Heterogeneous mass transfer equation
A mathematical model is established assuming that the particles are uniform, non-porous and almost spherical (radius r). The introduction of the shape factor φ corrects the increase in surface area due to deviation from spherical symmetry. The particles are believed to shrink with dissolution without shape change, and the surface is free of insoluble product layers. Mass transfer occurs through three mechanisms: diffusion, convection, and electromigration. The mass transfer flux of each mechanism can be considered separately, and then the three flux equations are combined to obtain a net flux. (1) Diffusion The driving force for diffusion is the chemical level gradient. The flux J d caused by diffusion alone can be expressed as B-absolute molar mobility in the formula: C-concentration; Γ-activity coefficient; Δ-δ operator of spherical coordinates; Îœ-chemical position. If the diffusion coefficient D is defined as Substituting it into the above formula yields the first law of diffusion J d =-Dâ–³c (3) Since the data of each substance considered in this system is not sufficient to calculate the D value, it is assumed here that D takes the value of the infinitely dilute solution regardless of the concentration. (2) Convection The convection term in the mass transfer equation is derived from the overall motion of the fluid relative to the solid particles. The flux caused by convection is J c =-vc (4) Where v is the flow rate of the fluid to the solid particles. The theoretical treatment of convective mass transfer of suspended particles in a stirred reactor has not been fully established, and the fluid along the particles may be in turbulent motion, and the relative velocity v is difficult to estimate. However, for particles in the Stokes sedimentation mode (the Reynolds number should be less than 1, and for ZnO particles this corresponds to a maximum particle size of 74 μm), v can be expressed as Where g-gravity acceleration; r t - the particle radius of time t; Ï l , Ï s - the density of the solution and the solid particles; η l - viscosity of the fluid. Drawn from the above two equations (3) Electromigration The presence of an electric field in the boundary layer causes electromigration of all charged substances. According to Ohm's law, the flux of electromigration is J e = ucE (7) Where u-ion mobility, cm 2 /(V·s); E- electric field strength. Ion mobility is closely related to concentration, but as with diffusion coefficients, the ion mobility in this model is also considered constant due to the lack of data for each ion component. The electric field is related to the concentration of the ion component by the Poisson equation: The dielectric constant of the ε-solution in the formula; The number of different ionic components present in the n-solution. I. Comprehensive mass transfer equation If all three mass transfer mechanisms are considered, the net flux is If the chemical reaction occurs only at the solid-liquid interface, it can be obtained from conservation of mass. If it is assumed that D, v and u are not related to concentration, then Second, the mathematical model Calculating the flux of each chemical substance using equation (11) for the established mathematical model can be simplified as follows. (a) quasi-steady approximation Although the solid-liquid interface moves at a finite rate as the solid particles dissolve and shrink, there is no steady state, but since the rate of interfacial movement is small relative to the rate at which the substance passes through the boundary layer, it is assumed that the steady state is reasonable. The sample is removed from equation (11) over time, so that for any substance, equation (11) can be rewritten as Equations (10) and (12) provide partial differential equations for the system. In principle, boundary conditions can be used to solve the change pattern and electric field of the concentration, and then the flux and reaction rate of each substance are calculated. However, these equations cannot be solved analytically, and it is time-consuming to find numerical solutions and difficulties. Assuming that it is electrically neutral in the boundary layer and using the semi-theoretical transformation of Sherwood, the partial differential equation can be transformed into a set of algebraic equations. (2) Electrically neutral determination of surface concentration The electric field gradient ΔE generally does not exceed 10 5 V∕cm. Substituting this value into equation (8) yields a value of ∑Z i C i of approximately 10 -12 equ·cm -3 , which is 7 to 9 orders of magnitude smaller than the ionic strength of the solution. Thus, the effect of charge on the interface between the actual concentration of the substance containing hydrogen and zinc can be ignored. Therefore, the total charge neutrality is assumed when calculating the concentration, ie It should be emphasized that although the effect of charge on the surface concentration of the material is negligible, the approximate electric field gradient causes significant migration and thus cannot be ignored in the mass transfer equation. (3) Uniform electric field Although the electric field E in the boundary layer is not a constant value in practice, it can be assumed to be a constant value for the mathematical convenience, and the uniform value of E is used in the equation (7) to obtain the electromigration flux of the substance i. (four) Sherwood transform The flux caused by diffusion and convection can be calculated by Sherwood semi-theoretical transformation The mass transfer coefficient K i of the substance i is defined as Where Re is the Reynolds number Where, Sc is the Schmidt number Combine these simplifications into equation (9) to get the total flux of each material as Multifunctional Oral Care,Oral Cleaner,Oral Cleaning,Oral Irrigator NINGBO LUBAN ELECTRIC POWER TOOLS CO.,LTD. , http://www.aye-group.com
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