Nusselt number is positive. The heat flux can be either positive or negative depending on the direction which has been chosen as heat in The First Law of Thermodynamics.

As the Nusselt number is h.x/k where h is the convection heat transfer coefficient and x is distance and k is thermal conductivity. Therefore, Nu cannot be negative. Check the formula for calculating h.

apparently there is something wrong with your simulation as a negative Nusselt number is meaningless.

Nusselt number cannot be negative as it leads to non practical conditions. Nu = f (Gr , Pr) assuming natural convection. In dominant radiation effect, the delta T in Gr number can be negative, please look for this while initializing

Hi Everyone....Thanks for all your suggestions. I have gone through the code and fixed the bug. Now the code is running properly and getting the positive value of Nu. Once again thanks a lot for your valuable responses...

Nusselt number can be negative, since it is defined as the dimensionless temperature gradient at the wall. The slop may be negative or positive, depending on the direction of heat flux.

Dear Ali JM,

Thanks for your comment. Can the negative magnitude of the Nu also due to non-uniform temperature b.cs?

Dimensionless Nusselt number depends on the coefficient of heat transfer by convection, thermal conductivity of the fluid and thermal characteristic length (all phy sical parameters). Convection coefficient is determined by integrating the heat convection equations - mass, momentum and energy conservation by defining the thermal boundary layer (Prandtl). It is impossible to derive negative values. Convection Heat Transfer book by A. Bejan is very useful for phenomenological analysis of heat transfer processes associated to mathematical modeling (Chapt.5 Natural Convection in Enclosures).

Dear Sankar Mani,

Can the negative magnitude of the Nu also due to non-uniform temperature b.cs?

I don't believe so.

Seemingly, you are now getting sensible answers in your simulation. So you do not need my reply. Yet, just for the sake of completeness, I add my response.

It is a fundamental law of thermodynamics that heat can only flow from a higher temperature to a lower temperature (an implication of the second law). By convention, heat flowing from a higher to lower temperature is taken to be positive; hence the negative sign in Fourier Law. This also requires that the temperature gradient be negative. If the direction of the space variable is taken to be such that the temperature gradient is positive, then heat flow, heat transfer coefficient and Nusselt number can, in principle, all have a negative sign. It however does not mean that heat is flowing from a lower to higher temperature.

An analogy can be seen in the question ‘Can friction factor be negative’? In separated regions of flow where recirculation occurs, the flow changes direction. The shear stress will come out negative here and so will the friction factor. If you need more clarity on this, consider flow over a bluff body such as a cylinder. In particular, look at the wall velocity gradient just before and after separation.

However, in your case, I don’t think this applies.

Dear Vijay,

Thanks a lot for your detailed response. Your responses are very informative and useful.

hey....your question is really a good one...never thought of it before....but after some reading..i find Vijay Raghavans argument to be convincing....its not valid to go after the formula Nu=(hL/k) therby say that no terms are negative and thereby Nu cant be negative...as per the above comment....its all just to balance the signs of the terms considering the sign of the temperature gradient and direction of length scale coordinate..

yes, Nusselt number is negative if h (convective heat transfer coeff.) is negative. This can happen since h is a very poor way to describe thermo-fluid-dynamic phenomena, expecially transient phenomena.

h is not a thermophysical propertiy and only represents the ratio between quantities defined with respect to different geometric scales, that is the heat flux q and temperature difference.

q=h*A*(Tw-Tinf) in case of a flat surface.

where q refers to a thermal power at the interface between solid and fluid (a local quantity, q=-kAdT/dx)), while the temperautre difference is between the surface and a far unperturbed fluid region temperature (an integral mean quantity int((dT/dx)*dx) ).

Now, the second law of thermodynamics certainly holds if we consider temperature derivatives and , in any case consistent quantities.

