Incompressible phases

Pressure drop and heat transfer in a single-phase incompressible flow. According to conventional theory, continuum-based models for channels should apply as long as the Knudsen number is lower than 0.01. For air at atmospheric pressure, Kn is typically lower than 0.01 for channels with hydraulic diameters greater than 7 pm. From descriptions of much research, it is clear that there is a great amount of variation in the results that have been obtained. It was not clear whether the differences between measured and predicted values were due to determined phenomenon or due to errors and uncertainties in the reported data. The reasons why some experimental investigations of micro-channel flow and heat transfer have discrepancies between standard models and measurements will be discussed in the next chapters. 

Flow through abrupt expansion Using the one-dimensional flow assumption for a single-phase incompressible fluid, the energy equation becomes... 

Flow through orifices For the liquid or vapor phase flow alone passing through the orifice, expressions similar to a single-phase incompressible flow case can be written for the mass flow rates of both phases ... 

The remaining approximations of liquid phase incompressibility and a discontinuous mass distribution can be removed through the use of the Yvon-Born-Green (YBG) equation (3), which is simply a... 

PRESSURE DROP CALCULATIONS FOR SINGLE-PHASE INCOMPRESSIBLE FLUIDS ... 

H. Herwig and S.P. Mahulikar, Variable property effects in single-phase incompressible flows through microchaimels. International Journal of Thermal Sciences 45, 977-981 (2006). 

We then discussed the modeling for single-fluid phase flow in porous media. In particular, the shear factor and permeability model of Liu et al. (32) is discussed in detail. The bounding wall effects are presented. This section completed the modeling requirements for single-phase incompressible flow in porous media. We showed how to solve the governing equations for flow in porous media and an approximate solution of the pressure drop for an incompressible flow through a cylindrically bounded porous bed was constructed. 

Vapor phase incompressibility that is, is constant during... 

This work is organized as follows the governing equations for two-phase incompressible flows and the corresponding VOF-based numerical methods are... 

Geometrically, Liouville s theorem means that if one follows the motion of a small phase volume in Y space, it may change its shape but its volume is invariant. In other words the motion of this volume in T space is like that of an incompressible fluid. Liouville s theorem, being a restatement of mechanics, is an important ingredient in the fomuilation of the theory of statistical ensembles, which is considered next. 

Surface waves at an interface between two innniscible fluids involve effects due to gravity (g) and surface tension (a) forces. (In this section, o denotes surface tension and a denotes the stress tensor. The two should not be coiifiised with one another.) In a hydrodynamic approach, the interface is treated as a sharp boundary and the two bulk phases as incompressible. The Navier-Stokes equations for the two bulk phases (balance of macroscopic forces is the mgredient) along with the boundary condition at the interface (surface tension o enters here) are solved for possible hamionic oscillations of the interface of the fomi, exp [-(iu + s)t + i V-.r], where m is the frequency, is the damping coefficient, s tlie 2-d wavevector of the periodic oscillation and. ra 2-d vector parallel to the surface. For a liquid-vapour interface which we consider, away from the critical point, the vapour density is negligible compared to the liquid density and one obtains the hydrodynamic dispersion relation for surface waves + s>tf. The temi gq in the dispersion relation arises from... 

Rapid approximate predictions of pressure drop for fully developed, incompressible horizontal gas/fiquid flow may be made using the method of Lockhart and MartineUi (Chem. Eng. Prog., 45, 39 8 [1949]). First, the pressure drops that would be expected for each of the two phases as if flowing alone in single-phase flow are calculated. The LocKhart-Martinelli parameter X is defined in terms of the ratio of these pressure drops ... 

As for any incompressible single-phase flow, the equivalent pressure P = p + pgz where g = acceleration of gravity z = elevation, may be used in place of p to account for gravitational effects in flows with vertical components. 

For isotropic homogeneous porous media (uniform permeability and porosity), the pressure for creeping incompressible single phase-flow may be shown to satisfy the LaPlace equation ... 

HEM for Two-Phase Pipe Discharge With a pipe present, the backpressure experienced by the orifice is no longer qg, but rather an intermediate pressure ratio qi. Thus qi replaces T o iri ihe orifice solution for mass flux G. ri Eq. (26-95). Correspondingly, the momentum balance is integrated between qi and T o lo give the pipe flow solution for G,p. The solutions for orifice and pipe now must be solved simultaneously to make G. ri = G,p and to find qi and T o- This can be done explicitly for the simple case of incompressible single-phase (hquid) inclined or horizontal pipe flow The solution is implicit for compressible regimes. 

If the mobile phase is a liquid, and can be considered incompressible, then the volume of the mobile phase eluted from the column, between the injection and the peak maximum, can be easily obtained from the product of the flow rate and the retention time. For more precise measurements, the volume of eluent can be directly measured volumetrically by means of a burette or other suitable volume measuring vessel that is placed at the end of the column. If the mobile phase is compressible, however, the volume of mobile phase that passes through the column, measured at the exit, will no longer represent the true retention volume, as the volume flow will increase continuously along the column as the pressure falls. This problem was solved by James and Martin [3], who derived a correction factor that allowed the actual retention volume to be calculated from the retention volume measured at the column outlet at atmospheric pressure, and a function of the inlet/outlet pressure ratio. This correction factor can be derived as follows. 

