Carnot cycle example

The experimental realization of a Carnot cycle to measure the temperature T is unusual. The coincidence of the thermodynamic temperature T with the temperature read by a gas thermometer, for example, allows the use of such thermometer to know T. As we shall see, also other laws of physics relating T with physical parameters other than heat can be used to get an absolute measure of T. 

For 6 to be a generally useful function we must remove any specifications as to the namre of the reversible cycle through which the substance is being carried. Let us represent a general reversible cycle by the example illustrated in Figure 6.5(a). This cycle also can be approximated in Carnot cycles, as illustrated in... 

Carnot s prescient pamphlet, apparently distributed only among a small circle of friends, remained unknown in the scientific literature for about a decade. Fortunately, its content and value were recognized by fellow French engineer Emile Clapeyron, who built on its concepts and extended its methods, including, for example, the first graphical PV representation of the Carnot cycle. Clapeyron s 1834 paper was the means by which Carnot s discoveries first became known to William Thomson, who made them the centerpiece of his own later work on thermodynamic theory. 

The entropy function S also simplifies the graphical depiction of the Carnot cycle. Consider, for example, the form of the Carnot cycle shown in the PV diagram of Fig. 4.4a. The corresponding ST diagram for the same Carnot cycle is shown in Fig. 4.4b. As can be seen, the ST representation of the Carnot cycle is a simple rectangle whose... 

So far only the energy requirement for a process in the form of work has been considered. Freezing, vapor compression, and reverse osmosis processes are examples of processes that require a work input. There are, however, other important processes, such as multiple-effect evaporation and flash evaporation, for which the energy input is in the form of heat. How does one relate the energy requirement of these processes to the minimum work of separation One method is to convert the heat requirement to a work equivalent by means of the Carnot cycle. If T is the absolute temperature of the heat source and T0 the heat-sink temperature, then one can use the familiar relation... 

In the development of the second law and the definition of the entropy function, we use the phenomenological approach as we did for the first law. First, the concept of reversible and irreversible processes is developed. The Carnot cycle is used as an example of a reversible heat engine, and the results obtained from the study of the Carnot cycle are generalized and shown to be the same for all reversible heat engines. The relations obtained permit the definition of a thermodynamic temperature scale. Finally, the entropy function is defined and its properties are discussed. 

For a proof of this result, we refer to the original papers [1,2], but we can make the existence of these two optimal temperatures plausible by the following reasoning. Suppose we wish to introduce heat into the Carnot engine with a rate Qin- This fixes the upper temperature Thc of the Carnot cycle, for example by the approximate relation... 

We have included the summation because there may be more than one heat reservoir in the surroundings that is involved in the process. For example, in the Carnot cycle engine, we remove heat from a hot reservoir and deposit heat in a cold reservoir. 

Students are reminded of the upper thermodynamic limit set on the efficiency of a heat engine, for example the internal combustion and gas-turbine engines. The ideal and totally unrealistic engine would operate on the so-called Carnot cycle where the working substance (e.g. the gas) is taken in at the high temperature (Th) and pressure and after doing external work is exhausted at the lower temperature (Tc) and lower pressure. The Carnot efficiency, /, is given by... 

As an example, if the hot temperature is 1273 K (1000 °C) and the cold temperature 373 K (100 °C) then the efficiency is approximately 70%. In practice the operation of a real engine does not follow the Carnot cycle and the efficiency is considerably lower. For a medium sized motor car with an internal combustion engine the fuel efficiency is about 12%, much of the wasted 88% demanding water cooling. There are continuous improvements made in petrol and diesel engine technologies and in the fuels and projections suggest that thermal efficiencies a little over 50% will eventually be achieved. 

