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Quantitative analysis of coupling effects in cross-flow membrane emulsification
Journal of Membrane Science 229 (2004) 199–209
Quantitative analysis of coupling effects in cross-flow membrane emulsification G. De Luca a,∗ , A. Sindona b , L. Giorno a , E. Drioli a a
Research Institute on Membrane Technology (ITM-CNR), via Pietro Bucci 18D, 87030 Rende, CS, Italy b Department of Physics, University of Calabria, via Pietro Bucci 30C, 87030 Rende, CS, Italy Received 21 March 2003; received in revised form 23 July 2003; accepted 22 September 2003
Abstract Membrane emulsification has attracted increasing experimental and theoretical interests over the last decade. On the experimental side, the parameters of this process have been thoroughly investigated and a linear relationship between the droplet size of an emulsion and the pore size of the membrane has been generally observed. Theoretical studies, however, have not provided an adequate description of many effects concerning droplet growth and detachment from membrane pores, such as the influence of the dynamic interfacial tension on the evolution of the droplet size. Most calculations have been based either on an algebraic torque balance equation or on a differential equation for the disperse phase flow rate through the membrane. The present paper offers a procedure for quantitative analysis which includes both the permeation of the disperse phase through the membrane pores and the mechanism of droplet detachment. The behavior of the droplet size of an emulsion versus the pore size of the membrane is determined with different sets of process parameters and it is compared satisfactorily with the experiments of Vladisavljevic and Schubert on Shirasu porous glass membranes. For the same membranes, the procedure is tested to pore sizes below 0.1 m and a linear correlation is observed, similarly to micropore sizes. © 2003 Elsevier B.V. All rights reserved. Keywords: Membrane emulsification; SPG membrane; Droplet formation; Process parameters; Transmembrane pressure
1. Introduction Emulsions, generally used in food, pharmaceutical and cosmetic products, are disperse systems of two (or more) immiscible liquids, such as water and oil, in which one phase (disperse phase) is distributed in form of droplets in the other phase (continuous phase). Most existing methods to prepare emulsions are based on the establishment of turbulent flows (turbulent eddies) in the fluid mixture consisting of the immiscible liquids. Both the size and the size distribution of droplets are important, since they determine the stability of emulsion against coalescence and its applications for intended uses. According to known techniques, the size of droplets, for a given pair of processed phases, is mainly determined by the size of the turbulent eddies and the times of exposure to these eddies [1]. In these methods, turbulence cannot be controlled or generated consistently throughout the volume of fluid; neither can the behavior ∗ Corresponding author. Tel.: +39-0984-492-014; fax: +39-0984-402-103. E-mail address: [email protected] (G. De Luca).
0376-7388/$ – see front matter © 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.memsci.2003.09.024
of any pair of immiscible phases be predicted on a large scale, based on tests performed in the laboratory [2]. The consequence is that energy is used inefficiently and, more importantly, it is not possible to directly control the size of droplets. For these reasons, much attention has been put in alternative emulsification processes. Membrane emulsification is a relatively new membrane technology, which allows the production of emulsion droplets under controlled conditions with a narrow size distribution. Both the size and the size distribution of droplets may be carefully controlled choosing suitable membranes and focussing on some fundamental process parameters. In cross-flow membrane emulsification, the disperse phase is pressed through a microporous membrane. Droplets formed at the membrane surface grow until their size reaches a critical value. Then, they are carried away with the continuous phase flowing through the membrane. Generally, systems with droplets of size ranging from 0.1 to 50 m are thermodynamically unstable; thus, a stable emulsion requires some additional substances, e.g., emulsifiers or stabilizers, to protect droplets against coalescence. Emulsifiers and stabilizers induce electrostatic or steric
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repulsions among droplets and generate a lower interfacial tension in the system. The interfacial tension yields a force at the border of the membrane pore, and, moreover, it influences the capillary pressure inside the droplet. In a membrane emulsification process, the stress to generate the droplets is much smaller with respect to traditional processes, because small droplets are directly formed at the micropores of a membrane, rather than by disruption in zones of high energy density. Accordingly, the energy needed is considerably less than that required by the conventional methods. Hence, the main advantage of membrane emulsification is the possibility to produce droplets of defined sizes, with a narrow size distribution and potentially lower energy demands [3]. Much experimental effort has been invested in determining the correlation between the droplet size of an emulsion and the pore size of the membrane [4,5]. A linear scaling law has been generally observed, with a slope ranging typically from 2 to 10, although a value near to 50, for particular operating conditions and emulsifier concentrations, has been found [6,7]. The reasons for this variation have not yet been well understood. The production of mono-disperse emulsions is essentially related to the size distribution of pores and their relative spatial distribution at the membrane surface. The formation and size of droplets depend on: (i) operating parameters, i.e., cross-flow velocity, transmembrane pressure and disperse phase flow; (ii) membrane parameters, i.e., pore size, active pores, membrane hydrophobicity/hydrophilicity; and (iii) phase parameters, i.e., interfacial tension, viscosity and density of the processed phases. Such quantities combine with different magnitudes, over the ranges of operating conditions, and many of them exhibit coupling effects. The aim of the article is to account for these correlations, using a torque balance model to describe the detachment of the droplet. In other words, combined action of the fundamental parameters controlling the evolution of droplets is studied. Estimation of coupling effects can provide a key for theoretical interpretation of some experimentally derived behaviors, such as the above-mentioned law between the droplet size and the pore size. The fundamental parameters are chosen on the basis of data that give evidence of their importance in controlling droplet formation. Other phenomena concerning coalescence among droplets, deformations in droplet shape and formation of droplet neck are not considered here. As reported by Joscelyne and Trägårdh [8], there is coalescence of growing droplets, at the membrane surface, because active pores are too close to one another. Due to the capillarity pressure, active pores may inhibit the flow of neighboring pores and greatly decrease the effective membrane porosity. Moreover, the distribution of the active pores changes as a function of the applied pressure. One approach that tries to overcome this complexity makes use of computational fluid dynamics, by which Abrahamse et al. [9] have recently determined the
shape and neck description of one oil droplet forming at a single cylindrical pore. Their work describes the influence of the droplet shape on the maximum porosity of the membrane and on the oil flux through the pore. In the present paper, theoretical calculations are applied to experiments in which the condition for unhindered droplet growth is satisfied for all membrane pores and, then, the above-mentioned issues are negligible. 2. Theoretical background Theoretical description of a membrane emulsification requires two main clues to be dealt with: one is permeation of the disperse phase through the membrane pores; the other is the mechanism of droplet detachment. Previous approaches [8,10] have treated these two aspects separately. However, both of them simultaneously contribute to droplet evolution and should be considered in the same framework. A main goal of this contribution is to theoretically support the observed linear behavior dd = xdp , (2.1) between the droplet size, dd , and the pore size, dp . The major factors affecting explicitly the slope of Eq. (2.1) are: (i) the wall shear stress, τw ; (ii) the dynamic interfacial tension, γ; and (iii) the disperse phase flux, Jd . Thus, from the theoretical point of view, x is a functional of the above introduced quantities x = x{τw (vc , µc , Di , ρc ), γ(t, µc , T), Jd [γ(t, µc , T), Ptm , dp , µc , L]}. (2.2)
