The Nusselt number is the ratio of the convective heat transfer with respect to the conductive heat exchange in the fluid. It measures the gain in heat exchange by making/letting the fluid move instead of keeping it still (in which case there will be only conduction). The Nusselt number usually used to compute the convective heat transfer coefficient for a specific geometry represents an average estimation (you must perform an integral average to calculate it) of the thermal and mechanical exchanges that take place in the so-called boundary and thermal boundary layer.

Thus, we must think locally to understand the physics. At the body surface, the fluid has zero velocity (viscous fluid) and the same temperature of the solid. In the bulk of the flow field, it has some reference temperature and some reference velocity. This difference in velocity and temperature creates a gradient, which typically takes place in a very small region close to the body surface, respectively the boundary layer and the thermal boundary layer (the size of the two is usually slightly different due to the difference in momentum and thermal diffusion properties of the fluid).

Locally, the fluid exchanges heat by diffusion. When the surface temperature is fixed, the fluid tries to reach an equilibrium between the plate's and the bulk flow's temperature. It means that the temperature gradient becomes smoother proceeding along the plate. The maximum heat exchange happens at the plate tip and then steadily decreases. When the heat flow is fixed, the temperature at the wall is not and its value is a consequence of a local thermal equilibrium. To keep the local thermal equilibrium, the fluid is forced to exchange the same amount of heat at each section along the plate's length, and the wall temperature is adjusted accordingly.

Thus, when you average the exchange over the full extension of the plate, you exchange more heat in the case of fixed heat exchange. Consequently, the Nusselt number is bigger for the case of fixed heat than for the case of fixed temperature.

Dear Luca Di Stasio,

Thanks for your answer.

Such a big difference (36%) is seen for laminar flow and for local Nusselt number only.

Under q=const, the temperature difference is increasing along the plate. A the point where local Nu is calculated the temperature difference is higher than that for T=const.

Thats why local Nu for q=const is 36% larger than that for T=const.

If you take Nu averaged along the plate, the difference will be minimal, if any.

Dear Dmitry Zaitsev

For the flow over a flat plate, we have no fully developed flow. First, there is a laminar boundary layer, next a transition and then the turbulent boundary layer.

This is an external flow and the temperature of the flow outside the boundary layer is always constant.

OK, I've corrected the text. But the key point remains the same:

Under q=const, the temperature difference is increasing along the plate.

Thats why local Nu for q=const is larger than that for T=const.

If you consider Nu number averaged along the plate (taking into account plate tip), the difference will be minimal, if any.

Dear Dmitry Zaitsev

Both the local and averaged Nusselt numbers are larger for q=const.

The results are available at: Introduction to Heat Transfer 6th Edition

by Frank P. Incropera (Author), et al.

Dear Morteza,

Again:

For local Nu the difference is appreciable (36%). Above I gave you a physical justification, as you requested.

For average Nu the difference is only 2%, i.e. there is virtually no difference. It's impossible to give "a physical justification" for an effect that is virtually absent.

Dear Dmitry

The averaged Nu is also 36% larger for the q=const in Laminar flow regime.

Dear Morteza,

>The averaged Nu is also 36% larger for the q=const in Laminar flow regime.

That's not true.

It's well known that avarage Nu for q=const is virtually the same as that for T=const. The difference is for local Nu only (36% for laminar flow).

Let me cite the book (by Incropera et al.) you mentioned above:

"This result [average Nu for laminar flow, q=const] is only 2% larger than that obtained by evaluating Equation 7.30 [average Nu for T=const]. Differences are even smaller for turbulent flow...".

(please see attached file, page 414, just after Eq. 7.49)

Dear Dmitry,

Thanks for your attention.

I wonder when at each point of the surface the local Nu is 36% larger for q=const. in Laminar flow regime, why the averaged Nu is only 2% larger??

Dear Morteza,

In [Incropera et al.] you can find that

Nuav,T=const=2Nuloc,T=const

Nuav,q=const=1.5Nuloc, q=const

Taking into account that Nuloc,q=const=1.36Nuloc,T=const , we find:

Nuav,q=const=1.5Nuloc, q=const=1.5*1.36Nuloc,T=const =2.04Nuloc,T=const

i.e. we get 2% difference between Nuav,q=const and Nuav,T=const

Constant heat flux (CHF) boundary condition is for the interface Biot number of zero, that is for an insulating boundary condition, which means less heat can be transferred across the interface. And vice versa for the fixed surface temperature (FST) boundary condition which is characterized by a Biot number of infinity, which means a conducing interface. You will see this difference very clearly for unsteady-state heat conduction in semi-infinite slab. The heat flux for FST surface is about 30% higher than that of a CHF one.

Dear Dmitry

Thanks for your answer.

Dear Kheng Ka Tan

I know that the T=const. equals to infinite Biot number, because it means that a fluid with a convective heat transfer of infinity is flowing over the surface. Why the q=const. equals a zero Biot number?

I will try to put it in simple words. Let's consider a cylindrical pipe (of wall temperature Tw) through which a fluid is flowing (of temperature Tm) and the pipe is surrounded by a fluid reservoir (flowing) of higher temperature (Tr>Tw). As the reservoir fluid temperature is higher than the pipe wall temperature, pipe gets heated by the convective heat transfer from Tr to Tw, which affects the axial variation in (Tw-Tm).

In simple words, Biot number can be defined as " the ratio of external convective heat transfer from a fluid and the internal conduction heat transfer within the body". When Biot number Bi=0, it indicates the poor thermal contact or no external heat transfer from reservoir to pipe. That means the temperature difference between the wall and inside fluid (Tw-Tm) does not vary axially which resembles the thermally fully developed flow with constant wall heat flux. So, Biot number zero is equals to q=constant.

Thank you.