4542
Ind. Eng. Chem. Res. 1998, 37, 4542-4550
Pressure Drop and Liquid Holdup in Trickle Flow Reactors: Improved Ergun Constants and Slip Correlations for the Slit Model Ion Iliuta,† Faı1c¸ al Larachi,* and Bernard P. A. Grandjean Department of Chemical Engineering & CERPIC, Laval University, Que´ bec, Canada G1K 7P4
The original and extended Holub phenomenological models for pressure drop and liquid holdup in trickle flow regime systematically under-predicted frictional pressure drops at elevated pressure and at high gas throughputs. On the basis of an extensive historic trickle flow regime database and Ergun bed constants (over 4000 measurements from 34 references between 1959 and 1998), state-of-the-art correlations for shear and velocity slip factors and Ergun singlephase flow bed constants (Blake-Kozeny-Carman and Burke-Plummer bed parameters) were developed. The correlations involved combination of feed-forward neural networks and dimensional analysis. The shear and velocity slip factors were expressed as a function of the six most expressive dimensionless groups (ReL, ReG, FrL, WeL, XL, StL), whereas Blake-KozenyCarman and Burke-Plummer bed parameters were correlated to particle equivalent diameter, sphericity factor, bed porosity, and column diameter. These correlations fed into Holub’s phenomenological model improved noticeably the prediction of frictional pressure drop and liquid holdup in trickle flow reactors. Introduction Trickle-bed reactors (TBR), which are catalytic randomly packed fixed-bed tubular reactors traversed cocurrently downward by gas and liquid, ushered in a variety of industrially important three-phase catalytic reactions such as in the petroleum and petrochemical sectors. With the constant market evolution of the products relying on TBR technology to meet concomitantly environmental regulation and economical constraints, any slight improvement of these reactors can translate into substantial benefits. Therefore, improving the understanding of the physicochemical phenomena taking place in trickle beds is vital for the accomplishment of better product quality, environmentally sustainable technologies, etc. The main parameters that impact TBR operation are liquid holdup and two-phase pressure drop which are closely interlinked with reaction conversion and selectivity, power consumption, interfacial mass transfer, etc. Therefore, a basic understanding of the hydrodynamics and mass transfer of TBRs at the operating conditions of interest is essential to their design, scale-up, scaledown, and performance prediction. A great deal of studies that appeared in the literature in the past four decades focused on the determination of liquid holdup and pressure drop in pilot as well as in laboratory scale TBRs. The majority of correlations thus far developed to predict these two parameters were entirely empirical and, as shown repeatedly in the literature, lacked to generality. Holub et al.1-3 developed a phenomenological pore-scale two-fluid separated-flow model in the form of an Ergun-like equation inside an inclined slit based on a simple physical sketch of the complex geometry of the porous medium. The model, valid in the trickle flow * Corresponding author. Telephone: 001-418-656-3566. Fax: 001-418-656-5993. E-mail:
[email protected]. † On leave from Department of Chemical Engineering, Faculty of Industrial Chemistry, University Politehnica of Bucharest, Polizu 1, 78126 Bucharest, Romania.
regime, viewed the gas-liquid flow as fully segregated with an outer liquid foil fully wetting the catalyst wall and an inner gas plug. With the shearless (gas-liquid interface hermetic to momentum transfer) and slipless (zero interfacial gas velocity) boundary conditions, these authors showed that this model predicted their own (and some literature) atmospheric pressure drop and liquid holdup data better than some recent empirical correlations.4-9 However, Al-Dahhan and Dudukovic10 showed that the slipless/shearless model did not perform well in the predictions of pressure drops at elevated pressures and high gas throughputs. The large underestimation of pressure drops was ascribed to the lack of an appropriate description of the interfacial interaction between gas and liquid. Al-Dahhan et al.11 extended therefore Holub’s model by developing two correlations for velocity and shear slip factors. In its last version, the model considered momentum transfer at the gasliquid interface as a plausible mechanism for enhancing pressure drop in trickle flow. In absence of an extensive database, only the experimental data of Al-Dahhan12 and Al-Dahhan and Dudukovic13 were used to assess the importance of the shear and velocity slip factors, which led to the derivation of narrow-range-based empirical shear and slip factor correlations. A large database is still needed for the development of more robust shear and velocity slip factors for high-pressure conditions as well as for moderate-to-high phase interaction within the trickle flow regime.11 Moreover, despite the model captures elegantly some features of the phenomenology of two-phase flow in TBRs, one of its reproaches lies in the a priori procurement of the two Ergun constants, i.e., the Blake-Kozeny-Carman constant (creeping flow contribution) and the BurkePlummer constant (inertial/turbulent contribution), which need to be fed into the model. The so-called Ergun constants are usually claimed to be complex lumps accounting for the bed geometry. But, to the best of our knowledge, no attempt was made to correlate these lumps in terms of bed and grain characteristics for the
10.1021/ie980394r CCC: $15.00 © 1998 American Chemical Society Published on Web 11/17/1998
