# Heat Transfer through slab - Conduction,Convection and Radiation

I have a problem here, attached the schematic diagram of the system

$Tu_0$ is cold air at $-20^{\circ}C$, so there exist convection to the $T_0$ (surface temperature of first slab). All the convection properties are set, heat transfer coefficient $h$, are set. Conduction happened from slab 1 to slab 5. Finally, radiation from $T_{black}$, a perfect black body with $T$ of $500^{\circ}C$, is coming to $T_6$ (surface temperature of slab 5). In between $T_{black}$ and $T_6$ is vacuum, and theory of radiation on enclosed black surfaces can be used (the usual Stefan Boltzman law with $T^4$).

I have done the problem, using finite difference technique encoded both in MATLAB and Excel as well. I just want to ask your opinion here about this problem. My MATLAB and Excel results shows T6 will converge into $250^{\circ}C$ after a while. But my friend argue that:

" Since $T_6$ is also assumed to be a black body, so there should be no heat loss from $T_{black}$, therefore $T_6$ should be almost equal to $T_{black}$, like $499^{\circ}C$. The convection from $Tu_0$ will be overcomed by the huge radiation from $T_{black}$, so that $T_0$ would not be even close to $-20 ^{\circ}C$, probably still closer to $500^{\circ}C$, like $450^{\circ}C$ or something"

I am having a hardtime to prove my friend argument to be wrong.... and he makes me unsure whether my MATLAB and excel is correct.

Thanks

• Hi Pratama, we use Mathjax manual on this site and I have added it to your post. Could you please read the link and use it in further questions. Thanks – user108787 Nov 30 '16 at 13:05
• Just a quick question but what do you mean by the temperature of a vacuum? A true vacuum has no temperature because it has no matter. Or are you talking about the equivalent radiation levels in this region? – honeste_vivere Nov 30 '16 at 14:27
• @honeste_vivere : Pratama is using $T_6$ to refer both to the vacuum region between slabs $T_5$ and $T_{black}$ and to the surface temperature of slab $T_5$. – sammy gerbil Nov 30 '16 at 15:13

Your friend's arguments are little more than vague opinions, guesses. I do not understand why you are listening to them. If he had backed up his arguments with his own model and calculations, it would be a different matter. You are under no obligation to prove your friend wrong : the onus is on him to prove he is right.

Perhaps he assumes that a temperature of $500^{\circ}C$ is more than 500 times as high as a temperature of $-20^{\circ}C$, and that after raising this temperature to the 4th power the heat flow would dominate the rate of convection and cause $T_0$ to rise. But that contradicts your statement that the convection properties are set.

If you have implemented the same model in MATLAB and Excel the fact that the answers agree suggests only that you have correctly coded the model, not that the model is correct. As KenG points out, it is a little suspicious that you get $T_6=\frac12T_{black}$ when measured in degrees C. It could just be coincidence.

Have you included in your model the fact that slabs $T_1$ and $T_5$ will also radiate according to Stefan's Law? But the effect might be negligible even for $T_5$.

• Thanks for your suggestion @sammy gerbil. Well the exercise was meant to be simplified ,and radiation from T1 and T5 is neglected. Like how it is suspicious that my T6 is half of my Tblack.. is it because its too low? – Tompel Nov 30 '16 at 16:08
• I don't see anything obviously wrong with your equilibrium value of $T_6$. If it is exactly 250C while the value of h is that for a real material, then this result looks unusual and might indicate an error in modelling or programming. But such coincidences do happen : $T_6$ has to be some value, and objectively a value of exactly 250C is as likely as any other. But your comment to KenG is that it is only approximately 250C, which makes the result less suspicious. – sammy gerbil Nov 30 '16 at 16:19
• Just checking it again in MATLAB, T6 is actually 268.02 C (sorry for my weary eyes in wrongly reading the MATLAB plot haha).... so I think nothing suspicious ey? – Tompel Nov 30 '16 at 17:35

Your friend is wrong, there is no requirement for a blackbody to be in radiative equilibrium. All you need is that the surface absorbs all light that impinges on it, if we say the blackbody is at some given temperature, we may assume it has whatever heat sources and/or sinks attached to it that are necessary to maintain that temperature. Also, you should regard T_0 as fixed, it is not allowed to come up to 500 C. Of course, one solution is that everything in the problem is at 500 C, if that were allowed, but it violates the conditions of the problem.

Also, since you didn't give the value of h your answer cannot be checked, but it seems odd that it came out 1/2 of the blackbody temperature in Celsius. Did you remember that you need to use Kelvin, so the blackbody temperature is really 773 K? That would require the other surface to be 523 K if your solution is correct, so it would just be a coincidence that this is 1/2 the other temperature in Celsius. I'm not saying I know it's wrong, it's just kind of an interesting coincidence so make sure you used Kelvin and make sure you raised it to the 4th power.

• Thanks a lot @KenG, Well I clearly use Kelvin in my MATLAB code. Well the value of h is 55 W/m2K and value of heat capacity Cp is 473 J/kh. Value of thermal conductivity of the plate is 45 W/mK. The other thing is that the T6 does not necessarily come out exactly half of the blackbody temperature. It just cameout around that value – Tompel Nov 30 '16 at 16:03
• OK, then all should be well, it's just a coincidence, and not exactly 250 C anyway. – Ken G Nov 30 '16 at 21:17