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New predictive correlation for mass transfer coefficient in structured packed extraction columns
chemical engineering research and design 9 0 ( 2 0 1 2 ) 615–621
Contents lists available at SciVerse ScienceDirect
Chemical Engineering Research and Design journal homepage: www.elsevier.com/locate/cherd
New predictive correlation for mass transfer coefficient in structured packed extraction columns Ahmad Rahbar-Kelishami a,∗ , Hossein Bahmanyar b a b
Faculty of Chemical Engineering, Iran University of Science & Technology (IUST), Narmak, Tehran, Iran College of Engineering, School of Chemical Engineering,University of Tehran, Iran
a b s t r a c t Developments in the area of packed columns, particularly structured packed columns, are ongoing, specifically in the area of liquid–liquid extractions in different industries. In the present study, mass transfer coefficients have been obtained experimentally in a structured packed extraction column to develop a new correlation for prediction of continuous phase Sherwood number. The experiments were carried out for toluene/acetic acid/water and n-butyl acetate/acetic acid/water systems with counter current flow in different heights of column. A new dimensionless parameter, d32 /h, is introduced in proposed equation. This number considers the effect of column height (h) and mean drop diameter (d32 ) jointly. The main advantage of this approach is that the principal effect of column height is considered in correlation without which the experimental data could not be fitted with a acceptable accuracy. © 2011 The Institution of Chemical Engineers. Published by Elsevier B.V. All rights reserved. Keywords: Liquid–liquid extraction; Packed column; Mass transfer coefficient; Sherwood number
1.
Introduction
Packed columns have found increasingly wide use in separation processes such as distillation and absorption due to lower pressure drop and lower liquid hold-up than tray columns. However, the design, analysis and scale up of packed columns today are still based mainly on empirical or semi-empirical correlation. The main reason for this is that some of the performance characteristics such as mass transfer efficiency cannot be predicted reliably in packed columns (Sun et al., 2000). Liquid extraction consists of separating one or several substances (solute) present in a solid or a liquid phase by the addition of another liquid phase in which these substances transferred preferentially. This mass transfer is often operated in countercurrent extractors. The efficiency of liquid–liquid contactors is primarily dependent on the degree of turbulence imparted to the system and the interfacial area available for mass transfer. Although many researchers consider that extraction is a fully mature technology lacking in potential for further improvement, there are still many questions that need to be solved urgently. One of them is to design commercial extractors safely with low costs. Nowadays, the scale-up of extractors still depends on large quantities of pilot experiments, which is expensive and time-consuming. Introduction of Sherwood
∗
number by measuring mass transfer coefficients is a promising method to solve the above problem (Jie and Weiyang, 2005). The fundamental process for the rate of mass transfer in extraction columns is still not sufficiently well understood nor adequately modeled (Korchinsky, 1994). The accurate specification of the mass transfer coefficient plays an important role in packed columns precise design. Therefore understanding of relevant fluid dynamics and the mass transfer coefficients in these columns is of paramount importance for the precise design. These parameters can be presented in the form of Sherwood number. Packing cause internal circulation of drops and increase mass transfer coefficient. The transfer of a solute between a rising drop and a continuous liquid phase has been widely studied, both theoretically and experimentally. The importance of this area is reflected in the vast number of works published over the years and is of interest in a variety of industrial processes, such as liquid extraction (Kumar and Hartland, 1999).
2. Theoretical predictions of mass transfer coefficients The mass transfer coefficient of dispersed or continuous phase is one of the fundamental and essential parameters
Corresponding author. Tel.: +98 021 77451505; fax: +98 021 77240495. E-mail address: [email protected] (A. Rahbar-Kelishami). Received 30 April 2011; Received in revised form 10 July 2011; Accepted 2 September 2011 0263-8762/$ – see front matter © 2011 The Institution of Chemical Engineers. Published by Elsevier B.V. All rights reserved. doi:10.1016/j.cherd.2011.09.004
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chemical engineering research and design 9 0 ( 2 0 1 2 ) 615–621
continuous-phase mass transfer coefficient (Friedlander, 1961; Baird and Hamielec, 1962; Lochiel and Calderbank, 1964):
Nomenclature C
solute concentration in dispersed phase (kmol m−3 ) equilibrium concentration of solute in dispersed phase (kmol m−3 ) initial concentration of solute in dispersed phase (kmol m−3 ) drop diameter (m) Sauter mean diameter of drop (m) diffusion coefficient (m2 /s) height of column (m) (getting sample) mass transfer coefficient based on dispersed phase (m s−1 ) mass transfer coefficient based on continuous phase (m s−1 ) contact time (s) slip velocity (m s−1 )
C* C0 d d32 D h Kd Kc t Vslip
1/3
Shc = 0.991 Pec
The upper limit of applicability of the above equation is approximately Pec = 1000, since Stokes-type flow does not occur above an Re of 1, and Scc is of the order of 1000 for liquids. Clift et al. give the following relationship for Shc valid for all Pec in Stokes regime (Clift et al., 1978): Shc = 1 + (1 + Pec )
Groups c V
d32
slip Reynolds number (Re = ) c c Schmidt number (Sc = c Dc ) continuous phase Sherwood number (Shc = Kc d32 Dc )
Re Sc Shc
in extraction column design. Mass transfer in the continuous phase around drops is generally affected by a combination of molecular diffusion and natural and forced convection in the continuous phase. The continuous-phase mass transfer coefficient, kc , will also depend upon whether the drop is internally stagnant or circulating. Much theoretical and experimental work has been done on mass transfer from single solid spheres, the results for which provide a useful basis for comparison with the data on circulating liquid drops. A variety of solutions for mass transfer has been obtained by using the equation of continuity for axisymmetric flow around a sphere, assuming constant physical properties. For a drop which is internally stagnant and when the continuous-phase velocity is zero, the expression for kc is: Kc = 2
Dc d
or Shc = 2
which is commonly ascribed to Langmuir. The above equation provides a limiting value of kc , any fluid motion will cause kc to be greater.
