Ind. Eng. Chem. Res. 2009, 48, 719–726
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Hydrodynamics in Sulzer SMX Static Mixer with Air/Water System Chandra Mouli R. Madhuranthakam, Qinmin Pan, and Garry L. Rempel* Department of Chemical Engineering, UniVersity of Waterloo, Waterloo, Ontario, N2L 3G1, Canada
A Sulzer static mixer with SMX internal structure is investigated for its performance with respect to flow behavior, gas hold-up, and mass transfer rates in an air/water system. The reactor consists of 18 such elements arranged in a line with the angle between successive elements being 90°. The void fraction in the reactor is 0.95. The length of the reactor is 90 cm, with an 6.3 cm internal diameter, and is operated cocurrently with vertical up-flow. The fluids used are air and water. In all the experiments performed in the static mixer, water is the continuous phase and air is the dispersed phase. Experiments are conducted with these fluids under laminar flow conditions. With 18 elements in the reactor, Peclet numbers of up to 100 were obtained in laminar flow regime, thus eliminating the parabolic velocity profile (which usually occurs in laminar flow) to a maximum extent. A gas hold-up of 15% was achieved at lower liquid flow rates and higher gas flow rates while mass transfer coefficients of up to 0.037 s-1 were achieved. Empirical correlations for Peclet number, gas hold-up, and overall mass transfer coefficient as a function of liquid-side and gas-side Reynolds numbers are obtained. From the experimental results, it was observed that a reactor with the Sulzer SMX internal structure would ensure plug flow behavior with enhanced radial mixing in addition to providing superior mass transfer coefficients and gas hold-up. Finally, the higher mass transfer coefficients achieved in the current study are compared to those obtained in a tube filled with Sulzer SMV packing and a dynamic mixer for identical power dissipation conditions. 1. Introduction Static mixers are tubular devices with static mixing elements aligned in a particular arrangement. There are many different mixer geometries for various purposes, and the performances of these are studied experimentally and theoretically.1 Static mixers were initially designed for mixing polymers of different viscosities. They are also used as reactors since they can ensure plug flow behavior and good mass and heat transfer. Other advantages include low maintenance cost, low space requirements, and easy installation. The performance and behavior of various types of static mixers are different based on the geometry of the internal structure.2-4 The various commercial static mixers are broadly classified into five categories viz., open designs with helices, open designs with blades, corrugated plates, multilayer designs, and closed designs with channels or holes.5 Knowledge about the residence time distribution in static mixers is very important when the mixer is used as a chemical reactor.6 Different residence time distribution (RTD) models for static mixers with helical elements have been proposed for both Newtonian and non-Newtonian flow.7-9 The most commonly used internal geometries include helical elements, open designs, and corrugated plates. These structures are highly efficient in producing large interfacial area when used for gas-liquid and liquid-liquid systems. Among these three types of structures, the open designs are used when very high viscous systems are encountered. The two important commercial open blade designs available and used for many applications in the chemical process industry are Sulzer SMX and Kenics KMX. The Sulzer SMX element has intricate open blade design with the blade being flat while the Kenics KMX design has identical structure except that the blades are concavely curved. With respect to mixing, Heniche et al.10 showed that the Kenics KMX structure is better than the Sulzer SMX at the expense of a higher pressure drop. Li et al.11 published an RTD model for flow of non-Newtonian fluids in static mixers with a Sulzer SMX internal structure. * To whom correspondence should be addressed. Tel.: (519)888456732702. Fax: (519)7464979. E-mail:
[email protected].
Considerably less literature is available on the hydrodynamic behavior in the static mixer (SM) with an open design structure when used for gas-liquid systems. Especially when high gas-liquid flow rates are to be used, a knowledge of the performance of the SM with respect to RTD, gas hold-up, mass transfer, pressure drop, and heat transfer is essential in order to design a real process. Furthermore, this information is also very important for scale up of the process. The gas hold-up in static mixers has not been accurately modeled in recent studies. Generally, the gas hold-up is calculated based on the “no slip velocity” condition which may only be true for limited cases.12 The volumetric mass transfer coefficient in a static mixer greatly depends on gas velocity and geometry of the internal element for a particular liquid velocity. The gas velocity further has a strong impact on the gas hold-up. Lakota et al.13 conducted experiments in an empty column and a column packed with Sulzer SMV elements for studying the mass transfer characteristics with an air-water system. They have examined the mass transfer characteristics over a wide range of gas and liquid velocities and used both a plug flow model and dispersion model to evaluate the volumetric mass transfer coefficient. In the present study, RTD, gas hold-up, and mass transfer characteristics in an SM with a Sulzer SMX internal structure are analyzed under high gas and liquid velocities. Empirical correlations are obtained for Peclet number (Pe), gas hold-up (εG), and the overall liquid-side mass transfer coefficient (KLa) in terms of gas- and liquid-side Reynolds numbers. 2. Experimental 2.1. Experimental Setup. The experimental setup consists of a static mixer with 18 elements, and the internal structure of the element has Sulzer SMX geometry. The reactor is configured vertically up, with the gas and liquid entering cocurrently at the bottom and leaving at the top. Figure 1 shows the reactor setup used for studying the RTD, hold-up, and mass transfer characteristics. The reactor is an
10.1021/ie801407y CCC: $40.75 2009 American Chemical Society Published on Web 12/09/2008
720 Ind. Eng. Chem. Res., Vol. 48, No. 2, 2009
Figure 1. Reactor setup used for performing RTD, hold-up, and mass transfer experiments.
