Chapter 1. Introduction to Modeling of Mass Transfer Processes

Mass transfer refers to a net movement of a species A from one location to another, usually in a multicomponent mixture containing species A, B, C, and so on, usually caused by a concentration difference. Consider, for example, a pollutant introduced in air. It gets transported from the source point to another location by the action of two common mechanisms of mass transport—namely, diffusion (a result of concentration difference between two points) and convection (caused by, for example, by wind velocity). Another common example of considerable industrial applications is the situation in which species A moves from one phase to the interface of a second phase and then crosses to the second phase; this is commonly referred to as interfacial transport.

Mass transport phenomena are ubiquitous in nature and in industrial applications. In process industries, most of the common separation processes involve the process of mass transfer from one phase to another. Penicillin production is not possible without rapid extraction of the product of fermentation (penicillin) by liquid extraction; otherwise, the penicillin will decompose quickly to unwanted products. The catalytic converter in a car removes NO_x, CO, and other hydrocarbons from combustion gases and involves transport of NO_x and other products to the walls of the converter, followed by a heterogeneous reaction at the catalytic walls. Transport and absorption of oxygen into blood is vital to life itself. Mass transfer with simultaneous reaction with hemoglobin plays an important role here. Plants draw water from soil by osmosis, which is an example of mass transport driven by a chemical potential gradient. The formation of acid rain is a consequence of mass transfer and reaction of sulfur dioxide in the rain droplets. More examples are provided in Section 1.5.

Development of a mathematical model to describe the process is central to the analysis and design of these processes. The model is expected to provide information on the concentration profile of the species being transported in the system and the rate at which the species is transported across the interface, system, or equipment. The goal of this book is to teach the methodology of modeling mass transport processes. You will learn how to set up the model equations and develop the skill set needed to use appropriate analytic and numerical tools to solve the resulting mathematical problem.

Analysis of mass transport phenomena is based on continuum approximation, which assumes that the matter is continuously distributed in space. This assumption permits us to assign a value to the concentration at each and every point in space. Differential equations for the concentration variation can then be developed based on this continuum model using a differential control volume. Such differential models are formulated using the species conservation laws coupled with some constitutive models for transport of mass by diffusion. The resulting model is the most detailed model in the context of the continuum assumption. The basic formalism for constructing such a model is introduced in this chapter.

Differential models are not always used and simpler but less descriptive models based on a larger sized control volume are often used. As you will learn from this chapter that models can also be developed at two levels (meso and maco) using a larger sized control volume and average concentrations are needed in these models rather than pointwise values. Three common averages used are the volume average, the cross-sectional average and the flow-weighted average and this chapter defines these and indicates in what context they are used and the additional information needed to close the models based on larger sized control volume.

This chapter provides a roadmap for subsequent chapters in the book. A careful study and understanding of the basic concepts and definitions described here will provide the needed vocabulary for further chapters.

The analysis of the phenomena of mass transfer is important in engineering design applications and in various other fields. Examples of mass transport are readily seen. A lump of sugar put in a coffee mug dissolves and spreads completely into the liquid. A perfume sprayed in the air freshens the whole room. A crystal of KMnO₄ dropped in a jar of water leads to the color spreading over a column of water placed above due to crystal dissolution and subsequent mass transport by diffusion.

Mass transfer is often studied as part of the subject of transport phenomena, which deals with momentum, heat, and mass transfer. What distinguishes mass transport from momentum and heat is that mass transfer by diffusion can occur only in a multicomponent mixture; it cannot occur in a single-component system, unlike transport of momentum and heat. The multicomponent nature of the system needs some additional considerations in the analysis.

Mass transfer is the underlying phenomenon for a large number of industrially important separation processes in unit operations. Consequently, some introductory treatment of mass transfer is provided in courses and in textbooks on unit operations. Most often, however, each unit operations (e.g., distillation, absorption) are studied individually, rather than as a comprehensive topic. Hence the common underpinning in the modeling of various separation processes is often not clear without a proper study of fundamentals of mass transfer.

Mass transfer is often accompanied by a chemical reaction. A detailed study of this coupling is important in heterogeneous reaction (systems with two or more phases) analysis. It turns out that most of the industrially important chemicals are made by a heterogeneous reaction in which two or more phases are contacted. Hence the central theme in reactor design and scale-up is, most often, the evaluation of the coupling between mass transfer rate and chemical reaction rate.

An important consideration here is that the concentration is a discontinuous function at the interface between the two phases (the concentration jump). Thus, if x = 0 is the location of the plane separating the two interfaces (for example, a gas phase and a liquid phase), the concentration of a species A at the gas side (say, x = 0^–) is different from that at x = 0⁺, the liquid side of the interface. As an example, consider the air–water interface. The oxygen concentration on the air side of the interface is much larger then the oxygen concentration on the water side of the interface, and the two are related by the thermodynamics of phase equilibrium or the solubility relationship. Hence thermodynamic relations are needed in interfacial mass transfer analysis. This is another distinguishing feature of mass transport. In contrast, in momentum and heat transport, the corresponding quantities—namely, velocity and temperature—are continuous functions at an interface for most common situations.

