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A computer program analysing transients in multistage pumping systems Schmitt, Klaus 1980

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A COMPUTER PROGRAM IN MULTISTAGE ANALYSING TRANSIENTS PUMPING SYSTEMS by KLAUS SCHMITT B.A.Sc. (Civ. Eng.) The U n i v e r s i t y of B r i t i s h Columbia, 1978 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENT FOR THE DEGREE OF MASTER OF APPLIED SCIENCE xn THE FACULTY OF GRADUATE STUDIES in the Department of C i v i l E ngineering We accept t h i s t h e s i s as conforming to the re q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA APRIL, 1980 (o) Klaus Schmitt, 1980 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of Brit ish Columbia, I agree that the Library shall make i t freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of CIVIL ENGINEERING The University of Brit ish Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date APRIL 10,1980 ( H i ) ABSTRACT T r a n s i e n t p r e s s u r e s s u b s e q u e n t t o s i m u l t a n e o u s powe r f a i l u r e a t a l l pumps o f a m u l t i s t a g e p u m p i n g s y s t e m a r e a n a l y s e d . D i s t r i b u t i n g p u m p i n g s t a t i o n s a l o n g a p i p e l i n e , r a t h e r t h a n p l a c i n g a l l o f t h e r e q u i r e d pumps w i t h i n one p u m p i n g s t a t i o n , s i g n i f i c a n t l y r e d u c e s t r a n s i e n t p r e s s u r e f l u c t u a t i o n s w i t h i n t h e s y s t e m . A c o m p u t e r p r o g r a m u s i n g t h e FORTRAN l a n g u a g e i s d e v e l o p e d t o a n a l y s e m u l t i s t a g e p u m p i n g s y s t e m s , w i t h a p p r o p r i a t e s u r g e c o n t r o l s , i n t h e e v e n t o f s u c h a powe r f a i l u r e . T h e s e s u r g e c o n t r o l s c o n s i s t o f v a l v e s , v a c u u m b r e a k e r s , a i r c h a m b e r s and r e s e r v o i r s ; w i t h o t h e r c o n t r o l s e a s i l y a d d e d a s t h e y d e v e l o p . B o u n d a r y c o n d i t i o n s d e t e r m i n i n g s y s t e m c o n t r o l s a r e n o t d e v e l o p e d a s a p a r t o f t h i s t h e s i s , b u t a r e d e s c r i b e d f o r c o m p l e t e n e s s . By c o m p a r i n g t h e maximum and m in imum t r a n s i e n t p r e s s u r e s o c c u r r i n g w i t h i n - s i n g l e s t a g e and m u l t i s t a g e s y s t e m s , t h e p r e m i s e t h a t m u l t i s t a g e s y s t e m s g i v e s i g n i f i c a n t l y l o w e r t r a n s i e n t p r e s s u r e s t h a n s i n g l e s t a g e s y s t e m s i s s u b s t a n t i a t e d . T h i s r e d u c t i o n i n t r a n s i e n t p r e s s u r e s a l l o w s f o r p o s s i b l e s a v i n g s i n c o s t s , a s p i p e w a l l t h i c k n e s s e s and t h e s i z e o f l a r g e , e x p e n s i v e c o n t r o l s t r u c t u r e s may be r e d u c e d . E x a m p l e s d e m o n s t r a t i n g t h e u s e o f t h e p r o g r a m a r e i n c l u d e d . ( i i i ) TABLE OF CONTENTS CHAPTER PAGE ABSTRACT i i LIST OF FIGURES v NOTATION v i i I. INTRODUCTION 1 I I . GENERAL THEORY 3 2.1 C h a r a c t e r i s t i c Equations 3 2.2 F i n i t e D i f f e r e n c e Equations 6 2.3 S t a b i l i t y Requirements 7 I I I . BOUNDARY CONDITIONS 8 3.1 Boundary C o n d i t i o n f o r a Pump 9 3.2 Check Valve Boundary C o n d i t i o n 15 3.3 Pressure R e g u l a t i n g Valve 15 3.4 A i r Chamber.at the Main Pumping S t a t i o n 21 3.5 Vacuum Breakers 24 3.6 S e r i e s J u n c t i o n of Two Pipes 26 3.7 Constant Head R e s e r v o i r at the Downstream End 27 3.8 Flow C o n t r o l Valve at the Downstream End 28 IV. THE PROGRAM 32 4.1 J u s t i f i c a t i o n and M o d i f i c a t i o n s 32 4.2 Assumptions 35 4.3 D e s c r i p t i o n of the Program 36 4.4 Program Usage and Data Input 36 4.5 Check of Accuracy 41 V. DISCUSSION 42 VI. CONCLUSIONS 45 BIBLIOGRAPHY 46 APPENDICES Appendix A: Program f o r M u l t i s t a g e Pumping Systems 47 Appendix B: Examples of Program Usage 59 Three Stage System 60 S i n g l e Stage System 68 Comparison of S i n g l e and Three Stage Systems 72 Check of Program Accuracy 74 (iv) LIST OF FIGURES FIGURE PAGE 2.1 C h a r a c t e r i s t i c s G r i d 5 3.1 Pump Boundary C o n d i t i o n with Check Valve 11 3.2 Check Valve Boundary C o n d i t i o n 16 3.3 N o t a t i o n at Pressure Regulating Valve-Pump System 18 3.4 Pressure R e g u l a t i n g Valve Boundary C o n d i t i o n 20 3.5 A i r Chamber at the Main Pump 22 3.6 A i r Chamber Boundary C o n d i t i o n 25 3.7 C o n t r o l valve 29 3.8 Flow C o n t r o l Valve Boundary C o n d i t i o n 31 4.1 T y p i c a l 2-Stage Pumping System 34 4.2 T y p i c a l Valve C l o s u r e Curves 40 B - l 65 B-2 > T r a n s i e n t Pressures i n a Three Stage System 66 B-3 67 B-4 T r a n s i e n t Pressures i n a S i n g l e Stage. System 71 B-5 Minimum and Maximum H y d r a u l i c Grade Lines f o r S i n g l e and Three Stage Systems 73 (v) ACKNOWLEDGEMENT The author wishes to thank h i s s u p e r v i s o r , Dr. E. Ruus, f o r h i s c o n s t r u c t i v e c r i t i c i s m s and suggestions during t h i s p r o j e c t , and Dr. W.F. C a s e l t o n f o r reviewing the manuscript. (vi) NOTATION a - speed of pressure pulse ( f t / s e c or m/sec) A - cr o s s s e c t i o n a l area of pipe ( f t or m) C - a i r chamber constant C + - p o s i t i v e c h a r a c t e r i s t i c s equation C~ - negative c h a r a c t e r i s t i c s equation C a - p i p e l i n e constant Cp,Cjj - known constants i n c h a r a c t e r i s t i c s equations C(3 - c o e f f i c i e n t of d i s c h a r g e C o r f - head l o s s through a i r chamber o r i f i c e ( f t or m) D - pipe diameter ( f t or m) f - Darcy-Weisbach f r i c t i o n f a c t o r F j - i d e n t i f i c a t i o n of a i r chamber equation FH,FB - dime n s i o n l e s s pump c h a r a c t e r i s t i c s g - a c c e l e r a t i o n of g r a v i t y ( f t / s e c or m/sec) h - dimensionless pressure head H - instantaneous pressure head ( f t or m) H^,Hg - instantaneous head at beginning of time i n t e r v a l Hp - instantaneous head at end of time i n t e r v a l H a i r ~ absolute pressure head i n a i r chamber ( f t or m) H o r f - head l o s s through a i r chamber o r i f i c e ( f t or m) Hp. - rated head f o r a pump ( f t or m) H R E S ~ e l e v a t i o n of r e s e r v o i r water s u r f a c e ( f t or m) H S U C ~ h E A 6 ! at s u c t i o n f l a n g e of a pump ( f t or m) H)-, - barometric pressure head ( f t or m) H e - head l o s s through c o n t r o l v a l v e ( f t or m) HQ - steady s t a t e head ( f t or m) ( v i i ) K - head l o s s c o e f f i c i e n t (Kv 2 /2g) K l - constant determined from pump c h a r a c t e r i s t i c s L - pipe l e n g t h ( f t or m) L l • - i d e n t i f i c a t i o n of momentum equation L 2 - i d e n t i f i c a t i o n of c o n t i n u i t y equation n - exponent i n p o l y t r o p i c gas equation np - number of p a r a l l e l pumps N - number of reaches i n a pipe N R - rated pump speed (rpm) Q - instantaneous d i s c h a r g e (ft-^/sec or m-Vsec) Q A . Q B - d i s c h a r g e at beginning of time i n t e r v a l Qo - steady s t a t e d i s c h a r g e Q P - d i s c h a r g e at end of time i n t e r v a l Qorf - d i s c h a r g e through a i r chamber o r i f i c e Q R - rated pump dis c h a r g e t - time (seconds) T - instantaneous pump torque ( l b - f t or N-m) TR - rated torque of pump V - dimenesionless d i s c h a r g e v e - estimated value of dimensionless d i s c h a r g e V P - dimensionless d i s c h a r g e at end of time i n t e r v a l V - average value of dimensionless d i s c h a r g e V - instantaneous v e l o c i t y i n p i p e l i n e ( f t / s e c or m/sec) V a i r - instantaneous a i r volume i n a i r chamber v 0 - steady s t a t e l i q u i d v e l o c i t y W R 2 - moment of i n e r t i a of i m p e l l a r ( l b - f t 2 or Kg-m2) z - height of a i r chamber water s u r f a c e above datum ( v i i i ) - known t i m e i n t e r v a l <x - d i m e n s i o n l e s s pump s p e e d o<e - e s t i m a t e d v a l u e o f d i m e n s i o n l e s s pump s p e e d o<p - d i m e n s i o n l e s s pump s p e e d a t end o f t i m e i n t e r v a l u - a v e r a g e d i m e n s i o n l e s s pump s p e e d ^ - d i m e n s i o n l e s s t o r q u e o f pump % - a v e r a g e d i m e n s i o n l e s s t o r q u e o f pump ?N - m u l t i p l i e r u s e d i n m e t h o d o f c h a r a c t e r i s t i c s t - r e l a t i v e o r d i m e n s i o n l e s s v a l v e o p e n i n g NOTE The f i r s t s u b s c r i p t P i n d o u b l e s u b s c r i p t e d v a r i a b l e s i n d i c a t e s c o n d i t i o n s a t t h e end o f t h e t i m e i n t e r v a l . 1 CHAPTER I. INTRODUCTION Pumping systems are being designed and c o n s t r u c t e d with ever i n c r e a s i n g c o m p l e x i t i e s , lengths and working p r e s s u r e s . In the past, engineers have not had s u f f i c i e n t t e c h n i c a l i n f o r m a t i o n to design systems whose o p e r a t i o n was e n t i r e l y p r e d i c t a b l e . S k i l l e d o p e r a t o r s were r e q u i r e d to ensure that u t i l i t i e s performed f r e e of inconvenient outages or c a t a s t r o p h i c f a i l u r e s . As more complex systems are developed i n the f u t u r e , the engineer w i l l be c a l l e d upon to design more f e a t u r e s f o r t r a n s i e n t c o n t r o l , r a ther than r e l y i n g on operator i n g e n u i t y . C a p a b i l i t i e s w i l l have to be developed to e f f e c t i v e l y analyse these systems. The philosophy behind most methods to p r o t e c t a g a i n s t waterhammer are s i m i l a r . In most cases, the o b j e c t i v e i s to decrease the r a t e of l i q u i d v e l o c i t y change, as t h i s d i r e c t l y a f f e c t s the magnitude of the pressure surge. In small systems no s p e c i a l p r e c a u t i o n s are taken to c o n t r o l waterhammer, as standard metal t h i c k n e s s e s i n component p a r t s are s u f f i c i e n t to withstand a p p r e c i a b l e t r a n s i e n t p r e s s u r e s . In l a r g e i n s t a l l a t i o n s , however, t h i s i s not the case and surge c o n t r o l s must be added. A computer program using the FORTRAN language i s developed to analyse m u l t i s t a g e pumping systems, with a p p r o p r i a t e c o n t r o l s , i n the event of a power f a i l u r e . The work i s an e x t e n s i o n to an e x i s t i n g program which analyses the t r a n s i e n t s f o r upstream pumps d i s c h a r g i n g i n t o a downstream r e s e r v o i r . T r a n s i e n t s subsequent to power f a i l u r e at a pump are u s u a l l y the most extreme a pumping system w i l l experience. Pumps i n 2 s e r i e s separated by some s i g n i f i c a n t pipe length dominate the system developed. T h i s c o n f i g u r a t i o n should experience lower t r a n s i e n t p r e s s u r e s than the s i t u a t i o n of two s e r i e s pumps i n the same l o c a t i o n , which has been e f f e c t i v e l y s o l v e d by o t h e r s . J u s t i f i c a t i o n f o r a system with.the pumps separated by a length of p i p e , as opposed to s e r i e s pumps s i t u a t e d together i n one pumping s t a t i o n or d i v i s i o n of the system i n t o separate systems by an intermediate sump, stems from a p o s s i b l e l a r g e saving i n pipe and surge c o n t r o l c o s t s . Any r e d u c t i o n i n pipe w a l l t h i c k n e s s or i n the s i z e of l a r g e c o n t r o l s t r u c t u r e s should be s i g n i f i c a n t . T h i s m u l t i s t a g e theme i s developed and t e s t s a computer a n a l y s i s which i s intended to permit the p o s s i b i l i t y f o r c o s t r e d u c t i o n s by reducing t r a n s i e n t pressures along a p i p e l i n e . The c a l c u l a t i o n technique employed i s the method of c h a r a c t e r i s t i c s as pioneered by V.L. S t r e e t e r . System c o n t r o l s c o n s i s t of v a l v e s , vacuum breakers, a i r chambers and r e s e r v o i r s ; with other c o n t r o l s e a s i l y added as they develop. The program developed i n t h i s study i n c o r p o r a t e s many assumptions. For t h i s reason, i t should not be used f o r design purposes, but r a t h e r as a check on t r a n s i e n t s i n systems with known component c h a r a c t e r i s t i c s . 