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A computer program analysing transients in multistage pumping systems Schmitt, Klaus 1980

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A COMPUTER PROGRAM ANALYSING IN  TRANSIENTS  MULTISTAGE PUMPING SYSTEMS  by  KLAUS B.A.Sc.  SCHMITT  ( C i v . Eng.) The U n i v e r s i t y  A THESIS SUBMITTED  of B r i t i s h  Columbia,  IN PARTIAL FULFILLMENT OF  THE REQUIREMENT FOR THE DEGREE OF MASTER OF APPLIED SCIENCE xn THE FACULTY OF GRADUATE STUDIES i n t h e Department of Civil  We  accept to  this  Engineering  thesis  the r e q u i r e d  as  conforming  standard  THE UNIVERSITY OF BRITISH COLUMBIA APRIL,  1980  (o) Klaus Schmitt,  1980  1978  In presenting this thesis in p a r t i a l fulfilment of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library shall make i t f r e e l y 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 f i n a n c i a l gain shall not be allowed without my written permission.  Department of  CIVIL  ENGINEERING  The University of B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5  Date  APRIL 10,1980  (Hi)  ABSTRACT Transient at  all  pressures  pumps  of  Distributing placing  a  multistage  pumping  all  of  significantly  the  subsequent  reduces  simultaneous  pumping  stations  required  to  along  system a  one  pressure  are  pipeline,  pumps w i t h i n  transient  power  failure  analysed. rather  pumping  fluctuations  than  station, within  the  system. A to  computer  program  analyse multistage  controls, controls  in  consist  reservoirs; Boundary  the  pumping  event  of  with  as  a  of  valves, other  conditions  developed  using  the  FORTRAN l a n g u a g e  systems,  s u c h a power  controls  easily  of  system  this  thesis,  developed  appropriate failure.  vacuum b r e a k e r s ,  determining  part  with  is  air  added  These  surge  chambers  as  they  controls but  surge  are  and  develop. are  not  described  for  completeness. By c o m p a r i n g occurring premise  withinthat  transient This  large,  single  pressures  in  in  costs,  expensive  Examples  maximum  single  transient  control  minimum and  systems  than  as  and  stage  multistage  reduction  savings  the  pipe  stage  the  is  use of  the  systems,  the lower  substantiated.  allows  thicknesses may be  pressures  significantly  systems  pressures  structures  demonstrating  multistage  give  wall  transient  for and  possible  the  size  of  reduced. program  are  included.  (iii)  TABLE OF CONTENTS CHAPTER  PAGE  ABSTRACT  i i  LIST OF FIGURES  v  NOTATION  v i i  I.  INTRODUCTION  1  II.  GENERAL THEORY  3  2.1 2.2  C h a r a c t e r i s t i c Equations F i n i t e Difference Equations  3 6  2.3  Stability  7  Requirements  III.  BOUNDARY CONDITIONS  IV.  3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 THE  B o u n d a r y C o n d i t i o n f o r a Pump Check V a l v e B o u n d a r y C o n d i t i o n Pressure Regulating Valve A i r t h e M a i n Pumping S t a t i o n Vacuum B r e a k e r s S e r i e s J u n c t i o n o f Two P i p e s C o n s t a n t Head R e s e r v o i r a t t h e Downstream F l o w C o n t r o l V a l v e a t t h e Downstream End PROGRAM  4.1 4.2 4.3 4.4 4.5  J u s t i f i c a t i o n and M o d i f i c a t i o n s Assumptions D e s c r i p t i o n o f t h e Program Program Usage and Data I n p u t Check o f A c c u r a c y  8  End  9 15 15 21 24 26 27 28 32 32 35 36 36 41  V.  DISCUSSION  42  VI.  CONCLUSIONS  45  BIBLIOGRAPHY  46  APPENDICES A p p e n d i x A: Program f o r M u l t i s t a g e Pumping Systems A p p e n d i x B: Examples o f Program Usage T h r e e S t a g e System S i n g l e S t a g e System C o m p a r i s o n o f S i n g l e and T h r e e S t a g e Systems Check o f Program A c c u r a c y  47 59 60 68 72 74  (iv)  L I S T OF  FIGURES  FIGURE  PAGE  2.1  C h a r a c t e r i s t i c s Grid  3.1  Pump B o u n d a r y  3.2  Check  3.3  Notation  at Pressure  3.4  Pressure  Regulating  3.5  A i r Chamber a t t h e M a i n Pump  22  3.6  A i r Chamber B o u n d a r y  25  3.7  Control valve  3.8  Flow C o n t r o l V a l v e  4.1  T y p i c a l 2-Stage  4.2  T y p i c a l Valve  Valve  5  C o n d i t i o n with  Boundary  Check  Valve  Condition Regulating Valve  16 Valve-Pump System  Boundary  Condition  Condition  18 20  29 Boundary  Pumping  Closure  Condition  System  31 34  Curves  40  B-l B-2  11  65 >  Transient Pressures  i n a T h r e e S t a g e System  B-3  66 67  B-4  Transient Pressures  i n a S i n g l e Stage. System  B-5  Minimum and Maximum H y d r a u l i c Grade S i n g l e and T h r e e S t a g e Systems  71  Lines f o r 73  (v)  ACKNOWLEDGEMENT The  author  wishes  t o thank h i s s u p e r v i s o r , D r . E . Ruus,  his  constructive criticisms  and  D r . W.F. C a s e l t o n  and s u g g e s t i o n s  f o r reviewing  during  the manuscript.  this  for  project,  (vi)  NOTATION a  - speed  A  - cross sectional  C  - a i r chamber  C  of pressure  pulse  area  ( f t / s e c o r m/sec)  of pipe  ( f t o r m)  constant  - positive  characteristics  equation  C~  - negative  characteristics  equation  C  - pipeline  constant  +  a  Cp,Cjj - known c o n s t a n t s C(3 C  o r  in characteristics  - c o e f f i c i e n t of discharge f  - head  loss  t h r o u g h a i r chamber o r i f i c e  D  - pipe diameter  f  - Darcy-Weisbach  Fj  - i d e n t i f i c a t i o n o f a i r chamber  FH,FB - d i m e n s i o n l e s s  friction  pump  factor equation  characteristics  - acceleration  h  - dimensionless  pressure  head  H  - instantaneous  pressure  head  of gravity  ( f t / s e c o r m/sec)  ( f t o r m)  H^,Hg - i n s t a n t a n e o u s  head  at beginning  Hp  head  a t end o f t i m e  - instantaneous  air  H  o r  f  ~ absolute - head  pressure  loss  Hp.  - rated  HRES  ~ elevation  H  SUC  ~  h  E A  6!  head  at  - barometric  H  - head  e  HQ  loss  - steady  through  head  ( f t o r m)  flange  head  interval  a i r chamber o r i f i c e  pressure  state  interval  ( f t o r m)  of reservoir  through  o f time  i n a i r chamber  f o r a pump  suction  H)-,  ( f t o r m)  ( f t o r m)  g  H  equations  ( f t o r m) water of  head  control  a  surface pump  ( f t o r m)  ( f t or  m)  ( f t o r m) valve  ( f t o r m)  ( f t o r m)  (vii)  K  - head  Kl  - constant  L  • -  Ll L  -  2  pipe  loss  determined  length  - number  of p a r a l l e l  N  - number  o f reaches  Q A . Q B  i n polytropic  -  rated  -  instantaneous  gas e q u a t i o n  i n a pipe  pump s p e e d (rpm) discharge  (ft-^/sec o f time  Qo  -  Q  - discharge  a t end o f t i m e  Qorf  - discharge  through  QR  -  rated  pump d i s c h a r g e  t  -  time  (seconds)  T  -  instantaneous  TR  -  rated  V  - dimenesionless  P  v  e  V  P  equation  pumps  - discharge at beginning steady  characteristics  equation  i d e n t i f i c a t i o n of continuity  np  Q  from pump  i d e n t i f i c a t i o n o f momentum  - exponent  R  2  ( f t o r m)  n  N  (Kv /2g)  coefficient  state  torque  - estimated  or  m-Vsec)  interval  discharge interval  a i r chamber  pump t o r q u e  orifice  (lb-ft  o r N-m)  o f pump discharge  value  of dimensionless  discharge  - dimensionless  d i s c h a r g e a t end o f t i m e  V  - average value  of dimensionless  V  -  instantaneous  velocity  Vair  -  instantaneous  a i r volume  v0  -  steady  WR  z  2  state  liquid  interval  discharge  i n pipeline  (ft/sec  o r m/sec)  i n a i r chamber  velocity  - moment o f i n e r t i a o f i m p e l l a r - h e i g h t o f a i r chamber  water  (lb-ft  2  o r Kg-m ) 2  s u r f a c e above datum  (viii)  -  known  -  dimensionless  -  estimated  o<p  -  dimensionless  u  - average  ^  -  dimensionless  %  -  average  ?N  -  multiplier  t  -  relative  <x o<  e  NOTE T h e indicates  first  time  interval pump  value  of  speed dimensionless  pump s p e e d a t  dimensionless torque  of  dimensionless used or  in  at  the  in end  time  interval  speed  torque  method  P  end o f  speed  pump  dimensionless  subscript  conditions  pump  pump  of  of  pump  characteristics  valve  opening  double of  the  subscripted  time  interval.  variables  1  CHAPTER I . INTRODUCTION Pumping increasing past,  have n o t  were r e q u i r e d  inconvenient  complex  rather  than  The  are  relying  developed  are  decrease  r a t e of  affects  the  standard  metal  withstand  that  most In  liquid  are  thicknesses  utilities  for  more  engineer  will  control,  ingenuity. Capabilities analyse  these to  change,  as  surge.  to  i n component transient the  this  directly systems  waterhammer,  p a r t s are  sufficient  pressures. case  against  o b j e c t i v e i s to  In s m a l l  control  will  systems.  protect  velocity pressure  to  free  As  transient  the  i s not  performed  f u t u r e , the  cases,  however, t h i s  the  failures.  methods  taken  In  pressures.  most  the  ever  predictable. Skilled  catastrophic i n the  with  technical information  entirely  operator  similar.  constructed  working  more f e a t u r e s  appreciable  installations,  and  In  surge  no as to  large controls  added. computer  analyse  program  multistage  controls,  in  extension  t o an  upstream  the  most  using  event  existing  of  a  FORTRAN l a n g u a g e systems,  a  power  into  t o power  failure.  pumping  a downstream failure  system w i l l  is  with  program w h i c h a n a l y s e s  pumps d i s c h a r g i n g  extreme  the  pumping  T r a n s i e n t s subsequent the  or  behind  precautions  and  sufficient  to ensure  on  the magnitude of  special  for  and  to e f f e c t i v e l y  philosophy  waterhammer  to  had  developed  upon t o d e s i g n  have t o be  A  lengths  outages  systems  called  must be  designed  s y s t e m s whose o p e r a t i o n was  operators  be  being  complexities,  engineers  design  of  systems are  developed  appropriate The  the  work  i s an  transients  reservoir.  a t a pump a r e experience.  usually Pumps i n  2  series  separated  system  developed.  transient the  This  pressures  same l o c a t i o n ,  Justification of  by some s i g n i f i c a n t  pipe,  station  by  intermediate  pipe  and  cost  to s e r i e s  or d i v i s i o n  should  o f two  i n the size  o f the system  series  which  into  Any  theme  r e d u c t i o n s by  reducing  calculation  technique  lower  pumps  solved  in  by o t h e r s .  by a  length  together  separate  reduction  is  i s intended  the  i n one systems  a possible large saving i n  of large control  This multistage  experience  pumps s i t u a t e d  costs.  dominate  pumps s e p a r a t e d  sump, stems from  or  analysis  length  w h i c h h a s been e f f e c t i v e l y  control  significant.  for  the s i t u a t i o n  surge  thickness  computer  than  opposed  pumping an  configuration  f o r a system w i t h . t h e  as  pipe  in  wall  s t r u c t u r e s should  developed to permit  transient  pipe  and  tests  be a  the p o s s i b i l i t y  pressures  along  a  pipeline. The  characteristics consist with  developed  a  controls  in this  i t should  check  on  characteristics.  study  easily  is  by V . L . S t r e e t e r .  o f v a l v e s , vacuum b r e a k e r s ,  other  reason,  as p i o n e e r e d  employed  added  the  method  System  transients  reservoirs;  as t h e y  The program  f o r design in  controls  a i r chambers and develop.  i n c o r p o r a t e s many a s s u m p t i o n s .  n o t be used  of  systems  purposes, with  For  this  b u t r a t h e r as  known  component  3  CHAPTER I I . GENERAL THEORY In  this  describing  chapter,  a  unsteady  flow  These e q u a t i o n s pair  of  general  numerical i n closed  solution  is  yet  equations  available.  