Question

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

Short answer

Based on the step-by-step solution given, the short answer is: The two-dimensional transient implicit finite difference equation for a general interior node (i, j) in rectangular coordinates with constant thermal conductivity and no heat generation is given by: \(\rho c_p\frac{T_{i,j}^{n+1}-T_{i,j}^n}{\Delta t} = k\left(\frac{T_{i+1,j}^{n+1} - 2T_{i,j}^{n+1} + T_{i-1,j}^{n+1}}{(\Delta x)^2} + \frac{T_{i,j+1}^{n+1}-2T_{i,j}^{n+1} + T_{i,j-1}^{n+1}}{(\Delta y)^2}\right)\) This equation is derived by performing an energy balance on a volume element, discretizing the heat diffusion equation using the finite difference method, and applying appropriate boundary conditions.

Step-by-step solution

Energy Balance on a Volume Element

To derive the two-dimensional transient heat diffusion equation, consider an infinitesimal rectangular volume element with dimensions \(\Delta x\), \(\Delta y\), and \(\Delta z\) in the x, y, and z directions. The energy balance on this volume element can be written as: Heat in - Heat out + Heat generated = Change in internal energy Since there is no heat generation, the equation simplifies to: Heat in - Heat out = Change in internal energy

Expressing Heat Flux Terms

Heat transfer occurs through the volume element's faces due to conduction. We can express the heat flux terms for each direction using Fourier's Law of Heat Conduction: \(q_x = -k\frac{\partial T}{\partial x}\) and \(q_y = -k\frac{\partial T}{\partial y}\) where \(k\) is the constant thermal conductivity and \(T\) refers to the temperature.

Writing the Energy Balance Equation

Now, we can substitute the heat flux terms into the energy balance and express the equation in terms of the partial temperature derivatives and volume element: \((-q_x\Delta y\Delta z) - (-q_x + \Delta q_x)\Delta y\Delta z + (-q_y\Delta x\Delta z) - (-q_y+\Delta q_y)\Delta x\Delta z = \rho c_pV\frac{\partial T}{\partial t}\) where \(\rho\) is the density, \(c_p\) is the specific heat, \(V\) is the volume of the element (\(\Delta x\Delta y\Delta z\)), and \(\Delta q_x\) and \(\Delta q_y\) are the differences in heat fluxes in the x and y directions, respectively.

Simplifying the Energy Balance Equation

Upon simplifying the above equation, we get the two-dimensional transient heat diffusion equation: \(\rho c_p\frac{\partial T(x, y, t)}{\partial t} = k\left(\frac{\partial^2T(x, y, t)}{\partial x^2} + \frac{\partial^2T(x, y, t)}{\partial y^2}\right)\)

Finite Difference Approximations

We will now discretize the heat diffusion equation using an implicit finite difference method, which takes into account temperatures from the next time step. We'll represent the partial derivatives with finite difference approximations, setting up a system of algebraic equations to solve iteratively. For a general interior node with coordinates \((i, j)\) and time step \((n+1)\), we make the following approximations: \(\frac{\partial^2 T}{\partial x^2} \approx \frac{T_{i+1,j}^{n+1} - 2T_{i,j}^{n+1} + T_{i-1,j}^{n+1}}{(\Delta x)^2}\) \(\frac{\partial^2 T}{\partial y^2} \approx \frac{T_{i,j+1}^{n+1}-2T_{i,j}^{n+1} + T_{i,j-1}^{n+1}}{(\Delta y)^2}\) \(\frac{\partial T}{\partial t}\approx \frac{T_{i,j}^{n+1}-T_{i,j}^n}{\Delta t}\)

Discretized Two-Dimensional Transient Implicit Finite Difference Equation

Substituting the finite difference approximations from Step 5 into the heat diffusion equation, we get the discretized equation for a general interior node: \(\rho c_p\frac{T_{i,j}^{n+1}-T_{i,j}^n}{\Delta t} = k\left(\frac{T_{i+1,j}^{n+1} - 2T_{i,j}^{n+1} + T_{i-1,j}^{n+1}}{(\Delta x)^2} + \frac{T_{i,j+1}^{n+1}-2T_{i,j}^{n+1} + T_{i,j-1}^{n+1}}{(\Delta y)^2}\right)\) This is the two-dimensional transient implicit finite difference equation for a general interior node in rectangular coordinates for the temperature distribution \(T(x, y, t)\), with constant thermal conductivity and no heat generation.

Boundary Conditions

In order to solve the above equation system, appropriate boundary conditions must be applied, such as Dirichlet (specified temperature), Neumann (specified heat flux), or mixed/Robin (combination of temperature and heat flux) boundary conditions. Boundary conditions might depend on the specific problem in which this equation is being applied.