Reaction overall resistance

There are multiple possible current pathways through a DSSC, as shown in Fig. 1, because the nanoporous cell consists of two interpenetrating, bicontinuous chemical phases. The relative conductivity of these two phases and of the connection between them, Rct, depends on the illumination intensity, applied potential, kinetics of the redox couple, and so forth. Therefore, the distribution of current pathways depends also on these variables. In the DSSC, the dark current will take the distributed path of least overall resistance (Sections III. A-III.C), meaning it will flow primarily through solution [50] under the expected conditions of Rct < / 2- The dark current is thus mainly a measure of reaction (5) in this potential range, even though reaction (4) is expected to be the dominant recombination... 

In the case where mass transfer of the reactants through the liquid film on the surface of the catalyst or enzyme particles is much slower than the reaction itself (Da 1), the apparent reaction rate becomes almost equal to the rate of mass transfer. This is analogous to the case of two electrical resistances of different magnitudes in series, where the overall resistance is almost equal to the higher resistance. In such a case, the apparent reaction rate is given by... 

And the overall resistance will be the sum of three resistances in series, the bubbles, the solid, and the reaction as ... 

If even larger particles are considered, repeating the above calculation shows that, although the gas-liquid resistance (a) decreases, both the liquid-solid resistance (b) and the reaction term (c) increase. As a result of these opposing tendencies, there must exist an optimum particle size for which the overall resistance is a minimum and therefore the overall reaction rate is a maximum. More detailed calculations 363 show that for the value of the rate constant k = 16.8 s 1 used above, the optimum particle size is indeed approximately 2 mm which is the size taken for the example. However, for lower values of the rate constant, the optimum shifts towards smaller particle sizes, the reason being that the gas-liquid interfacial area for mass transfer becomes less important. 

Equations 3.12, 3.13, and the final Equation 3.14 are all forms of the classic engineering expression [Rate = (Driving force)/Resistance] where the driving force is expressed as concentration differences. The overall resistance (1 /K ) can be controlled by a low value of either individual coefficient. The mass transfer coefficient (kd) controls crystallization when the reaction is very rapid relative to diffusion, but the reaction coefficient ( r) controls crystallization when diffusion is much more rapid than reaction. In such cases the overall coefficient K may be approximated by the smaller k value. However, the concentrations in the driving force remain measurable (c) or calculable (ceq) rather than non-measureable (c ). 

As shown in Fig. 3, the overall resistance to the reaction is considered to consist of three steps (i) gas phase mass transfer (ii) diffusion through the porous reacted zone (in reduction the solid products are generally smaller in volume than the reactants), and (iii) chemical reaction at the boundary. Additional assumptions usually made are... 

An ideal unpolarized cell would have R = 0 and infinite current an ideal polarized cell would have a fixed R independent of and thus a constant current. Reality is somewhere in between There are several sources of "polarization" that can be considered as finite contributions to the overall resistance R > 0 (or better, the impedance Z). The IR drop, from whatever source, is also called the overpotential t] (i.e., IR > 0), which always decreases the overall E remember that R is always a function of time and E. The causes of polarization are (1) diffusion-limited mass transfer of ions from bulk to electrode (2) chemical side reactions (if any), and (3) slow electron transfer at the electrode between the adsorbed species to be oxidized and the adsorbed species to be reduced. 

B. The extractable pH of many well-washed photographic prints has been in ihe range of 5.5 to 6.0 (only a medium permanence level by ASTM Specification D-3290-74), but well-washed photographic prints have endured for many years. Estimates of paper permanence based upon rates of change at several elevated temperatures assess the overall effect of degradative reactions and resistance to those reactions, including the effects of acidity. 

The overall reaction rate in the bed is proportional to Kr, so the reciprocal of can be viewed as an overall resistance to the reaction. The different terms and groups on the RHS of Eq. (11) can be viewed as individual resistances which can be arranged in series or parallel to give the overall resistance. 

The reaction products participate in the steps above but in reverse order (5 through 1). Each step may be thought of as a resistance to the overall rate of reaction These resistances are shown schematically in Figure 12-12. The concentration in the liquid phase is related to the gas-phase concentralion... 

Slurry Reactor Design In the previous material we diseussed the transport and reaction steps and developed an equation for the overall resistance. A rearrangement of Equation (12-81) gives... 

The overall mass-transfer rate of solute can be controlled by any of the chemical reaction-diffusion resistances in the three-liquid phases. 

Rate of disappearance of component / (kmol/kg.cat.h) Resistance to mass transfer Overall reaction-diflfusion resistance Radius of catalyst pellet (m)... 

Similarly, the overall resistance in a set of parallel branches of elementary reactions may be estimated for the linear case in analogy to Ohm s Law according to... 

We are now in a position to consider the rate of the overall reaction. First, we write a formal rate equation for the kinetics of the reduced reaction network, Fig. 7d, by employing the electrical circuit analogy and the linear rate law analogous to Ohm s law. Thus, the overall rate (overall current) is the ratio of the affinity of the OR and the overall resistance of the reaction network. The overall resistance of the reduced reaction network is... 

The importance of external mass transfer can be judged by comparing kgU and kt] or their reciprocals, which are resistances. For example, if kga = 10 and krj = 2 (any consistent units), Kq = 1.67 and external mass transfer contributes 0.1/0.6 or 17 percent of the overall resistance to the process of mass transfer plus reaction. 

Determining the values of h and from experiments is a challenging task, and a great many empirical correlations have been presented. Most of the data are for heat transfer without reaction, such as for heating air in a steam-jacketed pipe packed with spheres. For these tests, Q is measured from the change in sensible heat of the air, and U is calculated from the usual equation, Q = VAAT. The small steam-film and metal-wall resistances can be subtracted from the overall resistance to obtain an overall bed coefficient, ho. ... 

The reciprocal plot separates the catalyst resistance from the overall resistance, but it does not show the relative importance of external mass transfer and internal diffusion plus reaction. If the average particle size is known, a. ... 

Mass transfer, an important phenomenon in science and engineering, refers to the motion of molecules driven by some form of potential. In a majority of industrial applications, an activity or concentration gradient serves to drive the mass transfer between two phases across an interface. This is of particular importance in most separation processes and phase transfer catalyzed reactions. The flux equations are analogous to Ohm s law and the ratio of the chemical potential to the flux represents a resistance. Based on the stagnant-film model. Whitman and Lewis [25,26] first proposed the two-film theory, which stated that the overall resistance was the sum of the two individual resistances on the two sides. It was assumed in this theory that there was no resistance to transport at the actual interface, i.e., within the distance corresponding to molecular mean free paths in the two phases on either side of the interface. This argument was equivalent to assuming that two phases were in equilibrium at the actual points of contact at the interface. Two individual mass transfer coefficients (Ld and L(-n) and an overall mass transfer coefficient (k. ) could be defined by the steady-state flux equations ... 

We generally interpret the above equation as representing a series of resistances. The overall resistance, 1/ overall, equals the sum of the resistance to LR diffusion across the stagnant liquid film inside the micelle, the resistance to LR diffusion across the liquid film surrounding the micelle, and the resistance to LR reaction. Note, k Rp,... 

---