can  be  e q u a t i o n s by t h e  transformed  equations  2.1  then  be d e s c r i b e d .  equations)  form  a  e q u a t i o n s , t o w h i c h no  However,  transformed  method  equations  of  these  into  partial  four o r d i n a r y  characteristics.  integrated  to  yield  The  finite  e q u a t i o n s w h i c h c a n be m a n i p u l a t e d n u m e r i c a l l y .  CHARACTERISTIC EQUATIONS The  of  are  the  conduits w i l l  differential  differential  difference  to  ( t h e momentum and c o n t i n u i t y  hyperbolic partial  differential  solution  continuity  and momentum e q u a t i o n s a r e e x p r e s s e d  two d e p e n d e n t v a r i a b l e s ,  elevation, along  the  and  pipeline.  assumptions, motion. grade  two  velocity  independent Wylie  develop  and  simplified  Converting the l i q u i d  line  within  above an a r b i t r a r y  the  pipeline  liquid  to  h a s no l a t e r a l  be  and  hydraulic  variables, Streeter* , 0  equations  time by of  assuming  negligible  displacement,  and  grade  line  and d i s t a n c e incorporating  c o n t i n u i t y and  pressure to e l e v a t i o n datum,  i n terms  of hydraulic  density assuming  t h e momentum  changes that the  equation  is  e x p r e s s e d as L]_=g (c5H/ax)+aV/dt+ ( f / 2 D ) V | V | = 0 and  the c o n t i n u i t y  equation  (2.1)  i s e x p r e s s e d as  L =cWc)t+ (a /g)dV/dx=0.  (2.2)  2  2  The  absolute  ensures  that  value  sign  the f r i c t i o n a l  on t h e v e l o c i t y force  always  term  opposes  i n equation the  2.1  direction  of  liquid  motion.  Substituting of  equations  in  L=Li+  Q/A f o r V and t h e n  2.1 and 2.2 u s i n g  taking a linear  combination  an unknown m u l t i p l i e r  "X, r e s u l t s  L-2 o r l Q + ( A a ) ^ Q + A g A i H + ( l / > v ) £ H +(f/2DA)Q|Q|=0. dt ax ^>t ax  (2.3)  2  Since  H and Q a r e f u n c t i o n s o f x and t , t h e t o t a l  H(x,t)  and Q ( x , t )  may be e x p r e s s e d a s : dH^^H+^Hdx^ d t a t dxdt*  d_Q ^Q a2dx =  dt  +  dt <3xdt  In n o t i n g  that  identified  derivatives of  (2.4)  i f dx/dt=l/A=>\a , two p a i r s 2  by c o m p a r i n g  equation  2.3 w i t h  of equations equations  can  be  2.4:  dQ+qAdH+fQ|Q| n d t a d t 2DA  (2.5)  =  C+ (2.6)  dx/dt=a  if and  dQ.qAdH+fQlQUn dt  a dt  (2.7)  2DA  >  (2.8)  dx/dt=-a.  if Thus,  in  specifying  two  differential  equations  differential  equations.  The  solutions  independent (Fig.  +  for  and C~)  initiated  real  a  values  have been c o n v e r t e d  to  these  (x,t) variable  2.1), because  constant (C  C"  the  given  represent  equations, plane,  pressure pipe. the  w i t h i n the system.  plot wave  These travelling  of  A, t h e two p a r t i a l  into  as as  four  pictured straight  velocity  ordinary  on  the lines  is  usually  characteristic  lines  of  pressure  surges  5  —  A X  — —  i—  f  ^ t  1  P  0  B  A  i+1  CHARACTERISTICS GRID FIGURE  2.1  n+1  x  6  2.2  FINITE DIFFERENCE Integrating  respectively, constants  equation N  a  equation  finite  along  lines  AP  form  and  difference  and  isolating  and (2.10)  P  in  which:  Cp=Q +C H -(fAt/2DA)Q |Q |  (2.11)  C = Q - C H - ( f At/2DA)Q IQgI  (2.12)  C =gA/a.  (2.13)  N  a  B  a  A  A  B  A  B  a  The  values  grid  points  of  conditions.  2.10  yields  position  the  A and  state  dependent  B since  the  A simultaneous  the  conditions  designated  by p o i n t  variables analysis solution  at P.  H and Q a r e initiates  of  from  equations  a particular  known  time  P  N  2.9  and  determined  Thus, (2.14)  become and  the  so o n .  manner, at  the  the  characteristics available, introduced intervals. chapter.  from  the  end o f  not to  both. complete  These  the  time  (A a n d B)  boundaries  grid),  either  conditions  known v a l u e s At  and  spatial  P  a n d Hp c a n b e g e n e r a t e d this  at  steady  Q =0.5(C +C )  In  BP,  (2.9)  2.7,  A  the  to  and 2 . 7  P  2.5  Qp=C +C H  and  2.5  yields^: a  for  equations  converting  Qp=Cp-C H for  EQUATIONS  for  2.9  or  interior These  the  2.10. points  equation  time  1 a n d n+1 2.9  or  end c o n d i t i o n s  during  can  conditions  subsequent  positions  special  solution will  all  step.  either  Therefore,  conditions  at  (spatial  however,  the  equation  the  then step, on  the  2.10  is  must  subsequent  be d i s c u s s e d i n  be  be  time  following  7  2.3  STABILITY REQUIREMENTS Given  that  velocity,  the  stability  This  criteria  point  P must n o t  For  complex  in  series,  for  each  obtained number  at  the  specifies fall  i s much l e s s  solution is  assured^  that  outside  piping  This the  the  of  systems  of  the  that  ensures  probably  analysed.  adjustment allowable  cannot  Therefore,  i s not  1 0  Careful  line  the  that The  .  met  the  great  lines  for  the  length  will  the  wave speed  of  2.1).  one  conduit  be  constant  and  N  is  indicate that  majority  of a  accuracy,  to m a i n t a i n  are  solutions  wave speed w i t h i n  degree  through  solutions  most a c c u r a t e pipe  wave  (Fig.  increment  compatible  the  At/^x<l/a.  segment AB  time  reflection  since  known t o any (+15%) o f  be  if  i n v o l v i n g more t h a n  junctions.  reaches.  than  characteristic  i f /£>t = L / ( a N ) , where L i s t h e  condition  pipe  of  velocity  i t i s necessary  pipe.  generated  liquid  are the this  systems  particular a  a constant  slight ^t is  8  CHAPTER I I I . BOUNDARY CONDITIONS Waterhammer conduits rapid  i s defined  above  changes  or  i n the  o f pumps i n i t i a t e of  valves.  can  be  waterhammer  reduced  by  or  stopping  opening  and  closing  Slow c l o s i n g  3.  Pressure  which a l l o w  flow  the  opens  valve  valves,  the  pressure 4.  except  the  pressure  a pump  valve,  failure  can  or  the  produce  changes  can  be  f o l l o w i n g means:  to m a i n t a i n  low  steady  i f loss  state  of water  velocities. and  reverse  objectionable.  regulating  valves  bypass around  valve  starting  discharge  power  drastic  by  sudden  rapidly only  or  pressure  a pump. W i t h  and  opens as  then  relief  regulating  valves valves,  c l o s e s s l o w l y . With  f a r as n e c e s s a r y  relief  to minimize  the  surge.  Use  latter  during  the  control valves,  not  The  produced  opening  These  2.  flow.  waterhammer  slowly  System d e s i g n e d  of  c h a n g e s , as does t h e  s e v e r a l of  are  closed  and  1.  pump r o t a t i o n  in  starting  results. one  in pressure  velocity  produced  destructive  change  by  such  by  the  below n o r m a l c o n d i t i o n s c a u s e d  Whereas t h e  minimized  as  o f a i r chambers, a c c u m u l a t o r s ,  are  the  where t h e y  5. When breakers  separation  are  water  are  w a t e r column pressure  most  at  expensive the  rejoin. or  to c u s h i o n be  preset  from o c c u r r i n g .  the  used  surge  tanks.  h a r d l y ever  The  justified  transient control.  separation  They can  above  are  o n l y means o f  column  installed  and  or  is  unavoidable,  shock as  the  to m a i n t a i n  limits,  thus  vacuum  parts of the  the  pipeline  preventing  column  9  Pumps, the  reservoirs  boundary  boundary the  adjoining 2.10.  reaches to  specifies  This  chapter  boundary  3.1  during the  boundary  which  included  handled with  response  Each  appropriate  conditions  conditions  system  and  is  devoted  condition  to  this  transient  is  then  For  all  characteristics, conveyed  from  the  either  equation  2.9  or  requires  a second  between  the  determine  analysis.  of  by  some r e l a t i o n s h i p  devices  method  boundary  condition  in  the  the  control  head  development  and of  equation  discharge.  these  specific  relationships.  BOUNDARY C O N D I T I O N FOR A PUMP Voluntary  assuming  pump s t o p p i n g  the  pump s p e e d t o  stopping  or  must  included  be  torque  in  on a r o t a t i n g  developes flange  to  applied  to  pressure Since  starting  pumping  transient The  shaft,  is  flange.  decreased,  move u p s t r e a m  pertaining 2  Experience  of  2  homologous to  At  power  reducing  these  the from  and d o w n s t r e a m  to  upon  the  be t a k e n  into  pumps**  by e x e r t i n g impeller the  supplies  the  a  which suction  the  TDH a n d in  pump  parameters  failure, the  the  by  torque causing  pipeline.  pump  speed,  account. the  following  turbomachines:  = CONSTANT  with  dimensionless  have  to  increase  head and d i s c h a r g e depend  theory  H/(N D )  (TDH)  if  additional  conveys energy  head  speed changes w i l l  equations  voluntary,  analysed  However,  a n a l y s i s . T h e pump m o t o r ,  discharge  to  be c o n s t a n t l y .  be  the  the  waves  can  not  dynamic  shaft  starting  is  a total  the  and  Q/(ND ) 3  relationships  characteristics  a r e more  = CONSTANT. has  (3.1)  indicated  convenient  and  these  that are  10  defined  as:  h=H/H where and  the  3.2  ^=T/T  R  R refers  and  applying  dimensionless h/oc The terms  them  Marchal,  Flesh of  and  the  Suter^  complete  More  condition In  will  not  suction analysis)  setting using  following  and  both  vp  and  the  for  1  3.1  following  should (where  the  c a n be  handle, pump  by  all  h/(v +c*2) 2  conversion  a given  to  pump a s  two  where: x=TT+TAN  input are  - 1  v/o< .  (3.4)  and  holds  zero  simultaneously.  the  texts  but  this  aspect  by of  for  all  Wylie the  and  boundary  further.  condition  suction suction  the  as  operation.  using  this  for  plot,  may i n c l u d e  to  subscripted  to  for  a pump,  a number  b e c o n s i d e r e d . Some v a r i a t i o n s  long  hp e q u a l  (3.3)  in  with  boundary  situations  equations  use  w h e r e oc a n d v  Chaudhry ,  a  equations the  zone of  problem  1 0  computer  available  the  line  this  Streeter  be d e a l t  combinations  included);  and  by  developing  system short  and  1 0  the  2  is  in  difficult  characteristics  except  information  Streeter  the  and v  results  FB(x)=j9/(v +o<2)  preferable  of  are  co-ordinate  2  is  Combining  = CONSTANT.  upon  avoid  and  FH ( x ) = h / ( v + o < 2 )  form  unit  (3.2)  R  relationships:  depending  on a r e c t a n g u l a r  values  one  as w r i t t e n  h/o<2. w y l i e  represent  This  to  v/<*  sign  o<=N/N  R  quantities.  = CONSTANT  2  relationships  instead  rated  homologous  may c h a n g e  curves  to  v=Q/Q  R  line  c a n be  line pumps  in  it  indicated  written : 1  in  at  Figure  a in  must  parallel.  dimensionless values  notation  are:  neglected  (where  of  be By  the  pump  3.1,  the  i,n  i,n+1  i+1,1  i+1,2  PUMP BOUNDARY CONDITION WITH CHECK VALVE FIGURE  3.1  12  - 1. Pump c h a r a c t e r i s t i c  equation  h =K! (°<p +vp ) 2  (3.5)  2  P  where  i s determined  subscript interval  P p e r t a i n s to  from t h e pump c h a r a c t e r i s t i c s conditions  the  end  of  the  time  under c o n s i d e r a t i o n .  2. Head d i f f e r e n c e Hp=Hp  equation  -Hp i+1,1  a t t h e pump  +(check  v a l v e head  hp=h  as (3.7)  2  0  = C  i+1,1  N  R  characteristic + C  a  equation  P  where C = C / Q N  5  3  5  8  as (3.9)  P  and C = C  R  line < - )  H  =C +C h i+1,1 4  i+1,1  f o r the discharge  P i+1 i + 1 , 1  w h i c h may be n o n - d i m e n s i o n a l i z e d  4  (3.6)  -hp +KV /2gH . .i+1,1 i,n+l P  3. N e g a t i v e Qp  loss)  i,n+l  w h i c h may be n o n - d i m e n s i o n a l i z e d  v  at  and t h e  H /Q .  