Dimensionless groups equations

The prediction of such a curve for a specific system has defied mathematical analysis, except in the region of low inertial effects. Recourse must be made to the time-honored method of graphical empirical correlation. Klee and Treybal (K3) arrived at two equations, one for the region below the peak diameter and one for that above the peak. They warn that these are simply good approximations, and that their dimensionless-group equations describe their data and those of other authors fairly accurately. The field liquid in all systems used was one of low viscosity. 

Simulations of copper electrodeposition in sub-micron features in the presence of a leveling agent indicate that the formation of void-free deposits requires tight control of the operating conditions. For very small features, primarily one dimensionless group (equation 7) dictates the leveling capability of a process. Results also indicate that as feature size is reduced, the deposition tends to become conformal unless the additive chemistry is modified. It is proposed that conformal deposit is not desirable because random variations in deposition rate will lead to void formation in a statistically significant number of features on a wafer. 

Using the above dimensionless groups. Equation 8.8 was simplified to yield... 

The differential material balances contain a large number of physical parameters describing the structure of the porous medium, the physical properties of the gaseous mixture diffusing through it, the kinetics of the chemical reaction and the composition and pressure of the reactant mixture outside the pellet. In such circumstances it Is always valuable to assemble the physical parameters into a smaller number of Independent dimensionless groups, and this Is best done by writing the balance equations themselves in dimensionless form. The relevant equations are (11.20), (11.21), (11.22), (11.23), (11.16) and the expression (11.27) for the effectiveness factor. 

In section 11.4 Che steady state material balance equations were cast in dimensionless form, therary itancifying a set of independent dimensionless groups which determine ice steady state behavior of the pellet. The same procedure can be applied to the dynamical equations and we will illustrate it by considering the case t f the reaction A - nB at the limit of bulk diffusion control and high permeability, as described by equations (12.29)-(12.31). 

Three basic approaches have been used to solve the equations of motion. For relatively simple configurations, direct solution is possible. For complex configurations, numerical methods can be employed. For many practical situations, particularly three-dimensional or one-of-a-kind configurations, scale modeling is employed and the results are interpreted in terms of dimensionless groups. This section outlines the procedures employed and the limitations of these approaches (see Computer-aided engineering (CAE)). 

Simila.rityAna.Iysis, Similarity analysis starts from the equation describing a system and proceeds by expressing all of the dimensional variables and boundary conditions in the equation in reduced or normalized form. Velocities, for example, are expressed in terms of some reference velocity in the system, eg, the average velocity. When the equation is rewritten in this manner certain dimensionless groupings of the reference variables appear as coefficients, and the dimensional variables are replaced by their normalized relatives. If another physical system can be described by the same equation with the same numerical values of the coefficients, then the solutions to the two equations (normalized variables) are identical and either system is an accurate model of the other. 

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. 

It has been found that these dimensionless groups may be correlated well by an equation of the type... 

The dimensionless relations are usually indicated in either of two forms, each yielding identical resiilts. The preferred form is that suggested by Colburn ran.s. Am. In.st. Chem. Eng., 29, 174—210 (1933)]. It relates, primarily, three dimensionless groups the Stanton number h/cQ, the Prandtl number c Jk, and the Reynolds number DG/[L. For more accurate correlation of data (at Reynolds number <10,000), two additional dimensionless groups are used ratio of length to diameter L/D and ratio of viscosity at wall (or surface) temperature to viscosity at bulk temperature. Colburn showed that the product of the Stanton number and the two-thirds power of the Prandtl number (and, in addition, power functions of L/D and for Reynolds number <10,000) is approximately equal to half of the Fanning friction fac tor//2. This produc t is called the Colburn j factor. Since the Colburn type of equation relates heat transfer and fluid friction, it has greater utility than other expressions for the heat-transfer coefficient. 

Dukler Theory The preceding expressions for condensation are based on the classical Nusselt theoiy. It is generally known and conceded that the film coefficients for steam and organic vapors calculated by the Nusselt theory are conservatively low. Dukler [Chem. Eng. Prog., 55, 62 (1959)] developed equations for velocity and temperature distribution in thin films on vertical walls based on expressions of Deissler (NACA Tech. Notes 2129, 1950 2138, 1952 3145, 1959) for the eddy viscosity and thermal conductivity near the solid boundaiy. According to the Dukler theoiy, three fixed factors must be known to estabhsh the value of the average film coefficient the terminal Reynolds number, the Prandtl number of the condensed phase, and a dimensionless group defined as follows ... 

