New Model for Turbulent Mass Transfer and Its Application to the

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Ind. Eng. Chem. Res. 2006, 45, 3220-3229

New Model for Turbulent Mass Transfer and Its Application to the Simulations of a Pilot-Scale Randomly Packed Column for CO2-NaOH Chemical Absorption G. B. Liu, K. T. Yu, X. G. Yuan,* and C. J. Liu State Key Laboratory for Chemical Engineering, Chemical Engineering Research Center, and School of Chemical Engineering and Technology, Tianjin UniVersity, Tianjin, 300072, China

A complex model for simulating the mass transfer process in gas-liquid system is introduced by combining the computational fluid dynamics (CFD) model, the turbulent mass transfer model with the c2-c equations for its closure, and the heat balance equation. By the proposed complex model, the axial and radial concentration distributions along the packed column can be obtained without assuming the “turbulent Schmidt number” or experimentally measuring the turbulent mass transfer diffusivity. The validation of the proposed model is tested by its application to a pilot-scale randomly packed 7 m high chemical absorption column with a 0.1 m internal diameter packed with 1/2 in. ceramic Berl saddles for CO2 removal by NaOH aqueous solutions. The simulated results are compared with the experimental data by Tontiwachwuthikul et al. (Chem. Eng. Sci. 1992, 47 (2), 381-390). Satisfactory agreement between the simulation and experiment in terms of the temperature and concentration distributions along the column height is found. Furthermore, the present model is able to predict the axial and radial turbulent mass transfer diffusivities, which are found to vary with the height of the packed column. Introduction In the chemical industry, packed columns have been widely used in separation and purification processes involving gas and liquid, such as distillation and absorption, because of their high efficiencies, high capacities, and low pressure drops. Despite the success of applying structured packing in recent years,1,2 randomly packed columns are still commonly used in separation processes. The experimental work and mathematical modeling of the fluid flow and heat and mass transfer performance in randomly packed columns have been actively researched for decades. For the randomly packed column, especially in the case of column-to-packing diameter ratios (aspect ratios) lower than about 10, the usual plug-flow assumption is not applicable because of the nonuniform packed structure, higher porosity, and the wall flow.3-10 Over the past decades, the computational fluid dynamics (CFD) method and its relevant commercial software have been rapidly developed and extensively applied to chemical engineering research and design to give more visualized predictions of the velocity and temperature profiles in the processes.11-16 However, for the prediction of concentration profile, the empirical turbulent Schmidt number or the experimentally determined dispersion coefficient, usually obtained using the inert tracer technique with no mass transfer, is commonly employed.11,17 Nevertheless, the use of such approximate methods of computation is not always possible, especially for the cases where such empirical numbers or experimental data are not available. To overcome the uncertainties without relying on the expensive time-consuming experimental measurements, in this paper, a new c2-c model proposed by Liu18 is used to solve the unknown turbulent mass

transfer diffusivity, and satisfactory agreements were found between model prediction and experimental measurement. Gas absorption with chemical reaction has been extensively used in gas purification as well as in many chemical processes. Among them, the CO2 absorption in randomly packed columns by alkaline solutions such as NaOH, MEA, and MDEA aqueous solutions is commonly adopted in acid gas treatments, and many investigations of CO2 chemical absorption20-26 have been reported in the past decades. However, the industrial design procedure of such columns is conventionally based on macroscopic mass balances, without considering the uneven flow profile, temperature, and concentration distributions, which are inevitably as a result of nonideal momentum and heat and mass transfer in the randomly packed bed. In this paper, a complex model consisting of the quasi-singleliquid-phase CFD model, the turbulent mass transfer model with the c2-c equations for its closure, and the heat balance equation is proposed and used to simulate a randomly packed column for the chemical absorption of CO2 by NaOH aqueous solutions with consideration of the influence of the nonideal flow and nonuniform packing. As have been commonly employed by many researchers, two-dimensional (axial and radial) simulations are applied to the present work.7,9,11,12 Reaction Mechanism The absorption of CO2 and the reaction between CO2 and NaOH in the aqueous solution take place by the following steps HA

CO2,g98 CO2,L

(1)

transfer diffusivity. With the new c2-c model, Sun et al.19 simulated the performance of a commercial-size distillation column of sieve tray structure for the concentration profiles on the trays, the outlet concentration of each tray, and the overall tray efficiency without knowing in advance the turbulent mass

CO2,L + OH- 98 HCO3-

(2)

HCO3- + OH- 98 CO32- + H2O

(3)

* To whom correspondence should be addressed. Tel.: +86 22 27404732. Fax: +86 22 27404496. E-mail: [email protected].

