Flow  Meter  Discharge  Coefficient  Estimation  Thomas  Zhang  

 ABSTRCT.  Flow  meters  are  the  useful  and  common  devices  that  could  measure  the   mass   or   volumetric   flow   rate   of   fluid.   Understanding   how   to   estimate  coefficient   of   the   flow   meter   discharges   is   fundamental   for   a   huge   variety   of  applications   in   fluid   transportation.   This   experiment   was   designed   as   an  introduction   to   this   process   using   an   Edibon   flow   meter   module   to   estimate  discharge   coefficient   during   a   fluid   flowing   process.   This   simple   experiment  gives  a  glimpse  into  many  different  methods  that  can  be  utilized  to  determine  the  coefficient  of  flow  meter  discharges.  In  this  experiment,  two  valves  were  used  to  control  the  flow  of  water  through  the  system.  Flow  rates  could  be  indicated  by  a  metal   cone   and   were   changed   for   6   times.   The   volumetric   flow   rates   and   the  manometer   pressures   corresponding   to   taps   1-­‐7   could   be   measured   and  recorded   for   each   time.   The   result   showed   that  with   the   volumetric   flow   rate  increased,   the   energy   loss   of   fluid   flowing   would   increase   across   Venturi   and  Orifice   flow  meter   while   it   would   keep   constant   in   Rotameter.   Fluid   will   loss  more  energy  when  flowing  through  Orifice  flow  meter  than  through  Venturi  flow  meter.   The   discharge   coefficient   of   Venturi   and   Orifice   flow   meter   were  estimated  to  be  1.97  and  0.76  respectively.  However,  the  relative  errors  of  them  to  the  theoretical  values  were  100%  and  21%  respectively,  which  showed  a  bad  accuracy  of  the  measurement.    

Introduction  According   to   Bernoulli   equation,   the   pressures   of   a   flow   through   a   pipe   at  different   points   with   the   same   section   area   and   height   should   be   the   same.  However,   one   of   assumptions   of   Bernoulli   equation   is   that   the   friction   is  negligible,  which  could  seldom  be  set  up  in  real  life.  So  there  are  always  energy  losses  when   fluid   flows   through  a  pipe.  These  energy   losses  are   inevitable  and  could   have   a   significant   impact   on   fluid   flowing   process.   The   estimation   of  discharge   coefficient   is   of   great   importance   to   fluid   transportation   since   the  power  of  pump  needed  to  keep  the  fluid  flows  through  a  distance  depends  on  it.  Mannan  and  Frank   (2005)  described   that   in  a  nozzle  or  other   constriction,   the  discharge  coefficient  (also  known  as  coefficient  of  discharge)   is   the  ratio  of   the  actual  discharge  to  the  theoretical  discharge.  The  ratio  of   the  mass   flow  rate  at  the   discharge   end   of   the   nozzle   to   that   of   an   ideal   nozzle   which   expands   an  identical   working   fluid   from   the   same   initial   conditions   to   the   same   exit  pressures.    Flow   meters   are   important   and   indispensable   devices   for   fluid   measurement,  which   are   widely   used   in   fluid   mechanics   field.   So   it   is   really   necessary   to  measure  the  discharge  coefficient  of  flow  meter  in  order  to  obtain  the  actual  flow  

ratio.  Cengel  and  Cimbala  (2013)  also  stated  that  in  reality,  some  pressure  losses  due  to  frictional  effects  are  inevitable,  and  thus  the  velocity  will  be  less.  Also,  the  fluid   stream   will   continue   to   contract   past   the   obstruction,   and   the   vena  contracta  area   is   less   than   the   flow  area  of   the  obstruction.  Both   losses   can  be  accounted   for   by   incorporating   a   correction   factor   called   the   discharge  coefficient  Cd  whose  value  (which   is   less   than  1)   is  determined  experimentally.  Of  the  numerous  types  of  obstruction  meters  available,  those  most  widely  used  are  orifice  meters,  flow  nozzles,  and  Venturi  meters.    

Objectives  The   purposes   of   this   study   were:   (1)   To   learn   how   to   estimate   the   discharge  coefficients  for  venturi  and  orifice  flow  meters.  (2)  To  quantify  energy  losses  due  to  flow  through  flow  meters.  (3)  To  be  more  familiar  with  the  Bernoulli  equation  and  energy  equation.    

