Concentration polarization equation

When the mass-transport process cannot meet the demand for reactant, the IK drop in Equation 22-16 becomes smaller than the theoretical value, and a diffusion overvoltage appears that just offsets the decrease in H. Here, we consider an electrolytic cell to which we apply a negative voltage to produce a reduction at the cathode. We assume that the anode is nonpolarized, Thus, with the appearance of concentration polarization, Equation 22-16 becomes... 

Another equation, which is also widely used to obtain permeate flux in ultrafiltration, is the concentration polarization equation (Blatt, 1976) ... 

Reviews of concentration polarization have been reported (14,38,39). Because solute wall concentration may not be experimentally measurable, models relating solute and solvent fluxes to hydrodynamic parameters are needed for system design. The Navier-Stokes diffusion—convection equation has been numerically solved to calculate wall concentration, and thus the water flux and permeate quaUty (40). 

As the Nemst equation suggests, concentration variations in the electrolyte lead to potential differences between electrodes of the same kind. These potential differences are concentration polarizations or concentration overpotentials. Concentration polarizations can also affect the current distribution. Predicting these is considerably more difficult. If concentration gradients exist, equations 25 and 27 through 29 must generally be solved simultaneously. 

Electrode reactions are heterogeneous since they occur at interfaces between dissimilar phases. During current flow the surface concentrations Cg j of the substances involved in the reaction change relative to the initial (bulk) concentrations Cy p Hence, the value of the equilibrium potential is defined by the Nemst equation changes, and a special type of polarization arises where the shift of electrode potential is due to a change in equilibrium potential of the electrode. The surface concentrations that are established are determined by the balance between electrode reaction rates and the supply or elimination of each substance by diffusion [Eq. (4.9)]. Hence, this type of polarization, is called diffusional concentration polarization or simply concentration polarization. (Here we must take into account that another type of concentration polarization exists which is not tied to diffusion processes see Section 13.5.)... 

The polarization equation describes polarization as a fnnction of current density. In the case of concentration polarization, the form of the polarization eqnation is nnre-lated to the natnre of reaction or electrode. In the case of activation polarization, the parameters of the polarization eqnations depend decisively on the natnre of the reaction. At identical values of current density and otherwise identical conditions, the values of polarization for different reactions will vary within wide limits, from less than 1 mV to more than 2 or 3 V. However, these equations still have common features. A relatively simple set of equations is obtained for simple redox reactions of the type... 

Equations (6.9) and (6.10), which contain the rate constants, the electrode potential, and the concentrations, are equivalent to Eqs. (6.12) and (6.13), which contain the exchange CD and the electrode s polarization. But in the second set of equations the concentrations do not appear explicitly they enter the equations through the values of exchange CD and equilibrium potential. By convention, equations of the former type will be called kinetic equations, and those of the latter type will be called polarization equations. 

Polarization equations are convenient when (1) the measurements are made in solutions of a particular constant composition, and (2) the equilibrium potential is established at the electrode, and the polarization curve can be measured both at high and low values of polarization. The kinetic equations are more appropriate in other cases, when the equilibrium potential is not established (e.g., for noninvertible reactions, or when the concentration of one of the components is zero), and also when the influence of component concentrations on reaction kinetics is of interest. 

Equations for concentration polarization are often used in conjunction with the diffusion and kinetic equations hence, it will be more convenient here to use concentrations rather than activities.)... 

The kinetic and polarization equations described in Sections 6.1 and 6.2 have been derived under the assumption that the component concentrations do not change during the reaction. Therefore, the current density appearing in these equations is the kinetic current density 4. Similarly, the current density appearing in the equations of Section 6.3 is the diffusion current density When the two types of polarization are effective simultaneously, the real current density i (Fig. 6.6, curve 3) will be smaller than current densities and ij (Fig. 6.6, curves 1 and 2) for a given value of polarization. 

The kinetic equations describing the joint effects of activation and concentration polarization are very complex and we shall consider only the the case of a simple first-order reaction of the type (6.2) proceeding in the presence in the solntion of an excess of a foreign electrolyte. To simplify the appearance of these equations (which even in this case are very cnmbersome), in this section we use a more compact notation that contains two new kinetic parameters ... 

In the case considered in this section of a joint action of concentration and activation polarization, in the polarization equation (6.10) we must take into account the concentration changes of the rectants near the electrode surface ... 

Consider the case when the equilibrium concentration of substance Red, and hence its limiting CD due to diffusion from the bulk solution, is low. In this case the reactant species Red can be supplied to the reaction zone only as a result of the chemical step. When the electrochemical step is sufficiently fast and activation polarization is low, the overall behavior of the reaction will be determined precisely by the special features of the chemical step concentration polarization will be observed for the reaction at the electrode, not because of slow diffusion of the substance but because of a slow chemical step. We shall assume that the concentrations of substance A and of the reaction components are high enough so that they will remain practically unchanged when the chemical reaction proceeds. We shall assume, moreover, that reaction (13.37) follows first-order kinetics with respect to Red and A. We shall write Cg for the equilibrium (bulk) concentration of substance Red, and we shall write Cg and c for the surface concentration and the instantaneous concentration (to simplify the equations, we shall not use the subscript red ). 

