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UBC Theses and Dissertations

Air chamber design charts. 1973

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AIR CHAMBER DESIGN CHARTS by GALATIUBt, WILLIAM ROBERT B.Sc. (Civ. Eng.) The University of Manitoba Winnipeg, Manitoba, 1964 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENT FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in the Department of C i v i l Engineering We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA SEPTEMBER, 1973 In presenting this thesis in partial fulfillment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make i t freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of C i v i l Engineering The University of British Columbia Vancouver 8, Canada Date September 10, 1973 ABSTRACT The air chamber has certain advantages over both the open-top surge tank and the valve-type surge suppressor for controlling pressure surges in pump-discharge lines. The main purpose of this study was to produce charts which can be used for designing or checking the size of an air chamber required for a particular pumping installation. The characteristics method was used to convert the two partial differential equations of momentum and continuity into four total differential equations. The solution of the equations (finite-difference form) was carried through by d i g i t a l computer to provide the data required for the preparation of the charts. Results obtained on the d i g i t a l computer by the method of charac- ter i s t i c s are checked by the graphical method. Examples demonstrating the use Of the charts are included. TABLE OF CONTENTS CHAPTER PAGE ABSTRACT ( ( i i ) NOTATION (ix) INTRODUCTION 1 I. ASSUMPTIONS AND THEORY 3 1.1 Assumptions 3 1.2 General Theory 5 1.3 Parameters 5 1.4 Relationship between a* and p* 7 II. METHOD OF CHARACTERISTICS 8 2.1 General 8 2.2 Basic Equations for Unsteady Flow Through Pipes 8 2.3 General Characteristics Method 9 2.4 Convergence and Stability of the Method of Finite Differences 15 III. BOUNDARY CONDITIONS 16 3.1 The Air Chamber 16 3.2 Reservoir of Constant Water Level at the Downstream End 19 IV. THE PROGRAM 21 4.1 General 21 4.2 Check on the Program 21 4.3 Description of the Program 23 4.4 Approximation of Velocity of Flow out of the Chamber 24 V. THE CHARTS 26 5.1 Groups of Charts 26 5.2 No Head Loss, Frictionless Flow 28 5.3 Entire Head Loss Concentrated at the Orifice 28 5.4 Entire Head Loss Attributable to Distributed Friction 30 5.5 Head Loss Equally Divided Between Uniformly Distributed Wall Friction and Orifice Loss 32 5.6 Use of the Charts 32 VI. DISCUSSION 35 6.1 Volume of Air in the Chamber 35 6.2 Volume of the Air Chamber 36 6.3 Orifice Design 37 6.4 Water-Column Separation i n Pump Discharge Lines 40 (iv) CHAPTER PAGE VII. CONCLUSIONS 42 BIBLIOGRAPHY 44 THE CHARTS 45 Group I - No Head Loss, F r i c t i o n l e s s Flow 46 Group II - E n t i r e Head Loss Concentrated at the O r i f i c e 48 Group I I I - E n t i r e Head Loss A t t r i b u t a b l e to Di s t r i b u t e d F r i c t i o n 71 Group IV - Head Loss Equally Divided Between Uniformly D i s t r i b u t e d Wall F r i c t i o n and O r i f i c e Loss 82 APPENDICES Appendix A: Comparison of Charts and Numerical Examples 93 Appendix B: Graphical Checks on Program 106 Appendix C: Program f o r the E n t i r e Head Loss Concentrated at the O r i f i c e 113 Program f o r the En t i r e Head Loss A t t r i b u t a b l e to D i s t r i b u t e d F r i c t i o n 118 (v) LIST OF TABLES TABLES PAGE 5.1 Comparison of Results Obtained for the Powers 1.0, 1.2 and 1.4 29 5.2 Upsurges and Downsurges for 1:1 and 2.5:1 Orifices 31 (vi) LIST OF FIGURES FIGURE PAGE 1.1 Pipe Line with Air Chamber 4 2.1 Method of Specified Time Intervals 14 2.2 Characteristics at the Boundaries 14 3.1 Air Chamber 20 3.2 Reservoir at Downstream End 20 4.1 Program Flow Chart 22 5.1 Pipe Line with Air Chamber 34 6.1 Air Chamber Control Levels 38 6.2 Differential Orifice 38 B-la Schematic of Pipe Line 107 B-lb Transient-State Conditions for Total Head Loss Concentrated at the Orifice - Pipe Pressures 109 B-2a Schematic of Pipe Line Showing Line Friction Loss 1 1 0 B-2b Transient-State Conditions for Total Head Loss Attributable to Distributed Friction - Pipe Pressures 112 (vii) ACKNOWLEDGEMENT T h e a u t h o r w i s h e s t o e x p r e s s h i s g r a t i t u d e t o h i s s u p e r - v i s o r , D r . E . R u u s , f o r h i s v a l u a b l e g u i d a n c e , c o n s t r u c t i v e c r i t i c i s m a n d s u g g e s t i o n s , a n d t o D r . W . F . C a s e l t o n f o r r e v i e w i n g t h e w o r k . T h e s t u d y was s u p p o r t e d b y a g r a n t f r o m t h e N a t i o n a l R e s e a r c h C o u n c i l . ( v i i i ) NOTATION The following symbols are used in this thesis: , 2 A = cross-sectional area of pipe, in ft a = propagation velocity of waterhammer wave, in ft/sec. C Q = i n i t i a l volume of air in the air chamber at absolute pressure head H Q*, in f t 3 CQTf - o r i f i c e loss coefficient D = inside diameter of pipe, in f t f = Darcy-Weisbach f r i c t i o n factor 2 g = gravity acceleration, in f t / s e c H = transient-state piezometric pressure head above datum at the beginning of a time interval, in f t H Q = i n i t i a l steady-state piezometric pressure head above datum, in f t Hp = transient-state piezometric pressure head above datum at the end of a time interval, in f t H* = transient-state absolute piezometric pressure head above datum, in f t HQ* = i n i t i a l steady-state absolute piezometric pressure head above datum, in f t H Q r £ = o r i f i c e throttling loss corresponding to discharge q, in f t Horfo = o r i f i c e throttling loss corresponding to discharge q Q, i n f t K = coefficient relating total pipe line head loss due to f r i c t i o n to piezometric pressure head above datum L = length of pipe line, in f t m m = power used in pressure - volume relationship, H* v a ^ r = constant, for an air chamber ( ix) steady state discharge in the pipe line, in f t /sec. transient state o r i f i c e discharge, i n f t /sec. time, in seconds transient-state velocity in pipe at the beginning of a time interval, i n ft/sec. transient-state velocity in pipe at the end of a time interval, in ft/sec. i n i t i a l steady state velocity in pipe, in ft/sec. transient-state volume of air in air chamber at the beginning 3 of a time interval, in f t 3 i n i t i a l steady-state volume of air in air chamber, in f t transient-state volume of air in air chamber at the end of a time interval, i n f t J distance along pipe line, from pump, i n f t pipe line characteristic pipe line characteristic in terms of absolute pressure head parameter pertaining to a pump-discharge line having an air chamber, in terms of absolute pressure head angle the pipe makes with the horizontal At grid mesh ratio, time increment, i n seconds incremental distance along the pipe line, in f t (x) INTRODUCTION Sudden stopping or starting of large centrifugal pumps installed for irrigation, domestic water supply systems, pumped storage hydro- electric plants and other purposes cause transient pressures in the discharge lines. Starting control mechanisms can be designed to delay the starting up time sufficiently to prevent excessive over pressures. But sudden stopping in the event of power failure could result in objectionable waterhammer pressures in the pipe line. In small installations, no special precautions are taken to avoid high waterhammer effects. Standard pipes and fittings of small diameter have a wall thickness sufficient to withstand appreciable transient pressures. In large pumping installations various pressure-control devices may be used to reduce waterhammer pressures. Some of these devices include: (1) surge tanks, (2) air chambers, (3) surge suppressor valves, and (4) slow closing check valves. For controlling pressure surges in pump-discharge lines, the air chamber has certain advantages over both the open-top surge tank and the valve-type surge suppressor. For high head installations where the open surge tank is impractical, a properly designed air chamber provides good surge control. The air chamber can be near the pump whereas the surge tank can not always be so located. The air chamber can be designed to reduce the downsurges in a pump-discharge line, thus 1 2 preventing collapse of the line and water-column separation; ordinarily, surge-suppressor valves are not suitable for this important function. The main disadvantage of air chambers i s that the compressed air i s continuously being lost through dissolving in the water and possible leakage. Consequently the air must be replenished periodically. After i t has been decided that certain types of pressure- control devices w i l l meet design requirements, the f i n a l choice i s usually based on a cost study of the various devices. The cost of an air chamber is determined primarily by i t s size and inside pressure. In this thesis, charts are presented which provide for the rapid determination of air chamber sizes required to control waterhammer pressures in pump-discharge lines where the transient pressures are caused by rapid pump shut down or by power failure. The charts were prepared using the method of characteristics to convert the two partial differential equations of momentum and continuity into four total differential equations. The solution was done by d i g i t a l computer. Examples demonstrating the use of the charts are given i n Appendix A. CHAPTER I ASSUMPTIONS AND THEORY 1.1 ASSUMPTIONS For the purposes of this study, the following assumptions were made. (1) A check valve on the discharge side of the pump closes immedi- ately on power failure. This eliminates the need to consider pump characteristics but introduces an abrupt pressure wave which must be accounted for throughout the computations. (2) The air chamber is situated near the pump as shown in Fig. 1.1. The steady-state water surface in the chamber has an elevation equal to that of the center line of the pipe (see Fig. 3.1). The transient- state head difference between the chamber water surface and the pipe center line i s small and therefore neglected. The head loss through the or i f i c e , i f applicable, is taken into account in determining the absolute head, H*, in the tank. (3) The pressure-volume relationship for the air in the chamber is expressed as: 1 2 H* v . ' = a constant, air The power 1.2 is an average of the powers 1.0 and 1.4 for the isothermal and adiabatic expansions respectively. (4) The head loss, made up of surface f r i c t i o n and loss at the or i f i c e , varies with the square of the velocity. Two types of orifices, one simple and one differential, were considered in the study. The ratio 3 Sections FIG. 1.1 Sec. 1-1 Area of Pipe 5 of the total head loss for the same flow into and from the air chamber is 2.5:1 for the differential o r i f i c e , and 1:1 for the simple or i f i c e . (5) This study i s limited to cases i n which no water column separation occurs. This means that water vapour pressure i s not reached and the pipe stays f u l l of water at a l l times. (6) A reservoir of constant elevation serves as the downstream boundary condition. 