Figure 2. Geometry of the Sulzer SMX element. (a) Front view. (b) Side view (90°). (c) Side view (45°).
acrylic pipe with a diameter of 6.3 cm and a length of 90 cm. The reactor is filled with the Sulzer SMX elements, and the geometry of the internal element is shown in Figure 2. The internal element is made from circular discs of diameter 6.2 cm and thickness 1 mm. The circular discs are cut and welded together to form an open blade structure (similar to Sulzer SMX) as shown in Figure 2. The corresponding void fraction was found to be equal to 0.95. These elements are arranged in the reactor so that each element is at an angle 90° with its neighboring element. The intricate net shaped internal structure splits a single flow into several partial flows and these partial flows in turn mix together after one complete rotation which is obtained after every four elements in this type of reactor. A direct coupled rotary vane pump (supplied by GE motors and Industrial Systems) is used to pump the deionized water through the reactor while gas (air or oxygen free nitrogen supplied by Praxair) is delivered from corresponding compressed gas cylinders at the desired pressure. Liquid and gas flow rates are set using corresponding rotameters. At the exit of the reactor, a conductivity probe (which in turn is attached to a conductivity meter supplied by VWR) is connected to measure the liquid conductivity in RTD experiments. Similarly, an oxygen probe (supplied by Cole-Parmer, Model 52, YSI) is connected to measure the dissolved oxygen concentration which is used to
evaluate the mass transfer coefficients for various gas liquid flow rates. These probes are in turn connected to a data acquisition system thus making the measurement online. 2.2. Methods. Residence time distribution is usually studied by injecting a tracer as an impulse input or step input and obtaining the outlet concentration of the tracer at the exit of the reactor. An impulse input (close to the ideal impulse input because the time of injection is 1 s which is so small compared to the space time of the reactor which varied from 430-1380 s) of the tracer is introduced by arranging the tip of the needle exactly at the entrance of the reactor. A 0.25 N sodium chloride (supplied by Fisher Scientific) solution is used as a tracer and deionized water is pumped through the reactor. The outlet concentration of the tracer is measured using a conductivity probe and read using a conductivity meter. The measured data is recorded online (at time intervals of 1 s) through the data acquisition system. For determining gas hold-up, deionized water and air were used as working fluids in the static mixer. Deionized water is sent through the reactor at a particular flow rate. The initial height of the reactor is noted as (hi). Air is also sent from the bottom of the reactor at a particular gas flow rate. Once the flow becomes steady, the valves at the entrance and exit are suddenly closed and the liquid is allowed to settle. Final height of the liquid in the reactor is noted as (hf). The ratio of the difference in the liquid levels (hi - hf) to the original height (hi) gives the gas hold-up (εG). In the present study, an empirical correlation relating the liquid side Reynolds number, gas-side Reynolds number, and the number of elements (which is the same as the L/D ratio) to the gas hold-up is derived. Mass transfer characteristics are studied in the static mixer reactor with eighteen elements by using water and air as working fluids. Initially, deionized water in a tank is completely purged of oxygen using oxygen free nitrogen for thirty minutes to fifty minutes so that it becomes deficient of oxygen. Then, it is pumped through the reactor at a particular flow rate. Once, the flow is steady, air is introduced at the entrance of the reactor at a particular flow rate. The oxygen from the air is transferred to the oxygen deficient deionized water. The oxygen potential in the deionized water at the exit of the reactor is measured using an oxygen probe (Model 52, YSI) connected to a data acquisition system. The gas phase (in the form of bubbles) always had a significant effect on the dispersion in the liquid phase which is the cause for the nonideal flow in the reactor. A one-dimensional plug flow with axial dispersion model (PFAD) superimposed with the mass transfer rate is derived by applying a differential mass balance in the reactor for evaluating the mass transfer coefficient (KLa). The differential mass balance on the oxygen concentration in the liquid phase is given by eq 1. ∂CO2 ∂t
) Da
∂2CO2 2
∂z
j -U
∂CO2 ∂z
+ KLa(CO/ 2 - CO2)
(1)
where, Da is the axial dispersion, KLa is the liquid-side overall mass transfer coefficient, CO2 and C/O2 are the oxygen concentration in the deionized water at any time and at equilibrium respectively. Equation 1 is converted to dimensionless form by scaling the time with space time (τ), length with reactor length (L), and oxygen concentration with the corresponding equilibrium value. The resulting dimensionless model for normalized oxygen concentration (ξ) is shown by eq 2.
