Liquid Two-Phase Systems in Rotating Coiled Columns - Springer Link

Theoretical Foundations of Chemical Engineering, Vol. 37, No. 4, 2003, pp. 351–357. Translated from Teoreticheskie Osnovy Khimicheskoi Tekhnologii, Vol. 37, No. 4, 2003, pp. 378–384. Original Russian Text Copyright © 2003 by Kaminskii, Maryutina.

Behavior of Liquid–Liquid Two-Phase Systems in Rotating Coiled Columns V. A. Kaminskii* and T. A. Maryutina** * Karpov Institute of Physical Chemistry, Russian State Scientific Center, ul. Vorontsovo pole 10, Moscow, 105064 Russia ** Vernadsky Institute of Geochemistry and Analytical Chemistry, Russian Academy of Sciences, ul. Kosygina 19, Moscow, 119991 Russia Received January 20, 2003

Abstract—The distribution and motion of liquid–liquid two-phase systems in thin tubes under conditions modeling rotation (including planetary rotation) of chromatographic columns are considered. Criteria are derived that determine the conditions for transition between various types of distribution of the liquid phases in a column using different physicochemical and rotation parameters.

The spatial distribution of liquid–liquid two-phase systems plays an important role in various processes of interphase mass transfer. The mass transfer in twophase systems of immiscible liquids is enhanced by increasing the specific interfacial area and providing relative motion of the phases, e.g., by the stirring and dispersion of one or both phases, or by designing a film flow. For the last three decades in liquid chromatography, a method has been used that is based on the mass transfer in coiled separation columns between two liquid phases, one of which (mobile phase) is pumped through a column, and the other (stationary phase) is retained by the centrifugal force field without using a solid support [1]. This method is referred to as liquid– liquid partition chromatography without solid support [1]. Several variants of the design of this method have been proposed [2], and, on the basis of these, a large number of separation devices have been developed. Rotating coiled columns are most widely used. A column, which is a thin tube (capillary) coiled around a rigid core, executes synchronous planetary rotation, i.e., rotates about its axis and, simultaneously, revolves around the central axis of the device. Schematically, two types of such synchronous planetary rotation, I and J, are shown in Fig. 1 according to the Ito classification. In I-type devices, the directions of the rotation and revolution of the column are opposite to one another, and each point of the column assembly moves along a circular path, regardless of the value of the parameter β, which is the ratio of the radius of rotation of the column to the radius of its revolution. Such a design proves to be efficient for analytical separation in columns with a small inner diameter. If the directions of the rotation and revolution of the column are the same (in J-type devices), the path of each point of the column assembly depends on the value of the parameter β and ranges from circular (β = 0) to loop-shaped (β = 1). Synchronous planetary rotation ensures the convenient supply of the mobile phase to the column without using special

mobile connectors and, at the same time, gives rise to an unsteady-state centrifugal force field, under the action of which the distribution of the two-phase system along the column is established. The amount of the stationary phase retained in the column is characterized by the retention factor Sf of the stationary phase, which is equal to the ratio of the column volume occupied by the stationary phase to the total column volume and is an important index of the efficiency of the method considered. The possibility of retaining the liquid phase in coiled columns executing planetary rotation is a distinguishing feature and an important advantage of liquid– liquid partition chromatography without solid support. At the same time, this feature constitutes the main difficulty in studying such processes and determining the optimum process conditions at given physicochemical properties of a particular system, because the motion of the phases and their distribution along the tube are established in the course of the process and are a priori unknown. Although a lot of experimental studies have been performed, there have hitherto been no clear physical concepts of the behavior of two-phase systems in columns executing planetary rotation and the retention mechanism [3–10]. The complexity of problems arising in this case can be illustrated by the fact that, up to the present time, the effect of various parameters on the retention factor has been studied by multifactor regression analysis, which in no way takes into consideration the physical processes determining the behavior of a two-phase system [5]. Previously, several classifications were proposed for the behavior of two-phase liquid systems in columns executing planetary rotation. The most widely used one is Ito’s classification [4], according to which all of the systems are divided into hydrophilic, hydrophobic, and intermediate, depending on their ability to emulsify on being shaken and undergo the subsequent phase separation. In other words, Ito’s classification is based on the

