-
MODELING THE HYDRODYNAMICS OF KATAPAK-SP USING THE PARTICLE
APPROACH
Ricardo Macías-Salinas
Instituto Politécnico Nacional, ESIQIE, Departamento de
Ingeniería Química, Zacatenco, México, D.F. 07738 Tel.: (5)729-6000
ext. 55291, Fax: (5)586-2728,
E-mail: [email protected]
Abstract
Various experimental efforts have been reported in the
literature in an attempt to understand the hydrodynamic behavior of
a relatively new family of catalytic structured packings:
KATAPAK-SP. The only approach so far used to represent the
experimental results obtained for KATAPAK-SP 11 & 12 relies in
the channel model. The purpose of this work was to verify the
suitability of a particle model originally developed for random and
structured packings (Stichlmair et al., Gas Sep. Purif., 3, 19,
1989) by extending its applicability to catalytic structured
packings, particularly for the KATAPAK-SP family. In doing that, a
more suitable liquid holdup correlation for the open channels was
proposed. The question on how liquid splits into the open channels
and catalyst bags still remains partially unanswered. In an attempt
to solve this, a new calculation procedure has been devised here to
reasonably determine the liquid share between the open channels and
the catalyst bags. Model predictions in terms of pressure drop and
liquid holdup are presented and discussed in this work for the case
of KATAPAK-SP 11 & 12.
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Model Equations for the
Reactive Zone
-
Maximum Liquid Load through the Catalyst Bag
Modified Ergun equation:
fgd
U pCB
CBmaxL ⋅−
=ε
ε1
32
,
80.1180 +=hRe
f
valid for smooth spherical particles over 48.034.0 ≤≤ CBε
(Macdonald et al., 1979)
LCB
LpL,maxh
dURe
ηερ
)1( −=
An iterative procedure is required to obtain UL,max
Comparison with experimental UL,max (system: glass/water)
Source dp [mm] εCB [-] Experiment Model
Hoffmann et al. (2004) 0.7 0.370 3.07 2.87
Behrens (2006) 1.0 0.362 5.35 5.15
UL,max in mm/s
Macdonald et al., Ind. Eng. Chem. Fundam., 18, 199, (1979).
Hoffmann et al., Chem. Eng. Proc., 43, 383, (2004). Behrens, Ph.D.
Thesis, Technische Universiteit Delft (2006).
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Dynamic Liquid Holdup through the Catalyst Bag
Hoffmann et al. (2004) gives the following expression for
calculating the total liquid holdup inside the catalyst bags:
⎥⎥⎦
⎤
⎢⎢⎣
⎡
⎟⎟⎠
⎞⎜⎜⎝
⎛−−⋅=
2
,
,, 15.01
maxL
CBLCBCBCBT U
Uh εϕ
where CBϕ is the catalyst volume fraction, CBε is the void
fraction of the catalyst bed, UL,CB is the superficial velocity of
the liquid through the catalyst bed and UL,max is the maximum
liquid load through the catalyst bag. Using Hoffmann’s expression
for hT,CB, the dynamic portion of the liquid retained in the
catalyst bags can be readily calculated. At 0, =CBLU , the static
portion of the total liquid holdup can be extracted from the
Hoffmann’s expression
CBCBCBLHoffmann
CBTHoffmann
CBS Uhh εϕ ⋅=== 5.0)0( ,,,
(50% of the maximum liquid holdup remains static!!)
The portion of liquid holdup that behaves dynamic can be readily
calculated as follows
CBCB
HoffmannCBTCBD hh εϕ ⋅−= 5.0,,
hence
⎥⎥⎦
⎤
⎢⎢⎣
⎡
⎟⎟⎠
⎞⎜⎜⎝
⎛−−⋅=
2
,
,, 115.0
maxL
CBLCBCBCBD U
Uh εϕ
Hoffmann et al., Chem. Eng. Proc., 43, 383, (2004).
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Dynamic Liquid Holdup through the Catalyst Bag
Comparison with experimental hS,CB (system: glass/water)
Source dp [mm] εCB [-] Experiment Model
Sáez et al. (1991) 0.8 0.370 0.12 0.185
Behrens (2006) 1.0 0.362 0.20 0.181
CBModel
CBSh ε⋅= 5.0,
The value of hS;CB measured by Sáez et al. is much smaller than
model estimation because they used very smooth spherical particles
during their experiments thus ignoring contact angle effects
between the liquid and the particle. Conversely, Behrens did
include these effects in his static holdup measurements.
