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Optimum turbine gate operation to minimize speed change in an hydraulic turbine Bell, Peter Warren Wentworth 1966

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OPTIMUM TURBINE GATE OPERATION TO MINIMIZE SPEED CHANGE IN AN HYDRAULIC TURBINE by WARREN BELL B.A.Sc. ( E l e c t r i c a l E n g i n e e r i n g ) , U n i v e r s i t y o f B r i t i s h Columbia, 1962  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE  i n t h e Department of Civil  Engineering  We a c c e p t t h i s t h e s i s as c o n f o r m i n g required  to the  standard  THE UNIVERSITY OF BRITISH COLUMBIA April,  1966  In p r e s e n t i n g r e q u i r e m e n t s f o r an Columbia, for  granted It  the  gain  this  Head o f my  i s understood  financial  the  study.  copying of by  that  shall  Department o f C i v i l  April  27,  Library I  further  thesis  in p a r t i a l  not  be  f u l f i l m e n t of  University  of  make i t f r e e l y  agree that  by  his  p u b l i c a t i o n of  a l l o w e d w i t h o u t my  Columbia  the  British available  permission  for  f o r s c h o l a r l y p u r p o s e s may  Engineering  1966.  the  shall  Department or  copying or  The U n i v e r s i t y o f B r i t i s h V a n c o u v e r 8, C a n a d a Date  thesis  advanced degree at  I agree that  r e f e r e n c e and  extensive  this  be  representatives. this  thesis  written  for  permission  ABSTRACT  T h i s t h e s i s examines t h e i n f l u e n c e o f d i f f e r e n t gate o p e r a t i o n c u r v e s on t h e s u r p l u s o r d e f i c i e n c y o f energy i n p u t t o a h y d r a u l i c t u r b i n e accompanying a sudden change o f l o a d on t h e t u r b i n e . s o l u t i o n t o the problem i s obtained  by e v a l u a t i n g t h e energy i n p u t t o  the p e n s t o c k and t h e energy c o n v e r s i o n w i t h i n t h e p e n s t o c k transient conditions.  pressure  r e d u c t i o n i n the surplus o r d e f i c i e n c y o f  energy i n p u t t o t h e t u r b i n e c a n be o b t a i n e d operation curve.  during  The r e s u l t s show t h a t f o r g i v e n maximum  r i s e o r drop, a considerable  A general  by use o f a s u i t a b l e g a t e  A t t h e same time i t i s p o s s i b l e t o reduce t h e h y d r a u l i  o s c i l l a t i o n s i n t h e system.  111  TABLE OP CONTENTS  CHAPTER  II III  IV V  PAGE INTRODUCTION  1  ELEMENTS OF THE SCHNYDER-BERGERON GRAPHICAL SOLUTION OF THE A L L I E V I WATERHAMMER EQUATIONS '  .3  THEORY OF ENERGY CONVERSION I N THE PENSTOCK  4  APPLICATION OF ENERGY PRINCIPLES TO THE DETERMINATION OF OPTIMUM GATE OPERATION  22  EVALUATION OF DIFFERENT GATE OPERATION CURVES  45  CONCLUSIONS  75  SYMBOLS ABBREVIATIONS AND UNITS  83  APPENDIX I  IV  LIST OF FIGURES Number 1 2  Page 12  The b a s i c h y d r a u l i c system Conditions  i n t h e p e n s t o c k r e s u l t i n g from an 12  i n f i n i t e s i m a l wave A t y p i c a l g r a p h i c a l s o l u t i o n t o a waterhammer  15  problem An e n l a r g e d view o f t h e waterhammer c h a r t  showing 15  the c r e a t i o n o f a wave 5  T y p i c a l g r a p h i c a l s o l u t i o n s t o waterhammer problems  23  6  Power i n p u t t o t h e t u r b i n e d u r i n g t r a n s i e n t c o n d i t i o n s  23  7  The h e a d - v e l o c i t y c o n d i t i o n s f o r an i n f i n i t e s i m a l wave  24  Input v e l o c i t y t o the penstock  24  G r a p h i c a l s o l u t i o n s t o waterhammer problems F when  y  c  ' "•• n=Q+1 10  £>v n  - 1 )  (T  F  =o  n  _____ n=Q+1  C <^v (QL, - t ) = 0 n n F n v  37  Waterhammer c h a r t s and s u r p l u s energy p l o t s f o r t h e time i n t e r v a l t = T t o t = T Q, F  13  ^  - gate c l o s u r e  38  Waterhammer c h a r t s and s u r p l u s energy p l o t s f o r t h e time i n t e r v a l t = T  14  25  Waterhammer charts and s u r p l u s energy p l o t s f o r t h e time i n t e r v a l t = T„ t o t = T_  12  25  P l o t o f i n p u t v e l o c i t y t o the p e n s t o c k f o r t h e cond i t i o n  11  n  to t = T  - g a t e opening  39  Waterhammer c h a r t and g a t e c l o s u r e c u r v e f o r optimum gate operation  42  L i s t o f F i g u r e s (Cont'd) UTumber  _  Page  dh 15  Waterhammer c h a r t f o r g a t e c l o s u r e :  16  Waterhammer c h a r t and g a t e c l o s u r e c u r v e  2o >• —  ,yF  47  - instantaneous gate c l o s u r e 17  Waterhammer c h a r t s f o r d i f f e r e n t v a l u e s o f ^> 47  - instantaneous gate closure 18  42  Water v e l o c i t y a t t h e i n t a k e and a t t h e g a t e - optimum g a t e c l o s u r e  51  19  Waterhammer c h a r t f o r optimum g a t e c l o s u r e  51  20  Optimum g a t e c l o s u r e c u r v e  55  21  Water v e l o c i t y a t t h e i n t a k e - optimum g a t e o p e r a t i o n  55  22  Waterhammer c h a r t f o r t h e i n t e r v a l t = T  A  Q  t o t = T„ F  - optimum g a t e o p e r a t i o n 23  Waterhammer c h a r t and i n p u t v e l o c i t y f o r g a t e closure:  24  2p ^*  62  <y  Waterhammer c h a r t d e f i n i n g t h e r e g i o n s f o r optimum gate o p e r a t i o n :  25  t =  to t =  62  Comparison o f w i c k e t g a t e c l o s u r e c u r v e and optimum c l o s u r e curve  26  55  68  Waterhammer c h a r t comparing w i c k e t g a t e and optimum c l o s u r e curves  27  C l o s u r e curves f o r synchronized gate o p e r a t i o n  28  Comparison o f power .outputs f o r s i n g l e and s y n c h r o n i z e d gate o p e r a t i o n  68 72  72  L i s t of Figures  (Cont'd)  Number 29  Page Waterhammer c h a r t comparing s i n g l e and s y n c h r o n i z e d gate o p e r a t i o n  30  An upstream v a l v e  73 74  ACKNOWLEDGEMENT  The a u t h o r w i s h e s t o e x p r e s s h i s thanks t o h i s s u p e r v i s o r Dr. E. Ruus, f o r t h e v a l u a b l e c r i t i c i s m , guidance and encouragement. He a l s o w i s h e s t o e x p r e s s h i s thanks t o P r o f e s s o r J . F. M u i r f o r h i s v a l u a b l e a d v i c e and s u g g e s t i o n s .  1 INTRODUCTION  Many of the c o m p l i c a t e d  p r o c e s s e s of modern i n d u s t r y r e q u i r e  e l e c t r i c power whose f r e q u e n c y i s m a i n t a i n e d w i t h i n v e r y c l o s e (+ l / l O c y c l e ) .  I n any  l a r g e e l e c t r i c a l system, any major i n c r e a s e  d e c r e a s e i n the magnitude of the l o a d on the system can be i n s t a n t a n e o u s (such as would be caused by the l o s s of a line).  tolerances  F u r t h e r m o r e , the d i s t r i b u t i o n  or  considered  transmission  of t h i s l o a d change between the  v a r i o u s g e n e r a t o r s i s a l s o i n s t a n t a n e o u s when compared t o the speed of response of the t u r b i n e g o v e r n o r .  The  r e s u l t i n g change i n system f r e -  quency i s a p p r o x i m a t e l y p r o p o r t i o n a l to the square r o o t of the between the combined i n p u t t o the t u r b i n e s and connected g e n e r a t o r s ;  and  difference  the combined o u t p u t of  i n v e r s e l y p r o p o r t i o n a l t o the i n e r t i a of  whole system ( i . e . i n c l u d i n g connected l o a d ) .  To keep the  the  frequency  change ( o r c o r r e s p o n d i n g g e n e r a t o r speed change) w i t h i n s p e c i f i e d  limits  f o r an assumed major l o a d change i t i s n e c e s s a r y t o e i t h e r i n c r e a s e system i n e r t i a or d e c r e a s e the d i f f e r e n c e between t u r b i n e i n p u t g e n e r a t o r o u t p u t ; the c h o i c e b e i n g m a i n l y one In a hydraulic turbine-generator  to the t u r b i n e . t h i s input-output ditions. ponse, and and  the  and  of economics.  arrangement t h i s  d i f f e r e n c e i s a r e s u l t of the f i n i t e speed of response and o f the t u r b i n e g o v e r n o r and  the  input-output sensitivity  the i n e r t i a of the f l u i d s u p p l y i n g  T h i s paper i s concerned w i t h the p r o b l e m of  energy minimizing  d i f f e r e n c e i n a h y d r a u l i c t u r b i n e under t r a n s i e n t con-  I t i s assumed t h a t the g o v e r n o r s e n s i t i v i t y and the a l l o w a b l e p e n s t o c k p r e s s u r e  speed of r e s -  r i s e and drop a r e  specified  t h a t the o n l y v a r i a b l e q u a n t i t y i s the t u r b i n e g a t e c l o s u r e  I t i s f u r t h e r assumed t h a t the p e n s t o c k i s of c o n s t a n t  curve.  dimensions,  the  2 r e s e r v o i r e l e v a t i o n i s constant, the r a t e d speed a r e known and any  the t u r b i n e h e a d - d i s c h a r g e c u r v e s f o r i n f l u e n c e of t u r b i n e speed changes  on the h e a d - d i s c h a r g e r e l a t i o n s h i p s can be n e g l e c t e d change of + 1/10  (with a frequency  c y c l e t h i s l a s t assumption i s q u i t e j u s t i f i e d ) .  A l t h o u g h no optimum g a t e o p e r a t i o n c u r v e s are d e r i v e d i n t h i s p a p e r , an example i s worked showing t h a t the optimum c l o s u r e c u r v e f o r l i m i t i n g pressure  r i s e d e r i v e d by E. Ruus (6)  curve f o r minimizing The  the i n p u t - o u t p u t  i s a reasonably p r a c t i c a l  energy d i f f e r e n c e .  approach i n t h i s paper i s based on the g r a p h i c a l s o l u t i o n  o f the A l l i e v i c h a i n e q u a t i o n s d e v e l o p e d by Schnyder and B e r g e r o n ( 1 ) , combined w i t h an e v a l u a t i o n o f the energy c o n v e r s i o n  t a k i n g p l a c e i n the  p e n s t o c k under t r a n s i e n t c o n d i t i o n s . A l t h o u g h energy methods are  not  u s u a l l y used i n r a p i d l y v a r i e d f l o w problems because o f d i f f i c u l t y i n e v a l u a t i n g f r i c t i o n l o s s e s , i t i s shown t h a t i n t h i s case the l o s s e s be  neglected.  1.  Numbers i n the p a r e n t h e s i s  r e f e r to the  Bibliography.  can  3 CHAPTER I  ELEMENTS OP THE SCHNYDER-BERGERON GRAPHICAL SOLUTION OP THE A L L I E V I WATERHAMMER EQUATIONS  The t h e o r y d e r i v e d i n t h i s paper i s based on the g r a p h i c a l method o f waterhammer a n a l y s i s developed  by Schnyder and B e r g e r o n .  This  method i s a g r a p h i c a l s o l u t i o n o f t h e e q u a t i o n r e p r e s e n t i n g c o n d i t i o n s at  t h e t u r b i n e gate s i m u l t a n e o u s l y w i t h t h e c o n j u g a t e waterhammer equa-  t i o n s developed  by A l l i e v i .  A c c o r d i n g t o A l l i e v i the c o n d i t i o n s i n the penstock  a t the t u r -  b i n e gate c a n be r e p r e s e n t e d by V =  A ( i n r e l a t i v e form)  or ( o r X)  For d i f f e r e n t values of A  "the H-V ( o r h-v) p l o t i s a  parabola. The c o n j u g a t e waterhammer e q u a t i o n s developed  =+ -  H-H o or  h - 1  g  by A l l i e v i a r e  (v-V I  = + 2jo ( v - l )  ( i n r e l a t i v e form)  4 CHAPTER I I  THEORY OF ENERGY CONVERSION I N THE PENSTOCK  2.1  General I f c o n s e r v a t i o n o f energy i s a p p l i e d t o t h e system shown i n  f i g u r e 1, t h e n f o r any g i v e n time i n t e r v a l  E  where:  E  g = i E  +  E  c " fl E  - f2 E  (1  >  i s t h e energy o u t p u t t h r o u g h t h e gate i n t h e time i n t e r v a l ; E_^ i s t h e energy i n p u t t o t h e p e n s t o c k from the r e s e r v o i r i n  the time i n t e r v a l ; E  c  i s t h e change o f energy i n t h e p e n s t o c k , i . e . the d i f f e r e n c e  between t h e change i n k i n e t i c energy and t h e change i n energy, s t o r e d as s t r a i n energy, i n the f l u i d and i n t h e p e n s t o c k w a l l s .  (E_ i s p o s i t i v e  i f t h e n e t energy c o n t a i n e d i n t h e f l u i d and i n t h e p e n s t o c k w a l l s i s reduced); E^  i s the steady state f r i c t i o n  loss;  E_^2 i s t h e energy d i s s i p a t e d a s f r i c t i o n d u r i n g , and as a r e s u l t o f , any change o f energy form i n the p e n s t o c k . The s t e a d y s t a t e energy d i s s i p a t i o n E  i s usually a small  f r a c t i o n o f t h e t o t a l energy ( a t most 10^) and t h e r e f o r e c a n be n e g l e c t e d safely,  i . e . E ^ = 0 f o r a l l c a l c u l a t i o n s i n t h i s paper. Any energy changes i n t h e p e n s t o c k must be i n i t i a t e d from o u t -  s i d e t h e p e n s t o c k and must be i n t h e form o f e i t h e r p r e s s u r e o r v e l o c i t y changes.  A t t h e i n t a k e t h e r e i s no r e g u l a t i n g d e v i c e and the r e s e r v o i r  e l e v a t i o n i s assumed c o n s t a n t , t h e r e f o r e , any change i n the energy cont e n t o f t h e p e n s t o c k must be i n i t i a t e d by changes i n v e l o c i t y a t t h e gate  5 (which a r e a f u n c t i o n o f gate movement).  I f the v e l o c i t y a t the  gate  i s changed, then a c c o r d i n g to A l l i e v i t h e r e i s an a s s o c i a t e d p r e s s u r e change and t h i s p r e s s u r e change t r a v e l s , a t h i g h v e l o c i t y , intake.  toward the  To say t h e r e i s a v e l o c i t y change means the k i n e t i c energy o f  the f l u i d i s changed; s i m i l a r l y , a p r e s s u r e change means a change i n the amount o f energy s t o r e d as s t r a i n energy i n the f l u i d and i n the penstock w a l l s .  Any d i f f e r e n c e between the change i n k i n e t i c energy and  the change i n s t r a i n energy must appear a t the g a t e . is positive  T h i s energy, E ,  i f the net energy o f the f l u i d and p e n s t o c k w a l l s i s r e d u c e d .  (With t h i s c o n v e n t i o n a d e c r e a s e i n k i n e t i c energy i s p o s i t i v e  and a  de-  c r e a s e i n s t r a i n energy i s p o s i t i v e ) . F i n a l l y c o n s i d e r the term E ^  w h i c h r e p r e s e n t s the l o s s e s i n  the c o n v e r s i o n p r o c e s s w h i c h produces E .  I n a waterhammer p r o c e s s , w i t h  the e x c e p t i o n o f a s m a l l l e n g t h o f p e n s t o c k a d j a c e n t t o the g a t e ,  there  i s e s s e n t i a l l y no change i n the d i r e c t i o n o f f l o w o f a f l u i d p a r t i c l e , so t h a t the f l o w i s e s s e n t i a l l y v o r t e x f r e e . vortices  As i t i s the c r e a t i o n o f  i n a f l u i d w h i c h a l l o w s the d i s s i p a t i o n  p e r i o d , the f r i c t i o n l o s s e s due m a t e l y e q u a l zero ( i . e . E ^ The  ~  o f energy over a l o n g e r  t o the c o n v e r s i o n p r o c e s s must a p p r o x i -  0).  energy e q u a t i o n can t h e n he w r i t t e n t o a good  approximation  as: E  g  = E. + E 1  (2)  c  Example a = 3200 f t / s e c  H  A = 1/62.4 f t  V  L = 3200 f t  2  Q  Q  = 100 f t =  20  ft/sec  C o n s i d e r an i n c r e m e n t a l change o f v e l o c i t y o f A V = - 1 f t / sec The a s s o c i a t e d p r e s s u r e r i s e i s  AH = - — AV = - 2f°£  (_-,)  =  1 0 0  ft  32  g  The time r e q u i r e d f o r t h i s p r e s s u r e wave t o t r a v e l t h e l e n g t h o f t h e penstock i s t 13  At i n s t a n t  "t = ~  _ L - 1200 = 1 sec ~ a " 3200 1  S 6 C  the p r e s s u r e i s c o n s t a n t a l o n g t h e p e n s t o c k and i s + A H = 100 + 100 = 200 f t  H = H 0  The v e l o c i t y  i n the penstock i s  V = V  Q  + A V = 20 + (-1) = 19  ft/sec  The energy o u t p u t t h r o u g h t h e g a t e d u r i n g t h i s  t = —  seconds i s  cl  = w A H V ^ = 62.4  E  200 (19) 1 = 3800 f t - l b s  The energy i n p u t t o t h e p e n s t o c k d u r i n g t h i s time i s E. = w A H V i1 0 0  a  - = 6 2 . 4 •—— 100 (20) 1 = 2000 f t - l b s 62.4  The change i n t h e k i n e t i c energy o f t h e f l u i d i n t h e p e n s t o c k i s  AKE,  AKE  1  = J - A I ( vV  2 g  = \ &4  1  2  ^-L-  - ( vT  o  0  +  A.V) '  2 1  )  (52OO) ( 2 0 - 1 9 ) 2  2  = 1950 f t - l b s  The d i f f e r e n c e between t h e energy o u t p u t and t h e energy i n p u t i s E  A  g1  - E.„ = 3800 - 2000 = 1800 f t - l b s i1  T h i s i s d i f f e r e n t from t h e change i n k i n e t i c energy o f t h e f l u i d ; t h e r e f o r e , t h e s t r a i n energy s t o r e d i n t h e f l u i d and i n t h e p e n s t o c k w a l l i s  APE  1  = (E  - E  I 1  ) - A K E ^ = 1800 - 1950 = - 150 f t - l b s  (Note a n i n c r e a s e i n s t o r e d energy i s c o n s i d e r e d n e g a t i v e ) . Prom t = 1 t o t = 2 seconds t h e wave i s r e t u r n i n g t o t h e g a t e A t t = 2 seconds t h e c o n d i t i o n s i n t h e p e n s t o c k a r e ' H = H = 100 f t 0  V = V + 2 A V = 18 f t / s e c o ' The energy output' t h r o u g h t h e g a t e i n t h i s time i s E  g2  = w A H V - = a  38OO f t - l b s = E .  g1  The energy i n p u t t o t h e p e n s t o c k i n t h i s time i s  E  i2 =  w  A  H  o <o v  +  2  ^ I = -  A  62  4  <  1 0 0  18)  1  = 1800 f t - l b s The change i n t h e k i n e t i c energy o f t h e f l u i d i s  AKE  2  = 11  A L ( (V  = •_  62T4  + Av)  q  3  2  0  0  (  2  1S)2  -  -  (v  o  1 s 2  + 2 Av)  ) =  1 8  2  )  5° ft-lbs  S i n c e t h e change i n p r e s s u r e i s e q u a l and o p p o s i t e t o t h a t o f t h e f i r s t — time i n t e r v a l t h e change i n s t o r e d energy i s a APE  2  = - APE  1  = -  (-150) = 150 f t - l b s  As a check: E  = E.  g  i  E  +  E  c  =E.+  1  AKE  +  APE  = 3800 = 1800 + 1850 + 150 = 3800 f t - l b s  Note t h a t t h e n e t energy o u t p u t o f the wave i n t h e f i r s t — i n t e r v a l i s  cl E I n t h e second — a  = AKE  + APE  = 1950 + (-150) = 1800 f t - l b s  i n t e r v a l the net output i s  EQ2  =  AKE2  +  APE2  =  1850 + 150 = 2000 f t - l b s  The d i f f e r e n c e between t h e two wave o u t p u t s i s E  c2 0  - E  cl  = 2000 - 1800 = 200 f t - l b s  Also, E . - E._ = 1800 - 2000 = -200 f t - l b s i2 i l n  These two v a l u e s e x a c t l y c a n c e l so t h a t as f a r as t h e gate i s concerned, the  energy o u t p u t o f t h e i n c i d e n t and r e f l e c t e d waves from the r e s e r v o i r  i s constant.  ( A t t h e g a t e end o f t h e p e n s t o c k no v e l o c i t y change i s  n o t i c e d u n t i l t h e wave r e t u r n s t o t h e g a t e ) . 2.2  The Energy B a l a n c e for, an I n f i n i t e s i m a l I n c r e m e n t a l Gate Movement Any c o n t i n u o u s g a t e o p e r a t i o n can be approximated by a s e r i e s  of i n s t a n t a n e o u s i n f i n i t e s i m a l , i n c r e m e n t a l movements.  C o n s i d e r the  change i n power o u t p u t a t a p e n s t o c k gate due t o one o f these i n c r e m e n t s o c c u r r i n g a t time t , when t h e h e a d l a n d v e l o c i t y a t t h e g a t e a r e H and V respectively. steady s t a t e ) .  (Note t h a t the c o n d i t i o n s a r e n o t n e c e s s a r i l y those o f I f t h e time i n t e r v a l  it  i s made s m a l l enough so t h a t  no waves w h i c h have been r e f l e c t e d tance 2  from the r e s e r v o i r a r e w i t h i n a d i s -  S x = 2a 6 t o f the g a t e t h e n w i t h i n  t a n t d u r i n g the time i n t e r v a l the k i n e t i c • e n e r g y  £t.  o f the f l u i d  \  |  A  In this distance  _V)  2  ( Sx)  - | M V  (V  2  &x, t h e change i n  2  + 2V_,V +  - \ I A ( _ x ) (2v6v  S i n c e t h e change was  H and V w i l l be cons-  due t o the wave i s  6KE = | M (Y +  =  6x,  - V ) 2  +  6v 2 )  (V  6  (3)  infinitesimal  6 V <:«=: V  and as a r e s u l t  6v2  =o  and  6KE = |  A ( 6 i )  (4)  V)  As a d e c r e a s e i n k i n e t i c energy i s t o be c o n s i d e r e d p o s i t i v e ,  a minus  s i g n must be added t o make the energy and power o u t p u t s o f the wave o f correct sign.  T h e r e f o r e , t h e energy o u t p u t o f the wave due t o the  change i n k i n e t i c energy i s  <jE  = - <5KE = - -  A(C5X)  (V 6v)  (5)  10  The change i n s t o r e d energy (see f i g u r e 2) i n d i s t a n c e two p a r t s :  £x consists of  a) t h e work done i n e x p a n d i n g the p e n s t o c k and b) the work  done i n c o m p r e s s i n g the f l u i d . a)  The work done i n expanding t h e p e n s t o c k  From Hookes law  s E The work done on t h e p i p e w a l l i n e x p a n d i n g from E t o R +  6R i s e q u a l  t o t h e s t r a i n energy i n c r e a s e S  For i n f i n i t e -  stored i n the pipe w a l l .  g  s i m a l -values o f c>H and c$R t h i s i s S  S  b)  e  e  =F  = "  _R=w2TfR ^  6x H  2 T T R ( (bx) R s E  2  s E  _  H  2 ( H  6  h  v  )  ( 6 )  '  The work done i n c o m p r e s s i n g t h e f l u i d In the l e n g t h  <$x t h e l e n g t h o f the w a t e r column i s changed by  From Hookes law  (5^= I  6x) 6H  (  The work d o n e , S ^ , i n c o m p r e s s i n g t h e f l u i d i s e q u a l t o t h e s t r a i n energy F o r i n f i n i t e s i m a l v a l u e s o f &x^~ and  i n c r e a s e stored i n the f l u i d . this i s S  f  6x= 1  = F  wTT R  S =w TTH 4 2  f  2  £  2  H | ( 6x)  (H 6H)  6H  (7)  e>H  <5x r e s u l t i n g from t h e  The t o t a l change i n s t r a i n energy i n d i s t a n c e wave i s S = S + S_ T e f  /«\ (8)  By r e a r r a n g i n g and a d d i n g e q u a t i o n s 6 and 7 we o b t a i n  S =^w -n T  2  R  (cSx) f |  2  w TTE 2  +  2  (6x)  | ](H  6H)  2 F u r t h e r rearrangement and m u l t i p l i c a t i o n by ^  yields  S  S_ = — TT R (5x |(^ i ) " 4Jg° (H <$H) ' 2  2  T g  1  s E +K '  v  g  v  But by A l l i e v i t h e wave v e l o c i t y i s  1 a = g  E  K'  and t h e r e f o r e ,2 R  s E  IN +  K  Furthermore the c r o s s - s e c t i o n a l  A  Substitution  w _ 1 g ~ 2 ° a  area o f the penstock i s  = TT R  2  o f these v a l u e s y i e l d s  1  &  a^  12  Reservoir V-  Gade.  FIG 1  THE BASIC HYDRAULIC SYSTEM  -  Penstock Pressure  t  H  1 <J  (/  FIG 2  —  gx*a&t  4-  — >  CONDITIONS I N THE PENSTOCK RESULTING FROM AN INFINITESIMAL WAVE  13  W i t h a d e c r e a s e i n s t r a i n energy c o n s i d e r e d p o s i t i v e , t h e energy o u t p u t of the wave due t o t h e change i n s t r a i n energy i s  c5 = - s = - |A Ep  f_ (E6E)  T  (10)  The t o t a l energy o u t p u t o f t h e wave i s  <5E  = 6E^ + 6E „  (ll)  t  c  P  K  v  '  S u b s t i t u t i o n o f t h e r e s u l t s o f e q u a t i o n s 5 and 10 i n t o e q u a t i o n 11 y i e l d s  6E =-|A  (v6v  (C5X)  C  S u b s t i t u t i o n o f H = H h and V = V q  Q  f_ H6H)  +  (12)  V i n t o e q u a t i o n 12 and rearrangement  yields  =-^A (c$x) oH ov aa V(| a £6H g  _  E O-^c  v  y  With  _ V = - ^ t$H a  and  6v =  6v u  V  °  V  o  o -  +  ^ HV=^)  g  5  o  ;  Sv  =- ^ is a  V  o  a V with  2  »=  ^  i t follows that  6E  c  =| A H  Q  V  Q  J  (h -  2pv) 6v 6x  (13)  14  The power output o f the wave i s  n r " j ~ = - A H V « (h - 2 P v ) 6 v dt g o o a J ' v  i6 tf  But ^  x  6 *  - a  and t h e r e f o r e  6E T t  =  P  w =  w  A  H  o o < V  h  - 9^ 2  6  ^  v  C o n s i d e r now t h e g r a p h i c a l s o l u t i o n t o a waterhammer problem shown i n f i g u r e 3«  An i n c r e m e n t a l wave c r e a t e d a t the gate when c o n d i -  t i o n s t h e r e a r e r e p r e s e n t e d by p o i n t "b" i n t h e f i g u r e , w i l l , as i t t r a v e l s towards t h e r e s e r v o i r , encounter head and v e l o c i t y r e l a t i o n s h i p s t h a t f a l l on the l i n e b-c a t a l l t i m e s .  