0040-5795/03/3704-0351$25.00 © 2003 MAIK “Nauka /Interperiodica”

352

KAMINSKII, MARYUTINA (a)

(b)

(c) 5

y

4

ω

3

0

R 0'

Axis of rotation

1

1 0'

Axis of revolution

0

2 4 6 0

x

r ω

(d)

(e)

ω 0'

y

5 4

ω 0

(f)

3

Axis of revolution

R Axis of rotation

0

6 1 2

1 0'

4 x

0

r

Fig. 1. (a, d) Synchronous planetary rotation, (b, e) the distributions of centrifugal force vectors, and (c, f) the orbits of rotation of columns of the type (a–c) I and (d–f) J at β = (1) 0, (2) 0.1, (3) 0.25, (4) 0.50, (5) 0.75, and (6) 1.0.

assumption of the decisive role of dispersion and phase separation in rotating coiled columns. Such an approach is empirical, since the shaking conditions are undetermined and, moreover, no physical criteria characterizing the liquid breakup in a rotating column are used. Another approach for classifying two-phase systems was proposed by Menet [10] and is based on the concept of capillary wavelength. The capillary wavelength determines the size of drops which can form during the destabilization of the interface; it also governs the velocity of such drops in each of the phases, on which the rate of phase separation of the forming emulsion depends. This approach also ignores the actual conditions in a rotating coiled column, which determine the development of destabilization of the interface and the subsequent dispersion. In essence, Ito’s and Menet’s approaches are similar to one another, since both of them are based on the assumption of the decisive role of dispersion. The apparent simplicity of studying the behavior of two-phase systems in rotating coiled columns is related to the fact that the physicochemical parameters characterizing the systems under consideration have a clear meaning and are either known or are quite easy to determine from independent experiments. All of the parameters can be divided into three groups: physicochemical parameters (the densities ρ1 and ρ2 of the phases, the dynamic viscosities η1 and η2 of the phases, the interface tension γ, and the contact angle), geometrical or design parameters (the radius r of rotation, the radius R

of revolution, and the tube diameter d), and operating parameters (the velocity ω and direction of rotation, and the velocity and pumping direction of the mobile phase). Along with investigating hydrodynamic modes during planetary rotation, a study is also conducted of the distribution of two-phase systems in coiled columns that have closed ends and rotate in a vertical plane; in such columns, the phases are distributed under the simultaneous action of gravitational and the centrifugal forces (Fig. 2). Note that, in coiled columns executing planetary rotation, the turns rotate on a horizontal plane and, if the rotation is sufficiently fast, the force of gravity can be ignored. The expressions for the radial and tangential components of the acceleration of a liquid element in a rotating column have the following form [8]: for ordinary rotation in the field of gravity, a t = g sin θ,

a r = g ( cos θ + rω /g ); 2

(1a)

for J-type synchronous planetary rotation, a t = Rω sin θ, 2

a r = Rω ( cos θ + 4r/R ), 2

(1b)

for I-type synchronous planetary rotation, a t = Rω sin θ, 2

a r = Rω cos θ. 2

(1c)

Expressions (1) can readily be derived by analyzing the equations of motion of a selected element of a column in a Cartesian coordinate system (x, y) and using the

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(b)

353

(c)

2 ω1

ω ∆h

1

2

2

2

1

1

Fig. 2. (a, b) Open and (c) closed (partitioned) rings rotating in a vertical plane (the numbers 1 and 2 refer to the first and second liquid phases, respectively).

