# 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." '

### Famous quotes containing the words capacity, heat, solution, terms and/or object:

Has the art of politics no apparent utility? Does it appear to be unqualifiedly ratty, raffish, sordid, obscene, and low down, and its salient virtuosi a gang of unmitigated scoundrels? Then let us not forget its high capacity to soothe and tickle the midriff, its incomparable services as a maker of entertainment.
—H.L. (Henry Lewis)

For it is wretchedness that endures, shedding its cancerous light on all it approaches:
Words spoken in the heat of passion, that might have been retracted in good time,
All good intentions, all that was arguable.
John Ashbery (b. 1927)

Who shall forbid a wise skepticism, seeing that there is no practical question on which any thing more than an approximate solution can be had? Is not marriage an open question, when it is alleged, from the beginning of the world, that such as are in the institution wish to get out, and such as are out wish to get in?
Ralph Waldo Emerson (1803–1882)

They were pipes of pagan mirth,
And the world had found new terms of worth.
He laid him down on the sunburned earth
And raveled a flower and looked away.
Play? Play? What should he play?
Robert Frost (1874–1963)

Art expresses the one, or the same by the different. Thought seeks to know unity in unity; poetry to show it by variety; that is, always by an object or symbol.
Ralph Waldo Emerson (1803–1882)