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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 of 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 the Department of C i v i l E ngineering We accept t h i s t h e s i s as conforming to the r e q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA A p r i l , 1966 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced degree a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by t h e Head o f my Department o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n Department o f C i v i l E ngineering The U n i v e r s i t y o f B r i t i s h C o l umbia Vancouver 8, Canada Date A p r i l 27, 1966. ABSTRACT This t h e s i s examines the i n f l u e n c e of d i f f e r e n t gate opera-t i o n curves on the surplus or d e f i c i e n c y of energy input to a h y d r a u l i c t u r b i n e accompanying a sudden change of load on the t u r b i n e . A general s o l u t i o n to the problem i s obtained by e v a l u a t i n g the energy input to the penstock and the energy conversion w i t h i n the penstock during t r a n s i e n t c o n d i t i o n s . The r e s u l t s show that f o r given maximum pressure r i s e or drop, a considerable r e d u c t i o n i n the surplus or d e f i c i e n c y of energy input to the t u r b i n e can be obtained by use of a s u i t a b l e gate o p e r a t i o n curve. At the same time i t i s p o s s i b l e to reduce the h y d r a u l i o s c i l l a t i o n s i n the system. TABLE OP CONTENTS 111 CHAPTER I I I I I IV V INTRODUCTION ELEMENTS OF THE SCHNYDER-BERGERON GRAPHICAL SOLUTION OF THE ALLIEVI WATERHAMMER EQUATIONS ' THEORY OF ENERGY CONVERSION IN THE PENSTOCK APPLICATION OF ENERGY PRINCIPLES TO THE DETERMINATION OF OPTIMUM GATE OPERATION EVALUATION OF DIFFERENT GATE OPERATION CURVES CONCLUSIONS PAGE 1 .3 4 22 45 75 APPENDIX I SYMBOLS ABBREVIATIONS AND UNITS 83 IV Number 1 2 5 6 7 LIST OF FIGURES The b a s i c h y d r a u l i c system Conditions i n the penstock r e s u l t i n g from an 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 to a waterhammer problem An enlarged view of the waterhammer chart showing the c r e a t i o n of a wave 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 to waterhammer problems Power input to the tu r b i n e during t r a n s i e n t c o n d i t i o n s 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 Input v e l o c i t y to the penstock G r a p h i c a l s o l u t i o n s to waterhammer problems F when y c £>v (T - 1 ) = o ' "•• n n F n Page 12 12 15 15 23 23 24 24 25 n=Q+1 10 P l o t of input v e l o c i t y to the penstock f o r the cond i t i o n _____ n=Q+1 C <^ v (QL, - t ) = 0 n n v F n 25 11 Waterhammer charts and surplus energy p l o t s f o r the time i n t e r v a l t = T„ to t = T_ 37 12 Waterhammer charts and surplus energy p l o t s f o r the time i n t e r v a l t = T to t = T - gate c l o s u r e 38 Q, F 13 Waterhammer charts and surplus energy p l o t s f o r the time i n t e r v a l t = T to t = T - gate opening 39 14 Waterhammer chart and gate c l o s u r e curve f o r optimum ^ gate o p e r a t i o n 42 L i s t of Figu r e s (Cont'd) UTumber _ Page 42 dh 15 Waterhammer chart f o r gate c l o s u r e : 2o >• — ,y F 16 Waterhammer chart and gate c l o s u r e curve - instantaneous gate c l o s u r e 47 17 Waterhammer charts f o r d i f f e r e n t values of >^ - instantaneous gate c l o s u r e 47 18 Water v e l o c i t y at the intake and at the gate - optimum gate c l o s u r e 51 19 Waterhammer chart f o r optimum gate c l o s u r e 51 20 Optimum gate c l o s u r e curve 55 21 Water v e l o c i t y a t the i n t a k e - optimum gate o p e r a t i o n 55 22 Waterhammer chart f o r the i n t e r v a l t = T A to t = T„ Q F - optimum gate o p e r a t i o n 55 23 Waterhammer chart and input v e l o c i t y f o r gate c l o s u r e : 2p *^ <y 62 24 Waterhammer chart d e f i n i n g the regions f o r optimum gate o p e r a t i o n : t = to t = 62 25 Comparison of wicket gate c l o s u r e curve and optimum cl o s u r e curve 68 26 Waterhammer chart comparing wicket gate and optimum cl o s u r e curves 68 27 Closure curves f o r synchronized gate o p e r a t i o n 72 28 Comparison of power .outputs f o r s i n g l e and synchronized gate operation 72 L i s t of Figur e s (Cont'd) Number Page 29 Waterhammer chart comparing s i n g l e and synchronized gate o p e r a t i o n 73 30 An upstream valve 74 ACKNOWLEDGEMENT The author wishes to express h i s thanks to h i s s u p e r v i s o r Dr. E. Ruus, f o r the va l u a b l e c r i t i c i s m , guidance and encouragement. He a l s o wishes to express h i s thanks to P r o f e s s o r J . F. Muir f o r h i s v a l u a b l e advice and suggestions. 1 INTRODUCTION Many of the complicated processes 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 frequency i s maintained w i t h i n very c l o s e t o l e r a n c e s (+ 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 increase or decrease i n the magnitude of the l o a d on the system can be considered instantaneous (such as would be caused by the l o s s of a t r a n s m i s s i o n l i n e ) . Furthermore, the d i s t r i b u t i o n of t h i s l o a d change between the v a r i o u s generators i s a l s o instantaneous when compared to the speed of response of the t u r b i n e governor. The r e s u l t i n g change i n system f r e -quency i s approximately p r o p o r t i o n a l to the square root of the d i f f e r e n c e between the combined input to the t u r b i n e s and the combined output of the connected generators; and i n v e r s e l y p r o p o r t i o n a l to the i n e r t i a of the whole system ( i . e . i n c l u d i n g connected l o a d ) . To keep the frequency change (or corresponding generator speed change) w i t h i n s p e c i f i e d l i m i t s f o r an assumed major l o a d change i t i s necessary to e i t h e r increase the system i n e r t i a or decrease the d i f f e r e n c e between t u r b i n e input and generator output; the choice being mainly one of economics. I n a h y d r a u l i c turbine-generator arrangement t h i s input-output 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 s e n s i t i v i t y of the t u r b i n e governor and the i n e r t i a of the f l u i d s u p p l ying energy to the t u r b i n e . This paper i s concerned w i t h the problem of m i n i m i z i n g t h i s input-output 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-d i t i o n s . I t i s assumed that the governor s e n s i t i v i t y and speed of r e s -ponse, and the allowable penstock pressure r i s e and drop are s p e c i f i e d and that the only 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 gate c l o s u r e curve. I t i s f u r t h e r assumed that the penstock i s of constant 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 t u r b i n e head-discharge curves f o r the r a t e d speed are known and any i n f l u e n c e of t u r b i n e speed changes on the head-discharge r e l a t i o n s h i p s can be neglected (with a frequency change of + 1/10 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 ) . Although no optimum gate o p e r a t i o n curves are d e r i v e d i n t h i s paper, an example i s worked showing that the optimum c l o s u r e curve 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) i s a reasonably p r a c t i c a l curve f o r m i n i m i z i n g the input-output energy d i f f e r e n c e . The 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 of the A l l i e v i c h a i n equations developed by Schnyder and Bergeron (1 ) , combined w i t h an e v a l u a t i o n of the energy conversion t a k i n g place i n the penstock under t r a n s i e n t c o n d i t i o n s . Although 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 flow problems because of 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 that i n t h i s case the l o s s e s can be neglected. 1. Numbers i n the parenthesis r e f e r to the B i b l i o g r a p h y . 3 CHAPTER I ELEMENTS OP THE SCHNYDER-BERGERON GRAPHICAL SOLUTION OP THE ALLIEVI WATERHAMMER EQUATIONS The theory 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 of waterhammer a n a l y s i s developed by Schnyder and Bergeron. This method i s a g r a p h i c a l s o l u t i o n of the equation r e p r e s e n t i n g c o n d i t i o n s at the tur b i n e gate simultaneously w i t h the conjugate waterhammer equa-t i o n s developed by A l l i e v i . A ccording to A l l i e v i the c o n d i t i o n s i n the penstock at the t u r -bine gate can be represented by V = A or ( i n r e l a t i v e form) For d i f f e r e n t values of A (or X) "the H-V (or h-v) p l o t i s a parabola. The conjugate waterhammer equations developed by A l l i e v i are H - H =+ - ( v-V o g I or h - 1 = + 2jo ( v - l ) ( i n r e l a t i v e form) 4 CHAPTER I I THEORY OF ENERGY CONVERSION IN THE PENSTOCK 2.1 General I f c o nservation of energy i s a p p l i e d to the system shown i n f i g u r e 1, then f o r any given time i n t e r v a l E g = E i + E c " E f l - E f 2 ( 1> where: E i s the energy output through the gate i n the time i n t e r v a l ; E_^  i s the energy input to the penstock from the r e s e r v o i r i n the time i n t e r v a l ; E i s the change of energy i n the penstock, i . e . the d i f f e r e n c e c between the change i n k i n e t i c energy and the change i n energy, stored 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 . (E_ i s p o s i t i v e i f the net energy