a  R  R  i +1 4. P o s i t i v e Qp  characteristic  =Cp-C  a  i,n+l  equation  =C -C h  P  6  7  (3.10)  6  P  as (3.11)  P  i,n+l where C = C / Q  line  Hp i i,n+l  w h i c h may be n o n - d i m e n s i o n a l i z e d v  f o r the s u c t i o n  i,n+l R  and C = C 7  H /Q .  a  R  R  i For the  systems  consisting  o f pumps i n p a r a l l e l ,  pumps and t h e d i s c h a r g e m a n i f o l d  these  pipes are short. Equation n vp p  =C4+C h i+1,1 i+1,1 5  p  the pipes  c a n be n e g l e c t e d  3.9 s h o u l d  then  between  providing  be w r i t t e n a s (3.12)  13  and  equation  3.11  n vp  should  be w r i t t e n  as  =Cg-C7hp  p  i,n+l where np e q u a l s  t h e number  5. F l o w c o n t i n u i t y vp  (3.13) i,n+l of p a r a l l e l  pumps.  t h r o u g h t h e pumps  =vp  (3.14)  i,n+l and  vp=Vp  .  (3.15)  i + 1,1 6. Speed By  using  3.2,  change e q u a t i o n  the  f o r t h e pump  non-dimensional q u a n t i t i e s  t h e change  in rotational  specifies  s p e e d c a n be w r i t t e n  by  equation  as  p>=- (2irWR /60g) ( N / T ) dcx/dt.  (3.16)  2  R  Converting  this  into  R  finite  difference  form  yields  Ao<=c p  (3.17)  3  where JB i s t h e a v e r a g e o f t h e known t o r q u e  at the  the  t i m e A t , and  t i m e s t e p and  t h e unknown t o r q u e  after  beginning  C =-(30gAt/WR )(T /N ).  (3.18)  2  3  For in  R  3.13,  R  the system c o n s i s t i n g  parallel, 3.14  simultaneous  and  3.15  where line  1  P  equations  and  3.5  3.19, b u t C i = l / C However, initially information  thus  and 5  unknown, for  2  3.5,  with  pumps  3.7,  3.12,  (3.19)  2  1 (  0  1  I f the  C2=(C4C7+C5C6)/C5C7.  neglected,  3 . 1 2 would  and  since  of equations  line  ( K * p + C 2 - K V / 2 g ) ]/2K  1  and  C2=(C5+C7)/C5C7  i s short  suction  yields 2  P  of a long  solution  vp=[C n -/(C n ) -4K 1  of  yield  simultaneous  suction  solution  vp as s p e c i f i e d  by  of  equation  C2=C /C . 4  5  depends equation  solution.  upon 3.19  o<p would  Chaudhryl  and  vp,  require  presents  a  which  are  additional predictor-  14  corrector  iteration  procedure  for this  purpose. Given  and u a r e known a t t h e b e g i n n i n g o f t h e extrapolation  °<e  e  subscript  refer  3.19.  The  greater value  estimated  of  v  than a s p e c i f i e d calculated  determined  e  as  If  p  as f o l l o w s .  3.17  l°<p-«* l e  i s used  i s greater  from  i s less  v  than  from  equation  and i f I v - v I i s P  e  than  Kp+»< )  . 5 A < X J  the  a  L  _  and v p . Then t h e t h e t o l e r a n c e , <*  P  (3.21)  1  pump  specified  a g a i n . O t h e r w i s e , o<p and v  the  specified  tolerances  e  estimated  characteristics.  t o o b t a i n Ao( and <*p i s computed  e  pump  Setting  over  E  I  e  of  time  the  ( s a y 0.001), a new  average  v =Vp,  °< =0.5  be  from  and vp a r e compared  set  e  the p r e v i o u s  calculated  * = < * + 0  t o be o b t a i n e d  during  to  the  P  equation  (3.20)  yield  tolerance  v=0.5(v +v) allows  i-l  values  i s repeated. I f Ivp-v  procedure  linear  t o e s t i m a t e d v a l u e s ; A V J _ 1 and  refers  and a l l o w v  values  is  = 0 < + A o <  increments determined  These  characteristics  is  e  t o v and  interval.  interval,  v, h  gives  v =v+AVi_i where  time  that  tolerance  and do t h e whole p  as c*+Ac*.  ( s a y 0.001),  iteration  have been d e t e r m i n e d  and Hp c a n be c a l c u l a t e d  Now,  procedure to  from  within  equation  15  3.2 CHECK VALVE BOUNDARY If the  a check v a l v e  the crude  constant  for  t h e pump  just  3.3  through  is  The d i s c h a r g e  isolating  the  this  head d i v i d e d by t h e r a t e d  head  the  a t the closed  valve  which  f o r the  is  case  valve is  ignored where  to  closes  then  in  the  power  set  further  valve  a t t h e s u c t i o n s i d e o f t h e pump e x c e e d s  of  to reopen  made  subsequent  that  pump  Consequently,  the  check  the valve  f o r the check v a l v e  valve,  and  a  allow  i s presented  is that  p r o v i s i o n should  forward  flow.  A  be flow  i n F i g u r e 3.2.  PRESSURE REGULATING VALVE (PRV) The  which  pressure  i s operated  regulating by a  valve  servomotor.  is a pilot It  is  downstream o f t h e pump o r pump c h e c k v a l v e and  is  form  t h e pump r e v e r s e s  assumption  downstream  chart  loss  R  and t h e head  included  t h e head  2  calculations. closed  open. F o r such  and i n d i m e n s i o n l e s s  1 0  state velocity  0  the  zero,  flow  pump,  (KV /2gH ).  instantaneously. at  side of a  o f t h e pump keeps t h e v a l v e  a l l forward  When t h e f l o w failure,  at the discharge  a s s u m p t i o n may be made t h a t  becomes t h e s t e a d y of  i s placed  normal d i s c h a r g e  a valve,  CONDITION  discharges  Subsequent rapidly certain increase  to  t o power  and that  atmosphere failure  or  the  installations  1 0  danger .  Figure  is  installed  (provided into  not  of  Care should  opened  water  to  column  3.3 shows t h e  one e x i s t s )  valve  be e x e r c i s e d  soon,  as  separation  subscripted  valve  directly  the s u c t i o n  a t t h e pump m o t o r , t h i s  then c l o s e s s l o w l y . the v a l v e  back  controlled  line. opens t o be  this  may  in  some  notation  at  Check Boundary  Valve Cond i t i o n  YES  Open Valve  Call Pump Subroutine  Q=n  P  V QR P  YES Set  Q=0  NO C a l c u l a t e Hp From E Q . 3 . 8  (^CONTINUE J  CHECK V A L V E BOUNDARY C O N D I T I O N FIGURE  3.2  17  this  b o u n d a r y c o n d i t i o n and a l s o shows t h e r e l a t i v e  the  and  pump, PRV and c h e c k  valve.  Assuming  between t h e pump and t h e PRV t o be s h o r t  thus Q  the  pipe  neglected, =n Q  P  P  P  i+1,1  p  continuity  at this  section gives  -Qp v  (3.22)  where t h e s u b s c r i p t s p, v and i+1,1 r e f e r section, this  respectively.  section  is  characteristic at  positions of  The n e g a t i v e  expressed  equation  t h e pump d i s c h a r g e  by  head  a t the s u c t i o n f l a n g e during  the  head  loss  characteristic  equation  i s expressed  i s t h e head  through  the  t o pump, PRV and  steady  equation f o r  and  by e q u a t i o n  developed  check  3.8  pipe  the  pump  3.5. The head  by t h e pump p l u s t h e  state conditions  valve  (provided  minus  t h a t one  exists): Hp  =h H +H p  R  s u c  -KV  2 0  /2g.  (3.23)  i + 1,1 o<p and vp a r e s t i l l similar  iterative  boundary  unknown and  procedure  condition-'-.  illustrated  as  Estimating  previously Since  the values  side  3.23  are  of equation  that  be  computed  from  equation  for  f o r <Xp  t o be o b t a i n e d  and from  by  known, Hp  3.8 and Qp  a  t h e pump vp  as  t h e pump  o f a l l v a r i a b l e s on t h e  i+1,1 be  determined  proposed  values  allows  characteristics.  will  right  i s known, Qp can i+1,1  determined  from  v Qp =tpQn/Hp v tp  J  /H .  (3.24)  0  i+1,1  i s the r e l a t i v e  valve  opening  a t t h e end o f t h e t i m e  and QQ i s t h e s t e a d y s t a t e d i s c h a r g e IQp ( n p V p Q - Q p )| i s l e s s than i+1,1 v _  R  interval  under a head o f H Q . NOW, i f some specified tolerance  18  DATUM  NOTATION AT PRESSURE REGULATING VALVE-PUMP SYSTEM FIGURE 3.3  19  (say 0.001) the speed  i s obtained  presented value  estimated  for  of v  by  the  i s taken  e  applying pump  i s assumed  e  value, v , the  same  boundary  t o be  equal  as  v  procedure  Now,  is  check  valve  steady  state discharge  where AQ  = c  and  A  2  to  (Qp  -Qp  the  the 3.26  this  v  R  and  the  the  PRV  by  closing  by  3.25 0  to atmosphere.  i s given  The  by  and  other  i s the  coefficient  v a l v e opening  (Ay)  .  valve  opening  as T ^^Ay/C^An, and  dividing  in  t / H p V H o </ i+1,1  chart  (3.27)  simultaneously v  2  c  is  (3.26)  results  equation  (including  the  i+1,1  =0.5(-C +yc  flow  new  v  isolated  discharges  f o r any  P  which C y = ( Q r j t ) / H o a •  valve  PRV  through  discharge  relative  = Q  Qp  A  been  o f v a l v e opening  =C A /2gH y  i+1,1  3.8.  )/npQ  a  (3.25)  area  d  Qp  in  was  H  i s the  i+1,1  Solving  that  d oJ 9 0  Qp  equation  pump has  that  o f d i s c h a r g e . The  Defining  pump  repeated.  assume t h a t t h e  Qo  the  c o n d i t i o n . Otherwise,  i+1,1 procedure  and  p  2 v  with  equation  3.8  gives  +4C C ) v  Now,Hp can  f o r the  check v a l v e  (3.28)  P  be  determined  operation of  the  from  equation  pressure regulating  influences) i s presented  in  Figure  20  PRV B o u n d a r y Condition  YES  NO  Time to Open PRVZNO  YES NO  YES  Open PRV  NO  F i n d PRV T a u A f t e r t\t  NO  C a l l Pump Subroutine  YES Y  E  S  /  Check^ Closed?.  YES  NO NO  C a l l Pump Subrout ine  NO c =(Q r) /H C 2  v  0  0  YES Set Close  Q=0 Valve  C a l c u l a t e Hp F r o m E Q . 3.8  (^CONTINUE  Calculate From EQ.  J  P R E S S U R E R E G U L A T I N G V A L V E BOUNDARY C O N D I T I O N FIGURE  3.4  Qp 3.28  a  21  3.4 AIR CHAMBER AT THE MAIN PUMPING STATION An  a i r chamber  i s a closed v e s s e l with  liquid  i n i t s lower  liquid,  i n which case  replace  that  separated It  is  predetermined column  from  while  very  orifice,  to  the discharge  line  little  low  used  the  type  effective into  throttling  the  to  o r t h e a i r may be piston  exceeding  pressures,  or  cost  i f the reverse chamber  1 0  . a  water  this  same f l o w  throttled,  is  the  differential  o f c o n s t r u c t i o n may be p r o h i b i t i v e f o r  When  tested  of o r i f i c e  in  was found  the  laboratory,  t o g i v e a head  this  loss  as compared  ratio to the  l e a v i n g t h e chamber^.  i s common p r a c t i c e  chamber  Subsequent  to prevent to  power  t h a t assuming  power  failure  liquid  flow  flow  a check v a l v e through  t h e pump  upstream  of  (Fig.  3.5).  interruption,  the v a l v e c l o s u r e time  i s so  the check v a l v e  to close spontaneously  upon  is justified^.  i s either  of liquid  Assuming  to provide  reverse  into  w i t h i n t h e chamber  direction  is  flow of  f o r flow out o f the  2.5:1 f o r r e t u r n f l o w e n t e r i n g t h e chamber  short  provided  from  i s provided  to accomplish  a i r chambers.  It  be  membrane o r a  pressure  prevent  most  but  particular  air  and  the  are  chamber. A d e v i c e  the  by a f l e x i b l e  prevent  chambers  liquid  of  must  d i s s o l v e s i n the l i q u i d ,  to  value  compressor  separation.  These  small  an a i r  the l i q u i d  designed  i t s t o p and  p o r t i o n . The a i r may be i n c o n t a c t w i t h t h e  which  from  a i r at  T h u s , t h e pump i s i s o l a t e d  or out o f the  chamber.  The  expands and c o n t r a c t s d e p e n d i n g  and t h e trapped upon t h e  flow.  that the pressure-volume  changes  f o r the a i r i n the  7 AI R CAMBER  1,1  CHECK VALVE MAIN PUMP DATUM  AIR CHAMBER AT THE MAIN PUMP FIGURE  3.5  23  chamber  follow  Hp  Vp  and Vp  for  a  perfect  gas ,  then  1  (3.29)  are  the  absolute  pressure  head  and  is  often  the  volume  of  air  enclosed  air,  1.2  is  relation  air  air  be  polytropic  =C.  n  air Hp  the  and  n  G is  functioning If  flow  following  is  an e x p o n e n t  a constant  under  is  known  whose v a l u e  which steady  assumed p o s i t i v e  equations  is  can  determined  state  out  also  assumed  while  the  to  system  conditions.  of  the  air  be w r i t t e n  for  chamber,  the  the  enclosed  air  volume: Hp  =H  p  air V  P  +H -z-H 1 , 1 b  =V  a i r  =C  o r  (3.30)  p  orf  +0.5At(Q +Q)  (3.31)  p  air Hp Here,  orf  H5 i s  chamber  fQp  the  orf  I.Qp  barometric  water  surface  (3.32)  I-  orf  pressure  above  head,  datum  and  z is  the  Hp  is  height the  of  the  head  loss  orf through  the  orifice  for  a given  Qp  and  head  loss  coefficient  orf C  o  r  f.  