THE UNIVERSITY OF MICHIGAN INDUSTRY PROGRAM OF THE COLLEGE OF ENGINEERING THE EFFECT OF MINOR LOSSES ON WATERHAMMER PRESSURE WAVES Dinshaw M. Contractor A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the University of Michigan Department of Civil Engineering December, 1963 IP-651

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ACKNOWLEDGEMENTS The author is extremely grateful to Professor Victor Lo Streeter, chairman of the committee, for suggesting this interesting topic. The discussions held with him about various aspects of the thesis never failed to encourage the author in the solution of the problem. The other members of the committee are also to be thanked for their guidance and their cooperationo The author acknowledges the financial help given him by the University of Michigan by way of a fellowship for two academic years. The funds for this work were jointly borne by the Civil Engineering Department and a National Science Foundation research project directed by Professor Vo Lo Streeter. Since the computer was used for the theoretical solution of the problem, the author thanks the University of Michigan Computing Center for the use of this facility. The author is grateful to many of his colleagues with whom he has had stimulating discussions of the problem. The Industry Program has been very helpful for the typing and assembly of the thesis and the author acknowledges his indebtedness to the College of Engineering for this aido ii

TABLE OF CONTENTS Page ACKNOWLEDGEMENTS................o.......................ooo ii LIST OF FIGURES................................................. iv LIST OF PLATES.................................................. vi NOMENCLATURE.................................................... vii I HISTORICAL REVIEW....................................... 1 II INTRODUCTION............................................. 3 III ELEMENTARY SOLUTION................................. 5 Derivation....................................... 6 IV SOLUTION BY THE METHOD OF CHARACTERISTICS................ 13 Derivation of Characteristic Equations................. 14 Finite-Difference Equations......................... 18 V EXPERIMENTAL SET-UP...................................... 32 VI EXPERIMENTAL PROCEDURE................................... 44 VII THE COMPUTER PROGRAM.................................... 47 VIII DISCUSSION OF RESULTS.................................... 49 IX CONCLUSIONS.............................................. 71 APPENDIX I - COMPUTER PROGRAM AND PRINT OUT FOR WATER HAMMER IN A PIPELINE WITH A MINOR LOSS IN THE MIDDLE......................................... 72 APPENDIX II - COMPUTER PROGRAM AND PRINT OUT OF WATER HAMMER IN A STRAIGHT PIPELINE............................ 77 BIBLIOGRAPHY.................................................... 82 iii

LIST OF FIGURES Figure Page 1 Conditions Before and After a Water-Hammer Pressure Wave Encounters a Minor Loss................. 7 2 Characteristics................................... 21 3 Specified Time-Interval Method....................... 21 4 The (x,t) Plane for a Uniform Pipe.................... 22 5 The (x,t) Plane for a Left-Hand Boundary Condition.... 22 6 The (x,t) Plane for a Right-Hand Boundary Condition... 26 7 The (x,t) Plane for a Minor Loss Boundary Condition... 26 8 The (x,t) Plane for a Valve as a Right-Hand Boundary Condition................................ 31 9 Schematic Diagram of Experimental Set-Up.............. 33 10 Three Orifices Used in Producing Energy Loss.......... 34 11 Variation of K versus R of Three Closely-Spaced Orifices...................o.................... 35 12 Resistance of Solenoid Valve During Closure........... 38 13 Flow Diagram for Computer Program.................... 48 14 Water-Hammer Pressure-Time Diagram; Case l(a); One Cycle........................................ 59 15 Water-Hammer Pressure-Time Diagram; Case l(a); Four Cycles...................................... 60 16 Water-Hammer Pressure-Time Diagram; Case l(b); One Cycle.......................................... 61 17 Water-Hammer Pressure-Time Diagram; Case l(b); Four Cycles.......................................... 62 18 Water-Hammer Pressure-Time Diagram; Case l(c)......... 63 19 Water-Hammer Pressure-Time Diagram; Case l(d)...o..... 64 iv

LIST OF FIGURES (CONT'D) Figure Page 20 Water-Hammer Pressure-Time Diagram; Case 2(a)........ 65 21 Water-Hammer Pressure-Time Diagram; Case 2(b)........ 66 22 Water-Hammer Pressure-Time Diagram; Case 2(c)......... 67 V

LIST OF PLATES Plate Page I Compression Chamber and Air Pressure Regulator........ 42 II Solenoid Valve and Pressure Transducer..4........... 42 III Closely-Spaced Orifices Used. to Produce Energy Loss.. 43 IV Recording and Calibrating Instrumentation........... 43 V Water-Hammer Pressure-Time Diagram; Case l(a); One Cycle................................... 50 VI Water-Hammer Pressure-Time Diagram;Case l(a); Four Cycles...................................... 51 VII Water-Hammer Pressure-Time Diagram;Case l(b); One Cycle......................o........ 52 VIII Water-Hammer Pressure-Time Diagram;Case l(b); Four Cycles...................o.............. 53 IX Water-Hammer Pressure-Time Diagram; Case l(c)....... 54 X Water-Hammer Pressure-Time Diagram; Case l(d)......... 5 XI Water-Hammer Pressure-Time Diagram; Case 2(a),... 56 XII Water-Hammer Pressure-Time Diagram; Case 2(b)..o 57 XIII Water-Hammer Pressure-Time Diagram; Case 2(c) 58 vi

NOMENCLATURE Symbol Used In In Description Units Theory Computer Program Wave Celerity Ft/sec a A Acceleration Ft/sec2 A Pipe wall thickness Ft b B Inside diameter of pipe Ft D D Modulus of elasticity Lbs/ft2 E E Pressure wave height Ft FF1 F2 fl,f2 Friction factor f FR. Acceleration of gravity Ft/sec2 g Piezometric head Ft Hv Head at reservoir Ft HO HO Dimensionless piezometric head (H'/HO) H H or HP Loss coefficient K KOR. Bulk modulus of elasticity of water Lbs/ft2 K K1 Total length of pipe Ft L L Reynolds Number R R Time Secs. t' Dimensionless time (t'/(2L/A)) t T Time of valve closure Secs. tc TC Velocity in pipe Ft/sec VT Steady-state velocity Ft/sec VO VO Dimensionless velocity (V'/VO) V V,VP vii

NOMENCLATURE Symbol Used In In Description Units Theory Computer Program Distance from reservoir Ft xg Dimensionless distance from reservoir (x/L) x X Specific weight of water Lbs/ft3 Poisson's Ratio p Kinematic viscosity Ft2/sec v NU Mass density of water Slugs/ft3 p Ratio of effective gate opening to full gate opening T TAU viii

I. HISTORICAL REVIEW The first studies of water hammer date back to the beginning of the present century, when the contributions of Michaud(56), Joukowsky(36) and Allievi(l) pioneered the analytical treatment of this hydraulic problem. Joukowsky was the first to establish the rate of propagation of the wave, and prove that the head rise for instantaneous valve closure was equal to aV'/g, where a is the wave celerity, V' is the velocity in the pipe and g is the acceleration of gravity. Allievi developed the mathematical analysis of water hammer for uniform closures and presented charts for maximum pressure rise in simple conduits. Later authors extended Allievi's theory to consider changes of pipe diameter and pipe thickness, branch pipes, effects of air vessels and other complications. The solution to these different problems was obtained arithmetically by careful bookkeeping of the travel of the wave in the pipe system and the consequent changes in the head and the velocity of the water. This step by step procedure of determining the pressure in a pipe system was slow and laborious and was readily given up in favor of a graphical procedure. The graphical solution of water-hammer problems was (12) first suggested by Professor L. Bergeron2) and since then many authors (4,6,7,8,72) have extended this method to many complicated problems ranging from pipelines with surge tanks and air vessels to transient conditions following shutoff of a pump feeding a long pipeline. These graphical procedures, however, were derived from equations which neglected friction effects and kinetic energy terms. -1

-2Whenever friction in a pipeline became an important factor, the graphical methods were modified to take this into account by concentrating the loss of the entire pipe length at one or more points in the pipeline This approximation generally gave satisfactory results. Lately, however, interest in the solution of water-hammer problems has been revived because of the development of the method of characteristics(26'53) for solving hyperbolic partial differential equations and the availability of high speed electronic computers to carry out the numerous calculations associated with the solution. These developments enable one to take into account those terms of the water-hammer equations which were hitherto neglected. The terms that were formerly deleted from the equations were the non-linear terms, including those resulting from the friction in the pipe or from the minor losses occurring at one or more points in the pipeo Thus, a more complete and accurate picture of the wave profile is made available by this procedure. The versatility and flexibility of this method also enables one to provide solutions to more complicated problems than could be handled previously.

II. INTRODUCTION This thesis is concerned with the study of pressure-wave reflections produced when a water-hammer pressure wave encounters a device which produces an energy loss in the steady state. The device causing the energy loss may be a pipe bend, a tee joint, a valve that is open or partially closed or a restricting orifice. It is the intention of the author to determine the transmission and reflection coefficients of the water-hammer pressure wave and their relationship with the energy loss of the device. The theoretical relationships are first determined by a simplified theory neglecting friction effects, i.e., by using the solution to the classical wave equations. They are then confirmed by the use of a more complete and accurate theory. In this theory, the partial differential equations for water-hammer, including friction effects, are solved by the method of characteristics and a high speed digital computer, the IBM 7090, is utilized for their numerical integration. From the output of the computer program, it is possible to calculate the values of the approaching, reflected and transmitted pressure waves, and knowing the magnitude of the minor loss, to verify their inter-relationships. It is to be recognized at this point that this problem could be (12,132 solved by the graphical methods of Schnyder-Bergeron. 13) However, as with most graphical procedures, even though one can obtain the final solution one does not necessarily come to have a better understanding of the mechanics of the problem. Finally, the results of an experimental verification of the theoretical relationships are set forth. The experiment consists of -3

-4producing a water-hammer pressure wave in a pipeline with an orifice in the middle and recording the pressure-time history at two points of the pipeline. This pressure-time diagram is superimposed on the theoretical diagram, obtained from the computer program, for the sake of comparison. Thus, the validity of the assumptions made in the derivation of the theory is affirmed.

III. ELEMENTARY SOLUTION The theoretical study of water hammer reduces to the solution of two partial differential equations. As these equations have been derived in many texts(59,67,81) they will be used in this thesis directly. The first equation is derived from Newton's second law and is referred to as the condition of dynamic equilibrium. aH' = -1 V (1) ax' g at' where H' is the piezometric head in feet of water, t' is time in seconds and x' is distance in feet measured from the reservoir. The second equation is derived from considerations of continuity, in a horizontal pipe. H' -a2 V' (2 at' g ax' In Equation (2), a represents the wave velocity in the pipe and is obtained from the following formula. 2 -_ Klp 1 + C1 KD/Eb in which C1 is a constant(29,59) depending on the way in which the ends of the pipe are restricted and on the Poisson ratio of the pipe wall material, and the other symbols are as defined on page (vii). The simultaneous solution of these two equations is given by H' - HO = F(t' + ) + f(t' - ) (3) X' X' where F and f are arbitrary functions of (t' + -) and (t' - a ) respectively, aid V' - VO = {F(t' + -) -f(t' - -)} (4) -5

-6Thus the changes in head and velocity could be obtained if the functions F and f are known or these functions could be evaluated if the changes in head and velocity are knowno Using these formulae, it is possible to calculate the reflections for certain boundary conditions, as shown below. At a reservoir, H' - HO is zero at all times. Hence f(t' a ) = -F(t' + ) from Equation (3), and so V' -VO 2- F(t a a a a At a Dead-End. V' and VO are both zero at all times. Hence, f(t' x) F(t' + x-) from Equation (4). a a Thus, H' - HO = 2 F(t' + ) Similarly, when a water-hammer pressure wave encounters a change in pipe area and/or wave speed, the reflected wave is equal to (Al/a1 - A2/a2)/(Al/a1 + A2/a2) times the approaching wave and the transmitted wave is equal to (2A1/al)/(A1 + A) times the approaching wave. 1 a"2 a The magnitudes of the reflected and the transmitted waves will now be derived when a water-hammer pressure wave encounters a minor loss. Derivation In Figure 1, let points A and B be on either side of the loss-producing device, in this case an orifice. Let F1 be a water-hammer pressure wave approaching point A. Let fl be its reflection and F2 its transmission. Let the head and velocity before F1 reaches the orifice be HAO, HBO and VAO9 VBO Let the head and velocity after reflection and transmission be HAt, Ht, and VAts VBt

-7ENERGY GRADE LINE RESERVOIR MLOSSO HO VALVE Bx:xA Figure l(a). Conditions Before a Water-Hammer Pressure Wave Encounters a Minor Loss. MLOSS t |l A (MLOSS) BxlxA Figure l(b). Conditions After a Water-Hammer Pressure Wave Encounters a Minor Loss.

