**PART I**

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1 Introduction

All physical situations of interest to engineers and scientists are complex enough that a mathematical model of some sort is essential to describe them in sufficient detail for useful analysis and interpretation. Mathematical expressions provide a common language so different disciplines can communicate among each other more effectively. Models are very critical to chemical engineers, chemists, biochemists, and other chemical professionals because most situations of interest are molecular in nature and take place in equipment that does not allow for direct observation. Experiments are needed to extract fundamental knowledge and to obtain critical information for the design and operation of equipment. To do this effectively, one must be able to quantitatively analyze mass, energy, and momentum transfer (transport phenomena) at some level of complexity. In this text we define six levels of complexity, which characterize the level of detail needed in model development. The various levels are summarized in Table 1.1.

**Level Ⅰ, Conservation of Mass and/or Energy**. At this level of analysis the control volume is considered a black box. A control volume is some region of space, often a piece of equipment, that is designated for “accounting” purposes in analysis. Only the laws of conservation of mass and/or energy are applied to yield the model equations; there is no consideration of molecular or transport phenomena within the control volume. It is a valuable approach for the analysis of existing manmade or natural systems and is widely employed. The mathematical expressions needed to describe Level Ⅰ problems are algebraic or simple first-order differential equations with time as the only independent variable:

*A calculation of the flows and stream compositions to and from a continuously operating distillation column illustrates this level of analysis. A sketch of the system is shown as Figure 1.1.*. *A typical column has internals or some type of trays, a condenser, a reboiler, and pumps for circulation of liquids. The mass flow rate to the column, F, must be equal to the mass flow rate of the distillate, D, plus the mass flow rate of the bottoms product, B*,

F = D+B.

*This simple mass balance always holds and is independent of the column diameter, type, or number of trays or the design of the condenser or the reboiler. We must know two of the quantities in this equation before we can calculate the third. Such values can come from a distillation column already operating or we may specify the required quantities if considering a process design*.

Table 1.1. Level definitions

Level I | Level II | Level III | Level IV | Level V | Level VI | |
---|---|---|---|---|---|---|

Outcome | Overall mass and energy balances | Allows number of equilibrum stages to be determined but does not allow stage design to be achieved | Equipment design at the laboratory, pilot and commercial scale can be achieved | Molecular conduction and diffusion analysis are quantified; mass and heat transfer coefficient correlations are developed | ||

CONSERVATION OF MASS | Required | Required | Required | Required | Required | Required |

CONSERVATION OF ENERGY | Required | Required | Required | Required | Required | Required |

Conservation of momentum | Not required | Not required | Not required | Not required | Required | Required |

Equilibrium constitutive relations | Not required | Required | May be required | May be required | May be required | May be required |

Constitutive relations | ||||||

Rate equations | Not required | Not required | Required | Required | Not required | Not required |

CONDUCTION–DIFFUSION–viscosity relations | Not required | Not required | Not required | Not required | Required | Required |

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Figure 1.1. Distillation column.

*We can also write component mass balances for the column, and these are subsequently presented for a two-component system. The mass fraction of one component in the feed stream to the column is x _{F}, and the total amount of this component entering the column is Fx _{F}; similarly, the amounts leaving in the distillate and the column bottoms can be represented by Dx _{D}, and Bx _{B}. The component balance relation becomes*

Fx_{F} = Dx_{D}+Bx_{B}.

*These two simple equations are the first step in any analysis of a distillation column. Many other useful relations can be readily derived by selecting portions of the column as a control volume. A very clear discussion of this type of Level I* *analysis is presented in Chapter 18 of Unit Operations of Chemical Engineering by McCabe et al. (**1993**).*

**Level Ⅱ, Conservation of Mass and/or Energy and Assumption of Equilibrium.** At this level, transport phenomena are not considered. The analysis of molecular phenomena is simplified by assuming that chemical, thermal, or phase equilibrium is achieved in the control volume of interest. Chemical equilibrium is characterized by K_{eq}, a quantity that can be obtained from tabulated values of the free energy for many reactions. Thermal equilibrium is achieved in devices that exchange heat when all temperatures are the same (for a closed two-volume control system T_{1∞} = T_{2∞}). The constitutive relationships describing phase equilibria can be complex, but decades of research have produced many useful constitutive relationships for phase equilibria. The simplest for a liquid–liquid system is Nernst’s “Law,” which relates species concentration in one liquid to that in another by a constant called a distribution coefficient. Henry’s “Law” is another simple example relating concentration of a dilute species in a liquid phase to the concentration of the same species in a vapor phase by a single constant. Assuming phase equilibria has proven most valuable in determining the number of theoretical stages to accomplish some stated goal of mass transfer between phases, but does not allow the stage design to be specified. When equilibrium is assumed for reacting systems, the concentrations of product and reactant can be determined but not the volume of the reactor. Almost all situations of interest at Level Ⅱ require only algebraic equations.

