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Interfacial mass transfer in ternary liquid-liquid systems
Accepted Manuscript Interfacial mass transfer in ternary liquid-liquid systems Kai Fabian Kruber, Marius Krapoth, Tim Zeiner PII:
S0378-3812(17)30070-5
DOI:
10.1016/j.fluid.2017.02.013
Reference:
FLUID 11409
To appear in:
Fluid Phase Equilibria
Received Date: 30 December 2016 Revised Date:
17 February 2017
Accepted Date: 21 February 2017
Please cite this article as: K.F. Kruber, M. Krapoth, T. Zeiner, Interfacial mass transfer in ternary liquidliquid systems, Fluid Phase Equilibria (2017), doi: 10.1016/j.fluid.2017.02.013. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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Interfacial Mass Transfer in Ternary Liquid-Liquid Systems Kai Fabian Kruber1, Marius Krapoth1, Tim Zeiner2* Technical University Dortmund, Faculty of Bio-and Chemical Engineering, Emil-Figge Str. 70, 44137 Dortmund, Germany 2
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1
Technical University Graz, Institute of Chemical Engineering and Environmental Technology,
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Inffeldgasse 25C, 8010 Graz, Austria
KEYWORDS: Mass Transfer, Density Gradient Theory, Nitsch-Cell, Extraction, Interface
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*corresponding author: [email protected]
ABSTRACT
In this work, the interfacial mass transfer in two extraction systems, namely acetone-toluene-
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water (System I) and hexane-heptane-methanol (System II), was examined experimentally and theoretically. The interfacial mass transfer was experimentally examined by using a Nitsch-Cell. As theoretical approach the density gradient theory (DGT) in combination with the Koningveld-
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Kleintjens (KK) model was used. At first, the KK-model was used to model the liquid-liquid
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equilibrium of System I and System II. In combination of the KK-model with the DGT, the interfacial tension was modelled by fitting the influence parameter of the DGT. To estimate the required mutual mobility coefficients in each system, bulk diffusion coefficient coefficients were used. It was shown, that the DGT in combination with a thermodynamic model and experimental information of the bulk diffusion coefficients and the system’s interfacial tension is able to model the interfacial mass transfer. Moreover, it can be stated that the DGT predicts a high enrichment of acetone in System I and this enrichment has an influence on the mass transfer.
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1. Introduction For each separation operation in chemical engineering, the interfacial mass transfer is essential. Nevertheless, if it is a vapor-liquid interface in rectification/absorption or if it is a liquid-liquid
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interface in extraction, there is always a mass transfer across an interface. The characteristics of these interfaces have been debated by a number of scientists. On the one hand, some of them [1] postulated that these interfaces have a zero thickness; hereby, all physical quantities such as
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refraction index are assumed to be discontinuous across the interface. But other scientists such as Poisson [2], Maxwell [3], and Gibbs [4] assumed that the existence of an interfacial region
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describing a smooth transition of all properties between the two bulk fluid values. Based on this assumption Lord Rayleigh [5] and van der Waals [6] proposed gradient theories to describe this interfacial region on this basis of thermodynamics. By using his equation of state (EOS), van der Waals developed a theory of the interface. On the basis of this theory, he could calculate the
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thickness of vapor-liquid interfaces.
In the 1950’s, Cahn and Hillard [7] recovered these theories. On that basis, they developed the density gradient theory (DGT). The DGT is a mean field approach and it results in an expression
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of the Helmholtz free energy for an inhomogeneous system. This theory can be applied in equilibrated systems as well as in not equilibrated systems. So, the interfacial tension can be
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modelled by applying the stationary DGT in combination with an EOS or a Gibbs excess energy (gE)-model [8]. In literature there are several works using the DGT to model the interfacial properties of two phase systems [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19]. To calculate interfacial mass transfer or phase decomposition, the non-steady form of DGT can be applied [20, 21, 22]. To apply the DGT, a thermodynamic model has to be used. Here the DGT is combined with a
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gE-model as liquid-liquid systems are regarded, which are considered to be incompressible in low pressure ranges. Up to now, the mass transfer calculated by the DGT was not compared to experimental data. In
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literature there are two simple experimental apparatuses [23, 24] proposed. Constant interfacial area stirred cells [23] (Lewis-Cell) were applied to estimate the mass transfer properties in liquid-liquid systems with constant interfacial area. Unfortunately, this Lewis-cells do not
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provide efficient stirring conditions for regions close to the liquid-liquid interface. The reason is that the stirrers cause not enough interfacial turbulence to break diffusion films. To produce
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higher interfacial turbulence, Nitsch and Hillekamp [24] have developed an apparatus with flow breaker and stirrers with special blades (Nitsch-Cell). This new apparatus allows an efficient forced convection. Furthermore, it allows stirring the aqueous as well as the organic phases at high velocities without drop release from one phase to the other. For this reason, a Nitsch-Cell
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was used in this work to estimate the interfacial mass transfer in classical extraction systems. In this work, the interfacial mass transfer in two different liquid-liquid systems was analyzed. One system is the classical extraction system acetone-toluene-water (System I) and the other
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system is heptane-hexane-methanol (System II). Grunert et al. [25] have calculated the interfacial properties of System I by the DGT in combination with a Koningsveld-Kleintjens approach.
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They [25] found a high enrichment of acetone in the interface of System I. This raises the question, if this enrichment has an influence on the interfacial mass transfer of acetone. This question was answered by this work. Moreover, it was shown that the DGT can be used to model the interfacial mass transfer for the first time.
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2. Theoretical Framework To describe the interfacial mass transfer, the density gradient theory was applied. To reduce the number of adjustable parameters, the influence parameter was adjusted to interfacial tension
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data. Moreover, the mobility coefficients were adjusted to binary diffusion coefficients experimental data, so there is just one remaining fitting parameter ∆ω left which has to be
Koningsveld-Kleintjens (KK) approach was used. 2.1 Koningsveld-Kleintjens
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adjusted to the interfacial mass transfer experimental data. As thermodynamic model the
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To develop a thermodynamic model for non-ideal polymer solutions, Flory and Huggins independently suggested a simple lattice theory [26, 27]. The basic assumption of these theories is a lattice which is occupied by large molecules with different sizes. To account for the different sizes of molecules, the segment molar fraction of component i, φi , is introduced, which is defined
φi =
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as follows:
ri ni n
∑r n j
j
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j =1
Eq. 1
where ni is the number of molecules i , and ri is the number of segments of component i .
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Similar to the Flory-Huggins theory, the ideal-athermic mixture is chosen as a standard state in the Koningsveld-Kleintjens (KK) model. Thus, the KK model for an n-component mixture can be expressed as follows [28]: n n n ∆g KK φ = ∑ i ln(φi ) + ∑∑ χ ij φi φ j RT i =1 ri i =1 j >i
Eq. 2
Koningsveld and Kleintjens [28] developed a model in which the Flory-Huggins χ - parameter is expressed as follows:
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χ ij = α ij +
β ij (T ) 1 − γ ij φ j
Eq. 3
here the function β ij (T ) can be expressed as:
β ij ,1
Eq. 4
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β ij (T ) = β ij ,0 +
T
The KK-model has a physical meaning because the function β ij(T) describes the contact energy
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of the segments; the parameter γij expresses the ratio of the segment surfaces σi of the components:
σi σj
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γ ij = 1 −
Eq. 5
In this work, the chain length ri of each component is calculated by the molar mass that means that the smallest component in each system has the chain length 1 and the chain length of the
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other components is calculated by the division of their molar mass Mi by the molar mass of the smallest component. For this reason, the segment fraction is equal the mass weight fraction wi. 2.2 Density Gradient Theory
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In the theory of Cahn and Hilliard [7], the interfacial tension of inhomogeneous systems is proportional to the grand thermodynamic potential. To calculate the grand thermodynamic
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potential, the Helmholtz free energy was used [7]. If an incompressible system is assumed, the Gibbs energy gets equal to the Helmholtz free energy. In this case not the density gradient is calculated, but the concentration gradient of each component from one bulk phase to the other across the interface. This gradient can be used to calculate the interfacial tension. The grand thermodynamic potential ∆Ω of a multicomponent system corresponding to the LLE can be calculated as:
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n
∆Ω = ∆g (φi ) − ∑ µ i φi
Eq. 6
i =1
where µi
is the chemical potential of component i in equilibrium and ∆g(φi ) is the Gibbs
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energy at the corresponding composition. According to [14, 25], the integral between the bulk phase concentrations can be used to calculate the interfacial tension. The interfacial tension is given by φ ''
Eq. 7
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σ = ∫ 2κ12 ∆Ω(φ )dφ φ'
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where κ 12 is the influence parameter which must be fitted to one experimental interfacial tension data point per demixing subsystem. In addition to the interfacial tension, the concentration profile in equilibrium across the interface can also be calculated as follows [25]: φ*
Eq. 8
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2κ12 1 z − z0 = dφ ∫ ∆ω φ ' ∆Ω
mol where ∆ω 3 is a fitting parameter, because here the modified DGT is used and a gE-model m
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is not able to calculate the density at any time. This fitting parameter is of the same unit as the molar density but has no physical meaning. The reference coordinate z0 can be arbitrarily chosen
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and the segment fraction φ* varies between the bulk concentrations. Thus, any value of the segment fraction inside the interfacial region can be assigned to one spatial coordinate. The interfacial tension of a ternary system is given by:
σ=
φ1''
∫
2κ∆Ω(φ1,φ2 )dφ1
Eq. 9
φ1'
For ternary systems with only one binary demixing subsystem, Grunert et al. [25] demonstrated that the influence parameter of the ternary mixture is equal to that of the binary demixing 6
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subsystem. If there are two demixing subsystems, a combining rule for the influence parameters has to be used [29]. In addition to the interfacial tension, the concentration profile in equilibrium across the interface
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can also be calculated as follows [25]: φ*
Eq. 10
1 1 2κ z − zo = dφ1 ∫ ∆ω φ ' ∆Ω (φ1,φ2 ) 1
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The reference coordinate z0 can be arbitrarily chosen. To solve Eq. 9 and Eq. 10, an equation is required to calculate the related concentrations inside the interface of the second component.