-8- i/ From Equations (3) and (4), Ht - HA F1= +l (5 HBt - HBO F2 (6) At - VAO - ( - 7) VAt ~AO a - VBt - VBO - (8) From the condition of continuity, VAt = VBt and VAO VBO (9) Hence, from (7)9 (8), and (9) g F2 -g (F - fl) a a i.e. F = F1 - fl (10) Since a minor loss occurs at the orifice, HBO HAO MLOSS~ and HBt - HAt MLOSSt (11) From (5)9 (6), and (11), F2 - F1 f MLOSSt - MLOSSo A(MLOSS) (12) From (10), and (12), we have -2f1 A (MLOSS)

Therefore, f, - (MLOSS) (13) and from (10), F2 F + A(MLOSS) (14) 2 1 2 Thus, it can be seen that the reflection is dependent only upon the change in the minor loss before and after the passage of the wave. The transmitted wave is equal to the approaching wave plus half the change in the minor loss. The change in the minor loss is easily evaluated when the velocity behind the approaching wave F1 is zero or very nearly zero. This is so when instantaneous closure of the valve occurs, and the pipeline is considered frictionless. For this case, v2 A(MLOSS) = - K 1- (15) 2g Hence, 1 K VA0 fl-1 K 2 (16) and V2 F2 F - K A (17) However, when the velocity behind the wave F1 is not zero, Equations (16) and (17) do not apply and fl becomes a more complicated function of F1. This relationship is found in the following manner. Let MLOSS =K - K and MLOSS KK 2gK K V and MLOSSt = K g = Kg Therefore, A(MLOSS) - (V V- o) -2g ~At VAO)

-10From Equation (7), VAt - VAO -g {F1 - f} = -g {F1 + A(MLOSS)} VAt - VAO a a 2- o} - a ^ - ^ (^(~ - vA0)} Let AVA = VAt - VAO Th9 K Then AVA - F1 - a (VAt - VAO)(VAt + VAO). 1 9 K AVA (AVA + 2VA) a A A AO = g F1 - K(AVA) _ KV A a 4a 2a or W (AVA) + 4-VA (1 + K VAO) + gF - 0 Ta- 2a a Therefore 2a 1/7 KV KVA 2 K gFj A KvA o2a) (i4+-) 4aa AVA =K {-(1 + 2a a ) - 4 4a a2a ~ (K + vAo) 2 a2 4a2 + 2a_ VAO) ~ - - 2a- - + VAo) + y + /(AO)2 V - K - K K But AVA =- (F1 fl) Therefore g'2a 2a ~2 4gF1 - F1 - fl) = - (a-+ VAO) +( - VA) a K ( Therefore fl = F1 _ (2a VAO) - F (18) and a 2a - 2 (gF F2 = g- (K-+ VAO) + + VAO) - (19)

-11It is easy to prove that in Equation (18), the positive sign before the radical is the correct one. In the limiting case, when VAt = O, Equation (18) should reduce down to Equation (16)o When VAt =- 0 F = a VAO g From (18) aVAO 2a2 aVAO + a 2 aA 9 - " VgK K g 2a2 a /4a2+ 2 4aVAO 4aVAO gK g V K AO K K Expanding (-2 + V)/ by the Binomial Theorem, K2 AO = 2a2 + a (421/2 4a2- 1/2 o 1 4a2 Vi 2 +. } gK g. 2 ^ovA - 16 AK 2a2 2a2 2 K'Vo2 2 2a2 2a2 1 VAO K VA + = + K + +-K K —-- VA gK- gK- 2 2g 1283 2g - 1 2 = _ K iA., neglecting terms of smaller magnitude. Thus, - 2g the positive sign is correct in Equation (18) and hence the negative sign is the correct one in Equation (19)o The series expansion of ( a + Vo) /2 by the Binomial Theorem is convergent whenever 2_ < VA0 This condition is always fulfilled 2g g as the minor loss in a pipeline is generally much smaller than the waterhammer pressure wave height. The procedure is the same when analyzing the situation in which an fl wave approaches a minor loss, as in the case of flow establishment in a pipe. It can be shown that the reflected wave F1 = - A(MLOSS) and that the transmitted wave is equal to (f + )) and that the transmitted wave is equal to (fl + A(MLOSS)).

-12More complicated situations could also be handled in the same manner. For example, consider the situation in which an F1 wave approaches a minor loss from the right-hand side and an f2 wave approaches it from the left-hand side and both the waves encounter the minor loss at the same time. It can be shown that the reflected wave on the righthand side, fl = f2 - (MLOSS) and that the reflected wave on the left1ai dd 2 hand side is equal to F1 + A(MLOSS). 2

IV. SOLUTION BY THE METHOD OF CHARACTERISTICS The basic partial differential equations for water hammer in a pipe, taking into account friction effects, have been derived in Reference 78, and will be used directly in this thesis. The assumptions made in the derivation are listed below. 1. Uniform velocity and pressure distribution over area of pipe. 2. Pipe wall material and liquid in pipe are perfectly elastic and homogeneous. 3. The pipe remains full of liquid at all points and at all times. 4. The static pressure in pipe is above the vapor pressure of liquid at all times. 5. The elastic hysteresis of liquid is assumed negligible. The equations will be non-dimensionalized and their characteristic equations derived. The purpose of non-dimensionalizing is so that the same notation may be used here as is used in the conjugate equations of the graphical procedure. It will be seen later that these conjugate equations are a particular case of the characteristic equations derived here. These characteristic equations are then transformed into their finite-difference form and used for the solution of the head and velocity in the computer program. Next, the equations at the following boundary conditions are derived; at the reservoir end, at the valve end, both for time less than valve-closure time and for time after valve closure. Finally, the boundary conditions at a minor loss are derived in finitedifference form. -13 -

-14Derivation of Characteristic Equations The following two partial differential equations are the ones whose simultaneous solution gives the head and velocity (the dependent variables) in terms of the distance and time (the independent variables). These equations are quasi-linear and of the hyperbolic type. They include a term which takes into account the friction loss in the pipe, and those terms which were neglected in the classical water-hammer theory have been retained. The condition of equilibrium, 2 t D 2gf g (() + V aV). (20) 67" D 2g g 6tI 6' The condition of continuity, for horizontal pipes, aH'_ +. aVIH' a2 Avi (21) at' ax' g ax' In order to non-dimensionalize these equations, let Hy H V, -=x and t. HO VO L 2L7a AH 6H 6v 6v Also let H a = Ht, - Vx and Vt This transforms Equations (20) and (21) into HO Hx + fVO2 V2 1 aV Vt + V0 Vx} (22) L 2gD g 2L L and aHO H VOHO _ a2 VO L t + gv x Vx (23) 2L L g L

-15To determine the characteristic equations, let J1 =H Hx +f V2 V2 + VO {t V = 0 L 2gD gL 2 and aHO a2 J= Ht + VO HO V Hx + VO Vx = 0 ~2 ~g Multiplying J1 by 2gL/aVO and J2 by 2/aHO and putting 3 = aVO/2gHO; we have, Ji = Hx/ + fLVO V2 +t +2 V Vx = aD a J =Ht +2a V Hx + 4Vx = a - Combining J1 and J1 linearly with an undetermined multiplier X, 12 J - J+ X H + Y0 ++ 2 V xvv L + 2 x aD t a X + X Ht + a V Hx + 4VX} = Rearranging the terms, we have ____ 2fLVO o { + X2a } Hx + X Ht + a{ V + X 4 a Vx + Vt + fLVO = O (24) Now, since V and H are both functions of x and t, dH = Hx dx + Ht dt and dV = Vx dx + Vt dt

-16or dH H d + Ht dt dt and dV = Vx d + Vt dt dt With the last two equations in mind, Equation (24) could be reduced to the following simple form, AdH dV f L VO2 (25) dt t+ aD (25) if at the same time we make, dx {1 + X 2 VO V} (26) dt X - a and dx 2 VO V + X 4 } (27) dt a From Equations (26) and (27), 1 1 2 VO V1 2 VO V — ~- + X I +x4 k B a a Therefore, 2 1 4p2 or, X + (28) - 2p Substituting back in Equation (27), dx 2 VO V + 2 (29) dt a

-17Since the term 2 VO V is small compared with the other term, a we can approximate Equation (29) by dx + 2 (30) dt - and the characteristic equations for V and H become, dt J = dV +1 dH + f LVO V2 dt = 0 (31) 23 a D and dt J = dV - dH+ L VO V2 dt = (32) 23 a D Summing up, the simultaneous solution of the partial differential Equations (22) and (23) has been reduced to the solution of two total differential Equations (31) and (32), in the two directions given by Equations (30). The solution is thus seen to involve two planes, the (x,t) plane and the (V,H) plane. The lines in the (x,t) plane are straight lines as shown by Equations (30), whilst those in the (V,H) plane are curved because of the V2 term in Equations (31) and (32). These lines are called characteristics and will be referred to as the C+ and C_ characteristics. See Figure 2. Thus, along the C+ characteristic, dx - 2 dt = 0 (33) and dV + dH + f LVO V2 dt = O (34) 23 a D And, along the C_ characteristic, dx + 2 dt = 0 (35)

-18and dV - 1 dH + f L VO v2 dt = 0 (36) 23 a D From Equations (34) and (36), it can be seen that, when the friction term is neglected, the characteristic equations become dV = + - dH - 2 The above equations are precisely the conjugate equations used in the graphical analysis of water hammer and can be seen to be a particular case of the more general solution provided here. Finite-Difference Equations Since the general solution to hyperbolic, partial differential equations is yet unknown, particular solutions for individual cases under study are usually sought. A close examination of Equations (33) to (36) reveals that all the terms, except the friction terms, are exact differentials and hence can be integrated easily and without any error. Integration of the friction terms along their respective characteristics between two points 0 and 1 yields L VO f (f V2) dt aD o This integral can be approximated in finite-difference form in several ways. The first-order approximation assumes the function to have a constant value during the interval dt. Hence, L VO (f 2)dt = LVO (f V2)o (t,-to) (37) aD o aD

-19The second order approximation makes use of the trapezoidal rule to estimate the value of the integral. Hence, 1 L VO f (f V2)dt =L 0 l {(f V2) + (f V2)J (tl-to) (38) aD aD 2 The error involved in Equation (38) is smaller than that in Equation (37)o However, certain extrapolation procedures(53) are known whereby Equation (37) could be used and the error kept to the same order of magnitude as that of Equation (38). It will be shown later why Equation (37) was used in preference to Equation (38). Equations (33) to (36) will now be integrated along their respective characteristics from a point 0 at which V and H are known to another point 1 at which V and H are unknown. Thus, along C+, (xl-Xo) - 2(tl-to) = 0 (39) and (V1Vo) + (H1-Ho)/24 + (f V2)o(tl-to) 0 (4) Along C_, (Xl-xo) + 2(t1-to) = 0 (41) and (V1-Vo) - (Hl-Ho)/2p + L (f V2)o(tl-to) = 0 (42) aD The Equations (39) to (42) will now be used to obtain solutions to the head and velocity at specified points along a uniform pipe and at regular intervals of time. This procedure is known as the

-20Specified Time Interval Method. As illustrated in Figure 3 on page 21, the pipe is divided into N number of sections. The time interval 1 used is then [. From the initial, steady-state conditions in the pipe, the values of V and H at each point of the pipe will be known. From these values, it will be possible to calculate V and H for t = A t at the points marked with a black dot in Figure 3. From the known end conditions, it will be possible to determine V and H at the points marked with a cross for t = A t. This procedure is then repeated for t = 2 A t and so on, for any time length. Finite-Difference Equations for a Uniform Pipe Let the head and velocity at time t = t1 be known for all intervals of x. See Figure 4. Suppose the solution at P(xi,tj) is to be sought. Let the two characteristics through P. with Equations (39) and (41), cut the line t = tj_l at R and So By means of interpolation, the head and velocity at R and S could be determined from the values at A, B and Co Then from Equations (40) and (42) we have, (Vp-VR) + (Hp-HR)/2p + L VO (f V)R (tp-tR) 0 (43) aD and (Vp-Vs) - (Hp-HS)/23 + L O(f V2)s (tp-ts) 0 (44) Making (tp-tR) = (tp-tS) = A t, and solving the above two equations for Vp and Hp, we have, Vp = VR+VS - (H-HR)/4 - LVO {(f 2)R + (f V2)S (45) Hp =HR+HS (V-V)+L VO t (f v2)S - (f V2)} () 2 sR a

-21t H C- C- C - C- IV H), x,,t,),,) 0 O Ho.V) ( HoVe) /( Xoto) (xo, to. x ~, V 0 1.0 (x,t) PLANE (H,V) PLANE Figure 2. Characteristics. * POINTS OBTAINED FROM INITIAL CONDITIONS X POINTS OBTAINED FROM END CONDITIONS t 3 A i id -~ " q~' M'h od ~AX0 Fg. dk b dk KAh d d. 0 I Figure 5. Specified Time-Interval Method.

-22t ___A B tj-I -R C S tj-2at o t L ~X -. ".. I-I I I+ 1I Fue X, l X Xi+ I Figure 4. The (x,t) Plane for a Uniform Pipe. CC B At A —-x Xo XI X2 X$ Figure 5. The (x,t) Plane for a Left-Hand Boundary Condition.

