Unit Operations in Environmental Engineering

By Louis Theodore, Ryan R. Dupont and Kumar Ganesan

The authors have written a practical introductory text exploring the theory and applications of unit operations for environmental engineers that is a comprehensive update to Linvil Rich’s 1961 classic work, “Unit Operations in Sanitary Engineering”. The book is designed to serve as a training tool for those individuals pursuing degrees that include courses on unit operations. Although the literature is inundated with publications in this area emphasizing theory and theoretical derivations, the goal of this book is to present the subject from a strictly pragmatic introductory point-of-view, particularly for those individuals involved with environmental engineering.

This book is concerned with unit operations, fluid flow, heat transfer, and mass transfer. Unit operations, by definition, are physical processes although there are some that include chemical and biological reactions. The unit operations approach allows both the practicing engineer and student to compartmentalize the various operations that constitute a process, and emphasizes introductory engineering principles so that the reader can then satisfactorily predict the performance of the various unit operation equipment.

• Language: English
• Publisher: Wiley
• Release date: Aug 29, 2017
• ISBN: 9781119283683

Book preview

Part I

INTRODUCTION TO THE PRINCIPLES OF UNIT OPERATIONS

The purpose of this Part can be found in its title. The book itself offers the reader the principles of unit operations with appropriate practical applications, and serves as an introduction to the specialized and more sophisticated texts in this area. The reader should realize that the contents are geared not only toward practitioners in this field, but also students of science and engineering. Topics of interest to all practicing engineers have been included. It should also be noted that the microscopic approach of unit operations is not covered here. The approach taken in the text is to place more emphasis on real-world and design applications. However, microscopic approach material is available in the literature, as noted in the ensuing chapters.

The chapters in this Part provide an introduction and overview of unit operations. Part I chapter content includes:

1. History of Chemical Engineering and Unit Operations

2. Transport Phenomena versus Unit Operations Approach

3. The Conservation Laws and Stoichiometry

4. The Ideal Gas Law

5. Thermodynamics

6. Chemical Kinetics

7. Equilibrium versus Rate Considerations

8. Process and Plant Design

Topics covered in the first two introductory chapters include a history of chemical engineering and unit operations, and a discussion of transport phenomena versus unit operations. The remaining chapters are concerned with introductory engineering principles.

Chapter 1

History of Chemical Engineering and Unit Operations

A discussion of the field of chemical engineering is warranted before proceeding to some specific details regarding unit operations and the contents of this first chapter. A reasonable question to ask is: What is chemical engineering? An outdated, but once official definition provided by the American Institute of Chemical Engineers is:

Chemical Engineering is that branch of engineering concerned with the development and application of manufacturing processes in which chemical or certain physical changes are involved. These processes may usually be resolved into a coordinated series of unit physical operations (hence part of the name of the chapter and book) and chemical processes. The work of the chemical engineer is concerned primarily with the design, construction, and operation of equipment and plants in which these unit operations and processes are applied. Chemistry, physics, and mathematics are the underlying sciences of chemical engineering, and economics is its guide in practice.

The above definition was appropriate up until a few decades ago when the profession branched out from the chemical industry. Today, that definition has changed. Although it is still based on chemical fundamentals and physical principles, these principles have been de-emphasized in order to allow for the expansion of the profession to other areas (biotechnology, semiconductors, fuel cells, environment, etc.). These areas include environmental management, health and safety, computer applications, and economics and finance. This has led to many new definitions of chemical engineering, several of which are either too specific or too vague. A definition-proposed here is simply that chemical engineers solve problems. Unit operations is the one subject area that historically has been the domain of the chemical engineer. It is often present in the curriculum and includes fluid flow [1], heat transfer [2] and mass transfer [3] principles.

Although the chemical engineering profession is usually thought to have originated shortly before 1900, many of the processes associated with this discipline were developed in antiquity. For example, filtration operations were carried out 5,000 years ago by the Egyptians. MTOs such as crystallization, precipitation, and distillation soon followed. During this period, other MTOs evolved from a mixture of craft, mysticism, incorrect theories, and empirical guesses.

In a very real sense, the chemical industry dates back to prehistoric times when people first attempted to control and modify their environment. The chemical industry developed as did any other trades or crafts. With little knowledge of chemical science and no means of chemical analysis, the earliest chemical engineers had to rely on previous art and superstition. As one would imagine, progress was slow. This changed with time. The chemical industry in the world today is a sprawling complex of raw-material sources, manufacturing plants, and distribution facilities which supply society with thousands of chemical products, most of which were unknown only a century ago. In the latter half of the 19th century, an increased demand arose for engineers trained in the fundamentals of chemical processes. This demand was ultimately met by chemical engineers.

The first attempt to organize the principles of chemical processing and to clarify the professional area of chemical engineering was made in England by George E. Davis. In 1880, he organized a Society of Chemical Engineers and gave a series of lectures in 1887 which were later expanded and published in 1901 as A Handbook of Chemical Engineering. In 1888, the first course in chemical engineering in the United States was organized at the Massachusetts Institute of Technology by Lewis M. Norton, a professor of industrial chemistry. The course applied aspects of chemistry and mechanical engineering to chemical processes [4].

Chemical engineering began to gain professional acceptance in the early years of the 20th century. The American Chemical Society had been founded in 1876 and, in 1908, it organized a Division of Industrial Chemists and Chemical Engineers while authorizing the publication of the Journal of Industrial and Engineering Chemistry. Also in 1908, a group of prominent chemical engineers met in Philadelphia and founded the American Institute of Chemical Engineers [4].

The mold for what is now called chemical engineering was fashioned at the 1922 meeting of the American Institute of Chemical Engineers when A. D. Little’s committee presented its report on chemical engineering education. The 1922 meeting marked the official endorsement of the unit operations concept and saw the approval of a declaration of independence for the profession [4]. A key component of this report included the following:

Any chemical process, on whatever scale conducted, may be resolved into a coordinated series of what may be termed unit operations, as pulverizing, mixing, heating, roasting, absorbing, precipitation, crystallizing, filtering, dissolving, and so on. The number of these basic unit operations is not very large and relatively few of them are involved in any particular process … An ability to cope broadly and adequately with the demands of this (the chemical engineer’s) profession can be attained only through the analysis of processes into the unit actions as they are carried out on the commercial scale under the conditions imposed by practice.

