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Two-phase Flow of Partially Miscible Components in Submicron Channels
0263–8762/05/$30.00+0.00 # 2005 Institution of Chemical Engineers Trans IChemE, Part A, July 2005 Chemical Engineering Research and Design, 83(A7): 777–781
www.icheme.org/journals doi: 10.1205/cherd.04344
TWO-PHASE FLOW OF PARTIALLY MISCIBLE COMPONENTS IN SUBMICRON CHANNELS A. NIGAM and E. B. NAUMAN The Isermann Department of Chemical and Biological Engineering, Rensselaer Polytechnic Institute, New York, USA
T
his paper investigates creeping, two-phase flow of partially miscible components in submicron channels. A continuous gradient technique treats two-phase flow by coupling a single version of the Navier –Stokes equations to the modified Cahn –Hilliard equation (a form of the convective diffusion equation applicable to nonlinear diffusion). A body force based on gradient energy provides the coupling. Results for the cross-channel composition and velocity profiles are easily calculated at far downstream locations, including the case where the viscosity is a function of composition. The pressure drop versus flow rate in small channels is substantially different from that in large channels. Wall effects due to selective attractive forces are unimportant for the ranges of parameters studied. Two forms of entrance length problems are defined. In the simpler problem, the components are fed separately and form an equilibrium interface. The solution methodology for this problem can also be applied to fully miscible fluids that initially differ in composition and viscosity. The other problem envisions an initially homogeneous mixture that phase separates in the channel by spinodal decomposition. The necessary physical parameters are the properties of the pure components and an interaction parameter between the components (e.g., Flory – Huggins or regular solution theory). Potential applications include tertiary oil recovery and microreactors. Keywords: microfluidics; two-phase flow; channels; ducts; microreactor; Cahn– Hilliard; diffusion.
INTRODUCTION
as particle beam etching. Diffusion in the cross-channel direction is so fast that concentration gradients in that direction are largely eliminated. Diffusion in the axial direction may be so high that concentration gradients in that direction are reduced as well, adversely affecting reactor performance (Nauman and Nigam, 2004a). The limiting performance is that of a CSTR rather than a piston (plug) flow reactor. This is quite different from the conventionally sized tubular reactors where cross-channel (i.e. radial) diffusion is beneficial because it causes a closer approximation to piston flow. For two-phase flow, we are considering situations where the interfacial thickness is an appreciable fraction of the channel width and where strong attractions between the walls and some components of the fluid mixture could be significant. Figure 1 shows the type of device being considered. Two streams, respectively rich in components A and B, enter separately, merge together, and equilibrate via molecular diffusion while flowing downstream, possibly with reaction between A and B. This problem has been solved for the case where A and B (and possibly a solvent) form an ideal mixture and have parabolic velocity profiles before and after merging, hydrodynamic entrance effects being ignored (Nauman and Nigam, 2004b).
This paper is broadly concerned with mixing, diffusion and chemical reaction between miscible and partially miscible fluids in micron-scale channels. By micron-scale, we mean systems with cross-channel dimensions of about 0.1 to 10 micrometers. Most publications dealing with ‘micro-reactors’ have focused on sizes scales of about 100 micrometers and larger. These sizes are accessible by soft lithography where the device can be fabricated by controlled polymerizations and where the fluid mechanics are essentially identical to those of the bulk fluids. The dimensions are sufficiently small to ensure deep laminar flow, but not yet so small that cross-channel diffusion is fast. In fact, a variety of specialized, micro-scale static mixers have been devised to enhance cross-channel diffusion (Kobayashi et al., 2004). The smaller devices contemplated in this paper are more likely to be fabricated by hard lithographic techniques such
Correspondence to: Professor E. B. Nauman, The Isermann Department of Chemical and Biological Engineering, Rensselaer Polytechnic Institute, Troy, New York 12180, USA. E-mail: [email protected]
777
778
NIGAM and NAUMAN
Figure 1. Geometry of on open system.
Figure 2. Typical results for an open system.