1.2 GENERAL THEORY Normally, with the pump operating, the flow in the pipeline i s in the forward direction, toward the reservoir. The check valve closes simultaneously with pump failure. This creates a head differential across the air chamber outlet. The compressed air causes the water in the chamber to discharge into the pipeline to maintain the head and the flow. Water w i l l continue to flow out of the tank unt i l the head in the chamber becomes less than the head in the pipeline at the chamber outlet. At this instant, the water in the discharge line w i l l reverse i t s direction and flow into the air chamber. During this reverse flow condition, the retardation of the flow into the air chamber causes the pressure in the discharge line to increase to exceed normal operating head and w i l l produce the maximum head for the transient. Resurges in the pipeline w i l l occur with diminishing intensity. 1.3 PARAMETERS The pressure surges in a pipeline equipped with an air chamber depend on the two parameters, p * and a*, when f r i c t i o n i s not considered^. Because f r i c t i o n a l resistance i s essential to the efficient use of an 6 a i r chamber on a pump-discharge line, Evans and Crawford introduced a third variable, K, to account for f r i c t i o n a l losses. The variable K i s defined so that KHQ* is the total head loss for a reverse flow of Q0. QQ is the i n i t i a l rate of flow in the pipeline, in cubic feet per second ( f t - V s e c ) . The pipeline characteristic, p , i s defined as P (1.1) in which a is the propagation velocity of waterhammer waves in the pipeline, i n feet per second (ft/sec); V Q is the steady-state velocity in ft/sec; H Q i s the steady-state pressure head, in feet of water ( f t ) ; and g is gravity acceleration in feet per second per second (ft/sec.)- The characteristic p i s dimensionless and is a function of the ratio of the steady-state kinetic energy to the total potential energy in a unit length of conduit. In air chambers, the volume of the air is a function of the absolute pressure to which i t i s subjected. In terms of absolute pressure, the pipeline characteristic p becomes p* - — (1.2) 2gHQ* where HQ* is the normal absolute pressure head in the pipeline at the entrance to the a i r chamber. The parameter, a*, that i s characteristic to a pump-discharge line having an air chamber i s defined^ as 7 2gC0H0* ALVQ in which C Q is the i n i t i a l volume of air in the air chamber at absolute pressure head, HQ*, in cubic feet ( f t 3 ) ; A i s the cross-sectional 2 area of the pipe in square feet (ft ); and L is the length of the pipe i n feet. The parameter a* expresses the ratio of the steady- state potential energy of the air i n the air chamber to the steady- state kinetic energy of the water in the discharge line . 1.4 RELATIONSHIP BETWEEN o* AND p* From Eqs. (1.2) and (1.3) a*p* = — (1.4) ALVC or C q . a*p*QQL/a . (1.5) From Eq. (1.2) a V o 2p* = — — (1.6) gHQ* and the constant for a pipeline having an air chamber w i l l be defined as or (p * 0 * ) ALVQ C o = a (1.8) CHAPTER II METHOD OF CHARACTERISTICS 2.1 GENERAL g The characteristics method converts the two partial differential equations of momentum and continuity into four total differential equations. Non-linear f r i c t i o n i s retained, as well as the effect of the pipes being non-horizontal. The equations are expressed in finite-difference form, and the solution i s carried through by d i g i t a l computer. Advantages of the method are: - accuracy of results as non-linear terms are retained - there is proper inclusion of f r i c t i o n - i t affords ease in handling the boundary conditions and ease in programming complex piping systems - there is no need for large storage capacity in the computer - detailed results are completely tabulated. It i s by far the most general and powerful method for handling waterhammer. 2.2 BASIC EQUATIONS FOR UNSTEADY FLOW THROUGH PIPES The velocity and pressure of moving fluids in pipes are governed by the continuity and momentum equations. The momentum equation for flow through a pipe which is inclined or horizontal, tapered or straight, slightly or highly deformable, i s given by gH + V + W + M Z L = 0 , (2.1) e x t x 2D 8 9 in which g i s gravity acceleration, V i s f l u i d velocity, f i s the Darcy-Weisbach f r i c t i o n factor, H is the total pressure head above the datum line, D i s the inside diameter of the pipe, and fV|vl is the f r i c t i o n a l force of the f l u i d . The absolute sign i s introduced to ensure that the f r i c t i o n a l force w i l l always be opposite to the direction of velocity. The subscripts x and t indicate partial differentiation with respect to distance and time. For example, head in feet of water. Changes in the density of water may be neglected without intro- ducing significant error. Considering the density as constant, the continuity equation may be stated as in which 0 i s the angle the center line of the pipe makes with the horizontal axis (measured positive downwards), and a is the velocity of the waterhammer wave. 2.3 GENERAL CHARACTERISTICS METHOD In this section, a general solution for the continuity and mom- entum equations i s presented. For the complete treatment, see Ref. 9. A l l of the terms in the equations are retained. The method of specified 2D II _ 3 H H.. = , in which H is the total pressure (2.2) 10 time intervals which involves linear interpolation i s used. The momentum and continuity equations may be written as 4 = gnx + w x + v t + SjJvL - 0 (2.3) and L. = H + g x V +VH + V sin 9 = 0 (2.4) Multiplying Eq. (2.4) by X and adding i t to Eq. (2.3), one obtains ,2 L± + XL± = X [H x(V + f ) + H t ] + [ V x(V + |_ X) + V t] + X Vsin6 Let £ - V + * = V + 5 l A . dt X R Therefore, X a and — = v ± a . dt (2.5) (2.6) (2.7) (2.8) Through substitution of Equations (2.6), (2.7), and (2.8), Eq. (2.5) takes the form X + iii. + x VsinO dH _,_ dv ^ , „ „ , _ Q + fvlvl . 0 ( 2 .9) dt dt 2D It follows froms Eqs. (2.8) and (2.9) that a dt dt VsinS fyjvj + 2D 0, £ - v • > (2.10) C+ (2.11) 11 a dt dt a 2D - 0, (2.12) and dx dt V - a C- (2.13) Because V = V(x,t), the characteristic lines C+ and C-, given by Eqs. (2.11) and (2.13), plot as curves on the x-t plane (see Fig. 2*1). Eqs. (2.10) to (2.13) can be written in the following f i n i t e - difference forms: (V - V ) + & (H - H ) + & V sine (t - t ) + |- V IV 1 P R a p p/ a R P R ' 2D R1 R1 ( t p - t R ) = 0 (2.14) ( XP " XR ) = ( VR + a ) ( tP " V (2.15) (V p - V s) - f (Hp - Hg) - | V ssin6 ( t p - t g ) + f D V S|V S| - t s ) - 0 <XP " V = (vs " a ) ( tP " V (2.16) (2.17) Two techniques are commonly used for obtaining a numerical solution for the finite-difference equations (2.14) to (2.17). These are: (1) use of a grid of characteristics, (2) use of specified time intervals. In single pipe problems as covered by this study, these techniques are identical^. The parameters x p and t are assigned definite values 12 throughout the computation leaving only V p and as unknowns to be determined. In this study the technique of specified time intervals w i l l be used. Since the conditions at points A, B, and C (Fig. 2.1) are known, the conditions at R and S may be evaluated by linear inter- polation. Thus x - x V - V C R _ C R x - x A V - V C A C A But X p = x , and x - x = Ax. Therefore, the above equation takes the form v c - V R *P ~ XR = V r - V, A x • ( 2 ' 1 8 )  U A Since V i s much smaller than the waterhammer wave velocity a, •K V« may be deleted from Eq. (2.15) without incurring any serious loss of R accuracy. By combining the modified Eq. (2.15) with Eq. (2.18), one obtains V - V a At = Ax (2.19) C ' A The grid mesh ratio, 6', i s defined as At 6 ' " to ' Therefore, a 6' (V c - V A) = V c - V R, and VR = Vc ~ a 9 ' (Vc " V * (2.20) 13 Similarly, \ = ae' (H c - (2.21) v s = v c " ae- ( v c - (2.22) H s = ae« (H c - (2.23) Solve Eqs. (2.14) and (2.16) simultaneously to obtain: V P = ° - 5 [ V R + v s + i ( H R - V - I A t s i n e (\ - V - f ft (V R|V R| + V S|V S|)] (2.24) Hp •- 0.5 [ H R + H g + f (V R - V s) -At sine (V R + Vg> - § fj£ < V R I V R I - vsl vs'>> ( 2 - 2 5 ) At the boundary points (Fig. 2.2), either Eq. (2.14) or Eq. (2.16) or both are used together with the boundary conditions to solve for V and H. Eqs. (2.14) and (2.16) are termed the negative characteristic equation and the positive characteristic equation respectively and may now be written i n the following forms: The negative characteristic equation i s V = C + C„ H , (2.26) P 1 2 P where C = V - C„ H + C V sin 6At - FF V_I V_I (2.27) 1 S 2 S 2 S s s C = £ , (2.28) 2 a and FF - . (2.29) 14 t t Q + A t R S A C B METHOD OF SPECIFIED TIME INTERVALS FIG. 2.1 i Ax P / / t \ c -\ \ Ax A: CHARACTERISTICS AT THE BOUNDARIES FIG. 2.2 15 The positive characteristic equation is VP = S " °2 V where C = V + C H - C V At sine - FF V_1 V_I . 