It also went on to state that:

Chemical Engineering, as distinguished from the aggregate number of subjects comprised in courses of that name, is not a composite of chemistry and mechanical and civil engineering, but is itself a branch of engineering …

A classical approach to chemical engineering education, which is still used today, has been to develop problem solving skills through the study of several topics. One of the topics that has withstood the test of time is mass transfer operations (MTOs). In many MTOs, one component of a fluid phase is transferred to another phase because the component is more soluble in the latter phase. The resulting distribution of components between phases depends upon the equilibrium of the system. MTOs may also be used to separate products (and reactants) and may be used to remove byproducts or impurities to obtain highly pure products. Finally, they can be used to purify raw materials.

A time line of the history of chemical engineering between the profession’s founding to 2010 is shown in Figure 1.1 [4]. It can be seen from the time line that the profession has reached a crossroads regarding the future education/curriculum for chemical engineers. This is highlighted by the differences of Transport Phenomena and Unit Operations, a topic that is discussed in the next chapter.

Figure 1.1 Chemical Engineering time line [4].

References

1. Abulencia, P. and Theodore, L., Fluid Flow for the Practicing Chemical Engineer, John Wiley & Sons, Hoboken, NJ, 2009.

2. Theodore, L., Heat Transfer Applications for the Practicing Engineer, John Wiley & Sons, Hoboken, NJ, 2011.

3. Theodore, L. and Ricci, F., Mass Transfer Operations for the Practicing Engineer, John Wiley & Sons, Hoboken, NJ, 2010.

4. Serino, N., 2005 Chemical Engineering 125th Year Anniversary Calendar, term project, submitted to L. Theodore, Manhattan College, Bronx, NY, 2004.

5. Bird, R., Stewart, W., and Lightfoot, E., Transport Phenomena, 2nd Edition, John Wiley & Sons, Hoboken, NJ, 2002.

Chapter 2

Transport Phenomena versus the Unit Operations Approach

The history of unit operations is interesting. As indicated in the previous chapter, chemical engineering courses were originally based on the study of unit processes and/or industrial technologies. However, it soon became apparent that the changes produced in equipment from different industries were similar in nature, i.e., there was commonality in the mass transfer operations in the petroleum industry and the chemical. These similar operations became known as unit operations. This approach to chemical engineering was promulgated in the 1922 Little report discussed earlier, and has, with varying degrees and emphasis, dominated the profession to this day.

The unit operations approach was adopted by the profession soon after its inception. During the more than 135 years (since 1880) that the profession has been in existence as a branch of engineering, society’s needs have changed tremendously and so has chemical engineering.

The teaching of unit operations at the undergraduate level has remained relatively unchanged since the publication of several early – to mid-1900 texts. However, by the middle of the 20th century, there was a slow movement from the unit operation concept to a more theoretical treatment called transport phenomena or, more simply, engineering science. The focal point of this science is the rigorous mathematical description of all physical rate processes in terms of mass, heat, or momentum crossing phase boundaries. This approach took hold of the education/curriculum of the profession with the publication of the first edition of the Bird et al. book [1] in 1960. Some, including the authors of this text, feel that this concept set the profession back several decades since graduating chemical engineers were being trained more as applied physicists than traditional chemical engineers. There has fortunately been a return to the traditional approach to chemical engineering, primarily as a result of the efforts of ABET (Accreditation Board for Engineering and Technology). The more traditional approach replaced some theoretical material normally covered in transport phenomena courses in part with material emphasizing the solution of design and open-ended problems. This design-oriented approach is emphasized in this text.

The following paragraphs attempt to qualitatively describe the differences between the above two approaches. Both deal with the transfer of certain quantities (momentum, energy, and mass) from one point in a system to another. There are three basic transport mechanisms which can potentially be involved in a process. They are:

Radiation

Convection

Molecular Diffusion

The first mechanism, radiative transfer, arises as a result of wave motion and is not considered, since it may be justifiably neglected in most engineering applications. The second mechanism, convective transfer, occurs simply because of bulk motion. The final mechanism, molecular diffusion, can be defined as the transport mechanism arising as a result of gradients. For example, momentum is transferred in the presence of a velocity gradient; energy in the form of heat is transferred because of a temperature gradient; and, mass is transferred in the presence of a concentration gradient. These molecular diffusion effects are described by phenomenological laws [1].

Momentum, energy, and mass are all conserved. As such, each quantity obeys the conservation law within a system. The conservation law may be applied at the macroscopic, microscopic, or molecular level.

One can best illustrate the differences in these methods with an example. Consider a system in which a fluid is flowing through a cylindrical tube (see Figure 2.1) and define the system as the fluid contained within the tube between Points 1 and 2 at any time. If one is interested in determining changes occurring at the inlet and outlet of a system, the conservation law is applied on a macroscopic level to the entire system. The resultant equation (usually algebraic) describes the overall changes occurring to the system (or equipment). This approach is usually applied in Unit Operation (or its equivalent) courses, an approach which is highlighted in this and three companion texts [2–4].

Figure 2.1 Fluid flow through a cylinder tube.

In the microscopic/transport phenomena approach, detailed information concerning the behavior within a system is required; this is occasionally requested of and by the engineer. The conservation law is then applied to a differential element within the system that is large compared to an individual molecule, but small compared to the entire system. The resulting differential equation is then expanded via an integration in order to describe the behavior of the entire system.

The molecular approach involves the application of the conservation laws to individual molecules. This leads to a study of statistical and quantum mechanics – both of which are beyond the scope of this text. In any case, the description at the molecular level is of little value to the practicing engineer. However, the statistical averaging of molecular quantities in either a differential or finite element within a system can lead to a more meaningful description of the behavior of a system.

