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Air chamber design charts. Galatiuk, William Robert 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  i n 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 i n p a r t i a l f u l f i l l m e n t of the requirements  for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y available for reference study.  and  I further agree that permission for extensive copying of this  thesis f o r scholarly purposes may  be granted by the Head of  Department or by his representatives.  my  I t i s understood that copying  or publication of this thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission.  Department of  C i v i l Engineering  The University of B r i t i s h Columbia Vancouver 8, Canada  Date  September 10,  1973  ABSTRACT  The a i r chamber has certain advantages over both the open-top surge tank and the valve-type surge suppressor f o r c o n t r o l l i n g pressure surges i n pump-discharge l i n e s . The main purpose of this study was to produce charts which can be used f o r designing or checking the s i z e of an a i r chamber required for a p a r t i c u l a r pumping i n s t a l l a t i o n . The c h a r a c t e r i s t i c s method was used to convert the two p a r t i a l d i f f e r e n t i a l equations of momentum and continuity d i f f e r e n t i a l equations.  into four t o t a l  The solution of the equations ( f i n i t e - d i f f e r e n c e  form) was carried through by d i g i t a l computer to provide the data required f o r the preparation of the charts. Results obtained on the d i g i t a l computer by the method of charact e r 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  I.  II.  PAGE  ABSTRACT  ((ii)  NOTATION  (ix)  INTRODUCTION  1  ASSUMPTIONS AND THEORY  3  1.1 1.2 1.3 1.4  3 5 5 7  Assumptions General Theory Parameters Relationship between a* and p*  METHOD OF CHARACTERISTICS  8  2.1 2.2 2.3 2.4  8 8 9  General Basic Equations f o r Unsteady Flow Through Pipes General Characteristics Method Convergence and S t a b i l i t y of the Method of F i n i t e Differences  I I I . BOUNDARY CONDITIONS  15 16  3.1  The A i r Chamber  16  3.2  Reservoir of Constant Water Level at the Downstream End  19  IV.  THE PROGRAM  21  V.  4.1 General 4.2 Check on the Program 4.3 Description of the Program 4.4 Approximation of Velocity of Flow out of the Chamber THE CHARTS  21 21 23 24 26  5.1 5.2 5.3 5.4 5.5  26 28 28 30  5.6 VI.  Groups of Charts No Head Loss, F r i c t i o n l e s s Flow Entire Head Loss Concentrated at the O r i f i c e Entire Head Loss Attributable to Distributed F r i c t i o n Head Loss Equally Divided Between Uniformly Distributed Wall F r i c t i o n and O r i f i c e Loss  32  Use of the Charts  32  DISCUSSION  35  6.1 6.2 6.3 6.4  35 36 37 40  Volume of A i r i n the Chamber Volume of the A i r Chamber O r i f i c e Design Water-Column Separation i n Pump Discharge Lines (iv)  CHAPTER  PAGE  V I I . CONCLUSIONS  42  BIBLIOGRAPHY  44  THE CHARTS  45  Group I - No Head L o s s , F r i c t i o n l e s s Flow Group I I - E n t i r e Head Loss Concentrated a t the O r i f i c e Group I I I - E n t i r e Head Loss A t t r i b u t a b l e t o D i s t r i b u t e d Friction Group IV - Head Loss E q u a l l y D i v i d e d Between U n i f o r m l y D i s t r i b u t e d W a l l F r i c t i o n and O r i f i c e Loss  46 48 71 82  APPENDICES Appendix A: Appendix B: Appendix C:  Comparison o f C h a r t s and N u m e r i c a l Examples 93 G r a p h i c a l Checks on Program 106 Program f o r the E n t i r e Head Loss Concentrated a t the O r i f i c e 113 Program f o r the E n 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  5.1  5.2  PAGE  Comparison of Results Obtained f o r the Powers 1.0, 1.2 and 1.4  29  Upsurges and Downsurges f o r 1:1 and 2.5:1 O r i f i c e s  31  (vi)  LIST OF FIGURES FIGURE  PAGE  1.1  Pipe Line with A i r Chamber  2.1  Method of Specified Time Intervals  14  2.2  Characteristics at the Boundaries  14  3.1  A i r Chamber  20  3.2  Reservoir at Downstream End  20  4.1  Program Flow Chart  22  5.1  Pipe Line with A i r Chamber  34  6.1  A i r Chamber Control Levels  38  6.2  Differential Orifice  38  B-la  Schematic of Pipe Line  B-lb  Transient-State Conditions f o r Total Head Loss Concentrated  at the O r i f i c e - Pipe Pressures  B-2a  Schematic of Pipe Line Showing Line F r i c t i o n Loss  B-2b  Transient-State Conditions f o r Total Head Loss Attributable to Distributed F r i c t i o n - Pipe Pressures  (vii)  4  107  109 1  1  0  112  ACKNOWLEDGEMENT  The a u t h o r w i s h e s visor,  D r . E . Ruus,  criticism the  to  for his  and s u g g e s t i o n s ,  express valuable and t o  his  gratitude  guidance,  to h i s  super-  constructive  Dr. W.F. Caselton  for  reviewing  work. The s t u d y  was  s u p p o r t e d by  Council.  (viii)  a grant  from  the N a t i o n a l  Research  NOTATION  The following symbols are used i n this thesis: , 2  A  =  cross-sectional  a  =  propagation v e l o c i t y of waterhammer wave, i n f t / s e c .  =  i n i t i a l volume of a i r i n the a i r chamber at absolute pressure  C  Q  area of pipe, i n f t  head H *, i n f t  3  Q  Cf  -  o r i f i c e loss c o e f f i c i e n t  D  =  inside diameter of pipe, i n f t  f  =  Darcy-Weisbach f r i c t i o n factor  QT  2  g  =  gravity acceleration, i n f t / s e c  H  =  transient-state piezometric pressure head above datum at the beginning of a time i n t e r v a l , i n f t  H  Q  Hp  =  i n i t i a l steady-state piezometric pressure head above datum, i n f t  =  transient-state piezometric pressure head above datum at the end of a time i n t e r v a l , i n f t  H*  =  transient-state  absolute piezometric pressure head above datum,  in f t H*  =  Q  i n i t i a l steady-state absolute piezometric pressure head above datum, i n f t  H £  =  Q r  H  orfo  K  =  o r i f i c e t h r o t t l i n g loss corresponding to discharge q, i n f t o  =  r  i  f  i  c  e  t h r o t t l i n g loss corresponding  to discharge q , i n f t Q  c o e f f i c i e n t r e l a t i n g t o t a l pipe l i n e head l o s s due to f r i c t i o n to piezometric pressure head above datum  L  =  length of pipe l i n e , i n f t m  m  =  power used i n pressure - volume r e l a t i o n s h i p , H* v ^ a  for  an a i r chamber  (ix)  r  = constant,  steady state discharge i n the pipe l i n e , i n f t /sec. transient state o r i f i c e discharge, i n f t /sec. time, i n seconds transient-state v e l o c i t y i n pipe at the beginning of a time interval, i n ft/sec. transient-state v e l o c i t y i n pipe at the end of a time i n t e r v a l , in ft/sec. i n i t i a l steady state v e l o c i t y i n pipe, i n f t / s e c . transient-state volume of a i r i n a i r chamber at the beginning  3 of a time i n t e r v a l , i n f t  3 i n i t i a l steady-state volume of a i r i n a i r chamber, i n f t transient-state volume of a i r i n a i r chamber at the end of a time i n t e r v a l , i n f t  J  distance along pipe l i n e , f r o m pump, i n f t pipe l i n e c h a r a c t e r i s t i c pipe l i n e c h a r a c t e r i s t i c i n terms of absolute pressure head parameter pertaining to a pump-discharge  l i n e having an a i r  chamber, i n terms of absolute pressure head angle the pipe makes with the horizontal At grid mesh r a t i o , time increment, i n seconds incremental distance along the pipe l i n e , i n f t  (x)  INTRODUCTION Sudden stopping or s t a r t i n g of large c e n t r i f u g a l pumps i n s t a l l e d for i r r i g a t i o n , domestic water supply systems, pumped storage hydroe l e c t r i c plants and other purposes cause transient pressures i n the discharge l i n e s .  Starting control mechanisms can be designed to  delay the s t a r t i n g up time s u f f i c i e n t l y to prevent excessive over pressures.  But sudden stopping i n the event of power f a i l u r e could  r e s u l t i n objectionable waterhammer pressures i n the pipe l i n e . In small i n s t a l l a t i o n s , no s p e c i a l precautions are taken to avoid high  waterhammer e f f e c t s . Standard pipes and f i t t i n g s of small diameter  have a w a l l thickness s u f f i c i e n t to withstand appreciable transient pressures.  In large pumping i n s t a l l a t i o n s various pressure-control  devices may be used to reduce  waterhammer pressures.  Some of these  devices include: (1)  surge tanks,  (2)  a i r chambers,  (3)  surge suppressor valves, and  (4)  slow closing check valves.  For c o n t r o l l i n g pressure surges i n pump-discharge l i n e s , the a i r chamber has certain advantages over both the open-top surge tank and the valve-type surge suppressor.  For high head i n s t a l l a t i o n s where  the open surge tank i s impractical, a properly designed a i r chamber provides good surge control.  The a i r chamber can be near the pump  whereas the surge tank can not always be so located. can be designed  The a i r chamber  to reduce the downsurges i n a pump-discharge l i n e , thus  1  2 preventing collapse of the l i n e and water-column separation; o r d i n a r i l y , surge-suppressor valves are not suitable f o r this important function. The main disadvantage of a i r chambers i s that the compressed a i r i s continuously being l o s t through dissolving i n the water and p o s s i b l e leakage.  Consequently  the a i r must be replenished p e r i o d i c a l l y .  After i t has been decided that certain types of pressurecontrol 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. a i r chamber i s determined primarily by i t s  The cost of an  size and inside pressure.  