3 R 2 R 2 R R1 R1 C 2 and FF represent pipe constants. The values of and are constant during each time step. 2.4 CONVERGENCE AND STABILITY OF THE METHOD OF FINITE DIFFERENCES To be assured of st a b i l i t y and/or convergence of the solution^, i t is necessary that At (V + a) < Ax. Since V i s small relative to a, this may be stated as follows: T- <-  1 • Ax - a This indicates that i t i s important to select the grid mesh ratio so that the characteristics through P, C + and C~ w i l l not f a l l outside the 3 line segment AB (Fig. 2.1). The most accurate solutions are obtained when Ax " aAt . (2.30) (2.31) CHAPTER III BOUNDARY CONDITIONS 3.1 THE AIR CHAMBER (Fig. 3.1) Because of the assumption that the check valve closes simul- taneously with the pump failure, a l l the flow in the discharge pipe is either from or into the chamber. This assumption eliminates the pump characteristics from the waterhammer computations. The pressure and volume of air in the chamber follow the gas law 8 H* v a i r m = constant, (3.1) where H* and v . are the absolute pressure head and volume of air in air the chamber and m is the power 1.0 for isothermal expansion and 1.4 for adiabatic expansion. The or i f i c e in the chamber may be simple or of the differential type. The differential type of o r i f i c e throttles the reverse flow of water from the discharge pipe into the chamber while there i s very l i t t l e throttling of the flow out of the chamber. If there i s no or i f i c e in the chamber, the throttling loss is taken equal to zero. Flow out of the chamber is considered positive. For the transient condition, Eq. (3.1) may be written: [H + 34 + H 1 v m = C , (3.2) I P orf J Pair 10 in which H_ i s the transient pressure head (in ft) in the pipe at the 16 17 entrance to the chamber, H _ is the o r i f i c e resistance (in ft) orf corresponding to a discharge of q (ft 3/sec.) and v is the transient Pair 3 volume of air in the chamber (ft ). C is a constant given by: " l O " " . * V o a i r ' ° - 3 ) in which H * and v denote the i n i t i a l steady-state absolute o oair pressure head and volume of air in the chamber. For the transient state conditions at the junction of the chamber and the discharge pipe, the following equations can be written: The continuity equation: VA A t = v - v . (3.4) Pair air where V i s the velocity of flow in the pipe (in f t / s e c ) , A is the o cross-sectional area of the pipe (in ft ), At is the length of the time interval under consideration (in sees), v_ . i s the volume of air i n Pair the chamber (in f t 3 ) at the end of the time interval and v a i r is the volume of air in the chamber at the beginning of the time interval. Rearranging the terms, one gets: v_ . = v . + C At, (3.5) Pair air 11 in which C l l * V A ' The negative characteristic equation for the pipe i s : V p ( l ) " C x + C 2 Hpd), (3.6) where (1) designates section 1 on the pipe, i.e. at the air chamber. The o r i f i c e f r i c t i o n loss i s given by: H = C ^ E | £ q | q | (3.7) orf orf 2 1 1 q o in which COTf is the or i f i c e coefficient and H Q r^ o i s the head loss in the o r i f i c e (in ft) corresponding to a discharge of q Q. The absolute value of q ensures the correct sign on the head loss for changes in direction of flow through the o r i f i c e . For a simple or i f i c e , C o r j C = 1.0 for flow in either direction. For a differential o r i f i c e , G Q r£ = 1.0 when water flows out of the chamber, i.e. when V is positive, and C o r£ = when water flows into the chamber, i.e. when V i s nega- tive. The value of depends on the amount of throttling provided by the o r i f i c e . Substituting for q i n Eq. (3.7), one obtains: Horfo or H o r f " C o r f ~T VA' VA' 9n H * = C r C f C I C I (3.8) orf orf f 11' l l 1 in which H r = orfo C f 2 gives: Substitution of the values of v p a ^ r and ̂ Q t ^ into Eq. (3.2) h • 34 + C o r f C f C J C u | ( v a i r + C u At)* - C l o or C 1 Q HP = ( v a l r + C U At)* " 3 4 " C°rf C f C l l l C l l l Letting C . » v . + C„_ At, one obtains: air a ir 11 H p . 3 4 _ C o r f C f c n | C u | . (3.9) For each time increment, Hp can be determined from Eq. (3.9), V p from Eq. (3.6) and V p a i r from Eq. (3.5). 3.2 RESERVOIR OF CONSTANT WATER LEVEL AT THE DOWNSTREAM END (Fig. 3.2) At the junction of the pipe and the reservoir, Hp(ll) = H res The positive characteristic equation for section 11 i s given by: V p (11) = C 3 - C 2 H p ( l l ) . (3.10) From the above two equations, i t follows that: Vp(ll) = C 3 - C 2 H ^ . (3.11) 20 Section 1 1 2 x AIR CHAMBER FIG. 3.1 H res ( \ 10 Sec tion 11 i At 1 RESERVOIR AT DOWNSTREAM END FIG. 3.2 CHAPTER IV THE PROGRAM 4.1 GENERAL The program for this study designates to the computer a l l of the operations which must be performed to compute the maximum upsurges and downsurges for the transient phenomena. The flow chart for the program is given i n Fig. 4.1 and the entire programs for the entire head loss concentrated at the ori f i c e and entire head loss attributable to distributed f r i c t i o n are reproduced i n Appendix C. 4.2 CHECK ON THE PROGRAM Prior to proceeding with the actual study, the writer checked the validity of the program with several graphical analyses. These checks, presented in Appendix B, indicate that the program gives results which compare well with graphical solutions made by others. The graphical check for the total head loss concentrated at the o r i f i c e ^ shows that the program for this case gives valid results. See Fig. B-lb. The graphical check using several orifices to approximate head loss due to pipe wall f r i c t i o n ^ (graphical solution by E. Ruus) indicates that the program for distributed f r i c t i o n i s also valid. See Fig. B-2b . 21 Q START ^ * READ DATA COMPUTE At and Ax CHECK FOR CONVERGENCE 22 COMPUTE COEFFICIENTS AND CONSTANTS FOR ALL PIPES CALCULATE STEADY STATE CONDITIONS T - 0.0 V l " Vo H i " Ho PRINT T, H i , V i i CALCULATE V R, Vg, H 2, H s FOR ALL SECTIONS M - 0 I T «• T + DT M - M + 1 COMPUTE VP t and HPj AT INTERIOR POINTS COMPUTE VP and HP AT BOUNDARY POINTS V i - VPi H i = HPi TRUE ^ — ^ " ^ ^ FALSE M = MM PROGRAM FLOW CHART FIG. 4.1 23 4.3 DESCRIPTION OF THE PROGRAM The main functions of the program are as follows: i) Specification of the storage locations for the subscripted variables, the Dimension statement. i i ) Submission of data to the computer, i i i ) Computation of the time increment, iv) Check for convergence, v) Computation of constants, vi) Computation of steady-state values, v i i ) Computation of transient-state conditions, v i i i ) Check for maximum upsurges and downsurges. ix) Printout. The variables which the program reads in are: PLC the pipe line constant, 2p* TMAX — the length of time for which the transients are to be calculated, in seconds CPLAC - the constant for a pipe line with an air chamber adjacent to the pump, 2 p* a*. The remaining parameters are set in the Data statement. For any group, the only parameter which changes in the Data statement is the total head loss coefficient, CK. The programs are relatively efficient with a typical calculation taking approximately 12 to 13 seconds of computer use time. The programs for the four basic groups of charts, as listed i n Section 5.1, vary only slightly from each other. 24 Group I - No head loss For frictionless flow the program Data statement sets CK and the o r i f i c e loss equal to zero. The program automatically computes the f r i c t i o n factor, F, to be zero. Group II - Entire head loss concentrated at the or i f i c e The Data statement sets the f r i c t i o n factor, F, equal to zero, CK to some value between 0.1 and 1.0, and the or i f i c e inflow coefficient, CORFIN, to 1.0 or 2.5 depending on whether the or i f i c e i s simple or differential. Group III - Entire head loss attributable to distributed wall f r i c t i o n The Data statement sets the o r i f i c e loss, HORF, equal to zero. The program calculates the f r i c t i o n head loss, HF, and the f r i c t i o n factor, F, for the designated values of CK. Group IV - Head loss equally divided between uniformly d i s t r i - buted f r i c t i o n and or i f i c e loss The Data statement sets the o r i f i c e inflow (CORFIN) and the total head loss (CK) coefficients. The program computes the f r i c t i o n factor, F, the total steady-state f r i c t i o n loss and the total o r i f i c e loss for a flow of QQ into the chamber. The steady-state f r i c t i o n factor is used to calculate the f r i c t i o n head loss during the transient phase. 