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Dimensionless Models for Predicting the Effective Area, Liquid-Film, and Gas-Film Mass-Transfer Coefficients of Packing Chao Wang,†,‡ Di Song,† Frank A. Seibert,‡ and Gary T. Rochelle*,† †

Texas Carbon Management Program, McKetta Department of Chemical Engineering, The University of Texas at Austin, 200 E. Dean Keeton Street, C0400, Austin, Texas 78712-1589, United States ‡ Separations Research Program (SRP), Pickle Research Campus, The University of Texas at Austin, 10100 Burnet Road, Austin, Texas 78758, United States ABSTRACT: A total of 11 structured packings with surface areas (ap) from 125 to 500 m2/m3 and corrugation angles (θ) from 45 to 70° were studied to explore the effects of the packing geometry on mass transfer. A new concept, the mixing point density, was proposed to represent the effects of ap and θ. A generalized method to calculate the mixing point density from aP and θ was developed. Dimensionless mass-transfer models are developed to predict the effective area, liquid-film, and gas-film mass-transfer coefficients, with the mixing and Reynolds numbers imbedded into the correlations. Compared with literature models, the models developed in this work capture the effects of the packing geometry and predict the mass-transfer properties in aqueous systems.

1. INTRODUCTION Packing is widely used in distillation, stripping, and scrubbing processes because of its relatively low pressure drop, good mass-transfer efficiency, and ease of installation. The masstransfer performance of packing can be characterized by the gas-film and physical liquid-film mass-transfer coefficients (kG and kL) and the gas−liquid mass-transfer area (ae). In the amine scrubbing CO2 capture process, kG is the dominant masstransfer coefficient in the direct contact cooler and water wash; the liquid-film mass-transfer coefficient kg′ (which includes chemical reaction and the physical liquid-film mass-transfer coefficient kL) is the dominant mass-transfer coefficient for the absorber and stripper, respectively, and ae is important for all operations. Numerous mass-transfer models for packing have been developed and proposed in the literature. Onda et al.1 developed the first and still widely used mass-transfer area model based on the database from the absorption of CO2 by NaOH. However, the packings measured were mostly random. Rocha et al.2 developed a model for the effective area based on an extensive experimental database, mostly for structured packing. In his work, the gas-film mass-transfer coefficient is based on earlier investigations of wetted-wall columns (WWCs), while the liquid film is based on the penetration theory. Widely used mass-transfer correlations for random packing were developed by Billet and Schultes.3 The correlations for the gas and liquid mass-transfer coefficients were developed from the original formulation of Higbie.4 In general, these previous mass-transfer models have a common ground. The combination of mass-transfer coefficient and area (Ka) was measured, where K is the overall mass© XXXX American Chemical Society

transfer coefficient and a is the corresponding mass-transfer area. However, to separate K and a, either a theoretical assumption of area or proposed K models from other work were used. None of the mass-transfer values (kG, kL, and ae) was independently validated. In distillation systems, most cases only required the combination (Ka) values, where these models were acceptable. However, the design and optimization of different parts of the amine scrubbing CO2 capture system require separate values of kG, kL, and ae. Therefore, a consistent measurement of kG, kL, and ae at the same condition is required. The scope of this work will be focused on the consistent study and modeling of kG, kL, and ae. All of the experiments were made in a pilot-scale column in the SRP at The University of Texas at Austin. By the application of a direct methodology to obtain the area and the mass-transfer coefficients, the shortcomings of the previously discussed models are addressed. Details of this work are reported by Wang et al.5−9

2. EXPERIMENTAL SECTION A pilot-scale PVC column with an inner diameter of 0.428 m (16.8 in) and a total column height of 7.62 m (25 ft) capable of a maximum packed height of 3.05 m (10 ft) was utilized to measure the effective area, gas-film, and liquid-film masstransfer coefficients. The absorption of atmospheric CO2 by 0.1 g·mol/L NaOH (reaction-film-controlled) was used to measure the mass-transfer area. Air stripping toluene from water (liquidReceived: December 9, 2015 Revised: March 2, 2016 Accepted: April 12, 2016

A

DOI: 10.1021/acs.iecr.5b04635 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

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Industrial & Engineering Chemistry Research Table 1. Average Physical Properties for the Area Measurement temperature

pressure

DCO2

HCO2

kOH−

K 298

kPa 101.3

m2/s 2.28 × 10−9

m3·Pa/kmol 3.41 × 106

m3/kmol·s 1.08 × 104

units value

film-controlled) was used to measure the liquid-film masstransfer coefficient. The absorption of SO2 from air by 0.1 g· mol/L NaOH (gas-film-controlled) was used to measure the gas-film mass-transfer coefficient. The inlet and outlet gas CO2 was measured by a CO2 analyzer (Horiba VIA-510) with a typical range of 0−500 ppm. The inlet and outlet liquid toluene was measured by a gas chromatograph (Hewlett-Packard 5890A) with a typical range of 0−1000 ppm. Two ambientlevel pulsed fluorescence analyzers (Thermo Scientific model 43i) were used to measure the inlet and outlet gas SO2 at 0− 100 ppm of SO2 and 0−2000 ppb of SO2. The average physical properties for area, kL, and kG measurements are given in Tables 1−3. Detailed descriptions of the experimental methods and apparatuses can be found in previous papers.5−13 The complete experimental data set can be found in work by Wang9 (pp 172−195).

