7.9. BOUNDARY-LAYER FLOW AND TURBULENCE IN MASS TRANSFER

7.9A. Laminar Flow and Boundary-Layer Theory in Mass Transfer

In Section 3.10C an exact solution was obtained for the hydrodynamic boundary layer for isothermal laminar flow past a plate, and in Section 5.7A an extension of the Blasius solution was also used to derive an expression for convective heat transfer. In an analogous manner we use the Blasius solution for convective mass transfer for the same geometry and laminar flow. In Fig. 7.9-1 the concentration boundary layer is shown, where the concentration of the fluid approaching the plate is cA and cAS in the fluid adjacent to the surface.

Figure 7.9-1. Laminar flow of fluid past a flat plate and concentration boundary layer.


We start by using the differential mass balance, Eq. (7.5-17), and simplifying it for steady state where ∂cA/∂t = 0; RA = 0; flow only in the x and y directions, so νz = 0; and neglecting diffusion in the x and z directions to give

Equation 7.9-1


The momentum boundary-layer equation is very similar:

Equation 3.10-5


The thermal boundary-layer equation is also similar:

Equation 5.7-2


The continuity equation used previously is

Equation 3.10-3


The dimensionless concentration boundary conditions are

Equation 7.9-2


The similarity among the three differential equations (7.9-1), (3.10-5), and (5.7-2) is obvious, as is the similarity among the three sets of boundary conditions in Eq. (7.9-2). In Section 5.7A the Blasius solution was applied to convective heat transfer when (μ/ρ)/α = NPr = 1.0. We use the same type of solution for laminar convective mass transfer when (μ/ρ)/DAB = NSc = 1.0.

The velocity gradient at the surface was derived previously:

Equation 5.7-5


where NRe,x = ρ/μ. Also, from Eq. (7.9-2),

Equation 7.9-3


Differentiating Eq. (7.9-3) and combining the result with Eq. (5.7-5),

Equation 7.9-4


The convective mass-transfer equation can be written as follows and related to Fick's equation for dilute solutions:

Equation 7.9-5


Combining Eqs. (7.9-4) and (7.9-5),

Equation 7.9-6


This relationship is restricted to gases with a NSc = 1.0.

The relationship between the thickness δ of the hydrodynamic and δc of the concentration boundary layers where the Schmidt number is not 1.0 is

Equation 7.9-7


As a result, the equation for the local convective mass-transfer coefficient is

Equation 7.9-8


We can obtain the equation for the mean mass-transfer coefficient from x = 0 to x = L for a plate of width b by integrating as follows:

Equation 7.9-9


The result is

Equation 7.9-10


This is similar to the heat-transfer equation for a flat plate, Eq. (5.7-15), and also agrees with the experimental mass-transfer equation (7.3-27) for a flat plate.

In Section 3.10 an approximate integral analysis was performed for the laminar hydrodynamic and the turbulent hydrodynamic boundary layers. This was also done in Section 5.7 for the thermal boundary layer. An approximate integral analysis can also be done in exactly the same manner for the laminar and turbulent concentration boundary layers.

7.9B. Prandtl Mixing Length and Turbulent Eddy Mass Diffusivity

In many applications the flow in mass transfer is turbulent and not laminar. The turbulent flow of a fluid is quite complex as the fluid undergoes a series of random eddy movements throughout the turbulent core. When mass transfer is occurring, we refer to this as eddy mass diffusion. In Sections 3.10 and 5.7 we derived equations for turbulent eddy thermal diffusivity and momentum diffusivity using the Prandtl mixing length theory.

In a similar manner we can derive a relation for the turbulent eddy mass diffusivity, εM. Eddies are transported a distance L, called the Prandtl mixing length, in the y direction. At this point L the fluid eddy differs in velocity from the adjacent fluid by the velocity , which is the fluctuating velocity component given in Section 3.10F. The instantaneous rate of mass transfer of A at a velocity for a distance L in the y direction is

Equation 7.9-11


where is the instantaneous fluctuating concentration. The instantaneous concentration of the fluid is , where is the mean value and the deviation from the mean value. The mixing length L is small enough that the concentration difference is

Equation 7.9-12


The rate of mass transported per unit area is . Combining Eqs. (7.9-11) and (7.9-12),

Equation 7.9-13


From Eq. (5.7-23),

Equation 7.9-14


Substituting Eq. (7.9-14) into (7.9-13),

Equation 7.9-15


The term is called the turbulent eddy mass diffusivity εM. Combining Eq. (7.9-15) with the diffusion equation in terms of DAB, the total flux is

Equation 7.9-16


The similarities between Eq. (7.9-16) for mass transfer and heat and momentum transfer have been pointed out in detail in Section 6.1A.

7.9C. Models for Mass-Transfer Coefficients

1. Introduction

For many years mass-transfer coefficients, which were based primarily on empirical correlations, have been used in the design of process equipment. A better understanding of the mechanisms of turbulence is needed before we can give a theoretical explanation of convective mass-transfer coefficients. Several theories of convective mass transfer, such as the eddy diffusivity theory, have been presented in this chapter. In the following sections we present briefly some of these theories and discuss how they can be used to extend empirical correlations.

2. Film mass-transfer theory

The film theory, which is the simplest and most elementary theory, assumes the presence of a fictitious laminar film next to the boundary. This film, where only molecular diffusion is assumed to be occurring, has the same resistance to mass transfer as actually exists in the viscous, transition, and turbulent core regions. Then the actual mass-transfer coefficient is related to this film thickness δf by

Equation 7.9-17


Equation 7.9-18


The mass-transfer coefficient is proportional to . However, since we have shown that in Eq. (7.3-13) JD is proportional to (μ/ρDAB)2/3, then . Hence, the film theory is not correct. The great advantage of the film theory is its simplicity where it can be used in complex situations such as simultaneous diffusion and chemical reaction.

3. Penetration theory

The penetration theory derived by Higbie and modified by Danckwerts (D3) was derived for diffusion or penetration into a laminar falling film for short contact times in Eq. (7.3-23) and is as follows:

Equation 7.9-19


where tL is the time of penetration of the solute in seconds. This was extended by Danckwerts. He modified this for turbulent mass transfer and postulated that a fluid eddy has a uniform concentration in the turbulent core and is swept to the surface and undergoes unsteady-state diffusion. Then the eddy is swept away to the eddy core and other eddies are swept to the surface and stay for a random amount of time. A mean surface renewal factor s in s1 is defined as follows:

Equation 7.9-20


The mass-transfer coefficient is proportional to . In some systems, such as where liquid flows over packing and semistagnant pockets occur where the surface is being renewed, the results approximately follow Eq. (7.9-20). The value of s must be obtained experimentally. Others (D3, T2) have derived more complex combination film-surface renewal theories predicting a gradual change of the exponent on DAB from 0.5 to 1.0 depending on turbulence and other factors. Penetration theories have been used in cases where diffusion and chemical reaction are occurring (D3).

4. Boundary-layer theory

The boundary-layer theory has been discussed in detail in Section 7.9 and is useful in predicting and correlating data for fluids flowing past solid surfaces. For laminar flow and turbulent flow the mass-transfer coefficient . This has been experimentally verified for many cases.

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