Download Gas}liquid interfacial mass transfer in trickle-bed reactors: state

Chemical Engineering Science 54 (1999) 5633}5645
Gas}liquid interfacial mass transfer in trickle-bed reactors:
state-of-the-art correlations
Ion Iliuta , FamK c7 al Larachi *, Bernard P. A. Grandjean , Gabriel Wild
Department of Chemical Engineering and CERPIC, Laval University, Que& bec, Canada G1K 7P4
Laboratoire des Sciences du Ge& nie Chimique, CNRS-ENSIC, 1 Rue Grandville, BP 451, 54001 Nancy, France
Received 3 August 1998; received in revised form 12 February 1999; accepted 12 February 1999
Abstract
The state-of-the-art of the gas}liquid mass transfer characteristics in trickle-bed reactors was summarized and its quanti"cation methods were reevaluated based on a wide-ranging data base of some 3200 measurements. A set of three uni"ed whole#ow-regime dimensionless correlations for volumetric liquid- and gas-side mass transfer coe$cients, and gas}liquid interfacial
area, each of which spanned four-order-of-magnitude intervals, were derived. The correlations involved combination of arti"cial
neural networks and dimensional analysis. The dimensionless interfacial area, Sh and Sh were expressed as a function of the
*
%
most pertinent dimensionless groups: Re , Re , =e , =e , Sc , Sc , St , X , Mo , Fr , Eo , S . 1999 Elsevier Science Ltd.
*
%
*
%
*
%
* %
*
*
K @
All rights reserved.
Keywords: Trickle-bed reactor; Arti"cial neural networks; Dimensional analysis; Gas}liquid mass transfer; Correlations
1. Introduction
Trickle-bed reactors (TBRs) are randomly packed
"xed-bed tubular reactors in which gas and liquid
streams are processed in downward cocurrent two-phase
#ow. TBR technology is prevalent in diverse industrial
areas and spans a wide spectrum of applications including the manufacture of petroleum derivatives and fuels
(Chen and Tsai, 1997; Landau et al., 1998), the production of commodity and specialty chemicals (Westerterp
and Wammes, 1992; Chou et al., 1997), pharmaceuticals,
pesticides and herbicides (Khadilkar et al., 1998; Jiang
et al., 1998), pollution abatement and bio-scrubbing, etc.
(Cheng and Chuang, 1992; Pintar et al., 1997; Ravindra
et al., 1997; WuK bker et al., 1997). The permanent market
demands at meeting high-quality products while ful"lling
both environmental and economical requirements, drives
the researchers and developers to improve constantly
On leave from Department of Chemical Engineering, Faculty of
Industrial Chemistry, University Politehnica of Bucharest, Polizu 1,
78126 Bucharest, Romania.
* Corresponding author. Tel.: #1-418-656-3566; fax: #1-418-6565993.
E-mail address: #[email protected] (F. Larachi)
their knowledge about the physico-chemical phenomena
taking place in TBRs. However, in spite of considerable
research in the past 40 years, TBR mass transfer is still
not well understood and general correlations of mass
transfer parameters are still out of grasp.
Hence, it is well recognized that Achilles' heel of the
currently available TBR design tools lies in their lack of
generality inherent partly to the narrow experimental
windows upon which validation of most correlations/models is carried out, and partly to biased analysis
and inappropriate mathematical expressions. Yet a decade ago, a French initiative led to the so-called Ellman
TBR correlations (Ellman et al., 1988,1990; Wild et al.,
1992) which were considered among the best empirical
correlations as they covered the broadest database at
that time (ca 5000 measurements). Unfortunately, lack of
success of these correlations was repeatedly pinpointed in
the recent literature (Ratnam and Varma, 1992; Benkrid
et al., 1997; AndreH , 1997; Chen, 1998; Al-Dahhan et al.,
1998; Larachi et al., 1998a). In response, a renewed e!ort
to reach better predictability of TBR gas}liquid mass
transfer and hydrodynamic parameters was launched by
us with the endeavor of elaborating the most comprehensive two-phase #ow databases based on the worldwide literature published over the last half century on
cocurrent packed beds. There is a twofold goal behind
0009-2509/99/$ - see front matter 1999 Elsevier Science Ltd. All rights reserved.