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This segment fraction can be calculated for a LLE with one demixing subsystem as follows:
∂∆Ω (φ1,φ2 ) =0 ∂ φ 3 φ1 ,φ2
Eq. 11
In the case of two demixing subsystems, the segment fraction is computed as:
∂∆Ω (φ1,φ2 ) ∂∆Ω (φ1,φ2 ) = κ13 ∂φ3 ∂φ2 φ1 ,φ2 φ1 ,φ2
Eq. 12
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κ12
To calculate the interfacial properties, Eq. 11 respectively Eq. 12 can be solved by a root
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determination to find the composition of the second component. Thereby, the composition of the first component is given by the integration limits and the used numerical integration procedure to
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solve Eq. 9 or Eq. 10. Moreover, it is important that the component 1 is not enriched in the interface, otherwise this would lead to numerical problems.
2.2.1 Instationary Density Gradient Theory The commonly used law to model diffusion phenomena is Fick´s law:
J1,ideal = −D12∇c1
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Eq. 13
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where c1 is the concentration of component 1 and D12 is the mutual diffusion coefficient of component 1 in presence of component 2. This flux equation leads to the component continuity equation in a system at constant density and temperature: ∂c1 = ∇D12∇c1 ∂t
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Eq. 14
Fickian diffusion can just be applied on an ideal mixture. The Fickian diffusion acts to balance
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concentration differences and leads the system towards an equilibrium state where concentrations in each phase are uniform. Fick´s diffusion does not hold for a multiphase system
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because even at equilibrium, there will be concentration differences between the phases and there is no net diffusion. A more general form for Eq. 13 is based on chemical potential rather than concentration. The chemical potential is what becomes uniform everywhere in a non-ideal mixture at equilibrium [20].
Eq. 15
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J1,non −ideal = −B12φ1∇µ1
where B12 is a mobility coefficient. Eq. 15 can be taken as a phenomenological postulate or can be rationalized by non-equilibrium thermodynamics. The chemical potential in a binary system
∂g ∂φ1
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µ1 = g + (1 − φ1 )
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can be calculated as follows:
µ2 = g − φ1
∂g ∂φ1
Eq. 16
The continuity equation corresponding to Eq. 16 can be written as:
∂φ1 ∂ 2g = ∇ ⋅ B12φ1 (1 − φ1 ) 2 ∇φ1 ∂t ∂φ1
Eq. 17
where the second derivative of the Gibbs energy is multiplied by the contribution of an ideal mixture; thus, Eq. 17 is reduced to Eq. 14 in the case of ideal mixtures. However, Eq. 17 is only 8
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valid for a homogenous mixture. In the case of demixing liquids, the chemical potential must be used in the density gradient approach. The density gradient result of Eq. 17 is then [30]:
∂ 2g ∂φ1 2κ = ∇ ⋅ B12φ1 (1 − φ1 ) 2 ∇φ1 − 122 ∇3φ1 ∂t ∆ω ∂φ1
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Eq. 18
In the literature [20], Eq. 18 is frequently found without the coefficient 2 in front of the influence parameter κ12; however, in the original work of Cahn and Hillard [7] this coefficient appears.
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∂φ1 = ∇ B12φ1φ2∇ ( a1 − a2 ) + B13φ1 (1 − φ1 − φ2 ) ∇a1 ∂t
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For the application on a ternary system, Eq. 17 must be extended:
∂φ2 = ∇ B12φ1φ2∇ ( a2 − a1 ) + B23φ2 (1 − φ1 − φ2 ) ∇a2 ∂t
Eq. 19
Eq. 20
where a1 and a2 can be calculated as follows:
κ ∂g κ12 2 − ∇ φ1 − 132 ∇ 2φ2 2 ∂φ1 ∆ω ∆ω
a2 =
κ κ ∂g − 122 ∇ 2φ2 − 232 ∇ 2φ1 ∂φ2 ∆ω ∆ω
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a1 =
Eq. 21
Eq. 22
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In contrast to the He and Naumann [20], each pair of components has an individual mobility coefficient, which has to be fitted to binary diffusion coefficient experimental data. To solve the
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partial differential equations Eq. 18 and Eq. 19, an explicit finite differential method was used. Hereby, the differential quotients were approximated as shown in [31]. A fundamental assumption in this work is that the mobility coefficients in the bulk and in the interface are equal. For this reason, the mobility coefficients are calculated based on experimental fickian diffusion coefficients in the bulk. 2.3 Calculation of Bulk Diffusion
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In this section, the calculation of the Bulk Diffusion coefficients is shown. In diffusion experiments, the Fickian diffusion coefficients are estimated, but for the calculation of the mass transfer the mobility coefficients are required. For this, the mutual dependency of the mobility
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coefficients and the Fickian diffusion coefficients is needed. Fick’s law for ideal diffusion is defined as: J1,ideal = −D12 ρ∇x1
Eq. 23
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The concentration c1 is expressed by the product of the molar density ρ and the molar fraction x1 of component 1. It is assumed that the fluid is incompressible leading to a constant density. This
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is important since the specific free enthalpy is calculated with a gE-model based on the same assumption. Therefore, this assumption is taken for consistency for the whole model derivation. The mole fraction is described as a function of the segment fraction φ1 leading to an expression for the ideal flux of diffusion in Eq. 24. In contrast, the flux for the non-ideal diffusion, based on
J1,ideal = −D12 ρ
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Eq. 15 and Eq. 16, is shown in Eq. 25.
M1M2 ∇φ1 (φ1 (M2 − M1 ) + M1 )2
∂ 2 ∆g ∇φ1 ∂φ12
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J1,non −ideal = −B12φ1 (1 − φ1 )
Eq. 24 Eq. 25
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In Eq. 26, the connection of both fluxes is defined. Substitution of the fluxes with Eq. 24 and Eq. 25 leads to an expression which connects ideal (Fick) diffusion on the left side and non-ideal diffusion on the right side. Compared with Eq. 18, the second term including ∇ 3φ1 is missing. This can be explained by the small gradients in the bulk phase, because in this case the term with ∇3φ1 vanishes. As the density is not calculable by a gE-model, the coefficient B12 is a redefined
B12 by dividing the density and the molar masses. These simplifications lead to shortened Eq. 27.
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In both cases, the Fickian (ideal) diffusion and the assumption of non-ideal behavior, the same flux has to be calculated. So an expression for the dependency of the mobility coefficient and the Fickian diffusion can be derived:
J1,ideal ρ = J1,non −ideal
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D12 = B%12φ1 (1 − φ1 )
Eq. 26
∂ 2 ∆g ∂φ12
Eq. 27
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The mobility coefficient B12 is dependent on the diffusion coefficients at infinite dilution.
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Therefore, the mobility coefficient B12 is expressed by a simple approach by Darken to cover this dependency [32]. This approach is defined as a simple linear combining rule of two independent coefficients B1 and B2, respectively. Therefore, Eq. 27 has to equal the binary diffusion coefficients at infinite dilution. To cancel out the free enthalpy and the prefixed segment fractions, the linear mixing rule for the mobility coefficient was modified leading to Eq.
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28.
1 B%12 = (φ1B1r2 + (1 − φ1 ) B2r1 ) RT
Eq. 28
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Here, B1 and B2 are fitting parameters that have a physical meaning with respect to the state of infinite dilution. By the fitting the mobility coefficients on experimental data of the bulk
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diffusions, these coefficients can be used to predict the interfacial mass transfer.
3. Materials and Methods 3.1 Selection of Model Systems
The model systems for the investigation of the mass transfer, acetone-toluene-water (System I) and methanol-hexane-heptane (System II) were selected as they have different physical properties. 11
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System I was suggested by the EFCE (European Federation of Chemical Engineering) as one of the test-systems for extraction columns [33]. The system shows a high interfacial tension and a closed mixing gap; whereas the top phase (TP) in System I is the toluene-rich phase and the
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bottom phase (BP) is the water rich phase. Hereby, acetone is the transitioning component in System I. The partitioning of acetone in System I, considering weight fractions, is quite equal on both phases [33].
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On the other hand, System II shows a low interfacial tension in both demixing binary subsystems (methanol-hexane and methanol-heptane). The interfacial tension of System II is lower than the
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interfacial tension of System I. In contrast to System I, System II shows two demixing subsystems which offer the possibility of two different transitioning components, namely hexane and heptane. The partitioning of both transitioning components on the formed phases is quite unequal. As System II is quite different from System I, it was chosen as second system to test the
3.2 Materials
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DGT to calculate the interfacial mass transfer.
The materials which are used to estimate the interfacial mass transfer in the systems acetone-
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toluene-water and heptane-hexane-methanol are collected in Table 1. VE-water is demineralized water. The chemicals were used without further purification. The water content in methanol and
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acetone was controlled by Karl-Fisher titration and molecular sieves were added to dry them.
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Table 1: Chemicals used for the experiments and analytics with purities and supplier.
Purity
Supplier
toluene
>99,5%
VWR Chemicals
acetonitrile VE-water dibutylether
>99,8%
VWR Chemicals
>99,9%
VWR Chemicals
-
TU Dortmund
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acetone
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Chemical
99.3
Sigma Aldrich
>99,0%
VWR Chemicals
n-hexane
>95,0%
VWR Chemicals
methanol
>99,8%
VWR Chemicals
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n-heptane
3.3 Analytical Methods Quantification of organic component concentrations in the different phases was performed by gas chromatography (GC). A typical sample for GC analysis was composed of 800 mg solution, 13
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400 mg dibutylether as internal standard. All measurements were performed as triplets. The GC system Typ A0C-20i from Shimadzu was equipped with an INNOPEG-FFAP column (Polyethylen glycol, partially esterified with 2-Nitroterephthalsäure; 25 m, 0.32 mm, 0.5 µm) and
(Methrom 915 KF Ti-Touch, Methrom, Herisau, Swiss).
