DS -F(c.,u.i i r,r) s (B :) pt-1 ' The function F would have to be specified for particular problems depending on the nature of the heating, combustion or other appro-J priate irreversibility. As will be seen shortly it will not be - necessary 'to deal explicitly with f* in the present case. f Since equations (B12), (D13) and (B14) indicate how'the variables P Q and 5 vary along the curvou l i . u + a. g t Q_( de I

~... . ~ . ~. - - - - -..- - -.. 22 1 + j J y._. and w i dD .=respectively,the formulation of the problem in characteristic l terms is complete. Beforo continuing with the specific problem at hand it may i be worthwhile to return to the case of isentropic flow used earlier I as an example. The pertinent equations and curves can now be made l explicit. Since a significant portion of the flow is isentropic I the results for that case will be doubly useful. l In this, case equation (B14) is no longer necessary and equations (B12) and (B13) reduce to = causkmk A

  • M Q 's p=
0. 4-r4 l

O.-- u =. C. ins kmk %j and- ~^ R - A g g 1 The olution at point 3 in the sketch shown in terms of the properties at points 1 and 2 is thus quite simple. The characteristic form 1

  • to 3 has the positive slope so the solution would be 3

(B15) i k 11 ' * (. P, - Q u (B1s) 3 2. e i Using a different scale it is easy to sco how grid covering'an entiro \\ f \\p/ N b flow field can-be mapped out by repeatedly applying the above- "N,/ 4 .y scheme. In addition to point 3, l r

point 5
can be deternined from
  • (

Figure !!2 I

23 e 2 and 4. The results of these two calculations lead to the solution i at'6 and so forth. It is pertinent to note for what follows that in . the isentropic casesTh 9 (or Q as the case may be) is constant along a particular curve. In general the value of P will vary from one member to another of the f amily of characteristics with positive slope. In addition the values of u.and a.will usually vary along any given curve. This is because any given P -curve will be intersecting Q -curves.of changing value. The value of ((M-

0. - U-remains constant along any given p g, p Q Q curve however.

Resevoir Problem In the problem under consideration here the area to be rapped At' O' O the gas is at rest at the in is O$T6l )T20 t given initial temperature. The value of h c-v Q along any characteristic that terminates at any point on the 2-O Q i line is therefore known: L S O E p b'l Q Q = Figure B3 where O. is the sound speed in the initial, undisturbod state of r the gas. The flow is assumed to be initiated by instantaneously t openingtheendat.$5 O to the atmosphore (the orifice is located 5 = L ) causing the gas at that point to immediately take on at a finito velocity. The value of 'that velocity depends on the detailed boundary condition imposed cnd will be discussed below. As a result 1 v-

t 3

an entire family of Q -characteristics amanate from the {soj fso i

- point, one for each value of the velocity between zero and the Such a fan is called maximum imposed by the boundary condition. ? t a " centered expansion wave". ~ The bottom-most Q curve represents.the leading edge of.the wave. The value of i -Q along with curve is thus

2.. O o The topmost-Q

-~yg curve represents.the trailing edge of the wave. The fluid Figure B4 l 4 velocity there is the maximuth caused by the wave, call it-U. mag e It's value, again, is determined by the boundary for the present. condition. The.value of Q along this top-most curve is deter-mined from this known lamas and the fact that the value of any 25o line l intersectini; P - characteristic orginating on the i Designating an arbitrary point on this cruve by o( l is-known.. i i we have z 7 bot + MA Ft do .t

  • Q But lQ 5 Umax

, therefore Q, T4;g-tA ma.x ag = t 'and hd' o' ~ i L 1 h4 { ho ~ 1 (A MAK j l t ~ characteristic;at R is also known Q b] \\ The slope of the a %x - % i - at ' Q G d l

- -.~ - 25 ,i The values of IC? for the other characteristics of the expansion ['

fan, as well as their slopes, can be determined similarly.

The charactereistics thus c1carly show-the propagation of the ' wave.. As it moves down the - pipe t it spreads since the elements of .the expansion progressively lower the temperature of the gas and hence the sound speed behind them. l The profile of the wave at any time ~ such as those indicated by the dotted lines is given by the' values of R(3L

w.,

b 5*h Figure B5 and G. , and hence ), f, e.h. at the points of intersection of the characteristics and the 2 : C. i line. Although there are tn) characteristics through each point q I' -in the plane only those which delineate regions of change are i ? plotted. t [ The above conditions apply until the leading edge of the i wave strikes'the wall containing T 4 the orifice. The two character- 'istics through this point are / I ~shown'in Figure 6. It is seen r that the. C? -characteristic is p reflected as a {3-character-4 i istic atLthe orifice. The de-tails of the.reficction process j are again to be detcrnined from-s.m.. :.. ~: ' 'N the boundary conditions. In the Tsb E' O .timplent case of n'nolid wall this Figure DG l-i

o 26 . c 'wsuld ba-LA.s O Thus, egain dscignating tha point in question ~ t _' byj 0( we _.hav$ by cquations, (B15) and. (B16)'.. i ^ ' 94 = Qd. y y(PtQQ

  • "8 l

o t ? LQg,.is nou known' k and Og can bo determined..- Similarly f Since -3 when the 'second' characteristic. hits the wall at { 9 y og(f e,.)- [ 3 y y Cep og} -If the. parition at [- L contains an orifice the situation is' considerably more complicated. First of all the boundary con-I ~ dition is'not~so simply. stated. The velocity is not zero. The l Lincident expansion wave will induce a finite _ fluid velocity through Y Only part of-the incident wave will be reflected while the" orifice.. part will be transmitted through the opening. Secondly, the value_ p f This is o_f Q,on the incening characteristic is no longer known 7 because.the flow through the orifice is not isentropic and this j i results -in' a' change of Q as the orifice is approached as shown by I ) equation (B13). The detailed formulation of the boundary condition j t at the orifice will be.taken up shortly. The important point now is 1

  • that.the.proporties of the' reflected wave are determined from appro-4 priate boundary' conditions.

In the procedure itself the amount of-reflection and' transmission and all-other aspects of the interaction with the uall are distilled into the value of the characteristic ~ variabic 59 'which leaves the point-in question. /_ '

  • 4 1

9 1 4 1 a 4 'e s. . E ~ - <

-27 From'this point on the uave diagram ? I 'gets increasingly more complicated 4 as:a complex _ pattern of reflections. 1 andiinteractions builds up in time. If the upstream' side of the orifice acts as a reservoir. con- ' '.ditions there aro'of course'known. 'If theLpipe continues however the. . unsteady.flowrin;that. region:can be determined in a similar manner. The boundary condition consistent Dwith~the effects of the incident wave isJdetermined and the-calcu- $= L 5so lation_is beg 6n as before. In Figure B7

this instance however:the fluid is accelerated at a finite rate and I

the result is not a centered expansion wave. This means that the (3 - -character.istics comprising the expansion will not orig'inate f at a common point but will be distributed along that short segment of the.timo axis subtended by the incident wave. I Boundary Conditions The opening to the atmosphere at the right-hand end of tho' pipe is. treated as a short, isentropic nozzle. The assumption is based 1, on the. fact that the fluids in the corners.will be essentially t The flow pattern does not have the turbulent secondary stagnant._ 9 4 k' * '%g .. Wah.7,,!;*

28.

s'.

flous that noimally'bharacterize a region of' significant losses. "A - s s . An effective exit area uhich ^ s ~* j~ accounts for the usual over-s ? contraction of the flow should 1 be used in what follows however. y t l -s ,s O Figure B8 Since we take the fluid at position one to be accelerated l C occurs the flow in the l instantaneously when the opening at nozzle is computed assuniing steady state prevails. The pressure in the ambient gas is assumed known. As long as the flow is sub-k, - hambient. sonic the~ appropriate boundary condition at e is When the flow becomes choked this condition becomes Mc - L'. = I i o.e where M is the :5cch nunder. Mc ana the area ratio A./A, the. From either fc. or conditions at position 1 can be determined from the equations of one-dimensional isentropic. flow. If the flow is choked the k can be determined from YFl. ] '7 2 Lr-O 7,,

  • M, {l A'

l z. + y-( 1 l I f, on'the other hand, %.isknown isbotorminedfrom y t-i y' i 30'-0 j i + y-I M, &-(2 Ai i 2 vu = Ac.. Pc Mi ( 4 + m

_-. _.... ~ 29 } is-the initial pressure in the pipe.. Those equations-whern can be derived from the basic conservation laws and are developed (although in a slightly different form) in, for example, Shapiro l (3). The obvious algebraic complexity of these equations can be avoided by using the gas 'tablos such as those in Shapiro or Koenan i i and Yaye (8). The relations.between the various dependent variables in one-dimensional isentropic flow are tabulated there. The equations I can also be usefully plotted. The treatment'of the flow through the orifice is analogous to f that just described for the opening-at the end of the pipe. Flow l on the upstream side of the orifice is again.taken to be adequately i. 1-modeled by a short isentropic nozzle. Except for the use of an . effective orifice opening to allow for an over-contraction of the converging stream the losses in the orifice occur l'n the sudden 4 l expansion that occurs on the downstream side. The flow is again ~h i assumed to be steady. This .f implies that conditions at i ] Section 1 adjust'instantan-s eously to changes at Section -+ ~[ j 3. The flow between sections -9 . a, --e , 2~$l [ I-2 and 3 is not treated in detail. j .Instead a balance of mass, momen- _4 --e e j i tum and energy is imposed along -~ 0 (b

with the requirement that the flow at 1 and 3 be one-dimon-l '

sional. Such a spontaneous change will involve an entropy incroaco and the solution vill Figuro D9 4 1 e

30 reflect thic. A key.'escumption in this analysis 10.that tha pressure on the dounstream side of the orifice at s'e.'cMion.2 is l equal to the pressure at the centerline of the flow. This assump- - tion has been verified experimentally for all subsonic flows (9). The equations available to determine the conditions at Section 3 are the following. At Section 3 we have the incident c'haracter-istic function l O~ D (B17) h~yg S The conditions at 1 and 3 can also be related by the steady flow energy equation 2 W l h, & &= k 3+ 2. -z_ where k is the specific enthalpy of the fluid. Since for a perfect gas (l s Cp - Cy use of equation (B36) leads to Q. k

  • y-l Thus the energy equation can be written in terms convenient to this analysis in the form Y-(

t 2. s Q,' + yig u Os. F F_ % = t S t 1 J

31 t e In the case of an upstream reservoir LA g s o and Ag :Go the sound j 4 speed in the un' disturbed gas. Since this case is somewhat simpler i ' it will be used to demonstrate the method. The pipe problem can i be treated s'imilarly but' involves additional algebraic complica-tions. We thus write d. + (B18) ~ conservation of mass, momentuin and evergy between sections 2.and 3 imply U hE L L' fg S S (B19) 0 gk g4 gp g (B20) 3 4 and '1 0 +yI d 1 t 1 bz f p.( M 3 1 3 (B21) ~ i The final equation required to close this set comes from specifying that the flow to point 2 is isentropic. ~ 5,= 5 = So ~ g But since the gas was initially compressed isontropically from the ambient state 5,= Sa_. Furthermore, we are free to assign the value of k arbitrarily sinco only entropy difforences arc of significance. We thus obtain 5, = o 4 4

l J4 J* Tha most usoful form of thic: condition,howavar comse frcm the first 1 -'of equations (B5). Putting it in non-dimensional' form and integrat- ) ing between the. ambient state and state-2 yieldst- ~ ~ J Q - g Therefore-: i 4 I ^ 2P (Qs 7 (B22) l The'above entropy equation written'between 2 and 3 will also be i useful in th'e' following form s 7 ~ 1 1 3 -- l -{- (B23) b_ ]r-( y t r Equations (B17) ' -- (B22) plus the pair I RI= Y 0xd da, ' T es y j provide eight equations for the eight unknown variables W ia s, g Og, {kprovided Q is known. As mentioned hkM earlier however this is not the case. Thus the use of an iteration i l ' procedure is indicated. A value of Q is assumed and the other 4 s-variables calculated. Q is.then calculated from equation (D23). l r b A new value ofL-Q is then computed from equation (D13) uritten between the orifice wall and a previously; calculated point on the

Q

-characteristic icading to the wall such as point in r r r t

33 + i I YUb (D27) }.4..y b IM 3 = h 3. and t d f y.t Y-0 ik. lk 5 (B28) j I l4 2 = d g,. 1 i respectivaly. (B26) and (B27) are now combined to obtain G5 3 s. 2 2 Similarly (B26) and 4B28) yield k,3 O b ~ 3 3. 2 These are two coupled quadratics in the ratios g and RI in terms of M which has already been obtained. The utility 3 i 5 - of this formulation comes in when computing 3 from equation (B23). In the general case an iterative or graphical solution would be i required. A series solution valid for small Ng can be obtained analytically however. The formal solutions to these equations are [+ 4 f Y.dO A.) g f { {. k.d pt'- 2 2 3 Es -l t !s spu A, .b I

5. As M ;V?

a3 2}Y-1/1( 4 g, 3 A and f l+ Y M.) - bMMs 4 Ms T * @' M *~ L-2 A W $a 3 s. i a

34 r' Figure B10. If the iteration t' 4 ~ l is being used to determine the j conditions at d. Since the flow between K and 3 is l 3 ) isentropic Ds o Dt a and we can write Q4 - G@ = c (& -5p ) l Figure B10 i Unless'the fluid flowing from the orifice has reached f, 3 = 0 Thus Gg PQp h l hd, ' hp