PII: S 0 0 0 9 - 2 5 0 9 ( 9 9 ) 0 0 1 2 9 - 3
5634
I. Iliuta et al. / Chemical Engineering Science 54 (1999) 5633}5645
this initiative: (i) The "rst goal is to assess and to critically evaluate all available design tools, whether empirical
or theoretical; and (ii) while the second is to eventually
re-engineer these tools through sound approaches which
will lead to the most robust constitutive TBR design
predictors. We thus compiled over 30,000 experimental
data on #ow regime transitions, two-phase pressure
gradients, liquid (external, dynamic, static) holdups, gas}
liquid interfacial areas, and volumetric liquid- and gasside mass transfer coe$cients. Only with comprehensive
operating ranges covering every recess of the #ow operating map of two-phase #ow in packed beds, can state-ofthe-art correlations tangibly useful for industry scale-up
purposes be developed. Our approach rests on combining dimensional analysis with neural networks to train
neural network models based on huge data sets and then
make predictions based on such models. Two recent
articles illustrate the possibilities of such an approach for
packed bubble columns (Bensetiti et al., 1997; Larachi
et al., 1998b).
This work aims at providing researchers and engineers
with uni"ed, yet most accurate gas}liquid mass transfer
explicit and whole-#ow-regime correlations for interfacial areas, volumetric liquid-side and gas-side mass
transfer coe$cients in trickle-bed reactors based on beyond 3200 mass transfer data published in the literature
between 1963 and 1998 (a comprehensive bibliography
on TBR mass transfer consisting of over 70 references is
included in this report). The e!ectiveness of the developed neural correlations will be discussed through
comparisons against most popular literature mass transfer correlations. The methodology leading to these neural
network correlations is similar to the one already employed by Bensetiti et al. (1997). Therefore, for the sake of
brevity and to avoid repetitions, we will spare the reader
all the details which may be found in Bensetiti et al.
(1997) and Larachi et al. (1998b).
2. Need for new, improved gas}liquid mass transfer correlations
the reader is invited to consult the References section
where a meticulous report of the whole gas}liquid mass
transfer is given. Table 1 lists the ranges of variation of
the overall mass transfer parameters, #uid properties,
operating conditions, particle and bed geometrical properties, liquids and gases belonging to this database. The
di!erent mass transfer parameters are obtained using
operating conditions typical of cocurrent down#ow
trickle beds, packed absorption as well as desorption
columns. The corresponding solids inventories include
few cm size packings (Raschig and Pall rings, Intalox,
Berl saddles) of high bed porosity (e up to 94%), and few
mm size packings (porous and nonporous spheres, extrudates and pellets) of moderate bed porosity (e)45%).
The super"cial #uid velocity ratio, which is a measure of
gas}liquid interfacial slip and thus intensity of interaction, spans four-order-of-magnitude ranges for each mass
transfer parameter. These latter in turn vary by about the
same amplitudes. The gas}liquid interfacial area includes
conditions of partially (a(a ) to fully wetted (a*a )
Q
Q
packings. It is worth mentioning that the database encompasses the unusual conditions where gas-side mass
transfer resistance exceeds its liquid-side counterpart.
2.2. Dimensional analysis
From considerations stated elsewhere (Bensetiti et al.,
1997), it is proposed to analyze the impact of #uid velocities, densities, viscosities, di!usivities, surface tension,
gravitational acceleration, particle size and shape, bed
diameter and porosity, via their combinations in the
various dimensionless groups that may a!ect the
gas}liquid interfacial mass transfer parameters. Many
hundreds of sets of dimensionless groups were thus tested
by trial and error with the ultimate objective of identifying the optimal sets of groups best describing each of the
gas}liquid interfacial areas, a (via ad /(1!e)), the volF
umetric liquid-side mass transfer coe$cients, k a (via
*
Sh ), and the volumetric gas-side mass transfer coe$*
cients, k a (via Sh ). The choice of the best sets rests on
%
%
the ful"llment of the following criteria:
2.1. Data base compilation
The neural network gas}liquid mass transfer correlations were derived based on an extended mass transfer
database containing more than 3200 measurements compiled from about 70 references in the open literature
between 1963 and 1998. This database includes 52
gas}liquid systems, more than 60 packing sizes and geometries, 17 column diameters, high-pressure data, data
for coalescing, non-coalescing and pseudoplastic nonNewtonian aqueous and organic liquids, and data for
low and high interaction regimes (trickle, pulse, bubble
and dispersed bubble #ows, foaming and foaming-pulsing #ows). To avoid burdening the text with references,
E The optimal set must contain a minimum number of
selected dimensionless groups.
