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Boundary conditions for analysis of waterhammer in pipe systems Chaudhry, Mohammad Hanif 1968

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BOUNDARY CONDITIONS'FOR ANALYSIS OF WATERHAMMER IN PIPE SYSTEMS by CHAUDHRY MOHAMMAD HANIF B.Sc. (Hons.) ( C i v i l Engineering), West Pakistan U n i v e r s i t y of Engineering and Technology, Lahore, Pakistan, 1965 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF • THE REQUIREMENT FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n the Department of C i v i l Engineering. We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA A p r i l , 1968 i i i in p resent ing t h i s t h e s i s in p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that the L i b r a r y s h a l l make i t f r e e l y avai ]ab1e f o r re ference and s tudy . I f u r t h e r agree that permiss ion f o r e x t e n s i v e copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s represen -t a t i v e s . It i s understood that copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l ga in s h a l l not be a l lowed wi thout my w r i t t e n p e r m i s s i o n . Department of C i v i l Engineering The U n i v e r s i t y of B r i t i s h Columbia Vancouver 8 , Canada Date A p r i l 22, 1968 i i ABSTRACT The t r a n s i e n t f low i n p ipe networks i s represented by a p a i r of q u a s i - l i n e a r 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 q u a t i o n s . The method of c h a r a c t e r i s t i c s i s used to t r a n s f o r m these equat ions to a set of o r d i n a r y d i f f e r e n t i a l e q u a t i o n s , which are then solved, by a f i r s t o rder f i n i t e d i f f e r e n c e technique u s i n g s u i t a b l e boundary c o n d i t i o n s . The main purposes of these i n v e s t i g a t i o n s a r e : 1 ) To d e r i v e s u i t a b l e boundary c o n d i t i o n s or boundary c o n d i t i o n equat ions f o r v a l v e s , s p r i n k l e r s , surge tanks and a i r chambers, and 2) To i n v e s t i g a t e the e f f e c t of these boundary c o n d i t i o n s on the t r a n s i e n t f low i n p ipe systems. S e v e r a l n u m e r i c a l examples are so l ved on the d i g i t a l computer u s i n g the method of c h a r a c t e r i s t i c s . The r e s u l t s are compared w i t h those obta ined by the g r a p h i c a l method. A l though i n t h i s t h e s i s the developed boundary c o n d i t i o n s are used to study the t r a n s i e n t response of the i r r i g a t i o n p i p e systems, the boundary c o n d i t i o n s , w i thout any m o d i f i c a t i o n , can be used to determine the t r a n s i e n t c o n d i t i o n s i n water supply p ipe networks or i n p ipes c a r r y i n g o ther l i q u i d s . i v TABLE OF CONTENTS CHAPTER PAGE ABSTRACT i i NOTATION • x INTRODUCTION 1 I METHOD OF CHARACTERISTICS 1 . 1 B a s i c Equat ions f o r unsteady f l ow through the 4 p i p e s . 1 .2 B a s i c C o n s i d e r a t i o n s . 5 1 .3 C h a r a c t e r i s t i c E q u a t i o n s . 7 1.4 Genera l C h a r a c t e r i s t i c Method. 12 1 . 5 Convergence and s t a b i l i t y of the f i n i t e -d i f f e r e n c e t e c h n i q u e . 17 1.6 Time increment f o r complex p i p i n g systems. " 18 I I BOUNDARY CONDITIONS 2 . 1 R e s e r v o i r of Constant Water l e v e l . 21 2 .2 Dead end a t the downstream end. 23 2 . 3 S e r i e s C o n n e c t i o n . 24 2 .4 Branch C o n n e c t i o n . 26 2 . 5 S p r i n k l e r s . 29 2 .6 V a l v e s . 34 2.7 Surge tank . 47 2 .8 A i r chamber. 55 2 .9 C e n t r i f u g a l Pump. 62 I I I DESIGNATION OF THE PIPES 3 . 1 D e s c r i p t i o n of the system. 75 3 . 2 D e s i g n a t i o n of the p i p e s . 75 IV DESCRIPTION OF THE PROGRAMME 4 . 1 The MAIN Programme. 79 4 .2 Subrout ine STEADY. 81 4 . 3 Subrout ine INTER. 83 4 .4 Subrout ines BR1 and BR2. 83 4 . 5 Subrout ines CHAM1 and CHAM2. 84 4 . 6 Subrout ines SURGE1 and SURGE2. 85 4 .7 Subrout ines RESERU, RESERD and RESERI. 85 4 . 8 Subrout ine VALVE. 86 4 . 9 Subrout ines PUMPl, PUMP2 and PUMP3. 87 4 .10 Subrout ine SPRINK. 88 4 . 1 1 Subrout ine MVAL. 88 4 .12 Subrout ine PARAB. 89 4 .13 F u n c t i o n s VEL and HEAD. 90 V APPLICATIONS OF BOUNDARY CONDITIONS 5 . 1 Numer ica l Examples. " 9 1 5 .2 P ipe"System S t u d i e s . 91 5 . 3 Programme f o r an i r r i g a t i o n p ipe system. 92 CONCLUSIONS 93 BIBLIOGRAPHY 95 APPENDICES Appendix A : Numer ica l Examples. 97 Appendix B: P i p e System S t u d i e s . 118 Appendix C: Computer programme f o r a sample i r r i g a t i o n p ipe system. 130 Appendix D: Subrout ines 149 LIST OF FIGURES vx NO. 1.1 1.2 1.3 1.4 C h a r a c t e r i s t i c curves. Grid c h a r a c t e r i s t i c s . Method of s p e c i f i e d time i n t e r v a l s . C h a r a c t e r i s t i c s at the boundaries. PAGE 11 11 16 16 2.1 2.2 2.3 2.4 2.5 2 . 6 (a) Reservoir at the upstream end. (b) Reservoir at the downstream end. (c) Reservoir at an intermediate section. Dead end. Series connection. (a (b (a (b (a <b (c (d (e (f (8 (h Branch connection of m pipes. Branch connection of three pipes. Sprinkler at the ju n c t i o n of two pipes. Sprinklers on a pipe. T - t r e l a t i o n s h i p f o r a valve. T p r e l a t i o n s h i p for a valve. Valve at the downstream end of a pipe. Valve at the ju n c t i o n of two pipes. Valve at an intermediate s e c t i o n . O r i f i c e at an intermediate s e c t i o n . Valve at the upstream end of a pipe. Valve at a r e s e r v o i r . 22 22 22 25 25 28 28 33 33 36 36 40 40 43 43 46 46 (a) V e l o c i t y changes i n the d i s c h a r g e p i p e . 49 (b) Surge tank a t j u n c t i o n of two p i p e s . 49 (c) Surge tank near the pump. 54 (d) Surge tank of s m a l l c r o s s - s e c t i o n a l a r e a . 54 (e) Surge tank of v a r i a b l e c r o s s - s e c t i o n a l a r e a . 56 (a) A i r chamber a t the j u n c t i o n of two p i p e s . ^0 (b) A i r chamber near the pump. ^0 (a) Flow c h a r t f o r pump f a i l u r e ; check v a l v e c l o s e s upon f l o w r e v e r s a l . ^9 (b) Flow c h a r t f o r pump f a i l u r e ; f l ow r e v e r s e s through the pump. ^0 (c) Check v a l v e f a i l s to c l o s e upon f l o w r e v e r s a l ; r e f i e f v a l v e opens g r a d u a l l y . 72 (d) Flow c h a r t f o r pump f a i l u r e . R e l i e f v a l v e opens g r a d u a l l y , check v a l v e f a i l s to c l o s e upon f l ow r e v e r s a l . 74 (a) D e s i g n a t i o n of branch p i p e s . 77 (b) D e s i g n a t i o n of l a t e r a l s and d i s t r i b u t o r s . 77 (a) Flow char t f o r a complex p i p i n g sys tem. 80 (b) Flow c h a r t f o r an i r r i g a t i o n p i p e sys tem. 81 T r a n s i e n t s t a t e p r e s s u r e s and v e l o c i t i e s i n the p ipes i n s e r i e s . 102 (a) T r a n s i e n t s t a t e p r e s s u r e s and v e l o c i t i e s . Waterhammer waves i n the surge tank cons idered .105 (b) T r a n s i e n t s t a t e p r e s s u r e s and v e l o c i t i e s . Waterhammer waves i n the surge tank n e g l e c t e d . 106 V l l l A - 4 T r a n s i e n t s t a t e c o n d i t i o n s . A i r chamber near the pump. Check v a l v e c l o s e s upon pump f a i l u r e . 109 A - 5 (a) T r a n s i e n t s t a t e c o n d i t i o n s . Pump f a i l u r e ; check v a l v e c l o s e s when f l ow r e v e r s e s . H 3 (b) T r a n s i e n t s t a t e c o n d i t i o n s . Pump f a i l u r e ; f l ow r e v e r s e s through the pump. (c) T r a n s i e n t s t a t e c o n d i t i o n s . Pump f a i l u r e ; check v a l v e c l o s e s i n s t a n t l y and r e l i e f v a l v e opens g r a d u a l l y . .115 (d) T r a n s i e n t s t a t e c o n d i t i o n s . Pump f a i l u r e ; check v a l v e c l o s e s i n s t a n t l y and r e l i e f v a l v e opens g r a d u a l l y . - L J - D (e) G r a p h i c a l s o l u t i o n . Pump f a i l u r e ; check v a l v e c l o s e s i n s t a n t l y and r e l i e f v a l v e opens g r a d u a l l y . B - l (a) The branch p i p e s . 1 2 1 (b) E f f e c t of the s p r i n k l e r d i s c h a r g e on the w a v e - f r o n t . 1^1 B-2 (a) A sample i r r i g a t i o n p i p e system. (b) T r a n s i e n t s t a t e p r e s s u r e s at j u n c t i o n NO. 1 . 125 (c) T r a n s i e n t s t a t e p r e s s u r e s at j u n c t i o n NO. 3 . 126 (d) T r a n s i e n t s t a t e p r e s s u r e s a t j u n c t i o n NO. 5 . 127 (e) T r a n s i e n t s t a t e p r e s s u r e s a t j u n c t i o n NO. 7. 128 ( f ) T r a n s i e n t s t a t e p r e s s u r e s a t j u n c t i o n NO. 9 . 129 C - 1 T r a n s i e n t s t a t e c o n d i t i o n s a t the pump, check v a l v e and a i r chamber. 134 i x ACKNOWLEDGEMENT The author wishes to express h i s thanks to h i s s u p e r v i s o r , D r . E. Ruus, f o r h i s v a l u a b l e guidance and encouragement and to P r o f e s s o r J . F . M u i r and D r . M. C. Quick f o r t h e i r c o n s t r u c t i v e c r i t i c i s m and s u g g e s t i o n s . The study was sponsored by the Water Resources Research Centre of the Department of C i v i l E n g i n e e r i n g . The Centre i s supported by g rants from the Water I n v e s t i g a t i o n Branch of the Department of Lands , F o r e s t s and Water Resources of the Government of the P r o v i n c e of B r i t i s h Co lumbia . NOTATION The f o l l o w i n g symbols are used i n t h i s t h e s i s : 2 A = c r o s s - s e c t i o n a l a rea of p i p e , i n f t ; 2 -A = area of opening of a v a l v e , i n f t ; 8 2 = c r o s s - s e c t i o n a l a rea of surge tank , i n f t ; 2 A = c r o s s - s e c t i o n a l a rea of a i r chamber, i n f t ; a a = v e l o c i t y of p r e s s u r e wave, i n f t / s e c ; C- = c o e f f i c i e n t of d i s c h a r g e of v a l v e or s p r i n k l e r ; d D = i n s i d e d iameter of p i p e , i n f t ; 2 E -—= modulus of e l a s i c i t y , i n l b s / f t ; f = Darcy -Weisbach f r i c t i o n f a c t o r ; H = t r a n s i e n t s t a t e p i e z o m e t r i c p r e s s u r e head above datum at the beg inn ing of a t ime i n t e r v a l , i n f t ; HP = t r a n s i e n t s t a t e p i e z o m e t r i c p r e s s u r e head above datum at the end of a t ime i n t e r v a l , i n f t ; H = s teady s t a t e p i e z o m e t r i c p r e s s u r e head above datum, i n f t ; = f r i c t i o n l o s s i n a p i p e , i n f t ; h = r e l a t i v e head ; r i 0 H -- o r i f i c e t h r o t t l i n g l o s s co r respond ing to d i s c h a r g e q , i n f t ; o r f • • -H r = o r i f i c e t h r o t t l i n g l o s s cor respond ing to d i s c h a r g e q , i n f t ; OrfO o r 0 2 I = mass moment of i n e r t i a of pump and motor , i n l b - f t - s e c ; x i L = l e n g t h of p i p e , i n f t ; N = r o t a t i o n a l speed of pump, i n r e v o l u t i o n s / m i n u t e ; N = r a t e d r o t a t i o n a l speed of pump, i n rev/min ; 2 p = p r e s s u r e at a p o i n t , i n l b s / f t ; 3 QP = t r a n s i e n t s t a t e d i s c h a r g e , i n f t / s e c ; 3 Q q = i n i t i a l steady s t a t e d i s c h a r g e , i n f t / s e c ; 3 q = t r a n s i e n t s t a t e o r i f i c e d i s c h a r g e , i n f t / s e c ; T = pump i n p u t t o r q u e , i n l b - f t ; T = r a t e d pump input t o r q u e , i n l b - f t ; t = t i m e , i n seconds ; V = t r a n s i e n t s t a t e v e l o c i t y i n p i p e a t the beg inn ing of a t ime i n t e r v a l i n f t / s e c ; VP = t r a n s i e n t s t a t e v e l o c i t y i n p i p e a t the end of a t ime i n t e r v a l , i n f t / s e c ; V q = i n i t i a l steady s t a t e v e l o c i t y i n p i p e , i n f t / s e c ; v . = t r a n s i e n t s t a t e volume of a i r i n a i r chamber at the beg inn ing of a a i r 3 time i n t e r v a l , i n f t ; vP . = t r a n s i e n t s t a t e volume of a i r i n the a i r chamber a t the end of a a i r 3 • t ime i n t e r v a l , i n . f t ; 3 v = i n i t i a l s teady s t a t e volume of a i r i n a i r chamber, i n f t ; o . a i r WR^ = moment of i n e r t i a of r o t a t i n g p a r t s of motor , pump and e n t r a i n e d 2 w a t e r , i n l b - f t ; x = d i s t a n c e a long the p i p e l i n e , i n f t ; Z = e l e v a t i o n above datum, i n f t ; X l l -, • j N a = r e l a t i v e pump speed, rr— ; R T 3 = r e l a t i v e pump t o r q u e , — ; R 3 Y = s p e c i f i c weight of l i q u i d , i n l b s / f t ; 0 . = angle of s l o p e of p i p e ; n = e f f i c i e n c y of pump; n = e f f i c i e n c y of pump at r a t e d c o n d i t i o n s ; R 2. 4 SL = d e n s i t y of l i q u i d , i n l b - s e c / f t ; T = r a t i o of e f f e c t i v e gate opening to f u l l gate open ing ; and At =,. t ime i n c r e m e n t , i n seconds . S u b s c r i p t s : The s u b s c r i p t , j , r e f e r s to p i p e w h i l e 1 , 2 , . . . n , n+1 r e p r e s e n t p i p e s e c t i o n s . INTRODUCTION I f the f l o w i n a p i p e i s changed from one steady s t a t e c o n d i t i o n to a n o t h e r , the i n t e r m e d i a t e temporary unsteady f l o w i s d e f i n e d as t r a n s i e n t f l o w . Whi le i n v e s t i g a t i n g the t r a n s i e n t behav iour of a system i t i s u s u a l l y necessary to choose the main parameters f i r s t . The subsequent a n a l y s i s v e r i f i e s whether the t r a n s i e n t p r e s s u r e s a re w i t h i n the p r e s c r i b e d l i m i t s . I f not,some parameters a re a l t e r e d and the a n a l y s i s i s r e p e a t e d . Th is procedure i s cont inued u n t i l the d e s i r e d t r a n s i e n t response i s a c h i e v e d . The c o n t i n u i t y and momentum equat ions which govern the unsteady f l o w through p ipes form a set of n o n - l i n e a r , 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 q u a t i o n s . By making c e r t a i n s i m p l i f y i n g assumptions or n e g l e c t i n g the n o n - l i n e a r terms the f o l l o w i n g methods have been developed to s o l v e these e q u a t i o n s : a) A r i t h m e t i c a l Method: Th is m e t h o d 1 3 n e g l e c t s f r i c t i o n l o s s e s and assumes the p i p e l i n e h o r i z o n t a l . An account i s kept of a l l the r e f l e c t i o n s i n the system. Th is method i s a lmost o b s o l e t e because i t i s t e d i o u s , t ime-consuming and approx imate . b) G r a p h i c a l Method: Th is method has been d e v e l o p e d 3 ' 1 2 by n e g l e c t i n g the n o n - l i n e a r terms of the above-mentioned e q u a t i o n s . How-e v e r , the f r i c t i o n l o s s e s can be c o n s i d e r e d , i f d e s i r e d , by assuming a ** These numbers r e f e r to the B i b l i o g r a p h y . h y p o t h e t i c a l o b s t r u c t i o n l o c a t e d e i t h e r a t the upstream or the down-stream end c f the p i p e . The f r i c t i o n l o s s due to the o b s t r u c t i o n i s taken equa l to that of the e n t i r e p i p e l i n e . A l though more a c c u r a t e s o l u t i o n s can bt obta ined by assuming a number of o b s t r u c t i o n s a long the p i p e l i n e , t h i s compl i ca tes the g r a p h i c a l a n a l y s i s . The g r a p h i c a l method has the advantages that the phenomenon of waterhammer can be v i s u a l i z e d and s imp le systems can be e a s i l y a n a l -y z e d . The d isadvantages are as f o l l o w s : i ) Assumptions r e g a r d i n g f r i c t i o n l o s s e s are i n a c c u r a t e , i i ) Large systems cannot be a n a l y z e d , i i i ) Many boundary c o n d i t i o n s can be ana lyzed on ly by t r i a l and e r r o r . Because of these d i s a d v a n t a g e s , t h i s method i s be ing r e p l a c e d by n u m e r i c a l methods s u i t a b l e f o r computer a n a l y s i s . c ) : A n a l y t i c a l Methods: In these methods the c o n t i n u i t y and momentum equat ions are l i n e a r i z e d by n e g l e c t i n g the n o n - l i n e a r terms of l e s s e r importance (see S e c t i o n 1.2) and c o n s i d e r i n g the f r i c t i o n l o s s e s p r o p o r t i o n a l to v e l o c i t y . Wood 2 2 used H e a v i s i d e ' s o p e r a t i o n a l c a l c u l u s to s o l v e the r e s u l t i n g l i n e a r i z e d e q u a t i o n s . Th is method g i v e s on ly surge p ressures and v e l o c i t i e s . R i c h 1 4 s o l v e d the l i n e a r i z e d equat ions by u s i n g L a p l a c e - M e l l i n t r a n s f o r m a t i o n s . U n l i k e Wood's method, t h i s method g i v e s t o t a l p r e s s u r e s and v e l o c i t i e s . S t r e e t e r and L a i t r i e d to s o l v e the s e r i e s obta ined by t h i s method on the d i g i t a l computer. They r e p o r t e d 2 0 : "However, the re are s t i l l some d isadvantages i n u s i n g the o p e r a t i o n a l mathemat ics , even w i t h a i d of the d i g i t a l computer. The t r a n s f o r m a t i o n s and i n v e r s e t r a n s f o r m a t i o n s i n v o l v e d i f f i c u l t math -e m a t i c a l m a n i p u l a t i o n s and o f t e n r e s u l t i n ted ious s e r i e s . I t has c e r t a i n l i m i t a t i o n s because of necessary a p p r o x i m a t i o n s ; i t i s sometimes d i f f i c u l t to determine constants f o r some g i v e n boundary c o n d i t i o n s and consequent ly i s not f l e x i b l e i n a p p l i c a t i o n . " d) Method of C h a r a c t e r i s t i c s : In t h i s method a l l the n o n -l i n e a r terms of the c o n t i n u i t y and momentum equat ions are r e t a i n e d . The p a r t i a l d i f f e r e n t i a l equat ions are conver ted i n t o o r d i n a r y d i f f e r e n t i a l equat ions by the method of c h a r a c t e r i s t i c s 8 ' 2 0 ' 2 1 . These equat ions are then s o l v e d by a f i n i t e - d i f f e r e n c e t e c h n i q u e . Bes ides the r e l a t i v e ease to o b t a i n any d e s i r e d a c c u r a c y , t h i s method has the f o l l o w i n g advantages : i ) Large and complex systems can be a n a l y z e d , i i ) I t i s ve ry f a s t because the equat ions d e r i v e d can be e a s i l y s o l v e d on a d i g i t a l computer, i i i ) Once a programme has been w r i t t e n f o r a system, s i m i l a r systems can be ana lyzed by changing the data c a r d s . Because of the above advantages , t h i s method i s used i n t h i s s t u d y . Whi le i n t r o d u c i n g the method of c h a r a c t e r i s t i c s i n the f i e l d of waterhammer, S t r e e t e r and L a i 2 0 p resented boundary c o n d i t i o n s f o r a few s imp le systems. In t h i s t h e s i s , boundary c o n d i t i o n s are developed f o r r e s e r v o i r s , s p r i n k l e r s , v a l v e s , surge tanks and a i r chambers. T r a n s i e n t c o n d i t i o n s i n i t i a t e d by opening or c l o s i n g a v a l v e , by pump f a i l u r e or by changes i n the water s u r f a c e e l e v a t i o n of a r e s e r v o i r , a re a n a l y z e d . A number of examples are s o l v e d and the r e s u l t s presented i n Appendices A to C. CHAPTER I METHOD OF CHARACTERISTICS 1.1 BASIC EQUATIONS FOR UNSTEADY FLOW THROUGH PIPES: The v e l o c i t y and p r e s s u r e of moving f l u i d s i n p ipes a re governed by the momentum and c o n t i n u i t y e q u a t i o n s . The p ipes are cons ide red to be made up of "compartments" . The term compartment i s used to denote a l e n g t h of p i p e hav ing a un i fo rm d i a m e t e r . I t i s assumed that f l o w i s one d i m e n s i o n a l i n each compartment and the p r e s s u r e and v e l o c i t y are un i fo rm at the c r o s s s e c t i o n of a compart -ment. The s u b s c r i p t s x and t i n d i c a t e p a r t i a l d i f f e r e n c i a t i o n w i t h r e s p e c t to d i s t a n c e and t i m e . For example, x 3x ' H t " 8t ' i n which H i s t o t a l p r e s s u r e head i n f e e t of w a t e r . The momentum e q u a t i o n 1 1 f o r f l o w through a p i p e which i s i n c l i n e d or h o r i z o n t a l , - tapered or s t r a i g h t , s l i g h t l y or h i g h l y d e f o r -mable , i s g i v e n by gH + V + V V + M ^ J - = 0 , ( l . l ) b x t x 2D i n which g i s a c c e l e r a t i o n due to g r a v i t y , V i s v e l o c i t y of the f l u i d , f i s the Barch -Weisbach f r i c t i o n f a c t o r , H i s the t o t a l - p ressure head above the datum l i n e , D i s the i n s i d e diameter of the p i p e , and ^2p^^ i s the f r i c t i o n a l f o r c e of the f l u i d . The a b s o l u t e s i g n i s i n t r o d u c e d to tak?. i n t o account any change i n the d i r e c t i o n of v e l o c i t y . Thus the f r i c t i o n a l f o r c e i s always o p p o s i t e to the d i r e c t i o n of v e l o c i t y . 2 For a p ressure change of 100 l b s . / i n and f o r a waterhammer wave v e l o c i t y of 2000 f t / s e c . the c h a n g e 1 1 i n the d e n s i t y of water i s - 3 2 A approx imate ly 3 . 6 x 10 l b s - s e c / f t . Thus, changes i n the d e n s i t y of water can be n e g l e c t e d . C o n s i d e r i n g the d e n s i t y as c o n s t a n t , the c o n t i n -u i t y equat ion i s g i v e n by 2 — V + H + V [H + s i n ©] = 0 , (1 .2 ) g • x t x i n which 9 i s the ang le the c e n t r e l i n e of the p i p e makes w i t h the h o r i z -o n t a l a x i s (measured p o s i t i v e downwards), and a i s the v e l o c i t y of the waterhammer wave. 1 .2 BASIC CONSIDERATIONS: Eqs . (1 .1 ) and (1 .2 ) can be r e - w r i t t e n i n the form (V. + V V ) + gH +!=r V |V| = 0 (1 .3 ) and t x ' 6 x 2D 2 H + — V + V [H + s i n 9] = 0 . (1 .4 ) t • g x u x The above equat ions form a se t of s i m u l t a n e o u s , n o n - l i n e a r f i r s t order p a r t i a l d i f f e r e n t i a l e q u a t i o n s . S ince the n o n - l i n e a r terms 6. V V , V H and —r V Ivl L invo lve on ly the f i r s t power of the d e r i v a t i v e , x x 2D ' 1 J R the equat ions are c l a s s i f i e d as q u a s i - l i n e a r . These equat ions may be f u r t h e r c l a s s i f i e d as e l l i p t i c , p a r a b o l i c or h y p e r b o l i c as f o l l o w s : The most g e n e r a l r e p r e s e n t a t i o n 1 1 of a p a i r of f i r s t o r d e r , q u a s i - l i n e a r , p a r t i a l d i f f e r e n t i a l equat ions i n two independent v a r i a b l e s can be expressed by the f o l l o w i n g s i n g l e m a t r i x e q u a t i o n : - ( B ( V , H ) } The square m a t r i x [A] and the column v e c t o r {B} are f u n c t i o n s of the unknowns V and H . In the p a r t i c u l a r case represented by Eqs . ( 1 . 3 ) and ( 1 . 