Dimensionless numbers 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. 

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) ... 

N, IVc, IVnu IVp. K, IVh. IVs. N Proportionality coefficient, dimensionless group Grashof number, L p P Af/)U Nusselt number, hD/k or hL/k Peclet number, DGc/k Prandtl number, c A/k Reynolds number, DG/ l Stanton number, Number of sealing strips Dimensionless Dimensionless... 

The flow regime in natural corweetion is governed by a dimensionless number called the Grashof number, which represents the ratio of the buoyancy force to the viscous force acting on the fluid and is expressed as... 

Incidentally, Bowes discussed in his book, Self-Heating Evaluating and Controlling the Hazards , some dimensionless numbers, such as the Grashof number, the Reynolds number and the Prandtl number, which are all used in order to discuss the convective flow in fluid mechanics. 

Grashof number D -iPAPZ ) dimensionless number of sparger orifices or sites power number P JpO jN ), dimensionless bubble Reynolds number D fjiVdimensionless impeller Reynolds number (Dj Npjp, dimensionless Schmidt number (Pc/Pc ab) dimensionless Sherwood number (kiDy /DAB), dimensionless total pressure, N/m ... 

When the gas is motionless, the free convection is desoibed by three dimensionless numbers the Niisselt number, the Grashof (Gr) number, and the Prandtl (Pr) number ... 

The sedimentation of particles under inclined sur ces may be characterised by two dimensionless numbers Gr, a sedimentation Grashof number which represents the ratio of gravitational forces to viscous forces in a convective flow and R a sedimentation Reynolds number representing the signihcance of inertial forces to viscous forces in a convective flow [Acrivos and Erbholzeimer, 1979, 1981]... 

As discussed in Chapter 3, two dimensionless numbers are used to characterise the settling behaviour Gr, a sedimentation Grashof number and R, a sedimentation Reynolds number defined by the equations ... 

The Grashof number can be interpreted physically as a dimensionless number that represents the ratio of the buoyancy forces to the viscous forces in free convection and plays a role similar to that of the Reynolds number in forced convection. 

Presenting the process model as a mass transfer correlation is also conunon. This requires an understanding of the process s physical properties, namely, the density and viscosity of the SC-CO2 and the mass diffusion of the solute in SC-CO2. Dimensionless numbers, namely, Reynolds (Re) (Equation 5.16), which is related to fluid flow Schmidt (Sc) (Equation 5.17), which is related to mass diffusivity Grashof (Gr) (Equation 5.18), which is related to mass transfer via buoyancy forces due to difference in density difference between saturated SC-CO2 with solute and pure SC-CO2 and Sherwood (Sh) (Equation 5.19), which is related to mass transfer, are important in these correlations. In supercritical extraction, natural convection is not significant (Shi et al., 2007) and in this case, Shp is related only to Re and Sc, as shown in Equation 5.19. 

The dynamical response of two-phase flows can be commonly characterized successfully in terms of the dimensionless numbers [2]. Table 1 lists some force-related dimensionless numbers. These dimensionless numbers demonstrate competing phenomena of forces buoyancy, gravitational, inertial, viscous and interfacial forces. The Grashof number (buoyancy to viscous forces), the Bond number (gravitational to interfacial forces) and the... 

Heat and mass transfer coefficients are usually reported as correlations in terms of dimensionless numbers. The exact definition of these dimensionless numbers implies a specific physical system. These numbers are expressed in terms of the characteristic scales. Correlations for mass transfer are conveniently divided into those for fluid-fluid interfaces and those for fluid-solid interfaces. Many of the correlations have the same general form. That is, the Sherwood or Stanton numbers containing the mass transfer coefficient are often expressed as a power function of the Schmidt number, the Reynolds number, and the Grashof number. The formulation of the correlations can be based on dimensional analysis and/or theoretical reasoning. In most cases, however, pure curve fitting of experimental data is used. The correlations are therefore usually problem dependent and can not be used for other systems than the one for which the curve fitting has been performed without validation. A large list of mass transfer correlations with references is presented by Perry [95]. 

Grashof number A dimensionless number, Gr. It is used in the study of natural convection and represents the ratio of buoyancy forces to viscous forces ... 

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. 

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,... 

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... 

The other governing equations—the overall eontinuity equation, the speeies eontinuity equation, and the energy equation—are identieal to the dimensionless forms presented in Chapter 1. Two new dimensionless groups, a thermal Grashof number... 

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. 

Grashof number — The Grashof number (Ge) is a dimensionless parameter that relates the ratio of buoyancy forces to the viscous forces with a fluid solution. It is defined as ... 

NGt = Grashof number = L3p2gfi At/p2, dimensionless NPr = Prandtl number = cp(i/k, dimensionless Nr = number of rows of tubes across which shell fluid flows, dimensionless NRe = Reynolds number = DG/fi, dimensionless N, = total number of tubes in exchanger = number of tubes per pass X np, dimensionless... 

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