CO2,L + 2OH- 98 CO32- + H2O

(4)

HR,k2

10.1021/ie051184z CCC: $33.50 © 2006 American Chemical Society Published on Web 03/18/2006

Ind. Eng. Chem. Res., Vol. 45, No. 9, 2006 3221

Step 1 denotes the physical absorption of CO2 by water, accompanied by the heat of solution HA. Reaction 2 could be known as the rate controlling step because reaction 3 is a protontransfer reaction and is about 1000 times faster than reaction 2. So, the absorption of CO2 by an NaOH solution can be regarded as a gas absorption control process accompanied by a relatively slow second-order reaction,20 and thus the overall reaction is represented by formula 4. Experimental Data Experimental data used for comparisons with model predictions were taken from the work of Tontiwachwuthikul et al.21,22 Their experimental conditions were as follows: the absorption of CO2 from air by using NaOH solution at total pressure of 103.15 kPa was conducted in an absorption column 7.2 m high with a 0.1 m i.d. made of acrylic plastic and packed with 1/2 in. (∼1.27 cm) ceramic Berl saddles. The column consisted of six equal-height sections with a total packing height of 6.55 m. Assumptions for the Present Model. Four assumptions are made for the present simulation of the chemical absorption of CO2 by NaOH aqueous solution in randomly packed column. 1. Only the CO2 component in the gas phase is absorbed by the NaOH aqueous solution, and the solvent water does not transfer to gas phase. 2. The heat of absorption and reaction of the solute gas CO2 is all absorbed instantaneously by liquid phase. Therefore, the predicted temperature of the liquid phase should be little higher than the experimental measurements. 3. Adiabatic gas absorption is assumed, which means that there is no heat exchange between the system and environment. This is reasonable and justified by Pandya.27 4. The gas absorption operation is steady; the fluid is incompressible, and the flow is axis symmetric. Modeling Equation Set The equation set of the present model is consisted of the CFD modeling equations, the turbulent mass transfer modeling equations with c2-c equations for its closure, and the heat balance equation. (A) CFD Equations. Continuity Equation.

∂Fhu 1 ∂FhrV + )N ∂x r ∂r

(

)

∂2hu ∂p 1 ∂ ∂hu ∂Fhu ∂Fhu r + 2 ) - + FLG + +u - µeff ∂r ∂x r ∂r ∂r ∂x ∂x h(Fx,LS + Fg) (6a)

( )

((

)

∂2hV ∂p ∂FhV ∂ 1 ∂hrV ∂FhV + 2 ) - + hFr,LS +u - µeff ∂r ∂x ∂r r ∂r ∂r ∂x (6b)

)

µeff ) µ + µt µt ) FCµ

k2

) )

{ [(

) ( ) ( )] (

)}

Turbulent Dissipation Rate, , Equation.

(( ) ) (( ) ) { [( )

µt ∂h ∂Fhuk 1 ∂FhV ∂ µ+ + ∂x r ∂r ∂x σ ∂x µt ∂h ∂u 2 ∂V 2 V2 1 ∂ µ+ r ) c1hµeff 2 + + + r ∂r σ ∂x k ∂x ∂r r

(

( ) ( )]

)}

∂u ∂V 2 2 + - c2Fh (10) ∂r ∂x k

The parameters which appear in the k- model, eqs 8-10, are customarily chosen to be Cµ ) 0.09, σk ) 1.0, σ ) 1.3, c1 ) 1.44, and c2 ) 1.92. (B) Turbulent Mass Transfer Equations. Equation for Average OH- Mass Fraction C h in Liquid. The turbulent mass transfer equation expressed in time averaged concentration is

[

]

∂ ∂C h ∂C h D - ujc + S uj ) ∂xj ∂xj ∂xj

(11a)

where C h is the time average concentration, c is the average fluctuation of concentration in turbulent flow, D is the molecular diffusivity, and S is the source term. For the computation of the unknown term, -ujc, the usual method is the application of the Boussinisque’s postulation, by which -ujc is assumed to be proportional to the concentration gradient as

∂C h ujc ) Dt ∂xj

(11b)

where Dt is a coefficient, usually regarded as turbulent mass transfer diffusivity, and is considered to be a constant. If -ujc is not exactly proportional to the concentration gradient ∂C h /∂xj, such as that resulting from the change of the liquid velocity distribution during the process, the Dt value cannot be constant but should change somewhat from the initial value to make eq 11b valid. By substituting eq 11b into 11a for the system concerned in this paper, we have determined the following mass transfer equation in cylindrical coordinate system

h ∂FhuC 1 ∂FhrVC 1 ∂ ∂FhC h ∂FhC h h ∂ + ) rD + Deff +S r ∂r ∂x r ∂r eff ∂r ∂x ∂x (11c)

