6.1. INTRODUCTION TO MASS TRANSFER AND DIFFUSION

6.1A. Similarity of Mass, Heat, and Momentum Transfer Processes

1. Introduction

In Chapter 1 we noted that the various separation processes have certain basic principles which can be classified into three fundamental transfer (or “transport”) processes: momentum transfer, heat transfer, and mass transfer. The fundamental process of momentum transfer occurs in such operations as fluid flow, mixing, sedimentation, and filtration. Heat transfer occurs in conductive and convective transfer of heat, evaporation, distillation, and drying.

The third fundamental transfer process, mass transfer, occurs in distillation, absorption, drying, liquid–liquid extraction, adsorption, ion exchange, crystallization, and membrane processes. When mass is being transferred from one distinct phase to another or through a single phase, the basic mechanisms are the same whether the phase is a gas, liquid, or solid. This was also shown for heat transfer, where the transfer of heat by conduction followed Fourier's law in a gas, solid, or liquid.

2. General molecular transport equation

All three of the molecular transport processes—momentum, heat, and mass—are characterized by the same general type of equation, given in Section 2.3A:

Equation 2.3-1

This can be written as follows for molecular diffusion of the property momentum, heat, or mass:

Equation 2.3-2

3. Molecular diffusion equations for momentum, heat, and mass transfer

Newton's equation for momentum transfer for constant density can be written as follows in a manner similar to Eq. (2.3-2):

Equation 6.1-1

where τ_zx is momentum transferred/s · m², μ/ρ is kinematic viscosity in m²/s, z is distance in m, and v_x ρ is momentum/m³, where the momentum has units of kg · m/s.

Fourier's law for heat conduction can be written as follows for constant ρ and c_p:

Equation 6.1-2

where q_z/A is heat flux in W/m², α is the thermal diffusivity in m²/s, and ρc_p T is J/m³.

The equation for molecular diffusion of mass is Fick's law and is also similar to Eq. (2.3-2). It is written as follows for constant total concentration in a fluid:

Equation 6.1-3

where is the molar flux of component A in the z direction due to molecular diffusion in kg · mol A/s · m², D_AB the molecular diffusivity of the molecule A in B in m²/s, c_A the concentration of A in kg · mol/m³, and z the distance of diffusion in m. In cgs units is g mol A/s · cm², D_AB is cm²/s, and c_A is g mol A/cm³. In English units, is lb mol/h · ft², D_AB is ft²/h, and c_A is lb mol/ft³.

The similarity of Eqs. (6.1-1), (6.1-2), and (6.1-3) for momentum, heat, and mass transfer should be obvious. All the fluxes on the left-hand side of the three equations have as units transfer of a quantity of momentum, heat, or mass per unit time per unit area. The transport properties μ/ρ, α and D_AB all have units of m²/s, and the concentrations are represented as momentum/m³, J/m³, or kg mol/m³.

4. Turbulent diffusion equations for momentum, heat, and mass transfer

In Section 5.7C equations were given showing the similarities among momentum, heat, and mass transfer in turbulent transfer. For turbulent momentum transfer and constant density,

Equation 6.1-4

For turbulent heat transfer for constant ρ and c_p,

Equation 6.1-5

For turbulent mass transfer for constant c,

Equation 6.1-6

In these equations ε_t is the turbulent or eddy momentum diffusivity in m²/s, α_t the turbulent or eddy thermal diffusivity in m²/s, and ε_M the turbulent or eddy mass diffusivity in m²/s. Again, these equations are quite similar to each other. Many of the theoretical equations and empirical correlations for turbulent transport to various geometries are also quite similar.

6.1B. Examples of Mass-Transfer Processes

Mass transfer is important in many areas of science and engineering. Mass transfer occurs when a component in a mixture migrates in the same phase or from phase to phase because of a difference in concentration between two points. Many familiar phenomena involve mass transfer. Liquid in an open pail of water evaporates into still air because of the difference in concentration of water vapor at the water surface and the surrounding air. There is a “driving force” from the surface to the air. A piece of sugar added to a cup of coffee eventually dissolves by itself and diffuses to the surrounding solution. When newly cut and moist green timber is exposed to the atmosphere, the wood will dry partially when water in the timber diffuses through the wood to the surface and then to the atmosphere. In a fermentation process nutrients and oxygen dissolved in the solution diffuse to the microorganisms. In a catalytic reaction the reactants diffuse from the surrounding medium to the catalyst surface, where reaction occurs.

Many purification processes involve mass transfer. In uranium processing, a uranium salt in solution is extracted by an organic solvent. Distillation to separate alcohol from water involves mass transfer. Removal of SO₂ from flue gas is done by absorption in a basic liquid solution.

