heat flux through semi-infinite wall

[Thomson]

Hello everyone,I am a new user of the forum. I hope with the supportive people out there I can get help with my studies and if possible help others too.

 

To start with I have a question related to heat transfer through a wall. The question requires us to find the position x, using the expression of temperature T(x) in a semi-infinite wall for which the value of -Kdt/dx (it's delta in the question but unable to insert a symbol I replaced with d) is equal to given values of heat flux ( for eg; 3.4 W/m2).

I searched for an aid throughout but only found really complicated ones. My guess is that I need to first find an algebric expression of temperature T(x) for a semi-infinite wall and then find its derivative and then solve the equation for the flux. But I don't have 'aucune' clue for that.

Any help would be highly appreciated. I need to find the solution by tomorrow

[studiot]

Hello Thomson and welcome.

 

Please note that homework/coursework belongs in the homework section.

It is also wise to allow sufficient time to gather responses on a science website, one day is not enough.

 

Finally you need to post the actual question (don't worry about the greek letters we can help with that

plus some idea of your thoughts about the question.

 

Yes you need to derive a differential equation with respect to time for the heat flux.

What physical laws or theorems do you know in relation to heat transfer?

Hint they are associated with names of two of the giants of applied maths.

[Thomson]

Thank you for the reply. Sorry I didn't know that it was to be posted in another section. I will do so right now.

Regarding principles of heat transfer, I am only aware of 2: Fourier's law and that of Stephan Boltzman

 

The question is in fact last part of a series of questions. In the first few questions, we need to find the heat flux at the interior of the wall for conditions given: External temp (T) =Tint + Asinwt where Tint Internal temp = 19 degree and A(amplitude) = 10 degrees and period = 24 hours; thickness of wall = 20 cm and that of insulation = 12cm. Conductivity of wall = 2 SI and of insulation = 0.2 SI.

We need to first evaluate flux density for a given period ( between 14th and 15th day, etc.). That's not so difficult as we can employ software (like COMSOL).

Now the last question. It is in french so I try my best to translate:

 

' If we use theoretical expression, noted here T(x), of the temperature as obtained in the model of a semi infinite wall, at what position of x should we evaluate -Kdt/dx to find the value of flux density obtained in questions above ( I obtained 3,4 W/m2)

[studiot]

Quick reply, I will return again after I have digested your additions.

 

Yes Fourier and his law of heat transfer is one two names I had in mind.

 

Newton's law of cooling was the other.

 

Standard heat transfer calculations are based on these.

 

However you need to be careful if this is a building calculation because there are standardised approximations and assumptions made for the simplification of the calcs so they do not have to be worked from first principles every time. For building control purposes these standardised methos superceed first principle calcs.

 

You mentioned the complexity of some maths you have found.

Solutions to the equations have two aspects.

Steady state and transient. The transient solutions are much more complicated and these may be what you have found.

However Fourier's law does admit trigonometric series solutions for both cases.

 

Edit add the following material

Here is a good discussion of the maths.

 

https://www.thermalfluidscentral.org/encyclopedia/index.php/One-dimensional_transient_heat_conduction_in_semi-infinite_body

 

The first part of the article gives the basic time independent equations.

 

The part you will be interested in starts at equation 23 where a periodic heating function, as you seem to indicate, is introduced.

The analystical solution is then presented followed by a neumerical solution from equation 44 on.

Note that in the numerical soltion an approximating function is used (they used a cubic, equation 50).

Edited February 2, 2016 by studiot