Flooding Capacity in Packed Towers: Database, Correlations, and

Nov 30, 2000 - Artificial neural network modeling was then proposed to improve the broadness and accuracy in predicting the flooding capacity, which i...
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Ind. Eng. Chem. Res. 2001, 40, 476-487

CORRELATIONS Flooding Capacity in Packed Towers: Database, Correlations, and Analysis Simon Piche´ , Faı1c¸ al Larachi,* and Bernard P. A. Grandjean Department of Chemical Engineering and Center for Research on the Properties of Interfaces and Catalysis (CERPIC), Laval University, Que´ bec, Canada G1K 7P4

Experimental results on the flooding capacity of randomly dumped packed beds were collected from the literature to generate a working database. The reported measurements were first used to review the accuracy of existing predictive tools in that field. A total of 14 correlations were extracted from the literature and cross-examined with the database. Many limitations regarding the level of accuracy and generalization came to light with this investigation. Artificial neural network modeling was then proposed to improve the broadness and accuracy in predicting the flooding capacity, which is an important design parameter for packed towers. A combination of six dimensionless groups, namely, the Lockhart-Martinelli parameter (χ); the liquid Reynolds (ReL), Galileo (GaL) and Stokes (StL) numbers; the packing sphericity (φ); and one bed number (SB) outlining the tower dimensions were used as the basis of the neural network correlation. With an initial database containing 1019 measurements, the correlation yielded an absolute average relative error (AARE) of 16.1% and a standard deviation of 20.4%. Another database containing over 100 measurements on the flooding capacity was used to validate the correlation. The prediction based on these results yielded an AARE of 11.6% and a standard deviation of 13.7%. Through a sensitivity analysis, the Stokes number in the liquid phase was found to exhibit the strongest influence on the prediction, while the liquid velocity, gas density, and packing shape factor were determined to be the leading physical properties defining the flooding level. As a matter of fact, the neural correlation remains in accordance with the design recommendations and trends reported in the literature. 1. Historical Overview Packed towers operated in a two-phase countercurrent manner have maintained an important role in today’s chemical industry. Introduced early in the 20th century, the general concept has constantly evolved around the progress of various packing shapes that are stated to increase the tower’s efficiency as well as to reduce the operating costs. Currently, packed towers are used for several purposes, especially in distillation, absorption, and stripping processes. Because the operation of packed towers is considered to be cheap, they are of great interest for pollution control. However, their effectiveness will suffer if the design considerations do not suit the operating conditions. To improve that matter, several correlations, either empirical or semiempirical, were developed to gain precision over design parameters. Since the 1920s, many experiments have been carried out to understand the consequences of various operating parameters on the bed effectiveness. These studies can be divided into two distinctive categories, those related to the mass transfer efficiency and those related to the hydrodynamic phenomena. While the first topic implies the evaluation of volumetric mass transfer coefficients, * Corresponding author. Phone: 1-418-656-3566. Fax: 1-418656-5993. E-mail: [email protected].

the latter one regroups the studies of flow behavior, liquid hold-up, and pressure drop across the bed. One physical phenomenon emerges that must be investigated first, as its level of importance will inevitably influence every other parameter mentioned above. The flooding capacity consists essentially in a determination of the amount of fluid in the bed at which the liquid starts to overflow. Because the flooding capacity can only be observed and not measured, there is no accurate definition to describe this phenomenon, and it can be interpreted in many ways. For example, Silvey and Keller1 enumerated over 10 definitions proposed by different authors. From that list, two distinctive categories can be outlined. The first includes methods involving visual inspection of a column undergoing flooding. The second type, inferential in nature, that also proved to be effective relies on a graphical detection of “slope inflation” in pressure drop and liquid hold-up diagrams as a function of the gas flow rate. Hence, the operating point beyond which a tiny increase in gas velocity produces a substantially large change in pressure drop and liquid hold-up in the column is the basic definition for flooding. Ever since packed beds were introduced, ongoing efforts have been made by engineers in an effort to understand the phenomenology of the flooding capacity in packed beds. A comprehensive summary of all

10.1021/ie000486s CCC: $20.00 © 2001 American Chemical Society Published on Web 11/30/2000