Ind. Eng. Chem. Res., Vol. 37, No. 12, 1998 4543 Table 1. Description of Database fluid physical properties 650 e FL e 1204 3 × 10-4 e µL e 5.7 × 10-2 1.84 × 10-2 e σL e 7.77 × 10-2 0.16 e FG e 93.1 1.7 × 10-5 e µG e 2.4 × 10-5
operating conditions 10-6
limits of dimensionless groups
10-2
9× e vSL e 2.45 × 2 × 10-4 e vSG e 4.082 0.1 e P e 10 MPa 13 e T e 350 °C -8.52 e fs e -8 × 10-4 -133.36 e fv e 0
1.50 × 10-2 e ReL e 97.4 1.94 × 10-1 e ReG e 871 8.17 × 10-8 e FrL e 3.06 × 10-2 1.0 × 10-7 e WeL e 2.0 × 10-2 6.85 × 10-3 e XL e 3.06 × 102 3.33 × 10-7 e StL e 2.6 e 10-3
Geometrical Properties of Packings and Columns 1.14 × 10-3 e dp e 6.29 × 10-3 particle diameter, dp (m) column diameter, dc (m) 2.19 × 10-2 e dc e 3.0 × 10-1 bed porosity, 0.26 e e 0.49 shape factor, φ 0.75 e φ e 1.0 Ergun constants 35 e E1 e 380; 0.63 e E2 e 3.54 particle material glass, ceramics, porous alumina, polyethylene, polypropylene particle shape spherical, cylindrical and extrudates, raschig rings column geometry cylindrical Liquids Newtonian: water, water + 1% ethanol, water + 40% sugar, water + NaOH (2.4 N), water + NaOH (0.2 N), water + DIPA (1.8 M), water + Na2SO3 (0.8 M), water + 10% 2-propanol, water + 9% glycerol, water + 29% glycerol, methanol, cyclohexane, propylene carbonate, ethanol, ethanol + DEA (0.645 M), ethanol + MEA (0.183 M), ethylene-glycol, ethylene glycol + MEA (1.43 M), petroleum ether, toluene + CHA + 10% IPA, toluene + 10% IPA + DIPA, DMA, hexane, glycerol, soybean oil, gasoline, nondesulfurized gas oil, desulfurized gas oil, kerosene Non-Newtonian: water + 0.1 wt % CMC, water + 0.5 wt % CMC, water + 1.0 wt % CMC Gases air, argon, carbon dioxide, carbon dioxide + air, sulfur dioxide + air, helium, nitrogen, air + oxygen
conditions of interest to TBR operation based on a wide ranging historic database. In this work, state-of-the-art feed-forward neural network-based correlations for the prediction of shear and velocity slip factors and Ergun single-phase flow bed parameters are developed based on an extensive historic database containing beyond 4000 experimental liquid holdup and two-phase frictional pressure drop data belonging to the trickle flow regime. Two main objectives are sought by the current hybrid methodology: (i) to provide the most accurate predictions of liquid holdup and pressure drop in trickle flow regime using Holub’s phenomenological model1 aided with generalized neural network correlations; (ii) to retain much of the advantages of the phenomenological character of the model in order to provide meaningful simulations of the hydrodynamics within large-scale commercial trickle beds operating in trickle flow regime. Development of the Correlations Throughout this report, a procedure akin to that followed by Bensetiti et al.13 is adopted for the development of the shear and velocity slip factor correlations and Ergun bed constant correlations. It consists of the following steps. Database Compilation. Taking advantage of the abundant experimental information published worldwide between 1959 and 1998 in the TBR open literature, the most updated database of liquid holdup, frictional pressure drop for trickle flow regime, and Ergun parameters was set to derive general correlations. For the sake of homogeneity, the dynamic liquid holdups collected from some references were converted to external holdups by adding the external static holdup estimated by Saez and Carbonell correlation.14 For the porous particles, reported total liquid holdups were transformed into external holdups by resting the intraparticle liquid contribution. The total pressure gradient is comprised of three components, respectively, frictional, gravitational, and
acceleration/deceleration:
(∆P/H)t ) ∆P/H - (∆P/H)g + (∆P/H)a For whatever pressure gradients were reported in the literature, they were all converted into frictional pressure gradients using the above definition. Pressure variations occasioned by the acceleration/deceleration term were shown to be marginal and the corresponding pressure correction (third term in the above equation) was verified to be negligibly small for the trickle flow measurements of the database. Key information from the database is summarized in Table 1. It is composed of beyond 4000 experimental data collected from 34 references1-3,6-10,12,15-42 (almost all existing data for trickle flow regime) for different gas-liquid-solid systems. A wide range of liquid and gas velocities, fluid physical properties, and packing and column geometries is included: (i) 32 liquids, pure and mixed, aqueous and organic, Newtonian and pseudoplastic non-Newtonian, coalescing, and noncoalescing; (ii) eight pure and mixed gases from atmospheric pressure up to 10 MPa; (iii) cylindrical columns of 29 different diameters packed with 44 kinds of packing materials of different shapes. Even though the database is elaborate, it was not possible to include all the published data in such cases where (i) Ergun constants, E1 and E2, were not reported or (ii) some operating parameters were lacking. The database was arranged in a matrix form in which each row contains the liquid superficial velocity, liquid density, liquid dynamic viscosity, surface tension, gas superficial velocity, gas density, gas dynamic viscosity, bed porosity, particle equivalent diameter, sphericity factor, column diameter, Blake-Kozeny-Carman constant (E1), Burke-Plummer constant (E2), external liquid holdup, and frictional pressure drop. Ergun Constants. Ergun constants, E1 and E2, were correlated after analyzing cross-correlation effects (Rpq) with grain volume equivalent diameter, sphericity factor, bed porosity, and column diameter.
