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Resonance in pressurized piping systems Chaudhry, Mohammad Hanif 1970

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RESONANCE IN PRESSURIZED PIPING SYSTEMS by MOHAMMAD HANIF CHAUDHRY B.Sc(Hons.) ( C i v i l Engineering), West Pa k i s t a n U n i v e r s i t y of Engineering and Technology, P a k i s t a n , 1965 M.A.Sc ( C i v i l Engineering), U n i v e r s i t y of B r i t i s h Columbia, Canada, 1968 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n the department of CIVIL ENGINEERING We accept t h i s t h e s i s as conforming to the re q u i r e d standard. THE UNIVERSITY OF BRITISH COLUMBIA A p r i l , 1970 i i i In presenting 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 l m e n t of the requirements f o r an advanced degree at t h e . U n i v e r s i t y of B r i t i s h Columbia, I agree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r reference and study. I f u r t h e r agree that permission f o r extensive copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my department or by h i s r e p r e s e n t a t i v e s . I t i s understood that copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be allowed without my w r i t t e n permission. Department of C i v i l Engineering The U n i v e r s i t y of B r i t i s h Columbia Vancouver 8, Canada Date A B S T R A C T A new approach, based on the t r a n s f e r matrix method used i n the theory of v i b r a t i o n s , i s presented to analyze the s t e a d y - o s c i l l a t o r y f l o w s , and to determine the resonating c h a r a c t e r i s t i c s of p i p i n g systems. By l i n e a r i z i n g the f r i c t i o n l o s s term, c o n s i d e r i n g the system as d i s t r i b u t e d and assuming the discharge and pressure head f l u c t u a t i o n s as s i n u s o i d a l , f i e l d matrices f o r a simple p i p e l i n e and f o r a p a r a l l e l system are d e r i v e d . A numerical technique i s presented to determine the f i e l d matrix f o r a pipe having v a r i a b l e c h a r a c t e r i s t i c s along i t s l e n g t h . P o i n t matrices f o r o r i f i c e s , and f o r o s c i l l a t i n g valves are obtained by l i n e a r i z i n g the gate equation. Point matrices f o r the j u n c t i o n of the main and a branch having various boundary c o n d i t i o n s , e.g., dead end, r e s e r v o i r , o r i f i c e , o s c i l l a t i n g v a l v e , are a l s o d e r i v e d . A numerical procedure i s o u t l i n e d f o r computing the resonant frequen-c i e s of p i p i n g systems. Expressions are developed to determine the frequency response of systems having p e r i o d i c f o r c i n g f u n c t i o n s , such as f l u c t u a t i n g pressure head; f l u c t u a t i n g discharge; and o s c i l l a t i n g v a l v e . A number of systems commonly used i n waterpower development and water supply schemes are analyzed. The r e s u l t s obtained by the method presented h e r e i n are i n c l o s e agreement with those obtained experimentally by e a r l i e r i n v e s t i g a t o r s ; or determined by using the method of c h a r a c t e r i s t i c s , imped-ance theory, or energy concepts. i v TABLE OF CONTENTS Chapter D e s c r i p t i o n Page ABSTRACT ±± LIST OF TABLES v i LIST OF FIGURES v i i NOTATION i x ACKNOWLEDGMENT x i i i I INTRODUCTION 1 II TERMINOLOGY AND BLOCK DIAGRAMS 2.1 Terminology 11 2.2 Block diagrams 15 I I I DERIVATION OF TRANSFER MATRICES 3.1 F i e l d matrices 19 3.2 Poi n t matrices 32 IV RESONANT FREQUENCIES AND FREQUENCY RESPONSE OF PIPING SYSTEMS 4.1 Resonant frequencies 44 4.2 Frequency response 46 4.3 Pressure and discharge v a r i a t i o n along p i p e l i n e 54 V VERIFICATION OF TRANSFER MATRIX METHOD 5.1 Experimental r e s u l t s 61 5.2 Method of c h a r a c t e r i s t i c s 67 5.3 Impedance method 79 5.4 Energy concepts 87 V SUMMARY AND CONCLUSIONS 98 BIBLIOGRAPHY !00 APPENDICES A. Summary of t r a n s f e r matrices 106 B. Example 108 v i LIST OF TABLES No. D e s c r i p t i o n Page I C a l c u l a t e d and measured periods 66 I I Resonant frequencies of p i p i n g system o f F i g . 5. f4y g2 I I I Phase angles 86 IV V e r i f i c a t i o n by energy concepts 94 V Resonant frequencies of p i p i n g system of F i g . 5.17a 97 VI Scheme f o r m u l t i p l i c a t i o n of t r a n s f e r matrices -^.ll v i i LIST OF FIGURES No. Description Page 2.1 Instantaneous and mean discharge. 13 2.2 Single p i p e l i n e . 16. 2.3 Series connection. 16 2.4 Block diagram. 17 3.1 Pipe having v a r i a b l e c h a r a c t e r i s t i c s along i t s length. 26 3.2 Actual and substitute pipe. 26 3.3 P a r a l l e l system. 29 3.4 Valves. 35 3.5 Branch system. 38 4.1 Suction and discharge p i p e l i n e . 49 4.2 Designation of section on i ^ pipe. 56 4.3 Plot of tan w x/a£ ^ cox/a2- 59 5.1 Longitudinal p r o f i l e . 62 5.2 Plot of Residual ^ co for, Toulouse system. 65 5.3 Series system with dead end. 69 5.4 Series system. 70 5.5 Branch system (side branch having r e s e r v o i r ) . 71 5.6 Branch system (dead end side branch). 72 5.7 Branch system (side branch having o r i f i c e ) . 73 5.8 Branch system (side branch having o s c i l l a t i n g v a l v e ) . 74 5.9 P a r a l l e l system. 78 v i i i 5.10 E f f e c t of f r i c t i o n loss on frequency response of series system of F i g . 5.4a. 8 0 5.11 E f f e c t of mean discharge on frequency response of serie s system of F i g . 5.4a. g-^  5.12 h* "yt, q* vt,- and x*<\,t curves. 5.13 Series system. gg 5.14 Branch system (side branch having r e s e r v o i r ) . 5.15 Branch system (dead end side branch). 5.17 P i p e l i n e having v a r i a b l e c h a r a c t e r i s t i c s along i t s length. 89 90 5.16 Impedance diagram for p a r a l l e l system of F i g . 5.9a. ^\ 95 i x NOTATION The following symbols are used i n t h i s t h e s i s : A = cross-sectional area of pipe, i n sq. f t ; Ay. = gate opening of valve, i n sq. f t ; a = c e l e r i t y of water hammer wave, i n f t / s e c ; b = constant for pipe = l/a, i n sec; B = matrix given i n Eq. 3.22; C = constant f o r pipe = a/(gA), i n s e c / f t 2 ; = c o e f f i c i e n t of discharge; c 1 J c 2 = constants i n Eq. 3.14; C3,Ci4 = constants i n Eq. 3.27; D = insi d e diameter of pipe, i n f t ; E = energy, i n l b - f t ; e = base i n the natural logrithms; F = f i e l d matrix; Fp = f i e l d matrix f o r p a r a l l e l pipes; f = Darcy-Weisbach f r i c t i o n f actor; f1l>f12»f21>^22 = elements of f i e l d matrix; g = acceleration due to gravity, i n f t / s e c 2 ; I = i d e n t i t y or unit matrix; j = constant = /^T ; K = amplitude of f l u c t u a t i n g pressure head f o r c i n g function; k = amplitude of f l u c t u a t i o n of r e l a t i v e gate opening; H = instantaneous pressure head, i n f t ; H Q = mean or average pressure head, i n f t ; h = amplitude of pressure head f l u c t u a t i o n ; d e v i a t i o n of pressure head from mean, i n f t ; 2|h^ ,I/H , i . e . , r a t i o of twice the amplitude of I n+11 o r pressure f l u c t u a t i o n (twice the maximum d e v i a t i o n from mean) at v a l v e and s t a t i c head; h*/H o; length of pi p e , i n f t ; exponent of v e l o c i t y i n f r i c t i o n l o s s term; p o i n t matrix; p o i n t matrix f o r a s e r i e s connection; p o i n t matrix f o r a valve or o r i f i c e having constant gate opening; extended p o i n t matrix f o r o s c i l l a t i n g v a l v e ; p o i n t matrix f o r dead end side branch; p o i n t matrix f o r side branch having constant head r e s e r v o i r ; p o i n t matrix f o r side branch with o r i f i c e ; elements of poi n t matrix; instantaneous discharge, i n cu. f t / s e c ; mean or average discharge, i n cu. f t / s e c ; amplitude of discharge f l u c t u a t i o n ; d e v i a t i o n of instantaneous discharge from mean, i n cu. f t / s e c ; 2|q^+^|/QQ, i . e . , r a t i o of twice the amplitude of discharge f l u c t u a t i o n at val v e and mean discharge; q * / Q 0 ; constant i n Eq. 3 . 7 x i Tp = period of the f o r c i n g function, i n sec; T ^ = t h e o r e t i c a l period of piping system; t = time, i n seconds; U = o v e r a l l t r a n s f e r matrix; IT = o v e r a l l extended trans f e r matrix; u 1 1 J u 1 2 = elements of o v e r a l l t r a n s f e r matrix; x = distance along p i p e l i n e (positive i n the downstream d i r e c t i o n ) , i n f t ; Z = hydraulic impedance; Z c = c h a r a c t e r i s t i c impedance; z = state vector; z" = extended state vector; z^ = normalized impedance at value = | z | / Z . Greek l e t t e r s : Y = s p e c i f i c weight of f l u i d , i n l b s / f t 3 ; C,5,n = constants i n Eq. 3.38; X\,^2 = eigenvalues of matrix B£; u = constant i n Eq. 3.11; ,v = kinematic v i s c o s i t y of f l u i d , i n f t 2 / s e c ; T = r e l a t i v e gate opening of valve or o r i f i c e ; T Q = mean r e l a t i v e gate opening of valve or o r i f i c e ; T * = deviation of T from T ; o' ai = frequency, i n radians/sec; = frequency r a t i o = u/co ^; a) ^  = t h e o r e t i c a l frequency, i n radians/sec; \p = phase angle of the m harmonic of pump discharge curve; m r m and phase angle of the m**1 harmonic of pressure head x i i S u b s c r i p t s and s u p e r s c r i p t s : 1,2, ... n, n+l = number of a s e c t i o n on a pipe; i n = input q u a n t i t i e s ; L = l e f t hand side of a s e c t i o n ; o = mean, or average, value; out = output q u a n t i t i e s ; R = r i g h t hand side of a s e c t i o n ; s = steady s t a t e values; and ^ = q u a n t i t i e s f o r side branch. ACKNOWLEDGMENTS The author wishes to express h i s s i n c e r e thanks to h i s s u p e r v i s o r Dr. E. Ruus f o r guidance, i n s p i r a t i o n and encouragement provided through-out t h i s study; to the other members of the committee i n general and Prof. J.F. Muir i n p a r t i c u l a r f o r the va l u a b l e suggestions i n the prep-a r a t i o n of the t h e s i s ; and to the N a t i o n a l Research Council of Canada f o r f i n a n c i a l a s s i s t a n c e , indebtedness i s a l s o expressed to Dr. S. Cherry who introduced the t r a n s f e r matrix method to the author i n h i s graduate course on the theory of v i b r a t i o n s . S p e c i a l thanks are extended to Dr. E.B. Wylie of the U n i v e r s i t y of Michigan, Ann Arbor, f o r p r o v i d i n g the data and other in f o r m a t i o n f o r h i s a n a l y s i s of the p i p i n g systems by the impedance theory. C H A P T E R O N E INTRODUCTION I f a p i p i n g system i s subjected to a p e r i o d i c e x c i t a t i o n having a p e r i o d equal to one of the n a t u r a l periods of the system, severe o s c i l l a t i o n s of the pressure head and discharge develop. This may r e s u l t i n se r i o u s damage or f a i l u r e of the system. Several i n c i d e n t s and accidents caused by resonance have been reported [1, 27, 28, 32, 5 2 ] 1 i n the l i t e r a t u r e . Resonance i n pipeline's have been stud i e d by a number of i n v e s t i g a t o r s . A l l i e v i was the f i r s t to present, i n Note V of the Theory of Water Hammer [3] , the a n a l y s i s of reso n a t i n g c o n d i t i o n s i n a simple p i p e l i n e caused by the r y t h m i c a l opening and c l o s i n g of a valve at the downstream end of the pipe-l i n e . By using the general s o l u t i o n H - H q = F ( t - + f ( t + |) (1.1) and V " V o = " f { F ( t " a-* _ f ( t + 7 ) } ( 1 - 2 ) of the s i m p l i f i e d p a r t i a l d i f f e r e n t i a l equations governing the unsteady f r i c t i o n l e s s f l ow, namely 1 Numerals i n brackets r e f e r to corresponding items i n B i b l i o g r a p h y . 2. H = - - V (1.3) x g t a2 H = - - V (1.4) t g x he proved that the r e l a t i v e pressure head at the v a l v e , H^/H , at odd and 2 2 even periods approaches two l i m i t s Z-^ and Z 2. In the above equations, x = distance along the p i p e l i n e , measured p o s i t i v e i n the downstream d i r e c t i o n ; t = time; H = pressure head; /H^  =. s t a t i c head; V = v e l o c i t y of the f l u i d ; V q = steady s t a t e v e l o c i t y of the f l u i d ; a = c e l e r i t y of the pressure wave; F,f are i n t e g r a t i o n f u n c t i o n s ; and g = a c c e l e r a t i o n due to g r a v i t y . The s u b s c r i p t s x and t denote p a r t i a l d i f f e r e n t i a t i o n . For ry t h m i c a l v a l v e movements such that To = TT = T2T = • 1 * = C ' ( 1 - 5 ) TT/2 = T3T/2 = T5T/2 = • • • = 1 C1-6) 2 2 Zi and Z 2 are r e l a t e d by the equation 2 2 Zx + Z 2 = 2 (1.7) The t h e o r e t i c a l p e r i o d of the pi p e , T = 4£/a; and the r e l a t i v e gate opening of the v a l v e , T = (C^A ) / (C^A ) . In these expressions, £ = length of pipe-l i n e ; = c o e f f i c i e n t of discharge; A^ = area of the gate opening; and the s u b s c r i p t "o" denotes i n i t i a l steady s t a t e c o n d i t i o n . I t was demonstrated by A l l i e v i that as the amplitude of the val v e move-2 2 ment i n c r e a s e s , i . e . , as C'-O, Zy>0, and Z2>. Thus pressure v a r i a t i o n i n extreme c o n d i t i o n s i s between 0 and 2H q. Later on, however, Bergeron [5-7] proved g r a p h i c a l l y that f o r large values of A l l i e v i ' s parameter, p = (aV Q ) / ( 2 g H Q ) , i t i s p o s s i b l e to have c a v i t a t i o n at the val v e and to reach superpressures i n excess of the o r d i n a r i l y considered maximum of double the s t a t i c head. A l s o , Camichel [8] demonstrated that doubling of pressure head i s not p o s s i b l e unless H > aV /g. v o o & 3. Camichel conducted a number of experiments at the l a b o r a t o r y of Toulouse and Eydous performed f i e l d t e s t s on high head i n s t a l l a t i o n s . In most of these experiments, the resonating c o n d i t i o n s were produced by a r o t a t i n g cock d r i v e n by a motor which opened and close d the v a l v e r y t h m i c a l l y . Their experimental r e s u l t s , without d e t a i l e d t h e o r e t i c a l a n a l y s i s , are summarized i n two r e p o r t s [ 8 ] . Wylie [58, 59], and the author [12] have used these r e s u l t s f o r the v e r i f i c a t i o n of the t h e o r e t i c a l analyses. I f a pressure wave i s followed i n a compound p i p e l i n e , then the p e r i o d of the system, which Camichel et a l . c a l l e d the t h e o r e t i c a l p e r i o d , i s n * i T =4 7* _ th i = l a. (1.8) I V However, the p e r i o d of the maximum pressure surge, termed as the apparent  p e r i o d , was observed to be d i f f e r e n t from the t h e o r e t i c a l p e r i o d ; the former being g e n e r a l l y s h o r t e r . The experiments performed by Camichel et a l . demon-s t r a t e d the existence of the higher odd harmonics. I t was found that the p e r i o d of the higher odd harmonics are i n t e g e r f r a c t i o n s of the t h e o r e t i c a l p e r i o d , and the p e r i o d o f the fundamental i s equal to the apparent p e r i o d . These r e s u l t s are v a l i d f o r a simple p i p e l i n e but cannot be g e n e r a l i z e d f o r complex systems. I t was a l s o reported that through flow i s a minimum during resonating c o n d i t i o n s — a r e s u l t which w i l l be used i n the f o l l o w i n g chapters to determine the resonant frequencies. W i l k i n s [56] was the f i r s t to r e p o r t severe pressure o s c i l l a t i o n s com-bined w i t h machine v i b r a t i o n s i n h y d r o e l e c t r i c power p l a n t s i n 1923. Den Hartog [16] concluded from t h e o r e t i c a l a n a l y s i s that o b j e c t i o n a b l e v i b r a t i o n s are to be expected i n penstocks of F r a n c i s t u r b i n e i n s t a l l a t i o n s i f the number of vanes on the t u r b i n e runner i s one l e s s than the number of guide vanes. Although a number of s i m p l i f y i n g assumptions are made i n h i s a n a l y s i s , observations on eight d i f f e r e n t i n s t a l l a t i o n s show good agreement w i t h t h i s theory. By assuming a s i n u s o i d a l pressure v a r i a t i o n , u s i n g the general s o l u t i o n of the d i f f e r e n t i a l equations, and making use of the f a c t that during reson-ance v ,= 0, Jaeger [26, 28] derived the f o l l o w i n g equation f o r the fundamental frequency of two pipes i n s e r i e s having a constant head r e s e r v o i r at the upstream end: A oo&i A2 to £2 - i - tan — - = — cot (1.9) &l a.i a.2 &z The s u b s c r i p t s "1" and "2" r e f e r to the number of the p i p e . The s o l u t i o n of t h i s equation f o r 00 gives the frequency of the fundamental. S i m i l a r l y , the f o l l o w i n g equation was obtained f o r a t y p i c a l hydropower development scheme: A 2 a l a 2 a l i A 3 a l ^ 3 a l 1 1 ^ 1 ^ TT z 1 1 3 1 TT 3 1 1 TT 1 C O t -TT — i + ; — C O t — -z = tan TT a2^-1 2 a 2 & l 1 + e a 3 A l 2 a 3 ^ 1 l + e 2 l + e (1-10) i n which the s u b s c r i p t s "1", "2" and "3" r e f e r t o , r e s p e c t i v e l y , the penstock, surge tank and t u n n e l . A f t e r c a l c u l a t i n g e from Eq. 1.10, the apparent p e r i o d of the system can be determined from the equation ', 2a TT While a nalysing the higher harmonics Jaeger [26, 28] i m p l i c i t l y assumes that only odd harmonics are p o s s i b l e . He considers a p o i n t A i n a complex system such that the c r o s s - s e c t i o n of the p i p e l i n e i s constant between the p o i n t A and the gate. The c o n d i t i o n s at p o i n t A are represented by f = - « T F , (1.12) n n-1 n-1 J i n which a.