The equation to be solved is 1:5u
2 1 y2 @a @ a @2 a ¼ D þ ka(1 a) @z @y2 @z2 Y2
(1)
where Y is the half-width of the downstream channel and has different values prior to the mixing point. Figure 2 shows typical results. For the parameter set in the cited example, composition gradients in the cross-channel direction essentially vanish about 3Y downstream of the mixing point.
Equation (3) is appropriate for systems where van der Waals attractions dominate. The reaction term in equation (1) is arbitrary and will be ignored in what follows. There remain five constants, but most of these will be known when the components are specified. NA and NB are chain lengths for polymers and are normally set to 1 for small molecules. The basic thermodynamic constant is the Flory– Huggins interaction parameter, x, which is available in literature or can estimated from surface tension. When NA ¼ NB ¼ N, phase separation occurs if N x . 2. The gradient energy parameter, k, depends on x and on the radii of gyration of the molecules. A radius of gyration, RG, provides a convenient way to scale distance in the small systems under consideration. The theory applies to transients in both two-phase and miscible systems [i.e., the ‘miscible interface’ problem (Cavanaugh and Nauman, 1998)]. The remaining parameter in the model is the binary diffusion coefficient, D. As written, equation (2) allows D to be a function of concentration but it is assumed constant in what follows. The parameter R2G =D provides a convenient way to scale time in transient calculations. The application of equation (2) to quiescent systems has been extensively studied (Rogers et al., 1988; Rousar and Nauman, 1991). It is a non-linear diffusion equation that encompasses phase separation by spinodal decomposition. Extension to multicomponent systems is straightforward provided all components have similar diffusivities or provided there is one dominant component that acts as a solvent. Absent these approximations, the multicomponent problem remains challenging (He et al., 1996). Less recognized is the coupling of equation (2) to the equations of motion. The same concentration gradients that drive diffusion also generate body forces that drive flow (Hohenberg and Halperin, 1977; Tanaka and Araki, 1988). Although there are several apparently identical forms, the generally accepted formulation for the body force is FB ¼ ar
df kr2 a da
(4)
The Navier – Stokes equation is GENERALIZATION OF THE CONVECTIVE DIFFUSION EQUATION
r
Here we consider the case where components A and B are only partially miscible. If a solvent is present, it equally distributes between the phases. A suitable generalization of equation (1) for this pseudo-binary mixture is the modified Cahn – Hilliard equation (Nauman and Balsara, 1988): @a df 2 þ v ra ¼ r Da(1 a)r kr a @t da ka(1 a)
(2)
where f is the free energy of a homogeneous mixture of A and B. The Flory– Huggins form is assumed for the free energy (Flory, 1941): f ¼
a ln a (1 a) ln (1 a) þ þ xa(1 a) NA NB
(3)
@v þ v rv ¼ rP þ rmr v þ FB @t
(5)
which is subject to the continuity equation for an incompressible fluid rv¼0
(6)
Equations (2) through (6) constitute the general model, but the equations of motion will be simplified by dropping the momentum terms and using the unsteady version only to find steady-state solutions by the method of false transients. A further simplification is to reduce the dimensionality to one spatial dimension for the fully developed, far downstream flow and to two spatial dimensions for entrance length problems. There are two entrance length problems of interest for the case where the average concentrations give rise to two-phase behaviour.
Trans IChemE, Part A, Chemical Engineering Research and Design, 2005, 83(A7): 777–781
TWO-PHASE FLOW OF PARTIALLY MISCIBLE COMPONENTS IN SUBMICRON CHANNELS (1) Streams rich in A and B enter the reactor separately. They contact at the mixing point and diffuse together to form an equilibrium interface. Striated flow is assumed and stability questions are ignored. A slight variant on this problem assumes the two entering streams are already equilibrated, i.e., they are at the upper and lower binodal concentrations when they come in contact. Diffusion must still occur in the region of the interface to form an equilibrium concentration profile. In very small channels, the phase concentrations will be interior to the binodial concentrations and may even form a single phase (Nauman and Balsara, 1988). (2) A homogeneous but nonequilibrium mixture of A and B enters the channel as one stream but phase separates within the channel by spiniodal decomposition. Early on, particulate or co-continuous domains will be formed. These will ripen and, far downstream, are assumed to form a stable, striated flow.