  • Y

[ (B24) 7 where an average O-has been used. 1 To actually obtain a solution the following manipulations are.useful, espccially for lou Mach numbers at point 3. Equations (B17) and (B18) can be solved for 6 in. terms.of Q Ad Oo 3 Y e Y-\\ Q * ) .2.J. do ~ Ik u Y-t a-- b (B25) ) y-t j The "minus" solution corresponds to supersonic flow in the expansion M >can then be nnd is discarded. The velocity lag,and hence 5 obtained'from (17). Equations (D19), (B20) and (B21) are then rearranged with the help of the two equations of stato to obtain ~ a A3 o, N (n2G) Y Az a b $2 '?- 1 s A s

35 where only the appropriate sign of tha ridieni has chosen. If,- these radicals are.now expanded by the binomial theorem we obtain M) l !_) A3 [!J .~ I.h 0, =. \\ 4 T-j Hs /' } _ es, d.)) J g \\ 2-6f s KJ ' 't 3 pu th e 2 3 4<T A,fazY '1 fs l+TM {b Q-~,*~ 1.M 2-5 4 G,3 ~ 5 = 3 As 03 43 substituting one into the other, solving the resulting quadratic and expanding again leads to (+D -l h5 /A 3 ~ +.(gg, g((g_ Al } g3z2.MDds ItrA3 ,g ~~~ ~ ; p 3 { dJ Oz.! [ (At 3 (D30) L k s Os For low h the iteration scheme is simply: 3 k 1. Assume 2. compute 6 from (B25) 3 from (B17) 3. compute M3 4. compute Mg, 5. computo l'3/pu b *t I froni above. and 6. compute 5 from (B23) 7. compute a now Q from (D24) m

3g (. calculation of conditions at the orifice is now complete. The -characteristic at the wall point say M in Figure'10, is known from-l '7s kGg+R. d The calculation can be continued .,to point f. by'calcul'ation of Ko

  • s i

the value of-at that point / o from equation (B12). Again e ^ DS/pt s o so that a ~ k X. g Figure B11 Numerical Diamnle. i The following numerical example will serve to further clarify the.above procedure as uell as.to verify that the water-hammer analysis is quite adequate for waves of small amplitude. ' Ambient conditions are again'taken to be'15 psia and 60 degrees F. For simplicity the calculation is made for air with a Y of )

1. 4. - The ambient density is then (16 Pf

.o0742. $$w s 3 g fa. '.ETo_ (p:3 )(_52oX22 2_) The referenco sound speed is therefore 6 4)6NMc) J.(' 4 T -(, )i.-~ E,ew. \\ \\, ', r h = ( 00'1A Q j ga.. 6 4

g.4.-pf, 37 Tha initial cound sp:cd incida tha duct will.be taken to have thei value: z.o s c. 3 St/sec. Q, s 1.32.3 0.a. =. i The initial pressure is.therefore (cqu (22.)) l & T r-t fa. OA) = \\ l ~ 15 (1.72% ) 7 i 100 L 1 q c. u = v ie, vat psf. = Nondimensional variables will be used in the calculation. In r these terms j 6.g = l.0 do

  • l D h

', = u. 9 a 8 A h, .b. I 3 D I i t i e Y i._. i For numerical convenience the ratio of the area of the duct to the rarea of the opening at the right end is tal:en to bc 57.874. This cif =7,675 and allows D /d. =5.0. Since the pressure t means ratio across the opening is lower than the critical ratio o,f 0.528 l the flow at the c: cit uill be sonic. Thorofore, as described earlier, ..?L I 1 M. ))]) M*9 h4 \\ } l F :-( Y L 4 A. Mi / M i ? r i [

~ /. 38 -from which we~obtain D.Ol i '= I This'is the boundary conditon for the right end. As the blowdown _of.the pipe proceeds this uill eventually chango but there will be i a no need to go that far in this example. The structurelwave will be described by a set of five charact-aristics. The wave diagram is sketched below.- t . t g d' I1

L Lt t

to t. 1 s av G s s v ) ' ~~. =L {s( Figurc D12

39 The. calculation will be carried out for the 15 points danignat::d* on the-diagram. The resulting pressure at point 15 is the pressure .on the downstream side of the orifice aftor the first reflection a of'the wave. The Q -characteristics that make up the fan-are designated _I through V. The fluid velocity and sound speed along QI can be calculated from the Mach number at the right boundary and the i. a value of any intersecting 9 -characteristic which originates on f the T-O line. Since in this isentropic region both P and i 0-and u are also consthnt q are constant it follows that along each Q characteristic. When the Q -characteristics l intercept P -characteristics originating above point 1 however this is no longer true due to the influx of fluid through the orifice. I Thus along QY. we have. M = - =

0. 0 l

~ (6) h.h3 ) Qo s p 3. Q. t-LA. =- l f. s.n c - c a. + u-b =. l.3M s&z Q 1 a s.o n Q m. . 015 N Y We can also now calculate the value of 'QY Q~f,

  • S 0.1 ~ &
  • 0.O*]ElaIb The large number of decimr.1 places retained is due to the very small tc=porature changes occurring in this ucak wavo.

9

40 Since (A. and a.are constant along Q i until after point 5 g = l.f l 9 362. g g o.oiti94 j Calculation of R and 0-for CT ~, and hence point 4, pro-ceeds as follows. Take' l,14 0.'1 6" tr g 3

. 6. Ol%N s

The particular value chos_en only datormines the' size of the char- 'acteristic net. Again we use the fact that any 9 -characteristic crossing Q E between points zero and 4 has the same, known value. 0 4 & d.{ S.Ilf f-:: (Ao = Thus 0 7- {T2017j 4 There again follows' QE % 9.037709 l Similar calculations determine the values of the variables at the points 3, 2, and 1. The value of Q and the sound speed at point 2 are now used to calculate the vall point 6 using the iteration procedures outlined earlicr.- The actual calculation was programmed for a digital com- ,Puter. The results are shown in the following tabic. T'he method that has been d::scribcd allows the intersection of the charactor-istics'to actormino the grid points. Those locations can be cal-culated as follows. l t i i

...I p. e. m. p 2 Q G W d [' M 4 N & N 4 3 C o a M M G N M G~ r: w w

  • ..3
I I 2

I I I i iI i i i i I .l I

i. i,

{ w ' ml r ,o oi o to y f4 8 w-01 to N J. ,a ! p' wl Q

  • 3 in to !

t ce r' a d m mN c-N I' ol 1 La P W o m ; _0 w o! 4 .a. if l q 4 ,a

4) i c:

c4, M ;g ". ; *. f. 'n e e t% m m +

  • , (.*

n tr' a p 3 ca,;.M S. 1

  • x w

c,,, to - [ - l _. 2 ar-, i- -i - Y D fG! OI C ! lp ! c. j !,j i J C-Ni D ~ c v(d '"4-g e, g. r e c- ~a 4 p) v> r.c y a -l r i -n a en -l o c. o W G .0 N d M n1' b ~ j D b m'O b W W W u u sm s "d. v i 'W', T to ,e 0: d W, 'e '41 V! u u. 'B.W in j D, W i if. .i VI ) )

  • j
  • e N s [2 e t, O t,

d' h .D (g r i W J r 4 I~ u 8' ' l P S $o 5 ,n5'9 h' k 9 0 0 6 i ~ o e_ c. T ' h. g.' a. . c. ,s o O c J T. T w c. c. a-u, )Q e -e < c e.j u u a ,o w u_ c.

c-t =.

e ,e r x e 9 e T

r. i.c s

4 o a w ;. 4 to o J g d 5 d _a_ J o I o a 4 4 o o a 9 a o. o a a y g 3 a 4, o a 4 41 4 4 4 (d 0 ) I f. $ a.. d i i i T,, i &l hl o o % g ' +y ~5 d d 4 r w, i 0 t/' O.o; o o 0 q oj o o O .d 3p i o i el 3 o. I i> i i 0 0 t. i t T 0,1 N N .N (%l ~ M l' T I 4. rw ns a 0 r c! etc 5 l o- 01: 7: c4 c1 :r ~7

o. T e, ; M :

C". e Cl;..j r4

  • O!w.

a t o'c, r4 'd '4 1 c. ,i o c ..I d4 f4 4 g e to : -* t r4 r4 N e4 .d I.O * (ce ~~~~ cm M, ! '!4 M. ' 03 l ce t<3 t.o 4 ~ ~ * * ' ' * - >1. ;, ** i .;o 1 2 f2

)

..: i..:..: i

3 _3 i

e o r4,. Q l.) F^ J.-.' h.1 ~ __. ,J $e n' O 4 $3 li - p c) ,d O.. 79 g ;) - I 1, l C* lI Q g4 a

5 g

o . 'S. I:~:. o 4. .T p

h. yG*

c,- to r-t f' l. u. p1 A. i C'* ;' $. M 9., 4

  • ll" c

O. o,, .3. ;

0. ; c... Go. e, cll o.
o. - Q fi. o.a. t u
o.,.

c o Q. c~l J. 4 _. 4 a c '..!J 19 t'* a-c-: o ,3 O h,,e si a I ,O O Q ~ t,,, C' I Os N-P 4 9l ~ D"' f' .4 ' % '. i.!=* e f-6:a '/3 ' C-a;.3 r- ) .s =J 'd* I C* -l Mj,,, T f g,aJ r, p,,

  • a 4*

Q D pl l e~..o g 7*J pt F T 7 p F a o ). c

c. Q o

s1 r- ~> ; j [.*'D" N h, l h h8

  • - t L

3 ! ). ' ! D j r) ,,a ,- l ' ~. 5 } "' ' 4 I. ", T ";

  • i ci.

L c.4 g;* b: t*,. I.f:,',' U .,. (O r4 ej i It' l C" 8 ., *[ g '. g ~ i i. 3 p3 '. ". I "0,, :3. '~. ~. g

  • == ~. 's, l ~- l

.S i.- ~. '

  • .. [

j g.' j c j u.. :-,c',pi e.q if,

- 9 c

l l e s sst .g ) I et ".,g T. *,,' e r1l r}' ?I Oi G (- le D* ..l I 8 i ( *#' t I .,,i.. m.s .t '8 I l 4 I..4. 1

.=. ~ 41 I cinsidering ths'throce points ~ y shown in the sketch, the coor-- Je - u f a. Jr dinates of a can be detornincd g g s U. - c. M from the coordinates of : and k~ and the slopes of the char-3 K acteristics through } andk which intersect cI 1.The { ' general formula is Figure B13 ( w a.h t ; - @-a k 4 + fr - Te .t'. = (am); - @-o h {L j-j~ c' Y l J Translation of a few pertinent results to dimensional terns will facilitate a comparison with the first analysis. The wave has' been completely reflected at point 15. The pressure there is ~ 3,, - (c e. e ) ( i s) = us.sr and Q = (,012.3 4. (Ill 3 13*E Se e. and the initial pressure is 18.4 psi. Thus the difference between ac Doth figuros arc in c::cclient agrecacnt with the incompressible Oi,ec bg it') resultsf Since tihe. flow velocity behind the incident wave is only one per cent of the cound cpcod this is not unexpected. 4 m

42 The difference in time between points 1 and 15 gives an indication of how sharp the pressure rise is. If the orifice is 100 ft. from the open end of the pipe we have i ) h, = (,ssri9 r .E'h F46 )WlilE 4 i. goo 6,y-b = 0.074 seconds. g g i The above analysis does not use the empirical orifice coefficient k. It is, however, implied. It is perhaps of interest to use the results of the analysis to see what i value of k would have led to the same thing. We take ~ point 15 as an example 2 \\ Un~ 7-O- o gs If 2 1 2 2h b [ =. (A. 4 4 From the equation of state we obtain e (),, =.0% sin-( s j'.gn caos.o 6*) s

9.,

x' = Go4t ) Os.O =. W 9 +

t t e. This' agross' vary w211 with tha valus of 576 ussd in thz incomprassible cnal, j ysis. Again the agreement is to be expected with such' low fluid velocities. i l t .The calculation was not continued beyond this point since it was_ felt r that the-largest force on the orifice would occur just after the initial re- ~ -flection. If a longer time history is desired it would be preferable to set up a direct numerical calculation of the entire field. In this case a grid j is cho' en by selecting 'a regular array of points in space and a time incre-s ment. The intersection of the characteristics from a given point with the previous time line will not be at these pre-selected points and interpolation 1. l. r is necessary. Boundaries are handled as in the above. f i 4 I f 1 l i 4 ~ \\ 1 i e 4 4 4 4 b

_ -.. =. 44 ( t -c i L I C. Empirical' Compressible Analysis 1 f In cases where the obstruction is not an orifice it may not be possible to use a non-empirical approach. The,following com - prossible analysis involving the empirical loss factor is there-fore also provided. In addition to this generality, the follow-8 ing also turns out to be simpler to apply to what we have' referred l to as the pipe problem. i Reservoir Problem I The flow through the opening to the atmosphere is handled as s before. The treatment of the main flow away from the orifice using characteristics is also the same. In the case of the up-stream reservoir the orifice flow can be analyzed as follows. The empirical relation is 2. US ffL 5 [L-A ~ (C1) where subscript's L and 2. refer to the upstream (lef t) and down-stream (right)> sides of the orifice. The steady flow energy equation is \\ bc Og. t 3 (A g. (C2) i e 4 e 9 h.-- r

.. ~ 45: + ~ f*? i In addj. tion wo have the. equation.for. the.Q -characteristic on ^ the right, side and the equation of. state e Q g,, = Qg.,-- U m. (C3) c 'tr_ i s . Q g - s. Y (C4) l (#g..- A O,, g. and These four-equations are auf ficient to determine 4gg g !~ kg. for a given Qg.' Again an iteration is necessary. Once Jthe state on the right'is known the new.value of Qg,canbe'calcu-lated from: a 1 1 dyr. J 9., 5-5L g; w) g y{ sw g ~ ' ' g (Cs) g Qc.

