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Liquid phase residence time distribution for gas–liquid flow in Kenics static mixer
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Chemical Engineering and Processing 47 (2008) 2275–2280
Liquid phase residence time distribution for gas–liquid flow in Kenics static mixer Tirupati Reddy Keshav 1 , P. Somaraju 1 , K. Kalyan, A.K. Saroha, K.D.P. Nigam ∗ Department of Chemical Engineering, Indian Institute of Technology-Delhi, New Delhi 110016, India Received 24 February 2007; received in revised form 31 October 2007; accepted 27 December 2007 Available online 18 January 2008
Abstract The residence time distribution (RTD) of the liquid phase for co-current gas–liquid upflow in a Kenics static mixer (KSM) with air/water and air/non-Newtonian fluid systems was investigated. The effect of liquid and gas superficial velocities on liquid holdup and Peclet number was studied. Experiments were conducted in three KSMs of diameter 2.54 cm with 16 elements and 5.08 cm diameter with 8 and 16 elements, respectively, of constant Le /De = 1.5 for different liquid and gas velocities. A correlation was developed for Peclet number, in terms of generalized liquid Reynolds number, gas Froude number and liquid Galileo number, where as for liquid holdup, a correlation was developed as a function of gas Reynolds number. The axial dispersion model was found to be in good agreement with the experimental data. © 2008 Elsevier B.V. All rights reserved. Keywords: Two-phase flow; Residence time distribution; Kenics static mixer; Axial dispersion model
1. Introduction The use of in-line static mixers has been widely advocated for a variety of applications in process industry such as continuous mixing, heat and mass transfer processes and chemical reactions. Table 1 shows a variety of studies which have been reported in the literature [1–29]. It can be seen from Table 1 that the static mixers not only find applications in single phase flow but also for two-phase flow studies. A close analysis of literature shows that considerable amount of work has been reported in the literature for single phase flow studies while limited number of studies are reported for two-phase flow studies. A detailed literature summary has been reported recently by Thakur et al. [30] on static mixers. It can be seen from the review [30] that considerable amount of work on residence time distribution (RTD) for single phase flow has been reported, where as information on RTD of liquid phase for two-phase flow in Kenics static mixers’ (KSMs)
∗
Corresponding author. Tel.: +91 11 2685 3748; fax: +91 11 2659 1020. E-mail addresses: [email protected] (T.R. Keshav), [email protected] (K.D.P. Nigam). 1 Research Group of Higher Dimensional Sciences & Technology (HDST), G.V.P. College of Engineering, Visakhapatnam 530041, A.P., India. 0255-2701/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.cep.2007.12.015
is practically very limited. The present study provides the experimental data on RTD of liquid phase in two-phase air/Newtonian and air/non-Newtonian systems. 2. Experimental Experiments were conducted to measure the RTD of the liquid phase in two-phase flow through KSM using a stepresponse technique. Air was used as the gas phase, and the liquid phase was water and aqueous solutions of carboxymethyl cellulose (CMC) of different concentrations. The CMC solutions were prepared by dissolving CMC powder in water at 60 ◦ C. Formaldehyde was added to prevent biological degradation of solutions (0.5 ml of formaldehyde per liter of CMC solution). The rheological behavior of CMC solutions were described by the power law model: τ = Kγ n
(1)
The rheological properties of the solutions were measured with a RV 20 HAAKE viscometer. Table 2 summarizes the power law parameters obtained using a linear regression technique. All the experiments were conducted at ambient conditions of temperature and atmospheric pressure. The range of the superfi-
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Table 1 Applications of static mixers for single and two-phase flows Application
Author(s)
1. Residence time distribution studies in static mixers for single phase flow 2. Flow characteristics and mixing performance of the Kenics static mixers
Tung [1], Nauman [2], Nigam and Vasudeva [3], Nigam and Nauman [4] Byrde et al[5], Hobbs and Muzzio[6], Rauline et al. (2000), Szalai and Muzzio [7], van Wageningen et al. [8] van Wageningen et al. [9] Tajima et al. [10] Streiff and Rogers [11] Middink et al. (1980) Khinast et al. (2003) Bourne and Maire [14] Van Der Meer and Hoogendoorn [15] Gerard et al. (1994) Belyaeva et al. [17] Liu et al. [18] Qi et al. [19]
