Mass Transfer with Second-Order Chemical Reactions

2 Mass Transfer with Second-Order Chemical Reactions 

Like first-order reactions, second-order reactions can enhance interfacial mass transfer. Unlike the situation with first-order reactions, this enhancement eannot be easily calculated. Because second-order reactions are common and important, we resort to a variety of limiting cases to predict mass transfer coefficients in these situations. 

The reason that predictions are difficult for second-order reactions is again best illustrated by the film theory, as shown in Fig. 17.2-l(a). The mass balances in this film are 

Second-order chemical reaction increases mass transfer 

A more satisfying strategy is to consider three limiting cases. The most obvious limit, shown in Fig. 17.2-1 (b), occurs when reagent 2 is present in excess, so that the second-order reaction is equivalent to a first-order reaction. This limit was discussed in the previous section. 

---

Santiago,M. and I.H.Farina. "Mass transfer with second order chemical reaction. Numerical solution". Chem.Enqnq.Sci. 25 (1970) 744. 

Fig. 17.2-1. Mass transfer with second-order chemical reaction. The mass transfer coefficients for the general case (a) cannot be easily calculated. They can be found for the special cases (b) and (c).

The liquid-phase balances with interphase mass transfer and second-order irreversible chemical reaction provide four more equations that relate some of the... 

The sequence of equations presented below is required to solve the isothermal gas-liquid CSTR problem for the chlorination of benzene in the liquid phase at 55°C. After some simplifying assumptions, the problem reduces to the solution of nine equations with nine unknowns. Some of the equations are nonlinear because the chemical kinetics are second-order in the liquid phase and involve the molar densities of the two reactants, benzene and chlorine. The problem is solved in dimensionless form with the aid of five time constant ratios that are generated by six mass transfer rate processes (1) convective mass transfer through the reactor, (2) molecular transport in the liquid phase across the gas-liquid interface for each of the four components, and (3) second-order chemical reaction in the liquid phase. 

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... 

The experimental results of this work were analysed using the theory of mass transfer with chemical reaction. The data presented in this work have been obtained at conditions where the mathematical treatment of the problem was simplified assuming a pseudo-first-order assumption for kinetics. In this case, the concentration of the amine across the cross section of the liqitid boundary layer was assumed to be uniform. Thus, transforming the second-order reaction expression of carbon dioxide with amine into an approximated first-order expression. Hence, For piperazine ... 

A comparison as a gas-liqnid reactor was afforded by the Dow study of an unspecified reaction where the rate was known to be mass transfer limited. The rate of reaction was, specifically, controlled by the rate of reaction gas absorption into a solvent carrying the second reactant. However, when carried out in the RPB, the data output suggested that the reaction had become kinetically limited. Dow Chemicals stated that this allowed the exploration of the reaction chemistry and kinetics to provide a better understanding of the overall process. This was used to improve other mass transfer-limited reactors, and could lead to more applications for RPBs. The performance of the RPB reactor, illustrated in Figure 5.24, in comparison with an STR showed improvements in reaction rate of 3-4 orders of magnitude, with a 2-3 order increase in mass transfer. 

The Hatta criterion compares the rates of the mass transfer (diffusion) process and that of the chemical reaction. In gas-liquid reactions, a further complication arises because the chemical reaction can lead to an increase of the rate of mass transfer. Intuition provides an explanation for this. Some of the reaction will proceed within the liquid boundary layer, and consequently some hydrogen will be consumed already within the boundary layer. As a result, the molar transfer rate JH with reaction will be higher than that without reaction. One can now feel the impact of the rate of reaction not only on the transfer rate but also, as a second-order effect, on the enhancement of the transfer rate. In the case of a slow reaction (see case 2 in Fig. 45.2), the enhancement is negligible. For a faster reaction, however, a large part of the conversion occurs in the boundary layer, and this results in an overall increase of mass transfer (cases 3 and 4 in Fig. 45.2). 

It is obvious that re-atomization yields decrease the mean diameter of the liquid droplets and thus an increased interface area at the same time, it results in reduced average transfer coefficients, because heat and mass transfer coefficients between gas flow and particle or droplet are in positive correlation with the diameter of the particle or droplet, while coalescence of droplets yields influences opposite to those described above. In their investigation on the absorption of C02 into NaOH solution, Herskowits et al. [59, 60] determined theoretically the total interface areas and the mass transfer coefficients by comparing the absorption rates with and without reaction in liquid, employing the expression for the enhancement factor due to chemical reaction of second-order kinetics presented by Danckwerts [70],... 

The use of a stirred cell to simulate a packed column is widespread (D5, K7, D7, Cl 1, L5). Let us consider its use in predicting absorption rates at different levels of a column, when absorption occurs with a fast second-order irreversible chemical reaction and with mass-transfer resistance in both phases. 