E Each group must be highly cross-correlated with the
corresponding mass transfer parameter group (output
to be predicted).
E The optimal set must lead to the best output prediction, i.e. minimal absolute average relative error
(AARE) and variance.
E The neural network architecture must be of minimal
complexity, i.e. the least number of hidden neurons
giving the smallest errors both on the learning (70% of
the database) and the generalization "les (the remaining database).
I. Iliuta et al. / Chemical Engineering Science 54 (1999) 5633}5645
5635
Table 1
Description of database
Fluid physical properties
Operating conditions
805)o )1450
*
6.32;10\)k )4.72;10\
*
1.06;10\)p )7.77;10\ *
0.937)o )57.46
%
0.1)P)5.1 MPa
19)¹)30.53C
3;10\)< )1.488;10\
1*
1.289;10\)< )4.5
1%
<
k a: 6.3;10\) 1%)6.5;10
*
<
1*
1.66;10\)k )2.6;10\ %
Limits of gas}liquid mass transfer
parameters
2.13;10\)k a)7.04
*
4.76;10\)D )4.0;10\ *
6.6;10\)D )1.6;10\ %
Geometrical properties of packings and columns
Particle diameter, d (m)
N
Column diameter, d (m)
A
Bed porosity, e
Shape factor, Particle material
Particle shape
<
k a: 8;10\) 1%)1.8;10
%
<
1*
<
a: 0.23) 1%)1.5;10
<
1*
2.36;10\)k a)6.94
%
23.4)a)9070
5.4;10\)d )2.64;10\
N
2.3;10\)d )3.0;10\
A
0.263)e)0.94
0.133)
)1.0
Glass, ceramics, porous alumina, carbon, stainless,
polyethylene, polypropylene, CuO}ZnO catalyst
Spherical, cylindrical, extrudates, Raschig and
Pall rings, Intalox, Berl saddles
Cylindrical
Column geometry
Liquids
H O, H O#C H , H O#NaOH[0.1}2 N], H O#Na SO [0.8 M], H O#Na SO [1. M], H O#Na SO [0.5 M]#Co>, H O#Na SO
[0.8 M]#CoSO [5;10\ M], H O#NaOH [0.5}2 N]#Na SO , H O#K CO [0.8}1.3 M]#KHCO [0.65}1.2 M], H O#Na S O ,
H O#MEA [0.25}0.33 M], H O#DEA [1.5}2 M], H O#DIPA [2.4 M], H O#DEA [1.5 M]#ETG [20}40%], H O#40% CaCl ,
H O#naphthalene, H O#CHA [0.227}1.177 M]#toluene#10%IPA, H O#EtOH#MEA, EtOH#MEA [0.2}0.7 M], EtOH#DEA [0.6
0.8 M], IPA#toluene#CHA [0.1}0.7 M], toluene#10% IPA#DIPA, p-xylene#10% i-PrOH#CHA, n-C H OH#MEA, H O#0.1%
CMC, H O#0.5% CMC, H O#1.0% CMC
Gases
He, air, air#CO , air#SO , air#O , air#moisture, N #CO , N #O , CO
Common liquids (Lide, 1993); mixtures (Grunberg and Nissan method, Reid et al., 1987).
Macleod and Sugden method (Reid et al., 1987).
Ideal gas law ()0.1 MPa), if *0.1 MPa (L'Air Liquide, 1976).
Pure gases (L'Air Liquide, 1976); mixed gases (Wilke method: Reid et al., 1987).
Wilke and Chang method (Reid et al., 1987).
Takahashi method (Reid et al., 1987).
Finally, the most relevant dimensionless groups are:
E For Sh : liquid inertia-to-viscous forces ratio (Re ),
*
*
liquid inertia-to-capillary forces ratio, (=e ), gas-to*
liquid inertia forces ratio (X ), capillary forces ac%
counted for via a Morton number (Mo ), momentum
*
di!usivity-to-mass di!usivity ratio (Sc ), and the bed
*
correction function (S ).