Other parameters enter implicitly this relationship, i.e., the average velocity of the continuous phase flow, vc , which depends on the difference of continuous phase pressures at the ends of the membrane tube, the viscosities of the disperse and the continuous phases, µd and µc , the inner size of the membrane tube, Di , the density of the continuous phase, ρc , the thermodynamic temperature, T , the transmembrane pressure, Ptm , the pore size and the thickness of the membrane, L. Coupling effects arise because γ appears in x either explicitly or implicitly. The temperature and viscosity of the continuous phase influence the dynamic interfacial tension through the emulsifiers diffusion coefficients. The latter obey the Stokes–Einstein relation, De ∝ T/µc , implying that, as µc increases, the rate of variation of γ decreases, due to the decreasing of De . Below both µc and T are fixed to specific values so that γ can be parameterized as an explicit function of time. 2.1. Flux of the disperse phase The flux of the disperse phase through the pores of the membrane, Jd , may be assumed to follow Darcy’s law [2] Peff , (2.3) Jd = B µd L
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where B is determined by the membrane structure (pore size, shape and porosity) and Peff denotes the effective transmembrane pressure. A highly simplified picture is adopted in which the membrane has parallel cylindrical pores, perpendicular or oblique to the membrane surface, with lengths almost equal to the membrane thickness. When the pores have the same mean size, B takes the form B=
Np πdp4 128Am ξ
,
(2.4)
and Eq. (2.3) reduces to the Hagen–Poiseuille equation [11]. In Eq. (2.4), ξ is the pore tortuosity, Np the number of active pores and Am the membrane area. In the case of one active pore (Np = 1), the disperse phase flow rate, Qd = Jd Am , writes Qd =
πdp4 Peff . 128ξ µd L
(2.5)
The transmembrane pressure is defined as the difference of applied pressures between each side of the membrane and it consists of two parts: one is the capillary pressure, Pγ , due to the curvature of the interface and to the dynamic interfacial tension; the other is the effective transmembrane pressure Peff = Ptm − Pγ [γ(t), dd ],
(2.6)
or the drag of the disperse phase inside the pore which determines Qd . The droplet at a pore tends to form a spherical shape, whose radius of curvature, dd /2, is related to the height of the spherical droplet, h, and to the membrane pore radius, dp /2, by (dp /2)2 + h2 dd = . 2 2h
(2.7)
When the height of the growing droplet equals the pore radius, the droplet size reaches its minimum, dd = dp . Accordingly, the capillarity pressure, calculated from Laplace equation 4γ(t) Pγ [γ(t), dd ] = , dd
(2.8)
takes a critical value, offering the maximum resistance to the disperse phase flow. Finally, when h → 0 or h → ∞, the capillarity pressure goes to zero and the effective transmembrane pressure equals the transmembrane pressure. 2.2. Mechanism of droplet detachment Droplet formation at an individual pore involves two stages, i.e., droplet growth and droplet detachment [3]; the growth process ends when a balance among all mechanical torques, acting on the droplet, is reached. The droplet size, at this time, can be estimated from the algebraic equation that describes this balance. At the end of the first stage, the droplet is still connected to the pore through a neck; when
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the connection is broken, the detachment is completed [10]. The droplet neck may form either before or after a torque balance is established. In both cases, the detachment begins always after equilibrium, inducing modifications on the droplet. The final droplet size is determined by the droplet volume at the end of the detachment stage. Formation and shutting of the neck can be described by the parameters reported in Eq. (2.2). According to Peng and Williams [2,10], when the cross-flow shear force is small with respect to the interfacial tension force, the droplet forms a sphere at the pore mouth; on the other hand, as the cross-flow velocity increases, the droplet deforms from spherical symmetry. The main forces acting on a droplet to determine its growth and detachment, have been identified in literature [6,12] FD = kx 3πµc dd vc∞ ≈ 23 kx πτw dd2
(2.9)
is the drag force produced by the continuous phase flow, with kx the wall correction factor, e.g., kx = 1.7 for a single sphere touching an impermeable wall in a simple shear flow [10,13], and vc∞ the undisturbed tangential velocity of the continuous phase at the droplet center. In Eq. (2.9), the shear stress of the continuous phase is approximated to the shear at the membrane surface, τw , since small droplet sizes are considered. The relationships of the continuous phase velocity and τw are well known either for laminar or turbulent flows [12]. Fγ = πdp γ(t)
(2.10)
is the interfacial tension force; Fsp =
γ(t) 2 πdp dd
(2.11)
is the force due to the difference of pressure needed to overcome capillarity; Fdl =
0.761τw1.5 ρc0.5 3 dd µc
(2.12)
corresponds to the dynamic lift force [6]. The relative intensities of these global forces change as the droplet increases in size. Other forces, such as inertia and buoyancy forces, are approximately from six to nine orders of magnitude smaller than the FD , Fγ and Fsp , for droplet sizes of the order of ∼1 m [6]; therefore, such forces can be neglected in a torque balance model that applies to micropores. The droplet size can be, thus, estimated from the following torque balance equation dp dd FD = , (2.13) Fi 2 2 i
assuming that shape of the droplet is approximately spherical. Eq. (2.13) provides a relation between the membrane pore size and the droplet size at equilibrium. Several torque balance models may be defined on the basis of the forces Fi appearing in the torque balance equation. In either model,
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the torque balance equation establishes a non-linear relationship of dd versus dp .
3. Coupling effects and related models The following quantitative analysis considers Shirasu porous glass (SPG) membranes of pore sizes ranging from 0.4 to 6 m with the experimental conditions (Tween 80 emulsifier, transmembrane pressure, wall shear stress) recently reported by Vladisavljevic and Schubert [14]. This system is particularly suitable for the theoretical background discussed in Section 2, since SPG membranes have cylindrical, interconnected and uniform micropores and the disperse phase flow rate through the membranes follows Eq. (2.5). In other porous membranes, e.g., in ceramic membranes, Qd is better described by the Kozeny–Carman relation [19]. Attention must be paid to the wall shear stress in [14], τw is related to a tube with a turbulent Reynolds number. In any case, at very close distances from the membrane surface (sub-layer), the flow is laminar, thus, Stoke and Newton laws, leading to Eq. (2.9), are still valid. 3.1. Static torque balance models In setting up a static model based on the torque balance equation (Eq. (2.13)), one has to fix the dynamic interfacial tension at specific times. Fig. 1 shows the experimental values of the dynamic interfacial tension for the Tween 80 emulsifier, measured in [15], by the bursting membrane method [16]. An analytical expression for γ(t) is obtained from γF (t) =
γ0 + γm δ(t, t∗ , n) , 1 + δ(t, t∗ , n)
Fig. 1. Experimental values () of the dynamic interfacial tension for the Tween 80 emulsifier [15]. Data are linearly interpolated and their non-linear fitting function (3.1) is also reported.
(3.1)
with δ(t, t∗ , n) = (t/t∗ )n [17]. In γF , the values γ0 and γm are fixed by experiments, while t∗ and n are adjusted by a non-linear fitting routine to the values t∗ = 0.65 ± 0.01 s and n = 1.83 ± 0.07. Hereinafter, the dynamic interfacial function, is considered either at equilibrium, when t → ∞ and γF (t) approaches γm (Figs. 1 and 2 and Table 1), or at the experimental times of formation of the droplet, tf (Figs. 1 and 3 and Table 2). In the latter case, the values γ(tf ) are calculated from the experimental curve of Fig. 1 with a linear interpolation scheme. Table 1 Real solutions, ddi and ddii , for SM3 with the process parameters of Fig. 2 exp
dp (m)
ddi (m)
ddii (m)
dd
0.4 1.4 2.5 5.0 6.6
0.40 1.40 2.52 5.08 6.75
3.56 7.98 11.5 17.6 20.8
1.4 4.6 8.5 14.7 23.9
exp
dd
labels the experimental result of [14].
(m)
Fig. 2. Droplet size vs. pore size for SPG membrane in the static models SM1–SM3. The process parameters γm = 6.5 × 10−3 N/m, τw = 8 Pa, µc = 10−3 Pa s, ρc = 103 kg/m3 , kx = 1.7 are used.
G. De Luca et al. / Journal of Membrane Science 229 (2004) 199–209
Fig. 3. Droplet size vs. pore size for SPG membrane in the static models SM3. The interfacial tension is fixed either to γm (ddi and ddii , Fig. 2) or to γ(tf ) (d id and d iid , Table 2). Other process parameters are τw = 8 Pa, µc = 10−3 Pa s, ρc = 103 kg/m3 .
Eq. (2.13) becomes an algebraic equation for dd , whose real solutions can be compared with the experiments. The following models were tested: • static model 1 (SM1)considers only the action the interfacial tension force, i Fi = Fγ ; • in static model 2 (SM2), both the interfacial tension force and the static pressure difference force are included, i Fi = Fγ − Fsp ; • static model 3 (SM3) uses all the forces appearing in Eqs. (2.9)–(2.12), i Fi = Fγ − Fsp − Fdl . Fig. 2 shows the real solutions obtained from SM1–SM3, with the equilibrium interfacial tension γm . SM1 presents only one real solution for dd , whereas SM2 and SM3 offer two real solutions (Table 1). Table 2 Real solutions, d id and d iid , for SM3 with the process parameters of Fig. 3 dp (m)
tf (s)
γ(tf ) (mN/m)
d id (m)
d iid (m)
0.4 1.4 2.5 5.0 6.6
0.6 0.7 0.8 0.9 1.8
17.2 15.8 14.7 13.8 10.5
0.40 1.40 2.50 5.03 6.69
4.97 10.9 15.4 23.3 25.0
Experimental formation times tf are taken from [14] and γ(tf ) refers to a linear interpolation scheme (Fig. 1).