-23 - Whenever At is chosen to be exactly Ax/2, then the points R and S coincide with A and B respectively. This value of At then makes the interpolation procedure unnecessary and leads to a speedier solution. At this stage, it can be seen why Equation (37) is easier to use than Equation (38). If Equation (38) was used, the solution for Vp and Hp for a uniform pipe would be as follows. Vp = VR+VS - (HS-HR)/4 - L VO A t {(f V2)R + (f V2)S + 2(f V2)p} 2 4 aD Hp HR+HS - (VS-VR) + L VO t {(f V2)S - (f V2)R} 2 SR 2 aD It is clear that the first equation cannot be solved explicitly for Vp, as the friction factor f is a function of Vpo Thus, the solution would have to be carried out by an iterative procedure by assuming an initial value of Vp. Because of this and also since A t was very small, it was decided that Equation (37) provided sufficient accuracy. Boundary Conditions: a. A Reservoir at the Left-Hand End. Consider Figure 5 on page 22. Let a constant head reservoir exist at x0o Hence, at all times, Hp = 1o0 (47) Let the C_ characteristic through P cut t = tj_l in S, then from Equation (42) (Vp-Vs) - (l.0-Hs)/23 + L- (f V2)S A t = O or Vp = Vs + (1.0-HS)/2 - L VO (f V2)S t (47A) aD

-24b. A Valve at the Right Hand End. (i) For Time t Less Than Valve-Closure Time tc. Whenever t < tc, the head and velocity at the valve have to be determined from two conditions. The first is the water-hammer equation, AH - -2P A V and the second is the orifice equation Q = Cd. Avalve i2gH, where Hv is the head loss across the valve. The orifice equation is usually rewritten in the following form. V =T \/v where (Cd Avalve)t = t (Cd Avalve)t = O The values of T are determined experimentally as a function of time t. Using the appropriate value of T, solution of the two equations gives V and H at the valve. However, in this thesis, the conditions at a valve are treated differently. In the experimental set-up, the solenoid valve does not discharge into the atmosphere. Instead, the pipeline continues after the valve and hence the pressure downstream from the valve has also to be found. For this reason, the valve was treated as a device causing a sharp energy loss at a point in the pipeline. The equations for this condition are derived a little later in the thesis. The T values obtained experimentally are transformed into values of loss coefficients in the following manner. V = T HVv or = 2g V2 T2 2g = KSV V2

-25Therefore KSV = 2g T2 These values of loss coefficients are used in the equations derived later. (ii) For Time t Greater Than Valve-Closure Time tc. Refer to Figure 6 on page 26. When the valve is closed, the velocity at x = 1 is always zero. Let the C+ characteristic through P cut t = tjl in R. Then (Vp-VR) + (Hp-HR)/2p + L VO (f V2)R A t = 0 aD But VP = (48) Hence, Hp = HR + 23 VR - 2 LaD (f V2)R A t (9) c. At a Minor Loss In the Pipe Line. KV2 Refer to Figure 7 on page 26. Let a minor loss of - take 2g place at xi and let P1 and P2 be points just upstream and just downstream of xi respectively. Let the C+ characteristic through P1 cut the line t = tjl in R and the C_ characteristic through P2 cut t = tj_l in S. The four unknowns to be determined are Vpl, Vp2, Hpl and Hp2. The four available equations to determine them are the two characteristic equations, the continuity equation and equation of motion at t = ti.

-26tj tj _I I / At AX i,0 Figure 6. The (x,t) Plane for Right-Hand Boundary Condition. t' ~ ~Ai!\2 01 I I I I I I lCX X. X. LI I I+B Figure 7. The (x,t) Plane for a Minor Loss Boundary Condition.

-27From Equation (40), (Vpl-VR) + (Hpl-HR)/2p + VO (f V2)R A t = 0 (50) From Equation (42), (Vp2-VS) - (Hp2-H)/2p + VO (f V2)s A t = 0 (51) If the pipe diameter remains unchanged, continuity makes Vpl = Vp2 (52) The fourth equation is the equation of motion between the points P1 and P2 at time t = ti o The unsteady equation of motion for an incompressible, inviscid fluid in dimensional form is 2 + gHl + gZP1 + (v'p) dx _ (VP2 ) + gH{ + gZI{ + 2 o a(53t Now, Vp1 = Vp2 and if the pipe is horizontal, ZPl - ZP2 and Equation (53) reduces to gH =xgH' + f dx2 (V4) gH' = gH' + - - dx' (54) P1 P2' -t' XP1 Since the two points P1 and P2 can be chosen arbitrarily close, the last term can be made as small as to make it negligible. Also, since a minor loss takes place between P1 and P2, Equation (54) can be modified to take this into account.

-28Thus, gH'l = gH'p + K(V') /2 Or, Hl = H2 + K(Vl )2/2g Non-dimensionalizing the above equation, we have Hpl = Hp2 + K V02 (Vpl) (55) 2g HO Solving Equations (50), (51), (52) and (55) simultaneously, we get Hpl = (HR+HS)/2 - P(VS-VR) + {(f V2)S - (f V2)R 1K VO v2 + gO P1 (56) Hp2 = (HR+HS)/2 - P(VS-VR) + L VO A {f V)S - (f v2)R i K V02 V2P 2 2g HO P Vp1 = Vp2 = (VR+VS)/2 + (HR-HS)/4 - V t {(f V2)R + (f ) ) 1 o 1 KVO2 2 (58) 2: 2 2g HO pi Equation (58) is a quadratic and hence can be solved explicitly for Vpl. This value of Vp1 can then be used in Equations (56) and (57) to obtain the head before and after the minor loss. It should be recognized that these equations have been derived for positive velocities and hence when the velocities are negative, the friction and minor loss terms change sign. This is taken into account by replacing each V2 term by VIVI, which automatically changes the sign.

-29A comparison between the equation for Hpl and the equation for head in a uniform pipe [Equation (46)] shows that the difference between them is half the minor loss. This conclusion was also arrived at from the classical wave theory [Equation (13)]. In the use of Equations (56), (57) and (58), it should be determined, whether the loss coefficient should have the steady-state value K or whether an unsteady loss-coefficient Ku should be used. Experiments(l8'19) have shown that for accelerated flow, Ku is less than K and that Ku decreases with increasing acceleration. The value of Ku is shown to be a function of the acceleration parameter AcL/(V')2 Thus, Ku =K - C1 A L/(V')2 (59) where C1 is a constant of proportionality dependent upon the ratio of orifice area to pipe area. This thesis, however, made use only of the steady state loss coefficient K because of the following reasons. First, experimental values of C1 were not available for the low ratio of orifice to pipe area used in the experiment. Second, formula (59) would not be valid for the very high accelerations the fluid encounters upon passage of a water-hammer wave. For the case of instantaneous or rapid gate closure, the acceleration of the fluid approaches infinity each time a waterhammer wave passes by. This would imply from Equation (59) that Ku would become zero or even negative. For these reasons, it was decided that the steady-state loss coefficient should be used.

-30d. At a Valve Discharging Into the Atmosphere. This condition is a particular case of the conditions at a minor loss described in the last section. The pressure at P2 (refer Figure 8) is always atmospheric, i.e., Hp2 = 0. Hence, there are only three unknowns to be determined. They are Hpl, Vpl and Vp2. The following are the three equations to be used for their solution. From the C+ characteristic [Equation (40)] (Vpl-VR) + (Hpl-HR)/(2P) + L VO(f V2)R (tp-tR)/(a D) = 0 (60) From the condition of continuity, V1 = Vp2 (61) The minor loss equation gives Hpi = Hp2 + K VO0 V2p/(2g HO) = K Vo2 Vpi/(2g HO) (62) Solving Equations (60), (61) and (62) simultaneously, we obtain K V0 (Vpl) + 2 Vpl - HR - 2 VR - 2 L V t (fV2)R =0 (63) 2g HO a D Equation (63) is a quadratic in Vpl and.hence its value could be easily obtained. Substituting its value in Equation (62) gives Hp1 directly.

-31t A t ___T / CC2 ti_,l l lI-' R tj_-2 AX 1.0 Figure 8. TRhe x,t) Plane for a Valve as a RightHand Boundary Condition.

V. EXPERIMENTAL SET-UP The experimental set-up was organized in such a way that the reflection from a minor loss could be seen and compared with the minor loss. In order to make the reflection large, the minor loss also had to be made as large as possible. This was achieved by selecting a device with a very large loss coefficient. This device was placed in the middle of the pipeline and the pressure in the pipeline at two different points was recorded and compared with the theoretical results. A schematic diagram of the experimental set-up is shown in Figure 9. At one end of the pipeline is a compression chamber, half filled with water and with compressed air in the space above the water level. The pressure of the air in the compression chamber is maintained at a constant level by means of a pressure regulator placed between the chamber and the compressor. In this way, the air pressure stayed constant even when the level of the water in the chamber dropped during the course of the experiment. Twenty feet from the pressure chamber, three closely-spaced orifices were placed in the pipeline to produce a large energy loss (Figure 10, page 34 ) Steady state experiments were conducted to determine the loss coefficient of this device as a function of the Reynolds number The results of the steady state experiments are shown in Figure 11. The pipeline used was a hard copper tube, 1/2 inch inside diameter and with a wall thickness of 1/20 inch. Fittings could be -32

FROM AIR PRESSURE COMPRESSOR GAGE WATER FEED PRESSURE CHAMBER AIR PRESSURE REGULATOR PRESSURE GAGE SHORT RUBBER TRANSDUCER GLASS CSOLENOID VALVE WEIGHING TANK THREE ORIFICES GAE VALV < ~20*~^~~ ^ ~^~20' 20* PIPE SUPPORTS Figure 9. Schematic Diagram of Experimental Set-Up.

-34e\ ORIFICE PLATES SPACERS Figure 10. Three Orifices Used in Producing Energy Loss.

104'' 8 6 ^~~~~~~~~~d~~~~~~~~~ V2 4 HEAD LOSS ACROSS ORIFICES K,9 z 0 i. 1.5 L,~1 0 2 4 6 8 2 4 6 8 REYNOLDS NUMBER, R Figure 11. Variation of K vs. R of Three Closely-Spaced Orifices.

-36connected to it by means of soft-solder. The theoretical wave speed for water in this pipe was 4480 feet per second. Experiments were conducted to determine the Darcy-Weisbach friction factor of the pipe for different Reynolds numbers, From the plot of f versus R, it was seen that, in the turbulent range, the pipe behaved like a smooth 1/4 pipe and followed Blasius' law, f = 0.316/R. As only Reynolds numbers less than 105 were expected in the experiment, it was felt that this equation would suffice. For laminar flow, the points closely followed the law f = 64/R. Forty feet from the pressure chamber a solenoid valve was connected to the pipeline. This valve was of the normally closed type. It opened under the action of the solenoid and closed under the action of a coil spring. It was necessary to determine the characteristics of the valve, such as the rate and time of closure and the variation of the hydraulic resistance of the valve as it closed. The hydraulic resistance of the valve was determined by measuring the pressure drop across the valve as the sliding gate of the solenoid valve was depressed in small steps by means of a screw arrangement attached to the top of the valve. Thus, the variation of T with displacement of the valve was obtained. The other characteristic of the valve to be determined was the rate and time of closure of the valve. This was done by means of a metallic cantilever beam. The free end of the cantilever was attached to the solenoid bit. Close to the fixed end of the cantilever, two strain gages were glued on, one to the upper side and the other to the lower side of the beam. These two strain gages formed two arms of a Wheatstone bridge. The

-37output of this bridge was amplified and recorded by a polaroid camera from the screen of an oscilloscope. Calibration of the bridge was performed by depressing the solenoid by known displacements. Recordings were then made as the valve was closed with water in the pipeline. It was seen that the valve opened under solenoid action in about 8.0 milli sees. When closing, it took about 10.0 milli secs. for the solenoid to de-energize and another 12.0 milli secs. for the spring to close the valve. Combining these two characteristics of the valve, it is possible to determine the T-versus-time curve for the valve. Comparing the time of closure of the valve (12 milli secs.) with the return-travel time of the valve (2L/A = 18.0 milli secs.), it can be seen that we have a case of rapid closure of the valve. The T-versus-time curve is presented in Figure 12. Twenty feet downstream from the solenoid valve, the pipeline ends in a gate valve. This valve is operated in such a way as to elevate the static pressure in the pipeline and keep the velocity of the water low. This additional twenty feet of pipe was felt necessary to prevent any reflections from downstream travelling past the solenoid valve whilst it was closing. In this way, the pressure wave in the main pipeline was kept free of any extraneous disturbances. The discharge from the pipeline was measured gravimetrically. The water was allowed to accumulate in a tank resting on a weighing scale. The time required for a fixed weight of water to be discharged into the tank was determined with the help of a stop watch.

1.0 I 0.8 - <" \ 0.6 0 o < 0.4 Ld w \ U 0.2 n(I) LJ 0 2 4 6 8 10 12 TIME IN MILLISECS. Figure 12. Resistance of Solenoid Valve During Closure.

-39The pressure sensing device used was a Dynisco pressure transducer. The transducer was attached to the pipeline in such a way as to make the sensitive face of the transducer tangential to the inside circumference of the pipe. The transducer was connected to an Ellis bridge amplifier, which provided the input for the Wheatstone bridge in the transducer and which amplified the output. The bridge amplifier had an outlet for connection to an oscilloscope. The oscilloscope used was a Tektronix model. An attachment on it made it possible to mount a Du Mont camera in front of the screen and photograph traces on polaroid film. Brief specifications of the three pieces of equipment mentioned above are given below. Dynisco Pressure Transducer. Model No. PT25-1.5C Pressure Range. 0-150 psi. Safe Overload. 2x Full Scale Natural Frequency. 12500 cps. Configuration. 4 Active Arm Wheatstone Bridge. Bridge Resistance. 350 ohms + 10o. Ellis Associates Bridge Amplifier and Meter. Model BAM-1 The unit consists of a DC powered bridge circuit, DC transistor amplifier and static and dynamic output connections. By connection to a DC cathode ray oscilloscope, it measures dynamic signals over a frequency range of 0-20000 cps. Resistance transducers, 50-2000 ohms, with 2 or 4 external bridge arms could be used. Tektronix Dual-Beam Oscilloscope. Type 502 Frequency Response at 5 mV/cm -- 200 kc.