( (

The radial momentum equation is

V

(( ) ) ((

µt ∂hk ∂Fhuk 1 ∂FhVk ∂ 1 ∂ + µ+ µ+ ∂x r ∂r ∂x σk ∂x r ∂r µt ∂hk ∂u 2 ∂V 2 V2 ∂u ∂V 2 + r ) hµeff 2 + + + σk ∂x ∂x ∂r r ∂r ∂x Fh (9)

(5)

where F is the liquid density, h is the volume fraction of liquidphase based on pore space, and N is the source term of the continuity equation from the chemical absorption of CO2 from the gas phase. Momentum Equations. The axial momentum equation is

V

where µ, µt, and µeff represent the molecular, turbulent, and effective viscosity of the liquid, respectively. FLG is the interface drag force between the gas phase and liquid phases, FLS is the flow resistance created by the random packing, and FLS is the body force. Turbulent Kinetic Energy, k, Equation.

(7) (8)

)

(

))

where S is the OH- source term which comes from the chemical reaction between the absorbed CO2 and OH- in the aqueous solution; Deff is regarded as the effective diffusivity of OHand is defined by

Deff ) D + Dt

(12)

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Ind. Eng. Chem. Res., Vol. 45, No. 9, 2006

where Dt is the turbulent diffusivity for mass transfer, which can be solved by the c -c quent section, by the expression 2

model,18

as shown in the subse-

( )

k c2 Dt ≡ Ctk c

the terms of h, N, FLG, FLS, S, Rt, and Q should be first evaluated by the existing data and correlations as shown in the next section. Evaluation of Various Terms

1/2

(13)

The variance of concentration fluctuation c2 and its dissipation rate c are defined as

Volume Fraction of Liquid Phase, h. Under the condition of the gas-liquid two-phase flow, the volume fraction of the liquid-phase, based on pore space, can be determined from the total liquid holdup, Ht, and the porosity, γ, of the random packing.

h ) Ht/γ

c2 ≡ cc

(19)

(14)

( )

∂c ∂c c ≡ D ∂xj∂xj

(15)

where the total liquid holdup, Ht, is composed of static holdup, Hs, and the operating holdup (dynamic holdup), Hop

Ht ) Hs + Hop

(20)

As can be seen from eq 13, Dt is related to the flow parameter and the fluctuation of concentration. (C) Closure of Mass Transfer Equation. To solve the mass balance equation, the unknown parameter Dt should be determined either by experimental work or by the auxiliary closing

Shulman et al.29reported the value of the static holdup, Hs, as 0.0317 for 1/2 in. Berl saddles. Otake and Okada obtained a correlation for the operating holdup, Hop, which was reported by Sater et al.30 as

model. In this paper, the c2-c model proposed by Liu18 and Sun et al.19 is employed, and the final form of the model equations are given below.

Hop ) 1.295(Re)L0.676(Ga)L-0.44(adp)

2

Concentration Variance, c , Equation.

( ) (( ) ) (( ) ) [( ) ( ) ]

Dt ∂Fhc2 ∂ ∂Fhuc2 1 ∂FhrVc2 D+ + ∂x r ∂r ∂x σc ∂x

Dt ∂Fhc2 1∂ ∂C h 2 ∂C h 2 D+ r ) 2FhDt + - 2Fhc r ∂r σc ∂r ∂x ∂r (16) Dissipation Rate, Ec, Equation.

(

) (( ) )

) ) [( )

Dt ∂Fhc ∂Fhuc 1 ∂FhrVc ∂ + D+ ∂x r ∂r ∂x σc ∂x Dt ∂Fhc ∂C h 2 ∂C h 2 c 1 ∂ D+ r ) Cc1FhDt + r ∂r σc ∂r ∂x ∂r c2 c2 c Cc2Fh - Cc3Fh (17) k c2

((

where (Re)L ) dpL/µ is the Reynolds number of liquid phase, (Ga)L ) dp3gF2/µ2 is the Gallileo number of liquid phase, a is the surface area per unit volume of packed bed, dp is the nominal diameter of the packed particle, L is the flow rate of liquid which can be calculated from the local liquid velocity by L ) Fhγ|u|. Near the column wall, the porosity, γ, of the randomly packed bed is unevenly distributed in radial direction and changes from a constant at the axis to a maximum of about 1.0 in the wall, as observed by many experimental investigations.9,31,32 In this paper, the porosity distribution correlation reported by Liu12 is adopted

γ ) γ∞ +

cos

)

(

))

(

)

}

2π R-r + 0.3pd cγ + 1.6Er2 pddp

(22)

where γ∞ is the porosity in an unbounded packing, R is the radius of the column, r is the position in radial direction, and Er is the exponential decaying function, which is defined by

[

The constants in eqs 13-17 are28 Ct ) 0.11, Cc1 ) 1.8, Cc2 ) 2.2, Cc3 ) 0.8, σc ) 1.0, and σc ) 1.0 (D) Heat Balance Equation.