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It is evident that the first solutions of SM2 and SM3, labeled ddi , are well linearly correlated with a slope equal to one. On the other hand, the solution of SM1 and the second solutions of SM2 and SM3, labeled ddii , exhibit non-linear features. It may be concluded that the effect of the interfacial force alone in balancing the drag force of the continuous phase yields unsatisfactory results. In addition, slight differences are observed between SM2 and SM3, although the second solution of the latter is in better agreement with experiments. It is also worth underlining that the wall correction factor, kx , which contributes to establish the magnitude of the drag force, can be used as a fitting parameter in conjunction with effect of the droplet neck to reproduce experiments, however, in the present paper no adjustment of parameters is performed. Fig. 3 reports both real solutions of SM3, labeled d id and d iid , respectively, in which γ(t) is evaluated at the experimental formation times tf ; such times change with the pore size as displayed in Table 2. The solutions ddi and ddii , for the equilibrium interfacial tension γm , are also reported for comparison. No substantial differences are detected between ddi and d id , suggesting that the first linear solution of SM3 is independent on the time variation of the interfacial tension. This fact is not physically consistent, thus, such solution is not considered. On the contrary, the second non-linear solution of SM3 shows significant changes when the equilibrium interfacial tension, yielding ddii , is replaced with the dynamic interfacial tension, yielding d iid . The values d iid are larger than both ddii and the experimental droplet sizes. Looking at Fig. 1 and Table 2, it is clear that γm is the minimum interfacial tension for any given pore size, for which reason the values ddii are lower limits to the results that may be calculated from the second solution of SM3. 3.2. Disperse phase flow rate approach A dynamic model, which explicitly accounts for the time variation of the interfacial tension, is based on the disperse phase flow rate equation. The dynamic droplet volume Vd can be related to the droplet height from the surface membrane by the continuity equation d Vd [h(t)] = Qd [h(t), γ(t)], dt
(3.2)
in which Vd is the volume of a spherical cap of height h on a pore of size dp Vd [h(t)] = 16 πh(t)( 43 dp2 + h(t)2 ),
(3.3)
and Qd is disperse phase flow rate of Eq. (2.5). Eq. (3.2) is, then, arranged to the following differential equation dp4 dh 16γ(t)h Ptm . (3.4) = − 2 dt 16µd ξL dp2 + 4h2 (dp + 4h2 )2
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The latter can be solved numerically with a suitable set of initial conditions, discussed further, and, by Eq. (3.3), the evolution of Vd can be predicted. The dynamic droplet size can be obtained from Eq. (2.7), expressing the geometrical relation, in a spherical cap of height h(t), between the radius, dd (t)/2, and the base radius, dp /2. Alternatively, an average droplet size, dd , can be defined by approximating the droplet volume Vd [h(tf )], at the droplet formation time, to the volume of a sphere of diameter dd [14] dd = 3 h(tf )[ 43 dp2 + h(tf )2 ]. (3.5) It is observed that dd (tf ), calculated from Eqs. (2.7) and (3.4), differs negligibly from dd , at least in the experiment under consideration. Eqs. (2.7), (3.4) and (2.13) are a system of equations accounting for several correlations in process parameters; one of these is the effect of the dynamic interfacial tension on the dispersed flow rate at the droplet size. In addition, dd (t), calculated from Eqs. (2.7) and (3.4), can be introduced in the algebraic torque balance equation (Eq. (2.13)) with the dynamic interfacial tension of Fig. 1, to obtain a theoretical estimation of the formation times and of the final droplet size. This new procedure combines the torque balance approach with the disperse phase flow rate method, allowing a multivariate analysis of process parameters. The choice of an initial condition for h(t) is related to the considerations leading to Eqs. (2.7) and (2.8). Joscelyne and Trägårdh [18], have observed that the critical pressure changes during the process; in particular the difference of pressure from the beginning and the end of membrane emulsification is 260 and 40 kPa for ceramic membranes with pore size of 0.1 and 0.5 m, respectively. Then, for a given pore size and a given initial value of the capillarity pressure there corresponds an initial height h(0). On this basis, establishing a set of initial conditions compatible with experiments of [14] is a very hard task. To overcome this difficulty, the following scheme is proposed: • When h(0) = dp /2, Eq. (2.7) implies dd (0) = dp and, substituting γ(0) and dd (0) into Eq. (2.8), one obtains Pγ0
4γ(0) = Pγ [γ(0), dd (0)] = . dp
(3.6)
The transmembrane pressure, Ptm , must be slightly larger than Pγ0 , i.e., Ptm = Pγ0 + δP, with δP Pγ0 . • For values of h(0) larger than dp /2, Eq. (2.7) yields dd (0) = (dp2 /4h(0)) + h(0). Then, by Eq. (2.8), a relationship between the experimental capillarity pressures, Pγe , and the initial heights of the droplets is found Pγe = Pγ [γ(0), dd (0)] =
16γ(0)h(0) . dp2 + 4h(0)2
(3.7)
Fig. 4. Log–log representation of dd (t) (calculated from Eqs. (2.7) and (3.4)), with the initial conditions (3.6) and (3.7) and the process parameters of [14]. Comparison is made with the static case γ(t) ≈ γm .
In [14], Pγe is measured for all pore sizes reported in Tables 1 and 2 and the transmembrane pressure is set to Ptm = 1.1Pγe . Fig. 4 shows dd (t), calculated with the set of initial conditions (3.6) and (3.7), for the pore sizes reported in Table 1. The trend of the two set of solutions with respect to the pore size is similar, that is, dd increases with increasing dp . However, this monotonic behavior may change using different initial conditions, e.g., for certain values of h(0), the solutions corresponding to different pore sizes may cross at specific times. In any case, the functional relationship of dd versus t changes with changing the initial conditions; Nevertheless, such discrepancies are significant at times below 0.5 s and may influence the droplet formation when tf reaches shorter values than those reported in Table 2. Fig. 5 shows theoretical and experimental correlations between the droplet size and the membrane pore size.
G. De Luca et al. / Journal of Membrane Science 229 (2004) 199–209
Fig. 5. Theoretical behavior of dd (tf ) vs. dp from the disperse phase flow rate approach of Eqs. (2.7) and (3.4). Formation times are those of Table 2. The experimental result of [14] is also shown.
Theoretical values are obtained numerically from Eqs. (2.7) and (3.4) at the experimental formation times of Table 2 and using both conditions (3.6) and (3.7). Concerning the initial condition (3.7), a test is proposed in which the capillarity pressure is increased of ∼15% with respect to its measured value. This correction is because Eq. (3.4) requires, as input parameters, the initial capillarity pressures, Pγ0 , that are generally larger than the Pγe [8]. The same equations are also solved for the static interfacial tension γm , which corresponds to the approach of Peng and Williams [10]. It is observed that both dynamic solutions offer a more realistic trend with respect to the static torque balance models of Figs. 2 and 3. In addition, they are in better agreement with experiments, in comparison with the disperse phase flow method already solved in literature [10]. In order to investigate correlations between the transmembrane pressure and the droplet size, dd (tf ) is calculated from Eqs. (2.7) and (3.4), with the initial condition (3.7), at different values of Ptm . Fig. 6 reports the theoretical distribution dd (tf ) versus Ptm , for the different pore sizes of Tables 1 and 2. Values Ptm = αPγe , with 1.1 ≤ α ≤ 2.0 are considered. It is observed that dd increases
205
with increasing Ptm ; its rate increases as the pore size increases. Attention must be paid to sizes dp ≤ 2.5 m, for which small changes are detected in the droplet size over the whole range of the transmembrane pressure. Schröder et al. [6] have pointed out that this different trend depends on the relation between droplet formation times and the dynamic interfacial tension. They have suggested that, if droplet formation occurs before that the interfacial tension reaches its equilibrium value, the droplet size is correlated to the transmembrane pressure, while, in the other case, dd is almost independent on Ptm . Considering the formation times for two ceramic membranes, of pore sizes dp = 0.2 m and dp = 0.8 m, respectively, and the equilibrium time for the LEO-10 emulsifier, they have observed that Ptm influences the droplet size at the 0.8 m membrane, since the formation time of the droplet is shorter than the equilibrium time of the emulsifier; the same correlations are absent in the 0.2 m membrane, having a droplet formation time larger than the equilibrium time of the emulsifier. As shown in Fig. 6, the present work offers a ready to use quantitative method to estimate the effect of Ptm upon the droplet size, which accounts for the coupling of the transmembrane pressure, the disperse flow and the dynamic interfacial tension of the emulsifier. The type of membrane and pore sizes considered differ from the ones studied in [6] and the formation times of all droplets at the SPG membrane are shorter than the equilibrium time of the Tween 80 emulsifier. In this respect, this results allow a quantitative analysis of the transmembrane pressure effect in membrane emulsification. 3.3. Dynamic model For a full theoretical treatment of the experiment considered in the former subsection, a method to calculate droplet formation times at the SPG membrane pores is needed. One possibility is to use the dynamic solution, dd (t), of Eqs. (2.7) and (3.4), with the initial conditions (3.6) and (3.7), in the algebraic equation provided by the torque balance model SM3. The resulting equation 3 kx πτw dd3 (tfTH ) 2 =
πdp2 γ(tfTH ) −
γ(tfTH )
πdp3 dd (tfTH )
3/2 1/2
τw ρc − 0.761 dp dd3 (tfTH ) µc (3.8)
can be solved numerically to calculate the theoretical formation times tfTH . Fig. 7 shows dd versus dp for the theoretical formation times of Eq. (3.8) with both conditions (3.6) and (3.7). Such times are reported in Table 2 in comparison with the experimental times tf . It is noted the torque balance model in conjugation with the disperse phase flow based method provides a reasonable estimation of the dd versus dp distribution
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Fig. 7. The dd vs. dp distributions with both initial conditions (3.6) and (3.7). Either experimental, tf , or theoretical, tfTH , formation times are considered. Fig. 6. Theoretical behavior of dd (tf ) vs. Ptm , from the disperse phase flow rate approach of Eqs. (2.7) and (3.4) with the initial conditions (3.7).