The s l o p e o f t h e l i n e b-c i s 2p  and  t h e r e f o r e t h e e q u a t i o n o f t h e l i n e must be  h  2jDv + c o n s t a n t  h -  2jov = C  (c)  or  I n e q u a t i o n 14  ov i s a .-.-constant, s i n c e 6v = - £ a  and  (15)  6h  <5h i s unchanged as t h e wave t r a v e l s a l o n g t h e p e n s t o c k  of constant  15  THE CREATION OF A WAVE  16 w a l l thickness.  T h e r e f o r e s u b s t i t u t i o n o f e q u a t i o n 15 i n t o e q u a t i o n 14  yields P. = w A H ¥  0  V  0  C 6v  =  constant  (l6) v  '  The n e t d i f f e r e n c e between t h e change i n k i n e t i c energy and t h e change i n s t r a i n energy r e s u l t i n g from t h e wave t r a v e l l i n g toward t h e r e s e r v o i r i s a c o n s t a n t a t any p o i n t  i n the penstock.  At the r e s e r v o i r the c o n d i t i o n s 6t  f i g u r e J . I f t h e time i n t e r v a l  a r e r e p r e s e n t e d by p o i n t " c "  a f t e r t h e wave i s r e f l e c t e d from t h e  r e s e r v o i r i s made s m a l l enough so t h a t no waves t h a t have been c r e a t e d , a t the g a t e a r e w i t h i n a d i s t a n c e 6x, tance  2 o"x = 2 a i t o f t h e r e s e r v o i r , then w i t h i n  H and Y w i l l be c o n s t a n t d u r i n g t h e time i n t e r v a l 6t. 6x  In this dis-  t h e change i n t h e k i n e t i c energy o f t h e f l u i d due t o t h e r e -  f l e c t e d wave i s  KR = "I  6E  A  ( > 6 x  <&v  (17)  The energy o u t p u t due t o t h e change i n s t r a i n energy i s as b e f o r e  <*E = - | A ( 6x) f _ H 6 H P R  (18)  The n e t energy o u t p u t o f t h e r e f l e c t e d wave i s  <KR  =  <^PR t * K R E  By making t h e same s u b s t i t u t i o n s as b e f o r e , w i t h t h e e x c e p t i o n t h a t f o r a r e f l e c t e d wave  6v  = I 6H  we o b t a i n b y t h e a d d i t i o n o f e q u a t i o n s 17 and 18  17 6E  cR v  = £ A ( 6 x ) § H V (-h- 2pv) g a o o v  v  6v  (19)  and oR  P  WR  -  It"  =  W  A  H  o o ^" - ^ V  h  ^  ^ °) 2  The change i n power i n p u t f r o m t h e r e s e r v o i r due t o t h e a r r i v a l and r e f l e c t i o n  o f t h e wave i s  <$P„ = w A H V ( 2 h) 6 v R o o where  v  (21) J  h = 1 .  The wave l e a v i n g t h e r e s e r v o i r , where c o n d i t i o n s a r e g i v e n by p o i n t " c " , f i g u r e 3> w i l l , from the p r i n c i p l e s o f g r a p h i c a l a n a l y s i s , encounter 3.  a p r e s s u r e - v e l o c i t y r e l a t i o n s h i p t h a t f a l l s on l i n e c-d i n f i g u r e  The s l o p e o f t h i s l i n e i s - 2p and t h e r e f o r e i t s e q u a t i o n i s - h = 2pv + K  or - h - 2jDv = K S i n c e t h i s l i n e and t h e l i n e g i v e n by e q u a t i o n  (22) 15 pass t h r o u g h t h e common  p o i n t c " , ( f i g u r e 3) whose c o o r d i n a t e s a r e M  h = 1,  v = v  the v a l u e o f K can be d e t e r m i n e d i n terms o f C.  2pv = C - 1 2pv = K + 1 and  therefore  K = C - 2  Prom e q u a t i o n s  14 and 22,  18 The e q u a t i o n o f t h e l i n e c-d i s t h e n  (23)  - h - . 2pv = C-2 S u b s t i t u t i n g e q u a t i o n 23 i n t o e q u a t i o n 20,  (24)  (C-2) 6 v  P = w A H V WR o 0  v  Prom t h e same r e a s o n i n g as f o r t h e i n c i d e n t wave, P  must be a c o n s t a n t  as t h e r e f l e c t e d wave r e t u r n s t o t h e g a t e . Also  P - P. = w A H V f C 5v W WE 0 o v. lr  - (C-2)  P - P = w A H V W WE 0 0  '  v  m  6 v] J  (25)  ( 2&v)  ^  ~  '  K  '  But, i f . . h = l , t h e c o n d i t i o n f o r the r e s e r v o i r , i s s u b s t i t u t e d i n e q u a t i o n 21 t h e n :  6P  R  = wAH  o  V  o  (j>6v)  =  P  w  -  P  w  (26)  R  Thus t h e d i f f e r e n c e i n power o u t p u t s between the i n c i d e n t and r e f l e c t e d waves i s e x a c t l y compensated b y t h e change i n power i n p u t from t h e r e s e r v o i r , so t h a t as f a r as t h e g a t e i s concerned, wave i s a c o n s t a n t throughout  t h e power output o f t h e  i t s l i f e span o f 2- seconds. a  A t the gate  no change i n i n p u t r e s u l t i n g from t h i s p a r t i c u l a r wave i s n o t i c e d u n t i l t h i s r e f l e c t e d wave r e a c h e s t h e g a t e , a t w h i c h p o i n t t h e wave ceases t o e x i s t and a new wave i s formed. C o n s i d e r t h e wave as i t a r r i v e s a t the g a t e . e n l a r g e d g r a p h i c a l s o l u t i o n o f t h e problem  F i g u r e 4 shows an  f o r p o i n t "d" i n f i g u r e 3«  the i n s t a n t b e f o r e t h e wave a r r i v e s h and v a r e g i v e n by p o i n t d.  At  The  i n s t a n t t h e wave a r r i v e s t h e head i s a l t e r e d by 6~h. and t h e v e l o c i t y by  6v  so t h a t c o n d i t i o n s a t t h e gate a r e r e p r e s e n t e d  by p o i n t p.  At this  i n s t a n t t h e wave ceases t o e x i s t and t h e r e f o r e i t s power o u t p u t ceases also.  However c o n d i t i o n s a t p o i n t p a r e u n s t a b l e as t h e r e l a t i v e g a t e  o p e n i n g i s T and t h e head and d i s c h a r g e must be r e l a t e d by v = "J J h. Therefore  a new wave must be formed a t t h e g a t e so t h a t c o n d i t i o n s a r e  such as a r e g i v e n by " f l " i n f i g u r e 4> o r , i f a t t h i s i n s t a n t an i n c r e mental c l o s u r e  6"T  t a k e s p l a c e , by p o i n t "q" f i g u r e 4«  I  n  the new wave formed w i l l produce a c o n s t a n t power o u t p u t , 2L gate i s concerned, d u r i n g i t s — l i f e a The power i n c r e a s e  e i t h e r case  as f a r as t h e  span.  (which may be o f p o s i t i v e o r n e g a t i v e  sign)  a t t h e g a t e due t o t h e c e s s a t i o n o f t h e o l d wave and the c r e a t i o n o f t h e new wave i s g i v e n by t h e sum o f t h e change i n power due t o t h e c e s s a t i o n of. t h e o l d wave and t h e change i n power due t o t h e c r e a t i o n o f t h e new wave and i s  —v  A v  v  WN " (Where "W  (27)  WRO  denotes t h e new wave and "0" t h e wave t h a t j u s t From e q u a t i o n  ceased).  20 and f i g u r e 4 i t f o l l o w s t h a t  -P = w A H V ( h , + 2»v.) WRO 0 o d y d' x  8v„ R  (28)  If d  p  and v, d  =  v  p  t h e n f o r a wave c r e a t e d by a r e f l e c t i o n o n l y i t f o l l o w s from e q u a t i o n that  14  S u b s t i t u t i o n o f e q u a t i o n s 28 and 29 i n t o e q u a t i o n 27 y i e l d s  riP = w A H V J ( V . P V ( V V 6  2  6  <V.P d > * \ ]  +  2  Q  v  or *P = w A H  B u t , -2jO £ v — a  R  V  o  f (h -2j0v ) 6 v  Q  d  d  2p(2v ) 6 v d  1+  R  j  (30)  i s t h e p r e s s u r e wave "A" F" w h i c h o r i g i n a t e d from the gate  seconds e a r l i e r ,  6 P = w A H V. [ ( h - 2_pv) _>v - v o  d  d  1  2  d  (6 F)}  (3l)  S i m i l a r l y i f an incremental c l o s u r e takes p l a c e t o p o i n t q f i g u r e 4 6P = w A H  V {(h -  o  0  d  2pv ) d  Sv  2  - v 2  d  ( 6 F) }  (32)  6 v^ and <$"vv> a r e t h e n e t changes i n v e l o c i t y t a k i n g p l a c e a t the gate' a t time t . T h e r e f o r e n e g l e c t i n g s u b s c r i p t s and d i v i d i n g equat i o n s 31  a n  & 32 by d>t we o b t a i n  - f l = w A H  o  Y  Note:  . . A H  o  ( ( h - ^ v )  ^ - 2 v  (  5 3  )  0  or i n the l i m i t as t  f  o  T  o  j > - 2p,)  |f - 2 T g ]  (34)  T h i s r e s u l t c a n be o b t a i n e d d i r e c t l y b y t h e f o l l o w i n g d e r i v a t i o n : P = w A H V hv o 0  21  TT  at  = wA H  _  dt ~  ?  n  dv  o  -^at  Y h — o dt  „ dF  at  + v —  dt  [ J  22  CHAPTER I I I  APPLICATION OP ENERGY PRINCIPLES TO THE DETERMINATION OP OPTIMUM GATE OPERATION  3.1  General C o n s i d e r t h e h-v diagrams shown i n f i g u r e s 5  a  a n (  l 5b and t h e  c o r r e s p o n d i n g p l o t s o f power o u t p u t v e r s u s time shown i n f i g u r e s 6a and 6b, T  and T  a r e t h e i n i t i a l and f i n a l g a t e p o s i t i o n s .  V  and P  are the  S F o r i n i t i a l and f i n a l s t e a d y s t a t e power o u t p u t s t h r o u g h t h e t u r b i n e g a t e s . T i s the t o t a l C T  and T  and o  i s t h e time f o r t h e power o u t p u t t o f i r s t c r o s s t h e s t e a d y  s t a t e Pj, ' l i n e . are  time o f g a t e m o t i o n f o r a g a t e o p e r a t i o n between T"  g i v e n by  As t h i s p o i n t i s n o t reached u n t i l  t y v  then  p  T  >  p  T  c  c o n d i t i o n s a t the g a t e  .  The excess energy output r e s u l t i n g from a gate o p e r a t i o n i s r e p r e s e n t e d by t h e shaded a r e a s o f f i g u r e 6, w h i c h a r e  4E  g  =  j  T F  T / F  (P - P ) d t = F  j  P  at  -  P  T  F  (35)  F  '0 The d e r i v a t i v e o f e q u a t i o n 2  w i t h r e s p e c t t o time y i e l d s  dE dE. dE P = —£ = — + — at dt dt Therefore _ 1  p / 0  dE. d t  E=TE. +E ] g  L  i  (36)  c  l  C - I Q  /•  d  F  t  /  +  -  PT  F  dE d t  C  d  t  "  P  F  T  F  W  (38)  23  a) Gate C l o s u r e FIG 5  '".  b) Gate Opening  TYPICAL GRAPHICAL SOLUTIONS TO WATERHAMMER PROBLEMS  P  0  T, a) Gate C l o s u r e FIG 6  t  0  Tf b) Gate Opening  POWER INPUT TO THE TURBINE DURING TRANSIENT CONDITIONS  t  25  2j>v*Cf  a) Gate C l o s u r e FIG 9  b) Gate Opening  GRAPHICAL SOLUTIONS TO WATERHAMMER PROBLEMS F WHEN  ^  (T„ C £v„ Ov (T_ - t ) = 0 n n F n n  n=Q+1  II2 f lf„ II  a) Gate C l o s u r e F I G 10  b) Gate Opening  PLOT OF INPUT VELOCITY TO THE PENSTOCK FOR THE F CONDITION  C , 0 v (T - ' t ) = 0 n n F n n=Q+1  26  E, i s the t o t a l n e t energy output r e s u l t i n g from a l l waves c o r i g i n a t i n g a t t h e gate between t=0 and T=T_. I f a c o n t i n u o u s g a t e movea  ment i s approximated by a s e r i e s o f i n f i n i t e s i m a l i n c r e m e n t a l movements then  T F  P  E  T-  c 0  (*W  W n  +  n=l  where f o r a wave t r a v e l l i n g from the gate t o the r e s e r v o i r *Wn  (39)  " Wn P  and f o r a wave t r a v e l l i n g from t h e r e s e r v o i r t o t h e g a t e  E  t  WPn=  P  WRn  (* " ( * n l ) )  ^  +  i s t h e time o f o r i g i n o f t h e wave a t the g a t e .  Note t h a t P., Wn  = 0  i f  ft  -  t  )  >  n'  -  a  and P WRn i r o  =0  i f ( t - ( t + i n a'' v  )  )  v  >  ^  a  ( T h i s f o l l o w s from t h e o r i g i n a l assumptions made i n d e r i v i n g P  and P  3.2  Evaluation of E  All life  span o f ^  c  1 T O  )  WK  W  between t=0 and t=T.,-,, P  t h e waves c r e a t e d up t o p o i n t Q o f f i g u r e 5 w i l l have a f u l l sec so t h a t Ewn  = Wn  ^Wn  = WRn ( l )  P  P  <£>  ' ^2)  S u b s t i t u t i o n o f e q u a t i o n s 14 and 20 i n t o e q u a t i o n s 41  a  E ^ - w AH, V  (h-2fiv)  O  n  d  42 y i e l d s  o, \  (43) .  