Solve Conduction Problem for Refrigerator

[nonphysicsguy]

I'm trying to build a simple model to determine the temperature of a solution in a refrigerator assuming that the temperature in the refrigerator is constant. I'm also assuming that the solution in the fridge is the same temperature as it's vessel and that the vessel is solid and made only of the vessle material. I know, it's a gross over assumption but I want to start simple and build from there.

I'm assuming that I have a fridge that maintains a constant temp T1=50F. I assume that my vessel (solid glass k=0.6068 BTU/hr*ft*F) is 1"x1"x1" and has an initial temperature equal to T2=40F. Using Fourier's equation, I=dQ/dt=kAdT/dx, where k is the thermal conductivity of glass, A is 1 sq. in, and dx is 1", I = 0.50736 BTU/hr.

1) How can I calculate how long it will take to reach equilibrium with the air temp in the fridge?

2) If I wanted to make the problem more complicated and model the vessel as a glass cylinder that has a certain height, thickness and cross sectional area, can I say that the glass vessel is a series resistance with the contents inside (water) and repeat the calculation above? So, Req=Rglass + Rwater, Where Rwater = (vessel height)/(pi*r^2*k_water) and Rglass = (vessel height)*(pi*r1^2 - pi*r2^2), where r1-r2 is the thickness of the glass. Then, I = dT/Req. Once I have I, I can determine how long it will take the water in the vessel to cool when the fridge is set to a certain temp ... ? Make sense?

 

[Valerie Park]

1) To calculate how long it will take to reach equilibrium with the air in the fridge, you need to calculate the thermal resistance of the vessel and its contents. This can be done by using Fourier's Law of Heat Conduction: dQ/dt=kA(dT/dx), where k is the thermal conductivity of the vessel material, A is the cross-sectional area of the vessel, and dx is the thickness of the vessel. Once you have the thermal resistance, you can calculate the time it will take for the contents of the vessel to reach equilibrium with the air in the fridge. 2) Yes, you can make the problem more complicated by modeling the vessel as a glass cylinder with a certain height, thickness, and cross sectional area. You can then calculate the total thermal resistance of the glass cylinder and its contents by adding up the resistances of the two components. The resistance of the glass cylinder will be determined by (vessel height)*(pi*r1^2 - pi*r2^2), where r1-r2 is the thickness of the glass. The resistance of the contents in the vessel will be determined by (vessel height)/(pi*r^2*k_water), where k_water is the thermal conductivity of the water. Once you have the total thermal resistance, you can then use Fourier's Law to determine the time it will take for the contents of the vessel to cool when the fridge is set to a certain temperature.