For a single reversible process between two sets of fixed conditions, the work is independent of the reversible path. However, in a network of reversible processes, such as Figure A.l, alteration of the pressure and temperature of the isothermal enclosure alters the pressure ratio of, for example, the fuel isothermal expander. The power output of Figure A.l is therefore variable and not a constant, merely because it is reversible. The maximum power, the fuel chemical exergy, is obtained from an electrochemical reaction at standard temperature, Tq, and sum of reactant and product pressures, Pg, with isothermal expanders only and without a Carnot cycle. 

For given values of Tc and Tu, the highest possible value of co is attained for Carnot-cycle refrigeration. The lower values for the vapor-eompression eycle result from irreversible expansion in a tlrrottle valve and irreversible compression. The following example provides an indication of typical values for eoefficients of perfomianee. 

Any reversible cycle may be regarded as being made up of a number of Carnot cycles. Consider, for example, the cycle represented in Fig. 12 by the closed curve ABA imagine a series of isothermal and adiabatic curves drawn across the diagram, so that a number of Carnot cycles are indicated. 

The above value is not bad. In reality much more exergy is destroyed because of irreversibilities, particularly due to the cooling. For example, if we take into account only the hot utility for separation (feed preheating and reboiler) the total heat is g, = 138 + 1147 = 1285 kW. This could be extracted virtually from surroundings and converted by means of a Carnot cycle working between 298 and 406.15 K, for which the following work can been obtained ... 

Namely, we discuss two examples of equilibrium reversible processes the isothermal and then those which are adiabatic. Such processes with ideal gas (i.e., with real stable gas at sufficiently low pressures) are used in the Carnot cycle in Appendix A.l. 

Figure 4-6 A cycle composed of two mtercoimected Carnot cycles. Using enough reversible adiabatic paths close to each other, any reversible closed path can be represented by a series of Carnot cycles. Along such closed path, the integral of dQrev/T is zero (see Example...

We shall see, as an example, a simple counterflow cold exchanger connecting the cold and warm ends between (B) and (D) is the nearest realistic equivalent to a Carnot cycle. The idealized constant-mass flow system in a perfect counterflow heat regenerator operating with an idealized gas is thermodynamically equivalent to the adiabatic expansion paired with the adiabatic compression in the Carnot cycle, since the following intrinsic energy transfer is fulfilled in terms of a reciprocal isobaric transformation. (See Fig. 2)... 

Note the qualifier not necessarily. Sometimes the most direct pathway is the most economical. More to the point, the special pathways of Chapter 4 offer the maximum economy in one or more state variables. For isobaric, isochoric, adiabatic, and isothermal cases, Ix p, Ix<, and 0, respectively for every path of a closed system, Ix = 0. Note that special pathways afford straight-line representations in select planes, for example, isobaric and isochoric in pV, and adiabatic in the TS plane. A Carnot cycle appears as a square or rectangle when drawn in the TS plane. 

The advantage of converting the hydrocarbons in the fuel cell rather than converting the fuel to hydrogen first is evident. If steam reforming of natural gas takes place in an externally fired reformer, there is a loss in efficiency because of the high temperature created in the flame, which is not utilised fully for work because the waste heat ean only be recovered via the Carnot cycle as shown in Example 2.3. 

Heat engines have limited efficiencies which are determined by the Carnot cycle. Practical issues reduce the efficiency of steam engines, due to limits of ccaivective heat transfer and viscous flow (friction). There are also mechanical considerations, for example, limitations imposed by the materials such as nonideal properties of the working gas, thermal conductivity, tensile strength, creep, mpture strength, and melting point. 

A heat engine, as mentioned in Sec. 4.2, is a closed system that converts heat to work and operates in a cycle. A Carnot engine is a particular kind of heat engine, one that performs Carnot cycles with a working substance. A Camot cycle has four reversible steps, alternating isothermal and adiabatic see the examples in Figs. 4.3 and 4.4 in which the working substances are an ideal gas and H2O, respectively. 

Example 4.2 calculated AA for one step of a Carnot cycle. What is AA for the entire Carnot cycle ... 

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