THE' UNIVERSITY- OF- MICHIGAN INDUSTRY PROGRAM OF THE COLLEGE OF ENGINEERING RESONANCE IN; LIQUID ROCKET ENGINE SYSTEMS i...-: R. H. Fa'shb'augh V. tL. Stx'eeter...- 698 "-' " -..' -'(,. v,-IP-698

'S...

TABLE OF CONTENTS Page LIST OF FIGURES.... e * *. e!e. * * *....** t...*. V.INTRODUCTIONeeeee e4** **ooe oooooe *oooe*p**seoee*e * ***~* * 1 THE ROCKET ENGINE PROPELLANT SYSTEM....**...............0.......... 3 FLUID TRANSIENT EQUATIONS......-.................,.........**.. 7 PIPE END BOUNDARY EQUATIONS.o*....*.................................. 10 le Upstream End of Pipe 1,4*0.0~.* t * *** 10 2. Junction of Pipe 1 and Pipe 2............................. 10 3. Junction of Pipes 4, 5, and 6................... 11 4. Pulser Boundary Condition....................... 12 5. Downstream End of Pipe 7..............................., 12 TURBOPUMP REPRESENTATION........................... 14 THE COMPUTIER PROGRAM..O,000@*@*@ @~-^^ E., 19 RESULTS OF TE ANA......LYS................................22...... COMBINING THE PROPELLANT SYSTEMS WITH THE SUPPORTING STRUCTtJRE.. 27 CONCLUSIONS. oee*** ******29 APPENDIX A 30 REFERENCES ~.e 3 2 e * e * * o * ~ e o e e e * e ~,s eo0 32 i-i-i~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~90 0000 s00000o

LIST OF FIGURES Figure Page 1 Missile Rocket Engine Feed System...................... 4 2 Oxidizer Feed System Test Configaration................ ** 5 3 Oxidizer Pump Steady State Characteristic Curveso... 6 4 Method of Characteristics Mesh...............e... 8 5a Computer Flow Diagram.................................. 20 5b Computer Flow Diagram Continued........................ 21 6 Oxidizer Pump Transient Suction Pressure Versus Frequencyo.... @...oo...... 23 7 Oxidizer Pump Transient Pressure Ratio and Discharge system Transient Pressure Ratio versus Frequency....*.. 24 8 Fuel Pump Transient Suction Pressure versus Frequency..................................,*o.. 25 9 Fuel Pump Transient Pressure Ratio and Discharge System Transient Pressure Ratio versus Frequency... 26 10 Diagram of Combined Propellant System, Engine, and Structure Analysis,.......0............0..i... 28 veeo e o e e o e e o o o o.O. e o o

INTRODUCTION A development problem associated with the Titan II missile was a self-exited longitudinal vibration set up in the missile structure and the rocket engine propellant feed system. At one condition of operation a structure resonance developed which was reinforced by propellant system resonances. The motion of the feed tank and pump caused the outflow to the thrust chamber and the resultant thrust, to vary periodically at the natural frequency of the structural assembly. This analysis deals primarily with the transients in all the pipes of the engine feed system and especially focuses on the method of representing the turbopump transient characteristics analytically, including the effect of pump inlet impeller cavitation. The quasi-linear partial differential equations of continuity and momentum were solved by the methods of characteristics and The method of specified time intervals, and by use of finite difference methods placed in convenient form for use with a large digital computer. Non-linear friction losses in piping were included, and the turbopump transient pressure rise is included as a non-linear function of the transient flow rate and inlet pump static pressure. The propellant system analyzed is a full-scale ground test simulation of the missile propellant system. These results were compared with the actual ground tests which included the turbopumps and feed system. In these tests (and in the analytical representation) the engine feed system was hydraulically the same as the -1

-2actual missile system except that the engine fuel and oxidizer injectors were simulated by orifices and the engine thrust chamber and nozzle were similated by a cavitating venturi. The fuel and oxidizer systems were tested separately.

THE ROCKET ENGINE PROPELLANT SYSTEM A sketch of the rocket engine propellant system is shown in Figure 1. Both the fuel and oxidizer systpms include a propellant tank, a line from the tank to the turbopump inlet, the turbopump and a line from the turbopump discharge to the thrust chamber injector. Valves are located upstream of each pump and injector. A sketch of the oxidizer system ground test configuration which simulates the missile system is shown in Figure 2. The fuel system test simulation was similar. The hydraulic dynamic resistance for the cavitating venturi is twice that of the hydraulic dynamic resistance for the engine nozzle. This fact was taken into account when the test data were evaluated for the missile application, but is of no concern to this treatment since the analysis is of the test configuration. The fuel and oxidizer turbopump characteristic curves are shown in Figure 3. Since this is a dynamic problem, interest is primarily in the slope o~ these curves at a particular operating point, such as shown on the curves. The use of these curves as well as other parameters such as friction factors and pressure wave velocities, orifice and venturi coefficients are discussed subsequently. The fuel and oxidizer used in the test were the same as the missile propellants which are Aerogine 50 (50 Hydrazine, 50% UDMH) and nitrogen tetroxide. -3

-4OXIDIZER TANK STRUCTURE FUEL TANK PUMP T HRUST CHAMBER Figur 1 Missie NOZZLE Figure 1. Missile RocketEngine Feed System.

-5OXIDIZER TANK PIPE I FEED LINE PIPE 2 PIPE 3 VALV E DISCHARGE4 4./.PULSER-PIPE 5 LINE- PIPE 7, _ -', PUMP -PIPE 6 VALVE -- ORIFICE --—,CAVITATIN G VENTURI OUTFLOW F Figure 2. Oxidizer Feed System Test Configuration.

OPERATING POINT 1200 800 UL) U) w 1I000 20 60 100 460 500 540 580 NET POSITIVE SUCTION PRESSURE,psi WEIGHT FLOW, lb/sec Figure 3. Oxidizer Pump Steady State Characteristics.

FLUID TRANSIENT EQUATIONS The methods of calculating head and velocity of small time intervals for equally-spaced sections along a pipeline, with inclusion of friction, have been previously reported. (1,2,3) The important working equations are briefly summarized: For the interior sections along a pipe VP vR + V + g(HR - HS)/a - (fV2)C At/D (1) P 2 RR and H [H + H + a (V — V )/g] (2) in which Vp and H are velocity and elevation of a hydraulic gradeP p line at section P. With reference to Figure 4, it is considered that V and H at A,C, and B have previously been calculated. VR, VS, HR and HS are obtained by linear interpolation between A,,IC and B by the formulas VR VC + (V + a)C (VA - V) (3) HR =HC + (V + a)C (HA - HC) (4) Vs = Vc + (V- a) (V - V) (5) HS =HC + (V a)C (H C HB) (6) in which G = At/AX, the preselected mesh ratio for the calculations. C+ and C of Figure 4 are the characteristics. For the method to be stable, it is essential that

tat AX- I AX Figure 4. Characteristic Curves.

-91K. - -(7) vl + a in which'a' is the speed of a pressure pulse wave. aK= W 11/2 Eb (8) K is the bulk modulus of elasticity of the fluid, p the mass density, E the Young's modulus for pipe wall material, D the pipe diameter, and b the wall thickness. In the equations, g is the acceleration of gravity, f. the Darcy-Weisbach friction factor, and At the time increment used in the calculations. The subscript C denotes evaluation of the quanity at the section under consideration for the preceding time increment. At the end sections of a pipe, the characteristic equations yield one equation in the two unknowns Vp and Hp. For the downstream end V =V - g(H -H )/a - 2-t(v2)C/ (8) P R P R C 2 and for the upstream end Vp V + g(H - Hs)/a 1 At(fV2) /D (9) External conditions (pipe end boundary conditions) must supply the extra relationship so that V and H may be calculated. p p

PIPE END BOUNDARY EQUATIONS The pipe external or end boundary equations will be given for the oxidizer system pipes. The boundary equations for the fuel system pipes are derived in a similar manner. As shown in Figure 2 the oxidizer system test configuration was represented by 7 pipes; 1 pipe for the tank, 3 pipes for the pump feed line, 1 pipe to represent the pump, 1 pipe for the pump discharge line and 1 pipe to represent the pulser.. Each pipe was divided into sections as previously explained. The boundary equations for the end of each pipe in the system are as follows: l1. Upstream End of Pipe 1 The pressure head at the inlet to pipe 1 is Hp(1,l) = H (10) p g where H is the gas pressure head in the tank and Hp(ll) is the g p pressure head in pipe 1 of section 1 2. Junction of Pipe 1 and Pipe 2 The pressure head at the downstream end of pipe 1 (tank outlet) is equal to the pressure head at the upstream end of pipe 2, so that Hp(1,Nl) H (2,1) (11) P p whre ) indicates the pressure headHp(l in pipressure head in N (the last section) and Hp(2,1) indicates the pressure head in pipe 2, -10