2.1.
1/3
(2)
For rigid spheres at intermediate Reynolds numbers, Brauer advanced an empirical correlation on the basis of results from numerical solution to diffusion equation (Brauer, 1971), which follows:
Shc = 2 +
Greek letters viscosity ratio (dispersed/continuous phase) continuous phase viscosity (kg m−1 s−1 ) c d dispersed phase viscosity (kg m−1 s−1 ) difference of dispersed and continuous phase viscosity (kg m−1 s−1 ) c continuous phase density (kg m−3 ) dispersed phase density (kg m−3 ) d difference of dispersed and continuous densities (kg m−3 ) interfacial tension (N m−1 )
(1)
0.66 Scc + 1 + Scc 2.4 + Scc
0.79 1/6
Scc
Pe1.7 c 1 + Pe1.2 c
(3)
for 1 < Re < 100. For high Reynolds numbers, Lochiel and Calderbank (1964) assumed a quadratic velocity profile in the momentum boundary layer and calculated the thickness of this layer from Boltze’s application of the Navier–Stokes equation to the boundary layer around a sphere (Boltze, 1908). This approach resulted in: 1/3
Shc = 0.7 Re1/2 Scc
(4)
Clift et al. (1978) have correlated the available experimental data in the form: 1/3
Shc = 1 + 0.724 Re0.48 Scc
(5)
for 100 < Re < 2000, and Scc > 200. Sandoval-Robles et al. (1980) have shown that how the exponent on Re changes with the value of Re from 1/3 for Stokes regime to 1/2 for boundary-layer theory, and to 2/3 for a surface renewal mechanism or turbulent conditions.
2.2.
Circulating drops
For an internally circulating sphere with high Pec (the thin concentration boundary-layer approximation), the continuousphase mass-transfer coefficient for creeping flow (Re < 1) is given by (Lochiel and Calderbank, 1964):
Shc = 0.65
Pec 1+
(6)
which was also derived by Griffith (1960), and Ward et al. (1962) with multiplying constants of 0.67 and 0.61, respectively. For Re → ∞, an asymptotic formula for Shc can easily be derived (Clift et al., 1978): 2 1/2 Shc = √ (Pec )
(7)
Stagnant drops
For Stokes regime (Re < 1), Friedlander, Baird & Hamielec, and Lochiel & Calderbank derive the following expression for the
which is the well-known Boussinesq (1905) equation. For finite Reynolds numbers, Weber (1975) used the boundary layer velocities of Harper and Moore (1968), and correlated
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his numerical results by the following equation, provided that ≤ 2 and d /c ≤ 4:
Pe 1/2
Shc = 2
c
1−
1/2
1 (2.89 + 2.150.64 ) Re1/2
(8)
According to Weber (1975), this equation should be accurate for Re > 70. In a similar fashion, Clift et al. (1978) used the surface velocities calculated by Abdel-Alim and Hamielec (1975) to obtain Shc at intermediate Reynolds numbers. The numerical results for k < 2 were correlated by:
Pe 1/2
Shc = 2
c
2 + 3/3(1 + )
1/2
n1 1/n1 × 1− 1 + (2 + 3)Re1/2 /(1 + )(8.67 + 6.450.64 )
Fig. 1 – Schematic diagram of the apparatus. (9)
In which n1 = 43 + 3 Garner et al. (1959) suggested the following equation for 8 < Re < 800: Shc = −126 + 1.8 Re0.5 Scc0.42
(10)
Garner and Tayeban (1960) also proposed another correlation for determining Shc : Shc = 0.6 Pe0.5
(11)
Another correlation was provided by Thorsen and Terjesen (1962) which is similar to Garner equation. 1/3
Shc = −178 + 3.6 Re0.5 Scc
(12)
This equation should be accurate for 50 < Re < 800. Brauer (1971) presented the following equation for 4 < Re < 1000 and 130 < Sc < 23600: Shc = 2 + 0.0511 Re0.724 Scc0.7
(13)
For packed column, Seibert and Fair57 proposed the following correlation (Seibert and Fair, 1988): 0.5
Shc = 0.698 Re
Scc0.4 (1 − ϕ)
3.
Experimental
3.1.
Column set-up
(14)
Column consisted of a Pyrex glass tube (inside diameter 9.1 cm, height 125 cm) (Fig. 1). The column was packed with two pieces of Penta-Pak TM PS-500M1 structured packing (height 20 cm) (Fig. 2). These packings were stacked on top of each other (total height 40 cm). Each packing consisted of corrugated sheets in a cylindrical structure with a diameter equal to the internal diameter of the column. The inclination of the corrugation was 45◦ . Several valves along the column were used to collect drops at different heights. The drops were collected and analyzed at heights of 11, 24.5 and 39 cm. The specific area of packing was 500 m2 /m3 and the void fraction was 0.94.
Fig. 2 – Applied Structure packing. Three nozzles (diameters 0.6, 0.8 and 1 mm) were used with both columns to make drops of different sizes.
