Transient Finite Difference Boundary Conditions

[Publius190]

Hi everyone,

I am trying to find the temperature through an object of length L surrounded by air at one end and enclosed air at a different temperature on the other end. I am only considering one dimensional heat transfer through the object in the x direction. At x(0) I have the temperature increasing linearly to illustrate changing temperature throughout the year. At x(L) however I have a different case. I want to have T(L) initially at 273k (0 degrees Celsius), and calculate temperatures at each timestep based on convection into the surrounding enclosed air.

My one problem is that I don't know how to calculate the change in temperature of the enclosed air.

So far I have thought about somehow calculating a heat transfer rate "q" based on q=rho*V*A*dT/dt and q=hA(Tin-T(L)). However in order to use those equations I would need to know the change in temperature of the air for each timestep.

Does anyone have any advice on how to solve for that inside air temperature?

Also if anyone needs anything clarified I can probably draw up a diagram to show you.

 

[IRstuff]

" I would need to know the change in temperature of the air for each timestep"

That comes from the removal(addition) of heat during the previous time step

TTFN
faq731-376

 

[prex]

To calculate the change in temperature of air you need a further equation on the heat balance of the volume of air. This may be difficult to define correctly, depending on your setup, as it could require assumptions on the convective behavior of air.
One point is unclear to me in your description: I understand that the sides of the object (a bar I assume) are adiabatic and that the only heat loss is from the front area at x=L, correct? If so, it is a very simple problem and finite differences are not necessary to solve.

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[Publius190]

Well let me try to clarify a bit more. The bar is initially at 0 degrees C and is heated by the air at x=0. At x=L the bar will heat the enclosed air which is also initially at 0 degrees. The sides are adiabatic.

My question is how can I calculate the temperature of that enclosed air based on the volume, surface area etc.

 

[prex]

Again: to calculate the temperature of air at x=L you need one more equation for that enclosed space. This might account for the heat capacity of air (rarely relevant, but...) and should account for how heat goes out. If you can describe the geometry of your setup, possibly with a sketch, this would help.
And, sorry, I was wrong in saying you don't need finite differences for this problem: only the final steady state solution can be calculated by formula.

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[Publius190]

Basically the top bar is used as insulation for the system. I can compute the temperature at the bottom boundary T(L) using finite diff. analysis. I need some equation that can compute the temperature Tin. The inside of the chamber has a constant volume V and We can use the assumption that the temperature in the chamber changes uniformly. Thanks again for the help.

 

[prex]

The condition of zero heat transfer across the walls is unrealistic: the walls have mass and heat capacity as well as the insulating material, whatever it is (the heat capacity of air being likely negligible). So you need to account for the heat transfer from the inner face of the lid to the enclosed air and from the air to the walls, that will absorb heat until a steady state is reached with possibly a nearly zero net heat loss (at steady state).
Not an easy problem: a realistic solution requires an assumption on the convective behavior of air in the enclosed space (may vary a lot depending on shape and dimensions).
IMHO only an important research programme would justify a detailed analysis as compared to a test: I recall of such calculations done to estimate the heat loss through the penetrations of the cover of a nuclear reactor.

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