Optimal fin spacing for natural convection when Nu = 1

[manderson99]

Homework Statement

It is often stated that heat sinks, radiators, and the like require proper spacing of fins/pins/etc., especially when relying on natural convection. There is a formula from Rohsenow and Bar-Cohen that allows one to calculate the proper spacing between rectangular fins in a vertical arrangement when natural convection prevails; however, the formula makes reference to the Ra_L, which makes me wonder how useful it is when Gr_L and, consequently, Ra_L are so low that conductive heat transfer (that is, Nu_L = 1) is indicated.

Does this formula apply when conductive heat transfer is in effect due to low Gr_L values? If not, is there an alternative?

Homework Equations

S_opt = 2.714L/Ra_L^1/4

The Attempt at a Solution

The only attempt to a solution to this problem that I have made thus far is to mostly ignore this formula and instead examine mass flow as calculated at the end of any given plate, using a y-direction velocity equation and using half the distance between plates as the maximum possible value for δ (I noticed that the heat transfer lessons from my book on pipe flow indicate that the maximum boundary layer thickness is half the interior diameter of a pipe when fully developed, so I figured it would work about the same way between two vertical rectangular fins). Examining the expected amount of heat flow from any given fin, I use C_p values taken at the relevant fluid temperature used elsewhere in evaluating heat flow from the fin(s) to calculate approximately what mass flow would be required to convect the calculated amount of heat away per fin. If the mass flow value derived from using V and δ above does not equal the estimated required mass flow value, then I know my fins are spaced too closely to accommodate the expected/given base temperature (typically, the mass flow value increases as δ increases, though the thermal boundary layer equation taken from the same section of my book as the y-direction velocity equation for vertical flat plates tells me how high δ could ever become). This approach seems logical until one takes conductive (Nu_b) heat transfer situations into account. Even under these circumstances, the velocity formula seems to indicate that there is still movement of air, but the velocity and mass flow numbers are quite low.

 

[Chestermiller]

I don't know about anyone else, but I'm having trouble understanding the system you are describing. Let me say back what I think you are saying, and maybe you can confirm or correct it:

You have a flat horizontal plate at a perfectly uniform temperature, and there are vertical fins of length L in perfect contact with the plate. The thickness of the fins is b, and the spacing of the fins is δ. The width of the fins is ?? The geometric placement of the fins on the plate is ? There is air in the space above the plate, and natural convection is taking place, so that there is upward air flow near the hot fins, and downward air flow closer to the center of the spacing between the fins. The air flow turns around close to the surface of the plate. The fins are at the same temperature as the plate, with no variations along their length? You want to find the heat transfer coefficient from the fins to the air, based on the temperature difference between the plate and the bulk air above. You also want to determine the velocity profile of the air flow between the plates?

Chet

 

[manderson99]

Sorry this took me so long to get back to it . . .

I don't know about anyone else, but I'm having trouble understanding the system you are describing. Let me say back what I think you are saying, and maybe you can confirm or correct it:

Woops! I see that I could have included more detail here. Sorry about that.

You have a flat horizontal plate at a perfectly uniform temperature, and there are vertical fins of length L in perfect contact with the plate.

Yeah, that's the ticket. We'll say the plate is one meter in width and length. It has a
base temperature held at 301K.

The thickness of the fins is b, and the spacing of the fins is δ.

I'll add some "real" numbers here. Let's say the fins are .1 mm thick, 5 mm tall, and 1 m
wide. The spacing between the fins is something I'll address below. The fins are parallel
to the length dimension of the plate. Also, I'd kind of like to reserve δ for thermal
boundary layer thickness, if only because that's how I'm used to using it. I'll use S for
fin spacing.

There is air in the space above the plate, and natural convection is taking place, so that there is upward air flow near the hot fins, and downward air flow closer to the center of the spacing between the fins.

Yeah, that's right, though the convection currents will be minimal. We'll say that ambient air temperature is 300K (yes, that does result in an absurdly small ΔT, but . . .). This results in a Gr_L of ~8.3 and a Ra_L of ~5.88, assuming beta = 300.5^-1, v = 15.69 x 10^-6, Pr = .708, and ΔT = .5. It is my general assumption that a Ra_L this low (<2000) indicates conductive heat transfer, namely Nu_L = 1. Our h value here should equal ~5.25 (assuming k = .02624). Technically we could still say that natural convection is taking place, I guess.

Alternatively, if my assumption about conductive heat transfer is incorrect, we can fall
back on Churchill & Chu's correlation for vertical plates with Ra_L < 10^9:

Nu_L = .68 + (.670*Ra_L[sup(1/4)[/sup])/(1+((.492/Pr)^(9/16))^(4/9))

This produces an h value of ~6.43. Note that I'm treating this as a constant heat flux
situation since the fin temperatures are not technically uniform (only the plate base temp
is), but the difference between T_B and T_∞ is so small that assessing fluid properties at 300, 300.5, or 301K makes practically no difference. I'm just using fluid properties
from 300K and assuming the fin midpoint temperature is 300.5K (hence my use of 300.5^-1 as β and ΔT as .5 above).