dependence of the Cartesian coordinates of this element on the angle θ of rotation of the column. The ultimate purpose of studying the behavior of immiscible liquids in a rotating coiled column is to determine, for particular systems, the optimum modes for ensuring the retention of the stationary phase and efficient interphase mass transfer. Currently, even the direction of mobile-phase pumping is found from empirical rules, which can be formulated from the behavior of a two-phase system in a rotating coiled column with closed ends [8]. Information on the behavior of two-phase systems in rotating coiled columns can be provided not only by the experiment but also by direct numerical simulation based on the Navier–Stokes equation. However, on the one hand, such an approach is quite complex, since it leads to a three-dimensional unsteady-state problem of motion of a two-phase liquid system, in which the position of the interface is found in the course of solving the problem. On the other hand, this is merely a numerical experiment, which opens virtually no additional possibilities for determining new dimensionless criteria governing the critical conditions for transition between modes. Therefore, to derive explicit analytical relations characterizing hydrodynamic modes in two-phase systems in rotating coiled columns, it is of interest to analyze several simplified models. We further restrict our consideration to the laminar flow of two immiscible liquids in a rotating coiled column. We consider a single turn of the coiled column as an element which can be represented as a ring rotating in the field of gravity (Fig. 2). A coiled column with a closed end should be modeled by a ring with a transverse partition (Fig. 2c). Along the tube, plug flow and relative motion of liquids separated in the radial direction occur. Apparently, when plug flow takes places, the retention of the stationary phase during liquid-phase pumping is impossible. On the other hand, it is clear that, in the partitioned ring, only relative motion of the liquids is possible. Let us consider the behavior of immiscible liquids in an isolated ring without mobile-phase pumping (Fig. 2). The tube diameter is usually much smaller than the radius of rotation (d Ⰶ r); therefore, instead of a ring, a straight tube of length l = 2πr with the corresponding

distribution of forces acting on liquid elements along the tube can be considered. The immiscible liquids flow in the tube under the action of a force, which is the sum of the force of gravity and the centrifugal force. This force can be represented as a traveling wave propagating in the longitudinal direction x (0 < x < l). The components of this force in the longitudinal and normal directions are described by expressions (1a)–(1c), depending on the type of rotation. The boundary conditions at the tube ends are selected depending on whether an ordinary ring (hereinafter referred to as the open ring) or a partitioned (i.e., closed) ring is examined. Let us first analyze the behavior of the two-phase system in an open ring, which is filled with two liquids in a 1 : 1 volume ratio and rotates in a vertical plane in the field of gravity (Fig. 2a). In the ring at rest, the interfaces in the left and right parts of the ring are at the same height. As the ring rotates, the liquid is entrained by the rotating walls of the tube. An increase in the rotation rate causes an increase in the height difference ∆h, which determines the difference of the hydrostatic pressure in the lower section of the ring: ∆ph = g(∆h)(∆ρ), where ∆ρ = ρ1 – ρ2. The maximum ring rotation rate at which the steady-state distribution of the phases in the ring persists and the liquid does not rotate corresponds to the state shown in Fig. 2b, in which the heavier liquid completely fills the left half of the ring and the lighter liquid occupies the right half of the ring. In the general case, for an arbitrary distribution of the tangential acceleration at along the ring length as a function of the angle θ, the pressure difference ∆ph can be represented as 2π

∫

∆ p h = r a t ( θ )ρ ( θ ) dθ,

(2)

0

where ρ(θ) is the liquid density. The pressure difference causes a steady-state liquid flow, resistance to which is determined by the viscous dissipation. Assuming this flow to be a Poiseuille flow, we resort to the Hagen–Poiseuille equation

THEORETICAL FOUNDATIONS OF CHEMICAL ENGINEERING

32πr ω ( η 1 + η 2 ) ∆p = ----------------------------------------, 2 d 2

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KAMINSKII, MARYUTINA

which relates the pressure difference to the flow velocity. In Eq. (3), we used the expression for the average flow velocity 〈v〉 in the case of steady-state flow: 〈v〉 = ωr. For steady-state flow, the hydrostatic pressure is equal to the resistance to the flow. By equating ∆ph = ∆p, we obtain the expression ωr ( η 1 + η 2 ) N 1 = ----------------------------2 gd ( ∆ρ )

(4)

for the dimensionless number N1, which determines the condition for the transition to the mode of rotation of the two-phase liquid together with the rotating ring. The corresponding critical rotation rate ω1 (rad/s) is given by the expression ∆ρgd ω 1 = ----------------------------------- . 16πr ( η 1 + η 2 ) 2

(5)

