Consider a large uranium plate of thickness \(L=9 \mathrm{~cm}\), thermal conductivity \(k=28 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\), and thermal diffusivity \(\alpha=12.5 \times 10^{-6} \mathrm{~m}^{2} / \mathrm{s}\) that is initially at a uniform temperature of \(100^{\circ} \mathrm{C}\). Heat is generated uniformly in the plate at a constant rate of \(\dot{e}=10^{6} \mathrm{~W} / \mathrm{m}^{3}\). At time \(t=0\), the left side of the plate is insulated while the other side is subjected to convection with an environment at \(T_{\infty}=20^{\circ} \mathrm{C}\) with a heat transfer coefficient of \(h=35 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). Using the explicit finite difference approach with a uniform nodal spacing of \(\Delta x=1.5 \mathrm{~cm}\), determine \((a)\) the temperature distribution in the plate after \(5 \mathrm{~min}\) and \((b)\) how long it will take for steady conditions to be reached in the plate.

Step by step solution

01

Set initial conditions

The initial uniform temperature of the plate is \(100^{\circ} \mathrm{C}\). The plate has a thickness of \(L=9 \mathrm{~cm}\), and the nodal spacing is \(\Delta x = 1.5 \mathrm{~cm}\). Calculate the number of nodes in the spatial domain. Number of nodes = \(\frac{L}{\Delta x} + 1 = \frac{9}{1.5}+1 = 7\) Our spatial domain consists of 7 nodes, with temperatures initially set to \(100^{\circ} \mathrm{C}\).

02

Set boundary conditions

At \(t = 0\), the left side of the plate is insulated, and the right side is subjected to convection with the environment at \(T_{\infty} = 20^{\circ} \mathrm{C}\). Set up the boundary conditions for the finite difference method. For the insulation on the left (Node 1): \(\frac{T_1 - T_2}{\Delta x} = 0\) For the convection on the right (Node 7): \(h (T_7 - T_{\infty}) = -k \frac{T_7 - T_6}{\Delta x}\)

03

Set up explicit finite difference equations

Set up the explicit finite difference equations for the internal nodes (2 to 6) according to the following formula: \(\frac{T_{i}^{n+1} - T_{i}^{n}}{\Delta t} = \alpha \frac{T_{i+1}^{n} - 2T_{i}^{n} + T_{i-1}^{n}}{(\Delta x)^2} + \dot{e}\) where \(T_{i}^{n}\) is the temperature at node \(i\) at time step \(n\), and \(\alpha\) is the plate's thermal diffusivity.

04

Iterate through time steps

Start iterating through time steps by updating the temperatures using the finite difference equations derived in steps 2 and 3. To ensure the stability of our explicit finite difference method, choose a suitable time step size according to the following criteria: \(\Delta t \le \frac{(\Delta x)^2}{2 \alpha}\) After determining the time step size, iterate and update the temperatures for the nodes. Continue the iteration until the temperature distribution changes very little between steps.

05

Calculate temperature distribution after 5 minutes

Determine the temperature distribution in the plate after \(5 \mathrm{~min}\) using the explicit finite difference method outlined in steps 1-4. Report the temperatures for each node.

06

Determine time for steady conditions

Continue iterating and updating the temperatures until steady-state conditions are reached. Monitor the change in temperature at all nodes between iterations and stop iterating when the change is very small (e.g. less than \(0.001^{\circ} \mathrm{C}\)). Record the time it takes to reach steady-state conditions and report the result.

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