Carnot cycle with an ideal gas

Once one value of the thermodynamic temperature has been assigned a positive value, all other temperatures must be positive otherwise in some circumstances the Q s for the two reservoirs would have the same sign, resulting, as we have seen, in perpetual motion. 

From the characteristics of a particularly simple kind of heat engine, the Carnot engine, and from universal experience that certain kinds of engine cannot be constructed, we concluded that all reversible heat engines operating between the same two heat reservoirs have the same efficiency, which depends only on the temperatures of the reservoirs. Thus it was possible to establish the thermodynamic scale of temperature, which is independent of the properties of any individual substance, and to relate the efficiency of the engine to the temperatures on this scale  

If an ideal gas is used as the working substance in a Carnot engine, the application of the first law to each of the steps in the cycle can be written as in Table 8.3. The values of and FF3, which are quantities of work produced in an isothermal reversible expansion of an ideal gas, are obtained from Eq. (7.6). The values of AC are computed by integrating the equation dU = C dT. The total work produced in the cycle is the sum of the individual quantities. 

The two integrals sum to zero, as can be shown by interchanging the limits and thus 

Equation (8.20) can be simplified if we realize that the volumes V2 and F3 are connected by an adiabatic reversible transformation the same is true for V4 and V. By Eq. (7.57), 

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To illustrate the improper use of constraints, we deal with the Carnot cycle with an ideal gas as working system. In the isothermal expansion, the gas takes up thermal energy from the reservoir and does work of expansion to a further external system. Thus, the energy change of the working gas is... 

The derived expressions (4.9) and (4.10) can be shown to be general for reversible cyclic processes. Further, it can be shown that the reversible Carnot cycle with an ideal gas is the most efficient process possible for transforming heat... 

What value should Ti and T3 approach to obtain the efficiency r) 1 for a reversible Carnot cycle with an ideal gas ... 

The Carnot cycle is a reversible cyclic process during which a thermodynamic system cycles between two heat reservoirs of different temperatures. A Carnot process with an ideal gas will be described here as an illustration, but as will be shown later, there is no restriction on the nature of the system. 

In Chapter 2, we have analyzed one particular type of heat engine, the reversible Carnot cycle engine with an ideal gas as the working substance, and found that its efficiency is e = 1 — Tc/Th. For both practical and theoretical reasons, we ask if it is possible, with the same two heat reservoirs, to design an engine that achieves a higher efficiency than the reversible Carnot cycle, ideal gas engine. What can thermodynamics tell us about this possibility ... 

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)... 

The thermodynamic temperature scale is not related to any particular kind of substance and is therefore more fundamental than the ideal gas temperature scale. We now show that the thermodynamic temperature scale can coincide with the ideal gas temperature scale. Assume that the working fluid of a Carnot engine is an ideal gas with a constant heat capacity. For the first step of the Carnot cycle, from Ekj. (2.4-10)... 

The second law of thermodynamics stems from the studies in the 1800 s of heat engines, and in the this period s theories on the motive power of heat. Therefore, the second law is introduced, along with the concept of entropy, through the Carnot cycle for a heat engine operating with an ideal gas. The energy considerations used in the Carnot process are universal and thus they lead to general conditions of equilibrimn for thermodynamic systems. 

Figure 4.5. Reversible Carnot cycle abcda with an ideal gas the work performed on the system during the cyclic process corresponds to the shaded area in the pV diagram.

In the next chapter, we will return to the Carnot cycle, describe it quantitatively for an ideal gas with constant heat capacity as the working fluid in the engine, and show that the thermodynamic temperature defined through equation (2.34) or (2.35) is proportional to the absolute temperature, defined through the ideal gas equation pVm = RT. The proportionality constant between the two scales can be set equal to one, so that temperatures on the two scales are the same. That is, 7 °Absolute) = T(Kelvin).r... 

Figure 3.4 Carnot cycle for the expansion and compression of an ideal gas. Isotherms alternate with adiabats in a reversible closed path. The shaded area enclosed by the curves gives the net work in the cyclic process.

The reverse direction of a spontaneous process requires that some work must be done on the system. In 1824, Sadi Carnot, an engineer in Napoleon s army, presented an ideal engine in which the heat could not be completely converted into work. The engine had an ideal gas in a cylinder with a frictionless piston and employed a cyclic operation. In Figure 1.5, the pressure and volume are related to the four steps of the cycle. Initially, the engine contains an initial pressure of P, an initial volume of V, and an initial temperature of TH as the initial state A. 