at large pores, above ∼5.5 m. On the other hand, for pore sizes below ∼5 m, Eq. (3.8) overestimates the formation times of droplets and, accordingly, the droplet size. It is believed that such discrepancies are due to the highly simplified scheme of forces used in the torque balance model and it is hoped that a more realistic description of the mechanism of droplet detachment will provide more satisfactory results. A lot of empirical studies on SPG oil–surfactant–water systems can be found in literature. Thus, it would be interesting to improve the weight of evidence of the results of present section extending the same type of analysis to data from more than one source. However, particular attention must be paid to the availability of experiments reporting parameters compatible with the hypotheses of the model. For example, Yamazaki et al. [20] have studied Polytetrafluoroethylene (PTFE) membranes, detecting a linear behavior of dd with pore sizes estimated from the Laplace equation, rather than with nominal pore sizes. They have used a mixture of emulsifiers, corresponding to an equilibrium interfacial tension of 8.7 × 10−3 N/m (8.7 dyne/cm),
and have not reported the viscosity of the dispersed phase, due to a mixture of styrene, divinylbenzene and hexadecane at different concentrations. Katoh et al. [4] have also used a mixture of emulsifiers, for which the dynamic interfacial tension can be hardly obtained. Yasuno et al. [21] have recently published a characterization of SPG emulsification with a single emulsifier, SDS at 0.3 wt.%, and pore sizes equal to 16 m. The oil permeate flux was precisely controlled by the accurate syringe pump, although the transmembrane pressures, of crucial importance for the present analysis, have not been reported. The same authors have identified two droplet formation mechanism, explained by a spontaneous and a shear force formation model, which depend on the disperse and continuous phase flux velocities. In particular, they have observed that, when the dispersed phase flux is greater than ∼7.2 l m−2 h−1 and the continuous phase velocity above ∼0.5 m/s, droplets form because of the competing action of both mechanisms. The dispersed phase fluxes, presented in [14], are below 7.2 l m−2 and the reported continuous phase velocity is 1.4 m/s; thus, the emulsification is dominated by the shear force mechanism. Moreover the condition for unhindered droplet growth is satisfied for all membrane pores.
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Table 3 Experimental and theoretical formation times dp (m)
tf (s)
tfTH (s) (Eq. (3.6))
tfTH (s) (Eq. (3.7))
0.4 1.4 2.5 5.0 6.6
0.6 0.7 0.8 0.9 1.8
10.0 5.15 2.68 2.01 1.42
10.0 6.08 3.43 3.10 2.47
3.4. Theoretical prediction for SPG membrane with pore size below 0.1 µm Following the encouraging results presented in Sections 3.1–3.3, the next step is to extend the formulation for membranes with pore sizes below 0.1 m. However, at these sizes, attention must be paid to the morphology and tortuosity of pores and the thickness of the membrane. Eqs. (2.7) and (3.4) can be used only with the initial conditions (3.6), because experimental capillarity pressures are not reported for the considered type of membranes. The approach is tested to pore sizes of 0.01, 0.05 and 0.1 m. Correspondingly, the membrane thickness is reduced to L = 0.1 mm. Since the torque balance model is a highly simplified scheme to estimate droplet formation times (Table 3), the latter are extrapolated from experiments. Fig. 8 reports experimental formation times versus pore sizes taken from Table 3. Two possible extrapolation are shown: ta (dp ) = 0.12 ln(1+663dp ) is a logarithmic fitting curve and
Fig. 9. Distribution of dd vs. dp in log–log scale and its extrapolation to pore sizes below 0.1 m.
tb (dp ) a linear interpolating polynomial. Many other analytical functions passing through the shadowed region, ranging from dp = 0 and dp = 0.4 m and included between ta and tb , have been tested. The pore tortuosity, ξ, does not follow a monotonic trend versus dp , hence it is allowed to range from its minimum, ξ = 1, to its maximum, ξ = 2.1, observed values (Fig. 4). The results are summarized in Fig. 9, in which the error bar on the predicted values of dd accounts for: (i) the different kind of extrapolation used to estimate the formation times; (ii) the use of several values for the pore tortuosity in the range specified above; and (iii) the use of different membrane thickness, compatible with the considered pore sizes. It is noted that the experimentally observed linear behavior (Eq. (2.1)) extends also at the considered sizes.
4. Conclusions
Fig. 8. Experimental formation times vs. pore sizes. Two kinds of extrapolating functions, labeled ta and tb , are shown.
This work has presented a study of the influences of some process parameters, such as transmembrane and capillarity pressures, dynamical interfacial tension, membrane thickness and pore tortuosity, on the correlation between droplet size and pore size of an SPG membrane. These effects have been analyzed either in the framework of the current torque balance models (Eqs. (2.10)–(2.13)) or using a
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disperse phase flow based procedure (Eqs. (3.2)–(3.7)). The main aim was to propose a procedure, on the basis of some published material [2,3,6,8,9,14,17], that can connect droplet formation and detachment by means of simple, ready to use equations (Eqs. (3.4) and (3.8)). These equations take into account the dynamic interfacial tension and make use of experimental variables utilized in emulsification modules. At the moment, more realistic descriptions of the mechanism of droplet detachment are needed. Nevertheless, from the study presented here, some interesting conclusions on membrane emulsification can be drawn. In particular: (i) torque balance models SM1–SM3 (Eqs. (2.10)–(2.13)) are less reliable with respect to a disperse phase flow based description (Eqs. (3.2)–(3.7)); (ii) the correlation between transmembrane pressure and droplet size evidences the importance of using the dynamic interfacial tension for droplet with formation times below the emulsifier equilibrium time; the inclusion of effect of the dynamic interfacial tension can provide useful information to experimentalists concerning the type of emulsifier to be used in a membrane emulsification process in order to have droplets of well defined sizes; (iii) a combined use of two methods, disperse flow description and torque balance model (Eqs. (3.4) and (3.8)), is possible, although for more realistic estimation of droplet formation times the torque balance model should be replaced by a more accurate approach; (iv) extension of the same type of analysis on models of porous membranes with sizes below 0.1 m, reveals that strong linear correlations between droplet size and pore size are still present, if the membrane thickness and pore tortuosity are of the same order with respect to larger pores. Finally, the disperse phase flow approach should be tested with other flow equations in porous membranes; the influence of the wall shear stress on the droplet size and an energetic analysis of the droplet formation is also desirable.
Nomenclature Am dd dd dp Fdl FD Fsp Fγ h Jd kx L
membrane area droplet size average droplet size pore size dynamic lift force viscous phase force static pressure difference force interfacial tension force height of the spherical droplet disperse phase flux wall correction factor membrane thickness
Np Peff Ptm Pγ Pγ0 Pγe Qd ta , tb tf tfTH vc v∞ c Vd
number of active pores effective transmembrane pressure transmembrane pressure capillarity pressure theoretical critical pressure experimental critical pressure disperse phase flow rate extrapolating functions experimental formation times theoretical formation times continuous phase velocity continuous phase tangential velocity droplet volume
Greek letters γ(t) dynamic interfacial tension γF (t) fitting function of [17] γm equilibrium interfacial tension γ0 oil/water interface tension continuous phase viscosity µc µd disperse phase viscosity ξ pore tortuosity ρc continuous phase density τw wall shear stress
References [1] P. Walstra, Formation of emulsions, in: P. Belcher (Ed.), Encyclopedia of Emulsion Technology, vol. I, Dekker, New York, 1983. [2] R.A. Williams, S.J. Peng, D.A. Wheeler, N.C. Morley, D. Taylor, M. Whalley, D.W. Houldsworth, Controlled production of emulsions using a crossflow membrane. Part II. Industrial scale manufacture, Trans. IchemE 76 (1998) 894–901. [3] V. Schröder, H. Schubert, Einsatz querüberströmter, mikroporˇoser Membranen zum Emulgieren, DECHEMA-Jahrestagung, Band II, Wiesbaden, Germany, 1996, p. 247. [4] R. Katoh, Y. Asano, A. Furuya, K. Sotoyama, M. Tomita, Preparation of food emulsions using a membrane emulsification system, J. Membr. Sci. 113 (1996) 131–135. [5] Y. Mine, M. Shimizu, T. Nakashima, Preparation and stabilization of simple and multiple emulsions using a microporous glass membrane, Colloids Surf. B 6 (1996) 261–268. [6] V. Schröder, O. Behrend, H. Schubert, Effect of dynamic interfacial tension on the emulsification process using microporous ceramic membranes, J. Colloid Interface Sci. 202 (1998) 334–340. [7] L. Giorno, N. Li, E. Drioli, Preparation of oil in water emulsions using polyamide 10 kDa hollow fiber membranes, J. Membr. Sci., in press. [8] S.M. Joscelyne, G. Trägårdh, Membrane emulsification—a literature review, J. Membr. Sci. 169 (2000) 107–117. [9] A.J. Abrahamse, A. van der Padt, R.M. Boom, W.B.C. de Heij, Process fundamentals of membrane emulsification: simulation with CFD, AIChE J. 47 (2001) 1285–1291. [10] S.J. Peng, R.A. Williams, Controlled production of emulsions using a crossflow membrane. Part I. Droplet formation from a single pore, Trans. IchemE 76 (1998) 894–901. [11] M. Mulder, Basic Principles of Membrane Technology, Kluwer Academic Publishers, Dordrecht, 1991.