*In our example of the distillation column, detailed computer procedures that can handle complex equilibrium relations are available to determine the number of theoretical stages. To do so, mass and energy balances need to be derived by first selecting as a control volume an individual stage (tray) in the column. The set of such algebraic equations can become quite complex and requires a computer for solution. Widely used programs to do this are contained in the software package from Aspen Technology (see reference at the end of this chapter). However, even with a numerical solution, such an analysis is not able to specify the tray design or the number of actual trays needed*.

**Level Ⅲ, Conservation of Mass and/or Energy and Use of Constitutive Relationships.** This is the first level at which the rate of transport of mass and energy is considered. Momentum transfer is simplified by assuming simple single-phase fluid motions or no fluid motions within the chosen control volume. It is the level of analysis that is almost always employed for laboratory-scale experiments and is one in which pragmatic equipment design can be achieved. Only two types of limiting fluid motion are considered:

- well-mixed fluid motion,
- plug-flow fluid motion.

Well-mixed fluid motion within a control volume is easy to achieve with gases and low-viscosity liquids and almost always occurs in small-scale batch laboratory experimental apparatus. It is also relatively easy to achieve in pilot- or commercial-scale equipment. A very complete analysis of mixing and mixing equipment is presented in the *Handbook of Industrial Mixing* (Paul et al., 2004). By “well mixed,” we mean that one can assume there is no spatial variation in the measured variable within the control volume. This is easy to visualize with a batch system because we are not introducing any fluid into the vessel as the process takes place. However, there is a conceptual and sometimes a pragmatic difficulty when one considers a semibatch (sometimes referred to as a fed-batch) in which one fluid is introduced to the system over the time the process is taking place or a continuous-flow system in which fluids both enter and leave the vessel. The well-mixed assumption requires that any fluid introduced immediately reach the average property of the bulk fluid in the vessel. Of course, this is not exactly what occurs in a real system. Close to the point of introduction, the fluid in the vessel will have properties that are some intermediate value between that of the bulk fluid and that of the incoming fluid. In the majority of situations this will not materially affect the model equation behavior that is developed with the well-mixed assumption. There are a few cases with either chemical or biochemical reactors in which special attention must be paid to the fluid introduction. These are discussed in Chapters 2, 4, and 7. The well-mixed assumption also requires that any fluid leaving the vessel have the same properties as those of the well-mixed fluid in the vessel. This is almost always the case in well-designed vessels.

Plug-flow motion is the other extreme in fluid motion behavior, and the analysis assumes that changes occur in one spatial direction only. This is frequently a good assumption if one is concerned with chemical reaction, mass and/or heat transfer in pipes. A Level Ⅲ analysis is particularly valuable for the study of experiments to investigate molecular phenomena. Many simple chemistry experiments are carried out in well-mixed batch apparatus.

The models in a Level Ⅲ analysis of a fluid system are first-order ordinary differential equations with either time or one spatial dimension as the independent variable.

In a solid or for the case in which there is no fluid motion, one can model using time and all three spatial coordinates if required.

*In our distillation column example, a Level Ⅲ analysis is not useful because we are dealing with two phases, a liquid and a vapor. When more than one phase is present, we need a Level Ⅳ analysis*.

*A Level Ⅲ problem is illustrated in the discussion following the definition of levels*.

**Level Ⅳ, Level Ⅲ Equivalent for Multiple Phases.** In problems in which there is more than one phase, the fluid motions are often extremely complex and difficult to quantify. However, many significant problems can be solved by assuming simple two-phase fluid motions. The following fluid motion categories have proven useful:

- both phases well mixed,
- both phases in plug flow,
- one phase well mixed, one phase in plug flow.

Assuming these simplified fluid motions allows an analysis of experiment and pragmatic equipment design to be achieved.

*Level Ⅳ analysis of gas–liquid systems is illustrated in a series of papers that have been widely employed for analysis of experiment and equipment design (Cichy et al., 1969; Schaftlein and Russell, 1968)*.

**Level Ⅴ, Complex Analysis of Single-Phase Transport Phenomena.** This level considers the analysis of transport phenomena by considering time and, if required, all three spatial variables in the study of single-phase fluid motions. It is the first level in which we may need detailed fluid mechanics. Most problems of interest to us in this text will involve time and one spatial direction and are discussed in Part II of the text.

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Figure 1.2. The logic required for technically feasible analysis and design.

**Level Ⅵ, Complex Analysis of Multiple-Phase Transport Phenomena.** The analysis of Level Ⅵ is extended to multiphase systems and is an area of active research today. We will not consider problems at this level.

In Part I of this text we consider those physical situations that we can model at Levels Ⅰ through Ⅳ, with most of our analysis concentrating on Level Ⅲ. In Part II we solve some Level Ⅴ problems to gain insight into the small-scale fluid motions affecting heat and mass transfer.