4 ) , these m a t r i c e s a re f u n c t i o n of V on ly and are g i v e n by the f o l l o w i n g e q u a t i o n s : L~A(V)] = v g 2 ^ - V g and ( B ( V ) } = '2D V |v [V s i n 0j The e i g e n - v a l u e s , X , of m a t r i x A determine the type of the s e t of equat ions . The c h a r a c t e r i s t i c m a t r i x of A i s g i v e n by V - X g 2 5- V - X Hence, the c h a r a c t e r i s t i c equat ion of m a t r i x A can be w r i t t e n as , 2 ( V - A ) " J " g = 0. S o l v i n g t h i s equat ion f o r A, one o b t a i n s X1 = V + a and X 2 = V - a . S ince V and a a re r e a l , both the e i g e n - v a l u e s are r e a l and d i s t i n c t . Eqs . (1 .3 ) and (1 .4 ) a r e , t h e r e f o r e h y p e r b o l i c . 1 .3 CHARACTERISTIC EQUATIONS: To s i m p l i f y Eqs . (1 .3 ) and ( 1 . 4 ) , n o n - l i n e a r terms of l e s s e r importance are n e g l e c t e d i n t h i s s e c t i o n . The next s e c t i o n , however, d e a l s w i t h these equat ions w i thout n e g l e c t i n g any term. Eqs . (1 .3 ) and (1 .4 ) can be w r i t t e n i n a s i m p l i f i e d form as = 0 (1 .5 ) L l " 8 \ + V t + ^  V and 2 L „ = H + — V = 0 • (1 .6 ) 2 t g x M u l t i p l y i n g Eq . (1 .6) by an unknown m u l t i p l i e r A and adding the r e s u l t to Eq . ( 1 . 5 ) , one o b t a i n s L 1 + A L 2 = A [f H x + H t ] + [ v t + ^ V x ] + |p V |V| = 0 . (1 .7 ) Any two r e a l and d i s t i n c t v a l u e s of A g i v e two equat ions i n V and H which are e q u i v a l e n t to Eqs . (1 .5 ) and (1 .6 ) i n a l l r e s p e c t s . I f V = V ( x , t ) and H = H ( x , t ) 8. are s o l u t i o n s of Eqs . (1 .5 ) and ( 1 . 6 ) , then and Let so tha t and ^ = V ^ + V dt x dt t dt x d t t dx _ j> Xa dt X g ' X - ± * a dx • . . d F = ± 3 (1 .8 ) (1 .9 ) (1 .10) (1 .11) Then, i t f o l l o w s from Eqs . (1 .7 ) to (1 .10) tha t x dH dV f__ , , A d t d t + 2D V l V | U ' By s u b s t i t u t i o n of these v a l u e s of X i n t o Eq . ( 1 . 1 2 ) , the f o l l o w i n g c h a r a c t e r i s t i c equat ions are o b t a i n e d : ^ J H + dV f V |V[ = a dt dt 2D . dx (1 .12) d t = a . & M dV f V j V j ~ a dt d t 2D 0 dx dt - a . (1 .13) (1 .14) (1 .15) (1 .16) Eqs . (1 .13) and (1 .15) a re v a l i d a long the curves = a dx and -j-^- = - a . M a t h e m a t i c a l l y , these curves rep resent l i n e s a long which the d e r i v a t i v e s of v e l o c i t y and p r e s s u r e may have d i s c o n t i n u i t i e s . P h y s i c a l l y , they rep resent the paths on the x - t p lane a long which the waterhammer waves are propagated (See F i g . 1 . 1 ) . A f i r s t o rder f i n i t e - d i f f e r e n c e technique may be used to s o l v e Eqs . (1 .13) to ( 1 . 1 6 ) . Le t the t r a n s i e n t c o n d i t i o n s s t a r t a t t ime t = t . Then, a t the p o i n t s A and B ( F i g . 1 . 1 ) , V , H, x and t are equal to the known steady s t a t e v a l u e s . These q u a n t i t i e s a r e , however, unknown a t the p o i n t P , where c h a r a c t e r i s t i c s through the p o i n t s A and B i n t e r s e c t each o t h e r . Note tha t on the x - t p l a n e , the p o i n t P r e p r e s e n t s t ime t = t + A t , where At i s the t ime i n t e r v a l , o The s u b s c r i p t s A , B and P are used to denote q u a n t i t i e s a t the p o i n t s A , B and P. In the f i n i t e - d i f f e r e n c e fo rm, Eqs . (1 .13) to (1 .16) can be w r i t t e n a s , ( x p - x A ) = a ( t p - t A ) . ( v ' V = " a ( V t B ) • By s o l v i n g Eqs . (1 .17) to (1 .20) s i m u l t a n e o u s l y , four unknowns f o r the p o i n t P can be dete rmined . (1 .17) (1 .18) (1 .19) (1 .20) 10. The p i p e l e n g t h , L, i s d i v i d e d i n t o N equal r e a c h e s , so tha t L Ax N ' i n which Ax i s the l e n g t h of a r e a c h . Now, i f the t ime i n t e r v a l , A t , i s s e l e c t e d such tha t and A T = ^ " ' A ° S " h At - — , then the c h a r a c t e r i s t i c s l i n e s i n t e r s e c t one another at the i n t e r m e d i a t e s e c t i o n s , as shown i n F i g . 1 . 2 . The v e l o c i t y and p ressure head a t t ime t = t^ be ing known, and a t the i n t e r i o r s e c t i o n i a t t ime t = t + At can be determined from eqs . (1 .17) and ( 1 . 1 9 ) . S o l v i n g Eqs . (1 .17) and (1 .19) s i m u l t a n e o u s l y , one o b t a i n s V_ = 0 . 5 [V. + V . + & (H. - H - ) P . L l - l l + l a l - l l + l and LM ( v 2D k V i - l V i - 1 + V i+1 i+1 ) ] V = 0 - 5 [ H i - l + H i + 1 + I ( V i - l " W a f At (V. i - 1 - V i+1 V i+1 ) ] (1 .21) g 2D v i - 1 The v e l o c i t y and p r e s s u r e a t the end p o i n t s are computed from the boundary c o n d i t i o n s . Now, the v a l u e s of V and H a t N+l s e c t i o n s are known. By proceeding i n t h i s manner, the computat ion can be performed up to the s p e c i f i e d t i m e . (1 .22) 1 1 . o *^  At _4_. Ax Ax 9 - X CHARACTERISTIC CURVES FIG. I.I t = t. At At At • © O Initial boundary condition Determined by solving equations 1-17 and M9 simultaneously Determined by solving equation for C + and equation for end condition simultaneously Determined by solving equation for C~and end condition simultaneously GRID CHARACTERIST IC FIG. 1.2 12. 1 .4 GENERAL CHARACTERISTICS METHOD: In s e c t i o n 1 . 3 , s i m p l i f i e d c o n t i n u i t y and momentum equat ions were u s e d . Th is i s a lmost adequate f o r c a l c u l a t i n g the t r a n s i e n t c o n d i t i o n s i n m e t a l p i p e s . In t h i s s e c t i o n , a g e n e r a l s o l u t i o n f o r the c o n t i n u i t y -and momentum equat ions i s obta ined i n which a l l the terms of the a f o r e s a i d equat ions a re r e t a i n e d . For a p i p i n g system of two or more p i p e s , i t i s necessary tha t the same t ime increment be used f o r a l l the p i p e s , so that the end c o n d i t i o n s common to the p ipes may be obta ined from a se t of s imul taneous e q u a t i o n s . The method of s p e c i f i e d t ime i n t e r v a l s , which i n v o l v e s l i n e a r i n t e r p o l a t i o n , i s u s e d . T-he-momentum and c o n t i n u i t y equat ions may be w r i t t e n as L l = s H x + V V x + V t + H ^ M = 0 . ( 1 ' 2 3 ) and 2 L_ = H. + — V + V H + V s i n 0 = 0 . (1 .24) 2 t g x x M u l t i p l y i n g Eq . (1 .24) by X and then adding i t to Eq . ( 1 . 2 3 ) , one x>b t a i n s 2 a L r + x L 2 = x [ H x ( v + f> + V ; + - ( v x ( v + tx) + v t ] + X- V s i n 9 + f V2JV\ = 0 . . . (1 .25) - Le t 2 i _ d t ' ' X ' " g " *L = V + f = V + 5_ x so tha t = ± & a 1 3 . and It" = V 1 a -By u s i n g these r e l a t i o n s , Eq . (1 .25) takes the form •dH _, dV , , „ . n  A dt" "dt S i n +•f v l v L o. 2D I t f o l l o w s from Eqs . (1 .26) and (1 .27) that a dt dt a 2D ' > c + dx „ , dt " V + a * g dH J dV g T 7 . n , f V 1VI n a dt dt a 2D ' and dx dF = v " a" (1 .26) (1 .27) (1 .28) (1 .29) (1 .30) Because V = V ( x , t ) , the c h a r a c t e r i s t i c l i n e s C and C by Eqs . (1 .29) and ( 1 . 3 1 ) , p l o t as curves on the x - t p lane (See Eqs . (1 .28) to (1 .31) can be w r i t t e n i n the f o l l o w i n g d i f f e r e n c e fo rms: ( V P - v + i ( H P - v + f \ s i n 9 w f (1 .31) , g i v e n F i g . 1 . 3 ) . f i n i t e -+ l D \ R ( t p - t R ) - 0. ( v v - f ( v v - i v s s i n e ( t p - v (1 .32) (1 .33) 2D S ( t p - t s ) - 0. (1 .34) 14. ( x p - x s ) = (V s - a) ( t p - t - s ) . ( 1 > 3 5 ) Eq. (1 .32) to (1 .35) can be so l ved n u m e r i c a l l y by. u s i n g a g r i d of c h a r a c t e r i s t i c s or s p e c i f i e d t ime i n t e r v a l s . The former method i s advantageous when e i t h e r V or a v a r i e s c o n s i d e r a b l y w i t h x and t , as i n the case of h i g h l y deformable tubes . In the l a t t e r method x p and t are ass igned d e f i n i t e v a l u e s , l e a v i n g and V p as the two unknowns. Th is method i s u s e f u l f o r most of the waterhammer problems and i s used h e r e . S ince the c o n d i t i o n s a t the p o i n t s A , B and C ( F i g . 1.3) are known, they can be determined at the p o i n t s R and S by l i n e a r i n t e r p o l a t i o n . Thus X C - X R V C " \  X C - X A ~ V C - V A ' But xp = x c , and x p - x = Ax . By u s i n g these r e l a t i o n , the above equat ion takes the form V - V X P ' X R ° V ^ V ^ Ax . (1 .36) S i n c e V i s much s m a l l e r than the waterhammer wave v e l o c i t y , a , i t can R be n e g l e c t e d . By n e g l e c t i n g V i n Eq . (1 .33) and then combining i t w i t h Eq. (1 .36) K one o b t a i n s V - V a At = C _ R Ax . . . . (1 .37) C A 15. The g r i d mesh r a t i o , 0* , i s d e f i n e d by the e x p r e s s i o n 0 ' = Ax I t f o l l o w s from Eq . (1 .37) t h a t a 0 ' (V c - V A ) = V c - V R ) so tha t S i m i l a r l y , V R = V C - a 0 ' ( V C - V ' (1 .38) H R = H c - a 0 « ( H C - H A ) . (1 .39) (1 .40) (1 .41) By s o l v i n g Eqs . (1 .32) and (1 .34) s i m u l t a n e o u s l y , the f o l l o w i n g express ions f o r P V g = V c - a 0 ' (V c - V B ) H S " H C " a 9 ' ( H C " H B ) and Hp are o b t a i n e d V p = 0 . 5 [ V R + V s + f ( H R - H s ) - f A t s i n 0 (V R - V g ) f At 2D R (V. V + v„ V R S S )] (1 .42) ^ = 0 . 5 [ l ^ + H g + - (V R - V S ) - At s i n 0 (V R + V g ) a f At g 2D ( V R v - v„ V„ R S S ) ] (1 .43) At the boundary p o i n t s , e i t h e r Eq. (1 .32) or E q . ( 1 . 3 4 ) or both are used together w i t h the c o n d i t i o n s imposed by the boundary. Eqs . (1 .32) and ( 1 . 3 4 ) , ".(henceforth c a l l e d the n e g a t i v e c h a r a c t e r i s t i c equat ion and the 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 ) , can be r e w r i t t e n i n the f o l l o w i n g fo rms : t + At c / / / 1 \ \ \ A R C S B METHOD OF SPECIF IED TIME I N T E R V A L S FIG. 1.3 (o) (b) C H A R A C T E R I S T I C S AT THE BOUNDARIES FIG. 1.4 17. The 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 : i n which V p = CI + C 2 . H p , CI = V g - C2.Hg + C 2 . V g s i n 0 At - FF .V V C2 = - , g and FF = f At 2D The 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 : V p = C3 - C 2 . H p , V R ( l .<4) (1 .45) (1 .46) (1 .47) (1 .48) (1 .49) i n which C3 = V R + 0 2 . 1 ^ - C 2 . V R A t s i n 0 - F F . V R Note t h a t C2 and FF r e p r e s e n t p i p e - c o n s t a n t s . The v a l u e s of CI and C3 are constant d u r i n g each t ime s t e p . - 1 . 5 -CONVERGENCE AND STABILITY OF THE METHOD OF FINITE DIFFERENCES: The f i n i t e - d i f f e r e n c e s scheme developed i n the p receed ing s e c t i o n i s convergent i f the exact s o l u t i o n of the d i f f e r e n c e equat ions approaches that of the o r i g i n a l d i f f e r e n t i a l e q u a t i o n , as the g r i d mesh r a t i o tends towards z e r o . I f the r o u n d - o f f e r r o r due to r e p r e s e n t a t i o n of the i r r a t i o n a l numbers by a f i n i t e number of s i g n i f i c a n t d i g i t s grows or decays as the s o l u t i o n p r o g r e s s e s , the scheme i s s a i d to be u n s t a b l e or s t a b l e r e s p e c t i v e l y . I t has been determined^ that convergence i m p l i e s s t a b i l i t y and s t a b i l i t y i m p l i e s 18. convergence. Methods f o r de te rmin ing the convergence or s t a b i l i t y c r i t e r i a f o r the n o n - l i n e a r equat ions are ext remely d i f f i c u l t . However, a n a l y t i c a l s t u d i e s f o r the convergence and s t a b i l i t y can L .^ made by l i n e a r i z i n g the b a s i c e q u a t i o n s . S i n c e , the n o n - l i n e a r terms a re r e l a t i v e l y s m a l l , i t i s reasonable to assume that the c r i t e r i a a p p l i c a b l e to the s i m p l i f i e d equat ions are a l s o v a l i d f o r the o r i g i n a l n o n - l i n e a r e q u a t i o n s . Us ing the procedure proposed by O b r i e n 1 ^ and c o n s i d e r i n g l i n e a r i z e d e q u a t i o n s , P e r k i n s 1 1 has proved tha t f o r the process to be s t a b l e At < _1 • Ax a T h i s shows t h a t the c h a r a c t e r i s t i c s through the p o i n t P ( F i g . 1.3) shou ld not f a l l o u t s i d e the segment AB. For a n e u t r a l l y s t a b l e scheme, = — . The c r i t e r i a f o r convergence i n d i c a t e tha t the most accura te s o l u t i o n s are obta ined i n t h i s c a s e . Thus, the convergence and/or s t a b i l i t y c r i t e r i o n f o r the f i n i t e - d i f f e r e n c e scheme based on the method of c h a r a c t e r i s t i c s i s g i ven by the e x p r e s s i o n (1 .51) -1.-6 SELECTION OF THE TIME INCREMENT FOR A COMPLEX PIPING SYSTEM: For a complex p i p i n g system of two or more p i p e s , i t i s necessary tha t f o r a l l the p i p e s , the same t ime increment be used . Us ing the i n t e r p o l -a t i o n method o u t l i n e d above, i t i s t h e o r e t i c a l l y p o s s i b l e f o r one to merely 1 9 . s a t i s f y the l i m i t a t i o n on the g r i d mesh r a t i o i n each p ipe and proceed w i t h the s o l u t i o n s . However, w i t h the l i n e a r i n t e r p o l a t i o n , the l a r g e r the i n t e r -p o l a t i o n i . e . the l a r g e r the q u a n t i t y (Ax •- a A t ) , the l e s s accura te the s o l u t i o n i s l i k e l y to be . The convergence c o n s i d e r a t i o n s i n d i c a t e 1 1 t h a t the most a c c u r a t e s o l u t i o n s are ob ta ined when Ax = a A t . T h e r e f o r e , t ime inc rement , A t , i s s e l e c t e d such t h a t the computing t ime i s not e x c e s s i v e and the i n t e r -p o l a t i o n i s a p p l i e d over a minimum d i s t a n c e . The t ime increment and the number of r e a c h e s , N_. , i n each p i p e may be s e l e c t e d as f o l l o w s : i ) S e l e c t number of reaches i n the s h o r t e s t p ipe of l e n g t h L ^ . i i ) Compute the t ime inc rement , A t , from the e q u a t i o n , L l A t (V 1 + a x ) N 1 where and a^ r e p r e s e n t the v e l o c i t i e s of water and waterhammer wave i n the p i p e . i i i ) Then, the number of r e a c h e s , , i n the p ipe i s the s m a l l e s t i n t e g e r such that L . CHAPTER I I BOUNDARY CONDITIONS The boundary c o n d i t i o n f o r v a r i o u s p i p e l i n e geometr ies and appur -tenances i n the p i p i n g systems are developed i n t h i s c h a p t e r . The minor l o s s e s such as the ent rance l o s s , the l o s s at a change i n the c r o s s - s e c t i o n are n e g -l e c t e d . By s l i g h t m o d i f i c a t i o n s and a l t e r a t i o n s , these c a n , however, be c o n s i d e r e d . A l though i n the computer programme, developed i n the l a t e r s e c t i o n s , th ree d i m e n s i o n a l s u b s c r i p t e d v a r i a b l e s are used to d e s c r i b e the c o n d i t i o n s at a p o i n t ; f o r s i m p l i c i t y , on ly two s u b s c r i p t s are used i n t h i s , c h a p t e r . The f i r s t s u b s c r i p t denotes the p i p e number w h i l e the second r e f e r s to a s e c t i o n . The p i p e s are assumed h o r i z o n t a l . The c e n t r e - l i n e of a p ipe i s cons idered as the datum l i n e . The v e l o c i t y and the p i e z o m e t r i c p ressure head above the datum l i n e a t the beg inn ing and a t the end of a t ime i n t e r v a l are des ignated by V, H, VP and HP r e s p e c t i v e l y . To determine the v e l o c i t y , VP , and the p r e s s u r e head, HP, a t a boundary p o i n t , Eq . (1 .44) or Eq . (1 .48) or both are so l ved s i m u l t a n e o u s l y w i t h the c o n d i t i o n s imposed by the boundary. In the p i p i n g sys tem, the i n i t i a l s teady s t a t e f l ow d i r e c t i o n i s c o n -s i d e r e d p o s i t i v e . For a p i p e i n which i n i t i a l steady s t a t e v e l o c i t y i s z e r o , p o s i t i v e f low d i r e c t i o n i s a r b i t r a r i l y assumed. 21. An appurtenance on a p ipe should be l o c a t e d at the end of a reach i n t o which the p i p e i s d i v i d e d . I f i n the a c t u a l system such i s not the c a s e , i t i s assumed to be l o c a t e d a t the neares t end of the r e a c h . 2 . 1 RESERVOIR OF CONSTANT WATER LEVEL: a) R e s e r v o i r l o c a t e d at the upstream end ( F i g . 2 . 1 a ) : At the j u n c t i o n of the p ipe and the r e s e r v o i r , HP. . = H ] , 1 r e s i n which H i s the h e i g h t of the water s u r f a c e i n the r e s e r v o i r above the res G c e n t r e l i n e of the p i p e . The n e g a t i v e c h a r a c t e r i s t i c equat ion f o r s e c t i o n ( j , l ) i s g i ven by VP. - = C I . + C2. HP. .. , J . l 3 J J , l i n which C l^ and C2^ are the cons tants f o r p i p e J . For the express ions f o r these c o n s t a n t s , see Eqs . (1 .45) and ( 1 . 4 6 ) . From the above two e q u a t i o n s , i t f o l l o w s that VP. = C I . + C2. H . (2 .1 ) J , 1 J J r e s b) R e s e r v o i r a t the downstream end ( F i g . 2 . 1 - b ) : A t the j u n c t i o n of the p ipe and the r e s e r v o i r , HP. = H j , n + l r e s The p o s i t i v e c h a r a c t e r i s t i c equat ion f o r s e c t i o n ( j , n + l ) i s g i ven by VP. f 1 = C3 . - C2. HP. - . 22. H res Pipe J t+At (I) A * . At 1 (2) (I) 5 X (2) (a) Reservoir at the upstream end H res Pipe J t (n; (n+1) At R (b ) Reservoir at downstream end t + At (c) Reservoir at an intermediate section FIG. 2.1 23. From the above two e q u a t i o n s , i t f o l l o w s t h a t VP. , = C3 . - C2. H . . , . j , n + l j j r e s (2.?) c) R e s e r v o i r at an i n t e r m e d i a t e s e c t i o n ( F i g . 2 . 1 - c ) : In t h i s c a s e , HP. = H P . , , . = H . (2 .3 ) j , n + l 3+1,1 r e s The 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 f o r s e c t i o n ( j ,n+ l ) i s g i ven by VP. , , = C3. - C2. HP. , , . j , n + l j j J , n + 1 By v i r t u e of Eq . ( 2 . 3 ) , the above e q u a t i o n becomes VP. , , = C3 . - C2. H . (2 .4 ) j , n + l 3 3 r e s The 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 f o r s e c t i o n ( j+1,1) can be w r i t t e n as ^•.1.1 i = C 1 - J - i + c 2 - o . i i > J+1,1 J+ l J + l J+1,1 w h i c h , upon combining w i t h Eq . ( 2 . 3 ) , y i e l d s V P . , , i = C I . , , + C 2 . . . H . (2 .5 ) j + 1 , 1 j + l j + l res 2 .2 DEAD END AT THE DOWNSTREAM END ( F i g . 2 . 2 ) : The v e l o c i t y a t a dead end i s always z e r o . Hence, i n t h i s case ^ j . n + l - ° ' . ( 2 - 6 ) The v a l u e of HP. can now be determined from the f o l l o w i n g p o s i t i v e charac -j , n + l 6 t e r i s t i c e q u a t i o n f o r s e c t i o n ( j , n + l ) : VP. = C3 . - C2. HP. . (2 .7 ) J ,n+1 J 3 3 ,n+ l From Eqs. (2 .6 ) and ( 2 . 7 ) , i t f o l l o w s t h a t HP. - = C 3 . / C 2 . ( 2 . 0 3 . n + l 3 3 2 . 3 SERIES CONNECTION ( F i g . 2 . 3 ) : In t h i s s e c t i o n , boundary c o n d i t i o n s f o r a j u n c t i o n of two p ipes hav ing d i f f e r e n t d iameters are deve loped . However, w i thout any m o d i f i c a t i o n s , these r e s u l t s can be a p p l i e d to a j u n c t i o n of two p i p e s hav ing the same diameter but d i f f e r e n t s l o p e s , w a l l t h i c k n e s s e s or w a l l - m a t e r i a l or any combinat ion of these v a r i a b l e s . For the steady s t a t e p r e s s u r e head and v e l o c i t y a t the j u n c t i o n , the f o l l o w i n g equat ions can be w r i t t e n : The c o n t i n u i t y e q u a t i o n : VP. A . = V P . . . , A . , (2 .9 ) j , n + l j 3+1.1 j+1 i n which Aj and A_. +^ are the c r o s s - s e c t i o n a l areas of p i p e j and p i p e j + 1 . The e q u a t i o n f o r common p r e s s u r e head: HP. = HP. . (2 .10) j , n + l J+1,1 The 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 f o r s e c t i o n ( j , n + l ) : VP. , , = C3 . - C2. HP. . (2 .11) j , n + l j j J ,n+1 The 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 f o r s e c t i o n ( j + 1 , 1 ) : V P . , , , = C I . , , + C 2 . , 7 H P . . , . (2 .12) j + 1 , 1 j+1 j+1 J+1,1 I t f o l l o w s from Eqs. (2 .9 ) to (2 .12) tha t Pipe J At •Dead end DEAD END FIG. 2 . 2 Pipe J Pipe J+l * Ax. —>i j.n+l j+1,1. ** R s SERIES CONNECTION FIG. 2 .3 26. (C3. - C2. HP. . , ) A . = ( C I . . , + C 2 . . , HP. ) A . . . , 3 J J ,n+1 j 3+1 3+1 J ,n+1 J+1 so t h a t (C2. A . + C 2 . . , , A . . . ) HP. , , = C3. A . - C I . . , A . . , . J 3 J + 1 J+1 J ,n+1 3 3 3+1 j+1 S i m p l i c a t i o n of the above e q u a t i o n , y i e l d s C3. A . - C 1 . L 1 A . ^ , HP = 3 3 J + 1 J + 1 (2 13) H F j , n + l C2 . A . + C 2 . ^ , A . ^ , " K ' J J J J+1 J+1 Now the v a l u e s of HP. , , , , VP . , , and VP. ... , can be determined from 3+1,1 J ,n+1 J+1,1 Eqs . (2 .10) to ( 2 . 1 2 ) . 2 .4 BRANCH CONNECTION: The f o l l o w i n g d e r i v a t i o n a p p l i e s to a j u n c t i o n of m p i p e s . F i g . ( 2 . 4 - a ) shows the assumed i n i t i a l s teady s t a t e f low d i r e c t i o n s . A bar on the s u b s c r i p t i n d i c a t e s a p i p e i n which the assumed steady s t a t e f l ow i s towards the j u n c t i o n . For the v e l o c i t y and p r e s s u r e head a t the j u n c t i o n , the f o l l o w i n g equat ions can be w r i t t e n : The c o n t i n u i t y e q u a t i o n : A= V P T ,. + A^ VP^ ,- + ... + AT VPT ,-I l , n + l 3 3 ,n+ l 3 3 ,n+l = A„ V P 0 , + A. VP. , + ...+ A.,, VP.' , + ...+ 2 2 , 1 4 4 , 1 3+I 3+1,1 + A VP . /o 1 / \ m m , l (2 .14) 27-The e q u a t i o n f o r common p r e s s u r e head: HP- = HP_ = HP^ HP T • • • = HP . •. l , n + l 2,1 3 , n + l 3 , n + l m , l The p o s i t i v e c h a r a c t e r i s t i c equat ions f o r the p i p e s i n which the assumed steady s t a t e f l o w i s towards the j u n c t i o n : .I,i>.+1 1 1 l , n + l 3 , n + l 3 3 3 , n + l VPT = C3T - C2- HP T ,, 3 , n + l j j J , n+1 The n e g a t i v e c h a r a c t e r i s t i c equat ions f o r the p i p e s i n which the assumed steady s t a t e f l ow i s away from the j u n c t i o n : V P M - C l 2 + C 2 2 H P 2 F L . VP... = C I . + C2. HP... , 3 3 3+1,1 VP , = CI + C2- HP . m,± m m m , l S o l v i n g the above equat ions s i m u l t a n e o u s l y , one o b t a i n s 28-(a) BRANCH CONNECTION OF m - P I P E S (i.n) Cj,n+l) (J+1,2) Ax. Ax. P i p e J (j+2,1) f " ( j + 2 , 2 ) < i . Pipe J+1 Pipe J+2 ( b) B R A N C H CONNECTION OF 3 - P I P E S FIG. 2 . 4 Note : Arrows indicate init ial steady state flow direct ion. 29. HPr =' { (C3 T A T + C3^ + . . . + C 3 T AT +• ...) 1,-n+l 1 1 3 3 j J - ( C l 0 A ' + . . . + C I . - . , A... + . . . .+ . : CI A ) } / 2 2 j + l j + l m m {C2 T A T + C2„ A„ + ... + C2T AT + C 2 . j n A . ^ + ... 1 1 2 2 j j j + l j + l . . . + C2 A } . (2 .15) m m The remain ing (2m-1) unknownscan be determined from the remain ing (2m- l ) equat ions . Example: For a branch co nnec t io n of th ree p i p e s , i n which the i n i t i a l steady s t a t e f l ow d i r e c t i o n s are as shown i n F i g . ( 2 . 