Question

Cooling fins are used to increase the area available for heat transfer between metal walls and poorly conducting fluids such as gases. A rectangular fin is shown in the following figure.

Metal Wall Z-direction 2B Cooling Fin

To design a cooling fin and calculate the fin efficiency one must first calculate the temperature profile in the fin.

If L>>B, no heat is lost from the end or from the edges, and the heat flux at the surface is given by:

$q=\eta (T-Ta)$

in which the convective heat transfer coefficient ? is constant as is the surrounding fluid temperature Ta. Assume that the temperature T changes only along the z-direction.

a. Prove that at the steady state the governing differential equation describing T is as follows:

$\frac{\partial^2 T}{\partial z^2} = \frac{\eta}{kB}(T-Ta)$

where k is thermal conductivity of the fin, constant.

b. If the boundary conditions are as follows, classify the boundary conditions.

$T(0)=Tw$

$\frac{\mathrm{d}T }{\mathrm{d} z} at L = 0$

Using the following dimensionless parameters:

$\theta =\frac{T-Ta}{Tw-Ta}$

$x=\frac{z}{L}$

$H=\sqrt{\frac{nL^{2}}{kB}}$

Prove that the ODE with the boundary conditions in step a and b can be formulated as

$\frac{\partial^2\theta }{\partial x^2}=H^{2}\theta$

$\theta (0)=1$

$\frac{\mathrm{d}\theta }{\mathrm{d} x} at 1=0$

d. Write the method to solve the ODE from step c by Finite Difference method. Please provide all
the equations and related info by using N=3 (You don’t need to run it in programs.).

Answer 1

12 2 imcho

2 BK ス/レ-ー 0%, dR ス ん

だ dz: - 쓰 ldy 广dep -ㄧㄧㄧㄧㄧㄧ BK on T1

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