@
E For Sh : Re , X , Sc , S , gas inertia-to-capillary for%
* % % @
ces ratio (=e ), and liquid viscosity-to-gravity forces
%
ratio (St ).
*
E For ad /(1!e): Re , =e , X , S , gas inertia-to-visF
*
* % @
cous forces ratio (Re ), gravity-to-capillary forces ratio
%
(Eo ), and inertia-to-gravity liquid forces ratio (Fr ).
K
*
Cross-correlation coe$cients (a ) between the gas}
NO
liquid mass transfer parameters and the di!erent dimen-
sionless groups are listed in Table 2. This table shows
that: Sh increases as Re , =e , X and S increase
*
*
* %
@
and decreases with increasing Mo and Sc ; Sh in*
*
%
creases as Re , =e , Sc , X and S increase and de*
% % %
@
creases with increasing St ; ad /(1!e) increases as
* F
Re , Re , Eo , X and S increase and decreases with
*
%
K %
@
increasing =e and St . It is interesting to note that
*
*
(i) bed/particle geometrical properties, i.e. S , a!ect
@
strongly the gas}liquid mass transfer groups, and (ii) X
%
a!ects Sh and Sh the least, and so does Fr for
*
%
*
ad /(1!e).
F
2.3. Neural regression
A three-layer arti"cial neural network model was designed, using NN"t software (Cloutier et al., 1996), to
5636
I. Iliuta et al. / Chemical Engineering Science 54 (1999) 5633}5645
Table 2
Cross-correlation coe$cients of gas}liquid mass transfer parameters to dimensionless groups
Dimensionless group
Cross-correlation to Sh (%)
*
Cross-correlation to Sh (%)
%
Cross-correlation to
ad /(1!e) (%)
F
Re
*
Re
%
=e
*
=e
%
X
%
St
*
Eo
K
Fr
*
Mo
*
Sc
*
Sc
%
S
@
55.7
}
37.1
}
7.2
}
}
}
!8.9
!10.4
}
64.3
72.6
}
}
92.1
17.5
!19.2
}
}
}
}
66.5
91.3
35.5
49.0
!21.0
}
18.0
}
32.9
!10.1
}
}
}
81.4
derive dimensionless correlations for the gas}liquid mass
transfer parameters. The neural architecture is described
by Eqs. (1) and (2) (Table 3) that correlate the network
normalized output, S, to a set of normalized input variables, ; . In these equations, ; and H de"ne the input
G
and hidden layer vectors, and H
and ;
are the
(>
'>
bias constants set equal to 1, u and u are the weights or
GH
H
the "tting parameters of the neural network, and J is the
number of hidden nodes. The network weights are unknown a priori. They have to be determined using
a training algorithm by performing a nonlinear leastsquares regression over known pseudo-random set of
inputs/outputs (learning "le, ca. 70% of the database, i.e.,
2240 data). The weights are set to minimize the training
error on the training set using a quadratic objective
function which was minimized by the quasiNewton}Broyden}Fletcher}Goldfarb-Shanno algorithm
(Press et al., 1992). A good measure for the extrapolation
performance of a well-trained neural network is given
by the generalization error which should be close to
the training error in the case of inputs/outputs not
presented during the learning step of the neural network
(generalization "le, ca. 30% remaining data set, i.e., 960
data). The number of hidden neurons, J, was varied from
3 to 15. Hidden layers with 8 neurons (for Sh ), 13
%
neurons (for Sh ) and 11 neurons [for ad /(1!e)] were
*
F
found to be the optimal neural architectures leading to
the smallest average absolute relative errors (AARE)
and standard deviations on the learning as well as
the generalization "les. The complete sets of the
neural network equations for the gas}liquid mass
transfer parameters are given in Table 3, and Table 4
lists the "tted weights of each neural correlation.
The detailed form of the three correlations is freely accessible from the net at the following web address:
http://www.gch.ulaval.ca/&#arachi.
2.4. Performance of the neural network correlation, comparison with most general correlations
Figs. 1}3, respectively, are parity plots of the dimensionless overall liquid-side, gas-side mass transfer coe$cient and interfacial area correlations over the learning
and test sets from the database. The respective scatters
between neural network predictions and experimental
values are summarized in Table 5 for the learning "le, the
generalization (or test) "le, and the whole database. As
shown in these "gures, agreement between the experimental results and those predicted by the neural network
correlations is very good. Hence, respectively 55% (Fig.