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3.4 Nitsch-Cell
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a Flame Ionisation Detector. The water content was analyzed by Karl-Fisher titration in triplets
To estimate the interfacial mass transfer, a Nitsch-cell [34] composed of a double-walled glass
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cylinder, including a bottom and lid with several internal fittings to ensure the flow directions, was used. The Nitsch-cell had a diameter of 103.8 mm and a volume of 1.04 L. It was divided into top and bottom ranges. To provide a flow profile with a stable phase interface, two identical internal fittings were installed in the top and bottom ranges. These consisted of a 28 mm flow
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tube with an inner diameter of 72.5 mm and eight baffles with a height of 50 mm. The internal fittings were fixed with screws to the lid and the bottom of the Nitsch-cell, respectively. To mix the two separated phases, stirrers were integrated into the flow tubes located at a distance of 30
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mm to the bottom and lid. The counter-rotating stirrers were powered by Eurostar digital agitators from IKA-Werke. Figure 1 shows a schematic of the internal fittings, including the flow
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tube, baffles and integrated stirrer with the flow profile in the two phases. By employing a stirrer and internal fittings, a convective flow resulted from the phase interface to the stirrer. Furthermore, the liquid was turbulently mixed and guided through the flow tube to the bottom and top of the Nitsch-cell. There, the flow was deflected outward and then back to the interface.
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Figure 1: Schematic of the internals of the Nitsch cell with flow tubes and an integrated stirrer; the flow profile in the two
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phases.
In this region the baffles brought the radial flow direction into the axial flow direction. By adjusting the stirrer speed, the flow directions remained equal in both phases, enabling a stable interface. To hermetically seal the Nitsch-cell, rubber gaskets were inserted between the bottom
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and the glass cylinder, and between the lid and the glass cylinder. Additionally, the glass cylinder was screwed to the lid and the bottom via four threaded rods and corresponding spacers. At the bottom of the cell, a shut-off valve for emptying the cell was installed. In addition, a septum
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made of polytetrafluoroethylene acted as a sample extraction point. A pressure compensation valve, the upper agitator shaft and an identical sample extraction point were implemented on the
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lid. Pt 100 sensors for measuring temperature were applied to the lid and the bottom. The double jacket was connected to the C6 CS temperature control unit from MGW with silicone oil as a heat transfer medium. All experiments were conducted at 1,004 mbar. 3.5 Uncertainty of Measurements
Each experimental data points have an error due to statistical fluctuations which can be determined due to multiple measurements. To consider the statistical fluctuations all
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experimental data points were determined in triplicate and used to calculate averaged experimental data points s : Eq. 29
1 3 ∑ si 3 i =1
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s=
The error of each experimental data point is calculated by the Gaussian error propagation as follows:
Eq. 30
2
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dS ∆S = ∑ ∆X i i =1 dX i n
depends on the directly measured values
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where the error of the indirectly measured value ∆ was estimated by a series expansion und as well as from their errors ∆ .
For example the sample concentration depends on the areas in the chromatograms or the error by weighing the samples. Based on these errors, the error of the experimental data points will be
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estimated.
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4. Results and Discussion
In this work several experiments with different composition of System I and System II have
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been conducted to estimate the mass transfer in the individual system with different starting compositions. In Table 2 and Table 3 an overview on the mass transfer experiments is given.
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Table 2: Overview on the mass transfer experiments in System I; whereas the weight fractions given are the starting
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compositions of acetone in the corresponding phases.
Injection
w 'Ac
w 'Tol
w 'wat
w ''Ac
w ''Tol
w ''wat
Exp 1
BP
0.145
0.003
0.852
0
0.008
0.992
Exp 2
TP
0
0.004
0.996
0.20
0.007
0.793
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ID
is the transitioning component in Exp 4.
w 'hex
Injection
Exp 3
TP (hexane)
0
Exp 4
TP (heptane)
0.33
w 'h ep
w 'm eth
w ''hex
w ''hep
w ''meth
0.29
0.71
0.21
0.72
0.06
0.09
0.58
0.6
0.29
0.11
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ID
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Table 3: Overview on the mass transfer experiments in System II. Hexane is the transitioning component in Exp 3 and heptane
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4.1 System I: Acetone-Toluene-Water
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0.00 35
1.00
30
0.25
0.50
[-]
0.50
20 15 10
0.75 0.25
5 0
1.00 0.00 0.25
0.50
0.75
1.00
wtoluoene [-]
0.0
0.1
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0.00
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σ [mN/m]
wa ter
e ton
w
25
w ace
[-]
0.75
0.2
0.3
0.4
0.5
0.6
waceton [gew.%]
Figure 2: Phase and interfacial behavior of acetone-toluene-water. Left: Phase equilibrium of system I: squares are experimental data [36] and the lines are calculated by the KK-model at T=303.15K. Model parameters can be found in
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Table 4. Right: Interfacial tension of system I at T=303.15K: squares are experimental data taken from and the line is modelled by the DGT+KK.
At first, the experimental data of System I´s liquid-liquid equilibrium (LLE) was taken from
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literature [36] and the LLE was modelled by the KK approach. The model parameter can be found in Table 4. Figure 2 (left) shows that the LLE can be described in good accordance to
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experimental data at low acetone concentrations, but there are higher deviations between experimental data and the modelling results at high acetone concentrations. Table 4: Model parameters of the KK-model and mobility coefficients of System I.
18
1
i
J
β 0 ,ij
β1,ij K
γ ij
m2 Bi s
m2 Bj s
toluene
acetone
0.1
0
0.28
2.8 ⋅ 10-9
3.7 ⋅ 10-9
toluene
water
2.6
0
0.59
2.8 ⋅ 10-9
1.3 ⋅ 10-9
acetone
water
0.82
0
0.55
5.0⋅ 10-9
1.3 ⋅ 10-9
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By the combination of the KK-model and the DGT, it is possible to calculate the interfacial tension. To calculate the interfacial tension of a LLE, one experimental interfacial data point is required to determine the influence parameter.
an influence parameter κ = 6.3* 10−7
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Figure 2 (right) shows the result of the modelling of the interfacial tension of System I by using J * mol . The interfacial tension can be modelled in good m4
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agreement to experimental data. After fitting the influence parameters of the density gradient theory, the mobility coefficients are required to calculate the interfacial mass transfer. Therefore,
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binary diffusion coefficients were modelled, by fitting the two parameters B1 and B2 to experimental data. Figure 3 shows the binary diffusion coefficients for the sub-systems tolueneacetone (left) and acetone-water (right) as a function of the segment fraction, respectively. The experimental data was taken from literature [35]. Although the modeling of the LLE shows some
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deviations, the calculations of the binary diffusion coefficients are in good agreement with the experimental data of both subsystems. Since water and toluene are nearly not miscible, experimental data on the diffusion coefficient does not exist in literature. Therefore, the same
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mobility coefficients at infinite dilution were taken for the calculation of the interfacial mass transfer as in the modelled subsystems (Figure 3). All mobility coefficients at infinite dilution are
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shown in Table 4. These estimations of the influence parameter and the mobility coefficients are the basis for the modelling of the mass transfer of acetone across the interface.
19
6x10-9
6x10-9
5x10-9
5x10-9
4x10-9
4x10
-9
3x10-9
3x10-9
-9
2x10-9
1x10-9
1x10-9
2x10
0 0.0
0.2
0.4
0.6
0.8
0 0.0
1.0
0.2
0.4
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D [m2/s]
D [m2/s]
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0.6
0.8
1.0
wacetone [-]
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wtoluene [-]
Figure 3: Binary diffusion coefficients of the sub-systems toluene-acetone (left) and acetone-water (right) at 305.15 K. The solid line represents the calculated diffusion coefficients and the squares stand for the experimental data taken from
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literature [35].
Figure 4 (left) shows the concentration of acetone in the corresponding phases over the time. From the experiments it can be seen that the system is after about 250 min in equilibrium. Moreover, it can be seen that the interfacial mass transfer can be modelled by the DGT. These 4 calculations were performed with the remaining fitting parameter ∆ω = 2 ⋅ 10 ⋅ ( 2 κ ) . On the right
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2
hand side of Figure 4, the concentration profiles of acetone at different times can be seen. It can be stated, that acetone enriches almost immediately at the interface. This could be a large mass
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transfer resistance. To calculate the mass transfer, a concentration profile at t=0min has to be
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chosen. To minimize the number of parameters, a Boltzmann-function was used, where the acetone-concentrations at the beginning of the experiments were applied. The modelling could be improved by using another function which could account for the immediate enrichment of the acetone in the interface, but these would lead to a number of fitting parameters.
20
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0.16
0.5
0.14 0.4
0.10 0.08 0.06 0.04
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wacetone [ - ]
wacetone [ - ]
0.12
0.3
0.2
0.1
SC
0.02 0.0
0.00 0
100
200
300
400
500
0.0
0.2
0.4
0.6
0.8
z∆ω⋅(2κ)-2 [(mol/J)0.5]
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t [min]
Figure 4: Interfacial mass transfer of acetone in System I (Exp 1). Left: Time dependent bulk-phase composition: Filled squares are the weight fractions of acetone in the water-rich phase and the open squares are the weight fractions of acetone in the toluene-rich phase at T=303.15K. The lines are calculated by the DGT + KK. Right: Concentration profiles of acetone at different times: dotted-dashed line: t=0min; dashed line: t=5 min; dotted line: t=240 min. Errors are indicated by error bars.
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The mobility coefficients determined can be used to calculate the mass transfer of acetone at different starting concentrations on different tie lines. Figure 5 shows the time dependent acetone concentration at a different tie line (left) and the
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corresponding concentration profile across the interface (right). It can be concluded that by using the mobility coefficients determined by fitting binary diffusion coefficients, the interfacial mass
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transfer of acetone can be predicted.
21
0.5
0.20
0.4
0.15
0.10
0.3
0.2
0.1
0.05
0.0 0
100
200
300
400
0.0
500
0.2
0.4
0.6
0.8
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0.00
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0.25
wacetone [ - ]
wAceton [ - ]
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1.0
1.2
z∆ω⋅(2κ)-2 [(mol/J)0.5]
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t [min]
Figure 5: Interfacial mass transfer of acetone in System I (Exp 2). Left: Time dependent bulk-phase composition: Filled squares are the weight fractions of acetone in the water-rich phase and the open squares are the weight fractions of acetone in the toluene-rich phase at T=303.15K. The lines are calculated by the DGT + KK. Right: Concentration profiles of acetone at different times: dotted-dashed line: t=0min; dashed line: t=5 min; dotted line: t=240 min. Errors are indicated by error bars.
