Two  Dimensional    Steady  Conduc4on  

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Shape  Factors  •  Shape  factors  are  a  simple  means  of  accoun4ng  for  the  geometric  

factors  affec4ng  conduc4on  in  1D,  2D  and  3D  systems.    •  The  shape  factor  “S”  is  defined  such  that:  

 •  The  shape  factor  is  related  to  the  thermal  resistance  through:  

 

•  Shape  factors  are  determined  from  analy4c  solu4ons  to  Laplace’s  equa4on  and  using  the  temperature  field  to  calculate  S  or  R.  

•   Many  tabulated  results  exist  for  S.      

 

q = Sk(T1 −T2 )

€

R =1S⋅ k

€

∂ 2T∂x 2

+∂ 2T∂y 2

+∂ 2T∂z2

= 0

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Shape  Factors  •  Shape  factors  are  convenient  for  determining  thermal  resistance  in  simple  systems  such  as:  

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Shape  Factors  •  Shape  factors  are  convenient  for  determining  thermal  resistance  in  complex  systems  such  as:  

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Shape  Factors  •  Shape  factors  can  easily  be  used  in  thermal  circuits:  

where        Here  is  a  sample  of  shape  factors  from  your  text:  

 

€

RTotal =1hiAi

+1Sk

+1

hoAo

q = Ti −ToRTotal

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Shape  Factors  •  Many  shape  factors  are  tabulated  in  a  number  of  Heat  

Transfer  Handbooks.  Here  is  a  sample  of  some  useful  shape  factors  taken  from  the  Handbook  of  Heat  Transfer,  McGraw-­‐Hill,  1985:  

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Shape  Factors  9  

Shape  Factors  10  

Shape  Factors  •  Shape  factors  can  also  be  approximated  from  many  one  dimensional  solu4ons.    

•  Other  solu4ons  can  be  used  to  approximate  situa4ons  where  no  shape  factor  is  available.  

•  Two  rules  for  shape  factor  “equivalence”:  – Rule  #1  –  Preserve  the  smaller  surface  area          (inside  or  outside).  – Rule  #2  –  Preserve  the  volume  of  the  “conduc4ve”  material  in  the  system.    

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Example  -­‐  1  •  Calculate  the  heat  transfer  rate  from  the  cubical  oven  enclosure  having  an  outside  side  length  of  W=5  m,  and  a  wall  thickness  of  t=0.35  m.  The  thermal  conduc4vity  of   the  wall   is  1.4  W/mK,  and   there   is  an  outside  heat  transfer   coefficient   of   ho=5   W/m2k   and   an   ambient  temperature  of  25  C.  The  inside  surface   is  constant  at  1100  C.  Use  two  approaches:  

     –  1)   Find   appropriate   shape   factors   from   the   table   in   your  text  or  from  the  tables  provided.  

–  2)  Using  the  two  modelling  rules  approximate  the  cubical  enclosure  as  a  spherical  enclosure.  

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Example  -­‐  2  •  Compare   the   shape   factor   for   the   system  composed   of   a   circular   hole   in   a   square   bar,  with  that  approximated  using  a  circular  hole  in  a  circular  bar.    

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Example  -­‐  3  •  Compare   the   shape   factor   for   a   spherical  enclosure   in   an   infinite   medium   to   that   of   a  cubical   enclosure   in   an   infinite   medium.   Using  the   solu4on   for   the   spherical   enclosure  approximate   the   shape   factor   for   the   cubical  enclosure   and   compare   the  error.  Why?   So   that  we  can  test  the  two  rule  method!      

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Example  -­‐  4  •  You  are  asked  to  analyze  the  liquid  cooled  heat  sink,  which  is  produced  by  

machining   triangular   grooves   in   two   matching   plates   which   eventually  form  a  diamond  shaped  holes  when  the  plates  are  assembled  as  shown  in  the   sketch.   Calculate   the  overall   heat   transfer   coefficient   for   the   system  shown   below   assuming   that   the   internal   convec4ve   heat   transfer  coefficient  is  1000  W/m2K.  You  will  need  to  approximate  the  shape  factor  using   sound   physical   principles   as   no   exact   solu4on   is   available   to   this  configura4on.   Assume   that   the   plate   is   aluminum   with   a   thermal  conduc4vity  of  k  =  180  W/mK.  If  the  surface  temperature  is  limited  to  Ts  =  50  C  and  coolant  has  a  mean  bulk  temperature  of  30  C,  what  heat  transfer  rate  is  permissible,  when  the  sink  contains  four  channels  of  length  15  cm?    

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