Mass transfer equation dimensionless

A more recent review by Fahidy (FI) concerns the chemical engineering approach to electrochemical processes, such as fluidized-bed reactors, bipolar particulate reactors, pulsed electrochemical reactors, gas-phase electrochemical reactors, electrocrystallization and electrodissolution, and the enhancement of heat and mass transfer in electric fields. In this review, the author also discusses dimensionless mass-transfer equations applied in cell design. Such equations are reviewed in greater detail in Section VI. 

Dimensionless blend time method, 16 688 Dimensionless groups, 15 685, 686t, 687t Dimensionless mass transfer equation,... 

The strategy for obtaining the dimensionless mass transfer equation is as follows ... 

Divide the entire mass transfer equation by the scahng factor for diffusion (i.e ,mixCAo/i )- This is an arbitrary but convenient choice. Any of the r + 2 dimensional scaling factors can be chosen for this purpose. When the scaling factor for the diffusion term in the dimensional mass transfer equation is divided by ,mixCAo/T, the Laplacian of the molar density contains a coefficient of unity. When the remaining r - -1 scaling factors in the dimensional mass transfer equation are divided by i,mixCAo/T, the dimensionless mass transfer equation is obtained. Most important, r -h 1 dimensionless transport numbers appear in this equation as coefficients of each of the dimensionless mass transfer rate processes, except diffusion. Remember that the same dimensionless number appears as a coefficient for the accumulation and convective mass transfer rate processes on the left-hand side of the equation. 

The product of the Reynolds and Schmidt numbers, which counts as one dimensionless number, is equivalent to the Peclet number for mass transfer, PeMx- The Peclet number represents the ratio of the convective mass transfer rate process to the diffusion rate process of component, and it appears on the left-hand side of the dimensionless mass transfer equation for component i. The remaining r dimensionless transport numbers can be treated simultaneously because they represent ratios of scaling factors for the reactant-product conversion rate due to the jth independent chemical reaction relative to the rate of diffusion of component I. Hence,... 

For unsteady-state diffusion into a quiescent medium with no chemical reaction, the mass transfer Peclet number does not appear in the dimensionless mass transfer equation for species i because it is not appropriate to make variable time t dimensionless via division by L/ v) if there is no bulk fluid flow (i.e., (d) = 0). In this case, the first term on each side of equation (10-11) survives, which corresponds to the unsteady-state diffusion equation. However, the characteristic time for diffusion of species i over a length scale L, given by L /50i,mix. replaces L/ v) to make variable time t dimensionless. Now, the accumulation and diffusional rate processes scale as CAo i.mix/A, with dimensions of moles per volume per time. Since both surviving mass transfer rate processes exhibit the same dimensional scaling factor, there are no dimensionless numbers in the mass transfer equation which describes unsteady-state diffusion for species i in nonreactive systems. 

Consider several overlapping subsets of the dimensionless mass transfer equation from Section 10-2 which correspond to various combinations of convection, diffusion, and chemical reaction that may or may not exhibit transient behavior. 

Solve the dimensionless mass transfer equation (i.e., the mass balance for reactant A) with homogeneous one-dimensional diffusion and zeroth-order irreversible chemical reaction to obtain an expression for 4molar density of reactant A. 

Answer The dimensionless mass transfer equation in rectangular coordinates with one-dimensional diffusion and nth-order chemical reaction represents the starting point for a generic solution to part (a). The dimensionless molar density of reactant A must satisfy... 

A quantitative strategy is discussed herein to design isothermal packed catalytic tubular reactors. The dimensionless mass transfer equation with unsteady-state convection, diffusion, and multiple chemical reactions represents the fundamental starting point to accomplish this task. Previous analysis of mass transfer rate processes indicates that the dimensionless molar density of component i in the mixture I must satisfy (i.e., see equation 10-11) ... 

The important dimensionless parameter that determines the significance of external mass transfer resistance for nth-order irreversible chemical kinetics in packed catalytic tubular reactors was introduced in equation (30-63) as a = iS(CA.iniet)" Simple algebraic manipulation allows one to relate a to the interpellet Damkohler number, the effectiveness factor, the mass transfer Peclet number, and a few other dimensionless parameters. For example, let the coefficient of the chemical reaction term in the dimensionless mass transfer equation be defined as follows ... 

In terms of the coefficient of the chemical reaction term in the dimensionless mass transfer equation, which represents one of the independent parameters that strongly influences the simulation of non-ideal packed catalytic tubular reactors ... 

When symmetric membranes are used or when enzymes are fed to the spongy part of asymmetric membranes, enzyme immobilization results in either a uniform fixation of enzymes throughout the membrane wall, or in the formation of a carrier-enzyme insoluble network in the sponge of the membrane. Mass transfer through this solid phase must therefore be taken into account. A theoretical model neglecting radial convective transport and the dense layer in asymmetric membranes is available in the literature.81 The reacting solution is still assumed to be fed to the core of the hollow fibers. Steady state, laminar flow, and isothermal conditions are assumed. Moreover, the enzymes are assumed to be uniformly distributed and the membrane wall curvature is neglected. Differential dimensionless mass balance equations can be written as follows ... 