FAR DOWNSTREAM, STRIATED FLOW The simplest solution to the general model is presumptively stable, striated flow in the far downstream region. In this region there is only one component of the velocity profile, vz (y). Furthermore, @a=@z ¼ 0. This means that the velocity profile does not affect the concentration profile, which is identical to what would exist in a stagnant mixture between flat plates with the same spatial average composition. Figure 3 shows the results for various average concentrations for the case of N x ¼ 4. The channel width is 12 RG, and the dimensionless pressure drop in the axial direction is 0.01. It has been shown that the pressure drop is moderate for such systems (Nauman and Nigam, 2004a). The numerical calculations follow an established methodology and satisfy the so-called neutral boundary conditions at the wall (Nauman and Balsara, 1988). The extreme concentrations are (approximately) the upper and lower binodal concentration and there is a continuous gradient of concentration between the phases. The location of the interface, defined as the inflection point of the
Figure 3. Concentration profile for Nx ¼ 4 and various spatial mean concentrations.
779
Figure 4. Velocity profiles corresponding to concentration profiles of Figure 3 when the viscosity ratio is 10.
concentration profile, depends on the spatial average concentration across the tube. As will be seen, the spatial mean is different than the convected mean. For the case where A and B have equal viscosities, the concentration gradient causes no change in the axial velocity profile. It is identical to what would be expected for a single phase fluid. Figure 4 shows velocity profiles when the two phases differ by a factor of 10 in viscosity. Viscosity is assumed to vary linearly with concentration for values between the binodal concentrations. The distortions from a simple, parabolic profile are quite pronounced. Additionally, the results are different than would be expected based on the classical formulation for two-phase flow. The classical formulation assumes a sharp interface between the phases and matches stress at the interface. Velocity at the interface is also matched, but the first derivative of velocity is discontinuous. The continuous interface model allows a single, continuous version of the Navier – Stokes equations to be solved, giving continuity of both velocity and its first derivative at the interface. Figure 5 compares the velocity profile calculated using the continuous interface model to the classical result. At the same pressure drop, the average velocities can differ by a factor of 50%. Figure 6 shows the results when the low viscosity phase flows along both walls, lubricating the relatively viscous phase flowing at the centre of the channel. This situation
Figure 5. Velocity profiles calculated using the new method compared with the traditional method for a spatial mean concentration of 0.3 and Nx ¼ 4.
Trans IChemE, Part A, Chemical Engineering Research and Design, 2005, 83(A7): 777–781
780
NIGAM and NAUMAN
Figure 8. Concentration profile for the case of attracting walls. Figure 6. Lubrication flow: comparison of new and traditional method.
Figure 7. Convected versus spatial means concentrations. The continuous gradient cases are for Nx ¼ 4.
