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A computer program for surge tank analysis Parmley, Leslie James 1960

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A COMPUTER PROGRAM FOR SURGE TANK ANALYSIS by LESLIE JAMES PARMLEY B.A.Sc. (Civil Engineering), University of British Columbia, 1958 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in the Department of CIVIL ENGINEERING We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA December, I960 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 m e n t of the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree th a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r 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 g a i n s h a l l not be allowed without my w r i t t e n permission. Department of CIVIL ENGINEERING The U n i v e r s i t y of B r i t i s h Columbia, Vancouver 8, Canada. D a t e K i r c h 28, 1961 i i ABSTRACT A method of solving surge tank problems with an electronic digital computer is presented in this thesis. The basic differential equations governing the transient behavior of the water mass in a surge tank system are developed in dimensionless form. A program prepared for the ALWAC III E computer at the University of British Columbia, solves these equations by a numerical integration process of the Runge-Kutta type. This program is suitable for the solution of a wide range of surge tank problems involving a single surge tank either upstream or downstream of the power unit in a hydroelectric installation. Solutions for the three basic types of surge tanks (simple, restricted orifice, or differential tank) under any of the three conditions for turbine discharge (flow, gate or power variation) are possible. The surge tank geometry and system coefficients may be altered to suit any particular circumstance. Also, the program contains a procedure which allows the incorporation of a variable turbine efficiency term in the calculations. Several numerical examples are included to illustrate the use of this program. A copy of the program tape is on f i l e with the Department of Ci v i l Engineering at the University of British Columbia. V ACKNOWLDEGEMENT The author wishes t o thank h i s a d v i s o r , Dr. E. Ruus. He i s a l s o indebted to Dr. R. F. Hooley and the s t a f f of the Computing Centre at the U n i v e r s i t y of B r i t i s h Columbia f o r t h e i r a s s i s t a n c e i n the use of the computer. December, I960 Vancouver, B r i t i s h Columbia i i i TABLE OF CONTENTS PAGE INTRODUCTION 1 CHAPTER I SURGE TANKS 2 Part 1 Simple Surge Tank 3 Part 2 Restricted Orifice Surge Tank 5 Part 3 Differential Surge Tank 7 Part 4 Variation of Turbine Discharge 9 Part 5 Methods of Solution 10 CHAPTER II DIMENSIONAL ANALYSIS . 12 Part 1 Dimensionless Ratios 12 Part 2 Dimensionless Equations 13 CHAPTER III METHOD OF SOLUTION 17 Part 1 Numerical Integration 17 Part 2 Efficiency Grid 18 Part 3 Data 19 CHAPTER IV COMPUTER PROGRAM 22 Part 1 Program Outline 22 Part 2 Program Details 2 3 CHAPTER V APPLICATIONS 28 BIBLIOGRAPHY 32 TABLE OF CONTENTS (cont.) APPENDIX I NUMERICAL EXAMPLES Part 1 Simple Surge Tank Part 2 R e s t r i c t e d O r i f i c e Surge Tank Part 3 D i f f e r e n t i a l Surge Tank APPENDIX II DATA FORMAT APPENDIX I I I COMPUTER PROGRAM APPENDIX IV EFFICIENCY GRID APPENDIX V SUMMARY OF EQUATIONS APPENDIX VI FLOW CHART OF COMPUTER PROGRAM LIST OF FIGURES Figure ( l ) Simple Surge Tank Figure (2) R e s t r i c t e d O r i f i c e Surge Tank Figure (3) D i f f e r e n t i a l Surge Tank Figure (k) Turbine Performance Curve Figure (5) Types of Surge Tanks v i NOTATIONS General g Acceleration of gravity L Length of pipeline A Cross-sectional area of pipeline V Water velocity through pipeline Q Water discharge through pipeline CV2, Friction loss plus velocity head in pipeline, where C i s positive i f V i s positive a Cross-sectional area of penstock v Water velocity through penstock q Water discharge through penstock cv 2 Friction loss in penstock h Net head on turbine hj Gross head on turbine h 0 Turbine design head Q 0 Turbine discharge at design head and f u l l gate p Q Turbine power output at design head and f u l l gate e 0 Turbine efficiency at design head and f u l l gate E Effective turbine gate area E 0 Effective turbine gate area at design head and f u l l gate p Turbine power output K Friction loss across o r i f i c e , positive for outflow from tank K0 Friction loss across o r i f i c e , associated with Q 0 d| ,d 2 Height above or below reservoir w.s. at which cross-sectional area of tank changes. NOTATIONS (cont.) General (cont.) "Ft Cross-sectional area of tank below d, F<: F 2 Cross-sectional area of tank between d, and dz J?3 Cross-sectional area of tank above d 2 (Note: For differential tanks this excludes the cross-sectional area of the riser.) Simple and Restricted Orifice Tank x Height of w.s. in tank above reservoir w.s. y Conduit pressure head above reservoir w.s. at the surge tank d 3 Height of tank overflow crest above reservoir w.s. S Rate of overflow from tank s Rate of overflow from tank under a unit head Differential Tank x Height of w.s. in riser above reservoir w.s. y Height of w.s. in tank above-reservoir w.s. d 3 Height of riser overflow crest above reservoir w.s. S Rate of overflow from riser s Rate of overflow from riser under a unit head f Cross-sectional area of riser Abbreviations cr. Carriage return sp. Space w.s. Water surface 1 INTRODUCTION In hydroelectric installations a closed conduit between the reservoir and the turbine is subjected to waterhammer pressure when the turbine discharge i s altered. If the conduit is long the expense of pro-viding conduit strength to resist this pressure may be excessive. Water-hammer pressure may also seriously interfere with proper turbine regulation. To offset these conditions a surge tank is often- installed near the power unit. The function of this tank is to reduce waterhammer pressure in the conduit and to act as an auxiliary storage reservoir, supplying or accepting water according to the requirements of the turbine. The differential equations governing the transient behavior of the water mass in a surge tank system are non-linear and hence have no direct solution by integration except in a few special cases. Consequently, i t i s often necessary to resort to some form of numerical or graphical integration. The purpose of this thesis is to present a program for the ALWAC III E electronic computer at the University of British Columbia which w i l l solve surge tank problems by numerical integration. It is proposed that this program be suitable for the solution of any surge tank problem involving a single surge tank located either upstream or downstream of the power unit. 2 CHAPTER I SURGE TANKS The differential equations governing the transient action of the water mass in a surge tank system are based on two requirements: 1) continuity of flow at the junction of the conduit and surge tank, and 2) dynamic equilibrium of the water mass in the pipeline. The general assumptions involved in the derivation of these equations are as follows: 1) Water is incompressible and conduit walls are rigid. This results in pressure waves of infinite velocity. 2) Reservoir volume is large with a constant water surface elevation. 3) The mass of water in the surge tank is small and is usually neglected.^" 4) Transient friction losses are calculated as i f steady state 2 conditions prevailed. Charles Jaeger, Engineering Fluid Mechanics. Blackie e Son Limited, London, 1956, p.200. 2 Turgut Sarpkoya, "The Effect of Unsteadiness on Hydrodynamic Resistances and on Mass Oscillations in Surge Tanks," Transactions of the Engineering  Institute of Canada. Vol. 3, No. 3 (November 1959), pp.96-99. 