Comparative Study of Different Mathematical Models of OzoneMass Transfer in a Kenics Static Mixer

Mohamed Saad*

Chemical Engineering Department, Sirte University, Sirte, Libya* Corresponding Author: mohammed.saad37@yahoo.com

Abstract

Static mixers have been successfully employed in water and wastewater treatment, particularly in waterozonation for disinfection and oxidation purposes. Producing higher concentration of ozone requires newcontactors that operate efficiently at low gas/liquid ratio. The Kenics static mixer (KM) can meet theserequirements and therefore enhance the ozone mass transfer rate [1]. The main objective is to investigatethe correspondence between the transient BFCM of Romer and Durbin [2] and the axial diffusion model(ADM) and the continuous flow stirred tanks in series(CFSTR’s in series) in their prediction to the resi-dence time distribution of the gaseous solute inside a Kenics static mixer. The residences time distribution(RTD) curves produced from the experiment of Madhuranthaam et al. [3] have been used to validate andcompare the predicted RTD curves of the three models. The BFCM model provides accurate, reliable,flexible and easy design model to describe the back-mixing in the liquid phase inside the Kenics staticmixer.

Keywords: Static mixer; Back flow cell model; Axial diffusion model; Water ozonation.

1. Introduction

A lot of processes in the field of chemical engineer-ing are based on the chemical reaction between thesolute gas and soluble species present in liquid. Innature, gas diffuses into the liquid through gas–liq-uid interface due to the concentration difference ofthe gas between the liquid at the surface and theliquid bulk without mechanical energy. But, the gasmass transfer rate is low and the reaction rate is alsolow. Therefore, gas-liquid contactors such as staticmixers are used to obtain a higher gas mass transferrate and correspondingly faster reaction. These con-tactors enhance the mass transfer rate by increasingsurface area between the gas and liquid and increas-ing the intensity of turbulence [4, 5]. The ozonationprocess is practiced by dissolving gaseous ozone intothe liquid water so as to react with target contam-inants. Water ozonation is usually consists of foursteps: convection and back mixing of the liquid flow-ing through contacting chambers inside the staticmixer. These two processes occur simultaneouslywith two other processes: ozone gas mass transfer

from gas phase to the liquid phase, ozone decompo-sition and reactions of ozone with organic materialin the water [6]. The Kenics static mixer consists ofa series of fixed helical elements or blades installedinside tubular housing as shown in Figure 1.1. KMprovides continuous blending and dispersion of theflowing materials, with no moving parts, and no ex-ternal power or regular maintenance, by redirectingthe flow patterns present in the open pipe. The Ken-ics static mixer has an advantage over other typesof static mixer in that; it enhances the rate of masstransfer without wasting energy or material block-age. Moreover, the helical elements promote plugflow in continuous processes. The pressure drop in-creases along the mixer providing the energy needfor mixing process [7, 8, 9].Almost all the mathematical models that are devel-oped to predict the performance of the ozone contac-tors are based on one of the following two assump-tions: complete mixed flow or plug flow exist in theliquid phase. Applying these assumptions in model-ing the gas-liquid contactors will underestimate theperformance of the ozone contactor. Because of phe-

1

Figure 1.1: The structure of Kenics static mixer

nomena of the axial dispersion of the liquid phase,the real flow regime is closer to mixed flow thanthe plug flow, but it is not perfectly mixed flow.Thus, the back flow cell model (BFCM) has beendeveloped as an alternative way to describe the hy-drodynamics and mass transfer of the ozone insidethe Kenics static mixer [11]. The BFCM is a gen-eral form of stage-wise backmixing models and itcan be used to characterize the backmixing in theliquid phase for co-current or counter-current gas-liquid contactors at steady state or unsteady stateoperating conditions [12].

