Natural-convection dimensionless groups

The secondary flows from natural convection can become larger than the primary flow, so it seems likely that the secondary flows might become turbulent or nonsteady. Shown in Tables 1 and 2 are the dimensionless groups at the inlet and outlet, based on cup-average quantities, as well as the Reynolds numbers for the primary and secondary flows (Reynolds numbers defined in terms of the respective total mass flowrate, the viscosity and the ratio of tube perimeter to tube area). 

In Fig. 10.4 the sphere diameter, terminal velocity, and temperature difference each appear in only one dimensionless group. The effect of natural convection on is smaller at Pr = 10 because the region over which the buoyancy force acts is much thinner than for Pr = 1. As Pr oo the effect should disappear altogether. For Pr = 0, numerical solutions (W7) show effects about 50% larger than for Pr = 1. 

In this section the full equations of motion for the external problem sketched in Fig. 4.1a are simplified by using approximations appropriate to natural convection, and the resulting equations are nondimensionalized to bring to light the important dimensionless groups. Although... 

Natural convection currents will develop if there exists a significant variation in density within a liquid or gas phase. The density variations may be due to temperature differences or to relatively large concentration differences. Consider natural convection involving mass transfer from a vertical plane wall to an adjacent fluid. Use the Buckingham method to determine the dimensionless groups formed from the variables significant to this problem. 

As highlighted by Shah and London [2], a natural tendency exists to use in convection problems a large number of different sets of dimensionless groups based on the analyst s particular normalization of the differential equations and boundary conditions. An effort to standardize the definitions of dimensionless groups for laminar flows through channels was made by Shah and London [2] some years ago. In this section, the normalization of the convection problems proposed by Shah and London will be followed. 

Natural convection heat transfer outside a vertical plane. In the case of natural-convection heat transfer from a vertical plane wall of length L to an adjacent fluid, different dimensionless groups should be expected when compared to forced convection inside a pipe since velocity is not a variable. The buoyant force due to the difference in density between the cold and heated fluid should be a factor. As seen in Eqs. (4.7-1) and (4.7-2), the buoyant force depends upon the variables p, g, p, and AT. Hence, the list of variables to be considered and their fundamental units are as follows ... 

As we have seen, the critical dimensionless groups for free or natural convection are the Grashof and Prandtl numbers. In general (the h will be a mean value). 

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