4544 Ind. Eng. Chem. Res., Vol. 37, No. 12, 1998 Table 2. Extended Holub Model1 equations ΨL )
∆P/H +1) FLg
( )[ L,t
ΨG )
(
∆P/H +1) FLg
( )[ - L,t
ΨL ) 1 + Rei )
]
3
]
E1(ReG - fvGRei) E2(ReG - fvGRei)2 + GaG GaG
FG (Ψ - 1) FL G
0 < ηL < 5
Rei ) ΦL[-3.05 + 5 ln(ηL)] Rei ) ΦL[5.5 + 2.5 ln(ηL)] where:
ηL )
(5) 5 < ηL < 30
(6)
ηL > 30
( x x ( )(
10 νL (E1)0.75 νG
1 5(E1)0.25
(2)
(4)
νL(1 - )
ΦL )
(1)
(3)
viLdp
Rei ) ΦLηL
)
E1ReL E2ReL2 G FG + + fs 1- ΨL GaL GaL L,t FL
3
(7)
)
L,t G FG ΨG ΨLGaL 3 1 + fs L,t FL ΨL
ΨLGaL
L,t
3
(8)
)
G FG ΨG 1 + fs L,t FL ΨL
(9)
Shear and Velocity Slip Factors. Extended Holub model1 equations for pressure drop and liquid holdup are presented in Table 2. As mentioned earlier, in these equations fs (shear slip factor) and fv (velocity slip factor) characterize the degree of phase interaction at the gasliquid interface. Shear and velocity slip factors were evaluated by solving eqs 1 and 2 aided by eqs 3-9 (Table 2) for each row of the constructed database. In the case of fv, only the value which fulfilled Prandtl’s mixing length turbulence model1 was retained. Force Analysis. fv and fs being obtained for each measurement of the database, a force analysis was performed to identify the most meaningful forces that
impact the shear and velocity slip factors. Plausible forces to be considered are as follows: (a) liquid and gas inertial forces caused by turbulence or nonlinear laminar flows, scaling respectively as FiL ) FLvSL2 and FiG ) FGvSG2; (b) interfacial shear stress at the various liquid boundaries, which scales as FvL ) µLvSL/dp; (c) liquid gravitational force, FgL ) FLgdp; (d) capillary force, FcL ) σL/dp. The most significant forces being established, dimensional analysis was used to search for the best set of dimensionless groups that would intervene in the final shear and velocity slip factor correlations. Beyond one hundred sets of dimensionless numbers where tested by trial and error which led to the following most relevant groups: (a) liquid inertia-to-viscous forces ratio, ReL ) FLvSLdp/ [µL(1 - )]; (b) gas inertia-to-viscous forces ratio, ReG ) FGvSGdp/ [µG(1 - )]; (c) liquid inertia-to-gravity forces ratio, FrL ) vSL2/ [gdp]; (d) liquid inertia-to-capillary forces ratio, WeL ) vSL2dpFL/σL; (e) liquid-to-gas inertia forces ratio, XL ) vSLFL1/2/ [vSGFG1/2]; (f) liquid viscosity-to-gravity forces ratio, StL ) µLvSL/ [FLgdp2]. Neural Regression. Three-layer feed-forward neural network models were designed, using NNFit software,43 to derive three dimensionless correlations: one for the shear slip factor, one for the velocity slip factor, and one for both Ergun constants. The neural network model included synthetic neurons, grouped in specialized layers: input, hidden, and output (Figure 1). Every neuron acquired signals from neurons of precedent layer through its dendrites (known as weighted input connections), processed the sum of the weighted inputs according to its threshold function (a sigmoid function here), and conveyed the result to the subsequent layer through its axon (known as output connection). As an exception, the input layer was formed of neurons that had only one nonweighted dendrite. Their task was to
Figure 1. Architectures of the three-layer feed-forward neural network for fv, fs, and E1, E2.
Ind. Eng. Chem. Res., Vol. 37, No. 12, 1998 4545 Table 3. Set of Equations for the Neural Network Model 1
Sk )
J+1
1 + exp[-
for 1 e k e K
(10)
for 1 e j e J
(11)
∑ω H ] jk
j
j)1
1
Hj )
I+1
1 + exp[-
∑ω U ] ij
i
i)1
fs (I ) 6, J ) 6, K ) 1) and fv (I ) 6, J ) 4, K ) 1) S1 )
fs + 8.5230 8.5222
(
log U1 )
U4 )
)
ReL
)
WeL
1.0 × 10-7 5.301
fv + 133.3571 133.3571
(
log
1.506 × 10-2 3.81
(
log
S1 )
or
U2 )
(
log U5 )
(12)
)
ReG
1.944 × 10-1 3.651 XL
)
6.855 × 10-3 4.650
(
log U3 )
(
log U6 )
)
FrL
8.167 × 10-8 5.574 StL
(13)
)
3.333 × 10-7 3.894
(14)
U7 ) 1
(15) E1 and E2 (I ) 4, J ) 8, K ) 2)
S1 )
E1 - 35.0 345
U1 )
dp - 0.00114 0.00515
and
S2 )
U2 )
E2 - 0.63 2.91
dc - 0.0219 0.2781
(16) U3 )
φ - 0.75 0.25
U4 )
- 0.26 0.23
(17)
U5 ) 1
normalize the input data vector and to feed the hidden layer neurons. The neural architectures are described by the generic equations 10 and 11 (Table 3) that correlate the network outputs, Sk, to sets of normalized input variables, Ui. In these equations U and H define the input and hidden layer vectors, HJ+1 and UI+1 are the bias constants set equal to 1, ωij and ωjk are the weights or the fitting parameters of the neural network models, J is the number of nodes in the hidden layer, and K is the number of output nodes. The network fitting parameters are a priori unknown, and they have to be determined using a training algorithm by performing a nonlinear least-squares regression over known pseudo-random set of inputs/outputs (70% of the database). The weights are set as to minimize the training error on the training set using a quadratic objective function which is minimized by the quasi-NewtonBroyden-Fletcher-Goldfarb-Shanno algorithm.44 A good measure for the extrapolation performance of a well-trained neural network is given by the generalization error which should be comparable to the training error in the case of inputs/outputs not presented during the learning step to the neural network (i.e. 30% of the remaining data). Discussion Ergun Constants. Cross-correlation coefficients of Ergun constants, E1 and E2, with the bed and grain characteristics are shown in Table 4. E1 is affected the most by the particle volume equivalent diameter, and the least by the particle sphericity. E2 increases as dp, dc and are increased, and decreases as φ increases. E2 is affected the most by the bed porosity and the least by the column diameter.