3 With these assumptions, basic differential equations satisfying the two requirements previously mentioned are derived for the three common types of surge tanks. Part 1 Simple Surge Tank A diagram of a simple surge tank i s shown in Figure ( l ) . The connection between surge tank and conduit i s fashioned to allow the inflow and outflow of water without appreciable head loss. The head tending to accelerate the water mass in the pipeline forms gradually with the rise or f a l l of the water surface in the tank. This sluggish action although simplifying turbine regulation by the governor results i n a tank of greater volume than the other two types. The basic differential equations are: 1) Continuity Equation - (Q-q-S)-^- (l) 2) Dynamic Equation ^ = - (x + CV2) (2) when pipeline f r i c t i o n and velocity head are neglected the transient behavior of the system i s one of simple harmonic motion and direct integration i s possible. For this simplified case the variation of the water surface in the surge tank, i f the flow to the turbine i s suddenly rejected, is given by: From equation (3) the magnitude and time of occurrence of the f i r s t maximum upsurge are easily obtained as: 4 ^ s Surge Tank Figure ( l ) Simple Surge Tank 5 x, - V0 t (3b) (3a) These two expressions appear l a t e r i n connection with dimensionless r a t i o s . D i r e c t i n t e g r a t i o n i s a l s o p o s s i b l e w i t h p i p e l i n e f r i c t i o n and v e l o c i t y head 3 i n c l u d e d , but o n l y f o r the case of sudden fl o w r e j e c t i o n . Part 2 R e s t r i c t e d O r i f i c e Surge Tank A diagram of a r e s t r i c t e d o r i f i c e surge tank appears i n Figure (2). I t i s d i s t i n g u i s h e d from the simple surge tank by the r e s t r i c t i o n between the conduit and the tank. When water flows i n t o or out of the tank t h i s r e s t r i c t i o n develops considerable f r i c t i o n l o s s . Thus a l a r g e a c c e l e r a t i n g head i s formed immediately upon a l t e r a t i o n of the t u r b i n e discharge and conduit discharge i s r a p i d l y adjusted to meet t u r b i n e requirements. T h i s r e s u l t s i n a tank of r e l a t i v e l y s m a ll volume. The s i z e and shape of the o r i f i c e c o n t r o l s the magnitude of the i n i t i a l a c c e l e r a t i n g head. The b a s i c d i f f e r e n t i a l equations are: l ) C o n t i n u i t y Equation dx dt => ( Q - q - S ) - i - U) y = (x-K) (5) 2) Dynamic Equation J | - - ^ (x + CV 2- K) (6) where K = K 0 {^f Charles Jaeger, Engineering F l u i d Mechanics. B l a c k i e e Son L i m i t e d , London, 1956, p. 196. 6 0 a) Sudden Flow Rejection b) Sudden Flow Acceptance Figure (2) Restricted Orifice Surge Tank 7 Direct integration of these equations when pipeline friction and velocity head are considered, is possible only for the case of sudden flow rejection.^ Part 3 Differential Surge Tank The sudden formation of an accelerating head with a restricted orifice surge tank may complicate turbine regulation by the governor. Addi-tional inertia in the rotating elements of the power unit or a differential surge tank are possible solutions to this problem. The differential surge tank, a diagram of which appears in Figure (3), is basically a combination of the other two types of surge tanks. A small diameter riser or inner tank allows the rapid, but not instantaneous, formation of an accelerating head. The rate of rise or f a l l of the water surface in the main or outer tank is controlled by the size and shape of the orifices. The basic differential equations are: l) Continuity Equations ~ - (Q-q*Q0*J*^- - S)-jr~ (7) 2) Dynamic Equation ^ (x + CV2) (9) These equations have no direct solution by integration. It is noted that four possibilities for overflow between the tank and riser exist: *^ Josef Frank, Nichtstationare Vorgange in den Zuleitungs - und  Ableitungskanalen von Wasserkraftwerken, Springer-Verlag, Berlin, 1957, p.217. a) Sudden Flow Rejection b) Sudden Flow Acceptanc Figure (3) Differential Surge Tank 9 1) free overflow from riser to tank, 2) free overflow from tank to riser, 3) submerged overflow from riser to tank, and 4) submerged overflow from tank to riser. The equation S - a [(x-d 3f*- (y-d3f2' where (x-d3) =0 i f x < d 3, and (y-d3) =0 i f y < d 3 is a reasonable approximation for overflow under a l l these conditions.^ This equation is also valid for overflow from the simple or restricted / \ 3 / 2 orifice surge tank i f the term (y-d3) is omitted. (10) Part 4 Variation of Turbine Discharge The solution of the basic differential equations presented in the previous sections depends to a large extent upon the manner in which the turbine discharge is altered. There are three basic ways to alter turbine discharge: 1) Flow variation with time, where turbine discharge is a direct function of time. q - J(t) (11) 2) Gate variation with time, where turbine discharge is a function of the effective turbine gate area, and the net head. The dis-charge is assumed to follow the orifice law so that Horace William King, Handbook of Hydraulics. McGraw-Hill Book Company, Inc., New York, 1954, p.4-18. 10 q - E J2gh (12) where E = j{t) (12a) 3) Power v a r i a t i o n with time, where turbine discharge i s a function of turbine power output, net head and turbine e f f i c i e n c y . From the expression for turbine power output, the discharge becomes 8.8 p /, _ • <* • - e ~ r ( 1 3 J where p = j(t) (13a) The expression for the net head term used above i s V 2 h = h g + x + — - - K - c v 2 ilk) where K = 0 f o r simple and d i f f e r e n t i a l surge tanks. This expression may be e a s i l y v e r i f i e d by reference to Figures ( l ) , (2) and (3) . Part 5 Methods of S o l u t i o n As has been mentioned previously solutions f o r surge tank problems through d i r e c t i n t e g r a t i o n are possible f o r only a few s p e c i a l cases. However, approximate methods of s o l u t i o n have been developed f o r several types of surge tank problems. For convenience these methods are u s u a l l y presented i n the form of a graph or t a b l e d Graphical or numerical i n t e -gration procedures, however, are frequently used to check these approximate solutions or to solve problems for which no approximation e x i s t s . Several graphical methods of int e g r a t i o n have been devised f o r the s o l u t i o n of John Parmikan, Waterhammer Anal y s i s . P r e n t i c e - H a l l , Inc., New York, 1955, P. 128. 11 surge tank problems. These often become quite involved, however, and usually numerical integration procedures are more satisfactory. Both graphical and numerical integration methods are time consuming and laborious. Some means of automatic computation i s therefore desired. Charles Jaeger, Engineering Fluid Mechanics, Blackie e Son Limited, London, Glasgow, 1956, p. 255. Ibid.. p. 197. 