4 - b ) , Eq . (2 .15) reduces to C3 . A. - C I . . , A.,, - C l . , „ A.,„ HP ••• = J J J + 1 J + 1 J + 2 J + 2 (2 16) H P j , n + l C2. A. + C 2 . . . A... + C 2 . . . A... * U , L B ; 3 3 1+1 J+ l J + l J + l The v a l u e s of the remain ing unknowns can be determined from the f o l l o w i n g e q u a t i o n s : HP. . . = H P . , „ = HP. . (2 .17) j + 1 , 1 J+2,1 J ,n+1 VP. = C3 . - C2. HP. ^ . . (2 .18) J ,n+1 j J J ,n+1 VP-J-I i = C 1 - m i + C 2 - _ L I H P - _ L I i ' (2 .19) j + 1 , 1 j + l j + l J+1,1 VP-^o i = c l . j . 9 + c 2 - ± i HP-AO i • (2 .20) J + 2 , 1 J+2 j+2 J+2,1 2 . 5 THE SPRINKLERS: • "A s p r i n k l e r i s an o r i f i c e on an i r r i g a t i o n p i p e . The d i s t a n c e between two c o n s e c u t i v e s p r i n k l e r s on a p i p e should be e i t h e r more than or equa l to the v a l u e of Ax f o r the p i p e . 30. Boundary c o n d i t i o n s f o r the f o l l o w i n g cases are deve loped : a) S p r i n k l e r s l o c a t e d at the j u n c t i o n of two p i p e s hav ing d i f f e r e n t  d iameters and w a l l t h i c k n e s s e s ( F i g . 2 . 5 - a ) : For the steady s t a t e c o n d i t i o n s , Q = C, A /2gH , (2 :21) osp do sp osp i n which Q i s the steady s t a t e d i s c h a r g e of the s p r i n k l e r , A i s the c r o s s -x o s p sp s e c t i o n a l a rea of the s p r i n k l e r , C, i s the c o e f f i c i e n t of d i s c h a r g e and H r do ° osp i s the steady s t a t e p ressure head at the s p r i n k l e r . I f Q p g p denotes the t r a n -s i e n t s t a t e d i s c h a r g e through the s p r i n k l e r , HP g p l s the t r a n s i e n t s t a t e p r e s -sure a t the s p r i n k l e r and i f i t i s assumed that unsteady f low through the s p r i n k l e r f o l l o w s the steady s t a t e law , then QP = C, A /2gHP . (2 .22) sp d sp sp By d i v i d i n g Eq . (2 .22) by Eq . (2.2.1) and assuming 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 c o n s t a n t , one o b t a i n s QP = 5 ° § £ _ /HP L e t s p J r — s p osp Q . osp s p / i f Then, osp QP = C /HP . (2 .23) sp sp sp Th is equat ion i s c a l l e d the c h a r a c t e r i s t i c e q u a t i o n f o r a s p r i n k l e r . 3 1 . For the v e l o c i t y and p r e s s u r e head at s p r i n k l e r , the f o l l o w i n g equatioixs can be w r i t t e n : The 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 f o r s e c t i o n ( j , n + l ) : V P . ... = C3 . - C2. H P . , . . (2 .24) 3 , n + l j j J ,n+1 The 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 f o r s e c t i o n ( j + 1 , 1 ) : V P . , , = C I . , , + C2 H P . , , . (2 .25) j + 1 , 1 j + l j + l j + 1 , 1 The c o n t i n u i t y e q u a t i o n : V P . , , A . = OP + V P . , , , A . , , . (2 .26) j , n + l j ^ sp J+1,1 J + l Common p r e s s u r e head: H P . = H P . , , , = HP . (2 .27) j , n + l J+1,1 sp I t f o l l o w s from Eqs. (2 .23) to (2 .27) tha t (C3. - C2. H P . A . = C /HP7 7 7 + ( C I . . . + C 2 . , , H P . ) A . . . J 3 3 , n + l j sp J ,n+1 j + l j + l J ,n+1 j + l L e t H = ifW. 77 • sq . j , n + l Then, rearrangement of the terms of the above equat ion y i e l d s (C2. A . + C2. , A. , , ) H 2 + C H - (C3. A . - C l ' . A . = 0 • 3 3 3+1 J + 1 sq sp sq j j j + l j + l S o l v i n g t h i s q u a d r a t i c equat ion f o r H , one o b t a i n s sq H = [-C ± {C 2 + 4 (C3. A . - C l . . , A . ^ , ) ( C 2 . A. + C2 A . , , ) } ^ ] / sq L sp sp j j j + l j + l J J 3+1 3+1 {2(C2. A . + C2. A . , 7 ) } ' 3 3 J 3+1 Put C, = 2(C2. A . + C2 . , . A . , , ) / C k 3 3 3+1 3+1 sp I f the n e g a t i v e s i g n w i t h the r a d i c a l term i s d i s c a r d e d , the e x p r e s s i o n f o r H s 32. becomes H = ( - 1 + / l + 2C. (C3. A . - CI , A . , , ) / C )/C, . (2.28) sq k j j j+1 j+1 s p 7 k The v a l u e s of HP. , , and H P . ' , , , are obta ined from the f o l l o w i n g e q u a t i o n s : j , n + l J+1,1 HP. = HP . . . , ' = (H ) 2 . (2 .29) j , n + l J+1,1 sq Having determined HP. , , and H P . , , - , the v a l u e s of VP. , , and V P . , , , j , n + l J+1,1 J ,n+1 J+1,1 can be determined from Eqs. (2 .24) and ( 2 . 2 5 ) . b) S p r i n k l e r s on a p i p e of constant d iameter and w a l l t h i c k n e s s ( F i g . 2 . 5 - b ) : In order to have not more than th ree s u b s c r i p t s ( the maximum a l lowed on the IBM 7040/44) the s e c t i o n s on a d i s t r i b u t o r are des ignated as shown i n . F i g . 2 . 5 - b . I f the s e c t i o n on the upstream s i d e of a s p r i n k l e r i s des ignated as m, then the s e c t i o n downstream of i t i s des ignated as m+1. In t h i s c a s e , A . = A . J J+1 and C2 . = C 2 . ' J J+1 * • By v i r t u e of these r e l a t i o n s , Eq . (2 .28) reduces to ^ + /I + 2A. C, ( C 3 . - C l . ) / C H = ^ — J J SIL. ( 2 < 3 0 ) L k i n which 4 . C 2 . A . C = 7? ^ . Pipe J sprinkler J| , Pipe J+l , ^ n+l (I) '(2) t + At > s s s s s • s N N N N N s R (a) SPRINKLER AT THE JUNCTION OF TWO PIPES sprinkler § pipe J & jn -uki m-2 m-l m' "m-H m+2/ m+-t + At At I y v \ (b) S P R I N K L E R ON A PIPE FIG. 2.5 34. The v a l u e s of HP. , VP. and VP. , , can he. determined from the J . n j , n 3 , n + l f o l l o w i n g e q u a t i o n s : Hp. = HP. , = H 2 3 »n 3,n+l sq VP. C3. - C2. HP. . J . n J j J , n VP. , = C I . + C2. HP. j n . J . n + l 3 3 3 , n + l c) S p r i n k l e r at the downstream end of a p i p e : In t h i s c a s e , Aj + 1 ' ° » C 2 3 + l = ° ' By u s i n g these r e l a t i o n s , Eq. (2 .28) can be w r i t t e n as - 1 + VI + 2C, C3. A ./C k_ 'sq C H = ^ y y S p ' (2 .31) i n which 2 . C 2 . A . C, = J 1 k C sp Now the v a l u e s of HP. , , and VP. , , can be determined from the f o l l o w i n g 3 , n + l 3 , n + 1 e q u a t i o n s : HP. ^ = H 2 . (2 .32) 3 , n + l sq VP. = C3. - C2. HP. . (2 .33) 3 . n + 1 3 3 3 , n + 1 2.6 THE VALVES: The boundary c o n d i t i o n s f o r four d i f f e r e n t l o c a t i o n s of the v a l v e s 3 5 . a re developed i n t h i s s e c t i o n . V a r i o u s types and combinat ions of the v a l v e s used to l i m i t the t r a n s i e n t p ressures w i t r i n an a l l o w a b l e range upon pump f a i l u r e , are d i s c u s s e d i n S e c t i o n 2 . 9 . The e f f e c t i v e gate open ing , T , i s a f u n c t i o n of e i t h e r t i m e , t , or p r e s s u r e , p (hencefor th c a l l e d the x - t or x - p r e l a t i o n s h i p ) . Th is r e l a t i o n -s h i p i s g i v e n e i t h e r by a formula or by a se t of n u m e r i c a l v a l u e s . In the l a t t e r c a s e , the g i v e n v a l u e s are s t o r e d i n the computer and f o r an i n t e r m e d i a t e v a l u e of t or p, the v a l u e of x i s determined by p a r a b o l i c i n t e r p o l a t i o n . Formulae f o r the e f f e c t i v e gate open ing : (a) The x - t r e l a t i o n s h i p : For example, x - t r e l a t i o n s h i p f o r gate c l o s u r e may be g i v e n by / -i t ,m T = (1 - — ) , C i n v/hich i s the t ime r e q u i r e d f o r t a t a l gate c l o s u r e . (b) The x -p r e l a t i o n s h i p : Th is r e l a t i o n s h i p i s s p e c i f i e d i n the case of a s p r i n g loaded v a l v e . Th is type of v a l v e opens when the p r e s s u r e , p, a t the v a l v e exceeds P^ n » i s f u l l y open when p ^ P f j n > a n d c l o s e s a u t o m a t i c a l l y when p < P f ^ n ' The v a l u e s of p_j>n and depend upon the s t i f f n e s s of the s p r i n g . Assuming a x ^ p curve of degree m, ( F i g . 2 . 6 - b ) , x a t any p r e s s u r e p, ( p ^ n < P < P^^n)> i s g i v e n by x — k (p — p i n ) m , (2 .36) i n which k i s a c o n s t a n t . t 0.0 1.0 2.0 3.0 4 . 0 5 .0 6 .0 T 1.0 0.9 0 . 7 0 .5 0 .3 0.1 0 . 0 T - t RELATIONSHIP FOR A V A L V E FIG. 2 . 6 (a ) T - p RELATIONSHIP FOR A VALVE FIG. 2.6 (b ) 37. I t i s c l e a r from F i e . 2 . 6 - b that a t p = p . . ,x = 1 . Hence b r r i m 1 = k (v - p. ) m , r f m * i n so that k = (p_. - p. ) " m . (2 .37) ^ f i n r m E l i m i n a t i o n of k from Eqs . (2 .36) and ( 2 . 3 7 ) , y i e l d s T = m p - p. r r x n p , . - p. , f i n i n (2 .38) 2 I f the v a l u e s of p. and p_. are g i ven i n l b s / i n and the t r a n s i e n t p ressure i n f i n head , H, i s g i ven i n f e e t of w a t e r , then p = 0.433H , 2 i n which p i s the t r a n s i e n t p ressure i n l b s / i n . S u b s t i t u t i o n of t h i s v a l u e of p i n t o Eq. (2 .38) g i v e s 0.433H - p i n Pr • ~ P • f i n r m m (2 .39) CASE I. VALVE AT THE DOWNSTREAM END ( F i g . 2 . 6 - c ) : I f the v a l v e i s d i s c h a r g i n g i n t o atmospher ic p r e s s u r e , then f o r the steady s t a t e c o n d i t i o n s , the gate equat ion can be w r i t t e n as V . A , = (C, A ) /2gH . ~ , (2 .40) o j J d g o & o j , n + l i n which the s u b s c r i p t zero denotes the steady s t a t e v a l u e s , Ag i s the area of opening of the v a l v e , and C^ i s the c o e f f i c i e n t of d i s c h a r g e . By d e s i g -n a t i n g the t r a n s i e n t s t a t e v a l u e s by the s u b s c r i p t P, gate equat ion f o r the v a l v e f o r the t r a n s i e n t s t a t e c o n d i t i o n s can be w r i t t e n as 38. VP. A . = (C, A ) _ /2gHP" 77 • (2 .41) J J d g P J ,n+1 By d i v i d i n g Eq . (2 .41) by Eq . (2 .40) and making the s u b s t i t u t i o n ( = d V o one o b t a i n s T V . VP- .-i = ° J VW. 77 J »n+1 ^ J ,n+1 o j , n + l By s q u a r i n g both s i d e s of the above e q u a t i o n , one gets VP 2 J = T 2 C2. C HP. J . , (2 .42) j , n + l J v j , n + l i n which v 2. C v . c C2. H . -J o j ,n+ l . The p o s i t i v e c h a r a c t e r i s t i c equat ion f o r s e c t i o n ( j , n + l ) i s g i ven by VP . = C3 . - C2. HP. . (2 .43) J ,n+1 j j J ,n+1 2 Le t = x C^. Note that i s constant d u r i n g each time s t e p . Then, e l i m i n -a t i o n of HPj n + 1 from Eqs . (2 .42) and (2 .43) y i e l d s 7T- V P 2 . . + VP . . n - C3 . = 0 .. . c 4 J ,n+1 J ,n+1 3 S o l v i n g the above e q u a t i o n f o r VP^ and n e g l e c t i n g the n e g a t i v e s i g n w i t h the r a d i c a l te rm, one o b t a i n s VP. , = h C, ( - 1 + / l + 4 . C 3 . / C . ) . (2 .44) J » n + l fi 3 A 39. The v a l u e of HP , can now be determined from e i t h e r Eq. (2 .42) or Eq . j , n + l ( 2 . 4 3 ) . When the v a l v e i s f u l l y c l o s e d i . e . x = 0, then VP. = 0. By v i r t u e of the above remark, i t f o l l o w s from Eq . (2 .43) that ^ . n + l - c Y 1 • ' ( 2 - 4 5 ) J CASE I I . VALVE AT THE JUNCTION OF TWO PIPES ( F i g . 2 . 6 - d ) : C o n s i d e r i n g the p o s i t i v e f l o w d i r e c t i o n as shown i n F i g . 2 . 6 - d the boundary c o n d i t i o n s f o r the v a l v e are deve loped . Dur ing the t r a n s i e n t s t a t e , these can a l s o be used f o r the r e v e r s e f l o w . But t h i s r e q u i r e s s p e c i a l a t t e n t i o n i n the computer programme to a v o i d the p o s s i b i l i t y of t a k i n g the square roo t of a n e g a t i v e number. For the t r a n s i e n t s t a t e c o n d i t i o n s at the v a l v e , the f o l l o w i n g equat ions can be w r i t t e n : The 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 f o r s e c t i o n ( j , n + l ) : VP. = C3. - C2. HP. . (2 .46) J ,n+1 j j j , n + l The 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 f o r s e c t i o n ( j + 1 , 1 ) : V P . , . , , = C I . , , + C 2 . , , H P . , , ' . (2 .47) j + 1 , 1 j+1 j+1 J+1,1 The c o n t i n u i t y e q u a t i o n : VP. , , A . = V P . , , , A . . . . (2 .48) j , n + l j J+1,1 J+1 The gate e q u a t i o n : 4 0 . __£^°dy state A t Pipe J~ (j/n) Valve-, ( j ,n + l) H o j , n+l -^ •u— • ^-i i i • i i R -5=— X ( c ) V A L V E AT THE DOWNSTREAM END OF A PIPE Pipe J Valve -rJl Pipe J+l (j,n) .(j.ntl) (j + 1,2) At > C" I I C" n N 1 1 1 1 (d) VALVE AT THE JUNCTION OF TWO PIPES FIG. 2.6 4 1 . x V . VP- J.I ° J vST I T - - H P . . , , J » v^AH— J > n + 1 J + 1 . 1 (2 .49) i n which AH — H . . . — H . _ . o o j , n + l o j+1 ,1 From Eq . ( 2 . 4 9 ) , i t f o l l o w s t h a t VP 2 v 2 T V . 2 = 5 1 (HP - HP ) j , n + l AH V j , n + l ; j + l , r By combining Eqs. (2 .46) and (2 .47) w i t h the above equat ion , one o b t a i n s j , n + l - 4 H o | C2. C2 f • S u b s t i t u t i o n of V P , , , . i n terms of VP. from Eq . ( 2 . 4 8 ) , y i e l d s j + 1 , 1 J ,n+1 V P : 2 V 2 T. V . • 03 J C3. ,1 A H o I C2. .1 A - VP. J ,n+l _ _ j 3 ,n+l C2, •J+i " : +i CI C2 j + i (2 .50) Le t C4. J 2 2 T V . C2. AH J o and C4 2 M2 T V . 93 j+1 C 2 . , , AH J 3+I o Then Eq. (2 .50) takes the form VPT . , = C3 . C4. - C4. V P . . , - -TJ— C 4 . . , V P . . , J ,n+1 3 3 3 j , n + l A j + 1 3+I 3>n+l 42. so tha t - 2 A i VPT + (C4. + -r C 4 . . . ) VP. - (C3. C4. + C I . . , C4 . J _ 1 ) = 0. j , n + l 3 A + 1 j + l 3 ,n-. - l j 3 3+1 3+I S o l v i n g t h i s e q u a t i o n f o r VP .^ and n e g l e c t i n g the n e g a t i v e s i g n w i t h the r a d i c a l te rm, one o b t a i n s A . VP. ... = h <-(C4. + - T - J - C 4 . . . ) 3 ,n+ l I 3 A X 3+I + / (C4 . + — C 4 . J , ) 2 + 4(C3. C4. + C l . ^ , C 4 . ^ , ) 3 -i+i J + 1 J J J+1 3+1 1 (2 .51) Now the v a l u e s of HP. , , , H P . , , , and V P . , , , can be determined from Eqs. 3 , n + l ' 3+1,1 3+1,1 H (2 .46) to ( 2 . 4 8 ) . S i m i l a r l y , f o r the f l ow i n the n e g a t i v e d i r e c t i o n , one o b t a i n s A . VP. = h [ (C4 . + - T - J - C 4 . . , ) 3 ,n+ l j A j + 1 j + l ' - * 1 ^ C W 2 - 4 < c 4 j a i + c V i c V i ) ] (2 .52) I f (C4. C3 . + C 4 . , , C I . ) i s g r e a t e r t h a n . + l , the f low i s i n the 3 3 3+1 j normal d i r e c t i o n and Eq . (2 .51) i s u s e d ; o therwise Eq . (2 .52) i s used . SPECIAL CASE: V a l v e or o r i f i c e at an i n t e r m e d i a t e s e c t i o n of a p i p e or at the 4 3 . x . . . Valve (J ,n- I ) 1 , n j j , / ^ 1 ' 0 ( j ! + l , 2 ) Pipe J Pipe J+1 At i i i < ( e ) V A L V E AT JUNCTION OF TWO PIPES HAVING S A M E DIAMETER Pipe J Orif i ce Pipe J+1 ( J ,n-0 IT (1+1,2) t 4 «. j' * As "x * i i I i i 1 '* y e s . ' 1 C \ ! *-( f ) O R I F I C E AT JUNCTION OF TWO PIPES HAVING SAME D I A M E T E R FIG. 2 . 6 44. j u n c t i o n of two p i p e s hav ing the same d i a n e t e r , w a l l t h i c k n e s s and w a l l m a t e r i a l ( F i g . 2 . 6 - e ) : I n t h i s c a s e , A . = A # J 1 , C2. = C 2 . . . anc' C4. = C4. , . : J J + l J J+ l J J + l Hence, Eq. (2 .51) takes the form VP. , = - C4. + / ( C 4 . ) 2 + C 4 . ( C 3 . + C l . j n ) . (2 .53) J ,n+1 J J J J J + l S i m i l a r l y , f o r the f l o w i n the r e v e r s e d i r e c t i o n , Eq . (2.52) reduces to VP. J . = C4. - / ( C 4 . ) 2 - C 4 . ( C 3 . + C l . ^ ) . (2 .54) J »n+1 J J J J J+ l CASE I I I . VALVE AT THE UPSTREAM END OF A P IPE . THE VALVE DISCHARGES INTO  ATMOSPHERE (Fig. 2 . 6 - g ) : The boundary c o n d i t i o n s f o r a v a l v e a t the upstream end of a p ipe can be d e r i v e d by s o l v i n g the f o l l o w i n g two equat ions s i m u l t a n e o u s l y : The n e g a t i v e c h a r a c t e r i s t i c equat ion f o r s e c t i o n ( j , l ) : VP. = C I . + C2. HP.' . J , l J J J , l The gate equat ion f o r the v a l v e : V T vp. , = — = 3 — * /HPT7 • (2-5 5) o j , l Squar ing b o t h s i d e s of Eq. (2 .55) .and making the s u b s t i t u t i o n 4 5 . one o b t a i n s V P 2 = C2. C4. HP. , 3 , 1 3 3 3 A S u b s t i t u t i o n of the v a l u e of HP .^ ^ i n t o the above equat ion y i e l d s V P 2 - - C4. VP. , + C I . C4.•= 0 . 3,! J 3,1 J J S o l v i n g the above equat ion f o r VP. 1 and n e g l e c t i n g the n e g a t i v e s i g n w i t h the r a d i c a l te rm, one o b t a i n s VP. , = h (C4. + /z^T^T~ClTch'.) . (2 .56) 3 a 1 1 3 J 3 Now, the v a l u e of HP. , can be determined from Eq. ( 2 . 5 5 ) . J>1 CASE IV. VALVE AT A RESERVOIR ( F i g . 2 . 6 - h ) : The f o l l o w i n g two equat ions are s o l v e d to develop the boundary c o n d i t i o n s f o r the v a l v e at the r e s e r v o i r : The n e g a t i v e c h a r a c t e r i s t i c equat ion f o r s e c t i o n ( j , l ) : VP. '= C I . + C2. HP. . . 3 »1 J 3 J>1 The gate e q u a t i o n f o r the v a l v e : x V . VP. , =' fR - HP. n 3 , 1 ^ — res j , l ° J , 1 S o l v i n g the above two equat ions s i m u l t a n e o u s l y , making the s u b s t i t u t i o n C4. = 2 v2 x V . Q3 3 ,H . .. C2. 03 , 1 3 Check valve fully closed on flow reversal Pipe J -*=- steady [ f low L sui (j,2) rge relief valve or surge anticipator (g) V A L V E AT AN U P S T R E A M END OF A PIPE (h) V A L V E AT A RESERVOIR FIG. 2 . 6 4 7 . and n e g l e c t i n g the n e g a t i v e s i g n w i t h the r a d i c a l te rm, one o b t a i n s VP = \ \ - C4. + v /C42 + 4 . C 4 . (C2. H + C l . ) f . (2 .57) 3,1 3 3 3 3 res 3 ' Proceed ing i n a s i m i l a r manner, f o r f l o w i n the r e v e r s e d i r e c t i o n , VP. . = h 3 , 1 C4. + / C 4 2 - 4 . C 4 . (C2. H + C l . ) r ; (2 .58) J 3 3 3 r e s j 2 .7 THE SURGE TANKS: The boundary c o n d i t i o n s f o r an o r i f i c e surge tank are developed i n t h i s s e c t i o n . The r e s u l t s c a n , however, be used f o r a s imple surge tank by t a k i n g the o r i f i c e r e s i s t a n c e , H ^ ^ , e q u a l to z e r o . I t i s assumed that the t ime s teps are s m a l l and the v e l o c i t y of water i n the p i p e changes a t the end of each t ime s t e p , as shown i n F i g . 2 . 7 - a . The f o l l o w i n g two cases a re c o n s i d e r e d : CASE I. TANK OF LARGE CROSS-SECTIONAL AREA ( F i g . 2 . 7 - b ) : The c r o s s - s e c t i o n a l a rea of the tank i s assumed to be very l a r g e as compared to t h a t of the d i s c h a r g e p i p e . The e f f e c t s of the waterhammer waves i n the tank c a n , t h e r e f o r e , be n e g l e c t e d . The d i s c h a r g e , q , through the o r i f i c e of the tank i s cons idered p o s -i t i v e when water d i s c h a r g e s out of the tank . By d e s i g n a t i n g the c r o s s - s e c t i o n a l 48. area of the surge tank by and other v a r i a b l e s as shown i n F i g . 2 . 7 - b , f o r the t a n k , the f o l l o w i n g equat ions can be w r i t t e n : A (Z - ZP) = qAt . s HP. = 2P - H j , n + l o r f H •H "orf 2 q^ o r f o 2 q > (2 .59) (2 .60) (2 .61) i n which ^ o v £ Q 1 S the t h r o t t l i n g l o s s i n f e e t co r respond ing to a d i s c h a r g e of q o c u . f t / s e c . and H q i s the t h r o t t l i n g l o s s i n f e e t cor responding to a d i s -charge of q c u . f t / s e c . For the t r a n s i e n t s t a t e c o n d i t i o n s at the j u n c t i o n of the tank and the d i s c h a r g e p i p e , the f o l l o w i n g equat ions can be w r i t t e n : The e q u a t i o n f o r common p ressure head: H P . , , = HP. j + 1 , 1 J ,n+1 The c o n t i n u i t y e q u a t i o n : V. - A . + q = V. . , , A . . j , n + l j J+1,1 J+I The p o s i t i v e c h a r a c t e r i s t i c equat ion f o r s e c t i o n ( j , n + l ) VP. , = C3. - C2. HP. ' . The 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 f o r s e c t i o n ( j+1,1) V P . , , , = C I . , , + C 2 . , , H P . , , , . j + 1 , 1 j+1 j+1 j + 1 , 1 I t f o l l o w s from Eqs . (2 .61) and (2 .63) that H H o r f o (2 .62) (2 .63) (2 .64) (2 .65) o r f 2 (v. . , , A.„, - V. ., A.) j + 1 , 1 j+1 J ,n+1 2 j + 1 , 1 j+1 J ,n+1 j (2.66) 4 9 . o o > Assumed Actual A t 1) VELOCITY CHANGES ' IN THE DISCHARGE PIPE H o Z Z P j Orifice Steady state water surface Transient state W. S. at the beginning of time step •Transient state W.S. at the end of time step Pipe J ' ' Pipe J+1 (j,n) (i,n+l) (j+1,1) ( j -H .2 ) (b) SURGE TANK AT JUNCTION OF TWO P I P E S FIG. 2 . 7 50. E l i m i n a t i o n of q from Eqs . (2 .59) and (2.S3) y i e l d s (2 .67) S u b s t i t u t i n g these v a l u e s o f H ,and ZP i n t o Eq . (2 .60) one o b t a i n s ° or f and H o r f o 5 ^ + 1 . 1 A j + 1 " Vj,n + 1 V V j + l , l Aj+ 1 " V 3 , n + l A i L e t •> _ o r f o "5 " 2 (2 .68) At '6 A s C, = and C , = V . t 1 , A . , , - V . , , A . 7 J+1,1 J + l J ,n+1 j Then, ZP = Z - c 6 C ? (2 .69) HP. = ZP - C, J 5 6 (2 .70) The v a l u e s of H P . , , , , VP. ^ and V P . , , . can now be determined from j + 1 , 1 J ,n+1 J+1,1 Eqs. ( 2 . 6 2 ) , (2 .64) and (2 .65) 5 1 . SPECIAL CASE: The surge tank l o c a t e d ad jacent to the pump. The check v a l v e  c l o s e s s i m u l t a n e o u s l y w i t h pump f a i l u r e ; The check v a l v e c l o s e s i n s t a n t l y when the f l ow i n the d i s c h a r g e p i p e r e v e r s e s . The assumption that the check v a l v e c l o s e s s i m u l t a n e o u s l y w i t h the pump f a i l u r e i s f u l l y j u s t i f i e d f o r a pump of s m a l l moment of i n e r t i a . In t h i s c a s e , upon pump f a i l u r e , the f l o w r e v e r s e s a f t e r a shor t i n t e r v a l of t ime. Wi th t h i s assumpt ion , the pump c h a r a c t e r i s t i c s are e l i m i n a t e d from the w a t e r -hammer computat ions . In t h i s c a s e , V. , . = 0 and A . = 0 . J ,n+1 3 Hence, Eqs . (2 .67) and (2 .68) become ZP = Z - V . . . , , A. , A g j + 1 , 1 J+1 and 'At „ A H o r f o ,2 V. Le t ^ j + 1 , 1 c A j + 1 , 1 A j + 1 2 A j + 1 j + 1 , 1 " j + 1 , 1 C„ = 4^ A (2 .71) and 8 A j+1 ' s J = " o r f o 2 L 9 2 A j + 1 • q M o By u s i n g these r e l a t i o n s , the equat ions f o r ZP and H P , -. can be w r i t t e n as Z P " Z " C 8 V j + l , l > ( 2 ' 7 2 ) 52. and H V l , l = Z P - S V j + l , l Vj+l,l (2 .73) Now the v a l u e of V P , , can be determined from Eq . (2 .65) CASE I I . A SURGE TANK OF SMALL CROSS-SECTIONAL AREA ( F i g . 2 . 7 - d ) Because of the s m a l l e r c r o s s - s e c t i o n a l a rea of the tank as com-pared to tha t of the d i s c h a r g e p i p e , the waterhammer waves i n the tank cannot be n e g l e c t e d . The surge tank i s , t h e r e f o r e , t r e a t e d as a p i p e hav ing a f r e e water s u r f a c e . C o n s i d e r i n g the f l o w d i r e c t i o n s shown i n F i g . 2 . 7 - d as p o s i t i v e , the f o l l o w i n g equat ions can be w r i t t e n f o r the t r a n s i e n t s t a t e c o n d i t i o n s a t the j u n c t i o n of the tank and the d i s c h a r g e p i p e : The c o n t i n u i t y e q u a t i o n : VP. A . = V P . , , . A . , , +• V P . , 0 , A . , 0 . (2.74) j , n + l j J+1,1 J + l J+2,1 J+2 The e q u a t i o n f o r the common p r e s s u r e head: H P J , n + l = H P j + l , l = H P J + 2 , l ' ( 2 - 7 5 ) ' ( 2 - 7 6 ) The p o s i t i v e c h a r a c t e r i s t i c s e q u a t i o n f o r - s e c t i o n ( j , n + l ) : VP. , , = C3 . - C2. HP. , • . (2 .77) J ,n+1 3 3 3 ,n+l The 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 f o r s e c t i o n ( j + 1 , 1 ) : .'V.i-'V^W.i • ( 2 - 7 8 ) The 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 f o r s e c t i o n ( j + 2 , 1 ) : V P . ^ „ . = C l . , 0 + C 2 . , 0 H P . , „ . (2 .79) J+2,1 J+2 j+2 J+2,1 53. I t f o l l o w s from Eqs . (2 .74) to (2 .79) that (C3. - C2. HP. , , ) A . = ( C I . , , + C 2 . . . HP. ) A . . . J J J ,n+l j j+1 3+1 3 ,n+l 3+I + (CI 0 + C 2 . , 0 HP. ) A . ._ 3+2 3+2 3 ,n+l 3+2 . S i m p l i f i c a t i o n and rearrangement of the terms of the above equat ion y i e l d s (C2. A . + C 2 . . . A . , , + C 2 . , _ A . . . ) HP. 3 3 3+I j+1 3+2 3+2 J ,n+1 = C3.. A . - C I . , . A . . . - C 1 . . . ' A . . 