n  6v ±  E ^ - w A H . Y . (-h-2pv)  (44)  n  I f t h e c o n s t a n t terms h - 2 p v and -h-2_p v a r e e v a l u a t e d when h = l i . e . a t the r e s e r v o i r ,  then  F, + E = w A H V - ( l - 2 p v - 1 - 2 p v ) Sv Wn WRn o o a ~ s' ' n m  v  = w A H V \ (-4_Pv) 6 v Q  Q  (45)  n  F i g u r e 7 shows a n e n l a r g e d p i c t u r e o f t h e wave o c c u r r i n g a t t = t . The t o t a l energy o u t p u t o f t h i s wave i s g i v e n by e q u a t i o n 45-  B u t from  f i g u r e 7. <k =  -g-S  n  (46)  thus  ^  (*W  +  n=l  V  n  =  W  A  H  o oI V  ^  ("4p n=l  ) ^  v  (47)  and lim n—>oo  (~4j3v)  n=l n  - g — ^ = (-2/3 v dv /  v  s  =P (  V  2 S  S u b s t i t u t i n g e q u a t i o n 48 i n t o e q u a t i o n 47  a n (  -  V  F  2  )  l substituting jO =  (  4  8  )  aV 2gH  o  28  L w "W„ - fs o \ - V> ' (E  AL T  +  2  2  Alffi  m  n=l  T h i s i s t h e t o t a l k i n e t i c energy change o f the f l u i d "between the i n i t i a l and f i n a l s t e a d y s t a t e c o n d i t i o n s .  Therefore the t o t a l  energy  o u t p u t o f a l l waves produced up t o the waterhammer l i n e o f s l o p e 2 p ing  pass-  t h r o u g h t h e f i n a l s t e a d y s t a t e v , h^, p o i n t i s e q u a l t o t h e change i n p  k i n e t i c energy o f the f l u i d i n t h e p e n s t o c k i n g o i n g from t h e i n i t i a l t o the f i n a l s t e a d y s t a t e p o i n t s and i s independent o f t h e way i n w h i c h the gate i s operated.  T h i s may he seen more e a s i l y by r e f e r r i n g s i m u l t a n e o u s l y  to f i g u r e s 7 and 4«  I f f o r each wave r e a c h i n g t h e gate a f t e r t = t  p  -  2L —  the g a t e i s a d j u s t e d t o p o i n t p o f f i g u r e 4> no new waves w i l l be c r e a t e d at  the gate a f t e r t = T  2L - —  p  and t h e p r e s s u r e - d i s c h a r g e r e l a t i o n s h i p i <1E„ d  w i l l f a l l a l o n g l i n e Q-F o f f i g u r e 7. that i s  h  i  =h  S  =h, o'  At t=T , P p  ±  =  E  = P  g  =  TTT  v. = v l g  and t h e r e f o r e dE dt dE Q  W i t h — r — = 0 t h e r e can be no waves i n t h e p e n s t o c k and no a d d i t i o n a l dt s t r a i n energy. The t o t a l energy o u t p u t must t h e n be E  where  g  = E. + E 1  c  = E. + 1  AKE  A K E i s t h e change i n t h e k i n e t i c energy o f the f l u i d  from t h e v e l o c i t y change from v„ t o v„. O  I  1  Therefore  resulting  29  E  c=  A  K  =H ^  E  Vn  +  n=l  Now c o n s i d e r the energy o u t p u t p e r wave o f a l l the waves g i n a t i n g between t = t ^ and t = T  - —  p  -  T.  A l l the waves  R  ori-  produced  i n t h i s time i n t e r v a l w i l l he r e f l e c t e d by the i n t a k e b u t w i l l n o t have enough time t o r e t u r n a l l the way t o the g a t e .  w A H V (h-2Dv) o 0 ' '  =  Wn  Sv  v  Therefore  a  n  K  (50) ' '  and ^WRn  "  w  A  H  o o  ( " ^ ^  V  6  v  n  \ P ~ ^n T  ^  ~S[  +  The t o t a l energy o u t p u t up t o time T„ due t o waves o r i g i n a t i n g i n the time r  i n t e r v a l t = T„ t o t = T^ i s t h e n  L  R  R  ^  %  +  Wn  =  n=Q +1  w  A  H  K  o o V  n=Q+l  I  R L n=Q+l  W  A  H  o o V  ("h-2pv)c5v  [  n  V  The energy o u t p u t o f each wave o r i g i n a t i n g between T w h i c h r e a c h the r e s e r v o i r  \n  and  = w A HQ V  ( t  n +  *)}  ( 2) 5  and T„, none o f  i s g i v e n by  q  (h-2j?v) 6 v  n  (T  p  - t ) n  (53)  30 The t o t a l energy output of the waves o r i g i n a t i n g i n this time i n t e r v a l i s then F  F  ( E w - V n - H  L h=R+l  wAH  o  Y  (h-2^v)6v„  o  ( T - t„)  ( )  F  5 5  n=R+l  The t o t a l energy output of the waves created i n the time i n t e r v a l t = 0 to t = TT-, i s then F F E  c  = / L  !  W  n  n=l °  _Q  r  E  c  =  L %  R  +  W n  n=l  +  L  <W E  +  W n  +  H  n=Q+l  F ^  +  W n  ( 6) 5  n=R+l  Substitution of the r e s u l t s of equations 4 9 j 52 and 55 into equation 56 yields R  E = AKE + w A H V {[2 C  Q  ( - /> ) ^ (^) h  0  2  v  n=Q+l  R +  +  J2 (-h-2pv) 6v n=Q+l  ) _  n  (T - ( t p  n+  (h-2_Ov)-Jv (T -t )J n  F  n  S))  (57)  n=R+l Substituting f o r convenience the r e s u l t s of equations 15 and 2 3 , which are,  31 C  n  C  n  =  h - 2pv  and - 2 = -h-2jDy  i n t o e q u a t i o n 57 we g e t R E  = AKE + w A H  c  o  V ) ,} o £ <-—  C  n  v  ( c$ v ) n a /  n=Q+l R  K-  • L  >  2  K  < F - <*» T  1»  +  n=Q+l  +  E  K  :°n  (F - V 1  (58)  T  n=R+l Rearrangement o f t h e term _R )  i  v  ( C - 2 ) c_v n n 7  v  + S)) F(T_ - n ( t a'' v  n=Q+l yields R  c5~~  n=Q+l  v  (C - 2') £ v n (FT - n( t +^)) n a ' v  v  y  R + T~ i  n=Q+l  R  = Y~ £  v  n=Q+l  (-2)6T ' n v  F(T_, -n v  ( ta'' +^))  R C (Sv ( T n n F v  p  - t )  n'  + 5~ £ n=Q+l  C n  (5v  n  v  (-^) a 7  v  (59) '  32  Furthermore  y~ FL  c <^v  Z n=Q+l  n  n  v  c ov  (T_ - 1 ) + y F ir L  n  n=R+l  n  -1)  (T„ v  F  iv  =r  c ^  L— n n=Q+l  T  n  v  (%-t) F n'  (60)  S u b s t i t u t i o n o f t h e r e s u l t s o f e q u a t i o n s 5 9 and 6 0 i n t o e q u a t i o n 5 8 y i e l d s R E c  = AKE + w A H V ) ) o o( L—  N  (-2)  '  &v  n  v  (OL, - ( t + - ) ) F n a v  7 /  n=Q+l  + / £ n=Q+l 3.3  C (ST (T_, - t ) n n ,F n'  (61)  v  E v a l u a t i o n o f t h e energy i n p u t t o t h e P e n s t o c k f o r t h e time i n t e r -  val t =  t o t = LF .  0  The wave o r i g i n a t i n g a t t h e g a t e w i t h an a s s o c i a t e d v e l o c i t y change  6v  n  causes a v e l o c i t y change o f 2 6v^  a t the r e s e r v o i r .  Therefore,  r e f e r r i n g t o f i g u r e 8 , t h e energy i n p u t t o t h e p e n s t o c k d u r i n g t h e time interval t = 0 to t = T  E  i =  v  A  H  o  T  p  i s g i v e n by  o | V  S  V L  h  o < <K> 2  <F- n I » T  ( t  +  n=l  h (2<Jvn) ( T - ( t  +  Q  n=Q+l  F  n +  i)))  (62)  = T^, — ^ w i l l  Note t h a t any waves l e a v i n g t h e g a t e a f t e r -b =  have  no e f f e c t on t h e i n p u t t o the p e n s t o c k d u r i n g t h e time i n t e r v a l t = 0 t o t = [IL,. n = R.  As a r e s u l t , the l a s t term o f e q u a t i o n 62 i s o n l y summed t o  Remember t o o t h a t f o r g a t e c l o s u r e 6v i s negative. o n The f i r s t term o f e q u a t i o n 62 y i e l d s  w  A  H  \  o o V  V  T  F = S F P  ^  T  Furthermore, r e f e r r i n g to f i g u r e 7 note that  Q  L K 2  - F " S V  ^  V  n=l  ( T h i s means t h a t the f i n a l  s t e a d y s t a t e v e l o c i t y i s f i r s t r e a c h e d a t the  r e s e r v o i r a t time T„ - — ) . F a' w A H  o  Y  /  o  h  L  o  U s i n g t h i s r e s u l t we o b t a i n °  2 <Sv  n  v  (T_ - ( t + — ) ) F n a' J  n=l  w A Ho Vo J7 ho (T_. -a' -) Z) F  2t$vn  v  n=l Q  ) + I— n=l  = w A H  2h  o  o  V  ivn  (-t ) ] n' J  / h (T„ - -) o | o F a x  y  V  (v_ - v ) F S' Q  + n=l (P_, - P ) ( T _ - — ) + F S F a v  C  7  v  7  wAH  V o o  ) (1 n=l  h o  2 i v ( i t) n n v  J  \  (64b) ^  J  Letting h  = 1, w h i c h i s t h e c o n d i t i o n a t t h e r e s e r v o i r , and s u b s t i t u -  t i n g t h e r e s u l t s o f e q u a t i o n s 6 j and 64b i n t o e q u a t i o n 62 y i e l d s  E. = P T_ + (P_, - P ) (T„ 1 S ? P S P a Q  v  —) y  +  A H  W  j Y~  2 <Sv  V 0 0 / L n=l  n  v  (-t ) n'  2<$v ( T - ( t + ^ ) ) V n F n a.'' J  )  +  /  L  p  v  v  n=Q+l or  C = (P„ - P ) — + P_, T 1 S F a P P v  f  + w A H V o 0  y  Y •  I L  2 &v n  v  (-t ) n 7  n=l  +r  2 <£v  /—  3.4  n  (T_ - ( t + ^ ) ) v  F  v  E v a l u a t i o n o f t h e E x c e s s Energy  n  (65)  a ' \ y  Output  The excess energy o u t p u t A E  c a n now be found by s u b s t i t u t i n g  the r e s u l t s o f e q u a t i o n s 6 l and 65 i n t o e q u a t i o n J8 w h i c h i s  AE  =  E  and upon s u b s t i t u t i o n  AE  i  + E  Y  c  yields  =- p_, T_, + ( P - p_.) F F S F' a  + P„ T_,  q  g  P  P P  x  w A H V f) o o\L n=l  2 <Jv  n  v  (-t ) n 7  + LY~  2 iv  n  v  (T_,- ( t + F n a v  y /  -))l  J  n=Q+l (cont'd)  FL  AKE + w  +  A  H  V (^_  Q  (-2)  o  Sr  ( T  n  F  - (t  n+  £))  n=Q+l  C )_ nk^-*n)i n=Q+l  +  (66)  C a n c e l l a t i o n and rearrangement o f terms y i e l d s t h e g e n e r a l e q u a t i o n f o r AE  g  which i s Q  AE  g  = (P_ - P„) S F a v  + AKE  7  +  2 <Sv n  w A H V ]/ o o 11— n=l  (-t ) n'  F +y~ I h=Q+l  c  &V  n  n  ( T _ v  1  - 1 )  F  n'f  (67) v  /  J  I f t h e gate i s o p e r a t e d so t h a t d u r i n g t h e i n t e r v a l t  = T  P  2L' - — t o t = T „ no new waves a r e formed t h e n t h e h-v diagram w i l l he as a n F ^ shown i n f i g u r e 9.  I n t h i s case we have •F  = 0  6v n  and  Q+l  therefore  )  L—  C n  6vn  v  ( T  F  - tJ ) = 0  n  v  (68)  '  n=Q+l E q u a t i o n 67 i s t h e n m i n i m i z e d by m i n i m i z i n g t h e a b s o l u t e v a l u e o f ( r e f e r t o f i g u r e 10)  36  7"  2  6v  (-t )  n=l  w h i c h i s t h e shaded a r e a o f f i g u r e 10 and c a n he w r i t t e n as  II KI <v 2  L n=l  A l l o t h e r terms o f e q u a t i o n 67 a r e c o n s t a n t f o r a g i v e n gate o p e r a t i o n between T"g and 1"-p«  r  2  L— n=l  (Note i n t h e term  c^v ( - t ) n n'  the maximum v a l u e o f t i s n OT  t  = T„ = T - — n Q F a  and f o r gate c l o s u r e 6 v  /  £v  2  n  (1  v  i s negative.  As a r e s u l t  (-t ) n 7  n=l is positive).  L  n=l  O b v i o u s l y from f i g u r e 10 t h e term  H *K II <*„>  i s a minimum i f  |i 2 6v  n  || i s as l a r g e as p o s s i b l e when t  n  i s as s m a l l  37  a.) Gate C l o s u r e  b) Gate Opening F I G 11  WATERHAMMER CHARTS AND SURPLUS ENERGY PLOTS FOR THE TIME INTERVAL  t = T  0  TO  t = T.  38  FIG  a) L i n e  h = 20v  t o t h e L e f t o f T.  b) L i n e  h = 2pv  to the Right o f  7  12 WATERHAMMER CHARTS AND SURPLUS ENERGY PLOTS FOR THE TIME INTERVAL  t = T • TO  t = T„- - GATE CLOSURE  39  a) L i n e  h = 2$>v  to the L e f t of T  h  h) L i n e FIG  13  h = 2pv  t o t h e R i g h t o f 3"  F  WATERHAMMER CHARTS AND SURPLUS ENERGY PLOTS FOR THE TIME INTERVAL  t = T TO Q, n  t = T^ J)  - GATE OPENING  as p o s s i b l e  since  Av = -  7 ^  Ah  T h i s means that, t h e maximum v e l o c i t y and head changes s h o u l d be made as soon as p o s s i b l e f o r t h i s form o f g a t e o p e r a t i o n . Now c o n s i d e r t h e g e n e r a l e q u a t i o n 67 w i t h : n=F n _ n=Q+1 The o n l y d i f f e r e n c e i n  E  between t h i s case and t h e case where:  S  n=F  6,n  i s t h e term  =  0  n=Q+1  I  • (T^ - t ) . T h i s term i s .just t h e sum o f t h e n F n n n=Q+1 energy o u t p u t s o f a l l waves c r e a t e d between t = T^ and t = T , AS SEEN ^ n Q, n F BY THE GATE.  (Remember from t h e d i s c u s s i o n o f an i n c r e m e n t a l wave t h a t  the r a t e o f energy o u t p u t o f a s i n g l e wave i s as f a r as t h e g a t e i s conc e r n e d , a c o n s t a n t throughout i t s l i f e  If n=Q+1  C £ O vT n n  span).  iiss p l o t t e d a g a i n s t t i m e , t h e n c h a n g i n g t h e  axis of integration yields  \  J  ( Y ~  L n=Q+1  C c^v ) d t = n n  L n=Q+1  C £v (T - t ) n n F TY'  (69)  The q u a l i t a t i v e r e s u l t s o f such p l o t s f o r s e v e r a l g a t e o p e r a t i o n s a r e shown i n f i g u r e s 11, 12 and 1 J .  