The fins are at the same temperature as the plate, with no variations along their length?

Hmm. No, though they may as well be, since the fins could never drop below 300K in this
scenario. Still, I'm going to assume that the fin midpoint temperature is 300.5K as a guess.
I can correct for the actual value later via trial and error, though it shouldn't change h by
much. I think the maximum h value (assuming the entire fin is 301K) is ~6.97.

You want to find the heat transfer coefficient from the fins to the air, based on the temperature difference between the plate and the bulk air above. You also want to determine the velocity profile of the air flow between the plates?

Not exactly. I can find h, either assuming conductive heat transfer or not. You may have
noticed that I never specified the spacing S between fins. All my calculations of h assumed
that the fins were far enough apart that they would not interfere with one another.

I guess I should redo my questions:

1). Given the low Ra_L above (~5.88), should I just assume conductive heat transfer?
2). If the answer to 1). is "yes", then does the optimal fin spacing formula of:

S_opt = 2.714L/Ra^(1/4)

still apply? If not, is there a different formula that I should use for optimal fin spacing?

I only brought up velocity profiles and mass flows as an alternative to the above S_opt formula. Admittedly, it is a bit of a convoluted approach, and I never did bother using it to adjust h values. I only used it to see if fin spacing might be less-than-optimal.

 

[Chestermiller]

I don't even know how to begin to comment on this. Let me just start asking questions in random order. That's the only way I can think of.

1. You have a horizontal flat plate 1 meter square with cooler air above it. If there were no fins, when heat left the plate, where would it flow (assuming only conduction)? What would the heat flow lines look like? What will the heat flow look like far from the plate?

2. Is the heated flat plate embedded in an infinite insulated flat plane outboard of the heated plate?

3. Those fins seem very thin (0.004"). Do they have enough structural integrity to stand, and are they thick enough to conduct enough heat.

4. What are you trying to accomplish with this setup? Just to heat up some stagnant air above the plate as much as possible? For what reason?

5. Why are you using such a small temperature difference?

6. Are you aware that the equation you are using for the Nussult number applies to a vertical plate with no blockage at the bottom? The plate blocks the air flow from the bottom, and this will markedly affect the natural convection heat transfer rate.

7. I am familiar with the Nu number equation you used, but where did the 0.68 come from? Was this just Churchill and Chu's modification?

8. I am not familiar with the optimum fin spacing equation you presented. What geometric arrangement is this supposed to apply to?

9. Suppose you had to solve this problem as a straight heat conduction problem. How would you proceed? How far out would you solve for the temperature distribution?

10. I don't understand your statement that, for pure conduction, the Nussult number is equal to 1.0. Is this for some transient situation or for some steady state situation? What exactly is the geometry where Nu = 1? How is that geometry and heat flow paths related to the geometry and heat flow paths of your system?

Chet

 

[manderson99]

I don't even know how to begin to comment on this. Let me just start asking questions in random order. That's the only way I can think of.

That's fine with me.

1. You have a horizontal flat plate 1 meter square with cooler air above it. If there were no fins, when heat left the plate, where would it flow (assuming only conduction)? What would the heat flow lines look like? What will the heat flow look like far from the plate?

My guess is that, given a horizontal flat plate with only conduction, the lanes would flow nearly vertically away from the plate over the majority of its surface (especially towards its center), and that they would begin to flow outward at a slight angle near the edges.

2. Is the heated flat plate embedded in an infinite insulated flat plane outboard of the heated plate?

Yes, that would make things simpler in this example I suppose.

3. Those fins seem very thin (0.004"). Do they have enough structural integrity to stand, and are they thick enough to conduct enough heat.

I suppose it would depend on which material was used to construct them. For the purpose of this inquiry, we could consider them to be thicker if necessary.

4. What are you trying to accomplish with this setup? Just to heat up some stagnant air above the plate as much as possible? For what reason?

As far as this inquiry goes, I'm just trying to learn more about the implications of extremely low Rayleigh numbers when applied to heat sink design, particularly for heat sinks aimed at cooling small electronic chips with high thermal density (CPUs, etc). The setup is merely theoretical.

5. Why are you using such a small temperature difference?

It was an easy way to create a Rayleigh number below 10⁴(which appears to be the typical minimal Rayleigh number applicable to most flat plate Nusselt number correlations).

6. Are you aware that the equation you are using for the Nussult number applies to a vertical plate with no blockage at the bottom? The plate blocks the air flow from the bottom, and this will markedly affect the natural convection heat transfer rate.

Yes, though I do sometimes forget that point. It is wise of you to point that out.

7. I am familiar with the Nu number equation you used, but where did the 0.68 come from? Was this just Churchill and Chu's modification?

I think so, but I am not 100% sure about that. I got the equation from Chapter 8 of Schaum's Outline of Heat Transfer, Chapter 8 (near the end, in the Newer Correlations section, for what it's worth).

8. I am not familiar with the optimum fin spacing equation you presented. What geometric arrangement is this supposed to apply to?