( (

{

(1 - γ∞) Er (1 - 0.3pd) × 2

( )]

∂FcphT 1 ∂ 1 ∂FcphrVT ∂FcphuT + ) rReff + r ∂r ∂x r ∂r ∂r ∂FcphT ∂ Reff ∂x ∂x

(21)

( )]

Er ) exp -1.2pd

R-r dp

3/4

(23)

where pd is the period of oscillation normalized by the nominal particle size, equal to 0.94 for Berl saddles, and cγ is a constant depending on the ratio of the particle size to column size

+ Q (18)

where cp is the specific heat of liquid phase, Reff is the effective thermal diffusivity of liquid phase, defined as Reff ) R + Rt, in which R and Rt are the molecular and turbulent thermal diffusivities, respectively, and Q is the thermal source term reflecting the heat of absorption and reaction, as well as other thermal effects. Equations 5, 6, 9-11, and 16-18 are the mathematical model used to predict the momentum and heat and mass transfer behaviors in the absorption column. In the model equations,

cγ )

[

()]

2R R - 1.6exp -2.4pd nγpddp dp

3/4

(24)

where

nγ ) int

{

}

2 R 3/4 p d 1 + 1.6exp[-2.4pd(R/dp) ] d p

(25)

Source Term, N, in the Continuity Equation. For the chemical absorption of CO2 from gas phase, the source term, N, the quantity of mass transfer per unit volume and unit time, can be determined from the following equation based on a well-


known two-film model

N ) kLReffMCO2E(Xi,CO2 - XCO2)

(26)

where kL is the liquid-phase mass transfer coefficient without chemical reaction, Reff is the effective area for mass transfer between the gas phase and liquid phase, MCO2 is the molecular weight of CO2, E is the so-called enhancement factor, Xi,CO2 is the molar concentration of CO2 at interface, and XCO2 is the molar concentration of CO2 in the bulk liquid, which is assumed to be equal to zero. The liquid-phase mass transfer coefficient, kL, is determined from the correlation by Onda et al.33

kL ) 0.0051

()( )( µg F

L awµ

1/3

2/3

µ FDCO2,L

)

-1/2

(adp)0.4 (27)

where the wetted surface area, aw, is obtained from the following correlation

{ ()

σct aw ) 1 - exp -1.45 a σ

0.75

(aµL )

( ) ( )}

0.1

L 2a F2g

-0.05

L2 Faσ

0.2

(28)

the effective area According to the research of Onda et Reff is equal to Rw. The enhancement factor, E, is defined as the ratio of the mass transfer coefficient, kR,L, for absorption with chemical reaction to the mass transfer coefficient, kL, for physical absorption.34 For an irreversible second-order reaction such as the CO2NaOH reaction, Wellek et al.35 reported a correlation for the calculation of enhancement factor, E, as shown below with a deviation of less than 3%: where

{

-1.35

(Ei - 1)

Ei ) 1 + Ha )

1 + (E1 - 1)-1.35

}

1/1.35

DXOH2DCO2,LXCO2,i

DCO2,Lk2XOH-

E1 )

(kL)2

xHa tanh xHa

(29)

(30)

(31)

(32)

where XOH- denotes the molar concentration of OH- in liquid phase, DCO2,L represents the diffusivity of CO2 in the NaOH aqueous solution, and k2 is the second-order reaction rate constant for the CO2-NaOH reaction. The diffusivity of CO2 in the NaOH aqueous solution, DCO2,L, can be determined by modifying the diffusivity of CO2 in the pure water, DCO2,W, as follows23

DCO2,L ) DCO2,w(1 - ξXOH-)

(33)

where ξ is a constant depending on the aqueous solution. For the NaOH aqueous solution, we have ξ ) 0.129 m3 Kmol-1. The diffusivity of CO2 in the pure water, DCO2,W, is a function of liquid temperature correlated by Pohorecki and Moniuk36 as

log DCO2,w ) -8.1764 +

log k2 ) 11.895 -

712.5 2.591 × 105 T T2

(34)

2382 + 0.221Ic - 0.016Ic2 T

(35)

According to Pohorecki and Moniuk,36 the molar concentration of CO2 at the interface, Xi,CO2 can be described by Henry’s law

Xi,CO2 ) HptyCO2

(36)

where H is the Henry’s constant for CO2 in the NaOH solution, pt is the total pressure of gas phase, and yCO2 is the volume fraction of CO2 in the gas phase. For the evaluation of H, the correlation by Danckwerts34 is used

log

al.,33

E)1+

The diffusivity ratio of D/DCO2,L is 1.67 at 20 °C34 and is assumed to be independent of the temperature in the range of 20-40 °C. The second-order reaction rate constant, k2, for CO2-NaOH reaction was correlated by Pohorecki and Moniuk36 as a function of temperature and ionic strengths, Ic, of aqueous electrolyte solutions as

