# How to calculate rate of heat transfer of a body

To calculate the heat transfer of a hot object like steel to its surroundings, I thought it would be appropriate to use the Stefan-Boltzman equation. I did all of my work and my solution seemed correct, but then I realized that in the situation that hot steel was in water, then the rate of heat loss would be faster than if it was in the air; this solution doesn't take into account heat conduction.

I looked up an equation to calculate this and I found a lot of different equations which all confused me, like Newtons Law Of Cooling, The "heat equation", and this other one for heat conduction. I don't know which one to use. The last one was simple enough to understand and I thought it would work, except that this equation takes into account a barrier between the hot body and the surroundings. Since my hypothetical situation has no barrier, the equation does not work.

Is there any equation that I can use to simply calculate the initial rate of heat transfer between an object and the surroundings just like in my example of hot steel in water or hot steel in air?

Summarising the state of the art of heat transfer knowledge isn't easy, considering small libraries have been filled on this subject alone but I'll give it a shot.

Heat transfer occurs following three quite distinct mechanisms (or modes, if you prefer):

Jim's answer to this question deals adequately with this so I don't have to.

2. Heat conduction:

When temperature gradients (generally $\nabla T$) exist in an object then by Fourier's law, heat conduction will strive to minimise these gradients and uniformise the temperature throughout the object.

3. Convection:

Convection combines radiative and conductive heat transfer with mass transport of heat: a domestic radiator (for instance) heats up the air surrounding it, which then rises because its density has been lowered. Convection can also be forced by means of ventilators that force the air (or a liquid) to flow over the hot surface.

Real world heat transfer almost always combines the three modes although usually with emphasis on one of them (the predominant mode).

So, to answer the OP's question, which model (equation) to choose?

Let's look at a hot sphere to focus attention (ignore the 'dent' due to poor drawing skills): We're looking at a cut straight through the centre of a hot sphere (hotter than the surrounding temperature $T_{\infty}$).

The left hand side is a solid sphere and due to heat conduction the boundary of the sphere will be cooler than the core, as schematised by the temperature distribution curve $T(r,t)$, here at a specific time $t$.

The flow of heat inside the sphere is governed by Fourier's heat equation:

$$\frac{\partial T}{\partial t}=\frac{k}{\rho c_p}\nabla^2T$$

Where $\frac{k}{\rho c_p}=\kappa$ is the thermal diffusivity of the material.

Heat flow from the boundary of the sphere to the surrounding medium is then either through convection, radiation or a combination of both.

The right hand side is a hollow sphere filled with a liquid (or a fluid, more generally) that is constantly stirred. Due to this stirring there are no temperature gradients and thus no internal heat conduction:

$$\nabla^2T=0$$

Heat flow from the boundary of the sphere to the surrounding medium is then either through convection, radiation or a combination of both. Usually Newton's cooling law can be used to model that situation.

Choice of model:

The scientist/engineer will have to choose to model his real world system according to which of the two options, $\nabla^2T=0$ or $\nabla^2T\neq0$, best describes his system.

That choice will also be influenced by mathematical considerations: models that require use of Fourier's equation tend to be mathematically more demanding. Analytical solutions may not exist or be difficult to use. Some examples of using the Fourier equation for 1D heat transfer problems can be found in that link and illustrate the (relative) difficulty in obtaining analytical solutions. Numerical (computer) solutions may be preferred in many cases.

• I am sorry I did not mean a human body literally, I was just using the term synonymously with "object." I apologize for my poor term usage. – Ryan Aug 31 '16 at 0:19
• I see. Misunderstandings happen. Tomorrow I'll formulate a better answer, heat transfer is one of my strong suits. – Gert Aug 31 '16 at 0:31
• I've fully edited the answer to reflect your question. I hope this helps. – Gert Aug 31 '16 at 14:16