Both the microscopic and molecular approaches shed light on the physical reasons for the observed macroscopic phenomena. Ultimately, however, for the practicing engineer, these approaches may be valid but are akin to attempting to kill a fly with a machine gun. Developing and solving these differential equations (in spite of the advent of computer software packages) is typically not worth the trouble.

Traditionally, the applied mathematician has developed differential equations describing the detailed behavior of systems by applying the appropriate conservation law to a differential element or shell within the system. Equations were derived with each new application. The engineer later removed the need for these tedious and error-prone derivations by developing a general set of equations that could be used to describe systems. These have come to be referred to by some as the transport equations. In recent years, the trend toward expressing these equations in vector form has gained momentum (no pun intended). However, the shell-balance approach has been retained in most texts where the equations are presented in componential form, i.e., in three particular coordinate systems – rectangular, cylindrical, and spherical. The componential terms can be lumped together to produce a more concise equation in vector form. The vector equation can be, in turn, re-expanded into other coordinate systems. This information is available in the literature [1,5].

It should be noted that the macroscopic approach has been primarily adapted by undergraduate environmental engineering educators. Any attempt to include the microscopic approach has been essentially reserved solely for graduate studies in this area. Only the macroscopic approach is employed in this text.

References

1. Bird, R., Stewart, W., and Lightfoot, E., Transport Phenomena, 2nd Edition, John Wiley & Sons, Hoboken, NJ, 1960.

2. Theodore, L., Heat Transfer Applications for the Practicing Engineer, John Wiley & Sons, Hoboken, NJ, 2011.

3. Abulencia, P. and Theodore, L., Fluid Flow for the Practicing Chemical Engineer, John Wiley & Sons, Hoboken, NJ, 2009.

4. Theodore, L. and Ricci, F., Mass Transfer Operations for the Practicing Engineer, John Wiley & Sons, Hoboken, NJ, 2010.

5. Theodore, L., Introduction to Transport Phenomena, International Textbook Co., Scranton, PA, 1970.

Chapter 3

The Conservation Laws and Stoichiometry

This chapter is primarily concerned with the conservation laws, and to a lesser degree, chemical stoichiometry. As with all remaining chapters in Part I, there are several sections: an overview, specific technical topics, illustrative examples where appropriate, and references.

3.1 Overview

In order to better understand the design, as well as the operation and performance of equipment in the environmental industry, it is necessary for engineers (as well as applied scientists) to understand the fundamentals and principles underlying the conservation laws and stoichiometry. How can one predict what products will be emitted from effluent streams? At what temperature must a unit be operated to ensure the desired performance? How much energy in the form of heat is given off? Is it economically feasible to recover this heat? Is the design appropriate? The answers to these questions are rooted not only in the subject matter in this chapter but also in the various theories of chemistry, physics, and applied economics.

The remaining topics covered in this section include:

The Conservation Law

The Conservation Laws for Mass, Energy and Momentum

Stoichiometry

Note: the bulk of the material in this chapter has been drawn from the original work of Reynolds [1].

3.2 The Conservation Law

Mass, energy and momentum are all conserved. As such, each quantity obeys the general conservation law below, as applied within a system.

(3.1)

Graphic

Equation 3.1 may also be written on a time rate basis:

(3.2)

Graphic

The conservation law may be applied by the practitioner at the macroscopic, microscopic, or molecular level. The differences in these methods was illustrated with an example in the previous chapter in Figure 2.1 by considering fluid flow through a cylindrical tube. The microscopic approach is employed when detailed information concerning the behavior within the system is of interest. If one is interested in determining changes occurring at the inlet and outlet of the system, the conservation law is applied on a macroscopic level to the entire system. The resultant equation describes the overall changes occurring to the system without regard for internal variations within the system, and it is this approach that is usually applied by the practicing engineer. The macroscopic approach is primarily adopted and applied in this text, and little to no further reference to microscopic or molecular analyses will be made. This chapter’s aim, then, is to express the laws of conservation for mass, energy, and momentum in algebraic or finite difference form.

3.3 Conservation of Mass, Energy, and Momentum

The conservation law for mass can be applied to any process, equipment, or system. The general form of this law is given by Equations 3.3 and 3.4.

(3.3)

Graphic

or on a time rate basis by

(3.4)

Graphic

The law of conservation of mass states that mass can neither be created nor destroyed. Nuclear reactions, in which interchanges between mass and energy are known to occur provide a notable exception to this law. Even in chemical reactions, a certain amount of mass-energy interchange takes place. However, in normal environmental engineering applications, nuclear reactions do not occur and the mass-energy exchange in chemical reactions is so minuscule that it is not worth taking into account.

The law of conservation of energy, which like the law of conservation of mass, applies for all processes that do not involve nuclear reactions, states that energy can neither be created nor destroyed. As a result, the energy level of the system can change only when energy crosses the system boundary, i.e.,

(3.5)

Graphic

(Note: The symbol Δ means change in). Energy crossing the boundary can be classified in one of two different ways: heat, Q, or work, W. Heat is energy moving between the system and the surroundings by virtue of a temperature driving force, and heat flows from high temperature to low temperature. The entire system is not necessarily at the same temperature; neither are the surroundings. If a portion of the system is at a higher temperature than a portion of the surroundings and as a result, energy is transferred from the system to the surroundings, that energy is classified as heat. If part of the system is at a higher temperature than another part of the system and energy is transferred between the two parts, that energy is not classified as heat because it is not crossing the boundary. Work is also energy moving between the system and surroundings, but the driving force here is something other than temperature difference, e.g., a mechanical force, a pressure difference, gravity, a voltage difference, a magnetic field, etc. Note that the definition of work is a force acting through a distance. All of the examples of driving forces just cited can be shown to provide a force capable of acting through a distance [2].