In this thesis, charts are presented which provide for the rapid determination of a i r chamber sizes required to control waterhammer pressures i n pump-discharge l i n e s where the transient pressures are caused by rapid pump shut down or by power f a i l u r e .  The charts were  prepared using the method of c h a r a c t e r i s t i c s to convert the two p a r t i a l d i f f e r e n t i a l equations of momentum and continuity into four t o t a l d i f f e r e n t i a l 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 f a i l u r e .  This eliminates the need to consider pump  c h a r a c t e r i s t i c s but introduces an abrupt pressure wave which must be accounted (2)  f o r throughout  the computations.  The a i r chamber i s situated near the pump as shown i n  Fig. 1.1. The steady-state water surface i n the chamber has an elevation equal to that of the center l i n e of the pipe (see F i g . 3.1). The transientstate head difference between the chamber water surface and the pipe center l i n e i s small and therefore neglected.  The head loss through the o r i f i c e ,  i f applicable, i s taken into account i n determining the absolute head, H*, i n the tank. (3)  The pressure-volume relationship for the a i r i n the chamber  i s expressed as: 12 H* v . ' air  =  a constant,  The power 1.2 i s an average of the powers 1.0 and 1.4 f o r the isothermal and adiabatic expansions (4)  respectively.  The head loss, made up of surface f r i c t i o n and loss at the  o r i f i c e , varies with the square of the v e l o c i t y .  Two types of o r i f i c e s ,  one simple and one d i f f e r e n t i a l , were considered i n the study.  3  The r a t i o  Sections  FIG.  1.1  Sec. 1-1 A r e a of P i p e  5 of the t o t a l head loss f o r the same flow into and from the a i r chamber i s 2.5:1 for the d i f f e r e n t i a l o r i f i c e , and 1:1 for the simple o r 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 i n the pipeline i s i n  the forward d i r e c t i o n , toward the reservoir. simultaneously with pump f a i l u r e . across the a i r chamber outlet.  The check valve closes  This creates a head d i f f e r e n t i a l  The compressed a i r causes the water i n  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 u n t i l the head i n  the chamber becomes less than the head i n the pipeline at the chamber outlet.  At this instant, the water i n the discharge l i n e w i l l reverse  i t s d i r e c t i o n and flow into the a i r chamber.  During this reverse flow  condition, the retardation of the flow into the a i r chamber causes the pressure i n the discharge l i n e to increase to exceed normal operating head and w i l l produce the maximum head for the transient.  Resurges i n  the pipeline w i l l occur with diminishing i n t e n s i t y .  1.3  PARAMETERS The pressure surges i n a pipeline equipped with an a i r 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 e f f i c i e n t use of an  6  a i r chamber on a pump-discharge l i n e , Evans and Crawford introduced t h i r d v a r i a b l e , K, to account f o r f r i c t i o n a l losses.  a  The v a r i a b l e  K i s defined so that KH * i s the t o t a l head loss f o r a reverse flow Q  of Q .  Q  0  i s the i n i t i a l rate of flow i n the p i p e l i n e , i n cubic  Q  feet per second  (ft-Vsec).  The p i p e l i n e c h a r a c t e r i s t i c , p , i s defined as  (1.1)  P  i n which  a  i s the propagation v e l o c i t y of waterhammer waves i n the  p i p e l i n e , i n feet per second ( f t / s e c ) ; V in ft/sec; H and  g  Q  Q  i s the steady-state v e l o c i t y  i s the steady-state pressure head, i n feet of water ( f t ) ;  i s gravity acceleration i n feet per second per second ( f t / s e c . ) -  The c h a r a c t e r i s t i c p i s dimensionless and i s a function of the r a t i o of the steady-state k i n e t i c energy to the t o t a l p o t e n t i a l energy i n a unit length of conduit.  In a i r chambers, the volume of the a i r i s a  function of the absolute pressure to which i t i s subjected.  In terms  of absolute pressure, the p i p e l i n e c h a r a c t e r i s t i c p becomes  p*  -  —  2gH *  (1.2)  Q  where H * Q  i s the normal absolute pressure head i n the p i p e l i n e at the  entrance to the a i r chamber. The parameter, a*, that i s c h a r a c t e r i s t i c to a pump-discharge l i n e having an a i r chamber i s defined^ as  7  2gC H * 0  0  ALV  i n which C  Q  Q  i s the i n i t i a l volume of a i r i n the a i r chamber at absolute  pressure head, H *,  i n cubic feet  Q  (ft ); 3  A  i s the cross-sectional  L  i s the length of the  2 area of the pipe i n square feet ( f t ); and pipe i n feet.  The parameter a*  expresses the r a t i o of the steady-  state p o t e n t i a l energy of the a i r i n the a i r chamber to the steadystate k i n e t i c energy of the water i n the discharge l i n e . 1.4  RELATIONSHIP BETWEEN o* AND From Eqs. (1.2) and  p*  (1.3)  a*p*  =  — ALV  (1.4) C  or C  From Eq.  q  . *p*Q L/a . a  (1.5)  Q  (1.2) o — — gH * a V  2p*  =  (1.6)  Q  and the constant f o r a pipeline having an a i r chamber w i l l be defined as  or  ALV  (p*0*) C  o=  a  Q  (1.8)  CHAPTER I I  METHOD OF CHARACTERISTICS 2.1  GENERAL g  The c h a r a c t e r i s t i c s method  converts the two p a r t i a l d i f f e r e n t i a l  equations of momentum and continuity into four t o t a l d i f f e r e n t i a l equations.  Non-linear f r i c t i o n i s retained, as well as the e f f e c t  of the pipes being non-horizontal.  The equations are expressed i n  f i n i t e - d i f f e r e n c e 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 i s proper i n c l u s i o n of f r i c t i o n  -  i t affords ease i n handling the boundary conditions and ease i n programming complex piping systems  -  there i s no need f o r large storage capacity i n the computer  - detailed r e s u l t s are completely tabulated. I t i s by f a r the most general and powerful method f o r handling waterhammer.  2.2  BASIC EQUATIONS FOR UNSTEADY FLOW THROUGH PIPES The v e l o c i t y and pressure of moving f l u i d s i n pipes are governed  by the continuity and momentum equations. The momentum equation  f o r flow through a pipe which i s i n c l i n e d  or h o r i z o n t a l , tapered or s t r a i g h t , s l i g h t l y or highly deformable, i s given by gH e  x  + V  t  + W  8  x  +  M Z L = 0 ,  2D  (2.1)  9  i n which g i s gravity acceleration, Darcy-Weisbach f r i c t i o n factor, the datum l i n e ,  V i s fluid velocity,  f i s the  H i s the t o t a l pressure head above  D i s the inside diameter of the pipe,  i s the f r i c t i o n a l force of the f l u i d .  and  fV|vl 2D  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 d i r e c t i o n of v e l o c i t y . The subscripts  x and t  respect to distance and time. II  _  H.. =  indicate p a r t i a l d i f f e r e n t i a t i o n with For example, 3H  , i n which H i s the t o t a l pressure  head i n feet of water. Changes i n the density of water may be neglected ducing s i g n i f i c a n t error.  without  intro-  Considering the density as constant, the  continuity equation may be stated as  (2.2) i n which 0 i s the angle the center l i n e of the pipe makes with the horizontal axis (measured p o s i t i v e downwards), and  a  i s the v e l o c i t y  of the waterhammer wave.  2.3  GENERAL CHARACTERISTICS METHOD In this section, a general solution f o r the continuity and mom-  entum equations i s presented.  For the complete treatment, see Ref. 9.  A l l of the terms i n the equations are retained.  The method of s p e c i f i e d  10  time i n t e r v a l s which involves l i n e a r i n t e r p o l a t i o n i s used. The momentum and continuity equations may be written as  4 = gn + w + v + SjJvL - 0  (2.3)  L. = H +  (2.4)  x  x  t  and V +VH + V s i n 9 = 0 x  g  Multiplying Eq. (2.4) by X and adding i t to Eq. (2.3), one obtains L  ±  ,2 X [ H ( V + f ) + H ] + [ V ( V + |_ X) + V ] + X Vsin6  + XL = ±  x  t  x  t  (2.5) Let  £ dt  -V+*  Therefore, X and  (2.6)  =V+5lA .  X  R  (2.7)  a  = v ±a.  —  dt  (2.8)  Through substitution of Equations (2.6), (2.7), and (2.8), Eq. (2.5) takes the form X  dH + _,_ iii. dv ^ , „VsinO „,_ + x dt dt  Q +  fvlvl . 2D  0  (2  .9)  I t follows froms Eqs. (2.8) and (2.9) that  a  dt  VsinS  dt  +  fyjvj 2D  (2.10)  0,  > £  - v •  C+ (2.11)  11  a  dt  dt  a  (2.12)  - 0,  2D  C-  and dx dt Because  V - a  (2.13)  V = V(x,t), the c h a r a c t e r i s t i c l i n e s C+ and C-, given by Eqs.  (2.11) and (2.13), plot as curves on the x-t plane (see F i g . 2*1). Eqs. (2.10) to (2.13) can be written i n the following f i n i t e difference forms: (V  P  - V ) R  +  & (H - H ) a p p/  +  V sine ( t - t ) + |- V IV 1 ' 2D R R  & a  1  R  P  (t  ( X  (V  p  - V ) - f s  P " R X  )  =  (Hp - H ) - | g  (V  R  +  a  )  ( t  p  P "  (2.14)  - t ) = 0 R  V  V sin6 ( t - t ) + f s  1  R  p  g  (2.15)  D  V |V | S  S  - t )- 0  (2.16)  V  (2.17)  s  <P " X  V s" = (v  a  )  ( t  P "  Two techniques are commonly used f o r obtaining a numerical solution for the f i n i t e - d i f f e r e n c e equations (2.14) to (2.17). (1)  use of a g r i d of c h a r a c t e r i s t i c s ,  (2)  use of s p e c i f i e d time i n t e r v a l s .  These are:  In single pipe problems as covered by t h i s study, these techniques are identical^.  The parameters  x  p  and  t  are assigned d e f i n i t e values  12  throughout the computation leaving only V determined.  and  p  as unknowns to be  In this study the technique of s p e c i f i e d time intervals  w i l l be used.  Since the conditions at points A, B, and C ( F i g . 2.1)  are known, the conditions at R and S may be evaluated by l i n e a r i n t e r polation. Thus  x x  But  - x R - x  C  V  _  V  A  C  A  = x ,  X p  - V  C  - V  C  and  x  R A  - x  = Ax.  