Chemical Engineering Journal 4115 (2002) 1–8

A model for pressure drop and liquid saturation in gas–liquid cocurrent

upflow through packed beds
H. Sindhu, P.S.T. Sai∗
Department of Chemical Engineering, Indian Institute of Technology, Chennai 600036, India

Accepted 26 August 2002

Abstract
A macroscopic model for pressure drop and liquid saturation in cocurrent gas–liquid upflow through packed beds was proposed. The
three model parameters: two accounting for the effect of reduction in cross sectional area available for each phase due to the presence of
the other, the third accounting for the effect of bubble formation were evaluated from the experimental data of the earlier investigation.
The validity of the model for predicting the pressure drop was tested with an independent data reported in literature.
© 2002 Published by Elsevier Science B.V.
Keywords: Gas–liquid cocurrent; Countercurrent; Two-phase flow; Pressure drop; Liquid saturation

1. Introduction flowing continuously with liquid as film over the particles

and partly as droplets in the gas phase.
Simultaneous gas and liquid flow through packed beds is Pressure drop and liquid saturation are two important
commonly encountered in chemical processes for the trans- design parameters in packed beds. Khan et al. [1,2] summa-
fer of energy or mass with or without chemical reaction rized the correlations proposed by earlier authors for pres-
between the phases. The flow of the phases can be coun- sure drop and liquid saturation in gas–liquid upflow through
tercurrent or cocurrent downwards or cocurrent upwards. packed beds. These correlations are reported either in terms
While countercurrent is a mode of choice for mass transfer of Lockhart–Martinelli parameter or relating them to the
between the phases, cocurrent is for systems with chem- bed characteristics and the Reynolds number defined sepa-
ical reactions. Cocurrent upflow provides high interfacial rately for gas and liquid phases. Bensetiti et al. [3] proposed
area and radial mixing of the phases, improved gas–liquid a state-of-the art correlation for the prediction of external
mass transfer coefficients and higher liquid saturation as liquid saturation on the basis of a large data bank consisting
compared to cocurrent downflow of the phases. of more than 2600 experimental results published in the
Several flow patterns appear in cocurrent gas–liquid up- literature. Larachi et al. [4] derived a state-of-the art cor-
flow through packed beds depending upon the characteristics relation for the prediction of frictional gas–liquid pressure
of the packing as well as the physical properties and flow drop in cocurrent upflow fixed bed reactors based on a wide
rates of the two phases. The flow patterns for Newtonian hydrodynamic data bank of flooded packed bed reactors.
non-foaming liquids are classified as: (i) bubble flow (BF) An analysis of the literature indicates that reliable correla-
corresponding to low gas rates, characterized by continuous tions based on experiments exist for predicting the pressure
liquid flow and dispersed bubble flow; (ii) pulse flow (PF) drop and liquid saturation, but information on the model-
corresponding to high gas rates and moderate liquid rates, ing of the system is scarce. The complicated geometry and
characterized by liquid-rich and gas-rich portions passing the fundamental difficulties in the theoretical description of
through the column alternatively; and (iii) spray flow (SF) fluid flow have made rigorous treatment difficult. The gen-
corresponding to high gas and low liquid rates wherein gas eral approach has been semi-empirical through mathemati-
cal models based on analogy with simple systems.
Rao et al. [5] proposed a dynamic interaction model for
Abbreviations: BF, bubble flow; PF, pulse flow; SF, spray flow pressure drop and liquid saturation in cocurrent downflow
∗ Present address: School of Chemical Engineering, Engineering Cam-
pus, Universiti Sains Malaysia, 14300 Nibong Tebal, Seberang Perai Se-
through packed beds within the framework of the momen-
latan, Pulau Pinang, Malaysia. tum balance using the experimentally observed condition of
E-mail address: pstsmy@yahoo.com (P.S.T. Sai). no radial pressure gradients. The model includes the effect

1385-8947/02/$ – see front matter © 2002 Published by Elsevier Science B.V.