G. De Luca et al. / Journal of Membrane Science 229 (2004) 199–209 [12] Z. Wang, S. Wang, V. Schröder, H. Schubert, Effect of continuous phase viscosity on membrane emulsification, Chin. J. Chem. Eng. 8 (2000) 108–112. [13] M. O’Neil, A slow motion of viscous liquid caused by a slowly moving solid sphere, Chem. Eng. Sci. 23 (1964) 67–74. [14] G.T. Vladisavljevic, H. Schubert, Preparation and analysis of oil-in-water emulsions with a narrow droplet size distribution using Shirasu-porous-glass (SPG) membranes, Desalination 144 (2002) 167–172. [15] S. Tesch, Ch. Gerhards, H. Schubert, Stabilization of emulsions by OSA starches, J. Food Eng. 54 (2002) 167–174. [16] M. Stang, H. Karbstein, H. Schubert, Adsorption kinetics of emulsifiers at oil–water interfaces and their effect on mechanical emulsification, Chem. Eng. Process. 33 (1994) 307–311.
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[17] T. Gao, J.R. Milton, Dynamic surface tension of aqueous surfactant solutions, J. Colloid Interface Sci. 172 (1995) 242–248. [18] S.M. Joscelyne, G. Trägårdh, Food emulsions using membrane emulsification: conditions for producing small droplets, J. Food Eng. 39 (1999) 59–64. [19] P.C. Carmen, Flow of gases through porous media, Butterworths, London, 1956. [20] N. Yamazaki, H. Yuyama, M. Nagai, G.H. Ma, S. Omi, A brief review of SPG (Shirasu porous glass) membrane emulsification and introduction of poly(tetrafluorothylene) membrane as a new device, J. Dispersion Technol. Sci. 23 (1–3) (2002) 279–292. [21] M. Yasuno, M. Nakajima, S. Iwamoto, T. Maruyama, S. Sugiura, I. Kobayashi, A. Shono, K. Satoh, Visualization and characterization of SPG membrane emulsification, J. Membr. Sci. 210 (1) (2002) 29–37.
Quantitative analysis of coupling effects in cross-flow membrane emulsification G. De Luca a,∗ , A. Sindona b , L. Giorno a , E. Drioli a a
Research Institute on Membrane Technology (ITM-CNR), via Pietro Bucci 18D, 87030 Rende, CS, Italy b Department of Physics, University of Calabria, via Pietro Bucci 30C, 87030 Rende, CS, Italy Received 21 March 2003; received in revised form 23 July 2003; accepted 22 September 2003
Abstract Membrane emulsification has attracted increasing experimental and theoretical interests over the last decade. On the experimental side, the parameters of this process have been thoroughly investigated and a linear relationship between the droplet size of an emulsion and the pore size of the membrane has been generally observed. Theoretical studies, however, have not provided an adequate description of many effects concerning droplet growth and detachment from membrane pores, such as the influence of the dynamic interfacial tension on the evolution of the droplet size. Most calculations have been based either on an algebraic torque balance equation or on a differential equation for the disperse phase flow rate through the membrane. The present paper offers a procedure for quantitative analysis which includes both the permeation of the disperse phase through the membrane pores and the mechanism of droplet detachment. The behavior of the droplet size of an emulsion versus the pore size of the membrane is determined with different sets of process parameters and it is compared satisfactorily with the experiments of Vladisavljevic and Schubert on Shirasu porous glass membranes. For the same membranes, the procedure is tested to pore sizes below 0.1 m and a linear correlation is observed, similarly to micropore sizes. © 2003 Elsevier B.V. All rights reserved. Keywords: Membrane emulsification; SPG membrane; Droplet formation; Process parameters; Transmembrane pressure
1. Introduction Emulsions, generally used in food, pharmaceutical and cosmetic products, are disperse systems of two (or more) immiscible liquids, such as water and oil, in which one phase (disperse phase) is distributed in form of droplets in the other phase (continuous phase). Most existing methods to prepare emulsions are based on the establishment of turbulent flows (turbulent eddies) in the fluid mixture consisting of the immiscible liquids. Both the size and the size distribution of droplets are important, since they determine the stability of emulsion against coalescence and its applications for intended uses. According to known techniques, the size of droplets, for a given pair of processed phases, is mainly determined by the size of the turbulent eddies and the times of exposure to these eddies [1]. In these methods, turbulence cannot be controlled or generated consistently throughout the volume of fluid; neither can the behavior ∗ Corresponding author. Tel.: +39-0984-492-014; fax: +39-0984-402-103. E-mail address: [email protected] (G. De Luca).
0376-7388/$ – see front matter © 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.memsci.2003.09.024
of any pair of immiscible phases be predicted on a large scale, based on tests performed in the laboratory [2]. The consequence is that energy is used inefficiently and, more importantly, it is not possible to directly control the size of droplets. For these reasons, much attention has been put in alternative emulsification processes. Membrane emulsification is a relatively new membrane technology, which allows the production of emulsion droplets under controlled conditions with a narrow size distribution. Both the size and the size distribution of droplets may be carefully controlled choosing suitable membranes and focussing on some fundamental process parameters. In cross-flow membrane emulsification, the disperse phase is pressed through a microporous membrane. Droplets formed at the membrane surface grow until their size reaches a critical value. Then, they are carried away with the continuous phase flowing through the membrane. Generally, systems with droplets of size ranging from 0.1 to 50 m are thermodynamically unstable; thus, a stable emulsion requires some additional substances, e.g., emulsifiers or stabilizers, to protect droplets against coalescence. Emulsifiers and stabilizers induce electrostatic or steric
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repulsions among droplets and generate a lower interfacial tension in the system. The interfacial tension yields a force at the border of the membrane pore, and, moreover, it influences the capillary pressure inside the droplet. In a membrane emulsification process, the stress to generate the droplets is much smaller with respect to traditional processes, because small droplets are directly formed at the micropores of a membrane, rather than by disruption in zones of high energy density. Accordingly, the energy needed is considerably less than that required by the conventional methods. Hence, the main advantage of membrane emulsification is the possibility to produce droplets of defined sizes, with a narrow size distribution and potentially lower energy demands [3]. Much experimental effort has been invested in determining the correlation between the droplet size of an emulsion and the pore size of the membrane [4,5]. A linear scaling law has been generally observed, with a slope ranging typically from 2 to 10, although a value near to 50, for particular operating conditions and emulsifier concentrations, has been found [6,7]. The reasons for this variation have not yet been well understood. The production of mono-disperse emulsions is essentially related to the size distribution of pores and their relative spatial distribution at the membrane surface. The formation and size of droplets depend on: (i) operating parameters, i.e., cross-flow velocity, transmembrane pressure and disperse phase flow; (ii) membrane parameters, i.e., pore size, active pores, membrane hydrophobicity/hydrophilicity; and (iii) phase parameters, i.e., interfacial tension, viscosity and density of the processed phases. Such quantities combine with different magnitudes, over the ranges of operating conditions, and many of them exhibit coupling effects. The aim of the article is to account for these correlations, using a torque balance model to describe the detachment of the droplet. In other words, combined action of the fundamental parameters controlling the evolution of droplets is studied. Estimation of coupling effects can provide a key for theoretical interpretation of some experimentally derived behaviors, such as the above-mentioned law between the droplet size and the pore size. The fundamental parameters are chosen on the basis of data that give evidence of their importance in controlling droplet formation. Other phenomena concerning coalescence among droplets, deformations in droplet shape and formation of droplet neck are not considered here. As reported by Joscelyne and Trägårdh [8], there is coalescence of growing droplets, at the membrane surface, because active pores are too close to one another. Due to the capillarity pressure, active pores may inhibit the flow of neighboring pores and greatly decrease the effective membrane porosity. Moreover, the distribution of the active pores changes as a function of the applied pressure. One approach that tries to overcome this complexity makes use of computational fluid dynamics, by which Abrahamse et al. [9] have recently determined the
