Question

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.

Short answer

The criterion used to determine when steady-state conditions have been reached in the plate is when the maximum temperature change in the plate is less than a certain chosen threshold (typically 0.5% or 1% relative to the maximum temperature swing) for each time step.

Step-by-step solution

Mesh and Time Step Calculation

First, we need to calculate how many cells or nodes there are in the plate. Since the thickness L is 9 cm and the nodal spacing Δx is 1.5 cm, we will have 7 nodes. Next, the stability criteria for the explicit finite difference method is given by \(F=\frac{\alpha \Delta t}{(\Delta x)^{2}}\leq\frac{1}{2}\), where F is the Fourier Number, α is the thermal diffusivity, Δt is the time step, and Δx is the nodal spacing. Solving for Δt, we have $$ \Delta t \leq \frac{1}{2} \cdot (\Delta x)^{2} / \alpha = \frac{1}{2} \cdot (0.015 \, \text{m})^2 / 12.5 \times 10^{-6} \, \text{m}^2/\text{s} $$ Calculating the allowed time step Δt, we get a value of 9 seconds.

Explicit Finite Difference Approach

Now we apply the explicit finite difference approach for the 7 nodes. Use the formulas below that correspond to the initial and boundary conditions: For the initial condition (\(t=0\)), \(T(x) = 100 °C\) for all \(x\). For the left boundary condition (insulated), derivative vanishes, so \(T_1^{n+1}=T_2^{n+1}\) For the right boundary condition (convection), using the extra node method, \(T_{n+1}^{n+1}=T_{n-1}^{n+1} + 2\frac{h\Delta x}{k}\left(T_\infty - T_n^{n+1}\right)\) For the internal nodes, $$ T_i^{n+1} = T_i^n + F \left(T_{i+1}^n - 2T_i^n + T_{i-1}^n\right) + \frac{\Delta t \cdot \dot{e}}{k} $$

Temperature Distribution After 5 Minutes

Now, we apply the explicit finite difference approach for every node in the plate to find the temperature distribution after 5 minutes: 1. Start with the initial temperature distribution given by \(T(x) = 100 °C\). 2. Determine temperatures for each node at each time step using the explicit finite difference approach. 3. Repeat the process for 5 minutes or 300 seconds (300 s / 9 s per time step = 33 time steps). 4. After 33 time steps, we obtain the temperature distribution within the plate for each node.

Time to Reach Steady-State Conditions

Compute the maximum allowed time step, which is based on the threshold for when the temperature distribution changes less than a certain amount (typically 0.5% or 1% relative to the maximum temperature swing): 1. Apply the explicit finite difference approach until the maximum temperature change in the plate is less than the chosen threshold for each time step. 2. Note down the time taken for the temperature changes to decrease below the threshold. 3. The time taken to reach steady-state conditions is the time for the temperature changes to fall below the threshold. By following the above steps, we can determine the temperature distribution in the plate after 5 minutes, as well as the time it takes for steady-state conditions to be reached.