0 J J J+1 J+1 J+2 J+2 so tha t ' -(C3. A . - C I . , , A . . , - C I . A . ) H P = J J J + 1 J + 1 J + 2 J + 2 ( 2 80) ;j,n+l (C2. A . + C 2 . . . A . . , + C 2 . , , A . . , ) * ^ J 3 3 J+1 J+1 J + 2 J + 2 Now the v a l u e s of H p . , , K P . , 0 1 S VP. ,.. , V P . , , .. and V P . , „ .. can be deter -J+1,1 J + 2 , l J . n + 1 ' J+1,1 J + 2 , l mined from Eqs . (2 .75) to ( 2 . 7 9 ) . Moreover , a t the f r e e water s u r f a c e of the tank H Pj+ 2 , n + l = Z P > . ( 2 ' 8 1 ) ^j+Z.n+l = C 3j+ 2 ~ C 2j + 2 H Pj+ 2 , n + l ( 2 ' 8 2 ) and Hence (ZP - Z) A . . , = (VP. A. - V P . . , , A . , , ) At j+2 J ,n+1 j J+1,1 J+1 ZP = Z + 4^— (VP. A . - V P . . . , A . . , ) . (2 .83) j+2 J' 3 2 1 + 1 , 1 J + 1 The v a l u e s of H P . , . ' , , and V P . . . , , can be determined from Eqs . j+2 ,n+ l j+2 ,n+ l (2 .81) and ( 2 . 8 2 ) . 54 Steady state water surface Transient state W.S. at the beginning of time step Transient state W.S. at the end of time step -^ Ttcheck I Pipe J+l ' ~~ valve (J + '>') U+1,2) (c) SURGE T A N K N E A R THE P U M P Z P -Transient state W.S. at the end of time step •Transient state W.S. at the beginning of time step Initial steady state water surface . — j + 2,1 P i p e J (i.n+l) (i+U) P J P e J + ' . (d ) SURGE TANK OF SMALL C R O S S - S E C T I O N A L A R E A FIG. 2 . 7 55. SPECIAL CASE. A surge tank of v a r i a b l e G r o s s - s e c t i o n a l area ( F i g . 2 . 7 - e ) In t h i s case Eqs . (2 .81) and (2.83, become H P j + 2 , n + l = L j + 2 + Z P ( 2 - 8 4 > and ZP = Z + VT> , n A . , 0 4^  , (2 .85) j+2,n+l j+2 As i n which L j + 2 I s the l e n g t h of p ipe j+2. The r e s t of the equat ions are the same as f o r Case I I . 2 .8 THE AIR CHAMBER: The p r e s s u r e and volume of a i r i n the chamber f o l l o w the law H . v . = c o n s t a n t , a i r i n which H and v . a re the a b s o l u t e p r e s s u r e and volume of a i r i n the chamber, a i r For the a d i a b a t i c and i s o t h e r m a l expansion of a i r , the va lues of m are equa l to 1,4 and 1.0 r e s p e c t i v e l y . The o r i f i c e i n the chamber may be s imp le or of the d i f f e r e n t i a l t ype . The d i f f e r e n t i a l type of o r i f i c e t h r o t t l e s the reve rse f l o w of water from the d i s c h a r g e p ipe i n t o the chamber w h i l e very l i t t l e t h r o t t l i n g i s p rov ided f o r the f l o w out of the chamber. I f there i s no o r i f i c e i n the chamber, the t h r o t t l i n g l o s s i s taken equa l to z e r o . I t i s assumed t h a t the t ime i n t e r v a l s are s m a l l and the v e l o c i t y of water i n the d i s c h a r g e p i p e changes a t the end of each t ime i n t e r v a l , as shown i n F i g . 2 . 7 - a . 56 Z P j+2 Pipe J Transient state W.S. at the end of time interval - T rans ient state W.S. at the beginning of time interval S teady state W S. '(j + 2,n+l) Pipe J + 2 .(J^-2,1) pjpe J+| (j.n-M) (j+1,1) (e ) SURGE TANK OF V A R I A B L E C R O S S - S E C T I O N A L A R E A FIG. 2 . 7 57. Th°. f l ow out of the chamber i s cons idered p o s i t i v e . The f o l l o w i n g equat ions can be w r i t t e n f o r the volume of a i r i n the chamber: vP . = (Z - ZP) A , (2 .86) a i r a a ' i n which vP . i s the folume of a i r at the end of the t ime i n t e r v a l , Z and a i r a A rep resent the h e i g h t and the c r o s s - s e c t i o n a l a rea of the chamber, and ZP cL denotes the h e i g h t of water s u r f a c e i n the chamber above the c e n t r e l i n e of the d i s c h a r g e p i p e . HP. , - ZP + 34 + H . j ,n+l o r f < i r " C 1 0 • ( 2 ' 8 7 ) i n which H £ i s the o r i f i c e f r i c t i o n l o s s ( i n f t . . ) co r respond ing to a d i s -charge of q c u . f t / s e c . and C^Q i s a constant g i ven by c i n = H*'(v n . } m , 10 o o a i r i n which H and v , denote the i n i t i a l s teady s t a t e a b s o l u t e p ressure head o o a i r J r and volume of a i r i n the chamber. For the t r a n s i e n t s t a t e c o n d i t i o n s a t the j u n c t i o n of the chamber and the d i s c h a r g e p i p e , the f o l l o w i n g equat ions can be w r i t t e n : The c o n t i n u i t y e q u a t i o n : V . j , , A . j , At = V. j , A . At + (vP . - v . ) . j + 1 , 1 j+1 J ,n+1 j ^ a i r a i r 7 so tha t vP . = v . + C , , At , (2 .88) a i r a i r 11 58. i n which 11 J+1,1 J+ l J ,n+1 j The 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 f o r s e c t i o n " ( j » n + l ) : VP. , = C3. - C2. HP. ' . (2 .89) J .n - t - l j j j , n + l The 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 f o r s e c t i o n ( j + 1 , 1 ) : - . V,i-V t cViV,i • (2-90) Moreover , the p r e s s u r e heads a t the j u n c t i o n of the chamber and d i s c h a r g e p i p e are e q u a l , tha t i s HP. = H P . . . ' . (2 .91) j , n + l J+1,1 The o r i f i c e f r i c t i o n l o s s i s g i v e n by H H . = C . q- , o r f o r f 2 n o r f o 2 'o i n which C - i s the o r i f i c e c o e f f i c i e n t . For a s imple o r i f i c e , C . = 1 . 0 . o r f r o r f For a d i f f e r e n t i a l o r i f i c e : C _ = 1 .0 when water f l ows out of the chamber o r f i . e . when V . A . - V . A . > 0 ; j + 1 , 1 j + l J ,n+1 j C _ = k, when water f l o w s i n t o the chamber i . e . when o r f 1 V . x l A . . , - V . A . < 0 , j + l j + l J ,n+1 j where k^ depends upon the amount of t h r o t t l i n g p rov ided by the o r i f i c e . S u b s t i t u t i n g the v a l u e of q i n the above e q u a t i o n , one o b t a i n s H o r f o H o r f = C o r f ~~2~ ( V j + l , l A j + 1 " V j , n + 1 V " j+1 ,1 " j + l " j , n + l " j | » q o V . , A . - V. so tha t 59. H _ — C _ C.., o r f o r f f 11 11 (2 .92) i n which C f " 2 E l i m i n a t i o n of vP . from Eqs. (2 .86) and (2 .88) y i e l d s a i r (Z - ZP) A = V . + C n At , a a a i r 11 so tha t ZP = Z -v . C. - .At a i r 11 a A A a a (2 .93) S u b s t i t u t i n g the v a l u e s of ZP, vP . andH - i n t o Eq . ( 2 . 8 7 ) , one o b t a i n s o a i r orr n ' ' {HP. v . C At - Z + - 4 ^ + 7 + 34 i , n + l a A ' A J a a o r f f 11 J l l > { v a i r + C l l A t } m " C 1 0 ' S i m p l i f i c a t i o n and rearrangement of the terms of the above equat ion y i e l d s v . + C n At HP. ' = Z - — — — 34 - C r C . C . . j ,n+l a A o r f f 11 '11 \0 (v . + C n 1 At ) a i r 11 m so tha t HP . J » n + l '10 v . + C n i At a i r 11 (v . + C n i A t ) a i r 11 m + (34 - Z ) a + C , C . C . . o r f f 11 J l l 1 (2 .94) 6 0 . o INI O (SI (SI T lj,n) "~ / / / / Steady state water surface __ it Transient state W.S. at the beginning of time step Transient state W.S. at the -end of time step Ori f i ce Pipe J+l ( i+1,2) (a) AIR C H A M B E R AT JUNCTION OF TWO P I P E S o . ISI M N D_ 14 -Steady state water surface -Transient state W.S. at the beginning of the time step -Transient state W.S. at the end of the time step Orif i ce -Pipe - J + l check v a l v e (j + l,2) t - - -S 1 , NEAR THE P U M P F IG. 2 . 8 61. Let and Then C = 34 - Z ch a C . = v . + c. . . At a i r a i r 11 (2 .95) (2 .96) HP '10 j , n + l m a i r a i r . + C + C C, C f 1 A ch o r f f 11 Now the v a l u e s of VP VP, and HP. j , n + l ' " j + 1 , 1 ' " j + 1 , 1 '11 . (2 .97) can be determined from Eqs. (2 .89) to ( 2 . 9 1 ) . The v a l u e of vP . i s computed from the e q u a t i o n : 3.X1T vP . = V . + % a i r a i r A - J - I C V - J - I i + v p - ^ i i ) ~ A.'(V. + VP. ) J+1 J+1,1 J+1,1 J J ,n+1 J , n + r At (2 .98) SPECIAL CASE: The a i r chamber i s near the pump. The check v a l v e c l o s e s  s i m u l t a n e o u s l y w i t h pump f a i l u r e ( F i g . 2 . 8 - b ) : Because of the assumption t h a t the check v a l v e c l o s e s s imultaneous w i t h the pump f a i l u r e , a l l the f low i n the d i s c h a r g e p i p e i s e i t h e r from or i n t o the chamber. Th is assumption e l i m i n a t e s the pump c h a r a c t e r i s t i c s from the waterhammer computat ions . In t h i s c a s e , V. , = 0 and A. = 0 . Hence, Eq . (2 .97) reduces to J » n + l J C HP 10 j + 1 , 1 (C . ) a i r (C , C , C n 1 m o r f f 11 C l l + C c h + A a i r ) , (2 .99) 62. i n which c n = V i Aj+i • Now the v a l u e s of V P . , , , and vP . can be determined from Eqs. j + 1 , 1 a i r ^ (2 .90) and ( 2 . 9 8 ) . 2 .9 THE CENTRIFUGAL PUMP: A 6 21 In t h i s s e c t i o n the d i m e n s i o n l e s s homologous pump c h a r a c t e r i s t i c ' ' a re i n t r o d u c e d . An i t e r a t i o n technique f o r computing the t r a n s i e n t s t a t e r o t a t i o n a l speed, d i s c h a r g e and pumping head of the pump, upon pump f a i l u r e , i s p r e s e n t e d . A pump f a i l u r e r e s u l t s from any of the f o l l o w i n g causes : i ) Power outage or v o l t a g e d i p . i i ) Pump motor o v e r l o a d . ; i i i ) Emergency s t o p . 6 21 The f o l l o w i n g parameters ' a re used to s t o r e the pump c h a r a c t e r i s t i c s i n the computer memory: For the head: h v h_ ex 2 ^ a ' 2 * v a v and f o r the t o r q u e : 'v — and — 7 T ^ — 2 a 2 v ' a v i n which 6 3 . i n which H, Q, N and T are the t r a n s i e n t s t a t e head, d i s c h a r g e , r o t a t i o n a l speed and torque of the pump,. The r a t e d v a l u e s a re denoted by the s u b s c r i p t R. 7 The v a l u e s of the r a t i o s R n (equal tc —) and R„ (equal to — ) 1 c. 2. - v h h 8 8 range between +1 and -1. The v a l u e s of —^ , » 2 a n d 2 a r e s t o r e d i n t n e a v a v computer memory. For any v a l u e of a and v , v a l u e s of h and 8 are determined by p a r a b o l i c i n t e r p o l a t i o n . I f the s i g n convent ion of Table No. 1 f o r a and v i s kept i n mind and the r a t i o (R^ or ) which has a v a l u e l e s s than or equa l to u n i t y i s u s e d , on ly one c h a r a c t e r i s t i c curve and one p o i n t on i t are found . TABLE NO. 1 Zone of Operat ion S ign of R a t i o 1 1.0 Curve to be used a V v/ct a/v Normal + + X HAN** , BAN + + X HVN , BVN Energy d i s s i p a t i o n + - X HAD , BAD + - X HVD , BVD Turb ine o p e r a t i o n - - X HAT , BAT — — X HVT , BVT ** In the n o t a t i o n 2 1 , H r e f e r s to a head r a t i o , A means d i v i s i o n by a , and N r e p r e s e n t s the normal zone. V means d i v i s i o n by v 2 , D i s the zone of energy d i s s i p a t i o n , and T i s the zone of t u r b i n e o p e r a t i o n . 64. COMPUTATION OF THE TRANSIENT CONDITIONS CAUSED BY PUMP FAILURE: a) The r a t i o n a l speed of the pump: Upon pump f a i l u r e , an unbalanced t o r q u e , T, t h a t depends upon speed , w, and the d i s c h a r g e r a t i o , v , i s a p p l i e d to the r o t a t i n g p a r t s . I f i s the r a t e of speed change, then T = - I j± , (2 .100) i n which I i s the moment of i n e r t i a of the moving p a r t s . Eq . (2 .100) can be w r i t t e n i n the d i f f e r e n c e form as. TAt Ato = j - , i n which Aw i s the change i n the r o t a t i o n a l speed i n r a d i a n s / s e c d u r i n g a t ime i n t e r v a l of At seconds. The v a l u e s of the average d i s c h a r g e r a t i o , v\ , and the average speed r a t i o , a . , f o r the i t n t ime s tep may be es t imated by the f o l l o w i n g e q u a t i o n s : l and v f 0 ) = v . . + h Av. . (2.101) i l - l l - l a f 0 ) = a . . + h A a . , , (2 .102) l l - l l - l i n which Av^ ^ and Aa^_^ a r e t n e changes i n v and a d u r i n g the ( i - 1 ) t ime s t e p . The s u p e r s c r i p t r e f e r s to the number of i t e r a t i o n s per formed. For example, denotes the i n i t i a l es t imated d i s c h a r g e r a t i o f o r the i f c ^ t ime i n t e r v a l . The average d i s c h a r g e and speed r a t i o s f o r the i ^ t ime step a f t e r n i t e r a t i o n s are represented by v f n ^ and a^ n"* . 6 5 . As o u t l i n e d above, f o r these values of v ^ ' and a ^ \ torque 1 1 r a t i o , i s determined u s i n g p a r a b o l i c i n t e r p o l a t i o n . Now the change i n the r o t a t i o n a l speed of the pump, A a ^ ^ , can be computed from the fo rmula A (0) A a ; i 30 T R A t ~ T V r S ( 0 ) 3 i so tha t A a f 0 ) = C p 8 ? 0 ) l P a (2.103) i n which 30T R At (2 .104) Hence - a . , l l - l + % Aa (0) i i n which ex. , i s the r o t a t i o n a l speed at the end of ( i - 1 ) t ime i n t e r v a l . I f l - l - ( 1 ) - ( 0 ) a . - a . l l i n which i s the s p e c i f i e d t o l e r a n c e , then the above process i s repeated c o n s i d e r i n g as the i n i t i a l e s t i m a t e d v a l u e . Suppose that a f t e r n i t e r -a t i o n s - ( n ) - ( n - 1 ) l l Then the speed r a t i o , a ^ , a t the end of the i 1 " ^ t ime i n t e r v a l i s g i v e n by the equat ion 66. a . = a . , + A a f n ) . i (2 .105) 1 l - l I b) THE PUMPING HEAD AND DISCHARGE: The v e l o c i t y , V P ^ \ i n the d i s c h a r g e p i p e at the end of the i t n t ime s tep i s es t imated by the equat ion VP?°^ = V. . + AV. . l l - l l - l i n which V. , i s the v e l o c i t y a t the end of ( i - 1 ) t ime i n t e r v a l and AV. . l - l J . l - l denotes the change i n the v e l o c i t y d u r i n g the ( i - 1 ) time i n t e r v a l . The d i s c h a r g e r a t i o , v f ^ , i s g i ven by (0) ZV i n which A i s the c r o s s - s e c t i o n a l a rea of the d i s c h a r g e p i p e . By s u b s t i t u t i n g C = —— i n the above e q u a t i o n , one o b t a i n s pm Q R v < ° > » . C VP. . l pm l For t h i s v a l u e of v f ^ and f o r g i ven by Eq . ( 2 . 1 0 5 ) , i s i n t e r -p o l a t e d from the pump c h a r a c t e r i s t i c c u r v e s . The v a l u e of n o w be ing known, the v a l u e of H P ^ ^ i s computed by the equat ion <> X N . i n which i s the r a t e d head of the pump. The n e g a t i v e c h a r a c t e r i s t i c equat ion f o r the upstream end of d i s c h a r g e p i p e i s now used to determine V P ^ ^ , i . e . 67. V p f 1 ) = CI + C2 . H p f 0 ) , i n which CI and C2 are cons tants d e f i n e d by Eqs. (1 .45) and (1 .46) I f (2 .106) V P a ) - VP<°> > e 2 ' , ( D i n which e^ l s the s p e c i f i e d t o l e r a n c e , then c o n s i d e r i n g VP^ as the i n i t i a l es t imated v a l u e , the above process i s r e p e a t e d . I f a f t e r the n i t e r a t i o n , th then w ( n ) _ vptn-l) i i vp. = v p f n ) = "2 ' (2 .107) The f l ow c h a r t , g i ven i n F i g . 2 . 9 - b , i l l u s t r a t e s the procedure f o r computing the t r a n s i e n t s t a t e c o n d i t i o n s caused by a pump f a i l u r e The t r a n s i e n t s t a t e c o n d i t i o n s f o r the f o l l o w i n g cases are c o n s i d e r e d : ( i ) Check v a l v e c l o s e s i n s t a n t l y upon f l o w r e v e r s a l : The f l ow i n the d i s c h a r g e p i p e decreases as the speed of the pump decreases upon pump f a i l u r e . However, the speed of the pump reduces r a p i d l y to a p o i n t where no water can be d e l i v e r e d a g a i n s t the e x i s t i n g head. I f water i s pumped to a r e s e r v o i r , the f l o w r e v e r s e s a l though the pump may s t i l l be r o t a t i n g i n the normal d i r e c t i o n . The check v a l v e c l o s e s i n s t a n t l y upon f l ow r e v e r s a l . Th is prevents r e v e r s e f l ow through the pump. Thus, the t r a n s i e n t s t a t e can be d i v i d e d i n t o two phases : 68. (a) The f l ow i n the d i s c h a r g e p ipe ad jacent to the pump i s i n the p o s i t -i v e d i r e c t i o n . Dur ing t h i s p e r i o d , the puivp works i n the normal zone. The pump speed , head and d i s c h a r g e are determined as o u t l i n e d above. (b) The f l ow r e v e r s e s i n the d i s c h a r g e p.'pe and the check v a l v e c l o s e s i n s t a n t l y . Hence VP. . = 0 . (2.108) The p r e s s u r e head i s determined from the f o l l o w i n g 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 f o r the s e c t i o n ( j , l ) : VP. , = C I . + C2. HP. , By s u b s t i t u t i n g VP. , = 0 , one o b t a i n s H P j , i = - c T [ ' ( 2 ' 1 0 9 ) The procedure f o r computing the t r a n s i e n t s t a t e c o n d i t i o n s i s i l l u s t r a t e d by means of a f l ow c h a r t (See F i g . 2 . 9 - a ) ( i i ) No check v a l v e . Flow r e v e r s e s through the pump: Th is i s s i m i l a r to case ( i ) except tha t r e v e r s e f l o w through the pump i s a l l o w e d . Thus, the pump operates i n a l l the th ree zones of o p e r a t i o n , i . e . n o r m a l , energy d i s s i p a t i o n and t u r b i n e o p e r a t i o n . The f l ow c h a r t g i ven i n F i g . 2 . 9 - b i l l u s t r a t e s the procedure f o r de te rmin ing the t r a n s i e n t s t a t e c o n d i t i o n s . 69 START 1 a , = Oi S 10 Determine /3j from BAN A a ; = C p - £ , a i r a i - l + 2 " A a i > a, =ai_| + A a t i VP= VP O |VP = V i - | ^ i -u i = C p m V P 1 ,R = Determine h from HVN H P = h • H, V P •• C ( + C 2 - HP VP = . VP ' END FLOW CHART FOR PUMP FAl LURE , CHECK VALVE C L O S E S UPON FLOW R E V E R S A L FI G. 2 . 9 ( a ) R = Determine h from HAN Notes : C pm" Q RAT For expression for C p see Eqn ( 2.104) 70 CaMec! from MAIN Prog n Compjte V , H s ,C , V ; =V i_ r - A V M V, --( "pnrVi MM NN = 0 -o R -v< a . .. ; C A L L E D FROM SUB PUMP N - I R I • I M=6 1 |M= 5 ] |M- 4 M- 3 .,--2 [MV) M = I <^ "RETUR^ > SUBROUTINE MVAL YES R ; R " 0. '-BP - 2 -A a 7 ^ p a,' = Q : . , + - — A CI : 1*1 2 ' -a.U TOLER L > • ^ C a l l P A R A B B E Z V P P - V P I af^a V P P - V * A V , R= U , /QP, C A L L \ M V A L K K K = M / 2 C A L L P A R A B H H P I - h * 2 H » a p . R A T 1 ^ = B P * a/H' 6. VPj = HP, = VPI H P 1 RETURN CAL L f 5A:R A B H HPI - h - i H - a p V R 2 FLOW CHART FOR PUMP FAILURE , FLOW R E V E R S E S THROUGH THE P U M P FIG 2. 9 ( b) C A L L E D F R O M A SUB PUMP J M- X /AX A V = M 6 • ( X - A M » Ax)/&X B V T B A T 3 V O B A D B V N B A N or or or or Or or ( H V T ) ( H A T ) ( H V D ) ( H A D ) ( H V N ) ( H A N ) t t t t t t <^ R E n j R N > S U B R O U T I N E P A R A B B ( o r P A R 4 B H I N o t e s . 0 O u t p u t f r o m s u b M V A L i s v a l u « o f M (ii ) " " P A R A B i s B P ( i i , ) " " P A R A H i s h . ( i v ) F o r e x p r e s s i o n for C p o n d C p m s e e s e c t i o n 2-9 I 7 1 . ( i i i ) Upon f l e w r e v e r s a l the check Vclve c l o s e s i n s t a n t l y and the  r e l i e f v a l v e opens g r a d u a l l y ( F i ^ . 2 . 6 - g ) Up to the i n s t a n t of f l ow r e v e r s a l , computations are performed s i m i l a r to case ( i ) . A f t e r f l ow r e v e r s a l , due to the sudden c l o s u r e of the check v a l v e , the pump c h a r a c t e r i s t i c s are e l i m i n a t e d from the waterhammer computat ions . Now the r e l i e f v a l v e s t a r t s to open and the upstream end i s ana lyzed as a v a l v e . For t h i s purpose , the boundary c o n d i t i o n s developed i n s e c t i o n 2 . 6 , case I I I a re used. ( i v ) Upon f l ow r e v e r s a l , check v a l v e f a i l s to c l o s e , the r e l i e f v a l v e  opens at a r a p i d r a t e but c l o s e s s l o w l y a f te rwards ( F i g . 2 . 9 - c ) : The v e l o c i t y and d i s c h a r g e are cons idered p o s i t i v e as shown i n F i g . 2 . 9 - c . For the t r a n s i e n t s t a t e c o n d i t i o n s a t the j u n c t i o n the f o l l o w i n g equat ions can b e . w r i t t e n : The c o n t i n u i t y e q u a t i o n : QP = Q , . + Q . , (2 .110) pump v a l v e P x P e i n which QP , Q , and Q . a re the d i s c h a r g e s through the pump, r e l i e f pump ^va lve p i p e v a l v e and d i s c h a r g e p i p e . The 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 f o r s e c t i o n ( j , l ) : VP. . = C I . + C2. HP. . . (2 .111) J , l J 3 J , l The gate equat ion f o r the r e l i e f v a l v e : Pipe J+1 Relief valve Pump l&nj-»i,l) Pipe J Check valve CHECK V A L V E FAILS TO C L O S E U P O N - F L O W R E V E R S A L R E L I E F V A L V E O P E N S G R A D U A L L Y F I G . 2 . 9 73. Q T. QP . = — — SEP.,, , v a l v e jg- J+1,1 o so tha t i n which QP , = C x • H P . . , , , (2 .112) ^ v a l v e v J+1,1 • c v /r o and Q Q = d i s c h a r g e through the v a l v e under a p ressure head of H . The e q u a t i o n f o r the common p ressure head i s HP. . = H P . , . . . (2 .113) j , l J+1,1 Fur thermore , QP , = VP. . A p i p e j , l j i n which i s the c r o s s - s e c t i o n a l a rea of the d i s c h a r g e p i p e . On the b a s i s of Eqs . (2 .111) and ( 2 . 1 1 3 ) , t h i s equat ion takes the form QP . = (C I . + C2. H P . , . , ) A . . (2 .114) P i p e J J J+1,1 j From Eqs. ( 2 . 1 1 0 ) , (2 .112) and ( 2 . 1 1 4 ) , i t f o l l o w s tha t QP = C T /HP. n . + (C I . + C2. HP. ,' , ) A . . (2 .116) x pump v J+1,1 J J J+1,1 J Eq. (2 .116) cannot be so l ved e x p l i c i t l y because QP and H P . , , , . r J ' ^ pump j + 1 , 1 depend upon the pump c h a r a c t e r i s t i c s and T depends upon x ^ t r e l a t i o n s h i p . Th is i s s o l v e d by an i t e r a t i v e t e c h n i q u e , which i s i l l u s t r a t e d by means of a f low char t (See F i g . 2 . 9 - d ) . 7U ^ R O M MAIN PROG^ j ' C o • n p u t e V H , C 5- aj_,f MM : 0 NN = 0 R - L>; t t . . . C a l l M V A L F R O M SUB P U M P N - | R | -t- 1 | M ^ 6 M = 5 | M = 3 | M = 2 M = 1 < RETUR 5> S U B R O U T I N E M V A L I /3, : B P * a , 2 • A a - c C a l l P A R A B B ^ > I l - l i-l R - u /aP. i i I / Call MVAL > \ • ^ C ALL PARABhT^; < ^ C A L L P A R A B H y HPI =h*H R A T #aP | 2 , 2 HPI = h*H ' jK d!P / R R A T I Q P I = ( C l r C 2 « H P i ) » A 2 + C V * r # v / H P i aP. - a. + - A w i i - l 2 i - i —o Uj - QPI / Q R A T P R I N T I T E R A T I O N S F A I L E D T O L E R 3 > V P = C , + C . H P , H P- = H P, v\ z u'l • c RETURN B V T o r ( H V T T-F R O M S U B . P U M P 0 M - X / i X A M = M 8 = ( X - A M * A X ) / A X \ ^ Yes 0 / * M = M -M = 2 • < . - I B A T B V D B A D B V N o r o r o r o r ( H A T ) (l-IVO ) ( H A D ) ( H V N ) t , t . . t \ B A N or ( H A N ) <C R E T U R N > S U B R P A R A B B ( o r P A R A B H ) Notes : I) Output f rom subr. MVAL is value of M ii) Output from subr PARABB i s BP. H I ) Output from subr. P A R A B H is h. iv) For exp ress ions for C a C see sect ion 2.9-P v (d) FLOW CHART FOR PUMP F A I L U R E . Relief valve opens gradual ly , Check valve fails to close ! FIG. 2 . 