section 1. This notation will be used throughout the text. Another relation is derived from continuity of flow from pipe 1 to pipe 2 as V (l, N1 D2 2,l) (12) where D1 and D2 are the diameters of pipes 1 and 2 respectively. The above two equations along with the two boundary characteristic equations previously derived (one for each pipe) give the required four equations to solve for the four unknowns, Hp(l,Nl), Vp(l,Nl), Hp(2,1), and Vp(2,1). The boundary equations for pipe junctions 2 to 3, 3 to 4, and 6 to 7 are identical to the equations for junction 1 to 2 except for the subscripts. 3o Junction of Pipes 4, 5, and 6 As in the junction of pipes 1 and 2, the pressure head is equal, so that Hp(5,l) Hp(4,N4) (13) H (691) - Hp(4,N4) (14) From continuity we have, V (4.N4) = D512 Vp(5,1) + V (6,1) (15) These three equations along with three characteristic equations (one for each pipe) determine the six unknowns, Hp(4,N4), Vp(4,N4), Hp(591), V (5,1), H (6,1), and V (6,1). p p p

-124. Pulser Boundary Condition There are two unknowns at the pulser piston, Hp (5,N5) and V (5;N5). One characteristic equation is available and the other P necessary equation is simply Vp (5,N5) = Lo sin wt (16) p o where L is the pulser stroke and w is the angular frequency. The pulser inputs a sinusoidal variation in flow to the system. 5. Downstream End of Pipe 7 Since pipe 7 is terminated in an orifice the orifice equation is used to determine the boundary condition equation. If Ho is the pressure head downstream of the orifice and D7 and Do are pipe 7 and orifice diameters respectively, 2 22 p(7,N7)(D7) = D(Do g(Hp(7N7 - O) [\ 7p CD (t17) Vp(7;N7) = CDo (D)2 2g(Hp(7.N7 ) / Ho)(18) Since the pressure at the throat of the cavitating venturi is vapor pressure the pressure head, H, can be determined from 0 (77) V 2g(Ho - HV) (19) p D7 D 2 g LV(7,N7) (20) 2-P 2 + HV C'.DvDv/

-13Equations (18) and (20) yield the desired boundary equation ID \ 2g(Hp(7,N7) HV) v (7,N7) = CDO (p. (21) ~p 7 LI +D( 2 [Do 4 kCDV kDvi where CDO and CDV are orifice and cavitating venturi discharge coefficients, DV is the venturi throat diameter and HV is the vapor pressure of the propellant in feet of head.

TURBOPUMP REPRESENTATION Referring to characteristic Equation (1), the friction term can be written 1 C ~ C1 2. (fv'D0 AtD/D = g (X) At (22) where AH is the pressure head drop in the distance AX. Since internal losses in a pump are reflected in the pump head rise the friction term should not appear in the pump (pipe 6) equations. Instead, it is readily seen from Equation (22) that the term g Lp At (23) can be substituted for the friction term in the pipe 6 equations but with opposite sign to represent the pump head rise. Lo is the length of the pump flow path and AHp is the pump head rise. The characteristic equation for pipe 6 corresponding to Equation (1) is then P 2 (VR + VS) + g(HR( - HS)/ aC + 2g At (24) Equations (2) through (6) do not change because of the pump head rise and will apply to pipe:ii6.:The steady state pump head rise is a function of both the pump inlet pressure and flowrate as is shown on Figure 3. It was assumed that for transient conditions the head rise could be determined from these.steady state characteristic curves. It was also assumed that affects due to variations in the pump speed were negligible.

-15Equations for the characteristic curves (Figure 3), determined by curve fitting methods, were used to compute the head rise, AHp, for particular values of inlet pressure and flowrate. The equation for AH& is P (Hp(6,1 ) - HpO) AH *AH o - k(Vp(64) - %) - LBB j B + B3(Hp(6,1) (25) In this equation MHpo, BO, and VO are constants determined from the steady state head rise versus flowrate curve and Hpo, B1, B2, and B3 are constants determined from the steady state head rise versus inlet pressure curve. The cavitation of the pump inlet impeller has a large effect on the fluid transients in the suction system, Tests at the MartinMarietts Corporation (Denver Division) have shown that a region of cavitation can extend in the pump inlet pipe as far as from three to four feet upstream of the inlet to the oxidizer pump. This cavitation region drastically lowers the pressure-wave velocity in the inlet pipes and therefore lowers the quarter wave resonant frequency of the entire pump feed system. Since there is no way of computing the effect of this cavitation, the pressure wave velocity near the pump has to be approximated from test data. In this case, the quarter wave resonant frequency of each feed line used in the analysis was the value measured in the tests and pressure wave velocities for the feed system pipes were derived from