PII: S 1 3 8 5 - 8 9 4 7 ( 0 2 ) 0 0 2 2 6 - 7
2 H. Sindhu, P.S.T. Sai / Chemical Engineering Journal 4115 (2002) 1–8

Nomenclature computed using these model parameters is compared with

data reported by earlier investigators [8,9].
A interfacial area (m2 )
d0 orifice diameter (m)
dp effective particle diameter (m) 2. The model
g acceleration due to gravity (m s−2 )
G gas superficial mass velocity (kg m−2 s−1 ) A unified treatment based on the mechanical energy
h height of the packing (m) balance is presented using an internal flow model, based
L liquid superficial mass velocity (kg m−2 s−1 ) on analogy with flow through pipes, to describe the fluid

P pressure drop (N m−2 ) dynamic aspects of cocurrent gas–liquid upflow through

P0 pressure drop for single phase (N m−2 ) packed beds. The basic assumption in this treatment is that
mass, momentum and energy balances apply in each phase.
S free cross sectional area (= (πD2 /4) ε) (m2 )
A simplified schematic diagram of the system is shown in
V superficial velocity of the phase (m s−1 )
Fig. 1.
V actual velocity of the phase (m s−1 )
Assuming isothermality, the integral momentum balance
for the gas phase is as follows: [upward pressure force]
Greek letters
− [downward pressure force] + [downward gravity force]
αB non-ideality factor
= [total shear force between gas and liquid] + [total shear
αG Area of contact between gas and packing
force between gas and packing]
in two-phase flow to that in single-phase flow


αL area of contact between liquid and packing
(1 − β)P dS − (1 − β)P dS + ghS(1 − β)ρG
in two-phase to that in single-phase flow at Z=0
β total liquid holdup based on void volume

Z=h
γG dimensionless factor (1 − γL ) = τLG dALG + τGW dAGW (1)
γL dimensionless factor [=(L/ρL )/(L/ρL + G/ρG )]
δLG dimensionless frictional pressure drop Similarly, for the liquid



(=
PFLG /ρL gh)
δ0 dimensionless frictional pressure drop βP dS − βP dS + ghSβρL
at Z=0 Z=h
for single phase (=
P0 /ρL gh)


ε bed porosity = τLG dALG τGW dALW (2)
at Z=0
µ viscosity (kg m−1 s−1 )
ρ density (kg m−3 ) Adding the two equations and writing
P for pressure dif-
σ surface tension (N m−1 ) ference between inlet and outlet gives the following equation
τ shear force for two-phase flow:
ϕ energy dissipation (N m s−1 ) AGW ALW
X dimensionless factor (=σ/ρL gd02 )
PLG + [βρL + (1 − β)ρG ]gh = τGW + τLW
S S
(3)
Subscripts
F frictional It is assumed that
P is constant over the cross section and
G gas phase the shear stresses are constant over the entire area of contact
GW gas packing for performing the integrations.
L liquid phase
LG liquid–gas
W packing

of bubble formation on the pressure drop and liquid satu-

ration and provides a functional form for correlating pres-
sure drop and liquid saturation but some parameters have to
be determined by fitting experimental data. They obtained
the model parameters for air–water system. Sai and Varma
[6] obtained these parameters for Newtonian non-foaming,
Newtonian foaming and non-Newtonian systems in cocur-
rent gas–liquid downflow through packed beds.
In the present study, the above model is modified for con-
ditions of upflow and the model parameters are obtained
using the data reported in literature [7]. The pressure drop Fig. 1. Formulation of the flow of the phases.
 H. Sindhu, P.S.T. Sai / Chemical Engineering Journal 4115 (2002) 1–8 3

The mechanical energy balance equation for the gas is Two-phase pressure drop for upward flow is given by
given by

PLG =
PFLG + [βρL + (1 − β)ρG ] gh (15)
 

PLG (1 − β)SV G − ghS(1 − β)ρG V G The terms of order ρG /ρL can be neglected in comparison
 with δ, the above equation reduces as follows:
= τLG ALG V iG − φG (4)

PLG
Similarly, for the liquid = δLG + β (16)
ρL gh

PLG βSV L − ghSβρL V  L = −τLG ALG V  iL − φL (5)
where δLG =
PFLG /ρL gh. Substituting for
PLG ,
PFG
The rate of energy dissipation, φ, is related to τ W in single and
PFL of Eq. (11) from Eqs. (16), (12) and (13) respec-
phase as tively, the final equation obtained is as follows:
φ ρG
τW AW = − (6) δLG + β − γL − γG
V ρL
 
Assuming that there is no slip at the interface, Eqs. (4) and αG δG
0
−1 d 0
−γG + αB (6χ)2/3 (1.5δLG )1/3
(5) can be added to give (1 − β)3 dp