The phenomenon of diffusion is similar to heat conduction, in which a temperature difference causes a flow of heat from a higher-temperature location to a lower-temperature location. Diffusion may also be interpreted on the basis of the thermodynamics involved. If a chemical potential difference exists between two points, the natural tendency toward equilibrium is that the chemical potential should become the same at these points. Thus diffusion is caused by a gradient in chemical potential.

Convection refers to transport by bulk motion. If we add sugar to a mug of coffee and stir it, the dissolution rate is increased due to the fluid motion caused by stirring. Transfer of a pollutant from one location to another by wind is an another example of convection.

where V is the volume element and MA is the moles of species A present in that volume. In this book, the S.I. system of units is used, with kg, m, and s being the basic units. However, the basic unit of concentration, is mol/m³, where mol refers to g mol, rather than kg mol. The molarity, denoted as M, is also often used for concentration; it is equivalent to mol/L. Hence 1 M = 1000 mole/m³—a useful conversion factor. Note that units of kmol/m³ can also be used to represent concentration if convenient. In the final calculations, this will need to be converted to gmol/m³ to maintain consistency of units.

The total molar concentration of the mixture C (at any position and at any time) is defined as the summation of all the species concentrations:

where ns is the total number of species present in the mixture.

The total concentration for a gas mixture C_G and the total pressure for an ideal gas mixture are related as

CG=PRgT

Here R_g is the gas constant, which has a value of 8.314(Pa m³/mol K in the S.I. system of units. Note that the units of R_g can also be expressed as J/mol K.

For example, for a gas at 1 bar pressure and 300 K, the total concentration is calculated as 1 × 10⁵ Pa/[8.314(Pa m³/mol K) × 300 K] and has a value of 37 mol/m³.

Total concentration in liquid can be calculated if the partial molar volume of the components in the mixture solution is known. For dilute solutions, it can be based on the density of the solvent.

For example, for a dilute aqueous solution we can use the density of water as the representative density of the mixture. Hence ρ = 1000 kg/m³. The molecular weight of water is expressed as 18 × 10^–3 kg/mol rather than as 18 g/mol. Hence the total concentration in dilute solution can be approximated as 1000 kg/m³/18 × 10^–3 kg/mol = 55556 mol/m³.

Very often it is necessary to refer to two or more phases—for example, a gas and a liquid. In such cases we use C_AG to represent the concentration of A in the gas phase and C_AL to represent the concentration in the liquid. Thus we use the first subscript to represent the species by either by numbers i = 1, 2, . . . or simply by names as A, B, . . . , especially for binary mixtures. The second subscript is used to represent the phase (or the location) where it is present. Similarly, C_G rather than C is used for the total concentration in the gas phase and C_L for the total concentration in the liquid phase. The ratio of C_A to C is called the mole fraction, denoted by y_A for gas mixtures or x_A for liquid mixtures:

The average molecular weight of the mixture M¯ is calculated as

where M_A is the molecular weight of species A. Also note that

where ρ is the density of the mixture.

Note that ρ_A is the density of A in the mixture (same as mass of A per unit volume of the mixture) and should not be confused with the pure component density of A. The latter will be denoted in this book with a superscript pA∘ if we have to refer to both quantities in the same context.

The summation of this for all the species is the mixture density, ρ:

The ratio of ρ_A to ρ is the mass fraction of species A, denoted often by ω_A:

The average molecular weight of the mixture is calculated from the mass fraction values by the reciprocal weighting rule:

or

Partial pressure is therefore a direct measure of concentration in an ideal gas mixture and used commonly for such cases. The partial pressure difference can be used as a driving force for mass transfer instead of concentration differences for (ideal) gas-phase systems. The two sets of units can be interconverted easily, if needed, by the previous equations.

If we use the mass units, we can define a mass flux vector, which is usually denoted as n_A, with the units being kg A/m² s.

The flux vector is unique at a given point, whereas the components N_Ax and so forth depend on the orientation of the x-axis and other dimensions. Thus the components can change in magnitude if, for example, the coordinates are rotated. This will not alter the results in any way since the flux vector is frame indifferent. Similarly, the flux components can be defined in cylindrical, spherical, or other body-fitted coordinate systems and used for the specified geometry. The flux results both from diffusion and due to the flow rate across the control surface and hence the flux are referred to as combined flux. For modeling purposes, it is necessary to split this entity into diffusion flux and convection flux. Thus

Combined flux = convection flux + diffusion flux

This partitioning, which is illustrated in Figure 1.2, is not unique. Rather, it will depend on which value is assigned to the convection flux, as will be shown in the next section.

Convection flux vector = vC_A

Note that the mixture velocity in a multicomponent system is not uniquely defined, unlike in a single-component system. This topic is discussed further in Chapter 5. The point to note here is that the convection flux depends on which definition of the mixture velocity is used. The diffusion flux is correspondingly adjusted since the total flux at any point is fixed.

Here J_Ax is the diffusion flux of A in the x-direction. Note that the law implies that the diffusion occurs in the direction of negative concentration gradient— that is, from a region of higher concentration to a region of lower concentration. This is similar to heat flowing from a hot body to a cold body. The constant of proportionality D_A is called the diffusion coefficient of A. It has the units of m²/s.

