Question 7.11: The fin temperature distribution in the array of fins in Fig...

To proceed with the solution of Equation 7.37, the fin is divided into N small elements, so that X_i=i\Delta X and Z_j=j\Delta Z , where 0 ≤ i ≤ N and 0 ≤ j ≤ N. The second derivative is approximated by

\frac{d^2\vartheta }{dX^2}=\frac{\vartheta _{i+1}-2\vartheta _i+\vartheta _{i-1}}{(\Delta X)^2}

The integral on the right-hand side of Equation 7.37 can be approximated using the trapezoidal rule (Equation 7.31) as

\int_{z_0}^{z_N}{f(x,z)}dz\approx \Delta z\left[\frac{1}{2}f_0(x)+\sum\limits_{j=1}^{N-1}{f_j(x)+\frac{1}{2}f_N(x) } \right]                     (7.31)

\int_{0}^{1}{f(X,Z)}dZ\approx \Delta Z\left[\frac{1}{2}f_0(X)+\sum\limits_{j=1}^{N-1}{f_j(X)+\frac{1}{2}f_N(X) } \right]

where

f(X,Z)=\frac{B^2}{2}\left[-\mu (1-\epsilon )\frac{d^2\vartheta }{dZ^2} +\vartheta ^4(Z) \right]\frac{1}{[B^2+(Z-X)^2]^{3/2} }

Using ΔZ = ΔX and substituting the finite-difference form of the second derivative results in

f_j(X_i)=\frac{B^2}{2}\left[-\mu (1-\epsilon )\frac{\vartheta _{j+1}-2\vartheta _j+\vartheta _{j-1}}{(\Delta X)^2}+\vartheta ^4_j \right]\frac{1}{\left\{B^2+[(j-i)\Delta X]^2\right\}^{3/2} }

The two limits where j = 0 and j = N are evaluated by applying the boundary conditions. For j = i = 0, ϑ_0 = 1 and d^2 ϑ/dZ^2=0. The latter condition is needed because the ϑ_{i−1} term would otherwise be undefined; the condition arises because the energy entering the fin at the base is all by conduction, so the temperature gradient is linear at that location. Then, for j = 0,

f_0(X_i)=\frac{B^2}{2[B^2+(i\Delta X)^2]^{3/2}}

For j = N, ϑ_{N+1} = ϑ_{N− 1} from (d\vartheta /dx)_{X=1}=(\vartheta _{N-1}-\vartheta _{N-1})/(2\Delta X)=0, where N + 1 is a symmetric image point of N − 1. Then

f_N(X_i)=\frac{B^2}{2}\left[-\mu (1-\epsilon )\frac{2(\vartheta _{N-1}-\vartheta_N)}{(\Delta X)^2}+\vartheta ^4_N \right]\frac{1}{\left\{B^2+[(N-i)\Delta X]^2\right\}^{3/2} }

The energy equation (Equation 7.37) for element i at X_i = i ΔX can now be written in finitedifference form as

-\mu \frac{\vartheta _{i+1}-2\vartheta _{i-1}}{(\Delta X)^2}+\vartheta ^4_1=\frac{1}{2}\left\{\frac{1}{\left\{[B/(i\Delta X)]^2+1\right\}^{1/2} } \right\} +\Delta X\left[\frac{1}{2}f_0(X_i)+\sum\limits_{j=1}^{N-1}{f_j(X_i)+\frac{1}{2}f_N(X_i) } \right]                                     (7.38)

This equation is written for each element i for the range 1 ≤ i ≤ N, giving N equations for the N unknown temperatures ϑ_i . Each equation contains every unknown ϑ_i , which appear in the f_j  terms. If the resulting set of equations is written in matrix form, the coefficient matrix is full. This is in contrast to one-dimensional pure conduction problems where the coefficient matrix is usually tridiagonal. This illustrates that in radiative transfer problems the temperature of every element can be influenced by the temperature of all of the surrounding elements. Equation 7.38 is written for element i = 1 to give