Figure 1.2, modified from that presented in *Introduction to Chemical Engineering Analysis* by Russell and Denn (1972), identifies the critical issues in the analysis of mass and heat transfer problems that we discuss in this text. The logic diagram and the definition of levels guide us to the proper choice of model complexity.

Defining problem objectives and uncertainties, and identifying time constraints, are of critical importance but are often ignored. The objectives can have a great impact on the complexity of the model, which in turn affects time constraints on any analysis. One should always strive to have the simplest model that will meet the problem objectives (which almost always include severe time constraints and uncertainties). The problem complexity levels defined in Table are critical in this evaluation. There are many reasons for this, which we will illustrate as we proceed to develop our approach to heat and mass transfer.

Models of the physical situations encountered in heat and mass transfer are almost always a set of algebraic or differential equations. A straightforward application of the laws of conservation of mass, energy, and momentum allows one to derive the model equations if one attains a certain level of skill in selecting the right constitutive equation to use in the basic conservation law equations. Comparison of model behavior with data from laboratory experiments or other sources requires that model behavior be determined. One can do this by analytical procedures or by numerically solving equations using any one of a number of numerical software packages. It is often this part of analysis that is given the most attention because determining behavior follows well-established rules and is thus easier to teach even though it can be extremely tedious. Overconcentration on model behavior can take away from the time available to deal with the more critical issues of analysis, such as model development and evaluation of model uncertainty.

In our view, the most critical and indeed the most interesting part of analysis is deciding when a satisfactory agreement has been reached when one compares model behavior with experimental reality. *Model evaluation and validation is a nontrivial task and is the art of any analysis process*. Note that it is dependent on the problem objectives. It is a matter we will return to many times as we develop a logical approach to mass and heat transfer.

A verified model allows us to plan additional experiments if needed or to use the equations for the design, operation, and control of laboratory- pilot- or commercial-scale equipment.

*We can illustrate the analysis process most effectively with a simple example using a reaction and reactor problem. A single-phase liquid reacting system is the simplest Level Ⅲ problem chemical engineers and chemical professionals encounter, and its analysis effectively illustrates the crucial link among experiment, modeling, and design. We first must derive the model equations*.

Figure 1.3, modified from Figure 2.1 presented in *Introduction to Chemical Engineering Analysis* by Russell and Denn (1972), is a well-tested guide needed to obtain the model equations. We review in this book that part of model development critical to mass and heat transfer. A variety of physical situations that require analysis are discussed. In academic environments professors provide prose descriptions of the physical situation that students are then expected to use for model development. For any situation of reasonable complexity it is difficult to provide a completely adequate prose description, and this can lead to frustrations. In what is often called the “real world,” the physical situation is defined by dialogue with others and direct observation.

**EXAMPLE 1.1**. In our example reaction and reactor problem, the physical situation that must be modeled first is a batch reactor. We assume that our experiments will be carried out in a well-mixed 1-L glass flask that is kept at the temperature of interest by immersion in a temperature-controlled water bath. Figure 1.4 shows the same steps as in Figure 1.3 for this physical situation. For this example problem, a photo of this experimental apparatus is shown in Figure 1.5. We consider that a liquid-phase reaction is taking place in which a compound “A” is converted to a product “D.” Previous experiment has shown that the chemical equation is

A → D.

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Figure 1.3. Model development logic.

One may need one or more of the basic variables of mass, energy, and momentum to begin model development. These quantities are conserved within a given control volume. The word statement that allows us to begin an equation formulation is presented as Figure 1.6 [Russell and Denn (1972)]. For most situations encountered in mass and heat transfer we will make more frequent use of the conservation of mass and energy than of momentum. When it is not possible to measure mass or energy directly, we must express them with dependent variables that can be measured.

*Our conserved variable in the batch reacting system is mass. Because the total mass in the flask is constant we are concerned with the mass of the various species, which also satisfy the word statement of the conservation laws*.

The second block on the logic diagrams, Figures 1.3 and 1.4, selection of variables that can be measured, is very important in developing our quantitative understanding of any physical situation. It is easiest to visualize experiments for measuring mass. For example, if we are concerned with the flow of a liquid from a cylindrical tank mounted on a scale, m (mass) can be directly measured. If a scale were not available we could determine mass of liquid by measuring the height of liquid h, the liquid density ρ, (available in physical property tables), and the measured area of the tank A, thus expressing the mass of liquid in the tank by the combination ρAh. Temperature T and concentration of a species C are two of the most common measured state variables used in mass and heat transfer problems, but there are many other variables that may be needed. Some require a specially designed experiment, and some are obtained from physical property tables in handbooks or from software programs (experiments done by others and tabulated). The experiment to obtain any variable should be clearly understood ; an ability to do so is critical for effective interpretation of the model equations.

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Figure 1.4. Application of model development (Figure 1.3) to batch reaction example.