Number Grashof

Heat Exchangers Since most cryogens, with the exception of helium 11 behave as classical fluids, weU-estabhshed principles of mechanics and thermodynamics at ambient temperature also apply for ctyogens. Thus, similar conventional heat transfer correlations have been formulated for simple low-temperature heat exchangers. These correlations are described in terms of well-known dimensionless quantities such as the Nusselt, Reynolds, Prandtl, and Grashof numbers. 

Djj. The Grashof number Nq, = Dj pgpAto/p" were is equivalent diameter, g is acceleration due to gravity, p is coefficient of volumetric expansion, p is viscosity, p is density, and Atg is the difference between the temperature at the wall and that in the bulk fluid. Nq, must be calculated from fluid properties at the bulk temperature. 

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. 

The ratio Ap/p can be replaced by AT/T. Ar relates the influence of velocity and temperature of a jet when discharged into an environment of a different temperature. In some instances the Froude number, Galileo number, or Grashof number may replace the Archimedes number. 

Same as Peclet number except considered (entrance region). Grashof number... 

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 ... 

The constant a appears to be a function of the Grashof number, but approaches a value of about 2 as the Grashof number approaches zero. 

Several dimensionless groups characterize these equations. The Reynolds number Re = f O/Poo indicates the ratio of centrifugal forces to viscous forces. The ratio of the Grashof number and the Reynolds number to the 3/2 power,... 

Gr Grashof number D2pg Ap/p)/v2 Dp = particle diameter g = acceleration due to gravity Ap = solid-fluid density difference p = fluid density... 

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... 

GrL Grashof number = gli Ap/pv2 Gr Grashof number defined by Eq. (40) Gr Grashof number defined by Eq. (37) GrJ Grashof number defined by Eq. (39) / Ionic strength = c( z2 (moles liter"1)... 

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