might be achieved by using three inlet ports rather than the two ports shown in Figure 1, questions of stability being ignored. The results in Figure 6 show the average velocity to be substantially reduced compared to calculations using the conventional model for two-phase flow that assumes a sharp interface. The continuous interface between the phases means that regions of high viscosity extend on both sides of the interface. At Nx ¼ 2 there is no phase separation so that the mixture is homogeneous. This gives identical values for the convected and spatial means. Phase separation takes place when N x . 2, and the convected mean then differs from the spatial mean. Figure 7 shows the convected mean concentration as a function of the spatial mean concentration for a viscosity ratio of 1 so that the velocity profile is a simple parabolic. The 458 line is for a miscible fluid. The dotted line is for a completely immiscible fluid. The lines for the continuous gradient model are for N x ¼ 4. The binodals for this case are at a ¼ 0.979 and 0.021. However, in the confined geometry, phase separation does not occur when the minor phase concentration is 14% or less. The dashed line illustrates the effect of viscosity ratio, giving larger differences between convected and spatial means but not altering the limits of phase separation. EFFECTS OF WALL COMPOSITION The neutral boundary conditions used in connection with equation (2) assume that the wall attracts components A
and B equally. In principle, this can be achieved by using a frozen (or cross-linked) wall that has the same composition as the fluid adjacent to it. Since the two walls of the channel have different adjacent fluids, this suggests the walls should have different compositions. A more practical approach is to account for the effects of differential attraction. This is done by introducing special boundary conditions for equation (2) that are based on a boundary penalty model (Schmidt and Binder, 1985). A wall that strongly attracts component A will enrich A near the wall. The enrichment may cause a critical wetting transition where the fluid in contact with the wall achieves the upper, A-rich binodal composition even though the bulk fluid has the composition of the opposite binodal. Such transitions do not occur in very small channels because the energetics exceed plausible values for the wall interaction. Figure 8 illustrates the concentration profile in a system large enough to undergo such a transition. Hydrodynamic calculations, equation (5), show the effects to be minimal. Even when the wall is lubricated by a low viscosity phase, the average velocity is only slightly affected because the distance over which the lubrication occurs is small.
CONCLUSIONS Increasing complexity and decreasing sizes of micron scale devices will force the use of new models for twophase flow that include phenomena previously ignored. This paper has explored far downstream behavior. Because of rapid diffusion in submicron devices, the term ‘far downstream’ means only a few channel widths. Thus, for reasonably slow reactions, the far downstream results may be adequate for reaction engineering purposes. The results for flow versus pressure drop are different from those predicted by the conventional, sharp interface model. The convected mean concentrations are similar to those calculated using the sharp interface model except when confinement prevents phase separation. The material from which the walls are constructed will not markedly influence flow patterns, at least for weak van der Waals attractions. For fast reactions, we have defined two entrance length problems and the model that governs them. The coupling between the convective diffusion equation and the axial
Trans IChemE, Part A, Chemical Engineering Research and Design, 2005, 83(A7): 777–781
TWO-PHASE FLOW OF PARTIALLY MISCIBLE COMPONENTS IN SUBMICRON CHANNELS component of the equations of motion was not excited in the far downstream calculations, because there are no forces other than the usual pressure drop in the axial direction. The coupling will be excited in the entrance length problems.
NOMENCLATURE a A B D f FB k
volume fraction of species A species A species B coefficient of diffusion, m2 s21 dimensionless free energy body force term, kg m2 s22 reaction rate constant, 1 s for first order reaction, conc21 s21 for second order N degree of polymerization P pressure, Pa radius of gyration, m RG t time, s u mean axial velocity, m s21 n local velocity, m s21 y dimensionless coordinate, y/Y Y half height of the channel, m z axial coordinate Greek symbols k gradient energy parameter, kg m2 s22 x interaction parameter r density, kg m23 m viscosity, kg m s21
781
REFERENCES Cavanaugh, T.J. and Nauman, E.B., 1998, Particulate growth in phaseseparated polymer blends, J Polym Sci, Part B: Polyms Phys, 36: 2191. Flory, P.J., 1941, Therodynamics of high polymer solutions, J Chem Phys, 10: 51. He, D.Q., Kwak, S. and Nauman, E. B., 1996, On phase equilibria, interfacial tension and phase growth in ternary polymer blends, Macromol Theory Simul, 5: 801. Hohenberg, P.C. and Halperin, B.I., 1977, Theory of dynamic critical phenomena, Rev Mod Phys, 49: 435. Kobayashi, J., Mori, Y., Okamoto, K., Akiyama, R., Ueno, M., Kitamori, T. and Kobayashi, S., 2004, A microfluidic device for conducting gasliquid-solid hydrogenation reactions, Science, 304: 1305. Nauman, E.B. and Nigam, A., 2004a, Mixing in sub-micron ducts, Chem Eng Technol, 27: 293. Nauman, E.B. and Nigam, A., 2004b, On axial diffusion in laminar-flow reactors, Ind Eng Chem Res, ASAP article. Nauman, E.B. and Balsara, N.P., 1988, Phase equilibria and the LandauGinzburg functional, Fluid Phase Equil, 45: 229. Rogers, T.M., Elder, K.R. and Desai, R.C., 1988, Numerical study of the late stages of spinodal decomposition, Phys Rev B, 37: 9638. Rousar, I. and Nauman, E.B., 1991, Spinodal decomposition with surface tension driven flows, Chem Eng Comm, 105: 77. Schmidt, I. and Binder, K., 1985, Model calculations for wetting transitions in polymer mixtures, J Physique, 46: 1631. Tanaka, H. and Araki, T., 1988, Spontaneous double phase separation induced by rapid hydrodynamic coarsening in two-dimensional fluid mixtures, Phys Rev Lett, 81: 389. This paper was presented at the 7th World Congress of Chemical Engineering held in Glasgow, UK, 10–14 July 2005. The manuscript was received 15 December 2004 and accepted for publication 18 March 2005.
