Chapter 15 Mass transfer with chemical reactions (Material presented in this chapter are based on those in Chapter 16, ”Diffusion mass transfer in fluid systems” second edition by EL Cussler and Chapter 14, ”Fundamental of heat and mass transfer”, second edition by Incropera and DeWitt)
The liquid-phase mass transport across interface depends on the diffusion of the solute in the liquid phase. The diffusion of the solute depends on the chemical reaction, if any involving the solute in the interface. There are several examples in which simultaneous mass transport between two phases and chemical reactions are involved. For example, several polymerization reactions occur in emulsions, which consist of two liquid phases. One of the reactants is usually in a phase in which the reaction does not occur. Thus, the reactant has to cross the interface and react in another phase. Chemical reaction under certain conditions strongly influence the diffusion. This influence depends on whether the reaction is homogeneous or heterogeneous. In this chapter, mass transport during homogeneous chemical reactions will be characterized. The expressions for the coefficient with chemical reactions will be compared with those without chemical reactions. 191
192 CHAPTER 15. MASS TRANSFER WITH CHEMICAL REACTIONS
15.1
Mass diffusion with homogeneous reaction
Consider the system where a solute species present in the gaseous phase that transports into the liquid phase, maintained at a certain concentration where the solute gets converted into products via an irreversible reaction (Fig. 15.1)
Figure 15.1: Mass transfer with chemical reaction
A → P roducts
(15.1)
If the transport in the interface in the liquid-phase follows film theory, then the model that describes the simultaneous transport and reaction is given by the following differential balance D
d2 C A + NË™ A = 0 dx2
(15.2)
15.1. MASS DIFFUSION WITH HOMOGENEOUS REACTION
193
where, CA is the concentration of the species A, D is the mass diffusivity of the species in the liquid-phase, N˙ A is the rate of formation of the species A per unit volume, and x is the distance in the interface. If the reaction (Eq. 15.1) were to be a first order reaction, then the model (Eq. 15.2) can be written as d2 C A D − krxn CA = 0 dx2
(15.3)
where, krxn is the reaction rate constant. One boundary condition is the interface concentration at the contact point between the gas and liquid. There are two possible second boundary conditions for this model at the other end of the interface. If the bulk concentration of the speices is known then the constant boundary conditions at the edge of the interface can be used. If the bulk concentration is unknown, then somewhere just outside the interface, the bulk concentration should remain constant, that is, the flux or concentration gradient of the solute must be zero at the interface.
15.1.1
No flux boundary condition
The solution of the model (Eq. 15.3) subject to the boundary conditions CA (x = 0) = CAi ;
dCA |x=L = 0 dx
(15.4)
is given by CA (x) = CAi (cosh(mx) − tanh(mL) sinh(mx)) where, m = x = L is
q
krxn . D
(15.5)
The concentration at the end of the interface, that is, at
CA (L) = CAi
cosh2 2(mL) − sinh( mL) CAi = cosh(mL) cosh(mL)
(15.6)
and the flux at the gas-liquid contact point is given by NA” (x = 0) = DCAi m tanh(mL)
(15.7)
If the mass transport flux across the interface is defined as NA” = kL (CAi − CA (L)) then using Eqs (15.6) and (15.7), the effective mass transport coefficient across the interface can be found.
194 CHAPTER 15. MASS TRANSFER WITH CHEMICAL REACTIONS
15.1.2
Constant boundary condition
The solution of the model (Eq. 15.3) subject to the boundary conditions CA (x = 0) = CAi ; CA x = L = CAb
(15.8)
is given by CA = CAi
sinh
q
− x) sinh krxn L D krxn (L D
(15.9)
The flux at x = 0 is given by p dCA jA = −D |x=0 = CAi Dkrxn coth dx
r
krxn L2 D
(15.10)
If the mass transport flux across the interface is defined as NA” = kL (CAi − CA (L)) then, the effective mass transport coefficient across the interface is given by r p krxn L2 kL = Dkrxn coth (15.11) D Supposing the kL0 is the mass transfer coefficient when no chemical re, action occurs, then according to the film theory (section 14.1), kL0 = D L using which the ratio of mass transfer coefficients with and without reaction is given by s s ! kL = kL 0
Dkrxn coth 2 kL0
Dkrxn 2 kL0
(15.12)
The solution in both first and second cases shows that the concentration does not follow a linear profile as in the case observed during pure mass transport.
15.2
Mass transfer with second order reactions
Consider the case where one solute gets transported to the second phase, which reacts with another reactant in the second phase. Assume the reaction to be first order with respect to both components. If species A and B are
15.2. MASS TRANSFER WITH SECOND ORDER REACTIONS
195
the reactants to form a certain product via the chemical reaction A + B → P roducts, the model describing mass transport in the interface is given by d2 CA − krxn CA CB = 0 dx2 d2 CB DB − krxn CA CB = 0 dx2 DA
(15.13) (15.14)
B = 0 and z = subject to the boundary conditions, z = 0, CA = CAi , dC dz L, CA = 0, CB = CBi . Equations (15.14) when solved can provide an expression for the mass transfer coefficient.
196 CHAPTER 15. MASS TRANSFER WITH CHEMICAL REACTIONS