contained i n the f l u i d and i n the penstock w a l l s i s reduced); E ^ i s the steady s t a t e f r i c t i o n l o s s ; E_^ 2 i s the energy d i s s i p a t e d as f r i c t i o n d uring, and as a r e -s u l t of, any change of energy form i n the penstock. The steady s t a t e energy d i s s i p a t i o n E i s u s u a l l y a small f r a c t i o n of the t o t a l energy (at most 10^) and th e r e f o r e can be neglected s a f e l y , 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 the penstock must be i n i t i a t e d from out-side the penstock and must be i n the form of e i t h e r pressure or v e l o c i t y changes. At the intake there i s no r e g u l a t i n g device and the r e s e r v o i r e l e v a t i o n i s assumed constant, t h e r e f o r e , any change i n the energy con-tent of the penstock must be i n i t i a t e d by changes i n v e l o c i t y at the gate 5 (which are a f u n c t i o n of gate movement). I f the v e l o c i t y at the gate i s changed, then according to A l l i e v i there i s an a s s o c i a t e d pressure change and t h i s pressure change t r a v e l s , at h i g h v e l o c i t y , toward the i n t a k e . To say there i s a v e l o c i t y change means the k i n e t i c energy of the f l u i d i s changed; s i m i l a r l y , a pressure change means a change i n the amount of energy stored as s t r a i n energy i n the f l u i d and i n the pen-stock 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 at the gate. This energy, E , i s p o s i t i v e i f the net energy of the f l u i d and penstock w a l l s i s reduced. (With t h i s convention a decrease i n k i n e t i c energy i s p o s i t i v e and a de-crease 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 consider the term E ^ which represents the l o s s e s i n the conversion process which produces E . I n a waterhammer process, w i t h the exception of a small l e n g t h of penstock adjacent to the gate, 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 of flow of a f l u i d p a r t i c l e , so t h a t the flow i s e s s e n t i a l l y v ortex f r e e . As i t i s the c r e a t i o n of v o r t i c e s i n a f l u i d which allows the d i s s i p a t i o n of energy over a longer p e r i o d , the f r i c t i o n l o s s e s due to the conversion process must ap p r o x i -mately equal zero ( i . e . E ^ ~ 0). The energy equation can then he w r i t t e n to a good approximation as: E = E. + E (2) g 1 c Example a = 3200 f t / s e c H Q = 100 f t A = 1/62.4 f t 2 V Q = 20 f t / s e c L = 3200 f t Consider an incremental change of v e l o c i t y of AV = - 1 f t / sec The a s s o c i a t e d pressure r i s e i s AH = - — AV = - 2f°£ (_-,) = 1 0 0 f t g 32 The time r e q u i r e d f o r t h i s pressure wave to t r a v e l the l e n g t h of the penstock i s t _ L - 1200 = 1 sec 13 ~ a " 3200 1 S 6 C At i n s t a n t "t = ~ the pressure i s constant along the penstock and i s H = H + AH = 100 + 100 = 200 f t 0 The v e l o c i t y i n the penstock i s V = V Q + A V = 20 + (-1) = 19 f t / s e c The energy output through the gate during t h i s t = — seconds i s cl E = w A H V ^ = 6 2 . 4 200 (19) 1 = 3800 f t - l b s The energy i n p u t to the penstock during t h i s time i s E. = w A H V - = 62 .4 •—— 100 (20) 1 = 2000 f t - l b s i 1 0 0 a 62 . 4 The change i n the k i n e t i c energy of the f l u i d i n the penstock i s A K E , = J - A I ( V 2 - ( T + A.V) 2 ) 1 2 g v o v 0 ' 1 A K E 1 = \ & 4 ^-L- (52OO) ( 2 0 2 - 1 9 2 ) = 1950 f t - l b s The d i f f e r e n c e between the energy output and the energy input i s E A - E.„ = 3800 - 2000 = 1800 f t - l b s g1 i1 This i s d i f f e r e n t from the change i n k i n e t i c energy of the f l u i d ; there f o r e , the s t r a i n energy stored i n the f l u i d and i n the penstock w a l l i s A P E 1 = (E - E I 1 ) - AKE^ = 1800 - 1950 = - 150 f t - l b s (Note an increase i n stored energy i s considered n e g a t i v e ) . Prom t = 1 to t = 2 seconds the wave i s r e t u r n i n g to the gate At t = 2 seconds the c o n d i t i o n s i n the penstock are ' H = H = 100 f t 0 V = V + 2AV = 18 f t / s e c o ' The energy output' through the gate i n t h i s time i s E = w A H V - = 38OO f t - l b s = E . g2 a g1 The energy input to the penstock i n t h i s time i s E i 2 = w A H o <vo + 2 A ^ I = 62-4 1 0 0 <18) 1 = 1800 f t - l b s The change i n the k i n e t i c energy of the f l u i d i s A K E 2 = 11 A L ( (Vq + Av)2 - (vo + 2 Av)2 ) = •_ 62T4 3 2 0 0 ( 1 S ) 2 - 1 s 2 ) = 1 8 5 ° f t - l b s Since the change i n pressure i s equal and opposite to that of the f i r s t — time i n t e r v a l the change i n stored energy i s a A P E 2 = - A P E 1 = - (-150) = 150 f t - l b s As a check: E = E . + E = E . + A K E + A P E g i c 1 E = 3800 = 1800 + 1850 + 150 = 3800 f t - l b s Note that the net energy output of the wave i n the f i r s t — i n t e r v a l i s cl E = A K E + A P E = 1950 + (-150) = 1800 f t - l b s I n the second — i n t e r v a l the net output i s a E Q 2 = A K E 2 + A P E 2 = 1850 + 150 = 2000 f t - l b s The d i f f e r e n c e between the two wave outputs i s E 0 - E = 2000 - 1800 = 200 f t - l b s c2 c l A l s o , E. n - E._ = 1800 - 2000 = -200 f t - l b s i 2 i l These two values e x a c t l y cancel so that as f a r as the gate i s concerned, the energy output of the 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. (At the gate end of the penstock no v e l o c i t y change i s n o t i c e d u n t i l the wave r e t u r n s to the g a t e ) . 2.2 The Energy Balance for, an I n f i n i t e s i m a l Incremental Gate Movement Any continuous gate o p e r a t i o n can be approximated by a s e r i e s of instantaneous i n f i n i t e s i m a l , incremental movements. Consider the change i n power output at a penstock gate due to one of these increments o c c u r r i n g at time t , when the headland v e l o c i t y at the gate are H and V r e s p e c t i v e l y . (Note that the c o n d i t i o n s are not n e c e s s a r i l y those of steady s t a t e ) . I f the time i n t e r v a l it i s made small enough so that no waves which have been r e f l e c t e d from the r e s e r v o i r are w i t h i n a d i s -tance 2 Sx = 2a 6 t of the gate then w i t h i n 6x, H and V w i l l be cons-tant d u r ing the time i n t e r v a l £t. In t h i s d i s t a n c e &x, the change i n the k i n e t i c • e n e r g y of the f l u i d due to the wave i s 6KE = | M (Y + _ V ) 2 - | M V 2 = \ | A ( Sx) ( V 2 + 2V_,V + - V 2) - \ I A ( _x) (2v6v + 6v 2 ) (3) Since the change was i n f i n i t e s i m a l 6V <:«=: V and as a r e s u l t 6v2 = o and 6KE = | A ( 6 i ) (V 6 V) (4) As a decrease i n k i n e t i c energy i s to be considered p o s i t i v e , a minus s i g n must be added to make the energy and power outputs of the wave of c o r r e c t s i g n . Therefore, the energy output of the wave due to the change i n k i n e t i c energy i s <jE = - <5KE = - - A ( C 5 X ) (V 6v) (5) 10 The change i n stored energy (see f i g u r e 2) i n di s t a n c e £x c o n s i s t s of two p a r t s : a) the work done i n expanding the penstock and b) the work done i n compressing the f l u i d . a) The work done i n expanding the penstock From Hookes law s E The work done on the pipe w a l l i n expanding from E to R + 6R i s equal to the s t r a i n energy increase S g stored i n the pipe w a l l . For i n f i n i t e -s i m a l -values of c>H and c$R t h i s i s S = F _ R = w 2 T f R 6x H _ H e ^ s E S = " 2 2TTR ( (bx) R 2 ( H 6 h ) ( 6 ) e s E v ' b) The work done i n compressing the f l u i d I n the l e n g t h <$x the l e n g t h of the water column i s changed by From Hookes law (5^ = I ( 6x) 6H The work done,S^,in compressing the f l u i d i s equal to the s t r a i n energy increase stored i n the f l u i d . For i n f i n i t e s i m a l values of &x^~ and e>H t h i s i s S f = F 6x1= wTT R 2 H | ( 6x) 6H S f =w 2TTH 2 4 £ (H 6H) (7) The t o t a l change i n s t r a i n energy i n distance <5x r e s u l t i n g from the wave i s S = S + S _ /«\ T e f (8) By r e a r r a n g i n g and adding equations 6 and 7 we o b t a i n S T =^w 2 -n R 2 (cSx) f | + w2TTE2 (6x) | ] ( H 6H) 2 Furth e r rearrangement and m u l t i p l i c a t i o n by ^  y i e l d s S S_ = — TT R 2 (5x |(^ + i ) " 4 g 2 (H <$H) T g 1 v s E K ' g J ° v ' But by A l l i e v i the wave v e l o c i t y i s 1 a = g E K' and t h e r e f o r e ,2 R IN w _ 1 s E + K g ~ 2 ° a Furthermore the c r o s s - s e c t i o n a l area of the penstock i s A = TT R 2 S u b s t i t u t i o n of these values y i e l d s 1 & a^ 12 Reservoir V-Gade. FIG 1 THE BASIC HYDRAULIC SYSTEM Penstock Pressure -t H 1 <J — gx*a&t 4-(/ — > FIG 2 CONDITIONS IN THE PENSTOCK RESULTING FROM AN INFINITESIMAL WAVE 13 With a decrease i n s t r a i n energy considered p o s i t i v e , the energy output of the wave due to the change i n s t r a i n energy i s c5Ep = - s T = - |A f_ (E6E) (10) The t o t a l energy output of the wave i s <5E = 6E^ + 6Et„ ( l l ) c P K v ' S u b s t i t u t i o n of the r e s u l t s of equations 5 and 10 i n t o equation 11 y i e l d s 6EC =-|A (C5X) (v6v + f_ H6H) (12) S u b s t i t u t i o n of H = H q h and V = V Q V i n t o equation 12 and rearrangement y i e l d s _ E =-^ A (c$x) H v a (| £6H + ^ V= )^ O-^ c g v y o o a V V a g H ; ° o - 5 o With _ V = - ^  t$H a and 6v = V Sv o w i t h 6v = - ^  i s u a V o a V 2»= ^ i t f o l l o w s that 6E c = | A H Q V Q J (h - 2pv) 6v 6x (13) 14 The power output of the wave i s nr " j ~ = - A H V « (h - 2Pv) 6 v i f d t g o o a v J ' 6 t But ^ x - a 6 * and t h e r e f o r e 6E T t = Pw = w A H o V o < h - 29^ 6 v ^ Consider now the g r a p h i c a l s o l u t i o n to a waterhammer problem shown i n f i g u r e 3« An incremental wave created at the gate when condi-t i o n s there are represented by p o i n t "b" i n the f i g u r e , w i l l , as i t t r a -v e l s towards the 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 that f a l l on the l i n e b-c at a l l times. The slope of the l i n e b-c i s 2p and the r e f o r e the equation of the l i n e must be h 2jDv + constant (c) or h - 2jov = C (15) In equation 14 ov i s a .