Note i n these f i g u r e s t h a t  the optimum method o f g a t e o p e r a t i o n a f t e r t = T t i v e p o s i t i o n o f the l i n e h = 2 p v advantageous  depends upon t h e r e l a -  so t h a t i n c e r t a i n i n s t a n c e s i t i s  t o reduce t h e r a t e o f g a t e o p e r a t i o n i n t h e time i n t e r v a l *  t = T„ t o t = L i f AE 0, F fl g  3.5  n  i s t o be k e p t t o a minimum.  I d e a l Gate C l o s u r e Curves F o r g a t e c l o s u r e , f r o m t h e c o n d i t i o n t h a t t h e maximum head and  v e l o c i t y changes be made as soon as p o s s i b l e so as t o reduce t h e energy i n p u t , t h e maximum a l l o w a b l e head (h ) must be reached a t t = 2 ^ and m a i n t a i n e d a t l e a s t u n t i l t = T„.  y~  2  L— n=1 of  T h i s w i l l m i n i m i z e t h e term  L c-t) n  n  e q u a t i o n 67 ( r e f e r t o f i g u r e 1 0 a ) .  I f t h e l i n e h = 2tJ> v  i s t o the  l e f t o f p o i n t Q, on t h e h-v diagram, t h e n t h e term F )  C n  L  ivn  (T - t ) F n'  n=Q+1 w h i c h i s n e g a t i v e , i s maximized by d e c r e a s i n g t h e r a t e o f gate- c l o s u r e after t = T . for  F i g u r e 14 shows t h e h-v diagram and t h e g a t e c l o s u r e curve  such a c a s e .  Note t h a t t o m a i n t a i n a c o n s t a n t head a t t h e g a t e be-  tween t = 2— and t = T_, t h e c l o s u r e c u r v e i n t h i s time i n t e r v a l i s s p e c i a  0/  f i e d by t h e c l o s u r e c u r v e i n t h e time i n t e r v a l t = 0 t o t = 2^ ( i . e . a p o i n t X s p e c i f i e s p o i n t Q e t c . ) . I f the. extreme case o f f i g u r e 14b i s J  v  42  t a k e n , i t becomes a s e r i e s o f i n s t a n t a n e o u s t  closures taking place a t  = °' f ' V As i n s t a n t a n e o u s  c l o s u r e i s i m p o s s i b l e and a c l o s u r e o f t h e  f o r m shown i n f i g u r e l'4b e x t r e m e l y d i f f i c u l t t o d u p l i c a t e , t h e most r e a sonable,  f o r m o f g a t e c l o s u r e w o u l d seem t o be t h a t made up o f a s e r i e s  of s t r a i g h t l i n e segments j o i n i n g p o i n t s S, X, Q,, F o f f i g u r e 14b.  How-  e v e r t h i s c u r v e has t h e d i s a d v a n t a g e t h a t i f c l o s u r e s h o u l d be s t a r t e d from a p o i n t o t h e r t h a n Tg>  higher than allowable pressure  a  r i s e may  r e s u l t (6). I t i s p o s s i b l e t o d e r i v e a gate c l o s u r e curve f o r which the 2L maximum waterhammer o c c u r s a t t = — a  and f o r a l l o t h e r times  h . m  As t h e c l o s u r e s under c o n s i d e r a t i o n a r e always l e s s t h a n f u l l g a t e (and n e v e r t o *J"= 0) t h i s c u r v e c o u l d be a r e a s o n a b l e 3.6  solution.  I d e a l Gate Opening Curves F o r g a t e opening A E i s a n e g a t i v e g .AE  q u a n t i t y so t h a t t o make  as s m a l l as p o s s i b l e t h e term  T~  L— n=l  2 &v  n  (-t) v  n'  o f e q u a t i o n 67, w h i c h i s n e g a t i v e must be made as s m a l l as p o s s i b l e and the term F  I  C n  6vn (OL, - t ) F n' N  n=Q+l s h o u l d be made p o s i t i v e and as l a r g e as p o s s i b l e .  W i t h these f a c t o r s i n  mind, r e s u l t s o f s i m i l a r f o r m t o gate c l o s u r e may be o b t a i n e d .  3.7  dh I d e a l Gate O p e r a t i o n When 2jO > ( — ) For  j - _  .  c o n s i  many cases the s l o p e o f the waterhammer l i n e i s g r e a t e r  t h a n the s l o p e o f the 3"™  line.  Such a case i s shown i n f i g u r e  15.  P o i n t s Q and F c o i n c i d e so t h a t we have  ) L—  C  n  {T  n  ( L - t ) F n y  =  (70)  0  n=Q+l The g e n e r a l e q u a t i o n t h e n r e d u c e s to  AE  g  = (P_ - P J v  S  F  y  Q  - + A K E + Y~  a  L— n=l  2 iv  n  v  (-t ) n  J  v  .(,71) '  From f i g u r e 15 i t i s o b v i o u s t h a t the i n p u t ' v e l o c i t y i s reduced to the f i n a l s t e a d y s t a t e v e l o c i t y by r e a c h i n g 7"^  as r a p i d l y as p o s s i b l e .  W i t h g a t e opening the same r e s u l t s a p p l y so t h a t the minimum absolute value^of A E  i s o b t a i n e d by r e a c h i n g *~S  a s  r a  Pid-ly  a s  possible,  CHAPTER IV  EVALUATION OP DIFFERENT GATE OPERATION CURVES  4.1  Instantaneous P a r t i a l  Closure  I n f i g u r e l6b a v e r y r a p i d g a t e c l o s u r e i s shown as a s e r i e s of small increments.  If At  sidered instantaneous.  i s v e r y s m a l l t h e n t h e c l o s u r e can he con-  The p o i n t Q i s l o c a t e d as shown i n f i g u r e 16a.  F o r a l l t h e waves formed by t h e c l o s u r e t o J" , t  = At  p  the waves formed by t h e r e f l e c t i o n 2L W i t h TT-, = — f o r i n s t a n t a n e o u s ij a  ~ 0.  For a l l  2L o f r e t u r n i n g waves t ~ — + A t n a  c l o s u r e , we have i n t h e g e n e r a l  =  equation  Q  YL  2  ^ n v  ("V  =  ( 7 5 )  0  n=l  and P )  L  C iv (T n n F  - t ) =  n'  v  n=Q+l  L/  C  n  iv  n  v  (T-, - t )  F  n  n=Q+l ,F  c + y~ l__ n n=B+l  ~  2L a  7 i —  n=Q+l  JT v  (T  n  v  c i v  n  n  F  ra  -1) n  2L — a  7  (74)  The energy o u t p u t o f t h e waves formed between n = Q, + 1 and n = B can be e v a l u a t e d by r e f e r r i n g t o f i g u r e 16a and n o t i n g t h a t on t h e l i n e h  6vn  (75)  and C  n  =  1 - 2j3v  (76)  Prom t h i s we o b t a i n v 2L a  n  C  n  2L — a  v n  ( 1 - 2 / ) v)  n  ^  n=Q+l  n=Q+l  (77) or  B  ^ a  )  L a  C i v n n  i—  (78)  -p V  v  n=Q+l  v^ i s l o c a t e d as shown i n f i g u r e 16a (Note t h a t as A t -> 0 e q u a t i o n s 77 and 78 become e x a c t . S u b s t i t u t i o n o f the r e s u l t s o f e q u a t i o n s 73 and 78 i n t o e q u a t i o n 67 y i e l d s AE  = ( P - P_) - + A K E + S F' a c  g Purthermore  W  v  A H  V 0 0 a  v - j3v- j  (79) v,  we have P  v ) S  (80)  P„ = w A H V (h v„) F o 0 o F'  (81)  S c  = w A H  0  Y  o  v  (h 0  0 /  v  and  (82)  S u b s t i t u t i n g e q u a t i o n s 80, 81 and 82 i n t o equation.79  and l e t t i n g h  =  47  a) Waterhammer C h a r t  h) Gate C l o s u r e Curve  FIG 16 WATERHAMMER CHART AND GATE CLOSURE CURVE INSTANTANEOUS GATE CLOSURE  a) 2£>= 2 FIG 17  b) 2p = 1  WATERHAMMER CHARTS FOR DIFFERENT VALUES OF p  •'' ' .  - INSTANTANEOUS GATE CLOSURE  '  we g e t AE  g  = w A  H  o  V  L o a  v  / \ 1 w 2 , ? 2s (v_ - v j + % - A L V ( v *- v / ) S F' 2 g o S F v  y  v., +  w A  H  o  V  -  (8?)  v - o?  o a  Example A  =  1/62.4 f t  L  =  3220 f t  a  =  3220 f t / s e c  P  =  1  20 f t / s e c 1000 f t  H  =  1  F o r t h e g i v e n d a t a t h e s u r p l u s energy r e s u l t i n g from an i n s t a n t a n e o u s c l o s u r e w i l l he c a l c u l a t e d u s i n g e q u a t i o n 83. From t h e d a t a  w AH V - = l o  o a  2  - ^ 1000 (20)= 20,000 f t l b s .  62.4  From t h e h-v diagram shown i n f i g u r e 17a,v^ = .48. S u b s t i t u t i n g  values  i n t o e q u a t i o n 83 y i e l d s AE  = 20,000 (1 - .6)  g  +  +  I  ^  20,000 { ( . 4 8 - .48 ) - (.6 2  3220 ( 2 0 ) ( l 2  2  .6) 2  .6 )) 2  = 8,000 + 12,800 + 200 = 21,000 f t - l b s  AE  2L As a check t h e power output d u r i n g t = 0 t o t = — i s from f i g u r e 17a 2L a  P  B  = w AH V Q  q  (hg v ) = 2 (20,000) 1.52 (.74) = 45,000 f t l b s B  The d e s i r e d power output i n t h i s i n t e r v a l i s g i v e n by 9T  pT  If  P  = ~  F  w A H  V  Q  Q  (1^ v )  =2  F  (20,000) .6 = 24,000 f t - l b s  As a r e s u l t we have AE  = ^  g  (P  B  - P ) = 45,000 - 24,000 = 21,000 f t - l b s F  I f i n the above example an i n s t a n t a n e o u s c l o s u r e had been made to J =  T  N  = .675  a t t = 0 a.nd from Tr, = .675  then no waves would be formed a f t e r p o i n t Q. B ) L n=Q+l  C  n  Ov  n  to  = .60 a t t =  —  This gives  0  =  and as a r e s u l t AE  = 8,000 + 12,800 = 20,800 f t - l b s &  T h i s amounts to about a 1% s a v i n g i n A E , g  and a 25% r e d u c t i o n i n p r e s -  sure r i s e when compared t o the f i r s t c a s e .  Furthermore  s i d u a l o s c i l l a t i o n s i n the h y d r a u l i c system. h = 2J)v  was  t h e r e a r e no r e -  I n t h i s example the l i n e  e s s e n t i a l l y t o the l e f t of the a r e a r e p r e s e n t i n g the zone  o f gate o p e r a t i o n , thus p e r m i t t i n g the above mentioned s a v i n g s . Example A  =  2/62.4 f t  P  =  .5  V  2  =  10  ft/sec  The r e m a i n i n g d a t a i s the same as the p r e v i o u s example. f o r t h i s case i s shown i n f i g u r e Substituting  i n t o e q u a t i o n 83 we  AE  The h-v  diagram  17b. get  = 20,000 ( l - .6) + 10,000 ( l 2  .6 ) 2  (cont'd)  50 + 20,000 | (.58 - .5 (.38) )- (.6 - .5 (.6) )} 2  AE  2  = 8,000 + 6,400 - 2,200 = 12,200 f t - l b s  As a check, from f i g u r e 17b we  get  pT  AE  =  g  (P  B  - P )= s  40,000 ( l . 5 l ( . 6 9 )  - l ( - 6 ) ) = 12,200 f t - l b s .  t h i s example i f c l o s u r e i s made i n two s t e p s ; 7 t o ' T n t " f c = 0;  In  a  Q  ^  2L  Tn  t o T-n, a t t = — *i a a.  AE  €>  then the v a l u e o f A E  and  i s i n c r e a s e d by 16$ i . e . g  = 8,000 + 6,400 + 0 = 14,400 f t - l b s .  The e f f e c t o f b e i n g t o the l e f t o f the l i n e h = 2j0v i s v e r y marked on the term  > C L— n n=Q+l In  (  Ov  n  v  (T^ - t ). P n'  comparing the two examples w h i c h were f o r the same i n i t i a l  and f i n a l s t e a d y - s t a t e power o u t p u t s the e f f e c t o f r e d u c i n g the term by r e d u c i n g the i n i t i a l v e l o c i t y s h o u l d be n o t e d . crease i n penstock  hydraulic 4.2  A l s o the n e c e s s a r y i n -  a r e a accompanying the v e l o c i t y r e d u c t i o n , r e s u l t s i n  an i n c r e a s e d c a p a c i t y o f the p e n s t o c k further savings.  AKE  to' s t o r e energy-thus  resulting i n  These s a v i n g s a r e accompanied by the d i s a d v a n t a g e  of  oscillations.  Gate C l o s u r e w h i c h Y i e l d s the S m a l l e s t Value o f Maximum Waterhammer T h i s f o r m o f c l o s u r e was  given pipe l i n e constant p  developed  by E. Ruus (6), and f o r a  and c l o s u r e time T^, g i v e s the minimum p o s s i b l e  v a l u e o f maximum waterhammer.  I t i s shown i n the d e r i v a t i o n (6) t h a t the  change i n p e n s t o c k w a t e r v e l o c i t y a t the t u r b i n e gate and a t t h e i n t a k e  .  FIG  18  WATER V E L O C I T Y AT T H E I N T A K E AND AT THE GATE - -OPTIMUM G A T E  F I G . 19  -WATEPLHAMMER  CLOSURE  CHART FOR OPTIMUM GATE  CLOSURE  51  52  v a r y i n t h e time i n t e r v a l t = 0 t o t = diagram i s shown i n f i g u r e  as shown i n f i g u r e 18.  The h-v  19.  I f f o r t h e g e n e r a l form o f t h e waterhammer c h a r t shown i n f i g u r e 19,  p and h ^ a r e t h e s p e c i f i e d v a r i a b l e s ,  sure c u r v e can be o b t a i n e d .  then the c o r r e c t  shape o f c l o -  T h i s c u r v e has many o f t h e c h a r a c t e r i s t i c s  o f t h e optimum c u r v e f o r k e e p i n g A E  as low as p o s s i b l e .  A g e n e r a l de-  r i v a t i o n o f t h i s c u r v e f o r p a r t i a l g a t e movements i s g i v e n below. 