Problem-Solving Example #3

Heat Transfer in a Tapered Fin

(A Two Point Boundary Value Problem)

Consider a tapered fin of trapezoidal profile:

The governing equation for the temperature distribution in the fin is the non dimensionalized second order differential equation



where y = non dimensional temperature = (T - T_∞ ) / (T_w + T_∞ )
x = non dimensional distance =

(T_w - T_∞ )
where q is the uniform rate of internal heat generation per unit volume (via nuclear fission, electrical dissipation, chemical reaction, etc.) and k is the thermal conductivity.

. . .
where is the average convective heat transfer coefficient

Based upon the known physical conditions for a fin, it is usually necessary to specify the value of the boundary conditions at two points, unless both the temperature and the temperature gradient at x=0 are known. For example, in a fin with an adiabatic end, one can specify the value of Y at x=0 and the value of dy/dx at x=1. It is then necessary to find a value of dy/dx at x=0 which allows the resulting temperature profile to satisfy the end condition at x=1.

This is accomplished in Fortran Calculus using the statement:

  FIND dydx0; in bound; by Ajax(acon); to match dydx1

which invokes the solver AJAX to operate on the model Bound. Bound contains an integration process employing the solver ISIS. The process is initiated via the statements

  x=0 : y=1 : dydx = dydx0 : dx=.05  ! initial conditions
  INITIATE Isis; for diff; equations d2ydx2/dydx, dydx/y;
     &      of x; step dx; to xf;

where the model DIFF contains the differential equation:

Model Diff
  d2ydx2=((2*p4*y/cos(a)+(p3-p2)*dydx)/(p2+(p3-p2)*(1-x)))-p1
End

The process is integrated to the boundary and the boundary condition dydx1 is computed, via the statements:

  do while (xf .le. 1)
    INTEGRATE diff; by Isis
    print '(3e14.5)', x, y, dydx
    xf = xf + xp
  end do
  dydx1 = dydx   !  boundary condition

This program solves the two point boundary value problem of the 2nd order differential equation that describes the one dimensional, steady state temperature distribution in a tapered Fin, where y is a non-dimensional temperature and x is a non-dimensional distance.

The Fortran Calculus program for this problem contains 33 lines of text (Note: plots were removed for this demo), as follows:...

Problem Fin   ! Heat Transfer in a tapered fin
  common dydx0,d2ydx2,dydx,y,x,dydx1
  common/params/p1,p2,p3,p4,a
  ! Fortran problem ... sub'ing P's 4 N's
  p1=.1 : p2=.03 : p3=.08 : p4=.01
  a=atan(.5*(p3-p2))
  dydx0=-.5 !  initial guess
  FIND dydx0; in bound; by Ajax(acon); to match dydx1
end
controller acon(ajax)
  damp=0
end
model bound
  common dydx0,d2ydx2,dydx,y,x,dydx1
  data it/0/
  xp=.1 : xf=xp : it=it+1
  x=0 : y=1 : dydx=dydx0 : dx=.05  ! initial conditions
  INITIATE Isis; for diff; equations d2ydx2/dydx,dydx/y;
     &     of x; step dx; to xf;
  print '(a,i2/a)',' integration for iteration ',it,
     &    '     distance           y         dydx   '
  do while (xf.le.1)
    INTEGRATE diff; by isis
    print '(3e14.5)',x,y,dydx
    xf=xf+xp
  end do
  dydx1=dydx   !  boundary condition
  terminate diff
end
model diff
  common dydx0,d2ydx2,dydx,y,x,dydx1
  common/params/p1,p2,p3,p4,a
  d2ydx2=((2*p4*y/cos(a)+(p3-p2)*dydx)/(p2+(p3-p2)*(1-x)))-p1
End

A full discussion of this problem is given in Computer Aided Heat Transfer Analysis by J. A. Adams and D. F. Rogers, McGraw Hill, 1973, pp. 55-74. A BASIC program is given which contains 165 lines of text.

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