3.2. Extraction system chemical and physical properties Two chemical systems were selected to examine a range interfacial tension values. The systems studied were toluene–acetic acid–water (high interfacial tension) (T/A/W) and n-butyl acetate–acetic acid–water (medium interfacial tension) (B/A/W). Distilled water saturated with toluene and butyl acetate was used as the continuous phase, and toluene and butyl acetate saturated with distilled water acetic acid (5 vol.%) was used as the dispersed phase. The physical properties of these systems are given in Table 1. Table 1 – Physical properties of systems at 20 ◦ C. Physical property
System 1 T/A/W
System 2 B/A/W
c (kg/m3 ) d (kg/m3 ) c (mPa s) d (mPa s) (mN/m) Dc (m2 /s) Dd (m2 /s)
1009.7 882.7 1.016 0.611 27.5–30.1 1.10 × 10−9 2.92 × 10−9
1010.2 895.9 1.013 0.684 12.4–13.2 1.03 × 10−9 2.66 × 10−9
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chemical engineering research and design 9 0 ( 2 0 1 2 ) 615–621
1200
600
1000
500 400
h=11 cm h=24.5 cm
600
Shc
Shc
800
h=11 cm h=24.5 cm
300
h=39 cm
h=39 cm
400
200
200
100
0 0.0
0.5
1.0
1.5
2.0
2.5
3.0
0 0.0
3.5
0.5
1.0
Qd/Qc
1.5
2.0
Qd/Qc
Fig. 3 – Effect of Qd /Qc and packing height on Shc , T/A/W.
Fig. 4 – Effect of Qd /Qc and packing height on Shc , B/A/W.
The viscosities of both systems were measured using a LAUDA viscometer. The densities were determined using a scale (0.0001 g graduations). The interfacial tension values were obtained from the literature. DM (mixture molecular diffusivity) were calculated using the Wilke–Chang correlation (Bird et al., 2001).
systems, three various height of packed column (11 cm,24.5 cm and 39 cm), different dispersed phase flow rate and different size of nozzles to producing different size of drops, resulted in 72 different runs.
Procedure
a
Before carrying out the experiments, both phases were mutually saturated, after which solute was added to the dispersed phase. Samples of each phase were taken at their inlets to the column and used for the determination of the initial solute concentration. In this section Mass transfer coefficients were calculated from experimental measurements. Considering the mass balance equation for a single drop: Kod =
d −
6t
1200 1000
Experimental Shc
3.3.
800 600 400 200
ln(1 − E)
(15) 0
0
20
40
where: (16)
100
120
Clift, Eq. 2 Lochiel & Calderbank, Eq. 4
and C0 , C and C* are the solute concentration in the primary drop (before contact), at a specified height and in equilibrium with the continuous phase, respectively. The solute concentrations of collected drops were measured by titration with normal NaOH. Sherwood number of continuous phase was calculated using following equation: Kc d32 Shc = Dc
80
Calculated Shc
(17)
In each experiment and for different heights of the column by measuring Acetic Acid concentration, mean diameter and slip velocity of droplets and the contact time between two phases, Shc was calculated considering Eq. (17). For the calculations, Sauter mean drop diameters (d) were determined by photographing drops inside the column. To take into account the lens effect of the column a background reference size used inside the column. The sizes of drops were measured during steady state conditions using a highresolution Powershot G9 type camera followed by analysis with Photoshop software. At least 200 drops were analyzed for each determination. Applying mentioned chemical
b
1200 1000
Experimental Shc
C0 − C E= C0 − C∗
60
800 600 400 200 0
0
20
40
60
80
100
120
140
Calculated Shc Clift, Eq.5
Brauer, Eq. 3
Fig. 5 – (a) Comparison of experimental data with stagnant drop models. (b) Comparison of experimental data with stagnant drop models.
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chemical engineering research and design 9 0 ( 2 0 1 2 ) 615–621
c
1200 1000
Experimental Shc
Experimental Shc
a
800 600
1200 1000
800 600
400
400
200
200
0
0
0
100
200
300
400
500
600
0
100
200
Calculated Shc Lochiel, Eq.6
b
Clift, Eq.7
Garner, Eq.10
d
1200 1000
400
500
600
Garner, Eq.11
1200
1000
Experimental Shc
Experimental Shc
300
Calculated Shc
800
600 400 200
800
600 400
200
0 0
100
200
300
400
500
600
Calculated Shc Webber, Eq.8
Clift, Eq.9
0 -100
0
100
200
300
400
500
600
Calculated Shc Thorsen, Eq.12
Brauer, Eq.13
Fig. 6 – (a) Comparison of experimental data with circulating drop models. (b) Comparison of experimental data with circulating drop models. (c) Comparison of experimental data with circulating drop models. (d) Comparison of experimental data with circulating drop models.
4.
Results and discussion
4.1. Effect of flow rate ratio and packing height on continuous phase Sherwood number Two main operating parameters found to affect mass transfer performance of the column were the ratio of phase flow rates and packing height. The effect of these parameters is shown in Figs. 3 and 4. It is easy to see that the continuous phase Sherwood number decreases lightly by increase of flow rate ratio. The reason could be that when the dispersed phase flow rate increases the mean drop diameter does not increase significantly, the larger drops result in improved mass transfer. It appears that the numbers of drops in the column and consequently the values of holdup and the resistance for mass transfer will increase. This leads to less amounts of Sherwood number. According to Figs. 3 and 4 the Sherwood number is decreasing while the height is increasing. This happens due to the fact
that by approaching to the end of the column the decrease in solute concentration results in less mass transfer rate and less values Sherwood numbers. Comparing the Sherwood numbers in two systems shows that these values are larger overall in first system with greater value of interfacial tension. This is due to more internal circulations in larger drops in first system.
