The time it takes to reach 63.2% of the maximum temperature inside the a box.

Question

this is for my undergraduate thesis. I think this is a combination of thermodynamics and calculus. This really troubled me for quite a while. So heres the problem.

Given:
I have an incubator (box) with a volume of 1 m³. The box is made of plywood.
Initial temperature = 84° F.
Power of the heat rod = 200 W.

Question:
What time it takes to reach the air temperature at 63.2% of maximum temperature using the heat rod?

Please comment if I missed out any given or important information to solve this.

Answer 1

The maximum temperature of the box will depend on the rate of heat loss through the box walls. Otherwise the box will heat up indefinitely. To know the rate of heat loss (assuming there are no leaks) you need to know the thermal conductivity of the plywood and the surface area of the box (assume a perfect cube). There isn't one value for "plywood" as there are many different types and thicknesses. So, this problem as stated, is incomplete. Btw, the max temperature will be the point at which the heat loss is equal to 200W.

Answer 2

In nature we do not have adiabatic walls. So you will have a loss through the walls of your incubator.

We know volume (1 $m^3$) but not total area. Besides, are all 6 faces with equal thermal resistance?

For thermal characteristics, you can see http://www.australply.com.au/technical/thermal-properties.

In your thesis, are you working with a thermostat to control/measure internal temperature?

I found these two sites with models that you can take as examples or modify according to your needs:

http://lpsa.swarthmore.edu/Systems/Thermal/SysThermalModel.html https://www.mathworks.com/help/simulink/gs/define-system.html

The 63.2% refers to the time constant of your system. More at : https://en.wikipedia.org/wiki/Time_constant