Therefore, the above equation takes the form  v -  V  c  *P ~ R X  Since V  V  =  r  U  -  V,  R A  x  A  •  ( 2  '  1 8 )  i s much smaller than the waterhammer wave v e l o c i t y  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 a At  -  V  =  Ax C '  (2.19)  A  The g r i d mesh r a t i o , 6', i s defined as At 6  Therefore,  ' "  a 6' (V - V ) c  and  V  R  =  V  c  ~  a 9  A  '  (V  c  to  '  =  V  " V  c  - V , R  *  (2.20)  13  Similarly,  \  =  ae' ( H -  (2.21)  ae- ( v -  (2.22)  ae« ( H -  (2.23)  c  v  s  =  H  s  =  v  c "  c  c  Solve Eqs. (2.14) and (2.16) simultaneously to obtain:  V  P °=  5  [ R s i V  + v  +  R-V -I  ( H  A ts i n e (  \ - V - ft f  (V |V | + V |V |)] R  Hp •- 0.5  [H  R  + H  g  R  S  (2.24)  S  ( V - V ) -At sine ( V + V > - §  + f  R  s  - sl s'>> v  <  V  R  I  V  R  I  fj£  g  R  v  ( 2  -  2 5 )  At the boundary points ( F i g . 2.2), either Eq. (2.14) or Eq. (2.16) or both are used together with the boundary conditions to solve f o r V and H.  Eqs. (2.14) and (2.16) are termed the negative c h a r a c t e r i s t i c  equation and the p o s i t i v e c h a r a c t e r i s t i c equation respectively and may now be written i n the following forms: The negative c h a r a c t e r i s t i c equation i s V  = C P  where  C  1  = V C  and  FF -  2  S  1  + C„ 2  H ,  (2.26)  P  - C„ H + C V s i n 6At - FF V_I V_I 2 S 2 S s  = £ , a .  s  (2.27) (2.28) (2.29)  14  t  Ax  i  P  t +At Q  /  \ c \ \  /  t R A  S C  B  METHOD OF SPECIFIED TIME INTERVALS FIG.  2.1  Ax  A:  CHARACTERISTICS AT THE BOUNDARIES FIG.  2.2  15  The p o s i t i v e c h a r a c t e r i s t i c equation i s V  P  =  S " °2 V  (2.30)  where C C  2  3  = V + C H - C V At sine - FF V_1 V_I R 2 R 2 R R R 1  and FF represent pipe constants.  The values of  1  .  and  (2.31) are  constant during each time step.  2.4  CONVERGENCE AND STABILITY OF THE METHOD OF FINITE DIFFERENCES To be assured of s t a b i l i t y and/or convergence of the solution^, i t  i s necessary that At (V + a) < Ax.  Since V i s small r e l a t i v e to a,  this may be stated as follows:  T- <Ax  -  1  •  a  This indicates that i t i s important to select the grid mesh r a t i o so that the c h a r a c t e r i s t i c s through P, C  +  and C~ w i l l not f a l l outside the  3 l i n e segment AB ( F i g . 2.1).  The most accurate solutions are obtained  when Ax  "  aAt .  CHAPTER I I I  BOUNDARY CONDITIONS  3.1 THE AIR CHAMBER  ( F i g . 3.1)  Because of the assumption that the check valve closes simultaneously with the pump f a i l u r e , a l l the flow i n the discharge pipe i s either from or into the chamber.  This assumption eliminates the  pump c h a r a c t e r i s t i c s from the waterhammer  computations.  The pressure and volume of a i r i n the chamber follow the gas law  8  H*  v  m a i r  = constant,  (3.1)  where H* and v . are the absolute pressure head and volume of a i r i n air the chamber  and m  i s the power 1.0 f o r isothermal expansion and  1.4 f o r adiabatic expansion.  The o r i f i c e i n the chamber may be  simple or of the d i f f e r e n t i a l type.  The d i f f e r e n t i a l type of o r i f i c e  t h r o t t l e s the reverse flow of water from the discharge pipe into the chamber while there i s very l i t t l e t h r o t t l i n g of the flow out of the chamber.  I f there i s no o r i f i c e i n the chamber, the t h r o t t l i n g loss  i s taken equal to zero. Flow out of the chamber i s considered p o s i t i v e . For the transient condition, Eq. (3.1) may be written: [H  I P  + 34 + H  1  orf J  v = C , Pair 10 m  (3.2)  i n which H_ i s the transient pressure head ( i n f t ) i n the pipe at the  16  17  entrance to the chamber,  H  _ i s the o r i f i c e resistance ( i n f t ) orf corresponding to a discharge of q ( f t / s e c . ) and v i s the transient Pair 3 volume of a i r i n the chamber ( f t ). C i s a constant given by: 3  "lO"".* i n which H * o  and  v  oair  oair  V  '  °-  3 )  denote the i n i t i a l steady-state absolute  pressure head and volume of a i r i n 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 where  V  (3.4)  - v . air  Pair  i s the v e l o c i t y of flow i n the pipe ( i n f t / s e c ) ,  A i s the  o  cross-sectional area of the pipe ( i n ft ), At i s the length of the time i n t e r v a l under consideration ( i n sees), the chamber  (in f t )  v_ . i s the volume of a i r i n Pair  at the end of the time i n t e r v a l and v  3  a i r  i s the  volume of a i r i n the chamber at the beginning of the time i n t e r v a l . Rearranging  the terms, one gets: (3.5)  v_ . = v . + C At, Pair air 11 i n which C  ll  *  V A  '  The negative c h a r a c t e r i s t i c equation f o r the pipe i s : V (l)  " C  p  where (1)  x  + C  (3.6)  Hpd),  2  designates section 1 on the pipe, i . e . at the a i r chamber.  The o r i f i c e f r i c t i o n loss i s given by: H  orf  =  C  orf  ^E|£  2  q  o  q  |  1  q  |  1  (3.7)  i n which C f OT  the o r i f i c e  i s the o r i f i c e c o e f f i c i e n t and H ^ Q r  i s the head loss i n  o  ( i n f t ) corresponding to a discharge of q . Q  The absolute  value of q ensures the correct sign on the head loss f o r changes i n d i r e c t i o n of flow through the o r i f i c e . 1.0 f o r flow i n either d i r e c t i o n .  For a simple o r i f i c e , C  For a d i f f e r e n t i a l  o r j C  =  orifice, G £ = Q r  1.0 when water flows out of the chamber, i . e . when V i s p o s i t i v e , and C £ =  when water flows into the chamber, i . e . when V i s nega-  o r  tive.  The value of  depends on the amount of t h r o t t l i n g provided by  the o r i f i c e . Substituting f o r q i n Eq. (3.7), one obtains:  orfo ~T H  H  orf " orf C  VA  '  VA  '  9n or H *=C C orf orf r  ff  C I C I 11' l l  (3.8)  1  i n which r f  =  H  C  orfo 2  Substitution of the values of v  p a  ^  r  and ^  Q t  ^ into Eq. (3.2)  gives: • 34 + C  h  or  C H  P = (v  a l r  C  o r f  C J C  f  u  | (v  + C  a i r  u  At)* - C  l o  1 Q  + C  U  At)*  "  3  4  " °rf f l l l C  C  C  C  lll  Letting C . » v . + C„_ At, one obtains: air air 11  H  p  .  3  4  _  C  o  r  f  C  f  c  n  | C  u  | .  (3.9)  For each time increment, V  p  3.2  from Eq. (3.6) and V  Hp can be determined from Eq. (3.9),  from Eq. (3.5).  p a i r  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 p o s i t i v e c h a r a c t e r i s t i c equation f o r section 11 i s given by: V  p  (11) =  C  -  3  C H (ll). 2  (3.10)  p  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 f o r this study designates to the computer a l l of  the operations which must be performed to compute the maximum upsurges and downsurges f o r the transient phenomena.  The flow chart f o r the  program i s given i n F i g . 4.1 and the entire programs f o r the entire head loss concentrated at the o r i f i c e and entire head loss attributable to d i s t r i b u t e d f r i c t i o n are reproduced i n Appendix C.  4.2  CHECK ON THE PROGRAM P r i o r to proceeding with the actual study, the writer checked  the v a l i d i t y of the program with several graphical analyses.  These  checks, presented i n Appendix B, indicate that the program gives results which compare well with graphical solutions made by others. The graphical check f o r the t o t a l head loss concentrated at the o r i f i c e ^ shows that the program f o r this case gives v a l i d r e s u l t s . See F i g . B-lb. The graphical check using several o r i f i c e s 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 f o r d i s t r i b u t e d f r i c t i o n i s also v a l i d .  21  See F i g . B-2b .  Q  START  ^  * 22  READ DATA  COMPUTE At and Ax  CHECK FOR CONVERGENCE  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 , Vg, H , H R  2  FOR ALL SECTIONS  M - 0  I  T «• T + DT  M - M + 1  COMPUTE V P  t  and HPj  AT INTERIOR POINTS  COMPUTE VP and HP AT BOUNDARY POINTS  Vi - VPi Hi = HPi TRUE ^ — ^ " ^ ^ FALSE M = MM  PROGRAM FLOW CHART FIG. 4.1  s  23  4.3  DESCRIPTION OF THE PROGRAM The main functions of the program are as follows: i ) S p e c i f i c a t i o n of the storage locations f o r 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, v i ) Computation of steady-state values, v i i ) Computation of transient-state conditions, v i i i ) Check f o r maximum upsurges and downsurges. ix) Printout. The variables which the program reads i n are: PLC  the pipe l i n e constant, 2p*  TMAX —  the length of time f o r which the transients are to be calculated, i n seconds  CPLAC - the constant f o r a pipe l i n e  with an a i r chamber adjacent to  the pump, 2 p* a*. The remaining parameters are set i n the Data statement.  For any group,  the only parameter which changes i n the Data statement i s the t o t a l head loss c o e f f i c i e n t , CK. The programs are r e l a t i v e l y e f f i c i e n t with a t y p i c a l c a l c u l a t i o n taking approximately 12 to 13 seconds of computer use time. The programs f o r the four basic groups of charts, as l i s t e d i n Section 5.1, vary only s l i g h t l y from each other.  24  Group I - No head loss For f r i c t i o n l e s s  flow the program Data statement sets  the o r i f i c e loss equal to zero.  CK  and  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 o r 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 o r i f i c e inflow  c o e f f i c i e n t , CORFIN, to 1.0 or 2.5 depending on whether the o r i f i c e i s simple or d i f f e r e n t i a l . Group III - Entire head loss attributable to distributed wall friction The Data statement sets the o r i f i c e l o s s , HORF, equal to zero. The program calculates the f r i c t i o n head l o s s , 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 o r i f i c e loss The Data statement sets the o r i f i c e inflow (CORFIN) and the t o t a l head loss (CK) c o e f f i c i e n t s .  