# How to calculate the time to uniformly heat an object by natural convection?

I want to heat a piece of steel (~40 kg) in a muffle until I bring it to a uniform temperature. The idea is to preheat the muffle to a certain temperature and then place the piece in it and wait for it to get in thermal equilibrium.

The specific heat equation $$Q = cm\Delta T$$, tells me how much energy is necessary to increase the temperature of my part, and I guess I can use the convection equation, $$\dot Q = hA\Delta T$$, to compute how much time does it take to transfer that much energy into my part...

Would that be enough? Or should I also calculate the diffusion of heat inside my part. How can I estimate the heat transfer coefficient $$h$$?

• – Keith McClary Oct 17 '18 at 0:44
• It is hard to predict $h$ with any accuracy. What is the temperature? At some point, radiation becomes more important than convection. – Pieter Oct 27 '18 at 18:08
• @Pieter The temperature is 300 °C – user190081 Oct 29 '18 at 13:48
• That is a radiative intensity of 6 kW/m² in a black-body oven. – Pieter Oct 29 '18 at 14:54

## 2 Answers

I recommend you to look at the Biot number of your problem. It is a dimesionless number given by : Bi = Lc*h/k (where Lc is a characteristic length of your problem and k the conductivity of your material)

It is very useful to operate great simplifications in thermal transfer problems such as yours. It basically computes the ratio of "how fast you transfer heat to the surface" to "how fast you transfer heat inside the material".

A small Biot number (Bi<<1, tipically Bi<0.1) implies that you have a "thermally thin" body, which means that the heat diffusion process inside will occur much faster than the one to the surface. In this case, you can consider your piece of metal to be homogeneous in temperature for your problem.

In case your Biot number is big (Bi~1), then you should take into account internal diffusion, and model the inside of your material. The estimation of h can be done as proposed by pentane. This kind of measurement is anyway very common, and I think that a quick internet search on engineering websites will give you a whole variety of procedures you can pick from.

Hope that helped !

Assuming a relatively small piece of steel, the temperature at the surface of the metal will be the temperature of the bulk in less than a minute. No need to calculate conduction of heat inside your part unless you want the answer down to the second.

You can estimate $$h$$ by measuring the temperature of the steel after and before a certain amount of time in the muffle-- do this many times. Create a plot of the data, draw a linear fit, and the slope will be $$h$$.

• Thanks for the answer. I must clarify that my part is rather big (~40 kg), so I dont think that I can make the assumption that the temperature inside is the same that the surface temperature at a given time. – user190081 Oct 22 '18 at 15:53