More generally speaking, the gradients in the chemical potential can be viewed as causing mass transfer, such that use of these concepts leads to more complex expressions for the diffusive flux. These complexities are addressed in Chapter 7. One consequence of such models is that D_A may not have a constant value over the entire concentration range, but rather may be a function of concentration, especially for a non-ideal liquid mixture. Another consequence is that in some cases, the diffusion can occur counter to the direction of negative concentration gradient. These complex effects and models of them will be deferred to Chapter 7.

Note: A more precise definition of diffusion flux is provided in Chapter 5 and requires a more precise definition of the velocity of a multicomponent mixture. The point to note at this stage is that in a multicomponent system, the mixture velocity is not uniquely defined, unlike in a single-component system. The diffusion flux is defined correspondingly, with the mixture velocity as the frame of reference. Also, Fick’s law (shown by Equation 1.12) applies to a system in which the total concentration C remains constant at various points in the system. A more general representation and the topic of diffusion in systems with total constant density are deferred to Chapter 5. Nevertheless, for many applications the simple Fick’s model for diffusion shown here is widely used.

Generalization of Fick’s law to the 3-D case is accomplished by using the notion of the gradient of a scalar:

where ∇C_A is the concentration gradient vector.

Table 1.1 Order of Magnitude Values of the Diffusion Coefficient

System	DA Values in m2/s
Gas in gas	10–5 to 10–4
Gas in porous solid	10–6 to 10–5
Gas in narrow pores	10–8 to 10–6
Liquid in liquid	10–10 to 10–9
Liquid in porous solid	10–11 to 10–10
Solid in solid	10–12 to 10–17

Note that the convection flux is proportional to the concentration, while the diffusion flux is proportional to the concentration gradient.

Although convection and diffusion are the two fundamental modes of mass transport, additional phenomena such as turbulent diffusion and dispersion are also used. These are not independent mechanisms, but rather reflect the combined effects of diffusion and convection. A brief description is provided here, but detailed discussion of these topics is deferred to later chapters (4 and 12).

Diffusion can be viewed as manifestation of random molecular motion, while turbulent diffusion can be viewed as the result of random bulk motion of the eddies. Diffusion brings in species from a region of higher concentration to a region of lower concentration. Similarly, turbulent eddies bring in species from a region of higher concentration to a region of lower concentration and the process can be modeled by an equation similar to Fick’s law:

Flux due to turbulent eddied=−Dt∇C¯A

where D_t is the turbulent diffusivity, also known as eddy diffusivity. Turbulent diffusivity is a flow-dependent property rather than a molecular property. The C¯A is a time-averaged value of the fluctuating concentration, which is a characteristic of turbulent transport. Models to calculate D_t are provided in Chapter 12. (This quantity is commonly defined as the time average of the product of velocity fluctuation and the concentration fluctuation.)

where p_A is the partial pressure of the species in the gas phase and x_A is the mole fraction in the liquid. The constant H_A has the units of atm or Pa or bars here. The values of the Henry’s law constant for some common gases are shown in Table 1.2.

Table 1.2 Henry’s Law Constant for Some Common Gaseous Species in Water at 298.15 K

Gas	H, atm
Hydrogen	7.099 × 104
Oxygen	4.259 × 104
Nitrogen	8.65 × 104
Ozone	4570
Carbon dioxide	1630
Sulfur dioxide	440
Ammonia	30

The larger the value of the Henry’s law constant, H_A, the less soluble the gas is in the liquid phase. The effect of total pressure in the system is to increase the solubility in accordance with Henry’s law. In general, increasing the temperature decreases the solubility. Hence the Henry’s law constant is usually an increasing function of temperature. An exception occurs with hydrogen, which shows a retrograde behavior. Here the solubility increases at first with an increase in temperature, reaches a maximum, and then decreases thereafter.

where H_A,pc is the Henry’s law constant (unit of Pa m³/mol) with partial pressure, Pa, as the unit for the gas phase and mol/m³ as the concentration unit for the liquid phase, respectively. Another form is

where H_A,cp is the Henry’s law constant (unit of mol/Pa m³) and is the reciprocal of H_A,pc.

Since the concentration in the gas phase can also be used instead of the partial pressure unit, we have yet more ways of writing the Henry’s law relationship! Example 1.1 shows the application of the Henry’s law and the concentration jump at a gas–liquid interface.

Note: Mass transport is usually performed under non-equilibrium conditions. Hence Henry’s law should be applied only at the interface, which is assumed to still be at equilibrium. Concentrations in phase 1 and phase 2, away from the interface, will not be the equilbrium values. Otherwise, no mass transfer will occur.

The schematic of the concentration variation for interfacial mass transfer is shown in Figure 1.3. Although the concentration varies in both phases, the interfacial concentration values are related by thermodynamic considerations. The subscript i in Figure 1.3 refers to the interface and y_Ai and x_Ai are related by Henry’s law.

p_A,i = x_A,i p_vap,A

The vapor pressure p_vap varies as a function of temperature and is often correlated by the Antoine equation:

Values of the constants A, B, and C are tabulated in many books (for example, Reid et al., 1987) and websites. The units are often in mm Hg for vapor pressure and temperature (T) is in degrees Celsius rather than in standard S.I. units. Hence caution must be exercised when extracting these values from the literature.

y_A,i = m_Ax_A,i

where y_A,i is the interfacial mole fraction in one of the liquid phases and x_A,i is the interfacial mole fraction at the interface of the second liquid. The partition coefficient, in turn, is related to ratio of activity coefficients of A in the two phases:

mA=γAIγAII

The value of this coefficient is often predicted using thermodynamic models for activity coefficients.