3. Copper ion reduction using carbohydrates 4. Ocean disposal of CO2 5. Static mixers as bubble columns for hydration and for reactive distillation 6. Ultrafiltration of dairy liquids 7. Mass transfer enhancement by static mixers 8. Micromixing and fast chemical reactions in static mixers 9. Heat transfer coefficients for viscous fluids in a static mixer 10. Continuous production of hydroxypropyl starch in a static mixer reactor 11. Preparation of -carrageenan beads with the static mixer 12. Mixing of shear thinning fluids in a SMX static mixer 13. Enhancing heat transfer ability of drag reducing surfactant solutions with static mixers 14. Blending in static mixers 15. Gas–liquid flows in static mixers 16. Pressure drop correlations 17. Effect of static mixer on mass transfer in draft tube bubble column 18. Gas/liquid dispersions 19. Hydrodynamics and mass transfer studies 20. Droplet formation and drop breakage models
cial gas and liquid velocities investigated were 1.096–6.578 cm/s and 0.082–0.411 cm/s, respectively. The schematic diagram of the experimental set up is shown in Fig. 1. The set up consists of two reservoirs [1,2] containing liquid and tracer solutions, respectively. The liquid was pumped into the mixer and the switch of the feed from pure liquid reservoir to the reservoir with tracer solution was accomplished via a three-way stopcock S1 . The minimum possible distance was kept between the three-way stopcock (S1 ) and the inlet. Calibrated rotameters [5,6] were connected in the liquid lines to measure the liquid flow rates. Air, drawn from a compressor [9], passes through air saturation tank, pressure regulator, control valves and air rotameter [8] before entering the static mixer. The volume of liquid in the mixer was measured after each experimental run by simultaneously closing the inlet and outlet valves and draining the liquid through stopcock S2 . The values of the consistency index, K and the flow behavior index, n of each solution were frequently checked to ascertain that there is no change in the rheological properties of the solutions during the experiments. Step-response technique was used to measure the RTD. Congo-red dye in water or aqueous solution of CMC was used as the tracer. The dye concentration was kept low at 0.05 kg/m3 . Tracer samples were collected at the liquid outlet of gas–liquid separator [10], at regular time intervals and the tracer concentration was measured using a Shimadzu spectrophotometer at
Wadley and Dawson [20] Shah and Kale [21] Shah and Kale [22], Gokul Chandra et al. (1995), Lin and Fan [23] Goto and Gaspillo [24] Fradette et al. [25] Heyouni et al. [26], Ruivo et al. [27] Lemenand et al. [28], Das et al. [29]
a wavelength of 520 nm. A linear relationship between optical density and tracer concentration was experimentally verified which allows the direct evaluation of tracer concentration at the outlet using optical density measurements. A total of 180 step-response curves were obtained in three Kenics mixers of 2.54 cm diameter with 16 elements and 5.08 cm diameter with 16 and 8 elements, respectively, for different liquid and gas velocities. 3. Treatment of experimental data 3.1. Axial dispersion model The axial dispersion model was used to simulate the RTD data of liquid phase. The Peclet number was computed by minimizing
Table 2 Physical properties of liquids (28 ◦ C) Test liquid
Density (kg/m3 )
n
K (Pa sn )
0.5 g/l CMC in water 1.0 g/l CMC in water 2.0 g/l CMC in water Water
1010 1010 1020 1000
0.74 0.65 0.61 1.00
0.019 0.073 0.164 0.001
Fig. 1. Experimental set up. (1) Liquid storage tank, (2) tracer storage tank, (3, 4) centrifugal pumps, (5, 6) liquid rotameters, (7) Kenics static mixer, (8) air rotameter, (9) air compressor, (10) liquid outlet of gas–liquid separator, (11) gas–liquid separator.
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Fig. 2. Comparison of model predicted curve with experimentally measured RTD.
the square of the error between the experimental and fitted curve. A similar approach was used by Saxena et al. [31] for fitting the dispersion model to the liquid phase RTD for two-phase flow. The analytical solution used to fit the experimental data is given by Kramers and Westerterp [32]. The value of liquid holdup was calculated using the first moment of the curve. 3.2. Parameters evaluation The method used for the calculation of model parameters, Pe and ¯t , is based on the least square fit of the normalized experimental output response with the model predicted response. Marquardt-Levenberg algorithm was used to minimize the square error function (Fexp − Ftheor )2 . 3.3. Model comparison with experimental data The axial dispersionmodel was tested for all the 180 experimental RTD curves. |Fexp − Ftheor |θ represents the area between the theoretical and the experimental F-curves and is
Fig. 3. Variation of Peclet number with liquid velocity.
