Hydrodynamics of Trickle Bed Reactors with Catalyst Support Particle

Jan 21, 2014 - Characterization of the hydrodynamics enables operation of trickle bed reactors within the desired flow regime and under conditions for...
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Hydrodynamics of Trickle Bed Reactors with Catalyst Support Particle Size Distributions Gregory S. Honda,† Philip Gase,† Daniel A. Hickman,‡ and Arvind Varma*,† †

School of Chemical Engineering, Purdue University, 480 Stadium Mall Drive West Lafayette, Indiana 47907, United States Engineering and Process Science, The Dow Chemical Company, 1776 Building, Midland, Michigan 48674, United States

‡

S Supporting Information *

ABSTRACT: Characterization of the hydrodynamics enables operation of trickle bed reactors within the desired flow regime and under conditions for uniform distribution of gas and liquid, resulting in an essentially plug flow contacting pattern. Studies reported in the literature are typically restricted to systems of beds packed with catalyst supports of a uniform size. This work addresses the impact of supports with particle size distributions on reactor hydrodynamics. An experimental database of pressure drop and liquid holdup was developed and, by careful definition of the particle diameter, literature models were adapted to account for the particle size distribution. The resulting models give improved predictions for packing media with a particle size distribution while maintaining applicability to uniform systems. relatively low4 but sufficient for the wetting efficiency to approach unity.5 Pressure drop and liquid holdup remain relevant hydrodynamic parameters due to their use in design, correlations for heat and mass transfer coefficients, and determination of liquid residence time.6 Phenomenological hydrodynamic models are derived from mass and momentum balances and provide predictions for the pressure drop and liquid holdup of the packed bed.7−11 Publications in the literature related to the effect of operating conditions on hydrodynamics and subsequent influence on reactor performance have been restricted to ideal systems with uniform particle size, despite the use of granular activated carbons (typically with varying particle sizes) in industry for chemistries ranging from hydrogenations to selective oxidations.12,13 While publications in the literature on the effect of particle size distribution on trickle-bed hydrodynamics are limited, several studies for single phase flow are reported which contain pressure drop data for particles having a size and shape distribution.14,15 However, the models developed to predict pressure drop for these conditions are fitted and maintain the definition of effective particle size by Sauter diameter (DP,S = 6Vp/Ap). As a result, the weighting of particle size distribution is unchanged from the Ergun equation. Alternatively, for single-phase liquid flow at low Reynolds number, MacDonald et al. (1991) developed an improved method for weighting the particle size distribution.16 Their study focused on the influence of particle size distribution on pressure drop and permeability and developed a theoretical model following the Blake−Kozeny equation. For a binary mixture of meshes of particle sizes, the highest pressure drop

1. INTRODUCTION For trickle-bed reactors, characterization of the hydrodynamics facilitates the engineering of the uniform distribution of flow, efficient wetting of the catalyst, operation in the desired flowregime, and adequate heat and mass transfer. The CFD models describing these phenomena 1 and models of transient imbibition/drainage have been reported in the literature,2 while the ideal of a predictive and generally applicable model for maldistribution has yet to be developed. Precedent and foundational to these endeavors is the study and modeling of the hydrodynamics under uniform trickling flow. Accurate representation of the influence of bed and operating variables on the hydrodynamics (as described by pressure drop and liquid holdup) is necessary. Predictions for liquid holdup and pressure drop are used for design purposes, and the force interaction terms of the model are used in closure equations of CFD and transient models. To date, experimental studies on trickle-bed hydrodynamics have been restricted to beds packed with uniform catalyst supports. The previously published data omit catalyst supports having a particle size distribution, as is the case with granular activated carbon. The aim of this work is to account for the effect of particle size distribution on tricklebed reactor hydrodynamics under uniform flow conditions. Trickle-bed reactors are defined by cocurrent downflow of gaseous and liquid reactants over a packed bed of catalyst. The multiphase flow through the fixed bed results in hydrodynamics that significantly influence reactor design and performance. Gas−liquid flow through a packed bed is exemplified by the existence of several flow regimes, hysteresis effects, and complex interactions between operating and bed variables that affect values for wetting efficiency (fraction of particle surface wetted by flowing liquid), liquid holdup (fraction of bed volume occupied by the liquid), pressure drop, and other parameters that describe the trickle-bed hydrodynamics.3 Under industrially relevant conditions, trickle-bed reactors are generally operated in the trickling flow regime, where gas and liquid flow as continuous phases at superficial velocities that are © 2014 American Chemical Society