0.9869H ) - KsIc Hw

log Hw ) 9.1229 - 0.059044T + 7.8857 × 10-5T2

(37) (38)

where Hw is the Henry’s constant for CO2 in pure water and Ks is the salting parameter, denoting the sum of the contributions of the ions and gas molecules in the liquid. For the CO2-NaOH system, Ks ) 0.142. Interface Drag Force FLG. For irrigated packing, the pressure drop is greater than the dry bed pressure drop, ∆pd, because of the presence of liquid adhered to the packing surface so that the available cross section for gas flow is reduced, The part of the increased pressure drop, ∆pL, represents the interfacial drag force between gas phase and liquid phase. Robbins37 correlated the total pressure drop composed of ∆pd and ∆pL for irrigated packing. The correlations in international units are as follows:

∆pt ) ∆pd + ∆pL

(39)

∆pd ) p1Gf2 × 10p2Lf

(40)

∆pL ) 0.774

( ) Lf 20000

0.1

(p1Gf2 × 10p2Lf)4

(41)

where p1 ) 0.04002, p2 ) 0.0199, Gf is the gas loading factor, and Lf is the liquid loading factor. Gf and Lf can be determined by

Gf ) G(1.2/FG)0.5(Fpd/65.62)0.5

for pt e 1.0 atm

(42a)

Gf ) G(1.2/FG)0.5(Fpd/65.62)0.5 × 100.0187FG for pt > 1.0 atm (42b) Lf ) L(1000/F)(Fpd/65.62)0.5µ0.2 Lf ) L(1000/F)(65.62/Fpd)0.5µ0.1

for Fpd g 15 (43a) for Fpd < 15

(43b)

The term Fpd is a dry packing factor, specific for a given packing type and size. For 1/2 in. (∼1.27 cm) ceramic Berl saddles, Fpd ) 900 m-1, so eqs 43a and 42b are chosen in accordance with the operating conditions.

3224


Finally, the interface drag force FLG can be calculated by

FLG )

∆pL u |uslip| slip

(44)

where uslip is the slip velocity between gas phase and liquid phase

uslip ) uG - u

(45)

Body Force, FLS. The resistance to the liquid flow created by the presence of random packing can be treated as body force and can be modeled by the well-known Ergun38 equation by replacing the mean porosity, γ∞, with the porosity distribution function, γ.

(

(1 - γ)

FLS ) 150µ

2

2

γ de

| |)

(1 - γ) u u γde

2

+ 1.75F

(46)

where de is the equivalent diameter of random packing, defined by

de )

6(1 - γ∞) a

(47)

Source Term, S, in Equation 11c. The OH- concentration in the liquid phase decreases as the NaOH aqueous solution absorbs CO2, and the quantity of S can be calculated according to chemical reaction 4 and the amount of absorbed CO2

S)-

N × 17 × 2 44

N (H + HR) MCO2 A

c2 in the literature. However, according to the turbulent heat transfer experiments,41-43 the extent of temperature fluctuation induced by heating is about 0.082 times the heat flow. By the analogy between heat and mass transfer, we assume that

cinlet2 ) (0.082Cinlet)2

(50)

where HA is the heat of physical absorption, HA ) 1.9924 × 107 J kmol-1 of CO2 absorbed and HR denotes the heat of the chemical reaction, HR ) 1.67432 × 108 J kmol-1 of CO2 absorbed in reaction 4 at 25 °C, and it is assumed to be independent of temperature in the range of 20-40 °C. Boundary Conditions The computational domain and boundaries are shown in Figure 1. The boundary conditions for the model equation set are specified as follows. (A) Velocity at Inlet. At the top of the column, the “velocity inlet” boundary is set to be u ) uinlet, vinlet ) 0, T ) Tinlet, C h) Cinlet, and kinlet ) 0.005uinlet2,39 and inlet ) 0.09kinlet1.5/dH; these

(51)

The inlet condition specification for c could be obtained by the definitions of turbulent diffusivity for mass transfer Dt and the time scale ratio Rτ using eq 13 and

Rτ ≡

k/

(52)

c2/c

Combining eqs 13 and 52, we get

[ ]

k2 1 Rτ ) Ct Dt

(49)

Heat, Q. Because of the heat of absorption and reaction, the temperature of the liquid phase will be increased, and as a result, the properties of the liquid, such as H, k2, and DCO2,L, will also be changed; the absorption rate of CO2 will subsequently be affected. Therefore, the temperature change in liquid must be properly evaluated. According to the amount of CO2 absorbed by the NaOH aqueous solution per unit volume and unit time, the total heat effect, Q, can be calculated as

Q)

terms are commonly adopted in CFD computation. The term dH denotes the hydraulic diameter of random packing,40 which can be calculated by dH ) 4γ∞/a(1 - γ∞). There are no existing experimental measurements or empirical correlations for the inlet condition for the concentration variance

(48)

Turbulent Diffusivity for Heat Transfer, Rt. According to the Reynolds analogy law for turbulent heat and mass transfer, the turbulent heat diffusivity, Rt, is set to be equal to turbulent mass diffusivity, that is

Rt ) Dt

Figure 1. Boundary conditions and the simulation domain.

