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Stability analysis in multicomponent drying of homogeneous liquid mixtures
Chemical Engineering Science 54 (1999) 5823}5837
Stability analysis in multicomponent drying of homogeneous liquid mixtures F. Luna*, J. MartmH nez Department of Chemical Engineering and Technology, Royal Institute of Technology, (KTH), S-100 44 Stockholm, Sweden Received 27 July 1998; received in revised form 12 March 1999; accepted 16 March 1999
Abstract A stability analysis of the ordinary di!erential equations describing the process of convective gas-phase-controlled evaporation during drying is performed. Isothermal and non-isothermal as well as batch and continuous drying processes are considered. For isothermal evaporation of a ternary mixture into pure gas, the solutions of the di!erential equations are trajectories in the phase plane represented by a triangular diagram of compositions. The predicted ternary dynamic azeotropic points are unstable or saddle. On the other hand, binary azeotropes are stable when the combination of the selectivities of the corresponding components is negative. In addition, pure component singular points are stable when they are contained within their respective isolated negative selectivity zones. Under non-isothermal conditions, stable azeotropes are characterized by presenting maximum temperature values. Loading the gas with one or more of the components up to some value leads to a node-saddle bifurcation, where a saddle azeotrope and a stable azeotrope coalesce and disappear. The continuous drying process yields similar results for both #at and annular geometries. The singular points, in this case, are in"nite and represent dynamic equilibrium points whose stability is mainly dependent on the inlet gas-to-liquid #owrate ratio. As this ratio grows to in"nity, the phase portrait changes and the process approaches a batch behaviour so that the stability analysis for that case may be applied. The present stability analysis permits the prediction of trajectories and "nal state of a system in a gas-phase-controlled drying process. 1999 Elsevier Science Ltd. All rights reserved. Keywords: Azeotropic points; Convective drying; Di!erential equations; Multicomponent systems; Selectivity; Stability
1. Introduction One of the main concerns in multicomponent drying is the composition of the "nal moisture because of its crucial in#uence on product quality. Since separation processes normally do not run to completion, a residual moisture content will frequently remain in the solid after drying } due to either equipment limitations or process requirements. For instance, when pharmaceutical granulates are dried, the aim is often to remove primarily organic solvents because some residual water content is required in subsequent tableting of the granulates. If the moisture consists of water or another single solvent, the "nal liquid content after drying is determined by the equipment dimensions and the operating conditions. Moreover, when the moisture consists of a mixture, the "nal state of the drying process depends on the initial conditions. In multicomponent mixtures, these condi-
* Corresponding author. Tel.: 0046-8-7906569; fax: 0046-8-212-747.
tions can vary considerably and the number of possible alternatives increases with the number of components in the mixture. For this reason, a manner to predict all possible process paths from known system properties, and process kinetics would be useful. In general, the inherent complexity of convective drying of solids containing liquid mixtures does not allow for such kind of analysis. The achievement of particular solutions becomes a di$cult task due to the complex interaction of gas-phase di!usion, external heat transfer, phase equilibrium, mass and heat transfer within the solid with strong dependence on liquid content, composition and temperature, and the contact mode between the phases. However, if evaporation is gas-phase-controlled, a broad understanding of the process may be gained by a stability analysis of the ordinary di!erential equations that describe the dynamical system. Gas-phase-controlled drying occurs frequently at an initial stage when the solid is saturated. It may also occur during long drying periods, at low drying rates, drying of thin products } such as "lms or coated laminates } or of bulky
0009-2509/99/$ - see front matter 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 0 0 9 - 2 5 0 9 ( 9 9 ) 0 0 1 7 1 - 2
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materials, provided that the moisture is super"cially distributed as a thin layer throughout the drying process. These conditions may also be encountered during lowtemperature drying and in sections of continuous equipment where solids are in contact with an increasingly enriching vapour atmosphere. Considerable work has been devoted to the study of the dynamics of distillation processes. In a series of papers, Doherty and Perkins (1978a,b,1979) analysed mathematically the ordinary di!erential equations which describe the general characteristics of multicomponent, non-ideal distillation processes and generated a residue curve map to trace composition changes in a perfectly mixed liquid. Van Dongen and Doherty (1985) approximated by simple distillation}residue curves liquid composition pro"les in continuous staged or packed distillation columns. When the ideal solution model at sub-critical conditions is applied, non-elementary singular points cannot occur. In addition, for ternary mixtures they found that there is a topological relationship between the singular points known as azeotropes and singular points corresponding to pure components. Rules for constructing qualitative distillation region diagrams based only on pure component data and azeotrope boiling-point data were given by Foucher, Doherty and Malone (1991). The presence of an inert gas makes multicomponent drying by convection a more complex process than distillation since not only thermodynamic equilibrium is involved but also di!usion in gas phase which in mixtures of more than two components is usually interactive. Pakowski (1994a) analysed the stability of dynamic azeotropes in ternary mixtures evaporating into air assuming a perfect liquid-phase-mixing model. He found that trajectories tend towards the local minimum of the sum of absolute values of all individual molar evaporation #uxes. Luna and MartmH nez (1998) applied a stability analysis to an isothermal batch case with unchanged gas-phase composition. The purpose of this paper is to present a qualitative analysis of process dynamics in the cases where conditions lead to a gas-phase-controlled drying process. First, this study considers batch drying of a sample under constant gas conditions of a solid containing liquid mixtures. Then, attention is devoted to the continuous drying of a liquid mixture where the conditions of both liquid and gas phases change along the equipment. Although the analysis applies primarily to a liquid "lm, it is also valid for a solid containing a mixture as long as the process is gas-phase controlled.
2. Theory The design and choice of optimal operating conditions in convection dryers require information about moisture} solid equilibrium, drying rates as a function of moisture
content, and gas}solid contact pattern in the equipment. If the solid contains a multicomponent moisture, information about the changes in liquid composition during the process is additionally demanded. Composition curves are crucial because of the great in#uence of the remaining liquid on the properties and quality of the product. The variable which describes the ability of a species to be preferentially removed or not from the moisture is the selectivity. It is de"ned, in vector form, as follows: 1 s" G!x, G 2
(1)
where G is a column vector of molar drying rates, G is 2 the total molar drying rate, and x is a vector of molar fractions in the liquid phase. Depending on the values of the elements of s the following cases can be distinguished: s '0, component i is preferentially removed and its G molar fraction in the moisture decreases; s "0, G evaporation is non-selective and the molar fraction of the species i remains constant; s '0, component i is not G preferentially removed and its molar fraction increases. Like drying rates, drying selectivity depends on the intricate mechanisms behind the transport of solvents from within the solid to the solid}gas interface and subsequently into the gas phase. In gas-phase-controlled drying, selectivity depends only on the external conditions, that is, on vapour}liquid equilibrium, #ow pattern and gas conditions. External conditions have also a decisive in#uence on the appearance of internal resistance. During low intensity drying, internal resistance will normally not arise. In modelling gas-phase-controlled drying, the description of the dynamic system leads to ordinary di!erential equations. In this case, a deep understanding of the process may be obtained by a qualitative stability analysis of the governing ordinary di!erential equations. Two cases are important in this context: batch drying, where a certain amount of moist solids is dried periodically; and drying in continuously worked equipment. In the former case, the conditions of the solid change with time. In the latter, the conditions of the solid change as a function of the position along the dryer. 2.1. Batch drying Fig. 1 shows schematically batch drying of a wet solid, over which a hot gas stream of constant temperature #ows parallel to the exposed surface. When the gas #owrate is in excess compared to the vapours transferred between the phases, the changes of the gas conditions during the process can be neglected. This situation is achieved in batch drying experiments performed to determine drying characteristics of solids where very narrow samples are normally used. MartmH nez
F. Luna, J. Martn& nez / Chemical Engineering Science 54 (1999) 5823}5837
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If it is assumed that the phases are in equilibrium at the interface, the coupling between the gas composition y and the liquid composition x at the interface is given by the following expression: 1 y " pCx, * P Fig. 1. Schematic description of batch drying by convection of a solid wetted with a liquid mixture.
and Setterwall (1991) developed and veri"ed experimentally a model to describe this case. For a system consisting of n condensable components, the governing equations are: dn *"!AG, dt
(2)
A(q!+H !H ,2G) d¹ E * *" , (3) C2n #c m dt * * Q Q where n is a column vector of number of moles of each * species in the liquid mixture, A is the surface of the wet solid exposed to the gas stream, q is the heat #ux which reaches the drying surface by convection, H is a vector of molar enthalpies, C is a vector of molar heat capacities, c is a mass heat capacity; and m is mass. The subscripts g, ¸, and s denote gas, liquid and solid phase, respectively. The superscript ¹ denotes transposition. Eqs. (2) and (3) consist of a set of n#1 ordinary di!erential equations. These equations describe the variation of sample temperature, moisture content, and number of moles of the di!erent species in the liquid mixture. For a particular solution, the initial conditions are given by n (t )"n , ¹ (t )"¹ . (4) * * * * To integrate Eqs. (2) and (3), expressions for the mass and heat transfer rates must be provided. If di!usional interactions are considered, mass and heat #uxes may be calculated as follows: G"bkN+y !y,, (5) * q"aN (¹ !¹ ), (6) F E * where matrix b incorporates the convective contribution to mass transfer, k is the matrix of zero-#ux mass transfer coe$cients, N is the matrix of correction factors which accounts for the in#uence of "nite mass transfer rates upon the mass transfer coe$cients, N is the correspondF ing correction factor for the heat transfer coe$cient a, and y is the vector of molar gas compositions. The parameters in the #uxes may be calculated according to Taylor (1982). See Bird, Stewart and Lightfoot (1960) for the calculation of the correction factor for the heat transfer coe$cient.