12 CHAPTER I I DIMENSIONAL ANALYSIS Part 1 Dimensionless R a t i o s To f a c i l i t a t e c a l c u l a t i o n by e l e c t r o n i c computer the b a s i c d i f f e r -e n t i a l equations o u t l i n e d i n Chapter I are changed to dimensionless form. T h i s i s necessary to s i m p l i f y the equations and t o g e n e r a l i z e the c a l c u l a t i o n procedure t o i n c l u d e a wide range of surge tank geometry and flow c o n d i t i o n s . To accomplish t h i s , two a d d i t i o n a l parameters are s e l e c t e d : where F i s some assumed c r o s s - s e c i o n a l area of the surge tank. These w i l l be recognized from equations (3a) and (3b) as the expressions f o r the maximum upsurge and time of occurrence r e s p e c t i v e l y , f o r the case of sudden flow r e j e c t i o n w i t h f r i c t i o n l e s s flow. The ^ term i s omitted from equation (3b) f o r convenience. With these two parameters and the p r e v i o u s l y d efined value Qo the f o l l o w i n g dimensionless r a t i o s are formed: T -X - *_ x* Y = 2_ x* 13 Part 2 Dimensionless Equations Differentiation and combination of the dimensionless ratios pre-sented i n the previous section yield the expressions: dx dt xj? dX t * dT Qo _ dX F* dT dt d£ dt x*_ dY t* dT — dU t* dT Qo "7* dT These expressions are substituted into the basic differential equations outlined in Chapter I to form the following dimensionless equations: l ) For the simple surge tank equations (l) and (2) become § • (» - D - W) f (15) and dU - - (X • pU2) dT respectively, where D = ri— , CV<? and 0 -(16) 2) For the restricted o r i f i c e surge tank equations ( 4 ) , (5) and (6) become — - (U - D - W) — dT V ; F (17) Y = X - 6 (D - U) (l«) and dU = - [ X • 01T dT 6 (D - U)' respectively, where D = -r- , W = — , p Qo Qo CVo" Ko — i and b = —» X * X * (19) 14 3) For the differential surge tank equations (7), (8) and (9) become - /Y-X § - (U - D • - W) Y (20) b ' F (21) and dU dT (X + pu2) respectively, where D w CV<f 7TT and 6 (22) The general expression for surge tank overflow, equation (10), in its dimensionless form becomes W (X - aj/z- (Y a 3 ) 3 / 2 ] (23) where &2 Q, (x*) z and a, = —r The differential form of this equation is required for the numerical inte-gration procedure described later in Chapter III. Therefore, differenti-ating equation (23) with respect to T yields: dW dT dY dT (24) The expressions for the three types of turbine discharge yield the following dimensionless equations: l) Equation (11) for flow variation with time becomes D 3 _ Qo (T) (25) 15 2) Equation (12) f o r gate v a r i a t i o n w i t h time becomes D - a - - - Jjp (26) where T ™ 1 7 * j " ^ ' H = ~ i and H c = ^ 3) Equation (13) f o r power v a r i a t i o n w i t h time becomes 8.Bp Qo 8.8p 0 H e where P - 2- - " - ( ( T ) . H = ^ and H c - ^ P<> J x* x * For the purpose of t h i s t h e s i s the f u n c t i o n J ( T ) i s l i m i t e d t o a second degree expression j(T) - b,T 2 + b zT + b 3 (28) where b, , b 2 , and b 3 are a r b i t r a r y constants. The expression f o r the net head on the t u r b i n e , equation (14), i n dimension-l e s s form becomes where H, - -j, 0 - — a n d 9 = ( £ ) 2 ) , x* x * a x * These expressions f o r ^ ( T ) and H are s u b s t i t u t e d i n t o equations (26) and (27) t o y i e l d the f o l l o w i n g : 16 l ) Equation (26) f o r gate v a r i a t i o n with time becomes b. T +b aT+b 3 'Hg + X • 0U2 - 6(D-U) 2 - 6D 2 2) Equation (27) for power v a r i a t i o n with time becomes (b,T 2+b 2T +b 3) H c e, (30) [Hg+X+0U2-6(D-Uf - 6D 2] ( 3 D Again, the d i f f e r e n t i a l form of these l a s t two equations i s required f o r the numerical procedure described l a t e r i n Chapter I I I . D i f f e r e n t i a t i n g with respect to T and c o l l e c t i n g terms r e s u l t s i n the following expressions: l ) Equation (30) f o r gate v a r i a t i o n with time becomes dD dT (2b,T+b 2)H + ( b , T 2 + b 2 T + b 3 ) { M f ) + [0U + 6(D-U)] •§} + (b, T 2+b 2T+b 3) [S ( D - U ) + w] 2) Equation (31) f o r power v a r i a t i o n with time becomes (32) dD dT (2b,T+b 2)H - ^ (b,T 2+b 2T +b 3) H j j - - 2(b, T a+b 2T+b 3) [6(D-U) + 6D ] 2 ( b ( T 2 , b 2 T . b 3 ) { j ( g ) • [0U + 6 ( D - U ) ] - 2(b, T 2+b 2T+b 3) l"o(D-U) + 6D ] efl n_ U J dU dT (33) The necessary d i f f e r e n t i a l equations f o r automatic computation by the computer are now established. A resume of these equations and co-e f f i c i e n t s appears i n Appendix V. 1 7 CHAPTER III METHOD OF SOLUTION Part 1 Numerical Integration Many ingenious methods for solving differential equations by numerical integration have been devised. The method selected as most suit-9 able for the solution of surge tank problems is of the Runge-Kutta type. Methods of this type are self-starting. That is , no auxiliary device such as iteration or a Taylor1s-series method is required to determine additional starting values. This simplifies the computer program considerably. Also, with this type the interval between each step of the calculation may easily be altered at any time. The recurrance formula selected from those of the Runge-Kutta type is where ( X ) T _ ^ - ( X ) ^ . f (J*)^ (3ia) For this formula the trunciation error is of the order of (AT) . The differential equations presented in Chapter II are solved by this formula in the following manner: S.H.Crandall, Engineering Analysis, McGraw-Hill Book Company, Inc., Toronto, 1956, p.177. •• 18 1) The i n i t i a l values (X)T_(, (U)T_( etc., at the beginning of the time interval are used to calculate the slopes (^ )T_, J (CJT")t , etc., from the differential equations. 2) Average values (XL (/ , (U) , etc., are calculated from eouation (34a). dX 3) These. average values are used to calculate the slopes (^p)r_j/2 > ( ^ ) r ^ e t c , from the differential equations. 4) Final values (X)T , (U) etc., at the end of the time interval are calculated from equation (34). 5) These final values become the i n i t i a l values for the next cycle and the process is repeated. Thus the differential equations are integrated simultaneously and each curve is projected step by step for as many cycles as required. Part 2 Efficiency Grid Once values for the various constants and i n i t i a l conditions are known, the differential equations presented in Chapter II are readily solved de by the integration procedure just described. However, the terms e and p^j-r which appear in equation (33) cause some difficulty i f the turbine efficiency is not assumed constant. Provision is made for an efficiency grid which enables the use of a variable turbine efficiency in the calculations. This grid consists of spot efficiency values taken from a turbine performance curve similar to that shown in Figure (4). These values are taken at intervals of l& along both co-ordinate axes and are stored in a portion of the main memory of the computer. Appendix IV illustrates an efficiency grid based on an average performance curve for a Francis turbine 19 under a 250 f t . to 500 f t . head. For the sake of completeness this grid includes efficiency values for a considerable range of head and discharge. Also, note that the efficiency values extend slightly past the f u l l gate line. This prevents d i f f i c u l t i e s i n the table look-up scheme which selects the re-quired efficiency value. The table look-up scheme is incorporated in the computer program. After each g cycle of the integration the values D and jr- are computed and the four efficiency values closest to these co-ordinates are selected from the grid. Interpolation between these four values results i n a fi n a l value de for the turbine efficience. A value for the term is obtained by dividing the difference between this f i n a l efficiency value and the previous one by Although this is only approximate, the term ^  i s usually small and hence has l i t t l e influence on the calculations. Part 3 Data The constants and i n i t i a l conditions for the particular surge tank system under consideration must also be stored in the main memory of the com-puter. Appendix II l i s t s each item of data in i t s correct format together with i t s binary scaling, and working channel II address. To make the computer program as complete as possible i t was necessary to include 32 items of data. Some of these need additional explanation: F* F * F * 1) The terms TT-, TT-, and TT- provide for possible changes in the cross-sectional area of the surge tank. The terms a, and ot 2 R.E.Krueger, Selecting Hydraulic Reaction Turbines, Bureau of Reclaimation, 1954, P.38. 20 % Desgin Discharge Figure (i+) Turbine Performance Curve 21 enable the computer to select the correct cross-sectional area for each cycle of the calculations. 2) The terms 5, and 6 2 provide for the possibility of different orifice discharge coefficients for flow out from and into the surge tank respectively. For a simple surge tank 6, = 6 2 = 0. 