4.4 APPROXIMATION OF VELOCITY OF FLOW OUT OF THE CHAMBER I n i t i a l l y , the average velocity out of the chamber, VAVAPP, after the time interval was incremented, was set equal to the velocity in the pipe at Section (1) (Fig. 3.1) for the previous time interval. The computation was then followed through to the point where the actual average velocity of flow from the chamber was calculated. i , e * V(l) + VP(1) VAV = — v * 2 If the difference between the i n i t i a l assumed average velocity, VAVAPP, and the calculated average velocity, VAV, was less than or equal to 0.0001, the program continued the transient state computation. If the difference was greater than 0.0001, the values of HP(1) and VP(1) were recalculated using VAV as the new approximation for the velocity of flow out of the chamber. This iteration continued u n t i l the error criterion was met. The writer found that i f VAVAPP was set equal to V(l) from the previous time interval, the program would not converge to a solution, but in fact, the pressure surges would magnify increasingly causing the computer to terminate the program with an error message. CHAPTER V THE CHARTS 5.1 GROUPS OF CHARTS Four basic combinations of conditions were investigated in this study. These four combinations include: (1) No head loss, K = 0.0, (no wall f r i c t i o n , no or i f i c e loss) There i s only one chart in this group. (2) Entire head loss concentrated at the o r i f i c e , (no wall friction) There are ten charts in this group with K varying from 0.1 to 1.0 in increments of 0.1. Two orific e s , one differential and one simple, were investigated in this group. The differential o r i f i c e had an inflow to outflow head loss ratio of 2.5:1. That i s , for the simple o r i f i c e , the o r i f i c e resistance i s the same for inflow or outflow whereas for the differential o r i f i c e the inflow resistance i s 2.5 times the outflow resistance. Note that values of K = 0.7 to 1.0 are not practical but are included for the sake of completeness. Because of the great resistance to flow from the chamber for K = 0.7 to 1.0, large air chambers are needed to control the downsurges whereas the upsurges are not greatly reduced. (3) Entire head loss attributable to distributed f r i c t i o n , (no o r i f i c e loss) K varies from 0.1 to 1.0 in increments of 0.1. 26 (4) Head loss equally divided between uniformly distributed wall f r i c t i o n and ori f i c e loss K varies from 0.1 to 1.0 in increments of 0.1. The or i f i c e considered was a differential o r i f i c e with inflow to outflow loss ratio of 2.5:1. Under the conditions imposed by the assumptions, the entire transient following power interruption i s completely described by the variables K, 2p* and 2p* a*. In the charts, the maximum upsurges and downsurges have been plotted in terms of these variables. Maximum upsurges and downsurges at the pump, the midlength and the three- quarter point of the discharge line are plotted as percentages of HQ* for various values of these parameters. The normal range"*" of p* is from 0.25 to 2.0 and that of a* is from 2 to 30. This range i s covered in the charts. To use the individual charts, one must f i r s t determine the parameters K, 2p* and 2p* a* for the particular problem. With these known, one determines maximum upsurge by going upwards on the 2p* a* ordinate from the zero surge abscissa to the intersection with the 2p* curve. Similarly, the maximum downsurge i s found by going down- wards on the 2p* a* ordinate from the zero surge abscissa to the 2p* curve. To i l l u s t r a t e : Known: K = 0.1, 2p* o* = 10, 2p* = 4 No wall f r i c t i o n , Differential o r i f i c e 2.5:1 Required: Maximum upsurge and downsurge at midpoint. Solution: Maximum upsurge = 0.771 HQ* Maximum downsurge = 0.358 HQ* 28 5.2 NO HEAD LOSS, FRICTIONLESS FLOW The single chart in this category compares well with the chart for frictionless flow published by Evans and Crawford (Appendix A, Fig. A - l ) . Since frictionless flow would not occur in reality, this chart would be used for purposes of analysis but not for design problems. 5.3 ENTIRE HEAD LOSS CONCENTRATED AT THE ORIFICE Differential o r i f i c e - inflow to outflow head loss ratio 2.5:1 The graphs for K = 0.3, 0.5 and 0.7 compare well with the corresponding graphs published by Evans and Crawford as shown in Appendix A, Figures A-2, A-3, and A-4. The curves are generally well defined except for the lower values of 2p* and 2p* a* for the upsurge region. This i s in the range of very low velocities. The 2p* = 0.5 curves were eliminated for K=0.8 to K => 1.0 inclusive because the program would not converge to a solution. Two additional charts for K = 0.5 were included in this group. These were for powers of 1.0 and 1.4, the powers being the values of m in the equation H* v a i r m = constant. The Intent was to check the possible variation of results caused by using the power m as 1.0, 1.2 and 1.4. A comparison of the charts and a partial l i s t i n g of the results as shown in Table 5.1 indicate that the power 1.2 gives an approximate average for the upsurges and downsurges. The charts also indicate that one must accurately determine whether the system is isothermal or adiabatic when using the powers 1.0 and 1.4 because the resultant 29 TABLE 5.1 Comparison of Results Obtained for the Powers 1.0,1.2 and 1.4 2p* 2p*a* Point m = 1.0 m = 1.2 m = 1.4 UP D n U P D n "P D n 1 2 P .705 .572 .732 .615 .793 .649 M .435 .458 .527 .498 .669 .532 3/4 .235 .342 .290 .372 .343 .399 4 P .413 .452 .475 .499 .542 ..532 M .254 .355 .313 .386 .331 .414 3/4 .132 .264 .151 .283 .178 .302 10 P .173 .324 .208 .352 .240 .378 M .120 .250 .134 .270 .157 .287 3/4 .058 .200 .065 .210 .073 .219 30 P .061 .220 .073 .234 .085 .247 M .050 .185 .056 .194 .063 .201 3/4 .022 .165 .024 .169 .028 .172 4 8 P .782 .535 .902 .583 1.012 .623 M .435 .375 .504 .409 .575 .439 3/4 .211 .272 .249 .290 .278 .308 20 P .322 .385 .375 .421 .427 .454 M .191 .270 .220 .290 .248 .310 3/4 .089 .201 .104 .227 .118 .235 40 P .169 .286 .198 .313 .227 .339 M .102 .222 .121 .232 .137 .243 3/4 .049 .201 .056 .205 .064 .209 80 P .090 .225 .105 .234 .121 .249 M .056 .204 .065 .208 .075 .212 3 4 .025 .192 .031 .194 .035 .196 upsurges vary by as much as 50% and the downsurges vary by as much as 40%. The greater variation occurs generally for small 2p* a* values. Simple or i f i c e - inflow to outflow head loss ratio 1:1 The curves in this group are well defined except for some scatter in the range of low 2p* and 2p* a* values for upsurge only. The 2p* = 0.5 curves were eliminated for the range K = 0.7 to K = 1.0 inclusive because the program would not converge to a solution. Note that for the higher values of K, 2p*, and 2p* a*, the upsurges at the mid-point of the line become higher than the upsurges at the pump. Comparison of upsurges and downsurges for 2.5:1 and 1:1 orifices The f r i c t i o n factor, K, is based on inflow to the air chamber. To compare the upsurges and downsurges for the two orifi c e s , one differential with a 2.5:1 inflow to outflow head loss ratio and the other simple, assume that the inflow losses are equal. Therefore, the outflow loss for the simple o r i f i c e w i l l be 2.5 times greater than the outflow loss for the differential o r i f i c e . It follows that the down- surges w i l l be equal for the following f r i c t i o n factors: (1) Differential, K = 0.5; Simple, K » 0.2; and (2) Differential, K = 1.0; Simple, K = 0.4. Table 5.2 does in fact verify this, except for isolated instances. 5.4 ENTIRE HEAD LOSS ATTRIBUTABLE TO DISTRIBUTED FRICTION As the total head loss increases, the distributed f r i c t i o n significantly reduces the upsurges, and, to a lesser extent, the down- surges. The downsurges are affected to a greater degree away from the TABLE 5.2 UPSURGES AND DOWNSURGES FOR 1;1 AND 2.5:1 ORIFICES 2p * 0.5 0.5 0 . 2 2p*c* 1 2 3 4 6 8 10 15 UP h i .688 .561 .460 .377 .295 .236 .193 .137 .587 .314 .311 .320 .213 .160 .148 .103 .330 .184 .198 .162 .093 .080 .067 .048 DN M • 505 .442 .403 .371 .325 .293 .269. .231 .442 • 371 • 325 .293 .252 .227 .209 .183 .360 .284 .244 .220 .192 .175 .165 .150 0.4 UP .608 .488 .376 .298 .224 .177 .139 .092 M i .568 . 271 .326 .153 .258 .165 .254 .128 .182 .083 .141 .061 .116 .050 .085 .035 DN M .487 . 437 . 360 .437 .370 .296 .399 . 330 .264 .369 . 303 - 246 .329 . 270 . 224 .302 . 251 -213 .284 . 244 . 206 .254 . 241 .202 K - 0 . 5 UP M .593 . 528 .478 . 308 .365 . 243 .289 . 217 .215 .145 .162 .111 .127 .092 .083 .O64 .250 .146 .147 .110 .071 .052 .040 .027 DN P M .486 . 442 .442 .371 .403 . 325 .371 .293 .325 .252 .293 .227 .269 . 209 .231 .183 .360 .284 .244 .220 .192 .175 .165 .ISO 1.0 UP EN 1.0 1.0 2 3 4 6 10 15 20 30 .834 .737 .651 .489 .325 .232 .181 .123 .710 .459 .456 .276 .213 .164 .135 .101 .405 .243 • 233 .142 .099 .073 .057 .045 .614 • 546 .497 .430 .352 .300 .269 .234 .498 • 431 .385 .328 .270 .234 .215 .194 .372 .317 .233 .245 .210 .190 .180 .169 .725 .622 .494 .349 .221 • 153 .116 .081 .641 .314 .359 .184 .362 .177 .247 .107 .174 .076 .131 .058 .105 .046 .081 .033 .625 .522 .415 .564 .463 .369 .521 .425 .342 .462 .378 .311 .397 .354 .329 .330 .283 .302 . 269 .298 .266 .302 . 295 . 262 .729 .528 .607 .349 .476 .323 .336 .199 .208 .134 .141 .096 .108 .076 .073 .056 .290 .188 .151 .101 .065 .045 .035 .024 .614 .498 .546 .431 .497 . 385 .430 .328 .352 . 270 .300 .234 .269 .215 .234 .194 .372 .317 .233 .245 .210 .190 .180 .169 .571 .398 .183 .400 .277 .131 .299 .202 .100 .203 .149 .061 .121 .099 .041 .081 .087 .032 .063 .079 .026 .043 .064 .022 .625 .522 .415 .564 . 463 . 369 .521 .425 -342 .462 . 378 . 311 .397 . 330 . 283 • 354 . 302 . 269 .329 . 298 . 266 .302 .295 .262 1.191 - 7 0 1 .e67 .502 .572 .353 .4Ci .259 .318 .206 .227 .155 .178 .126 .126 ,r»3 2.0 2.0 4 6 10 15 20 30 40 60 .365 .257 .174 .123 .097 .071 .057 .040 •599 .461 • 516 .391 • 4 » .319 .364 .322 .275 .251 .225 .275 .251 .226 .212 .198 333" .288 .245 .221 .203 .195 .189 .182 .887 .621 .393 .271 .210 .146 .112 .078 .510 .263 .417 .184 .278 .126 .208 . 092 .166 .073 .122 .054 .098 .043 .074 .031 .665 .543 .591 .481 .506 .451 .418 .381 .361 .340 .419 .382 .362 .340 .338 .337 .434 .393 .356 .336 .326 .318 .316 .314 .878 .611 .379 .261 .200 .137 .105 • 072 .470 .327 .235 .162 .128 .091 .070 .250 I l 6 6 .109 .077 .060 .043 .033 .010 -02? .599 .516 .429 .364 .322 .275 .251 .225 .461 .391 .319 .275 .251 .226 .212 • 198 .335 .288 .245 .221 .208 .195 .189 .182 .551 -338 .369 -235 .222 .151 .150 .114 .113 -090 .077 .073 .058 .065 .039 .059 .171 .113 .070 .048 .036 .027 .023 .019 .665 .543 .591 .481 .506 .419 .451 .382 .418 .362 • 381 .340 • 361 .338 •340 .337 • 434 .393 • 356 • 336 .326 .318 .316 • 314 4 . 