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





RE: "NON"-Constant surface heat flux in Internal Forced Convection in Pipe
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.
RE: "NON"-Constant surface heat flux in Internal Forced Convection in Pipe
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.
RE: "NON"-Constant surface heat flux in Internal Forced Convection in Pipe
RE: "NON"-Constant surface heat flux in Internal Forced Convection in Pipe
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RE: "NON"-Constant surface heat flux in Internal Forced Convection in Pipe
RE: "NON"-Constant surface heat flux in Internal Forced Convection in Pipe
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RE: "NON"-Constant surface heat flux in Internal Forced Convection in Pipe
RE: "NON"-Constant surface heat flux in Internal Forced Convection in Pipe
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(aTw+b)-hπD(Tw-Tf)=0
(equal to zero because of the steady state). This gives a direct relationship between Tw and Tf.
The heat balance for the fluid is
cpwdTf=hπDdx(Tw-Tf)
that gives a solvable differential equation for the temperature profile in the fluid.
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RE: "NON"-Constant surface heat flux in Internal Forced Convection in Pipe
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.
RE: "NON"-Constant surface heat flux in Internal Forced Convection in Pipe
RE: "NON"-Constant surface heat flux in Internal Forced Convection in Pipe
RE: "NON"-Constant surface heat flux in Internal Forced Convection in Pipe
The fact that the resistance varies is only of interest to the heater controller designer.
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RE: "NON"-Constant surface heat flux in Internal Forced Convection in Pipe
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 :)
RE: "NON"-Constant surface heat flux in Internal Forced Convection in Pipe
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.
RE: "NON"-Constant surface heat flux in Internal Forced Convection in Pipe
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.