Polarization equations of the type (14.35) or (14.38) contain the mean values of true current density. However, the rate-determining step is more often concentrated at just a few segments of the electrode the true working area changes continuously and an exact determination of this area is practically impossible. This gives rise to difficulties in an interpretation of polarization data. 

The shape of polarization curves for metals with low polarizability depends primarily on concentration polarization. In the case of highly polarizable metals, where activation polarization can be measured sufficiently accurately, the polarization curve can usually be described by an equation of the type (6.3) (i.e., by a Tafel equation). For metals forming polyvalent ions, slope b in this equation often has values between 30 and 60 mV. 

When only taking into account the concentration polarization in the pores (disregarding ohmic potential gradients), we must use an equation of the type (18.15). Solving this equation for a first-order reaction = nFhjtj leads to equations exactly like (18.18) for the distribution of the process inside the electrode, and like (18.20) for the total current. The rate of attenuation depends on the characteristic length of the diffusion process ... 

Concentration Polarization The equations governing cross-flow mass transfer are developed in the section describing ultrafiltration. 

In the practice of electrolysis one mostly deals with altering and even exhausting redox concentrations at the electrode interface, so-called concentration polarization this has been considered already on pp. 100-102 for exhaustion counteracted by mere diffusion. The equations given for partial and full exhaustion (eqns. 3.3 and 3.4) can be extended to the current densities ... 

Upon substituting Equations (2-42) in (2-45), the concentration polarization is given by the equation... 

Concentration in the permeate is expressed by that in the feed as Equation (5), using solution-diffusion model assuming no concentration polarization. 

Equation (4) states that the linear deposition rate vj is a diffusion controlled boundary layer effect. The quantity Ac is the difference in foulant concentration between the film and that in the bulk flow and c is an appropriate average concentration across the diffusion layer. The last term approximately characterizes the "concentration polarization" effect for a developing concentration boundary layer in either a laminar or turbulent pipe or channel flow. Here, Vq is the permeate flux through the unfouled membrane, 6 the foulant concentration boundary layer thickness and D the diffusion coefficient. 

Assuming that the concentration polarization, pressure drop and back mixing are negligible, module model can be expressed simply in terms of the transport equation and the material balance of each component. 

Both activation and concentration polarization typically occur at the same electrode, although activation polarization is predominant at low reaction rates (small cnrrent densities) and concentration polarization controls at higher reaction rates (see Fignre 3.10). The combined effect of activation and concentration polarization on the cnrrent density can be obtained by adding the contribntions from each [Eqs. (3.26) and (3.28)], with appropriate signs for a redaction process only to obtain the Butler-Volmer equation ... 

The structure of the present subsection is as follows. First, the governing b.v.p. is formulated and reduced, following the scheme of 4.2, to a system of algebraic equations. Then, two important limit cases are discussed counterion selectivity near equilibrium and selectivity at high concentration polarization. Finally, we present and discuss the results of a numerical solution of the above algebraic system for the intermediate range of deviations from equilibrium. 

Equation 8.7 [6] was obtained to correlate the experimental data on membrane plasmapheresis, which is the MF of blood to separate the blood cells from the plasma. The filtrate flux is affected by the blood velocity along the membrane. Since, in plasmapheresis, all of the protein molecules and other solutes will pass into the filtrate, the concentration polarization of protein molecules is inconceivable. In fact, the hydraulic pressure difference in plasmapheresis is smaller than that in the UF of plasma. Thus, the concentration polarization of red blood cells was assumed in deriving Equation 8.7. The shape of the red blood cell is approximately discoid, with a concave area at the central portion, the cells being approximately 1-2.5 pm thick and 7-8.5 pm in diameter. Thus, a value of r (= 0.000257 cm), the radius of the sphere with a volume equal to that of a red blood cell, was used in Equation 8.7. 

Concentration polarization occurs when the concentrations of reactants or products are not the same at the surface of the electrode as they are in bulk solution. For Reaction 17-1, the Nemst equation should be written... 

The discussion of concentration polarization so far has centred on the depletion of electroactive material on the electrolyte side of the interface. If the metal deposition and dissolution processes involve metastable active surface atoms, then the rate of formation or disappearance of these may be the critical factor in the overall electrode kinetics. Equation (2.69) can be rewritten for crystallization overvoltage as... 

The right hand side of Equation (A3.18) shows two terms, the first representing the anodic concentration polarization and the second the cathodic one ... 

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