This  orifice  major  situations:  C  for  o  r  f  where  C  o  r  a  f  for  = C  inflow in  (i.e.  include  bend as  coefficient  orifice o  r  f  and  for  = 2.5  the  negligible  ignored  pipeline,  no  inflow  Fluctuations have  loss  times  these  between  losses  C  f  r  o  r  water  upon  z = constant). losses  o  = 0,  outflow  chamber  effect  C  c a n be c a t e g o r i z e d  Also, the  may w e l l  and f  for  a plate  care air  three  orifice  a differential  where orifice  outflow.  surface  future  by  are  assumed  calculations should chamber  exceed the  be and  losses  to  and  are  taken  to  the  through  main the  24  orifice. Using with  equations  the negative  five  F  3.30,  characteristic  equations  variable  3.29,  and  five  substitution,  equation  unknows.  these N  ] [V  b  conjunction  (Eq. 3.8)  By e q u a t i o n  equations  l = [ d / C ) (Qp-C )+H' -z-Hp a  3.31 and 3 . 3 2 i n  results  c o m b i n a t i o n and  c a n be s i m p l i f i e d  to  + * t (Qp+Q)/2] - C = 0  (3.33)  n  a i r  in  orf which  is  a  non-linear  Newton-Raphson  method  estimated  of  value  equation is  used  Qp b y u s i n g  in to  the find  variable a  Q .  The  p  correction  t o an  the expression  F]+(dF /dQ )AQ=0 1  (3.34)  p  where d F l = l_ v i r dQ C  + A  a  P  a  Consequently, time  where  Qp i s  A flow  3.5  equations  for  only)  a specified  by A Q a f t e r  the a i r  chamber  is presented  breakers  are  subatmospheric  pressure  at  drops  the  valve  the p i p e l i n e  vacuum b r e a k e r The but  than  (3.35)  p  iterated  tolerance  (say 0.001);  each u n s u c c e s s f u l iteration  until  iteration.  process  (upstream  in Figure 3.6.  VACUUM B R E A K E R S Vacuum  in  r  3 . 3 3 , 3.34 and 3 . 3 5 a r e  less  incremented  chart  boundary  +  as A Q i s  such  nCAt 2(V,,- +At(Q +Q)/2)  t(Qp Q) 2  for  in  and open to admit  below atmospheric  pipelines  simply  acts  as a  condition  purposes  of  for this  to  a i r when t h e  pressure.  i n c r e a s e s above the opening  boundary the  installed  As the  setting  prevent pressure pressure  again,  the  vent. these  valves  analysis  is  rather  three  complex,  simplifying  A i r Chamber Boundary C o n d i t i o n Estimate Q =Q P  I Determine A i r Volume From EQ. 3.31  'Flow i n t o ^ Chamber' YES NO  C -f-2.5xC f o r  o r  -  Determine  Hp  orf From EQ. 3. 32  1  F i From EQ. 3 .33 AQ From EQ. 3 .34 dF/dQ From 3 .35 NO Qp=Q +AQ P  YES v  air  = v  air  + 0  - *t(Q Qp) 5  +  CONTINUE  AIR CHAMBER BOUNDARY CONDITION FIGURE  3.6  26  assumptions are  made:  1. Vacuum flanges of 2. As  the  inline the  the v a l v e at  breakers  Once  setting  pressure  drops  volume  of  air  liquid  volume  vacuum  breaker  Since pressure  the  the  the  breaker  above t h e  opening  affecting  the  because  the  is valid,  the  reach  near  the  and  be  given  setting  i s assumed  to  remain  expelled. pump w i l l simply  datum.  a constant  Once  head this  be  known a p r i o r i ,  becomes t h e  height  the v a l v e  i s open,  the  of  i t is  (the pump e l e v a t i o n ) and head. For  (atmospheric  any  pressure  p r e s s u r e ) , the  the  the head  vacuum  i s ignored.  SERIES JUNCTION OF  TWO  PIPES  A boundary c o n d i t i o n f o r the could  pressure  with  system  opening  pressure,  i s u s u a l l y s m a l l when compared  i t can  is calculated  rises  assumption  e l e v a t i o n of each  to m a i n t a i n  discharge above  where  suction  atmospheric  a i r i s released without  a l l o w i n g v a l v e opening  above  assumed  in a given  the  pump.  last  passed  at  atmospheric  a t the v a l v e  trapped  condition. This  below  a i r to m a i n t a i n  pressure  any  positioned  pumps.  sufficient  the  again,  transient  3.6  only  s u c t i o n f l a n g e of the  3.  pump  (booster)  line  passes  are  be  necessary  because  friction  factor,  or w a l l  conduits  and  l o s s e s at the  the  of  series a  junction  change  in  of pipe  t h i c k n e s s . I f the v e l o c i t y j u n c t i o n are  two  assumed  pipes  diameter,  heads  in  the  negligible,  then Hp  =H i,n+l  .  p  i+1,1  (3.36)  27  Since  no f l u i d  relation Q  is lost  =Q  P  Applying  the  continuity  (3.37) i+1,1  =C  - C  P  i,n+l  i  Qp  =C i+1,1  follows  +C  =(C  other  i  to pipe  i+1 (3.39)  i+1,1 manipulation  )/(C  N  i +1  unknowns  equation  Hp  a  i+1  - C  P  i (3.38)  equation  i ,n+l  to pipe  i,n+l  characteristic N  equation  Hp  i+1  from  Hp  a  characteristic  i  the negative  The  the j u n c t i o n ,  P  the p o s i t i v e  Qp  it  at  gives i,n+l  and  or stored  can  +C  a  a  i  that  ). i +1  (3.40)  now b e d e t e r m i n e d  from  equations 3.36  through 3 . 3 9 .  3.7  CONSTANT HEAD R E S E R V I O R A T THE DOWNSTREAM END The d o w n s t r e a m  pipes  is  drains  into  velocity  at  a pipeline  t h e n+1 s e c t i o n  a reservoir  with  head a r e n e g l i g i b l e ,  Hp=H and  end o f  R E S  = CONSTANT  Qp=C -C H P  a  R E S  .  of  with  the l a s t  constant  a specified pipe.  If  number  the  head and t h e e x i t  of  pipeline l o s s and  then (3.41) (3.42)  28  3.8  FLOW CONTROL VALVE AT Transient  c o n t r o l by  precise  knowledge o f  t i m e , as  the  the  valve  a given  valve  opening  has  sectional with  area.  r e d u c e d . The  control  as  reasonable  degree of  For  flow  to  a  valve  t o 35  line the  percent  containing  should of  be  valve  long  p i p e l i n e , the  the  smaller  upon power control  c l o s u r e of  cannot  be  assumed  instantaneous  and  0.75  The pipeline 3.7b).  t i m e . The  s e c o n d s and  control  valve  discharging For  a given  pipe  cross valve  control  so  is  butterfly  p i p e l i n e , the  to  through  valve  is substantially  opening  a short  0.15  upon  butterfly the  to  effective  the  selected  For  from  directly  of  closing  slowly.  closure  respect  that  the  known w i t h  a  accuracy.  through  initial  with  requires  discharge  control valve  instantaneously  this  valves  opening  o p e n i n g . The  a quick  of  function  close  bypassed  15  bypass  valve  a relatively  assumed  valve  to  cost  discharge  control  waterhammer d e p e n d s  installing  the  END  s u b s t a n t i a l l y a f f e c t e d once the  diameter  3.7a),  the  reduced  By  flow  relative  t i m e and  been  a reduced  (Fig.  the  i s only  DOWNSTREAM  means o f  s e v e r i t y of  closure  THE  is  time  the  butterfly  to be  s i t u a t e d at  Qp,  which  the  closure  the  head  closes  made f o r  a straight  constant  valve  may  downstream  range line.  end  of a  head  loss across  be  with a l l  valve  for this  i s assumed  can  failure,  a l l o w a n c e must be  i n t o a r e s e r v o i r with discharge  valve  (Fig.  the  valve  is H =H Qp2/(Q E  0  where H Q i s the discharge  Q  0  a  n  o r )  2  (3.43)  steady d  ^  i s  s t a t e head t  h  e  loss  dimensionless  across valve  the  valve  o p e n i n g . The  for head  a)  CONTROL  VALVE-BUTTERFLY  VALVE  COMBINATION  VALVE LOSS  VALVE LOSS  HG.L.  H  H RES  r  H  r  HRES  FLOW  FLOW CONTROL VALVE  CONTROL VALVE '  DATUM  DATUM  b)  CONTROL  V A L V E AT CONSTANT FIGURE  3.8  HEAD  RESERVOIR  30  at  the upstream  two  f l a n g e o f t h e c o n t r o l v a l v e c a n be s p e c i f i e d  by  cases: 1. F l o w H  P  = H  into  RES  + H  t h e r e s e r v o i r and  e  (3.44)  2. F l o w o u t o f t h e r e s e r v o i r and H =H P  R E S  -H .  (3.45)  e  Substituting  f o r Hp  (Eq. 3.10) i n t o  from t h e p o s i t i v e  equation  characteristic  equation  3.44 y i e l d s  K Qp +Qp-K =0  (3.46)  2  2  where  3  K =HoC /(QQTT) 2  negative  and  2  a  K =C H s-Cp. 3  sign i n the r a d i c a l  a  In  R E  neglecting  t e r m , Qp becomes  Qp=(-1+^1-4K K )/2K2 2  and  the  (3.47)  3  Hp c a n be d e t e r m i n e d from Hp=H  +H Qp|Qp|/(Q t) . e  0  The  absolute  value  head  i s added  or  reservoir, discharge chart  (3.48)  2  R E S  s i g n on t h e d i s c h a r g e  subtracted  respectively.  This  c o n s t r a i n t s imposed  f o r the c o n t r o l valve  f o r flow  term e n s u r e s t h a t t h e into  effectively  by e q u a t i o n s  i s presented  or  out  of the  satisfies  the  3.44 and 3.45. A f l o w  i n F i g u r e 3.8.  31  Control Valve Boundary C o n d i t i o n  Get S l o p e B/F C l o s u r e YES  Determine Tau  ^Tau<TAUMIN  J NO TAU> TAURED?  DETERMINE Tau  NO  YES  Tau=TAUMIN  YES  Calculate C ,K ,K  NO  p  2  Tau<TAUMIN  2  From EQ. 3 .47 Q Hp From EQ. 3 .48  Qp=0.0  P  ^CONTINUE  Hp From EQ. 3.10  FLOW CONTROL VALVE BOUNDARY CONDITION FIGURE  3.8  32  CHAPTER The  program  developed  existing  program  Hydraulic  Transients'  program the  was  required  as c o n v e r g e n c e  proved  pumps  with  conditions  f o r a pumping  long  which  is  user  a computer  input.  These  o r System  analysing  short  suction  i s described  in  of  this  and  some o f  a  system  lines.  to determine  Pumping conditions,  the  The  boundary  detail  within  parameters system  to the  specific  are discussed  examples  t y p e . The  computer. system  entire  be  Prior  parameters  in detail  presented in  (SI) u n i t s may  performed  in this  Appendix  B.  used, with  the  program  i s presented  A.  systems  beyond An  The  MODIFICATIONS  consisting  require  installations.  importance.  conditions  run, though,  the u n i t  J U S T I F I C A T I O N AND  although  when  an  'Applied  modification  designates a l l computations  International  specifying  i n Appendix  4.1  of  existed  used  system  d e v e l o p e d program  c h a p t e r , w i t h pumping English  and  draft  to  3.  t o commencing be  extension  problems  cumbersome  in d e t e r m i n i n g the t r a n s i e n t  must  Some  1  procedure  The  s t u d y i s an  by C h a u d h r y .  iteration  Chapter  in this  PROGRAM  o b t a i n e d from a p r e l i m i n a r y  equations  containing  IV. THE  careful  of long  pipelines  consideration  system  layout,  component  the scope  of t h i s  thesis,  example  importance of these  s h o u l d be  factors.  with high  lift  to ensure  economical  selection  and  are  sufficient  of  costs,  fundamental  to i l l u s t r a t e  the  33  Codrington supply  system  and W i t h e r e l l  pump l o c a t e d  upstream  pressure.  up  The  the higher  the l i n e ,  allowable provided  limits  pressures of  not  transient  the value  be  surge  costly  s o l u t i o n would  installation  The  within  thesis  in  Chapter  as a p a r t  of this  pumping  o f a main pumping This  sumps o r  main pump, vacuum b r e a k e r s pressure  regulating  thesis)  reservoir  have been added system  by less and  b a s i c a p p r o a c h was  take  and  discharging  the  boundary conditions  combine pumping  into a  by a d d i n g  them system  constant  a number o f  s t a t i o n s , an a i r chamber a t t h e  a t the booster  valves,  valving  3 (the boundary  s y s t e m was m o d i f i e d  s t a t i o n s . Then, t o t h e s e  This  to  equipment  o f minimizing the  s y s t e m . The o r i g i n a l station  the  configuration  proposition. A  a i r chamber. T h i s  this  illustrated  s t e p by s t e p .  This  expensive  was  reservoir.  pumping  t o be  method  purpose  of  to  pump p a r t way  be t h e s e l e c t i o n o f a d e q u a t e an  pipe  selected control  One  i s a very  the  costs.  thesis.  