1a, and b), 75% (Fig. 2a and b) and 50% (Fig. 3a and b) of
the mass transfer data fell within the $15% limits of the
Sh , Sh and ad /(1!e) correlations. Also, we compared
*
%
F
the experimental data with the predictions from the majority of the literature correlation (AndreH , 1997). Table
6 excerpts, among these literature correlations, the ones
which exhibited the lesser scatters (AndreH , 1997). AAREs
given by the neural network correlations (Table 5) and by
the most general literature correlations (Table 7), indicate
that prediction of gas}liquid mass transfer parameters
were improved noticeably. The former outperforms the
latter by up to a factor eight for Sh , a factor nine for Sh
%
*
and a factor xve for gas}liquid interfacial area. Despite
gas}liquid mass transfer parameters are strongly dependent on #ow regimes, the present correlations are capable
of predictions of these mass transfer parameters regardless of the prevailing #ow regime.
3. Closing remarks
Based on the largest gas}liquid interfacial mass
transfer database (beyond 3200 data), new, state-of-art
I. Iliuta et al. / Chemical Engineering Science 54 (1999) 5633}5645
5637
Table 3
Set of equations for the neural network correlations
1
S"
1#exp [!(>u H ]
H H H
(1)
1
H"
H 1#exp [!'>u ; ]
G GH G
Sh (I"6, J"13)
*
sh
*
log
0.3359
S"
5.2428
(2)
(3)
Re
=e
X
*
*
%
log
log
log
1.994;10\
1.851;10\
2.636;10\
, ; "
, ; "
; "
4.898
6.298
4.918
Mo
Sc
S
*
*
@
log
log
log
2.545;10\
44.33
2.552
; "
, ; "
, ; "
, ; "1
10.114
4.042
1.9011
(5)
Sh (I"6, J"8)
%
Sh
%
log
1.311;10\
S"
6.752
(6)
Re
St
=e
*
*
%
log
log
log
2.089;10\
4.323;10\
1.711;10\
, ; "
, ; "
; "
2.6751
3.615
6.8823
X
Sc
S
%
%
@
log
log
log
8.288;10\
2.328;10\
1.740
; "
, ; "
, ; "
, ; "1
0.55765
4.359
2.0205
ad /(1!e) (I"7, J"11)
F
ad /(1!e)
F
log
9.952;10\
S"
4.741
(4)
(7)
(8)
(9)
Re
Re
=e
*
%
*
log
log
log
1.007;10\
1.709
1.275;10\
; "
, ; "
, ; "
4.077
3.4213
5.2032
(10)
Fr
X
Eo
*
%
K
log
log
log
1.088;10\
1.078;10\
2.674;10\
, ; "
, ; "
; "
5.0172
3.651
4.906
S
@
log
1.655
; "
, ; "1
2.1742
explicit and whole-#ow-regime dimensionless correlations for gas}liquid mass transfer parameters in trickle
beds were derived upon combination of dimensional
analysis and arti"cial neural networks. The overall
(11)
(12)
result was a signi"cant improvement in predicting
overall liquid-side and gas-side volumetric mass
transfer coe$cients and gas}liquid interfacial area.
