Calculation of Nusselt # in thermally developing flow

[gpsimms]

Hey y'all.

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!

 

[Chestermiller]

Hey y'all.

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!

It's not clear to me what you are asking. In the thermal entrance region, the Nusselt number is inversely proportional to the distance x to the 1/3 power. This has to be included in the integration of the heat balance equation. Are you doing a numerical integration to get the average temperature? You also implied that the wall temperature may be a function of x. This can be taken into account, but you need to know what that variation is. Also, that large temperature difference between the wall and gas suggests that radiative heat transfer might be important to consider.

Chet

 

[gpsimms]

Hey Chet,

Thanks for the quick reply.

To give more information concerning whether radiative heat transfer could be important:

The flow is N2 through a quartz tube. The tube is heated inside a heavily insulated furnace with Nichrome heating wire. There are thermocouples placed very near the quartz tube to control temperature for each 4" long heating element. The entire furnace if 16" long (4 circular heating elements with ID of 2" and OD of 6"). I did not initially intend to ignore the radiative effects, but then when I made measurements of the flow inside the tube, I found that Newton's law of Cooling could almost EXACTLY model the temperature of the flow inside the tube, so I decided convection/conduction had to be by far the dominant mode of heat transfer. Do you think this is likely a bad assumption? I should note that most of the "ramping up" section of the tube is heavily insulated as well, so there should be no radiative heat transfer to the gas until it is already "pretty warm."

To respond to the integration question:

I am handling this problem in 1-D. The ID of the quartz tube is 3.98 mm. At each time step, I consider the temperature of the flow inside the tube to be uniform. The equation I am using is simply:

mdot*cp*(Tout - Tin) = h(Twall - Tin)

I am, at each timestep, accounting for the change in thermal conductivity and specific heat of N2.

To respond to the wall temperature question:

Yes, I have the wall temp as a function of x. To get this, I measured the temp. at the inside of the tube without flow at different locations, giving a long time for the system to reach equilibrium. I thought this method might be slightly suspect, except when I ran the code, to within a constant "fudge factor" the measurements fit the model *very* well.

To help detail my explanation, I have attached a .pdf of some of my code results.

In each graph, the dark blue line is the "wall temperature" measurement as described above. The light blue line is the temperature as calculated by the code. The red dots are the temperatures measured in the flow.

The thin blue line and the yellow line you can ignore.

The first slide and second slide show the "best possible fit." To get the best fit, I used a constant thermal conductivity (based on the set temperature), a constant Nusselt number, and introduced a "fudge factor." Note that the "fudge factor" had to be different for the high (T=1100K) and low (T=900K) case.

The third and fourth slide show that when I introduce a varying thermal conductivity (based on fluid mean temperature), it is not possible to get a "perfect fit." I found that one fudge factor would get a perfect fit in the low temperature region, but a different one was needed to get a perfect fit in the high temperature region.

The fifth and sixth slide show that if I instead base thermal conductivity on the wall temperature (which may make sense, since that is the temperature of the boundary layer), I can get a "decent but not perfect" fit for all cases, with the same 'fudge factor.'

My guess is that the 3rd and 4th slide, where I get a good fit early but not late (or vice versa) is the closest thing to "correct," except I need to introduce the varying Nusselt number, which leads to my question, as I am not exactly sure how to implement it.


Sorry this got so long, I am just trying to give as much detail as possible. Thanks again for reading!

 

[gpsimms]

Just read about the Graetz number, which I think is what you are referencing. Will play with it for a bit...

 

[Chestermiller]

Have you calculated the Reynolds number for the flow? How do you know that the flow is laminar? Also, I don't like your heat balance equation. I would have preferred:

WC_p\frac{dT}{dx}=hπD(T_{wall}-T)

Also, what is this about timesteps? This isn't a transient problem, is it?

Chet

 

[gpsimms]

Yes, the flow is relatively low Re number. 500 for the higher end mass flow cases and 300 for lower end.

Timestep was a bad word choice. Everything is resolved in space only.

Is W mdot? My right hand side is actually

hA(ΔT)

where h=\frac{Nu*k}{D}

and A=\pi*D_{tube}*dx

so that, all together my equation is:

Wc_{p}\frac{dT}{dx}=kNu\pi(T_{wall}-T_{fluid})

I was just being lazy in writing the first equation, since I was pretty sure I at least had that part correct.

The newest/closest iteration I currently have is I am using a Nusselt correlation from Hausen for the uniformly heated wall section (ie wall temp is approximately constant..Hausen gives a correlation for thermally developing region with constant temp wall), and for the ramping section, using a correlation from Shah which is 1.302/x*^(1/3), which is meant to be used in the constant heat flux case (which I think is a decent assumption for the ramping section, given the fluid temp line and wall temp line are close to parallel)...

It's pretty close now. Maybe 'pretty close' is the best fit I can get. Let me know if you can think of anything else...

Thanks again!