Table 5. Random Packing

temperature

pressure

Dtoluene

μL

ρL

K 298

kPa 101.3

m2/s 8.6 × 10−10

kg/(m·s) 1.002 × 10−3

kg/m3 998

units values

pressure

DSO2

μG

ρG

K 298

kPa 101.3

m2/s 1.31 × 10−5

kg/(m·s) 1.98 × 10−5

kg/m3 1.204

corrugation angle (deg)

MP 125Y RSP 200X MP 2X MP 250Y MP 250X RSP 250Y GT-PAK 350Y GT-PAK 350Z A 350Y B 350X GT-PAK 500Y

125 200 205 250 250 250 350 350 350 350 500

45 60 60 45 60 60 45 70 45 60 45

96 97 98

315 250 180

δ Nusselt =

3u filmμL ρL g sin θ

=

3

⎛Q ⎞ ⎜ ⎟ ρL g sin θ ⎝ L P ⎠ 3μL

(2)

Thus, the dimensionless number group can be expressed by WeLFrL−1/3 =

4/3 ⎛ ρL ⎞ 1/3⎛ Q ⎞ ⎜ ⎟g ⎜ ⎟ ⎝σ⎠ ⎝ LP ⎠

(3)

where Q is the volumetric liquid flow rate (m /s) and LP is the wetted perimeter (m). For structured packing, the wetted perimeter can be calculated from the channel dimensions: 3

Table 4. Structured Packing surface area (m2/m3)

15 20 25

where WeL is the liquid-phase Weber number, ρLuL2δL/σ, and FrL is the liquid-phase Froude number, uL2/gδL. In the Tsai model, the liquid-film thickness (δL) was used as the characteristic length. To calculate the liquid-film thickness, the classic Nusselt film thickness assumption13 was used:

3. MASS-TRANSFER EFFECTIVE AREA MODEL The effective mass-transfer area model was developed based on the experimental data.5−8 The stainless steel structured packings (Table 4) were manufactured by Sulzer ChemTech,

packing name

surface area (m2/m3)

The effective mass-transfer area model was developed based on the dimensionless model by Tsai et al. (eq 1).11,12 According to Tsai’s experiments as well as the effective area measurements conducted in this work, the effective area is a function of the liquid flow rate, liquid density, and surface tension and is independent of the gas flow rate and liquid-phase viscosity. This model is supported by the majority of the experimental data, although at some conditions, the effective area changes slightly with the gas flow rate. ae = 1.34(WeLFrL−1/3)0.116 aP (1)

Table 3. Average Physical Properties for the kG Measurement temperature

void fraction (%)

RSR#0.3 RSR#0.5 RSR#0.7

Table 2. Average Physical Properties for the kL Measurement units value

nominal size (mm)

LP = A

4S Bh

(4) 2

where A is the column cross-sectional area (m ), S is the packing channel side (m), B is the packing channel base (m), and h is the packing crimp height (m). However, with a larger scope including random packing and hybrid packing such as the Raschig Super-Pak family, the original form of the Tsai model is not applicable. In those situations where channel dimensions are not known or hardly defined, using the liquid superficial velocity over the packing total area (uL/aP) instead of Q/LP is a good alternative. The mass-transfer area model in this work is developed based on the Tsai model and utilizes uL/ap as the liquid flow rate per wetted perimeter. The experimental coefficient is changed from 1.34 to 1.41, which provides a better fit of the larger database.

GTC Technology, and Raschig. Every packing surface except those of the two RSP packings was perforated. The packing surface area varied from 125 to 500 m2/m3, while the corrugation angle varied from 45 to 70°. Three random packings in the Raschig Super-Ring family were also included in the database (Table 5).

0.116 ⎡ ρ ⎛ uL ⎞4/3⎤ ae ⎛ ⎞ 1/3 L = 1.41⎢⎜ ⎟g ⎜ ⎟ ⎥ ⎢⎣⎝ σ ⎠ aP ⎝ aP ⎠ ⎥⎦

B

(5)


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Industrial & Engineering Chemistry Research

Figure 1. Mass-transfer area of structured packing.

Figure 2. Effects of the packing surface area and corrugation angle on kL.

