Transient Heat Transfer Equations 4

[Thane]

I am designing an insulative fire blanket to be used in fire rated conditions for commercial buildings. I will run my design through a transient thermal analysis in ANSYS, but I want to use some transient Heat Transfer equations I can set-up in a spreadsheet to help verify my FEA results. I have been researching some of my old college notes and text book, but am not having much luck in finding the equations I think I need.

The hot side of the system is 1900 F for the duration. The temperature of the cold side starts at 60 F. I need to determine the cold side temperature at the end of 2 hours. I will need to consider conduction through the solid materials and natural convection through the air. I have the specific heat, thermal conductivity, and conductive heat transfer coefficient for each material. The sides are assumed to be adiabatic and the air inside the system is contained within the system. The cold side is bordered by ambient air with natural convection only. The air on the hot side is bordered by adiabatic walls with heat transfer to the insulative blanket only. Below is a simple sketch of the system.

Cold Side
ppppppppppppp
AAAAAAAAAAA
AAAAAAAAAAA
IIIIIIIIIIIIIIIIIIIIII
SSSSSSSSSSS
FFFFFFFFFFFF
Hot Side

p = thin aluminum foil on the cold side
A = air between insulative blanket and p
I = insulative material
S = Stainless Steel Foil
F = hot air at given temperature

Thank you in advance for any assistance you can provide.

 

[pmover]

while i do not have my heat xsfer textbook, i vaguely recall two techniques(?) for transient heat xsfer - explicit and implicit. any heat xsfer textbook should have that information.

you need to generate a diagram and include nodes throughout the system. then write out all the equations for heat xsfer in/out of each node.

this is from memory and i had heat xsfer 16 years ago.

good luck!
-pmover

 

[imok2]

Thane,
maybe this can help you

http://www.nscee.edu/Publications/N...mulationPredictionOfTransientHeatTransfer.pdf

 

[corus]

ANSYS already has been verified with many examples. A comparison with a spreadsheet won't be of much use because you haven't been able to verify the mathematics of your spreadsheet. Besides which your results will depend largely on the assumptions you are making in the model which would apply equally to ANSYS and your spreadsheet.
If you want verification of your results then look to some actual measurements of similar situations and see if your modelling assumptions give reasonably accurate results.

corus

 

[davefitz]

I think that with a 1900F hot side, the radiation heat transfer will dominate over convection, at least from the hot side to the blanket. You will need to obtain the emmissivity of the 2 foil layers , preferably as a function of wavelength.

A simplified 1 dimensional form of the Laplacian harmonic diff eq is:
dT/dt = a* d2T/dx2, where a=k/(Cp*d), Cp= heat cap, d= density.

The initial conditions and boundary conditons need to be correctly defined in order to solve the DE.

The DE can be easily transformed into the self adjoint form and solved using finite element methods; this has the advantage of also simultaneously solving the biharmonic DE of stress in the cold side metal and thus result in a determination of the thermal stresses.

 

[magicme]

if i did my algebra correctly, the temperature of the contained internal air varies with time as follows....

Tcoolside = Thotside - (Thotside - 60F) * exp ( - U * area * time / airmass / airCp)

where

U = overall HTC from hotair to coolair
U = 1 / (1/h1 + t1/k1 + t2/k2 +.... + 1/h2)

Area = the aurface area exposed to hotside gas

time = seconds

airmass = mass of contained air

airCP = specific heat.

this equation ignores the themal inertia of the layer masses and will give a quicker inside air temperature rise than the ansys results which i believe will include the time it takes each layer of mass to heat up.

you could however write an equation similar to this for each layer and piece them together in a spreadsheet and iterate a solution at each time step. it might not be as bad as it sounds, and its always good to have some kind of sanity check on finite element results.

daveleo

 

[corus]

Heat transfer in the air gap will be largely governed by radiation, not by conduction, and so daveleo's approximate solution would be incorrect. You should include this internal radiation in your ANSYS model as well as the radiation from the hot and cold surfaces. To solve this by a spreadsheet would require an iterative finite difference solution. For a regular mesh a finite element model gives the same solution as a finite difference model, in terms of the error from the discretization, and thus there would be no benefit in attempting to check ANSYS.

corus

 

[magicme]

thane
your original plan to do a backup or alternate calculation is an excellent idea, and you should continue down that path. despite the fact that ansys itself has been verified endlessly (though it still has many bugs and many, many quirks ......read the error reports) , it also generates garbage until your specific model itself is fully developed and somehow verified outside of ansys. if the right answer is important to your business, an alternate calculation is money well spent.
as far as the equation i posted (regarding corus's remarks about it) .... it's a generic textbook formula that i re-arranged. of course it would have to be modified for any specific application...... but you asked for an equation, to get you started and there it is.
an excellent example of a transient numerical method (great for a spreadsheet through multiple layers) is given in the book "conduction heat transfer" by P J Schneider in chapter 12.

regards

daveleo

 

[Thane]

Thank you all for your input.

I agree with daveleo. I think it is wise to have a sanity check to help validate your FEA model. In using any type of FEA analysis (thermal or structural) software, garbage in = garbage out.

I will also add thermal radiation into my sanity check.

Thane

 

[pyroris]

Thane:
The solution of the differential equation of davefitz for a semi-infinite solid under transient conditions is as follows:

BC: T(0,t)=T(hot) (Constant surface temperature)

[T(x,t)-T(hot)]/[T(cold)-T(hot)]=erf[x/(2.SQRT(a.t))]

a=thermal diffusivity
erf=Gausian error function

You can find the surface heat flux using:

q=k(T(hot)-T(cold))/SQRT(pi*a*t)

Radiation seems to be important in your case. You can find the transient solution of the radiation heat transfer in your "insulative material" using Rosseland Equation:

q=-R*dT/dx

where: R=16*sigma*T3/3*B

R: rosseland mean extinction coefficient
sigma=staphen boltzman coeff.
B=extinction coeff.

I am working on a similar problem. The material that I am working on forms its own insulator layer. In my case, radiation is the main mode of heat transfer. In your case stainless steel layer will apparently shield a significant amount of the incident radiation. If you find the temperature profile inside your material, you would have a better idea about the extent of radiation heat transfer.

 

[AkhanThermal]

My report presented at the 15th Symposium at Boulder, CO (NIST) may be useful (unfortunately with no infrared radiation, which of course should be taken into account at such high temperatures):

http://www.lasercomp.com/List of Papers.html