# How hot will the exterior get?

Before getting into the material science of cement, let’s take a step back here. What’s the fundamental nature of the furnace? That is, what are the simplest, most fundamental parts and functions of a metal-melting furnace? You need:

• A source of heat. In this case, fire.
• An enclosure around the heat with low thermal conductivity. That is, some way of preventing the heat from escaping. If heat is being produced faster than it can escape then by definition the temperature is going up.

That’s it; a furnace is an enclosure with a fire in it. It could be a hole in the ground, and of course, for many centuries, a hole in the ground with a fire in it was perfectly adequate. The area surrounding my summer cottage in Ontario is dotted with the ruins of 19th century lime kilns, which were basically holes in the ground in which limestone was turned into quicklime by the application of intense heat.

So what we need is a substance that first, has a low thermal conductivity; that is, heat energy moves slowly through it, and therefore builds up the temperature on the hot side.

The thermal conductance is the quantity of heat energy that is emitted by a unit area in a unit time (and energy per time is of course power) when there is a unit temperature difference between the two sides separated by a unit thickness of material. A typical refractory cement will transmit on the order of 2 joules of heat energy per second (so, two watts of power) through a square meter of cement if the cement is 1 cm thick and the temperature difference between the sides is one degree Fahrenheit.

In our case, the furnace will have a total surface area of about 0.4 square metres, be about 5 cm thick, and have a temperature delta of about 1350°F. Multiply that all out and that’s about 5400 watts, which seems like rather a lot! That’s the surface of the furnace emitting the same heat as that of eleven 500W shop lights, which is more than I’d like to touch, even wearing gloves.

But we’ve forgotten something important in this back-of-the-envelope calculation: the thermal conductance that we’ve just worked out is the steady state. That is, if we left the furnace running indefinitely, burning charcoal such that the interior was exactly 1400°F, then of course eventually the exterior would get quite hot as it attempted to radiate that heat out into the air. But we’re not going to stoke the furnace with fresh charcoal indefinitely; it’ll typically run for less than half an hour. Also we have completely neglected the fact that a huge amount of heat is going to leave via the top when it is opened. And that when the crucible full of molten metal is removed, all of that heat disappears from the furnace.

Actually solving analytically how hot the surface of the furnace will get is more calculus than I’d care to do; it has been a long time since my 3A term Partial Differential Equations One, and we only solved the Heat Equation in one dimension. I suppose I could write a program to simulate it; all you’d need to know is the specific heat of all the parts (that is, how much energy a unit volume of a substance possesses at a given temperature), and then use a combination of conservation of energy and Fourier’s rule for heat transfer to work it out.

I seem to have strayed somewhat from my original topic, which was what properties the body of the furnace needs to have. More on that next time.