Fluid flow critical pressure ratio

For compressible fluids one must be careful that when sonic or choking velocity is reached, further decreases in downstream pressure do not produce additional flow. This occurs at an upstream to downstream absolute pressure ratio of about 2 1. Critical flow due to sonic velocity has practically no application to liquids. The speed of sound in liquids is very liigh. See Sonic Velocity later in this chapter. 

It will be seen when the pressure ratio Pi/Pi is less than the critical value (wr — 0.607) the flow rate becomes independent of the downstream pressure P2. The fluid at the orifice is then flowing at the velocity of a small pressure wave and the velocity of the pressure wave relative to the orifice is zero. That is the upstream fluid cannot be influenced by the pressure in the downstream reservoir. Thus, the pressure falls to the critical value at the orifice, and further expansion to the downstream pressure takes place in the reservoir with the generation of a shock wave, as discussed in Section 4.6. 

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... 

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. 

In the above equation, is the critical velocity (m/s), K is the ratio of specific heats (Cp/C ) at inlet conditions, P is the pressure in the restriction at critical flow conditions (KPa, absolute - Note that this term is known as the critical flow pressure ), and p, is the density of the fluid at the critical flow temperature and pressure (kg/m ). 

The majority of fixed-bed adsorbers are cylindrical, vertical vessels. However, when large volumes of fluid are being treated by small inventories of adsorbent, tbe pressure drop may become excessive unless the bed height (or depth) is very small. If cylindrical vessels are needed for pressure or ease of fabrication, (hey heve been installed horizontally in such cases with vertical flows. Because of the low pressure drop and (he low ratio of bed height to the horizontal dimension, flow distribution is much more critical ihan in vertical vessels. 

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