But in the above Newton formula q and (Tw-Tinf) do not refer to the same object so it is possible (expecially during a transient) for q and (Tw-Tinf) to have different sign. See the attached image.

for guys who just wanna post something just like that....see i am amazed.....really...why did Thermodynamics second law come into picture at all.....who said temp is flowing from a lower to higher level without any external agent....? the simple and proper explanation given by Vijay Raghavan is very convincing....plz refer to that..it all about the directionality and sign convention.....

Satish, I voted up your answer as it was 'interesting' and something totally new. Perhaps you uploaded your answer before you gave it deeper thought.

I am sure you have read all the above responses. What did you conclude?

It is true that when k is large, h is large (similarly, low k gives low h). But have you considered the role of the characteristic length? You don't need to look very far. Consider a point close to the leading edge of a flat plate in external flow!

I chose to add this answer only because in your line of work on nuclear reactors, heat transfer plays a pivotal role.

Frederico gives and interesting situation of when this can be true, but this kind of situation is somewhat unusual. Typically when δT/δx (with δ standing for a partial derivative) is calculated, with x pointing from the wall into the fluid, the heat flux is calculated from

q” = - k δT/δx

If the temperature gradient is positive, the heat flux is negative. The heat transfer coefficient defined as

h = q” /(T_w – T_a)

with T_w the wall temperature and T_a the the ambient, is positive since T_a > T_w and the difference is negative. On the other hand, if the temperature gradient is negative, the heat flux is positive and then T_w > T_a and the difference is positive and again the heat transfer coefficient is positive.

The reason why the unusual temperature distribution given by Frederico does not occur is that the convective flow is the result of an evolution from some initial state which often the fluid is either heated or cooled by the wall and then the temperature distribution tends to monotonic. Of course on may heat the fluid upstream in some unusual way and this could lead to the special situation.

Seppo

the situation I described is unusual, but not so much.

Think at a transient case where the temperature of the wall, for some reason, suddenly start to increase starting from a condition in which Tw

The bottom wall exchanges energy with the particles in the main stream and since its temerature is not fixed there is tendency of unnusual temperature increase of the bottom wall due to varied velocity of the particles leading to fluctuation in temperature which propagates into the main stream from the wall. Therefore see how you can keep the bottom wall fixed at your choosen temperature; this you can do by using a spring system for damping! Another issue is also insulating the top surface; what temperature are you asuuming for the insulation or is the insulation temp at absolute zero :) even though there is no exchange of energy with the wall, this is another recipe for more fluctuations and increase in temperature in the main stream what i am reasoning is that there is no need to insulate the wall; you can also use a spring on it, cos insulating the top wall means you have only one wall which is the botom wall that interacts with the main fluid and there will be no profile at all talkless of a linear profile! since the fluid temperature is higher than the wall temperature because most of the particle end up in the fluid causing an increase of its temperature relative to the wall a negative nusselt number is not far from been recorded!

Hope I do not sound too authoritative; pardon my choice of words also a learner!

Michael,

Your second response was better than the first one!! Your first response as well as that of many others to this question can be explained in the rather simple explanations that I had posted earlier. I hope you had a look.

Obviously Sankar's negative Nu number was the result of some numerical error that has long since been corrected by him. However, his question has raised an issue that should help many more participants on this forum understand the flow of heat from a more fundamental point of view. Let us thank him for that.

To Surya's question as to how or "why did Thermodynamics second law come into picture at all.....who said temp is flowing from a lower to higher level",

heat flow is a consequence of the second law. A proper introductory chapter in a Heat Transfer book should start with the relationship of Thermodynamics to Heat Transfer. But it is rarely seen in English text books. Text books in German, fastidious and thorough just like the people, do that.

