Solving internal flow exiting temp w/ conjugate HT BCs.

[pm272]

Hi. There is a problem that I have been working on and I seem to be getting somewhat unrealistic results. Can anyone critique my modeling method?

Problem: Heated air enters a duct of length L at temp T_h. The outside of the thin walled duct will have convection and radiation both being important. I am assuming duct wall thickness is infinitely conductive. Find the exiting air temp.

My logic is as follows: Model this in "n" thin slices, dx, along the duct. Where n = L/dx. For the first slice simply use T_entering = T_h and set up two energy balances: First at the air: m_dot*c*(T_ent-T_exit) = h_in*A_in*(T_fluid-T_wall) and second at the wall: h_in*A_in*(T_fluid-T_wall) + q_solar*A_out = h_out*A_out*(T_wall-T_amb) + sigma*epsilon*(T_wall^4-T_amb^4).

T_fluid = (T_ent + T_exit)/2

q_solar is only applicable for outdoor conditions. I have assumed T_amb = T_surr for thermal radiation.

With these equations I should be able to begin at n = 1 by guessing a T_exit, then solving for T_wall. Then using the sum of the two energy balance equations above as follows, just for simplicity: m_dot*c*(T_ent-T_exit) + q_solar*A_out = h_out*A_out*(T_wall-T_amb) + sigma*epsilon*(T_wall^4-T_amb^4).

and rearranging to:
m_dot*c*(T_ent-T_exit) + q_solar*A_out - h_out*A_out*(T_wall-T_amb) - sigma*epsilon*(T_wall^4-T_amb^4) = 0

finding the residual: r = m_dot*c*(T_ent-T_exit) + q_solar*A_out - h_out*A_out*(T_wall-T_amb) - sigma*epsilon*(T_wall^4-T_amb^4)

Then guessing a new T_exit and repeat again to get a new r. Use the T_exit value that gives r closest to 0.

Then , n = n + 1 and the new T_ent = the previous T_exit.

When I set all this up in a script, my result seems to indicate that the fluid temp drops extreme amounts in the first meter (When L = 200m for instance and entering temp of 600 K). I have compared this to a quick and dirty autodesk CFD model that leads me to believe that my script does indeed appear unrealistic. On a side note, the autodesk CFD doesn't allow me to include a solar heat flux value in addition to the convection and thermal radiation - which is a bit limiting.

Does this set-up appear complete? Am I omitting anything here?

Thank you for your time.

-D

 

[Chestermiller]

What does the differential equation for this system look like, in terms of $mC\frac{dT}{dx}$? Please use LaTex to make your equations more readable.

Do you think that the temperature profile will be uniform within the air in the duct? Is this accounted for in you equation?

 

[pm272]

What does the differential equation for this system look like, in terms of $mC\frac{dT}{dx}$? Please use LaTex to make your equations more readable.

Do you think that the temperature profile will be uniform within the air in the duct? Is this accounted for in you equation?

The temperature of the air will vary axially (let's say z-coordinate) and radially (r-component). No, I am not considering this. I simply am saying that the air flow is abstracted with an average temperature which is passing energy to the duct wall via convection (again using an abstracted convection coefficient h, which I am drawing from the equation Nusselt correlation for internal, turbulent flow and depends on Re and Pr). Perhaps in this case, the devil is in the details and it is exactly this abstraction that is giving me unrealistic results.

 

[Chestermiller]

The temperature of the air will vary axially (let's say z-coordinate) and radially (r-component). No, I am not considering this. I simply am saying that the air flow is abstracted with an average temperature which is passing energy to the duct wall via convection (again using an abstracted convection coefficient h, which I am drawing from the equation Nusselt correlation for internal, turbulent flow and depends on Re and Pr). Perhaps in this case, the devil is in the details and it is exactly this abstraction that is giving me unrealistic results.

So you have something like $$mC\frac{dT}{dx}=-h_{in}A(T-T_W)$$with $T_W$ determined from $$h_{in}(T-T_W)=h_{out}(T_W-T_{amb})+\sigma \epsilon(T_W^4-T_{amb}^4)$$

 

[pm272]

So you have something like $$mC\frac{dT}{dx}=-h_{in}A(T-T_W)$$with $T_W$ determined from $$h_{in}(T-T_W)=h_{out}(T_W-T_{amb})+\sigma \epsilon(T_W^4-T_{amb}^4)$$

So continuing along this path leads us to separating and integrating the first equation which leaves us with: $$T_W = (T_{i+1} - T_ie^N)/(1 - e^N)$$ where $$N = -\frac{h_{in}D\pi}{mC}\Delta x$$

So, now at each iteration, I should know my $T_i$ but I will be guessing $T_{i+1}$ Then I have wall temp. Arranging that second equation you have listed: $$-h_{in}(T-T_W)+h_{out}(T_W-T_{amb})+\sigma \epsilon(T_W^4-T_{amb}^4)=r$$ Should indicate how close my initial guess is depending on how close r is to zero. but what about $T$ in that equation? I take that to be an average between $T_{i+1}$ and $T_i$ which could be reasonable for small values of $\Delta x$, I think. These results seem to indicate an under-prediction of temperature for the first few iterations while the remaining temperature is almost unchanging through the majority of the remaining duct.

Does there seem to be anything incorrect about my analysis here?

Thank you again