Trans IChemE, Part A, Chemical Engineering Research and Design, 2005, 83(A7): 777–781
www.icheme.org/journals doi: 10.1205/cherd.04344
TWO-PHASE FLOW OF PARTIALLY MISCIBLE COMPONENTS IN SUBMICRON CHANNELS A. NIGAM and E. B. NAUMAN The Isermann Department of Chemical and Biological Engineering, Rensselaer Polytechnic Institute, New York, USA
T
his paper investigates creeping, two-phase flow of partially miscible components in submicron channels. A continuous gradient technique treats two-phase flow by coupling a single version of the Navier –Stokes equations to the modified Cahn –Hilliard equation (a form of the convective diffusion equation applicable to nonlinear diffusion). A body force based on gradient energy provides the coupling. Results for the cross-channel composition and velocity profiles are easily calculated at far downstream locations, including the case where the viscosity is a function of composition. The pressure drop versus flow rate in small channels is substantially different from that in large channels. Wall effects due to selective attractive forces are unimportant for the ranges of parameters studied. Two forms of entrance length problems are defined. In the simpler problem, the components are fed separately and form an equilibrium interface. The solution methodology for this problem can also be applied to fully miscible fluids that initially differ in composition and viscosity. The other problem envisions an initially homogeneous mixture that phase separates in the channel by spinodal decomposition. The necessary physical parameters are the properties of the pure components and an interaction parameter between the components (e.g., Flory – Huggins or regular solution theory). Potential applications include tertiary oil recovery and microreactors. Keywords: microfluidics; two-phase flow; channels; ducts; microreactor; Cahn– Hilliard; diffusion.
INTRODUCTION
as particle beam etching. Diffusion in the cross-channel direction is so fast that concentration gradients in that direction are largely eliminated. Diffusion in the axial direction may be so high that concentration gradients in that direction are reduced as well, adversely affecting reactor performance (Nauman and Nigam, 2004a). The limiting performance is that of a CSTR rather than a piston (plug) flow reactor. This is quite different from the conventionally sized tubular reactors where cross-channel (i.e. radial) diffusion is beneficial because it causes a closer approximation to piston flow. For two-phase flow, we are considering situations where the interfacial thickness is an appreciable fraction of the channel width and where strong attractions between the walls and some components of the fluid mixture could be significant. Figure 1 shows the type of device being considered. Two streams, respectively rich in components A and B, enter separately, merge together, and equilibrate via molecular diffusion while flowing downstream, possibly with reaction between A and B. This problem has been solved for the case where A and B (and possibly a solvent) form an ideal mixture and have parabolic velocity profiles before and after merging, hydrodynamic entrance effects being ignored (Nauman and Nigam, 2004b).