On a similar note for fluid–solid systems, linear adsorption equilibria are often used, although these are actually the limiting case of nonlinear relations (the classical Langmuir equation and other isotherms are described in Section 29.3). The linear relation is a good approximation for dilute systems. More generally, the Langmuir isotherm is used to represent equilibrium for gas–solid and liquid–solid systems for concentrated solutions. For gases that undergo a reaction in the liquid, the solubility has to be viewed as the first step for the dissolution equilibrium constant—for example, A(gas) to A(aq) equilibrium. Dissolved gas may further react; hence the quantity of gas absorbed will also depend on the equilibrium constant for these reactions, which are accounted for separately. This value can exceed that calculated using the solubility parameter alone. An example of such system is SO₂ in water, where the dissolved SO₂ undergoes reaction to form HSO3− or SO3−− ions depending on the pH of the solution.

Table 1.3 Examples of Heterogeneous Reactions Based on the Phases Being Contacted

Phases Contacted	Example
Gas + solid catalyst	oxidation of ethylene
Gas + solid reactant	combustion of coal
Gas–liquid	removal of CO2 by reactive solvent
Liquid–liquid	biodiesel production
Gas–liquid + solid catalyst	removal of sulfur compounds from diesel
Gas–liquid + solid reactant	carbonation of lime
Gas–liquid–liquid	production of hydroxylamine from nitrobenzene

Two examples of reacting systems where mass transfer plays an important role are shown here. Other examples are taken up in Part II.

Table 1.4 Common Mass Transfer–Based Separation Processes and the Phases Being Contacted

Phase 1	Phase 2	Operation
Vapor	Liquid	Distillation
Gas	Liquid	Absorption
Liquid	Gas	Stripping
Liquid	Liquid	Extraction
Gas	Solid	Adsorption
Liquid	Solid	Adsorption
Wet solid	Gas	Drying

An example of a unit operation is liquid–liquid extraction. The schematic of a simple single-stage extraction unit is shown in Figure 1.5, and we will use this unit as a prototype example for modeling stage contactors. In this unit operation, two phases are intimately mixed in the mixer section of the contactor to create an emulsion or dispersion, which promotes a high rate of mass transfer. The two-phase mixture is then allowed to settle in the settler section, and the enriched solvent and the lean solutions are separated. Determining the degree of mass transfer that can be achieved for a specified rate of agitation is one of the objectives of the mass transfer calculation.

The degree of separation that can be achieved in a single-stage contactor is usually limited; hence a multistage cascade is used, with the lean solution being treated further with fresh or recycled solvent. A designer may want to know how many stages are needed to achieve a certain level of purity. Modeling of such a multistage cascade is used to provide the answer.

Part III of the book covers the application of mass transfer principles to design some of the various unit operations listed in Table 1.3. The key idea is that the modeling methodology is common to all the unit operations and can be approached in an unified format. Thus the various aspects of individual unit operation, although very important on their own, can be brought together under one umbrella of modeling of mass transfer processes. Seader, Henley, and Roper (2011) offer valuable insights into separation process principles and provide a detailed analysis of the various operations.

The book by Harrison, Todd, Rudge, and Petrides (2003) is a good introduction to this field. The book by Seader et al. (2011) has also considerable information on this topic. A review article published by Harrison (2014) provides an introductory reading, with the author stressing the importance of understanding the basic principles and theory as a prelude to design and control of purity of products obtained in bioprocessing.

A number of processing steps in this field involve chemical vapor deposition, in which a species (precursor) is transported from a vapor and reacts and forms a deposit or a film on a (substrate) surface. This process again involves mass transport of reactants, with control of the deposited material’s properties being affected by the rate of transport. Oxidation of silicon to form an insulating layer is another example of mass transfer in fabrication of metal oxide semiconductor (MOS) devices. The book by Middleman and Hochberg (1993) is a classic in this area and a must-read for students who wish to get involved in this field.

• Pharmacokinetics analysis, distribution, and metabolism of drugs in the body

• Understanding of transport of oxygen in the lungs and tissues

• Tissue engineering, including development of artificial organs

• Design of assistive devices such as dialysis units

Some application examples are briefly described in this book in Chapter 23. An early book by Lightfoot (1974) and the more recent books by Sharma (2010); Truskey, Yuan, and Katz (2004); and Fournier (2011) are illustrative of the mass transport applications in this field.

An example of drug release from a capsule is shown in Figure 1.6. In this case, if the drug has a uniform concentration inside the capsule, the release rate reaches its maximum in the beginning and decreases with time. Ideally we want a (nearly) constant rate, like that shown in the figure. The design of the pore structure of the capsule to achieve this type of steady release rate is an important application of transient mass diffusion principles.

In the energy sector, there is a need to store solar energy generated during nonpeak hours and to develop improved batteries for electric cars. A commonly used type of battery is the lithium-ion battery shown in Figure 1.7. Here Li ions are transported across the electrolyte separating cathode and anode. During the charging cycle, Li ions are transported and stored in the carbon matrix. During the discharging cycle, the transport takes place in the opposite direction, such that Li is stored in the metal oxide matrix. Mass transport considerations are an important component in the simulation of the performance of this device; a mass transfer–based model for this system is discussed in Chapter 24.

