Charge Transport Equations

The charge transport equations in electrolyte, electrodes, and current collector or bipolar plates are derived based on charge balance. 

Considering current flow owing to potential gradient only and neglecting the diffusion and convection terms, the charge transport equations are given as follows  

The charge transport equation in an electrolyte with solid or stationary immobilized liquid electrolyte can be derived on the basis of charge balance and assuming steady-state diffusion of charge particles based on ohm s law as 

In the anode and cathode electrodes, electrons transfer from the electrodeelectrolyte interfaces to the current collector plate. Considering electronic diffusion and current source in the active region, the charge transport equations in the electrode are given as 

In Equations 7.17a and 7.18a, charged transfer current densities 4 and are given on the basis of Butler-Volmer charge transfer kinetics described in Chapter 5 as follows  

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In the case of redox sites covalently bound to a polymer backbone, when only Dg contributes to charge transport. Equation 2.12 has systematically failed to explain the dependence of D pp with the concentration of redox sites. Blauch and Saveant have shown that for completely immobile centers, charge transport is basically a percolation process random distribution of isolated clusters of electrochemically coimected sites [33,40]. Only by dynamic rearrangements can these clusters become in contact and charge transport occur, giving rise to the concept of bound diffusion where each... 

The charge-transport equation includes the electrochemical kinetics for both anode and cathode catalyst layers. If we assume an infinitely large electric conductivity of the electronic phase, the electrode becomes an equipotential line, such that... 

With the evaluated site coverage and pore blockage correlations for the effective ECA and oxygen diffusivity, respectively, and the intrinsic active area available from the reconstructed CL microstructure, the electrochemistry coupled species and charge transport equations can be solved with different liquid water saturation levels within the 1-D macrohomogeneous modeling framework,25,27 and the cathode overpotential, q can be estimated. 

The interpretation of measured data for Z(oi) is carried out by their comparison with predictions of a theoretical model based either on the (analytical or numerical) integration of coupled charge-transport equations in bulk phases, relations for the interfacial charging and the charge transfer across interfaces, balance equations, etc. Another way of interpretation is to use an -> equivalent circuit, whose choice is mostly heuristic. Then, its parameters are determined from the best fitting of theoretically calculated impedance plots to experimental ones and the results of this analysis are accepted if the deviation is sufficiently small. This analysis is performed for each set of impedance data, Z(co), measured for different values of external parameters of the system bias potentials, bulk concentrations, temperature... The equivalent circuit is considered as appropriate for this system if the parameters of the elements of the circuit show the expected dependencies on the external parameters. 

Analytical Model. The internal heat transfer and electric charge transportation are analyzed using the thermal diffusion and charge transportation equations. The differential equations for the heat conduction at each finite element volume are solved on the basis of energy conservation. The heat and electric charge balances in a control volume are shown in Fig. 2, where q and J denote the heat and current density, respectively. 

In the catalyst layer, the same equations of liquid water (Equation 7-37) and charge transport (Equation 7-40) in the electrolyte phase apply, however, with effective properties, Kp , and to account for a porous nature of the catalyst layer where the electrolyte phase occupies only a portion, of the total volume. 

An important step toward the understanding and theoretical description of microwave conductivity was made between 1989 and 1993, during the doctoral work of G. Schlichthorl, who used silicon wafers in contact with solutions containing different concentrations of ammonium fluoride.9 The analytical formula obtained for potential-dependent, photoin-duced microwave conductivity (PMC) could explain the experimental results. The still puzzling and controversial observation of dammed-up charge carriers in semiconductor surfaces motivated the collaboration with a researcher (L. Elstner) on silicon devices. A sophisticated computation program was used to calculate microwave conductivity from basic transport equations for a Schottky barrier. The experimental curves could be matched and it was confirmed for silicon interfaces that the analytically derived formulas for potential-dependent microwave conductivity were identical with the numerically derived nonsimplified functions within 10%.10... 

