Change of variables in heat equation

In summary, the conversation discusses a method for changing the equation dTB/dt = -k(TB-TM) so that it can be integrated with respect to Q instead of TB, with TM being held constant. This can be achieved through a simple linear substitution, resulting in the equation dQ/dt = -kQ.
  • #1
tickle_monste
69
1
dTB/dt = -k(TB-TM).
TM is held constant.

(TB-TM) = Q, so:
dTB/Q = -kdt.

How would I change this equation so that instead of integrating wrt to TB, I can instead integrate wrt Q?
 
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  • #2
tickle_monste said:
dTB/dt = -k(TB-TM).
TM is held constant.

(TB-TM) = Q, so:
dTB/Q = -kdt.

How would I change this equation so that instead of integrating wrt to TB, I can instead integrate wrt Q?
That's a simple "linear" substitution. Since TB= Q+ TM, dTB/dt= dQ/dt. Your equation becomes just dQ/dt= -kQ.
 
  • #3
Wow, I really should've been able to do that myself. Thanks a lot though, that's a big help
 

Related to Change of variables in heat equation

1. What is a change of variables in the heat equation?

A change of variables in the heat equation refers to the process of transforming the independent and dependent variables in the equation to a new set of variables. This is usually done to simplify the equation or to make it easier to solve.

2. Why is a change of variables necessary in the heat equation?

A change of variables is necessary in the heat equation because it allows us to solve more complex problems and obtain more accurate solutions. It also helps in understanding the physical behavior of heat transfer in different scenarios.

3. How is a change of variables performed in the heat equation?

A change of variables in the heat equation can be performed using various techniques such as substitution, transformation, or integration. The specific method used depends on the form of the original equation and the desired outcome.

4. What are the benefits of a change of variables in the heat equation?

The benefits of a change of variables in the heat equation include simplifying the equation, making it more manageable to solve, and providing deeper insights into the behavior of heat transfer. It also allows for a wider range of problems to be solved and more accurate solutions to be obtained.

5. Are there any limitations to using a change of variables in the heat equation?

While a change of variables can be a powerful tool in solving heat transfer problems, it may not always be applicable or effective. It may also introduce new complexities or uncertainties in the solution. Therefore, it is important to carefully consider the appropriateness and effectiveness of using a change of variables in each specific case.

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