Sáez et al., AIChE J., 37, 1733 (1991). Behrens, Ph.D. Thesis,
Technische Universiteit Delft (2006).
-
Model Equations for the
Separation Layers
-
Dry Pressure Drop
Stichlmair et al. (1989) developed the following expression
based on the particle approach for calculating the pressure loss of
a gas flow through a packed bed at dry conditions:
eq
GG
p
p
dry dUf
ZP ρ
εε 265.40
143 −
⋅=⎟⎠⎞
⎜⎝⎛
ΔΔ
where UG is the superficial velocity of the gas, Gρ is the gas
density, pε is the void fraction of the packing and eqd is the
equivalent diameter of the packing ( )/)1(6 ppeq ad ε−= . For
packings of low porosity such as KATAPAK-SP, Stichlmair´s
expression may not be suitable: the term 65.4pε may yield
over-weighted pressure drop estimations, particularly those at wet
conditions. To alleviate this, an Ergun-type expression was chosen
instead:
eq
GG
p
p
dry dUf
ZP ρ
εε 230
1−⋅=⎟
⎠⎞
⎜⎝⎛
ΔΔ
The parameter f0 in the above equation is the friction factor
for gas-solid interactions. The following relation originally
proposed by Stichlmair et al. (1989) was chosen for f0:
321
0 CRe
CReCf
GG
++= and G
GGeqG
UdRe
ηρ⋅
=
C1, C2 & C3 are packing-specific parameters that should be
determined by fitting experimental dry pressure drop data
Stichlmair et al., Gas Sep. & Pur., 3, 19 (1989).
-
Relationship between the friction factors at dry and wet
conditions
The expression for the friction factor handling the gas-solid
interactions is
321
0 CRe
CReCf
GG
++=
In general, a simplified form of the friction factor can be as
follows
βRef ∝0 or
βα Ref ⋅=0 or Ref lnlnln 0 ⋅+= βα
Differentiating the above expression with respect to Reln
β=Redfd
lnln 0 or dRe
dffRe 0
0
⋅=β
therefore
⎥⎥⎦
⎤
⎢⎢⎣
⎡+⋅−=
GG Re
CReC
f21
0 211β
Introducing a friction factor for gas-liquid interactions:
β
β
ηρ
αα ⎟⎟⎠
⎞⎜⎜⎝
⎛ ⋅⋅=⋅=
G
GGweq UdRef ,1
Dividing f1 over f0 we finally obtain a relationship between the
friction
factors at dry and wet conditions:
β
⎥⎥⎦
⎤
⎢⎢⎣
⎡=
eq
weteq
dd
ff ,
0
1
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Wet Pressure Drop
The same type of model previously proposed for ΔPdry can be also
used for calculating the pressure loss of the gas through the wet
packing
weteq
GG
wetp
wetp
wet dUf
ZP
,
2
3,
,1
1 ρε
ε−⋅=⎟
⎠⎞
⎜⎝⎛
ΔΔ
where f1 is the friction factor of the gas interacting with the
liquid. This parameter differs from that at dry conditions (f1) as
follows
β
⎥⎥⎦
⎤
⎢⎢⎣
⎡⋅=
eq
weteq
dd
ff ,01
where
3/1
,3, 1
1⎟⎟⎠
⎞⎜⎜⎝
⎛
−
−⋅=
p
wetpeqweteq dd ε
ε
⎥⎥⎦
⎤
⎢⎢⎣
⎡+⋅−=
GG Re
CReC
f21
0 211β
and
SLLpwetp h ,, −= εε
In the above equation, hL,SL is the volumetric liquid holdup
through the separation layers. Dividing ΔPwet over ΔPdry we
obtain:
3,
3/)2(
, )/1(1
)/1(1 −+
−⋅⎥⎥⎦
⎤
⎢⎢⎣
⎡
−
−−=
ΔΔ
pSLLp
pSLLp
dry
wet hh
PP
εε
εεβ
where
-
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛ΔΔ
⋅+⋅=2
0,,, 1 gZPChh
L
wetSLLSLL ρ
Introducing the following dimensionless pressure drop
expressions:
gZP
L
wetwet ρΔ
Δ=Φ
gZP
L
drydry ρΔ
Δ=Φ
therefore
[ ] 320,,3/)2(2
0,, /)1(11
)1(1 −+
Φ⋅+⋅−⋅⎥⎥⎦
⎤
⎢⎢⎣
⎡
−
Φ⋅+⋅+−⋅Φ=Φ pwetSLL
p
wetSLLpdrywet Ch
Chε
εε
β
where the above equation must be solved iteratively for wetΦ
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Dynamic Liquid Holdup below the Gas Loading Point
A simple and sufficiently accurate particle-based correlation
for dynamic liquid holdup in the preloading region (hL,SL,0) was
that proposed by Stichlmair et al. (1989):
3/1,0,, 555.0 SLLSLL Frh =
where
65.4
2
,p
pSLSLL g
aUFr
ε⋅
=
The same expression for hD,SL,0 was used in this work except
that the coefficient and the exponent of the Froude number were
treated as adjustable parameters:
2,10,,
αα SLLSLL Frh ⋅=
Additionally, to preserve model consistency, the Froude number
was calculated as follows:
3
2
,p
pSLSLL g
aUFr
ε⋅
=
ΝΟΤΕ: α1 & α2 are packing-specific parameters that should be
determined by fitting experimental dynamic holdup data at stagnant
gas conditions.