3) The term x 0 gives the position at which turbine gates are blocked to prevent further opening. If gates are not blocked T 0 ° 1. 4) The terra T 0 represents the time at which the turbine flow, gate position, or power output reaches its final steady state value 5) The term — represents the cross-sectional area of the riser in a differential surge tank. For a simple or restricted orifice 6) Turbine efficiency at design head and f u l l gate is given by the term e„. If e e is input as zero the table look-up scheme for the efficiency grid and the calculation of turbine power output are omitted in the computer program sequence. This speeds up the calculations appreciably. If a constant turbine efficiency is required a one word program change must be made. (Channel 78, address 37, word 00000000.) The remaining items of data have been explained previously. 22 CHAPTER IV COMPUTER PROGRAM Part 1 Program Outline The program proper, consisting of 12 channels, is stored in main memory channels 70 to 7b inclusive. An auxiliary routine whose function is explained later i s stored in main memory channel 7c. A l l program channels operate in working channel I. Data i s stored in working channel II and intermediate calculations are stored in working channel III. Subroutines used include the decimal typewriter input routine, the decimal typewriter output routine, and the square root routine A-17. These a l l operate in work-ing channel IV. Routine A-17 is stored in main memory channel 7d and a copy is included on the program tape. The input and output routines are permanently stored in the computer. I n i t i a l data is stored in main memory channel 80. Current data is stored in main memory channel 81 at the end of each cycle of calculations. A copy of the computer program is provided in Appendix III. A flow chart giving the sequence of computer operations i s found in Appendix VI. Briefly, the main steps in the calculations are as follows: 1) Efficiency values and data are input, converted to binary form, and stored. 2) Column headings T, U, X, Y, D and P are typed out; followed by the i n i t i a l values of these variables. F* 3) The correct values of y and 6 are selected, and the values dX dU dY ^ and -jjp are calculated for the appropriate type of surge dW tank. The value -yr is also calculated i f required. 2 3 4 ) The value is then determined for flow variation, gate vari-ation, or power variation depending on the setting of jump switches #1 and #2. (See Part 2 , Program Details.) 5 ) From equation (34a) the values U)T.Vz, (U)T_^ , ( Y ) T _ ^ , (D)r_//2 , and (W)T_y2 are calculated and stored. 6 ) The net head term H is calculated and the appropriate efficiency value determined. This permits the calculation of the power output de term P and the value - j ^ . 7) The process is repeated from step 3 ) except that in step 5 ) the values (X) T , (U) , (Y) T , ( D ) t , and (W)T are calculated from equation ( 3 4 ) . These values are stored in channel 81 to complete one cycle of the calculations. 8) The next cycle starts from step 3 ) using the values obtained in step 7). After every five cycles the values of the variables T, U, X, Y, D, and P are typed out. The time required to complete 5 cycles of calculations and type out the values of the 6 variables varies between 30 seconds and 50 seconds. This depends on the type of the turbine discharge and whether or not the efficiency grid is used. To change the number of cycles of calculations before the values of the variables are typed out the required number of cycles, scaled ( 2 ) / 6 must be placed in addresses 37 and 3 b of channel 7b. Part 2 Program Details 1) Start I, Efficiency Values The entry code for start I is 7000 cr. The computer then normally expects an inpit of 800 efficiency values. This input may be reduced to 480 24 values i f only the central portion of the efficiency grid between the maximum and minimum allowable head is required. A short correction tape at the end of the program tape makes the necessary program alterations for the input of this smaller grid. In either case the sequence of input for the efficiency values must be from the lower left corner to the upper right corner, and from left to right across the grid. (See Appendix III.) As each efficiency value is input i t is converted to binary form and stored in the appropriate location in the main memory. When the efficiency values have been input for the first time, a reload tape for channels 50 to 68 inclusive (54 to 62 for the smaller grid) should be made. Efficiency values may thereafter be input directly with this tape thus eliminating the time consuming conversion to binary form. 2) Start II, Data The entry code for start II is 7018 cr. The computer then expects the input of 32 items of data which are converted to binary form and stored in channel 80. The program sequence as previously outlined is then followed. 3) Additional Starts The entry code for start III is 7188 cr. The computer utilizes the ini t i a l data previously stored in channel 80 and continues through the program sequence. This allows the calculations to be restarted without the input of the data items. The entry code for start IV is 718a cr. The computer utilizes the current data previously stored in channel 81 and continues through the program sequence except that no column headings are typed out. Data changes can there-fore be made after any cycle of the calculations. Also calculations may be 25 stopped and then continued at a later date i f a reload tape of channel 81 is made. 4) Switches #1 and #2 The position of these switches determines the type of turbine dis-charge to be used in the calculations. For flow variation with time both switches are at the normal position. For gate variation with time switch #1 is at the jump position and switch #2 at the normal position. For power variation with time switch #1 is at the normal position and switch fi2 at the jump position. 5) Switch #3 With switch #3 at the normal position the computer halts after the i n i t i a l type out. This provides a chance to make the required settings for switches #1 and #2. If switch #3 is now moved to the proceed position the program sequence continues with AT = 0.002. If switch #3 is moved to the jump position the program sequence continues with AT = 0.01. These two time intervals are included primarily for calculations involving the differential surge tank. When X -> Y the equations are very sensitive and the solution becomes erratic i f the smaller time interval is not used. Therefore, with a differential tank i t i s advisable to change to the smaller time interval when Y - X < 0.005 approximately. The smaller time interval should also be used during surge tank overflow because of the large increase in the overflow dis-charge due to a relatively small increase in head. Better accuracy i s obtained i f the small time interval i s used for a few cycles of calculations at sudden breaks in the continuity of the variables or when these variables are changing rapidly. The position of switch #3 should normally be altered during the type out period of the program sequence. The computer w i l l halt at the end of the 26 type out i f the switch i s moved to the normal position. This allows time to examine the typed values and decide on the appropriate time interval. AT AT The values — = 0.005 and -y =» 0.001 are stored in channel 72, address 37 and 3b, respectively. These may be easily altered i f desired through the use of the auxiliary input-output routine described later. The scaling for the time intervals i s (.2; 6) Flags Flag #1 i s automatically turned on when the computer changes from the case of power variation to the constant gate case. This occurs when the power output required cannot be met under the available head, even at maximum gate opening. (See Figure (4)-) When the available head is sufficient for the change back to power variation, flag #1 is turned off. Flag #2 i s automatically turned on when the net head increases or decreases beyond the range of the efficiency grid. When this happens the last value of e obtained from the grid i s used in the calculations u n t i l the net head is again in the efficiency grid range. Flag #2 is then turned off. If a constant turbine efficiency i s used flag #2 w i l l be on during the entire calculation. 