# Nu (Nusselt) number for vertical rectangular enclosures

## Main Question or Discussion Point

hi Everyone,
I have a couple of books on heat transfer (Heat Transfer, A Practical Approach by Cengel and another one) but none of them contain any information on Nu number for vertical rectangular enclosures. I am trying to find convection coefficient (for free convection) and need Nu to determine h. I know that Nu depends on H (height), L (width or space between walls), Pr (Prandtl number) and RaL (Rayleigh number). What I am intertested in is situations for 8 < H/L < 70, Pr < 1 (for air) and
RaL > 1000. Do you know any books that contain experimental data for Nu under such conditions. If you know of any table with experimental data anywhere would be appreciated. I have a general idea that Nu must be less then 10 and greater then 2 but no idea which value. CFD and FEA software is normally used to calculate Nu for certain simulations as well.

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Astronuc
Staff Emeritus
The Nusselt number (Nu) is the ratio of convective heat transfer to conduction in a fluid slab of thickness (or characteristic length or depth) of L (or D).

Nu = $$\frac{h\,L}{k}$$, where h is the convective heat transfer coefficient defined by q"= h$\Delta\,T$, L (or D) is the characteristic distance, and k is the thermal conductivity.

If one knew h and k, one could calculate Nu.

On the other hand, various correlations have been developed for Nu based on Re (Reynolds number) and Pr (Prandtl number).

One such correlation is Nu = 0.023 Re0.8Pr0.4, but I am not sure if it appropriate for the stated conditions. It is used in this report
http://gltrs.grc.nasa.gov/reports/2000/TM-2000-209772.pdf [Broken]

Consider this paper:
Liqiu Wang and K. C. Cheng
Flow transitions and combined free and forced convective heat transfer in rotating curved channels: The case of positive rotation
Physics of Fluids -- June 1996 -- Volume 8, Issue 6, pp. 1553-1573
http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=PHFLE6000008000006001553000001&idtype=cvips&gifs=yes [Broken]
See if the journal is in your library.

Meanwhile I will browse my books to see if there are additional correlations. Likely, one has to look in older journals for such data, or correlations for specific geometries and conditions.

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Thanks,
In my case I have to determine Nu based on geometry conditions and medium. I am using Nu to calculate h. I have about 4 different ways of determining Nu knowing RaL and Pr if certain condition are satisfied, but unfortunately my model does not satisfy one of the conditions on each of the 4 cases. The other problem is that in this case Nu is dependent on temperature of the walls and medium because the movement of the air inside the enclosure is driven by boyancy effects. So to calculate Reynolds number one has to first find velocity which means that temperatures and medium properties (viscosity, density etc.) are the factors which velocity of air will depend on. Usually Nu is found experimentally for different geometries. I will see if those books you recomended are available.

Astronuc
Staff Emeritus
What are temperature and velocity ranges of your problem?

The Prandtl number is weakly temperature dependent (since kinematic viscosity and thermal diffusivity have similar temperature dependence, at least for gases), but the Reynolds number is more so.

Temperature difference between walls must be greater then 50 C and less then 300 C, and velocity I do not know. If I knew velocity then I can calculate Nu. Thanks.

Astronuc
Staff Emeritus
When you say temperature between the walls, is that the temperature difference across the fluid, or between the fluid and walls. I am trying to visualize the flow channel - which you mentioned as square - and the heat flux/temperature on the 4 walls.

I would expect that one could to a mass balance (continuity equation) and heat balance to get velocity, i.e. $\dot{m}_{in} \,=\, \dot{m}_{out}$ and $\dot{m}_{out}\,h_{out}\,-\,\dot{m}_{in}\,h_{in}\,=\,Q$, where Q = q"A = heat input into the channel.

Is this driven by natural convection? Then the driving force is the bouyancy of hot fluid (air) with respect to cold fluid.

What you are discribing is not possible. I am trying to determine Nu then h then Q lost due to natural convection. In the first post I think it is clear the set up of the problem and the variables. It si a rectangular enclosure. Temperature difference is between walls.
thanks

Astronuc
Staff Emeritus
OK, yes, you did mention free convection (natural circulation) in the OP.

I am still somewhat puzzled by the heat transfer across the wall, because some heat will be transferred from the hottest wall to the fluid and some to the coldest wall. Do you have some boundary conditions on the walls, or you must determine those?

Yep, I do know the temperatures of both opposing vertical walls. Any difference of temperature from 50 to 300 C will do. And the temperature range is from 20 to 350.
thanks

Astronuc
Staff Emeritus