Flow through ideal nozzles

The widespread occurrence in process plant of nozzles, intentional and unintentional, means that it is necessary for the control engineer to have an understanding of their physical principles for modelling purposes. We will consider in this chapter the features of an ideal, frictionless nozzle. Results from this simplified model will be put to immediate use in Chapter 6 in order to account for the unavoidable nozzle formed at beginning of a gas pipe. Moreover, the idealized approach provides the groundwork for the more complex models of Chapter 14, where turbine nozzles with friction are considered in some detail. 

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The formula for semi-ideal gas flow through a nozzle is commonly used to obtain the flow capacity of a safety valve14 -... 

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... 

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... 

Equation (9-45) represents flow through an ideal nozzle, i.e., an isentropic constriction. 

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 ... 

With the P-v relation given by Eq. (23-103), Eq. (23-98) can be integrated to give a generalized equation for flow through an ideal nozzle... 

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. 

When the fluid flowing through the valve is a compressible gas or a vapor, then the design must consider whether critical flow is achieved in the nozzle of the valve. The critical flow rate is the maximum flow rate that can be achieved and corresponds to a sonic velocity at the nozzle. If critical flow occurs, then the pressure at the nozzle exit cannot fall below the critical flow pressure Pcf, even if a lower pressure exists downstream. The critical flow pressure can be estimated from the upstream pressure for an ideal gas using the equation... 

Flow-through relief valves have traditionally been analyzed by treating the valve as a convergent noz-2ie [i 29.30] discharge coefficient (kactual mass flow rate to the theoretical ideal mass flow rate through a one-dimensional convergent nozzle of exit area equal to the nominal bore of the tested valve. Measured discharge coefficients are derated by 10% (Aidr) to specify valve capacity. 

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... 

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 ... 

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. 

However, at the microscale, the performance of the microrocket is limited by viscous losses due to the low Reynolds number of the expanding flow. These losses are characterized by the thrust efficiency which compares the observed momentum flux to the momentum flux predicted for an ideal (zero viscosity) fluid. Computations and experiments (e.g., [1]) cOTifirmed supersonic exit flow and have found efficiencies ranging from 10 % to 80 %, depending on the Reynolds number of the system, which in turn depends directly on the size of the throat and the fluid temperature and pressure as it passes through the nozzle, prior to supersonic expansion. The performance of the system can be improved by raising the temperature of the gas. Examples include using an electrical heater in the plenum upstream of the throat [2] and using a chemical reaction, such as combustion [3]. 

The flow through an orifice plate, a nozzle, or a valve is rednced from the ideal theoretical value hy a discharge coefficient ... 

To maximise rinsing efficiency, it is necessary to keep the cake flooded with rinse liquor, but not to add excess unless it flows through the cake, rather than over it. The ideal location for admitting the rinse, therefore, is at the junction between the wet and the dry beach. For optimum location of the rinse nozzle(s) it is necessary to have a good appreciation of the cake profile around the wet and dry beach junction. 

The sizing coefficient for steady-state conditions of an ideal gas through the nozzle without friction and heat exchange with the wall is written as (isentropic flow of an ideal gas)... 

As a consequence, the mass flow rate through a nozzle or even a safety valve can no longer be determined by simple relations for ideal gases. 

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