The program computes the f r i c t i o n  factor, F, the t o t a l steady-state f r i c t i o n loss and the total o r i f i c e loss f o r a flow of Q  Q  into the chamber.  The steady-state f r i c t i o n factor i s 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 v e l o c i t y out of the chamber, VAVAPP, after  the time i n t e r v a l was  incremented, was  pipe at Section (1) (Fig. 3.1)  set equal to the v e l o c i t y i n the  for the previous time i n t e r v a l .  The  computation was then followed through to the point where the actual average v e l o c i t y of flow from the chamber was  i , e  *  VAV  =  calculated.  V ( l ) + VP(1) — * 2 v  If the difference between the i n i t i a l assumed average v e l o c i t y , VAVAPP, and the calculated average v e l o c i t y , 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 f o r the v e l o c i t y of flow out of the chamber. error c r i t e r i o n was  This i t e r a t i o n continued u n t i l the  met.  The writer found that i f VAVAPP was set equal to V ( l ) from the previous time i n t e r v a l , the program would not converge to a solution, but i n f a c t , 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 i n this  study.  These four combinations include: (1)  No head l o s s ,  K = 0.0, (no wall f r i c t i o n , no o r i f i c e loss)  There i s only one chart i n this group. (2)  Entire head loss concentrated  at the o r i f i c e , (no wall f r i c t i o n )  There are ten charts i n this group with 1.0 i n increments of 0.1.  K  varying from 0.1 to  Two o r i f i c e s , one d i f f e r e n t i a l and one simple,  were investigated i n this group.  The d i f f e r e n t i a l o r i f i c e had an inflow  to outflow head loss r a t i o of 2.5:1.  That i s , f o r the simple o r i f i c e ,  the o r i f i c e resistance i s the same f o r inflow or outflow whereas f o r the d i f f e r e n t i a l 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 p r a c t i c a l but are included f o r the sake of completeness. to flow from the chamber for  Because of the great resistance  K = 0.7 to 1.0, large a i r chambers are  needed to control the downsurges whereas the upsurges are not greatly reduced. (3)  Entire 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 ,  (no o r i f i c e loss) K varies from 0.1 to 1.0 i n increments of 0.1.  26  (4)  Head loss equally divided between uniformly distributed  wall f r i c t i o n and o r i f i c e loss K varies from 0.1 to 1.0 i n increments of 0.1. The o r i f i c e considered was a d i f f e r e n t i a l o r i f i c e with inflow to outflow loss r a t i o 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 i n terms of these variables.  Maximum  upsurges and downsurges at the pump, the midlength and the threequarter point of the discharge l i n e are plotted as percentages of H * f o r various values of these parameters. Q  The normal range"*" of p*  i s from 0.25 to 2.0 and that of a* i s from 2 to 30. This range i s covered i n the charts. To use the i n d i v i d u a l charts, one must f i r s t determine the parameters K, 2p* and 2p* a* f o r the p a r t i c u l a r 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.  S i m i l a r l y , 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 , D i f f e r e n t i a l  o r i f i c e 2.5:1  Required: Maximum upsurge and downsurge at midpoint. Solution: Maximum upsurge = 0.771 H * Q  Maximum downsurge = 0.358 H * Q  28  5.2  NO HEAD LOSS, FRICTIONLESS FLOW The single chart i n this category compares well with the  chart for f r i c t i o n l e s s flow published by Evans and Crawford (Appendix A, F i g . A - l ) .  Since f r i c t i o n l e s s flow would not occur i n  r e a l i t y , this chart would be used f o r purposes of analysis but not f o r design problems.  5.3  ENTIRE HEAD LOSS CONCENTRATED AT THE ORIFICE D i f f e r e n t i a l o r i f i c e - inflow to outflow head loss r a t i o  2.5:1  The graphs f o r K = 0.3, 0.5 and 0.7 compare well with the corresponding graphs published by Evans and Crawford as shown i n Appendix A, Figures A-2, A-3, and A-4.  The curves are generally well  defined except f o r the lower values of 2p* and 2p* a* f o r the upsurge region.  This i s i n the range of very low v e l o c i t i e s .  curves were eliminated f o r K = 0 . 8  to  The 2p* = 0.5  K => 1.0 i n c l u s i v e because the  program would not converge to a solution. Two additional charts f o r K = 0.5 were included i n this group. These were f o r powers of 1.0 and 1.4, the powers being the values of m i n the equation H* v  m a i r  = constant.  The Intent was to check the  possible v a r i a t i o n of r e s u l t s caused by using the power  m  as 1.0,  1.2 and 1.4. A comparison of the charts and a p a r t i a l l i s t i n g of the r e s u l t s as shown i n Table 5.1 indicate that the power 1.2 gives an approximate average f o r the upsurges and downsurges.  The charts also indicate that  one must accurately determine whether the system i s isothermal or adiabatic when using the powers 1.0 and 1.4 because the resultant  29  TABLE 5.1 Comparison of Results Obtained f o r the Powers 1.0,1.2 and 1.4  2p* 1  4  2p*a*  Point  m = 1.0  m = 1.4  2  P M 3/4  .705 .435 .235  4  P M 3/4  .413 .254 .132  .452 .355 .264  .475 .313 .151  .499 .386 .283  .542 .331 .178  ..532 .414 .302  10  P M 3/4  .173 .120 .058  .324 .250 .200  .208 .134 .065  .352 .270 .210  .240 .157 .073  .378 .287 .219  30  P M 3/4  .061 .050 .022  .220 .185 .165  .073 .056 .024  .234 .194 .169  .085 .063 .028  .247 .201 .172  8  P M 3/4  .782 .435 .211  .535 .375 .272  .902 .504 .249  .583 .409 .290  1.012 .575 .278  .623 .439 .308  20  P M 3/4  .322 .191 .089  .385 .270 .201  .375 .220 .104  .421 .290 .227  .427 .248 .118  .454 .310 .235  40  P M 3/4  .169 .102 .049  .286 .222 .201  .198 .121 .056  .313 .232 .205  .227 .137 .064  .339 .243 .209  80  P M 3 4  .090 .056 .025  .225 .204 .192  .105 .065 .031  .234 .208 .194  .121 .075 .035  .249 .212 .196  P  U  m = 1.2  D n .572 .458 .342  U  n  P .732 .527 .290  n  "P  .615 .498 .372  .793 .669 .343  .649 .532 .399  D  D  upsurges vary by as much as 50% and the downsurges vary by as much as 40%.  The greater v a r i a t i o n occurs generally f o r small 2p* a* values. Simple o r i f i c e - inflow to outflow head loss r a t i o  1:1  The curves i n this group are w e l l defined except for some scatter in the range of low 2p* and 2p* a* values f o r upsurge only. 0.5 curves were eliminated for the range K = 0.7  to K = 1.0  because the program would not converge to a solution.  The 2p* = inclusive  Note that for the  higher values of K, 2p*, and 2p* a*, the upsurges at the mid-point of the l i n e 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, i s based on inflow to the a i r chamber. To compare the upsurges and downsurges for the two o r i f i c e s , one d i f f e r e n t i a l with a 2.5:1  inflow to outflow head loss r a t i o 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 d i f f e r e n t i a l o r i f i c e .  I t follows that the down-  surges w i l l be equal for the following f r i c t i o n factors: (1)  D i f f e r e n t i a l , K = 0.5;  Simple, K » 0.2;  (2)  D i f f e r e n t i a l , K = 1.0;  Simple, K =  and  0.4.  Table 5.2 does i n fact v e r i f y t h i s , except for isolated instances.  5.4  ENTIRE HEAD LOSS ATTRIBUTABLE TO DISTRIBUTED FRICTION As the t o t a l head loss increases, the distributed f r i c t i o n  s i g n i f i c a n t l y reduces the upsurges, and, to a lesser extent, the downsurges.  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  0.2 UP 2p * 0.5  0.5 1.0  1.0 2.0  2.0 4.0  1 2 3 4 6 8 10 15 2 3 4 6 10 15 20 30 4 6 10 15 20 30 40 60 8 10 15 20 30 40 80 100  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  • 505 .442 .403 .371 .325 .293 .269. .231  .442 • 371 • 325 .293 .252 .227 .209 .183  .360 .284 .244 .220 .192 .175 .165 .150  .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  .461 .391 .319 .275 .251 .226 .212 .198  333"  2p*c*  .365 1.191 - 7 .e67 .502 .257 .572 .353 .174 . 4 C i .259 .123 .318 .206 .097 .227 .155 .071 .178 .126 .057 .126 ,r»3 .040 0 1  K - 0.5  0.4 DN  •599 • 516 •4» .364 .322 .275 .251 .225  1.303 .718 T36T .583 1.061 .581 .293 .543 .737 413 .204 .470 335 .165 .421 • 371 244 .119 .355 .404 .317 194 .095 .313 .177 .113 .055 .234 .147 .095 .045 .222  .288  .245 .221 .203 .195 .189 .182  DN  M  i  .608 .488 .376 .298 .224 .177 .139 .092  .568 .326 .258 .254 .182 .141 .116 .085  . 271 .153 .165 .128 .083 .061 .050 .035  .487 .437 .399 .369 .329 .302 .284 .254  . 437 . 360 .370 .296 . 330 .264 . 303 - 246 . 270 . 224 . 251 -213 . 244 . 206 . 241 .202  .593 .478 .365 .289 .215 .162 .127 .083  . 528 . 308 . 243 . 217 .145 .111 .092  .725 .622 .494 .349 .221 • 153 .116 .081  .641 .359 .362 .247 .174 .131 .105 .081  .314 .184 .177 .107 .076 .058 .046 .033  .625 .564 .521 .462 .397 .354 .329 .302  .522 .463 .425 .378 .330 .302 .298 . 295  .415 .369 .342 .311 .283 . 269 .266 . 262  .887 .621 .393 .271 .210 .146 .112 .078  .510 .417 .278 .208 .166 .122 .098 .074  .263 .184 .126 . 092 .073 .054 .043 .031  .665 .591 .506 .451 .418 .381 .361 .340  .434 .393 .356 .336 .326 .318 .316 .314 .430 .413 .389 .376 .363 .357 .351 .350  M  .409 .290 .914 .376 .270 .737 .323 .241 .504 .290 .227 .387 .268 .253 .213 .232 .205 .207 .208 .194 .112 .203 .192 .092  1.0  UP  DN  EN  P  M  .O64  .250 .146 .147 .110 .071 .052 .040 .027  .486 .442 .403 .371 .325 .293 .269 .231  . 442 .371 . 325 .293 .252 .227 . 