9 CHAPTER I I I DESIGNATION OF PIPES 3 . 1 DESCRIPTION OF THE IRRIGATION PIPING SYSTEM: An i r r i g a t i o n p i p i n g system c o n s i s t s of the m a i n , the b ranches , the l a t e r a l s and the d i s t r i b u t o r s ( F i g . 3 . 2 ) . Water i s pumped from the main i n t o the b ranches , from the branches i n t o the l a t e r a l s and from the l a t e r a l s i n t o the d i s t r i b u t o r s . The s p r i n k l e r s are l o c a t e d on the d i s -t r i b u t o r s . In the p i p i n g system under c o n s i d e r a t i o n , a l l the l a t e r a l s and d i s t r i b u t o r s have the same diameter and the d i s t a n c e s between any two c o n s e c u t i v e s p r i n k l e r s on a d i s t r i b u t o r are e q u a l . The l a t e r a l s and the d i s t r i b u t o r s are assumed to be h o r i z o n t a l . The branch p ipes may be i n c l i n e d . An a i r chamber or a surge tank may be p rov ided to keep the w a t e r -hammer p r e s s u r e s , caused by pump f a i l u r e , w i t h i n an a l l o w a b l e range. A number of v a l v e s a re i n s t a l l e d to cut o f f the water supply to a c e r t a i n p o r t i o n of the sys tem, i f so d e s i r e d . Surge s u p p r e s s o r s , r e l i e f v a l v e s , surge a n t i c i -p a t o r s , b y - p a s s v a l v e s and p ressure reducers may be present i n the system. 3 .2 DESIGNATION OF PIPES: ' , A new method i s dev ised to des ignate the p i p e s and p i p e s e c t i o n s by th ree d i m e n s i o n a l and two d imens iona l s u b s c r i p t e d v a r i a b l e s . Th is method i s not on ly h e l p f u l i n d e s i g n a t i n g the p ipes and s t o r i n g the input data i n the computer, but a l s o makes i t p o s s i b l e to ana lyze a l l the boundary p o i n t s of the.same type by a d o - l o o p . Three d i m e n s i o n a l s u b s c r i p t e d v a r i a b l e s (maximum a l lowed on the IBM 7040/44) are used to rep resent the steady s t a t e and t r a n s i e n t s t a t e c o n d i t i o n s at d i f f e r e n t p i p e s e c t i o n s . The j u n c t i o n s of the branch and l a t e r a l s , and of the l a t e r a l s and d i s t r i b u t o r s , a re numbered i n an ascending order i n the steady s t a t e f l ow d i r e c t i o n ( F i g . 3 . 2 - a ) . I f two or more p ipes of the same type meet at a j o i n t , they are numbered i n the a n t i - c l o c k w i s e d i r e c t i o n (see F i g . 2 . 4 - a ) . The upstream end (accord ing to the steady s t a t e f l ow d i r e c t i o n ) of a p i p e i s des ignated as s e c t i o n 1 . The l e t t e r s ' B ' , ' L ' and ' D ' a t the end of the names of the v a r i a b l e s r e f e r to the b r a n c h , the l a t e r a l and the d i s t r i b u t o r . For example, DB, DL and DD r e p r e s e n t the d iameter of the b r a n c h , l a t e r a l and d i s t r i b u t o r . I f an appurtenance, to be cons idered as a boundary d u r i n g the t r a n -s i e n t s t a t e c o n d i t i o n s , i s l o c a t e d a t an i n t e r m e d i a t e s e c t i o n of a p i p e , then the s e c t i o n on each s i d e of the boundary i s des ignated s e p a r a t e l y . For example, i f the s e c t i o n on the upstream s i d e of the appurtenance i s des ignated by , then the s e c t i o n on the downstream s i d e i s des ignated by m+1 (see F i g . 2 . 5 - b ) . The procedure f o r d e s i g n a t i o n of d i f f e r e n t p ipes i s as f o l l o w s : ( i ) B ranches : Two d i m e n s i o n a l and three d i m e n s i o n a l s u b s c r i p t e d v a r i a b l e s are used 77. , LB(I,2) LB(i,3) LB(I ,4) . LB(I,5) H •  •+= : H*- . f - — ^ C J © ro Distributor 7 0 Branch no. I a _J (a ) DESIGNATION OF B R A N C H P I P E S I t h BRANCH 2 -3 4--5 (I .J .2K) (I.J.2K+I) <5> I(J,2£) -KJ .2 ,9) t Distributor No. I Distributor No.2 Distributor No. 3 I (J ,K,2m) J Distributor No. K I( J , K , 2 m +1 ) (b) DESIGNATION OF L A T E R A L S AND DISTRIBUTORS F IG . 3 . 2 Note 1 Numbers in circles represent the number of junction. 78. to des ignate a branch p i p e and a s e c t i o n on i t . For example, D B ( I , J ) denotes the d iameter c f the J t h p i p e of the I1-*1 b r a n c h , and V B ( I , J , K ) r e p r e s e n t s the v e l o c i t y at the IC*"*1 s e c t i o n of the J*"*1 p i p e of the I*-*1 b r a n c h . In other words , I i s the number of the j u n c t i o n of the main and the branch and J i s the number of the next j u n c t i o n of the branch and the l a t e r a l ( F i g . 3 . 2 - a ) . ( i i ) L a t e r a l s : Because a l l the l a t e r a l s i n the system have the same diameter and w a l l t h i c k n e s s , i t i s not necessary to s p e c i f y s e p a r a t e l y p ipe cons tants f o r each l a t e r a l . The f o l l o w i n g example i l l u s t r a t e s how to des ignate the c o n d i t i o n s at a s e c t i o n of a l a t e r a l . The p ressures a t the s e c t i o n s on the upstream and on the downstream s i d e of the K1-*1 j u n c t i o n of the J f c ^ l a t e r a l of the 1^ branch i s des ignated by H L ( I , J , 2 K ) and HL( I , J ,2K+1) r e s p e c t i v e l y ( F i g . 3 . 2 - b ) . ( i i i ) D i s t r i b u t o r s : S ince every d i s t r i b u t o r i n the system has the same d i a m e t e r , w a l l t h i c k n e s s and s p r i n k l e r s p a c i n g , on ly the p i p e s e c t i o n s on the d i s t r i b u t o r s are to be s p e c i f i e d . The f o l l o w i n g example i l l u s t r a t e s how to des ignate the t r a n -s i e n t c o n d i t i o n s a t d i f f e r e n t s e c t i o n s of a d i s t r i b u t o r . The v e l o c i t i e s a t the s e c t i o n s upstream and downstream of the s p r i n k l e r on the d i s t r i b u t o r of the J t h l a t e r a l of the 1 t h branch i s denoted by VDi(J ,K,2*M) and VDi(J ,K,2*M+l) r e s p e c t i v e l y ( F i g . 3 . 2 - b ) . I f a p i p i n g system of on ly one branch i s to be ana lyzed the s u b s c r i p t i may be d e l e t e d . CHAPTER IV DESCRIPTION OF THE PROGRAMME The computer programme comprises of the main programme and a number of s u b r o u t i n e s . I t i s d e s c r i b e d by d i s c u s s i n g the major f u n c t i o n s of the main programme and the i n d i v i d u a l s u b r o u t i n e s . A FORTRAN IV programme to study the t r a n s i e n t response of the i r r i g a t i o n p i p e system shown i n F i g . B-2 (a) i s p r e -sented i n Appendix C; the f l o w c h a r t f o r the programme i s g i ven i n F i g . 4 . 1 - b . A number of s u b r o u t i n e s used to study the t r a n s i e n t s a t d i f f e r e n t appurtenances i n a p i p e system, are presented i n Appendix D. 4 . 1 THE MAIN PROGRAMME: The main programme serves p r i m a r i l y as a dev ice f o r l i n k i n g the v a r i o u s s u b r o u t i n e s . I t performs c e r t a i n book -keep ing and t e s t computations a t the beg inn ing and a t the end of each t ime i n t e r v a l . I t s p r i n c i p a l f u n c t i o n s may be l i s t e d as f o l l o w s : ( i ) S p e c i f i c a t i o n of the s to rage l o c a t i o n s f o r the s u b s c r i p t e d v a r i a b l e s (Dimension s t a t e m e n t ) . ( i i ) A l l o c a t i o n of the same s torage l o c a t i o n s f o r the v a r i a b l e s common to the main programme and to the s u b r o u t i n e s (Common s t a t e m e n t ) . ( i i i ) P r i n t i n g the steady s t a t e v a l u e s . START "CO. Read Data Compute.At a Nj Notes : U) The subscript k denotes *he section k on the pipe i,j ( i i ) MM = no. of time steps afte-which values are to be printed ( i i i ) Tj_as| is the time upto which transient computations are to be performed Compute coeff. a consts.for all pipes Calculate steady state conditions T = 0.0 Vi,j,k = Voi,j Print T.Vj^k ,HLJJ< |M= 0 • — * — ^ S T O P ^ ComputeHPj^ f tVP - j k at interior points (sub. INTER) X Compute HP,VP at boundary points (subs, for B.C.) H = VP = HP Yes :M=M No FLOW CHART FOR A PIPING S Y S T E M FIG. 4.1 8 1 . ( i v ) Computation of the v a l u e of c o n s t a n t s . (v) C a l l i n g the s u b r o u t i n e s . ( v i ) P r i n t i n g the t r a n s i e n t s t a t e c o n d i t i o n s . The f l o w c h a r t g i ven i n F i g . 4 . 1 - a i l l u s t r a t e s the sequence of o p e r a t i o n s i n a main programme. 4 . 2 SUBROUTINE STEADY: Purpose : Th is s u b r o u t i n e i s used to determine the steady s t a t e v e l o c i t i e s and p r e s s u r e s i n the system. D e s c r i p t i o n : The steady s t a t e p r e s s u r e i s u s u a l l y known a t a j u n c t i o n of the branch p i p e and the l a t e r a l . The s p r i n k l e r d i s c h a r g e i s a f u n c t i o n of the p ressure at the s p r i n k l e r . I n i t i a l l y n e i t h e r the s p r i n k l e r d i s c h a r g e nor the p ressure i s known. These are determined by the i t e r a t i v e procedure g i ven below: Assuming the p ressure head at a l l the s p r i n k l e r s on the l a t e r a l equa l to i t s known v a l u e a t the j u n c t i o n , the s p r i n k l e r d i s c h a r g e i s computed. Hence the v e l o c i t i e s and the f r i c t i o n l o s s e s between d i f f e r e n t s e c t i o n s of the d i s -t r i b u t o r s and l a t e r a l a re computed. Having determined the f r i c t i o n l o s s e s , the p ressures at the s p r i n k l e r s a re c o r r e c t e d and the d i s c h a r g e s of the s p r i n k l e r s cor respond ing to these p ressures a re determined . Now, the d i f f e r e n c e between the i n i t i a l guessed v a l u e and the c o r r e c t e d v a l u e of p ressure f o r every s p r i n k l e r i s computed. I f f o r any s p r i n k l e r , the n u m e r i c a l v a l u e of t h i s d i f f e r e n c e i s more than the s p e c i f i e d t o l e r a n c e , the process i s repeated t a k i n g the c o r r e c t e d v a l u e s of p ressure as the i n i t i a l es t imated v a l u e s . Th is i t e r a t i v e procedure i s cont inued u n t i l r e s u l t s of r e q u i r e d accuracy are o b t a i n e d . ^ S T A R T ^ Read Data Com put e At, Nj Compute coefficients and constants for pipes Call sub STEADY Compute constants for pump & chamber T =0.0 P r i nt T , V i > j | k  Hi,j,k . a . Z P and V air M = 0 6 T = T + DT M= M+l II Call INTER < Call sub. " B R l " Cal l sub."BR2" Call sub."PUMP" Call sub. CHAM2 Call sub"RESERD I Da = a P - a vir,k = vPi.j.k H-'J,k = H P i ,J,k Voir = v p a i r a = Q P -=_<M : M (b ) FLOW CHART FOR AN IRRIGATION PIPE S Y S T E M FIG. 4.1 (b ) 8 3 . Now, the d i s c h a r g e and f r i c t i o n l o s s i n the branch p i p e are computed and hence the p r e s s u r e at the j u n c t i o n of the next l a t e r a l and the branch p ipe i s determined . The p ressures and d i scharges of the s p r i n k l e r s on t h i s l a t e r a l ar°. found as o u t l i n e d above. By proceeding i n t h i s manner, the steady s t a t e v e l o c i t i e s and p r e s s u r e s i n the p ipe system are determined . How to Use: The s u b r o u t i n e i s c a l l e d by CALL STEADY (NBR,NDIST,NSPR) i n which NBR i s the number of the b r a n c h , NDIST i s the number of d i s t r i b u t o r s on a l a t e r a l , and NSPR i s the number of s p r i n k l e r s on a d i s t r i b u t o r . 4 . 3 SUBROUTINE INTER: Purpose : Th is s u b r o u t i n e i s used to determine the v e l o c i t i e s and p ressures at the i n t e r m e d i a t e s e c t i o n s of a p i p e . D e s c r i p t i o n : Eqs . (1 .42) and (1 .43) a re used to determine the t r a n s i e n t s t a t e v e l o c i t i e s and p ressures at the i n t e r m e d i a t e s e c t i o n s of a p i p e . A l l s e c t i o n s on a p i p e except the boundary p o i n t s are c a l l e d i n t e r m e d i a t e s e c t i o n s . How to Use: The s u b r o u t i n e i s c a l l e d by CALL INTER ( I , J , K 1 , K 2 ) i n w h i c h I i s the number of the b r a n c h , and K l and K2 are the f i r s t and the l a s t i n t e r m e d i a t e s e c t i o n on p i p e J . 4 . 4 SUBROUTINES BR l AND BR2: Purpose : These s u b r o u t i n e s are used to determine the t r a n s i e n t s t a t e v e l o c i t y 84. and p r e s s u r e a t a j u n c t i o n . BR1 i s used f o r c o n n e c t i o n of a branch p i p e and a l a t e r a l and BR2 f o r the connec t ion of a l a t e r a l and a d i s t r i b u t o r . D e s c r i p t i o n : Boundary c o n d i t i o n s developed i n S e c t i o n 2.4 are used to determine the t r a n s i e n t s t a t e c o n d i t i o n s a t a branch c o n n e c t i o n . How to Use: These subrout ines are c a l l e d by CALL BR1 ( I , J , K ) CALL BR2 ( I , NLAT, NDIST) i n which I i s the branch number, J and K a re the p ipe and i t s l a s t s e c t i o n , NLAT i s the number of l a t e r a l s on the branch and NDIST i s the number of d i s t r i -b u t o r s on a l a t e r a l , 4 . 5 SUBROUTINE CHAMl AND CHAM2: Purpose : These s u b r o u t i n e s compute the t r a n s i e n t s t a t e c o n d i t i o n s at an a i r chamber. CHAMl i s used when the chamber i s s i t u a t e d near the pump and the . check v a l v e c l o s e s i n s t a n t l y upon pump f a i l u r e . CHAM2 i s used when the chamber i s l o c a t e d at some i n t e r m e d i a t e p o i n t i n the p ipe system. D e s c r i p t i o n : Boundary c o n d i t i o n s developed i n S e c t i o n 2.8 are used i n these s u b r o u t i n e s . The v a l u e s of the cons tants f o r the a i r chamber are determined i n the main programme and are p rov ided i n the s u b r o u t i n e s through a COMMON s t a t e -ment . How to Use: The subrout ines are c a l l e d by CALL CHAMl ( 1 , 1 , 1) CALL CHAM2 ( I , J , K, M, N) i n which I i s the branch number. The chamber i s l o c a t e d a t the j u n c t i o n of 8 5 . p i p e J and M. The f i r s t s e c t i o n on p i p e M and the l a s t s e c t i o n on p^pe J are rep resented by N and K r e s p e c t i v e l y . 4 . 6 SUBROUTINES SURGE1 AND SURGE2: Purpose : These s u b r o u t i n e s are used to determine the t r a n s i e n t s t a t e c o n d i t i o n s a t an o r i f i c e surge tank . SURGE1 i s used when the waterhammer waves i n the tank are n e g l e c t e d ; SURGE2 i s used i f the waves i n the tank are to be c o n -s i d e r e d . D e s c r i p t i o n : The boundary c o n d i t i o n s developed i n S e c t i o n 2 .7 are used to determine the t r a n s i e n t s t a t e c o n d i t i o n s . The cons tants f o r the tank are c a l c u l a t e d i n the main programme, and are p rov ided i n the s u b r o u t i n e by a COMMON s ta tement . For a s imp le surge t a n k , o r i f i c e t h r o t t l i n g l o s s i s taken equa l to z e r o . How to Use: For a surge t a n k , s i t u a t e d at the j u n c t i o n of p ipe J and M, the s u b r o u t i n e s are c a l l e d by CALL SURGE1 ( I , J , K, M, N) CALL SURGE2 ( I , J , K, M, N) i n which I i s the number of the b r a n c h , and N and K a re the f i r s t and the l a s t s e c t i o n on p i p e M and J . 4 .7 SUBROUTINES RESERU, RESERD AND RESERI: Purpose : These s u b r o u t i n e s are used to determine the t r a n s i e n t s t a t e c o n d i t i o n s 36. a t a r e s e r v o i r of constant water s u r f a c e e l e v a t i o n . RESERU, RESERD and RESERI are used when the r e s e r v o i r i s a t the upstream end, a t the downstream end and at some i n t e r m e d i a t e p o i n t of the system. D e s c r i p t i o n : The boundary c o n d i t i o n s developed i n S e c t i o n 2 . 1 are used i n these s u b r o u t i n e s . How to Use: The s u b r o u t i n e s RESERU and RESERD are c a l l e d by CALL RESERU ( 1 , 1 , 1) CALL RESERD ( I , J , K) i n which I i s the number of "the b r a n c h , J i s the l a s t p i p e i n the b r a n c h , and K i s the l a s t s e c t i o n on the p i p e J . For a r e s e r v o i r at the j u n c t i o n of p i p e J and M, s u b r o u t i n e RESERI i s c a l l e d by CALL RESERI ( I , J , K, M, N) i n which I i s the number of the b r a n c h , K i s the l a s t s e c t i o n on p i p e J and N i s the f i r s t s e c t i o n of p i p e M. 4 . 8 SUBROUTINE VALVE: Purpose : Th is s u b r o u t i n e i s used to determine the t r a n s i e n t s t a t e c o n d i t i o n s caused by opening or c l o s i n g a v a l v e . D e s c r i p t i o n : The boundary c o n d i t i o n s developed i n S e c t i o n 2 .6 are used i n t h i s s u b r o u t i n e . The e f f e c t i v e g a t e - o p e n i n g (T) ^ t ime ( t ) curve i s s t o r e d i n the computer by s t o r i n g the v a l u e s of T. For any i n t e r m e d i a t e v a l u e of t , the v a l u e of T i s determined by p a r a b o l i c i n t e r p o l a t i o n . For t h i s , s u b r o u t i n e PARAB i s c a l l e d . 87. Note t h a t when the v a l v e i s f u l l y c l o s e d , boundary c o n d i t i o n s f o r a dead end , developed i n S e c t i o n 2 . 2 , are used to determine the t r a n s i e n t s t a t e c o n d i t i o n s at the v a l v e . How to Use: The s u b r o u t i n e i s c a l l e d by CALL VALVE ( I , J , K) i n which I i s the number of the b r a n c h , J i s the p ipe and K i s the l a s t s e c t i o n on p i p e J . 4 . 9 SUBROUTINES PUMPl, PUMP2 AND PUMP3: Purpose : These s u b r o u t i n e s are used to determine the t r a n s i e n t s t a t e c o n d i t i o n s at a c e n t r i f u g a l pump caused by a p u m p - f a i l u r e . Subrout ine PUMPl i s used when there i s a check v a l v e near the pump which c l o s e s upon f l o w r e v e r s a l ; PUMP2 i s used when r e v e r s e f l o w through the pump i s a l l o w e d . Subrout ine PUMP3 i s used when there i s a check v a l v e which c l o s e s i n s t a n t l y and a r e l i e f v a l v e which opens g r a d u a l l y upon f l o w r e v e r s a l . D e s c r i p t i o n : An i t e r a t i v e p rocedure , d e s c r i b e d i n s e c t i o n 2 . 9 , i s used to detemrine the t r a n s i e n t s t a t e c o n d i t i o n s . Flow c h a r t s g i ven i n F i g . 2 .9 ( a ) , (b) and (c) i l l u s t r a t e the sequence of computat ions i n the s u b r o u t i n e s PUMPl PUMP2 and PUMP3. The pump c h a r a c t e r i s t i c s a re s t o r e d i n the computer as o u t l i n e d i n s e c t i o n 2 . 9 . By c a l l i n g the s u b r o u t i n e MVAL, the a p p r o p r i a t e c h a r a c t e r i s t i c curve i s s e l e c t e d . The s u b r o u t i n e s PARABH and PARABB are c a l l e d to determine the i n t e r m e d i a t e v a l u e s of the p r e s s u r e head r a t i o , h , and torque r a t i o , 3 , by p a r a b o l i c i n t e r p o l a t i o n . 8 8 . How to Use; The s u b r o u t i n e s are c a l l e d by the f o l l o w i n g s ta tements : CALL PUMP1 ( I , J , . K) CALL PUMP2 ( I , J , K) CALL PUMP3 ( I , J , K) i n which I i s the number of the b r a n c h , J i s the p ipe and K i s the f i r s t s e c t i o n on p i p e J . 4 .10 SUBROUTINE SPRINK: Purpose : Th is s u b r o u t i n e i s used to determine the t r a n s i e n t s t a t e c o n d i t i o n s a t a s p r i n k l e r . D e s c r i p t i o n : The boundary c o n d i t i o n s developed i n S e c t i o n 2 .5 are used i n t h i s s u b r o u t i n e . The constants f o r the s p r i n k l e r are computed i n the main programme and are l i n k e d w i t h the s u b r o u t i n e through a COMMON statement . How to Use: Th is s u b r o u t i n e i s c a l l e d by the statement CALL SPRINK (NDIST, NSPR) i n which NDIST i s the number of d i s t r i b u t o r s on a l a t e r a l and NSPR i s the number of s p r i n k l e r s on a d i s t r i b u t o r . 4 . 1 1 SUBROUTINE MVAL: Purpose : Th is s u b r o u t i n e i s used to s e l e c t the a p p r o p r i a t e pump c h a r a c t e r -i s t i c cu rve . D e s c r i p t i o n : A number of c o n d i t i o n a l s tatements a re used to s e l e c t the a p p r o p r i a t e cu rve . Flow c h a r t of F i g . 2 . 9 - d i l l u s t r a t e s the sequence of 89. computat ions . How to Use: Th is s u b r o u t i n e i s c a l l e d by CALL MVAL (A, B, C, M) i n which A i s the d i s c h a r g e r a t i o , v, B i s equa l to v / a , C i s the speed r a t i o a , and M i s the output from the s u b r o u t i n e . The v a l u e of M d e f i n e s the pump c h a r a c t e r i s t i c c u r v e . 4 .12 SUBROUTINE PARAB: Purpose : Th is s u b r o u t i n e determines the i n t e r m e d i a t e v a l u e s of a dependent v a r i a b l e by p a r a b o l i c i n t e r p o l a t i o n . The v a l u e s of the dependent v a r i a b l e at e q u a l increments i n the independent v a r i a b l e are s t o r e d i n the computer. D e s c r i p t i o n : The v a l u e s of the dependent v a r i a b l e and the v a l u e of the i n t e r -v a l of the independent v a r i a b l e are g i ven i n the s u b r o u t i n e by a COMMON s tatement . The i n i t i a l v a l u e of the independent v a r i a b l e shou ld be zero or made e q u a l to z e r o . For example, the e f f e c t i v e g a t e - o p e n i n g ,T , v a l u e s f o r a v a l v e are g i ven at a t ime i n t e r v a l of z seconds . I f the v a l v e s t a r t s to c l o s e a t t ime t = t = 1 0 . 0 seconds , then the s u b r o u t i n e i s c a l l e d by the o • J statement CALL PARAB (T - 1 0 . 0 , TAU) to determine the v a l u e of TAU at t ime t = T. How to Use: The s u b r o u t i n e i s c a l l e d by CALL PARAB (X, Y) 90. i n which X i s the independent v a r i a b l e , and Y i s the dependent v a r i a b l e . 4 .13 FUNCTIONS VEL AND HEAD: Purpose : I f the v a l u e s of p r e s s u r e and v e l o c i t y are known a t two p o i n t s , then t h e i r v a l u e s a t any i n t e r m e d i a t e p o i n t a re determined by l i n e a r i n t e r p o l a t i o n by these f u n c t i o n subprogrammes. D e s c r i p t i o n : Eq. (1 .38) to (1 .41) are used i n these subprogrammes. How to Use: The f u n c t i o n subprogrammes are c a l l e d by V E L ( I , J , K, L, M) HEAD(I , J , K, L, M) i n which I i s the number of the b r a n c h , K and L are the s e c t i o n s of p ipe J a t which the v a l u e s of p r e s s u r e and v e l o c i t y are known, and M i s the type of p i p e J . For the branch p i p e , l a t e r a l and d i s t r i b u t o r , the v a l u e of M i s e q u a l to 1 , 2 and 3 r e s p e c t i v e l y . 