(7)
where p and C are diagonal matrices of pure component saturated vapour pressures and activity coe$cients, respectively; and P is the total pressure. Note that the analysis is not restricted to this particular geometry. As long as the process is gas-phase controlled and the conditions of the gas do not change appreciably during the process, the analysis also applies to other drying methods, such as #uidized bed or spray drying, provided that suitable transfer coe$cients are used to calculate the mass and heat transfer rates. For the geometry described in Fig. 1 and conditions of streamline #ow, the results of the boundary layer theory may be used to calculate the mass and heat transfer coe$cients at zero mass transfer rates. For turbulent #ow, the Chilton}Colburn analogy is applicable (see Coulson & Richardson, 1996). 2.2. Continuous drying In a continuous drying process, the conditions of both phases change along the contact path. It can be seen in Figs. 2 and 3 where co-current contact mode of the phases in a #at and in an annular geometry are schematically depicted. Vidaurre and MartmH nez (1997) developed a model for the #at geometry. In both cases, mass and energy balances along the evaporation path are described by the same set of di!erential equations if the solid #ow is set to zero for the annular geometry dF *"!aG, dz
(8)
dF E"aG, dz
(9)
d¹ q!+H !H ,2G *"a E * , (10) dz C2F #c S * * Q d¹ !q E"a , (11) dz C2F E E where F is the vector of the number of moles passing through the dryer per unit time and cross section; a is the interfacial area per unit dryer volume; and S is the #ow of solid material associated to the liquid #ux. Eqs. (8)}(11) represent a set of 2n#2 ordinary nonlinear di!erential equations. They describe an initial-value problem where the location is related to the residence time in the contact device by the linear velocity of the phases and inlet
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conditions correspond to the initial values. If the liquid and gas have a uniform composition at z"0, the following conditions are applicable: F "F , F "F , ¹ "¹ , ¹ "¹ , * * E E * * E E S"S at z"0. (12) The calculation of mass and heat transfer rates in the continuous case is similar to the batch case, except for the transfer coe$cients and the progressively changing gas conditions. Therefore, Eqs. (5) and (6) are applicable in both cases provided that temperature and composition of the gas are regarded as functions of z. The method used in this work to analyse stability applies to autonomous di!erential equations, that is, when the right-hand sides do not depend explicitly on the independent variable. This is satis"ed by the batch case and the annular geometry. On the other hand, the
method to calculate the transfer coe$cients in a #at geometry with uncon"ned #ow leads to non-autonomous di!erential equations since they are functions of the position z along the dryer. However, if the transfer coe$cients are calculated at some average position along the dryer, using similar relationships as those for the batch case, an approximate analysis, still preserving the main feature of a continuous contact mode may be performed even in the #at geometry. 2.3. Singular points The vector "eld described by the ordinary di!erential equations (2)}(3) or (8)}(11) vanishes at the singular, equilibrium or critical points, which make the right-hand side of the system of di!erential equations equal to zero. For the sake of simplicity, the isothermal batch case will be analysed "rst. 2.3.1. Isothermal batch drying If the net energy #ow in Eq. (3) is negligibly small or the wet solid has a large mass and heat capacity, the process can be considered isothermal and only Eq. (2) need be analysed. The singular points can be obtained by making the right-hand side of Eq. (2) equal to zero, resulting in G"0.
Fig. 2. Schematic description of continuous drying of a solid wetted with a liquid mixture in the following geometries: (a) #at, (b) annular.
(13)
One solution to this system of non-linear equations is n "0, since starting from any initial point n , as time * * tends towards in"nity the liquid mixture will completely evaporate and n will tend towards zero, and there will be * no explicit information on the composition of the mixture before it dries out. We can rewrite Eq. (2) as a function of molar fractions x and the total number of
Fig. 3. Geometrical representation of the phase plane. Zero-selectivity curves.
F. Luna, J. Martn& nez / Chemical Engineering Science 54 (1999) 5823}5837
moles n * dx A " +xG !G,. 2 dt n *
(14)
From the de"nition of the molar fractions, the elements of the solution vector x are constrained in the interval [0, 1] and satisfy 12x"1. From this, it results that the elements of the vector equation (14) are linearly dependent and only the "rst n!1 equations need to be solved to describe the process. The singular points can be determined by making the right-hand side of Eq. (14) equal to zero, i.e., xG !G"0. 2
(15)
The roots of this non-linear equation can be in"nite. Simple analysis shows that compositions corresponding to the pure components will satisfy Eq. (15) for a set of speci"c conditions. Thus, during isothermal drying of a solid containing any homogeneous liquid mixture of n condensable solvents, there will be at least n singular points. In a ternary system, these singular points will lie at the vertices of the diagram. We will refer to these singular points as unary nodes. Other singular points will appear on the axes of the composition diagram, in which case the points will be binary nodes; or within the diagram, in which case the points will be ternary nodes. Graphically, the nodes can be easily determined by the intersection of the three-dimensional surfaces represented by the individual elements of the right-hand side vector function in Eq. (14) over the whole range of compositions. These singular points represent the dynamic azeotropic points of the system at a given temperature. 2.3.2. Non-isothermal batch drying If conditions are such that temperature variations cannot be neglected, Eq. (3) must be included in the analysis. Similar to the isothermal case, the singular points are given by the simultaneous solution of Eq. (15), and the right-hand side of Eq. (3) equal to zero: q!+H !H ,2G"0. E *
(16)
Note that this equation gives a wet-bulb temperature surface over the whole range of liquid compositions. 2.3.3. Continuous drying Similarly, as in the batch case, Eqs. (8)}(11) can be rewritten as functions of the total molar #ow rates and molar fractions in the liquid and gas phases dx a " (xG !G), 2 dz F *
(17)
dy a " (G!y G ), 2 dz F E
(18)
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d¹ a q!+H !H ,2G *" E * , (19) dz C2x#c S/F F * Q * * d¹ a !q E" . (20) dz F C2y E E The singular points can be determined by making the right-hand sides of Eqs. (17)}(20) equal to zero. The roots of this non-linear set of equations can be in"nite. Notice that when F tends towards zero, i.e., when the liquid * dries out, Eqs. (17) and (19) approach a discontinuity. 2.3.4. Relationship between system dynamics and selectivity If we rewrite Eq. (1) as !G s"xG !G, (21) 2 2 the relationship between selectivity and system dynamics becomes apparent [see Eq. (15)]; consequently, the singular points of the system can also be determined by the intersection of the zero-selectivity curves. Mathematically, the zero-selectivity curves are the null isoclines of the system of di!erential equations.
3. Stability Depending on process conditions, the nodes can be stable, unstable or saddle points. The stability characteristics of the nodes will depend on the thermodynamical and di!usional interactions of the components. Owing to this dependence, nodes di!erent from those of pure components are in some cases called dynamic azeotropes (SchluK nder, 1984). One approach is to study geometrically the locus of the singular points; another approach is to study the eigenvalues of the Jacobian, evaluated at the singular points. Pakowski (1994b) uses an entropy production rate approach to determine the stability of singular points. First, we will determine the stability by studying the geometrical signi"cance of the phase plane in relation to the concept of selectivity by building graphically a direction "eld. Then, we will derive general expressions for the Jacobian in each case to elucidate the dynamics of the process. Finally, we will present a type of bifurcation (in a mathematical sense) occurring when the gas is pre-loaded with vapour of one of the components in the batch case and when the ratio between the inlet gas to liquid #ow rates is changed in the continuous process. 3.1. Geometrical determination of the stability The zero selectivity curves divide the composition diagram for each component into negative and positive selectivity zones (see Fig. 3). Since a negative selectivity indicates a retention of any component, then the direction of a trajectory would be in increasing the
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concentration of that component. For the mixture acetone}chloroform}methanol mixture in Fig. 3(a), the arrows show the movement of the three components. The direction resulting from the combination of the selectivity zones suggests that the trajectories would move towards the diagonal and the left lower vertex. This behaviour implies that the binary node on the aceton}chloroform line and the pure methanol vertex may be stable nodes. In the case of the water}ethanol}ethyl acetate mixture, Fig. 3(b), each negative selectivity zone points out higher concentrations in the direction of the respective vertices. This suggests that the vertices may be stable and the singular point represented by the ternary azeotropes may be unstable. In such a case, the singular points represented by the binary azeotropes may be saddle points. In the areas resulting by the intersection of the negative selectivity zones, the components will compete to attract the trajectories towards their respective vertices. According to this behaviour, there must be borderlines separating the areas where individual components exert their attraction. 3.2. Mathematical determination of the stability 3.2.1. Isothermal batch drying Eq. (14) can be parametrized by de"ning an arbitrary variable q(t) (see Doherty & Perkins, 1978a) provided that q(t"0)"0 and A dq" dt. (22) n * Thus, Eq. (14) and its respective initial conditions may be rewritten as dx "xG !G, 2 dq
(23)
x(q )"x . (24) The systems of equations (14) and (23) represent the same vector "eld; this can be easily proven by dividing the individual equations in each system by their respective nth equations. In the neighbourhood of a singular point, the nonlinear equations can be approximated by their linearized equivalent equations. Thus, the Jacobian of Eq. (23) results in the following expression: *G J(x)"G I#+x12!I, . 2 *x
(25)
For a node to be stable, the eigenvalues of the Jacobian in Eq. (25) must have negative real parts. An analytical solution of the characteristic equation is almost impossible for multicomponent systems even if the evaporation #uxes could be expressed simply. By substracting the
diagonal matrix of the total #uxes from both sides of Eq. (25), it is possible to consider the matrix as a perturbation to the Jacobian: *G J(x) "+x12!I, . M *x
(26)
The singularity of this Jacobian can be determined by means of the product of its eigenvalues. The matrix multiplying the partial derivatives can be written as s"x12!I. This is the liquid-phase factor matrix whose eigenvalues, for a multicomponent system with n condensables, are
L !1, !1, 2 !1, x !1 ("0) . (27) G L G Note that the eigenvalues are negative, except for the largest which is zero, i.e., the Jacobian is singular. Thus, it is only necessary to consider the "rst n!1 equations. For the reduced matrix, the eigenvalues are [!1 !1 2 !x ] . (28) L L\ By reducing the original matrix to n!1 components, it is easily seen that the corresponding reduced matrix of partial derivatives of the evaporation #uxes with respect to the molar fractions is decisive for the stability of the evaporation process and the following expression holds: *G +R K R\, "RKR\, Q Q Q *x
(29)
where K is a diagonal matrix of eigenvalues and R is an eigenvector matrix. The subscript s represents the factor matrix for the liquid phase, and the non-subscripted matrices correspond to the perturbed Jacobian, Eq. (26). All matrices in Eq. (29) are of the order n!1. For a ternary system, the eigenvector and eigenvalue matrices for Eq. (29) are
!1 0 K" , Q 0 !x !1 1 " . !1 V Q V
(30)
(31)
3.2.2. Non-isothermal batch drying To simplify the qualitative analysis, the presence of the solid will be neglected since it has little in#uence on the position of the singular points. Thus, after parametrizing Eq. (3) and rearranging terms, we obtain dh q!+H !H ,2G *" E * , (32) dq ¹ C2x * * where h is a dimensionless temperature. Adding an additional dimension augments the order of the Jacobian
F. Luna, J. Martn& nez / Chemical Engineering Science 54 (1999) 5823}5837
matrix, Eq. (25). The Jacobian may be considered as a block matrix whose blocks correspond to the individual mass and heat transfer e!ects (main diagonal blocks) and their coupling e!ects (o!-diagonals):
*G G I#s 2 *x
*G s *h E
*G . *q *H2 * G!+H !H ,2 # E * *h *h *h * * * ¹ 0C2x * *
[02]
(33)
The eigenvalues for this matrix are given by the eigenvalues of the individual block matrices, which for the liquid side are given by the eigenvalues of the Jacobian in Eq. (25). The corresponding eigenvalue for the temperature is given by the second block in the main diagonal which in this case is a scalar. The sign of the eigenvalue will be determined by the terms in the numerator. 3.2.3. Continuous evaporation of a liquid xlm As in the case of drying under constant gas conditions, Eqs. (17)}(20) can be parametrized by de"ning an arbitrary variable f(z), provided that f(z"0)"0 and a df " dz, (34) * F * a df " dz. (35) E F E Thus, Eqs. (17)}(20) and their respective initial conditions may be rewritten, after simplifying by making S"0 and rearranging terms, as dx "xG !G, (36) 2 df * dy 1 " (G!yG ), (37) 2 df R * 1 q!+H !H ,2G dh *" E * , (38) df ¹ C2x * * * dh 1 !q E" , (39) df ¹ R C2y * E E x(f )"x ; y(f )"y ; h (f )"h (f )"1. (40) * * * * E * Note that when the gas-to-liquid #owrate ratio R"F /F increases towards in"nity, that is, the #ow of E * gas is in excess compared to the #ow of liquid, Eqs. (37) and (39) tend towards zero. When this happens, the concentration and temperature of the gas can be considered constant and the system behaves similarly to the case when drying under constant gas conditions.