^ ^ i s an unknown f u n c t i o n which depends on the geometry of the conduit and on the movement of the gate. He shows that i f a = 1, resonance w i l l occur. When a i s equal to one, a nodal p o i n t e x i s t s at A w i t h a c o r r e s -ponding loop at the v a l v e . In the d e r i v a t i o n of t h i s theory, i t i s assumed 5. that the h a l f p e r i o d of each pipe i n the system i s an i n t e g e r m u l t i p l e of the h a l f p e r i o d of the pipe from the valve to the p o i n t A. Since by t h i s method only those periods are p r e d i c t e d which are r e l a t e d to the theoret-i c a l p e r i o d by i n t e g e r s , Jaeger concluded that the higher periods of a complex system are r e l a t e d to the t h e o r e t i c a l p e r i o d while the fundamental i s a s s o c i a t e d w i t h the apparent p e r i o d . This i s true i n a simple pipe but cannot be g e n e r a l i z e d f o r complex systems. Jaeger [27, 28] has compiled data of a number of h y d r o e l e c t r i c power p l a n t s and pumping schemes i n which extensive damage or f a i l u r e of the system occurred. Out of twenty such cases, twelve were caused by resonance or s e l f - e x c i t e d v i b r a t i o n s . Bergeron [5-7] introduced the g r a p h i c a l method to analyze resonant c o n d i t i o n s . In h i s l a t e s t work [ 7 ] , g r a p h i c a l a n a l y s i s of the resonant c o n d i t i o n s caused by r e c i p r o c a t i n g pumps i s a l s o i n c l u d e d . Angus [4] reported that valves having a loose connection between the stem or operating s p i n d l e and the piece that performs the f u n c t i o n of changing the area, or having a s l i g h t p l a y of the s p i n d l e or stem, or having s p i n d l e s which bend under pressure, are subjected to c h a t t e r i n g when p a r t i a l l y closed and, q u i t e f r e q u e n t l y , when n e a r l y c l o s e d . He demonstrated g r a p h i c a l l y the a m p l i f i c a t i o n of the pressure head due to c h a t t e r i n g of v a l v e s . I t i s shown that the shape of the r e l a t i v e pressure head versus r e l a t i v e v e l o c i t y diagram depends upon the value of the A l l i e v i ' s para-meter and on the s e t t i n g of the v a l v e at which the c h a t t e r i n g occurs. Schhyder [44] showed, by the g r a p h i c a l method, the a m p l i f i c a t i o n of the pressure head at a l e a k i n g v a l v e . Rocard [43] was the f i r s t to introduce the concept of h y d r a u l i c im-pedance, l a t e r used by Paynter [38], Waller [53-55], and Wylie (58-60], to 6. analyze a u t o - o s c i l l a t i o n s i n the Lac Blanc-Lac N o i r pumped storage scheme. His s o l u t i o n y i e l d e d lengthy equations which were p r a c t i c a l l y impossible to solve by hand. To overcome t h i s d i f f i c u l t y , he used a pipe having equivalent diameter i n s t e a d of the a c t u a l p i p i n g system. E v a n g e l i s t i [18] derived equations f o r resonant frequencies of three system types, namely, a simple p i p e l i n e ; two pipes i n s e r i e s ; and two branch pipes by applying the admittance method commonly used i n e l e c t r i c a l engineering. He d i d not, however, solve any example or compare the r e s u l t s with experimental values. By u s i n g the general s o l u t i o n of A l l i e v i ' s equations, assuming a s i n u s o i d a l pressure v a r i a t i o n and c o n s i d e r i n g that a discharge node i s formed at the v a l v e during resonance, Favre [19] developed the f o l l o w i n g expression f o r the p e r i o d of the fundamental and of the higher harmonics of a pipe having l i n e a r l y v a r i a b l e c h a r a c t e r i s t i c s — d i a m e t e r and wave v e l o c i t y — a l o n g i t s length: tan^-£= - i J L A l (1.13) a a a m m i n which a = [1 + . v / 2 ) [ y ( l +'v/2) + v] ; v = (a - a )/a ; and y = (D^ - D )/D . The s u b s c r i p t s o, m and A r e f e r to values at the v a l v e , at the mid-point, and at the r e s e r v o i r end of the pipe. As the c o n i c a l pipe approaches a c y l i n d e r , a approaches zero. Favre a l s o demonstrated that f o r continuous p e r i o d i c v a l v e motion, the apparent p e r i o d , and f o r r a p i d l y c l o s i n g and opening a v a l v e at r e g u l a r i n t e r v a l s , the t h e o r e t i c a l p e r i o d produces severe resonating c o n d i t i o n s . Paynter [38] introduced the idea of e l e c t r i c a l - h y d r a u l i c analogy. By using the concept of h y d r a u l i c impedance, he s t u d i e d the surges caused by r y t h m i c a l motion of a v a l v e at the downstream end of a p i p e l i n e and 7. presented the r e s u l t s i n a g r a p h i c a l form showing the p e r i o d and amplitude of gate movement, t h e o r e t i c a l p e r i o d of the p i p e l i n e , A l l i e v i ' s parameter, and amplitude and phase angle of pressure f l u c t u a t i o n s . The r e s u l t s are, however, v a l i d only f o r small pressure f l u c t u a t i o n s . Surges i n o i l p i p e l i n e s connected to r e c i p r o c a t i n g pumps were s t u d i e d by Waller [53-55] by using the impedance concepts. He decomposed the pump discharge curve (which he derived by n e g l e c t i n g the c o m p r e s s i b i l i t y of the l i q u i d i n the pump) i n t o a set of harmonics and then determined the surge pressures from the computed values of the impedances. He als o presented the design of a few c o r r e c t i v e devices f o r the system. Abbott, Gibson, and McCaig [1] and McCaig and Gibson [32] reported measurements of s e l f - e x c i t e d v i b r a t i o n s . In the f i r s t example a s l i g h t leak i n the twelve foot diameter penstock v a l v e , caused by a re d u c t i o n of the s e a l pressure, generated s e l f - s u s t a i n e d v i b r a t i o n s of the v a l v e . The valv e v i b r a t i o n s and the pressure o s c i l l a t i o n s were found to be s i n u s o i d a l . Opening a bypass valve reduced the v i b r a t i o n s to zero. The second i n c i d -ent occurred when a ten-inch diameter spring-cushioned check v a l v e i n a pump discharge l i n e leaked under s t a t i c head. Measurements showed that the r e s u l t a n t pressure o s c i l l a t i o n s were approximately s i n u s o i d a l with sharp impulses of large magnitude imposed every t h i r d c y c l e . The v i b r a t i o n s were prevented by i n s t a l l i n g a weaker cushioning s p r i n g i n the check v a l v e and removing the a i r valve from the p i p e l i n e . D'Souza and Oldenburger [17] presented t r a n s f e r f u n c t i o n s r e l a t i n g the pressure and v e l o c i t y v a r i a b l e s at two cross s e c t i o n s of a p i p e l i n e . The v i s c o s i t y and the c o m p r e s s i b i l i t y of the f l u i d were taken i n t o consid-e r a t i o n i n the a n a l y s i s . However, only laminar flow and a s i n g l e pipe of constant area were considered and the e l a s t i c i t y of the pipe w a l l s was 8. neglected. The t h e o r e t i c a l r e s u l t s were v e r i f i e d by comparing them with the experimental values obtained by Roberts [42]. Oldenburger and Goodson [34] i n t h e i r a n a l y s i s of the dynamics of h y d r a u l i c l i n e s replaced the transcendental f u n c t i o n s r e p r e s e n t i n g the t r a n s f e r f u n c t i o n s by the i n f i n i t e products of l i n e a r f a c t o r s and t h e i r approximations. I t was shown that s a t i s f a c t o r y r e s u l t s are obtained by r e t a i n i n g only a few terms. The l o n g i t u d i n a l v i b r a t i o n s and the v i s c o s i t y e f f e c t s were, however, neglected i n the a n a l y s i s . Oldenburger and Donelson [35] conducted frequency response t e s t s at the Apalachia h y d r o e l e c t r i c power p l a n t to check the v a l i d i t y of express-ions d e r i v e d by Paynter. These t e s t s confirmed that the expressions i n general are v a l i d under a l l major operating c o n d i t i o n s . Although the idea of impedance has been used s i n c e 1937, Wylie [50, 58-60] was the f i r s t to extend and systematize t h i s concept f o r s o l -v i n g complex systems, such as, s e r i e s , p a r a l l e l , and branch systems with the side branch having an o r i f i c e , r e s e r v o i r or a dead end. He assumed a s i n u s o i d a l pressure and flow v a r i a t i o n , l i n e a r i z e d the n o n l i n e a r terms and computed the r a t i o of the d e v i a t i o n of the discharge, q, and pressure head, h (both expressed as complex numbers), from the mean. By using the known boundary c o n d i t i o n s he computed the sending end impedance, Z g, and then p l o t t e d an impedance diagram, u) versus |Z |. Frequencies at which |Z | i s a maximum are the resonant frequencies of the system. Because of the lengthy a l g e b r a i c equations i n v o l v e d , the method i s s u i t a b l e f o r com-puter a n a l y s i s only. For a p a r a l l e l p i p i n g system a procedure i s suggested i n which a number of simultaneous equations are to be solved. This becomes cumbersome i f there are many pipes i n p a r a l l e l . For example, f o r two pipes i n p a r a l l e l e i g h t simultaneous equations have to be solved. 9. The impedance theory was v e r i f i e d by Wylie by comparing the r e s u l t s w i t h experimental values or with those obtained by the method of cha r a c t e r -i s t i c s . The resonating c o n d i t i o n s at the Bersimis No. 2 system [60] were studi e d and c l o s e agreement was found between the measured and the computed values. S t r e e t e r and Wylie [49] st u d i e d the resonating c h a r a c t e r i s t i c s of a system having a r e c i p r o c a t i n g pump by using the impedance theory. The kinematics and dynamics of flow through r e c i p r o c a t i n g pumps were discussed. The pump discharge was decomposed i n t o a set of harmonics and the express-io n f o r pressure f l u c t u a t i o n s on the s u c t i o n side of a pump having a simple s u c t i o n l i n e was developed. In the a n a l y s i s , c o m p r e s s i b i l i t y of the water i n the pump was taken i n t o c o n s i d e r a t i o n . An e x c e l l e n t agreement i s shown to e x i s t between the measured values and those c a l c u l a t e d by the impedance theory and by the method of c h a r a c t e r i s t i c s . S t r e e t e r and Wylie [48] were the f i r s t to use the impedance theory f o r the s t a b i l i t y a n a l y s i s of governed hydro-systerns. Other methods [13, 15, 23] are a v a i l a b l e f o r t h i s type of a n a l y s i s . Recently Z i e l k e , Wylie, and K e l l e r [61] used the impedance theory to study the forced and s e l f - e x c i t e d o s c i l l a t i o n s i n the p r o p e l l a n t feed systems of rocket engines. The f l u i d system c o n s i s t e d of a s u c t i o n p i p e , c e n t r i f u g a l pump, and a discharge pipe. The adequacy of the a n a l y t i c a l model as w e l l as the usefulness of the l i n e a r i z a t i o n procedure was v e r i f i e d e x p e r i m e n t a l l y . H o l l e y [21, 22] i n v e s t i g a t e d , experimentally and t h e o r e t i c a l l y , the surging i n p i p e l i n e s . He considered the water as incompressible and the wa l l s of the p i p e l i n e as r i g i d . A f t e r n o r m a l i z a t i o n the governing d i f f e r -e n t i a l equation was solved by the f o u r t h order Runge-Kutta method.-While 10. d i s c u s s i n g R ef. 27, t h e w r i t e r [10] n o r m a l i z e d t h e g o v e r n i n g e q u a t i o n i n terms o f p h y s i c a l l y i n t e r p r e t a b l e p a r a m e t e r s and e l u c i d a t e d some o f t h e l i m i t a t i o n s o f t h e t h e o r e t i c a l a n a l y s i s p r e s e n t e d by t h e a u t h o r . I n t h i s t h e s i s a new a p p r o a c h , based on t h e t r a n s f e r m a t r i x method o f t h e t h e o r y o f m e c h a n i c a l v i b r a t i o n s [40, 41, 51] o r f o u r p o l e - p a r a m e t e r s o f e l e c t r i c a l e n g i n e e r i n g , i s p r e s e n t e d f o r d e t e r m i n i n g t h e f r e q u e n c y r e s -ponse and r e s o n a t i n g c h a r a c t e r i s t i c s o f p i p i n g systems w i t h v a r i o u s boun-d a r y c o n d i t i o n s and h a v i n g one o r more f o r c i n g f u n c t i o n s . The method p r e s e n t e d h e r e i n i s s u i t a b l e f o r b o t h hand and d i g i t a l c o m p u t a t i o n s . F o r u s i n g t h i s method, t h e r e a d e r s h o u l d have some bac k g r o u n d i n t h e t h e o r y o f complex v a r i a b l e s , and an e l e m e n t a r y knowledge o f t h e o r d i n a r y m a t r i x o p e r a t i o n s . T e r m i n o l o g y i s f i r s t i n t r o d u c e d ; t h e n the b l o c k diagrams a r e d i s c u s s e d . T h i s i s f o l l o w e d by a d e r i v a t i o n o f f i e l d m a t r i c e s f o r a s i m p l e p i p e , f o r a p i p e h a v i n g v a r i a b l e c h a r a c t e r i s t i c s , and f o r a system o f p a r a l l e l p i p e s . T h e r e a f t e r t h e p o i n t m a t r i c e s f o r v a l v e s , o r i f i c e s , s e r i e s c o n n e c t i o n s , and the j u n c t i o n o f a b r a n c h and a main a r e p r e s e n t e d . F o r d e t e r m i n i n g t h e r e s o n a t i n g f r e q u e n c i e s , a n u m e r i c a l p r o c e d u r e f o r complex o r s i m p l e systems and e x p r e s s i o n s f o r s i m p l e systems a r e p r e s e n t e d i n C h a p t e r 4. Then equa-t i o n s a r e d e v e l o p e d f o r f r e q u e n c y r e s p o n s e o f p i p i n g systems h a v i n g v a r i o u s f o r c i n g f u n c t i o n s , e.g. f l u c t u a t i n g d i s c h a r g e , f l u c t u a t i n g p r e s s u r e and o s c i l l a t i n g v a l v e . T h i s i s f o l l o w e d by a d e r i v a t i o n o f e q u a t i o n s f o r d e t e r m i n i n g t h e l o c a t i o n o f p r e s s u r e nodes and a n t i n o d e s . The v a l i d i t y o f the method i s d e m o n s t r a t e d by comparing t h e r e s u l t s w i t h t h o s e o b t a i n e d by e x p e r i m e n t , t h e method o f c h a r a c t e r i s t i c s , t h e impedance method and energy c o n c e p t s . C H A P T E R T W O TERMINOLOGY AND BLOCK DIAGRAMS In t h i s chapter, the terms f r e q u e n t l y used i n t h i s t h e s i s are def i n e d , block diagrams are introduced and the advantages of usin g these diagrams are discussed. 2.1 TERMINOLOGY The terminology e s t a b l i s h e d by Camichel et a l . [8] and l a t e r used by Jaeger [24 - 29] and Wylie [58 - 60] i s followed h e r e i n . STEADY-OSCILLATORY FLOW A flow i n which a permanent regime i s e s t a b l i s h e d such that the con-d i t i o n s at a p o i n t , e.g., pressure, discharge, are p e r i o d i c f u n c t i o n s of time i s c a l l e d s t e a d y - o s c i l l a t o r y flow. In the theory of v i b r a t i o n s , steady s t a t e o s c i l l a t i o n s r e f e r to o s c i l l a t i o n s of constant amplitude. However, the term s t e a d y - o s c i l l a t o r y i s used h e r e i n to avoid confusion w i t h steady flow i n which c o n d i t i o n s at a p o i n t are constant w i t h respect to time. INSTANTANEOUS AND MEAN DISCHARGE AND PRESSURE HEAD In a s t e a d y - o s c i l l a t o r y flow, the instantaneous discharge, Q, and the instantaneous pressure head, H, can be d i v i d e d i n t o two p a r t s : Q = Q 0 + q* (2.1) H = H + h* (2.2) i n which Q q = average, or mean, discharge; q* = discharge d e v i a t i o n from the mean, (see F i g . 