The energy level of a system has three contributions: kinetic energy, potential energy, and internal energy. Any body in motion possesses kinetic energy. If the system is moving as a whole, its kinetic energy, Ek, is proportional to the mass of the system and the square of the velocity of its center of gravity. The phrase as a whole indicates that motion inside the system relative to the system’s center of gravity does not contribute to the Ek term, but rather to the internal energy term. The terms external kinetic energy and internal kinetic energy are sometimes used here. An example would be a moving railroad tank car carrying liquid waste. (The liquid waste is the system). The center of gravity of the waste is moving at the velocity of the train, and this constitutes the system’s external kinetic energy. The liquid molecules are also moving in random directions relative to the center of gravity, and this constitutes the system’s internal energy due to motion inside the system, i.e., internal kinetic energy. The potential energy, Ep, involves any energy the system as a whole possesses by virtue of its position (more precisely, the position of its center of gravity) in some force field, e.g., gravity, centrifugal, electrical, etc., that provides the system with the potential for accomplishing work. Again, the phase as a whole is used to differentiate between external potential energy, Ep, and internal potential energy. Internal potential energy refers to potential energy due to force fields inside the system. For example, the electrostatic force fields (bonding) between atoms and molecules provide these particles with the potential for work. The internal energy, U, is the sum of all internal kinetic and internal potential energy contributions [2].

The law of conservation of energy, which is also called the first law of thermodynamics, may now be written as:

(3.6)

Graphic

or equivalently as

(3.7)

Graphic

It is important to note the sign convention for Q and W adapted for the above equation. Since any term is always defined as the final minus the initial state, both the heat and work terms must be positive when they cause the system to gain energy, i.e., when they represent energy flowing from the surroundings to the system. Conversely, when the heat and work terms cause the system to lose energy, i.e., when energy flows from the system to the surroundings, they are negative in sign. This sign convention is not universal and the reader must take care to check what sign convention is being used by a particular author when referring to the literature. For example, work is often defined in some texts as positive when the system does work on the surroundings [2, 3].

The application of the conservation laws to both environmental equipment and process design and analysis is presented in Chapter 8.

3.4 Stoichiometry

When chemicals react, they do so according to a strict proportion. When oxygen and hydrogen combine to form water, the ratio of the amount of oxygen to the amount of hydrogen consumed is always 7.94 by mass and 0.500 by moles. The term stoichiometry refers to this phenomenon, which is sometimes called the chemical law of combining weights. The reaction equation for the combining of hydrogen and oxygen is:

(3.8)

Graphic

In chemical reactions, atoms are neither generated nor consumed, merely rearranged with different bonding partners. The manipulation of the coefficients of a reaction equation so that the number of atoms of each element on the left of the equation is equal to that on the right is referred to as balancing the equation. Once an equation is balanced, the whole number molar ratio that must exist between any two components of the reaction can be determined simply by observation; these are known as stoichiometric ratios. There are three such ratios (not counting the reciprocals) in the above reaction. These are:

2 mol H2 consumed/mol O2 consumed

1 mol H2O generated/mol H2 consumed

2 mol H2O generated/mol O2 consumed

The unit mole represents either the gmol or the lbmol. Using molecular weights, these stoichiometric ratios (which are molar ratios) may easily be converted to mass ratios. For example, the first ratio above may be converted to a mass ratio by using the molecular weights of H2 (2.016) and O2 (31.999) as follows:

(2 gmol H2 consumed) (2.016 g/gmol) = 4.032 g H2 consumed

(1 gmol O2 consumed) (31.999 g/gmol) = 31.999 g O2 consumed

The mass ratio between the hydrogen and oxygen consumed is therefore:

4.032/31.999 = 0.126 g H2 consumed/g O2 consumed

These molar and mass ratios are used in material balances to determine the amounts or flow rates of components involved in chemical reactions.

Multiplying a balanced reaction equation through by a constant does nothing to alter its meaning. The reaction used as an example above is often written:

(3.9)

Graphic

In effect, the stoichiometric coefficients of Equation (3.8) have been multiplied by 0.5. There are times, however, when care must be exercised because the solution to a problem may depend on the manner or form the reaction is written. This is the case with chemical equilibrium problems and problems involving thermochemical reaction equations. These are addressed in Chapters 5 and 6.

There are two different types of material balances that may be written when a chemical reaction is involved: the molecular balance and the atomic balance. It is a matter of convenience which of the two types is used. Each is briefly discussed below.

The molecular balance is the same as that described earlier. Assuming a steady-state continuous reaction, the accumulation term, A, is zero for all components involved in the reaction, Equation 3.3 then becomes:

(3.10)

Graphic

If a total material balance is performed, the above form of the balance equation must be used if the amounts or flow rates are expressed in terms of moles, e.g., lbmol or gmol/h, since the total number of moles can change during a chemical reaction. If, however, the amounts or flow rates are given in terms of mass, e.g., kg or lb/h, the G and C terms may be dropped since mass cannot be gained or lost in a chemical reaction. Thus,

(3.11)

Graphic

In general, however, when a chemical reaction is involved, it is usually more convenient to express amounts and flow rates using moles rather than mass.

A material balance that is not based on the chemicals (or molecules), but rather on the atoms that make up the molecules, is referred to as an atomic balance. Since atoms are neither created nor destroyed in a chemical reaction, the G and C terms equal zero and the balance once again becomes:

(3.11)

Graphic

As an example, consider once again the combination of hydrogen and oxygen to form water:

(3.8)

Graphic

As the reaction progresses, O, and H, molecules (or moles) are consumed while H2O molecules (or moles) are generated. On the other hand, the number of oxygen atoms (or moles of oxygen atoms) and the number of hydrogen atoms (or moles of hydrogen atoms) do not change. Care must also be taken to distinguish between molecular oxygen and atomic oxygen. If, in the above reaction, one starts out with 1000 lbmol of O2 (oxygen molecules), one may replace this with 2000 lbmol of O (oxygen atoms).

Thus, a chemical equation provides a variety of qualitative and quantitative information essential for the calculation of the quantity of reactants reacted and products formed in a chemical process. As noted, a balanced chemical equation must have the same number of atoms of each type in the reactants on the left hand side of the equation and in the products on the right hand side of the equation. Thus a balanced equation for butane combustion (reaction with oxygen to form oxidized end products CO2 and H2O) is:

(3.12)

Graphic

Note that:

Number of carbons in reactants = number of carbons in products = 4

Number of oxygens in reactants = number of oxygens in products = 13

Number of hydrogens in reactants = number of hydrogens in products = 10

Number of moles of reactants is 1 mol C4H10 + 6.5 mol O2 = 7.5 mol total

Number of moles of products is 4 mol CO2 + 5 mol H2O = 9 mol total

The reader should note that although the number of moles on both sides of the equation do not balance, the masses of reactants and products (in line with the conservation law for mass) must balance.

