Thermodynamics - Free Convection and Temperature Gradient Problem

[HarryWang713]

Hello everyone,

So I am an Aero-Thermo Intern at Pratt and Whitney, and my supervisor gave me the following problem to set up a mathematical Excel model for the temperature gradient inside an enclosed box with a heat source underneath it to be used in thermal analysis of an engine component. He detailed the problem exactly as follows:

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Problem Statement:
To calculate the temperature profile of an enclosure wall with a fire (heat flux) input at the bottom.

Assume:
• The box is 10” by 8” by 21”, made of cres 345
• The fire is at the bottom of 10” by 8” floor.
• The outside temperature is 59ºF
• The heat flux is 4,500 BTU/min
• The temperature is uniform in the horizontal plane ( 1D problem in Z direction)

Physics:
• The primary form of heat transfer is to the outside environment.
• Use natural convection for a flat plate on the outside.
• The primary driver is buoyancy force for the velocity to calculate the heat transfer coefficient. (Can assume flat plate to start with)
• The heat balance enables to calculate
o We can calculate the wall temperature
o The temperature of the air inside the box

Method:
• Initially cut the box into variable # of horizontal slices
• Calculate the wall area and volumes
• Calculate inside vertical velocity of air due to buyoncy.
• Calculate the internal and external heat transfer coefficients
• Assume an initial internal air temperature.
• Calculate the heat flux.
• Calculate the wall temperature and new air temperature for the next slice
• Continue till the last slice.
• Sum the heat flux and iterate the initial temperature to get the right total heat flux.

Results:
• Plot the temperature profile of the wall temperature and air temperature.
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I'm only just now taking my first thermodynamics class, and the conditions and method of solving this problem are foreign to me. My supervisor is very old school and could not explain how to do this very well, and if there is an easier method of obtaining the temperature profile of the walls and air then by all means inform me :). I'm familiar with the physics and equations for conduction and convection, but am not sure how to solve this problem. Thanks for your help!

 

[Mech_Engineer]

This is a pretty challenging problem to solve, especially if you haven't taken a heat transfer class which covers natural convection. Be that as it may, what you're basically looking for is convection/advection in a rectangular enclosure/cavity. Up to a certain point the buoyant forces in the fluid will not overcomeviscous forces and no advection will occur, and then at the critical Rayleigh number (greater 1708 from what my textbook tells me) advection will occur.

The heat transfer textbook I have from school, "Introduction to Heat Transfer" by Incropera et. al, has a short chapter on convection in enclosures in which it recommends an empirical correlation developed by Globe and Dropkin in the Journal of Heat Transfer.

Basically the correlation states that for a specific range of Rayleigh numbers you can use this correlation:

$Nu_{L}=\frac{h*L}{k}=0.069*Ra_{L}^{\frac{1}{3}}*Pr^{0.074}$

For Rayleigh Numbers:
$3*10^{5}\leq Ra_{L}\leq7*10^{9}$

My first inclination would be to use FEA for this problem, but this project is probably more to test your capability and though processes, in which case this correlation might be able to give you a good start, or at least steer you in the right direction.

 

[shushu97]

Read chapter 8 here:
http://web.mit.edu/lienhard/www/ahtt.html