Introduction to Modeling of Mass Transfer Processes
- 1.1 What Is Mass Transfer?
- 1.2 Preliminaries: Continuum and Concentration
- 1.3 Flux Vector
- 1.4 Concentration Jump at Interface
- 1.5 Application Examples
- 1.6 Basic Methodology of Model Development
- 1.7 Conservation Principle
- 1.8 Differential Models
- 1.9 Macroscopic Scale
- 1.10 Mesoscopic or Cross-Section Averaged Models
- 1.11 Compartmental Models
- Review Questions
Introduces the basic methodology for modeling of mass transport processes and indicates three hierarchical levels of models, namely differential, macroscopic, and mesoscopic models.
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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 NOx, CO, and other hydrocarbons from combustion gases and involves transport of NOx 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.
1.1 What Is Mass Transfer?
As indicated in the preamble, mass transfer refers to a net movement of a species A from one location to another in a multicomponent mixture containing species A, B, C, and so on. Note that mass can also be transported in a single-component system—for example, in water being pumped from one location to other—but these problems are more pertinent to hydraulics or fluid dynamics and not considered in the realm of mass transfer.
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.
1.1.1 What Is Interfacial Mass Transfer?
Interfacial mass transfer refers to a situation in which a species A crosses from one phase to another across the interface separating the two phases. Examples are widespread in the field of unit operations. In liquid–liquid extraction, for instance, species A in a mixture of A and B in one phase can be separated by contact with an immiscible solvent in which A is preferentially more soluble, resulting in an interfacial transport of A from the first phase to the solvent phase.
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.
1.1.2 What Causes Mass Transfer?
Mass transfer is caused by a combined process of diffusion and convection. Diffusion is an effect of molecular- or atomic-level interactions. For example, in a gas, the molecules are in a state of random motion, so there is tendency for concentration to equalize. Thus, if a concentration difference for species A exists between two points, then the molecular motion causes a net transport of A from a region of higher concentration to a region of lower concentration. (Note: There are a few cases in which mass transfer can occur from lower to higher concentration; these are discussed in Chapter 7.)
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.