Multicomponent multiphase film-like systems: A modelling approach

Computers" chem. Engng, Vol. 21, Suppl., pp. $355-$360, 1997

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Multicomponent multiphase film-like systems: A modelling approach E.Y. Kenig Dortmund University, Chemical Engineering Department, 44221 Dortmund, Germany

Abstract - In this work a general approach is proposed for modelling of multicomponent multiphase film-like systems. It integrates into a single whole the Maxwell-Stefan formulation of multicomponent diffusion and partial differential equations of continuum mechanics. The former enables multicomponent diffusional interactions (crosseffects) to be taken into account, the latter makes it possible to avoid design trouble caused by the employment of mass-transfer and heat-transfer coefficients. The proposed approach is applicable to a variety of multicomponent separation processes where phase boundaries and interfaces can be properly determined. In the present work attention is focused on the application to the processes with the film-like hydrodynamics and geometry. These include both film flows and more complicated structures allowing hydrodynamic analogy between simple and complex flow to be established, in particular, porous media and structured packing flows. The application is illustrated with an example of multicomponent liquid film pertraction - a three-phase membrane extraction in falling liquid films. INTRODUCTION The separation processes in current industrial applications are extremely complicated. They combine into a single whole a number of various phenomena, such as fluid flows and physicochemical interactions, phase transitions and interface development, chemical reactions, etc. Therefore, their proper modelling is a very difficult challenge. The process equipment itself gives rise to the major difficulties: actually, in terms of practical tasks we are interested in the most developed contact area. However it results in so much complicated flow patterns of the process that they are often very difficult to imagine, and so it is impossible to locate physical boundaries, to define flow fields, and eventually to apply the classical equations of continuum mechanics. This is the most important reason why separation equipment design is usually accomplished using simplified theoretical considerations like the equilibrium stage model (see, e.g., Henley and Seader, 1981, Taylor and Krishna, 1993). The key feature of the equilibrium stage model is that the gas or vapour streams leaving any arbitrary stage of the apparatus are at equilibrium with the leaving liquid streams. In reality separations almost never reach the equilibrium, and the improvement of the results is achieved by the use of stage efficiences responsible for hydrodynamics, mass transfer, interface interactions, etc. However, if we face multicomponent multiphase systems, and this is practically always the case when we deal with industrial separation, things are much more complicated. Diffusion interactions lead to unusual phenomena, like, for example, mass transport of the component in the direction opposite to its own driving

force - the so-called reverse diffusion (Tool 1964). For multicomponent systems stage efficiencies for different components differ from one another and may range from - o o to +oo. Therefore the use of the equilibrium stage model cannot be recommended in this case. It is necessary to operate sufficiently closer to the essence of the separation process having regard to actual mass transfer rates and relevant phenomena. Multicomponent nature requires that mutual diffusional interactions of components taking place within the contacting phases be taken into account via the Maxwell-Stefan relationships which in turn result in matrix-form coupled mass transfer equations for the theoretical description (Tool 1964, Stewart and Prober, 1964, Taylor and Krishna, 1993). Because of multiphase character of separation processes it is necessary to take a proper account of the interface phenomena. In the case of multicomponent systems the interface relationships connecting the parameters of two neighbouring phases acquire a matrix coupled form as well (Taylor and Krishna, 1993, Kenig, 1994). Hence general theoretical description of multicomponent multiphase mass transport processes is based on strongly coupled matrix mathematical models. The range of accuracy is closely related to the hydrodynamic assumptions. For many industrial processes where the hydrodynamic pattern cannot be described by the rigorous equations of continuum mechanics, it is substituted by the two-film, penetration or other similar simplified model pictures (Taylor and Krishna, 1993, Kenig, 1994). In this case it is necessary to make use of gross coefficients like mass- and heat transfer coefficients or HTU (height of a transfer unit). For the separation processes taking place in