Fig. 4. Variation of Peclet number with gas velocity.
equal to twice the fraction of fluid that is assigned an incorrect residence time by the model. In the present study, the value of |Fexp − Ftheor |θ for all the 180 measured RTDs was less than 0.034. This means that <1.7% of the liquid phase has been assigned an incorrect residence time by the model. A typical model fit to the experimental data is shown in Fig. 2. There is an excellent agreement between the model and experimental data for all the 180 RTD curves. 4. Results and discussion Fig. 3 shows the variation of Peclet number with liquid velocity for a constant gas velocity. The results obtained from the two different diameter mixers indicate that Peclet number increases with increase in liquid velocity ul for both Newtonian and non-Newtonian liquids. The values of Peclet number with air/non-Newtonian systems are higher than those with air/Newtonian system. For all the gas–liquid systems studied in two different diameter mixers, it was found that the variation
Fig. 5. Comparison of Peclet number in two different diameter mixers.
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Fig. 6. Parity plot of experimental and calculated liquid holdup.
Fig. 7. Parity plot for Peclet number.
of Peclet number with increasing gas velocity is very small. No specific trend was observed. Fig. 4 shows the variation in Pe with gas velocity, ug . As the flow behavior index of the CMC solutions decreased (with increasing CMC concentration) the value of Peclet number increased over the range of liquid and gas velocities studied. This indicates that as the flow behavior index decreases (with increasing CMC concentration) the effect of buoyancy forces decreases. Fig. 5 shows the comparison of Peclet number in the two different diameter mixers. It was found that the values of Peclet number in 2.54 cm diameter mixer are higher than the values of Peclet number in 5.08 cm diameter mixer in the range of liquid and gas velocities studied. However, for the same value of mean residence time in the two mixers, Peclet number values are higher in 5.08 diameter mixer. Even though the values of Pe with 16 elements are higher than those with 8 elements, the difference is very low, as shown in Fig. 6. Liquid hold up, (Hl ) was found to be independent of liquid velocity (ul ) over the entire range of the gas and liquid velocities studied. However, it decreased with increase in gas velocity ug . Liquid holdup is the volume of liquid per unit volume of the
static mixer. Liquid holdup was calculated from model predicted mean residence time, ¯tmodel as follows: Hl =
¯tmodel Q Vm
(10)
Fig. 7 shows the comparison of experimental liquid holdup values and calculated liquid holdup values. 5. Correlations From the experiments carried out in this work, the following correlation describing the dependence of Peclet number on the generalized liquid Reynolds number, gas Froude number and Galileo number was developed: Pe = 39.79(Rel gen )0.164 (Frg )−0.015 (Gal )−0.175
(11)
A parity plot of Pe is shown in fig. 8. The confidence limit was ±25%.