(18) Table 4. Cross-Correlation Coefficients of E1 and E2 with Fixed-Bed Parameters solid-phase parameter
cross-correlation to E1 (%)
cross-correlation to E2 (%)
dp dc φ
43.1 16.2 1.00 16.9
21.0 7.10 -10.5 43.2
Force Analysis. Cross-correlation coefficients between the phase interaction parameters (fv and fs) and the different forces, and the corresponding dimensionless groups are listed in Table 5. fv increases as all forces considered increase, while fs increases with increasing gas inertial and liquid capillary forces and decreases with increasing liquid inertial, gravitational and viscous forces. Gas inertia affects fv and fs the most, whereas liquid gravitational force affects fv and, in particular, fs the least. Table 5 also shows that fv increases as all relevant dimensionless numbers increase, while fs increases only if ReG is increased. Neural Regression. For each of fs, fv and (E1, E2) neural network, the number of hidden neurons, J, was varied from 3 to 15. Hidden layers with six neurons (for fs), four neurons (for fv) and eight neurons (for E1, E2) were found to be the optimal neural architectures leading to the smallest average absolute relative errors (AARE) and standard deviations (Table 6) on the training and the generalization sets. The complete sets of the neural network equations for shear and velocity slip factors as well as for Ergun constants are given in Table 3, while Tables 7-9 list the corresponding fitted weights. The neural network correlations for fs, fv, and
4546 Ind. Eng. Chem. Res., Vol. 37, No. 12, 1998 Table 5. Cross-Correlation Coefficients of fv and fs with Forces and Dimensionless Groups force
cross-correlation to fv (%)
cross-correlation to fs (%)
dimensionless group
cross-correlation to fv (%)
cross-correlation to fs (%)
FiG FiL FGL FVL FC
32.9 17.9 4.8 6.1 17.2
20.0 -7.5 -0.8 10.6 -14.7
ReL ReG FrL WeL XL StL
23.2 15.7 16.8 16.9 8.6 7.4
-14.6 26.3 -4.8 -17.0 -22.2 -13.1
Table 6. Statistical Tests of Neural Regression of Phase Interaction Functions (fv, fs) and Ergun Constants statistical parameter
learning data
AARE (%) σ (%)
8.7 (7.2) 21.0 (13.8)
statistical tests: fs (fv) generalization data 9.5 (8.0) 25.0 (26.0)
whole data
learning data
8.9 (7.5) 22.0 (18.0)
12.5 (5.6) 17.0 (6.8)
statistical tests: E1 (E2) generalization data 13.5 (5.8) 16.0 (7.0)
whole data 12.7 (5.7) 16.8 (6.9)
Table 7. Fitting Parameters of the Neural Network Model for the Shear Slip Factor fs ωij
1
2
3
4
5
6
1 2 3 4 5 6 7
-10.6566 11.0997 19.6306 -18.2336 5.0963 31.5273 -18.5781
5.3764 16.1433 1.6994 -1.6442 -17.8860 -4.4686 -5.2485
4.1366 18.5459 11.3146 -9.5024 -19.1944 11.8720 0.4091
-14.6786 5.8315 4.4774 -11.4508 -9.4076 16.3982 -10.1241
4.9670 14.6081 8.8917 0.1292 -19.1432 3.8392 -7.2616
-2.2693 23.5468 -0.0237 20.1003 -17.4383 -2.0279 11.6020
ωj1
1
2
3
4
5
6
7
0.7094
1.4189
-0.1645
-1.5378
4.9176
-3.7952
0.4387
pressure gradient via the proposed hybrid methodology. Although this methodology mixes phenomenology and empiricism, it helps the extended Holub phenomenological model1 (Table 2) with robust neural network correlations to provide, as will be shown later, the most accurate predictions of liquid holdup and pressure drop in trickle flow regime. Also, since Holub’s phenomenological model and fs and fv correlations are formulated with dimensionless numbers, it is expected that by preserving the phenomenological character to the model, large-scale trickle bed reactor hydrodynamics can be forecast more confidently. Performance of Hybrid Methodology, Comparison with the Extended Holub Model, and the Most Used Correlations. Figures 3a and 4a show the lack of fit of the extended Holub model1 (E1 and E2 from the literature) coupled with the shear and velocity slip correlations of Al-Dahhan et al.11 (Table 10) to predict
Table 8. Fitting Parameters of the Neural Network Model for the Velocity Slip Factor fv ωij
1
2
3
4
1 2 3 4 5 6 7
2.1977 -14.1504 19.1520 -7.7351 -26.0550 18.8997 -2.4689
12.1498 -16.7830 -7.0664 -12.5719 2.7179 -22.6449 -1.6521