Equation 46 is a general expression that may be applied to the treatment of experimental data to evaluate exponent a. This, however, is a cumbersome approach that can be avoided by rewriting the equation in dimensionless form. Equation 42 shows that there are n = 5 dimensional values, and the number of values with independent measures is m = 3 (m, kg, sec.). Hence, the number of dimensionless groups according to the ir-theorem is tc = 5 - 3 = 2. As the particle moves through the fluid, one of the dimensionless complexes is obviously the Reynolds number Re = w Upl/i. Thus, we may write ... 

Coefficient A and exponent a can be evaluated readily from data on Re and T. The dimensionless groups are presented on a single plot in Figure 15. The plot of the function = f (Re) is constructed from three separate sections. These sections of the curve correspond to the three regimes of flow. The laminar regime is expressed by a section of straight line having a slope P = 135 with respect to the x-axis. This section corresponds to the critical Reynolds number, Re < 0.2. This means that the exponent a in equation 53 is equal to 1. At this a value, the continuous-phase density term, p, in equation 46 vanishes. 

There are nine variables and three primary dimensions, and therefore by Buekingham s theorem. Equation 7-1 ean be expressed by (9-3) dimensionless groups. Employing dimensional analysis. Equation 7-1 in terms of the three basie dimensions (mass M, length L, and time T) yields Power = ML T. ... 

The objectives are not realized when physical modeling are applied to complex processes. However, consideration of the appropriate differential equations at steady state for the conservation of mass, momentum, and thermal energy has resulted in various dimensionless groups. These groups must be equal for both the model and the prototype for complete similarity to exist on scale-up. 

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.

A differential equation deseribing the material balanee around a seetion of the system was first derived, and the equation was made dimensionless by appropriate substitutions. Seale-up eriteria were then established by evaluating the dimensionless groups. A mathematieal model was further developed based on the kineties of the reaetion, deseribing the effeet of the proeess variables on the eonversion, yield, and eatalyst aetivity. Kinetie parameters were determined by means of both analogue and digital eomputers. 

Solutions to Fourier s equation are in the form of infinite series but are often more conveniently expressed in graphical form. In the solution the following dimensionless groups are used. 

Referenced to 806 data points for binary systems. Equation 8-70A gives absolute deviation of 13.2%, which is about as accurate, or perhaps more so, than other efficiency equations. Equation 8-70B uses the same data and has an absolute average deviation of 10.6%. See Example 8-13 for identification of dimensionless groups. 

In 1953, Rushton proposed a dimensionless number that is used for scale-up calculation. The dimensionless group is proportional to NRe as shown by the following equation 2,3... 

Equation (49) can be cast into the following dimensionless groupings ... 

Comparing equations 1.6 and 1.10, it is seen that a relationship between six variables has been reduced to a relationship between three dimensionless groups. In subsequent sections of this chapter, this statement will be generalised to show that the number of dimensionless groups is normally the number of variables less the number of fundamentals. 

The need for dimensional consistency imposes a restraint in respect of each of the fundamentals involved in the dimensions of the variables. This is apparent from the previous discussion in which a series of simultaneous equations was solved, one equation for each of the fundamentals. A generalisation of this statement is provided in Buckingham s n theorem(4) which states that the number of dimensionless groups is equal to the number of variables minus the number of fundamental dimensions. In mathematical terms, this can be expressed as follows ... 

By making use of this theorem it is possible to obtain the dimensionless groups more simply than by solving the simultaneous equations for the indices. Furthermore, the functional relationship can often be obtained in a form which is of more immediate use. 

It is important to recognise the differences between scalar quantities which have a magnitude but no direction, and vector quantities which have both magnitude and direction. Most length terms are vectors in the Cartesian system and may have components in the X, Y and Z directions which may be expressed as Lx, Ly and Lz. There must be dimensional consistency in all equations and relationships between physical quantities, and there is therefore the possibility of using all three length dimensions as fundamentals in dimensional analysis. This means that the number of dimensionless groups which are formed will be less. 

On the assumption that the power required for mixing in a stirred tank is a function of the variables given in equation 7.12, obtain the dimensionless groups which are important in calculating power requirements for geometrically similar arrangements. 

With nine parameters and five dimensions, equation 9.55 may be rearranged in four dimensionless groups. 

This general equation involves the use of four dimensionless groups, although it may frequently be simplified for design purposes. In equation 9.56 ... 

Equations (3.11) and (3.12) show that the friction factor of a rectangular micro-channel is determined by two dimensionless groups (1) the Reynolds number that is defined by channel depth, and (2) the channel aspect ratio. It is essential that the introduction of a hydraulic diameter as the characteristic length scale does not allow for the reduction of the number of dimensionless groups to one. We obtain... 

In these equations, vz is the average axial velocity at the die entrance, and r i is the radius of the die hole at the entrance. The other dimensionless groups are ... 

Using the dimensionless groups defined in notation, Equations 1-3 are transformed into. 

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