How does the temperature of a solid sphere change when exposed to a hot environment?

Question

I need to know how the temperature of an object will change with time when exposed to a time dependant increase in the temperature of the environment.

I only require an approximation and I thought I could use Newton's Law of Cooling.

$$\frac{dQ}{dt}=\alpha A\left(T\left(t\right)-T_E\right)$$

We can use the well know equation that relates heat and temperature;

$$\delta Q=mC_P\delta T$$

Then we can rewrite Newton's law;

$$\frac{dT}{dt}=\frac{\alpha A}{mC_P}\left(T\left(t\right)-T_E\right) $$

This differential equation has a well known solution;

$$T\left(t\right)= T_E+\left(T_0-T_E\right)e^{-kt}$$

Where $k=\frac{\alpha A}{mC_P}$

The environmental temperature ($ T_E $) varies with time but can be considered independent of the object temperature ($ T\left(t\right) $) in this case, I'm assuming that I can replace $ T_E $ with an appropriate $f\left(t\right)$.

Assuming my thinking above is correct, my question is; how do I find the $\alpha$, the heat transfer coefficient? I'd like to use a temperature independent approximation if possible.

Assume the following:

• The Biot number is small.
• That the sphere is floating in hot air with no fixings.
• There is no bulk motion of the air relative to the object.
• The bounding temperature range is 0 to 350C.
• Changes in the environment occur slowly enough as to not complicate things.

Many thanks.

Answer 1

The integral used to solve the differential equations works this way only if $T_E$ is a constant. Otherwise you need to assume a specific time dependence and solve it from scratch. You may need to do a numerical simulation rather than looking for analytical solutions.

Answer 2

For the problem to be interesting, you must assume that the temperature of the air around the sphere can vary spatially and temporally. Otherwise, there is no answer, or rather, the implication from a minuscule Biot number of the sphere is that the entire sphere instantly reaches $T_E$, as there's no part of the system that can sustain gradual heat transfer.

If the surrounding air could move, which is the more typical assumption, then you'd have a natural-convection problem, where $\alpha$ would be the convective coefficient. By eliminating air movement, your case seems more like a that of a sphere suspended in a solid (i.e., an inclusion) or in a sufficiently viscous liquid that precludes convection.

You could treat the air as a semi-infinite medium with constant material properties. The relevant constitutive equation is the heat equation $$\nabla^2 T=\frac{1}{D}\frac{\partial T}{\partial t}.$$

where $D$ is the air thermal diffusivity. If the sphere is large enough to assume a fairly flat surface, the Laplacian $\nabla^2 T\approx \frac{\partial^2T}{\partial r^2}$, where $r$ is the radial direction. If not, the Laplacian $\nabla^2 T=\frac{1}{r}\frac{\partial}{\partial r}\left(\frac{\partial T}{\partial r}\right)$. (These are the Cartesian and spherical results for rotational symmetry.)

Now you need to analytically or numerically solve the constitutive equation for your boundary conditions $T(r\ge R,t=0)=T_E$ and $T(r=R,t)=T_S(t)$, where $R$ is the sphere radius and $T_S$ is the uniform temperature of the sphere. This temperature is coupled to the air temperature through an energy balance at the surface of the sphere, giving

$$q|_{r=R}=-kA\nabla T(r=R)=mC_P\frac{dT_S(t)}{dt}.$$

where $k$ is the air thermal conductivity and $A$ is the sphere surface area.

If $T_E$ is time-dependent, you're almost certainly going to need to solve this problem numerically.

(Things would get much simpler if you allowed the air to exhibit natural convection. I'm curious why you disallowed this.)

Does this all make sense?

Answer 3

If the Biot number is low, conduction resistance inside the sphere is negligible, and the Biot number approaches 2 (if the external temperature is changing very slowly), based on the sphere diameter and the air conductivity. The time variation of the external temperature can be taken into account in the solution to the differential equation: $$\frac{dT}{dt}=-k(T-T_E(t))$$which has the solution: $$T=T_0e^{-kt}+k\int_0^t{e^{-k(t-\tau)}}T_E(\tau)d\tau$$