Transient heat transfer problem in cylinderical coordinates

[johnsonb.engr]

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Could some one please help me. My professor for a lower level programming class (in labview) has given the class the following assignment. This is not a heat transfer class and he has not spoken/taugh us anything related to heat transfer. He emailed us a few pages from a heat transfer book about bessel-fourier series solutions to differential equations but the text is pretty technical plus pretty low quality scanned image.

Could some one please help me/point me in the right direction to solve this problem?
(note i just need to help setting up the equations, the programming of this is going to be an eternally different beast!)

"Using the method of separation of variables, provide a VI that predicts, starting from a fixed initial bulk temperature, the transient temperature in a finite length section of a thick-walled tube, subject to insulated boundaries at the outer radius and both ends, and a prescribed temperature or heat flux on the inner surface. Make the heat capacity, thermal conductivity, coefficient of heat transfer, dimensions, and internal BC all variables. For a fixed inner temp, predict heat flux, for a fixed heat flux, predicted temperature, each as a function of time.

Superposition allows the steady state solution of the radial and axial directions "

Your help would be greatly appreciate, I have taken heat transfer about a year ago but i don't remember ever having to solve a transient state finite cylinder problem. ps. sense this class is lower division i would say over half the class has not taken heat transfer
yet!

Thank you for your time, any help would be greatly appreciated!

 

[johnsonb.engr]

I know my last post was a little long winded, but i here a summary of my problem. I have to use separation of variables to find the Tx in a cylinder which is heated from the a whole along its axis and the out side of the cylinder is completely insulated.

In class we look at bessel function which comes with labview. As always we never, he went off on a tangent and started taking about temp fluctuations in He3 (which is interesting but not relevant to our class).

I know that the bessel function can be used to model surface progation in drums but do not have a clue how to related the bessel function to a transiatie state heat problem.

If some one could help me get stated on this it would be greatly appreated. I don't need anyone to give me the anwser but I have no idea how to even start this problem. thank you very much

 

[Mapes]

Start by writing an energy balance on an element of the tube, selecting an appropriate coordinate system. Assume that the temperature can be expressed as the product of a time-dependent function and a location-dependent function. Apply the method of separation of variables. Apply the Bessel-Fourier series solutions to the differential equations that arise.

Incropera and DeWitt's Fundamentals of Heat and Mass Transfer should be of some help with the first part; unfortunately, they don't model this particular system.

 

[johnsonb.engr]

ok thank you very much for the reply. I looked up the conduction equation for spherical eq and got. Can u understand this, i don't really known how to work with latex, but i can reenter it needed.

1/r(d/dr)(kr(dT/dr)) + 1/r^2(d/dO)(k(dT/dO) + d/dz(kdT/dz) + q = pc(dT/dt)

once again the problem is an finite cylinder which is perfectly insulated, and their is a whole in the center of the cylinder.

By applying the approate boundry conditions I can cancel the following: q (not heat generation in cylinder only in hole), d/D0 because it is a cylinder and the same in all dimensions.


so that leaves us with

1/r(d/dr)(kr(dT/dr)) + d/dz(kdT/dz) + = pc(dT/dt)


so then my next step would be to find the solutions to each of these differential equations separately for both sides then my overal solution would be the production of both of these solutions?


second question, can i cancel the z if the system is has insulators on the top and bottom of the cylinder?

third questions, why do i need to use the bessel functionn for this problem (in this class we have never talked about a heat transer problem, or diff eq for that matter. so i am unsure why i need to use the bessel)


Thank you very much for your help, you guys rock!

 

[Mapes]

so then my next step would be to find the solutions to each of these differential equations separately for both sides then my overal solution would be the production of both of these solutions?

Yes, where temperature is separated into functions of location and time.

second question, can i cancel the z if the system is has insulators on the top and bottom of the cylinder?

Only if the boundary conditions on the inside are independent of z.

third questions, why do i need to use the bessel functionn for this problem (in this class we have never talked about a heat transer problem, or diff eq for that matter. so i am unsure why i need to use the bessel)

Bessel functions often arise as solutions in cylindrical coordinate systems.

 

[johnsonb.engr]

Ok thanks for the help. I am having a problem find the currect solutions to this problem, from a previous post of PF https://www.physicsforums.com/archive/index.php/t-204570.html

i found that u(r,z) = u_0 \cdot \sum_{n=1}^{\infty}\frac{sinh(\lambda_n z) \cdot J_0(\lambda_n r)}{\lambda_n \cdot sinh(4 \lambda_n) \cdot J_1(2 \lambda_n)}

******I am unsure how to get latex to work, could someone please send me a link or tell me what i am doing wrong*****************

I am not sure how to use equation.

 

[Mapes]

Try enclosing between {tex} and {/tex} (change braces to brackets).