3 CHAPTER I I . GENERAL THEORY In t h i s chapter, a numerical s o l u t i o n to the equations d e s c r i b i n g unsteady flow i n c l o s e d c o n d u i t s w i l l be d e s c r i b e d . These equations (the momentum and c o n t i n u i t y equations) form a p a i r of h y p e r b o l i c p a r t i a l d i f f e r e n t i a l e quations, to which no gene r a l s o l u t i o n i s yet a v a i l a b l e . However, these p a r t i a l d i f f e r e n t i a l equations can be transformed i n t o four o r d i n a r y d i f f e r e n t i a l equations by the method of c h a r a c t e r i s t i c s . The transformed equations are then i n t e g r a t e d to y i e l d f i n i t e d i f f e r e n c e equations which can be manipulated n u m e r i c a l l y . 2 . 1 CHARACTERISTIC EQUATIONS The c o n t i n u i t y and momentum equations are expressed i n terms of two dependent v a r i a b l e s , v e l o c i t y and h y d r a u l i c grade l i n e e l e v a t i o n , and two independent v a r i a b l e s , time and d i s t a n c e along the p i p e l i n e . Wylie and S t r e e t e r * 0 , by i n c o r p o r a t i n g assumptions, develop s i m p l i f i e d equations of c o n t i n u i t y and motion. Converting the l i q u i d pressure to e l e v a t i o n of h y d r a u l i c grade l i n e above an a r b i t r a r y datum, assuming d e n s i t y changes w i t h i n the l i q u i d to be n e g l i g i b l e and assuming that the p i p e l i n e has no l a t e r a l displacement, the momentum equation i s expressed as L]_=g (c5H/ax)+aV/dt+ (f / 2 D)V|V | = 0 ( 2 . 1 ) and the c o n t i n u i t y equation i s expressed as L 2 = c W c)t+ (a 2/g)dV/dx= 0 . ( 2 . 2 ) The absolute value s i g n on the v e l o c i t y term i n equation 2 . 1 ensures that the f r i c t i o n a l f o r c e always opposes the d i r e c t i o n of l i q u i d motion. S u b s t i t u t i n g Q/A f o r V and then t a k i n g a l i n e a r combination of equations 2.1 and 2.2 using an unknown m u l t i p l i e r "X, r e s u l t s i n L=Li+ L-2 or lQ+(Aa 2)^Q >^t ax +AgA iH+(l/ > v)£H dt ax +(f/2DA)Q|Q|=0. (2 .3 ) Since H and Q are f u n c t i o n s of x and t , the t o t a l d e r i v a t i v e s of H(x,t) and Q(x,t) may be expressed as: (2.4) d_Q=^ Q+a2dx dt dt <3xdt dH^^H+^Hdx^ dt at dxdt* In n o t i n g that i f dx/dt=l/A=>\a 2, two p a i r s of equations can be i d e n t i f i e d by comparing equation 2.3 with equations 2.4: (2.5) i f and i f dQ+qAdH+fQ|Q|=n dt a dt 2DA dx/dt=a dQ.qAdH+fQlQUn dt a dt 2DA dx/dt=-a. C+ (2.6) (2.7) > C" (2.8) Thus, i n s p e c i f y i n g two r e a l values of A, the two p a r t i a l d i f f e r e n t i a l equations have been converted i n t o four o r d i n a r y d i f f e r e n t i a l e quations. The s o l u t i o n s to these equations, as p i c t u r e d on the independent (x,t) v a r i a b l e plane, p l o t as s t r a i g h t l i n e s ( F i g . 2.1), because the pressure wave v e l o c i t y i s u s u a l l y constant f o r a given p i p e . These c h a r a c t e r i s t i c l i n e s ( C + and C~) represent the t r a v e l l i n g of pressure surges i n i t i a t e d w i t h i n the system. 5 0 — A X — — i — P f ^ t 1 A B i+1 n+1 x CHARACTERISTICS GRID FIGURE 2.1 6 2 . 2 F I N I T E D I F F E R E N C E EQUATIONS I n t e g r a t i n g e q u a t i o n s 2 . 5 and 2 . 7 a l o n g l i n e s AP and B P , r e s p e c t i v e l y , c o n v e r t i n g t o f i n i t e d i f f e r e n c e f o r m and i s o l a t i n g c o n s t a n t s y i e l d s ^ : Q p = C p - C a H P ( 2 . 9 ) f o r e q u a t i o n 2 . 5 and Q p = C N + C a H P ( 2 . 1 0 ) f o r e q u a t i o n 2 . 7 , i n w h i c h : C p = Q A + C a H A - ( f A t / 2 D A ) Q A | Q A | ( 2 . 1 1 ) C N = Q B - C a H B - ( f A t / 2 D A ) Q B I Q g I ( 2 . 1 2 ) and C a = g A / a . ( 2 . 1 3 ) The v a l u e s o f t h e d e p e n d e n t v a r i a b l e s H and Q a r e known a t t h e g r i d p o i n t s A and B s i n c e t h e a n a l y s i s i n i t i a t e s f r o m s t e a d y s t a t e c o n d i t i o n s . A s i m u l t a n e o u s s o l u t i o n o f e q u a t i o n s 2 . 9 and 2 . 1 0 y i e l d s t h e c o n d i t i o n s a t a p a r t i c u l a r t i m e and s p a t i a l p o s i t i o n d e s i g n a t e d by p o i n t P . T h u s , Q P = 0 . 5 ( C N + C P ) ( 2 . 1 4 ) and Hp c a n be g e n e r a t e d f r o m e i t h e r e q u a t i o n 2 . 9 o r 2 . 1 0 . I n t h i s m a n n e r , t h e c o n d i t i o n s a t a l l i n t e r i o r p o i n t s c a n be d e t e r m i n e d a t t h e end o f t h e t i m e s t e p . T h e s e c o n d i t i o n s t h e n become t h e known v a l u e s (A a n d B) f o r t h e s u b s e q u e n t t i m e s t e p , and so o n . A t t h e b o u n d a r i e s ( s p a t i a l p o s i t i o n s 1 a n d n+1 on t h e c h a r a c t e r i s t i c s g r i d ) , h o w e v e r , e i t h e r e q u a t i o n 2 . 9 o r 2 . 1 0 i s a v a i l a b l e , n o t b o t h . T h e r e f o r e , s p e c i a l end c o n d i t i o n s m u s t be i n t r o d u c e d t o c o m p l e t e t h e s o l u t i o n d u r i n g s u b s e q u e n t t i m e i n t e r v a l s . T h e s e c o n d i t i o n s w i l l be d i s c u s s e d i n t h e f o l l o w i n g c h a p t e r . 7 2.3 STABILITY REQUIREMENTS Given that the l i q u i d v e l o c i t y i s much l e s s than the wave v e l o c i t y , s t a b i l i t y of the s o l u t i o n i s assured^ i f At/^x<l/a. T h i s c r i t e r i a s p e c i f i e s t h a t the c h a r a c t e r i s t i c l i n e s through p o i n t P must not f a l l o u t s i d e of the l i n e segment AB ( F i g . 2.1). For complex p i p i n g systems i n v o l v i n g more than one conduit i n s e r i e s , i t i s necessary that the time increment be constant f o r each p i p e . T h i s ensures that compatible s o l u t i o n s are generated at the j u n c t i o n s . The most accurate s o l u t i o n s are obtained i f /£>t = L/(aN), where L i s the pipe length and N i s the number of reaches. C a r e f u l r e f l e c t i o n w i l l i n d i c a t e that t h i s c o n d i t i o n probably cannot be met f o r the m a j o r i t y of systems analysed. T h e r e f o r e , s i n c e the wave speed w i t h i n a p a r t i c u l a r pipe i s not known to any g r e a t degree of accuracy, a s l i g h t adjustment (+15%) of the wave speed to maintain a constant ^ t i s a l l o w a b l e 1 0 . 8 CHAPTER I I I . BOUNDARY CONDITIONS Waterhammer i s d e f i n e d as the change i n pressure i n c l o s e d c o n d u i t s above or below normal c o n d i t i o n s caused by sudden or r a p i d changes i n the v e l o c i t y of flow. The s t a r t i n g and stopping of pumps i n i t i a t e such changes, as does the opening and c l o s i n g of v a l v e s . Whereas the waterhammer produced by s t a r t i n g a pump can be minimized by s l o w l y opening the d i s c h a r g e v a l v e , the waterhammer produced during power f a i l u r e can produce d e s t r u c t i v e r e s u l t s . These d r a s t i c pressure changes can be reduced by one or s e v e r a l of the f o l l o w i n g means: 1 . System designed to maintain low steady s t a t e v e l o c i t i e s . 2. Slow c l o s i n g c o n t r o l v a l v e s , i f l o s s of water and reverse pump r o t a t i o n are not o b j e c t i o n a b l e . 3 . Pressure r e g u l a t i n g v a l v e s or pressure r e l i e f v a l v e s which allow flow bypass around a pump. With r e g u l a t i n g v a l v e s , the v a l v e opens r a p i d l y and then c l o s e s s l o w l y . With r e l i e f v a l v e s , the v a l v e o n l y opens as f a r as necessary to minimize the pressure surge. 4. Use of a i r chambers, accumulators, or surge tanks. The l a t t e r are the most expensive and are h a r d l y ever j u s t i f i e d except where they are the o n l y means of t r a n s i e n t c o n t r o l . 5. When water column s e p a r a t i o n i s unavoidable, vacuum breakers are i n s t a l l e d to cushion the shock as the p a r t s of the water column r e j o i n . They can be used to maintain the p i p e l i n e pressure at or above p r e s e t l i m i t s , thus p r e v e n t i n g column s e p a r a t i o n from o c c u r r i n g . 9 P u m p s , r e s e r v o i r s and a p p r o p r i a t e c o n t r o l d e v i c e s d e t e r m i n e t h e b o u n d a r y c o n d i t i o n s i n c l u d e d i n t h i s a n a l y s i s . F o r a l l b o u n d a r y c o n d i t i o n s h a n d l e d w i t h t h e m e t h o d o f c h a r a c t e r i s t i c s , t h e s y s t e m r e s p o n s e d u r i n g t h e t r a n s i e n t i s c o n v e y e d f r o m t h e a d j o i n i n g r e a c h e s t o t h e b o u n d a r y by e i t h e r e q u a t i o n 2 . 9 o r 2 . 1 0 . E a c h b o u n d a r y c o n d i t i o n t h e n r e q u i r e s a s e c o n d e q u a t i o n w h i c h s p e c i f i e s some r e l a t i o n s h i p b e t w e e n h e a d and d i s c h a r g e . T h i s c h a p t e r i s d e v o t e d t o t h e d e v e l o p m e n t o f t h e s e s p e c i f i c b o u n d a r y c o n d i t i o n r e l a t i o n s h i p s . 3 . 1 BOUNDARY CONDIT ION FOR A PUMP V o l u n t a r y pump s t o p p i n g and s t a r t i n g c a n be a n a l y s e d by a s s u m i n g t h e pump s p e e d t o be c o n s t a n t l y . H o w e v e r , i f t h e pump s t o p p i n g o r s t a r t i n g i s n o t v o l u n t a r y , a d d i t i o n a l p a r a m e t e r s m u s t be i n c l u d e d i n t h e a n a l y s i s . The pump m o t o r , b y e x e r t i n g a t o r q u e on a r o t a t i n g s h a f t , c o n v e y s e n e r g y t o t h e i m p e l l e r w h i c h d e v e l o p e s a t o t a l d y n a m i c h e a d (TDH) i n c r e a s e f r o m t h e s u c t i o n f l a n g e t o t h e d i s c h a r g e f l a n g e . A t powe r f a i l u r e , t h e t o r q u e a p p l i e d t o t h e s h a f t i s d e c r e a s e d , r e d u c i n g t h e TDH and c a u s i n g p r e s s u r e w a v e s t o move u p s t r e a m and d o w n s t r e a m i n t h e p i p e l i n e . S i n c e p u m p i n g h e a d and d i s c h a r g e d e p e n d upon t h e pump s p e e d , t r a n s i e n t s p e e d c h a n g e s w i l l h a v e t o be t a k e n i n t o a c c o u n t . The t h e o r y o f h o m o l o g o u s pumps** s u p p l i e s t h e f o l l o w i n g e q u a t i o n s p e r t a i n i n g t o t u r b o m a c h i n e s : H / ( N 2 D 2 ) = CONSTANT Q / ( N D 3 ) = CONSTANT. ( 3 . 1 ) E x p e r i e n c e w i t h t h e s e r e l a t i o n s h i p s h a s i n d i c a t e d t h a t d i m e n s i o n l e s s c h a r a c t e r i s t i c s a r e more c o n v e n i e n t and t h e s e a r e 10 d e f i n e d a s : h = H / H R ^ = T / T R v = Q / Q R o<=N/N R ( 3 . 2 ) w h e r e t h e R r e f e r s t o r a t e d q u a n t i t i e s . C o m b i n i n g e q u a t i o n s 3 . 1 and 3 . 2 and a p p l y i n g them t o one u n i t r e s u l t s i n t h e f o l l o w i n g d i m e n s i o n l e s s h o m o l o g o u s r e l a t i o n s h i p s : h/oc2 = CONSTANT v/<* = CONSTANT. ( 3 . 3 ) The r e l a t i o n s h i p s a s w r i t t e n a r e d i f f i c u l t t o h a n d l e , a s a l l t e r m s may c h a n g e s i g n d e p e n d i n g upon t h e z o n e o f pump o p e r a t i o n . M a r c h a l , F l e s h and S u t e r ^ a v o i d t h i s p r o b l e m by u s i n g h / ( v 2 + c * 2 ) i n s t e a d o f h/o<2. w y l i e and S t r e e t e r 1 0 u s e t h i s c o n v e r s i o n t o r e p r e s e n t t h e c o m p l e t e c h a r a c t e r i s t i c s f o r a g i v e n pump a s two c u r v e s on a r e c t a n g u l a r c o - o r d i n a t e p l o t , w h e r e : FH (x) = h / (v 2 +o<2 ) FB(x )= j9 / ( v 2 + o <2) x = T T + T A N - 1 v/o< . ( 3 . 4 ) T h i s f o r m i s p r e f e r a b l e f o r c o m p u t e r i n p u t and h o l d s f o r a l l v a l u e s o f and v e x c e p t w h e r e oc and v a r e z e r o s i m u l t a n e o u s l y . M o r e i n f o r m a t i o n i s a v a i l a b l e i n t h e t e x t s b y W y l i e and S t r e e t e r 1 0 and by C h a u d h r y 1 , b u t t h i s a s p e c t o f t h e b o u n d a r y c o n d i t i o n w i l l n o t be d e a l t w i t h f u r t h e r . I n d e v e l o p i n g t h e b o u n d a r y c o n d i t i o n f o r a pump, a number o f s y s t e m c o m b i n a t i o n s s h o u l d be c o n s i d e r e d . Some v a r i a t i o n s a r e : a s h o r t s u c t i o n l i n e ( w h e r e t h e s u c t i o n l i n e c a n be n e g l e c t e d i n t h e a n a l y s i s ) and a l o n g s u c t i o n l i n e ( w h e r e i t m u s t be i n c l u d e d ) ; b o t h s i t u a t i o n s may i n c l u d e pumps i n p a r a l l e l . By s e t t i n g vp and hp e q u a l t o t h e d i m e n s i o n l e s s v a l u e s a t t h e pump and u s i n g t h e s u b s c r i p t e d n o t a t i o n i n d i c a t e d i n F i g u r e 3 . 