Modifying the heat equation for multiple sources

In summary, the heat/diffusion equation describes the distribution of temperature with respect to radius and time for a single hot wire. However, for multiple wires, the equation must be modified to account for all of them. One approach is to consider symmetry, such as in the case of two heated wires where there is a zero temperature gradient at the center. This concept is known as the "Method of Images" and can be applied to various problems involving heat transfer and potential fluid flow. Chet suggests searching for more information on this method online.
  • #1
babagoslow
13
0
If I have a hot wire, the distribution of its temperature with respect to radius (from the center of the wire) and time follows the heat/diffusion equation.

However, now consider two wires, or even an array of many such wires. Say we can ignore the z coordinate and treat them as a point source in cylindrical polar coordinates. How would one modify the heat equation to account for all of them?

One way that I have thought about in this direction is considering symmetry. Due to the symmetry between two heated wires, there must be a zero temperature gradient in the geometrical centre between the two wires. But then you would have the problem of extending this to the case of N arbitrary heat sources.
 
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  • #2
What you seem to be (cleverly) reinventing is the "Method of Images." This can be used in many potential flow problem involving heat transfer and potential fluid flow (including flow in porous underground geological formations containing arrays of injection or production wells). Try Googling.

Chet
 

FAQ: Modifying the heat equation for multiple sources

How does modifying the heat equation account for multiple sources?

Modifying the heat equation involves adding additional terms to account for multiple heat sources. These terms represent the heat generated by each individual source and are included in the overall heat equation to accurately model the system.

Can the heat equation be modified for an arbitrary number of sources?

Yes, the heat equation can be modified to account for any number of heat sources. The number of sources will determine the number of additional terms that need to be added to the equation.

What factors should be considered when modifying the heat equation for multiple sources?

When modifying the heat equation, it is important to consider the strength and location of each heat source, as well as the material properties and boundary conditions of the system. These factors will impact the overall heat transfer and must be accounted for in the modified equation.

How do multiple sources affect the overall heat distribution in a system?

The presence of multiple heat sources can lead to complex heat distribution patterns in a system. The modified heat equation takes into account the individual heat sources and their contributions to the overall heat transfer, resulting in a more accurate representation of the system's heat distribution.

Are there any limitations to modifying the heat equation for multiple sources?

One limitation of modifying the heat equation for multiple sources is that it assumes the system is in a steady state, meaning that the temperature distribution does not change over time. This may not accurately model systems that are experiencing transient conditions or have rapidly changing heat sources.

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