2.1); H q = average, or mean, pressure head; and h* = pressure head d e v i a t i o n from the mean. Both h* and q* are fu n c t i o n s of time, t , and d i s t a n c e , x. I t i s assumed that h* and q* are s i n u s o i d a l i n time which, i n p r a c t i c e , i s o f t e n true or a s a t i s f a c t o r y approximation [1, 26, 28, 32]. Hence, by using complex al g e b r a , one can w r i t e q* = Re [q(x) e^ w t ] (2.3) h* = Re [h(x) ei w t ] (2.4) i n which OJ = frequency, i n r a d / s e c , j = /^T; h and q are complex v a r i a b l e and are f u n c t i o n s of x only; and "Re" stands f o r r e a l p a r t of the complex v a r i a b l e . THEORETICAL PERIOD For a s e r i e s p i p i n g system n A. T = 4 .1 — (2.5) t h i = l a. v l and 2TT th (2.6) i n which T ^ = t h e o r e t i c a l p e r i o d , i n sec; co^ = t h e o r e t i c a l frequency, i n rad/sec; n = number of pip e s ; and a = c e l e r i t y of water hammer wave. The "th s u b s c r i p t i denotes q u a n t i t i e s f o r the i p i p e . In a branch system, Z^/a i s c a l c u l a t e d along the main p i p e l i n e . RESONANT FREQUENCY The frequency corresponding to the fundamental or one of the higher harmonics of the system i s c a l l e d the resonant frequency. STATE VECTORS The q u a n t i t i e s of i n t e r e s t at a s e c t i o n , i , of a p i p e l i n e are h and 13. Fig. 2 I Instantaneous and mean d i s c h a r g e . 14. q which can be combined i n ma t r i x n o t a t i o n as The column vec to r z . i s known as the s t a t e v e c t o r at the s e c t i o n i . The —l s t a t e v e c t o r s j u s t to the l e f t and to the r i g h t o f a s e c t i o n are des igna ted L by the s u p e r s c r i p t s "L" and " R " r e s p e c t i v e l y . For example, z_^  r e f e r s to "th the s t a t e v e c t o r j u s t to the l e f t of the i s e c t i o n ( F i g . 2.2). To combine the mat r i x terms i n some cases (see S e c t i o n 3.2-2) the s t a t e v e c t o r i s de f i ned as z • = (2.8) Because of the a d d i t i o n a l element w i t h u n i t v a l u e , the column vec to r z ' i s — i c a l l e d the extended s t a t e v e c t o r . A prime i s used to des igna te an extended s t a t e v e c t o r . TRANSFER MATRICES A ma t r i x r e l a t i n g two s t a t e vec to r s i s c a l l e d a t r a n s f e r m a t r i x . The upper case l e t t e r s F, P, and U are used to des igna te t r a n s f e r m a t r i c e s ; the cor respond ing lower case l e t t e r s w i t h double s u b s c r i p t s r e f e r to the e l e -ments o f the m a t r i x : the f i r s t s u b s c r i p t rep resen ts the row, and the second, the column of the e lement. For example, the element i n the second row and the f i r s t column of the ma t r i x U i s rep resen ted by u 2 i . T rans fe r ma t r i ces are of th ree t ypes : (1) F i e l d t r a n s f e r m a t r i x , o r f i e l d m a t r i x , F . Th is ma t r i x r e l a t e s the s t a t e v e c t o r s at two ad jacent s e c t i o n s . For example i n F i g . 2.2, L R z . = F. z. (2.9) —l + 1 l —l K J i n which F^ = f i e l d ma t r i x f o r the i 1"* 1 p i p e . (2) Po in t t r a n s f e r m a t r i x , or Po in t m a t r i x , P. The s t a t e v e c t o r s j u s t to the l e f t and to the r i g h t of a d i s c o n t i n u i t y , such as at a s e r i e s connection ( F i g . 2.3) or at a v a l v e , are r e l a t e d by t h i s matrix. The type of the d i s c o n t i n u i t y i s designated by s p e c i f y i n g a s u b s c r i p t w i t h the l e t t e r , "P". For example, i n F i g . 2.3 . z?, = P z L , (2.10) —x+1 sc —L+l i n which P„ i s the p o i n t m a t r i x f o r a s e r i e s connection, sc v (3) O v e r a l l t r a n s f e r m a t r i x , U. This matrix r e l a t e s the s t a t e v e c t o r at one end of a system, or a si d e branch, to that at the other end. For example i f n+l i s the l a s t s e c t i o n , then i n = u i i ( 2 - 1 1 ) i n which U = o v e r a l l t r a n s f e r matrix. This i s obtained by an ordered m u l t i p l i c a t i o n of a l l the intermediate f i e l d and p o i n t matrices. 2.2 BLOCK DIAGRAMS A block diagram i s a schematic r e p r e s e n t a t i o n of a system i n which each component, or a combination of components, o f i ' t h e system i s repres-ented by a "black box". The box rep r e s e n t i n g a p i p e l i n e of constant c r o s s -s e c t i o n a l area, w a l l t h i c k n e s s , and w a l l m a t e r i a l i s c h a r a c t e r i z e d by a f i e l d m a t r i x , w h i l e that r e p r e s e n t i n g a d i s c o n t i n u i t y i n the system geo-metry, by a poi n t matrix. The block diagram f o r a system can be s i m p l i f i e d by representing a block of i n d i v i d u a l boxes by a s i n g l e box. This procedure i s i l l u s t r a t e d i n the f o l l o w i n g s e c t i o n s by a number of t y p i c a l examples. A s e c t i o n on a block diagram i s shown by a small c i r c l e on the l i n e j o i n i n g the two boxes. The number of the s e c t i o n i s w r i t t e n below the c i r c l e and the l e f t and r i g h t hand s i d e of the s e c t i o n are designated by w r i t i n g the l e t t e r s "L" and "R" above the c i r c l e . For example, i n F i g . 2.4, 16. + X L R P i p e i 11 i + 1 Fig. 2 2 S i n g l e p ipe l ine . P i p e i L R P i p e i + 1 i + 1 F i g . 2-3 Ser ies connect ion . 17. F i g . 2 - 4 B lock d iagram 18. i and 1+1 denote the number of the s e c t i o n s ; and "L" and "R", l e f t and r i g h t hand of the s e c t i o n . In the case of a branch p i p e , the number of the s e c t i o n i s w r i t t e n to the r i g h t of the c i r c l e and the l e f t and r i g h t hand secti o n s are i d e n t i f i e d by w r i t i n g the l e t t e r s "BL" and "BR" to the l e f t of the c i r c l e (see F i g . 3.5). The block diagrams are of great help f o r : ( i ) the concise and o r d e r l y f o r m u l a t i o n and a n a l y s i s of problems i n v o l v i n g complex systems; ( i i ) an easy understanding of the i n t e r a c t i o n of d i f f e r e n t parts of the system; and ( i i i ) determining the sequence of m u l t i p l i c a t i o n of t r a n s f e r matrices w h i l e doing the c a l c u l a t i o n s by hand or w r i t i n g a computer program. C H A P T E R T H R E E DERIVATION OF TRANSFER MATRICES To analyze the s t e a d y - o s c i l l a t o r y flows and to determine the reson-a t i n g c h a r a c t e r i s t i c s of a p i p i n g system by the method presented h e r e i n , i t i s necessary that the t r a n s f e r matrices of the elements of the system be known. In t h i s chapter f i e l d matrices f o r a simple p i p e l i n e and f o r a system of p a r a l l e l loops are de r i v e d . A numerical procedure i s pres-ented to determine the f i e l d m atrix f o r a pipe having v a r i a b l e character-i s t i c s along i t s length. Then poi n t matrices f o r a s e r i e s connection, f o r v a lves and o r i f i c e s , and f o r the j u n c t i o n of a branch (branch having v a r i o u s boundary cond i t i o n s ) and the main are developed. A l l the t r a n s f e r matrices derived i n t h i s chapter are summarized i n Appendix A f o r easy reference. 3.1 FIELD MATRICES 2 1. SINGLE PIPELINE The f i e l d matrix f o r a p i p e l i n e having a constant c r o s s - s e c t i o n a l 2 Most of the m a t e r i a l presented i n Secti o n 3.1-1, 3.1-3, and 3.2 has been reported by the w r i t e r i n two papers [11, 12]. To f a c i l i t a t e c r o s s -reference, the n o t a t i o n used h e r e i n i s the same as that used i n these papers except as otherwise i n d i c a t e d by a footnote. 20. area, constant w a l l thickness and the same w a l l m a t e r i a l i s derived i n t h i s s e c t i o n . In the d e r i v a t i o n , the system i s considered as d i s t r i b u t e d and the f r i c t i o n l o s s term i s l i n e a r i z e d . Flow through pipes i s governed by the f o l l o w i n g two equations which i n a s i m p l i f i e d form [50] are: C o n t i n u i t y equation: Q + = 0 (3.1) a Momentum equation: H + -jL Q + -^L. =o (3.2) X § A 1 2gDAn i n which A = the c r o s s - s e c t i o n a l area of the p i p e l i n e ; g = a c c e l e r a t i o n due to g r a v i t y ; D = the i n s i d e diameter of the p i p e l i n e ; f = Darcy-Weisbach f r i c t i o n f a c t o r ; n = exponent of v e l o c i t y i n the f r i c t i o n l o s s term; x = distance along the p i p e l i n e measured p o s i t i v e i n the downstream d i r e c t i o n (see F i g . 2.2); and t = time. The s u b s c r i p t s x and t denote p a r t i a l d i f f e r e n t i a t i o n w i t h respect to x and t . Since (Q ) = (Q ) = (H ) = 0, i t f o l l o w s from Eqs. 2.1 and '2.2 that o x o t o Q = q* Q. = q* x x n x x t n t H = h* H = (H ) + h* t t X O X X Note that due to f r i c t i o n losses (H ) ^ 0. For t u r b u l e n t flow o x f 0 n (H ) = - r.^o (3.4a) 2gDAn and f o r laminar flow (H ) = - 3 2 v Q n (3.4b) gAD 2 i n which y = kinematic v i s c o s i t y of the f l u i d . I f q* << Q , then Q n = CQn + q * ) n « Q n + n Q n _ 1 q* (3.5) ° o o i n which higher order terms are neglected. (3.3) 21. I t f o l l o w s from Eqs. 3.1 through 3.5 that q* + St. h* = 0 (3.6) n X o t h* + —- q* + R q* = 0 (3.7) x gA n t n i n which R = (n f Q n _ 1 ) / ( 2 g D A n ) f o r tu r b u l e n t flow and R = (32 v)/(gAD 2) o f o r laminar flow. The f i e l d matrix f o r a pipe can be derived by e i t h e r of the f o l l o w i n g two methods (both the methods give i d e n t i c a l f i e l d m a t r i x ) : (a) By using the separation of v a r i a b l e technique: In t h i s method, Eqs. 3.6 and 3.7 are solved by the technique of sep-a r a t i o n of v a r i a b l e s [57] assuming a s i n u s o i d a l s o l u t i o n i n time and sub-s t i t u t i n g the boundary c o n d i t i o n s at the i ^ , and at the ( i + l ) ^ s e c t i o n . The e l i m i n a t i o n of h* from Eqs. 3.6 and 3.7 y i e l d s «xx= - T < t t + M ! «t a a Now, i f i t i s assumed that the v a r i a t i o n of q* i s s i n u s o i d a l 'with respect to t , then on the b a s i s of Eq. 2.3, Eq. 3.8 takes the form *?S = (_ a i + J§ A " R ) q (3.9) dx 2 a 2 a 2 or ^3. - y 2 q = o (3.10 3) d x 2 i n which y 2 = - o j 2 / a 2 + jgA OJ R/a 2 (3.11) The s o l u t i o n of Eq. 3.10 i s q = ci s i n h px + c 2 cosh yx (3.12) i n which c i and c 2 are a r b i t r a r y constants. 3 In Ref. 12, y 2 = - U ) 2 ^ 2 / a 2 + j gAoi£ 2 R/a 2. 22. I f h* i s a l s o assumed s i n u s o i d a l i n t , then by s u b s t i t u t i n g Eqs, 3.12 and 2.4 i n t o Eq. 3.6 and s o l v i n g f o r h, one obtains h = - —%—^- f c i cosh yx + c 0 sinh yx) th (3.13) th The f i e l d matrix r e l a t i n g the s t a t e vectors at the i and at the (i+1) s e c t i o n on the i ^ reach (see F i g . 2.2) of length £^ i s to be d e r i v e d . I t th R R . i s known that at the i s e c t i o n (at x = 0), h = I K and q = q^. Hence, i t f o l l o w s from Eqs. 3.11 and 3.13 that JgA. OJ and Cn = C2 R a? y. l l (3.14) th L L In a d d i t i o n , at the (i+1) s e c t i o n (at x = £^), h = h^ +^ and q = The s u b s t i t u t i o n of these values of h and q, and C j and c 2 from Eq. 3.14 i n t o Eqs. 3.12 and 3.13 y i e l d s L L R R = (cosh V i &/) q i - ( 1 / Z c ) ( s i n h P i i/) R R h i + 1 = -Z c ( s i n h y.. g,/) q± + (cosh v ± l/) h± (3.15) i n which c h a r a c t e r i s t i c impedance [50] f o r the pi p e , Z^ = (y^ a ? ) / ( j OJ gA ) These equations can be expressed i n matrix n o t a t i o n as i+1 cosh y. £ . l l - Z s i n h y. £. c 1 1 (1/Z ) s i n h y . £ . c 1 1 cosh u. I. l l (3.16) or z. . = F. z. — l + l l —I th Hence, f i e l d matrix f o r the i pipe i s F. = I cosh V l ± £ i Z s i n h y. £. c 1 1 (1/Z ) s i n h y. £. cJ 1 1 cosh u. £. l l (3.17) 23. I f f r i c t i o n i s neglected, i . e . , = 0, then F^ becomes cos b. <JJ 1 TT s i n b. a) C. I I j C. s i n b. w cos b. w J l l l (3.18) i n which b. = &./a. and C. = a./(gA.). Note that b. and C. are constants i i i I I ^ 6 iJ I I f o r a pipe and are not f u n c t i o n s of a>, and that i s the c h a r a c t e r i s t i c th impedance [58, 59] f o r the i pipe i f f r i c t i o n i s neglected, (b) By using the Cayley-Hamilton Theorem S u b s t i t u t i o n of Eqs. 2.3 and 2.4 i n t o Eqs. 3.6 and 3.7 y i e l d s d3_ + 3§ A <" Y dx dh dx 0 + ( R + I J £ } q = Q Eqs. 3.19 and 3.20 can be w r i t t e n i n matrix form as dz_ 3— = Bz dx — i n which z_ i s the column ve c t o r as given by Eq. 2.7 and f0 -jgAw/a 2' (3.19) (3.20) (3.21) B (R + ) gA 0 (3.22) The s o l u t i o n of the set of d i f f e r e n t i a l equations, Eq. 3.21, i s given by (3.23) e z = F z -o —o i n which z = i n i t i a l s t a t e v e c t o r at x = 0: and F = f i e l d m a trix. F can —o ' be expanded i n an i n f i n i t e s e r i e s as Bx 1 F(x) = e = I + Bx + — (Bx) + (3.24a) i n which I = i d e n t i t y or u n i t matrix. I t f o l l o w s from the above equation that the t r a n s f e r matrix f o r a pipe of length H i s (3.24b) The c h a r a c t e r i s t i c equation f o r the matrix B& i s given by the determinant F = e B £ = I + BA + — (B£) 2 + jgA co£ 24. R + 3 a gA which upon s i m p l i f i c a t i o n becomes a" , 2 .( <LJL ) 2 + J g A <*> & s , 2 (3.25) Hence X x ,A2 = ± y£ i n which y i s as defined i n Eq. 3.11. Since B£ i s a 2x2 square matrix, Eq. 3.24b can be w r i t t e n [50] as B& F = e = c q l + c uB£ (3.26) i n which C 3 and c^ are constants. By s u b s t i t u t i n g the eigenvalues ^1 and A 2 f o r B£ i n t o the above equation, and s o l v i n g f o r c 3 and c^, one obtains ( e y £ + e " y £ ) / 2 = cosh y£ (3.27) Cu = 1 (e y£ e"^) si n h y£ 2y£ ^ 1 y£ By s u b s t i t u t i n g these values of C 3 and c^ i n t o Eq. 3.26 and s i m p l i f y i n g the r e s u l t i n g equation y i e l d the f i e l d matrix f o r the i ^ pipe as cosh y. %. 1 1 j g "> Aj y 2 *. a. 1 1 sinh y. I. -:' 1 1 — (R. + J - r ^ ) s i n h y. I. cosh y. I. y- 1 gA. J 1 1 1 1 (3.28) I t can be seen that the elements of matrix F^ i n Eq. 3.17 and 3.28 are i d e n t i c a l except f 2 i - By m u l t i p l y i n g f 2 l i n Eq. 3.17 by y^ i n the numerator and denominator and then s u b s t i t u t i n g f o r y^ from Eq. 3.11, i t can be shown that the element f 2 1 i n Eq. 3.17 and 3.28 are als o i d e n t i c a l . I f o)£^ /a^  << 1 then the system may be analyzed as a lumped system. 25. In t h i s case, f o r a f r i c t i o n l e s s system, f i e l d matrix becomes F. = 1 a? 1 2 . i w l 1A-(3.29) and 'X; which f o l l o w s from Eq. 3.18 si n c e cos (w^/a^) ^ 1 s i n (w&./a.) ^ u)£./a. f o r small values of u)£./a.. l l i i i i While doing the a n a l y s i s one f i r s t c a l c u l a t e s the elements of the f i e l d matrix f o r each pipe. A comparison of the f i e l d matrices of Eqs. 3.18 and 3.29 shows that t h i s i d e a l i z a t i o n of a d i s t r i b u t e d system does not r e s u l t i n much s i m p l i f i c a t i o n of the computations. 2. PIPE HAVING VARIABLE CHARACTERISTICS ALONG ITS LENGTH The c o n t i n u i t y , and momentum equation f o r the flow i n a p i p e l i n e having v a r i a b l e c h a r a c t e r i s t i c s , e.g., A, a, w a l l t h i c k n e s s , w a l l m a t e r i a l , along i t s length are [39], Q + l A W H = 0  X a 2 ( x ) 1 X gA(x) V t (3.30a) i n which A(x) and a 2 ( x ) denote that A and a 2 are fu n c t i o n s of x. In these equations n o n l i n e a r terms of higher order and f r i c t i o n are neglected. By s u b s t i t u t i n g Eqs. 2.1 through 2.4 i n t o the above equations and s i m p l i f y i n g , one obtains + jgADOu) h = 0 h + a 2 ( x ) 3 " (3.30b) 0 x gA(x) q These equations can be expressed i n the matrix n o t a t i o n as dz dx Bz (3.31) 26. i i+1 A = A ( x ) a = a ( x ) F i g . 3 I P ipe hav ing var iab le c h a r a c t e r i s t i c s along its length. F i g . 