Development of Unified Correlations for Volumetric Mass-Transfer

4216

Ind. Eng. Chem. Res. 2009, 48, 4216–4236

KINETICS, CATALYSIS, AND REACTION ENGINEERING Development of Unified Correlations for Volumetric Mass-Transfer Coefficient and Effective Interfacial Area in Bubble Column Reactors for Various Gas-Liquid Systems Using Support Vector Regression Ankit B. Gandhi,† Prashant P. Gupta,† Jyeshtharaj B. Joshi,*,† Valadi K. Jayaraman,‡ and Bhaskar D. Kulkarni‡ Institute of Chemical Technology, UniVersity of Mumbai, Matunga, Mumbai-400 019, India, and Chemical Engineering & Process DeVelopment DiVision, National Chemical Laboratory, Pune-411008, India

The objective of this study was to develop a unified correlation for the volumetric mass-transfer coefficient (kLa) and effective interfacial area (a) in bubble columns for various gas-liquid systems using support vector regression (SVR-) based modeling technique. From the data published in the open literature, 1600 data points from 27 open sources spanning the years 1965-2007 for kLa and 1330 data points from 28 open sources spanning the years 1968-2007 for a were collected. Generalized SVR-based models were developed for the relationship between kLa (and a) and each design and operating parameters such as column and sparger geometry, gas-liquid physical properties, operating temperature, pressure, superficial gas velocity, and so on. Further, these models for kLa and a are available online at http://www.esnips.com/web/UICT-NCL. The proposed generalized SVR-based correlations for kLa and a have prediction accuracies of 99.08% and 98.6% and average absolute relative errors (AAREs) of 7.12% and 5.01%, respectively. Also, the SVR-based correlation provided much improved predictions compared to those obtained using empirical correlations from the literature. 1. Introduction Bubble columns are extensively used in the chemical, petrochemical, biochemical, and metallurgical industries to carry out a variety of different unit processes such as oxidation, chlorination, alkylation, polymerization, hydrogenation, FischerTropsch synthesis, fermentation, biological wastewater treatment processes, and so on. Bubble columns are preferred to other multiphase reactors because they require less maintenance owing to the absence of moving parts, offer high values of heat- and mass-transfer coefficients, allow for better handling of solids and hence higher durability of solid catalysts, require less floor space and hence lower costs, and are easier to operate. The present work concentrates on gas-liquid contacting, whereby a discontinuous gas phase in the form of bubbles is introduced through a gas distributor situated at the bottom of the column that then moves relative to a continuous liquid phase. In practice, gas-liquid contacting might or might not be accompanied by chemical reaction. However, in all cases, knowledge of the mass-transfer coefficient (kLa) and/or effective interfacial area (a) helps in determining the overall reaction rate per unit volume, the volume of the column, and the design and scaleup parameters of the bubble column. In a bubble column, variations in kLa are primarily due to variations in a. Since the values of a are known to be a function of both the gas holdup and the Sauter mean bubble diameter, the mass-transfer coefficient is also dependent on these two parameters. In addition, kLa also depends on the values of kL, which is governed by the fluid mechanics in the vicinity of gas-liquid interface. Qualitatively, kL can be viewed * To whom correspondence should be addressed. Tel.: +91-224145616. Fax: +91-22-4145614. E-mail: [email protected]. † University of Mumbai. ‡ National Chemical Laboratory.

to depend on the extent of slip at the gas-liquid interface and the surface renewal rate. The physics of slip and surface renewal continues to attract the attention of scientists, even though efforts in the past have improved our understanding substantially. However, the present status of knowledge forces the estimation of kL (and to a large extent a) to be largely empirical. Over the years, extensive experimental research on kLa and a has been carried out in bubble column reactors, and numerous empirical correlations have been proposed for mass-transfer coefficient in various gas-liquid systems. Some of the important correlations for kLa and a for various gas-liquid systems are listed in Tables 1 and 2, respectively. Variable definitions can be found in the Nomenclature section. It has now been established in the literature that a large number of factors affect the volumetric mass-transfer coefficient (kLa), with each affecting this coefficient in a different way. For instance, kLa and a vary with changes in sparger type and geometry, gas-liquid properties, operating pressure and temperature, column dimensions, and so on. A large number of experiments have been reported in the literature for various gas-liquid systems (including gas-water, gas-organic, gas-aqueous organic, gas-aqueous viscous Newtonian, gas-aqueous electrolyte, and gas-liquid systems operated under high temperature and pressure), and a number of systemspecific correlations are available. Most of these correlations are in dimensionless numbers. When these correlations and the correlation constants are carefully analyzed, the correlation constants are often found not to remain constant as envisaged in the correlations. For instance, the exponent on a variable such as VG, µL, FG, or σL strongly depends on the range of the variable itself, as well as the ranges of the other variables. As a result, all published correlations are insufficient to meet the objectives of design engineers. For instance, Figures 1 and 2 show plots of all of the published correlations, with deviations up to 1000%. This is mainly

10.1021/ie8003489 CCC: $40.75  2009 American Chemical Society Published on Web 03/25/2009

Ind. Eng. Chem. Res., Vol. 48, No. 9, 2009 4217 Table 1. Summary of Correlations for Volumetric Mass-Transfer Coefficient for Various Gas-Liquid Systemsa researchers

gas-liquid system

correlation

( )( ) gD2FL σL

( )

kLa_D2 µL ) 0.6 DL FLDL

Deckwer et al.2

gas-water/aqueous electrolyte

kLa ) b0(VG)b1 b1 ) 0.78-1.28, b0 ) 0.0086-0.043

Wang and Fan3

gas-water

kLa_ ) 0.137(VL)0.44

Nakanoh and Yoshida4

gas-water/aqueous viscous Newtonian solution


Hikita et al.5

gas-water/aqueous viscous Newtonian/ organic/aqueous electrolyte solution

0.5

(

( )( )

kLa_ )

gFLD2 σL

0.5

( )

14.9gf VGµL VG σL

1.76

( ) µL4g

( )( ) 0.39

µL

-0.248

VG

√gD

2

( )( )

gas-water/aqueous organic solution


Seno et al.8

gas-aqueous viscous Newtonian/organic solution

kLa_D2 VG ) 0.6 DL VG + VL

gas-water/aqueous viscous Newtonian


0.5

gFLdB2 σL

( ) ( ) µG µL

µL FLDL

0.243

-0.604

0.33

( ) gFL2dB3

(√ ) ( )

0.29

2

µL

VG

0.68

gdB

FG FL

( ) ( ) ( )( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) () ( )( )

kLa_VG VGµL ) 16.9 g σL

0.5

gDFL σL

-0.39

0.5

2.14

µL4g

gas-water/aqueous organic/aqueous viscous Newtonian solution

Dewes and Schumpe11

aqueous electrolyte solution

kLa ) VG0.90µL-0.55FG0.46

Kojima et al.12

gas-water/aqueous buffered enzyme solution

kLa_ ) C(G)D

gas-aqueous organic solution

kLa ) CVG0.91FG0.24

gas-water gas-water/organic/aqueous electrolyte/ aqueous viscous Newtonian/high-pressure system

kLa ) 0.00062(VG)0.4(µL)-1(D)0.5

( ) VGD2FL

0.62

gD3FL2

gD2FL2

0.5

G1.1

0.62

µL2

0.20

gD3FL2

FLσL

3

( )

µG µL

gD3FL2

0.31

µL2

0.62

VG

G1.1

0.51

√gD

µL2

-0.518

0.04

0.31

µL2

µL FLDL

gFLD2 σL

Sotelo et al.10

a

)

gD3FL2

FLσL

Posarac and Tekic7

Bando et al. Lau et al.15

G1.1

f ) 1 for nonelectrolytes, f ) 100.068I for I < 1 f ) 1.114 × 100.02I for I > 1 kLa_dB2 µL ) 0.62 DL FLDL

14

0.75

3

gas-water/organic

Jordan and Schumpe13

0.31

µL2

VL 2.81(VG + VL) + 25.8

Ozturk et al.6

Suh et al.9

0.62

gD3FL2

gas-water/organic/aqueous electrolyte/ aqueous viscous Newtonian solution

Akita and Yoshida1

0.074

µL FLDL

-0.038

do D

0.908

-0.016-0.122

3

σLdo

P P0

0.033-0.155

kLa ) 1.77σL-0.22 exp(1.65VL - 65.3µL)G1.2

dB ) bubble diameter (m), P0 ) standard atmospheric presence (Pa).

because of the assumed constancy of empirical constants over the ranges considered. To overcome this problem, we thought it desirable to use the technique of support vector regression (SVR). In this context, it was the objective of this study to propose unified SVR-based data-driven correlations for kLa and a in bubble column reactors for all of the above-mentioned systems.