Materials  and  Methods  Flow  Meter  Module  A   flow   meter   is   an   instrument   used   to   measure   linear,   nonlinear,   mass   or  volumetric  flow  rate  of  a  liquid  or  a  gas.  The  flow  meter  module  which  was  used  in   this   experiment   consists   of   three   kinds   of   flow  meters:   Venture,  Orifice   and  Rotameter  flow  meters.    

 Figure  1.  FNE18  Flow  Meter  Module  From  Edibon  (by  taking  photo)  

 

Venturi  meter  Herschel   (1898)   mentioned   that   a   Venturi   meter   constricts   the   flow   in   some  fashion,   and   pressure   sensors   measure   the   differential   pressure   before   and  within  the  constriction.  This  method  is  widely  used  to  measure  flow  rate  in  the  transmission  of  gas   through  pipelines,   and  has  been  used  since  Roman  Empire  times.  The  coefficient  of  discharge  of  Venturi  meter  ranges  from  0.93  to  0.97.  The  first   large-­‐scale   Venturi   meters   to   measure   liquid   flows   were   developed   by  Clemens  Herschel  who  used  them  to  measure  small  and  large  flows  of  water  and  wastewater  beginning  at  the  end  of  the  19th  century.  

 Figure  2.  The  working  principal  of  Venturi  meter    

(http://commons.wikimedia.org/wiki/File:  Venturifixed2.PNG)    Orifice  meter  Lipták   (1993)   discussed   that   an   orifice   plate   is   a   plate  with   a   hole   through   it,  placed  in  the  flow;  it  constricts  the  flow,  and  measuring  the  pressure  differential  across  the  constriction  gives  the  flow  rate.  It  is  basically  a  crude  form  of  Venturi  meter,  but  with  higher  energy  losses.  There  are  three  type  of  orifice:  concentric,  eccentric,  and  segmental.  

 Figure  3.  Orifice  plate    

(fpunktz,  6  October  2006,  http://www.ilkdresden.de/iso5167/index.html)    Rotameter  Holman   (2001)   defined   rotameter   as   a   device   that   measures   the   flow   rate   of  liquid  or  gas  in  a  closed  tube.  It  belongs  to  a  class  of  meters  called  variable  area  meters,  which  measure   flow   rate  by  allowing   the   cross-­‐sectional   area   the   fluid  travels  through,  to  vary,  causing  a  measurable  effect.  It  consists  of  a  tapered  tube,  typically  made  of   glass,  with   a   float   inside   that   is   pushed  up  by   fluid   flow   and  

pulled   down   by   gravity.   As   flow   rate   increases,   greater   viscous   and   pressure  forces  on  the  float  cause  it  to  rise  until  it  becomes  stationary  at  a  location  in  the  tube  that  is  wide  enough  for  the  forces  to  balance.  Rotameters  are  available  for  a  wide  range  of  liquids  but  are  most  commonly  used  with  water  or  air.  They  can  be  made  to  reliably  measure  flow  down  to  1%  accuracy    Experimental  method  Frank   (2003)   showed   that   mathematically   the   discharge   coefficient   might   be  related   to   the   mass   flow   rate   of   a   fluid   through   a   straight   tube   of   constant  cross-­‐sectional  area  through  the  following  equation:  

        (1) Where:

= Discharge Coefficient through the constriction (unit-less).

= Cross-sectional area of flow constriction (unit length squared).

= Mass flow rate of fluid through constriction (unit mass of fluid per unit time).

= Density of fluid (unit mass per unit volume).

= Pressure drop across constriction (unit force per unit area).

In  this  experiment,  two  valves  were  used  to  control  the  flow  of  water  through  the  system.  Flow  rates  could  be  indicated  by  the  metal  cone  and  were  changed  for  6  times.   The   volumetric   flow   rates   of   the  water   flow  were  measured  with   timer  and  the  dump  valve  system.  The  manometer  pressures  corresponding  to  taps  1-­‐7  were  also  recorded  for  each  time.  For  analysis,  the  graphs  of  the  energy  losses  at  the   various   flow   rates   due   to   flow   through   each  of   the  meters  were  plotted   in  order  to  observe  the  relationship  between  them  with  the  data  collected  in  the  lab.  In  addition,   the  graphs  of  volumetric   flow  rates  (Q)  vs.  square  root  of  pressure  drop  across  the  three  flow  meters  for  each  time  could  also  be  plotted.  Thus  the  equation  given  above  could  finally  be  simplified  and  transformed  into:  

 

                              (2)  

 

Slope =CdA22

ρ 1− A2A1

"

#$

%

&'

2(

)**

+

,--

       

Table  1.  Cross-­‐section  areas  of  Venturi  and  Orifice  Flow  Meters    The  discharge  coefficient  was  obtained  from  equation  (1)  using  the  slopes  of  the  graphs  and  the  cross-­‐sectional  areas  of  Venturi  and  Orifice  flow  meter.  