7) Channel 7d, Input-Output This auxiliary routine is included to provide a convenient means for the input of data changes or the output of additional data in decimal form. The format for the input of a decimal number i s : 7c00 cr. mm sp. nn sp. oo sp. (input a decimal number) sp. cr. nn sp. oo sp. (input second number) sp. cr. etc., 27 and the format f o r the output of a decimal number i s : 7c02 c r . mm sp. nn sp. oo sp. (output of decimal number) cr. nn sp. oo sp. (output of second number) cr. etc., where mm i s the channel number, nn i s the s c a l i n g , and oo i s the working channel II address. Appendix II gives the required s c a l i n g andaidress f o r each item of data. 28 CHAPTER V APPLICATIONS The computer program as outlined in Chapter IV is suitable for the solution of a wide variety of surge tank problems. Figure (5) illustrates the various types of surge tanks which this program is capable of handling. The geometry of the tank may be varied according to the values of F, , F 2, F 3, d, and d 2. The value of d 3 indicates whether or not overflow of the surge tank will occur. Calculations may be based on any of the three types of turbine discharge; flow variation, gate variation or power variation. Also any i n i t i a l conditions or system constants may be included in the data input. For a surge tank located downstream of the power unit a slight modification of the computer program is required. In a system of this type the pipeline section of the conduit is located downstream of the power unit. The dimensionless equations for a system of this type are as follows: U - W) |* (35) W1) (36) 2) Restricted Orifice Surge Tank § - (D - U - W) (37) l) Simple Surge Tank dX dT (D dU dT (X Y = X - 6 (D - U) 2 (38) 29 Differential Tank Differential Tank Combined Simple (no overflow to outer tank) e Differential Tank Figure (5) Types of Surge Tanks 30 S± - X - ?UZ - &(U - D) z (39) 3) Differential Surge Tank § - (X - PU2) (42) 4 ) Net Head H o H9 - X - 0J2 + 6(U - D ) 2 - 0DZ (43) 5 ) Gate Variation (2b,T+b4)H - (b,T2+baT+b3){Mff) • [0U-6(U-D)] § dD D ^HH7 + (b, T2+b2T+b3) [6(U-D) + 6D] 6) Power Variation dD (2b,T+b2)H - ^ (b,T2+b2T+b3)|f dT ~ z - 2(b,T +b2T+b3) &(U-D)+6D e 0n 0 L. j 2(b,T2+b2T+b3) * [0U-6(U-D)] § } 2(b,TZ+bJLT+b3) [&(U-D) + 6D] eH2 e0H0 (44) ( 4 5 ) These equations differ only slightly from those for the upstream surge tank system. Hence, the computer program is easily modified to accommodate these differences. This is accomplished by means of a correction tape at the end of the program tape. The extent of the computer program is in no way limited 31 by this modification. Several surge tank problems have been solved with the computer program. A description of each problem, together with the necessary steps for computer operation and a plot of the calculated variables appears in Appendix I. Values for the maximum or minimum surges as calculated with the computer were compared whenever possible to corresponding values obtained by direct integration. Where direct integration was impossible check values were obtained from graphs of approximate solutions or by numerical integration by hand. A comparison of these values is included in Appendix I. A copy of the program tape and an efficiency grid tape for the efficiency values given in Appendix III are on f i l e with the Department of Civil Engineering at the University of British Columbia. It is hoped that this program will prove useful for solving those surge tank problems not readily solved by conventional methods. 32 BIBLIOGRAPHY 1. Crandall, Stephen H. Engineering Analysis. McGraw-Hill Book Company, Inc., New York, 1956. 2. Frank, Josef. Nichtstationare Vorgange i n den Zuleitungs - und Ableitungskanalen von Wasserkraftwerken. Springer - Verlag, B e r l i n , 1957. 3. Jaeger, Charles. Engineering F l u i d Mechanics. Blackie e Son Limited, London, 1956. 4. King, Horace William. Handbook of Hydraulics. McGraw-Hill Book Company, Inc., New York, 1954. 5. Krueger, R.E. S e l e c t i n g Hydraulic Reaction Turbines. Bureau of Reclamation, Denver, 1954.(Engineering Monographs No. 20.) 6. Parmakian, John. Waterhammer Analysis. P r e n t i c e - H a l l , Inc., New York, 1955. 7. Rich, George R. Hydraulic Transients. McGraw-Hill Book Company, Inc., New York, 1951. 8. Sarpkaya, Turgut. "The E f f e c t of Unsteadiness on Hydrodynamic Resistances and on Mass O s c i l l a t i o n s i n Surge Tanks." Transactions of the Engineering I n s t i t u t e of Canada. Vol. 3, No. 3 (November 1959), pp. 96-99. 33 APPENDIX I NUMERICAL EXAMPLES Several numerical examples have been calculated to check the computer program and to determine the r e l i a b i l i t y of the r e s u l t s . These examples also i l l u s t r a t e the v e r s a t i l i t y of the computer program as well as serving as a guide for subsequent c a l c u l a t i o n s . For s i m p l i c i t y , the same basic data are used i n each example. Basic Data A = a = 200 f t ! C = c = 0.075 s e c f / f t . L = 6440 f t . V 0 - 26 2/3 f t . / s e c . Qo = 5333 1/3 c . f . s . h 9 = 400 f t . h 0 = 346 f t . P o = 183,485 hp. e Q = 0.875 g = 32.2 f t . / s e c f I f F* i s assumed to be 6 4 O O f t ? the parameters x* and t * become: 34 These two parameters are used throughout the following calculations. When the data is in the dimensionless form outlined in Appendix II, a data tape with the entry code 701R cr. is prepared. To start the calcula-tions the program tape and the efficiency grid tape are loaded into the computer with a l l switches at the normal position. Both these tapes have check sums and any channel which is input incorrectly will be indicated via the flexowriter. When an error does occur the channel should be input a second time or until the input is correct. For a neat tabulation of the results by the flexowriter, five tab stops must be suitably arranged. The data tape is then input. The computer will type out the column headings and the corresponding i n i t i a l values. It then halts until switch #3 is moved to either the proceed or jump position depending upon the size of time interval required. Switches #1 and #2 should be moved to their appropriate positions depending upon the type of turbine discharge variation required, before switch #3 is moved. The calculations then proceed with the results typed out after every five cycles. Results as obtained from the computer for a differential surge tank under a sudden load increase are shown on the following page. In the following numerical examples the preparation of data and the steps in computer operation are indicated. Also, graphs of the functions computed and a comparison of these results with those obtained by some other method are included. The examples are divided into three sections, depending on the type of surge tank involved. COMPUTER RESULTS DIFFERENTIAL SURGE TANK - SUDDEN LOAD ACCEPTANCE T U X Y D P .000 .3750 -.1125 -.1125 .7078 .7903 .010 .375*+ -.1923 -.1180 • 7178 .7895 .020 .3765 -.2440 -.1267 .7251 .7895 .030 • 3780 -.2831 -.1368 .7308 .7895 .01+0 .3799 -.31^7 -.1478 .7356 .7895 .050 .3820 -o3Ul2 -.1595 .7396 .7895 .060 . 3 ^ 3 -.3641 -.1715 • 7*+32 .789!