Design a two-phase gas-liquid CSTR for the chlorination of benzene at 55°C by calculating the total volume that corresponds to an operating point where r/X = 500 on the horizontal axis of the CSTR performance curve in Figure 24-1. The time constant for convective mass transfer in the liquid phase is r. The time constant for second-order irreversible chemical reaction in the liquid phase is If the liquid benzene feed stream is diluted with an inert, then 7 increases. The liquid-phase volumetric flow rate is 5 gal/min. The inlet molar flow rate ratio of chlorine gas to liquid benzene... 

Answer Two. The thermal energy balance is not required when the enthalpy change for each chemical reaction is negligible, which causes the thermal energy generation parameters to tend toward zero. Hence, one calculates the molar density profile for reactant A within the catalyst via the mass transfer equation, which includes one-dimensional diffnsion and multiple chemical reactions. Stoichiometry is not required because the kinetic rate law for each reaction depends only on Ca. Since the microscopic mass balance is a second-order ordinary differential eqnation, it can be rewritten as two coupled first-order ODEs with split boundary conditions for Ca and its radial gradient. 

Quantitative results in Table 30-1 reveal that one achieves maximum conversion of reactants to products in ideal (i.e., 30%) and non-ideal (i.e., 25%) packed catalytic tubular reactors when the mass transfer Peclet number is approximately 6 for second-order irreversible chemical kinetics with an interpeUet porosity of 50%. Specific values for PeMT and the corresponding maximum conversion are sensitive to the simple mass transfer Peclet number and the chemical reaction coefficient, where the latter is defined by the product of the effectiveness factor, the interpeUet Damkohler number, and the catalyst filling factor. For example, when Pesimpie is 50 and the chemical reaction coefficient is 5 for second-order irreversible chemical kinetics, the critical value of PeMT [i e., (Re Sc)criacai] is approximately 30, whereas maximum conversion is obtained when PeMT is only 6. Hence, one concludes that the ideal simulations in Table 30-1 with a 0,... 

An analysis of chemical desorption has recently been published (Chem.Eng.Sci., 21 0980)), which is based on a number of simplifying assumptions the film theory model is assumed, the diffusivities of all species are taken to be equal to each other, and in the solution of the differential equations an approximation which is second order with respect to distance from the gas-liquid interface is used this approximation was introduced as early as 1948 by Van Krevelen and Hoftizer. However, the assumptions listed above are not at all drastic, and two crucial elements are kept in the analysis reversibility of the chemical reactions and arbitrary chemical mechanisms and stoichiometry.The result is a methodology for developing, for any given chemical mechanism, a highly nonlinear, implicit, but algebraic equation for the calculation of the rate enhancement factor as a function of temperature, bulk-liquid composition, interface gas partial pressure and physical mass transfer coefficient The method of solution is easily gene ralized to the case of unequal diffusivities and corrections for differences between the film theory and the penetration theory models can be calculated. 

An analysis equivalent to that proposed above has been carried out by Hoffman et al (18). They used data relevant to the chlorination of n-decane with m = n = 1 i.e. the reaction is first order with respect to each component. For a single backmixed gas-liquid reactor (equivalent to the element of Fig. 2), it was demonstrated that the interaction of mass transfer and chemical reaction gave rise to the possibility of up to 5 steady states for a single overall second order reaction. In their quantitative treatment, they made use of a reaction factor E., which is related to the normal enhancement factor by... 

Step), and leave the reaction area into bulk solution (second mass transfer). The mass transfer step, as well as the electrochemical one, are always present in any electrochemical transformation. Importantly, the electrochemical step is always accompanied by transfer of a charged particle through the interface. That is why this step is called the transfer step or the discharge-ionization step. Other complications are also possible. They are related to the formation of a new phase on the electrode (surface diffusion of adatoms, recombination of adatoms, formation of crystals or gas bubbles, etc.). The transfer step may be accompanied by different chemical reactions, both in bulk and on the electrode surface. A set of all the possible transformations is called the electrode process. Electrochemical kinetics works with the general description of electrode processes over time. While related to chemical kinetics, electrochemical kinetics has several important features. They are specific to the certain processes, in particular - the discharge-ionization step. Determination of a possible step order and the slowest (rate-determining) step is crucial for the description of the specific electrode process. 

The overall rate of a homogeneous reaction hke that of ammonia is determined by a nonlinear combination of effects of diffusion and chemical reaction. The effects of such a reaction on the rates of mass transfer are analyzed in the first two sections of this chapter. In Section 17.1, we describe the simplest case, that of a first-order irreversible chemical reaction. We also summarize extensions of this case, extensions that produce significant gains only at the cost of major effort. In Section 17.2, we describe some results for second-order reactions. In Section 17.3, we apply these ideas to a specific case, that of H2S scrubbing with amines. 

---