Figure 13. Numerically calculated PMC potential curves from transport equations (14)—(17) without simplifications for different interfacial reaction rate constants for minority carriers (holes in n-type semiconductor) (a) PMC peak in depletion region. Bulk lifetime 10" s, combined interfacial rate constants (sr = sr + kr) inserted in drawing. Dark points, calculation from analytical formula (18). (b) PMC peak in accumulation region. Bulk lifetime 10 5s. The combined interfacial charge-transfer and recombination rate ranges from 10 (1), 100 (2), 103 (3), 3 x 103 (4), 104 (5), 3 x 104 (6) to 106 (7) cm s"1. The flatband potential is indicated.

This expression applies to the transport of any conserved quantity Q, e.g., mass, energy, momentum, or charge. The rate of transport of Q per unit area normal to the direction of transport is called the flux of Q. This transport equation can be applied on a microscopic or molecular scale to a stationary medium or a fluid in laminar flow, in which the mechanism for the transport of Q is the intermolecular forces of attraction between molecules or groups of molecules. It also applies to fluids in turbulent flow, on a turbulent convective scale, in which the mechanism for transport is the result of the motion of turbulent eddies in the fluid that move in three directions and carry Q with them. 

Assumption 5. Transfer processes as within the cell have been regarded as quasistationary. The typical time of the processes in the electrode (time of a charging or discharging being Ur-I04 s) is longer than the time of the transitional diffusion process in the elementary cell tc Rc2/D 10"1 s (radius of the cell is Rc 10"5 m, diffusion coefficient of dissolved reagents is D 10"9 m2/s). Therefore, the quasistationary concentration distribution is quickly stabilized in the cell. It is possible to neglect the time derivatives in the transport equations. 

Numerical solution of Chazelviel s equations is hampered by the enormous variation in characteristic lengths, from the cell size (about one cm) to the charge region (100 pm in the binary solution experiments with cell potentials of several volts), to the double layer (100 mn). Bazant treated the full dynamic problem, rather than a static concentration profile, and found a wave solution for transport in the bulk solution [42], The ion-transport equations are taken together with Poisson s equation. The result is a singular perturbative problem with the small parameter A. 

The zero-order solution reproduces the transport equations for an electroneutral solution. At length scales where A is close to unity, space charges become significant, and the transport equations can be expanded in powers of A. The concentration and... 

A fundamental fuel cell model consists of five principles of conservation mass, momentum, species, charge, and thermal energy. These transport equations are then coupled with electrochemical processes through source terms to describe reaction kinetics and electro-osmotic drag in the polymer electrolyte. Such convection—diffusion—source equations can be summarized in the following general form... 

Much effort has been expended in the last 5 years upon development of numerical models with increasingly less restrictive assumptions and more physical complexities. Current development in PEFC modeling is in the direction of applying computational fluid dynamics (CFD) to solve the complete set of transport equations governing mass, momentum, species, energy, and charge conservation. 

If the electric conductivity of electrode matrixes and plates is limited, an additional equation governing charge transport in the electronic phase would have to be solved. This issue is separately addressed in section 3.4. 

To calculate the electron-transport effect through GDL and flow plate, the charge conservation equation for the electronic phase must be solved additionally, namely... 

Fig. 13 Experimental (symbols) and theoretical (lines) data for the current-density as a function of applied voltage for a polymer film of a derivative of PPV under the condition of space-charge-limited current flow. Full curves are the solution of a transport equation that includes DOS filling (see text), dashed lines show the prediction of Child s law for space-charge-limited current flow assuming a constant charge carrier mobility. From [96] with permission. Copyright (2005) by the American Institute of Physics...

The general solution of the system of transport equations for electrons and holes permits the photopotential of an open circuit to be calculated. The assumption that the total potential change due to illumination occurs in the space-charge region of a semiconductor, i.e., equilibrium value of , and that the exchange currents and... 

Fick s second law, Eqn. (4.42), is a partial differential equation for matter transport. Equations which describe the equilibration in space and time of heat, electrical charge, or momentum (dissipative processes) are of the same type and reflect the action of a local field. 

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