7.2. CONVECTIVE MASS-TRANSFER COEFFICIENTS

7.2A. Introduction to Convective Mass Transfer

In the previous sections of this chapter and in Chapter 6 we have emphasized molecular diffusion in stagnant fluids or fluids in laminar flow. In many cases the rate of diffusion is slow, and more rapid transfer is desired. To do this, the fluid velocity is increased until turbulent mass transfer occurs.

To have a fluid in convective flow usually requires the fluid to be flowing past another immiscible fluid or a solid surface. An example is a fluid flowing in a pipe, where part of the pipe wall is made by a slightly dissolving solid material such as benzoic acid. The benzoic acid dissolves and is transported perpendicular to the main stream from the wall. When a fluid is in turbulent flow and is flowing past a surface, the actual velocity of small particles of fluid cannot be described clearly as in laminar flow. In laminar flow the fluid flows in streamlines and its behavior can usually be described mathematically. However, in turbulent motion there are no streamlines; instead there are large eddies or “chunks” of fluid moving rapidly in seemingly random fashion.

When a solute A is dissolving from a solid surface, there is a high concentration of this solute in the fluid at the surface, and its concentration, in general, decreases as the distance from the wall increases. However, minute samples of fluid adjacent to each other do not always have concentrations close to each other. This occurs because eddies having solute in them move rapidly from one part of the fluid to another, transferring relatively large amounts of solute. This turbulent diffusion or eddy transfer is quite fast in comparison to molecular transfer.

Three regions of mass transfer can be visualized. In the first, which is adjacent to the surface, a thin, viscous sublayer film is present. Most of the mass transfer occurs by molecular diffusion, since few or no eddies are present. A large concentration drop occurs across this film as a result of the slow diffusion rate.

The transition or buffer region is adjacent to the first region. Some eddies are present, and the mass transfer is the sum of turbulent and molecular diffusion. There is a gradual transition in this region from the transfer by mainly molecular diffusion at one end to mainly turbulent at the other end.

In the turbulent region adjacent to the buffer region, most of the transfer is by turbulent diffusion, with a small amount by molecular diffusion. The concentration decrease is very small here since the eddies tend to keep the fluid concentration uniform. A more detailed discussion of these three regions is given in Section 3.10G.

A typical plot for the mass transfer of a dissolving solid from a surface to a turbulent fluid in a conduit is given in Fig. 7.2-1. The concentration drop from c_A1 adjacent to the surface is very abrupt close to the surface and then levels off. This curve is very similar to the shapes found for heat and momentum transfer. The average or mixed concentration is shown and is slightly greater than the minimum c_A2.

7.2B. Types of Mass-Transfer Coefficients

1. Definition of mass-transfer coefficient

Since our understanding of turbulent flow is incomplete, we attempt to write the equations for turbulent diffusion in a manner similar to that for molecular diffusion. For turbulent mass transfer for constant c, Eq. (6.1-6) is written as

Equation 7.2-1

where D_AB is the molecular diffusivity in m²/s and ε_M the mass eddy diffusivity in m²/s. The value of ε_M is variable and is near zero at the interface or surface and increases as the distance from the wall increases. We therefore use an average value since the variation of ε_M is not generally known. Integrating Eq. (7.2-1) between points 1 and 2,

Equation 7.2-2

The flux is based on the surface area A₁ since the cross-sectional area may vary. The value of z₂ − z₁, the distance of the path, is often not known. Hence, Eq. (7.2-2) is simplified and is written using a convective mass-transfer coefficient :

Equation 7.2-3

where is the flux of A from the surface A₁ relative to the whole bulk phase, is (D_AB + )/(z₂ − z₁), an experimental mass-transfer coefficient in kg mol/s · m² · (kg mol/m³) or simplified as m/s, and c_A2 is the concentration at point 2 in kg mol A/m³ or, more usually, the average bulk concentration This definition of a convective mass-transfer coefficient is quite similar to the convective heat-transfer coefficient h.