2

(53)

For the ratio Rτ, numerous values have been reported, which, according to Launder,44 may not be a universal constant and may strongly depend on the nature of the flow field. As an approximation, we may allow Rτ to be a definite value only at the inlet boundary, and this value can be determined from the experimental data obtained under the “end conditions” at C h ) 0. The diffusivity, Dt, at C h ) 0 for a randomly packed column with 1/2 in. (∼1.27 cm) ceramic Berl saddles has been measured using the iodine-131 tracer by Sater et al.30 and is correlated to the “coefficient of dispersion” of the liquid phase by the equation

( )

( )

Udp dpL ) 7.58 × 10-3 Dt µ

0.703(0.238

(54)

where U is the liquid superficial velocity. According to eq 54, the order of magnitude for Dt is about 10-4-10-3. Later, Michell and Furzer45 obtained the following dispersion coefficient correlation according to their own experiment and some published data:

(

)( )

u{int}erdpF |u|dp ) 1.00 Dt µ

0.7

dp3gF2 µ2

-0.32

(55)

The order of magnitude for Dt in eq 55 is about 10-3. By taking the average value of Dt in the above correlations and substituting


Figure 2. Comparison between computed results and experiment T1.



it into eq 53, the Rτ value was found to be about 0.4. Therefore, the boundary condition of c at the inlet with Rτ ) 0.4 is

c,inlet ) 0.4

( )

inlet c 2 kinlet inlet

(56)

(B) Outflow. The “outflow” boundary is set at the bottom of the column, where the exit flow is close to the fully developed condition, that is ∂Φ/∂x ) 0. (C) Axis. The “axis” boundary is set at the centerline of the column, which means that all variables, Φ, are mathematically symmetric and ∂Φ/∂r ) 0 at r ) 0. (D) Wall. The usual “standard wall functions” are employed at the near-wall region.46 Numerical Procedure The model equations were solved numerically using the commercial software FLUENT 6.1 and the finite volume method. The well-known SIMPLEC algorithm is used to solve the pressure-velocity coupling problem in the momentum equations. Result and Discussion Twelve sets of experimental data for the chemical absorption of CO2 by NaOH aqueous solution in randomly packed columns were reported in the work of Tontiwachwuthikul et al.22 including variation of OH- concentration and temperature in

the liquid phase and the CO2 concentration in the gas phase along the height of the column. In this paper, only experiments T1, T2, T7, T8, T11, and T12 are simulated as examples for comparison. (A) Radial Average OH- Concentration and Temperature Distributions in the Liquid Phase along the Axial Direction. In Figures 2a-7a, the squares and circles represent the experimentally measured OH- concentration in liquid phase and CO2 concentration in gas phase, respectively, and in Figures 2b-7b, the circles represent the experimentally measured liquid temperature. The solid lines indicate the simulated results which were obtained after the radial average. As can be seen from Figures 2a-7a, the agreement between the simulations and experimental results for OH- concentration in the liquid phase and CO2 concentration in the gas phase is satisfactory, thereby confirming the validity of the turbulent mass transfer model presented in this paper. Generally, the temperature profiles along the column predicted by simulation are somewhat higher than the experimental measurements as shown in Figures 2b-7b, especially at the bottom of the column. There are several reasons: first, the quasisingle-liquid-phase model adopted in this paper ignores the cooling of descending liquid by the entering gas; second, we neglected the evaporation of solvent water in the liquid phase, leading to an overestimate of the liquid temperature; and finally, the assumption of adiabatic operation means the neglect of heat exchange between the column and environment.

3226





Figure 8. Liquid relative axial velocity profile along the radial direction in T12: (a) liquid relative axial velocity profile and (b) two-dimensional distribution of liquid relative to the axial velocity.

(B) Liquid Velocity Profile along the Radial Direction. Because of the nonuniformity of the packed structure and higher porosity at the wall, the fluid flow seriously deviates from the plug flow. As can be seen from Figure 8, serious “wall flow” appears near the wall region, and the flow becomes relatively uniform only about 2dp away from the wall. Similar results have been observed by many investigators.9,12 (C) Turbulent Diffusivity, Dt, for Mass Transfer Profiles along the Radial Direction. The turbulent mass transfer diffusivity profiles along the radial direction at different axial positions are presented in Figure 9. The uneven Dt distributions are related to the uneven flow fields, as shown in Figure 8. Lower values of Dt appear at the region around column wall,

Figure 9. Turbulent mass transfer diffusivity profiles along the radial direction in T12.


which is likely to be caused by the higher velocity near the wall region (Figure 8). The predicted profile of Dt along the radial direction at a fixed axial position is consistent with the experimentally measured results by Fahien and Smith47 and Dorweiler and Fahien,48 who examined it using a tracer technique for the variations of diffusivity and the Peclet number with radial positions in different diameter pipes packed randomly with different sizes of spheres. Rapid increases of Dt are found at the region around the liquid inlet because of the quick change of the liquid velocity profile from uniform velocity at the inlet boundary to uneven flow immediately after the liquid enters the column. (D) Radial Average Enhancement Factor, E, Profile Along the Column. The enhancement factor, E, is an important parameter for chemical absorption, and accurate estimation of the enhancement factor, E, is essential for proper modeling of the mass transfer process. This factor is influenced by many aspects such as the physical properties of liquid and gas, OHconcentration in liquid phase, CO2 concentration in gas phase, flow field, and reaction rate, and therefore it is not constant along the column. As can be seen from Figures 2b-7b, the enhancement factor, E, varies along the column height. In T11, for example, where E increases from about 20 at the column bottom to about 100 at the column top, the rate of chemical absorption is about 20-100 times higher than that of physical absorption. Conclusions The following conclusions can be drawn from the present work. (1) A new model is proposed for mass transfer process by combining the turbulent mass transfer model with the c2 - c two-equation model for its closure, the accompanying CFD formulation, and the heat balance. The validity of the proposed model is tested by its application to a pilot-scale randomly packed column for the chemical absorption of CO2 from air at atmospheric pressure by NaOH aqueous solutions. The results of the simulation on the temperature and the concentration distributions along the height of the column are in satisfactory agreement with the experimental data reported by Tontiwachwuthikul et al.22 (2) The results of computations reveal that both the axial and radial turbulent mass transfer diffusivities vary with the height of the packed column. (3) The enhancement factor of chemical absorption predicted from the proposed model varies significantly along the column. (4) The feature of the proposed model is able to predict simultaneously the velocity, temperature, and concentration profiles, as well as the turbulent mass transfer diffusivity, in a randomly packed column for chemical absorption. Acknowledgment The authors acknowledge the financial support of the National Natural Science Foundation of China (Contract 20136010) and the assistance from the staff in the State Key Laboratories for Chemical Engineering (Tianjin University). Nomenclature a ) surface area per unit volume of packed bed (m-1) aeff ) effective area for mass transfer between the gas phase and liquid phase (m-1) aw ) wetted surface area (m-1)

c2 ) concentration variance C h ) average concentration of mass fraction Cµ, c1, c2 ) model parameters in the k- model equations Ct, Cc1, Cc2, Cc3 ) model parameters in c2-c model equations cp ) liquid-phase specific heat (J kg-1 K-1) cγ ) a constant depending on the ratio of the particle size to the column size DCO2,L ) diffusivity of CO2 in the NaOH aqueous solution (m2 s-1) DCO2,W ) diffusivity of CO2 in pure water (m2 s-1) de ) equivalent diameter of random packing (m) dH ) hydraulic diameter of random packing (m) dp ) nominal diameter of the packed particle (m) Deff ) effective diffusivity of OH-1 (m2 s-1) Dt ) turbulent diffusivity for mass transfer (m2 s-1) E ) enhancement factor Er )exponential decaying function, defined by eq 23 FLG ) interface drag force between gas phase and liquid phase (N m-3) FLS ) flow resistance created by the random packing (N m-3) Fpd ) dry packing factor (m-1); for 1/2 in. (∼1.27 cm) ceramic Berl saddles, Fpd ) 900 m-1 g ) acceleration due to gravity (m s-2) G ) gas-phase flow rate per unit cross-section area (kg m-2 s-1) (Ga)L ) Gallileo number of the liquid phase Gf ) gas loading factor (kg m-2 s-1) h ) volume fraction of liquid-phase based on pore space H ) Henry’s constant in the NaOH solution (kmol m-3 atm-1) HA ) physical absorption heat of mol CO2 absorbed (J kmol-1) Hop ) operating holdup HR ) chemical reaction heat of mol CO2 absorbed (J kmol-1) Hs ) static holdup Ht ) total liquid holdup Hw ) Henry’s constant in pure water (kmol m-3 atm-1) Ic ) ionic strengths of aqueous electrolyte solutions (kmol m-3) k ) turbulent kinetic energy (m2 s-2) k2 ) second-order reaction rate constant (m3 kmol-1 s-1) kL ) liquid-phase mass transfer coefficient without chemical reaction (m s-1) kR,L ) liquid-phase mass transfer coefficient with chemical reaction (m s-1) Ks ) salting-out parameter (L mol-1); for the CO2-NaOH system, Ks ) 0.142 L mol-1 L ) liquid flow rate per unit cross-section area (kg m-2 s-1) Lf ) liquid loading factor (kg m-2 s-1) N ) CO2 quantity absorbed by the aqueous solution (kg m-3 s-1) MCO2 ) molecular weight of CO2 (kg kmol-1) p1, p2 ) constants pd ) period of oscillation normalized by the nominal particle size; pd ) 0.94 for Berl saddles pt ) total pressure of gas phase (atm) ∆pd ) dry-bed pressure drop per meter packing (N m-3) ∆pL ) wet-bed pressure drop per meter packing (N m-3) ∆pt ) total pressure drop per meter packing (N m-3) Q ) thermal source term of temperature equation (J m-3 s-1) r ) position in radial direction (m) R ) radius of the column (m) (Re)L ) Reynolds number of liquid phase Rτ ) velocity to concentration time scale ratio S ) sink term of OH- conservation equation (kg m-3 s-1) T ) liquid temperature (K)

