Determining the final surface temperature and final gas temperature flow through a tube in a room

Question

I am trying to calculate the final temperatures as stated above. There is hot gas flowing through a tube that is insulated and inside of a room (no wind). I will try to walk you through my thought process in doing so while typing out the formulas I'm using.

I'm starting with a visual resistor method (shown below). The main issues I believe I'm having is with both convective forces (inside the tube and outside, assuming there is convection inside). The formula for hrad (shown below) is one I found online but I'm not sure if that is correct.

NOTE: I'm not using the Nusselt equations shown as I did find one for a cylinder in cross flow that I'm using.

I am using the Rayleigh number for the external convection while also trying to use radiation as a factor. The problem arises when I try to run the system without insulation the outer surface temperature is MUCH lower than it should be (900F inside, 100F outside , its showing 550F surface temp). I know that this should be near the 900F mark as metal is a terrible insulator. I'm losing most of my heat transfer from my convection internally so I'm assuming that I'm doing something wrong there but can't figure it out.

Working on this today and when I completely remove the convection from the inside flow the temperatures seem to be a lot better. Another question that now arises is do the k-values for steel change with temperature? I'm assuming this is a definite yes but now am looking for a correlation between the temp and k values. This may be digging too deep now and basically need a full experiment to do but maybe its something that's already done? Thank you!

I've been trying to create an excel document that will be able to calculate these variables for any inputs for the system.

I was hoping that someone who knows more than me about this can look over what I'm using and hopefully tell me whether I'm using the correct equations/values.

If anyone could explain my wrong assumptions/thinking that would help immensely.

Answer 1

The formula for $h_{rad}$ (shown below) is one I found online but I'm not sure if that is correct.

You can try turning off radiation entirely to see the effect. You can also try fixing the surface temperature and estimating the relative heat flows by convection compared to radiation, since the two equations are well-defined. Use that estimate to substitute a multiplying factor $M$ for the outer resistivity. Rather than

$$\frac{1}{R_{outer}} = \frac{1}{R_{conv}} + \frac{1}{R_{rad}} $$

write as

$$\frac{1}{R_{outer}} = M\frac{1}{R_{conv}} $$

I'm losing most of my heat transfer from my convection internally so I'm assuming that I'm doing something wrong there but can't figure it out.

You have one of two approaches for the case that you remove the insulation. Are you holding the internal temperature constant and allowing the total heat flow to the gas in the tube to increase? Or are you fixing the heat flow to the gas in the tube and thereby observing that the internal temperature decreases?

Another question that now arises is do the k-values for steel change with temperature?

Yes. See one of these two links and/or search also for thermal conductivity versus temperature for the steel that you use.

https://physics.stackexchange.com/questions/330158/why-does-the-thermal-conductivity-of-pure-metals-decrease-with-increase-in-tem?rq=1

https://nvlpubs.nist.gov/nistpubs/jres/12/jresv12n4p441_A2b.pdf

I was hoping that someone who knows more than me about this can look over what I'm using and hopefully tell me whether I'm using the correct equations/values.

Your approach appears to be correct for a specific case. You have not accounted for the temperature change of the gas inside the tube. An example of the formulation for this is shown at this link. An example of the equation for heating a fluid in a solar concentrator is shown at this link.