3 2 A c t u a l and S u b s t i t u t e p i p e 27. i n which z i s the column ve c t o r as defined i n Eq. 2.7 and 0 JgA(x) co a 2 ( x ) B = (3.32) 3 a gAlxJ 0 Since the elements of the matrix B are f u n c t i o n s of x, the procedure out-l i n e d i n the l a s t s e c t i o n to determine the f i e l d m atrix f o r the pipe f a i l s . To analyze such cases, recourse i s made to e i t h e r of the f o l l o w i n g pro-cedures : ( i ) The a c t u a l p i p e l i n e i s replaced by a s u b s t i t u t e p i p e l i n e having piecewise constant elements (see F i g . 3.2), and the system i s analyzed by u s i n g the f i e l d matrices derived i n the l a s t s e c t i o n . This gives s a t i s -f a c t o r y r e s u l t s at low frequencies (see Chapter 5). ( i i ) A numerical procedure i s adopted to determine the elements of the f i e l d matrix. The determination of the f i e l d matrix f o r a p i p e l i n e having v a r i a b l e c h a r a c t e r i s t i c s i s equivalent to i n t e g r a t i n g the d i f f e r e n t i a l equation, Eq. 3.31. This may be done by the Runge-Kutta method as f o l l o w s . The p i p e l i n e i s d i v i d e d i n t o n reaches as shown i n F i g . 3.1. F i r s t the f i e l d matrix f o r each reach i s computed and then the f i e l d matrix f o r the p i p e l i n e i s determined by m u l t i p l y i n g these matrices i n a proper sequence. I f the length of the reach between the s e c t i o n s i and i+1 i s s, then the fourth-order Runge-Kutta method gives [40] z. -* — l + l = z. -1 + F Qio + 2 h + 2*2 + *3> (3.33) i n which 28. (3.34) ho = s ^ l i k x = s B(x. +s/2)(z. + k^/2) k 2 = s B ( x i + s / 2 ) ( z . + kj/2) k 3 = s B ( x . + 1 ) ( z . + k 2) i n which B ( x ^ ) , B ( x ^ + ^ ) , and B(x^+s/2) are, r e s p e c t i v e l y , the values of the matrix B(x) at s e c t i o n i , at i+1, and at the s e c t i o n between i and i+1. By s u b s t i t u t i n g Eqs. 3.34 i n t o Eq. 3.33, one obtains z. = F z. (3.35) — l + l vc—1 i n which the f i e l d matrix f o r a pipe having v a r i a b l e c h a r a c t e r i s t i c s along i t s length i s F y c = I + J [B(x.) + 4 B(x.+s/2) + B ( x . + 1 ) ] + | 2[B(x.+s/2)B(x.) + B(x. + 1)B(x.+s/2) + B 2(x.+s/2)] + 17 [B 2(x.+s/2)B(x.) + B ( x . + 1 ) B 2 ( x . + s / 2 ) ] + [ B ( x i + 1 ) B 2 ( x . + s / 2 ) B ( x . ) ] (3.36) 3. PARALLEL SYSTEM Let there be n loops i n p a r a l l e l ( F i g . 3.3) whose o v e r a l l t r a n s f e r matrices are (m) = F W P W < e i > p(m) (m) = n' n' 2' 1' ' ' m m m m The s u p e r s c r i p t i n the parenthesis r e f e r s to the number of the loop. The matrix r e l a t e s the s t a t e v e c t o r s at the l | S t and at the (n'+l) t' 1 m m s e c t i o n of the m loop (see F i g . 3.3b), i . e . , (m)L = (m) (m)R -n'+l —1' m m A prime on the s u b s c r i p t denotes a s e c t i o n on the p a r a l l e l loop. The elements of the f i e l d matrix, F , f o r the p a r a l l e l loops r e l a t i n g R L the s t a t e vectors and ( F i g . 3.3b) can be determined from the f o l l o w i n g equations [33]: ( a ) Piping s y s t e m . F i g . 3-3 Para l le l s y s t e m p 2' t \ (2) P ( 2 ) Jn) (n) r p2' o < 2 ) n 2 ° 2 (n) (n) nn R 'n-l (n-l) U R (n) In U nn-l L -o-n, ( b ) B l o c k d i a g r a m . Fig . 3 -3 P a r a l l e l s y s t e m 31. f i 2 = (Cc/n) - n f 2 i = l / n •22 i n which (3.37) n- Cm) , „(m) m=l •5 = u l T / u 2 1 5 Cm) , (m) ? = mil U 2 2 / u21 (3.38) In d e r i v i n g the above expressions use has been made of the r e l a t i o n that U Cm) = 1 m=l,2,...,n i . e . , (m) (m) (m) (m) 0 ,_ u l l u 2 2 " u 1 2 u 2 1 = 1 m=l,2,...,n (3.39) Note that Eq. 3.37 i s v a l i d only i f the elements of the o v e r a l l t r a n s f e r matrix f o r each p a r a l l e l loojj; s a t i s f y Eq. 3.39. I t i s known from the theory"of matrices t h a t , f o r square ma t r i c e s , the determinant of the prod-uct of matrices i s equal to product of the determinants of matrices. 1, k = 1,2,.., m = 1,2, n, then |U,'"V| = 1 . I t i s c l e a r from Eqs. 3.17 and 3.18 that | F | = 1 . Furthermore, the determinants of the p o i n t matrices f o r s e r i e s connections, and f o r valves and o r i f i c e s are a l s o u n i t y (see the p o i n t matrices derived i n the f o l l o w i n g s e c t i o n s ) . Thus i f there i s a discon-t i n u i t y other than a s e r i e s connection, v a l v e , or o r i f i c e i n any of the p a r a l l e l loops the determinant of the poin t matrix f o r the d i s c o n t i n u i t y should be checked to ensure i t has u n i t value before using Eq. 3.37. k Hence i f | P1V'"-' | = 1, k = 2,3, ...n and |F m (m) k .n , f o r m' ,Cm) 32. 3.2 POINT MATRICES At a s e c t i o n where there i s a d i s c o n t i n u i t y i n the geometry of the system, e.g., s e r i e s connection, o r i f i c e , v a l v e , branch connection, one has to de r i v e a poi n t matrix r e l a t i n g the s t a t e v e c t o r to the l e f t of the d i s c o n t i n u i t y w i t h that to the r i g h t . This p o i n t matrix i s r e q u i r e d i n the c a l c u l a t i o n of the o v e r a l l t r a n s f e r matrix f o r the system which i s then used to determine the resonant frequencies and/or frequency response of the system. Poi n t matrices f o r various boundary c o n d i t i o n s u s u a l l y found i n hydro-power, and i n water supply schemes are derived i n the f o l l o w i n g s e c t i o n s . 1. SERIES CONNECTION A j u n c t i o n of two pipes having d i f f e r e n t diameters (see F i g . 2.3), d i f f e r e n t w a l l t h i c k n e s s , d i f f e r e n t w a l l m a t e r i a l , or any combination of these v a r i a b l e s i s c a l l e d a s e r i e s connection. I t f o l l o w s from the c o n t i n u i t y equation that q R = q\ (3.40) In a d d i t i o n h R = h L (3.41) i i i f losses at the j u n c t i o n are neglected. These two equations can be ex-pressed 'in matrix n o t a t i o n as z R = P z L (3.42) —1 sc—1 i n which the p o i n t matrix f o r the s e r i e s connection i s ri o] p sc 0 (3.43) Since P s c i s a u n i t m a t r i x , i t can be incorporated i n t o the f i e l d matrix while doing the c a l c u l a t i o n s . 33. 2. VALVES AND ORIFICES The po i n t matrices f o r valves and o r i f i c e s are der i v e d by l i n e a r i z i n g the gate equation. As w i l l be c l e a r a f t e r completing the d e r i v a t i o n , t h i s l i n e a r i z a t i o n does not introduce serious e r r o r s i f the pressure r i s e at the v a l v e i s much l e s s than the s t a t i c head. For an o s c i l l a t i n g v a l v e , a s i n u s o i d a l v a l v e motion i s assumed. I t i s p o s s i b l e , however, to analyze non- s i n u s o i d a l p e r i o d i c valve motions by t h i s method. The p e r i o d i c motion i s reduced to a set of harmonics by F o u r i e r a n a l y s i s [57] , and the system response i s determined f o r each harmonic. Then the i n d i v i d u a l responses are superimposed to determine the t o t a l response f o r the given v a l v e motion (since a l l the equations are l i n e a r , the p r i n c i p l e of s u p e r p o s i t i o n can be a p p l i e d ) . (a) O s c i l l a t i n g v a l v e d i s c h a r g i n g i n t o atmosphere The instantaneous and mean discharge through a valve ( F i g . 3.4a) are given by the equations Q L . = C,A (2gH L J 1 / 2 (3.44) x n + l d v ^ 5 n+l J ..  v ' Q = (C.A ) (2gH ) 1 / 2 (3.45) o di v o 6 o i n which C^ = c o e f f i c i e n t of discharge; and A y = area of the val v e opening. D i v i s i o n of Eq. 3.44 by Eq. 3.45 y i e l d s r„L n+l H o 1/2 (3.46) i n which the instantaneous r e l a t i v e gate opening, T = (C^A )/(C^Ay) g; and the mean r e l a t i v e gate opening, T q = (C^A^) /(C^A^) . The s u b s c r i p t " s " denotes steady s t a t e r e f e r e n c e , or index, values. The r e l a t i v e gate opening may be considered to be made up of two p a r t s , i . e . , T = x + T * (3.47) o 34. i n which x * = d e v i a t i o n of the r e l a t i v e gate opening from the mean ( F i g . 3.4b). S u b s t i t u t i o n of Eqs. 2.1, 2.2, and 3.47 i n t o Eq. 3.46 y i e l d s 1 + n n + l (1 + T — ) (1 T o h n ; i \ / 2 (3.48) O O O I f the valve motion i s assumed s i n u s o i d a l , then x* = Re(ke ; i w *) (3.49) i n which k = amplitude of valve motion. The phase angle between any other f o r c i n g f u n c t i o n i n the system and the o s c i l l a t i n g valve can be taken i n t o c o n s i d e r a t i o n by making k a complex number; otherwise k i s r e a l . By expanding Eq. 3.48, n e g l e c t i n g terms of higher order ( t h i s i s v a l i d only i f I h * ^ ! << H q ) , and s u b s t i t u t i n g Eqs. 2.3, 2.4, and 3.49 i n t o the r e s u l t i n g equation, one obtains 2H 2H k o n+1 R (3.50) Since h , = 0, on the b a s i s of Eq. 3.50, one can w r i t e n+1 ' n > 2H R , L o i , =- h , + n+1 n+1 *n+l o "<o In a d d i t i o n , from the c o n t i n u i t y equation i t f o l l o w s that R ln+l L q n + l (3.51a) (3.51b) Eqs. 3.51a and 3.51b may be expressed i n matrix n o t a t i o n as q R '1 o' q L ' 0 ' h n+1 2H o 1 V 1, • h n+1 2H k o The two matrix terms on the r i g h t hand s i d e may be combined as f o l l o w s L R r q <h = i n+1 1 2H 0 0 2H k f N q 1 0 - h T 0 0 1 . , i . (3.53) n+1 Valve P i p e n ( a ) V a l v e at d o w n s t r e a m end of p i p e l i n e . ( b ) S i n u s o i d a l v a l v e m o t i o n . i—O r i f i c e L R P i p e i w J_LL_ Z ipe i+1 + 1 ( c ) Cri t ic© a t i n t e r m e d i a t e s e c t i o n F i g . 3-4 V a l v e s 36, Note that expansion of Eq. 3.53 y i e l d s Eqs. 3.51a, 3.51b, and 1 = 1 . Thus the a d d i t i o n a l element one i n the column ve c t o r aids i n w r i t i n g the r i g h t hand side of Eq. 3.52 i n a compact form. As defined i n Chapter 2, Eq. 2.8, the column v e c t o r w i t h one as an a d d i t i o n a l element i s c a l l e d an extended s t a t e v e c t o r , z_' . The extended s t a t e vectors and extended t r a n s f e r matrices are denoted by a prime. On the b a s i s of Eq. 2.8, Eq. 3.53 may be w r i t t e n as z i R = p t Z . L —n+1 ov —n+1 (3.54) i n which P' = the extended p o i n t matrix f o r an o s c i l l a t i n g valve and i s ov r b given by P» = ov 1 2H 0 1 0 2H k' o To 1 ; (3.55) (b) Valve having constant gate opening d i s c h a r g i n g i n t o atmosphere In t h i s case, k = 0. Hence, Eq. 3.52 takes the form (1 n+1 0) 2H • > h (3.56) n+1 or R D L z , = P z ., —n+1 v -^n+1 (3.57) i n which P^ = the p o i n t matrix f o r a valve or o r i f i c e d i s c h a r g i n g i n t o atmosphere, and i s given by (1 0) v 2H (3.58) Note that P i s not an extended p o i n t m a t r i x , v r I f a valve of constant gate opening, or an o r i f i c e , i s at an i n t e r -mediate s e c t i o n ( F i g . 3.4c) then Eq. 3.58 becomes 37. vi 2 A H (3.59) i n which A H q = the mean head l o s s across the valve corresponding to the mean discharge, Q . 3. BRANCH PIPELINES In the p i p i n g systems shown i n F i g . 3.5a, p i p e l i n e abc i s the main and bd, the side branch. The t r a n s f e r matrix f o r the p i p e l i n e ab can be computed by usin g the f i e l d and p o i n t matrices derived above. To c a l c u l -ate the o v e r a l l t r a n s f e r matrix f o r abc, the p o i n t matrix at the j u n c t i o n b, r e l a t i n g the s t a t e vectors to the l e f t and to the r i g h t of the j u n c t i o n , must be known. This matrix can be obtained i f the boundary c o n d i t i o n s at po i n t d are s p e c i f i e d . Point matrices f o r the j u n c t i o n of the main and the branch having v a r i o u s boundary c o n d i t i o n s are der i v e d i n t h i s s e c t i o n . Let C f be the o v e r a l l t r a n s f e r matrix f o r the branch ( r e f e r to F i g . 3.5b), i . e . , (3.60) ^L -n+l U i L l or L a. > (-MR q q u n U12 ft = n+l <\, ( u21 'X, a. 'Xi <\> 'Xi <\J (3.61) i n which U = F P . n n . P 3 F 2 P 2 F i . The q u a n t i t i e s r e l a t i n g to the branch are designated by a t i l d e ' 4 . Expansion of Eq. 3.61 y i e l d s ^L < V l a, ^ £ft u n q i + ui 2 h . i (3.62) In Re. 12, s u p e r s c r i p t B i s used to designate q u a n t i t i e s r e l a t i n g to the branch. (a) Pi ping system . F i g . 3 - 5 B r a n c h s y s t e m ( b ) B l o c k d i a g r a m . F i g . 3 - 5 B r a n c h s y s t e m 40. n+1 u 2 i q i + u 2 2 h 1 (3.63) I f the flow d i r e c t i o n i s assumed p o s i t i v e as shown i n F i g . 3.5a and the losses at the j u n c t i o n are neglected, the f o l l o w i n g equations can be w r i t t e n L h L I R <\,R qj- + q i h. = h n (3.64) (3.65) By s u b s t i t u t i n g appropriate boundary c o n d i t i o n s i n Eqs. 3.62 and 3.63 and making use of Eqs. 3.64 and 3.65, the p o i n t matrices at the j u n c t i o n of the main and the branch can be d e r i v e d . The f o l l o w i n g examples i l l u s -t r a t e the procedure, (a) Dead end branch In t h i s case, q ^ = 0. Hence, i t f o l l o w s from Eqs. 3.62 and 3.65 that ^R q i = 12 S u b s t i t u t i o n of t h i s equation i n t o Eq. 3.64 y i e l d s R L U 1 2 L q. = q. + h. n i n i ^ l (3.66) (3.67) u Eq. 3.65 can be w r i t t e n as h R I n L , L 0 q. + h. n i l (3.68) Eqs. 3.67 and 3.68 can be expressed i n matrix n o t a t i o n as ,hJ R u 1 2 / u u q j > h (3.69) or R N L — l bde — l (3.70) i n which P. bde the p o i n t matrix f o r the s i d e branch w i t h dead end and i s given by bde 'X, 'Xi U 1 2 / U 1 1 (3.71) (b) Branch with constant head r e s e r v o i r ' V L In t h i s case, h n + j = 0. Hence, i t fo l l o w s from Eqs. 3.63 through 3.65 that R L ^ h L 1 'Xi U21 (3.72) Eqs. 3.72 and 3.68 can be expressed i n matrix n o t a t i o n as L q -R f 1 'Xi /Xi - i U 2 2 / U 2 1 • q h 0 1 . (3.73) or R n L z. = P, z. —1 bres —1 (3.74) i n which P, = the p o i n t matrix f o r the branch w i t h constant head r e s e r -bres r v o i r and i s given by <Vi /V. -v 1 U 2 2 / U 2 1 •v - (3.75) bres J 0 1 J (c) Branch having o s c i l l a t i n g valve at the downstream end I f the frequency of the o s c i l l a t i n g v a l v e on the branch i s the same as t h a t of the f o r c i n g f u n c t i o n on the main (these may not be i n phase), the system can be analyzed by using the p o i n t matrix d e r i v e d i n t h i s sec-t i o n . In case the frequencies of the f o r c i n g f u n c t i o n s are not the same, then the system i s analyzed c o n s i d e r i n g each f o r c i n g f u n c t i o n at a time and the r e s u l t s are superimposed to determine the t o t a l response. Since an extended o v e r a l l t r a n s f e r matrix r e l a t i n g the s t a t e v e c t o r at the f i r s t , and at the l a s t s e c t i o n on the branch i s r e q u i r e d , the ex-tended t r a n s f e r matrices w i l l a l s o have to be used f o r the main. Let the extended o v e r a l l t r a n s f e r matrix f o r the branch be 42. rO/ Oj U12 % u 2 1 OJ u 2 2 0 0 IV. (3.76) f u n c t i o n on the branch; i f t h i s i s not the case, then at l e a s t one of the Oj Oj O J O J elements U13, u 2 3 , U31, and u 3 2 i s not equal to zero and the p o i n t matrix derived i n t h i s s e c t i o n should be modified a c c o r d i n g l y . For the branch p i p e l i n e 2'*. = V' z ' \ (3.77) -ii+1 ov —n+l and 2>\ = CV- £ { R (3.78) -n+l — 1 By s u b s t i t u t i n g from Eq. 3.55 and from Eq. 3.78 i n t o Eq. 3.77, and expanding the r e s u l t i n g equation, one obtains OJR O J OJR O J OJR q n + l = U l i q i + U l 2 h l n+l 2fr ,0/ O % . C ^ R ( u 2 1 " u l l ) q i + ( u 2 2 2& 2fir ft-at U 1 2)'h r, + -5-$ ^ To (3.79) (3.80) o^ "o A l l the notations defined i n the previous s e c t i o n s apply except that the - Oj '• • t i l d e ; '(^) r e f e r s to the branch. For example, T q = the mean r e l a t i v e gate opening of the va l v e on the branch. Any phase s h i f t between the va l v e on the branch and the f o r c i n g f u n c t i o n on the main can be taken i n t o consid-Oj Oj e r a t i o n by making k a complex number; otherwise k i s r e a l . o R OjR L Since h , = 0, and h-, = h., i t f o l l o w s from Eq. 3.80 that n+l ' 1 1' M OJR i n which 13 (3.81) 1 2 u 2 2 - 2$ u12/$ Oj O J O J O J u 2 i - 2H oun/Q o (3.82) 43. and "u 'Vi 'VI 2H k / x o o r i o 'Vi 'VI OI , ' V 1 3 u 2 i - 2H un/Q 'VJR By s u b s t i t u t i n g q from Eq. 3.81 i n t o Eq. 3.64, one obtains R L . L q. = q. + p, h. + p, „ Moreover, one can w r i t e 1 = 0 q L + 0 h L + 1 n i l Eqs. 3.84, 3.68, and 3.85'can be expressed i n matrix n o t a t i o n as L q R 1 P 1 2 P i s ' q • h = 0 1 0 • h . i . i 0 0 1 . lj (3.83) (3.84) (3.85) (3.86) or ;' = P I z' -i • bov — A (3.87) i n which Pv' = the t r a n s f e r matrix at the j u n c t i o n of the side branch 'bov J having an o s c i l l a t i n g valve and i s given by bov r 1 P p 12 0 1 0 0 0 1 13 (3.88) I f there i s an o r i f i c e , or a valve having constant gate opening at the downstream end of the branch, then k = 0. Hence, p =0, and the p o i n t 1 3 matrix f o r the branch can be w r i t t e n as f l p. b o r f 12 (3.89) 1° 1 Note that t h i s i s not an extended p o i n t matrix. C H A P T E R F O U R RESONANT FREQUENCIES AND FREQUENCY RESPONSE OF PIPING SYSTEMS This chapter deals w i t h determining the resonant frequencies and f r e -quency response of f r i c t i o n l e s s p i p i n g systems. F i r s t a numerical pro-cedure i s presented to determine the resonant frequencies. Then, the ex-pressions f o r the resonant frequencies of a number of simple systems are given. This i s followed by d e r i v a t i o n of equations f o r determining the frequency response of p i p i n g systems having f l u c t u a t i n g pressure, o s c i l l -a t i n g v a l v e s , and f l u c t u a t i n g discharge as the f o r c i n g f u n c t i o n . 