1 , t h e f o l l o w i n g e q u a t i o n s c a n be w r i t t e n 1 : i,n i,n+1 i+1,1 i+1,2 PUMP BOUNDARY CONDITION WITH CHECK VALVE FIGURE 3.1 12 - 1. Pump c h a r a c t e r i s t i c equation h P=K! (°<p 2+vp 2) (3.5) where i s determined from the pump c h a r a c t e r i s t i c s and the s u b s c r i p t P p e r t a i n s to c o n d i t i o n s at the end of the time i n t e r v a l under c o n s i d e r a t i o n . 2. Head d i f f e r e n c e equation at the pump Hp=Hp -Hp +(check v a l v e head l o s s ) (3.6) i+1,1 i,n+l which may be non-dimensionalized as hp=h P -hp +KV 0 2/2gH R. (3.7) .i+1,1 i,n+l 3. Negative c h a r a c t e r i s t i c equation f o r the di s c h a r g e l i n e Qp = C N + C a H P < 3- 8) i+1,1 i+1 i+1,1 which may be non-dimensionalized as v P =C 4+C 5h P (3.9) i+1,1 i+1,1 where C 4=C N/Q R and C 5=C a H R/Q R. i + 1 4. P o s i t i v e c h a r a c t e r i s t i c equation f o r the s u c t i o n l i n e Qp =Cp-C a Hp (3.10) i,n+l i i,n+l which may be non-dimensionalized as v P =C 6-C 7h P (3.11) i,n+l i,n+l where C 6=C P/Q R and C 7=C a H R/Q R. i For systems c o n s i s t i n g of pumps i n p a r a l l e l , the pipes between the pumps and the di s c h a r g e manifold can be neglected p r o v i d i n g these pipes are s h o r t . Equation 3.9 should then be w r i t t e n as n pvp =C4+C 5h p (3.12) i+1,1 i+1,1 13 and equation 3.11 should be w r i t t e n as n pvp =Cg-C7hp (3.13) i,n+l i,n+l where np equals the number of p a r a l l e l pumps. 5. Flow c o n t i n u i t y through the pumps vp =vp (3.14) i,n+l and vp=Vp . (3.15) i + 1,1 6. Speed change equation f o r the pump By using the non-dimensional q u a n t i t i e s s p e c i f i e s by equation 3.2, the change i n r o t a t i o n a l speed can be w r i t t e n as p>=- (2irWR2/60g) (N R/T R) dcx/dt. (3.16) Converting t h i s i n t o f i n i t e d i f f e r e n c e form y i e l d s A o <=c 3 p (3.17) where JB i s the average of the known torque at the beginning of the time s t e p and the unknown torque a f t e r time A t , and C 3 = - ( 3 0 g A t / W R 2 ) ( T R / N R ) . (3.18) For the system c o n s i s t i n g of a long s u c t i o n l i n e with pumps i n p a r a l l e l , simultaneous s o l u t i o n of equations 3.5, 3.7, 3.12, 3.13, 3.14 and 3.15 y i e l d s v p = [ C 1 n P - / ( C 1 n P ) 2 - 4 K 1 (K 1 (*p 2+C2-KV 0 2/2g) ]/2K1 (3.19) where C 2 = ( C 5 + C 7 ) / C 5 C 7 and C 2 = ( C 4 C 7 + C 5 C 6 ) / C 5 C 7 . I f the s u c t i o n l i n e i s shor t and thus n e g l e c t e d , simultaneous s o l u t i o n of equations 3.5 and 3.12 would y i e l d vp as s p e c i f i e d by equation 3.19, but C i = l / C 5 and C2=C 4/C 5. However, s i n c e depends upon o<p and vp, which are i n i t i a l l y unknown, equation 3.19 would r e q u i r e a d d i t i o n a l i n f o r m a t i o n f o r s o l u t i o n . Chaudhryl presents a p r e d i c t o r -14 c o r r e c t o r i t e r a t i o n procedure f o r t h i s purpose. Given that v, h and u are known at the beginning of the time i n t e r v a l , l i n e a r e x t r a p o l a t i o n g i v e s v e = v + A V i _ i °<e = 0 < + A o <i-l (3.20) where s u b s c r i p t e r e f e r s to estimated v a l u e s ; A V J _ 1 and r e f e r to v and increments determined during the previous time i n t e r v a l . These estimated values y i e l d from the pump c h a r a c t e r i s t i c s and allow v p to be c a l c u l a t e d from equation 3.19. The values of v e and vp are compared and i f I v P - v e I i s gr e a t e r than a s p e c i f i e d t o l e r a n c e (say 0.001), a new estimated value i s c a l c u l a t e d as the average of v e and vp. Then the procedure i s repeated. I f I vp-v e I i s l e s s than the t o l e r a n c e , <*P i s determined as f o l l o w s . S e t t i n g v=0.5(v P+v) * = < * + 0 . 5 A < X J L _ 1 (3.21) allows to be obtained from the pump c h a r a c t e r i s t i c s . Now, equation 3.17 i s used to o b t a i n Ao( and <*p i s computed as c*+Ac*. I f l°<p-«*el i s g r e a t e r than a s p e c i f i e d t o l e r a n c e (say 0.001), set v e=Vp, °<E=0.5 Kp+»<e) and do the whole i t e r a t i o n procedure over again. Otherwise, o<p and v p have been determined to w i t h i n the s p e c i f i e d t o l e r a n c e s and Hp can be c a l c u l a t e d from equation 15 3.2 CHECK VALVE BOUNDARY CONDITION If a check v a l v e i s placed at the di s c h a r g e s i d e of a pump, the normal d i s c h a r g e of the pump keeps the va l v e open. For such a v a l v e , the crude assumption may be made that the head l o s s i s constant f o r a l l forward f l o w 1 0 and i n dimensionless form t h i s becomes the steady s t a t e v e l o c i t y head d i v i d e d by the rated head of the pump ( K V 0 2 / 2 g H R ) . When the flow through the pump reverses subsequent to power f a i l u r e , the assumption i s made that the v a l v e c l o s e s i n s t a n t a n e o u s l y . The di s c h a r g e at the c l o s e d v a l v e i s then set at zero, i s o l a t i n g the pump which i s ignored i n f u r t h e r c a l c u l a t i o n s . Consequently, f o r the case where the v a l v e i s c l o s e d and the head at the s u c t i o n s i d e of the pump exceeds that j u s t downstream of the check v a l v e , a p r o v i s i o n should be inc l u d e d to reopen the va l v e and allow forward flow. A flow c h a r t f o r the check v a l v e i s presented i n Fig u r e 3.2. 3.3 PRESSURE REGULATING VALVE (PRV) The pressure r e g u l a t i n g v a l v e i s a p i l o t c o n t r o l l e d v a l v e which i s operated by a servomotor. I t i s i n s t a l l e d d i r e c t l y downstream of the pump or pump check v a l v e (provided one e x i s t s ) and d i s c h a r g e s to atmosphere or back i n t o the s u c t i o n l i n e . Subsequent to power f a i l u r e at the pump motor, t h i s v a l v e opens r a p i d l y and then c l o s e s s l o w l y . Care should be e x e r c i s e d to be c e r t a i n that the va l v e i s not opened to soon, as t h i s may in c r e a s e the danger of water column s e p a r a t i o n i n some i n s t a l l a t i o n s 1 0 . F i g u r e 3.3 shows the s u b s c r i p t e d n o t a t i o n at C h e c k V a l v e B o u n d a r y Cond i t i o n Q= n P V P Q R NO C a l c u l a t e Hp F rom E Q . 3 . 8 ( ^CONTINUE J YES S e t Q=0 YES Open V a l v e C a l l Pump S u b r o u t i n e CHECK V A L V E BOUNDARY CONDIT ION F I G U R E 3 . 2 17 t h i s boundary c o n d i t i o n and a l s o shows the r e l a t i v e p o s i t i o n s of the pump, PRV and check v a l v e . Assuming the pipe between the pump and the PRV to be sh o r t and thus n e g l e c t e d , c o n t i n u i t y at t h i s s e c t i o n g i v e s Q P =n PQ P -Qp (3.22) i+1,1 p v where the s u b s c r i p t s p, v and i+1,1 r e f e r to pump, PRV and pipe s e c t i o n , r e s p e c t i v e l y . The negative c h a r a c t e r i s t i c equation f o r t h i s s e c t i o n i s expressed by equation 3.8 and the pump c h a r a c t e r i s t i c equation i s expressed by equation 3.5. The head at the pump di s c h a r g e i s the head developed by the pump plus the head at the s u c t i o n f l a n g e during steady s t a t e c o n d i t i o n s minus the head l o s s through the check valve (provided that one e x i s t s ) : Hp =h pH R+H s u c-KV 0 2/2g. (3.23) i + 1,1 o<p and vp are s t i l l unknown and w i l l be determined by a s i m i l a r i t e r a t i v e procedure as that proposed f o r the pump boundary condition-'-. E s t i m a t i n g values f o r <Xp and vp as i l l u s t r a t e d p r e v i o u s l y allows to be obtained from the pump c h a r a c t e r i s t i c s . Since the value s of a l l v a r i a b l e s on the r i g h t s i d e of equation 3.23 are known, Hp i s known, Qp can i+1,1 i+1,1 be computed from equation 3.8 and Qp determined from v Qp =tpQn/Hp /H 0. (3.24) v J i+1,1 t p i s the r e l a t i v e v a l v e opening at the end of the time i n t e r v a l and QQ i s the steady s t a t e d i s c h a r g e under a head of HQ. NOW, i f IQp _(npVpQ R-Qp )| i s l e s s than some s p e c i f i e d t o l e r a n c e i+1,1 v 18 DATUM NOTATION AT PRESSURE REGULATING VALVE-PUMP SYSTEM FIGURE 3.3 19 (say 0.001) the estimated v a l u e , v e , i s taken as v p and the pump speed i s obtained by a p p l y i n g the same procedure that was presented f o r the pump boundary c o n d i t i o n . Otherwise, a new value of v e i s assumed to be equal to (Qp -Qp )/npQ R and the i+1,1 v procedure i s repeated. Now, assume that the pump has been i s o l a t e d by c l o s i n g the check v a l v e and that the PRV d i s c h a r g e s to atmosphere. The steady s t a t e d i s c h a r g e through the PRV i s given by Q o = c d A o J 2 9 H 0 (3.25) where AQ i s the area of v a l v e opening and i s the c o e f f i c i e n t of d i s c h a r g e . The d i s c h a r g e f o r any other v a l v e opening (Ay) i s Qp =C dA v/2gH P . (3.26) i+1,1 y i+1,1 D e f i n i n g the r e l a t i v e v a l v e opening as T ^^Ay/C^An, and d i v i d i n g equation 3.26 by 3.25 r e s u l t s i n Qp = Q 0 t / H p V H o - (3.27) i+1,1 </ i+1,1 S o l v i n g t h i s equation s i m u l t a n e o u s l y with equation 3.8 g i v e s Qp =0.5(-C v + y c v 2+4C vC P) (3.28) i n which Cy=(Qrjt) 2 / H o ca• Now,Hp can be determined from equation 3.8. A flow c h a r t f o r the o p e r a t i o n of the pressure r e g u l a t i n g v a l v e ( i n c l u d i n g check v a l v e i n f l u e n c e s ) i s presented i n F i g u r e 20 YES Open PRV PRV Bounda ry -C o n d i t i o n T i m e t o Open PRVZ-NO NO YES YES YES Y E S / Check^ C l o s e d ? . YES S e t Q=0 C l o s e V a l v e NO C a l l Pump S u b r o u t i n e NO C a l l Pump S u b r o u t i n e NO NO NO NO F i n d PRV T a u A f t e r t\t YES C a l c u l a t e Hp F rom E Q . 3.8 c v = ( Q 0 r ) 2 / H 0 C a C a l c u l a t e Qp F rom E Q . 3.28 ( ^CONTINUE J PRESSURE REGULATING V A L V E BOUNDARY CONDIT ION F I G U R E 3.4 21 3.4 AIR CHAMBER AT THE MAIN PUMPING STATION An a i r chamber i s a c l o s e d v e s s e l with a i r at i t s top and l i q u i d i n i t s lower p o r t i o n . The a i r may be i n c o n t a c t with the l i q u i d , i n which case an a i r compressor must be provided to re p l a c e t hat which d i s s o l v e s i n the l i q u i d , or the a i r may be separated from the l i q u i d by a f l e x i b l e membrane or a p i s t o n 1 0 . I t i s designed to prevent the pressure from exceeding a predetermined value and to prevent low p r e s s u r e s , or water column s e p a r a t i o n . These chambers are most e f f e c t i v e i f the reverse flow of l i q u i d from the di s c h a r g e l i n e i n t o the chamber i s t h r o t t l e d , while very l i t t l e t h r o t t l i n g i s provided f o r flow out of the chamber. A de v i c e used to accomplish t h i s i s the d i f f e r e n t i a l o r i f i c e , but the co s t of c o n s t r u c t i o n may be p r o h i b i t i v e f o r small a i r chambers. When t e s t e d i n the l a b o r a t o r y , t h i s p a r t i c u l a r type of o r i f i c e was found to g i v e a head l o s s r a t i o o f 2.5:1 f o r r e t u r n flow e n t e r i n g the chamber as compared to the same flow l e a v i n g the chamber^. I t i s common p r a c t i c e to provide a check v a l v e upstream of the chamber to prevent reverse flow through the pump ( F i g . 3.5). Subsequent to power i n t e r r u p t i o n , the va l v e c l o s u r e time i s so sho r t that assuming the check v a l v e to c l o s e spontaneously upon power f a i l u r e i s j u s t i f i e d ^ . Thus, the pump i s i s o l a t e d and the l i q u i d flow i s e i t h e r i n t o or out of the chamber. The trapped a i r w i t h i n the chamber expands and c o n t r a c t s depending upon the d i r e c t i o n of l i q u i d flow. Assuming that the pressure-volume changes f o r the a i r i n the 7 A I R CAMBER 1,1 CHECK VALVE MAIN PUMP DATUM AIR CHAMBER AT THE MAIN PUMP FIGURE 3.5 23 c h a m b e r f o l l o w t h e p o l y t r o p i c r e l a t i o n f o r a p e r f e c t g a s 1 , t h e n Hp Vp n =C. ( 3 . 2 9 ) a i r a i r Hp and Vp a r e t h e a b s o l u t e p r e s s u r e h e a d and t h e v o l u m e o f a i r a i r e n c l o s e d a i r , n i s an e x p o n e n t whose v a l u e i s o f t e n a s s u m e d t o be 1 . 