Isentropic flow equation

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. 

From equation (14) a simple and explicit formula for V2 (and therefore for /gp) can be drived if a one-component gas with a constant specific heat flows through the nozzle. Since h = constant CpT in this case, /iq 2 = CpTo(l — T2/T0). In place of the parameter T2, it is generally more convenient to use P2 because the atmospheric pressure (to which P2 has been equated above) is usually specified in discussing propellant performance. For isentropic flow [equation (13)] of a one-component ideal gas with Cp — constant, it is well known and readily derived from equations (6), (7), and (1-9) that T2/T0 = (P2/PoY where the ratio of specific heats is y = Cplc (Cj, = specific heat at constant volume). Since... 

Why does l(Dwering the downstream pressure not cause the fluid to flow faster in the nozzle Suppose we attach an observer to a balloon and let her ride along with the fluid through the nozzle. When she gets to the nozzle throat, she observes that the downstream pressure is lower than she had anticipated fromjthe isentropic-flow equations, and she shouts back to those of us who are behind her to come faster. Figure 8.10 shows the fate of her shout. 

The liquid-nitrogen flow rate was measured with a turbine flowmeter. The total pressure of the jet exhaust Ptj was measured at an orifice in the model plenum chamber by a Precision Pressure Balance transducer. The total temperature of the CO2 exhaust was measured by a thermocouple in the line leading into the vacuum chamber. This thermocouple was downstream of the pressure-regulating valve and therefore measured the total temperature of the CO2 after the expansion to near the jet total pressure. The total temperature and the total pressure of the jet were used in a one-dimensional isentropic flow equation for choked nozzles to calculate the mass flow of CO2. The specific heat ratio used was that corresponding to the total temperature and pressure of the jet. 

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). 

Equation (6-128) does not require fric tionless (isentropic) flow. The sonic mass flux through the throat is given by Eq. (6-122). With A set equal to the nozzle exit area, the exit Mach number, pressure, and temperature may be calculated. Only if the exit pressure equals the ambient discharge pressure is the ultimate expansion velocity reached in the nozzle. Expansion will be incomplete if the exit pressure exceeds the ambient discharge pressure shocks will occur outside the nozzle. If the calculated exit pressure is less than the ambient discharge pressure, the nozzle is overexpanded and compression shocks within the expanding portion will result. 

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 ... 

For isentropic flow, the energy equation can be written as follows, noting that the addition of internal and flow energies can be written as the enthalpy (h) of the fluid ... 

To analyze compressible flow through chokes it is assumed that the entropy of the fluid remains constant. The equation of isentropic flow is... 

Because the gas viscosity is not highly sensitive to pressure, for isothermal flow the Reynolds number and hence the friction factor will be very nearly constant along the pipe. For adiabatic flow, the viscosity may change as the temperature changes, but these changes are usually small. Equation (9-15) is valid for any prescribed conditions, and we will apply it to an ideal gas in both isothermal and adiabatic (isentropic) flow. 

In the case of adiabatic flow we use Eqs. (9-1) and (9-3) to eliminate density and temperature from Eq. (9-15). This can be called the locally isentropic approach, because the friction loss is still included in the energy balance. Actual flow conditions are often somewhere between isothermal and adiabatic, in which case the flow behavior can be described by the isentropic equations, with the isentropic constant k replaced by a polytropic constant (or isentropic exponent ) y, where 1 < y < k, as is done for compressors. (The isothermal condition corresponds to y= 1, whereas truly isentropic flow corresponds to y = k.) This same approach can be used for some non-ideal gases by using a variable isentropic exponent for k (e.g., for steam, see Fig. C-l). 

The foregoing equations are based on flow coefficients determined by calibration with air. For application with other gases, the difference between the properties of air and those of the other gas must be considered. The gas density is incorporated into the equations, but a correction must be made for the specific heat ratio (k = cp/cv) as well. This can be done by considering the expression for the ideal (isentropic) flow of a gas through a nozzle, which can be written (in engineering units ) as follows ... 

For isentropic flow with negligible change of elevation and no shaft work, equation 6.11 reduces to... 

To understand the difference in stagnation pressure losses between subsonic and supersonic combustion one must consider sonic conditions in isoergic and isentropic flows that is, one must deal with, as is done in fluid mechanics, the Fanno and Rayleigh lines. Following an early NACA report for these conditions, since the mass flow rate (puA) must remain constant, then for a constant area duct the momentum equation takes the form... 

The number of depende nt variables is reduced by various assumptions on the form of solution. If the adiabatic flow equations 2.1.1 to 2.1.4 onp 131 are simplified to a pair of eqs in two dependent and two independent variables by assuming one-dimensional, home-otropic (uniformly isentropic) flow, eqs 2.2.1 to 2.2.7... 

The differential energy balances of Eqs. (6.10) and (6.15) with the friction term of Eq. (6.18) can be integrated for compressible fluid flow under certain restrictions. Three cases of particular importance are of isentropic or isothermal or adiabatic flows. Equations will be developed for them for ideal gases, and the procedure for nonidcal gases also will be indicated. 

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. 

This relation is seemingly identical to the classical Bernoulli equation (1.238) along a given streamline. It is emphasized that the Bernoulli equation (1.251), as derived from the energy balance (1.96), is restricted to steady-, incompressible- and isentropic flows. 

Since we have just verified that both the viscous stresses and the heat conduction terms vanish for equilibrium flows, the constitutive stress tensor and heat flux relations required to close the governing equations are determined. That is, substituting (2.233) and (2.234) into the conservation equations (2.202), (2.207) and (2.213), we obtain the Euler equations for isentropic flow ... 

A very important special case of the polytropic flow equation (5.25) is that describing a reversible, adiabatic expansion, where no heat is exchanged with the surroundings, i.e. an isentropic expansion. In this case, the ratio of specific heats, y, is substituted for n in the mass-flow equation ... 

The gas flow through the control valve as a whole will be very close to adiabatic, and, as just noted, the expansion as far as the valve throat will incur only a small frictional loss, implying a process that is approximately isentropic. Accordingly we may substitute the ratio of the gas s specific heats, y = Cp/c,. for the polytropic index, n, in the flow equations (5.25) and (5.26) derived for an ideal nozzle in Chapter 5. Combining those two equations gives the mass flow, W, as ... 

Equation (9.2) holds at low valve pressure ratios, when flow is choked and where the value of Cg comes from manufacturer s experimental data. But from our model of the valve as an ideal nozzle undergoing isentropic, choked flow, equation (9.5) should also apply. Since the two equations should give the same flow, it follows that the SI limiting gas conductance Cg must be given by... 

Having embarked on the quest of simplification, we show that the steam tables may be represented in the regions of interest to steam turbines by relatively simple approximating functions, examples of which are given. Further, it is shown that small changes in efficiency have little effect on the calculated mass flow, and it will often be sufficient for control engineering purposes to use the isentropic mass flow equation. 

EQUATIONS FOR ISENTROPIC FLOW. The phenomena occurring in the flow of ideal gas through nozzles are described by equations derivable from the basic equations given earlier in this chapter. 

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