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geometrically simple flows like films, jets, drops, etc. physical boundaries of the contacting phases are spatially localized and hence can be well described by the relevant boundary conditions (Kenig, 1994). In this case partial differential equations of continuum mechanics (Sedov, 1971) allow the most rigorous models of the process hydrodynamics to be applied. By employing partial differential transport equations we can determine all the necessary process parameters on the basis of local data and avoid using the gross coefficients mentioned above. APPLICATION OF THE HYDRODYNAMIC ANALOGIES Perhaps of even greater importance is the opportunity to use such rigorous methods also for some complex hydrodynamic conditions, by application of the idea of hydrodynamic analogy between complex and simple flows. Different authors, e.g. Zogg (1973) and Rocha et al. (1993) made use of this technique describing the perfomance of structured packings. Zogg (1973) developed a model based on geometrical consideration of a Sulzer packing. His analysis allows the calculation of the actual film thickness (according to hydrodynamic relationships) and the length of the undisturbed laminar liquid flow (given by the packing geometry). Based on his geometric model and the Sherwood analogy, Zogg (1973) determined the liquid-phase mass transfer coefficients and the average vapour side Sherwood number. Similar, Rocha et al. (1993) developed a mechanistically-based model for the description of structured packings of the corrugated plate type. It considers the interaction of falling liquid films with upflowing vapour (wetted-wall flow channel) and thus takes into account the flow and physical property characteristics of the system as well as geometric variables. In such a manner the flow description is reduced to a combination of falling liquid films. Rigorous modelling of the latter is well established concerning both hydrodynamics (Levich, 1962; Grossman, 1986) and mass and heat transport (Kenig, 1994; Kenig, 1995a). Another application is presented by porous media flows (Lazarova and Boyadzhiev, 1992; Tschernjaew et al., 1996). Again, the calculation methods for multicomponent mass and heat transfer based on the partial differential transport equations can be directly applied to the modelling of complex seperation units (see Kenig and G6rak (1996)). Thus the essence of the hydrodynamic analogy idea is to use a simplified hydrodynamic presentation so that to make a transfer from a complex flow pattern to a filmlike (or other similar) one. Further, to apply matrixbased calculation methods developed for various flow regimes and physicochemical conditions and employing partial differential equations of convective mass and heat transport (Kenig, 1994). In this case the exactness of the modelling is directly connected with the accuracy of the hydrodynamic reduction and, of course, with the

correct estimation of the model parameters like the film thickness, the length of the undisturbed laminar liquid flow, etc. In this work an illustration of this approach is given for an example of multicomponent liquid membrane separation. DESCRIPTION OF THE PROCESS Membrane separation is one of the most promising modern technological processes characterized by high separation efficiency. This process is particularly profitable when it proceeds in moving systems. In this case mass transfer occurs between fluid phases, their motion is defined by specified flow conditions. Separation is achieved due to the difference in thermodynamic and diffusional properties of the components. The simplest system of this sort consists of three flowing phases, the middle one playing a role of a moving fluid membrane (Boyadzhiev, 1990, Lazarova and Boyadzhiev, 1992). More complex processes may contain more phases separated by the relevant interfaces. Let us consider one of the most known membrane separation processes, namely pertraction in falling liquid films (Lazarova and Boyadzhiev, 1992). This process presents, essentially, a combination of extraction and re-extraction between three liquid films flowing cocurrently (Figure 1). The middle film is a moving liquid membrane. Diffusion from the first liquid

0

Yl 0

0

Y2 Y3

~V

~T

Ul

U2

U3

hi

h2

h3 ~1 ~

I

IP

H

Figure 1. Schematic diagram of the pertraction process

film (raffinate) through the second one (membrane) to the third film (extract) r~sults in the separation of the species involved in the process. Once we consider a multicomponent process, we obtain three moving liquid multicomponent phases separated by the two interfaces. Correct theoretical description of such a process can be given by closely coupled mathematical models (Kenig, 1994).