2L A t t h e gate d u r i n g t h e f i r s t — 18),  time i n t e r v a l ( r e f e r t o f i g u r e  t h e r a t e o f change o f v e l o c i t y i s g i v e n by  If - *  «*>  The h e a d - d i s c h a r g e r e l a t i o n f o r t h e g a t e i s  v  jy  =  h  (85)  F u r t h e r m o r e f r o m A l l i e v i ' s f i r s t c h a i n e q u a t i o n we have h - 1  = -2p  (v - v )  (86)  s  Combining t h e r e s u l t s o f e q u a t i o n s 8 5 and 8 6 we o b t a i n v Integration 87  o f e q u a t i o n 8 4 and s u b s t i t u t i o n o f t h e r e s u l t i n t o e q u a t i o n  yields Kt  + v  =  s  TiJ 1 - 2p  (Kt + v  s  - v )  (88)  g  or Kt + v  T-  c  S  i  1  "  2JO K t  K i s d e t e r m i n e d f r o m e q u a t i o n 8 6 by s u b s t i t u t i n g h = h 2L d i v i d i n g t h e e q u a t i o n by — . The r e s u l t i s 3*  (89)  2L at t = —  and  53  h  v - v  m  s  "MI  dv dt  =  K  a or  - 1  h K  (90)  =  2L F o r the c l o s u r e between time t = — a v  =  J  and t = T„ we have C  (V m  -  . (91)  and t h e r e f o r e  If -  - If J^T  2K  <*>  or  dT dt  =  2K  (93)  I n t e g r a t i o n o f e q u a t i o n 93 w i t h r e s p e c t t o time y i e l d s  T-  •§_<*__)  y '  +  T,  (94)  h  m  2L J, i s determined by s u b s t i t u t i n g t = — i n t o e q u a t i o n 8 9 .  /T"  -1-  El  The r e s u l t  yields jr-  2L  _ 2L  m F i n a l l y s u b s t i t u t i o n o f t h i s r e s u l t i n t o e q u a t i o n 94 y i e l d s  (96)  The g a t e c l o s u r e time T^ i s d e t e r m i n e d as f o l l o w s . t o t a l v e l o c i t y change a t t h e g a t e i n time T  p  From f i g u r e 18 the  i s g i v e n by  From f i g u r e 19 we have  V  B  =  IF  v  s  =  T |/ = T  I m  ^  h  and 1  S  (99)  S  Combining t h e r e s u l t s o f e q u a t i o n s 97? 98 and 99 we o b t a i n  T^ F V V - T§•  =  T  F  V  S  m  +  -  (100)  2K  C  The g e n e r a l form o f t h e c l o s u r e c u r v e as g i v e n by e q u a t i o n s 89 and 96 i s ' shown i n f i g u r e 20.  The r e s u l t s a r e g e n e r a l and a p p l y t o g a t e o p e n i n g  as w e l l as g a t e c l o s u r e . different, exceeded  However, f o r each v a l u e o f "X, t h e c u r v e i s T 2L  ( i f c l o s u r e were s t a r t e d from  \ ( — ) o f f i g u r e 18, h  would be  (6).) To e v a l u a t e A E  s a r y t o know T„. V  f o r t h i s form o f g a t e o p e r a t i o n i t i s n e c e s -  From f i g u r e 19 we have  Q " F ° 2^ V  (h  m  "  1 }  or ^  Q,  = TTTT  2jO  v  (h  m  - l)  '  + v.,  F  y  (101)  '  also F  =T  Q  =T  V  V  (102)  F  Q  K  (103)  -'OPTIMUM GATE OPERATION  56 Combining the r e s u l t s  o f e q u a t i o n s 101, 102 and 103 we o b t a i n  ( h  '  * "  X )  +  T  F>  ( 1  °  4 )  m  But from e q u a t i o n 96 we have  T  = (2 K  Q  T  KI  - 2  Q  v )  +  (105)  i  s  « m E q u a t i n g e q u a t i o n s 104 and 105 we o b t a i n  Yp  K  -1-  -2L_  -^  +  %  =  2  K  T  Q "  2  K  \  +  v  s  < ) 106  or 2jO  2K  ^ 2K  +  =  T  _L Q a  ( 1 0 7 ) U W  /  Substituting h  2 K  =  J2  - 1  _  a i n t o the f i r s t term o f e q u a t i o n 107 * i d r e a r r a n g i n g we g e t a  I t i s now p o s s i b l e t o e v a l u a t e A E gate o p e r a t i o n .  Y~ Ln=l  f o r t h i s optimum form o f  C o n s i d e r the term  2 iv  (-t ) n  v  n'  )f t h e g e n e r a l e q u a t i o n f o r <A,E .  According to figure  21 t h i s a r e a i s  2 iv  Y £ n=l  n  v  (-t ) n'  =  (v v  (109)  v j  -  q  S  F  y  2  ( R e c a l l f r o m e q u a t i o n 64a t h a t Q 2 iv  y~  L—  n  n=l  S u b s t i t u t i n g e q u a t i o n 108 i n t o 109 and l e t t i n g  V  S  =  V  F  =  T  S  K  we o b t a i n  Q  XT * k < - * n >  (%^)  =  =- - # (  )  (no)  2  n=l The term F  y~  'c^v  L  n  w  n  v  (T_  - 1 )  F  n  y  n=Q+l o f t h e g e n e r a l e q u a t i o n c a n be e v a l u a t e d i f i t i s assumed t h a t s t r a i g h t l i n e between the p o i n t s h^, v ^ and h , v^,.  isa  (For A h ^  .6 t h i s  i s a r e a s o n a b l e a p p r o x i m a t i o n . ) T h i s t erm can be w r i t t e n as  Y  F  ii  n=Q+l  ,  B  C Qv (UL, - t ) = Y ~ n n F n' L v  C n  QV  C  Ov  n  (T„ - t ) F n  v  /  n=Q+l F  T~ n=B+l  n  n  v  (T„ - t ) F n'  ( i l l ) ' v  58 The term  ) I  JT  C  n  ( L -  n  F  v  t  )  n  y  n=Q+l can he e v a l u a t e d the f a c t s  a l o n g t h e l i n e h = 1 ( r e f e r t o f i g u r e 22)by c o n s i d e r i n g  that  n  =  1 - 2pv  n  =  dv -  C ( 6 v  Furthermore, since —  =  K, we have from geometry ( r e f e r t o f i g u r e 22)  T - T T - t = T_ - (T„ + — & F n F Q, v_, - v„ w  v  ±s  y,  (v - v j ) Q/ y  v  (112) J  Furthermore from geometry we have  V  l " F = B " Q V  v - v  V  F  V  (114)  = v - v  Combining e q u a t i o n s 112, 113 and 114 we o b t a i n T - T T„ - t = T_ - (T_ + — & F n F Q v - vv  v  n  1  (v - v j ) F''  (115)  \ *> 1  £  As a r e s u l t , we have B  T~  v.  /  C  dv  (T„ - t ) S  n=Q+l  (  T -T T^ - (T„ + - ^ — ^ ( v - v j )  j.  (1 - 2 p v )  v_  (116) S i m i l a r l y , since  ($v i s a c o n s t a n t between n = B + 1 and n = F (as a  r e s u l t o f the s t r a i g h t l i n e approximation)  dv  2  59  C  iv  n  n=B+l  n  (T_  - t )  F  rr  v  T -T j  ( i  F  -  2  pv)  | T  - ( T  F  C  v.,  (117) Rearrangement o f e q u a t i o n 117  and r e v e r s a l o f t h e l i m i t s o f i n t e g r a t i o n  yields F  v^  6v  C  E  n  n  v  T  (!  - t ) n  ( T  F  7  -  2  f v)  —  -T  (v - v ) F  •r F  -  (118)  v  n=B+l  vF  A d d i n g e q u a t i o n s 116  Y  and 118 and c a n c e l l i n g terms we g e t  T  c  iv  £ n n=Q+l  n  v  ([IL, - t ) "= F n J  v  -T  2  V  1" F V  v„ I"  (l-2j0v)  dv  V-  F  F 2  . \  (l-2pv)  ( i~V v  v dv  (119)  Upon s u b s t i t u t i o n o f  F ~  2L Q, ~ a  the s o l u t i o n o f e q u a t i o n 119  E n=Q+l  c n  i v n  ( T _  F  becomes  - t ) ^ — (v n a 1 y  v  n  v, J  P [ 2 - 5  ri  ( 2  V  F  +  V.  (120)  From f i g u r e 22 we have  F  = T J  F  (121)  60 (122)  S u b s t i t u t i o n o f t h e r e s u l t s o f e q u a t i o n s 121 and 122 i n t o e q u a t i o n 120 yields _L  )  c  L n=Q+l  ( J~h  i v ( L - t ) : n  n  v  F  L  a  n'  F  v  V  - i)  m  ;  - - i - (h  - I)  2j3 "m 2p ™ ) v  v  x  1 JL 2  or F  y £— n=Q+l  c  ^ n  (L . t) n F n v  =  y  L / V m a I2p  x  - 1  h  v  m  h  m  + 2  (124)  E q u a t i o n 124 i s v a l i d f o r gate c l o s u r e and gate opening o f t h e form g i v e n by e q u a t i o n s 89 and 96 p r o v i d e d  2* a and dh dv  2p  F F o r t h e r e m a i n i n g terms i n t h e g e n e r a l e q u a t i o n f o r A E  we have  S ( s p  - V  I  =  w  A H  , 0^0 7a  (T v  q  'S  - T , /j  F  (125)  and AKE  1 w 2 g  TJ  A L V  2  0  ( V - V )  (126)  61 or A K E - w A H  Furthermore  o  V  o  (p)  i  (T  s  2  -T  p  )  2  (127)  upon s u b s t i t u t i o n o f - 1  h K  =  m  -4  P  i n t o e q u a t i o n 110 we o b t a i n  -T  m  J  n=l  (128)  y  F  ;  S u b s t i t u t i o n o f t h e r e s u l t s o f e q u a t i o n s 124) 125? 127 and 128 i n t o the general equation f o r AE  which i s Q  AE  = (P_ S  g  v  -  v)  F  y  a  +  2I  AKE + w A H V f 7 . o o J n=l '  n  —  v  (-t ) n  C d v (T - t ) n n F n'  7  JI  (129)  n=Q+l yields  AE  =  W  A H  g  V o  +  o a  { (T -T ) S  F  +  \F -1 f rl i p - I IP^F A -  P  (T -T ) E4T ( ^ - V S  2  P  2  +  m  1  (130)  62  FIG 24  WATERHAMMER CHART DEFINING THE REGIONS FOR OPTIMUM GATE OPERATION:  t = T~- TO  t = T.  63 T h i s e q u a t i o n f o r AE  i s subject to the l i m i t a t i o n s p r e v i o u s l y  mentioned.  F o r the case where  AE  c a n be e v a l u a t e d from f i g u r e 23 where t h e a s s u m p t i o n has a g a i n been  made t h a t  i s a straight line.  The shaded a r e a o f f i g u r e 23b i s as be-  fore Q A r e a mpv..  =  °  2  ^r)  v n  n=l Furthermore A r e a mpxv  ( ( v - v ) + (v _ - v ) )  = ± (T - ^ )  1  Q  Area  V-jXyVg  = ^  ((v  Area  Vgyzv^  = |  (( g V  1  -  g  v^)  +  p  (v  -  2  - v ) + (v^ p  (152)  ?  ]  v j )  (133)  v j )  (154)  From f i g u r e 23a we have  V  JL  l - F V  v . - v„ = 2 F 2  V  - 1)  ( h l  (^  h  V  v^3 - v"F T  etc.  =  I  (  H  I  _ D  . _L. (  +  I)  ( h i  - 1) + i ( h _ ! ) = ( ! ! )  - !)  2 " F = 2 f (2  +  x  +  ~ 2p~ < 2 - 1)  I  h  +  <2 - > = £ h  X  +  _ 1)  ( 1 5 5  )  ( _ - 1) (136) h]  2^) < 2 ~  ^  h  7 ( h - 1) = (J " 2 ^ ) ( h - 1) (138) 2  2  D i v i d i n g e q u a t i o n 138 by e q u a t i o n 137 and e q u a t i o n 136 by e q u a t i o n 135 we o b t a i n  V  2  _  V  F  (7 2f)  =  (^-D  +  ' ( l ^ ) ^ -  1  ) " "  1  ' ^  " (139)  T h e r e f o r e we have v^ - v  F  = R (v - v )  (140)  v  F  = R (v  (141)  2  - v  2  1  F  - v ) F  S u b s t i t u t i o n o f the r e s u l t s o f e q u a t i o n s 140 and 141 i n t o e q u a t i o n 134 yields A r e a v^v^  = ^ R  = R (Area  (142)  - v ) + (v - v )) p  2  p  (143)  v-jXyVg)  etc. T h e r e f o r e we have  J~  2 iv  n  (-t ) = A r e a mpxv-j^ + A r e a v-j^xyVg ( ^  n=l  j=0  However 2p  •> s  and as a r e s u l t R<  1  R^)  (144)  65  T h i s means t h a t  R  = 1 + R + R  J  2  +....+ R  = ^  J  (  1 4 5  )  j = o and  /  ( - " t ) = A r e a mpxi^ + -—^ A r e a v x y v  2  n  £  n = 1 R e f e r r i n g t o f i g u r e 23a w i t h h, = h  1  we have m  i - \ A . if - >  T  (h  +  1  m  and  so t h a t  i  <-*„> k < c - ^  k  2  :  (T  T  A  +  +  .  a  0  .  2  T  F )  »  n=1 +  ii>< :>VA.- > 2  1  046)  where  - 1  v/h  T  1 h  R =  ^  =  P %  2j)T -(A -M) F  _ 1  1  h, - 1 m  2  ?P  1  X Am -  1  ,  +  2pT  F +  m  (/h  m +  1)  (147)  2p  +  \  (/h  m  +  l)  TT^C—^— 2 ( / h  m  +  D  ( 1 4 8 )  6 6  and  TF f\  2L_  m  C  a  "Ts  L a  2K  (H9)  {( FA -T ) ( g ^ ) - i ]  L a  T  B  S  S u b s t i t u t i n g these v a l u e s i n t o the equation f o r A E  (150)  which i s i n t h i s  case  S g i v e n by e q u a t i o n Jl) we o b t a i n  AE  g  = WAH  o  £ I|  V• o a  W  (T  - T  ' F  S  J  ) + P ( T y S K  J  TpK-V 2 p  T  + A  F  /h  V  m  m  + i  + 1  (Tj V  F'  2  - T J  (/h m  v v  2  F  2 T  ;  )  F  - i  >]}  (151)  I t c a n be shown t h a t when + 1  v/h J  F  ~  1  (152)  2j3  e q u a t i o n s 130 and 151 c o i n c i d e .  A + •'F  1  2jD  e q u a t i o n 130 a p p l i e s . v  If  (153)  ' If m  + 1 (154)  67 e q u a t i o n 151  applies.  From e q u a t i o n s 124  a.nd 152  F /  Sv  C  L—  n  i t i s seen t h a t f o r any g a t e o p e r a t i o n  ( L  n  -  t )  F  v  = 0  n'  n=Q+l if  rr  + 1  /h  2©  F ^  K  D D >  or i f  F o r v a l u e s o f *T„  /h  g i v e n by  +  m  2p  27  2  ^  \  xr-  1  F *  >  +  2^(/h  2  m  + 2)  .  < ) 157  (-t ) i s a p o s i t i v e q u a n t i t y f o r g a t e c l o s u r e and a n e g a t i v e  n=l F o r v a l u e s o f '3"_ g i v e n by  q u a n t i t y f o r gate o p e n i n g . +  2  25. (yff  +  h T  ^  F  <  m  2)  (158  )  2 (bv^ ( ~ t ) i s a n e g a t i v e q u a n t i t y f o r g a t e c l o s u r e and a p o s i t i v e n  n=l q u a n t i t y f o r gate opening.  These r e s u l t s are shown i n f i g u r e 24.  