9 1 . CHAPTER V  APPLICATIONS OF BOUNDARY CONDITIONS 5". l . NUMERICAL EXAMPLES : A number of examples are s o l v e d to i l l u s t r a t e how to use the boundary c o n d i t i o n s . The d e s c r i p t i o n of each problem and the r e s u l t s obta ined on the d i g i t a l computer by the method of c h a r a c t e r i s t i c s and by the g r a p h i c a l method, are presented i n Appendix A . The p r o g r a m m e - l i s t i n g of the s u b r o u t i n e s used i n - s o l v i n g these examples i s g i v e n i n Appendix D. 5 .2 PIPE SYSTEM STUDIES: The t r a n s i e n t s t u d i e s , l i s t e d be low, are made . fo r the i r r i g a t i o n p i p e system shown i n F i g . B - 2 ( a ) . The p ipes are assumed h o r i z o n t a l and the f r i c t i o n l o s s e s are n e g l e c t e d . T r a n s i e n t c o n d i t i o n s are caused by a p r e s s -ure wave whose magnitude remains constant at the upstream end f o r t ' l 0 . Th is i s e q u i v a l e n t to a sudden r i s e i n the water s u r f a c e e l e v a t i o n i n r e s e r v o i r . The f o l l o w i n g s t u d i e s are made: a) Comparison of a t r a n s i e n t p ressure a t d i f f e r e n t p o i n t s of the sys tem, when i t i s ana lyzed assuming the s p r i n k l e r s , i ) on the d i s t r i b u t o r s , 92. i i ) on the l a t e r a l s and i i i ) on the branch p i p e s , b) The e f f e c t s of changes i n the s p r i n k l e r d i s c h a r g e on the magni -tude of the waterhammer w a v e - f r o n t . In t h i s c a s e , i t i s assumed that the s p r i n k l e r s are l o c a t e d on the branch p i p e . The r e s u l t s f o r each case a re presented i n Appendix B. 5 . 3 COMPUTER PROGRAMME FOR AN IRRIGATION PIPE SYSTEM The l i s t i n g of the computer programme to study the t r a n s i e n t c o n d i t i o n s i n the i r r i g a t i o n p ipe system shown i n F i g . B -2(a) i s p r e -sented i n Appendix C. The t r a n s i e n t c o n d i t i o n s are caused by a pump-f a i l u r e . Flow char t f o r the programme i s g i v e n i n F i g . 4 . 1 - b . In the programme, the s l o p e s of the branch p ipes and f r i c t i o n l o s s e s i n the p ipes are c o n s i d e r e d . CONCLUSIONS (1) Liquations r e p r e s e n t i n g the boundary c o n d i t i o n s as d e r i v e d i n t h i s t h e s i s can be used to ana lyze t r a n s i e n t f l o w c o n d i t i o n s i n any p i p e system. (2) The v a l i d i t y of these boundary c o n d i t i o n s i s demonstrated by comparing the r e s u l t s ob ta ined by the method of c h a r a c t e r i s t i c s w i t h those obta ined by the g r a p h i c a l method. (3) To ana lyze the t r a n s i e n t s t a t e c o n d i t i o n s i n a complex system w i t h l a r g e f r i c t i o n l o s s e s , the method of c h a r a c t e r i s t i c s i s recom-mended i n s t e a d of the g r a p h i c a l method because the former method i s f a s t e r , more a c c u r a t e and v e r s a t i l e . However, f o r a s imp le system w i t h s m a l l f r i c t i o n l o s s e s , (up to about 8% of t o t a l h e a d ) , the g r a p h i c a l method i s p r e f e r a b l e f o r i t s c l a r i t y , ease of c o n s t r u c t i o n and because lumping the l o s s e s a t c e r t a i n p o i n t s does not cause l a r g e e r r o r s . (4) In the method of c h a r a c t e r i s t i c s , l i k e a l l o ther a v a i l a b l e methods, the number of computat ions i s c o n s i d e r a b l y i n c r e a s e d i f the d i s t a n c e between two boundary p o i n t s i s s h o r t . (5) I f f r i c t i o n l o s s e s are n e g l e c t e d , the s o l u t i o n s obtained by the method of c h a r a c t e r i s t i c s c l o s e l y agree w i t h those obta ined by the g r a p h -i c a l method. 94. (6) The method developed to d e s i g n a t e the p i p e s and p i p e s e c t i o n s i n an i r r i g a t i o n p ipe system i s h e l p f u l i n s t o r i n g the i n p u t data i n the computer. A l s o , by t h i s method i t i s p o s s i b l e to ana lyze s i m i l a r types of boundary p o i n t s by means of a d o - l o o p . (7) A p r e s s u r e wave i n c r e a s e s i n magnitude as i t t r a v e r s e s a p i p e w i t h s t e p - w i s e decrease i n d iameter (see F i g . B - l f o r Q g ^ = 0 . 0 ) . A s i m i l a r s u r f a c e wave phenomenon exper ienced i n open channels i s c a l l e d f u n n e l l i n g e f f e c t . (8) The magnitude of a p r e s s u r e wave i s reduced when i t i s r e f l e c t e d or t r a n s m i t t e d a t a s p r i n k l e r . The magnitude of the t r a n s m i t t e d or r e f l e c t e d wave decreases as the s p r i n k l e r d i s c h a r g e i n c r e a s e s . (9) I f the l a t e r a l s and d i s t r i b u t o r s a re d e l e t e d and the s p r i n k l e r s a re cons idered as lumped on the branch p i p e , waterhammer p r e s s u r e s are under -e s t i m a t e d . The i r r i g a t i o n p ipe system shown i n F i g . B -2 was ana lyzed by making t h i s assumpt ion . I t r e s u l t e d i n an u n d e r e s t i m a t i o n of the w a t e r -hammer p ressure by 34%. The system was ana lyzed by d e l e t i n g the d i s t r i b u t o r s and c o n s i d e r i n g the s p r i n k l e r s as lumped on the l a t e r a l s . Th is a n a l y s i s underest imated the p ressur by on ly 4%. 9 5 . BIBLIOGRAPHY 1 . A l l i e v i , L . , " A i r chambers f o r D ischarge P i p e s " T rans . ASME, V o l . 5 9 , Paper H y d - 5 9 - 7 , November 1937, pp . 651 -659 . 2 . Angus, R. W. , " A i r chambers and Va lves i n r e l a t i o n to Waterhammer", T rans . ASME, V o l 5 9 , Paper H y d - 5 9 - 8 , November 1937, pp. 661 -668 . 3 . Be rgeron , L . , Waterhammer i n p i p e s and Wave Surges i n E l e c t r i c i t y , John Wi ley and Sons, I n c . , New Y o r k . 4 . Donsky, B . , "Complete Pump C h a r a c t e r i s t i c s and the E f f e c t s of S p e c i f i c Speeds on H y d r a u l i c T r a n s i e n t s " , J o u r n a l of B a s i c E n g i n e e r i n g , Dec. 1961, pp. 6 8 5 - 6 9 9 . 5 . Fox , P . , "The s o l u t i o n 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 Equat ions by D i f f e r e n c e Methods" . Chapter i n M a t h e m a t i c a l Methods f o r D i g i t a l Computers E d i t e d by R a l s t o n and H. W i l f , John Wi ley and Sons, I n c . , New Y o r k , 1960. 6. K i t t r e d g e , C . , " H y d r a u l i c T r a n s i e n t s i n C e n t r i f u g a l Pump Systems" , T rans . ASME, August 1956, pp. 1307-1322. 7. L e s c o v i c h , J . , "The C o n t r o l of Waterhammer by Automat ic V a l v e s " , J o u r n a l AWWA, May 1967, pp. 632 -644 . 8 . L i s t e r , M . , "The Numer ica l S o l u t i o n 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 q u a t i o n s , by the Method of C h a r a c t e r i s t i c s , i n A . R a l s t o n and H . W i l f , (eds . ) M a t h e m a t i c a l Methods f o r D i g i t a l Computers , John Wi ley and Sons, I n c . , New Y o r k , 1960. 9 . McCracken and Dorn , Numer ica l Methods and FORTRAN Programming, John W i l e y and Sons, I n c . , New Y o r k . 10. O ' B r i e n , G. G . , Hyman, M. A . , and K a p l a n , S . , "A Study of the Numer ica l S o l u t i o n of P a r t i a l D i f f e r e n t i a l E q u a t i o n s , " J . Math and P h y s i c s , N o . 29 (1951) , pp . 2 2 3 - 2 5 1 . 1 1 . P e r k i n , Tedro , Eag leson and Ippen , "Hydro-Power P l a n t T r a n s i e n t s P a r t I I , Response to Load R e j e c t i o n " , Hydrodynamics Laborato ry Report No. 7 1 , Massachuset ts I n s t , of T e c h . , Cambridge 3 9 , Mass . 12 . Parmakien , J . , Waterhammer A n a l y s i s , Dover P u b l i c a t i o n s , I n c . New York (1963) . 13 . R i c h , G . , H y d r a u l i c T r a n s i e n t s , 1st E d i t i o n , E n g i n e e r i n g S o c i e t i e s  Monographs, M c G r a w - H i l l Book C o . , I n c . , New Y o r k , 1951. (Dover R e p r i n t ) . 96. 14. R i c h , G . , "Waterhammer A n a l y s i s by the La P l a c e - M e l l i n T r a n s f o r m a t i o n s " T rans . ASME, 1945. 15 . Ruus, E . , "Bestimmung von S c h l i e s s f u n h t i o n e n , Welche den K l e i n s t e n Wert des Maximalen Druckstosses e rgeben" , Thes is submit ted i n 1957 to the T e c h n i c a l U n i v e r s i t y of K a r l s r u h e (Germany) i n p a r t i a l f u l f i l m e n t of the requi rements f o r the degree of Doctor of E n g i n e e r i n g . 16. S t r e e t e r , V. L. , "Waterhammer A n a l y s i s of P i p e l i n e s " , J o u r n a l of the  H y d r a u l i c s D i v i s i o n , ASCE, V o l . 90 , No. H Y - 4 , J u l y , 1964, pp. 1 5 1 - 1 7 2 . 17. S t r e e t e r , V. L. , "Computer s o l u t i o n s of Surge P r o b l e m s " , Symposium on  Surges i n P i p e l i n e s , I n s t , of Mech. E n g i n e e r s , Proceedings 1 9 6 5 - 6 6 , v o l . 180, P a r t 3E, pp. 6 2 - 8 2 . 18. S t r e e t e r , V. L. , "Va l ve S t r o k i n g to C o n t r o l Waterhammer", J o u r n a l of  H y d r a u l i c D i v i s i o n ASCE 89 , No. H Y - 2 , March 1963. 19. S t r e e t e r , V. L. , "Water-hammer a n a l y s i s of d i s t r i b u t i o n sys tems" , P r o c . of the ASCE, J o u r n a l of H y d r a u l i c D i v i s i o n , Sept . 1967. pp. 1 8 5 - 2 0 1 . 20. S t r e e t e r and L a i , Waterhammer A n a l y s i s I n c l u d i n g F l u i d F r i c t i o n , " J o u r n a l of the H y d r a u l i c D i v i s i o n , ASCE, v o l . 8 8 , no . H Y - 3 , May, 1962, pp .79 -112 2 1 . S t r e e t e r and W y l i e , H y d r a u l i c T r a n s i e n t s , M c G r a w - H i l l Book Company, New Y o r k , 1967, 22. Wood, F. M . , "The A p p l i c a t i o n of H e a v i s i d e s O p e r a t i o n a l C a l c u l u s to the S o l u t i o n of Problems i n Waterhammer", T rans . ASME, v o l . 5 9 , Paper H y d - 5 9 - 1 5 , November 1937, pp. 7 0 7 - 7 1 3 . 97. APPENDIX A NUMERICAL EXAMPLES A - l . A p i p e l i n e w i t h a r e s e r v o i r a t the upstream end. A v a l v e c l o s e s a t the downstream end. A - 2 . P i p e s i n s e r i e s w i t h a r e s e r v o i r a t the upstream end. A v a l v e c l o s e s at the downstream end. A - 3 . Surge tank . A - 4 . A i r chamber. A - 5 . Pump f a i l u r e : i ) Check v a l v e c l o s e s i n s t a n t l y upon f l o w r e v e r s a l , i i ) Reverse f l o w through the pump a l l o w e d , i i i ) Check v a l v e c l o s e s i n s t a n t l y upon f l o w r e v e r s a l . R e l i e f 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 . 98. APPENDIX A - l F E D C B A T r U PROBLEM: Determine the t r a n s i e n t s t a t e p r e s s u r e s and v e l o c i t i e s at the p o i n t s A , B, C, D, E, and F. The t r a n s i e n t c o n d i t i o n s a re caused by c l o s i n g a v a l v e a t the p o i n t A . E f f e c t i v e gate o p e n i n g - t i m e r e l a t i o n s h i p i s g i ven by the equat ion , T = (1 - | - ) , C 1 i n which t i s the t ime of c l o s u r e , c DATA: L Length of the p i p e l i n e (L) *= 4253.5 f t . 9 9 . S t a t i c head (H Q) = 300.0 f t . V e l o c i t y of water i n p i p e l i n e (V Q ) = 3 . 3 f t / s e c . I n s i d e diameter of the p i p e l i n e (D) = 3 . 0 f t . F r i c t i o n f a c t o r ( f ) = 0.019 Waterhammer wave v e l o c i t y (a) = 3963. f t / s e c . Time of c l o s u r e (t ) = 5 . 9 s e c . c CHECK: "•' The computed v a l u e s agree w i t h those obta ined by S t r e e t e r on the d i g i t a l computer (Reference 2 1 , page 2 8 ) . 100. APPENDIX A - 2 PIPES IN SERIES W.S. PIPE 1 PIPE 2 A V- t PROBLEM: Determine the t r a n s i e n t s t a t e c o n d i t i o n s caused by c l o s i n g a v a l v e at the p o i n t A . DATA: P i p e 1 P i p e 2 Length of the p i p e , i n f t . 1500 1800 V e l o c i t y of the waterhammer wave, i n f t / s e c . 3000 3600 I n s i d e diameter of the p i p e , i n f t . 1 2 . 9 1 10 .0 I n i t i a l steady s t a t e v e l o c i t y of w a t e r , i n f t / s e c 5.37 8.94 S t a t i c head (H Q) - = 500 f t . 101. F r i c t i o n f a c t o r = 0 . 0 Gate c l o s u r e t ime r e l a t i o n s h i p i s shown above. CHECK: R e s u l t s . d e t e r m i n e d g r a p h i c a l l y by P a r m a k i a n 1 2 (page 55) a re i n c l o s e agreement w i t h those obta ined on the d i g i t a l computer, u s i n g the method of c h a r a c t e r i s t i c s (see F i g . A - 2 ) . 102. Graphical method (Parmakian pp55) Method of characteristics 103. W.S. APPENDIX A - 3 SURGE TANK PIPE 1 D _ 3 t -Surge tank PIPE 2 H PROBLEM: Determine the t r a n s i e n t s t a t e c o n d i t i o n s caused by c l o s i n g a v a l v e a t the p o i n t E. Cons ider the changes i n the e l e v a t i o n of the water s u r f a c e i n the tank . DATA: Length of the p i p e , i n f t . I n s i d e diameter of the p i p e , i n f t . P i p e 1 P i p e 2 Surge Tank 6000.0 1500 10.0 5 . 0 11.55 104. Waterhammer wave v e l o c i t y , i n f t / s e c . 3000.0 3000.0 2000.0 I n i t i a l steady s t a t e v e l o c i t y of w a t e r , i n f t / s e c 5.37 21.466 0 . 0 Neg lec t the f r i c t i o n l o s s e s , x ^ t re la t ions/h ip f o r the v a l v e i s g i v e n above. CHECK: i ) C o n s i d e r i n g the waterhammer waves i n the t a n k : R e s u l t s ob ta ined by P a r m a k i a n 1 2 (page 108) by the g r a p h i c a l method agree w i t h those obta ined on the d i g i t a l computer, u s i n g the method of c h a r a c t e r i s t i c s (See F i g . A - 3 (a)) i i ) N e g l e c t i n g waterhammer waves i n the surge t a n k : No check i s a v a i l a b l e . R e s u l t s obta ined are presented i n F i g . A - 3 ( b ) . 105. TRANSIENT STATE PRESSURES AND V E L O C I T I E S (Water-hammer waves in the surge tank considered) FIG. A - 3 ( a ) 106. TRANSIENT STATE P R E S S U R E S AND VELOCIT IES (Water -hammer waves in the surge tank neglected) F IG. A - 3 ( b ) 107. APPENDIX A - 4 AIR CHAMBER PROBLEM: Determine the t r a n s i e n t s t a t e p r e s s u r e s and v e l o c i t i e s , volume of a i r and water s u r f a c e e l e v a t i o n i n the a i r chamber. The t r a n s i e n t con-d i t i o n s are caused by pump f a i l u r e . I n i t i a l s teady s t a t e water s u r f a c e r e s B C v W.S, check v a l v e DATA: Check v a l v e c l o s e s immediate ly upon pump f a i l u r e . Length of the p i p e l i n e (L) = 16080 f t . 108. C r o s s - s e c t i o n a l a rea of the p i p e (A) = 4 .0 f t 2 C r o s s - s e c t i o n a l a rea of the chamber (A ) = 20 .0 f t a Height of the chamber (Z ) = 12 .0 f t . . 3. Atmospher ic p ressure a t C = 3 4 . 0 f t . of water I n i t i a l steady s t a t e water s u r f a c e e l e v a t i o n i n the a i r chamber (Z Q ) = 10 .0 f t . Waterhammer wave v e l o c i t y (a) = 3216 f t / s e c . I n i t i a l s teady s t a t e v e l o c i t y of water i n the p i p e l i n e (V ) = 4 .0 f t / s e c . O r i f i c e t h r o t t l i n g l o s s = 0 . 0 f t . A i r expansion i n the a i r chamber i s i s o t h e r m a l , i . e . p* . v . = constant , r a i r i n which p* and v . a re the a b s o l u t e p r e s s u r e and volume of a i r i n the r a i r r chamber. Neg lect the f r i c t i o n l o s s e s . CHECK: R e s u l t s obta ined on the d i g i t a l computer u s i n g the method of c h a r a c t e r i s t i c s are c l o s e to those ob ta ined by Angus 2 (page 661) by the g r a p h i c a l method (See F i g . A - 4 ) . 109. FIG. A - 4 110. APPENDIX A - l PUMP FAILURE w.s. H o (t-j^-«i£ 3 pumps D PROBLEM: Determine the t r a n s i e n t s t a t e v e l o c i t y and p r e s s u r e i n the d i s -charge p i p e and r o t a t i o n a l speed of the pump f o r the f o l l o w i n g th ree cases i ) C e n t r i f u g a l pump w i t h a check v a l v e , i i ) C e n t r i f u g a l pump w i t h o u t a check v a l v e . • i i i ) C e n t r i f u g a l pump w i t h a check v a l v e and a r e l i e f v a l v e . The t r a n s i e n t c o n d i t i o n s are caused by pump f a i l u r e . DATA: Length of the p i p e l i n e (L) = 3940 f t . I n s i d e diameter of the p i p e (D) = 32 .0 i n . Waterhammer wave v e l o c i t y (a) = 2820 f t / s e c . I n i t i a l steady s t a t e v e l o c i t y i n the p ipe (V ) = 5 . 8 1 f t / s e c ( f o r 3 pumps S t a t i c head (H ) = 220 f t . 111. WR of each pump and motor = 384.9 l b / f t . Rated pump speed (N ) = 1760 p.m. Pump e f f i c i e n c y (n^) at r a t e d v a l u e s = 84.7 % R Neg lec t the f r i r t i o n l o s s e s . Pump c h a r a c t e r i s t i c c u r v e s " a re g i v e n by the f o l l o w i n g t a b l e : - or - 0 . 0 0 . 1 0 . 2 0 . 3 0 .4 0 . 5 0 .6 0 .7 0 . 8 0 . 9 1 .0 a v HAN 1.080 1. 081 1 . 082 1 . 085 1 . 090 1. 095 1.10 1 . 097 1 . 080 1.050 1.000 HVN _ *** - - - - - 0.112 0 . 328 0 . 560 0 .761 1.000 HAD 1.080 1 . 085 1. 100 1. 120 1 . 150 1. 210 1.310 1 . 450 1 . 580 1.730 1.890 HVD 0.780 0 . 830 0 . 900 0 . 960 1 . 050 1 . 120 1.280 1 . 420 1 . 560 1.750 1.940. HAT - - - - - 0 . 660 0 .700 0 . 750 0 . 800 0.860 0.950 HVT - - - - 0 . 050 0 .160 0 . 290 0 . 410 0.580 0 .730 BAN 0.480 0 . 520 0 . 550 0 . 590 0 . 640 0 . 700 0.755 0 . 820 0 . 880 0.940 1.000 BVN - - - - - 0.585 0 . 745 0 . 920 0.980 1.000 BAN 0.480 0 . 460 0 . 430 0 . 425 0 . 445 0 . 490 0.600 0 . 770 0 . 988 1.170 1.420 BVD 1.130 1 . 180 1 . 210 1 . 250 1 . 280 1 . 310 1.320 1. 350 1 . 440 1.550 1.620 BAT - - - 0 . 400 0 . 200 0 . 140 0.205 0 . 250 0 . 800 0.770 0.180 BVT _ _ _ 0 . 899 o. 880 0 . 875 0 .820 0 . 812 0 . 800 0.770 0.720 In Reference 1 2 , (page 79-81) these curves are p l o t t e d as h ^ v . For t e r m i n o l o g y , r e f e r to S e c t i o n 2 . 9 . For the zone of energy d i s s i -p a t i o n v/a and a/v a re n e g a t i v e . *** Not a v a i l a b l e . 112. CHECK: Case ( i ) and case ( i i ) a re s o l v e d by the g r a p h i c a l method i n Reference 12 (page 8 3 ) . G r a p h i c a l s o l u t i o n of case ( i i i ) i s g i v e n i n F i g . A - 5 ( e ) . The r e s u l t s ob ta ined on the d i g i t a l computer u s i n g the method of c h a r a c t e r i s t i c s and by the g r a p h i c a l method are presented i n F i g . A - 5 ( a ) , (b) , ( c ) , and (d ) . The r e s u l t s obta ined by the two methods are i n c l o s e agreement. 1 1 3 . \ \ X V* \ *Y\ \ v*\ \ * \ \ V \ \ \ \ X W • IP Graphical method ( Parmakian pp 83) Method of characterist ics \ \ \ \ \ \ v \ \ i * \ \ \ \ I \ \ i \ \ N Vl ft \\ \ \ ^ ^ ^ ^ \ \ \ \ \ \ \\ \\ \\ \\ \\ \ \ \\ \\ \\ \ \ \ . \ X \ \ \ \ \ \ vv a \ \ \\ \\ / / / * / / / t / ' / / // // // if "\ \ \ \ \ VS. * \ \ \ * \ "\ \X >\ \\ \ \ \ \ \ ~ - . l \ \ \ »\ \ \ %\ * \ \ \ \ \ 1 / \ \ \ \ \ \ \ \ \ \ V 1/ 1 1 1 2 TIME (SEC) 3 4 K Checl 5 . valve closes instantly ( a ) TRANSIENT STATE CONDITIONS ( Pump failure , check valve closes when flow reverses') F I G . A - 5 0 . 2 - 0 . 4 Method of characteristics Graphical method (See Fig A - 5 ( f ) ) T - t curve for the relief valve (c ) T R A N S I E N T STATE CONDITIONS (Pump failure, check valve closes instantly and relief valve opens gradually upon flow reversal) FIG. A - 5 1 1 6 . (d ) . . TRANSIENT STATE CONDITIONS (Pump failure , Check valve closes instantly and relief valve opens gradually upon flow reversaI ) FIG. A-5 1 1 7 . Linear t - r relationship (Fig. A - 5 (d) ) Non- l inear t - T relationship (F ig . A - 5 (c) ) ( Graphical construction up to time 4.54 sec. has been done by Parmak ian '2 page 83 ) oo a> d d i i? 6 A ° • A 2I .3I \ A \ \% \ "0.3 0.2 0.1 V A \ \ A \ A\ yj \\ A / / ? „ . 5 3 \ \ VA V* A\ ' \ \ A ^ / D 8.73\ \ / \ A 1 / \)A / \ V A / Y» A /V'i / i V i ' . A A 3 2 4 8 •X'\ A 3 5 •'Aw 28 y \ j 3 6 . 6 8 \fWl2\^ DI9.9I 'A ^ \ / \ /A\\ 22.70 >f\\ \ \\\ ' / ' / y A \ D 2 5 5 0 \ D 28.24 ' > \ '' \ ^  \ \ \ > \ » to ; D — * \ ^ 31.04', 33.88 \ 1 I • > i i 1 I i . t 1 1 \ \ •V "AY \ x \ V N^ ^ \ \ \ \ \ /••• ^ \ \ \ \ i i • • \ i i i i i \ \ N \ \ \ \ \ \ A N \ V \ \ V \ \ \ « * \ \.\ \ \ \ \ x\ ! \ > • \ 1 • \\ ! \ A \ \ \ \ v » w > N \ \ \ \ \ X 0 \ V N A 4.54 1.2 '14.32 1.0 '12.92 0;8 0.6 0 .4 0 . 2 - 1.0 - 0 . 8 - 0.6 - 0 . 4 - 0.2 V > (e).GRAPHICAL SOLUTION (Pump fai lure , check valve closes instantly and relief valve opens gradually upon flow reversa I ) , • . F I G . A - 5 APPENDIX B PIPE SYSTEM STUDIES E f f e c t of the change i n the s p r i n k l e r d i s c h a r g e on the waterhammer w a v e - f r o n t . T r a n s i e n t s t a t e p r e s s u r e s f o r d i f f e r e n t l o c a t i o n s of the s p r i n k l e r s . 119. APPENDIX B - l EFFECT OF THE CHANGE IN THE SPRINKLER DISCHARGE ON THE WATERHAMMER WAVE-FRONT PROBLEM: Assuming the s p r i n k l e r s on the branch p i p e s determine the magnitude of the waterhammer w a v e - f r o n t as i t t r a v e r s e s the branch p i p e s shown i n F i g . B - l ( a ) . DATA: Lengths and d iameters of the branch p i p e s are shown " i n F i g . B - l (a)'. I n i t i a l steady s t a t e e l e v a t i o n of the water s u r f a c e i n the r e s e r v o i r = 100 f t . A t t ime t = 0 . 0 , the e l e v a t i o n of the water s u r f a c e i n the r e s e r v o i r suddenly r i s e s to 150.0 f t . V e l o c i t y of the waterhammer wave = 4000 f t / s e c . The s p r i n k l e r d i s c h a r g e , (Q ) , i n c u . f t . / s e c . i s : sp i ) 0 .384 i i ) 0 .192 i i i ) 0 .096 i v ) 0 . 0 120. REMARKS: The r e s u l t s are presented i n F i g . B-2 (b ) . I t i s c l e a r from the f i g u r e t h a t the magnitude of a p r e s s u r e rave i s reduced when i t i s r e f l e c t e d or t r a n s m i t t e d at a s p r i n k l e r . The magnitude of the t r a n s m i t t e d or r e f l e c t e d wave decreases as the s p r i n k l e r d i s c h a r g e i n c r e a s e s . 1 2 1 . Reservoir Q s p Q s P Q s P Q S P Q S P Q S P Q S P Q S P Q S P Q S P Q S P Q s p  4 * * * * t_ t t f t t_ t I6;'OS I4"OS 12" IO"co r8"gS 190 180 170 -160 H Lt_ < 150 U J X ui cn CO cn UJ 140 rr a. 120 FIG. B - l (a ) / Q s p = 0 0 Q =0.096 cfs r JL. * • — i Qsp= 0.192 cfs L_ 1 _ . ~l,Q s p= 0.384 cfs r L_. MO 0 1000 3 0 0 0 5 0 0 0 DISTANCE (FT) 7 0 0 0 9 0 0 0 11,iO00 |0  (b ) E F F E C T OF THE S P R I N K L E R D I S C H A R G E ON THE WAVE FRONT FIG. B - l 122. APPENDIX B-2 TRANSIENT STATE PRESSURES FOR DIFFERENT LOCATION OF SPRINKLERS PROBLEM: Compare the t r a n s i e n t s t a t e p ressures a t d i f f e r e n t p o i n t s of the system shown i n F i g . B-2 ( a ) , c o n s i d e r i n g the s p r i n k l e r s on t h e : i ) d i s t r i b u t o r s . , i i ) l a t e r a l s , i i i ) branch p i p e s . T r a n s i e n t c o n d i t i o n s are caused by r a i s i n g the e l e v a t i o n of the water s u r f a c e i n the r e s e r v o i r by 50 f t . DATA: Lengths and d iameters of the p i p e s a re shown i n F i g . B-2 ( a ) . V e l o c i t y of the waterhammer wave = 4000 f t / s e c . D ischarge of a s p r i n k l e r when i t i s on t h e : i ) d i s t r i b u t o r = 0 .005 c u . f t / s e c i i ) l a t e r a l = 0 . 