PSE'97-ESCAPE-7 Joint Conference GOVERNING EQUATIONS Let us take a look at a system containing n components. Mass transfer in each phase is described by partial differential convective diffusion equation in matrix formulation. For the case of downward liquid film flow this equation takes the following form (Levich, 1962, Grossman, 1986) OC k ~2C k uk(Yk) ~ = [Ok]'.~ 2 ;

k=1,2,3

(1)

oY k

The boundary conditions for the system (1) are: - at the entrance x = 0,

(2)

Ck= C0k

- at the left wall of the channel bC~ Yl = 0,

OYl - 0

(3)

- at the right wall of the channel Y3 = 0,

bC3 0Y3 - 0

(4)

At the both interfaces the fundamental laws of equilibrium and mass conservation are observed (Yl = h l ) & ( Y 2 = 0),

(5)

C2= [M 1] C 1 0C 2

(6)

(a)

(7)

C2= [M3] C 3 t)C 2

This is a very importnat point which differentiates socalled one-phase problems from two-phase or multiphase ones. If a problem is formulated as a onephase problem, then the interface can be descibed as if there were no real interactions between the contacting phases. In terms of mathematical principles, the description of the interface is accomplished via the boundary conditions of the first, second or third kind (Bird et al., 1960, Luikov, 1968). In such cases the equations and boundary conditions covering a problem involve the same potentials. Regarding coupled mass transfer equations, this means that they can be reduced to a completely uncoupled form, for example via the technique suggested by Toor (1964) and Stewart and Prober (1964). Unfortunately, this is not the case when the problem is considered as really multiphase (Kenig, 1994). On the other hand most of actual chemical engineering processes are at least two-phase processes, and this does result in multiphase formulations of the relevant mathematical models. Thus for the process considered in this work there is no way to obtain completely uncoupled system, and a special technique must be worked out to overcome this difficulty. The hydrodynamic part of the problem is tackled independently of the mass transfer part (1)-(8) to give the velocity profiles uk(Yk) and the liquid film thicknesses hk. The problem is reduced to the system of nonlinear algebraic equations which is solved by the Newton-Raphson method. Then the values of uk and hk are substituted in the system (1)-(8).

OC 1

[D2I Oy---~= [D 11 Oy---~

(Y2 = h2) & (Y3 = h3)'

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t~C3

[DE] 3y---~= -[D3] ~

6

(8)

//~0.1

U° cln/$

The vectors and matrices entering into eqns. (1)-(8) are of dimension (n-l). The minus sign in eqn. (8) is due to the opposite directions of the Y2-axis and the Y3-axis. The velocity profiles in eqn. (1) are determined by the system of Navier-Stokes equations in the film-flow approximation (Levich, 1962, Grossman, 1986).

12 u , ¢ffl/$

SOLUTION Basically, the difficulties in solving the system (1)-(8) result from the coupled nature of the transport equations (1) and conjugate form of the interface boundary conditions (5)-(8). Eqns. (5)-(8) constitute a matrix generalization of the so-called boundary conditions of the forth kind (Luikov, 1968). The solution of the system (1)-(8) is based on the matrix transformation of eqns. (1) to an uncoupled form. However, conjugated boundary conditions at the interface (5)-(8) cannot be uncoupled simultaneously with eqns. (1)-(4) since they connect parameters and variables of two different phases.

cm

h~

__

h2

_ ~ h3 .

Figure 2. The flow velocity patterns in the pertractor at different liquid film densities and viscosities.

Mass transfer equations (1) are rearranged to an uncoupled form with the use of the diagonalization technique (Toot, 1964, Stewart and Prober, 1964), and the transformed system is solved by the modification of the so-called sweep method (implicit finite-difference method which is well suited for parabolic partial differential equations). The sweep method is similar to

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the Thomas algorithm called also TDMA- TriDiagonalMatrix Algorithm (see Patankar (1980)). However, application of the sweep method is hampered by the presence of the two interfaces simultaneously. Actually, there is no way to begin the ,,back-substitution" (Patankar, 1980), since the boundary conditions at the both interfaces are of a conjugated nature (the boundary conditions of the forth kind - see above). For this reason a special iterative procedure is developed to recalculate interracial concentrations of the components. It is worth noting that instead of eqns. (5) and (7) we can apply the equilibrium relationships at the interface written more generally (Kenig et al. 1993), for example

C2= fk (Ck) ;

k=l,3

(9)

important points of the process under study, thus obtaing a crucial information on the process development (see Kenig, 1995b). MORE COMPLICATED SYSTEMS The proposed approach to the modelling of three-phase membrane extraction can be modified to describe other multicomponent multiphase separation processes. It is possible, for example, to model three-phase membrane extraction taking place between two moving liquid films through an immobile liquid phase. It is also possible to C+, kmol/m 3 2 - I

