Mass-Transport Transfer Function

The mass-transfer problem for a rotating disk electrode at a constant rotation speed is presented in Section 11.6.2. Under modulation of the rotation speed of the electrode, equation (11.80) becomes 

Remember 15.3 The phase shift of the hydrodynamic transfer function f tends toward —45° as the dimensionless frequency p = co/Ci tends toward infinity. 

By using the same dimensionless position and the same dimensionless frequency Ki as was used to solve equation (11.83), equation (15.24) can be written in the form 

The solution of equation (15.25) can be obtained by the technique of reduction of order by setting Ci = A( )0i( ), where 9i( ) is a solution of the homogeneous equation satisfying the boundary conditions (11.88) (see Section 11.6) and A satisfies 

The expansion for 0, is given by equation (11.92) in terms of powers of pSc. The expansion for Wf is more complicated because, while the expansion of 0, depends on pSc /3 f/(p) depends on p without expansion for Wf takes the 

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Figure 15.3 Dimensionless mass-transport transfer-function Zc in Bode representation a) modulus versus the dimensionless frequency and b) phase shift versus the dimensionless...

When the Schmidt number is infinitely large, W,- is reduced to f p) h + jh) and appears as the product of a hydrodynamic transfer fimction f p) and a mass-transport transfer function Zc = fi -I- jt2- The mass-transport transfer function Zc is presented in Figure 15.3. It is eeisily verified that W,- approaches 0.5 when the frequency tends toward zero, in agreement with the exponent of the rotation speed in the Levich equation. This value of 0.5 is also verified if the complete expression of W,- is used. The complex function 2W,- is presented in Figure 15.4 in... 

In which the first term corresponds to the charge of the interface, the second term expresses the temperature dependency of the diffusion coefficient and YC is the TEC mass transport transfer function. According to the theory developed by Aaboubi et al. (Aaboubi, Citti,... 

When two reactants in a catalytic process have such different solubiUty properties that they can hardly both be present in a single Hquid phase, the reaction is confined to a Hquid—Hquid interface and is usually slow. However, the rate can be increased by orders of magnitude by appHcation of a phase-transfer catalyst (40,41), and these are used on a large scale in industrial processing (see Catalysts, phase-TRANSFEr). Phase-transfer catalysts function by faciHtating mass transport of reactants between the Hquid phases. Often most of the reaction takes place close to the interface. 

For quasi-reversible systems (with 10 1 > k" > 10 5 cm s1) the current is controlled by both the charge transfer and mass transport. The shape of the cyclic voltammogram is a function of k°/ JnaD (where a = nFv/RT). As k"/s/naD increases, the process approaches the reversible case. For small values of k°/+JnaD (i.e., at very fast i>) the system exhibits an irreversible behavior. Overall, the voltaimnograms of a quasi-reversible system are more drawn-out and exhibit a larger separation in peak potentials compared to those of a reversible system (Figure 2-5, curve B). 

As reversible ion transfer reactions are diffusion controlled, the mass transport to the interface is given by Fick s second law, which may be directly integrated with the Nernst equation as a boundary condition (see, for instance. Ref. 230 232). A solution for the interfacial concentrations may be obtained, and the maximum forward peak may then be expressed as a function of the interfacial area A, of the potential scan rate v, of the bulk concentration of the ion under study Cj and of its diffusion coefficient D". This leads to the Randles Sevcik equation [233] ... 

Van Campen et al. [31] developed models describing the rate of moisture uptake above RH0 that consider both the mass transport of water to the solid substance and the heat transfer away from the surface. For the special case of an environment consisting of pure water vapor (i.e., initial vacuum conditions), the Van Campen et al. model is greatly simplified since vapor diffusion need not be considered. Here, only the rate at which heat is transported away from the surface is assumed to be an important factor in limiting the sorption rate, W. For this special case, an expression was derived to express the rate of moisture uptake solely as a function of RHj, the relative humidity of the environment, and RH0. 

To learn that when the rate of electron transfer is slow, a useful approach is to construct Koutecky-Levich plots of (/(current) against l/Tafel plot from these mass-transport-limiting values of the current. 

Instruction Calculate the current density values as a function of overpotential (in a range of -0.200 to 0.200 V) assuming that the reaction is under mass transport control and under mixed mass transport and charge-transfer control determine the error of the approximation and plot i-T) dependencies. (Gokjovic)... 

In general, traditional electrode materials are substituted by electrode superstructures designed to facilitate a specific task. Thus, various modifiers have been attached to the electrode that lower the overall activation energy of the electron transfer for specific species, increase or decrease the mass transport, or selectively accumulate the analyte. These approaches are the key issues in the design of chemical selectivity of amperometric sensors. The long-term chemical and functional stability of the electrode, although important for chemical sensors as well, is typically focused on the use of modified electrodes in energy conversion devices. Examples of electroactive modifiers are shown in Table 7.2. 

