Lateral pressure ratio

The normal pressure p (mean value around the perimeter) is deemed to be related to the mean vertical stress q through a lateral pressure ratio K (Figure 3.4) as... 

The values of the wall friction coefficient /x and the lateral pressure ratio K may be measured in control tests on the particular solid being stored (see Chapter 1). 

The asymptotic value of pressure po is actually more robust than the pressure distribution according to Janssen, because it does not need the assumption of a lateral pressure ratio. At great depth, conditions are stable, and neither the mean vertical stress q nor the mean wall pressure p changes. The equilibrium of a simple slice then simply equates the weight of the slice to the support given by wall friction, which becomes (adopting r = pp). 

In the first use of Janssen s theory (Koenen 1895), it was assumed that the solid in a silo after filling was in a Rankine active state, giving a low value of lateral pressure ratio K, and leading to smaller pressures. However, after extensive damage to many silos, it was widely recognised by the 1960s that this was an underestimate of K. 

It is best to measure the lateral pressure ratio K directly (see Chapter 1), but it has long been common to estimate it from the measured angle of internal friction (pi. Accounting for the above effects, the European standard EN 1991-4 (2007) defines the filling value of Kt for design purposes as... 

Purpose Wall friction coefficient W Lateral pressure ratio (K) Angle of internal friction ([Pg.114]

Since the state was to pass from filling (close to an active stress state) to passive, the pressure just below an abrupt step is easily determined as the Janssen value multiplied by the ratio of passive to filling values of lateral pressure ratio K /Ki). The ratio of peak symmetrical discharge pressure to symmetrical filling pressure is a very widely used variable, and its origins can be seen here to have some foundation in mechanics. This ratio is so important in silo design that it is given a symbol and defined as... 

Figure 3.13 Consequences of a switch in lateral pressure ratio at different levels (the switch is taken to occur at depth Zs)-...

Figure 3.15 Effect on pressures of a smooth change in lateral 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. 

The [CBT]ig efficiency is replotted in Fig. 3.14, against (Tt,ITx) with pressure ratio as a parameter. There is an indication in Fig. 3.14 that there may be a limiting maximum temperature for the highest thermal efficiency, and this was observed earlier by Horlock et al. [8] and Guha [9]. It is argued by the latter and by Wilcock et al. [10] that this is a real gas effect not apparent in the a/s calculations such as those shown in Fig. 3.9. This point will be dealt with later in Chapter 4 while discussing the turbine cooling effects. 

The original solids-conveying model developed by Darnel and Mol [7] assumed that the pressure (or stress) in the solid bed is isotropic. This assumption was made to simplify the mathematics and because of the lack of stress data for solid bed compacts. Previous research, however, showed that stresses in solid compacts are not isotropic [8]. Anisotropic stresses can be represented by the lateral stress ratio. It is defined as the ratio of the compressive stress in the secondary direction to the compressive stress in the primary direction, as shown in Fig. 4.7 and Eq. 4.1. 

The lateral stress ratio depends on the resin type and shape, surface treatments such as additives, temperature, and pressure. The ratio is measured using a compaction cell [2], as shown in Fig. 4.8. This cell is very similar to one shown in Fig. 4.3 except the piston for the lateral stress ratio cell is octagonal in cross section and a pressure sensor is mounted in the cylinder wall. The stress ratio is calculated by dividing the pressure measured at the side of the cylinder by the calculated pressure in the axial direction at the height of the sensor. The calculation method can be found elsewhere [2j. The lateral stress ratio for select resins at 25°C and 2.5 MPa are provided in Table 4.1. 

The lateral stress ratio Is defined for particle systems that are consolidated under pressure, and thus the particles are essentially locked In place and unable to move relative to the neighboring particles. This type of consolidation occurs In the later... 

Here, is the lateral stress ratio, and oy are the stresses in the x and y directions, and ff, in the case of the single-screw extruder, is assumed to be the local downstream pressure P. Lateral stress ratios were discussed in Section 4.2. 

