The lateral pressure ratio

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. 

The value for Kq has long been approximately related to the angle of internal friction (pi of the solid (Jaky 1948) as 

The background to this equation may be read in Muir Wood (1990). 

The ideal Kq relates to conditions in which the vertical and horizontal stresses are principal stresses and both uniform. Since the state of the silo after filling has both a non-uniform vertical stress pattern and shear stresses against the wall, it is best here to assign the value K for the filling state, noting that K( K, but Kq. 

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 

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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 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. 

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 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). 

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)-...

The oil and gas samples are taken from the appropriate flowlines of the same separator, whose pressure, temperature and flowrate must be carefully recorded to allow the recombination ratios to be calculated. In addition the pressure and temperature of the stock tank must be recorded to be able to later calculate the shrinkage of oil from the point at which it is sampled and the stock tank. The oil and gas samples are sent separately to the laboratory where they are recombined before PVT analysis is performed. A quality check on the sampling technique is that the bubble point of the recombined sample at the temperature of the separator from which the samples were taken should be equal to the separator pressure. 

In a later, but less detailed analysis of flow in the intermediate pressure range, Hiby and Pahl [37] suggested that the minimum should be absent when the length/diameter ratio of a capillary is less than about sixteen. Since it is likely that this is the case for the channels in most porous media of... 

Another commonly used elastic constant is the Poisson s ratio V, which relates the lateral contraction to longitudinal extension in uniaxial tension. Typical Poisson s ratios are also given in Table 1. Other less commonly used elastic moduH include the shear modulus G, which describes the amount of strain induced by a shear stress, and the bulk modulus K, which is a proportionaHty constant between hydrostatic pressure and the negative of the volume... 

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. 

In practice the clamping pressure will also depend on the geometry of the cavity. In particular the flow ratio (flow length/channel lateral dimension) is important. Fig. 4.42 illustrates typical variations in the Mean Effective Pressure in the cavity for different thicknesses and flow ratios. The data used here is typical for easy flow materials such as polyethylene, polypropylene and polystyrene. To calculate the clamp force, simply multiply the appropriate Mean Effective Pressure by the projected area of the moulding. In practice it is... 

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. 

Acid anhydrides have been employed with, and without the use of a base catalyst. For example, acetates, propionates, butyrates, and their mixed esters, DS of 1 to ca. 3, have been obtained by reaction of activated cellulose with the corresponding anhydride, or two anhydrides, starting with the one with the smaller volume. In all cases, the distribution of both ester groups was almost statistic. Activation has been carried out by partial solvent distillation, and later by heat activation, under reduced pressure, of the native cellulose (bagasse, sisal), or the mercerized one (cotton linters). No catalyst has been employed the anhydride/AGU ratio was stoichiometric for microcrystalhne cellulose. Alternatively, 50% excess of anhydride (relative to targeted DS) has been employed for fibrous celluloses. In all cases, polymer degradation was minimum, and functionalization occurs preferentially at Ce ( C NMR spectroscopic analysis [52,56,57]). 

Although the above demonstrated that product control could be achieved in scC02, the difference in selectivity was relatively small. However, later work using a Lewis acid catalyst, scandium triflate, on the Diels-Alder reaction of n-butyl acrylate and cyclopentadiene (Scheme 7.7) showed that the endo exo ratio was again found to rise to a maximum and then decrease again as the pressure, and hence density, was increased (Figure 7.3) [19]. 

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