Nusselt numbers in the lower limit

Factual Questions
1

Is the lower limit for Nusselt numbers 0 or 1? I think it should be 0, but there seems to be confusion about it.

As I understand it, and consistent with my reading of several references (the wikipedia entry for “Nusselt number” is one), the idea is that heat can be conducted through a fluid, and more heat can be transferred if the fluid moves. On a small scale it is all conduction, but if the fluid moves, heat conducts into it in one location and out of it in another, which accomplishes moving more heat. So, if you divide the heat that is carried by motion over the heat that would be conducted if the fluid was stagnant, you’d have a Nusselt number. If the floor is hot and the ceiling is cold, there can be convection that naturally occurs because the hot air is relatively buoyant, and more heat would go from floor to ceiling, than would have gone in the same situation with the ceiling hot and the floor cold. In this picture, there would be a nonzero Nusselt number when the floor was hot, and the Nusselt number would be zero when the floor was cold.

What seems inconsistent is that sometimes Nusselt numbers appear to be the heat transfer coefficient times the distance of the transfer, over the conductivity of the fluid. This would evaluate to 1 in the lower limit, not 0.

It doesn’t matter in the situations where the transfer is practically all convective. But I’ve been considering a situation around the onset of buoyant convection, and evaluating published Nusselt correlations (which are empirical relationships that are often a power law function of Reynolds number or Rayleigh number). Understanding whether there is a convective contribution requires being clear whether the Nusselt number is 0 or 1 in the lower limit.

Thanks, anybody!

2

My understanding is that the “convection” term is the heat transfer that occurs with fluid motion rather than only the heat transfer contributed by the fluid motion. So if you considered laminar flow in a pipe, the top term would be the heat transfer with flow and the bottom term would be the heat transfer without flow. As the fluid flowrate approaches zero, the rate of heat transfer with flow would approach the rate without flow and the Nusselt number would approach one.

3

The equation from my heat transfer text is:

Nu = hL/k

where:

h = convective transfer coefficient
L = characteristic length (see below)
k = conductive transfer coefficient of the fluid medium

L, the characteristic length, varies depending on the scenario. For a flat plate subjected to parallel flow, it’s the length of the plate parallel to the direction of flow. For flow over a sphere or cylinder, it’s the diameter. Note that it is NOT the thickness of the fluid layer through which conduction is taking place; if it was, then you could say, yes, Nu approaches 1 for stagnant flow. But L could be huge, for example the diameter of a blimp or hot air balloon. If you were somehow able to prevent movement of the air surrounding a hot air balloon, you’d be dealing with purely conductive transfer:

Q = k * d * A * (T[sub]surface[/sub]-T[sub]infinity[/sub])

where d = the thickness of the magically stagnant air layer around the balloon
and A = surface area of the hot air balloon

this would be equal to

Q = h * A * (T[sub]surface[/sub]-T[sub]infinity[/sub])
so h = k * d

and Nu = h * L/k (where L = diameter of balloon)

so Nu = kd L/k

Nu = d/L

where, again, d = the thickness of the magically stagnant boundary layer, and L = the characteristic length associated with the flow geometry (sphere diameter, flat plate length, etc.). Note that d and L don’t have any constraints relative to each other. Have a tiny sphere in the middle of a big room filled with magically stagnant air; d is huge, L is tiny, the opposite of the hot-air balloon scenario. IOW, I don’t think the lower bound of Nu is 1.

I would think the local Nu could approach zero (as the boundary layer temperature approaches the local surface temperature (example: creeping/laminar flow over a flat plate, at a point far downstream from the edge of the plate), but never actually reach zero; Nu = 0 implies that there is a situation in which you could have a temperature difference between ambient-fluid and surface, and have zero heat transfer, which is not possible.

4

Thanks.

Baracus, my reading of Incropera and Dewitt’s textbook is consistent with what you say. If you check out the Wikipedia article, though, it says the h in “Nu = h L / k” is the “convective heat transfer coefficient”. Thus I think Wikipedia contradicts Incropera. I realize it would be hard to say people should disbelieve Incropera because Wikipedia contradicts him. It’s also not perfectly clear that “convective” is exclusive of conduction in this context, though that seems most reasonable.

Joe Fricking Friday, you seem to agree with Wikipedia. Yeah, that bit about L not necessarily being the distance that conduction must follow bothered me too. How about this? In a few places I’ve found a Nusselt correlation published for the situation where a rectangular enclosure has a dimension L that separates a hot wall from a cold wall, and there is also a dimension H perpendicular to it, and this L by H rectangle is at some angle relative to H running vertically. The hot wall is lower than the cold wall (unless the angle makes them both vertical). There’s also a dimension perpendicular to the plane of the drawing that is assumed to be huge. In this situation, k/L actually would be the exact conductive heat transfer coefficient. The correlation I have found is 1 if conditions do not favor convection (there are three “max(x,0)” terms that are evaluating to 0 in such conditions). This seems to support the view that it’s the total h and not just that associated with convection.

I am pretty sure I found several references on each side of this, but Incropera and Wikipedia are the only two I have right this minute.