Implement neumann boundary condition for nondimensional energy equation

new_at_this · February 24, 2012, 17:29

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

FMDenaro · February 24, 2012, 18:59

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

new_at_this · February 25, 2012, 14:10

How would I know Tw? The only thing that I have defined at the boundaries is q.

FMDenaro · February 25, 2012, 14:40

Quote:

Originally Posted by new_at_this

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

new_at_this · February 27, 2012, 14:44

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?

mettler · February 27, 2012, 16:56

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

FMDenaro · February 27, 2012, 17:49

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) ...

mazhar · February 29, 2012, 14:41

Energy equation is an elliptic differential equation. Therefore, you will not be able to solve by defining neuman conditions on all boundaries.

sbaffini · February 29, 2012, 17:01

I guess his T0 at inlet is used in a Dirichlet sense

mazhar · February 29, 2012, 21:19

Then he must be solving a thermally developing flow

new_at_this · March 9, 2012, 12:15

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?

FMDenaro · March 9, 2012, 12:50

your temperature field should be bounded 0<= theta<=1, this is if the correct reference quantities are chosen

serene6390 · April 23, 2018, 19:52

Hi, peter,

Could you tell me more about how did you deal with non dimensional energy equation? When you set your boundary condition, you set your temperature as a number between 0 and 1? And you talked about using an UDF, what did you use the UDF to define? Boundary condition?

Thanks a lot.

Rami · May 17, 2018, 10:56

... A bit late, but still...

If you normalize the temperature that way, than the (normalized flux) = flux*(Tw-T0)/K. It therefore follows that the dimensional prescribed flux is replace by the normalized BC: (prescribed normalized flux) = d(theta) / dn where n is the normal coordinate.

February 24, 2012, 17:29	Implement neumann boundary condition for nondimensional energy equation	#1
new_at_this Member Peter Join Date: Oct 2011 Posts: 52 Rep Power: 15	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

March 9, 2012, 12:15		#11
new_at_this Member Peter Join Date: Oct 2011 Posts: 52 Rep Power: 15	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=mc_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 12:35.

April 23, 2018, 19:52	How to deal with non dimensional energy equation?	#13
serene6390 New Member Jing Join Date: Feb 2013 Posts: 25 Rep Power: 13	Hi, peter, Could you tell me more about how did you deal with non dimensional energy equation? When you set your boundary condition, you set your temperature as a number between 0 and 1? And you talked about using an UDF, what did you use the UDF to define? Boundary condition? Thanks a lot.

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February 24, 2012, 18:59		#2
FMDenaro Senior Member Filippo Maria Denaro Join Date: Jul 2010 Posts: 6,849 Rep Power: 73	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

February 25, 2012, 14:10		#3
new_at_this Member Peter Join Date: Oct 2011 Posts: 52 Rep Power: 15	How would I know Tw? The only thing that I have defined at the boundaries is q.

February 27, 2012, 14:44		#5
new_at_this Member Peter Join Date: Oct 2011 Posts: 52 Rep Power: 15	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?

February 27, 2012, 16:56		#6
mettler Senior Member Andrew Join Date: Mar 2009 Location: Washington, DC Posts: 211 Rep Power: 18	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

February 27, 2012, 17:49		#7
FMDenaro Senior Member Filippo Maria Denaro Join Date: Jul 2010 Posts: 6,849 Rep Power: 73	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) ...

February 29, 2012, 14:41		#8
mazhar New Member Mazhar Iqbal Join Date: Mar 2009 Posts: 10 Rep Power: 17	Energy equation is an elliptic differential equation. Therefore, you will not be able to solve by defining neuman conditions on all boundaries.

February 29, 2012, 17:01		#9
sbaffini Senior Member Paolo Lampitella Join Date: Mar 2009 Location: Italy Posts: 2,190 Blog Entries: 29 Rep Power: 39	I guess his T0 at inlet is used in a Dirichlet sense

February 29, 2012, 21:19		#10
mazhar New Member Mazhar Iqbal Join Date: Mar 2009 Posts: 10 Rep Power: 17	Then he must be solving a thermally developing flow

March 9, 2012, 12:50		#12
FMDenaro Senior Member Filippo Maria Denaro Join Date: Jul 2010 Posts: 6,849 Rep Power: 73	your temperature field should be bounded 0<= theta<=1, this is if the correct reference quantities are chosen

May 17, 2018, 10:56		#14
Rami Senior Member Rami Ben-Zvi Join Date: Mar 2009 Posts: 155 Rep Power: 17	... A bit late, but still... If you normalize the temperature that way, than the (normalized flux) = flux*(Tw-T0)/K. It therefore follows that the dimensional prescribed flux is replace by the normalized BC: (prescribed normalized flux) = d(theta) / dn where n is the normal coordinate.