Ideal gases, isentropic processes

The Brayton cycle in its ideal form consists of two isobaric processes and two isentropic processes. The two isobaric processes consist of the combustor system of the gas turbine and the gas side of the HRSG. The two isentropic processes represent the compression (Compressor) and the expansion (Turbine Expander) processes in the gas turbine. Figure 2-1 shows the Ideal Brayton Cycle. 

Figure 15.5 shows the ideal open cycle for the gas turbine that is based on the Brayton Cycle. By assuming that the chemical energy released on combustion is equivalent to a transfer of heat at constant pressure to a working fluid of constant specific heat, this simplified approach allows the actual process to be compared with the ideal, and is represented in Figure 15.5 by a broken line. The processes for compression 1-2 and expansion 3-4 are irreversible adiabatic and differ, as shown from the ideal isentropic processes between the same pressures P and P2 -... 

The above relations apply for an ideal gas to a reversible adiabatic process which, as already shown, is isentropic. 

It has been seen in deriving equations 4.33 to 4.38 that for a small disturbance the velocity of propagation of the pressure wave is equal to the velocity of sound. If the changes are much larger and the process is not isentropic, the wave developed is known as a shock wave, and the velocity may be much greater than the velocity of sound. Material and momentum balances must be maintained and the appropriate equation of state for the fluid must be followed. Furthermore, any change which takes place must be associated with an increase, never a decrease, in entropy. For an ideal gas in a uniform pipe under adiabatic conditions a material balance gives ... 

To evaluate the integral in Equation B.l requires the pressure to be known at each point along the compression path. In principle, compression could be carried out either at constant temperature or adiabatically. Most compression processes are carried out close to adiabatic conditions. Adiabatic compression of an ideal gas along a thermodynamically reversible (isentropic) path can be expressed as ... 

A general ideal gas adiabatic (isentropic) compression process is given by ... 

If there is no heat transfer or energy dissipated in the gas when going from state 1 to state 2, the process is adiabatic and reversible, i.e., isentropic. For an ideal gas under these conditions,... 

Theoretical methods allow making such calculations for ideal and real gases and gas mixtures under isothermal and frictionless adiabatic (isentropic) conditions. In order that results for actual operation can be found it is neecessary to know the efficiency of the equipment. That depends on the construction of the machine, the mode of operation, and the nature of the gas being processed. In the last analysis such information comes from test work and its correlation by manufacturers and other authorities. Some data are cited in this section. 

A closed system cannot perforin an isentropic process without performing work. Example (Fig. 3) A quantity of gas enclosed by an ideal, tfictionless, adiabatic piston in an adiabatic cylinder is maintained at a pressure p by a suitable ideal mechanism, so that Gl = pA (A being the area of piston). When the weight G is increased (or decreased) by an infinitesimal amount dG, the gas will undergo an isentropic compression (or expansion). In this case,... 

Example 5.2 For an ideal gas with constant heat capacities undergoing a reversible adiabatic (and therefore isentropic) process, we found earlier that... 

Determine the dependence of T on P and on V for isentropic processes using an ideal gas. Note the resulting expressions. 

A small adiabatic air compressor is used to ptunp air into a 20-ni insulated tank. The tank initially contains air at 298.15 K (25°C)and 101.33 kPa, exactly tlie conditions at wliich air enters tlie compressor. The pumping process continues until tlie pressure in tile tank reaches 1000 kPa. If tlie process is adiabatic and if compression is isentropic, wliat is tile shaft work of tlie compressor Assume air to be an ideal gas for wliich Cp = (7/2)P and Cy = (5/2)P. 

For a steady flow of an incompressible fluid in a uniform pipe, the only property that varies along the pipe is pressure. However, for a compressible fluid when the pressure varies (i.e., drops), the density also drops, which means that the velocity must increase for a given mass flow. The kinetic energy thus increases, which results in a decrease in the internal energy and the temperature. This process is usually described as adiabatic, or locally isentropic, with the effect of friction loss included separately. A limiting case is the isothermal condition, although special means are usually required to achieve constant temperature. Under isothermal conditions for an ideal gas. 

Cp/Cy for a constant-entropy (isentropic) process in an ideal gas of constant heat capacity... 

In the case of an isentropic process, we have to use the adiabatic equation instead of the ideal gas law. Thus to allow an isothermal process, we have to place the column into a thermostat. In this case, not the energy is equated to zero, but rather the free energy... 

Example 9.2. We draw now the Carnot process for an ideal gas in an energy diagram. We draw the steps in the T5—pV diagram. First, observe that pV = n/ 7 for an ideal gas. Thus, we may use a parametric plot with x = nRT and y = 75. At constant mol number, in the isothermal steps, the temperature is constant and we have a vertical straight line. In the isentropic steps, the entropy 5 is constant, and we have a straight line passing the origin. 

There is still another approach to clarify the representation in Fig. 9.6. For the isothermal process, pV = nRT, which is a constant. Thus, the process must move parallel to the ordinate. For the isentropic process, we may consider the parametric representation (.t = pV = nRT, y = TS), where S is constant. Even for an isotropic process, the ideal gas law holds. On resolving the parametric representation, y/x = S/R, which indicates that the extrapolated lines cut the origin. 

The entropy of ideal gas is given in eg. (d.22). For an isentropic process, AS s = o. Applying this condition to eg. fd.22) we obtain a relationship between pressure and temperature along the isentropic path ... 

Process simulation programs are preferred to compute the theoretical and brake horsepower requirements, as well as the exit temperature, of a compressor because the ideal gas law is not usually applicable for pressures above two atmospheres. However, Eq. (16.30) can be used to obtain a preliminary estimate of the brake horsepower. An estimate of the exit temperature, including the effect of compressor efficiency, can be made with the following modification of the equation for the isentropic exit temperature ... 

The values of P and are tabulated for the air against temperature in Appendix E, Table E4. Equations (4.8)-(4.10) account the variation of specific heats with temperature and are valid only for isentropic processes of ideal gases. Equations (4.9) and (4.10) are useful in the analysis of compression and gas power cycles operating with isentropic processes. 

To determine the entrainment factors, mass flow rates and heat of the various components of such machine, we use the same equations as for the cascade AGAG, except the compressor s modeling, which must be studied sep>arately. In fact, to determine the compressor power, we consider the ammonia at the generator exit (GEi) as an ideal gas. For an isentropic process Laplace relation gives ... 

Table 2.3 demonstrates the impact of the various thermodynamic paths on a total energy balance for an open system. For the isenthalpic case, AT = 0 for an ideal gas since the enthalpy is a function of temperature only. For the isentropic case, Q = 0 since dS = dQ/T. For the isothermal case, AH = 0 since the enthalpy for an ideal gas is a function of temperature only. For the adiabatic case, AS = 0 for a reversible process only. For both the isentropic and adiabatic cases, the shaft work determined is a maximum for reversible processes. 

If the process of water vapor compression is described like that for an ideal gas undergoing an adiabatic process, then the reversible work required for the compression is given by that for an isentropic process ... 

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