Editing Lumped-element model (section)

==== Solution in terms of object heat capacity ====

If the entire body is treated as lumped-capacitance heat reservoir, with total heat content which is proportional to simple total [[heat capacity]] <math>C</math>, and <math>T</math>, the temperature of the body, or <math>Q = C T</math>.  It is expected that the system will experience [[exponential decay]] with time in the temperature of a body.

From the definition of heat capacity <math>C</math> comes the relation <math>C = dQ/dT</math>. Differentiating this equation with regard to time gives the identity (valid so long as temperatures in the object are uniform at any given time):  <math>dQ/dt  = C (dT/dt)</math>. This expression may be used to replace <math>dQ/dt</math> in the first equation which begins this section, above. Then, if <math>T(t)</math> is the temperature of such a body at time <math>t</math>, and <math>T_\text{env}</math> is the temperature of the environment around the body:
<math display="block"> \frac{d T(t)}{d t} = - r (T(t) - T_{\text{env}}) = - r \Delta T(t) </math>
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>t_0 = 1/r = -\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:
<math display="block"> T(t) = T_{\mathrm{env}} + (T(0) - T_{\mathrm{env}}) \ e^{-r t}. </math>

If:

: <math> \Delta T(t) \quad </math> is defined as : <math> T(t) - T_{\mathrm{env}} \ , \quad </math> where <math> \Delta T(0)\quad </math> is the initial temperature difference at time 0,

then the Newtonian solution is written as:
<math display="block"> \Delta T(t) = \Delta T(0) \ e^{-r t} = \Delta T(0) \ e^{-t/t_0}. </math>

This same solution is almost immediately apparent if the initial differential equation is written in terms of <math>\Delta T(t)</math>, as the single function to be solved for.
<math display="block"> \frac{d T(t)}{d t} = \frac{d \Delta T(t)}{d t} = - \frac{1}{t_0} \Delta T(t) </math>