Special Issue: Massimo Morbidelli Festschrift Received: Revised: Accepted: Published: 9027

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did not correspond to the minimum porosity or a bed containing only the smallest mesh particles. Rather, the highest pressure drop was observed at a high volume ratio (3:1) of fines to coarse. Other published studies related to the influence of particle size distribution on reactor performance are an experimental investigation of the effect of incorporating inert fines (one-tenth the active catalyst particle diameter) in tricklebed reactors17 and investigations on single-phase gas pressure drop in coal gasifiers.18 In the first case, uniform fines (DP = 210 μm) were incorporated at different volume ratios to the active catalyst (DP = 2 mm). Inclusion of fines resulted in a broader tracer response in RTD tests as well as increases in wetting efficiency, pressure drop, and liquid holdup. In the coal gasifier study, beds with broader particle size distributions were shown to have significantly higher pressure drops, even with all other bed variables held constant. Additionally, maldistribution effects were significant in the case of larger particle size distribution, resulting in both hot-spots and nonreactive stagnant zones. Although the single phase system in a coal gasifier differs significantly from that of a multiphase trickle-bed reactor, the fundamental effect of particle size distribution on fluid flow is translatable. Resulting maldistribution and deviations from predicted pressure drop suggest difficulties in operation and scale-up when the effect of particle size distribution is not adequately addressed. The phenomenological models reported in the literature incorporate some aspect of the Ergun equation in their definition. In their adaptation of the Ergun equation, these models define the relative effect of pore geometry and bed variables. In light of this, the definition of the pore geometry in the Ergun equation will be revisited. For the case of laminar, single-phase liquid flow through a packed bed with a particle size distribution, MacDonald et al. have shown that appropriate definition of the hydraulic radius accounts for the effect of the particle size distribution on pressure drop.16 By applying an analogous definition of pore geometry to the case of multiphase flow, the effect of particle size distribution on the hydrodynamics of trickle-bed reactors is accounted for in this present work.

Figure 1. Experimental setup consisting of (1) mass flow controller and mass flow meter, (2) Masterflex peristaltic pump, (3) rotameter, (4) gas−liquid distributor, (5) upper portion of vessel packed with 2 mm ceramic beads for all runs, (6) lower portion of vessel packed with material of interest, (7) 0−30 psid differential pressure gauge, (8) gas− liquid separator, (9) suspended liquid reservoir, and (10) balance for measuring liquid holdup.

external liquid holdup of liquid in the packing of interest. For porous particles, the internal pore volume was assumed to be occupied by liquid and this value was also subtracted from the measured mass (M2) to calculate the liquid holdup external to the particles. Also note that the liquid reservoir is suspended separately so as to not rest on the balance. To verify that pressure drop did not impact the reading on the balance, a valve was placed at the outlet of the vessel and set to be slightly open. Air flow rate was then varied to fluctuate the pressure drop across the valve up to 20 psid. No change in reading on the balance was observed, so pressure drop was determined not to impact the balance reading. Lastly, although the setup was freestanding on the balance, it was attached by flexible hoses so as not to impact the balance reading. This was confirmed by attaching all feed hoses to the vessel and then adding aliquots of known mass of water to the vessel and observing that the balance agreed with mass added within 1%. Operational variables of gas and liquid superficial velocities were varied over a range covering the trickling flow regime (up to 10 and 0.50 kg/m2s, respectively). The liquid flow rate was controlled by the variable speed peristaltic pump and confirmed by reading a rotameter. The gas flow rate was controlled by a mass flow controller for gas fluxes up to 0.030 kg/m2s and a mass flow meter and manual control valve for higher fluxes. As the focus of this work was on investigating the effect of bed variables, the system was operated under atmospheric pressure at room temperature with air and water as the gas and liquid, respectively. In order to avoid hysteresis effects and ensure repeatability, set points were approached from pulsing flow with decreasing liquid flow rate. Packing materials included a range of mesh sizes of activated carbons and glass beads that were sieved and recombined in different ratios to vary the particle size distribution (Table 1). The activated carbon used was Norit Darco (4 × 12, 12 × 20, and 20 × 40 mesh prior to sieving). Particle size was characterized by a Horiba laser-scattering particle size analyzer (model LA-950). An example of the breadth of particle size distribution for activated carbon is shown in Figure 2. It should be noted that, owing to the nonuniform shape of activated carbon, particle sizes are observed both above and below the actual mesh size.