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For liquid holdup the following correlation has been developed, in terms of gas Reynolds number: Hl = 1.131(Reg )−0.0614
References
The axial dispersion model represents the experimental RTD curves of the liquid phase in two-phase gas–liquid system. Peclet number increases with increase in liquid velocity for air/Newtonian and non-Newtonian systems. For air–CMC systems the Peclet number values are higher than those for air–water system. Peclet number increases with decrease in the value of power law index. With increase in Kenics mixer diameter, Peclet number decreases. A generalized correlation has been reported to predict the value of Peclet number and liquid holdup for two-phase flows in KSMs. Appendix A. Nomenclature
c C Co De Dz D F Frg g Gal Hl K Le L
dimensionless tracer concentration, C/Co tracer concentration (kmol/m3 ) feed tracer concentration (kmol/m3 ) diameter of Kenics mixing element (m) axial dispersion coefficient (m2 /s) diameter of the Kenics mixer (m) dimensionless tracer concentration, C/Co gas Froude number, ug 2 /g D acceleration due to gravity (m/s2 ) 2 2−n K −2 82n−2 liquid Galileo number, D2+n ρlg liquid holdup consistency index (Pa sn ) length of the Kenics mixing element (m) length of the mixer (m) i |Hlexp −Hlcalc | MARE mean absolute relative error, 1i Hl 1
¯t u ul ug Vl Vm X Z
exp
flow behavior index number of Kenics elements in a mixer Peclet number, uL/Dz volumetric flow rate of liquid (m3 /s) gas Reynolds number, ug Dρg /μg generalized liquid Reynolds number, 2−n 2−n n n−1 D ul ρl /K8 εl mean residence time of the liquid phase (s) actual liquid velocity (m/s) superficial liquid velocity (m/s) superficial gas velocity (m/s) volume of liquid in the mixer (m3 ) volume of the mixer (m3 ) dimensionless distance in axial direction, z/L distance in axial direction (m)
Greek letters γ shear rate (s−1 ) θ dimensionless time, t/¯t μl liquid viscosity (kg/m s)
liquid density (kg/m3 ) shear stress (Pa)
(12)
6. Conclusions
n N Pe Q Reg Relgen
ρl τ
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[1] T.T. Tung, Low Reynolds number entrance flow: a study of a motionless mixer, Ph.D. Thesis, University of Massachusetts, 1976. [2] E.B. Nauman, Reactions and residence time distributions in motionless mixers, Can. J. Chem. Eng. 60 (1982) 136–140. [3] K.D.P. Nigam, K. Vasudeva, Residence time distribution in static mixer, Can. J. Chem. Eng. 58 (1980) 543–544. [4] K.D.P. Nigam, E.B. Nauman, Residence time distributions of power fluids in motionless mixers, Can. J. Chem. Eng. 63 (1985) 519–521. [5] O. Byrde, M.L. Sawley, Optimization of a Kenics static mixer for noncreeping flow conditions, Chem. Eng. J. 72 (1999) 163–169. [6] D.M. Hobbs, F.J. Muzzio, Reynolds number effects on laminar mixing in the Kenics static mixer, Chem. Eng. J. 70 (1998) 93–104. [7] E.S. Szalai, F.J. Muzzio, Fundamental approach to the design and optimization of static mixers, AIChE J. 49 (2003) 2687–2699. [8] W.F.C. Van Wageningen, D. Kandhai, R.F. Mudde, H.E.A. van den Akker, Dynamic flow in a Kenics static mixer: an assessment of various CFD methods, AIChE J. 50 (2004) 1684–1696. [9] W.F.C. Van Wageningen, R.F. Mudde, H.E.A. van den Akker, Numerical simulation of growing Cu particles in a Kenics static mixer reactor in which Cu2+ is reduced by carbohydrates, Chem. Eng. Sci. 59 (2004) 5193– 5200. [10] Hideo Tajima, Akihiro Yamasaki, Fumio Kiyono, AIChE J. 50 (2004) 