Biorenewable Resources

CHAPTER 2

Fundamental Concepts in Engineering Thermodynamics

2.1 Introduction

Engineering thermodynamics provides the foundation in mass and energy balances essential to understanding bioenergy and biobased products. Accounting for these balances is more complicated than for energy conversion processes that do not include chemical reaction because chemical constituents change and energy is released from the rearrangement of chemical bonds.

This chapter is designed to introduce or reacquaint readers, as appropriate, to fundamental concepts in engineering thermodynamics. The treatment does not pretend to be exhaustive; readers requiring additional background are directed to the list of reference materials at the end of this chapter.

2.2 General Concepts in Mass and Molar Balances

In the absence of chemical reaction, the change in mass of a particular constituent within a control volume is equal to the difference in net mass flow of the constituent entering and exiting the control volume. Figure 2.1 illustrates mass balance for a system consisting of five inlets and five exits. In general, the mass balance for a given chemical constituent can be written in the form:

(2.1)

where mCV is the amount of mass contained within the control volume; and are, respectively, the rates at which mass enters at i and exists at e, where we allow for the possibility of several inlets and exits. For steady flow conditions, the net quantity of mass in the control volume is unchanging with time, and Equation 2.1 can be written as:

(2.2)

FIG. 2.1 Mass balance on steady-flow control volume with five inlets and five exits.

However, when chemical reaction occurs, chemical compounds are not conserved as they flow through the system. For example, methane (CH4) and oxygen (O2) entering a combustor are consumed and replaced by carbon dioxide (CO2) and water (H2O):

(2.3)

Accordingly, mass balances cannot be written for methane and oxygen using either Equation 2.1 (unsteady flow) or Equation 2.2 (steady flow). Although chemical compounds are not conserved, the chemical elements making up these compounds are conserved; thus, elemental mass balances can be written.

In the case of the reaction of CH4 with O2, mass balances can be written for the chemical elements carbon (C), hydrogen (H), and oxygen (O). However, because chemical compounds react in distinct molar proportions, it is usually more convenient to write molar balances on the elements.

Recall that a mole of any substance is the amount of mass of that substance that contains as many individual entities (whether atoms, molecules, or other particles), as there are atoms in 12 mass units of carbon-12. For engineering systems, it is usually more convenient to work with kilograms as the unit of mass; thus, for this measure kilomole (kmol) will be employed instead of the gram-mole (gmol) that often appears in chemistry books. The number of kilomoles of a substance, n, is related to the number of kilograms of a substance, m, by its molecular weight, M (kg/kmol):

(2.4)

On a molar basis, it is straightforward to account for the mass changes that occur during chemical reactions: an overall chemical reaction is written that is supported by molar balances on the elements appearing in the reactant and product chemical compounds.

Example: One kilogram of methane reacts with air. (a) If all of the methane is to be consumed, how many kilograms of air will be required? (b) How many kilograms of carbon dioxide, water, and nitrogen will appear in the products?

One kilogram of methane, with a molecular weight of 16, is calculated to be 1/16 kmol using Equation 2.4. Air is approximated as 79% nitrogen and 21% oxygen on a molar basis. The overall chemical reaction can be written as:

where a is the number of kilomoles of oxygen required to consume 1/16 kilomole of CH4 and x, y, and z are the kilomoles of CO2, H2O, and N2, respectively, in the products. The unknowns in this equation can be found from molar balances on the elements C, H, O, and N:

A check shows that 18.2 kg of methane and air are converted into 18.2 kg of products in the form of carbon dioxide, water, and nitrogen, as expected from mass conservation.

Mixtures of reactants or products are conveniently described on the basis of either mass fractions or mole fractions. If a mixture consists of N constituents, then the total mass, m, and total number of moles, n, are given by:

(2.5)

(2.6)

The mass fraction, yi, of the ith constituent of a mixture is equal to:

(2.7)

Mass fractions are sometimes presented as percentages by multiplying by 100 and assigning units of weight percent (wt%). The mole fraction, xi, of the ith constituent of a mixture is equal to:

(2.8)

Mole fractions are sometimes presented as percentages by multiplying by 100 and assigning units of mole percent (mol%).

Mole fractions are useful in calculating partial pressures, pi, of the constituents of a gas mixture:

where p is the total pressure of the mixture:

The apparent molecular weight of a mixture, M, can be calculated from the molecular weights of each of the constituents, Mi:

(2.9)

It is often useful to convert from mass fractions to mole fractions and vice versa:

(2.10)

(2.11)

Example: The combustion of 1 kg of methane requires 17.2 kg of air (4 kg of oxygen and 13.2 kg of nitrogen). As shown in the previous example, the products of combustion are 2.75 kg of carbon dioxide, 2.25 kg of water, and 13.2 kg of nitrogen. Calculate the mass fractions of products. From the mass fractions, calculate the mole fractions. Use the mole fractions to calculate the apparent molecular weight of the product mixture.

Mass of products:

Mass fractions of products:

Mole fractions of products from the mass fractions calculated above:

Apparent molecular weight from the mole fractions calculated above:

Mass and molar balances are extremely important in evaluating the progress of chemical reactions and in designing chemical reactors. A number of different measures have been devised for evaluating reactant ratios and the extent of chemical reactions.

2.2.1 Mass and Molar Balances Applied to Combustion and Gasification

For combustion and gasification processes, it is useful to compare the actual oxygen provided to the fuel to the amount theoretically required for complete oxidation (the stoichiometric requirement). The fuel–oxygen ratio, F/O, is defined as the mass of fuel per the mass of oxygen consumed (a molar fuel–oxygen ratio is also sometimes defined). Another frequently used ratio is the equivalence ratio, :

(2.12)

This ratio is less than unity for fuel-lean conditions and greater than unity for fuel-rich conditions. For combustion reactions, two other relationships are also useful, which can be calculated on either mass or molar bases:

(2.13)

(2.14)

2.2.2 Mass and Molar Balances Applied to Reaction Conversion, Yield, and Selectivity

In evaluating the changes that actually take place during chemical reaction, three quantities are particularly useful: conversion, yield, and selectivity. Conversion is the amount of reactant that is transformed into products during a reaction. The relative conversion, X (not to be confused with mole fractions xi), is the ratio of the change in the amount of reactant to the initial amount of reactant. It is readily calculated by comparing the initial mass of reactant (mr initial) to the final mass of reactant (mr final):

(2.15)

Notice that both the numerator and the denominator of Equation 2.15 are based on amounts of reactant, which means it could also be calculated from the initial moles of reactant (nr initial) and the final moles of reactant (nr final):

(2.16)

Conversion is often presented on a percentage basis by multiplying by 100. Of course, a reaction may involve more than one reactant, in which case the reactant which limits the extent of reaction is considered in calculating conversion.

Yield is the amount of a particular product formed from a reaction. The relative yield Y (not to be confused with mass fractions yi) can be calculated on either a mass or molar basis, but different operational definitions are required in these two cases, because they involve both reactants and products in their calculation. Relative mass yield is calculated as the ratio of the mass of product, mp, to the mass of reactant, mr:

(2.17)

This ratio is often presented as a percentage by multiplying by 100 and assigning dimensions of wt% to make clear that it is on a mass basis. Mass yields are straightforward to calculate from gravimetric yield data. They are also frequently preferred when calculating processing costs.

The relative molar...