Chapter 15. Introduction to Diffusion and Mass Transfer

What makes chemical engineering different from other engineering disciplines? The key to the uniqueness of chemical engineers is that we understand, design, and operate chemical reactors and separators. The fundamental knowledge necessary to understand chemical reactors is mass and energy balances, kinetics, reaction equilibrium, and mass transfer. The fundamental knowledge required to understand separators is mass and energy balances, phase equilibrium, and mass transfer. Thus mass transfer is fundamental for understanding the two unique parts of chemical engineering. In addition mass transfer is important in pollution (e.g., movement of pollutants in soil and movement of pollution plumes in air), the controlled release of medicines, safety and fire control, and the very functioning of life itself. Without mass transfer processes you would be unable to move nutrients and oxygen to your cells and to remove carbon dioxide and other waste products.

Except for the short introductory Section 1.3, to this point the entire analysis of separation processes has been equilibrium based. Effects of nonequilibrium operation have been lumped into either stage efficiency (Sections 4.11, 10.2, 10.12, and 12.5) or height equivalent of a theoretical plate (HETP) (Sections 10.9 and 10.12). We need to study mass transfer if we want to predict values of the stage efficiency and the HETP (Chapter 16), to determine crystal size distributions (Chapter 17), to study membrane separators (Chapter 18), or to study sorption separations (Chapter 19). This chapter presents the fundamentals of diffusion and mass transfer in sufficient detail to make analyses in the remaining chapters understandable. Additional information on mass transfer is presented as needed in Chapters 16 to 19. If you have already studied mass transfer and diffusion, most, but probably not all, of this chapter is a review.

Mass transfer is movement of mass caused by species concentration differences in a mixture. Diffusion is mass transfer caused by molecular movement, while convection is mass transfer caused by bulk movement of mass. Diffusion always causes convection, and if convection is significant it must be included in the analysis.

Mass transfer is often considered a difficult subject to understand for several reasons:

• Although mass transfer is extremely important, in everyday life mass transfer processes tend to be hidden and are much less familiar than fluid flow or heat transfer.

• There are at least five different models for mass transfer. Since they all look at the same phenomena, ultimate predictions of mass transfer rates and concentration profiles of the different models should be similar. However, each of the five has its place: they are useful in different situations and for different purposes.

• Mass transfer requires logical analysis and cannot be reduced to a plug-and-chug procedure.

• The mathematics, which includes the solution of ordinary and partial differential equations, can be formidable. Classical mathematical solutions are useful for constant coefficient systems with simple boundary conditions and are invaluable for benchmarking numerical methods.

• Because most real mass transfer problems involve varying parameters and/or complex geometries, numerical solution methods are often required.

This chapter covers three mass transfer models (Fickian, linear driving force, and Maxwell-Stefan) in detail. Difficult classical mathematics involving solutions of partial differential equations and of coupled systems are beyond the scope of this introductory treatment. Numerical methods are limited to solution of ordinary differential equations. We start in Section 15.1 with a nonmathematical molecular picture of mass transfer (based on the kinetic theory of gases) that is useful for understanding basic concepts.

For robust correlation of mass transfer rates with different materials, we need a parameter—the diffusivity that is a fundamental measure of the ability of solutes to transfer in different fluids and solids. Since this parameter is not directly measured, we need a model that allows calculation of diffusivity from experimental data. In Section 15.2 we discuss the Fickian diffusion model, which is the diffusivity model usually taught in undergraduate chemical engineering courses. Typical values and correlations for Fickian diffusivity are discussed in Section 15.3. The Fickian model is convenient for binary mass transfer but has severe limitations for multicomponent mass transfer.

In Section 15.4 an engineering approach to mass transfer, the linear driving-force model introduced in Eq. (1-4), is explored in more detail. This approach is widely applied for mass transfer between two phases. Correlations for mass transfer coefficients can be developed based on dimensional analysis, and constants in the correlations can be fit to experimental data. The linear driving force model is used in Section 15.4.3 to analyze unsteady shrinkage or expansion of bubbles and other objects. In Section 15.5 correlations for mass transfer coefficients are presented. Additional correlations are presented when needed in Chapters 16 to 19.

If you have taken a chemical engineering mass transfer course, Sections 15.1 to 15.5 will contain familiar material. If you have not taken a mass transfer course, Sections 15.1 to 15.5 are the minimum material required to proceed to Chapters 16 to 19.

Section 15.6 describes the deficiencies in the Fickian model and points out why an alternative model is needed. The Maxwell-Stefan model of mass transfer and diffusivity explored in Section 15.7 has advantages for nonideal systems and multicomponent mass transfer but has a reputation of being computationally difficult. Inclusion of the Maxwell-Stefan model in an introductory treatment is unusual, but I believe undergraduate chemical engineers need to learn both Fickian and Maxwell-Stefan models. Fortunately the difference equation approximation used for solving problems makes mathematical manipulations quite tractable (Wesselingh and Krishna, 1990, 2000).

The final mass transfer model, irreversible thermodynamics (deGroot and Mazur, 1984; Ghorayeb and Firoozabadi, 2000; Haase, 1990), is beyond the scope of this introductory treatment. The advantages, disadvantages, and relationships among the models are delineated in Section 15.8. Applications of mass transfer theories to separations are covered in Chapters 16 to 19.

1. Explain qualitatively how molecular motion leads to diffusion

2. Describe Fick’s model of diffusion in words and equations, and use the model to solve steady-state binary diffusion problems without convection

3. Choose an appropriate reference velocity v_ref, and solve Fick’s model for steady-state binary diffusion with convection

4. Estimate diffusivity of gases and liquids in binary systems

5. Explain and use the linear driving-force model to solve steady-state problems

6. Derive and solve unsteady mass balances for expansion or contraction of objects controlled by pseudo steady-state mass transfer

7. Estimate mass transfer coefficients

8. Explain the deficiencies in the Fickian model of diffusion

9. Describe how the Maxwell-Stefan model differs from the Fickian model, and use the Maxwell-Stefan model to solve ideal and nonideal binary and ideal ternary diffusion problems

The number of molecules present in a volume (e.g., 1.0 ml) can easily be estimated by remembering that a gram mole consists of Avogadro’s number (6.023 × 10²³) of molecules. If we have liquid water at 20°C (molecular mass = 18.016 and density = 0.998 g/ml), there are 3.35 × 10²² molecules/ml—truly a large number! For an ideal gas (say, nitrogen with molecular weight [MW] = 28.0), a mole occupies 22.4 L at STP (0°C and 1.0 atm). In this case there are 2.69 × 10¹⁹ molecules/ml—fewer but still a huge number. Under normal pressures, so many molecules are present that collisions with other molecules are much more likely than with the wall (an exception is Knudsen diffusion, discussed in Section 18.6.1). With an enormous number of molecules, the change in number density of molecules caused by the huge number of collisions appears to be continuous instead of discontinuous. This apparently continuous behavior allows us to use our normal continuous (differential) mathematics.

If we introduce a different type of molecule at one place in the container (e.g., a bit of helium in the nitrogen), both helium and nitrogen molecules move randomly because of thermal energy. As a result of the huge number of collisions that occur both helium and nitrogen spread randomly throughout the container. At equilibrium there is an equal density of total molecules everywhere in the container and an equal density of helium molecules everywhere. The net result of this random movement is that helium molecules on average move from high concentrations to low concentrations. This process of movement of molecules from a region of high concentration to a region of low concentration is called molecular diffusion.

Although oversimplified, this picture gives a reasonable starting point for studying binary diffusion. Since the velocity of molecules increases with higher thermal energy, we would expect that diffusion rates (however defined and measured) increase as temperature increases. Since gases have fewer molecules per volume than liquids, random movement of the molecules are less impeded in a gas. Thus we would expect higher diffusion rates in gases than in liquids.

If we consider a slightly more complex arrangement, we could continually flow a stream relatively concentrated in ethanol on one side of the space and flow a stream relatively less concentrated in ethanol on the other side of the space, which is essentially what we do in many separation processes. If we flow fairly rapidly, we will prevent the entire space from ever reaching equilibrium (same ethanol concentration everywhere). However, there will be a net movement of ethanol from the concentrated region to the dilute region. You have experienced a somewhat analogous situation when you want to move through a crowd of people and have to bob and weave and sometimes step backward. The difference between the situations is that people have a purpose to their movement, while molecules do not have a purpose and their motions are random. Thus in describing our models we must try to avoid assigning a purpose to molecular diffusion.

Even though this picture is quite simple, it is relatively easy to consider complicating factors and qualitatively predict their effect on molecular diffusion. For example, if the system is quite concentrated, diffusion of a large number of molecules occurs. This diffusion leads to convection (movement by flow). Although diffusion always causes convection, in dilute systems convection is small enough that it can be ignored (Sections 15.2.1 and 15.2.2). In more concentrated systems the coupling of diffusion and convection complicates the analysis (Section 15.2.3).

Consider another complication by analogy. Suppose you want to move through a crowd, but you have to take your little sister with you. You take her firmly by the hand, and the two of you zigzag through the crowd. Since the two of you together require a larger space to squeeze between people, your motion is slower than if you were alone. A similar effect occurs if molecules agglomerate or stick together (but remember that the molecules do not have a purpose for their movement). The larger group of molecules moves more slowly, so measured diffusivity is lower. This situation is discussed in Section 15.3.

Another situation we can explore by analogy occurs when you want to go in one direction, but a number of people are headed in another direction. The contact, or friction, with these other bodies tends to carry you in the direction they are going, and if there is a sufficient number of them, you may be swept along with them. The molecular equivalent can occur in a flow situation. Assume we have a water stream with a modest amount of methanol and a small amount of ethanol on the left and another water stream with a small amount of methanol and a fairly large amount of ethanol on the right. If we allowed the system to come to equilibrium, we would have equal methanol concentrations everywhere. In the nonequilibrium flow situation, we can continually transfer ethanol from the right to the left. We (the people—not molecules, which just move randomly) would expect random fluctuations to move methanol from the left to the right, but if there is sufficient ethanol movement, we may observe the reverse transfer direction of methanol. This case is discussed in Section 15.7.

The word pictures painted in this section have been quantified for gases in the kinetic theory of gases. A brief introduction to this theory is presented in Section 15.7.1.

In Chapter 1 of this book we presented Eqs. (1-5a) and (1-5b) that relate the rate of mass transfer/volume to a mass transfer coefficient, the area/volume, and a driving force. This mathematical model is an attempt to quantify a complicated situation. Although useful, this model and the other models presented later in this chapter can also be misleading. The term driving force implies purpose or desire to transfer, and as noted there is no purpose—the molecules are just moving randomly. With this caveat, let’s look at the various models used to analyze mass transfer, starting with the Fickian diffusion model.

Defining the proportionality constant as the thermal conductivity k_conduction, the definition becomes

If Q_z is the heat-transfer rate by conduction, kJ/s (= kW), in the z direction for a material with an area for heat transfer of A m² over a distance measured in m and a thermal gradient with units °C/m, then k_conduction has units kJ/(s m °C). It was later realized that a slightly more general form of this equation is

where thermal diffusivity α_thermal = k_conduction/(ρC_p), and (ρC_pT) is the system’s thermal energy. With density ρ in kg/m³ and heat capacity C_p in kJ/(kg °C), thermal diffusivity α_thermal has units m²/s. For additional information on heat transfer, see Hottel et al. (2008) and Incropera et al. (2011).

Fick showed that for molecular diffusion of a dilute solution of solute A in solvent B (a binary mixture) with no convection in the z direction,

Defining the proportionality constant as the molecular diffusivity D_AB, the definition for D_AB becomes

Since J_A,Z is the molecular flux by diffusion, (mole A)/(s m²), in the z direction over a distance measured in meters and concentration gradient has units (mole A)/m³, D_AB has units of m²/s. Then the total amount transferred is

To calculate the total transferred we also need to know the m² of area for mass transfer.

For a given dilute binary system the Fickian diffusivity D_AB depends, as expected, on temperature but is approximately independent of concentration. Typical values of D_AB for gas systems at atmospheric pressure are 10^–5 m²/s and for liquids are 10^–9 m²/s. Experimental values and correlations for predicting D_AB are presented in Section 15.3.

Note that the thermal diffusivity and the molecular diffusivity have identical units. In addition if we consider that 1/α_thermal is the resistance to transferring energy and 1/D_AB is the resistance to transferring mass, then Eqs. (15-1c) and (15-2b) can both be written in the form

For mass transfer the flux of moles of solute A is J_A,Z and the driving force is (dC_A/dz).