f o r g a t e o p e r a t i o n i n t o r e g i o n B o f f i g u r e 24 would be advantageous  Only to  68  '• F I G 25  COMPARISON OF WICKET GATE CLOSURE CURVE AND OPTIMUM CLOSURE CURVE  FIG 26 WATERHAMMER CHART COMPARING WICKET GATE AND OPTIMUM CLOSURE CURVES  69 reduce the r a t e o f g a t e o p e r a t i o n a f t e r t = T . n  F o r gate o p e r a t i o n i n t o  r e g i o n s A o r C o f f i g u r e 24 g a t e o p e r a t i o n s h o u l d be a t the maximum r a t e . It  i s i n t e r e s t i n g t o note from f i g u r e 24 t h a t f o r a l l but low-  v a l u e s o f j (p^-  1«5)  the optimum g a t e o p e r a t i o n i s t h a t w h i c h l e a v e s no  h y d r a u l i c o s c i l l a t i o n s i n the system.  (The normal range o f g a t e o p e r a t i o n  i s between T=  .3 the machine i s r u n n i n g a t speed  no  .3 and 7=  !•  Below X=  load). C l o s e e x a m i n a t i o n o f e q u a t i o n s 130 and 151  ofoon  i s v e r y pronounced  shows t h a t the e f f e c t  w h i l e that of h^ i s l e s s important.  There-  f o r e i n d e s i g n a r e d u c t i o n i n <y> ( a l t h o u g h i n c r e a s i n g the d i a m e t e r o f a p e n s t o c k ) may p o s s i b l y r e s u l t i n economic s a v i n g s because o f the r e d u c t i o n i n head r i s e f o r the same s p e c i f i e d A E  ( p a r t i c u l a r l y i f decreased steady  s t a t e f r i c t i o n l o s s e s are t a k e n i n t o a c c o u n t ) . The r e l a t i v e importance o f of  AE  and h ^ show t h a t the w o r s t v a l u e s  w i l l be o b t a i n e d a t r a t e d head. g  4.3  T u r b i n e W i c k e t Gate C l o s u r e Curve A t y p i c a l t u r b i n e w i c k e t g a t e c l o s u r e curve i s shown i n f i g u r e  25 and the a s s o c i a t e d h - v i n f i g u r e 26. J^"  2 (Sv  n  diagram f o r c l o s u r e from f u l l g a t e i s shown  T h i s type o f c l o s u r e b e g i n s s l o w l y so t h a t the term  ( t ) o f the g e n e r a l e q u a t i o n f o r A E ^ -  n  i s much l a r g e r t h a n  n=l i t would be f o r the optimum c l o s u r e c u r v e shown i n f i g u r e 25. ing  v a l u e s i n the e q u a t i o n f o r A E  The  remain-  would be r e a s o n a b l y s i m i l a r f o r the g  two c l o s u r e s .  Note t h a t f o r g a t e o p e r a t i o n s from f u l l g a t e to p a r t i a l g a t e ,  (which a r e i m p o r t a n t f o r t h e i r e f f e c t s on the e l e c t r i c a l f r e q u e n c y even i f t h e y do not cause maximum speed d e v i a t i o n ) , the e f f e c t o f the  initial  slowness o f the w i c k e t gate c l o s u r e i s even more m a g n i f i e d when the r a t i o s  70 o f the v a l u e s o f A E  f o r each type o f c l o s u r e a r e c o n s i d e r e d . This i s g ^ because most o f the d i f f e r e n c e between the two v a l u e s of A E occurs i n g 2L the f i r s t 3 (~)  seconds i n the example shown.  curve can r e s u l t i n s a v i n g s o f 20 - 30$  Use  o f a good c l o s u r e  i n the v a l u e o f A E ^ f o r the same  head r i s e . 4-4 Use o f an Upstream Gate i n a C l o s i n g  Operation  D u r i n g the f i r s t — seconds a f t e r the i n i t i a t i o n o f gate c l o s u r e ° a &  the power i n p u t to the p e n s t o c k i s unchanged.  I f c l o s u r e were i n i t i a t e d  a t the upstream end o f the p e n s t o c k , s i m u l t a n e o u s l y w i t h c l o s u r e a t the turbine,  A E ^ c o u l d be s u b s t a n t i a l l y r e d u c e d a l o n g w i t h a r e d u c t i o n i n h^.  F i g u r e s 27,  28 and 29 show the g a t e c l o s u r e c u r v e s , power i n p u t t o the  p e n s t o c k , and power o u t p u t from the p e n s t o c k , and the h - v  diagram f o r  a case where c l o s u r e i s made i n 12 — seconds a t the downstream g a t e a ° 7 ^ seconds ( t o .05 g a t e ) a t the upstream g a t e .  and  (The upstream c l o s u r e i s  more r a p i d because the i n i t i a l e f f e c t a t l a r g e g a t e openings i s s m a l l ) . The head d i s c h a r g e e q u a t i o n s upstream gate are g i v e n h -  i  = v  f o r d i f f e r e n t g a t e openings o f the  by: -  2  r)  2  &  (159)  where V i s the r e l a t i v e gate o p e n i n g (3). I n f i g u r e 29, used i s 2.5  the head r i s e when the downstream v a l v e o n l y i s  times g r e a t e r t h a n the head r i s e due  stream c l o s u r e s .  I n f i g u r e 28,  t o combined up and down-  A E ^ i s approximately  20$  l e s s f o r the  case o f combined c l o s u r e s when compared to downstream c l o s u r e o n l y . Use  o f an upstream v a l v e i n the c l o s i n g o p e r a t i o n o b v i o u s l y  produce a l a r g e r e d u c t i o n i n A E ^ and i n waterhammer and may use o f a p r e s s u r e r e g u l a t o r .  The b u t t e r f l y v a l v e and  avoid  the  s p h e r i c a l valve  can  are not s u i t e d t o t h i s type o f o p e r a t i o n and a n e e d l e v a l v e would p r e sent tremendous  d e s i g n problems.  F i g u r e 30  shows an a l t e r n a t i v e w h i c h  s h o u l d e l i m i n a t e most o f the problems o f the o t h e r v a l v e s and might warr a n t some i n v e s t i g a t i o n . A s i d e from the problem o f a s u i t a b l e v a l v e t h e r e a r e s e v e r a l o p e r a t i n g problems; p a r t i c u l a r l y t h a t o f f a i l u r e o f the upstream v a l v e .  A p o s s i b l e s o l u t i o n t o t h i s problem i s t o mount an  e l e c t r i c a l c o n t a c t on the upstream v a l v e t h a t would be connected t o a s o l e n o i d c o n t r o l l i n g p a r t o f the f l u i d s u p p l y t o the downstream s e r v o motor p i s t o n .  I f the upstream v a l v e f a i l e d t o o p e r a t e ,  the e l e c t r i c a l  c o n t a c t would r e m a i n open, the s o l e n o i d would t h e n r e m a i n c l o s e d and the r a t e o f c l o s u r e o f the downstream v a l v e would be l i m i t e d t o a s a f e v a l u e . A n o t h e r p r o b l e m i s the a c t u a l o p e r a t i n g sequence o f the two v a l v e s , ( i t would be uneconomic position)/  t o l e a v e the upstream g a t e i n i t s p a r t i a l l y c l o s e d  I f the downstream g a t e were c l o s e d t o i t s f i n a l s t e a d y s t a t e  p o s i t i o n and the upstream v a l v e c l o s e d t o say Y=  .05,  t h e n by m a i n t a i n i n g  the upstream v a l v e i n t h i s p o s i t i o n u n t i l the i n i t i a l v a l u e of AE  were  n e a r l y c a n c e l l e d by the l o s s e s due t o the upstream v a l v e , the upstream v a l v e c o u l d t h e n be s l o w l y opened t o f u l l g a t e .  The n e t speed change o f  the t u r b i n e would be zero and h y d r a u l i c o s c i l l a t i o n s would be r e d u c e d .  .9 Tun >ine  at 6  i i tree 177  (Jl  re. < C/osc  lost// 2 Cut  \\  ^-^ W rsec)  O l 2 2 4 S £ 7 8 9 10 H /Z FIG 27 CLOSURE CURVES FOR SYNCHRONIZED GATE OPERATION  Ot  treaVn  Turbine closure  t ot  it  Turbine  teoas Upstraam am Closure  4»  i (sec)  FIG 28 COMPARISON OF POWER OUTPUTS FOR SINGLE AND SYNCHRONIZED GATE OPERATION  FIG. 29  WATERHAMMER CHART COMPARING SINGLE AND SYNCHRONIZED GATE OPERATION  H *H, 0  l/ohe  in Open  Position  l/oli/e in Partially  From  the &orc/a Loss  Mz~H  2  A% AT  =/Vz\  2  « relative  ^  Closed  Position  l/z = V$  Equation  (±-l) qate.  H =H >H  Z  Z  area  J  FIG 50 AN: UPSTREAM VALVE  0  3  75  CHAPTER  V  CONCLUSIONS  5-1  The Use o f the Energy Method o f S o l u t i o n . A l t h o u g h many t e x t - h o o k s s t a t e t h a t energy methods a r e n o t  a p p l i c a b l e t o problems o f r a p i d l y v a r i e d f l o w , i t cannot be d e n i e d t h a t i n c e r t a i n i n s t a n c e s energy methods can l e a d t o v a l u a b l e interpretations.  I f the p r o c e s s c a u s i n g the r a p i d f l o w v a r i a t i o n s  i s a d i a b a t i c (as i t u s u a l l y i s - even i n t h e case o f an h y d r a u l i c jump) and i f the a c c e l e r a t i o n s o f t h e f l o w t a k e p l a c e i n the d i r e c t i o n o f the s t r e a m - l i n e s (so t h a t no t u r b u l e n c e - w h i c h i s the major s o u r c e o f energy l o s s - t a k e s p l a c e ) t h e n the energy l o s s e s , w h i c h u s u a l l y make s o l u t i o n s by energy methods i m p o s s i b l e , can be n e g l e c t e d . such i s the case the use o f energy methods i s j u s t i f i a b l e .  If  The  waterhammer p r o c e s s meets the above r e q u i r e m e n t s and so energy problems may be s o l v e d d i r e c t l y . Use o f t h e energy method f o r an i n c r e m e n t a l g a t e movement shows t h a t as the p r e s s u r e wave t r a v e l s a l o n g the p e n s t o c k , any change i n k i n e t i c energy i s accompanied by a change i n the energy, s t o r e d as s t r a i n energy, i n the f l u i d and i n the p e n s t o c k , and the n e t change o f energy a t any p o i n t i n t h e p e n s t o c k i s a c o n s t a n t f o r a g i v e n wave as i t t r a v e l s i n one d i r e c t i o n a l o n g the p e n s t o c k .  5.2  The Importance o f the L i n e h = 2p v on the Waterhammer C h a r t . The energy method e x p l a i n s why, under c e r t a i n c o n d i t i o n s o f  gate o p e r a t i o n , t h e change i n power o u t p u t i s i n i t i a l l y o p p o s i t e t o t h a t desired.  I f t h e p o i n t r e p r e s e n t i n g t h e c o n d i t i o n s a t the s t a r t o f a g a t e  76  o p e r a t i o n i s t o the r i g h t o f the l i n e h = 2pv  on the waterhammer c h a r t ,  t h e n any waves r e l a t e d t o an o r i g i n a t i n g p o i n t l o c a t e d to the r i g h t o f t h i s l i n e w i l l cause a change i n the k i n e t i c energy o f the f l u i d i s g r e a t e r t h a n the amount o f energy t h a t the p e n s t o c k can s t o r e .  that  and the  fluid  T h e r e f o r e , i f the gate o p e r a t i o n i s one o f c l o s u r e , t h e r e  w i l l he an excess o f energy e q u a l t o the d i f f e r e n c e between the a b s o l u t e v a l u e o f the change i n k i n e t i c energy and the a b s o l u t e v a l u e o f the change i n s t o r e d energy.  T h i s excess appears a t the gate and  r e s u l t i s an i n c r e a s e d power o u t p u t . o f opening,  the  I f the gate o p e r a t i o n i s one  an energy d e f i c i e n c y e q u a l t o the d i f f e r e n c e between the  a b s o l u t e v a l u e o f the change i n k i n e t i c energy and the a b s o l u t e v a l u e o f the change i n s t o r e d energy w i l l r e s u l t .  I n l e s s p i s very small  t h i s means t h a t f o r most gate o p e r a t i o n s the i n i t i a l power change w i l l be o p p o s i t e t o t h a t d e s i r e d .  T= v^h —  .  + 1  2  on the Waterhammer C h a r t .  F o r any g i v e n r e l a t i v e head h t h i s l i n e d e f i n e s where the s l o p e o f the  T  curve i s e q u a l t o the s l o p e o f the waterhammer l i n e (2o  T h i s l i n e i s always t o the r i g h t o f the l i n e h = 2 p v .  the maximum v a l u e o f  d  —  T  T  T  dt  T= —  \/h + 1 —  Any  ).  gate  must m a i n t a i n  t h a t i s c o n s i s t e n t w i t h the maximum a l l o w a b l e  head d e v i a t i o n , i f the a b s o l u t e v a l u e o f  AE  i s t o be k e p t t o a minimum.  F (The term /  Wl  C n  iv  (T n  F  - t ) i n the g e n e r a l e q u a t i o n f o r A E n  T  \/h + 1  = ~~2p  does not g  77  5.4  The Importance  o f the A r e a on the Waterhammer C h a r t Between the  L i n e s h = 2 Q v and  T=  ^  I n t h i s a r e a any waves c r e a t e d w i l l cause an energy opposite to that desired.  