209 .183  .360 .284 .244 .220 .192 .175 .165  .729 .607 .476 .336 .208 .141 .108 .073  .528 .349 .323 .199 .134 .096 .076 .056  .290 .188 .151 .101 .065 .045 .035 .024  .614 .546 .497 .430 .352 .300 .269 .234  .498 .431 . 385 .328 . 270 .234 .215 .194  .372 .317 .233 .245 .210 .190 .180 .169  .571 .400 .299 .203 .121 .081 .063 .043  .398 .277 .202 .149 .099 .087 .079 .064  .183 .131 .100 .061 .041 .032 .026 .022  .625 .564 .521 .462 .397 • 354 .329 .302  .522 . 463 .425 . 378 . 330 . 302 . 298 .295  .415 . 369 -342 . 311 . 283 . 269 . 266 .262  .878 .611 .379 .261 .200 .137 .105 • 072 .902 .725 .491 .375 .258 .198 .115 .086  .470 .327 .235 .162 .128 .091 .070 .010 504 .409 .286 .220 .155 .121 .065 .053  .250 Il66 .109 .077 .060 .043 .033 -02? .249 .199 .136 .104 .072 .056 .031 .025  .599 .516 .429 .364 .322 .275 .251 .225 .583 .543 .470 .421 .355 .313 .234 .222  .461 .391 .319 .275 .251 .226 .212 • 198 .409 .376 • 323 .290 .253 .232 .208  .335 .288 .245 .221 .208 .195 .189 .182 .290 .270 .241 .227 .213 .205 .194 .192  .551 .369 .222 .150 .113 .077 .058 .039  -338 -235 .151 .114 -090 .073 .065 .059  .171 .113 .070 .048 .036 .027 .023 .019  .665 .591 .506 .451 .418 • 381 • 361 •340 .636 .594 .528 .488 .444 .420 .-380 .371  .543 .481 .419 .382 .362 .340 .338 .337 .519 .489 -A45 . 420 .394 .380 .365 .364  • 434 .393 • 356 • 336 .326 .318 .316 • 314 430 .413 .389 .376 • 363 .357 .351 .350  M  .543 .481 .419 .382 .362 .340 .338 .337 .557 .260 .636 .519 .466 .218 .594 . 489 .332 .153 .528 .445 .365 .120 .488 .420 .191 .084 . 444 .394 .152 .066 .420 . 380 .086 .038 .380 .365 .072 .031 .371 .364  UP  M  .203  .ISO  .541 .331 .158 .432 .269 .128 .289 .185 .087 .219 .142 .067 .148 .100 .046 .112 .079 .035 .058 .052 .019 .046 .048 .017  pump.  This i s l o g i c a l because the distributed f r i c t i o n i s i n e f f e c t  over a longer distance.  For K = 0.4 and above, upsurges have been  eliminated while the downsurges are divided into three d i s t i n c t groups, at the pump, the mid-point, and the three-quarter point. above 0.6,  For K values  the downsurges for the various values of 2p* i n 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 i n c l u s i v e , the upsurges disappear  completely and the downsurges are segregated into three d i s t i n c t 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 i n 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 i n the a i r chamber, the charts can be used for f i n a l design as w e l l as preliminary design and checking  purposes.  Usually when an a i r chamber i s being designed f o r a pump discharge l i n e , the values of L, a, V , Q , A, H * and g q  From these values, 2p* can be computed.  q  w i l l be known  The allowable maximum surge  values may be dictated by s p e c i f i c a t i o n s , operating conditions, or the p r o f i l e of the discharge l i n e .  For the computed value of 2p* and the  s p e c i f i e d maximum allowable surges, values of K and 2p* a* can be  chosen from the charts such that the surge l i m i t a t i o n s are met.  If  the allowable surge conditions can not be s a t i s f i e d by data from the charts, probably some means other than an a i r chamber should be used to control the surges. When 2 p * a* has been determined, C Eq. (1.5)  can be computed from  q  (i.e.): C  o  =  p*  a *  Q  -  oa  .  Numerical examples demonstrating the use of the charts are given i n Appendix A. Figure 5.1 shows the configuration of the pump, a i r chamber, pipeline and reservoir. The Charts begin on page 45.  Pipe Wall F r i c t i o n PIPELINE WITH AIR CHAMBER FIG. 5.1  H  o*  - Steady-state absolute pressure head at pump  H  o  - Steady-state pressure head at pump - Total head loss  H  o r  f  0  - Head loss due to o r i f i c e resistance  Upsurges and Downsurges measured from this l i n e (For uniformly distributed f r i c t i o n only) Measured to transient ordinary pressure head  W.S..  Atmospheric Pressure Head  Reservoir  Pipeline  4-1  C •rl  O PM  I  T3 •rl  a  c •H  O PM  I  ca  ro  erf  CHAPTER VI  DISCUSSION  6.1  VOLUME OF AIR IN THE CHAMBER Since a*, the parameter pertaining to a pump discharge l i n e  having an a i r chamber, i s d i r e c t l y proportional to C , 0  the i n i t i a l  volume of a i r i n the chamber, the i n i t i a l volumes of a i r and water i n the tank must be maintained within certain l i m i t s to ensure proper operation of the chamber.  The compressed a i r which dissolves i n the  water or i s l o s t through leakage must be continually replaced.  Some  means of automatic shut down of the pump or pumps must be provided should the proper water l e v e l i n the tanks not be maintained.  The  minimum controls required are shown schematically i n F i g . 6.1. The following items^ should be considered when f i x i n g the compressor "on" and " o f f " l e v e l s :  and  (a)  capacity of the compressor,  (b)  size of the a i r chamber,  (c)  frequency of s t a r t i n g and stopping of the compressor,  (d)  d a i l y temperature variations that might actuate the controls,  (e)  how  quickly the system i s to be put back into operation  a f t e r a prolonged shutdown. The emergency levels can be at nominal distances above and below the compressor operating levels on i n s t a l l a t i o n s having only one pump or that provide manual starting or stopping f o r i n d i v i d u a l pumps on the same l i n e .  I f automatic  s t a r t i n g and stopping of the i n d i v i d u a l pumps 35  on the same l i n e are required, the emergency levels should be s u f f i c i e n t l y removed from the compressor operating levels to contain the surges produced by s t a r t i n g 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 l e v e l s .  6.2  TOTAL VOLUME OF THE AIR CHAMBER Once 2p* a* has been determined from the charts, C  culated by using Eq. 1.8.  Q  can be c a l -  The volume of the a i r chamber i s then deter-  mined by considering that the chamber must contain adequate a i r above the upper emergency l e v e l to control the surges to desirable l i m i t s , and enough water below the lower emergency l e v e l to prevent unwatering. With allowance f o r the volume between the upper and lower emergency l e v e l s , the t o t a l required volume of the a i r chamber can be computed. The minimum volume of a i r that must be maintained i n the chamber to control the pressure surges i s the volume of the chamber above the upper emergency l e v e l .  This volume can be designated C'  which i s numerically equal to the volume C . Q  By adding to this  quantity the volume of the chamber between the upper and lower emergency l e v e l s , one determines the i n i t i a l volume of a i r i n the chamber that w i l l r e s u l t i n the lowest water-surface l e v e l following pump shut down.  This new volume of a i r becomes C" equal to C' plus  the volume of a i r between the upper and lower emergency l e v e l s . The downsurge at the pump with this i n i t i a l volume of a i r can be determined from the curves by computing a new value of 2p* a* based  on C" instead of C'. Assuming that this expansion i s isothermal , the t o t a l volume of the a i r chamber becomes C" Ho* H * - downsurge at pump q  Under favorable conditions, the a i r 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 l i n e s 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 a i r volume i s generally about 40 percent of the tank volume.  6.3  ORIFICE DESIGN Since the function of an a i r chamber i s to decrease both the  upsurges and the downsurges on pump f a i l u r e , i t i s necessary to t h r o t t l e the reverse flow of water from the discharge l i n e into the chamber while providing l i t t l e t h r o t t l i n g f o r flow out of the chamber. An e f f e c t i v e device f o r producing a high head loss for inflow while keeping the e x i t head loss at a minimum i s a d i f f e r e n t i a l o r i f i c e as shown i n F i g . 6.2. The design i s e s s e n t i a l l y a bellmouth f o r flow from the chamber and a re-entrant tube f o r flow into the chamber.  This  design w i l l give discharge c o e f f i c i e n t s of 1.0 and 0.5 for outflow and inflow respectively.  The inflow head loss f o r a s p e c i f i e d rate of flow  would be approximately four times as great as the outflow headloss. However, this head loss r a t i o of 4:1 i s d i f f i c u l t to obtain i n p r a c t i c e .  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 o f 2.5:1 i s more r e a l i s t i c . I f head l o s s i n the p i p e l i n e due t o w a l l f r i c t i o n i s c o n s i d e r e d as c o n c e n t r a t e d a t t h e o r i f i c e , t h i s assumption.  the o r i f i c e d e s i g n s h o u l d a l l o w f o r  F o r example, d e s i g n a d i f f e r e n t i a l o r i f i c e f o r an  i n f l o w l o s s o f 60 p e r cent o f H * and o u t f l o w l o s s o f 30 p e r cent o f q  H * f o r an i n f l o w and o u t f l o w o f Q . o o  F o r a flow o f Q , t h e p i p e l i n e o  s u r f a c e - f r i c t i o n l o s s i s 10 p e r cent o f H *. o  The o r i f i c e s h o u l d be  r  f o r a head l o s s o f 50 per cent o f H * f o r an i n f l o w of Q and  designed  q  q  a head l o s s o f 20 p e r cent o f H * f o r an o u t f l o w o f Q . o o  The a c t u a l  r  o r i f i c e d e s i g n , head l o s s r a t i o o f inward  flow t o outward f l o w s h o u l d  be 2.5:1. An o r i f i c e may be designed through  t o g i v e a maximum i n i t i a l head l o s s  the o r i f i c e e q u a l t o the maximum downsurge.  known as normal t h r o t t l i n g ^ .  