# Need help for heat conduction problem with multiple BC

• bancux
In summary, the speaker is seeking help with solving a problem involving heat conduction within a rectangular area. They have provided relevant equations and boundary conditions, but are struggling with the differential equation. They also mention that the heat transfer coefficient is dependent on x-location and time and will be simulated as a periodic function.
bancux
Dear all, I have difficulty in solving this problem (see the figure in attached thumbnails)

I have a rectangular shape with length/height of L and the thickness/width of $$\delta$$
Within the rectangular area, a heat conduction occurred. I would like to determine the temperature profile within the rectangular.

Based on the right hand side figure, I take a small cell with size of dx times dy. Applying the energy balance within the cell.

I have attached all the equation that I can think of, and also the boundary conditions. I have difficulty in solving the differential equation.

note:

$$\lambda$$ is thermal conductivity
$$\alpha$$ is heat transfer coefficient, it is suppose to be dependent on x-location and time. I am going to simulate the heat transfer coefficient as a periodic function.

#### Attachments

• schema1.jpg
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• equation1.jpg
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• BC.jpg
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Hmmm...

## 1. What is heat conduction and how does it work?

Heat conduction is the transfer of thermal energy from one point to another within a solid or between two solids that are in direct contact. It occurs due to the movement of heat from higher temperature regions to lower temperature regions until an equilibrium is reached. This process is governed by the laws of thermodynamics.

## 2. What are the common boundary conditions for heat conduction problems?

The most common boundary conditions for heat conduction problems are constant temperature, constant heat flux, and adiabatic (no heat transfer) conditions. Other boundary conditions may include convective heat transfer, radiation heat transfer, and phase change (e.g. melting or boiling) conditions.

## 3. How do you handle multiple boundary conditions in a heat conduction problem?

Multiple boundary conditions can be handled by dividing the problem into smaller sub-problems, each with a different set of boundary conditions. Each sub-problem can then be solved separately and the overall solution can be obtained by combining the solutions from each sub-problem.

## 4. What are the key equations used to solve heat conduction problems?

The key equations used to solve heat conduction problems are Fourier's law of heat conduction, which relates heat flux to temperature gradient; the heat conduction equation, which describes the rate of heat transfer within a solid; and the boundary conditions, which specify the conditions at the boundaries of the problem.

## 5. What are some common methods for solving heat conduction problems?

Some common methods for solving heat conduction problems include analytical methods, such as separation of variables and integral transforms, and numerical methods, such as finite difference and finite element methods. The choice of method depends on the complexity of the problem and the accuracy required for the solution.

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