multistage  consisted head  This  of  were n p t d e v e l o p e d a  of properly  80  the booster  i n the development o f t h i s  conditions  into  schedule  pipe.  i n pipe  minimize  i s by t h e p l a c e m e n t o f i n t e r m e d i a t e  surge tanks.  followed  40  to  had been t o l o c a t e a  were d e t e r m i n e d  underestimated.  using  the  requiring  schedule  o f a main pump and  line  proposal  I n moving  a s u b s t a n t i a l saving  Similarly, should  head,  pressure.  system  the  original  t h e a n a l y s i s o f a water  consisted  p a r t way up  pump a t t h e w e l l  handle  described  t o a mine. The l a y o u t  a booster  single  2  and  a  pumps,  flow  check  control valve  ( F i g . 4 . 1 ) . Thus, a system was t h e n  analysed.  was  valves, at the built  RESERVOIR CONTROL VALVE  TYPICAL  2-STAGE PUMPING SYSTEM FIGURE  4.1  35  4.2 ASSUMPTIONS The  pumping  stations  in  system  under c o n s i d e r a t i o n  series containing  system  transient  t o be a n a l y s e d . S i n c e  analysis  rather  of  a v a r i e t y of control  T h e s e c o n t r o l mechanisms c a n be added the  consists  than  pumping  mechanisms.  o r removed d e p e n d i n g  this  upon  program was d e v e l o p e d f o r  system  design,  the  following  a s s u m p t i o n s a r e made: 1.  The  failure 2. of  analysis  o c c u r s a t a l l pumps A check v a l v e  a pump c l o s e s  the  (provided  future  The main pump  directly  from  neglected).  is  the  lines  flange  pressure this  at this  elevation.  atmospheric 5.  and  of  pressure  at the d i s c h a r g e  reversal. This  side  eliminates  as  i f  ( i . e . the  i t were  suction  booster  line  discharging i s s h o r t and  s t a t i o n s are modelled  and  once t h e h y d r a u l i c valves  do  not  having  i n computations.  they e x i s t )  pumps  as  are s i t u a t e d  maintain gradeline affect  a t the  atmospheric drops  below  pressures  above  pressure. (provided  immediately  elevation  one e x i s t s )  A check v a l v e  pump. The s t e a d y s t a t e specified  treated  pumping  These  pump o n l y .  closes  upon f l o w  (provided  point  An a i r chamber  upstream  one e x i s t s )  w h i c h c a n n o t be n e g l e c t e d  4. Vacuum b r e a k e r s suction  t o t h e s i t u a t i o n where power  calculations.  sump  A l l booster  suction  only  simultaneously.  immediately  pump from most 3.  long  applies  upon water  above  i s included  power  in  centerline  f l u c t u a t i o n s due t o c h a n g e s  with  failure,  surface the  i s situated  in this  the  near the chamber  eliminating  the  chamber  of level  this  has  t h e pump, ignored.  a  with  36  6.  Head  reservoir 7.  l o s s e s a t the  are  This  separation  study  is limited  occurs.  The  DESCRIPTION OF The  main  this  THE  to cases  occurrence program  f u n c t i o n s of t h i s  1.  Specification  2.  Reading  of unit  input data  to the  the  same t i m e  Initializing  6.  Computation  of constants  7.  Computation  of  8.  S t o r i n g maximum and  9.  Writing transient  control  PROGRAM USAGE AND program  may  divided into  The are first  to  type be  data  contain  maximum  card  the  and  to be  able  to  column  produces  a  handle.  as f o l l o w s :  pipe  data.  f o r each  pipe.  steady  parameters.  state conditions.  transient conditions. minimum p r e s s u r e  conditions after  values.  specified  time  steps.  DATA INPUT will  analyse  pumping  a t most t w e n t y  specified  phenomena  operating  number o f s e r i e s  of u n i t s  water  computer.  device  number o f s e r i e s  the  i s not  interval  as p r e s e n t e d  five;  of t h i s  s y s t e m , pump and  5.  be  downstream  type.  Ensuring  s y s t e m s . The  the  i n w h i c h no  program a r e  4.  The  and  PROGRAM  3. W r i t i n g o f g e n e r a l  4.4  junctions  negligible.  d i s c o n t i n u i t y which  4.3  pipe  (either by  the  read  pipes  pumping  i s t e n and  each  to pipe  reaches.  user. This the  of  stations is restricted  E n g l i s h or  into  a variety  System  International)  i s accomplished  computer. T h i s  words ENGLISH o r SI w i t h i n t h e  first  with  the  card  should  seven  columns  37  of  this  to  card,  beginning  The  remaining  be  examined.  i n the f i r s t  input  parameters apply  should  determine data  area  within  t h e program  to rated  values ER  inertia,  includes  which  of  entry  liquid  to steady  is  held  the  within  parallel  the  head  given datum  and with  by  angular  F correspond  to  pumping  length,  station  discharge  and  the  the  number  FH and FB s p e c i f y f o r each  between  pumping  t h a t a r e t o be  these  points  and HLC i s t h e head  is  above loss  i n t h e s y s t e m , and L, D, A and  diameter,  i s given  pressure  wave  f o r each p i p e .  speed  i s specified  within  and  The l o c a t i o n o f  by t h e number o f t h e p i p e  number 1 i s a t t h e main pumping  a t the  the vector  LOCAT.  Pipe  the  pipe.  This  station.  i s t h e number o f r e a c h e s w i t h i n  should  and  valves.  respectively,  station  curves  and h e a d ,  o f DTH. ELBO i s t h e h e i g h t  t h e number o f p i p e s  factor,  e a c h pumping  term  these  The pump u n i t  is  t o t h e number o f p o i n t s  interval  f o r t h e check  gives  NRLP  of  by WR2. NSPUMP i s  The a r r a y s  The s e p a r a t i o n  f o r each b o o s t e r  friction  the input  impeller  s t a t i o n s and NPPUMP  characteristic  NPC e q u a l  an  coefficient  pump  torque  pump  i t , i s designated  i n each s t a t i o n .  f o r each c u r v e .  NP  is  and NR, QR and HR  pump e f f i c i e n c y .  the motor, the  of  pumps  state values  rated  number o f s e r i e s pumping  input  by e x a m i n i n g  (Appendix A ) . The d e f i n i t i o n s  the  station,  patterns  o f turbomachine speed, d i s c h a r g e  respectively.  mass  that  follow.  NO, QO and HO r e f e r refer  to the system  These p a r a m e t e r s a r e s y s t e m d e p e n d e n t and t h e  user  parameters  column.  be c h o s e n c a r e f u l l y ,  as i t i s used  last  to determine the  38  calculation  time  interval  pipe. Very short is  selected The  be  or v e r y  of  calculated is parameter  chosen  will  result  costly,  in  the  printed  each the its  a  when  NRLP  t r a n s i e n t phenomena  i s to  IPRINT  Both of short  prevent  IPRINT the  printing increments  terms  should  a calculation of  the  period  very  during  a  time  these  record  analysing  Thus,  is  number o f  calculation  generated  are  transient  specified  presence of a valve absence. the at  breakers,  the  pose p r o b l e m s  i n each if  TLAST;  results.  long  results  to  by  the  S e l e c t i n g too  too  v a r i e t y of  indicate  or  reaches  transient could  large  each  time  should  be  period  prove  systems.  iteration  may  given  due  inordinate generation  of  useless  output.  pump  valves  number o f  could  insignificant  significant.  consideration  The  an  while  Similarly,  f o r which  printed  particularly  be  pipes  w h i c h c o n t r o l s the  judiciously.  phenomena,  not  time  specified  between s u c c e s s i v e be  long  thus the  unwisely.  length  control  and  following  of  o f an  by  upstream  outlet the  that  and  a  PRV,  PRV  parameters are  and  available a  inputting a  PRCHK, CHECK and  pressure  valves  VAC  indicates are  only The  a i r chamber, d i f f e r e n t i a l  NDIFF and  does  exist  required  as  flow  used  valves,  e a c h s e r i e s pump s t a g e .  at  *1' i n d i c a t e s '0'  regulating  check  downstream  t e r m s AIR,  mechanisms  indices. Inputting  valves,  r e s p e c t i v e l y , at  chamber  Given  presence  by  or d e v i c e  vectors  regulating  absence  specified  The  control  control  and  to  check vacuum  presence orifice  at  valve  is  NVAL, r e s p e c t i v e l y . at  a  input:  specified  pump,  the  39  PRVTAU NTAU  ..  the  array  containing  ....  the  number o f d a t a  DTIME  ...  time  TOPEN  ...  time at which  Similar control VALVE  interval  gives  DTVAL  ...  p o i n t s on data  the  begins  valve  time  one  TAURED .. e f f e c t i v e  the  f o r these  area  TAUMIN  .. g a t e o p e n i n g when v a l v e  VALOSS  .. s t e a d y  The This not  c l o s u r e of  the  i s done by required)  time a t which b u t t e r f l y  TBFVAL  ..  i s the above  f o r the  mentioned  points points  movement  may  valve  butterfly  closure  curves  noted  the  coefficient  determined  experimentally.  from  valve  the  the PRV  and  discharge  area  f o r the  or  are  of  valve included.  a 0  (closure  of  valve  with  valve  setting  results  closure  4.2  be  close.  a f u n c t i o n of opening.  discharge opening  is and  u s u a l l y has  should shows the  be  the It not the  to  be  available  notation  for  curves.  a i r chamber a t t h e  f o l l o w i n g parameters should  close  valve  given  These  to to  any  manufacturer. Figure  control valve  Similarly, the  the  for  of  butterfly  valve  should  variation  ceases  begins  the  constant  is closed  a l s o have t o be  and  necessarily  valve  the  of discharge that  downstream  curve  coefficient be  the  INCL. Then,  ..  The  for  (closure required)  TCLOSE  time  open  butterfly  valve  inputting a 1 term  to  l o s s through  butterfly  f o r the  curve  points  data  once the  s t a t e head  closure  input  tau  number o f d a t a  interval  curve  exists:  containing  the  closure  the  f o r these  providing  a vector  NVT  valve  p a r a m e t e r s must a l s o be  valve,  ...  the  input:  main pumping  station,  TIME  a)  PRESSURE  b)  REGULATING  VALVE CLOSURE  CONTROL V A L V E C L O S U R E  FIGURE  4.2  CURVE  CURVE  41  HBAR  ....  barometric  CORF  ....  head  loss  pressure  head  coefficient  ALINE  ...  area of  CLINE  ...  head l o s s c o e f f i c i e n t  AIRVOL  ..  steady  EN ...  The  pipe  state  exponent  ZSURF  the  the  for  liquid  of  the  volume  downsurge  to  should  polytropic  air  chamber  charts  4.5  CHECK OF A C C U R A C Y  the  results  was  performed  followed system. for  by  the  E.  E.  pumping  Ruus  system  and  validity  gas  equation above  datum  draining  the  the  prior  using  to  as w e l l  of  the  solution  as  tank  analysis,  of  analysis.  others,  the  program,  present  the  validity  waterhammer  solution  results  generated  for  obtained the  both  programs  computer  generated  These c h e c k s ,  presented  the  during  this  simultaneously developed his  favorable.  the  pipeline  chamber  examined. A g r a p h i c a l  proved indicate  main  purpose.  A comparison of  be  to  chamber  and  development  Ruus  the  maximum p r e s s u r e  3  independently. to  the  control  and R u u s ,  computer  Also,  to  totally  this  the  were  in  determined  and G a l a t i u k  of  pipe  elevation  volume  avoid  for  this  the  Parmakian^,  of  for  surface  out  from chamber  volume  air  be  Upon c o m p l e t i n g  section  flow  air  chamber water size  for  developed  program.  closely  same  pumping  own  program  being  developed  results in  also  Appendix  B,  42  CHAPTER V. This give  discussion  partial  or  deals  inaccurate  this  thesis.  Accurate  t o be  analysed  i s very  physical  constraints  Accurate various  significantly  system pressure  into  should  periods  flanges  the The  system  the  as of  of and  validity  of  analysis  is  Attention  should  be  associated  with  the  results  should  paid  be  applicable  to  interchangeable. is  that  program d e v e l o p e d  pumping  system  system with  Even a s m a l l and  addition  as  that well  in is as  cycle.  