2. Mathematical Models of Ozone MassTransfer in a Kenics Static Mixer

2.1. Transient Back Flow Cell Model(TBFCM)

The developed models for designing gas-liquid reac-tors must describe the flow and mixing conditionsinside the reactor. The most common ideal reactorsare that used to design ozone contactor are the plugflow reactor (PFR) and the continuous flow stirredtank reactor (CFSTR). The PFR approaches theplug flow conditions. Therefore, the mixing betweenthe adjacent flow cells is not permitted whereas, theCFSTR is considered to be perfectly mixed and hasuniform concentration along the column [13]. Due tobackflow, short-circuiting and stagnant zone, thesetwo ideal flows are no longer applicable to describethe real flow inside the ozone contactors. The resi-dence time distribution can be used to analyse thesecomplicated flow characteristic in the ozone contac-tor [11]. The ordinary stirred tanks in series modelwhich assume perfect mixed cells has been employedto describe the mixing process. However, this modeldoes not take into consideration the upstream mix-ing of material i.e. the mixing in direction oppositeto the direction of the main flow. In order to over-come this problem, the backflow cell model has beendeveloped by Mecklenburg and Hartland [12]. The

BFCM introduces the backflow between the cells in-side the gas-liquid contactor. The backflow ratio,B is the main parameter in the BFCM. The perfor-mance of the back flow cell model varies according tothe value of the backflow ratio from ordinary tanksin series (B → 0) to single stirred tank (B → ∞)[2, 14, 15]. The BFCM is a mathematical modelthat is applied to characterise the performance ofthe Kenics ozone static mixer. In order to describethe axial dispersion in the liquid phase, the BFCMassumes a back flow between the cells in directionopposite to the main liquid flow and exchange flowin the same direction of the main liquid flow. Thesetwo flows have been expressed as back flow ratio (B)and exchange flow ratio (B) and both of them areassumed to be equal and constant along the mixer.Generally, BFCM is composed of two series of equalnumber of completely mixed cells in which one se-ries describe the liquid phase and the other describesthe gas phase [16]. In this model, the backmixingin the gas phase was assumed to be negligible be-cause of the large buoyancy of the gas bubbles, gasand liquid flow rates, interfacial and gas hold-up areconstant along the contactor. Roemer and Durbin[2] have developed very efficient TBFCM to describethe residence time distribution inside the chemicalreactors as shown in the Figure 2.1 below.This model consists of number of completely mixedcells (N) that have an equal volume

(Vc =

V TN

)with

equal backflow rates between the cells. The cells:(0 and N + 1) have negligible volume or hold-upand they allow the boundary conditions to be easilydetermined. The material balance has been carriedout around each cell with respect to the inert tracerand the following equations are produced:

δ(t) = (1 +B)E0 −Bg1 ⇒n = 0 (2.1)1

N

dEn

dt= (1 +B)En−1

− (1 + 2B)En +BEn+1 ⇒1 ≤ n ≤ N(2.2)

0 = EN−1 − EN ⇒n = N(2.3)

These equations have been transformed to the fol-lowing equations:

Et(θ) =

N∑i=1

[Ai exp(siθ)] (2.4)

Where E(θ) is the impulse response of the Nth cell

2

Figure 2.1: The transient back flow cell model

at time θ,(

Ci(θ)C∗

i

)For (0 < B < ∞ or 0 < λ < 1), the distinct poles ofthe transfer function can calculated as:

si =

(N

1− λ

)[2λ0.5Cos(θi)− (1 + λ)

](2.5)

Where 1 ≤ i ≤ N

Ai =(−2Nλ−N/2

)(Sin2(θi)

D′(θ)

)(2.6)

D′(θi) is the derivative of the function D(θi) whichis equal to:

D(θ) =λ0.5Sin ((N + 1) θ)− 2Sin(Nθ)

+ λ0.5Sin(N − 1)θ (2.7)

The above equations were solved using Matlabr Ver-sion R2014 in order to produce the impulse residencetime distribution of the ozone solute and also to in-vestigate the effect of the cells number and back flowratio on the concentration of the tracer inside theKenics static mixer. Before measuring the impulseresponse, the roots, θi of the function D(θ) havebeen determined by developing a Matlab code basedon the Newton-Raphson method. Newton methodis modified Taylor series method and uses iterativetechniques to solve the non-linear algebraic equationD(θ) = 0. Initial guess of the θi has to be made inthe interval (0 < θi < π). Newton method is veryfast and efficient and it has the following generalformula:

θ(i+ 1) = θ(i) +D(θi)

D′(θi)(2.8)