Ind. Eng. Chem. Res., Vol. 48, No. 2, 2009 721 2
1 ∂ ξ ∂ξ ∂ξ ) + St(1 - ξ) ∂θ Pe ∂λ2 ∂λ CO2 jL z t L U with ξ ) / , Pe ) , λ , θ ) ) , St ) KLaτ j D L τ U CO2 a
(2)
In eq 2, St is the Stanton number from which KLa is determined. The Peclet number (Pe) for the flow conditions used in the mass transfer experiments is obtained from the RTD results which are discussed in the next section. Open-open boundary conditions (described by eq 6 in the next section where ψ is replaced by ξ) are used for solving eq 2. Equation 2 is discretized in the space using central differences for the first and second derivates. Thus, the original partial differential equation is converted to an ordinary differential equation (ODE). The resulting ODEs are solved using the stiff solvers in MATLAB with KLa as the parameter to be estimated. The liquid-side overall mass transfer coefficient (KLa) is estimated by nonlinear least-squares fitting obtained by minimizing the sum of squares of error (by using the lsqcurVefit routine in MATLAB optimization toolbox). The error corresponds to the difference between the experimental data and the data obtained from the proposed model (given by eq 2). Figure 3 shows a comparison of the normalized concentration computed using the proposed model and experimental data. It is clear that the experimental results could be satisfactorily fitted by using the proposed model with simultaneous estimation of KLa. 3. Results and Discussion Static mixers are designed for a particular process residence time when used for chemical reactions and mass transfer processes. Residence time distribution in static mixers greatly depends upon the geometry of the internal structure. Nigam et al.14 conducted experiments in a Kenics mixer (which has a helical type of internal structure) with water and a 70% diethylene glycol solution in water. They found that the RTD is narrower than those for laminar flow in empty straight tubes as well as those for low Reynolds number flow in helical coils. Later, Bayer et al.15 quantified the RTD in static mixers with any internal geometry with the Peclet number Pe, which is a measure of the width of residence time distribution according to the Danckwerts dispersion model.16 The RTD data thus
obtained from our reactor is analyzed by plotting the normalized exit age distribution. The experimental data are modeled using a plug flow with axial dispersion model with open-open boundary conditions.17 3.1. Plug Flow with Axial Dispersion (PFAD) Model. The plug flow with axial dispersion model is valid when there is flow where the radial dispersion is negligible and some dispersion occurs in the axial direction. This is the most widely used model for chemical reactors and other contacting devices. A balance on the molar flow rate, FT, for the tracer, T, in the liquid at concentration, CT, gives eq 3. ∂CT ∂2CT ∂CT j ) Da -U 2 ∂t ∂z ∂z
(3)
j is the where Da is the dispersion coefficient (m2 s-1), and U average velocity of the liquid. Equation 3 can be represented in dimensionless form by using appropriate scale factors and the resulting equation is represented by eq 4. 1 ∂2ψ ∂ψ ∂ψ ) ∂θ Pe ∂λ2 ∂λ CT jL t L U z , Pe ) , λ) , θ) , τ) with ψ ) j CTO Da L τ U
(4)
j L/Da, is a measure of convection The Peclet number defined as, U rate of transport to dispersion transport rate and is also referred to as Bodenstein number (especially in RTD experiments, because Da can be replaced with the diffusion in mass transfer). The Peclet number is the one dimensionless parameter that represents the PFAD model. In determining the Peclet number from the actual measurements of the reactor dispersion, several combinations of the boundary conditions are used based on the flow conditions used in the real system. The two most important types of the boundary conditions usually encountered in a chemical processes are closed-closed boundary condition and an open-open boundary condition. In the closed-closed system, the tracer before entering and after leaving the reactor will not have dispersion or diffusion and is assumed to have plug flow only. While, for the open-open system, the tracer at the entrance and the exit of the reactor has the same dispersion or diffusion as inside the reactor. These two boundary conditions are mathematically represented by the following equations: For a closed-closed system, At θ ) 0 and λ > 0 ψ(0+, 0) ) 0 CTO 1 ∂ψ +ψ) )1 At λ ) 0 Pe ∂λ CT ∂ψ )0 At λ ) 1 ∂λ
(5)
For an open-open system, At θ ) 0 and λ > 0 ψ(0+, 0) ) 0 1 ∂ψ 1 ∂ψ At λ ) 0 + ψ(0-, θ) ) + ψ(0+, θ) Pe ∂λ λ)0Pe ∂λ λ)0+ ψ(0-, θ) ) ψ(0+, θ) 1 ∂ψ 1 ∂ψ At λ ) 1 + ψ(1-, θ) ) + ψ(1+, θ) Pe ∂λ λ)1Pe ∂λ λ)1+ ψ(1-, θ) ) ψ(1+, θ) (6) Figure 3. Comparison of the normalized oxygen concentration computed using the proposed model and experimental results (UL ) 0.012 cm s-1, UG ) 0.166 cm s-1, ne ) 18, Pe ) 5.05).