Data-driven modeling has been finding increasing relevance and importance in chemically reacting systems. Two techniques based on data-driven modeling that are gaining popularity are artificial neural networks (ANN) and support vector regression (SVR). Of the two, ANN is more commonly used. Applications of ANN in the context of reactor design have been described in the literature

4218 Ind. Eng. Chem. Res., Vol. 48, No. 9, 2009 Table 2. Summary of Correlations for Effective Interfacial Area for Various Gas-Liquid Systemsa researchers

gas-liquid system

Akita and Yoshida16

Dierendonck et al.17

correlation

gas-water/organic/aqueous viscous Newtonian/aqueous electrolyte solution

a_D )

gas-water/organic/aqueous electrolyte solution

a_ )

( )( )

2 1 gD FL 3 σL

0.5

gD3FL2 µL2

( ) ( )(

6 σL √c FLg

0.1

-0.5

µLVG σL

1⁄4

G1.13

FLσL3 gµL4

)

1⁄8

G

c ) 6.25 for nonelectrolyte solutions c ) 2.1 for electrolyte solutions H gas-water/organic/aqueous viscous Newtonian and a_ ) 26 non-Newtonian/aqueous electrolyte solution/high-pressure system D

Gestrich and Krauss18

-0.3

()

G

( ) FLσL3

-0.003

gµL4

Tabei et al.19

gas-aqueous alkaline solution

a ) 2100G1.25(1 - G)0.75

Serizawa and Kataoka20

gas-water/aqueous viscous Newtonian

a ) 1030G0.87VG0.2

Kulkarni et al.21

gas-aqueous electrolyte solution

a ) 225(VG)0.635(VL)-2.05(σL)-0.11

gas-aqueous electrolyte solution

a ) 0.140(VG)0.81(µ)-0.22

gas-aqueous electrolyte solution system

a ) 1967VG0.856(1000do)-0.458

Quicker and Deckwer22 23

Popovic and Robinson

Schumpe and Deckwer24 gas-aqueous electrolyte solution system

a ) 651VG0.87µeff-0.24

Kawase and Young25

a_ ) 0.399n

a

gas-water/aqueous viscous Newtonian solution

1.85

(

VG0.73g0.23FL0.6 σL0.6D0.16

)

H ) clean liquid height (m), µ ) liquid viscosity (Pa · s), D ) shean index.

since the early 1990s. For instance, Xie et al.27 reported flow regime classification in multiphase flows using ANN, Shaikh and AlDahhan28 correlated overall gas holdup in bubble column reactors using ANN, and Lemoine et al.29 successfully correlated the gas-liquid volumetric mass-transfer coefficient in surface aerators and gas-inducing reactors using ANN.

In this study, unified data-driven models based on support vector regression (SVR) for correlating kLa and a in bubble column reactors are proposed. The applicability of SVR-based modeling in the field of chemical engineering and the fundamentals of SVR have been described earlier by Nandi et al.30 and Desai et al.31 This technique is based on structural risk

Figure 1. Parity plot based of experimental data on volumetric mass-transfer coefficient against the values estimated from literature correlations and SVRbased generalized model for kLa.

Figure 2. Parity plot based of experimental data on interfacial area against the values estimated from literature correlations and SVR-based generalized model.

Ind. Eng. Chem. Res., Vol. 48, No. 9, 2009 4219

solution systems, and gas-liquid systems operated over a wide range of temperature and pressure were considered. 2. SVR-Based Modeling

Figure 3. Nature of the objective functions: (A) SVR, (B) ANN.

2.1. Mathematical Modeling. Support vector regression (SVR) is an adaptation of a recently introduced statistical/ machine learning theory known as support vector machines.32 The objective here is to build an epsilon-SVR model32 to fit a regression function, y ) f(x), such that it accurately predicts the outputs {yi} corresponding to a new set of input examples {xi}. In the epsilon-SVR model, epsilon (ε) represents (loss function) the radius of the tube located around the regression function, f(x) (Figure 4), and the region enclosed by the tube is known as the ε-insensitive zone.33 The loss function assumes a value of 0 in this zone, and as a result, it does not penalize the prediction errors with magnitudes smaller than ε. To fulfill the stated goal, SVR considers the following linear estimation function in the high-dimensional feature space34 f(x, w) ) (w · φ(x) + b) (1) where φ(x) is a function termed a feature and w · φ(x) is the dot product in feature space, F, such that φ(x) f F and w ∈ F. Thus, after algebraic transformation, the objective function (eq 1) is converted to convex optimization problem.32-34 The primal form of the optimization problem is given as N

Maximize L(R*i,j) )

N

∑ y (R - R ) - ε∑ (R + R ) i

* i

i

* i

i

i)1

i)1

N

N

∑ ∑ (R - R )(R - R )[φ(x ) · φ(x )] (2)

1 2 i)1

* i

i

* j

j

i

j

j)1

subject to the constraints Figure 4. Schematic illustration of SVR using ε-sensitive loss function by Nandi et al.30

minimization as opposed to empirical risk minimization, which forms the basis of conventional data-driven techniques. Thus, SVR minimizes training data error and also accounts for model complexity unlike other conventional data-driven techniques, which minimize the training data error only; as a result, SVR shows high predictive capabilities for both training and unseen test data sets. Thus, the solutions obtained from conventional data-driven techniques can become trapped in local minima as opposed to finding global minima, whereas SVR finds global minima only. Figure 3A,B shows how the solutions for the objective function are obtained for SVR and conventional datadriven techniques, respectively. Also, SVR facilitates the execution of linear regression in a high-dimensional feature space for problems where the inputs are correlated nonlinearly with the output. However, to avoid computationally intractable problems (high-dimensional transformations), the appropriate kernel functions are used to carry out the computations in the input space itself. SVR also exhibits properties such as good generalization ability of the regression function, robustness of the solution, and sparseness of the regression. The sparseness property allows the final regression function to be expressed in terms of a limited number of support vectors that represent a subset of a given training data. Thus, the objective of this study was to develop unified SVRbased correlations for the values of kLa and a in bubble column reactors based on all data available for various gas-liquid systems in the literature. For the purpose of establishing unified correlations, various gas-liquid systems such as gas-water system, gas-aqueous organic liquid systems, gas-aqueous viscous Newtonian solution systems, gas-aqueous electrolyte

N

C g Ri, R*i g 0, and

∑ (R - R )y ) 0 * i

i

i

i)1

where C is the cost function employed to obtain a tradeoff between the flatness of the regression function and the amount to which deviations larger than ε can be tolerated. Solving this problem (eq 2) by convex quadratic programming (QP) gives the value of the coefficients Ri and R*i . Owing to the specific character of the above-described quadratic programming problem, only some of the coefficients, (Ri - R*i ), are nonzero, and the corresponding input vectors, xi, are called support vectors (SVs). These SVs are known to be the most informative data points that compress the information content of the training set, thereby representing the entire SVR function (in a simpler case of empirical correlation, these are proportionality constants and exponents over various variables). The coefficients Ri and R*i have an intuitive interpretation as forces pushing and pulling the regression estimate, f(x), toward the measurements, yi. It can be seen in Figure 4 that SVs are depicted as points lying on the surface of the tube and the regression function can be fully characterized by these support vectors. Owing to this characteristic, the final regression model can be defined with relatively small numbers of input vectors. These SVs, xi, and the corresponding nonzero Lagrange multipliers, Ri and R*i , give the value of the weight vector, w, followed by the expanded form of the SVR N

w)