As an example we would try to conceive of a Carnot s cycle without using ideal gas as the system. Let the container in Fig. 4.1 contain water and steam in equilibrium instead of an ideal gas. Here pressure would be 1 atmosphere if temperature was 100 °C. If heat is supplied to the system, more of water would get converted to steam. It is easy to conceive that, if heat is transferred infinitesimally slowly, heat transfer can be carried out in thermodynamically reversible manner with corresponding increase in volume. By releasing pressure, also in a reversible manner, more of water gets vaporised and volume increases further to point C (Fig. 6.6) at reduced pressure and temperature. Thereafter the system can be made to lose heat reversibly at the lower temperature, which would make some steam to condense to liquid water with reduction in volume. 

Assume that the system used to carry out the Carnot cycle is an amount of ideal gas contained in a cylinder fitted with a frictionless piston. The concept of an ideal gas is introduced in Section 1.2. Of consequence at this point is the premise that for an ideal gas the internal energy, U, is a function of temperature only. The Carnot cycle consists of reversible isothermal and adiabatic processes. An isothermal process is one in which the system temperature is kept constant. An adiabatic process requires that no heat be transferred between the system and its surroundings. The steps are as follows ... 

Let ns nse an ideal gas in a Carnot cycle and find the efficiency of the cycle by using ideal-gas properties in ennmerating the changes in the four steps of the cycle. Let us designate the intial state of step 1 with the subscript A, the initial state of step 2 with B, and so on. The high temperature at step 1, which is i on the thermodynamic temperatnre scale, will be Tj on the ideal-gas temperature scale. The low temperature of step 3 will be T2, corresponding to 02. The work and heat terms of a step will be designated with subscripts 1, 2, 3, or 4. 

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. 

Let us tabulate the thermodynamic quantities for the four steps of the Carnot cycle in Figure 3.3. We will assume that the engine operates with an ideal monatomic gas (i.e., that U = 3nRT/2). For steps 1 and 3, the work is obtained by integrating -PdV with the ideal gas substitution, P = nRT/V, as in Equation 3.2. For steps 2 and 4, the change in L/ is used to find the work. 

The thermal efficiency of an ideal simple cycle is decreased by the addition of an intercooler. Figure 2-7 shows the schematic of such a cycle. The ideal simple gas turbine cycle is 1-2-3-4-1, and the cycle with the intercooling added is -a-b-c-2- i-A-. Both cycles in their ideal form are reversible and can be simulated by a number of Carnot cycles. Thus, if the simple gas turbine cycle 1-2-3-4-1 is divided into a number of cycles like m-n-o-p-m,... 

The ideal gas temperature scale is of especial interest, since it can be directly related to the thermodynamic temperature scale (see Sect. 3.7). The typical constant-volume gas thermometer conforms to the thermodynamic temperature scale within about 0.01 K or less at agreed fixed points such as the triple point of oxygen and the freezing points of metals such as silver and gold. The thermodynamic temperature scale requires only one fixed point and is independent of the nature of the substance used in the defining Carnot cycle. This is the triple point of water, which has an assigned value of 273.16 K with the use of a gas thermometer as the instrument of measurement. 

The existence of a finite heat transfer in the isothermal processes is affected with the assumption of a non-endoreversible cycle with ideal gas as working substance. Power output and ecological function have also an issue that shows direct dependence on the temperature of the working substance. Expressions obtained with the changes of variables have the virtue of leading directly to the shape of the efficiency through Z, function. Thus, in classical equilibrium thermodynamics, the Stirling cycle has its efficiency like the Carnot cycle efficiency in finite time thermodynamics, this cycle has an efficiency in their limit cases as the Curzon-Ahlborn cycle efficiency. 

In the remaining part of this AppendixA.1, we obtain the important result (A.9) using an ideal cyclic process from subset C of Sect. 1.2, namely the Carnot cycle [1, 2, 4, 5]. Carnot cycle is a cyclic process with (fixed number of mols, n, of) uniform ideal gas composed from isothermal and adiabatic (no heat exchange) expansions followed by isothermal (at lower temperature) and adiabatic compressions back to the starting state. All these processes pass the equilibrium (stable) states and they are reversible (cf. definition in Sect. 1.2), see also Rem. 48 in Chap. 3. 

The heat exchanged with the surroundings in the steps of fhe Carnot cycle can be related to the temperature of fhe heat baths. Again assuming an ideal monatomic gas, the expression for U,U = 3nRT/2, leads to the differential relation... 

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