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The Jacobian of Eqs. (36)}(39), evaluated at some singular point, results in the following block matrix:
*G *G s s *x *y *G *G t t *x *y 02
02
02
02
*G s *h * *G t *h * *q 1 *h C2x * E *q 1 *h C2y * E
*G s *h E *G t *h E , *q 1 *h C2x E E *q 1 *h C2y E E
(41)
where t"I!y12. The eigenvalues of this Jacobian are the eigenvalues of the block elements of the main diagonal of the matrix represented by Eq. (41). This leads us to consider the eigenvalues of the individual blocks in the main diagonal separately. This eigenvalues corresponding to the contribution of the energy balances result in *q j " "!aN ¹ , F * * *h *
(42)
*q j " "!aN ¹ , F E E *h E
(43)
where j is an eigenvalue. Since both eigenvalues are negative, the stability of the system would be determined by the e!ect of the mass transfer interactions. By adding Eqs. (8) and (9), it is easily seen that the sum of the individual liquid and gas #uxes F #F "F is constant. * E Thus, to study the stability of the system there is only need to consider the block matrix corresponding to either the liquid or the gas phase. Since the system is gas-phase controlled, it would be advisable to consider the block matrix of the liquid phase. Notice that this matrix has the same structure as the perturbed Jacobian in Eq. (26). Similarly, the matrix is singular and it is only necessary to consider n!1 species to determine the stability.
4. Results and discussion In the simulations that follow, the activity coe$cients were calculated using the Wilson model with parameter values from Gmehling and Onken (1982). The ordinary di!erential equations for each case were solved using an adaptive 4th}5th-order Runge}Kutta solver. In all cases, model ternary liquid mixtures were used to wet a solid for the isothermal and non-isothermal batch cases and the falling liquid "lm. Two di!erent model ternary mixtures, acetone}chloroform}methanol (ACM), and water} ethanol}ethyl acetate (WEE) were considered in this study.
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4.1. Batch isothermal case Fig. 4 shows the evaporation of solids containing the mixtures ACM and WEE, respectively. Figs. 4(a) and (c) present the vector "eld described by Eq. (23) when drying into pure nitrogen, whereas Figs. 4(b) and (d) present the corresponding trajectories. The vector "elds in Figs. 4(a) and (c) originate in the unstable nodes and end up at the stable nodes. Note that vectors grow bigger for the component, or combination of components exhibiting the larger attraction zone. For instance, in Figs. 4(a) and (b), both the binary azeotrope acetone}chloroform and the pure methanol vertices attract the trajectories in the phase plane. The attraction zone for the azeotrope acetone}chloroform is larger than the attraction zone for the pure methanol node. This suggests that the negative selectivity zones of the involved components combine to give the attraction zone of a stable node. Note that the
ternary azeotrope in Fig. 4(a) corresponds to a saddle point. For the case in Figs. 4(c) and (d), the ternary system WEE exhibits three stable nodes in the vertices of the diagram. The node corresponding to pure water has a larger attraction zone. In general, it is observed that in the case of pure stable nodes } as in the WEE system } the direction of a trajectory will tend towards any node where the component tends to be preferentially retained, i.e., it has negative selectivity. Note that the attraction zones are delimited by the separatrices, particularly stable separatrices which de"ne the regions and which originate or terminate at saddle azeotropes. For example, the ACM system [Fig. 4(b)] exhibits a stable separatrix which originates from the two unstable binary azeotropes (chloroform}methanol and methanol}acetone) and ends up on the saddle ternary azeotrope. This separatrix divides the diagram in two attraction zones, one for each stable node. In the WEE
Fig. 4. Evaporation into pure gas during drying of a solid wetted with a selected ternary mixture. P "1.013 bar, ¹ "303.15 K, ¹ "323.15 K and 2 * E u"0.1 m s\.
F. Luna, J. Martn& nez / Chemical Engineering Science 54 (1999) 5823}5837
system, three stable separatrices originate in the ternary unstable azeotrope and end up on the saddle binary azeotropes; these separatrices divide the diagram in three attraction zones, one for each of the pure stable nodes. From the "gures, it is easy to see which points are stable, unstable or saddle. The eigenvalues calculated from the Jacobian, Eq. (25), were used to con"rm the stability of those points. Since the right-hand side of Eq. (23) is highly non-linear in the #uxes, it is practically impossible to obtain an analytical expression for the Jacobian matrix without making further simpli"cations that would be valid mainly for ideal cases; moreover, the numerical determination of the nodes impose an additional limitation. Eq. (25) shows that the Jacobian, and consequently its eigenvalues, is highly in#uenced by the #uxes. This agrees with the suggestion by Pakowski (1994a) that the magnitude of the total #ux will determine the stability of the nodes. Both the Jacobian and its eigenvalues were calculated numerically. Fig. 5 shows a map of eigenvalues for the ACM and WEE mixtures. The curves in the diagrams separate the negative from the positive eigenvalue zones. In the case of the ACM system, the negative eigenvalue zone for chloroform and the positive eigenvalue zone for acetone are indicated. The pure methanol node and the binary chloroform}acetone node fall within both negative eigenvalue zones, and they behave as stable nodes. The two binary nodes (chloroform}methanol and methanol} acetone) fall within both positive zones, resulting in unstable nodes. The ternary node falls within the chloroform's negative zone and the acetone's positive zone, resulting in a saddle point. For the WEE system, the three pure component nodes are within the negative zones, resulting in stable nodes. The binary nodes are saddle points since they fall within the positive zone for
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water and negative zone for ethanol. The ternary node is in both positive zones and thus it is unstable. 4.2. Non-isothermal case Simulations were performed for the same geometry using the same set of constant conditions as in the isothermal case, except that the value of the liquid temperature here is initial and di!erent for every initial point. Fig. 6 shows the trajectories for di!erent initial conditions. All trajectories reach the wet-bulb temperature and the evaporation continues along the wet-bulb temperature towards the stable azeotropes. As in the isothermal case, the direction of the trajectories will depend on the location of the initial point. In Figs. 6(a) and (b), the trajectories are projected onto the concentration diagram only. The stable separatrices de"ne the attraction zones of the corresponding stable points. In both cases, trajectories tend towards increasing temperature in the concentration diagram. For example, in the system ACM, the two stable azeotropic points correspond to the maximum temperatures. In the case of the WEE mixture, these maximum temperatures lie on the vertices. This behaviour corresponds to the fourth theorem in Doherty and Perkins (1978a), where they prove that the temperature surface is a Liapunov function, an increasing function, indicating stability. Note that other systems may behave di!erently. By looking at the second block in the main diagonal in Eq. (33), for any given value of liquid composition and as long as trajectories lie on the temperature surface, the heat #ux partial derivative with respect to temperature is negative, since as temperature increases less heat is required to supply energy for the evaporation. As moisture liquid is being evaporated, an increase of the temperature results in
Fig. 5. Map of eigenvalues during evaporation into a pure gas.
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Fig. 6. Non-isothermal evaporation into pure nitrogen during drying of a solid wetted with a liquid ternary mixture.
Fig. 7. Evaporation during drying into pure gas. Map of eigenvalues for (a) ACM and (b) WEE.
higher evaporation #uxes; then the partial derivative with respect to the temperature will be positive. The enthalpy of evaporation, which is the di!erence between the gas and the liquid enthalpies, is always positive. Consequently, the third term will be negative. The liquid enthalpy increases when the temperature is raised giving a positive change. The product of this change with the #uxes gives a small term compared to the net e!ect of the other two terms and can be neglected. Therefore, since the denominator is always positive, the eigenvalue is negative as long as the temperature increases. Thus, the non-isothermal case can be analysed by determining the stability of the system as if it is an isothermal case at the maxima of the temperature surface. Fig. 7 shows the maps of eigenvalues of the systems ACM and WEE. Similarly, as in the isothermal case, the stable singular
points lie in the negative eigenvalue zones. For the mixture ACM, the binary acetone}chloroform azeotrope and the pure methanol vertex result in stable points. In the case of the mixture WEE, all three pure component nodes are stable. 4.3. Bifurcation in the batch case Since the di!erence in concentration in the interface and the bulk gas provides the driving force, gas preloading in batch drying will have a tremendous in#uence on the evaporative process. As the gas is loaded with one of the components in the liquid mixture, the phase portrait changes gradually as the concentration parameter for that component varies and at some value the phase portrait changes abruptly.