To  breakers  wavespeed from  only  characteristics  well  documented  paid  to  problem,  the  as  can  of  the  within  the  given  passage be  method  in  inaccurate  at  the  of  air  avoided.  applied  the  system  low  system  positioned  a i r p o c k e t s can  in  air  solution during  pumps. T h u s , t h e of  of  magnitude  maintain  are  formation  a i r trapped  pocket  the  the  a i r i s released  the  transient  the  will  considered.  booster  the  the  Time c o n s t r a i n t s  timing  the  the  of  of a p i p i n g  vacuum  generate  be  is difficult.  change  characteristics suction  important.  change  p r i n c i p a l f a c t o r s which  r e s u l t s from  f l u c t u a t i o n s . In may  the  simulation  simulation  locations  pressure  with  DISCUSSION  to  literature.  characteristics modelling  will  for a d i f f e r e n t problem. P a r t i c u l a r a t t e n t i o n  to  valve  motions.  a v a r i e t y of One  s y s t e m s , as  advantage of  additional  The  the  as  well  as  program  is  boundary c o n d i t i o n s  are  method  boundary c o n d i t i o n s  they develop. Steady s t a t e  developed  of  can  characteristics  easily  transient  be  added  conditions  as are  modelled. Time  constraints  may  dictate  the  use  of  a  different  43  calculation  technique  to generate  an a c c u r a t e  could  be  f o rvery  long p i p e l i n e s .  representation  prohibitive  when  considering  of  The t i m e  the  required  valve  closure  the required  computer  time. Devices  provided  a r e assumed intended. the  i n the system  t o be p r o p e r l y d e s i g n e d Criteria  which  u s e r . T h i s program  o f low d i s c h a r g e  converge  to a s o l u t i o n .  carried  numerical  o u t by hand;  diverged below  A  from  this  (near The  zero)  problem  solution  of  being  are  design are l e f t to  f o r the  selection  of  i n some s i t u a t i o n s  may  approximation.  No s o l u t i o n  the o r i g i n a l  exists such  in  an  the  occurence  that successive  approximation,  approximation  pump s u b r o u t i n e where t h e program run  proper  n o t be used  the r e s u l t  the o r i g i n a l  problem. Using  pressures  the system.  Periods  subroutine.  transient  and t o f u n c t i o n a s t h e y  constitute  should  t h e components c o m p r i s i n g  not  t o reduce  alternating  above  point  terminates w i l l  was  iterations  has been found at the  pump  and  for this i n the  a l l o w the next  to continue. Air  inlet  incorporated  valves  into  the system  column s e p a r a t i o n . The main 1. may  in  correspondingly The  an  A pump w i t h flow  large pressure  rise.  l e n g t h o f t h e pumping  the pressure  time  required  a i r chamber  the occurrence  i n f l u e n c e s on column  instantaneous  which  upstream  to minimize  Rate o f f l o w s t o p p a g e .  result  2.  and  i s allowed  and  s y s t e m . The t i m e  to decrease  o f water  separationare:  low r o t a t i o n a l  stoppage  were  inertia  produce  a  period during  i s a function  o f the  f o r a p r e s s u r e wave t o t r a v e r s e up and down t h e  44  system. A l l other pipe  the g r e a t e r 3.  Normal  the  pressure  pressure  can  be  Steady  velocity  be  influences  size  a i r entrainment  characteristics representative reason,  and o f the  w a t e r column  this  research.  phenomena  a  points. If  at a p r e s e t  or  limit,  the  i f the water  As  the  steady  vacuous space, the  local  state  the  velocity  pressure  rise  increased.  the  the  interdependent,  system  s e p a r a t i o n should the  with  system. T h i s changes  generated  original  into  the  problem.  and  all  into  longer  t h e maximum d e c r e a s e  o f the  stoppage  are  the  critical  velocity.  the v o i d c o l l a p s e are  These  of  flow  at  maintained  liquid  i n c r e a s e s , the  change a t sudden during  not  equal,  drop.  than  artificially  state  being  pressures  is greater  s e p a r a t i o n would  4.  being  pressure  operating  original  column  characteristics  results being be  system c o u l d  the the  system  no  longer  are  analysed.  avoided.  result  The  For  this  inclusion  produce v i a b l e  future  45  CHAPTER V I . CONCLUSIONS This  program  for multistage air  chamber  within single  was d e v e l o p e d  pumping at  significantly  reduced  minimum extreme  transient Pipe  wall  structures  distributing  pumping  transient  pressures  has been  in  multistage  costs.  within  between  the system.  i s the  which  Thus,  along the  substantiated,  main  system  an  valving to  maximum  This  reduction  advantage  could  pipeline  of  and of  this  expensive  result  in  premise will  subsequent  as was e x p e c t e d .  a  configuration  the  the o r i g i n a l  the  by  system  and t h e s i z e o f l a r g e ,  reduced,  stations  are c o n t r o l l e d  B), the multistage  thickness  may be  savings  a  the d i f f e r e n c e  fluctuations  substantial  failure  (Appendix  analysis  s t a t i o n and a d e q u a t e  comparing  pipeline pressures within  approach. control  main pumping  t h e s y s t e m . When system  the t r a n s i e n t  systems. T r a n s i e n t s  the  stage  to provide  to  a  that  reduce power  46  BIBLIOGRAPHY  C h a u d h r y , M.H., A p p l i e d H y d r a u l i c T r a n s i e n t s , Van Nostrand R e i n h o l d Company, New Y o r k , pp. 1-103,302-331, 1979. C o d r i n g t o n , J . B . , and R.G. W i t h e r e l l , "The Use o f Impedence C o n c e p t s and D i g i t a l M o d e l l i n g T e c h n i q u e s i n t h e S i m u l a t i o n o f P i p e l i n e T r a n s i e n t s " , Second I n t e r n a t i o n a l C o n f e r e n c e on Pressure Surges, Paper A2, The C i t y U n i v e r s i t y , London, E n g l a n d , pp. 15-44, 1976. G a l a t i u k , W.R., A i r Chamber Design Charts, A thesis submitted in partial f u l f i l l m e n t o f the requirements f o r the d e g r e e o f M a s t e r o f A p p l i e d S c i e n c e a t t h e University o f B r i t i s h C o l u m b i a , 1973. M a r c h a l , M., G. Flesh, and P. S u t e r , "The C a l c u l a t i o n o f Waterhammer P r o b l e m s by Means o f the D i g i t a l Computer", P r o c . I n t . Symp. Waterhammer Pumped s t o r a g e P r o j e c t s , ASME, C h i c a g o , 1965. P a r m a k i a n , J . , Waterhammer I n c . , New Y o r k , 1963.  Analysis,  S t e p a n o f f , A . J . , C e n t r i f u g a l and A x i a l J o h n W y l i e and S o n s , I n c . , New York, 1957.  Dover  Publications,  F l o w Pumps, 2nd e d . , pp. 269-292,425-458,  S t e p h e n s o n , D., Pipeline Design f o r Water Engineers, E l s e v i e r S c i e n t i f i c P u b l i c a t i n g Company, New Y o r k , pp. 5383, 1976. S t r e e t e r , V.L., F l u i d M e c h a n i c s , 3rd ed., M c G r a w - H i l l Company, New Y o r k , pp. 343-358, 1962. T u l l i s , J.P. (ed.), C o n t r o l of Colorado State U n i v e r s i t y , Fort 314,543-557, 1971.  Book  Flow i n Closed Condui t s , C o l l i n s , C o l o r a d o , pp. 229-  W y l i e , E.B., and V . L . S t r e e t e r , F l u i d T r a n s i e n t s , McGrawH i l l Book Company, New Y o r k , pp. 1-117,180-189, 1978.  APPENDIX  PROGRAM FOR M U L T I S T A G E  A  PUMPING  SYSTEMS  ANALYSIS OF TRANSIENTS IN A PIPELINE CAUSED BY PUMPS **************************************************************** REAL L,NO(5), NR(5),TYPE*8(2)/'ENGLISH', SI '/,UNITS*8 REAL K2,K3 INTEGER OPEN(5),PRV(5),CLOSED(5),CHECK(5),PRCHK(5),VAC(5),AIR DIMENSION F(10),AREA(10),A(10),L(10),D(10),HMAX(10),HMIN(10), 1 LOCAT(5),NPPUMP(5),HDIS(5),ER(5),WR2(5),SHFTRQ(5),ELBO(5), 2 TM(5),TOPENf5),TMAX(10),TMIN(10),HLC(5) COMMON /AREA1/ FH(5,90),FB(5,90),PRVTAU(5,90),VALVE(90) COMMON /AREA2/ N (10) ,Q(10,20) ,H (10,20) ,CA (10) ,CF(10) COMMON /OUT2/ QP(10,20),HP(10,20) COMMON /AREA3/ ALPHA(5),V(5),DALPHA(5),DV(5),CVHL(5),DTH COMMON /OUT3/ ALPOUT(5),VPOUT(5) COMMON /PRV3/ OPEN,HR(5),H0(5),QR(5),QO,CN,CA2,TAUPRV(5),HSUC(5) DATA PI/3.141593/ ***** DETERMINATION OF UNIT TYPE - INPUT AS 'ENGLISH' OR 'SI' 1  READ(5,1) UNITS,OUTPUT IF(UNITS.EQ.TYPE(2)) GO TO 22 G=32.22 SPECWT=62.4 GO TO 33 22 G=9.820 SPECWT=9802. ' ***** READING OF INPUT DATA 33 READ(5,2) NP,NSPUMP,NRLP,NPC,NTAU,IPRINT,AIR,NDIFF,NVAL,NVT,INCL READ(5,3) QO,DTH,TLAST READ(5,3) (L(I),D(I),A(I),F(I),I=1,NP) DO 44 1=1,NSPUMP READ(5,3) (FH ( I , J ) , J = l,NPC) READ(5,3) (FB(I,J),J=1,NPC) READ(5,3) ELBO(I),NO(I),OR(I) ,HR (I) ,NR(I),ER (I) ,WR2(I),HLC(D READ (5,2) LOCAT(I) ,NPPUMP(I) ,PRV(I) ,PRCHK(I) ,CHECK(I) ,VAC(I) IF(PRV(I).EQ.O) GO TO 44 READ(5,3) TOPEN(I),DTIME,(PRVTAU(I,J),J=1,NTAU) 44 CONTINUE IF(AIR.EQ.O) GO TO 55 READ (5,3) HBAR,AIRVOL,CORF,EN,ZSURF,CLINE,ALINE 55 IF(NVAL.EQ.O) GO TO 66 READ(5,3) TCLOSE,VALOSS,TAURED,TBFVAL,TAUMIN,DTVAL READ(5,3) (VALVE(I),1=1,NVT) 1 FORMAT (A7,1X,A4) 2 FORMAT (1216) 3 FORMAT (12F6.0) ***** WRITING OF GENERAL DATA 66 WRITE(6,10) UNITS,NP,NRLP,NSPUMP,QO,TLAST,IPRINT,NPC,DTH 10 FORMAT('1 ,4X,'*** GENERAL SYSTEM DATA ***'//12X,'UNITS USED IN TH IIS PROGRAM ARE '.A7/12XNUMBER OF PIPES=',13/12X,'NUMBER OF REACH 2ES IN THE LAST PIPE=13/12X,'NUMBER OF SERIES PUMPS=' ,I 3/12X,'SYS 3TEM STEADY STATE DISCHARGE=',F6.2//12X,'TIME FOR WHICH TRANSIENT C ,  40NDITI0NS ARE CALCULATED IS',F6.2,' SEC'/12X,'PRINTING INTERVAL IS 5 EVERY' 12 ' TIME INTERVALS'//12X,'NUMBER OF POINTS ON CHARACTERIS 6TIC CURVES=',I4/12X,'THETA INTERVAL FOR STORING CHARACTERISTIC CUR 7VE=',F7.4//5X,'*** INPUT PUMP DATA ***'/) ***** WRITING OF PUMP DATA DO 77 I=1,NSPUMP WRITE(6,11) I,NPPUMP (I),QR (I),HR (I) ,ELBO (I) ,NR (I) ,ER (I) ,WR2(I) , 1HLC(I),(FH(I,J),J=1,NPC) 11 FORMATC ' , 9X , 'SERIES PUMP #',11/12X,'NUMBER OF PARALLEL PUMPS=', 1I3/12X,'RATED DISCHARGE=',F6.2/12X,'RATED HEAD=',F6.1,27X,'PUMP EL 2EVATION=',F6.1/12X,'RATED PUMP SPEED=*,F7.1/12X,'PUMP EFFICIENCY=' 3,F5.2/12X,'PUMP UNIT INERTIA (WR**2)=',F7.1,1IX,'HEAD LOSS COEFFIC 4IENT FOR CHECK VALVE=',F7.4//12X,'POINTS ON HEAD CHARACTERISTIC' /6 5 (12X,15F7.3/)) WRITE(6,12) (FB(I,J) ,J = l,NPC) 12 FORMAT(12X,'POINTS ON TORQUE CHARACTERISTIC'/6(12X,15F7.3/)) 77 CONTINUE ***** WRITING OF PIPE DATA WRITE(6,13) 13 FORMAT(/5X,'*** INPUT PIPE DATA ***'//12X,'PIPE NO.' , 4 X, 'LENGTH' , 4 IX,'DIAM.', 4X,'WAVE VEL.',4X,'FRICTION FACTOR') WRITE(6,14) (I,L(I),D(I),A(I),F(I),I=1,NP) 14 FORMAT(14X,I3,6X,F7.1,3X,F5.1,5X,F7.1,1IX,F5.3) WRITE(6,15) 15 FORMAT(/12X,'PIPE NO.',SX,'ADJUSTED WAVE VEL',5X,*NO. OF REACHES') ***** CALCULATION OF PIPE CONSTANTS DT=L(NP)/(NRLP*A(NP)) T=0.0 DO 88 1=1,NP AN=L(I)/(DT*A(I)) N(I)=AN *** ADJUSTING WAVE VELOCITY AS NECESSARY IF((AN-N(I)).GE.0.5) N(I)=N(I)+1 A(I)=L(I)/(DT*N(I)) WRITE(6,16) I,A(I),N(I) 16 FORMAT(14X,I3,12X,F7.1,18X,I2) AREA(I)=(PI/4.)*D(I)**2 CA(I)=G*AREA(I)/A(I) CF(I)=F(I)*DT/(2.*D(I)*AREA(I)) F(I)=F(I)*L(I)/(2.*G*D(I)*N(I)*AREA(I)**2) TMAX (I)=0 . 0 TMIN (I)=0 . 0 88 CONTINUE ***** WRITING OF VALVE AND SURGE CONTROL DEVICE DATA 17 FORMAT^/5xl'*** VALVES AND SURGE CONTROL DEVICES ***'//12X,'A ONE 1INDICATES PRESENSE OF VALVE OR DEVICE'//l2X,'PUMP NO.',3X, PUMP LO 2CATION',3X,'PRV',3X,'PRV-CHECK VALVE',3X,'CHECK VALVE',3X,'VACUUM  3BREAKER ,3X,'U/S AIR CHAMBER',3X , 'D/S CONTROL VALVE'/101X, 4I1,18X,I1) WRITE (6,18) (J,LOCAT(J) ,PRV(J) ,PRCHK(J) ,CHECK (J) ,VAC(J) ,J = 1,NSPUMP) 18 FORMAT(16X,I1,12X,I1,10X,I1,11X,I1,15X,I1,14X,I1) IF(AIR.EQ.O) GO TO 99 WRITE(6,19) HBAR,NDIFF,AIRVOL,CORF,EN,ZSURF.CLINE,ALINE 19 FORMAT(/12X,'BAROMETRIC PRESSURE=*,F5.1,57X,'DIFFERENTIAL ORIFICE 1... ',I1/12X,'INITIAL AIR VOLUME IN AIR CHAMBER=',F5.1/12X,'HEAD L 20SS COEFFICIENT FOR CHAMBER ORIFICE=',F6.3/12X,'POLYTROPIC GAS CON 3STANT=',F4.1/12X,'ELEVATION OF CHAMBER WATER SURFACE ABOVE DATUM=' 4,F7.2/12X,'HEAD LOSS COEFFICIENT FROM CHAMBER TO MAIN LINE=',F5.2/ 512X,'AREA OF PIPE FROM CHAMBER TO MAIN LINE=',F5.2) 99 IF (NVAL.EQ.O) GO TO 100 WRITE(6,20) TCLOSE,INCL,TBFVAL,VALOSS,TAURED,TAUMIN,NVT,DTVAL 20 FORMAT(/12X,'TIME AT WHICH BUTTERFLY VALVE BEGINS TO CLOSE=' , F5.?, 131X,'BUTTERFLY VALVE CLOSURE ...',12/ 212X,'TIME REQUIRED TO CLOSE BUTTERFLY VALVE=',F5.2/12X,'STEADY STA 3TE HEAD LOSS THROUGH BUTTERFLY VALVE=',F6.3/12X,'RELATIVE AREA OF 4BYPASS LINE=',F6.3/12X,'MINIMUM RELATIVE CONTROL VALVE OPENING=',F 56.3/12X,'NUMBER OF DATA POINTS ON CONTROL VALVE CURVE=',13/12X, ' TI 6ME INTERVAL BETWEEN THESE DATA POINTS=',F6.3) ***** INITIALIZE VALVE AND SURGE DEVICE CONTROLLING PARAMETERS 1  100 DO 150 I=1,NSPUMP OPEN(I)=0 CLOSED(I)=l TM (I)=0.0 OVERFLOW PRINTING FIELD TO INDICATE ABSENCE OF VALUE. TAUPRV(I)=111111. ALPOUT(I)=llllll. VPOUT(I)=llllll. 