0.00
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4.2 System II: Hexane-Heptane-Methanol
1.0
EP
1.00
0.8
0.25
σ [mN/m]
tha no l me
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0.50
[-]
w
e an
0.50
w hex
[-]
0.75
0.6
0.4
0.75
0.25
0.2
1.00
0.00
0.25
0.50
wheptane [-]
0.00 0.75
1.00
0.0 280
290
300
310
320
330
T [K]
Figure 6: Phase and interfacial behavior of hexane-heptane-methanol. Left: Phase equilibrium of system II: squares are experimental data [36] and the lines are calculated by the KK-model at T=303.15K. Right: Interfacial tension of binary
22
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subsystems of system II: squares are experimental data of heptane-methanol [37] and circles are experimental data of hexane-methanol [38] and the lines are modelled by the DGT+KK.
Similar as in 4.1, the first step is the modelling of the LLE [36] of System II. Figure 6 (left)
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shows the LLE of methanol-hexane-heptane. It can be seen that the LLE is in good accordance with experimental data at low concentrations of hexane, but there are higher deviations at higher hexane concentrations. The modelling parameter can be found in Table 5. The KK-parameter of
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heptane-methanol and hexane-methanol were fitted on LLE data of the particular subsystem. Based on the LLE of the binary subsystems, the interfacial tensions of these systems were
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calculated.
Table 5: Model parameters of the KK-model and mobility coefficients of System II. J
β 0 ,ij
heptane
methanol
-2.96
hexane
methanol
-2.66
heptane
hexane
-0.05
β1,ij K
γij
1700
1
m2 Bi s
m2 Bj s
-1.17
2.2 ⋅ 10-9
2.8 ⋅ 10-9
1500
-1.3
2.2 ⋅ 10-9
2.8 ⋅ 10-9
0
-1.1
3.3 ⋅ 10-9
3.6 ⋅ 10-9
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i
Figure 6 (right) shows the interfacial tensions´ modelling of the binary subsystems methanol-
parameters
which
were
fitted
on
one
experimental
data
point
are:
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influence
EP
hexane and methanol-heptane by the DGT in combination with KK-model. The required
κ methanol − hep tan e = 5.3* 10 -9
J * mol J * mol and κ methanol − hex an e = 9.4* 10 -9 . This influence parameter 4 m m4
could be used to predict the interfacial tension of System II. But up to now, there is no experimental data of the interfacial tension of System II in literature. In Figure 7, the binary diffusion coefficients for the sub-systems hexane-heptane (left) and methanol-hexane (right) are plotted against the segment fraction. The experimental data are taken from literature [39, 40]. The calculations of the binary diffusion coefficients show good 23
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agreements with respect to the experimental data over the whole range of composition. As there is no available data for the binary sub-system methanol-heptane, the parameters were taken from the fitting of the sub-system methanol-hexane. The corresponding parameters are listed in Table
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5. So, these parameters are used to calculate the interfacial mass transfer of heptane and hexane in System II, respectively. 3x10-9
D [m2/s]
3x10
2x10-9
-9
2x10-9
1x10-9
1x10-9 0 0.0
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D [m2/s]
4x10-9
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5x10-9
0.2
0.4
0.6
0.8
whexane [-]
1.0
0 0.0
0.2
0.4
0.6
0.8
1.0
wmethanol [-]
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Figure 7: Binary diffusion coefficients of the sub-systems hexane-heptane at 300 K (left) and methanol-hexane at 305.65 K (right). The solid line represents the calculated diffusion coefficients, the dashed line is the diffusion coefficient inside the
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unstable two-phase area and the squares stand for the experimental data taken from literature [39, 40].
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Figure 8 (left) shows the concentration of hexane over the time and the modelling of the interfacial mass transfer by the DGT in combination with KK-model. By fitting the remaining parameter to ∆ω = 2 ⋅ 104 ⋅ ( 2 κ ) , the concentration progress over time could be modelled in good 2
accordance to experimental data points. Moreover, it can be stated that the system is after approximately 80 min in equilibrium.
24
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Figure 8 (right) shows the concentration profile of hexane. It can be stated that the enrichment of hexane in the interface of System II is much lower than the enrichment of acetone in System I (Figure 4, right). So, a possible explanation for the longer equilibration time of System I could be
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the mass transfer resistance of the enrichment in the interface. This could also be a first experimental hint for the existence of this enrichment, which cannot be proofed by experiments
0.25
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0.30
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because of the small dimensions of the interface.
0.20
wheptane [ - ]
whexane [ - ]
0.25
0.15
0.10
0.20 0.15 0.10
0.05
0.05
0.00 20
40
60
80
100
120
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0
t [min]
0.00 0.0
0.1
0.2
0.3
0.4
0.5
z∆ω(2κ)−2 [mol/m²]
Figure 8: Interfacial mass transfer of hexane in System II (Exp 3). Left: Time dependent bulk-phase composition: Filled
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squares are the weight fractions of hexane in the methanol-rich phase and the open squares are the weight fractions of hexane in the heptane-rich phase at T=303.15K. The lines are calculated by the DGT + KK. Right: Concentration profiles
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of hexane at different times: dotted-dashed line: t=0min; dashed line: t=20 min; solid line: t=60 min. Errors are indicated by error bars.
25
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0.25
0,3
0,2
0,1
0.15
0.10
0.05
0.00
0,0 20
40
60
80
100
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
z∆ω(2κ)−2 [mol/m²]
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0
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whexane [ - ]
wheptane [ - ]
0.20
t [min]
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Figure 9: Interfacial mass transfer of heptane in System II (Exp 4). Left: Time dependent bulk-phase composition: Filled squares are the weight fractions of heptane in the methanol-rich phase and the open squares are the weight fractions of heptane in the hexane-rich phase at T=303.15K. The lines are calculated by the DGT + KK. Right: Concentration profiles of hexane at different times: dotted-dashed line: t=0min; dashed line: t=20 min; solid line: t=60 min. Errors are indicated by error bars.
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Based on the calculated mobility coefficients and the fitting parameter, the interfacial mass transfer of heptane can be predicted (Figure 9, left). It can be stated that the concentration progress over the time can be predicted in good accordance to experimental data. Deviations
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between the calculated concentration progress and the experimental one can be explained by the deviation between the KK-model and LLE experimental data at high hexane concentration.
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Figure 9 (right) shows the concentration profiles of heptane at different times. It shows that there is no enrichment of hexane in the interface. Comparing Figure 9 (left) and Figure 8 (left), it can be seen that the equilibration of the heptane concentration is faster than the equilibration of the hexane concentration. 5. Conclusion The interfacial mass transfer of two different extraction systems was investigated in this work. As theoretical model the density gradient theory was combined with the Koningsveldt-Kleintjens 26
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(KK) model. In addition to the theoretical investigations, the interfacial mass transfer was also experimentally estimated using a Nitsch-Cell. The interfacial mass transfer was investigated in the ternary system acetone-toluene-water (System I) and methanol-heptane-hexane (System II).
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The first step was the modelling of the liquid-liquid equilibrium (LLE) of each system. Based on the LLE the interfacial tension of each system was modelled by fitting the influence parameter of the DGT on one experimental data point of each system. These influence parameters were
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required to calculate the interfacial mass transfer in each system. Additionally to the influence parameters, mobility coefficients of each component pair were required. These parameters were
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gained by deriving an approach to connect bulk and interfacial diffusion and successfully fit this approach to experimental data of binary diffusion coefficients. Adjusting a last fitting parameter to one experimental set of data of interfacial mass transfer, it was possible to predict the concentration progress at other compositions of the systems. Moreover, the DGT in combination
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with the KK-model calculates a high enrichment of acetone in the interface of System I; whereas hexane has a very low enrichment in the interface of System II. It was found, that this
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enrichment has a large impact on the mass transfer of the transitioning component.
Acknowledgements
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This work is part of the Collaborative Research Centre "Integrated Chemical Processes in Liquid Multiphase Systems" coordinated by the Technische Universität Berlin. Financial support by the Deutsche Forschungsgemeinschaft (DFG) is gratefully acknowledged (TRR 63).
Symbols
27
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Symbol
Meaning
Unit
Concentration
/ /
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Gibbs energy Gibbs excess energy
/
Gibbs mixing enthalpy
∆
/
Number of experiments
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Time
SC
Mol number
−
Weight fraction
−
Mole number
−
Spatial coordinate
Symbol
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Reference coordinate
Meaning
Mobility coefficient
Unit ³/( ⋅ ⋅
Bi
EP
Mobility coefficient at infinite /
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dilution
28
Diffusion coefficient
/
Chain length Universal gas constant
/(
⋅ )
Indirect experimental value
−
∆
Error of S
−
!
Temperature
−
)
ACCEPTED MANUSCRIPT
−
Error of X
−
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∆
Direct experimental value
Meaning
" (!)