-.-constant, since 6 v = - £ 6h a and <5h i s unchanged as the wave t r a v e l s along the penstock of constant 15 THE CREATION OF A WAVE 16 w a l l t h i c k n e s s . Therefore s u b s t i t u t i o n of equation 15 i n t o equation 14 y i e l d s P. = w A H V C 6v = constant (l6) ¥ 0 0 v ' The net 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 r e s u l t i n g from the wave t r a v e l l i n g toward the r e s e r v o i r i s a constant at 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 are represented by p o i n t "c" f i g u r e J . I f the time i n t e r v a l 6t a f t e r the wave i s r e f l e c t e d from the r e s e r v o i r i s made small enough so that no waves that have been created, at the gate are w i t h i n a distance 2 o"x = 2 a i t of the r e s e r v o i r , then w i t h i n 6x, H and Y w i l l be constant during the time i n t e r v a l 6t. In t h i s d i s -tance 6x the change i n the k i n e t i c energy of the f l u i d due to the r e -f l e c t e d wave i s 6EKR = "I A ( 6 x> <&v (17) The energy output due to the change i n s t r a i n energy i s as before <*EPR = -| A ( 6x) f _ H 6 H (18) The net energy output of the r e f l e c t e d wave i s < K R = <^PR t * EKR By making the same s u b s t i t u t i o n s as before, w i t h the exception that f o r a r e f l e c t e d wave 6v = I 6H we o b t a i n by the a d d i t i o n of equations 17 and 18 17 6E v = £ A ( 6 x ) § H V (-h- 2pv) 6 v (19) cR g v a o o v and o R PWR - I t " = W A H o V o ^"h- ^ ^ ^ 2°) The change i n power input from the r e s e r v o i r due to the a r r i v a l and r e f l e c t i o n of the wave i s <$P„ = w A H V ( 2 h) 6 v (21) R o o v J where h = 1 . The wave l e a v i n g the r e s e r v o i r , where c o n d i t i o n s are given 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 of g r a p h i c a l a n a l y s i s , en-counter 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 that f a l l s on l i n e c-d i n f i g u r e 3. The slope of t h i s l i n e i s - 2p and t h e r e f o r e i t s equation i s - h = 2pv + K or - h - 2jDv = K (22) Since t h i s l i n e and the l i n e g i ven by equation 15 pass through the common po i n t M c " , ( f i g u r e 3) whose coordinates are h = 1, v = v the value of K can be determined i n terms of C. Prom equations 14 and 22, 2pv = C - 1 2pv = K + 1 and t h e r e f o r e K = C - 2 18 The equation of the l i n e c-d i s then - h - . 2pv = C-2 (23) S u b s t i t u t i n g equation 23 i n t o equation 20, P = w A H V (C-2) 6v (24) WR o 0 v Prom the same reasoning as f o r the i n c i d e n t wave, P must be a constant as the r e f l e c t e d wave r e t u r n s to the gate. A l s o P l r - P. m = w A H V f C 5v - (C-2) 6 v ] W WE 0 o v. v ' J P - P = w A H V ( 2&v) (25) W WE 0 0 ^ ~ ' K ' But, if..h=l, the 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 equation 21 then: 6P R = w A H o V o (j>6v) = P w - P w R (26) Thus the d i f f e r e n c e i n power outputs 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 by the change i n power input from the r e s e r -v o i r , so t h a t as f a r as the gate i s concerned, the power output of the wave i s a constant throughout i t s l i f e span of 2- seconds. At the gate a no change i n input 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 reaches the gate, at which p o i n t the wave ceases to e x i s t and a new wave i s formed. Consider the wave as i t a r r i v e s at the gate. Figure 4 shows an enlarged g r a p h i c a l s o l u t i o n of the problem f o r p o i n t "d" i n f i g u r e 3« At the i n s t a n t before the wave a r r i v e s h and v are gi v e n by p o i n t d. The i n s t a n t the wave a r r i v e s the head i s a l t e r e d by 6~h. and the v e l o c i t y by 6v so that c o n d i t i o n s at the gate are represented by p o i n t p. At t h i s i n s t a n t the wave ceases to e x i s t and th e r e f o r e i t s power output ceases a l s o . However c o n d i t i o n s at p o i n t p are unstable as the r e l a t i v e gate opening i s T and the head and discharge must be r e l a t e d by v = "J J h. Therefore a new wave must be formed at the gate so that c o n d i t i o n s are such as are given by " f l " i n f i g u r e 4> or, i f at 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 takes p l a c e , by p o i n t "q" f i g u r e 4« I n e i t h e r case the new wave formed w i l l produce a constant power output, as f a r as the 2L gate i s concerned, during i t s — l i f e span. a The power increase (which may be of p o s i t i v e or negative sign) at the gate due to the c e s s a t i o n of the o l d wave and the c r e a t i o n of the new wave i s gi v e n by the sum of the change i n power due to the c e s s a t i o n of. the o l d wave and the change i n power due to the c r e a t i o n of the new wave and i s A v — v v WN " WRO (27) (Where "W denotes the new wave and "0" the wave that j u s t ceased). From equation 20 and f i g u r e 4 i t f o l l o w s that -P = w A H V (h, + 2»v.) 8v„ (28) WRO 0 o x d y d' R I f d p and v, = v d p then f o r a wave created by a r e f l e c t i o n o n ly i t f o l l o w s from equation 14 that S u b s t i t u t i o n of equations 28 and 29 i n t o equation 27 y i e l d s riP = w A H Q V J (V 2 . P V ( 6 V 6 V + <V2.P v d > * \ ] or *P = w A H o V Q f ( h d - 2 j 0 v d ) 6 v 1 + 2 p ( 2 v d ) 6 v R j (30) But, -2jO £ v R i s the pressure wave "A" F" which o r i g i n a t e d from the gate — seconds e a r l i e r , a 6P = w A H o V. [ ( h d - 2_pvd) _>v1 - 2 v 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 place to p o i n t q f i g u r e 4 6P = w A H o V 0 { ( h d - 2 p v d ) Sv2 - 2 v d ( 6 F) } (32) 6 v^ and <$"vv> are the net changes i n v e l o c i t y t a k i n g place at the gate' a t time t . Therefore 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 equa-t 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 o ( ( h - ^ v ) ^ - 2 v ( 5 3 ) or i n the l i m i t as t 0 f . . A H o T o j > - 2p,) |f - 2 T g ] (34) Note: This r e s u l t can be obtained d i r e c t l y by the 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 T T = w A H Y h — + v — [ at o o dt dt J _ ? n d v „ dF dt ~ -^at at 22 CHAPTER I I I APPLICATION OP ENERGY PRINCIPLES TO THE DETERMINATION OP OPTIMUM GATE OPERATION 3.1 General Consider the h-v diagrams shown i n f i g u r e s 5 a a n ( l 5b and the corresponding p l o t s of power output versus time shown i n f i g u r e s 6a and 6b, T and T are the i n i t i a l and f i n a l gate 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 steady s t a t e power outputs through the t u r b i n e gates. T i s the t o t a l time of gate motion f o r a gate o p e r a t i o n between T" and C o T and T i s the time f o r the power output to f i r s t cross the steady s t a t e Pj, ' l i n e . As t h i s p o i n t i s not reached u n t i l c o n d i t i o n s at the gate are g i ven by t y v p then T p > T c . The excess energy output r e s u l t i n g from a gate operation i s represented by the shaded areas of f i g u r e 6, which are T T F / F 4 E g = j (P - P F) dt = j P at - P F T F (35) '0 The d e r i v a t i v e of equation 2 w i t h respect to time y i e l d s dE dE. dE P = — £ = — 1 + — c (36) at dt dt Therefore _ /p dE. /• F dE d t l d t + / d t C d t " P F T F W 0 E = T E . + E ] F - P T (38) g L i C - I Q 23 a) Gate Closure '". b) Gate Opening FIG 5 TYPICAL GRAPHICAL SOLUTIONS TO WATERHAMMER PROBLEMS P 0 T, t 0 Tf t a) Gate Closure b) Gate Opening FIG 6 POWER INPUT TO THE TURBINE DURING TRANSIENT CONDITIONS 25 2j>v*Cf a) Gate Closure b) Gate Opening FIG 9 GRAPHICAL SOLUTIONS TO WATERHAMMER PROBLEMS F WHEN ^ C n £v„ (T„ -n=Q+1  O  _  t ) = 0 n n F n II2 f lf„ II a) Gate Closure b) Gate Opening FIG 10 PLOT OF INPUT VELOCITY TO THE PENSTOCK FOR THE CONDITION F C ,0v (T - ' t ) = 0 n n F n n=Q+1 26 E, i s the t o t a l net 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 at the gate between t=0 and T=T_. I f a continuous gate move-a ment i s approximated by a s e r i e s of i n f i n i t e s i m a l incremental movements then T F P E c T- (*W + W n 0 n=l where f o r a wave t r a v e l l i n g from the gate to the r e s e r v o i r *Wn " PWn (39) and f o r a wave t r a v e l l i n g from the r e s e r v o i r to the gate EWPn= PWRn (* " ( * n + l ) ) ^ t i s the time of o r i g i n of the wave at the gate. Note that P., = 0 i f f t - t ) > -Wn n' a and P i r o =0 i f ( t - ( t + i ) ) > ^ WRn v v n a'' a (This f o l l o w s from the o r i g i n a l assumptions made i n d e r i v i n g P and P 1 T O) W WK 3.2 E v a l u a t i o n of E between t=0 and t=T.,-,, c P A l l the waves created up to p o i n t Q of f i g u r e 5 w i l l have a f u l l l i f e span of ^  sec so that Ewn = PWn <£> ' ^ W n = PWRn ( l ) ^ 2 ) S u b s t i t u t i o n of equations 14 and 20 i n t o equations 41 a n d 42 y i e l d s E ^ - w AH, V O (h-2fiv) o,n\ (43) . E ^ - w A H . Y . (-h-2pv) 6vn± (44) I f the constant terms h - 2 p v and -h-2_p v are evaluated when h=l i . e . at the r e s e r v o i r , then F, + E m = w A H V - ( l - 2 p v - 1 - 2 p v ) Sv Wn WRn o o a v ~ s' ' n = w A H Q V Q \ (-4_Pv) 6 v n (45) Figure 7 shows an enlarged p i c t u r e of the wave o c c u r r i n g a t t=t . The t o t a l energy output of t h i s wave i s gi v e n by equation 45- But from f i g u r e 7. < k n = - g - S (46) thus and ^ (*W + V n = W A H o V o I ^ ("4p v ) ^ (47) n=l n=l l i m ( ~ 4 j 3 v ) -g—^ = (-2/3 v dv n—>oo n / n=l v s = P ( V S 2 - V F 2 ) ( 4 8 ) aV S u b s t i t u t i n g equation 48 i n t o equation 47 a n ( l s u b s t i t u t i n g j O = 2gH o 28 L (Ew + "W„ - fs AL To2 \2 - V> ' Alffi m n=l This i s the t o t a l k i n e t i c energy change of the f l u i d "between the i n i t i a l and f i n a l steady s t a t e c o n d i t i o n s . Therefore the t o t a l energy output of a l l waves produced up to the waterhammer l i n e of slope 2 p pass-i n g through the f i n a l steady s t a t e v p , h^, p o i n t i s equal to the change i n k i n e t i c energy of the f l u i d i n the penstock i n going from the i n i t i a l to the f i n a l steady s t a t e p o i n t s and i s independent of the way i n which the gate i s operated. This may he seen more e a s i l y by r e f e r r i n g simultaneously 2L to f i g u r e s 7 and 4« I f f o r each wave reaching the gate a f t e r t = t p - — the gate i s adjusted to p o i n t p of f i g u r e 4> no new waves w i l l be created 2L at the gate a f t e r t = T p - — and the pressure-discharge r e l a t i o n s h i p d E i <1E„ w i l l f a l l along l i n e Q-F of f i g u r e 7 . At t=T p , P± = = P g = TTT that i s h = h = h , v. = v i S o' l g and t h e r e f o r e dE dt dE Q With — r — = 0 there can be no waves i n the penstock 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 output must then be E = E. + E = E. + A K E g 1 c 1 where AKE i s the change i n the k i n e t i c energy of the f l u i d r e s u l t i n g from the v e l o c i t y change from v„ to v„. Therefore O I1 29 E c = A K E = H ^ + Vn n=l Now consider the energy output per wave of a l l the waves o r i -g i n a t i n g between t=t^ and t = T p - — - T R. A l l the waves 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 intak e but w i l l not have enough time to r e t u r n a l l the way to the gate. Therefore Wn = w A H V (h-2Dv) Sv - (50) o 0 v ' ' n a K' ' and ^WRn " w A H o V o ( " ^ ^ 6 v n \ T P ~ ^ n + ~S[ ^ The t o t a l energy output up to time T„ due to 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„ to t = T^ i s then R R ^ % + W n = L w A H o V o K I n=Q +1 n=Q+l R L W A H o V o ("h - 2 p v ) c 5 v n [ V ( t n + * ) } ( 52) n=Q+l The energy output of each wave o r i g i n a t i n g between T and T„, none of which reach the r e s e r v o i r i s given by \n = w A H Q V q (h-2j?v) 6 v n ( T p - t n ) (53) and 30 The total energy output of the waves originating in this time interval is then F F L ( E w - V n - H w A H o Y o (h-2^v )6v„ ( T F - t„) ( 5 5) h=R+l n=R+l The total energy output of the waves created in the time interval t = 0 to t = TT-, i s then F F E c = / L ! W n n=l ° r _Q R F E c = L % + W n + L <EW + W n + H ^ + W n (56) n=l n=Q+l n=R+l Substitution of the results of equations 4 9j 52 and 55 into equation 56 yields R E C = A K E + w A H Q V0{[2 (h-2/>v) ^ (^) n=Q+l R + J2 (-h-2pv) 6vn (T p - ( t n + S)) n=Q+l + ) _ (h-2_Ov)-Jv n(T F-t n ) J (57) n=R+l Substituting for convenience the results of equations 15 and 23, which are, 31 C n = h - 2 p v and C n - 2 = -h-2jDy i n t o equation 57 we get R E = AKE + w A H V ) ,} C ( c$ v ) -c o o £ <-— n v n / a n=Q+l R • L K- 2> K <TF - <*» +1» n=Q+l + E :°n K (TF - V 1 ( 5 8 ) n=R+l Rearrangement of the term _R ) ( C - 2 ) c_v (T_ - (t + S)) i v n 7 n v F v n a'' n=Q+l y i e l d s R R 5~~ (C -2 )£v ( T - ( t +^ )) = Y~ (-2)6T (T_, - (t + ^ ) ) c v n ' n v F v n a y ' £ v ' n v F v n a'' n=Q+l n=Q+l R R + T~ C (Sv ( T p - t ) + 5~ C (5v (-^) (59) i n n v F n' £ n n v a 7 v ' n=Q+l n=Q+l 3 2 Furthermore FL y~ c <^ v (T_ - 1 ) + y c o v (T„ - 1 ) = r c ^ T ( % - t ) Z n n v F i r L n n v F iv L— n n v F n' n=Q+l n=R+l n=Q+l ( 6 0 ) S u b s t i t u t i o n of the r e s u l t s of equations 5 9 and 6 0 i n t o equation 5 8 y i e l d s R E c = AKE + w A H V ) ) ( - 2 ) &v (OL, - (t + - ) ) o o( L— N ' n v F v n a 7 / n=Q+l + / C (ST (T_, - t ) £ n n v ,F n' n=Q+l ( 6 1 ) 3 . 3 E v a l u a t i o n of the energy input to the Penstock f o r the time i n t e r -v a l t = 0 to t = L . F The wave o r i g i n a t i n g at the gate w i t h an a s s o c i a t e d v e l o c i t y change 6vn causes a v e l o c i t y change of 2 6v^ at the r e s e r v o i r . Therefore, r e f e r r i n g to f i g u r e 8 , the energy input to the penstock during the time i n t e r v a l t = 0 to t = T p i s g i v e n by E i = v A H o T o | V S V L h o <2 <K> <TF - ( t n + I » n=l + h Q (2<Jvn) ( T F - ( t n + i))) ( 6 2 ) n=Q+l Note that any waves l e a v i n g the gate a f t e r -b = = T^ , — ^  w i l l have no e f f e c t on the input to the penstock d u r i n g the time i n t e r v a l t = 0 to t = [IL,. As a r e s u l t , the l a s t term of equation 62 i s only summed to n = R. Remember too that f o r gate c l o s u r e 6v i s negative. o n The f i r s t term of equation 62 y i e l d s w A H o V o \ V T F = P S T F ^ Furthermore, r e f e r r i n g to f i g u r e 7 note that Q L 2 K - V F " V S ^ n=l (This means that the f i n a l steady s t a t e v e l o c i t y i s f i r s t reached a t the r e s e r v o i r a t time T„ - — ) . Using t h i s r e s u l t we o b t a i n F a' ° w A H Y / h 2 <Sv (T_ - ( t + — ) ) o o L o n v F n a' J n=l w A H V J h (T_. - -) ) 2t$v o o 7 o v F a' Z n n=l Q ) 2h iv (-t ) ] I— o n n' J + n=l = w A H V / h (T„ - -) (v_ - v Q ) o o | o x F a y V F S' + n=l (P_, - P C ) ( T _ - — ) + w A H V ) h 2 i v ( i t ) (64b) v F S 7 v F a 7 o o (1 o n v nJ \ ^ J n=l L e t t i n g h = 1, which i s the c o n d i t i o n at the r e s e r v o i r , and s u b s t i t u -t i n g the r e s u l t s of equations 6j and 64b i n t o equation 62 y i e l d s E. = P T_ + (P_, - P Q) (T„ - —) + W A H V j Y~ 2 <Sv (-t ) 1 S ? P S / v P a y 0 0 / L n v n' n=l + ) 2<$v ( T p - (t + ^)) V L n v F v n a.'' J n=Q+l or C = (P„ - P ) — + P_, T + w A H V f Y • 2 &v (-t ) 1 v S F y a P P o 0 I L n v n 7 n=l + r 2 <£v (T_ - (t + ^ ) ) /— n v F v n a y ' \ (65) 3.4 E v a l u a t i o n of the Excess Energy Output The excess energy output A E can now be found by s u b s t i t u t i n g the r e s u l t s of equations 6 l and 65 i n t o equation J8 which i s AE = E + E i c Y P and upon s u b s t i t u t i o n y i e l d s A E =- p_, T_, + (P q - p_.) - + P„ T_, g F F x S F' a P P w A H V f) 2 <Jv (-t ) + Y~ 2 i v (T_,- (t + -))l o o\L n v n 7 L n v F v n a y / J (cont'd) n=l n=Q+l FL + A K E + w A H Q V o ( ^ _ (-2) Srn ( T F - ( t n + £)) n=Q+l + )_ Cnk^-*n)i ( 6 6 ) n=Q+l C a n c e l l a t i o n and rearrangement of terms y i e l d s the general equation f o r A E which i s g Q A E = (P_ - P„) - + AKE + w A H V ] / 2 <Sv (-t ) g v S F 7 a o o 11— n n' n=l F +y~ c &V ( T _ - 1 ) 1 (67) I n n v F n ' f v / h=Q+l J I f the gate i s operated so that d uring the i n t e r v a l t = T P 2L' - — to t = T „ no new waves are formed then the h-v diagram w i l l he as a n F ^ shown i n f i g u r e 9. In t h i s case we have 6 v n •F = 0 Q+l and t h e r e f o r e ) C 6v ( T - t ) = 0 (68) L— n n v F nJ v ' n=Q+l Equation 67 i s then minimized by mi n i m i z i n g the absolute value of ( r e f e r to f i g u r e 10) 36 7" 2 6v (-t ) n=l which i s the shaded area of f i g u r e 10 and can he w r i t t e n as L II2 KI <v n=l A l l other terms of equation 67 are constant 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« (Note i n the term r 2 c^v ( - t ) L— n n' n=l the maximum value of t i s n OT t = T„ = T - — n Q F a and f o r gate c l o s u r e 6v i s negative. As a r e s u l t / 2 £v (-t ) (1 n v n 7 n=l i s p o s i t i v e ) . Obviously from f i g u r e 10 the term L H *K II <*„> n=l i s a minimum i f |i 2 6v || i s as l a r g e as p o s s i b l e when t i s as small n n 37 a.) Gate Closure b) Gate Opening -FIG 11 WATERHAMMER CHARTS AND SURPLUS ENERGY PLOTS FOR THE TIME INTERVAL t = T 0 TO t = T. 38 a) Li n e h = 20v to the L e f t of T. b) Li n e h = 2 p v to the Right of 7 FIG 12 WATERHAMMER CHARTS AND SURPLUS ENERGY PLOTS FOR THE TIME INTERVAL t = T • TO t = T„- - GATE CLOSURE 39 a) Li n e h = 2$>v to the L e f t of T h h) Line h = 2 p v to the Right of 3"F  FIG 13 WATERHAMMER CHARTS AND SURPLUS ENERGY PLOTS FOR THE TIME INTERVAL t = T n TO t = T^ - GATE OPENING Q, J) as p o s s i b l e s i n c e Av = - 7 ^ Ah This means that, the maximum v e l o c i t y and head changes should be made as soon as p o s s i b l e f o r t h i s form of gate o p e r a t i o n . Now consider the general equation 67 w i t h : n n=F _ n=Q+1 The only d i f f e r e n c e i n E between t h i s case and the case where: S n=F 6, n = 0 n=Q+1 i s the term I n=Q+1 n • (T^ - t ) . This term i s .just the sum of the n F n energy outputs of a l l waves created between t = T^ and t = T , AS SEEN ^ n Q, n F BY THE GATE. (Remember from the d i s c u s s i o n of an incremental wave that the r a t e of energy output of a s i n g l e wave i s as f a r as the gate i s con-cerned, a constant throughout i t s l i f e span). I f n=Q+1 £ v is C OT i s p l o t t e d a gainst time, then changing the n n a x i s of i n t e g r a t i o n y i e l d s \ ( Y ~ C c^v ) dt = C £v (T - t ) J L n n L n n F TY' (69) n=Q+1 n=Q+1 The q u a l i t a t i v e r e s u l t s of such p l o t s f o r s e v e r a l gate opera-t i o n s are shown i n f i g u r e s 11, 12 and 1J. Note i n these f i g u r e s that the optimum method of gate o p e r a t i o n a f t e r t = T n depends upon the r e l a -t i v e p o s i t i o n of the l i n e h = 2 p v so that i n c e r t a i n instances i t i s advantageous to reduce the r a t e of gate o p e r a t i o n i n the time i n t e r v a l * t = T„ to t = L i f A E i s to be kept to a minimum. 