2. Mass-transfer coefficient for equimolar counterdiffusion

Generally, we are interested in N_A, the flux of A relative to stationary coordinates. We can start with the following equation, which is similar to that for molecular diffusion with the term ε_M added:

Equation 7.2-4

For the case of equimolar counterdiffusion, where N_A = −N_B, and integrating at steady state, calling ,

Equation 7.2-5

Equation (7.2-5) is the defining equation for the mass-transfer coefficient. Often, however, we define the concentration in terms of mole fraction if a liquid or gas, or in terms of partial pressure if a gas. Hence, we can define the mass-transfer coefficient in several ways. If y_A is mole fraction in a gas phase and x_A in a liquid phase, then Eq. (7.2-5) can be written as follows for equimolar counterdiffusion:

Equation 7.2-6

Equation 7.2-7

All of these mass-transfer coefficients can be related to each other. For example, using Eq. (7.2-6) and substituting y_A1 = c_A1/c and y_A2 = c_A2/c into the equation,

Equation 7.2-8

Hence,

Equation 7.2-9

These relations among mass-transfer coefficients, together with the various flux equations, are given in Table 7.2-1.

Table 7.2-1. Flux Equations and Mass-Transfer Coefficients

Flux equations for equimolar counterdiffusion
Flux equations for A diffusing through stagnant, nondiffusing B
Conversions between mass-transfer coefficients
Gases:
Liquids:
(where ρ is density of liquid and M is molecular weight)
Units of mass-transfer coefficients
	(preferred)

3. Mass-transfer coefficient for A diffusing through stagnant, nondiffusing B

For A diffusing through stagnant, nondiffusing B, where N_B = 0, Eq. (7.2-4) gives for steady state

Equation 7.2-10

where the x_BM and its counterpart y_BM are similar to Eq. (6.2-21) and k_c is the mass-transfer coefficient for A diffusing through stagnant B. Also,

Equation 7.2-11

Rewriting Eq. (7.2-10) using other units,

Equation 7.2-12

Equation 7.2-13

Again all the mass-transfer coefficients can be related to each other and are given in Table 7.2-1. For example, setting Eq. (7.2-10) equal to (7.2-13),

Equation 7.2-14

Hence,

Equation 7.2-15

EXAMPLE 7.2-1. Vaporizing A and Convective Mass Transfer A large volume of pure gas B at 2 atm pressure is flowing over a surface from which pure A is vaporizing. The liquid A completely wets the surface, which is a blotting paper. Hence, the partial pressure of A at the surface is the vapor pressure of A at 298 K, which is 0.20 atm. The has been estimated to be 6.78 × 10−5 kg mol/s · m2 · mol frac. Calculate NA, the vaporization rate, and also the value of ky and kG. Solution: This is the case of A diffusing through B, where the flux of B normal to the surface is zero, since B is not soluble in liquid A. pA1 = 0.20 atm and pA2 = 0 in the pure gas B. Also, yA1 = pA1/P = 0.20/2.0 = 0.10 and yA2 = 0. We can use Eq. (7.2-12) with mole fractions: Equation 7.2-12 However, we have a value for which is related to ky from Table 7.2-1 by Equation 7.2-16 The term yBM is similar to xBM and is, from Eq. (7.2-11), Equation 7.2-11 Substituting into Eq. (7.2-11), Then, from Eq. (7.2-16), Also, from Table 7.2-1, Equation 7.2-17 Hence, solving for kG and substituting knowns, For the flux, using Eq. (7.2-12), Also, Using Eq. (7.2-12) again, Note that in this case, since the concentrations were dilute, yBM is close to 1.0 and ky and differ very little.

7.2C. Mass Transfer Coefficients for General Case of A and B Diffusing and Convective Flow Using Film Theory

Equation (7.2-4) can be integrated assuming a simplified film theory where the mass transfer is assumed to occur through a thin film next to the wall of thickness δ_f and by molecular diffusion. Then the experimental value of for dilute solutions is used to determine the film thickness δ_f:

Equation 7.2-18

Rewriting Eq. (7.2-4),

Equation 7.2-19

The convective term is x_A(N_A + N_B). Rearranging and integrating,

Equation 7.2-20

Equation 7.2-21

For N_B = 0, Eq. (7.2-21) reduces to Eq. (7.2-10).