Koningsveld-Kleintjens parameter (Eq.4)
−
" (i, j)
Koningsveld-Kleintjens parameter (Eq.4)
−
"' (i, j)
Koningsveld-Kleintjens parameter (Eq.4)
(
Koningsveld-Kleintjens parameter (Eq.3)
)
Influence parameter of the DGT
*
Chemical potential of component i
∇
Nabla operator
-
+
Density
/ ³
∆+
Density difference/ fitting parameter
,
Interfacial Tension
/0
Segment fraction of component i
−
ΔΩ
Grand potential
−
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⋅
BP
Bottom phase
DGT
Density Gradient theory European
Federation
Engineering EOS
Equation of state
GC
Gaschromatography
of
/
/ ³ -/ ²
Meaning
EFCE
29
−
TE D
AC C
Abbreviation
EP
Abbreviations
Unity
SC
Symbol
Chemical
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Koningsveld-Kleintjens
LLE
Liquid-liquid equilibrium
TP
Top phase
AC C
EP
TE D
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SC
RI PT
KK
30
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References
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S0378-3812(17)30070-5
DOI:
10.1016/j.fluid.2017.02.013
Reference:
FLUID 11409
To appear in:
Fluid Phase Equilibria
Received Date: 30 December 2016 Revised Date:
17 February 2017
Accepted Date: 21 February 2017
Please cite this article as: K.F. Kruber, M. Krapoth, T. Zeiner, Interfacial mass transfer in ternary liquidliquid systems, Fluid Phase Equilibria (2017), doi: 10.1016/j.fluid.2017.02.013. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
ACCEPTED MANUSCRIPT
Interfacial Mass Transfer in Ternary Liquid-Liquid Systems Kai Fabian Kruber1, Marius Krapoth1, Tim Zeiner2* Technical University Dortmund, Faculty of Bio-and Chemical Engineering, Emil-Figge Str. 70, 44137 Dortmund, Germany 2
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1
Technical University Graz, Institute of Chemical Engineering and Environmental Technology,
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Inffeldgasse 25C, 8010 Graz, Austria
KEYWORDS: Mass Transfer, Density Gradient Theory, Nitsch-Cell, Extraction, Interface
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*corresponding author: [email protected]
ABSTRACT
In this work, the interfacial mass transfer in two extraction systems, namely acetone-toluene-
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water (System I) and hexane-heptane-methanol (System II), was examined experimentally and theoretically. The interfacial mass transfer was experimentally examined by using a Nitsch-Cell. As theoretical approach the density gradient theory (DGT) in combination with the Koningveld-
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Kleintjens (KK) model was used. At first, the KK-model was used to model the liquid-liquid
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equilibrium of System I and System II. In combination of the KK-model with the DGT, the interfacial tension was modelled by fitting the influence parameter of the DGT. To estimate the required mutual mobility coefficients in each system, bulk diffusion coefficient coefficients were used. It was shown, that the DGT in combination with a thermodynamic model and experimental information of the bulk diffusion coefficients and the system’s interfacial tension is able to model the interfacial mass transfer. Moreover, it can be stated that the DGT predicts a high enrichment of acetone in System I and this enrichment has an influence on the mass transfer.
1
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1. Introduction For each separation operation in chemical engineering, the interfacial mass transfer is essential. Nevertheless, if it is a vapor-liquid interface in rectification/absorption or if it is a liquid-liquid
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interface in extraction, there is always a mass transfer across an interface. The characteristics of these interfaces have been debated by a number of scientists. On the one hand, some of them [1] postulated that these interfaces have a zero thickness; hereby, all physical quantities such as
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refraction index are assumed to be discontinuous across the interface. But other scientists such as Poisson [2], Maxwell [3], and Gibbs [4] assumed that the existence of an interfacial region
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describing a smooth transition of all properties between the two bulk fluid values. Based on this assumption Lord Rayleigh [5] and van der Waals [6] proposed gradient theories to describe this interfacial region on this basis of thermodynamics. By using his equation of state (EOS), van der Waals developed a theory of the interface. On the basis of this theory, he could calculate the
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thickness of vapor-liquid interfaces.
In the 1950’s, Cahn and Hillard [7] recovered these theories. On that basis, they developed the density gradient theory (DGT). The DGT is a mean field approach and it results in an expression
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of the Helmholtz free energy for an inhomogeneous system. This theory can be applied in equilibrated systems as well as in not equilibrated systems. So, the interfacial tension can be
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modelled by applying the stationary DGT in combination with an EOS or a Gibbs excess energy (gE)-model [8]. In literature there are several works using the DGT to model the interfacial properties of two phase systems [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19]. To calculate interfacial mass transfer or phase decomposition, the non-steady form of DGT can be applied [20, 21, 22]. To apply the DGT, a thermodynamic model has to be used. Here the DGT is combined with a
2
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gE-model as liquid-liquid systems are regarded, which are considered to be incompressible in low pressure ranges. Up to now, the mass transfer calculated by the DGT was not compared to experimental data. In
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literature there are two simple experimental apparatuses [23, 24] proposed. Constant interfacial area stirred cells [23] (Lewis-Cell) were applied to estimate the mass transfer properties in liquid-liquid systems with constant interfacial area. Unfortunately, this Lewis-cells do not
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provide efficient stirring conditions for regions close to the liquid-liquid interface. The reason is that the stirrers cause not enough interfacial turbulence to break diffusion films. To produce
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higher interfacial turbulence, Nitsch and Hillekamp [24] have developed an apparatus with flow breaker and stirrers with special blades (Nitsch-Cell). This new apparatus allows an efficient forced convection. Furthermore, it allows stirring the aqueous as well as the organic phases at high velocities without drop release from one phase to the other. For this reason, a Nitsch-Cell
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was used in this work to estimate the interfacial mass transfer in classical extraction systems. In this work, the interfacial mass transfer in two different liquid-liquid systems was analyzed. One system is the classical extraction system acetone-toluene-water (System I) and the other
EP
system is heptane-hexane-methanol (System II). Grunert et al. [25] have calculated the interfacial properties of System I by the DGT in combination with a Koningsveld-Kleintjens approach.
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They [25] found a high enrichment of acetone in the interface of System I. This raises the question, if this enrichment has an influence on the interfacial mass transfer of acetone. This question was answered by this work. Moreover, it was shown that the DGT can be used to model the interfacial mass transfer for the first time.
3
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2. Theoretical Framework To describe the interfacial mass transfer, the density gradient theory was applied. To reduce the number of adjustable parameters, the influence parameter was adjusted to interfacial tension
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data. Moreover, the mobility coefficients were adjusted to binary diffusion coefficients experimental data, so there is just one remaining fitting parameter ∆ω left which has to be
Koningsveld-Kleintjens (KK) approach was used. 2.1 Koningsveld-Kleintjens
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adjusted to the interfacial mass transfer experimental data. As thermodynamic model the
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To develop a thermodynamic model for non-ideal polymer solutions, Flory and Huggins independently suggested a simple lattice theory [26, 27]. The basic assumption of these theories is a lattice which is occupied by large molecules with different sizes. To account for the different sizes of molecules, the segment molar fraction of component i, φi , is introduced, which is defined
φi =
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as follows:
ri ni n
∑r n j
j
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j =1
Eq. 1
where ni is the number of molecules i , and ri is the number of segments of component i .
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Similar to the Flory-Huggins theory, the ideal-athermic mixture is chosen as a standard state in the Koningsveld-Kleintjens (KK) model. Thus, the KK model for an n-component mixture can be expressed as follows [28]: n n n ∆g KK φ = ∑ i ln(φi ) + ∑∑ χ ij φi φ j RT i =1 ri i =1 j >i
Eq. 2
Koningsveld and Kleintjens [28] developed a model in which the Flory-Huggins χ - parameter is expressed as follows:
4
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χ ij = α ij +
β ij (T ) 1 − γ ij φ j
Eq. 3
here the function β ij (T ) can be expressed as:
β ij ,1
Eq. 4
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β ij (T ) = β ij ,0 +
T
The KK-model has a physical meaning because the function β ij(T) describes the contact energy
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of the segments; the parameter γij expresses the ratio of the segment surfaces σi of the components:
σi σj
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γ ij = 1 −
Eq. 5
In this work, the chain length ri of each component is calculated by the molar mass that means that the smallest component in each system has the chain length 1 and the chain length of the
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other components is calculated by the division of their molar mass Mi by the molar mass of the smallest component. For this reason, the segment fraction is equal the mass weight fraction wi. 2.2 Density Gradient Theory
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In the theory of Cahn and Hilliard [7], the interfacial tension of inhomogeneous systems is proportional to the grand thermodynamic potential. To calculate the grand thermodynamic
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potential, the Helmholtz free energy was used [7]. If an incompressible system is assumed, the Gibbs energy gets equal to the Helmholtz free energy. In this case not the density gradient is calculated, but the concentration gradient of each component from one bulk phase to the other across the interface. This gradient can be used to calculate the interfacial tension. The grand thermodynamic potential ∆Ω of a multicomponent system corresponding to the LLE can be calculated as:
5
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n
∆Ω = ∆g (φi ) − ∑ µ i φi
Eq. 6
i =1
where µi
is the chemical potential of component i in equilibrium and ∆g(φi ) is the Gibbs
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energy at the corresponding composition. According to [14, 25], the integral between the bulk phase concentrations can be used to calculate the interfacial tension. The interfacial tension is given by φ ''
Eq. 7
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σ = ∫ 2κ12 ∆Ω(φ )dφ φ'
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where κ 12 is the influence parameter which must be fitted to one experimental interfacial tension data point per demixing subsystem. In addition to the interfacial tension, the concentration profile in equilibrium across the interface can also be calculated as follows [25]: φ*
Eq. 8
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2κ12 1 z − z0 = dφ ∫ ∆ω φ ' ∆Ω
mol where ∆ω 3 is a fitting parameter, because here the modified DGT is used and a gE-model m
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is not able to calculate the density at any time. This fitting parameter is of the same unit as the molar density but has no physical meaning. The reference coordinate z0 can be arbitrarily chosen
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and the segment fraction φ* varies between the bulk concentrations. Thus, any value of the segment fraction inside the interfacial region can be assigned to one spatial coordinate. The interfacial tension of a ternary system is given by:
σ=
φ1''
∫
2κ∆Ω(φ1,φ2 )dφ1
Eq. 9
φ1'
For ternary systems with only one binary demixing subsystem, Grunert et al. [25] demonstrated that the influence parameter of the ternary mixture is equal to that of the binary demixing 6
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subsystem. If there are two demixing subsystems, a combining rule for the influence parameters has to be used [29]. In addition to the interfacial tension, the concentration profile in equilibrium across the interface
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can also be calculated as follows [25]: φ*
Eq. 10
1 1 2κ z − zo = dφ1 ∫ ∆ω φ ' ∆Ω (φ1,φ2 ) 1
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The reference coordinate z0 can be arbitrarily chosen. To solve Eq. 9 and Eq. 10, an equation is required to calculate the related concentrations inside the interface of the second component.
