Modeling the Effect of Relative Humidity on Adsorption Dynamics of

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Modeling the Effect of Relative Humidity on Adsorption Dynamics of Volatile Organic Compound (VOC) onto Activated Carbon Imranul Laskar, Zaher Hashisho, John H. Phillips, James E Anderson, and Mark Nichols Environ. Sci. Technol., Just Accepted Manuscript • DOI: 10.1021/acs.est.8b05664 • Publication Date (Web): 07 Feb 2019 Downloaded from http://pubs.acs.org on February 8, 2019

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Environmental Science & Technology

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Modeling the Effect of Relative Humidity on Adsorption Dynamics of Volatile Organic Compound (VOC) onto Activated Carbon

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Imranul I. Laskar1, Zaher Hashisho1, John H. Phillips2, James E. Anderson3, and Mark Nichols3

5 6

1University

7

2Ford

8 9

3Ford

of Alberta, Department of Civil and Environmental Engineering, Edmonton, AB T6G

2R3 Motor Company, Environmental Quality Office, Dearborn, Michigan 48126, United States

Motor Company, Research and Advanced Engineering, Dearborn, Michigan 48121, United States

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ABSTRACT

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A two-dimensional heterogeneous mathematical model was developed and validated to study

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the effect of relative humidity on volatile organic compound (VOC) adsorption onto activated

14

carbon. The dynamic adsorption model consists of the macroscopic mass, momentum, and

15

energy conservation equations, and includes a multicomponent adsorption isotherm to predict

16

the competitive adsorption equilibria between VOC and water vapor, which is described by an

17

extended Manes method. Experimental verifications show that the model predicted the

18

breakthrough profiles during competitive adsorption of the studied VOCs (2-propanol, acetone,

19

n-butanol, toluene, 1,2,4- trimethylbenzene) at relative humidity range 0-95% with an overall

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mean relative absolute error (MRAE) of 11.8% for dry (0% RH) conditions and 17.2% for humid

21

(55% and 95% RH) conditions, and normalized root-mean-square error (NRMSE) of 5.5% and

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8.4% for dry and humid conditions, respectively. Sensitivity analysis was also conducted to test

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the robustness of the model in accounting for the impact of relative humidity on VOC

24

adsorption by varying the adsorption temperature. A good agreement was observed between

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the experimental and simulated results with an overall MRAE of 12.4% and 7.1% for the

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breakthrough profiles and adsorption capacity, respectively. The model can be used to quantify

27

the impact of carrier gas relative humidity during adsorption of contaminants from gas streams,

28

which is useful when optimizing adsorber design and operating conditions.

29 30

INTRODUCTION

31

Activated carbon adsorption is a widely used method to capture volatile organic compound

32

(VOC) emissions from industrial gas streams.1-4 In such streams, water vapor tends to be

33

ubiquitous, and may compete with VOCs during the separation/ purification/ recovery

34

process.5-8 Typical relative humidity (RH) values ranged from 0 to 95%.3, 8-10 For instance, in the

35

automotive manufacturing process, painting operations take place in spraybooths, which

36

include a spraying section to apply paint to the vehicle and a water scrubber system that

37

captures paint overspray.2 However, contact between the VOC-laden air stream and the water

38

scrubber system increases the humidity of the air stream.

39

The presence of water vapor in an adsorption stream has a detrimental effect on the

40

performance of adsorbents such as activated carbon.5 This is because water vapor can compete

41

with VOCs for adsorption onto the carbon, reducing the adsorbent’s capacity for VOCs

42

adsorption, particularly at high RH.11-15 For this reason, some adsorption-based VOC capture

43

systems utilize air preheating in order to reduce RH. The majority of experimental work from

44

the literature observed similar effects.4-6, 12-20

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Theoretical studies on the effect of water vapor on VOCs’ adsorption are limited and therefore,

46

mathematical models describing the competitive adsorption equilibria and dynamics between


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water vapor and VOCs are scarce.5, 6, 8, 12 In addition, the majority of adsorption dynamics models

48

are one-dimensional and focus only on the axial variation of the adsorption process

49

parameters. They fail to analyse important factors such as radial dispersion and channeling

50

effects in an adsorption column and therefore have limitations in comprehensively simulating a

51

fixed bed adsorption process.16-20 In recent years, a few studies have carried out two-

52

dimensional (axial and radial) mathematical modeling to predict the transport processes (mass,

53

momentum, and energy) in a fixed bed adsorber, with good agreement between experimental

54

and modeled results (overall mean relative absolute error = 6%).21-24 However, these models did

55

not consider the impact of water vapor on VOC adsorption.