This paper is broadly concerned with mixing, diffusion and chemical reaction between miscible and partially miscible fluids in micron-scale channels. By micron-scale, we mean systems with cross-channel dimensions of about 0.1 to 10 micrometers. Most publications dealing with ‘micro-reactors’ have focused on sizes scales of about 100 micrometers and larger. These sizes are accessible by soft lithography where the device can be fabricated by controlled polymerizations and where the fluid mechanics are essentially identical to those of the bulk fluids. The dimensions are sufficiently small to ensure deep laminar flow, but not yet so small that cross-channel diffusion is fast. In fact, a variety of specialized, micro-scale static mixers have been devised to enhance cross-channel diffusion (Kobayashi et al., 2004). The smaller devices contemplated in this paper are more likely to be fabricated by hard lithographic techniques such
Correspondence to: Professor E. B. Nauman, The Isermann Department of Chemical and Biological Engineering, Rensselaer Polytechnic Institute, Troy, New York 12180, USA. E-mail: [email protected]
777
778
NIGAM and NAUMAN
Figure 1. Geometry of on open system.
Figure 2. Typical results for an open system.
The equation to be solved is 1:5u
2 1 y2 @a @ a @2 a ¼ D þ ka(1 a) @z @y2 @z2 Y2
(1)
where Y is the half-width of the downstream channel and has different values prior to the mixing point. Figure 2 shows typical results. For the parameter set in the cited example, composition gradients in the cross-channel direction essentially vanish about 3Y downstream of the mixing point.
Equation (3) is appropriate for systems where van der Waals attractions dominate. The reaction term in equation (1) is arbitrary and will be ignored in what follows. There remain five constants, but most of these will be known when the components are specified. NA and NB are chain lengths for polymers and are normally set to 1 for small molecules. The basic thermodynamic constant is the Flory– Huggins interaction parameter, x, which is available in literature or can estimated from surface tension. When NA ¼ NB ¼ N, phase separation occurs if N x . 2. The gradient energy parameter, k, depends on x and on the radii of gyration of the molecules. A radius of gyration, RG, provides a convenient way to scale distance in the small systems under consideration. The theory applies to transients in both two-phase and miscible systems [i.e., the ‘miscible interface’ problem (Cavanaugh and Nauman, 1998)]. The remaining parameter in the model is the binary diffusion coefficient, D. As written, equation (2) allows D to be a function of concentration but it is assumed constant in what follows. The parameter R2G =D provides a convenient way to scale time in transient calculations. The application of equation (2) to quiescent systems has been extensively studied (Rogers et al., 1988; Rousar and Nauman, 1991). It is a non-linear diffusion equation that encompasses phase separation by spinodal decomposition. Extension to multicomponent systems is straightforward provided all components have similar diffusivities or provided there is one dominant component that acts as a solvent. Absent these approximations, the multicomponent problem remains challenging (He et al., 1996). Less recognized is the coupling of equation (2) to the equations of motion. The same concentration gradients that drive diffusion also generate body forces that drive flow (Hohenberg and Halperin, 1977; Tanaka and Araki, 1988). Although there are several apparently identical forms, the generally accepted formulation for the body force is FB ¼ ar
df kr2 a da
(4)
The Navier – Stokes equation is GENERALIZATION OF THE CONVECTIVE DIFFUSION EQUATION
r
Here we consider the case where components A and B are only partially miscible. If a solvent is present, it equally distributes between the phases. A suitable generalization of equation (1) for this pseudo-binary mixture is the modified Cahn – Hilliard equation (Nauman and Balsara, 1988): @a df 2 þ v ra ¼ r Da(1 a)r kr a @t da ka(1 a)
(2)
where f is the free energy of a homogeneous mixture of A and B. The Flory– Huggins form is assumed for the free energy (Flory, 1941): f ¼
a ln a (1 a) ln (1 a) þ þ xa(1 a) NA NB
(3)
@v þ v rv ¼ rP þ rmr v þ FB @t
(5)
which is subject to the continuity equation for an incompressible fluid rv¼0
(6)
Equations (2) through (6) constitute the general model, but the equations of motion will be simplified by dropping the momentum terms and using the unsteady version only to find steady-state solutions by the method of false transients. A further simplification is to reduce the dimensionality to one spatial dimension for the fully developed, far downstream flow and to two spatial dimensions for entrance length problems. There are two entrance length problems of interest for the case where the average concentrations give rise to two-phase behaviour.
