Critical pressure ratio

The velocity in the narrowest cross section in a safety valve may reach the sound velocity at maximum. Simultaneously, the pressure decreases to its minimum value, the fluid dynamic critical pressure, and the mass flow rate is maximized. 

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  

An estimation of the critical pressure ratio is necessary to calculate the average values of (IT) and (k). The critical pressure ratio of an ideal gas is a good first estimate for the 

The critical pressure ratio i] for calculating the mass flow rate, cf. Eqs. (15.2) and (15.3) (ideal gas) and Eqs. (15.2) and (15.21) (real gas), corresponds to the maximum of either the back pressure ratio t] = r] = Pi,/po or the critical pressure ratio r] = = Pcrit/Po- 

For most gases in practice, the specific heat capacity Cp is not available depending on temperature and pressure. But the specific heat capacity for the ideal gas is known -it depends only on temperature. Hence, the integration of Eq. (15.12) should follow the following integration path [14, chapter 5]  

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Note that under choked conditions, the exit velocity is V = V = c = V/cKTVM not V/cKT(/M, . Sonic velocity must be evaluated at the exit temperature. For air, with k = 1.4, the critical pressure ratio p /vo is 0.5285 and the critical temperature ratio T /Tq = 0.8333. Thus, for air discharging from 300 K, the temperature drops by 50 K (90 R). This large temperature decrease results from the conversion of internal energy into kinetic energy and is reversible. As the discharged jet decelerates in the external stagant gas, it recovers its initial enthalpy. 

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. 

The critical pressure ratio r can be obtained from the following theoretical equation, which assumes a perfect gas and a frictionless nozzle ... 

To determine the critical pressure ratio for gas sonic velocity across a nozzle or orifice use... 

Kn, = Valve recovery coefficient (see Table 3) r, = Critical pressure ratio (see Figures 1 and 2)... 

Critical Pressure Ratios For Liquids Other Than Water... 

Figure 2. Determine the vapor pressure/critical pressure ratio by dividing the liquid vapor pressure at the valve inlet by the critical pressure of the liquid. Enter on the abscissa at the ratio just calculated and proceed vertically to intersect the curve. Move horizontally to the left and read r< on the ordinate (Reference 1).

For gases with specific heat ratios of approximately 1.4, the critical pressure ratio is approximately 0.5. For hydrocarbon service, this means that if the back-pressure on the relief valve is greater than 50% of the set pressure, then the capacity of the valve will be reduced. In other words, if the pressure in the relief piping at the valve outlet is greater than half (he set pressure, then a larger relief valve will be required to handle the same amount of fluid. 

As long as the pressure ratio exceeds the critical-pressure ratio, the throughput will vary with the inlet pressure and be independent of outlet pressure. For example, a relief valve set at 100 psi will have the same gas flow through it as long as the back-pressure is less than approximately 50 psi. 

There are two flow regimes corresponding to sonic (or choked) flow for liigher pressure drops and subsonic flow for lower pressure drops. The transition between the two flow regimes occurs at tlie dimensionless critical pressure ratio, Ter,I, which is related to tlie gas lieiit capacity ratio y via... 

These conditions are similar to flow through orifices, nozzles, and venturi tubes. Flow through nozzles and venturi devices is limited by the critical pressure ratio, r,. = downstream pressure/upstream pressure at sonic conditions (see Figure 2-38C). 

For nozzles and venturi meters, the flow is limited by critical pressure ratio and the minimum value of Y to be used. 

Figure 2-38C. Critical Pressure Ratio, r, for compressible flow through nozzles and venturi tubes. By permission, Crane Co., Technical Paper 410, 1957. Also see 1976 edition. See note at Figure 2-18 explaining details of data source for chart. Note P = psia p= ratio of small-to-large diameter in orifices and nozzles, and contractions or enlargements in pipes.

If the ratio of backpressure to inlet pressure to valve exceeds the critical pressure ratio, Pc/Pj,... 

Vhen Pj is increased, the flow through an open disk increases and the pressure ratio, P2/P1, decreases when P2 does not change, until a value of Pj is reached, and there is no further increase in mass flow through the disk. The value of Pi becomes equal to P, and the ratio is the critical pressure ratio, and the flow velocity is sonic (equals the speed of sound). 

Actual pressure ratio, P /Pi, less than critical pressure ratio, flow is sonic or critical. 

Important note when actual system ratio, P9/P1, is less than critical pressure ratio calculated above by Equation 7-40, flow is sonic. Wlien actual P2/P1 ratio is greater than critical pressure ratio, flow is subsonic. 

Steam Rupture disk sonic flow ailical pressure = 0.55 and P2/P1 is less than critical pressure ratio of 0.55. 

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

R = temperature, absolute, degrees Rankin r = rc = ratio of back pressure to upstream pressure, P2/Pi, or critical pressure ratio, Pc/Pi rj = relative humidity, percent S = maximum allowable stress in vessel wall, from ASME Code, psi., UCS-23.1-23.5 UHA-23, UHT-23... 

The critical pressure ratio for discharge through the valve... 

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