Math 2.0

Classical thermodynamics, heat transfer Example

Newton's law of cooling

Builds on Initial value problemExponential growth and decay

Example

Let T(t)T(t) be the temperature of a body at time tt. Let TaT_a be the constant temperature of the surroundings, and let k>0k > 0 be a constant. Newton's law of cooling is the equation T′(t)=−k(T(t)−Ta).T'(t) = -k \left(T(t) - T_a\right). With the initial condition T(0)=T0T(0) = T_0, its unique solution is T(t)=Ta+(T0−Ta) e−kt.T(t) = T_a + (T_0 - T_a)\, e^{-k t}.

Remarks

The difference D=T−TaD = T - T_a satisfies D′=−kDD' = -k D. This is exponential decay, so D(t)=D(0)e−ktD(t) = D(0) e^{-kt}. As t→∞t \to \infty, T(t)→TaT(t) \to T_a. If T0>TaT_0 > T_a the body cools, and if T0<TaT_0 < T_a it warms. The law is a good model when the temperature difference is small and the body has a nearly uniform temperature.

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Differential equation, concept, no evidence levelDifferential equationconceptFunction, concept, no evidence levelFunctionconceptInterval, concept, no evidence levelIntervalconceptOrdinary differential equation, definition, no evidence levelOrdinary differential…definitionSolution of an ordinary differential equation, definition, no evidence levelSolution of an ordina…definitionInitial value problem, definition, no evidence levelInitial value problemdefinitionExponential growth and decay, example, no evidence levelExponential growth an…exampleNewton's law of cooling, example, no evidence levelNewton's law of cooli…example
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  8. Newton's law of cooling example

Further reading

  1. Web https://en.wikipedia.org/wiki/Newton%27s_law_of_cooling, First-order transient response of lumped-capacitance objects
About this page
Tag02HB5F40VR
URIm2://02HB5F40VR
Kindexample
WritingW2 usable
MSC80A19
Instance ofLinear first-order equation, Separable equation
UsesInitial value problem, Exponential growth and decay
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