0, F fl g 3.5 I d e a l Gate Closure Curves For gate c l o s u r e , from the c o n d i t i o n that the 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 to reduce the energy i n p u t , the maximum all o w a b l e head (h ) must be reached a t t = 2^ and maintained a t l e a s t u n t i l t = T„. This w i l l minimize the term y~ 2 L c-t) L— n n n=1 of equation 67 ( r e f e r to f i g u r e 10a) . I f the l i n e h = 2tJ> v i s to the l e f t of p o i n t Q, on the h-v diagram, then the term F ) C iv (T - t ) L n n F n' n=Q+1 which i s negative, i s maximized by decreasing the r a t e of gate- c l o s u r e a f t e r t = T . Figure 14 shows the h-v diagram and the gate c l o s u r e curve f o r such a case. Note that to maintain a constant head at the gate be-tween t = 2— and t = T_, the c l o s u r e curve i n t h i s time i n t e r v a l i s speci-a 0/ f i e d by the c l o s u r e curve i n the time i n t e r v a l t = 0 to t = 2^ ( i . e . J a v 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 of f i g u r e 14b i s 42 taken, i t becomes a s e r i e s of instantaneous c l o s u r e s t a k i n g place at t = °' f ' V As instantaneous c l o s u r e i s impossible and a cl o s u r e of the form shown i n f i g u r e l'4b extremely d i f f i c u l t to d u p l i c a t e , the most r e a -sonable, form of gate c l o s u r e would seem to be that made up of 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 of f i g u r e 14b. How-ever t h i s curve has the disadvantage that i f c l o s u r e should be s t a r t e d from a p o i n t other than Tg> a higher than allowable pressure r i s e may r e s u l t (6). I t i s p o s s i b l e to d e r i v e a gate c l o s u r e curve f o r which the 2L maximum waterhammer occurs at t = — and f o r a l l other times h . a m As the c l o s u r e s under c o n s i d e r a t i o n are always l e s s than f u l l gate (and never to *J"= 0) t h i s curve could be a reasonable s o l u t i o n . 3.6 I d e a l Gate Opening Curves For gate opening A E i s a negative q u a n t i t y so that to make .AE g as small as p o s s i b l e the term T~ 2 &v ( - t ) L— n v n' n=l of equation 67, which i s negative must be made as small as p o s s i b l e and the term F I n=Q+l C 6v (OL, - t ) n n N F n' should 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 . With these f a c t o r s i n mind, r e s u l t s of s i m i l a r form to gate c l o s u r e may be obtained. dh 3.7 I d e a l Gate Operation When 2jO > ( — ) j - _ c o n s i . For many cases the slope of the waterhammer l i n e i s g r e a t e r than the slope of the 3"™ l i n e . 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 that we have ) C { T ( L - t ) = 0 (70) L— n n F n y n=Q+l The general equation then reduces to Q A E = (P_ - P J - + AKE + Y~ 2 i v (-t ) .(,71) g v S F y a L— n v nJ v ' n=l From f i g u r e 15 i t i s obvious that the i n p u t ' v e l o c i t y i s reduced to the f i n a l steady s t a t e v e l o c i t y by reaching 7"^ as r a p i d l y as p o s s i b l e . With gate opening the same r e s u l t s apply so that the minimum absolute value^of A E i s obtained by reaching *~S a s r a P i d - l y a s p o s s i b l e , 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 very r a p i d gate c l o s u r e i s shown as a s e r i e s of small increments. I f A t i s very small then the c l o s u r e can he con-s i d e r e d instantaneous. 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. For a l l the waves formed by the c l o s u r e to J" p , t = A t ~ 0. For a l l 2L 2L the waves formed by the r e f l e c t i o n of r e t u r n i n g waves t ~ — + A t = — n a a 2L With TT-, = — f o r instantaneous c l o s u r e , we have i n the general equation ij a Q YL 2 ^ v n ("V = 0 ( 7 5 ) n=l and P ) C i v (T - t ) = / C i v (T-, - t ) L n n v F n' L n n v F n n=Q+l n=Q+l ,F + y~ c JT (T r a - 1 ) l__ n v n v F n 7 n=B+l ~ 2L a 7 c i v i — n n (74) n=Q+l The energy output of the waves formed between n = Q, + 1 and n = B can be evaluated by r e f e r r i n g to f i g u r e 16a and n o t i n g that on the l i n e h 6v and n C n = 1 - 2j3v (75) (76) Prom t h i s we o b t a i n v n 2L a n 2L C v — n n a ( 1 - 2 / ) v) ^ n=Q+l n=Q+l (77) or B ^ ) C i v a i — n n L a v -p V (78) 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 that as A t -> 0 equations 77 and 78 become exact. S u b s t i t u t i o n of the r e s u l t s of equations 73 and 78 i n t o equation 67 y i e l d s A E = ( P c - P_) - + AKE + W A H V -g v S F' a 0 0 a v - j3v- j (79) v, Purthermore we have P c = w A H Y (h v 0 ) S 0 o v 0 S/ (80) P„ = w A H V (h v„) F o 0 v o F' (81) and (82) S u b s t i t u t i n g equations 80, 81 and 82 i n t o equation.79 and l e t t i n g h = 47 a) Waterhammer Chart h) Gate Closure Curve FIG 16 WATERHAMMER CHART AND GATE CLOSURE CURVE INSTANTANEOUS GATE CLOSURE a) 2£>= 2 b) 2p = 1 FIG 17 WATERHAMMER CHARTS FOR DIFFERENT VALUES OF p • ' ' ' . - INSTANTANEOUS GATE CLOSURE ' we get L / \ 1 w 2 , ? 2s AE = w A H V - (v_ - v j + % - A L V ( v * - v / ) g o o a v S F' 2 g o v S F y + w A H V -o o a v - o? Example v., (8?) A = 1/62.4 f t L = 3220 f t a = 3220 f t / s e c P = 1 H 20 f t / s e c 1000 f t = 1 For the gi v e n data the surplus energy r e s u l t i n g from an instantaneous 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 equation 83. From the data w A H V - = l 2 - ^ 1000 (20)= 20,000 f t l b s . o o a 62.4 From the 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 equation 83 y i e l d s A E g = 20,000 (1 - .6) + I ^ 3220 (20 2) ( l 2 - .62) + 20,000 {(.48 - .482) - (.6 - .62)) A E = 8,000 + 12,800 + 200 = 21,000 f t - l b s 2L As a check the power output during t = 0 to t = — i s from f i g u r e 17a 2L a P B = w A H Q V q (hg v B ) = 2 (20,000) 1.52 (.74) = 45,000 f t l b s 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 given by p T 9 T If P F = ~ w A H Q V Q (1^ v F ) =2 (20,000) .6 = 24,000 f t - l b s As a r e s u l t we have A E g = ^ ( P B - P F ) = 45,000 - 24,000 = 21,000 f t - l b s I f i n the above example an instantaneous c l o s u r e had been made to J = T N = .675 at t = 0 a.nd from Tr, = .675 to = .60 at t = — then no waves would be formed a f t e r p o i n t Q. This g i v e s B ) C Ov = 0 L n n n=Q+l and as a r e s u l t A E = 8,000 + 12,800 = 20,800 f t - l b s & This amounts to about a 1% saving 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 to the f i r s t case. Furthermore there are no r e -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. I n t h i s example the l i n e h = 2J)v was e s s e n t i a l l y to the l e f t of the area r e p r e s e n t i n g the zone of gate o p e r a t i o n , thus p e r m i t t i n g the above mentioned savings. Example A = 2/62.4 f t 2 V = 10 f t / s e c P = .5 The remaining data i s the same as the previous example. The h-v diagram f o r t h i s case i s shown i n f i g u r e 17b. S u b s t i t u t i n g i n t o equation 83 we get A E = 20,000 ( l - .6) + 10,000 ( l 2 - .62) (cont'd) 50 + 20,000 | (.58 - .5 (.38) 2)- (.6 - .5 (.6) 2)} AE = 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 p T A E g = ( P B - P s)= 40,000 (l . 5 l ( . 6 9 ) - l(-6))= 12,200 f t - l b s . I n t h i s example i f c l o s u r e i s made i n two steps; 7 Q t o ' T n a t " f c = 0; and ^ 2L Tn to T-n, at t = — then the value of A E i s increased by 16$ i . e . *i a a. g A E = 8,000 + 6,400 + 0 = 14,400 f t - l b s . €> The e f f e c t of being to the l e f t of the l i n e h = 2j0v i s very marked on ( the term > C Ov (T^ - t ). L— n n v P n' n=Q+l In comparing the two examples which were f o r the same i n i t i a l and f i n a l steady-state power outputs the e f f e c t of reducing the term AKE by reducing the i n i t i a l v e l o c i t y should be noted. A l s o the necessary i n -crease i n penstock area 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 increased c a p a c i t y of the penstock to' s t o r e energy-thus r e s u l t i n g i n f u r t h e r savings. These savings are accompanied by the disadvantage of h y d r a u l i c o s c i l l a t i o n s . 4.2 Gate Closure which Y i e l d s the Smallest Value of Maximum Waterhammer This form of c l o s u r e was developed by E. Ruus (6), and f o r a g i v e n pipe l i n e constant p 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 value of maximum waterhammer. I t i s shown i n the d e r i v a t i o n (6) that the change i n penstock water v e l o c i t y at the t u r b i n e gate and at the i n t a k e . 51 F I G 18 WATER V E L O C I T Y AT THE INTAKE AND AT THE GATE - -OPTIMUM GATE CLOSURE F I G . 19 -WATEPLHAMMER CHART FOR OPTIMUM GATE CLOSURE 5 2 vary i n the time i n t e r v a l t = 0 to t = as shown i n f i g u r e 18. The h-v diagram i s shown i n f i g u r e 19. I f f o r the general form of the waterhammer chart shown i n f i g u r e 19, p and h^ are the s p e c i f i e d v a r i a b l e s , then the c o r r e c t shape of c l o -sure curve can be obtained. This curve has many of the c h a r a c t e r i s t i c s of the optimum curve f o r keeping A E as low as p o s s i b l e . A general de-r i v a t i o n of t h i s curve f o r p a r t i a l gate movements i s given below. 