• Fate and contaminant transport in the atmosphere is usually simulated by dividing the system into four (air, water, soil, and biota) or more compartments and considering transport and reaction in each of the compartment. See, for example, Figure 1.14.

• Groundwater transport is another example. Leakage of contaminants from nuclear waste tanks into rivers could be a major problem, and some of these scenarios can be analyzed by transport models to provide information on the rate of leakage and measures needed to alleviate the problem.

• Transport of excess nutrients to water bodies leads to algae growth and destruction of other organisms, a process known as eutrophication. The rate of transport in such systems is needed to determine further remediation actions.

• Carbon dioxide sequestration in underground mines is contemplated as a solution to reduce the impact of global warming. Mass transfer analysis is needed to predict the leakage and long-term feasibility of this solution.

The book by Clark (1996) provides a nice introduction to some of the problems mentioned here.

Next, we discuss the general methodology involved in setting up models for mass transfer processes.

Control volume can be of any size and shape. Typically, models at three levels are developed based on the size of the control volume: differential models, macroscopic (or macroscale) models, and mesoscopic (or mesoscale) models. These three types of models are explained sequentially in the following sections. On the topmost level is the differential control volume and the differential models. This type of model contains the local information at each point in the continuum. However, for design of process equipment, macro or meso levels may also be used. Although models at these levels are not as detailed as the differential models, they are easier to solve and to apply in practical designs. Hence modeling at all three levels is covered in this book.

For modeling very large or complex systems, such as transport of a pollutant in the environment or drug metabolism in the human body, the system is divided into a number of interconnected compartments and a macroscale model is applied to each compartment. Applications of these so-called compartment models are provided in Chapter 13.

Conservation laws alone are not sufficient to complete the model, since the model will contain terms for the mass crossing into a control surface. Hence additional closure relations are needed. For example, a differential model using the conservation principle alone will contain the flux vector as a parameter. This system has to be closed and a flux versus concentration relation has to be applied using a constitutive model. Similarly, the macro- and meso-level models may contain terms such as the mass crossing a larger-size control surface or from one phase to another. Suitable transport laws must be applied to close these terms, thereby completing the model. Hence coupling the conservation laws together with suitable transport law is the basic methodolgoy in developing mass transfer models.

In+Generation=Out+Depletion+Accumulation

Species A can cross into the control volume from a part of the control surface (the in term) and leave out of the rest of control surface.

Species may also be consumed (depletion) or produced (generation) within the control volume by chemical reaction.

Finally, the species concentration in the control volume may change with time due to the accumulation of the species within the control volume. The various processes involved are shown schematically in Figure 1.8.

The depletion is normally taken as a negative generation term and is often not included separately. In such cases, the generation is to be understood as net generation, which is equal to generation – depletion.

Similarly the out – in term is often written as a net efflux term. Hence another form of the conservation statement is

(Net) Generation = Net efflux + Accumulation

We now illustrate the use of this principle for the three levels of models discussed earlier and indicate which additional relations are needed to complete the model.

The simplest differential control volume is the one constructed in Cartesian coordinate volumes, which is shown in Figure 1.9. The net generation and accumulation depend on the control volume and, therefore, are known as volumetric terms. The net efflux term, known as the surface term, depends on the combined flux at that control surface. Application of the conservation principle, then, leads to the species mass balance equation involving the unknown fluxes. Nevertheless, this equation alone is not sufficient: We need additional relations because we ultimately want to solve for the concentration field. The flux term in the equation is the combined flux, which is the sum of the convection flux and the diffusion flux. The convection flux depends on the local velocity and local concentration, so it can be calculated if the velocity profile in the system is known. The diffusion flux results from the molecular interactions, which are not modeled at the continuum level. Hence a closure equation—that is, a constitutive model for diffusion— is needed to close the model. The most commonly used constitutive model is some form of Fick’s law. Note that for multicomponent diffusion in mixtures, the Stefan-Maxwell model for diffusion is also used; it is discussed in detail in Chapter 15.

In summary, mass transfer models at the differential levels are built through the combination of the conservation law and the constitutive law. The differential equations of mass transfer based on this approach are derived using this methodology in Chapter 5 and have general applicability. Many simpler problems can be deduced by simplification of these equations. In essence, this method is similar to that used in fluid mechanics, where the Navier-Stokes equations have general applicability for modeling flow of Newtonian fluids and where solutions to specific problems such as pipe flow can be obtained by simplification of these general equations. Likewise, we can go from the general to the particular in modeling of mass transfer. However, for simpler cases where the concentration varies only along one coordinate direction, it is instructive to derive and solve the equation by starting directly from the conservation laws and then using Fick’s law, rather than by directly using the general differential equations. This approach is illustrated in Chapter 2 for some common problems in mass transfer. A careful study of Chapter 2 will provide you with a basic understanding of the process of setting up and solving differential models.

Information, on length (or time) scales lower than that at which analysis is done, is lost and has to be supplemented in some suitable manner.