PLG [βV  L + (1 − β)V  G ] δ0L

−αL γL = 0 (17)
1 β3
 
−[βρL V L + (1 − β)ρG V G ]gh = − (φL + φG ) (7)
S where
Since the pressure drop for two-phase flow equals the pres- L/ρL
γL = and γG = 1 − γL (18)
sure drop in each of the two phases, this equality for upward (L/ρL ) + (G/ρG )
flow may be expressed as follows: Substituting for
PLG and
PFL in Eq. (8), the final ex-
pression is as follows:

PLG =
PL =
PG =
PFL + ρL gh =
PFG + ρG gh
αL δ0L
(8) δLG + β − −1=0 (18)
β3
Using this equation with along with equation Eq. (7) an
Eqs. (17) and (18) are the final dimensionless expressions
expression for (φL + φG ) is
for pressure drop and liquid saturation.
1 The factor αL and αG are the ratios of the equivalent area

PFL βV  L +
PFG (1 − β)V  G = − (φL + φG ) (9)
S of contact between the phases (liquid and gas, respectively)
Expressing the frictional dissipation per unit area in terms of and packing in two-phase flow to that in a single-phase flow.
superficial mass velocities of gas and liquid and the frictional The variable, αB is a non-ideality factor accounting for the
pressure losses of the individual phases fact that in practice bubbles may not break or reform quite
so often, and is expected to assume values between 0 and 1.
L G 1 At low liquid rates corresponding to spray flow, the liquid

PFL +
PFG = − (φL + φG ) (10)
ρL ρG S saturation is unaffected by the gas flow rate and thus αL =
1. This is true provided the gas flow rate is not too high to
Substituting Eq. (10) in Eq. (7)
cause significant entrainment.
 
L G G L At intermediate and high liquid flow rates, the gas flow

PLG + − [L + G] gh =
PFG +
PFL reduces the liquid saturation. At low gas flow rates corre-
ρL ρG ρG ρL
(11) sponding to bubble flow, the formation and coalescence of
bubbles result in filling up many of the voids, thus reduce
As per the geometric interaction model [5], the frictional the area of contact between liquid and packing when com-
pressure drop
PFG for flow of gas due to the presence of pared with single-phase flow. It is assumed that the reduc-
liquid and bubble formation can be written as tion is equal to the liquid saturation itself, that is αL = β.
 1/3 The factor αB should be maximum and lies between 0 and

PFG
0 α
G σ 1.5δ−1 h 1. The variables αG and αB are determined by fitting the

PFG = + 6αB LG
(12)
(1 − β)3 d0 6χ dp experimental data with the model equations.
Alternate pulses of low density and high density character-

PFL αL ize the pulse flow. The bubble formation in the high-density
= (13)

PFL
0 β3 pulses reduces αL , though the reduction is less than in bubble
flow. The increased gas flow rate, however, causes spread-

PFG αG ing of the liquid film over a large area of packing. The effect
= (14)

PFG
0 (1 − β)3 of bubbles will be more significant in reducing αL . By trial
4 H. Sindhu, P.S.T. Sai / Chemical Engineering Journal 4115 (2002) 1–8

and error, a value of 0.7 was assigned to give predictions Table 3

with experiment. The values of αG and αB should be less Magnitude of different contributions to the total frictional pressure drop
as predicted by the present model for typical mass flow rate of the phases,
than that for bubble flow, with the former being greater than
dp = 6.2, ε = 0.5 and D = 91.0
1. In summary, of the nine parameters, values are assigned
a priori to three parameters as follows: L G Dimensionless pressure drop Flow

Geometric interaction Contribution

αL = 1 (spray flow);
from bubble
αL = 0.7 (pulse flow); Liquid phase Gas phase formation
αL = β (bubble flow). 43.564 0.841 0.3889 3.5404 1.5359 BF
41.727 0.841 0.3291 3.4553 1.5338 BF
39.122 0.577 0.3129 2.9401 0.9754 BF
3. Results and discussion 34.851 0.521 0.2583 2.8522 0.9583 BF
24.430 0.124 0.1164 1.1360 0.8951 BF
17.596 1.029 0.0405 2.4966 0.7364 PF
The unknown parameters αG and αB for each of the iden- 17.596 0.239 0.0475 1.0216 0.2250 PF
tified flow regimes, i.e. spray flow, pulse flow and bubble 8.542 0.577 0.0136 1.6677 0.5737 PF
flow are determined using the two-phase pressure drop data 26.138 1.029 0.1233 2.7276 0.8560 PF
[7]. Using the experimental pressure drop, liquid saturation 39.122 0.978 0.3807 2.9977 0.8793 PF
8.542 0.773 0.0215 1.3499 0.1414 SF
is calculated from Eq. (18). Then a linear least square anal-
8.542 0.611 0.0216 1.0122 0.1142 SF
ysis of Eq. (17) using experimental δLG and the calculated 10.677 0.611 0.0392 1.2095 0.1776 SF
β gives the best values of constants αG and αB for each re- 6.406 0.521 0.0167 0.8419 0.0857 SF
gion. The values of αG and αB thus obtained are listed in 10.677 0.577 0.0398 1.1179 0.1285 SF
Table 1 along with the values of αL .
Once the parameters are estimated, δLG and β are calcu-
lated from input data on packing characteristics, fluid prop- are as stated by Khan [7]. The column and packing charac-
erties and flow rates of the two phases by solving Eqs. (17) teristics and physical properties used in the experiment are
and (18) using an iterative procedure. The details of this cal- listed in Table 2.
culation as well as the method of parameter estimation are The results of the simulation are presented in Table 3,
described in Appendix A. and in Figs. 2 and 3. For the present study, the operating
The model is simulated for air–water and air–MEA sys- conditions as used by Rao et al. [5] for downflow are con-
tems in cocurrent upflow through a packed bed with differ- sidered to make meaningful comparisons. Fig. 2 shows a
ent types of packing. The details of the experimental set-up plot of pressure drop versus gas flow rate for different liq-
uid flow rates covering all regions of flow. Fig. 3 is similar
plot for liquid saturation. In simulation the bed and column
Table 1 characteristics are dp = 6 mm, ε = 0.5 and D = 91 mm.
Model parameters obtained in the present study Each of these plots correspond to the whole range of opera-
Flow pattern Model parameters tion of cocurrent upflow in terms of the variables liquid and
gas flow rates. It is assumed in the theoretical development,
αL αG αB
that the region of operation is known a priori. The theoreti-
Spray flow 1.0 1.1961 0.0976 cal model for each region is then used to develop the curves
Pulse flow 0.70 1.2446 0.5757 shown in Figs. 2 and 3. Different lines, indicating the tran-
Bubble flow β 2.4534 0.8493
sition between regions, identify the flow regions.

Table 2
Input parameters to the model
Packing Nominal size (mm) Equivalent particle diameter (m) Porosity

Packing characteristics
Ceramic spheres 6.0 0.00620 0.39
Ceramic raschig rings 6.0 0.00600 0.48
Ceramic berl saddles 6.0 0.00857 0.62
Fluid Density (kg m−3 ) Viscosity (kg m−1 s−1 ) Surface tension (N m−1 )

Physical properties of the fluids

Water 1000 0.001 0.072
MEA 1020 0.015 0.049
Air 1.165 0.000018 –

Column characteristics: diameter (m), 9.1 × 10−2 ; packing height (m), 1.0; liquid flow rate (kg m−2 s−1 ), 1.067–46.00; gas flow rate (kg m−2 s−1 ),
0.075–1.47.
 H. Sindhu, P.S.T. Sai / Chemical Engineering Journal 4115 (2002) 1–8 5

Fig. 2. Variation in pressure drop with mass flow rate of the phases as predicted by the present model.

At low liquid rates (e.g. L = 0.1), the flow pattern cor- spray flow regions, the pressure drop is increasing rapidly
responds to bubble flow (0 < G < 0.2), pulse flow (0.2 < with an increase in gas flow rate while the liquid saturation
G < 0.6) and spray flow (G > 0.6). It is seen from the decreases. The slope of the pressure drop curve is less in
Fig. 2 that the calculated pressure drop is nearly independent spray flow region indicating greater effect of the gas flow
of the gas flow rate for values G < 0.2. There is a step rise rate on the pressure drop on the latter region.
in pressure drop of the gas flow rate for values of G > 0.2. At high liquid rates (L = 100) flow pattern indicates
Fig. 3 shows the total liquid saturation at low liquid rates the two major identified regions of flow, i.e. bubble flow
is similarly independent of the gas flow rate for G < 0.5. (0 < G < 1.6) and pulse flow (G > 1.6). Figs. 2 and 3
Thereafter, it decreases as gas flow rate increases. show that in bubble flow region, both the pressure drop
At intermediate flow rates of the liquid (L = 5), the flow and liquid saturation are independent of gas flow rate for
pattern corresponds to bubble flow (0 < G < 0.2), pulse G < 0.003. Thereafter, as G is increased, the two phase
flow (0.2 < G < 1.5) and spray flow (G > 1.5). In bubble pressure drop decrease at first slowly (0.003 < G < 0.3)
flow region both the pressure drop and liquid holdup are and then rapidly (G > 0.3). For G > 0.3, the slope of
nearly independent of the gas flow rate. In the pulse and the pressure drop curve is greater in pulse flow region

Fig. 3. Variation in liquid saturation with mass flow rate of the phases as predicted by the present model.
6 H. Sindhu, P.S.T. Sai / Chemical Engineering Journal 4115 (2002) 1–8

Fig. 4. Comparison of the two-phase experimental pressure drop data for bubble flow with data obtained using the present model.

than in bubble flow region indicating a greater effect of the to cocurrent downflow. On comparison of αB values for up-
two-phase pressure drop on the former region. These obser- flow with that of downflow, it is observed that the contri-
vations are, in general, same as that reported for downflow bution towards total pressure drop due to bubble formation
[5] excepting that the ranges are slightly different. is more for upflow compared to that of downflow [5] indi-
Figs. 4 and 5 compare the experimental two-phase pres- cating the reduction of the gas phase pressure drop due to
sure drop with the pressure drop predicted using the model geometric interaction. It is also observed that the contribu-
for bubble flow and pulse flow respectively. The experimen- tion to the total pressure drop due to bubble formation is
tal liquid saturation is compared, typically for bubble flow, significant in both bubble flow and pulse flow regions and
with that predicted using the model in Fig. 6. in spray flow region it is less compared to the other two re-
Table 3 indicates the relative magnitude of different con- gions. On the other hand, there is no significant variation in
tributions to the total frictional pressure drop in two-phase the values of αL and αG for upflow and downflow.
flow for typical values of G and L in three different regions. The validity of the present model is tested with exper-
It is clear that the contribution to the total pressure drop due imental data obtained from two independent sources viz.
to bubble formation may be small but significant in both Srinivas and Chhabra [8]and PERC [9]. While PERC [9]
bubble flow and pulse flow regions. data covers low liquid rates and high gas rates thus yielding
From the model it is predicted that the total pressure drop spray and pulse flows, the data due to Srinivas and Chhabra
and liquid saturation are high for cocurrent upflow compared [8] cover high liquid rates and low gas rates yielding bub-
ble and pulse flows. The experimental pressure drop data is

Fig. 5. Comparison of the two-phase experimental pressure drop data for Fig. 6. Comparison of the two-phase experimental liquid saturation data
pulse flow with data obtained using the present model. for bubble flow with data obtained using the present model.
 H. Sindhu, P.S.T. Sai / Chemical Engineering Journal 4115 (2002) 1–8 7