The usual form for writing Fick’s law with no convection in the z direction is

This form or equivalent Eq. (15-2b) is normally used to analyze experimental data to determine values of the Fickian diffusivity D_AB. However, Sherwood et al. (1975) and Bird et al. (2006) point out that a fundamentally more correct form is

In this equation y_A is mole fraction of A, and C_m is mixture concentration (mol total mixture)/m³. The equations are identical for dilute, isobaric, and isothermal gases because C_A = y_AC_m and C_m is constant. However, in nonisothermal situations, Eq. (15-4b) is the correct form (see Problem 15.A1).

Since Fick cast his equation in a familiar form and since Eqs. (15-2b) and (15-4a) fit data for isothermal dilute binary systems very well, this equation rapidly became enshrined as Fick’s law (sometimes called Fick’s first law). However, problems arose when other researchers extended Fick’s work to more concentrated systems. In Section 15.2.3 we will see that when there is significant convection in the diffusion direction, Fick’s law needs to be modified. This picture becomes more complicated, but Fick’s law remains valid. As we shall see later, when extended to concentrated, nonideal systems or to multicomponent systems, Fick’s law often requires very large adjustments of molecular diffusivity—sometimes with negative values—as a function of concentration to predict behavior. Said in clearer terms Fick’s law no longer applies. We should not blame Fick for this lack of agreement. His law works fine for the conditions for which he developed it.

Figure 15-1 illustrates classic steady-state diffusion across a thin film at constant pressure and temperature with no convection in the direction of diffusion (z direction). At steady state there is no accumulation in the film, and the concentration profile does not change with time. This simple geometry can be used to experimentally determine diffusivity based on the definition derived from the Fickian model, Eq. (15-2b).

Example 15-1 is probably the simplest diffusion problem possible. Reread part C of this example noting the method used to determine the concentration profile. If data to determine flux, J_A,z, had not been given, we would have substituted Eq. (15-8d) into Eq. (15-4a) and calculated the flux:

Note that with the problem defined with both concentrations and the diffusion length, L, given, the concentration profile does not depend on diffusivity, although the flux does depend on diffusivity. This result is not universal, and it is easy to pose problems in which the concentration profile does depend on diffusivity (e.g., Problem 15.D.1).

Equations very similar to Eqs. (15-8) and (15-9) occur in a number of situations, such as steady-state evaporation (Example 15-2) and steady-state permeation of gases and liquids in membranes.

For a segment of thickness Δz, the mass balance per unit area is Accumulation = Input – Output. The amount of material in a segment of thickness Δz at any time t is C_A(t) Δz. Since accumulation is the change in this amount of material, the mass balance becomes

Dividing by Δz and taking the limit as Δz → 0 (we need to assume this limit exists) we obtain

Since terms are now functions of both t and z, partial derivatives are required. Assuming constant diffusivity and substituting in Fick’s law, Eq. (15-4a), we obtain

This equation is often called Fick’s second law. Note that for steady state, Eq. (15-12c) becomes Eq. (15-8b). As expected, a very similar equation can be derived for unsteady-state heat conduction in an infinite slab (Incropera et al., 2011).

Boundary conditions for an infinitely thick slab are

Defining the dimensionless distance ς (Greek letter zeta),

the solution for Eq. (15-12c) that satisfies the boundary conditions is

where erf is the error function. The definition and properties of the error function are discussed in Section 19.7.1 in conjunction with the solution for unsteady dispersion in adsorption columns. Since the error function is supported by Excel, calculations with a spreadsheet are straightforward. If C_A,0 > C_A,∞, a convenient form of Eq. (15-14b) for plotting results is

This equation was used to plot Figure 15-3 for diffusion of sucrose in liquid water.

For different geometries it can be useful to write Fick’s second law in cylindrical coordinates for transfer in radial direction only:

and for spherical coordinates for transfer in radial direction only

Except for this section and Section 18.7 solutions of unsteady diffusion equations in one to three dimensions are beyond the scope of this book. Solutions to Eqs. (15-12c), (15-12d), and (15-12e), corresponding two- and three-dimension equations, and equivalent heat conduction equations have been extensively studied for a variety of boundary conditions (e.g., Crank, 1975; Cussler, 2009; Hines and Maddox, 1984; Incropera et al., 2011). Readers interested in unsteady-state diffusion problems should refer to these or other sources on diffusion. However, solutions to unsteady problems using a linear driving-force model are much more tractable and are analyzed in Sections 15.4.3 and 15.7.7.

A useful analogy is to consider a sugar molecule moving in your body as you walk across a room. Overall movement of sugar will be dominated by your movement in the room (the room serves as a fixed frame of reference, and flux measured in this frame is N). To study the movement of the sugar, we first look at flux in your body using the body as the reference frame (meaning sugar movement is measured with respect to the body, not the room). Once we know flux J with respect to the moving reference frame, we can add flux J to the movement of your body to find N.

Since both convection and diffusion occur, we need to separate terms. In the Fickian model there are multiple correct methods to separate mass transfer into diffusion and convection terms. As a result, values of these two terms can be different if the calculation is done in different, yet correct, ways. However, the final answer, the total amount of mass transferred, will be the same. The usual assumption is that we can find a reference frame with diffusion and no convection. The diffusion problem is then solved in this reference frame. Then we add the effects of convection in a fixed reference frame to obtain the total flux, N_A mol/(m²s).

This step certainly makes sense in the analogy with your body. Fick’s law is applied in a reference frame (your body) with no convection, which allows use of the procedures of Section 15.2.2. Since we are doing a steady-state analysis the total fluxes, N_A and N_B [mol/(m²s)] are constant (e.g., not functions of z); however, the diffusive and convective fluxes may depend on z. To use this separation of terms, choose a reference velocity v_ref (z) so that there is no convection in the reference frame. This is always possible in a steady-state system. The convective flux is defined in terms of the reference or basis velocity v_ref times the molar concentrations.

where C_A and C_B are in mol/m³ and v_ref is in m/s. Since v_ref, C_A, and C_B can depend on z, the convective fluxes can depend on z.

The total flux of components A and B (due to both diffusion and convection) can be defined in terms of currently unknown component transfer velocities v_A(z) and v_B(z):

The diffusive flux of A is given by Eq. (15-4a), since it is calculated in a reference frame with no convection, and there is a similar equation for B. The diffusive flux of A can also be determined by subtracting Eq. (15-15c) from (15-15e) because J_A is the difference between the total flux and the convective flux (in the analogy, this step allows us to look at only what is happening within your body). Comparing the diffusive flux found from Eqs. (15-15c) and (15-15e) with the result from Eq. (15-4a), we have

In a similar fashion we obtain

Solving for the component transfer velocities,

If we expand Eqs. (15-15e) and (15-15f)

N_A = C_A v_A = C_A(v_A – v_ref) + C_A v_ref

N_B = C_B v_B = C_B(v_B – v_ref) + C_B v_ref

and combine the expanded terms with Eqs. (15-16a) and (15-16b), we obtain the total fluxes:

To find the total fluxes we have to decide on an appropriate reference coordinate system and reference or basis velocity. Because the reference coordinate system is defined as a system in which there is no convection, the net flux of A plus B must be zero; otherwise there would be convection. Thus in the reference coordinate system moving at reference velocity v_ref, by definition,

If we add Eqs. (15-16e) and (15-16f), apply Eq. (15-17a) to remove the diffusivity terms, and note that mole fractions in a binary gas system are

(C_m is total molar concentration [e.g., kmol/m³]). The molar reference velocity is

In a liquid system replace the symbol y with x. If we want to calculate a mass flux while retaining the molar reference velocity, we can multiply Eqs. (15-16e) and (15-16f) by MW_A and MW_B, respectively.

Following steps similar to those used to derive the molar reference velocity, we can develop the mass reference velocity:

For a volumetric flux divide Eqs. (15-16e) and (15-16f) by the molar densities of A and B, respectively, and proceed with the same steps in the derivation to develop the volumetric reference velocity:

Remembering that fractions are defined so that

Eq. (15-17c1) can be written in terms of the molar fluxes:

There are similar equations for v_ref,vol and v_ref,mass. Note that reference velocities v_ref,vol, v_ref,mol, and v_ref,mass can be identical or different depending on whether volume, mole, or mass fractions are the same. For example, volume and mole fractions are identical in an ideal gas, and v_ref,vol = v_ref,mol. If the molecular weights are different, v_ref,mol ≠ v_ref,mass. Why have we introduced three reference velocities? And how do we decide which reference velocity v_ref should be used?

Use the reference velocity that makes the problem as simple as possible! The choice of reference velocity and the solution of Fickian diffusion problems are best illustrated with examples. A reference velocity of zero is often a good choice; however, in the analogy with your body the reference velocity was not zero, but the solution was not difficult. In this analogy we would probably use mass fractions, with A as the sugar and B as everything else, to calculate v_ref,mass. Since there is a lot more mass of the body than mass of the sugar, x_B,ref = x_{Body,wt frac} >> x_A,ref = x_{sugar,wt frac}, x_B,ref ∼ 1, and v_ref,mass ∼ v_Body,mass. So to find the total flux of sugar, we add the velocity at which you walk across the room to the diffusive flux of sugar within your body. We could use v_ref,vol in this analogy with your body, but mass is more familiar.

For a second example assume we have two fixed, equal-volume chambers (Figure 15-4) that contain different ideal gases but are at the same pressure and temperature. With ideal gases at constant pressure and temperature, there is no volume change on mixing. Since J_A = –J_B, there will be equal and opposite molar flows of gases A and B, and the molar reference velocity v_ref,mol = 0. This case is an example of equimolar counter diffusion, which obviously simplifies the equations. Equations (15-16e) and (15-16f) become

Use of a volume average reference velocity also results in v_ref,vol = 0, but if molecular weights are different, a mass average reference velocity will not be zero. Even if the gas is not ideal, the use of molar or volumetric reference velocities will often make an approximate solution easier.

Calculation of diffusion in distillation columns tends to be easier if the molar average reference velocity v_ref,mol is used. In distillation, constant molal overflow is often valid or close to valid (Section 4.2). The resulting equimolar counter diffusion results in N_A = –N_B, and there is no convection in the reference frame with v_ref,mol = 0. If we choose the reference velocity as molar average velocity, the result is Eq. (15-18). A similar simplification occurs for linear driving force analysis in Section 15.4 and for analysis of distillation in Section 16.1.

A third example that frequently occurs in evaporation, absorption, and stripping is binary diffusion through a stagnant fluid. In this case N_B = 0. Setting N_B = 0 in Eq. (15-17d), we find v_ref,mol = N_A/C_m. Substitution of this result into Eq. (15-16e), defining gas mole fraction y_A = C_A/C_m, and solving for N_A, we obtain the flux equation for A:

This result is derived in a different way in Example 15-3.

Most liquids have no or very little volume change on mixing—one of the ways to estimate the density of a liquid mixture is to assume volumes are additive. Even liquid systems with large volume changes on mixing rarely have more than a 10% change. Molar and mass densities are usually significantly less constant. In this case the volumetric reference velocity v_ref,vol is the simplest reference velocity to use.

In separations it is common to use v_ref,mol for distillation and absorption, v_ref,mass for extraction, and v_ref,vol for gas permeation through membranes. In fluid dynamics the mass average velocity is usually chosen as the reference velocity. On the other hand, Cussler (2009) states that use of the volumetric average velocity v_ref,vol often results in the simplest diffusion problem.

Diffusion problems are almost always solved as special cases. These solutions are tabulated in significant detail in Crank (1975), Cussler (2009), Hines and Maddox (1984), and Incropera et al. (2011). Example 15-3 presents the special case of steady-state diffusion of component A through a stagnant layer of B. This case is important for evaporation, absorption, and stripping.

Students often find this section confusing. The choice of reference velocity based on volume, mole, or mass appears arbitrary and may result in different values for convective flux. This in turn results in different values for diffusive flux. Even though total fluxes of A and B do not change, how is this correct? In physical situations there is only total flux. Separating this flux into convective and diffusive terms is a human invention. The method works because diffusive flux is defined with respect to a reference frame in which convective flux is zero. If we change v_ref, we change convective and diffusive fluxes by the exact amounts required to keep total fluxes unchanged. If we report diffusive flux and convective flux, we need to be clear whether a volume, mole, or mass basis was used. Another way to think about this calculation is total fluxes of A and B are state functions—they do not depend on the calculation path chosen. Diffusive and convective fluxes are path functions and depend on how the calculations are done.

Please do not consider the tables as an opportunity for a reading break. Instead of skipping Table 15-1, study the numbers and search for patterns. Obviously D_AB increases as temperature increases. By comparing data for different alcohols in air at 298 K the pattern that emerges is D_AB values are larger for molecules with lower molecular weights. This is generally true even for very different molecules if there is a big difference in molecular weights.