Trans IChemE, Part A, Chemical Engineering Research and Design, 2005, 83(A7): 777–781
TWO-PHASE FLOW OF PARTIALLY MISCIBLE COMPONENTS IN SUBMICRON CHANNELS (1) Streams rich in A and B enter the reactor separately. They contact at the mixing point and diffuse together to form an equilibrium interface. Striated flow is assumed and stability questions are ignored. A slight variant on this problem assumes the two entering streams are already equilibrated, i.e., they are at the upper and lower binodal concentrations when they come in contact. Diffusion must still occur in the region of the interface to form an equilibrium concentration profile. In very small channels, the phase concentrations will be interior to the binodial concentrations and may even form a single phase (Nauman and Balsara, 1988). (2) A homogeneous but nonequilibrium mixture of A and B enters the channel as one stream but phase separates within the channel by spiniodal decomposition. Early on, particulate or co-continuous domains will be formed. These will ripen and, far downstream, are assumed to form a stable, striated flow.
FAR DOWNSTREAM, STRIATED FLOW The simplest solution to the general model is presumptively stable, striated flow in the far downstream region. In this region there is only one component of the velocity profile, vz (y). Furthermore, @a=@z ¼ 0. This means that the velocity profile does not affect the concentration profile, which is identical to what would exist in a stagnant mixture between flat plates with the same spatial average composition. Figure 3 shows the results for various average concentrations for the case of N x ¼ 4. The channel width is 12 RG, and the dimensionless pressure drop in the axial direction is 0.01. It has been shown that the pressure drop is moderate for such systems (Nauman and Nigam, 2004a). The numerical calculations follow an established methodology and satisfy the so-called neutral boundary conditions at the wall (Nauman and Balsara, 1988). The extreme concentrations are (approximately) the upper and lower binodal concentration and there is a continuous gradient of concentration between the phases. The location of the interface, defined as the inflection point of the
Figure 3. Concentration profile for Nx ¼ 4 and various spatial mean concentrations.
779
Figure 4. Velocity profiles corresponding to concentration profiles of Figure 3 when the viscosity ratio is 10.
concentration profile, depends on the spatial average concentration across the tube. As will be seen, the spatial mean is different than the convected mean. For the case where A and B have equal viscosities, the concentration gradient causes no change in the axial velocity profile. It is identical to what would be expected for a single phase fluid. Figure 4 shows velocity profiles when the two phases differ by a factor of 10 in viscosity. Viscosity is assumed to vary linearly with concentration for values between the binodal concentrations. The distortions from a simple, parabolic profile are quite pronounced. Additionally, the results are different than would be expected based on the classical formulation for two-phase flow. The classical formulation assumes a sharp interface between the phases and matches stress at the interface. Velocity at the interface is also matched, but the first derivative of velocity is discontinuous. The continuous interface model allows a single, continuous version of the Navier – Stokes equations to be solved, giving continuity of both velocity and its first derivative at the interface. Figure 5 compares the velocity profile calculated using the continuous interface model to the classical result. At the same pressure drop, the average velocities can differ by a factor of 50%. Figure 6 shows the results when the low viscosity phase flows along both walls, lubricating the relatively viscous phase flowing at the centre of the channel. This situation
Figure 5. Velocity profiles calculated using the new method compared with the traditional method for a spatial mean concentration of 0.3 and Nx ¼ 4.
Trans IChemE, Part A, Chemical Engineering Research and Design, 2005, 83(A7): 777–781
780
NIGAM and NAUMAN
Figure 8. Concentration profile for the case of attracting walls. Figure 6. Lubrication flow: comparison of new and traditional method.
