Math 2.0

Classical thermodynamics, heat transfer Definition

Lumped-element thermal model

Builds on Heat transfer coefficientHeat capacityBiot numberOrdinary differential equation

Definition

Let a solid body have volume VV, surface area AA, density ρ\rho and specific heat capacity cc. Let it exchange heat with a fluid at constant temperature T∞T_\infty. Let h>0h > 0 be the heat transfer coefficient on its surface. The lumped-element thermal model assumes that the temperature of the body is uniform, equal to T(t)T(t) at time tt. It is the ordinary differential equation ρ c V dTdt=−hA(T−T∞).\rho \, c \, V \, \frac{dT}{dt} = -h A \left(T - T_\infty\right).

Remarks

The solution with T(0)=T0T(0) = T_0 is T(t)=T∞+(T0−T∞)e−t/τT(t) = T_\infty + (T_0 - T_\infty) e^{-t / \tau}, with time constant τ=ρcV/(hA)\tau = \rho c V / (h A). Indeed T′(t)=−(T0−T∞)e−t/τ/τ=−(T−T∞)/τT'(t) = -(T_0 - T_\infty) e^{-t/\tau} / \tau = -(T - T_\infty) / \tau, which is the equation divided by ρcV\rho c V.

Example: a ball with τ=100\tau = 100 seconds, T0=80T_0 = 80 and T∞=20T_\infty = 20 degrees Celsius reaches 20+60e−1≈4220 + 60 e^{-1} \approx 42 degrees Celsius after 100100 seconds.

Non-example: a thick block of wood cooled in strong wind has a large Biot number. Its centre stays much hotter than its surface, so the uniform temperature assumption fails.

The usual criterion for the model is Bi<0.1\mathrm{Bi} < 0.1 with length V/AV / A.

Reading path

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Biot number, concept, no evidence levelBiot numberconceptHeat capacity, concept, no evidence levelHeat capacityconceptHeat transfer coefficient, concept, no evidence levelHeat transfer coeffic…conceptOrdinary differential equation, concept, no evidence levelOrdinary differential…conceptLumped-element thermal model, definition, no evidence levelLumped-element therma…definition
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  1. Biot number concept
  2. Heat capacity concept
  3. Heat transfer coefficient concept
  4. Ordinary differential equation concept
  5. Lumped-element thermal model definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Lumped-element_model
About this page
TagMZPYYMSVEJ
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WritingW0 stub
MSC80A19, 34A30
DefinesLumped-element thermal model
UsesHeat transfer coefficient, Heat capacity, Biot number, Ordinary differential equation
Changed3 November 2026

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