22.9972 -33.0181 10.5337 0.7644 22.5920 -4.0852 13.1155
3.9277 6.0222 0.0253 -1.4378 9.3031 -4.3526 -7.6978
ωj1
1
2
3
4
5
1.7660
8.6471
6.1057
4.5899
-6.0959
(E1, E2) are available on the net and accessible at the following web address: http://www.gch.ulaval.ca/∼flarachi. Figure 2 details a step-by-step calculation algorithm leading to the external liquid holdup and frictional
Table 9. Fitting Parameters of the Neural Network Model for Ergun Constants ωij 1 2 3 4 5
ωj1 ωj2
1
2
3
4
5
6
7
8
-33.3862 -11.7138 11.4772 -35.0969 16.0804
16.0868 -25.8297 2.5285 -32.9885 4.3801
-44.9879 1.4835 -7.1554 -59.4323 29.5109
-28.5733 -17.8114 4.6443 -21.6510 14.3006
-52.2660 -39.9019 50.0182 52.3405 -4.8844
-4.3879 -19.8660 29.9996 -4.8121 -14.6521
-19.7709 -39.6969 11.6226 0.7970 -13.3545
-43.5459 8.4842 26.9603 -48.6549 28.6635
1
2
3
4
5
6
7
8
9
-11.7547 2.5634
19.4895 57.3052
-22.3442 -61.2694
10.2955 -3.2538
6.2857 41.1309
2.7051 0.0496
11.7742 34.3030
-1.9215 -2.5728
-6.6653 -39.3581
Table 10. Statistical Tests of the Extended Holub Models fv and fs as prescribed by Al-Dahhan et al.11 a
a
fv and fs calculated with the relations developed using neural network and database
statistical parameters
E1, E2 (literature) L,t ∆P/H
E1, E2 (literature) L,t ∆P/H
E1, E2 (neural network) L,t ∆P/H
AARE (%) σ (%)
13.4 12.6
9.5 9.1
12.1 10.7
68.0 22.0
fs ) -4.4 × 10-2ReL0.15ReG0.15, fv ) -2.3ReL-0.05ReG0.05.
36.0 20.0
35.4 26.2
Ind. Eng. Chem. Res., Vol. 37, No. 12, 1998 4547
Figure 2. Synoptic guide to the integration of the neural network correlations for fs, fv, and (E1, E2) into the extended Holub model for the computation of external liquid holdup and frictional pressure gradient.
adequately the whole frictional pressure gradient measurements in the elaborated trickle flow database. A better fit is achieved when the extended Holub model1 is coupled with the shear and velocity slip neural network correlations developed in this work; see Figures 3b, 4b (E1 and E2 from the literature), and Figure 5 (E1 and E2 predicted from neural network). The respective scatters between experimental measurements and model predictions are summarized in Table 10. The differences in predicting liquid holdup are not significant whether the correlations of Al-Dahhan et al.11 (fv, fs) (AARE ) 13.4%) or of this work are employed (AARE ) 9.5% and 12.1%). This is in accordance with AlDahhan et al.,11,12 who already showed that Holub’s model does not require precise correlations for fs and fv to satisfactorily forecast the liquid holdup. However, a net improvement by almost a factor of 2 reduction on the error is attained for the frictional pressure drop when using the fs, fv neural network correlations along with the tabulated or the neural-network computed E1, E2 bed constants (Table 10). The improved performance of the present approach over some general literature correlations to predict holdups and pressure drops is illustrated in Table 11, which shows the poor confidence of the empirical correlations to be general. Conclusion New state-of-the-art correlations for shear and velocity slip factors, and Ergun single-phase flow bed constants were developed using feed-forward neural networks, dimensional analysis, and an extensive historic trickle flow regime database. The database, built from 34 sources published between 1959 and 1998, consisted of more than 4000 liquid holdup and frictional pressure drop measurements (specific to the trickle flow regime) and beyond 100 sets of Ergun bed constants obtained with 32 liquids, 8 gases up to 10 MPa, 44 packing materials, and 29 column diameters. On the basis of this large historic data set, a set of six most expressive dimensionless groups (ReL, ReG, FrL, WeL, XL, StL) accounting for the effects of the dominant forces in
Figure 3. Whole database parity plot. Predicted vs experimental external liquid holdup (E1 and E2 reported in the literature), for the extended Holub model: (a) with Al-Dahhan et al. fs, fv correlations;11 (b) with the present fs, fv neural network correlations.