2 and G i s a c o n s t a n t w h i c h i s d e t e r m i n e d w h i l e t h e s y s t e m i s f u n c t i o n i n g u n d e r known s t e a d y s t a t e c o n d i t i o n s . I f f l o w i s a s s u m e d p o s i t i v e o u t o f t h e a i r c h a m b e r , t h e f o l l o w i n g e q u a t i o n s c a n a l s o be w r i t t e n f o r t h e e n c l o s e d a i r v o l u m e : Hp = H p + H b - z - H p ( 3 . 3 0 ) a i r 1 , 1 - o r f V P = V a i r + 0 . 5 A t ( Q p + Q ) ( 3 . 3 1 ) a i r Hp = C o r f Q p I.Qp I- ( 3 . 3 2 ) o r f o r f o r f H e r e , H5 i s t h e b a r o m e t r i c p r e s s u r e h e a d , z i s t h e h e i g h t o f t h e c h a m b e r w a t e r s u r f a c e a b o v e d a t u m and Hp i s t h e h e a d l o s s o r f t h r o u g h t h e o r i f i c e f o r a g i v e n Qp and h e a d l o s s c o e f f i c i e n t o r f C o r f . T h i s o r i f i c e l o s s c o e f f i c i e n t c a n be c a t e g o r i z e d by t h r e e m a j o r s i t u a t i o n s : no o r i f i c e and C o r f = 0 , a p l a t e o r i f i c e w h e r e C o r f f o r i n f l o w = C o r f f o r o u t f l o w and a d i f f e r e n t i a l o r i f i c e w h e r e C o r f f o r i n f l o w = 2 . 5 t i m e s C o r f f o r o u t f l o w . F l u c t u a t i o n s i n t h e c h a m b e r w a t e r s u r f a c e a r e a s s u m e d t o h a v e a n e g l i g i b l e e f f e c t upon f u t u r e c a l c u l a t i o n s and a r e i g n o r e d ( i . e . z = c o n s t a n t ) . A l s o , c a r e s h o u l d be t a k e n t o i n c l u d e b e n d l o s s e s b e t w e e n t h e a i r c h a m b e r and t h e m a i n p i p e l i n e , a s t h e s e l o s s e s may w e l l e x c e e d t h e l o s s e s t h r o u g h t h e 24 o r i f i c e . U s i n g e q u a t i o n s 3 . 2 9 , 3 . 3 0 , 3 . 3 1 and 3 . 3 2 i n c o n j u n c t i o n w i t h t h e n e g a t i v e c h a r a c t e r i s t i c e q u a t i o n ( E q . 3 . 8 ) r e s u l t s i n f i v e e q u a t i o n s and f i v e u n k n o w s . By e q u a t i o n c o m b i n a t i o n and v a r i a b l e s u b s t i t u t i o n , t h e s e e q u a t i o n s c a n be s i m p l i f i e d t o F l = [ d / C a ) (Qp-CN)+H'b-z-Hp ] [ V a i r + * t (Qp+Q)/2] n - C = 0 ( 3 . 3 3 ) o r f w h i c h i s a n o n - l i n e a r e q u a t i o n i n t h e v a r i a b l e Q p . The N e w t o n - R a p h s o n m e t h o d i s u s e d t o f i n d a c o r r e c t i o n t o an e s t i m a t e d v a l u e o f Qp b y u s i n g t h e e x p r e s s i o n F ] + ( d F 1 / d Q p ) A Q = 0 ( 3 . 3 4 ) w h e r e d F l = l_ d Q P C a C o n s e q u e n t l y , e q u a t i o n s 3 . 3 3 , 3 . 3 4 and 3 . 3 5 a r e i t e r a t e d u n t i l s u c h t i m e a s A Q i s l e s s t h a n a s p e c i f i e d t o l e r a n c e ( s a y 0 . 0 0 1 ) ; w h e r e Qp i s i n c r e m e n t e d by A Q a f t e r e a c h u n s u c c e s s f u l i t e r a t i o n . A f l o w c h a r t f o r t h e a i r c h a m b e r i t e r a t i o n p r o c e s s ( u p s t r e a m b o u n d a r y o n l y ) i s p r e s e n t e d i n F i g u r e 3 . 6 . v a i r + A t ( Q p + Q ) 2 n C A t 2 ( V , , - r + A t ( Q p + Q ) / 2 ) ( 3 . 3 5 ) 3 . 5 VACUUM BREAKERS Vacuum b r e a k e r s a r e i n s t a l l e d i n p i p e l i n e s t o p r e v e n t s u b a t m o s p h e r i c p r e s s u r e a n d o p e n t o a d m i t a i r when t h e p r e s s u r e a t t h e v a l v e d r o p s b e l o w a t m o s p h e r i c p r e s s u r e . A s t h e p r e s s u r e i n t h e p i p e l i n e i n c r e a s e s a b o v e t h e o p e n i n g s e t t i n g a g a i n , t h e v a c u u m b r e a k e r s i m p l y a c t s a s a v e n t . The b o u n d a r y c o n d i t i o n f o r t h e s e v a l v e s i s r a t h e r c o m p l e x , b u t f o r t h e p u r p o s e s o f t h i s a n a l y s i s t h r e e s i m p l i f y i n g A i r Chamber Boundary C o n d i t i o n Estimate QP=Q I Determine A i r Volume From EQ. 3.31 'Flow into^ Chamber' YES C o r - f - 2 . 5 x C o r f NO Determine Hp - o r f From EQ. 3. 32 1 F i From EQ. 3 .33 AQ From EQ. 3 .34 dF/dQ From 3 .35 NO Qp=Q P+AQ YES v a i r = v a i r + 0 - 5 * t ( Q + Q p ) CONTINUE AIR CHAMBER BOUNDARY CONDITION FIGURE 3.6 26 assumptions are made: 1. Vacuum breakers are o n l y p o s i t i o n e d at the s u c t i o n f l a n g e s of i n l i n e (booster) pumps. 2. As the l i n e pressure drops below atmospheric p r e s s u r e , the v a l v e passes s u f f i c i e n t a i r to maintain atmospheric pressure at the s u c t i o n f l a n g e of the pump. 3. Once the pressure at the v a l v e r i s e s above the opening s e t t i n g again, any trapped a i r i s r e l e a s e d without a f f e c t i n g the t r a n s i e n t c o n d i t i o n . T h i s l a s t assumption i s v a l i d , because the volume of a i r passed i s u s u a l l y small when compared with the l i q u i d volume i n a given reach and i s assumed to remain near the vacuum breaker where i t can be e x p e l l e d . Since the e l e v a t i o n of each pump w i l l be known a p r i o r i , the pressure a l l o w i n g v a l v e opening simply becomes the height of the pump above the system datum. Once the v a l v e i s open, i t i s assumed to maintain a constant head (the pump e l e v a t i o n ) and the d i s c h a r g e i s c a l c u l a t e d given t h i s head. For any pressure head above the opening s e t t i n g (atmospheric p r e s s u r e ) , the vacuum breaker i s ignored. 3.6 SERIES JUNCTION OF TWO PIPES A boundary c o n d i t i o n f o r the s e r i e s j u n c t i o n of two pipes could be necessary because of a change i n pipe diameter, f r i c t i o n f a c t o r , or w a l l t h i c k n e s s . I f the v e l o c i t y heads i n the c o n d u i t s and the l o s s e s at the j u n c t i o n are assumed n e g l i g i b l e , then Hp =Hp . (3.36) i,n+l i+1,1 27 S i n c e no f l u i d i s l o s t o r s t o r e d a t t h e j u n c t i o n , t h e c o n t i n u i t y r e l a t i o n g i v e s Q P =Q P ( 3 . 3 7 ) i , n + l i + 1 , 1 A p p l y i n g t h e p o s i t i v e c h a r a c t e r i s t i c e q u a t i o n t o p i p e i Qp = C P - C a Hp ( 3 . 3 8 ) i , n + l i i i , n + l and t h e n e g a t i v e c h a r a c t e r i s t i c e q u a t i o n t o p i p e i+1 Qp = C N + C a Hp ( 3 . 3 9 ) i + 1 , 1 i+1 i+1 i + 1 , 1 i t f o l l o w s f r o m e q u a t i o n m a n i p u l a t i o n t h a t Hp = ( C P - C N ) / ( C a + C a ) . ( 3 . 4 0 ) i , n + l i i + 1 i i + 1 The o t h e r unknowns c a n now be d e t e r m i n e d f r o m e q u a t i o n s 3 . 3 6 t h r o u g h 3 . 3 9 . 3 . 7 CONSTANT HEAD R E S E R V I O R AT THE DOWNSTREAM END The d o w n s t r e a m end o f a p i p e l i n e w i t h a s p e c i f i e d number o f p i p e s i s a t t h e n+1 s e c t i o n o f t h e l a s t p i p e . I f t h e p i p e l i n e d r a i n s i n t o a r e s e r v o i r w i t h c o n s t a n t h e a d and t h e e x i t l o s s and v e l o c i t y h e a d a r e n e g l i g i b l e , t h e n H p = H R E S = CONSTANT ( 3 . 4 1 ) and Q p = C P - C a H R E S . ( 3 . 4 2 ) 28 3.8 FLOW CONTROL VALVE AT THE DOWNSTREAM END T r a n s i e n t c o n t r o l by means of flow c o n t r o l v a l v e s r e q u i r e s p r e c i s e knowledge of the r e l a t i v e v a l v e opening with r e s p e c t to time, as the s e v e r i t y of the waterhammer depends d i r e c t l y upon the v a l v e c l o s u r e time and v a l v e opening. The d i s c h a r g e through a given v a l v e i s o n l y s u b s t a n t i a l l y a f f e c t e d once the e f f e c t i v e opening has been reduced to 15 to 35 percent of the pipe c r o s s s e c t i o n a l area. By i n s t a l l i n g a quick c l o s i n g b u t t e r f l y v a l v e with a reduced diameter bypass l i n e c o n t a i n i n g the c o n t r o l v a l v e ( F i g . 3.7a), the c o s t of the c o n t r o l v a l v e i s s u b s t a n t i a l l y reduced. The c o n t r o l v a l v e should be s e l e c t e d so that the d i s c h a r g e as a f u n c t i o n of v a l v e opening i s known with a reasonable degree of accuracy. For a r e l a t i v e l y long p i p e l i n e , the b u t t e r f l y v a l v e can be assumed to c l o s e i n s t a n t a n e o u s l y upon power f a i l u r e , with a l l flow bypassed through the s m a l l e r c o n t r o l v a l v e which c l o s e s s l o w l y . For a s h o r t p i p e l i n e , the c l o s u r e of the b u t t e r f l y v a l v e cannot be assumed instantaneous and allowance must be made f o r t h i s i n i t i a l c l o s u r e time. The time fo r t h i s c l o s u r e may range from 0.15 to 0.75 seconds and i s assumed to be a s t r a i g h t l i n e . The c o n t r o l v a l v e i s s i t u a t e d at the downstream end of a p i p e l i n e d i s c h a r g i n g i n t o a r e s e r v o i r with constant head ( F i g . 3.7b). For a given d i s c h a r g e Qp, the head l o s s across the v a l v e i s H E = H 0 Q p 2 / ( Q o r ) 2 (3.43) where H Q i s the steady s t a t e head l o s s across the v a l v e f o r d i s c h a r g e Q 0 a n d ^ i s t h e d i m e n s i o n l e s s v a l v e opening. The head a) CONTROL VALVE-BUTTERFLY VALVE COMBINATION VALVE LOSS H r FLOW CONTROL VALVE DATUM H RES VALVE LOSS H G . L . H r H RES FLOW CONTROL VALVE ' DATUM b) CONTROL VALVE AT CONSTANT HEAD RESERVOIR F IGURE 3.8 30 at the upstream f l a n g e of the c o n t r o l v a l v e can be s p e c i f i e d by two cases: 1. Flow i n t o the r e s e r v o i r and H P = H R E S + H e (3.44) 2. Flow out of the r e s e r v o i r and H P=H R E S-H e. (3.45) S u b s t i t u t i n g f o r Hp from the p o s i t i v e c h a r a c t e r i s t i c equation (Eq. 3.10) i n t o equation 3.44 y i e l d s K 2Qp 2+Qp-K 3=0 (3.46) where K 2=HoC a/(QQTT) 2 and K 3=C aH R Es-Cp. In n e g l e c t i n g the negative s i g n i n the r a d i c a l term, Qp becomes Qp=(-1+^1-4K 2K 3)/2K2 (3.47) and Hp can be determined from Hp=H R E S+H eQp|Qp|/(Q 0t) 2. (3.48) The absolute value s i g n on the di s c h a r g e term ensures that the head i s added or su b t r a c t e d f o r flow i n t o or out of the r e s e r v o i r , r e s p e c t i v e l y . T h i s e f f e c t i v e l y s a t i s f i e s the dis c h a r g e c o n s t r a i n t s imposed by equations 3.44 and 3.45. A flow c h a r t f o r the c o n t r o l v a l v e i s presented i n Figu r e 3.8. 31 C o n t r o l Valve Boundary C o n d i t i o n Get Slope B/F Closure Determine Tau TAU> TAURED? YES ^Tau<TAUMIN J NO NO DETERMINE Tau YES Tau=TAUMIN YES Tau<TAUMIN C a l c u l a t e C p,K 2,K 2 NO Q P From EQ. 3 .47 Hp From EQ. 3 .48 Qp=0.0 ^CONTINUE Hp From EQ. 3.10 FLOW CONTROL VALVE BOUNDARY CONDITION FIGURE 3.8 32 CHAPTER IV. THE PROGRAM The program developed i n t h i s study i s an e x t e n s i o n to an e x i s t i n g program obtained from a p r e l i m i n a r y d r a f t of 'Applied H y d r a u l i c T r a n s i e n t s ' by Chaudhry 1. Some m o d i f i c a t i o n of t h i s program was r e q u i r e d as convergence problems e x i s t e d and some of the equations proved cumbersome when a n a l y s i n g a system c o n t a i n i n g pumps with long and shor t s u c t i o n l i n e s . The i t e r a t i o n procedure which i s used to determine the boundary c o n d i t i o n s f o r a pumping system i s d e s c r i b e d i n d e t a i l w i t h i n Chapter 3. The developed program designates a l l computations performed i n determining the t r a n s i e n t c o n d i t i o n s to the computer. P r i o r to commencing a computer run, though, s p e c i f i c system parameters must be i n p u t . These parameters are d i s c u s s e d i n d e t a i l i n t h i s chapter, with pumping system examples presented i n Appendix B. E n g l i s h or System I n t e r n a t i o n a l (SI) u n i t s may be used, with the user s p e c i f y i n g the u n i t type. The e n t i r e program i s presented i n Appendix A. 4.1 JUSTIFICATION AND MODIFICATIONS Pumping systems c o n s i s t i n g of long p i p e l i n e s with high l i f t c o n d i t i o n s , r e q u i r e c a r e f u l c o n s i d e r a t i o n to ensure economical i n s t a l l a t i o n s . The system l a y o u t , component s e l e c t i o n and c o s t s , although beyond the scope of t h i s t h e s i s , are of fundamental importance. An example should be s u f f i c i e n t to i l l u s t r a t e the importance of these f a c t o r s . 