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Fig. 8 shows the phase portrait of the system ACM when the gas is loaded with methanol at di!erent concentrations. At a concentration di!erent from zero, see Fig. 8(b), the original binary node acetone}chloroform is shifted to a ternary node, which remains stable. Now the system presents two ternary azeotropes one stable and one saddle. At some value of methanol concentration (see Figs. 8(c) and (d)) the two azeotropes coalesce and disappear. This phenomenon occurs for relatively higher concentrations in the gas when loaded with acetone or chloroform alone. When the gas is loaded with acetone (for example see Figs. 9(a) and (b)), the binary point acetone}chloroform is shifted down along the binary mixture diagonal towards the acetone vertex. In this case, the stable pure methanol point is shifted into a binary acetone}methanol azeotrope moving towards the acetone vertex. At some value of acetone concentration,
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the ternary saddle azeotrope and the binary acetone}methanol node coalesce and disappear. When the gas is loaded with all three components (Figs. 9(c) and (d)) the pure component and binary singular points are shifted into the diagram becoming ternary stable azeotropes. By increasing the concentrations beyond some value, the saddle azeotropic point and the stable one, originating from the methanol vertex, coalesce and disappear leaving a unique ternary stable azeotrope. The system WEE presents a similar behaviour when the gas is loaded with one of the components. Fig. 10 shows the case for water-loaded nitrogen. At some value of vapour concentration in the gas, the ternary unstable azeotrope disappears leaving a large attraction zone for the pure water stable node. Both systems exhibit what constitutes a saddlenode bifurcation. The coalescence and disappearances of two singularities, like in the two previous cases, is
Fig. 8. Bifurcation in the evaporation of the mixture ACM into nitrogen loaded with methanol vapour.
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Fig. 9. Bifurcation in the evaporation into vapour-loaded nitrogen of the mixture ACM: (a) and (b) acetone; (c) and (d) all three solvents.
comprehensible if one considers the null-isoclines along which the vector "eld is horizontal and vertical and whose intersection gives the zeroes of the vector "eld. If these isoclines are dependent on a parameter being the gas concentration of one of the components in the mixture, the number of intersections will remain constant unless the parameter passes through a value where the isoclines are tangent, and where, in general, two intersection points coalesce and disappear. 4.4. Continuous drying Since in the continuous case the gas and liquid conditions change simultaneously along the dryer length, it is impossible to represent all independent variables in a unique phase portrait of dimension RL>. However, the ternary diagram may be used as a projection of the
trajectories. Results of simulations starting from an initial common point for di!erent values of the ratio R "F /F are shown in Fig. 11(a) for both cases for E * the liquid mixture ACM, the corresponding temperatures are shown in Fig. 11(b). Notice that the trajectories follow a similar behavior. For each initial point, trajectories end up at an equilibrium point corresponding to the saturation point of the gas at some "nite length in the dryer. At this point, the system reaches a dynamic equilibrium. These points constitute the singular points of the system. Note that these points are in"nite and their position will depend on the inlet conditions. Since continuous evaporation of the liquid will load the gas along the dryer length, the driving force, represented by the di!erence between the gas and liquid interface composition, will decrease correspondingly. This will ensure low drying rates
F. Luna, J. Martn& nez / Chemical Engineering Science 54 (1999) 5823}5837
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Fig. 10. Bifurcation in the evaporation into water-loaded nitrogen of the mixture WEE.
Fig. 11. Continuous evaporation into nitrogen of a liquid "lm for the system ACM.
and reinforce the assumption of gas-phase-controlled conditions. 4.5. Bifurcation in the continuous case In Fig. 12 trajectories are plotted for di!erent inlet ratios R . Observe that as R increases the equilibrium point displaces through the composition diagram tending towards the position of the singular point of a batch case with the same process conditions. The inlet total gas #ow rate is the only parameter varied. As this ratio increases, see Eq. (39), the gas temperature tends towards the constant inlet gas temperature. In the case of the initial points B and C, at some value of R the equilib
rium point tending towards the pure methanol vertex disappears and the trajectories change direction at some value of R . For larger values of R , increasing towards in"nity, the equilibrium point tends towards the binary acetone}chloroform, as in the batch case. Depending on the position of the initial conditions, the system may or may not exhibit a bifurcation for increasing values of R as in the case of point A. Increasing R towards in"nity does not abruptly change the direction of the trajectory and the equilibrium point is shifted towards the pure methanol vertex. The stability zones, corresponding to the negative zones of the map of eigenvalues, are displaced along with the respective nodes and depend upon the corresponding
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Fig. 12. In#uence of the inlet #owrate ratio F /F on the position of a singular point: Stability zones for di!erent inlet process conditions. E *
inlet conditions as well. Fig. 12 shows the stability zones for di!erent equilibrium conditions. For small values of R the zones remain negative. By increasing the ratio, positive zones begin to appear and grow (Fig. 12(a)). When R tends to in"nity (Fig. 12(b)), the process ap proaches a batch behaviour and the resulting map of eigenvalues is similar to the non-isothermal batch case (Fig. 7(a)).
5. Conclusions The stability of a drying process is characterized by the stability of the system's singular points, determined by the dynamics of the process. Particularly, we have presented the stability analysis of the models for batch and continuous processes. In the case of the batch process, when isothermal conditions are assumed, the phase portrait is represented by the diagram of compositions. The stable nodes are contained in attraction zones, delimited by stable separatrices passing through the saddle azeotropes. The extent of these zones is related to the net e!ect of the negative selectivity zones of the components in the mixture. The stability of the azeotropes may be determined by calculating the eigenvalues of the n!1 components in the mixture. In general, trajectories will tend towards the point that gives a component or mixture of components which are preferentially retained, that is, where selectivities will be predominantly negative for the component or components involved. Thus, azeotropic points may result in stable or unstable nodes or saddles. For a non-isothermal batch process, the analysis may be done individually for the mass transfer and heat transfer processes. In general, when trajectories tend in the
direction of increasing temperature, the eigenvalue corresponding to the temperature is negative. Thus, the analysis may be carried out following the same approach as in the isothermal case. In the batch case, the phase portrait may be changed by loading the gas with one or more components of the system. This may cause a bifurcation of the saddle}node type, where a node and a saddle coalesce and disappear at some particular value of the gas concentration. Consequently, by loading the gas, additional nodes may appear, and some may disappear. In the continuous case the conditions change in both phases along the dryer; consequently, trajectories will depend on the inlet conditions. For small values of the inlet gas-to-liquid #owrate ratio, R , the system exhibits in"nite singular points. Each point corresponds to a dynamic equilibrium state and results in stable nodes. As the ratio increases, tending towards in"nity, the process approximates a batch behaviour, and an analysis of this case may be applied.
Notation a A c C
speci"c area, m m\ area, m mass heat capacity, kJ kg\ K\ molar heat capacity vector at constant pressure, kJ kmol\ K\ F molar #ux vector, kmol m\ s\ F total molar #ux, kmol m\ s\ G molar evaporation rate vector, kmol m\ s\ G total molar evaporation rate, kmol m\ s\ 2 H molar enthalpy vector, kJ kmol\ I indentity matrix
F. Luna, J. Martn& nez / Chemical Engineering Science 54 (1999) 5823}5837
J k
Jacobian matrix matrix of zero-#ux mass transfer coe$cients, kmol m\ s\ m mass, kg n number of moles vector, kmol n number of condensable components n total number of moles in liquid phase, kmol * p saturated vapour pressure vector, bar P total pressure, bar q heat #ux by convection, kW m\ R gas-to-liquid #owrate ratio s vector of selectivities S solid mass #ux, kg m\ s\ t time, s ¹ temperature, K x vector of molar fractions in the liquid, kmol kmol\ y vector of molar fractions in the gas, kmol kmol\ z length, m 0 column vector of zeroes 1 column vector of ones Greek letters
a b C f h j K N N R q s t
heat transfer coe$cient, kW m\ K\ Bootstrap matrix diagonal matrix of activity coe$cients arbitary, length-related variable, Eqs. (34) and (35) dimensionless temperature ("¹/¹ ) eigenvalue eigenvalue matrix mass transfer coe$cients correction matrix, Eq. (5) heat transfer correction factor, Eq. (6) eigenvector matrix arbitary, time-related variable, Eq. (22) liquid-phase factor matrix ("x12!I) gas-phase factor matrix ("I!y12)
Subscripts g h i ¸ s ¹ 0 s h o
gas heat transfer index liquid solid total initial or inlet related to liquid-phase factor matrix, Eq. (29) related to dimensionless temperature perturbed matrix, Eq. (26)
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Superscripts T 0
transpose saturated
Acknowledgements The authors gratefully acknowledge the "nancial support provided by the Swedish Research Council for Engineering Sciences (TFR) for this work. References Bird, R. B., Stewart, W. E., & Lightfoot, E. N. (1960). Transport phenomena. New York: Wiley. Coulson, J. M., & Richardson, J. F. (1996). Coulson & Richardson's chemical engineering, vol. 1 (5th ed). London: Butterworth-Heinemann. Doherty, M. F., & Perkins, J. D. (1978a). On the dynamics of distillation processes } I. The simple distillation of multicomponent non-reacting homogeneous liquid mixtures. Chemical Engineering Science, 33, 281}301. Doherty, M. F., & Perkins, J. D. (1978b). On the dynamics of distillation processes } II. The simple distillation of model solutions. Chemical Engineering Science, 33, 569}578. Doherty, M. F., & Perkins, J. D. (1979). On the dynamics of distillation processes } III. The topological structure of ternary residue curve maps. Chemical Engineering Science, 34, 1401}1414. Foucher, E. R., Doherty, M. F., & Malone, M. F. (1991). Automatic screening of entrainers for homogeneous azeotropic distillation. Industrial Engineering and Chemical Research, 30, 760}772. Gmehling, J., & Onken, U. (1982). Vapor}liquid equilibrium data collection. DECHEMA. Luna, F., MartmH nez, J. (1998). Stability of the dynamical system describing gas-phase-controlled drying of ternary mixtures. Drying Technology Journal, 6 (9}10). MartmH nez, J., & Setterwall, F. (1991). Gas-phase controlled convective drying of solids wetted with multicomponent liquid mixtures. Chemical Engineering Science, 46(9), 2235}2252. Pakowski, Z. (1994a). Drying of solids containing multicomponent mixture: recent developments. Proceedings of the 10th International Drying Symposium, vol. A. Gold Coast, Brisbane, Australia (pp. 27}38). Pakowski, Z. (1994b). Stationary states in evaporation of multicomponent liquid droplets to inert gas stream. Zesz. Nauk. 716. Lodz Tech. University, pp. 1}91 (In Polish). SchluK nder, E. U. (1984). Einfu( hrung in die Stowu( bertragung (pp. 56}77). Georg Thieme Verlag, Stuttgart. Taylor, R. (1982). Film models for multicomponent mass transfer: Computational methods } II. The linearised theory. Computers in Chemical Engineering, 6, 69}75. Van Dongen, D. B., & Doherty, M. F. (1985). Design and synthesis of homogeneous azeotropic distillations. 1: Problem formulation for a single column. Industrial Engineering and Chemical Fundamentals, 24, 454. Vidaurre, M., & MartmH nez, J. (1997). Continuous drying of a solid wetted with ternary mixtures. A.I.Ch.E. Journal, 43(3), 681}692.