For typical values of the parameters involved in expression (5), ∆ρ = 300 kg/m3, d = 1.5 × 10–3 m, r = 5 × 10–2 m, η1 ≈ η2 = 10–3 kg/(m s), we obtain ω1 = 2.3 rps. As follows from expressions (1) for planetary rotation of both types (I and J), a dimensionless number can be obtained by replacing the acceleration due to gravity with centrifugal acceleration. This yields the expressions for the dimensionless complex N2, r ( η1 + η2 ) N 2 = ------------------------2 Rd ω ( ∆ρ )

(6)

and the critical rotation rate ω2, 16πr ( η 1 + η 2 ) -. ω 2 = ---------------------------------2 ( ∆ρ )Rd

(7)

For the above parameter values, at R = 0.1 m, we have ω2 = 0.2 rps. The effect of interfacial tension on the distribution of a two-phase system has not hitherto been considered. Under the action of the radial force component, the system tends to undergo phase separation in the radial direction. Phase separation decreases the potential energy of the system in the centrifugal force field and, simultaneously, increases the surface energy. To determine the conditions for phase separation, we consider a ring rotating in a horizontal plane without taking the force of gravity into account. Let us compare the energies for two cases of distribution of the twophase system in a horizontal ring. In the first case, the distribution corresponds to that shown in Fig. 2 when there is a minimum interfacial area. In the second case, the distribution corresponds to a phase-separated structure, in which the heavier liquid fills the outer part of the ring. In the first case, the surface energy can be (1) neglected: E γ ≈ 0 (in comparison with the Er value for the phase-separated structure) and the surface energy is (1)

E p = π r d ω ( ρ 1 + ρ 2 )/8. 2 2 3

2

In the second case, we find (2)

Eγ

= 2πrdγ ,

4 4 2 2 3 2 (2) E p = π r d ω   1 – ------ ρ 1 +  1 – ------ ρ 2 /8.   3π 3π  The critical condition for phase separation is derived by equating the change in the total energy of the system to zero: (2)

(2)

(1)

(1)

∆E = E p + E γ – E p – E γ

= 0.

As a result, we obtain the expression γ N 3 = -------------------------2 2 rd ω ( ∆ρ )

(8)

for the dimensionless number N3, which determines the condition for phase separation, and the corresponding expression for the critical rotation rate ω3: 12γ  1/2 - . ω 3 =  ------------------ rd 2 ( ∆ρ )

(9)

Let us estimate ω3 by supplementing the above parameters with the characteristic value of interfacial tension γ = 3 × 10–2 N/m; this gives us ω3 ≈ 5 prs. Taking into account the force of gravity for such a rotation leads to a change in the direction and magnitude of the total acceleration (it is this parameter that causes phase separation). The centrifugal acceleration must be replaced by the expression ((rω2)2 + g2)1/2. For the parameter values used above, the ratio g/(rω2) = 0.2; therefore, neglecting the force of gravity is quite a satisfactory approximation. For planetary rotation, according to expressions (1), account must be taken of the dependence of the radial component of the acceleration on the angle θ, which leads to some difference from expression (8). Note, however, that, for planetary rotation, the radial force depends on the angle of rotation; therefore, the equilibrium (at each moment of time) distribution of the phases can be maintained only by the continuous relative motion of phases along the tube. The equilibrium shape of the interface is determined by equipotential levels. Under the action of only the force of gravity, these levels are horizontal planes. In the above case of phase separation in the ring rotating in a horizontal plane, equipotential surfaces are cylindrical (if the force of gravity is neglected) or conic (if the force of gravity is taken into account). Let us determine the shape of equipotential levels as the function re(θ) for a ring rotating in a vertical plane, with allowance made simultaneously for gravitational and centrifugal forces. We find the function re(θ) from the condition that the total acceleration along an equipotential surface be equal to zero: a r dr + a t rdθ = 0.