Federico, I should have perhaps said that in your illustration in which you correctly point out heat transfer coefficient to be negative, the use of heat transfer coefficient is rendered quite meaningless. In calculations of transients, such as in your example the wall heat capacity and conduction in the wall ought to be also taken into account and then in this conjugate heat transfer situation the proper wall condition as you know is the continuity of temperature and the heat flux across the interface. This raises the interesting question of how applicable are the heat transfer correlations in unsteady situations. In external convection they would seem to be fine, but in natural convection less so, for then the temperature and the fluid flow fields are tightly coupled. Nevertheless we use the steady correlations because we have nothing better to substitute for them.

The situation is somewhat similar in the use of incompressible friction factor results for internal compressible flows. These are forced on the analyst even as he or she knows that in compressible the flow is evolving along the conduit.

Hello,

the heat flux can be either positive or negative depending on the direction which has been chosen as a positive axis. Concerning the heat transfer coefficient, we generally choose a positive sign because we choose a positive axis for heat flux corresponding to a positive temperature gradient. If the axis positive direction is a priori chosen whatever the sense of the heat flux and whatever the sign of the temperature gradient one can get, for a given experiment, positive or negative heat flux and correlatively negative or positive heat tranfer coefficient (therefore of the Nusselt number).

Il must be reminded that all the correlations for the Nusselt number are given (i) for heat fluxes in the positive direction (ii) for steady state regime. In fact, in transient situations the correlations determined in steady state are generally used but it is a misuse which is admitted by the scientific community. In a publication, the sign problem must be correctly addressed and made clear because some researchers are not familiar to these notions.

Andre Bontemps

salam

I worked on the configuration (top: adiabatic. Bottom: isothermal Th. sides: twall Th = (1-y)).

I drew lines of heat (heatlines). some lines of the heat rises from the bottom to the highest points of the vertical walls . fluid heated from below warms the highest parts of the walls verticals. Finally, I think the very hot fluid heats up vertical walls and consequently may result in a negative Nusselt.

@Vijay Raghavan...i wasnt referring to your post while talking about the relevance about the Second Law here..there are some deletd posts....i just meant the whole mess here could also be dealt with out bringing it (II law) out here..one guys starts a hifi.all others deviate..and we can have this discussion for ever going on like this....coming to your posts....i am thankful,u make me think for a while... :)

This is my opinion as physician, If you have a look on Nusselt number you can see that it depends on some physical characteristics that can not be negative! So I think that Nu can not be negative ? Am I right?

Lyes, I know less about medicine than you must be knowing as a physician. Therefore it is remarkable that you know about Nusselt number and everything. There is a very brief and quite simple explanation which i hope you will find convincing. Please see the attached file. I only wanted to help. I hope I do not sound superior.

Surya, messages and emails can sometimes convey what is not intended. I am guilty of not reading your post properly and not responding with the right words :-(

Hello and thank you for the answer I would like for to apologize for the confusion I mean by physician not a specialist in medicine but in Physics! thank you

I saw an answer from Morocco in which that Nu can be negative can he explain to me what mean in physics point of view a negative Nu? Thank you

the local Nusselt number is dependent on the temperature gradient. it is a quantity which can be negative at specific points of the wall (the case of non-uniform distribution of temperature).

Hello, Nusselt number is;

Nu=h.l/k

Negative means one of these parameters is negative, which one?

salam

please here is two articles. the authors found a number of negative Nusselt.

see (page 33 Azzi2007) and (page 293 Akhter).

we found positive and negative Nusselt numbers.

This is due to the variation of fluid temperature near the wall. The rate of heat transfer exchange ITS sign in case of fluid temperature near the wall Becomes Higher or Lower than the wall temperature Which May Occur due to the Conditions Imposed on the problem. (If you want, that is h is negative (calculated), but it keeps the value positive (by definition) and reverses the thermal tranfer).

by analogy, electricity, electric current calculation can give a negative value, in this case we keep the positive (by definition) but reverses the direction of current in the circuit ..

Thank you Shiva this is what i try to say! Physics point of view there is no sence to find a negative Nu! But using mathematical computing we can find many unphysical results.