-40Triggering signal sources -- Upper beam, lower beam, external or line. Internal triggering -- a signal producing 2 mm vertical deflection on either lower or upper beam. External triggering -- 0.2 to 10 volts on either polarity. The oscilloscope was capable of being triggered in several ways. It was necessary for this experiment to trigger the trace in two different ways. First, use was made of a microswitch, depressing which a single trace would travel across the screen. This method was used for registering the constant pressures before and after closure of the solenoid valve. The other arrangement triggered a trace by the use of the same switch which operated the solenoid valve. This method was used to trigger the water-hammer pressure wave. The sweep rate of the oscilloscope could be adjusted in steps from 5 secs/cm to 200 p.secs/cm. Two sweep rates were most satisfactory for these experiments. First, when registering only one cycle of the wave on the screen, a sweep rate of 5 milli-secs/cm. was used. However, when four cycles of the wave were required, 2 photographs were made, each registering 2 consecutive cycles using a sweep rate of 10 milli-secs/cm. The accuracy of these sweep rates was checked by means of an audio oscillator. It was found that there was an error of about 6% in the sweep rates and so the actual time scales on the photographs were 1 cm. = 4.7 milli secso and 1 cm. = 9.4 milli secs. The camera mounted on the oscilloscope was a Du Mont Oscilloscope Camera Type 450. A few adjustments had to be made before good, clear photographs could be taken with it. First of all, the back

-41of the polaroid camera was opened up and a ground-glass plate taped exactly where the film would be held. A steady trace was produced on the screen by means of an audio oscillator. This trace was brought into focus on the ground-glass plate by means of the focussing knob. This position of the knob was marked, so that it could be brought back to this position at a later time. In this position, it was seen that the graticule was not in focus since the graticule and the screen are not in the same plane. Hence, after illuminating the graticule, it was brought into focus on the ground glass plate and this position of the focussing knob was also marked. For clear, well-focussed pictures, these two different positions had to be used when photographing the graticule and traces on the screen. Photographs of the experimental set-up and the instruments are presented in Plates I to IV, on pages 42 and 43.

-42-;u: Gil-i Plate I. Compression Chamber and Air Pressure Regulator. Plate II. Solenoid Valve and Pressure Transducer.

-43Plate III. Closely-Spaced Orifices Used to Produce Energy Loss. Plate IV. Recording and Calibrating Instrumentation.

VI. EXPERIMENTAL PROCEDURE The following different experiments were conducted, 1) Pipeline with a Minor Loss in the Middle. a) Turbulent Flow. Pressure transducer at x' = 40 fto b) Turbulent Flow. Pressure transducer at x' = 30 ft. c) Laminar Flow. Pressure transducer at x' = 40 ft. d) Laminar Flow. Pressure transducer at x! = 30 ft. 2) Straight Pipeline. a) Turbulent Flow. Pressure transducer at x' = 40 ft. b) Turbulent Flow. Pressure transducer at x' = 30 ft. c) Laminar Flow. Pressure transducer at x' = 30 ft. Each of these experiments followed the same procedure, which will be described below. First of all, the electronic instruments were switched on and allowed to warm up for an hour or more, so that there is no drift in the trace. Next, the pressure chamber was filled with water and the air pressure regulator set to a constant pressure at about 80 psi. With the solenoid valve open, the gate valve is regulated so that the required velocity in the pipe is obtained. This is accomplished by trail and error. The required velocity in the pipe was chosen so that the pressure in the pipe did not at any time of the experiment fall below atmospheric pressure. This was to reduce all chances of water column separation. With the electronic instruments warmed up, the pressure transducer was connected to a dead-weight gage tester. Dead weights -44

-45equivalent to a pressure of 90 psi were applied and the vertical deflection of the beam on the oscilloscope adjusted to be exactly 3 cm. In this way, a pressure scale of 1 cm. = 30 psi was obtained. The transducer was then connected to the appropriate place in the pipeline. Every effort was made to remove all the free air that may have been trapped in the pipeline. This was accomplished by letting the water flow in the pipeline with a high velocity for a sufficiently long time. The time scale on the oscilloscope was adjusted so that one or two cycles of the pressure wave were accommodated on the screen. The intensity of the beam was adjusted so that it was not too bright and yet all the fine details of the wave registered on the photograph. The camera was now attached to the oscilloscope. The screen was brought into focus and the lens opening increased to its maximum value. The shutter-timing mechanism was set to the'bulb' position. With the shutter lever depressed, a trace was triggered across the screen by pressing the micro-switch. This straight line represented the static pressure at that point under steady-state conditions. Then, the switch operating the solenoid valve was closed and another trace was thereby triggered across the screen. This was the water-hammer pressure versus time curve. The third trace to be triggered across the screen was the constant pressure HO causing the flow. The last thing to be done was to illuminate the graticule, bring it into focus, reduce the lens opening to f = 11, adjust the shutter time to be 1/25 second and depress the shutter release. This provided a background of horizontal and vertical lines spaced one cmo apart in both directions.

-46These lines were helpful when transferring the trace on the photograph to another graph. The polaroid film was then developed in the 10 secs. coated with the permanentizing solution and was ready for comparing with the theoretical curve.

VII. THE COMPUTER PROGRAM The computer program was written for an IBM 7090 computer installed at the University of Michigan Computing Center. The language in which the program has been written is known as the Michigan Algorithm (9) Decoder,( or in short, the MAD Language. A flow diagram, illustrated in Figure 13, indicates the procedure used in the program. It also helps to break up the long program into short and simple parts which make it easier to understand. The main steps in the program are as follows; reading-in the data, setting up of function subroutines, calculation of certain constants, setting up the initial values in the pipe and calculation of pressure and velocity for as long a time as may be desired. The last part is the real core of the program. It is broken up into two parts. The first part involves the calculation of V and H for time t less than tc and the other part of time t greater than tc When the time t exceeds a specified limit, the program is terminated, Three function statements are defined at the beginning of the program. The first determines the friction factor f for a given velocity. The second determines the loss coefficient of the orifice for a given velocity. The last function determines the head and velocity at a boundary condition, involving a minor loss. This last function is called upon in the main part of the program by means of an "Execute Function" Statement. The program and a part of the program output is given in Appendices I and II. The head and velocity in the pipeline at 5-foot intervals are printed out, even though calculations were made at intervals of one foot, -47

-48Read Internal Internal Internal Calculate Data unction Function Function Constants for ~ for Minor -, for Minor A, DELX, riction Loss Loss DELT, BETA ctor Coefficient Boundary and Print Condition Out Values Eqns. (56), (57), (58) Set up Initial Print Out Set Time Values of HP(I)' Initial Counter VP(I) along Conditions J-l Pipeline, from EX - 0 to bO ~\ Time Whenever Set * a T~T=J*DELT ~T less than'V(I) VP( 8.0, H(I) HP(I) Transfer to Calculate Calculate HP(X) Yess HP(I), VP(I) VP(I) at Reservoir along Uniform EX-0 Eqns.(47),(47A) Pipe, Eqns. (45), (46), for EX I Execute Internal Function 1-19, 21-39, C for HP(I), VP(I), at Whenev1___41-59 Minor Loss. EX = 20 -T less | |Execute Internal Calculate JJ+l than | Function C for HP(I), VP(t) HP(I), VP(I) at at Gate Valve. Transfer Solenoid Valve. EX 60 to IEX 0w 40 Eqns.(63)(64) I a No Calculate Calculate HP(I), HP(I), VP(I) ~4 VP(1) at Reservoir along Uniform I EX O0. Eqns.(47)(4A Pipe, for EX - 1-19, 21-39. Execute Internal Function Eqns.(45),(46) C for HP(X), VP(I), at.... -___ Minor Loss. EX X 20 Calculate HP(I), VP(I), J I+Transfer at Solenoid Valve. to EiX w 40. Eqns.(48),(49) Co a F 1 EnDd of Program I Figure 13. Flow Diagram for Computer Program.

VIIIo DISCUSSION OF RESULTS The results of the experiments are shown in their original photographic form in Plates (V) to (XIII), on pages 50 through 58. Comparisons of the experimental and theoretical wave forms can be made in Figures 14 to 22, For all cases, four cycles of the pressure wave have been reproduced. In addition to this, pressure-time diagrams of one cycle have also been presented for the case of the pipeline with a minor loss in the middle. In this way it is possible to see the magnitude of the reflection in the one-cycle diagram and also observe its influence on the decay of the pressure wave in the four-cycle diagram. It was the intention of the author to obtain a wave form with as few disturbances as possible so that the reflection of the wave could be noticed easily. It was for this reason that a pump was not used at the upstream end and instead a compression chamber with compressed air was thought necessary. However, there were some disturbances that could not be eliminated, The way in which the solenoid valve closed, produced one such disturbance that appears in every cycle of the wave and was taken into account in the theoretical program, It can be seen that in every case, the experiment and theory agree in the first half of the first cycle. There is agreement both in magnitude and form of the wave. In Figures 14 and 16 the reflection from the minor loss can be seen and compared with the steady-state minor loss. Thus, the validity of Equations (56), (57), (58) and hence of Equations (13) and (15) is upheld. It is only in this part of the -49

-50PIPE WITH MLOSS Plate V. Water-Hammer Pressure-Time Diagram. Case l(a), 1 cycle VO = l.20'/sec Temp. 78~F Pressure Scale 1 cm = 30 psi HO = 194' Time Scale 1 cm = 5 m.secs.

PIPE WITH MLOSS PIPE WITH MLOSS 1 2 Plate VI. Water-Hammer Pressure-Time Diagram. Case l(a) 4 cycles VO = 1.2'/sec Temp. = 78~F Pressure Scale 1 cm = 30psi HO = 1940' Time Scale 1 cm = 10 m.secs.

-52PIPE WITH MLOSS Plate VII. Water-Hammer Pressure-Time Diagram. Case 1 (b) 1 cycle VO = l.2'/sec. Temp. = 750F Pressure Scale 1 cm = 30 psi HO = 193' Time Scale 1 cm = 5 m.secs.

PIPE WITH MLOSS PIPE WITH MLOSS i~~~~~~~~~~~~~~ 2 Plate VIII. Water-Hammer Pressure-Time Diagram. Case 1(b) I cycles VO = 1.2'/sec Temp. = 750F Pressure Scale 1 cm = 30 psi HO = 197.0' Time Scale 1 cm = 10 m.secs.

PIPE WITH TMLOSS LAMINAR FLOW PIPE WITH MLOSS LAMINAR FLOW i 2 Plate IX. Water-Hammer Pressure-Time Diagram. Case l(c) VO = O.00'/sec Temp. = 770F Pressure Scale 1 cm = 8 psi HO 4= 66 Time Scale 1 cm = 10 m.secs.

PIPE WITH MLOSS LAMINAR FLOW PIPE WITH MLOSS LAMINAR FLOW 1 2 Plate X. Water-Hammer Pressure-Time Diagram. Case l(d) VO =.330'/sec Temp. = 760F Pressure Scale 1 cm = 8 psi HO = 49.0* Time Scale 1 cm = 10 m.secs.

STRAIGHT PIPE STRAIGHT PIPE i~~~~~~~~~~~~~ 2 Plate XI. Water-Hammer Pressure-Time Diagram. Case 2(a) VO = 1.212'/sec. Temp. = 790F Pressure Scale 1 cm = 30 psi HO = 194.-0 Time Scale 1 cm = 10 m.secs.

STRAIGHT PIPE STRAIGHT PIPE 2 Plate XII. Water-Hammer Pressure-Time Diagram. Case 2(b) VO = I.20'/sec Temp. = 75 0F Pressure Scale I cm = 30 psi HO = 194.0* Time Scale 1 cm = 10 m.secs.

STRAIGHT PIPE LAMINAR FLOW STRAIGHT PIPE LAMINAR FLOW -~~~~~~~~~~~~~1 2 Plate XIII. Water-Hammer Pressure-Time Diagram. Case 2(c) VO = 0.299'/sec Temp. = 77F Pressure Scale 1 cm = 8 psi HO = 47s.0 Time Scale 1 cm = 10 m.secs.

400 EXPERIMENTAL — _ — — THEORETICAL,,....... ______ HEAD CAUSING FLOW HO I I~^ I i_ __-_- STATIC HEAD Z ^I 300 0 hz C_ 2.00 _ CL)W~~~~~~~~ |1~~ ~ A~ 5~M,.LOSS 03 I 1.00 I L00 I I I I I I I 0 5 10 15 20 25 30 35 40 TIME IN M.SECS. CASE I(a) Figure 14. Water-Hammer Pressure-Time Diagram. At x 3 L/4 Pipe with Minor Loss in Middle. Turbulent Flow. HO = 194.0'; VO = 1.20'/sec.; Temp. = 78F.

EXPERIMENTAL 400 - THEORETICAL -________ - HEAD CAUSING FLOW HO 0 - _ STATIC HEAD 300 300 I:., Q ~: t; ^1 \ i V 1. 100 0 20 40 60 so 100 120 140 16 n~.l l BI II I I ] TIME IN M. SECS. CASE l(a) in Middle. Turbulent Flow. H 1.0; V 1.20/se. Temp. ~I 100 M L

400- EXPERIMENTAL ___ T_ -THEORETICAL,/ — ^' _.__ ___.. _ _HEAD CAUSING FLOW HO |- __ _- __- __STATIC HEAD 300 eT I 200 cr.~~~~~~ —,1.,.Mi - Fo mm 1T c: w' I a. 0.) ~ 0 5 10 15 20 25 30 35 40 45 TIME IN M.SECS. CASE I(b) Figure 16. Water-Hammer Pressure-Time Diagram. At x. L. Pipe with Minor Loss in Middle. Turbulent Flow. HO = 193.0'; VO = 1.20'/sec.; Temp. = 750F.