  • 0. S b 5

it - (C6),. The form of the above most useful for the calculation are as follows. Equation C2 can be written -I m dre 5 N 2. 4 I p (C7) s. ' Equations (C1) and (C4) yield Yk' Mr. / -l y 2 l.. f .j< (C8) 2 s u 9 9 9 e

46 [ . Dividing equation-(C3) by O we obtain Ge. Qa /au ~ (C9) 1 g (( E. 1 Ife now ' substitute (C9) into (C7) to obtain Mr&2h)&+hhi[=o -i or f.?. (Yj Qh 1 h hz)Li 1. f p, t 4: ; _ L 2. g3 - 12 gy }k % ~ -(C10) ~ ~ L* Q7 .i The. scheme is then to calculate Mg. from the assumed value F/k (and. Ok /a[ of Qn, in (C10). Then obtain and hence and ag. ) from (C9) and (C8). S. can now be i g s o ), and Qre. calculated obtained from (CS) (recall that Sg l from (C6) is checked with the assumed value. If they are not* ) the same it is suggested that the new value of Qg. be obtained i from CU (c.) ~ Q(c,iH) k.- * ) Q s:. = e

  1. Qs.

f l where: g} value of kg. for next iteration . Qg., = value of Qg.,, from last iteration = sf Q[) value of gy. calculated from = .a a e a a e.1c x i %S i

  • t t

~ The same numerical example used in the previous two analyses was also used with this approach. Again the iteration was carried out on a digital computer. The most important results for the 4 wall points, (see Figure B12) 6, 10, 13 and 15, are listed below. [5 e c. ) Po l M'i~ Q h h5I) U g 9 109919 { oc o. /, 6,6 C (O 910l,920 j9 E. & T,0 l Is 4.104 f61 490.6 ll.66 15 9. l 01 C'2.6 $ i 6. 2. 13.9 TAGLE cl (ti,ce. kil) These results are the same as those of the previous analysis A., (Volocities in ft/sec are obtained by multiplying the non-dimen-sional results in Table B1 by 1118.0. The analogous factor for pressures is 15.0.) Pipe Problem The same analysis can be applied to the pipe problem as follows. The pressure drop is again given by 2 NS 1

E-T (C11)
t. " ff
  • a

G Conservation of snas cnd energy through thm crifica rsquira I A e p, u g, Ug (C12) a. and l ~ l O. +- M ~ L F-(C13) t l In addition we now have two characteristics relations l' i 2. hiz. R R F-ME (C14) 4 (C15) t* g i q and'two equations of state G s y !f (C16) j j. ~~ g fL (A j f Finally the flow on the left is isentropic so t}}at 5s.2 S. : C. Therefore by (B22) i 2%~ l b I 9 L / (C17) j The computation scheme proceeds somewhat differently'in this ) 6fandG[from(C11) using (C16) and case. We first eliminate -(C12) 4 f (C18) i 4

49-i .,s.. ) Equations ~ (C14) and (C15).'are then used to eliminate' O.u and 14 ) - from. both - (C18) and (C13). The result'is.the following two i u. and hc coupled quadratics for s UE 2.(QdPc)ug t.Q[-NPuc ' I ( N.g t E'- (A +. i (C19) i i s. ,2,e d= t --(A[~2.- he Mc h ~Q~AQt LAR. - ~ ft (C20) L = O- , ith assumed-values of Qg. ( kisknown)and LAs*/4 j W i we - first calculate LA g. from (C19). (C20) then yields ut and l the results are compared with the assumed velocity ratio. When i this iteration loop is satisfied the remaining variablescan be calculated as follows. The upstream sound speed is obtained l 1 hu. is then obtained from (C17). then i from (C15). follows using (C16), and fg. from (C12). The downstream pressure is now available from (C11), and as, from (C13). With all the --variables thus determined we again go to (C5) and (C6) to check the assumed value of Qg and begin that iteration. Results of this analysis using the same initial wave, etc., as in the previous examples ~are shown in Table B2. The " points" again' refer.to Figure B12). .TA Bt.E c.2 PotuT bu (psQ pp (.bst) t.te((%cd ui' (&/m.) L to o o.9s 999.4 3.97 3.'17 (c. 499.o 994.4 .I. 13 c.E6 l ?, qT1.4 q?9.I T. ( 3 (i. c ) t 9 (,. o - 9 F t.6-11 2 I ti. E c-b- 4 L m .mm

So 4 4 ,~ Point 15 is seen to agree very well with the results of the incompressible analysis applied to this same case (see pages 10 and 13). h e e e a, e e e 4 e e a b ~ ~.. - - -

o* d*4 References ~ s Contractor, D. N., "The Rcflection of Waterhammer Pressure 1. Waves From Minor Losses", Journal of Basic Engineering, Trans ASME, p 445 (June 1965). 2. Jobson, D. A,, "On the Flow of a Compressible Fluid through Orifices", Proceedings, Institution of Mechanical Engineers, 169, 37 (1955). 3. Shapiro, A. S., Compressible Fluid Flou, VI, Ronald Press Co., New York (1953). 4. Rudingcr, G., Have Diagrams for Nonsteadv Flow in Ducts, D. Van Nostrand Co., Inc., New York (1955). 5. von Mises, R., Mathematical Theory of Comoressible Fluid Flow, Academic Press Inc., New York (1958). Anderson, G. D., and Band, W., " Compressible Fluid Flow and 6. the Theory of Characteristics", American Journal of Physics, p 831-837 (1964). 7.

Courant, R.,

and Friedrichs, K. O., Suoersonic Flow and Shock ~ Waves, Interscience Publishers Inc., New York (194 8). 8. Keenan, J. H., and Kaye, J., Gas Tables, John Wiley & Sons i Inc., New York (1945). Hall, W. B., and Orme, E. M., " Flow of a Compressible Fluid 9. Through a Sudden Enlargement in a Pipe", Proceedings, Insti-tution of Mechanical Engineers, 169, 1007 (1955). 10. Shames, I. H., Mechanics of Fluids, McGraw-Hill Book Company, Inc., Haw York (1962). e' O

ps, ma mvs. mm_ ps,c zivouceuruct 7 V ~ fr(s The Reflection of Waterhammer

7. UuHe iM) g. = 45-45a.

^^ i. Pressure Waves from Minor Losses D. N. CONTRACTOR v.-- This paper deals with the reflection prods.ced sehen a waterhammer pressure wave Q% neseares sesennse, nyeronavnes, Inc., encounters any device that produces a sudden ene gy loss such as an orifce, a valve, or an

  • ,n -c g

Laurel. Md. elboto. The classical wave theory is used to determine the magnitudes of the reffected and the transmitted ucres. .fn The scalerhammer equations, with the friction term included, are solved b conditson to these equattons. ~ @ 3. R method of characteristics. y ,e G-p q, ~Q,_ Agreement between theoretical and experimental pressure. time diagrams is :ujicient to validate the theory. jf + m >< R., o 'v intf(il!UCllon continuous distribution of fluid friction in the pipe and provide O {D 1 3 $*% [ 42 / ip f~ garent!AMMEa in a pipe system can be analysed in J. j the pressure and velocity at regular intervals along the pipe as a function of time. A program for a high-spe-d computer can be .? many ways. There is the numerical method in which the head. written with ease using the characteristic equations and appro-h and VIlocity at a particular section in the pipeline are obtained by priate boundary conditions. In working out the equations at a ~N,. f,. 2 9 f a careful bookkeeping of the passage of waterhammer waves and minor loss as a boundary condition, it will be seen that the con-. ' their reflections past that section. In this elementary form of clusions reached earlier are verified. w;terhammer analysis, it is imper stive to know the magnitude of t,

  • IM the reflection of waterhammer waves at certain boundary condi-Elementafy So!Utlen tions, such as changes la pipe diameter, pipe-wall thickness, and The theoretical study of waterhammer reduces to the solution y*

6 pipe-wall material. The reflections at certain end conditions must of two partial differential equations. These equations have been to, 1 also be known, such as at a constant head reservoir or at a dead end. These reflections have been known and used for a long time derived by many writers [1,2,3] and will be used directly in this yg 1, 2).* In this paper, we examine the reflection and transmis-paper. The first equation is derived from the condition of dy. eq q "" c['on that occur when a waterhanuner wave encounters a device namic equilibrium: ey that produces a sudden energy loss during steady flow, such as at BH' 1 BF' (l) gN 3 4 n orifice, a pipe bend, or a partly closed valve. b' a bt' The second method of waterhammer analysis is graphical in g, asture. In this method, the slopes of two characteristic 11nes are where the notation used is presented at the teginning of the pa-(p

  • 1 determined, and, knowing the initial and boundary conditions of per.

7 the pipeline, the head and velocity can be determined as a fune-The second equation is derived from considerations of con-tion of time. This method automatically takes account of the tinuity in a horizontal pipe: reflection and transmission of waterhammer waves at any BH' -a' B V' (2) D ' boundary or end conditions. Thus this paper does not present bt, p bz, .~ anything that is useful to this method. In fact, the conditions l [g *at a minor los's can be taken into account by this method of solu-where a represents the wave celerity in the pipe. g tfon. Ilowever, despite the fact that the finalsolution is obtained, The simultaneous solution of these two equations is given by - i d.'the graphical method does not give one a clear insight into the \\ + a / + /{\\ , jnechar.ia of the problem. H'-HO=T l' t' - (3) a/ r.,.s. The most recent and versatile method of obtaining solutions 4 3,,"of_the votahammer equations makes use of the method of and i 4., ch ;a= *ristice. This pn.eedure is able to take into account a ~ "~ ~ ~ k hu.Aers in trackets rienignate Iteferences at end of papec. r'c.w%ted & the Fluids Engineering Dhiaion and presented at In the foregoing equations, T and d l can be evaluated at po. ts where i,f,are arbitrary functions e. the %ter.imain) Steeting. New York, N. Y., November 29" Dece:nber 3.1064. ed Tar. Aur.mc.m Socicry or AttenAMCAt. Ewas-and H, are known and then m Pctt. 4n".1pt received at ASN!E Headquarters. August 8. used to find F' and H' at other points in the same pipe. These 9 ,3,9,64. P..rer

9. 64-WA/FE-16.

f M l!T.SilCl8tufC l ' .q. o = wave rcierity in pipe H = dimensionless piezometric FO = steady-state velocity ,6' 4 .'rurssertional area of pipe head = N'/HO z' = distance from reservoir HO = head causing flow z = dimensionless distance = 4, = area of gate valve 8'/U I 3h Cs = oefliticnt of discharge L " ength p1Peh.ne S = waterhammer constant ^ = t D = diameter of P Iie 3! LOSS = minorloss PO/2 HO ' i 7 f = Darey Weichach friction t'== t me A = difference or change factor i = dimensionless time = waterhammer, pressure g'/2L/a r = ratio of effective gate open-f./t. ft, e f. M, wsve heights r' = velocity of water in pipe ing to full gate opening (C A ls " s J ~ gravitationalacceleration F = dimensionless velocity d e = = plesometric head V'/VO (CaA,% = s f JUNE 1965 / 445 b 9t!N I N,lt ED[lD!8ilD[ W ~

$f,dk' ~ A ue , t h.,i s a lq e n C. equations are used to find the reBection and transedesion of a 7 / ame, ""I'7"4_' is

  • j ti.

sunt t=c a pressure wave when it encounters a minor loss. - In Fig.1,let points A and B be on either side of the loss-pro. a o f Let Fi e a waterhammer ,,,,,X b I ducing device, for example an urifice. l pressure wave approaching point J. Iet fi be its reflection and"* Let the head and velocity before Ti reaches +5 a Ts its transmission. the ori6ce be H'm, H's and I"u, F's.; and let the head and og velocity after reflection and transmission be H'u..H',, and-s V a, Y,,. ] 92 1 ,,, ^, I h e' From equations (3) and (4) (5) ,,,,,,,,,t,,y,,,, H'u - H'u = F +fi f,a m. (6) H ',, - N',. = F. v M(L.ce 1 The l (7) ~ usoss,1 1"o - F'u = -f (Fi - fa) d'!"?$5L _.k _ - E;NW. 4. u. V's,- V's,- ~~a F, (8) w .g. a Qef3 oo If the pipe diameter is the same before and after the orifice, then e,;.. . 9" continuity requires condi,ien, before and efter a waterhemmer pressure wave sua: P'u = I",, and V'u = 1"a (9) rig. 3 encevnters a minerieu pr Since a minor loss occurs at the oriSce Using me fomgoing relationships and solving forfi and Fs p}u H's. - H'u - MLOSS. and H',, - H'u = MLOSS, k From the above equations,it can be shown that ' f, = F, - + ru + + Yu and (IE -A(MLOSS) (10) ) '(i 1 f 2 (MLOSS. - MLOSS.) =2 and E k + Vu k^'ilth ~ F, = - A + Vu Jh[ I The procedure is the same when analyzing the situation in 4(MLOSS) } ^ which an f, wave approaches a minor lose, as in the case of flaw I' ~ I' Y j 2 ? establishment in a pipe. It can be shown that the reflected wave C Thus it is seen that the reflection is dependent only upon the - A(MLOSS) a d that the transmitted wave f = fi + ,gi ch:nge in the minor loss before and after the passage of the wave. I The transmitted wave is equal to the approaching wave plus 2 T I ) A(MLOSS) half the change in the minor loss.

ear The change in the minor loss is easily evaluated when the 2

v;1ocity behind the approaching wave Ti s sero or very nearlyM re e mplicated situations can also be handled in the same am i This is so when instantaneous closure of the valve occurs For exarnple, consider the situation m which an F

manner, i?

