Flow rate superposition

There is an additional pressure drop across the cake, developed by electroosmosis, which leads to increased flow rates through the cake and further dewatering at the end of the filtration cycle. The filtration theory proposed for electrofiltration assumes the simple superposition of electroosmotic pressure on the hydraulic pressure drop. 

All of the data in Fig 7.5 were analyzed using linear regression. The summation of the helix and core-regressed flow rates are plotted in Fig. 7.5 as the red dotted line. The experimental superposition for the flows induced by the screw elements essentially overlaid the regression line for the screw configuration rate. Thus, it was concluded that the helix is the pump in the single-screw extruder, and core rotation reduces the flow by dragging the fluid back toward the extruder inlet. 

Equation (6.1.4) asserts that the volumetric flow rate is a superposition of two components. They are the electro-osmotic component proportional to the electric field intensity (voltage) with the proportionality factor u> and the filtrational Darcy s component proportional to —P with the hydraulic permeability factor i>. Teorell assumed both w and t> constant. Finally another equation, crucial for Teorell s model, was postulated for the dynamics of instantaneous electric resistance of the filter R(t). Teorell assumed a relaxation law of the type... 

Equation E2.5-9 further indicates that, in the absence of a pressure drop, the net flow rate equals the drag flow rate. Note that qp is positive if Pq > PL and pressure flow is in the positive z direction and negative when Pp > Po- The net flow rate is the sum or linear superposition of the flow induced by the drag exerted by the moving plate and that caused by the pressure gradient. This is the direct result of the linear Newtonian nature of the fluid, which yields a linear ordinary differential equation. For a non- Newtonian fluid, as we will see in Chapter 3, this will not be the case, because viscosity depends on shear rate and varies from point to point in the flow field. 

Equations 6.3-19 is the well-known isothermal Newtonian extrusion theory. Since it was obtained by the solution of a linear differential equation, it is composed of two independent terms, the first representing the contribution of drag flow Q,j, and the second, the pressure flow, Qp. The net flow rate is the linear superposition of the two. 

In Eq. 6.6-16 the first term on the right-hand side is the drag flow and the second term is the pressure flow. The net flow rate is their linear superposition, as in the case of the Newtonian model in single screw extrusion. The reason that in this case this is valid for non-Newtonian flow as well is because the drag flow is simply plug flow. 

The Superposition Correction Factor Combined drag and pressure flow between parallel plates (or concentric cylinders17) of a Newtonian fluid at isothermal conditions leads to a flow-rate expression that is the linear sum of two independent terms, one for drag flow and another for pressure flow ... 

The mass-transfer term is proportional to the flow rate, which means that the faster the flow, the greater the band-broadening. Superposition of all three processes is shown schematically in Figure 2-3. 

Similarly, we apply the superposition principle to the species mass flow rates to get... 

Since these experiments were not carried out under ideally defined flow conditions the dependence of corrosion rate on flow rate will be discussed only in a qualitative manner. Under laminar flow conditions and mass transfer control one would have expected the corrosion rate to increase with the square root of the velocity while under turbulent conditions proportionality would prevail. However, in Fig.15 one finds that the corrosion rate varies approximately with the 0.2 to 0.3 power of the flow rate. It appears therefore that the observed dependence on the flow rate does not obey conventional mass transfer theory. A flow effect might be expected in uninhibited hydrochloric acid because hydrogen bubbles, formed on the surface of the metal, are faster and more easily removed at higher flow rates. While this argument could be applied in discussing Fig.15, we find in Fig.16 that the flow effect at similar corrosion rates is much less pronounced under deaerated conditions. We therefore have to conclude that the observed flow effect is not mechanical and cannot be related to pure mass transfer control either. In Fig.17, the flow dependence of the corrosion rate is shown for 2-butyne-l,4-diol in deaerated UN hydrochloric acid. Note that the corrosion rate appears to be noticeably affected only at the higher flow rates. Finally, in Fig.18, we observe that increased flow rate can either increase or decrease the corrosion rate in the presence of an inhibitor. This effect was observed reproducibly only in 6N hydrochloric acid with 2-butyne-l,U-diol under deaerated conditions for 0.2% and 0.1% inhibitor concentration. This behavior indicates that the corrosion rate is controlled by the superposition of two partial reaction rates each of which is mass transfer dependent to a certain extent. In terms of the model delineated in Table 6, it is suggested that the three-dimensional polymeric layer made up by inhibitor molecules is in fact a three-dimensional chelate made up of iron ions and inhibitor molecules. The corrosion rate is then... 