Temperature gradients. Let us talk in terms of that. It appears as if many recent respondents have not taken the trouble to read the earlier posts at all. The question and all the answers have been well understood; and now these simplistic posts!!

Nobody tries to develop a correlation for negative Nu number. I thought the issue was settled but these simple answers keep reappearing.

Elabdallaoui had given an analogous situation in electricity. I can also give some similar situations from fluid flow analogy, but I will spare the readers for now. If this comedy persists, I may enter the fray again.

No way to close a topic?

Vijay Raghavane considers the electrical analogy simple reasoning! . but the electrical patterns were always a support that help to illustrate the heat transfer. the great writers like Incropera, Duffie & Beckman, have used it.

Elabdallaoui, peace! If you read my post again slowly, you will see that I said no such thing as you assume. The 'simplistic post' referred to some other responses.

I termed your example as 'analogous', not as 'simple'.

I endorse what Federico had said about closing a topic. I think the discussion has now descended from the intellectual to the ludicrous. Thanks Federico.

Nusselt number is positive. The heat flux can be either positive or negative depending on the direction which has been chosen as heat in The First Law of Thermodynamics.

In case of Forced convection Nu is function of Re and Pr (with some power). Re is function of rho, V, Dh and nue (and mu). Pr is f(Cp, mu and k). None of these thermophysical property is negative, velocity and Dh are also non negative. The powers are also non negative. Hence Nu can not be negative. In simulation if vortex is obtained or flow reversal take place then the velocity vector may have been taken as (-ve). This may lead to (-ve) Re and hence (-ve) Nu.

Similarly it can also be checked in Natural convection, whether there is some how negative acceleration due to gravity(??????) is coming into picture!!!

Dear Sankar: no, it does not have anything to do with the type of heating, or the nondimensional relations, or electrical analogy ... or anything else. The explanation why the Nusselt number can not be negative is quite simple. Nusselt number is defined as: Nu = hL/k, where h is the convection heat transfer coefficient, L is a length scale, and k is a thermal conductivity. All these parameters are ALWAYS positive. In particular, h and k, which are defined using Newton's Law of Cooling and Fourier's Law of Heat Diffusion, respectively, must be positive quantities, otherwise the heat flow associated with them would invalidate the Second Law of Thermodynamics (more precisely, the Clausius Theorem that states heat - no matter the mode - must always flow from high to low temperature). If the value of Nu you are getting is negative, then there is an error in the way you have defined it or calculated it.

Your problem is not yet clear. Are you taking both the side walls as linear heating?

What is the range of temperature in your case. are you getting perpendicular isotherm lines on top walls?

Simply speaking, negative Nu implies the presumed direction of the heat flux should be reversed. Nu should always be equal to or greater than ONE. It is the ratio of the heat transferred by convection and that by conduction when the fluid is assumed to be stationary. Thinking about the heat transferred by convection and that by conduction, they must be in the same direction under the same dT. Hence, same direction means positive Nu. Moreover, convective heat transfer (V>=0) must be equal to or greater than the conductive heat transfer (V=0). That's why Nu must be equal to or greater than 1.

Correction: Nu should be greater than 1. As convection is defined as the heat transferred between a surface and a moving fluid, velocity must not be 0.

The conduction hea flux at the wall is

q_w = k(dT/dn)|_w (1)

here n is the direction normal to the wall

We cal also write q_w in terms of a film coefficient

q_w = h (T_w - T_0) ( 2)

with T0 a reference temperature

Thus

h (T_w - T_0) = k(dT/dn)|_w (3)

let us non dimensionalize the variables as

T* = (T_w - T0) / (T1 - T0)

where T1 - T0 is a characteristic temperatu re diffenece

(3) becomes :

h (T_w - T_0) = k (dT/dn)

= k (T_w-T_0)/L_0 (dT*/dn*)

divide both sides by (T_w - T_0)

h = k /L_0 (dT*/dn*)

the nondimensional temperature gradient at the wall

is now:

dT*/dn* = (h L_0)/k which is precisely the Nusselt number

Thus Nu = (dT/dx)* Nusselt is the dimensionless temperature gradient at the wall

Thus depending on the sign of dT*/dn* ( i.e dt/dn ) the Local value of the Nusselt number can indeed be negative. It simply tells you whether heat is flowing in or out

of the domain. In most boundary layer flows the wall heat flux will not

change sign.

