Nusselt number 4.36 validation in a tube in constant heat flux situation
 

Hi,guys.

According to classic heat transfer knowledge ,the Nusselt number in a fully developed tube with constant heat flux should be 4.36 .

CFD code like fluent is so powerful ,so I want to validate the problem in fluent .And if I model right ,I should get the right Nusselt Number.

But maybe there is something wrong in my model process , I just can not get the Nusselt number as described above .

These is the process below.


according to experience ：the length of entry region l=0.05Re×Pr。

Material is water ： and I set Re=100.
Re=Rho×velocity×D/viscousity. then I get inlet veloctiy=0.1m/s.

Pr（30Celsius-60Celsius）is about 3 .then we get the entry length l=0.05×100×3=0.15m.


I build a cylinder in Gambit with the dimension below：
D=0.1m,L=0.3m.

Boundary condition:

inlet velocity :v=0.1m/s .T=300K. outflow in outlet。
heat flux of wall =q=400W/m^2.
the conductivity of water :lamda=0.6.

I use the 3D single precise solver to solve the problem. Set the residual to 10-4. At the same time ,I set a temperature monitor point in the middle point at the tube .


The calculation of Nusselt number:

Nu=h*D/Lamda.
h=q/DletaT=q/(Twall-Tref).
Twall is the temperature of wall of the tube ,Tref is the average temperature of the section I choose in the fully developed region.

I create a section in fully developed region .

plot temperature coutour of this section ,and I get max temperature in the wall is 300.948K .

The difficulty comes as I want to calculate the average temperature of the section I created.

According to heat transfer textbook ,Tref=Tm=(Intergral of（u×T）in seciton ）/（pi×(D/2)^2×Veloctiy）。

The result of the section is 0.002341306。So Tref=Tm=0.002341341/（3.1415926×0.05^2×0.1）=298.2555K.

Then h=q/(Twall-Tref)=400/（300.948-298.2555 ）k=148.560817084494 w/(m^2.k).

At last ,I get Nu=h×D/Lamda=148.560817084494*0.01/0.6=2.4760136180749 ！= 4.36.

I don't know what was wrong with the above process .Can somebody so kind to point the error in my process ?

Is your heat flux heating up or cooling the fluid? Check your calculation of mean temperature. For heating, the temperature cannot be less than 300K, neither can the wall temperature be less than 300K. Either way your bulk temperature must be between your inlet temperature and wall temperature, anything outside that range is non-physical and your simulation must be giving wrong results.

Lastly, you can more easily simulate fully developed flows using the periodic boundary conditions. If you don't care about entry length effects, I recommend you do fully developed flows this way. Just wanted to inform you that the feature exists.

It is heating the fluid in the tube.

Thank you very much .
In fact I want to investigate the Nusselt number along the axial direction.
I am not very sure about the calculate method of average temperature in the section of the tube . Fluent provide so many method to calculate the average temperature in a section ,like facet average ,vertex average and area-weighted temperature and so on . But I don't know which is right for my problem. I use different methods and I get different answers . I was really confused by this.

You already showed me the correct equation for bulk temperature! The bulk temperature is calculated using the mass weighted average. Use a surface integral on a plane that cuts the cross-section of your pipe and calculate the mass-weighted average temperature to get the bulk temperature.

Thanks much to your suggestions . I use periodic boundary condition now and iterate again and at last I get a good result. I am not sure about the length of entry region .

I set the bulk temperature of inflow of tube as 300K,and transfer the velocity 0.1 m/s to mass flow rate as 0.0078437714588745 kg/s.

Here is my result :

bulk temperature is 300.0098 and wall temperature is 300.4001 .then I can get Nusselt number as 4.259307.

Now I want to use double precision solver to solve it again to get Nusselt number ,maybe it is closer to 4.36.

Thank you very much for your help again. You are really a nice guy.

Then how will find the heat flux for constant wall temperature to find the nusselt number

The report function will provide us the information of heat flux on bondary area.