Question

Consider a large uranium plate of thickness \(5 \mathrm{~cm}\) and thermal conductivity \(k=34 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\) in which heat is generated uniformly at a constant rate of $\dot{e}=6 \times 10^{5} \mathrm{~W} / \mathrm{m}^{3}$. One side of the plate is insulated, while the other side is subjected to convection to an environment at \(30^{\circ} \mathrm{C}\) with a heat transfer coefficient of $h=60 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\(. Considering six equally spaced nodes with a nodal spacing of \)1 \mathrm{~cm},(a)$ obtain the finite difference formulation of this problem and (b) determine the nodal temperatures under steady conditions by solving those equations.

Short answer

Answer: The governing equation for this problem, considering convection and heat generation, is given by: \(\frac{d^2 T}{d x^2} - \frac{\dot{e}}{k} = 0\)

Step-by-step solution

Understanding the Finite Difference Method (FDM)

FDM is a numerical technique that can be used to approximate solutions of partial differential equations (PDEs) arising from heat transfer problems. It discretizes the problem domain into a finite number of nodes and represents the PDEs at these nodes using Taylor series expansions. By solving the resulting linear algebraic system, we can compute the nodal values.

Discretize the Uranium Plate with Nodes

Divide the uranium plate into six equally spaced nodes, with a spacing of 1cm. Let's denote the nodal spacing as \(\Delta x\). The nodal temperatures are denoted by \(T_i\), where \(i\) is the node number (\(i=1,2,3,4,5,6\)), and \(T_1\) corresponds to the insulated side of the plate.

Derive the Finite Difference Equation of the Problem

For this problem, considering convection and heat generation, the governing equation is given by: \(\frac{d^2 T}{d x^2} - \frac{\dot{e}}{k} = 0\) Applying the finite difference formulation for the second-order derivative of temperature with respect to x, we have: \(\frac{T_{i+1} - 2T_i + T_{i-1}}{(\Delta x)^2} - \frac{\dot{e}}{k} = 0\) for \(i = 2, 3, 4, 5\) Except at \(i=1\) and \(i=6\) nodes, where there are boundary conditions. At the insulated side (node 1), the boundary condition is \(-k(dT_1/dx) = 0\), which results in \(T_{1} = T_{2}\). At the convective side (node 6), the boundary condition is \(-k(dT_{6}/dx) = h(T_6 - T_{\infty})\), where \(T_{\infty}\) is the environment temperature. Applying finite difference approximation, we have: \(-k \frac{T_6 - T_5}{\Delta x} = h(T_6 - T_{\infty})\) Now, we have derived the finite difference equations for the problem.

Formulate the Linear Algebraic System

Express the finite difference equations for all six nodes in terms of unknown nodal temperatures: (1) \(T_{1} = T_{2}\) (2) \(T_{2} - 2T_3 + T_4 = \frac{\dot{e}(\Delta x)^2}{k}\) (3) \(T_{3} - 2T_4 + T_5 = \frac{\dot{e}(\Delta x)^2}{k}\) (4) \(\frac{k(T_6 - T_5)}{\Delta x} = h(T_6 - T_{\infty})\)

Solve the Linear Algebraic System

Solving these equations simultaneously, we can obtain the nodal temperatures under steady-state conditions. You can use a scientific calculator, software, or even manual calculation to solve this linear system. Knowing the provided values: \(k = 34 \mathrm{~W~m^{-1}~K^{-1}}\), \(\dot{e} = 6 \times 10^5 \mathrm{~W~m^{-3}}\), \(\Delta x = 0.01 \mathrm{~m}\), \(h = 60 \mathrm{~W~m^{-2}~K^{-1}}\), and \(T_{\infty} = 30^{\circ} \mathrm{C}\), you can plug in these values and find the nodal temperatures. With this, the finite difference formulation of the problem is obtained (part a), and the nodal temperatures under steady conditions are determined by solving these finite difference equations (part b).