This condition i s  G r e a t e r or s m a l l e r t h r o t t l i n g  l o s s e s may  be s a i d t o g i v e o v e r - t h r o t t l i n g or u n d e r - t h r o t t l i n g , r e s p e c t i v e l y . minimum head i n the p i p e w i l l correspond  The  t o the maximum a i r expansion  i n the chamber f o r the c o n d i t i o n s o f normal t h r o t t l i n g and underthrottling. be used  F o r o v e r - t h r o t t l i n g the minimum head i n the p i p e can n o t  t o determine  the maximum a i r expansion  i n the chamber.  F o r the  c o n d i t i o n o f o v e r - t h r o t t l i n g , the minimum p r e s s u r e i n the tank must be known i n o r d e r t o determine Large c o m p u t a t i o n a l  t h e maximum a i r expansion. e r r o r s r e s u l t when f r i c t i o n i s i g n o r e d .  The  i n c l u s i o n o f d i s t r i b u t e d w a l l f r i c t i o n i n c r e a s e s the accuracy o f the maximum and minimum p r e s s u r e s and c o r r e s p o n d i n g maximum a i r expansion i n t h e chamber.  Thus the c h a r t s i n c l u d i n g d i s t r i b u t e d w a l l f r i c t i o n  give h i g h l y 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 i n the development  of one of the most destructive types of waterhammer surge i n pumpdischarge pipe l i n e s .  Following pump f a i l u r e , the sudden pressure  drop downstream might be severe enough to bring about a temporary vapour pressure condition, and possibly the formation of a void i n the pipe l i n e .  The subsequent closure of this void often results i n v i o l e n t  l o c a l surges well above any possible transient pressure r i s e s i n a continuous water column.  The extent of pressure r i s e i s proportional  to the f l u i d v e l o c i t y destroyed at the instant of vacuous space closure. The four major f a c t o r s ^ 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)  v e l o c i t y of flow.  (1)  For pumps that have small r o t a t i o n a l i n e r t i a s the r e s u l t i s  complete pump stoppage from within a f r a c t i o n of a second to a very few seconds a f t e r pump f a i l u r e .  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 p o s i t i v e pressure waves r e f l e c t e d from the f a r end of the l i n e counteract the pressure drop.  A long l i n e  with a pump having a small r o t a t i o n a l i n e r t i a very often w i l l  experience  water - column separation on pump f a i l u r e . (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 l i n e passes, a  41  s l i g h t interruption of flow may r e s u l t i n a drop to vapour pressure and r e s u l t i n g column separation. (4)  The fourth major element i n water-column separation i s the  v e l o c i t y of water i n the pipe l i n e preceding the cause of perturbation. As the steady state v e l o c i t y increases, the size of the vacuous space, the reverse flow v e l o c i t y , and the f i n a l surges following the void collapse a l l become greater. A l l of these elements are i n t e r - r e l a t e d .  For example, extensive  water-column separation may occur even with a very low v e l o c i t y i f the pipe l i n e i s long enough and the steady state pressure head i s low. An a i r chamber i s one means of preventing or c o n t r o l l i n g watercolumn separation f o r medium to high-head systems.  An example i s given  i n Appendix A indicating the manner i n which the charts can be used to determine the p o s s i b i l i t y 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 f o r the water-column separation phase of waterhammer.  CHAPTER VII  CONCLUSIONS (1)  Since the program was evolved from the basic d i f f e r e n t i a l equations  for momentum and continuity to give the absolute pressure i n the a i r chamber, nonlinear terms are retained and f r i c t i o n i s included, the charts can be used f o r f i n a l design (2)  purposes*.  The v a l i d i t y of the charts i s demonstrated by comparing the  results obtained by the method of c h a r a c t e r i s t i c s with those obtained by the graphical method. (3)  I t i s important to analyze the system properly and to use the group  of charts which most closely approximate the system i n order to get v a l i d results.  In some cases i t might be advantageous to interpolate between  For example, K might be in the range 0.0  graphs within the same group.  to 0.1. (4)  I t i s important to determine whether the expansion and compression  of a i r i n the chamber i s adiabatic or isothermal because the r e s u l t s vary s i g n i f i c a n t l y f o r the powers m = 1.0 power i n the equation H * v ^ a  2p*  o* =  m =  1.4  8,  m r  = constant.  and m = 1.4,  where  i s the  For example, for 2p* = 4 and  the upsurge at the pump f o r m =  1.0  i s 0.782 H*  i s 1.012 H* , the downsurge at the pump for m = Q  m  1.0  q  and for i s 0.535 H*  and for m = 1.4 i s 0.623 H* . Q  * The charts produced by Evans and Crawford are to be used only f o r preliminary design purposes. The authors stress that " f o r f i n a l design of an i n s t a l l a t i o n having an a i r chamber, i n d i v i d u a l solutions s i m i l a r to that shown by Mr. Angus ('Air Chambers and Valves i n Relation to Water-Hammer', Transactions, ASME, V o l . 59, 1937, p.661) should be made to ensure that the a i r chamber w i l l f u l f i l l design requirements". 42 -  Q  The power m = 1.2 gives an approximate average f o r the upsurges and downsurges.  For the same p i p e l i n e constants as above, f o r m = 1.2,  the upsurge at the pump i s 0.902 H*  0  and the downsurge at the pump i s  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 characteristics.  The accuracy of the charts produced i n this work i s enhanced  by the i n c l u s i o n of f r i c t i o n and nonlinear terms. The charts presented i n 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 F r i c t i o n , Line F r i c t i o n Only No O r i f i c e Loss, and F r i c t i o n Loss Equally Distributed between O r i f i c e Loss and wall F r i c t i o n , the range of K i s from 0.1 to 1.0. (6)  Bergeron's method of graphical analysis considering l i n e f r i c t i o n  concentrated at f i v e points i s quite accurate as demonstrated by the computer check using the method of c h a r a c t e r i s t i c s . (7)  The number of sections used i n analyzing the pipe system i s  important.  I f N i s 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, f o r instance, N was set equal to ten and gave good r e s u l t s . For N equals f i v e , 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., " A i r chambers f o r Discharge Pipes", Trans. ASME, Vol. 59, Paper Hyd-59-7, November 1937, pp. 651-659.  2.  Bergeron, L., Water Hammer i n Hydraulics and Wave Surges i n E l e c 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 f o r Analysis of Water Hammer i n Pipe Systems, A thesis submitted i n p a r t i a l f u l f i l l m e n t of the requirements f o r the degree of Master of Applied Science at the University of B r i t i s h Columbia, 1968.  4.  Evans, W.E., and Crawford, C.C., "Design Charts f o r A i r Chambers on Pump Lines", Trans. ASCE, September 1954, pp. 1025-1036.  5.  Parmakian, J . , Waterhammer Analysis, New York, 1963.  6.  Paynter, H;M., Discussion of "Design Charts f o r A i r Chambers on Pump Lines", Trans. ASCE, September 1954, pp. 1039-1045.  7.  Richards, R.T., "Water-Column Separation i n Pump Discharge Lines", Trans. ASME, Paper No. 55-A-74, 1955, pp. 1297-1304.  8.  Ruus, E., and Chaudhry, M.H., "Boundary Conditions f o r A i r Chambers and Surge Tanks", Trans. EIC, November 1969, EIC-69-HYDEL 22, V o l . 12, No. C-6.  9.  Streeter, V.L., and Wylie, E.B., Hydraulic Transients, McGraw-Hill Book Company, New York, 1967.  Dover Publications, Inc.  45  THE CHARTS  GROUP I  NO HEAD LOSS, FRICTIONLESS FLOW (No wall f r i c t i o n , no o r i f i c e loss)  M A X . DOWNSURGE  MAX.UPSURGE  GROUF I I  ENTIRE HEAD LOSS CONCENTRATED AT THE ORIFICE (no wall f r i c t i o n ) A.  DIFFERENTIAL ORIFICE, RATIO  B.  SIMPLE ORIFICE, RATIO  1:1  2.5:1  MAX.  DOWNSURGE  MAX.UPSURGE  H *  H *  0  o  o  o  p  ui  p  ^  o  w  p  ro  o  —  O .  p  —  p  K>  p  OJ  p  A  p  01  p  p  ff>  p  p  co  co  — P  — '—  — ro  —  01  4^  to  MAX. UPSURGE  MAX. DOWNSURGE  H,  H;  o  o  o  o  o  p  o  ho  o  o  o bi  o cr>  O  o  co  o  Hi?  0  o  o  o  o  o Ln  '1  MAX.UPSURGE  MAX. DOWNSURGE H*  o  O ro  O  o  o  o  0>  o  o  co  o to  4—  ^  i  T>  -0 TJ  I  I I  I  2  'I to  MAX. DOWNSURGE  MAX. UPSURGE  MAX. DOWNSURGE H*  MAX. UPSURGE H*  GROUP I I I  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 c h a r a c t e r i s t i c s and those produced by Evans and Crawford. Example on the design of an a i r chamber f o r a short p i p e l i n e of large diameter. Example on checking the maximum upsurges and downsurges f o r 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  No f r i c t i o n loss  FIGURE A-2  Total f r i c t i o n loss  0.3 H *  (orifice  loss)  FIGURE A-3  Total f r i c t i o n loss  0.5 H *  (orifice  loss)  FIGURE A-4  Total f r i c t i o n loss  0.7 H *  (orifice  loss)  Q  Q  Q  FIG  Al  96  FIG. A-?  F I G . A-3  98  1.3 1.2  i—r  PUMP  I.I  MI0LENGTH  1.0  PUMP  . _ .. T  r  } |  Evans ft Crawford  Computer study  MIDLENGTH  0.9  UJ  0.8  CO * o CL X Z) X  0.7  o or z>  <  0.6  0.5  0.4  0.3  0.2  4 1  5 !  6 I  H  7  , (-  6 9 10  UJ  o or 0.3 r> co * o z:  X < 2  20  ....)._ . -L...  0.2  o Q  15  ' i : . . 2 :/?V.V 1  0.4  0.5  0.6  0.7 FIG. A-4  30  H  r-  !  !  40  50 60 70 8090 M  I i  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 a i r chamber which w i l l l i m i t the waterhammer surges to the s p e c i f i e d  limits.  Atmospheric Pressure E l . 400 "  DATA Check valve closes immediately Length of p i p e l i n e  on pump f a i l u r e .  (L) = 3220 feet.  Area of pipe (A) - 3.142 f t  2  Steady-state discharge (Q ) = 18.5 cu.ft Q  per sec.  