In equation (37), for an incompressible mobile phase, the kinetic dead volume is (Vi(m)) which is the volume of moving phase only. Consequently, at a flow rate of... 

D. E. Martire, Unified Approach to the Theory of Chromatography Incompressible Binary Mobile Phase (Liquid Chromatography) in Theoretical Advancement in Chromatography and Related Separation Techniques (Ed. F. Dondi, G. Guiochon, IGuwer, Academic Publishers, Dordrecht, The Netherlands,(l993)261. 

It should be noted that all the equations assume that the mobile phase is incompressible which will not be true for equations (23) and (24). It follows that equations (23) and (24) will require modification in order to be applicable to practical situations. It will also be shown in a later chapter that, from experimental data, (oo)... 

The mobile phase in LC is considered incompressible from the point of view of dispersion and, so, the equation will not contain the variable (y). Thus,... 

When the cake structure is composed of particles that are readily deformed or become rearranged under pressure, the resulting cake is characterized as being compressible. Those that are not readily deformed are referred to as sem-compressible, and those that deform only slightly are considered incompressible. Porosity (defined as the ratio of pore volume to the volume of cake) does not decrease with increasing pressure drop. The porosity of a compressible cake decreases under pressure, and its hydraulic resistance to the flow of the liquid phase increases with an increase in the pressure differential across the filter media. 

The estimation of the diffusional flux to a clean surface of a single spherical bubble moving with a constant velocity relative to a liquid medium requires the solution of the equation for convective diffusion for the component that dissolves in the continuous phase. For steady-state incompressible axisym-metric flow, the equation for convective diffusion in spherical coordinates is approximated by... 

The concepts of interface rheology are derived from the rheology of three-dimensional phases. Characteristic for the interface rheology is the coupling of the motions of an interface with the flow processes in the bulk close to the interface. Thus, in interface rheology the shear and dilatational stresses of the interface are in equilibrium with the corresponding shear stress in the bulk. An important feature is the compressibility of the adsorption layer of an interface in contrast, the flow elements of the bulk are incompressible. As a result, compression or dilatation of the adsorption layer of a soluble surfactant is associated with desorption and adsorption processes by which the interface tends to reinstate the adsorption equilibrium with the bulk phase. 

In Chap. 3 the problems of single-phase flow are considered. Detailed data on flows of incompressible fluid and gas in smooth and rough micro-channels are presented. The chapter focuses on the transition from laminar to turbulent flow, and the thermal effects that cause oscillatory regimes. 

The problems of micro-hydrodynamics were considered in different contexts (1) drag in micro-channels with a hydraulic diameter from 10 m to 10 m at laminar, transient and turbulent single-phase flows, (2) heat transfer in liquid and gas flows in small channels, and (3) two-phase flow in adiabatic and heated microchannels. The smdies performed in these directions encompass a vast class of problems related to flow of incompressible and compressible fluids in regular and irregular micro-channels under adiabatic conditions, heat transfer, as well as phase change. 

For adiabatic, steady-state, and developed gas-liquid two-phase flow in a smooth pipe, assuming immiscible and incompressible phases, the essential variables are pu, pG, Pl, Pg, cr, dh, g, 9, Uls, and Uas, where subscripts L and G represent liquid and gas (or vapor), respectively, p is the density, p is the viscosity, cr is the surface tension, dh is the channel hydraulic diameter, 9 is the channel angle of inclination with respect to the gravity force, or the contact angle, g is the acceleration due to gravity, and Uls and Ugs are the liquid and gas superficial velocities, respectively. The independent dimensionless parameters can be chosen as Ap/pu (where Ap = Pl-Pg), and... 

In capillary flow with a distinct meniscus separating the regions of pure liquid and pure vapor flows, it is possible to neglect the change in densities of the phases and assume po and pu are constant. For flow of incompressible fluid (p = const., p = 0, duildx = 0) the substitution of (11.8) in Eqs. (11.1-11.3) leads, in a linear approximation, to the following system of equations... 

Depart from Geometric Similarity. Adding length to a tubular reactor while keeping the diameter constant allows both volume and external area to scale as S if the liquid is incompressible. Scaling in this manner gives poor results for gas-phase reactions. The quantitative aspects of such scaleups are discussed... 

The stationary phase matrices used in classic column chromatography are spongy materials whose compress-ibihty hmits flow of the mobile phase. High-pressure liquid chromatography (HPLC) employs incompressible silica or alumina microbeads as the stationary phase and pressures of up to a few thousand psi. Incompressible matrices permit both high flow rates and enhanced resolution. HPLC can resolve complex mixtures of Upids or peptides whose properties differ only slightly. Reversed-phase HPLC exploits a hydrophobic stationary phase of... 

When the two phases in contact are condensed phases and the entire volume is taken up by incompressible substances, positive adsorption of one component must be attended by negative adsorption (desorption) of other components. This phenomenon is called adsorptive displacement. 

The above equation can be divided by cross-sectional area to give the balance in terms of superficial velocities. Since the liquid phases are incompressible... 

The general approach for kinetic optiaization of open i tubular columns has been to adopt the familiar Golay equation T (equation 1.34) and to assuae that the aobile phase can be approximated by an incompressible fluid with ideal gas properties, (44-50). Circumstances that are approximate at best but serve adequately to demonstrate some of the fundamental characteristics of open tubular columns operated at low fluid densities. The column plate height equation can be written in the form given in M equation (6.1)... 

---