Dimensionless groups table

For purposes of data correlation, model studies, and scale-up, it is useful to arrange variables into dimensionless groups. Table 6-7 lists many of the dimensionless groups commonly founa in fluid mechanics problems, along with their physical interpretations and areas of application. More extensive tabulations may oe found in Catchpole and Fulford (Ind. Eng. Chem., 58[3], 46-60 [1966]) and Fulford and Catchpole (Ind. Eng. Chem., 60[3], 71-78 [1968]). 

Electrochemical Systems Scaling, Dimensionless Groups, Table 1 Boundary conditions corresponding to Eqs. 1 and 2... 

Electrochemical Systems - Scaling, Dimensionless Groups, Table 3 Dimensionless parameters required to characterize an electrochemical reaction taking place on the surface of a cylinder as described in Eqs. 1, 2, 3, and 4... 

Electrochemical Systems - Scaling, Dimensionless Groups, Table 4 Summary of kinetic parameters and bulk concentrations for the reactions (13) note the several orders of magnitude differences in the reaction rate constants and the concentration values... 

Equation (3.35), i.e., the Sh-Re-Sc form, is usually modified to suit different equipment and condition of mass transfer by adding extra-geometric term or dimensionless group. Table 3.3 is given some examples. 

Dimensionless numbers are not the exclusive property of fluid mechanics but arise out of any situation describable by a mathematical equation. Some of the other important dimensionless groups used in engineering are Hsted in Table 2. 

Dimensional Analysis. Dimensional analysis can be helpful in analyzing reactor performance and developing scale-up criteria. Seven dimensionless groups used in generalized rate equations for continuous flow reaction systems are Hsted in Table 4. Other dimensionless groups apply in specific situations (58—61). Compromising assumptions are often necessary, and their vaHdation must be estabHshed experimentally or by analogy to previously studied systems. 

Table 4. Dimensionless Groups in Chemical Reaction Systems...

Table 5. Representative Dimensionless Groups for Agitated Reactors...

TABLE 3-8 Dimensionless Groups in the Engineering System of Dimensions... 

TABLE 6-7 Dimensionless Groups and Their Significance Concluded]... 

Dimensionless groups for a proeess model ean be easily obtained by inspeetion from Table 13-2. Eaeh of the three transport balanees is shown (in veetor/tensor notation) term-by-term under the deseription of the physieal meanings of the respeetive terms. The table shows how various well-known dimensionless groups are derived and gives the physieal interpretation of the various groups. Table 13-3 gives the symbols of the dimensions of the terms in Table 13-2. 

Tables 13-2 and 13-3 elueidate how the eommon dimensionless groups are derived. The boundary eonditions governing the differential equations eombined with the relative size of the system should be eonsidered when determining dimensionless parameters. Using Table 13-2 to determine the dimensionless groups for any of the three equations, divide one set of the dimensions into all the others ineluding the boundary eonditions.

Some of the important dimensionless groups used in Chemical Engineering are listed in Table 1.3. 

The secondary flows from natural convection can become larger than the primary flow, so it seems likely that the secondary flows might become turbulent or nonsteady. Shown in Tables 1 and 2 are the dimensionless groups at the inlet and outlet, based on cup-average quantities, as well as the Reynolds numbers for the primary and secondary flows (Reynolds numbers defined in terms of the respective total mass flowrate, the viscosity and the ratio of tube perimeter to tube area). 

Table 1.3 Comparison of miscellaneous dimensionless groups characterizing different hydrodynamic regimes in macroscopic vessels and micro reactors.

There are 21 parameters in Eq. (25) (Table I). These provide the five dimensionless groups given in Table II. A 10-dimensional process variation space is needed to characterize the slow coke preformance of the two-zone air lift TCC kilns, even when a is constant for each zone. It is not surprising that engineers and operators had problems in understanding the responses observed in the In. 

Using the parameters defined in Table II, the ODE system characterizing the ammonia converter can be reduced to dimensionless form with the following dimensionless groups ... 

We will assume forced convection in order to calculate dimensionless groups. Physical properties of air must be estimated, and we will use the ambient air temperature to estimate them. The following physical properties can be found in tables or calculated ... 

The basis of the scale-of-agitation approach is a geometric scale-up with the power law exponent, = 1 (Table 1). This provides for equal fluid velocities in both large- and small-scale equipment. Furthermore, several dimensionless groups are used to relate the fluid properties to the physical properties of the equipment being considered. In particular, bulk-fluid velocity comparisons are made around the largest blade in the system. This method is best suited for turbulent flow agitation in which tanks are assumed to be vertical cylinders. 

In addition to the Peclet number, one can also define other dimensionless groups that compare either relevant time scales or energies of interaction. Using some of the concepts previewed in Section 4.7c and Table 4.4, one can define an electrostatic group (in terms of the zeta potential f and relative permittivity cr of the liquid) as... 

The values of the above dimensionless groups for the kinetic data from Table 2.1 are listed in Table 3.1, together with some of the specific results we will derive in the following sections. 

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