Stability analysis in multicomponent drying of homogeneous liquid mixtures F. Luna*, J. MartmH nez Department of Chemical Engineering and Technology, Royal Institute of Technology, (KTH), S-100 44 Stockholm, Sweden Received 27 July 1998; received in revised form 12 March 1999; accepted 16 March 1999
Abstract A stability analysis of the ordinary di!erential equations describing the process of convective gas-phase-controlled evaporation during drying is performed. Isothermal and non-isothermal as well as batch and continuous drying processes are considered. For isothermal evaporation of a ternary mixture into pure gas, the solutions of the di!erential equations are trajectories in the phase plane represented by a triangular diagram of compositions. The predicted ternary dynamic azeotropic points are unstable or saddle. On the other hand, binary azeotropes are stable when the combination of the selectivities of the corresponding components is negative. In addition, pure component singular points are stable when they are contained within their respective isolated negative selectivity zones. Under non-isothermal conditions, stable azeotropes are characterized by presenting maximum temperature values. Loading the gas with one or more of the components up to some value leads to a node-saddle bifurcation, where a saddle azeotrope and a stable azeotrope coalesce and disappear. The continuous drying process yields similar results for both #at and annular geometries. The singular points, in this case, are in"nite and represent dynamic equilibrium points whose stability is mainly dependent on the inlet gas-to-liquid #owrate ratio. As this ratio grows to in"nity, the phase portrait changes and the process approaches a batch behaviour so that the stability analysis for that case may be applied. The present stability analysis permits the prediction of trajectories and "nal state of a system in a gas-phase-controlled drying process. 1999 Elsevier Science Ltd. All rights reserved. Keywords: Azeotropic points; Convective drying; Di!erential equations; Multicomponent systems; Selectivity; Stability
1. Introduction One of the main concerns in multicomponent drying is the composition of the "nal moisture because of its crucial in#uence on product quality. Since separation processes normally do not run to completion, a residual moisture content will frequently remain in the solid after drying } due to either equipment limitations or process requirements. For instance, when pharmaceutical granulates are dried, the aim is often to remove primarily organic solvents because some residual water content is required in subsequent tableting of the granulates. If the moisture consists of water or another single solvent, the "nal liquid content after drying is determined by the equipment dimensions and the operating conditions. Moreover, when the moisture consists of a mixture, the "nal state of the drying process depends on the initial conditions. In multicomponent mixtures, these condi-
* Corresponding author. Tel.: 0046-8-7906569; fax: 0046-8-212-747.
tions can vary considerably and the number of possible alternatives increases with the number of components in the mixture. For this reason, a manner to predict all possible process paths from known system properties, and process kinetics would be useful. In general, the inherent complexity of convective drying of solids containing liquid mixtures does not allow for such kind of analysis. The achievement of particular solutions becomes a di$cult task due to the complex interaction of gas-phase di!usion, external heat transfer, phase equilibrium, mass and heat transfer within the solid with strong dependence on liquid content, composition and temperature, and the contact mode between the phases. However, if evaporation is gas-phase-controlled, a broad understanding of the process may be gained by a stability analysis of the ordinary di!erential equations that describe the dynamical system. Gas-phase-controlled drying occurs frequently at an initial stage when the solid is saturated. It may also occur during long drying periods, at low drying rates, drying of thin products } such as "lms or coated laminates } or of bulky
0009-2509/99/$ - see front matter 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 0 0 9 - 2 5 0 9 ( 9 9 ) 0 0 1 7 1 - 2
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materials, provided that the moisture is super"cially distributed as a thin layer throughout the drying process. These conditions may also be encountered during lowtemperature drying and in sections of continuous equipment where solids are in contact with an increasingly enriching vapour atmosphere. Considerable work has been devoted to the study of the dynamics of distillation processes. In a series of papers, Doherty and Perkins (1978a,b,1979) analysed mathematically the ordinary di!erential equations which describe the general characteristics of multicomponent, non-ideal distillation processes and generated a residue curve map to trace composition changes in a perfectly mixed liquid. Van Dongen and Doherty (1985) approximated by simple distillation}residue curves liquid composition pro"les in continuous staged or packed distillation columns. When the ideal solution model at sub-critical conditions is applied, non-elementary singular points cannot occur. In addition, for ternary mixtures they found that there is a topological relationship between the singular points known as azeotropes and singular points corresponding to pure components. Rules for constructing qualitative distillation region diagrams based only on pure component data and azeotrope boiling-point data were given by Foucher, Doherty and Malone (1991). The presence of an inert gas makes multicomponent drying by convection a more complex process than distillation since not only thermodynamic equilibrium is involved but also di!usion in gas phase which in mixtures of more than two components is usually interactive. Pakowski (1994a) analysed the stability of dynamic azeotropes in ternary mixtures evaporating into air assuming a perfect liquid-phase-mixing model. He found that trajectories tend towards the local minimum of the sum of absolute values of all individual molar evaporation #uxes. Luna and MartmH nez (1998) applied a stability analysis to an isothermal batch case with unchanged gas-phase composition. The purpose of this paper is to present a qualitative analysis of process dynamics in the cases where conditions lead to a gas-phase-controlled drying process. First, this study considers batch drying of a sample under constant gas conditions of a solid containing liquid mixtures. Then, attention is devoted to the continuous drying of a liquid mixture where the conditions of both liquid and gas phases change along the equipment. Although the analysis applies primarily to a liquid "lm, it is also valid for a solid containing a mixture as long as the process is gas-phase controlled.
2. Theory The design and choice of optimal operating conditions in convection dryers require information about moisture} solid equilibrium, drying rates as a function of moisture
content, and gas}solid contact pattern in the equipment. If the solid contains a multicomponent moisture, information about the changes in liquid composition during the process is additionally demanded. Composition curves are crucial because of the great in#uence of the remaining liquid on the properties and quality of the product. The variable which describes the ability of a species to be preferentially removed or not from the moisture is the selectivity. It is de"ned, in vector form, as follows: 1 s" G!x, G 2
(1)
where G is a column vector of molar drying rates, G is 2 the total molar drying rate, and x is a vector of molar fractions in the liquid phase. Depending on the values of the elements of s the following cases can be distinguished: s '0, component i is preferentially removed and its G molar fraction in the moisture decreases; s "0, G evaporation is non-selective and the molar fraction of the species i remains constant; s '0, component i is not G preferentially removed and its molar fraction increases. Like drying rates, drying selectivity depends on the intricate mechanisms behind the transport of solvents from within the solid to the solid}gas interface and subsequently into the gas phase. In gas-phase-controlled drying, selectivity depends only on the external conditions, that is, on vapour}liquid equilibrium, #ow pattern and gas conditions. External conditions have also a decisive in#uence on the appearance of internal resistance. During low intensity drying, internal resistance will normally not arise. In modelling gas-phase-controlled drying, the description of the dynamic system leads to ordinary di!erential equations. In this case, a deep understanding of the process may be obtained by a qualitative stability analysis of the governing ordinary di!erential equations. Two cases are important in this context: batch drying, where a certain amount of moist solids is dried periodically; and drying in continuously worked equipment. In the former case, the conditions of the solid change with time. In the latter, the conditions of the solid change as a function of the position along the dryer. 2.1. Batch drying Fig. 1 shows schematically batch drying of a wet solid, over which a hot gas stream of constant temperature #ows parallel to the exposed surface. When the gas #owrate is in excess compared to the vapours transferred between the phases, the changes of the gas conditions during the process can be neglected. This situation is achieved in batch drying experiments performed to determine drying characteristics of solids where very narrow samples are normally used. MartmH nez
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If it is assumed that the phases are in equilibrium at the interface, the coupling between the gas composition y and the liquid composition x at the interface is given by the following expression: 1 y " pCx, * P Fig. 1. Schematic description of batch drying by convection of a solid wetted with a liquid mixture.
and Setterwall (1991) developed and veri"ed experimentally a model to describe this case. For a system consisting of n condensable components, the governing equations are: dn *"!AG, dt
(2)
A(q!+H !H ,2G) d¹ E * *" , (3) C2n #c m dt * * Q Q where n is a column vector of number of moles of each * species in the liquid mixture, A is the surface of the wet solid exposed to the gas stream, q is the heat #ux which reaches the drying surface by convection, H is a vector of molar enthalpies, C is a vector of molar heat capacities, c is a mass heat capacity; and m is mass. The subscripts g, ¸, and s denote gas, liquid and solid phase, respectively. The superscript ¹ denotes transposition. Eqs. (2) and (3) consist of a set of n#1 ordinary di!erential equations. These equations describe the variation of sample temperature, moisture content, and number of moles of the di!erent species in the liquid mixture. For a particular solution, the initial conditions are given by n (t )"n , ¹ (t )"¹ . (4) * * * * To integrate Eqs. (2) and (3), expressions for the mass and heat transfer rates must be provided. If di!usional interactions are considered, mass and heat #uxes may be calculated as follows: G"bkN+y !y,, (5) * q"aN (¹ !¹ ), (6) F E * where matrix b incorporates the convective contribution to mass transfer, k is the matrix of zero-#ux mass transfer coe$cients, N is the matrix of correction factors which accounts for the in#uence of "nite mass transfer rates upon the mass transfer coe$cients, N is the correspondF ing correction factor for the heat transfer coe$cient a, and y is the vector of molar gas compositions. The parameters in the #uxes may be calculated according to Taylor (1982). See Bird, Stewart and Lightfoot (1960) for the calculation of the correction factor for the heat transfer coe$cient.