-16these measured values. It is also possible to determine these resonant frequencies from a spectrum analysis of pump inlet pressure measured during engine test firings. The fuel pump feed pipe is short and it was assumed therefore that the cavitation affected the pressure wave velocity in the entire pipe. The length of pipe from the tank outlet to the pump inlet will be a quarter wave length. The average pressure wave velocity in the pipe is therefore af 4 fnfLf (26) where ff is the quarter wave resonant frequency and Lf is the length of the fuel pump feed pipe. The oxidizer feed pipe is long and the cavitation will only affect a small portion of the pipe. The pipe was therefore considered as three pipes each with a different pressure wave velocity. In this case they are denoted pipe 2, 3, and 4. The pressure wave velocity in pipe 2 was computed using Equation (8), the pressure velocity in pipe 3 was taken as one half the value of pipe 2, and the pressure wave velocity in pipe 4 was determined so that the system resonant frequency would be equal to the measured value. Since the friction loss is small, the impedance relations for a lossless fluid pipe were used to determine the pressure wave velocity in pipe 4. The surge impedance for pipe 2 is

-17a2 Zo2 a2 (27) ~ ]2 g and the impedance of the downstream end of pipe 2, since the impedance of the tank end is very small, is Z2 = j Zo2 tan P2 L2 (28) where 2ir f P2 fno (29) 2 =- a2 and L2 is the length of pipe 2 and fno is the quarter wave resonant: frequency. Similarly, the relations for pipes 3 and 4 are a3 Zo3 - 2 D 17 D3 g (30) z2 + j Zo3 tan 3 L3 (31) Z3 Z03 3 + j Z2 tan L (31) 2it fno a = no (32) 3 Z a4 (33) o4 D 4 Z3 + j Zo tan 4 L4 4 ~ Z4 LZo4 J 3 tan P4 L4)i 274 ano (35) 4a 4

-18Since Z4 represents the impedance of the entire feed system at the pump inlet, this impedance will be maximum at the quarter wave resonant, frequency fn. Therefore, the denominator of Equation (34) will be zero at the frequency f which provides the expression from which the n pressure wave velocity a4 can be obtained. Zo4 + j Z3 tan 4 L4 = O (36) Using the first two terms of the tangent series and the expressions for Z4 and 4 Equation (36) becomes 4 2 3tD gfoLf)a 4 z31 (n4fgfno3L) = (37) which yields 4 no 4D9- V 31 +1 + 1 (38) This is an approximation for the effect of the cavitation. The most important aspect is that the pump feed lines in the analysis have the proper quarter wave resonant frequency.

THE COMPUTER PROGRAM An abbreviated flow diagram for the solution of the fluid transients in the oxidizer system is illustrated in Figures 5a and 5b. The steady-state pressure heads, velocities, and the pressure wave velocity for each pipe are computed, stored, and printed for time t = O. The velocity and pressure head for the interior sections of each pipe are computed from the same block of equations and the boundary points between the pipes are then computed from the boundary equations for time t + At. The time increment At is chosen so that the interpolation points R and S lie within points A and C of Figure 4. The computed values are printed and stored for the next computation. The result is a time history of pressure head and velocity at each section of each pipe in the system. -19

READ INPUT DATA-NPIPE, N, L, D PRINT DO TO ( START b, EQf, t AHpB0, B1, B, B3, INPUT FOR I11 Hp, P, K, 0 D D., IP DATA TO NPIPE ________) __(N_____-_ L(J)(V(I)) 7r/4~ 2(I)e AHPA1) f(I) D(l)(2g) 7r/4 (D (D2 (I DObTO O K COMPUTE a(3), a(4) FOR I:1 a(I)= 2 BASED ON MEASURED +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ TO NPIPE P E+ FEED LINE FREQ. —fno VENTURI INLET HEAD ORIFICE INLET HEAD PRINT a(I), 8e ) ( D(7) 2 +H (vcr))2 D 4T) Hoi 2 DV Dv 2gc 2 Do o PIPE 7 INLET HEAD INLET HEAD-PIPE 1.2.3.4,5,6. H(ItJ)=H(I)- IHf Figure 5a.Coputer Flow Diagram.I) H(7)= H(79 N7) +aHf(7) H(1)=H(7qN7)+j:A& f (I) -A H~ — NIE c1 a Figure 5a. Computer Flow Diagram.

