Solving 2D Conduction Problem with Matlab

[sys86]

Hi. Everyone

I have stuck in the problem of 2D conduction problem by using matlab, here is the following question:

Consider a long bar of square cross section (1.0 m to the side) and of thermal conductivity 2 W/m.K. Three of these sides are maintained at a uniform temperature of 300°C. The fourth side is exposed to a fluid at 100°C for which the convection heat transfer coefficient is 10 W/m2.K.

a)Using an appropriate numerical technique with a grid spacing of 0.2m, determine the midpoint temperature and heat transfer rate between the bar and the fluid per unit length of the bar.
b)Reducing the grid spacing by a factor of 2, determine the midpoint temperature and heat transfer rate. Plot the corresponding temperature distribution across the surface exposed to the fluid. Also, plot the 200 and 250°C isotherms.

Hopefully that anyone can give me some tips to do it. Thanks.

 

[lofgran]

Hello,

Thank you for sharing your question with us. Solving 2D conduction problems using numerical techniques can be challenging, but with the right approach and tools, it can be done effectively. Here are some tips to help you solve this problem in MATLAB:

1. Use the finite difference method: The finite difference method is a commonly used numerical technique for solving conduction problems. It involves discretizing the domain into a grid and using finite differences to approximate the derivatives in the governing equations. This method is well-suited for 2D problems and can be easily implemented in MATLAB.

2. Define the problem parameters: Before starting your code, make sure to define all the necessary parameters such as the dimensions of the bar, thermal conductivity, and convection heat transfer coefficient. This will make it easier to modify your code if needed.

3. Create a grid: As mentioned earlier, the finite difference method involves discretizing the domain into a grid. In this case, you can create a grid with a spacing of 0.2m using the "meshgrid" function in MATLAB.

4. Set boundary conditions: In this problem, three sides of the bar are maintained at a uniform temperature of 300°C, while the fourth side is exposed to a fluid at 100°C. These are known as boundary conditions and need to be specified in your code.

5. Define the governing equations: The governing equation for heat conduction is the heat diffusion equation, which can be written as ∂T/∂t = α (∂^2T/∂x^2 + ∂^2T/∂y^2), where T is the temperature, t is time, α is the thermal diffusivity, and x and y are the spatial coordinates. In this case, since the problem is steady-state, the time derivative can be ignored.

6. Implement the finite difference method: Using the defined grid and boundary conditions, you can now implement the finite difference method to solve for the temperature distribution in the bar. This can be done using a for loop or vectorized operations in MATLAB.

7. Calculate heat transfer rate: Once you have solved for the temperature distribution, you can calculate the heat transfer rate using the heat transfer equation, Q = hA(Tb - Tf), where Q is the heat transfer rate, h is the convection heat transfer coefficient, A is the surface area, Tb is the temperature of the bar, and Tf is