Diffusivities for gas pairs can be fairly accurately predicted from kinetic theories. A simple kinetic theory for hard spheres predicts (Cussler, 2009):

T is absolute temperature in Kelvin, is an average molecular weight, p_tot is total absolute pressure in atmospheres, and σ is average diameter of the spherical molecules in Å.

The more detailed and accurate Chapman-Enskog kinetic theory is valid for nonpolar molecules to about 70 atm. This equation with D_AB in m²/s (Cussler, 2009; Geankoplis, 2003; Sherwood et al., 1975; Wankat and Knaebel, 2008) is

The collision diameter σ_AB is determined from the Lennard-Jones potential parameters,

and dimensionless Ω_D is a collision integral that is a function of k_BT/ε_AB, where k_B is Boltzmann’s constant (1.38066 × 10^–23 J/K) and ε_AB is the Lennard-Jones energy of interaction, which can be calculated from values for the two molecules:

Table 15-2 is a brief list of Lennard-Jones parameters and values of Ω_D. Detailed tables are available for a variety of compounds to calculate these parameters (e.g., Cussler, 2009; Hirschfelder et al., 1964). An empirical fit for Ω_D is given in Eq. (15-21e) (Wankat and Knaebel, 2008):

Because the collision integral is temperature dependent, gas diffusivities are proportional to T² at low temperatures and to T^1.66 at high temperatures. Wankat and Knaebel (2008) summarize other methods to predict gas diffusivities.

The kinetic theory of gases can also be used to predict the dimensionless Schmidt number, Sc_V = µ/(ρD_AB), which is useful because the Schmidt number is used in correlations for the mass transfer coefficient (see Section 15.5). For diffusion of component A at low concentrations in gas B, Sc_V can be estimated from (Sherwood et al, 1975)

Ω_V is the viscosity collision integral. Values of Ω_D/Ω_V given in Table 15-2 can be calculated from k_Bε_B/T.

Although prediction methods are available, it is always better to have an experimental value for D_AB and then adjust for pressure or temperature differences. Equations (15-21a) and (15-21b) predict the same inverse dependence of diffusivity on pressure but a slightly different dependence on temperature. If a single experimental value is available, D_AB can be accurately predicted at any temperature or pressure. This is illustrated in Example 15-4.

*Cussler (2009); Demirel (2013); Geankoplis (2003); Poling et al. (2008); Sherwood et al. (1975); Treybal (1980); Tyn and Calus (1975)

There are a number of theories for predicting diffusivity of liquids at infinite dilution (Cussler, 2009; Kirwan, 1987; Poling et al., 2001; Sherwood et al., 1975; Wankat and Knaebel, 2008). Many of these theories use the Stokes-Einstein equation as a starting point (Bird et al., 2006; Cussler, 2009; Wankat and Knaebel, 2008).

In this equation µ_B is solvent viscosity, and is the solute radius, which is assumed to be a rigid sphere with gravity as the only body force. The Stokes-Einstein equation works best for unhydrated molecules with a molecular weight greater than 1000, and even there it is not very accurate.

The most popular theory for diffusivity of liquids is Wilke-Chang theory (Cussler, 2009; Sherwood et al., 1975; Wankat and Knaebel, 2008), which uses the Stokes-Einstein equation as a starting point and predicts infinite dilution diffusivity of solute A in solvent B, .

In this equation V_A is molar volume of solute in m³/kmol at its normal boiling point, T is in Kelvin, solvent viscosity µ_B is in Pa • s [kg/(m s)], φ_B is a solvent interaction parameter, and D°_AB is in m²/s. There is some disagreement in the literature on appropriate values of φ_B, particularly for water. The following values are recommended for different solvents (Wankat and Knaebel, 2008): water, φ_B = 2.26 (other authors such as Geankoplis [2003] recommend φ_B = 2.6); methanol, φ_B = 1.9; ethanol, φ_B = 1.5; propanol, φ_B = 1.2; other solvents, φ_B = 1.0.

Although not obvious from the Wilke-Chang equation’s form, because of variation of viscosity with temperature, it predicts an Arrhenius dependence on temperature (Kirwan, 1987):

Typical values for the activation energy E_o are approximately 10,000 J/mol and R = 8.314 J/(mol K). This equation can also be used at constant mole fractions instead of infinite dilution if data are available (see Problem 15.D5). Alternatively, if the effect of temperature on solvent viscosity is known, we can write Eq. (15-22b) for both temperatures and take the ratio of these equations:

In more concentrated solutions Eq. (15-22d) becomes

which although not exact is still useful.

Unfortunately, except in quite dilute mixtures, liquid diffusivity is rarely constant, and very large changes in D_AB are often observed going from a trace of A in almost pure B (infinite dilution limit) to a trace of B in almost pure A (other infinite dilution limit). For example, compare the change in D_AB in Table 15-3 for a relatively ideal chlorobenzene–bromobenzene system at 283.3 K to the change in D_AB for a nonideal ethanol–water system at 298.15 K. The former values increase modestly, while the latter values go through a minimum as water mole fraction increases.

If the diffusivity is not constant, we would at least hope that there is a relatively simple relationship that allows calculation of D_AB at any concentration based on the two infinite dilution limits, and . The Vignes correlation based on the activity coefficient γ_A is reasonably accurate (Kirwan, 1987; Sherwood et al., 1975; Treybal, 1980) for moderately nonideal systems:

The last term is a thermodynamic correction factor for nonideal solutions. Equation (15-22f) predicts that D_AB is not constant even for ideal systems [term in brackets in Eq. (15-22f) has a value of 1.0]. Equation (15-22f) can also predict a negative diffusion coefficient, which is an indication of the formation of two liquid phases.

The most accurate results are obtained not by predicting diffusivity but by obtaining experimental values at two, preferably more, temperatures and then adjusting for temperature differences. Activity coefficient data is required to adjust for concentration changes in nonideal systems.

For systems at constant temperature and pressure, only the concentration effects on diffusivity need to be considered. In addition, total molar density C_m can be a function of concentration. The effects of concentration on diffusivity in liquid systems are considered in detail in section 15.3.2. In this section we assume that experimental values of diffusivity are available. A simple numerical solution method for steady-state diffusion problems with convection when D_AB and C_m depend on concentration is illustrated.

Note that the estimate done initially with D_EW and C_m calculated at x_E = 0 was off by more than a factor of two.

As noted in Section 15.1, the use of driving force can be misleading, since the molecules have no brain or intention to transfer, and in molecular diffusion processes transfer occurs because of random collisions. However, since driving force is embedded in the jargon of chemical engineering, we use the term here. The strength of Eq. (15-24a) is that it is very broad and flexible and can be applied to a wide variety of situations. The weakness of this equation is that it is empirical, and its theoretical background is based on film theory (Section 15.4.1), which is not always applicable. In some practical situations a few experiments will suffice to provide the mass transfer coefficient in the range of interest. If temperature, pressure, and concentrations vary significantly in the separator, successful application of Eq. (15-24a) requires a correlation for the mass transfer coefficient based on extensive experimental data.

Equation (15-24a) can be written in terms of concentrations as

If the flux is desired in (kmol A)/(s m²), the area across which the mass transfer occurs is measured in m², and concentration is in (kmol A)/m³, then the mass transfer coefficient k_c has units of velocity, m/s. In single-phase systems the concentrations C_A,2 and C_A,1 are the (kmol A)/m³ at locations 1 and 2, which are usually the system boundaries. For mass transfer from one phase to another, one of the concentrations would be the concentration at the interface. Typical values for the mass transfer coefficient are 0.1 m/s for gases and 10^–4 m/s for liquids (Wesselingh and Krishna, 2000).

We can also write the mass transfer equation in terms of liquid mole fractions:

In terms of vapor mole fractions the equation is

In these cases mass transfer coefficients k_x and k_y have units (kmol A)/[m² s (mole fraction A)] or equivalently (kmol fluid mixture)/(m² s). Occasionally, flux is written in terms of a partial pressure driving force, p_A = y_Ap_tot,

In this case mass transfer coefficient k_p = k_y/p_tot has units (kmol fluid mixture)/(m² s bar).

In Section 15.2.1 the analogy between heat and mass transfer was discussed. This analogy also extends to linear driving force models as long as heat transfer by radiation can be neglected. The heat-transfer equation analogous to Eq. (15-24a) is

The heat-transfer equation analogous to Eqs. (15-24b) to (15-24e) is

Here Q_z/A is the heat flux in J/(m²s), h_{heat transfer} is the heat-transfer coefficient in J/(m²s • K), and T₂ and T₁ are temperatures at the two system boundaries, K. The analogy between heat and mass transfer will prove useful in determining correlations for the mass- and heat-transfer coefficients in Section 15.5.4.

Since film theory postulates that mass transfer occurs across a stagnant film, we can use the Fickian diffusion solution from Eq. (15-9) for diffusion across a thin layer of fluid. If this thin layer is a gas film of unknown thickness Δ and we convert from concentrations to mole fractions, we obtain

For systems that are dilute or have equimolar counter transfer N_A,z = J_A,z. Since film thickness Δ_gas is unknown, the mass transfer coefficients k_c,gas and k_y are defined as

Then the equation for mass transfer for dilute or equimolar counter transfer is

Because it uses the Fickian diffusion solution for diffusion across thin layers to define the mass transfer equations and mass transfer coefficients, the linear driving-force model has Fickian genes. Thus the linear driving-force model inherits the good traits (e.g., simplicity, analogous to heat transfer, works well for binary systems, vast quantities of data) and the bad traits (nonideal diffusivities are difficult to predict, theory does not extend easily to multicomponent systems, see Section 15-6) of Fickian diffusion.

For equilibrium staged and sorption separations, we are interested in mass transfer from one phase to another. This is illustrated schematically in Figure 15-5 for transfer of component A from the liquid to the vapor phase. x_A,I and y_A,I are interfacial mole fractions. For dilute absorbers and strippers and for distillation where there is equimolar counter transfer, mass transfer Eqs. (15-26d) and (15-26e) can be written for each stage in the following different forms:

or

where k_y and k_x are the individual mass transfer coefficients with mole fraction driving forces for vapor and liquid phases, respectively, and k_c,gas and k_c,liq are the individual mass transfer coefficients with concentration driving forces for vapor and liquid phases, respectively. At steady state the rates of mass transfer in the gas and liquid phases have to be equal, and N_A,z,gas = N_A,z,liq. Additionally, the assumption is usually made that the gas and liquid at the interface are in equilibrium. If there is a resistance to obtaining equilibrium at the interface (e.g., if a surface-active agent is present), then an equation for this resistance must be included.

Unfortunately, there are two major problems with these equations when they are applied to vapor-liquid and liquid-liquid contactors. First, the interfacial area A_I between the two phases is very difficult to measure. This problem is usually avoided by writing Eqs. (15-27) as

or

where a is the interfacial area per unit volume (m²/m³). Since a is no easier to measure than AI, we often measure and correlate the products k_ya, k_xa, k_c,vapa, and k_c,liqa. Typical units for k_ya and k_xa are kmol/(s m³) or lbmol/(h ft³), and for k_ca typical units are [(m/s)(m²/m³)] = s^–1.

The second problem is that the interfacial mole fractions are also very difficult to measure. To avoid this problem, mass transfer calculations often use a driving force defined in terms of hypothetical equilibrium mole fractions.

These equations, which are a repeat of Eqs. (1-5), define the overall mass transfer coefficients K_y and K_x. Typical units for K_ya and K_xa are kmol/(s m³) and lbmol/(h ft³). is the vapor mole fraction, which would be in equilibrium with the bulk liquid of mole fraction x_A, and is the liquid mole fraction that would be in equilibrium with the bulk vapor of mole fraction y_A.

To obtain the relationship between the overall and individual coefficients, we begin by assuming there is no resistance to mass transfer at the interface. This assumption implies that x_I and y_I must be in equilibrium. The mole fraction difference in Eq. (15-29a) can be written as

where m is the average slope of the equilibrium curve (y_A versus x_A) at x_A and x_Ai.

Combining Eq. (15-30a) with Eqs. (15-28) and (15-29a), we obtain

which leads to the result

Similar manipulations starting with Eq. (15-29b) lead to

If there is a resistance at the interface, then this resistance must be added to Eqs. (15-31a), (15-31b), and (15-31c). This sum-of-resistances model shows that the overall coefficients will not be constant, even if k_x and k_y are constant, if the equilibrium is curved and m varies. In binary distillation m is the tangent to the equilibrium curve at x_A,avg = (x_A,I + x_A)/2, and since the equilibrium curve is never straight, m has to vary. For example, if we have a constant relative volatility system with equilibrium given by Eq. (2-22b), m from Eq. (15-30b) is

As an example of the variation in m, if the relative volatility α = 2.5, at x_A,avg = 0.01, m = 2.43; at x_A,avg = 0.5, m = 0.82; and at x_A,avg = 0.99, m = 0.40. At the very least, average values of m need to be determined separately for the stripping and enriching sections.