"for end the pipeline is considered frictionless. For tids case wave approaches a minor loss from the right-hand side and an f

ro.

i vio wave approaches it from t ie left-hand side and both the waves --ane encounter the minor loss at the same time. It can be show (12) 4 (MLOSS) = -Kru'/2p P sh< the reflected wave on the right-hand sidef = f - 4(MLOSS)/2k and that the reflected wave on the left-hand side F, = F +.1 Hs Hence (13) f,-ggym'/2g (MLOSS)/2. p; Solution of Waterhammer Equations by the Method !? ad 1 O;I' F, = F, - gy,./2, (lo of Chafactefistics I 3 However, when the velocity behind the wave Fi s not zero, The partial differential equat.mns for waterhammer m a pme,g i equations (13) and (14) do not apply and fi ecomes a more takmg mto account fluid-friction effects, ha q4pg. b These equatmr.,

4 where [4, 5] and wdl be used d,rectly here.

complicated function of Fi. This relationship is found in the i are quasi-linear, hyperbolic partial differential equations, and rg. Rf following manner: since they have two real characteristics, t he method of characteru-h geg ties can be used for their solution. These equations are gitro

g "

Fa.' b=K as follows: h MLOSS, - K 2p 2p The condition of dynamic equilibrium: h DY' + y, DE.' (g) and D1 / U")' I MLOSS, = K h - K h bz' 4 D 2g g bt' bz' ~.

  • g

~g The condition of continuity for horizontal pipes: 0 Thenfore a' by' u < ;.a bH' + v' bH' A(MLOSS) = 29K lr t - Yu'l bt' bz' g bz' l ' >;: ..y u Tfansactions of the ASf'Eg 446 / suwe i,6s .w2

.h m a s ,.,y--- ,[" ~% U.' I 8' i /."=2 k 8Q s y maws , 3., 3 n A' / \\ / 5-8 n. c n ,,,,,,o,, "e

        • =a maa c

'"8""*"' gg v.= = c. - s.-- saw as ,,g,},,, C

a a

w i i.: ,, a Ma,. Ed a n, _, Fig. 2 The (n,r) piene for a unifeem pipe 4 \\ /.,eeg,vg p.,g 3 These equations are made dimensionless by the use of certain pi,, a s,g...,i, gio,,,,,, o,,3. ,,i w,, constants. The dimensionless parameters are given sm P' the characteristic equations (20) and (22). The other two are the g, y, and g. continuity and energy equations between Pi and Ps. The final 8, H = HO,'F = r0-L, 2L/s - solutions are given as follows: 1s shown by previous writers {4, 5], the partial diNerential Hr. - (Hn + H,)/2 - #(Y, - F,) t'e equations (17) and (18) can be replaced by four total differential LVOAt 1 K YO' y equations that describe two characteristics. By making the as- +g aD 2 2gHO sumption that F0 < a and writing # = a V0/2g#0, these equa. d9D tions can be written in finite-difference form by lategrating from

    1. ' = (H, + H,)/2 - #W, - F,)

yq point 0 to point 1 along each characteristic. t LFOAi l(f18), - (/F')a! .1 Kr08 e,

  • y Thus, along the Ce haracteristic c

I,ri' (26) N +# aD

' 2gNO (zi - ze) - 2(fi - f.) = 0 (19)

Q '~ and Fr = Fra = (F, + F,)/2 + (U, - H,)/4# { *. h LV068 1 1 KYO' Yh (Vi - F.) + (# - H.)/2# + LVOI ' ~ ) ~ I ~ 2aD 2# 2 2gNO e }" + ~- aD ions the C. characteristic It can be seen that the first three terms of equations (25) and a a h. (26) are identical with equation (23) for a uniform pipeline. The 'h (z - s.) + 2(f - f.) = 0 (21) last term is equal to half the dimensionless minor loss and g,g ' s and accounts for the reflection that takes place. The magnitude of the reflection is seen to be the same as indicated in equation (13). (Ya - F.) - (Hs - II.)/2# + 7pgaD (ff8)(in - t.) = 0 (22) Expefiniental Vefilication i 'a I 4 y ~ If the friction term is neglected in equations (20) and (22), it An experiment was devised to check the validity of the equa-i 8 ' ean be seen that these are the equations used in the graphical tions derived for conditions at a minor loss. The experiment was . ' analysis of waterhammer. organized in such a way that the reflected wave could be re- ,l The equations for the pressure and velocity at a point in a uni. corded and compared with the minor loss. In order to make the form pipe have been derived from equations (19) to (22) by pre-reflection large, the minor loss also had to be made as large as vious authors [4,5]. Let P be the point at which the pressure possible and, hence, a device was chosen with a very large loss j and velocity are to be determined. (See Fig. 2). Then it can be coefficient. This device was placed in the middle of a pipeline shown that and the pressure at different points was recorded and compared with the theoretical pressure-time history. N[ = ($ + U,)/2 - (Y, - r )# A schematic of the experimental setup is shown in Fig. 3. At 2 n "4" LFOAg one end of the pipeline is a compression chamber, half filled with ~ j )nl (23) water and with compressed air above it. The pressure of the air ui. * +# aD i ~'"? is always maintained at a constant level by means of a pressure Ir - (Ya + V,)/2 - (H, - H,)/4# regulator placed between the chamber and the compressor. grogg Twenty it from the pressure chamber, three closely spaced ( V'}' + (IV'}') (24) orifices are placed in the pipeline to produce a large loss. Steady. ,'aD state experiments conducted on the orifices determined the loss ilmilar equat' ens can be derived for the end points of a pipe-coeflicient to be 1150 over a Reynolds number range of 600-line by combining the appropriate characteristic with the par-10,000. This steady-state loss coefficient was used in the com-ticularcnd conditions. These equations will not be presented here puter program for the calculation of the theoretical pressures and velocities. f as they are not ofimmediate interest. However, conditions at the location of a minor loss in a uniform pipeline will be studied in A quick-acting solenoid valve, placed twenty ft from the Nail. ~ orifices, was used to produce the waterhammer waves. For use in the theoretical calculation of pressures and velocities in the pipe, it was necessary to determine su-h characteristics of the colindafy Conditions at a Minor Loss valve as rate and tune of closure and the variation of the hy. Consider two points Pi and P just upstream and downstream draulic resistance of the valve as it closed. The hydraulic re-d a nunos loss. The vclocity and pressure at these two points sistance of the valve was determined statically by measuring the can be dete: mined by solving four equations. Two of them are precsure drop acroes the valve as the sliding gate of the solenoid ) hornal of Bas!c Engineering suwe i,6s / m

'~ 4 4 ~, i !I; ping emi utms (r e,. l 1 18 l \\ , J.s. J-r, c { w y ~ a l l 9 j g o.. rieung s. warts.Mauuta entssung.rius etcostoiene av ee st/s k cast atoi, e cycts vo e Ls0 fps f } mssuas scaa i =. so au y*., e g no. n r rius scats.. sa secs no. ise rr 4 g Fle. S Waterhemmer pressure-time ruarding; case f(e): ene cycle I j k. e s a s. e no an m,,, Tout an uwstcs. F3e. 4. seistense of solenoid velve during closure 7 7y / y l 1 A

  1. '. _...ic.;/'.

valve was depressed in small steps. The rate and time of closure y - - , w/ '1 c' s were determined electronically. The hydraulic resistance of the ~ Z ~' .4-valve as a function of time is given in Fig. 4. It can be seen that ~ 1' N s ~ 4 r the valve closed in 12.0 millisec. The return-travel time of the - A ~ -Q wave,24/a, for this pipeline is 18.0 millisee.; hence, this was a ~ 4 ~ i came of rapid closure of the valve. l Twenty it downstream from the solenoid valve, the pipeline / was terminated by a gate valve. This valve was operated in such r'avas s. ars= asa raissuas na. asceana ar.. sse. i a way as to elevate the static pressure in the pipeline and keep . w,..,ca. ,e te m the velocity of the water low. This additional pipeline was felt misvas==s "a += *.a** necessary to prevent any reflections from downstream traveling e' past the solenoid valve while it was closing. In this way, the waterhemmer pressure 41me rwerdings cose 1(ab four sysi s Fig. 6 J pressure wave in the main pipeline was kept free of any ex. I traneous disturbances. The pressure transducer used was of the strain gage type and Disclissiollof Results was mounted so as to make the sensitive face tangential to the Inside of the pipe. The transduEer was connected to an Ellis The experimental and theoretical pressure wave fonna are For all en=e+. bridge arnplifier, which also provided the input for the Wheat-superimposed in Figs. 7 to 12 for comparison. In stone bridge in the transducer. The output of the amplifier was four cycles of the pressure wave have been reprod fed into p oscilloscope, and a polaroid camera mounted on the addition to this, pressure-time diagrams of one c been presented for the case of the pipeline with a minurlossin the 3 f cecilloscope recorded the wave pattern, Three pressures were recorded during each experiment: first, middle. In this way, it is possible to see the m the head HO causing flow in the pipeline; second, the static reflection in the one-cycle diagram and compare it with t pressure at that point under steady-flow conditions; and last, loss and also observ'e its influence on the decay o wave in the four-cycle diagrarn. l the waterhammer pressure in the pipe due to sudden closing of the i solenoid valve. The difference in the two static pressures is the It was the intention of the uthor to obtain a wave form whh l loss that occurs at the orifices. This can be compared to the as few disturbanees as possible so that the reflection of the could be noticed easily. It was for this reason that a pump wu l, wave reflected from the orifices. Typical recordings of such not used at the upstream end and a compression chamber with experiments are given in Figs. 5 and 6. compressed air was thought necessary, llowever, there were j The following exoeriments were conductedt l some disturbances that could not be eliminated. The way in )k 1 Pipeline with a minor loss in the middle which the solenoid valve closed produced one such disturl,ans i (a) Turbulent flow: pressure transducer at z' = 3L/4 = 30'. that appears in every cycle of the wave and was taken ini" (b) Turbulent flow: pressure transducer at s' = L = 40'. account in the theoretical program. 1 (c) I aminar flow: pressure transducer at z' = 3L/4 - 30'. It can be seen that, in every case, the experiment and theorv p T. agree in the first half of the first cycle. There le agreement hub lo 2 Straight pipeline in magnitude and form of the wave. In Figs. 7 and 9 the refle" (p (a) Turbulent flow: pressure transducer at z' = 3E/4 - 30'. tion from the minor loss can be seen and compared with tb It is only in this part of the diagram steady-state minor loss. that the theoretical program exactly depicts the experimental ' fe Computef Pfogfam , g 8' conditions. L'ains the equations derived from the method of characteristics When the pressure in the pipeline falls below the static hen 1 w d and the experimentally determined characteristics of the solenoid NO for the first time, air that was dissolved in the water at the p ir valve, the theoretical pressure and velocity at regular intervals statie head HO is liberated and begins to cushion the wave fronn i along the pipe were calculated as a function of time. A high-This effect can be seen in all of the remainian cycles by th" speed electronic computer (IBM 7000) was used for the arith. discrepancy which develops between the experimental airi metical integration of the waterhammer equations. The results n This situation is more pronounced in the ra* theoretical traces. from the computer program were plotted with the experimental S of the pipeline with a minor loss than in the straight pilwh"C tl I results for comparison. Tfansactions of the ASfnE h f .448 / Jung i,4s a

f. h 7 F-A f.* *, { .w p's , y,I - h s N, *1r c.. 6i o. , P. t'. - 400 ta m tum. . 4 ...a(* g.%

  • _, e _,.o.

! l 1_ MEAD CAusaNG FLDe NO 8 Stanc naa0 ""J"Q Y + y s B g I C' t 3 ,._____3_.- S 800

  • ~ ~ ~ ~ ~

e. r m _._.._._._._._._. 30 0 E 9. .c. - N* O 10 e to N 30 at e6 Test M M secs. ',..f

fd S.g.,

CW He) t-. Fig. 7 Waterhammer pressure.Hme diagroms esse 1(a): one syslo '.h ' N. e.,... o gr, 'I s u pt esassis,at , e.,,m g t e .... rue. a t .e M.;. 3,anc asas .h.Id 4 I ..Q t ( (f $,,9, son. r e N% { .h' '.A.._._7.g___.__.. ._..7.__.,r,,, g y.... g so. 4%. _... i p fb. 4 I I 41 l " h?id /. ";. see. T-k I I J Nx e

  • O

= v-t g ,.3...-*

A e

so se se ao ico no eso soo 9 3>g Tiut M M. secs. Og e s. ' e4. h Cast He) RIN ?

  • tt.

F*g. 8 Waterhammer pressure.Ilme diagrams case 1(a): four tysles B, I g%vrr< -l i s %,.;,w - 8 ~ eco s. <aen. awe. ,, m nsas caus-e Ptos no i .,A..J M. :- ' f&.H ':. @no, f ~ ) __ s,,n ,,..e I l y wye 3.~ g l beM ' B

~ k. (;,

g .._ _l._ v 6 1 soo. _ _. _. - M 8N ' ' 8 // I .,dn <..?a'_'. ._._? i A :q[;p@h. M*

  • v t

1.:q ; G g --...x 1 i@-.- 'o io e no n so u rikt et M.stCs. Cast its)

y. 4 h,....