+ .070 .3869 -.3845 - . I838 .7*+64 .7895 .060 .3896 -.U029 -.1964 • 7*+96 .7895 .090 .3925 -.4199 -.2091 .7526 .7895 .100 .3956 -.4358 -.2219 • 7555 .7895 .110 .3967 -.4508 -.231+8 .7583 • 7895 .120 .4020 -.4650 -.2477 .7610 • 7895 .130 .4054 -.4788 -.2607 .7638 .7895 • l40 .4090 -.4921 -.2737 .7665 • 7895 .150 .4126 -.5050 -.2867 .7692 .7895 .200 .1+323 -.5650 -.351*+ .7819 .7895 .250 .4540 -.6197 -.4150 • 79*+9 .7891+ .300 .1+776 -.6707 -.»+771 .8082 .789!+ .350 .5028 -.7182 -.5372 .8212 • 7895 .1*00 • 5291 -.7623 -.5952 .8347 • 789»+ .450 .5565 -.8049 -.6509 . 8 5 U .7893 .500 • 581+9 -.8493 -.7045 .8719 .7892 .550 . .6141 -.8973 -.7569 .9020 .7885 .560 .6201 -.9090 -.7673 .9131 .7884 .570 .6261 -.9225 . - .7778 • 9217 .7866 .580 .6323 -.9331+ -.7884 .9209 .7843 .590 .6384 -.9h\9 -.7990 .9202 .7826 .600 .6446 -.9^88 -.8094 • 9197 .7812 .650 .6752 -.9719 -.8589 .9182 .7771 .700 .7052 -.9890 -.9027 .9173 .771+5 .750 • 73^3 -1.0041 -.9406 .9166 .7724 .800 .7624 -1.0176 -.9728 .9160 .7707 .«50 .7895 -1.0296 -.9995 .9155 .7695 .900 .8155 -I .0398 -1.0211 .9152 .7687 • 950 .8403 -1.0482 -1.0377 .9151 .7683 .960 .8451 -1.U496 -1.0404 .9151 .7682 • 970 .8498 -I.O509 -1.0430 .9151 .7682 .980 . 8 5 ^ -1.0522 -1.0454 .9151 .7683 .990 .8592 -I.O534 -I.0476 •9151 .7683 1.000 .8638 -I.0545 -I .0496 .9151 .7683 36 Part 1 Simple Surge Tank 1) Sudden Flow Acceptance 2) Sudden Flow Rejection 3) Gradual Gate Opening 4) Sudden Load Acceptance 5) Sudden Load Acceptance (gates blocked) 6) Linear Load Acceptance 7) Sudden Flow Rejection (cross-section change and overflow) 8) Sudden Flow Acceptance (downstream surge tank) 37 1) Sudden Flow Acceptance D e t a i l s : F, = F2 = F 3 = 1600 f t . q varies from zero to 4000 c.f.s, instantaneously. Data: T 0 U = 0 X = 0 Y 3= 0 D = 0.75 P = 0.8451 W = 0 =r 0 a2 = 0 a 3 BE 2.0 0 = 0.1656 To 1 .0 e = 0.92 = 0.875 b 3 = 0 bo = 0.75 F * F * 4.0 P 0.8 e 0.8 F t XX 4 .0 F 2 Z3 F * Fa zs 4.0 *>. 0 = 0 To = 0 3 F * H = 5-55 H 0 = 5.19 H 9 6.0 f S3 0 P 0 b 2 0 de dT = 0 e= 0 Switches: Results: Timing: Remarks: Switch ill at normal Switch #2 at normal Switch ;'/3 at jump x, = -1.5474(66*4) = - 103.16 f t . 40 sec. f o r 5 complete cycles. x, = -103.2 f t . as determined from graphs. U D P, 39 2) Sudden Flow Rejection Details: F, = F 2 = F 3 - 1600 f t . q varies from 4000 c.f.s. to zero instantaneously. Data: T = 0 U = 0.75 D = 0 P - 0 az = 0 a3 = 2.0 € ; = 0 e 0 - 0 0 = 0.8 6 = 0 F* TT » k.O * 3 6, = 0 H - 0 H c = 0 f > - 0 b 2 - 0 Switches: Switch #1 at normal Switch #2 at normal Switch #3 at jump Results: X ! = 1.2097(66%) - 81.08 f t . X2 = -0.9158(66%) - -61.05 f t . X3 = 0.7357(66%) - 49.05 f t . x4 = -0.6144(66%) =-40.96 f t . X w 0 de dT -0.45 0 0 0.75 4.0 0 0 0 Y =-0.45 a. = 0 1.0 b e - 0 F * fr~ - 4.0 T c I* f b, 0 0 0 Timing: 30 sec. for 5 complete cycles. Remarks: x, = 81.1, x 2 = - 6 l . l , x 3 = 49.1, x + =-41.0 as determined by direct integration. 3 ) Gradual Gate Opening Details: F , = F 2 - F 3 - 1600 f t f T varies parabolically from zero to 1.0 in 20 sec. T ss 0 u S3 0 X = 0 Y SS 0 D = 0 p O 0 w - 0 ss 0 az = 0 A3 cz 2.0 0 a 0.1656 To = 1.0 e ss 0 S3 0.875 b3 83 0 b 0 ss 1.0 F * F a E= =s 0.8 4.0 e &. ss SB 0.8 0 F * F, =3 S3 4.0 0 F * F 2 To za 4.0 0.25 .3 H ss 6.0 Ho s= 5.19 Hg = 6.0 F* f = 0 P as 0 b* OS 0.8 de - dT S3 0 S3 12.8 Switches: Switch #1 at jump Switch #2 at normal Switch if3 at proceed T < 0.05 at jump T > 0.05 Results: x, - -1.8007 (66%) = - 120.05 f t . Timing: 48 sec. for 5 complete cycles. Remarks: x, = - 120.0 f t . as determined by numerical integration by hand using a time interval of 5 sec. A3 4) Sudden Load Acceptance Details: F, - F 2 F 3 - 1600 f t . p varies from 69,000 hp. to 145,000 hp. instantaneously. Data: T = 0 D = 0.7078 az =• 0 e = 0.92 0 - 0.8 F* F " = A.O *3 H = 5.5095 p - 0 U P u a e 0.375 0.7903 2.0 0.875 0.8 0 5.19 0 X =-0.1125 W » 0 0 = 0.1656 b 3 - 0.3761 F* fr- = 4.0 b2 = 0 Hg = 6.0 ^ - 0 dT U Y =-0.1125 o, » 0 T 0 = 1.0 0 . 7 9 0 3 F* I" • A.O T 0 = 0 ^ = 0 b, = 0 Switches: Switch #1 at normal Switch #2 at jump Switch #3 at jump Results: x, = -1.2209(66 %) - - 81.39 f t . Timing: 49 sec. for 5 complete cycles. Remarks: x, = - 79.0 f t . as determined by numerical integration by hand using a time interval of 5 sec. 45 5) Sudden Load Acceptance (gates blocked) Details: LZ F, = F 2 - F 3 = 1600 f t f p varies from 69,000 hp. to 145,000 hp., instantaneously. Gates are blocked at 90$ of f u l l gate. Data: T 0 U 0.375 X -0.1125 Y -0.1125 D = 0.7078 P = 0.7903 W = 0 - 0 a z = 0 a3 2.0 0 es 0.1656 D 0.9 e = 0.92 = 0.875 b 3 ts 0.3761 bo = 0.7903 e 0.8 F * F* P Et 0.8 F, S3 4.0 F 2 S3 4.0 F* S3 4.0 * i 0 *z S3 0 To S3 0 F* H = 5.5095 Ho = 5.19 H, S3 6.0 f SB 0 P 0 b 2 = 0 de dT = 0 b, = 0 Switches: Switch Hi at normal Switch #2 at .jump Switch #3 at jump Results: x, = -1.0935(66%) = - 72.90 f t . Timing: 49 sec. for 5 complete cycles Remarks: x ( = - 71.0 f t . as determined by numerical integration by hand using a time i n t e r v a l of 5 sec. -0.2 -0.8 H D P 47 6) Linear Load Acceptance D e t a i l s : F l = F2 = F3 = 1 6 0 0 f t -p v a r i e s from 69,000 hp. to 145,000 hp. i n 16 sec. Data: Switches: Results: Timing: Remarks: T D P Fj* F3 H P 0 0.375 0 0.785 0.8 4.0 5.7975 0 U P e 0.375 0.3761 2.0 0.875 0.8 6, = 0 5.19 bz = 2.071 X = -0.1125 w = 0 0.1656 0.3761 4.0 0 6.0 0 0 F * F. 6 Z H 9 de dT Y = -0.1125 a ( = 0 T Q = 1.0 b 0 = 0.7903 F * F l = ^ ° T 0 = 0.2 F * ~ = 0 b, = 0 Switch #1 at normal Switch #2 at jump Switch #3 at proceed T < 0.20 at jump T > 0.20 x, = -1.2179(66*4) = - PI.19 f t . 49 sec. f o r 5 complete c y c l e s . No check value was determined f o r t h i s case. However, the r e s u l t can be compared with x, = - 81.39 f t . f o r sudden load acceptance. 49 7) Sudden Flow Rejection (cross-section change and overflow) D e t a i l s : 2 F, = 3200 f t . ; F 2 » 800 f t . F 3 = 4000 f t . 2 ; s = 500 f t . 1 / 2 / s e c . d, = - 50 f t . ; d-> = 70 f t . d3 - 80 f t . q varies from 4000 c . f . s . to zero 1 I instantaneously. Data: Switches: Results: Timing: Remarks: T = 0 U =3 0.75 D = 0 P = 0 a2 = 1.05 a3 = 1.2 e = 0 = 0 P = 0.8 e = 0 F * *'s = 1.6 = 0 H = 0 = 0 P 2= 51.03 = 0 6, = H 3 de dT •0.45 0 0 0.75 2.0 0 0 0 Y - -0.45 a, = -0.75 T 0 = 1.0 b G = 0 8.0 T Q = 0 f = ° b, = 0 Switch Hi at normal Switch #2 at normal Switch r/3 at jump x, = 1.2146(66%) - 80.97 f t . 30 sec. for 5 complete cycles X| = 81.7 f t . as determined by numerical i n t e g r a t i o n by hand using a time i n t e r v a l of 2 sec. 0 0 . 1 0 . 2 0 . 3 0.4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1 . 0 jF 51 8) Sudden Flow Acceptance (downstream surge tank) D e t a i l s : F, = F2 = F3 = 1600 f t . 2 q varies from zero to 4000 c . f . s . instantaneously. Date Switches: Results: Timing: Remarks: T = D = az = e P -H = P = 0 0.75 0 0.92 0.8 4-0 5.55 0 U = 0 P = 0.8451 a3 = 2.0 e c - 0.875 0 = 0.8 6 , - 0 H 0 - 5-19 b, = 0 Switch #1 at normal Switch H2 at normal Switch fij at proceed at jump X = 0 W = 0 0 = 0.1656 b, - 0 F| = 4.0 - 0 = 6.0 4s = 0 dT u T < 0.20 T > 0.20 Y = 0 a, = 0 T 0 = 1.0 b 0 - 0.75 = 4.0 F 2 T 0 = 0 F* F = ° b , = 0 x, = 1.5474(66 %) = 103.16 f t . 40 sec. f o r 5 complete cycle s . These r e s u l t s correspond exactly to those f o r case l ) , Sudden Flow Acceptance except f o r the change of sign and a s l i g h t d i f -ference i n the power curve due to the reversed e f f e c t of the v e l o c i t y head term. X Y Iii 53 Part 2 R e s t r i c t e d O r i f i c e Surge Tank 1) Sudden Flow Acceptance (balanced design) 2) Sudden Flow Acceptance 3) Sudden Flow Rejection (balanced design) 4) Sudden Flow Rejection 5) Linear Gate Closure 6) Linear Load Acceptance 7) Sudden Flow Rejection (combined o r i f i c e and simple tank) 54 l ) Sudden Flow Acceptance (balanced design) D e t a i l s : F, = F 2 = F 3 - 1600 ftf K 0 (outflow) = 132.7 f t . K 0 (inflow) = 144.0 f t . q varies from zero to 4000 c.f.s, instantaneously. Data: Switches: Results: Timing: Remarks: T = 0 U = 0 X = = 0 D 0.75 P ss 0.6659 W = 0 az = 0 a3 B3 2.0 0 = 0.1656 e = 0.91 e e S3 0.875 = 0 P 0.8 6 = 0.8 F * F. = 4.0 p* = 4.0 = 1.99 &2 = 2.16 H — 4.4306 = 5.19 H, = 6.0 P = 0 b 2 = 0 de dT - 0 Switch #1 at normal Switch ft2 at normal Switch #3 at jump x, = -1.1178(66z/3) - - 74.52 f t . 