2L At the gate during the f i r s t — time i n t e r v a l ( r e f e r to f i g u r e 18), the r a t e of change of v e l o c i t y i s g i v e n by If - * «*> The head-discharge r e l a t i o n f o r the gate i s v = jy h ( 8 5 ) Furthermore from A l l i e v i ' s f i r s t c h ain equation we have h - 1 = -2p (v - v s ) ( 8 6 ) Combining the r e s u l t s of equations 8 5 and 8 6 we o b t a i n v I n t e g r a t i o n of equation 8 4 and s u b s t i t u t i o n of the r e s u l t i n t o equation 8 7 y i e l d s Kt or + v s = TiJ 1 - 2p (Kt + v s - v g ) ( 8 8 ) Kt + v c T- S ( 8 9 ) i 1 " 2JO Kt 2L K i s determined from equation 8 6 by s u b s t i t u t i n g h = h at t = — and 2L d i v i d i n g the equation by — . The r e s u l t i s 3* 53 h m v - v s "MI a dv dt = K or h - 1 K = (90) 2L For the c l o s u r e between time t = — and t = T„ we have a C v = J (V - . (91) m and t h e r e f o r e If - 2K - If J^ T <*> or dT = 2K dt (93) I n t e g r a t i o n of equation 93 w i t h respect to time y i e l d s T- • § _ < * _ _ ) +T, (94) y h ' m / T " 2L J, i s determined by s u b s t i t u t i n g t = — i n t o equation 8 9 . The r e s u l t -1- El y i e l d s jr- 2L _ 2L m F i n a l l y s u b s t i t u t i o n of t h i s r e s u l t i n t o equation 94 y i e l d s (96) The gate c l o s u r e time T^ i s determined as f o l l o w s . From f i g u r e 18 the t o t a l v e l o c i t y change at the gate i n time T p i s given by From f i g u r e 19 we have V B = IF I hm ^ and v s = TS |/ 1 = TS (99) Combining the r e s u l t s of equations 97? 98 and 99 we o b t a i n TFVV - TS T = ^ F V m §• + - (100) C 2K The general form of the cl o s u r e curve as g i v e n by equations 89 and 96 i s ' shown i n f i g u r e 20. The r e s u l t s are general and apply to gate opening as w e l l as gate c l o s u r e . However, f o r each value of "X, the curve i s T 2L d i f f e r e n t , ( 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 exceeded (6).) To evaluate AE f o r t h i s form of gate o p e r a t i o n i t i s neces-sary to know T„. From f i g u r e 19 we have VQ " V F ° 2 ^ ( hm " 1 } or a l s o ^ = TTTT (h - l ) + v., (101) Q, 2jO v m ' F y ' VF =TF (102) VQ =TQ K (103) -'OPTIMUM GATE OPERATION 56 Combining the r e s u l t s of equations 101, 102 and 103 we o b t a i n ( h * " X ) + TF> ( 1 ° 4 ) ' m But from equation 96 we have TQ = (2 K T Q - 2 K I + v s ) i (105) « m Equating equations 104 and 105 we o b t a i n Yp K - ^  + % = 2 K TQ " 2 K \ + vs < 106) or - 1 - -2L_ + ^ = T _ L ( 1 0 7 ) 2jO 2K 2K Q a U W / S u b s t i t u t i n g h - 1 2 K = J2 _ a i n t o the f i r s t term of equation 107 a * i d r e a r r a n g i n g we get I t i s now p o s s i b l e to evaluate A E f o r t h i s optimum form of gate o p e r a t i o n . Consider the term Y~ 2 i v (-t ) L n v n' n=l )f the general equation f o r <A,E . According to f i g u r e 21 t h i s area i s Y 2 i v (-t ) = ( v q - v j £ n v n' v S F y 2 n=l ( R e c a l l from equation 64a that (109) Q y~ 2 i v L — n n=l S u b s t i t u t i n g equation 108 i n t o 109 and l e t t i n g V S = T S V F = K we o b t a i n Q XT * k<-*n> = (%^ ) = - ( - # ) 2 (no) n=l The term F y~ ' c ^ v (T_ - 1 ) L n w n v F n y n=Q+l of the general equation can be evaluated i f i t i s assumed that i s a 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^,. (For A h ^ .6 t h i s i s a reasonable approximation. ) This t erm can be w r i t t e n as F , B Y C Qv (UL, - t ) = Y ~ C Q V (T„ - t ) ii n n v F n' L n n v F n / n=Q+l n=Q+l F T~ C Ov (T„ - t ) ( i l l ) n n v F n' v ' n=B+l 58 The term ) C JT ( L - t ) I n n v F n y n=Q+l can he evaluated along the l i n e h = 1 (refer to f i g u r e 22)by c o n s i d e r i n g the f a c t s that C n = 1 - 2pv ( dv 6 v n = -Furthermore, since — = K, we have from geometry ( r e f e r to f i g u r e 22) T - T T w - t = T_ - (T„ + — & (v - v j ) (112) F n F v Q, v_, - v„ Q/y v J ±s y, Furthermore from geometry we have V l " V F = VB " VQ v - v F = v - v (114) Combining equations 112, 113 and 114 we o b t a i n T - T T„ - t = T_ - (T_ + — & (v - v j ) (115) F n F v Q v n - v- v F'' \ *> 1 1 £ As a r e s u l t , we have B v. / j. T -T T~ C dv (T„ - t ) S ( (1 - 2pv) T^ - (T„ + - ^ — ^ (v - v j ) n=Q+l v_ (116) S i m i l a r l y , since ($v i s a constant between n = B + 1 and n = F (as a r e s u l t of the s t r a i g h t l i n e approximation) dv 2 59 n=B+l C i v (T_ - t ) n n v F rr jF ( i - 2pv) | T F - ( T C v., T -T (117) Rearrangement of equation 117 and r e v e r s a l of the l i m i t s of i n t e g r a t i o n y i e l d s F E C 6v ( T - t ) n n v F n 7 n=B+l v^ T - T (! - 2 f v) — (v - v F ) -• r vF (118) v F Adding equations 116 and 118 and c a n c e l l i n g terms we get Y c i v ([IL, - t ) "= £ n n v F nJ 2 n=Q+l T - T v V 1" VF v„ I" ( l - 2 j 0 v ) dv V-F Upon s u b s t i t u t i o n of F . \  2 ( v i ~ V ( l - 2 p v ) v dv (119) F ~ Q, ~ 2L a the s o l u t i o n of equation 119 becomes E c i v ( T _ - t ) ^  — (v n n n F n y a v 1 v, r i P J [ 2 - 5 ( 2 V F + V. (120) n=Q+l From f i g u r e 22 we have = T F J F (121) 60 (122) S u b s t i t u t i o n of the r e s u l t s of equations 121 and 122 i n t o equation 120 y i e l d s _L ) c i v ( L - t ) : - ( J~h - i) - - i - (h - I ) L n n v F n' a L F v V m ; 2 p v ™ x ) 1 JL 2 n=Q+l 2j3 v"m or F y c ^ ( L . t ) = £ — n n v F n y L / V m x n=Q+l a v I2p h - 1 m h + 2 m (124) Equation 124 i s v a l i d f o r gate c l o s u r e and gate opening of the form given by equations 89 and 96 provided and 2p 2 * a dh dv F For the remaining terms i n the general equation f o r A E we have S ( p s - V I =w A H , ^ 7 ( T q - T j and AKE 1 w TJ - A L V 2 g 0 0 0 a v 'S ,F/ 2 ( V - V ) (125) (126) 61 or A K E - w A H o V o i (p) ( T s 2 - T p 2 ) (127) Furthermore upon s u b s t i t u t i o n of K = h - 1 m -4 i n t o equation 110 we o b t a i n P m - T y J F ; n=l (128) S u b s t i t u t i o n of the r e s u l t s of equations 124) 125? 127 and 128 i n t o the general equation f o r A E which i s Q A E = (P_ - v ) - + AKE + w A H V g v S F y a . o o f 7 2 I (-t ) J ' — n v n 7 n=l C d v (T - t ) I n n F n' J (129) n=Q+l y i e l d s A E = W A H V -g o o a { ( T S - T F ) + P ( T S2 - T P 2 ) + E 4 T ( ^ - V m \F -1 f r-+ l i p - I IP^F A - 1 (130) 62 FIG 24 WATERHAMMER CHART DEFINING THE REGIONS FOR OPTIMUM GATE OPERATION: t = T~- TO t = T. 63 This equation 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. For the case where AE can be evaluated from f i g u r e 23 where the assumption has again been made that i s a s t r a i g h t l i n e . The shaded area of f i g u r e 23b i s as be-f o r e Q Area mpv.. = 2 ° v n ^r) n=l Furthermore Area mpxv 1 = ± (T Q - ^ ) ( ( v g - v p ) + (v]_ - v ? ) ) (152) Area V - j X y V g = ^ ( ( v 1 - v ^ ) + ( v 2 - v j ) (133) Area V g y z v ^ = | ( ( V g - v p ) + (v^ - v j ) (154) From f i g u r e 23a we have J L ( h l - 1) + I ( H I _ D . (_L. + I) ( h i _ 1) ( 1 5 5 ) V l - V F v. - v„ = - (^ - 1) + i ( h x _!) = ( ! ! ) (h]_ - 1) (136) 2 F 2 V2 " V F = 2 f ( h2 - !) + I <h2 - X> = £ + 2^) <h2 ~ ^ 3 "F = ~ 2p~ <h2 - 1) + 7 ( h 2 - 1) = (J " 2^) ( h 2 - 1) (138) v^ - v T e t c . D i v i d i n g equation 138 by equation 137 and equation 136 by equation 135 we o b t a i n V 2 _ V F = ( 7 + 2 f ) (^-D ' ( l ^ ) ^ - 1 ) " " 1 ' ^ " (139) Therefore we have v^ - v F = R ( v 2 - v F ) (140) v 2 - v F = R (v 1 - v F ) (141) S u b s t i t u t i o n of the r e s u l t s of equations 140 and 141 i n t o equation 134 y i e l d s Area v^v^ = ^  R - v p ) + ( v 2 - v p ) ) (142) = R (Area v - j X y V g ) (143) e t c . Therefore we have J ~ 2 i v n (-t ) = Area mpxv-j^  + Area v-j^xyVg ( ^  R^) (144) n=l j=0 However 2 p •> s and as a r e s u l t R < 1 65 This means that j = o R J = 1 + R + R 2 +....+ R J = ^ ( 1 4 5 ) and / 2 ( - " t n ) = Area mpxi^ + -—^ Area v xyv £ n = 1 R e f e r r i n g to f i g u r e 23a w i t h h, = h we have 1 m T i - \ A . + if ( h m -1 > and so that i 2 k <-*„>: k <Tc - ^  (T A + + a . 0 . 2 T F ) n=1 » + i i > < 2 : > V A . - 1 > 0 4 6 ) where T v/h - 1 , 1 1 ? P 2 j ) T F - ( A m - M ) h R = X A m - 1 _ 1 2 p T F + ( / h m + 1) (147) h, - 1 + 2p m \ 1 2 P % + ( / h m + l ) ^ = T T ^ C — ^ — ( 1 4 8 ) 2 ( / h m + D 6 6 and m 2L_ C a TF f\ " T s 2K L a (H9) L a {(TFAB-TS) ( g ^ ) - i ] (150) S u b s t i t u t i n g these values i n t o the equation f o r A E which i s i n t h i s case S given by equation Jl) we o b t a i n A E = WAH V• £ I ( T - T ) + P ( T 2 - T 2 ) g o o a | W S ' FJ y K J S J F ; TpK-V - 2 T F 2 p T F + A m + i / h + 1 V m ( T j (/h - i V F' v v m >]} (151) I t can be shown that when v/h + 1 J F ~ 1 2j3 (152) equations 130 and 151 c o i n c i d e . I f A + 1 •'F 2jD (153) equation 130 a p p l i e s . ' I f v m + 1 (154) 67 equation 151 a p p l i e s . From equations 124 a.nd 152 i t i s seen that f o r any gate o p e r a t i o n F i f / C Sv ( L - t ) = 0 L— n n v F n' n=Q+l rr / h + 1 F ^ 2© K D D > or i f For values of *T„ given by / h m + 1 x r - \ + 2 2 p > F * 2 ^ ( / h m + 2) . < 1 5 7 ) 27 2 ^ (-t ) i s a p o s i t i v e q u a n t i t y f o r gate c l o s u r e and a negative n=l q u a n t i t y f o r gate opening. For values of '3"_ given by h m + 2 T F < 25. (yff + 2) ( 1 5 8) ^ 2 (bv^ (~t n) i s a negative q u a n t i t y f o r gate c l o s u r e and a p o s i t i v e 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. Only f o r gate o p e r a t i o n i n t o r e g i o n B of f i g u r e 24 would be advantageous to 68 '• FIG 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 of gate o p e r a t i o n a f t e r t = T n. For gate o p e r a t i o n i n t o regions A or C of f i g u r e 24 gate o p e r a t i o n should be at the maximum r a t e . I t i s i n t e r e s t i n g to note from f i g u r e 24 that f o r a l l but low-values o f j (p^ - 1«5) the optimum gate o p e r a t i o n i s that which leaves 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 of gate o p e r a t i o n i s between T= .3 and 7= !