Appreciation and awareness of this principle are important in modeling. In particular, for the macroscale analysis discussed in this section, we lose information on the point-to-point concentration values and the point-wise variation of the flux over a given control surface. This loss must be countered by making some assumptions and using some closure models. In the following sections we discuss qualitatively the closures commonly used by examining some specific examples. More fully worked-out examples of macroscopic balances are provided in Chapter 3 and continued in Chapter 13.

in – out + generation = 0

where all the terms are in moles A per unit time.

Now let’s use mathematical terms to describe this system. In and out can be formulated in terms of the volumetric flow rate times the corresponding concentration. The main assumption needed is for the generation term. This term is the total generation over the entire control volume and is the volume integral of the local rate of reaction (mole/m³ local volume sec) R_A. The local rate, in turn, is a function of the local concentration. We assume that the required relation is known from a kinetic model. Now the generation over the entire volume is the volume integral of the local rate:

Here R_A (local) is some specified function of the local or pointwise concentration.

This local concentration is not known (not being modeled) in the macroscopic model, so it is not possible to evaluate precisely the generation term. Hence some assumption about the mixing level in the reactor (the concentration variation in the reactor) must be made. These details are studied in Chapters 3 and 13.

A common simplifying assumption is that the reactor is well mixed and the reactants present at a uniform concentration. This permits us to close the model, which is then referred to as a completely backmixed model. Many other models for closure may be used depending on the anticipated mixing level in the system (e.g., dividing the tank into a mixed zone and a dead zone or dividing the tank into two well-mixed subcompartments), and some of these models are explained in more detail in Chapter 13. The main point to note here is that some assumption about the mixing pattern must be made and the level of mixing needs to be tested and quantified by suitable experiments (e.g., by tracer methods).

in (zero) – out (transferred from solid to gas) = accumulation

The rate of transfer from solid to gas is precisely the surface integral of the radially directed outward flux at the surface of the solid:

transfer to gas=∫A(NA⋅er)dA=∫A(NAs)dA

where N_As is the radial component of the flux vector at the surface.

This flux vector can be computed only if a differential model for the gas phase is solved, including the convection and diffusion of the solute in the gas phase. It is not always possible to do this (or such details may not be important) since this endeavor is computationally intensive and the velocity profile in the gas phase may not be precisely known for complex geometries and for turbulent flow. Obviously, we need a closure for the unknown term, which done by introducing the mass transfer coefficient, k_m. Using this parameter, we model the transfer rate as a product of the mass transfer coefficient times a concentration difference:

C_As – C_Ab is taken as a driving force for mass transfer from the solid to the bulk gas. Note that C_As is the concentration of A in the gas phase at the gas side of the solid–gas interface, just near the solid. C_Ab is the concentration in the gas phase at a point far away from the solid, known as the bulk gas. This concentration is usually taken as zero.

Equation 1.19 is actually a definition, rather than a fundamental law. The mass transfer rate is given per unit area of transfer. Balancing the unit, we find that the mass transfer coefficient has the units of m/s. Values of the mass transfer coefficient are provided by numerous sources. For example, they may be empirically fit to experimental data or fit to detailed computational models. For many practical cases the values of the coefficient have previously been published; as a consequence, the concept of the mass transfer coefficient is widely used in practical design calculations. The main point here is that the use of the mass transfer coefficient completes the missing flux information in the context of macro-level (as well as meso-level) models.

We now look at mesoscopic models.

The conservation law (in – out = 0) is now represented as

in from flow at z + in by transported from the walls –out from flow at z + Δz = 0

To calculate the in and out terms due to flow, we must recognize that the velocity can vary across the cross-section. Concentration also varies along the pipe, with the maximum concentration found at the dissolving pipe wall. The flow can be laminar or turbulent. Laminar flow of a Newtonian fluid has a parabolic profile, whereas the profile is very steep in turbulent flow as shown in Figure 1.13. The velocity for the turbulent flow is the time-averaged value, a concept that is discussed more fully in Chapter 12.

A cross-sectionally averaged velocity is an useful quantity and is defined as

which is the integral average of the local velocity. The volumetric flow rate is then equal to 〈v〉 A; v is the axial velocity.

The plug flow shown in the Figure 1.13 is an idealization that is often used to simplify the models. Here the velocity is assumed to be the same at each local radial position. Consequently, the cross-sectional average velocity is also equal to the local velocity.

The in and out terms are the integral average of the axial flux values. The flux component in the axial direction is primarily due to convection and is locally equal to vC_A. For a small differential area, ΔA, the moles of A crossing is therefore equal to vC_A ΔA. Hence the moles crossing over the entire tube area is the integral of the product of local velocity and the local concentration:

in, convection=∫AvCAdA=〈vCA〉A

where the notation 〈...〉 is used as shorthand to indicate the averaged value of any quantity within the brackets. Thus we define

We need 〈vC_A〉 to evaluate the in and out terms, which cannot be calculated because local concentration values are not available at the meso-level. Hence we define and use an average concentration. This average concentration is called the cup mixing average concentration, of the flow weighted average. It is denoted as C_Ab and defined as follows:

Hence

The in and out terms are then represented simply as A 〈v〉 C_Ab. Thus the cup mixing concentration is used as the representative concentration variable in the context of our mesoscopic model. The local concentration has no relevance because the local information is not available. Note: In the context of the meso-level model, the notation C_A is used in many books as a simpler notation; by implication, this is the cup mixed average value and should be interpreted accordingly. The results in such models are the variation of the cup mixing concentration with the axial distance.