The iteration stops when the function D(θ) valuesatisfies the following conditions:D(θi) ≤ δabs (tolerance) [17]. It should be notedthat the number of the roots, θi of the function D(θ)

is equal to the number of cells used and their valuesdepends on the cell number and the back flow ratiowhich is expressed in this model by the term (λ).After estimating the roots of the function D(θ), aMatlabr code is built to produce the impulse andstep residence time distribution, E(θ) and F (θ) re-spectively. Another aim of this code is to investigatethe effects of the number of cells and the backflowratio on the residence time distribution of the ozonesolute inside the Kenics static mixer. Other objec-tives of this code are to compare the prediction of thetransient BFCM, the ADM and CFSTR’s in seriesat different operating conditions and also to validatethe models with experiment data. After implement-ing the code in Matlab software, several graphs havebeen produced as shown later in this research.

2.2. The Axial Dispersion Model (ADM)

This model simply characterizes the backmixing byone-dimensional and diffusion process superimposedon plug flow equation which is expressed by theFick’s law [18]. The ADM uses only one parame-ter (Peclet number) to characterize the back mix-ing. Thus, ADM became simple and widely usedmodel [11]. If an ideal tracer is injected to the reac-tor, it will spread as it travels through the column.The dispersion coefficient D (m2/s) exemplifies thespreading process. According to the dispersion co-efficient value, we have three cases: firstly, when Dis equal to zero, no spreading, thus plug flow. Sec-ondly, small D results in slow spreading of the tracercurve. Thirdly, large D results in rapid spreading[13]. The dispersion coefficient can be representedby the dimensionless Peclet number (Pe = D

uL ) inorder to characterize the spreading in whole reac-tor, Pe is used to define the degree of backmixing.When (Pe = 0), we have complete backmixing, andif (Pe = ∞), plug flow exists. For the ADM, theresidence time distribution (Eθ) can be expressedby [11]:

3

Eθ =1

2√π/Pe

exp

(−Pe(1− θ)2

4

)(2.9)

2.3. Continuous Stirred Tank in Series Model(CSTR’s in series)

It is the simplest among stage-wise models for char-acterization backmixing in the multiphase reactors.In the CSTR’s in series model, the reactor is viewedas a series of completely mixed stages. The degree ofbackmixing is determined by stage’s number. Thesmaller the number of stages, the more significantis the backmixing [11]. Generally, CSTR’s in seriesmodel is more reliable than the ADM at high valuesof the dispersion coefficient [19]. The residence timedistribution (Eθ) can be determined by [7]:

Eθ =NCFSTR(NCFSTR θ)(NCFSTR−1)

(NCFSTR − 1) !(2.10)

2.4. The Relationship Between ADM andTBFCM

The transient BFCM is easier to formulate and solvethan the ADM. This is because the BFCM producesnon-linear algebraic equations whereas; the ADMproduces non-linear differential equations which haveto be converted to non-linear algebraic equations.Both transient BFCM and ADM characterize thebackmixing of the liquid phase but in different ways.BFCM describes the backmixing by using the back-flow ratio (B) and number of the cells (NBFCM)whereas the backmixing in the ADM is character-ized by the dimensionless Peclet number. The backflow ratio (B) and Peclet number (Pe) can be inter-related by the following equation:

B =NBFCM

PeL− 0.5 =

DLεLNBFCM

uLL− 0.5 (2.11)

λ =B

1 +B(2.12)

Where: NBFCM = cells number, B = Back flowratio = exchange flow ratio, DL = axial disper-sion coefficient in the liquid phase (m2/s), uL =superficial liquid velocity, εL = liquid phase hold-up[14, 16, 18].

3. Results and Discussion

3.1. Transient Back Flow Cell Model (TBFCM)

3.1.1. Effete of Number of Cells on ResidenceTime Distribution of TBFCM

Figure 3.1 depicts the effect of the number of cells atconstant back flow ratio on the RTD curves of thetransient BFCM.

