Natural convection induced by heating a part of a vertical ylinder of finite length was investigated.
The heat transfer to a laminar fluid flow inside cylinders has already been studied by many authors since Gratz. Recently, Sellars and his coworkers analyzed this problem by the use of W-K-B approximation, and Whiteman and Drake extended the application of this method to a flow with a profile expressed by u
0 (1-γ+
m). However, the W-K-B method does not give a good approximation to problems where the eigenvalues are not large and several incipient terms are dominant for the solution. Moreover, most of the authors have assumed a constant value for the density of the fluid. This will cause a big error when we deal with a flow caused only by natural convection.
In this report, three fundamental equations for fluid flow were analyzed for a system represented by Fig. 1. Temperature dependence of the density of air was taken into account by employing the relation given by Eq.(4). From the energy equation (3), combined with the continuity equation (1) and the boundary conditions of the system, the temperature distribution functions in the cylinder were obtained as given by Eqs.(14), (15) and (16). These distribution functions were applied to the momentum equation (2), and integrated for one complete cycle of the convection, as given by Eq.(18). The result is given by Eq.(19), (where the abbreviations of Eqs. from (20) to (25) are employed) by which the rate of convection, or the Reynolds number Re
0 can be estimated as a function of T
w-T
0 andthe geometrical structure of the system.
The W-K-B approximation and the estimation by a digital computer were applied to the solution. Values of several incipient terms of eigenvalues λ
n and coefficients C
n are given in Table 1, and are compared with the values obtained by Abramowitz, who tried to give accurate values by expanding the differential equation by Bessel function. Only those results estimated for
m=2 were tabulated, though other values of
m were examined, too. The values obtained by the computer show good coincidence with those by Abramowitz. The former may be more reliable than the latter. The W-K-B method gave poor result
In Figs. 3, 4 and 5, the areal mean rate of convection is shown as a function of wall temperature. Good coincidence is mostly found between experimental and computed values. In Figs. 4, experimental values for large
Tw-T0 To are somewhat higher than estimated values. In this region, the Reynolds number exceeds 1, 500 and the parabolic flow assumption might be inapplicable. On the other hand, in Figs. 3 and 5, the observed rates are lower than those estimated for large values of
Tw-T0. This might be attributed to a dominant effect of the use of ρ
0 and β
0 instead of ρ and β in the estimation of the rates by means of Eq.(19).
In Figs. 6 and 7, the effect of the location of the heating section on the wall temperature is shown, by taking the areal mean velocity u
o as a parameter. It will be seen that the height of the heating section has a predominant effect on the rate of convection. Radius of cylinder also produces a large effect on the rate of convection. When the radius is small, the viscosity of the gas retards the flow rate, and when it is large, prolonged heat transfer retards the rate, too. There must exist an optimum radius for an assigned rate of convection. Fig. 8 shows this situation. The chain line in the figure shows the optimum radius for obtaining the highest rate of convection by a given temperature and a given height of the heating section.
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