4.2. Comparing of previous equations with experimental results As mentioned before the previously proposed correlations can be categorized according to drop internal situation into stagnant and circulating models. In this section the experimental data have been compared with the mentioned correlations in previous sections and then a new correlation for prediction of mass transfer coefficient in form of Sherwood number has been proposed.
chemical engineering research and design 9 0 ( 2 0 1 2 ) 615–621
1200
1200
1000
1000
Experimental Shc
Experimental Shc
620
800 600 400
+15%
800 -15%
600 400
200 200
0 0
40
80
120
160
200
0
Calculated Shc
0
200
400
600
800
1000
1200
Shc Calculated by model
Seiber and Fair, Eq.14
Fig. 7 – Comparison of experimental data with Seibert and Fair model.
Fig. 8 – Comparison of experimental results with calculated values.
where h is the height of sampling point in packed column and:
4.2.1. Comparison of experimental data with stagnant drop models
Re =
Fig. 5 shows the comparison of calculated Sherwood numbers from proposed correlations with experimental ones. The average absolute error of these models was 82.7%. Since these models does not consider any circulation in drop and practically there are some movements in droplets so the predicted Sherwood numbers of these models are considerably lower than experimental ones.
4.2.2. Comparison of experimental data with circulating drop models Fig. 6 shows the comparison of calculated Sherwood numbers from proposed correlations with experimental ones. The average absolute error of these 8 models was 50.7%. Because these models consider internal circulation of drops they give better results than stagnant models and the calculated values are closer to the experimental ones. However, they do not predict Sherwood number precisely. Comparison of experimental Sherwood numbers with those obtained from Seibert and fair model which has been developed for packed column is presented in Fig. 7. This model has an average absolute error of 72%.
4.3.
Prediction of new correlation
Trying to obtain a new correlation which is able to predict Sherwood number precisely leads to introduction of a new dimensionless factor, d32 /h. This number considers the effect of column height (h) and mean drop diameter (d32 ) jointly. As discussed before mass transfer coefficient and Sherwood number increase with an increased in drop diameter and a decrease in height of packing. So this new parameter, d32 /h, can consider these two effects mutually without which the experimental data would not be fitted precisely anyway. The obtained equation for continuous phase Sherwood number for both systems and in different heights of packed column is:
Shc = −55.6 + 6.21 ∗ 10−5 Re0.433 Scc2.26
d 0.4 32
h
(18)
c Vslip d32 c
,
Sc =
c c Dc
The comparison of experimental results of Shc with those calculated by the proposed model is shown in Fig. 8. This figure indicates that the suggested equation can estimate the dispersed phase mass transfer coefficient with high accuracy. The dispersed phase mass transfer coefficient calculated with this model reproduces the experimental data with an average error of 13.62%. Thus the proposed correlation can predict the behavior of continuous phase mass transfer coefficients perfectly.
5.
Conclusions
Design procedures for packed extraction column still depend on having large amounts of experimental data which are difficult and expensive to obtain. The aim here was to try to reduce the experimental effort by proposing experimental correlations for prediction of mass transfer coefficient in form of continuous phase Sherwood number, Shc . The effects of two significant variables, flow ratio and height of packing, on Shc were investigated. The results showed that Shc decreased by increasing flow ratio and height of packing, h. Many correlations are published for prediction of Shc . Although many of them have a good theoretical basis, they do not practically have accurate results. Even though circulating drop models give better results than stagnant drop models as they consider internal circulation of drops, we can obtain the mass transfer coefficient with much better precision by considering the effect of packing height in Sherwood number equation. Because Shc is a dimensionless number a new dimensionless parameter, d32 /h, is introduced in proposed correlation. This novel conceptual parameter shows that Shc is increased with an increase of mean drop diameter which results in more internal circulation and consequently more mass transfer and also it presents that Shc decreases with an increase of packing height. It is for the first time that the effect of packing height is considered in Sherwood number. This correlation is recommended for calculation of Shc and consequently for final sizing of the column height.
chemical engineering research and design 9 0 ( 2 0 1 2 ) 615–621
References Abdel-Alim, A.H., Hamielec, A.E., 1975. A theoretical and experimental investigation of the effect of internal circulation on the drag of spherical droplets falling at terminal velocity in liquid media. Ind. Eng. Chem. Fundam. 14, 308–312. Baird, M.H.I., Hamielec, A.E., 1962. Forced convection transfer around spheres at intermediate Reynolds numbers. Can. J. Chem. Eng. 40, 119–121. Bird, R.B., Stewart, W.E., Lightfoot, E.N., 2001. Transport Phenomena. Wiley, New York, p. 530. Boltze, E., 1908. Grenzschichten an Rotationskörpen. PhD Thesis, University of Göttingen, Germany. Boussinesq, M.J., 1905. Calcul du pouvoir refroidissant des courants fluides. J. Math. Pure Appl. Ser. 6 1, 285–332. Brauer, H., 1971. Stoffaustausch einschliesslich chemischer Reaktionen. Verlag Sauerla¨nder, Aarau, Switzerland. Clift, R., Grace, J.R., Weber, M.E., 1978. Bubbles, Drops and Particles. Academic Press, New York. Friedlander, S.K., 1961. A note on transport to spheres in Stokes flow. AIChE J. 7, 347–348. Garner, F.H., Foord, A., Tayeban, M., 1959. Mass transfer from circulating drops. J. Appl. Chem. 9, 315–323. Garner, F.H., Tayeban, M., 1960. The importance of the wake in mass transfer from both continuous and dispersed phase systems. I. Ann. R. Soc. Esp. Fis. Quim. Ser. B 56, 479–490. Griffith, R.M., 1960. Mass transfer from drops and bubbles. Chem. Eng. Sci. 12, 198–213.