Figure 3.1: Impulse residence time distribution of ozone atconstant back flow ratio

From Figure 3.1, it has been concluded that as thenumber of cells increases and therefore the Pecletnumber increases, the impulse RTD curves narrowand become more symmetrical and also the peaksof the impulse RTD curves increase. This can beexplained by the fact that the liquid phase flow ap-proaches the plug flow regime at high values of Pecletnumber. The RTD curves at small cell numbersare quite broad and non-symmetrical and this canattributed to the fact that the pulse tracer slowlypasses through the mixer. At constant back flow ra-tio and according to this relationship,(B = NBFCM

PeL− 0.5

), as the cells number increases

and therefore the Peclet number increases, the liquidphase flow is approaching the plug flow regime andthe RTD curves are becoming more symmetrical.

3.1.2. The Effect of Backmixing Ratio on Res-idence Time Distribution of TBFCM

Figure 3.2 shows the impulse RTD curves for 24 cellsat different back flow ratio which is expressed interm of

(λ = B

1+B

).

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Figure 3.2: The effect of the backflow ratio on Impulseresponse of the ozone solute

It can be clearly seen that as the back flow ratio de-creases, the impulse RTD curves are becoming moresymmetrical with higher and late peaks and shortertails. At back flow ratio value equal 0.8, the RTDcurve is symmetrical with respect to dimensionlesstime equal to 1 and also has no tail. These phe-nomena suggest that the fluid elements have uniformtime distribution and no stagnant zones at this smallback flow ratio respectively.

3.2. Testing of Transient BFCM with Exper-imental Data

The experimental tracer study of Madhuranthaamet al. [3] was used for testing the applicability of theTransient BFCM to characterize the hydrodynamicof mass transfer in a Kenics static mixer (KMX).The experimental setup used in this study consistedof a static mixer with 24 mixing elements with an in-ternal diameter of 3.8 cm and total length of 0.98mand was operated concurrently with vertical up-flow.The fluids used were hydrogen representing the gasphase and monochlorobenzene representing the liq-uid phase. RTD experiment was conducted at roomtemperature and atmospheric pressure, and the hy-drogen gas flow rate varied from 46 to 564 ml/minand liquid flow rate varied from 23 to 98 ml/minFigure 3.3 shows the correspondence between the ex-perimental data and the prediction Transient BFCMfor gas-liquid system. There is slight difference be-tween the experimental and predicted RTD curvesand they are much more symmetrical than asym-metrical and this suggests that all the fluid elementshave uniform history. Furthermore, as it can be seen

Figure 3.3: Model validation with experimental data

from the graph we have single peaks and short tails,thus it can be said that there are no stagnant zonesor channelling respectively. The Transient BFCMpredictions slightly deviate from the experimentaldata and this has been attributed to the fact thatof using a step response method and this problemcan be avoided by using pulse input, but this is notapplicable in this experiment due to the of the lowconductivity of organic salt [3]. Even though theTransient BFCM has not been validated with ex-perimental data of ozonation process, this does notmean this model is insufficient for this process.

3.3. Comparison Between TBFCM, ADM andCFSTR’s and Testing Experimental Data

In order to compare the prediction of three modes:Transient BFCM, ADM and CFSTR’s model, theRTD curves of the three models plotted togetheragainst the experimental data of the Kenics staticmixer of 24 cells. In this Analysis, ADM1 refers tothe model equation of Fogler that has been reportedin the Madhuranthakam study [3] and the ADM2refers to the model equation of Levenspiel [13]whichhas been cited by Gamal El-Din [19].

ADM1 Eθ =1

2√

πθ3

Pe

exp

(− (1− θ)2

4θPe

)(3.1)

ADM2 Eθ =1

2√

πPe

exp

(−Pe (1− θ)2

4

)(3.2)

Figure 3.4 clearly shows that the Transient BFCMand axial dispersion (ADM1) can almost predict the

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Figure 3.4: Comparison among TBFCM, ADM and CF-STR’s in predicting RTD’s in the Kenics static mixer