4548 Ind. Eng. Chem. Res., Vol. 37, No. 12, 1998
Figure 4. Whole database parity plot. Predicted vs experimental two-phase frictional pressure drop (E1 and E2 reported in the literature), for the extended Holub model: (a) with Al-Dahhan et al. fs, fv correlations;11 (b) with the present fs, fv neural network correlations.
trickle flow regime were selected to establish the neural network relationships for shear and velocity slip factors. Moreover, a neural network correlation was provided in which the Blake-Kozeny-Carman and the BurkePlummer bed parameters were expressed as a function of particle equivalent diameter, sphericity factor, bed porosity, and column diameter. The three proposed correlations fed into Holub’s phenomenological slit model allowed to achieve improved predictions of the frictional pressure drops and the external liquid holdups in the trickle flow regime. As more experimental data will be released in the literature for new unexplored operating domains, it is expected that the proposed database will be permanently upgraded and the constitutive correlations for fs, fv, and (E1, E2) be recalibrated (if needed) in order to provide the designers with the most accurate tool to estimate pressure drop and liquid holdup in trickle flow reactors. Acknowledgment We are indebted to many contributors who helped us building up this data base, among them we are thankful to Profs Al-Dahhan (Washington University), J. Levec (University of Ljubljana), A. Lakota (University of Ljubljana), G. Wild (INP Lorraine, Nancy database). Dr. Bensetiti contribution in collecting the database is
Figure 5. Whole database parity plot. Predicted vs experimental data, for the extended Holub model and (E1, E2), fs, and fv from neural networks: (a) external liquid holdup; (b) two-phase pressure drop. Table 11. Statistical Tests of Some TBR Pressure Drop and Liquid Holdup Correlations Ellman et al. correlations3,4 statistical parameters AARE (%) σ (%)
L,t ) L,da + L,stb
∆P/H
Larachi et al. correlations5 L,t ∆P/H
25.0 29.6
70.0 26.8
19.5 22.0
72.0 33.0
a The Ellman et al.5 correlation was used to calculate dynamic liquid holdup. b The Saez and Carbonell14 correlation was used to calculate static liquid holdup.
acknowledged. Financial support from the Natural Sciences and Engineering Research Council of Canada (NSERC), the Fonds pour la Formation de Chercheurs et d’Aide a` la Recherche (Que´bec), and the Program de Coope´ration Scientifique et Technologique FranceQue´bec is gratefully acknowledged. Glossary AARE:
average absolute relative error, AARE ) N 1/N∑i)1 |(ycalc,i - yexp,i)/yexp,i|
dc:
column diameter (m)
dp:
grain volume mean diameter (m)
Ind. Eng. Chem. Res., Vol. 37, No. 12, 1998 4549 E1, E2:
Ergun constants for single-phase flow
st:
static total
fs:
shear slip factor
t:
fv:
velocity slip factor
Abbreviations
FrL:
liquid Froude number, FrL ) vSL2/gdp
g:
gravity acceleration (m/s2)
GaR:
Galileo number, GaR ) dp3FR2g3/|µR2(1 - )3|
H:
hidden-layer vector
J:
number of nodes in the hidden layer
N:
number of data
P:
pressure (MPa)
∆P/H:
two-phase frictional pressure drop (Pa/m)
Rei:
interfacial Reynolds number
ReR:
Reynolds number, ReR ) vSRFRdp/[µR(1 - )]
Literature Cited
Sk:
node k network output
StL:
liquid Stokes number, StL ) µLvSL/FLgdp2
T:
temperature (°C)
(1) Holub, R. A. Hydrodynamics of trickle bed reactors. Ph.D. Thesis, Washington University, St. Louis, MO, 1990. (2) Holub, R. A.; Dudukovic, M. P.; Ramachandran, P. A. A phenomenological model of pressure drop, liquid holdup and flow regime transition in gas-liquid trickle flow. Chem. Eng. Sci. 1992, 47, 2343. (3) Holub, R. A.; Dudukovic, M. P.; Ramachandran, P. A. Pressure drop, liquid holdup and flow regime transition in trickle flow. AIChE J. 1993, 39, 302. (4) Ellman, M. J.; Midoux, N.; Laurent, A.; Charpentier, J. C. A new improved pressure drop correlation for trickle-bed reactors. Chem. Eng. Sci. 1988, 43, 2201. (5) Ellman, M. J.; Midoux, N.; Wild, G.; Laurent, A.; Charpentier, J. C. A new improved liquid holdup correlation for tricklebed reactors. Chem. Eng. Sci. 1990, 45, 1677. (6) Larachi, F.; Laurent, A.; Midoux, N.; Wild, G. Experimental study of a trickle-bed reactor operating at high pressure: Twophase pressure drop and liquid saturation. Chem. Eng. Sci. 1991, 46, 1233. (7) Larachi, F.; Laurent, A.; Wild, G.; Midoux, N. Liquid saturation data in trickle beds operated under elevated pressure. AIChE J. 1991, 37, 1109. (8) Wammes, W. J. A.; Westerterp, K. R. Hydrodynamics in a pressurized cocurrent gas-liquid trickle-bed reactor. Chem. Eng. Technol. 