0 8 10 15 20 30 40 80 100 1 .303 .718 1.061 .581 .737 • 371 .404 .317 413 335 244 194 .177 .113 .147 .095 T36T .293 .204 .165 .119 .095 .055 .045 .583 .409 .543 .376 .470 .323 .421 .290 .355 .253 .313 .232 .234 .208 .222 .203 .290 .270 .241 .227 .213 .205 .194 .192 .914 .737 .504 .387 .268 .207 .112 .092 .557 .260 .466 .218 .332 .153 .365 .120 .191 .084 .152 .066 .086 .038 .072 .031 .636 .519 .594 . 489 .528 .445 .488 .420 . 444 .394 .420 . 380 .380 .365 .371 .364 .430 .413 .389 .376 .363 .357 .351 .350 .902 .725 .491 .375 .258 .198 .115 .086 504 .409 .286 .220 .155 .121 .065 .053 .249 .199 .136 .104 .072 .056 .031 .025 .583 .543 .470 .421 .355 .313 .234 .222 .409 .376 • 323 .290 .253 .232 .208 .203 .290 .270 .241 .227 .213 .205 .194 .192 .541 .331 .432 .269 .289 .185 .219 .142 .148 .100 .112 .079 .058 .052 .046 .048 .158 .128 .087 .067 .046 .035 .019 .017 .636 .519 .594 .489 .528 -A45 .488 . 420 .444 .394 .420 .380 .-380 .365 .371 .364 430 .413 .389 .376 • 363 .357 .351 .350 pump. This i s logical because the distributed f r i c t i o n i s i n effect over a longer distance. For K = 0.4 and above, upsurges have been eliminated while the downsurges are divided into three distinct groups, at the pump, the mid-point, and the three-quarter point. For K values above 0.6, the downsurges for the various values of 2p* in each group become so closely spaced as to almost merge. 5.5 HEAD LOSS EQUALLY DIVIDED BETWEEN UNIFORMLY DISTRIBUTED WALL FRICTION AND ORIFICE LOSS For K values of 0.7 to 1.0 inclusive, the upsurges disappear completely and the downsurges are segregated into three distinct groups i.e. at the pump, the mid-point, and the three-quarter point. 5.6 USE OF THE CHARTS The downsurge charts produced by Evans and Crawford are based on the minimum head in the pipeline. For this reason they stated that the charts were for preliminary design purposes only. Since this program was derived to give the actual absolute pressure in the air chamber, the charts can be used for f i n a l design as well as preliminary design and checking purposes. Usually when an air chamber is being designed for a pump - discharge line, the values of L, a, V , Q q , A, H q* and g w i l l be known From these values, 2p* can be computed. The allowable maximum surge values may be dictated by specifications, operating conditions, or the profile of the discharge line. For the computed value of 2p* and the specified maximum allowable surges, values of K and 2p* a* can be chosen from the charts such that the surge limitations are met. If the allowable surge conditions can not be satisfied by data from the charts, probably some means other than an air chamber should be used to control the surges. When 2 p * a* has been determined, C q can be computed from Eq. (1.5) ( i . e . ) : C = p * a * Q - . o o a Numerical examples demonstrating the use of the charts are given in Appendix A. Figure 5.1 shows the configuration of the pump, air chamber, pipeline and reservoir. The Charts begin on page 45. PIPELINE WITH AIR CHAMBER FIG. 5.1 Pipe Wall Friction - Steady-state absolute pressure head at pump - Steady-state pressure head at pump - Total head loss H o r f 0 - Head loss due to or i f i c e resistance Ho* Ho Upsurges and Downsurges measured from this line (For uniformly distributed f r i c t i o n only) Measured to transient ordinary pressure head Pipeline W.S.. Atmospheric Pressure Head Reservoir 4-1 C •rl O PM I T3 •rl a c •H O PM I ro ca erf CHAPTER VI DISCUSSION 6.1 VOLUME OF AIR IN THE CHAMBER Since a*, the parameter pertaining to a pump discharge line having an air chamber, is directly proportional to C 0, the i n i t i a l volume of air in the chamber, the i n i t i a l volumes of air and water in the tank must be maintained within certain limits to ensure proper operation of the chamber. The compressed air which dissolves in the water or is lost through leakage must be continually replaced. Some means of automatic shut down of the pump or pumps must be provided should the proper water level in the tanks not be maintained. The minimum controls required are shown schematically in Fig. 6.1. The following items^ should be considered when fixing the compressor "on" and "off" levels: (a) capacity of the compressor, (b) size of the air chamber, (c) frequency of starting and stopping of the compressor, (d) daily temperature variations that might actuate the controls, and (e) how quickly the system is to be put back into operation after a prolonged shutdown. The emergency levels can be at nominal distances above and below the compressor operating levels on installations having only one pump or that provide manual starting or stopping for individual pumps on the same line. If automatic starting and stopping of the individual pumps 35 on the same line are required, the emergency levels should be sufficiently removed from the compressor operating levels to contain the surges produced by starting or stopping the largest of the pumps under the most c r i t i c a l i n i t i a l conditions. The charts can be usefully employed to check the locations of the emergency levels. 6.2 TOTAL VOLUME OF THE AIR CHAMBER Once 2p* a* has been determined from the charts, C Q can be cal- culated by using Eq. 1.8. The volume of the air chamber is then deter- mined by considering that the chamber must contain adequate air above the upper emergency level to control the surges to desirable limits, and enough water below the lower emergency level to prevent unwatering. With allowance for the volume between the upper and lower emergency levels, the total required volume of the air chamber can be computed. The minimum volume of air that must be maintained in the chamber to control the pressure surges i s the volume of the chamber above the upper emergency level. This volume can be designated C' which is numerically equal to the volume C Q. By adding to this quantity the volume of the chamber between the upper and lower emergency levels, one determines the i n i t i a l volume of air in the chamber that w i l l result in the lowest water-surface level following pump shut down. This new volume of air becomes C" equal to C' plus the volume of air between the upper and lower emergency levels. The downsurge at the pump with this i n i t i a l volume of air can be determined from the curves by computing a new value of 2p* a* based on C" instead of C'. Assuming that this expansion is isothermal , the total volume of the air chamber becomes C" H * o H q* - downsurge at pump Under favorable conditions, the air tank volume i s about one to two percent of the conduit volume. A conservative approximation of tank size would be two to four percent of the conduit volume. Favourable conditions would be interpreted as long pipe lines with high f r i c t i o n losses and no high points of topography. The i n i t i a l air volume is generally about 40 percent of the tank volume. 6.3 ORIFICE DESIGN Since the function of an air chamber is to decrease both the upsurges and the downsurges on pump failure, i t is necessary to throttle the reverse flow of water from the discharge line into the chamber while providing l i t t l e throttling for flow out of the chamber. An effective device for producing a high head loss for inflow while keeping the exit head loss at a minimum is a differential o r i f i c e as shown in Fig. 6.2. The design is essentially a bellmouth for flow from the chamber and a re-entrant tube for flow into the chamber. This design w i l l give discharge coefficients of 1.0 and 0.5 for outflow and inflow respectively. The inflow head loss for a specified rate of flow would be approximately four times as great as the outflow headloss. However, this head loss ratio of 4:1 is d i f f i c u l t to obtain in practice. Upper Emergency Level Compressor On Compressor Off Lower Emergency Level AIR CHAMBER CONTROL LEVELS FIG; 6.1 DIFFERENTIAL ORIFICE FIG. 6.2 39 A r a t i o of 2.5:1 i s more r e a l i s t i c . I f head loss i n the p i p e l i n e due to w a l l f r i c t i o n i s considered as concentrated at the o r i f i c e , the o r i f i c e design should allow f o r th i s assumption. For example, design a d i f f e r e n t i a l o r i f i c e f o r an inflow loss of 60 per cent of H q* and outflow loss of 30 per cent of H * for an inflow and outflow of Q . For a flow of Q , the p i p e l i n e o o o s u r f a c e - f r i c t i o n loss i s 10 per cent of H *. The o r i f i c e should be r o designed f o r a head loss of 50 per cent of H q* for an inflow of Q q and a head loss of 20 per cent of H * for an outflow of Q . The actual r o o o r i f i c e design, head loss r a t i o of inward flow to outward flow should be 2.5:1. An o r i f i c e may be designed to give a maximum i n i t i a l head loss through the o r i f i c e equal to the maximum downsurge. This condition i s known as normal t h r o t t l i n g ^ . Greater or smaller t h r o t t l i n g losses may be s a i d to give o v e r - t h r o t t l i n g or under-throttling, r e s p e c t i v e l y . The minimum head i n the pipe w i l l correspond to the maximum a i r expansion i n the chamber f o r the conditions of normal t h r o t t l i n g and under- t h r o t t l i n g . For o v e r - t h r o t t l i n g the minimum head i n the pipe can not be used to determine the maximum a i r expansion i n the chamber. For the condition of o v e r - t h r o t t l i n g , the minimum pressure i n the tank must be known i n order to determine the maximum a i r expansion. Large computational errors r e s u l t when f r i c t i o n i s ignored. The i n c l u s i o n of d i s t r i b u t e d w a l l f r i c t i o n increases the accuracy of the maximum and minimum pressures and corresponding maximum a i r expansion i n the chamber. Thus the charts including d i s t r i b u t e d w a l l f r i c t i o n give highly accurate r e s u l t s . 