33 Codrington and W i t h e r e l l 2 d e s c r i b e d the a n a l y s i s of a water supply system to a mine. The lay o u t c o n s i s t e d of a main pump and a booster pump l o c a t e d p a r t way up the l i n e to minimize the upstream p r e s s u r e . The o r i g i n a l proposal had been to l o c a t e a s i n g l e pump at the w e l l head, r e q u i r i n g schedule 80 pipe to handle the higher p r e s s u r e . In moving the booster pump pa r t way up the l i n e , system pressures were determined to be w i t h i n the al l o w a b l e l i m i t s of schedule 40 pi p e . Th i s c o n f i g u r a t i o n provided a s u b s t a n t i a l saving i n pipe c o s t s . S i m i l a r l y , the value of p r o p e r l y s e l e c t e d c o n t r o l equipment should not be underestimated. One method of minimizing the t r a n s i e n t surge i s by the placement of int e r m e d i a t e sumps or by using surge tanks. T h i s i s a very expensive p r o p o s i t i o n . A l e s s c o s t l y s o l u t i o n would be the s e l e c t i o n of adequate v a l v i n g and the i n s t a l l a t i o n of an a i r chamber. This b a s i c approach was followed i n the development of t h i s t h e s i s . The purpose of t h i s t h e s i s was to take the boundary c o n d i t i o n s i l l u s t r a t e d i n Chapter 3 (the boundary c o n d i t i o n s were npt developed as a part of t h i s t h e s i s ) and combine them i n t o a m u l t i s t a g e pumping system. The o r i g i n a l pumping system c o n s i s t e d of a main pumping s t a t i o n d i s c h a r g i n g i n t o a constant head r e s e r v o i r . T h i s system was modified by adding a number of pumping s t a t i o n s . Then, to these s t a t i o n s , an a i r chamber at the main pump, vacuum breakers at the booster pumps, check v a l v e s , pressure r e g u l a t i n g v a l v e s , and a flow c o n t r o l v a l v e at the r e s e r v o i r have been added ( F i g . 4.1). Thus, a system was b u i l t s tep by st e p . Th i s system was then analysed. RESERVOIR CONTROL VALVE TYPICAL 2-STAGE PUMPING SYSTEM FIGURE 4.1 35 4.2 ASSUMPTIONS The pumping system under c o n s i d e r a t i o n c o n s i s t s of pumping s t a t i o n s i n s e r i e s c o n t a i n i n g a v a r i e t y of c o n t r o l mechanisms. These c o n t r o l mechanisms can be added or removed depending upon the system to be analysed. Since t h i s program was developed f o r t r a n s i e n t a n a l y s i s r a t h e r than system d e s i g n , the f o l l o w i n g assumptions are made: 1. The a n a l y s i s a p p l i e s only to the s i t u a t i o n where power f a i l u r e occurs at a l l pumps si m u l t a n e o u s l y . 2. A check v a l v e (provided one e x i s t s ) at the di s c h a r g e s i d e of a pump c l o s e s immediately upon flow r e v e r s a l . T h i s e l i m i n a t e s the pump from most f u t u r e c a l c u l a t i o n s . 3. The main pump i s t r e a t e d as i f i t were d i s c h a r g i n g d i r e c t l y from the sump ( i . e . the s u c t i o n l i n e i s sh o r t and n e g l e c t e d ) . A l l booster pumping s t a t i o n s are modelled as having long s u c t i o n l i n e s which cannot be neglected i n computations. 4. Vacuum breakers (provided they e x i s t ) are s i t u a t e d at the s u c t i o n f l a n g e of booster pumps and maintain atmospheric pressure at t h i s p o i n t once the h y d r a u l i c g r a d e l i n e drops below t h i s e l e v a t i o n . These v a l v e s do not a f f e c t p r e s s u r e s above atmospheric p r e s s u r e . 5. An a i r chamber (provided one e x i s t s ) i s s i t u a t e d near the upstream pump on l y . A check v a l v e i s inc l u d e d with the chamber and c l o s e s immediately upon power f a i l u r e , e l i m i n a t i n g t h i s pump. The steady s t a t e water s u r f a c e i n the chamber has a s p e c i f i e d e l e v a t i o n above the c e n t e r l i n e of the pump, with pressure f l u c t u a t i o n s due to changes i n t h i s l e v e l ignored. 36 6. Head l o s s e s at the pipe j u n c t i o n s and the downstream r e s e r v o i r are n e g l i g i b l e . 7. T h i s study i s l i m i t e d to cases i n which no water column s e p a r a t i o n o c c u r s . The occurrence of t h i s phenomena produces a d i s c o n t i n u i t y which t h i s program i s not able to handle. 4.3 DESCRIPTION OF THE PROGRAM The main f u n c t i o n s of t h i s program are as f o l l o w s : 1. S p e c i f i c a t i o n of u n i t type. 2. Reading input data to the computer. 3. W r i t i n g of g e n e r a l system, pump and pipe data. 4. Ensuring the same time i n t e r v a l f o r each p i p e . 5. I n i t i a l i z i n g c o n t r o l d e v ice o p e r a t i n g parameters. 6. Computation of constants and steady s t a t e c o n d i t i o n s . 7. Computation of t r a n s i e n t c o n d i t i o n s . 8. S t o r i n g maximum and minimum pressure v a l u e s . 9. W r i t i n g t r a n s i e n t c o n d i t i o n s a f t e r s p e c i f i e d time s t e p s . 4.4 PROGRAM USAGE AND DATA INPUT The program as presented w i l l analyse a v a r i e t y of pumping systems. The number of s e r i e s pumping s t a t i o n s i s r e s t r i c t e d to f i v e ; the maximum number of s e r i e s pipes i s ten and each pipe may be d i v i d e d i n t o at most twenty reaches. The type of u n i t s ( e i t h e r E n g l i s h or System I n t e r n a t i o n a l ) are to be s p e c i f i e d by the user. T h i s i s accomplished with the f i r s t data card to be read i n t o the computer. This card should c o n t a i n the words ENGLISH or SI w i t h i n the f i r s t seven columns 37 of t h i s c a r d , beginning i n the f i r s t column. The remaining input parameters apply to the system that i s to be examined. These parameters are system dependent and the user should determine data e n t r y p a t t e r n s by examining the input area w i t h i n the program (Appendix A). The d e f i n i t i o n s of these parameters f o l l o w . NO, QO and HO r e f e r to steady s t a t e values and NR, QR and HR r e f e r to rated values of turbomachine speed, d i s c h a r g e and head, r e s p e c t i v e l y . ER i s the rated pump e f f i c i e n c y . The pump u n i t i n e r t i a , which i n c l u d e s the motor, the pump i m p e l l e r and the mass of l i q u i d held w i t h i n i t , i s designated by WR2. NSPUMP i s the number of s e r i e s pumping s t a t i o n s and NPPUMP i s the number of p a r a l l e l pumps i n each s t a t i o n . The a r r a y s FH and FB s p e c i f y the head and torque c h a r a c t e r i s t i c curves f o r each pumping s t a t i o n , with NPC equal to the number of p o i n t s that are to be input f o r each curve. The s e p a r a t i o n between these p o i n t s i s give n by an angular i n t e r v a l of DTH. ELBO i s the height above datum f o r each booster pumping s t a t i o n and HLC i s the head l o s s c o e f f i c i e n t f o r the check v a l v e s . NP g i v e s the number of pipes i n the system, and L, D, A and F correspond to l e n g t h , diameter, pressure wave speed and f r i c t i o n f a c t o r , r e s p e c t i v e l y , f o r each p i p e . The l o c a t i o n of each pumping s t a t i o n i s given by the number of the pipe at the pump di s c h a r g e and i s s p e c i f i e d w i t h i n the v e c t o r LOCAT. Pipe number 1 i s at the main pumping s t a t i o n . NRLP i s the number of reaches w i t h i n the l a s t p i p e . T h i s term should be chosen c a r e f u l l y , as i t i s used to determine the 38 c a l c u l a t i o n time i n t e r v a l and thus the number of reaches i n each p i p e . Very s h o r t or very long pipes could pose problems i f NRLP i s s e l e c t e d unwisely. The length of time f o r which the t r a n s i e n t phenomena i s to be c a l c u l a t e d i s s p e c i f i e d by TLAST; IPRINT i s a p r i n t i n g c o n t r o l parameter which c o n t r o l s the number of time increments between s u c c e s s i v e p r i n t e d r e s u l t s . Both of these terms should be chosen j u d i c i o u s l y . S e l e c t i n g too s h o r t a c a l c u l a t i o n p e r i o d w i l l r e s u l t i n an i n s i g n i f i c a n t record of the t r a n s i e n t phenomena, while too long a c a l c u l a t i o n p e r i o d could prove c o s t l y , p a r t i c u l a r l y when a n a l y s i n g very l a r g e systems. S i m i l a r l y , the r e s u l t s generated dur i n g each time i t e r a t i o n may not be s i g n i f i c a n t . Thus, IPRINT should be g i v e n due c o n s i d e r a t i o n to prevent the i n o r d i n a t e g e n e r a t i o n of u s e l e s s p r i n t e d output. The v a r i e t y of t r a n s i e n t c o n t r o l mechanisms a v a i l a b l e at each pump are s p e c i f i e d by i n d i c e s . I n p u t t i n g a *1' i n d i c a t e s the presence of a v a l v e or device and i n p u t t i n g a '0' i n d i c a t e s i t s absence. The v e c t o r s PRV, PRCHK, CHECK and VAC are used to i n d i c a t e the presence of pressure r e g u l a t i n g v a l v e s , check v a l v e s at r e g u l a t i n g v a l v e s , check v a l v e s o n l y and vacuum breakers, r e s p e c t i v e l y , at each s e r i e s pump stage. The presence or absence of an upstream a i r chamber, d i f f e r e n t i a l o r i f i c e at the chamber o u t l e t and downstream flow c o n t r o l v a l v e i s s p e c i f i e d by the terms AIR, NDIFF and NVAL, r e s p e c t i v e l y . Given that a PRV does e x i s t at a s p e c i f i e d pump, the f o l l o w i n g parameters are r e q u i r e d as i n p u t : 39 PRVTAU .. the a r r a y c o n t a i n i n g the v a l v e c l o s u r e curve NTAU .... the number of data p o i n t s on the c l o s u r e curve DTIME ... time i n t e r v a l f o r these data p o i n t s TOPEN ... time at which the v a l v e begins to open S i m i l a r parameters must a l s o be input f o r the downstream c o n t r o l v a l v e , p r o v i d i n g one e x i s t s : VALVE ... a v e c t o r c o n t a i n i n g the tau curve NVT g i v e s the number of data p o i n t s DTVAL ... time i n t e r v a l f o r these data p o i n t s TAURED .. e f f e c t i v e area once the b u t t e r f l y v a l v e i s c l o s e d TAUMIN .. gate opening when va l v e movement ceases VALOSS .. steady s t a t e head l o s s through the b u t t e r f l y v a l v e The c l o s u r e of the b u t t e r f l y v a l v e may a l s o have to be i n c l u d e d . T h i s i s done by i n p u t t i n g a 1 ( c l o s u r e required) or a 0 ( c l o s u r e not required) f o r the term INCL. Then, TCLOSE .. time at which b u t t e r f l y v a l v e begins to c l o s e TBFVAL .. i s the time f o r the b u t t e r f l y v a l v e to c l o s e . The above mentioned c l o s u r e curves are a f u n c t i o n of the c o e f f i c i e n t of d i s c h a r g e and the area of v a l v e opening. I t should be noted that the c o e f f i c i e n t of d i s c h a r g e i s not n e c e s s a r i l y constant f o r any given v a l v e opening and the v a r i a t i o n of the d i s c h a r g e with v a l v e s e t t i n g u s u a l l y has to be determined e x p e r i m e n t a l l y . These r e s u l t s should be a v a i l a b l e from the v a l v e manufacturer. F i g u r e 4.2 shows the n o t a t i o n f o r the PRV and c o n t r o l v a l v e c l o s u r e curves. S i m i l a r l y , f o r the a i r chamber at the main pumping s t a t i o n , the f o l l o w i n g parameters should be input: TIME a) PRESSURE REGULATING VALVE CLOSURE CURVE b) CONTROL V A L V E CLOSURE CURVE F IGURE 4.2 41 HBAR . . . . b a r o m e t r i c p r e s s u r e h e a d CORF . . . . h e a d l o s s c o e f f i c i e n t f o r f l o w o u t o f t h e c h a m b e r A L I N E . . . a r e a o f t h e p i p e s e c t i o n f r o m c h a m b e r t o m a i n p i p e l i n e C L I N E . . . h e a d l o s s c o e f f i c i e n t f o r t h i s p i p e A I R V O L . . s t e a d y s t a t e a i r v o l u m e i n t h e c h a m b e r EN e x p o n e n t f o r t h e p o l y t r o p i c g a s e q u a t i o n ZSURF . . . c h a m b e r w a t e r s u r f a c e e l e v a t i o n a b o v e d a t u m The s i z e o f t h e a i r v o l u m e t o c o n t r o l maximum p r e s s u r e and t h e l i q u i d v o l u m e t o a v o i d t o t a l l y d r a i n i n g t h e t a n k d u r i n g t h e d o w n s u r g e s h o u l d be d e t e r m i n e d p r i o r t o u s i n g t h i s a n a l y s i s . P a r m a k i a n ^ , and G a l a t i u k and R u u s 3 , a s w e l l a s o t h e r s , p r e s e n t a i r c h a m b e r c h a r t s f o r t h i s p u r p o s e . 4 . 