Steady-state v e l o c i t y (V ) = 5.9 f t per sec. Q  100  Steady-state head at pump (H ) = 300 f t Q  Water hammer wave v e l o c i t y (a) = 3660 f t Atmospheric pressure = 34.0 f t  per sec.  of water.  Neglect l i n e 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  H* Q  At midlength - allowable downsurge = 400-350+20 =0.21  H* Q  At three-quarter point - allowable downsurge = 400-370+20 = 0.15  2 o  * =  2 p  1S7  =  (3660)(5.9) (32.2)(334)  =  0  2 2  ,  0  From the charts i n Group I I , E n t i r e Head Loss Concentrated at the O r i f i c e , D i f f e r e n t i a l O r i f i c e 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 a i r i n the chamber w i l l vary d i r e c t l y as a*, so the smallest value of 2p* o* w i l l be used. K =  0.3  2p* a* - 22 2p* =  2.0  For the values,  H* D  101 At the pump:  Maximum upsurge = 0.26 H * Q  Maximum downsurge = 0.32 H * Q  At the midlength:  Maximum upsurge = 0.155 H * Maximum downsurge = 0.21 H * Q  At the three-quarter point:  Maximum upsurge = 0.07 H * Q  Maximum downsurge = 0.15 H * Q  The d i f f e r e n t i a l o r i f i c e should be designed to provide a head loss of (0.3)(334) = 100 f t f o r a flow of 18.5 cu.ft  per sec. into the chamber.  From Eq. 1.8, (2p* a*) ALV  C °  Q  2a =  (22) (3.142)(3220)(5.9) (2)(3660)  C'=C  0  =  179 cu.ft  -  179 cu.ft  Assume: Volume between upper and lower emergency l e v e l s i s 20% of C'. Then, C" = 1.20 C and  = 215 cu.ft  2p* a* = 1.2 x 22 - 26.4. The maximum downsurge at the pump becomes 0.295 H *. Q  Total a i r chamber volume  =  C"H * 0  H * - downsurge at pump Q  -  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 t h i s example at the midlength and three-quarter point.  The design  ensures that water-column separation w i l l not occur.  I f 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 p i p e l i n e .  Atmospheric Pressure Head = 34 f t - ^ 7  o  z  II  W 4-1 o m ro II  VO  'rH  ro II O  sa  •7  d. 9192 f t  DATA Check valve closes immediately  on pump f a i l u r e .  Length of p i p e l i n e (L) = 9192 f t Area of pipe (A) =0.79 f t  2  Steady-state discharge (Q ) =5.0 f t / s e c 3  Q  Steady-state v e l o c i t y (V ) = 6.3 ft/sec Q  Steady-state head at pump (H ) = 316 f t . Q  Line f r i c t i o n loss (Hp) = 70 f t . Waterhammer  wave v e l o c i t y (a) = 3660 f t / s e c .  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 . o 3  SOLUTION (A) No o r i f i c e loss 2C a  2(50)(3660)  Q  2p* a* =  = ALV  K  -  2p*  =  70 —  = 8.0 (0.73)(9192)(6.3)  C  - 0.20  aV  (3660)(6.3) =  gH * Q  = 2.04 (32.2)(350)  From the charts i n Group I I I , Entire Head Loss Attributable to Distributed (1)  Friction: At pump:  Maximum upsurge = 0.285 H * Q  Maximum downsurge = 0.55 H * Q  (2)  At midlength:  Maximum upsurge =0.15 H * Q  Maximum downsurge = 0.32 H * Q  (3)  At three-quarter point:  Maximum upsurge = 0.075 H * Q  Maximum downsurge = 0.175 H (B)  Q  Total head loss evenly divided between l i n e f r i c t i o n loss  and o r i f i c e loss (2.5:1 d i f f e r e n t i a l 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 H * Q  Maximum downsurge = 0.32 H * Q  (3)  At three-quarter  point:  Maximum upsurge = 0.14 H * Q  Maximum downsurge = 0.19 H * Q  REMARKS I t i s obvious from the foregoing r e s u l t s that care must be exercised i n s e l e c t i n g the charts to best approximate the actual physical condition. The surge r e s u l t s (especially the upsurges) vary considerably f o r d i f f e r e n t types of head l o s s .  106  APPENDIX - B  GRAPHICAL CHECKS ON PROGRAM  B-1  Check f o r t o t a l head loss concentrated at the o r i f i c e .  B-2  Check f o r t o t a l 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 v e l o c i t i e s p i p e l i n e adjacent to the pump at A.  i n the  The transient conditions are caused  by pump f a i l u r e .  Atmospheric I ^B  Throttling  = ±50 f t for Q = ±20 f t / s e c . 3  FIG. B-la  DATA  Check valve closes immediately  on pump f a i l u r e .  Length of pipe l i n e (L) = 3220 f t . 3  Steady-state discharge (Q ) « 20.0 f t /sec. Q  Steady-state v e l o c i t y Waterhammer wave  (V ) ~ 5.00 Q  velocity  ft/sec.  (a) = 3220 ft/sec.  Pipe l i n e constant (2p*) =2.00 Constant f o r a pipe l i n e having an a i r chamber  2C a (2p* a*  =  -^-j-)  =  10.0  o Atmospheric pressure = 34.0 f t O r i f i c e t h r o t t l i n g loss =  of water.  ± 50 f t f o r Q  = + 20 f t / s e c . 3  q  A i r expansion i n the chamber i s given by H* v . air  1 2  = a constant, i n which H* and v . air  are the absolute pressure and volume of a i r i n the chamber. Neglect l i n e f r i c t i o n losses. CHECK Results obtained on the d i g i t a l computer using the method of c h a r a c t e r i s t i c s are close to those obtained by Parmakian^ (page 135) by the graphical method (see F i g . B-3b ).  103  APPENDIX B-2  CHECK FOR TOTAL HEAD LOSS ATTRIBUTABLE TO DISTRIBUTED FRICTION  PROBLEM Determine the transient state pressures and v e l o c i t i e s i n the pipe l i n e adjacent to the pump at B.  The transient conditions are caused  by pump f a i l u r e .  FIG. B-2a DATA (Water supply l i n e f o r c i t y of T r a i l ) . Check valve closes immediately  on pump f a i l u r e .  Length of p i p e l i n e (L) = 9150 f t .  3 Steady-state discharge (Q ) = 4.0 f t /sec. Q  Steady-state v e l o c i t y (V ) = 5.1 f t / s e c . Pipe l i n e constant (2p*) =1.38  2C a o Q L -)= 5 o x  Constant f o r a pipe l i n e having an a i r chamber (2p* a*  Ill  Atmospheric pressure = 33.0 f t . of water Total f r i c t i o n loss (H^,) = 90.0 f t . of water. Pressure head at the pump (H ) = 387.0 f t . of water Q  3  Steady-state volume of a i r i n the chamber (C ) = 25.0 f t Q  1 2 A i r expansion i n the chamber i s given by H* C = a constant, i n which H* and C are the absolute pressure and volume of a i r i n the chamber. 1.2 H* C This may be written as h* c = 1 where h* = rr~ and c = ~r~ • o o There i s no loss f o r flow into or out of the chamber. H  c  The f r i c t i o n loss i n 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 c h a r a c t e r i s t i c s are close to those ( F i g . B-2b) obtained by Eugen Ruus, who analyzed this system by the method of graphical water hammer analysis concentrating the pipe l i n e wall f r i c t i o n at f i v e points as shown i n F i g . B-2a.  pump.  Check v a l v e c l o s e s 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 O R I F I C E  $LIST  ?  AIRCHAMIO WATEPHAMfER PROG P AM. PUMP AT UPSTREAM END WITH AIR CHAMBER ADJACENT 2 C TU THE PUMP. RESERVOIR AT DOWNSTREAM END. 3 C CHECK VALVE CLOSES I MM EC I AT EL V ON PUMP F A I L U R E . 4 C NO LIME F R I C T I O N . HEAR LOSS CONCENTRATED AT O R I F I C E . 5 C NO MI NCR L O S S E S . 5. 5 C 6 01 KENS ION V ( 2 0 ) , V P ( 2 0 ) , H ( 2 0 ) H P ( 20 ) , V P ( 2 0 ) , V S ( 2 0 ) , H R ( 2 0 ) , H S { 2 0 ) , 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) DATA N / 1 0 / , V A / 3 2 1 6 . / , G / 3 2 . 1 6 / , F L / 3 2 1 6 . / , P M / l . 2 / , M M / l / , 9 10 IF/0.0/,CORFIN/2.5/.AP/3./, 2CK/0.1/, 11 3VO/3.5/ 12 WRITE! 6, 15) N»VA,G,FL,PM,N'M,F,CORFIN,AP,CK,VO 13 14 1S» FORMAT !/• THE PARAMETERS ARE NOW...'/ 15 1' N= »,I5,' VA= ' , F 8 . 2 , ' G= »,F6.2,' FL= ' , F 8 . 2 / 16 2' PM= ' • F 6 . 2 , ' MM = • I 5,' F= " ^ 6 . 3 ^ C 0 R F I N = «,F6.2/ 17 3* AP= ' t F8 .2 t' CK= ' , F 6 . 2 , ' V0= «,F8.2) 18 27 W R I T E ( 6 , 3 0 ) 19 30 FORM AT( ' PLC TMAX CPLAC UPSANS(l) UPAN10 UP SANS(6) UPAN3Q'/ 20 119X,' D N S A N S ( l ) DN AN 10 DNS ANS(6 ) DNAN3Q * ) 6 READ!5,10) PLC,TMAX,CPLAC 21 10 F O R M A T ( 3 F 8 . 3 ) 22 I F ( C P L A C . L E . O . O ) GO TO 110 23 24 C COMPUTE DT • DT = FL/ ( (VO + V A ) * F L O A T ( N ) ) 25 26 c CHECK FOR CONVERGENCE. DX=F L / F L O A T ( N ) 27 T HET A= DT / DX 28 I F < T H E T A . L E . ( 1 . / V A ) ) GO TO 2 0 ^ " 29 17 GO TO 110 30 31 c COMPUTE C O E F F I C I E N T S AND CONSTANTS FOP. ALL P I P E S . 20 A P I = 3 . 1 4 2 32 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 . H0ABS=VC/(C2*PLC) 37 37. 5 c H0= HEAD AT RE SE RVOIR. HC=H0ABS-HF-34. 38 c H0RF0= O R I F I C E HEAD LOSS FOR FLOW QO FROr* TANK. 38.5 39 HORFO= (C.K*HOABS-HF) /CORFIN 39. 5 c VOAIR= I N I T I A L AIR VOLUME IN TANK. 40 VOAIR=(CPLAC*V0*AP*FL)/(2.*VA) 40 .b c 00= STEADY STATE DISCHARGE. CO=VO*AP 41 FF=F*DT/(7.*0P) 42 CF=HOPFC/(CG*CO) 43 44 C10=HOA6S*V0AIR**PM c STEADY' STATE C A L C U L A T I O N S . 45 46 DHF=F*FL*VO*VO/(2.*G*DP*FLOAT(N)) NN=N+1 47 CO 25 1=1,NN 48 VI I ) =V0 49 • TE MP=MM-1 50 51 H( I ) = HO +TFVP*OHF t  t  \.  115  <  52 • 52.5 53 54 55 56 57 58 59 60 61 62 63 63. 5 64 65 65. 5 66 67 68 69 70 ' 71 72 73 74 75 76 77 78 7 9 80 81 82 83 84 85 86 87 88 88. 5 89 90 91 92 93 94 95 96 97 98 99 100 10 1 101. 5 102 103 104 105 106  C  C C C C  C C  C  25 CONTINUE 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 . DO 2 6 IM,6,5 H M A X U )=H( I ) H M J N l I )=H( I ) H S S ( I )=H< I ) 26 C O N T I N U E S U M S S I 1) =H< 3 ) -»H( 4 ) SUM.SS ( 2 ) = H ( 8 ) + H I 9 ) SUM A X ( 1 ) = H ( 3 ) + H ( 4 ) ' ' ~ SUM IN ( 1 ) = H( 3 ) + H ( 4 ) SUM A X ( 2 )=H(tt ) « H < 9 ) SUM I N I 2 ) = H ( 8 ) + H { 9 ) TIME INITIALIZATION. T= 0 . 0 VAIK=V0A1R PRINTOUT INTERVAL INITIALIZATION. K=0 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 . INTERIOR SECTIONS. 4 0 DO 5 0 I = 2 , N VR( I )=V( I ) - V A * T H E T A * l V I I ) - V < I ~ l ) ) " " HR( I ) = H( I ) - V A * T N E T A * { H I I ) - H ( I - 1 ) ) VS( I ) = V ( I ) - V A * T H E T A * ( V I I ) - V ( 1 + 1) ) HS( I ) = H( I ) - V A * T H E T A * < H( I ) - H ( I + 1) ) 5 0 CONTINUE BOUNDARY S E C T I O N S . RESERVOIR. V R ( N + 1 )=V(N+1 > - V A * T F , E T A * ( V ( N + l ) - V ( N ) ) HHIN+1)=H(N+1)-VA*THETA*(H(N+1)-H(N)) C 3 = V R ( N + l > + C 2 * H R ( N + l ) - F F * V R ( N + l ) * A B S < VR(N+ 1)) AIR CHAMBER. 