The thermal conductivity k inside the definition of Nu = h L / k is of the moving fluid, not of a solid wall. So I cannot agree with Prof. Pelletier.

in what i wrote, k is the conductivity of the fluid.

k dT/dn is the heat flux at the wall in the fluid.

the same goes for Nu.

I am afraid you are mistakeen

Dominique Pelletier:

what if normalize T with T0-T1? Could it turn dT*/dn* back to positive?

Sure that Nu should be positive! The definition of a "Parameter" has to serve the purpose of measuring the physical characteristic. Heating or cooling can be handle by some other ways. We don't need to consider that in defining Nu.

Be careful when we define a dimensionless parameter. It should come from the non-dimensionalization of governing equations and boundary condition. Once you do that, you will never find a negative Nu.

In addition, if Nu is negative, it indicates the directions of heat transfer by conduction and convection in the fluid are opposite. Is it possible physically?

_w = k(dT/dn)|_w (1)

There should be a negative sign on the right side of this equation for the heat to flow from the high temperature source to the low temperature sink. This app[lies for both heating and cooling. Please refer to books on Heat Transfer by Conduction for example by Arpaci..

@dominique,

You put forward a general confusion about these two equations like fourier law and Nu, But keep in mind the way you are going to define something. So be sure that when you are on conduction, defining heat going from say solid to other anything then it s always thermal conductivity of solid and in case of Nu, you are in convection so it is thermal conductivity of fluid.

The answer is (No) even there's non uniform heating. Simply because Nu=h L*/k which represents the heat transfer coefficient (h) in dimensionless form with the always positive characteristic length (L*) and the fluid thermal conductivity (k).

Negative Nusselt No. means negative (h) and this doesn't mean anything in a the physical basis.

Nusselt number is defined positive. It can happen that it turns out to be negative: in this case it looses its meaning intended to appropriately describe the heat transfer @ hand. Such cases can happen, for instance, when in presence of volumetric heat generation which can heat the fluid "from inside" inspite of the cooling from the wall. In these cases the tangent has a different signum from the secant...

Nusselt is ratio of convective heat transfer by conductive heat transfer. In general both convective heat and conductive heat transfer is in same direction. only in case direction both mode of heat transfer opposite each other nusselt number is negative

Nusselt number is positive. Convective heat transfer is determined by Nu*k/L*deltaT and the direction is with deltaT, that is from high temperature to low temperature. Of cause the change of T from the wall to your T_fluid must be monotonic.

I suggest to check the temperature distribution on the walls .

if the distribution shows that the surface temperature becomes lower than the surrounding, you will have a negative temperature difference which may give a negative values of NU.

However, I believe this case is not possible if the case set properly (the model, properties, boundary conditions etc) the boundary conditions.

Nusselt number can never be negative. It ca be either less than 1 or greater than 1.

Nusselt number is always positive in any case. Because heat transfer coefficient, thermal conductivity should never negative in any case. So, forget about negativeness of Nusselt number and find out the reason behind why it comes negative.

The question is meaningless. Nusselt number indicates the ratio of convective heat transfer to conduction and nothing more.

Simple and clear answer from N.C.Markatos. Nusselt number by definition can't be negative. You could also argue about Biot numer being negative. Nusselt number describes the ratio of convective and diffusive fluxes on the same side of a surface.

The Nusset number is always positive; if it come negative; I thing you have to investigate the physics behind it.

