Maximum flow and critical pressure ratio

In the flow of a gas through a nozzle, the pressure falls from its initial value F 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 F2/F1, 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 F2/F1 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 

Case I. Back-pressure Pg quite high. Curves I show how pressure and velocity change along the nozzle. The pressure falls to a minimum at the throat and then rises to a value Pe = Pg. The velocity increases to maximum at the throat (less than sonic velocity) and then decreases to a value of ue at the exit of the 

Case II. Back-pressure reduced (curves II). The pressure falls to the critical value at the throat where the velocity is sonic. The pressure then rises to P 2 = the exit. The velocity rises to the sonic value at the throat and then falls to ue2 at the outlet. 

Case III. Back-pressure low, with pressure less than critical value at the exit. The pressure falls to the critical value at the throat and continues to fall to give an exit pressure Pez = Pb- The velocity increases to sonic at the throat and continues to increase to supersonic in the diverging cone to a value ue3- 

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. 

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It should be noted that equations 6.16 and 6.17 apply provided that P2/P1 is greater than the critical pressure ratio wc. This subject is discussed in Chapter 4, where it is shown that when Pi/P < wc, the flowrate becomes independent of the downstream pressure P2 and conditions of maximum flow occur. 

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,... 

In an emergency relief situation releasing compressible media, critical and sub-critical pressure and flow conditions must be discriminated. The gas flow rate is limited to a maximum value of sonic velocity. This flow is called critical. In order to insert the correct flow function into Equ.(7-2/3), the critical pressure ratio between internal pressure, existing in the vessel to be vented, and the counter pressure, which in most cases is equivalent to ambient pressure, has to be evaluated ... 

The maximum velocity of flow that can be attained is the speed of sound. It is reached for the critical pressure ratio Wi ji, and true as well if w < Wknt- The flow rate then only depends on the pressure inside the enclosure, pi, and not on the external pressure, p2. Equation (7.20) hence applies for discharge processes with velocities below the speed of sound and Eq. (7.23) for all other cases. The latter is usually required if gas under high pressure is discharged. 

Since the maximum fluid velocity obtainable in a converging nozzle is speed of sound, a nozzle of this kind can deliver a constant flow rate into a regi of variable pressure. Suppose a compressible fluid enters a converging nozzle pressure Pi and discharges from the nozzle into a chamber of variable press P2. If this discharge pressure is P)t the flow is zero. As P2 decreases below the flow rate and velocity increase. Ultimately, the pressure ratio P2/Pi reach a critical value at which the velocity in the throat is sonic. Further reduction i P2 has no effect on the conditions in the nozzle. The flow remains constant, ah the velocity in the throat is that given by Eq. (7.21), regardless of the value P2/P , provided it is always less than the critical value. For steam, the criti value of this ratio is about 0.55 at moderate temperatures and pressures. 

Following the procedure of Section 5.3, we may differentiate equation (9.3) with respect to throat pressure ratio and set the result to zero to show that the maximum flow occurs when the throat pressure ratio has decreased to a critical value given by ... 

While process design and equipment specification are usually performed prior to the implementation of the process, optimization of operating conditions is carried out monthly, weekly, daily, hourly, or even eveiy minute. Optimization of plant operations determines the set points for each unit at the temperatures, pressures, and flow rates that are the best in some sense. For example, the selection of the percentage of excess air in a process heater is quite critical and involves a balance on the fuel-air ratio to assure complete combustion and at the same time make the maximum use of the Heating potential of the fuel. Typical day-to-day optimization in a plant minimizes steam consumption or cooling water consumption, optimizes the reflux ratio in a distillation column, or allocates raw materials on an economic basis [Latour, Hydro Proc., 58(6), 73, 1979, and Hydro. Proc., 58(7), 219, 1979]. 

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