4.1 RESONANT FREQUENCIES In a f r i c t i o n l e s s system having a constant head r e s e r v o i r at the up-stream end and an o s c i l l a t i n g v a l v e at the downstream end, the amplitude of discharge f l u c t u a t i o n at the val v e i s zero during r e s o n a t i n g c o n d i t i o n s at the fundamental or one of the higher odd harmonics. This was observed by Camichel et a l . [ 8 ] , and reported to be true by Jaeger [26, 28, 29]. The frequency response diagrams obtained by t h e o r e t i c a l a n a l y s i s of a number of s e r i e s , p a r a l l e l , and branch systems (a branch system w i t h a side branch having an o r i f i c e or an o s c i l l a t i n g v alve being an exception) done by Wylie [58-60], and by the author [12] confirm t h i s r e s u l t . Expressions f o r the resonant frequencies of simple f r i c t i o n l e s s systems and t h e i r numerical values f o r simple or complex systems can be determined by u s i n g the r e s u l t as f o l l o w s . Let U be the o v e r a l l t r a n s f e r matrix f o r a system having a constant head r e s e r v o i r at the upstream end ( s e c t i o n 1) and an o s c i l l a t i n g v alve at the downstream end ( s e c t i o n n+1), i . e . , z L = U z R (4.1) -n+1 —1 v J By expanding Eq. 4.1 and noting that hj = 0 (constant head r e s e r v o i r ) , and q|j +^ = 0 (discharge f l u c t u a t i o n node) at a resonant frequency, one obtains u u qf = 0 (4.2) R e c a l l that i s the element i n the f i r s t row and the f i r s t column of the matrix U. Since f o r n o n - t r i v i a l s o l u t i o n , q 1 f- 0, t h e r e f o r e U n = 0 (4.3) To determine the resonant f r e q u e n c i e s , the value of u x l i s computed A.'V f o r d i f f e r e n t t r i a l values of to, and the u^^to curve i s p l o t t e d . The p o i n t s of i n t e r s e c t i o n of t h i s curve and the co-axis are the resonant f r e -quencies (see F i g . 5.2). I f the chosen value of oo i s equal to one of the resonant frequency, u l x = 0. This requirement i s not normally met by the f i r s t guess f o r u) and the r e s u l t i n g numerical value of u ^ i s r e f e r r e d to as the r e s i d u a l . The expressions f o r u-^ f o r simple systems are d e r i v e d and then by using Eq. 4.3, equations f o r the resonant frequencies are obtained. The f o l l o w i n g example i l l u s t r a t e s the procedure: For a s e r i e s system having two pipes ( F i g . 2.3) U = F^F-L (4.4) By s u b s t i t u t i n g F 2 and from Eq. 3.18 and P 2 from Eq. 3.43, m u l t i p l y i n g 46. the matrices and using Eq. 4.3, one obtains 2 cos b i u cos b 2 w a 2 D 2. s i n bjo) s i n b 2oj = 0 (4.5a) i n which b-^  and b 2 are constants as defined i n Eq. 3.18. Proceeding s i m i l a r l y , the f o l l o w i n g equation f o r three pipes i n s e r i e s i s obtained. a cos b-j^ oj cos b 2oj cos b 3aj - — D 2 2 s i n b i & s i n b2co cos b 3oj a 2 a l fD 312 cos b^oj s i n b 2oj s i n b 3o) a 3 Pi s i n b 1OJ cos b 2aj s i n b 3oj = 0 (4.5b) S o l u t i o n of Eqs. 4.5 f o r OJ gives the resonant frequencies of the sys-tem. Note that t h i s equation i s only v a l i d f o r a f r i c t i o n l e s s system. By proceeding i n a s i m i l a r manner, equations f o r four or more pipes i n s e r i e s , f o r branch systems, and f o r p a r a l l e l systems can be d e r i v e d . These equations are cumbersome and i t i s b e t t e r to f o l l o w the numerical procedure o u t l i n e d above r a t h e r than to d e r i v e the equations and then solve them. 4.2 FREQUENCY RESPONSE The method presented h e r e i n may be used to determine the frequency response of a system having one or more than one p e r i o d i c f o r c i n g f u n c t i o n . I f a l l the f o r c i n g f u n c t i o n s are s i n u s o i d a l and have the same frequency then the equations derived i n t h i s chapter can be used without any modif-i c a t i o n . However, i f the frequencies are d i f f e r e n t , then the system i s analyzed c o n s i d e r i n g each f o r c i n g f u n c t i o n at a time and the r e s u l t s are superimposed to determine the t o t a l response. Non-harmonic p e r i o d i c f u n c t i o n s are decomposed i n t o d i f f e r e n t harmonics by F o u r i e r a n a l y s i s [57]. By c o n s i d e r i n g each harmonic at i t s p a r t i c u l a r frequency, the system response i s determined. The r e s u l t s are then superimposed to f i n d the t o t a l response. 47. Expressions to determine the frequency response of t y p i c a l p i p i n g systems having three common types of f o r c i n g f u n c t i o n s — f l u c t u a t i n g press-ure head, o s c i l l a t i n g v a l v e , and f l u c t u a t i n g d i s c h a r g e — a r e derived i n the f o l l o w i n g s e c t i o n s . By proceeding i n a s i m i l a r manner, expressions f o r other types of systems can be deri v e d . 1. FLUCTUATING PRESSURE HEAD Consider the system shown i n F i g . 5.3a having a dead end at the r i g h t end. A wave on the surface of the r e s e r v o i r produces pressure o s c i l l a t i o n s i n the system. Due to the wave, the pressure head at s e c t i o n 1 f l u c t u a t e s s i n u s o i d a l l y about the mean pressure head. Let t h i s pressure head v a r i a t i o n be given by h f R = Re [ h R e^ w t ] = K cos CD t = Re [K " t ] (4.6) st and U be the t r a n s f e r matrix r e l a t i n g the s t a t e v e c t o r s at the 1 and , , ., t h (n+1) s e c t i o n , i . e . , z ^ + 1 = U z R (4.7) I t i s assumed that there i s no other f o r c i n g f u n c t i o n i n the system; other-wise the extended t r a n s f e r matrix, U' w i l l have to be used. Expansion of Eq. 4.7 y i e l d s q n + l = U n q * + U l 2 h l t-4'8-1 h j j + 1 = u 2 i q? + u 2 2 h? (4.9) Since q^ +^ = 0 (dead end), i t f o l l o w s from Eq. 4.8 th a t = " ^ ~ h R (4.10) which on the b a s i s of Eq. 4.6 becomes q R = - U 1 2 K/un C 4 . l l ) S u b s t i t u t i o n of Eq. 4.11 i n t o Eq. 4.9 and s i m p l i f i c a t i o n of the r e s u l t i n g 48, equation gives T " 12^21 h . = ( u ? ? - — - )K (4.12) n+l ^ 1 1 u n ' ^ 1 Hence the amplitude of the pressure head f l u c t u a t i o n at the dead end i s L "12^21 a 1 n+l 1 un ^ = • ^ 1 , 1 = 1 ^ 2 2 - _ _ — )K| (4.13) The amplitude of the pressure head at the dead end may be nondimen-s i o n a l i z e d by d i v i d i n g the amplitude of pressure f l u c t u a t i o n s at the reser-v o i r end, i . e . , u 1 2 u 2 i h = h /K = u 22 r ' a' ' • . " u n (4.14) 2. FLUCTUATING DISCHARGE P e r i o d i c flows are encountered on the s u c t i o n , and on the discharge s i d e of a r e c i p r o c a t i n g pump. These f l u c t u a t i o n s can be decomposed i n t o a set of harmonics. Severe pressure o s c i l l a t i o n s may develop i f any of these harmonics has a pe r i o d equal to one? of the n a t u r a l periods of the s u c t i o n or discharge p i p e l i n e . Expressions are derived i n t h i s s e c t i o n to determine, by the t r a n s f e r m a t r i x method, the frequency response of systems having a r e c i p r o c a t i n g pump.. The s u c t i o n and the discharge p i p e l i n e may have stepwise changes i n diameter and/or w a l l thickness and may have branches w i t h r e s e r v o i r s , dead ends, or o r i f i c e s , (a) Suction l i n e . s t Let the t r a n s f e r matrix r e l a t i n g the s t a t e vectors at the 1 and th (n+l) s e c t i o n of the s u c t i o n l i n e ( F i g . 4.1) be U, i . e . , z L = U z R (4.15) -n+l — 1 By expanding Eq. 4.15 and no t i n g that h-^  = 0, one obtains V i = u n q l i ( 4 A 6 ) F i g . 4 1 S u c t i o n and d i s c h a r g e p ipe l ines . 50. and n+1 Hence, R U 2 i q i u 2 i (4.17) (4.18) n+1 u n Mn+1 The inflowvtime curve f o r one pe r i o d can be decomposed i n t o a set of th harmonics by F o u r i e r a n a l y s i s [57] . Let the discharge f o r the m harmonic be q. n+1 A' s i n ( m o j t + ijj) m m (4.19) or. q £ , = Re [ A e^m U *] ^n+1 m i n which A = A' exp [j C4> m m ^ L J m angle f o r the m^ harmonic; and to = frequency of the fundamental. I t f o l l o w s from Eqs. 2.3 and 4.20 that , = A i n which A i s a complex n n n + l m m r (4.20) ~-)] ; A 1 and <J  are the amplitude and the phase 2J J ' m m r v constant. S u b s t i t u t i o n of t h i s r e l a t i o n s h i p i n t o Eq. 4.18 y i e l d s , L n+1 u 2 i A un m (4.21) Hence the amplitude of pressure head f l u c t u a t i o n at the s u c t i o n flange i s , ' 11 L j . . i | = u 2 i A /uj i m 'n+l'm I Z i m i i l and the phase angle f o r the pressure head i s •1 m tan Im(hL J /Re(h L J ^ n+l'm *• n+1 m (4.22) (4.23) The pressure headv-time curve may be obtained by v e c t o r i a l l y adding the pressure^time curve f o r each harmonic. For the m^ harmonic n+1 Re h ej (m w t + O m (4.24) or h* , = h cos(m to t + <b ) n+1 m K ym . Hence the pressure head^time curve may be computed from the equation h* n+1 nv Z, h cos(m to t + d) ) = 1 m ^ Y n r (4.25) (4.26) 51. i n which M = number of harmonics i n t o which inflow^time curve f o r the pump i s decomposed. (b) Discharge l i n e . By proceeding i n a s i m i l a r manner and noting that h^ +^ = 0, the f o l l o w -ing equation i s obtained f o r the pressure head^time curve at the discharge s i d e of the pump: R M h* = h' cos(m OJ t + V ) (4.27) l m=l m nr J i n which i R , | u 2 1 A m h» = hi = | 1 r- (4.28) m 1 1'm | u i 2 u 2 1 ~ u l l u 2 2 l b' = t a n " 1 m Im(hR) /Re(h R) (_ ^ m ^ Jmj (4.29) and A m = complex amplitude of m^n harmonic of the discharge time curve of the pump. 3. OSCILLATING VALVE In t h i s case the area of the opening of the gate or v a l v e i s v a r i e d p e r i o d i c a l l y . Since the gate equation, Eq. 3.44, r e l a t i n g the head, d i s -charge, and the gate area i s n o n l i n e a r , t h i s case i s more d i f f i c u l t to analyze than the preceeding ones. However, as discussed i n s e c t i o n 3.2-2, t h i s equation can be l i n e a r i z e d i f h << H . In the d e r i v a t i o n of express-ions i n t h i s s e c t i o n , the p o i n t matrix of Eq. 3.55 i s used. This matrix i s derived by assuming the valve movement as s i n u s o i d a l and l i n e a r i z i n g the gate equation. Thus to use the expressions derived h e r e i n f o r the nonharmonic p e r i o d i c v a l v e movements, the valve motion i s decomposed i n t o a set of harmonics by F o u r i e r a n a l y s i s , the system response i s determined c o n s i d e r i n g each harmonic at i t s frequency, and then the t o t a l system r e s -ponse i s c a l c u l a t e d by superimposing the i n d i v i d u a l responses. 52. Let U' be the extended o v e r a l l t r a n s f e r matrix r e l a t i n g the s t a t e vec-t o r s at the 1 s t and the ( n + l ) ^ s e c t i o n of the system, i . e . , z' 1, = U 1 z' R (4.30) —n+l —1 v In a d d i t i o n z | R , = P' z' L . (4.31) -^i+l ov — n+l Hence f . R = pi ui z i R (4.32) -n+l ov —1 By s u b s t i t u t i n g from Eq. 3.55, m u l t i p l y i n g the matrices P ^ and U', ex-R R L R panding, and n o t i n g that h * = 0, h n + ^ = 0, and q n +-^ = ^+1' o n e obtains u 2 3 - (211 /QJu'h +(2H k/x ) u 3 3 K _ o <y o o 1 (A 7 « q i - " u 2 1 - ( 2 H o / Q o ) u n + ( 2 H o k / x o ) u 3 i l 4 ' " J V i = u l l 11 + u 1 3 ( 4 - 3 4 ) i n which u l l 5 u 1 2 , > u 3 3 a r e t n e elements of the matrix, U'. By ex-panding Eq. 4.30 and n o t i n g that h j = 0 one obtains h n + i = u 2 1 <U + "23 C4.35) Extended f i e l d and p o i n t matrices are f i r s t computed. Then the exten-ded o v e r a l l t r a n s f e r matrix i s determined by m u l t i p l y i n g the f i e l d and p o i n t matrices s t a r t i n g at the downstream end, i . e . , U' = F' P' P'o F\ (4.36) n n 1 1 R L L The value of q, i s determined from Eq. 4.33 and q and h .. are computed M l n Mn+1 n+l r from Eqs. 4.34 and 4.35. The absolute values of h .. and q .. are the am-n n+l n n + l p l i t u d e s of pressure head and discharge f l u c t u a t i o n s at the v a l v e , and t h e i r arguments are, r e s p e c t i v e l y , the phase angles between head and j*, and between discharge and T*. I f there i s no other f o r c i n g f u n c t i o n except the o s c i l l a t i n g valve at the downstream end of the system, o r d i n a r y f i e l d and p o i n t matrices may be used i n s t e a d of the extended ones. In t h i s case u 1 3 = u 2 3 = u 3 1 = 0 and u 3 3 = 1 i n Eqs. 4.33-4.35. 53. 4. PROCEDURE FOR DETERMINING THE FREQUENCY RESPONSE The frequency response of p i p i n g systems at a p a r t i c u l a r p o i n t may be determined as f o l l o w s : ( i ) Draw the block diagram and then the s i m p l i f i e d block diagram f o r the system. In the case of simple systems, t h i s step may be omitted. ( i i ) In the case of a non-harmonic p e r i o d i c f o r c i n g f u n c t i o n , de-compose i t i n t o a set of harmonics by F o u r i e r a n a l y s i s and consider one harmonic at a time. For the s p e c i f i e d frequency, compute the p o i n t and f i e l d m atrices. A summary of f i e l d and p o i n t matrices derived i n Chapter 3 i s presented i n Appendix A f o r an easy reference. To w r i t e an extended t r a n s f e r m a t r i x , simply add the f o l l o w i n g elements to the t r a n s f e r matrix l i s t e d i n Appendix A: u 1 3 = u 2 3 = u 3 1 = u 3 2 = 0 and u 3 3 = 1. Note that extended t r a n s f e r matrices are used only i f there i s more than one f o r c i n g f u n c t i o n i n the system. ( i i i ) C a l c u l a t e the o v e r a l l t r a n s f e r matrix by an ordered m u l t i p l i -c a t i o n o f the po i n t and f i e l d m atrices. For t h i s c a l c u l a t i o n , the block diagram of step ( i ) i s very h e l p f u l . For m u l t i p l i c a t i o n of matrices the scheme o u t l i n e d i n appendix B may be followed i f the c a l c u l a t i o n i s done by hand, s l i d e r u l e or desk c a l c u l a t o r . This scheme w i l l c o n s i d e r a b l y r e -duce the amount of computations. (i v ) Use the expressions developed i n Sections 4.2-1 through 4.2-3 to determine the frequency response. (v) I f a "frequency response diagram i s to be p l o t t e d repeat the above procedure by t a k i n g d i f f e r e n t frequencies. Computations to determine the frequency response at the valve end of a branch system having an o s c i l l a t i n g v a l v e at the downstream end are pre-sented i n Appendix B to i l l u s t r a t e the above procedure. 