EXPERIMENTAL 400 ---------------- THEORETICAL 0, _______ HEAD CAUSING FLOW HO I - _- STATIC HEAD LL 0 0:LL - -. I — z I I IL W~~~~~~~~~~~~~~~~~~~~~~ I I I~~~~ CL 100 ~~~~I~~ ~ I I ~ ~ ~ ~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ I 0 20 40 60 80 100 120 1401 I I I II ~TIME IN M. SECS.MLOSS CASE I I (b) Figure. Water-Hammer Pressure-Time Diagram. At x L. Pipe with Minor ossI in Middle. Turbulent Flow. HO = 197.0'! VO = 1.2* Isee.; Temp. = 750. tn 100 I I I'-' Cl) 0. I O 20 40 60 80 100 120 140 160 TIME IN M. SECS. CASE 1(b) Figure 171. Water-Hammer Pressure-Time Diagram. At x - L. Pipe with Min~or L~oss in M~4die. Thrbulent F~low. HO = 197.0'; VO - 1.2'/sec.; Temp. - 750F.

100- EXPERIMENTAL -. — ~THEORETICAL HEAD CAUSING FLOW HO rf\^ -__-_ _-_STATIC HEAD 80 I 80 I I I~~~~~ QZ I I~~~~ I I;Q: 80 i \ 0-I\ i r " C0 20 40 60 s \/ L 00L I I ^ A'i \;'~~~~~~~~~~~~~~~~~~~~~~~~~L~ ) v I^.iIM i i L * III I\I I' I^ J 0 20 40 60 80 100 (20 140 160 TIME IN M.SECS. CASE 1(0 Figure 18. Water-Hammer Pressure-Time Diagram. At x ~ 31/4- Pipe vitn Minor L~oss in Middle. Laminar Flow. HO ~ 46.6*; VO ~ 0.300*/sec.; Temp. ~ 77^.

EXPERIMENTAL 100 -00 - THEORETICAL...HEAD CAUSING FLOW HO __ _STATIC HEAD I III i~~~~~~~~~~~~~~~~ 80, I I II' 80 )-!I;II I ~~~~~~~~~~~~~~I ~;Q I ~ ~~~ ~~~I I1 I L 60! o ~ ~ ~~~ ~ ~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~I I - I1 I I I W~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ La.. I ~ ~ ~ ~ ~ ~ ~ ~ ~~~~~~~~~~~~I i I II W~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ tF -__ I~~ ~ ~ ~ ~ ~ ~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~. I - u,o trl c~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~' W~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ cr 20 I I ~I I I 0 20 40 60 so 100 120 140 6 TIME IN M SECS. I I I~~' I I CASE I(d) Figure 19. Water-Hamnmer Pressure-Time DiaLgram. At x L. Pipe with Minor Loss in Middle. Laminar Flow. HO = 49.01; VO = 0.330'/sec.; Temp. = T60F. Cl) ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~I I rv 20I I II 0~~ ~ ~ ~ ~ ~ ~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ I! 0 20 40 60 80 100 120 140 160" TIME~ ~ ~~~~~~~~''- IN".SCS CASE 1(d)~~~~~~~~~~~~~~~~~~~~. Figure~~ ~ 19... Wae-Hme Prssr-TieDarm tx- L. PiewthMnr"osi Midl. amna Fow O i9.I;VO-.30/sc. Tmp%-76F

EXPERIMENTAL - ^-. —--— THEORETICAL 400- ---- -__ -_HEAD CAUSING FLOW HO a:: I I I. cr~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~c LL. I I I 300 s *,_ I \ / I I 0~~~~~~~~~~~~~~~~~ I ii LCI ^/ I I / z 3: k! <v LI^I II i I^\ ---- L-~~ - - - w200 _ 01 ~~ 1~1~1~1~~1~~ Cr.~~r I~~~~~~~~ I ~ U)\ I I I (r100 0 20 40 60 80 100 120 140 160 TIME IN M.SECS. CASE 2(a) Figure 20. Water-Hammer Pressure-Time Diagram. At x 3 L/. Straignt Pipeline; Turbulent Flow. HO = 194.0; VO = 1.212'/sec.; Temp. a 79F.

..EXPERIMENTAL -~ ~.^THEORETICAL 400 -....HEAD CAUSING FLOW HO - l I -l/- -,/I IJ.I I — r-...... I -., I I I i I I I, Li 0 300 LL I i LI.! I * z I w 200'C _b f J oI I u, I I 0 20 40 60 s0 100 120 140 160 TIME IN M.SECS. CASE 2(b) Figure 21. Water-Hammer Pressure-Time Diagram. At x - L Straight Pipeline; Turbulent Flow. HO - 194.0'; VO 1.20'/sec.; Temp. = 75>F.

EXPERIMENTAL THEORETICAL r.... —- -.......,. —-...._.-.-.- HEAD CAUSING FLOW, I! i' H ^~~~~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~ I I ""ti 100 I'' i 0 8 I N~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ II I I I Ij ~/ I m. I o 80i I I I I I~ ~~~~~~ I U. u0 i i z " \I I i'' ^{' 1~~~~-.-.1 1*~ -- -- - I i~ A/'I V\ ^ *^' 60 - w i 2:~~ ~ ~~~~~~ JI I I I 1 4 0 - UJ<3 I I Cl) ~~~ ~ ~~~~~~~~~~~~~~~~~~~~~~I! a. I~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ u.I~~ ~~~~~~~~~~~ I I I I I 20 i 0 I ~ ~~~~~~~~~~~~~~~~~~~~~ I I i I O 20 ~~~~40 6'0 80 100 120 14016 TIME IN M. SECS. CASE 2(c) Figure 22. Water-Hammer Pressure-Time Diagram. At x - 3L/4 Straight Pipeline; Laminar Flow.

-68 - diagram that the theoretical program exactly depicts the experimental conditions. When the pressure in the pipeline falls below the static head HO for the first time, air that was dissolved in the water at the static head HO, is liberated and begins to cushion the wave front. This effect can be seen in all of the remaining cycles, by the discrepancy which developes between the experimental and theoretical traces. This situation is more pronounced in the case of the pipeline with a minor loss than in the straight pipeline. This is so because the constriction at the orifices produces a far lower pressure than in the case of the straight pipeline and consequently, more dissolved air is set free. There are several reasons to make one believe that the difference between the theoretical and experimental curves was due to gas liberation alone. First, the water used in the experiment was drawn from the sump in the laboratory. This water is far from pure as many additives are added for various purposes, such as rustpreventives and algae inhibitors. These compounds, like Chlorox and dilute Hydrochloric acid, when dissolved in water introduce gases like Chlorine, making the water more susceptable to gas liberation when subject to low pressures. Second, at the end of each experiment, bleeding of the pipeline at the orifice and at the valve would indicate the presence of small bubbles. Third, the discrepancy occurs only after the first drop in pressure below static pressure. The discrepancy cannot be attributed to an incorrect value of the steadystate loss coefficient or to the fact that the unsteady loss coefficient

-69was not used since the same type of discrepancy exists in the case of the straight pipeline. Finally, the nature of the discrepancy itself leads one to recognize it as one of air liberation, In nearly all the cases, air bubbles act as a spring cushioning the pressure change, allowing the maximum pressure to be reached after some time. This phenomenon of air liberation could not be handled analytically because of several reasons. First, the mass of air set free would have to be determined by trial and error, and its distribution along the pipe length would also have to be assumed. Also, even if the above two factors were known, new equations would have to be developed to depict this phenomenono The equations used (3,16,59) for large air chambers' would not be applicable in these circumstances for the following reason. Since the volume of air is so small, such large pressure changes are bound to produce heat, which would be dissipated in the water. This additional form of energy loss would have to be taken into account in the new equations. Because of these difficulties, the theoretical program did not take the effects of the liberated air into account. Every care was taken to remove all the free air from the pipeline before the experiment began. However, it was impracticable to modify the experiment so that there was no air liberated from solution. Since a closed circuit was not feasible, use of distilled water would not have made any difference, It was also felt that use of an oil would not have helped, since most oils have volatile components which would vaporize at low pressures.

-70There are some diagrams in which the theoretical and experimental wave speeds differ. This discrepancy is believed due to errors in the sweep rate of the oscilloscopeo It was found that there was a 6% error in the sweep rates of the oscilloscope., Calibration with an audio oscillator reduced the error to 1%, which error is noticeable in the traces. For the case of laminar flow, it can be seen that the calculated and actual wave speeds do not differ by more than lo% This is in contrast to the experimental results reported in Reference 780 The magnitude of the pressure wave also agrees fairly well with theoretical results, despite the assumption of uniform velocity distribution over the pipe area. However, it is not possible to see the reflection from the minor loss, because of the disturbances of the free air. A more elaborate theoretical study of waterhammer in laminar flow is presented by Rouleau.(71) However, in the experimental verification of his theory he used oil flowing in a pipeline, and because of the volatile constituents of the oil, the experimental and theoretical traces did not match well.

IXo CONCLUSIONS Whenever a water-hammer wave encounters a device causing a sharp energy loss, a reflection wave is sent back from it. The A(MLOSS) magnitude of this reflection is equal to - (2. The wave transmitted on is equal to the value of the approaching wave plus (MLOSS) Although the solution to the problem of a minor loss 2 in a pipeline can be obtained graphically, it does not give one a clear idea of the mechanics of the problem. The method of characteristics is a very simple and efficient way to provide particular solutions for the water-hammer equations, including friction effects. When setting up a computer program for water hammer in a pipeline with a minor loss in it, conditions at the minor loss must be treated as a boundary condition. The equations at the boundary condition verify the conclusions about the magnitude of the reflection wave, reached earlier. It is sufficiently accurate to use the steady-state loss coefficients of the device, at least for the case of instantaneous and rapid gate closures, even though water hammer is a markedly unsteady phenomenon. An experiment conducted to verify the theoretical equations provided fairly good agreement between the experimental pressure diagram and the theoretical curve, thus validating the theory. -71

APPENDIX I COMPUTER PROGRAM AND PRINT OUT FOR WATER HAMMER IN A PIPELINE WITH A MINOR LOSS IN THE MIDDLE -72

MAD ( 9 JAN 1963 VERSION) PROGRAM LISTING......... WATERHAMMER IN A PIPE WITH A MINOR LOSS IN THE MIDDLE OF PIPE LINE. CASE OF RAPID CLOSURE OF VALVE. TIME OF CLOSURE TC=12.OMILLI SECS. -- --------- INOR LOSS CREATED BY 3 ORIFICES. KOR= _. READ DATA 031 -_ PRINT COMMENT S1WATERHAMMER IN A PIPE WITH A MINOR LOSS IN TH -- _____ 002 IE MIDDLE OF PIPELINE S *002 PRINT COMMENT S CASE OF RAPID CLOSURE. TIME OF CLOSURE TC=12. *003 10 MILLI SECS.S *003 PRINT'COMMENT S MINOR LOSS CREATED BY 3 ORIFICES. KOR=F(R)._ S -004 PRINT RESULTS HO,D,E,Kl,B,N,L,NU,VO,TEMP,P,TC *005 INTERNAL FUNCTION (S) - 006 P.CTOWSTTEEW i I NTERND*.AL__f UNCS*Vt O.)NU_ X S ) ______________________________________________,_,___________________________ ENTRY TO FR, *007 FUNCTION STATEMENT R=D.ABS. ( S*V)/NU *008 WHENEVER R.L. 1.0 *009 FOR FRICTION FACTOR i- F=64.Q0 00 _ —-------------------------------------- F=64./R _ __ __ ___ _- _0 12 FONRE0./(R.P.__________ ____________________________ ________________________________________ ___________________Q___1_ OR WHENEVER R.L. 100000..013 F=0.316/(R.P. 0.25).014 END OF CONDITIONAL.015 _ FUNCTION RETURN F -... rEND OF FUNCTION *017 INTERNAL FJ.TLIQON i- - — ________________ —----— __-__.__-___ ___-o ----— _ —-— _ —----- ENTRY TO KOR. *019 WHENEVER R.L. 250. *021 FUNCTION STATEMENT KORIF=1500.____ -- ----- ------------ A02OR WHENEVER R.L. 600. *023 FOR LOSS COEFFICIEN KORIF=4700./(R.P. 0.237) -024 KORIF=1390./IR.P. 0.0205)_ -____________________________*026 OR WHENEVER R.L. 10000.0 END OF CONDITIONAL *027 FUNCTION RETURN KORIF _____ 028. END OF FUNCTION *029 INTERNAL FUNCTION (ILCOEF) ____030 _ _ _ ENTRY TO BCOND. *031 FUNCTION STATEMENT C4=LCOEF*VO/VO/(8..32.2*HO*BETA) *332 C5=(V(I-1)+V(1+2))/2.+(H(I-1)-H(1+2))/(4.*BETA) *033 FOR MINOR LOSS 1-L*VO*DELT*(FR.(V(1-1))*V(I-1)*.ABS.V(I-1)+FR.(V(I+2)) 033 2*V(I+2)*.ABS.V(1+2) )/(2.A*D) *033 BOUNDARY CONDITION VA=O.- — 3 WHENEVER LCOEF.G. 100000. *035 VP(I)=SQRT.(C5/C4-VA/C4).036 OTHERWISE *037 VP(I)=C5-C4"VA*.ABS.VA *038 END OF CONDITIONAL *039 WHENEVER.ABS.(VP(I)-VA).L..ABS.(VA/200.), TRANSFER TO WES 0____ 40 VA=VP(I) *041 TRANSFER TO MISS *042 VP(I+1)=VP(I) *043 MLOSS=LCOEFVVO*VO*VP(I)-.ABS.VP(I)/(64.4*HO) *044 HP(I)=BETA-(V(I-l)-V(I+2))-BETA~L*VO*DELT*(FR.(V(I-1))*V(I-1) *045 -----------------------— 4 5 —--------------------