Similarly to any other available correlation in literature,
ad /(1!e)
F
u
GH
1
2
3
4
5
6
7
8
u
G
Sh
%
u
GH
1
2
3
4
5
6
7
u
H
Sh
*
u
GH
1
2
3
4
5
6
7
u
H
1
1.5435
2.2434
2.5155
2.5649
0.9739
1.4736
3.0631
3.8628
1
3.0432
1
!1.6356
2.1719
!1.4329
0.0279
!3.7197
1.9894
0.6121
1
!13.571
1
1.7352
3.7209
!3.1528
10.8069
!5.7494
9.2444
!5.0819
1
5.2505
2
!3.9356
2.5744
0.3614
0.1710
!1.7651
!8.0268
!4.1966
2.4074
2
5.0432
2
0.4413
!4.7885
4.2847
!4.7665
10.2979
!13.042
0.7693
2
!4.4635
2
!1.0859
4.3527
!5.9518
0.3045
!0.8630
1.2997
5.1608
2
!3.9571
3
4.1974
!0.5456
5.6068
!8.1754
0.4768
!6.6128
1.4328
1.4903
3
5.3856
3
!2.6556
3.9519
!0.1984
!2.2429
9.0856
2.0753
!2.0236
3
!4.4251
3
4.4131
!0.9842
1.4652
2.5158
6.0466
10.0231
!6.9573
3
7.6703
4
2.1391
11.0825
!8.5112
!5.0699
!10.376
!10.457
2.5269
9.3200
4
!1.9445
4
!1.6555
!3.0385
!3.2826
!4.6367
0.3069
3.3017
0.7602
4
8.4609
4
5.99 93
2.2940
!5.1562
4.6663
3.2448
8.7118
!3.3028
4
4.3998
Table 4
Fitting parameters of the neural network correlations
5
1.8422
1.0471
8.6948
!12.646
!2.8720
!7.1529
!2.2898
7.5124
5
!5.7894
5
5.1509
!1.4101
!7.2576
6.8145
!0.5302
1.5895
!0.8027
5
5.1391
5
5.53 25
!1.2809
!0.7164
!3.2695
2.0916
!5.8084
3.9598
5
9.9828
6
7.4930
!14.600
0.0311
!6.3510
5.3100
!1.8612
!5.9076
!2.9321
6
!8.4580
6
1.2263
3.4532
!5.5895
4.7365
0.5461
!2.1867
!2.9046
6
8.0094
6
!0.85 89
1.5063
!0.6652
4.2675
!7.2309
!5.6044
5.6439
6
!8.1581
7
7.4055
!11.867
11.3668
!6.5566
11.0910
!1.7484
!2.1967
!9.1011
7
!6.6317
7
8.0269
!2.0732
!8.6778
10.6797
!0.9721
!4.2163
!2.7202
7
!10.884
7
2.7195
!2.1694
!1.2966
!4.6230
0.4930
2.3693
2.6890
7
6.3469
8
!4.6897
!28.205
11.0779
17.1914
32.2031
!1.1837
7.3281
!17.957
8
!1.1868
8
7.1496
!3.7079
!3.8304
7.2395
!4.8951
!3.5211
1.5878
8
8.8342
8
1.8965
5.1352
4.2572
!3.7013
!1.9482
11.4437
!5.2396
8
!4.3963
9
!8.8114
6.4693
12.4214
!0.4853
1.9989
!1.8698
9.4666
!0.2705
9
!8.2298
9
0.9308
9
!0.4475
2.4182
!1.2258
3.0663
0.1645
9.7706
!1.6727
9
!7.0899
10
!3.4613
!6.2165
!1.4482
2.9275
!5.9464
!5.2761
!3.3217
!0.0944
10
5.0896
10
!1.7098
!3.1443
0.5253
!0.7606
4.6749
!6.4817
!2.3656
10
3.4881
11
!4.2452
!5.0921
16.6859
!3.8874
9.3558
1.3218
15.0035
!6.0079
11
4.9539
11
1.9772
!7.2618
!4.1405
0.1795
2.0291
3.1763
4.6354
11
!6.1047
12
3.4090
12
!1.8640
6.3021
!2.4334
2.5706
1.1673
4.8084
!1.7145
12
!4.3143
13
6.8217
7.4821
2.5210
0.0143
0.6759
8.9547
1.2650
13
5.5141
14
!4.4515
5638
I. Iliuta et al. / Chemical Engineering Science 54 (1999) 5633}5645
I. Iliuta et al. / Chemical Engineering Science 54 (1999) 5633}5645
5639
Table 5
Statistical tests of neural regression of gas}liquid mass transfer parameters
Gas}liquid mass transfer dimensionless number
Neural regression stage
Statistical parameter
AARE (%)
p (%)
Sh
*
Learning data
Generalization data
Whole data
Learning data
Generalization data
Whole data
Learning data
Generalization data
Whole data
22.0
22.9
22.3
10.1
13.1
10.9
27.8
29.1
28.1
24.9
26.4
25.3
8.9
13.5
10.4
39.0
34.0
37.0
Sh
%
ad /(1!e)
F
Fig. 1. Neural network predictions versus experimental liquid Sherwood numbers: (a) training set; (b) comparison with other data (test
set). Dotted lines represent $15% envelopes.