3.5 Limiting and Excess Reactants

Limiting and excess reactants involve an extension of the stoichiometric calculations provided above. Consider the following example.

When methane is combusted completely, the stoichiometric equation for the reaction is:

(3.13)

Graphic

The stoichiometric ratio of the oxygen to the methane is:

0.5 mol methane consumed/mol oxygen consumed

If one starts out with 1 mol of methane and 3 mol of oxygen in a reaction vessel, only 2 mol of oxygen would be used up, leaving an excess of 1 mol of oxygen in the vessel. In this case, the oxygen is called the excess reactant and methane is the limiting reactant. The limiting reactant is defined as the reactant that would be completely consumed if the reaction went to completion. All other reactants are excess reactants. The amount by which a reactant is present in excess of stoichiometric requirements (i.e., the exact number of moles needed to react completely with the limiting reactant) is defined as the percent excess, and is given by Equation 3.14:

(3.14)

Graphic

where n = number of moles of the excess reactant at the start of the reaction; and ns = the stoichiometric number of moles of the excess reactant.

In the example above, the stoichiometric amount of oxygen is 2 mol, since that is the amount that would react with the 1 mol of methane. The excess amount of oxygen is 1 mol, which is a percentage excess of 50% or a fractional excess of 0.50. These definitions are employed in Chapter 6, which is concerned with chemical kinetics.

A detailed and expanded treatment of stoichiometry is available in references [3] and [4].

References

1. Reynolds, J., Material and Energy Balances, A Theodore Tutorial, East Williston, NY, originally published by the USEPA/APTI, RTP, NC, 1992.

2. Theodore, L., and Reynolds, J., Thermodynamics, A Theodore Tutorial, East Williston, NY, originally published by the USEPA/APTI, RTP, NC, 1991.

3. Green, D., and Perry, R., (Ed.), Perry’s Chemical Engineers’ Handbook, 8th Edition, McGraw-Hill, New York City, NY, 2008.

4. Theodore, L., Chemical Engineering: The Essential Reference, McGraw-Hill, New York City, NY, 2014.

Chapter 4

The Ideal Gas Law

There are numerous environmental science and engineering applications that involve air pollution. These applications require an understanding of the role the Ideal Gas Law plays in fundamental calculations and relationships related to gas volumes, densities, and concentrations.

4.1 Overview

Observations based on physical experimentation often can be synthesized into simple mathematical equations called laws. These laws are never perfect and hence are only an approximate representation of reality. The Ideal Gas Law (IGL) is one such law derived from experiments in which the effects of pressure and temperature on gaseous volumes were measured over moderate temperature and pressure ranges. This law works well in the pressure and temperature ranges that were used in collecting the data; extrapolations outside of the ranges have been found to work well in some cases and poorly in others. As a general rule, this law works best when the molecules of the gas are far apart, i.e., when the pressure is low and the temperature is high. Under these conditions, the gas is said to behave ideally, i.e., its behavior is a close approximation to the so-called perfect or ideal gas: a hypothetical entity that obeys the IGL exactly. For environmental calculations, the ideal gas law is often assumed to be valid since it generally works well (usually within a few percent of the correct result) up to the highest pressures and down to the lowest temperatures used in many industrial environmental applications [1].

The two precursors of the ideal gas law were Boyle’s Law and Charles’ Law. Boyle found that the volume of a given mass of gas is inversely proportional to its absolute pressure if the temperature is kept constant:

(4.1)

Graphic

where V1 = volume of gas at absolute pressure P1 and temperature T; and V2 = volume of gas at absolute pressure P2 and absolute temperature T.

Charles found that the volume of a given mass of gas varies directly with the absolute temperature at constant pressure:

(4.2)

Graphic

where V1 = volume of gas at pressure P and absolute temperature T1; and V2 = volume of gas at pressure P and temperature T2.

Boyle’s and Charles’ laws may be combined into a single equation in which neither temperature nor pressure need be held constant:

(4.3)

Graphic

For Equation 4.3 to hold, the mass of gas must be constant as the conditions change from (P1, T1) to (P2, T2). This equation indicates that for a given mass of a specific gas, PV/T has a constant value. Since, at the same temperature and pressure, volume and mass must be directly proportional, this statement may be extended to:

(4.4)

Graphic

where m = mass of a specific gas and C = a constant that depends on the gas.

Moreover, experiments with different gases showed that Equation 4.4 can be expressed in a far more generalized form. If the number of moles (n) is used in place of the mass (m), the constant is the same for all gases:

(4.5)

Graphic

where R = the universal gas constant. Equation 4.5 is referred to as the Ideal Gas Law. Numerically, the value of R depends on the units used for P, V, T and n (see Table 4.1).

Table 4.1 Values of R in various units.

4.2 Other Forms of the Ideal Gas Law

Other useful forms of the IGL are shown in Equations 4.6 and 4.7. Equation 4.6 applies to gas flow rather than to gas confined in a container.

(4.6)

Graphic

where P = absolute pressure (psia); q = gas volumetric flow rate (ft³/h); Graphic = molar flow rate (lbmol/h); R = 10.73 psia-ft³/lbmol-°R; and T = absolute temperature (°R).

Equation 4.7 combines n and V from Equation 4.6 to express the law in terms of density:

(4.7)

Graphic

where MW= molecular weight of the gas (lb/lbmol) and ρ = density of the gas (lb/ft³).

