Heat Transfer Diff. Equation: Fixed vs Free Extremities

In summary, there are two types of boundary conditions for heat transfer: temperature-based and heat transfer-based. For more complex cases, numerical solutions are needed, often utilizing series analysis. Solutions for periodic heat changes can sometimes be found exactly, but for non-periodic cases, a numerical solution is typically necessary. These topics are typically not covered in undergraduate texts and require a more advanced understanding of heat transfer and differential equations.
  • #1
Jairo
5
0
I know the heat transfer diferential equation, but I only saw it being used when the temperatures of the extremities of the material are fixed. Is there any phormula that aplies for situations where at least one of the extremities is free to change its temperature?

Ex.: the left side of an iron bar with dimentions A,B,C is heated to T1 while the right side starts with T2. Ignoring radiation loss, when the temperature of the right side will be 90% of T1? And if we consider radiation?

If it´s too hard to solve, is there any numerical aproximation? Thanks.
 
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  • #2
Yes there are two typical kinds of boundary conditions. One type defines the temperature on the boundaries the other defines the heat transfer on the boundaries. A common heat-transfer type of boundary is the insulating boundary where no heat is allowed to transfer in or out of the boundary. I think for most general cases that it requires numerical solution to the PDE.

-Dale
 
  • #3
There is a second order, transient differential equation, which is typically solved with series analysis. Oftentimes, if the heat change is perdioc, one can find an exact solution to the problem. If not, then a numeric solution is in order. You will likely not find this in an undergraduate text, as non-transient heat transfer solutions are not addressed until you take a graduate level heat transfer course. I suggest you look in a college library or a graduate heat transfer text or a graduate diff E book for Fourier series analysis of 2nd order transiet diff equations.
 

1. What is the difference between fixed and free extremities in heat transfer differential equations?

Fixed extremities refer to boundaries or surfaces where the temperature is held constant, while free extremities refer to boundaries or surfaces where the temperature is allowed to vary.

2. How do fixed and free extremities affect heat transfer in a system?

The type of extremity present in a heat transfer system affects the flow of heat through the system. Fixed extremities create a constant temperature gradient, while free extremities allow for variations in temperature that can impact the overall heat transfer rate.

3. Can fixed and free extremities coexist in the same heat transfer system?

Yes, it is possible for a heat transfer system to have both fixed and free extremities. This can occur when there are multiple boundaries or surfaces present, some of which are fixed and others that are free.

4. How are fixed and free extremities represented in heat transfer differential equations?

Fixed extremities are typically represented using boundary conditions, which specify the temperature at that boundary. Free extremities are represented using the Neumann boundary condition, where the temperature gradient at the boundary is taken into account.

5. What are some real-life examples of fixed and free extremities in heat transfer?

An example of a fixed extremity would be the surface of a pot on a stove, where the temperature of the surface is held constant. A free extremity could be the surface of a cup of hot coffee, where the temperature can vary due to heat loss to the surrounding environment.

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