Question

The unsteady forward-difference heat conduction for a constant-area \((A)\) pin fin with perimeter \(p\), when exposed to air whose temperature is \(T_{0}\) with a convection heat transfer coefficient of \(h\), is $$ \begin{aligned} T_{m}^{*+1}=& \frac{k}{\rho c_{p} \Delta x^{2}}\left[T_{m-1}^{*}+T_{m+1}^{*}+\frac{h p \Delta x^{2}}{A} T_{0}\right] \\\ &-\left[1-\frac{2 k}{\rho c_{p} \Delta x^{2}}-\frac{h p}{\rho c_{p} A}\right] T_{m}^{*} \end{aligned} $$ In order for this equation to produce a stable solution, the quantity $\frac{2 k}{\rho c_{p} \Delta x^{2}}+\frac{h p}{\rho c_{p} A}$ must be (a) negative (b) zero (c) positive (d) greater than 1 (e) less than 1

Short answer

Answer: The stability condition for the given equation is that the sum of the terms (S) must be positive and less than 1 (option e).

Step-by-step solution

Identify given equation and terms

The given equation is: $$ T_{m}^{*+1} = \frac{k}{\rho c_{p} \Delta x^{2}}\left[T_{m-1}^{*}+T_{m+1}^{*}+\frac{h p \Delta x^{2}}{A} T_{0}\right] -\left[1-\frac{2 k}{\rho c_{p} \Delta x^{2}}-\frac{h p}{\rho c_{p} A}\right] T_{m}^{*} $$ where - \(T_m^*\) represents the temperature at location \(m\) and time \(t^*\), - \(\rho\), \(c_p\), \(k\) are the material properties (density, specific heat, and thermal conductivity, respectively), - \(\Delta x\) is the spatial step size, - \(h\) is the convection heat transfer coefficient, - \(p\) is the perimeter of the pin fin, and - \(A\) is the constant-area of the pin fin. The stability condition is related to the sum of the following two terms: $$ \frac{2k}{\rho c_p \Delta x^2} + \frac{hp}{\rho c_p A} $$

Simplify and compute the sum of the terms

Let's simplify the sum of the terms given and compute its value: $$ S = \frac{2k}{\rho c_p \Delta x^2} + \frac{hp}{\rho c_p A} $$ Factor out common terms: $$ S = \frac{2k + hp\Delta x^2 / A}{\rho c_p \Delta x^2} $$ Now, we need to determine the condition for stability.

Evaluate the stability condition

To produce a stable solution, we must find whether the sum of these terms \(S\) fulfills one of the following conditions: (a) negative (b) zero (c) positive (d) greater than 1 (e) less than 1 Notice that \(2k\), \(hp\Delta x^2 / A\), \(\rho\), and \(c_p \Delta x^2\) are all positive since they represent physical properties and steps. Thus, the sum \(S\) must be positive as the sum of positive terms. Therefore, the only possible conditions are (c), (d), or (e). It tells us that condition (a) and (b) are incorrect.

Eliminate remaining options

We have left options (c), (d), and (e). Since all terms in the denominator and numerator are positive, that means that the sum must be greater than 0. So, it is not zero, nor can it be negative. Now, let's look at the term within the brackets of the given equation: $$ 1 - \frac{2k}{\rho c_p \Delta x^2} - \frac{hp}{\rho c_p A} $$ In order to produce a stable solution, this term must be positive if the given equation converges to correct values. So, given that \(1 > \frac{2k}{\rho c_p \Delta x^2} + \frac{hp}{\rho c_p A}\), it's clear that the sum must be less than 1. From above, the sum \(S\) is positive and less than 1. This leads us to the conclusion that the stability condition is given by option (e), less than 1.