Volumetric flow rates are often not given at the actual conditions of pressure and temperature but at arbitrarily chosen standard conditions (STP, standard temperature and pressure). To distinguish between flow rates based on the two conditions, the letters a and s are often used as part of the unit. The units acfm and scfm represent actual cubic feet per minute and standard cubic feet per minute, respectively. The ideal gas law can be used to convert from standard to actual conditions, but, since there are many standard conditions in use, the STP employed must be known. Standard conditions most often used in environmental applications are shown in Table 4.2. The reader is cautioned on the incorrect use of acfm and/or scfm. Employing standard conditions is a convenience; when predicting the performance of or designing equipment, the actual conditions must be employed. Designs based on standard conditions can lead to disastrous results, with the unit usually under designed. For example, for a gas stream at 2140 °F, the ratio of acfm to scfm (standard temperature = 60 °F) is 5.0. Equation 4.8, which is a form of Charles’ law, can be used to correct flow rates from standard to actual conditions:

Table 4.2 Common standard conditions.

(4.8)

Graphic

where qa = volumetric flow rate at actual conditions (ft³/h); qs = volumetric flow rate at standard conditions (ft³/h); Ts = standard absolute temperature (°R); and Ta = actual absolute temperature (°R). The reader is again reminded that absolute temperatures and absolute pressures must be employed in all IGL calculations.

In engineering practice, mixtures of gases are more often encountered than single or pure gases. The IGL is based on the number of molecules present in the gas volume; the type of molecules is not a significant factor, only the number. The IGL applies equally well to mixtures and pure gases alike. Dalton and Amagat both applied the IGL to mixtures of gases. Since pressure is caused by gas molecules colliding with the walls of a container, it seems reasonable that the total pressure of a gas mixture is made up of pressure contributions due to each of the component gases. These pressure contributions are called partial pressures. Dalton defined the partial pressure of a component as the pressure that would be exerted if the same mass of the component gas occupied the same total volume alone at the same temperature as the mixture. The sum of these partial pressures equals the total pressure:

(4.9)

Graphic

where P = total pressure; pi = partial pressure of component i; and n = number of components in the mixture.

Equation 4.9 is known as Dalton’s law of partial pressures. Applying the ideal gas law to one component (A) only, yields:

(4.10)

Graphic

where nA = number of moles of component A.

Eliminating R, T, and V between Equations 4.5 and 4.10 yields:

(4.11)

Graphic

where yA = mole fraction of component A.

Amagat’s law is similar to Dalton’s law. Instead of considering the total pressure to consist of partial pressures where each component occupies the total container volume, Amagat considered the total volume to consist of the partial volumes in which each component is at (or is exerting) the total pressure. The definition of the partial volume is therefore the volume occupied by a component gas alone at the same temperature and pressure as the mixture. For this case:

(4.12)

Graphic

Applying Equation 4.10 as before, one obtains:

(4.13)

Graphic

where VA = partial volume of component A.

It is common in environmental engineering practice to describe low concentrations of components in gaseous mixtures in parts per million by volume, ppmv. Since partial volumes are proportional to mole fractions, it is necessary only to multiply the mole fraction of the component by 1 million to obtain the concentration in parts per million. For liquids and solids, parts per million (ppm) is also used to express concentration, although it is usually on a mass basis rather than a volume basis. The terms ppmv and ppmw are sometimes used to distinguish between the concentration of a component on a volume or mass basis, respectively.

4.3 Non-Ideal Gas Behavior

Some environmental applications require that deviations from ideality be included in the analysis. Many of the non-ideal correlations involve the critical temperature Tc, the critical pressure Pc, and a term defined as the acentric factor, w. An abbreviated list of these properties is available in the literature [2, 3]. These reduced quantities find wide application in thermodynamic analyses of non-ideal systems (see also Chapter 5).

The critical temperature and pressure are employed in the calculation of the reduced temperature, Tr, and the reduced pressure, Pr, as provided in Equations 4.14 and 4.15:

(4.14)

Graphic

(4.15)

Graphic

Both reduced properties are dimensionless and play important roles in non-ideal gas behavior.

Many physical and chemical properties of elements and compounds can be estimated from models (equations) that are based on the reduced temperature and pressure of the substance in question. These reduced properties have also served as the basis for many equations that are employed in practice to describe non-ideal gas (and liquid) behavior. Although a rigorous treatment of this material is beyond the scope of this book, information is available in the literature [2,3]. Highlights of this topic are presented below.

No real gas conforms exactly to the IGL, but it can be used as an excellent approximation for most gases at pressures about or less than 5 atm and near ambient temperatures. One approach to account for the previously mentioned deviations from ideality is to include a correction factor, Z, which is defined as the compressibility coefficient or compressibility factor. The ideal gas law is then modified to the following form:

(4.16)

Graphic

Note that Z approaches 1.0 as P approaches 0.0. For an ideal gas, Z is exactly unity. This equation may also be written as:

(4.17)

Graphic

where v is now the specific molar volume (not the total volume) with units of volume/mole. Regarding gas mixtures, the ideal gas law can be applied directly for ideal gas mixtures. However, the molecular weight of the mixture is based on a mole fraction average, Graphic , of the n components:

(4.18)

Graphic

One approach to account for deviations from ideality is to assume the aforementioned compressibility coefficient for the mixture, Graphic , is a linear mole fraction combination of the individual component Z values:

(4.19)

Graphic

Furthermore, Kay [4] has shown that the deviations arising in using this approach can be reduced by employing pseudocritical values for T and P where:

(4.20)

Graphic

(4.21)

Graphic

In lieu of other information, the authors suggest employing Equation 4.22 for the pseudocritical value of v:

(4.22)

Graphic

These pseudocritical values – Tc, Pc, and Graphic – are then employed in the appropriate pure component equation of state. This approach has been defined by some as Kay’s rule, an approach that has unfortunately been abandoned in recent years.

Another equation of state available to account for observations from ideality is that proposed by van der Waal. This equation attempts to correct for intermolecular forces of attraction (a/V²) and the volume occupied by the molecules themselves (b) in the following manner [1]:

(4.23)

Graphic

where a = (27R²Tc²)/(64 Pc), and b = (RTc)/(8Pc).

References

1. Theodore, L., Ricci, F., and VanVliet, T., Thermodynamics for the Practicing Engineer, John Wiley and Sons, Hoboken, N.J., 2009.

