Choking coefficient

When choking occurs in the valve, the upstream and downstream pressures are related by a condition very similar to the equation pair (7.11) for cavitation, but now using equation (7.16) for the throat pressure, rather than (7.IS), and replacing the cavitation coefficient, Kc, by a new choking coefficient, K ... 

K is sometimes referred to as simply the pressure recovery coefficient . The two coefficients are identical for globe valves K = Kc, and often only K Is quoted. The choking coefficient is somewhat larger than the cavitation coefficient for a rotary valve, with the full-open value, Ku, taking a value of typically... 

Discharge Coefficients and Gas Discharge A compressible fluid, upon discharge from an orifice, accelerates from the puncture point and the cross-sec tional area contracts until it forms a minimum at the vena contracta, If flow is choked, the mass flux G, can be found at the vena contrac ta, since it is a maximum at that point, The mass flux at the orifice is related to the mass flux at the vena contracta by the discharge coefficient, which is the area contraction ratio (A at the vena contracta to Ay at the orifice) ... 

For critical flow the discharge coefficient is dependent upon the geometry of the choke and its diameter or the ratio P of its diameter to that of the upstream pipe (see Figure 2-24). 

In subcritical flow the discharge coefficient is affected by the velocity of approach as well as the type of choke and the ratio of choke diameter to pipe diameter. Discharge coefficients for subcritical flow are given in Figure 2-24 as a function of the diameter ratio and the upstream Reynolds number. Since the flow rate is not initially known, it is expedient to assume C = 1, calculate Q, use this Q to calculate the Reynold s number, and then use the charts to find a better value of C. This cycle should be repeated until the value of C no longer changes. 

Now it appears that the value of may be estimated by using the loss coefficient K determined at choking provided K is not too small. This is unlikely since in most valves effective flow control occurs at very small throat area when the valve is in the 10-30% open range. The loss coefficient is determined from the pressure loss across the valve and the velocity in the upstream pipe at choking. 

Just as for isothermal flow, this is an implicit expression for the choke pressure (P ) as a function of the upstream pressure (Pi), the loss coefficients (J] Kf), and the isentropic exponent (7c), which is most easily solved by iteration. It is very important to realize that once the pressure at the end of the pipe falls to P and choked flow occurs, all of the conditions within the pipe (G = G, P2 = P, etc.) will remain the same regardless of how low the pressure outside the end of the pipe falls. The pressure drop within the pipe (which determines the flow rate) is always Pt — P when the flow is choked. 

Typical manufacturer s values of Cv to be used with Eq. (10-29) require the variables to be expressed in the above units, with hv in ft. [For liquids, the value of 0.658 includes the value of the density of water, pw = 62.3 lbm/ft3, the ratio g/gc (which has a magnitude of 1), and 144 (in./ft)2]. For each valve design, tables for the values of the flow coefficients as a function of valve size and percent of valve opening are provided by the manufacturer (see Table 10-3, pages 318-319). In Table 10-3, Km applies to cavitating and flashing liquids and C applies to critical (choked) compressible flow, as discussed later. 

The flow coefficient Cv is determined by calibration with water, and it is not entirely satisfactory for predicting the flow rate of compressible fluids under choked flow conditions. This has to do with the fact that different valves exhibit different pressure recovery characteristics with gases and hence will choke at different pressure ratios, which does not apply to liquids. For this reason, another flow coefficient, Cg, is often used for gases. Cg is determined by calibration with air under critical flow conditions (Fisher Controls, 1977). The corresponding flow equation for gas flow is... 

For sharp-edged orifices with Reynolds numbers greater than 30,000 (and not choked), a constant discharge coefficient C0 of 0.61 is indicated. However, for choked flows the discharge coefficient increases as the downstream pressure decreases.9 For these flows and for situations where C0 is uncertain, a conservative value of 1.0 is recommended. 

For vapor flows that are not choked by sonic flow the area is determined using Equation 4-48. The downstream pressure P is now required, and the discharge coefficient C0 must be estimated. The API Pressure Vessel Code4 provides working equations that are equivalent to Equation 4-48. 

Search with values of Pi until G is maximized. The choke pressure, PcU is the value of Pi that produces a maximum value of mass flux Gmax. The discharge rate w is given from the mass flux, a discharge coefficient CD, and the orifice cross-sectional area A as... 

The capacity of the relief system can be obtained from a two-phase flow calculation for nozzle flow. If the flow is not choked, then the Omega method (see Annex 8) or suitable computer code must be used to calculate flow capacity. For choked flow a larger range of methods may be applicable, e.g. ERM for vapour pressure systems (see 9.4.2) or Tangren et al. s method for gassy systems (see 9.4.3), together with the application of a discharge coefficient. The capacity can then be obtained from ... 

Suitable values of discharge coefficient for different situations are given in BS 2915[13] for single phase liquid or gas flow. CCPS1141 indicate that a gas discharge coefficient should be used for two-phase flow provided the flow chokes, otherwise a liquid discharge coefficient should be used. 

In the absence of a discharge coefficient, the most accurate way of estimating a flow reduction factor is to use the Omega method (see Annex 8). Alternatively, a discharge coefficient can be estimated from the following equation which applies for single-phase non-choked flow ... 

CHOKED FLOW DATA FOR GASES AS A FUNCTION OF ISENTROPIC COEFFICIENT... 

Because of the AP, the flow is not choked, but subsonic or subcritical. This obviously has an effect on the sizing of the valve (coefficient). Subsonic can occur at 25% to 30% backpressure Always check first ... 

For two-phase flow through pipes, an overall dimensionless discharge coefficient, /, is applied. Equation 12-11 is referred to as the equilibrium rate model (ERM) for low-quality choked flow. Leung [28] indicated that Equation 12-11 be multiplied by a factor of 0.9 to bring the value in line with the classic homogeneous equilibrium model (HEM). Equation 12-11 then becomes... 

One would think a solution to increase the productivity would be to use oxygen enrichment, not just during the choke-out period but also during the remainder of the fermentation in order to sustain the productivity. This does not work because of broth viscosity and gas holdup problems. In highly mycelial systems, a 15% increase in cell mass doubles the viscosity. The volumetric oxygen mass transfer coefficient and the bubble rise velocity—... 

A graph of discharge coefficient ( m) plotted against dimensionless lift L/D) is known as a flow characteristic. Fig. 9 shows an example of flow characteristics plotted for various pressure ratios relieving from an air receiver, where Pb is the total backpressure at the valve exit port and Pq the reservoir total pressure.According to ideal one-dimensional compressible flow theory, choked flow should occur at pressure ratios below 0.528. In practice, choked flow will occur at lower pressure ratios because of friction losses upstream of the curtain area. Nevertheless, the curves in Fig. 9 are close-together at pressure ratios below 0.333, and have collapsed to almost a single curve for ratios less than 0.25. Other work shows a similar effect. 

Typical flow coefficient values are shown in Table 5.14, in which K , applies to cavitating and flashing liquids and Ci applies to critical (choked) compressible flow, as discussed below. 

Figure 8.3 Showing transients for valve travel, x, valve opening, /, and choking and cavitation coefficients Km and Kc.

Figure 9.3 Valve pressure ratio at the onset of choking versus squared friction coefficient comparison of equation (9.21) with Fisher direct data.

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