When solving thermodynamic problems, usually the best approach is to begin by applying the mass and energy balances. The entropy balance provides another balance, but it is not always necessary for every problem. In this chapter, we have introduced some new terms which can specify additional constraints when used in the problem statement, e.g., “isentropic,” “reversible,” “internally reversible,” “irreversible,” “thermal efficiency,” and “turbine/expander or compressor/pump efficiency.”
The entropy balance is useful to calculate maximum work available from a process or to evaluate reversibility. The entropy balance should be introduced with care because it is often redundant with the energy balance when simplified with information from step 5 from the strategies of Section 2.13. (For example, the entropy balance applied to Example 2.11 on page 75 results in the same simplified equation as the energy balance.) In general, if the pressures and temperatures of the process are already known, the entropies at each point, and the entropy changes, can be determined without direct use of the entropy balance. However, if either the pressure or the temperature is unknown for a process, the entropy balance may be the key to the solution.
In some cases, T and P are known, so S can be determined without the entropy balance in a pure system.
Before beginning more examples, it is also helpful to keep in mind those processes which generate entropy. This is important because, in the event that such processes arise, the entropy-generation term cannot be set to zero unless we modify the process to eliminate the source of the generation. Entropy is generated by the following processes.
1. Heat conduction along a temperature gradient.
2. Diffusion along a concentration gradient.
3. Mixing of substances of different composition.
4. Adiabatic mixing at constant system volume of identical substances initially at different molar entropies due to (T, P) differences.
5. Velocity gradients within equipment. This is accounted for in pipe flow by the friction factor developed in textbooks on fluid flow.
6. Friction.
7. Electrical resistance.
8. Chemical reactions proceeding at measurable rates.
In an open system, irreversibilities are always introduced when streams of different temperatures are mixed at constant pressure (item 4 above) because we could have obtained work by operating a heat engine between the two streams to make them isothermal before mixing. If the streams are isothermal, but of different composition, mixing will still generate entropy (e.g., see Eqn. 4.8 on page 138), and we have not yet devised a general method to obtain work from motion on this molecular scale.
Problem statements will rarely explicitly point out entropy generation, so you will need to look for causes.
As chemical engineers, it is important to recognize that all chemical reactions proceeding at a finite rate generate entropy. The fundamental proof of this is provided in Section 17.16 and requires discussion of the chemical potential discussed much later in the text. Like the other processes listed here, reactions proceed spontaneously toward an equilibrium state due to finite driving forces. It is possible to calculate the rate of entropy generation if the chemical potentials are known at the reaction conditions. However, do not be deceived into thinking that a realistic reaction is thermodynamically reversible. Reaction engineering literature refers to reactions as reversible if the equilibrium constant (to be discussed later) is near 1 because the reaction can go “in either direction” (forward/backward) depending on the driving forces. This type of reversibility is not the same as thermodynamic reversibility. The reaction proceeding in either direction at measurable rates will generate entropy and be thermodynamically irreversible.
When a situation requires the minimum work input, or the maximum work output, the system is designed to minimize entropy generation, or make it zero if possible considering the limitations discussed here. As we work examples, recall the comments from Section 3.4 which we repeat here:
Note: For irreversible processes, note that we do not apply the entropy balance to find entropy changes. We always calculate the entropy change by an alternate reversible pathway that reaches the final state, then we apply the entropy balance to find how much entropy was generated, or we find the reversible work, apply an efficiency factor, and identify the final state via the energy balance.
Alternatively, for reversible processes, we do apply the entropy balance because we set the entropy generation term to zero.
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