150 CONTINUE IF(AIR.EQ.O) AIRVOL = l l l l l l . VALTAU=1.0 IF(NVAL.EQ.O) VALTAU=111111 . TVAL=0.0 RESID=0.0 ***** COMPUTATION OF CONSTANTS FOR PUMPS DO 200 I=1,NSPUMP SHFTRQ(I)=(30.*SPECWT*HR(I)*QR(I))/(PI*NR(I)*ER(I)) ALPHA(I)=NO(I)/NR(I) V(I)=QO/(NPPUMP(I)*QR(I)) DV(I)=0.0 DALPHA(I)=0.0 *** DETERMINE HEAD LOSS AT CHECK VALVES (=K*V**2/2G) NPB=LOCAT(I) IF(I.EQ.NSPUMP) NPE=NP IF(I.NE.NSPUMP) NPE=LOCAT(I+l)-l CVH0=HLC(I)*Q0*Q0/(2.*G*AREA(NPB)**2) CVHL(I)=CVHO/HR(I) ***** CALCULATION OF STEADY STATE CONDITIONS  51  IF(V(I).EQ.O.O) GO TO 210 VO=V(I) ALPHAO=ALPHA(I) THETA=57.296*(PI+ATAN2(VO,ALPHAO)) GO TO 220 210 THETA=0.0 220 CALL PARAB(THETA,1,1,DTH,Z) HO (I)=Z*HR(I)*(ALPHA0**2+V0**2) H (NPB,1)=H0(I)+RESID-CVHO HSUC(I)=RESID HDIS(I)=H(NPB,1) DO 240 J=NPB,NPE NN=N(J)+l DO 230 K=1,NN H(J,K)=H(J,1)-(K-1)*F(J)*Q0**2 IF(J.NE.NPE.AND.K.EQ.NN) H(J+l,1)=H(J,NN) Q(J,K)=Q0 230 CONTINUE HMAX(J)=H(J,1) HMIN(J)=H(J,NN) 240 CONTINUE RESID=H(NPE,NN) 200 CONTINUE NN=N(NP)+1 IF(NVAL.EQ.O) VALOSS=0.0 HRES=H(NP,NN)-VALOSS C AIR CHAMBER CONSTANTS AT STEADY STATE CONDITIONS IF(AIR.EQ.O) GO TO 24 HPORF=0.0 HPAIR=H(1,1)+HBAR-ZSURF-HPORF C=HPAIR*AIRVOL**EN C ***** WRITING TIME,PRESSURES AND DISCHARGES C 24 WRITE(6,25) HRES 25 FORMAT('l',4X,****** HEAD AT THE DOWNSTREAM RESERVOIR=',F7.2) WRITE(6,30) 30 FORMAT(///IX,'SERIES',1X,'TIME',4X,'V,4X,'ALPHA',IX,'PIPE ' , 19X , 1'REACHES ALONG PIPES',44X, PRVTAU',2X,'AIR',4X,'VALVE'/2X,'PUMP', 2114X,'VOLUME',3X,'TAU'/) 300 ICOUNT=0 DO 350 1=1,NSPUMP NPB=LOCAT(I) IF((CHECK(I).EQ.O).AND.(CLOSED(I).EQ.l)) GO TO 320 IF(I.NE.l) GO TO 310 IF(Q(1,1).LE.O.O) WRITE(6,35) I,T,TAUPRV(I),AIRVOL,VALTAU 35 FORMATC ' , 2X , 12 , 2X , F5 . 2 , IX , ' VALVE 1 CLOSED', 87X, F4 . 2, 2X, F6 .1, 2X , F 15.3) IF(Q(1,1).GT.O.O) WRITE(6,40) I,T,V(I),ALPHA(I),TAUPRV(I),AIRVOL 1,VALTAU 40 FORMAT(' ',2X,12,2X,F5.2,2(IX,F6.3),88X,F4.2,2X,F6.1,2X,F5.3) GO TO 330 310 IF(Q(NPB,1).LE.O.O) WRITE(6,45) I,I,TAUPRV(I) 45 FORMAT(' ,2X,I2,8X,'VALVE',12,' CLOSED',87X,F4.2) IF(Q(NPB,1).GT.O.O) WRITE(6,50) I,V(I),ALPHA(I),TAUPRV(I) 50 FORMATC ' , 2X, 12, 7X, 2 (IX, F6. 3) , 88X, F4 . 2) c  1  1  52  C C C  GO TO 330 320 IF(I.EQ.l) WRITE(6,40) I,T,V(I),ALPHA(I),TAUPRV(I),AIRVOL,VALTAU IF(I.NE.l) WRITE(6,50) I,V(I),ALPHA(I),TAUPRV(I) 330 IF(I.EQ.NSPUMP) NPE=NP IF (I.NE.NSPUMP) NPE = LOCAT(I + l ) - l DO 340 J=NPB,NPE NN=N(J)+1 WRITE(6,60) J , (H(J,K) ,K = 1,NN) 60 FORMAT(* ' , 27X,12,IX,'H=',10F8.2/33X,10F8.2) WRITE(6,70) (Q(J,K),K=1,NN) 70 FORMAT(' ' , 30X,'Q=' , 10F8.2/33X,10F8.2) 340 CONTINUE 350 CONTINUE 400 T=T+DT ICOUNT=ICOUNT+l IF(T.GT.TLAST) GO TO 990  C C C  C C  C C C C C C C C  ***** DETERMINATION  OF CONSTANTS AT PUMP BOUNDARY CONDITION  DO 500 I=1,NSPUMP NPP=NPPUMP(I) NPB=LOCAT(I) IF(I.EQ.NSPUMP) NPE=NP IF(I.NE.NSPUMP) NPE=LOCAT(I+l)-l CN = Q (NPB,2)-CA(NPB)*H(NPB,2)-CF(NPB)*Q(NPB,2)*ABS(Q(NPB,2)) C4=CN/QR (I) C5=CA(NPB)*HR(I)/QR(I) *** SHORT SUCTION LINE SUCTION LINE AT U/S PUMP ASSUMED SHORT AND NEGLECTED DATUM ASSUMED TO BE PUMP CENTER LINE IF(I.NE.l) GO TO 510 Cl=l./C5 C2=C4/C5 GO TO 520 *** LONG SUCTION LINE REQ'D TO ANALYSE BOUNDARY CONDITION AT IN LINE PUMP 510 MM=NPB-1 NN=N(MM) CP=Q (MM,NN)+CA(MM)*H(MM,NN)-CF(MM)*Q(MM,NN)*ABS(Q(MM,NN)) C6=CP/QR(I) C7=CA(MM)*HR(I)/QR(I) C1=(C5+C7)/(C5*C7) C2= (C4*C7+(C5*C6))/(C5*C7) 520 C3=- (30.*G*SHFTRQ(I)*DT)/(PI*NR(I)*WR2(I)) ***** PUMPS INCLUDING OTHER BOUNDARY CONDITIONS *** PRV (PRESSURE REGULATING VALVE) AT PUMP DISCHARGE WITH OR WITHOUT CHECK VALVE IF(PRVd) .EQ.O) GO TO 550 CA2=CA(NPB) IS THE PRV OPEN? YES:GO TO (1), NO:GO TO (2) IF(OPEN(I).EQ.0) GO TO 542 (1) HAS THE PRV SUBSEQUENTLY CLOSED? IF(TAUPRV(I).EQ.0.0) GO TO 540 NO, IT HAS NOT.  53  TIME=TM(I) CALL PARAB(TIME,I,3,DTIME,TAU) TAUPRV(I)=TAU TM(I)=TM(I)+DT DOES THE CHECK VALVE EXIST? c IF(PRCHKd) .EQ.O) GO TO 570 IS THE CHECK VALVE CLOSED? c IF(CLOSED(I) EQ.O) GO TO 546 GO TO 552 CLOSE THE PRV. 540 OPEN(I)=0 GO TO 544 (2) IS IT TINE TO OPEN PRV? 542 IF(T.LT.TOPEN(I)) GO TO 544 TAUPRV(I)=0.00001 TM(I)=TM(I)+DT OPEN(I)=l IS THERE A CHECK VALVE AT THE PUMP DISCHARGE? 544 IF(PRCHK(I).EQ.O) GO TO 570 CLOSED CHECK VALVE - FLOW THROUGH PRV? IF(CLOSED(I).NE.O) GO TO 552 QP(NPB,1)=0.0 546 GO TO 580 CV=(QO*TAUPRV(I))**2/(HDIS(I)*CA(NPB)) QP(NPB,1)=0.5*(CV-SQRT(CV**2-4.*CN*CA(NPB))) GO TO 580 *** CHECK VALVE (ONLY) AT PUMP DISCHARGE 550 IF(CHECK(I).EQ.O) GO TO 560 IS THE CHECK VALVE CLOSED? IF(Q(NPB,1).EQ.0.0) GO TO 554 552 CALL PUMP(Cl,C2,C3,NPP,I,T) QP (NPB,1)=NPP*VPOUT(I)*QR(I) CHECK VALVE CLOSES IF DISCHARGE LESS THAN ZERO IF(QP(NPB,1).LT.0.0) QP(NPB,1)=0.0 IF( (PRCHK (I) .EQ.l) .AND. (QP(NPB,1) .EQ.0.0)) CLOSED(I)=0 GO TO 580 C IF HEAD AT PUMP SUCTION GREATER THAN PUMP DISCHARGE C - CHECK VALVE REOPENS 554 IF(I.EQ.l) GO TO 556 556 IF(H(MM,NN+1).GT.H(NPB,1)) GO TO 570 QP (NPB,1)=0.0 GO TO 580 C *** AIR CHAMBER (INCLUDING CHECK VALVE) AT MAIN PUMP C 560 IF( (AIR.EQ.O) .OR. (I.NE.l)) GO TO 570 ITER=0 QP(1,1)=Q(1,1) 562 VPAIR=AIRVOL+0.5*DT*(QP(1,1)+Q(1,1)) IF(VPAIR.LT.0.00001) VPAIR=0.00001 ORIFICE LOSS COEFFICIENT - 3 TYPES C 1) NO ORIFICE - CORF=0 C 2) PLATE ORIFICE - CORF(FLOW IN)=CORF(FLOW OUT) C C 3) DIFFERENTIAL ORIFICE - CORF(Q IN)=2.5*CORF(Q OUT) IF(NDIFF.EQ.O.OR.QP(1,1).GT.O.O) GO TO 564 CORFIN=2.5*CORF  54  C C  HPORF=CORFIN*QP(1,1)*ABS(QP(1,1)) GO TO 566 564 HPORF=CORF*QP(1,1)*ABS(QP(1,1)) 566 HPORF=HPORF+CLINE*QP(1,1)*ABS(QP(1,1))/(2.*G*ALINE**2) Fl=((QP (l,l)-CN)/CA(l)+HBAR-ZSURF-HPORF)*VPAIR**EN-C DF1DQP=EN*DT*C*0.5/VPAIR+VPAIR**EN/CA(1) DELTAQ=-F1/DF1DQP IF(ABS(DELTAQ).LE.0.001) GO TO 568 QP (1,1)=QP(1,1)+DELTAQ ITER=ITER+1 IF(ITER.GT.20) GO TO 980 GO TO 562 568 AIRVOL=AIRVOL + 0.5*DT*(QP(1,1)+Q(1,1) ) IF(AIRVOL.LT.O.O) AIRVOL=0.0 GO TO 580  *** PUMP BOUNDARY CONDITION PUMP(C1,C2,C3,NPP,I,T) C DISCHARGE SIDE OF PUMP QP(NPB,1)=NPP*VPOUT(I)*QR(I) 580 HP(NPB,1)=(QP(NPB,1)-CN)/CA(NPB) IF(I.EQ.l) GO TO 590 C SUCTION SIDE OF PUMP QP(MM,NN+1)=QP(NPB,1) HP(MM,NN + 1) = (CP-QP(MM,NN+1))/CA (MM) C C *** VACUUM BREAKER AT SUCTION FLANGE OF IN LINE PUMP IF(VAC(I).EQ.O) GO TO 590 IF(HP(MM,NN+1).GE.ELBO(I)) GO TO 590 HP(MM,NN+l)=ELBO(I) CP=Q(MM,NN)+CA(MM)*H(MM,NN)-CF(MM)*Q(MM,NN)*ABS(Q(MM,NN)) QP (MM,NN + 1)=CP-CA(MM)*HP(MM,NN+1) C C *** INTERIOR POINTS 590 CALL INTER(NPB,NPE) ALPHA(I)=ALPOUT(I) V (I)=VPOUT(I) 500 CONTINUE C ***** FLOW CONTROL VALVE AT CONSTANT HEAD RESERVOIR C IF(NVAL.EQ.O) GO TO 700 IF (T.LE.TCLOSE) GO TO 650 NN=N(NP)+1 IF(VALTAU.LE.TAURED) GO TO 610 IF(INCL.EQ.O) GO TO 600 C NO INSTANTANEOUS CLOSURE - SHORT PIPE LINE SLOPE=(TAURED-1.)/TBFVAL VALTAU=1.+SLOPE*(T-TCLOSE) IF(VALTAU.GE.TAURED) GO TO 630 600 TVAL=T-(TCLOSE+TBFVAL) C FLOW THROUGH BY-PASS WITH CONTROL VALVE CLOSED SLOWLY 610 IF(VALTAU.EQ.O.O) GO TO 620 IF(VALTAU.EQ.TAUMIN) GO TO 630 CALL PARAB(TVAL,I,4,DTVAL,VALTAU) IF(VALTAU.EQ.O.O) GO TO 620 c  570 CALL  IF(VALTAU.LE.TAUMIN) VALTAU=TAUMIN GO TO 630 620 CP=Q(NP,NN-1)+CA(NP)*H(NP,NN-1)-CF(NP)*Q(NP,NN-1)*ABS(Q(NP,NN-1)) QP(NP,NN)=0.0 HP (NP,NN)=(CP-QP(NP,NN))/CA(NP) GO TO 800 CALCULATE HEAD AND DISCHARGE AT CONTROL SECTION 63 0 CP=Q(NP,NN-1)+CA(NP)*H(NP,NN-1)-CF(NP)*Q(NP,NN-1)*ABS(Q(NP,NN-1)) K2=VALOSS*CA(NP)/(QO*VALTAU)**2 K3=CA(NP)*HRES-CP IF((l.-4.*K2*K3).LT.0) GO TO 650 QP(NP,NN)=(-1.+SQRT(1.-4.*K2*K3))/(2.*K2) 64 0 HP(NP,NN)=HRES+VALOSS*QP(NP,NN)*ABS(QP(NP,NN))/(Q0*VALTAU)**2 TVAL=TVAL+DT IF(VALTAU.GT.TAURED) TVAL=0.0 GO TO 8C0 650 NN=N (NP)+1 HP(NP,NN)=HRES+VALOSS GO TO 750 ***** CONSTANT HEAD RESERVIOR AT DOWNSTREAM END 700 NN=N(NP)+1 HP (NP,NN)=HRES 750 CP=Q(NP,NN-1)+CA(NP)*H(NP,NN-1)-CF(NP)*Q(NP,NN-1)*ABS(Q(NP,NN-1) QP(NP,NN)=CP-CA(NP)*HP(N P,NN) ***** STORING MAX. AND MIN. VALUES FOR NEXT TIME STEP 800 DO 900 1=1,NP NN=N(I)+l DO 920 J=1,NN Q(I,J)=QP(I,J) H(I,J)=HP(I,J) IF(H (I,J) .LT.HMAX(I)) GO TO 910 HMAX (I)=H(I,J) TMAX(I)=T 910 IF (H (I,J) .GT.HMIN(I)) GO TO 920 HMIN (I)=H(I,J) TMIN(I)=T 920 CONTINUE 900 CONTINUE IF(ICOUNT.EQ.IPRINT) GO TO 300 GO TO 400 980 WRITE(6,80) 80 FORMAT(/5X,'*** ITERATIONS IN AIR CHAMBER FAILED ***'/) STOP 990 WRITE(6,90) 90 FORMAT(//10X,'PIPE NO. ' ,5X,'MAX. PRESS. AT TIME',5X,'MIN. PRESS. A IT TIME'/) WRITE(6,95) (I,HMAX(I) ,TMAX(I) ,HMIN (I) ,TMIN(I) ,1 = 1,NP) 95 FORMAT(11X,I3,10X,F7.1,6X,F5.2,6X,F7.1,6X,F5.2) STOP END ********************** ********************** SUBROUTINE PARAB  SUBROUTINE PARAB(X,I,N,DX,Z) COMMON /AREA1/ FH(5,90) ,FB(5,90)>PR (5,90) ,VL(90) QUADRATIC CURVE FITTING UTILIZING THREE POINTS  10 20 30 40  J=X/DX R=(X-J*DX)/DX IF(J.EQ.O) R=R-1 J=J + 1 I F ( J . L T . 2 ) J=2 K=J + 1 L=J-1 GO TO (10,20,30,40) ,N Z=FH(I,J)+.5*R*(FH(I,K)-FH(I,L)+R*(FH(I,K)+FH(I,L)-2.*FH(I,J))) RETURN Z=FB(I,J)+.5*R*(FB(I,K)-FB(I,L)+R*(FB(I,K)+FB(I,L)-2.*FB(I,J))) RETURN Z = PR(I,J) + .5*R*(PR(I,K)-PR(I,L)+R*(PR(I,K)+PR (I,L)-2.*PR ( I , J ) ) ) RETURN Z=VL(J) + .5*R*(VL(K)-VL(L)+R*(VL(K)+VL(L)-2.*VL ( J ) ) ) RETURN END ********************** SUBROUTINE INTER ********************** SUBROUTINE INTER (NPB,NPE) COMMON /AREA2/ N(10) ,Q (10,20) ,H (10,20) ,CA (10) ,CF(10) COMMON /OUT2/ QP (10,20) ,HP (10,20) INTERIOR POINTS  DO 20 I=NPB,NPE NN=N (I) DO 10 J=2,NN CN=Q(I,J+1)-CA(I)*H(I,J+1)-CF(I)*Q(I,J+1)*ABS(Q(I,J+1)) CP=Q(I,J-1)+CA(I)*H(I,J-1)-CF(I)*Q(I,J-1)*ABS(Q(I,J-l)) QP(I,J)=0.5*(CP+CN) HP(I,J)=(CP-QP(I,J))/CA(I) 10 CONTINUE 20 CONTINUE SERIES JUNCTION IF((NPE-NPB).EQ.O) GO TO 40 NPEMIN=NPE-1 DO 30 I=NPB,NPEMIN N1=N(I) NN=N(I)+l CN=Q(I+1,2)-CA(I+1)*H(I+1,2)-CF(I+1)*Q(I+1,2)*ABS(Q(I+1,2)) CP=Q(I,N1)+CA(I)*H(I,N1)-CF(I)*Q(I,N1)*ABS(Q(I,N1)) HP (I,NN)=(CP-CN)/(CA(I)+CA(I + 1)) HP(I+1,1)=HP(I,NN) QP(I,NN)=CP-CA(I)*HP(I,NN) QP(I+1,1)=CN+CA(I+1)*HP(1+1,1) 30 CONTINUE  57  C Q C  40 RETURN END **********************  COMPUTATION OF PUMP DISCHARGE  10 15  20 25  30 35 36  C C C  ***********************  SUBROUTINE PUMP(Cl,C2,C3,NPP,II,T) REAL Kl INTEGER OPEN(5) COMMON /AREA3/ ALPHA(5),V(5),DALPHA(5),DV(5),CVHL(5),DTH COMMON /OUT3/ ALPOUT(5),VPOUT(5) COMMON /PRV3/ OPEN,HR (5) ,HO(5),QR(5) ,QO,CN,CA,TAUPRV(5) ,HSUC(5)  C C C  C  SUBROUTINE PUMP  PI=3.141593 VE=V(II)+DV(II) ALPHAE=ALPHA(II)+DALPHA(II) JJ=0 KK=0 LL=0 IF(VE.NE.O.O) GO TO 20 IF(ALPHAE.GE.O.O) TH=0.0 IF(ALPHAE.LT.0.0) TH=180. GO TO 25 TH=57.296*(PI+ATAN2(VE,ALPHAE)) CALL PARAB(TH,II,1,DTH,K1) PRESSURE REGULATING VALVE AT PUMP DISCHARGE? IF(OPEN(II).EQ.O) GO TO 35 HP = K1*HR (II)*(VE**2+ALPHAE**2)+HSUC(II)-CVHL(II) QP=CN+CA*HP QPV=TAUPRV(II)*QO*SQRT(HP/HO(II)) IF(ABS(QP-(VE*NPP*QR(II)-QPV)).LE.0.001) GO TO 30 VE=(QP-QPV)/NPP*QR(II) LL = LL+1 IF(LL.GE.20) GO TO 90 GO TO 15 VP=VE GO TO 40 ARGUE=(NPP*C1)**2-4.*Kl*(Kl*ALPHAE**2+C2-CVHL(II)) VP=(C1*NPP-SQRT(ARGUE))/(2.*Kl) IF(ABS(VP-VE).LE.0.001) GO TO 40 VE=0.5*(VP+VE) KK=KK+1 IF(KK.GE.20) GO TO 80 GO TO 15 COMPUTATION OF PUMP SPEED  40 VM=0.5*(V(II)+VP) ALPHAM=ALPHA(II)+0.5*DALPHA(II) IF(VM.NE.O.O) GO TO 50 IF(ALPHAM.GE.O.O) THM=0.0 IF(ALPHAM.LT.O.O) THM=180. GO TO 60 50 THM=57.296*(PI+ATAN2(VM,ALPHAM)) 60 CALL PARAB(THM,11,2,DTH,BETAM)  58  DALPHA(II)=C3*BETAM*(ALPHAM**2+VM**2) ALPHAP=ALPHA(II)+DALPHA(II) DV(II)=VP-V(II) IF(ABS(ALPHAP-ALPHAE).LE.0.001) GO TO 70 ALPHAE=0.5*(ALPHAP+ALPHAE) VE=VP JJ=JJ+1 I F ( J J . G E . 