(7)
where p and C are diagonal matrices of pure component saturated vapour pressures and activity coe$cients, respectively; and P is the total pressure. Note that the analysis is not restricted to this particular geometry. As long as the process is gas-phase controlled and the conditions of the gas do not change appreciably during the process, the analysis also applies to other drying methods, such as #uidized bed or spray drying, provided that suitable transfer coe$cients are used to calculate the mass and heat transfer rates. For the geometry described in Fig. 1 and conditions of streamline #ow, the results of the boundary layer theory may be used to calculate the mass and heat transfer coe$cients at zero mass transfer rates. For turbulent #ow, the Chilton}Colburn analogy is applicable (see Coulson & Richardson, 1996). 2.2. Continuous drying In a continuous drying process, the conditions of both phases change along the contact path. It can be seen in Figs. 2 and 3 where co-current contact mode of the phases in a #at and in an annular geometry are schematically depicted. Vidaurre and MartmH nez (1997) developed a model for the #at geometry. In both cases, mass and energy balances along the evaporation path are described by the same set of di!erential equations if the solid #ow is set to zero for the annular geometry dF *"!aG, dz
(8)
dF E"aG, dz
(9)
d¹ q!+H !H ,2G *"a E * , (10) dz C2F #c S * * Q d¹ !q E"a , (11) dz C2F E E where F is the vector of the number of moles passing through the dryer per unit time and cross section; a is the interfacial area per unit dryer volume; and S is the #ow of solid material associated to the liquid #ux. Eqs. (8)}(11) represent a set of 2n#2 ordinary nonlinear di!erential equations. They describe an initial-value problem where the location is related to the residence time in the contact device by the linear velocity of the phases and inlet
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conditions correspond to the initial values. If the liquid and gas have a uniform composition at z"0, the following conditions are applicable: F "F , F "F , ¹ "¹ , ¹ "¹ , * * E E * * E E S"S at z"0. (12) The calculation of mass and heat transfer rates in the continuous case is similar to the batch case, except for the transfer coe$cients and the progressively changing gas conditions. Therefore, Eqs. (5) and (6) are applicable in both cases provided that temperature and composition of the gas are regarded as functions of z. The method used in this work to analyse stability applies to autonomous di!erential equations, that is, when the right-hand sides do not depend explicitly on the independent variable. This is satis"ed by the batch case and the annular geometry. On the other hand, the
method to calculate the transfer coe$cients in a #at geometry with uncon"ned #ow leads to non-autonomous di!erential equations since they are functions of the position z along the dryer. However, if the transfer coe$cients are calculated at some average position along the dryer, using similar relationships as those for the batch case, an approximate analysis, still preserving the main feature of a continuous contact mode may be performed even in the #at geometry. 2.3. Singular points The vector "eld described by the ordinary di!erential equations (2)}(3) or (8)}(11) vanishes at the singular, equilibrium or critical points, which make the right-hand side of the system of di!erential equations equal to zero. For the sake of simplicity, the isothermal batch case will be analysed "rst. 2.3.1. Isothermal batch drying If the net energy #ow in Eq. (3) is negligibly small or the wet solid has a large mass and heat capacity, the process can be considered isothermal and only Eq. (2) need be analysed. The singular points can be obtained by making the right-hand side of Eq. (2) equal to zero, resulting in G"0.
Fig. 2. Schematic description of continuous drying of a solid wetted with a liquid mixture in the following geometries: (a) #at, (b) annular.
(13)
One solution to this system of non-linear equations is n "0, since starting from any initial point n , as time * * tends towards in"nity the liquid mixture will completely evaporate and n will tend towards zero, and there will be * no explicit information on the composition of the mixture before it dries out. We can rewrite Eq. (2) as a function of molar fractions x and the total number of
Fig. 3. Geometrical representation of the phase plane. Zero-selectivity curves.
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moles n * dx A " +xG !G,. 2 dt n *
(14)
From the de"nition of the molar fractions, the elements of the solution vector x are constrained in the interval [0, 1] and satisfy 12x"1. From this, it results that the elements of the vector equation (14) are linearly dependent and only the "rst n!1 equations need to be solved to describe the process. The singular points can be determined by making the right-hand side of Eq. (14) equal to zero, i.e., xG !G"0. 2
(15)
The roots of this non-linear equation can be in"nite. Simple analysis shows that compositions corresponding to the pure components will satisfy Eq. (15) for a set of speci"c conditions. Thus, during isothermal drying of a solid containing any homogeneous liquid mixture of n condensable solvents, there will be at least n singular points. In a ternary system, these singular points will lie at the vertices of the diagram. We will refer to these singular points as unary nodes. Other singular points will appear on the axes of the composition diagram, in which case the points will be binary nodes; or within the diagram, in which case the points will be ternary nodes. Graphically, the nodes can be easily determined by the intersection of the three-dimensional surfaces represented by the individual elements of the right-hand side vector function in Eq. (14) over the whole range of compositions. These singular points represent the dynamic azeotropic points of the system at a given temperature. 2.3.2. Non-isothermal batch drying If conditions are such that temperature variations cannot be neglected, Eq. (3) must be included in the analysis. Similar to the isothermal case, the singular points are given by the simultaneous solution of Eq. (15), and the right-hand side of Eq. (3) equal to zero: q!+H !H ,2G"0. E *
(16)
Note that this equation gives a wet-bulb temperature surface over the whole range of liquid compositions. 2.3.3. Continuous drying Similarly, as in the batch case, Eqs. (8)}(11) can be rewritten as functions of the total molar #ow rates and molar fractions in the liquid and gas phases dx a " (xG !G), 2 dz F *
(17)
dy a " (G!y G ), 2 dz F E
(18)
5827
d¹ a q!+H !H ,2G *" E * , (19) dz C2x#c S/F F * Q * * d¹ a !q E" . (20) dz F C2y E E The singular points can be determined by making the right-hand sides of Eqs. (17)}(20) equal to zero. The roots of this non-linear set of equations can be in"nite. Notice that when F tends towards zero, i.e., when the liquid * dries out, Eqs. (17) and (19) approach a discontinuity. 2.3.4. Relationship between system dynamics and selectivity If we rewrite Eq. (1) as !G s"xG !G, (21) 2 2 the relationship between selectivity and system dynamics becomes apparent [see Eq. (15)]; consequently, the singular points of the system can also be determined by the intersection of the zero-selectivity curves. Mathematically, the zero-selectivity curves are the null isoclines of the system of di!erential equations.
3. Stability Depending on process conditions, the nodes can be stable, unstable or saddle points. The stability characteristics of the nodes will depend on the thermodynamical and di!usional interactions of the components. Owing to this dependence, nodes di!erent from those of pure components are in some cases called dynamic azeotropes (SchluK nder, 1984). One approach is to study geometrically the locus of the singular points; another approach is to study the eigenvalues of the Jacobian, evaluated at the singular points. Pakowski (1994b) uses an entropy production rate approach to determine the stability of singular points. First, we will determine the stability by studying the geometrical signi"cance of the phase plane in relation to the concept of selectivity by building graphically a direction "eld. Then, we will derive general expressions for the Jacobian in each case to elucidate the dynamics of the process. Finally, we will present a type of bifurcation (in a mathematical sense) occurring when the gas is pre-loaded with vapour of one of the components in the batch case and when the ratio between the inlet gas to liquid #ow rates is changed in the continuous process. 3.1. Geometrical determination of the stability The zero selectivity curves divide the composition diagram for each component into negative and positive selectivity zones (see Fig. 3). Since a negative selectivity indicates a retention of any component, then the direction of a trajectory would be in increasing the
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concentration of that component. For the mixture acetone}chloroform}methanol mixture in Fig. 3(a), the arrows show the movement of the three components. The direction resulting from the combination of the selectivity zones suggests that the trajectories would move towards the diagonal and the left lower vertex. This behaviour implies that the binary node on the aceton}chloroform line and the pure methanol vertex may be stable nodes. In the case of the water}ethanol}ethyl acetate mixture, Fig. 3(b), each negative selectivity zone points out higher concentrations in the direction of the respective vertices. This suggests that the vertices may be stable and the singular point represented by the ternary azeotropes may be unstable. In such a case, the singular points represented by the binary azeotropes may be saddle points. In the areas resulting by the intersection of the negative selectivity zones, the components will compete to attract the trajectories towards their respective vertices. According to this behaviour, there must be borderlines separating the areas where individual components exert their attraction. 3.2. Mathematical determination of the stability 3.2.1. Isothermal batch drying Eq. (14) can be parametrized by de"ning an arbitrary variable q(t) (see Doherty & Perkins, 1978a) provided that q(t"0)"0 and A dq" dt. (22) n * Thus, Eq. (14) and its respective initial conditions may be rewritten as dx "xG !G, 2 dq
(23)
x(q )"x . (24) The systems of equations (14) and (23) represent the same vector "eld; this can be easily proven by dividing the individual equations in each system by their respective nth equations. In the neighbourhood of a singular point, the nonlinear equations can be approximated by their linearized equivalent equations. Thus, the Jacobian of Eq. (23) results in the following expression: *G J(x)"G I#+x12!I, . 2 *x
(25)
For a node to be stable, the eigenvalues of the Jacobian in Eq. (25) must have negative real parts. An analytical solution of the characteristic equation is almost impossible for multicomponent systems even if the evaporation #uxes could be expressed simply. By substracting the
diagonal matrix of the total #uxes from both sides of Eq. (25), it is possible to consider the matrix as a perturbation to the Jacobian: *G J(x) "+x12!I, . M *x
(26)
The singularity of this Jacobian can be determined by means of the product of its eigenvalues. The matrix multiplying the partial derivatives can be written as s"x12!I. This is the liquid-phase factor matrix whose eigenvalues, for a multicomponent system with n condensables, are
L !1, !1, 2 !1, x !1 ("0) . (27) G L G Note that the eigenvalues are negative, except for the largest which is zero, i.e., the Jacobian is singular. Thus, it is only necessary to consider the "rst n!1 equations. For the reduced matrix, the eigenvalues are [!1 !1 2 !x ] . (28) L L\ By reducing the original matrix to n!1 components, it is easily seen that the corresponding reduced matrix of partial derivatives of the evaporation #uxes with respect to the molar fractions is decisive for the stability of the evaporation process and the following expression holds: *G +R K R\, "RKR\, Q Q Q *x
(29)
where K is a diagonal matrix of eigenvalues and R is an eigenvector matrix. The subscript s represents the factor matrix for the liquid phase, and the non-subscripted matrices correspond to the perturbed Jacobian, Eq. (26). All matrices in Eq. (29) are of the order n!1. For a ternary system, the eigenvector and eigenvalue matrices for Eq. (29) are
!1 0 K" , Q 0 !x !1 1 " . !1 V Q V
(30)
(31)
3.2.2. Non-isothermal batch drying To simplify the qualitative analysis, the presence of the solid will be neglected since it has little in#uence on the position of the singular points. Thus, after parametrizing Eq. (3) and rearranging terms, we obtain dh q!+H !H ,2G *" E * , (32) dq ¹ C2x * * where h is a dimensionless temperature. Adding an additional dimension augments the order of the Jacobian
F. Luna, J. Martn& nez / Chemical Engineering Science 54 (1999) 5823}5837
matrix, Eq. (25). The Jacobian may be considered as a block matrix whose blocks correspond to the individual mass and heat transfer e!ects (main diagonal blocks) and their coupling e!ects (o!-diagonals):
*G G I#s 2 *x
*G s *h E
*G . *q *H2 * G!+H !H ,2 # E * *h *h *h * * * ¹ 0C2x * *
[02]
(33)
The eigenvalues for this matrix are given by the eigenvalues of the individual block matrices, which for the liquid side are given by the eigenvalues of the Jacobian in Eq. (25). The corresponding eigenvalue for the temperature is given by the second block in the main diagonal which in this case is a scalar. The sign of the eigenvalue will be determined by the terms in the numerator. 3.2.3. Continuous evaporation of a liquid xlm As in the case of drying under constant gas conditions, Eqs. (17)}(20) can be parametrized by de"ning an arbitrary variable f(z), provided that f(z"0)"0 and a df " dz, (34) * F * a df " dz. (35) E F E Thus, Eqs. (17)}(20) and their respective initial conditions may be rewritten, after simplifying by making S"0 and rearranging terms, as dx "xG !G, (36) 2 df * dy 1 " (G!yG ), (37) 2 df R * 1 q!+H !H ,2G dh *" E * , (38) df ¹ C2x * * * dh 1 !q E" , (39) df ¹ R C2y * E E x(f )"x ; y(f )"y ; h (f )"h (f )"1. (40) * * * * E * Note that when the gas-to-liquid #owrate ratio R"F /F increases towards in"nity, that is, the #ow of E * gas is in excess compared to the #ow of liquid, Eqs. (37) and (39) tend towards zero. When this happens, the concentration and temperature of the gas can be considered constant and the system behaves similarly to the case when drying under constant gas conditions.