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This segment fraction can be calculated for a LLE with one demixing subsystem as follows:
∂∆Ω (φ1,φ2 ) =0 ∂ φ 3 φ1 ,φ2
Eq. 11
In the case of two demixing subsystems, the segment fraction is computed as:
∂∆Ω (φ1,φ2 ) ∂∆Ω (φ1,φ2 ) = κ13 ∂φ3 ∂φ2 φ1 ,φ2 φ1 ,φ2
Eq. 12
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κ12
To calculate the interfacial properties, Eq. 11 respectively Eq. 12 can be solved by a root
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determination to find the composition of the second component. Thereby, the composition of the first component is given by the integration limits and the used numerical integration procedure to
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solve Eq. 9 or Eq. 10. Moreover, it is important that the component 1 is not enriched in the interface, otherwise this would lead to numerical problems.
2.2.1 Instationary Density Gradient Theory The commonly used law to model diffusion phenomena is Fick´s law:
J1,ideal = −D12∇c1
7
Eq. 13
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where c1 is the concentration of component 1 and D12 is the mutual diffusion coefficient of component 1 in presence of component 2. This flux equation leads to the component continuity equation in a system at constant density and temperature: ∂c1 = ∇D12∇c1 ∂t
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Eq. 14
Fickian diffusion can just be applied on an ideal mixture. The Fickian diffusion acts to balance
SC
concentration differences and leads the system towards an equilibrium state where concentrations in each phase are uniform. Fick´s diffusion does not hold for a multiphase system
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because even at equilibrium, there will be concentration differences between the phases and there is no net diffusion. A more general form for Eq. 13 is based on chemical potential rather than concentration. The chemical potential is what becomes uniform everywhere in a non-ideal mixture at equilibrium [20].
Eq. 15
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J1,non −ideal = −B12φ1∇µ1
where B12 is a mobility coefficient. Eq. 15 can be taken as a phenomenological postulate or can be rationalized by non-equilibrium thermodynamics. The chemical potential in a binary system
∂g ∂φ1
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µ1 = g + (1 − φ1 )
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can be calculated as follows:
µ2 = g − φ1
∂g ∂φ1
Eq. 16
The continuity equation corresponding to Eq. 16 can be written as:
∂φ1 ∂ 2g = ∇ ⋅ B12φ1 (1 − φ1 ) 2 ∇φ1 ∂t ∂φ1
Eq. 17
where the second derivative of the Gibbs energy is multiplied by the contribution of an ideal mixture; thus, Eq. 17 is reduced to Eq. 14 in the case of ideal mixtures. However, Eq. 17 is only 8
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valid for a homogenous mixture. In the case of demixing liquids, the chemical potential must be used in the density gradient approach. The density gradient result of Eq. 17 is then [30]:
∂ 2g ∂φ1 2κ = ∇ ⋅ B12φ1 (1 − φ1 ) 2 ∇φ1 − 122 ∇3φ1 ∂t ∆ω ∂φ1
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Eq. 18
In the literature [20], Eq. 18 is frequently found without the coefficient 2 in front of the influence parameter κ12; however, in the original work of Cahn and Hillard [7] this coefficient appears.
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∂φ1 = ∇ B12φ1φ2∇ ( a1 − a2 ) + B13φ1 (1 − φ1 − φ2 ) ∇a1 ∂t
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For the application on a ternary system, Eq. 17 must be extended:
∂φ2 = ∇ B12φ1φ2∇ ( a2 − a1 ) + B23φ2 (1 − φ1 − φ2 ) ∇a2 ∂t
Eq. 19
Eq. 20
where a1 and a2 can be calculated as follows:
κ ∂g κ12 2 − ∇ φ1 − 132 ∇ 2φ2 2 ∂φ1 ∆ω ∆ω
a2 =
κ κ ∂g − 122 ∇ 2φ2 − 232 ∇ 2φ1 ∂φ2 ∆ω ∆ω
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a1 =
Eq. 21
Eq. 22
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In contrast to the He and Naumann [20], each pair of components has an individual mobility coefficient, which has to be fitted to binary diffusion coefficient experimental data. To solve the
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partial differential equations Eq. 18 and Eq. 19, an explicit finite differential method was used. Hereby, the differential quotients were approximated as shown in [31]. A fundamental assumption in this work is that the mobility coefficients in the bulk and in the interface are equal. For this reason, the mobility coefficients are calculated based on experimental fickian diffusion coefficients in the bulk. 2.3 Calculation of Bulk Diffusion
9
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In this section, the calculation of the Bulk Diffusion coefficients is shown. In diffusion experiments, the Fickian diffusion coefficients are estimated, but for the calculation of the mass transfer the mobility coefficients are required. For this, the mutual dependency of the mobility
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coefficients and the Fickian diffusion coefficients is needed. Fick’s law for ideal diffusion is defined as: J1,ideal = −D12 ρ∇x1
Eq. 23
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The concentration c1 is expressed by the product of the molar density ρ and the molar fraction x1 of component 1. It is assumed that the fluid is incompressible leading to a constant density. This
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is important since the specific free enthalpy is calculated with a gE-model based on the same assumption. Therefore, this assumption is taken for consistency for the whole model derivation. The mole fraction is described as a function of the segment fraction φ1 leading to an expression for the ideal flux of diffusion in Eq. 24. In contrast, the flux for the non-ideal diffusion, based on
J1,ideal = −D12 ρ
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Eq. 15 and Eq. 16, is shown in Eq. 25.
M1M2 ∇φ1 (φ1 (M2 − M1 ) + M1 )2
∂ 2 ∆g ∇φ1 ∂φ12
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J1,non −ideal = −B12φ1 (1 − φ1 )
Eq. 24 Eq. 25
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In Eq. 26, the connection of both fluxes is defined. Substitution of the fluxes with Eq. 24 and Eq. 25 leads to an expression which connects ideal (Fick) diffusion on the left side and non-ideal diffusion on the right side. Compared with Eq. 18, the second term including ∇ 3φ1 is missing. This can be explained by the small gradients in the bulk phase, because in this case the term with ∇3φ1 vanishes. As the density is not calculable by a gE-model, the coefficient B12 is a redefined
B12 by dividing the density and the molar masses. These simplifications lead to shortened Eq. 27.
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In both cases, the Fickian (ideal) diffusion and the assumption of non-ideal behavior, the same flux has to be calculated. So an expression for the dependency of the mobility coefficient and the Fickian diffusion can be derived:
J1,ideal ρ = J1,non −ideal
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D12 = B%12φ1 (1 − φ1 )
Eq. 26
∂ 2 ∆g ∂φ12
Eq. 27
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The mobility coefficient B12 is dependent on the diffusion coefficients at infinite dilution.
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Therefore, the mobility coefficient B12 is expressed by a simple approach by Darken to cover this dependency [32]. This approach is defined as a simple linear combining rule of two independent coefficients B1 and B2, respectively. Therefore, Eq. 27 has to equal the binary diffusion coefficients at infinite dilution. To cancel out the free enthalpy and the prefixed segment fractions, the linear mixing rule for the mobility coefficient was modified leading to Eq.
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28.
1 B%12 = (φ1B1r2 + (1 − φ1 ) B2r1 ) RT
Eq. 28
EP
Here, B1 and B2 are fitting parameters that have a physical meaning with respect to the state of infinite dilution. By the fitting the mobility coefficients on experimental data of the bulk
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diffusions, these coefficients can be used to predict the interfacial mass transfer.
3. Materials and Methods 3.1 Selection of Model Systems
The model systems for the investigation of the mass transfer, acetone-toluene-water (System I) and methanol-hexane-heptane (System II) were selected as they have different physical properties. 11
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System I was suggested by the EFCE (European Federation of Chemical Engineering) as one of the test-systems for extraction columns [33]. The system shows a high interfacial tension and a closed mixing gap; whereas the top phase (TP) in System I is the toluene-rich phase and the
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bottom phase (BP) is the water rich phase. Hereby, acetone is the transitioning component in System I. The partitioning of acetone in System I, considering weight fractions, is quite equal on both phases [33].
SC
On the other hand, System II shows a low interfacial tension in both demixing binary subsystems (methanol-hexane and methanol-heptane). The interfacial tension of System II is lower than the
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interfacial tension of System I. In contrast to System I, System II shows two demixing subsystems which offer the possibility of two different transitioning components, namely hexane and heptane. The partitioning of both transitioning components on the formed phases is quite unequal. As System II is quite different from System I, it was chosen as second system to test the
3.2 Materials
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DGT to calculate the interfacial mass transfer.
The materials which are used to estimate the interfacial mass transfer in the systems acetone-
EP
toluene-water and heptane-hexane-methanol are collected in Table 1. VE-water is demineralized water. The chemicals were used without further purification. The water content in methanol and
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acetone was controlled by Karl-Fisher titration and molecular sieves were added to dry them.
12
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Table 1: Chemicals used for the experiments and analytics with purities and supplier.
Purity
Supplier
toluene
>99,5%
VWR Chemicals
acetonitrile VE-water dibutylether
>99,8%
VWR Chemicals
>99,9%
VWR Chemicals
-
TU Dortmund
EP
acetone
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Chemical
99.3
Sigma Aldrich
>99,0%
VWR Chemicals
n-hexane
>95,0%
VWR Chemicals
methanol
>99,8%
VWR Chemicals
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n-heptane
3.3 Analytical Methods Quantification of organic component concentrations in the different phases was performed by gas chromatography (GC). A typical sample for GC analysis was composed of 800 mg solution, 13
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400 mg dibutylether as internal standard. All measurements were performed as triplets. The GC system Typ A0C-20i from Shimadzu was equipped with an INNOPEG-FFAP column (Polyethylen glycol, partially esterified with 2-Nitroterephthalsäure; 25 m, 0.32 mm, 0.5 µm) and
(Methrom 915 KF Ti-Touch, Methrom, Herisau, Swiss).