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u ) interstitial axial velocity of liquid (m s-1) u ) interstitial velocity vector of liquid (m s-1) uG ) gas velocity vector (m s-1) uslip ) slip velocity vector between gas phase and liquid phase (m s-1) U ) liquid superficial velocity (m s-1) V ) interstitial radial velocity of liquid (m s-1) XCO2 ) molar concentration of CO2 in the liquid bulk (kmol m-3) Xi,CO2 ) molar concentration of CO2 at interface (kmol m-3) XOH- ) molar concentration of OH- in liquid phase (kmol m-3) yCO2 ) volume fraction of CO2 in gas phase Greek Symbols R, Reff, Rt ) molecular, turbulent, and effective thermal diffusivities, respectively (m2 s-1) ) turbulent dissipation rate (m2 s-3) c ) turbulent dissipation rate of concentration fluctuation (s-1) Φ ) variable γ ) porosity distribution of the random packing bed γ∞ ) porosity in an unbounded packing µ, µt, µeff, ) liquid molecular, turbulent, and effective viscosity, respectively (kg m-1 s-1) F ) liquid density (kg m-3) σ ) surface tension of liquid (N m-1) σct ) critical surface tension of the packing material (N m-1); for ceramic, σct ) 0.061 N m-1 σc, σc ) model parameters in c2-c model equations σk, σ ) model parameters in k- model equations ξ ) a constant depending on aqueous solution (m3 kmol-1) Subscripts eff ) effective G ) gas i ) interface L ) liquid R ) reaction slip ) slip S ) solid τ ) time scale Literature Cited (1) Gualito, J. J.; Cerino, F. J.; Cardenas, J. C.; Rocha, J. A. Design method for distillation columns filled with metallic, ceramic, or plastic structured packings. Ind. Eng. Chem. Res. 1997, 36 (5), 1747-1757. (2) Spiegel, L.; Meier, W. Distillation columns with structured packings in the next decade. Chem. Eng. Res. Des. 2003, 81 (A1), 39-47. (3) Lerou, J. J.; Froment, G. F. Velocity, temperature and conversion profiles in fixed-bed catalytic reactors. Chem. Eng. Sci. 1977, 32 (8), 853861. (4) Zhang, Z. T. Study on liquid flow characteristics and mass transfer model in packed column. Ph.D. Thesis, Tianjin University, Tianjin, China, 1986. (5) Zhang, Z. T.; Yu, K. T. Stochastic simulation of liquid flow distribution in packed column. J. Chem. Ind. Eng. (China) 1988, 39 (2), 162-169. (6) Ziolkowska, I.; Ziolkowski, D. Fluid-flow inside packed-beds. Chem. Eng. Process. 1988, 23 (3), 137-164. (7) Yuan, X. J.; Li, F. S.; Yu, K. T. Distribution of liquid phase in a large packed column. J. Chem. Ind. Eng. (China) 1989, 40 (6), 686-692. (8) Bey, O.; Eigenberger, G. Fluid flow through catalyst filled tubes. Chem. Eng. Sci. 1997, 52 (8), 1365-1376. (9) Giese, M.; Rottschafer, K.; Vortmeyer, D. Measured and modeled superficial flow profiles in packed beds with liquid flow. AIChE J. 1998, 44 (2), 484-490. (10) Wen, X.; Shu, Y.; Nandakumar, K.; Chuang, K. T. Predicting liquid flow profile in randomly packed beds from computer simulation. AIChE J. 2001, 47 (8), 1770-1779.

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ReceiVed for reView October 25, 2005 ReVised manuscript receiVed January 15, 2006 Accepted February 27, 2006 IE051184Z

New Model for Turbulent Mass Transfer and Its Application to the

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