∑ (R - R )φ(x ) * i

i

(3)

i

i)1

Nsv

f (x, Ri, R*i ) )

∑ (R - R )[φ(x ) · φ(x )] + b i

i)1

* i

i

j

(4)

4220 Ind. Eng. Chem. Res., Vol. 48, No. 9, 2009

However, for the aforementioned optimization problem (eq 2), with an increase in the input dimensions, the dimensions in the high-dimensional feature space further increase many fold, thus resulting in a computationally intractable problem. Such a problem can be overcome by defining appropriate kernel functions in place of the dot product of the input vectors in high-dimensional feature space K(xi, xj) ) φ(xi) · φ(xj)

(5)

The advantage of a kernel function is that the dot product in the feature space can now be computed without actually mapping the input vectors, xi, into high-dimensional feature space. Thus, when a kernel function is used, all necessary computations can be performed implicitly in input space instead of in feature space. As a consequence, the dual optimization problem (eq 2) is revised to the following form N

Maximize L(R*i,j) )

∑

N

yi(Ri - R*i ) - ε

i)1

∑ (R + R ) * i

i

i)1

N

N

∑ ∑ (R + R )(R - R )K(x , x ) (6)

1 2 i)1

* i

i

* i

j

i

j

j)1

subject to the constraints N

C g Ri, R*i g 0, and

∑ (R - R )y ) 0 * i

i

i

i)1

Thus, the basic SVR formulation takes the form Nsv

f (x, Ri, R*i ) )

∑ (R - R )K(x , x ) + b i

* i

i

j

(7)

i)1

Also, the bias parameter, b, can be computed by applying Karush-Kuhn-Tucker (KKT) conditions, which state that, at the optimal solution, the product between dual variables and constraints has to vanish, thus giving b ) {yi - (Ri - R*i )K(x, xi) - ε} for Ri ∈ 〈0, C〉 b ) {yi - (Ri - R*i )K(x, xi) + ε} for R*i ∈ 〈0, C〉

(8)

where xi and yi denote the ith SV and the corresponding target output, respectively. There exist several choices of kernel function, K, including linear, polynomial, and Gaussian radial basis function. The most commonly used kernel function is the Gaussian radial basis function (RBF).32,33 It is defined as

(

K(xi, xj) ) exp

-||xi - xj||2

2σ2 where σ denotes the width of the RBF.

)

(9)

3. Results and Discussion The process of development of the SVR-based correlation can be divided into three stages, namely: (i) collection of data sets, (ii) calculation of the various parameters for establishing the regression function, and (iii) prediction of kLa and a. 3.1. Collection of Data Sets. During the past 40 years, the measurement of kLa and a has received very wide attention. The values of kLa and a depend on the design parameters (column diameter, column height, and sparger design), operating parameters (superficial gas and liquid velocities, temperature, and pressure), physical properties of both phases (density and viscosity), surface tension, and gas solute diffusivity in liquid. All of the published information on the measurement of kLa and a is summarized in Tables 3 and 4, respectively. The ranges

of parameters covered in these investigations are listed in Table 5. A careful analysis of all of the data brings out the following points: (1) We have now assembled a list of various methods used by various authors for the estimation of kLa and a in the form of tables (Tables 3 and 4). These tables also include the ranges of design and operating parameters, as well as the physical properties of the gas and liquid phases. (2) For the measurement of kLa, two methods have been widely used: physical (transient absorption/desorption) and chemical. Chemical methods have also been used for the measurement of a. (3) The physical methods include (a) photography (direct and optical-fiber-based), (b) electrical conductivity/resistivity measurement, and (c) light scattering. Chemical methods have been applied to a large number of gas-liquid systems such as CO2-NaOH, O2-sodium dithionite, CO2-diethanolamine, and so on. Critical analyses of these techniques have been published.25,43 On the other hand, estimation of a using physical methods (electrical conductivity/resistivity measurement) involves intrusion of the measurement probe within the system, thereby affecting the flow of the bubbles. Therefore, for the sake of simplicity, the photography technique is usually used to measure bubble size (Sauter diameter) through a transparent column. However, the estimation of bubble size using photography is restricted only to very low gas velocities. Also, nonspherical bubbles and radial maldistributions of bubbles can result in small bubble diameters, i.e., an interfacial area that is too large even at homogeneous flow conditions.25 (4) For the measurement of kLa, and a, some investigators have assumed the liquid phase to be completely back-mixed, whereas some have included the extent of liquid-phase axial mixing during the calculations. (5) In the estimation of kLa and a by chemical methods and of kLa by physical methods, knowledge of axial mixing is needed. However, this might not be important in kLa measurements, as the conversion with respect to gas-phase species is normally less than 10%. For the selection of data for a that originate from the chemical method, care has to be taken to ensure that gas-phase conversion is limited and, hence, that the role of gas-phase back-mixing becomes unimportant. (6) The chemical methods invariably involve the use of electrolyte solutions, whereas the physical methods generally involve nonelectrolytes as liquids, such as water, aqueous solutions, or organic liquids. It is known that the average bubble diameter decreases with increasing electrolyte concentration and, hence, kLa also attains higher values. (7) Bubble columns operate in either of the two characteristic flow regimes: homogeneous and heterogeneous. In the homogeneous regime, the bubble concentration is uniform throughout the column, particularly in the transverse direction. The bubbles are also uniform in size, and they rise practically without any breakup or coalescence. In this regime, the fractional gas holdup (G) is almost proportional to the superficial gas velocity (VG). The heterogeneous regime is characterized by a wide bubble size distribution and the occurrence of a parabolic holdup profile with a maximum at the center. The heterogeneous regime is also characterized by intense liquid circulation with an upward flow in the central region and downflow near the column wall. The fractional gas holdup varies as VG0.5-0.7. The wide size distribution of bubbles is known to create difficulties in the measurement of a, particularly by chemical methods.67,68 (8) The presence of adventious impurities and surface-active agents affects fractional gas holdup (G) and, hence, kLa and a.

0.077/0.152/0.3/ 0.66 -----90-3.50

0.152 -----3.0

0.15/0.20 -----3.52/7.23

0.102 -----0.76

0.14 -----3.12

5.5 -----9

0.15/0.30/1.0 -----0.6/1.2

0.145 -----1.1

Yoshida and Akita35

Akita and Yoshida1

Deckwer et al.2

Wang and Fan3

Buchholz et al.26

Kataoka et al.36

Kastanek et al.37

Nakanoh and Yoshida4

investigators

column diameter (m) -----column height (m)

single hole -----0.004 -----1

spider-type/perforated plate -----0.004/0.0016 -----250/1954

perforated plate -----0.03 -----200

sintered plate -----0.000175 -----75112

single hole -----0.019 -----1

cross sparger/sintered plate -----0.001/0.00015 -----56/75112

single hole -----0.05 -----1

single hole -----0.0025-0.040 -----1

sparger type -----hole diameter (m) -----number of holes

air-water/aqueous sucrose -----303 -----101.3

0.016-0.085 ------

0.0048-0.025 ------

0.024-0.048 ------

air + CO2-water -----583 -----101.3 air-water -----298 -----101.3

0.0074-0.068 -----0.014

0.030-0.30 -----0.067-0.40

O2-water -----293 -----101.3 air-water -----298 -----101.3

0.0025-0.067 ------

0-0.032 -----0.0072-0.043

0.0064-0.25 ------

range of VG (m/s) -----range of VL (m/s)

air-water/aqueous electrolyte soln -----289 -----101.3

gases: oxygen/air liquids: water/aqueous Na2SO3 soln -----293 -----101.3

air-aqueous Na2SO3 soln -----293 -----101.3

gas-liquid system -----temperature (K) -----pressure (kPa)