1991, 14, 406. (9) Wammes, W. J. A.; Mechielsen, S. J.; Westerterp, K. R. The influence of pressure on the liquid hold-up in a cocurrent gasliquid trickle-bed reactor operating at low gas velocities. Chem. Eng. Sci. 1991, 46, 409. (10) Al-Dahhan, M. H.; Dudukovic, M. P. Pressure drop and liquid holdup in high-pressure trickle-bed reactors. Chem. Eng. Sci. 1994, 49, 5681. (11) Al-Dahhan, M. H.; Khadilkar, M. R.; Wu, Y.; Dudukovic, M. P. Prediction of pressure drop and liquid holdup in highpressure trickle-bed reactors. Ind. Eng. Chem. Res. 1998, 37, 793. (12) Al-Dahhan, M. H. Effects of high pressure and fines on the hydrodynamics of trickle-bed reactors. D.Sc. Thesis, Washington University, St. Louis, MO, 1993. (13) Bensetiti, Z.; Larachi, F.; Grandjean, B. P. A.; Wild, G. Liquid saturation in cocurrent upflow fixed-bed reactors: a stateof-the-art correlation. Chem. Eng. Sci. 1997, 52, 4239. (14) Saez, A. E.; Carbonell, R. G. Hydrodynamic parameters for gas-liquid cocurrent flow in packed-beds. AIChE J. 1985, 31, 52. (15) Azzaz, M. S. Re´acteurs gaz-liquide-solide a` lit fixe: Re´actions catalytiques, hydrodynamique et transfert de matie`re. Ph.D. Thesis, Institut National Polytechnique de Lorraine, Nancy, France, 1984. (16) Charpentier, J. C. Hydrodynamique des colonnes a` garnissage avec e´coulement de gaz et de liquide a` co-courant et a` contre-courant. Aspects statique et dynamique de la texture du liquide. Ph.D. Thesis, University of Nancy, Nancy, France, 1968. (17) Charpentier, J. C.; Prost, C.; LeGoff, P. Chute de pression pour des e´coulements a` co- courant dans les colonnes a` garnissage arrose´: Comparaison avec le garnissage noye´. Chem. Eng. Sci. 1969, 24, 1777.
Ui:
normalized input variables
viL:
interfacial liquid velocity (m/s)
vSG:
superficial gas velocity (m/s)
vSL:
superficial liquid velocity (m/s)
XL:
Lockhart-Martinelli number, XL ) vSLFL1/2/vSGFG1/2
y:
hydrodynamic parameter (y ) L,t, ∆P/H)
w:
parameter
WeL:
liquid Weber number, WeL ) vSL2dpFL/σL
Greek Letters R:
subscript meaning gas (G) or liquid (L)
Rpq:
cross-correlation coefficient between column p and column q of the database, Rpq ) N p col p q col q (wcol wmean )(wcol wmean )/ ∑n)1 i i N p col p 2 N q col q 2 [x∑i)1 (wcol - wmean ) x∑i)1 (wcol - wmean )] i i
:
bed void fraction
L:
liquid holdup
φ:
sphericity factor
µR:
viscosity of R phase (kg/m‚s)
νR:
kinematic viscosity of R phase (m2/s)
ω:
weights
FR:
density of R phase (kg/m3)
σ:
standard
deviation,
σ
)
N [|(ycalc,i - yexp,i)/yexp,i| - AARE]2/(N - 1) x∑i)1
σL:
surface tension (N/m)
ΨG :
dimensionless body force on gas phase
ΨL:
dimensionless body force on liquid phase
Subscripts calc:
calculated
col:
column
d:
dynamic
exp:
experimental
i:
gas-liquid interface
G:
gas
L:
liquid
CMC:
carboxymethylcellulose
DEA:
diethanolamine
DIPA:
diisopropanolamine
DMA:
dimethylamine
CHA:
cyclohexylamine
IPA:
isopropropanolamine
MEA:
monoethanolamine
4550 Ind. Eng. Chem. Res., Vol. 37, No. 12, 1998 (18) Charpentier, J. C.; Favier, M. Some liquid holdup experimental data in trickle-bed reactors for foaming and nonfoaming hydrocarbons. AIChE J. 1975, 21, 1213. (19) Charpentier, J. C. New trends in gas-liquid reaction engineering. Int. Chem. Eng. 1988, 28, 285. (20) Dudukovic, M. P.; Devanathan, N.; Holub, R. Multiphase reactors: Models and experimental verification. Rev. Inst. Franc¸ . Pe´ tr. 1991, 46, 439. (21) Iliuta, I. Hydrodynamics and mass transfer in multiphase fixed bed reactors. Ph.D. Thesis, Universite´ Catholique de Louvain, Belgium, 1996. (22) Iliuta, I.; Thyrion, F. C.; Muntean, O. Hydrodynamic characteristics of two-phase flow through fixed beds: Air/Newtonian and non-Newtonian liquids. Chem. Eng. Sci. 1996, 51, 4987. (23) Iliuta, I.; Thyrion, F. C.; Muntean, O. Residence time distribution of the liquid in two-phase cocurrent downflow in packed beds: Air/Newtonian and non-Newtonian liquid systems. Can. J. Chem. Eng. 1996, 74, 783. (24) Lakota, A. Hydrodynamics and mass transfer characteristics of trickle-bed reactors. Ph.D. Thesis, University of Ljubljana, Ljubljana, Slovenia, 1990. (25) Lakota, A.; Levec, J. Solid-liquid mass transfer in packed beds with cocurrent downward two-phase flow. AIChE J. 1990, 36, 1444. (26) Larachi, F. Les re´acteurs triphasiques a` lit fixe a` e´coulement a` co-courant vers le bas et vers le haut de gaz et de liquide. E Ä tude de l’influence de la pression sur l’hydrodynamique et le transfert de matie`re