.1 Fig. 9 Waterhemmer pressure. time diagrams esse 1(b): one sysle . This is so because the constriction at the orifices produces a far bubbles. Third, the cay y occurs only after the first drop lova pressure than in the case of the straight pipeline and, conse. in pressure below static pressure. The discrepancy cannot be l ' quently, more dissolved air is set free. attributed to an incorrect value of the steady. state loss coefficient ' There are several reasons to make one believe that the dif-or to the fact that the unsteady loss coefficient was not used, '9 arence between the theoretical and experimental curves was due since the same type of discrepancy exists in the case of the f ? ' to su liberation alone. First, the water used in the experiment straight pipeline. Finally, the nature of the discrepancy itself M. was drawn from the sump in the laboratory. This water is far leads one to recognize it as one of gas liberation. In nearly all frorn pure, as unany additives are added for various purposes, the cases, gas bubbles act as a spring that cush. ions the pressure such as rust. preventives and algae inhibitors. These com. change, allowing the maximum pressure to be reached only after pounds, like chlorox and dilute hydrochloric acid, when dis. a me time. Oed in water introduce games like chlorine, making the water For the case of laminar flow, it can be seen that the calculated "'s susceptible to gas liberation when subject to low pressures. Secoed, at the end of each experiment, bleeding of the pipeline at and actual wave speeds do not differ by more than 1 percent. 1 the or13ce and at the valve would ludicate the presence of small. Thisis in contrast to the experimental results reported in reference !ssfiial.ol Basic Engineefing su ur i,6s / 449

jpg e b a j ~ taseastsa6 ' ". D eq snessessava6 sas causme psew me ese. 'f stars acae de m _. eb / C*' ^ r~ S., ph i s ~ l g J_ y% At _ _i 08- ..-.d..; .I. ..I., .d. .'F' qss

  • m-m.

i t A. c i i~... n.. liu j r.. .T! ,,e p y m. qw 33 h e a 0 e 0 01 Tius ne M. srcs. tast tibb Waterhommer pressure-elmo disgram; sese 1(bh fewe sycles s Re Fig.10 h ) j'r - EsPtamagura6 lI tutoutca6 gao, .wq=J,] _ __ _ _ a ~. rs .e I ,p f7 T'~' r^',' [ Ili Ja,. i I l ! b-E 5 l ' U 'I"' h so I 6 6 h --[ ) k b .( q u, vol -.-m--. e g..-

q Q

s i E ? i I .a { I I l N ' l 6 I I 1~ w E k E. .s. 0, go ao se ao too tuo 840 fee .t

  • ,gjp 7 tut se W sECa wol

. Cast t(cl jng Waterhammer pressere-rime 4*g sm; ceta 1(c)

  • mest Fig.11 i the tartsw*wta6

.,,aa' h clu< lyFOg'., : ;pa, ......... var 0atrica' }. 4 - -. ~. - 8'E 60 C8#8'a8 FLO" "O -~.. p.* %'g.}% .oo. 7 I g suit j $ 300 g 'a .: *he as l,00 c- -. - :u-, - } a E e Jr { eco id !J .d ,3 W he .y 5 .4 s 9 to 40 60 00 60 0 53 0

  1. 40 14 0

); TIME si M 8tC3 CASE tis) an d Waterhemmer pressure-time diagrams cose 2(e) 1 t Fig.12 Conc lUslOrs Thn magnitude of the pressure wavT also agrees fairly well Whenever a waterhammer wave encounters a device enus I 15]. with theoretical results, despite the assumption of uniform The me Ilowever, st as not possi-a sharp energy loss, a reflection is sent back from it.

3. The v:locity distribution over the pipe area.

ble to see the reflection from the minor loss because of the small rutude of th.is reflect. ion is equal to .i(.\\ FLOSS)/2. The nu magnitude of the !*s. A more elaborate theoretical study of transmitted on is equal to the value of the approaching man w is presented by Itouleau (6}. plus MESS)A Iforever, in the experimental verification of his theory, he usedThe method of characteristics is a very simple and efficient m a) {,- waterbarnmer in laminar flo oil flowing in a pipelina, and because of the volatife constituents of to provide pstrticular solutions for the waterhammer ecsisti the <>il, the experimental and theoretical truca did not match including friction effects. When setting up a computer r***'"' well. Tfansactions of th ASME

1

.,3.u,

3, v ec 1 ) D '( [,' C Continuity condi-constants cnd current values of the vari 1bles. ,@. oint waterhammer in c pipeline trith o minor loss b it, conditions ] - at the minor loss must be treated as a boundary condition. The tions at the minorloss yields e ~ l V equations at the boundary condition verify the conclusions about Fr/s = Fr.As (32)

  • the magnitude of the reflected wave, reached earlier. It is suf.
  • '*witly accurate to use the steady-state loss coeflicients of the-and the minor loss equation gives l

1, at least for the case of instantaneous and rapid gate N ces, even though waterhammer is a markedly unsteady g,, u g,, 4 g p,,s (33) phenomenon. 2pHO - or Acknowledgments This work was carried out under the sponsorship of the Na-Hr, = Hr, + KC.Fr,' (34) tional Science Foundation Grant GP-340 at the University of The set of s.multaneou,. equat. ions (2S), (29), (32), and (34) can ~T!ichigm. be solved explicitly for one of the veh. cities and then for each of The author wisha to express his gratitude to Prof. V. L. the other three unknowns. The solutmn of the quadratic equa-Streeter for suggesting this problem,' for his support and en-uou resultmg from a comb,mation of these equations is couragement. The author thanks the University of Michigan j, Computing Center for the use of the IBM 7000 computer. 3 Vr, = KCs Si + As$s ~' References A,#[/ + 2KC.(Ca& + C.S:)" AJ (35) l 1 J. Parmakian. Water-Hamener Analysis, Pantice Hall Inc., 4 44 ). New York. N. Y.,1955. 2 O. R. Rich. Hydraulic Transients. SteGraw-Hill Book Com- ", pany, Inc., New York. N. Y.. first edition.1951. The form of equations (2S) and (20)is identical to that used at 3 Sympoelum on Water-Emmer. ASaf E ASCE.1933 any boundary condition. Therefore, this set of equations la 4 T. L. Streeter " Valve Stroking to Control Water Hammer '. somewhat more dem.rable than equat. ions (25), (26), and (27), ,{ Jevenal of the Hydraulics Dirision. Proc. of the ASCE vol. 89. no. HY2. nf arch.1963. which are written for a minor loss in a constant diameter 8a e V. L. Streeter and Lal Chintu. " Water. Hammer Analysis In-pipe. Identical results would be obtained using either set of P" cluding Fluid Friction." ASCE Proc., Paper 3135. 31sy,1962. equations for a case of minor lors in a pipe of uniform character. rz ? e W. T. Rouleau, ' Pressure Surges in Pipelines Carrying Viscous ist,cs. i Liquida.** Jocawat.or Basic Ewcrwxrarwa TaANa. ASSIE, Series D. One of the assumptions in the development of the geners! d g. vot 83,1960 pp. 912-930. waterhammer equations (1) and (2) is that the velocity head is a. negligible when compared with the pressure changes. Thus, the k

  • '*1 h'*d ""'d i" 'h"9"*'i "' ' ' 'h' hY '*" i' 5'*d' li"*

d 7" DISCUSSION rather than the energy grade line. Using the characteristics e C,g,gjjgt solution on the computer, it is not necessary to make this assump-e ti n. In m st practical situations, the assumption leads to no Q. l-The author has made a significant contribution to the literature appreciable error; however, if a case is being treated m which g; ith this presentation of theory and experiment on the topic of min r I s8es are sismScant, it also may be true that the velocity p, minor losses in pipe lines subjected to waterhammer. The writer head should be considered. ' could like to elaborate on the conclusions of the paper by presect-Two examples follow to illustrate the treatment of boundary 1 Ing the equations for the computer nolution in a different and c nditions with the velocity head taken into consideration. The more convenient form. Supplemental in!ornation concerning firSt enniple is an entrance condition with little or no minor loss; k the inclusion of the velocity bead in the computations also is in, the second is an abrupt expansion with the appmpriate minorloss I cluded herein. included. ,f;" In the solution of a practical problem, the conditions at a minor ..g loss are treated as a boundary condition. As pointed out by the pathor, equations (23) and (24) can be combined, yielding a form ,. hy,. p suitable for use at boundary conditions. They can be written in rc=cnov snaot uw - the following dimensionless form. GronInuuc anaos unt y Fr = Cs - Hr. (28) no w', E a ( 1 Yr, = C + 2# Hr. (29) r dhere - - ) ris. 3 smeeth.nere......dm. C = F, + y, g,vo,,3,(fps), (30) .A ciDi 'P("'a s'I/' cnd femenov snaoc t.s.c se,f + cnvonEuc onaos unc C. = F, ,Hs L VO,.St (fFs), (31) e .S a2D ae s The notationis the same as that of the authors, with the subscripts ( ' tad 2 referring to the locations upstream and downstream of a T ior loss or boundary condition. The magnitude of C. and C. ~~ Qbe evaluated during each computational period from system PIPE 3 ' Research Associate Civil Engineering Dept University of h!ichi. saa. Ann Arbor.Stich. Fig. 84 sudden empenslen Jeafsal of Basic Engineefing' au we i 96s / 451

p,Q ~ r (39)

  1. r, + C Fr,8 - #r, + Ca r'8 + KU.Fr,' -

h.F The boundary condition shown in Fig.13, illustrating a smooth ~ W entrance condition, can be described by equation (29) and an ex-These four equations can be solved easily for the four unknowns(Y.TG pression of the energy relationship at the entrance: at the junction. The provision to handle a flow reversal also enn HO = ##' + (M (36) be included without difficulty. A similar approach can be used to obtain a solution including 2g the velocity head at any boundary condition or minorloss. .'d+/ In dimenuionless form, this equation becomes ~ 9 (37) y

  1. r = 1 - C.Fr*

Auth0!'s Cl050f 6 b The abrupt expansion shown in Fig.14 can be described by The author w,ahes to thank alr. E. B. Wylie for his discu=-ion. equations (2S), (29), (32), and the Bernoulli equation, inc!nding The velocity head term could easily be handled ir the equation = Like the minor yFb as shown by him in equai, ions (37) and (39). N the minorloss, loss terms the velocity-head terms are taken into account withou , + (Pr,')* + K(Fr/)* approximation or linearization, and this fact is one of 11 e s f[d(, (38) , + (Fr/)8 ~ Nf8 N#5 2p 2g 2g vantages of the method of characteristics. In dimensionless form, the last equation becomes >W 5 cw p .,? I N w' x' m: i n i J j I i ,8 .. j n ll H di f ~ -Q < {i o -Q' ~

m.,.

^ 4i 'b'. -=. f l e. 1 i ) 4 4' l 1' i i e i F i 1) Transactions of the ASME L;- lU 452 / J u a e i 9 6 s P L

SEhT BY:ESL Info Sves

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ESL-, 70351502$51#3/13 & ~' g-j pyg, 1 l Q E. P y g.y thf :3 7 /f 3-W~5-l NOT UBRARY COPY l l On the Flow of a Compres;ible Fluid through Orifices By D. A Jobsoit* By making certain beele assumptione, the autho has detesumined a theoretkal expression for the contreccon coedicient, C, apptopriate to en ordh when aansmitting a compressible Suid, either i above or below the critical pressure ratio, provider that the comepending value for isc ,. ible Aow, Ca be known. INTsoDUCTION norsie (which always Sows full) as dierb from an orifice j When c Suld is discharged through a convergent acazle, me. (through which contraction of the jet occure), was,noted and ditions across the exit secnon are scnerally assuined, with litik Y c'plained by Stenson (1926 who nadacated that J ermor, Qo be sensibly uniform. The sait velocity may therefore be thsont of the jet me be expected to in siac as the predhi frosa one-disa aaaaa I smanidersticas, thus enebing back preamsele the masse Sow to be deternuned. This approachie not appropriate By postukting, a,s adds' tion, that ,bility effects may to the exit conditions across an ori6ce, asoce the str*==h=== are be neglected la the apptoech meta ao orifice, quearirative d I geoeselly still contracting, so that the flow pattern le essentially caprmions for these phea=*a= are dada ~d Probably the two-or thsee dha*a aaalia characterla this If,however, most le of the above sesumpoons is thst conocrmag aaaAleba mey be sHumed to be Acerly acrose somBC aceoes the throat for supercrkscal Sows, enace the -di l other esction, such as at a taene comraers or at a throat, the outermost layer must be at the back pressure. However, the ' he of essesy, ran=entum, and continuky may enable curvature of the streamlines, which will tie most marked towards the eine of the jet and the conditions across it to be simp)y the edge of the ist,lasplies a transverse pressure gradicat la this determined at this section. region. This will give rise to a pressure distribution acrose the j Buch principles, when applied to the discharge of sa la. throat that is somewhat as indicated in Tag.1.

