Finding power needed to heat an oven through radiation and conduction

[swilly]

I was given an oven that comprises of several cylinders that fit inside one another. The innermost cylinder is a Molybdenum oven that will hold a sample of metal. A ceramic cylinder fits over this. A tungsten wire heating element is wrapped around this cylinder. Two more ceramic cylinders fit around this, and then lastly a Molybdenum shield. I am going to be raising the temperature of the Tungsten wire to 900°C. My question is this: how can I estimate how much current to run through the Tungsten wire to reach 900°C?

I know that P=I^2R. I know the resistance of the Tungsten wire. The total power needed is going to equal the sum of the heat transfers (I think?). There is heat transfer by thermal radiation and thermal conduction. I have included a picture of the oven for reference. (Note that in the picture, the wire is not wrapped around the ceramic with threads). Please help me!

 

[mfb]

If you know heat conductivity and capacity of all of your components, it is possible to evaluate that. The current will not be constant, as heat loss depends on the temperature of the surrounding material.

Another idea: The resistance of the tungsten wire depends on the temperature. Assuming that the temperature is the same in the whole wire: if you measure voltage and current, you can calculate the resistance and use this as thermometer.

 

[Khashishi]

You might need a feedback control. That sort of thing isn't very easy to get accurate. The folks in the mechanical engineering forum might know more about how to solve the thermal resistances.

 

[swilly]

So, as it heats up, I will need less and less current?
Basically, I'm just trying to estimate the current required to keep the sample of metal at the center at 900 deg C. I'm confused on how I calculate thermal radiation and conductivity. I realize that every part of the system is going emit thermal radiation, and that there's going to be conduction through several layers...

I believe we will be purchasing a PID temperature controller that can control the current, but I wanted an idea of what kind of current supply we would need.

 

[mfb]

Without any numbers, it is hard to tell. I would guess that conductivity of the ceramic cylinders is the most important part.

So, as it heats up, I will need less and less current?

Right.

but I wanted an idea of what kind of current supply we would need.

Depends on the time you have to heat the whole setup. The current required to keep the whole system warm should be small compared to the current required to heat it up quickly (i.e. with tungsten at constant 900°C).

 

[swilly]

The ceramic cylinders are made of Alumina, so I can find the conductivity. I have a lot of time to heat it up. Can anyone help me with the math to estimate the current?

 

[mfb]

A sheet of material with thermal conductivity \sigma, area A, thickness x and temperature difference \Delta T between both sides will have an internal flow of P=\frac{\sigmaA\Delta T}{x}. For cylinders, you need an integration to get an exact formula, but if the radius is large compared to x the difference is negligible.
Multiple materials can be treated with an equation system, or by adding their "heat resistance" (inverse value of the conductivity).

You could try to find the thermal conductivity of your ceramic cylinders and their thickness, this would help to get a rough estimate.

 

[swilly]

Okay, so I treated it as a system and added the heat resistances, or R-values:

\dot{Q} = \frac{1173K-Tb}{Rtotal}

I calculated Rtotal by adding the R-values for the conductive layers with the R-value for radiation (radiation from the outside of the oven to the walls of the vacuum).

I used Mathematica to solve the Temperature of the outside of the oven (which came out as 798°C), which I then plugged into the original equation to find \dot{Q} . My end result was 120 Watts of energy transfer by heat.

How can I use this result to find the current I need to run through the Tungsten wire?

 

[mfb]

The power in the Tungsten wire is P = V I = R I² = V²/R
You need the resistance to calculate required current and voltage.