Dimensional analysis groups

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. 

Friction Factor and Reynolds Number For a Newtonian fluid in a smooth pipe, dimensional analysis relates the frictional pressure drop per unit length AP/L to the pipe diameter D, density p, and average velocity V through two dimensionless groups, the Fanning friction factor/and the Reynolds number Re. 

Turbomachines can be compared with each other by dimensional analysis. This analysis produces various types of geometrically similar parameters. Dimensional analysis is a procedure where variables representing a physical situation are reduced into groups, which are dimensionless. These dimensionless groups can then be used to compare performance of various types of machines with each other. Dimensional analysis as used in turbomachines can be employed to (1) compare data from various types of machines—it is a useful technique in the development of blade passages and blade profiles, (2) select various types of units based on maximum efficiency and pressure head required, and (3) predict a prototype s performance from tests conducted on a smaller scale model or at lower speeds. 

Dimensional analysis leads to various dimensionless parameters, wliieli are based on the dimension s mass (M), length (L), and time T). Based on these elements, one ean obtain various independent parameters sueh as density (p), viseosity (/i), speed (A ), diameter ( )), and veloeity (V). The independent parameters lead to forming various dimensionless groups, whieh are used in fluid meehanies of turbomaehines. Reynolds number is the ratio of the inertia forees to the viseous forees... 

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. ... 

Pavlushenko et al. (P4) in their dimensional analysis considered Ks, the volumetric mass transfer coefficient, to be a function of pc, pc, L, Dr, N, Vs, and g. They determined the following relationship for the dimensionless groupings ... 

The requirement of dimensional consistency places a number of constraints on the form of the functional relation between variables in a problem and forms the basis of the technique of dimensional analysis which enables the variables in a problem to be grouped into the form of dimensionless groups. Since the dimensions of the physical quantities may be expressed in terms of a number of fundamentals, usually mass, length, and time, and sometimes temperature and thermal energy, the requirement of dimensional consistency must be satisfied in respect of each of the fundamentals. Dimensional analysis gives no information about the form of the functions, nor does it provide any means of evaluating numerical proportionality constants. 

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. 

A number of important dimensionless groups have been arrived at by dimensional analysis and by other means. The numerical value of such a dimensionless group for a given case is independent of the units chosen for the primary quantities as long as consistent units are used within that group. The units used in one group need not be consistent with those used in another. 

It must be emphasized that dimensional analysis is used to find the minimum number of dimensionless groups of all the variables known to be relevant to the description of a... 

In the Rayleigh method of carrying out a dimensional analysis the dependent variable is assumed to be proportional to the product of the independent variables raised to different powers. By equating dimensions, the number of independent dimensionless groups and one set of their possible forms can be obtained. By way of illustration two examples may be considered. 

Dimensional analysis of the variables characteristic of mass transfer under flow conditions suggests that the following dimensionless groups are appropriate for correlating mass transfer data. 

It is important to realize that the process of dimensional analysis only replaces the set of original (dimensional) variables with an equivalent (smaller) set of dimensionless variables (i.e., the dimensionless groups). It does not tell how these variables are related—the relationship must be determined either theoretically by application of basic scientific principles or empirically by measurements and data analysis. However, dimensional analysis is a very powerful tool in that it can rovide a direct guide for... 

The method we will use to illustrate the dimensional analysis process is one that involves a minimum of manipulations. It does require an initial knowledge of the variables (and parameters) that are important in the system and the dimensions of these variables. The objective of the process is to determine an appropriate set of dimensionless groups of these variables that can then be used in place of the original individual variables for the purpose of describing the behavior of the system. The process will be... 

The procedure for performing a dimensional analysis will be illustrated by means of an example concerning the flow of a liquid through a circular pipe. In this example we will determine an appropriate set of dimensionless groups that can be used to represent the relationship between the flow rate of an incompressible fluid in a pipeline, the properties of the fluid, the dimensions of the pipeline, and the driving force for moving the fluid, as illustrated in Fig. 2-1. The procedure is as follows. 

It should be emphasized that the specific relationship between the variables or groups that is implied in the foregoing discussion is not determined by dimensional analysis. It must be determined from theoretical or experimental analysis. Dimensional analysis gives only an appropriate set of dimensionless groups that can be used as generalized variables in these relationships. However, because of the universal generality of the dimensionless groups, any functional relationship between them that is valid in any system must also be valid in any other similar system. 

We have stated that dimensional analysis results in an appropriate set of groups that can be used to describe the behavior of a system, but it does not tell how these groups are related. In fact, dimensional analysis does not result in any numbers related to the groups (except for exponents on the variables). The relationship between the groups that represents the system behavior must be determined by either theoretical analysis or experimentation. Even when theoretical results are possible, however, it is often necessary to obtain data to evaluate or confirm the adequacy of the theory. Because relationships between dimensionless variables are independent of... 

It should be noted that a dimensional analysis of this problem results in one more dimensionless group than for the Newtonian fluid, because there is one more fluid rheological property (e.g., m and n for the power law fluid, versus fi for the Newtonian fluid). However, the parameter n is itself dimensionless and thus constitutes the additional dimensionless group, even though it is integrated into the Reynolds number as it has been defined. Note also that because n is an empirical parameter and can take on any value, the units in expressions for power law fluids can be complex. Thus, the calculations are simplified if a scientific system of dimensional units is used (e.g., SI or cgs), which avoids the necessity of introducing the conversion factor gc. In fact, the evaluation of most dimensionless groups is usually simplified by the use of such units. 

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