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3.4 Nitsch-Cell
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a Flame Ionisation Detector. The water content was analyzed by Karl-Fisher titration in triplets
To estimate the interfacial mass transfer, a Nitsch-cell [34] composed of a double-walled glass
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cylinder, including a bottom and lid with several internal fittings to ensure the flow directions, was used. The Nitsch-cell had a diameter of 103.8 mm and a volume of 1.04 L. It was divided into top and bottom ranges. To provide a flow profile with a stable phase interface, two identical internal fittings were installed in the top and bottom ranges. These consisted of a 28 mm flow
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tube with an inner diameter of 72.5 mm and eight baffles with a height of 50 mm. The internal fittings were fixed with screws to the lid and the bottom of the Nitsch-cell, respectively. To mix the two separated phases, stirrers were integrated into the flow tubes located at a distance of 30
EP
mm to the bottom and lid. The counter-rotating stirrers were powered by Eurostar digital agitators from IKA-Werke. Figure 1 shows a schematic of the internal fittings, including the flow
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tube, baffles and integrated stirrer with the flow profile in the two phases. By employing a stirrer and internal fittings, a convective flow resulted from the phase interface to the stirrer. Furthermore, the liquid was turbulently mixed and guided through the flow tube to the bottom and top of the Nitsch-cell. There, the flow was deflected outward and then back to the interface.
14
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Figure 1: Schematic of the internals of the Nitsch cell with flow tubes and an integrated stirrer; the flow profile in the two
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phases.
In this region the baffles brought the radial flow direction into the axial flow direction. By adjusting the stirrer speed, the flow directions remained equal in both phases, enabling a stable interface. To hermetically seal the Nitsch-cell, rubber gaskets were inserted between the bottom
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and the glass cylinder, and between the lid and the glass cylinder. Additionally, the glass cylinder was screwed to the lid and the bottom via four threaded rods and corresponding spacers. At the bottom of the cell, a shut-off valve for emptying the cell was installed. In addition, a septum
EP
made of polytetrafluoroethylene acted as a sample extraction point. A pressure compensation valve, the upper agitator shaft and an identical sample extraction point were implemented on the
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lid. Pt 100 sensors for measuring temperature were applied to the lid and the bottom. The double jacket was connected to the C6 CS temperature control unit from MGW with silicone oil as a heat transfer medium. All experiments were conducted at 1,004 mbar. 3.5 Uncertainty of Measurements
Each experimental data points have an error due to statistical fluctuations which can be determined due to multiple measurements. To consider the statistical fluctuations all
15
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experimental data points were determined in triplicate and used to calculate averaged experimental data points s : Eq. 29
1 3 ∑ si 3 i =1
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s=
The error of each experimental data point is calculated by the Gaussian error propagation as follows:
Eq. 30
2
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dS ∆S = ∑ ∆X i i =1 dX i n
depends on the directly measured values
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where the error of the indirectly measured value ∆ was estimated by a series expansion und as well as from their errors ∆ .
For example the sample concentration depends on the areas in the chromatograms or the error by weighing the samples. Based on these errors, the error of the experimental data points will be
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estimated.
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4. Results and Discussion
In this work several experiments with different composition of System I and System II have
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been conducted to estimate the mass transfer in the individual system with different starting compositions. In Table 2 and Table 3 an overview on the mass transfer experiments is given.
16
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Table 2: Overview on the mass transfer experiments in System I; whereas the weight fractions given are the starting
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compositions of acetone in the corresponding phases.
Injection
w 'Ac
w 'Tol
w 'wat
w ''Ac
w ''Tol
w ''wat
Exp 1
BP
0.145
0.003
0.852
0
0.008
0.992
Exp 2
TP
0
0.004
0.996
0.20
0.007
0.793
SC
ID
is the transitioning component in Exp 4.
w 'hex
Injection
Exp 3
TP (hexane)
0
Exp 4
TP (heptane)
0.33
w 'h ep
w 'm eth
w ''hex
w ''hep
w ''meth
0.29
0.71
0.21
0.72
0.06
0.09
0.58
0.6
0.29
0.11
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EP
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ID
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Table 3: Overview on the mass transfer experiments in System II. Hexane is the transitioning component in Exp 3 and heptane
17
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4.1 System I: Acetone-Toluene-Water
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0.00 35
1.00
30
0.25
0.50
[-]
0.50
20 15 10
0.75 0.25
5 0
1.00 0.00 0.25
0.50
0.75
1.00
wtoluoene [-]
0.0
0.1
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0.00
SC
σ [mN/m]
wa ter
e ton
w
25
w ace
[-]
0.75
0.2
0.3
0.4
0.5
0.6
waceton [gew.%]
Figure 2: Phase and interfacial behavior of acetone-toluene-water. Left: Phase equilibrium of system I: squares are experimental data [36] and the lines are calculated by the KK-model at T=303.15K. Model parameters can be found in
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Table 4. Right: Interfacial tension of system I at T=303.15K: squares are experimental data taken from and the line is modelled by the DGT+KK.
At first, the experimental data of System I´s liquid-liquid equilibrium (LLE) was taken from
EP
literature [36] and the LLE was modelled by the KK approach. The model parameter can be found in Table 4. Figure 2 (left) shows that the LLE can be described in good accordance to
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experimental data at low acetone concentrations, but there are higher deviations between experimental data and the modelling results at high acetone concentrations. Table 4: Model parameters of the KK-model and mobility coefficients of System I.
18
1
i
J
β 0 ,ij
β1,ij K
γ ij
m2 Bi s
m2 Bj s
toluene
acetone
0.1
0
0.28
2.8 ⋅ 10-9
3.7 ⋅ 10-9
toluene
water
2.6
0
0.59
2.8 ⋅ 10-9
1.3 ⋅ 10-9
acetone
water
0.82
0
0.55
5.0⋅ 10-9
1.3 ⋅ 10-9
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By the combination of the KK-model and the DGT, it is possible to calculate the interfacial tension. To calculate the interfacial tension of a LLE, one experimental interfacial data point is required to determine the influence parameter.
an influence parameter κ = 6.3* 10−7
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Figure 2 (right) shows the result of the modelling of the interfacial tension of System I by using J * mol . The interfacial tension can be modelled in good m4
SC
agreement to experimental data. After fitting the influence parameters of the density gradient theory, the mobility coefficients are required to calculate the interfacial mass transfer. Therefore,
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binary diffusion coefficients were modelled, by fitting the two parameters B1 and B2 to experimental data. Figure 3 shows the binary diffusion coefficients for the sub-systems tolueneacetone (left) and acetone-water (right) as a function of the segment fraction, respectively. The experimental data was taken from literature [35]. Although the modeling of the LLE shows some
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deviations, the calculations of the binary diffusion coefficients are in good agreement with the experimental data of both subsystems. Since water and toluene are nearly not miscible, experimental data on the diffusion coefficient does not exist in literature. Therefore, the same
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mobility coefficients at infinite dilution were taken for the calculation of the interfacial mass transfer as in the modelled subsystems (Figure 3). All mobility coefficients at infinite dilution are
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shown in Table 4. These estimations of the influence parameter and the mobility coefficients are the basis for the modelling of the mass transfer of acetone across the interface.
19
6x10-9
6x10-9
5x10-9
5x10-9
4x10-9
4x10
-9
3x10-9
3x10-9
-9
2x10-9
1x10-9
1x10-9
2x10
0 0.0
0.2
0.4
0.6
0.8
0 0.0
1.0
0.2
0.4
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D [m2/s]
D [m2/s]
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0.6
0.8
1.0
wacetone [-]
SC
wtoluene [-]
Figure 3: Binary diffusion coefficients of the sub-systems toluene-acetone (left) and acetone-water (right) at 305.15 K. The solid line represents the calculated diffusion coefficients and the squares stand for the experimental data taken from
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literature [35].
Figure 4 (left) shows the concentration of acetone in the corresponding phases over the time. From the experiments it can be seen that the system is after about 250 min in equilibrium. Moreover, it can be seen that the interfacial mass transfer can be modelled by the DGT. These 4 calculations were performed with the remaining fitting parameter ∆ω = 2 ⋅ 10 ⋅ ( 2 κ ) . On the right
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2
hand side of Figure 4, the concentration profiles of acetone at different times can be seen. It can be stated, that acetone enriches almost immediately at the interface. This could be a large mass
EP
transfer resistance. To calculate the mass transfer, a concentration profile at t=0min has to be
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chosen. To minimize the number of parameters, a Boltzmann-function was used, where the acetone-concentrations at the beginning of the experiments were applied. The modelling could be improved by using another function which could account for the immediate enrichment of the acetone in the interface, but these would lead to a number of fitting parameters.
20
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0.16
0.5
0.14 0.4
0.10 0.08 0.06 0.04
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wacetone [ - ]
wacetone [ - ]
0.12
0.3
0.2
0.1
SC
0.02 0.0
0.00 0
100
200
300
400
500
0.0
0.2
0.4
0.6
0.8
z∆ω⋅(2κ)-2 [(mol/J)0.5]
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t [min]
Figure 4: Interfacial mass transfer of acetone in System I (Exp 1). Left: Time dependent bulk-phase composition: Filled squares are the weight fractions of acetone in the water-rich phase and the open squares are the weight fractions of acetone in the toluene-rich phase at T=303.15K. The lines are calculated by the DGT + KK. Right: Concentration profiles of acetone at different times: dotted-dashed line: t=0min; dashed line: t=5 min; dotted line: t=240 min. Errors are indicated by error bars.
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The mobility coefficients determined can be used to calculate the mass transfer of acetone at different starting concentrations on different tie lines. Figure 5 shows the time dependent acetone concentration at a different tie line (left) and the
EP
corresponding concentration profile across the interface (right). It can be concluded that by using the mobility coefficients determined by fitting binary diffusion coefficients, the interfacial mass
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transfer of acetone can be predicted.
21
0.5
0.20
0.4
0.15
0.10
0.3
0.2
0.1
0.05
0.0 0
100
200
300
400
0.0
500
0.2
0.4
0.6
0.8
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0.00
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0.25
wacetone [ - ]
wAceton [ - ]
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1.0
1.2
z∆ω⋅(2κ)-2 [(mol/J)0.5]
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t [min]
Figure 5: Interfacial mass transfer of acetone in System I (Exp 2). Left: Time dependent bulk-phase composition: Filled squares are the weight fractions of acetone in the water-rich phase and the open squares are the weight fractions of acetone in the toluene-rich phase at T=303.15K. The lines are calculated by the DGT + KK. Right: Concentration profiles of acetone at different times: dotted-dashed line: t=0min; dashed line: t=5 min; dotted line: t=240 min. Errors are indicated by error bars.