I come back to my previous answer where I "declared" the Nu number by definition being positive. I am thinking about high speed heat transfer in boundary layer. In this situation we have a surface relatively cold as well as the ambient environment (this latter is colder than the surface). But inside the boundary layer we have high temperature due to kinetic heating. This means that the overall temperature difference between the surface and the ambiance if positive (i.e. the surface is cooled down, but it is true mainly by radiation), so the convective heat flux is from "surface to ambiance". But because there is somewhere in the BL higher temparature than on the surface, there will be diffusive flux towards the surface. So in this situation both fluxes may have opposite directions. Am I right?

Nusselt number does not...care about directions ! It is not its purpose.

..

Nusselt number is positive. you need to make sure that the temperastures measured accuratley and you follow the correct procedure in calculating heat transfer coefficient.

OK, I did not mean by my last answer that Nu number is negative. I just responded to some previous comments that both fluxes / convective and diffusive must be the same direction. So I still am on the position of the positive Nu. :-)

Tom Diller

Please look to the peer reviewed literature for the answers. This current situation is addressed in Moffat, R.J., "What's new in convective heat transfer?," International Journal of Heat and Fluid Flow, Vol. 19, 1998, pp. 90-101.

I fervently hope that Tom Diller's response will bring the discussion on this to a close. Far too much energy has already been spent on it.

Come on, it's obvious that Nu can be negative in a confined domain. It's sign just depends on the choice of the bulk temperature . So, if the heat flux enters the domain in a point, it means the temperature derivative at the wall is negative at that point. This does not imply that the (arbitrarily defined - mean, bulk, central, etc) fluid temperature is less than the local wall temperature. So the heat flux can enter and the temperature difference is negative. This can happen in heat transfer in pipes under specific circumstances, and it's absolutely correct and physically sound.

Nusselt number cant be negative whether it is cooling or heating application.Rate of heat transfer could be negative or positive depending on the sign onvection assumed.From the basic definition of Nusselt number both denominator and numerator will be come either positve or Negative thus making it always postive.

If you get a negative value for Nu pl check the sign convention followed and correct it.

I think N number refers to steady state conditions or instantaneous values and not transients.

Leaving this topic for awhile and recently reading Prof. Jicha's case. It's quite interesting. Consider infinitesimal layers between the surface and the fluid with higher temperature. During transient state, the direction of heat transfer may change at one of the layer, say, Layer n-1 left, Layer n no heat transfer; Layer n+1 right. Hence for each individual layer, Nu is always positive.

When radiation heat transfer is involved, it's good to calculate convection and radiation heat transfers separately because they are different mechanism. Don't treat radiation as convection.

kuttappan: the nusselt number can be defined in transient flow and unstable and unsteady flow with time.

The nusselt number is always positive

As concept Nusselt neumber is positive, but efectevely the rate of heat transfer could be negative or positive depending on the sign convection assumed.

Nusselt number is always positive, as it is h. Just because of its definition. If you get a negative h value, this means that either the conduction heat flux at the quiescent fluid layer adjacent to the solid wall has a "wrong" sign or the temperature difference (fluid-wall) has a "wrong" sign. Summing up: if the heat leaves the wall, fluid heats up; if the heat enters the wall, fluid cools down. In any case h is positive as well as Nusselt.

There are 82 contributions to this question.

After, say, the 20th , answers are a duplication of a previous one.

Is it possible to close this discussion?

Or let me know if it is used only to augment the RG score.

The nusselt can not be negative.

Identical situation is with the Reynolds number. Can be Re negative ? - no problem just put U negative - but you should not do that. You shuld put positive defined value for velocity reference U.

Dear Federico,

Very good remark!!

