# Revisiting the power calculation in the Lugano report

• So..., in other words, since the ribbed device is actually so effective at radiating power, the effective emissivity when calculating the radiated power based on area, when assuming it is a diffuse cylinder, must be very high (?)

Or the device must be cooler than they said it was, since it is so efficiently radiant it could not get so hot?

• Good to have you back in the discussion again !

So..., in other words, since the ribbed device is actually so effective at radiating power, the effective emissivity when calculating the radiated power based on area, when assuming it is a diffuse cylinder, must be very high (?)

In essence we have two effects here

First of all is that you should use in the calculation of radiated energy the real surface area,

That means the area of the tube with fins, not that of the tube without fins.

However thermal radiation of a fin is partly blocked by the opposing fin. That makes the area of the fin less effictive and thus you need to calculate with an effective area of Af * Fbg in which Af is the area of the fin and Fbg the view factor to the background.

The second effect effect is that due to the reflection between the fins the effective emissivity changes (Infinite reflection method). The effective emissivity becomes

The total effect is Af x e x Fbg x { 1/(1 - (1-Fbg)(1-e))} , which is the formula shown in

an earlier post.

This makes the finned area more effective in radiating thermal power then without fins. However less effective then the total area of the fins without view factor correction would suggest and not very high, but surely more effective then without fins.

Or the device must be cooler than they said it was, since it is so efficiently radiant it could not get so hot

I have currently no idea if the device was cooler then what the testers reported. However having found the correct view factor is a first step in recalculating the results and see where that ends. After having solved this and other issues it gives me the opportunity to restart my thermal simulations of the dogbone and possibly solve the question between the different results of Lugano, MFMP and the thermal simulations

However there is also possibly another issue with the calculation of the convected power of the stacked rods , which I want to work on before doing new simulations.

Another issue I currently am investigating is why, if I remember well, there was such a large temperature difference reported between top and bottom of a fin. I do not know where and by whom that was reported, maybe you know and can point me to the information ?

LDM

• The hot valley vs cooler tip of the ribs is easily replicated. At about 800 C, about 50-80 C difference is typical. Testing the tip temperature is tricky, though. The valleys are much easier to get a thermocouple tightly affixed. A glob of cement to hold the thermocouple skews things significantly when ribs are involved.

Since the heat source is internal (important), there is a conduction temperature gradient from the center to the outside, modified by the increasing volume outwards. This is again modified by the rib shape, where the volume decreases from the base to the tip, offsetting the diameter-Pi volume increase.

Convection here, however, is the main rib temperature gradient maker. The valleys convect poorly (probably very rarified air there, in the extreme valley V location, due to the high heat and small volume) the tips are increasingly in denser, cooler air towards the tips. The ribs also conduct heat semi-poorly, being alumina, a poor heat conductor in general, so that the highly convecting outer rib area is in competition with conducted heat replenishment. As well as radiated heat transfer to the fins.

• The hot valley vs cooler tip of the ribs is easily replicated. At about 800 C, about 50-80 C difference is typical. Testing the tip temperature is tricky, though. The valleys are much easier to get a thermocouple tightly affixed. A glob of cement to hold the thermocouple skews things significantly when ribs are involved.

I wonder if the thermocouples are partly the cause of the difference in measured temperature. The connecting wires to the thermocouple tip are good thermal conductors, and are thus conducting thermal energy away from the tip, thereby lowering the tip temperature.

Since at the bottom of the fin there is more thermal mass then at the top of the fin the temperature drop due to this thermal conductance through the thermocouple leads will be less at the bottom of the fin then at the top. This will probably cause an (unknown) temperature difference in the thermocouple measurements.

Note also that at the MFMP dogbone retest for a temperature of about 970 degree C, the thermocouple showed around 840 degree C, a 130 degree C difference ! In my opinion also caused by the draining away of heat through the thermocouple wires.

Convection here, however, is the main rib temperature gradient maker.

In general in heat exchangers where ribs are involved, convection will indeed be the major source of heat transfer and thus the cause of the temperature gradient on the fin. Fins in heat exchangers have normally a larger ratio between height and base width of the fin then the fins on the dogbone. Due to this radiation is largely blocked and thus convection will indeed become the major source for heat transfer. (Also because in heat exchangers there is often a forced air flow)

In Lugano the hight/base width ratio is small and thus radiation can play a much larger role then for the larger ribs used in heat exchangers.

