Boundary conditions for convective heat and mass transfer + wall Temperature

In summary, the conversation discusses the use of boundary conditions in finite difference formulations for a pipe containing fluid. The boundary condition at x = x1 is given by the effective thermal conductivity of the fluid, T is the temperature at any point x, hw is the wall heat transfer coefficient, and Tw is the temperature at the wall. It is clarified that T at x = x1 is the bulk temperature and not the wall temperature, and the problem arises because the temperature of the fluid is not constant along the spatial dimension. The conversation also mentions the use of Tb as the exterior imposed temperature at the right side of the wall in a transient problem, and how it gets eliminated when combined with the transient energy balance equation.
  • #1
maistral
240
17
I am operating via finite differences.

Say for example, I have this pipe that contains a fluid. I have the boundary condition at x = x1:
deriv.png

k is the effective thermal conductivity of the fluid, T is the temperature of the fluid at any point x, hw is the wall heat transfer coefficient, and Tw is the temperature at the wall.

I know how to 'plug in' such boundary conditions in finite difference formulations. My question is which value of T do I use? Is it T at x = x1? But isn't that Tw already, since x1 is the wall?

I keep on seeing that T here is supposed to be the bulk temperature. The problem is since I'm doing this by finite difference, the temperature of the fluid is not constant along the spatial dimension. In this case, which point to use exactly? Is it the point at the fluid beside the wall point?

Or is it that at the 'wall point', there are 'two operating temperatures' - the 'wall temperature' and the 'bulk fluid temperature' at the wall?

Any help is appreciated. Thank you!
 
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  • #2
EDITED; Additional info:

The first equation written below is the boundary condition I lifted directly from a book. If I express the boundary condition in terms of finite differences, I write it as the second equation, am I correct?

1574704848241.png


But if I do write it as the second equation, isn't T(r) just equal to Tw(r), since this boundary condition is applied at the wall? Or am I missing something here?

Or Tw(r) is indeed, the temperature at the wall (r = d/2)? If then, what is T in the RHS of the equation, and at which value of r do I evaluate it? Certainly it should not be r = d/2 since it would just cancel out?
 
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  • #3
This graphic should sum up my rather, annoying problem that makes me look like someone who wants to constipate. Please help.
reactor help!.png
 
  • #4
You are aware that Tb is supposed to be the exterior imposed temperature of the surrounding medium, outside the thermal boundary layer at the wall. It has nothing to do with the wall temperature. I assume you are solving a transient problem, correct? Your boundary condition equation should read: $$-\lambda\frac{(T_5-T_3)}{2\Delta x}=h_w(T_b-T_4)$$This should be combined with your transient differential equation at point 4 to eliminate ##T_5##.
 
  • #5
Allow me to reinforce my understanding.

In your equation, this means Tb is the bulk temperature at the right side (outside) of the wall, yes?
 
  • #6
maistral said:
Allow me to reinforce my understanding.

In your equation, this means Tb is the bulk temperature at the right side (outside) of the wall, yes?
Yes. Of course, T5 is a fictitious temperature that gets eliminated by combining with the discretized transient energy balance equation at point 4.
 
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1. What are boundary conditions for convective heat and mass transfer?

The boundary conditions for convective heat and mass transfer refer to the conditions that must be specified at the interface between two media, such as a solid surface and a fluid, in order to accurately model the transfer of heat and mass between them. These conditions typically include the temperature and concentration of the fluid at the interface, as well as the heat and mass transfer coefficients.

2. How do boundary conditions affect convective heat and mass transfer?

The boundary conditions play a crucial role in determining the rate of convective heat and mass transfer between two media. By specifying the temperature and concentration at the interface, as well as the transfer coefficients, we can accurately model the transfer of heat and mass and predict the behavior of the system.

3. What is the significance of wall temperature in convective heat and mass transfer?

The wall temperature is a critical boundary condition in convective heat and mass transfer, as it represents the temperature of the solid surface in contact with the fluid. This temperature can greatly influence the rate of heat and mass transfer, and it is important to accurately measure and control it in order to optimize the process.

4. How do you determine the wall temperature in convective heat and mass transfer?

The wall temperature can be determined through a variety of methods, such as direct measurement using thermocouples or infrared cameras, or through numerical simulations using computational fluid dynamics (CFD) software. The method used will depend on the specific system and the desired level of accuracy.

5. Can boundary conditions change during convective heat and mass transfer?

Yes, boundary conditions can change during convective heat and mass transfer, particularly in dynamic systems where the temperature and concentration of the fluid may change over time. It is important to account for these changes and adjust the boundary conditions accordingly in order to accurately model the process.

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