Convection Grashof number

See also -> convection, -> Grashof number, - Hagen-Poiseuille, -> hydrodynamic electrodes, -> laminar flow, - turbulent flow, -> Navier-Stokes equation, -> Nusselt number, -> Peclet number, -> Prandtl boundary layer, - Reynolds number, -> Stokes-Einstein equation, -> wall jet electrode. 

The dimensionless numbers are important elements in the performance of model experiments, and they are determined by the normalizing procedure ot the independent variables. If, for example, free convection is considered in a room without ventilation, it is not possible to normalize the velocities by a supply velocity Uq. The normalized velocity can be defined by m u f po //ao where f, is the height of a cold or a hot surface. The Grashof number, Gr, will then appear in the buoyancy term in the Navier-Stokes equation (AT is the temperature difference between the hot and the cold surface) ... 

Model experiments where free convection is the important part of the flow are expressed by the Grashof number instead of the Archimedes number, as in Eq. (12.61). The general conditions for scale-model experiments are the use of identical Grashof number, Gr, Prandtl number, Pr, and Schmidt number,, Sc, in the governing equations for the room and in the model. 

Follow steps 7 (Gilmour method), etc., of the procedure for vaporization only. If baffles are added for sensible heat (not assumed in free convection), then pressure drop will be affected accordingly. Gr is the Grashof number using properties at average fluid temperature, = Dj pgP At/p. ... 

For conditions in which only natural convection occurs, the velocity is dependent on the buoyancy effects alone, represented by the Grashof number, and the Reynolds group may be omitted. Again, when forced convection occurs the effects of natural convection are usually negligible and the Grashof number may be omitted. Thus ... 

Another case of free convection with some complications, but amenable to solution, is that due to combined temperature and concentration differences. De Leeuw den Bouter et al. (DIO) experimented with such combined free convective transfer, assuming complete analogy of heat and mass transfer if the Grashof number employed is of the form... 

Marchiano and Arvia (M3) also measured mass transfer by thermal and diffusional free convection at a vertical plate. They derived on theoretical grounds a combined Grashof number as follows ... 

The Grashof number given by Eq. (40) appears to have a weaker theoretical basis than that given by Eq. (37), since it is based on an analysis that approximates the profile of the vertical velocity component in free convection, for example, by a quadratic function of the distance to the electrode. The choice of an appropriate Grashof number, as well as the experimental conditions in the work of de Leeuw den Bouter et al. (DIO) and Marchiano and Arvia (M3), has been reviewed critically by Wragg and Nasiruddin (W10). They measured mass transfer by combined thermal and diffusional, turbulent, free convection at a horizontal plate [see Eq. (31) in Table VII], and correlated their results satisfactorily with the Grashof number of Eq. (37). 

Combined thermal and diffusional free convection at a horizontal cylinder was investigated by Weder (W3a). He used a Grashof number of the type of Eq. (37), but corrected for the higher than 0.25 exponent of Gr found in... 

Weder s experiments were carried out with opposing body forces, and large current oscillations were found as long as the negative thermal densification was smaller than the diffusional densification. [Note that the Grashof numbers in Eq. (41) are based on absolute magnitudes of the density differences.] Local mass-transfer rates oscillated by 50%, and total currents by 4%. When the thermal densification dominated, the stagnation point moved to the other side of the cylinder, while the boundary layer, which separates in purely diffusional free convection, remained attached. 

Tobias and Hickman (T2), the only investigators to date to study combined free and forced convection in horizontal channel flow, found a remarkably sharp separation between forced- and free-convection dominated mass transfer. In forced convection, the critical Grashof number, based on the diffusion layer thickness, is... 

Dimensional analysis shows that, in the treatment of natural convection, the dimensionless Grashof number, which represents the ratio of buoyancy to viscous forces, is often important. The definition of the Grashof number, Gr, is... 

This type of equation comes from dimensional analysis. The coefficient and exponents are found by experiment. If forced convection is used, the Reynolds number has the conventional meaning of DVp/p. If free convection is used, the Reynolds number is replaced by the Grashof number, which can be shown to have a meaning of a Reynolds number, owing to free convection (B8). 

In cases where Gr Re2, free convection currents set in, being responsible for the transport processes. In packed beds of seeds, the particle Reynolds number is less than 10 to 50. The Grashof number represents the squared Reynolds number for the velocity of the buoyancy flow [18]. Therefore, the ratio of eqn. (3.4-68) is a comparison of the convective flow owing to buoyancy to the flow owing to circulation in terms of their respective squared Reynolds numbers. 

In free convection at small Grashof numbers (Gr) we have laminar selfsimilarity of the Peclet number (Pe), which is no longer a governing parameter proportional to Gr, and once again in the limit of small Gr... 

Gr = Grashof number = free convection/viscous forces = cPDrg/v... 

Fortunately, most cryogens, with the exception of helium II, behave as classical fluids. As a result, it has been possible to predict their behavior by using well-established principles of mechanics and thermodynamics applicable to many room-temperature fluids. In addition, this has permitted the formulation of convective heat transfer correlations for low-temperature designs of simple heat exchangers that are similar to those used at ambient conditions and utilize such well-known dimensionless quantities as the Nusselt, Reynolds, Prandtl, and Grashof numbers. 

