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In our courses of differential equations we usually find Newton's Law of Cooling stated and written as

The rate of change of temperature of a body is directly proportional to the difference in temperature of the body and the surrounding at a time $$ \frac{dT}{dt} \propto (T_{surrounding} - T_{Body}) \\ \frac{dT}{dt} = k (T_{surroudning}-T_{Body})$$

But in textbooks on thermodynamics and even on wikipedia the mathematical form of the law is written as $$ \frac{dQ}{dt}= k (T_{surrounding}-T_{Body})$$.

So, which one was the originally given by the Sir Issac Newton? And which one is more useful? Please explain the difference in the two forms of the equation.

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    $\begingroup$ The introductory section of the Wikipedia article you linked seems to answer the question. $\endgroup$ – hiccups Jan 4 '20 at 12:10
  • $\begingroup$ Exactly where did you get the highlighted equations? $\endgroup$ – Bob D Jan 4 '20 at 13:55
  • $\begingroup$ @hiccups No, that doesn’t answer my question. You see in that page it is written that “Sir Isaac Newton did not originally state his law in the above form in 1701, when it was originally formulated. Rather, using today's terms, Newton noted after some mathematical manipulation that the rate of temperature change of a body is proportional to the difference in temperatures between the body and its surroundings”. $\endgroup$ – user240696 Jan 4 '20 at 13:57
  • $\begingroup$ @BobD In the OCW lectures on differential equation (by Prof. Arthur Mattuck). $\endgroup$ – user240696 Jan 4 '20 at 13:58
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    $\begingroup$ @Knight They appear to be based on the verbal description by Wiki. But personally i have never seen the actual original equations. $\endgroup$ – Bob D Jan 4 '20 at 14:38
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So, which one was the originally given by the Sir Issac Newton? And which one is more useful? Please explain the difference in the two forms of the equation.

I've never seen the actual original equations, but it appears that the highlighted ones were probably the original based on the Wiki discussion. However as Wiki points out:

"Newton's law behavior, when stated in terms of temperature change in the body, also requires that internal heat conduction within the object be large in comparison to the loss/gain of heat by surface transfer (conduction and/or convection), which is the condition where the Biot number is less than about 0.1".

That makes the usefulness of the original equations limited, and makes the second equation, with $\frac{dQ}{dt}$ on the left, more useful as less restrictions apply.

As pointed out in the Wikipedia article, one of the reasons for Newton's original formulation had $\frac{dT}{dt}$ on the left instead of $\frac {dQ}{dt}$ was because he, and others at the time, confused temperature with heat. A situation requiring another century to untangle.

As an aside, a more practical engineering version of Newton's Law of Cooling for steady heat flow is the following because it identifies the components of the proportionality constant.

$$\dot Q=hA(T_{w}-T_{∞})$$

where

$\dot Q$ = the heat transfer rate, $h$ = the heat transfer coefficient of the fluid, $A$ = the convection surface area, $T_w$ = the wall surface temperature and $T_∞$= the bulk fluid temperature.

Hope this helps.

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