-\mu \frac{\vartheta _{2}-2\vartheta _{1}+1}{(\Delta X)^2}+\vartheta ^4_1=\frac{1}{2}\left\{1-\frac{1}{\left\{[B/(\Delta X)]^2+1\right\}^{1/2} } \right\} +\frac{\Delta XB^2}{2} \left\lgroup\frac{1}{2}\frac{1}{[B^2+(\Delta X)^2]^{3/2}} \right\rgroup -\sum\limits_{j=1}^{N-1}{\left[\mu (1-\epsilon )\frac{\vartheta _{j+1}-2\vartheta _j+\Theta _{j-1}}{(\Delta X)^2} -\Theta _j^4\right] } \frac{1}{\left\{B^2+[(j-1)\Delta X]^2\right\}^{3/2} } +\frac{1}{2}\left[-\mu (1-\epsilon )\frac{2(\vartheta _{N-1}-\vartheta_N)}{(\Delta X)^2}+\Theta ^4_N\right]\frac{1}{\left\{B^2+[(N-1)\Delta X]^2\right\}^{3/2} }                       (7.39)

For the infinite array of fins, ϑ_i = ϑ_{j =1} so that, gathering terms that have ϑ_j and ϑ_ j^ 4 and using ϑ_0 = 1, Equation 7.39 is written as (a repeated subscript denotes a summation over the values of that subscript, e.g.,  A _{1j}  ϑ_j=\sum\limits_{j=1}^{N}{A _{1j}  ϑ_j})

A _{1j}  ϑ_j+{B _{1j}  ϑ_j^4}=C_1                       (7.40)

where [let \ \widetilde{B}\equiv B^2\mu (1-\epsilon ) /2\Delta X ]

A_{11}=\widetilde{B}\left\{\frac{4}{(1-\epsilon )\Delta XB^2}-\frac{2}{B^3}+\frac{1}{[B^2+(\Delta X)^2]^{3/2}} \right\}

A_{12}=\widetilde{B}\left\{\frac{-2}{(1-\epsilon )\Delta XB^2}+\frac{1}{B^3}-\frac{2}{[B^2+(\Delta X)^2]^{3/2}}+ \frac{1}{[B^2+(2\Delta X)^2]^{3/2}} \right\}

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A_{1j}=\widetilde{B} \left\lgroup \frac{1}{\left\{B^2+[(j-2)\Delta X]^2\right\}^ {3/2}}-\frac{2}{\left\{B^2+[(j-1)\Delta X]^2\right\}^ {3/2}}+\frac{1}{\left[B^2+(j\Delta X)^2\right]^{3/2} } \right\rgroup          2< j< N-1

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A_{1(N-1)}=\widetilde{B} \left\lgroup \frac{1}{\left\{B^2+[(N-3)\Delta X]^2\right\}^ {3/2}}-\frac{2}{\left\{B^2+[(N-2)\Delta X]^2\right\}^ {3/2}}+\frac{1}{\left\{B^2+[(N-1)\Delta X]^2\right\}^{3/2} } \right\rgroup

A_{1N}=\widetilde{B} \left\lgroup \frac{1}{\left\{B^2+[(N-2)\Delta X]^2\right\}^ {3/2}}-\frac{2}{\left\{B^2+[(N-1)\Delta X]^2\right\}^ {3/2}} \right\rgroup

B_{1j}=\delta _{1j}-\frac{B^2\Delta X}{\beta _j\left\{B^2+[(j-1)\Delta X]^2 \right\}^{3/2} } ;      \begin{matrix} \beta _j=2, \ 1\leq j\lt N \\ \beta _j=4, j=N \end{matrix}

C_1=\frac{1}{2} \left\{1-\frac{\Delta X}{[B^2+(\Delta X)^2]^{1/2}} \right\} +\frac{\mu }{(\Delta X)^2}+\frac{B^2(\Delta X)}{4[B^2+(\Delta X)^2]^{3/2}} -\frac{\mu (1-\epsilon )}{2B(\Delta X)}

This is done for each element i, 1 ≤ i ≤ N, and a matrix equation is generated of the form

[A_{ij}][\vartheta _j]+[B_{_{ij}}][\vartheta ^4_j]=[C_i]         (7.41)

This is a set of nonlinear algebraic equations for the unknown temperatures ϑ_i = ϑ_{j=i}. Solution methods are in Section 7.8.