995-1230 -----0.000894-0.01 -----0.071

997 -----0.00089 -----0.072

677.5 -----0.00079 -----0.014

1006-1008 -----0.46-1.9 -----0.069-0.072

998 -----0.00107 -----0.072

998-1038 -----0.00097-0.00127 -----0.071-0.073

1000-1015 -----0.00097-0.0011 -----0.072-0.073

1015 -----0.0011 -----0.073

range of liquid properties -----density (kg/m3) -----viscosity (Pa · s) -----surface tension (N/m)

Table 3. Summary of the Published Literature on Experimental Details: Range of Variables Covered and Measurement Techniques for kLa

1.16 -----0.000018

1.18 -----0.000018

1.19 -----0.000029

1.18 -----0.000018

1.26 -----0.000018

1.22 -----0.000017

1.2 -----0.000018

1.2 -----0.000018

range of gas properties -----density (kg/m3) -----viscosity (Pa · s)

dynamic oxygen absorption technique, DO probe method, CSTR model

oxygen absorption technique, oxygen probe, CSTR model

dynamic carbon dioxide absorption technique, chromatography, CSTR model (plug flow and perfectly mixed with respect togas)

dynamic oxygen absorption technique, optical oxygen probe (relatively fast response time), axial dispersion model (ADM)

steady-state oxygen absorption technique, oxygen probe CSTR model and plug-flow model (PFM)

steady-state oxygen absorption technique, polarographic electrode, axial dispersion model (ADM)

dynamic oxygen absorption technique, DO probe, CSTR model

steady-state sulfite oxidation technique

measurement technique and working principle


0.15 -----0.66-4.3

0.19 -----1.7

Merchuk and Ben-Zvi 38

39

0.15 -----2.32

Suh et al.9

Grund et al.

0.1 -----2

0.095 -----0.85

6

Ozturk et al.

Posarac and Tekic7

0.10-0.19 -----1.5-2.4

5


Hikita et al.

investigators

Table 3. Continued


ring sparger -----0.001 -----64

sintered plate -----0.0005 -----676000

single hole -----0.004 -----1

single hole -----0.003 -----1

single hole -----0.009-0.0362 -----1


air-water/methanol/ ligroin/toulene -----293 -----101.3

air-water/aqueous glycerol/aqueous CMC soln -----298 -----101.3

gas: air liquids: aqueous sucrose -----301 -----101.3

air-water, aqueous alcohol/soln, (methanol, ethanol, propanol, and butanol) -----296 -----101.3

gases: air, He, H2, N2 liquids: methanol/ propnaol/butnaol/glycol/ xylene/decalin/CCl4/ ligroin/acetone/aniline/ nitrobenzene/ dichlroethane/ nitrobenzene/ethyl acetate 293 -----101.3

air-water/aqueous electrolyte soln -----295 -----101.3


0.0076-0.196 ------

0.01-0.10 ------

0.034-0.31 ------

0.0071-0.21 ------

0.0063-0.09 ------

0.040-0.37 -----0.013


714-998 -----0.00047-0.001 -----0.020-0.072

1070-1137 -----0.0024-0.011 -----0.066-0.068

1175-1264 -----6.2-2.2 -----0.071-0.074

793-991 -----0.000858-0.00097 -----0.021-0.071

714-1593 -----0.00032-0.02 -----0.0204-0.053

998-1130 -----0.00083-0.00096 -----0.0725-0.0759


dynamic oxygen absorption, DO probe, CSTR model




1.26-1.34 dynamic oxygen absorption -----technique, DO probe, CSTR 0.000016-0.000018 model; steady-state oxygen absorption technique, DO probe, axial dispersion model (ADM)

1.18 -----0.000018

1.17 -----0.000018

1.27-1.85 dynamic oxygen absorption -----technique, DO probe, CSTR 0.000015-0.000018 model


1.19-2.61 -----0.000018



0.115 -----1.37

0.2 -----0.5-2

(14)

Bando et al.

0.15 -----0.97

0.045 -----1.2

13

12

Jordan et al.

Letzel et al.42

Kojima et al.

0.115 -----1.09-1.37

Dewes and Schumpe11

0.04/0.08 -----1.2-1.6

0.158 -----1.5

10


Wilkinson et al.40

Sotelo et al.

investigators

Table 3. Continued

single hole -----0.013 -----1



single hole -----0.00138 -----1


ring sparger -----0.01 -----19

porous plate -----0.00003 -----711112


0.027-0.29 ------

N2-water -----298 -----101.3-400

air-water -----298 -----101.3

0.01/0.08 ------

0.01-0.21 ------

0.008-0.10 ------

N2 + O2-water/aqueous buffered enzyme soln -----298 -----101.3-1000

gases: He/N2/O2 liquids: aqueous ethanol soln/1-butanol/toluene/ decalin -----293 -----22-101.3

0.03-0.08 ------

0.001-0.14 ------

0.00006-0.0049 ------


air-aqueous sodium sulfate soln -----289-303 -----101.3-794

air-sodium sulfite -----293 -----101.3-400

CO2-water/ethanol/ ligroin/toulene -----298 -----101.3


1000 -----0.001 -----0.072

793-884 -----0.0012-0.0029 -----0.022-0.032

997 -----0.00089 -----0.072

1000-1025 -----0.00089-0.0010 -----0.072-0.063

1094 -----0.00135 -----0.076

1086 -----0.0015 -----0.075

928-1147 -----0.001-0.004 -----0.029-0.073

range of liquid properties -----density (kg/m3) -----viscosity (Pa · s) -----surface tension (N/m) measurement technique and working principle

(1) dynamic oxygen absorption technique, DO probe, CSTR model; (2) pressure-step method


steady-state sulfite oxidation technique

1.18 -----0.000018


0.26-1.22 dynamic oxygen absorption -----technique, optical oxygen 0.000019-0.000022 probe (relatively fast response time), CSTR model

1.13 -----0.000017

1.16 -----0.000018

1.12-9.3 -----0.000018

1.18-9.53 staedy-state sulfite oxidation -----technique 0.000017-0.000018




0.06-0.1 -----1.25

0.1-6.3 ------

0.058-0.1 -----2.23

0.2 -----1.6

0.162 -----1.81

Vandu and Krishna44

Lau et al.15

Chaumat et al.45

Han and Al-Dahhan46


Jin et al.43

investigators

Table 3. Continued

ring/perforated plate sparger -----0.0005-0.0025 -----4-163

ring sparger -----0.0005-0.001 -----260-1037


perforated plate -----0.0005 -----199-2750



0.01-0.60 ------

0.041-0.13 -----0.04-0.08

CO2-water/cyclohexane -----293 -----101.3 air-water -----298 -----101.3

0.03-0.2 -----0.0008-0.0026

0.007-0.4 ------

0.034-0.067 ------


N2/air-water/paratherm -----293-364 -----101.3-4240

-----293 -----101.3

air-water -----298-383 -----1000-3000 air-water/ tetradecane/paraffin oil/tellus oil