40 6.4 WATER-COLUMN SEPARATION IN PUMP DISCHARGE LINES Water - column separation^ i s the f i r s t phase in the development of one of the most destructive types of waterhammer surge in pump- discharge pipe lines. Following pump failure, the sudden pressure drop downstream might be severe enough to bring about a temporary vapour pressure condition, and possibly the formation of a void in the pipe line. The subsequent closure of this void often results in violent local surges well above any possible transient pressure rises i n a continuous water column. The extent of pressure rise i s proportional to the f l u i d velocity destroyed at the instant of vacuous space closure. The four major factors^ influencing water - column separation are: (1) rate of flow stoppage, (2) length of system, (3) normal operating pressure at c r i t i c a l points, (4) velocity of flow. (1) For pumps that have small rotational inertias the result i s complete pump stoppage from within a fraction of a second to a very few seconds after pump failure. This very much aggravates the downsurge problem. (2) The length of the system determines the length of time the pressure w i l l continue to f a l l before positive pressure waves reflected from the far end of the line counteract the pressure drop. A long line with a pump having a small rotational inertia very often w i l l experience water - column separation on pump failure. (3) Points of low pressure are c r i t i c a l . At points of low pressure such as the crests of h i l l s over which a pipe line passes, a 41 slight interruption of flow may result in a drop to vapour pressure and resulting column separation. (4) The fourth major element in water-column separation i s the velocity of water i n the pipe line preceding the cause of perturbation. As the steady state velocity increases, the size of the vacuous space, the reverse flow velocity, and the f i n a l surges following the void collapse a l l become greater. A l l of these elements are inter-related. For example, extensive water-column separation may occur even with a very low velocity i f the pipe line i s long enough and the steady state pressure head is low. An air chamber i s one means of preventing or controlling water- column separation for medium to high-head systems. An example is given in Appendix A indicating the manner in which the charts can be used to determine the possibility of water-column separation. Although these charts cannot be used to analyze the water-column separation condition, the high degree of accuracy does enhance the a b i l i t y of being able to predict i f water-column separation w i l l occur. Further studies could be carried out to attempt to determine maximum and minimum pressures occurring for the water-column separation phase of waterhammer. CHAPTER VII CONCLUSIONS (1) Since the program was evolved from the basic differential equations for momentum and continuity to give the absolute pressure i n the ai r chamber, nonlinear terms are retained and f r i c t i o n i s included, the charts can be used for f i n a l design purposes*. (2) The validity of the charts i s demonstrated by comparing the results obtained by the method of characteristics with those obtained by the graphical method. (3) It i s important to analyze the system properly and to use the group of charts which most closely approximate the system in order to get valid results. In some cases i t might be advantageous to interpolate between graphs within the same group. For example, K might be in the range 0.0 to 0.1. (4) It is important to determine whether the expansion and compression of a i r i n the chamber i s adiabatic or isothermal because the results vary significantly for the powers m = 1.0 and m = 1.4, where m is the power in the equation H*v a^ r m = constant. For example, for 2p* = 4 and 2p* o* = 8, the upsurge at the pump for m = 1.0 is 0.782 H* q and for m = 1.4 is 1.012 H*Q, the downsurge at the pump for m = 1.0 is 0.535 H*Q and for m = 1.4 i s 0.623 H*Q. * The charts produced by Evans and Crawford are to be used only for preliminary design purposes. The authors stress that "for f i n a l design of an installation having an air chamber, individual solutions similar to that shown by Mr. Angus ('Air Chambers and Valves in Relation to Water-Hammer', Transactions, ASME, Vol. 59, 1937, p.661) should be made to ensure that the air chamber w i l l f u l f i l l design requirements". 42 - The power m = 1.2 gives an approximate average for the upsurges and downsurges. For the same pipeline constants as above, for m = 1.2, the upsurge at the pump is 0.902 H*0 and the downsurge at the pump is 0.584 H*D. (5) The charts produced by Evans and Crawford are quite accurate as shown by the computer check on these charts using the method of charac- t e r i s t i c s . The accuracy of the charts produced in this work is enhanced by the inclusion of f r i c t i o n and nonlinear terms. The charts presented in this thesis cover a much wider range of variables than those published by Evans and Crawford. For each group the following charts are presented: No Line Friction, Line Friction Only No Orifice Loss, and Friction Loss Equally Distributed between Orifice Loss and wall Friction, the range of K is from 0.1 to 1.0. (6) Bergeron's method of graphical analysis considering line f r i c t i o n concentrated at five points i s quite accurate as demonstrated by the computer check using the method of characteristics. (7) The number of sections used in analyzing the pipe system is important. If N is too large, excessive computer time w i l l be required; i f N i s too small, the program w i l l not converge to a solution. In this program, for instance, N was set equal to ten and gave good results. For N equals five, the program would not always converge to a solution. (8) For high values of K, 2p*, and 2p* a* the upsurges at the mid-point can be higher than those at the pump. 44 BIBLIOGRAPHY 1. A l l i e v i , L., "Air chambers for Discharge Pipes", Trans. ASME, Vol. 59, Paper Hyd-59-7, November 1937, pp. 651-659. 2. Bergeron, L., Water Hammer in Hydraulics and Wave Surges in Elec t r i c i t y , John Wiley and Sons, Inc. New York, Copyright 1961 by the ASME. 3. Chaudhry, M.H., Boundary Conditions for Analysis of Water Hammer in Pipe Systems, A thesis submitted in partial fulfillment of the requirements for the degree of Master of Applied Science at the University of British Columbia, 1968. 4. Evans, W.E., and Crawford, C.C., "Design Charts for Air Chambers on Pump Lines", Trans. ASCE, September 1954, pp. 1025-1036. 5. Parmakian, J., Waterhammer Analysis, Dover Publications, Inc. New York, 1963. 6. Paynter, H;M., Discussion of "Design Charts for Air Chambers on Pump Lines", Trans. ASCE, September 1954, pp. 1039-1045. 7. Richards, R.T., "Water-Column Separation in Pump Discharge Lines", Trans. ASME, Paper No. 55-A-74, 1955, pp. 1297-1304. 8. Ruus, E., and Chaudhry, M.H., "Boundary Conditions for Air Chambers and Surge Tanks", Trans. EIC, November 1969, EIC-69-HYDEL 22, Vol. 12, No. C-6. 9. Streeter, V.L., and Wylie, E.B., Hydraulic Transients, McGraw-Hill Book Company, New York, 1967. 45 THE CHARTS GROUP I NO HEAD LOSS, FRICTIONLESS FLOW (No wall f r i c t i o n , no or i f i c e loss) MAX. DOWNSURGE MAX.UPSURGE GROUF II ENTIRE HEAD LOSS CONCENTRATED AT THE ORIFICE (no wall friction) A. DIFFERENTIAL ORIFICE, RATIO 2.5:1 B. SIMPLE ORIFICE, RATIO 1:1 M A X . D O W N S U R G E M A X . U P S U R G E H 0 * H * o o p p o p o p p p p p p p p p — — — — o ui ^ w ro — O . — K> O J A 01 ff> co co P '— ro 01 4^ to         MAX. DOWNSURGE MAX. UPSURGE H; H, o o o o o p o ho o o o b i o cr> O o co o MAX. DOWNSURGE MAX.UPSURGE H0* Hi? o o o o o Ln o O ro O o o o 0> o o co o to '1 4 — ^ i 2 T> -0 TJ I I I I 'I to MAX. DOWNSURGE MAX. UPSURGE MAX. DOWNSURGE H* MAX. UPSURGE H*          GROUP III ENTIRE HEAD LOSS ATTRIBUTABLE TO DISTRIBUTED FRICTION (no o r i f i c e loss)    75       GROUP IV HEAD LOSS EQUALLY DIVIDED BETWEEN UNIFORMLY DISTRIBUTED WALL FRICTION AND ORIFICE LOSS           APPENDIX A COMPARISON OF CHARTS AND NUMERICAL EXAMPLES Comparison of the charts derived by the method of characteristics and those produced by Evans and Crawford. Example on the design of an air chamber for a short pipeline of large diameter. Example on checking the maximum upsurges and downsurges for a long pipeline. APPENDIX A-l COMPARISON OF THE CHARTS DERIVED BY THE METHOD OF CHARACTERISTICS AND THOSE PRODUCED BY EVANS AND CRAWFORD FIGURE A-l FIGURE A-2 FIGURE A-3 FIGURE A-4 No f r i c t i o n loss Total f r i c t i o n loss Total f r i c t i o n loss Total f r i c t i o n loss 0.3 HQ* (orifice loss) 0.5 HQ* (orifice loss) 0.7 HQ* (orifice loss) FIG A l 96 FIG. A-? FIG. A-3 98 UJ o or z> CO CL Z) X < 1.3 1.2 I.I 1.0 0.9 0.8 * o 0.7 X 0.6 0.5 0.4 0.3 0.2 0.2 UJ o or r> co z: o Q X < 2 0.3 * o 0.4 0.5 0.6 0.7 i — r . T _ r . . PUMP MI0LENGTH PUMP MIDLENGTH } Evans ft Crawford | Computer study 4 5 6 1 ! I H , (- 7 6 9 10 15 1 ' i : . . 2 :/?V.V 20 ....)._ . -L... 30 H r- 40 50 60 70 8090 ! ! M I i FIG. A-4 APPENDIX A-2 EXAMPLE ON DESIGN OF AN AIR CHAMBER FOR A SHORT PIPELINE OF LARGE DIAMETER PROBLEM Given the following data, design the most economical air chamber which w i l l limit the waterhammer surges to the specified limits. DATA Atmospheric Pressure E l . 