2. Smith, J., Van Ness, H., and Abbott, M., Introduction to Chemical Engineering Thermodynamics, 6th Edition, McGraw-Hill, New York City, NY, 2005.

3. Green, D., and Perry, R., (Ed.), Perry’s Chemical Engineers’ Handbook, 8th Edition, McGraw-Hill, New York City, NY, 2008.

4. Kay, W., Density of hydrocarbon gases and vapors. Ind. Eng. Chem., 28, 1014, 1936.

Chapter 5

Thermodynamics

Prior to undertaking the writing of this text, one of the authors co-authored a text entitled Thermodynamics for the Practicing Engineer [1]. It soon became apparent that some overlap existed between thermodynamics (the subject of this chapter) and heat transfer (the subject of –Part III of this text). Even though the former topic is broadly viewed as engineering science, heat transfer is one of the unit operations and can justifiably be classified as an engineering subject. But what are the similarities and what are the differences?

The similarities that exist between thermodynamics and heat transfer are grounded in the three conservation laws: mass, energy, and momentum. Both are primarily concerned with energy-related subject matter and both, in a very real sense, supplement each other. However, thermodynamics deals with the transfer of energy and the conversion of one form of energy into another (e.g., heat into work), with consideration generally limited to systems in equilibrium. The topic of heat transfer deals with the transfer of energy in the form of heat; the applications almost exclusively occur within heat exchangers that are employed in chemical, petrochemical, petroleum (refinery) and environmental engineering processes and applications.

5.1 Overview

Thermodynamics was once defined as the science that deals with the inter-transformation of heat and work. The fundamental principles of thermodynamics are contained in the first, second, and third laws of thermodynamics. These principles have been defined as pure or theoretical thermodynamics. These laws were developed and extensively tested in the latter half of the 19th Century and are essentially based on experience. (The third law was developed later in the 20th Century).

Practically all thermodynamics, in the ordinary meaning of the term, is applied thermodynamics in that it is essentially the application of these three laws, coupled with certain facts and principles of mathematics, physics, and chemistry, to problems in engineering and science. The fundamental laws are of such generality that it is not surprising that these laws find application in other disciplines, including physics, chemistry, plus environmental, chemical and mechanical engineering.

The first law of thermodynamics is a conservation law for energy transformations. Regardless of the types of energy involved in processes – thermal, mechanical, electrical, elastic, magnetic, etc. – the change in energy of a system is equal to the difference between energy input and energy output. The first law also allows free convertibility from one form of energy to another, as long as the overall energy quantity is conserved. Thus, this law places no restriction on the conversion of work into heat, or on its counterpart – the conversion of heat into work.

Because work is 100% convertible to heat, whereas the reverse situation is not true, work is a more valuable form of energy than heat. This leads to an important second-law consideration – i.e., that energy has quality as well as quantity. Although it is not as obvious, it can also be shown through the second-law principles and arguments that heat has quality in terms of its temperature. The higher the temperature at which heat transfer occurs, the greater the potential for energy transformation into work. Thus, thermal energy stored at higher temperatures is generally more useful to society than that available at lower temperatures. While there is an immense quantity of energy stored in the oceans and the earth’s core, for example, its present availability to society for performing useful tasks is essentially nonexistent. Theodore et al. [1] provide additional qualitative reviews of the second law.

The choice of topics to be reviewed in this chapter was initially an area of debate, and after some deliberation, it was decided to provide an introduction to five areas that many have included in this broad engineering subject. These are detailed below:

The First Law of Thermodynamics

Enthalpy Effects

Second Law Calculations

Phase Equilibrium

Chemical Reaction Equilibrium

The reader should note that the bulk of the material in this chapter has been drawn from L. Theodore, Thermodynamics, A Theodore Tutorial, originally published by the USEPA/APTI, RTP, NC in 1991 [2].

5.2 The First Law of Thermodynamics

For many environmental processes, the energy requirement represents a major item of the cost of operation and one cannot arrive at a proper systems analysis and/or economic evaluation without performing an energy balance as well as a material balance. Just as practicing engineers rely on the law of conservation of mass for a material balance, they depend on the law of the conservation of energy for energy balance calculations.

These energy balance considerations are based on thermodynamics, the branch of science founded on laws of experience which deal with both energy and its conversion, as well as the transfer of energy in terms of heat and work as a system passes from one equilibrium state to another. Joule’s experiments cleared the way for the enunciation of the first law of thermodynamics; namely, when a closed system is taken through a cyclic process, the work done on the surroundings equals the heat absorbed from the surroundings.

This law for batch processes, can be represented as:

(5.1)

Graphic

where potential, kinetic, and other energy effects have been neglected and E (often denoted as U), is the internal energy of the system, ΔE is the change in the internal energy of the system, Q is energy in the form of heat transferred across the system boundaries, and W is energy in the form of work transferred across system boundaries. In accordance with a recent change in convention, both Q and W are treated as positive terms if added to the system.

For practical purposes, the total work term, W, in the first law may be regarded as the sum of shaft work, Ws, and flow work, Wf.

(5.2)

Graphic

where Ws is work done on the fluid by some moving solid part within the system such as the vanes of a centrifugal pump. Note that in Equation 5.2, all other forms of work such as electrical, surface tension, and so on are neglected. The first law of thermodynamics for steady-state flow processes is then:

(5.3)

Graphic

where H is the enthalpy of the system and ΔH is the change in the system’s enthalpy.

The internal energy and enthalpy in Equations 5.1 and 5.2, as well as other equations in this section may be on a mass basis, on a mole basis, or represent the total internal energy and enthalpy of the entire system. They may also be written on a time-rate basis as long as these equations are dimensionally consistent. For the sake of clarity, upper case letters (e.g., H, E) represent properties on a mole basis, while lower-case letters (e.g., h, e) represent properties on a mass basis. Properties for the entire system will rarely be used and therefore require no special symbols.