997 -----0.00089 -----0.072

780-998 -----0.00089-0.001 -----0.024-0.072

804-997 -----0.0005-0.051 -----0.016-0.072

763-998 -----0.001-0.075 -----0.023-0.072

956-997 -----0.00026-0.00089 -----0.0618-0.072


1.18 -----0.000018

1.54 -----0.00014

0.95-51.6 -----0.000018

1.12 -----0.000017

1.18 -----0.000018


dynamic oxygen absorption technique, optical oxygen probe (relatively fast response time), axial dispersion model (ADM)

steady-state carbon dioxide absorption/desorption technique, chromatography, plug-flow model (PFM)

dynamic oxygen desorption technique, optical oxygen probe, CSTR model and also


dynamic oxygen absorption technique, chromatography, CSTR model



Ind. Eng. Chem. Res., Vol. 48, No. 9, 2009 4225 Table 4. Summary of the Published Literature on Experimental Details: Range of Variables Covered and Measurement Techniques for a

investigators



Voyer and Miller47

0.14 -----1.26


Burckhart and Deckwer 48

0.15-0.20 -----3.5-7.5

Oels et al.49


range of gas properties -----density (kg/m3)


0.09-0.63 -----0.015

997 -----0.00089 -----0.072

1.14

absorption accompanied by chemical reaction of CO2 in NaOH soln

ring sparger/ gas: oxygen sintered plate liquids: water/ -----aqueous NaCl/ 0.001-0.00015 aqueous Na2SO4 -----soln 56-7512 -----289 -----101.3

0.0027-0.058 ------

999-1038 -----0.0009-0.0012 -----0.071-0.073

1.22

photographic observations (Sauter mean diameter approach)

0.14 -----3.2

perforated gases: air, He, O2, plate/sintered CO2 liquids: water/ plate aqueous NaCl ejector/ soln/CCl4/ injector type aqueous glycol/ -----0.0000175-0.004 aqueous Na2SO3/ aqueous methanol -----soln 1-470000 -----289-298 -----101.3

0.001-0.073 -----0.012-0.022

985-1089 -----0.00083-0.0010 -----0.065-0.072

1.18-1.20


0.010 -----3.2

ejector type -----0.004 -----1

O2-aqueous Na2SO3 soln -----293 -----101.3

0.010-0.42 -----0.001-0.008

1086 -----0.0015 -----0.075

1.33

sulfite oxidation method

Schumpe and Deckwer51

0.10-0.14 -----1.88-2.16

perforated O2-aqueous plate/sintered Na2SO3 soln plate ----------298 0.001-0.00015 ----------101.3 73-2989

0.004-0.18 -----0.006-0.012

1080 -----0.001 -----0.073

1.18


Quicker and Deckwer22

0.095 -----1

single hole -----0.0009 -----1

air-xylene/decalin/ C10-C14/ vestowax -----301 -----101.3

0.0016-0.08 ------

1070 -----0.00099 -----0.072

1.17


Schumpe and Deckwer24

0.14 -----2.16


air-aqueous sucrose/aqueous CMC/aqueous sodium polyacrylate soln -----298 -----101.3

0.003-0.048 -----0.006

1080 -----0.001 -----0.073

1.18

(1) photographic observations (Sauter mean diameter approach), (2) sulfite oxidation method

Kulkarni et al.21

0.075 -----2.12

ring sparger -----0.001 -----18


0.003-0.022 -----0.195-0.34

1080 -----0.001 -----0.073

1.18-1.30


Bucholz et al.52

0.15 -----2.72

perforated air-distilled water, plate/ejector air-aqueous type methanol soln ----------0.0005-0.001298 298 ----------1-181 101.3

0.011-0.08 -----0.01

985-997 1.18 -----0.000086-0.000089 -----0.069-0.072

Tomida et al.

50

air-water -----298 -----101.3



electrical conductivity/ resistivity measurements

4226 Ind. Eng. Chem. Res., Vol. 48, No. 9, 2009 Table 4. Continued

investigators








Schumpe et al.53

0.095 -----1

single hole -----0.0009 -----1

oxygen-water, oxygen-aqueous Na2SO3 soln -----298 -----101.3

0.008-0.07 ------

1100 -----0.00126 -----0.074

1.18

s

Sedelies et al.54

0.3 -----2.92

perforated plate -----0.001 -----440-1261

gas: oxygen liquids: aqueous Na2SO3 soln -----293 -----101.3

0.007-0.75 -----0.006-0.015

1086 -----0.0015 -----0.075

1.20


Popovic and Robinson23

0.557 -----1.88



0.006-0.09 ------

1086 -----0.0015 -----0.075

1.20

(1) photographic observations (Sauter mean diameter approach); (2) light scattering method,. (3) sulfite oxidation method

Popovic and Robinson55

0.557 -----1.88


O2-aqueous CMC soln -----293 -----101.3

0.01-0.1 ------

1102-1100 -----0.032-0.538 -----0.075

1.20


Sada et al.56

0.078 -----1.2


oxygen-aqueous Na2SO3 soln -----298 -----101.3

0.016-0.17 ------

1080 -----0.001 -----0.073

1.30


air/He-water/ aqueous ethanol soln/methanol/ acetone/ethanol. -----284-293 -----101.3-5000

0.01-0.03 ------

791-1000 -----0.00035-0.003 -----0.022-0.073

1.18-63.2

electrical conductivity/ resistivity measurement

Idogawa et al.

57

0.05 -----0.66


Oyevaar et al.

58

0.081 -----0.81

perforated CO2-DEA plate/sintered -----plate 298 ----------0.00003-0.0004 146-8000 -----21-41345

0.01-0.1 ------

1023 -----0.00197 -----0.072

1.69-78.89


Patel et al.59

0.05 -----2.4

perforated plate/single hole -----0.002 -----1-19

air-water -----298 -----101.32

0.01-0.09 ------

997 -----0.00089 -----0.072

1.18

Sauter mean diameter approach

Wolff et al.60

0.3 -----8.8

ring sparger -----0.001 -----316

air-water -----293 -----101.32

0.008-0.09 ------

998 -----0.001 -----0.072

1.20



air-water -----293 -----101.32

0.009-0.03 ------

998 -----0.001 -----0.072

1.18


Jamialahmadi and 0.152 Muller-Steinhagen61 -----1.6

Ind. Eng. Chem. Res., Vol. 48, No. 9, 2009 4227 Table 4. Continued

investigators








Grund et al.39

0.15 -----0.66-4.3


gas: air liquids: water/ methanol/toluene/ ligroin -----293 -----101.3

0.014-0.178 ------

714-998 -----0.00047-0.001 -----0.020-0.072

1.28-1.34


Wilkinson et al.40

0.158-0.4 -----1.46-1.5

ring sparger/ perforated plate -----0.01-0.001 -----19/16

gases: He/N2/Ar/ CO2/SFl6 liquids: sodium sulfite -----293 -----101.3-800

0.001-0.12 ------

1086 -----0.0015 -----0.075

1.18-9.53

(1) photographic observations (Sauter mean diameter approach), (2) sulfite oxidation method

Stegeman et al.62

0.156 -----0.64


CO2/ nitrogen-water/ DEA/ethylene glycol -----298 -----1100-6600

0.01-0.05 ------

1045-1090 -----0.0026-0.0090 -----0.049-0.055

12.25-67.07


0.15-0.2 -----2

sintered plate/ perforated plate -----0.0003 -----32-4012

air-water -----298 -----101.3

0.0072 ------

997 -----0.0009 -----0.072

1.18


0.28 -----1.48

ring sparger -----0.0015 -----24

air-water -----298 -----101.3

0.1-0.3 ------

997 -----0.00093 -----0.072

1.18

based on literature correlations

0.31 -----2.24

ring sparger -----0.005 -----108

N2, air-toluene/ benzoic acid/ benzaldehyde -----300 -----176-800

0.05-0.15 ------

862-900 -----0.00054-0.00083 -----0.027-0.028

1.97-9.30

-

Dursun and Akosman65

0.05 -----0.64

single hole -----0.002-0.004 -----1

air-water/aqueous glycerol/aqueous glucose soln -----298 -----101.3

0.00025-0.0006 997-1010 ----------0.012-0.025 0.00089-0.001 -----0.071-0.073

1.18


Majumder et al.66

0.05 -----1.28

ejector type -----0.005 -----1

air-water -----302 -----101.3

0.0017 -----0.07-0.14

995 -----0.00082 -----0.071

1.17


Han and Al-Dahhan46

0.162 -----1.81

perforated plate air-water ----------0.0005-0.00132 298 ----------163 101.3-1000

0.02-0.6 ------

997 -----0.00089 -----0.072

1.18-11.72


Bouaifi et al.