Stichlmair et al., Gas Sep. & Pur., 3, 19 (1989).
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Dynamic Liquid Holdup above the Gas Loading Point
The equation proposed by Stichlmair et al. (1989) is perhaps the
best approach for correcting the value of hD,SL,0 above the gas
loading point
)1( 20,,, Φ⋅+⋅= Chh SLDSLD
gZP
L
wet
ρΔΔ
=Φ
where C is usually constant for a family of packings. For
example, Stichlmair et al. (1989) found that for random and
conventional structured packings:
20=C
For the case of KATAPAK-SP packings, it was found that the value
of C varied with packing porosity and the number of separation
layers per catalytic bags (NSL) as follows:
33
20
pp
Stichlmair
NSLNSLCC
εε ⋅=
⋅=
Stichlmair et al., Gas Sep. & Pur., 3, 19 (1989).
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Flooding Point Calculation
On the basis of the particle approach, the present pressure drop
model is given by the following expression:
[ ]320,,3/)2(
20,,
/)1(1)1(1
1pwetSLL
wetSLLp
p
wet
dry ChCh
εε
εβ
Φ⋅+⋅−⋅⎥⎥⎦
⎤
⎢⎢⎣
⎡
Φ⋅+⋅+−
−=
Φ
Φ+
The pressure drop at the flooding point ( fΦ ) can be determined
by means of the following relation
0=Φ∂
Φ∂
wet
dry
Differentiating the hydraulic model with respect to wetΦ , one
obtains
0)1(
6)1(1
322
20,,
0,,2
0,,
0,,2 =
Φ⋅+⋅−
⋅−
Φ⋅+⋅+−
+⋅⋅
−Φ−fSLLp
SLL
fSLLp
SLL
f ChhC
Ch
hC
εε
β
The above expression contains two unknowns, namely fΦ and UG,f
(flooding velocity of the gas) from the β term. To solve for these
two variables, one needs one more independent equation, for
example:
[ ] 0/)1(1)1(1
1 320,,
3/)2(
20,,
=Φ⋅+⋅−⋅⎥⎥⎦
⎤
⎢⎢⎣
⎡
Φ⋅+⋅+−
−−
Φ
Φ+
pfSLLfSLLp
p
f
dry ChCh
εε
εβ
-
where
)( , fGdrydry UΦ=Φ and )( , fGUββ =
Therefore, the two framed equations should be solved
simultaneously in order to determine the flooding variables ( fΦ
& UG,f) from the pressure drop model.
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Calculation Procedure to Estimate
Liquid Split through the Packing
-
New Liquid Split Procedure
Assuming that the inlet liquid load (UL) is well distributed
over the top surface of the packing, an initial portion of it flows
through the separation layers (USL,0) while the other portion goes
into the catalyst bags (UCB,0). As the overall liquid flows
downwards, liquid velocities through the separation layers and the
catalyst bags change (before reaching steady state conditions) due
to flow interactions (USL,CB) between these two liquid flow
passages (see figure below). The level of interaction is greatly
dictated by the amount of liquid that gradually fills the voids of
the catalytic structure until its saturation point is reached.
Therefore, any attempt to estimate how the liquid splits into the
separation layers and catalyst bags should take into account the
maximum load point exhibited by the catalyst particles.