56

This paper focuses on the development and validation of a mathematical model to predict the

57

effects of carrier gas relative humidity (RH) on VOC adsorption dynamics. For this purpose, a

58

comprehensive mathematical model encompassing competitive adsorption kinetics

59

(macroscopic mass, energy, and momentum conservation equations) was developed and

60

coupled to a water vapor-VOC adsorption isotherm. The model was validated and used to

61

evaluate the effect of temperature on the adsorber performance.

62 63

MODEL DEVELOPMENT AND VALIDATION METHOD

64

Variable and Parameters Definition

65

The model parameters and variables are defined in Table 1.

66

Physical Model

67

The simulated bench-scale adsorber consisted of a cylindrical stainless-steel tube with a 7.87-

68

mm inner radius (R), containing 13.3 g of beaded activated carbon (BAC) particles (mean



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diameter = 0.75 mm) resulting in a 115 mm-long fixed bed (L) of BAC. The BAC is mainly

70

microporous (micropore volume = 0.51 cm3/g, total pore volume = 0.57 cm3/g) and has a

71

Brunauer-Emmett-Teller (BET) area of 1390 m2/g.22 For dry (0% RH) and humid (55% and 95%

72

RH) conditions, a 10-SLPM air stream at 298 K (25 °C) containing 1,000 ppmv of VOC entered

73

from the top of the fixed-bed adsorber tube at a superficial velocity (us) of 0.856 m/s and exited

74

from the bottom of the tube. For the humid conditions, a dry air stream was humidified and

75

then mixed with the VOC, to maintain the RH of the inlet stream at 55% or 95%. The VOCs

76

tested in this study are typically present in automotive painting operations and were selected

77

to provide a range of water miscibility and polarity; with toluene, n-butanol, and 1,2,4-

78

trimethylbenzene (TMB) being the non-polar VOCs and acetone and 2-propanol the polar ones.

79

The adsorbent bed effluent VOC concentration and relative humidity were measured using a

80

flame ionization detector (FID) (Baseline Mocon, Series 9000) and a RH sensor (Vaisala

81

HMT330), respectively. To evaluate the effect of temperature on adsorption during dry and

82

humid conditions, the temperature of the adsorbent bed was raised to 305 K (32 °C) using

83

heating and insulation tapes (instead of the baseline 298 K), while keeping the dry/humid inlet

84

air stream at room temperature of 298 K. The bed temperature was measured using a 0.9 mm

85

type K thermocouple (Omega) that was inserted at the center of the tube and at a distance of

86

75 mm from the bed bottom. More details on the setup of the adsorber are provided in the

87

supporting information (SI).

88

Major assumptions used for the proposed model development include negligible variation of

89

flow properties in the angular direction of the cylindrical tube, negligible adsorption of the

90

carrier gas (air), ideal gas behaviour, and axisymmetric flow conditions. These assumptions


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simplified the model geometry representing the adsorber into a two-dimensional (2D)

92

axisymmetric geometry (Figure 1), which reduces the overall computation cost.

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94 95

Figure 1. Simplification of the simulated adsorption unit into a 2D axisymmetric geometry.


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Table 1. Model variables and parameters Symbol description main variables 𝑪𝒊 gas-phase concentration 𝑪𝒔,𝒊 adsorbed-phase concentration 𝑪𝒔𝒆,𝒊 equilibrium adsorbedphase concentration body force 𝑭 gas pressure 𝑷 radial distance 𝒓 adsorption time 𝒕 fixed-bed temperature 𝑻 gas velocity vector 𝒖 resultant gas velocity |𝒖| axial distance 𝒛 Input variables 𝑩𝒑,𝒊 pore Biot number 𝑪𝒐, 𝒊

𝑪𝒔𝒐, 𝒊

𝑪𝒆𝒇𝒇 𝑪𝑭

𝑪𝒑,𝒇 𝑪𝒑,𝒑 𝑫𝒃 𝒅𝒑

inlet gas concentration of the ith component

value/formula

𝜌𝑏(𝑞𝑣 𝑜𝑟 𝑞𝑚𝑣); 𝜌𝑏(𝑞w 𝑜𝑟 𝑞𝑚w) 𝑔 ∗ 𝜌𝑓

Table 2; (𝑘𝑒𝑥,𝑖𝑑𝑝) (2𝐷𝑒𝑓𝑓,𝑖) 1000 (VOC) 0, 55 or 95 (relative humidity) 0, 18000 or 31000 (relative humidity) 𝜌𝑏(𝑞𝑣 𝑜𝑟 𝑞𝑚𝑣); 𝜌𝑏(𝑞w 𝑜𝑟 𝑞𝑚w)

units

reference(s)

kg/m3

eq. (1)

kg/m3

eq. (6)

kg/m3

eq. (7) & (8)