For elastic solids Hooke s law is valid only at small strains, and Newton s law of viscosity is restricted to relatively low flow rates, as only when the stress is proportional either to the strain or the strain rate is analysis of the deformation feasible in simple form. A comparable limitation holds for viscoelastic materials general quantitative predictions are possible only in the case of linear viscoelasticity, for which the results of changing stresses or strains are simply additive, but the time at which the change is made must be taken into account. For a single loading process there will be a linear relation between stress and strain at a given time. Multistep loading can be analysed in terms of the Boltzmann superposition principle (Section 4.2.1) because each increment of stress can be assumed to make an independent contribution to the overall strain. 

Nonlinear superposition. Very often in pressure transient testing, the pressure (or flow rate) is changed in time for liquids, flow rate (or pressure) response is obtained by linear superposition of elementary solutions. For gases, superposition is not possible because nonlinear solutions are not linearly additive. How does one calculate the response when pressure or flow rate at the well vary, say stepwise, in time Fortunately, the governing equations can be numerically integrated with respect to t. It remains for us to represent stepwise changes in any particular variable using convenient mathematical devices. 

Figure 22. Typical power-time curve for the action of cadmium intoxication on a freshwater snail Planorbis comeus). At the arrow, contaminated water is introduced to the sorption chamber at a higher flow rate to guarantee for a quick and effective water exchange. The bar near the ordinate indicates the standard deviation for a superposition of 3 individual traces [182].

Whilst the flow curves of materials have received widespread consideration, with the development of many models, the same cannot be said of the temporal changes seen with constant shear rate or stress. Moreover we could argue that after the apparent complexity of linear viscoeleastic systems the non-linear models developed above are very poor cousins. However, it is possible to introduce a little more phenomenological rigour by starting with the Boltzmann superposition integral given in Chapter 4, Equation (4.60). This represents the stress at time t for an applied strain history ... 

Viscoelastic behavior is classified as linear or non-linear according to the manner by which the stress depends upon the imposed deformation history (SO). Insteady shear flows, for example, the shear rate dependence of viscosity and the normal stress functions are non-linear properties. Linear viscoelastic behavior is obtained for simple fluids if the deformation is sufficiently small for all past times (infinitesimal deformations) or if it is imposed sufficiently slowly (infinitesimal rate of deformation) (80,83). In shear flow under these circumstances, the normal stress differences are small compared to the shear stress, and the expression for the shear stress reduces to a statement of the Boltzmann superposition principle (15,81) ... 

The maximum strain rate (e < Is1) for either extensional rheometer is often very slow compared with those of fabrication. Fortunately, time-temperature superposition approaches work well for SAN copolymers, and permit the elevation of the reduced strain rates kaj to those comparable to fabrication. Typical extensional rheology data for a SAN copolymer (h>an = 0.264, Mw = 7 kg/mol,Mw/Mn = 2.8) are illustrated in Figure 13.5 after time-temperature superposition to a reference temperature of 170°C [63]. The tensile stress growth coefficient rj (k, t) was measured at discrete times t during the startup of uniaxial extensional flow. Data points are marked with individual symbols (o) and terminate at the tensile break point at longest time t. Isothermal data points are connected by solid curves. Data were collected at selected k between 0.0167 and 0.0840 s-1 and at temperatures between 130 and 180 °C. Also illustrated in Figure 13.5 (dashed line) is a shear flow curve from a dynamic experiment displayed in a special format (3 versus or1) as suggested by Trouton [64]. The superposition of the low-strain rate data from two types (shear and extensional flow) of rheometers is an important validation of the reliability of both data sets. 

Birefringence setups can be designed to characterize molten materials undergoing isothermal homogeneous flow. The ranges of strains and strain rates also often coincide with those of rheometers, and consequently may be limited relative to those used in fabrication. Similarly, time-temperature superposition approaches may be used to expand the rate window. State-of-the-art setups suitable for rapid screening of new materials with research-scale quantities (5-20 g) are available for shear flow [72] and startup of uniaxial extensional flow [73,74]. 

The experimental ranges of strain rates (or strains) are summarized in Table 2 for the various types of experiments. Time-temperatiire superposition was successfully applied on the various steady shear flow and transient shear flow data. The shift factors were foimd to be exactly the same as those obtained for the dynamic data in the linear viscoelastic domain. Moreover, these were found to be also applicable in the case of entrance pressure losses leading to an implicit appUcation to elongational values. 

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