shape and neck description of one oil droplet forming at a single cylindrical pore. Their work describes the influence of the droplet shape on the maximum porosity of the membrane and on the oil flux through the pore. In the present paper, theoretical calculations are applied to experiments in which the condition for unhindered droplet growth is satisfied for all membrane pores and, then, the above-mentioned issues are negligible. 2. Theoretical background Theoretical description of a membrane emulsification requires two main clues to be dealt with: one is permeation of the disperse phase through the membrane pores; the other is the mechanism of droplet detachment. Previous approaches [8,10] have treated these two aspects separately. However, both of them simultaneously contribute to droplet evolution and should be considered in the same framework. A main goal of this contribution is to theoretically support the observed linear behavior dd = xdp , (2.1) between the droplet size, dd , and the pore size, dp . The major factors affecting explicitly the slope of Eq. (2.1) are: (i) the wall shear stress, τw ; (ii) the dynamic interfacial tension, γ; and (iii) the disperse phase flux, Jd . Thus, from the theoretical point of view, x is a functional of the above introduced quantities x = x{τw (vc , µc , Di , ρc ), γ(t, µc , T), Jd [γ(t, µc , T), Ptm , dp , µc , L]}. (2.2)
Other parameters enter implicitly this relationship, i.e., the average velocity of the continuous phase flow, vc , which depends on the difference of continuous phase pressures at the ends of the membrane tube, the viscosities of the disperse and the continuous phases, µd and µc , the inner size of the membrane tube, Di , the density of the continuous phase, ρc , the thermodynamic temperature, T , the transmembrane pressure, Ptm , the pore size and the thickness of the membrane, L. Coupling effects arise because γ appears in x either explicitly or implicitly. The temperature and viscosity of the continuous phase influence the dynamic interfacial tension through the emulsifiers diffusion coefficients. The latter obey the Stokes–Einstein relation, De ∝ T/µc , implying that, as µc increases, the rate of variation of γ decreases, due to the decreasing of De . Below both µc and T are fixed to specific values so that γ can be parameterized as an explicit function of time. 2.1. Flux of the disperse phase The flux of the disperse phase through the pores of the membrane, Jd , may be assumed to follow Darcy’s law [2] Peff , (2.3) Jd = B µd L
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where B is determined by the membrane structure (pore size, shape and porosity) and Peff denotes the effective transmembrane pressure. A highly simplified picture is adopted in which the membrane has parallel cylindrical pores, perpendicular or oblique to the membrane surface, with lengths almost equal to the membrane thickness. When the pores have the same mean size, B takes the form B=
Np πdp4 128Am ξ
,
(2.4)
and Eq. (2.3) reduces to the Hagen–Poiseuille equation [11]. In Eq. (2.4), ξ is the pore tortuosity, Np the number of active pores and Am the membrane area. In the case of one active pore (Np = 1), the disperse phase flow rate, Qd = Jd Am , writes Qd =
πdp4 Peff . 128ξ µd L
(2.5)
The transmembrane pressure is defined as the difference of applied pressures between each side of the membrane and it consists of two parts: one is the capillary pressure, Pγ , due to the curvature of the interface and to the dynamic interfacial tension; the other is the effective transmembrane pressure Peff = Ptm − Pγ [γ(t), dd ],
(2.6)
or the drag of the disperse phase inside the pore which determines Qd . The droplet at a pore tends to form a spherical shape, whose radius of curvature, dd /2, is related to the height of the spherical droplet, h, and to the membrane pore radius, dp /2, by (dp /2)2 + h2 dd = . 2 2h
(2.7)
When the height of the growing droplet equals the pore radius, the droplet size reaches its minimum, dd = dp . Accordingly, the capillarity pressure, calculated from Laplace equation 4γ(t) Pγ [γ(t), dd ] = , dd
(2.8)
takes a critical value, offering the maximum resistance to the disperse phase flow. Finally, when h → 0 or h → ∞, the capillarity pressure goes to zero and the effective transmembrane pressure equals the transmembrane pressure. 2.2. Mechanism of droplet detachment Droplet formation at an individual pore involves two stages, i.e., droplet growth and droplet detachment [3]; the growth process ends when a balance among all mechanical torques, acting on the droplet, is reached. The droplet size, at this time, can be estimated from the algebraic equation that describes this balance. At the end of the first stage, the droplet is still connected to the pore through a neck; when
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the connection is broken, the detachment is completed [10]. The droplet neck may form either before or after a torque balance is established. In both cases, the detachment begins always after equilibrium, inducing modifications on the droplet. The final droplet size is determined by the droplet volume at the end of the detachment stage. Formation and shutting of the neck can be described by the parameters reported in Eq. (2.2). According to Peng and Williams [2,10], when the cross-flow shear force is small with respect to the interfacial tension force, the droplet forms a sphere at the pore mouth; on the other hand, as the cross-flow velocity increases, the droplet deforms from spherical symmetry. The main forces acting on a droplet to determine its growth and detachment, have been identified in literature [6,12] FD = kx 3πµc dd vc∞ ≈ 23 kx πτw dd2
(2.9)
is the drag force produced by the continuous phase flow, with kx the wall correction factor, e.g., kx = 1.7 for a single sphere touching an impermeable wall in a simple shear flow [10,13], and vc∞ the undisturbed tangential velocity of the continuous phase at the droplet center. In Eq. (2.9), the shear stress of the continuous phase is approximated to the shear at the membrane surface, τw , since small droplet sizes are considered. The relationships of the continuous phase velocity and τw are well known either for laminar or turbulent flows [12]. Fγ = πdp γ(t)
(2.10)
is the interfacial tension force; Fsp =
γ(t) 2 πdp dd
(2.11)
is the force due to the difference of pressure needed to overcome capillarity; Fdl =
0.761τw1.5 ρc0.5 3 dd µc
(2.12)
corresponds to the dynamic lift force [6]. The relative intensities of these global forces change as the droplet increases in size. Other forces, such as inertia and buoyancy forces, are approximately from six to nine orders of magnitude smaller than the FD , Fγ and Fsp , for droplet sizes of the order of ∼1 m [6]; therefore, such forces can be neglected in a torque balance model that applies to micropores. The droplet size can be, thus, estimated from the following torque balance equation dp dd FD = , (2.13) Fi 2 2 i
assuming that shape of the droplet is approximately spherical. Eq. (2.13) provides a relation between the membrane pore size and the droplet size at equilibrium. Several torque balance models may be defined on the basis of the forces Fi appearing in the torque balance equation. In either model,
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the torque balance equation establishes a non-linear relationship of dd versus dp .
3. Coupling effects and related models The following quantitative analysis considers Shirasu porous glass (SPG) membranes of pore sizes ranging from 0.4 to 6 m with the experimental conditions (Tween 80 emulsifier, transmembrane pressure, wall shear stress) recently reported by Vladisavljevic and Schubert [14]. This system is particularly suitable for the theoretical background discussed in Section 2, since SPG membranes have cylindrical, interconnected and uniform micropores and the disperse phase flow rate through the membranes follows Eq. (2.5). In other porous membranes, e.g., in ceramic membranes, Qd is better described by the Kozeny–Carman relation [19]. Attention must be paid to the wall shear stress in [14], τw is related to a tube with a turbulent Reynolds number. In any case, at very close distances from the membrane surface (sub-layer), the flow is laminar, thus, Stoke and Newton laws, leading to Eq. (2.9), are still valid. 3.1. Static torque balance models In setting up a static model based on the torque balance equation (Eq. (2.13)), one has to fix the dynamic interfacial tension at specific times. Fig. 1 shows the experimental values of the dynamic interfacial tension for the Tween 80 emulsifier, measured in [15], by the bursting membrane method [16]. An analytical expression for γ(t) is obtained from γF (t) =
γ0 + γm δ(t, t∗ , n) , 1 + δ(t, t∗ , n)
Fig. 1. Experimental values () of the dynamic interfacial tension for the Tween 80 emulsifier [15]. Data are linearly interpolated and their non-linear fitting function (3.1) is also reported.
(3.1)
with δ(t, t∗ , n) = (t/t∗ )n [17]. In γF , the values γ0 and γm are fixed by experiments, while t∗ and n are adjusted by a non-linear fitting routine to the values t∗ = 0.65 ± 0.01 s and n = 1.83 ± 0.07. Hereinafter, the dynamic interfacial function, is considered either at equilibrium, when t → ∞ and γF (t) approaches γm (Figs. 1 and 2 and Table 1), or at the experimental times of formation of the droplet, tf (Figs. 1 and 3 and Table 2). In the latter case, the values γ(tf ) are calculated from the experimental curve of Fig. 1 with a linear interpolation scheme. Table 1 Real solutions, ddi and ddii , for SM3 with the process parameters of Fig. 2 exp
dp (m)
ddi (m)
ddii (m)
dd
0.4 1.4 2.5 5.0 6.6
0.40 1.40 2.52 5.08 6.75
3.56 7.98 11.5 17.6 20.8
1.4 4.6 8.5 14.7 23.9
exp
dd
labels the experimental result of [14].
(m)
Fig. 2. Droplet size vs. pore size for SPG membrane in the static models SM1–SM3. The process parameters γm = 6.5 × 10−3 N/m, τw = 8 Pa, µc = 10−3 Pa s, ρc = 103 kg/m3 , kx = 1.7 are used.
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Fig. 3. Droplet size vs. pore size for SPG membrane in the static models SM3. The interfacial tension is fixed either to γm (ddi and ddii , Fig. 2) or to γ(tf ) (d id and d iid , Table 2). Other process parameters are τw = 8 Pa, µc = 10−3 Pa s, ρc = 103 kg/m3 .