RTD experimental data. However, Transient BFCMis a slightly better than the ADM1 in predicting theexperimental data and this mainly because the tran-sient BFCM characterize the backmixing in the liq-uid phase using two mixing parameters: Back flowratio and cells number, whereas, ADM1 uses onlyone parameter which is the Peclet number to de-scribe the backmixing in the liquid phase. The threemodels: transient BFCM, ADM1 and CFSTR’s pre-dicted symmetrical impulse RTD curves which haveidentical spread. However, the peak predicted bythe CFSTR’s model is around two times higher thanthat of the transient BFCM and ADM1. This ismaybe because of the fact that this model does notcharacterize the backmixing in the liquid phase andalso the cells number have higher impact than thebackmixing.From the figure, it is evident that the ADM2 is in-adequate to characterize the hydrodynamic of thestatic mixer. This model represents slight devia-tion from the plug flow regime and it is not ac-curate. This is may be due to the fact that realflow regime inside the mixer is closer to the com-pletely mixed flow than the plug flow regime and alsothis model represents open-open boundary condi-tions and therefore it is not suitable because the realconditions are usually closed-closed or closed-openconditions [19]. The ADM2 predicted the broadest,the latest and the lowest peak of the RTD curveamong all the other models. This can be attributedto the fact that the value of the Peclet number (PeL =10) is very small. Thus, the assumption that theliquid flow approaches the plug flow regime is notapplicable and therefore the ADM2 under-predict

the backmixing in the liquid phase under such con-ditions. However, the CFSTR’s in series model pre-dicted the highest peak and the RTD curve is sym-metrical. This is because the CFSTR’s in seriesmodel does not consider the backmixing betweenthe cells, which is has large value at such low Pecletnumber and it assumes completely mixed cells [13,19].

3.4. Number of Cells Effect on the Theoreti-cal Residence Time Distribution ofTBFCM, ADM and CFSTR’s in Series

Figure 3.5with its subfigures shows the effect of in-creasing the cells number of the static mixer on thecorrespondence between the three models and alsotheir predictions of the RTD curves. As shown in thegraphs above, and as the number of cells is increased,the Transient BFCM approaches or converges to theaxial dispersion model. As shown in Figure 3.5a atsmall number of cells (N = 6), there is large differ-ence between the transient BFCM and the ADM inpredicting the RTD curves. However, as you cansee in Figures 3.5b, 3.5c and 3.5d that the deviationbetween these two models is becoming smaller andsmaller and also the RTD curves are getting moresymmetrical as the number of cells is increased. Asshown in figure 10, the best correspondence betweentransient BFCM and ADM was at a number of cellsequal to 24 which is the same number of cells ofthe Kenics static mixer that used in the experimen-tal study. There is long tails at small number ofcells, especially at N=6 or 10, which suggest thepresence of stagnant zones. Therefore, this smallnumber of cells is insufficient to provide adequatemixing between the two phases. For the CFSTR’s,it has poorly predicted the RTD curves comparedto other two models, and this is very noticeable athigh cells’ number. This is due to the fact that thismodel does not account for the backmixing in theliquid phase. RTD curves of the CFSTR’s in seriesmodel were always symmetrical and therefore thissuggests uniform distribution of the solute concen-tration inside the mixer of completely mixed cells.

4. Conclusion

Three different models have been used to charac-terize the performance of the gas-liquid system in-side Kenics static mixer: the transient back flow cellmodel of Roemer and Durbin, [2], the axial disper-sion model (ADM) and the continuous flow stirred

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(a) RTD curves of TBFCM, ADM and CFSTR’sin series model at N = 6

(b) RTD curves of TBFCM, ADM and CFSTR’sin series model at N = 10

(c) RTD curves of TBFCM, ADM and CFSTR’sin series model at N = 20

(d) RTD curves of TBFCM, ADM and CFSTR’sin series model at N = 24

Figure 3.5: Number of cells effect on theoretical residence time distribution (RTD)

tank in series (CFSTR’s in series). The three modelshave been validated with experimental data Mad-huranthaam et al. [3] and they have been used tosimulate the impact of different parameters: cellsnumber and back flow ratio on the performance ofthe mixer.As a result of this comparison, both BFCM andADM have proved to be accurate, sufficient and re-liable in their predictions of the performance of theozone Kenics static mixer. However, the transientBFCM have provided slightly better results than theADM at a high number of cells, around 24. Thisis because the transient BFCM uses two mixing pa-rameters to characterise the backmixing in the liquidphase: the number of cells and the backmixing ratiowhereas the ADM uses only one parameter which is

the Peclet number. Interestingly, it was found thatthe Transient BFCM approaches or converges to theaxial dispersion model at a higher number of cells.

