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: rms@ipn.mx

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.

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).

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).

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

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

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).

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).

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.

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.

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

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

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

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.