Li et al.

63

64

Lemoine et al.

29

These additives cannot be easily analyzed quantitatively or even detected. Further, procedures are not yet evolved for the measurement of modifications in the physical properties of the liquid phase in the presence of these impurities and surfaceactive agents.

(9) The gas-phase distributors are mainly of three types: perforated plates (or pipes, rings, multiple rings, and spiders), sintered plates, and ejectors/injectors. The distributor design governs the generation of the bubble size distribution. Sintered plates, ejectors, and injectors generate finer bubbles because of

4228 Ind. Eng. Chem. Res., Vol. 48, No. 9, 2009 Table 5. Range of Input and Output Parameters of SVR-Based Correlation for Volumetric Mass-Transfer Coefficient and Effective Interfacial Area parameters range of output parameters

kLa (1/s) 0.000245-0.7465

a (1/m) 1.278-8737.4

Operating Conditions pressure (kPa) temperature (K) superficial gas velocity (m/s) superficial liquid velocity (m/s)

22.65-4240 289-583 0.003-0.602 0-0.399

100-8093.9 284-456 0.000125-0.76539 0-0.34

Gas-Liquid Properties gas density (kg/m3) viscosity (Pa · s) molecular weight liquid density (kg/m3) viscosity (Pa · s) surface tension (N/m) ionic strength of electrolyte solution (kion/m3) diffusivity of gas into liquid (m2/s)

0.187-51.6 (1.33 × 10-5)-(2.954 × 10-5) 4-44

0.16709-78.8921 2.0158-44.0098

677.52-1593 (7.98 × 10-5)-22 0.0146-0.080 1-7 (2.29 × 10-10)-(3.21 × 10-7)

714-1593 0.000309-0.0096 0.0182-0.0755 1-3 -

Reactor Geometry column diameter (m) liquid height (m) sparger type sparger hole diameter (m)

0.04-5.5 0.542-7 ring, single nozzle, multiple nozzle, perforated plate, sintered plate, spider, cross, toroidal (1.7 × 10-5)-0.05

the high pressure drop and high energy dissipation rates across these distributors. In any case, whatever bubble size distribution prevails near the sparger gets modified as the bubbles rise as a result of breakup and coalescence. Finally, the equilibrium bubble size distribution is attained depending on the energy dissipation rate in the bulk, the surface tension, and the coalescing property of the liquid phase. The distance over which the equilibrium bubble size distribution is attained (from that prevailing near the sparger) is called the sparger region. Obviously, the height of the sparger region depends on the sparger design and has relatively high values for sintered plates and ejectors/injectors as compared to perforated plates. Therefore, for perforated spargers and coalescing-type systems, the sparger region spans up to only a height-to-diameter ratio of 3. For sintered plates, ejectors, and injectors, this value can go up to 12. Further, for any distributor, the sparger region becomes longer as the coalescing property of the liquid phase decreases. All of these points become important when selecting the sources of data for kLa and a. In particular, for a given case, one needs to examine the role of the sparger region in the overall performance of the column as represented by the measured values of kLa and a. This point is particularly important when the column diameter is smaller than 0.1 m, and it is generally believed that the data from smaller columns are not scalable. The foregoing discussion brings out important points that the measured values of kLa and a depend on a large number of parameters. To clarify this point, all of the reported measurement over the past 40 years are shown in Figure 5 (kLa) and Figure 6 (a). In the past, large numbers of empirical and semiempirical correlations were developed for kLa and a, as listed in Tables 1 and 2, respectively. The performance of these correlations in terms of their predictive capabilities has already been shown in Figures 1 and 2, respectively. It is known that correlations lose their capabilities when (i) all of the important variables are not included in the correlation, (ii) limited data are used for the development of the correlation, and (iii) estimations are made outside the range of variables for which the correlation was originally developed.

0.05-0.557 0.1633-8.8 ring, single nozzle, multiple nozzles, spider, perforated plate, sintered plate, ejector and injector (1.75 × 10-5)-0.01

Apart from the above-mentioned reasons, the most important reason that these correlations lose their capabilities is that the effect of any variable on kLa and a depends on the ranges of many other parameters (which are 17 and 15 in number for kLa and a, respectively). For instance, Figure 7 shows the large variation in the effects of superficial gas velocity (Figure 7A), pressure (Figure 7B), and liquid viscosity (Figure 7C) at various values of VG. From this information, one can imagine the complexities arising from the interdependencies among the variables and, hence, the possibility of success for any empirical or semiempirical correlation. In view of the foregoing discussion about the current status of such efforts, it was thought desirable to employ the SVR strategy, as described in the following section. The subject of the selection of good data sources is also covered. It can be pointed out that, in the present work, systems containing non-Newtonian liquids have not been considered. Also, a data set for viscous Newtonian systems for a could not be obtained, and hence, such systems are not considered in the present study. 3.2. Procedure for Estimating Regression Function Correlation for kLa and a_. A careful analysis of the literature data indicates that the following parameters (17 in number) affect kLa: superficial gas velocity (VG), superficial liquid velocity (VL), sparger type, number of holes in the sparger (No), sparger hole diameter (do), viscosity of the liquid (µL), surface tension of the liquid (σL), density of the liquid (FL), ionic strength of the liquid phase, diffusivity of the solute gas into the liquid (DL), viscosity of the gas (µG), density of the gas (FG), molecular weight of the gas (MG), column diameter (D), liquid height (HL), operating pressure (P), and operating temperature (T). Similarly for a, 15 parameters were identified as having significant effects: superficial gas velocity (VG), superficial liquid velocity (VL), sparger type, number of holes in the sparger (No), sparger hole diameter (do), viscosity of the liquid (µL), surface tension of the liquid (σL), density of the liquid (FL), ionic strength of the liquid phase, density of the gas (FG), molecular weight of the gas (MG), column diameter (D), liquid height (HL), operating


Figure 5. Volumetric mass-transfer coefficient as a function of superficial gas velocity for (inset showing the amplified view for VG < 0.1 m/s).

Figure 6. Interfacial area as a function of superficial gas velocity (inset showing the amplified view for VG < 0.1 m/s).

pressure (P), and operating temperature (T). The ranges of these variables are listed in Table 5 for kLa and a. For the purpose of building comprehensive data sets for kLa and a, an extensive literature search was done spanning the years 1965-2008. The data sets were collected from 27 sources for kLa, giving 1600 experimental data points, and from 28 open sources for a, giving a total of 1330 experimental data points. In addition to the care needed while selecting the data sets, the following additional “quality check” was also incorporated. In the case of manual data collection (reading the data points from the graphs of published literature), care was taken that