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Harper, J.F., Moore, D.W., 1968. The motion of a spherical drop at high Reynolds number. J. Fluid Mech. 32, 367–391. Jie, Y., Weiyang, F., 2005. Hydrodynamics and mass transfer in a pulsed packed column. Can. J. Chem. Eng. 78, 1040–1045. Korchinsky, W.J., 1994. In: Godfrey, S., Slater, M.J. (Eds.), Liquid–Liquid Extraction. Wiley, Chichester (Chapter 9). Kumar, A., Hartland, S., 1999. Correlation for prediction of mass transfer coefficients in single drop system and liquid–liquid extraction columns. Chem. Eng. Res. Des. 77 (Part A), 372–384. Lochiel, A.C., Calderbank, P.H., 1964. Mass transfer in the continuous phase around axisymmetric bodies of revolution. Chem. Eng. Sci. 19 (7), 471–484. Sandoval-Robles, J.G., Riba, J.P., Couderc, J.P., 1980. Mass transfer around a sphere. Trans. IChemE 58, 132–134. Seibert, A.F., Fair, J.R., 1988. Hydrodynamics and mass transfer in spray and packed liquid–liquid extraction columns. Ind. Eng. Chem. Res. 27, 470–481. Sun, C.G., Yin, F.H., Afacan, A., Nandakumar, K., Chaung, K.T., 2000. Modeling and simulation of flow maldistribution in random packed columns with gas–liquid countercurrent flow. Trans. IChemE 78 (Part A), 378–388. Thorsen, G., Terjesen, S.G., 1962. On the mechanism of mass transfer in liquid–liquid extraction. Chem. Eng. Sci. 17, 137–148. Ward, D.M., Trass, O., Johnson, A.I., 1962. Mass transfer from fluid and solid spheres at low Reynolds numbers. Part II. Can. J. Chem. Eng. 40, 164–168. Weber, M.E., 1975. Mass transfer from spherical drops at high Reynolds numbers. Ind. Eng. Chem. Fundam. 14, 365–366.
Contents lists available at SciVerse ScienceDirect
Chemical Engineering Research and Design journal homepage: www.elsevier.com/locate/cherd
New predictive correlation for mass transfer coefficient in structured packed extraction columns Ahmad Rahbar-Kelishami a,∗ , Hossein Bahmanyar b a b
Faculty of Chemical Engineering, Iran University of Science & Technology (IUST), Narmak, Tehran, Iran College of Engineering, School of Chemical Engineering,University of Tehran, Iran
a b s t r a c t Developments in the area of packed columns, particularly structured packed columns, are ongoing, specifically in the area of liquid–liquid extractions in different industries. In the present study, mass transfer coefficients have been obtained experimentally in a structured packed extraction column to develop a new correlation for prediction of continuous phase Sherwood number. The experiments were carried out for toluene/acetic acid/water and n-butyl acetate/acetic acid/water systems with counter current flow in different heights of column. A new dimensionless parameter, d32 /h, is introduced in proposed equation. This number considers the effect of column height (h) and mean drop diameter (d32 ) jointly. The main advantage of this approach is that the principal effect of column height is considered in correlation without which the experimental data could not be fitted with a acceptable accuracy. © 2011 The Institution of Chemical Engineers. Published by Elsevier B.V. All rights reserved. Keywords: Liquid–liquid extraction; Packed column; Mass transfer coefficient; Sherwood number
1.
Introduction
Packed columns have found increasingly wide use in separation processes such as distillation and absorption due to lower pressure drop and lower liquid hold-up than tray columns. However, the design, analysis and scale up of packed columns today are still based mainly on empirical or semi-empirical correlation. The main reason for this is that some of the performance characteristics such as mass transfer efficiency cannot be predicted reliably in packed columns (Sun et al., 2000). Liquid extraction consists of separating one or several substances (solute) present in a solid or a liquid phase by the addition of another liquid phase in which these substances transferred preferentially. This mass transfer is often operated in countercurrent extractors. The efficiency of liquid–liquid contactors is primarily dependent on the degree of turbulence imparted to the system and the interfacial area available for mass transfer. Although many researchers consider that extraction is a fully mature technology lacking in potential for further improvement, there are still many questions that need to be solved urgently. One of them is to design commercial extractors safely with low costs. Nowadays, the scale-up of extractors still depends on large quantities of pilot experiments, which is expensive and time-consuming. Introduction of Sherwood
∗
number by measuring mass transfer coefficients is a promising method to solve the above problem (Jie and Weiyang, 2005). The fundamental process for the rate of mass transfer in extraction columns is still not sufficiently well understood nor adequately modeled (Korchinsky, 1994). The accurate specification of the mass transfer coefficient plays an important role in packed columns precise design. Therefore understanding of relevant fluid dynamics and the mass transfer coefficients in these columns is of paramount importance for the precise design. These parameters can be presented in the form of Sherwood number. Packing cause internal circulation of drops and increase mass transfer coefficient. The transfer of a solute between a rising drop and a continuous liquid phase has been widely studied, both theoretically and experimentally. The importance of this area is reflected in the vast number of works published over the years and is of interest in a variety of industrial processes, such as liquid extraction (Kumar and Hartland, 1999).
