Grashof

American engineers are probably more familiar with the magnitude of physical entities in U.S. customary units than in SI units. Consequently, errors made in the conversion from one set of units to the other may go undetected. The following six examples will show how to convert the elements in six dimensionless groups. Proper conversions will result in the same numerical value for the dimensionless number. The dimensionless numbers used as examples are the Reynolds, Prandtl, Nusselt, Grashof, Schmidt, and Archimedes numbers. 

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. ... 

However, as the Grashof and Prandtl numbers can be different to determine, the following formula, which gives a close approximation, can be used for cylinders freely exposed to air ... 

Grashof Gr Coefficient of expansion of fluid Density of fluid Viscosity of fluid Force of gravity Temperature difference Dimension of surface... 

Gr Grashof l3p2flgAT P2 Heat transfer by natural convection... 

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. 

For large values of z a fully developed case is reached in which the velocities are only functions of r and 0. In the fully developed case the weight fraction polymer increases linearly in z with the same slope for all r and 0. An implicit finite difference scheme was used to solve the model equations, and for the fully developed case the finite difference method was combined with a continuation method in order to efficiently obtain solutions as a function of the parameters (see Reference 14). It was determined that except for very large Grashof... 

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,... 

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