1*.ABS.VII-1)-FR.(V(1+2))*V(1+2)*.ABS.V(1+2))/(A.D)_+IHJI-:1.1 045 2(1+2) )/2.+MLOSS/2. 045~ ~-~-Hi~ —~-~J G~T —-------------------- ------ ------------------------ -- ----------- -----— I —--- -------- -- ------------------------ ------— ~ HP( I+)=HP( I )+-MLOSS.046 FUNCTION RETURN.047 END OF FUNCTION _._048 A-SQRT.(l(32.2.KL/62.4)/(1.+(KlD5.O592)/(E'B)))'049._QkLA:.~ 1~/N -QNQ DIMENSION V O H VP O0 HP O0 EX TAU O) 051 CALCULATION OF INTEGER ItJtNtP *052 BETA=(A.VO)/(64.4.HO).053 CONSTANTS DELT=0.5/N _..-_ — --- *Q RO=D*VO/NU.055 DXFEET=L*DELX.056 DTSECS=DELT*2.~L/A ~057 TCLOSE=TC*A/( 2.*L) *.____________________________________058____________ PRINT RESULTS A, OELX,DELTBETARODXFEET,DTSECSTCLOSE ~059 MLOSS=KOR.(VO).*VO.VO/(64.4HO) _ _ _ _ _ 060 _ HLOSS=VO~ VO/ I1.04~HO) -061 KGV=64.4*HO/(VO0VO)-KOR.(VO)-(FR.(VO))*L/D-62.0.062 INITIAL EXO)=O ------------—. __ __ 066 - HP (0)=1 *0067 CONWITIONS VPtOJI.O_ -- 068 - THROUGH BOSTON, FOR 1=1,1,1.G.P 69 VP(I)=1.0.070 HP(1)-HP(1-1)-HF o071 — EXUI)=EX(1-1)+UELX -- - -— _ — ~ -- _072 — WHENEVER I -E. 21 ~073 HP__( _ _I __) ____P LU M O S H. _... _. ~. _.... _.. _ ~. _.. _... ~. _ ~.._ 074....... _.......... _ ~.........^ HP(I)=HP(IJ-MLOSSHF —-0-4 EX( I)=EX(1I)-DELX.075 OR WHENEVER I *E. 42.076 EX<I)=EXII)-DELX ~077 HPII )HP(I)+HF-HLOSS _078 END OF'CONDITIONAL ~379 CONTINUE _080 TIME=T*2.LL/A ~081 PRINT RESULTS TIME.J.TAU(J).082 PRINT FORMAT RSULT1,L*EX(O),L*EX(5),L*EX(IG),L.EX(I5),.083 IL*EX(20)_,LEX(21),L.EX(26),L.EX(31),L*EX136),LEX(41) 0 -— _.-______-.-_________-0_83_~PRINT FORMAT RSULT2,HO.HPIO),HO*HP(5),HO*HP(10),HO.HPli5),~I' 084 IHO*HP(20)1H0.HP(.21)_tHO.HP(26 )_HO.HP Ii(31) HOHP(36)_tHO.HPI(41) _084 PRINT FORMAT RSULT3,VO*VPIO),VO*VP(5),VO*VPIlG),VO'VP(15)~.085 IVO.VP(20),VO*VP(21),VO*VP(26),VO*VP(31),VO.VP(36),VO1 VP?41).085 VECTOR VALUES RSULTI=$IH,6H EX=IOF1O.2.$.086 VECTOR VALUES RSULT2=$IH t6H HEAD~IOF10.4 _$.087 VECTOR VALUES RSULT3=SIH,6H VEL. =10FIO.4*$ ~088 JT=~J-O-LT ~090 ~ —------------------ ----- ------ TIME=T*2.*L/A.091 WHENEVER T.G.6.0, TRANSFER TO AOK.092 WHENEVER T L1. TCLOSE ~093 T THROUGH MYSORE, FOR I=OilIG.P'094 H( I)=HP( — ) ~095 -------------------------------- V(I)=VP(I).096 THROUGH NUYORK, FOR I=1,1,I.E.P *097 WHENEVER I.NE.20.OR. I.NE.21.OR. I.NE.41.OR. I.NE.42.098 HP(I)=(H(I+1)+H- I-1))/2.-BETA(_(Vl+1)-V(l-l))+BETA*VO.LDELT* ----------- _099 -------------------------------- l(FR.(V( 1+1) )*V(f+1)*.ABS.V(l+1)-FR. (V(1-1)').*V(-1)i.ABS.V(I-1 l099

2))/IA~D) ~099 VP2 I)={VAD - 1)+V(1+1))/2.-IHI1+1)-HI1-1))/(4..BETA)-VOeL.DELT ~100 VP uVpi~viI-1)4v( i*100~^)Hi^)/4^77v^T ET~"~~~~io~~~~~~~ 1*(FR.(V(I-1))*V(I-1)e.ABS.V(I-1)+V(I+1)*FR.(V(I+I1))*.ABS.V(I+.100 21))/12.cA~D) ~100 END OF CONDITIONAL. _ 101 CONTINUE.102 VP(O)=V(1i)+1.-H(I))/(2.'BETA)-FR.(V(1))'L~VO*DELT'V(1)~.ABS.'104 1V(1)/(A~D) *104 K3=KOR.(VI20)) m105 CA.cUULrnON OF _ i Qr._UTE jBQN_ K___3)..Q- - - - - CALCU~m OM OP ~^ ^ U L^ CQN~ l20^ K3J~..~..~~~...~~.............~.._....~...__~~ __...~~__.~._.Ji_..__._._._._...__ K3=KOR.(VP(20)) *107 VANOM FOR EXECUTE BCOND.(20, K3) 108 KSV-62.0/(TAU(J)*TAU(J,)) *109 TiMETLEES EXECUTE BCOND.(41,KSV) ~110 C1=KGVTVOVO/(4.*32.2oHO*BETA) ~111 THAN TC. C2=L~VODELT*V(P-I)*.ABS.V(P-1)*(FR.(V(P-1)))/(A~D)-V(P-1)-H -112 - ~ ~ ~ ~ - ~ -- ~ - - - - ~ - ~ - - ~ ~ _ ~~~~~~~~ —-- -._ _- ~- ~ —- - - - - - -- - - - -— _ —_-_ - _ —-- ----- I(P-1))/(2..mBETA) ~112 VP(P)=(SQRT.(I1.-4.*C2.CI)-1.)/(2.*C1).113 WHENEVER VP(P).1.0..OR. VP(P).G. 1.01.113 Cl=-Cl.115 VP(P)=(1.-SQRT.(1.-4.*C2*Cl))/(2.-Cl) 0116 END OF CONDITIONAL ~117 HP(P)=KGV.VO*VO*VP(P)*(.ABS.VP(P))/(2.*32.2*HO) -118 PRINT RESULTS J,TIME,TAUIJ) 119 PRINT FORMAT RSULT1,L*EX(O)tLEX(5),L.EX(1C),L.EX(15), 1201 IL.EX(20),L.EX(21).L*EX(26)tL-EX(31),L~EX(36)tL~EXI41) e120 PRINT FORMAT RSULT2,HO*HP(O),HO*HP(5),HO*HP(10),HO.HP(15),.121 IHO*HLP 120J) H0_HP(2. ).LHO.HP26) (HO.HP 31J)H0.HPf36),HO*HP(41I_ —-..._-_ —.-._ —_-_-tJ.. -- - - PRINT FORMAT RSULT3,VO.VP(O),VO.VP(5),VO.VP(10),VQOVP(15),.122 VQ*YP (20) I. VOQ*VLOl-LLQ!Y-.JL2, P-IYQ-P-L.Ly-QeJ______ - zz-L --- -- OTHERWISE.123 - THROUGH BARODA! FOR 1=0.1I.I.G.41 *124 H(I)=HP(1) ~125 V(I)=VP(I) -— 126 THROUGH ANNARB, FOR *i1, iI.4E.4 ~127 WHENEVER I.NE.20.OR. I.NE.21 *128 I HP(I)=(H(I+1)+H(Il-1))/2.-BETA(V(I+)-V(I - i ))+BETAVOLDELT 129 1(FR.(V(1+1))*V(1+1)*.ABS.V(1+1)-FR.(V(1-1))~V(1-11*.ABS.V(I-11 129 2))/(A.D) *129 VP(I)=(V(I-1)+V(1+1))/2.-(IH(I+1)-H(I-1))/(4.*BETA)-VO*L.OELT _130 1lIFR.(V(I-1)))V(I-1)~.ABS.V(I-1)+V(I+I)~FR.(v(I+1)).oABS.v(I+ ~130 21) )/(2.-ADD) * 130 END ~OF CONDIT[IONAL 131 CONTINUE *132 HP(0)=1.0.133 CALCULATION OF VP(O)=V(1)+(O.-H1))/(2.*BETA)FR.(V(1))}L.VO*DELTV(1).ABS. 134 1V(1)I(A*D).134 V!AND H FOR! K3=KOR.(V(20)) -135 EXECUTE BCOND.(20, K3) ~136 TIME T GREATER K3=KOR.(VP(20)).137 EXECUTE BCOND.(20, K3).138 THAN TC. VP(41)/-. _! - -1.9 — ---- — _._.._... HP(41)=H(40)+2..BETA.V140)-2..BETA.L.VO.FR. (V(40)).DELTV40) 140 1 *(.ABS.V(4C))/(A*D) _140 PRINT RESULTS J,TIME ~141 PRINT FORMAT RSULTLL.EX(0),L*EX(5)tL.EX(10),L.EX(15),.142 IL*EX(20),L*EX(21)tLEX(26),L*EX(31),L*EX(36),L*EX(41) e142 PRINT FORMAT RSULT2,HO*HP(O),HOHP(5),HO.HP(1)_HOl__ _*HP(15), ____ _____._143_ iHO.HP(20),HOQHP(21),HOHP(26),HO*HP(31),HOHP(36),HO*HP(41) -143 ____-~- PRINT FORMAT RSULT3yVOeVP(O)tVO*VP(5)gV0VP(IIlOJLyVOVPlei5),_._ — -.144 IVO.VP(20),VO.VP(21),VOVP(26),VO.VP(31),VO*VP(36),VO*VP(41) *144 - ~___~_____!_____ _END OF CONDITIONAL.145 J=J+!.146 TRANSFER TO BOMBAY.147 AOK CONTINUE ~148 END OF PROGRAM 49 THE FOLLOWING NAMES HAVE OCCURRED ONLY ONCE IN THIS PROGRAM. COMPILATION WILL CONTINUE. TEMP.- -....