Equations (15-31a), (15-31b), and (15-31c) also show the effect of equilibrium on the controlling resistance. If m is small, then from Eq. (15-31b) K_y ∼ k_y and the gas-phase resistance controls. If m is large, then Eq. (15-31c) gives K_x ∼ k_x and the liquid-phase resistance controls. Absorption of sparingly soluble gases that can have very large Henry’s law constants (Section 12.1) are an example of a liquid-phase, resistance-controlled separation, while absorption of very soluble gases (e.g., HCl in water) often have gas-phase resistance controlling. In distillation, m often varies from greater than 1.0 at low concentrations of MVC to less than 1.0 at high concentrations of MTZ. As a result, both resistances are important.

Although Figure 15-5 and Eqs. (15-31a), (15-31b), and (15-31c) were specific for a gas-liquid interface, the geometry and resulting mass-transfer equations apply to a number of situations in which two media are in intimate contact. For example, in Section 16.7.1 Eq. (16-31c) is applied to liquid-liquid extraction. This analysis can also be applied to solid-liquid interfaces in leaching or crystallization (Section 17.6.1) and to membranes in series (Problem 18.C7).

By now it should be obvious that you need to be careful to keep capital and lowercase letters clearly different from each other. This is also true in Chapter 12 where mole fractions and mole ratios are different. You must also be careful with subscripts. One reason that engineering is challenging is that you not only must understand the big picture but also must get the details correct.

The solution obtained in Example 15-3, Eq. (15-20a), can be written for transfer between phases as

In this equation y_A,I is the mole fraction of A at the interface (see Figure 15-6), and y_A,bulk is the mole fraction of A in the bulk of the gas. Equation (15-32b) can also be written as

The log mean differences (1 – y_A)_lm and (y_B)_lm are defined as

Note that as y_A → 0 (very dilute systems), (1 – y_A)_lm → 1, and Eq. (15-32c) simplifies to (15-26a).

The reason for doing this rather convoluted algebra is that for concentrated systems with N_B = 0, if we define the concentrated mass transfer coefficient as

then the flux becomes

which mimics the flux for dilute systems, Eq. (15-26b). This analysis can also be done based on partial pressure differences and a log-mean partial pressure difference (see Problem 15.C3). The concentrated system analysis is used in Section 16.4.

In this section we consider situations in which only mass transfer outside the object is important. This will be the case if the mass transfer rate inside the object is much greater than the rate outside the object or if the object is a pure compound (e.g., a bubble of oxygen). Chapter 16 explores interphase mass transfer where transfer rates on both sides of the interface are important.

The mass balance for any object with no reaction occurring is

With a single mass transfer process this is

where m is the mass of the object, kg, and is the flow rate of mass due to mass transfer, kg/s. For inlet flow of mass > 0, and for outlet flow of mass < 0. The mass of the object can be determined from the volume and the density of the object, and the mass flow rate can be determined from the flux and the area for mass transfer:

The volume and areas for mass transfer depend on the geometry (see Table 15-5).

If the solute being transferred is dilute in the external fluid, then the mass flux is

where ρ_fluid, x_A,bulk, and x_A,I are mass density, bulk mass fraction of A, and interfacial mass fraction of A in the fluid external to the object. Determination of the value of interfacial mass or mole fractions depends on the physical situation. If x_A > x_A,I, the flux is into the object, and if x_A < x_A,I, the flux is out of the object.

Shrinking sphere. Small bubbles will be spherical, and the formulas for volume and mass transfer area are given in Table 15-5. The mass balance equation for a shrinking sphere is

If ρ_sphere is constant, after taking the derivative and switching the mole fraction difference to the positive quantity (x_AI – x_A,bulk), we obtain

Simplifying

Separating variables and taking the integral of both sides

Assuming that k, ρ_sphere, ρ_solution, and (x_A,i – x_A,bulk) are constant and integrating

When we integrated, we assumed that (x_A,i – x_A,bulk) is not a function of time. In other words we assumed concentration profiles are established much more quickly than the time required for significant movement of the sphere boundary. This assumption is called the pseudo-steady-state assumption because the concentration profile was assumed to be at a steady state, although we know that it takes a finite amount of time to establish concentration profiles. Kmit and Shah (1996) analyzed when assuming pseudo-steady state is valid. The question becomes one of comparing time scale for shrinkage of the sphere to time scale for establishing the concentration profile. Concentration profiles develop rapidly if mass transfer coefficients are relatively high (gas systems or liquid systems with stirring or natural convection) and if the difference between x_A,I and x_A,bulk is small, which is automatically true in dilute systems but can also be valid in concentrated systems. Shrinkage (or growth) of the sphere will be slow if sphere density is much greater than the density of the continuous phase or if interfacial concentration is very low (low solubility). In this section and in Section 9.7.7 we analyze problems where pseudo-steady state is likely to be valid.

Dilute Gas Bubble. Up to this point the analysis has been general and is applicable for any shrinking sphere of pure A in a dilute fluid that satisfies the assumptions. Consider a bubble of pure, slightly soluble gas such as oxygen immersed in a nonvolatile liquid (e.g., cold water) that is not saturated with oxygen. Oxygen will transfer from the bubble to the liquid. If pressure and temperature are constant, the sphere density ρ_sphere will be constant. Gas and liquid are assumed to be in equilibrium at the interface. Equilibrium for sparingly soluble gases is given by Henry’s law, Eq. (12-1b), y_A,mol,I = (H_A/p_tot)x_A,mol,I written in terms of interfacial mole fractions.

Since the problem has been presented in mass units, we need to convert Henry’s law to mass units. For sparingly soluble gas A in the liquid B the liquid will be almost pure B so that MW_solution = MW_B, and ρ_solution = ρ_B = constant. Then

and Henry’s law becomes

For a bubble of pure gas y_A,mol,I = 1. The mass density of the sphere is the gas density, which for an ideal gas is

In Section 15.5.2 we see that for simple physical situations such as steady-state mass transfer across a stagnant film, mass transfer coefficients can be determined analytically by solving the appropriate diffusion equations. Because actual processes often do not match assumptions required by theories, the resulting theoretical correlations often are not accurate. However, with a modest amount of data coefficients can be adjusted to provide significantly more accurate semi-empirical correlations. Many common and practical situations are complex and do not allow analytical solutions of the diffusion equation. For these cases empirical correlations for k or ka are often developed on the basis of dimensional analysis and experimental data. Dimensional analysis can predict probable forms for correlations, and experimental data is used to determine the exponents on the dimensionless groups and the value of the coefficient. A few mass transfer correlations are presented in this section, and a large number are summarized in Section 5B of Perry’s Handbook of Chemical Engineering (Wankat and Knaebel, 2008).

In simple geometries determination of area A in Eq. (15-1b), (15-1c), (15-2a), or (15-2b) is straightforward and is illustrated in this section. Membrane separators (Chapter 17) have areas A and areas per volume a that can be determined from straightforward geometric calculations. For more complex systems with interfacial mass transfer the value of interfacial area per volume a in Eqs. (15-3) and (15-4) and in the height of transfer unit (HTU) terms in Table 16-1 is more difficult to determine. Although a can be determined separately, it is more common to determine the product (ka). Correlations for (ka) in packed beds for distillation and absorption are discussed in Section 16.3. Correlations to calculate k and a separately in extraction mixers are discussed in Section 16.7. Correlations to estimate k in membrane separators are given in Section 18.4.3. Correlations for (ka) in adsorption, chromatography, and ion exchange are discussed in Section 19.6.3.

Mass transfer correlations are often reported as the Sherwood number Sh, which is defined as

The Sherwood number is the ratio of the actual rate of mass transfer to the mass transfer due to diffusion alone. The appropriate length to use can be a film thickness, a diameter, or a plate length. The mass transfer coefficient can be either a local coefficient (e.g., at a given distance) or an average coefficient obtained by integrating the local coefficient over the entire distance. The dimensionless group defined in the first equation of Eq. (15-46a) is defined in terms of k_c, which uses rate Eq. (15-24b). The second and third equations of Eq. (15-46a), which use k_x or k_y from rate Eqs. (15-24c) and (15-24d), need the total molar concentration C_m in the definition of the Sherwood number to make it dimensionless.

Mass transfer coefficients often depend on velocity, represented as a Reynolds number Re.

The Reynolds number is the ratio of inertial forces to viscous forces. In pipes the Reynolds number has the familiar form Re = Dv_bρ/µ, but in other configurations, such as the flow of a liquid film down a flat plate, the definition can look superficially very different [e.g., see Eq. (15-56a)].

In correlations of gas or liquid data in separators, we often use the Peclet number Pe, which is the ratio of flow velocity to the diffusion velocity.

Forms of the Peclet number are shown with both column length L and particle diameter d_p because both forms are used. It is obviously important to know whether L or d_p is being employed in the definition of Pe. Sometimes a dimensionless group called the Stanton number St, which is the ratio of mass transfer velocity to flow velocity, is used.

The effect of the gas or liquid properties on mass transfer are taken into account in correlations by Schmidt number Sc, which can be considered the ratio of momentum diffusivity to mass diffusivity.

In Section 15.5.4 we consider the analogy between heat and mass transfer. The dimensionless group for heat transfer analogous to Sc is the Prandtl number Pr, which is the ratio of momentum diffusivity to thermal diffusivity.

The dimensionless group for heat transfer analogous to Sh is the Nusselt number Nu, which is the ratio of the actual rate of heat transfer to heat transfer due to thermal diffusion alone.

Since these dimensionless groups contain terms that depend on temperature and concentration, dimensionless groups depend on both temperature and concentration. In dilute systems concentration dependence is often negligible, and temperature-dependent solvent properties can be used.

As a simple first example consider diffusion in a stagnant film (N_B = 0) that was solved in Example 15-2 and a solution for constant total flux through the film that was given in modified form in Eq. (15-32c). Since total flux determined from diffusion and linear driving-force models, Eq. (15-32f), have to be equal, setting the equations equal for a film of known thickness δ we obtain

Solving for k_y we obtain the simple result that

Substituting this result into Eq. (15-46a) the Sherwood number correlation for stagnant films is

Use of a linear driving-force model instead of a diffusion model [Eq. (15-32f) or for dilute systems Eq. (15-26b)] for this single-phase system would be unusual, but it could certainly be done. This development is for a gas film, but repeating it for a liquid film with liquid properties and liquid-phase, mass transfer coefficient k_x is straightforward.

Generally, a linear driving force model is used when there is transfer between phases. The applications in Section 15.4.3 are examples. A classic problem that practically every mass transfer book includes is mass transfer from a gas to a falling laminar liquid film (Figure 15-8). This problem has practical significance, since falling films can occur in absorption, distillation, and stripping. In addition, the basic problem is mathematically interesting because it can be solved exactly with a series solution, and simple solutions are available for short residence times. Moreover, mass transfer becomes more complicated, requiring a semi-empirical correlation, as fluid velocity increases and ripples or waves form on the liquid surface. Finally, at even higher velocities, when the film becomes turbulent, empirical correlations are required. The theoretical model is considered in this section, and the semi-empirical and empirical models are considered in Section 15.6.3.

The diffusion equation for a falling liquid film (Figure 15-8) can be solved theoretically for absorption or stripping with the following rather restrictive set of assumptions:

1. Liquid flow is fully developed and laminar (will be relaxed in Section 15.6.3).

2. The film surface is flat (will be relaxed in Section 15.6.3).

3. Entering liquid has a constant, uniform concentration C_A,init.

4. Gas-phase mass transfer rate is much greater than liquid-phase mass transfer rate. Thus surface concentration (at z = 0) is constant at C_A,surf (will be relaxed in Section 15.6.3).

5. Diffusion rate in the liquid in the horizontal z direction is small, and convection in the z direction can be neglected.

6. The rate of convection due to the liquid flow in the vertical (y) direction is much greater than diffusion, and diffusion in the y direction can be neglected.

7. Plate width w is large enough that side-end effects can be ignored; thus velocity and concentration do not depend on the horizontal direction.

8. Diffusion coefficient D_AB is constant.

9. The system is very dilute (this assumption is not necessary but simplifies the solution).

Solution for fluid velocity in a falling film is well known (Geankoplis, 2003; Treybal, 1980).