7.2D. Mass Transfer Coefficients under High Flux Conditions

The final Eq. (7.2-21) is a result of assuming the film theory where molecular diffusion occurs across the film δ_f. This assumes that this film thickness is unaffected by high fluxes and bulk or convective flow (diffusion-induced convection). As a result, other definitions of the mass-transfer coefficient that include this effect of diffusion-induced convection have been derived. One common method by Bird et al. (B1) and Skelland (S8) is given as follows for the case of A diffusing through stagnant, nondiffusing B where diffusion-induced convection is present.

Rewriting Eq. (7.2-19) for the flux N_A at the surface z = 0 where x_A = x_A1,

Equation 7.2-22

Note that the convective term is defined in terms of x_A1 and not the average value.

Defining a mass-transfer coefficient in terms of the diffusion flux,

Equation 7.2-23

Substituting Eq. (7.2-23) into (7.2-22) and solving for N_A,

Equation 7.2-24

At low concentrations and fluxes, approaches for no bulk flow, or

In general, a coefficient k_c may be defined without regard to convective flow:

Equation 7.2-25

Combining (7.2-24) and (7.2-25),

Equation 7.2-26

The relationship between or k_c for high flux and for low flux will be considered. These corrections or relationships will be derived using the film theory for the transfer of A by molecular diffusion and convective flow, with B being stagnant and nondiffusing. This has been done previously and is obtained by setting N_B = 0 in Eq. (7.2-21) to obtain the following, which is identical to Eq. (7.2-10):

Equation 7.2-27

Hence, for the film theory,

Equation 7.2-28

Combining Eqs. (7.2-26) and (7.2-28),

Equation 7.2-29

For this film theory, Eqs. (7.2-28) and (7.2-29) are independent of the Schmidt number. These correction factors predicted by the film theory give results that are reasonably close to those using more complex theories—the penetration theory and boundary layer theory— which are discussed later in Section 7.9C. An advantage of this simple film theory is that it is quite useful in solving complex systems.

EXAMPLE 7.2-2. High Flux Correction Factors Toluene A is evaporating from a wetted porous slab by having inert pure air at 1 atm flowing parallel to the flat surface. At a certain point the mass-transfer coefficient for very low fluxes has been estimated as 0.20 lb mol/hr · ft2. The gas composition at the interface at this point is xA1 = 0.65. Calculate the flux NA and the ratios kc/ or kx/ and / or to correct for high flux. Solution: For the flux NA, using Eq. (7.2-11), Using Eq. (7.2-10) in terms of where , Using Eq. (7.2-28), Then, kx = 1.616 = 1.616(0.20) = 0.323 lb mol/hr · ft2. Using Eq. (7.2-29), and = 0.565(0.200) = 0.113 lb mol/hr · ft2.

7.2E. Methods for Experimentally Determining Mass-Transfer Coefficients

Many different experimental methods have been employed to obtain mass-transfer coefficients. In determining the mass-transfer coefficient to a sphere, Steele and Geankoplis (S3) used a solid sphere of benzoic acid held rigidly by a rear support in a pipe. Before the run the sphere was weighed. After flow of the fluid for a timed interval, the sphere was removed, dried, and weighed again to give the amount of mass transferred, which was small compared to the weight of the sphere. From the mass transferred and the area of the sphere, the flux N_A was calculated. Then the driving force (c_AS − 0) was used to calculate k_L, where c_AS is the solubility and the water contained no benzoic acid.

Another method used is to flow gases over various geometries wet with evaporating liquids. For mass transfer from a flat plate, a porous blotter wet with the liquid serves as the plate.

If the solution is quite dilute, then the mass-transfer coefficient measured is and the dilute low flux coefficient or is experimentally obtained.