- cosapeessible Suid through a Bords
  • , for example, 3

show that,if friction and gravity iney be , the let has a Apeasaatttocut paa-sensues monas.: vnoose contraction cocihcient of 05. By the of shnilar l 7'"' reasommy it will be shown that a si,. - 'n foe the 4 manecnoa==%. c, any w daemined, aperopeine w = t esiSce when i,- =: ileg a ---- '" Suid esther above or 4 below the critical pneeure ratio, o'serided that tin corneponding j vehas for inoatapmeihie now, Q, he known. 'D .m j

d

- _ wkh such c A a.s as those of Staason ~"C i (1926)t,Schiller 1933) and Perry (1949) indicates that the mese 1 sow through a sh(arp-edged orisce may be predicted to wkhin a s i e- -g one type of oriace,pressem ratio. These eests were carried out on Q ".iocu, fewpercent at any ekher air or superbested mesa, mureo d. ,,%,uo,, ) emu mwn hist the theory le more in cherecter, sad fearther com-parisons with teste are needed, menues aceoss an vs oove.caen an i The pree.st analysie <= enders tases la which the effect of A i Idction is ersall, greWy and heet transfer may be neglected, and Ms.1. Throat Paad**iaan for Supercritical Flows oosh,4 rhanges of state may be reptesented over the range j u by a law of the typc' It is beyond the scope of a one-damensional tnatment to

  1. = mamar

'"Ell"wm be -~ be. d.e maican.eesu,e. msuch r*A=-are and,in the w m.ho.e.-=pm., ww.h an e.o,ted in most nos.ie and he adece problems, are generally found la practice to give adequate tror satroduced by hu.hlar star equation of motion on this mooeracy, divided imo twowithout makang the analysis czoenolvely indious. The sais will be largely Nfeet by tta coeroependans so 1 theory a

in the Srst, the pressure ratio be velocky is sonic scross the whole section, whereas in practice acmes the criaceis to he grosser than the critscal value,
must be ;-

'T towards the edge of the jct,if the effect of to that the Sows are Sow is - J a subsoniciin the second, choked incosity is negligible. Fince the latter factor must tend to - "d'fhe foranct case semanes a a===r con-caerd the outer layers of the jet, some uncertainty,in any event, ditione aeroes the swee swerrects which the jet near be espected miste ma-niat = dad-in this regson. tidsmamely to form, and the letter case makes e aisaller asseseption conneeniusthe Arettiment whe esoniccondhinnearepostulated. j Neresien. h contraczios M is deemed as che ratio or the above '1 Projecnod area of ori6ce* Sainianasa ar-se to the assainel projeceed area of the crieer to the jet Conoscoed area of ja.

    • " *ammem may) be since at the critkal pressure ratio the

? Contraa'an==meie=r of on6ce. daad wkh the erst thsoat, the two i en Maa m, mea,a, of oriSce for incomprenible Sow * ,deAaltions are aa p.aa* fee the borderlina case dividing the 4 V,o cosity corarcdon facsor. a* s/ sis. is it a peria e wo* aedag dds stage d t c "g,_, sapercrincel e,enditions the now le choked,in so far defe ca reservoir w Force defect m. mci,ar are sonic at the Aret throat, the sness Sow does not f Gasvhetiommt ecomieration. of tie downstream pressure. 'nais charoc. .T Total h>desube hand la reservoir. 4 i I h benma the behaviour of the Aow through a J 'nissestical mese-sow memed== eof orirwe. 19hThe MS.ef this paper wee realved et the festitwice en 6th Auguse ., Theorvaant annee-Sow coedldent of noaale.

  • nmoevocal sness now (= IF/g),

o A_ _. A Laboronary, Royal Evel Collrse. Osorneich. t. Actual snese now. a I ^* -Y " i ' list of seicnaues h given la Appendaa 11. lades,ofleeanopic==paaalaa (= y for a perfect gas). e 1

8-30-94;7
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ESL-+ 703519b225i#4/13 SENT BY:ESL Info Sycs

  • 1 768 ON THE FLOW OF A ;OMPRE53!B!.! FLUID THROUGH ORIFICES pe P essure in reservoir.

of its e energy. The tailing-of of pressure due.to this I p Bad pressure, sink adds to the driving force as ladir=e=d, akanriar the Freeme ratio p/pe., --rum Sun the 6-~ea irs. r r, Critica1 pressure reno. Por an oriace the velocacy will be comparatively antall Velocity through contracted area. along the seservoir walls, and hemos the esset of compressibility m on the Sow pattern actueDy within the reservoet may bc==W Critical velocity, / time. This assumption will be lacreasingly valid as the ori5ce shape i u. Descharge, weight W 4 r., Dacharme through _...,~ ding aa-L appseeches that of a norda a._.c % from a nozzle-aike fann, se, Specine weight in reservoir. If, however, the eSect of compreseElityon the force defect sosy y Ratio of speca$c hests, c,/c.. he neglected,it should be possible to express it in terms of a p Density at contracted area. Newtonima force r-hr which is a constant. That is, it pe Density la reservoer (= og/t), the Suid sney be assumed seambly incompressible within the reservoie, the force defect F, should be a unsque funcoca of the f ty pe, b stu & &, a MW Q foe i avaCRITICAL PLOW AMD TH: FORCE DIFsCT b "**E*** " cogpyICABNT espresse'd in terms of the'rames Sow,s. This presuppoems that the Suppose that a compecesible fluid iseuce frosa i reservoir in streamilne pattern withis the reservoir is not spadanad to any which mnditens are steady, through an oriEce f aving a pro-appreciable extent by chanses in the boundary randreiaan down-jected area. A, normal to the amis of the Guid jet, a i fadir=ted in stream, secociated with, for====ple, compreenbelay eSects on Fig. 2a. By equating the resultant force acting on the fluid the form of the emergent jet. If theos assumpoone are valld: F " M5e As pe) e a {m' J Dimensional considerations suggest that the relation between lA theoc variables must be of the form I / u s j1 F = l g...... (2) ~ - ~ ~ - - 1 where /is e dieneastan1*ss coesident having a value -f-;="== I I only on the form of the orifice. It will subsequently be referred \\ to es the force defect coef5cient. Since f may be espected to have -} p the same value, whatever the Mach number of the Sow its vahas 4 may be inferred from low-speed or incompressible. flow theory ~ or tests. If the auntraction MA' of the orifum is denoted by Q for incompressible flow, the -.,~ ding volumetric ~ C-I now rate: Q = 44u....... (3) j \\ )poact,# L where w it the velocity of the entracted jet which, neslecting 1 friction,is given by: i .g u = V(2sH) - 'a

  • I1is e tion, when re-espressed in terms of the notauon of the PSPue s For subcritical gu.

"N a= ..... (4) 8b----4 In practice, it is found that the resulting expression foe the I U l discharge requires factoring by a oo.cnned coelhcient c(velocity $c th [i y slig y than unhy rhmuchout the s.cse uent amlysis of am,ressue - the t ( ~ effect of friction will b,e neglected. it being r ca Ad \\ k 7I that the t theoretical mass flow snay require a correction foe the efect of vncoury. This will be to be of the same order as the velocity coefEcient foe mpressible now. l _$q } Equation (3) may be re.espressed in terms of the mass Sow as : j k = poQAu t ' and,when the theoretical erpression for the velocity la substituted g

  1. y into this from equation (4), it becomes :

poacs 4 - p,qAfp"0 l These eaust, % enable the Arst and last terms in the equaden i f. e ~#a of motion (1) to t e -,.. d foe incompressible flow as: 6 6 For supercritical candir-1 A2 ~ "2T*M Fig. 2. Forces on Control Volume 1 s8 contained within the boundaries AA and BB to the cor- "" "' *d,

  • M responding Sux of momentum through them, a a equation of o,,n=W thue subrist=eian=; tW with that foi @

anotion is stained: force defect given by equation (2),f is found to be uniquely .. (1) related to Q: (pe-p)A+F - m.u..

  • "'he arst term represents the direct driving f6 ne suces the 1

st At 1 si orince arca, and the second term the integral e l the resolved fip p+f.p " 5,* p I-components of the defects of pressere along tk r walls of the reservoir surroundes the aperture. This latter f rce defeer, F, that is, p.., 3 3 is associated with the velocity of the Suid as it t pproaches the j,, D _,2D * *

  • g<

j ori5ce, the incrcsee of kinetic emergy occurring t the expenec g ( l l

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ESL-* 703519ON2b$h5/13 l l [ l ON THE FLOW OF A COMPRE$t15LE FLUID THROUGH OR1FICE5 769 Sinoe the la==h " '- contractaan==8deat lies consrection <= mdear It reinaias to dewtop an espremior. betwena the values for a Bords "- ' - for which C# = 05. " ; with equation (11), to cover those cases in which and for a nossic, for wkkh Q =%,that the force defect the loisi hinch number suschte unity la the jet. Such Bows,

==ad-ar willlie between the limits 0<f<05.Re /is sem which will be ederred an as supercritical, are investigated la the ser a Bords --f,

  • A how been aaddp=aad basa nest - '==i phyeleet oossidseations, since this case the niossey the surascatitCAL Ft,0Ws e of the mesvoir tends m are mywheat (D 1t has been assumed abow that the Sow was both swersible ana -a be a+=d. aa phed that im heat k

it has est.ey..i. e,mr, be arr,a forward - '2 as - w*h emi nas " ' - Aow enough ee er:Sce. suppost this asemapuas for high-egoed comptessible Bows.,1f, analysis In order to use the equation of manion to determine the dis-howm, de h pense se less than a arrtana enucal charge for conspressible Sows in terms of, which will now be pmouse, p. so that se amm asey become supersome, the 88' by a theid ad,a,,ac as,an, ion, se 3,im. yen,a, en,eal,meea,f,n,e -s nis.d.e.oseim oei.e.w. - yf -..a 4 appppnme, and this is, for subcedeal news. recompeusmus aAer over. cap-dtne downstream of the Arst throat. Evidence of this alseraative espansaan and casopression 2n f. ". ~ !, is provided by acidieren and " ". fght -A ha of Suld ."q. 1-r* I ". (6) jets, and thsee wave p6=na=== a be fastseen from s= da thaaretkal canalderadoes4 tor esseple, E anson 1926). where r denotes the pressure ratio, p/h, and n is the laden which It le sharefore preferable to analyse supercritical Bows la terms best sepsusents isentsvpic - p=adaa= by a law of the tFpe

  • of a assurance seeana across the Aret throat, rather thaa one pfps constaat acsoes the section at widch abe Suid has espanded to the down-be ideaaand wkh same pmem. This corresponds m ee use of ee emot as the i

Por a asse-perfect y ofhe Mon h h condidons asst

==i ly equetions of' by C ee man u tenas of the boundaries t.aouen wit there. fore bc==*=h6hed

  • =,CA.

=m d ia ns..ne - d.uioory serose m im are I "~ I' considered so have vahaes . _ ; to sonic conditions. If 2n = CA papa,-1. rUa. I 1-r ' I (7) thses cdtical values are denoted by sums e the equation of 4 4 tuotionis: By substituting the value C = 1 the w., widely used p,A-pA(1-C)-ACA+F = sim, l espeemium for a riozzle is obtained. Th sussests that the neIch-hand side of this equadon consists of the direct driving corsesponding mass-Sow m md=r for a nozzle, K., may be form aaem de eres of ee ortfice, sosemer wis me force defect, i seed as a convealeist substitution: as before. In ele mee, however, shbough ec ibece on boundary i f a.:P AAis C M the back psecourt across the eres of the oci6ce 2ss 1 K. - -rW=l 1-r* I (3) A,is now considesed to coasset of two parts, the critical pressure, "-I k 4 g, beiss====~8 o act across the threat area (e = CA) sad the t Hence, for a nozzle: so act over the aree : - 8ai-- downstroom pressure,p,diria== the espressione for t5s thmat i ds = K.AV(pops) por 2'=* ma and for en orince niadty and sness Sow are obtained by subsututing the critical j $ = CK.AV(pape)....... (9) wheee e inao eqususa @) and G), a for a ggg The velocity thmuch the contracted ares of the ist is,la either 7 i case, given by equation (6) which on substituting foe K, seduces 2. 1tg 2n 1 p j 1-r,7 l 4.g l to: Me " j w = hpo. U ... (10) s - CA pre. g. r,8/=. ll-r, "" Il 2s I l Hence, for subcritical compressible sowe, the last teren in the /i j#'P" ~'Ei)"M j equados of motion (1) for an orbe ba==*=: 2 " Ud i C(K.)'Ah l f j rL/= 4 { The force defect tasy similarly be espmeed la terms of K as : since r, = h A )

  1. "I* N"#

If the substitution K. is introduoed as before to denote the j = /C'(Ka)'Ah mass-sow==mdme of the corresponding nogale which, being so that the equenom of motion gives, with those subetitutions, a -hah =4 han the value: quadsseic espacesare for Ct r .-t 1 a+ 1 0-r) Ape +fC2(K )Mpe - 1 M I ~ 02) ee nir u, and A boom: m_ 1 ~ "I" =0 g (l3) r % s,e r. re it 4 = CK.,Av(sva).... (14) C= y,ig,1-qt-(2, ump (i.-r)f. 01) These equenone are analogous to equeuans (9) and (10) i akhousin la this case r is repleosd by the conecant value r, so that Simot it will be acted that depends only on r,and n. the K is now indepeadcat of the pressure ratio r. The flux of Contraction coeE&st is thus erardned for @.L1 Sows es a function of r, se, andf the latter being found from the ,um and the fora defect sasy now be expressed as: . ;ble-Sow ea =a-t_ --md--*; Q. das, = C(K.FAh The theoretical mass Opw faay then be deduced bv factorieg f*N" the Salm-Vennot and Wanael equenon for the teses dow by the F = /C2(K.) tap, ~ 4

SENT BY:ESL ltifo Sycs

8-30-34;7
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~d35190225i#'6/13 7 t no CN THE FLOW OF A ::OMPRES$1BLE FLUID THROUGH ORIFICES so that the equation of motion for choked dows e : obtamed in this ney and plotted in Fig. 4. They indicare iRt, 8 C( as the back pressure is reduced, the jet expands prosfussiveif, (1-r)Ah+(r-r,)CAh+/CNK.)8Ah = K,,)',Ah thus a cantimaous incnase in the Sow rete.This '8n continues the n.,-...A ! Aow range at n ' Ilia assin yicids a quadratic espression for C; decrusing rase) so that even though the Sow is choked, in the 3 1_ sense that conditions at the throat sernaise constant, the sness j /C8 T3-1 +(e,_r)r,t> C+g, i nO Sow continues to increase. This fact is emphasized by de6ning (g y ' a theoredcal mass-dow medicient for an oriEce K such that: C = 2frAalt + '(K.)2 " " A (Ape) N 1 Curves showing the theoretical variation of K with presrure 34(r,-r)rM _ 2r#=)8(1 'rl/ '

  • * (15) tatio how been ploued.in Fig. 3, which inchades the cor.