The average velocity and film thickness δ are

where q is the volumetric flow rate divided by the width of the plate [(m³/s)/m = m²/s]. The average residence time of fluid on the plate in seconds is

The steady-state mass balance states that (In – Out) of solute due to convection (in the y direction) equals (In – Out) due to diffusion (in the z direction). With constant diffusivity, this is

Taking the limits as Δy → 0 and Δx → 0 (assuming the limits exist), this equation becomes

Mathematically, this problem has a number of interesting characteristics. It has been solved for two different sets of boundary conditions. For relatively slow diffusion rates or short residence times (short length), solute from the liquid surface does not have time to penetrate to the wall. Boundary conditions are

These boundary conditions are the same as in Eq. (15-13) for unsteady diffusion. The answer is similar to Eq. (15-14b) and to Eq. (19-69) for adsorption. This result (modified from Geankoplis, 2003; and Cussler, 2009) is

Error function is defined in Eq. (19-70), and values are tabulated in Table 19-7. Based on the following properties of the error function, Eq. (15-51a) satisfies boundary conditions in Eq. (15-50):

For a dilute system flux can be calculated from Fick’s law, Eq. (15-4a). From Eqs. (15-41), (15-51a), and (15-51b), at z = 0,

In general, for a system with N_B,z = 0, the relationship between N_A and J_A is given by Eq. (15-32a). In dilute systems this simplifies to N_A = J_A, and Eq. (15-52a) becomes

For dilute systems linear driving force equations simplify to

Setting Eqs. (15-52b) and (15-52c) equal, we obtain for dilute systems at z = 0,

or in terms of the Sherwood number with length chosen as the film thickness δ, at z = 0 this is

The local mass transfer coefficient depends on the value of y and is undefined at y = 0. Integrating Eq. (15-52b) over the range from y = 0 to y = L (the plate length), we obtain the total amount of solute transferred into the liquid at the surface of the liquid (z = 0):

We can calculate the total transfer rate as N_A,avg × (Lw). Setting this value equal to Eq. (15-53a) and solving for the average mass transfer coefficient and the average Sherwood number, we obtain

The alternative solution (Treybal, 1980; Sherwood et al., 1975) to Eq. (15-49b) does not assume that contact time is short and uses the boundary conditions

The last boundary condition at the solid wall implies there is no diffusion through the solid wall. Equation (15-49b) is more difficult to solve with this set of boundary conditions. The solution is an infinite series for the average concentration, leaving the film at y = L.

The total amount absorbed is the amount of solute exiting in the liquid minus the amount entering in the liquid:

This result can be related to N_A,avg × (Lw), where the average flux for dilute systems is determined from Eq. (15-52d). Since C_A,avg,L is given by an infinite series, this calculation is complicated. However, we can find limiting solutions for short contact times (large Reynolds number) and long contact times (small Reynolds number), where the Reynolds number for a falling liquid film is

For large values of Re, the result is Eq. (15-53b). Thus both analyses agree for short contact times. For long contact times (Re < 100 and large L), the average mass transfer coefficient and the Sherwood number for a system with no surface ripples are

Remembering that both δ and the Reynolds number increase as q increases, this result predicts that k_c,liq,avg decreases as the Reynolds number increases. This is the opposite of what normally occurs, which is k_c,liq,avg increases as the Reynolds number increases, as shown, for example, by Eq. (15-58). Thus in laminar flow, increasing velocity may not increase mass transfer coefficients or rates. Equation (15-53b) shows that at short contact times, the average mass transfer coefficient is proportional to , while Eq. (15-56b) shows that at long contact times k_c,liq,avg is proportional to D_AB.

If ripples are suppressed with a surfactant, these results are valid at least to Re = 250. Treybal (1980) reports that Eq. (15-56b) is valid to Re = 1200 if ripples are suppressed, while Sherwood et al. (1975) report that transition to turbulence occurs before a Reynolds number of 1200. This type of disagreement between sources is not unusual in mass transfer studies.

For long contact times (large L) Eq. (15-56b) will be valid if a surfactant is added or if Re < 20 (Wankat and Knaebel, 2008). For 20 < Re < 100 with no surfactant, the effect of ripples can be approximated by increasing k_c,liq,avg by 40% to 100%. Then the coefficient in Eq. (15-56b) ranges from 4.77 to 6.82. With no other information, the average value can be used.

For short contact times (small L), Eq. (15-53b) can be used with ripples by multiplying k_c,liq,avg by an average value of 1.7.

Obviously, a significant error is possible (±18%) for Eqs. (15-57a) and (15-57b) from just the multiplier for ripple formation. This type of error is not unusual for semi-empirical and empirical mass transfer correlations. These equations are semi-empirical because the basic form is from theoretical analysis, but the coefficient is from experimental data.

Sherwood et al. (1975) report that transition from laminar to turbulent occurs in the Reynolds number range from 250 to 500. Treybal (1980) reports the following correlation for a liquid film with constant surface concentration at a somewhat higher range of Reynolds numbers:

The Reynolds number Re_{liquid film} for a liquid film is defined by Eq. (15-56a), and the Schmidt number that adjusts mass transfer coefficient for changes in liquid properties is defined in Eq. (15-46e). In dilute systems solvent properties can be used to calculate the Schmidt number.

If concentration of the liquid surface is not constant, there will be mass transfer and a mass transfer resistance on the gas side also. In separation processes, the gas-phase resistance often controls because gas-phase mass transfer coefficients are often significantly smaller than liquid-phase mass transfer coefficients. For turbulent flow of gas in a wetted wall tube, Gilliland reported the following correlation (see Sherwood et al., 1975; or Wankat and Knaebel, 2008).

Reevaluation of the data showed a better fit with (Wankat and Knaebel, 2008)

Both of these equations use a log mean gas mole fraction driving force, Eq. (15-32d). For mass transfer with a rippled surface, the gas-phase mass transfer correlation (Sherwood et al., 1975; Wankat and Knaebel, 2008) is

This equation agrees with Eq. (15-46a) when Re_liq = 1000.

One additional common use of wetted-wall columns has been to study binary distillation. Johnstone and Pigford (1975) studied systems in which the gas mass transfer coefficient controlled. If partial pressure is used as the driving force, their correlation is

The modified Reynolds number is defined on the basis of the gas velocity relative to the surface of the liquid film. For countercurrent flow of gas and liquid, this is

The term v_liq,y,max is defined in Eq. (15-48b). Equation (15-60a) was used as the basis for mass transfer in each flow element of a structured packing to develop the first successful correlation for mass transfer in structured packings (Bravo et al., 1985).

A very large number of correlations are available in the sources referenced in this and later chapters. The purpose of showing these correlations is to show the types of forms that result, the need to know the range of validity, the need to know what difference or driving force is required, and the need to be very careful about the definition of terms.

Since I could not improve on it, I have closely followed Cussler’s (2009) development of Reynolds. Since eddies depend on fluid velocity, the easiest functional form is to assume eddy diffusion is linearly dependent on velocity. Then the mass transfer equation is

And in a similar fashion the equation for heat transfer is

For fluid flow, we introduce the friction factor f to find shear stress τ from the momentum ρv:

In highly turbulent flow, a terms (molecular diffusion) are much less than b terms (eddy diffusion) and can be neglected. Cussler notes Reynolds now took an amazing intuitive leap and concluded

This step says that transfer of mass, heat, and momentum by eddies is the same and requires

Quite logically, this is called Reynolds analogy. If we know either the heat-transfer coefficient h or friction factor f, we can estimate the mass transfer coefficient k. For gases Reynolds analogy is reasonably accurate, but for liquids it often fails.

Chilton and Colburn (1934) were able to empirically modify Reynolds analogy into a more robust analogy that works for gases and most liquids. Realizing that heat transfer would depend on fluid properties through the Prandtl number, Eq. (15-46f), and that mass transfer would depend on fluid properties through the Schmidt number, Eq. (15-46e), they included powers of Prandtl and Schmidt numbers in the first two terms of Eq. (15-62b). Fitting the resulting equation to experimental data showed that a power of 2/3 was appropriate and easy to use in calculations (very important in slide rule days). The result is

This equation is also written as

The definition of j_D (the mass transfer term) is j_D = (k_c/v)(Sc)^2/3, and the heat-transfer term j_H is defined as j_H = [h_{heat transfer}/(ρC_pv)](Pr)^2/3. Even when the analogy is not being used, literature sources often give mass transfer correlations in terms of j_D. If the definitions of the dimensionless groups from Eqs. (15-46a) and (15-46e) to (15-46g) are substituted in, the analogy can also be written as

Equations (15-63) apply to fully turbulent flow (although the equations also fit data for some laminar systems, such as flow past a flat plate) with a range of Schmidt number from 0.6 ≤ Sc ≤ 3000. Although not derived theoretically, there is some theoretical justification for these equations (Sherwood et al., 1975).

Because it is often easier to measure heat transfer than mass transfer, there are more heat-transfer correlations (usually in terms of Nusselt number) than mass transfer correlations (e.g., Incropera et al., 2011). Thus a major application of the Chilton-Colburn analogy is to estimate mass transfer coefficients or Sherwood numbers from existing heat-transfer correlations. In distillation mass transfer is more studied than heat transfer, and the Chilton-Colburn analogy is used to estimate heat-transfer coefficients in rate-based distillation models (Section 16.8).

Although very useful, the Chilton-Colburn analogy, as with all analogies, breaks down when the phenomena of heat transfer, mass transfer, and momentum transfer become different. For example, if radiation becomes important and is not separated from the heat-transfer term, then j_D ≠ j_H. If both skin friction and form drag (due to flow past blunt objects) occurs, then j_D ≠ f/2 and j_H ≠ f/2.

There is an additional irritating problem. For liquid binary systems D_AB usually depends on concentration, and although Eq. (15-22f) is often reasonably accurate, activity coefficient data may not be readily available. Even if activity coefficients are known, accurate prediction of infinite dilution diffusivities is difficult with current models (e.g., Sherwood et al., 1975; Wankat and Knaebel, 2008). Thus a significant amount of data may be required to accurately determine concentration dependence of binary diffusion coefficients in nonideal systems. Although this difficulty is significant, it can be overcome by dedicated laboratory analysis.

Finally, empirical extension of Fick’s law to systems with more than two components leads to logical inconsistencies and major calculation difficulties (Taylor and Krishna, 1993; Wesselingh and Krishna, 2000). For example, in ternary systems Fickian diffusion coefficients are not symmetrical, D_i,j ≠ D_j,i; thus additional constants are required. In addition, values of D_i,j depend on the value chosen for v_ref. Even worse, ternary systems can have a component diffuse into a more concentrated region or have no diffusion despite the presence of a concentration gradient (Taylor and Krishna, 1993). Accurately fitting this experimental data requires negative diffusivities.

The conclusions are that use of the Fickian model for binary diffusivities is reasonable, but the Fickian model is not suitable for systems with more than two components.

The proponents of Maxwell-Stefan theory claim it is a better model than Fickian theory. What makes one model better than another model? First, it is nice to have a model tied to the basic physics or chemistry. A simple explanation based on the physics or chemistry should explain the basic behavior. This is illustrated for the Maxwell-Stefan model in Section 15.7.1. Second, the model should not conflict with well-accepted laws, such as the first or second law of thermodynamics. The Maxwell-Stefan model incorporates thermodynamics for the analysis of nonideal systems. Third, we want a model that can explain and/or predict data and that can be extrapolated. Except for ideal gas behavior diffusion models invariably have to use measured constants (diffusivity values). A good model minimizes the number of constants required and the variation of these “constants.” If the constants vary (say with concentration), the variation should be monotonic and preferably be close to linear. The constants for multicomponent systems should be predictable from binary pairs, and no impossible or improbable values (e.g., negative values) of the constants should be required to predict the data. Based on these criteria, the Maxwell-Stefan theory is a better theory than the Fickian model.

My advice is to learn to use both models. Learn the Fickian model in order to use the vast collections of solutions and data and to be able to communicate with other engineers. Learn the Maxwell-Stefan theory because its extra power allows relatively simple solutions to problems that are very difficult to solve with the Fickian model (e.g., nonideal binary systems and multicomponent systems).