In this case a double iterative algorithm must be incorporated for solving the problem. EXAMPLE OF CALCULATION By way of example, let us consider pertraction process in a three-component model system. Once physicochemicai and hydrodynamic parameters of the system have been defined the process behaviour may be analyzed. First, the hydrodynamic part of the problem must be solved. The solution not only provides us with the velocity profiles and film thicknesses necessary for mass transfer calculations, but also gives a way of flow stability estimation. In particular, it is found that when otgH3-Qv2 ,

0, 001 < c~<0, 015

(10)

with H = hl+h2+h 3, and Q - volumetric flow rate related to the channel depth, it is possible to ensure stable gravitational three-film flow in the pertractor. Figure 2 presents examples of the velocity distribution in the phases obtained for Re I = 30, Re 2 = 100, and Re 3 = 20, for two different combinations of the phase densities and viscosities. It can be seen in Figure 2a that the left and right films "accelerate" the middle one, whereas in the Figure 2b the flow character is different-looking, namely the left and right films "slow down" the middle film. Both examples satisfy the relation (10). Figure 3 demonstrates the results of mass transfer calculation, namely, local concentration profiles of the two transferring components. These local distributions constitute the basis of all the relevant information about the process (average concentrations, mass fluxes, etc.). As examples, profiles of the average concentrations (integral mean over the film thickness values) and boundary concentrations are demonstrated in Figures 4 (raffinate phase) and 5 (extract phase). These profiles are obtained by direct processing of the local data shown in Figure 3. Consequently, mass transfer process behaviour is completely characterized by the obtained local concentration fields. For example, it is possible to locate the change in the transport direction of a component, extrema and other

2

O~

~

2+

(o'>

0

1 0 2

~0 .__o

d

Figure 3. Distributions of local concentrations of the components over the liquid film thicknesses at different distances from the pertractor entrance (solid lines, 1st component; dash lines, 2nd component): ~ = 10-! (a); 100 (b); 101 (c); 102(d); 103(e); 104(f), where ~ = 4xl(Relhl), Re I is Reynolds number for the 1st film. consider processes including more contacting phases and, correspondingly, more interfaces. In this case an iterative procedure for the calculation of the interface parameters gets more complicated, but the key features of the suggested method remain unchanged. Countercurrent flow regimes (Boyadzhiev, 1990) lead to further complication in the process description, nevertheless they can be dealt with in a similar manner.

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Cli, k m o l / m 3

222 S55525 2 0

"

'

10-1

~

101

r

,

,

103

~I

105

Figure 4. Component concentrations in the phase 1 (raffinate) as functions of the pertractor length (solid lines conform to average values, dash lines to values at the interface and dash-and-dot lines to values at the left wall: 1 - 1st component, 2 - 2nd component.

nature of the separation processes requires also rigorous account of the interface phenomena. Partial differential equations of continuum mechanics together with the idea of hydrodynamic analogy provide a good alternative to simplified process models. The proposed approach should be verified using some relevant practical applications. By way of example, the process of pertraction (threephase membrane extraction) in falling liquid films is considered. A mathematical model is developed on the basis of partial differential transport equations in matrix conjugate formulation. A relevant calculation method is suggested which includes the Newton-Raphson method (hydrodynamic part) and a combination of the sweep method and iterative procedure (mass transfer part). The solution of the problem considered enables the distributions of local concentrations of all components in a multicomponent mixture to be determined; the values can then be used for the direct calculation of any arbitrary characteristic of the process. In addition, the suggested calculation method gives a clue to the modelling of further multicomponent membrane separation processes in film-like systems. NOTATION = column vector of molar concentrations of

Ck multicomponent multi-layer or multi-film separation processes.