(K.)2 (Ke)8 respondas curve for a noule (K.). -* of the crince This empreseson, which doestmines the consrcreion a-m,** e., of a choked orince,is seen to reduce to equation (11) when the Q;Iighp,g overall pressure ratio, r, is equal to the critical isloe, r The \\ two equations toestber deterrane the contractiin ea=mnsar w \\ ih.# the whoie,ange of

  • mdo... The hesic m

Y8., assuropeian is that the essce of on the 8erce g i defect alces the reservoir walls sney be asylaredi ; this saay be \\ espected to be valid so long as the osince does not appensch a i noexle form. Por a Borda mouth-the force <$sisetisaeso N A \\. m and in this case the theory is, in sense, esne:. For auch a W es \\y,,, \\ k mouth it is convenient to return to the origdmal quadretie 5, es for C, since theos reduce to timent equations in the Iting case whcu f is sero. For a Borde suoiah-the expressions for its cmocraction,,.me=at, Co, thur to: g e.e 'g -- Co r8ta(1-r) . (16) g - gg,), for subcritical flows, ther is, when r,<r<!, and ,3 I_ C' " rea(r,-(r)+r) l rMa1-III) d \\ (K.)3 i O \\ for choked flows, that ja, when 0<r <r 8 si By su,bstituting the chr&~t value of K., from q=r><= (12) E in equatmo (17), the latter may be czpressed more samply, after some algebrale manipulation, m: Co " r,(1+s)-r 4 i This theoretical expresnan presupposes that the mouth-piece is short enough for the emergent jet to clear the outerlip of the l crisce,ahhough.ofcourse notsoshortthatthereis oappreciable ~ approach velocity along the reservoir well. e ,e 3 eg es es 8s rsI15URE RAtlQ, s Fig. 3. Theretiant hSow Coedicient Pioned as a i suMMAny AND TassomaticA1. CURVss Functaan of Pressure Ratio for a== 14 i The purpose of the paper is to present a methxi of Anding the discharge 'through any ori6ce, when transmitting a corn-pressib!c Buid, either above or below the critical srusum ratio. t It is presumed that the incumpressible-Sou. contraction cordicient, C,is known either freen theory or esperimenti for o j example, Q = 0 5 for a Bords.M.";;he and G + 2) gg

== i

== 0 611 for a plane slit. From Cv the strength of tbe s' effect z along the reservoir walle may be infernd. Its intensity determines f the snagnitude of a force defect M9 a,;f, such that: g \\ 1 1 u /ua 1" x w l On the assumption that the Sow pattern wrthinthe reservoir V is very little innuenced by the efect of compnosibility,fenablee 8 theoretical rzpressions for the contraction ~+P-L R, C, to be f e., \\ 8 l 4 deduced for compressible Sows, both above and bebw the critical w pressure ratio. This cretraction evwmele=* enables the saaes on Sow to be determined from the expresuon: l ds - C K.AV(pape) e4 For a nozzle C = 1 and hence K. is the wide'i used men. Sow cocindent for a,nossle, based on the Sala.Venant and .- H Wansel equation. 'Ihis mass-6ow coediciene is ein we dotted in i Fig.3 for the typicalcase of a== 14.The contract m coedicient et o ea H es H 64 j atany suretodosasybefoundby Art deterinining /and sad tuting their values into either ;quation (11) estisunt sato e J 4 or equataca (15) accmeding to whether r is greate or less then Fig. 4. Theoretical Contraction Coef6cient Plotted as a the critical value e,. Some N,- Gwe curvi 3 have been Function of Pressure Ratio for n = 14 ) \\ s l k 1

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ESL-* 703 519 0225;# 7/13 -i l ON THE FLOW OF A COMPRESS (BLE FLUID THROUGH ORIFICES 771 and nosde curves indicates the d.e.e..dtic ddlerence between an ovesall carrection for viscossey effects, asther than dash as 5e and & the theoretical curves heve timelt behaviour, the ~ of the forince

  • emaatple d a veloczy coedEcient. In without the factor C, = 0982. It is been drawn, both with amasked the vanutiuo of with r. A num '

s!venia dis 1. seen that the 2 ' i resuhs in each instance lie between the two cueves, making 1===ric as to whether such a correction is worth a either lastance the agreement COMPARISON WITM TM5 NODOoAAFM s0LUTION appears marienwemy for snoet pur emnamernng either air or seensa. poses for sharp-edged or fices FOR A PLANS 2L17 nisosetical ranks for incomprasible now thavugh ori6ces j aan under certain *6'- be estendrd to comptesable Sows, es ^ using a hadograph n=rhewi diowarth 1953). The resulte shown '.lgh a la Table 1 for the contraction coeficsent of a dit la a plane well j l TAes,s 1. CoNTsActioN Courrtenorr or St.sv tw Ptmes M Watt. r C 1900 & 611 &932 0623 0 567 & 636 0 805 4 650 0747 &665 ,. 3 0-092 0481 0639 &699 8 0 590 0 717 r 0543 E738 k l 0 528 0745 = \\ have been computed, using tables due to Persuson and Ushthill k + (1947), i 5, ',' g { The results in Table I are appropriate to y = 14 and the nethod apphes to subsonic flows only >0328). i I The corresponding values obtained the inethod outhned in the paper are se given in Table 2, at roughly comparable e TAbt.B 2. CopassroNDING VAINEs Pos con 13ACr1005 f e i I l i i i C mynenorr OsTA!Nea aT MsTMoo DUrt.DED IN THs PArsu 5 s6 k %e q h ,.3 b 100 & 611 &932 &622 -0001 4 &a65 0 635 -.0001 2 Oa05 4 64s -0 002 2 1 6745 0463 -0402 0 690 0 679 -0 002 e,e 0640 0696 -0403 l 0 590 & 714 -0903 1 i 0545 0 734 -O 004 4 0 528 0741 -0006 i 04 l values of r, together with the ' error' tneting the hodograph octuuas se 'esact'. It con be seen that the present snethod slightir underestimates a 1 the anses the d 6 , increasing progresolvel 5 per czat7the entical pressure rudo, y frasa sero so about 4i CIAsFARtsDN WITM BIF5m! MENT, AND CONCLUS10M8 eI i the teost e=h==tively tested ort 6ce le that of the I type. The ruoults of Perry.(1949 and Schiller 1 such an ociace to tranendt nar and)stesse, (19 ,have been l j -.ueLotted la Fig. Se and 6. Since the j .e - r is.n de,ed to be 0 i. e. e., e.e e.s e.e t.e each instance, this corres to a velocity =amei=w, as need emessuna mata r ue

11. I would 4 5. hk Rm correction factor C, for the th curves the et pressure ratios. The actual sness Sow for a
Said, a Por a = 14 1

m., then predicted by: -o- % tears, unies air. (m = l 4, Q = w/(w+2)). UM = C,CKn - C,K & Por n = t 3. -o-sdduer's tests, using steeni. It is perhaps preferable to consider C, la the general case as --- - neery (a = l 3, q = w/(w+2)). 4

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ESL-. 703 519 0225;S 8/13 772 CN THE FLCW CF A COMPRESSIBLE FLUID THROUGH ORIPICES Further of the theory wkh espcriment are Neglecting any velocity of approach ce ' carry over' effects, the required to esta the full range over wlitek the assumptions nonic mass-$ow cortilcicot is, for subcritical dows : 1 of the theory are valid. Since the neglect of coinpressdshey .: 3 within the reservoir ename be when the arda tends g,. _2n_ 3,,7 to a nonle-like form, en upper of C, = &7 fo' applicecon a-1 of the theory is tentatively suggested. For .L ;.". eseem at the critical pressure ratio (r r, j = 0 546, af a = l 3): l acawowtsnosusur Km = 0 667 The author's thanks are due to the staff of the Atrodynamics For values of r less than &546. Kn semains constant at this Department of the Royal Aarcrsh L*=Miakmat Panaborough, value. I ~ for their advice and criJcises during the preparstion of the Suppose that the discharge arrespondang to a pressure rado paper. It is published by pernues,an of the Admarilty, but the of 0 546 is required. From equation (a) (or equation (12)): views espassed att personal to the author. 1 (2r81=)2(1-r)f[ l i j ,2frt/=. t (K )2 J, 1 ,x = U 2 x&2M xt546 APPBNDIX I* I-(2 x0 'x 0 454 x 0278 i womsm1 CAL 31AMFts Many engineers prefer to work in terms of the weight flow. F. I" 04672 J' l and speaac weight, w, rather than the mass Sow, d, and density, p, to which they are plated by: - 2 865(1 - 0,744] l - 4 732 y,g l ,, g The equivalent noule discharge may be obtamed from: From equadon (9) the discharge in terms of F lecoascs: Fm " Kw'4Y(iflFs) I i I y. cg,agy(p,p,) or by any other preferred esethod. If the secem is initially at. foe that is' example,200 I#in* and has 100 =P of superheat,its total hear, l ,25s B I specEe volume l W = CK Av'(ffs g) (18) {en a 9 j For a nonle C = 1, so that the coceanondmg nomIc discharte 1 1 l - = 1253x,258-335 A'/Lb 200 F, = K,4V(spo o) (19) so a = &377 Lb/As I A number of methods, using charts, tables, or weking ru!es, that is, s have been developed by engineers for ~t~tada: F,. As a Therefore, l typical exampic ce a choked nonle discharsm,may be quoted Nepier's squation 1 F, = 0-667 x 01985 in* V(32 2 ft/ sect g steam 8 x200 Lb/ int x0377 Lb/ft )[1 ft/12 in) F,== A# (20) in which F.Is the diediarse in Lb. per sec; A is the throat arts = 4516 Wecc in in8; and is the reservoir pressure in Lb/in1 Hence, the d=4arge through the erince is, fmally : i The through an onfice is related Io the cor* F== CF I mp ga disc arse by: i = 0 732 x0 516 Lb/see F = CF, (21) = S378 lb/sec I l C is the contraction ~4t-ar, which may be obeined from Pig. 4 for a specined pressure ratio, r, and incom This How rate correspads with a reservoir p,ressure of 200 contraction coemcicat, Q, as used in hydrautka.ptrssible-flow Lb/ lot and a bei pm,sure of 0 546x200 Lblin* that is,109 i I'ig. 4 refers strictly to an capansion indes a = I 4, but the y of C does Lb/in2. Any reduction a the latter does not affect Fn but it ) not appear to be critically dependent on n. The co ding does increesc C, thus locatesing the dischatsc, F, even though l algebraic expressions for C ase equations (11) and ( 5) for sub. the ori6ce is choked. critical and choked Sows, respectively. In the fo expression I Km as given by equation p) and in the latter by (12). If the equivalent noule discharge has been dete ed without l reference to these equations, Ku may akernatively deduced from Fn by use of egnarnaa (19). APPENDIXII l i As a typscal numerkal example the leakage rate an 1 annular sht which is 6 inches in diametcr and 00 fach wide asiaa8HCRs will be estimated. It has an arts: (5"===, H. L. 1939 e r.it-a., Steam Tables, A = r x 6 in x 001 in = 01885 Ins Fahrenheit Units' Arnold and Co.). p Frsousow D. F., and Ltoerrua.r, M. J. 1947 Proc. Roy. Soc., and, if it were one stage of a labryrinth seal, for ample, it vol. A192, p.135, 'The Hodcaraph Transformanon in would probably have a hydraulic (that is, incom 'ble-Sow) Trans sonic Flow'. I contraction mencient, Q = 0 6. Thus the a m force HOWAms, E. L. 1953 ' Modern Developments in Fluid defect coemcient is, by equation (5): Dynantics: High Speed Plow', vol.1, p. 222 (Oxford j 1 -1 University Press). / = Q 2@ P'"' WHow Through" 2'*' Ta"' A.s.u.a., vol. 71, p. 757, OriSces'. 1 1 Scun. Lam, W. 1933 Forzheng Gebiete der bqrmeinr* f j "0T 2xt6a=422 anseur, vol. 4, p. Im, tiedse Entspanm:na kosopressibler Fhissigkeiten'.

  • The nomenclature used la e appendia was ed at the STA) trow, T. E.

