Mass Transfer with First-Order Chemical Reactions

1 Mass Transfer with First-Order Chemical Reactions 

Chemical reaction increases the rate of interfacial mass transfer. The reaction reduces the reagent s local concentration, thus increasing its concentration gradient and its flux. Because chemical reaction rates ean be very fast, the increase in mass 

In this section we want to calculate the increased mass transfer caused by a first-order, irreversible chemical reaction. This special case is the limit with which more elaborate calculations are compared. As a result, we shall go over the calculation in considerable detail so that its nuances are explicitly stated. 

One might wonder why we make such a fuss over first-order reactions. After all, these reactions are uncommon. Real chemical reactions involve two reagents, like sodium hydroxide plus hydrochloric acid or methane plus oxygen. This focus on first-order reactions may seem a scientific ploy, emphasizing problems we can solve rather than problems that are important. 

This skepticism has some justification, for there certainly are important reactions that are not first order. However, in many cases, all but one of the reagents will be present in excess in stoichiometric terms, only one of the reagents is limiting. In this case, we can accurately approximate the reaetions as first order. For the examples given earlier, we might have 

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Mass transfer with first-order chemical reaction... 

Fig. 6. Comparison of the computed numerical results ( ) with the results obtained from the approximate analytical expression ( ), given by eqs (24) and (31)-(34) for mass transfer with first-order chemical reaction, vs reaction rate constant. Parameter values are given in Table 4.

N. Turbulent Mass Transfer with First-Order Chemical Reaction... 

Ratcliff and Holdcroft (Rl), 1961 Mass transfer into liquid film flowing on sphere with first-order chemical reaction (theory and experiment). 

TABLE 27-3 Numerical Results for Coupled Heat and Mass Transfer with First-Order Irreversible Exothermic Chemical Reaction in Porous Catalysts with Rectangular Symmetry"... 

A slightly different approach was taken by Gill (G15), who considered the case of a bubble moving through a stationary liquid with mass transfer accompanied by simultaneous first-order chemical reaction. His assumptions were as follows ... 

A first-order chemical reaction occurs isothermally in a reactor packed with spherical catalyst pellets of radius R. If there is a resistance to mass transfer from the main fluid stream to the surface of the particle in addition to a resistance within the particle, show that the effectiveness factor for the pellet is given by ... 

Table 4. Parameter values used to calculate the data points shown in Figs 6 and 7 for mass transfer with, respectively, first-order chemical reaction and (l,l)-order chemical reaction...

In the framework of this description an attempt to model an effect of spatial non-uniformity of real catalytic systems was made (Bychkov et al., 1997). It was assumed that reaction proceeds in a heterogeneous system represented by two active infinite plane surfaces and in the gas gap between them. Surface chemistry was treated as for the Li/MgO catalyst (see Table III). Because of substantial complexity of the kinetic scheme consisting of several hundred elementary steps, the mass-transfer was described in this case as follows. The whole gas gap was divided into several (up to 10) layers of the same thickness, and each of them was treated as a well-stirred reactor. The rate of particle exchange between two layers was described in terms of the first-order chemical reaction with a rate constant ... 

The mass balance with diffusion and first-order chemical reaction, given by (24-12), is classified as a frequently occurring second-order linear ordinary differential equation (i.e., ODE) with constant coefficients. It is a second-order equation because diffusion is an important mass transfer rate process that is included in the mass balance. It is linear because the kinetic rate law is first-order or pseudo-first-order, and it is ordinary because diffusion is considered only in one coordinate direction—normal to the interface. The coefficients are constant under isothermal conditions because the physicochemical properties of the fluid don t change... 

Figure 2.12 Concentration profiles for mass transfer with pseudo first order chemical reaction (fiim model) (a) slow chemical reaction Ha <03 (b) moderate chemical...

Figure 34 Conversion vs. dimensionless rate constant from simple one-dimensional, two-region, axially uniform model for first-order chemical reaction with different overall voidages of 0.90 and 0.96 and different values of the dimensionless core-annulus interregion mass transfer coefficient.