the data were extracted from the literature with an error of not more than 2% in any case. This can be explained on the basis of the resolution of the WINDIG 2.5 software considered for the extraction of the data sets. The resolution of the software stands to be 550 × 400 pixels, constituting the total area of the plot used for the data extraction. The data from each of these plots were extracted with a deviation not exceeding 5 × 5 pixels, thus maintaining the acceptable standards of data extraction error. Further, the quality of the data being collected was clearly reflected in terms of the results/predictions given by the final SVR-based models for various gas-liquid systems. After comparing the performance of the SVR-based models with the unknown test data sets (data not considered for training of the model), it can be seen (section 3.3) that the SVRbased models showed substantially improved agreement with the actual values of the test data set. Thus, in a way, such a fine agreement between the models and the actual values shows the quality of the data being considered for training of the SVR-based models. For the estimation of the regression function, an SVR implementation known as epsilon-SVR in the LIBSVM software library70 was used to develop the SVR-based models for kLa and a. The LIBSVM package ensures fast and efficient solutions for large quadratic programming problems by making use of sequential minimal optimization. The procedure for using the LIBSVM software is described in detail by Gandhi et al.71 For the regression function developed using the epsilon-SVR-based formalism, the best values of C, ε, and γ were obtained by using the standard k-fold crossvalidation procedure. Thus, for a given combination of width of the RBF kernel (σ), C, and ε, the training data set is randomly divided into k equal-sized subsets. After this, k models are constructed by leaving out a different subset each time, with the remaining (k - 1) subsets collectively representing the training set. An average of the error corresponding to the left-out subsets, known as crossvalidation error, gives an estimate of the model performance if a large-sized data set were available for building the model. After evaluation of the model for the wide range of model parameters, a grid search methodology gives the optimal values of C, γ, and ε. The resulting optimal values of the three model parameters are represented in Table 6 for kLa and a. Thus, for these model parameters, the corresponding numbers of support vectors were 1357 for kLa and 1201 for a. Thus, all of the data sets without any scrutiny gave SVRbased correlations with average absolute relative errors (AAREs) of 10.32% for kLa and 6.32% for a. It was found that some data points resulted in systematic errors, particularly when the column diameter was smaller than 0.1 m and when a sintered sparger was used. Perforated plates having holes with inner diameters of less than 0.5 mm also resulted in systematic errors. Therefore, the corresponding data sets were eliminated,10,12,44 and the SVR-based correlations were developed on the basis of the remaining data sets. After the data sets giving rise to systematic errors had been removed, the resulting AAREs were reduced to 7.12% from 10.32% for kLa and to 6.32% from 5.01% for a. 3.3. Performance by SVR-Based Correlation for Test Data Sets. For the validation of the SVR-based technique, the models were also compared against data sets for which the model was not trained (i.e., test data sets). The simulation conditions were selected so that the effects of various design and operating conditions on kLa and a could be observed.


Figure 7. Plots showing the effect of various parameters on volumetric mass-transfer coefficient. (A) Effect of superficial gas velocity at various pressures. (B) Effect of pressure at various superficial gas velocities. (C) Effect of viscosity at various superficial gas velocities. Table 6. Parameter Selection for Generalized SVR-Based kLa and a Models

and sparger type on the kLa values were well reflected by the SVR model (see Table 7).

number of number of support training vectors data points

Along similar lines, the SVR model for a was tested for the data set of Botton and Charpentier,73 the simulation conditions of which are listed in Table 7. Here, the effects of changes in the superficial liquid velocity, sparger geometry, and column diameter on the a values are well reflected by the corresponding SVR model. Even though the column diameter was below 0.1 m for the simulation conditions under consideration, the results obtained with the SVR model were in good agreement with the actual values (see Table 7).

model

C

volumetric mass-transfer 5 coefficient (kLa, 1/s) effective interfacial 46000 area (a, 1/m)

γ ) 1/2σ 5 188

2

ε 0.001

1357

1600

0.96

1201

1330

For this purpose, SVR simulations were carried out using the test data sets for systems from the open literature, and the ranges of these test data sets along with the performance of the SVR-based models are reported in Table 7. The kLa simulations were carried out at the experimental conditions of Eickenbusch.72 The effects of changing the column diameter

Thus, from the foregoing discussions, the SVR models for kLa and a demonstrate their utility as design estimation tools for bubble column reactors.

Ind. Eng. Chem. Res., Vol. 48, No. 9, 2009 4231 Table 7. Simulation Conditions and Results for Various Gas-Liquid Systems for kLa and a details

SVR model for kLa

SVR model for a

72

Botton and Charpentier73 (air-electrolyte) 101.325 293 0.00012-0.042 0-0.025 1.20 28.84 1086 0.0015 0.075 3 0.02/0.075 0.2-3.44 perforated plate 3-12 0.0005-0.007 0.98 2.59

author(s) (gas-liquid system)

Eickenbusch

pressure (kPa) temperature (K) superficial gas velocity (m/s) superficial liquid velocity (m/s) gas density (kg/m3) gas viscosity (Pa · s) gas molecular weight liquid density (kg/m3) liquid viscosity (Pa · s) liquid surface tension (N/m) ionic strength of electrolyte solution (kion/m3) diffusivity of the gas into the liquid (m2/s) column diameter (m) liquid height (m) sparger type number of sparger holes sparger hole diameter (m) correlation coefficient (CC) AARE (%)

101.325 298 0.019-0.09 0 1.20 0.0000182 28.84 997 0.00089 0.072 1 0.0000000024 0.19/0.29/0.6 2.8-5.75 ring, perforated plate 8-89 0.002 0.98 3.14

(air-water)

Table 8. Performance Indicators for Previous Literature Correlations for Various Gas-Liquid Systems for kLa and a correlation for kLa 1

Akita and Yoshida Deckwer et al.2 Wang and Fan3 Nakanoh and Yoshida4 Hikita et al.5 Ozturk et al.6 Posarac and Tekic7 Seno et al.8 Suh et al.9 Sotelo et al.10 Dewes and Schumpe11 Kojima et al.12 Jordan et al.13 Bando et al.14 Lau et al.15 SVR (this work)

CC 0.83 0.21 0.19 0.71 0.68 0.66 0.52 0.90 0.57 0.27 0.48 0.58 0.42 0.62 0.40 0.99

AARE (%) 32.13 95.15 97.35 52.75 55.23 51.34 65.12 28.13 68.34 87.34 68.23 69.10 74.34 71.23 78.34 7.12

3.4. Comparison of the SVR-Based Correlation to Literature Correlations. To check the applicability and performance of the unified model against the literature correlations for various gas-liquid systems, data for the gas-liquid systems for which the literature correlations were applicable were subjected to tests against the SVR-based models. The simulation results for kLa and a are reported in Table 8. It can be seen that the SVR-based models perform much better than any of the correlations proposed in the literature by various authors. Figure 5 shows a comparison between the generalized SVR-based unified model for kLa and the correlations proposed by various researchers (as mentioned in Table 1), and Figure 6 shows a comparison between the generalized SVR-based unified model for a and the correlations proposed by various researchers (as mentioned in Table 2). It can be seen from Figures 5 and 6, as well as from the values reported in Table 8, that the SVRbased models give much better correlations than the various correlations reported for the available data sets. 3.5. Parametric Sensitivity Analysis. Parametric sensitivity analyses of the proposed model were carried out by checking the effects of the input parameters on kLa and a in bubble column reactors. The parametric sensitivity analyses involved examining the effects of individual parameters on the values of kLa and a obtained with the models. Thus, to check the effects of individual parameters on the values of kLa, simulations were performed at a constant gas velocity of 0.065 m/s for the homogeneous regime and at a constant gas

correlation for a 16

Akita and Yoshida Dierendonck et al.17 Gestrich and Krauss18 Tabei et al.19 Serizawa and Kataoka20 Kulkarni et al.21 Quicker et al.22 Popovic and Robinson23 Schumpe and Deckwer24 Kawase et al.25 SVR (this work)

CC

AARE (%)

0.41 0.45 0.28 0.43 0.33 0.51 0.25 0.27 0.28 0.22 0.98

62.20 54.70 86.20 57.13 60.35 51.40 89.32 87.34 87.30 92.42 5.01

velocity of 0.14 m/s for the heterogeneous regime. Figures 8A and 9A show the various data on kLa with respect to superficial gas velocity for the homogeneous and heterogeneous regimes, respectively. Along similar lines, Figures 10A and 11A show the various data on a with respect to superficial gas velocity for the homogeneous and heterogeneous regimes, respectively. From the four plots, it can be seen that, for a particular value of VG, multiple values of kLa and a exist. This is due to the effects of individual parameters such as sparger type, number of holes in sparger, hole diameter, operating pressure and temperature, physical properties of the gas and liquid, and column diameter on kLa and a. All of the effects mentioned above are well captured by the SVR model, as shown by the parity plots in Figures 8B and 9B for kLa and in Figures 10B and 11B for a. 3.6. Establishment of the SVR-Based Model for Homogeneous and Heterogeneous Regimes. Better performance is expected when the correlations are developed separately for homogeneous and heterogeneous regimes. Therefore, the total points mentioned above (1600) were classified into two separate SVR-based models for the homogeneous (992) and heterogeneous (485) regimes. The resulting AAREs were found to be 10.12% and 4.21% for the homogeneous and heterogeneous regime models, respectively. Thus, one can see a substantial improvement in the correlation based on data for


Figure 8. Parametric sensitivity analysis for volumetric mass-transfer coefficient (VG ) 0.065 m/s). (A) Volumetric mass-transfer coefficient as a function of superficial gas velocity. (B) Parity plot showing the predictions by the SVR-based model.