2. Theoretical predictions of mass transfer coefficients The mass transfer coefficient of dispersed or continuous phase is one of the fundamental and essential parameters
Corresponding author. Tel.: +98 021 77451505; fax: +98 021 77240495. E-mail address: [email protected] (A. Rahbar-Kelishami). Received 30 April 2011; Received in revised form 10 July 2011; Accepted 2 September 2011 0263-8762/$ – see front matter © 2011 The Institution of Chemical Engineers. Published by Elsevier B.V. All rights reserved. doi:10.1016/j.cherd.2011.09.004
616
chemical engineering research and design 9 0 ( 2 0 1 2 ) 615–621
continuous-phase mass transfer coefficient (Friedlander, 1961; Baird and Hamielec, 1962; Lochiel and Calderbank, 1964):
Nomenclature C
solute concentration in dispersed phase (kmol m−3 ) equilibrium concentration of solute in dispersed phase (kmol m−3 ) initial concentration of solute in dispersed phase (kmol m−3 ) drop diameter (m) Sauter mean diameter of drop (m) diffusion coefficient (m2 /s) height of column (m) (getting sample) mass transfer coefficient based on dispersed phase (m s−1 ) mass transfer coefficient based on continuous phase (m s−1 ) contact time (s) slip velocity (m s−1 )
C* C0 d d32 D h Kd Kc t Vslip
1/3
Shc = 0.991 Pec
The upper limit of applicability of the above equation is approximately Pec = 1000, since Stokes-type flow does not occur above an Re of 1, and Scc is of the order of 1000 for liquids. Clift et al. give the following relationship for Shc valid for all Pec in Stokes regime (Clift et al., 1978): Shc = 1 + (1 + Pec )
Groups c V
d32
slip Reynolds number (Re = ) c c Schmidt number (Sc = c Dc ) continuous phase Sherwood number (Shc = Kc d32 Dc )
Re Sc Shc
in extraction column design. Mass transfer in the continuous phase around drops is generally affected by a combination of molecular diffusion and natural and forced convection in the continuous phase. The continuous-phase mass transfer coefficient, kc , will also depend upon whether the drop is internally stagnant or circulating. Much theoretical and experimental work has been done on mass transfer from single solid spheres, the results for which provide a useful basis for comparison with the data on circulating liquid drops. A variety of solutions for mass transfer has been obtained by using the equation of continuity for axisymmetric flow around a sphere, assuming constant physical properties. For a drop which is internally stagnant and when the continuous-phase velocity is zero, the expression for kc is: Kc = 2
Dc d
or Shc = 2
which is commonly ascribed to Langmuir. The above equation provides a limiting value of kc , any fluid motion will cause kc to be greater.
2.1.
1/3
(2)
For rigid spheres at intermediate Reynolds numbers, Brauer advanced an empirical correlation on the basis of results from numerical solution to diffusion equation (Brauer, 1971), which follows:
Shc = 2 +
Greek letters viscosity ratio (dispersed/continuous phase) continuous phase viscosity (kg m−1 s−1 ) c d dispersed phase viscosity (kg m−1 s−1 ) difference of dispersed and continuous phase viscosity (kg m−1 s−1 ) c continuous phase density (kg m−3 ) dispersed phase density (kg m−3 ) d difference of dispersed and continuous densities (kg m−3 ) interfacial tension (N m−1 )
(1)
0.66 Scc + 1 + Scc 2.4 + Scc
0.79 1/6
Scc
Pe1.7 c 1 + Pe1.2 c
(3)
for 1 < Re < 100. For high Reynolds numbers, Lochiel and Calderbank (1964) assumed a quadratic velocity profile in the momentum boundary layer and calculated the thickness of this layer from Boltze’s application of the Navier–Stokes equation to the boundary layer around a sphere (Boltze, 1908). This approach resulted in: 1/3
Shc = 0.7 Re1/2 Scc
(4)
Clift et al. (1978) have correlated the available experimental data in the form: 1/3
Shc = 1 + 0.724 Re0.48 Scc
(5)
for 100 < Re < 2000, and Scc > 200. Sandoval-Robles et al. (1980) have shown that how the exponent on Re changes with the value of Re from 1/3 for Stokes regime to 1/2 for boundary-layer theory, and to 2/3 for a surface renewal mechanism or turbulent conditions.
2.2.
Circulating drops
For an internally circulating sphere with high Pec (the thin concentration boundary-layer approximation), the continuousphase mass-transfer coefficient for creeping flow (Re < 1) is given by (Lochiel and Calderbank, 1964):
Shc = 0.65
Pec 1+
(6)
which was also derived by Griffith (1960), and Ward et al. (1962) with multiplying constants of 0.67 and 0.61, respectively. For Re → ∞, an asymptotic formula for Shc can easily be derived (Clift et al., 1978): 2 1/2 Shc = √ (Pec )
(7)
Stagnant drops
For Stokes regime (Re < 1), Friedlander, Baird & Hamielec, and Lochiel & Calderbank derive the following expression for the
which is the well-known Boussinesq (1905) equation. For finite Reynolds numbers, Weber (1975) used the boundary layer velocities of Harper and Moore (1968), and correlated
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his numerical results by the following equation, provided that ≤ 2 and d /c ≤ 4:
Pe 1/2
Shc = 2
c
1−
1/2
1 (2.89 + 2.150.64 ) Re1/2
(8)
According to Weber (1975), this equation should be accurate for Re > 70. In a similar fashion, Clift et al. (1978) used the surface velocities calculated by Abdel-Alim and Hamielec (1975) to obtain Shc at intermediate Reynolds numbers. The numerical results for k < 2 were correlated by:
Pe 1/2
Shc = 2
c
2 + 3/3(1 + )
1/2
n1 1/n1 × 1− 1 + (2 + 3)Re1/2 /(1 + )(8.67 + 6.450.64 )
Fig. 1 – Schematic diagram of the apparatus. (9)
In which n1 = 43 + 3 Garner et al. (1959) suggested the following equation for 8 < Re < 800: Shc = −126 + 1.8 Re0.5 Scc0.42
(10)
Garner and Tayeban (1960) also proposed another correlation for determining Shc : Shc = 0.6 Pe0.5
(11)
Another correlation was provided by Thorsen and Terjesen (1962) which is similar to Garner equation. 1/3
Shc = −178 + 3.6 Re0.5 Scc
(12)
This equation should be accurate for 50 < Re < 800. Brauer (1971) presented the following equation for 4 < Re < 1000 and 130 < Sc < 23600: Shc = 2 + 0.0511 Re0.724 Scc0.7
(13)
For packed column, Seibert and Fair57 proposed the following correlation (Seibert and Fair, 1988): 0.5
Shc = 0.698 Re
Scc0.4 (1 − ϕ)
3.