0.00
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4.2 System II: Hexane-Heptane-Methanol
1.0
EP
1.00
0.8
0.25
σ [mN/m]
tha no l me
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0.50
[-]
w
e an
0.50
w hex
[-]
0.75
0.6
0.4
0.75
0.25
0.2
1.00
0.00
0.25
0.50
wheptane [-]
0.00 0.75
1.00
0.0 280
290
300
310
320
330
T [K]
Figure 6: Phase and interfacial behavior of hexane-heptane-methanol. Left: Phase equilibrium of system II: squares are experimental data [36] and the lines are calculated by the KK-model at T=303.15K. Right: Interfacial tension of binary
22
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subsystems of system II: squares are experimental data of heptane-methanol [37] and circles are experimental data of hexane-methanol [38] and the lines are modelled by the DGT+KK.
Similar as in 4.1, the first step is the modelling of the LLE [36] of System II. Figure 6 (left)
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shows the LLE of methanol-hexane-heptane. It can be seen that the LLE is in good accordance with experimental data at low concentrations of hexane, but there are higher deviations at higher hexane concentrations. The modelling parameter can be found in Table 5. The KK-parameter of
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heptane-methanol and hexane-methanol were fitted on LLE data of the particular subsystem. Based on the LLE of the binary subsystems, the interfacial tensions of these systems were
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calculated.
Table 5: Model parameters of the KK-model and mobility coefficients of System II. J
β 0 ,ij
heptane
methanol
-2.96
hexane
methanol
-2.66
heptane
hexane
-0.05
β1,ij K
γij
1700
1
m2 Bi s
m2 Bj s
-1.17
2.2 ⋅ 10-9
2.8 ⋅ 10-9
1500
-1.3
2.2 ⋅ 10-9
2.8 ⋅ 10-9
0
-1.1
3.3 ⋅ 10-9
3.6 ⋅ 10-9
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i
Figure 6 (right) shows the interfacial tensions´ modelling of the binary subsystems methanol-
parameters
which
were
fitted
on
one
experimental
data
point
are:
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influence
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hexane and methanol-heptane by the DGT in combination with KK-model. The required
κ methanol − hep tan e = 5.3* 10 -9
J * mol J * mol and κ methanol − hex an e = 9.4* 10 -9 . This influence parameter 4 m m4
could be used to predict the interfacial tension of System II. But up to now, there is no experimental data of the interfacial tension of System II in literature. In Figure 7, the binary diffusion coefficients for the sub-systems hexane-heptane (left) and methanol-hexane (right) are plotted against the segment fraction. The experimental data are taken from literature [39, 40]. The calculations of the binary diffusion coefficients show good 23
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agreements with respect to the experimental data over the whole range of composition. As there is no available data for the binary sub-system methanol-heptane, the parameters were taken from the fitting of the sub-system methanol-hexane. The corresponding parameters are listed in Table
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5. So, these parameters are used to calculate the interfacial mass transfer of heptane and hexane in System II, respectively. 3x10-9
D [m2/s]
3x10
2x10-9
-9
2x10-9
1x10-9
1x10-9 0 0.0
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D [m2/s]
4x10-9
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5x10-9
0.2
0.4
0.6
0.8
whexane [-]
1.0
0 0.0
0.2
0.4
0.6
0.8
1.0
wmethanol [-]
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Figure 7: Binary diffusion coefficients of the sub-systems hexane-heptane at 300 K (left) and methanol-hexane at 305.65 K (right). The solid line represents the calculated diffusion coefficients, the dashed line is the diffusion coefficient inside the
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unstable two-phase area and the squares stand for the experimental data taken from literature [39, 40].
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Figure 8 (left) shows the concentration of hexane over the time and the modelling of the interfacial mass transfer by the DGT in combination with KK-model. By fitting the remaining parameter to ∆ω = 2 ⋅ 104 ⋅ ( 2 κ ) , the concentration progress over time could be modelled in good 2
accordance to experimental data points. Moreover, it can be stated that the system is after approximately 80 min in equilibrium.
24
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Figure 8 (right) shows the concentration profile of hexane. It can be stated that the enrichment of hexane in the interface of System II is much lower than the enrichment of acetone in System I (Figure 4, right). So, a possible explanation for the longer equilibration time of System I could be
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the mass transfer resistance of the enrichment in the interface. This could also be a first experimental hint for the existence of this enrichment, which cannot be proofed by experiments
0.25
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0.30
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because of the small dimensions of the interface.
0.20
wheptane [ - ]
whexane [ - ]
0.25
0.15
0.10
0.20 0.15 0.10
0.05
0.05
0.00 20
40
60
80
100
120
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0
t [min]
0.00 0.0
0.1
0.2
0.3
0.4
0.5
z∆ω(2κ)−2 [mol/m²]
Figure 8: Interfacial mass transfer of hexane in System II (Exp 3). Left: Time dependent bulk-phase composition: Filled
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squares are the weight fractions of hexane in the methanol-rich phase and the open squares are the weight fractions of hexane in the heptane-rich phase at T=303.15K. The lines are calculated by the DGT + KK. Right: Concentration profiles
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of hexane at different times: dotted-dashed line: t=0min; dashed line: t=20 min; solid line: t=60 min. Errors are indicated by error bars.
25
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0.25
0,3
0,2
0,1
0.15
0.10
0.05
0.00
0,0 20
40
60
80
100
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
z∆ω(2κ)−2 [mol/m²]
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0
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whexane [ - ]
wheptane [ - ]
0.20
t [min]
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Figure 9: Interfacial mass transfer of heptane in System II (Exp 4). Left: Time dependent bulk-phase composition: Filled squares are the weight fractions of heptane in the methanol-rich phase and the open squares are the weight fractions of heptane in the hexane-rich phase at T=303.15K. The lines are calculated by the DGT + KK. Right: Concentration profiles of hexane at different times: dotted-dashed line: t=0min; dashed line: t=20 min; solid line: t=60 min. Errors are indicated by error bars.
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Based on the calculated mobility coefficients and the fitting parameter, the interfacial mass transfer of heptane can be predicted (Figure 9, left). It can be stated that the concentration progress over the time can be predicted in good accordance to experimental data. Deviations
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between the calculated concentration progress and the experimental one can be explained by the deviation between the KK-model and LLE experimental data at high hexane concentration.
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Figure 9 (right) shows the concentration profiles of heptane at different times. It shows that there is no enrichment of hexane in the interface. Comparing Figure 9 (left) and Figure 8 (left), it can be seen that the equilibration of the heptane concentration is faster than the equilibration of the hexane concentration. 5. Conclusion The interfacial mass transfer of two different extraction systems was investigated in this work. As theoretical model the density gradient theory was combined with the Koningsveldt-Kleintjens 26
ACCEPTED MANUSCRIPT
(KK) model. In addition to the theoretical investigations, the interfacial mass transfer was also experimentally estimated using a Nitsch-Cell. The interfacial mass transfer was investigated in the ternary system acetone-toluene-water (System I) and methanol-heptane-hexane (System II).
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The first step was the modelling of the liquid-liquid equilibrium (LLE) of each system. Based on the LLE the interfacial tension of each system was modelled by fitting the influence parameter of the DGT on one experimental data point of each system. These influence parameters were
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required to calculate the interfacial mass transfer in each system. Additionally to the influence parameters, mobility coefficients of each component pair were required. These parameters were
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gained by deriving an approach to connect bulk and interfacial diffusion and successfully fit this approach to experimental data of binary diffusion coefficients. Adjusting a last fitting parameter to one experimental set of data of interfacial mass transfer, it was possible to predict the concentration progress at other compositions of the systems. Moreover, the DGT in combination
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with the KK-model calculates a high enrichment of acetone in the interface of System I; whereas hexane has a very low enrichment in the interface of System II. It was found, that this
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enrichment has a large impact on the mass transfer of the transitioning component.
Acknowledgements
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This work is part of the Collaborative Research Centre "Integrated Chemical Processes in Liquid Multiphase Systems" coordinated by the Technische Universität Berlin. Financial support by the Deutsche Forschungsgemeinschaft (DFG) is gratefully acknowledged (TRR 63).
Symbols
27
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Symbol
Meaning
Unit
Concentration
/ /
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Gibbs energy Gibbs excess energy
/
Gibbs mixing enthalpy
∆
/
Number of experiments
M AN U
Time
SC
Mol number
−
Weight fraction
−
Mole number
−
Spatial coordinate
Symbol
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Reference coordinate
Meaning
Mobility coefficient
Unit ³/( ⋅ ⋅
Bi
EP
Mobility coefficient at infinite /
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dilution
28
Diffusion coefficient
/
Chain length Universal gas constant
/(
⋅ )
Indirect experimental value
−
∆
Error of S
−
!
Temperature
−
)
ACCEPTED MANUSCRIPT
−
Error of X
−
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∆
Direct experimental value
Meaning
" (!)
Koningsveld-Kleintjens parameter (Eq.4)
−
" (i, j)
Koningsveld-Kleintjens parameter (Eq.4)
−
"' (i, j)
Koningsveld-Kleintjens parameter (Eq.4)
(
Koningsveld-Kleintjens parameter (Eq.3)
)
Influence parameter of the DGT
*
Chemical potential of component i
∇
Nabla operator
-
+
Density
/ ³
∆+
Density difference/ fitting parameter
,
Interfacial Tension
/0
Segment fraction of component i
−
ΔΩ
Grand potential
−
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⋅
BP
Bottom phase
DGT
Density Gradient theory European
Federation
Engineering EOS
Equation of state
GC
Gaschromatography
of
/
/ ³ -/ ²
Meaning
EFCE
29
−
TE D
AC C
Abbreviation
EP
Abbreviations
Unity
SC
Symbol
Chemical
ACCEPTED MANUSCRIPT
Koningsveld-Kleintjens
LLE
Liquid-liquid equilibrium
TP
Top phase
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EP
TE D
M AN U
SC
RI PT
KK
30
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