871–878. [11] F.A. Streiff, J.A. Rogers, Don’t overlook static-mixer reactors, Chem. Eng. Prog. (1994) 76–82. [12] J. Hiddink, D. Kloosterboer, S. Bruin, Evaluation of static mixers as convection promoters in the ultra filtration of dairy liquids, Desalination 35 (1980) 149–167. [13] G.K. Johannes, B. Alexander, B. David, P. Anthony, Mass-transfer enhancement by static mixers in a wall-coated catalytic reactor, Chem. Eng. Sci. 58 (2003) 1063–1070. [14] J.R. Bourne, H. Maire, Micromixing and fast chemical reactions in static mixers, Chem. Eng. Process. 30 (1991) 23–30. [15] T.H.H. Van Der Meer, C.J. Hoogendoorn, Heat transfer coefficients for viscous fluids in a static mixer, Chem. Eng. Sci. 33 (1978) 1277– 1282. [16] G. Lammers, A.A.C.M. Beenackers, Heat transfer and the continuous production of hydroxypropyl starch in a static mixer reactor, Chem. Eng. Sci. 49 (1994) 5097–5107. [17] E. Belyaeva, D. Della Valle, D. Poncelet, Immobilization of ␣chymotrypsin in -carrageenan beads prepared with the static mixer, Enzyme Microb. Technol. 34 (2004) 108–113. [18] S. Liu, A.N. Hrymak, P.E. Wood, Laminar mixing of shear thinning fluids in a SMX static mixer, Chem. Eng. Sci. 61 (2006) 1753–1759. [19] Y. Qi, Y. Kawaguchi, R.N. Christensen, J.L. Zakin, Enhancing heat transfer ability of drag reducing surfactant solutions with static mixers and honeycombs, Int. J. Heat Mass Transfer 46 (2003) 5161–5173. [20] R. Wadley, M.K. Dawson, LIF measurements of blending in static mixers in the turbulent and transitional flow regimes, Chem. Eng. Sci. 60 (2005) 2469–2478. [21] N.F. Shah, D.D. Kale, Two-phase, gas–liquid flows in static mixers, AIChE J. 38 (1992) 308–310. [22] N.F. Shah, D.D. Kale, Pressure drop for laminar flow of non-Newtonian fluids, Chem. Eng. Sci. 46 (1991) 2159–2161. [23] S.T. Lin, L.T. Fan, Pressure drop of two-phase flow through circular tubes with in-line static mixers accompanied by condensation—simple stochastic modeling of the data, Int. J. Heat Mass Transfer 24 (1981) 1851– 1853. [24] S. Goto, P.D. Gaspillo, The effect of static mixer on mass transfer in draft tube bubble column and in external loop column, Chem. Eng. Sci. 47 (1992) 3533–3539.
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[25] L. Fradette, H.-Z. Li, L. Choplin, P. Tanguy, Gas/liquid dispersions with a SMX static mixer in the laminar regime, Chem. Eng. Sci. 61 (2006) 3506–3518. [26] A. Heyouni, M. Roustan, Z. Do-Quang, Hydrodynamics and mass transfer in gas–liquid flow through static mixers, Chem. Eng. Sci. 57 (2002) 3325–3333. [27] R. Ruivo, A. Paiva, P.C. Simoes, Hydrodynamics and mass transfer of a static mixer at high pressure conditions, Chem. Eng. Proc. 45 (2006) 224–231. [28] T. Lemenand, D. Della Valle, Y. Zellouf, H. Peerhossaini, Droplets formation in turbulent mixing of two immiscible fluids in a new type of static mixer, Int. J. Multiphase Flow 29 (2003) 813–840.
[29] P.K. Das, J. Legrand, P. Moranc¸ais, G. Carnellea, Drop breakage model in static mixers at low and intermediate Reynolds number, Chem. Eng. Sci. 60 (2005) 231–238. [30] R.K. Thakur, Ch. Vial, K.D.P. Nigam, E.B. Nauman, G. Djelveh, Static mixers in the process industries—a review, Trans. IChemE 81 (2003) 787–826. [31] A.K. Saxena, K.D.P. Nigam, A. Schumpe, W.D. Deckwer, Liquid phase residence time distribution for two phase flow in coiled tubes, Can. J. Chem. Eng. 74 (1996) 861–866. [32] H. Kramers, K.R. Westerterp, Elements of Chemical Reactor Design and Operation, Academic Press, 1963, pp. 70–72.