Best

Nikos

The Nusselt number is |h|*L/k; therefore it should not be negative. The software generally calculate h using Q/{A*(Twall-Tfluid)}. Therefore, h can be negative or positive depending on whether Twall>Tfluid or Twall

No dr. Kumar.! We are not considering what a computer does, bur if Nu can be negative under a physical ground. The answer is yes. It means that the heat flow does opposite to the temperature difference. It does not happen frequently, but can happen. An example: consider the flow in a pipe where you have internal heat generation in the core, and the initial fluid temperature is less than the wall temperature. SAfter a while it can happen that the bulk fluid temperature is higher than the wall temperature because of internal generation, but the temperature gradient at the wall remains positive.

Then Nu < 0 locally.

Heat transfer coefficient is not a physical parameter but only a concept to ease the heat transfer calculations. It is defined near the wall by kf/tf, where kf is fluid conductivity and tf is the thickness of fluid film near the wall; therefore it should not be negative. However, if the program is calculating h by h=Q/{A(Twall-Tfluid)}, then it can be positive or negative. Therefore, Nu also can also be positive or negative. By looking at the problem it appears possible that heat transfer is taking place from the fluid to the side walls as the fluid is getting hot and moving up due to bottom wall boundary condition. In this process it starts heating the side walls. Therefore it is possible that the calculated Nu is negative. However, if the wall material conductivity is high, then the side walls may reach a temperature which is higher than the fluid temperature near the side walls and a reverse situation (+ve Nu) may develop.

I got enough inputs to this question and could able to overcome this problem too, which I have mentioned previously. As Federico Scarpa mentioned, some of the responses are repeating, Now, its time for stopping this discussion. Once again, I express my sincere thanks to everyone who contributed the answers to my question.

Is the heat exchange well defined ? what is th fluid reference temperature (bulk, ...)? all is possible depending the adopted definition ?

Nusselt Number is a Non-dimensional number, which will remain positive always. The direction of heat transfer dicatates whether the heat flow into the system if taken as positive, the heat removed from the system will be negative. It should be never confused with the negative nusselt number.

Is this ridiculous discussion still going on?

I agree with Markatos. Really!!! are you guys serious in discussing sign conventions?

Nusselt number is a ratio between heat transfer due to convection (advection+conduction) to heat transfer by conduction (stationary fluid). Can be negative, the answer yes, as discussed before, depend on the your definition (geometry) and direction of heat flow. at the wall q=kdT/dn, n-direction normal to the wall. I assume you approximating dT/dx= (Tw-Tf)/dx, where Tw is the wall temperature and Tf is the fluid temperature very near the wall and at distance dx from the wall.

What happen if you have a vertical wall and temperature linearly decreases along the wall. Hot stream due to buoyancy rises and there is a possibility that the fluid temperature becomes higher than the wall temperature, and dT/dx become negative, i.e. local Nu become negative. However, average Nu must not.

Nusset number is part of applied physics that facilitates the approximation of heat transfer between solid surface and moving fluid. It has proved to be an excellent engineering tool and not a divine entity that deserves endless probing. As suggested by many it is time to close this debate that has more than served the purpose. I may be disconnected from this senseless debate.

Negative /positive Nusselt number implies outgoing / incoming (respec.) heat fluxes.

Please refer the book" Advances in Heat Transfer" ( Academic Press) page #32-58. It may be useful stuff for this debate

Nusselt number could be negative if the fluid is stratified initially. Suppose, in a rectangular cavity fluid is stratified and the wall temperature is within that range.

-k(dT/dn)=h DT ===>  hL/k=-dT/dn*   n*=n/L

Hence Nu=-dT/dn* (n - distance normal to the wall). The sign of Nu depends on the sign of dT/dx. see my comments abov.

Nusselt number and the convective heat transfer coefficient are always positive regardless of the sign of temperature difference between the surface and undisturbed fluid. Therefore, the direction of heat transfer is only determined by the sign of temperature difference Q=hA(Ts-Tf), i.e. the surface rejects heat to the fluid if Ts>Tf and absorbs heat if Ts