Dimensional analysis and dimensionless

Mass transfer coefficients, dimensional analysis, and dimensionless numbers... 

Similarity Hypothesis, Dimensional Analysis, and Dimensionless Numbers... 

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. 

Dimensionless correlations based on dimensional analysis and on a qualitative analysis of the two-phase flow. 

Often, the same expressions from this model are applied to the reductions of pellets, in which cases such structural factors as particle size distribution, porosity and pore shape, its size distribution, etc. should really affect the whole kinetics. Thus, the application of this model to such systems has been criticised as an oversimplification and a more realistic model has been proposed [6—12,136] in which the structure of pellets is explicitly considered to consist of pores and grains and the boundary is admitted to be diffusive due to some partly reduced grains, as shown in Fig. 4. Inevitably, the mathematics becomes very complicated and the matching with experimental results is not straightforward. To cope with this difficulty, Sohn and Szekely [11] employed dimensional analysis and introduced a dimensionless number, a, given by... 

For instance, the processes of motion in the living world are perfectly describable by dimensional analysis and from these correlations valuable information is obtained about a similar process concerning another, larger or older species. The scale-invariance of dimensionless representation is an advantage for the living world, which should not be underestimated The relevant dimensions of length span here over a whole of eight decades. 

This revolution is the application of the methods of asymptotic analysis to problems in transport phenomena. Perhaps more important than the detailed solutions enabled by asymptotic analysis is the emphasis that it places on dimensional analysis and the development of qualitative physical understanding (based upon the m. thematical structure) of the fundamental basis for correlations between dependent and independent dimensionless groups [cf. 7-10]. One major simplification is the essential reduction in detailed geometric considerations, which determine the magnitude of numerical coefficients in these correlations, but not the form of the correlations. Unlike previous advances in theoretical fluid mechanics and transport phenomena, in these developments chemical engineers have played a leading role. 

Dimensionless analysis — Use of dimensionless parameters (-> dimensionless parameters) to characterize the behavior of a system (- Buckinghams n-theorem and dimensional analysis). For example, the chronoampero-metric experiment (-> chronoamperometry) with semiinfinite linear geometry relates flux at x = 0 (fx=o, units moles cm-2 s-1), time (t, units s-1), diffusion coefficient (D, units cm2 s-1), and concentration at x = oo (coo, units moles cm-3). Only one dimensionless parameter can be created from these variables (-> Buckingham s n-theorem and dimensional analysis) and that is fx=o (t/D)1/2/c0C thereby predicting that fx=ot1 2 will be a constant proportional to D1/,2c0O) a conclusion reached without any additional mathematical analysis. Determining that the numerical value of fx=o (f/D) 2/coo is 1/7T1/2 or the concentration profile as a function of x and t does require mathematical analysis [i]. 

In many cases of industrial importance, heat is transferred from one fluid, through a solid wall, to another fluid. The transfer occurs in a heat exchanger. Section 11 introduces several types of heat exchangers, design procedures, overall heat-transfer coefficients, and mean temperature differences. Section 3 introduces dimensional analysis and the dimensionless groups associated with the heat-transfer coefficient. 

As long as one is aware of such limitations, excellent results are obtained with the scaling technique. Another problem comes up, however, when one cannot keep all material properties constant. Perhaps a different metal must be used or a different explosive, or some other property changed. In that case, a different scaling technique must be used, and for that we turn to dimensional analysis and scaling by means of dimensionless groups. 

Dimensional analysis and application of the Buckingham theorem lead to four dimensionless quantities that adequately describe the process Ne, T Re, Fr, and h/d. As before, a relationship of the form... 

Power consumption and blend times in the mixing and agitation of Newtonian and non-Newtonian fluids are not equivalent and, in fact, blend times can be much longer for non-Newtonian fluids when comparing fluids with comparable apparent viscosity values. Through dimensional analysis, the dimensionless blend or mix time, m, is expressed as ... 

So far, we have studied a number of illustrative examples for two-phase laminar heat transfer following the analytical approach we used in Chapter 5. For two-phase turbulent heat transfer we use an approach based on two-length scale dimensional analysis and the correlation of experimental data in terms of dimensionless numbers resulting from this analysis. 

For the single-reaction cases, we performed dimensional analysis and found a dimensionless number, the Thiele modulus, which measures the rate of production divided by the rate of diffusion of some component. A complete analysis of the first-order reaction in a sphere suggested a general approach to calculate the production rate in a pellet in terms of the rate evaluated at the pellet exterior surface conditions. This motivated the definition of the pellet effectiveness factor, which is a function of the Thiele modulus. 

The matrix formulation of dimensional analysis and the availability of free-for-use matrix calculators on the Internet solve the algebraic issues for chemical engineers and provide a rapid method for determining the dimensionless parameters best describing a chemical process. 

Heat and mass transfer coefficients are usually reported as correlations in terms of dimensionless numbers. The exact definition of these dimensionless numbers implies a specific physical system. These numbers are expressed in terms of the characteristic scales. Correlations for mass transfer are conveniently divided into those for fluid-fluid interfaces and those for fluid-solid interfaces. Many of the correlations have the same general form. That is, the Sherwood or Stanton numbers containing the mass transfer coefficient are often expressed as a power function of the Schmidt number, the Reynolds number, and the Grashof number. The formulation of the correlations can be based on dimensional analysis and/or theoretical reasoning. In most cases, however, pure curve fitting of experimental data is used. The correlations are therefore usually problem dependent and can not be used for other systems than the one for which the curve fitting has been performed without validation. A large list of mass transfer correlations with references is presented by Perry [95]. 

Many assumptions underlie the method of dimensional analysis and the use of dimensionless numbers. The most general assumptions are of course not specific... 

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. 

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. 

Damkdhler (1936) studied the above subjects with the help of dimensional analysis. He concluded from the differential equations, describing chemical reactions in a flow system, that four dimensionless numbers can be derived as criteria for similarity. These four and the Reynolds number are needed to characterize reacting flow systems. He realized that scale-up on this basis can only be achieved by giving up complete similarity. The recognition that these basic dimensionless numbers have general and wider applicability came only in the 1960s. The Damkdhler numbers will be used for the basis of discussion of the subject presented here as follows ... 

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. 

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