Potter & Somerton - Chapter 6: Entropy

Following on the Second Law developed in Chapter 5 we consider the Clausius Inequality leading to the definition of a new property Entropy (s) and the Increase in Entropy Principle. Various methods are developed to evaluate the change of entropy (Ds), and this enables us to evaluate the "isentropic" efficiency of various components, in particular turbines, compressors, and nozzles. The entire chapter is required reading, and we go over all the important derivations in class. Section 6.4 (Ideal Gas with Variable Specific Heats) is important for evaluating systems with large temperature variations such as the Diesel cycle engine. We present an alternative approach to that of Potter by using a table of Specific Heat Capacities of Air. We also include an Entropy Summary Sheet, an Isentropic Processes Summary Sheet, and an Adiabatic Efficiency Summary Sheet of all the relevant equations pertaining to this section, for convenience.

Section 6.5 mentions the two property diagrams involving entropy, the temperature-entropy (T-s) and enthalpy-entropy (h-s) "Mollier" diagrams. We will find that the h-s diagram is extremely useful for evaluating adiabatic turbines and compressors, and complements the P-h diagram which we used in Chapter 4 to evaluate entire steam power plants or refrigerator systems. The h-s diagram for steam is presented below:

Recommended Supplementary Problems for this section - as many as possible of the following SI problems 6.25 through 6.69 - at least one from each group. (All answers to the Supplementary Problems are given at the end of the chapter)

• Isentropic process -
  • Ideal gas: 6.25, 6.26, 6.37, 6.38.
  • Steam: 6.42.
• Rigid container -
  • Ideal gas: 6.27, 6.34, 6.35, 6.54.
  • Insulated, ruptured membrane (see Example 6.7): 6.33 (Note - in problem 6.33 the initial temperature (not specified) is 20°C)
  • Steam: 6.56
• Piston-cylinder -
  • Ideal gas: 6.29, 6.30, 6.31. (Note - the only way that problem 6.30 can relate to the answer given is if the cylinder height is 20cm (not 20mm) and the initial temperature 47°C (and not 27°C). Even then the answer for work is incorrect. The correct answers for 6.30 (air) are: work = 2.58 kJ, entropy change = 15.4 J/K)
  • Steam: 6.39, 6.55
• Nozzle - 6.59
• Turbine -
  • Gas: 6.60.
  • Steam: 6.62, 6.63, 6.64 (Note - I was wrong! The turbine in 6.64 is NOT adiabatic, thus the correct answer is as presented by Potter - h_T = 39.9%. The main reason for my confusion is that one does not normally associate an adiabatic efficiency with a non-adiabatic turbine, and I will certainly not do so in any quiz. On evaluating the heat flow we find that a significant amount of heat is added to the system (1.44 MW) - this is totally unrealistic, since one normally expects a heat loss.
• Otto cycle 6.66, Diesel cycle 6.69 (Correct answer for problem 6.69 is 55%)
• Rankine cycle 6.67 - this is an important problem, however it is confusing as worded, and the associated T-s diagram in Potter (Fig 6-20) is incorrect (can you see why?) We reword the problem as follows:
  Assuming that the turbine process is isentropic, calculate the efficiency of the Rankine cycle shown, if P₄ = 20kPa, P₁ = P₂ = 4MPa, and T₂ = 600°C. Draw the complete cycle on a P-h diagram clearly indicating all 4 processes. The Rankine cycle (as described) is shown schematically as follows:
  
  We augment problem 6.67 as follows: Consider again the above steamplant schematic, in which the (ideal) isentropic turbine is replaced by an actual adiabatic turbine such that the outlet at Station (3) is now 20 kPa, 150°C.
  • a) Augment the P-h diagram plot above with the actual turbine process.
  • b) Using the steam tables calculate the thermal efficiency of this cycle using the actual turbine specific work to replace the isentropic turbine specific work. [h_th = 26%]
  • c) Plot both the actual and the isentropic turbine processes (2)-(3) on the enthalpy-entropy h-s "Mollier" diagram above. Indicate the specific work done by both the actual and isentropic turbines on the h-s diagram.
  • d) Using the steam tables determine the actual turbine adiabatic efficiency (h_T) [72%]
• Additional Problem 6.1 - Rankine Cycle Steam Power Plant for Athens, Ohio

Once again we were surprised that no problems were presented involving refrigeration and heat pump systems using R134a refrigerant, or complete gas turbine systems such as those used in aircraft jet engines. Thus we augment the above problem set with additional problems to fill in this gap, in particular making use of h-s diagrams. We have also provided an R134a enthalpy-entopy (h-s) diagram which we find useful for evaluating adiabatic compressors that are normally found in refrigeration, air-condition and heat pump systems. We will also extend the h-s diagram into the ideal gas region and use it to advantage when we consider jet engine systems.

• Additional Problem 6.2 - Recall in Chapter 4b that we provided Additional Problems 4.1 & 4.2 concerning a home refrigerator, and examining it's performance before and after adding an internal heat exchanger. We wish to augment the problem statement with the following additional requirements:
  • a) Consider the case of replacing the actual adiabatic compressor with an isentropic compressor, and augment the P-h diagram drawn to include this change.
  • b) Using the R134a tables determine the specific work done on the isentropic compressor, and determine the new Coefficient of Performance (COP_R) of the system both with and without the internal heat exchanger.
  • c) Plot the actual and the isentropic compressor processes on the enthalpy-entropy (h-s) diagram provided, for both cases - with and without the internal heat exchanger.
  • d) Using the R134a tables determine the actual compressor adiabatic efficiency for both cases (h_C).
• Additional Problem 6.3 - Recall in Chapter 4b that we provided Additional Problem 4.4 concerning an innovative home air conditioner and hot water heating system, in which we determined the COP for both the air conditioning and the water heating systems. Do sections a) through d) specified in Additional Problem 6.2 above on the compressor of this home air conditioning system.

We take a somewhat simpler approach for liquids in that we consider them to be incompressible. This leads us to

• Additional Problem 6.4 - evaluating the adiabatic efficiency (h_P) of a power plant feedwater pump.

Aircraft Gas Turbine Engines

There are many different forms and modifications of aircraft gas turbine engines, and in this course we discuss two variants - the ideal turbojet engine, and the gas turbine engine for usage in helicopters.

The ideal turbojet engine shown schematically in the above figure comprises the series connection of five components - diffuser, compressor, combustor, turbine, and nozzle. The analysis of the complete system, is best done in terms of the h-s (enthalpy-entropy) diagram, which we will develop in class. Throughout the system we assume that the fluid is pure air, and the combustors are considered to be constant-pressure heat-addition devices. Notice that the sole purpose of the turbine is to drive the compressor, the nozzle providing the final kinetic energy increase to drive the aircraft.

The gas turbine engine for usage in helicopters is shown below:

In this case we see that there is no diffuser or nozzle, and that the turbine section has been replaced by two independent turbines - a "gassifier" turbine to drive the compressor, and an output turbine to drive the helicopter blades. A typical gas turbine engine of this type is the General Electric T700 engine shown below, which is used in the Army Black Hawk helicopter.

Dr. Tom Scott of the Industrial Technology department, who previously worked in the Allison Gas Turbines company on gas turbine engines (now Rolls Royce Allison), has a fullsize cutaway model of a T700 engine which he will demonstrate and discuss with us in one of our classes.

This leads us to:

• Additional Problem 6.5, in which we wish to do a thermodynamic analysis of the General Electric T700 Gas Turbine Engine

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