In the previous section, the velocity and concentration distributions have been established and two transfer functions were considered. The explicit form of the third function which relates the fluctuating interfacial concentration or concentration gradient to the potential or the current at the interface, requires to make clear the kinetic mechanism composed of elementary steps with at least one of them being in part or wholly mass transport controlled. 

When the surface concentration of species Csoi cannot be considered as constant the analysis of the electrochemical response that arises from reaction scheme (6.X) becomes much more complex since the process is of second order and the value of the surface concentration of Csoi will be a function of the kinetics of the catalytic reaction and also of the mass transport (and therefore of the electrode geometry). Due to this higher complexity, only the current-potential response in CV will be treated with the additional simplification of fast surface charge transfer. 

The overall mass-transfer rates on both sides of the membrane can only be calculated when we know the convective velocity through the membrane layer. For this, Equation 14.2 should be solved. Its solution for constant parameters and for first-order and zero-order reaction have been given by Nagy [68]. The differential equation 14.26 with the boundary conditions (14.28a) to (14.28c) can only be solved numerically. The boundary condition (14.28c) can cause strong nonlinearity because of the space coordinate and/or concentration-dependent diffusion coefficient [40, 57, 58] and transverse convective velocity [11]. In the case of an enzyme membrane reactor, the radial convective velocity can often be neglected. Qin and Cabral [58] and Nagy and Hadik [57] discussed the concentration distribution in the lumen at different mass-transport parameters and at different Dm(c) functions in the case of nL = 0, that is, without transverse convective velocity (not discussed here in detail). 

The curve shown in Fig. 3 cannot proceed indefinitely in either direction. In the cathodic direction, the deposition of copper ions proceeds from solution until the rate at which the ions are supplied to the electrode becomes limited by mass-transfer processes. In the anodic direction, copper atoms are oxidized to form soluble copper ions. While the supply of copper atoms from the surface is essentially unlimited, the solubility of product salts is finite. Local mass-transport conditions control the supply rate so a current is reached at which the solution supersaturates, and an insulating salt-film barrier is created. At that point the current drops to a low level further increase in the potential does not significantly increase the current density. A plot of the current density as a function of the potential is shown in Fig. 5 for the zinc electrode in alkaline electrolyte. The sharp drop in potential is clearly observed at -0.9 V versus the standard hydrogen electrode (SHE). At more positive potentials the current density remains at a low level, and the electrode is said to be passivated. 

Thermodynamic non-idealities are taken into account while calculating necessary physical properties such as densities, viscosities, and diffusion coefficients. In addition, non-ideal phase equilibrium behavior is accounted for. In this respect, the Elec-trolyte-NRTL model (see Section 9.4.1) is used and supplied with the relevant parameters from Ref. [50]. The mass transport properties of the packing are described via the correlations from Refs. [59, 61]. This allows the mass transfer coefficients, specific contact area, hold-up and pressure drop as functions of physical properties and hydrodynamic conditions inside the column to be determined. 

Studies with porous catalyst particles conducted during the late 1930s established that, for very rapid reactions, the activity of a catalyst per unit volume declined with increasing particle size. Mathematical analysis of this problem revealed the cause to be insufficient intraparticle mass transfer. The engineering implications of the interaction between diffusional mass transport and reaction rate were pointed out concurrently by Damkohler [4], Zeldovich [5], and Thiele [6]. Thiele, in particular, demonstrated that the fractional reduction in catalyst particle activity due to intraparticle mass transfer, r, is a function of a dimensionless parameter, 0, now known as the Thiele parameter. 

Figure 6.21 shows the AC impedance spectra for the cathodic ORR of the cell electrodes prepared using the conventional method and the sputtering method. It can be seen that the spectra of electrodes 2 and 3 do not indicate mass transport limitation at either potentials. For electrode 1, a low-frequency arc develops, due to polarization caused by water transport in the membrane. It is also observable that the high-frequency arc for the porous electrode is significantly depressed from the typical semicircular shape. Nevertheless, the real-axis component of the arc roughly represents the effective charge-transfer resistance, which is a function of both the real surface area of the electrode and the surface concentrations of the species involved in the electrode reaction. 

Mean free path indicates the transfer of momentum, energy, or mass a distance. In the steady state, the net transport equals zero. These forces, or transfer functions, always act to bring a system back to the steady state. This implies (1) diffusion from high concentration to low concentration, (2) heat conduction from hot to cold, and (3) momentum flow—mass motion energy to molecular motion (hence accompanied by a rise in temperature of the gas). 

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