Consider an ideal gas at constant pressure and temperature consisting of gases A and B. The entire system is moving at some average molar velocity v_z,ref,mol. The coordinate system and control volume shown in Figure 15-9 are being translated at v_z,ref,mol so that there is no net flow. As noted in Section 15.1 at normal pressures there are a huge number of molecules, and collisions between molecules occur repeatedly. If the two gases are moving through each other in the z direction, we can do a force balance for a control volume of thickness Δz shown in Figure 15-9. There are forces on molecule A due to partial pressure of A (p_A × Area) at z, and at z + Δz,

and there is a friction force caused by molecules of B flowing past molecules of A. Forces other than pressure and friction are assumed to be negligible but are easily included in the continuum mechanics treatment (Datta and Vilekar, 2010). We know that friction force is proportional to velocity differences of the two gases and amounts of each gas per volume of the segment. For an ideal gas, molar densities (amounts per volume) are

Then friction force is

Setting pressure force equal to friction force and taking limit as Δz → 0, we obtain

Noting p_A = y_Ap_tot and p_B = y_Bp_tot, and for constant total pressure and constant temperature incorporating term p_tot/(RT)² in a constant of proportionality f_A,B this becomes

Term f_A,B is a friction coefficient between molecules A and B, which are moving at average velocities v_A and v_B, respectively. Inverse of friction coefficient, , is the Maxwell-Stefan diffusivity. Since system pressure is constant, the left-hand side becomes –dy_A/dz, and we obtain

The left-hand side of Eq. (15-65a) is the driving force for diffusion. The equation for component B is

If we substitute in the molar fluxes,

we obtain

It can be shown that these equations are equivalent to

Since the control volume in Figure 15-9 was set to move at the average molar velocity v_z,ref,mol, there is no net flow, J_B,z = –J_A,z, which means

This equation requires

Thus binary ideal gas Maxwell-Stefan diffusivities are symmetric.

Equations (15-65) are different forms of the one-dimensional Maxwell-Stefan equations for a binary ideal gas at constant pressure and temperature. The equation can be written in three dimensions by replacing derivatives with gradients and velocities with vectors. For ideal binary systems, Section 15.7.5 shows that results obtained from the Maxwell-Stefan equations with is identical to results obtained with the Fickian diffusion equations.

For ideal gases, the driving force on the left side of these equations reduces to (1/P)dp_i/dz, which for constant pressure systems is equivalent to dy_i/dz, and Eqs. (15-67) simplify to Eqs. (15-65a) and (15-65b). For nonideal systems chemical potential can be written in terms of activity coefficients γ_A and γ_B for liquids or the fugacity coefficients for gases. The results for component A for liquids are

or the equivalent form,

Since v_A = N_A,z/(x_Aρ_m) and v_B = N_B,z/(x_Bρ_m) where ρ_m is molar density, Eq. (15-68b) becomes

The corresponding equation for component B is

These forms are often useful in solving mass transfer problems. Obviously, detailed activity coefficient data or accurate correlations are required.

Situation 1. Equimolar counter diffusion. In this case total flux N_tot,z = 0, v_z,ref,mol = 0, and N_j,z = J_j,z. For a binary system N_B,z = –N_A,z and Eq. (15-69a) becomes

Situation 2. Distillation. If constant molar flow (CMO) is valid, then N_tot,z = 0, v_z,ref,mol = 0, and N_i,z = J_i,z. For a binary system with CMO, Eq. (15-70a) is valid. If CMO is not valid, a reasonable approximation is ΣN_i,zi = 0, which for a binary is

Multicomponent distillation is treated in detail by Taylor and Krishna (1993).

Situation 3. Flow in a stagnant fluid. This situation can occur for condensation and evaporation when there is a noncondensing gas and for absorption and stripping. For binary systems the additional equation is N_B,z = 0. This relationship leads to N_A = J_A/x_B, which is the same result as for Fickian diffusion. For a ternary system, see Example 15-10.

Situation 4. Trace component is stagnant. This happens for dilute membrane permeation with concentration polarization. The result is N_A,z = 0 and v_z,ref,mol = 0.

Situation 5. Flux ratios are specified. If N_A,z = (frac_A)N_tot,z, then for a binary system, Eq. (15-69a) becomes

Situation 6. Chemical reaction. Reaction stoichiometry determines relationship. This topic is outside the scope of this text. See Taylor and Krishna (1993) or Wesselingh and Krishna (1990, 2000).

The bars over terms mean they should be evaluated at the average conditions. If these equations are applied to mass transfer across a film, Δz = δ where δ is film thickness, we can define the average mass transfer coefficient in Maxwell-Stefan terms as . The resulting equations are

Two additional useful forms can be obtained by substituting v_A = N_A/(x_AC_m) and v_B = N_B/(x_BC_m) (where C_m is the total molar concentration) into Eqs. (15-71a) to (15-71d):

and

If the system is ideal, γ_A = 1, and the equations become

The forms in terms of the fluxes can be convenient in solving mass transfer problems.

The difference formula approximation can be made more accurate by doing several steps; however, since we need to know concentrations for the ends and center of each step, the usual method of dividing Δz into equal segments is awkward. The calculation is easier if segments are selected to give known mole fractions of A and each step size is unknown (see Problem 15.D18).

For binary systems the relationships are

Since in the limit of infinite dilution γ_A = 1, Eqs. (15-72a) and (15-72b) require the infinite dilution limits of Maxwell-Stefan and Fickian diffusivities and mass transfer coefficients are equal.

Maxwell-Stefan diffusivities for approximately three-fourths of binary systems follow or approximately follow the empirical Vignes relationship (Wesselingh and Krishna, 1990):

This equation is Eq. (15-22f) converted to Maxwell-Stefan diffusivities. Comparison of this equation with the corresponding equation for Fickian diffusivity, Eq. (15-22f), shows that the equation for Maxwell-Stefan diffusivity is simpler and, for nonlinear systems where γ ≠ 1, Eq. (15-73a) will be closer to linear, which makes interpolation easier. Fewer data points are required to accurately fit an almost linear function than a more nonlinear one.

In ideal systems γ_A = 1 and

Ideality is most likely in low-pressure gas systems.

Substituting in

we obtain

In difference notation Eqs. (15-77a) and (15-77b) become

The bars over the terms mean that average values should be used. In the film model for mass transfer the mass transfer coefficient is . Dividing the denominators of Eq. (15-65a) by Δz, we obtain

Unlike the Fickian diffusivities for ternary systems, the Maxwell-Stefan diffusivity is symmetric and is greater than or equal to zero for both ideal and nonideal systems. In ideal ternary systems, the Stefan-Maxwell diffusivities have the same values as the binary pairs, and as we already saw, the binary pairs have the same values as the Fickian binary pairs.

Unfortunately, the same condition does not hold for Fickian diffusivities. In an ideal ternary system, values for the Fickian diffusivities are not equal to values of the Fickian binary pairs.

For the limiting case of an ideal ternary that is quite dilute in one component (arbitrarily called component A), an effective Fickian diffusion coefficient for dilute component A can be calculated:

Substitution of this equation to replace D_AB in Eq. (15-4a) allows calculation of the diffusion of dilute component A. Note that the effective diffusion coefficient depends on the concentrations of the other species. Although diffusion of the dilute component can be calculated accurately, diffusion of the concentrated components B and C probably will not be accurate (see Problem 15.H3b). Even for this special limiting case (ideal ternary with dilute component A), use of the Maxwell-Stefan equations is certainly preferable and not any more difficult than the Fickian method.

For ternary mass transfer the Maxwell-Stefan procedure with difference equations is the easiest approach to develop approximate answers that can be quite accurate. In this section we will assume ideal gases. The total mole balance for an expanding/contracting object was Eq. (15-33b),

which is still valid, but rate of mass transfer can be due to any combination of the three components. In molar units the terms are

Equations (15-78c) and (15-78d) are used to determine flux. Terms in these equations can be estimated

Often the average mole fraction of component C will be found from

At this point we need a specific problem so that we can calculate interfacial mole fractions, densities, mass transfer coefficients, and the bootstrap relationship.

Expanding chemical potential µ in terms of activity coefficients, these equations become

If we note that v_A = N_A/(x_Aρ_m) and v_B = N_B/(x_Bρ_m) where ρ_m is molar density, we obtain

Converting these equations to difference equation form is left as an exercise (Problem 15.C10).

Multicomponent solutions are considered in detail by Krishna and Wesselingh (1997), Taylor and Krishna (1993), and Wesselingh and Krishna (1990, 2000). The Maxwell-Stefan matrix approach to multicomponent mass transfer is used in the Aspen Plus simulator to solve distillation problems. Use of computer simulators to obtain rate-based solutions of distillation problems is considered in Section 16.8 and in the appendix of Chapter 16.

Molecular model. A molecular model has the advantage of explaining physically what is happening during diffusion, but the detailed theory is complex. The results [e.g., Eqs. (15-21a) to (15-21f)] are useful for predicting the diffusivity of gases. In its current state of development this model needs to be used in conjunction with another model for complicated mass transfer calculations. Use this model for physical understanding.

Fickian model. The Fickian model is a widely accepted model for diffusion, which means most engineers expect you to use it. It works well for ideal and close-to-ideal binary systems and can be used for nonideal binary systems if data are available. Most diffusivity data and correlations for mass transfer coefficients are based on the Fickian model. This model is very difficult to use for concentrated ternary systems and can require negative diffusion coefficients to predict data.

Linear driving force mass transfer model. This model is a widely accepted (by chemical engineers), empirical formulation that can be extended to very difficult mass transfer problems if empirical correlations for the mass transfer coefficient are available. The linear driving-force model based on Fickian diffusion is the model most commonly used for separation problems, but it can fail where the Fickian model fails. For ternary systems the linear driving-force model based on Maxwell-Stefan diffusion is more robust.

Maxwell-Stefan model. The Maxwell-Stefan model is generally agreed to be a better model than the Fickian model for nonideal binary and all ternary systems. However, it is not as widely understood by engineers, data collected in terms of Fickian diffusivities need to be converted to Maxwell-Stefan values, and the model can be more difficult to use. Use this model, coupled with a mass transfer model, when the Fickian model fails or requires an excessive amount of data.

Irreversible thermodynamics model. The irreversible thermodynamics model is useful in regions where phases are unstable and can split into two phases (deGroot and Mazur, 1984; Ghorayeb and Firoozabadi, 2000; Haase, 1990). However, this model is beyond the scope of this introductory treatment.

Bravo, J. L., J. A. Rocha, and J. R. Fair, “Mass Transfer in Gauze Packings,” Hydrocarb. Proc., 64 (1), 91, (1985).

Chilton, Y. H., and Colburn, A. P., “Mass Transfer (absorption) Coefficients Prediction from Data on Heat Transfer and Fluid Friction,” Ind. Eng. Chem., 26, 1183 (1934).

Crank, J., The Mathematics of Diffusion, 2nd ed., Clarendon Press, Oxford, 1975.

Cussler, E. L., Diffusion. Mass Transfer in Fluid Systems, 3rd ed., Cambridge University Press, Cambridge, 2009.

Datta, R., and S. A. Vilekar, “The Continuum Mechanical Theory of Multicomponent Diffusion in Fluid Mixtures,” Chem. Engr. Sci., 65, 5976–5989, (2010).

deGroot, S. R., and P. Mazur, Non-Equilibrium Thermodynamics, Dover, New York, 1984.

Dean, J. A. (Ed.), Lange’s Handbook of Chemistry, 13th ed., McGraw-Hill, New York, 1985.

Demirel, Y., Nonequilibrium Thermodynamics, Transport and Rate Processes in Physical, Chemical and Biological Systems, 3rd ed., Elsevier, Amsterdam, 2013.

Geankoplis, C. J., Transport Processes and Separation Process Principles (Includes Unit Operations), 4th ed., Prentice Hall, Upper Saddle River, NJ, 2003.

Ghorayeb, K., and A. Firoozabadi, “Molecular, Pressure, and Thermal Diffusion in Nonideal Multicomponent Mixtures,” AIChE J, 46 (5), 892–900 (2000).

Haase, R. Thermodynamics of Irreversible Processes, Dover, New York, 1990.

Hines, A. L., and R. N. Maddox, Mass Transfer: Fundamentals and Applications, Prentice Hall, Upper Saddle River, NJ, 1985.

Hirschfelder, J., C. F. Curtiss, and R. Bird, Molecular Theory of Gases and Liquids, Revised Edition, Wiley, New York, 1964.

Hottel, H. C., J. J. Noble, A. F. Sarofim, and G. D. Silcox, “Heat Transfer,” in D. W. Green and R. H. Perry (Eds.), Perry’s Handbook of Chemical Engineering, 8th ed., McGraw-Hill, New York, 2008, Section 5a.

Incropera, F. P., D. P. DeWitt, T. L. Bergman, and A. S. Lavine, Fundamentals of Heat and Mass Transfer, 7th ed., Wiley, New York, 2011.

Johnstone, H. F., and R. L. Pigford, “Distillation in a Wetted Wall Column,” Trans. AIChE, 38, 25 (1942).

Kmit, J. M., and D. B. Shah, “Application of Pseudo-Steady-State Approximation in Solving Chemical Engineering Problems,” Chem. Engr. Educ., 30 (1), 14-18 (Winter 1996).

