The Buckingham Method

Use the Buckingham method to determine the dimensionless groups significant to a given mass-transfer problem. 

Most practically useful mass-transfer situations involve turbulent flow, and for these it is generally not possible to compute mass-transfer coefficients from theoretical considerations. Instead, we must rely principally on experimental data. The data are limited in scope, however, with respect to circumstances and situations as well as to range of fluid properties. Therefore, it is important to be able to extend their applicability to conditions not covered experimentally and to draw upon knowledge of other transport processes (of heat, particularly) for help. A very useful procedure toward this end is dimensional analysis. 

In dimensional analysis, the significant variables in a given situation are grouped into dimensionless parameters which are less numerous than the original variables. Such a procedure is very helpful in experimental work in which the very number of significant variables presents an imposing task of correlation. By combining the variables into a smaller number of dimensionless parameters, the work of experimental data reduction is considerably reduced. 

Dimensional analysis predicts the various dimensionless parameters which are helpful in correlating experimental data. Certain dimensions must be established as fundamental, with all others expressible in terms of these. One of these fundamental dimensions is length, symbolized L. Thus, area and volume may dimensionally be expressed as L2 and L3, respectively. A second fundamental dimension is time, symbolized t. Velocity and acceleration may be expressed as L/t and Lit1, respectively. Another fundamental dimension is mass, symbolized M. The mole is included in M. An example of a quantity whose dimensional expression involves mass is the density (mass or molar), which would be expressed as Mil . 

If the differential equation describing a given situation is known, then dimensional homogeneity requires that each term in the equation have the same units. The ratio of one term in the equation to another must then, of necessity, be dimensionless. With knowledge of the physical meaning of the various terms in the equation we are then able to give some physical interpretation to the dimensionless parameters thus formed. A more general situation in which dimensional analysis may be profitably 

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The Buckingham method is based on the Buckingham Pi Theorem, which states... 

The dimensional matrix is simply the matrix formed by tabulating the exponents of the fundamental dimensions M, L, and t, which appear in each of the variables involved. The rank of a matrix is the number of rows in the largest nonzero determinant which can be formed from it. An example of the evaluation of r and i, as well as the application of the Buckingham method, follows. 

Natural convection currents will develop if there exists a significant variation in density within a liquid or gas phase. The density variations may be due to temperature differences or to relatively large concentration differences. Consider natural convection involving mass transfer from a vertical plane wall to an adjacent fluid. Use the Buckingham method to determine the dimensionless groups formed from the variables significant to this problem. 

Table 2.1 presents a non-exhaustive list of expressions of the characteristic times corresponding to the most commonly used phenomena involved in chemical reactors. The previous definitions unfortunately do not always enable one to build the expressions presented in Table 2.1. Various methods can be used such as a blind dimensional analysis, similar to the Buckingham method used for dimensionless numbers [7], which can be applied to the list of fundamental physical and chemical properties. Nevertheless, the most relevant method consists in extracting the expressions from a mass/heat/force balance.

The method of obtaining the important dimensionless numbers from the b sic differential equations is generally the preferred method. In many cases, however, we are not able to formulate a differential equation which clearly applies. Then a more general procedure is required, which is known as the Buckingham method. In this method the listing of the important variables in the particular physical problem is done first. Then we determine the number of dimensionless parameters into which the variables may be combined by using the Buckingham pi theorem. 

The same dimensionless functionality is apparent as from the Buckingham pi method. 

From the point of view of the Buckingham formula (Equation (2.23)) only the effect of long-range electrostatic and induction interactions crE of the solvent molecule with the reaction field is included in the traditional methods of the (n) group (continuum models). Contrary to that, the supermolecular approach (I) or combined MD/QM methods (III) includes the short-range term cra and the long-range crw and some of the [Pg.132]

Rizzo reviews in a unitary framework computational methods for the study of linear birefringence in condensed phase. In particular, he focuses on the PCM formulation of the Kerr birefringence, due to an external electric field yields, on the Cotton-Mouton effect, due to a magnetic field, and on the Buckingham effect due to an electric-field-gradient. A parallel analysis is presented for natural optical activity by Pecul Ruud. They present a brief summary of the theory of optical activity and a review of theoretical studies of solvent effects on these properties, which to a large extent has been done using various polarizable dielectric continuum models. 

The intermolecular interactions are usually assumed to be pair-additive functions such as the Lennard-Jones 12-6 or 9-6 potentials or the Buckingham expontential-6 type of potentials and are parameterized using methods similar to those described in the previous paragraph to reproduce the crystallographic structure and the lattice energy. For the case of liquid systems the parameterization of non-bonded interactions can be done to reproduce the liquid densities and the heats of vaporization. 

This theorem provides a method to obtain the dimensionless groups which affect a process. First, it is important to obtain an understanding of the variables that can influence the process. Once you have this set of variables, you can use the Buckingham Pi Theorem. The theorem states that the number of dimensionless groups (designated as n, ) is equal to the number (n) of independent variables minus the number (m) of dimensions. Once you obtain each n, you can then write an expression ... 

Note that this contribution to overall energy does not include other through-space effects such as van der Waals interactions. To take account of these effects, interactions between atoms which are separated from each other by greater than 1,4 distances are usually split into van der Waals and electrostatic components. There are many ways of describing van der Waals interactions the most common methods employ either the 6-12 (Lennard-Jones) potential or the Buckingham potential as shown below ... 

The method of perturbed stationary state (PSS) was applied to the heavy particle collision of symmetrical resonance by Buckingham and Dalgarno, Matsuzawa and Nakamura, and Kolker and Michels. Although the essential idea of our method is the same as in these papers, except for the inclusion of four levels, we include a brief description of the PSS method for later discussion. 

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