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 $h=7.9 \mathrm{Btu} / \mathrm{h} \cdot \mathrm{ft}^{2}{ }^{\circ} \mathrm{F}$. 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

Please provide a short answer to this heat transfer problem, given that the temperatures at the nodes are as follows: Node 1: 130°F Node 2: 142°F Node 3: 139°F Node 4: 157°F Node 5: 146°F Node 6: 148°F Node 7: 160°F Node 8: 150°F Node 9: 154°F Use the heat transfer coefficients provided earlier to calculate the total heat loss from the bar through a 1-ft-long section.

Step-by-step solution

Convert heat generation rate to appropriate units

We are given the heat generation rate (\(\dot{e}\)) in Btu/h·ft³. To match the other given units, we will convert it to Btu/h·in³: \(\dot{e} = 0.19 \times 10^{5} \frac{\mathrm{Btu}}{\mathrm{h} \cdot \mathrm{ft}^3} \times \frac{1 \, \mathrm{ft}}{12 \, \mathrm{in}} \times \frac{1 \, \mathrm{ft}}{12 \, \mathrm{in}} \times \frac{1 \, \mathrm{ft}}{12 \, \mathrm{in}} = \frac{0.19 \times 10^{5}}{(12)^3} \frac{\mathrm{Btu}}{\mathrm{h} \cdot \mathrm{in}^3}\)

Set up the finite difference equations

Given the solid bar is of square cross-section(0.5 ft x 0.5 ft), the mesh size \(\Delta x = \Delta y = 0.25\) ft, we have 9 nodes and require 9 equations for each node. For interior nodes, we can apply 2D heat conduction equation: \(\left(\frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2}\right) = \frac{\dot{e}}{k}\) For boundary nodes, convective boundary condition will be used: \(q = h(T - T_{\infty})\), where \(q = -k \frac{\partial T}{\partial n}\)

Solve the system of linear equations

Now we have 9 finite difference equations for the 9 nodes. Set up a system of linear equations with corresponding node temperatures as unknowns and use the most convenient method (Gaussian elimination, Cholesky decomposition, etc.) to solve them.

Calculate heat loss

To calculate the total heat loss from the bar through a 1-ft-long section, we need to find the heat flux at each of the four sides, considering convection. Heat loss from a side: $$Q_{\text{side}} = h \, \text{(Side Area)} (T_{\text{node}} - T_{\infty})$$ $$Q_{\text{total}} = \sum_{i=1}^{4} Q_{\text{side}_i} = h \cdot (4 \times 0.5 \times 1 \text{ft}^2) \sum_{i=1}^{4}(T_{\text{node}_i} - T_{\infty})$$ The calculated temperatures from step 3 at the nodes of each side can be plugged into the equation to find the total heat loss from the bar through a 1-ft-long section.