0 5 c u . f t / s e c . i i i ) branch p ipe = 0 .50 c u . f t / s e c . Diameter of the l a t e r a l s = 4 II 10 Laterals © 1 0 0 0 ' 123 3 6 IOO 1 0 0 0 1 0 0 0 Branch pipe © 1 0 0 0 - I 1 -lOlSprinklers (a) 100'c/c 10 Sprinklers (a) 100 I" »• h — H *t • » — * • K - W c/c ® H ' f O ' > l _ a> u> a» 1 0 0 0 ® ' + ' • • • * 1 • I • f 4 • " H t * t l | 1 1 | *• — 4 1 1 1 * - "* ' * 1 • 1 1 ' 1 ' * 1 ' ' ' ' 10 Sprinklers (oj 100 c/c K r« H •( M © ® © ® © © ® © ® o "O g © O a Pump 01 > o > u o IS Air Chamber •-I00 © I000; I4"0> jj (SO x» v D i 0 0 0 ' , l 0 " o S ©jo°S--(23.0) (23.0) DATUM LINE SECTION A A A SAMPLE IRRIGATION PIPE SYSTEM FIG. B-2 (a) T 100' Reservoir (9) 1000', 6'>, (40.0) Notes :(DNumbers in the circles represent the number of the junction. (2) All the laterals and distributors are horizontal. (3) Reservoir has constant water level. (4) Numbers in the brackets are the heights of the junctions above the datum line. 124. Diameter of the d i s t r i b u t o r s = 2" A l l p i p e s aro h o r i z o n t a l . Neg lect the f r i c t i o n l o s s e s . REMARKS: The r e s u l t s are p resented i n F i g . B -2(b) to ( f ) . I t i s c l e a r t h a t t r a n s i e n t p r e s s u r e s ob ta ined by c o n s i d e r i n g the s p r i n k l e r s on the d i s t r i b u t o r s are c l o s e to those ob ta ined by d e l e t i n g the d i s t r i b u t o r s and assuming the s p r i n k l e r s on the l a t e r a l s . However, the a n a l y s i s of the system by d e l e t i n g the l a t e r a l s and d i s t r i b u t o r s and c o n s i d e r i n g the s p r i n k l e r s on the branch p ipes gave t o t a l l y d i f f e r e n t r e s u l t s . 125. 160 150 u_ o < U J Jl C O C O U J rr 0-140 Spr inklers on distr ibutors Spr inklers on laterals Sprinklers on branch pipe 120 T IME (SEC) (b) T R A N S I E N T S T A T E P R E S S U R E S AT JUNCTION NO. I F I G . B - 2 126. Sprinklers on distributors Sprinklers on laterals Sprinklers on branch pipe 150 140 130 120 1 A 1 \ K\ A h \\ V /~\ V ft Y/1 — T — -~rt \» / \ f \ | \ I N> ! V 1 "T\ / 1 ' / 1 '/ 1 '/ 1 '/ V r~ i \ / ' \ l i \ /' W \ /' w \ V \1 1 / ' -l l ' r « I ft 1 ft V' ¥ 1 '1 1 '/ 1 ll i • TIME (SEC) ( c) TRANSIENT S T A T E P R E S S U R E S AT J U N C T I O N NO. 3 FIG. B - 2 _ Spr i nk iers S p r i n k l e r s S p r i n k l e r s on on on d is t r ibutors la tera ls branch p i p e 1 2 7 . 150 FIG. B - 2 128. S p r i n k l e r s on b r a n c h p ipe S p r i n k l e r s on l a t e r a l s — S p r i n k l e r s on d i s t r i b u t o r s 150 140 < 130 120 2 V T I M E ( S E C ) ( e ) T R A N S I E N T S T A T E P R E S S U R E S AT J U N C T I O N N O . 7 F I G . B - 2 150 140 130 < X 120 110 TIME (f ) TRANSIENT STATE P R E S S U R E S AT J U N C T I O N NO. 9 F I G . B APPENDIX C COMPUTER PROGRAMME FOR A SAMPLE IRRIGATION PIPE SYSTEM 131. APPENDIX C PROBLEM: Determine the t r a n s i e n t s t a t e p r e s s u r e s and v e l o c i t i e s i n the sys tem, r o t a t i o n a l speed of the pump, and volume of a i r and e l e v a t i o n of the water s u r f a c e i n the chamber. The p i p e system i s shown i n F i g . B -2(a) The t r a n s i e n t c o n d i t i o n s are caused by pump f a i l u r e . DATA: C e n t r i f u g a l pump: Rated head of the pump = 1 7 7 . 8 ' Rated d i s c h a r g e of the pump = 6.07 c u . f t / s e c . 2 2 WR of the pump and motor = 250 l b - f t Rated speed of the pump ' = 1400 r . p . m . Pump e f f i c i e n c y at r a t e d speed and head = 0 . 8 4 Values f o r the pump c h a r a c t e r i s t i c curves are g i v e n i n Appendix A - 5 . A i r chamber: Diameter of the a i r chamber = 5 . 0 f t , Height of the a i r chamber = 1 5 . 0 f t . I n i t i a l a i r volume = 100 c u . f t . 132. O r i f i c e t h r o t t l i n g l o s s co r respond ing to a d i s c h a r g e of 5 c u . f t / s e c = 25 .0 f t . T h r o t t l i n g c o e f f i c i e n t when water f l o w s out of chamber = 1 . 0 T h r o t t l i n g c o e f f i c i e n t when water f l o w s i n t o the chamber = 1 .5 1 . 2 A i r expansion f o l l o w s the law p* . va'±x ~ c o n s t a n t -Branch p i p e s : L e n g t h s , d iameters and e l e v a t i o n s are shown i n F i g . B -2 ( a ) . F r i c t i o n f a c t o r ( f ) = 0 . 0 1 Waterhammer wave v e l o c i t y = 4000 f t / s e c . L a t e r a l s : Diameter = 4 " F r i c t i o n f a c t o r = 0 . 0 1 Waterhammer wave v e l o c i t y = 4000 f t / s e c . A l l l a t e r a l s are h o r i z o n t a l D i s t r i b u t o r s : Diameter = 2 " F r i c t i o n f a c t o r = 0 . 0 1 Waterhammer wave v e l o c i t y = 4000 f t / s e c . A l l d i s t r i b u t o r s are h o r i z o n t a l . 133. S p r i n k l e r s : S p r i n k l e r d i s c h a r g e cor respond ing to a head of 100 f t . — 0 .005 c u . f t / ^ e c . REMARKS: The r e s u l t s a re p resented i n F i g . C - ( l ) . When check v a l v e i s c l o s e d , i t becomes a dead end. The p r e s s u r e head f l u c t u a t i o n f o r the dead ends are c l e a r from the g raph . The IBM 7044 takes about 2 7 . 5 minutes to compute t r a n s i e n t s t a t e c o n d i t i o n s f o r 10 seconds . 134. TRANSIENT STATE CONDITIONS AT THE P U M P , CHECK VALVE AND AIR C H A M B E R FIG. C - I APPENDIX C SJOB 1 7 0 5 7 C . M . H A N I F $T I ME 30 SPAGE 4 0 0 '  ~$TB"FTC MAIN I ; C A N A L Y S I S OF WATERHAMMER IN THE I R R I G A T I O N P I P E SYSTEM SHOWN IN C ' F IGURE B - 2 . ~C rR'AN'STE'N T S~ "C A US E D~~ BT~ TOM " P - FATTJTJ ^ T ~ C H E C K VALVE CLOSES INSTAN f LY~ C UPON FLOW R E V E R S A L . REAL L B > L L » L D » L B S » N R A T DTM'E'N"SraN"HB (~2 » 1QTT5T* V B ( 2 » 10 » 1 5 ) >HPB ( 2 »10 il5) » V P B ( 2 » 1 0 »1 5 ) » 1 L B ( 2 » 1 0 ) » D B ( 2 » l O ) » E L B f 2 * 1 0 ) » F B ( 2 » 1 0 ) » F F B ( 2 » 1 0 ) -2 A B ( 2 » 1 0 ) » A R B ( 2 » 1 0 ) » T H B ( 2 » 1 0 ) » C 2 B ( - 2 » 1 0 ) »HFB ( 2 » l 0 ) , "3 Q " 0 B ( ' 2 » 1 0 ) * A N ( 2 » 1 0 ) » L B S ( 2 » 1 0 ) » N K ( 2 » 1 0 ) » C I N C ( 2 * 1 0 ) » 4 H L ( 2 » l O » 2 0 ) t V L ( 2 » l O » 2 0 ) » H P L C 2 » l O , 2 0 ) » V P L ( 2 • 1 0 » 2 0 ) * 5_ HD ( 1.0 >10 » ?0 ) |Vp ( 1 0 > 1 0 , ?0 ) >HPD( 1 0 - 1 0 » ?0 ) » V P D ( 1 . 0 Q O > ? 0 ) , "6 : " S T N X Q " , T O T T W N ( I D * flATTT 11) • BVN ( 13. ) , BAN ( 11) COMMON / C M 1 / V B > H B » A B * T H B /CM2 / VPB HPB • C 2 B » FFB , ARB 1 / C M 3 / V L » H L » A L » T H L / C M 4 / V P L » H P L » C 2 L * F F L , A R L 12 _U 10 9 "2" 7CM 57VTJ7HTJVXD7TTTTJ" • C 2 D ,"FF D , A RD 3 /CM7/HSPEC»ELB»FB»FL»FD»NK»SINA,LBS 4 /CM8/CSP-CINC»NLAT,HRES ~5 /T7M"9TCT01 CF , CORT , CCH , Z A * A A , EXP , VPA I R,VAIR»ZP,DT 6 /CM10/ALPHA»ALPHAP» CP»HRAT,QRAT,DV*DALPHA 7 /CM11/DX,BAN»BVN,HAN»HVN D_A~TA F~0~R~~T7-fE~PTPTfS~i ; DATA L L » D L » A L » ' L D * D D » AD/ ' lOO. 0 » 4 . 0 , 4 0 0 0 . 0 , 1 0 0 . 0 , 2 . 0 , 4 0 0 0 . 0 / » 1 F D » F L » N L A T , N D I S T , N S P R / 2 * . 0 1 * 3 * 1 . 0 / » H S P E C » Q S P E C / 1 0 0 . , . 0 0 5 / , D~AT7A~F0"R THE AT'R CTTA I^TJETTi : : : 2 V O A I R » E X P , Z A * D A » Q O » C O R F , H F O / 1 0 0 . * 1 . 2 » 1 5 . » 5 . 0 0 * 5 . » 1 . 5 , 2 5 . / , DATA FOR THE CENTR I FUGAL P U M P . 3 WR2 >NRAT ,EFFR/2 50. ,1400. , . 84/"»DX7TLA~ST7. i TTO . /" ~ Lo C THE PUMP CHARACTERISTICS ARE STORED IN THE COMPUTER BY THE C FOLLOWING READ STATEMENT. R-|T-A;D 8~»"HVNTHA~NTB"VNTB"A"N : " 8 FORMAT(11F6.0) DT=LL/AL Z = 3 . 1 4 1 ( 5 / 4 . 0 1=1 DO 11 J = l >10 . R~EATJ 9 » DB ( I » J ) » E L B ( I » J ) »LB- ( . I *J) » A B ( I . » J ) » F B ( I » J ) C E L B = E L E V A T I O N OF THE DOWNSTREAM END OF THE P I P E ABOVE THE DATUM. 9 FORMAT ( 5 F 1 0_._0J D"B"riT^T^D811TjT7T2T0 : C COMPUTATION OF THE P I P E C O N S T A N T S . A R 8 ( I » J ) = Z * D B M > J ) * * 2 F F B ( I * J ) = F B ( I » J ) * D T / ( 2 . 0 * D B ( I » J ) ) NK( J ) = L B ( I , J ) / ( D T * A B ( I *J) ) "C NO"ITJ7~=~N0"MBER~"OF~"R"EAClTES INTO WHICH P I P E J IS D I V I D E D . '. ~ A N ( I » J ) = N K ( I , J ) THB( I > J )=AN( I >J)*DT/|_B( I » J ) rB"'Sl"ITDT=TrBTTTJT77rRTTTTn : N K ( I » J ) = N K ( I , J ) +1 F B ( I » J ) = ( F B ( I » J ) * L B S ( I » J ) ) / ( 6 4 . 4 * D B ( I » J ) ) r 'FTJTEOTl ) GO TO~T0 ' ' 1 ' : S I NA (-I » J ) = ( ELB ( I » J - l ) - E L B ( I • J ) ) / LB ( I » J ) C INC! I • J ) =S I NA ( I * J )*C2B( I >J)*DT  G - 0 — r - 0 — n : ; S I N A t I •• J ) = ELB( I » J ) / L B ( I , J ) ' C I N C ( I » J ) = S I N A ( I » J ) * C 2 B ( T * J ) * D T N KT'ITJ" r=WriTJT~+1 : CONTINUE NSPR=NUMBER OF S P R I N K L E R S ON A D I S T R I B U T O R . K~M=T*N~SP"K ~ : ' : ; NDIST=NUMBER OF D I S T R I B U T O R S ON A L A T E R A L . KL=2*NDIST . COM P0'T"A"TT"0.'N~0F THE PT~P~r^WS'T A~NTTF6TTTHE~LATERALS AND D I S T R I B U T O R S . ~ DD = D D / 1 2 . 0 DL = D L / 1 2 . 0 ARL = Z*DT*"-*'2 :  ARD=Z*DD**2 C 2 L = 3 2 . 2 / AL C2D=3"2T2~~7~~ATj : ' £' THL = DT/LL THD=OT/LD ' F F L = F IT* D T T (TTU^TJ L") FFD=FD*DT / C 2 . 0 * D D ) F L = ( F L * L L ) / ( 6 4 . 4 * D L ) 10 1 1 C 10 F D = ( F D * L D ) / ( 6 4 . 4 * D D ) C S P = Q S P E C / S Q R T ( H S P E C ) SUBR0UTI_NE STEADY COMPUTES THE STEADY STATE PRESSURES AND " V E L O C I T I E S IN THE " P I P I N G S Y S T E M . C A L L S T E A D Y ( 1 , 1 0 , 1 0 ) H R E S = H B ( 1 » 1 0 , 1 1 ) "C'O'M P'U T A I "0"N~ OFTONS'T?TNT"S"-TOR~THT~~ATR CHAMBER • ' FOR DES IGNAT ION OF THE V A R I A B L E S SEE SECT ION 2 . 8 . AA=Z*QA**2 CCH=3~4;-Z7A"~ : CF=HF0/<Q0**2) Z P = Z A - V 0 A I R/AA H 0 A IR = H B ( I , 1 , 2 ) - Z P + 3 4 . VAIR=VOAIR C 1 0 = H O A I R * ( V O A I R * # E X P ) CO"WPTJ"T7CTT0T\r~0F CTTITSTTANTS FOR THE SUBROUTINE PUMPT FOR DESIGNAT ION OF THE V A R I A B L E S , SEE SECT ION 2 . 9 . HRAT = HB( 1» 1 , 1 ) O R ^ A T W B l T T " l ~ , l T^R " B T l T T ) ' C P M = A R B ( 1 » 1 ) / Q R A T T R = ( 6 0 . * 6 2 . 4 * H R A T * Q R A T ) / ( 2 . * 3 . 1 4 1 6 # N R A T * E F F R ) •Or=zr(3 oV*T2T2*TTT»D~ T )/ (3 .1416*NRAT*WR2~) ~ A L P H A = 1 . V P B ( 1 , 1 , 1 ) = V B ( 1 » 1 » 1 ) A"L~PTTAP"=~arc P"H A : : D V = 0 . 0 DALPHA = 0 . 0 H O : : : MM = 0 W R I T E ( 6 » 1 3 ) T , A L P H A , V A I R , Z P FOR M'A'TTIXT5"HT"rM"E =TF8~.~3~77 5X , 6H ACPTTA= , F 8 . 3 / 5X » 5HVA I R= , F 8 . 3 / 5 X , 3 H Z P = , F 8 . 3 / / 2 X , 1 1 H B R A N C H P I P E /) NLAT=NUMBER OF L A T E R A L S ON THE BRANCH P I P E . D"0~T5~J"STTNC7n W R I T E ( 6 » 1 4 ) J » ( H B ( I » J - K ) » K = 1 » 1 1 ) » ( V B ( I » J » K ) » K = 1 » 1 1 > F O R M A T ( 3 X , 1 4 , 1 1 ( 4 X , F 7 . 2 ) / 7 X , 1 1 ( 4 X , F 7 . 2 ) ) CONTINUE I F t T . G T . l . ) GO TO 22 W R I T E ( 6 » 1 6 ) FO'RM'ATI_4TCTTHL7A"TER7AT_ /) ' : DO 18 J = 1 , N L A T KK=2*NDIST " R " W R I T E ( 6 » 1 7 ) J » <HL( I , J , K ) , K = 1 , K K , 2 ) , < V L ( I , J , K ) , K = 1 , K K , 2 > 17 F 0 R M A T ( 3 X , I 4 , 1 0 ( 4 X , F 7 . 2 ) / 7 X , 1 0 ( 4 X , F 7 . 2 ) ) 18 CONTINUE b O ^ T ^ T T T T E X r : : : : W R I T E ( 6 » 1 9 ) J 19 F O R M A T ( / / 4 X » 8 H L A T E R A L = , I 4 / M X , 1 1 H D I S T R I B U T O R /) 12 ..11 10 DO 2 1 K = T T N D 1 S T " X K = 2 * N S P R W R I T E ( 6 » 2 0 ) K » ( H D ( J » K » M ) , M = 1 » K K » 2 > » < V D ( J » K » M ) , M = 1 , K K , 2 ) 2 0 F O R M A T ( 4 X » 1 4 , 1 0 ( 4 X , F 7 . 2 ) / 8 X , 1 0 ( 4 X , F 7 . 2 ) ) 2 1 C O N T I N U E 2 2 T =T + DT TLAST=T IME UP TO WHICH TRANSIENT C O N D l t l O N S ARE TO BE D E T E R M I N E D . . I F ( T . G T • T L A S T ) GO TO 27 MM=MM+1 "C P U M P I S " S I T U A T E D AT T H E F I R S T S E C T I O N OF T H E F I R S T P I P E O F T H E F I R S T C B R A N C H . C A L L P U M P 1 ( 1 , 1 , 1 ) "C A I " R C H A M B E R I S S I T U A T E D AT T H E S E C O N D S E C T I O N OF T H E F I R S T P I P E C O F T H E F I R S T B R A N C H . C A L L C H A M 2 ( 1 , 1 , 2 , 1 , 3 ) , ^ c _ 2 1 _ _ J _ T . r N . [ : _ . T . . K K = N K ( I , J ) M = 2 I T CD . T ^ Q V I T M =~4~ ' C A L L I N T E R ( I , J » M , K K - 1 ) C A L L BR 1 ( I , J , K K ) ~ZT CO'NTTNTJ E ~ : C A L L R E S E R D ( 1 , 1 0 , 1 1 ) C A L L BR2( I , N L A T , N D I ST) CATTL^SPR.I N K ( NDTSTTN'SP'R") D V = V P B ( 1 , 1 , 1 ) - V B ( 1 , 1 , 1 ) D A L P H A = A L P H A P - A L P H A D-0"2 '6 -^J=TT -NrAT K K = N K ( I , J ) DO 24 K = 1 , K K V B T r r j n a = v P B ( r"»"jTK~) — : : _  HB( I , J , K ) = H P B < I , J , K ) oo 2 4 . C O N T I N U E D - 0 _ 2 . 5 . . _ K . . _ r _ 1 < t _ . . . V L ( I , J , K ) = V P L ( I , J , K ) H L ( I , J , K ) = H P L ( I , J , K ) 25 CONTINUE DO 26 K = 1 » N D I S T DO 26 KS = 1 , K M [ ; ; ~ V D T J " » K » " K S ) = V P D ( J » K » K S ) H D ( J » K » K S ) = H P D ( J » K » K S ) 26 CONTINUE A LPH"A~=ALPHA"P' 3 ' ' V A I R = V P A I R I F ( M M . L T . 4 ) GO TO 22 G r ( r _ n o _ r 2 : . 2 7 STOP END $TB"FTC~ST E'A D'Y ~ SUBROUTINE S T E A D Y ( N 3 R » N D I S T , N S P R ) REAL L B S ; DTM EITSTO'N RBT2 » 1 U » 1 5 > » V B < 2 » 1 0 » 1 5 ) » Q O B < 2 » 1 1 ) * H F B ( 2 » 1 0 ) , A R B ( 2 » 1 0 ) T " 1 H L ( ? * ] . 0 » 2 0 ) » V L ( 2 ' 10 »20 ) • HPL ( 2 »1 0 , ? 0 ) , 00 L M 0 , 1 1 ) • HFL ( 10 , 10 ) , 2 H D < 1 0 »10 , 2 0 ) , VD ( 1 0 , 1 0 , 2 0 ) , HPD( 1 0 , 1 0 , ? 0 ) , 0 0 D ( 1 0 , 1 1 ) , HFD ( 7 . 0 , 1 0 ) , 3 F'BlT"nr0_)_,~ErBT"2Tri01"TL_B~S"r2 , 1 0 ) , " Q 0 S P ( 1 0 , 10 , 2 0") , N K ( 2 , 1 0 ) , 4 DB< 2 »10 > » A N ( 2 » 1 0 ) , S I N A ( 2 » 1 0 ) » L B ( 2 » 1 0 ) * A B ( 2 » 1 0 ) » V P B ( 2 » 1 0 » 1 5 ) » 5 T H B ( 2 , 1 0 ) ,C2B (2 » 1 0 ) , F F B ( 2 , 1 0 ) T H P B ( 2 > 1 0 » 1 5 ) » C I N C ( 2 » 1 0 ) »  '6 v T ' H X * TCTTTCTTW'DTTOTI 0 , 2 0 ) COMMON / C M 1 / V B , H B , A B , T H B / C M 2 / V P B , H P B , C 2 B , F F B , A R B 1 / C M 3 / V L , H L , A L , T H L /CM4/VPL , HPL , C2L , F F L , ARL 2 7~CM5VV"DXflDT7TDTT^ ^ 3 /CM7/ H S P E C , E L B , F B , F L , F D , N K , S I N A , L B S . • 4 / C M 8 / C S P , C I N C , N L A T , H R E S DA I A I0TT£R7'0T0T7 ; C PRESSURE HEAD S P E C I F I E D AT THE DOWNSTREAM END OF THE BRANCH P I P E . H S P E C = H S P E C + E L B ( N B R , N L A T ) D - Q - r 5 r r r= i - rNUAT : : I =IMLAT-I 1 + 1 MM = 0 ICM"A"X="Z*'N'STJ1R : DO 5 J = 1 , N D I S T DO 5 K = 2 , K M A X , 2 H D ( I 7 J , K ) = H SPTTC " 5 CONTINUE • C COMPUTATION OF THE V E L O C I T I E S AND F R I C T I O N L O S S E S IN THE ~C DI3TRTBCTTOirS^NTXnTrE^ 6 DO 7 J = 1 , N D I S T DO 7 K = 1 » N S P R . K K = ( N S P R - K + 1 ) * 2 K K B 2 = K K / 2 Q 0 S P ( I > J , K K ) = C S P * S Q R T ( H D < I » J » K K ) ) F F ( 1 < X " . " f r Q V l < l ^ A ^ ~ 0 W { T r 7 ^ 1 < B 2 + l ) = (5T0 : Q O D ( J » K K B 2 ) = O 0 D ( J » K K B 2 + 1 ) + O 0 S P ( I » J , K K ) V D ( I » J » K K ) = Q O D ( J » K K B 2 ) / A R D v r D " r r T J T K K ^ i " ) " W D ~ n » J » KK ) H F D ( J » K K B 2 ) = F D * V D ( I » J , K K ) * * 2 • 7 C O N T I N U E C CA"rcrjUATrO"N~OT~TPfE. V E L O C I T Y A N D F R I C T I O N L O S S I N T H E L A T E R A L . D O 8 J = 1 » N D I S T J J = N D I S T - J + 1 : r r t ^TE'QTNUTSTIXTOTTTTJ J + I J =O . o ~ Q O L ( I » J J ) = Q O L U » J J + l ) + Q O D ( J J » l ) V L < N B R > I > 2 * J J ) = Q O L < I > J J ) / A R L v r r N B ^ n T 2 * T r j ^ r ) ^ r r N " B " R » i . 2 * J J I H F L ( I » J J ) = F L * V L ( N B R » I » 2 * J J ) * * 2 8 C O N T T N U E ~ H P T 7 ( W R T l T l ) = H S ~ P E C ~ C C O R R E C T I O N O F P R E S S U R E S . I N T H E L A T E R A L A N D D I S T R I B U T O R S . D O _10 J = l » _ N D I S T R P " n W R ~ r T 2 ^ T T W r ^ T J B R » I , 2 * J - 1 ) - H F L ' ( I » J ) ' H P D - ( I • J» 1 ) = H P L < NBR • I » 2 * J ) I F ( J . E Q . N D I S T ) G O T O 9 ~ ~ ¥ P L 1 N ' B ' R T l V 2 * " J + l l " " = H P r r N ¥ R T r r 2 " * ^ l J 9 D O 1 0 , K = 2 » K M A X » 2 K B 2 = K / 2 H P " D " ( - T T " J T I C " r ^ H P " D ~ n » J • K - l ) - H F D ( J , KB21 :  I F ( K . . E Q . K M A X ) G O T O 10 H P D ( I » J » K + 1 ) = H P D ( I » J » K ) T O C O N T I N U E : " D O 1 1 J = 1 » N D I S T D O 1 1 K = 1 » K M A X • XX = T f D T T r j T l C T - ~ H P " D ( I » J » KTJ I F ( A B S ( X X ) , G T . T O L E R ) G O T O 1 0 0 1 1 C O N T I N U E . . G . 0 _ T . 0 . . _ r 3 . 1 0 0 D O 12 J = 1 » N D I S T H L ( N B R » I , 2 * J ) = H P L ( N B R » I > 2 * J ) H L ~ ( 1 N B R T ^ T - 2 " * " T - T T = H P ' L l " N r E r R T T l ~ 2 ^ i r ^ l ~ T : :  D O 1 2 K = 1 » K M A X H D ( I > J * K ) = H P D ( I » J » K ) 12 CONTINUE GO TO 6 13 N N = N K ( N B R , I ) R'BlNBRTr7irNT=TTS~PEC : I F ( I . E O . N L A T ) Q 0 B ( N B R , 1 + 1 ) = 0 . 0 Q 0 B ( N B R > I ) = Q 0 B ( N B R » I + 1 ) + Q 0 L ( 1 , 1 ) • VB""("NBRTT-rN"N"")"W0ffrN"B"R"rr)"7WBl^BRTn : HFBf NBR • I ) = F B ( N B R , I ) * V B ( N B R , I , N N ) * * 2 DO 14 J = 2 , N N : JJ=NW=rr+T : '• V B ( N B R » I , J J ) = V B ( N B R » I , N N ) I F ( I . ' EQ . 1 . AND. J J . E 0 . 2 ) GO TO 16 HBTNBRTTTJT)~ = HBTNBTR~riVJ^^ : ~ GO TO 14 16 H B ( N B R , I » J J ) = H B ( N B R , I » J J + 1 ) T 4 ClONTrNTJE : HSPEC = H8(N B R , I , 1 ) 15 CONTINUE RETURN : " : END S I B F T C INTER SUBRUUTTNL 1 N I E R ( I , J , K 1 » K 2 ) : C SUBROUTINE FOR COMPUTING THE TRANSIENT CONDITIONS AT THE C INTERMEDIATE SECT IONS OF A BRANCH P I P E . DTME"NST01OTPF( 2 , 1 0 , 1 5 ) » VPB ( 2 » 1 C",T5lTc2rJT2TfOT", FFB ( 2 »T0 "T, 1 C I N C ( 2 , 1 0 ) , A R B ( 2 , 1 0 ) C O M M O N / C M 2 / V P B , H P B , C 2 B , F F B , A R B / C M 8 / C S P , C I N C , N L A T , H R E S DTJ-T7"-X^l<CrTK"2 ~ H R = H E A D ( I , J , K , K - 1 , 1 ) V R = V E L ( I » J , K , K - 1 , 1 ) vs=-v E rrrruTKTK + T T D • H S = H E A D ( I , J , K , K + 1 , 1 ) VPB'( I » J » K ) = 0 . 5 * ( VR+VS+ ( H R - H 5 ) » C 2 B ( I » J ) - FFB ( I , J ) * ( V R * A B 5 ( VR ) + I — VS ^ A~B~S ( V 5 ) ) - TTNTCrTT)*! V R - V S ) ) ~~ HPB( I , J , K ) = 0 . 5 * < H R + H S + ( V R - V S ) / C 2 B ( I , J ) - ( F F B ( I , J ) / C 2 B ( 1 , J ) ) * 1 ( V R * A B S ( V R ) - V S * A B S ( V S ) ) - C I N C ( I , J ) * ( V R + V S ) / C 2 B ( I , J ) ) T 7 CO'NTTN U'E RETURN END "S"rB"FTC~"R'E'STiRT5 : " SUBROUTINE R E S E R D ( I , J , K ) C SUBROUTINE FOR A RESERVOIR OF CONSTANT WATER SURFACE E L E V A T I O N . C. RESERVOIR IS S I T U A T E D AT THE DOWNSTREAM END OF THE S Y S T E M . D IMENSION V P ( 2 » 1 0 » 1 5 ) * H P ( 2 » 1 0 » 1 5 ) » C 2 ( 2 * 1 0 ) » F F ( ? t l O > » A R ( 2 » I O ) » 1 _ C INC( 2 » 10 ) C C W W N / C M 2 / V T ^ T P T C 2 > F F » A R : '. 1:- / C M 8 / C S P » C I N C » N L A T , H R E S H P ( I » J , K ) = H R E S ~ R-R~=fl E X D T I T J T K » K - 1 * n ' : VR =VEL ( I » J * K > K - 1 » 1 ) C3=VR + C2( I » J ) * H R - C I N C ( I . J ) * V R - F F ( I » J ) * V R * A B S ( V R ) VP~l I » J , KT^CT^CS U » J ) *HPTES : RETURN END "$TB"FTC ~CR~A~M2 : SUBROUTINE C H A M 2 ( I * J » K » M » N ) C A_I R E_X P_A N S I ON FOLLOWS THE LAW H*V AI R » * 1 . 2 = CO NSTANT Dl"METTSTC"N-V"P"B"( 2 »10 »15 ) »HPB ( 2 » 10» 15 ) » V B ( 2 * 1 0 T f 5 T 7 H B ( 2 » 10 • 15 ) » 1 C 2 B ( 2 » 1 0 ) » F F B ( 2 » l O ) * A R B ( 2 » l O ) . t H B ( 2 » l O ) » A B { ? r l O ) , 2 C I N C < 2 » 1 0 J "TDlWCfN 7~CWr/~VSTHQl~ABTJHB /~CWT7VPWjTiPW^/2WiTWWV^Q 1 / C M 8 / C S P » C I N C » N L A T » H R E S 2 / C M 9 / C 1 0 , C F , C 0 R F > C C H , Z A , A A , E X P , V P A I R , V A I R , Z P » D T V ^ V ' E T r r T T - M TNTN"+ rVT] :  HS = HEAD ( I »M»N» 'N+1 » 1 ) V R = V E L ( I » J » K » K - 1 * 1 ) H R-=HEW(TTXJTKTK^rTT) : ! C 1 = V S - C 2 B ( I » M ) * H S - F F B ( I » M ) * V S * A B S ( V S - ) + C I N C ( I » M ) * V S C 3 = V R + C 2 B ( I » J ) * H R - F F B ( I » J ) * V R * A B S ( V R ) - C I N C ( I » J ) * V R C T 1 - ^ V B - ( X 7 f ; r r N T ^ r A - R - B - T i , M) - V B T T^Trr^ T ^ A R ^ T l T j - ) : ~. ' C A I R = ( V A I R + C 1 1 * D T ) . H P B ( I , J » K ) = C ! . 0 / ( C A I R**EXP ) - ( CORF*CF*C 11*ABS ( C I 1 ) +CCH + CA I R /AA ) FPBTITMTN - ) =H'P"B1"TTJTK~)' : ' V P B ( I » J » K ) = C 3 - C 2 B ( I » J ) * H P B ( I » J » K ) V P B ( I » M » N ) =C1+C2B( I » M)*HPB( I ? M » N ) vvmit=\rrXrRTQT5^TirrvTwrrv 1 ( V B ( I * J » K ) + V P B ( I » J » K ) ) # A R B ( I * J ) ) Z P = Z A - V P A I R / A A RETURN J > END - M S I B F T C BR 1 isuHR^TjTrNF^HRTrrrTj ixT : C SUBROUTINE FOR CONNECTION OF THE BRANCH P I P E AMD L A T E R A L . D IMENSION H P B ( 2 » 1 0 » 1 5 ) * V P B ( 2 » 1 0 » 1 5 ) > C 2 B ( 2 • 1 0 ) » F F B ( 2» 10) , 1 A R B ( 2 » 1 0 ) , V P L < 2 , 1 0 , 2 0 ) , H P L ( 2 , 1 0 , 2 0 ) » C I N C ( 2 , 1 0 ) COMMON / C M 2 / V P B , H P B , C 2 B » F F B , A R B / C M 4 / V P L , H P L » C 2 L » F F L , A R L 1 / C M 8 / C S P , C I N C , N L A T , H R E S HRB= H E A D ( I , J » K » K - 1 » 1 ) VSL= VEL ( I » J » 1 » 2 » 2 ) HSL= HEAD ( T " » ' J , 1 , 2 , 2 ) C3B = VRB + C2B( I , J ) * H R B - FFB( I , J ) * V R B * A B S ( V R B ) - C I N C ( I , J ) * V R B C1L=VSL - C 2 L * H S L - F F L •* VSL * A B S ( V S L ) ~T"F"QTCTTNITAT1 G~0 TO T5~ : C AS RESERVOIR OF CONSTANT WATER SURFACE ELEVAT ION IS S I T U A T E D AT C . THE DOWNSTREAM END OF THE 10TH BRANCH P I P E , S P E C I A L BOUNDARY C CONDTT I ONS ARE R'E'G'U I RED FOR TH I S "SECT I 0>i . ~ HPB( I , J » K ) = H R E S GO TO 16 T 5 \TSB=~VEL ( 1 , J +1 , 1 , 2 , 1 ) * HSB= H E A D ( I , J + l , 1 , 2 , 1 ) C l B = VSB - C2B( I » J + l ) * H S B - F F B ( I , J + l ) * V S B * A B 5 ( V 5 B ) + C I N C ( I , J + l ) * V S B HPBTITJTK") = CC3T3~*7kRT3TTTj~) : r ^TB~~ : ^R73T7T j+Tf ) - C1L * ARL) 7 : ~ 1 ( C2B ( I , J ) *ARB ( I , J )+C2B ( I , J + l ) -""ARB ( I , J + l ) + C 2 L * A R L ) H P B ( I , J + l , 1 ) = H P B ( I , J « K ) •  VPBlTTZI"+T^T~CrB~"C2Tfr i , J +1 ) * R W T T , J + 1 , 1 ) : 16 H P L ( I , J , 1 ) = H P B ( I , J , K ) V PL ( I , J , 1 )• =C I L + C 2 L * H P L ( I , J , 1 ) - V"P"B""(TTJTK~) = C3T3~~^CTBTITJ7*HPB ( I", J , K ) RETURN END . S IBF I ' C~ER2 ~ SUBROUTINE B R 2 ( I , N L A T , N D I ST) C SUBROUTINE FOR CONNECTION OF THE L A T E R A L AND D I S T R I B U T O R S , ~DTMEWST07r~V'PT"(TTll)^^ ) , H P D ( 1 0 , 1 0 , 2 0 ) COMMON / C M 4 / V P L , H P L , C 2 L , F F L , A R L / C M 6 / V P D , H P D , C 2 D , F F D , A R D DO 18 J=1*NLAT "DTJ—r8~LT=-i~7N'Dl~Sl K = 2*L VRL = V E L ( I , J , K , K - 1 » 2 ) • -H"Rr_'=HEAD"'("rrjTKTK^r7"2~) VSD =VEL ( J , L » 1 , 2 , 3 ) HS-D = H E A D ( J » L » 1 » 2 » 3 ) ~C ' J L - "=~V RXTT-"~C2 TTfRRL - F F U^V R L * A B S rVRCT : C1D =VSD - C2D*HSD - F F D * V S D * A B S ( V S D ) IF ( L . L T . N D I S T ) GO TO 16 HPL( I » J » K ) = ( C 3 L * A R L - ~ C 1 D * A R D ) / ( C 2 L * A R L + C?0*A~RD) GO TO 17 16 VSL = VLL ( I > J> K + l » K + 2 » 2) R"S"L = H E W ( 1 » J » K + l , K + 2T21 : ~ C1L =VSL - C 2 L * H S L - F F L *VSL * A B S ( V S L ) H P L U » J » K ) = ( C 3 L * A R L - C l L * A R L - C 1 D * A R D ) ' / ( 2 . 0*C2L*ARL+C2D*ARD) . R - p - r r i ~ - jTIC+"r)^"H"Pn"I » J , K ) : VPL( I » J » K+ 1 ) = C1L +• C 2 L * H P L ( I » J » K + 1.) 