Throat critical pressure ratio

We may assume that the nozzle will be designed to produce a supersonic discharge velocity, and that the design pressure ratio will coincide with the lower critical discharge pressure ratio for the design inlet conditions, i.e. (pi/por)lo = (Pimi2/Por)lo-The nozzle will be choked at this point, and this implies that the pressure at the throat will be at its critical value. The associated throat critical pressure ratio is given by equation (14.55) ... 

Critical Flow Nozzle For a given set of upstream conditions, the rate of discharge of a gas from a nozzle will increase for a decrease in the absolute pressure ratio po/pi until the linear velocity in the throat reaches that of sound in the gas at that location. The value of po/pi for which the acoustic velocity is just attained is called the critical pressure ratio r. The actual pressure in the throat will not fall below even if a much lower pressure exists downstream. 

Thus there is a critical pressure ratio beyond which the flow at the throat is always sonic. This is termed critical flow. 

In the flow of a gas through a nozzle, the pressure falls from its initial value Pi to a value P2 at some point along the nozzle at first the velocity rises more rapidly than the specific volume and therefore the area required for flow decreases. For low values of the pressure ratio P2/P1, however, the velocity changes much less rapidly than the specific volume so that the area for flow must increase again. The effective area for flow presented by the nozzle must therefore pass through a minimum. It is shown that this occurs if the pressure ratio P2/P1 is less than the critical pressure ratio (usually approximately 0.5) and that the velocity at the throat is then equal to the velocity of sound. For expansion... 

With a converging-diverging nozzle, the velocity increases beyond the sonic velocity only if the velocity at the throat is sonic and the pressure at the outlet is lower than the throat pressure. For a converging nozzle the rate of flow is independent of the downstream pressure, provided the critical pressure ratio is reached and the throat velocity is sonic. 

The value of w given by equation 4.43 is the critical pressure ratio wc given by equation 4.26a. Thus the velocity at the throat is equal to the sonic velocity. Alternatively, equation 4.42 may be put in terms of the flowrate (G/A2) as ... 

This is illustrated by conditions (a) to (c). In each case the pressure PE at the exit plane is equal to the back pressure PB. Flow is subsonic throughout the nozzle. This type of behaviour in which the flow rate increases as the back pressure is reduced (P0 held constant) continues until a critical value of the pressure ratio P/P0 is reached at the throat of the nozzle, condition (d). At the critical pressure ratio PJPo, the gas reaches the speed of sound at the throat. It will be shown that PJPo is a function of y only. 

Thus, the gas speed at the throat is less than the sonic speed there. The flow is subsonic throughout the nozzle. This result is to be expected because the pressure ratio is 0.7 and the critical pressure ratio (for y = 1.39) is 0.53. 

Figure 6-23 shows a converging/diverging nozzle. When p2/p0 is less than the critical pressure ratio (p a/p ), the flow will be subsonic in the converging portion of the nozzle, sonic at the throat, and supersonic in the diverging portion. At the throat, where the flow is critical and the velocity is sonic, the area is denoted A. The cross-sectional... 

On inverting equation set (9.23) and introducing the critical pressure ratio limit in the valve throat, Prc(y)/Pi. the corresponding equations for the throat pressure ratio in terms of the valve pressure ratio when the latter is low become ... 

The full set of the above equations cea.ses to be valid when the flow in the valve goes sonic, since at this point there is a physical decoupling of the conditions upstream and downstream of the valve. In particular, the relationship between the ratios of throat and valve outlet pressures to valve inlet pressure given in equation (10.40) will hold no longer. Hence it will be necessary to detect the onset of sonic flow, which will occur when p,/p,., as calculated from equation (10.39), is equal to the critical pressure ratio, rc = (2/(y+ ))> / > ". We may then use the procedure outlined in Section 10.2.3. to calculate flow and conditions when the flow in the valve has become sonic. This is exactly the same procedure as... 

The mass flow rate is obtained by a progressive variation of the state conditions in the nozzle throat at a constant stagnation enthalpy - or approximately in the case of isentropic flow (no heat transfer, = 0) - until the back pressure or a maximum of the mass flow rate (critical pressure ratio) is reached. 

The fluid dynamic critical pressure and the critical pressure ratio r) = Pait/po in a nozzle throat may be calculated by differentiating the sizing coefficient, Eq. (15.21), with respect to the pressure ratio and setting the result equal to zero. Enuther mathematical transformations lead to the critical pressure ratio of a real gas ... 

From Equation 2.12, it is clear that for a given set of conditions, the flow through the orifice/Venturi will increase for a decrease in absolute pressure ratio P2/P1, xmtil a linear velocity in the throat reaches the velocity of soimd. The value of P2/P1 for which the acoustic flow is just attained is called the critical pressure ratio r, and flow at such a condition is called critical or choked flow. Under choked flow situations, the flow through the orifice depends only on the upstream pressure. 

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