gaz-liquide. Ph.D. Thesis, Institut National Polytechnique de Lorraine, Nancy, France, 1991. (27) Larachi, F.; Laurent, A.; Wild, G.; Midoux, N. Some experimental liquid saturation results in fixed-bed reactors operated under elevated pressure in cocurrent upflow and downflow of the gas and the liquid. Ind. Eng. Chem. Res. 1991, 30, 2404. (28) Lazzaroni, C. L.; Keselman, H. R.; Figoli, N. S. Trickle bed reactors. Multiplicity of hydrodynamic states. Relation between the pressure drop and the liquid holdup. Ind. Eng. Chem. Res. 1989, 28, 119. (29) Lazzaroni, C. L.; Keselman, H. R.; Figoli, N. S. Trickle bed reactors. Relation between the pressure drop and the liquid holdup with the wetting efficiency. Latin Am. Res. J. 1990, 20, 203. (30) Levec, J.; Sa´ez, A. E.; Carbonell, R. G. The hydrodynamics of trickling flow in packed Beds. Part II: Experimental observations. AIChE J. 1986, 32, 369. (31) Morsi, B. I. Hydrodynamique, aires interfaciales et coefficients de transfert de matie`re gaz-liquide dans les re´acteurs catalytiques a` lit fixe arrose´: Les re´sultats obtenus en milieu liquide aqueux acade´mique sont-ils encore repre´sentatifs en milieu organique industriel?. Ph.D. Thesis, Institut National Polytechnique de Lorraine, Nancy, France, 1982. (32) Morsi, B. I.; Midoux, N.; Charpentier, J. C. Flow patterns and some holdup experimental data in trickle-bed reactors for foaming, nonfoaming, and viscous organic liquids. AIChE J. 1978, 24, 357. (33) Morsi, B. I.; Midoux, N.; Laurent, A.; Charpentier, J. C. Hydrodynamics and interfacial areas in downward cocurrent gas-
liquid flow through fixed beds. Influence of the nature of the liquid. Int. Chem. Eng. 1982, 22, 142. (34) Morsi, B. I.; Laurent, A.; Midoux, N.; Barthole-Delanauy, G.; Storck, A.; Charpentier, J. C. Hydrodynamics and gas-liquidsolid interfacial parameters of co-current downward two-phase flow in trickle-bed reactors. Chem. Eng. Comm. J. 1984, 25, 267. (35) Purwasasmita, M. Contribution a` l’e´tude des re´acteurs gazliquide a` lit fixe fonctionnant a` co-courant vers le bas a` fortes vitesses du gaz et du liquide. Hydrodynamique, transfert de matie`re et de chaleur pour des liquides aqueux et organiques. Ph.D. Thesis, Institut National Polytechnique de Lorraine, Nancy, France, 1985. (36) Sa´ez, A. E.; Levec, J.; Carbonell, R. G. The hydrodynamics of trickling flow in packed Beds. Part I: Conduit models. AIChE J. 1986, 32, 353. (37) Sa´ez, A. E. Hydrodynamics and lateral thermal dispersion for gas-liquid cocurrent flow in packed beds (Holdup, permeability, meniscus). Ph.D. Thesis, University of California, Davis, California, USA, 1984. (38) Vivarie´, A. E Ä tude de l’hydrodynamique et des aires interfaciales gaz-liquide associe´es a` des e´coulements diphase´s a` cocourant vers le bas dans un re´acteur pilote a` garnissage. Comparaison avec les re´sultats de la litte´rature obtenus a` l’aide de travaux avec des colonnes de laboratoire. CNAM thesis, Conservatoire National des Arts et Me´tiers de Paris, Nancy, France, 1981. (39) Yaı¨ci, W. Mise au point de nouveaux syste`mes d′absorption gaz-liquide avec re´action chimqiue en milieux liquides aqueux et organique en vue de leur application a` la de´termination par me´thode chimique de la conductance de transfert de matie`re en phase gazeuse dans un re´acteur catalytique a` lit fixe arrose´. Ph.D. Thesis, Institut National Polytechnique de Lorraine, Nancy, France, 1985. (40) Yaı¨ci, W.; Laurent, A.; Midoux, N.; Charpentier, J. C. De´termination des coefficients de transfert de matie`re en phase gazeuse dans un re´acteur catalytique a` lit fixe arrose´ en pre´sence de phases liquides aqueuses et organiques. Bull. Soc. Chim. Fr. 1985, 6, 1032. (41) Yaı¨ci, W.; Laurent, A.; Midoux, N.; Charpentier, J. C. Determination of gas-side mass transfer coefficients in trickle-bed reactors in the presence of an aqueous or an organic liquid phase. Int. Chem. Eng. 1988, 28, 299. (42) Charpentier, J. C. Hydrodynamics of two-phase flow. J. Powder Bulk Solids Technol. 1978, 2, 68. (43) Cloutier, P.; Tibirna, C.; Grandjean, B. P. A.; Thibault, J. NNFit, logiciel de re´gression utilisant les re´seaux a` couches. http:// www.gch.ulaval.ca/∼nnfit. (44) Press: W. H.; Teukolsky, S. A.; Vetterling, W. T.; Flannery, B. P. Numerical recipes in Fortran. The art of scientific computing. 2nd ed.; Cambridge University Press: Cambridge, MA, 1992.
Received for review June 22, 1998 Revised manuscript received September 18, 1998 Accepted October 8, 1998 IE980394R
