Heat transfer problem for a long rectangular bar

In summary, the problem involves determining the temperature distribution in a bar with a rectangular cross section, in steady state, with imposed flux at one face, convection at the opposing face, and imposed temperature at the remaining walls. The differential equation and four boundary conditions are provided, but the analytical solution has not yet been found using the superposition method. The suggested approach is to use three superpositions, with the first case being where the temperature is equal to the imposed temperature everywhere and the flux ends are both insulated. The other two cases can be solved using separation of variables, with sine series in the x direction and hyperbolic sines and cosines in the y direction.
  • #1
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Moved from a technical forum, so homework template missing
Summary:: Determine the temperature distribution in a bar (very long– 2D) with rectangular cross section, in steady state, with imposed flux at one face, convection at the opposing face (Tinf, h), and imposed temperature (T1) at the two remaining walls.

I am trying to find the analytical solution to the following problem :

Determine the temperature distribution in a bar (very long– 2D) with rectangular cross section, in steady state, with imposed flux at one face, convection at the opposing face (Tinf, h), and imposed temperature (T1) at the two remaining walls.

I have the differential equation and four boundary conditions , however I can't find the temperature distribution even using the supeposition method ( trying to find the solution with one of the walls adiabatic and the other transferring energy by convection and then finding the solution for one of the walls adiabatic and the other with imposed flux).

Has anyone came across a problem like this?
 

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  • #2
Your figure matches the problem statement, but there are no adiabatic walls mentioned in the problem statement.
 
  • #3
Chestermiller said:
Your figure matches the problem statement, but there are no adiabatic walls mentioned in the problem statement.
I was describing how i was using the superposition method
 
  • #4
I like your approach. How do you handle the other two boundaries in each of the two cases?
 
  • #5
You are on the right track. Instead of using 2 superpositions, use three. The first case is where T is equal to T1 everywhere and the flux ends are both insulated. The other two cases are the ones you already described, but with zero temperatures where T1 had been. These two other cases can be solved by separation of variables, with sine series in the x direction and hyperbolic sines and cosines in the y direction.
 

1. What is heat transfer?

Heat transfer is the movement of thermal energy from one object or system to another. This can occur through three main mechanisms: conduction, convection, and radiation.

2. How does heat transfer occur in a long rectangular bar?

In a long rectangular bar, heat transfer occurs through conduction, as thermal energy is transferred from one end of the bar to the other through direct contact between the molecules of the bar.

3. What factors affect heat transfer in a long rectangular bar?

The main factors that affect heat transfer in a long rectangular bar are the temperature difference between the two ends of the bar, the thermal conductivity of the material the bar is made of, and the cross-sectional area of the bar.

4. How can heat transfer be calculated for a long rectangular bar?

The rate of heat transfer for a long rectangular bar can be calculated using the equation Q = kA(T2-T1)/L, where Q is the heat transfer rate, k is the thermal conductivity of the material, A is the cross-sectional area of the bar, T1 and T2 are the temperatures at the two ends of the bar, and L is the length of the bar.

5. What are some practical applications of heat transfer in long rectangular bars?

Heat transfer in long rectangular bars is important in many engineering and industrial applications, such as in the design of heat exchangers, cooling systems, and thermal insulation. It is also essential in everyday objects, such as cooking utensils and heating systems.

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