An LDPE resin was used for this study. The resin had a melt index of 2.0 dg/min (2.16 kg, 190 °C) and a solid density of 0.922 g/cmT The shear viscosity was reported previously [37] thermal properties are provided in Chapter 4 bulk density as a function of temperature and pressure is provided in Fig. 4.4 and the coefficients of dynamic friction are provided in Appendix A5. The lateral stress ratio was measured at 0.7 [38] using the device shown in Fig. 4.8. 

If the effect of mixture ratio upon the characteristic velocity and thrust coefficient is examined, additional information is gained over that obtained from consideration of the specific impulse alone. Since the characteristic velocity bears the same dependence upon combustion temperature and molecular weight as does the specific impulse, the optimum characteristic velocity and optimum specific impulse would be expected and are observed to occur at approximately the same mixture ratio, see figure V. A. 6. The failure of these two performance parameters, c andcF. to have maximum values at the same propellant mixture ratio is traceable to their differing dependencies upon the specific heat ratio. The same effect is reflected in the dependency of the thrust coefficient upon the mixture ratio. For optimum expansion and a fixed pressure ratio across the nozzle, the thrust coefficient depends only upon the specific heat ratio. The dependence of the specific heat upon the mixture ratio in turn results in a dependence of the thrust coefficient upon the mixture ratio. Since thrust coefficient increases with decreasing specific heat ratio for a fixed pressure ratio, the maximum thrust coefficient should occur at the mixture ratio of maximum specific heat. The specific heat of the products increases both with temperature and with complexity of the product species. The maximum specific heat is found near the stoichiometric mixture ratio. For equilibrium expansion the specific heat also includes the effect of exothermic recombinations. This later effect is a maximum at the condition of maximum dissociation in the chamber which similarly occurs at the maximum combustion temperature or near the stoichiometric mixture ratio. 

Phase coexistence in lipid bilayers may be an important physical property for membranes of cells. When two phases coexist in a bilayer, depending upon the relative mass fractions of the phases and the shapes of their domains, one of the phases is percolative (physically continuous) and the other is nonpercola-tive (physically discontinuous or dispersed as isolated domains). Changes in the physico-chemical properties of the membrane (lateral pressure, temperature, and chemical composition are the most relevant for biological membranes) result in interconversion between the two phases—one phase grows at the expense of the other. In phase-separated systems of this type, a critical mass ratio of phases called the percolation threshold, at which the previously continuous phase becomes discontinuous and the previously discontinuous phase becomes continuous, becomes... 

When the hole is created in the pipe surface, as is said, the pressure ratio to the ambient pressure below the hole, which is in the pipe, is more than the CPR in the later time steps. Therefore, the flow is checked in the hole location and the outflow of the leak location is calculated by the equation below [3] ... 

The gas turbine cycle pressure ratio corresponding to peak thermal efficiency remains nearly identical at around 2.0 for both the 850°C and 950°C core outlet temperature cycles. This is the basis for the baseline and growth cycles to employ a similar line of gas turbines, which is another topic to be discussed later in Section 4. 

This will apply to a high degree of accuracy to the flow of liquid through a nozzle. It will be shown later that it serves reasonably well also as an approximation for gas flow at low pressure ratios. 

There are two extreme states First, Qp = 0 the extruder is open, and there are no die or pressure drops. In this case Q = q. Second, Q = 0 the extruder is closed (and may be blocked). In this case qp = qj. This enables calculation of the ultimate pressure, as will be shown later. The ratio a = qjq, possesses an operational significance, and its boundaries are 0 < a < 1. 

Another possible source of nonideal behavior and large pressure fluctuations would be boundary layer separation caused by the interaction with the reflected shock wave. Boundary layer separation and bifurcated reflected shock waves are observed under certain conditions in shock tubes with nonreactive flows. Mark formulated a simple model that predicts the occurrence of bifurcation shock bifurcation and boundary layer separation will occur when the pressure jump across the reflected shock exceeds the maximum stagnation pressure possible in the cold boundary layer fluid. Numerical calculation for the present situation reveals bifurcation would not be expected when the detonation first reflects. This is a situation peculiar to detonations and is due to the much lower reflected-shock pressure ratio relative to that which would be produced by reflecting a shock wave of comparable strength. Consideration of the reflected shock motion at later times indicates that bifurcation would not occur until after the shock had reflected from the far end of the tube. 

---