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

 

[ravenprp]

Hi there,

Solving 2D conduction problems in Matlab can be tricky, but with the right approach, you can find a solution. Here are some tips to help you solve this specific problem:

1. Understand the problem: Before jumping into coding, make sure you understand the problem and its requirements. In this case, you have a long bar with three sides at a constant temperature and the fourth side exposed to a fluid with a known convection heat transfer coefficient. The goal is to determine the midpoint temperature and heat transfer rate between the bar and the fluid.

2. Choose an appropriate numerical technique: There are various numerical techniques that can be used to solve conduction problems, such as finite difference, finite element, and boundary element methods. For this problem, it is recommended to use the finite difference method, as it is relatively easy to implement in Matlab.

3. Define the grid: As mentioned in the problem, the grid spacing is given as 0.2m. You can define a grid using the linspace function in Matlab, with the appropriate number of points to cover the length of the bar.

4. Set boundary conditions: Since three sides of the bar are at a constant temperature, you can set these as boundary conditions. The fourth side, exposed to the fluid, will have a convection boundary condition. You can use the built-in function convection2d to define this boundary condition.

5. Write the governing equations: In this problem, you will need to solve the 2D heat conduction equation, which can be written as a system of linear equations. You can use the built-in function laplacian2d to construct the coefficient matrix for the system.

6. Solve the equations: Once you have defined the grid, boundary conditions, and the governing equations, you can solve the system of equations using the backslash operator (\) in Matlab. This will give you the temperature at each grid point.

7. Calculate the heat transfer rate: To calculate the heat transfer rate, you can use the built-in function gradient2d to calculate the temperature gradient at each grid point. Then, you can use the convection heat transfer coefficient to calculate the heat transfer rate at the exposed side of the bar.

8. Repeat with a finer grid: To answer part b) of the problem, you will need to repeat the above steps with a finer grid spacing. This will give you a more accurate solution, but it will also require more computational time.