I've seen it mentioned on several resources available on the web for rectangular profile fins on a heat sink. Here is a decent example: http://www.sfu.ca/~mbahrami/ENSC 388/Notes/Natural Convection.pdf

Check page 6 of the .pdf, if you have the time. If not, I certainly understand.

Note that I have not chosen to use the formula for h in this document:

h = 1.31k/S_Opt

because I am not sure what to make of it just yet. In this circumstance, it might work better than the flat plate correlation I used above.

9. Suppose you had to solve this problem as a straight heat conduction problem. How would you proceed? How far out would you solve for the temperature distribution?

I would guesstimate the midpoint temperature of the fin, take fluid properties at that temperature, use that to determine h using Churchill Chu above and then (sinh(n(x-L)) + (h/nk)(cosh(n(x-L))))/(sinh(nL) + (h/nk)(cosh(nL))), which I would then equate to (T_B - T_x0/(T_B - T_∞) and solve for T_x, where x = L/2. I would then plug the new temperature value back into the process at least two or three times, or until I stopped noticing any change beyond maybe three decimal places (the temperature differences are already very small, but still). Once I had settled on my midpoint temperature, I'd determine h, then solve for q using q = kAn(T_x - T_∞)(sinh(nL) + (h/nk)(cosh(nL)))/(cosh(nL) + (h/nk)(sinh(nL))).

10. I don't understand your statement that, for pure conduction, the Nussult number is equal to 1.0. Is this for some transient situation or for some steady state situation? What exactly is the geometry where Nu = 1? How is that geometry and heat flow paths related to the geometry and heat flow paths of your system?

Steady state, and I'm afraid I am only operating on statements I have seen by trying to find situations where Nu could be less than 1; that is, many seem to be of the opinion that a Nusselt number below 1 "makes no sense", and that Nu = 1 implies no convection (only conduction). An example, if a bit of a messy one:

http://www.researchgate.net/post/Can_the_Nusselt_No_be_negative_for_non-uniform_temperature_boundary_conditions

See post by Seng-Kin Lao, post #8 under "all answers". I can't claim this to be authoritative, but I'm kind of groping for information here. And he also corrects himself shortly thereafter and claims that Nu must be greater than 1 . . . hmm.

Anyway, I've seen others acting under supposed authority agreeing with his statement, more-or-less saying that Nu never gets lower than 1, and that Nu = 1 means conduction. Strangely, many of the common correlations can produce Nu < 1 if the Rayleigh and Graetz numbers are low enough. That doesn't mean the result is accurate, though.

Regardless of all this, I thought I'd ask here to get a more definitive answer. I would trust the information here more than the scraps I've dug up elsewhere.

Thanks for your patience in all this.

 

[Chestermiller]

The way to proceed is to first dope out what is going to happen if there are no fins (and with pure conduction). Why? Because if you can't do that, you certainly won't be able to do the case in which fins are present.

First of all, you are going to have a steady state situation, in which the temperatures are not changing with time, and the heat fluxes are constant with time. The region above the plate (and surrounding insulated table) is going to extend to infinity in the vertical direction. If radial distances are measured from the very center of the plate in all directions (spherical coordinates), then at radial locations far from the center of the plate, the heat flow lines are going to be purely radial, and the isotherms are going to be spherical surfaces. If Q is the total heat flow rate from the plate, then at large values or r,
$$-4πr^2k\frac{dT}{dr}=Q$$
This gives:
$$T→T_∞+\frac{Q}{4πrk}$$
where T_∞ is the temperature at infinity.

The temperature profiles near the heated plate need to match up with this relationship at radial locations far from the heated plate. Practically speaking, this would happen on the order of a couple of meters from the heated plate. There would have to be a smooth transition from the constant temperature at the heated flat plate (and the insulated region outboard) to this temperature variation.

Now for the case of fins. The fins are only 0.5 cm high. So, at distances above the plate more than about 1.5 cm, the shape of the temperature profiles are going to be virtually the same as if no fins were present. So there is going to be an isotherm about 1.5 cm above the plate where the temperature profile is essentially parallel to the plate, and this parallel region will extend within a few cm inboard of the edges of the plate on all the sides. The situation would essentially be the same as if the plate were infinite in all lateral directions, and there were a second plate above the heated plate that is a few degrees cooler than the heated plate. This would reduce the situation to a 2D heat flow problem, and you could identify a unit cell for the heat flow. The unit cell would consist of one vertical insulated planar boundary half-way between two adjacent fins, and a second insulated planar vertical boundary running up the center of one of the two fins. One would have to solve for the temperature profiles within the space between these two parallel vertical boundaries. The best you could do is if the fin is the same temperature as the heated plate below. This would give you the boundary conditions for the conduction heat flow problem, and would tell you the best you could do with fins. This could be coupled to the intermediate region between 1.5 cm, and about 2 meters, and this, in turn would be coupled to the outer region of purely radial heat flow. This is how I would go about setting up this complicated heat conduction problem (numerically).

chet