The entire Chapter is required reading (except for the last section on Transient Flow) and forms the major content of our course. This first part covers Sections 4.1 through 4.7 including mainly Closed Systems using Gas Power Cycles as examples, and the second part Sections 4.8 and 4.9 will cover Control Volumes using Steam Power Plants, Refrigeration Systems and Gas Turbines as examples.
Recommended Supplementary Problems for this section - as many as possible from the SI problems 4.31 through 4.78, however at least one from each of the following groups of problems:
Problems 4.42 and 4.44 should have 1.5 MPa replaced with 1.4 MPa (Answer: T2 = 833°C). The correct answer to Problem 4.60b: T2 = 285°C.
In order to do a first Law analysis of ideal gasses it is necessary to develop equations to determine internal energy and enthalpy changes of ideal gasses. This is done by Potter & Somerton in Sections 4.6, however we prefer to develop the equations in terms of the differential form of the energy equation, as in the following web page:
Section 4.7 of Potter & Somerton covers the First Law applied to various processes, including Constant-Temperature, Constant-Volume, Constant-Pressure and Adiabatic processes. As before, rather than evaluating simple processes occurring in isolated components, such as piston/cylinder or constant volume devices, we prefer to relate these to actual complete systems. We will use the two heat engines which we introduced in Chapter 3 (Stirling and Diesel engines) as examples covering all four of these processes. In Chapter 3 we evaluated the work done for each process, and in what follows we will evaluate the heat transferred leading to an evaluation of the thermal efficiency of these engines.
The ideal Stirling cycle includes two constant temperature processes 1-2 and 3-4 and two constant volume displacement processes 2-3 and 4-1, as shown below. The two constant volume processes are formed by holding the piston in a fixed position, and shuttling the gas between the hot and cold spaces by means of the displacer. During process 4-1 the hot gas gives up its heat QR by passing through a regenerator matrix, which is then completely recovered during the process 2-3.
We will find in Chapter 5 that this is the maximum theoretical efficiency that is achievable from a heat engine, and usually referred to as the "Carnot" efficiency.
Note that if no regenerator is present the heat QR must be supplied by the heater. Thus the efficiency will be significantly reduced to hth = Wnet / (Qin + QR). Furthermore the cooler will then have to reject the heat that is normally absorbed by the regenerator, thus the cooling load will be increased to Qout + QR. Recall that Q2-3 = QR = -Q4-1.
One important aspect of Stirling cycle machines that we need to consider is that the cycle can be reversed - if we put net work into the cycle then it can be used to pump heat from a low temperature source to a high temperature sink. Sunpower, Inc. of has been actively involved in the deveplopment of Stirling cycle refrigeration systems and has begun to manufacture Stirling cycle croygenic coolers for liquifying oxygen. Global Cooling is a licencee of Sunpower, mainly in order to develop free-piston Stirling cycle coolers for home refrigerator applications. These systems, apart from being significantly more efficient than regular vapor-compression refrigerators, have the added advantage of being compact, portable units using environmentally benign helium as the working fluid. We are fortunate to have obtained two M100B state-of-the-art coolers from Global Cooling. The one is used as a demonstrator unit, and is shown in operation in the following photograph. The second unit is set up as a ME Senior Lab project in which we evaluate the performance of the machine under various specified loads and temperatures.
A schematic diagram of the cooler is shown below.
Conceptually the cooler is an extremely simple device, consisting essentially of only two moving parts - a piston and a displacer. The displacer shuttles the working gas (helium) between the compression and expansion spaces. The phasing between the piston and displacer is such that when the most of the gas is in the ambient compression space then the piston compresses the gas while rejecting heat to the ambient. The displacer then displaces the gas through the regenerator to the cold expansion space, and then both displacer and piston allow the gas to expand in this space while absorbing heat at a low temperature. The complete cycle is demonstrated in a delightful animated schematic of the cooler by Global Cooling.
In the Air-Standard Diesel cycle engine the heat input Qin occurs by combusting the fuel which is injected in a controlled manner, ideally resulting in a constant pressure expansion process 2-3 as shown below. At maximum volume (bottom dead center) the burnt gasses are simply exhausted and replaced by a fresh charge of air. This is represented by the equivalent constant volume heat rejection process Qout = -Q4-1. Both processes are analyzed as follows:
At this stage we can conveniently determine the engine efficiency in terms of the heat flow as follows:
We notice from the above analysis that we cannot complete the analysis before evaluating the pressures and temperatures at each end state in the cycle. State 1 and the various volumes are defined, however we need to do an adiabatic process analysis in order to determine the conditions at States 2, 3, and 4. This is done in Potter and Somerton in Section 4.7 (refer: "The Adiabatic Process") and is essentially redeveloped in the following web page for convenience:
The analysis results in the following three general forms representing an adiabatic process:
Thus (for example), given the conditions at state 1 and the compression ratio of the engine, in order to determine the pressure and temperature at state 2 (at the end of the adiabatic compression process) we have:
We present the following exercise to summarize this section, We will go over the solution in class, however we strongly suggest that you attempt to solve it independently:
Exercise An ideal
air-standard Diesel cycle engine has a compression ratio of 18
and a cutoff ratio of 2. At the beginning of the compression process,
the working fluid is at 100 kPa, 27°C (300 K). Determine the
temperature and pressure of the air at the end of each process,
the net work output per cycle [kJ/kg], and the thermal efficiency.
(Answers: T2 = 889 K, P2 = 5336 kPa, T3 =
1778 K, T4 = 884 K, P4 = 295 kPa, wnet
= 612 kJ/kg, Thermal efficiency hth = 58%)
Note that the nominal values used for air at 300K are CP = 1.00 kJ/kg.K, Cv = 0.717 kJ/kg.K,, and k = 1.4. However they are all functions of temperature, and with the extremely high temperature range experienced in Diesel engines one can obtain significant errors. For this exercise you should use the temperature dependence table of Specific Heat Capacities of Air.
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