Experimental
3.1.
Column set-up
(14)
Column consisted of a Pyrex glass tube (inside diameter 9.1 cm, height 125 cm) (Fig. 1). The column was packed with two pieces of Penta-Pak TM PS-500M1 structured packing (height 20 cm) (Fig. 2). These packings were stacked on top of each other (total height 40 cm). Each packing consisted of corrugated sheets in a cylindrical structure with a diameter equal to the internal diameter of the column. The inclination of the corrugation was 45◦ . Several valves along the column were used to collect drops at different heights. The drops were collected and analyzed at heights of 11, 24.5 and 39 cm. The specific area of packing was 500 m2 /m3 and the void fraction was 0.94.
Fig. 2 – Applied Structure packing. Three nozzles (diameters 0.6, 0.8 and 1 mm) were used with both columns to make drops of different sizes.
3.2. Extraction system chemical and physical properties Two chemical systems were selected to examine a range interfacial tension values. The systems studied were toluene–acetic acid–water (high interfacial tension) (T/A/W) and n-butyl acetate–acetic acid–water (medium interfacial tension) (B/A/W). Distilled water saturated with toluene and butyl acetate was used as the continuous phase, and toluene and butyl acetate saturated with distilled water acetic acid (5 vol.%) was used as the dispersed phase. The physical properties of these systems are given in Table 1. Table 1 – Physical properties of systems at 20 ◦ C. Physical property
System 1 T/A/W
System 2 B/A/W
c (kg/m3 ) d (kg/m3 ) c (mPa s) d (mPa s) (mN/m) Dc (m2 /s) Dd (m2 /s)
1009.7 882.7 1.016 0.611 27.5–30.1 1.10 × 10−9 2.92 × 10−9
1010.2 895.9 1.013 0.684 12.4–13.2 1.03 × 10−9 2.66 × 10−9
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1200
600
1000
500 400
h=11 cm h=24.5 cm
600
Shc
Shc
800
h=11 cm h=24.5 cm
300
h=39 cm
h=39 cm
400
200
200
100
0 0.0
0.5
1.0
1.5
2.0
2.5
3.0
0 0.0
3.5
0.5
1.0
Qd/Qc
1.5
2.0
Qd/Qc
Fig. 3 – Effect of Qd /Qc and packing height on Shc , T/A/W.
Fig. 4 – Effect of Qd /Qc and packing height on Shc , B/A/W.
The viscosities of both systems were measured using a LAUDA viscometer. The densities were determined using a scale (0.0001 g graduations). The interfacial tension values were obtained from the literature. DM (mixture molecular diffusivity) were calculated using the Wilke–Chang correlation (Bird et al., 2001).
systems, three various height of packed column (11 cm,24.5 cm and 39 cm), different dispersed phase flow rate and different size of nozzles to producing different size of drops, resulted in 72 different runs.
Procedure
a
Before carrying out the experiments, both phases were mutually saturated, after which solute was added to the dispersed phase. Samples of each phase were taken at their inlets to the column and used for the determination of the initial solute concentration. In this section Mass transfer coefficients were calculated from experimental measurements. Considering the mass balance equation for a single drop: Kod =
d −
6t
1200 1000
Experimental Shc
3.3.
800 600 400 200
ln(1 − E)
(15) 0
0
20
40
where: (16)
100
120
Clift, Eq. 2 Lochiel & Calderbank, Eq. 4
and C0 , C and C* are the solute concentration in the primary drop (before contact), at a specified height and in equilibrium with the continuous phase, respectively. The solute concentrations of collected drops were measured by titration with normal NaOH. Sherwood number of continuous phase was calculated using following equation: Kc d32 Shc = Dc
80
Calculated Shc
(17)
In each experiment and for different heights of the column by measuring Acetic Acid concentration, mean diameter and slip velocity of droplets and the contact time between two phases, Shc was calculated considering Eq. (17). For the calculations, Sauter mean drop diameters (d) were determined by photographing drops inside the column. To take into account the lens effect of the column a background reference size used inside the column. The sizes of drops were measured during steady state conditions using a highresolution Powershot G9 type camera followed by analysis with Photoshop software. At least 200 drops were analyzed for each determination. Applying mentioned chemical
b
1200 1000
Experimental Shc
C0 − C E= C0 − C∗
60
800 600 400 200 0
0
20
40
60
80
100
120
140
Calculated Shc Clift, Eq.5
Brauer, Eq. 3
Fig. 5 – (a) Comparison of experimental data with stagnant drop models. (b) Comparison of experimental data with stagnant drop models.