5 CHECK OF ACCURACY Upon c o m p l e t i n g t h e d e v e l o p m e n t o f t h e p r o g r a m , t h e v a l i d i t y o f t h e r e s u l t s w e r e e x a m i n e d . A g r a p h i c a l w a t e r h a m m e r s o l u t i o n was p e r f o r m e d by E . R u u s and t h e r e s u l t s o b t a i n e d c l o s e l y f o l l o w e d t h e c o m p u t e r s o l u t i o n g e n e r a t e d f o r t h e same p u m p i n g s y s t e m . A l s o , E . R u u s s i m u l t a n e o u s l y d e v e l o p e d h i s own p r o g r a m f o r p u m p i n g s y s t e m a n a l y s i s , b o t h p r o g r a m s b e i n g d e v e l o p e d i n d e p e n d e n t l y . A c o m p a r i s o n o f c o m p u t e r g e n e r a t e d r e s u l t s a l s o p r o v e d t o be f a v o r a b l e . T h e s e c h e c k s , p r e s e n t e d i n A p p e n d i x B , i n d i c a t e t h e v a l i d i t y o f t h e d e v e l o p e d p r o g r a m . 4 2 CHAPTER V. DISCUSSION Th i s d i s c u s s i o n d e a l s with the p r i n c i p a l f a c t o r s which w i l l g i v e p a r t i a l or i n a c c u r a t e r e s u l t s from the program developed i n t h i s t h e s i s . Accurate s i m u l a t i o n of the pumping system that i s to be analysed i s very important. Time c o n s t r a i n t s as w e l l as p h y s i c a l c o n s t r a i n t s should be c o n s i d e r e d . Accurate s i m u l a t i o n of a p i p i n g system with a i r trapped i n v a r i o u s l o c a t i o n s i s d i f f i c u l t . Even a small pocket of a i r can s i g n i f i c a n t l y change the timing and the magnitude of the pressure f l u c t u a t i o n s . In a d d i t i o n the wavespeed w i t h i n the system may change as a i r i s r e l e a s e d from s o l u t i o n d u r i n g low pressure p e r i o d s of the c y c l e . To maintain g i v e n system c h a r a c t e r i s t i c s vacuum breakers are only p o s i t i o n e d at the s u c t i o n f l a n g e s of the booster pumps. Thus, the passage of a i r i n t o the system and the formation of a i r pockets can be avoided. The v a l i d i t y of the c h a r a c t e r i s t i c s method a p p l i e d to t r a n s i e n t a n a l y s i s i s w e l l documented i n the l i t e r a t u r e . A t t e n t i o n should be paid to the system c h a r a c t e r i s t i c s a s s o c i a t e d with the problem, as i n a c c u r a t e m o d e l l i n g w i l l generate r e s u l t s f o r a d i f f e r e n t problem. P a r t i c u l a r a t t e n t i o n should be paid to v a l v e motions. The developed program i s a p p l i c a b l e to a v a r i e t y of systems, as boundary c o n d i t i o n s are i n t e r c h a n g e a b l e . One advantage of the method of c h a r a c t e r i s t i c s i s t h a t a d d i t i o n a l boundary c o n d i t i o n s can e a s i l y be added as they develop. Steady s t a t e as w e l l as t r a n s i e n t c o n d i t i o n s are modelled. Time c o n s t r a i n t s may d i c t a t e the use of a d i f f e r e n t 43 c a l c u l a t i o n technique f o r very long p i p e l i n e s . The time r e q u i r e d to generate an accurate r e p r e s e n t a t i o n of the va l v e c l o s u r e could be p r o h i b i t i v e when c o n s i d e r i n g the r e q u i r e d computer time. Devices provided i n the system to reduce t r a n s i e n t p r e s s u r e s are assumed to be p r o p e r l y designed and to f u n c t i o n as they are intended. C r i t e r i a which c o n s t i t u t e proper design are l e f t to the user. T h i s program should not be used f o r the s e l e c t i o n of the components comprising the system. P e r i o d s of low di s c h a r g e (near zero) i n some s i t u a t i o n s may not converge to a s o l u t i o n . The problem e x i s t s i n the pump sub r o u t i n e . A numerical s o l u t i o n of such an occurence was c a r r i e d out by hand; the r e s u l t being that s u c c e s s i v e i t e r a t i o n s d iverged from the o r i g i n a l approximation, a l t e r n a t i n g above and below t h i s approximation. No s o l u t i o n has been found f o r t h i s problem. Using the o r i g i n a l approximation at the p o i n t i n the pump subroutine where the program terminates w i l l a llow the next run to co n t i n u e . A i r i n l e t v a l v e s and an upstream a i r chamber were in c o r p o r a t e d i n t o the system to minimize the occurrence of water column s e p a r a t i o n . The main i n f l u e n c e s on column s e p a r a t i o n are: 1 . Rate of flow stoppage. A pump with low r o t a t i o n a l i n e r t i a may r e s u l t i n instantaneous flow stoppage and produce a c o r r e s p o n d i n g l y l a r g e pressure r i s e . 2. The len g t h of the pumping system. The time p e r i o d d u r i n g which the pressure i s allowed to decrease i s a f u n c t i o n of the time r e q u i r e d f o r a pressure wave to t r a v e r s e up and down the 44 system. A l l other c h a r a c t e r i s t i c s being equal, the longer the pipe the g r e a t e r the pressure drop. 3 . Normal o p e r a t i n g pressures at c r i t i c a l p o i n t s . I f the o r i g i n a l pressure i s g r e a t e r than the maximum decrease or i f the pressure can be a r t i f i c i a l l y maintained at a p r e s e t l i m i t , water column s e p a r a t i o n would not be a problem. 4. Steady s t a t e l i q u i d v e l o c i t y . As the steady s t a t e v e l o c i t y i n c r e a s e s , the s i z e of the vacuous space, the v e l o c i t y change at sudden flow stoppage and the l o c a l pressure r i s e d u r i n g the v o i d c o l l a p s e are i n c r e a s e d . These i n f l u e n c e s are a l l interdependent, with the r e s u l t being a i r entrainment i n t o the system. This changes the system c h a r a c t e r i s t i c s and the generated r e s u l t s are no longer r e p r e s e n t a t i v e of the o r i g i n a l system being analysed. For t h i s reason, water column s e p a r a t i o n should be avoided. The i n c l u s i o n of t h i s phenomena i n t o the system could produce v i a b l e f u t u r e r e s e a r c h . 45 CHAPTER VI. CONCLUSIONS This program was developed to provide the t r a n s i e n t a n a l y s i s f o r m u l t i s t a g e pumping systems. T r a n s i e n t s are c o n t r o l l e d by an a i r chamber at the main pumping s t a t i o n and adequate v a l v i n g w i t h i n the system. When comparing a m u l t i s t a g e system to a s i n g l e stage system (Appendix B), the m u l t i s t a g e c o n f i g u r a t i o n s i g n i f i c a n t l y reduced the d i f f e r e n c e between the maximum and minimum p i p e l i n e pressures w i t h i n the system. T h i s r e d u c t i o n of extreme t r a n s i e n t f l u c t u a t i o n s i s the main advantage of t h i s approach. Pipe w a l l t h i c k n e s s and the s i z e of l a r g e , expensive c o n t r o l s t r u c t u r e s may be reduced, which could r e s u l t i n a s u b s t a n t i a l savings i n c o s t s . Thus, the o r i g i n a l premise that d i s t r i b u t i n g pumping s t a t i o n s along the p i p e l i n e w i l l reduce t r a n s i e n t p r e s s u r e s w i t h i n the system subsequent to power f a i l u r e has been s u b s t a n t i a t e d , as was expected. 46 BIBLIOGRAPHY Chaudhry, M.H., A p p l i e d H y d r a u l i c T r a n s i e n t s , Van Nostrand Reinhold Company, New York, pp. 1-103,302-331, 1979. Codrington, J.B., and R.G. W i t h e r e l l , "The Use of Impedence Concepts and D i g i t a l M o d e l l i n g Techniques i n the S i m u l a t i o n of P i p e l i n e T r a n s i e n t s " , Second I n t e r n a t i o n a l Conference on  Pressure Surges, Paper A2, The C i t y U n i v e r s i t y , London, England, pp. 15-44, 1976. G a l a t i u k , W.R., A i r Chamber Design Charts, A t h e s i s submitted i n p a r t i a l f u l f i l l m e n t of the requirements f o r the degree of Master of A p p l i e d Science at the U n i v e r s i t y of B r i t i s h Columbia, 1973. Marchal, M., G. F l e s h , and P. Suter, "The C a l c u l a t i o n of Waterhammer Problems by Means of the D i g i t a l Computer", Proc. I n t . Symp. Waterhammer Pumped storage P r o j e c t s , ASME, Chicago, 1965. Parmakian, J . , Waterhammer A n a l y s i s , Dover P u b l i c a t i o n s , Inc., New York, 1963. Stepanoff, A.J., C e n t r i f u g a l and A x i a l Flow Pumps, 2nd ed., John Wylie and Sons, Inc., New York, pp. 269-292,425-458, 1957. Stephenson, D., P i p e l i n e Design f o r Water Engineers, E l s e v i e r S c i e n t i f i c P u b l i c a t i n g Company, New York, pp. 53-83, 1976. S t r e e t e r , V.L., F l u i d Mechanics, 3rd ed., McGraw-Hill Book Company, New York, pp. 343-358, 1962. T u l l i s , J.P. (ed.), C o n t r o l of Flow i n Closed Condui t s , Colorado State U n i v e r s i t y , F o r t C o l l i n s , Colorado, pp. 229-314,543-557, 1971. Wylie, E.B., and V.L. S t r e e t e r , F l u i d T r a n s i e n t s , McGraw-H i l l Book Company, New York, pp. 1-117,180-189, 1978. A P P E N D I X A PROGRAM FOR MULT ISTAGE PUMPING SYSTEMS ANALYSIS OF TRANSIENTS IN A PIPELINE CAUSED BY PUMPS **************************************************************** REAL L,NO(5), NR(5),TYPE*8(2)/'ENGLISH',1 SI '/,UNITS*8 REAL K2,K3 INTEGER OPEN(5),PRV(5),CLOSED(5),CHECK(5),PRCHK(5),VAC(5),AIR DIMENSION F(10),AREA(10),A(10),L(10),D(10),HMAX(10),HMIN(10), 1 LOCAT(5),NPPUMP(5),HDIS(5),ER(5),WR2(5),SHFTRQ(5),ELBO(5), 2 TM(5),TOPENf5),TMAX(10),TMIN(10),HLC(5) COMMON /AREA1/ FH(5,90),FB(5,90),PRVTAU(5,90),VALVE(90) COMMON /AREA2/ N (10) ,Q(10,20) ,H (10,20) ,CA (10) ,CF(10) COMMON /OUT2/ QP(10,20),HP(10,20) COMMON /AREA3/ ALPHA(5),V(5),DALPHA(5),DV(5),CVHL(5),DTH COMMON /OUT3/ ALPOUT(5),VPOUT(5) COMMON /PRV3/ OPEN,HR(5),H0(5),QR(5),QO,CN,CA2,TAUPRV(5),HSUC(5) DATA PI/3.141593/ ***** DETERMINATION OF UNIT TYPE - INPUT AS 'ENGLISH' OR 'SI' READ(5,1) UNITS,OUTPUT IF(UNITS.EQ.TYPE(2)) GO TO 22 G=32.22 SPECWT=62.4 GO TO 33 22 G=9.820 SPECWT=9802. ' ***** READING OF INPUT DATA 33 READ(5,2) NP,NSPUMP,NRLP,NPC,NTAU,IPRINT,AIR,NDIFF,NVAL,NVT,INCL READ(5,3) QO,DTH,TLAST READ(5,3) (L(I),D(I),A(I),F(I),I=1,NP) DO 44 1=1,NSPUMP READ(5,3) (FH (I,J),J = l,NPC) READ(5,3) (FB(I,J),J=1,NPC) READ(5,3) ELBO(I),NO(I),OR(I) ,HR (I) ,NR(I),ER (I) ,WR2(I),HLC(D READ (5,2) LOCAT(I) ,NPPUMP(I) ,PRV(I) ,PRCHK(I) ,CHECK(I) ,VAC(I) IF(PRV(I).EQ.O) GO TO 44 READ(5,3) TOPEN(I),DTIME,(PRVTAU(I,J),J=1,NTAU) 44 CONTINUE IF(AIR.EQ.O) GO TO 55 READ (5,3) HBAR,AIRVOL,CORF,EN,ZSURF,CLINE,ALINE 55 IF(NVAL.EQ.O) GO TO 66 READ(5,3) TCLOSE,VALOSS,TAURED,TBFVAL,TAUMIN,DTVAL READ(5,3) (VALVE(I),1=1,NVT) 1 FORMAT (A7,1X,A4) 2 FORMAT (1216) 3 FORMAT (12F6.0) ***** WRITING OF GENERAL DATA 66 WRITE(6,10) UNITS,NP,NRLP,NSPUMP,QO,TLAST,IPRINT,NPC,DTH 10 FORMAT('1,,4X,'*** GENERAL SYSTEM DATA ***'//12X,'UNITS USED IN TH IIS PROGRAM ARE '.A7/12XNUMBER OF PIPES=',13/12X,'NUMBER OF REACH 2ES IN THE LAST PIPE=13/12X,'NUMBER OF SERIES PUMPS=' ,I 3/12X,'SYS 3TEM STEADY STATE DISCHARGE=',F6.2//12X,'TIME FOR WHICH TRANSIENT C 40NDITI0NS ARE CALCULATED IS',F6.2,' SEC'/12X,'PRINTING INTERVAL IS 5 EVERY' 12 ' TIME INTERVALS'//12X,'NUMBER OF POINTS ON CHARACTERIS 6TIC CURVES=',I4/12X,'THETA INTERVAL FOR STORING CHARACTERISTIC CUR 7VE=',F7.4//5X,'*** INPUT PUMP DATA ***'/) ***** WRITING OF PUMP DATA DO 77 I=1,NSPUMP WRITE(6,11) I,NPPUMP (I),QR (I),HR (I) ,ELBO (I) ,NR (I) ,ER (I) ,WR2(I) , 1HLC(I),(FH(I,J),J=1,NPC) 11 FORMATC ' , 9X , 'SERIES PUMP #',11/12X,'NUMBER OF PARALLEL PUMPS=', 1I3/12X,'RATED DISCHARGE=',F6.2/12X,'RATED HEAD=',F6.1,27X,'PUMP EL 2EVATION=',F6.1/12X,'RATED PUMP SPEED=*,F7.1/12X,'PUMP EFFICIENCY=' 3,F5.2/12X,'PUMP UNIT INERTIA (WR**2)=',F7.1,1IX,'HEAD LOSS COEFFIC 4IENT FOR CHECK VALVE=',F7.4//12X,'POINTS ON HEAD CHARACTERISTIC' /6 5 (12X,15F7.3/)) WRITE(6,12) (FB(I,J) ,J = l,NPC) 12 FORMAT(12X,'POINTS ON TORQUE CHARACTERISTIC'/6(12X,15F7.3/)) 77 CONTINUE ***** WRITING OF PIPE DATA WRITE(6,13) 13 FORMAT(/5X,'*** INPUT PIPE DATA ***'//12X,'PIPE NO.' , 4 X, 'LENGTH' , 4 IX,'DIAM.', 4X,'WAVE VEL.',4X,'FRICTION FACTOR') WRITE(6,14) (I,L(I),D(I),A(I),F(I),I=1,NP) 14 FORMAT(14X,I3,6X,F7.1,3X,F5.1,5X,F7.1,1IX,F5.3) WRITE(6,15) 15 FORMAT(/12X,'PIPE NO.',SX,'ADJUSTED WAVE VEL',5X,*NO. OF REACHES') ***** CALCULATION OF PIPE CONSTANTS DT=L(NP)/(NRLP*A(NP)) T=0.0 DO 88 1=1,NP AN=L(I)/(DT*A(I)) N(I)=AN *** ADJUSTING WAVE VELOCITY AS NECESSARY IF((AN-N(I)).GE.0.5) N(I)=N(I)+1 A(I)=L(I)/(DT*N(I)) WRITE(6,16) I,A(I),N(I) 16 FORMAT(14X,I3,12X,F7.1,18X,I2) AREA(I)=(PI/4.)*D(I)**2 CA(I)=G*AREA(I)/A(I) CF(I)=F(I)*DT/(2.*D(I)*AREA(I)) F(I)=F(I)*L(I)/(2.*G*D(I)*N(I)*AREA(I)**2) TMAX (I)=0 . 0 TMIN (I)=0 . 