"NON"-Constant surface heat flux in Internal Forced Convection in Pipe

[nrhoades]

I am an electrical engineer and I just finished reading "Introduction to Heat Transfer" by Incropera and DeWitt. I am new to heat transfer. I want to find what the internal surface temperature of a pipe is as a function of axial distance from the inlet (assuming incompressible fluid with constant properties, fully developed flow).

This seems easy with the assumption that either the surface heat flux, or, the surface temperature stays constant.

However, the heat source in my problem comes from Ohmic heating of the pipe walls, in which the resistance is a function of temperature.

Therefore, I need to solve this problem using a NON-CONSTANT surface heat flux/surface temperature (since the surface heat flux is dependent on the surface temperature, and vice versa.)

This isn't treated in the textbook I have. Any ideas?

Thanks

 

[zekeman]

Fairly complicated problem that can be solved piecewise and iteratively.

A general solution is not possible; therefore We need to know the pipe dimensions, the flow rate, the length of pipe, the source of heat (e.g. voltage or current source,the winding data, the pipe environment, etc.

 

[flash3780]

Great book. I use mine all of the time. I think he's got all of the tools in there to solve your problem. Matlab would be a good tool to hone in on a solution.

Really, it's two problems:
1. Heat leaving the pipe & the associated pipe dT
and
2. Heat entering the fluid & the associated fluid dT

You'll need to balance the energies.

I think that you could write a program to descretize the pipe and find the heat transfer rates for the discrete sections, and hence the temperature profile. It's kind of like a mini-fea problem.

 

[flash3780]

Can you post a diagram of your particular problem?

 

[IRstuff]

If you have Incropera, then you have everything you need, starting in Chapter 5, coupled with Chapter 9.

TTFN

FAQ731-376
Chinese prisoner wins Nobel Peace Prize

 

[nrhoades]

Chapter 9 is Natural Convection. I did not read this chapter yet. I will, and then I'll post an illustration of my problem on Monday (tomorrow).

 

[nrhoades]

I've posted a diagram of my problem.

http://files.engineering.com/getfil...-a6fc-4a02c19207b4&file=pipe_heattransfer.pdf

See attachment...

 

[ione]

Divide your pipe length in intervals where you can consider the heat flux as a constant (what affects the heat generated is the electrical resistivity, which can be considered as a constant in an appropriate interval). Then assume a trial average fluid temperature in the each interval and determine the heat transfer coefficient htc. Impose steady state condition (energy generated = energy transferred to the fluid) and iterate until results converge. Iterate the procedure above for each interval.

 

[prex]

If you want to arrive at a solution by analysis, then you should start by simplifying the problem. The assumption of thick wall with the associated radial temperature profile can only be treated via FEM.
Simplifying assumptions may be:
-thin wall behavior for the pipe = uniform radial temperature
-also uniform radial temperature in the fluid
-no axial conductance in the wall nor in the fluid
and of course steady state!
Under these assumptions, by taking an axial slice of pipe of length dx, the heat balance of pipe wall is
(S is pipe's cross section and a and b are in suitable units)
S(aT_w+b)-h[π]D(T_w-T_f)=0
(equal to zero because of the steady state). This gives a direct relationship between T_w and T_f.
The heat balance for the fluid is
c_pwdT_f=h[π]Ddx(T_w-T_f)
that gives a solvable differential equation for the temperature profile in the fluid.


prex

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[nrhoades]

prex,

Making the assumption of radial temperature is acceptable. I'll re-work my posted diagram to make sure I understand your advice. Thanks.

Once I get this part down I'll try to use FDM to move forward.

 

[ione]

If you have a temperature variation that affects the heat flux, it should also affect the heat transfer coefficient. So htc (or h in prex’s post) won’t be a constant for the whole length of pipe. Each pipe interval (or slice) will have its own heat transfer coefficient.

 

[nrhoades]

Ugh. It bothers me that there is no analytical solution.

 

[IRstuff]

Looking back at your original posting, you seem to have separated the resistance/flux, but a typical heater configuration is designed to provide a certain amount of power, i.e., heat flux, regardless of what the heater resistance is. If this is your case, then the problem reverts to a constant heat flux condition.

The fact that the resistance varies is only of interest to the heater controller designer.

TTFN

FAQ731-376
Chinese prisoner wins Nobel Peace Prize

 

[nrhoades]

IRstuff,

On the contrary, I am designing a microfluidic temperature/flow sensor... it acts kind of like a solid-state hot wire anemometer.

Imagine fluid flowing through a channel cut into a material with a high positive-temperature-coefficient. On each side of the channel are two pairs of metal pads. The pads are connected to a power source so that current flows around the channel, normal to the channel. Here is how the physics works, assuming that the fluid temperature is cold...

1. Pad pair 1 heats up the material under the channel due to Joule heating.
2. Heat from the material transfers to the fluid.
3. The fluid moves the heat by bulk advection.
4. The fluid is hotter between pads #2 than pads #1 because of step 3.
5. The material under the channel between pads #2 becomes slightly hotter than the material under the channel between pads #1.
6. The resistance between pads #2 increases relative to the resistance between pads #1 because of the PTC.
7. There is now a current differential between the two pairs of pads that can be mapped to a unique temperature/flow-rate pair.

Clearly, this needs to be modeled using FEM. I use COMSOL to do this, and I am successful.

However, even with a simple model using assumptions and approximations, I ATLEAST need to include

1. PTC effect
2. Material heat transfer through conduction
3. Fluid heat transfer through convection

Using just these three, and using the simplest geometry (a pipe or tube), it appears that a solution still does not exist (without using numerical methods). I don't like this reality

 

[ione]

Seems to be kind of a colorimetric flow meter.
In a common calorimetric flow meter two thermistors, one thermally insulated from the other, have their sensing tip immersed in a fluid. One thermistor (that placed upstream) is continuously heated, whilst the second is not. With no flow a temperature difference between the two thermistors establishes. When fluid flows heat is carried away from the heated thermistor and the temperature difference decreases as fluid velocity increases.
If so you have to use a high thermal conductivity fluid to get a swift response. Moreover the thermal inertia of the flowing fluid should be that high that stream temperature shouldn’t sense any temperature increase.

 

[nrhoades]

I've made some progress with this problem. I've developed two solutions: one is an algebraic extension of constant heat flux, and the other is an FDM solver for the governing PDE.

Question:

Assuming laminar flow, incompressible fluid, the PDE is

v0*cp*rho*(1 - r^2/R^2)dT/dx = k*Laplacian(T)

where the Laplacian is cylindrical and the angular and conductance terms go away.

How do I figure out what the "heat outflow" boundary is at the end of the pipe? Do I have to extrapolate it or something? How do I do that?

Thanks.