input to the fsolve function, two values of β are guessed

between 0 and 1 such that the function values of equa-
tion is in opposite sign, corresponding to these values.
By linear interpolation between these values an approx-
imate value of β is obtained. This β value can be used
as an initial guess.
7. Since δLG and β are known, Eq. (17) is in the form Y =
αG X1 + αB X2 where Y, X1 and X2 are known quantities.
The present scheme for determining αG and αB assumes
that αL is known. This is necessary for using simple linear
least square technique to determine αG and αB . However,
αL has been varied by assigning different values to it. The
values of 1 and 0.7 for spray and pulse flow regions were
arrived by selecting the best values of αL . For bubble
flow, αL is assumed to be equal to the liquid saturation.
Fig. 7. Comparison of the literature two-phase experimental pressure drop Now, αL is specified for all regions, αG and αB are to be
data with data obtained using the present model.
determined for these regions. Y, X1 and X2 are calculated
for all experimental data. The αG and αB are obtained by
compared with that predicted using the present model satis- minimizing the sum of the squared deviations between
factorily in Fig. 7. the left and right-hand sides of the above equation.
The procedure for the direct calculation of δLG and β is as
4. Conclusions follows: once the parameters have been estimated, Eqs. (17)
and (18) can be solved simultaneously for δLG and β, given
The model, which was developed for cocurrent downflow, the flow rates of the phases, the packing characteristics and
was modified to cocurrent upflow and satisfactorily predicts the physical properties of the fluids.
the pressure drop and liquid saturation. However, the model 1. Calculate the single-phase pressure drop of liquid (δ0L )
requires the knowledge of single-phase pressure drops and
and gas (δ0G ) from Ergun’s equation.
these are obtained using Ergun’s equation [10]. The use of
2. Calculate the orifice diameter using Eq. (A.1).
this expression alters the model parameters very lightly, but
3. Identify the flow regime by following the equations pro-
does not alter the overall agreement between the experiment
posed by Khan [7].
and the model. This stresses the validity and merits of the
Transition from bubble flow to pulse flow
present model.
∗
1.4e0.15L
G∗ = 0.19 + ∗
500 + e0.15L
Appendix A
where
The algorithm for the parameter estimation in the model     0.33
∗ 1−ε ∗ 1−ε µL
is as follows. G =G and L =L
ε ε µW
1. Input the physical properties of gas and liquid (viscosity, (A.2)
density and surface tension).
2. Input the packing characteristics (porosity and effective Transition from pulse flow to spray flow
particle diameter).  
µL 0.33
3. Read the experimental frictional pressure drop data with G∗ = 0.6 + 0.083L (A.3)
µW
mass flow rate of the two phases.
4. Calculate single-phase pressure drop of the phases by 4. Eq. (18) can be solved directly to give β as a function of
using Ergun’s equation. δLG using the model for αL
5. Calculate the equivalent orifice diameter using the fol-  1/2
lowing equation: δ0L
bubble flow : β =
dt − Ndp δLG + (β − 1)
d0 = (A.1)  1/3
N 0.7δ0L
pulse flow : β = (A.4)
where D is the column diameter and N an integer calcu- δLG + (β − 1)
 1/3
lated as the integral part of (D/dp ) minus one; δ0L
6. Calculate the liquid saturation from Eq. (18) using MAT- spray flow : β =
δLG + (β − 1)
LAB fsolve function. For finding initial guess to give
8 H. Sindhu, P.S.T. Sai / Chemical Engineering Journal 4115 (2002) 1–8

5. Eq. (17) can be written as an equation containing only [3] Z. Bensetiti, F. Larachi, B.P.A. Grandjean, G. Wild, Liquid saturation
δLG as an unknown quantity by substituting the above in cocurrent upflow fixed-bed reactors: a state-of-the-art correlation,
result for β and using (␦LG ␦LG +␤) for δLG . The resulting Chem. Eng. Sci. 52 (1997) 4239.
[4] F. Larachi, A. Laurent, G. Wild, N. Midoux, Some experimental
equation is solved by fsolve function in MATLAB. liquid saturation results in fixed bed reactors operated under elevated
For finding initial guess, two values of δLG are guessed pressure in cocurrent upflow and downflow of the gas and the liquid,
Ind. Eng. Chem. Res. 30 (1991) 2404.
such that the function values of the equation are in opposite [5] V.G. Rao, M.S. Ananth, Y.B.G. Varma, Hydrodynamics of two-phase
sign corresponding to these values. A linear interpolation concurrent downflow through packed beds, AIChE J. 29 (1983)
for f = 0 between these two values is used as initial guess 467.
input to fsolve function. From the calculated value of δLG [6] P.S.T. Sai, Y.B.G. Varma, Pressure drop in gas–liquid downflow
using Eq. (A.4), β value is calculated. Thus, δLG and β are through packed beds, AIChE J. 33 (1987) 2027.
[7] A. Khan, Flow pattern of the phases, phase holdup and pressure drop
estimated simultaneously. in concurrent gas–liquid upflow through packed beds, Ph.D Thesis,
Indian Institute of Technology Madras, India, 1998.
[8] K.V. Srinivas, R.P. Chhabra, Pressure drop in two-phase cocurr-
References ent upward flow in packed beds, Can. J. Chem. Eng. 72 (1994)
1085.
[1] A. Khan, A.A. Khan, Y.B.G. Varma, Flow regime identification and [9] Pittsburgh Energy Research Centre (PERC) quarterly report
pressure drop in cocurrent gas–liquid upflow through packed beds, 1975–1976 (as cited in Y.T. Shah, Dynamics of the cocurrent
Bioprocess Eng. 16 (1997) 355. upflow fixed-bed column, in: Gas–Liquid–Solid Reactor Design,
[2] A. Khan, A.A. Khan, Y.B.G. Varma, An analysis of the phase McGraw-Hill, New York, USA, 1979, p. 230.)
holdup in concurrent gas liquid upflow through packed beds using (I) [10] Ergun, Fluid Flow Through Packed Columns, Chem. Eng., Prog. 48
drift-flux model, and (II) slip ratio, Bioprocess Eng. 22 (2000) 165. (1952) 89.