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I am working on a real application to the common textbook problem: fluid through a pipe with constant temperature wall. Find mean temperature profile of the fluid.

In the textbook version, we assume the fluid is cold and it enters the hot walled region suddenly. We might also assume the flow profile becomes thermally developed very quickly and use Nu=3.66.

In the real case, I have a tube reactor in a furnace which is heated to 1100K. I am flowing 98% N2 gas at various pressures and mass flow rates.

Using a constant Nusselt number gives me a very close model of the real temperatures (I took measurements inside the tube with a long thermocouple), but it requires me to introduce a 'fudge factor' to make the fit very good.

So I was reading about the problem in 'Convective Heat and Mass Transfer' in Kays/Crawford, and they say that the Nusselt number is higher than 3.66 in the space where the flow is still developing thermally.

The give the equation Nu=[1/(2x*)]ln(1/theta), where x* and theta are non-dimensional space and temperature variables defined as follows:

x*=(x/r)/(Re*Pr) and theta = (Twall - Tfluid)/(Twall - Tentry)

Now, the formula for the Nusselt number is hard to incorporate in code for a couple of reasons.

(1) The Nusselt# --> inf as x* --> 0. So, in the beginning of the warm region, Nusselt number is very large.

(2) The definition of x* is sort of nebulous, because I have a gradually ramping up heating section. Technically, everywhere Twall is still changing, we kind of have to "reset" x* back to 0, right?

(3) For my flow with its RePr, x* gets large very fast, but with my 1 mm steps forward in space in each iteration, the ln(1/theta) term cannot grow fast enough to keep up with it, and my temperature remains unchanging.

So, my main question is: this seems like a very classical type problem. There must be somewhere someone has worked out these details, right? Is there a correlation for Nu that I am missing for this type of problem?

Thanks so much for reading and thanks in advance for any help I get!

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# Calculation of Nusselt # in thermally developing flow

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