Isentropic Relations

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). 

These equations are consistent with the isentropic relations for a per-... 

Derive the following isentropic relation for ideal gases with constant specific heats. 

These equations are consistent with the isentropic relations for a perfect gas p/po= (p/po) T/To = (p/po) 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. 

For this type of flow, the mass balance, energy, and perfect-gas equations take the same form as for steady, isentropic flow of a perfect gas. However, in Sec. 8,2 we used the isentropic relations from App. D here we cannot, because the effect of friction is to increase the entropy of the flowing gas. In their place we use the momentum balance (Eq. 7.13), written for two points dx... 

Assuming that a sound wave causes an isentropic pressure rise of 10" psia in air, calculate the temperature rise caused by such a wave. Isentropic relations for perfect gases (e.g., air) are shown in App. D. 

For this particular case it is necessary to introduce a flctitious convergent whose ttoat area is the same as the whole areas of the various holes. Isentropic relations are used to find the Mach number in the section A... 

At t - 0 there is isentropic flow in this region with the flow variables satisfying the isentropic relations and the area Mach number relation. It is also assumed that the flow Mach number ahead of the shock M (x) is nowhere transonic and the flow... 

Because the Griineisen ratio relates the isentropic pressure, P, and bulk modulus, K, to the Hugoniot pressure, P , and Hugoniot bulk modulus, K , it is a key equation of state parameter. 

Ruoff (1967) first showed how the coefficients of the shock-wave equation of state are related to the zero pressure isentropic bulk modulus, and its first and second pressure derivatives, K q and Kq, via... 

Moreover, upon comparing (4.32) with (4.14), it can be seen that (Jeanloz and Grover, 1988) the Birch-Murnaghan equation (4.32) with a2 = 0 describes the isentropic equation of state provided the linear shock-particle velocity relation (4.5) describes the Hugoniot. In combination, these require that... 

Figure 4.21. Pressure-volume paths used to relate the slope of Hugoniot (dP/dV) to isentropic sound speed G (4.57).

The polytropic efficiency in a turbine can be related to the isentropic efficiency and obtained by combining the previous two equations... 

As alternatives to the isentropic efficiencies for the turbomachinery components, tjt and Tjc. which relate the overall enthalpy changes, small-stage or poly tropic efficiencies (Tjpj and Tjpc) are often used. The pressure-temperature relationship along an expansion line is then p T = constant, where z = [y y OtJpt]-... 

Clearly, if A is zero (no heat transfer), then the normal polytropic relation holds. A point of interest is that if Tjp = (1 — A) then rj = 1 and the expansion becomes isentropic (but not reversible adiabatic). 

Initial shock-wave overpressure can be determined from a one-dimensional technique. It consists of using conservation equations for discontinuities through the shock and isentropic flow equations through the rarefaction waves, then matching pressure and flow velocity at the contact surface. This procedure is outlined in Liepmatm and Roshko (1967) for the case of a bursting membrane contained in a shock tube. From this analysis, the initial overpressure at the shock front can be calculated with Eq. (6.3.22). This pressure is not only coupled to the pressure in the sphere, but is also related to the speed of sound and the ratio of specific heats. 

A reversible adiabatic process is known as isentropic. Thus, the two conditions are directly related. In actual practice compressors generate friction heat, give off heat, have valve leakage and have piston ring leakage. These deviations... 

Solution In a reversible adiabatic expansion, 6qrev = T dS = 0. Thus, the process is isentropic, or one of constant entropy. To obtain an equation relating p, V and T, we start with... 

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

The velocity uw = fkP2v2 is shown to be the velocity of a small pressure wave if the pressure-volume relation is given by Pifi = constant. If the expansion approximates to a reversible adiabatic (isentropic) process k y, the ratio of the specific heats of the gases, as indicated in equation 2.30. 

The gas continues to expand isentropically and the pressure ratio w is related to the flow area by equation 4,47. If the cross-sectional area of the exit to the divergent section is such that >r 1 = (10,000/101.3) = 98.7, the pressure here will be atmospheric and the expansion will be entirely isentropic. The duct area, however, has nearly twice this value, and the flow is over-expanded, atmospheric pressure being reached within the divergent section. In order to satisfy the boundary conditions, a shock wave occurs further along the divergent section across which the pressure increases. The gas then expands isentropically to atmospheric pressure. 

Equation 23.7 is based on the actual change in steam enthalpy across the turbine. Although both Equations 23.6 and 23.7 have the same form, their coefficients have completely different meanings. Comparing Equations 23.6 and 23.7, it becomes apparent that the slope of the linear Willans Line (Equation 23.6) is related to the isentropic enthalpy change and turbine isentropic efficiency9. 

The isentropic power from a steam turbine can be related to the maximum shaft power through the overall turbine efficiency ... 

The isentropic and polytropic efficiencies can be related by taking the ratio of Equations B.15 and B.35 for the same... 

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