Dimensionless Groups and Their Relationships

Inspecting the remaining variable, we observe that [k d ] = L T i. Then, by multiplying kjd by 1/D we obtain the third pi group  

The formulation of the fourth pi group (II4) takes into account the observation that k and Dp/5 present the same dimensional formula and that their ratio is therefore dimensionless  

The last pi group (II5) can be obtained by multiplying k d (which has L T i dimension) by 1/Dj , which also results in a dimensionless formula  

The last three pi groups are well known in chemical engineering (113 is recognized as the Fourier reaction number (Fo ), 114 is the famous Biot diffusion number (Bi(j) and Hs is the Sherwood number (Sh)). 

Relationship (6.98) shows the last result of this particularized case of dimensional analysis. 

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Common Dimensionless Groups and Their Relationships 507 for example, the Fourier heat-conduction equation qi = —XVt will be replaced by ... 

It is, therefore, considered more appropriate in this book to use the chemical engineering approach of dimensionless groups and their semi-empirical relationships, as given in section 6.6.6. 

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 ... 

Every physical relationship between n physical quantities can be reduced to a relationship between m = n - r mutually independent dimensionless groups, whereby r stands for the rank of the dimensional matrix, made up of the physical quantities in question and generally equal to (or in some few cases smaller than) the number of the base quantities contained in their secondary dimensions. 

Governing equations are the continuity equation, the chemical reactions and their thermodynamic relationships, and the heat, mass, and momentum equations. Elastic behavior of an expanding bed of particles sometimes must be included. These equations can be many and complex because we are dealing with both multiphase and multicomponent systems. Correlations are often in terms of phase-based dimensionless groups such as Reynolds numbers, Froude numbers, and Weber numbers. 

Many concentrated suspensions follow non-Newtonian behavior and considerations ought to be made as the viscosity is not a constant anymore, and such fluids should be characterized properly to use their parameters of characterization instead of viscosity in calculations using some dimensionless groups. Most of the theory developed for non-Newtonian fluid flow through tubes, apply to laminar flow round smooth tubes. The most studied non-Newtonian fluids are the power-law fluids and there are some relationships available for pressure-loss-flow rate in purely viscous and viscoelastics flows principally. Relations for thixotropic and rheopectic systems appear not to be available. 

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