Separated flows, defined

For nonlinear systems, however, the evaluation of the flow rates is not straightforward. Morbidelli and co-workers developed a complete design of the binary separation by SMB chromatography in the frame of Equilibrium Theory for various adsorption equilibrium isotherms the constant selectivity stoichiometric model [21, 22], the constant selectivity Langmuir adsorption isotherm [23], the variable selectivity modified Langmuir isotherm [24], and the bi-Langmuir isotherm [25]. The region for complete separation was defined in terms of the flow rate ratios in the four sections of the equivalent TMB unit ... 

Porosity of a separator is defined as the ratio of void volume to apparent geometric volume. High porosity is desirable for unhindered ionic current flow. 

In the separated flow models presented in Sections 7.7 and 7.8, the method of calculating the frictional component of the pressure gradient involves use of the two-phase multiplier 4>1 2 3 4 defined by... 

Reverse-Osmosis Experiments. All reverse-osmosis experiments were performed with continuous-flow cells. Each membrane was subjected to an initial pure water pressure of 2068 kPag (300 psig) for 2 h pure water was used as feed to minimize the compaction effect. The specifications of all the membranes in terms of the solute transport parameter [(Dam/ 6)Naci]> the pure water permeability constant (A), the separation, and the product rate (PR) are given in Table I. These were determined by Kimura-Sourirajan analysis (7) of experimental reverse-osmosis data with sodium chloride solution at a feed concentration of 0.06 m unless otherwise stated. All other reverse-osmosis experiments were carried out at laboratory temperature (23-25 °C), an operating pressure of 1724 kPag (250 psig), a feed concentration of 100 ppm, and a feed flow rate >400 cmVmin. The fraction solute separation (/) is defined as follows ... 

From Eqs. (7-1) and (7-2), it follows that the separation factor is purely based on the compositions of the entering and exit streams regardless of their flows. Another measure of the separation efficiency of a membrane process is the extent of separation proposed by Rony [1968]. In the context of applying this index of separation efficiency between two comfionents, it is assumed that there is no difficulty in separating the third component Thus the segregation fractions, fiy, are obtained from the molar flow rates of the permeate and retentate streams on the basis of only two components. The extent of separation is defined as the absolute value of a determinant of a binary separation matrix consisting of the segregation fractions as follows ... 

For air entering a valve at room temperature R = 284kJ/kg/K, 7 = 1.4, and iJ/ = 0.04042. However, real relief valves suffer from area contraction due to separated flow patterns, and the flow is neither one-dimensional nor isentropic. Consequently, the flow rate (m) will differ from that in an ideal duct having the same minimum section. A general form of discharge coefficient can then be defined by... 

The most widely used method for estimating the pressure drop due to friction is that proposed by Lockhart and Martinelli [1949] and subsequently improved by Chisholm [1967]. It is based on a physical model of separated flow in which each phase is considered separately and then the interaction effect is introduced. In this method, the two phase pressure drop due to friction (—ApTp), is expressed in terms of dimensionless drag ratios, defined by the following equations ... 

The two main approaches used to model frictional pressure gradients in macro- and microscale two-phase gas-liquid flow are the homogeneous model and the separated flow model. The homogeneous model is based on the assumption that the two-phase fluid behaves as a pseudo single-phase fluid with pseudo-properties that are weighted relative to the vapor and Uquid flow fraction. Different ways to define the pseudo-properties, usually called mixture properties, have been proposed that are well detailed by Collier and Thome [54] and Thome [55]. 

For investigations with defined pore pressures, a separate system and sintered pistons or separate flow channels are used. Sophisticated equipments have in addition transducer-receiver systems for simultaneous ultrastmic measurements of dynamic properties (Fjaer et al., 2008). 

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