___ WATEHAMMER. _ P.PEl ITH NOR LOSS IN THE MIDDLE OF PIPELINE CASE OF RAPID CLOSURE. TIME OF CLOSURE TC=12.0 MILLI SECS. MINOR LOSS CREATED BY 3 ORIFICES. KOR=FIR). _ HO = _46.T60000Q0, __________ D_ = 7.040Q00D____.___________ E = 2.450000E-09,__ —---------- Ki_ 4.630000E 07 VO =.299000. TEMP = 77.000000. P 61. TC =.012000 RO = 135T.688324 ~ DXFEET =... ___ 1.000000_3 D —EC 4..a...__TL __-... TIME =.000000, J = O. TAUJO) - 1.000000 EX-.00 5.00 10.00 15.00 20.00 20.00 25.00 30.00 35.00 40.00 HEAD- 46.6000 46.5751 46.5503 46.5254 46.5005 44.9289 44.9040 44.8792 44.8543 44.8294 VEL.-.299.2 990. 2 990.2 990.2990.2990.2990.2990.2990.2990 J = 1, TIME = 2.231153E-04, TAU(1) =.997000 EX=.00 5.00 10.00 15.00 20.60 20.00 25.00 30.00 35.00 40.00 HEAD- 46.6000 46.5751 46.5503 46.5254 46.5453 44.8841 44.9040 44.8792 44.8543 44.8297 VEL.-.2990.2990.2990.2990.2987.2987.2990.2990.2990.2990 J = 2, TIME = 4462306E-04_ TAU(2) =.993000 EX-.00 5.00 10.00 15.00 20.00 20.00 25.00 30.00 35.00 40.00 HEAD- 46.6000 46.5751 46.5503 46.5254 46.5454 44.8840 44.9040 44.8792 44.8543 44.8301 VEL.-.2991.2991.2991.2~91.2987.2987.2991.2991.2991.2990 J = 3, TIME = 6.693460E-04, TAU(3! =.992000 ~ EX-.00 5.O0 O. 000 _____5_,0_____2Q -_- 25 0 30.00 35 ----—.-0..O _QQ _ I HEAD- 46.6000 46.5751 46.5503 46.5254 46.5456 44.8838 44.9040 44.8792 44.8543 44.8302 VEL.=.2991.2991.2991.2991.2988.2988.2991.2991.2991.2991 ~J = —~~ 4~a — TIME - 8.9246113E-.04.JL....-. —-TAUI4-~_ - --- 95-QYQ... 99QQ...... -__ —_ —----- EX-.00 5.00 10.00 15.00 20.00 20.00 25.00 30.00 35.00 40.00 HEAD= 46.6000 46.5751 46.5503 46.5254 46.5457 44.8837 44.9040 44.8792 44.8543 44.8305 VEL.=.2991.2991.2991.'2991.2988.2988.2991.2991.2991.2991 J = 5, TIME = 1.115577E-03, TAU(5) -.985000 EX=.00 5.00 10.00 15.00 20.00 20.00 25.00 30.00 35.00 40.00 HEAD- 46.6000 46.5751 46.5503 46.5254 46.5458 44.8836 44.9040 44.8792 44.8543 44.8308 _ VEL.=.2991.2991.2991.2991.2988.2988.2991.....991 -.......2991 — ____-._ — _. J = 6. TIME - 1.338692E-03. TAU(6) =.980000 EX-.00 5.00 10.00 15.00 20.00 20.00 25.00 30.00 35.00 40.00 HEAD- 46.6000 46.5751 46.5503 46.5695 46.5460 44.8834 44.8599 44.8792 44.8546 44.8313....... VEL.=.2992.2992.2992.2988.2988.2988.2988.2992.2991.2991 J = 7, TIME = 1.561807E-03, TAUM7! =.975000 EX=.00 5.00 10.00 15.00 20.00 20.00 25.00 30.00 35.00 40.00 HEAD0 46.6000 46.5751 46.5503 46.5696 46.5461 44.8833 44.8598 44.8792 44.8550 44.8317 VEL.=.2992.2992.2992.2989.2989.2989.2989.2992.2992.2992 -— _ -J - --- j = 8, —-— ~~-... TIME = 1.784923E-03_L...T.AU - 9 Q —-------------------------------- EX=.00 5.00 10.00 15.00 20.00 20.00 25.00 30.00 35.00 40.00 HEAD= 46.6000 46.5751 46,5503 46.5697 46.5462 44.8832 44.8597 44.8792 44.8551 44.8322 VEL.=.2992.2992.2992.2989.2989.2989.2989.2992.2992.2992 J = 9 TIME = 2.0G8038E-03, TAU(9) -.960000

APPENDIX II COMPUTER PROGRAM AND PRINT OUT OF WATER HAMMER IN A STRAIGHT PIPELINE -77

------- C I — J ^MP~ —--- ----- ----- ----- ----- ----- ----- ----- ---------- 004211 04/09/63 1 2 4 MAO D 9 JAN 1963 VERSION) PROGRAM LISTING......... WATERHAMMER IN A STRAIGHT PIPELINE CASE OF RAPID CLOSURE OF VALVE. TIME OF CLOSURE=12.0 M.SECS. __________________________________________________ PRINT COMMENT $l WATERHAMMER IN A STRAIGHT PIPELINE S'0:1~ -^~~~~~^~~.~.-.eRINT.Cg~MMN_-A-CASE —OF__RAPID.O.CLQSURE.L TIME OF CLOSURE TC=12. UL2 IC MILLI SECS. S.'0)2 READ DATA.0 3 PRINT RESULTS HOOEiKlPB,NpLvNUtVOTEMP,P,TC *.004' INTERNAL FUNCTION (S) * 0.5 ENTRY TO FR. ~006 FUNCTION STATEMENT R=D*.ABS.(S*VO)/NU 0.7 WHENEVER R.L.1.0 *008 FOR PIPE FRICTION F=64..009 OR WHENEVER R.L. 2000.'010 FACTOR F=64./R 011 OR WHENEVER R.L. 100000.'012 F=C.316/(R.P. 0.25).013 ENO OF CONUITIONAL"" "014 —--------- --------- -------------- FUNCTION RETURN F ______- -_ — - - -_______ - - -_ - - ---.015 END OF FUNCTION'016 INTERNAL FUNCTION (ILCOEF).017 ENTRY TO BCOND. ~018 C4=LCOEF*VO*VO/(8.*32.2.HO.BETA) *019 C5=(V(I-1)+V(1+2))/2.+(H(i-1)-H(1+2f)/(4.*BETA) 0 — FUNCTION STATEMENT 1-L*VO*DELT*(FR.(V(1-1))*V(I-1)~.ABS.V(1-1)+FR.(V(I+2)) *020 "0 2.V(I+2)..ABS.V(1+2))/(2..A.D).020 FOR MINOR LO SS VA=O..021 WHENEVER LCOEF.G. 100000..022 BOUNDARY CONDITION VP(I)=SQRT.(C5/C4-VA/C4).023 OTHERWISE -024 VP(I)=C5-C4*VA*.ABS.VA _025'END OF CONDITIONAL'026 WHENEVER.ABS.(VP(1)-VA).L..ABS.(VA/200.), TRANSFER TO WES.027 VA=VP(I).028 TRANSFER TO MISS.029 VP( 1i+i)=VP(l) 0030 MLOSS=LCOEF*VO*VOgVP(I)*.ABS.VP(I)/(64.4*HO).031 HP(I)=BETA.(V(i-1)-V(i+2))-BETA*L*VO.DELT*(FR.(V(I.-i))*V(i-1).032 1.ABS.V(I-1)-FR.(V(1+2))*V(1-2)*.ABS.V(142))/(A*D) +(H(1-1)+H *032 21 I2))/2.+MLOSS/2.'.32 HP(I+1)=HP(I)-MLOSS 0u33 FUNCTION RETURN' 034 END OF FUNCTION J035 A=SQRT.((32.2*Ki/62.4)/(i.+(KI*0*0.92)/(E*B))) *036 DIMENSION V(10C),H(100),VP(100),HP(100),EX(100),TAU(100)._37 ___________________________ DELX=1./N *038 CALCULATION OF INTEGER I,J,N,P.039 BETA=(A.VO)/(64.4'HO) *040 CONSTANTS DELT=C.5/N *041 RO=DOVO/NU.042 DXFEET=L*DELX _043 UTSECS=DELT*2.*L/A " ^44 TCLOSE=TC*A/(2.*L) *045 PRINT RESULTS A,D ELX-,- ELTaETAROOX'FEETDTSECSTCLOSE 046

L y.gi.HO0.._. _.._....... _..... KGV=2.*32.2-HO/(VOeVO)- (FR.(VO))*L/D-62.0 @048 T=0. -049 J=C -050 HF=FR.(VO).L-*VOVO/64.4-HO*ND). 051 EX(O)=O. *052 __~eLQJ_.0___.053 o^ ~ —------------------------- - ----- ------------------------------------ 053 VP(O)=1.0.054 MTIAL THROUGH BOSTONI FOR 1I1l1l1.G.P *055 VP(I)=l.O.056 CONDITIONS HP IHIt_ )}-IHF.-____.___. _._.....__. 057 EX(I)=EX(I-1)+DELX -058 WHENEVER I.E.41i 059 EX(I)=EX(I)-DELX'060 HP(I)=HP(II)+HF-HLOSS.061 END OF CONDITIONAL ~062 CONTINUE.063 TIME=T-2.L/A'064 PRINT RESULTS JjTI ME.TAU(J).065 PRINT FORMAT RSULTi,LEEX(O ), LEXI5)TLaEXiO t iiEX(1Ei5), 066 1L*EX(20),LeEX(25),LeEX(30)}LeEX(35)}tLeEX(40) e066 PRINT FORMAT RSULT2,HOeHP(O)tHO~HP(5)~HO.HP(10),HO~HP(15),.067 IHOHP_(20)tHO*HP(_25),HOeHP(30)_tHO-HP (35)tHO-HP(40) -____ —.067 PRINT FORMAT RSULT3,VO-VP(O),VOeVPI(5)VO.VP(10)VP( O VP(15),.068 IVOVP(20) tVOeVP( 25),VOVP(30).,VODVP(35),VODVP(40) -068 VECTOR VALUES RSULTt=$1H,6H EX=9F10.2'$ ~069 VECTOR VALUES RSULT2=$1H,6H HEAD=9F10.4*$S 070 VECTOR VALUES RSULT3=S 1H,6H VEL.=9F10.4-$.071 J=_.072 T=JDOELT.073 - WHENEVER T.G.6.OtTRANSF.ER TO AOK - __________.___ — 074 ___ -_.._ _ __. \O WHENEVER T.L. TCLOSE.AND. TAUIJ).G. 0.01'075 THROUGH MYSORE,FOR I=0,,I.G.P -076 H(I)=HPII).077 V(II)VP(I) _078 THROUGH NUYORK, FOR 1=l,I,I.E.P.079 WHENEVER I.NE.40.OR. I.NE.41 080 HP(I)=(H(I+I)+H(I-1))/2.-BETA-(VII+r)-V(I-}))+BET'AVO L.DELT. "081 I(FR.IV(I+I))*V(I+I)e.ABS.V(I+1)-FR.I(V(I-1))} V(I-1).ABS.ViC[-1.081 2))/(A~0) ~081 VP(I)=(V_(I-I)+V(I+1))/2.-(H(I+1)-H(I-1))/(4.*BETA)-VO.L*DELT.082 1.(FR.(V ^I-1)~(VII-1)*..ABS.V(-1i)+V(I+1)fFR.(VII+1))-.ABS.V(I+ -082 21L) /(2.-A.) 082 END OF CONOITIONAL -083 CONTINUE.084 HPIO)=1.0 0-5 CALCUMATION OF VP(0)=V(I)+(IHP(O)-H(1))/(2.-BETA)-L*VO"DELT~FR.(V(1)).V(1)~.086 1.ABS.V(I1)/ (Ae0 -386 V AND H ^PR TIME=T-2.eL/A.087 KSV=62.0/(TAU(J)*TAUIJ)) -088 TIME T L.. EXECUTE BCOND.(40,KSV).089 Cl=KGV-VOeVO/(4.e32.2*HO.BETA).090 THAN TC. C2=LEVOODELT*V(P-1).ABS.((VP- FR.(V(P-))/AD)-V(P-)-(H 091 i(P-i))/(2..BETA).-.C...... -091 WHENEVER (1.-4.-C2-CL).L.0., CI=-CL _092 VPT(P)=SQRT.(.-4.C2WCI)-I.J/(2.C1)...... -093 HP(P)=KGVeVO*VO*VP(P)*(.ABS.VP(P))/(2.*32.2-HO).094 PRINT RESULTS J,TIMEtTAU(J).095 PRINT FORMAT RSULTtLeEX(OI,L*EX(5),LeEX(10),L-EX( 5)t, 096 IL.EX(20),L.EX{ 2) )LeEX LEX(30),LEX(351)L.EX(40) -096

.- P^JL^I-FJIB^AJ-AT__PSYL~2JHO_~,, ~1PB-SI L,0~- -OAJ Q~~I -------------------------- - --- IHO.HP(20), HOHP(25)HO.HP(30),HOOHP (35),HO.HP(40) ~097 PRINT FORMAT RSULT3.VO.VP(O).VO*VP(5).VO.VP(1IO).VO*VP(15)._. 098 1VO.VP(20),VO*VP(25), VO.VP (30)),VV VOVP(35 VOVP3(40) 098 -0 99 - ---------------- THROUGH BARODA, FOR I=Oil, I.G.4C *100.H llJ }^ ^ JLL -..~_ ~_ -..__ ____..~__ ~ -__ _~ - - - -. -^ _ _ _ _ __ _ _ _ _ __ _ _ ___-~ _. -. - ~ ~ ~_.~ - _ ~ ~ _ -~ ^ LI _ - _ _ _ __ - - ~ ~ - ~ — - _ _ V(I)=VP(I).102 THROUGH ANNARB. FOR I=11,1I.E.40_ *103 HP(I)=(H(I+1)+H(I-1))/2.-BETA*(V(141)-V (1-1)l+BETAVO.L*DELT* *104 ItFA^YiI~IitLI~Y-L.+lltABeSYIIJfl-l-PYLI:UIItyi-^AB.SsVU- - ~ - JI-_-._~.__ —-- ---------- 2))/4A~O).104 VP (1) VII)_I~ ~ /_-(HIi)-(l UL(.-EAlv ~D.y__..___ ~ _ 1.5~._____ _._____~__.._..________.._...__ YP(I= VLL1 ) (H-( I 1 I - II-/ 14. * BE A)- VO*L.ODE L T.105 —--------------- 1*(FR.(V(I-1) V( I-l)*.ABS.V(l-1)+V(1+l)*FR.(V(l+1))*.A6S.V(I+ 135 21))/(2.*AOD).105 HP(O)~1.O *106 CALCU~lfOW OF VP ( 0 )_~LV ( 1 ) * ( HP_(.0 ) -Hi(1 _)_/_(_2~ 1^1 )B -' L~VO~UDELT ~~FR._( V ( l ) ) ~V ( l}__._.._ -~.~IP7__ _._~.-_~ ----- --- CALCUL~tiON OF VP(0)=V( 1)+(HP(O)-H(l1) )/(2.'BETA)-L*VO*OELT*FR.(V(1))V(1) -- - 107i.ABS.V(i) 1 A1) A ) 107 VAND H FOR VP40 -- ---—.____~. - -- - -— ___ ___-_-_4-_._.~_ —--— 8___~___ -— __ - -____.. -— ___.._ __ —- -- *108 —------------------ HP(40)=H(39)+2.*BETA*V(39)-2.*BETA*L.VO*FR.(V(39))*DELT.V(39) 109 TIME T OGREATER l(.ABS.V(39))/(A.O).109 TIME=T*2.*L/A'110 THAN TC. PR INT!1RESLS_Jj --— T —— _ —-11- -_-_- - ----------— _ —-------- - --- ---— i ll.-. —-.. —-------- ----------- PRINT FORMAT RSULT1,L*EX(O),L.EX(15),LEX(10),L.EX(15), 112 ILLEX (20),L*EX(25), LEX(30,L*EX(35),LEX(40) 112 PRINT FORMAT RSULT2,HO*HP(O),HO*HP(5),HO*HP(10),HO*HP(15), 113 IHO.HP(20),HO*HP(25),HO*HP(30),HOHP3(35)HO*HP(40) 113 0 PRINT FORMAT RSULT3,VO*VP(0),VO~VP(5),VO.VP(10),VO*VP(15),.114 ivo*VP(20)-tVQ*V 25vV- -- ---— o-vp(2_ —).3_5_. i --- ---------------- -(*VP! (35L, 9VO*VP (40) -114 ENO OF CONOITIONAL.115 t -- ---- --- --- - -- ------- --- --- ---- --- --- ---- --- --- --- ----- -— _- - - - - -- - - --- -- TRANSFER TO BOMBAY.117 AOK CONTINUE H118 ENO OF PROGRAM.119' THE FOLLOWING NAMES HAVE OCCURRED ONLY ONCE IN THIS PROGRAM." - COMPILATION WILL CONTINUETEMP _~_^OM~lL~ON.^ILLCO~NUE^__~~_..___-..___~.-. —. -.-. —------------------— ~ —-------------------------------------------------------------------------------- __ m__ —--------------------- - --- ----------------------------------------------------------------------------------------------------------------- -`-~~~' ~ -~~ —- -~` ~~~ -------- ---- ------- -------- 11 - - -----—` ----------------- - - - - - - - - - - - -- - - - - - - - - - - -- - - - - - - - - - - - - - - - - - - - - - - -

