TheoriesMT

Chapter 14 Theories for interface mass transfer (Material presented in this chapter are based on those in Chapters 13, ”Diffusion mass transfer in fluid systems” second edition by EL Cussler) Fluid-fluid interface is commonly found in several applications such as absorption towers, cooling towers, distillation columns, multiphase reactors.

Characterization of the fluid-fluid interface is an important aspect for designing these practical applications. In this chapter, models for characterization of the fluid-fluid interface will be presented. The following three interface mass transfer theories will be developed • The film theory • Penetration theory • Surface-renewal theory

14.1

The film theory

The simplest theory that characterizes interface mass transport is the film theory. Consider a stagnant interface between liquid and gas (Fig. 14.1).

Assume the partial pressure of the solute to be P1 and the concentration of solute at the interface and in bulk liquid to be C1i and C1 . Assume the length of the hypothetical interface to be L. 183

184

CHAPTER 14. THEORIES FOR INTERFACE MASS TRANSFER

Figure 14.1: Film theory model Film theory assumes that the solute under high dilution diffuses slowly across the interface. Film theory neglects the diffusion induced convection in the interface. Under steady-state conditions, the flux of solute across the interface is given by N1 = k(C1i − C1 ) (14.1) where k is the mass transfer coefficient.

Shell balance across the interface for pure diffusion of the solute in the interface is d2 C1z =0 (14.2) D dz 2 the solution of which when subject to boundary conditions C1z (z = 0) = C1i and C1z (z = L) = C1 is C1z = C1i + (C1 − C1i )

z L

(14.3)

and the flux of mass transport in the interface is j1 = −D

dC1 D = (C1i − C1 ) dz L

(14.4)

which is constant across the interface. The concentration profile in the interface is linear.

14.2. PENETRATION THEORY

185

Comparing Eqs. (14.1) and (14.4), the mass transport coefficient for transport of solute across the interface is k=

kL D ⇒ = Sh = 1 L D

(14.5)

Equation (14.5) suggests that the mass transport coefficient in the interface is proportional to the diffusivity of the solute in the liquid phase. Although, film theory holds true under several situations, stagnant fluid assumption may not be valid always. Film theory may not hold under several practical situations. In order to reconcile this, penetration theory and its alternative the surface-renewal theory were developed.

14.2

Penetration theory

Penetration theory assumes that the solute is being transported from a wellmixed gas phase into the liquid phase and that the liquid is falling at a certain velocity (Fig. 14.2). That is, the solute from the gas phase penetrates into the liquid phase. The theory assumes that falling film is very thick and that the diffusion is important in the direction of transport and convection dominates diffusion in the flow direction.

Figure 14.2: Penetration theory

186

CHAPTER 14. THEORIES FOR INTERFACE MASS TRANSFER

The flux of mass transfer across the interface is given by N1 = k(C1i − C1 )

(14.6)

Shell balance across the interface leads to the following model equation vmax

∂c1 ∂ 2 c1 = ∂x ∂z 2

(14.7)

Model (Eq. 14.7) is subjected to the boundary conditions c1 (x = 0, z) = 0, c1 (x > 0, z = 0) = c1 (sat), c1 (x, z = L) = 0. Supposing that the film is exposed only for a very short time, then the solute can diffuse only for a short distance. If a wall exists at z = ∞, then the wall will not affect the concentration profile as the film is exposed only for a short time. Therefore the model can be considered as diffusion into a semi-infinite falling film. As a result the boundary condition, c1 (x, z = L) = 0 can be replaced with c1 (x, z = ∞) = 0.