 

Results  and  Discussion  The  graphs  of  the  energy  losses  vs.  flow  rates  were  plotted  for  each  in  order  to  observe  the  relationship  between  them.  The  energy  losses  of  them  are  obtained  from   the   pressure   drop   between   the   inlet   and   outlet   of   each   flow   meter  respectively.  

 

 Graph  1.  Energy  loss  vs.  volumetric  flow  rate  of  Venturi  flow  meter  

 With   the  volumetric   flow  rate   increased,   the  pressure  drop  of   the   fluid   flowing  through  venturi  flow  meter  would  increase  as  well.  It  means  that  when  the  fluid  flows  faster,   it  will  have  more  energy   loss  because  of   the   friction.  This  result   is  reasonable  since  the  friction  between  fluid  and  surface  of  pipe  is  mainly  caused  by  viscosity:    

  τ =ηdvdy   (3)

 

Where:               τ:  the  shear  stress  of  fluid  (unit  force  per  unit  area)               v:  the  velocity  of  fluid  (unit  length  per  unit  time)               y:  the  distance  of  the  pipe  surface  to  the  counting  point  (unit  length)               η:  the  viscosity  of  fluid  (unit  pressure  times  unit  time)  

0  10  20  30  40  50  60  70  80  90  100  

0   0.0001   0.0002   0.0003   0.0004  

Pressure  drop  (pa)

Flow  rate  (m3  /s)

Venturi  energy  loss  

A1（m2） A2（m2）

Venturi 8.04 10－4 3.14 10－4

Orifice 9.62 10－4 2.83 10－4

× ×× ×

 Since   viscosity   and   radius   of   fluid   is   constant,   with   the   velocity   increases,   the  shear   stress   of   fluid  would   increase   as  well,  which  will   cause   energy   loss.  And  through  observing   the  graph  1,   in  Venturi   flow  meter,   the  energy   loss  will   rise  almost  linearly  with  the  volumetric  flow  rate  increases.      

 Graph  2.  Energy  loss  vs.  volumetric  flow  rate  of  Rotameter  

 However,  in  Rotameter,  the  pressure  drop  would  stay  nearly  the  same  with  the  volumetric   flow   rate   increased.   This   result  may   has   something   to   do  with   the  variable  cross-­‐sectional  area,  which  could  influence  the  shear  stress  and  make  it  constant,  as  a  result,  keep  the  energy  loss  in  a  same  level.  

 

 Graph  3.  Energy  loss  vs.  volumetric  flow  rate  of  Orifice  flow  meter  

 The  energy   loss  of   the   fluid   flowing   through  Orifice   flow  meter  would   increase  with  the  volumetric  flow  rate  increased,  which  was  similar  to  Venturi  flow  meter.  The  difference  between  them  was  that   the  energy   loss  had  a  powerful   trend  of  rising   instead  of   linear   trend.  As  a  result,   it  would   lead  to  a  higher  energy   loss.  This  was  mainly  caused  by  the  hole   in   the  center  of  Orifice  meter,  which  had  a  

2284  

2286  

2288  

2290  

2292  

2294  

2296  

2298  

0   0.0001   0.0002   0.0003   0.0004  

Pressure  drop  (pa  )

Flow  rate  (m3  /s)

Rotameter  energy  loss  

0  100  200  300  400  500  600  700  800  900  

0   0.0001   0.0002   0.0003   0.0004  

Pressure  drop  (pa)

Flow  rate  (m3  /s)

Orifice  energy  loss  

much  smaller  radius  compared  with  Venturi  flow  meter.  Because  if  the  velocity  and  viscosity  of  fluid  keep  constant,  a  smaller  radius  could  lead  to  a  larger  shear  stress,  which  makes  the  fluid  loss  more  energy.    