Eq. (2.13) becomes an algebraic equation for dd , whose real solutions can be compared with the experiments. The following models were tested: • static model 1 (SM1)considers only the action the interfacial tension force, i Fi = Fγ ; • in static model 2 (SM2), both the interfacial tension force and the static pressure difference force are included, i Fi = Fγ − Fsp ; • static model 3 (SM3) uses all the forces appearing in Eqs. (2.9)–(2.12), i Fi = Fγ − Fsp − Fdl . Fig. 2 shows the real solutions obtained from SM1–SM3, with the equilibrium interfacial tension γm . SM1 presents only one real solution for dd , whereas SM2 and SM3 offer two real solutions (Table 1). Table 2 Real solutions, d id and d iid , for SM3 with the process parameters of Fig. 3 dp (m)
tf (s)
γ(tf ) (mN/m)
d id (m)
d iid (m)
0.4 1.4 2.5 5.0 6.6
0.6 0.7 0.8 0.9 1.8
17.2 15.8 14.7 13.8 10.5
0.40 1.40 2.50 5.03 6.69
4.97 10.9 15.4 23.3 25.0
Experimental formation times tf are taken from [14] and γ(tf ) refers to a linear interpolation scheme (Fig. 1).
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It is evident that the first solutions of SM2 and SM3, labeled ddi , are well linearly correlated with a slope equal to one. On the other hand, the solution of SM1 and the second solutions of SM2 and SM3, labeled ddii , exhibit non-linear features. It may be concluded that the effect of the interfacial force alone in balancing the drag force of the continuous phase yields unsatisfactory results. In addition, slight differences are observed between SM2 and SM3, although the second solution of the latter is in better agreement with experiments. It is also worth underlining that the wall correction factor, kx , which contributes to establish the magnitude of the drag force, can be used as a fitting parameter in conjunction with effect of the droplet neck to reproduce experiments, however, in the present paper no adjustment of parameters is performed. Fig. 3 reports both real solutions of SM3, labeled d id and d iid , respectively, in which γ(t) is evaluated at the experimental formation times tf ; such times change with the pore size as displayed in Table 2. The solutions ddi and ddii , for the equilibrium interfacial tension γm , are also reported for comparison. No substantial differences are detected between ddi and d id , suggesting that the first linear solution of SM3 is independent on the time variation of the interfacial tension. This fact is not physically consistent, thus, such solution is not considered. On the contrary, the second non-linear solution of SM3 shows significant changes when the equilibrium interfacial tension, yielding ddii , is replaced with the dynamic interfacial tension, yielding d iid . The values d iid are larger than both ddii and the experimental droplet sizes. Looking at Fig. 1 and Table 2, it is clear that γm is the minimum interfacial tension for any given pore size, for which reason the values ddii are lower limits to the results that may be calculated from the second solution of SM3. 3.2. Disperse phase flow rate approach A dynamic model, which explicitly accounts for the time variation of the interfacial tension, is based on the disperse phase flow rate equation. The dynamic droplet volume Vd can be related to the droplet height from the surface membrane by the continuity equation d Vd [h(t)] = Qd [h(t), γ(t)], dt
(3.2)
in which Vd is the volume of a spherical cap of height h on a pore of size dp Vd [h(t)] = 16 πh(t)( 43 dp2 + h(t)2 ),
(3.3)
and Qd is disperse phase flow rate of Eq. (2.5). Eq. (3.2) is, then, arranged to the following differential equation dp4 dh 16γ(t)h Ptm . (3.4) = − 2 dt 16µd ξL dp2 + 4h2 (dp + 4h2 )2
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The latter can be solved numerically with a suitable set of initial conditions, discussed further, and, by Eq. (3.3), the evolution of Vd can be predicted. The dynamic droplet size can be obtained from Eq. (2.7), expressing the geometrical relation, in a spherical cap of height h(t), between the radius, dd (t)/2, and the base radius, dp /2. Alternatively, an average droplet size, dd , can be defined by approximating the droplet volume Vd [h(tf )], at the droplet formation time, to the volume of a sphere of diameter dd [14] dd = 3 h(tf )[ 43 dp2 + h(tf )2 ]. (3.5) It is observed that dd (tf ), calculated from Eqs. (2.7) and (3.4), differs negligibly from dd , at least in the experiment under consideration. Eqs. (2.7), (3.4) and (2.13) are a system of equations accounting for several correlations in process parameters; one of these is the effect of the dynamic interfacial tension on the dispersed flow rate at the droplet size. In addition, dd (t), calculated from Eqs. (2.7) and (3.4), can be introduced in the algebraic torque balance equation (Eq. (2.13)) with the dynamic interfacial tension of Fig. 1, to obtain a theoretical estimation of the formation times and of the final droplet size. This new procedure combines the torque balance approach with the disperse phase flow rate method, allowing a multivariate analysis of process parameters. The choice of an initial condition for h(t) is related to the considerations leading to Eqs. (2.7) and (2.8). Joscelyne and Trägårdh [18], have observed that the critical pressure changes during the process; in particular the difference of pressure from the beginning and the end of membrane emulsification is 260 and 40 kPa for ceramic membranes with pore size of 0.1 and 0.5 m, respectively. Then, for a given pore size and a given initial value of the capillarity pressure there corresponds an initial height h(0). On this basis, establishing a set of initial conditions compatible with experiments of [14] is a very hard task. To overcome this difficulty, the following scheme is proposed: • When h(0) = dp /2, Eq. (2.7) implies dd (0) = dp and, substituting γ(0) and dd (0) into Eq. (2.8), one obtains Pγ0
4γ(0) = Pγ [γ(0), dd (0)] = . dp
(3.6)
The transmembrane pressure, Ptm , must be slightly larger than Pγ0 , i.e., Ptm = Pγ0 + δP, with δP Pγ0 . • For values of h(0) larger than dp /2, Eq. (2.7) yields dd (0) = (dp2 /4h(0)) + h(0). Then, by Eq. (2.8), a relationship between the experimental capillarity pressures, Pγe , and the initial heights of the droplets is found Pγe = Pγ [γ(0), dd (0)] =
16γ(0)h(0) . dp2 + 4h(0)2
(3.7)
Fig. 4. Log–log representation of dd (t) (calculated from Eqs. (2.7) and (3.4)), with the initial conditions (3.6) and (3.7) and the process parameters of [14]. Comparison is made with the static case γ(t) ≈ γm .
In [14], Pγe is measured for all pore sizes reported in Tables 1 and 2 and the transmembrane pressure is set to Ptm = 1.1Pγe . Fig. 4 shows dd (t), calculated with the set of initial conditions (3.6) and (3.7), for the pore sizes reported in Table 1. The trend of the two set of solutions with respect to the pore size is similar, that is, dd increases with increasing dp . However, this monotonic behavior may change using different initial conditions, e.g., for certain values of h(0), the solutions corresponding to different pore sizes may cross at specific times. In any case, the functional relationship of dd versus t changes with changing the initial conditions; Nevertheless, such discrepancies are significant at times below 0.5 s and may influence the droplet formation when tf reaches shorter values than those reported in Table 2. Fig. 5 shows theoretical and experimental correlations between the droplet size and the membrane pore size.
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Fig. 5. Theoretical behavior of dd (tf ) vs. dp from the disperse phase flow rate approach of Eqs. (2.7) and (3.4). Formation times are those of Table 2. The experimental result of [14] is also shown.
Theoretical values are obtained numerically from Eqs. (2.7) and (3.4) at the experimental formation times of Table 2 and using both conditions (3.6) and (3.7). Concerning the initial condition (3.7), a test is proposed in which the capillarity pressure is increased of ∼15% with respect to its measured value. This correction is because Eq. (3.4) requires, as input parameters, the initial capillarity pressures, Pγ0 , that are generally larger than the Pγe [8]. The same equations are also solved for the static interfacial tension γm , which corresponds to the approach of Peng and Williams [10]. It is observed that both dynamic solutions offer a more realistic trend with respect to the static torque balance models of Figs. 2 and 3. In addition, they are in better agreement with experiments, in comparison with the disperse phase flow method already solved in literature [10]. In order to investigate correlations between the transmembrane pressure and the droplet size, dd (tf ) is calculated from Eqs. (2.7) and (3.4), with the initial condition (3.7), at different values of Ptm . Fig. 6 reports the theoretical distribution dd (tf ) versus Ptm , for the different pore sizes of Tables 1 and 2. Values Ptm = αPγe , with 1.1 ≤ α ≤ 2.0 are considered. It is observed that dd increases
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with increasing Ptm ; its rate increases as the pore size increases. Attention must be paid to sizes dp ≤ 2.5 m, for which small changes are detected in the droplet size over the whole range of the transmembrane pressure. Schröder et al. [6] have pointed out that this different trend depends on the relation between droplet formation times and the dynamic interfacial tension. They have suggested that, if droplet formation occurs before that the interfacial tension reaches its equilibrium value, the droplet size is correlated to the transmembrane pressure, while, in the other case, dd is almost independent on Ptm . Considering the formation times for two ceramic membranes, of pore sizes dp = 0.2 m and dp = 0.8 m, respectively, and the equilibrium time for the LEO-10 emulsifier, they have observed that Ptm influences the droplet size at the 0.8 m membrane, since the formation time of the droplet is shorter than the equilibrium time of the emulsifier; the same correlations are absent in the 0.2 m membrane, having a droplet formation time larger than the equilibrium time of the emulsifier. As shown in Fig. 6, the present work offers a ready to use quantitative method to estimate the effect of Ptm upon the droplet size, which accounts for the coupling of the transmembrane pressure, the disperse flow and the dynamic interfacial tension of the emulsifier. The type of membrane and pore sizes considered differ from the ones studied in [6] and the formation times of all droplets at the SPG membrane are shorter than the equilibrium time of the Tween 80 emulsifier. In this respect, this results allow a quantitative analysis of the transmembrane pressure effect in membrane emulsification. 3.3. Dynamic model For a full theoretical treatment of the experiment considered in the former subsection, a method to calculate droplet formation times at the SPG membrane pores is needed. One possibility is to use the dynamic solution, dd (t), of Eqs. (2.7) and (3.4), with the initial conditions (3.6) and (3.7), in the algebraic equation provided by the torque balance model SM3. The resulting equation 3 kx πτw dd3 (tfTH ) 2 =
πdp2 γ(tfTH ) −
γ(tfTH )
πdp3 dd (tfTH )
3/2 1/2
τw ρc − 0.761 dp dd3 (tfTH ) µc (3.8)
can be solved numerically to calculate the theoretical formation times tfTH . Fig. 7 shows dd versus dp for the theoretical formation times of Eq. (3.8) with both conditions (3.6) and (3.7). Such times are reported in Table 2 in comparison with the experimental times tf . It is noted the torque balance model in conjugation with the disperse phase flow based method provides a reasonable estimation of the dd versus dp distribution
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Fig. 7. The dd vs. dp distributions with both initial conditions (3.6) and (3.7). Either experimental, tf , or theoretical, tfTH , formation times are considered. Fig. 6. Theoretical behavior of dd (tf ) vs. Ptm , from the disperse phase flow rate approach of Eqs. (2.7) and (3.4) with the initial conditions (3.7).