components in kth phase, kmol/(m 3 s) [Dk] = matrix of multicomponent diffusion

CONCLUDING R E M A R K S The problems arising in modelling of multicomponent mixture separation are analyzed. For the model development an approach based on the kinetic description of mass transfer via the Maxwell-Stefan relationships should be recommended. The multiphase

fk

coefficients in kth phase, m2/s = nonlinear vector functions (eqn. (9))

g = gravity, m/s 2 hk = the thickness of kth phase, m [M1],[M3] = matrices consisting of equilibrium parameters Q -- overall volumetric flow rate related to the channel depth, m2/s = velocity of directional motion of kth phase, rn/s = cartesian coordinates, m x, Yk = constant parameter (eqn. (10)) = kinematic viscosity of the 2nd liquid film, v2

C3i, k m o l / m 3

Uk(Yk)

/

m2/s

Subscripts: i k 0

= ith component = kth phase = entrance value

i

10-1

101

103

105

Figure 5. Component concentrations in the phase 3 (extract) as functions of the pertractor length (solid lines conform to average values, dash lines to values at the interface and dash-and-dot lines to values at the right wall: 1 - 1st component, 2 - 2nd component.

REFERENCES Bird, R. B., Stewart, W. E. and Lightfoot, E. N., 1960, Transport Phenomena. Wiley, New York. Boyadzhiev, L., 1990, Liquid pertraction or liquid membranes - state of the art. Separ. Sci. Technol. 25, 187-205. G. Grossman, 1986, in N.P. Cheremisinoff (ed.), Handbook of Heat and Mass Transfer, Vol. H, Gulf Publishing Company Book Division, Houston, 211-232. Henley, E.J. and Seader, J.D., 1981, Equilibrium Stage Separation Operations in Chemical Engineering. Wiley,

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New York. Kenig, E.Y., 1994, Studies into kinetics of mass and heat transfer during the separation of multicomponent mixtures: Part I. Theor. Found Chem. Engng 28, 199216. Kenig, E.Y., 1995a, Combined processes in multicomponent fluid systems: simulation and design. Computers chem. Engng 19, Suppl., $287-$292. Kenig, E.Y., 1995b, Mass transfer-reaction coupling in two-phase multicomponent fluid systems. Chem. Engng Journal 57, 289-204. Kenig, E. Y. and G6rak, A., 1996, Proper modeling of multicomponent multiphase systems: is it possible? Proc. 5th World Congress of Chemical Engineering, San-Diego, V. 1, p. 365-370. Kenig, E.Y., Kholpanov, L.P. and Malyusov, V.A., 1993, Equilibrium relationships in conjugate multicomponent heat and mass transfer problems. Proc. of the Acad. Sci., Chem. Technol. Section 330, 17-21. Lazarova, Z. and Boyadzhiev, L., 1992, Liquid film pertraction - a liquid membrane preconcentration technique. Talanta 39, 931-935. Levich, V.G., 1962, Physicochemical Hydrodynamics. Prentice Hall, Englewood Cliffs, New Jersey.

Luikov, A.V., 1968, Analytical Heat Diffusion Theory. Academic Press, New York. Patankar, S.V., 1980, Numerical Heat Transfer and Fluid Flow. Hemisphere Publ. Corp., McGraw-Hill, New York. Rocha, J.A., Bravo, J.L. and Fair, J.R., 1993, Distillation columns containing structured packing: a comprehensive model for their performance. 1. Hydraulic models. Ind. Eng. Chem. Res., 32, 641-651. Sedov, L.I., 1971, A Course in Continuum Mechanics. Wolters-Noordhoff Publ., Groningen. Stewart, W.E. and Prober, R., 1964, Matrix calculation of multicomponent mass transfer in isothermal systems. Ind. Eng. Chem., Fundam. 3, 224-235. Taylor, R. and Krishna, R., 1993, Multicomponent mass transfer. Wiley, New York. Toor, H.L., 1964, Solution of the linearized equations of multicomponent mass transfer. II. Matrix methods. AIChE Journal 10, 460-465. Tschernjaew, J., Kenig, E.Y. and G6rak, A., 1996, Mikrodestillation von Mehrkomponentensystemen. Chem.-Ing.-Techn., 68, 272-276. Zogg, M., 1973, Stoffaustausch in der Sulzer Gewebepackung. Chem.-Ing.-Techn., 45, 67-74.