Chemical Engineering and Processing 47 (2008) 2275–2280
Liquid phase residence time distribution for gas–liquid flow in Kenics static mixer Tirupati Reddy Keshav 1 , P. Somaraju 1 , K. Kalyan, A.K. Saroha, K.D.P. Nigam ∗ Department of Chemical Engineering, Indian Institute of Technology-Delhi, New Delhi 110016, India Received 24 February 2007; received in revised form 31 October 2007; accepted 27 December 2007 Available online 18 January 2008
Abstract The residence time distribution (RTD) of the liquid phase for co-current gas–liquid upflow in a Kenics static mixer (KSM) with air/water and air/non-Newtonian fluid systems was investigated. The effect of liquid and gas superficial velocities on liquid holdup and Peclet number was studied. Experiments were conducted in three KSMs of diameter 2.54 cm with 16 elements and 5.08 cm diameter with 8 and 16 elements, respectively, of constant Le /De = 1.5 for different liquid and gas velocities. A correlation was developed for Peclet number, in terms of generalized liquid Reynolds number, gas Froude number and liquid Galileo number, where as for liquid holdup, a correlation was developed as a function of gas Reynolds number. The axial dispersion model was found to be in good agreement with the experimental data. © 2008 Elsevier B.V. All rights reserved. Keywords: Two-phase flow; Residence time distribution; Kenics static mixer; Axial dispersion model
1. Introduction The use of in-line static mixers has been widely advocated for a variety of applications in process industry such as continuous mixing, heat and mass transfer processes and chemical reactions. Table 1 shows a variety of studies which have been reported in the literature [1–29]. It can be seen from Table 1 that the static mixers not only find applications in single phase flow but also for two-phase flow studies. A close analysis of literature shows that considerable amount of work has been reported in the literature for single phase flow studies while limited number of studies are reported for two-phase flow studies. A detailed literature summary has been reported recently by Thakur et al. [30] on static mixers. It can be seen from the review [30] that considerable amount of work on residence time distribution (RTD) for single phase flow has been reported, where as information on RTD of liquid phase for two-phase flow in Kenics static mixers’ (KSMs)
∗
Corresponding author. Tel.: +91 11 2685 3748; fax: +91 11 2659 1020. E-mail addresses: [email protected] (T.R. Keshav), [email protected] (K.D.P. Nigam). 1 Research Group of Higher Dimensional Sciences & Technology (HDST), G.V.P. College of Engineering, Visakhapatnam 530041, A.P., India. 0255-2701/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.cep.2007.12.015
is practically very limited. The present study provides the experimental data on RTD of liquid phase in two-phase air/Newtonian and air/non-Newtonian systems. 2. Experimental Experiments were conducted to measure the RTD of the liquid phase in two-phase flow through KSM using a stepresponse technique. Air was used as the gas phase, and the liquid phase was water and aqueous solutions of carboxymethyl cellulose (CMC) of different concentrations. The CMC solutions were prepared by dissolving CMC powder in water at 60 ◦ C. Formaldehyde was added to prevent biological degradation of solutions (0.5 ml of formaldehyde per liter of CMC solution). The rheological behavior of CMC solutions were described by the power law model: τ = Kγ n
(1)
The rheological properties of the solutions were measured with a RV 20 HAAKE viscometer. Table 2 summarizes the power law parameters obtained using a linear regression technique. All the experiments were conducted at ambient conditions of temperature and atmospheric pressure. The range of the superfi-
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Table 1 Applications of static mixers for single and two-phase flows Application
Author(s)
1. Residence time distribution studies in static mixers for single phase flow 2. Flow characteristics and mixing performance of the Kenics static mixers
Tung [1], Nauman [2], Nigam and Vasudeva [3], Nigam and Nauman [4] Byrde et al[5], Hobbs and Muzzio[6], Rauline et al. (2000), Szalai and Muzzio [7], van Wageningen et al. [8] van Wageningen et al. [9] Tajima et al. [10] Streiff and Rogers [11] Middink et al. (1980) Khinast et al. (2003) Bourne and Maire [14] Van Der Meer and Hoogendoorn [15] Gerard et al. (1994) Belyaeva et al. [17] Liu et al. [18] Qi et al. [19]
3. Copper ion reduction using carbohydrates 4. Ocean disposal of CO2 5. Static mixers as bubble columns for hydration and for reactive distillation 6. Ultrafiltration of dairy liquids 7. Mass transfer enhancement by static mixers 8. Micromixing and fast chemical reactions in static mixers 9. Heat transfer coefficients for viscous fluids in a static mixer 10. Continuous production of hydroxypropyl starch in a static mixer reactor 11. Preparation of -carrageenan beads with the static mixer 12. Mixing of shear thinning fluids in a SMX static mixer 13. Enhancing heat transfer ability of drag reducing surfactant solutions with static mixers 14. Blending in static mixers 15. Gas–liquid flows in static mixers 16. Pressure drop correlations 17. Effect of static mixer on mass transfer in draft tube bubble column 18. Gas/liquid dispersions 19. Hydrodynamics and mass transfer studies 20. Droplet formation and drop breakage models