4. MIXING POINT DENSITY DEVELOPED FROM THE PACKING GEOMETRY 4.1. Packing Geometry Effects on kL and kG. The liquidfilm and gas-film mass-transfer coefficients increase with the packing surface area and decrease with the packing corrugation angle. Figure 2 illustrates the effects of the packing surface area (aP) and corrugation angle (θ) on the liquid-film mass-transfer coefficient, kL. For packing with the same corrugation angle, kL increases by 33% as the surface area increases from 350 to 500 m2/m3. For packing with the same surface area, kL decreases by 28% as the corrugation angle increases from 45 to 70°. Figure 3 illustrates the effects of the packing surface area (aP) and corrugation angle (θ) on the gas-film mass-transfer coefficient, kG. For packing with a 45° angle (Y), kG increases as the surface area increases from 125 to 500 m2/m3. The same trend also applies for packing with a 60° angle (X). For packing

Figure 1 shows the fractional mass-transfer area as a function of the dimensionless number group WeLFrL−1/3 at a moderate gas velocity (0.99 m/s). The mass-transfer area model predicted by eq 5 is shown by the black solid line in the middle. The database includes 14 packings measured in this work and contains a large variety of packing types (structured, random, and hybrid). The model shows a good fit with most data. GT-PAK 500Y shows a lower effective area than that predicted because of the difficulty of effectively wetting the large surface area. The hybrid packing RSP200X shows a higher value than that predicted. The model accurately captures the dependence of the mass-transfer area on the liquid rate. The average deviation of this area model is 10.5%, which is quite acceptable considering the broad scope of the packing type. C


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Figure 3. Effects of the packing surface area and corrugation angle on kG.

with the same surface area, kG decreases as the corrugation angle increases. 4.2. Mixing Number. In the model development, a new concept, the mixing point density (M), was introduced to account for the effect of the packing geometry on kL and kG. The mixing point density is the number of mixing points per cubic meter in the packing and can be calculated by eq 6a. The dimensionless form of the mixing point density, mixing number Mi, is used in the mass-transfer correlations in this work. Mi is a transformation of the corrugation angle that should be better behaved in quantitative correlations. Mi can be calculated by eq 6b.

M=

Mi =

3aP3 sin θ cos θ 32 2

3 sin θ cos θ 32 2

calculate the mixing point density from the packing surface area aP and corrugation angle θ is developed in this work. On the basis of the solid geometry of structured packing, the relationship between B, h, aP, and θ is given by

h=

2 2 aP

(6a)

M= (6b)

6 BhB tan θ

M = Npyramid × mixing points per pyramid =

4 2 aP sin θ

(9)

(10)

A detailed description of the packing geometry study can be found in the Appendix section of this paper. Equation 8 can be expressed by aP and θ: 3a 3 sin θ cos θ 6 = P BhB tan θ 32 2

(11)

Finally, the dimensionless form, mixing number Mi, can be calculated by

Mixing points are the contact points of metal sheets in structured packings and are believed to enhance mass transfer.7 The mixing point density (M) is the number of mixing points per cubic meter. The mixing point density M will be greater with a reduced corrugation angle or a smaller geometric dimension (larger surface area), which explains the larger values of kL and kG in those packings. In previous work,7 a quantitive method was developed to calculate the mixing point density for regular structured packings:

M=

B=

Mi =

M 3 sin θ cos θ = 3 32 2 aP

(12)

It needs to be mentioned that B and S used in this work are measured from the top cross section of the packing channel (Figure 4), which are different from the b and s used in

(7)

6 BhB tan θ (8)

where B is the packing channel base (m), h is the packing crimp height (m), and θ is the packing corrugation angle (deg). However, the mixing point density calculated by eq 7 needs the specific information on the structured packing geometry: channel base B and crimp height h. These data are usually held proprietary to the packing vendors. A generalized method to

Figure 4. B and S measured from the top cross section of the packing channel (this work). D


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Industrial & Engineering Chemistry Research Olujic’s14 and Basden’s15 work (measured from the vertical cross section of the packing channel; Figure 5). The relationship between these two different definitions is discussed in the Appendix.

Table 6. Standard Error kL model kG model

5. DIMENSIONLESS kL AND kG MODELS Fundamental models utilize the dimensionless form of the velocity (Reynolds number, Re), the dimensionless form of liquid- or gas-phase physical properties (Schmidt number, Sc), and the dimensionless form of packing geometries (mixing number, Mi) as the independent variables. The dimensionless form of kL or kG (Sherwood number, Sh) is used as the dependent variable. Thus, the model can be written as