If the falling film is flowing at a constant velocity, accordingly the solution is ! z c1 (14.8) = 1 − erf p c1,sat 4Dx/vmax and the mass transfer flux at the interface (at z = 0) is given by p j1 |z=0 = Dvmax /(πx)(c1i − c1 )

(14.9)

Average mass transport flux is given by averaging the local mass transfer flux over the mass transfer cross-section, that is x = 0, P and y = 0, W , which is 1 N1 = WP

ZP ZW 0

p N1 |z=0 dxdy = 2 Dvmax /(πP )(c1i − c1 )

(14.10)

0

Comparing Eqs (14.6) and (14.10), the mass transfer coefficient is given by p k = 2 Dvmax /(πP ) (14.11) P Note that vmax is called the contact time. Unlike in the case of film theory, the mass transport coefficient goes as square root of the diffusivity. Equation (14.11) can be re-written in terms of the dimensionless quantities as below 12 0 12 1 12 1 1 6 Pv 6 kP ν 2 Sh = = = Re 2 Sc 2 (14.12) D π ν D π

14.3. SURFACE-RENEWAL THEORY

14.3

187

Surface-renewal theory

An alternative to the penetration theory is the surface-renewal theory. Assume well-mixed gas phase and well-mixed liquid phase. The theory assumes that at local length scales, that is in the ”interfacial” region mass transfer occurs via the penetration theory model. Additionally, in the liquid phase, the fluid elements near the interface continously gets exchanged with the bulk fluid and this process is called ”surface-renewal” process (Fig. 14.3).

Figure 14.3: Surface-renewal theory As the exchange of fluid element occurs in the interface, the surfacerenewal depends on the amount of time small fluid element spends in the interface. The quantity that characterizes the time the fluid element spends in the interface is the residence time distribution. Residence time distribution is defined as the distribution of the probability that a given surface element will be at the surface at a given time or age of an element θ.

Suppose if the area of surface occupied by elements between the ages θ and θ + dθ is φ(θ)dθ. Then the surface area that is transported from age element θ − dθ and θ is given by φ(θ − dθ)dθ(1 − sdθ) where s is the rate at which the fractional surface exchanges with bulk for fresh element. As the new occupied surface between ages θ and θ + dθ must be those that arrives from the elements between ages θ − dθ and θ, a differential area balance is φ(θ)dθ = φ(θ − dθ)dθ(1 − sdθ)

(14.13)

188

CHAPTER 14. THEORIES FOR INTERFACE MASS TRANSFER

Using Taylor series expansion upto first order terms and neglecting all other higher order terms, the area balance (Eq. 14.13) will become dφ = −sφ(θ) ⇒ φ(θ) = C exp(−sθ) dθ

(14.14)

where C is the integration constant. Sum of all the age probabilities must be unity, that is, Z∞ φ(θ)dθ = 1 (14.15) 0

using which the integration constant C can be found to be equal to s. Therefore, the age distribution is given by φ(θ) = s exp(−sθ)

(14.16)

The rate of mass transport is given by the product of area and surface flux. As the transport through the element is governed by penetration theory, the flux is governed by that for penetration theory model. Therefore, the rate of mass transport is given by r Dvmax s exp −sθ (14.17) Rate = ∆C πP P Note that vmax = θ is the residence time or the age (see section (14.2). As the total area is given by Eq. (14.15), the mass transfer flux is given by r D N = ∆C s exp −sθ (14.18) πθ

Total flux is given by the sum of flux of mass transport over fluids of all ages.

Therefore, total flux Ntot is given by Z∞ Ntot =

r ∆C

D s exp −sθdΘ πθ

0

√

Z∞

= ∆C D

∆C 0

s exp −sθ √ dΘ πθ √ = ∆C Ds (14.19)

14.3. SURFACE-RENEWAL THEORY

189

By definition, if the total flux is given by Ntot = kL ∆C, then the mass transfer coefficient across the interface is given by √ kL = Ds (14.20) As s is the rate at which the fractional surface exchanges with bulk for fresh element, 1s = τ is the average residence time, therefore, the overall interface mass transport coefficient is given by r D (14.21) kL = τ

190

CHAPTER 14. THEORIES FOR INTERFACE MASS TRANSFER