cial gas and liquid velocities investigated were 1.096–6.578 cm/s and 0.082–0.411 cm/s, respectively. The schematic diagram of the experimental set up is shown in Fig. 1. The set up consists of two reservoirs [1,2] containing liquid and tracer solutions, respectively. The liquid was pumped into the mixer and the switch of the feed from pure liquid reservoir to the reservoir with tracer solution was accomplished via a three-way stopcock S1 . The minimum possible distance was kept between the three-way stopcock (S1 ) and the inlet. Calibrated rotameters [5,6] were connected in the liquid lines to measure the liquid flow rates. Air, drawn from a compressor [9], passes through air saturation tank, pressure regulator, control valves and air rotameter [8] before entering the static mixer. The volume of liquid in the mixer was measured after each experimental run by simultaneously closing the inlet and outlet valves and draining the liquid through stopcock S2 . The values of the consistency index, K and the flow behavior index, n of each solution were frequently checked to ascertain that there is no change in the rheological properties of the solutions during the experiments. Step-response technique was used to measure the RTD. Congo-red dye in water or aqueous solution of CMC was used as the tracer. The dye concentration was kept low at 0.05 kg/m3 . Tracer samples were collected at the liquid outlet of gas–liquid separator [10], at regular time intervals and the tracer concentration was measured using a Shimadzu spectrophotometer at
Wadley and Dawson [20] Shah and Kale [21] Shah and Kale [22], Gokul Chandra et al. (1995), Lin and Fan [23] Goto and Gaspillo [24] Fradette et al. [25] Heyouni et al. [26], Ruivo et al. [27] Lemenand et al. [28], Das et al. [29]
a wavelength of 520 nm. A linear relationship between optical density and tracer concentration was experimentally verified which allows the direct evaluation of tracer concentration at the outlet using optical density measurements. A total of 180 step-response curves were obtained in three Kenics mixers of 2.54 cm diameter with 16 elements and 5.08 cm diameter with 16 and 8 elements, respectively, for different liquid and gas velocities. 3. Treatment of experimental data 3.1. Axial dispersion model The axial dispersion model was used to simulate the RTD data of liquid phase. The Peclet number was computed by minimizing
Table 2 Physical properties of liquids (28 ◦ C) Test liquid
Density (kg/m3 )
n
K (Pa sn )
0.5 g/l CMC in water 1.0 g/l CMC in water 2.0 g/l CMC in water Water
1010 1010 1020 1000
0.74 0.65 0.61 1.00
0.019 0.073 0.164 0.001
Fig. 1. Experimental set up. (1) Liquid storage tank, (2) tracer storage tank, (3, 4) centrifugal pumps, (5, 6) liquid rotameters, (7) Kenics static mixer, (8) air rotameter, (9) air compressor, (10) liquid outlet of gas–liquid separator, (11) gas–liquid separator.
T.R. Keshav et al. / Chemical Engineering and Processing 47 (2008) 2275–2280
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Fig. 2. Comparison of model predicted curve with experimentally measured RTD.
the square of the error between the experimental and fitted curve. A similar approach was used by Saxena et al. [31] for fitting the dispersion model to the liquid phase RTD for two-phase flow. The analytical solution used to fit the experimental data is given by Kramers and Westerterp [32]. The value of liquid holdup was calculated using the first moment of the curve. 3.2. Parameters evaluation The method used for the calculation of model parameters, Pe and ¯t , is based on the least square fit of the normalized experimental output response with the model predicted response. Marquardt-Levenberg algorithm was used to minimize the square error function (Fexp − Ftheor )2 . 3.3. Model comparison with experimental data The axial dispersionmodel was tested for all the 180 experimental RTD curves. |Fexp − Ftheor |θ represents the area between the theoretical and the experimental F-curves and is
Fig. 3. Variation of Peclet number with liquid velocity.
Fig. 4. Variation of Peclet number with gas velocity.
equal to twice the fraction of fluid that is assigned an incorrect residence time by the model. In the present study, the value of |Fexp − Ftheor |θ for all the 180 measured RTDs was less than 0.034. This means that <1.7% of the liquid phase has been assigned an incorrect residence time by the model. A typical model fit to the experimental data is shown in Fig. 2. There is an excellent agreement between the model and experimental data for all the 180 RTD curves. 4. Results and discussion Fig. 3 shows the variation of Peclet number with liquid velocity for a constant gas velocity. The results obtained from the two different diameter mixers indicate that Peclet number increases with increase in liquid velocity ul for both Newtonian and non-Newtonian liquids. The values of Peclet number with air/non-Newtonian systems are higher than those with air/Newtonian system. For all the gas–liquid systems studied in two different diameter mixers, it was found that the variation
Fig. 5. Comparison of Peclet number in two different diameter mixers.