• Below X= .3 the machine i s running at speed no l o a d ) . Close examination of equations 130 and 151 shows that the e f f e c t o f o o n i s very pronounced w h i l e that of h^ i s l e s s important. There-f o r e i n design a r e d u c t i o n i n <y> (although i n c r e a s i n g the diameter of a penstock) may p o s s i b l y r e s u l t i n economic savings because of 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 taken i n t o account). The r e l a t i v e importance of and h^ show that the worst values of A E w i l l be obtained at r a t e d head. g 4.3 Turbine Wicket Gate Closure Curve A t y p i c a l t u r b i n e wicket gate 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 diagram f o r c l o s u r e from f u l l gate i s shown i n f i g u r e 26. This type of c l o s u r e begins s l o w l y so that the term J^ " 2 (Svn ( - t n ) of the general equation f o r AE^ i s much l a r g e r than n=l i t would be f o r the optimum c l o s u r e curve shown i n f i g u r e 25. The remain-i n g values i n the equation f o r AE would be reasonably s i m i l a r f o r the g two c l o s u r e s . Note that f o r gate operations from f u l l gate to p a r t i a l gate, (which are important 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 frequency even i f they do not cause maximum speed d e v i a t i o n ) , the e f f e c t of the i n i t i a l slowness of the wicket gate c l o s u r e i s even more magnified when the r a t i o s 70 of the values of A E f o r each type of c l o s u r e are considered. This i s g ^ because most of the d i f f e r e n c e between the two values of A E occurs i n g 2L the f i r s t 3 (~) seconds i n the example shown. Use of a good cl o s u r e curve can r e s u l t i n savings of 20 - 30$ i n the value of A E ^ f o r the same head r i s e . 4-4 Use of an Upstream Gate i n a C l o s i n g Operation During the f i r s t — seconds a f t e r the i n i t i a t i o n of gate c l o s u r e ° a & the power input to the penstock 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 of the penstock, simultaneously w i t h c l o s u r e at the t u r b i n e , A E ^ could be s u b s t a n t i a l l y reduced along 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 gate c l o s u r e curves, power input to the penstock, and power output from the penstock, 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 at the downstream gate and a ° 7 ^ seconds (to .05 gate) at the upstream gate. (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 at l a r g e gate openings i s s m a l l ) . The head discharge equations f o r d i f f e r e n t gate openings of the upstream gate are given by: h - i = v 2 - r)2 & (159) where V i s the r e l a t i v e gate opening (3). In f i g u r e 29, the head r i s e when the downstream va l v e only i s used i s 2.5 times g r e a t e r than the head r i s e due to combined up and down-stream c l o s u r e s . In f i g u r e 28, A E ^ i s approximately 20$ l e s s f o r the case of combined c l o s u r e s when compared to downstream c l o s u r e only. Use of an upstream va l v e i n the c l o s i n g operation o b v i o u s l y can 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 avoid the use of a pressure 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 s p h e r i c a l valve are not s u i t e d to t h i s type of o p e r a t i o n and a needle v a l v e would pre-sent tremendous design problems. Figure 30 shows an a l t e r n a t i v e which should e l i m i n a t e most of the problems of the other valves and might war-rant some i n v e s t i g a t i o n . Aside from the problem of a s u i t a b l e valve there are 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 that of f a i l u r e of the upstream v a l v e . A p o s s i b l e s o l u t i o n to t h i s problem i s to mount an e l e c t r i c a l contact on the upstream valve that would be connected to a s o l e n o i d c o n t r o l l i n g p a r t of the f l u i d supply to the downstream servo-motor p i s t o n . I f the upstream v a l v e f a i l e d to operate, the e l e c t r i c a l contact would remain open, the s o l e n o i d would then remain close d and the r a t e of c l o s u r e of the downstream va l v e would be l i m i t e d to a safe value. Another problem i s the a c t u a l o p e r a t i n g sequence of the two v a l v e s , ( i t would be uneconomic to leave the upstream gate i n i t s p a r t i a l l y c l o s e d p o s i t i o n ) / I f the downstream gate were closed to i t s f i n a l steady 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 to say Y= .05, then 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 value 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 to the upstream v a l v e , the upstream valve could then be s l o w l y opened to f u l l gate. The net speed change of 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 reduced. .9 it Tun >ine at 6 C/osc re. < i i tree 177 (Jl lost// 2 Cut \ \ -^^  O l 2 2 4 S £ 7 8 9 10 H /Z FIG 27 CLOSURE CURVES FOR SYNCHRONIZED GATE OPERATION W rsec) Ot Turbine treaVn closure t ot 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 H0*H, l/ohe in Open Position l/oli/e in Partially Closed Position From the &orc/a Loss Equation l/z = V$ Mz~H2 =/Vz\2 ^ (±-l)Z H Z = H 0 > H 3 A% « relative qate. area AT J FIG 50 AN: UPSTREAM VALVE 75 CHAPTER V CONCLUSIONS 5-1 The Use of the Energy Method of S o l u t i o n . Although many text-hooks s t a t e that energy methods are not a p p l i c a b l e to problems of r a p i d l y v a r i e d f l o w , i t cannot be denied that i n c e r t a i n instances energy methods can l e a d to v a l u a b l e i n t e r p r e t a t i o n s . I f the process causing the r a p i d flow 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 the case of 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 of the flow take place i n the d i r e c t i o n of the stream-lines (so that no turbulence - which i s the major source of energy l o s s - takes place) then the energy l o s s e s , which u s u a l l y make s o l u t i o n s by energy methods imp o s s i b l e , can be neglected. I f such i s the case the use of energy methods i s j u s t i f i a b l e . The waterhammer process meets the above requirements and so energy problems may be solved d i r e c t l y . Use of the energy method f o r an incremental gate movement shows that as the pressure wave t r a v e l s along the penstock, any change i n k i n e t i c energy i s accompanied by a change i n the energy, stored as s t r a i n energy, i n the f l u i d and i n the penstock, and the net change of energy at any p o i n t i n the penstock i s a constant 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 along the penstock. 5.2 The Importance of the Line h = 2p v on the Waterhammer Chart. 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 of gate o p e r a t i o n , the change i n power output i s i n i t i a l l y opposite to that d e s i r e d . I f the p o i n t r e p r e s e n t i n g the c o n d i t i o n s at the s t a r t of a gate 76 o p e r a t i o n i s to the r i g h t of the l i n e h = 2pv on the waterhammer c h a r t , then any waves r e l a t e d to 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 of 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 of the f l u i d t hat i s g r e a t e r than the amount of energy that the penstock and the f l u i d can s t o r e . Therefore, i f the gate o p e r a t i o n i s one of c l o s u r e , there w i l l he an excess of energy equal to the d i f f e r e n c e between the absolute value of the change i n k i n e t i c energy and the absolute value of the change i n stored energy. This excess appears at the gate and the r e s u l t i s an increased power output. I f the gate o p e r a t i o n i s one of opening, an energy d e f i c i e n c y equal to the d i f f e r e n c e between the absolute value of the change i n k i n e t i c energy and the absolute value of the change i n stored 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 that f o r most gate operations the i n i t i a l power change w i l l be opposite to that d e s i r e d . T v^ h + 1 . = — 2 on the Waterhammer Chart. For 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 slope of the T curve i s equal to the slope of the waterhammer l i n e (2o ). This l i n e i s always to the r i g h t of the l i n e h = 2 p v . Any gate T \/h + 1 = — must maintain d T the maximum value of — T T — that i s c o n s i s t e n t w i t h the maximum all o w a b l e dt head d e v i a t i o n , i f the absolute value of A E i s to be kept to a minimum. F (The term / C i v (T - t ) i n the general equation f o r A E does not W l n n F n g T \/h + 1 = ~~2p 77 5.4 The Importance of the Area on the Waterhammer Chart Between the Lines h = 2 Q v and T= ^ I n t h i s area any waves created w i l l cause an energy output