6. COMPARISON WITH LITERATURE AREA MODELS The mass-transfer area model developed in this work is compared with previous mass-transfer area models (Olujic et al.,14 Billet and Schultes,3 Bravo et al.,18 Rocha et al.,19 Hanley and Chen,20 and Linek et al.21). The effective area correlations are listed in the Appendix. Figure 8 shows a comparison between the area model developed in this work and the literature models. A packing (GT-OPTIMPAK 250Y) with a surface area of 250 m2/m3 and corrugation angle of 45° was used in the comparison as an example. GT-OPTIMPAK 250Y is a packing studied by the SPR at The University of Texas at Austin. These data were not used in the development of models. The results show that the models developed in this work can predict not only the masstransfer properties for packings within the databank of this work but also packings beyond the databank. The differences between the model developed in this work and the literature models are quite distinct. The differences are small for some recent literature models: 11% for Olujic et al.,14 13% for Linek et al.,21 36% for Hanley and Chen.20 The differences become large for models based on hydrocarbon systems or based mostly on random packing: 45% for Bravo et al.,18 59% for Billet and Schultes,3 and 73% for Rocha et al.19 The model developed in this work has better consistency with the experimental data than all literature models, showing its good treatment of aqueous systems. The model developed by Linek et al.21 shows a good match with the model developed in this work because it was based on a similar system (absorption of 1% CO2 in air with a 1 g·mol/ NaOH solution). The deviation is due to the larger gas-phase resistance. The Olujic model shows a good comparison in the absolute value. However, it does not predict the effect of the liquid superficial velocity on the mass-transfer area well with an exponent of 0.011, which is lower than the exponent predicted by all other models.

(13)

The conclusions of other researchers are used for the effect of the Schmidt number on the Sherwood number because the Schmidt number influence is not yet explored in this work. For the gas phase, Mehta and Sharma’s16 conclusion is used in this model, which is that ShG depends on ScG to the power of 0.5. For the liquid phase, Mangers and Ponter’s17 conclusion is used with a dependence of ShL on ScL to the power of 0.5. The dimensionless kL and kG models for structured packings are ShL = 21ReL 0.78Sc L 0.5 Mi1.11,

ShG = 14ReG

0.59

1.1

0.5

Mi ScG ,

kL = ShLaPDL

(14)

k G = ShGaPDG

(15)

Mi amounts to a transformation of the corrugation angle that gives values of the X, Y, and Z packings for 45°, 60°, and 70°, respectively. Sh, Re, and Sc are defined as Sh =

Re =

kleq D

=

ρuleq μ

k DaP

=

μ υ = Sc = ρD D

ρu μaP

Mi 1.11 ± 0.287 1.1 ± 0.162

packing geometry on mass-transfer coefficients. The dependence of ShL on the liquid rate is 0.78, an average of the dependence for each individual packing. This dependence is larger than that of previous models developed from penetration theory, which will be discussed in the model comparison section. The dependence of ShG on the gas rate is 0.59, which falls in the range of literature models (the dependence varies from 0.5 to 1). The kL and kG models developed in this work are compared with the literature models in section 7. The deviation between the experimental data and model value is 25% for ShL, while the deviation between the experimental data and model value is 12% for ShG, showing good consistency between the model developed in this work and the experimental data.

Figure 5. b and s measured from the vertical cross section of the packing channel.

ShL/G = CReL/G mSc L/G n Mi p

Re 0.78 ± 0.043 0.59 ± 0.031

(16)

(17) (18)

Table 6 shows the standard error for the regressed power of Re and Mi. Comparisons between the experimental data and values predicted by dimensionless kL and kG models are shown in Figures 6 and 7. The kL model shows good prediction for most packings, except for RSP200X and RSP250Y, which show larger values than those predicted. The mass-transfer models accurately predict the effects of the liquid or gas velocity and

7. COMPARISON WITH LITERATURE kL AND kG MODELS The models of the liquid-film and gas-film mass-transfer coefficient models (eqs 14 and 15) are compared with the literature models (Olujic et al.,14 Billet and Schultes,3 Bravo et al.,18 Rocha et al.,19 Hanley and Chen,20 and Linek et al.21). E


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Figure 6. Comparison between the experimental ShL and ShL predicted by the dimensionless model, eq 14.

Figure 7. Comparison between the experimental ShG and ShG predicted by the dimensionless model, eq 15.

The analytical kL equation (Pigford22) used in the WWC calculation (Dugas23) was also compared with the model developed in this work. The correlations are listed in the Appendix. Figures 9 and 10 show comparisons between the literature kLa and kGa models and models developed in this work. Because most literature models were developed from measured kLa and kGa values with a theoretical assumption of the area, the reasonable comparison is with the respective ka. In the kLa comparison, most literature models use the assumption of penetration theory4 with different expressions of equivalent liquid velocity (u) and characteristic length. The difference between the model developed in this work (absorption systems) and models from distillation systems (Olujic et al.,14 Bravo et al.,18 and Rocha et al.19) is from 30% to 40%. The difference becomes smaller (20−30%) when compared with the model developed from absorption data