5. Further Work

The transient BFCM should be improved to includethe effect of the gas and liquid flow rates, the back-mixing in the gas phase, variable backmixing ratioin the liquid phase across the static mixer.

Acknowledgment

I gratefully acknowledge the financial support givenfor this work by Al-Mergib University.

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References

[1] Munter, R. Mathematical Modelling and Sim-ulation of Ozonation Process in a DownstreamStatic Mixer with Sieve Plates, Ozone Science& Engineering, 2004, 26, pp 227 – 236.

[2] Romer, M.H.; Durbin, L.D. Transient Responseand Moments Analysis of Backflow Cell Modelfor Flow Systems with Longitudinal Mixing,Ind. Eng. Chem. Fundam, 1967, 6, pp 120–129.

[3] Madhuranthakam, C. M.; Pan, Q.; Rempel,G. L. Residence Time Distribution and LiquidHoldup in Kenics KMX Static Mixer with Hy-drogenated Nitrile Butadiene Rubber Solutionand Hydrogen Gas System, Chemical Engineer-ing and Processing, 2009, 64, pp 3320–3328.

[4] Gaddis, E. S. Mass Transfer in Gas-liquid Con-tactors, Chemical Engineering and Processing,1999, 38, pp 503 – 510.

[5] Sanchez, C.; Couvert, A.; Laplanche, A.; Ren-ner, C. Hydrodynamic and Mass Transfer ina New Co-current Two-phase Flow Gas–liq-uid Contactor. Chemical Engineering Journal,2007, 131, pp 49–58.

[6] Zhou, H.; Smith D. W. Modelling of Dis-solved Ozone Concentration Profiles in BubbleColumns. Journal of Environmental Engineer-ing, 1995, 120, pp 821 – 840.

[7] Coker, K. A. Modeling of Chemical Kinet-ics and Reactor Design. Boston: Butterworth-Heinemann. 2001

[8] Thakur, R. K.; Vial C.; Nigam, K.; Nauman,E. B.; Djelveh, D. Static Mixers in the ProcessIndustries, Trans IChemE, 2003, 81, pp 787 –826.

[9] Ghanem, A.; Lemenand, T.; Valle, D. D. StaticMixers: Mechanisms, Applications and Charac-terization Methods, Chemical Engineering Re-search and Design. 2014, 92, pp 205 – 228.

[10] Chemineer, Inc., Kenics: Static Mixing Tech-nology. Bulletin, 800 Commercial Documenta-tion. 1998.

[11] Shah, T. Y.; Stiegel, J. G.; Sharma, M. M.Backmixing in gas-liquid reactor. The Amer-ican institute of Chemical Engineering, 197824:3, pp 369-400.

[12] Mecklenburgh, J.C.; Hartland, S. Theory ofBackmixing. London: John Wiley & Sons.1975.

[13] Levenspiel, O. Chemical Reaction Engineering.New York: John Wiley & Sons. 1999.

[14] Tizaoui, C.; Zhang, Y. The Modeling of OzoneMass Transfer in Static Mixers Using Back-flow Cell Model. Chemical Engineering Journal,2010, pp 164, 557 – 564.

[15] Baldwin, J. T.; Durbin L. D. The Back-flowCell Model of Isothermal First Order Flow Re-actors with Axial Dispersion. The CanadianJournal of Chemical Engineering, 1966, pp 151– 157.

[16] El-Din, M. G.; Smith D. W. Theoretical Anal-ysis and Experimental Verification of OzoneMass Transfer in Bubble Column. FundamentalTechnology, 2001c 23, pp 135 – 147.

[17] Beers, K. J. Numerical Methods for Chemi-cal Engineering: Application in Matlab. Cam-bridge: Cambridge University Press. 2001

[18] Baawain, M. S.; El-Din, M. G.; Clarke.,Katie and Smith; Daniel W. Impinging-JetOzone Bubble Column Modeling: Hydrody-namics, Gas Hold-up, Bubble Characteristics,and Ozone Mass Transfer. Ozone: Science &Engineering, 2007, 29: 4, 245 — 259

[19] El-Din, M. G.; Smith, D. W. Development ofTransient Back Flow Cell Model (BFCM) forBubble Columns. Ozone: Science & Engineer-ing, 2001a, 23, pp 313 – 326.

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