Lumped-element model: Difference between revisions

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where

<math>r = hA/C</math> is a positive constant characteristic of the system, which must be in units of <math>s^{-1}</math>, and is therefore sometimes expressed in terms of a characteristic [[time constant]] <math>t_0</math> given by: <math>rt_0 = 1/t_0r = -\Delta T(t)/(dT(t)/dt)</math>. Thus, in thermal systems, <math>t_0 = C/hA</math>. (The total [[heat capacity]] <math>C</math> of a system may be further represented by its mass-[[specific heat capacity]] <math>c_p</math> multiplied by its mass <math>m</math>, so that the time constant <math>t_0</math> is also given by <math>mc_p/hA</math>).

The solution of this differential equation, by standard methods of integration and substitution of boundary conditions, gives: