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Ind. Eng. Chem. Res. 2008, 47, 3274-3284
CORRELATIONS Seamless Mass Transfer Correlations for Packed Beds Bridging Random and Structured Packings Faı1c¸ al Larachi,* Ste´ phane Le´ vesque, and Bernard P. A. Grandjean Department of Chemical Engineering, LaVal UniVersity, Que´ bec, Canada G1K 7P4
A unifying correlative approach for the gas-liquid mass transfer in both structured- and random-packing containing towers was developed based on a two-correlation kernel. Two databanks consisting of 861 experiments for structured packings and 4291 experiments for random packings were merged and concerned the volumetric local and overall gas- and liquid-side mass transfer coefficients kGaw, kLaw, KGaw, and KLaw, the effective gas-liquid interfacial area, aw, and the height equivalent to a theoretical plate, HETP. The three-phase systems were representative of absorption, desorption and distillation applications. Two correlations have emerged, the first to evaluate the local gas- or liquid-side mass transfer coefficient (kγ), the second to correlate the effective gas-liquid interfacial area (aw). A reconciliation method was used to calibrate and validate the two-correlation kernel (kγ,aw) owing to a broad domain of applicability and embracing indifferently both structured- and random-packing columns. The results proved satisfactory and statistical analysis yielded, respectively, 22.1% and 17.6% for the mean and standard deviation for the absolute relative error (ARE) regarding the mass transfer parameters applicable to structured packings. These correlations had also the capacity to predict the same parameters for the random packings with a mean and standard deviation for ARE, respectively, of 26.3% and 24.4%. 1. Introduction The mass transfer correlative approaches for structured and random packings evolved hitherto as two separate realms leading often to sets of ad hoc correlations exclusively applicable either to structured or to random packings.1 The different treatment so diagnosed is even more perplexing as similar foundational assumptions, e.g., the penetration theory or the falling film analogy, have fueled the development of correlations pertaining to both packing families.1 Yet columns containing either packing types as internals experience qualitatively similar gas-liquid flow patterns (i.e., pre-loading, loading, and flooding). Hence, the dichotomy drawn between structured and random packings as reflected through these ad hoc correlations appears in essence to be accessory and prompted by the commodity of the assumptions subtending the empirical or semiempirical correlations to be devised. Therefore, there is in principle no objection for the mass transfer in packed columns to be correlated/modeled indifferently by means of one single set using some sort of canonical formulation valid for both categories of packings. Billet and Schultes, throughout incremental upgrades of their mass transfer model,2 were among the very few researchers who paved the way by recognizing such indifference between structured and random packings and who chose to represent the gas-liquid mass transfer characteristics by means of a structurally similar mathematical model. To take advantage of their approach and to fully define the hydrodynamic and mass transfer states of the column for given operating conditions, a set of six packing-sensitive (size, shape, and material) constants as well as liquid holdup must be known a priori for each packing. This may represent an obstacle for the generalization of the approach when such constants are unavailable or when * To whom correspondence should be addressed. Phone: 1-418656-3566. Fax: 1-418-656-5993. E-mail:
[email protected].
an insufficient number of experiments for a given packing prevent the adjustment of the model to extract the needed packing-sensitive constants. In our previous works, a correlative methodology was developed for the prediction of the gas-liquid interfacial area (aw) and the pure local mass transfer coefficients (kγ, kγ ) kL or kG) using a comprehensive databank consisting of gas-liquid interfacial area and volumetric mass transfer coefficients (kγaw, Kγaw) and representing an up-to-date repository for randompacking containing columns.3-5 The database was exclusive to absorption and stripping while the modeling approach revolved around the use of a reconciliation procedure combining actually measured interfacial areas with pseudo interfacial areas intuited from the actually measured volumetric mass transfer coefficients. Having demonstrated good level of precision and robustness in terms of phenomenological consistency, this empirical modeling approach was later extended to correlate HETP for random packings5 by including as an appropriate discriminatory variable for nonaqueous systems the so-called Marangoni factor adopted by Billet and Schultes.2 The correlation thus far developed was extended and validated to embrace absorption, stripping and distillation applications alike for random packings. Pollock and Eldridge6 spotting particularly the HETP data for structured packings in the distillation context devised a correlation that they validated over nine types of structured packings. As a natural extension to our previous works and considering the lack of unification between the structured and random packings regarding the gas-liquid mass transfer, we proposed in the present contribution to merge two comprehensive databanks consisting of 861 experiments for structured packings and 4291 experiments for random packings concerning the volumetric local and overall gas- and liquid-side mass transfer coefficients kGaw, kLaw, KGaw, KLaw, the gas-liquid interfacial area, aw, and the height equivalent to a theoretical plate, HETP.
10.1021/ie070718o CCC: $40.75 © 2008 American Chemical Society Published on Web 03/26/2008
Ind. Eng. Chem. Res., Vol. 47, No. 9, 2008 3275 Table 1. Number Distributions of Mass Transfer Parameters among Random and Structured Packings in the Databanks aw kGaw kLaw kL KGaw KLaw HETP total a
random
structured
138 487 997 0 1477 0a 1192 4291
198 230 81 24 169 32 127 861
Included with KGaw.
A reconciliation procedure was used to devise a two-correlation (kγ,aw) kernel for predicting any mass transfer parameters applicable to distillation, stripping and absorption, and valid irrespective of structured or random packing containing columns by using an identical set of independent variables. In addition, HETP predictions from the newly developed correlations were compared to some manufacturers’ curves and which were purposely hidden during the construction process of the correlations to test the robustness of the proposed approach. 2. Strategy for Developing Unified Two-Correlation (kγ,aw) Kernel 2.1. Database Overview. The cited literature pertaining to the absorption/stripping/distillation databank and relevant to the following mass transfer characteristics: kGaw, kLaw, KGaw, KLaw, aw, and HETP for random packings was described in our previous publications.3-5 A similar effort has been spent in this work to elaborate a comprehensive databank relevant to the structured packings and reported in the open literature.7-39 Table 1 displays the partition inside both databanks as used in this work along with Table 2 which summarizes the breadth of operating variables characteristic of these six mass transfer parameters for random and structured packings. The two databanks cannot be merged directly because of some differences between the random and structured packing. Such differences stand in the first place in the characterization of the packing. For instance, while the sphericity factor is a more suitable geometrical variable for random packings, the corrugation angle is only meaningful for structured packings. We will
discuss in the subsequent sections of this work about the method employed to merge together these two databanks. The distillation data deserved special treatment in order to reconstruct the HETP values from the mass transfer parameters kγ and aw, the binary-mixture partition coefficient, m, and the superficial vapor and liquid velocities and their corresponding molar flow rates. This part of the databanks consisted of 1192 HETP experiments for random packings and 127 experiments for structured packings. All the distillation experiments were conducted at total molar reflux with standard binary mixtures such as chlorobenzene/ethylbenzene, ethylbenzene/styrene, benzene/toluene, methanol/ ethanol, etc. (see the work of Piche´ et al.5). The liquid physical properties were established on an equimolar composition basis. If information on the liquid physical properties were unavailable in the sources, the following methods were employed to estimate the liquid density (weighted average), liquid viscosity (Grunberg and Nissan method), surface tension (Macleod-Sugden method) and liquid diffusion coefficient (Siddiqi-Lucas correlation); see the work of Reid et al.40 The resulting gas-phase equilibrium composition was also used as basis for the calculation of gas density (ideal gas law), gas viscosity (Wilke correlation), gas diffusion coefficient (Chapman-Enskog theory), and partition coefficient (m). In addition, a customized relative stability index, I, representing the ratio between the compositional surface tension gradient with respect to the most volatile component and the mixture surface tension was added (eq 1). This index was computed for an equimolar composition of the binary mixture. Furthermore, it was introduced to account for the Marangoni effects on the gasliquid interfacial areas in distillation conditions:
I)
(
dσL 1 (0.5) σL(0.5) dxvol
)
The Marangoni effect (dσL/dxvol) was evaluated using a simple finite difference method around xvol ) 0.5 with respect to the mole fraction of the most volatile compound. For binary nonaqueous solutions, the index will be either negative or positive depending on the liquids to be separated, whereas I equals zero for aqueous solutions or in the case of absorption and stripping mass transfer experiments.
Table 2. Variable Ranges Describing the Mass Transfer Databanks temperature (K) pressure (kPa) liquid mass flow rate (kg/m2‚s) gas mass flow rate (kg/m2‚s) packing bed height (m) column diameter (m) porosity (-) bed specific surface area (m2/m3) sphericity (-) corrugation angle (deg) liquid density (kg/m3) liquid viscosity (Pa‚s) surface tension (N/m) Liquid diffusion coefficient (m2/s) gas density (kg/m3) gas viscosity (Pa‚s) gas diffusion coefficient (m2/s) effective interfacial area (m2/m3) volumetric gas phase-film mass transfer coefficient (1/s) volumetric liquid phase-film mass transfer coefficient (1/s) volumetric overall gas phase side mass transfer coefficient (1/s) volumetric overall liquid phase-side mass transfer coefficient (1/s) height equivalent to a theoretical plate (m)
(1)
random packing
structured packing
246-1000 0.667-10300 0.074-95.2 0.0005-7.43 0.1-5.9 0.04-2.13 0.44-0.98 87-1150 0.056-1 N/A 600-1160 0.00023-0.0185 0.0071-0.487 1 × 10-10-1.65 × 10-8 0.175-42 6.76 × 10-6-1.43 × 10-5 8.3 × 10-7-6.51 × 10-5 7.2-244 0.753-14.9
276-603 1.3-11500 0.032-136.3 0.12-154.9 0.28-4.57 0.043-1.83 0.70-0.98 125-1920 not applicable (N/A) 45-60 381-1190 0.000054-0.0077 0.00089-0.0896 6.7 × 10-10-2.08 × 10-8 0.179-48.4 6.64 × 10-6-2.12 × 10-5 6.67 × 10-9-8.4 × 10-5 39-633 0.09-109.6
0.00074-0.0828
0.0035-0.062
0.000044-29.1
0.00099-9.85
0.000032-35.3
0.001-0.094
0.172-1.14
0.06-1.38
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Figure 1. Flow diagram of kγ-correlation construction.
2.2. Methodology. Past works from the research group3-5 revealed the success of the approach consisting in the prediction with artificial neural networks of the volumetric mass transfer coefficients for random packings by using only two correlations: one for predicting the local mass transfer coefficient, both gas and liquid side, i.e., kγ, and the other one for the effective gas-liquid interfacial area, aw. We will adopt here the same philosophy for deriving a more general two-correlation kernel (kγ,aw) embracing the structured and random packings alike. Two steps are required in order to build these two kernel correlations. The first step’s objective is to formulate preliminary or initial kernel correlations for kγ and aw by interrogating separately the individual databanks for random and structured packings. The second step will reinforce those correlations by optimizing their weights over the merged databanks simultaneously using a reconciliation method. Figure 1 illustrates the modeling organization chart to generate the initial correlation for the gas and liquid side mass transfer coefficient, kγ. This initial kγ correlation consists of an artificial neural network (ANN) with still sub-optimal weights, i.e., ANNkγ (random + structured). The rectangular boxes in Figure 1 stand for databanks, whereas the oval boxes correspond to the ANN generated correlations. From the separate databanks containing only the effective interfacial area, independent correlations are generated for the random and structured packings, respectively, ANN-aIwa and ANN-aIws. Those correlations will not be optimized thereafter and are used only to dissociate the effective interfacial area from the local mass transfer coefficient, kγ. The left stream in Figure 1, corresponding to the random packings, allows thence harvesting pseudo mass transfer coefficients kγ obtained from the ratio between experimental volumetric mass transfer coefficients (gas and liquid alike) and ANN-aIwa correlation for the interfacial area. The utility of ANN-aIwa is indeed to split the individual contributions of kγ and aw from kγaw in the same conditions. Similarly, the right stream in Figure 1, starting from the actual structured packing volumetric mass transfer coefficients and interfacial areas, and ANN-aIws correlation for structured packings, yields a kγ pseudo databank. It thereafter becomes possible to combine both pseudo databanks together, i.e., pseudo kγ from random packings, and pseudo kγ from structured packings and eventually experimentally measured kγ for structured packings found in the literature (Figure 1). From this databank, a first initial correlation is established, ANN-kγ (random + structured). Figure 2 illustrates the modeling organization chart to derive the second initial correlation of the kernel that is ANN-aw
Figure 2. Flow diagram of aw-correlation construction.
Figure 3. Reconciliation between kγ and aw databanks and ANN correlations using the Powell algorithm.
(random + structured). Recall that in the previous step, two dissimilar ANN correlations were required for the effective interfacial area to establish the packing-indifferent single initial ANN-kγ correlation. Exploiting now the experimental data regarding the overall volumetric mass transfer coefficients, Kγaw, the local volumetric mass transfer coefficients, kγaw, and the HETP values from both databanks enables harvesting a pseudo databank for effective interfacial areas by taking out the estimated kγ values from the initial ANN-kγ correlation. This databank, which gathers pseudo interfacial areas for both random and structured packings, is enriched with the same experimental data for the effective interfacial areas used in Figure 1 to derive the packing-indifferent initial ANN-aw (random + structured) correlation. Having at disposal two initial ANN-kγ and ANN-aw correlations, these can now be refined using the reconciliation method sketched in Figure 3. Until now, several pseudo kγ and aw data were estimated from preliminary correlations including voluntarily quite a few data during their learning. In addition, the
Table 3. ANN-kγ Neural Network Weights and Input Variables
log S)
(
)
kγ
9.22 × 10-6
Normalized Output (H11 ) 1)a
(
)
4.194
1 11
1 + exp -
∑ω H j
j)1
j
) Hidden Layer Function (1 e J e 10 and U10 ) 1)
(
Hj )
1 10
∑ω U
1 + exp -
U1 )
ij
i
j)1
- 0.44 0.54
) log
U2 )
() aP
87 1.344
log U3 )
( ) DC
0.043 1.629
U4 )
FL - 381.4 808.6
log U5 )
(
Normalized Inputsa UL
6.05 × 10 3.042
)
log
-5
( ) FG
0.175 2.442
U6 )
U7 )
log
µG - 6.64 × 10-6 1.456 × 10
U8 )
-5
( ) UG
0.016 2.853
log U9 )
(
)
Dγ
1 × 10-10 5.924
ANN-kγ Optimized Weigths 1
2
3
4
5
6
7
8
9
10
1 2 3 4 5 6 7 8 9 10
-2.429 -7.349 1.108 0.6437 1.154 7.691 -0.1282 0.2091 5.299 2.075
-11.17 -13.83 0.6154 18.22 1.420 -7.682 7.243 3.346 -10.78 0.2926
-4.797 -1.912 5.638 0.2923 -4.990 -0.4008 -0.9116 3.503 -5.184 1.857
-3.105 2.114 -11.50 3.708 1.454 -4.831 2.393 1.855 -4.557 6.128
-4.715 -2.851 -0.4615 0.00614 2.766 1.418 3.665 0.7889 -13.23 3.371
17.89 1.798 -10.89 7.61 21.95 6.341 -28.06 0.8184 28.24 4.109
-2.620 -4.040 -0.7848 -3.323 0.3409 -3.119 1.587 3.151 5.209 -1.319
28.07 15.08 2.518 -0.2952 0.2537 -7.517 5.231 30.50 -9.654 1.867
-7.782 -5.747 -1.363 -3.971 -0.2075 -1.802 2.900 -1.767 -0.9400 14.00
-9.546 0.8170 -22.33 20.95 2.504 1.066 4.576 1.439 -4.781 1.132
ωj
1
2
3
4
5
6
7
8
9
10
11
8.855
-2.421
-1.385
-2.897
1.809
6.584
5.647
-6.570
-2.679
3.708
-7.108
0.44 e e 0.98 a
87 e aP e 1920
0.043 e DC e 1.829
381.4 e FL e 1190
6.05 × 10-5 e UL e 0.0667
Ranges of Normalization
0.175 e FG e 48.407
6.64 × 10-6 e µG e 2.12 × 10-5
0.016 e UG e 11.405
For the normalization of input and output variables (S and Ui), depending on breadth of intervals, either log(variable) or the variable itself is normalized.
1 × 10-10 e Dγ e 8.4 × 10-5
9.22 × 10-6 e kγ e 0.144
Ind. Eng. Chem. Res., Vol. 47, No. 9, 2008 3277
ωij
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Table 4. ANN-aw Neural Network Weight and Input Variables
( )
Normalized Output (H9 ) 1)a
R
R)
log
aw aP
S)
w
0.15
1
(∑ )
)
0.9994
9
1 + exp -
ωjHj
j)1
Hidden Layer Function (1 e J e 8 and U7 ) 1)
1
(∑ )
Hj )
8
1 + exp -
ωijUi
j)1
log U1 ) ReLe )
( ) ReLe 1.098
log U2 )
2.955 FLUL,eff aPµL
FrLe )
(
FrLe
Normalized Inputsa
)
log
1.96 × 10-7 5.998
aPUL,eff g
U3 ) F Lg
2
EoL )
2
aP σL
( ) EoL 0.118 3.02
K4 )
U4 )
DCaP 6(1 - )
I )
( )
K4 9.812 3.077
log
dσL/dxvol σL
U5 ) χ )
log
I + 0.4359 1.411
x
UG UL
FG FL
U6 )
UL,eff )
χ (0.0339 )
5.048 UL sin R
ANN-aw Optimized Weights ωij
1
2
3
4
5
6
7
8
1 2 3 4 5 6 7
3.040 -3.180 0.8071 -12.39 -3.33 -2.044 8.978
9.297 -5.767 -4.312 11.65 0.5374 2.858 -0.1945
-11.33 3.181 5.219 -10.93 -20.7 -8.370 10.94
4.322 1.261 -6.547 38.00 -13.74 0.1547 2.758
-1.342 11.45 18.99 16.93 24.31 18.04 -36.54
-1.848 7.083 -3.263 -6.934 -4.020 4.191 5.998
-4.612 -7.620 9.517 10.11 -0.645 -7.533 4.284
-16.88 -1.283 22.19 8.760 -0.06859 -17.31 -3.124
ωj
1.098 e ReLe e 990.2 a
1
2
3
4
5
6
7
8
9
2.286
10.58
3.516
-2.061
2.498
-4.082
-2.919
-2.069
-4.680
1.96 × 10-7 e FrLe e 0.195
0.118 e EoL e 123.6
Ranges of Normalization -0.436 e I e 0.975 9.81 e K4 e 11.7 × 103
0.034 e χ e 3.79 × 103
For the normalization of input and output variables (S and Ui), depending on breadth of intervals, either log(variable) or the variable itself is normalized.
0.150 e R e 1.499
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Figure 4. Parity plot diagrams for the random packing mass transfer parameters (a) volumetric gas-side mass transfer coefficients, (b) volumetric liquidside mass transfer coefficients, (c) overall volumetric gas mass transfer coefficients, (d) HETP values, and (e) gas-liquid interfacial areas. Table 5. Statistical Comparison of ANN Correlations versus Measured Mass Transfer Parameters random packing mass transfer coefficients interfacial area, aw local gas-phase, kGaw local liquid-phase kLaw overall gas-side, KGaw overall liquid-side, KLaw HETP total
structured packing
mean (ARE) (%)
σ (ARE) (%)
no. of data
mean (ARE) (%)
σ (ARE) (%)
138 487 997
24.1 27.7 20.9
20.8 20.2 16.6
198 230 81
28.2 23.6 23.7
17.8 20.9 19.5
1,477
27.3
27.8
1,192 4,291
29.2 26.3
29.0 24.4
169 32 127 837
14.3 25.2 18.2 22.1
11.9 22.4 16.5 17.6
no. of data
two correlations were set consecutively. Thus, a way to improve those correlations is by readjusting their weights by using the two initial correlations and the whole databank simultaneously. Powell’s algorithm41 was used to minimize an objective criterion, Q (eq 2), by adjusting the weights of the two kernel
correlations. The objective function was defined as the sum of the squared logarithmic residuals between experimental and ANN-estimated mass transfer characteristics irrespective of the databanks along with weighting factors, ωi as defined in eq 3. The initial guesses were the reciprocal of the corresponding
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Figure 5. Parity plot diagrams for the structured packing mass transfer parameters (a) volumetric gas-side mass transfer coefficients, (b) volumetric liquidside mass transfer coefficients, (c) overall volumetric gas mass transfer coefficients, (d) overall volumetric liquid mass transfer coefficients, (e) HETP values, and (f) gas-liquid interfacial areas.
number of data, imax, i.e., λ ) 1. The weighting factors were then readjusted from trial and errors to balance the optimization over each databank evenly (0.4 e λ e 6). 138
Q ) ω1‚
1477
(log aawexp - log aawcal)2 + ω2‚ ∑ ∑ i)1 i)1 1192
(log KGaawexp - log KGaawcal)2 + ω3‚
(log HETPaexp ∑ i)1
487
log HETPacal)2 + ω4‚
(log kGaawexp - log kGaawcal)2 + ∑ i)1
997
ω5 ‚
198
(log kLaawexp - log kLaawcal)2 + ω6‚ ∑ (log aswexp ∑ i)1 i)1 169
log aswcal)2 + ω7‚
(log KGaswexp - log KGaswcal)2 + ∑ i)1
32
ω 8‚
∑ i)1
127
(log KLaswexp - log KLaswcal)2 + ω9‚
∑ i)1
230
(log HETPsexp - log HETPscal)2 + ω10‚
(log kGaswexp ∑ i)1
81
log kGaswcal)2 + ω11‚
(log kLaswexp - log kLaswcal)2 ∑ i)1 ωi )
λ imax
(2) (3)
Figures 1 and 2 indicate that by postulating that the interfacial area (aw), the volumetric liquid- and gas-side mass transfer coefficients (kLaw, kGaw), the volumetric overall gas- and liquidside mass transfer coefficients (KGaw, KLaw), and the heightequivalent-to-a-theoretical plate (HETP) all depend on two natural variables, namely, aw and kγ, it is possible to avoid the problem of under-representativity for the structured packing
databanks (see Table 1) versus the random packing databanks. As an illustration, besides the 198 aw data for structured packings, it is possible to enrich the interfacial area databank with pseudo aw coming from the volumetric coefficients and HETP databanks. This amounts in a total of 837 actual plus and pseudo interfacial areas only for structured packings (Figure 2). Using similar argument, the number of mass transfer coefficients, kγ (actual and virtual) serving the structured packing databank amounts also to 837 (Figure 1). This number is enough representative to propose robust canonical correlations for the structured packings. 3. Results and Discussion After testing several dozens model structures with dimensional and dimensionless input variables, optimal correlations for ANN-kγ and ANN-aw were identified. For the sake of brevity, the details of the model building approach and the input variables selection were not exposed here. The interested readers can consult our several previous works on these aspects.4,42,43 The best ANN-kγ correlation used nine purely dimensional input variables along with ten hidden nodes for a total of 111 weights. The definition of the normalized input variables and the values of the weights were given in Table 3. This correlation predicts indifferently either local gas-side (kG) or liquid-side (kL) mass transfer coefficient for any packing belonging to the applicability domain of the selected input variables in the merged mass transfer databanks as shown in Table 2. The chosen input variables, listed in Table 3, are: bed porosity, packing specific area, column diameter, liquid density, liquid superficial velocity, gas density, gas viscosity, gas superficial velocity and γ-phase molecular diffusion coefficient, Dγ. It is this latter variable that discriminates between the gas- or liquid-side mass transfer coefficients to be estimated. The best ANN-aw correlation used six dimensionless numbers as input variables and consisted of eight hidden nodes giving a total of 65 weights. The definition of the dimensionless numbers, their normalized formulation and the numerical values of the
Ind. Eng. Chem. Res., Vol. 47, No. 9, 2008 3281 Table 6. Statistical Results of Separate ANN Correlations for Structured Packings and for Merged Structured + Random Packings Databanks structured + random
structured only
aw kGaw kLaw KGaw KLaw HETP total
no. of data
mean (ARE) (%)
σ (ARE) (%)
no. of data
mean (ARE) (%)
σ (ARE) (%)
198 230 81 169 32 127 837
63.4 23.8 43.3 15.6 20.5 16.9 32.2
106.4 24.8 45.4 12.9 21.0 15.3 42.1
198 230 81 169 32 127 837
28.2 23.6 23.7 14.3 25.2 18.2 22.1
17.8 20.9 19.5 11.9 22.4 16.5 17.6
weights for this correlation were listed in Table 4. The liquid Reynolds number (ReLe), the liquid Froude number (FrLe), the Eotvo¨s number (EoL), the bed hydraulic diameter dimensionless number (K4), the modified relative stabilizing index (I),5 and finally the Lockhart-Martinelli group (χ) are the input variables for the ANN-aw. The effective liquid superficial velocity was incorporated into the liquid Reynolds and Froude numbers using the packing porosity and corrugation angle. For the random packing, the value of the corrugation angle must be replaced by the dummy variable value 90°.
At this stage, a caution must be exerted on what is meant by normalization in the ANN correlations tabulated in Tables 3 and 4. As a matter of fact, the normalizing procedure is based on the extent of the input variables relevant to all the experimental mass transfer characteristics except those of HETP. Readers interested in the developed correlations can download the simulator from the following web address (http://www.gch.ulaval.ca/bgrandjean or http://www.gch.ulaval.ca/flarachi). 3.1. Assessment and Validation of the Two-Correlation (kγ,aw) Kernel. The quality of fit of the two-correlation (kγ,aw)
Figure 6. Simulating the effect of gas and liquid superficial velocities on Mellapak 250Y mass transfer (a) gas-liquid interfacial area, (b) liquid-, and (c) gas-side volumetric mass transfer coefficients.
Figure 7. Simulating the effect of packing specific area on (a) gas-liquid interfacial area, (b) liquid-, and (c) gas-side volumetric mass transfer coefficients, Mellapak and Montz packings.
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Figure 9. Comparison of ANN predicted HETP with Jaeger manufacturer’s performance data.13
Figure 8. Simulating the effect of corrugation angle on (a) gas-liquid interfacial area, (b) liquid-, and (c) gas-side volumetric mass transfer coefficients, Montz packings.
kernel was judged through visualization of the parity diagrams as well as assessed through statistical analysis. Figure 4a-e shows respectively the parity plots of kGaw, kLaw, KGaw, HETP, and aw, measured versus predicted for the databank of random packings. Similarly, Figure 5a-f illustrates parity plot comparisons in the case of the structured packing databank. In general, the two-correlation kernel captures pretty well the behavior of the six mass transfer parameters (kGaw, kLaw, KGaw, KLaw, aw, and HETP) for both random and structured packings. Note that the parity plots for the structured packings look sparser because the size of the random packing databank exceeds by almost a factor five that of the structured packings. The quality of fit of the two correlations can also be analyzed more quantitatively using statistical descriptors as shown in Table 5. For the random packings, the average absolute relative error (see nomenclature) values varied between 20.9% and 29.2% among the six mass transfer parameters, kGaw, kLaw, KGaw, KLaw, aw, and HETP, with a mean value for ARE of 26.3%. This latter overall error compares pretty well with the performance prediction given by Piche´ et al.5 correlations valid only for the random packings. The results for the structured
packings were better, varying between 14.3% and 28.2% with a mean value of 22.1%. Those results for the structured packings look better because the weighting factors, ωi, in eq 3 above were set to ameliorate the goodness of fit with respect to the structured packing data, despite their sparseness, without sacrificing too much the correlations predictability over the random packing data. The merit of the strategy in which two sets of separate correlations (one for structured packing and one for random packing) was assessed versus one single set of aw and kγ correlation embracing the two types of packings. The ARE of structured packings in the merged databanks are most often much lower than if they were calculated had separate structured packing databanks been used for correlating the structured packing mass transfer parameters alone, see Table 6. Definitely, the merger with the random packing databanks improves significantly the robustness of the structured packing mass transfer estimations. 3.2. Simulating the Influence of Operational Parameters on Structured Packings. The capability of the ANN correlations to restore phenomenologically sound predictions for wide ranges of operating conditions can be verified with the help of a sensitivity analysis on physical properties and operating variables of the mass transfer parameters. The interest in the new correlations concerns in the first place their behavior with regard to the structured packing. Several ANN-aw and ANN-kγ simulations were therefore performed to simulate the effect of superficial gas and liquid velocities on the reduced effective interfacial area, aw/aP, and the local volumetric liquid and gasside mass transfer coefficients, kGaw and kLaw, for structured packings. For the simulations shown in Figure 6, the chlorobenzene/ethylbenzene distillation system at a pressure of 40 kPa with Mellapak 250Y structured packing was employed. Note that the gas superficial velocity range interrogated [0.5-2 m/s] coincides with a tendency for the interfacial area to level off (Figure 6a). This is coherent with the experimental trends as shown for instance by Wang et al.44 where the influence of gas superficial velocity tends to vanish. However, as the liquid holdup and effective interfacial area are monotonically interdependent, an increase in liquid superficial velocity induces a more remarkable increase of the gas-liquid interfacial area than an increase in gas superficial velocity (Figure 6a). The weak dependency between UG and kL as noticed by Piche´ et al.4 for the random packings appears also to prevail for structured packings and explains thus the UG weaker influence on kLaw (Figure 6b). The kLaw values on the other hand are increasing function of the liquid superficial velocity. Regarding kGaw (Figure 6c), the gas and liquid superficial velocities bring about
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an ameliorative impact on aw via UL, as in Figure 6a, and on kG via UL and UG as previously observed in the case of random packings.4 Figure 7a-c illustrates the incidence of changing the packing specific area on aw/aP, kGaw and kLaw, in a gas-liquid absorption test at atmospheric pressure. The simulated packings were Mellapak 250HC (aP ) 250 1/m) and Montz BSH 400.6 (aP ) 386 1/m) with an identical corrugation angle of 60° and practically the same porosity, respectively, 0.975 and 0.97. The simulated range suggests the effective interfacial area increases proportionally with the packing specific area thus reflecting in an improvement of the volumetric mass transfer coefficients, kGaw and kLaw, when slecting Montz BSH400.6 over Mellapak 250HC. Finally, the impact of the structured packing corrugation angle can be seen in Figure 8a-c for the effective interfacial area and the volumetric liquid- and gas-side mass transfer coefficients, kLaw and kGaw. The Montz B1-400 and B1-400.6 packings both exhibit aP ≈ 400 1/m and porosity ≈ ca. 96%. However, the corrugation angle of Montz B1-400 packing is 45° whereas it is 60° for the Montz B1-400.6. The steeper walls in the case of the latter packing retain less liquid holdup with respect to the 45° packing with the result that less fractional packing area, aw/aP, is wetted for the 60° inclination; the same is true for kLaw and kGaw presumably for the same reason. These above analyses indicate that the established correlations reflect satisfactorily the actual mass transfer behavior for structured packings. Thorough parametric analyses regarding other operating variables can be found elsewhere.45 Other performance curves, such as the HETP curves, have been simulated in order to compare the results with those given by some manufacturers. These data were not incorporated in the databanks that served in elaborating the above mass transfer correlations, giving thus a further opportunity for testing alien conditions. HETP values were thus computed using the above two-correlation (kγ,aw) kernel (Tables 3 and 4) along with an estimation of the partition coefficient, m, and the vapor and liquid superficial velocities and molar flow rates. Figure 9 shows the evolution of HETP as a function of the capacity factor for the cyclohexane/n-heptane distillation system at a pressure of 5 psia in a 4 ft column diameter containing the Maxpak structured packing of the Jaeger company.13 As can be seen from the trends predicted from the ANN correlations, the simulated HETP are close to the experimental ones except when the column operation nears the flooding region. Acknowledgment Financial support from the Natural Sciences and Engineering Research Council of Canada is gratefully acknowledged. The authors acknowledge the discussions and help with Dr. S. Piche´. Nomenclature aP ) packing-specific surface area, m2/m3 ARE ) average relative error, ARE ) |X experimental - Xcalculated|/X experimental
aw ) gas-liquid interfacial area, m2/m3 CS ) capacity factor, (m/s (kg/m3))0.5 Dγ ) γ-phase diffusion coefficient, m2/s DC ) column diameter, m EoL ) liquid-phase Eotvo¨s number, FrLe ) effective liquid-phase Froude number, g ) gravitational acceleration, m/s2 HETP ) height equivalent to a theoretical plate, m
I ) relative stabilizing index, Kγ ) overall γ-phase-side mass transfer coefficient, m/s kγ ) local γ-phase film mass transfer, m/s K4 ) packing number, m ) thermodynamic partition coefficient, Q ) optimized cost function, eq 2 ReLe ) effective Reynolds number, S ) normalized output variable, U ) normalized input variable, Uγ ) γ-phase superficial velocity, m/s UL,eff ) effective liquid superficial velocity, m/s x ) molar fraction in liquid phas, Greek Letters R ) corrugation angle, deg or interfacial area per packing specific surface area, aw/ap, χ ) Lockhart-Martinelli parameter, ) packing porosity, µγ ) γ-phase dynamic viscosity, kg/m/s Fγ ) γ-phase density, kg/m3 σ ) standard deviation, σL ) surface tension, N/m ωi ) weighting factor, eq 3 ωj ) ANN fitting parameter ωij ) ANN fitting parameter Subscripts cal ) calculated value exp ) experimental value G ) gas phase γ ) liquid (L) or gas (G) phase L ) liquid phase Superscripts a ) random packing s ) structured packing Acronyms ANN ) artificialneural network Literature Cited (1) Wang, G. Q.; Yuan, X. G.; Yu, K. T. Review of mass-transfer correlations for packed columns. Ind. Eng. Chem. Res. 2005, 44, 8715. (2) Billet, R.; Schultes, M. Prediction of mass transfer columns with dumped and arranged packings. Trans IChemE 1999, 77, 498. (3) Piche´, S.; Grandjean, B. P. A.; Iliuta, I.; Larachi, F. Interfacial mass transfer in randomly packed towers: A confident correlation for environmental applications. EnV. Sci. Technol. 2001, 35, 4817. (4) Piche´, S.; Grandjean, B. P. A.; Larachi, F. Reconciliation procedure for gas-liquid interfacial area and mass-transfer coefficient in randomly packed towers. Ind. Eng. Chem. Res. 2002, 41, 4911. (5) Piche´, S.; Le´vesque, S.; Grandjean, B. P. A.; Larachi, F. Prediction of HETP for randomly packed towers operation: Integration of aqueous and non-aqueous mass transfer characteristics into one consistent correlation. Sep. Purif. Technol. 2003, 33, 145. (6) Pollock, G. S.; Eldridge, R. B. Neural network modeling of structured packing height equivalent to a theoretical plate. Ind. Eng. Chem. Res. 2000, 39, 1520. (7) Uresti-Melendez, J.; Rocha, J. A. Pressure drop in ceramic structured packings. Ind. Eng. Chem. Res. 1993, 32, 2247. (8) Rocha, J. A.; Bravo, J. L.; Fair, J. R. Distillation columns containing structured packings: A comprehensive model for their performance. 1. Hydraulic models. Ind. Eng. Chem. Res. 1993, 32, 641. (9) Potthoff, R.; Sweeney, P.; Krishnamurthy, K. R. Structured packing liquid hold-up in cryogenic systems. Int. J. Refrig. 1997, 20, 63. (10) Moritz, P.; Hasse, H. Fluid dynamics in reactive distillation packing KatapaK-S. Chem. Eng. Sci. 1997, 54, 1367.
3284
Ind. Eng. Chem. Res., Vol. 47, No. 9, 2008
(11) Ellenberger, J.; Krishna, R. Counter-current operation of structured catalytically packed distillation columns: pressure drop, holdup and mixing. Chem. Eng. Sci. 1997, 54, 1339. (12) Billet, R.; Mackowiak, J. Application of modern packings in thermal separation processes. Chem. Eng. Technol. 1988, 11, 213. (13) Jaeger products, Inc. Structured packing efficiencies. http:// www.jaeger.com/MaxEffcies.pdf., product bulletin-500, p. 4. (Accessed March 22, 2008). (14) Gierczycki, A. T.; Stanek, V.; Vychodil, P. On the hydrodynamic behavior of the counter-current regularly stacked bed columns. Chem. Biochem. Eng. Q. 1989, 3, 17. (15) Suess, P.; Spiegel, L. Hold-up of Mellapak structured packings. Chem. Eng. Process. 1992, 31, 119. (16) Chung, T.-W.; Ghosh, T. K.; Hines, A. L. Comparison between Random and structured packings for dehumidification of air by lithium chloride solutions in a packed column and their heat and mass transfer correlations. Ind. Eng. Chem. Res. 1996, 35, 192. (17) Subawalla, H.; Gonzalez, J. C.; Seibert, A. F.; Fair, J. R. Capacity and efficiency of reactive distillation bale packing: Modeling and experimental validation. Ind. Eng. Chem. Res. 1997, 36, 3821. (18) Fitz, C. W., Jr.; Kunesh, J. G.; Shariat, A. Performance of structured packing in a commercial-scale column at pressures of 0.02-27.6 bar. Ind. Eng. Chem. Res. 1999, 38, 512. (19) Billet, R. Packed Column Analysis and Design; Rhur University, Bochum, Germany 1989. (20) Billet, R. Packed Towers in Processing and EnVironmental Technology; VCH: Germany, 1995. (21) Spagnolo, D. A.; Chuang, K. T. Hydraulic, mass transfer and heat transfer performance comparison between ordered bed packing and sieve trays. Can. J. Chem. Eng. 1986, 64, 62. (22) Schmitz, D.; Petritsch, G.; Mewes, D. Tomographic measurement of the liquid hold-up in a structured packing. Proc. ASME Heat Transfer DiV. 1998; Vol. 5 (HTD-Vol.361-5), p 371. (23) Lopez-Toledo, J.; Rocha, J. A. U. Effect of channels, punctures, and engraving in structured packings for distillation service. Inst. Chem. Eng. Symp. Ser. 1997, 142, 961. (24) Kolev, N.; Istatkova, E.; Darakchiev, R. A redistribution layer for packed columns intended for high flow velocities. Chem. Eng. Process. 1987, 21, 77. (25) Svinukhov, A. G.; Klykov, M. V. Hydraulic resistance and dynamic holdup of liquid in beds of screen packing. Chem. Technol. Fuels Oils 1978, 12, 900. (26) Buhlmann, U. Garnissages de colonnes Rombopak. Inf. Chimie 1989, 305 (May), 179. (27) Brunazzi, E.; Paglianti, A. Mechanistic pressure drop model for columns containing structured packings. AIChE J. 1997, 43, 317. (28) Brunazzi, E.; Nardini, G.; Paglianti, A.; Petarca, L. Interfacial area of Mellapack packing: Absorption of 1,1,1-trichloroethane by Genosorb 300. Chem. Eng. Technol. 1995, 18, 248. (29) Fair, J. R.; Bravo, L. Distillation columns containing structured packings. Chem. Eng. Prog. 1990, January, 19.
(30) Olano, S.; Nagura, S.; Kosuge, H.; Asanom, K. Mass transfer in binary and ternary distillation by a packed column with structured packing. J. Chem. Eng. Jpn. 1995, 28, 750. (31) Pondebat, C.; Peytavy, J. L.; Huor, M. H.; Prevost, M.; Laurent, A. Hydrodynamics of a gas-liquid column equipped with a structured packing. Inst. Chem. Eng. Symp. Ser. 1992, 128, B253. (32) Meier, W.; Hunkeler, R.; Stoecker, W. D. Performance of the new regular tower packing “Mellapack”. Inst. Chem. Eng. Symp. Ser. 1979, 56, 3.3/1. (33) Spiegel, L; Meier, W. Correlations of the performance characteristics of the various Mellapak types. Inst. Chem. Eng. Symp. Ser. 1987, 104, A203. (34) Wu, K. Y.; Chen, G. K. Large-scale pilot columns and packed column scale-up. Inst. Chem. Eng. Symp. Ser. 1987, 104, B225. (35) Weiland, R. H.; Ahlgren, K. R.; Evans, M. Mass transfer characteristics of some structured packings. Ind. Eng. Chem. Res. 1993, 32, 1411. (36) Stockfleth, R.; Brunner, G. Hydrodynamics of a packed countercurrent column for the gas extraction. Ind. Eng. Chem. Res. 1999, 38, 4000. (37) Kister, Z. H. Packing Capacity and pressure drop GPDC interpolation charts atlas. In Distillation-Design-; Kister, Z. H., Ed.; Mc Graw-Hill: New York, 1992; Chapter 10. (38) Meier, W. Sulzer Tech. ReV. 1970, 3. (39) McNulty, K. J.; Hsieh, C. L. Hydraulic performance and efficiency of Koch Flexipac structured packing. Proceedings of the AIChE Annual Meeting, Los Angeles, CA, November 14-19, 1982. (40) Reid, R. C.; Prausnitz, J. M.; Poling, B. E. The properties of gases and liquids, 4th ed.; McGraw Hill Inc.: New York, 1987. (41) Press, W.; Flannery, B. P.; Teukolsky, S. A.; Vetterling, W. T. Numerical recipes: The art of scientific computing; Cambridge University Press: New York, 1988. (42) Tarca, L. A.; Grandjean, B. P. A.; Larachi, F. Reinforcing the phenomenological consistency in artificial neural network modeling of multiphase reactors. Chem. Eng. Process. 2003, 42, 653. (43) Tarca, L. A.; Grandjean, B. P. A.; Larachi, F. Feature selection methods for multiphase reactors data classification. Ind. Eng. Chem. Res. 2005, 44, 1073. (44) Wang, G. Q.; Yuan, X. G.; Yu, K. T. A method for calculating effective interfacial area of structured packed distillation columns under elevated pressure, Chem. Eng. Process. 2006, 45, 691. (45) Le´vesque, S. Transfert de matie`re dans les garnissages structures: De´veloppement de corre´lations ge´ne´rales. MSc thesis, Laval University, Quebec, Canada, April, 2006.
ReceiVed for reView May 19, 2007 ReVised manuscript receiVed February 13, 2008 Accepted February 25, 2008 IE070718O