Perhaps the most important thermodynamic function the engineer works with is the above mentioned enthalpy. This is a term that requires additional discussion. The enthalpy is defined by the equation

(5.4)

Graphic

where P is once again the pressure of the system and V is the volume of the system. The terms E and H are state or point functions. By fixing a certain number of variables upon which the function depends, the numerical value of the function is automatically fixed; that is, it is single-valued. For example, fixing the temperature and pressure of a one-component single-phase system immediately specifies its enthalpy and internal energy.

5.3 Enthalpy Effects

There are many different types of enthalpy effects. These include:

Sensible (temperature)

Latent (phase)

Dilution (with water), e.g., HCl with H2O

Solution (nonaqueous), e.g., HCl with a solvent other than H2O

Reaction (chemical)

This section is only concerned with Effects 1, 2 and 5. Details on Effects 3 through 4 are available in the literature [1–3].

5.3.1 Sensible enthalpy effects

Sensible enthalpy effects are associated with temperature. There are methods that can be employed to calculate these changes. These methods include the use of:

enthalpy values

average heat capacity values

heat capacity as a function of temperature

If enthalpy values are available, the enthalpy change is given by

(5.5)

Graphic

(5.6)

Graphic

If average molar heat capacity data are available,

(5.7)

Graphic

where Graphic = average molar value of Cp in the temperature range ΔT. Average molar heat capacity data are provided in the literature [1–5].

A more rigorous approach to enthalpy calculations can be provided if the heat capacity variation with temperature is available. If the heat capacity is a function of the temperature, the enthalpy change is written in differential form:

(5.8)

Graphic

If the temperature variation of the heat capacity is given by

(5.9)

Graphic

Equation 5.8 may be integrated directly between some reference or standard temperature (To) and the final temperature (T1) employing Equation 5.9:

(5.10)

Graphic

(5.11)

Graphic

Equation 5.8 may also be integrated if the heat capacity is a function of temperature of the form:

(5.12)

Graphic

The enthalpy change is then given by

(5.13)

Graphic

Tabulated values of α, β, γ, and a, b, c for a host of compounds (including some chlorinated organics) are available in the literature [2].

5.3.2 Latent Enthalpy Changes

It has been observed that there is absorption of heat at constant temperature and pressure that accompanies the transition or equilibrium phase change from solid to liquid and liquid to gas. In terms of molecular theory, this latent enthalpy represents the energy required to overcome inter-molecular forces of attraction and to permit molecules to pass from a more highly restrained condensed phase to a more mobile phase. This relationship indicates that the latent heat is the sum of a change in internal energy and energy of expansion (or contraction).

Common types of transition and the terminology applied to them are:

The last case represents a change from one crystalline modification to another. The enthalpy change involved is small compared with those accompanying the other three types, and rarely finds application in environmental engineering.

Since there is a relatively small difference in volume between a solid and the liquid to which it melts, the heat of fusion represents mainly an increase in internal energy. If the molar heat of fusion is not known, it may be approximated from the absolute temperature of fusion, Tf, using the following equation:

(5.14)

Graphic

The ratio may be expressed in any consistent molar units, such as cal/(gmol-K) or Btu/(lbmol-°R).

Since there is a very substantial increase in volume in passing from the liquid to the vapor state, usually from a few hundred to over a thousand-fold, the heat of vaporization is large and important in many environmental engineering applications involving water. Information on properties of water are available in steam tables in the literature [1, 2] and in Appendix C.

5.3.1 Chemical Reaction Enthalpy Effects

The standard enthalpy (heat) of reduction can be calculated from standard enthalpy of formation data. To simplify the presentation that follows, examine the authors’ favorite equation:

(5.15)

Graphic

If the above reaction is assumed to occur at a standard (or reference) state, the standard enthalpy of reaction, ΔHo, is given by:

(5.16)

Graphic

where (ΔHf°)i.= standard enthalpy of formation of species i.

Thus, the (standard) enthalpy of a reaction is obtained by taking the difference between the (standard) enthalpy of formation of products and reactants multiplied by their respective stoichiometric coefficients. If the (standard) enthalpy of reaction or formation is negative (exothermic), as is the case with most combustion reactions, then energy is liberated due to the chemical reaction. Energy is absorbed and ΔHo is positive (endothermic).

Tables of enthalpies of formation and reaction are available in the literature (particuiariy thermodynamics text/reference books) for a wide variety of compounds [1]. It is important to note that these are valueless unless the stoichiometric equation and the state of the reactants and products are included.

Theodore, et al. [1, 2] provide equations to describe the effect of temperature on the enthalpy of reaction. For heat capacity data in α, β, γ form:

(5.17)

Graphic

For the reaction presented in Equation 5.15:

(5.18)

Graphic

(5.19)

Graphic

(5.20)

Graphic

For heat capacity in a, b, c form,

(5.21)

Graphic

5.4 Second Law Calculations [2]

The law of conservation of energy has already been defined as the first law of thermodynamics. Its application allows calculations of energy relationships associated with a wide variety of processes. The limiting law is called the second law of thermodynamics. Applications involve calculation of the maximum power output from a power plant and equilibrium yields of chemical reactions. In principle, this law starts that water cannot flow uphill and heat cannot flow from a cold to a hot body of its own accord. Other defining statements for this law that have appeared in the literature are provided below:

Any process, the sole net results of which is the transfer of heat from a lower temperature level to a higher one, is impossible.

No apparatus, equipment, or process can operate in such a way that its only effect (on system and surroundings) is to convert heat absorbed completely into work.

It is impossible to convert the heat taken into a system completely into work in a cyclical process.

The second law also serves to define another important thermodynamic function called entropy. It is normally designated as S. The change in S for a reversible adiabatic process is always zero:

(5.22)

Graphic

For liquids and solids, the entropy change for a system undergoing an absolute temperature change from T1 to T2 is given by:

(5.23)

Graphic

The entropy change of an ideal gas undergoing a physical change of state from P1 to P2 at a constant temperature T is given by:

(5.24)

Graphic

The entropy change of one mole of an ideal gas undergoing a physical change of state from absolute temperatures T1 to T2 at a constant pressure is given by:

(5.25)

Graphic

Correspondingly, the entropy change for an ideal gas undergoing a physical change from (P1, T1) to (P2, T2) is

(5.26)

Graphic

Some