Gas—Isentropic

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, 

That is, the temperature drops linearly as the elevation increases. 

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Thus, from Equation B.4 for an adiabatic ideal gas (isentropic) compression ... 

To obtain the temperature rise for an ideal gas isentropic compression, substitute P = RT/V in Equation B.5 to... 

Equation B.21 assumes the compression to be adiabatic ideal gas (isentropic) compression. In practice, the compression will be neither perfectly adiabatic nor ideal. To allow for this, the gas compression can be assumed to follow a polytropic compression represented by the empirical expression ... 

For real gas, its isentrople expansion coefficient (n) is tfifferent from its ideal gas isentropic expansloii coefficient (k). Most hydrocarbon vapor at high pressure, its heat capacity (Cp) is equal to its ideal gas Cp plus some correction. 

Figure A2.1.4. Adiabatic reversible (isentropic) paths that do not intersect. (The curves have been calculated for the isentropic expansion of a monatomic ideal gas.)...

Calculation of Actual Work of Compression For simplicity, the work of compression is calciilated by the equation for an ideal gas in a three-stage reciprocating machine with complete intercoohng and with isentropic compression in each stage. The work so calculated is assumed to represent 80 percent of the actual work. The following equation may be found in any number of textbooks on thermodynamics ... 

These equations are consistent with the isentropic relations for a perfect gas p/po = (p/po), T/To = p/poY. Equation (6-116) is valid for adiabatic flows with or without friction it does not require isentropic flow However, Eqs. (6-115) and (6-117) do require isentropic flow The exit Mach number Mi may not exceed unity. At Mi = 1, the flow is said to be choked, sonic, or critical. When the flow is choked, the pressure at the exit is greater than the pressure of the surroundings into which the gas flow discharges. The pressure drops from the exit pressure to the pressure of the surroundings in a series of shocks which are highly nonisentropic. Sonic flow conditions are denoted by sonic exit conditions are found by substituting Mi = Mf = 1 into Eqs. (6-115) to (6-118). 

Refrigerating capacity is the product of mass flow rate of refrigerant m and refrigerating effect R which is (for isobaric evaporation) R = hevaporator outlet evaporator mJef Powei P required foi the coiTipressiou, necessary for the motor selection, is the product of mass flow rate m and work of compression W. The latter is, for the isentropic compression, W = hjisehatge suction- Both of thoso chai acteristics could be calculated for the ideal (without losses) and for the ac tual compressor. ideaUy, the mass flow rate is equal to the product of the compressor displacement per unit time and the gas density p m = p. 

Similar to volumetric efficiency, isentropic (adiabatic) efficiency T is the ratio of the work required for isentropic compression of the gas to work input to the compressor shaft. The adiabatic efficiency is less than one mainly due to pressure drop through the valve ports and other restricted passages and the heating of the gas during compression. 

Efficiency for a turboexpander is calculated on the basis of isentropic rather than polytropic expansion even though its efficiency is not 100 percent. This is done because the losses are largely introduced at the discharge of the machine in the form of seal leakages and disk friction which heats the gas leaking past the seals and in exducer losses. (The exducer acts to convert the axial-velocity energy from the rotor to pressure energy.)... 

As stated earlier, turboexpanders are normally used in cryogenic processes to produce isentropic expansion to cool down the process gas. Two common applications are natural gas processing plants and chemical plants. In natural gas processing plants, turboexpanders are installed to liquify heavier hydrocarbon components and produce lean natural gas with specified dew point limits to meet required standards. 

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. 

With incompressibile fluids, the value of the acoustic speed tends toward infinity. For isentropic flow, the equation of state for a perfect gas can be written ... 

In the gas turbine (Brayton cycle), the compression and expansion processes are adiabatic and isentropic processes. Thus, for an isentropic adiabatic process 7 = where Cp and c are the specific heats of the gas at constant pressure and volume respectively and can be written as ... 

It has played a dual role, one in Equation 2.18 on specific heat ratio and the other as an isentropic exponent in Equation 2.53. In the previous calculation of the speed of sound. Equation 2.32, the k assumes the singular specific heat ratio value, such as at compressor suction conditions. When a non-perfect gas is being compressed from point 1 to point 2, as in the head Equation 2.66, k at 2 will not necessarily be the same as k at 1. Fortunately, in many practical conditions, the k doesn t change very... 