40 sec. for 5 complete cycles. x. = - 74.7 f t . as determined from graphs. Y = -1.1194 a, = 0 T 0 = 1.0 b 0 = 0.75 F* T 0 = 0 F* f = ° b, = 0 56 2) Sudden Flow Acceptance D e t a i l s : F, = F 2 = F 3 = 1600 f t . 2 K 0 (outflow) = 106.7 f t . K 0 (inflow) - 144.0 f t . q varies from zero to 4000 c . f . s . __TT__ instantaneously. Data: Results: Timing: Remarks: T = 0 D = 0.75 az = 0 e =0.92 p = 0.8 F * F- = A.O *3 H = 4.65 p = 0 Switches: U -P = 3^ = e D = 6 = 6, = H„ = 0 0.7065 2.0 0.875 0.8 1.6 5-19 0 Switch #1 at normal Switch #2 at normal Switch #3 at jump x, = -1.1742(66%) • X = 0 W = 0 0 - 0.1656 b 3 = 0 F * 4.0 6 2 = 2.16 H 9 = 6.0 dT w 78.28 f t . 40 sec. f o r 5 complete cycles x, = - 78.0 f t . as determined from graphs. Y = -0.9 a, = 0 T„ = LO b D = 0.75 F l -F * b . = 0 U D 58 3) Sudden Flow Rejection (balanced design) D e t a i l s : F, . F 2 = F 3 = 1600 f t . 2 K 0 (outflow) - 132.7 f t . K„ (inflow) = 144.0 f t . q varies from 4000 c . f . s . to zero instantaneously. Data : Switches: Results: Timing: Remarks: T D P Fa H P 0 0 0 0 0.8 4.0 0 0 u = p = a3 = e c = e = s. = Ho = 0.75 0 2.0 0 0 1.99 0 0 X = 0 = b 3 = F j F, 6 2 = H, -de dT -0.45 0 0 0.75 4.0 2.16 0 0 0.765 Switch #1 at normal Switch il2 at normal Switch #3 at jump x, = 0.7644(66%) - 50.96 f t . 30 sec. f o r 5 complete cycl e s . x, = 51.8 f t . as determined from graphs. a, = 0 F 2 F * f = 1.0 = 0 = 4.0 = 0 = 0 = 0 U D P 60 4) Sudden Flow Rejection D e t a i l s : F i = F2 = F 3 - 1600 f t . 2 K 0 (outflow) - 132.7 f t . K G (inflow) = 213.3 f t . q varies from 4000 c . f . s . to zero instantaneously. ( T = 0 u = 0.75 X = -0.45 Y = 1.35 D = 0 p ss 0 w S3 0 a \ = 0 a2 0 a3 = 2.0 0 = 0 To 1.0 e ss 0 = 0 b 3 = 0.75 bo = 0 = 0.8 e 0 p * F7 4.0 F* F 2 ss 4.0 F* = 4.0 ss 1.99 62 - 3.2 To = 0 F3 Ii 0 Ho = 0 H 9 = 0 F* f ss 0 P = 0 b 2 0 de dT 0 b , = 0 Switches: Switch #1 at normal Switch #2 at normal Switch #3 at jump Results: x, = 0.6449(66%) = 42.99 f t . Timing: 30 sec. f o r 5 complete cycles. Remarks: x, = 43.2 f t . as determined from graphs. 62 5) Linear Gate Closure D e t a i l s : F, = F 2 = F3 = 1600 f t . 2 K 0 (outflow) « 132.7 f t . Ko (inflow) - 144.0 f t . ZZ1 I T varies from 0.75 to zero i n 20 sec. Data: Switches: Results: Timing: Remarks: T = 0 U = 0.75 D s 0.75 P = 0.7903 *2 - 0 a 3 - 2.0 e 0.924 = O.S75 P = 0.8 e = 0.8 F* Fa 4.0 & , = 1.99 H = 5.19 Ho = 5.19 P = 0 b 2 = -3.0 X - -0.45 W = 0 0 = 0.1656 Y = F, 0.75 4.0 6 2 = 2.16 H, = 6.0 ^ = 0 dT b 0 = Kl = To = F* = f b, = -0.45 0 1.0 0 4.0 0.25 0 0 Switch #1 at jump Switch //2 at normal Switch #3 at proceed T < 0.30 at jump T > 0.30 x, = 0.7945(66^3) - 52.97 f t . 48 sec. f o r 5 complete cycles. No check value was determined f o r t h i s case. However, the r e s u l t s can be compared with x ( =* 50.96 f t . f o r sudden flow r e j e c t i o n at balanced design. Note that max-positive surge does not occur at sudden gate closure. 6 4 6) Linear Load Acceptance D e t a i l s : F, - F 2 F 3 = 1600 f t . K c (outflow) - 132.7 f t . K 0 (inflow) = 144.0 f t . p varies from 69,000 hp. to 145,000 hp. i n 16 sec. Data: Switches: Results: Timing: Remarks: T = 0 D = 0.375 a2 = 0 e = 0.7*5 0 = 0.8 F* F~ = 4.0 H = 5.7975 p = 0 U = 0.375 P = 0.3761 a 3 - 2.0 e„ = 0.875 6 = 0.8 6, - 1.99 H Q = 5.19 bz = 2.071 Switch #1 at normal Switch #2 at jump Switch #3 at proceed at jump X = -C0.125 W = 0 0 = 0.1656 b 3 = 0.3761 F * 6 2 = 2.16 H 3 = 6.0 ^ - 0 dT u -0.1125 T < 0.25 T > 0.25 a, = 0 T 0 - 1.0 b„ = 0.7903 F* F- - 4.0 T G - 0.2 F * 1 "° b. = 0 x> = -1.0818(665^ ) = - 72.12 f t . 49 sec. for 5 complete cycles . No check value was determined for t h i s case. However, i t i s inter -esting to compare t h i s r e s u l t with x ( = -81.19 f t . , t h e r e s u l t f o r the same case with a simple surge tank. 66 7) Sudden Flow Rejection (combined o r i f i c e and simple tank) D e t a i l s : F, = 3200 f t ! ; F2 =- 500 f t . 2 F 3 . 2000 f t ? ; d, =-66.7 f t . d 2 => 0 q v a r i e s from 4000 c . f . s . to zero instantaneously. I I K 0 - 133.3 f t . Data: Switches: Results: Timing: Remarks: T = D = a 2 = e = P -• 111 = F 3 H = P = 0 0 0 0 0.8 3.2 0 0 U = 0.75 P = 0 a 3 = 2.0 e 0 = 0 0 = 0 6, = 2.0 H 0 = 0 b 2 = 0 0.45 Switch #1 at normal Switch ff2 at normal Switch #3 at proceed at jump X W = 0 0 = 0 b 3 = 0.75 "L? ^ 1 &2 = 2.0 H Q = 0 de dT = 0 T < 0.10 T > 0.10 a Y = -0.45 -1.0 x 0 = 1.0 b 0 = 0 * 2 o = 0 b, = 0 x, = 0.8095(66%) = 53.97 f t . 30 sec. f o r 5 complete cycles. F * No check value was determined for t h i s case. Note that TT- must b 1 be input as a negative value to d i s t i n g u i s h t h i s type of tank. 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K0 (outflow) = 138.0 f t . Kc (inflow) = 266.7 f t . d 3 « 46.7 f t . q varies from zero to 4000 c.f.s. instantaneously. Data: T 0 u = 0 X 0 Y = 0 D = 0.75 p = 0.8451 W = 0 a , = 0 a2 = 0 "3 - 0.7 0 = 0.1656 To = 1.0 e = 0.92 e 0 = 0.875 =3 0 b e s 0.75 F * F* 3 0.8 6 0.8 F| XX 4.0 F* m 4.0 F* F 3 = 4.0 & , = 2.07 &a ES 4.0 To a 0 H SB 5.55 H 0 5.19 H 9 6.0 F * f SB 40.0 P ££ 0 b 2 = 0 de dT => 0 IS 0 Switches: Switch #1 at normal Switch #2 at normal Switch #3 at proceed T < 0.20 at jump 0.20 < T < 0.75 at proceed T > 0.75 Results: x, - -1.126l(66 2^) = - 75.07 f t . Timing: 45 sec for 5 complete cycles. Remarks: x, = -75-9 f t . as determined from graphs. x Y I U D P 71 2) Sudden Flow Rejection (balanced design) D e t a i l s : F, = F 2 1600 f t : f = 160 f t . K 0 (outflow) - 133.0 f t . K 0 (inflow) = 266.7 f t . d 3 = 46.7 f t . ; s = 150 f t l / 2 / s e c . q v a r i e s from 4000 c . f . s . to zero instantaneously. Data: Switches: Results: Timing: Remarks: T 0 U 0.75 D = 0 P = 0 az = 0 a3 E3 0.7 e = 0 eo = 0 0 = 0.8 6 0 F * F 3 = 4.0 6 . = 2.07 H = 0 Ho = 0 P = 15-31 b 2 o • X = -0.45 W = 0 0 = 0 b 3 = 0.75 = 4.0 4.0 H 3 = 0 ^ = 0 dT u Switch f/l at normal Switch #2 at normal Switch #3 at proceed x{ = 0.7308(66%) = 48.72 f t . Y = -0.45 a, = 0 T 0 = 1.0 b 0 - 0 F - - k.o To = 0 Y =40.0 bj - 0 36 sec. f o r 5 complete cycles. No check value was determined for t h i s case. However, the r e s u l t can be compared with x ( = 50.96 f t . , the r e s u l t f o r the same case with an o r i f i c R surge tank at balanced design. 0.1 0.2 0.3 0 7 3 3 ) Sudden Flow Rejection D e t a i l s : ZD LZ F, - F 2 = F 3 = f = 1 6 0 f t . 2 K 0 (outflow) = K D (inflow) d 3 = 5 0 . 0 f t . 1 6 0 0 f t . 1 3 8 . 0 f t . 2 6 6 . 7 f t . ; s = 1 5 0 f t i V s e c . q varies from 4000 c.f.s to zero instantaneously. Data: T = 0 U ss 0 . 7 5 X ss - 0 . 4 5 Y ss - 0 . 4 5 D = 0 P = 0 W S3 0 a . ss 0 a2 = 0 a 3 = 0 . 7 5 0 = 0 To S3 1 . 0 e = 0 e 0 S3 0 b3 S3 0 . 7 5 b 0 - 0 F* F* 4 . 0 P = 0 . 8 •e SB 0 F, SS 4 . 0 F 2 S3 F * ss 4.0 6 < =3 2 . 0 7 bz = 4 o 0 To = 0 F* 40.0 H = 0 H 0 SS 0 Hg SB 0 f S3 P = 1 5 . 3 1 b 2 = 0 de dT S3 0 b, = 0 Switches: Switch # 1 at normal Switch # 2 at normal Switch # 3 at proceed Results: x, = 0 . 6 9 5 0 ( 6 6 % ) = 4 6 . 3 3 f t . Timing: 3 6 sec. f o r 5 complete cycles. Remarks: No check value was determined f o r t h i s case. x Yn U D P 75 4) Linear Gate Opening Details: Z H F, - F 2 = F 3 1600 f t . f = 160 f t . K„ (-outflow) - 138.0 f t . K Q (inflow) = 266.7 f t . d 3 - 46.7 f t . x varies from zero to 1.0 in 20 sec. Data: T -D » a , = 0 0 0 e - 0 0 = 0.8 * = 4.0 - 6.0 p = 0 H U - 0 P - 0 a3 - °- 7 e D = 0.875 0 = 0.8 6, = 2.07 H 0 = 5-19 4.0 X = 0 w = 0 0 - 0.1656 b 3 = 0 F * TT- = 4.0 62 = 4.0 Hg = 6.0 ^ - 0 dT u Y =0 a, = 0 x 0 = 1.0 b G = 1.0 T c - 0.25 F * f 40.0 b, = 0 Switches: Results: Timing: Remarks: Switch fil at jump Switch #2 at normal Switch //3 at proceed T < 0.30 at jump 0.30 < T < 0.90 at proceed T > 0.90 -1.3336(66 %) - 88.91 f t . 