Convective Heat Transfer - Newton's Law of Cooling - Solution in Terms of Object Heat Capacity

Solution in Terms of Object Heat Capacity

If the entire body is treated as lumped capacitance thermal energy reservoir, with a total thermal energy content which is proportional to simple total heat capacity, and, the temperature of the body, or, it is expected that the system will experience exponential decay with time in the temperature of a body.

From the definition of heat capacity comes the relation . Differentiating this equation with regard to time gives the identity (valid so long as temperatures in the object are uniform at any given time): . This expression may be used to replace in the first equation which begins this section, above. Then, if is the temperature of such a body at time, and is the temperature of the environment around the body:

where

is a positive constant characteristic of the system, which must be in units of, and is therefore sometimes expressed in terms of a characteristic time constant given by: . Thus, in thermal systems, . (The total heat capacity of a system may be further represented by its mass-specific heat capacity multiplied by its mass, so that the time constant is also given by ).

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

If:

is defined as : where is the initial temperature difference at time 0,

then the Newtonian solution is written as:

This same solution is more immediately apparent if the initial differential equation is written in terms of, as a single function of time to be found, or "solved for." '