Lecture 32, March 24, 2004 • Apologies about the tutorial yesterday –

questions will be posted • Reminder Quiz next Tuesday covers from the last

quiz up until intro to radiation (Friday/Monday) Natural Convection Contd.. Consider a horizontal cavity heated from below (tau = 0o). It is the Raleigh number that we rely on to characterize the flow regime, in comparison to the Reynolds number in the case of forced convection. Below RaL=1708, buoyant forces are not strong enough to overcome viscous forces and there is no convection. This means that NuL = 1, and heat transfer is by conduction only. The next flow regime is that of laminar roll cells. The number of cells that develop can depend on many things including the aspect ratio (H/L) of the enclosure. Usually in this regime though, the roll cells are almost circular with a diameter very near L. Above RaL = 5e4 the regular cells break down and the flow is turbulent inside the cavity. Next consider a vertical cavity (tau = 90o). The aspect ratio now becomes very important since the heated fluid is carried all the way along the hot side of Length H before turning and (cooling) along the opposite

side. The hotter this fluid gets before turning towards the cool side, the less will be the heat transfer. For RaL < 1000 flow is weak and heat transfer is very close to conduction only. Refer to the text for the appropriate correlations for the correct range of RaL. As always be very careful not to use correlations outside their range of applicability. Let’s attempt to explore natural convection in the window systems that we looked at way back in chapter 3. At that time, we assumed that the air gaps transferred heat by conduction only (Nu = 1). First we need to look at several of the correlations presented in your text.

And, for the lower aspect ratios,

Alternatively,

and for the lower aspect ratios,

These correlations are all out of your text and are explained more therein.

The windows we considered were double and triple pane windows with H = 0.8m. The gap width, L was 10mm for the double pane window and 3mm for the triple pane window. This results in an aspect ratio H/L = 80 for the double pane, and H/L=267 for the triple pane window. We are clearly well above the range of aspect ratios for all the correlations presented above. Next we need to determine the Ra number in the gaps, so that we can determine NuL and hence complete a thermal resistance network. This should be done at the average T for the enclosure, and again we will approximate this with the total temperature range. The average temperature is then 5oC, or 278K. Properties Air Argon k = 0.0263 W/mK 0.0186 W/mK mu = 2.27e-5 Ns/m2 1.85e-5 Ns/m2

rho=1.68 kg/m3 1.16 kg/m3

Pr = 0.633 0.707 alpha=2.19e-5 m2/s 2.25e-5 m2/s Be very careful –we are outside the range of H/L for these correlations. We predict NuL for the triple pane

window which are less than 1. This is very unphysical and should be discarded immediately. The heat transfer in the gaps of these triple pane windows will be conduction only (Nu=1). Note that with natural convection that Argon performs slightly worse than air. If you calculate the Nusselt number for the double plane windows with blinds in the cavity which have a spacing of 10mm, you find that NuL for the air filled window increases to 4.55, and to 4.74 for the Argon filled window. Essentially we have made many small enclosures which are very effective at carrying heat from the hot window to the cool window. We have decreased the window performance by more than a factor of 2! Please read chapter 9, and notice the correlations for such things as horizontal cylinders, plates, and for cylindrical and spherical enclosures.