Flow through Frictionless Nozzle

Adiabatic Frictionless Nozzle Flow In process plant pipelines, compressible flows are usually more nearly adiabatic than isothermal. Solutions for adiabatic flows through frictionless nozzles and in channels with constant cross section and constant friction factor are readily available. 

The mass velocity G = w/A, where w is the mass flow rate and A is the nozzle exit area, at the nozzle exit is given by 

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

The exit Mach number M may not exceed unity. At )/ = 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). 

When it is desired to determine the discharge rate through a nozzle from upstream pressure p0 to external pressure p2, Equations (6-115) through (6-122) are best used as follows. The critical pressure is first determined from Eq. (6-119). If p2 p , then the flow is subsonic (subcritical, unchoked). Then p, = p2 and M, may be obtained from Eq. (6-115). Substitution of Mx into Eq. (6-118) then gives the desired mass velocity G. Equations (6-116) and (6-117) may be used to find the exit temperature and density. On the other hand, if p2 p , then the flow is choked and M = 1. Then j = p , and the mass velocity is G obtained from Eq. (6-122). The exit temperature and density may be obtained from Eqs. (6-120) and (6-121). 

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Example 7 Flow through Frictionless Nozzle Air at po and temperature To = 293 K discharges through a frictionless nozzle to atmospheric pressure. Compute the discharge mass flux G, the pressure, temperature, Mach mimher, and velocity at the exit. Consider two cases (1) po = 7 X 10 Pa absolute, and (2) po = 1.5 x 10 Pa absolute. 

The adiabatic flow of an ideal gas flowing through a frictionless conduit or a constriction (such as an orifice nozzle, or valve) can be analyzed as follows. The total energy balance is... 

In flow through a frictionless nozzle, there is a critical pressure ratio, r, which will just cause choking. The critical pressure ratio is the ratio of the downstream back pressure to the upstream pressure, both in absolute pressure units. If the actual back pressure (e.g. atmospheric) is less than the critical pressure at which choking occurs, then there will be a pressure discontinuity at the end of the nozzle the pressure just inside the nozzle will be the critical pressure for choking, and that just outside the nozzle will be the actual back pressure, which is normally atmospheric. See Figure 9.2. -. ... 

Run times can be significantly reduced by using a (frictionless) nozzle model for flow through the relief, system. Most such models incorporate a discharge coefficient, and the value supplied can be used to approximately take account of the friction which will actually occur Alternatively, the discharge coefficient can be set equal to 1 to... 

However, equation (A8.13) must be solved by trial and error and it is therefore more convenient to use the plot of the solutions to the above equations, given in Figure A8.2, of Gc and r c versus . r c can be used to check whether the flow through a frictionless nozzle would be choked. Leung13,4] has also published some equations which are numerical solutions for Gc and r c f°r vapour pressure systems, but these make use of a slightly different definition of Omega (see A8.4.1). 

The ratio of the actual flow rate (which is reduced by friction etc.) to the calculated flow rate through ah ideal frictionless nozzle. 

The other method is the velocity head method. The term V2/2g has dimensions of length and is commonly called a velocity head. Application of the Bernoulli equation to the problem of frictionless discharge at velocity V through a nozzle at the bottom of a column of liquid of height H shows that H = V2/2g. Thus II is the liquid head corresponding to the velocity V. Use of the velocity head to scale pressure drops has wide application in fluid mechanics. Examination of the Navier-Stokes equations suggests that when the inertial terms dominate the viscous terms, pressure gradients are expected to be proportional to pV2 where V is a characteristic velocity of the flow. 

This fonnula is the same for frictionless flow through the venturi and nozzle meters. 

Moving on to compressible flow, it is first of all necessary to explain the physics of flow through an ideal, frictionless nozzle. Chapter S shows how the behaviour of such a nozzle may be derived from the differential form of the equation for energy conservation under a variety of constraint conditions constant specific volume, isothermal, isentropic and polytropic. The conditions for sonic flow are introduced, and the various flow formulae are compared. Chapter 6 uses the results of the previous chapter in deriving the equations for frictionally resisted, steady-state, compressible flow through a pipe under adiabatic conditions, physically the most likely case on... 

Chapter 5 presented a mathematical model for adiabatic flow through a frictionless nozzle, and we shall use some of that chapter s results. It will be helpful for our present purpose if we derive the Mach number at station 2 , where the Mach number, M, is defined as the ratio of the velocity to the local speed of sound. Recalling equation (5.47), the Mach number at station 2 will be ... 

Example 5.1. The tank in Fig. 5.4 is full of air at 70°F, The air is flowing put at a steady rate through a smooth, frictionless nozzle to the local atmosphere. What is the flow velocity for various tank pressures ... 

In the tank in Fig. 5.24 water is under a layer of compressed air, which is at a pressure of 20 psig. The water is flowing out through a frictionless nozzle, which is 5 ft below the water surface. What is the velocity of the water ... 

A cylindrical tank, shown in Fig. 7.29, is sitting on a platform with absolutely frictionless wheels on a horizontal plane. There is no air resistance. At time 0, the level in the tank is 10 ft above the outlet, and the whole system is not moving. Then the outlet is opened, and the system is allowed to accelerate to the left. The flow through the outlet nozzle is frictionless. What is the final velocity, assuming that (fl) the mass of the tank and cart is zero and (b) the mass of the tank and cart is 3000 Ibm ... 

Qm,nozzle Dischargeable mass flow through an ideal frictionless nozzle (kg/s)... 

Nitrogen contained in a large tank at a pressure P = 200000 Pa and a temperature of 300 K flows steadily under adiabatic conditions into a second tank through a converging nozzle with a throat diameter of 15 mm. The pressure in the second tank and at the throat of the nozzle is P, = 140000 Pa. Calculate the mass flow rate, M, of nitrogen assuming frictionless flow and ideal gas behaviour. Also calculate the gas speed at the nozzle and establish that the flow is subsonic. The relative molecular mass of nitrogen is 28.02 and the ratio of the specific heat capacities y is 1.39. 

Equation (6-128) does not require frictionless (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. 

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