I have an object of temperature $T$ inside a medium of temperature $T_a$.

What does it means the Newton's cooling law in this form: $$q\cdot n=h (T-T_a) \ \ \ \ \ \ \ \ \ \ (1)$$

I got this equation in a problem but I didn't understand it.

I already searched in google about the Newton's cooling law, what I found were the following statements

$$\dfrac{dQ}{dt}=hA(T-T_a)\ \ \ \text{ and }\ \ \ dQ=mcdT\ \ \ \ \ \ \ \ \ (2)$$

In the latter $Q=Q(t)$ is the amount of heat transferred (or loss) by the object, $A$ is the contact surface of the object with the medium, $h$ is the coefficient of heat transfer, $m$ is the mass of the object, $T=T(t)$ is the temperature of the body and $T_a$ is the temperature of the medium.

¿What is the relation between equation (1) and equations (2)?

  • $\begingroup$ Probably it has to do with the rate of heat flux in the direction $\hat{n}$. $\endgroup$ – Sayan Mandal Oct 3 '17 at 23:05
  • $\begingroup$ I think, but I can not imagine how is it. $\endgroup$ – GuadalupeAnimation Oct 3 '17 at 23:18
  • $\begingroup$ @Chester Miller has given a nice answer. $\endgroup$ – Sayan Mandal Oct 3 '17 at 23:55

The left hand side of the equation in question is the vector dot product of the heat flux vector $\mathbf{q}$ with a unit outwardly directed normal to the surface of the object $\mathbf{n}$. So it physically represents the local rate of heat flow per unit area over a differential section of the object's boundary. If this is integrated over the entire boundary of the object, one obtains the total rate at which heat is leaving the object: $$\frac{dQ}{dt}=-\int{\mathbf{q}\centerdot \mathbf{n}dA}$$ (There should be a minus sign on the right hand side of your Eqn. 2). Inside the object, the heat flux vector $\mathbf{q}$ is related to the temperature gradient by:

$$\mathbf{q}=-k\nabla T$$ This relationship is used in conjunction with the differential heat balance equation within the object in cases where, rather than the temperature within the object being nearly uniform, the temperature varies with spatial position (and time).


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