Figure 7. Convected versus spatial means concentrations. The continuous gradient cases are for Nx ¼ 4.
might be achieved by using three inlet ports rather than the two ports shown in Figure 1, questions of stability being ignored. The results in Figure 6 show the average velocity to be substantially reduced compared to calculations using the conventional model for two-phase flow that assumes a sharp interface. The continuous interface between the phases means that regions of high viscosity extend on both sides of the interface. At Nx ¼ 2 there is no phase separation so that the mixture is homogeneous. This gives identical values for the convected and spatial means. Phase separation takes place when N x . 2, and the convected mean then differs from the spatial mean. Figure 7 shows the convected mean concentration as a function of the spatial mean concentration for a viscosity ratio of 1 so that the velocity profile is a simple parabolic. The 458 line is for a miscible fluid. The dotted line is for a completely immiscible fluid. The lines for the continuous gradient model are for N x ¼ 4. The binodals for this case are at a ¼ 0.979 and 0.021. However, in the confined geometry, phase separation does not occur when the minor phase concentration is 14% or less. The dashed line illustrates the effect of viscosity ratio, giving larger differences between convected and spatial means but not altering the limits of phase separation. EFFECTS OF WALL COMPOSITION The neutral boundary conditions used in connection with equation (2) assume that the wall attracts components A
and B equally. In principle, this can be achieved by using a frozen (or cross-linked) wall that has the same composition as the fluid adjacent to it. Since the two walls of the channel have different adjacent fluids, this suggests the walls should have different compositions. A more practical approach is to account for the effects of differential attraction. This is done by introducing special boundary conditions for equation (2) that are based on a boundary penalty model (Schmidt and Binder, 1985). A wall that strongly attracts component A will enrich A near the wall. The enrichment may cause a critical wetting transition where the fluid in contact with the wall achieves the upper, A-rich binodal composition even though the bulk fluid has the composition of the opposite binodal. Such transitions do not occur in very small channels because the energetics exceed plausible values for the wall interaction. Figure 8 illustrates the concentration profile in a system large enough to undergo such a transition. Hydrodynamic calculations, equation (5), show the effects to be minimal. Even when the wall is lubricated by a low viscosity phase, the average velocity is only slightly affected because the distance over which the lubrication occurs is small.
CONCLUSIONS Increasing complexity and decreasing sizes of micron scale devices will force the use of new models for twophase flow that include phenomena previously ignored. This paper has explored far downstream behavior. Because of rapid diffusion in submicron devices, the term ‘far downstream’ means only a few channel widths. Thus, for reasonably slow reactions, the far downstream results may be adequate for reaction engineering purposes. The results for flow versus pressure drop are different from those predicted by the conventional, sharp interface model. The convected mean concentrations are similar to those calculated using the sharp interface model except when confinement prevents phase separation. The material from which the walls are constructed will not markedly influence flow patterns, at least for weak van der Waals attractions. For fast reactions, we have defined two entrance length problems and the model that governs them. The coupling between the convective diffusion equation and the axial
Trans IChemE, Part A, Chemical Engineering Research and Design, 2005, 83(A7): 777–781
TWO-PHASE FLOW OF PARTIALLY MISCIBLE COMPONENTS IN SUBMICRON CHANNELS component of the equations of motion was not excited in the far downstream calculations, because there are no forces other than the usual pressure drop in the axial direction. The coupling will be excited in the entrance length problems.
NOMENCLATURE a A B D f FB k
volume fraction of species A species A species B coefficient of diffusion, m2 s21 dimensionless free energy body force term, kg m2 s22 reaction rate constant, 1 s for first order reaction, conc21 s21 for second order N degree of polymerization P pressure, Pa radius of gyration, m RG t time, s u mean axial velocity, m s21 n local velocity, m s21 y dimensionless coordinate, y/Y Y half height of the channel, m z axial coordinate Greek symbols k gradient energy parameter, kg m2 s22 x interaction parameter r density, kg m23 m viscosity, kg m s21
781
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Trans IChemE, Part A, Chemical Engineering Research and Design, 2005, 83(A7): 777–781