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BEHAVIOR OF LIQUID–LIQUID TWO-PHASE SYSTEMS

Substitution of the expressions for ar and at yields the equation for the dimensionless ratio m = re /r: dm ( m + α cos θ ) ------- – αm sin θ = 0, dθ

2

1

(10)

where the dimensionless parameter α = g/(rω2) is introduced. The solution of Eq. (10) has the form m ( θ ) = α ( cos ( θ ) + A – cos ( θ ) ),

2

355

(11)

where A is an arbitrary constant determined from the value of m(θ) at a given θ. It is easy to show that the equipotential levels in this case are circles, the centers of which have coordinates (0, αr), provided that the center of the ring coincides with the coordinate origin. When analyzing the derived expression for m(θ) and the shape of equipotential levels, note that the shape of the interface inside the tube is considered, i.e., m can vary only in the range

Fig. 3. Relative motion of the (1) first and (2) second phases in a plane channel.

placement of the phases, whose dynamics is found by solving the Navier–Stokes equation for the two-phase system. An approximate estimate of the critical condition for transition to the relative motion of the phases in the plane horizontal channel under the action of the force of gravity is made under the assumption that the deformed interface is an inclined plane (Fig. 3). The difference of the potential energies for the state with the deformed interface and the initial state is ∆E p = – ( ∆ρ )gδh /12, 3

1 – d/ ( 2r ) < m < 1 + d/ ( 2r ).

and the difference of the interfacial tension energies is

Equipotential levels for J-type planetary rotation are found from expressions (10) and (11) by substituting the variables α 1/(4β). Equipotential levels for I-type planetary rotation are determined from the levels that correspond to rotation in the field of gravity, if one first neglects the centrifugal force (α ∞) and then substitutes variables g Rω2. Let us now consider the behavior of a two-phase liquid system in a partitioned ring, which models a turn of a coiled column with closed ends (Fig. 2c). As noted above, in such a ring, plug flow of the liquid is impossible and the only motion possible is the relative displacement of liquid layers, belonging to different phases, due to the difference of their densities under the action of the buoyancy force. Such a displacement is accompanied by a change in the potential energy of the system in a field of external forces (the force of gravity and/or centrifugal force) and by a change in the surface energy due to a change in the interfacial area. It is clear that, if the tube is sufficiently thin, the interfacial tension can prevent the relative displacement of the phases along the tube, as follows from expressions (8) and (9) derived in analyzing the conditions for phase separation. To make an approximate analysis of the initial stage of the relative displacement of the phases in the tube with closed ends, we consider a plane horizontal channel of height h (Fig. 3). At the initial moment of time, the liquid phases are separated by the plane vertical interface, under the assumption that the adhesion of the liquids to the channel walls is the same. Under the action of the force of gravity, the interface deforms, which tends to prevent the relative displacement of the liquids. The rigorous solution of this problem involves the determination of the exact equilibrium shape of the deformed interface. The absence of a solution describing the immobile interface, i.e., corresponding to a certain equilibrium state, means that there is relative dis-

∆E γ = γ h ( 1 + ( δ/h ) – 1 ), 2

2

where δ is the length of the transition zone, in which both liquids separated by the inclined surface are located. Equating the total energy of the system to zero, ∆E = ∆Ep + ∆Eγ = 0, we obtain, at δ/h Ⰶ 1, the dimensionless number N4: γ N 4 = --------------------. 2 gh ( ∆ρ )

(12)

After substituting the variables g rω2, N4 appears as N2; the critical value of N2 is 1/12. Passing from the plane channel to the tube and replacing the channel height h by the tube diameter d, at the critical diameter value dcr we obtain 12γ 1/2 d cr =  --------------- .  g ( ∆ρ )

(13)

For the simplest air–water system, the critical tube diameter is dÒr ≈ 8.5 × 10–3 m. The table presents the values of the parameters involved in expression (13) and the corresponding dcr values for some two-phase systems used in rotating coiled columns. Expression (13) is the upper-bound estimate of the condition for the relative displacement of the liquids, because it was derived under the assumption that the inclined-plane shape of the interface an remains unchanged until δ/h > 1. In all likelihood, the relative displacement in the form of plane layers actually begins at δ/h ≈ 1, which leads to the coefficient 5 instead of 12 in expression (13) and, correspondingly, to the value dcr = 5.5 × 10–3 m for the air–water system. Further experimental verification of the derived expressions characterizing the critical conditions for transition between different modes for model systems

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KAMINSKII, MARYUTINA

Parameters of some two-phase systems used in rotating coiled columns γ × 103, N/m