2 0 ) GO TO 80 GO TO 10 70 CONTINUE ALPOUT(II)=ALPHAP VPOUT(II)=VP RETURN 80 WRITE(6,1) T,KK,JJ ,ALPHAE,VP 1 FORMAT (/5X,'*** ITERATIONS IN PUMP SUBROUTINE FAILED ***'/lOX,'T=' 1,F6.2,5X,'KK=',I3,5X,'JJ = *,13/1 OX,'ALPHA= ,F6.3,1 OX,'VP=',F6.3) STOP 90 WRITE(6,2) 2 FORMAT(/5X,'*** ITERATIONS IN PRV SECTION OF PUMP SUBROUTINE FAILE ID ***'/) STOP END 1  59  APPENDIX B  EXAMPLES  OF PROGRAM  B-l  Example  o f 3 - s t a g e Pumping  B-2  Example  of Single  B-3  Comparison o f S i n g l e  B-4 Check o f Program  USAGE  System  S t a g e Pumping  System  and T h r e e S t a g e Pumping  Accuracy  Systems  60  APPENDIX B - l  EXAMPLE  OF A THREE STAGE PUMPING SYSTEM  PROBLEM: Given at  the following  the three  conditions  pumping  data, determine  stations  a r e t o be c a l c u l a t e d  following  DATA: o f pump s t a t i o n s  Number  of pipes = 3  Steady s t a t e Transients  discharge  calculated  power  fora specified  Pump  Number  the t r a n s i e n t  = 3  = 35.0 c f s f o r 5 seconds  conditions  failure. period  These  o f time.  61  M A I N PUMP S T A T I O N : The c h e c k  valve  the  pump.  this  pumping  Pump  data:  pumps,  closes  All  following  Rated  head = 398.0  Rated  pump s p e e d = 1 7 6 0 . 0  Head  loss  chamber  = 35.0  =  inertia  Pump e l e v a t i o n  into  power  or  power  out  valve failure,  of  the  air  eliminating chamber  failure.  cfs  rpm  0.85 = 660.0  = 0.0  lb-ft  2  ft for  check  valve  =2.0  data:  Initial  pressure  air  Polytropic  volume  = 33.9 in  of  Head  loss  coefficient  Area  of  Head  loss  pipe  chamber  from chamber  = 960.0  Friction  for  for  pump s t a t i o n  to  ft chamber  surface  factor  =  0.014  cu.  ft.  above datum  orifice  =  to  pipe  main  this 1st  ft = 3130.0  = 60.0  1.2  ft  = 2.0  Wave v e l o c i t y  =  water  coefficient  from main  Diameter  air  gas c o n s t a n t  Elevation  Length  upon  check  ft  coefficient  Barometric  Pipe  then  discharge  Pump u n i t  Air  is  Rated  Pump e f f i c i e n c y  chamber,  instantly  flow  station  air  ft/sec  pipe  =  0.0  =  booster  = 2.0  ft  0.0 station:  0.0  at  FIRST Pump  BOOSTER S T A T I O N :  Rated  = 35.0  Pump u n i t  loss  from 1st Length  = 660.0  coefficient booster  valve  ft  rpm  lb-ft  2  for  check to  valve  2nd  =  2.0  booster  station:  ft  = 2.0  Friction  Rated head = 350.0  ft  station  ft  Wave s p e e d = 3 1 3 0 . 0 factor  =  ft/sec  0.014  SECOND BOOSTER S T A T I O N :  pumps,  vacuum b r e a k e r ,  check  valve  data: Rated d i s c h a r g e Rated  = 35.0  Pump u n i t  loss  from  2nd  Length  = 660.0  coefficient booster  = 2.0  Wave v e l o c i t y factor  Rated head = 3 5 0 . 0 rpm  0.85  = 680.0  = 1050.0  Diameter  Frition  =  inertia  Pump e l e v a t i o n Head  cfs  pump s p e e d = 1 7 6 0 . 0  Pump e f f i c i e n c y  Pipe  check  0.85  = 360.0  = 990.0  Diameter  =  inertia  Pump e l e v a t i o n Head  cfs  pump s p e e d = 1 7 6 0 . 0  Pump e f f i c i e n c y  Pump  vacuum b r e a k e r ,  data: Rated d i s c h a r g e  Pipe  pumps,  to  lb-ft  ft for  check  control  ft ft  = 3130.0 =  0.014  2  ft/sec  valve  valve:  =  2.0  ft  63  CONTROL V A L V E D A T A : Time  at  Time  required  Steady  which  state  Relative Minimum Number Time  of  loss  of  bypass  data  valve  line  control  points  valve  opening  =  points  closure  curve:  reservoir  units).  once b u t t e r f l y  these  0.0481 0.0419  with  the  close  butterfly  between  0.1089  not  data  instantaneous = 0.25  = 0.15  curve  0.0542  (gpm  valve  the  is  to  on c l o s u r e  0.1161  stations,  begins  through  0.1215  of  valve  closure  close butterfly  head  relative  valve  sec.  sec. valve  = 7.35  closed =  0.025  = 16 = 0.070  sec.  0.0790  0.0705  0.0620  0.0367  0.0286  0.0249  0.0216  0.0328  = 1040.5  are  specific  ft  0.1215  0.0991 0.0892  T h e pump c h a r a c t e r i s t i c s  pumping 1800  to  area  Elevation  NOTE:  butterfly  interval  Control  butterfly  ft  the speed  same for  the  for  all  pumps  three  equal  to  64  SOLUTION: The  data  1 2 3 4 5-12, 13-20, 21 22 23-30, 31-38, 39 40 41-48, 49-56, 57 58 59 60  cards  f o r t h e program  a r e as f o l l o w s :  ENGLISH 3,3,9,89,0,1,1,0,1,16,1 35.0,4.0909,5.0 960.0,2.0,3130.0,0.014,990.0,2.0,3130.0,0.014,1050.0, 2.0,3130.0,0.014 Head c h a r a c t e r i s t i c f o r main pump (89 v a l u e s ) T o r q u e c h a r a c t e r i s t i c f o r main pump (89 v a l u e s ) 0.0,1760.0,35.0,398.0,1760.0,0.85,660.0,2.0 1,1,0,0,0,0 Head c h a r a c t e r i s t i c f o r 1 s t b o o s t e r pump T o r q u e c h a r a c t e r i s t i c f o r 1 s t b o o s t e r pump 360.0,1760.0,35.0,350.0,1760.0,0.85,660.0,2.0 2,1,0,0,1,1 Head c h a r a c t e r i s t i c f o r 2nd b o o s t e r pump T o r q u e c h a r a c t e r i s t i c f o r 2nd b o o s t e r pump -680.0,1760.0,35.0,350.0,1760.0,0.8 5,660.0,2.0 3,1,0,0,1,1 33.9,60.0,0.0,1.2,0.0,0.0,2.0 0.25,7.35,0.1215,0.15,0.02 5,0.070  61-62  C o n t r o l valve c l o s u r e curve  NOTE:  The  data for  c a r d s . Spacing actual  available The transient hydraulic this  numbers  data  i n the f i r s t  i n the l i s t i n g above  data  columns a r e t h e numbers o f t h e  o f the e n t r i e s  entry into  the  i s n o t t h e same as t h e f o r m a t  program.  o f t h e program  was  entered  c o n d i t i o n s determined  into  This  information  is  (Appendix A ) . the  program  f o r a period of 5  gradeline fluctuations  period are plotted  (14 v a l u e s )  and t h e  seconds.  a t t h e pumping s t a t i o n s  i n F i g u r e s B - l t o B-3.  The  during  TRANSIENT P R E S S U R E S IN 1st PIPE T H R E E STAGE S Y S T E M  DEVIATIONS OF R E S U L T S OBTAINED BY RUUS FROM T H O S E OBTAINED IN THIS THESIS  _,  2 TIME (SECONDS)  ,  r-  3  4  TRANSIENT PRESSURES IN 2nd PIPE THREE STAGE SYSTEM  • HEAD AT SUCTION FLANGE OF BOOSTER * 2  _  TIME(SECONDS)  DEVIATIONS OF RESULTS OBTAINED BY RUUS FROM THOSE OBTAINED IN THIS THESIS  T R A N S I E N T PRESSURES JN 3rd PIPE THREE S T A G E S Y S T E M  14T  i  0  —  ,  1  ,  , 3  2 TIME(SECONDS)  r  4  68  B-2  APPENDIX  EXAMPLE  OF A SINGLE STAGE PUMPING SYSTEM  PROBLEM: This except in  problem that  there  series will  same  head  an a i r chamber  of  two  pumping  supplied  pumping  i n Appendix B - l ,  s t a t i o n s . Three  station  by t h e t h r e e  stage  station  to  and a c h e c k v a l v e  at  will  be  supply  the  discharging will  pumping  into  consist of  station.  i n place  o f t h e two  example.  Air Chamber Check Valve  960 *  990'  To  booster  Reservoir  Main Pump  a  two s y s t e m s , s e r i e s j u n c t i o n s  positioned  s t a t i o n s i n the previous  the  pumps  system. Thus, the  r e s e r v o i r . The c o n t r o l e q u i p m e n t  the comparison of these  pipes  presented  i n one pumping  c o n s i s t o f one pumping  head  facilitate  a r e no b o o s t e r  be p l a c e d  as t h a t  system w i l l constant  i s t h e same as t h a t  1050  69  DATA: Number o f pump s t a t i o n s  = 1  Number o f p i p e s = 3 Steady  state  Transients  discharge  calculated  MAIN PUMP STATION: pumps, The  check  valve  closes  the  pumps. A l l f l o w  =35.0 c f s f o r 5 seconds.  a i r chamber, c h e c k  i n s t a n t l y upon power  i s then  into  valve failure,  eliminating  o r o u t o f t h e a i r chamber.  Pump d a t a : Rated  discharge  = 35.0 c f s  Rated  head = 1084.8 f t  Rated  pump s p e e d  = 1760.0 rpm  Pump e f f i c i e n c y = 0.85 Pump u n i t  inertia  Pump e l e v a t i o n Head Air  loss  chamber  = 0.0 valve  = 2.0  a i r chamber a s i n t h e t h r e e  stage  pumping s t a t i o n  system.  to r e s e r v o i r :  = 2.0 f t  Wave speed Friction  2  data:  from  Diameter  lb-ft  c o e f f i c i e n t f o r check  Identical Pipeline  = 1980.0  = 3130.0  factor  ft/sec  = 0.014  Pipe  length  from  pumps t o 1 s t j u n c t i o n  Pipe  length  from  1st junction  t o 2nd j u n c t i o n  Pipe  length  from  2nd j u n c t i o n  to r e s e r v o i r  Reservoir  elevation  = 1040.5 f t .  = 960.0 f t = 990.0 f t  = 1050.0 f t  70  SOLUTION: The  data  cards  f o r t h e program a r e as f o l l o w s :  1 2 3 4  ENGLISH 3,1,9,89,0,1,0,0,0,0,0 35.0,4.0909,5.0 960.0,2.0,3130.0,0.014,990.0,2.0,3130.0,0.014,1050.0, 2.0,3130.0,0.014 5-12, Head c h a r a c t e r i s t i c f o r pumps (89 v a l u e s ) 13-20, T o r q u e c h a r a c t e r i s t i c f o r pumps (89 v a l u e s ) 21 0.0,1760.0,35.0,108 4.8,1760.0,0.85,1980.0,2.0 22 1,1,0,0,1,0 23 33.9,60.0,0.0,1.2,0.0,0.0,2.0 NOTE: The numbers data  cards.  for  actual  available The transient hydraulic the  i n the f i r s t  Spacing data  entry  data  was  conditions gradeline  two s e r i e s  numbers  of  the  o f t h e e n t r i e s i s n o t t h e same as t h e f o r m a t  i n the l i s t i n g above  columns a r e t h e  into  the program. T h i s  information i s  o f t h e program (Appendix A ) . entered  determined  into  the  and  the  f o r a p e r i o d o f 5 s e c o n d s . The  f l u c t u a t i o n s a t t h e pumping  junctions of pipes  program  station  and  are p l o t t e d i n Figure  B-4.  at  TRANSIENT  HEAD  P R E S S U R E S IN A SINGLE  AT RESERVOIR  TIME(SECONDS)  STAGE SYSTEM  72  APPENDIX  COMPARISON OF SINGLE AND  The  comparison  performed and  Appendix  are in  the system  SYSTEMS  s t a g e and a t h r e e examples  p r e s e n t e d i n Appendix B - l  from  t h e computer  system  gives  lines  o u t p u t s and t h e s e a r e p l o t t e d  B-5. I t i s o b v i o u s when e x a m i n i n g  multistage  stage system i s  B-2. The maximum and minimum h y d r a u l i c g r a d e  determined Figure  THREE STAGE  of a single  by u s i n g  B-3  significantly  lower  this  figure  transient  that the pressure  fluctuations. It  should  system does Figure since  be  noted, that  not occur w i t h i n  maximum  simultaneously within the  the 5 second  B-4, b u t o c c u r s s h o r t l y the  hydraulic  grade  maximum and minimum  and  t h e maximum f o r t h e s i n g l e  after minimum  (at  time 6.82  are  plotted  shown i n  seconds).  pressures  each p i p e o f the systems lines  interval  do  being  stage  not  Also, occur  examined,  f o r t h e t i m e when t h e  p r e s s u r e s occur i n pipe 1 o f each  system.  0  10  20 DISTANCE  (FEETxIOO)  FIGURE B-5  30  74  APPENDIX  CHECK OF  PROGRAM ACCURACY  The  check  of  accuracy  transients  f o r the  The  characteristics  system  entered  into  developed  by  by  Ruus from  the  with  by  Slightly  the  curve  two  by  an  approximates p o i n t s on 2.  different  developed  by  this  and  can  control  p r o g r a m s . The equation, while  in  stage this  results  are  are  s t u d y and  that  presented  results  attributed  approximates  t h e program d e v e l o p e d by  interpolating  in  obtained lines.  to:  valve closure curves  Ruus program  curve  system  s t u d y shown as d a s h e d be  the  utilized the  actual  in this  thesis  between  known  the c l o s u r e c u r v e .  Slightly  differences different  three  examining  i n Appendix B - l .  the d e v i a t i o n s o f the  obtained  the a c t u a l  by  presented  this  generated  These d e v i a t i o n s are s l i g h t 1.  performed  system  for  program  E. Ruus. The  those  is  three stage  B - l to B - 3 ,  Figures  B-4  a r e a t most  time  Neither  different  intervals  of these  t h e maximum and  1.2  wave per  speeds  within  each  c e n t ) . T h i s i s due  calculated  w i t h i n the  changes produces  two  significant  minimum p r e s s u r e s w i t h i n t h e  to  pipe  (the  slightly  programs. differences  system.  in  


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