17 H P D ( J , L * 1) = HPL( I » J » K) WO ( J , L • 1 ) =—CTD + C 2 D * H P D ( J » L » 1 ) ' V P L ( I » J , K ) = ' C 3 L - C 2 L * H P L ( I » J » < ) • 18 CONTINUE . RTTURN : : : END S I B F T C S P R I N K S W R W f l NE S P R I N K ( NDI S T > N S P R ) [ DIMENSION V P D ( 1 0 » 1 0 » 2 0 ) » H P D ( 1 0 » 1 0 » 2 0 ) » C I N C ( 2 » i n ) C O M M O N / C M 6 / V P D » H P D » C 2 D » F F D » A R D / C M 8 / C S P , . C I N C » N L A T , H R E S CK=C"2"D""*7AR"D~7T"S"P ~~- • " ! KK=2*NSPR DO 13 I = 1»NLA.T DITTT-.T=TTN'D1ST : DO 13 K=2»KK>2 VR= V E L ( I » J » K » K - 1 , 3 ) . H R _ R c A D ( j ,j7]<7K -1T3~) : C 3 = V R + C 2 D * H R - F F D * V R * A B S ( V R ) C S P E C I A L BOUNDARY CONDIT IONS' ARE REQUIRED FOR THE LAST S P R I N K L E R . TT"rKTUTVKKT~GO"TO — Tl \ :  H S Q = ( - 1 . + S Q R T ( 1 . + A . * C K * C 3 * A R D / C S P ) ) / ( 2 « * C K ) H P D ( I » J » K ) = H S Q * A B S ( H S Q ) Q-Q—J Q — ^ 11 VS=VEL ( I * J > K+1,K + 2 » 3) HS = HEAD( I * J » K + 1 , K + 2 , 3 ) CT^S~-C2V^S-=FTT)'^V^A-BSTVS1 :  I F ( C 3 . G T . C 1 ) GO TO 100 PRINT 1 6 , C 3 » C 1 > H R , H S , V R , V S , F F D » I » J * K T6~. F"0RMAT"(~"""6TT0."3TFr2"."6T31"8") : : £ 100 CONTINUE f-H S Q = ( - 1 . + S Q R T ( 1 . + 8 . * C K * ( C 3 - C 1 ) * A R D / C S P ) ) / ( 4 . * C K ) _ ^ . p ^ ^ - ^ ^ - ^ ^ . ^ ^ ^ . ^ ^ ^ . : H P D ( I • J » K + 1 ) = H P D ( I » J » K ) V P D ( I , J > K + 1 ) = C 1 + C 2 D * H P D ( I » J » K + l ) 12 V P D < I , J » K ) = C 3 - C 2 D * H P D ( I » J , K > 13 CONTINUE . RETURN  END S I B F T C VEL FUNCTION V E L ( I > J , K , K K , N )  ~C TH E""V A L I f E OF N DlElF TTTES T H E TYPE~OF P I P E . FOR BRANCH P I P E , L A T E R A L C AND D I S T R I B U T O R S IS EQUAL TO 1 , 2 AND 3 R E S P E C T I V E L Y . DI MENS ION V ( 2 » 1 0 • 1 5 ) » H ( 2 * 1 0 , 1 5 ) , T H ( 2 , 1 0 ) , A ( 2 , 1 0 ) , V L ( 2 , 1 0 , 2 0 ) , I rTr( _2T"rOTr0T7VTJTT0 , 1 0 , 2 0 ) , HDTTUTTGTZQl COMMON / C M 1 / V , H , A , T H / C M 3 / V L , H L , A L , T H L / C M 5 / V D , H D , A D , T H D GO TO ( 1 5 , 1 6 , 1 7 ) , N T 5 V F L ^ V T I T T ~ r K T - T H r i T D T * X r r ^ ' RETURN 16 V E L = V L ( I , J » K ) - T H L * A L * ( V L ( I , J , K ) - V L ( I , J , K K ) ) RTfTCTR'N : 17 V E L = V D ( I » J , K ) - T H D * A D * ( V D ( I , J , K ) - V D < I , J » K K ) ) RETURN EWD : S I B F T C HEAD . FUNCTION H E A D ( I , J , K , K K , N ) C " T H E ^ V A ' D J E - 0 f - N DbFTNES ! HE TYPE OF P T P E . ' ; DIMENSION V ( 2 » 1 0 , 1 5 ) , H ( 2 , 1 0 , 1 5 ) , T H ( 2 » 1 0 > , A ( 2 , 1 0 ) , V L ( ? , 1 0 , 20 ) , 1 H L ( 2 , 1 0 , 2 0 ) , V D ( 1 0 , 1 0 , 2 0 ) , H D ( 1 0 , 1 0 , 2 0 ) : cuMMo;N~~rc>Ti7~\rriTr^^ GO TO ( 1 5 , 1 6 , 1 7 ) , N 15 HEAD = H( I , J , K ) - T H ( I , J ) * A < I , J ) * < H ( + , J , K ) - H ( I , J , K K ) ) R'ETUR'N" ' : ~~ 16 H E A D = H L ( I » J » K ) - T H L * A L * < H L ( I , J » K ) - H L ( I , J » K K ) ) RETURN T 7 FEA~D=HDXrrJ7!T)^rHTJ^^ ' •RETURN END " i I B M C~"P"UMP1 : SUBROUTINE P U M P K I , J , K ) C CHECK V A L V E CLOSES INSTANTLY UPON FLOW R E V E R S A L . D T M E N S T O N ~ " V C 2 V l W 1 3 T , ~ H T 2 " r r O ^ £ 1 F F ( 2 , 1 0 ) , A R ( 2 * 1 0 ) , A ( 2 » 1 0 ) , T H ( 2 * 10 ) , C I N C ( 2 , 1 0 ) ^ COMMON / C M 1 / V , H , A , T H / C M 2 / V P , H P , C 2 , F F , A R 1 7"CM"8TC3PTCT7^CTNOTTPWE"S 2 / C M 1 0 / A L P H A , A L P H A P , C P , H R A T , Q R A T , D V , D A L P H A DATA T O L E R 1 , T O L E R 2 / . 0 1 , 0 . 0 4 / V S = V E L < I » J »K » K + l • 1 ) H S = H E A D < I » J » K » K + 1 » 1 ) C 1 = VS_- C 2 ( I » J ) * H S - F F ( I > J ) * V 5 * A B S ( V S ) + C I N C < I » J > * V S rrrv^TTjTKTTGT, O . o J G O TO 8 : : V P ( I , J » K ) = 0 . 0 H P ( I » J » K ) = - C l / C 2 ( I * J ) ~ R"E T'O'R'N - — 8 V P M = V ( I » J » k ) + 0 . 5 * D V A L P H A 1= A L P H A + 0 . 5 * D A L P H A v-R-H-s V~PWA"RTT , J ) /ORATT : 9 R AT I 0 = V R M / A L P H A 1 I F ( R A T I O . L E . 1 . 0 ) GO TO 1 0 ^RAT"I10irrr0^7"RA"TIO~ : : M = 2 GO TO 11 TO (5|-=i ; : • 1 1 C A L L P A R A B ( R A T I O » M » B P ) I F ( M . E w . l ) B E T T A = 8 P * A L P H A 1 * * 2 r F r M T E Q V 2 T 6 " E T T 7 \ W P ^ ^ D A L P H A = C P * B E T T A A L P H A 2 = A L P H A + 0 . 5 * D A L P H A IT'rAB'^rTrLTJ'fTA^'^^ATTPHA 1 ) . L E . T O L E R 1 ) GO TO 12 A L P H A 1 = A L P H A 2 GO TO 9 T 2 - A C p - R . A - p - = 7 r [ ; p - H A _ D _ A ^ p T r A _ : . V P P = V ( I » J , K ) + D V 1 3 V R P = V P P * A R ( I , J ) / O R A T R " A"ni5= ^ R P — 7 ' A T P T T A P ' : : I F ( R A T I O . L E . l . O ) G O TO 1 4 R A T I O = 1 . 0 / R A T I O M^4_ : GO TO 15 1 4 M = 3 T 5 CA"LT^T>7fR7fBTR-ATT U » M * HR PI : !  I F ( M . E Q . 3 ) H P 1 = H R P * H R A T * ( A L P H A P * * 2 ) I F ( M . E Q . 4 ) H P 1 = H R P * H R A T * ( A L P H A P * * 2 ) / ( R A T I 0 * * 2 ) - v'P"l~=~~C!~T-C"2Tr7 J"T*~H PT " £" I F ( A B S ( V P 1 - V P P ) . L E . T 0 L E R 2 ) GO TO 1 6 V P P = V P 1 -G'0""T0~1"3 : 1 6 I F ( V P 1 . L T . 0 . 0 ) V P 1 = 0 . 0 V P ( I , J , K ) = V P 1 H P ( I , J , K ) = H P I RETURN END TrBTTC~'PTR7U3 S U B R O U T I N E P A R A B ( X » MM•Y) D I M E N S I O N Y B A N ( 1 1 ) » Y B V N ( 1 1 ) , YHAN ( 1 1 ) , , Y H V N ( 1 1 ) X"OMMON7'CM1T'7"DXTYBXN"»YWNTYHAN , YHVN . ~ M= X/DX AM = M "THET"A = (~X-AM*DX) /D"X : I F ( M . E O . O ) THETA= THE T A - 1 . M=M+1 _ r F _ ( . H _ _ 1 _ T _ . 2 . . } _ M _ 2 GO TO ( 6 , 7 , 8 , 9 ) , M M 6 Y = Y B A N ( M ) + 0 . 5 * T H E T A * ( Y B A N ( M + l ) - Y B A N ( M - l ) + T H E T A * ( YB A N ( M + l ) + I YTJTA"NT|5F1 ) - - (M) ) ) : ~~ RETURN 7 Y = Y B V N ( M ) + 0 . 5 * T H E T A * ( Y B V M ( M + 1 ) - Y B V N ( M - 1 ) + T H E T A * ( Y B V N ( M + l ) + r Y B i / 7 n i T ^ _ ^ T r * T B w r M T r j RETURN - o 8 Y = Y H A N ( M ) + 0 . 5 * T H E T A * ( Y H A N ( M + l ) - Y H A N ( M - l ) + T H E T A * ( Y H A N ( M + l ) + 1 : rKKffCjJFY) - 2.* YHAN( M ) ) ) : RETURN 9 Y = Y H V N ( M ) + 0 . 5 * T H E T A * ( Y H V N ( M + l ) - Y H V N ( M - l ) + T H E T A * ( Y H V N ( M + l ) + YHVN ( M-1 ) - 2 . * YHVN ( M ) ~)~j RETURN END S t N T R Y . 1 1 2 . 3 2 8 . 5 6 . 7 6 1 1 . 1.08 1 . 0 8 1 1 . 0 8 2 1 . 0 8 5 1 . 0 9 1 . 0 9 5 1 . 1 0 1 . 0 9 7 1 . 0 8 1.0 5 1 . . 5 8 5 . 7 4 5 . 9 2 .98 1 . .48 . 5 2 . 5 5 . 5 9 . 6 4 .70" . 7 5 5 .82 • 8 8 .94 1 . 1 4 . 0. 1 0 0 0 . 4 0 0 0 . .0 1 1 4 . 2 0 . 1 0 0 0 . 4 0 0 0 . . 0 1 1 4 . 3 0 . 1 0 0 0 . 4 0 0 0 . . 0 1 1 2 . 1 8 . 1 0 0 0 . 4 0 0 0 . . 0 1 LZ . 4 0 . "1 00"0 . 4 0 0 0 . TO'l i—* 1 0 . 2-3 . 1 0 0 0 . 4 0 0 0 . . 0 1 1 0 . 2 3 . 1 0 0 0 . 4 0 0 0 . . 0 1 8 . [•iZ. 1UUO. "40"0"0 . . 0 1 8. 4 0 . 1 0 0 0 . 4 0 0 0 . . 0 1 6. 4 0 . 1 0 0 0 . 4 0 0 0 . . 0 1 APPENDIX D SUBROUTINES D-•1. For r e s e r v o i r (RESERU, RESERI) D-•2. For A i r Chamber (CHAMl) D-•3. For Surge Tank (SURGE1, SURGE2) D-•4. For Va lve (VALVE, PARAB) D-- 5 . For R e l i e f Va lve (RELIEF, PARAB) D-•6. For Pump (PUMP2, PARABB, PARABH) A P P E N D I X D - 1 ~$TBFTC~R~E~SERTJ SUBROUTINE RESERU < I » J > K) C T H I S SUBROUTINE IS FOR THE RESERVOIR AT THE UPSTREAM END OF THE "c STsrmi : ~ ~~ DIMENSION V P ( 2 » 1 0 - , 1 5 ) * H P ( 2 » 1 0 » 1 5 ) » C 2 ( 2 » 1 0 ) , F F ( 2 » 1 0 ) » A R ( 2 » 1 0 ) » 1 C I N C 1 2 » 1 0 ) COMMON /CM27VP »H"PTC2TFFTAR 1 / C M 8 / C S P » C I N C » N L A T , H R E S HS = HEAD( I »• J.» K • K +1 » 1 ) v"S="V"EirriTjTKTKTTri ~ H P ( I » J , K ) • =HRES C1 = V S - C 2 ( I , J ) * H S - F F ( I > J ) * V S * A B 5 ( V S ) + C I N C ( . I >J)*VS  VP"( " i T J T K T ^ C T + C T T n J ) # H P ( I , J » K ) RETURN END Y T B F T C — R " E 5 r R ~ 7 ; : ~ SUBROUTINE R E S E R I < I » J , K » M » N ) C TH I S _ S UBROUTINE IS FO R_ THE RESERVOIR- ANYWHERE IN THE S Y S T E M ,  C S EC T I ON ( I » J • K.) IS J U S T ' U P S T R E A M AND SECT ION ( I , M , N ) JUST C DOWNSTREAM OF THE R E S E R V O I R . D IMENSION V P t 2 » l O » 1 5 ) » H P ( 2 » 1 0 » 1 5 ) > C ? ( 2 * 1 0 ) » F F ( 7 * 1 0 J , A R ( ' ? , 1 0 ) , T~ C I N C ( 2 » 1 0 ) COMMON / C M 2 / V P » H P , C 2 » F F » A R 1 / C M 8 / C S P » C I N C » N L A T , H R E S RS = H~EA~D~(T» M » N » N +1 »1 ) V S = V E L ( I » M » N » N + 1 > 1 ) V R = V E L ( I » J » K . K - 1 » 1 ) TTR=WA~DTTTJTKTK - I TU C3=VR + C 2 ( I » J ) * H R - F F ( I > J ) * V R * A B S ( V R ) - C I N C ( I • J ) * V R C 1 = V S - C 2 ( I »M ) * H S - F F ( I » M ) * V S * A B S ( V S ) + C I N C ( I » M ) * V S ~ HP (I > J > :< ) = HRES " : H P ( I » M , N ) = H R E S V P ( I » J , K ) = C 3 - C 2 ( I » J ) * H P ( I , J » K ) VPT ' ITMTNT^CITC ITTMT^TTP ( I »M , N l : RETURN -END A P P E N D I X D - 2 S I B F T C CHAMl . ^S'UBROUTTNE^CHAMrriTjTK") C CHAMBER IS CLOSE TO PUMP , CHECK V A L V E CLOSES S IMULTANEOUSELY C WITH POWER F A I L U R E "C ATR~rXP7A"N rSXO iN~TS~l:"S^THE R M A L : DIMENSION V P ( 2 » 1 0 » 1 5 ) » H P ( 2 » 1 0 » 1 5 ) » C 2 ( 2 * 1 0 ) , F F ( 2 » 1 0 ) » A R ( 2 » 1 0 ) » . 1 C I NC ( 2 »' 1 0 ) »V ( 2 * 1 0 »1 5 ' »H ( 2 > 10 •> 1 5 ) > A ( 2 » 1 0 ) » TH ( 7 , 1 0 ) COM M W 7 C MT7 W H TATT fl ' : 1 /CM2/' - V P , H P » C 2 » F F t A R / CM 1 2/VPA I R » VA IR » Z P 2 / C M 1 3 / C 5 , C P » C F » C C H , Z A » A A 3 7"C fv r87r5TTCTNCTWOTTF^ ! VS = V E L ( I » J » K » K + 1 » 1 ) HS = HEAD ( I 9 J » K > K + 1  " C T ^ 7 S ^ C 2 T I , J )*H"S~ FF ( I » J ) * V S * A B S ( VS ) +C INC ( I » J > *VS C V A L U E S OF THE COSTANTS C 5 > C P , C F , C C H ARE COMPUTED IN THE MAIN C P R0 G R AMM E » FOR E X P R E S S I O N S FOR THE CONSTANTS*REFER TO S E C T I O N 2 . 8 . C"V="VA~IR + C P * V ( f , J , K ) H P ( I , J » K ) = C5/CV - CCH - C F * V ( I » J , K ) * A B S ( V ( I • J , K ) ) - CV/AA V P ( I » J » K J = C I + C 2 ( I » J ) * HP( I >J>K ) VPATR~"=~VATR + 0 . 5 * T V ( I , J »K T~+~\T>TlTjTkTT* C P • C ZA=HEIGHT OF THE A IR CHAMBER. C _AA=CROSS-SECTIONAL AREA OF THE CHAMBER.  RETURN END A P P E N D I X D - 3 TI^FTC~3^CJFT~ ' — • — SUBROUTINE SERGE 1<I , J , K » K K ) C T H I S SUB ROU TIN E IS FOR THE SURGE TANK AT T HE JUNOTION OF ~C ~P' I"P"E~JTA"NTTTl P E J + 1 . K = L A S T ^ F X T T O N OF P I P E ~ j T K K = SECT I ON NUMBER C OF THE FREE SURFACE IN THE SURGE T A N K . S U R G E TANK OF OR I F I C E T Y P E . C WA T ERHAMMER WAVES IN THE 5URGE TANK ARE CONS I PER E__D_. D I MEN SI ON VP"( 2 » TO • 1 5 ) "• HP ( 2 » 1 0TT5TTC2 ( 2 * 1 0 ) , FWTJTTOTTKRVIVTOU 1 C I N C ( 2 » 1 0 ) » V ( 2 * 1 0 » 1 5 ) * H ( 2 » 1 0 » 1 5 ) » A ( ? » 1 0 ) » T H ( ? » 1 0 ) -C 0 M M 0 N / C M 1 / V , H , A , T H / C M 2 / V P , H P , C 2 , F F , A R  1 / C M 1 5 / Z P , Z , C S , C S U 2 / C M 8 / C S P » C I N C » N L A T , H R E S V/R = VEL ( 1 ' J ' K » K - l i l )  H~R = HEADl I » J » K , K - 1 , 1 ) "" V S = V E L ( I , J + 1 , 1 , 2 , 1 ) H S = H E A D ( I » J + l , 1 , 2 • 1 ) V"S:s=VErTlTU+?T~T7y7IT H S S = H E A D ( I , J + 2 , 1 , 2 , 1 ) C3 = VR + C 2 < I , J ) * H R - F F ( I » J ) # V R # A B S ( V R ) - C INC ( I • J )*VR. CT="V"S~^^C2TTTJ+Tr*H~S^F"n , J ) *VS C l S = V S S - C 2 < I , J + 2 > * H S S - F F ( I , J + 2 ) * V S S * A 8 S ( V S S ) C _VALUES OF THE CONSTANTS CS AND C S U ARE DETERMINED I N THE_MAIN "C P'RTOGRXM'M"ETFDT^TXPRTSSIONS FoTTTl^TTolIFtTFrfTrs^E S FcTToN~T77 H P ( I » J » K) = (C3*AR( I * J ) - C 1 * A R ( I * J + l ) - C I S * A R ( I *J + 2) )/CSU H P ( I , J + l , 1) = HP( I , J , K ) H"P""CITJ + 2Tr)"=HPl"TTJ 'rKT ~ : " : V P ( I , J , K ) = C3 - C 2 ( I , J ) * H P ( I , J , K ) V P ( I , J + l , 1 ) =C1 + C2( I , J + l ) * H R ( I , J + l , 1 ) : V T : C T T U + ^ T I T ^ C I 3 T O ZP=Z+CS* < V P ( I , J , K ) * A R ( I , J ) - V P ( I , J + l , 1 ) * A R ( I , J + l ) ) H P ( I , J + 2 , K K ) = Z P V R 3 = V E l " ( l " V J + ^ K K 7 X K - l T " l " ) ' £ H R S = H E A D ( I , J + 2 , K K , K K - 1 , 1 ) • C3S= VRS+C2( I , J + 2 ) * H R 5 - F F ( I , J + 2 ) * V R S * A B S t VRS ) ' ' V'PTTT J+'2TK~KT"=CXS~-TTTITJ , KK) : RETURN • END S I B F T C SERGE2 ~~ SUBROUTINE S.URGE2 ( I » J » K ) C SUBROUTINE FOR THE SURGE TANK AT THE JUNCTION OF P I P E J » A N D J + 1 . C SURGE TANK OF O R I F I C E T Y P E . WT ERHAMMER WAVES IN THE TANK N E G L E C T E D . DIMENSION V PT2TT0TT5T7H P~( 2~>TO * 15 ) * C 2T2TTO > , F F ( ? » l 6 ) i A R ( 2 » l O ) » 1 C I N C ( 2 » 1 0 ) » V ( 2 » 1 0 » 1 5 ) » H ( 2 » 1 0 » 1 5 ) » A ( 2 » 7 0 ) » T H ( 2 » T 0 ) COMMON / CM. 1 / V • H » A » T H / C M 2 / V P , H P , C ? , F F , A R 2 / C M 8 / C S P , C I N C , N L A T , H R E S C 1 4 = V( I » J + 1 » 1 ) * A R ( I » J + 1 ) - V( I » J » K ) * A R ( I » J ) "C VTALUE~OF~T"H'E~CONSTTA N T 3 " ~ C T l T C T 3 "CWPWETTTN THE MAIN PROGT^A^MF. C FOR E X P R E S S I O N S FOR THE CONSTANTS , S E E SECT ION 2 . 7 . Z P = Z - C 1 3 * C 1 4 PTPXITT rKT^ZP^C 1 2 * C 1 4 * A B S ( C 1 4 l " ' HP ( I » J + 1 » 1 )' = H P ( I , J » K ) VR=VEL( I » J , K j K - l ) ' HR = HEAD T l • J K • K - 1 ) V S = V E L ( I » J + 1 , 1 » 2 ) HS = H E A D ( I , J + 1 , 1 , 2 ) C W R ~ T C 2 ( I , J ) #HR - F F ( I , J ) * V R * A B S ( V R ) C1 = VS - C 2 ( I , J + 1 ) * H S - F F ( I * J + 1 ) * V S * A B S ( V S ) V P ( I j O , K ) = C3 - C2 ( I » J ) * H P ( I , J > K )  VP riTjTDTT^CT+ C 2 T lHTJ + 1 ) * H P ( I » J+1•1) RETURN END $ T S F T C V A L V E S U B R O U T I N E V A L V E ( I , J » K ) c T H I S SUBROUTINE IS FOR A V A L V E AT THE DOWNSTREAM END OF A P I P E . c THE V A L V E D ISCHARGES INTO ATMOSPHERIC P R E S S U R E . D IMENSION V P ( 2 » 1 0 » i 5 ) » H P ( 2 » 1 0 » 1 5 ) » - C 2 ( 2 » 1 0 ) » F F ( ? » 1 0 J , ' A R ( 2 » l 0 ) » 1 C I N C ( 2 » 1 0 ) .COMMON / C M 2 / V P » H P » C 2 » F F » A R / C M 1 6 / T » T C » C V , T A U 1 / C M 8 / C S P » C I N C » N L A T » H R E S V R = V E L ( I > J » K » K - 1 » 1 ) H R = H E A D ( I » J » K » K - 1 » 1 ) C3=VR + C 2 ( I » J ) * H R - F F ( I » J ) * V R * A B S ( V R ) - C I N C ( I » J ) * V R c WHEN THE V A L V E IS F U L L Y C L O S E D » I T BECOMES A DEAD END AND IS c ANALYSED AS S U C H . I F ( T . G E . T C ) GO TO 6 c SUBROUTINE PARAB DETERMINES THE VALUE OF TAU BY P A R A B O L I C I N T E R -c P O L A T I O N . C A L L P A R A B ( T » TAU) c VALUE OF CV IS COMPUTED IN THE MAIN PROGRAMME.FOR E X P R E S S I O N FOR c THIS CONSTANT, SEE SECTION 2 . 5 . C4 = CV* TAU**2 V P_(_I_, Jj» K ?= 0 . 5 * C 4 * ( - 1 . 0 + SORT ( 1 . 0 + 4..Q*C3/CA-) ) • HP ( I , J,:< ) = (C3 - V P ( I » J » ' K ) ) / C 2 ( I » J ) RETURN 6 V P ( I > J , K ) = _ 0 «_0_ T T u = l 3 V o " H P ( I , J » K ) = C 3 / C 2 ( I » J ) RETURN E'N'D" S I B F T C PARAB SUBROUTINE P A R A B ( X , Y Y ) D"T MEN SI ON Y ( 1 0 ) " COMMON / C M 1 1 / D X , Y M =X/DX THETA = ( X - A M * D X ) / D X I F ( M . E Q . O ) THETA= T H E T A - 1 . M = M+1 I F ( M - . L T . 2 > M = 2 YY= Y (M) + 0 . 5 » T H E T A * ( Y ( M + D - Y ( M - D + T H E T A * ( Y ( M + 1 ) + Y ( M - D -1 2" .*Y (M ) ) ) R E T U R N END T 12 _ JLL 10 9 5 S ' 4 3 A P P E N D I X D - 5 S I B F T C R E L I E F SUBROUTINE R E L I E F ( I , J » < • ) C R E L I E F V A L V E ADJACENT TO THE PUMP.UPON FLOW REVERSAL CHECK V A L V E C CLOSES I N S T A N T L Y , R E L I E F VALVE OPENS G R A D U A L L Y . C T I M E - T A U CURVE STORED IN THE COMPUTER. INTERMEDIATE V A L U E S C • INTERPOLATED P A R A S O L I C A L L Y BY S U B R . P A R A B . D IMENSION V P ( 2 > 1 0 , 1 5 ) , H P ( 2 , 1 0 , 1 5 ) , C ? ( 2 , 1 0 ) , F F ( 7 , 1 0 ) , A R ( ? , 1 0 ) , 1 C I N C ( 2 , 1 0 ) COMMON / C M 2 / V P , H P , C 2 » F F , A R /CM 1 7 / T , T 1 , C V , T A U 1 T c m T C S ? ^ X J W C ^ N L a T T F m r s- • -DATA T 2 » T 3 , T 4 / 4 . , 1 0 . , 3 0 . / H S = H E A D ( I , J , K » K + 1 , 1 ) . V"S"= V E T f l T J , K, K + T T T 1 ~~ C 1 = V S - C 2 ( I , J ) * H S - F F ( I , J > * V S * A B S ( V S ) + C I N C ( I , J ) * V S C T 1 = TI ME WHEN FLOW R E V E R S E S . . c T~2~= Tll^HnR"E131JTR_E"D~F O'R" FTjfLT~0P E NIN G . ' 7" C T 3 - T 2 = V A L V E REMAINS OPEN FOR TH IS P E R I O D . "" C T 4 - T 3 = TIME RQU I RED FOR FULL CLOSURE OF THE V A L V E * . T t _ t „ . t _ _ . . I F ( T T . G T . T 2 . A N D . T T . L T . T 3 ) GO TO 7 ; I F I T T . G E . T 3 ) GO TO 6 . ^ A - _ _ _ _ p . _ A R . ^ ^ ^ . . . " ' GO TO 8 6 I F ( T T . G E . T 4 ) GO TO 9 GO TO 8 1 " 1 2 7 T A U = 1 . _ _ _ _ . _ U C vTO7E~0"F~C'v' IS COMPTJTED IN THE MAIN PROGRAMME. FOR E X P R E S S I O N FOR 2 10 C C V , SEE SECT ION 2 . 5 . 8 C4=CV*TAU**2 3 s VP'( T "7J ","K~) "= 0 .5*C4*' ( "17 - S Q R ~ T 1 1 7 ^ T * C T 7 C4"f) gj . i H P ( I , J , K ) = ( V P ! I , J , K ) - C l ) / C 2 ( I , J ) ' ^ t 6 RETURN _ c ^ ^ . ^ . ^ ^ ^ ^ 5 4 9 T A U = 0 . 0 i V P { I , J , K ) = 0 . 0 V o H P ( I » J , K ) = - C 1 / C 2 ( I » J ) RETURN _E_ND . ; __. $ IBFTC" PARAB C THIS SUBROUTINE D E T E R M I N E S , BY P A R A B O L I C I N T E R P O L A T I O N , I N T E R M E D I -C ATE V A L U E S THE PUMP C H A R A C T E R I S T I C CURVES* THE CURVES ARE C STORED' IN THE "COMPUTER BY G I V I N G VALUES OF Y AT UNIFORM C I N T E R V A L S OF X . SUBROUTINE P A R A B ( X , M M , D X , Y ) DTME'N'STO'N 7HA"N"( 11 ) » Y B VN ( 11 ) , Y H A N l l l ) , Y H V N ( 1 1 ) , T A U O ( 1 1 ) , T A U C ( 1 1 ) COMMON / C M 1 8 / Y B A N , Y B V N , Y H A N » Y H V N , T A U O , T A U C M= X/DX A~M=M ' : THETA = ( X - A M * D X ) /DX I F ( M . E Q . O ) THETA= THE T A - 1 . • '• M"=M+T : I F ( M « L T . 2 ) M = 2 GO TO ( 6 , 7 , 8 , 9 , 1 0 , 1 1 ) , M M ; 6 Y ' ^YBTrTn^JT^^^ ) - Y B A N ( M - 1 ) +THETA* (YB AN t M + l ) + 1 Y B A N ( M - l ) - 2 . * Y B A N ( M ) ) ) RETURN 2 . 10 7 Y = Y B V N ( M ) + 0 . 1 Y B V N ( M - l ) RETURN 5^-THETA* ( YBVN (M+l ) -- 2 . * Y B V N ( M ) ) ) Y B V N ( M - 1 ) + T H E T A * ( Y B V N ( M + l ) + 8 Y = Y H A N ( M ) + 0 . 1 Y H A N ( M - l ) RETURN 5 * T H E T A * ( Y H A N ( M + l ) -- 2 . * Y H A N (M ) )•) Y H A N ( M - 1 ) + T H E T A * ( Y H A N ( M + l ) + 9 Y =YHVN(M)+0 . 1 Y H V N ( M - l ) RETURN 5 * T H E T A * ( Y H V N ( M + l ) -- 2 . * Y H V N ( M ) ) ) Y H V N ( M - 1 ) + T H E T A * ( Y H V N ( M + l ) + 10 Y =TAUO(M)+0 . 1 T A U O ( M - l ) RETURN 5 * T H E T A * ( T A U O ( M + l ) -- 2 .*TAUO(M) ) ) T A U O ( M - 1 ) + T H E T A * ( T A U O ( M + l ) + 11 Y =TAUC<M)+0. 1 T A U C ( M - l ) RETURN 5 * T H E T A * ( T A U C ( M + 1 ) -- 2 . * T A U C ( M ) ) ) T A U C ( M - 1 ) + T H E T A * ( T A U C ( M + l ) + END 156. '5 4 3 A P P E N D I X D - 6 $T8TTC~PTfMP2 ' SUBROUTINE P U M P 2 ( I * J » K ) C TH IS_ SUBROUT INE IS FOR THE PUMP OPERATING IN THE THREE ZONES "C OPERA"? TON. PUMP C H A R A C T E R I S T I C S ARE STORED IN THE COMPUTER C . THE INTERMEDIATE V A L U E S ARE INTERPOLATED P A R A B O L I C A L L Y . D IMENSION V P < 2 » 1 0 » 1 5 ) » H P ( 2 » 1 0 » 1 5 ) » C 2 < 2 » 1 0 > , F F ( ? » 1 0 > , A R ( 2 » 1 0 ) , 1 C I N C ( 2 » 1 0 ) » V ( 2 » 1 0 » 1 5 ) » H ( 2 » 1 0 COMMON / C M 1 / V » H . A » T H / C M 2 / V P , H P 1 . / C M 1 9 / A L P H A » A L P H A P » C P » H R A T , Q R A T » 1 5 ) » A ( 2 * 10) , T H ( 2 » 1 0 ) , C 2 » F F , A R » D V , D A L P H A » C P M 2 / C M 8 / C S P , C I N C , N L A T , H R E S DATA TOLER1•TOLER2 / . 0 2 5 , . 2 / MM = 0 NN = 0 V S = V E L ( I , J , K , K + 1 » 1 ) H S = H E A D ( I » J > K » K + 1 • 1 ) C1 = VS-C2"< H J ) * H S - F F ( I » J ) * V S * A B S ( V S ) + C I N C ( I » J ) * V S C COMPUTATION OF THE TRANSIENT SPEED OF THE P U M P . VPM = V ( I , J , K ) + 0 . 5 * DV ^Al"PHA"F='~AXPHXT^r5WATP"RA^ : VRM=VPM*CPM 6 R A T I O = V R M ' / A L P H A 1 NN^'NN^l I F ( N N . G T . 2 5 ) GO TO 20 SUBROUT_I_N_E_ _MVAL_G IVES THE VALUE OF M,WHICH D E F I N E S THE C H A R A C T . CURVE TO BE U S E D . C A L L M V A L < V R M » R A T I 0 , A L P H A l , M ) KKK=M/2 "IT T'MTE QTT*Xi< KT) GlT~TO~T3 : CALL P A R A B B ( R A T I O , M , B P ) B E T T A = B P * A L P H A 1 * * 2 GO TO 14 R A T I O = l . / R A T I O CALL P A R A 6 B ( R A T I O , M , B P ) BTTrA"=BP*7CCPT^^^ ~ DALPHA= C P * B E T T A . ALPHA2= ALPHA + 0 . 5 * DALPHA 4-12 J l 10 Q C ~C~ 12 13 14 I F ( A B S ( A L P H A 2 - ALPHA 1 ) . L E . T O L E R 1 ) GO TO 15 A L P H A 1 = ALPHA2 GO TO 6 2 0 W R I T E ( 6 » 2 1 ) A L P H A , A L P H A 2 21 F 0 R M A T ( 5 X , 1 6 H I T E R A T I 0 N F A I L E D , 2 F 1 2 • 3 ) 15 ALPHAP=ALPHA+DALPHA C_ COMPUTATION OF THE TRANSIENT STATE PUMPING H E A D . VPP = V ( I , J » K ) + DV 16 VRP=VPP*CPM MM=MM+1 I F ( M M . G T . 2 5 ) GO TO 22 RATIO= VRP / A L P H A P C A L L M V A L ( V K P , R A T 1 0 , A L P H A P , M ) KKK=M/2 IF (M•EQ•2*KKK) GO TO 17 -CALL P A R A B H ( R A T I 0 , M , H R P ) HP1=HRP*HRAT*ALPHAP**2 GO TO 18 17 R A I 1 0 = 1 . / K A I 1 0 CALL P A R A B H ( R A T I O , M , H R P ) HP1=HRP*HRAT*ALPHAP**2 / R A T I 0 * * 2 18 " V P l = C I + C 2 ( I » J ) » H P I I F ( A B S ( V P l - V P P ) . L E . T 0 L E R 2 ) GO TO 19 VPP =VP1 GO TO 16 22 W R I T E ( 6 » 2 1 ) V ( 1 , 1 , 1 ) , V P P 19 V P ( I » J , K ) = V P l H P ( I , J , K ) = HP 1 RETURN ' . END T S I B F T C PARABH "5UBRDUTTNE PAFTABH ( X » J J » Y ) : ' D IMENSION HAN <11) ,HVN( 1 1 ) , H A D ( 1 1 ) , H V D ( 1 1 ) , H A T ( 1 1 ) , H V T ( 1 1 ) COMMON / C M 2 0 / H A N , H V N , H A D , H V D , H A T , H V T DATA D X / . 1/ M= X/DX AM = M TH ETA _ ^7r=A7Wy5rr~7"D"X : I F ( M . E Q . O ) THETA= THE T A - 1 . M = M+1 I F ( M . L T . 2 ) M = 2 GO TO ( 6 , 7 , 8 , 9 , 1 0 * 1 1 ) » J J 6 Y=HAN(M)+ .5*THETA*(HAN(M+1) - H A N ( M - 1 )+THETA*(HAN(M+l ) + H A N ( M - 1 ) - 2 . 1 *HAN(M ) ) ) RETURN 7 Y = H V N ( M ) + . 5 * T H E T A * ( H V N ( M + l ) - H V N ( M - 1 ) + T H E T A * ( H V N ( M + l ) + H V N ( M - 1 ) - 2 . 1 * H V N ( M ) ) ) RETURN 8 Y = HAD< M ) + . 5 * T H E T A * ( H A D ( M + l ) - H A D ( M - 1 ) + T H E T A * ( H A D ( M + l ) + H A D ( M - 1 ) - 2 . 1 *HAD(M) ) ) RETURN 9 Y=HVD<M)+ .5*THETA*(HVD(M+l ) - H V D ( M - 1 ) +THETA*;( HVD( M+l ) + H V D ( M - 1 ) - 2 . 1 * H V D ( M ) ) ) RETURN 10 Y = H A T ( M ) + . 5 * T H E T A * t H A T ( M + l ) - H A T ( M - 1 ) + T H E T A * ( H A T ( M + l ) + H A T ( M - 1 ) - 2 . 1 *HAT (M) ) ) RETURN 1 1 Y = H V T ( M ) + . 5 * T H E T A * ( H V T ( M + l ) - H V T ( M - 1 ) + T H E T A * ( H V T ( M + l ) + H V T ( M - 1 )• 1 * H V T ( M ) ) ) RETURN END S I B F T C PARABB "SUBR"OUTrN"E~P"A"RXBBTXTJI)TYT " TH IS SUBROUTINE DETERMINES ,BY P A R A B O L I C INTERPOLAT I O N , INTERMEDI ATE TORQUE -RAT IOS FROM THE PUMP C H A R A C T E R I S T I C C U R V E S . T H E CURVES ARE " 3 T 0 ' R T D ~ ' T ! T T ' f i E ~ C 0 W ^ U X l T O R M : " I N T E R V A L S OF X . D IMENSION B A N ( 1 1 ) *BVN( 1 1 ) » B A D ( 1 1 ) » B V D ( 1 1 ) » B A T ( 1 1 ) » B V T ( 1 1 ) COMMON /CM217 BAN • BVNVBAD» B V D » B A T »'?. VT DATA D X / . l / M= X/DX A'M-M : : ~ ! : : THETA = ( X - A M * D X ) /DX I F ( M . E Q . O ) . THETA= THE T A - 1 • ' M=M+1 : - R -I F ( M . L T . 2 ) M = 2 GO TO ( 6 * 7 * 8 * 9 * 1 0 * 1 1 ) » J J '6 Y = B"AN~(^n+^^TTHTrr7C^TBW(^,+l ) - B A~NT M - l ) + T H E T A ^ B l W ^ + T T + ~ T i 7 V N T M - l ) - 2 . 1 * B A N ( M ) ) ) RETURN C C "C c 

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