Thus there are three modifications to the a/s efficiency analysis, involving (i) the specific heats ( and n ), (ii) the fuel-air ratio / and the increased turbine mass flow (I +/), and (iii) the pressure loss term S. The second of these is small for most gas turbines which have large air-fuel ratios and / is of the order of l/IOO. The third, which can be significant, can also be allowed for a modification of the a/s turbine efficiency, as given in Hawthorne and Davis [I]. (However, this is not very convenient as the isentropic efficiency tjt then varies with r and jc, leading to substantial modifications of the Hawthome-Davis chart.)... 

But another approach to multi-step cooling [8, 9] involves dealing with the turbine expansion in a manner similar to that of analysing a polytropic expansion. Fig. 4.4 shows gas flow (1 + ijj) at (p,T) entering an elementary process made up of a mixing process at constant pressure p, in which the specific temperature drops from temperature T to temperature T, followed by an isentropic expansion in which the pressure changes to (p dp) and the temperature changes from T to (7 - - dT). 

Fig. 4.10 shows more fully calculated overall efficiencies (for turbine cooling only) replotted against isentropic temperature ratio for various selected values of Tj = T,.,. This figure may be compared directly with Fig. 3.9 (the a/s calculations for the corresponding CHT cycle) and Fig. 3.13 (the real gas calculations of efficiency for the uncoooled CBT cycle). The optimum pressure ratio for maximum efficiency again increases with maximum cycle temperature T. ... 

The flow field in front of an expanding piston is characterized by a leading gas-dynamic discontinuity, namely, a shock followed by a monotonic increase in gas-dynamic variables toward the piston. If both shock and piston are regarded as boundary conditions, the intermediate flow field may be treated as isentropic. Therefore, the gas dynamics can be described by only two dependent variables. Moreover, the assumption of similarity reduces the number of independent variables to one, which makes it possible to recast the conservation equations for mass and momentum into a set of two simultaneous ordinary differential equations ... 

For an ideal gas, pV is constant for isentropic expansion (that is, without energy addition or energy loss). Therefore, V2 is ... 

At the instant a pressure vessel ruptures, pressure at the contact surface is given by Eq. (6.3.22). The further development of pressure at the contact surface can only be evaluated numerically. However, the actual p-V process can be adequately approximated by the dashed curve in Figure 6.12. In this process, the constant-pressure segment represents irreversible expansion against an equilibrium counterpressure P3 until the gas reaches a volume V3. This is followed by an isentropic expansion to the end-state pressure Pq. For this process, the point (p, V3) is not on the isentrope which emanates from point (p, V,), since the first phase of the expansion process is irreversible. Adamczyk calculates point (p, V3) from the conservation of energy law and finds... 

When a Mollier chart is available for the gas involved the first method, which is illustrated by Figure 12-12A is the most convenient. On the abscissa of Figure 12-12A four enthalpy differences are illustrated. (Hg — Hj) is the enthalpy difference for the isentropic path. (Hg — Hi°) is the ideal gas state enthalpy difference for the terminal temperatures of the isentropic path. The other AH values are the isothermal pressure corrections to the enthalpy at the terminal temperatures. A generalized chart for evaluating these pressure corrections was presented previously. 

In adiabatic compression or expansion, no release or gain of heat by the gas occurs, and no change occurs in entropy. This condition is also known as isentropic and is typical of most compression steps. Actual conditions often cause a realistic deviation, but usually these are not sufficiently great to make the calculations in error. Table 12-4 gives representative average k values for a few common gases and vapors. 

Adiabatic compression (termed adiabatic isentropic or constant entropy) of a gas in a centrifugal machine has the same characteristics as in any other compressor. That is, no heat is transferred to or from the gas during the compression operation. The characteristic equation... 

A Mollier Diagram is useful for the expansion of a specific gas/vapor or multicomponent vapor fluid. See Figure 12-91 for comparison of (1) constant enthalpy (Joule-Thompson effect), isenthalpic, and (2) isentropic (constant entropy), which provides the colder temperature. Note that the expander indicated on the figure is somewhere between isenthalpic and isentropic or polytropic. See Figure 12-92. ... 

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 -... 

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