52 sec. for 5 complete cycles. No check value was determined for this case. 77 5) Sudden Load Acceptance D e t a i l s : F i = F2 = F3 = 1600 f t . f - 160 f t . K e (outflow) - 138.0 f t . K 0 (inflow) = 266.7 f t . d 3 = 46.7 f t . p varies from 69,000 hp. to 145,000 hp. instantaneously. Data: Switches: Results: Timing: Remarks: T = 0 U 0.375 1 = -0.1125 D = 0.7078 P = 0.7903 W = 0 az S3 0 a3 S3 0.7 0 = O.I656 e = 0.92 e c = 0.875 S3 0.3761 (3 0 0.8 e = 0.8 F* Fj = 4.0 F * F3 S3 4.0 &. = 2.07 &2 = 4.0 H S3 5.5095 Ho = 5.19 H, = 6.0 P = 0 b 2 33 0 de dT = 0 Switch / / l at normal Switch #2 at jump Switch #3 at proceed T < 0.15 at jump 0.15 < T < 0.95 at proceed T > 0.95 x, = -1.0611(66%) = - 70.74 f t . 53 sec. f o r 5 complete c y c l e s . jo -0.1125 0 1.0 0.7903 v~ C A.O ^2 T 0 = 0 F* ^ = 40.0 b, = 0 x, = -68.5 f t . as determined by numerical i n t e g r a t i o n by hand using a time i n t e r v a l of 5 sec. -0.2 79 6) Sudden Flow Rejection (combined d i f f e r e n t i a l and simple tank) D e t a i l s : F, = 3200 f t . F 3 = 1600 f t f d, = -66.7 f t . d 3 = 46.7 f t . K G(inflow) = 0 F 2 = 500 f t ! f = 160 f t . d 2 = 0 K 0(outflow) = 138.0 f t . 150 f t . /sec q v a r i e s from 4000 c . f . s . to zero instantaneously. Data: T = 0 U = 0.75 D - 0 P = 0 a2 = 0 a 3 = 0.7 e = 0 = 0 0 = 0.8 e 0 F* F 3 = 4.0 & ! = 2.07 H = 0 Ho = 0 P SB 15.31 = 0 X = -0.45 Y = 0 w = 0 a i = -1.0 0 = 0 T o SS 1.0 b 3 " 0.75 bo = 0 F * F. -2.0 F * ?Z = 12.8 0 To = 0 H 9 = 0 F * f = 40.0 de dT 0 = 0 Switches: Switch ftl at normal Switch §2 at normal Switch #3 at proceed Results: x, « 0.9149(66%) = 60.99 f t . Timing: 36 sec. f o r 5 complete cycles. F* Remarks: No check value was determined f o r t h i s case. Note that r r -must be input as a negative number to d i s t i n g u i s h t h i s type of tank. 81 7) Sudden Flow Rejection (no overflow to outer tank) Details: Data: F, = F 2 = F 3 = 1600 f t i f = 160 f t . K 0 (outflow) = 138.0 f t . K c (inflow) = 266.7 f t . d 3 = 46.7 f t . ; s = 150 ft.'Vsec. q varies from 4000 c.f.s. to zero instantaneously. '/2 T = 0 u 0.75 X -0.45 Y - -0.45 D 0 p = 0 w = 0 a, = 0 a z 0 a 3 0.7 0 S3 0 To " 1.0 e = 0 e c = 0 b3 = 0.75 bo = 0 F * 4.0 F* 4.0 P SS 0.8 6 = 0 F, S3 —— ss F 2 F * S3 -4.0 & , 2.07 & a EE 4.0 To = 0 F * H S3 0 Ho = 0 H9 S3 0 — S3 f 40.0 P S3 15.31 b z 0 de dT SS 0 0 Switches: Switch #1 at normal Switch if2 at normal Switch //3 at proceed Results: Timing: Remark s: x, = 0.6283(66%) 41.89 f t . 36 sec. for 5 complete cycles. No check value was determined for this case. However, the result can be compared with x( = 48.72 f t . for sudden flow rejection with overflow to the outer tank. 0.1 0.2 0.3 0.4 0.5 0;i 83 APPENDIX II STORAGE OF DATA 28 T 20 29 U 21 29 X 22 29 Y 29 D 24 29 P 25 29 W 26 29 a . 29 28 29 «3 29 29 0 2a 29 To 29 e 2c 29 e 0 2d 29 2e 29 b 0 28 P i-< It 30 28 6 31 28 F* F, 32 28 F* F 2 28 F* F3 34 28 35 28 62 36 28 To 26 H 38 26 49 26 H, 3a 26 F* f 24 P 3c 24 b 2 3d 24 de dT 3e 22. 23 27 2b 2f 33 3 7 3b 31-KEY : 28 T 20 indicates that the value T is scaled (2) 2 8 from the right and that its working channel II address is 20. APPENDIX I I I COMPUTER PROGRAM 230f TO 871e5717 570f5b0b c317571f 11604830 5blb l l6o 17815707 481+01708 5b031t60 57178a69 1+838170d 1701+0000 571e5bla 871e5bl3 lt6o483c l l 604920 17955716 70 bd285d99 5bl21l6o OOlcOOll U83fl702 00080000 817H lOO 001d0005 OOOOOOOO OOOfOOOO 00180006 001c009c 00030000 00190000 001a0099 OOldOOOc OOOl+OOOO 00200000 2 3 0 f 71 5 b l f l l 6 0 7 9 0 7 f 7 0 1 l * 9 3 f O 0 0 0 8 7 1 f 5 b 0 3 8 b 8 0 8 3 8 0 7 9 2 0 1 1 6 0 7 9 1 b f 7 0 5 7 9 0 b f 7 0 1 7 9 1 7 f 7 0 5 5 7 l e 7 8 2 5 7 9 1 3 f 7 0 5 5 b l a l l 6 o 7 9 0 f f 7 0 5 7 9 0 b f 7 0 1 7 9 0 b f 7 0 5 1 7 9 1 7 9 2 5 7 1 a U 9 2 a 6 e 7 5b161160 02lc030d 7907f70l 5a000000 8b8l838l 5800l2dO 8 5 8 I O O O O 7 1 6 O O O l a 81721100 01bd8000 oiidoUo6 00069600 01id04l9 520019d8 00040000 00160004 230f 72 571flb88 49U879I42 791b^7 a5032eOO Il8c7917 6748U9U8 49U7877d 79U167W+ 79I4.1U130 674649I+9 0c00e7l+l 79^26728 a30330O0 Id9d793-+ e7^la301 U8a79-+2 72 3d63f22c 67271dOa 79^26528 7933118a Id0f2800 79322c00 8173119c 494b793b 81731100 I913I4.II4.9 81731105 e7-+ba301 00l47ael 49497932 000U1893 l d 0 3 H 0 f 00020000 85 230f 73 79I1467UI 4948111f 494aldOc 79436728 79351110 Id8d793^  79362eOO U997943 494f4l4a 67271dl9 e74aa302 79331199 3000e74f 79322cOO a3026l48 49537943 73 2602aabO 6742494a 7 9 4 a l l l 7 Id8a7935 79286742 I l0e7936 ld le794b 494f191a a502494b 2800494c 79436728 8174118c 2d0eal07 79321d07 8 1 7 4 H 0 0 793b494b 8174119a 230f 74 eb4fc54c 67461185 7917II6O 794c2f30 4l4a310c e753a301 c5Uc79U9 494d2800 6l4c3000 494f4l4d e74ba303 e747a307 4949793I4. 61234143 ld05794c 49437934 74 5fffX)5a8 Id9a3000 6729MI3 6729ld9a a50l494d a50l494e 7 9 m 160 791bll60 8175H00 30C0e74d 8175H88 a302494e 004c0008 U9e2800 004e0012 49^7942 004d000f 230f 75 3000e749 49467940 a302H0c 67371 d91 00002800 2800493d 674e3000 493f792f e747a307 492el3l6 3000e71f dl !315l6 OOOOe73c 79476120 33076126 49403000 75 eec8ac94 e73fa306 e740a306 6l3d3000 6l3d3000 e740a309 e740a309 6l2e4953 6l2e495l 81771lOe 79416544 4l40e73f 3000e74f a3076l 3d 8176HOO 4950413f OOOOOOO3 230f 76 a3024952 49527949 4l31e744 & 5026l52 a3026l52 4955dl8d 49544141 I5l3791f OOOOOOOO 11604956 e72aa301 791b1160 61523OOO 3000e756 e748a304 aiOl4956 76 97fc8285 1386792b a301eb54 49514154 3OOO6157 e75la301 49570000 6156495^ 81771 Hd 4l51e755 4l51e754 a!03eb54 81771100 c557dlOf OO38OOI9 4l38e750 OO390091 S6 230f 77 a30lU956 e755a!03 I*-I38e738 eb543000 eb39e72c 65574957 eb2d30O0 4l38e751 a502675^ eb2ce73e 49510+150 eb5l+3000 e738eb5l+ 65574957 c5574l51 79206147 77 8d3d6487 49U04147 6553^955 e757a307 3000e755 612M+953 a3023000 4l47e749 e7-+fa302 a3076l22 U958I+153 49424147 e753a304 e748a307 3000e731 6121495b 81781100 230f 78 61584959 8 l 7 a l l 9 8 Ul5be75b 79590eOO a3033000 a!02eb39 e72a6l3a c55a!30a 67594959 dlOa791f 79-+2a503 H603000 61594959 e72ba301 792dl905 65531dOa 78 8c7b3b2e d9010000 a3060000 81751113 81791100 Ul5ae71b 8l7all00 l+95ca!06 00000000 c55d3000 34000000 65171dOb 94000000 795d67l3 00000064 Id0b3128 005a0095 230f 79 6l l fUd9d 4d9a795c 6llb4d92 0e00al06 4l53e717 3OOO655d a30l495e a706495c a!06c55f 795eOeOO 795e6U3 a!063000 4d824d96 6551 .^706 6llb4d86 495e8700 79 8e3efl+ll d9207900 e75e6l5f 495d7900 675d3000 675d3000 e75c6l5d e75e6l5d 8l7all84 l+95d8700 00000060 00007900 00000064 1+9517900 00000001 675f3000 00000050 230f 7a d910792c 49441395 0000495f 158d672f 0e00al02 Id95d902 eb2de75a ll957945 a3023000 672fld95 e753a302 d902793b ^5795b 198e7958 49417953 a7036542 7a f87276bf 49437929 Id071l8f 67421d8f 79286743 28004946 Id8f7928 Il8f7929 49437959 6742ld9e 4938171b 79296743 8l7bli01 Id9e2800 8l7bllOO 49467932 OOOOOOOO 230f 7b 792dl984 571f781+7 U94795f 48271781 672cOeOO 792dl98d al09eb47 H9d795f c53e795f 672cOeOO 492c79^7 al08ebU7 a70lU947 c53e795f 8172118c l+92c7917 7b d738df53 67134917 OOOOOOOO 1992791b OOOOOOOO 4917897b OOOOOOOO 81711105 OOOOOOOO 0000897b OOOIOOOO 8171110a 00050000 oooooooo 00050000 oooooooo 00070000 230f 7c 871efl82 00008b00 4d944d8l 11080000 f1812800 OOOOOOOO f lc2f5a4 OOOOOOOO 4dldf l83 OOOOOOOO 4d9c8300 OOOOOOOO 00005bId OOOOOOOO 11604900 0000009c 7c b8fb828c 871ff182 OOOOOOOO 4d96fl8l OOOOOOOO 2800flc2 OOOOOOOO f5a44dlf OOOOOOOO fl834d9a OOOOOOOO f 1818300 OOOOOOOO 5bIf7900 OOOOOOOO U601186 02OOO49e 230f 7d 4d644del 02001100 79001f6l 5a7dc32f Id6l l9ec OOOOOOOO lle!3129 OOOOOOOO 31ab3aOO a30130OO alOl4df5 3000a 100 c5'693aOO llelOOOO a5011l62 b504f334 7d 0290d5b7 49636165 OOOOOOOO 4l673be5 6ad4ffff 6l6b496d 995c1743 796fe971 3635e6o6 3OOO656d bd7fal01 3a007963 79691975 e97b2800 e77dll71 c77f l l73 OOOOOOOO 

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