54. - - - 4.3 PRESSURE AND DISCHARGE VARIATION• "ALONG PIPELINE The previous sections dealt with the determination of the pressure and discharge o s c i l l a t i o n s at the end sections of a system. Sometimes i t i s , however, necessary to determine the amplitudes of the discharge and pressure f l u c t u a t i o n s along the length of the p i p e l i n e . In t h i s section, a procedure to determine the amplitudes of discharge and pressure f l u c -tuations along the length of the p i p e l i n e i s f i r s t o utlined. Then ex-pressions to f i n d the loc a t i o n of nodes and antinodes are derived. To analyze a piping system two of the four q u a n t i t i e s — d i s c h a r g e and pressure or t h e i r r e l a t i o n s h i p at eit h e r end of the system—must be known. The other two quantities can then be calculated by using the equations der-ived i n the l a s t section. The amplitudes of the discharge and pressure fl u c t u a t i o n s at the upstream end being known, t h e i r amplitudes along the pi p e l i n e may be determined as follows. The procedure i s i l l u s t r a t e d by discussing a system having a r e s e r v o i r at the upstream end and an o s c i l -l a t i n g valve at the downstream end. S i m i l a r l y , equations f o r other systems having d i f f e r e n t boundary conditions can be developed. Suppose that the amplitudes of discharge and pressure at the sec-t h t i o n on the i pipe (see F i g . 4.2a) are to be determined. Let the trans-fe r matrix r e l a t i n g the state vectors at the f i r s t section of the f i r s t and of the i ^ pipe be designated by W, i . e . , ( z R ) . = W(z R). (4.37) and the f i e l d matrix r e l a t i n g the state vectors at the f i r s t and the k ^ t h section of the i pipe by F , i . e . , ( z h . = F ( z ^ . (4.38) —k l x — V i The matrix W i s computed by multiplying the point and f i e l d matrices f o r the f i r s t ( i - 1 ) pipes i n a proper sequence (see the block diagram of Fi g . 4.2b), i . e . , 55. W = P. F. . P. . F, (4.39) 1 l - l i - i 1 and the matrix F^ i s c a l c u l a t e d by r e p l a c i n g £ w i t h x i n Eq. 3.18. Note that the elements of the matrix W f o r a s p e c i f i e d frequency are constants while those of the matrix F^ depend upon the value of x as w e l l . I t f o l l o w s from Eqs. 4.37 and 4.38 that (z£). = S f z ^ j (4.40) i n which S = F W = F P. F. . P. . F, (4.41) x x i 1-1 i - i L p The value of (q^)^ i s c a l c u l a t e d from Eq. 4.33. Furthermore i t i s known p that ( h 1 ) 1 = 0. S u b s t i t u t i o n of these values i n t o the expanded form of Eq. 4.40 y i e l d s (qjp. = s n Cq?)i (4-42) and (hjp. = s 2 1, ( q ^ ! (4.43) The amplitudes of the discharge and pressure f l u c t u a t i o n s at any other s e c t i o n can be determined by proceeding i n a s i m i l a r manner. LOCATION OF PRESSURE NODES AND ANTINODES " The l o c a t i o n of pressure nodes and antinodes i s an important aspect of the a n a l y s i s of resonance i n p i p e l i n e s at higher harmonics. The amplitude of the pressure f l u c t u a t i o n i s a minimum at a node and a maximum at an antinode. For a f r i c t i o n l e s s system, the amplitude of pressure f l u c t u a t i o n s at the node i s zero. At the antinodes the p i p e l i n e s may be subjected to severe pressure f l u c t u a t i o n s . Thus the pipe may b u r s t due to pressure i n excess of the design pressure or may c o l l a p s e due to subatmospheric pressure. A surge tank becomes i n o p e r a t i v e i n preventing the appearance of pressure waves upstream of the tank i f a node i s formed at i t s base. Jaeger explained P i p e I P i p e 2 P i p e i - -}i P i p e i P i p e n h ki ( a ) P i p i n g s y s t e m . P 2 F i - I Pi kj II w Fx A II s ( b ) B l o c k d i a g r a m . F i g . 4 - 2 D e s i g n a t i o n of k t h s e c t i o n on i t h p i p e . U l the development of f i s s u r e s i n the Kandergrund tunnel [27, 28] due to the establishment of a pressure node at the tank which made the tank inopera-t i v e although i t was over-designed. The l o c a t i o n s of the nodes and antinodes may be determined as f o l l o w s : Eq. 4.43 gives the amplitude of the pressure f l u c t u a t i o n at a p o i n t . By making use of the f a c t that f o r a f r i c t i o n l e s s system, the amplitude of pressure f l u c t u a t i o n s at a nodal p o i n t i s zero and q 1 f 0 f o r n o n - t r i v i a l s o l u t i o n s , one obtains The s o l u t i o n of t h i s equation f o r x gives the l o c a t i o n of the nodes on *u •th . the I pipe. The amplitude of pressure f l u c t u a t i o n i s a maximum at the antinodes. The l o c a t i o n of these p o i n t s may be determined by d i f f e r e n t i a t i n g Eq. 4.44 with respect to x, equating the r e s u l t to zero and then s o l v i n g f o r x, i . e . , the roots of the equation give the l o c a t i o n of the antinodes. By making use of Eqs. 4.44 and 4.45 expressions f o r the l o c a t i o n of nodes and antinodes i n simple systems can be de r i v e d . The procedure i s i l l u s t r a t e d below by d e r i v i n g expressions f o r a simple pipe and f o r two pipes i n s e r i e s . Expressions f o r complex systems may be der i v e d i n a sim-i l a r manner. However, i t i s b e t t e r to solve Eq. 4.44 and 4.45 nu m e r i c a l l y r a t h e r than to d e r i v e the expressions and then solve them. On the b a s i s of the Eq. 3.18, f o r a f r i c t i o n l e s s s i n g l e p i p e l i n e having constant c r o s s - s e c t i o n a l area, Eq. 4.44 becomes s 2 1 (x) = 0 (4.44) T 7 s 2 1 (x) = 0 (4.45) - j C.sin (oo x/a.) = 0 l l (4.46) or 58. s i n (OJ x/a i) = 0 (4.47) whose so l u t i o n gives x = n ^ a i / o j (n = 0, 1, 2, ) (4.48) The values of x > £\ represent the locations of the .imaginary nodes which are discarded. It follows from Eqs. 4.45 and 4.47 that cos (OJ x/a i) = 0 (4.49) The s o l u t i o n of t h i s equation gives the locations of antinodes, i . e . , x = (n + 1/2)TT a./w (n = 0, 1, 2, ) (4.50) Again the values of x > £\ are the locations of imaginary nodes and are discarded. Eqs. 4.47 and 4.49 show that a standing wave i s formed along the length of the p i p e l i n e . It i s clear from Eq. 4.46 that the pressure v a r i -ation i s si n u s o i d a l . Series,system: In a series system having two pipes (Fig. 2.3), the locations of nodes and antinodes i n the pipe leading from the re s e r v o i r are given by Eqs. 4.47 and 4.49. However, t h e i r location i n the second pipe can be deter-mined by using Eqs. 4.44 and 4.45. By su b s t i t u t i n g the expressions f o r F , F j , and p,2 into Eq. 4.39, multiplying the matrices, and using Eq. 4.44, one obtains - C 2 s i n ( o j x/a 2) cos ( o j£i/ai) - C j c o s ( o j x / a 2 ) s i n ( o j £ 1 / a i ) = 0 (4.51) which upon s i m p l i f i c a t i o n becomes OJ£I tan — = j - tan (4.52) Note that Eqs. 4.51 and 4.52 are v a l i d f o r a f r i c t i o n l e s s system only. Since a1> a 2 , A 1 ? and A 2 are p o s i t i v e constants, the r i g h t hand side of Eq. 4.50 i s p o s i t i v e or negative i f tan (o j£i/ai) i s negative or p o s i t i v e r e s p e c t i v e l y . Hence, i t f o i l ows from F i g . 4.3 that no node i s established 60. i n the i n t e r v a l s 0 < x <_ Tra2/2o), Tra2/u) <_ x <^  3Tra2/2u), ... i f 0 < u i ^ / a i <TT/2 or TT < co£i/ai < 3TT/2; and' i n the i n t e r v a l s Tra2/2w x Tra2/o), 3Tra2/2o) < x < 2 I T / O J , i f TT/2 < to^/ai < TT or 3TT/2 < ooii/a! < 2TT. C H A P T E R F I V E VERIFICATION OF TRANSFER MATRIX METHOD To demonstrate the v a l i d i t y of the method presented h e r e i n , the r e s u l t s obtained by the t r a n s f e r matrix method are compared with experimental v a l -ues and with those determined by the method of c h a r a c t e r i s t i c s , by the impedance theory, and by energy concepts. A b r i e f d e s c r i p t i o n of each method, i t s advantages and l i m i t a t i o n s are a l s o presented 5.1 EXPERIMENTAL RESULTS Except f o r the l a b o r a t o r y and f i e l d t e s t s reported by Camichel and h i s c o l l a b o r a t o r s [8], few experimental r e s u l t s on the reso n a t i n g character-i s t i c s of pipes are a v a i l a b l e i n the l i t e r a t u r e . In the t e s t s reported by Camichel et a l . , resonating c o n d i t i o n s i n s e r i e s pipes were e s t a b l i s h e d by a r o t a t i n g cock located at the downstream end of the p i p e l i n e . Each system had a r e s e r v o i r of constant head at the upstream end. The data f o r these systems are given i n F i g . 5.1. The values of the periods of the fundamental and higher harmonics determined experimentally and by the procedure o u t l i n e d i n s e c t i o n 4.1 are given i n Table I. As can be seen, c l o s e agreement i s found between the experimental values and those determined by the t r a n s f e r matrix method. I = 2 0 1 6 3 m D = 8 0 m m a 3 I 3 0 0 m / s e c . I = 1 0 5 8 5 m D = 4 0 m m a = I 3 5 6 m / s e c . ( a ) T o u l o u s e p i p e l i n e . P i p e I P i p e 2 I = 2 2 7 - 8 m I s 2 3 4 9 m 0 = 0 - 6 m D = 0 - 5 m a = I 0 7 5 m / s e c . a = I 2 5 6 m / s e c . ( b ) F u l l y p i p e l i n e . Fig . 5 1 L o n g i t u d i n a l p r o f i l e Reservoir ( c ) P i p e l i n e C 4 . ON OJ F i g . 51 Longitudinal p r o f i l e Thickness in mm L e n g t h in m A3 _L_'i._ 93 82 1020 {J090 Fig. 5 R e s e r v o i r Total length = 346-5lm (d) P i p e l i n e P 3 . Longitudinal prof i le F ig . 5 - 2 Plot of r e s i d u a l ~ou for Tou louse system TABLE I. CALCULATED AND MEASURED PERIODS System CD No. of Pipes C2) T V * n -v* n +• Periods i n Seconds Ineoret-i c a l p e r i o d Fundamental 3rd Harmonic 5th Harmonic 7th Harmonic 9th Harmonic 11th Harmonic i n seconds (3) c a l c . (4) meas. (5) c a l c . " C6) meas. (7) c a l c . (8) meas. (9) c a l c . Cio) meas. (11) c a l c . (12) meas. (13) c a l c . (14) meas. (15) Toulouse 2 0.932 0.708 0.69 0.311 0.31 0.198 0.19 - - - - - -F u l l y 2 15.96 13.719 13.50 - - - - - - - - - -C. 15 2.008 1.887 1.882 - - - - - - - - - -F 3 9 1.464 1.405 1.368 0.502 0.505 0.296 0.310 0.2117 0.2150 0.1650 0.1667 0.1338 0.1420 For purposes of i l l u s t r a t i o n the p l o t of r e s i d u a l ^ used to determine the resonant frequencies of the Toulouse system i s presented i n F i g . 5.2. 5.2 METHOD OF CHARACTERISTICS The method of c h a r a c t e r i s t i c s [9, 31, 45-47, 50] i s w e l l e s t a b l i s h e d f o r the a n a l y s i s of the t r a n s i e n t s t a t e , and s t e a d y - o s c i l l a t o r y flows i n p i p i n g systems. The unsteady flow i s represented by n o n l i n e a r , h y p e r b o l i c p a r t i a l d i f f e r e n t i a l equations. Without n e g l e c t i n g or l i n e a r i z i n g the non-l i n e a r terms, these p a r t i a l d i f f e r e n t i a l equations are converted i n t o o r d i n a r y d i f f e r e n t i a l equations which are then solved by a f i n i t e - d i f f e r e n c e technique. To ensure convergence and s t a b i l i t y of the numerical procedure/ [9, 50], the time i n t e r v a l , At, must be s e l e c t e d such that At < Ax/a, i n which A x i s the length of one of the reaches i n t o which a pipe i s d i v i d e d . To analyze the s t e a d y - o s c i l l a t o r y flows by t h i s method, the i n i t i a l steady s t a t e v e l o c i t y and pressure head are assumed equal to t h e i r mean values. The p r e s c r i b e d f o r c i n g f u n c t i o n i s then imposed as a boundary c o n d i t i o n and the system i s analyzed by c o n s i d e r i n g one frequency at a time. The amplitudes of the pressure head and of the discharge f l u c t u a t i o n s are determined when a s t e a d y - o s c i l l a t o r y regime i s e s t a b l i s h e d , i . e . , when the i n i t i a l t r a n s i e n t s have vanished. The process of convergence to steady-o s c i l l a t o r y c o n d i t i o n s i s slow and r e q u i r e s a considerable amount of com-puter time thus making i t uneconomical f o r general s t u d i e s . In a d d i t i o n , the "round o f f " e r r o r introduced i n the computations during each time i n t e r v a l might outweigh the accuracy of the method r e s u l t i n g from the i n c l u s i o n of the n o n l i n e a r terms. A number of s y s t e m s — s e r i e s , p a r a l l e l and branch systems with the side branch having various boundary c o n d i t i o n s — a r e analyzed us i n g the t r a n s f e r matrix method and the method of c h a r a c t e r i s t i c s . The data f o r these systems and the frequency response diagrams are presented i n F i g . 5.3 through 5.9. The frequency response diagrams are presented i n non-dimensional form. The frequency r a t i o , to , i s defined as t o / t o ^ ; the pressure head r a t i o , h , as 2|h^ , I/H : and the discharge r a t i o , q , as 2|q^ , 1/Q . ' r ' 1 n+11 o 6 ' ^ r | n n + l ! x o The values of h^ and q^ determined by the method of c h a r a c t e r i s t i c s rep-resent the amplitude of the swing from the minimum to the maximum value. The frequency of the f o r c i n g f u n c t i o n i s designated by to. The o s c i l l a t i n g v a lves are the f o r c i n g f u n c t i o n s i n a l l the systems except the dead end s e r i e s system of F i g . 5.3 i n which the f l u c t u a t i n g pressure head at the upstream end i s the f o r c i n g f u n c t i o n . The v a l v e move-ment i s taken as s i n u s o i d a l with x =1.0 and k = 0.2. The f l u c t u a t i n g o & pressure head i n F i g . 5.'3 i s a l s o s i n u s o i d a l w i t h K = 1.0. In the branch systems of F i g . 5.8, T q = 1.0 and k = 0.2. The s e r i e s system of F i g . 5.4 i s analyzed by the t r a n s f e r matrix method to study the e f f e c t of the f r i c t i o n losses and of the v a r i a t i o n i n the mean discharge on the frequency response. Frequency response diagrams f o r the system having f r i c t i o n l osses about 35% of the s t a t i c head and having no f r i c t i o n losses i s shown i n F i g . 5.10. The data are the same as given above except that Qq = 9.42 f t / s e c . For c a l c u l a t i n g the system response by the method of c h a r a c t e r i s t i c s the f r i c t i o n l o s s e s are taken p r o p o r t i o n a l to the square of v e l o c i t y . The resonant frequencies deter-mined by c o n s i d e r i n g the system as f r i c t i o n l e s s and by t a k i n g the f r i c t i o n l o sses i n t o c o n s i d e r a t i o n are recorded i n Table I I . I t i s c l e a r from F i g . 5.10 and Table I I that the f r i c t i o n l o s s e s , even as high as 35%, have a n e g l i g i b l e e f f e c t on the values of resonant frequencies. Hence, i n the a n a l y s i s of other systems presented i n t h i s study, f r i c t i o n losses are neglected. This s i m p l i f i e s the programming and reduces the amount of 69. Reservoir Dead endN l = IOOOft. I = 2 0 0 0 f t . I = 1600ft. I = 2150ft. D=40ft. D=3 5ft. D= 30f t . D= 2-5ft. a=4000ft^ec. a = 2000ft/sec. a = 3000ft>sec. a = 4300f t/sec. (a) Piping system 4 0 30 2 0 10 5 6 OJ (Radians/sec.) (b) Frequency response d i a g r a m . F i g . 5-3 S e r i e s s y s t e m with d e a d end _2_ R e s e r v o i r 70. H« Pipe I P i p e 2 ^Oscillating valve -r i I, = 2 0 0 0 f t . 2 l 2 = 7 5 0 ft. 3 a , = 4 0 0 0 f t . / s e c . a 2 = 3 0 0 0 f t . / s e c . 0| = 2ft . 0 2 = Iff (o) P i p i n g s y s t e m . -— i — T r a n s f i • M e t h o d h r = 2 | h \x matr ix m 1 of c h a r a c k | / H 0 ; qr = J e t h o d t e r i s t i c s MqL3|/Q0 /< r u l\ 1 \ 1 \ ' 1 1 > 1 1 1 /cr-w A A / \ / \ / \ / \ / V / \ / \ / \ / \ 1 \ ' €\ 1 \ 1 \ 1 \ / \ \ <^  \ 4 1 1 1 1, 1 i ! i i i 1! / i / \ / \ i i . , 1 1 1 \ 0 - 8 0 - 6 0 - 4 0 2 0 0 u> r S OJ /cu t h (b) F r e q u e n c y r e s p o n s e d i a g r a m . F ig . 5 - 4 Ser ies system I2= 750ft -I, = ' 2 0 0 0 ft-a, = 4 0 0 0 f t . / s e c a~= 3000f t . /sec. 0, = 2ft. D2= Iff ( a ) P i p i n g s y s t e m . — » — T r c o Me 2 | h 3 | / H o i q msfer m a t r i thod of cha r = 2 | q s | / Q 0 x method r a c t e r i s t i c s A A hr / 1 7° 1 / i / W= / , , ... \ i i - — - ... / \ / \ / \ / * f o\ \ / / * / \ / \ / * / \ / \ i \ \ s \s A i \ i \ i \ i \ i \ i \ • \ 1 \ i \ i \ 1 / \ / » / \ / * / \ I \ 1 \ I i / i / i / \ / i I i i \ i \ i \ i \ 0 8 0 6 0-4 0-2 0 0 0 - 5 5 6 0 ) R = <o/C0th (b) F r e q u e n c y r e s p o n s e d i a g r o m Fig . 5 5 Branch system(side branch having r e s e r v o i r ) 72. Dead end Oscillating valve 1,= 2000f t . a," 4000ft. /sec. D,= 2ft. I2= 750 ft. a 2= 3000f t . /sec. 0 2 = Ift. ( a ) P i p i n g s y s t e m i e Method of chara — > Transfer matrix i cteristics method /< < h r = 2 | h L J / H 0 Qo r o • J -- — —t v /i ~ •*« i \ / \ / \ / \ / \ / » • / V /OV '* \ / \ ' \ / \ I \ / \ ; \ — • — o — i \ Ii i i /1 11 I \ i \ i \ i \ 1 \ J\ M 1 \ / I g \ f \ } t 1 • \ / \ i \ i , i s i i i i i 1 /1 /1 \ / c1 \ 1 \ 1 \ 1 \ 1 \ i \ 1 \ J\ i0-8 0 6 0-4 0 2 0 0 C O r = £ O / 0 U t h (b) F r e q u e n c y r e s p o n s e d i a g r a m . F i g . 