|
|
|
|
The Peclet number in these two types of systems can be evaluated from the mean residence time (tm) and the variance (σ2) of the experimental residence time distribution (first and
722 Ind. Eng. Chem. Res., Vol. 48, No. 2, 2009 Table 1. Mean Residence Time and Peclet Number for Closed-Closed and Open-Open Systems18 system
mean residence time (tm)
equation for Peclet number
closed-closed
τ
2 σ2 2 ) (1 - e-Pe) tm2 Pe Pe2
(1 + Pe2 )τ
8 2 σ2 + ) tm2 Pe Pe2
open-open
second moments, respectively) modeled by solving eq 4 with appropriate boundary conditions. Table 1. shows the equations used to obtain the Peclet number for closed-closed and open-open systems. In the case of closed-closed systems, eq 4 has to be solved by using numerical methods while for the open-open systems an analytical solution is possible and is given by eq 7.17 E(θ) ) ψ(1, θ) )
1 2√πθ3/Pe
[
exp
-(1 - θ)2 4θ/Pe
]
(7)
Residence time distribution experiments are conducted by varying the liquid flow rate (which corresponds to NRe varying from 44 to 213) and the number of elements (6-18 in increments of 6). The RTD experimental results are modeled using eq 4. Raines et al.19 suggested that in packed beds whose length to diameter ratio is more than 20, the closed-closed and open-open boundary conditions give the same performance while the difference is significant when the Peclet number is less than two. Figure 4 shows a comparison of RTD between the experimental data and model for a static mixer with 18 elements. For this case, Pe ) 88, τ ) 4.7 min, and σ2 ) 0.07. Plug flow behavior can be achieved for higher values of Pe. For the same mean residence time, the variation of Pe with the number of elements was studied. Figure 5 shows the comparison of Peclet numbers obtained in the experimental results and the relation obtained by Himmler.20 The slope of the fit between the Peclet number and the number of elements is observed to be 4.45 and 4.0 in our experiments and in Himmler’s20 results, respectively, in the static mixer reactor with SMX internal geometry under
Figure 4. Residence time distribution in static mixer with 18 elements, Pe ) 88, τ ) 4.7 min, σ2 ) 0.07.
Figure 5. Peclet number vs number of elements in the SM at laminar flow (tmean ) 4.7 min).
laminar flow conditions. The Peclet number in the reactor is approximately 4.45 times the number of internal elements. This is an important relation that is required to roughly design a reactor for a particular mean residence time. The significance of this relation is important even to compare the performances of different reactors. For example, to achieve a Peclet number of 100, a static mixer with 22 elements is required, while to achieve the same residence time distribution in continuously stirred tank reactors, a cascade of 50 (NC ∼ Pe/2) reactors in series is required. 3.2. RTD in Air/Water Systems. The RTD experiments with the air/water system were carried out using the same technique used for the water system alone. After the gas and liquid flows are set and become steady, the tracer is injected at the inlet of the reactor. The corresponding tracer concentration at the outlet was measured using the method described in the previous section. The experiments were carried out with varying numbers of internal elements (6, 12, and 18). The experimental results showed that increase in the gas flow rate resulted in the decrease of Peclet number. With the increase in the liquid flow rate, the Peclet number increased which is more obvious because the rate of convection would be more than the rate of dispersion. But, with the introduction of air, the reactor’s flow behavior approached that of a CSTR at high gas flow rates and low liquid flow rates. Figure 6 shows the typical RTD obtained in SM reactor with air/water system with UL ) 0.315 cm s-1, UG ) 0.535 cm s-1, and 18 elements and the fitted plug flow with axial dispersion model. The dispersion model could satisfactorily predict the experimental RTD in the SM reactor with very little deviation. The RTD curve is not smooth in some places because of the gas bubbles hitting the probe and then collapsing. This could possibly effect the calculation of corresponding Peclet number (which is obtained from the variance and mean residence time of the corresponding RTD). Figure 7 shows the comparison between the RTD obtained in the SM reactor with water and air/water systems. In Figure 7, with the water system alone for UL ) 0.315 cm s-1, the Peclet number in the reactor was found to be 32.26 while with the air/water system at the same liquid flow rate and at an air flow rate of UG ) 0.535 cm s-1; the Peclet number was 6.21. For this typical case, the Peclet number in the air/water system had decreased by five times of that of water system alone. Figure 8 shows the effect of gas/liquid
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Figure 6. Comparison of the experimental RTD and the fitted dispersion model for air/water system with UL ) 0.315 cm s-1, UG ) 0.535 cm s-1, and 18 elements (tm ) 4.2 min, σθ2 ) 0.529). Figure 8. Effect of gas and liquid Reynolds numbers on Peclet number in air/water system. Table 2. Parameter Values and the Confidence Intervals Corresponding to Equation 8 system
β1
β2
β3
R2
water + air 2.9080 0.1636 0.0533 0.81 water
Figure 7. Comparison of RTD between liquid alone and gas/liquid system with UL ) 0.315 cm s-1, UG ) 0.535 cm s-1, and 18 elements.