Fig. 2. Neural network predictions versus experimental gas Sherwood
numbers: (a) training set; (b) comparison with other data (test set).
Dotted lines represent $15% envelopes.
5640
I. Iliuta et al. / Chemical Engineering Science 54 (1999) 5633}5645
Table 6
The most general gas}liquid mass transfer literature correlations
Gas}liquid mass
transfer dimensionless number
Sh
*
Correlation
Flow regime
a d Q F
Sh "2.8;10\ XRe=eSc
*
%
*
*
* 1!e
a d Q F
Sh "0.091 XRe=eSc
*
%
*
*
* 1!e
Transition
Wild et al. (1992)
Low interaction
High interaction and transition
a d Q F
Sh "0.067 XRe=eSc
%
%
*
%
% 1!e
a d Q F
Sh "0.123 XRe=eSc
%
%
*
%
% 1!e
d Sh "0.049(a d )\ReReSc N
%
Q N
%
*
% d
A
a/a
Q
Low interaction
a d Q F
Sh "0.45 XRe=eSc
*
%
*
*
* 1!e
Sh
%
Reference
a
a d Q F
"10 X Re\=e
% *
* 1!e
a
Q
High interaction
Wild et al. (1992)
All regimes
YamK ci (1985)
Low interaction
Transition
High interaction
a d \ a
Q F
"21.3 X Re\=e
% *
* 1!e
a
Q
a
a d \ Q F
"1550 X Re\=e
% *
* 1!e
a
Q
Wild et al. (1992)
Table 7
Statistical tests of most general literature correlations
Gas}liquid mass transfer
dimensionless number
Correlation
Sh
*
Wild et al. (1992)
Sh
%
Wild et al. (1992)
a/a
Q
YamK ci (1985)
Wild et al. (1992)
Flow regime
Low interaction
Transition
High interaction
Low interaction
Transition
High Transition
All regimes
Low interaction
Transition
High interaction
Statistical parameter
AARE (%)
p (%)
119.0
196.0
137.0
89.4
41.0
37.2
54.9
148.0
71.6
70.8
233.0
421.0
272.0
266.0
65.0
167.4
141
356.0
83.0
71.1
I. Iliuta et al. / Chemical Engineering Science 54 (1999) 5633}5645
5641
Notation
a
a
Q
gas}liquid interfacial area, m/m
external area of particles and wall per unit
reactor volume, ("6(1!e)/
d #4/d ), m/m
N
A
AARE average absolute relative error
d
A
d
F
d
N
D
?
EoK
K
Fr
*
g
H
J
ka
?
Mo
*
N
P
Re
?
S
S
@
Sc
?
Sh
?
Sh
%
St
*
¹
;
G
v
1%
v
1*
X
%
y
Fig. 3. Neural network predictions versus experimental dimensionless interfacial areas ad /(1!e): (a) training set; (b) compaF
rison with other data (test set). Dotted lines represent $15%
envelopes.
w
=e
?
Greek letters
a
a
NO
the neural network correlations proposed here are
basically interpolation correlations. Hence, to prevent unacceptable deviations, it is suggested to verify
a priori that the conditions to be predicted fall within the
validity range of the dimensionless groups shown in
Table 1.
(y
!y )
G
G
"(1/N),
y
G
G
column diameter, m
Krischer and Kast hydraulic diameter
("d (16e/9n(1!e))
N
grain equivalent diameter, diameter of sphere
having same volume as grain, m
di!usivity in a-phase, m/s
modi"ed EoK tvos number ("o gd
e/
* N
p (1!e))
*
liquid Froude number ("v /gd )
1* N
gravitational acceleration, m/s
hidden-layer vector
number of nodes in hidden layer
a-phase side volumetric mass transfer coe$cient,
s\
liquid Morton number ("gk/o p)
* * *
number of data
pressure, MPa
a-phase Reynolds number ("v o d /k )
1? ? N ?
network output
bed correction function ("a d /(1!e)
Q F
a-phase Schmidt number, ("k /D o )
? ? ?
a-phase Sherwood number ("k ad/D )
? F ?