Figure 9. Parametric sensitivity analysis for the volumetric mass-transfer coefficient (VG ) 0.14 m/s). (A) Volumetric mass-transfer coefficient as a function of superficial gas velocity. (B) Parity plot showing the prediction by the SVR-based model.

the heterogeneous regime. A separate correlation for the homogeneous regime did not result in significant improvement. 3.7. Web-Based SVR Models for kLa and a. Because the numbers of support vectors are large (1357 for kLa and 1201 for a), the support vectors are made available on the Web. For this purpose, an SVR simulation tool was prepared and made available on the Web for the prediction of kLa for bubble column reactors. This tool is named SVR _MTC_BC and can be downloaded from the URL http://www.esnips.com/web/UICTNCL. This tool is in the form of a Microsoft Excel spreadsheet wherein the sheet of interest is named SVR model kLa. Here, one can insert the desired input features (17), and the cell named predicted volumetric mass-transfer coefficient gives the value of the predicted output (kLa). The prediction is made on the

basis of the procedure described in the earlier sections. The support vectors are listed in their scaled format in the sheet named support vectors kLa. The calculation of kernel elements using eq 9 is done in a sheet named kernel elements kLa. This sheet is password-protected to ensure that the calculations are not tampered with, as the sheet contains vital information for the model. The summation of the product of the abovementioned kernel elements and the corresponding nonzero Lagrange multipliers for each of the support vectors with addition of the bias term gives the predicted output as per eq 7. This prediction is shown in the cell named predicted volumetric mass-transfer coefficient for kLa in the sheet labeled SVR model kLa. Also, similar models can be found for the homogeneous regime (SVR_homoMTC_BC) and the heterogeneous regime (SVR_hetroMTC_BC).


Figure 10. Parametric sensitivity analysis for interfacial area (VG ) 0.065 m/s). (A) Interfacial area as a function of superficial gas velocity. (B) Parity plot showing the predictions by the SVR-based model.

Figure 11. Parametric sensitivity analysis for interfacial area (VG ) 0.14 m/s). (A) Interfacial area as a function of superficial gas velocity. (B) Parity plot showing the predictions by the SVR-based model.

4. Conclusions Following a similar methodology, a tool for a has been developed and can be downloaded from the same URL. This particular tool is named SVR_AREA_BC and is in the form of a Microsoft Excel spreadsheet wherein the sheet of interest is named SVR model a. The cell named predicted effective interfacial area gives the value of the predicted output (a) based on desired input features (15). The support vectors are listed in their scaled format in the sheet named support vectors a, and the calculation of the kernel elements using eq 9 is done in the sheet named kernel elements a. Along similar lines, separate models have been proposed for the homogeneous and heterogeneous regimes for kLa and a (SVR_homoMTC_BC, SVR_ hetroMTC_BC, SVR_homoAREA_BC, and SVR_hetroAREA_ BC).

(1) In the case of bubble columns, all of the published correlations for mass-transfer coefficient (kLa, of which there are 15) and effective interfacial area (a, of which there are 10) have been analyzed. A critical assessment has been made regarding their predictive capabilities, as shown in Figures 1 and 2. It has been shown that the published correlations have four limitations: (a) all of the governing parameters are not considered, (b) data sets have been selected over a limited range of parameters, (c) a procedure was needed to confirm the quality of data sets used for developing the correlations, and (d) all of the published correlations do not consider the interdependence of parameters on kLa and a. (2) A discussion has been provided on the selection of data sets (section 3.1). Further, a stepwise procedure has been


provided on checking the quality of the selected data sets (section 3.2). (3) The SVR-based correlations show remarkable improvement for the prediction of kLa and a for bubble column reactors, as compared to other types of empirical and semiempirical correlations available in the literature. The generalized SVRbased correlations for the prediction of kLa and a yield AAREs of 7.12% and 5.01%, respectively, with corresponding percent prediction accuracies of 99.08% and 98.6%, which are far better than those obtained through the selected literature correlations. (4) The SVR-based correlations give enhanced and more accurate predictions for a variety of gas-liquid systems over a wide range of operating pressures, operating temperatures, superficial gas and liquid velocities, column diameters, and liquid heights. Hence, the proposed SVR-based correlations are expected to be useful in the design and scale-up of bubble column reactors. (5) We have provided the SVR-based correlations on the Web site http://www.esnips.com/web/UICT-NCL. We invite the users of these correlations to communicate their deviations, which will enable the improvement of the data-driven model. Acknowledgment P.P.G. is grateful to the University Grants Commission, Government of India, New Delhi, India, for financial support. Thanks are also given to Prof. Dr. Adrian Schumpe for sending the articles. A.B.G. thanks the Council of Scientific and Industrial Research (CSIR), Government of India, New Delhi, India, for a Senior Research Fellowship (SRF). V.K.J. acknowledges financial support received from the Department of Science and Technology, Government of India, New Delhi, India. Nomenclature a ) effective interfacial area, 1/m AARE ) average absolute relative error ) N

(1⁄N) ∑ |ypredicted - yexperimental / yexperimental | i)1 b ) bias term C ) cost function CC ) correlation coefficient ) N

⁄

∑ [yexperimental(i) - yexperimental(mean)][ypredicted(i) - ypredicted(mean)]

i)1

N

∑ [yexperimental(i) - yexperimental(mean)]2

i)1

N

∑ [ypredicted(i) - ypredicted(mean)]2

i)1

D ) column diameter, m DL ) diffusivity of solute gas into liquid, m2/s do ) sparger hole diameter, m f(x) ) regression function f ) correction factor for the effect of the presence of electrolyte in water g ) acceleration due to gravity, m/s2 I ) ionic strength of the electrolyte solution, g of ions/m3 kL ) liquid mass-transfer coefficient, m/s kLa ) volumetric mass-transfer coefficient, 1/s K(xi,xj) ) kernel function Kd ) sparger distribution coefficient L ) Lagrangian function (dual form) MG ) molecular weight of gas, kmol/kg Nsv ) number of support vectors P ) operating pressure, kPa

R ) input space V ) superficial gas velocity, m/s w ) weight vector x ) superficial gas velocity/pressure/liquid viscosity (Figure 7A-C) xi ) ith input vector y ) volumetric mass-transfer coefficient (Figure 7) yi ) target output corresponding to the ith vector * Ri,j ) Lagrange multipliers γ ) gamma [1/(2σ2)] ε ) loss function G ) overall gas holdup µ ) viscosity, Pa · s ξi* ) slack variables σ ) width of radial basis function (RBF) kernel σW ) surface tension of water, N/m F ) density, kg/m3 φ (xi) ) mapping function to high-dimensional feature space for input vector Subscripts G ) overall gas phase L ) overall liquid phase Superscript N ) number of training data points

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ReceiVed for reView March 3, 2008 ReVised manuscript receiVed January 12, 2009 Accepted January 15, 2009 IE8003489

Development of Unified Correlations for Volumetric Mass-Transfer

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