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The Jacobian of Eqs. (36)}(39), evaluated at some singular point, results in the following block matrix:
*G *G s s *x *y *G *G t t *x *y 02
02
02
02
*G s *h * *G t *h * *q 1 *h C2x * E *q 1 *h C2y * E
*G s *h E *G t *h E , *q 1 *h C2x E E *q 1 *h C2y E E
(41)
where t"I!y12. The eigenvalues of this Jacobian are the eigenvalues of the block elements of the main diagonal of the matrix represented by Eq. (41). This leads us to consider the eigenvalues of the individual blocks in the main diagonal separately. This eigenvalues corresponding to the contribution of the energy balances result in *q j " "!aN ¹ , F * * *h *
(42)
*q j " "!aN ¹ , F E E *h E
(43)
where j is an eigenvalue. Since both eigenvalues are negative, the stability of the system would be determined by the e!ect of the mass transfer interactions. By adding Eqs. (8) and (9), it is easily seen that the sum of the individual liquid and gas #uxes F #F "F is constant. * E Thus, to study the stability of the system there is only need to consider the block matrix corresponding to either the liquid or the gas phase. Since the system is gas-phase controlled, it would be advisable to consider the block matrix of the liquid phase. Notice that this matrix has the same structure as the perturbed Jacobian in Eq. (26). Similarly, the matrix is singular and it is only necessary to consider n!1 species to determine the stability.
4. Results and discussion In the simulations that follow, the activity coe$cients were calculated using the Wilson model with parameter values from Gmehling and Onken (1982). The ordinary di!erential equations for each case were solved using an adaptive 4th}5th-order Runge}Kutta solver. In all cases, model ternary liquid mixtures were used to wet a solid for the isothermal and non-isothermal batch cases and the falling liquid "lm. Two di!erent model ternary mixtures, acetone}chloroform}methanol (ACM), and water} ethanol}ethyl acetate (WEE) were considered in this study.
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4.1. Batch isothermal case Fig. 4 shows the evaporation of solids containing the mixtures ACM and WEE, respectively. Figs. 4(a) and (c) present the vector "eld described by Eq. (23) when drying into pure nitrogen, whereas Figs. 4(b) and (d) present the corresponding trajectories. The vector "elds in Figs. 4(a) and (c) originate in the unstable nodes and end up at the stable nodes. Note that vectors grow bigger for the component, or combination of components exhibiting the larger attraction zone. For instance, in Figs. 4(a) and (b), both the binary azeotrope acetone}chloroform and the pure methanol vertices attract the trajectories in the phase plane. The attraction zone for the azeotrope acetone}chloroform is larger than the attraction zone for the pure methanol node. This suggests that the negative selectivity zones of the involved components combine to give the attraction zone of a stable node. Note that the
ternary azeotrope in Fig. 4(a) corresponds to a saddle point. For the case in Figs. 4(c) and (d), the ternary system WEE exhibits three stable nodes in the vertices of the diagram. The node corresponding to pure water has a larger attraction zone. In general, it is observed that in the case of pure stable nodes } as in the WEE system } the direction of a trajectory will tend towards any node where the component tends to be preferentially retained, i.e., it has negative selectivity. Note that the attraction zones are delimited by the separatrices, particularly stable separatrices which de"ne the regions and which originate or terminate at saddle azeotropes. For example, the ACM system [Fig. 4(b)] exhibits a stable separatrix which originates from the two unstable binary azeotropes (chloroform}methanol and methanol}acetone) and ends up on the saddle ternary azeotrope. This separatrix divides the diagram in two attraction zones, one for each stable node. In the WEE
Fig. 4. Evaporation into pure gas during drying of a solid wetted with a selected ternary mixture. P "1.013 bar, ¹ "303.15 K, ¹ "323.15 K and 2 * E u"0.1 m s\.
F. Luna, J. Martn& nez / Chemical Engineering Science 54 (1999) 5823}5837
system, three stable separatrices originate in the ternary unstable azeotrope and end up on the saddle binary azeotropes; these separatrices divide the diagram in three attraction zones, one for each of the pure stable nodes. From the "gures, it is easy to see which points are stable, unstable or saddle. The eigenvalues calculated from the Jacobian, Eq. (25), were used to con"rm the stability of those points. Since the right-hand side of Eq. (23) is highly non-linear in the #uxes, it is practically impossible to obtain an analytical expression for the Jacobian matrix without making further simpli"cations that would be valid mainly for ideal cases; moreover, the numerical determination of the nodes impose an additional limitation. Eq. (25) shows that the Jacobian, and consequently its eigenvalues, is highly in#uenced by the #uxes. This agrees with the suggestion by Pakowski (1994a) that the magnitude of the total #ux will determine the stability of the nodes. Both the Jacobian and its eigenvalues were calculated numerically. Fig. 5 shows a map of eigenvalues for the ACM and WEE mixtures. The curves in the diagrams separate the negative from the positive eigenvalue zones. In the case of the ACM system, the negative eigenvalue zone for chloroform and the positive eigenvalue zone for acetone are indicated. The pure methanol node and the binary chloroform}acetone node fall within both negative eigenvalue zones, and they behave as stable nodes. The two binary nodes (chloroform}methanol and methanol} acetone) fall within both positive zones, resulting in unstable nodes. The ternary node falls within the chloroform's negative zone and the acetone's positive zone, resulting in a saddle point. For the WEE system, the three pure component nodes are within the negative zones, resulting in stable nodes. The binary nodes are saddle points since they fall within the positive zone for
5831
water and negative zone for ethanol. The ternary node is in both positive zones and thus it is unstable. 4.2. Non-isothermal case Simulations were performed for the same geometry using the same set of constant conditions as in the isothermal case, except that the value of the liquid temperature here is initial and di!erent for every initial point. Fig. 6 shows the trajectories for di!erent initial conditions. All trajectories reach the wet-bulb temperature and the evaporation continues along the wet-bulb temperature towards the stable azeotropes. As in the isothermal case, the direction of the trajectories will depend on the location of the initial point. In Figs. 6(a) and (b), the trajectories are projected onto the concentration diagram only. The stable separatrices de"ne the attraction zones of the corresponding stable points. In both cases, trajectories tend towards increasing temperature in the concentration diagram. For example, in the system ACM, the two stable azeotropic points correspond to the maximum temperatures. In the case of the WEE mixture, these maximum temperatures lie on the vertices. This behaviour corresponds to the fourth theorem in Doherty and Perkins (1978a), where they prove that the temperature surface is a Liapunov function, an increasing function, indicating stability. Note that other systems may behave di!erently. By looking at the second block in the main diagonal in Eq. (33), for any given value of liquid composition and as long as trajectories lie on the temperature surface, the heat #ux partial derivative with respect to temperature is negative, since as temperature increases less heat is required to supply energy for the evaporation. As moisture liquid is being evaporated, an increase of the temperature results in
Fig. 5. Map of eigenvalues during evaporation into a pure gas.
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Fig. 6. Non-isothermal evaporation into pure nitrogen during drying of a solid wetted with a liquid ternary mixture.
Fig. 7. Evaporation during drying into pure gas. Map of eigenvalues for (a) ACM and (b) WEE.
higher evaporation #uxes; then the partial derivative with respect to the temperature will be positive. The enthalpy of evaporation, which is the di!erence between the gas and the liquid enthalpies, is always positive. Consequently, the third term will be negative. The liquid enthalpy increases when the temperature is raised giving a positive change. The product of this change with the #uxes gives a small term compared to the net e!ect of the other two terms and can be neglected. Therefore, since the denominator is always positive, the eigenvalue is negative as long as the temperature increases. Thus, the non-isothermal case can be analysed by determining the stability of the system as if it is an isothermal case at the maxima of the temperature surface. Fig. 7 shows the maps of eigenvalues of the systems ACM and WEE. Similarly, as in the isothermal case, the stable singular
points lie in the negative eigenvalue zones. For the mixture ACM, the binary acetone}chloroform azeotrope and the pure methanol vertex result in stable points. In the case of the mixture WEE, all three pure component nodes are stable. 4.3. Bifurcation in the batch case Since the di!erence in concentration in the interface and the bulk gas provides the driving force, gas preloading in batch drying will have a tremendous in#uence on the evaporative process. As the gas is loaded with one of the components in the liquid mixture, the phase portrait changes gradually as the concentration parameter for that component varies and at some value the phase portrait changes abruptly.
F. Luna, J. Martn& nez / Chemical Engineering Science 54 (1999) 5823}5837
Fig. 8 shows the phase portrait of the system ACM when the gas is loaded with methanol at di!erent concentrations. At a concentration di!erent from zero, see Fig. 8(b), the original binary node acetone}chloroform is shifted to a ternary node, which remains stable. Now the system presents two ternary azeotropes one stable and one saddle. At some value of methanol concentration (see Figs. 8(c) and (d)) the two azeotropes coalesce and disappear. This phenomenon occurs for relatively higher concentrations in the gas when loaded with acetone or chloroform alone. When the gas is loaded with acetone (for example see Figs. 9(a) and (b)), the binary point acetone}chloroform is shifted down along the binary mixture diagonal towards the acetone vertex. In this case, the stable pure methanol point is shifted into a binary acetone}methanol azeotrope moving towards the acetone vertex. At some value of acetone concentration,
5833
the ternary saddle azeotrope and the binary acetone}methanol node coalesce and disappear. When the gas is loaded with all three components (Figs. 9(c) and (d)) the pure component and binary singular points are shifted into the diagram becoming ternary stable azeotropes. By increasing the concentrations beyond some value, the saddle azeotropic point and the stable one, originating from the methanol vertex, coalesce and disappear leaving a unique ternary stable azeotrope. The system WEE presents a similar behaviour when the gas is loaded with one of the components. Fig. 10 shows the case for water-loaded nitrogen. At some value of vapour concentration in the gas, the ternary unstable azeotrope disappears leaving a large attraction zone for the pure water stable node. Both systems exhibit what constitutes a saddlenode bifurcation. The coalescence and disappearances of two singularities, like in the two previous cases, is
Fig. 8. Bifurcation in the evaporation of the mixture ACM into nitrogen loaded with methanol vapour.