output  As a r e s u l t i t i s . d e s i r a b l e t o c r e a t e the  2L minimum number o f waves p o s s i b l e i n the — ^ a  seconds b e f o r e the f i n a l  s t e a d y s t a t e power o u t p u t i s f i r s t r e a c h e d .  (Up t o the p o i n t T  the energy o u t p u t i s f i x e d by the t o t a l change i n the k i n e t i c  r  -  2L — a  energy o f  the f l u i d between the i n i t i a l and f i n a l s t e a d y s t a t e s , and the energy i n p u t to the p e n s t o c k .  Thus i t i s d e s i r a b l e t o d e c r e a s e the c r e a t i o n 2L  of waves o n l y i n the l a s t — output.)  T h i s i s e q u i v a l e n t t o r e d u c i n g the r a t e o f gate o p e r a t i o n i n  this interval. of  AE;'  seconds b e f o r e r e a c h i n g s t e a d y s t a t e power  A l t h o u g h the r e s u l t a n t r e d u c t i o n i n the a b s o l u t e v a l u e  due t o the reduced c l o s i n g r a t e may be s m a l l , the h y d r a u l i c  o s c i l l a t i o n s i n the system w i l l be g r e a t l y r e d u c e d , thus l e a d i n g to g r e a t e r system s t a b i l i t y .  The example o f i n s t a n t a n e o u s g a t e c l o s u r e  demonstrates the i d e a t h a t i n c a s e s where a p a r t i a l g a t e o p e r a t i o n t a k e s 2L p l a c e i n a time l e s s t h a n —  seconds, i t may be q u i t e advantageous  from  the p o i n t o f head r i s e as w e l l as energy o u t p u t and h y d r a u l i c s t a b i l i t y reduce the r a t e o f gate c l o s u r e a f t e r a c e r t a i n  ,3  fc^^fci,  to  interval.  C. - O „ « . L  c  n  n  v  F  n'  n=Q+l Equation f o r A E  .  I n the d i s c u s s i o n o f the g a t e c l o s u r e w h i c h y i e l d s the s m a l l e s t v a l u e o f maximum waterhammer yP ~  L  c  n=Q+l  iv  n  -1)  (T  n  P  n  78  was e v a l u a t e d t o be: F  7"  C  Z  n=Q+l  n  Sv n  (T - t ) F n  "V  =  12 P  20 X jf F  - (yC + 1) y  (h  + 2)  m  -  2f  %  (/  m  + 2)  2L I f t h e v a r i a t i o n s i n head d u r i n g t h e l a s t —  i n t e r v a l of closure are  n o t t o o g r e a t t h e n t h i s e q u a t i o n would be a r e a s o n a b l e a p p r o x i m a t i o n f o r any type o f g a t e o p e r a t i o n .  I f so t h e n i t i s i m m e d i a t e l y  t h a t u n l e s s ^ i s v e r y s m a l l o r s/^ -~  s v e r  m  obvious  y l a r g e t h i s term i s o f  n e g l i g i b l e v a l u e when compared t o terms o f t h e o r d e r o f (0(1^,2  - Tj, ) 2  w h i c h i s r e p r e s e n t a t i v e o f t h e change i n k i n e t i c energy between the i n i t i a l and f i n a l s t e a d y s t a t e s .  T h e r e f o r e the main advantage o f making  the a b s o l u t e v a l u e o f t h i s term as s m a l l as p o s s i b l e i n t h e area, between v/h+i j .. = ~2Q— ° ^ the waterhammer c h a r t , i s t h e r e d u c t i o n o f h y d r a u l i c o s c i l l a t i o n s . Even t o t h e l e f t o f t h e l i n e h = 2 o v  T  the advantages  o f reduced h y d r a u l i c o s c i l l a t i o n s may w e l l outweigh any dT  r e d u c t i o n o f A E ^ t h a t would be o b t a i n e d by k e e p i n g  at a high value.  Q 5.6 for  The Importance  o f t h e Term ^  2 o"v (~^ ) i n t h e G e n e r a l E q u a t i o n n  n  n=l  AE . g  The l a s t s e c t i o n showed t h a t f o r a wide range:  y~ L n=Q+l I f such i s the case t h e n :  c 6v n  n  v  F  - 1 ) n y  79 Q AE  S  =  (p„ - p_) S F' a  + AKE  +  [  v x (-t ) f 7(WAH V )( J n n' ) ( o o x  n  =  1  From t h i s e q u a t i o n ,  t h e importance o f c h a n g i n g t h e energy i n p u t t o t h e Q. p e n s t o c k as r a p i d l y as p o s s i b l e i s o b v i o u s (WAH V > 2 d v (-t ) i s o ^Z]_ n n 0  the v a r i a b l e term i n t h e energy i n p u t e q u a t i o n ) .  Furthermore t h i s  e q u a t i o n g i v e s an approximate method f o r comparing the r e l a t i v e  merits  of d i f f e r e n t c l o s u r e c u r v e s s i n c e the r a t e o f change o f v e l o c i t y a t the i n t a k e i s r e l a t e d t o t h e r a t e o f change o f v e l o c i t y a t the g a t e . We c a n t h e n w r i t e Q  YLZ  2 iv  n=l  n  (-t) = f v  n'  K  (4f) dt '  T h e r e f o r e those c l o s u r e c u r v e s w h i c h have a h i g h i n i t i a l v a l u e o f —r-r° dt can be e x p e c t e d t o produce s m a l l e r v a l u e s o f A E ^ .  I n f a c t the p l o t  o f i n p u t power' i s u s u a l l y v e r y s i m i l a r i n form t o the g a t e c l o s u r e c u r v e (compare f i g u r e s 28 and 27 f o r t h e case o f no upstream c l o s u r e ) .  5.7  The Use o f an Upstream Gate i n C l o s i n g The  Operations.  example showing t h e e f f e c t o f combined up and downstream  g a t e c l o s u r e showed t h a t many advantages may be g a i n e d by the use o f t h i s type o f s y n c h r o n i z e d  c l o s i n g operation.  As t h e r e i s u s u a l l y a  v a l v e on t h e upstream end o f a p e n s t o c k , i t seems w e l l w o r t h w h i l e t o make g r e a t e r use o f i t , p a r t i c u l a r l y i n cases where a p r e s s u r e i s necessary.  regulator  The p o s s i b l e economic advantages o f one l e s s v a l v e and  i t s accompanying energy d i s s i p a t o r , a l o n g w i t h reduced p e n s t o c k c o s t s and  energy s a v i n g s make f u r t h e r i n v e s t i g a t i o n o f t h i s p o s s i b i l i t y o f  definite  usefulness.  80 5.8  G e n e r a l Comments  p  Because of the g e n e r a l l y s m a l l v a l u e of ^ C <^v (TL. - t ) n n F n n=Q+l v  7  n  much s t a b i l i t y can be g a i n e d w i t h n e g l i g i b l e  effect  on  AE  i f turbine  g a t e s are always r e g u l a t e d i n a manner such t h a t the h y d r a u l i c a r e r e d u c e d t o as n e a r zero as p o s s i b l e .  To t h i s end  oscillations  i t might be of some  advantage t o g o v e r n h y d r a u l i c t u r b i n e s on the b a s i s o f the change i n l o a d on the g e n e r a t o r i n s t e a d o f on the b a s i s of speed d e v i a t i o n .  (For  s m a l l l o a d changes the t u r b i n e gate i s o f t e n r e g u l a t e d on the b a s i s of l o a d - i n such cases the t u r b i n e speed g o v e r n o r i s not i n a c t i o n . ) g o v e r n i n g were based on the amount of l o a d change f o r l a r g e  If  load  v a r i a t i o n s a f o r m of programmed g a t e o p e r a t i o n - m i g h t be used ( i . e . f o r each l o a d change t h e r e would be a p r e d e t e r m i n e d g a t e o p e r a t i o n The  r e s u l t would be b e t t e r speed r e g u l a t i o n and Good speed r e g u l a t i o n i n g e n e r a l  increased  pattern).  system s t a b i l i t y .  i s d e t e r m i n e d by the  ratio  o f the t o t a l energy s u p p l i e d t o the prime-mover, t o the k i n e t i c  energy  o f m o t i o n of the f l u i d s u p p l y i n g the energy t o the prime-mover.  Thus  a steam t u r b i n e i s c o m p a r a t i v e l y  easy to g o v e r n - the r a t i o of  the  t o t a l energy o f a pound o f steam t o i t s k i n e t i c energy of m o t i o n i s usually f a r greater high pressure,  ( p a r t i c u l a r l y w i t h the p r e s e n t h i g h  temperature,  steam p l a n t s ) t h a n t h a t o f the h i g h e s t head hydro-power  plants. I n a hydro-power p l a n t the t o t a l energy s u p p l i e d v a r i e s as 2 the p r o d u c t HV w h i l e the k i n e t i c energy o f the f l u i d v a r i e s as V . The  r a t i o o f these two  energies i s :  T o t a l Energy K i n e t i c Energy  cy;  HV y2  =  H y  81  •n , But  1 -r— =  0  ——  9 Therefore  o  p i s a measure o f t h e r a t i o o f t h e t o t a l energy o f the f l u i d  t o t h e k i n e t i c energy o f t h e f l u i d . the l i n e h = 2ov  Furthermore, from the f a c t that  i n d i c a t e s under what c o n d i t i o n s the d e l i v e r y system  ( i . e . t h e p e n s t o c k ) can s t o r e an amount o f energy g r e a t e r t h a n an accompanying change o f k i n e t i c energy, ( a t w h i c h p o i n t good g o v e r n i n g becomes p o s s i b l e ) i t i s seen t h a t tp i s a v e r y i m p o r t a n t f a c t o r in'turbine governing.  ( T h i s was demonstrated i n t h e d i s c u s s i o n o f the  g a t e c l o s u r e c u r v e w h i c h y i e l d s the minimum v a l u e o f maximum waterhammer). A low v a l u e o f  i s n o t o n l y i n d i c a t i v e o f a h i g h t o t a l energy t o k i n e t i c  energy r a t i o , b u t a l s o i n c r e a s e s  the a r e a on t h e waterhammer c h a r t i n  w h i c h good g o v e r n i n g i s p o s s i b l e . ' head r i s e f o r a g i v e n gate o p e r a t i o n l o w e r t h e maximum p r e s s u r e  Because o f i t s e f f e c t on t h e p r e s s u r e (the l o w e r t h e v a l u e o f <p , t h e  deviation) increasing p  may be more  e c o n o m i c a l t h a n u s u a l l y b e l i e v e d , p a r t i c u l a r l y i f t h e s a v i n g s due t o decreased steady s t a t e f r i c t i o n l o s s e s are i n c l u d e d .  82 BIBLIOGRAPHY  1.  B e r g e r o n , L. Water Hammer i n H y d r a u l i c s and Wave Surges i n E l e c t r i c i t y , New Y o r k : John W i l e y and Sons, 1961.  2.  J a e g e r , C. E n g i n e e r i n g F l u i d M e c h a n i c s , L i m i t e d , 1957-  3-  J a e g e r , C. "Water Hammer Caused b y Pumps," J u l y , 1959.  4«  K i r c h m a y e r , L. K. Economic C o n t r o l o f I n t e r c o n n e c t e d Systems, New Y o r k : John W i l e y and Sons, 1959  5.  Parmakian, J . Waterhammer A n a l y s i s , New Y o r k : t i o n s , I n c . , 1963  6.  Ruus E. D e t e r m i n a t i o n o f C l o s u r e Curves Which Y i e l d t h e S m a l l e s t V a l u e o f Maximum Waterhammer, T r a n s l a t e d by E. Ruus, K a r l s r u h e : 1957-  7.  Stevenson, W. D. Elements o f Power S y s t e m . A n a l y s i s , New Y o r k : M c G r a w - H i l l Book Company, I n c . , 1955.  London: B l a c k i e and Son Water Power,  Dover P u b l i c a -  83 APPENDIX I SYMBOLS, ABBREVIATIONS AND UNITS  a  -  waterhammer'wave v e l o c i t y  ft/sec 2  A  -  cross-sectional  D  -  penstock diameter  ft  E  -  modulus o f e l a s t i c i t y o f p e n s t o c k w a l l  lbs/ft  E  -  energy'  ft-lbs  P  -  force  lbs  g  -  acceleration  H  -  t o t a l p r e s s u r e head  ft  q  -  s t e a d y s t a t e p r e s s u r e head  ft  iH  -  i n f i n i t e s i m a l change i n p r e s s u r e head  ft  AH  -  t o t a l change i n p r e s s u r e head  ft  h  -  r e l a t i v e p r e s s u r e head = —  H  area of penstock  of g r a v i t y  ft  ft/sec  ti o  -  b u l k modulus o f f l u i d  L  -  penstock l e n g t h  P  -  power  R  -  penstock r a d i u s  ft  S  -  work  ft-lbs  t  -  time - g e n e r a l  sec  T  -  s p e c i f i e d time i n t e r v a l  sec  V  -  v e l o c i t y o f the f l u i d i n the p e n s t o c k  ft/sec  -  f u l l gate v e l o c i t y a t H = H  ft/sec  V  ft ft-lbs/sec  o  0  v  lbs/ft  2  K  -  V relative velocity = —  q  -z  w ^ ~ 1  -  u n i t weight of water y  -  p i p e l i n e c o n s t a n t = TJ-J^0  -  r e l a t i v e gate area  a  lbs/ft  

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