For natural convection, the Grashof number, Gr, is found instead of the Reynolds number for forced convection correlations. The Grashof number may be shown to be equivalent to a ratio of forces ... 

Compare the heat-transfer coefficients for laminar forced and free convection over vertical flat plates. Develop an approximate relation between the Reynolds and Grashof numbers such that the heat-transfer coefficients for pure forced convection and pure free convection are equal. 

Gryzagoridis, J., Natural Convection from a Vertical Rat Plate in the Low Grashof Number Range , Int. J. Heat Mass Transfer, Vol. 14, pp. 162-164, 1971. 

Variation of Nussell number with Reynolds number in forced convection and in assisting and opposing convection for a Grashof number of 2500. 

Consider assisting combined convective flow over a body. If the Grashof number is kept constant and the Reynolds number is varied, the variation of Xu with Re would resemble that shown in Fig. 9.17. This type of result could be obtained by considering a body of fixed size, kept at a fixed temperature (this would keep the Grashof number constant), that is placed in a fluid flow in which the velocity could be varied (this would allow the Reynolds number to be varied). 

For this value of the Grashof number, standard correlation equations for natural convection from a horizontal cylinder indicate that for a Prandtl number of 0.7 ... 

This chapter has been concerned with flows in wb ch the buoyancy forces that arise due to the temperature difference have an influence on the flow and heat transfer values despite the presence of a forced velocity. In extemai flows it was shown that the deviation of the heat transfer rate from that which would exist in purely forced convection was dependent on the ratio of the Grashof number to the square of the Reynolds number. It was also shown that in such flows the Nusselt number can often be expressed in terms of the Nusselt numbers that would exist under the same conditions in purely forced and purely free convective flows. It was also shown that in turbulent flows, the buoyancy forces can affect the turbulence structure as well as the momentum balance and that in turbulent flows the heat transfer rate can be decreased by the buoyancy forces in assisting flows whereas in laminar flows the buoyancy forces essentially always increase the heat transfer rate in assisting flow. Some consideration was also given to the effect of buoyancy forces on internal flows. 

Joye, D.D., Comparison of Correlations and Experiment in Opposing Flow, Mixed Convection Heat Transfer in a Vertical Tube with Grashof Number Variation , Int. J. Heat and Mass Transfer, Vol. 39, No. 5, pp. 1033-1038, 1996. 

Nakai, S., and T. Okazaki Heat Transfer from a Horizontal Circular Wire at Small Reynolds and Grashof Numbers—1 Pure Convection, Int. J. Heat Mass Transfer, vol. 18, p. 387, 1975. 

The Grashof number may be interpreted physically as a dimensionless group representing the ratio of the buoyancy forces to the viscous forces in the free-convection flow system. It has a role similar to that played by the Reynolds number in forced-convection systems and is the primary variable used as a criterion for transition from laminar to turbulent boundary-layer flow. For air in free convection on a vertical flat plate, the critical Grashof number has been observed by Eckert and Soehngen [1] to be approximately 4 x 10". Values ranging between 10" and 109 may be observed for different fluids and environment turbulence levels. ... 

The foregoing analysis of free-convection heat transfer on a vertical flat plate is the simplest case that may be treated mathematically, and it has served to introduce the new dimensionless variable, the Grashof number, which is important in all free-convection problems. But as in some forced-convection problems, experimental measurements must be relied upon to obtain relations for heat transfer in other circumstances. These circumstances are usually those in which it is difficult to predict temperature and velocity profiles analytically. Turbulent free convection is an important example, just as is turbulent forced convection, of a problem area in which experimental data are necessary however, the problem is more acute with free-convection flow systems than with forced-convection systems because the velocities are usually so small that they are very difficult to measure. Despite the experimental difficulties, velocity measurements have been performed using hydrogen-bubble techniques [26], hot-wire anemometry [28], and quartz-fiber anemometers. Temperature field measurements have been obtained through the use of the Zehnder-Mach interferometer. The laser anemometer [29] is particularly useful for free-convection measurements because it does not disturb the flow field. 

The characteristic dimension to be used in the Nusselt and Grashof numbers depends on the geometry of the problem. For a vertical plate it is the height of the plate L for a horizontal cylinder it is the diamter d and so forth. Experimental data for free-convection problems appear in a number of references, with some conflicting results. The purpose of the sections that follow is to give these results in a summary form that may be easily used for calculation purposes. The functional form of Eq. (7-25) is used for many of these prer sentations, with the values of the constants C and m specified for each case. 

The free-convection flow phenomena inside an enclosed space are interesting examples of very complex fluid systems that may yield to analytical, empirical, and numerical solutions. Consider the system shown in Fig. 7-10, where a fluid is contained between two vertical plates separated by the distance 5. As a temperature difference AT,. = T - T> is impressed on the fluid, a heat transfer will be experienced with the approximate flow regions shown in Fig. 7-11, according to MacGregor and Emery [18]. In this figure, the Grashof number is calculated as... 

At very low Grashof numbers, there are very minute free-convection currents and the heat transfer occurs mainly by conduction across the fluid layer. As the Grashof number is increased, different flow regimes are encountered, as shown, with a progressively increasing heat transfer as expressed through the Nusselt number... 

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