Question

Starting with an energy balance on the volume element, obtain the three- dimensional transient explicit finite difference equation for a general interior node in rectangular coordinates for \(T(x, y, z, t)\) for the case of constant thermal conductivity and no heat generation.

Short answer

The explicit finite difference equation for a general interior node in rectangular coordinates for temperature \(T(x, y, z, t)\) for the case of constant thermal conductivity and no heat generation can be written as: \(T(i, j, k, n+1) = T(i, j, k, n) + \Delta t \alpha(\frac{T(i+1, j, k, n) - 2T(i, j, k, n) + T(i-1, j, k, n)}{(\Delta x)^2} + \frac{T(i, j+1, k, n) - 2T(i, j, k, n) + T(i, j-1, k, n)}{(\Delta y)^2} + \frac{T(i, j, k+1, n) - 2T(i, j, k, n) + T(i, j, k-1, n)}{(\Delta z)^2})\) Here, \(\Delta t\) is the time step, \(\alpha\) is the thermal diffusivity, and \(i\), \(j\), and \(k\) are indices of grid points in the x, y, and z directions, respectively. The equation describes how the temperature at an interior node \((i, j, k)\) changes from time step \(n\) to \(n+1\).

Step-by-step solution

Write down the transient heat conduction equation

The transient heat conduction equation for constant thermal conductivity (\(k\)) and no heat generation is given as: \(\frac{\partial T}{\partial t} = \alpha(\frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2} + \frac{\partial^2 T}{\partial z^2})\) where \(\alpha = \frac{k}{\rho c_p}\) is the thermal diffusivity, \(\rho\) is the density, and \(c_p\) is the specific heat capacity at constant pressure.

Discretize the domain using grid points

We discretize the domain into a grid with uniform spacing \(\Delta x\), \(\Delta y\), and \(\Delta z\) along x, y, and z directions respectively. The grid points can be represented as \((i, j, k)\), corresponding to the spatial coordinates \((x_i, y_j, z_k)\) where \(i=1, 2, ..., N_x\), \(j=1, 2, ..., N_y\), and \(k=1, 2, ..., N_z\). The time variable is also discretized into time steps \(\Delta t\).

Apply central finite differences to the spatial second derivatives

We apply central finite differences to approximate the second derivatives of the temperature with respect to x, y, and z. For example, the second derivative with respect to x can be approximated as follows: \(\frac{\partial^2 T}{\partial x^2} \approx \frac{T(i+1, j, k, n) - 2T(i, j, k, n) + T(i-1, j, k, n)}{(\Delta x)^2}\) Similarly, the second derivatives with respect to y and z can be approximated as: \(\frac{\partial^2 T}{\partial y^2} \approx \frac{T(i, j+1, k, n) - 2T(i, j, k, n) + T(i, j-1, k, n)}{(\Delta y)^2}\) \(\frac{\partial^2 T}{\partial z^2} \approx \frac{T(i, j, k+1, n) - 2T(i, j, k, n) + T(i, j, k-1, n)}{(\Delta z)^2}\)

Apply forward finite difference to the time derivative

We apply forward finite difference to approximate the time derivative of the temperature: \(\frac{\partial T}{\partial t} \approx \frac{T(i, j, k, n+1) - T(i, j, k, n)}{\Delta t}\)

Substitute the finite difference approximations into the transient heat conduction equation

Substitute the finite difference approximations obtained in Steps 3 and 4 into the transient heat conduction equation: \(\frac{T(i, j, k, n+1) - T(i, j, k, n)}{\Delta t} = \alpha(\frac{T(i+1, j, k, n) - 2T(i, j, k, n) + T(i-1, j, k, n)}{(\Delta x)^2} + \frac{T(i, j+1, k, n) - 2T(i, j, k, n) + T(i, j-1, k, n)}{(\Delta y)^2} + \frac{T(i, j, k+1, n) - 2T(i, j, k, n) + T(i, j, k-1, n)}{(\Delta z)^2})\)

Rearrange the equation to find \(T(i, j, k, n+1)\)

We can rearrange the above equation to obtain the temperature at the next time step for the interior node \((i, j, k)\): \(T(i, j, k, n+1) = T(i, j, k, n) + \Delta t \alpha(\frac{T(i+1, j, k, n) - 2T(i, j, k, n) + T(i-1, j, k, n)}{(\Delta x)^2} + \frac{T(i, j+1, k, n) - 2T(i, j, k, n) + T(i, j-1, k, n)}{(\Delta y)^2} + \frac{T(i, j, k+1, n) - 2T(i, j, k, n) + T(i, j, k-1, n)}{(\Delta z)^2})\) This is the three-dimensional transient explicit finite difference equation for a general interior node in rectangular coordinates for \(T(x, y, z, t)\) for the case of constant thermal conductivity and no heat generation.