Superposition flow

Mewis, J. and Biebaut, G. 2001. Shear thickeiting in steady and superposition flows effect of particle... 

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. 

The discussion of the interaction of air jets supplied at some angle to each other shows that application of the method of superposition of the interacting jets momentums and surplus heat to predict velocity and temperatures in the combined flow results in inaccuracy when two unequal jets are supplied at a right angle. A different approach was undertaken in the studies of interaction of the main stream with vertical directing jets. Ti i... 

Superposition of Flows Potential flow solutions are also useful to illustrate the effect of cross-drafts on the efficiency of local exhaust hoods. In this way, an idealized uniform velocity field is superpositioned on the flow field of the exhaust opening. This is possible because Laplace s equation is a linear homogeneous differential equation. If a flow field is known to be the sum of two separate flow fields, one can combine the harmonic functions for each to describe the combined flow field. Therefore, if d)) and are each solutions to Laplace s equation, A2, where A and B are constants, is also a solution. For a two-dimensional or axisymmetric three-dimensional flow, the flow field can also be expressed in terms of the stream function. 

This function expresses a volume production during flame passage which starts slowly, speeds up, and gradually declines again. The flow field generated at time t upon ignition somewhere in the environment can be computed by superposition... 

Regarding this relationship, when designing the mold it is necessary to know the flow direction. To obtain this information, a simple flow pattern construction can be used (Fig. 3-28) via computer analysis. However, the flow direction is not constant. In some cases the flow direction in the filling phase differs from that in the holding phase. Here the question arises of whether this must be considered using superposition. 

FIGURE 5.6 Heat balance in a CSTR (a) heat generated by reaction (b) heat removed by flow and transfer to the environment (c) superposition of generation and removal curves. The intersection points are steady states, (d) Superposition of alternative heat removal curves that give only one steady state. 

Similar convection-diffusion equations to the Navier-Stokes equation can be formulated for enthalpy or species concentration. In all of these formulations there is always a superposition of diffusive and convective transport of a field quantity, supplemented by source terms describing creation or destruction of the transported quantity. There are two fundamental assumptions on which the Navier-Stokes and other convection-diffusion equations are based. The first and most fundamental is the continuum hypothesis it is assumed that the fluid can be described by a scalar or vector field, such as density or velocity. In fact, the field quantities have to be regarded as local averages over a large number of particles contained in a volume element embracing the point of interest. The second hypothesis relates to the local statistical distribution of the particles in phase space the standard convection-diffusion equations rely on the assumption of local thermal equilibrium. For gas flow, this means that a Maxwell-Boltzmann distribution is assumed for the velocity of the particles in the frame-of-reference co-moving with the fluid. Especially the second assumption may break dovm when gas flow at high temperature or low pressure in micro channels is considered, as will be discussed below. 

Flow boiling of sodium Noyes and Lurie (1966) attempted to correlate experimental data of flowing sodium by the method of superposition,... 

Measurement of the equilibrium properties near the LST is difficult because long relaxation times make it impossible to reach equilibrium flow conditions without disruption of the network structure. The fact that some of those properties diverge (e.g. zero-shear viscosity or equilibrium compliance) or equal zero (equilibrium modulus) complicates their determination even more. More promising are time-cure superposition techniques [15] which determine the exponents from the entire relaxation spectrum and not only from the diverging longest mode. 

As shown in Fig. 5.4, the flow domain can be denoted by 2 with inlet streams at Ain boundaries denoted by 3 2, (/el,..., Ain). In many scalar mixing problems, the initial conditions in the flow domain are uniform, i.e., cc(x, 0) = 40). Likewise, the scalar values at the inlet streams are often constant so that cc(x e 3 2, t) = c(f for all / e 1,..., Nm. Under these assumptions,38 the principle of linear superposition leads to the following relationship ... 

Time Dependence in Flow and the Boltzmann Superposition Principle... 

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 ... 

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. 

The calculation method and equations presented in the previous sections are for Newtonian fluids such that the flow due to screw rotation and the downstream pressure gradient can be solved independently, that is, via the principle of superposition. Since most resins are highly non-Newtonian, the rotational flow and pressure-driven flow in principle cannot be separated using superposition. That is, the shear dependency of the viscosity couples the equations such that they cannot be solved independently. Potente [50] states that the flows and pressure gradients should only be calculated using three-dimensional (3-D) numerical methods because of the limitations of the Newtonian model. 

As discussed in Section 7.4 and using the principle of superposition, the flow components were separated into rotational flow and pressure flows. The equation for the total flow and the components are as follows ... 

The principle of superposition is used to break the complicated flow into the component velocities. These component velocities will be derived in the next sections. 

The coefficients Lu, L2A, and L34 describe the viscous flow contributions of the transport of all three species in a total pressure gradient totai- Because a pressure gradient also imposes a chemical potential gradient on each species (eq 24), experimentally, there is always a superposition of diffusive and viscous flow e.g., for the description of the water flux in a total pressure gradient, all coefficients must be included, i.e.. 

While the external electrical field approach is a method directly modifying the zeta-potential of the capillary wall, it is not applicable with commercial apparatuses. The back-pressure technique, on the other hand, has the disadvantage that the flat electroosmotic flow profile is disrupted by superposition of a pressure-driven laminar flow profile hence, the efficiency of separation deteriorates. 

Closed Recirculation System. If we introduce a 8 signal into an N stage system, as shown in Fig. 14.5, the recorder will measure tracer as it flows by the first time, the second time, and so on. In other words it measures tracer which has passed through tanks, tanks, and so on. In fact it measures the superposition of all these signals. 

Shock Relationships and Formulas, which include Changes During Steady Reversible Compressible Flow (61-4) Pressure-Velocity Relationship (65-6) Irreversibility and Degradation (66-8) Derivation of Formulas (68-70) Pressure Efficiency Factor and Recovery Factor (70-2) and Oblique Shocks in Air (72). Shock Wave Interaction, which includes Strong Shock Waves (81) Superposition of Plane Shock Waves (81-2) ... 

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