Kirwan, D. J., “Mass Transfer Principles,” in R. W. Rousseau (Ed.), Handbook of Separation Process Technology, Wiley-Interscience, New York, 1987, Chap. 2.

Krishna, R., and J. A. Wesselingh, “The Maxwell-Stefan Approach to Mass Transfer,” Chemical Engineering Science, 52, 861–911 (1997).

Marrero, T. R., and E. A. Mason, “Gaseous Diffusion Coefficients,” J. Phys. Chem. Ref. Data, 1, 3–118 (1972).

Poling, B. E., J. M. Prausnitz, and J. P. O’Connell, The Properties of Gases and Liquids, 5th ed., McGraw-Hill, New York, 2001.

Poling, B. E., G. H. Thomson, and D. G. Friend, “Physical and Chemical Data,” in D. W. Green and R. H. Perry (Eds.), Perry’s Handbook of Chemical Engineering, 8th ed., McGraw-Hill, New York, 2008, Section 2, Tables 2-324 and 2-325.

Sherwood, T. K., R. L. Pigford, and C. R. Wilke, Mass Transfer, McGraw-Hill, New York, 1975.

Taylor, R., and R. Krishna, Multicomponent Mass Transfer, Wiley-Interscience, New York, 1993.

Treybal, R. E., Mass Transfer Operations, 3rd ed., McGraw-Hill, New York, 1980.

Tyn, M. T., and W. F. Calus, “Temperature and Concentration Dependence of Mutual Diffusion Coefficients of Some Binary Liquid Systems,” J. Chem. Eng. Data, 20, 310–316 (1975).

Wankat, P.C., and K. S. Knaebel, “Mass Transfer,” in D. W. Green and R. H. Perry (Eds.), Perry’s Handbook of Chemical Engineering, 8th ed., McGraw-Hill, New York, 2008, Section 5B, pp. 5-1, 5-2, 5-43 to 5-83.

Wesselingh, J. A., and R. Krishna, Mass Transfer, Ellis Horwood, Chichester, England, 1990.

Wesselingh, J. A., and R. Krishna, Mass Transfer in Multicomponent Mixtures, Delft University Press, Delft, Netherlands, 2000.

A1. Suppose we have a volume of nitrogen plus a small amount of water vapor at 1.0 atm. The walls of the container are at 25°C, and there is a hot pipe at 105°C running through the volume. Explain the behavior predicted by Eq. (15-4a), the behavior predicted by Eq. (15-4b), and the reasons that Eq. (15-4b) more closely predicts reality.

A2. When is J_A = N_A, and when are they not equal?

A3. Explain why infinite dilution Fickian diffusivities for binary liquid systems are not equal,

A4. The constant in Eq. (15-22b) is 1.173 × 10^–16, which agrees with Geankoplis (2003). However, Cussler (2009) and Wankat and Knaebel (2008) use a constant of 7.4 × 10^–8. Both are correct. Explain.

A5. What is a controlling resistance? How do you determine which resistance, if either, is controlling?

A6. In Problem 15.H5 changing the pressure changes the diffusivities but does not change the Henry’s law constant of ammonia. However, changing the pressure does change the surface concentrations. Explain.

B. Generation of Alternatives

B1. In Example 15-1 operation is at a pseudo-steady state. Brainstorm alternative designs for this diffusion measurement.

B2. Although the additive approach traditionally used for coupling Fickian diffusion with convection appears logical and works for calculating total fluxes of A and B, this is not the only way one could tackle the problem. For binary systems this approach has the advantages of forcing D_AB = D_BA (see Problem 15.C1), and D_AB is the same for any choice of v_ref (see Problem 15.C2). Instead of treating diffusion and convection terms as additive Eqs. (15-15a) and (15-15b), what other approaches could be used to analyze simultaneous convection and diffusion?

B3. Think of an experiment to measure diffusion coefficients that you could set up at home or in your apartment. Detail the equipment list. Estimate the amount of change you will observe. Will you be able to make accurate measurements? What could go wrong that will skew results?

B4. You have been invited to give a talk on mass transfer at the local high school. You want to show a live demonstration. Brainstorm at least five different demonstrations that you can develop with very simple apparatus—preferably made from standard cooking equipment.

B5. Develop roleplays to illustrate:

a. The difference between ordinary and Knudsen diffusion.

b. The difference between Fickian and Maxwell-Stefan diffusion.

C. Derivations

C1. For binary diffusion with convection, use Eqs. (15-16a) to (15-16c) and equivalent equations for component B to show that D_AB = D_BA.

C2. For binary diffusion, show that D_AB does not depend on the choice of v_ref.

C3. Derive the equation that is equivalent to Eqs. (15-32d) and (15-32e) in terms of a partial pressure driving force and a log mean partial pressure difference:

C4. Derive Eq. (15-44b).

C5. For binary Maxwell-Stefan diffusion through a stagnant layer of B, show that J_A = N_A/y_B.

C6. Starting with Eqs. (15-74a) and (15-74b), derive Eq. (15-74c). Note: Because x_W = 1 – x_E, x_W is not a constant.

C7. For binary distillation with CMO, v_ref,mol = 0. If CMO is valid, show that v_ref,mass ≠ 0 if MW_A ≠ MWB_A, and calculate the functional form for v_ref,mass that will make convection zero in this reference frame. Do this for diffusion in the vapor assuming an ideal gas.

C8. Derive Eq. (15-39b).

C9. For the dissolution of platelets in dilute solution show that

h = –2k_flat ρ_liquid (x_{A,solubility,mass} – x_A,bulk,mass)/ρ_solid + h_initial

L = –2k_{growth sides} ρ_liquid (x_{A,solubility,mass} – x_A,bulk,mass)/ρ_solid + L_initial

C10. Write the nonideal Maxwell-Stefan difference equations for a ternary system with the following Maxwell-Stefan mass transfer coefficient for each binary pair:

One approach is to pattern the steps after the development of the nonlinear binary Maxwell-Stefan equations [from Eqs. (15-68a) and (15-68b) to (15-71a) and (15-71b) to (15-71c) and (15-71d) to (15-71e) to (15-71h)]. The third component can be included following the pattern used for the ternary ideal Maxwell-Stefan Eqs. (15-76a) to (15-78d).

C11. Derive equation , assuming that is valid.

D. Problems

D1. We have steady-state diffusion of 1-propanol across a liquid water film that is 0.10 mm thick. On one side of the film the 1-propanol concentration is 1.2 kg/m³. We desire a 1-propanol flux rate of 0.2 × 10^–5kg/(m²s). The apparatus is at 25°C.

a.* What is the concentration at the other side of the film? Since the answer depends on the direction of transfer, there are two answers.

b. What are the equations for the concentration profiles?

Diffusivity value is given in Table 15-3.

D2. For the same system as in Problem 15.D1, the high concentration C_A,0 = 1.2 kg/m³ and C_A,L = 0.9701 kg/m³, which is same as in Problem 15.D1, but we want a flux rate = 0.35 × 10^–5 kg/(m²s). Apparatus temperature can be adjusted. Estimate the Fickian infinite dilution diffusivity by adjusting the value from Table 15-3 for the temperature difference by assuming that E_o = 3000 cal/mol in Eq. (15-22c). What temperature is required?

D3. A column arrangement similar to Figure 15-2 is used for organic liquids. A large reservoir of water is under the column. The water is stirred and solute concentration in the water is constant. The organic liquid is immiscible in water and floats on the water surface at z = 0. Mole fractions x of the two layers in equilibrium are related by K = x_{solute,organic}/x_solute,water. At z = L = 20 cm (top of the column) mole fraction of solute in the organic liquid is zero. The entire apparatus is at 25°C and 1.0 bar. The column cross-sectional area is 0.55 cm². Diffusion of solute, benzoic acid (C₆H₅COOH), in toluene (C₆H₅CH₃) is studied. Assume toluene is totally immiscible in water. The mole fraction of benzoic acid in the water is kept constant at = 0.00212 mol%. Calculate benzoic acid flux in mol/(m²s).

Data: Diffusion coefficient of benzoic acid in toluene at 25°C = 1.5 × 10^–9 m²/s.

MW toluene = 92.14 g/mol, MW water = 18 g/mol, MW benzoic acid = 122.12 g/mol. Density toluene = 865 kg/m³, density water = 1000 kg/m³.

K(25°C) = 123.1 = x_{benzoic_acid_in_toluene}/x_{benzoic_acid_in_water}.

D4. a. Estimate the Fickian diffusivity of a binary mixture of benzene and air at 298.2 K and 1.0 atm pressure using Chapman-Enskog theory and Table 15-2.

b. Compare your result with the experimental value in Table 15-3.

c. Estimate the Fickian diffusivity at 273 K and 0.5 atm.

d. Estimate Maxwell-Stefan diffusivity at 298.2 K and 1.0 atm.

D5. What is the Fickian diffusivity of chlorobenzene in liquid bromobenzene at 300 K when the mole fraction of chlorobenzene is 0.0332? Assume that the diffusivity follows an Arrhenius form and use the data in Table 15-3 to determine E_o. Also report the value of E_o in J/mol.

D6. Water at 60°C and 0.95 bar is evaporating into a 12.0-cm-long tube (also at 60°C) and diffusing through a stagnant layer of air. The device is illustrated in Figure 15-2.

a. Calculate the flux of water vapor if convection is included.

b. Calculate the value of the molar reference velocity.

c. If the tube is 1.5 cm in diameter, calculate the grams of water evaporated per hour.

d. Calculate flux of water vapor if convection is not included (a calculation that is expected to be incorrect for this example).

Data: At 60°C: D_{water – air} = 3.05 × 10^–5 m² / s, VP_water = 149.5 mm Hg

D7. Assuming that the mixture is ideal, estimate infinite dilution Fickian diffusivities at 283.3 K for chlorobenzene in liquid bromobenzene and for bromobenzene in liquid chlorobenzene from data in Table 15-3.

D8. Use the Wilke-Chang theory to estimate infinite dilution Fickian diffusivity of methanol in liquid water at 293.16 K. Data are available at www.engineeringtoolbox.com and www.cheric.org/research/kdb.

a. Use a value of φ_B = 2.26.

b. Use a value of φ_B = 2.6.

D9. Determine the modified Sherwood number for gas-side-controlled mass transfer for distillation of ethanol-water in a wetted wall column. Column diameter is 10.0 cm. Measurements are at very low ethanol concentrations where flowing liquid can be assumed to be pure water. Total pressure is 1.0 bar. Liquid water at its boiling temperature is flowing down a vertical column at a volumetric flow rate per meter of circumference of q = 0.0000075 m²/s. Ethanol is diffusing through an upward-flowing vapor (almost pure water) at 1.0 bar and its boiling temperature. The upward vapor velocity is 0.81 m/s. Densities and viscosities of pure water liquid and pure water vapor are available in Perry’s Handbook of Chemical Engineering. Use parameters in Table 15-2 to estimate diffusivity of ethanol-water in the vapor.

D10. A pipeline containing ammonia gas is vented to ambient air via a 20-m long, 3-mm diameter tube. What is the mass flow (g/day) of ammonia into the atmosphere? What is the mass flow of air into the pipeline (g/day)? Temperature and pressure everywhere are 25°C and 101,325 Pa, respectively. Assume pure ammonia at the pipeline and pure air at the air end of the tube. D_AB = 2.8 × 10^–5 m²/s (ammonia-air at 25°C). Assume ideal gas.

D11. We have steady-state diffusion of ammonia in air across a 0.033-mm-thick film. On one side of the film ammonia concentration is 0.000180 kmol/m³, and on the other side it is 0.000257 kmol/m³. What operating temperature is required to have an ammonia flux rate of 9.60 × 10^–5 kmol/(m²s)? Pressure = 0.90 atm. Use the Chapman-Enskog theory to estimate diffusivity.

D12.* Water at 20°C is flowing down a 3.0-m long vertical plate at a volumetric flow rate per meter of plate width of q = 0.000005 m²/s. Entering (y = 0) water is pure. The water is in contact with carbon dioxide gas that is saturated with water vapor at 20°C (no water is evaporating). Carbon dioxide transfers into the water. At the water-gas surface (z = 0) water and carbon dioxide are in equilibrium at the solubility limit of carbon dioxide at C_CO2,surf. Estimate Fickian infinite dilution diffusivity by adjusting the value from Table 15-3 for the temperature difference by assuming that E_o = 2000 cal/mol in Eq. (15-22c). At 20°C the solubility of carbon dioxide in water is 1.7 g/kg water.

a. Determine film thickness δ, average vertical velocity of film, and Reynolds number.

b. Is flow laminar? If no surfactant is added, do you expect ripples on the surface? If surfactant is added, do you expect ripples on the surface?

c. With no surfactant, determine average mass transfer coefficient and average Sherwood number.

d. Per meter of plate width, at what rate (kg/s) is carbon dioxide transferred into water over the length of the plate?

