Help needed transport problems involving bessel eqn

[aelisha1079]

Hi everybody... i would like to seek help for the problem below. the assumptions I've considered is that transfer is radial only since it is a very long cylinder (infinitely long) that transfer in z direction is negligible, thermal radiation is zero, and wood properties are constant. Starting from the general energy equation I've come with a form similar to a modified bessel function of zero order but I'm stuck after that. i hope you may be able to help me. thanks a lot in advance.


Problem : Burning of a wooden dowel

In a fire research experiment, a long wooden dowel ( around rod), diameter 1.00 inch, temperature 74°F, is placed in a stream of 1,500°F air. The heat transfer coefficient between gas and surface is 5 Btu/h-ft2-F. Estimate the exposure time up to ignition of the rod on the assumption that ignition takes place when the surface temperature reaches 750°F.
Wood Properties at 74°F
Ρ= 50 lb/ft3
k = 0.10 Btu/h-ft-F
cp = 0.6 Btu/lb-F

assume that the wood properties are constant up to 750°F. Neglect the possible role of thermal radiation.

 

[Mapes]

Hi aelisha1079, welcome to PF. I would check first if conduction within the rod can be ignored (i.e., calculate the Biot number). If it can, then a lumped capacitance approach can be used. If not, then the standard approach is to start with the time-dependent solution for the heat equation in cylindrical geometry; there's no reason to try to re-derive it.

 

[aelisha1079]

hi mapes! thanks for the reply. I've tried to derive the equation because i can't find a reference with the same case as unsteady heat conduction in an infinitely long cylinder. I would appreciate if you can provide me some referance for the problem. thanks so much

 

[Mapes]

Check Incropera and Dewitt's Fundamentals of Heat and Mass Transfer.