The solution of this equation would then follow as described in Section 9.4. The modifications of the mass-transfer equations for the different cases generally follow those used in voltammetric methods as shown in Table 12.2.1. Appropriate dimensionless parameters are listed in Table 12.3.1. It is usually not possible to solve these equations analytically, so various approximations (e.g., the reaction layer approach, as described in Section 1.5.2), digital simulations, or other numerical methods must be employed. The behavior of systems at the RDE can be analyzed by means of the zone diagrams employed for voltammetry (Section 12.3) by redefining the parameter A. This is accomplished by re-... 

Liu and Agarwal [1974] presented their data for the inertia regimes in terms of the dimensionless mass transfer coefficient K and the dimensionless relaxation time T. The equation has the form ... 

DIMENSIONLESS FORM OF THE GENERALIZED MASS TRANSFER EQUATION WITH UNSTEADY-STATE CONVECTION, DIFFUSION, AND CHEMICAL REACTION... 

The objective of this section is to identify the dimensionless transport numbers that appear in the mass transfer equation for component i. Order-of-magnitude estimates of the importance of one mass transfer rate process relative to another... 

DIMENSIONLESS FORM OF THE GENERALIZED MASS TRANSFER EQUATION 267... 

As mentioned above, the dimensionless transport numbers in the mass transfer equation are generated from ratios of dimensional scaling factors. If one divides the scaling factor for convective mass transfer by the scaling factor for diffusion, the result is... 

If one constructs the appropriate dimensionless equation that governs the molar density profile fi for component i, then xj/i depends on all the dimensionless independent variables and parameters in the governing equation and its supporting boundary conditions. Geometry also plays a role in the final expression for in each case via the coordinate system that best exploits the summetry of the macroscopic boundaries, but this effect is not as important as the dependence of on the dimensionless numbers in the mass transfer equation and its boundary conditions. For example, if convection, diffusion, and chemical reaction are important rate processes that must be considered, then the governing equation for transient analysis... 

What important dimensionless number(s) appear in the dimensionless partial differential mass transfer equation for laminar flow through a blood capillary when the important rate processes are axial convection and radial diffusion ... 

Since both of these terms in the mass transfer equation have units of moles per volume per time, it should be obvious that the ratio of diffusion to convection is dimensionless. This suggests that the relative importance of the two terms should be expressed as a dimensionless number for mass transfer. The desired ratio is... 

Effect of Flow Regime on the Dimensionless Mass Transfer Correlation. For creeping flow of an incompressible Newtonian fluid around a stationary solid sphere, the tangential velocity gradient at the interface [i.e., g 9) = sin6>] is independent of (he Reynolds number. This is reasonable because contributions from accumulation and convective momentum transport on the left side of the equation of motion are neglected to obtain creeping flow solutions in the limit where Re 0. Under these conditions. 

Dimensionless Molar Density. The final form of the mass transfer equation for Cp, y, t), which will be used to calculate the concentration profile and boundary layer thickness of species A in the liquid phase, is... 

The corresponding temperatures at the solid-liquid interface and in the bulk fluid are Tinterface and Tbuik, respectively. Since each term in the mass transfer equation is linear with respect to Ca, the concentration driving force (i.e., Ca, equilibrium — CA.buik) does not appear in the partial differential equation for the dimensionless profile ... 

The mass transfer equation for the dimensionless molar density profile of mobile component A is... 

Combination of variables will be successful if the mass transfer equation can be written exclusively in terms of f. For example, if one substitutes the three previous partial derivatives of the dimensionless molar density profile into the mass transfer equation for species A, then the following equation is obtained after multiplication by S ... 

The primary focus of this chapter is to analyze the dimensionless equation of motion in the laminar flow regime and predict the Reynolds number dependence of the tangential velocity gradient at a spherical fluid-solid interface. This information is required to obtain the complete dependence of the dimensionless mass transfer coefficient (i.e., Sherwood number) on the Reynolds and Schmidt numbers. For easy reference, the appropriate correlation for mass transfer around a solid sphere in the laminar flow regime, given by equation (11-120), is included here ... 

The mass transfer equation described in Chapters 9 and 10 was developed from first principles by considering a generic volume element and accounting for all the mass transfer rate processes that contribute to the mass of component i in this element of volume. The mass balance for component i is written in dimensional and dimensionless form as... 

This second-order ordinary differential equation given by (16-4), which represents the mass balance for one-dimensional diffusion and chemical reaction, is very simple to integrate. The reactant molar density is a quadratic function of the spatial coordinate rj. Conceptual difficulty arises for zeroth-order kinetics because it is necessary to introduce a critical dimensionless spatial coordinate, ilcriticai. which has the following physically realistic definition. When jcriticai which is a function of the intrapellet Damkohler number, takes on values between 0 and 1, regions within the central core of the catalyst are inaccessible to reactants because the rate of chemical reaction is much faster than the rate of intrapellet diffusion. The thickness of the dimensionless mass transfer boundary layer for reactant A, measured inward from the external surface of the catalyst,... 

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