H (6 J) HW + (A HI)) 1 r v(LJv(I) a I PRINT t=t +aI Nu IH I -6+( NPIHPE HU J( J a I -— a- N (6) 1U=1~~~~~IUIU+1 N(6;-l- J-9N V(I J) OSTART DOCTO OMPUTE COMPUTE IF_ _ _ _ _ _ _ _ _ _ _V_ _ _ _ N D T VSVR HD (, NI), H(2I H(N2)Ha2 -(3) N M-3I HSNHR ), H(4J) I I — s- NPIPEL COM PUTE COMCUTE COMPUTE COMPUTE VP (I 9 ) -Vp(IjNI Vp(20I) 7 VP (2 9N2) qVP(39 1 ) Vp (3 t N 9 V(49 1) H091~) HP(l v NO) Hp(2 J) Hp (2 q N2) Hp(3 91) Hp(3tN3)9 Hp(4j)) COMPUTE COMPUTE COMPUTE OPT 01Vp(49N4),, VP(5j0),Vp(6 1) Vp(5 9 N5) &-Lowsinwt VP (6 1 N 6), VP(7t1) Vp (To N7 Hp(4, N4), Hp(5,I), Hp(6, I) Hp(51N5) H (6,N6),Hp(7,1) H (t7 N7) (V(7N7) _D (Li DOTO H J)=H (I GO TO Ho 2g C H DvI I NPIPEJ) JIMI-*=N(I) Figure 5b. Computer Flow Diagram (Continued).

RESULTS OF THE ANALYSIS The results of the analysis compared with test results are shown in Figures 6 through 9. Figures 6 and 8 illustrate the ratio of the amplitude of the pump inlet pressure oscillation to the corrected acceleration amplitude of the pulser versus frequency. The pulser acceleration amplitude is corrected by the ratio of the flow area of the pulser to the flow area of the pump inlet pipe. These curves show a good agreement of the analysis results to test data. Curves are shown in Figures 7 and 9 which compare the ratio of the amplitude of the pump discharge oscillation to the amplitude of the pump inlet oscillation. These ratios depend upon the pump dynamic characteristics. The agreement is fairly good which shows that the pump steady state characteristic curves are adequate for use in dynamic problems. The test data was quite non-linear for the fuel pump around the resonant frequency of 12 cps which could account for the deviation of the analysis results from the data around this frequency. Curves are also shown in Figures 7 and 9 which compare the pump discharge pressure oscillation amplitude to the amplitude of the venturi inlet pressure oscillation. The analytical results compare well with the data which substantiates the analytical representation of the pump discharge systems. -22

40 ANALYSIS -O — TEST DATA 30 _ 200 u 20 15C~~~~~~~~~~~~~~~ -i~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~t w~~~~~~~~~~~~~~~~~~~ 20 150 C,) 0 5C J o _ _ _ _ _ _ _ _ _ _ r:) 10 2 46 8 10 0 3 10~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~I b)~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~. z 0 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~Lu (I ~~~~~~~~~~~~~~~~as I ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~Lu (0 2 4 6 8 tO203 FREQUENCY CPS Figure 6. Oxidizer Pump Transient Suction Pressure.

1.0 0.8 w A0 0.6 w 0 U. 0.4 o o I I 0 0.2 0 ANALYSIS --—'- TEST DATA 1.2 eJ 0 -_ w 0.8,' - I 0 Z 0 a. a. -. C 0 4 —~~~0 8 10 12 14 16 18 20 22 FREQUENCY~ CPS Figure 7. Oxidizer Pump Pressure Ratio and Discharge System Pressure Ratio.

20 ANALYSIS ---- TEST DATA IO 200 - 0......I5)0,.) 1:13~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ w 1 1 w010 C3 -I0 2 I00 /,,3L 0~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~I.5 I::z. ~ ~ ~~ REUEC - CPS -20...........50 1 003 2 4 6 8 0t 15 20 30 50 FREQUENCY - CPS Figure 8. Fue] Pump Transient Suction Pressure.