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Fig. 6. Parity plot of experimental and calculated liquid holdup.
Fig. 7. Parity plot for Peclet number.
of Peclet number with increasing gas velocity is very small. No specific trend was observed. Fig. 4 shows the variation in Pe with gas velocity, ug . As the flow behavior index of the CMC solutions decreased (with increasing CMC concentration) the value of Peclet number increased over the range of liquid and gas velocities studied. This indicates that as the flow behavior index decreases (with increasing CMC concentration) the effect of buoyancy forces decreases. Fig. 5 shows the comparison of Peclet number in the two different diameter mixers. It was found that the values of Peclet number in 2.54 cm diameter mixer are higher than the values of Peclet number in 5.08 cm diameter mixer in the range of liquid and gas velocities studied. However, for the same value of mean residence time in the two mixers, Peclet number values are higher in 5.08 diameter mixer. Even though the values of Pe with 16 elements are higher than those with 8 elements, the difference is very low, as shown in Fig. 6. Liquid hold up, (Hl ) was found to be independent of liquid velocity (ul ) over the entire range of the gas and liquid velocities studied. However, it decreased with increase in gas velocity ug . Liquid holdup is the volume of liquid per unit volume of the
static mixer. Liquid holdup was calculated from model predicted mean residence time, ¯tmodel as follows: Hl =
¯tmodel Q Vm
(10)
Fig. 7 shows the comparison of experimental liquid holdup values and calculated liquid holdup values. 5. Correlations From the experiments carried out in this work, the following correlation describing the dependence of Peclet number on the generalized liquid Reynolds number, gas Froude number and Galileo number was developed: Pe = 39.79(Rel gen )0.164 (Frg )−0.015 (Gal )−0.175
(11)
A parity plot of Pe is shown in fig. 8. The confidence limit was ±25%.
T.R. Keshav et al. / Chemical Engineering and Processing 47 (2008) 2275–2280
For liquid holdup the following correlation has been developed, in terms of gas Reynolds number: Hl = 1.131(Reg )−0.0614
References
The axial dispersion model represents the experimental RTD curves of the liquid phase in two-phase gas–liquid system. Peclet number increases with increase in liquid velocity for air/Newtonian and non-Newtonian systems. For air–CMC systems the Peclet number values are higher than those for air–water system. Peclet number increases with decrease in the value of power law index. With increase in Kenics mixer diameter, Peclet number decreases. A generalized correlation has been reported to predict the value of Peclet number and liquid holdup for two-phase flows in KSMs. Appendix A. Nomenclature
c C Co De Dz D F Frg g Gal Hl K Le L
dimensionless tracer concentration, C/Co tracer concentration (kmol/m3 ) feed tracer concentration (kmol/m3 ) diameter of Kenics mixing element (m) axial dispersion coefficient (m2 /s) diameter of the Kenics mixer (m) dimensionless tracer concentration, C/Co gas Froude number, ug 2 /g D acceleration due to gravity (m/s2 ) 2 2−n K −2 82n−2 liquid Galileo number, D2+n ρlg liquid holdup consistency index (Pa sn ) length of the Kenics mixing element (m) length of the mixer (m) i |Hlexp −Hlcalc | MARE mean absolute relative error, 1i Hl 1
¯t u ul ug Vl Vm X Z
exp
flow behavior index number of Kenics elements in a mixer Peclet number, uL/Dz volumetric flow rate of liquid (m3 /s) gas Reynolds number, ug Dρg /μg generalized liquid Reynolds number, 2−n 2−n n n−1 D ul ρl /K8 εl mean residence time of the liquid phase (s) actual liquid velocity (m/s) superficial liquid velocity (m/s) superficial gas velocity (m/s) volume of liquid in the mixer (m3 ) volume of the mixer (m3 ) dimensionless distance in axial direction, z/L distance in axial direction (m)
Greek letters γ shear rate (s−1 ) θ dimensionless time, t/¯t μl liquid viscosity (kg/m s)
liquid density (kg/m3 ) shear stress (Pa)
(12)
6. Conclusions
n N Pe Q Reg Relgen
ρl τ
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