∆ρ × 10–3, kg/m3

dcr × 102, m**

(5% D2EHPA solution in CHCl3)–(10% (NH4)2SO4 solution in water)

4.2

0.40

0.36

(5% D2EHPA solution in CHCl3)–(5% (NH4)2SO4 solution in water)

5.0

0.43

0.38

(5% D2EHPA solution in CHCl3)–(2.5% (NH4)2SO4 solution in water)

6.8

0.44

0.43

CHCl3–(10% (NH4)2SO4 solution in water)

7.8

0.43

0.47

System*

CHCl3–(5% (NH4)2SO4 solution in water)

8.8

0.46

0.48

(5% D2EHPA solution in CHCl3)–(15% NH4Cl solution in water)

11.0

0.43

0.56

CHCl3–(2.5% (NH4)2SO4 solution in water)

13.0

0.47

0.58

(5% D2EHPA solution in CHCl3)–H2O

15.0

0.45

0.64

CCl4–H2O

22.0

0.59

0.68

(5% D2EHPA solution in CCl4)–(10% (NH4)2SO4 solution in water)

18.0

0.48

0.68

CHCl3–H2O

19.0

0.48

0.70

(5% D2EHPA solution in CCl4)–H2O

21.0

0.53

0.70

CCl4–(10% (NH4)2SO4 solution in water)

22.0

0.54

0.71

(5% TBP solution in n-decane)–H2O

14.0

0.26

0.81

(5% TOA solution in n-decane)–H2O

17.4

0.27

0.89

(5% D2EHPA solution in n-decane)–H2O

18.0

0.26

0.92

CH2Cl2–H2O

27.0

0.31

1.03

n-C6H14–H2O

45.0

0.32

1.31

n-C10H22–H2O

39.0

0.27

1.33

1,4-Dichlorobutane–H2O

21.0

0.14

1.35

(5% D2EHPA solution in n-hexane)–H2O

34.0

0.20

1.44

* D2EHPA, di-2-ethylhexyl phosphoric acid; TBP, tributyl phosphate; TOA, trioxylamine. ** Results of calculations by expression (13).

will give deeper insight into the physical processes determining the behavior of two-phase liquid systems in rotating coiled columns. ACKNOWLEDGMENTS This work was supported by INTAS, project no. 2000-00782. NOTATION at, ar—accelerations of a liquid in the tangential and radial directions, respectively, m/s2; d—tube diameter, m; g—acceleration due to gravity, m/s2; Ep, Eγ—potential energy of a two-phase system and the interfacial tension energy, respectively, J; h—height of a plane channel, m; m = re /r;

R—radius of revolution of a column, m; r—radius of rotation of a column, m; re—radial coordinate of an equipotential surface, m; t—time, s; 〈v〉—flow velocity averaged over a tube section, m/s; ph—hydrostatic pressure, N/m2; α = g/(rω2); β = r/R; γ—interfacial tension, N/m; η1, η2—dynamic viscosities of the liquid phases, Pa s; ρ1, ρ2—densities of the phases, kg/m3; ∆ρ = ρ1 – ρ2, kg/m3; θ—angle of rotation of a column, rad; ω—circular rotation rate, rad/s.

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Chromatography, Chromatographic Science Series, New York: Marcel Dekker, 1999, vol. 82, p. 29. Conway, W.D., Countercurrent Chromatography. Preface, J. Chromatogr., 1991, vol. 538, no. 1, p. 1. High-Speed Countercurrent Chromatography, Chemical Analysis Series, Ito, Y. and Conway, W.D., Eds., New York: Wiley, 1996, vol. 132. Countercurrent Chromatography: Apparatus, Theory, and Applications, Conway, W.D., Ed., New York: VCH, 1990. Ito, Y., High-Speed Countercurrent Chromatography, CLC Critical Reviews in Analytical Chemistry, 1986, vol. 17, no. 1, p. 65. Menet, J.-M., Thiebaut, D., and Rosset, R., Classification of Countercurrent Chromatography Solvent Systems on the Basis of the Capillary Wavelength, Anal. Chem., 1994, vol. 66, no. 1, p. 168.

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