Newton's cooling law application

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  1. INTRODUCTION

Temperature differences in any situation result from energy flow into a system (heating by electrical power, contact to thermal bath, absorption of radiation, e.g. microwaves, sun radiation, etc) or energy flow from a system to the surrounding. The former leads to heating, whereas the latter results in cooling of an object. The cooling of objects is usually considered to be due to three fundamental mechanisms: conduction of heat, convection and radiative transfer of energy . Although these three mechanisms of energy flow are quite different from each other, one often finds a very simple law for their combined action to describe the cooling curves of hot objects if temperature differences are small.

When hot bodies are left in the open they are found to cool gradually. Newton found that the rate of cooling was proportional to the excess of temperature of the body over that of the surroundings. This observation is what is called Newton’s law of Cooling. It is not known if Newton attempted any theoretical explanation of this phenomenon. But it is unlikely because the concepts about heat were not clear in those times. But what is important is that the original statement of the conditions for the validity of Newton’s law included the presence of a draught.

Let us see now the process of cooling of a solid suspended in a fluid. When a body with mass m and specific heat s cools, the rate of loss of heat is given by

,

Where Q is the temperature of the body. But when a solid cools its temperature has a gradient from its center to its surface. Hence there is no unique temperature of the body during the cooling process. Perhaps that is the reason why Newton’s law of cooling is studied by observing the temperature fall of some liquid kept in a container. (The temperature of a liquid can be made uniform throughout its mass by stirring it.) During cooling process there has to be some temperature gradient across the wall of the container and the conduction through the wall leads to the loss of heat of the liquid. If K be the conductivity of the material of the container and A and d the area and thickness of the wall of the container, the rate of conduction of heat across this thickness is given by

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Where

and

 are the temperatures of the inner and outer surfaces of the container respectively. The heat lost by the body by conduction is equal to the heat carried away by the fluid in convection. Hence

ms

Therefore,

This appears similar to Newton’s law of cooling because  

 is constant.

So we have this equation of Newton’s cooling law

In detail, earlier work on Newton’s law of cooling dealt with

  • Undergraduate lab experiments to demonstrate exponential cooling curves
  • Comparison of the cooling of solids ...

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