opposite to that d e s i r e d . As a r e s u l t i t i s .desirable to create the 2L minimum number of waves p o s s i b l e i n the — seconds before the f i n a l ^ a 2L steady s t a t e power output i s f i r s t reached. (Up to the p o i n t T - — r a the energy output 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 energy of the f l u i d between the i n i t i a l and f i n a l steady s t a t e s , and the energy input to the penstock. Thus i t i s d e s i r a b l e to decrease the c r e a t i o n 2L of waves only i n the l a s t — seconds before reaching steady s t a t e power output.) This i s equivalent to reducing the r a t e of gate o p e r a t i o n i n t h i s i n t e r v a l . Although the r e s u l t a n t r e d u c t i o n i n the absolute value of AE;' due to 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 reduced, 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 of instantaneous gate c l o s u r e demonstrates the i d e a that i n cases where a p a r t i a l gate o p e r a t i o n takes 2L p l a c e i n a time l e s s than — seconds, i t may be q u i t e advantageous from the p o i n t of head r i s e as w e l l as energy output and h y d r a u l i c s t a b i l i t y to reduce the r a t e of gate c l o s u r e a f t e r a c e r t a i n i n t e r v a l . , 3 fc^^fci, C. - O „ «. c L n n v F n' n=Q+l Equation f o r A E . In the d i s c u s s i o n of the gate c l o s u r e which y i e l d s the smallest value of maximum waterhammer P y ~ c i v ( T - 1 ) L n n P n n=Q+l 78 was evaluated to be: F 7" C Sv (T - t ) = " V 2 0 X - (yC + 1) Z n n F n 12 P jf F y m n=Q+l (h + 2) - 2 f % (/m + 2) 2L I f the v a r i a t i o n s i n head d u r i n g the l a s t — i n t e r v a l of c l o s u r e are not too great then t h i s equation would be a reasonable approximation f o r any type of gate o p e r a t i o n . I f so then i t i s immediately obvious that unless ^ i s very small or s/^ m -~s v e r y l a r g e t h i s term i s of n e g l i g i b l e value when compared to terms of the order of (0(1^,2 - Tj,2) which i s r e p r e s e n t a t i v e of the 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 steady s t a t e s . Therefore the main advantage of making the absolute value of t h i s term as small as p o s s i b l e i n the area, between T v/h+i j .. = ~2Q— ° ^ the waterhammer cha r t , i s the r e -du c t i o n of h y d r a u l i c o s c i l l a t i o n s . Even to the l e f t of the l i n e h = 2 o v the advantages of 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 d T r e d u c t i o n of A E ^ that would be obtained by keeping at a high v a l u e . Q 5 . 6 The Importance of the Term ^ 2 o"vn (~^n) i n the General Equation n=l f o r A E . g The l a s t s e c t i o n showed that f o r a wide range: y~ c 6v - 1 ) L n n v F n y n=Q+l I f such i s the case then: 79 Q A E = (p„ - p_) - + AKE + v (-t ) f 7(WAH V )( S S F' a [ n = 1 nx n' ) ( x o oJ From t h i s equation, the importance of changing the energy input to the Q. -penstock as r a p i d l y as p o s s i b l e i s obvious (WAH V > 2 d v (-t ) i s o 0 ^Z]_ n n the v a r i a b l e term i n the energy input equation). Furthermore t h i s equation g i v e s an approximate method f o r comparing the r e l a t i v e m e rits of d i f f e r e n t c l o s u r e curves since the r a t e of change of v e l o c i t y at the i n t a k e i s r e l a t e d to the r a t e of change of v e l o c i t y at the gate. We can then w r i t e Q YL 2 i v ( - t ) = f (4f) Z n v n' K dt ' n=l Therefore those c l o s u r e curves which have a high i n i t i a l value of —r-r-° dt can be expected to produce smaller values of AE^. I n f a c t the p l o t of input power' i s u s u a l l y very s i m i l a r i n form to the gate c l o s u r e curve (compare f i g u r e s 28 and 27 f o r the case of no upstream c l o s u r e ) . 5.7 The Use of an Upstream Gate i n C l o s i n g Operations. The example showing the e f f e c t of combined up and downstream gate c l o s u r e showed that many advantages may be gained by the use of t h i s type of synchronized c l o s i n g o p e r a t i o n . As there i s u s u a l l y a valve on the upstream end of a penstock, i t seems w e l l worthwhile to make gr e a t e r use of i t , p a r t i c u l a r l y i n cases where a pressure r e g u l a t o r i s necessary. The p o s s i b l e economic advantages of one l e s s valve and i t s accompanying energy d i s s i p a t o r , along w i t h reduced penstock costs and energy savings make f u r t h e r i n v e s t i g a t i o n of t h i s p o s s i b i l i t y of d e f i n i t e u s e f u l n e s s . 80 5.8 General Comments p Because of the g e n e r a l l y small value of ^ C <^ v (TL. - t ) n n n v F n 7 n=Q+l much s t a b i l i t y can be gained w i t h n e g l i g i b l e e f f e c t on A E i f t u r b i n e gates are always r e g u l a t e d i n a manner such that the h y d r a u l i c o s c i l l a t i o n s are reduced to as near zero as p o s s i b l e . To t h i s end i t might be of some advantage to govern h y d r a u l i c t u r b i n e s on the b a s i s of the change i n load on the generator i n s t e a d of on the b a s i s of speed d e v i a t i o n . (For small 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 governor i s not i n a c t i o n . ) I f governing were based on the amount of l o a d change f o r l a r g e l o a d v a r i a t i o n s a form of programmed gate operation-might be used ( i . e . f o r each lo a d change there would be a predetermined gate o p e r a t i o n p a t t e r 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 increased system s t a b i l i t y . Good speed r e g u l a t i o n i n general i s determined by the r a t i o of the t o t a l energy s u p p l i e d to the prime-mover, to the k i n e t i c energy of motion of the f l u i d s u p p l ying the energy to the prime-mover. Thus a steam t u r b i n e i s comparatively easy to govern - the r a t i o of the t o t a l energy of a pound of steam to i t s k i n e t i c energy of motion i s u s u a l l y f a r g r e a t e r ( p a r t i c u l a r l y w i t h the present hi g h temperature, h i g h pressure, steam p l a n t s ) than that of the highest head hydro-power p l a n t s . 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 product HV w h i l e the k i n e t i c energy of the f l u i d v a r i e s as V . The r a t i o of these two energies i s : T o t a l Energy cy; H V = H K i n e t i c Energy y2 y 8 1 •n , 1 0 But -r— = — — 9 o Therefore p i s a measure of the r a t i o of the t o t a l energy of the f l u i d to the k i n e t i c energy of the f l u i d . Furthermore, from the f a c t that the l i n e h = 2ov 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 . the penstock) can store an amount of energy g r e a t e r than an accompanying change of k i n e t i c energy, (at which p o i n t good governing becomes p o s s i b l e ) i t i s seen that tp i s a very important f a c t o r in'-t u r b i n e governing. (This was demonstrated i n the d i s c u s s i o n of the gate c l o s u r e curve which y i e l d s the minimum value of maximum waterhammer). A low value of i s not only i n d i c a t i v e of a high t o t a l energy to k i n e t i c energy r a t i o , but a l s o increases the area on the waterhammer chart i n which good governing i s po s s i b l e . ' Because of i t s e f f e c t on the pressure head r i s e f o r a gi v e n gate o p e r a t i o n (the lower the value of <p , the lower the maximum pressure d e v i a t i o n ) i n c r e a s i n g p may be more economical than 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 the savings due to 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. Bergeron, 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 York: John Wiley and Sons, 1961. 2. Jaeger, C. Engineering F l u i d Mechanics, London: B l a c k i e and Son Li m i t e d , 1957-3- Jaeger, C. "Water Hammer Caused by Pumps," Water Power, J u l y , 1959. 4« Kirchmayer, L. K. Economic C o n t r o l of Interconnected Systems, New York: John Wiley and Sons, 1959 5 . Parmakian, J . Waterhammer A n a l y s i s , New York: Dover P u b l i c a -t i o n s , Inc., 1963 6 . Ruus E. Determination of Closure Curves Which Y i e l d the Smallest Value of Maximum Waterhammer, Trans l a t e d by E. Ruus, K a r l s r u h e : 1957-7. Stevenson, W. D. Elements of Power System.Analysis, New York: McGraw-Hill Book Company, Inc., 1955. 83 APPENDIX I SYMBOLS, ABBREVIATIONS AND UNITS a - waterhammer'wave v e l o c i t y f t / s e c 2 A - c r o s s - s e c t i o n a l area of penstock f t D - penstock diameter f t E - modulus of e l a s t i c i t y of penstock w a l l l b s / f t E - energy' f t - l b s P - f o r c e l b s g - a c c e l e r a t i o n of g r a v i t y f t / s e c H - t o t a l pressure head f t H q - steady s t a t e pressure head f t i H - i n f i n i t e s i m a l change i n pressure head f t AH - t o t a l change i n pressure head f t h - r e l a t i v e pressure head = — ti o 2 K - bulk modulus of f l u i d l b s / f t L - penstock l e n g t h f t P - power f t - l b s / s e c R - penstock r a d i u s f t S - work f t - l b s t - time - general 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 of the f l u i d i n the penstock f t / s e c V - f u l l gate v e l o c i t y at H = H f t / s e c 0 o V v - r e l a t i v e v e l o c i t y = — q -z w - u n i t weight of water a y l b s / f t ^ - pipe l i n e constant = TJ-J^ ~ 01 - r e l a t i v e gate area 

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