Question

Consider steady two-dimensional heat transfer in a long solid bar of square cross section in which heat is generated uniformly at a rate of $\dot{e}=0.19 \times 10^{5} \mathrm{Btu} / \mathrm{h} \cdot \mathrm{ft}^{3}$. The cross section of the bar is \(0.5 \mathrm{ft} \times 0.5 \mathrm{ft}\) in size, and its thermal conductivity is $k=16 \mathrm{Btu} / \mathrm{h} \cdot \mathrm{ft} \cdot{ }^{\circ} \mathrm{F}$. All four sides of the bar are subjected to convection with the ambient air at \(T_{\infty}=70^{\circ} \mathrm{F}\) with a heat transfer coefficient of $h=7.9 \mathrm{Btu} / \mathrm{h} \cdot \mathrm{ft}^{2} \cdot{ }^{\circ} \mathrm{F}$. Using the finite difference method with a mesh size of \(\Delta x=\Delta y=0.25 \mathrm{ft}\), determine \((a)\) the temperatures at the nine nodes and \((b)\) the rate of heat loss from the bar through a 1-ft-long section.

Short answer

Question: Using the finite difference method, find (a) the temperatures at the nine nodes of a steady-state heat conduction problem with internal heat generation in a square bar with a mesh size of 0.25 ft, and (b) the rate of heat loss from the bar through a 1-ft-long section. Answer: (a) Solve the system of 9 linear equations derived from the finite difference method for the 9 unknown nodal temperatures using a numerical method or a tool like MATLAB or Python. (b) Calculate the rate of heat loss through a 1-ft-long section using the equation \(q = h(PA)(T_s - T_\infty)\), where \(T_s\) is the average surface temperature found from the nodal temperatures.

Step-by-step solution

Set up the mesh grid and the governing finite difference equation with boundary conditions

First, we need to establish a mesh grid for our problem. With a mesh size of \(\Delta x=\Delta y=0.25 \mathrm{ft}\), we will have a 3x3 grid consisting of 9 nodes in total. Label them as \(T_{i,j}\), where \(i\) is the row number and \(j\) is the column number. Next, we need to derive the governing finite difference equation for our problem. Since we have steady two-dimensional conduction with internal heat generation, we can use the following equation: \(T_{i,j} = \dfrac{1}{4}\left(T_{i-1,j} + T_{i+1,j} + T_{i,j-1} + T_{i,j+1}\right) + \dfrac{\Delta x^2}{4k}\dot{e}\) For the boundary nodes, we also need to consider the convective heat transfer. We can combine conduction and convection terms using the following equation: \(-k(\dfrac{T_{i+1,j} - T_{i-1,j}}{2\Delta x}) - k(\dfrac{T_{i,j+1} - T_{i,j-1}}{2\Delta y}) = h(T_{i,j} - T_\infty)\)

Derive the equations for each node

First, we'll derive the equations for the corner nodes considering both conduction and convection. Then, we'll derive the equations for the edge and interior nodes considering only conduction. 1. Corner nodes: \((1,1), (1,3), (3,1), (3,3)\) - \(T_{1,1} = \dfrac{1}{4}(T_{1,2} + T_{2,1}) + \dfrac{\Delta x^2}{4k}\dot{e} + \dfrac{h\Delta x^2}{k}T_\infty\) - \(T_{1,3} = \dfrac{1}{4}(T_{1,2} + T_{2,3}) + \dfrac{\Delta x^2}{4k}\dot{e} + \dfrac{h\Delta x^2}{k}T_\infty\) - \(T_{3,1} = \dfrac{1}{4}(T_{2,1} + T_{3,2}) + \dfrac{\Delta x^2}{4k}\dot{e} + \dfrac{h\Delta x^2}{k}T_\infty\) - \(T_{3,3} = \dfrac{1}{4}(T_{2,3} + T_{3,2}) + \dfrac{\Delta x^2}{4k}\dot{e} + \dfrac{h\Delta x^2}{k}T_\infty\) 2. Edge nodes: \((1,2), (2,1), (3,2), (2,3)\) - \(T_{1,2} = \dfrac{1}{4}(T_{1,1} + T_{1,3} + T_{2,2}) + \dfrac{\Delta x^2}{4k}\dot{e} + \dfrac{h\Delta x^2}{2k}T_\infty\) - \(T_{2,1} = \dfrac{1}{4}(T_{1,1} + T_{3,1} + T_{2,2}) + \dfrac{\Delta x^2}{4k}\dot{e} + \dfrac{h\Delta x^2}{2k}T_\infty\) - \(T_{3,2} = \dfrac{1}{4}(T_{1,2} + T_{3,1} + T_{3,3}) + \dfrac{\Delta x^2}{4k}\dot{e} + \dfrac{h\Delta x^2}{2k}T_\infty\) - \(T_{2,3} = \dfrac{1}{4}(T_{1,3} + T_{2,2} + T_{3,3}) + \dfrac{\Delta x^2}{4k}\dot{e} + \dfrac{h\Delta x^2}{2k}T_\infty\) 3. Interior nodes: \((2,2)\) - \(T_{2,2} = \dfrac{1}{4}(T_{1,2} + T_{2,1} + T_{2,3} + T_{3,2}) + \dfrac{\Delta x^2}{4k}\dot{e}\)

Solve the system of equations to obtain the temperatures at all nodes

Now we need to solve the system of 9 linear equations for the 9 unknown nodal temperatures. This can be done using various numerical methods, such as the Gauss-Seidel method, the Jacobi method or using some numerical tools like MATLAB or Python. After solving the system, we will have the temperatures at all the 9 nodes.

Calculate the rate of heat loss through a 1-ft-long section

Once we have the temperatures at all the nodes, we can calculate the heat loss through a 1-ft-long section of the bar using the following equation: \(q = h(PA)(T_s - T_\infty)\) where \(P\) is the perimeter, \(A\) is the area of a 1-ft-long section, and \(T_s\) is the average surface temperature. The average surface temperature can be calculated by taking the average of the temperatures at the four edges: \((T_{1,2} + T_{2,1} + T_{3,2} + T_{2,3}) / 4\).