 The  graph  of  volumetric   flow  rate  vs.  square  root  of  pressure  drop  across  each  flow  meter  was   plotted   in   order   to   estimate   the   discharge   coefficient   of   these  three  flow  meters.    

 Graph  4.  Volumetric  flow  rate  vs.  square  root  of  pressure  drop  across  each  flow  

meter    

Through  observing  the  graph  4,   the  slope  of   the  graph  of  Venturi   flow  meter   is  equal   to  0.00003;   the  slope  of  Orifice   flow  meter   is  equal   to  0.00001.  Since  the  energy   loss   of   Rotameter   does   not   change   with   the   volumetric   flow   rate  increases,  the  slope  of  it  is  infinite  and  the  discharge  coefficient  of  it  could  not  be  obtained   through   calculation.  Put   the   slope  of   the   graph  of  Venturi   and  Orifice  flow  meter  as  well  as  the  cross-­‐section  areas  into  the  equation  (2),  Cd  =  1.97  for  Venturi   flow   meter   and   Cd   =   0.76   for   Orifice   flow   meter   could   be   obtained.  Compared   them   with   the   theoretical   values   which   are   0.982   and   0.627  respectively,  the  discharge  coefficient  of  orifice  flow  meter  are  similar,  which  has  a  relative  error  of  21%.  However,  the  discharge  coefficient  of  Venturi  flow  meter  got   from   lab   is   far   away   from   the   theoretical   value,  which   has   a   relative   error  over  100%.  So  it  shows  a  bad  accuracy  of  the  estimation  of  discharge  coefficient  in   this   lab.   It  may   have   something   to   do  with   the   imprecise   reading   or  wrong  operation.  

Conclusion

The   analysis   of   the   collected   data   from  discharge   coefficient   estimation   in   this  experiment  allowed  for  a  sufficient  overview  of  the  rules  of  fluid  transportation  

y  =  3E-­‐05x

y  =  1E-­‐05x

0  

0.00005  

0.0001  

0.00015  

0.0002  

0.00025  

0.0003  

0.00035  

0   10   20   30   40   50   60  

Flow

 rate  (m

3  /s)

Square  root  of  pressure  drop  [pa1/2]

Venturi  

Rotameter  

Orifice  

and  measurement.   The   energy   loss   of   fluid   flowing   across   Venturi   and   Orifice  flow  meter  would   increase  with   the   volumetric   flow   rate   increased.  While   the  fluid   flowed   through   Orifice   flow   meter   would   have   a   higher   energy   loss.  However,   the   energy   loss   would   keep   constant   with   the   volumetric   flow   rate  increased  in  Rotameter.  Through  calculation,  the  discharge  coefficient  of  Venturi  and  Orifice   flow  meter  were  1.97   and  0.76   respectively.  However,   it   showed   a  bad  accuracy  since  the  relative  errors  of  them  to  the  theoretical  values  were  100%  and   21%   respectively.   The   precision   of   reading   and   operation   need   to   be  improved   to  reduce  errors.  Generally  speaking,   the  energy   loss  of   fluid   flowing  through  a  pipe  must  be  taken  into  account  when  working  on  fluid  transportation  during  engineering  process.  And  when  using  a  flow  meter  to  measure  flow  rate  of   a   fluid,   it   is   also   necessary   to   take   the   discharge   coefficient   of   it   into  calculation.    

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Hazard   Identification,   Assessment   and   Control,   Volume   1,   Elsevier  Butterworth  Heinemann.  ISBN  978-­‐0750678575.  

Cengel,  Y.  A.  and  Cimbala,  J.  M.  (2014).  Fluid  Mechanics  Fundamentals  and             Applications,  3rd  edition.  New  York:  McGraw  Hill.  88  p.  Herschel,  C.  (1898).  Measuring  Water,  Providence,  RI:  Builders  Iron  Foundry  Lipták,  B.  G.  (1993).  Flow  Measurement,  Chilton  Book  Company,  p.  85  Holman,   J.  A.   (2001).  Experimental  methods   for   engineers,  Boston:  McGraw-­‐Hill.  

ISBN  978-­‐0-­‐07-­‐366055-­‐4.  Frank   M.   W.   (2003).   Fluid   Mechanics,   7th   Edition,   McGraw-­‐Hill   Series   in  

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