Change of variables in heat equation

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SUMMARY

The discussion focuses on the change of variables in the heat equation represented as dTB/dt = -k(TB-TM), where TM is a constant. By substituting (TB-TM) with Q, the equation can be transformed to dQ/dt = -kQ. This linear substitution simplifies the integration process, allowing for easier manipulation of the equation. Participants confirmed the effectiveness of this method, highlighting its utility in solving differential equations related to heat transfer.

PREREQUISITES
  • Understanding of differential equations
  • Familiarity with the heat equation
  • Knowledge of linear substitution methods
  • Basic concepts of thermodynamics
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  • Study the derivation and applications of the heat equation in thermodynamics
  • Learn about linear substitution techniques in solving differential equations
  • Explore the implications of the first-order linear ordinary differential equations
  • Investigate numerical methods for solving heat transfer problems
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Students and professionals in physics, engineering, and applied mathematics who are working with heat transfer equations and differential equations.

tickle_monste
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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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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.
 
Wow, I really should've been able to do that myself. Thanks a lot though, that's a big help
 

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