5-6 B r a n c h system(s ide branch having d e a d end). Reservoi r 73. H, Pipe I 1,= 200 0 ft. a,= 4 0 0 0 f t . / s e c . D,= 2 ft. Orif ice Oscillating valve a 2 = 3 0 0 0 f t . / s e c . 0, = I ft. ( a ) P i p i n g s y s t e m . 0 8 1 o Method — j— - T r a n s f hr = 2|hU of characte er matrix n ' I V . q r = 2 | r ist ics let hod H s l / Q . hr—J (\ » t r •* \ 1 \ 1 \l fl ft / 1 / 1 » / » J % I 1 1 1 l / * / \ / \ / \ / \ / \ / % / \ / * \ # ' / \ i \ 1 \ » \ 0 \ » \ t \ § \ 1 \ i \ 1 \ L£ °*"-*NN \ \ \ 1 I / / X I \ I \ 1 \ 1 \ \ \\ \\ \\ 1 \ 1 \ 1) f \ 1 \ I 0 6 0 4 0-2 0 0 0-5 10 2 0 SO 4 0 5 0 6 0 M J r = a > / C O t h 7 0 ( b ) F r e q u e n c y r e s p o n s e d i a g r a m . F i g . 5 - 7 B r a n c h s y s t e m ( s i d e b r a n c h hav ing o r i f i c e ) . 74. O s c i l l a t i n g valve No.2 O s c i l l a t i n g va lve No.I I, = 2 0 0 0 f t . a, = 4 0 0 0 f t . / s e c . I 2 = 7 5 0 ft. a , = 3 0 0 0 f t . / s e c . D, * 2ft . D 2 = I ft . ( a ) P i p i n g s y s t e m . F i g 5 -8 B r a n c h sys tem(s ide branch having oscillating valve) 75. 0 4 0-2 — T r a r © Method h r * 2 | h L 3 | / l • 1 s fer m a t r i x of character H. i Q r = 2 | method is t ics q L 3 l/Qo X^ V 1 V / \ f / \ 5 1 / 1 1 1 1 / / \ / \ / % / \ J \ P * / \ / t / \ w A / \ / \ / \ l \ ' \ J " 0 ~ ^ s A /1 / i / i / 1 / i / i A ' 1 1 \ 1 \ • \ 1 \ 1 \ i i r — / N I \ l l / \ / 1 i / i / i / i / / 1 \ 1 \ 1 \ 1, \ / j 0-5 1 2 3 4 5 6 7 CUR = C O / O J t h — ( b ) F r e q u e n c y r e s p o n s e d i a g r a m ( v a l v e No.I and N o . 2 are in p h a s e ) . F i g . 5-6 B r a n c h s y s t e m ( s i d e b r a n c h having osci l la t ing v a l v e ) . 76. T r a n s f e r m a t r i x m e t h o d Mo in B r a n c h M e t h o d of c h a r a c t e r i s t i c s 0-5 1 2 3 4 5 6 7 G U r = G U / U J t h 9» ( c ) F r e q u e n c y r e s p o n s e d i a g r a m ( v a l v e N o . 2 lagging No.I by 9 0 ° ) . F i g . 5 - 8 B r a n c h s y s t e m ( s i d e b r a n c h having oscil lating v a l v e ) 77. ( d ) F r e q u e n c y r e s p o n s e d i a g r a m ( v a l v e N o . 2 l a g g i n g No.I by 1 8 0 ) F i g . 5 - 8 Branch s y s t e m ( s i d e b r a n c h having osc i l l a t ing va lve ) R e s e r v o i r 1= 2 2 0 0 f t -a = 4 2 5 0 f t . / s e c . D= Iff. i Osci l lat ing v a l v e T i ! 1 i ' I = 1 1 0 0 f t . 2 1 = II 0 0 f t . 3 i 1= 2 5 0 0 f t . 4 78. a • 4 2 5 0 f t . / s e c . D = 2 f t . a = 4 2 5 0 f t . / s e c . D= 2 f t . a = 3 6 0 0 f t . / s e c . D = 2 f t . ( a ) P i p i n g s y s t e m 8 h r = 2 ! h 4 l / H q r = 2 I h ^ l / Q At 0 5 I 5 6 C U r =tO/CA>th-t b ) F r e q u e n c y r e s p o n s e d i a g r a m F i g . 5 - 9 P a r a l l e l s y s t e m . computer time r e q u i r e d . Frequency response diagram f o r the s e r i e s system f o r two d i f f e r e n t d i s c h a r g e s— Q Q = 9.42, and 0.314 cu. f t / s e c — i s shown i n F i g . 5.11. F r i c t i o n losses are neglected i n c a l c u l a t i n g the response. A n a l y s i s of the f r i c t i o n a l system by the method of c h a r a c t e r i s t i c s showed that the amplitudes of the p o s i t i v e swing of pressure head and negative swing of the discharge are l a r g e r than the corresponding negative and p o s i t i v e swings. This i s caused by the no n l i n e a r terms, e s p e c i a l l y the f r i c t i o n l o s s term, of the governing d i f f e r e n t i a l equations. In the t r a n s f e r matrix method, however, amplitudes of the p o s i t i v e and negative o s c i l l a t i o n s are equal because a s i n u s o i d a l s o l u t i o n i s assumed. To check the values of the phase angles between d i f f e r e n t q u a n t i t i e s of i n t e r e s t , the o s c i l l a t o r y discharge and pressure head at the valve are computed by usin g the method of c h a r a c t e r i s t i c s . The q*^t, h * ^ t , and x*^t curves are p l o t t e d i n F i g . 5.12. In t h i s diagram, h* = h*/H Q and q^ = q*/Q0- T n e phase angles determined by the t r a n s f e r matrix method and by the method of c h a r a c t e r i s t i c s are presented i n Table I I I . Close agree ment i s found between the r e s u l t s obtained by the two methods. 5.3 IMPEDANCE METHOD The concept of impedance was introduced by Rocard [43] and l a t e r used by Paynter [ 3 8 ] , Waller [53-55] and Wylie [ 5 8 - 6 0 ] . The pressure head and the discharge f l u c t u a t i o n s are assumed s i n u s o i d a l , n o n l i n e a r r e l a t i o n s h i p s are l i n e a r i z e d and the r a t i o of the o s c i l l a t o r y pressure head and dis c h a r g i s termed the h y d r a u l i c impedance. For the given boundary c o n d i t i o n s of the system, the terminal impedance i s c a l c u l a t e d and the impedance diagram i s p l o t t e d . Those frequencies at which t e r m i n a l impedance i s a maximum are the n a t u r a l frequencies of the system. I f the value of e i t h e r h or q i s known, the other can be c a l c u l a t e d from the computed value of the 8 0 . F r i c t i o n c o n s i d e r e d F r i c t i o n n e g l e c t e d F i g . 5- 10 Ef fect of friction losses on f r e q u e n c y r e s p o n s e of s e r i e s s y s t e m of F i g . 5 -4 a . Fig . 511 Ef fect of mean d i s c h a r g e on f r e q u e n c y r e s p o n s e of s e r i e s s y s t e m of F i g . 5 - 4 a . TABLE I I . RESONANT FREQUENCIES OF PIPING SYSTEM OF FIG. 5.4a r ' t h System Funda- 3rd 5th 7 t h 9th 11th ; mental Harm. Harm. Harm. Harm. Harm. (1) (2) (3) (4) (5) (6) (7) F r i c t i o n considered 1.259 3.000 4.741 7.259 9.000 10.741 F r i c t i o n neglected 1.260 3.000 4.740 . 7.259 9.000 10.741 I 2 r 0 2 10 - 0 0 0 8 l-4r 0-4 1-2 0-2 10 0 0 0 8 -0-2 0 6 L - 0 4 (i)c^r = 2-5 (ii)cor= 3 0 ( a ) S e r i e s s y s t e m of Fjg. 5 - 4 a . F i g . 5 12 h * ~ t , q * ~ t , o n d T ~ t c u r v e s 00 ( i ) C O r = 2 5 ( b ) B ranch system of F i g . 5 6 a F i g . 5 1 2 l -4r 0-4" to O" 1-2- 0-2 10 - 0 0 0 8 - -0-2 (i) ojr =2-5 0-6L- -0-4 (i i) cx>r = 3 0 (c) B r a n c h s y s t e m of F ig . 5 - 8 a ( V a l v e No.I and 2 in p h o s e ) . 00 F i g . 5 12 h ~ t , q * ~ t , a n d r * ~ t c u r v e s 86. TABLE I I I . PHASE ANGLES System (1) Phase Angles, <b, i n degrees between h and x* between q and x* rPeCjUcTicy r a t i o , w r (2) Transfer M a t r i x Method (3) Method of Character-i s t i c s (4) Transfer M a t r i x Method (5) Method of Character-i s t i c s (6) S e r i e s ( F i g . 5.4a) 2.5 3.0 -110.99 -180.01 -110.50 -180.00 -20.99 -270.01 -20.5 Branch ( F i g . 5.6a) (Side branch w i t h dead end) 2.5 3.0 -117.90 -180.01 -119.00 -180.00 -27.90 -270.01 -29.50 Branch ( F i g . 5.8a) (Side branch with o s c i l l a t i n g valve) 2.5 3.0 -117.13 -180.01 -118.00 -180.00 -18.17 -270.01 -18.00 88. I, = 1 9 0 0 f t . I 2= 3 6 0 0 f t . (b) I m p e d a n c e d i a g r a m . Fig . 5 1 3 S e r i e s s y s t e m \ = l o o o f t . 2 1 l 2 = IOOO ft. 3 a, = 4 0 0 0 f t . / s e c . o 2 s 4000ft . / s ec . D, = 3ft. D 2 = 2 ft. ( a ) P i p i n g s y s t e m . ( b ) I m p e d a n c e d i a g r a m . F i g . 5 1 4 B r a n c h s y s t e m ( s i d e b r a n c h having reservoi r ) . Reservoi r I, = lOOOft . Dead end 90. Oscil lating Va lve l 2 - lOOOft. a, = 4000 f t . / sec . a 2 = 4 0 0 0 f t . D, = 3ft. D 2= 2 ft. (a) P i p i n g s y s t e m . u N \ N II w N 0-5 10 2 0 3 0 4 0 5 0 6 0 co r«aj/cu f h 7 0 ( b ) I m p e d a n c e d i a g r a m F i g . 5 15 B r a n c h s y s t e m ( s i d e b r a n c h hav ing d e a d e n d ) . 91. Fig . 5 1 6 Impedance d iagram for para l le l system of F i g . 5 - 9 a 92. Energy en t e r i n g the system during the time i n t e r v a l At, i s E. = v Q H A t (5.1) m ' x i n which y = s p e c i f i c weight of the f l u i d and the s u b s c r i p t " i n " r e f e r s to the input q u a n t i t i e s . S u b s t i t u t i o n of Eqs. 2.1 and 2.2 i n t o Eq. 5.1 and expansion of the r e s u l t i n g equation y i e l d A E. = v (Q H + q* H + h* Q + q* h* ) A t (5.2) i n ' o o n m o m xo n m i n Let q* and h* be s i n u s o i d a l , i . e . , n m i n ' ' h* = h! cos ca t (5.3) m m q* = q! cos(ojt - <b. ) . (5.4) n m n m ^ i n • i n which <b. = phase angle between q* and h? and h! and q! are the i n r 6 n m m m n m am-p l i t u d e s of pressure and discharge f l u c t u a t i o n s . Note that both h | and q j n are r e a l q u a n t i t i e s . The energy input during one c y c l e may be c a l c u l -ated by s u b s t i t u t i n g Eqs. 5.3 and 5.4 i n t o Eq. 5.2 and i n t e g r a t i n g the r e s u l t i n g equation over one p e r i o d , T. This process gives r,T E. = Y Q H T + Y q . ' h ! cos oi t cos (a* - <t>. )dt (5.5) i n H o o M m m J 0 v m^ I f there i s a r e s e r v o i r of constant l e v e l at the upstream end, then h! =0. Hence, Eq. 5.5 becomes i n ^ E. = y Q H T (5.6) i n x o o By proceeding i n a s i m i l a r manner E = Y Q H T + Y h ' q ' out o o out n o u t • cos OJ t cos(wt - <f> ,_)dt (5.7) out "P The s u b s c r i p t "out" designates output q u a n t i t i e s . I f the losses i n the system are neglected, then E. = E . f o r estab-J 6 m out lishment of s t e a d y - o s c i l l a t o r y c o n d i t i o n s . Hence, i t f o l l o w s from Eqs. 5.6 and 5.7 that f T cos CD t cos fast - <b ^ I d t = 0 (5.8) n out J u which y i e l d s d> = 90° (5.9) y out For a l l the systems analyzed i n t h i s study, $ Q U ^ w a s 90°• The only exceptions were the branch systems with the side branch having an o r i f i c e or an o s c i l l a t i n g v a l v e . A clos e examination of the above d e r i v a t i o n r e v e a l s that Eqs. 5.8 and 5.9 do not hold i n these cases because there i s energy output at more than one p o i n t . By f o l l o w i n g a s i m i l a r procedure as above, one obtains T E = Y Q h T + y h ' q ' out < xom om ' out nout cos co t cos (cot - <|> .)dt o o u t + v 3 & T + Y. &» 3- ' T ' x> o 1 out nout i n which a t i l d e . , designates' q u a n t i t i e s f o r the side branch and the sub-s c r i p t "m", the downstream end of the main. For a s t e a d y - o s c i l l a t o r y c o n d i t i o n to e x i s t i n a f r i c t i o n l e s s system, = E . Hence, by comparing Eqs. 5.6 and 5.10, noting that Y Q 0H QT = y Q Q mH o mT + y Q 0H QT and s i m p l i f y i n g the r e s u l t i n g equation, one obtains h' .q' cos * . + fr' .q' .cos ^ = 0 (5.11) out^out vout out nout Yout This equation i s used h e r e i n to v e r i f y the numerical values of d i f f e r e n t q u a n t i t i e s determined by using the t r a n s f e r matrix method. The computations f o r the branch system of F i g . 5.8 are recorded i n Table IV. "5.5 STUDIES ON PIPELINE HAVING VARIABLE CHARACTERISTICS The resonating c h a r a c t e r i s t i c s of a p i p e l i n e - h a v i n g l i n e a r l y v a r i a b l e c h a r a c t e r i s t i c s — A and a — a l o n g i t s length, a constant head r e s e r v o i r at the upstream end, and an o s c i l l a t i n g v a l v e at the downstream end, F i g . 5.17a, are studied by using the t r a n s f e r matrix method. Frequency response cos co t cos (cot - <f> ^ ) d t (5.10) 0 out TABLE IV. VERIFICATION BY ENERGY CONCEPTS Branch system of F i g . 5.8 System r out f t . q ' o u t c f s v o u t degrees h' -out q out cfs 'V, T o u t degrees cos di out cos cb out h'q' cos ^out + h'q cos v o u t Valve No. 1 and \T /—\ / i >-» l-\ o f* /-\ 2.0 4.4 .0620 98.92 2.2 .0637 72 57 -.1552 .3000 -.0425 + .0425 a. 0. 0 iNO. z i n pnase 3,2 23.0 .0430 104.69 9.9 .0669 67 83 -.2536 .3773 -.2520 + .2510 a/ 0. 0 Valve No. 1 leading Mo ? Kir Q D ° 2.0 9.3 .0672 66.94 4.5 .0559 169 00 .3909 -.9816 .2445 - .2465 a. 0. 0 NO. z Dy yu 3.5 18.5 .0458 112.44 9.5 .0688 60 53 -.3816 .4919 -.3230 + .3220 a. 0. 0 Valve No. 1 leading 2.0 12.5 .0605 86.85 5.9 .0609 96 62 .0548 -.1161 .0416 - .0417 0. 0 viO . z Dy l o U i 3.6 20.6 .0562 85.72 9.8 .0587 98 65 .0746 -.1504 .0865 - .0866 a. 0. 0 Reservoir Data for actual pi pe ' 0 ( x j 5 4 ""250 a(x) - 3 2 0 0 + I6X -Substitute pipe 95. ^Oscillating valve 100ft. 100ft. 100ft. 100ft. 100ft. D s 3-8ft. 3-4ft. 30f t . 2-6ft. 2-2ft. a - 3280ft./sec. 3440ft . /sec 3600ft./feec. 3760ft.^ec. 3920ft. /sec. 0 8 0-6 0 4 0-2 ( a ) P i p i n g s y s t e m . O G Actual pip Substitute | e )ipe — 7^ TsOr If If If li / / w II II II II il \\ \ \ \\ \\ » \ \\ \\ i 1 1 1 V 11 11 If * \ q\ \ \ \ \ \ \ \ \ \\ * \ • IA— / ' \ i i \ / ' \ / ' \ / / \ —H> \— It— // a i i-f II II II Ij if II II II II \\ 4 \\ u \\ I 1 11 V II II II 1 Jw ' / \\ \ \ \ \ \\ i \\ / ' I / ' \ / ' I / ' I / ' I / * i /' 1 1' 1 J f - L \ ' \ A f\ v M r# * \ ,7 VV ii \ \ . / \ \ . / * \ il o \ t \ i f i / w Jf i \ \ / 1 \ v / 1 T A — / " ; — III >4 III * •1 * if i \ '/ ' \ V • l // i} if 1 // i a » \ n v \ '1 x \ •l \ \ 11 v \ ii * \ / °* 11 * n 1 * *J v i \ i \ il \ v \ \ ' / 1 i \ » / 1 \ °J 0-5 10 2 0 3 0 4 0 7 0 OJR - CU/co t h ( b ) F r e q u e n c y r e s p o n s e d i a g r a m Fig. 5 17 Pipeline having var iable characteristics along its length i s determined by using the t r a n s f e r matrix derived i n S e c t i o n 3.1-2. Then the a c t u a l pipe i s replaced by a s u b s t i t u t e pipe having stepwise changes i n c h a r a c t e r i s t i c s as shown i n F i g . 5.17a. The expressions presented i n S e c t i o n 4.2-3 are used to determine the frequency response. To compute to , the t h e o r e t i c a l p e r i o d i s c a l c u l a t e d from the equation T , = 4A/a (5.12) th m i n which a^ = v e l o c i t y of water hammer wave at the midpoint of the pipe-l i n e . The r e s u l t s f o r both the cases are presented i n F i g . 5.17b. Resonant frequencies f o r the system of F i g . 5.17a were determined by c o n s i d e r i n g the pipe per se and then r e p l a c i n g i t wit h a s u b s t i t u t e pipe (shown dotted i n F i g . 5.17a). Favre's expression-, Eq. 1.13, was used to compute the resonant frequencies. The r e s u l t s are tabulated i n Table V. Close agreement i s found between the r e s u l t s obtained i n these cases up to the f i f t h harmonic. The higher harmonics can be p r e d i c t e d to a reasonable degree of accuracy by i n c r e a s i n g the number of reaches i n t o which the p i p e l i n e i s d i v i d e d . TABLE V RESONANT FREQUENCIES OF PIPING SYSTEM OF FIG. 5.17a Resonant Frequencies, i n rad/sec Mode Favre's expression Transfer M a t r i x Method A c t u a l Pipe S u b s t i t u t e Pipe CD (2) (3) (4) Fundamental 15.127 15.075 14.905 T h i r d 35.683 35.702 35.001 F i f t h 57.647 57.856 56.375 f / Seventh 79.963 74.130 77.749 SUMMARY AND CONCLUSIONS A new method which i s known as the t r a n s f e r matrix method has been presented to analyze resonating c h a r a c t e r i s t i c s of p i p i n g systems. This method i s s u i t a b l e f o r hand or d i g i t a l computations. Following the i n i t i a l d e r i v a t i o n of t r a n s f e r matrices, one does not have to deal w i t h d i f f e r e n t -i a l equations or lengthy a l g e b r a i c equations or take i n t o c o n s i d e r a t i o n r e f l e c t i o n and transmission c o e f f i c i e n t s i n complex systems. In the der-i v a t i o n of t r a n s f e r matrices, the f r i c t i o n l o s s term and the gate equation, r e l a t i n g the head and the discharge through a v a l v e , are l i n e a r i z e d and the pressure head and discharge f l u c t u a t i o n s are assumed s i n u s o i d a l . The v a l i d i t y of the method presented h e r e i n has been v e r i f i e d by comparing the r e s u l t s w i t h those obtained e x p e r i m e n t a l l y , by using the. method of charac-t e r i s t i c s , by the impedance theory, and by the energy concepts. The f o l l o w i n g conclusions can be drawn from the a n a l y s i s of a number of system types commonly found i n waterpower development and water supply schemes: 1. The f r i c t i o n losses have a n e g l i g i b l e e f f e c t on the values of reson-ant frequencies of a system. A n a l y s i s of the f r i c t i o n a l system by the method of c h a r a c t e r i s t i c s shows that the amplitude of the p o s i t i v e swing of pressure head and negative swing of discharge are l a r g e r than the corresponding negative or p o s i t i v e swings. This i s due to the n o n l i n e a r f r i c t i o n term of the governing d i f f e r e n t i a l equation. In the t r a n s f e r matrix method, however, amplitudes of the p o s i t i v e and negative o s c i l l a -t i o n s are equal because a s i n u s o i d a l s o l u t i o n i s assumed i n the d e r i v a t i o n of t r a n s f e r matrices. 