54 V S ( 1 ) = V ( 1 ) - V A * T H E T A * < V ( 1 ) ~ V l 2 ) ) HS(1)=HI 1)—VA*THET A * ( H ( I ) - H ( 2 ) }  "  "  C1=VSI 1)-C2*HS(i)-FF*VS(1)*ABS(VS(1)) T=T+.OT M=M+ 1 I F I T . G E . T M A X l GO TO 1 0 7 T I M E I NCR EMFNT E D . BOUNDARY CONDITIONS. AIR CHAMBER. H O A B S * V O A l R * * P N =CONSTANT.' ~ 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 . VAVAPP=V(1) GO TO 2 1 0 200 VAVAPP=VAV 210 C11=VAVAPP*AP CAIR = VAIR + C 1 1 * 0 T " ~ . IFIC11) 53,5 2,51 51 C O R F = 1 . 0 GO TO 59 52 CORF=0.0 GO TO 5 9 5 3 C(iKF=2.5 59 H0RF=C0RF*CF*C11*ABSICl1) 60 HP( 1 ) = ( C 1 C / C A I R * * P ' - 1 ) - H 0 R F - 3 4 . NEGATIVE CHARACTERISTIC EQUATION. VPI1)=C1+C2*HP(1) V A V = ( V I 1 ) + V P ( 1) Ml. VEHR=VAVAPP-VAV IF(ABSlVERR).LE.0.0001) GO TO 2 3 0 2 2 0 GO TO 2 0 0  "  "  38  C C C  C  "  ~  116 107 108 109 109.5. 110 11 1 112 113 114 115 116 117 118 119 120 121 122 123 123.5 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 146.5 147 148 149 150 151 152 153 154 155 156 157 T58 159 160 161 162 163  C C C  C  C  C  2 JO VPAIR = VAIRMAP*DT*VAV) RESERVOIR AT DOWNSTREAM END. HP(N+l)=hO P O S I T I V E C H A R A C T E R I S T I C EQUATION. VP( N+l )=C3-C2*HP(N«-1) COMPQTATICN OF INTERIOR POINTS. 00 5 5 1=2,M V P ( I ) = 0 . 5 * I V R ( I ) + V S ( I ) + C2*(HR( I )-HS( I ) )-FF * ( V R ( I ) * A B S ( V R ( I ) ) + 1VS(I)*ARS(VS(I)))) H P ( 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 ) l * A F S ( V S ( I ) ) )/C2) 5 5 CONTINUE CONVERT V ( I ) = V P ( I ) , AND H ( I ) = H P ( I ) FOR ALL S E C T I O N S . 80 DO 90 1=1,NN VI I >=VP( I ) H ( I ) = HP( I ) 90 CONTINUE VAIR = VPAIR TABULATION OF MAX. AND MIN. HEADS. DO 95 1=1,6,5 I F ( H ( I ) . L T . H M I N l I ) ) GO TO 123 121 I F ( H ( I ) . G T . H M A X ( I ) ) GO TO 125 122 GO TO 95 123 HM IN(I ) = H(I ) GO TO 95 125 H M A X ( I ) = H { I ) 95 CONTINUE I F ( ( H ( 3 ) + H ( 4 ) ) . L T . S U M I N ( 1 ) ) G O T O 13 5 131 I F ( ( H ( 3 ) + H ( 4 ) ) . G T . S U M A X ( 1 ) ) G O T O 137 132 GO TO 140 135 SUM INI 1 ) = H ( 3 ) + H ( 4 ) 136 GO TO 140 137CSUMAXI 1) = H( 3 H H ( 4 ) 140 COMTINUF ' I F ( ( H ( 8 ) + H ( 9 ) ) . L T . S U M I N I 2 ) ) . G O TO 147 142 IF( ( H ( 8 ) + H ( 9 ) ) . GT .SUM AX ( 2 ) ) GO TO 150 144 GO TO 155 147 S U M I N ( 2 ) = H ( 8 ) + H ( 9 ) 149 GO TO 155 150 S U M A X I 2 ) = H ( 8 ) + H ( 9 ) 155 CONTINUE GO TO 40 COMPUTATION OF MAX. UPSURGES AND DOWNSURGES. 107 DO 170 1=1,6,5 UPS MAX(I ) = H M A X ( I ) - H S S ( I ) ONSMAXII)=HSS(Il-HMINII) UPS ANS(I)=U PSMAX(I)/HOABS ON SANS ( 1 ) =DNSMAX ( I ) /HOABS 170 CONTINUE HMAX1C=SUMAX(1)/2. HMIN1Q=SUMIN(1)/2. HMAX30=SUHAXl 2 ) 1 2 . HMIN3U=SUMIN(2)/2. HSS1Q=SUMSS( 1 ) /2. HSS3Q=SU,MSS( 2 1/2. UPMA10=H^AX1^-HSS1Q DNMA1G=HSS10-HMINIO UPKA30 = HMAX3G-HSS3Q ----DNMA3G=HSS3C-HMIN3Q UPAM 10 = UPMA10/HOABS  117 164 165 166 167 168 169 170 171 172 173 END OF  tCOPY  -  FILE  *SKIP  DNA NL C = D M " A 1 C / H O A B S U P A N 3 0 = UPKA 3 0 / H O A B S DN AN 30 = 0 Pv M.A 3 QI HO A 8 S W R I T E ( 6 , 1 8 0 ) PL C , T'AAX , C PL AC , U PS ANS ( 1 ) , U P AN 1 0 , UP S AN S ( 6 ) , UP AN 30 , 1 0 N S A N S I 1 ) » D NA N 1 0 , 0 N S A N S ( 6 ) , ONAN3Q 180 FCRN ; AT (/ F4 . 1, 2X , E 5 . 1, 2 X , F 5 . 1, 4 X , f b . 3 , 3X , F 6 . 3 , 4 X , F 6 . 3 , 3 X , F 6 . 3 / 1 2 2 X , F 6 . 3 , ? . X , F 6 . 3 , 4 X , F 6 . 3 t 3X , F 6 . 3 ) GO TO 6 110 STOP END  *SINK*  118 2.  ENTIRE HEAD LOSS ATTRIBUTABLE TO DISTRIBUTED FRICTION  $LIST  AI RCHA.M1 1 WATERHAMMFP PROGRAM. PUMP AT UPSTRFAM END WITH AIR CHAMBER ADJACENT C C TO THE P U M P . Rt-SERVO IK AT DOWNSTREAM END. C CHECK VALVE CLOSES IMMEDIATELY CN PUMP F A I L U R E . 5 C L I N E F R I C T I O N ONLY. NO O R I F I C E LOSS. NO MINOR L O S S E S . 6 DIM ENS ION V ( 2 0 ) , V P ( 2 0 ) , H ( 2 0 ) , H P ( 2 0 ) , V R ( 2 0 ) , V S ( 2 C ) ,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./, 2CK/1./, ll 12 3V0/3.5/ 13 WRI TE ( 6, 15) N,VA,G,FL,PN',r'i ,H0RF,AP,CK,VO 14 15 FORMAT!/• THE PARAMETERS ARE NOW...'/ 15 1* N= ' , 1 5 , ' VA= ' , F 8 . 2 , ' G= ' , F 6 . 2 , ' FL= ' , F 8 . 2 / 16 2' PM= » , F 6 . 2 , ' MM= ',15,' HQRF = ',F6.3/ 31 AP= ' ^ 8 . 2 , ' CK= , F 6 . 2 , ' V0= S F 8 . 2 ) 17 2 7 WRITE(6,30) 18 19 UPAN30' / 30 FORMAT( * PLC TMAX CPLAC UPSANS(l) UPAN1Q UPS ANS(6) 20 119X, DNSANS(l) CN AN 1Q DN SANSt 6) DNAN30') 6 READ(5,1C) PLC,TMAX,CPLAC 21 10 FORMAT(3F8.3) 22 I F ( C P L A C . L E . 0 .0) GO TO 110 23 C COMPUTE DT 24 25 DT= F L / ( ( V O +VA ) AFLOAT(N)) 26 C CHECK FOR CONVERGENCE. 27 DX=FL/FLOAT(N) 28 THETA= 29 I F I T H E T A . L E . ( 1 . / V A ) ) GO TO 20 30 17 GO TO 110 31 C COMPUTE C O E F F I C I E N T S AND CONSTANTS FOR ALL P I P E S . 20 API =3. 142 32 DP=S0RT(4.*AP/API) 33 C2 = G/VA 34 H0ABS=H0+HF+34. 36 C 37 H0ABS=V0/(C2*PLC ) " ' 37.1 HEAD LOSS FOR FLOW INTO CHAMBER. C HF = CK*HOAP. S 37.2 3 7.5 HEAD AT RESERVOIR. c H0= H0=H0ABS-HF-34. 38 38.5 c F= F R I C T I O N FACTOR. F=(HF*2.*G*0P )/(FL*VO*VO) " " —39 39. 5 c VOAIR= I N I T I A L AIR VOLUME IN TANK. 40 VCAIR=(CPLAC*VO*AP*FL)/(2.*VA) " ' 41 CO=VO*AP FF=F*DT/(2.*DP) 42 C10=H0ABS*V0A IR**PM 44 ' " 45 c STEADY STATE C A L C U L A T I O N S . OHF=HF/FLOAT!N) 46 47 NN=N+1 DO 25 I=1,NN 48 49 V(I)=VO 50 EM P = NN- I H ( I )=HC+TEMP*OHF 51 25 CONTIlsMJE 52 52.5 c I N I T I A L I Z A T I O N OF MAX. AND MIN. HEADS. 2 3 4  v  ,  •  1  DT/DX  T  V  f  ?  53 54 55 56 57 58 59 60 61 62 63 63. 5 64 65 65. 5 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 88. 5 89 90 91 92 93 101 10.1 .5 102 103 104 105 106 107 108 109 109. 5 110 111 112 113 114  119 DO 26 1=1,6,5 HMAX < I ) = M 1 ) HMINJI)=H(I) HSSI I )=H(1 ) 26 CONTINUE SU^SSI1)=H(3)+H{4) SUN SSI 2 ) = H ( 8 ) +H( 9) SUMAX ( 1 ) = H ( 3 H H ( 4 ) SUM IN( 1 ) = H ( 3 ) + H ( 4 ) SUf'AXI 2)=H( 8) +H( 91 SUMIM(2)-=H(8)-»H(9) TIME I N I T I A L I Z A T I O N . C T=0. 0 VAIR=VCAIR PRINTOUT INTERVAL I N I T I A L I Z A T I O N . c M.= 0 COMPUTATION OF VR,VS,HR,HS FOR ALL S E C T I O N S . c INTERIOR S E C T I O N S . c 40 DO 50 I=2,N V R ( I ) = V ( I ) - V A * T H E T A * ( V ( I )-V( I-1) ) HR ( I )=H( I ) - V A * T H E T A * ( H ( I ) - H ( I - D ) VS( I ) = V( I ) ~ V A * T F E T A * ( V ( I )-V( 1 + 1) ) " ' ~ ~ HS(I)=H(I)-VA*THETA*IH(I)-H11+1)) 50 CONTINUE BOUNDARY SECT IGNS. c RESERVOIR. c VR{N+l ) = V ( N + l ) - V A * T H E T A * ( V I N + 1 ) " V ( N ) ) HRIN + l ) = H(N+1 ) - V A * T H E T A * ( H ( N + 1 ) - H ( N ) ) C3= VR(N+ 1)+C 2 *HR(N+1)—FF * V R ( N + l ) * A B S ( V R ( N + l ) ) AIR CHAMBER. c 54 V S ( 1 ) = V ( 1 ) - V A * T H E T A * I V ( 1 ) - V ( 2 ) ) HS( 1)=H( 1 )-VA*THETA*(H( 1 J-H12) ) C1=VS(1 ) - C 2 * h S ( 1 ) - F F * V S ( 1 > * A B S I V S ( 1>) ; _ 38 T= T + DT P = M+1 I F ( T . G E . T M A X ) GO TO 107 TIME INCREMENTED. BOUNDARY CONDITIONS. c AIR CHAMBER. hOABS*VGAIR**PM=CONSTANT. c LOOP ( 89, 106) TO APPROX. AVE. VELOCITY FROM CHAMBER. . __ c VAVAPP=V(1) GO TO 210 200 VAVAPP=VAV 210 Cl1=VAVAPP*AP CAIR=VAIR+C11*DT 60 H P ( l ) = ( C 1 0 / C A I R * * P M ) - H 0 R F - 3 4 . NEGATIVE C H A R A C T E R I S T I C EQUATION. c VP( 1)=C1 + C 2 * H P ( 1 ) VAV=(V( 1 ) + VP( 1 ) ) / 2 . VERR=VAVAPP-VAV I F ( A B S ( V E R R ) . L E . 0 . 0 0 0 1 ) GO TO 230 220 GO TO 200 230 VPAIR=VAIR+<AP*0T*VAV) RESERVOIR AT DOWNSTREAM END. c HP(N+l)=H0 P O S I T I V E CHARACTERISTIC EWUATION. c VP(N+J)=C5-C2*HP(N+1) COMPUTATION OF INTERIOR POINTS. c DO 55 1=2,N VP( I ) = 0. 5*( VP. (I 1 + V S I I ) +C2*(HR( I ) -HS ( I ) )-F F* ( VR ( I )* ABS (VR I 1VS( I ) * A B S ( V S ( I ) ) ))  120  (  ?  115 116 117 118 119 120 121 122 123 123.5 124 125 126 127 12 8 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 146. 5 147 148 149 150 151 152 153 154 155 15(3  157 158 159 160 161 162 163 164 165 166 16 7 168 16 9 170 171 172  HP( I 1 = 0 . 5 * ( H R ( I ) + H S I I ) + ( V R ( I ) - V S ( I ) ) / C 2 - F F * < V R ( I ) * A B S ( V R ( I ) ) - V S ( I ) 1 * A B S ( V S ( I ) ) )/C2 ) 5 5 CONTINUE CONVFPT V ( I ) = V P ( I ) , AND H ( I ) = H P ( I ) FOR ALL S E C T I O N S . C 80 DO 90 I-l.NM V(I)-VP(I) H ( I ) = HP ( I ) 90 CONTINUE VAlR=VPAIR TABULATION OF MAX. AND MIN. HEADS. C DO 95 1 = 1 ,6 ,5 I E ( H ( I ) . L T . H M I N I I ) ) GO TO 123 121 I F ( H ( I).GT.HM A X( I ) ) GO TO 125 122 GO TO 95 123 HMINl I )=H(I ) GO TO 95 125 U M A X ( I ) = H ( I ) 95 CONTINUE IF( <H<3>+H(4)).LT.SUM IN( 1) ) GO TO 135 131 I F ( ( H ( 3 ) + H ( 4 ) ) . G T . S U M A X ( 1 ) ) G O T O 137 132 GO TO 140 135 SUMIN(1) = H(3 ) *H<4 ) 136 GO TO 140 1370SUMAX(1)=H(3)+H(4) 140 CONTINUE IF( (H( 8 ) + H ( 9 ) ) . L T . S U M I N ( 2 ) ) GO TO 147 142 I F ( ( H ( 8 ) + H ( 9 ) ) . G T . S U M A X ( 2 ) ) GO TO 150 144 GO TO 155 147 S U M I N ( 2 ) = H ( 8 ) + H ( 9 ) 149 GO TO 155 150 S U K A X ( 2 ) = H ( 8 ) + H ( 9 ) 155 CONTINUE GO TO 40 COMPUTATION OF MAX. UPSURGES AND DOWNSURGES. C 107 DO 170 1=1,6,5 UPS MAX(I )=HMAX( I J - H S S ( I ) DNS MAX(I )=H S S I I ) - H M I N ( I ) UPSANStI )=UPSMAX( I ) /HOABS DNS A N S ( I ) = DNSMAX( I)/HOABS 170 CONTINUE ~ " HMAX1Q=SUMAX(1)/2. HMIN10=SUMIN(l)/2. HMAX3Q=SUMAX(2)/2. HM IN 30= SUM I N ( 2 ) / 2 . HSS10=SUMSS(1)/2. HSS3Q=SUMSS(2)/2. UPMA10=HMAX1Q-HSS1Q DNMA10=HSS1 G-HMIMQ UPN A 30 = H Mi A X 30 -H S S 3Q DNMA3G= HSS 3Q-HMIN3Q UPA N10= UPMA10/HO ABS DNAN 1Q= DNMA IC /HOAB S UPAM3 0=U PMA30/H0ABS DNAN 30 = 0NMA30/HOABS W R I T E ( 6 , 1 8 0 ) P L C , T M A X , C F L A C , UP SANS(1) ,UPAN10,UPSANS(6) ,UPAN30 , 1 DNSANS(1) ,DNAN1Q,DNS ANS(6),DNAN3Q 180 F 0 P M A T I / F 4 . 1,2X,F5. 1 , 2 X , F 5 . 1 , 4 X , F 6 .3,3X,F6.3 , 4 X , F 6 . 3 , 3 X , F 6 . 3 / 1 2 2 X , F 6 . 3 , 3 X , r 6.3,4X,F6.3,3X,F 6.3) GO TO 6 110 STOP  12) 173  END  y  OF  $CCPY  END FILE  *SKIP  *SINK*  

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