The overall mass transfer coefficient is a rate constant, a measure of how fast the process occurs. It is a close parallel to the rate constant of a first-order chemical reaction, or to the half life of radioactive decay. It is different from these chemical rate constants in two important ways. First, K is defined per unit area, and chemical rate constants are normally defined per unit volume. One consequence is that we will sometimes work with Ka, where a is the interfacial area per system volume. The product Ka has the same units of reciprocal time as a first-order chemical rate constant. 

We are interested in how a first-order chemical reaction alters the mass transfer in industrial equipment. For example, imagine that we are scrubbing ammonia out of air with water, using equipment like that shown in Fig. 10.2-1. To increase our equipment s capacity, we are considering adding small amounts of hydrogen chloride to the water. We want to predict the effect of this acid. However, the a-priori prediction of mass transfer in a scrubber is a tremendously difficult problem, requiring expensive numerical calculation. 

Danckwerts et al. (D6, R4, R5) recently used the absorption of COz in carbonate-bicarbonate buffer solutions containing arsenate as a catalyst in the study of absorption in packed column. The C02 undergoes a pseudo first-order reaction and the reaction rate constant is well defined. Consequently this reaction could prove to be a useful method for determining mass-transfer rates and evaluating the reliability of analytical approaches proposed for the prediction of mass transfer with simultaneous chemical reaction in gas-liquid dispersions. 

The mechanisms considered above are all composed of steps in which chemical transformation occurs. In many important industrial reactions, chemical rate processes and physical rate processes occur simultaneously. The most important physical rate processes are concerned with heat and mass transfer. The effects of these processes are discussed in detail elsewhere within this book. However, the occurrence of a diffusion process in a reaction mechanism will be mentioned briefly because it can lead to kinetic complexities, particularly when a two-phase system is involved. Consider a reaction scheme in which a reactant A migrates through a non-reacting fluid to reach the interface between two phases. At the interface, where the concentration of A is Caj, species A is consumed in a first-order chemical rate process. In effect, consecutive rate processes are occurring. If a steady state is achieved, then... 

In these electrode processes, the use of macroelectrodes is recommended when the homogeneous kinetics is slow in order to achieve a commitment between the diffusive and chemical rates. When the chemical kinetics is very fast with respect to the mass transport and macroelectrodes are employed, the electrochemical response is insensitive to the homogeneous kinetics of the chemical reactions—except for first-order catalytic reactions and irreversible chemical reactions follow up the electron transfer—because the reaction layer becomes negligible compared with the diffusion layer. Under the above conditions, the equilibria behave as fully labile and it can be supposed that they are maintained at any point in the solution at any time and at any applied potential pulse. This means an independent of time (stationary) response cannot be obtained at planar electrodes except in the case of a first-order catalytic mechanism. Under these conditions, the use of microelectrodes is recommended to determine large rate constants. However, there is a range of microelectrode radii with which a kinetic-dependent stationary response is obtained beyond the upper limit, a transient response is recorded, whereas beyond the lower limit, the steady-state response is insensitive to the chemical kinetics because the kinetic contribution is masked by the diffusion mass transport. In the case of spherical microelectrodes, the lower limit corresponds to the situation where the reaction layer thickness does not exceed 80 % of the diffusion layer thickness. 

Intraparticle Diffusion and External Mass-Transfer Resistance For typical industrial conditions, external mass transfer is important only if there is substantial intraparticle diffusion resistance. This subject has been discussed by Luss, Diffusion-Reaction Interactions in Catalyst Pellets, in Carberry and Varma (eds.), Chemical Reaction and Reactor Engineering, Dekker, 1987. This, however, may not be the case for laboratory conditions, and care must be exerted in including the proper data interpretation. For instance, for a spherical particle with both external and internal mass-transfer limitations and first-order reaction, an overall effectiveness factor r, can be derived, indicating the series-of-resistances nature of external mass transfer followed by intraparticle diffusion-reaction ... 

The appropriate mass transfer coefficient in the boundary layer on the liquid side of spherical interfaces with first-order or pseudo-first-order irreversible chemical reaction predominantly in the liquid phase is... 

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