Combined radiation with conductive and convective heat transfer

I have a tube(only the cross-section displayed in the figure!), where gas flow is used to heat up the water. The water flows inside the steel tube, where the walls have been marked red. The heat transfer coefficient between steel tube and gas is known, as well as the heat transfer coefficient between water and steel and the conductive heat transfer coefficient for the steel. To calculate the heat transfer between the gas and water is pretty straightforward, using Fouriers and Newtons law.

The question is:

What happens if we consider the radiation heat transfer from the gas as well? The walls have emissivity e2 (the outer steel wall) and e3. My thought was that this could simply be added to the convective + conductive heat transfer, but I am not sure if it is that easy.

Thanks!!

What I have done, heat transfer from gas to water per length:

The radiation from the gas can be expressed as, following Stefan Boltzmann laws:

Here T2 represents the surface temperature of the steel pipe (corresponding to r2) while T3 represents the surface temperature of the outer layer(corresponding to r3).

My question is:::

What is the TOTAL heat transfer from the gas to the liquid, when radiation is considered as well? Can I simply say Qtot = Qconv+cond + Qrad ?

• These slides starting with "Radiation with Participating Media" may be of use. Is your strategy to solve the problem (analytically or numerically) in polar coordinates employing axisymmetry? – Chemomechanics Apr 23 '18 at 23:47
• Thank you, but I did not find a similar example. I was planning to solve the problem by using cartesian coordinates. – Jmei Apr 24 '18 at 6:24
• The entire slide set addresses radiative flux. Do you know how to write an energy balance for a differential element? This is generally the first step to solving a heat transfer problem. Why do you think your gas will emit substantial radiation (e.g., does it contain high-temperature water vapor)? Please consider editing your question to add these details and to show where you are in your analysis and where you're stuck. The question is currently too vague because yes, we can simply add radiation, but no, it's not clear why this is necessary and how we would solve the resulting problem. – Chemomechanics Apr 24 '18 at 17:26
• The question has been updated now – Jmei Apr 25 '18 at 7:01
• Somebody able to help me? :) – Jmei Apr 27 '18 at 12:47

You initially asked about radiation from the gas (i.e., emitted radiation, as would be relevant in the case of carbon dioxide or water vapor, for example, at very high temperatures) but then added an expression for radiative heat transfer from the outer wall through the gas. So I'll assume you meant the latter—that the gas itself is not radiating significantly but that the heat of the outer wall is so high that radiation from it is relevant.

You cannot add the heat flux in your first equation to the heat flux in your second equation because they were derived under different conditions. Specifically, the first equation was derived under the condition that only convective heat transfer affects the outside of the steel tube. Let's look at this derivation.

The heat flow from the steel tube to the water is

$$Q=2\pi r_1Lh_1(T_1-T_\mathrm{water}).$$

The heat flow through the inner tube is

$$Q=\frac{2\pi Lk_\mathrm{steel}(T_2-T_1)}{\ln(r_2/r_1)}.$$

The heat flow from the gas and wall to the steel tube, with the radiation mechanism added, is

$$Q=2\pi r_2 L\left(h_2(T_\mathrm{gas}-T_2)+\frac{\sigma(T_\mathrm{wall}^4-T_2^4)}{\frac{1}{\epsilon_\mathrm{steel}}+\frac{1-\epsilon_\mathrm{wall}}{\epsilon_\mathrm{wall}}\left(\frac{r_2}{r_\mathrm{wall}}\right)}\right).$$

(The radiation term should be equivalent to yours; it's derived, for example, in Incropera's table "Special Diffuse, Gray, Two-Surface Enclosures".)

These terms are all equal at steady state. Now, if the radiation term weren't present, then you could take advantage of the fact that these equivalent heat fluxes are linear in $T$ and construct an equivalent thermal resistance, as you do in your question.

As far as I know, however, the presence of the radiation term precludes a tidy analytical solution for the general case.

Nevertheless, if you can make two assumptions, you can recover the simple thermal resistance form. One is that $T_\mathrm{wall}\approx T_\mathrm{gas}$, and the other is that $T_\mathrm{wall}$ isn't much higher than $T_2$. In such a case, you can linearize $T_\mathrm{wall}^4-T_2^4$ as $4T_\mathrm{wall}^3(T_\mathrm{wall}-T_2)$, turning the third equation above into

$$Q=2\pi r_2 L\left(h_2+\frac{4\sigma T_\mathrm{wall}^3}{\frac{1}{\epsilon_\mathrm{steel}}+\frac{1-\epsilon_\mathrm{wall}}{\epsilon_\mathrm{wall}}\left(\frac{r_2}{r_\mathrm{wall}}\right)}\right)(T_\mathrm{wall}-T_2)=2\pi r_2 L\left(h_2+R\right)(T_\mathrm{wall}-T_2),$$

where $R$ captures all that radiative heat transfer information. In this case, the heat transfer from wall to water per unit length is

$$\frac{Q}{L}=\frac{2\pi r_1(T_\mathrm{wall}-T_\mathrm{water})}{\frac{1}{h_1}+\frac{r_1}{(h_2+R)r_2}+\frac{\ln(r_2/r_1)}{k}}.$$

• Thank you very much! that was very helpful and cleared it up for me. Your assumption regarding the gas is correct, sorry for a badly formulated question. Final question I have is why are you not including the convective heat transfer between the wall and the gas? I.e. in your third equation. – Jmei Apr 29 '18 at 13:43
• You certainly could incorporate convective heat transfer between the wall and the gas, but then you could no longer use the thermal resistance model, which assumes that all heat transfer occurs in series. If the wall and the gas don't have the same temperature, then convective transfer from the wall to the gas and from the gas to the tube is essentially parallel to the radiative heat transfer from the wall to the tube. The model would get much more complex and probably wouldn't be analytically tractable. Try playing with the terms, though—maybe you could make some simplifying assumptions. – Chemomechanics Apr 29 '18 at 18:49