(Linek et al.21 and WWC23) or models developed from distillation and absorption systems (Billet et al.3 and Hanley and Chen20). The difference between kLa values predicted by different models is smaller than what would be expected. The kL and ae models developed by the same author should be used together because the errors from the kL and ae models frequently cancel out. Another finding is that the liquid-rate dependence of the kL models developed from penetration theory (Bravo et al.,18 Rocha et al.,19 Billet and Schultes,3 Olujic et al.,14 and WWC23) is smaller than that of the kL models developed based on the experimental data (Linek et al.,21 Hanley and Chen,20 and Wang et al.8). Penetration theory assumes a 0.5 power of the liquid-rate dependence of kL (kL ∼ uL0.5). However, when penetration theory was applied, most authors used the effective liquid velocity (uLE) instead of uL. Equation 20a shows the effective liquid velocity form used by Bravo et al.,18 Rocha et F


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Figure 8. Comparison of the area models.

Figure 9. Comparison of the literature kLa and kLa model in this work calculated from eqs 14 and 5

al.,19 and Billet and Schultes,3 and eq 20b shows the effective liquid velocity form used by Olujic et al.14 u uLE = C L hL (20a) uLE = C

uL δL

Laso et al.25), the average liquid-rate dependence of kL is between 0.5 and 0.7, which means the previous kL models using penetration theory and the effective liquid velocity (uLE) underpredict the liquid-rate dependence. Most recent experimental data (Linek et al.,24 Laso et al.,25 and Wang et al.8) show that the liquid flow pattern in packings does not follow penetration theory completely, especially in the high liquid flow region where the liquid flow pattern changes from laminar flow to turbulent flow. In the kGa comparison, the model developed in this work is higher than literature models by 40−80%. One possible reason could be that all literature models have been developed from distillation systems where equilibrium is critical to establishing the driving force. Gas and liquid back-mixing and maldistribution may play a critical role in commercial distillation

(20b)

The effective liquid velocity uLE has the liquid hold-up term (hL) or liquid-film thickness term (δL) at the bottom and either hL or δL is a function of the liquid velocity uL. Thus, the actual liquid-rate dependence of these models using the effective liquid velocity is between 0.2 and 0.35, which is even smaller than the power predicted by penetration theory. From the experiments conducted in this work or the experiments conducted by other authors (Linek et al.24 and G


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Figure 10. Comparison of the literature kGa and kGa models in this work calculated from eqs 15 and 5

Figure 11. Comparison of the literature kL and kL models in this work from eq 14.

to underpredict the effective area for aqueous systems, which would result in higher values of kL and kG.

separations that is not observed with the NaOH/SO2 system used in this work. One of the contributions of this work is to provide a consistent method to measure ae, kL, and kG. Thus, kL and kG can be separated from measured ka values and directly measured ae values, instead of using assumed ae models to separate ka as most researchers did. Figures 11 and 12 show comparisons between the literature kL and kG models with models developed in this work. The literature kL and kG models are developed by dividing kLa and kGa correlations with proposed ae models from the same author. As discussed in the area model comparison section, the area models developed from hydrocarbon systems (Bravo et al.18 and Rocha et al.19) or based mostly on random packings (Billet and Schultes3) tend

8. CONCLUSIONS In this paper, three dimensionless mass-transfer models are developed. The models capture the effects of the operating conditions and packing geometries on mass transfer. Instead of vendor proprietary geometric data (packing channel base B and crimp height h), generalized packing information (surface area aP and corrugation angle θ) is used in the models. The database includes 11 structured packings with surface areas ranging from 125 to 500 m2/m3 and corrugation angles from 45 to 70° and three random packings from the Raschig Super-Ring family. The experimental systems use absorption/desorption from aqueous solvents with liquid physical properties close to those H


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Figure 12. Comparison of the literature kG and kG models in this work from eq 15. 0.75 ⎛ ⎞−0.2 ⎛ uL 2ρL dh ⎞ ⎛ uL 2 ⎞−0.45 ae −0.5 uLdh ⎟⎟ ⎜ ⎟ = 1.5(aPdh) ⎜ ⎟ ⎜⎜ aP ⎝ νL ⎠ ⎝ σ ⎠ ⎝ gdh ⎠

of pure water. Details regarding the experimental work can be found in refs 5−9. The three dimensionless mass-transfer correlations developed in this work are ⎡ ρ ⎛u ⎞ ae ⎛ ⎞ = 1.41⎢⎜ L ⎟g 1/3⎜ L ⎟ ⎢⎣⎝ σ ⎠ aP ⎝ aP ⎠ ShL = 21ReL

0.78

0.5

Bravo et al. (1985):

⎥ ⎥⎦

1.11

⎛ σ 0.5 ⎞ ae = 0.498⎜ 0.4 ⎟(CaLReG)0.392 aP ⎝Z ⎠

,

kL = ShLaPDL

ShG = 14ReG 0.59 Mi1.1ScG 0.5 ,

k G = ShGaPDG

Mi =

Sc L Mi

(A2)