2. EXPERIMENTAL SECTION Pressure drop and liquid holdup were measured in a cylindrical vessel (dr = 5.08 cm), Figure 1. Gas and liquid flow downward through a distributor, packed entrance length, packed bed containing the packing of interest, and a gas−liquid separator. The exit gas (air) flows to the atmosphere while liquid (water) is collected in a suspended reservoir and recycled. The distributor was comprised of 13 evenly spaced annular inlets for gas and liquid. An additional 30 cm entrance length packed with 2 mm ceramic spheres was used to further ensure even distribution. The total length of the section containing the packing material of interest was 26.7 cm, and pressure drop was measured using a differential pressure gauge over a 15 cm length. Liquid holdup, external to the porous particles, was measured based on reading the mass of the setup on a balance. In order to account for the holdup in the distributor, the 2 mm beads section of bed, hoses, and the gas−liquid separator, the mass of the setup (M1) containing 2 mm beads packed in the upper portion and empty in the lower portion was recorded for each operational point of gas and liquid flow rate. For each data point, the corresponding mass of the setup packed only with 2 mm in the upper portion (M1) was subtracted from the measured mass of the total setup (M2) to determine the 9028

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Table 1. Materials Used in the Experimental Studya mesh activated carbon

ceramic glass beads

a

14−16 16−18 16−18|30−35 16−18|30−35 20−25 20−25|16−18 20−25|30−35 20−40 30−35 1 mm 12−14 12−14|20−30 12−14 | 20−30 12−14|14−20 12−14|14−20 14−20 20−30 14−20|20−30

ratio

1:4 4:1 1:1 1:1

1:1 2:1 1:1 2:1

1:1

DP,S

DP,M

void fraction

1.50 1.29 0.73 1.11 0.87 1.08 0.76 0.76 0.66 1.00 1.45 0.95 1.07 1.09 1.19 0.87 0.70 0.78

1.34 1.14 0.61 0.88 0.76 0.92 0.65 0.67 0.58 1.00 1.41 0.80 0.89 1.01 1.10 0.86 0.66 0.74

0.42 0.45 0.40 0.40 0.38 0.39 0.40 0.44 0.44 0.35 0.36 0.33 0.33 0.35 0.35 0.36 0.37 0.36

or multiphase data. The models used are noted in Table 2, along with relevant information regarding the databases on Table 2. Models Used in This Study, Sources, and Respective Database Informationa model two-fluid slit

relative permeability algebraic

source

database

Attou et al. (1999)7 Boyer et al. (2007)22 Holub et al. (1992)8 Al-Dahhan et al. (1998)19 Iliuta et al. (1998)20 Iliuta et al. (2002)21 Saez and Carbonell (1985)9 Nemec and Levec (2005)23 Alopaeus (2006)24 Lappalainen (2008)25

external internal external external external external external (air−water) internal (air−water) external external

a

External and internal refer to the source of the database used in the cited work.

which they were developed. For the slit models, fixed literature Ergun constants are used. The simplified slit model of Holub et al.8 and the extended slit model with corrections for high pressure by Al-Dahhan et al.19 were both considered. As noted by Larachi et al., using fixed Ergun constants provides similar levels of prediction as fitting the constants to single phase data.11 The remaining models included additional variations on the slit model by Iliuta,20,21 as well as the two-fluid models of Attou et al.7 and Boyer et al.,22 the relative permeability models by Saez and Carbonell9 and Nemec and Levec,23 and the algebraic models by Alopaeus et al.24 and Lappalainen et al.25 Evaluating multiple model variations allows for comparison of the impact of the model assumptions and databases on which they were developed. Phenomenological hydrodynamic models follow mass and momentum balances through the bed. Models differ in their definition of force interaction or drag force terms. In general, force interaction terms are defined following an Ergun-like equation, with additional balances made to weight relative contributions of gas and liquid. While a number of different arguments have been used to arrive at the Ergun equation as representative of force interactions, the ultimate result is the same. Therefore, the commonly used friction-factor argument was followed in this work.26 In this definition of the Ergun equation, the packed bed is represented as a bundle of tubes having a particular hydraulic radius. For multiphase flow, this idea is extended to each phase-interaction (which vary between different models), so that a general hydraulic radius can be defined, Rh,v. This hydraulic radius is defined as the cross sectional area available for flow of a given phase divided by the wetted perimeter. More often, the hydraulic radius is taken as the volume available for flow divided by the wetted surface area. In the friction-factor based development of the Ergun equation for multiphase flow, definition of the hydraulic radius by volume results in an effective diameter defined by the Sauter diameter, DP,S, as follows:

DP,S and DP,M are defined in eqs 2 and 4, respectively.

