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Implement neumann boundary condition for nondimensional energy equation

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Old   February 24, 2012, 16:29
Default Implement neumann boundary condition for nondimensional energy equation
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Peter
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I have written a code to solve the non dimensional temperature distribution in a 2D plane for a given velocity field. I have defined the non dimensional temperature as:

\theta = \frac{T_w-T}{T_w-T_0}

T_w = wall temperature

T(x,y) = temperature

T_0 = Inlet temperature

I have successfully simulated dirichlet boundary conditions. However I am not sure how to implement neumann boundary conditions for a constant heat flux. If someone could offer some pointers I would appreciate it.

Thanks
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Old   February 24, 2012, 17:59
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Filippo Maria Denaro
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Consider the Neumann condition (dimensional)

q = -k dT/dn, at a surface with n unit normal outward to surface, then from your definition you simply get:

q = k (Tw-To) dTheta/dn

The quantity q /k (Tw -To) is a known term of your problem so you have hust to discretize the derivative at the surface
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Old   February 25, 2012, 13:10
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How would I know Tw? The only thing that I have defined at the boundaries is q.
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Old   February 25, 2012, 13:40
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Quote:
Originally Posted by new_at_this View Post
How would I know Tw? The only thing that I have defined at the boundaries is q.

you have to know (Tw -To) from data of your physical problem
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Old   February 27, 2012, 13:44
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So i'm missing something. Say i'm solving for a flow through parallel plates. The initial condition is that theta is 1 everywehere.

If I am solving for a constant wall temperature, then I can set the boundary conditions for theta to be 0 at the surface of the plates. I haven't used Tw or To anywhere.

So for the constant heat flux case, shouldn't I be able to do the same by defining some boundary condition on the wall that does not involve dimensional values? Also Tw will change as the flow progresses, shouldn't I be solving for this in the code instead of inputing it?
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Old   February 27, 2012, 15:56
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Can you define Tin? I'll check when I get a chance, but I think I solved your model knowing Tin and then backing out the Tw based on the q..It's been a long time. But, I do think you will have to have a temp defined somewhere
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Old   February 27, 2012, 16:49
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the quantity q /k (Tw -To) is substantially the Nusselt number, therefore you can fix an arbitrary Nusselt value for your numerical purposes without taking care of your physical values.
Of course you cannot say what is the physical problem until you specify (Tw-To) ...
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Old   February 29, 2012, 13:41
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Energy equation is an elliptic differential equation. Therefore, you will not be able to solve by defining neuman conditions on all boundaries.
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Old   February 29, 2012, 16:01
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I guess his T0 at inlet is used in a Dirichlet sense
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Old   February 29, 2012, 20:19
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Then he must be solving a thermally developing flow
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Old   March 9, 2012, 11:15
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So I am able to define Tin, however I am trying to find the nusselt number for this particular flow. Therefore I can't define the temperature at the wall but only the gradient. I have made some progress but I am getting kinda stuck.

I have redefined my non dimensional temperature to be
\theta = \frac{T_L-T}{T_L-T_{in}}

Where T_L is the the temperature at the end of the heated section calculated by q=m*c_p*(T_L-Tin). As a result I find that

T_L = T_{in} + \frac{q'' L_H}{\rho H L_D C_P}
\frac{\theta_1 - \theta_w}{\Delta y} = \frac{q''}{k(T_L-T_{in})}

This way I should be able to define a neumann condition at the boundary. This logically makes sense however I think that my dimensionless temperature field should be roughly bounded by 0, however my code seems to run to negative theta which is confusing me at the moment.

Alternately I have seen some texts define theta as
\theta = \frac{T-T_{in}}{q''\frac{H}{k}}

When I work out the boundary condition with this definition it is strange because the gradient no longer becomes dependent on q'' so I am confused. Could someone perhaps verify that the first formulation I have for theta is correct and that my code is just doing something wrong?

Last edited by new_at_this; March 9, 2012 at 11:35.
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Old   March 9, 2012, 11:50
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your temperature field should be bounded 0<= theta<=1, this is if the correct reference quantities are chosen
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