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Fig. 9. Bifurcation in the evaporation into vapour-loaded nitrogen of the mixture ACM: (a) and (b) acetone; (c) and (d) all three solvents.
comprehensible if one considers the null-isoclines along which the vector "eld is horizontal and vertical and whose intersection gives the zeroes of the vector "eld. If these isoclines are dependent on a parameter being the gas concentration of one of the components in the mixture, the number of intersections will remain constant unless the parameter passes through a value where the isoclines are tangent, and where, in general, two intersection points coalesce and disappear. 4.4. Continuous drying Since in the continuous case the gas and liquid conditions change simultaneously along the dryer length, it is impossible to represent all independent variables in a unique phase portrait of dimension RL>. However, the ternary diagram may be used as a projection of the
trajectories. Results of simulations starting from an initial common point for di!erent values of the ratio R "F /F are shown in Fig. 11(a) for both cases for E * the liquid mixture ACM, the corresponding temperatures are shown in Fig. 11(b). Notice that the trajectories follow a similar behavior. For each initial point, trajectories end up at an equilibrium point corresponding to the saturation point of the gas at some "nite length in the dryer. At this point, the system reaches a dynamic equilibrium. These points constitute the singular points of the system. Note that these points are in"nite and their position will depend on the inlet conditions. Since continuous evaporation of the liquid will load the gas along the dryer length, the driving force, represented by the di!erence between the gas and liquid interface composition, will decrease correspondingly. This will ensure low drying rates
F. Luna, J. Martn& nez / Chemical Engineering Science 54 (1999) 5823}5837
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Fig. 10. Bifurcation in the evaporation into water-loaded nitrogen of the mixture WEE.
Fig. 11. Continuous evaporation into nitrogen of a liquid "lm for the system ACM.
and reinforce the assumption of gas-phase-controlled conditions. 4.5. Bifurcation in the continuous case In Fig. 12 trajectories are plotted for di!erent inlet ratios R . Observe that as R increases the equilibrium point displaces through the composition diagram tending towards the position of the singular point of a batch case with the same process conditions. The inlet total gas #ow rate is the only parameter varied. As this ratio increases, see Eq. (39), the gas temperature tends towards the constant inlet gas temperature. In the case of the initial points B and C, at some value of R the equilib
rium point tending towards the pure methanol vertex disappears and the trajectories change direction at some value of R . For larger values of R , increasing towards in"nity, the equilibrium point tends towards the binary acetone}chloroform, as in the batch case. Depending on the position of the initial conditions, the system may or may not exhibit a bifurcation for increasing values of R as in the case of point A. Increasing R towards in"nity does not abruptly change the direction of the trajectory and the equilibrium point is shifted towards the pure methanol vertex. The stability zones, corresponding to the negative zones of the map of eigenvalues, are displaced along with the respective nodes and depend upon the corresponding
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F. Luna, J. Martn& nez / Chemical Engineering Science 54 (1999) 5823}5837
Fig. 12. In#uence of the inlet #owrate ratio F /F on the position of a singular point: Stability zones for di!erent inlet process conditions. E *
inlet conditions as well. Fig. 12 shows the stability zones for di!erent equilibrium conditions. For small values of R the zones remain negative. By increasing the ratio, positive zones begin to appear and grow (Fig. 12(a)). When R tends to in"nity (Fig. 12(b)), the process ap proaches a batch behaviour and the resulting map of eigenvalues is similar to the non-isothermal batch case (Fig. 7(a)).
5. Conclusions The stability of a drying process is characterized by the stability of the system's singular points, determined by the dynamics of the process. Particularly, we have presented the stability analysis of the models for batch and continuous processes. In the case of the batch process, when isothermal conditions are assumed, the phase portrait is represented by the diagram of compositions. The stable nodes are contained in attraction zones, delimited by stable separatrices passing through the saddle azeotropes. The extent of these zones is related to the net e!ect of the negative selectivity zones of the components in the mixture. The stability of the azeotropes may be determined by calculating the eigenvalues of the n!1 components in the mixture. In general, trajectories will tend towards the point that gives a component or mixture of components which are preferentially retained, that is, where selectivities will be predominantly negative for the component or components involved. Thus, azeotropic points may result in stable or unstable nodes or saddles. For a non-isothermal batch process, the analysis may be done individually for the mass transfer and heat transfer processes. In general, when trajectories tend in the
direction of increasing temperature, the eigenvalue corresponding to the temperature is negative. Thus, the analysis may be carried out following the same approach as in the isothermal case. In the batch case, the phase portrait may be changed by loading the gas with one or more components of the system. This may cause a bifurcation of the saddle}node type, where a node and a saddle coalesce and disappear at some particular value of the gas concentration. Consequently, by loading the gas, additional nodes may appear, and some may disappear. In the continuous case the conditions change in both phases along the dryer; consequently, trajectories will depend on the inlet conditions. For small values of the inlet gas-to-liquid #owrate ratio, R , the system exhibits in"nite singular points. Each point corresponds to a dynamic equilibrium state and results in stable nodes. As the ratio increases, tending towards in"nity, the process approximates a batch behaviour, and an analysis of this case may be applied.
Notation a A c C
speci"c area, m m\ area, m mass heat capacity, kJ kg\ K\ molar heat capacity vector at constant pressure, kJ kmol\ K\ F molar #ux vector, kmol m\ s\ F total molar #ux, kmol m\ s\ G molar evaporation rate vector, kmol m\ s\ G total molar evaporation rate, kmol m\ s\ 2 H molar enthalpy vector, kJ kmol\ I indentity matrix
F. Luna, J. Martn& nez / Chemical Engineering Science 54 (1999) 5823}5837
J k
Jacobian matrix matrix of zero-#ux mass transfer coe$cients, kmol m\ s\ m mass, kg n number of moles vector, kmol n number of condensable components n total number of moles in liquid phase, kmol * p saturated vapour pressure vector, bar P total pressure, bar q heat #ux by convection, kW m\ R gas-to-liquid #owrate ratio s vector of selectivities S solid mass #ux, kg m\ s\ t time, s ¹ temperature, K x vector of molar fractions in the liquid, kmol kmol\ y vector of molar fractions in the gas, kmol kmol\ z length, m 0 column vector of zeroes 1 column vector of ones Greek letters
a b C f h j K N N R q s t
heat transfer coe$cient, kW m\ K\ Bootstrap matrix diagonal matrix of activity coe$cients arbitary, length-related variable, Eqs. (34) and (35) dimensionless temperature ("¹/¹ ) eigenvalue eigenvalue matrix mass transfer coe$cients correction matrix, Eq. (5) heat transfer correction factor, Eq. (6) eigenvector matrix arbitary, time-related variable, Eq. (22) liquid-phase factor matrix ("x12!I) gas-phase factor matrix ("I!y12)
Subscripts g h i ¸ s ¹ 0 s h o
gas heat transfer index liquid solid total initial or inlet related to liquid-phase factor matrix, Eq. (29) related to dimensionless temperature perturbed matrix, Eq. (26)
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Superscripts T 0
transpose saturated
Acknowledgements The authors gratefully acknowledge the "nancial support provided by the Swedish Research Council for Engineering Sciences (TFR) for this work. References Bird, R. B., Stewart, W. E., & Lightfoot, E. N. (1960). Transport phenomena. New York: Wiley. Coulson, J. M., & Richardson, J. F. (1996). Coulson & Richardson's chemical engineering, vol. 1 (5th ed). London: Butterworth-Heinemann. Doherty, M. F., & Perkins, J. D. (1978a). On the dynamics of distillation processes } I. The simple distillation of multicomponent non-reacting homogeneous liquid mixtures. Chemical Engineering Science, 33, 281}301. Doherty, M. F., & Perkins, J. D. (1978b). On the dynamics of distillation processes } II. The simple distillation of model solutions. Chemical Engineering Science, 33, 569}578. Doherty, M. F., & Perkins, J. D. (1979). On the dynamics of distillation processes } III. The topological structure of ternary residue curve maps. Chemical Engineering Science, 34, 1401}1414. Foucher, E. R., Doherty, M. F., & Malone, M. F. (1991). Automatic screening of entrainers for homogeneous azeotropic distillation. Industrial Engineering and Chemical Research, 30, 760}772. Gmehling, J., & Onken, U. (1982). Vapor}liquid equilibrium data collection. DECHEMA. Luna, F., MartmH nez, J. (1998). Stability of the dynamical system describing gas-phase-controlled drying of ternary mixtures. Drying Technology Journal, 6 (9}10). MartmH nez, J., & Setterwall, F. (1991). Gas-phase controlled convective drying of solids wetted with multicomponent liquid mixtures. Chemical Engineering Science, 46(9), 2235}2252. Pakowski, Z. (1994a). Drying of solids containing multicomponent mixture: recent developments. Proceedings of the 10th International Drying Symposium, vol. A. Gold Coast, Brisbane, Australia (pp. 27}38). Pakowski, Z. (1994b). Stationary states in evaporation of multicomponent liquid droplets to inert gas stream. Zesz. Nauk. 716. Lodz Tech. University, pp. 1}91 (In Polish). SchluK nder, E. U. (1984). Einfu( hrung in die Stowu( bertragung (pp. 56}77). Georg Thieme Verlag, Stuttgart. Taylor, R. (1982). Film models for multicomponent mass transfer: Computational methods } II. The linearised theory. Computers in Chemical Engineering, 6, 69}75. Van Dongen, D. B., & Doherty, M. F. (1985). Design and synthesis of homogeneous azeotropic distillations. 1: Problem formulation for a single column. Industrial Engineering and Chemical Fundamentals, 24, 454. Vidaurre, M., & MartmH nez, J. (1997). Continuous drying of a solid wetted with ternary mixtures. A.I.Ch.E. Journal, 43(3), 681}692.