modi"ed
gas-phase
Sherwood
number
("k ad /a D )
% N Q %
liquid Stokes number ("k v /o gd)
* 1* * N
temperature, 3C
normalized input variables
super"cial gas velocity, m/s
super"cial liquid velocity, m/s
Lockhart}Martinelli number ("v o/v o)
1% % 1* *
gas}liquid
mass
transfer
parameter
(y"k a, k a,a)
* %
parameter
a-phase Weber number ("v d o /p )
1? N ? *
subscript meaning gas (G) or liquid (L)
cross-correlation coe$cient between column
p and column q of the database,
a "
NO
, (w N!w N ) (w O!w O )
G
L G
(, (w N!w N )(, (w O!w N )
G G
G G
e
bed void fraction
5642
k
?
u
o
?
p
I. Iliuta et al. / Chemical Engineering Science 54 (1999) 5633}5645
sphericity factor
a-phase dynamic viscosity, kg/ms
weights
a-phase density, kg/m
standard deviation,
p" ,
G
p
*
y
!y
G
G !AARE /(N!1)
y
G
surface tension, N/m
Subscripts
calc
exp
G
¸
calculated
experimental
gas
liquid
Abbreviations
CHA cyclohexylamine
CMC carboxymethylcellulose
DEA diethanolamine
DIPA diisopropanolamine
DMA dimethylamine
ETG ethylene glycol
EtOH ethanol
IPA
isopropanolamine
MEA monoethanolamine
i-PrOH isopropanol
Acknowledgements
Financial support from the Natural Sciences and Engineering Research Council of Canada (NSERC), the
Fonds pour la Formation de Chercheurs et d'Aide à la
Recherche (QueH bec) is gratefully acknowledged. Part of
this work was done in the frame of the `CoopeH ration
scienti"que franco-queH becoisea (Programme Conception
assisteH e par ordinateur - GeH nie des ProceH deH s). The "nancial support of the Ministère des A!aires Etrangères
(France), the Ministère des Relations Internationales
(QueH bec, Canada) and the organizational help of Prof.
Carrière are gratefully aknowledged. We also express our
appreciation to Dr. Z. Bensetiti and Ms A. AndreH for the
collection of the database. We are also indebted to many
contributors who provided us with theses and reports,
among them we are thankful to Profs A. Lakota and
J. Levec from University of Ljubljana (Slovenia),
Prof. W. P. M. van Swaaij from Twente University (The
Netherlands) and Drs K. Tahraoui and W. J. A. Wammes.
The author would also like to acknowledge use of the
following literature:
Azzaz, 1984; Bakos et al., 1980; Blok et al., 1984; Boyes
et al., 1995; Charpentier, 1976,1986; Dodds et al., 1960;
Frank, 1996; Fukushima and Kusaka, 1977a,1977b,1978;
Gianetto et al., 1970,1973; Goto and Smith, 1975; Goto
et al., 1975; Guangda and Dudukovic, 1993; Hirose et al.,
1974; Iliuta, 1996; Iliuta and Thyrion, 1997; Iliuta et al.,
1997; Lakota, 1990; Larachi, 1991; Larachi et al., 1992,
1997; Lara-MaH rquez et al., 1992; Mahajani and Sharma,
1979,1980; Martin et al., 1980; Merchuk, 1980; Midoux
et al., 1984,1986; Mitchell and Perona, 1979; Morsi et al.,
1980a,1980b; Morsi, 1982; Morsi et al., 1982,1984; Morsi,
1989; Nagel and KuK rten, 1972; Patil and Sharma,
1981,1983; Purwasasmita, 1985; Ratnam and Varma,
1991; Ratnam et al., 1994; Reiss, 1967; Sato et al., 1972;
Satter"eld, 1975; Seira" and Smith, 1980; Shende and
Sharma, 1974; Shiying et al., 1989; Sylvester and
Pitayagulsarn, 1975; Tahraoui, 1990; Tahraoui et al.,
1992; Turek and Lange, 1981; U!ord and Perona, 1973;
Versteeg, 1987; Versteeg and van Swaaij, 1988; VivarieH ,
1981; Wammes, 1990; Wammes et al., 1991; Wammes
and Westerterp, 1991; Wang et al., 1994,1997; Wen et al.,
1963; YamK ci et al., 1985; YamK ci et al., 1988.
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