Diffusion dimensionless form

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). 

For the same reaction in a pellet of finely porous structure, where Knudsen diffusion controls, the appropriate dynamical equations sre (12.20) and (12.21) if we once more adopt approximations which are a consequence of Che large size of K. These again have a dimensionless form, which may be written... 

Asymptotic Solution Rate equations for the various mass-transfer mechanisms are written in dimensionless form in Table 16-13 in terms of a number of transfer units, N = L/HTU, for particle-scale mass-transfer resistances, a number of reaction units for the reaction kinetics mechanism, and a number of dispersion units, Np, for axial dispersion. For pore and sohd diffusion, q = / // p is a dimensionless radial coordinate, where / p is the radius of the particle, if a particle is bidisperse, then / p can be replaced by the radius of a suoparticle. For prehminary calculations. Fig. 16-13 can be used to estimate N for use with the LDF approximation when more than one resistance is important. 

Solution The problem requires solution of the convective diffusion equation for mass but not for energy. Rewriting Equation (8.71) in dimensionless form gives... 

In this section we establish the equation of the forward scan current potential curve in dimensionless form (equation 1.3), justify the construction of the reverse trace depicted in Figure 1.4, and derive the charge-potential forward and reverse curves, also in dimensionless form. Linear and semi-infinite diffusion is described by means of the one-dimensional first and second Fick s laws applied to the reactant concentrations. This does not imply necessarily that their activity coefficients are unity but merely that they are constant within the diffusion layer. In this case, the activity coefficient is integrated in the diffusion coefficient. The latter is assumed to be the same for A and B (D). 

In order to compare data obtained with otherwise similar chromatographic systems in which only the particle size of the column packing and solute diffusivity may vary, Eq. (21) should be written in dimensionless form. Using an approach taken from chemical engineering, Knox (7, 34) has shown that a corresponding reduced plate height equation is given by Eq. (22)... 

The equations were transformed into dimensionless form and solved by numerical methods. Solutions of the diffusion equations (7 or 13) were obtained by the Crank-Nicholson method (9) while Equation 2 was solved by a forward finite difference scheme. The theoretical breakthrough curves were obtained in terms of the following dimensionless variables... 

The transfer number B in Equations 3 and 4 is Spalding s contribution. It is the driving force for mass transfer in dimensionless form. With diffusion controlling (Equation 3) ... 

As might be expected, the dispersion coefficient for flow in a circular pipe is determined mainly by the Reynolds number Re. Figure 2.20 shows the dispersion coefficient plotted in the dimensionless form (Dl/ucI) versus the Reynolds number Re — pud/p(2Ai). In the turbulent region, the dispersion coefficient is affected also by the wall roughness while, in the laminar region, where molecular diffusion plays a part, particularly in the radial direction, the dispersion coefficient is dependent on the Schmidt number Sc(fi/pD), where D is the molecular diffusion coefficient. For the laminar flow region where the Taylor-Aris theory18,9,, 0) (Section 2.3.1) applies ... 

ZD is the convective diffusion impedance with its dimensionless form -1/0J(O) [41]. 

This is written out in order to make the next point. The first and last equation in the set are superfluous, because the boundary concentrations Co and C/y I are not subject to diffusion changes, but to other conditions. Also, where the boundary values appear in the other equations, they must be replaced with what we can substitute for them. The outer boundary value, C/y I, is (almost always) equal to the initial bulk concentration C, usually equal to unity in its dimensionless form. This means that the last term in each equation separates out as a constant term and makes for a constant vector [Hgw+iC II 2,n+iC. .. H jv+iC ]7, which will be called Z here. The concentration at the electrode Co is handled according to the boundary condition. For Cottrell, for example, it is set to zero throughout and thus simply drops out of the set. For other conditions, for example constant current or an irreversible reaction, a gradient C is involved, as described in Chap. 6. In that chapter, the gradient was expressed as a possibly multipoint approximation,... 

If u0 is the average velocity in a system where both molecular diffusion and convective diffusion are taking place, L is a characteristic length, and c0 is a representative concentration, then Eq. 10.16 can be put in dimensionless form by making the following substitutions Ux = uju0, X = x/L, C = c/c0, etc. Equation 10.16 becomes... 