2. The values of the resonant frequencies are independent of the mean discharge or mean pressure head. A change i n any of these v a r i a b l e s r e s -u l t s i n a q u a n t i t a t i v e , not q u a l i t a t i v e , m o d i f i c a t i o n of the frequency response diagram. 3. The pressure and discharge nodes and antinodes are not n e c e s s a r i l y located at the p o i n t s of geometric changes i n the system. 4. The l i n e a r i z a t i o n of the gate equation i s v a l i d only i f h^ i s s m a l l . For example, f o r h^ = 0.8 (amplitude of pressure o s c i l l a t i o n s i n 0.4 H Q ) , t h i s l i n e a r i z a t i o n r e s u l t e d i n an overestimation of h by 3.5%. r ' 5. The periods of higher harmonics of a system are not n e c e s s a r i l y i n t e g r a l f r a c t i o n s of the t h e o r e t i c a l p e r i o d . 6. The frequency response of p i p e l i n e s having v a r i a b l e c h a r a c t e r i s t i c s along i t s length can be c a l c u l a t e d by r e p l a c i n g the a c t u a l pipe by a s u b s t i t u t e pipe having stepwise changes. To p r e d i c t the higher harmonics a c c u r a t e l y , the p i p e l i n e should be d i v i d e d i n t o a l a r g e r number of reaches. B I B L I O G R A P H Y Abbot, H.F., Gibson, W.L., and McCaig, I.W., "Measurements o f A u t o - O s c i l l a t i o n s i n a H y d r o e l e c t r i c Supply Tunnel and Penstock System," Trans. ASME, V o l . 85, D e c , 1963, pp. 625 - 630. A l l i e v i , L., "Teoria Generale Del Moto Perturbate Dell'Acqua Nei Tubi i n Pressione," A n n a l i D e l i a S o c i e t a D e g l i Ingegneri ed  A r c h i t e t t i I t a l i a n i , M i l a n , 1903. . , Theory of Water Hammer, Translated by E.E. Halmos, Ricardo Garoni, Rome. I t a l y , 1925. Angus, R.W., " A i r Chambers and Valves i n R e l a t i o n to Water Hammer," Trans. ASME, V o l . 59, Nov., 1937, pp. 661 - 668. Bergeron, L., "Etude des Coups de B61ier dans l e s Conduites, Nouvel Expose" de l a Methode Graphique," La Technique Moderne, V o l . 28, Nos. 2 and 3, 1936. , "Etude des V a r i a t i o n s de Regime dans l e s Conduites d'Eau," Revue Generale de L'Hydraulique, P a r i s , V o l . I , 1935, pp. 12. , "Water Hammer i n Hydraulics and Wave Surges i n E l e c t r i c i t y , " Translated under the sponsorship of ASME, John Wiley and Sons, New York, 1961. Camichel, C , Eydoux, D., and G a r i e l , M., "Etude Theorique et Experimentale des Coups de B e l i e r , " P a r i s , Dunod, 1919. Chaudhry, M.H., "Boundary Conditions f o r A n a l y s i s of Water Hammer i n Pipe Systems," t h e s i s presented to the U n i v e r s i t y of B r i t i s h Columbia, Vancouver, i n 1968, i n p a r t i a l f u l f i l l m e n t of the requirements f o r the degree of Master of A p p l i e d Science. , D i s c u s s i o n , "Surging i n Laboratory P i p e l i n e s with Steady Inflow," Paper No. 6573, Jour., Hyd. Div., ASCE, Jan 1970, pp. 294 - 296. , "Resonance i n Pipe Systems," Water Power, London ( i n press.) , "Resonance i n Pressure Conduits," Jour., Hyd. Div., ASCE ( i n p r e s s ) . , "Governing S t a b i l i t y of a H y d r o e l e c t r i c Power P l a n t , " Water Power~" London, A p r i l , 1970, pp. 131 - 136. 101. 14. , "Frequency Response of P i p e l i n e s Having V a r i a b l e C h a r a c t e r i s t i c s , " Jour., Basic Engg., ASME (under p r e p a r a t i o n ) . 15. Chaudhry, M.H. and Ruus, E., " A n a l y s i s of Governing S t a b i l i t y of a Hydropower P l a n t , " Trans., Engg. I n s t , of Canada ( i n p r e s s ) . 16. Den Hartog, J.P., "Mechanical V i b r a t i o n s i n Penstocks of H y d r a u l i c Turbine I n s t a l l a t i o n s , " Trans., ASME, V o l . 51, 1929, pp. 101 - 110. 17. D'Souza, A.F. and Oldenburger, R., "Dynamic Response of F l u i d L i n e s , " Trans. ASME, Se r i e s D, Sept. 1964. pp. 589. 18. E v a n g e l i s t ! , G., "Determinazione Operatoria D e l l e Frequenze d i Risonanza Nei Sistemi I d r a u l i c i i n Pressione," L e t t a A l i a R.  Accademia D e l i a Scienze D e l l ' I n s t i t u t o d i Bologne, Feb., 1940. 19. Favre, H., "La Resonance des Conduites a C h a r a c t e r i s t i q u e s Lineairement V a r i a b l e s , " B u l l e t i n Technique de l a Suisse Romande, V o l . 68, No. 5., Mar. 1942, pp. 49 - 54. 20. Gaden, D., C o n t r i b u t i o n s a 1'etude des re g u l a t e u r s de V i t e s s e  Considerations sur l e probleme de l a s t a b i l i t e , Lausanne, S w i t z e r l a n d , E d i t i o n s La Concorde, 1945. 21. H o l l e y , E.R., "Surging i n a Laboratory P i p e l i n e w i t h Steady Inflow," Report No. HYD-580, U.S. Bureau of Reclamation, Denver, Colorado, Sept., 1967. 22. H o l l e y , E.R., "Surging i n Laboratory P i p e l i n e s with Steady Inflow", Jour., H y d r a u l i c s Div., ASCE, May, 1969, pp. 961 - 980. 23. Hovey, L.M., "Optimum Adjustment of Governors at the Hydro Generating S t a t i o n s of the Manitoba H y d r o - E l e c t r i c Board," Engg. I n s t , of Canada, V o l . 43, Nov., 1960, pp. 64 - 71. 24. Jaeger, C , Theorie Generale du Coup de B e l i e r , P a r i s , Dunod, 1933. 25. , "Note sur l e s Pheonomenes Periodiques dans l e s Conduites Forcees a C a r a c t e r i s t i q u e s M u l t i p l e s , " La H o u i l l e Blanche, V o l . 35, May-June, 1936, pp. 71 - 75; and July-Aug., 1936, pp. 97 - 101. 26. , "Theory of Resonance i n Pressure Conduits," Trans. ASME, V o l . 61, Feb., 1939, pp. 109 - 115. 27. , "Water Hammer E f f e c t s i n Power Conduits," C i v i l Engineering and P u b l i c Works Review, V o l . 23, Nos. 500 - 503, Feb. - May, 1948. 28. , "The Theory of Resonance i n Hydropower Systems. D i s c u s s i o n of Incidents and Accidents Occuring i n Pressure Systems," Trans. ASME, Se r i e s D, D e c , 1963, pp. 631 - 640. 102. 29. , "Theory of P e r i o d i c Motion and Resonance i n P i p e l i n e s , " Engineering F l u i d Mechanics, Translated from German, B l a c k i e and Sons L t d . , London, 1961, pp. 315 - 323. 30. Kersten, R.D., and Wall e r , E.J., " P r e d i c t i o n of Surge Pressures i n O i l P i p e l i n e s , " Jour., P i p e l i n e Div., ASCE, V o l . 83, March, 1957. 31. L i s t e r , M., "The Numerical S o l u t i o n s of Hyperbolic P a r t i a l D i f f e r e n t i a l Equations by the Method of C h a r a c t e r i s t i c s , " i n A. Ralston and H.S. W i l f (eds.) : Mathematical Methods f o r D i g i t a l Computers, John Wiley and Sons, Inc., New York, 1960. 32. McCaig, I.W., and Gibson, W.L., "Some Measurements of A u t o - O s c i l l a t i o n s I n i t i a t e d by Valve C h a r a c t e r i s t i c s , " I n t e r n a t i o n a l A s s o c i a t i o n f o r  H y d r a u l i c Research, Tenth General Assembly, London, 1963. 33. Molloy, C.T., "Use of Four-Pole Parameters i n V i b r a t i o n C a l c u l a t i o n s , " Jour., Accous.tical S o c i e t y of America, V o l . 29, No. 7, J u l y , 1957, pp. 842 - 853. 34. Oldenburger, R., and Goodson, R.E., " S i m p l i f i c a t i o n of Hy d r a u l i c Line Dynamics by Use of I n f i n i t e Products," Trans. ASME, Paper No. 62-WA-55, Jour., Basic Engg., 1963. 35. Oldenburger, R., and Donelson, J . , "Dynamic Response of H y d r o e l e c t r i c P l a n t , " Trans., AIEE, Power App. and Systems, V o l . 81, Oct. 1962, pp. 403 - 418. 36. Parmakian, J . , Waterhammer A n a l y s i s , Dover P u b l i c a t i o n s , Inc., New York, 1963. 37. Parmakian, J . , " V i b r a t i o n and Noise i n Hy d r a u l i c Turbines and Pumps," symposium, I n s t , of Mech. Engrs., V o l . 181, Pt. 3A, 1966-67, pp. 74 - 83. 38. Paynter, H.M., "Surge and Water Hammer Problems," Trans., ASCE, V o l . 118, 1963, pp. 962 - 1009. 39. Paynter, H.M., and E z e k i e l , F.D., "Water Hammer i n Non-uniform Pipes as an Example of Wave Propagation i n Gradually Varying Media," Trans., ASME, V o l . 80, 1958, pp. 1585 - 1595. 40. P e s t e l , E.C., and La c k i e , F.A., M a t r i x Methods i n Elastomechanics, McGraw H i l l Book Co., New York, 1963. 41. P r e n t i s , J.M. and La c k i e , F.A., Mechanical V i b r a t i o n s - - A n I n t r o d u c t i o n  to M a t r i x Methods, Longmans, Canada, 1963. 42. Roberts, W.J., "Experimental Dynamic Response of F l u i d L i n e s , " M.S. t h e s i s , Purdue U n i v e r s i t y , Jan., 1963. 103. 43. Rocard, Y., Les Phenomenes d ' A u t o - O s c i l l a t i o n s dans l e s I n s t a l l a t i o n s  Hydrauliques, Hermann, P a r i s , 1937. 44. Schnyder, 0., "Considerations sur l e coup de B e l i e r , " B u l l e t i n Technique de l a Suisse Romande, Mar., 1936. 45. S t r e e t e r , V.L., "Water Hammer A n a l y s i s of P i p e l i n e s , " Jour., H y d r a u l i c s Div., ASCE, V o l . 90, J u l y 1964, pp. 151 - 172. 46. , "Water Hammer A n a l y s i s w i t h Non-Linear F r i c t i o n a l R esistance," Proc., F i r s t A u s t r a l a s i a n Conference on H y d r a u l i c s and F l u i d Mechanics, Pergamon Press, 1963. 47. S t r e e t e r , V.L., and L a i , C , "Water Hammer A n a l y s i s I n c l u d i n g F l u i d F r i c t i o n , " Jour., H y d r a u l i c s Div., ASCE, V o l . 88, May, 1962, pp. 79 - 112. 48. S t r e e t e r , V.L., and Wylie, E.B., "Resonance i n Governed Hydro P i p i n g Systems," Proc. I n t e r n . Symp. Waterhammer i n Pumped Storage P r o j e c t s , ASME, Chicago, Nov., 1965. 49. , "Hydraulic Transients Caused by R e c i p r o c a t i n g Pumps," Paper No. 66-WA/;FE ASME, Nov., 1966. 50. , H y d r a u l i c T r a n s i e n t s , McGraw H i l l Book Company, New York, 1967. 51. Thomson, W.T., V i b r a t i o n Theory and A p p l i c a t i o n s , P r e n t i c e - H a l l Inc., 1965. 52. " V i b r a t i o n s i n H y d r a u l i c Pumps and Turbines," Symposium I n s t , of  Mech. Engrs. V o l . 181, Pt. 3A, 1966-67. _ 53. W a l l e r , E.J. " P r e d i c t i o n of Pressure Surges i n P i p e l i n e s by T h e o r e t i c a l and Experimental Methods," P u b l i c a t i o n No. 101, Oklahoma State U n i v e r s i t y , S t i l l w a t e r , June, 1958. 54. , "Pressure Surge Control i n P i p e l i n e Systems," P u b l i c a t i o n No. 102, Oklahoma State U n i v e r s i t y , S t i l l w a t e r , Jan., 1959. 55. , "Problems of Pressure Surge A n a l y s i s of P o s i t i v e Displacement Pump Systems," P u b l i c a t i o n No. 107, Oklahoma State U n i v e r s i t y , S t i l l w a t e r , Aug., 1959. 56. W i l k i n s , R., "A Study of I r r e g u l a r i t y of Reaction i n F r a n c i s Turbines," Trans., Am. I n s t , of E l e c t . Engrs., V o l . 42. 1923, pp. 1001 -57. Wylie, C.R., Advanced Engineering Mathematics, T h i r d ed., McGraw H i l l Book Co., New York, 1966. 104. 58. Wylie, E.B., "Resonance i n P r e s s u r i z e d P i p i n g Systems," Thesis presented to the U n i v e r s i t y of Michigan, Ann Arbor, i n 1964, i n p a r t i a l f u l f i l l m e n t of the requirements f o r the degree of Doctor of Philosophy. 59. , "Resonance i n P r e s s u r i z e d P i p i n g Systems," Jour., Basic Ingg., Trans. ASME, V o l . 87, No. 4, D e c , 1965, pp. 960 - 966. 60. Wylie, E.B., and S t r e e t e r , V.L., "Resonance i n Bersimis No. 2 P i p i n g System," Jour. Basic Engg., Trans. ASME, V o l . 87, No. 4, D e c , 1965, pp. 925 - 931. 61. Z i e l k e , W., Wylie, E.B., and K e l l e r , R.B., "Forced and S e l f - E x c i t e d O s c i l l a t i o n s i n P r o p e l l a n t L i n e s , " Paper No. 69-FE-6, Trans., ASME. APPENDIX A SUMMARY OF TRANSFER MATRICES The t r a n s f e r matrices derived i n Chapter 3 are summarized h e r e i n . The n o t a t i o n and equation numbers of Chapter 3 are used here to f a c i l i t a t e c r o s s - r e f e r e n c e . A . l FIELD MATRICES (1) For i^1 pipe having constant p r o p e r t i e s along i t s length, a. F r i c t i o n taken i n t o c o n s i d e r a t i o n cosh u.I. I l •=• s inh p . I. Z i i c -Z s i n h V i cosh u.£. K C 1 1 1 1 b. F r i c t i o n neglected F. = I cos b. OJ l s i n b. OJ C. l l - jC. s i n b. OJ cos b. OJ ^ J l l l (2) For pipe having v a r i a b l e c h a r a c t e r i s t i c s along i t s length F v c , f I + f [B(x.) + 4 B(x. +s/2) + B ( x . + 1 ) ] + ^ [B(x. +s/2)B(x.) + BCx. + 1)B(x. +s/2) + B 2(x.+s/2)] + f^- [ B 2 ( x . + s / 2 ) B ( x . ) + B ( x . + 1 ) B 2 ( x . + s / 2 ) ] + |£ [B(x. + 1) B 2 ( x . + s / 2 ) B ( x . ) ] (3) P a r a l l e l pipes r5 / n ( ? ? / n ) - r i l / n ? / n (3.17) (3.18) (3.36) (3.37) 106. A.2 POINT MATRICES (1) S e r i e s connection '1 0' P sc 0 1 V J (2) Valves and o r i f i c e s a. O s c i l l a t i n g valve d i s c h a r g i n g i n t o atmosphere 1 0 0 1 0 0 1 b. O r i f i c e d i s c h a r g i n g i n t o atmosphere 1 0 -2H /Q o x o P' = ov -2H /Q o x 0 2H k / t o o c. O r i f i c e at intermediate s e c t i o n 1 0) vi -2 AH /Q o x o (3) J u n c t i o n of branch and main a. Dead end branch 1 bde U i . - 2 / u n . 0 1 b. Branch with constant head r e s e r v o i r I 1 u 2 2 / u 2 1 bres (3.43) (3.53) (3.58) (3.59) (3.71) (3.75) 107. c. Branch having o s c i l l a t i n g v a l v e 1 fP/ 1 2 bov 0 1 0 (3 0 0 1 d. Branch having o r i f i c e b o r f (3.89) APPENDIX B EXAMPLE The branch system shown i n F i g . 5.6a i s analyzed h e r e i n to i l l u s t r a t e the procedure of c a l c u l a t i n g the frequency response by usi n g a s l i d e r u l e or a desk c a l c u l a t o r . By t a k i n g d i f f e r e n t values of O J ^ and f o l l o w i n g the procedure o u t l i n e d below, the frequency response diagram shown i n F i g . 5.6b can be p l o t t e d . DATA OJ = 2.0 L , = 3.0 sec. r th R = 0.0 k 0.2 x = 1.0 H = 100 f t " o o Q = 0.314 cu f t / s e c x o Dimensions of the system are shown i n F i g . 5.6a. Components of t r a n s f e r matrices w^ u = 2 T T / 3 . = 3.094 rad/sec th co = OJ O J . . = 2. x 2.094 = 4.189 rad/sec r t h Pipe 1: b l = & i / a l = 2000/4000 = 0.5 sec A x = TT bf*/4 = TF (2) 2/4 = 3.1416 sq f t Ci = al/(gA1) = 4000/(32.2 x 3.1416) = 39.542 s e c / f t 2 S u b s t i t u t i o n of these values i n t o Eq. 3.18 y i e l d s f l l = f22 = cos(0.5 x 4.189) = - 0.5 f 2 1 = - 39.542 sin ( 0 . 5 x 4.189) j = -34.244 j f 2 1 = - j sin ( 0 . 5 x 4.189)/39.542 = - 0.022 j 109. Proceeding i n a s i m i l a r manner, the f o l l o w i n g f i e l d m a t r i x , F 2 , f o r pipe 2 i s obtained 0.500 - 0.007 j ' F 2 = Branch pipe: 102.732 j 0.500 Since the branch pipe i s made up of a s i n g l e pipe U = F Proceeding s i m i l a r l y as above, the f o l l o w i n g values of the elements of the f i e l d matrix f o r the branch are obtained: 'X, u i i = 0.5 U 1 2 = - 0.0154 j S u b s t i t u t i o n of these values i n t o Eq. 3.71 y i e l d s the f o l l o w i n g p o i n t matrix f o r the j u n c t i o n of the branch and the main 1.0 : - 0.0308 j ' bde 0.0 1.0 I t i s c l e a r from the block diagram shown i n F i g . B that U = F * P b d e F l These matrices may be m u l t i p l i e d i n a schematic manner as shown i n Table R VI. Since hi = 0 (constant head r e s e r v o i r ) the second column i n the matrices F i , P ^ g F l > a n < i ^ P ^ d e ^ 1 i s m u l t i p l i e d by zero. Thus the elements i n the second column of these matrices are unnecessary and t h e r e f o r e may be dropped. The unnecessary elements i n Table VI are i n d i c a t e d by a h o r i z o n t a l dash. Note that o r d i n a r y t r a n s f e r matrices have been used because there i s only one f o r c i n g f u n c t i o n . Hence, ui 3, U23, and U31 are zero and U33 i s B L o 2 R B R ' L — © — R -o— L 3 L R ^bde •"a 2 2 I 3 U = F 2 P b d e F 1 L 3 F i g . B . B l o c k d iagram for branch system of F ig . 5 6 a u n i t y i n Eqs. 4.33 through 4.35. S u b s t i t u t i o n of these values and those f o r u n and u 2 i c a l c u l a t e d i n Table VI i n t o Eq. 4.33 y i e l d s R q-j = - 0.0584 + 0.0127 j Hence, i t f o l l o w s from Eqs. 4.34 and 4.35 that q 3' = 0.0600 - 0.0131 j and h 3 = - 1.8134 - 8.3215 j Hence 111. h = 2 h 3 /H = 0.170 r 1 3 ' o q r = 2 | qg | /QQ = 0.390 The phase angle between head and the r e l a t i v e gate opening = t a n - 1 [ ( - 8.3215)/(- 1.8134)] = - 102.29° The phase angle between discharge and r e l a t i v e gate opening = t a n - 1 [ ( - 0.0131)/0.0600] = - 12.29° TABLE VI. SCHEME FOR MULTIPLICATION OF TRANSFER MATRICES bde 1.000 0.000 0.500 -0.031 j 1.000 -0.0073J 102.732J 0.500 - 0.500 -34.244 j - 1.555 -34.244 j - 1.0273 142.595 j = 12 13 

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