In general, total liquid load (LT) is divided in two main
portions
SLCBT LLL +=
CatalystBags
USL,CB
UL
SeparationLayers
UCB,0 USL,0
UCB
UL
USL
-
where CBL and OCL stand for the amount of liquid flowing through
the catalyst bags (CB) and the separation layers (SL),
respectively. In terms of superficial velocities, the above
equation becomes:
SLCBL UUU ⋅+⋅= εϕ where ϕ is the catalyst volume fraction and ε
is the void fraction of the separation layers. A good initial
distribution of the liquid load is to assume equal superficial
velocities through CB and SL:
0,0, SLCB UU =
In terms of known variables:
εϕ += LCB
UU 0,
εϕ)( 0,
0,
⋅−= CBLSL
UUU
The above equations apply only below the load point of the
catalytic structures. The assumption of equal velocities, however,
may yield low values of UCB particularly for those liquid loads
well below the saturation point of CB were capillary effects are
dominant bringing more liquid from SL to CB. Accordingly, the
following equation is proposed to adjust the value of UCB,0 to a
more realistic one (UCB):
CBSLCBCB UUU ,0, +=
where USL,CB is the liquid share between OC and CB. Its value
depends on the load point of CB (ULP), namely:
0, >CBSLU when LPL UU < 0, =CBSLU when LPL UU = 0,
A simple expression for USL,CB (that satisfies the above
boundary conditions) is proposed as follows:
-
0,, 1 SLLP
LCBSL UU
UU ⎟⎟⎠
⎞⎜⎜⎝
⎛−⋅= λ
For example, below the load point of CB, the term )/1( LPL UU−⋅λ
signifies a fraction of liquid that is taken off from USL,0 and
then added to UCB,0 to obtain a more realistic value of liquid
flowing through CB. How realistic? … it depends on λ, an adjustable
parameter that is introduced in order to preferentially fit
experimental dynamic liquid holdup data, or eventually pressure
drop data. Knowing that
maxCB
CB
LP
L
UU
UU
,
= and 0,0, CBSL UU =
therefore
0,,
0, 1 CBmaxCB
CBCBCB UU
UUU ⎟⎟⎠
⎞⎜⎜⎝
⎛−⋅+= λ
Solving for UCB:
0,
,,
1
CB
maxCBmaxCBCB
UUUU
+
+⋅=λ
λ for maxCBCB UU ,0, ≤
maxCBCB UU ,= for maxCBCB UU ,0, >
and
εϕ)( ⋅−
= CBLSLUUU
Important notes: (1) the parameter λ does not vary with liquid
load (it is a constant) below the load point of CB, (2) UCB,max is
the maximum liquid load through the CB.
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Model Correlation Results
-
Geometric Characteristics of the 4 Packings Under Study
† Brunazzi, DICCSIM, Universitá di Pisa (2005). ‡ Ratheesh &
Kannan, Chem. Eng. J., 104, 45 (2004). * Behrens, Ph.D. Thesis,
Technische Universiteit Delft (2006).
K-SP 11† K-SP 12‡ K-SP 11* K-SP 12*
Packing Diameter [mm] 100 100 450 450
Packing Surface Area [m2/m3] 203 325 300.2 341.2
Packing Void Fraction 0.409 0.549 0.55 0.70
Catalyst Volume Fraction 0.397 0.295 0.46 0.34
Catalyst-Bag Void Fraction 0.37 0.37 0.362 0.362
Particle Diameter [mm] 1 1 1 1
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Optimized Dry-Pressure-Drop Parameters & AAD Values
Packing C1 C2 C3 AAD, % K-SP 11, 100 mm ID
K-SP 12, 100 mm ID
K-SP 11, 450 mm ID
K-SP 12, 450 mm ID
66.16
96.35
665.6
33.64
2.613
-1.725
-34.98
9.820
0.129
0.353
1.170
0.426
1.21
1.23
3.92
1.95
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Variation of experimental & calculated dry pressure drops
with f -factor
f gas [Pa0.5]
0 1 2 3 4
Pres
sure
Dro
p [m
bar/m
]
0
5
10
15
20
Brunazzi (2005), KSP-11, 100-mm IDPresent ModelBehrens (2006),
KSP-11, 450-mm IDPresent ModelRatheesh & Kannan (2004), KSP-12,
100-mm IDPresent ModelBehrens (2006), KSP-12, 450-mm IDPresent
Model
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Optimized hL,SL,0 Parameters & AAD Values