4/3⎤0.116

Rocha et al. (1996): ⎛ ρL ⎞0.15 ae 29.12uL 0.4νL 0.2S 0.359 = FSE ⎜ ⎟ aP (1 − 0.93 cos γ )(sin α)0.3 ε 0.6 ⎝ σg ⎠

M 3 sin θ cos θ = 32 2 aP3

ae 1−Ω = aP 1 + A /uLS B

(A5)

Hanley and Chen (2011): ⎛ ρ ⎞−0.033 am 0.145 0.2 −0.153 −0.2 = 0.539ReV ReL WeL FrL ⎜⎜ V ⎟⎟ ad ⎝ ρL ⎠ ⎛ μ ⎞0.090⎛ cos(θ) ⎞4.078 × ⎜⎜ V ⎟⎟ ⎜ ⎟ ⎝ cos(π /4) ⎠ ⎝ μL ⎠

Linek et al. (2011) ae = 1.343uL 0.104 aP

APPENDIX

(A6)

(A7)

Literature kL and kG Correlations

Billet and Schultes (1993):

Literature Effective Area Correlations

Onda et al. (1968): ⎡ ⎤ ⎛ σ ⎞0.75 ae = 1 − exp⎢ − 1.45⎜ C ⎟ ReL 0.1FrL−0.05WeL 0.2 ⎥ ⎢⎣ ⎥⎦ aP ⎝ σL ⎠

(A4)

Olujic et al. (1999):

The effective area model uses the liquid superficial velocity over the packing total area (uL/aP) as the liquid flow rate per wetted perimeter. Thus, the applied range of this area model is extended to include hybrid and random packings. The average deviation is 10.5% for the entire data bank, with a total of 14 packings including hybrid and random packings. The liquid-film and gas-film mass-transfer coefficient models are built in dimensionless form. The mass-transfer coefficients are dependent on the liquid or gas superficial velocity (uL or uG), packing corrugation angle (θ), and packing surface area (aP). Through combination, two dimensionless numbers (Re and Mi) are used to represent these three variables. The average deviation is 25% for the kL model and 12% for the kG model.

■

(A3)

(A1)

Billet and Schultes (1993): I

kG =

2 π

DG

uG (ε − hL)lπ

(A8)

kL =

2 π

DL

uL hLlπ

(A9) DOI: 10.1021/acs.iecr.5b04635 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

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Industrial & Engineering Chemistry Research Bravo et al. (1985): k Gdeq DG

⎡ deqρ (uG,eff + uL,eff ) ⎤0.77 ⎛ μ ⎞0.33 G ⎥ ⎜⎜ G ⎟⎟ = 0.0328⎢ ⎢⎣ ⎥⎦ ⎝ ρG DG ⎠ μG (A10)

DL uL,eff

kL = 2

πS

(A11)

Olujic et al. (1999): kG =

k G,lam 2 + k G,tur 2

(A12)

ShG,lam = 0.664ScG1/3 ReGrv

ShG,tur

Figure A1. b and s measured from the vertical cross section of the packing channel.

dhG lG,pe

which is the cross-sectional area of the channel multiplied by the length of the channel. The packing specific area is now the ratio of the packing area and packing volume: A packing 4s aP = = Vpacking bh (A24)

(A13)

⎡ ⎛ d ⎞2/3⎤ ⎢ ⎜ hG ⎟ ⎥ = ⎢1 + ⎜ l ⎟ ⎥ ζGLφ 2/3 ⎝ G,pe ⎠ ⎦ 1 + 12.7 (ScG − 1) ⎣ 8 ReGrvScG

ζGLφ 8

Figure A2 shows the structured packing with packing channel angle α. b, h, and s have the following relationships:

(A14)

DL uLe π 0.9dhG

kL = 2

dhG =

(A15) (bh − 2δs)2 bh

⎡ ⎢⎣

2

( bh −2h2δs ) + (

0.5 bh − 2δs 2 ⎤ ⎥ b ⎦

)

+

bh − 2δs 2h

⎛ 3μ uLS ⎞1/3 L ⎟⎟ δ = ⎜⎜ ⎝ ρL ga sin α ⎠

⎛α⎞ b = 2s sin⎜ ⎟ ⎝2⎠

(A25)

⎛α⎞ h = s cos⎜ ⎟ ⎝2⎠

(A26)

(A16)

(A17)

Hanley and Chen (2011): ⎛c D ⎞ kx = 0.33ReLSc L1/3⎜ L L ⎟ ⎝ de ⎠

(A18)

⎛ c D ⎞⎛ cos(θ) ⎞ k y = 0.0084ReV Sc V1/3⎜ V V ⎟⎜ ⎟ ⎝ de ⎠⎝ cos(π /4) ⎠

Figure A2. Packing channel with channel angle α.