at large pores, above ∼5.5 m. On the other hand, for pore sizes below ∼5 m, Eq. (3.8) overestimates the formation times of droplets and, accordingly, the droplet size. It is believed that such discrepancies are due to the highly simplified scheme of forces used in the torque balance model and it is hoped that a more realistic description of the mechanism of droplet detachment will provide more satisfactory results. A lot of empirical studies on SPG oil–surfactant–water systems can be found in literature. Thus, it would be interesting to improve the weight of evidence of the results of present section extending the same type of analysis to data from more than one source. However, particular attention must be paid to the availability of experiments reporting parameters compatible with the hypotheses of the model. For example, Yamazaki et al. [20] have studied Polytetrafluoroethylene (PTFE) membranes, detecting a linear behavior of dd with pore sizes estimated from the Laplace equation, rather than with nominal pore sizes. They have used a mixture of emulsifiers, corresponding to an equilibrium interfacial tension of 8.7 × 10−3 N/m (8.7 dyne/cm),
and have not reported the viscosity of the dispersed phase, due to a mixture of styrene, divinylbenzene and hexadecane at different concentrations. Katoh et al. [4] have also used a mixture of emulsifiers, for which the dynamic interfacial tension can be hardly obtained. Yasuno et al. [21] have recently published a characterization of SPG emulsification with a single emulsifier, SDS at 0.3 wt.%, and pore sizes equal to 16 m. The oil permeate flux was precisely controlled by the accurate syringe pump, although the transmembrane pressures, of crucial importance for the present analysis, have not been reported. The same authors have identified two droplet formation mechanism, explained by a spontaneous and a shear force formation model, which depend on the disperse and continuous phase flux velocities. In particular, they have observed that, when the dispersed phase flux is greater than ∼7.2 l m−2 h−1 and the continuous phase velocity above ∼0.5 m/s, droplets form because of the competing action of both mechanisms. The dispersed phase fluxes, presented in [14], are below 7.2 l m−2 and the reported continuous phase velocity is 1.4 m/s; thus, the emulsification is dominated by the shear force mechanism. Moreover the condition for unhindered droplet growth is satisfied for all membrane pores.
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Table 3 Experimental and theoretical formation times dp (m)
tf (s)
tfTH (s) (Eq. (3.6))
tfTH (s) (Eq. (3.7))
0.4 1.4 2.5 5.0 6.6
0.6 0.7 0.8 0.9 1.8
10.0 5.15 2.68 2.01 1.42
10.0 6.08 3.43 3.10 2.47
3.4. Theoretical prediction for SPG membrane with pore size below 0.1 µm Following the encouraging results presented in Sections 3.1–3.3, the next step is to extend the formulation for membranes with pore sizes below 0.1 m. However, at these sizes, attention must be paid to the morphology and tortuosity of pores and the thickness of the membrane. Eqs. (2.7) and (3.4) can be used only with the initial conditions (3.6), because experimental capillarity pressures are not reported for the considered type of membranes. The approach is tested to pore sizes of 0.01, 0.05 and 0.1 m. Correspondingly, the membrane thickness is reduced to L = 0.1 mm. Since the torque balance model is a highly simplified scheme to estimate droplet formation times (Table 3), the latter are extrapolated from experiments. Fig. 8 reports experimental formation times versus pore sizes taken from Table 3. Two possible extrapolation are shown: ta (dp ) = 0.12 ln(1+663dp ) is a logarithmic fitting curve and
Fig. 9. Distribution of dd vs. dp in log–log scale and its extrapolation to pore sizes below 0.1 m.
tb (dp ) a linear interpolating polynomial. Many other analytical functions passing through the shadowed region, ranging from dp = 0 and dp = 0.4 m and included between ta and tb , have been tested. The pore tortuosity, ξ, does not follow a monotonic trend versus dp , hence it is allowed to range from its minimum, ξ = 1, to its maximum, ξ = 2.1, observed values (Fig. 4). The results are summarized in Fig. 9, in which the error bar on the predicted values of dd accounts for: (i) the different kind of extrapolation used to estimate the formation times; (ii) the use of several values for the pore tortuosity in the range specified above; and (iii) the use of different membrane thickness, compatible with the considered pore sizes. It is noted that the experimentally observed linear behavior (Eq. (2.1)) extends also at the considered sizes.
4. Conclusions
Fig. 8. Experimental formation times vs. pore sizes. Two kinds of extrapolating functions, labeled ta and tb , are shown.
This work has presented a study of the influences of some process parameters, such as transmembrane and capillarity pressures, dynamical interfacial tension, membrane thickness and pore tortuosity, on the correlation between droplet size and pore size of an SPG membrane. These effects have been analyzed either in the framework of the current torque balance models (Eqs. (2.10)–(2.13)) or using a
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disperse phase flow based procedure (Eqs. (3.2)–(3.7)). The main aim was to propose a procedure, on the basis of some published material [2,3,6,8,9,14,17], that can connect droplet formation and detachment by means of simple, ready to use equations (Eqs. (3.4) and (3.8)). These equations take into account the dynamic interfacial tension and make use of experimental variables utilized in emulsification modules. At the moment, more realistic descriptions of the mechanism of droplet detachment are needed. Nevertheless, from the study presented here, some interesting conclusions on membrane emulsification can be drawn. In particular: (i) torque balance models SM1–SM3 (Eqs. (2.10)–(2.13)) are less reliable with respect to a disperse phase flow based description (Eqs. (3.2)–(3.7)); (ii) the correlation between transmembrane pressure and droplet size evidences the importance of using the dynamic interfacial tension for droplet with formation times below the emulsifier equilibrium time; the inclusion of effect of the dynamic interfacial tension can provide useful information to experimentalists concerning the type of emulsifier to be used in a membrane emulsification process in order to have droplets of well defined sizes; (iii) a combined use of two methods, disperse flow description and torque balance model (Eqs. (3.4) and (3.8)), is possible, although for more realistic estimation of droplet formation times the torque balance model should be replaced by a more accurate approach; (iv) extension of the same type of analysis on models of porous membranes with sizes below 0.1 m, reveals that strong linear correlations between droplet size and pore size are still present, if the membrane thickness and pore tortuosity are of the same order with respect to larger pores. Finally, the disperse phase flow approach should be tested with other flow equations in porous membranes; the influence of the wall shear stress on the droplet size and an energetic analysis of the droplet formation is also desirable.
Nomenclature Am dd dd dp Fdl FD Fsp Fγ h Jd kx L
membrane area droplet size average droplet size pore size dynamic lift force viscous phase force static pressure difference force interfacial tension force height of the spherical droplet disperse phase flux wall correction factor membrane thickness
Np Peff Ptm Pγ Pγ0 Pγe Qd ta , tb tf tfTH vc v∞ c Vd
number of active pores effective transmembrane pressure transmembrane pressure capillarity pressure theoretical critical pressure experimental critical pressure disperse phase flow rate extrapolating functions experimental formation times theoretical formation times continuous phase velocity continuous phase tangential velocity droplet volume
Greek letters γ(t) dynamic interfacial tension γF (t) fitting function of [17] γm equilibrium interfacial tension γ0 oil/water interface tension continuous phase viscosity µc µd disperse phase viscosity ξ pore tortuosity ρc continuous phase density τw wall shear stress
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