1926 Proc. Roy. Soc., vol. All!, p. 306,'The specist request of the author. Flow of Gases at High Speeds'. .p. 4

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ESL-+ 703 519 0225:* 9/13 ) 4' 773 l 1 1 j Commmdcations Mr. C. H. BosAnquer ("n" M) wrote that the author's Since a veloczy

  • of 140 was assumed, the contrac-mesdeed of calculat discherse coemeients i--

tion -m~wt referred to in the popcr was equivalent to a dis-en the : "h the constancy off, using hSa A4 entirely chasse

  • They had smessured the discharge coeScient formula

-Cs of various nossles air at sure ratios fromr equal dC to 0412 to 0033. All nostles a throat of a normaal 7 =1-C diameter of ) inch. Comparative discharge co,mcients for Valene of C appionehing usky asut therefore be very smsitive venous noaales had been obinined by timin i to amen variations off. reservoir passeure under similar conditions. g the rate of fall of For a penfect nosase C - 1 deAaition but iff - 05 and . 6 showed the efect of the entry radius on the discharge a== 14 then C, which was y unity, feu rapidly with of a supersonic coevergent-diversent nozzle. Since ~, 4., - f-r and poseed through a adnimum value of 0 852 when the strenn was wi 4. downstreamof thethroat, r = 0 7.It then rom to 0 888 at the critical ratio and 0459 for k muld not in8uence upstream. In other words,in 3 diedentse into a v--i any supersonic nosale the upstream flow was unasected by the If C was seewned constant then / = 05 for sanall preneure divergent secnon and. in particaer, the discharge cosmcient was diferenose and roes to &571 at and beyond the critical ratio. unaleered by the supersonic espannian section. That was indeed i Poe internwdiate condisione it versed nearly linearly with dritkal data for other values of a were given in Table 3. 23 73 / // / f TAas.a 3. CatTrcAr. DATA 70s VAa10tts VAX,UIs or a f fa, ifa gaf-o,.e Ie 06065 11564 11 05:47 11561 12 cSat$ l1500 / 13 o5497 11662 f h 1,4 o52s3 1Hn ,yyy[ 7/f l667 04s71 1!MI 414 - s. i A suggested snod Ararian of the saethod was to multiplyfe by a factor which depended on the mean maso Aow la the plane of the oriace. The modiSed value off was given by [ = 1+(f-1%' [" Densities and v tarielaa were assumed the same as for a aosate i wkh the ensae mass Sow. The value of pas in the orifice plane 5 ' f on C so that k was noosasary to solve by trial.The cal-l cuIntion of f wee facilitated by employing the useful and j adequately accurate appe--- e. I ^ (h+([ = 1 At and beyond the crkkal sedo the $rst term was simply equal 3, so C. .mui o,,,nay, ,f,4 Table I save values which fall almost canct half waybetween 6 l those calculated by the two authods so that cocepasison was laconclusive. For ammu dischasse cosmeiears and subeenic Fig. 6. Effect of Radius on Disdiarse Coefficient of Sows the anodiacatica did not make much difference. For values Supersonic vergent-divergens Nozzle of C g

  • t-5 unky the modiaod marhad gave resuhs which were et least possibie; cuentantf did not. The =a'imad method i

theetfoot appeared psefernbic la spite ofits greater casapissary. R/D 0 05 24 Mr. R. P. PkAsat and Dr. P. N. Rows (I.aedoel wrote that as a C. H60 N 8esult of work carried out in the High Speed Pluid Rinatka of the ! Scatace and (Psener and Rowe 1 ,Preser,and Rowe the csee in for experunents had shown that the dis-(to be pu , and Couber 1 ) k was pnaaihla to can-charpe was independent of the divergent section telbune some deta to congere with calpredictions of withm the limits of m-- -- --=' ertor (1041). Thus the the paper la the :

rense, nossie wkh R/D, = 0 was, in the supersonie (i.e., the author's
-- ') region, equivalent to an oriSce where Q = 0 6.

Paesse, R. P., and Rown. P. N. 195411. of the ImperielCoucse dinc~with R/Dr== 24 corresponded to a nossic where Q = 14. ,., vol. 8, p. I, 'A Method of Measur=g Very Large The outlior had predicted a h in discharge coemeiess .M' O., PaAsm. R. P.. ead Rows F. N. (to be publiebed), between those two nestles of about 014 (Fig. 4) whereas they .A into the Des of Superosaie Nosales for Rocaets'. had found 0 047. However, as increased contraction cae8cian' Cosi.taa, M. O.1 FILD. Thress. London Universky, June, for '-. - 4 Sow (which the en necteocy of Supweenic Nemales*. bring those results more into line. y had not measured) would

^ ~)) ~{[~ h[" 8-30-94;7:49di ESL-703 519 0225;*10/13 ] 3 $ I. 774-COMMUNICATION 8 ON THE FLOW OF A COMPSISSIDLE FLUID THROUGH ORIFICES MW ppproach was 0908 (:l:041N). (Unfoetuantel 'h Is ad6 tion to - _. A the discharge mag,they had I measured the stuvec reaction scacreted try the Jcts luom various had no date for the axiventional ori6cr treaud in ik y. aanW Since the thrust wee proportional to the onduct of dis-The discharge coef6cient for the radiused noale was charse rete and the acceleranon occurring la the scale, the f:om the data of Fig. 6 (0 906) and so its velocity cor&4cne was thrust coefEoent was equal to the product of dscharge cn. 0975/0 986 - 0 989 which iusti6ed the author's assumpo'on. la j efEcient and velocity 9"' Por the case of a convergent-that instance, that the wlocity

  • was unity. If for the criace with the mod: Sed apprusth the coef5cient fee i

j a noaale of R De = 0 was used 0 9 the ty coedEcient for that was 906/00939 = 0 7 to in that instance the assumption of a wbcity W of unhy was touch less i 'm/m //wf, y accurate. i / The author appeared to have assumed that there was a osas resseracts in cosopressible Bow at small pteasme ratios U.e. high 1 reservoir pressures). Espansion at the est plane acco. ding to the Prandtl-Meyer theory wdd produce inesnediate divergence as I l e Radissed nousic. 6 Orince with modi 6etapproach. 3 j Pag. 7. Two Noazles divergent rmale cut off at the throat (Le., without a suprionic j espansson section), the velocity coef5cient w.wvv.f o that i t i of the author which he had assumed to be uniry. Thrust racesurements snade on the two nogaleg of Fig. 7 l showed that the thrust coef5cient for that with tme radiused approach (R/D. - 20) was 0975 and that for the o ce with a l 8 l I IN - l e se ~ [ 4 A 7 rc i 4l E ~ an u j smongtegc 4,fwopa8 N 'Lowa l I \\ l 1 r 1 3 i I \\, l 1 5 en h ge e. - l l i l i i i \\, l i en P ,t g l k! i e* ntEttat"mcLa 5 trienca ct.boccEss g i Fag. 8. Pnssure Ratio Plotted Against Resultant of a l 1 Emergence e r perim :.: pow. ru, ram.e.d noise. Fig. 9. Shadow Photographa o Esperunantal A for ori6ce with mdi6ed r = 0 GM. j 4 g i 1

m i

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ESL-703 519 0225:*11/13 ) l 4 COMMUNICATIONS ON THE PLOW 0F A COMPRESSIBLE FLUID THROUGH ORIFICES 775 ~ shown in Fig. 8, in which the Att line showed the theoretical efect. "Ihat was precisely the effect the author had been con. angle that would result from saw through a nozz!c sidering but the result was a net enlargtment, not a contraction. I so a pressus, rocio, r. The treatment was for two-The annergence angles for their criSce indicated a redactaou 1 A= mad==1 flow, y = l 40, and it was assumed that the flow ofthe Prend Mleyer angle. Thus, a parallei jet would cmcrge { ' 1 the throat perpendicularl. Those angles were at sense pressure ratio and at higher reuos the 3ct might contract ac6ieved la practice for the it.cc "--- y---' flow c(aar through outside tae mosale and the sonic plane would enove downstream. j a sediased mostle at difkrcat pressure ratios as was seen from Fig. 10 showed that a substantially parallclict emerged at a j she shadow photograph in Fig. 9s. 'Ilie emergeoce angle had pressure ratio of about 013 and that above that pressure ratio j been ansseured from photographs of the jet at diferent pressure contraction occurred. It was not possible to measure those i ados and, for a tadaused nocale, they agreed dusely with the eincrgence angles very accurstely but, undoubtedly, the order theoretical angle as would be seen la Fag. 8. ofthe eSect was as they had==== rad 1 j Unfortunately, in the standard criSce the thmet p_ lane was not They were of the opinion that the author's theoretical treat-aormally visible, widch ted photographic u.Ction ment asust be modaned for supersonic Sow, particularly at high 4 of the emergence angle, the ori6cc with a n.odined appmech, rese voir pressures, to take fato account the espansion effect. i k would be seen that the emergence angle had been reduced They hoped in the near future to publish a w v..S dve ac-(Figs. 8 and 96)its now amis whicit produced a cxatractangbecause the said now had radial velocity count of their esp 1 duscred sowards andRowe (to be pubhshed)). 1 Mr. D. H. TAarram (Associers Man >6er wrote that the znass 4 of a compassible Guid thavush an on) ace had been given in l Sow equation (7) as i r w. 1-r * }l f tt' u l m=CA p,y. r l In that an innan ty large sucrvuir to crioce area rado was con-j sidered. Many applications were -7= ad with an orince in a pipeline 1 when the rado was much smaller and an equation which took ) e Per radiused ace le with 6 For oriAce with maMWI thatinto account was given by i r = 0 23. opproach and r = 023. / 'v-l j 2sPsW1 1 5 T '* Q = DAs v ,a ~ 8" whcre Q was the mass flow; D, the coef5cacnt of discharge; i Apthe ares of orince; s, the ratio of area of pipeline to area of nt:6er; ps, the upstream pressure; pa, the downstream pressure [ Futhe speci6c wcight of Auld; y, the ratio of spectAc heats ; ana p' g, the accelesation due to grarity. Equation (7) was obtained from it of course when s = m. j However, in the practical aaaa'=laa_. the mass flow would depend on the contracnort cdthcient C, theoretical values of i e For radiused nasale with 4 For oriace with modated which had been given la Fig. 4. He would like to know, however, r r a= 013. approach eed r = 013. whetbar any recent information had been established on those cxP8Erients in addition to the work of Perry (1949) and Schiller l Pig.10. Emergcas ut a substantsally Paralle! Jet (1933) and how those results agreed with t 2.1 values. l i I i i 1 i i l i

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ESL-. 703 519 0225;#12/13 \\ ~ m -s l Author's Reply Mr. D. A. Jonsons wsoce, in reply to the - ^ - that he agreed that the force defect cocaicientf could - piccci la that narance also the emergent )ct might not always I ,in general, clear the outerlip downstream. be consideml to be a constaat. It was,in sosne , simular In regard to the shadowgraphs he was interested to note that i to a dras coedicient; just as the latter would on the they had recorded Jet contraction at r equal to 0 23, even with ~ Mach number, sof depded on the mass-Aow lent and,.an opening which was heavily countersunk upstream. had hence, the pressure raus r. It was only for criSce-openings aho shown that the emergcat angle of a noaale jet, men from j that the (=As=ar* cs compressibdity om f might be capacted to the axis of the latter, had cortelstad well with the theoretical be small. The Dorde mouth-piccc represented one for Frand Hdeyer angle. Poe an nri8ce, aonic canditions at the lip i which / was always scru, whatever the value of r. the other (i.e. M equal to 1 in Fig.1) were reached by Suid which was e j endof thescale,seMr. had pointed out, fora nmk axwing radiauy inwards. Hence, the theoretical curve of Fig. 8 varied abnost hacerly with r equal to 0 5 t the typecal would, in that instance, be shifted back 90 deg., ladicatang i value of f equaito 0 571 fora to 14,thereaft itmnained contraction, even at pressure ratios lower than 001. The corre. I i constaat for choked Sows. IIence the ratio //fe always be spending shift for the moddled oriSce tested by them would be j expected tolie between thelimits of to and 114 that value much less, as the countersink implied that the nuid would ) Wn seasonable approach thelip obliquely. That would be even less than cert,and it could thereforc always be estimated wit e l by soone such seethod as he had suspeeted. would that of the counsersink,owing to separation at the aboulder, a enable theory to be applied beyond the limit C,- valto07 so that the upstream condiuons were approaching those for a whidi he himself had suggested, but he doubted hether the nozzle. added -- ;- - =- was justinable below that vat For the extreme case of an orince discharging into a vacuum, He noted that Mr. Praser and Dr. Rowe measured Frandt!-Meyer thecey suggested an inner contracu'na jet, dhcharges on en ori6ce atted with an exit cone, whi discharges surrounded by a very smallinnes of Suid which splayed out. were numewhat higher than thooc obtained and for a wards as in Fig. 46 of Howar A (1953). 'ne motion of the latter simple anfice. It was his own belief that that laterest anomaly would in practice be comiderably modified by growth of j t be traced to the fact that the initial con ~ of the jet, boundary layer along the rtservoir wall and he was unable to foi by the subsequent overcapans6on down suggest any method of allowing for the expansion efect in that well, if enclosed in an exit cone, trap a region of

air, very limited region.

pressurs in that would be diferent from the pressure exit,and in the matter of recent infor. nation on contraction coefEcients i i would therefore rnodify the contraction coccicient ,hence, ratscd by Mr. Tentam,he hos.ed shortly to publish a further the mass Sow. He had referred to that possibility ' the paper paper, which would enable velocity of apprunch effects to be under Supercritical Plows,in -* wnh the da noouth-accounted for. t l 4 i .i i 4 4 I i l e P I t l t 4 i o 4 -}}

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