−7.15

When eqs A24−A26 are combined, b, s, and h can be expressed by 2 s= α α aP sin 2 cos 2 (A27)

(A19)

Linek et al. (2011):

kLa = 0.562uL 0.668

() ()

(A20)

Dugas (2009): ⎛ 31/321/2 ⎞⎛ Q1/3h1/2W 2/3 ⎞⎛ gρ ⎞1/6 ⎟⎜ ⎟ DCO21/2 kL° = ⎜ 1/2 ⎟⎜ A ⎠⎝ ⎝ π ⎠⎝ μ ⎠

b= (A21)

h=

( α2 )

(A29)

2

Figure A3 shows the relationship between the top and vertical cross sections. The top section channel base B, vertical section base b, and ridge of the packing channel D form a rightangle triangle in the longitudinal section (Figure A3). Thus, channel base B can be related with b:

(A22)

where Lchannel is some arbitrary channel length. The volume of this triangular channel is Vpacking

(A28)

aP sin

Figure A1 shows the vertical cross section of the packing channel. For a packing channel with a channel length L, the surface area is given by

⎛ bh ⎞ = ⎜ ⎟Lchannel ⎝2⎠

( α2 )

aP cos

Calculating the Mixing Point Density from aP and θ

A packing = 2sLchannel

4

B=

b sin θ

(A30)

When eqs A28 and A30 are combined, B can be expressed by aP, θ, and α:

(A23) J


Article


ae = the effective mass transfer area, m2/m3 aP = total surface area, m2/m3 B = packing channel base (measured from the top cross section), m b = packing channel base (measured from the vertical cross section), m C = constant in eqs 20a and 20b D = diffusivity, m2/s g = gravity constant, 9.8 m/s2 H = packing crimp height (measured from the top cross section), m h = packing crimp height (measured from the vertical cross section), m hL = fractional liquid hold-up, m3/m3 K = overall mass-transfer coefficient, m/s kL = physical liquid-film mass-transfer coefficient, m/s kG = gas-film mass-transfer coefficient, m/s kg′ = liquid-film mass-transfer coefficient including chemical reactions, m/s leq = packing equivalent length (1/aP), m LP = wetted perimeter, m M = mixing point density, pts/m3 Npyramid = number of square pyramids formed by crossed corrugated metal sheets in structured packing per cubic meter, m−3 S = packing channel side (measured from the top cross section), m s = packing channel side (measured from the vertical cross section), m uG = superficial gas velocity, m/s uL = superficial liquid velocity, m/s

Figure A3. Longitudinal section of structured packing channel (I).

4

B= aP cos

( α2 ) sin θ

(A31)

Finally, the mixing point density can be calculated from the packing surface area aP, corrugation angle θ, and packing channel angle α: M=

( α2 ) cos2( α2 )

3aP3 sin θ cos θ sin 6 = BhB tan θ 16

(A32)

Mi =

α 2

()

3 sin θ cos θ sin M = 3 16 aP

α cos 2 2

() (A33)

Normally, the packing channel angle α is made to 90°. Thus, eq A32 can be simplified to eq A34:

Dimensionless Groups

Ar = Archimedes number (gρ2/μ2aP3) FrL = liquid-phase Froude number (uL2/gδL) Mi = mixing number (M/aP3) Re = Reynolds number (ρu/aPμ) Sc = Schmidt number (μ/ρd) Sh = Sherwood number (k/aPD) WeL = liquid-phase Weber number (ρLuL2δL/σ)

3

M=

3aP sin θ cos θ 32 2

(A34)

The dimensionless form of the mixing point density, mixing number Mi, can be expressed by Mi =

■

M 3 sin θ cos θ = 3 32 2 aP

(A35)

Greek Letters

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

■

The authors declare the following competing financial interest(s): One author of this publication consults for Southern Company and for Neumann Systems Group on the development of amine scrubbing technology. The terms of this arrangement have been reviewed and approved by The University of Texas at Austin in accordance with its policy on objectivity in research. The authors have financial interests in intellectual property owned by The University of Texas that includes ideas reported in this paper..

δ = liquid film thickness, m μ = viscosity, kg/(s m) θ = packing corrugation angle, deg ρ = density, kg/m3 σ = surface tension, N/m

REFERENCES

(1) Onda, K.; Takeuchi, H.; Okumoto, Y. Mass transfer coefficients between gas and liquid phases in packed columns. J. Chem. Eng. Jpn. 1968, 1, 56. (2) Rocha, J. A.; Bravo, J. L.; Fair, J. R. Distillation Columns Containing Structured Packings: A Comprehensive Model for Their Performance. 2. Mass-Transfer Models. Ind. Eng. Chem. Res. 1996, 35, 1660. (3) Billet, R.; Schultes, M. Predicting mass transfer in packed column. Chem. Eng. Technol. 1993, 16, 1. (4) Higbie, R. The rate of absorption of a pure gas into a still liquid during short periods of exposure. Trans. Am. Inst. Chem. Eng. 1935, 31, 365. (5) Wang, C.; Perry, M.; Rochelle, G. T.; Seibert, A. F. Packing characterization: Mass Transfer Properties. Energy Procedia 2012, 23, 23.