Let us first outline the theoretical background of the evaluation of both the charge and potential of two interacting diffuse electric layers. It is well known that the charge and potential distribution in the diffuse layer can be represented with a sufficient degree of accuracy using the Poisson-Boltzman (PB) approximation [e.g. 246]. For a planar film from aqueous symmetrical electrical electrolyte of valence z, the respective equation can be written in dimensionless form as... 

In the limiting case of mass transfer from a single sphere resting in an infinite stagnant liquid, a simple film-theory analysis122 indicates that the liquid-solid mass-transfer coefficient R s is equal to 2D/JV, where D is the molecular diffusivity of the solute in the liquid phase and d is the particle diameter. In dimensionless form, the Sherwood number... 

Undeniably, the speed vector, by its size and directional character, masks the effect of small displacements of the particle. Another difference comes from the different definition of the diffusion coefficient, which, in the case of the property transport, is attached to a concentration gradient of the property it means that there is a difference in speed between the mobile species of the medium. A second difference comes from the dimensional point of view because the property concentration is dimensional. When both equations are used in the investigation of a process, it is absolutely necessary to transform them into dimensionless forms [4.6, 4.7, 4.37, 4.44]. 

In Fig. lD(a) the concentration profile is presented in dimensionless form, as the relative concentration, C/C° as a function of the dimensionless distance, xl(4Dt). This is an elegant way of representing the results calculated from a function that depends on several parameters. Thus, this curve is independent of the initial concentration in solution, of the diffusion coefficient of (he reacting species, and of time. A great deal of information is condensed into a single curve, which can then be used to calculate values of the concentration as a function of distance and time for any specific system. 

But how far is one unit of dimensionless distance in real terms That depends on the diffusion coefficient and on time. After 1 ms it is about 1.6 im from the electrode surface, but after 1 s it corresponds to 50 pm. Figure lD(b) shows concentration profiles calculated for the same system, but plotted versus the distance in centimeters Each curve corresponds to a different time after application of the potential pulse, and the evolution of the concentration profile with time is presented. All the information contained in the curves in Fig. lD(b) exists, of course, in the single curve shown in Fig. lD(a). This is both the advantage and the disadvantage of presenting the data in dimensionless form ... 

Solution of the transport problem when the process is controlled by both the interfacial electron transfer and the steady-state or linear diffusion of reactants was derived by Samec [181, 182]. These results represent the basis for the kinetic analysis, e.g., in dc polarography or convolution and potential sweep voltammetry. Under the conditions of steady-state diffusion, Eq. (60) can be transformed into a dimensionless form [181],... 

Write down in dimensionless form the material balance equation for a laminar flow tubular reactor accomplishing a first-order reaction and having both axial and radial diffusion. State the necessary conditions for solution. 

We note that all the equations above are already written in dimensionless form in order to highlight the key parameters characterizing diffusion, reaction, and convection effects. Equations (8)-(14) form a coupled system of ordinary and partial differential equations in the unknowns X , Y , and Z (n = 2, N). Many additional details on the dimensional form and... 

Dimensionless form of equations describing diffusion and reaction... 

The term Zd represents the convective-diffusion impedance with its dimensionless form -1/0.(O) (see Chapter 11). Considering now equations (14.27), (14.28) and (15.47), one obtains the relation between observable quantities corresponding to the general form of equation (14.22), i.e.. 

By examining these characteristic dimensionless numbers, it is possible to appreciate possible interactions of different processes (convection, diffusion, reaction and so on) and to simplify the governing equations accordingly. A typical dimensionless form of the governing equation can be written (for a general variable, (p) ... 

The dimensionless form of the equation contains one dimensionless parameter as a multiplier of the first term of the right-hand side and maybe some additional dimensionless parameters, which may appear within the dimensionless source term, S. Depending on the general variable, 0, the effective diffusion coefficient, F, appearing in this dimensionless number will be different, leading to different dimensionless numbers. For the species mass fraction, momentum and enthalpy transport equations, the effective diffusion coefficient will be molecular diffusion coefficient, the kinematic viscosity of the fluid and the thermal diffusivity of the fluid respectively. The corresponding dimensionless numbers are, therefore, defined as follows. 

The classical formulation of the GRM includes a mass balance equation in each of the two fractions of the mobile phase, axial dispersion, intraparticle diffusion, and the kinetics of adsorption-desorption. In dimensionless form, the following set of equations is written for each component of the system, as follows. 

Consider diffusion with a first order isothermal reaction in a rectangular pellet.[ll] [8]. The governing equation and boundary conditions for concentration in dimensionless form are ... 

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