0 88 CONTINUE ***** WRITING OF VALVE AND SURGE CONTROL DEVICE DATA 17 FORMAT^/5xl'*** VALVES AND SURGE CONTROL DEVICES ***'//12X,'A ONE 1INDICATES PRESENSE OF VALVE OR DEVICE'//l2X,'PUMP NO.',3X, PUMP LO 2CATION',3X,'PRV',3X,'PRV-CHECK VALVE',3X,'CHECK VALVE',3X,'VACUUM 3BREAKER1,3X,'U/S AIR CHAMBER',3X , 'D/S CONTROL VALVE'/101X, 4I1,18X,I1) WRITE (6,18) (J,LOCAT(J) ,PRV(J) ,PRCHK(J) ,CHECK (J) ,VAC(J) ,J = 1,NSPUMP) 18 FORMAT(16X,I1,12X,I1,10X,I1,11X,I1,15X,I1,14X,I1) IF(AIR.EQ.O) GO TO 99 WRITE(6,19) HBAR,NDIFF,AIRVOL,CORF,EN,ZSURF.CLINE,ALINE 19 FORMAT(/12X,'BAROMETRIC PRESSURE=*,F5.1,57X,'DIFFERENTIAL ORIFICE 1... ',I1/12X,'INITIAL AIR VOLUME IN AIR CHAMBER=',F5.1/12X,'HEAD L 20SS COEFFICIENT FOR CHAMBER ORIFICE=',F6.3/12X,'POLYTROPIC GAS CON 3STANT=',F4.1/12X,'ELEVATION OF CHAMBER WATER SURFACE ABOVE DATUM=' 4,F7.2/12X,'HEAD LOSS COEFFICIENT FROM CHAMBER TO MAIN LINE=',F5.2/ 512X,'AREA OF PIPE FROM CHAMBER TO MAIN LINE=',F5.2) 99 IF (NVAL.EQ.O) GO TO 100 WRITE(6,20) TCLOSE,INCL,TBFVAL,VALOSS,TAURED,TAUMIN,NVT,DTVAL 20 FORMAT(/12X,'TIME AT WHICH BUTTERFLY VALVE BEGINS TO CLOSE=' , F5.?, 131X,'BUTTERFLY VALVE CLOSURE ...',12/ 212X,'TIME REQUIRED TO CLOSE BUTTERFLY VALVE=',F5.2/12X,'STEADY STA 3TE HEAD LOSS THROUGH BUTTERFLY VALVE=',F6.3/12X,'RELATIVE AREA OF 4BYPASS LINE=',F6.3/12X,'MINIMUM RELATIVE CONTROL VALVE OPENING=',F 56.3/12X,'NUMBER OF DATA POINTS ON CONTROL VALVE CURVE=',13/12X, ' TI 6ME INTERVAL BETWEEN THESE DATA POINTS=',F6.3) ***** INITIALIZE VALVE AND SURGE DEVICE CONTROLLING PARAMETERS 100 DO 150 I=1,NSPUMP OPEN(I)=0 CLOSED(I)=l TM (I)=0.0 OVERFLOW PRINTING FIELD TO INDICATE ABSENCE OF VALUE. TAUPRV(I)=111111. ALPOUT(I)=llllll. VPOUT(I)=llllll. 150 CONTINUE IF(AIR.EQ.O) AIRVOL = l l l l l l . VALTAU=1.0 IF(NVAL.EQ.O) VALTAU=111111 . TVAL=0.0 RESID=0.0 ***** COMPUTATION OF CONSTANTS FOR PUMPS DO 200 I=1,NSPUMP SHFTRQ(I)=(30.*SPECWT*HR(I)*QR(I))/(PI*NR(I)*ER(I)) ALPHA(I)=NO(I)/NR(I) V(I)=QO/(NPPUMP(I)*QR(I)) DV(I)=0.0 DALPHA(I)=0.0 *** DETERMINE HEAD LOSS AT CHECK VALVES (=K*V**2/2G) NPB=LOCAT(I) IF(I.EQ.NSPUMP) NPE=NP IF(I.NE.NSPUMP) NPE=LOCAT(I+l)-l CVH0=HLC(I)*Q0*Q0/(2.*G*AREA(NPB)**2) CVHL(I)=CVHO/HR(I) ***** CALCULATION OF STEADY STATE CONDITIONS 51 IF(V(I).EQ.O.O) GO TO 210 VO=V(I) ALPHAO=ALPHA(I) THETA=57.296*(PI+ATAN2(VO,ALPHAO)) GO TO 220 210 THETA=0.0 220 CALL PARAB(THETA,1,1,DTH,Z) HO (I)=Z*HR(I)*(ALPHA0**2+V0**2) H (NPB,1)=H0(I)+RESID-CVHO HSUC(I)=RESID HDIS(I)=H(NPB,1) DO 240 J=NPB,NPE NN=N(J)+l DO 230 K=1,NN H(J,K)=H(J,1)-(K-1)*F(J)*Q0**2 IF(J.NE.NPE.AND.K.EQ.NN) H(J+l,1)=H(J,NN) Q(J,K)=Q0 230 CONTINUE HMAX(J)=H(J,1) HMIN(J)=H(J,NN) 240 CONTINUE RESID=H(NPE,NN) 200 CONTINUE NN=N(NP)+1 IF(NVAL.EQ.O) VALOSS=0.0 HRES=H(NP,NN)-VALOSS C AIR CHAMBER CONSTANTS AT STEADY STATE CONDITIONS IF(AIR.EQ.O) GO TO 24 HPORF=0.0 HPAIR=H(1,1)+HBAR-ZSURF-HPORF C=HPAIR*AIRVOL**EN C c ***** WRITING TIME,PRESSURES AND DISCHARGES C 24 WRITE(6,25) HRES 25 FORMAT('l',4X,****** HEAD AT THE DOWNSTREAM RESERVOIR=',F7.2) WRITE(6,30) 30 FORMAT(///IX,'SERIES',1X,'TIME',4X,'V,4X,'ALPHA',IX,'PIPE ' , 19X , 1'REACHES ALONG PIPES',44X,1PRVTAU',2X,'AIR',4X,'VALVE'/2X,'PUMP', 2114X,'VOLUME',3X,'TAU'/) 300 ICOUNT=0 DO 350 1=1,NSPUMP NPB=LOCAT(I) IF((CHECK(I).EQ.O).AND.(CLOSED(I).EQ.l)) GO TO 320 IF(I.NE.l) GO TO 310 IF(Q(1,1).LE.O.O) WRITE(6,35) I,T,TAUPRV(I),AIRVOL,VALTAU 35 FORMATC ' , 2X , 12 , 2X , F5 . 2 , IX , ' VALVE 1 CLOSED', 87X, F4 . 2, 2X, F6 .1, 2X , F 15.3) IF(Q(1,1).GT.O.O) WRITE(6,40) I,T,V(I),ALPHA(I),TAUPRV(I),AIRVOL 1,VALTAU 40 FORMAT(' ',2X,12,2X,F5.2,2(IX,F6.3),88X,F4.2,2X,F6.1,2X,F5.3) GO TO 330 310 IF(Q(NPB,1).LE.O.O) WRITE(6,45) I,I,TAUPRV(I) 45 FORMAT(' 1,2X,I2,8X,'VALVE',12,' CLOSED',87X,F4.2) IF(Q(NPB,1).GT.O.O) WRITE(6,50) I,V(I),ALPHA(I),TAUPRV(I) 50 FORMATC ' , 2X, 12, 7X, 2 (IX, F6. 3) , 88X, F4 . 2) 52 GO TO 330 320 IF(I.EQ.l) WRITE(6,40) I,T,V(I),ALPHA(I),TAUPRV(I),AIRVOL,VALTAU IF(I.NE.l) WRITE(6,50) I,V(I),ALPHA(I),TAUPRV(I) 330 IF(I.EQ.NSPUMP) NPE=NP IF (I.NE.NSPUMP) NPE = LOCAT(I + l ) - l DO 340 J=NPB,NPE NN=N(J)+1 WRITE(6,60) J, (H(J,K) ,K = 1,NN) 60 FORMAT(* ' , 27X,12,IX,'H=',10F8.2/33X,10F8.2) WRITE(6,70) (Q(J,K),K=1,NN) 70 FORMAT(' ' , 30X,'Q=' , 10F8.2/33X,10F8.2) 340 CONTINUE 350 CONTINUE 400 T=T+DT ICOUNT=ICOUNT+l IF(T.GT.TLAST) GO TO 990 C C ***** DETERMINATION OF CONSTANTS AT PUMP BOUNDARY CONDITION C DO 500 I=1,NSPUMP NPP=NPPUMP(I) NPB=LOCAT(I) IF(I.EQ.NSPUMP) NPE=NP IF(I.NE.NSPUMP) NPE=LOCAT(I+l)-l CN = Q (NPB,2)-CA(NPB)*H(NPB,2)-CF(NPB)*Q(NPB,2)*ABS(Q(NPB,2)) C4=CN/QR (I) C5=CA(NPB)*HR(I)/QR(I) C *** SHORT SUCTION LINE C SUCTION LINE AT U/S PUMP ASSUMED SHORT AND NEGLECTED C DATUM ASSUMED TO BE PUMP CENTER LINE IF(I.NE.l) GO TO 510 Cl=l./C5 C2=C4/C5 GO TO 520 C *** LONG SUCTION LINE C REQ'D TO ANALYSE BOUNDARY CONDITION AT IN LINE PUMP 510 MM=NPB-1 NN=N(MM) CP=Q (MM,NN)+CA(MM)*H(MM,NN)-CF(MM)*Q(MM,NN)*ABS(Q(MM,NN)) C6=CP/QR(I) C7=CA(MM)*HR(I)/QR(I) C1=(C5+C7)/(C5*C7) C2= (C4*C7+(C5*C6))/(C5*C7) 520 C3=- (30.*G*SHFTRQ(I)*DT)/(PI*NR(I)*WR2(I)) C C ***** PUMPS INCLUDING OTHER BOUNDARY CONDITIONS C C *** PRV (PRESSURE REGULATING VALVE) AT PUMP DISCHARGE C WITH OR WITHOUT CHECK VALVE IF(PRVd) .EQ.O) GO TO 550 CA2=CA(NPB) C IS THE PRV OPEN? YES:GO TO (1), NO:GO TO (2) IF(OPEN(I).EQ.0) GO TO 542 C (1) HAS THE PRV SUBSEQUENTLY CLOSED? IF(TAUPRV(I).EQ.0.0) GO TO 540 C NO, IT HAS NOT. 53 c c TIME=TM(I) CALL PARAB(TIME,I,3,DTIME,TAU) TAUPRV(I)=TAU TM(I)=TM(I)+DT DOES THE CHECK VALVE EXIST? IF(PRCHKd) .EQ.O) GO TO 570 IS THE CHECK VALVE CLOSED? EQ.O) GO TO 546 CLOSE THE PRV. 540 542 544 546 550 552 C C C C 554 556 560 562 C C C C IF(CLOSED(I) GO TO 552 OPEN(I)=0 GO TO 544 (2) IS IT TINE TO OPEN PRV? IF(T.LT.TOPEN(I)) GO TO 544 TAUPRV(I)=0.00001 TM(I)=TM(I)+DT OPEN(I)=l IS THERE A CHECK VALVE AT THE PUMP DISCHARGE? IF(PRCHK(I).EQ.O) GO TO 570 CLOSED CHECK VALVE - FLOW THROUGH PRV? IF(CLOSED(I).NE.O) GO TO 552 QP(NPB,1)=0.0 GO TO 580 CV=(QO*TAUPRV(I))**2/(HDIS(I)*CA(NPB)) QP(NPB,1)=0.5*(CV-SQRT(CV**2-4.*CN*CA(NPB))) GO TO 580 *** CHECK VALVE (ONLY) AT PUMP DISCHARGE IF(CHECK(I).EQ.O) GO TO 560 IS THE CHECK VALVE CLOSED? IF(Q(NPB,1).EQ.0.0) GO TO 554 CALL PUMP(Cl,C2,C3,NPP,I,T) QP (NPB,1)=NPP*VPOUT(I)*QR(I) CHECK VALVE CLOSES IF DISCHARGE LESS THAN ZERO IF(QP(NPB,1).LT.0.0) QP(NPB,1)=0.0 IF( (PRCHK (I) .EQ.l) .AND. (QP(NPB,1) .EQ.0.0)) CLOSED(I)=0 GO TO 580 IF HEAD AT PUMP SUCTION GREATER THAN PUMP DISCHARGE - CHECK VALVE REOPENS IF(I.EQ.l) GO TO 556 IF(H(MM,NN+1).GT.H(NPB,1)) GO TO 570 QP (NPB,1)=0.0 GO TO 580 *** AIR CHAMBER (INCLUDING CHECK VALVE) AT MAIN PUMP IF( (AIR.EQ.O) .OR. (I.NE.l)) GO TO 570 ITER=0 QP(1,1)=Q(1,1) VPAIR=AIRVOL+0.5*DT*(QP(1,1)+Q(1,1)) IF(VPAIR.LT.0.00001) VPAIR=0.00001 ORIFICE LOSS COEFFICIENT - 3 TYPES 1) NO ORIFICE - CORF=0 2) PLATE ORIFICE - CORF(FLOW IN)=CORF(FLOW OUT) 3) DIFFERENTIAL ORIFICE - CORF(Q IN)=2.5*CORF(Q OUT) IF(NDIFF.EQ.O.OR.QP(1,1).GT.O.O) GO TO 564 CORFIN=2.5*CORF 54 HPORF=CORFIN*QP(1,1)*ABS(QP(1,1)) GO TO 566 564 HPORF=CORF*QP(1,1)*ABS(QP(1,1)) 566 HPORF=HPORF+CLINE*QP(1,1)*ABS(QP(1,1))/(2.*G*ALINE**2) Fl=((QP (l,l)-CN)/CA(l)+HBAR-ZSURF-HPORF)*VPAIR**EN-C DF1DQP=EN*DT*C*0.5/VPAIR+VPAIR**EN/CA(1) DELTAQ=-F1/DF1DQP IF(ABS(DELTAQ).LE.0.001) GO TO 568 QP (1,1)=QP(1,1)+DELTAQ ITER=ITER+1 IF(ITER.GT.20) GO TO 980 GO TO 562 568 AIRVOL=AIRVOL + 0.5*DT*(QP(1,1)+Q(1,1) ) IF(AIRVOL.LT.O.O) AIRVOL=0.0 GO TO 580 C C *** PUMP BOUNDARY CONDITION 570 CALL PUMP(C1,C2,C3,NPP,I,T) C DISCHARGE SIDE OF PUMP QP(NPB,1)=NPP*VPOUT(I)*QR(I) 580 HP(NPB,1)=(QP(NPB,1)-CN)/CA(NPB) IF(I.EQ.l) GO TO 590 C SUCTION SIDE OF PUMP QP(MM,NN+1)=QP(NPB,1) HP(MM,NN + 1) = (CP-QP(MM,NN+1))/CA (MM) C C *** VACUUM BREAKER AT SUCTION FLANGE OF IN LINE PUMP IF(VAC(I).EQ.O) GO TO 590 IF(HP(MM,NN+1).GE.ELBO(I)) GO TO 590 HP(MM,NN+l)=ELBO(I) CP=Q(MM,NN)+CA(MM)*H(MM,NN)-CF(MM)*Q(MM,NN)*ABS(Q(MM,NN)) QP (MM,NN + 1)=CP-CA(MM)*HP(MM,NN+1) C C *** INTERIOR POINTS 590 CALL INTER(NPB,NPE) ALPHA(I)=ALPOUT(I) V (I)=VPOUT(I) 500 CONTINUE C c ***** FLOW CONTROL VALVE AT CONSTANT HEAD RESERVOIR C IF(NVAL.EQ.O) GO TO 700 IF (T.LE.TCLOSE) GO TO 650 NN=N(NP)+1 IF(VALTAU.LE.TAURED) GO TO 610 IF(INCL.EQ.O) GO TO 600 C NO INSTANTANEOUS CLOSURE - SHORT PIPE LINE SLOPE=(TAURED-1.)/TBFVAL VALTAU=1.+SLOPE*(T-TCLOSE) IF(VALTAU.GE.TAURED) GO TO 630 600 TVAL=T-(TCLOSE+TBFVAL) C FLOW THROUGH BY-PASS WITH CONTROL VALVE CLOSED SLOWLY 610 IF(VALTAU.EQ.O.O) GO TO 620 IF(VALTAU.EQ.TAUMIN) GO TO 630 CALL PARAB(TVAL,I,4,DTVAL,VALTAU) IF(VALTAU.EQ.O.O) GO TO 620 IF(VALTAU.LE.TAUMIN) VALTAU=TAUMIN GO TO 630 620 CP=Q(NP,NN-1)+CA(NP)*H(NP,NN-1)-CF(NP)*Q(NP,NN-1)*ABS(Q(NP,NN-1)) QP(NP,NN)=0.0 HP (NP,NN)=(CP-QP(NP,NN))/CA(NP) GO TO 800 CALCULATE HEAD AND DISCHARGE AT CONTROL SECTION 63 0 CP=Q(NP,NN-1)+CA(NP)*H(NP,NN-1)-CF(NP)*Q(NP,NN-1)*ABS(Q(NP,NN-1)) K2=VALOSS*CA(NP)/(QO*VALTAU)**2 K3=CA(NP)*HRES-CP IF((l.-4.*K2*K3).LT.0) GO TO 650 QP(NP,NN)=(-1.+SQRT(1.-4.*K2*K3))/(2.*K2) 64 0 HP(NP,NN)=HRES+VALOSS*QP(NP,NN)*ABS(QP(NP,NN))/(Q0*VALTAU)**2 TVAL=TVAL+DT IF(VALTAU.GT.TAURED) TVAL=0.0 GO TO 8C0 650 NN=N (NP)+1 HP(NP,NN)=HRES+VALOSS GO TO 750 ***** CONSTANT HEAD RESERVIOR AT DOWNSTREAM END 700 NN=N(NP)+1 HP (NP,NN)=HRES 750 CP=Q(NP,NN-1)+CA(NP)*H(NP,NN-1)-CF(NP)*Q(NP,NN-1)*ABS(Q(NP,NN-1) QP(NP,NN)=CP-CA(NP)*HP(N P,NN) ***** STORING MAX. AND MIN. VALUES FOR NEXT TIME STEP 800 DO 900 1=1,NP NN=N(I)+l DO 920 J=1,NN Q(I,J)=QP(I,J) H(I,J)=HP(I,J) IF(H (I,J) .LT.HMAX(I)) GO TO 910 HMAX (I)=H(I,J) TMAX(I)=T 910 IF (H (I,J) .GT.HMIN(I)) GO TO 920 HMIN (I)=H(I,J) TMIN(I)=T 920 CONTINUE 900 CONTINUE IF(ICOUNT.EQ.IPRINT) GO TO 300 GO TO 400 980 WRITE(6,80) 80 FORMAT(/5X,'*** ITERATIONS IN AIR CHAMBER FAILED ***'/) STOP 990 WRITE(6,90) 90 FORMAT(//10X,'PIPE NO. ' ,5X,'MAX. PRESS. AT TIME',5X,'MIN. PRESS. A IT TIME'/) WRITE(6,95) (I,HMAX(I) ,TMAX(I) ,HMIN (I) ,TMIN(I) ,1 = 1,NP) 95 FORMAT(11X,I3,10X,F7.1,6X,F5.2,6X,F7.1,6X,F5.2) STOP END ********************** SUBROUTINE PARAB ********************** SUBROUTINE PARAB(X,I,N,DX,Z) COMMON /AREA1/ FH(5,90) ,FB(5,90)>PR (5,90) ,VL(90) QUADRATIC CURVE FITTING UTILIZING THREE POINTS J=X/DX R=(X-J*DX)/DX IF(J.EQ.O) R=R-1 J=J + 1 IF(J.LT.2) J=2 K=J + 1 L=J-1 GO TO (10,20,30,40) ,N 10 Z=FH(I,J)+.5*R*(FH(I,K)-FH(I,L)+R*(FH(I,K)+FH(I,L)-2.*FH(I,J))) RETURN 20 Z=FB(I,J)+.5*R*(FB(I,K)-FB(I,L)+R*(FB(I,K)+FB(I,L)-2.*FB(I,J))) RETURN 30 Z = PR(I,J) + .5*R*(PR(I,K)-PR(I,L)+R*(PR(I,K)+PR (I,L)-2.*PR (I,J))) RETURN 40 Z=VL(J) + .5*R*(VL(K)-VL(L)+R*(VL(K)+VL(L)-2.*VL (J))) RETURN END ********************** SUBROUTINE INTER ********************** SUBROUTINE INTER (NPB,NPE) COMMON /AREA2/ N(10) ,Q (10,20) ,H (10,20) ,CA (10) ,CF(10) COMMON /OUT2/ QP (10,20) ,HP (10,20) INTERIOR POINTS DO 20 I=NPB,NPE NN=N (I) DO 10 J=2,NN CN=Q(I,J+1)-CA(I)*H(I,J+1)-CF(I)*Q(I,J+1)*ABS(Q(I,J+1)) CP=Q(I,J-1)+CA(I)*H(I,J-1)-CF(I)*Q(I,J-1)*ABS(Q(I,J-l)) QP(I,J)=0.5*(CP+CN) HP(I,J)=(CP-QP(I,J))/CA(I) 10 CONTINUE 20 CONTINUE SERIES JUNCTION IF((NPE-NPB).EQ.O) GO TO 40 NPEMIN=NPE-1 DO 30 I=NPB,NPEMIN N1=N(I) NN=N(I)+l CN=Q(I+1,2)-CA(I+1)*H(I+1,2)-CF(I+1)*Q(I+1,2)*ABS(Q(I+1,2)) CP=Q(I,N1)+CA(I)*H(I,N1)-CF(I)*Q(I,N1)*ABS(Q(I,N1)) HP (I,NN)=(CP-CN)/(CA(I)+CA(I + 1)) HP(I+1,1)=HP(I,NN) QP(I,NN)=CP-CA(I)*HP(I,NN) QP(I+1,1)=CN+CA(I+1)*HP(1+1,1) 30 CONTINUE 57 40 RETURN END C Q ********************** SUBROUTINE PUMP *********************** C SUBROUTINE PUMP(Cl,C2,C3,NPP,II,T) REAL Kl INTEGER OPEN(5) COMMON /AREA3/ ALPHA(5),V(5),DALPHA(5),DV(5),CVHL(5),DTH COMMON /OUT3/ ALPOUT(5),VPOUT(5) COMMON /PRV3/ OPEN,HR (5) ,HO(5),QR(5) ,QO,CN,CA,TAUPRV(5) ,HSUC(5) C C COMPUTATION OF PUMP DISCHARGE C PI=3.141593 VE=V(II)+DV(II) ALPHAE=ALPHA(II)+DALPHA(II) JJ=0 10 KK=0 LL=0 15 IF(VE.NE.O.O) GO TO 20 IF(ALPHAE.GE.O.O) TH=0.0 IF(ALPHAE.LT.0.0) TH=180. GO TO 25 20 TH=57.296*(PI+ATAN2(VE,ALPHAE)) 25 CALL PARAB(TH,II,1,DTH,K1) C PRESSURE REGULATING VALVE AT PUMP DISCHARGE? IF(OPEN(II).EQ.O) GO TO 35 HP = K1*HR (II)*(VE**2+ALPHAE**2)+HSUC(II)-CVHL(II) QP=CN+CA*HP QPV=TAUPRV(II)*QO*SQRT(HP/HO(II)) IF(ABS(QP-(VE*NPP*QR(II)-QPV)).LE.0.001) GO TO 30 VE=(QP-QPV)/NPP*QR(II) LL = LL+1 IF(LL.GE.20) GO TO 90 GO TO 15 30 VP=VE GO TO 40 35 ARGUE=(NPP*C1)**2-4.*Kl*(Kl*ALPHAE**2+C2-CVHL(II)) VP=(C1*NPP-SQRT(ARGUE))/(2.