Figure 2. Particle size distribution for 20−40 mesh Norit Darco activated carbon.

Bed void fraction was determined based on the measured mass of particles added to the 2 in. diameter vessel. Aliquots of particles were added in increments (50 mL) with tapping to avoid segregation of particle size. The mass added was measured on a separate balance and then confirmed by the balance on which the vessel rests. The volume of particles added to the vessel was determined by multiplying the mass added by the envelope volume (skeletal volume plus internal pore volume per particle mass). Void fraction was then calculated by dividing the difference between vessel volume and particle volume by the vessel volume. Envelope volume was determined by measurements with a Micromeritics GeoPyc 1360 and mercury porosimeter for porous particles. For the nonporous particles, envelope volume was measured by water displacement. Internal particle pore volume was determined by the difference between envelope and skeletal volume, with the latter measured by helium pycnometry using a Micromeritics AccuPyc II 1340 as well as being confirmed by mercury porosimetry.

3. THEORY Phenomenological models reported in the literature were adapted to better account for particle size distribution. The selected models were those requiring no fitting to single phase 9029

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Vf /Vu σw /Vu V V = ϵv u w Vw σw

which holds for either definition of diameter and can be applied in any of the models considered.

R h, v =

=

=

Fv = E1 0 3 x q 0 (x ) ∞ 0 2 x q 0 (x ) ∞

(1 + δv ,S) ∫ ϵv (1 − ϵv ) 6 ∫

DP, i 2 1 (1 + δv)

dx dx

(1 + δv ,S) ϵv DP,S (1 − ϵv) 6

DP,S

(1) 0

∫∞ x−1q3(x) dx)−1 (2)

where the subscript v designates properties specific to a given drag force, and areal porosity is assumed equivalent to bed porosity. Given multiple drag force interactions specific to each model, terms are defined generally such that ϵv is the area available for flow of a particular phase per unit area, (1 − ϵv) is the area occupied by particles and other phases per unit area, and δv,S is a correction for a fractional increase in observed particle size when the particle is covered with a liquid film and is also specific to the volume definition of the hydraulic radius (denoted by subscript S). The particle size distribution is represented by the number based distribution qr(x) (r = 0), where x is the particle diameter and qr(x) is the quantity (based on the r distribution) of particles having the diameter x. For convenience, the number based distribution is converted to a volume based distribution (q3(x)) as this is the normal representation of particle size distribution reported by most particle size analyzers. While DP,S is commonly used in the literature, definition by cross section has been shown to be more accurate for laminar, single phase flow.16 Their method considers a slice through the bed that contains a distribution of circles including all possible slices through the particles of the original size distribution. In the derivation by MacDonald, the hydraulic radius is then used to directly derive the laminar term of the Ergun equation. Ultimately, derived constants were dropped in favor of a global Ergun constant (E1 = 180, after Macdonald et al.14) for the laminar term. Consequently, the more direct approach of the friction-factor argument may be followed to define the hydraulic radius, effective particle diameter, and subsequent force interactions containing both laminar and turbulent terms.

δGL ,S =

0

DP,M =

0

∫∞ xq0(x) dx

(3)

0

=

∫∞ x−1q3(x) dx 0

∫∞ x−2q3(x) dx

1 − αϵ −1 1−ϵ

δGL ,M =

1 − αϵ −1 1−ϵ

4. RESULTS AND DISCUSSION Examples of experimental results are reported in Figures 3 and 4. Note that gas mass flux varies slightly, with the number of significant figures appropriately accounted for in figure legends and captions. Figure 3 shows the effect of gas and liquid mass fluxes on (a) dimensionless pressure drop and (b) liquid holdup. Pressure drop increases with gas and liquid flux, while liquid holdup increases with liquid flux and decreases with increasing gas flux. Figure 4 shows the effect of different mesh sizes of activated carbon on (a) dimensionless pressure drop and (b) liquid holdup at a fixed gas mass flux. Comparing results with Table 1, the highest pressure drop occurs for the mixture of 16−18 and 30−35 mesh in a ratio of 1:4, which interestingly gave a lower void fraction than 30−35 mesh alone. Pressure drop decreases with material having progressively higher void fraction and larger particle size, with the impact of void fraction being greater than that of particle size. For liquid holdup, materials with higher void fraction have a higher liquid holdup; whereas, particle size plays a more limited role. All additional data are reported in Supporting Information Table 1 for reference. The predicted values for pressure drop and liquid holdup from the various models, with both definitions of particle diameter (Sauter and MacDonald), were compared to the experimental data. Representative parity plots of predicted and

0

∫∞ x 2q0(x) dx

3

For the remaining models and drag forces, there is assumed to be no film thickness (δv = 0), so the terms outside of the particle size remain unchanged. Models were solved following recommendations reported in the literature. Where appropriate, correlations specific to air−water systems and spherical particles were used.

A f /A u Sw /A u A A = ϵv u w A w Sw

∞

(5)

(6)

R h, v =

2 ϵv (1 + δv) ∫∞ x q0(x) dx = 0 (1 − ϵv) 4 ∫ xq0(x) dx

ϵv 3

2

ρ vα ,0 (1 − ϵv) 1 + E2 α 2 DP, i ϵv 3 (1 + δv)

Where DP,i is a general effective particle diameter which is substituted for either DP,S or DP,M. All parameters take on those values or representations recommended for the specific interaction forces of each particular model. This includes relative permeabilities or other correlated parameters which may act as coefficients. The only deviations in structure of the force interaction terms, excluding the different definitions of DP,i, occur for the two-fluid models of Attou et al.7 and Boyer et al.22 In the twofluid model, force interaction terms are defined for the gas and liquid which include gas−solid, liquid−solid, and gas−liquid terms. Within the gas−solid and gas−liquid terms, a uniform liquid film thickness results in an increase in the observed particle size. Accounting for this film thickness must follow the same assumptions as defined for the particle size. As a result, the relative film thickness parameters associated with DP,S and DP,M and specific to the gas−liquid and gas−solid interaction forces are defined as follows:

0

⎛ Vp ⎞ ∫∞ x 3q0(x) dx ⎜ ⎟ = 6⎜ ⎟ = 0 =( ⎝ A p ⎠ ∫ x 2q0(x) dx ∞

μα vα ,0 (1 − ϵv)2

(4)

Where DP,M is referred to as the MacDonald or corrected particle diameter. A general Ergun equation can then be defined 9030

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Figure 3. Example results for 20−40 mesh activated carbon for (a) dimensionless pressure drop (ψL) and (b) liquid holdup (ϵL).

Figure 5. Example parity plot for predictions and experimental results of a) dimensionless pressure drop (ψL) and b) liquid holdup (ϵL) for granular activated carbon using the model of Nemec et al.23

Sauter and MacDonald diameters. MRE values for dimensionless pressure drop and liquid holdup are reported in Tables 3 Table 3. Mean relative error of predictions for dimensionless pressure drop (ψL) using both definitions of particle diameter, Sauter (S) and MacDonald (M), and using measured, −5%, and +5% the measured void fraction Void fraction, ε Particle diameter, DP,i Attou et al. (1999) Boyer et al. (2007) Holub et al. (1992) Al-Dahhan et al. (1998) Iliuta et al. (1998) Iliuta et al. (2002) Saez and Carbonell (1985) Nemec and Levec (2005) Alopaeus (2006) Lappalainen (2008)

Measured ε

−5% ε

+5% ε

S

M

S

M

S

M

0.33 0.53 0.58 0.57 0.73 0.78 0.34

0.26 0.41 0.50 0.49 0.67 0.75 0.25

0.27 0.45 0.50 0.48 0.76 0.81 0.27

0.20 0.31 0.41 0.39 0.71 0.79 0.22

0.42 0.60 0.64 0.63 0.71 0.76 0.44

0.34 0.50 0.58 0.57 0.65 0.72 0.34

0.31 0.34 0.41

0.25 0.25 0.67

0.28 0.27 0.67

0.26 0.19 1.03

0.38 0.43 0.28

0.30 0.33 0.40

Figure 4. Example results for beds of activated carbon of different mesh sizes for (a) dimensionless pressure drop (ψL) and (b) liquid holdup (ϵL).

and 4 respectively, which show that predictions for dimensionless pressure drop is improved when using the MacDonald diameter for all models, even with ±5% the measured void fraction, with the exception of the Lappalainen model. Bias (B), standard deviation (S), and fractional improvement in prediction (IMP), as well as MRE for each material type (activated carbon, 1 mm ceramic beads, and glass beads) were also determined and are reported in Supplemental Table 2. The effectiveness of changing the particle size definition from the Sauter to the MacDonald diameter for each model can be represented by the fractional improvement in prediction (IMP).