Reynolds numbers on the Peclet number achieved in the SM reactor with air/water system. Because of the small range of the gas and liquid flow rates, a clear trend was not observed with respect to the effect of the gas side Reynolds numbers on the Peclet number. It was observed that there was an increase in the Peclet number with an increase in the gas flow while in some cases it decreased also. An increase in the gas flow rate would decrease the liquid hold-up in the reactor and hence, the j /(1 - εG)) increases. Hence, the superficial liquid velocity (U rate of convection (the velocity term in the definition of Peclet number) would increase while the axial dispersion decreases which results in achieving higher Peclet numbers. An empirical correlation for Peclet number, as a function of the gas- and liquid-side Reynolds numbers is derived and the correlation is given by eq 8. Pe ) β1ReLβ2ReGβ3
(8)
where β1,β2, and β3 are constants, ReL and ReG are Reynolds numbers on the liquid- and gas-side, respectively. The values of the parameters and the corresponding confidence intervals are shown in Table 2. The parameter corresponding to ReG shows that with an increase in the gas flow rate the Peclet
1.7852 0.8531 0
0.96
95% confidence interval on the parameters 1.5973 0.1154 -0.0625 -1.8860 0.1890
4.2187 0.2118 0.1691 5.4565 1.5172
number increases but does not exhibit such a significant effect as that of the effect ReL has on the Peclet number. Further investigation at low liquid flow rates could be advantageous in order to study the effect of these individual phase flows on the Peclet number. 3.3. Gas Hold-up. Gas hold-up is an important design factor to be considered in designing a static mixer for two phase operation. The choice of the gas and liquid flow rates was based on specified space time and gas hold-up. Little data related to gas hold-up in static mixers is available in the literature. Couvert et al.21 carried out experiments in a static mixer with Sulzer SMX which has an open blade internal structure. In their experiments, they used gas as the continuous phase and liquid as the distributed phase. They estimated the gas hold-up to be the ratio of gas flow rate to total flow rate. Heyouni et al.12 also used the same method for the estimation of gas hold-up in Lightnin static mixers where liquid is the continuous phase. This method of estimating gas hold up involves usage of “no slip velocity” in a homogeneous flow regime where the relative velocity between the gas and the liquid phases (which is called the slip velocity) is zero. The same method of estimating the gas hold-up in our system led to large errors. The probable reason for that comes from the fact that the “no slip velocity” condition cannot be used in a system where the gas and the liquid phases have large velocity differences. In the current study, the gas and liquid velocities correspond to both the laminar regime and turbulent regime. In SM reactors, ReL-h < 20 corresponds to laminar regime where, ReL-h is the Reynolds number based on hydraulic mean diameter. The hydraulic mean diameter of the SMX element used in the present study is 2.2 cm. Experiments were carried out with the static mixer by varying gas and liquid flow rates and also the number of elements in the reactor. Gas hold-up data obtained from these
724 Ind. Eng. Chem. Res., Vol. 48, No. 2, 2009
Figure 9. Gas hold-up vs gas Reynolds number for different liquid Reynolds number in a static mixer with eighteen elements.
