Dimensionless diffusion flux

For brevity, throughout the book (where it cannot lead to a misunderstanding) we use the terms concentration and diffusion flux instead of dimensionless concentration and dimensionless diffusion flux. ... 

By differentiating this expression with respect to z followed by setting z = 0, we obtain the dimensionless diffusion flux to the disk surface ... 

For each w satisfying condition (4.4.27), the leading terms of the asymptotic expansions for Eqs. (4.4.26) and (4.4.28) with the same boundary conditions coincide in the inner and outer regions. Therefore, as Pe - 0, in the diffusion equation one can replace the actual fluid velocity field v by w. This fact allows one to use the results presented later on in Section 4.11. Namely, as w we take the velocity field for the potential flow of ideal fluid past the cylinder. This approximation yields an error of the order of Pe in the inner expansion. By retaining only the leading terms in (4.11.15), we obtain the dimensionless diffusion flux at small Peclet numbers in the form... 

By differentiating (5.5.3) with respect to x and by setting x = 0, we find the expression for the dimensionless diffusion flux of the substance through the interface ... 

Convective mass and heat transfer to a plate in a longitudinal flow of a non-Newtonian fluid was considered in [443]. By solving the corresponding problem in the diffusion boundary layer approximation (at high Peclet numbers), we arrive at the following expression for the dimensionless diffusion flux ... 

The equation for the time development of macroscopic concentrations formally coincides with the law of mass action but with dimensionless reaction rate K(t) = K(t)/ AnDr ) which is, generally speaking, time-dependent and defined by the flux of the dissimilar particles via the recombination sphere of the radius tq, equation (5.1.51). Using dimensionless units n(t) = 4nrln(t), r = t/tq, t = Dt/r, and the condition of the reflection of similar particles upon collisions, equation (5.1.40) (zero flux through origin), we obtain for the joint correlation functions the equations (6.3.2), (6.3.3). Note that we use the dimensionless diffusion coefficients, a = 2k, IDb = 2(1 — k), k = Da/ Da + Dq) entering equation (6.3.2). 

Here, (ux) is the average velocity in the pipe, a the radius, and Dm the molecular diffusivity of the species. [Remark In writing the boundary condition at x — 0, we have assumed that L/a S> 1. If this is not the case, it can be modified to include the diffusive flux. Also, when L/a is of order unity, we need boundary conditions at the inlet (x — 0) as well as exit (x — L).] Defining dimensionless variables... 

To normalize the governing equations, we introduce a dimensionless position, z = x/a, and two dimensionless dependent variables,/ =/// and u = ua/DD. Note that the normalized velocity m is equivalent to a local Peclet number, indicating the relative magnitudes of the advective and diffusive fluxes of the reactive species. Applying these definitions to the transport equations yields the dimensionless governing equations... 

Here Pes is the interface Pecletnumber defined in terms ofthe interface diffusivity Ds. The diffusive flux on the right-hand side is equal to the net rate of adsorption (minus desorption). Because we have adopted Langmuir kinetics according to (2-150), the dimensionless form of the balance between diffusion to the interface and adsorption-desorption takes the form... 

In a similar way, by substituting the first term of the expansion (3.2.5) into (3.2.9), we obtain the dimensionless local diffusion flux... 

The dimensionless total diffusion flux interval from 0 to a is equal to... 

All facts established for a fluid film remain valid for a majority of problems on the diffusion boundary layer. Namely, near a gas-fluid or fluid-fluid interface, the dimensionless thickness of the layer is proportional to Pe-1 2 (for the diffusion flux we have j Pe1 2), and near the fluid-solid interface the thickness of the boundary layer is proportional to Pe-1 3 (the diffusion flux is j Pe1 3). 

For a particle of arbitrary shape in a translational flow, the first three terms of the asymptotic expansion of the dimensionless total diffusion flux as Pe — 0 have the form [62]... 

Mass transfer between a circular cylinder of radius a and a simple shear flow (G 2 = 1, the other Gkm = 0) was studied in [132]. For the dimensionless total diffusion flux per unit length of the cylinder, the following expression was obtained as Pe 0 ... 

Let us calculate the dimensionless local diffusion flux to the surface of the drop ... 

Formulas for calculating diffusion fluxes. The dimensionless total diffusion flux to a part of the surface of a particle (drop or bubble) between neighboring stagnation lines (points) rjk and rjk+i can be calculated by the formula [166]... 

The dimensionless total diffusion flux corresponding to a potential flow past an ellipsoidal bubble (Re = oo) is calculated at high Peclet numbers by the formula [174]... 

In the diffusion boundary layer approximation with allowance for the corrections (with respect to the Reynolds number) to the potential flow past the bubble, one can obtain the following two-term expansion of the dimensionless total flux I ... 

Diffusion flux. To approximate the dimensionless local diffusion flux j = (dc/dy)y=o on the disk surface, it is convenient to use the cubic equation... 

One can see from (5.2.5) that the diffusion flux decreases with the increase of the exponent n and increases with the decrease of the dimensionless reaction rate constant fev-... 

For nth-order volume reactions with n- 1/2, 1, 2, a numerical solution of problem (5.2.1), (5.2.2) was obtained in [357]. The maximum error of the cubic equation for the diffusion flux (5.2.3) in the entire range of the dimensionless reaction constant kv does not exceed 3%. 

The distribution of the dimensionless local diffusion flux along the plate, j = -(dc/dy)y=Q, can be approximately found by solving the cubic equation... 

Dimensionless diffusion deposition parameter or mass flux away from reaction zone... 

In conclusion, it is worthwhile to cite two additional dimensionless parameters describing process of heat and mass transfer at convective flow of a medium. Determine first the heat-transfer coefEdent a as ratio of thermal flux to the temperature difference, that is q = oAT, and then the mass-transfer coefficient as the ratio of the diffusion flux to difference of concentrations j = AC. Then one can introduce two dimensionless Nusselt s numbers... 

Putting the expressions for h, Dj and F into the equation (13.82) for diffusion flux of drops of type 2 toward the drop of type 1, we get the following relation for the dimensionless flow ... 

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