Ind. Eng. Chem. Res., Vol. 40, No. 1, 2001 477

published correlations on the flooding capacity is presented in Table 1. It is not before 1935 that the first results on flooding capacity in beds of Raschig rings were published.18 Following this undertaking, Sherwood et al.2 introduced the first known correlation on that subject. Using data on the air-water system exclusively, a curve representing the flooding capacity was drawn on a specific set of axis parameters. Such an idea has helped to form the so-called generalized pressure drop correlation (GPDC) concept, introduced in the mid 1950s by Leva.8 Based on a large experimental survey, several isobaric curves were sketched on a set of axis parameters similar to Sherwood’s diagram in order to provide an accurate tool for determining the pressure drop in packed beds. The upper line on the chart represented the flooding capacity of the bed occurring at a pressure drop of around 2.5 and 3.0 in. of water per foot of bed for packing elements of the first generation like Raschig rings and Berl saddles. Similar charts were developed to cope with the inconsistencies as it was considered the only reliable source to determine the pressure drop and flooding capacity.10,13,15,19-21 Later, new packing shapes appeared on the market, stated to reach better performance. As an example of higher efficiency, the Pall ring, a secondgeneration packing, could reach a pressure drop of 1.5 in. of H2O/ft at flooding22 which is lower than previous results for less porous beds of first-generation packings. Consequently, generalized pressure drop correlations were limited to first-generation packings as the flooding curve was implemented for pressure drops of 2.5-3.0 in H2O/ft. It is only much later that Kister and Gill13 introduced the idea of correlating the pressure drop at flooding in order to again make accessible the use of GPDCs for predicting the flooding capacity in packed beds containing an arbitrary packing. A second group of correlations, emanating from the German school, focuses more on theoretical considerations of two-phase flow through packed beds.12,16 In most cases, these correlations calculate the superficial gas velocity at flooding for a given set of conditions. A drawback inherent to these correlations is that they require empirical constants specific to the packing under consideration.17 Unfortunately, these constants were not established for every packing element, which clearly illustrates a lack of generalization. Considering all aforementioned limitations, the present work aims at providing a new correlation in order to increase the broadness and accuracy in predicting the flooding capacity essential for design purposes of randomly dumped counter-current packed columns. Consequently, perceptron-like artificial neural network (ANN) modeling was employed here to identify the forces and packing parameters that are involved in describing the flooding phenomenon in terms of the most appropriate set of Buckingham Π and other characteristic dimensionless groups. This model could then become an ultimate tool for cases involving new packing shapes and/or different fluid packages showcasing seldom physical properties. A complete database containing experimental results on the flooding capacity published in the literature since 1935 was first built and then used to evaluate the strengths and weaknesses of all important correlations developed on that subject. Thereafter, a dimensionless ANN correlation is presented, and its effectiveness discussed through statistical comparisons against the most important empirical

and semiempirical correlations from the literature (Table 1). The methodology leading to the neural network correlation will not be detailed in this work as it has already been discussed previously.23,24 2. Databases Overview Two databases containing 1019 and 101 sets of experimental results were used to develop the neural network correlation. They represent well the broad range of packing shapes available on the market, from the common Raschig ring to high-throughput packings such as the IMTP ring22 or the Jaeger ring,25 just to name a few. Similarly, a variety of operating conditions is also exposed in the databases, spanning a broad range from vacuum distillation to supercritical fluid conditions. While the first database, referred to as the correlation database, was expressly used to build the ANN correlation, the second and smaller one, the validation database, served as a validation tool. This practice was established here to confirm the effectiveness of the correlation in predicting the flooding capacity in specific cases such as prediction of flooding for packing shapes alien to the correlation database. Both databases together exhibit results on 27 packing types that were tested at least once for flooding considerations (Table 2). Unfortunately, most of the runs were carried out between 1935 and 1950, when needs for global designing tools were being felt. In fact, near 75% of the correlation database contains results on what are called first-generation packings (Raschig rings, Berl saddles, spheres, coke particles, and wire helices). Second-generation packings such as Pall and Hiflow rings constitute the remaining 25%. On the other hand, the validation database contains mostly results on thirdgeneration packings that are more complex in shape and are not present in the correlation database. Following the same approach, several gas and liquid systems were collected to assess the impact of physical properties such as the density, viscosity, and surface tension. A total of 23 single, binary, and aqueous liquids and 18 gas mixtures are tabulated in the databases. Table 3 offers a more detailed description of the correlation database (1019 data sets) relative to the extent of the operating variables that are susceptible to having an effect on the flooding capacity. These parameters include properties of the liquid (density, viscosity, surface tension, superficial velocity) and gas (density, viscosity, superficial velocity), as well as the packing and bed properties (nominal diameter, porosity, bed-specific surface area, column diameter, bed height). Table 4 offers an equivalent summary for the validation database. In that case, all 101 experiments were carried out with air and water exclusively. To avoid burdening the text with the references from which the experimental data were extracted and incorporated in the databases, the reader can refer to Table 5 for the data sources. This table also contains some work specifications established by the authors. 3. Artificial Neural Network Correlation It has been shown in our previous works23,24 that the clustering of physical properties and operating parameters into dimensionless Buckingham Π groups renders empirical modeling methods such as neural network computing more tractable. Hence, the formation of over 20 dimensionless Π groups was proposed to cover all

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Table 1. Comprehensive Summary of Published Correlations on the Flooding Capacitya

Ind. Eng. Chem. Res., Vol. 40, No. 1, 2001 479 Table 1 (Continued)

a The corresponding correlations (indicated by asterisks) are in a graphical form. Therefore, they are represented with the best polynomial fit for the Y parameter as a function of the X parameter.

possible force ratios or external effects that a randomly packed bed can undergo. For the liquid phase itself, nine representative Π numbers were established. They are the Reynolds (ReL(1)), Froude (FrL(2)), Weber (WeL(3)), Morton (MoL(4)), Eo¨tvos (EoL(5)), Galileo (GaL(6)), Stokes (StL(7)), Capillary (CaL(8)), and Ohnesorge (OhL(0)) numbers. These groups cover the complete range of force ratios that could possibly influence the flooding capacity

such as the inertial-to-viscous (1), inertial-to-gravitational (2, 7), inertial-to-capillary (3), viscous-to-capillary (4, 8, 9), gravitational-to-capillary (4, 5, 6), and gravitational-to-viscous (6, 7) forces. As for the gas phase, a total of four dimensionless groups were established, namely, the Reynolds (ReG), Froude (FrG), Galileo (GaG), and Stokes (StG) numbers. Some packing and bed properties converted into

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Table 2. Databases’ Packing Elements and Corresponding Number of Runs packing type

correlation database

validation database

Raschig ring Berl saddle Pall ring Hiflow ring Sphere NOR PAC (NSW) ring Coke particles Dixon ring Wire helices Intalox saddle Mitsui Nutter ring Novalox saddle Cascade mini ring (CMR) Bialecki ring

467 190 85 79 33 31 30 16 14 13 10 10 9 5

0 0 2 28 0 0 0 0 0 0 0 18 0 0

packing type

correlation database

validation database

McMahon ring Hackettes Glitsch 30Pmk ring McMahon saddle Hy-Pak Glitsch 30P ring IMTP ring VSP ring Intalox Snowflake Jaeger ring Jaeger saddle Lantec Nupac Jaeger Tri-pack

5 4 4 4 4 3 3 0 0 0 0 0 0

0 0 0 0 0 0 0 4 3 14 11 6 15

Table 3. Intervals of Operating Conditions for the Correlation Database classifications

properties

min 0.066 -27

median

pressure, P (atm) temperature, T (°C)

packing & bed properties

nominal diameter, dN (mm) bed porosity,  (%) bed-specific surface area, aT (m-1) tower diameter, DC (m) bed height, Z (m)

liquid properties

density, FL (kg/m3) viscosity, µL (Pa s) surface tension, σL (N/m) superficial velocity, UL (mm/s)

486.1 9.95 × 10-5 1.15 × 10-2 0.153

997.1 1.07 × 10-3 7.30 × 10-2 6.600

13550.0 3.51 × 10-2 4.87 × 10-1 69.400

gas properties

density, FG (kg/m3) viscosity, µG (Pa s) superficial velocity, UG,FL (m/s)

0.082 6.56 × 10-6 0.001

1.190 1.75 × 10-5 0.503

541.000 1.86 × 10-5 7.020

3.2 41.0 57 0.03 0.20

1.000 20

max

operating conditions

15.0 72.5 303 0.16 1.22

102.000 211 88.9 98.4 1148 1.22 5.50

Table 4. Intervals of Operating Conditions for the Validation Database classification

properties

min

median

max

operating conditions

pressure, P (atm) temperature, T (°C)

1.000 21

1.000 21

1.000 21

packing & bed properties

nominal diameter, dN (mm) bed porosity,  (%) bed-specific surface area, aT (m-1) tower diameter, DC (m) bed height, Z (m)

3.2 73.0 66 0.430 1.52

15.0 93.0 125 1.000 3.00

88.9 98.0 623 1.000 3.05

liquid properties

density, FL (kg/m3) viscosity, µL (Pa s) surface tension, σL (N/m) superficial velocity, UL (mm/s)

1000.0 1.00 × 10-3 7.30 × 10-2 3.190

1000.0 1.00 × 10-3 7.30 × 10-2 13.600

1000.0 1.00 × 10-3 7.30 × 10-2 41.600

gas properties

density, FG (kg/m3) viscosity, µG (Pa s) superficial velocity, UG,FL (m/s)

1.151 1.78 × 10-5 0.401

1.186 1.78 × 10-5 2.283

1.188 1.78 × 10-5 4.432

dimensionless groups were also included in the study, contributing to an enlargement of possibilities for a better correlation. The bed porosity () and the packing sphericity (φ), two dimensionless parameters, describe the characteristics of the packing itself. Also, it is known from other considerations that small columns (DC < 0.2 m) which represent the bulk of our databases, have a discrepancy-like effect on the pressure drop. Consequently, there is a possibility that this same uncertainty will influence the flooding capacity. With that in mind, two wall factors and one bed correction factor (SB) were added to the study in order to depict the column factor in the neural network correlation, if necessary. Finally, two mixed groups were also included in the analysis consisting of the Lockhart-Martinelli parameter (χ) and the Reynolds number ratio (ReG/L), both describing the gas-liquid interaction factor.

Several hundred sets of dimensionless groups were tested by using three-layer artificial neural network modeling processed by the NNfit software.53 Several aspects must be taken into consideration to decide whether a set of dimensionless groups presents acceptable results in order to make some further analysis with validating data. The choice of the best sets rests on the fulfillment of the following criteria: (1) The optimal set must lead to the best output prediction, with a minimum absolute average relative error and standard deviation, as well as a maximum correlation coefficient. (2) The optimal set must contain a minimum number of selected dimensionless groups. (3) The neural network architecture must be of minimal complexity, with the least number of hidden neurons without deterioration in the prediction. (4) The optimal set of dimensionless groups

Ind. Eng. Chem. Res., Vol. 40, No. 1, 2001 481 Table 5. Neural Correlation Prediction Results for Each Data Source authors

process type

phases

bed properties (DC:Z)

number of data sets (AARE)

White18

cold model (NTP)

water-air

0.152:1.52 0.076:1.52

18 (38.6%) 12 (17.7%)

Uchida and Fujita26

cold model (NTP)

water-air

0.260:1.40

09 (27.3%)

Uchida and Fujita27

cold model (NTP)

water-air

0.260:0.50 0.260:1.40 0.260:1.70 0.260:3.15

13 (17.5%) 08 (23.3%) 12 (15.1%) 08 (11.5%)

Sherwood et al.2

cold model (NTP) absorption (NTP)

0.050:1.22

stripping (NTP)

water-air water-hydrogen water-carbon dioxide water/glycerol-air water/butyric acid-air methanol-air water/methanol-air

23 (16.3%) 09 (5.2%) 06 (7.7%) 23 (6.5%) 15 (16.4%) 07 (13.6%) 12 (4.5%)

Elgin and Weiss3

cold model (NTP)

water-air

0.073:1.42

62 (22.3%)

Sarchet28

cold model (NTP)

water-air

0.220:0.61

09 (14.5%)

Bain and Hougen29

cold model (NTP)

light oil-air light oil-hydrogen light oil-carbon dioxide

0.220:1.24

65 (16.2%) 07 (24.1%) 17 (33.0%)

Schoenborn and Doherty30

cold model (NTP)

0.220:0.61

Spector and Dodge31

absorption (NTP)

water-air light oil-air water/NaOH-air/CO2

0.305:0.20

42 (21.7%) 85 (15.0%) 01 (22.2%)

Lerner and

Grove32

cold model (NTP)

water-air

0.152:1.22

09 (37.7%)

Minard and Winning33

cold model (NTP) absorption (NTP)

0.152:1.52

stripping (NTP)

water-air water-nitrogen water-carbon dioxide carbon tetrachloride-air

07 (13.1%) 03 (9.6%) 04 (4.7%) 04 (62.7%)

Morton et al.34

cold model (NTP) stripping (NTP)

water-air chlorobenzene-air

0.051:0.61

06 (15.1%) 06 (18.3%)

Billet35

cold model (NTP) stripping (NTP) distillation

water-air ethylene glycol-air water/2-propanol cyclohexane/n-heptane i-butane/n-butane

0.435:1.65 1.200:5.50

03 (3.3%) 03 (2.7%) 04 (7.6%) 08 (33.4%) 03 (5.2%)

cold model (NTP)

water-air

0.102:0.30

64 (25.2%)

absorption (NTP)

mercury/nitrogen mercury/helium

0.051:0.60

13 (14.3%) 02 (41.1%)

Standish and Drinkwater36 Szekely and

Mendrykowski37

Mangers and Ponter38

absorption (NTP)

water/glycerol-CO2

0.100:0.50

04 (12.9%)

Krehenwinkel and Knapp39

absorption (HP)

water-nitrogen methanol-nitrogen

0.155:0.80 0.155:0.80 0.155:1.75 0.155:1.00 0.086:1.00 0.155:0.80

06 (27.1%) 74 (14.2%) 30 (3.7%) 36 (14.1%) 13 (8.3%) 17 (5.1%)

ethylene glycol-nitrogen Rathkamp et

al.40

extraction (HP)

water/2-propanol-co2

0.025:0.61

03 (58.5%)

Billet and Mac´kowiak41

absorption (NTP)

water-air/ammonia

0.450:1.90

02 (12.4%)

Stichlmair et al.42

cold model (NTP)

water-air

0.150:1.50

05 (21.0%)

Billet et

al.43

cold model (NTP)

water-air

0.450:2.00

04 (10.9%)

Billet44

cold model (NTP)

water-air

0.300:1.45 0.300:0.86 0.300:1.20

24 (19.0%) 05 (7.4%) 04 (6.0%)

Miyahara et al.14

stripping (NTP)

water/ethanol-air water/glycerol-air

0.155:0.30 0.155:0.30

22 (13.9%) 12 (20.4%)

Kister22

cold model (NTP) distillation

water-air water/methanol water/acetone p-xylene/m-xylene chlorobenzene/ethylbenzene ethanol/metanol ethylbenzene/styrene toluene/i-octane c-decalin/t-decalin n-hexane/n-heptane

0.229 < DC < 1.219 0.61 < Z < 5.50

76 (7.2%) 01 (4.9%) 01 (19.0%) 02 (12.6%) 09 (17.5%) 03 (14.4%) 04 (6.1%) 15 (14.4%) 02 (11.0%) 02 (9.4%)

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Table 5 (Continued) process type

phases

bed properties (DC:Z)

number of data sets (AARE)

Billet45

cold model (NTP)

water-air

0.300: 1.20

11 (4.0%)

Billet46

cold model (NTP)

water-air

0.300:1.40

12 (19.0%)

authors

Benadda et

al.47

cold model (Vac)

water-air

0.100: 0.80

18 (7.3%)

Macı´as-Salinas and Fair48

cold model (NTP)

water-air

0.430:2.00

02 (4.0%)

Wu and Chen49

cold model (NTP)

water-air

0.914:1.22

02 (2.4%)

cold model (NTP)

water -air

1.000:3.00a

62 (8.9%)

cold model (NTP)

water -air

1.000:3.00a

03 (11.0%)

cold model (NTP)

water -air

1.000:3.00a

06 (57.6%)

cold model (NTP)

water -air

1.000:3.00a

28 (8.4%)

Jeager

Products25

Norton Chemical50 Lantec

Company51

Rauschert Industries52 a

The bed properties were not given on the leaflets, and therefore it was assumed that the column diameter would be approximately 1 m or at least be greater than 0.5 m.

tions can be calculated using the values of the normalized inputs and the corresponding weights (ωij), which are presented in Table 6. The hidden layer contains 13 neurons.

Hj )

1 6

1 e j e 13

(2)

ωjHj) ∑ i)1

1 + exp(-

Figure 1. Schematic of the three-layered feed-forward perceptron artificial neural network.

must be closely associated with the phenomenology of the flooding capacity. One good measure of performance of the trained network is known as generalization ability. It consists of investigating how closely the actual output of the neural network approximates the desired output data for an input that has never been presented as a learning pattern. Each ANN was systematically trained using 714 learning instances (learning set) selected randomly from the correlation database. Once an ANN had been trained, a test set of 305 vectors (i.e., 1019-714) was used to test the correlation robustness. After thorough analysis, six relevant dimensionless numbers were identified to describe appropriately the flooding phenomenon. These characteristic numbers are the Lockhart-Martinelli parameter (χ) as the ANN output and the liquid Reynolds number (ReL), the liquid Stokes number (StL), the liquid Galileo number (GaL), the packing sphericity (φ), and the bed correction number (SB) as the ANN inputs. The general architecture of the actual ANN correlation is presented in Figure 1. The appropriate procedure for the application of this ANN implies first the normalization of the inputs using the boundaries of the dimensionless numbers within the working database. For example

U1 )

φ - φmin φ - 0.045 w U1 ) φmax - φmin 1.00 - 0.045

(1)

The normalization can also be done in the logarithmic scale, especially for cases in which the maximum-tominimum ratio factor reaches 10 or higher. The five normalized input functions (Ui) are summarized in Table 6. At this point, the hidden layer transfer func-

Thereafter, the output transfer function is employed to determine the normalized output function (S) according to another set of weights (ωj) presented in Table 6. The output function is connected to the 13 neurons and one bias value via

S)

1 14

1 e j e 14

(3)

ωjHj) ∑ j)1

1 + exp(-

Finally, the dimensionless output value being the Lockhart-Martinelli parameter is extracted from the normalized output, S, using the following relationship:

χ ) 9.3925 × 10(4.22524S-3)

(4)

4. Performance of Flooding Capacity Correlations 4.1. Statistical Analysis. A statistical evaluation of several correlations summarized in Table 1, including the ANN correlation, is presented in Table 7. This table depicts the number of data sets that were applied to the corresponding correlation with some statistical figure-of-merit parameters, such as the correlation coefficient (R), the absolute average relative error (AARE), and the AARE standard deviation (σ). More emphasis has been put on the analysis of the AARE as it is very representative of the quality of prediction. On the other hand, small values of AARE standard deviations (respectively, high R values) are tantamount to low scatters of the residuals around the average error, thus indicating a good predictive model. The variation in the amount of data from one correlation to the other is mainly due to the fact that only the data fulfilling the specifications of the tested correlation is retained. Three correlations exhibit excellent precision in evaluating the flooding capacity in packed beds. The semiempirical correlations of Billet and Schultes16 and

Ind. Eng. Chem. Res., Vol. 40, No. 1, 2001 483 Table 6. ANN Normalized Input and Output Functions and the Corresponding Weightsa Required in Eqs 2-4b

a Ranges of applicability in brackets. b A “user-friendly” spreadsheet of the neural correlation is accessible on the net at the web address http://www.gch.ulaval.ca/∼flarachi.

Table 7. Statistical Analysis of Correlations on Flooding Capacity number of data sets

AARE author correlation

AARE neural correlation

σ author correlation

σ neural correlation

R author correlation

R neural correlation

Sherwood et al.2 Elgin and Weiss3 Bertetti4 Lobo et al.5 Zenz and Eckert7 Leva8,9 Eckert10 Takahashi et al.11 Maækowiak12 Kister and Gill13 Miyahara et al.14 Leva15 Billet and Schultes16

1104 272 1114 1102 1120 972 710 1120 164 195 1120 645 226

43.8% 46.4% 178.6% 38.4% 58.9% 46.2% 45.5% 50.3% 11.1% 11.4% 101.3% 40.1% 10.8%

15.3% 16.3% 15.5% 15.3% 15.7% 15.2% 15.4% 15.7% 13.7% 11.6% 15.7% 15.3% 12.2%

64.1% 28.0% 2649.0% 53.8% 132.3% 56.1% 52.7% 107.7% 6.2% 6.2% 179.4% 38.8% 7.1%

18.5% 11.7% 18.7% 18.3% 19.5% 17.7% 17.1% 19.5% 6.9% 6.4% 19.5% 16.7% 7.2%

0.943 0.790 0.939 0.942 0.941 0.911 0.959 0.942 0.968 0.980 0.940 0.963 0.978

0.971 0.969 0.971 0.971 0.966 0.965 0.970 0.966 0.930 0.967 0.966 0.969 0.942

this work-learning data this work-generalization data this work-validation data this work-overall data

714 305 101 1120

15.3% 18.1% 11.6% 15.7%

authors

Mac´kowiak,12 as well as the GPDC of Kister and Gill,13 all showcase AAREs of ca. 10%, which is excellent given the circumstances. However, information including the packing factor (Q) and packing constants (CFO, K1, K2) necessary for their implementation are not routinely available for the majority of flooding data (Table 7), which points out an inherent limitation of these tools. Only 20% or less of the database was applicable in those cases. On the other hand, correlations such as those of Sherwood et al.,2 Zenz and Eckert,7 and Takahashi et al.11 are not satisfactory. Although the entire database was applicable, the predictions yielded AAREs of only 40%, which is inadequate for proper design. The neural network correlation produces an AARE of 15.7% based on all of the data, i.e., both correlation and validation databases, with a standard deviation of 19.5%. Figure 2a demonstrates the almost uniform level of accuracy of the neural correlation on both the learning

18.2% 24.4% 13.7% 19.5%

0.978 0.974 0.855 0.966

and generalization files of the correlation database. The validation database, containing results on packing elements that have complex shapes (e.g., Jeager Tri-packs), was then applied to the correlation and yielded results as shown in Table 7 and Figure 2b. An AARE of 11.6% and a standard deviation of 13.7% came out of the analysis, securing further the applicability of the neural correlation to broad situations. Furthermore, Table 5 exposes some statistical results of the ANN correlation for every source composing the databases in this work. The phases’ description associated with a process type and the tower’s size help to illustrate the strength of the correlation on these important aspects. Each individual grouping presents good averaging results except for one set of six data sets in which flooding results were collected for the Nupac packing51 (AARE ) 58%). One possible explanation for this sudden discrepancy resides in the unusual configuration of this packing element.

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Figure 2. (a) Learning (b) and generalization (O) files parity plot on the correlation database. (b) Validation (b) file parity plot. Predicted versus measured gas superficial velocity in flooding conditions. (Dotted lines represent (30% envelopes).

Figure 3. Sensitivity analysis on the effect of the dimensionless variables used as inputs in the neural correlation. The AARE (9) and standard deviation (0) were evaluated on the basis of the whole database (1120 data sets).

Being shaped as a porcupine, it is possible that the sphericity and all other normal packing properties encountered in the neural correlation are not appropriate for the Nupac packing. 4.2. Sensitivity Analysis. A statistical validation of the proposed neural network correlation has been done in order to ensure that it has the potential to predict flooding for any conditions. However, one might have some concern on the actual capability of the correlation to represent the exact phenomenology of the flooding capacity. For instance, each correlation input group must exert a nonnegligible effect on the correlation output, which is the Lockhart-Martinelli parameter in this case. For that matter, a simple analysis to verify the sensitivity of each input variable was performed. Statistical information on five alternate neural network simulations allowing one input to be withdrawn from the original neural correlation is presented in Figure 3. It clearly shows that all five input groups improve by all means the level of precision of the output. The group having the largest sensitivity on the output is the Stokes number (StL), followed by the Reynolds number (ReL) and the bed correction number (SB). Figure 2 shows that the elimination of the Stokes number from

the original neural correlation would result in an unacceptable increase of the AARE (by +8%) and especially of the standard deviation (by +38%). Similarly, the effect of the Galileo number (GaL) and the sphericity (φ) cannot be ruled out of the correlation as the level of precision would also suffer from this action. In fact, the elimination of the less influential input, the packing sphericity, from the original neural correlation would nevertheless result in a 5% increase of the AARE and 8% increase of the standard deviation. 4.3. Influence of Physical Properties on the Flooding Phenomenon. To provide more insight into the exact influence of operating variables on flooding, several ANN simulations were performed by attributing different values for one studied variable while all of the others were held constant. For a set of physical properties, the gas velocity at flooding was calculated at different superficial liquid velocities and compared with the simulations representing other physical properties. From that analysis, six physical properties can be confirmed as being important factors in defining the flooding capacity level. Four diagrams, showcasing some neural network simulations, are proposed to demonstrate their actual level of significance (Figure 4). The corresponding physical properties of a 1-in. metal Pall ring and an air-water system were used for the simulations unless otherwise stated. Most obviously, larger liquid flow in packed beds resulting in greater liquid hold-up increases the obstruction for proper gas flow. It results that the gas velocity at flooding will continuously decrease as the liquid flow rate increases. This trend is clearly shown in Figure 4a-d. Another factor of equivalent importance is the gas density, which depends on the operating pressure of the packed tower. Figure 4a shows how a greater density (e.g., supercritical conditions) will promote lower gas velocity at flooding. With a 10-fold increase in the gas density (0.5 kg/m3 f 50 kg/m3), the flooding capacity decreases by a factor of 10. However, this trend of inverse proportionality is not clearly exhibited in the figure. For example, the case of a 4-fold increase in the air density near the atmospheric pressure (0.5 kg/m3 f 2.0 kg/m3) produces only a 2-fold decrease in the flooding capacity. Therefore, it can be concluded that variations in the gas density near atmospheric pressure have less of an influence on flooding compared to

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Figure 4. ANN simulations describing the effect of operating variables on the flooding capacity: (a) gas density, (b) packing properties, (c) liquid viscosity, (d) column diameter. All simulations were performed with 1-in. metal Pall rings and the air-water system unless otherwise stated. The dots represent the single predictions of the ANN, while the lines produce a comparative point of view.

situations when the variations occur under supercritical conditions, for example. The packing shape properties, which include the bed porosity and the packing surface area, also have notable effects on the flooding level. Four simulations on different packing elements were thus performed. For one, Figure 4b shows that the bed porosity has a pronounced effect on flooding, compared to the Tellerette and CMR simulations for which the surface area is almost identical. This result exposes the fact that high porosity induces larger flooding capacity, which is admittedly reported in the literature. The gas superficial velocity at flooding for the CMR simulation is greater by 1 m/s across the whole liquid velocity range. For that case, the factor of porosity variation is 1.1 (0.96/0.86). Similarly, a 2-fold reduction of the surface area (Intalox ring, aT ) 245 m-1; Flexiring, aT ) 102 m-1), inducing, in all likelihood, less friction by the fluids on the packing also induces a greater flooding capacity by approximately 1 m/s. Clearly, the variation in the bed porosity becomes an all-important physical property for better control of the flooding capacity level. With the same flooding capacity increase (1 m/s), the bed porosity variation factor needed for that purpose was only 1.1, whereas it was 0.5 for surface area case. The last two operating variables are showcased in Figure 4c,d. The column diameter presents a discrepancy-like effect on flooding. While larger diameters

create a marginal shift on flooding, the wall effect becomes more persistent when the ratio of the column diameter (DC) to the nominal particle diameter (dN) falls below 10. Similarly, the liquid viscosity plays a limited role in defining the flooding level. As depicted in Figure 4d, more viscous liquids, even by a factor of 100, inhibit only slightly the facility of the gas phase to flow through the bed interstice, consequently reducing the flooding capacity. 5. Closing Remarks A flooding capacity correlation was developed by combining artificial neural network computing and dimensional analysis. With the correlation database (1019 data sets), the neural network correlation yielded an absolute average relative error (AARE) of 16.1% with a standard deviation of 20.4%. On the other hand, the validation database (101 data) yielded an AARE of 11.6% and an AARE standard deviation of 13.7%. Because of the broadness and diversity of the databases, the proposed correlation has the capability of simulating the flooding capacity in randomly dumped packed beds for any purposes such as absorption and distillation. The correlation is also consistent for cases with compact and high-throughput packings, with special gas-liquid systems presenting alien operating conditions. The identification of six relevant physical properties influencing

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the flooding capacity was confirmed. They are the liquid superficial velocity, the liquid viscosity, the gas density, the bed porosity, the packing surface area, and the column diameter. Consequently, the neural correlation remains in accordance with the design recommendations and trends reported in the literature.

Acknowledgment Financial support from the Natural Sciences and Engineering Research Council of Canada (NSERC) and the Fonds pour la Formation de Chercheurs et d’Aide a` la Recherche (Que´bec) is gratefully acknowledged.

Notation AARE ) Average absolute relative error, AARE ) N |y 1/N∑i)1 exp(i) - ypred(i)/yexp(i)| (-) aS ) External area of particle and wall per unit volume, aS ) aT + 4/DC (m2/m3) aT ) Bed specific surface area (m2/m3) CaL ) Liquid Capillary number, CaL ) (µLUL)/(σL) (-) DC ) Column diameter (m) dh ) Krischer and Kast hydraulic diameter, dh ) dPV[163/ 9π(1 - )2]1/3 (m) dN ) Packing nominal diameter (m) dP ) Sphere diameter equivalent with the particle specific area, dP ) 6(1 - )/aT (m) dPV ) Sphere diameter equivalent with the particle volume, dPV ) 6(1 - )/(φaT) (m) dT ) Sauter mean droplet diameter (m) EoL ) Liquid Eo¨tvos number, EoL ) (gFLd2P)/σL (-) FrG ) Gas Froude number, FrG ) U2G,Fl/(2gdP) (-) FrL ) Liquid Froude number, FrL ) U2L/(2gdP) (-) g ) Gravitational constant (m/s2) G ) Gas mass flow rate (kg m-2 s-1) GaG ) Gas Galileo number, GaG ) (gF2Gd3P)/µ2G (-) GaL ) Liquid Galileo number, GaL ) (gF2Ld3P)/m2L (-) hS ) Static hold-up (-) hT,Fl ) Total liquid hold-up at flooding (-) K1, K2, CFO ) Constants specific to the packing element (-) L ) Liquid mass flow rate (kg m-2 s-1) MoL ) Liquid Morton number, MoL ) (gµ4L)/(FLσ3L) (-) N ) Number of data sets (-) NP ) Number of particles per unit volume of bed (m-3) OhL ) Liquid Ohnesorge number, OhL ) xµ2L/(σLdPFL) (-) P ) Pressure (Pa) Q ) Packing factor (m2/m3) R ) Correlation coefficient (-) ReG ) Gas Reynolds number, ReG ) (FGUG,Fl)/[aT(1 - )µG] (-) ReG/L ) Reynolds number ratio, ReG/L ) ReG/ReL (-) ReL ) Liquid Reynolds number, ReL ) (FLUL)/[aT(1 - )µL] (-) S ) Normalized output variable (-) SB ) Bed correction number, SB ) (aSdh)/(1 - ) (-) StL ) Liquid Stokes number, StL ) (µLUL)/(FLgd2P) (-) StG ) Gas Stokes number, StG ) (µGUG,Fl)/(FGgd2P) (-) T ) Temperature (K) UG,Fl ) Gas superficial velocity at flooding (m/s) Ui ) Normalized input variables (-) UL ) Liquid superficial velocity (m/s) WeL ) Liquid Weber number, WeL ) (U2LdPFL)/(2σL) (-) Z ) Bed height (m)

Greek Letters χ ) Lockhart-Martinelli parameter, χ ) UG,FlxFG/FL/UL (-) ∆PFl ) Pressure drop at flooding per unit length of bed (Pa/m)  ) Bed porosity (-) φ ) Particle sphericity, φ ) (πNP/aT)[6(1 - )/πNP]2/3 (-) µa ) Air viscosity (kg m-1 s-1) µG ) Gas viscosity (kg m-1 s-1) µL ) Liquid-phase viscosity (kg m-1 s-1) µW ) Water viscosity (kg m-1 s-1) Fa ) Air density (kg/m3) FL ) Liquid-phase density (kg/m3) FG ) Gas-phase density (kg/m3) FW ) Water density (kg/m3) σ ) Standard deviation, σ ) 2 N [|y x∑i)1 exp(i) - ypred(i)/yexp(i)| - AARE] /(N - 1) (-) σL ) Liquid surface tension (N/m) σW ) Water surface tension (N/m) ωij, ωj ) Neural net fitting parameters (-) ψF ) Drag coefficient (Mac´kowiak’s correlation) (-) ψL ) Resistance coefficient at flooding (Billet’s correlation) (-)

Abbreviations ANN ) Artificial neural network GPDC ) Generalized pressure drop correlation HP ) High pressure NTP ) Normal temperature and pressure Vac ) Vacuum w/o ) Without Subscripts G ) gas L ) liquid

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Received for review May 15, 2000 Revised manuscript received September 20, 2000 Accepted October 3, 2000 IE000486S