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c
1200 1000
Experimental Shc
Experimental Shc
a
800 600
1200 1000
800 600
400
400
200
200
0
0
0
100
200
300
400
500
600
0
100
200
Calculated Shc Lochiel, Eq.6
b
Clift, Eq.7
Garner, Eq.10
d
1200 1000
400
500
600
Garner, Eq.11
1200
1000
Experimental Shc
Experimental Shc
300
Calculated Shc
800
600 400 200
800
600 400
200
0 0
100
200
300
400
500
600
Calculated Shc Webber, Eq.8
Clift, Eq.9
0 -100
0
100
200
300
400
500
600
Calculated Shc Thorsen, Eq.12
Brauer, Eq.13
Fig. 6 – (a) Comparison of experimental data with circulating drop models. (b) Comparison of experimental data with circulating drop models. (c) Comparison of experimental data with circulating drop models. (d) Comparison of experimental data with circulating drop models.
4.
Results and discussion
4.1. Effect of flow rate ratio and packing height on continuous phase Sherwood number Two main operating parameters found to affect mass transfer performance of the column were the ratio of phase flow rates and packing height. The effect of these parameters is shown in Figs. 3 and 4. It is easy to see that the continuous phase Sherwood number decreases lightly by increase of flow rate ratio. The reason could be that when the dispersed phase flow rate increases the mean drop diameter does not increase significantly, the larger drops result in improved mass transfer. It appears that the numbers of drops in the column and consequently the values of holdup and the resistance for mass transfer will increase. This leads to less amounts of Sherwood number. According to Figs. 3 and 4 the Sherwood number is decreasing while the height is increasing. This happens due to the fact
that by approaching to the end of the column the decrease in solute concentration results in less mass transfer rate and less values Sherwood numbers. Comparing the Sherwood numbers in two systems shows that these values are larger overall in first system with greater value of interfacial tension. This is due to more internal circulations in larger drops in first system.
4.2. Comparing of previous equations with experimental results As mentioned before the previously proposed correlations can be categorized according to drop internal situation into stagnant and circulating models. In this section the experimental data have been compared with the mentioned correlations in previous sections and then a new correlation for prediction of mass transfer coefficient in form of Sherwood number has been proposed.
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1200
1200
1000
1000
Experimental Shc
Experimental Shc
620
800 600 400
+15%
800 -15%
600 400
200 200
0 0
40
80
120
160
200
0
Calculated Shc
0
200
400
600
800
1000
1200
Shc Calculated by model
Seiber and Fair, Eq.14
Fig. 7 – Comparison of experimental data with Seibert and Fair model.
Fig. 8 – Comparison of experimental results with calculated values.
where h is the height of sampling point in packed column and:
4.2.1. Comparison of experimental data with stagnant drop models
Re =
Fig. 5 shows the comparison of calculated Sherwood numbers from proposed correlations with experimental ones. The average absolute error of these models was 82.7%. Since these models does not consider any circulation in drop and practically there are some movements in droplets so the predicted Sherwood numbers of these models are considerably lower than experimental ones.
4.2.2. Comparison of experimental data with circulating drop models Fig. 6 shows the comparison of calculated Sherwood numbers from proposed correlations with experimental ones. The average absolute error of these 8 models was 50.7%. Because these models consider internal circulation of drops they give better results than stagnant models and the calculated values are closer to the experimental ones. However, they do not predict Sherwood number precisely. Comparison of experimental Sherwood numbers with those obtained from Seibert and fair model which has been developed for packed column is presented in Fig. 7. This model has an average absolute error of 72%.
4.3.
Prediction of new correlation
Trying to obtain a new correlation which is able to predict Sherwood number precisely leads to introduction of a new dimensionless factor, d32 /h. This number considers the effect of column height (h) and mean drop diameter (d32 ) jointly. As discussed before mass transfer coefficient and Sherwood number increase with an increased in drop diameter and a decrease in height of packing. So this new parameter, d32 /h, can consider these two effects mutually without which the experimental data would not be fitted precisely anyway. The obtained equation for continuous phase Sherwood number for both systems and in different heights of packed column is:
Shc = −55.6 + 6.21 ∗ 10−5 Re0.433 Scc2.26
d 0.4 32
h
(18)
c Vslip d32 c
,
Sc =
c c Dc
The comparison of experimental results of Shc with those calculated by the proposed model is shown in Fig. 8. This figure indicates that the suggested equation can estimate the dispersed phase mass transfer coefficient with high accuracy. The dispersed phase mass transfer coefficient calculated with this model reproduces the experimental data with an average error of 13.62%. Thus the proposed correlation can predict the behavior of continuous phase mass transfer coefficients perfectly.
5.
Conclusions
Design procedures for packed extraction column still depend on having large amounts of experimental data which are difficult and expensive to obtain. The aim here was to try to reduce the experimental effort by proposing experimental correlations for prediction of mass transfer coefficient in form of continuous phase Sherwood number, Shc . The effects of two significant variables, flow ratio and height of packing, on Shc were investigated. The results showed that Shc decreased by increasing flow ratio and height of packing, h. Many correlations are published for prediction of Shc . Although many of them have a good theoretical basis, they do not practically have accurate results. Even though circulating drop models give better results than stagnant drop models as they consider internal circulation of drops, we can obtain the mass transfer coefficient with much better precision by considering the effect of packing height in Sherwood number equation. Because Shc is a dimensionless number a new dimensionless parameter, d32 /h, is introduced in proposed correlation. This novel conceptual parameter shows that Shc is increased with an increase of mean drop diameter which results in more internal circulation and consequently more mass transfer and also it presents that Shc decreases with an increase of packing height. It is for the first time that the effect of packing height is considered in Sherwood number. This correlation is recommended for calculation of Shc and consequently for final sizing of the column height.
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