Dimensionless groups Schmidt number

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. 

Both methods yield dimensionless groups, which correspond to dimensionless numbers (1), e.g.. Re, Reynolds number Fr, Froude number Nu, Nusselt number Sh, Sherwood number Sc, Schmidt number etc. (2). The classical principle of similarity can then be expressed by an equation of the form ... 

In this example, we recognize two well-known dimensionless groups, i.e. the Reynolds (Re = ul/v) and Schmidt (Sc = v/D) numbers. In contrast, the third dimensionless group in the last equation is not usually used. Instead, the Sherwood number is more useful (Sh = kl/D). The Sherwood number can result from multiplying both sides of the original functional form with the Sc number. The final relationship is... 

Knowing the viscosity and density of the reaction mixture, the flow channel diameter, void fraction of the bed, and the superficial fluid velocity, it is possible to determine the Reynolds number, estimate the intensity of dispersion from the appropriate correlation, and use the resulting value to determine the effective dispersion coefficient Del or I). Figures 8-32 and 8-33 illustrate the correlations for flow of fluids in empty tubes and through pipes in the laminar flow region, respectively. The dimensionless group De l/udt = De l/2uR depends on the Reynolds number (NRe) and on the molecular diffusivity as measured by the Schmidt number (NSc). For laminar flow region, DeJ is expressed by ... 

The mass diffusivity Dt], the thermal diffusivity a = k/pCp, and the momentum diffusivity or kinematic viscosity v = fi/p, all have dimensions of (length)2/time, and are called the transport coefficients. The ratios of these quantities yield the dimensionless groups of the Prandtl number, Pr, the Schmidt number, Sc, and the Lewis number, Le... 

This relation is analogous to the expression for the heat transfer by forced convection given earlier. The dimensionless group kd/D corresponds to the Nusselt group in heat transfer. The parameter rj/pD is known as the Schmidt number and is the mass-transfer counterpart of the Prandtl number. For example, the evaporation of a thin liquid film at the wall of a pipe into a turbulent gas is described by the equation... 

The basic similarity hypothesis states simply that the turbulent transport processes of momentum, heat and mass are caused by the same mechanisms, hence the functional properties of the transfer coefficients are simiiar. The different transport coefficients can thus be related through certain dimensionless groups. The closure problem is thus shifted and henceforth consist in formulating sufficient parameterizations for the turbulent Prandtl Pr )- and Schmidt (Sct) numbers. 

The first model suggested for these dimensionless groups is named the Reynolds analogy. Reyuolds suggested that in fully developed turbulent flow heat, mass and momentum are transported as a result of the same eddy motion mechanisms, thus both the turbulent Prandtl and Schmidt numbers are assumed equal to unity ... 

After suitable non-dimensional variables are substituted into the equations, following the same procedure as outlined in sect 1.2.5, the important dimensionless groups are obtained for the problem in question. These are the Reynolds number, the Schmidt number, the Peclet number, Pe = Re Sc = ul/D, and the Damkohler number, Daj = Ir/u. The u and I are the characteristic velocity and length scales, respectively, for the velocity field, and r denotes a characteristic chemical reaction rate. 

Average transport coefficients between the bulk stream and the particle surface in a fixed-bed reactor can be correlated in terms of dimensionless groups which describe the flow conditions For mass transfer the group kj p(G is a function of the Reynolds number dpGIfi and the Schmidt number pIp. Chilton and Colburn suggested plotting 7 vs dpGfp., where... 

The particle transfer coefficient k has dimensions of velocity and is often called the deposition velocity. At a given location on the collector surface the dimensionless group kL/D, known as the Sherwood number, is a function of the Reynolds. Peclet, and interception numbers. Rates of particle deposition measured in one fluid over a range of values of Pe, Re, and R can be u.sed to predict deposition rate.s from another fluid at the same values of the dimensionless groups. In some cases, it is convenient to work with the Schmidt number Sc = u/D = Pe/Re in place of Pe as one of the three groups, because Sc depends only on the nature of the fluid and the suspended particles. 

This equation represents the usual functional relationship between the dimensionless groups (Sherwood, Reynolds, and Schmidt numbers) plus the assumption that the effective packing element... 

Unlike the feed side, flow on the dia lysate side of the membrane is oflen turbo lent. For turbulent flow, the dialysate-side mess transfer coefficieni can be described in terms of a Sherwood number which can be related to two other dimensionless groups, the Reynolds number (Re) and the Schmidt number (Sc), by a correlation of the form ... 

The dimensionless groups in Eqs. (21.46) and (21.47) have been given names and symbols. The group k DjD is called the Sherwood number and is denoted by ATg,. This number corresponds to the Nusselt number in heat transfer. The group pjpD is the Schmidt number, denoted by It corresponds to the Prandtl number. Typical values of are given in Appendix 19. The group DGjp is, of course, a Reynolds number, IVr,. 

Preparatory work for the steps in the scaling up of the membrane reactors has been presented in the previous sections. Now, to maintain the similarity of the membrane reactors between the laboratory and pilot plant, dimensional analysis with a number of dimensionless numbers is introduced in the scaling-up process. Traditionally, the scaling-up of hydrodynamic systems is performed with the aid of dimensionless parameters, which must be kept equal at all scales to be hydrodynamically similar. Dimensional analysis allows one to reduce the number of variables that have to be taken into accoimt for mass transfer determination. For mass transfer under forced convection, there are at least three dimensionless groups the Sherwood number, Sh, which contains the mass transfer coefficient the Reynolds number. Re, which contains the flow velocity and defines the flow condition (laminar/turbulent) and the Schmidt number, Sc, which characterizes the diffusive and viscous properties of the respective fluid and describes the relative extension of the fluid-dynamic and concentration boundary layer. The dependence of Sh on Re, Sc, the characteristic length, Dq/L, and D /L can be described in the form of the power series as shown in Eqn (14.38), in which Dc/a is the gap between cathode and anode Dw/C is gap between reactor wall and cathode, and L is the length of the electrode (Pak. Chung, Ju, 2001) ... 

Additional analysis reveals that the dimensionless mole fraction is a function of various groups (r, 6, the Reynolds number) and an additional term, the Schmidt number ... 

Because of the change in the Reynolds number, calculated values of kg change however, because of the relationship between two other dimensionless groups, the Prandtl and Schmidt numbers, assume the kg/h ratio is constant at 9.0 x 10 cm K/cal. 

The frame when a sub-frame is in use The mass transfer rale per unit area per unit concentration difference A dimensionless group which is a function of Reynolds and Schmidt numbers, used in tiia.ss t ransfer work. 

It is particularly important in the case of dimensionless group correlation, in order to provide a meaningfuJ value of the constants K, a and b in the expression Sh= where Re is the Reynolds number, 5c is the Schmidt number and Sh is the Sherwood number. 

Two additional dimensionless groups, the Peclet number and the Stanton number, are also used, although with lesser frequency. Both of these numbers are composites of other dimensionless groups, which frequently occur in unison. Thus, the Reynolds and Schmidt numbers often crop up combined as a product, which leads to the Peclet number ... 

This is fhe more common form of dimensionless grouping seen in the literature and states that the Sherwood number Sh, which contains the mass transfer coefficient as tiie dependent variable, is a function of both Reynolds and Schmidt numbers. 

---