400 " Check valve closes immediately on pump failure. Length of pipeline (L) = 3220 feet. Area of pipe (A) - 3.142 f t 2 Steady-state discharge (QQ) = 18.5 cu.ft per sec. Steady-state velocity (V Q) = 5.9 f t per sec. 100 Steady-state head at pump (HQ) = 300 f t Water hammer wave velocity (a) = 3660 f t per sec. Atmospheric pressure = 34.0 f t of water. Neglect line f r i c t i o n losses. ALLOWABLE HEADS Maximum at pump = 400 f t of water. Maximum negative heads at midlength and three-quarter point = 20 f t . of water (sub-atmospheric). SOLUTION The allowable surges are: At pump - allowable upsurge = 400-300 =0.30 HQ* At midlength - allowable downsurge = 400-350+20 =0.21 HQ* At three-quarter point - allowable downsurge = 400-370+20 = 0.15 HD* 2 o * = = (3660)(5.9) = 2 0 2 p 1S7 (32.2)(334) 2 , 0 From the charts in Group II, Entire Head Loss Concentrated at the Orifice, Differential Orifice 2.5:1, the surge conditions can be met using the values: K = 0.1, 2p* o* = 35 K = 0.2, 2p* a* = 24 K = 0.3, 2p* a* - 22 K = 0.4, 2p* a* =60 The volume of air in the chamber w i l l vary directly as a*, so the smallest value of 2p* o* w i l l be used. For the values, K = 0.3 2p* a* - 22 2p* = 2.0 101 At the pump: Maximum upsurge = 0.26 HQ* Maximum downsurge = 0.32 HQ* At the midlength: Maximum upsurge = 0.155 H * Maximum downsurge = 0.21 HQ* At the three-quarter point: Maximum upsurge = 0.07 HQ* Maximum downsurge = 0.15 HQ* The differential o r i f i c e should be designed to provide a head loss of (0.3)(334) = 100 f t for a flow of 18.5 cu.ft per sec. into the chamber. From Eq. 1.8, (2p* a*) ALVQ C ° 2a = (22) (3.142)(3220)(5.9) (2)(3660) = 179 cu.ft C'=C0 - 179 cu.ft Assume: Volume between upper and lower emergency levels i s 20% of C'. Then, C" = 1.20 C = 215 cu.ft and 2p* a* = 1.2 x 22 - 26.4. The maximum downsurge at the pump becomes 0.295 HQ*. Total air chamber volume = C"H0* HQ* - downsurge at pump - 215 1-.295 = 305 cu.ft. REMARKS The c r i t i c a l points with respect to water-column separation occur for this example at the midlength and three-quarter point. The design ensures that water-column separation w i l l not occur. If the problem were one of analysis, the maximum downsurge at the c r i t i c a l points would be determined. A pressure of -34 f t or less would indicate the formation of a vacuous space. 103 APPENDIX A-3 EXAMPLE ON CHECKING THE MAXIMUM UPSURGES AND DOWNSURGES FOR A LONG PIPELINE PROBLEM Given the following data, determine the maximum upsurges and downsurges at the pump, the midlength and the three-quarter point of the pipeline. o II W Atmospheric Pressure Head = 34 ft - ^ 7 o m ro II 4-1 VO ' r H ro II O sa •7 d. z 9192 f t DATA Check valve closes immediately on pump failure. Length of pipeline (L) = 9192 f t Area of pipe (A) =0.79 f t 2 Steady-state discharge (QQ) =5.0 ft 3/sec Steady-state velocity (V Q) = 6.3 ft/sec Steady-state head at pump (HQ) = 316 f t . Line f r i c t i o n loss (Hp) = 70 f t . Waterhammer wave velocity (a) = 3660 ft/sec. Atmospheric pressure = 34.0 f t . of water. I n i t i a l a i r volume i n chamber (C ) = 50 f t 3 . o SOLUTION (A) No or i f i c e loss 2C Qa 2(50)(3660) 2p* a* = = = 8.0 ALVC (0.73)(9192)(6.3) 70 K - — - 0.20 aV (3660)(6.3) 2p* = = = 2.04 gHQ* (32.2)(350) From the charts in Group III, Entire Head Loss Attributable to Distributed Friction: (1) At pump: Maximum upsurge = 0.285 HQ* Maximum downsurge = 0.55 HQ* (2) At midlength: Maximum upsurge =0.15 HQ* Maximum downsurge = 0.32 HQ* (3) At three-quarter point: Maximum upsurge = 0.075 HQ* Maximum downsurge = 0.175 H Q (B) Total head loss evenly divided between line f r i c t i o n loss and o r i f i c e loss (2.5:1 differential o r i f i c e ) . (1) At pump: Maximum upsurge = 0.50 H Q* Maximum downsurge = 0.515 H Q 105 (2) At midlength: Maximum upsurge = 0.28 HQ* Maximum downsurge = 0.32 HQ* Maximum upsurge = 0.14 HQ* Maximum downsurge = 0.19 HQ* (3) At three-quarter point: REMARKS It i s obvious from the foregoing results that care must be exercised in selecting the charts to best approximate the actual physical condition. The surge results (especially the upsurges) vary considerably for different types of head loss. 106 APPENDIX - B GRAPHICAL CHECKS ON PROGRAM B-1 Check for total head loss concentrated at the o r i f i c e . B-2 Check for total head loss attributable to distributed f r i c t i o n . APPENDIX B-l CHECK FOR TOTAL HEAD LOSS CONCENTRATED AT THE ORIFICE PROBLEM Determine the transient state pressures and velocities in the pipeline adjacent to the pump at A. The transient conditions are caused by pump failure. Atmospheric DATA I ^B Throttling = ±50 f t for Q = ±20 ft 3/sec. FIG. B-la Check valve closes immediately on pump failure. Length of pipe line (L) = 3220 f t . 3 Steady-state discharge (QQ) « 20.0 f t /sec. Steady-state velocity (V Q) ~ 5.00 ft/sec. Waterhammer wave velocity (a) = 3220 ft/sec. Pipe line constant (2p*) =2.00 Constant for a pipe line having an air chamber CHECK 2C a (2p* a* = -^-j-) = 10.0 o Atmospheric pressure = 34.0 f t of water. Orifice throttling loss = ± 50 f t for Q q = + 20 ft 3 / s e c . Air expansion in the chamber is given by 1 2 H* v . = a constant, in which H* and v . air air are the absolute pressure and volume of a i r in the chamber. Neglect line f r i c t i o n losses. Results obtained on the d i g i t a l computer using the method of characteristics are close to those obtained by Parmakian^ (page 135) by the graphical method (see Fig. B-3b ). 103 APPENDIX B-2 CHECK FOR TOTAL HEAD LOSS ATTRIBUTABLE TO DISTRIBUTED FRICTION PROBLEM Determine the transient state pressures and velocities in the pipe line adjacent to the pump at B. The transient conditions are caused by pump failure. DATA FIG. B-2a (Water supply line for city of T r a i l ) . Check valve closes immediately on pump failure. Length of pipeline (L) = 9150 f t . 3 Steady-state discharge (QQ) = 4.0 f t /sec. Steady-state velocity (V ) = 5.1 ft/sec. Pipe line constant (2p*) =1.38 Constant for a pipe line having an air chamber (2p* a* 2C a o Q L xo -)= 5 I l l Atmospheric pressure = 33.0 f t . of water Total f r i c t i o n loss (Ĥ ,) = 90.0 f t . of water. Pressure head at the pump (HQ) = 387.0 f t . of water 3 Steady-state volume of air in the chamber (C Q) = 25.0 f t 1 2 Air expansion in the chamber i s given by H* C = a constant, in which H* and C are the absolute pressure and volume of air in the chamber. 1.2 H* C This may be written as h* c =1 where h* = rr~ and c = ~r~ • Ho co There i s no loss for flow into or out of the chamber. The f r i c t i o n loss in the pipe i s considered concentrated at the orifices shown on the diagram. CHECK Results calculated on the d i g i t a l computer using the method of characteristics are close to those (Fig. B-2b) obtained by Eugen Ruus, who analyzed this system by the method of graphical water hammer analysis concentrating the pipe line wall f r i c t i o n at five points as shown in Fig. B-2a. pump. Check valve closes on pump f i i l u r e ) .;. -< ! ; 1 V ) APPENDIX - C PROGRAM FOR THE ENTIRE HEAD LOSS CONCENTRATED AT THE ORIFICE PROGRAM FOR THE ENTIRE HEAD LOSS ATTRIBUTABLE TO DISTRIBUTED FRICTION 114 1 . ENTIRE HEAD LOSS CONCENTRATED AT THE ORIFICE $LIST AIRCHAMIO 2 C WATEPHAMfER PROG P AM. PUMP AT UPSTREAM END WITH AIR CHAMBER ADJACENT ? 3 C TU THE PUMP. RESERVOIR AT DOWNSTREAM END. 4 C CHECK VALVE CLOSES I MM EC I AT EL V ON PUMP FAILURE. 5 C NO LIME FRICTION. HEAR LOSS CONCENTRATED AT ORIFICE. 5. 5 C NO MI NCR LOSSES. 6 01 KENS ION V ( 2 0 ) , V P ( 2 0 ) , H ( 2 0 ) t H P ( 20 ),VP(20),VS(20) ,HR(20),HS{20) , 7 1HMAXI I C ) , H MI N (1 C) ,HSS( 10) . SUM AX ( 10 ) , S U M I N (1 0 ) , SUNSS ( 10 ) , 8 2UPSMAX( 10) , IJNSMAXI 10),UPSANS( 10) ,DNSANS< 10) 9 DATA N/10/,VA/3216./,G/32.16/,FL/3216./,PM/l.2/,MM/l/, 10 IF/0.0/,CORFIN/2.5/.AP/3./, 11 2CK/0.1/, 12 3VO/3.5/ 13 WRITE! 6, 15) N»VA,G,FL,PM,N'M,F,CORFIN,AP,CK,VO 14 1S» FORMAT !/• THE PARAMETERS ARE NOW...'/ 15 1' N= »,I5,' VA= ',F8.2,' G= »,F6.2,' FL= ',F8.2/ 16 2' PM= '•F6.2,' MM = • t I 5,' F= " ^ 6 . 3 ^ C0RFIN= «,F6.2/ 17 3* AP= ' t F8 .2 t' CK= ',F6.2,' V0= «,F8.2) 18 27 WRITE(6,30) 19 30 FORM AT( ' PLC TMAX CPLAC UPSANS(l) UPAN10 UP SANS(6) UPAN3Q'/ 20 119X,' DNSANS(l) DN AN 10 DNS ANS(6 ) DNAN3Q * ) 21 6 READ!5,10) PLC,TMAX,CPLAC 22 10 FORMAT(3F8.3) 23 IF(CPLAC.LE.O.O) GO TO 110 24 C COMPUTE DT • 25 DT = FL/ ( (VO + VA)*FLOAT(N) ) 26 c CHECK FOR CONVERGENCE. 27 DX=F L/FLOAT(N) 28 T HET A= DT / DX 29 IF<THETA.LE.(1./VA)) GO TO 2 0 ^ " 30 17 GO TO 110 31 c COMPUTE COEFFICIENTS AND CONSTANTS FOP. ALL PIPES. 32 20 API=3.142 33 DP=SQRT(4.*AP/API) 34 C2 = G/VA 35 HF=(F*FL*VO*VO)/(2.*G*DP) 36 c HOABS= HO + HF + 34 . 37 H0ABS=VC/(C2*PLC) 37. 5 c H0= HEAD AT RE SE RVOIR. 3 8 HC=H0ABS-HF-34. 38.5 c H0RF0= ORIFICE HEAD LOSS FOR FLOW QO FROr* TANK. 39 HORFO= (C.K*HOABS-HF) /CORFIN 39. 5 c VOAIR= INITIAL AIR VOLUME IN TANK. 40 VOAIR=(CPLAC*V0*AP*FL)/(2.*VA) 40 . b c 00= STEADY STATE DISCHARGE. 41 CO=VO*AP 42 FF=F*DT/(7.*0P) 43 CF=HOPFC/(CG*CO) 44 C10=HOA6S*V0AIR**PM 45 c STEADY' STATE CALCULATIONS. 46 DHF=F*FL*VO*VO/(2.*G*DP*FLOAT(N)) 47 NN=N+1 48 49 CO 25 1=1,NN VI I ) =V0 50 • TE MP=MM-1 \. 51 H( I ) = HO +TFVP*OHF 115 52 25 C O N T I N U E • 5 2 . 5 C I N I T I A L I Z A T I O N 1 f: F M A X . A NO M I N . H E A D S . 53 DO 2 6 I M , 6 , 5 54 H M A X U )=H( I ) 55 H M J N l I )=H( I ) 56 H S S ( I )=H< I ) 57 26 C O N T I N U E 58 SUMSSI 1) =H< 3 ) -»H( 4 ) 5 9 SUM.SS ( 2 ) = H ( 8 ) + H I 9 ) 60 SUM AX( 1 ) = H ( 3 ) + H ( 4 ) ' ' ~ " 6 1 SUM IN ( 1 ) = H( 3 ) + H ( 4 ) 62 SUM A X ( 2 )=H(tt ) « H < 9 ) 63 SUM I N I 2)=H ( 8 ) + H { 9 ) 6 3 . 5 C T I M E I N I T I A L I Z A T I O N . 6 4 T = 0 . 0 6 5 V A I K = V 0 A 1 R 6 5 . 5 C P R I N T O U T I N T E R V A L I N I T I A L I Z A T I O N . 6 6 K = 0 6 7 C C O M P U T A T I O N OF V K , V S , H R , H S FOR A L L S E C T I O N S . 6 8 C I N T E R I O R S E C T I O N S . 