CASE OF RAPID CLOSURE* TIME OF CLOSURE TC12.0 MILLI SECS. HO 194.000000, 0. ~0440000 E 2.450000E 09, KI - 4.630:)0E.7 -----—, —---- ^ —--—, —-~ —- ----- - - - - - - - - - - - - - -- -- - - - - - - - - - - - - -- - - - - - - - - - - - - - - -- - - - -- - -- -,- - -- - - -- - - - - - - - - 8 = 4.0400OOE-03, N = 60, L 60.000000, NU 2 9.69000E- 6 ---- --- ----- - -- --------- ----------- ------- ---------------- --------- - -----—. —0-0 —------------— E —Vo 2 1.2000009 TEMP - 77.000000, P z 61, TC 2.2C A = 4481.987122, DELX a.016667, DELT - 8.333333E-03, BETA.430491 RQ = 5448.916260, OXFEET a 1.000000, DTSECS a 2.231153E-04, TCLOSE *.448199 J = 0, TIME =.000000, TAU10) - 1.000000 EX2. 00 o 5.00 10.00oo 15.00 20.00 25.00 30.00 35.00 40.00 HEAD* 194.0000 193.9107 193.8214 193.7321 193.6428 193.5535 193.4642 193.3749 193.2856 ~~VEL.- ~1.2Q0 120 00 1OO ~_.200 14Q0~~1.2000 1OL~.2.000 1.2000 1.^2000)~1.2000 2 TI —--------- -------------- - ME -- 2 2 11 3 - -AU-1- _.......... 997000 ------------------ EX-.00 5.00 l0.00 15.00 20.00 25.00 30.00 35.00 40.00 HEAD* 194.0000 193.9107 193.8214 193.7321 193.642g 193.5535 193.4642 193.3749 193,2906 VEL.- 1.20CO 1.2000 1.2000 1.2000 1.2000 1.2000 1.2000 1.2000 1.1999 ------------------------------------------------ - - ------------— ^- -- - --- - - ------------------------------------------------------------------------ - --- ----- -------------- ----- 3 2 2, TIME - 4.462306E-04* TAU(21 =.993000 EX:.00 5.00 10.00 15.00 20.00 25.00 30.00 35.00 40.00 HEADO 194.0000 193.9107 193.LZ214 193.7321 193.6428 193.5535 193.4642 193.3749 193.2961 VEL. 1.2000ooo 1.2000 1.2000 1.2000 1.2000 1.2000 1.2000 1.2000 1.1999 _3 _- 3,.. TIME 2 6.693460E-04, TAU(3).992000 --------------- — ~ -------- -~- ----------—' —---------— ~~~ -- - - - - - EX-.00 5.00 10.00 15.00 20.00 25.00 30.00 35.00 40.00 IliD LA.__HQQAD~~i l^^0^ 90Q3I9A& ^1l4~1 193.7321_ 193.428 1935525_193 4642 P3749 193.2975VEL.- 1.2000 1.2000 1.2000 1.2000 1.2000 1.2000 1.2000 1.2000 1.1999 J 2 4, TIME - 8.924613E-04, TAU(4) -.989000 EX=.00 5.00 10.00 15.00 20.00 25.00 30.00 35.00 40.00 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~__ -----.-. —------ - ----------- ---------- "HEAD- 194. 0000 - 193.9107 193.8214 2 193.7321 193.6428 193.5535 193.4642 193.3749 193.3018 VEL.2 1.2000 1.2000 1.2000 1.2000 1.2000 1.2000 1.2000 1.2000 1- 1998 J - 51 - _ TIME L.115577E-037 TAU(5) =.985000 EX=.00 5.00 10.00 15.00 20.00 25.03 30.00 35.00 40.00 HEAO- 194.0000 193.9107 193.8214 193.7321 193.6428 193.5535 193.4642 193.3749 193.3075 — VEL.2"-i o 1.2000 1.2000 1.2000 1.2000 1.2000 12000 12000 1.2000 1.1998 - 6- TIME L-.338692E-03, - TAU(6) -.980000 EX=.00 5.00 10.00 15.00 20.00 25.03 30.00 35.00 40.00 HEAD- 194.0000 193.9107 193.8214 193.7321 193.6428 193.5535 193.4642 193.3828 193.3147 VEL._ 1.2000 1.2000 1.2000 1.2000 1.2000 1.2000 1.2000 1.1999 1.1997 --------—' —---—'-^" " ------ - -- -- --- - -- -- - -- --- -- -- -- -- --- -- - -- -- -- -7________-__J- *- - 7.. ~- TIME - 1.561807-03-_ —--- TAU —1T7I) 2.975000 EX= 00 1.00 15.0 20.00 2.00 36. 35 --- --------— ~ —~ —--------- --- -~ —6 — --- J ~ —--------— 40.-O ------------------- EX-.00 5.00 10.00 15.00 20.00 25.00 30.00 35.00 40.00 HEAD= 194.0000 193.9107 193.8214 193.7321 193.6428 193.5535 193.4642 193.3884 193.3221 VEL.= 1,2000 1.2000 1.2000 1.2000 1.2000 1.2000 1.2000 1.1999 1.1997 J 8,p TIME - 1.784923E-039 TAU(8) 2.970000 EX=.00 5.00 10.00 15.00 20.00 25.00 30.00 35.00 40.00 H A 9 6-4 —----— 76 —----- 4 —-l — HEAD- 1i9.COOO -93.9'i07 193.8214 193.7321 193.6428 193.5535 193.4642 193.3898 193.3295 VEL.= 1.2000 1.2000 1.2000 112000 1.2000 1.2000 1.2000 1.1998 1.1996 J 9, TIME = 2.00803-E-03 - TAU(9) 2.960000 EX= --.00 5.00 10.00 15.00 20.00 25.00 30.00 35.00 40.00

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-85 - 47. Kessler, L. Ho "Speed of Water-Hammer Pressure Wave in Transite Pipe." ASME Trans. (1939), 11; Discussion Trans. (1939), 451 48. Kittredge, Co Po "Hydraulic Transients in Centrifugal Pump Systems." Trans. ASME (1956) 1307. 49. Knapp, F. "Operation of Emergency Shutoff Valves in Pipe Lines." ASME Trans. (1937), 679; Discussion Trans. (1939), 75. 50. Knapp, R. T. "Complete Characteristics of Centrifugal Pumps and Their Use in the Prediction of Transient Behaviour." ASME Trans., 59, Paper Hyd-59-11 (Nov.., 1937), 683-689. 51. Le Conte, J. N. "Experiments and Calculations of Resurge Phase of Water Hammer." ASME Trans. (1937), 691; Discussion Trans. (1939), 440. 52. Li, Wen-Hsiung. "Mechanics of Pipe Flow Following Column Separation." ASCE, Jour. of Engrg. Mech., 88 (Aug., 1962), 97. 53. Lister, M. "The Numerical Solutions of Hyperbolic Partial Differential Equations by the Method of Characteristics." Chapter in, Mathematical Methods for Digital Computers, Edited by Ralston, A. and Wilf, H. S. John Wiley and Sons, Inc., 1960. 54. McCaig, I. W. and Jonker, F. H. "Applications of Computer and Model Studies to Problems Involving Hydraulic Transients." ASME, Jour. of Basic Engrg., 81 (1959), 433. 55. McNown, J. S. "Surges and Water Hammer." Chapter in, Engineering Hydraulics, Edited by H. Rouse. John Wiley and Sons, Inc., New York, 1950. 56. Michaud, J. Coup de belier dans les conduites. Bull. de la Soc. vaudoise des Ing. et Arch. Lausanne, 1878. 57. Parmakian, J. "Pressure Surge Control at Tracy Pumping Plant.'" Proc. ASCE, 79 (Dec., 1953), Separate No. 361. 58. "Pressure Surges in Pump Installations." ASCE Trans., 120 (1955) 69759. Water-Hammer Analysis. Prentice-Hall Inc., New York, 1955. 60. "Water Hammer Design Criteria." Proc. ASCE (Apr., 1957), Paper 1216 in Power Journal. 61. "One-way Surge Tanks for Pumping Plants." ASME Trans., 80 (1958), 1565. Trans., 80 (1958), 1585.

-8663. Peabody, R. M. "Typical Analysis of Water Hammer in Pumping Plant of Colorado River Aqueduct." ASME Trans. (1939), 117. 64. Quick, R. S. "Comparison and Limitations of Various Water-Hammer Theories." Mech. Engrg. (1927), 524; Discussion Mech. Engrg. (1927), 1219. 65. "Development of Water Hammer Theory and Its Applications." Mech. Engrg. (1930), 376. 66. Rich, G. R. "Water-Hammer Analysis by the Laplace-Mellin Transformation." ASME Trans. (1945), 361. 67~ "Basic Hydraulic Transients." Jour. of Boston Society of Civil Engrgs., 1948. 68.'"Hydraulic Transients." 1st Ed. Engineering Societies Monographs. McGraw-Hill Book Co., Inc., New York, 1951. 69. "Water Hammer. " Chapter in Handbook of Hydraulics. Edited by Davis, C. V., 2nd Edition, McGraw-Hill Book Co., Inc., New York, 1951. 70. Richards, R. T. "Water Column Separation in Pump Discharge Lines." ASME Trans. (1956), 1297. 71. Rouleau, W. T. "Pressure Surges in Pipelines Carrying Viscous Liquids." ASME, Jour. of Basic Engrg. (Dec., 1960), 912-920. 72. Schnyder, 0. "Comparisons Between Calculated and Test Results on Water Hammer in Pumping Plants." ASME Trans. (1937), 695. 73. Schoklitsch, A. "Hydraulic Structures." ASME Trans., II (1937), 867-883. Translation. 74. Skalak, R. "An Extension of the Theory of Water Hammer." ASME Trans. (1956), 105. 75. Stepanoff, A. J. "Elements of Graphical Solution of Water-Hammer Problems in Centrifugal Pump Systems." ASME Trans. (1949), 515. 76. Streeter, V. L. Fluid Mechanics. McGraw-Hill Book Co., Inc., New York, 1958. 77, "Valve Stroking to Control Water Hammer." Jour. of the Hydraulics Div., Proc. of the ASCE, 89, No. Hy2, Mar. 1965. 78. Streeter, V. L. and Lai, Chintu. "Water-Hammer Analysis Including Fluid Friction. " ASCE Proc., Paper 3135, May, 1962.

-8779. Strowger, E. B. "Relation of Relief Valve and Turbine Characteristics In Determination of Water Hammer. " ASME Trans. (1937), 701; Discussion Trans. (1938), 608. 80. "Water-Hammer Problems in Connection with the Design of Hydro-Electric Plants. " ASME Trans. (1945), 377. 81. Symposium on Water Hammer. ASME - ASCE, 1933. 82. Walker, M. L., Kirkpatrick, E. T. and. Rouleau, W. T. "Viscous Dispersion in Water Hammer." ASME Trans. Jour. of Basic Engrg., 82, (Dec., 1960), 759-765. 83. White, I. M. "Application of the Surge Suppressor in Water Systems." Water Works Engrg., March 25, 1942. 84. Wood, F. M. "Application of Heaviside's Operational Calculus to Solution of Problems in Water Hammer." ASME Trans. (1937), 707; Discussion Trans. (1938), 682.

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