Additivity of resistances

If resistance to transfer is present in both phases, Nqq may be expressed as an addition of resistances using equations 54, 47, 41, and 44 ... 

This is the important rule of additivity of resistances. In practice, and are often of the same order of magnitude, but the distribution coefficient m can vary considerably. For solutes which preferentially distribute toward solvent B, m is large and the controlling resistance Hes in phased. Conversely, if the distribution favors solvent A the controlling mass-transfer resistance Hes in phase B. 

In either equation, /c is given by Eq. (16-84) for parallel pore and surface diffusion or by Eq. (16-85) for a bidispersed particle. For nearly linear isotherms (0.7 < R < 1.5), the same linear addition of resistance can be used as a good approximation to predict the adsorption behavior of packed beds, since solutions for all mechanisms are nearly identical. With a highly favorable isotherm (R 0), however, the rate at each point is controlled by the resistance that is locally greater, and the principle of additivity of resistances breaks down. For approximate calculations with intermediate values of R, an overall transport parameter for use with the LDF approximation can be calculated from the following relationship for sohd diffusion and film resistance in series... 

Secondary Control of Wound-Rotor Motors Wound-rotor motors may be effectively reduced-voltage-started or have their speed controlled by using external secondaiy resistance. The addition of resistance into the secondaiy circuit of a wound-rotor motor reduces the starting current and affects the speed under load conditions. 

The summation in the denominator represents the additivity of resistances for the three series processes. From 9.2-17 and -18, we obtain cA in terms of pA and cB ... 

Comment This result shows that Hki and 1// " are additive resistances. It so happens that the addition of resistances to obtain on overall resistance is permissible only when the rate is a linear function of the driving force and when the processes occur in series. 

However, if heat flows perpendicular to the sheet surfaces, then the thermal conductivity of the composite is given by linear additivity of resistances ... 

When a molecule passes across an interface without chemical reaction, it encounters a total resistance R which is the sum of three separate diffusional resistances. These originate in phase 1, in the interfacial region (perhaps lOA thick) and in phase 2 (see Fig. 1). This additivity of resistances is expressed by ... 

Equation (6.1-50) is another equation for the additivity of resistances. It is subject to the same considerations as Equation (6.1-16) plus some others. 

The change in concentration of the phase would depend on the flow-rate, even if the transfer coefficients and rate of transfer did not. This equation is not truly one for the additivity of resistances, for each resistance has a scale factor, LM or GM. The ratio of GM to Lm is usually dictated by the plant production the individual values GM and LM are normally controlled by the physical capacity of the equipment rather than by mass transfer considerations, which control the contacting length rather than the flow-area of the equipment. 

If the material at the junction of the walls is neglected, the temperature in the wall varies linearly with distance, and the addition of resistances gives... 

The overall process rate involves the addition of resistances for mass transfer and chemical reaction. With respect to chemical rate, Orejas [11] proposes the following equations obtained by the regression of industrial data ... 

Ah ex situ methods discussed above have one important aspect in common. As in microelectronic applications, contamination issues are of extreme importance in catalysis. Ah ex situ preparation steps, which involve additions of resist layers, liftoff steps, etc., constitute potential contamination risks. This problem is shared with self-assembly methods, which usually involve some organic carrier of the inorganic nanoparticles to be deposited. 

For binary systems all matrices contain must one element and we have no difficulty in deriving the formula for the addition of resistances... 

Equations 7.3.11 and 7.3.14 are used in the development of expressions for modeling mass transfer in multicomponent distillation, a topic we consider in Chapter 12. The addition of resistances concept has seen use in distillation models by Krishna et al. (1981a), Burghardt et al. (1983, 1984) and by Gorak and Vogelpohl (1985). 

Krishna, R. and Standart, G. L., Addition of Resistances for Non-Isothermal Multicomponent Mass Transfer, Letts. Heat Mass Transfer, 3, 41-48 (1976b). 

In general, one can write contributions to the total impedance corresponding to the resistance as R, the capacitance as 1/sC, and the inductance as sL. The addition of impedances is analogous to the addition of resistances. Knowledge of the system s impedance allows an easy solution of the problem. 

Equation (2.3-55) is in the form of a rate being governed by two resistances in series—diffusion and chemical reaction. If I k SIOAB (fast surface reaction), die rale is governed by diffusion, while if Ilk 6/Dar (slow reaction rate), the rate is governed by cheraical kinetics. This additivity of resistances is only obtained when linear expressions relate rates and driving forces and wonld not be obtained, for example, if Ihe surface reaction kinetics were second order. More complex kinatic situations can be analyzed in a similar fashion where reaction stoichiometry at the surface provides information on (be flux ratio of various species. 

Addition of Resistances for Bubbles That Translate (i.e., Rise) and Expand in a Quiescent Liquid. Consider the addition of mass transfer resistances when gas bubbles translate and expand simultaneously. The simplest analysis of this rather... 

Addition of Resistances When Knudsen Diffusion and Ordinary Molecular Diffusion Are Operative in Binary and Multicomponent Gas Mixtures... 

The net diffusivity of component A within the pores of a catalytic pellet is obtained by adding mass transfer resistances for Knudsen diffusion and ordinary molecular diffusion, where convection reduces the resistance due to ordinary molecular diffusion but Knudsen flow occurs over length scales that are much too small for convective mass transfer to be important. This addition of resistances is constructed to simulate resistances in series, not parallel. Consider the trajectory of a gas molecule that collides with the walls of a channel or other gas molecules. In the pore-size regime where Knudsen and ordinary molecular diffusion are equally probable, these collisions occur sequentially, which suggests that gas molecules encounter each of these resistances in series. Hence, for binary mixtures. 

Rigorously, for a mixture of n components, the net diffusivity of component i is calculated via addition of resistances in series in the following manner ... 

From additivity of resistances, an equivalent expression yielding identical results is... 

Again, Eq. 3.6.d-4 clearly shows the additivity of resistances for first-order reactions. Note that in the asymptotic re on, where 0 is large. 

The surface renewal models only consider the liquid phase. In Sec. 6.3 on the film model the resistances of both gas and liquid phase were combined into one single expression like Eq. 6.3.b-5. The same can be done here Danckwerts [24] has shown that in most cases the surface renewal models combined with a gas side resistance lead to the same rules for the addition of resistances as the two-film theory. 

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