*Kl) 36 IF(ABS(VP-VE).LE.0.001) GO TO 40 VE=0.5*(VP+VE) KK=KK+1 IF(KK.GE.20) GO TO 80 GO TO 15 C C COMPUTATION OF PUMP SPEED C 40 VM=0.5*(V(II)+VP) ALPHAM=ALPHA(II)+0.5*DALPHA(II) IF(VM.NE.O.O) GO TO 50 IF(ALPHAM.GE.O.O) THM=0.0 IF(ALPHAM.LT.O.O) THM=180. GO TO 60 50 THM=57.296*(PI+ATAN2(VM,ALPHAM)) 60 CALL PARAB(THM,11,2,DTH,BETAM) 58 DALPHA(II)=C3*BETAM*(ALPHAM**2+VM**2) ALPHAP=ALPHA(II)+DALPHA(II) DV(II)=VP-V(II) IF(ABS(ALPHAP-ALPHAE).LE.0.001) GO TO 70 ALPHAE=0.5*(ALPHAP+ALPHAE) VE=VP JJ=JJ+1 IF(JJ.GE.20) GO TO 80 GO TO 10 70 CONTINUE ALPOUT(II)=ALPHAP VPOUT(II)=VP RETURN 80 WRITE(6,1) T,KK,JJ ,ALPHAE,VP 1 FORMAT (/5X,'*** ITERATIONS IN PUMP SUBROUTINE FAILED ***'/lOX,'T=' 1,F6.2,5X,'KK=',I3,5X,'JJ = *,13/1 OX,'ALPHA=1,F6.3,1 OX,'VP=',F6.3) STOP 90 WRITE(6,2) 2 FORMAT(/5X,'*** ITERATIONS IN PRV SECTION OF PUMP SUBROUTINE FAILE ID ***'/) STOP END 59 APPENDIX B EXAMPLES OF PROGRAM USAGE B - l Example of 3-stage Pumping System B -2 Example of S i n g l e Stage Pumping System B -3 Comparison of S i n g l e and Three Stage Pumping Systems B-4 Check of Program Accuracy 60 APPENDIX B - l EXAMPLE OF A THREE STAGE PUMPING SYSTEM PROBLEM: Given the f o l l o w i n g data, determine the t r a n s i e n t c o n d i t i o n s at the three pumping s t a t i o n s f o l l o w i n g power f a i l u r e . These c o n d i t i o n s are to be c a l c u l a t e d f o r a s p e c i f i e d p e r i o d of time. Pump DATA: Number of pump s t a t i o n s = 3 Number of pipes = 3 Steady s t a t e d i s c h a r g e = 35.0 c f s T r a n s i e n t s c a l c u l a t e d f o r 5 seconds 61 MAIN PUMP S T A T I O N : p u m p s , a i r c h a m b e r , c h e c k v a l v e The c h e c k v a l v e c l o s e s i n s t a n t l y upon power f a i l u r e , e l i m i n a t i n g t h e pump. A l l f l o w i s t h e n i n t o o r o u t o f t h e a i r c h a m b e r a t t h i s p u m p i n g s t a t i o n f o l l o w i n g power f a i l u r e . Pump d a t a : R a t e d d i s c h a r g e = 3 5 . 0 c f s R a t e d h e a d = 3 9 8 . 0 f t R a t e d pump s p e e d = 1 7 6 0 . 0 rpm Pump e f f i c i e n c y = 0 . 8 5 Pump u n i t i n e r t i a = 6 6 0 . 0 l b - f t 2 Pump e l e v a t i o n = 0 . 0 f t Head l o s s c o e f f i c i e n t f o r c h e c k v a l v e = 2 . 0 A i r c h a m b e r d a t a : B a r o m e t r i c p r e s s u r e = 3 3 . 9 f t I n i t i a l a i r v o l u m e i n a i r c h a m b e r = 6 0 . 0 c u . f t . P o l y t r o p i c g a s c o n s t a n t = 1 . 2 E l e v a t i o n o f c h a m b e r w a t e r s u r f a c e a b o v e d a t u m = 0 . 0 Head l o s s c o e f f i c i e n t f o r o r i f i c e = 0 . 0 A r e a o f p i p e f r o m c h a m b e r t o m a i n p i p e = 2 . 0 f t Head l o s s c o e f f i c i e n t f o r t h i s p i p e = 0 . 0 P i p e f r o m m a i n pump s t a t i o n t o 1 s t b o o s t e r s t a t i o n : L e n g t h = 9 6 0 . 0 f t D i a m e t e r = 2 . 0 f t Wave v e l o c i t y = 3 1 3 0 . 0 f t / s e c F r i c t i o n f a c t o r = 0 . 0 1 4 F I R S T BOOSTER S T A T I O N : p u m p s , v a c u u m b r e a k e r , c h e c k v a l v e Pump d a t a : R a t e d d i s c h a r g e = 3 5 . 0 c f s R a t e d h e a d = 3 5 0 . 0 f t R a t e d pump s p e e d = 1 7 6 0 . 0 rpm Pump e f f i c i e n c y = 0 . 8 5 Pump u n i t i n e r t i a = 6 6 0 . 0 l b - f t 2 Pump e l e v a t i o n = 3 6 0 . 0 f t Head l o s s c o e f f i c i e n t f o r c h e c k v a l v e = 2 . 0 P i p e f r o m 1 s t b o o s t e r s t a t i o n t o 2nd b o o s t e r s t a t i o n : L e n g t h = 9 9 0 . 0 f t D i a m e t e r = 2 . 0 f t Wave s p e e d = 3 1 3 0 . 0 f t / s e c F r i c t i o n f a c t o r = 0 . 0 1 4 SECOND BOOSTER S T A T I O N : p u m p s , v a c u u m b r e a k e r , c h e c k v a l v e Pump d a t a : R a t e d d i s c h a r g e = 3 5 . 0 c f s R a t e d h e a d = 3 5 0 . 0 f t R a t e d pump s p e e d = 1 7 6 0 . 0 rpm Pump e f f i c i e n c y = 0 . 8 5 Pump u n i t i n e r t i a = 6 6 0 . 0 l b - f t 2 Pump e l e v a t i o n = 6 8 0 . 0 f t Head l o s s c o e f f i c i e n t f o r c h e c k v a l v e = 2 . 0 P i p e f r o m 2nd b o o s t e r t o c o n t r o l v a l v e : L e n g t h = 1 0 5 0 . 0 f t D i a m e t e r = 2 . 0 f t Wave v e l o c i t y = 3 1 3 0 . 0 f t / s e c F r i t i o n f a c t o r = 0 . 0 1 4 63 CONTROL V A L V E D A T A : b u t t e r f l y v a l v e c l o s u r e i s n o t i n s t a n t a n e o u s T i m e a t w h i c h b u t t e r f l y v a l v e b e g i n s t o c l o s e = 0 . 2 5 s e c . T i m e r e q u i r e d t o c l o s e b u t t e r f l y v a l v e = 0 . 1 5 s e c . S t e a d y s t a t e h e a d l o s s t h r o u g h t h e b u t t e r f l y v a l v e = 7 . 3 5 f t R e l a t i v e a r e a o f b y p a s s l i n e o n c e b u t t e r f l y c l o s e d = 0 . 1 2 1 5 M i n i m u m r e l a t i v e c o n t r o l v a l v e o p e n i n g = 0 . 0 2 5 Number o f d a t a p o i n t s on c l o s u r e c u r v e = 16 T i m e i n t e r v a l b e t w e e n t h e s e d a t a p o i n t s = 0 . 0 7 0 s e c . C o n t r o l v a l v e c l o s u r e c u r v e : 0 . 1 2 1 5 0 . 1 1 6 1 0 . 1 0 8 9 0 . 0 9 9 1 0 . 0 8 9 2 0 . 0 7 9 0 0 . 0 7 0 5 0 . 0 6 2 0 0 . 0 5 4 2 0 . 0 4 8 1 0 . 0 4 1 9 0 . 0 3 6 7 0 . 0 3 2 8 0 . 0 2 8 6 0 . 0 2 4 9 0 . 0 2 1 6 E l e v a t i o n o f r e s e r v o i r = 1 0 4 0 . 5 f t N O T E : The pump c h a r a c t e r i s t i c s a r e t h e same f o r a l l t h r e e p u m p i n g s t a t i o n s , w i t h t h e s p e c i f i c s p e e d f o r t h e pumps e q u a l t o 1800 ( g p m u n i t s ) . 64 SOLUTION: The data cards f o r the program are as f o l l o w s : 1 ENGLISH 2 3,3,9,89,0,1,1,0,1,16,1 3 35.0,4.0909,5.0 4 960.0,2.0,3130.0,0.014,990.0,2.0,3130.0,0.014,1050.0, 2.0,3130.0,0.014 5-12, Head c h a r a c t e r i s t i c f o r main pump (89 values) 13-20, Torque c h a r a c t e r i s t i c f o r main pump (89 values) 21 0.0,1760.0,35.0,398.0,1760.0,0.85,660.0,2.0 22 1,1,0,0,0,0 23-30, Head c h a r a c t e r i s t i c f o r 1st booster pump 31-38, Torque c h a r a c t e r i s t i c f o r 1st booster pump 39 360.0,1760.0,35.0,350.0,1760.0,0.85,660.0,2.0 40 2,1,0,0,1,1 41-48, Head c h a r a c t e r i s t i c f o r 2nd booster pump 49-56, Torque c h a r a c t e r i s t i c f o r 2nd booster pump 57 -680.0,1760.0,35.0,350.0,1760.0,0.8 5,660.0,2.0 58 3,1,0,0,1,1 59 33.9,60.0,0.0,1.2,0.0,0.0,2.0 60 0.25,7.35,0.1215,0.15,0.02 5,0.070 61-62 C o n t r o l v a l v e c l o s u r e curve (14 values) NOTE: The numbers i n the f i r s t columns are the numbers of the data c a r d s . Spacing of the e n t r i e s i s not the same as the format f o r a c t u a l data e n t r y i n t o the program. This i n f o r m a t i o n i s a v a i l a b l e i n the l i s t i n g of the program (Appendix A). The above data was entered i n t o the program and the t r a n s i e n t c o n d i t i o n s determined f o r a p e r i o d of 5 seconds. The h y d r a u l i c g r a d e l i n e f l u c t u a t i o n s at the pumping s t a t i o n s during t h i s p e r i o d are p l o t t e d i n F i g u r e s B - l to B-3. TRANSIENT PRESSURES IN 1st PIPE THREE STAGE SYSTEM DEVIATIONS OF RESULTS OBTAINED BY RUUS FROM THOSE OBTAINED IN THIS THESIS _, , r-2 3 4 TIME (SECONDS) TRANSIENT PRESSURES IN 2nd PIPE  THREE STAGE SYSTEM • HEAD AT SUCTION FLANGE OF BOOSTER * 2 _ DEVIATIONS OF RESULTS OBTAINED BY RUUS FROM THOSE OBTAINED IN THIS THESIS TIME(SECONDS) 14T TRANSIENT PRESSURES JN 3rd PIPE THREE STAGE SYSTEM i — , , , r 0 1 2 3 4 TIME(SECONDS) 68 APPENDIX B-2 EXAMPLE OF A SINGLE STAGE PUMPING SYSTEM PROBLEM: This problem i s the same as that presented i n Appendix B - l , except that there are no booster pumping s t a t i o n s . Three pumps i n s e r i e s w i l l be placed i n one pumping s t a t i o n to supply the same head as that s u p p l i e d by the three stage system. Thus, the system w i l l c o n s i s t of one pumping s t a t i o n d i s c h a r g i n g i n t o a constant head r e s e r v o i r . The c o n t r o l equipment w i l l c o n s i s t of an a i r chamber and a check v a l v e at the pumping s t a t i o n . To f a c i l i t a t e the comparison of these two systems, s e r i e s j u n c t i o n s of two pipes w i l l be p o s i t i o n e d i n place of the two booster pumping s t a t i o n s i n the previous example. R e s e r v o i r Check Valve A i r Chamber Main Pump 960 * 9 9 0 ' 1050 69 DATA: Number of pump s t a t i o n s = 1 Number of pipes = 3 Steady s t a t e d i s c h a r g e =35.0 c f s T r a n s i e n t s c a l c u l a t e d f o r 5 seconds. MAIN PUMP STATION: pumps, a i r chamber, check v a l v e The check v a l v e c l o s e s i n s t a n t l y upon power f a i l u r e , e l i m i n a t i n g the pumps. A l l flow i s then i n t o or out of the a i r chamber. Pump data: Rated d i s c h a r g e = 35.0 c f s Rated head = 1084.8 f t Rated pump speed = 1760.0 rpm Pump e f f i c i e n c y = 0.85 Pump u n i t i n e r t i a = 1980.0 l b - f t 2 Pump e l e v a t i o n = 0.0 Head l o s s c o e f f i c i e n t f o r check v a l v e = 2.0 A i r chamber data: I d e n t i c a l a i r chamber as i n the three stage system. P i p e l i n e from pumping s t a t i o n to r e s e r v o i r : Diameter = 2.0 f t Wave speed = 3130.0 f t / s e c F r i c t i o n f a c t o r = 0.014 Pipe le n g t h from pumps to 1st j u n c t i o n = 960.0 f t Pipe length from 1st j u n c t i o n to 2nd j u n c t i o n = 990.0 f t Pipe le n g t h from 2nd j u n c t i o n to r e s e r v o i r = 1050.0 f t Res e r v o i r e l e v a t i o n = 1040.5 f t . 70 SOLUTION: The data cards f o r the program are as f o l l o w s : 1 2 3 4 ENGLISH 3,1,9,89,0,1,0,0,0,0,0 35.0,4.0909,5.0 960.0,2.0,3130.0,0.014,990.0,2.0,3130.0,0.014,1050.0, 2.0,3130.0,0.014 Head c h a r a c t e r i s t i c f o r pumps (89 values) Torque c h a r a c t e r i s t i c f o r pumps (89 values) 0.0,1760.0,35.0,108 4.8,1760.0,0.85,1980.0,2.0 1,1,0,0,1,0 33.9,60.0,0.0,1.2,0.0,0.0,2.0 5-12, 13-20, 21 22 23 NOTE: The numbers i n the f i r s t columns are the numbers of the data c a r d s . Spacing of the e n t r i e s i s not the same as the format f o r a c t u a l data e n t r y i n t o the program. This i n f o r m a t i o n i s a v a i l a b l e i n the l i s t i n g of the program (Appendix A ) . The above data was entered i n t o the program and the t r a n s i e n t c o n d i t i o n s determined f o r a p e r i o d of 5 seconds. The h y d r a u l i c g r a d e l i n e f l u c t u a t i o n s at the pumping s t a t i o n and at the two s e r i e s j u n c t i o n s of pipes are p l o t t e d i n Fig u r e B-4. TRANSIENT PRESSURES IN A SINGLE STAGE SYSTEM HEAD AT RESERVOIR TIME(SECONDS) 72 APPENDIX B-3 COMPARISON OF SINGLE AND THREE STAGE SYSTEMS The comparison of a s i n g l e stage and a three stage system i s performed by using the system examples presented i n Appendix B - l and Appendix B-2. The maximum and minimum h y d r a u l i c grade l i n e s are determined from the computer outputs and these are p l o t t e d i n F igure B-5. I t i s obvious when examining t h i s f i g u r e t h a t the m u l t i s t a g e system g i v e s s i g n i f i c a n t l y lower t r a n s i e n t pressure f l u c t u a t i o n s . I t should be noted, that the maximum f o r the s i n g l e stage system does not occur w i t h i n the 5 second time i n t e r v a l shown i n Figure B-4, but occurs s h o r t l y a f t e r (at 6.82 seconds). A l s o , s i n c e the maximum and minimum pressures do not occur s i m u l t a n e o u s l y w i t h i n each pipe of the systems being examined, the h y d r a u l i c grade l i n e s are p l o t t e d f o r the time when the maximum and minimum pressures occur i n pipe 1 of each system. 0 10 20 30 DISTANCE (FEETxIOO) FIGURE B-5 74 APPENDIX B-4 CHECK OF PROGRAM ACCURACY The check of accuracy i s performed by examining the t r a n s i e n t s f o r the three stage system presented i n Appendix B - l . The system c h a r a c t e r i s t i c s f o r t h i s three stage system are entered i n t o the program developed i n t h i s study and that developed by E. Ruus. The generated r e s u l t s are presented i n F i g u r e s B - l to B - 3, with the d e v i a t i o n s of the r e s u l t s obtained by Ruus from those obtained by t h i s study shown as dashed l i n e s . These d e v i a t i o n s are s l i g h t and can be a t t r i b u t e d t o : 1 . S l i g h t l y d i f f e r e n t c o n t r o l v a l v e c l o s u r e curves u t i l i z e d by the two programs. The Ruus program approximates the a c t u a l curve by an equation, while the program developed i n t h i s t h e s i s approximates the a c t u a l curve by i n t e r p o l a t i n g between known p o i n t s on the c l o s u r e curve. 2 . S l i g h t l y d i f f e r e n t wave speeds w i t h i n each pipe (the d i f f e r e n c e s are at most 1 . 2 per c e n t ) . T h i s i s due to s l i g h t l y d i f f e r e n t time i n t e r v a l s c a l c u l a t e d w i t h i n the two programs. N e i t h e r of these changes produces s i g n i f i c a n t d i f f e r e n c e s i n the maximum and minimum pressures w i t h i n the system. 

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