Isentropic flow

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 ... 

Apply Eq. (9.77) to solve for the reaction ratio in isentropic flow ... 

In Eq. (9.90), C2 is the tangential component of the absolute velocity at the exit if the flow is exactly in the blade direction. Since the slip factor is ieSs than 1, the total pressure increase will decrease according to Eq. ( 9.72) for the same impeller and isentropic flow. 

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. 

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

It has been shown above that when the pressure in the diverging section is greater than the throat pressure, subsonic flow occurs. Conversely, if the pressure in the diverging section is less than the throat pressure the flow will be supersonic beyond the throat. Thus at a given point in the diverging cone where the area is equal to At the pressure may have one of two values for isentropic flow. 

Obtain an expression for the maximum flow for a given upstream pressure for isentropic flow through a horizontal nozzle. Show that for air (ratio of specific heats y = 1.4) the critical pressure ratio is 0,53 and calculate the maximum flow through an orifice of area 30 mm2 and coefficient of discharge 0.65 when the upstream pressure is 1.5 MN/m2 and the upstream temperature 293 K,... 

Response time constant 403 Rkster. S. 6-13,655 Return bends, heat exchanger 505 Reversed flow 668 Reversibility, isothermal flow 143 Reversible adiabatic, isentropic flow 148... 

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). 

In isentropic flow (just as in isothermal flow), the mass velocity reaches a maximum when the downstream pressure drops to the point where the velocity becomes sonic at the end of the pipe (e.g., the flow is choked). This can be shown by differentiating Eq. (9-25) with respect to P2 (as before) or, alternatively, as follows... 

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... 

At specified mass flow rate and inlet conditions Py and V), Eq. (6.68) predicts a relation between the area ratio A2IAX and the pressure ratio P-JPy when isentropic flow prevails. It turns out that, as the pressure falls, the cross section at first narrows, reaches a minimum at which the velocity becomes sonic then the cross section increases and the velocity becomes supersonic. In a duct of constant cross section, the velocity remains sonic at and below a critical pressure ratio given by... 

Flow Toward the Sampling Orifice. The assumption that flow toward the sampling orifice is unlikely to significantly distort the reactor flow or to be a factor in reactant sampling can be justified with compressible flow arguments. For isentropic flow of an ideal gas the temperature and pressure are related to the Mach number by Eqs. 7 and 8 [40] ... 

The gas flow toward the pinhole is a converging flow. The ratio of the flow cross-sectional area to the area in the pinhole, for isentropic flow of an ideal gas is... 

Solving Eq. 9 for M = 0.11 gives A/A = 5.415. For continuum isentropic flow the gas will reach M = 1.0 at the minimum cross-sectional area located at the pinhole. If the effect of boundary layers is ignored and the flow region is bounded by a hemisphere centered on the pinhole, A/A is... 

This assumption is required in order to assume isentropic flow. Losses are generally small for most practical rocket motors. 

Substituting this expression into Eq. (7.20) for isentropic flow gives the throat... 

For the near-equilibrium flow case, it is also necessary to calculate Te by trial and error. Each assumed value of Te will have a unique gas composition. The correct value will satisfy the criteria for isentropic flow,... 

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