Example 1.2 shows a calculation of the cup mixing concentration to clarify the definition. Here we assume that the radial variation is known from other methods, such as from a differential model or by experimental measurements.

Having defined the in and out terms using the cup mixing concentration, we need an expression for the transport rate to the walls so as to close the model. This is done by defining and using a mass transfer coefficient. It is common practice to use the cup mixing concentration as the representative concentration away from the solid; the driving force for mass transfer is then defined using (C_Ab – C_Aw) as the driving force. Hence the mass transfer rate is defined as

N_Aw = k_m(C_Ab – C_Aw)

The mass transfer coefficients are available for a large number of flows (such as pipe flows) and, in turn, this approach to modeling is widely used in practice. The mesoscopic formulation can be cast into mathematical framework to account for variation of the concentration (cup mixing) as a function of axial position. Further details and the mathematical details follow in Chapter 4. At this stage, however, you should be able to appreciate the essential concepts that go into the construction of mesoscopic models.

Let us start with the conservation principle in words:

in – out + generation = 0

The generation is the average of the local rate:

generation = (A Δx) 〈R_A〉

where

〈RA〉=1A∫ARAdA

For a first-order reaction, for example, R_A = –k₁C_A, where C_A is the local concentration. Hence

−〈RA〉=1A∫Ak1CAdA

To incorporate this into the conservation law, it is customary to define and use the the cross-sectional average concentration:

〈CA〉=1A∫ACAdA

Hence the generation term (first order) to be used in the reactor model is –(AΔx)k₁ 〈C_A〉.

In and out terms are A 〈v〉 C_Ab as for the solid dissolution. Thus the conservation law in mathematical terms is

In the limit Δx → 0, this produces the following differential equation:

We find that both the cup mixing and the cross-sectional average appear in the reactor model. Some closure relationship is needed, however, and the commonly used closure models are the plug flow model and the axial dispersion model.

〈v〉ddx〈CA〉=−k1〈CA〉

The first term on the right side is the mole flow crossing a cross-section if the plug flow prevails. The second term is a correction term. This system is modeled as though some diffusion type of mechanism is superimposed on the plug flow value. Hence

correction term=−ADEddx〈CA〉

where D_E is called the axial dispersion coefficient. This is only a model, and not a fundamental principle, but nevertheless it provides an important crutch to proceed further in modeling these systems. The axial dispersion coefficient is a commonly used parameter when modeling both reactors and separation processes. Details of the origin of this term and the calculation and application of this concept are discussed in Section 14.2. For now, recognize that the dispersion model closure is

The key points to appreciate from this section are the various definitions and closures that are needed in the context of meso-scale models.

Now we discuss models at an even higher level of simplicity—namely, the compartmental models.

Compartmental models are also widely used in biomedical systems modeling. A simple two-compartment model to represent the body is shown in Figure 1.15. A simulation example based on this model is provided in Chapter 13.

Examples and solutions to compartmental models are taken up in Sections 13.7 and 13.8. It is common practice to simplify each compartment to be a well-mixed system with a constant composition. In addition, an exchange parameter needs to be assigned to model the transfer between the various parameters. In turn, the compartmental models typically use many simplifying (perhaps crude) assumptions. Despite this background, they have proven to have value in environmental and biomedical engineering.

1.2 What is meant by the continuum assumption and what are its implications in model development?

1.3 Indicate some situations where the continuum models are unlikely to apply.

1.4 Give an example of a system in which the total concentration is (nearly) constant.

1.5 Give an example of a system in which the mixture density is nearly constant.

1.6 Can average molecular weight be a function of position?

1.7 Which information is missing in the differential models based on the continuum assumption?

1.8 Why are constitutive models needed in the context of differential models?

1.9 Does the Fick’s law apply universally to all systems?

1.10 Write a form of Fick’s law using partial pressure gradient as the driving force.

1.11 What is meant by the invariant property of the flux vector?

1.12 The combined flux is partitioned into the sum of the convection flux and the diffusion flux. Is this partitioning unique?

1.13 State the units of the gas constant if the pressure is expressed in bar instead of Pa, (ii) if expressed in atm.

1.14 Express the dissolved oxygen concentration in Example 1.1 in parts per million (ppm).

1.15 What is a macroscopic-level model? Which information needs to be added in such a model in addition to the conservation principle?

1.16 How is the mass transfer coefficient defined? Why is it needed?

1.17 What is meant by a cross-sectional average concentration? What is a cup mixing or the bulk concentration?

1.18 Which assumptions are involved in a plug flow model?

1.19 Which assumptions are involved in a completely backmixed model?

1.20 Which additional closure is needed for mass transport in turbulent flow? Why?

1.21 What is an ideal stage contactor? How do you correct the model if the stage is not ideal?

1.22 What is the dispersion coefficient and where is it needed?

Derive an expression for dy_i as a function of dω_i values. Do this for a binary mixture. Expression for multicomponent mixture becomes rather unwieldy!

1.2 Mole fraction to mass fractions. Show that mole fractions can be converted to mass fractions by the use of the following equation:

Derive an expression for dω_i as a function of dy_i values for a binary mixture.