Figure 10. Gas hold-up data obtained from model (eq 9) vs experimental hold-up.
types of reactors can be correlated using one of the two important types proposed in the literature. The first type comprises correlating the gas hold-up directly to either the gas velocity or the liquid velocity.22 The second type correlates the gas hold-up with dimensionless numbers such as a gas side Reynolds number or a liquid side Reynolds number. Figure 9 shows the effect of the gas and liquid Reynolds numbers on gas hold-up in our system. The experimental gas hold-up data obtained by varying gas and liquid flow rates and also the number of elements (6, 12, and 18) could be correlated by using eq 9. εG ) ANReLRNReGβ exp(nγ)
(9)
where n is the number of elements, NReL and NReG are Reynolds numbers based on the liquid side and gas side, respectively. The parameters in the above equation are given by A ) 2.12 × 10-5; R ) -0.06; β ) 1.18; γ ) 0.01. The negative power on the liquid side Reynolds number and the positive power on the gas side Reynolds number are in agreement with the fact that the gas hold-up decreases with the an increase in liquid flow rate and it increases with an increase in gas flow rate. It also indicates that the gas hold-up is more affected by the gas flow rate rather than by the liquid flow rate. The parameter γ also has the same effect as the parameter β, suggesting that with an increase in the number of elements, the gas hold up increases though it does not have a stronger influence compared to the effect of parameter β. Also, the number of elements is indirectly related to the aspect ratio (length to diameter) of the reactor which is the ratio of the length to corresponding diameter of the reactor. Figure 10 shows the validity of the model with respect to the experimental data. 3.4. Volumetric Mass Transfer Coefficient (KLa). The liquid-side mass transfer coefficient obtained from experiments for various gas and liquid flow rates is shown in Figure 11. The effect of gas and liquid flow rates on the mass transfer coefficient is correlated using eq 10. KLa ) ANReLRNReGβ
(10)
the constants being A ) 0.003 s-1; R ) 0.099; β ) 0.307. The values obtained for the parameters in eq 10 show that the gas side Reynolds number has more impact than the liquid Reynolds number. For a given liquid flow rate, an increase in
Figure 11. Effect of gas and liquid Reynolds numbers on KLa in a static mixer with 18 elements.
the gas flow rate increases the interfacial area for mass transfer, thus increasing the mass transfer coefficient. The coefficient A and the parameter R depend on the viscosity, surface tension, density, and density difference of the working fluids. Experimental results show that trend for the effect of mass transfer coefficient with respect to gas/liquid Reynolds numbers is similar to that obtained by Lakota et al.13 except that the geometry of the internal element is different. Also, the coefficients in eq 10 depend on the regime in which gas and liquid flow rates are operated. Figure 12 shows a comparison between the empirical model used to estimate KLa and experimental mass transfer coefficient. For higher gas flow rates and lower liquid flow rates, all positive coefficients are reported when gas is used as the continuous phase.21 Also, they obtained almost equal parameters for the gas and liquid side superficial velocities. Heyouni et al.12 reported high KLa values in a different geometry static mixer where they varied UL from 70-130 cm s-1 and UG from 1.6-43.7 cm s-1. In this range (turbulent regime) of gas and liquid superficial velocities, they achieved KLa in the range of 0.1-2.4 s-1. Also, in their case, a strong dependence of KLa on the liquid velocity was observed. Higher volumetric mass transfer coefficients are obtained in their case because of very high gas and liquid velocities. In the present study, the mass transfer characteristics
Ind. Eng. Chem. Res., Vol. 48, No. 2, 2009 725
Figure 12. Comparison of the experimental volumetric mass transfer coefficient with the theoretical KLa estimated using eq 10 (R2 ) 0.92).
at high gas velocities and low liquid velocities were studied. The velocities, UL, were varied from 0.06 to 0.33 cm s-1, and UG was varied from 0.56 to 1.047 cm s-1. Despite the low liquid velocities, higher volumetric mass transfer coefficients (0.0270.038 s-1) are achieved. Owing to the moderate dependence of UL on the mass transfer, KLa (s-1) can be dimensionally correlated to UG (cm s-1) alone according to eq 11. KLa ) AUGβ
(11)
where the constants, A ) 0.737 and β ) 0.92, with R2 ) 0.9. With a similar system (air/water), Lakota et al.13 studied mass transfer characteristics in an empty column and a column packed with Sulzer SMV elements (UL in the range of 0.3-5.4 cm s-1 and UG in the range of 0.91-9 cm s-1). They also found that the effect of superficial gas velocity on KLa is high compared to the superficial liquid velocity. The coefficient A in their study was equal to 0.0184 and the parameter β ) 0.86. The volumetric mass transfer coefficients obtained in our study are higher than those reported by Lakota et al.13 for the same system while the internal structure of the static mixer is different. The performance of the static mixer with Sulzer SMX internal structure with respect to mass transfer is compared with dynamic mixer and SM with Sulzer SMV internal structure for same energy dissipation condition. For a typical dynamic mixer which has a Rushton turbine, with air-water system, the mass transfer coefficient is correlated with the superficial gas velocity and the power dissipated, which is represented by eq 12.23 KLa ) 2.148 × 10-3(Ptot/VL)1.13Ug0.627Po0.287
(12)
where Ptot is the total energy consumption, Po is the power number (for Rushton turbine Po ) 5). The power dissipation (ε˙ ) in the static mixer is calculated by using eq 13. ε˙ ) ∆PG/L
( ) QL VL
(13)
where ∆PG/L is the gas-liquid pressure drop, QL is the liquid volumetric flow rate, and VL is the volume of liquid. The gas-liquid pressure drop in SM reactors is estimated using the method proposed by Lockhart and Martinelli24 where the gas-liquid pressure drop is calculated from the individual pressure drops on the gas and liquid sides. The gas liquid
Figure 13. Comparison of the KLa values obtained from the model, packed column and dynamic mixer (eq 11) with the experimental KLa obtained in a static mixer with Sulzer SMX internal structure.