D13. Repeat all parts of Problem 15.D12 but with a water rate of q = 0.000015 m²/s.

D14. Repeat Problem 15.D12 but for q = 0.0015 m²/s.

a. Determine film thickness δ, average vertical velocity of film, and Reynolds number.

b. Determine average mass transfer coefficient and average Sherwood number.

c. Per meter of plate width, at what rate (kg/s) is carbon dioxide transferred into water over the length of the plate?

D15. We are measuring the diffusivity of water in air at 42°C. A tube is placed with one end in the water and the other end in a stream of dry air. The air column in the tube is 22-cm long, and we assume mole fraction of water is zero at the end in the air stream, y_water (z = 22) = 0. System pressure = 0.95 atm and is constant. At 42°C vapor pressure of water is 0.1098 atm, and diffusivity of water in air is D_W-air = 2.88 × 10^–5 m²/s. Assume ideal gases. Use the Maxwell-Stefan theory in difference equation form to determine flux of water at steady state.

D16. CO₂ is being scrubbed out of a gas using water flowing through a packed bed. At the top of the bed CO₂ is absorbed at a rate of 2.3 × 10^–6 mol/cm²s. Inlet liquid is pure water, CO₂ is present at a partial pressure of 10 atm, the Henry’s law constant is H_conc= 10,800 atm cm³/mol if Eq. (12-1a) is written as p_B = C_BH_B,conc. The diffusion coefficient of CO₂ in water is 1.9 × 10^–5 cm²/s. Find the film thickness (δ). Assume that liquid phase resistance controls.

D17. Repeat Example 15-10 but with a mass transfer coefficient that is 10 times larger (use δ = 0.001 m). Report x_NH3, y_NH3,surface, N_water, and N_NH3.

D18.*

a. Repeat Example 15-9 part c, but divide the film into two parts (from 0 to Δz and from Δz to δ = 0.000012 m). Δz is unknown but is selected so that x_E = 0.155 (the previous average point) at z = Δz. Now the average point for the first interval is x_E = 0.0825 and for the second is x_E = 0.2275. Since operation is at steady state, N_E has to be the same in both intervals. Write Eq. (15-61e) for each interval (with thickness Δz for the first and δ – Δz for the second), and solve the resulting two equations for unknowns Δz and N_E.

b. Divide the film into three parts (0 to Δz₁, Δz₁ to Δz₂, and Δz₂ to δ). Δz₁ is unknown but is selected so that x_E = 0.0825 at z = Δz₁. Δz₂ is also unknown but is selected so that x_E = 0.2275 at z = Δz₁ + Δz₂. Since operation is at steady state, N_E has to be the same in all three intervals. Write Eq. (15-61e) for each interval (with thickness Δz₁ for the first, Δz₂ for the second, and δ – Δz₁ – Δz₂ for the third), and solve the resulting three equations for unknowns Δz₁, Δz₂, and N_E.

c. Divide the film into four parts, and determine N_E.

*Answers are in Example 15-7, part c.

D19.* (Optional). Plot Figure 15-3 for the following unsteady diffusion problem. A thick layer of pure water (C_initial = C_∞= 0) at 25°C has a constant concentration of sucrose, C₀ = 0.002 mol/L, placed on one surface at time t = 0. Assume the diffusivity (Table 15-3) is constant at the average concentration of 0.001 mol/L.

a.* Graph a plot C/C₀ versus z (in cm) for t = 10,000 s. Selected values are in the answers at the back of the book.

b. If the measurement threshold of a zero concentration is 1.0 × 10^–6 mol/L, what is the minimum thickness the layer must have to appear infinite at t = 100,000 s?

c. If the measurement threshold of a zero concentration is 1.0 × 10^–6 mol/L, what is maximum exposure time the layer can have to appear infinite if layer thickness is 0.1 cm?

D20. Calculate the value of Maxwell-Stefan diffusivity for ethanol water at 40°C for ethanol mole fractions of 0.0, 0.2, 0.3, 0.4, 0.7, and 1.0. The Fickian diffusivities are available in Table 15-4. Which set of diffusivities are closer to linear?

D21. NaCl is crystallizing from an aqueous (water) liquid solution onto a crystal particle of pure NaCl at 18°C. Assume particle growth is controlled by mass transfer, and the particle is spherical. The aqueous solution is supersaturated at a mass fraction of NaCl = 0.30. If the initial particle diameter is D = 0.1 mm, find D after 3600 s.

Data: Solubility of NaCl in water at 18°C = 0.2647 mass fraction.

Density pure solid NaCl = 2.163 g/cm³. Molecular weight NaCl = 58.45.

Density of aqueous solution of NaCl = 1.20 g/cm³ (assume constant).

Density of pure water = 1.0 g/cm³. Molecular weight pure water = 18.

D_NaCl-water at 18°C = 1.24 × 10^–9 m²/s. Mass transfer coefficient k = 2.0 × 10^–6 m/s.

D22. A 2-cm-diameter, 19-cm-long tube is placed touching a pool of liquid. The end away from the liquid pool (z = 0.19 m) is in an air stream (component C) so that it is pure air, y_C (z = 0.19 m) = 1.0. The liquid is 80 mol% component A and 20 mol% component B. The ratio of these two components in the gas at z = 0 is the same as in the liquid, y_A,i/y_B,i = 4. The total vapor pressure of the liquid is 13,005 Pa. Thus 13,005/P_total = (y_A,i + y_B,i). Total pressure is 107,404 Pa, and temperature is 40°C. Assume ideal gas. At 40°C and 107,404 Pa, D_AB = 2.45 × 10^–5 m²/s, D_BC = 2.69 × 10^–5 m²/s, and D_AC = 1.01 × 10^–5 m²/s. MW B = 18, MW air = 28.9, MW A = 60.1. A partial solution is N_A = 0.000228074 mol/(m²s). What are the molar fluxes, mol/(s m²), of B and C (air) at steady state?

D23. A crystal particle of pure NaCl is dissolving in an aqueous liquid (water) solution at 18°C. The dissolution of the particle is controlled by mass transfer. The aqueous solution is at a mass fraction of NaCl = 0.25.

Data: Solubility of NaCl in water at 18°C = 0.2647 mass fraction. MW NaCl = 58.45.

Density pure solid NaCl = 2.163 g/cm³.

Density of aqueous solution of NaCl = 1.20 g/cm³ (assume constant).

Density of pure water = 1.0 g/cm³. MW pure water = 18.

Diffusivity NaCl-water at 18°C = 1.24 × 10^–9 m²/s. Mass transfer coefficient k = 1.8 × 10^–6 m/s.

a. Assume the particle is spherical with an initial particle diameter D = 1.64 mm. Find the particle diameter after 7200 s.

b. Assume the particle is a cube with an initial particle length L = 1.64 mm. Find the particle length after 7200 s.

D24. Solve Example 15-7 using the difference equation form of the Maxwell-Stefan equations.

D25. Repeat Example 15-11 for the following conditions:

a. 0.03 g CO₂/1000 g water in the drop. y_{water, bulk} = 0, y_CO2,bulk = 0.05.

b. 0.00 g CO₂/1000 g water in the drop. y_{water, bulk} = 0, y_CO2,bulk = 0.0007.

c.* 0.03 g CO₂/1000 g water in the drop. y_{water, bulk} = 0, y_CO2,bulk = 0.0007.

* Answer is in a spreadsheet in Figure 15-A4.

D26.* Estimate Sc_V for a saturated vapor mixture that is 80 mol% ethanol and 20 mol% water at 1.0 atm.

E. More Complex Problems

E1. A short connecting pipe between two tanks is clogged with a plug of NaCl crystals. The plug formed as a cylinder of circular cross-sectional area with a constant diameter D = 2.0 cm and an initial length of 1.0 cm. The pipe is 2.0 cm in diameter and prevents the plug from increasing its diameter. However, the plug can grow or shrink from the two ends (it gets longer or shorter but has no change in diameter). The pipe is 10-cm long. Initially, the crystal is in the middle of the pipe from z = 4.5 to z = 5.5 cm. Tank 1 on the z = 0 side of the pipe contains pure water and is well mixed so that the bulk mass fraction of NaCl, x_NaCl,1 = 0. Assume the solution density from z = 0 to the crystal plug is the density of water. Tank 2 on the z = 10 cm side contains a well-mixed, aqueous solution of NaCl with a mass fraction NaCl x_NaCl,2 = 0.37. The rate of growth is controlled by mass transfer.

a. Find the length of the crystal plug after 10⁴ seconds.

b. How far is the center of the plug from tank 1 after 10⁴ seconds?

Data: Solubility of NaCl in water = 0.2647 mass fraction. MW NaCl = 58.45.

Density pure solid NaCl = 2.163 g/cm³.

Density of aqueous solution of NaCl with x = 0.37 is 1.22 g/cm³ (assume constant).

Density of pure water = 1.0 g/cm³. MW pure water = 18.

Mass transfer coefficient on side 1 (pure water) k₁ = 5.5 × 10^–6 m/s.

Mass transfer coefficient on side 2 (x₂ = 0.37) k₁ = 2.4 × 10^–6 m/s.

The plug is cylindrical. Diameter is constant, and L varies.

H. Spreadsheet Problems

H1. Two identical, large glass bulbs are filled with gases and connected by a capillary tube that is δ = 0.0100 m long. Bulb 1 at z = 0 contains the following mole fractions: y_air = 0.520, y_H2 = 0.480, and y_NH3 = 0.000. Bulb 2 at z = δ contains y_air = 0.540, y_H2 = 0.000, and y_NH3 = 0.460. Since the bulbs are quite large, operation is at pseudo- (or quasi-) steady state. (In other words assume mole fractions at the boundaries are constant [e.g., y_air = 0.520 at z = 0, and y_air = 0.540 at z = δ so that Δy_air = 0.02]) and total flux of air + hydrogen + ammonia is zero.] Pressure is uniform at 2.00 atm, and temperature is uniform at 273 K. Diffusivity values can be determined from Table 15-1. Assume ideal gases. Estimate fluxes of the three components using difference equation formulation of the Maxwell-Stefan method.

H2. Repeat Example 15-10 but for a bulk gas that is 40 mol% air, 15 mol% NH₃, and 45 mol% water. Report x_NH3, y_NH3,surface, N_water, and N_NH3.

H3.

a. Repeat Problem 15.H1 (use the Maxwell-Stefan equations), but bulb 1 at z = 0 contains the following mole fractions: y_air = 0.500, y_H2 = 0.500, and y_NH3 = 0.000. Bulb 2 at z = δ contains y_air = 0.499, y_H2 = 0.499, and y_NH3 = 0.002.

b. Solve this problem using the effective Fickian diffusion coefficient for NH₃, Eq. (15-67b), and usual Fickian equations for a very dilute system with no convection. Although there is not a simple approach for the two concentrated components, one is tempted to treat these as a binary with the Fickian binary diffusion coefficient for air and hydrogen. Try this method and then compare to solution of part a.

H4. Repeat Problem 15.H1, but bulb 1 at z = 0 contains the following mole fractions: y_air = 0.520, y_H2 = 0.480, and y_NH3 = 0.000. Bulb 2 at z = δ contains y_air = 0.520, y_H2 = 0.000, and y_NH3 = 0.480.

H5. Repeat Example 15-8 but for a pressure of 1.1 atm. Note that the diffusivities depend on pressure. Also answer problem 15.A6.

H6. A pipeline containing 99.0mol% ammonia and 1.0mol% hydrogen gas is vented to ambient air via a 15-m-long, 3.5-mm-diameter tube. Temperature and pressure everywhere are 0°C and 101,325 Pa, respectively. Assume 99.0mol% ammonia and 1.0 mol% hydrogen at the pipeline and pure air at the air end of the tube. Ideal gas, D_AB = 1.98 × 10^–5 m²/s (A = ammonia, B = air at 0°C), D_CB = 6.11 × 10^–5 m²/s (C = hydrogen, B = air at 0°C), D_CA = 7.48 × 10^–5 m²/s (A = ammonia, C = hydrogen at 0°C).

What are the molar fluxes, mol/(s m²), of ammonia, hydrogen, and air?

The spreadsheets for Example 15-11 are shown in Figures 15-A3 (with formulas) and 15-A4 (with numerical results). In Figures 15-A3 and 15-A4 check in B15 is Eq. (15-78d) written in functional form (subtract right hand side so that equation = 0) × 1000 to decrease error in Goal Seek.