Packing α1 α2 λ∗ AAD, % K-SP 11, 100 mm ID
K-SP 12, 100 mm ID
K-SP 11, 450 mm ID
K-SP 12, 450 mm ID
0.129
0.172
0.088
0.139
0.281
0.175
0.070
0.090
0.93
1.86
1.09
2.04
3.94
1.36
4.69
2.99
* Assuming that 90% of the total liquid goes into the CB at 1
m3/m2-h
Total dynamic liquid holdup leaving the packing element:
0,,, SLLCBDD hhh +=
-
Variation of experimental & calculated dynamic liquid
holdups at zero gas with liquid load
Liquid Load [m3/m2-h]
0 10 20 30
Dyn
amic
Liq
uid
Hol
dup
[-]
0.03
0.08
0.13
0.18
0.23
Brunazzi (2005), KSP-11, 100-mm IDPresent ModelBehrens (2006),
KSP-11, 450-mm IDPresent ModelRatheesh & Kannan (2004), KSP-12,
100-mm IDPresent ModelBehrens (2006), KSP-12, 450-mm IDPresent
Model
-
Calculated dynamic liquid holdup behavior exhibited by the
catalyst bags. System: glass beads/water
Liquid Load [m3/m2-h]
0 10 20 30
Dyn
amic
Liq
uid
Hol
dup
[-]
0.02
0.04
0.06
0.08
0.10
KATAPAK-SP 11, 100-mm IDKATAPAK-SP 11, 450-mm IDKATAPAK-SP 12,
100-mm IDKATAPAK-SP 12, 450-mm ID
-
Model estimations of the fraction of liquid that goes into the
catalyst bags with liquid load
Liquid Load [m3/m2-h]
0 10 20 30
L CB /
LTO
TAL [
-]
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
KATAPAK-SP 11, 100-mm IDKATAPAK-SP 11, 450-mm IDKATAPAK-SP 12,
100-mm IDKATAPAK-SP 12, 450-mm ID
-
Model Prediction Results
-
Comparison between experimental & predicted wet pressure
drops for KATAPAK-SP 11, 100-mm ID
f gas [Pa0.5]
1
Pres
sure
Dro
p [m
bar/m
]
1
10
DryPresent ModelL = 5 m3/m2-hPresent ModelL = 10 m3/m2-hPresent
ModelL = 15 m3/m2-hPresent ModelL = 20 m3/m2-hPresent ModelL = 30
m3/m2-hPresent Model
-
Comparison between experimental & predicted wet pressure
drops for KATAPAK-SP 12, 100-mm ID
f gas [Pa0.5]
1
Pres
sure
Dro
p [m
bar/m
]
1
10
DryPresent ModelL = 3.82 m3/m2-hPresent ModelL = 8.89
m3/m2-hPresent ModelL = 12.7 m3/m2-hPresent ModelL = 19.07
m3/m2-hPresent Model
-
Comparison between experimental & predicted dynamic liquid
holdups as a function of f-factor for KATAPAK-SP 12,
100-mm ID
f gas [Pa0.5]
1
Frac
tiona
l Liq
uid
Hol
dup
[-]
0.05
0.10
0.15
0.20
0.25
L = 5.09 m3/m2-hPresent ModelL = 8.89 m3/m2-hPresent ModelL =
12.7 m3/m2-hPresent ModelL = 15.26 m3/m2-hPresent Model
-
Comparison between experimental & predicted dynamic liquid
holdups as a function of f-factor for KATAPAK-SP 11,
450-mm ID
f gas [Pa0.5]
0.1 1
Dyn
amic
Liq
uid
Hol
dup
[-]
0.05
0.10
0.15
0.20
0.25
0.30
L = 2.5 m3/m2-hPresent ModelL = 7.5 m3/m2-hPresent ModelL = 15
m3/m2-hPresent ModelL = 30 m3/m2-hPresent Model
-
Comparison between experimental & predicted dynamic liquid
holdups as a function of f-factor for KATAPAK-SP 12,
450-mm ID
f gas [Pa0.5]
1
Dyn
amic
Liq
uid
Hol
dup
[-]
0.05
0.10
0.15
0.20
0.25
0.30
L = 2.5 m3/m2-hPresent ModelL = 7.5 m3/m2-hPresent ModelL = 15
m3/m2-hPresent ModelL = 30 m3/m2-hPresent Model
-
(Very) Brief Discussion of Results
• The present model correlated remarkably well the experimental
dry pressure drops for 4 KATAPAK-SP packings of two different
diameters (100 & 450 mm) despite the fact that the model
explicitly ignores gas-gas fluid interactions, particularly for
KATAPAK-SP 12.
• The adopted liquid holdup correlations for both the reaction
zone & the
separation layers performed quite well. The suitability of the
assumed liquid split between the catalyst bags & the separation
layers was also confirmed.
• The present model was able to satisfactorily predict wet
pressure drop
& dynamic liquid holdup. However, near the flooding point,
model largely under-predicts flooding conditions in most cases.