Critical mass flow

This method employs a theoretical critical mass flow based on an ideal nozzle and isothermal flow condition. For a pure gas, the mass flow can be determined from one equation ... 

The actual mass flow rate through a pipe, G, in lbs per sec per sq ft is a function of critical mass flow G,-i, line resistance, N, and the ratio of downstream to upstream pressure. These relationships are plotted in Figure 19. In the area below the dashed line in Figure 19, the ratio of G to G,. remains constant, which indicates that sonic flow has been established. Thus, in sizing flare headers the plotted point must be above the dashed line. The line resistance, N, is given by the equation ... 

In the other case the vapour mass fiaction at the relief device entrance must be estimated taking into account the flushing flow hypothesis. This allows the calculation of the so-called entropy parameter ro, which is required for the determination of the critical mass flow density G. ... 

From this diagram it can be seen that 5% vapour mass fraction already correspond to 65% of volumetric vapour fraction on average. Actually this value of 5 % may be regarded as a good first estimate if a rigorous design is acceptable. For such cases, with a low vapour mass fractions the critical mass flow density may be approximated with the help of the following equation. 

This critical mass flow density G has to be corrected with a flow correction value a, as was the case for single phase flow. The model uncertainties, which are still inherent to all two-phase flow models, are arbitrarily compensated by this parameter. Therefore a range for a of... 

The critical mass flow density is referenced to the cross-sectional area of the pipe in this case. In order to be able to calculate the friction ctor f for the pipe, the Rey-nolds-number must be known. For the special case of a homogeneous two-phase pipe flow, this can be calculated according to ... 

Where volumetric accuracy and repeatability are critical, mass flow calibration can be advantageous. 

Otherwise there is a limitation on flow ( choked flow ) and we have for the critical mass flow rate... 

To determine the maximum possible flow through an actuated pressure relief device, use of the phenomenon that a maximum mass flux exists which only depends on the state in the vessel (see Figure 14.4) is made. For a given pressure Po in the vessel, the mass flow is determined by the size of the narrowest cross-flow area A] in the line. When the outlet pressure Pi is decreased, the mass flow increases. However, if Pi falls below a certain value, no further increase of the mass flow takes place (Figure 14.5). The maximum mass flow is called the critical mass flow merit ... 

To compare performance between vanes and screen channels, expulsion efficiency and resultant PMD mass were determined for both PMDs across a wide range of demand flow rates. Based on the maximum achievable expulsion efficiency and resultant PMD mass, the critical mass flow rate range for the small-scale LH2 tank was determined to be 4 X 10 -1.4 X 10 kg/s, below which vanes are the optimal PMD, above which screen channels are the optimal choice. Finally, comparison of PMD mass shows that, depending on the desire to minimize PMD mass or maximize expulsion efficiency, vanes may be used to supply flow rates as high as 1.4 x 10 kg/s for a small-scale LH2 tank in a continuous outflow. The analysis also showed that there exists a limit on the mass flow rate that a single vane can deliver due to choking. The analytical models and rationale developed here can be used to size and trade PMD performance for any sized propellant tank in any gravitational or thermal environment. 

Because mass flow bins have stable flow patterns that mimic the shape of the bin, permeabihty values can be used to calculate critical, steady-state discharge rates from mass flow hoppers. Permeabihty values can also be used to calculate the time required for fine powders to settle in bins and silos. In general, permeabihty is affected by particle size and shape, ie, permeabihty decreases as particle size decreases and the better the fit between individual particles, the lower the permeabihty moisture content, ie, as moisture content increases, many materials tend to agglomerate which increases permeabihty and temperature, ie, because the permeabihty factor, K, is inversely proportional to the viscosity of the air or gas in the void spaces, heating causes the gas to become more viscous, making the sohd less permeable. 

To be consistent with a mass flow pattern in the bin above it, a feeder must be designed to maintain uniform flow across the entire cross-sectional area of the hopper outlet. In addition, the loads appHed to a feeder by the bulk soHd must be minimised. Accuracy and control over discharge rate ate critical as well. Knowledge of the bulk soHd s flow properties is essential. 

Testers are available to measure the permeabihty and compressibiUty of powders and other bulk soflds (6). Erom such tests critical, steady-state flow rates through various outlet sizes in mass flow bins can be calculated. With this information, an engineer can determine the need for changing the outlet size and/or installing an air permeation system to increase the flow rate. Furthermore, the optimum number and location of air permeation levels can be deterrnined, along with an estimate of air flow requirements. 

The area of influence of a vibrating discharger is limited to a cylinder, the diameter of which is roughly equal to the top diameter of the discharger. Hence, if a vibrating discharger is mounted onto a conical hopper section, flow is confined predominantly to a central flow channel located directly above the discharger. This is tme unless the slope and smoothness of the static cone meet requirements for mass flow, or the diameter of the flow channel exceeds the critical rathole diameter for the material. 

As normally designed, vapor flow through a typical high-lift safety reliefs valve is characterized by limiting sonic velocity and critical flow pressure conditions at the orifice (nozzle throat), and for a given orifice size and gas composition, mass flow is directly proportional to the absolute upstream pressure. 

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). 

For rupture disks, pressure ratio is greater than critical pressure mass flow pounds/hr... 

Wmax = maximum mass flow at critical or choked conditions, lb/sec... 

The effect on the propellant burning rate of gas flow parallel to the propellant surface has received considerable experimental and theoretical attention. These studies have generally shown that there is little effect of gas flow parallel to the propellant surface, provided the flow rate is below a certain critical level. However, once this critical value has been exceeded, the burning rate increases with increasing mass flow rate. 

A third approach has been suggested by Jaroudi (Jl), who points out that one necessary condition to prevent reignition of the propellant is to ensure that the gas temperature resulting from thermal equilibrium between the injected fluid and the combustion products is below the propellant autoignition temperature. This approach leads to the conclusion that the ratio of coolant mass flow to propellant mass flow is the critical correlating parameter. 

The phenomenon of critical flow is well known for the case of single-phase compressible flow through nozzles or orifices. When the differential pressure over the restriction is increased beyond a certain critical value, the mass flow rate ceases to increase. At that point it has reached its maximum possible value, called the critical flow rate, and the flow is characterized by the attainment of the critical state of the fluid at the throat of the restriction. This state is readily calculable for an isen-tropic expansion from gas dynamics. Since a two-phase gas-liquid mixture is a compressible fluid, a similar phenomenon may be expected to occur for such flows. In fact, two-phase critical flows have been observed, but they are more complicated than single-phase flows because of the liquid flashing as the pressure decreases along the flow path. The phase change may cause the flow pattern transition, and departure from phase equilibrium can be anticipated when the expansion is rapid. Interest in critical two-phase flow arises from the importance of predicting dis-... 

Figure 6.1 Critical heat flux versus mass flow rate for constant pressure drop. (From Maulbetsch and Griffith, 1965. Reprinted with permission of Massachusetts Institute of Technology, Cambridge, MA.)... 

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