Conduction, heat transfer circular cylinder

The basic solutions for the infinite plates and infinitely long cylinders can be used to obtain solutions for multidimensional systems such as long rectangular plates, cuboids, and finite circular cylinders with end cooling. The texts on conduction heat transfer [4,11, 23, 29, 38, 49,56, 87] should be consulted for the proofs of the method and other examples. 

For example, to determine Nu, for the case where the body is a very long horizontal isothermal circular cylinder of diameter D, the relevant heat transfer would then be that by heat conduction across a cylindrical annulus of inner diameter D, inner temperature T , outer diameter D + 2A, and outer temperature 7U (assumed constant). Calculating this heat transfer by standard methods, substituting Eq. 4.16, and converting to a Nusselt number yields... 

Figure 4.31 shows a plot of Eq. 4.89 for a circular cylinder cavity with perfectly conducting walls and various values of D/L. As is clear from the graph, the Nusselt number rises very steeply with Ra after initiation of convection, and very rapidly approaches the value of Nu for the horizontally extensive cavity. This behavior is consistent with the conduction layer model at high Ra, the conduction layers on the walls at the sides are so thin that they have no effect on the heat transfer at sufficiently low Ra, they are so thick that they overlap (even though those on the horizontal plates do not), so that their presence governs the condition for a stationary fluid. 

J. C. Buell and I. Catton, The Effect of Wall Conduction on the Stability of a Fluid in a Right Circular Cylinder Heated From Below, J. Heat Transfer (105) 255-260,1983. 

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