Resistance to flow

The flowrate of oil into the wellbore is also influenced by the reservoir properties of permeability (k) and reservoir thickness (h), by the oil properties viscosity (p) and formation volume factor (BJ and by any change in the resistance to flow near the wellbore which is represented by the dimensionless term called skin (S). For semisteady state f/owbehaviour (when the effect of the producing well is seen at all boundaries of the reservoir) the radial inflow for oil into a vertical wellbore is represented by the equation ... 

A basic theme throughout this book is that the long-chain character of polymers is what makes them different from their low molecular weight counterparts. Although this notion was implied in several aspects of the discussion of the shear dependence of viscosity, it never emerged explicitly as a variable to be investi-tated. It makes sense to us intuitively that longer chains should experience higher resistance to flow. Our next task is to examine this expectation quantitatively, first from an empirical viewpoint and then in terms of a model for molecular motion. 

The term essentially a drag coefficient for the dust cake particles, should be a function of the median particle size and particle size distribution, the particle shape, and the packing density. Experimental data are the only reflable source for predicting cake resistance to flow. Bag filters are often selected for some desired maximum pressure drop (500—1750 Pa = 3.75-13 mm Hg) and the cleaning interval is then set to limit pressure drop to a chosen maximum value. 

The specific cake resistance is the most troublesome parameter ideally constant, its value is needed to calculate the resistance to flow when the amount of cake deposited on the filter is known. In practice, it depends on the approach velocity of the suspension, the degree of flow consoHdation that the cake undergoes with time, the feed soHds concentration, and, most importantly, the appHed pressure drop Ap. This changes due to the compressibiHty of most cakes in practice. often decreases with the velocity and the feed concentration. It may sometimes go through a maximum when it is plotted against soHds concentration. The strongest effect on is due to pressure, conventionally expressed as ... 

A key factor determining the performance of ultrafiltration membranes is concentration polarization due to macromolecules retained at the membrane surface. In ultrafiltration, both solvent and macromolecules are carried to the membrane surface by the solution permeating the membrane. Because only the solvent and small solutes permeate the membrane, macromolecular solutes accumulate at the membrane surface. The rate at which the rejected macromolecules can diffuse away from the membrane surface into the bulk solution is relatively low. This means that the concentration of macromolecules at the surface can increase to the point that a gel layer of rejected macromolecules forms on the membrane surface, becoming a secondary barrier to flow through the membrane. In most ultrafiltration appHcations this secondary barrier is the principal resistance to flow through the membrane and dominates the membrane performance. 

Solids nd Colloids. Suspended soHds can accumulate at the membrane surface, creating an additional resistance to flow through the membrane as well as a possible feed channel, such as that for a spiral-wound module plugging and subsequently a decrease in flux. Prevention of this type of fouling lies in the removal of the suspended soHds, which can be accompHshed using filters and screens prior to arrival at the RO unit. 

Dispersion of a soHd or Hquid in a Hquid affects the viscosity. In many cases Newtonian flow behavior is transformed into non-Newtonian flow behavior. Shear thinning results from the abiHty of the soHd particles or Hquid droplets to come together to form network stmctures when at rest or under low shear. With increasing shear the interlinked stmcture gradually breaks down, and the resistance to flow decreases. The viscosity of a dispersed system depends on hydrodynamic interactions between particles or droplets and the Hquid, particle—particle interactions (bumping), and interparticle attractions that promote the formation of aggregates, floes, and networks. 

Transport Properties. Viscosity, themial conductivity, the speed of sound, and various combinations of these with other properties are called steam transport properties, which are important in engineering calculations. The speed of sound (Fig. 6) is important to choking phenomena, where the flow of steam is no longer simply related to the difference in pressure. Thermal conductivity (Fig. 7) is important to the design of heat-transfer apparatus (see HeaT-EXCHANGETECHNOLOGy). The viscosity, ie, the resistance to flow under pressure, is shown in Figure 8. The sharp declines evident in each of these properties occur at the transition from Hquid to gas phase, ie, from water to steam. The surface tension between water and steam is shown in Figure 9. 

If expression is effective, it reduces the permeabiHty of the cake being compacted and, as a consequence, the resistance to flow of the Hquid increases considerably (27). The effectiveness of expression is governed by cake thickness, specific resistance, consoHdation properties, and shear forces. 

For a trumpet-shaped rounded entrance, with a radius of rounding greater than about 15 percent of the pipe diameter (Fig. 6-13Z ), the turbulent flow loss coefficient K is only about 0.1 (Vennard and Street, Elementary Fluid Meehanies, 5th ed., Wiley, New York, 1975, pp. 420-421). Rounding of the inlet prevents formation of the vena eontraeta, thereby reducing the resistance to flow. 

Friction Head This is the pressure required to overcome the resistance to flow in pipe and fittings. It is dealt with in detail in Sec. 5. 

There are many reasons for division into trays. The best known reason is to limit the temperature change by having a heat exchanger between trays. The other is to give an opportunity to inject a reactant, the concentration of which was limited by safety or selectivity reasons. The final reason is to compensate for uneven flow distribution, the result of uneven catalyst packing across a bed, which happens during catalyst charging. The channels of low resistance to flow have a tendency to extend themselves. 

Viscosity is a measure of the resistance to flow and is important in the design of fuel pumping systems. 

Permeability is the conductance of the medium and has direct relevance to Darcy s law. Permeability is related to the pore size distribution, since the distribution of the sizes of entrances, exits and lengths of the pore walls constitutes the primary resistance to flow. This parameter reflects the conductance of a given pore structure. 

In an oriented porous medium, the resistance to flow differs depending on the direction. Thus, if there is a pressure gradient between two points and a particular fluid particld is followed, unless the pressure gradient is parallel to oriented flow paths, the fluid particle will not travel from the original point to the point which one would expect. Instead, the particle will drift. 

Filtrate volume, Xq can be expressed in terms of the ratio of the mass of solid particles settled on the filter plate to the filtrate volume (Xw) and instead of r, a specific mass cake resistance r is used. That is, r represents the resistance to flow created by a uniformly distributed cake, in the amount of 1 kg/m. Replacing units of volume by mass, the term r, Xq in Equation 9 changes to r rX, . Neglecting filter plate resistance (Rf = 0), and taking into account Equation 7, we obtain from Equation 3 the following expression ... 

The resistance to flow method (K[ ), which has been employed by industry for years, has now been adopted by the ASME Code for rupture disc. Sizing is performed on a relief system basis and not by capacity of individual components. The key elements of tills metliod are ... 

The rupture disc is treated as another component in the relief system that provides some resistance to flow. 

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