measured values for liquid holdup, ϵL, and dimensionless pressure drop, ψL = (− (ΔP/ρLgL) + 1), are presented in Figure 5. In general, similar trends were observed for each material. For trickle beds, the hydrodynamics are particularly sensitive to the void fraction of the bed. As such, model predictions may vary significantly with small changes in measured void fraction. To verify that the observed trends are independent of possible error in measured void fraction, model predictions were also determined for void fractions of ±5% of the measured value. To compare model predictions, mean relative error (MRE) was determined using both the 9031

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having little impact on liquid holdup predictions, which are satisfactory by either method (MRE < 20%). To further verify that the particle size distribution is satisfactorily represented by DP,M, the relative improvement in prediction (IMP) for models with IMP > 20% is plotted against the relative particle size correction. The latter can be taken as an indicator of particle size distribution, since the difference between DP,S and DP,M widens as the breadth of the particle size distribution increases. DP , S − DP , M relative particle size correction = DP , S (8)

Table 4. Mean relative error of predictions for liquid holdup (ϵL) using both definitions of particle diameter, Sauter (S) and MacDonald (M), and using measured, −5%, and +5% the measured void fraction Void fraction, ε

Measured ε

Particle diameter, DP,i Attou et al. (1999) Boyer et al. (2007) Holub et al. (1992) Al-Dahhan et al. (1998) Iliuta et al. (1998) Iliuta et al. (2002) Saez and Carbonell (1985) Nemec and Levec (2005) Alopaeus (2006) Lappalainen (2008)

IMP =

−5% ε

+5% ε

S

M

S

M

S

M

0.20 0.17 0.29 0.30 0.35 0.15 0.14

0.22 0.16 0.30 0.32 0.37 0.16 0.14

0.16 0.14 0.24 0.25 0.28 0.14 0.11

0.17 0.13 0.25 0.26 0.30 0.14 0.11

0.25 0.21 0.33 0.35 0.41 0.19 0.17

0.27 0.19 0.35 0.36 0.44 0.20 0.18

0.17 0.18 0.25

0.18 0.18 0.25

0.14 0.14 0.20

0.14 0.14 0.20

0.21 0.22 0.30

0.22 0.23 0.31

Figure 6 shows a clear trend of increasing improvement in prediction of pressure drop with increase in relative particle

MRE DPS − MRE DPM MRE DPS

(7)

Of the models considered, the Nemec model with the corrected particle size (DP,M) gives the best prediction based on the MRE for the dimensionless pressure drop. However, since the Nemec model is based solely on data for air and water, care should be taken when applying it to other systems. In these circumstances, we suggest that the Saez, Nemec, and Alopaeus models be used and all three predictions compared. For all materials, the Lappalainen model significantly overpredicts pressure drop. Decreasing the represented particle size, by use of DP,M, exacerbates this result. Analysis of the model revealed that the E1 constant produced by the correlation used in the Lappalainen model was more than twice that determined in the Alopaeus model. The Lappalainen model was developed to account for wetting efficiency. In doing so, the liquid tortuosity was redefined, leading to subsequent differences in the Ergun constant and IMP, even with wetting efficiencies at or near unity. In general, the slit models gave poor predictions relative to experimental results. For the Holub model and Al-Dahhan models, the predictions may have been improved by fitting constants to single phase data; however, this would have removed the impact of alternative definitions of particle size. Within the two Iliuta models, neural network correlations for the Ergun constants gave results which were considerably lower than the normal global constants (E1 = 180 and E2 = 1.8, after Macdonald et al.14), resulting in significant under-prediction. The results for the ceramic beads (CB), where the particle sizes are uniform, show no change in MRE between DP,S and DP,M for models that do not consider a change in film thickness. For the two-fluid models of Attou and Boyer, MRE changes because of the alteration in definition of film thickness relative to cross sectional area when using DP,M. This compromises the general applicability of the two-fluid models to other systems when corrected for DP,M. Similarly, this change in the film thickness definition appears to impact the prediction of liquid holdup for the two-fluid models. Excluding the Lappalainen and slit models, using DP,M improves predictions for pressure drop by 20−30% and improves bias. Following this definition brings predictions into a suitable range of error for pressure drop (MRE < 30%), matching predictions for the uniform ceramic beads while

Figure 6. IMP vs relative particle size correction for predictions and experimental results of a) dimensionless pressure drop (ψL) and b) liquid holdup (ϵL).

diameter, therefore demonstrating the success of the improved particle size definition. Additionally, little change in IMP for liquid holdup is observed, for which the MRE is satisfactory (