-260.8 w W 0.6 o. 0.4 0 _ _ _ E Z: 0.2 ANALYSIS -0 — TEST DATA!0 brJ 8 -w ~4_ _ 0 2 8 10 12 14 16 18 20 22 FREQUENCY- CPS Figure 9. Fuel Pump Pressure Ratio and Discharge System Pressure Ratio.

COMBINING THE PROPELLANT SYSTEMS WITH THE SUPPORTING STRUCTURE A simplified block diagram to illustrate how the propellant system analysis can be combined with the supporting structure is shown in Figure 10. The fuel and oxidizer injector outflow oscillations as computed above are converted to a thrust oscillation by use of the engine thrust relations. The pump and tank motions which excite the propellant systems are computed from the structure dynamic equations of motion from this known thrust oscillation. From the pump motion the injector outflow oscillation is computed which completes the closed loop self excited system. To determine the stability of this system the loop can be opened at the input to the structure and a conventional open loop stability analysis can be done. Since the scope of this paper is the analysis of the propellant systems, the stability problem will not be presented in any further detail. -27

STAGE I OXIDIZER SYSTEM 1 IIIIH HI1 FEED DISCHAR TANK V IE vPUMP v IEv INJECTOR Xto X po EXCITATION v H CLOSED LOOP OPTTION Tin utou MECHANICAL THRUST STRUCTURE CHAMBER STAGEC I FUL YSE OPEN LOOP 0 OPTION Ppf vv H Xtf H H H H FEED ~~~~~~DISCHARGE TANK FEDPUMP INJECTOR v LINE V V LINE v STAGE I FUEL SYSTEM Figure 10. Diagram of Com~bined Prop~ellant System, Engine., and Structure Analysis.

CONCLUSIONS The transient flow in a liquid rocket engine propellant feed system including turbopumps can be predicted when the effect of the pump cavitation on the resonant frequency of the system is known. The steady state pump characteristic curves (head rise as functions of flow rate and inlet pressure) were found adequate for the transient problem. The pump transient pressure ratio, defined as the amplitude of the discharge pressure oscillation per unit of inlet pressure oscillation, is predictableo -29

APPEND IX A NOTATION Symbol Description Units a velocity - pressure wave ft/sec b wall thickness - pipe ft B0 B1, B2, B3 constants - pump curve CDO discharge coefficient - orifice CDV discharge coefficient - venturi D diameter - pipe ft D diameter - orifice ft o DV diameter - venturi throat ft E Young's modulus - pipe wall lb/ft2 f Darcy-Weisbach friction coefficient f quarter wave resonant frequency cps n g gravitation constant ft/sec2 H pressure head ft ZH pressure head differential ft,pL' constant - pump curve ALH pressure head rise - pump ft p Ho pressure head - orifice discharge ft Hg pressure head - tank gas ft HV vapor pressure head - fluid ft H constant - pump curve ft H (c,d) pressure head, pipe c, section d ft -30

-31K bulk modulus - fluid lb/ft2 Lo pulser stroke ft L pump flow path length ft L pipe length ft Nl, N2, etco number of sections in pipe 1, pipe 2, etc. t time sec. At time increment sec. V velocity - fluid ft/sec V constant - pump curve 0 V (c,d) velocity-fluid, pipe c, section d ft/sec AX distance increment ft Z surge impedance sec/ft2 Z impedance sec/ft2 Xo angular frequency - pulser rad/sec phase velocity p mass density - fluid slugs/ft3 9 At/AX sec/ft Subscripts Denotes * f fuel system 0 oxidizer system 1, 2, 3 etc. pipe number A, B, C, P points on characteristics R, S interpolated points

REFERENCES 1. "Water-Hammer Analysis Including Fluid Friction", by V. L. Streeter and Chintu Lai, Trans. ASCE, Vol. 128, Part I, (1963), pp. 1491-1552. 2. "Waterhammer Analysis with Nonlinear Frictional Resistance", by V. L. Streeter, Proc. 1st Australasian Conf. on Hydraulics and Fluid Mechanics, 1962. Pergamon Press, 1963, pp. 431-452. 3. "Waterhammer Analysis of Pipelines", by V. L. Streeter, J. Fod. Div. Proc. ASCE, paper 3974, (July 1964), pp. 131-172. -32

UNIVERSITY OF MICHIGAN II 9015 02826 7220Ill llIll,3 90'15 02826 7220