Question

Starting with an energy balance on the volume element, obtain the two- dimensional transient explicit finite difference equation for a general interior node in rectangular coordinates for \(T(x, y, t)\) for the case of constant thermal conductivity and uniform heat generation.

Short answer

Explain, in your own words, the process of deriving a two-dimensional transient explicit finite difference equation for a general interior node in rectangular coordinates for temperature T(x, y, t), considering constant thermal conductivity and uniform heat generation. First, the heat diffusion equation with constant thermal conductivity and uniform heat generation for temperature in rectangular coordinates is written down. Then, we discretize the temperature function using grid spacings in x and y directions, and time steps. After that, we discretize the derivatives in the heat diffusion equation using finite differences for both the time derivative and the second-order spatial derivatives. Finally, we substitute the discretized derivatives back into the original heat diffusion equation and reorganize the terms to obtain the transient explicit finite difference equation for a general interior node in rectangular coordinates.

Step-by-step solution

1. Heat Diffusion Equation

The heat diffusion equation with constant thermal conductivity \(k\) and uniform heat generation \(Q\) for temperature \(T(x, y, t)\) in rectangular coordinates is given by: \[ \frac{\partial T}{\partial t} = \frac{k}{\rho c_p}\left(\frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2}\right) + \frac{Q}{\rho c_p} \] Here, \(\rho\) is the density and \(c_p\) is the specific heat capacity.

2. Finite Difference Discretization

Now, we will discretize the temperature function \(T(x, y, t)\), considering the grid spacing \(\Delta x\) in the \(x\)-direction and \(\Delta y\) in the \(y\)-direction: \[i = 0, 1, \dots, N-1 \rightarrow x_i = i \Delta x\] \[j = 0, 1, \dots, M-1 \rightarrow y_j = j \Delta y\] And considering the time steps \(\Delta t\): \[n = 0, 1, \dots \rightarrow t_n = n \Delta t\] Then, \(T(x, y, t)\) can be discretized as \(T(i \Delta x, j \Delta y, n \Delta t) \approx T_{i,j}^n\).

3. Discretizing the Derivatives

Now we can rewrite the heat diffusion equation using finite differences for the derivatives: a) For the time derivative (first order forward-difference), \[\frac{\partial T}{\partial t} \approx \frac{T_{i,j}^{n+1} - T_{i,j}^n}{\Delta t}\] b) For the second-order spatial derivatives (central difference), \[\frac{\partial^2 T}{\partial x^2} \approx \frac{T_{i-1,j}^n - 2T_{i,j}^n + T_{i+1,j}^n}{(\Delta x)^2} \] \[\frac{\partial^2 T}{\partial y^2} \approx \frac{T_{i,j-1}^n - 2T_{i,j}^n + T_{i,j+1}^n}{(\Delta y)^2} \]

4. Combining the Discretized Derivatives

Now we can substitute the discretized derivatives back into the original heat diffusion equation: \[\frac{T_{i,j}^{n+1} - T_{i,j}^n}{\Delta t} = \frac{k}{\rho c_p}\left(\frac{T_{i-1,j}^n - 2T_{i,j}^n + T_{i+1,j}^n}{(\Delta x)^2} + \frac{T_{i,j-1}^n - 2T_{i,j}^n + T_{i,j+1}^n}{(\Delta y)^2}\right) + \frac{Q}{\rho c_p} \]

5. Transient Explicit Finite Difference Equation

Finally, we can reorganize the terms to obtain the transient explicit finite difference equation for a general interior node \((i,j)\) in rectangular coordinates: \[T_{i,j}^{n+1} = T_{i,j}^n + \Delta t \frac{k}{\rho c_p} \left(\frac{T_{i-1,j}^n - 2T_{i,j}^n + T_{i+1,j}^n}{(\Delta x)^2} + \frac{T_{i,j-1}^n - 2T_{i,j}^n + T_{i,j+1}^n}{(\Delta y)^2}\right) + \Delta t \frac{Q}{\rho c_p}\] This is the final two-dimensional transient explicit finite difference equation for \(T(x, y, t)\) considering constant thermal conductivity and uniform heat generation.