The valleys convect poorly (probably very rarified air there, in the extreme valley V location, due to the high heat and small volume) the tips are increasingly in denser, cooler air towards the tips.

This sounds logical, however I have not found this confirmed in documents on fin heat transfer.

The ribs also conduct heat semi-poorly, being alumina, a poor heat conductor in general, so that the highly convecting outer rib area is in competition with conducted heat replenishment. As well as radiated heat transfer to the fins.

While alumina is not a very good heat conductor, it is also not a very bad one either.

I have simulated a single Lugano style rib in the finite element thermal simulation program I use, and for a fin base temperature of 750 degree C the difference between top and bottom of the fin is only 20 degree C, indeed indicating that the thermal conductance is not too bad.

However that simulation does not take into account the possible difference in air flow between bottom and top of the fin you suggest.

Maybe I will be able to adjust that similation for difference in conducted heat transfer from bottom to top and run it again.

• If the tips of the fins were only 20 C cooler, then nobody would use them. Radiant heat transfer is the dominant heat transfer mode, over around 750 C. Below that rough temperature, convection does the bulk, and fins are the primary convection improver.

• If the tips of the fins were only 20 C cooler, then nobody would use them.

That is a misconception.

Newton's law states that :

-----Q = h x A x (Ts - Tamb)

h being the heat transfer coefficient, A the surface area , Ts the surface temperature and Tamb the ambient temperature.

Now consider two surfaces of the same amount of area, the first surface at the bottom of the fin, the second at the top.

If the temperature at the top of the fin is lower then at the bottom of the fin, Qtop will be lower then Qbottom.

If the temperature at the top of the fin is equal to the temperature at the bottom of the fin then Qtop will be equal to Qbottom.

In the last case where the temperatures at the bottom and the top are equal Qtop + Qbottom will be larger then when the temperature at the tip is lower then that at the bottom.

Otherwise stated, the convected heat transfer of the fin becomes better if the temperature drop between bottom of the fin and top of the fin is smaller.

That is one of the reasons why fins in heat exchangers are made of materials with good thermal conductivity such as aluminum and copper. The good thermal conductivity keeps the temperature difference between top and bottom of the fin as small as possible. The other reason being that heat transfer from the base into the fin is large.

More important factors for a good conductive heat transfer are a high heat transfer coefficient and a high surface area.

Radiant heat transfer is the dominant heat transfer mode, over around 750 C. Below that rough temperature, convection does the bulk, and fins are the primary convection improver.

Agreed that at higher temperatures radiation is dominant for heat transfer.

At lower temperatures it depends on the way how the heat is exchanged. With larger fins (fin height 3 cm) there is good conductive heat transfer and less good radiated heat transfer due to the fins largely blocking the radiation.

For the Lugano fins, which have only a small height and where radiation is less blocked and due to the small surface area convective heat transfer is limited. As a result even at the temperatures of about 450 degree C, the radiated heat transfer is still dominant. (See Lugano dummy run data)

• I see your point. If the fin tips are too cool compared to the base, then that could indicate that the ribs are probably too tall to be maximally efficient.

On the other hand, if there is no base to tip rib temperature gradient, then heat will not flow into the ribs and convection will not be improved by ribs. (The case of being in a vacuum would be like this). It would merely be a surface area increase radiation improvement.

• New

I see your point. If the fin tips are too cool compared to the base, then that could indicate that the ribs are probably too tall to be maximally efficient.

It is indeed so that if you increase the fin dimensions that the temperature drop between bottom and top becomes larger and thus the convected heat near the top becomes less. However this is (partly) offset by the increase in surface area.

In practice it is about finding the optimum dimensions, meaning where the convected heat per surface area devided by the material/production cost per surface area is optimal.

On the other hand, if there is no base to tip rib temperature gradient, then heat will not flow into the ribs and convection will not be improved by ribs. (The case of being in a vacuum would be like this). It would merely be a surface area increase radiation improvement.

Totally agree with you

The (heat) current has to flow, however to get a high heat flow at a low temperature drop you use materials with a high termal conductivity.