pressure drop thus obtained is used to calculate the power dissipation using eq 13. Figure 13 shows a comparison of mass transfer coefficients obtained in the static mixer with open blade internal structure with a column packed with SMV elements and a dynamic mixer with a Rushton turbine for the same power dissipation and superficial gas velocity. It shows that the mass transfer coefficients obtained in the static mixer are superior to those for the dynamic mixer and a column packed with SMV elements under identical operating conditions. 4. Concluding Remarks The residence time distribution in a Sulzer SMX static mixer (with open blade internal structure) is modeled using a PFAD model and shows that plug flow behavior can be obtained for an increased number of internal elements for a given mean residence time. A functional relationship is achieved between the Peclet number and the number of internal elements and is compared with the value reported in the literature. The Peclet number for a air/water system is studied for at very high gas and liquid flow rates. An empirical correlation for the Peclet number as a function of gas and liquid side Reynolds number was derived. An empirical model is derived for calculating the gas hold-up (air/water system) from the corresponding gas and liquid side Reynolds numbers. Mass transfer coefficients were evaluated by fitting the PFAD model (superimposed with rate of mass transfer) to the experimental data. Empirical correlation for mass transfer coefficient (KLa) as a function of the gas/ liquid side Reynolds numbers was derived. KLa values in the SM reactor with Sulzer SMX internal geometry are found to be high compared to other conventional reactors (column packed with Sulzer SMV internal structure, dynamic mixer) under identical energy dissipation conditions. Acknowledgment Support from the Natural Sciences and Engineering Research Council of Canada (NSERC) and LANXESS Inc. is gratefully acknowledged. Nomenclature A ) constant Bo ) Bodenstein number (UL/D) C ) oxygen concentration (mg/L)
726 Ind. Eng. Chem. Res., Vol. 48, No. 2, 2009 C* ) equilibrium concentration (mg/L) C0 ) initial concentration (mg/L) D ) diameter of the reactor (m) Dh ) hydraulic mean diameter (m) Da ) dispersion coefficient (m2/s) E(θ) ) normalized mean residence time hi ) initial height of the column (m) ho ) final height of the column (m) KLa ) liquid side mass transfer coefficient (s-1) L ) length of the reactor (m) n ) number of elements NC ) number of CSTRs NReL ) liquid-side Reynolds number (DUF/µ)|Liquid NReL-h ) hydraulic liquid-side Reynolds number (DhUF/µ)|Liquid NReG ) gas-side Reynolds number (DUF/µ)|Gas Ptot ) total energy consumption (W) Po ) power number Pe ) peclet number (Da/UL) ∆PG/L ) gas-liquid pressure drop (N m-2) ∆PG ) gas phase pressure drop (N m-2) ∆PL ) liquid phase pressure drop (N m-2) QL ) volumetric flow rate of liquid (m3 s-1) t ) time (s) UL ) velocity of liquid (m s-1) Ug ) superficial gas velocity (m s-1) VL ) volume of liquid (m3) X ) square root of the ratio of liquid to gas phase pressure drops Greek Symbols θ ) dimensionless time (t/τ) σ ) standard deviation τ ) mean residence time (s) ε˙ ) power dissipated (W/m-3) εG ) gas hold-up φL ) liquid phase correction factor φG ) gas phase correction factor R, β, γ ) constants AbbreViations CSTR ) continuous stirred tank reactor PFR ) plug flow reactor PFAD ) plug flow with axial dispersion RTD ) Residence Time Distribution SM ) static mixer
Literature Cited (1) Visser, J. E.; Rozendal, P. F.; Hoogstraten, H. W.; Beenackers, A. C. M. Three-Dimensional Numerical Simulation of Flow and Heat Transfer in the Sulzer SMX static mixer. Chem. Eng. Sci. 1999, 54, 2491– 2500. (2) Baker, J. R. Motionless mixers stir up new uses. Chem. Eng. Prog. 1991, 87, 32–38. (3) Cybulski, A.; Werner, K. Static mixers-criteria for applications and selection. Int. Chem. Eng. 1986, 26, 171–180.