■

ACKNOWLEDGMENTS The authors gratefully acknowledge financial support from the Texas Carbon Management Program and the Process Science and Technology Center of The University of Texas at Austin.

■

NOMENCLATURE A = column cross-sectional area, m2 K


Article

Industrial & Engineering Chemistry Research (6) Wang, C.; Perry, M.; Seibert, F.; Rochelle, G. T. Characterization of novel structured packing for CO2 capture. Energy Procedia 2013, 37, 2145. (7) Wang, C.; Perry, M.; Seibert, F.; Rochelle, G. Packing characterization for post combustion CO2 capture: mass transfer model development. Energy Procedia 2014, 63, 1727. (8) Wang, C.; Seibert, F.; Rochelle, G. T. Packing characterization: Absorber economic analysis. Int. J. Greenhouse Gas Control 2015, 42, 124. (9) Wang, C. Mass Transfer Coefficients and Effective Area of Packing. Ph.D. Thesis, The University of Texas at Austin, Austin, TX, 2015. (10) Wilson, I. Gas−Liquid Contact Area of Random and Structured Packing. M.S. Thesis, The University of Texas at Austin, Austin, TX, 2004. (11) Tsai, R. E.; Schultheiss, P.; Kettner, A.; Lewis, J. C.; Seibert, A. F.; Eldridge, R. B.; Rochelle, G. T. Influence of Surface Tension on Effective Packing Area. Ind. Eng. Chem. Res. 2008, 47, 1253. (12) Tsai, R.cE.; Seibert, A.cF.; Eldridge, R.cB.; Rochelle, G.cT. A Dimensionless Model for Predicting the Mass-Transfer Area of Structured Packing. AIChE J. 2011, 57, 1173. (13) Tsai, R. E. Mass Transfer Area of Structured Packing. Ph.D. Thesis, The University of Texas at Austin, Austin, TX, 2010. (14) Olujic, Z.; Kamerbeek, A. B.; de Graauw, J. A Corrugation Geometry Based Model for Efficiency of Structured Distillation Packing. Chem. Eng. Process. 1999, 38, 683. (15) Basden, M. Characterization of Structured Packing via Computational Fluid Dynamics. Ph.D. Thesis, The University of Texas at Austin, Austin, TX, 2014. (16) Mehta, V. D.; Sharma, M. M. Effect of diffusivity on gas-side mass transfer coefficient. Chem. Eng. Sci. 1966, 21, 361. (17) Mangers, R. J.; Ponter, A. B. Effect of viscosity on liquid film resistance to mass transfer in a packed column. Ind. Eng. Chem. Process Des. Dev. 1980, 19, 530. (18) Bravo, J. L.; Rocha, J. A.; Fair, J. R. A comprehensive model for the performance of columns containing structured packings. ICHEME Symp. Ser. 1992, 128, 489. (19) Rocha, J. A.; Bravo, J. L.; Fair, J. R. Distillation Columns Containing Structured Packings: A Comprehensive Model for Their Performance. 2. Mass-Transfer Models. Ind. Eng. Chem. Res. 1996, 35, 1660. (20) Hanley, B.; Chen, C. New Mass-Transfer Correlations for Packed Towers. AIChE J. 2012, 58, 132. (21) Valenz, L.; Rejl, F. J.; Sima, J. L.; Linek, V. Absorption MassTransfer Characteristics of Mellapak Packing Series. Ind. Eng. Chem. Res. 2011, 50, 12134. (22) Pigford, R. L. Counter-Diffusion in a Wetted Wall Column. Ph.D. Thesis, The University of Texas at Austin, Austin, TX, 1941. (23) Dugas, R. E. Carbon Dioxide Absorption, Desorption and Diffusion in Aqueous Piperazine and Monoethanolamine. Ph.D. Thesis, The University of Texas at Austin, Austin, TX, 2009. (24) Linek, V.; Moucha, T.; Prokopova, E.; Rejl, J. F. Simultaneous Determination of Vapour and Liquid-Side Volumetric Mass Transfer Coefficients in Distillation Column. Chem. Eng. Res. Des. 2005, 83, 979. (25) Laso, M.; de Brito, M. H.; Bomio, P.; von Stockar, U. Liquidside mass transfer characteristics of a structure packing. Chem. Eng. J. Biochem. Eng. J. 1995, 58, 251.

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