1.3 Average molecular weight. At a point in a methane reforming furnace, we have a gas of the composition CH₄ = 10%, H₂ = 15%, CO = 15%, and H₂O = 10% by moles. Find the mass fractions and the average molecular weight of the mixture. Find the density of the gas.

1.4 Average molecular weight variations. Two bulbs are separated by a 20-cm-long capillary tube. One bulb contains hydrogen and the other bulb contains nitrogen. The mole fraction profile varies in a linear manner along the length of the capillary. Calculate the mass fraction profile and show that the variation is not linear. Also calculate the average molecular weight as a function of the length along the capillary.

1.5 Mass fraction gradient. For a diffusion process across a stagnant film, the mole fraction gradient of the diffusing species was found to be constant. What is the mass fraction gradient? Is it also linear? The mixture is benzene–air.

1.6 Total concentration in a liquid mixture. Find the total molar concentration and species concentrations of 10% ethyl alcohol by mass in water at room temperature.

1.7 Effect of coordinate rotation on flux components. In Figure 1.1, the flux vector is 2e_x + e_y. Now consider a coordinate system that is rotated by an angle θ. Find this angle such that the flux component N_Ay is zero. What is the value of N_Ax in this coordinate system?

1.8 Flux vector in cylindrical coordinates. Define flux vector in terms of its components in cylindrical coordinates. Sketch the planes over which the components act. Show the relations between these components and the components in Cartesian coordinates.

1.9 Flux vector in spherical coordinates. Define the flux vector in terms of its components in spherical coordinates. Sketch the planes over which the components act. Show the relations between these components and the components in Cartesian coordinates.

1.10 Different forms of the Henry’s law constant. Express the Henry’s law constants reported in Table 1.2 as H_i,pc and H_i,cp.

1.11 Henry’s law constants: Unit conversions. Henry’s law constants for O₂ and CO₂ are reported as 760.2 L ˙ atm/mol and 29.41 L ˙ atm/mol. Which form of Henry’s law is being used? Convert these constants to values for the other forms of Henry’s law shown in the text.

1.12 Solubility of CO₂. The Henry’s law constant values for CO₂ are shown below as a function of temperature.

Temperature K	280	300	320
H. bar	960	1730	2650

Fit an equation of the type

ln H = A + B/T

What is the physical significance of the parameter B? Find the solubility of pure CO₂ in water at these temperatures.

1.13 Vapor pressure calculations: The Antoine equation. The Antoine constants for water are A = 8.07131, B = 1730.63, C = 233.426 in units of mm Hg for pressure and degrees Celsius for temperature. Convert these constants to a form where pressure is in Pa and temperature is in kelvins. Also rearrange the Antoine equation to a form where temperature can be calculated explicitly. This represents the boiling point at that pressure. What is the boiling point of water at Denver, Colorado (the “Mile High City”)?

1.14 Vant Hoff relation. Given the Antoine constants for a species, can you calculate the heat of vaporization of that species? Find this value for water from the data given in Problem 1.13.

1.15 Concentration jump. A solid rock of NaCl is in contact with water. Calculate the concentration of NaCl on the water side and salt side of the interface in mol/m³.

1.16 A well-mixed reactor: Mass balance calculation. Consider a well-stirred reactor where a reaction A → B is taking place. The volumetric flow rate is 1 m³/s and the reactor volume is 0.3 m³. The inlet concentration of A is 1000 mol/m³ and the exit concentration is 200 mol/m³. What is the rate of reaction of A in the system? If the reaction is first order and the contents are well mixed, what is the value of the rate constant?

1.17 Mass transfer coefficient calculation. A naphthalene ball (M_A = 128 g/mol and ρ_A = 1145 kg/m³) is suspended in a flowing stream of air at 347 K and 1 atm pressure. The vapor pressure of naphthalene is 666 Pa for the given temperature. The diameter of the ball was found to change from 2.1 cm to 1.9 cm over a time interval of 1 hour. Estimate the mass transfer coefficient from the solid to the flowing gas.

1.18 A model for VOC loss from a holding tank. Wastewater containing a VOC at a concentration of 10 mol/m³ enters an open tank at a volumetric flow rate of 0.2 m³/min and exits at the same rate. The tank has a diameter of 4 m and the depth of liquid in the tank is 1 m. The concentration of VOC in the exit stream and the rate of release of VOC is requested by the EPA. Use the conservation law to set a up a model. State further assumptions you may need to complete the model. List the parameters needed to solve the problem.

1.19 Averaging. Velocity profile in laminar flow is

v_z(r) = v_max[1 – (r/R)²]

What does v_max represent? How is it related to the pressure drop? Find the average velocity.

The concentration distribution at a given axial position for a solid dissolving from the wall is given by Equation 1.24 Find the cross-sectional average concentration and compare the value with the cup mixing concentration calculated in Example 1.2.

1.20 Cup mixing versus cross-sectional average. The variation of scaled concentration in a laminar flow tubular reactor was measured and fitted to the following equation at a specified axial position:

c_A = 0.5[1 – (r/R)² + (r/R)⁴/2]

Calculate the center, wall, cup mixing, and cross-sectional average concentrations.

1.21 Turbulent flow velocity profile. Velocity profile in turbulent flow is commonly modeled by the 1/7-th law:

v_z(r) = v_c[1 – (r/R)]^1/7

What does v_c represent? Find the average velocity. Compare the average velocity and the center line velocity.