(4) Myers, K. J.; Bakker, A.; Ryan, D. Avoid agitation by selecting static mixers. Chem. Eng. Progr. 1997, (June), 28–38. (5) Thakur, K. J.; Vial, Ch.; Nigam, K. D. P.; Nauman, E. B.; Djelveh, G. Static Mixers in the Process Industries-A Review. Trans. Inst. Chem. Eng. Part A 2003, 81, 787–826. (6) Streiff, F. A.; Rogers, J. A. Don’t Overlook Static Mixer Reactors. Chem. Eng. 1994, June, 76–82. (7) Nigam, K. D. P.; Naumann, E. B. Residence time distribution of power law fluids in motionless mixers. Can. J. Chem. Eng. 1985, 63, 519– 521. (8) Pustelnik, P. Investigation of Residence Time Distribution in Kenics Static Mixers. Chem. Eng. Process. 1986, 20, 147–154. (9) Hobbs, D. M.; Muzzio, F. J. The Kenics Static Mixer: a three dimensional chaotic flow. Chem. Eng. J. 1997, 67, 153–166. (10) Heniche, M.; Tanguy, P. A.; Reeder, M. F.; Fasano, J. B. Numerical Investigation of Blade Shape in Static Mixing. Am. Inst. Chem. Eng. J. 2005, 51 (1), 44–58. (11) Li, H. Z.; Fasol, C.; Choplin, L. Residence time distribution of rheologically complex fluids passing through a Sulzer SMX static mixer. Chem. Eng. Commun. 1998, 165, 1–15. (12) Heyouni, A.; Roustan, M.; Do-Quang, Z. Hydrodynamics and Mass Transfer in Gas-Liquid Flow Through Static Mixers. Chem. Eng. Sci. 2002, 57, 3325–3333. (13) Lakota, A.; Jazbec, M.; Levec, J. Impact of Structured Packing on Bubble Column Mass Transfer Characteristics. Part 2. Analysis of GasLiquid Mass Transfer Measurements. Acta Chim. SloVinica 2002, 49, 587– 604. (14) Nigam, K. D. P.; Vasudeva, K. Residence Time Distribution in Static Mixer. Can. J. Chem. Eng. 1980, 58, 543–544. (15) Bayer, T.; Himmler, K.; Hessel, V. Don’t be Baffled by Static Mixers. Chem. Eng. 2003, 5, 50–57. (16) Danckwerts, P. V. Continuous Flow Systems-Distribution of Residence Time. Chem. Eng. Sci. 1953, 2, 1–3. (17) Froment, G. F.; Bischoff, K. B. Chemical Reactor Analysis and Design, 2nd ed.; Wiley Publisher: New York, 1990; pp 529-533. (18) Scott Fogler, H. Elements of Chemical Reaction Engineering, 2nd ed.; Prentice Hall PTR: Saddle River, NJ, 1992; pp 769-773. (19) Raines, G. E.; Corrigan, R. E. The use of axial dispersion model to predict conversion of first and second order reactions. Chem. Eng. Progr. Symp. Ser. 1967, 72, 90–96. (20) Himmler, K. In-line dispergierung und stoffaustauch mit statischen mischern. Tech. Rundschau Sulzer 1977, 3. (21) Couvert, A.; Peculier, M.; Laplanche, A. Pressure Drop and Mass Transfer Study in Static Mixers with Gas Continuous Phase. Can. J. Chem. Eng. 2002, 80, 727–733. (22) Thanos, A. M.; Galtier, P. A.; Papyannakos, N. G. Liquid Flow Non-Idealities and Hold-up in a Pilot Scale Packed Bed Reactor with Cocurrent Gas-Liquid Upflow. Chem. Eng. Sci. 1996, 51, 2709–2714. (23) Moucha, T.; Linek, V.; Prokopova, E. Gas hold-up, mixing time and gas-liquid volumetric mass transfer coefficient of various multipleimpeller configurations: Rushton turbine, pitched blade and techmix impeller and their combinations. Chem. Eng. Sci. 2003, 58, 1839–1846. (24) Lockart, R. W.; Martinelli, R. C. Proposed correlation of data for two phase, two-component flow in pipes. Chem. Eng. Progr. 1949, 45, 39– 48.
ReceiVed for reView September 18, 2008 ReVised manuscript receiVed October 24, 2008 Accepted November 4, 2008 IE801407Y
