Mathematical Methods (10/24.539)
VIII. Special Functions
Example 8.3 -- Analytical Solution to the Circular Fin Problem
Problem Description:
With the figure, general notation, and the model development given previously (see
Section V), analytically determine the temperature and temperature gradient profiles for the circular fin problem given the following numerical data:|
rw = 1 in. |
rs = 1.5 in. |
d = 0.0625 in. |
|
Tw = 200 ° F |
T ¥ = 70 ° F |
|
|
h = 20 BTU/hr-ft2 - °F |
k = 75 BTU/hr-ft- °F |
|
Evaluate and plot the normalized temperature and gradient profiles and determine the absolute fin edge temperature. Also determine the total heat loss from the fin and compute the fin efficiency,
h , where
Problem Solution:
The dimensionless form of the steady state energy balance for combined heat conduction and convection for a cylindrical fin configuration is given as (see the formal development with appropriate limitations and definitions in
Section V of these notes):
with boundary conditions,
With our recent discussion concerning Bessel functions, one recognizes this as a special form of the modified Bessel’s equation,
with general solution
Comparing this standard system to the energy balance for the problem of interest shows that 
and the general solution for the normalized temperature profile is
Applying the boundary conditions to this general solution gives:
1. For u(a) = 1, we have
2. For u’(b) = u’(1) = 0, we have
and letting x = 1 gives
Thus, the two boundary conditions give two coupled equations for c1 and c2, or
From the second equation, we see that
and putting this into the first equation in the set gives
Thus, the two coefficients become
and
Putting these constants into the general solution gives an explicit formulation for the normalized temperature profile, or
The temperature gradient can also be evaluated to give
These analytical expressions are evaluated using the parameter specifications given above within the Matlab file
CYLFINA.M. This file is listed in Table 8.2 and the resultant temperature and gradient profiles are plotted in Fig. 8.2. Note that the results here are exactly the same as those developed using the numerical techniques discussed in Section V (see Example 5.3A and Example 5.3B). As before, we also compute the fin efficiency as
where the ideal energy transfer is computed assuming that the fin temperature is constant at the wall value, or
The actual energy transferred from the fin can be computed from the conduction representation at the wall, or
Table 8.3, which contains a listing of the output file from CYLFINA.M, shows that the numerical values for the heat transfer and fin efficiency from the analytical solution are exactly as computed in Example 5.3A and Example 5.3B (using numerical methods). The overall efficiency of about 93% and a tip temperature of almost 188 F - a drop of only 12 F from the wall temperature - indicate a fairly efficient overall fin arrangement.
This problem represents a good illustration of the use of Bessel functions in finding analytical solutions to a real problem. It also serves as a good example of how to apply the general analytical solution scheme, including the evaluation of the boundary conditions to determine the unknown coefficients in the general solution. In addition, the Matlab m-file associated with this problem, CYLFINA.M, can be used as another example of the evaluation of the Bessel functions within the Matlab environment. Finally, the combination of this example and those given in Section V (Example 5.3A and Example 5.3B) also represents a series of applications that contrast the various techniques generally available for solving linear boundary value problems (BVPs).
Fig. 8.2 Solution profiles for the circular fin problem (Analytical Solution).
|
|
|
|
|
10/24.539 Lecture Notes by Dr. J. R. White, UMass-Lowell (updated October 1998).
