Understanding Newton's Law of Cooling and Its Derivation | dQ/dt = KA(dT/dx)"

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Discussion Overview

The discussion centers on the derivation and interpretation of Newton's law of cooling and its relationship to Fourier's law of heat transfer. Participants explore the definitions and implications of the variables involved in both equations, particularly focusing on the meanings of Q and K in different contexts.

Discussion Character

  • Technical explanation
  • Debate/contested

Main Points Raised

  • Some participants assert that Newton's law of cooling is expressed as dQ/dt = KA(θ - θo), which applies to the surface interface between two materials.
  • Others introduce the equation dQ/dt = KA(dT/dx), suggesting it is related to Fourier's law, which pertains to heat transfer within a material.
  • A participant questions how dQ/dt can have different dimensions in the two equations.
  • It is noted that K has different meanings in the context of the two formulas.
  • Some participants propose that Q represents different concepts in each law, with Q in Newton's law of cooling referring to temperature and in Fourier's law referring to heat.
  • One participant elaborates that Fourier's law relates to heat transfer per unit time across a surface and connects to temperature changes through the continuity equation, involving heat capacity, volume, and density.

Areas of Agreement / Disagreement

Participants express differing views on the definitions of Q and K in the context of the two laws, indicating that multiple competing interpretations remain unresolved.

Contextual Notes

Participants highlight the need for clarity in definitions and the implications of the equations, but do not resolve the differences in interpretation or the dimensionality concerns raised.

andyrk
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Newton's law of cooling is: dQ/dt = KA(θ - θo). Then where does the equation dQ/dt = KA(dT/dx) come from?
 
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andyrk said:
Newton's law of cooling is: dQ/dt = KA(θ - θo).

This generally applies to the surface interface between two materials.

andyrk said:
Then where does the equation dQ/dt = KA(dT/dx) come from?

This is Fourier's law (or something reminiscent of it, you really should define what you mean by Q). It applies to the heat transfer within a material.
 
Orodruin said:
This generally applies to the surface interface between two materials.
This is Fourier's law (or something reminiscent of it, you really should define what you mean by Q). It applies to the heat transfer within a material.
How can dQ/dt have two different dimensions?
 
K means different things in the two formulas.
 
I would say that Q means different things too. Q in Newton's law of cooling is temperature whereas in Fourier's law it is heat.
 
andyrk said:
I would say that Q means different things too. Q in Newton's law of cooling is temperature whereas in Fourier's law it is heat.
Well, Fourier's law is actually just a statement on the current. What appears in the left hand side is the heat transfer per unit time across a surface. This can be related to an actual change in temperature (or heat, they are related by heat capacity, volume, and density) through the continuity equation.
 

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