6 9 4 0 DO 5 0 I = 2 , N 7 0 ' VR( I )=V( I ) - V A * T H E T A * l V I I ) - V < I ~ l ) ) " " " " " ~ 71 HR( I ) = H( I ) - V A * T N E T A * { H I I ) - H ( I - 1 ) ) 72 VS( I ) = V ( I ) - V A * T H E T A * ( V I I ) - V ( 1 + 1) ) 7 3 HS( I ) = H( I ) - V A * T H E T A * < H( I ) - H ( I + 1) ) 7 4 5 0 C O N T I N U E 7 5 C BOUNDARY S E C T I O N S . 7 6 C R E S E R V O I R . 77 V R ( N + 1 )=V(N+1 > - V A * T F , E T A * ( V ( N + l ) - V ( N ) ) 78 H H I N + 1 ) = H ( N + 1 ) - V A * T H E T A * ( H ( N + 1 ) - H ( N ) ) 7 9 C 3 = V R ( N + l > + C 2 * H R ( N + l ) - F F * V R ( N + l ) * A B S < V R ( N + 1 ) ) 80 C A I R C H A M B E R . 81 54 V S ( 1 ) = V ( 1 ) - V A * T H E T A * < V ( 1 ) ~ V l 2 ) ) 82 H S ( 1 ) = H I 1 ) — V A * T H E T A * ( H ( I ) - H ( 2 ) } 8 3 C 1 = V S I 1 ) - C 2 * H S ( i ) - F F * V S ( 1 ) * A B S ( V S ( 1 ) ) 84 38 T=T+.OT 8 5 M = M + 1 86 I F I T . G E . T M A X l GO TO 107 87 C T I M E I NCR EMFNT E D . BOUNDARY C O N D I T I O N S . 88 C A I R C H A M B E R . H O A B S * V O A l R * * P N = C O N S T A N T . ' ~ 8 8 . 5 C LOOP ( 8 9 , 1 0 6 ) TO A F P R O X . A V E . V E L O C I T Y FROM C H A M B E R . 8 9 V A V A P P = V ( 1 ) 9 0 GO TO 2 1 0 9 1 2 0 0 V A V A P P = V A V 92 2 1 0 C 1 1 = V A V A P P * A P 9 3 C A I R = V A I R + C 1 1 * 0 T " ~ . " 9 4 I F I C 1 1 ) 5 3 , 5 2 , 5 1 9 5 51 C O R F = 1 . 0 9 6 GO TO 59 9 7 52 C O R F = 0 . 0 9 8 GO TO 59 9 9 5 3 C ( i K F = 2 . 5 100 59 H 0 R F = C 0 R F * C F * C 1 1 * A B S I C l 1 ) 10 1 60 HP( 1 ) = ( C 1 C / C A I R * * P ' - 1 ) - H 0 R F - 3 4 . 1 0 1 . 5 C N E G A T I V E C H A R A C T E R I S T I C E Q U A T I O N . 102 V P I 1 ) = C 1 + C 2 * H P ( 1 ) 1 0 3 VAV=( VI 1 )+VP( 1) Ml. 1 0 4 V E H R = V A V A P P - V A V 105 I F ( A B S l V E R R ) . L E . 0 . 0 0 0 1 ) GO TO 2 3 0 < 106 2 2 0 GO TO 2 0 0 116 107 2 JO VPAIR = VAIRMAP*DT*VAV) 108 C RESERVOIR AT DOWNSTREAM END. 109 HP(N+l)=hO 109.5. C POSITIVE CHARACTERISTIC EQUATION. 110 VP( N+l )=C3-C2*HP(N«-1) 1 1 1 C COMPQTATICN OF INTERIOR POINTS. 112 00 5 5 1=2,M 113 VP(I)=0.5*I VR(I)+VS(I ) + C2*(HR( I )-HS( I ) )-FF *(VR(I)*AB S(VR(I)) + 114 1 V S ( I ) * A R S ( V S ( I ) ) ) ) 115 HP(I ) = C.5*(HR (I)+HS( I ) M V R ( I ) - V S ( I ) ) / C 2 - F F * ( V R ( I ) * A R S ( V R ( I ) ) - V S ( I ) 116 l*AFS(VS( I)) )/C2) 117 5 5 CONTINUE 118 C CONVERT V ( I ) = V P ( I ) , AND H(I)=HP(I) FOR ALL SECTIONS. 119 80 DO 90 1=1,NN 120 VI I >=VP( I ) 121 H(I) = HP( I ) 122 90 CONTINUE 123 VAIR = VPAIR 123.5 C TABULATION OF MAX. AND MIN. HEADS. 124 DO 95 1=1,6,5 125 IF(H( I ) .LT.HMINlI)) GO TO 123 126 121 IF(H(I).GT.HM AX(I)) GO TO 125 127 122 GO TO 95 128 123 HM IN(I ) = H(I ) 129 GO TO 95 130 125 HMAX(I)=H{I) 131 95 CONTINUE 132 IF( (H(3)+H(4)).LT.SUMIN( 1)) GOTO 13 5 133 131 IF((H(3)+H(4)).GT.SUMAX(1)) GOTO 137 134 132 GO TO 140 135 135 SUM INI 1)=H(3)+H(4) 136 136 GO TO 140 137 137CSUMAXI 1) = H( 3 H H ( 4 ) 138 140 COMTINUF ' 139 IF((H(8)+H(9)).LT.SUMINI2)).GO TO 147 140 142 IF( (H(8)+H(9) ) . GT .SUM AX ( 2 ) ) GO TO 150 141 144 GO TO 155 142 147 SUMIN(2)=H(8)+H(9) 143 149 GO TO 155 144 150 SUMAXI2)=H(8)+H(9) 145 155 CONTINUE 146 GO TO 40 146.5 C COMPUTATION OF MAX. UPSURGES AND DOWNSURGES. 147 107 DO 170 1=1,6,5 148 UPS MAX(I )=HMAX(I)-HSS(I) 149 ONSMAXII)=HSS(Il-HMINII) 150 UPS ANS(I)=U PSMAX(I)/HOABS 151 ON SANS ( 1 ) =DNSMAX ( I ) /HOABS 152 170 CONTINUE 153 HMAX1C=SUMAX(1)/2. 154 HMIN1Q=SUMIN(1)/2. 155 HMAX30=SUHAXl 2)12. 156 HMIN3U=SUMIN(2)/2. 157 HSS1Q=SUMSS( 1 ) /2. T58 HSS3Q=SU,MSS( 2 1/2. 159 UPMA10=H^AX1^-HSS1Q 160 DNMA1G=HSS10-HMINIO 161 UPKA30 = HMAX3G-HSS3Q ----- 162 DNMA3G=HSS3C-HMIN3Q 163 UPAM 10 = UPMA10/HOABS 117 164 DNA NL C = D M " A 1 C / H O A B S 165 UPAN3 0 = UPKA 3 0 / H O A B S 166 DN AN 30 = 0 Pv M.A 3 QI HO A 8 S 1 6 7 - W R I T E ( 6 , 1 8 0 ) PL C , T 'A AX , C PL AC , U PS ANS ( 1 ) , U P AN 1 0 , UP S AN S ( 6 ) , UP AN 30 , 168 1 0 N S A N S I 1 ) » D NA N 1 0 , 0 N S A N S ( 6 ) , ONAN3Q 169 180 FCRN ; AT (/ F4 . 1, 2X , E 5 . 1, 2 X , F5 . 1, 4 X , f b . 3 , 3X , F 6 . 3 , 4 X , F 6 . 3 , 3 X , F 6 . 3 / 170 122X , F 6 . 3 , ? . X , F 6 . 3 , 4 X , F 6 . 3 t 3X , F 6 . 3 ) 171 GO TO 6 172 1 1 0 S T O P 173 END END OF FILE tCOPY *SKIP *SINK* 118 2. ENTIRE HEAD LOSS ATTRIBUTABLE TO DISTRIBUTED FRICTION $LIST AI RCHA.M1 1 2 C WATERHAMMFP PROGRAM. PUMP AT UPSTRFAM END WITH AIR CHAMBER ADJACENT 3 C TO THE PUMP. Rt-SERVO IK AT DOWNSTREAM END. 4 C CHECK VALVE CLOSES IMMEDIATELY CN PUMP FAILURE. 5 C LINE FRICTION ONLY. NO ORIFICE LOSS. NO MINOR LOSSES. 6 DIM ENS ION V(20),VP(20),H(20),HP(20) ,VR(20),VS(2C) ,HR(20) ,HS(20) , 7 1 H M A X ( 1 0 ) , H M I M 1 0 J , HS S (10) .SUM AX ( 10 ) , SUM. IN ( 1 0 ) . SUM SSI 10) , 8 2UPSMAX( 10),DNSMAX(10) ,UPSANS( 10) ,0NSANS(10) q CAT A N/10/,VA/3216./,G/32.16/,FL/3216./,PM/1.2/,MM/1/, 10 1H0RF/0.0/,AP/3./, l l 2CK/1./, 12 3V0/3.5/ 13 WRI TE ( 6, 15) N,VA,G,FL,PN',r'iv,H0RF,AP,CK,VO 14 15 FORMAT!/• THE PARAMETERS ARE NOW...'/ 15 1* N= ',15,' VA= ',F8.2,' G= ',F6.2,' FL= ',F8.2/ 16 2' PM= »,F6.2,' MM= ',15,' HQRF = ',F6.3/ 17 31 AP= ' ^ 8 . 2 , ' CK= ,,F6.2,' V0= S F 8 . 2 ) 18 2 7 WRITE(6,30) • 19 30 FORMAT( * PLC TMAX CPLAC UPSANS(l) UPAN1Q UPS ANS(6) UPAN30' / 20 119X, 1 DNSANS(l) CN AN 1Q DN SANSt 6) DNAN30') 21 6 READ(5,1C) PLC,TMAX,CPLAC 22 10 FORMAT(3F8.3) 23 IF(CPLAC.LE.0 .0) GO TO 110 24 C COMPUTE DT 25 DT= FL/((VO +VA ) AFLOAT(N)) 26 C CHECK FOR CONVERGENCE. 27 DX=FL/FLOAT(N) 28 THETA= DT/DX 29 IFITHETA .LE.(1./VA) ) GO TO 20 30 17 GO TO 110 31 C COMPUTE COEFFICIENTS AND CONSTANTS FOR ALL PIPES. 32 20 API =3. 142 33 DP=S0RT(4.*AP/API) 34 C2 = G/VA 36 C H0ABS=H0+HF+34. 37 H0ABS=V0/(C2*PLC ) " ' 37.1 C HEAD LOSS FOR FLOW INTO CHAMBER. 37.2 HF = CK*HOAP. S 3 7.5 c H0= HEAD AT RESERVOIR. 38 H0=H0ABS-HF-34. 38.5 c F= FRICTION FACTOR. 39 F=(HF*2.*G*0P )/(FL*VO*VO) " " - —- 39. 5 c VOAIR= INITIAL AIR VOLUME IN TANK. 40 VCAIR=(CPLAC*VO*AP*FL)/(2.*VA) " ' 41 CO=VO*AP 42 FF=F*DT/(2.*DP) 44 C10=H0ABS*V0A IR**PM 45 c STEADY STATE CALCULATIONS. ' " 46 OHF=HF/FLOAT!N) 47 NN=N+1 48 DO 25 I=1,NN 49 V(I)=VO 50 T EM P = NN- I 51 H ( I )=HC+TEMP*OHF 52 25 CONTIlsMJE V 52.5 c INITIALIZATION OF MAX. AND MIN. HEADS. 119 f 53 DO 26 1=1,6,5 54 HMAX < I )=M 1 ) 55 HMINJI)=H(I) 56 HSSI I )=H(1 ) 57 26 CONTINUE 58 SU^SSI1)=H(3)+H{4) ? 59 SUN SSI 2)=H(8) +H( 9) 60 SUMAX (1 ) = H ( 3 H H ( 4 ) 61 SUM IN( 1)=H(3)+H(4) 62 SUf'AXI 2)=H( 8) +H( 91 63 SUMIM(2)-=H(8)-»H(9) 63. 5 C TIME INITIALIZATION. 64 T=0. 0 6 5 VAIR=VCAIR 65. 5 c PRINTOUT INTERVAL INITIALIZATION. 66 M.= 0 67 c COMPUTATION OF VR,VS,HR,HS FOR ALL SECTIONS. 68 c INTERIOR SECTIONS. 69 40 DO 50 I=2,N 70 VR(I)=V(I)-VA*THETA*(V( I )-V( I-1) ) 71 HR ( I )=H( I ) - V A * T H E T A * ( H ( I ) - H ( I - D ) 72 VS( I ) = V( I)~VA*TFETA*(V( I )-V( 1 + 1) ) " ' ~ ~ 73 HS(I)=H(I)-VA*THETA*IH(I)-H11+1)) 74 50 CONTINUE 75 c BOUNDARY SECT IGNS. 76 c RESERVOIR. 77 VR{N+l ) = V(N+l)-VA*THETA*(VIN+1)"V(N)) 78 HRIN + l ) = H(N+1 )-VA*THETA*(H(N+1)-H(N)) 79 C3= VR(N+ 1)+C 2 *HR(N+1)—FF *VR(N+l)* ABS(VR(N+l)) 80 c AIR CHAMBER. 81 54 VS(1)=V(1)-VA*THETA*IV(1)-V(2)) 82 HS( 1)=H( 1 )-VA*THETA*(H( 1 J-H12) ) 83 C1=VS(1 )-C2*hS(1)-FF*VS(1>*ABSIVS( 1>) ; _ 84 38 T= T + DT 85 P = M+1 86 IF(T.GE.TMAX) GO TO 107 87 c TIME INCREMENTED. BOUNDARY CONDITIONS. 88 c AIR CHAMBER. hOABS*VGAIR**PM=CONSTANT. 88. 5 c LOOP ( 89, 106) TO APPROX. AVE. VELOCITY FROM CHAMBER. . __ 89 VAVAPP=V(1) 90 GO TO 210 91 200 VAVAPP=VAV 92 210 Cl1=VAVAPP*AP 93 CAIR=VAIR+C11*DT 101 60 HP(l)=(C10/CAIR**PM)-H0RF-34. 10.1 . 5 c NEGATIVE CHARACTERISTIC EQUATION. 102 VP( 1)=C1 + C2*HP(1) 103 VAV=(V( 1 ) + VP( 1))/2. 104 VERR=VAVAPP-VAV 105 IF(ABS(VERR).LE.0.0001) GO TO 230 106 220 GO TO 200 107 230 VPAIR=VAIR+<AP*0T*VAV) 108 c RESERVOIR AT DOWNSTREAM END. 109 HP(N+l)=H0 109. 5 c POSITIVE CHARACTERISTIC EWUATION. 110 VP(N+J)=C5-C2*HP(N+1) 111 c COMPUTATION OF INTERIOR POINTS. 112 DO 55 1=2,N 113 VP( I ) = 0. 5*( VP. (I 1 + VSII ) +C2*(HR( I ) -HS ( I ) )-F F* ( VR ( I )* ABS (VR I I ) ) + 114 1VS( I )*ABS(VS( I)) )) 120 ( 115 HP( I 1 = 0.5* ( H R (I )+HSII)+(VR(I ) - V S ( I ) ) / C 2 - F F * <VR(I)*ABS(VR(I))-VS(I ) 116 1 *ABS(VS( I ) ) )/C2 ) 117 5 5 CONTINUE 118 C CONVFPT V ( I ) = V P ( I ) , AND H(I)=HP(I) FOR ALL SECTIONS. 119 80 DO 90 I-l.NM 120 V ( I ) - V P ( I ) ? 121 H ( I ) = HP ( I ) 122 90 CONTINUE 123 VAlR=VPAIR 123.5 C TABULATION OF MAX. AND MIN. HEADS. 124 DO 95 1 = 1 ,6 ,5 125 IE(H (I).LT.HMINII)) GO TO 123 126 121 I F(H( I).GT.HM A X( I)) GO TO 125 127 122 GO TO 95 12 8 123 HMINl I )=H(I ) 129 GO TO 95 130 125 UMAX(I)=H(I ) 131 95 CONTINUE 132 IF( <H<3>+H(4)).LT.SUM IN( 1) ) GO TO 135 133 131 IF((H(3)+H(4)).GT.SUMAX(1)) GOTO 137 134 132 GO TO 140 135 135 SUMIN(1) = H(3 ) *H<4 ) 136 136 GO TO 140 137 1370SUMAX(1)=H(3)+H(4) 138 140 CONTINUE 139 IF( (H( 8)+H(9)).LT.SUMIN(2)) GO TO 147 140 142 IF((H(8)+H(9)).GT.SUMAX(2)) GO TO 150 141 144 GO TO 155 142 147 SUMIN(2)=H(8)+H(9) 143 149 GO TO 155 144 150 SUKAX(2)=H(8)+H(9 ) 145 155 CONTINUE 146 GO TO 40 146. 5 C COMPUTATION OF MAX. UPSURGES AND DOWNSURGES. 147 107 DO 170 1=1,6,5 148 UPS MAX(I )=HMAX( I J-HSS(I ) 149 DNS MAX(I )=H S S I I)-HMIN(I) 150 UPSANStI )=UPSMAX( I ) /HOABS 151 DNS ANS(I) = DNSMAX( I)/HOABS 152 170 CONTINUE ~ " 153 HMAX1Q=SUMAX(1)/2. 154 HMIN10=SUMIN(l)/2. 155 HMAX3Q=SUMAX(2)/2. 15(3 HM IN 30= SUM IN(2)/2. 157 HSS10=SUMSS(1)/2. 158 HSS3Q=SUMSS(2)/2. 159 UPMA10=HMAX1Q-HSS1Q 160 DNMA10=HSS1 G-HMIMQ 161 UPN A 30 = H Mi A X 30 -H S S 3Q 162 DNMA3G= HSS 3Q-HMIN3Q 163 UPA N10= UPMA10/HO ABS 164 DNAN 1Q= DNMA IC /HOAB S 165 UPAM3 0=U PMA30/H0ABS 166 DNAN 30 = 0NMA30/HOABS 16 7 WRITE(6,180) PLC,TMAX,CFLAC , UP SANS(1) ,UPAN10,UPSANS(6) ,UPAN30 , 168 1 DNSANS(1) ,DNAN1Q,DNS ANS(6),DNAN3Q 16 9 180 F0PMATI/F4. 1,2X,F5. 1,2X,F5.1,4X,F6 .3,3X,F6.3 ,4X,F6.3,3X,F6.3/ 170 122X,F6.3,3X,r 6.3,4X,F6.3,3X,F 6.3) 171 GO TO 6 172 110 STOP 1 7 3 END OF FILE END y $CCPY *SKIP *SINK* 12)

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