We can treat mass transfer in a manner somewhat similar to that used in heat transfer with Fourier's law of conduction. However, an important difference is that in molecular mass transfer, one or more of the components of the medium is moving. In heat transfer by conduction, the medium is usually stationary and only energy in the form of heat is being transported. This introduces some differences between heat and mass transfer that will be discussed in this chapter.

6.1C. Fick's Law for Molecular Diffusion

Molecular diffusion or molecular transport can be defined as the transfer or movement of individual molecules through a fluid by means of the random, individual movements of the molecules. We can imagine the molecules traveling only in straight lines and changing direction by bouncing off other molecules after collision. Since the molecules travel in a random path, molecular diffusion is often called a random-walk process.

In Fig. 6.1-1 the molecular diffusion process is shown schematically. A random path that molecule A might take in diffusing through B molecules from point (1) to (2) is shown.If there are a greater number of A molecules near point (1) than at (2), then, since molecules diffuse randomly in both directions, more A molecules will diffuse from (1) to (2) than from (2) to (1). The net diffusion of A is from high- to low-concentration regions.

As another example, a drop of blue liquid dye is added to a cup of water. The dye molecules will diffuse slowly by molecular diffusion to all parts of the water. To increase this rate of mixing of the dye, the liquid can be mechanically agitated by a spoon and convective mass transfer will occur. The two modes of heat transfer, conduction and convective heat transfer, are analogous to molecular diffusion and convective mass transfer.

First, we will consider the diffusion of molecules when the whole bulk fluid is not moving but is stationary. Diffusion of the molecules is due to a concentration gradient. The general Fick's law equation can be written as follows for a binary mixture of A and B:

Equation 6.1-7

where c is total concentration of A and B in kg mol A + B/m³, and x_A is the mole fraction of A in the mixture of A and B. If c is constant, then since c_A = cx_A,

Equation 6.1-8

Substituting into Eq. (6.1-7) we obtain Eq. (6.1-3) for constant total concentration:

Equation 6.1-3

This equation is the one more commonly used in many molecular diffusion processes. If c varies some, an average value is often used with Eq. (6.1-3).

EXAMPLE 6.1-1. Molecular Diffusion of Helium in Nitrogen A mixture of He and N2 gas is contained in a pipe at 298 K and 1 atm total pressure which is constant throughout. At one end of the pipe at point 1 the partial pressure pA1 of He is 0.60 atm and at the other end 0.2 m (20 cm) pA2 = 0.20 atm. Calculate the flux of He at steady state if DAB of the He–N2 mixture is 0.687 × 10−4 m2/s (0.687 cm2/s). Use SI and cgs units. Solution: Since total pressure P is constant, then c is constant, where c is as follows for a gas according to the perfect gas law: Equation 6.1-9 Equation 6.1-10 where n is kg · mol A plus B, V is volume in m3, T is temperature in K, R is 8314.3 m3 · Pa/kg · mol · K or R is 82.057 × 10−3 m3 · atm/kg mol · K, and c is kg mol A plus B/m3. In cgs units, R is 82.057 cm3 · atm/g mol · K. For steady state the flux in Eq. (6.1-3) is constant. Also, DAB for a gas is constant. Rearranging Eq. (6.1-3) and integrating, Equation 6.1-11 Also, from the perfect gas law, pAV = nART, and Equation 6.1-12 Substituting Eq. (6.1-12) into (6.1-11), Equation 6.1-13 This is the final equation to use, which is in a form easily used for gases. Partial pressures are pA1 = 0.6 atm = 0.6 × 1.01325 × 105 = 6.08 × 104 Pa and pA2 = 0.2 atm = 0.2 × 1.01325 × 105 = 2.027 × 104 Pa. Then, using SI units, If pressures in atm are used with SI units, For cgs units, substituting into Eq. (6.1-13),

Other driving forces (besides concentration differences) for diffusion also occur because of temperature, pressure, electrical potential, and other gradients. Details are given elsewhere (B3).

6.1D. Convective Mass-Transfer Coefficient

When a fluid is flowing outside a solid surface in forced convection motion, we can express the rate of convective mass transfer from the surface to the fluid, or vice versa, by the following equation:

Equation 6.1-14

where k_c is a mass-transfer coefficient in m/s, c_L1 the bulk fluid concentration in kg mol A/m³, and c_Li the concentration in the fluid next to the surface of the solid. This mass-transfer coefficient is very similar to the heat-transfer coefficient h and is a function of the system geometry, fluid properties, and flow velocity. In Chapter 7 we consider convective mass transfer in detail.