N/m3 kPa m s K m/s m/s m

25, 26

1

27

ppmv %

boundary condition

eq. (5) N/A N/A eq. (20) eq. (16) eq. (18) N/A

ppmv

adsorbed phase concentration of the ith component in equilibrium with its inlet gas phase concentration effective volumetric heat capacity empirical correction 0.55(1 ― 5.5(𝑑𝑝 𝐷𝑏)) factor for Forchheimer’s drag coefficient calculation heat capacity of air

kg/m3

boundary condition

J/(m3.K)

22

1

25

J/(kg.K)

heat capacity of BAC tube inner diameter average BAC particle diameter

J/(kg.K) m m

COMSOL material database

706.7 0.01575 7.5 * 10-4


22

measured 28


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8

𝑫𝒂𝒙,𝒊 𝑫𝑨𝑩,𝒊 𝑫𝒆𝒇𝒇,𝒊 𝑫𝒊 𝑫𝒑,𝒊 𝑫𝒓,𝒊 𝑫𝒔,𝒊 𝑫𝒔𝒐

𝒈 𝒉 ∆𝑯𝒗𝒂𝒑,𝒊 ∆𝑯𝒂𝒅,𝒊 ∆𝑯𝒔𝒐𝒍,𝒊

axial dispersion coefficient molecular diffusivity effective diffusion coefficient symmetric mass dispersion tensor pore diffusion coefficient radial dispersion coefficient surface diffusion coefficient surface diffusion constant acceleration of gravity adsorber wall heat transfer coefficient adsorbate heat of vaporization heat of adsorption heat of dissolution

ionization potential shear stress

𝒌𝒂𝒙

axial thermal conductivity stagnant bed thermal conductivity effective thermal conductivity tensor external mass transfer coefficient air thermal conductivity

𝒌𝒆𝒇𝒇 𝒌𝒆𝒙, 𝒊 𝒌𝒇 𝒌𝒊𝒏, 𝒊 𝒌𝒐𝒗, 𝒊

eq. (4)

m2/s m2/s

eq. (5) eq. (13)

m2/s

eq. (2)

m2/s

eq. (15)

m2/s

eq. (3)

Table 2

m2/s

eq. (14)

1.1 * 10-8

m2/s

29

Table 2

𝑰𝑷𝒊 𝑱

𝒌𝒃

m2/s

9.81 m2/s 𝑝 𝐷𝑏)30 1 ― (2𝑑.K) (2.4 𝑑𝑝)𝑘𝑝 + 0.054(𝑘𝑓 𝑑𝑝)(W/(m )𝑅𝑒𝑝𝑃𝑟1/3 Table S1 17.42 (acetone-water system) 210.35 (2-propanolwater system) Table S1

Table 2

internal mass transfer coefficient overall mass transfer coefficient


kJ/mol

eq. (22)

kJ/mol kJ/mol

eq. (22)

kJ/mol

32, 33

eV N/m2

eq. (22), 34 eq. (17)

W/(m.K)

eq. (27)

W/(m.K)

eq. (28)

W/(m.K)

eq. (25)

m/s

eq. (11)

W/(m.K) 1/s

COMSOL material database eq. (12)

1/s

eq. (10)

31

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9

0.17

𝒖𝒔

BAC particle thermal conductivity radial thermal conductivity tube bed length molecular weight of adsorbate molecular weight of air mass of BAC in tube molecular Peclet number for heat transfer Prandtl number equilibrium adsorption capacity for watermiscible VOC in a mixture gas flow rate equilibrium adsorption capacity for water vapor in a mixture equilibrium adsorption capacity for VOC in a mixture equilibrium adsorption capacity for water vapor in a mixture containing water-miscible VOC particle Reynolds number average BAC pore radius momentum sink Schmidt number mass sink of the gas phase heat source adsorbate boiling point adsorber wall temperature superficial velocity

𝑽𝒑𝒐𝒓𝒆

BAC pore volume

𝒌𝒑 𝒌𝒓 𝑳 𝑴𝑨,𝒊 𝑴𝑩 𝒎𝑩𝑨𝑪 𝑷𝒆𝒐

𝑷𝒓 𝒒𝒎𝒗

𝑸 𝒒𝒘

𝒒𝒗

𝒒𝒎𝒘

𝑹𝒆𝒑 𝒓𝒑𝒐𝒓𝒆 𝑺 𝑺𝒄𝒊 𝑺𝒎,𝒊 𝑺𝒉,𝒊 𝑻𝒃,𝒊 𝑻𝒘

W/(m.K)

35

W/(m.K)

eq. (26)

0.115 Table S1

m g/mol

measured eq. (5), 34

29.0 13.3 (𝑢𝑠𝜌𝑓𝐶𝑝,𝑓𝑑𝑝) 𝑘𝑓

g/mol g 1

(𝜇𝑓𝐶𝑝,𝑓) 𝑘𝑓

1 kg/kg

eq. (7)

SLPM kg/kg

measured eq. (8)

kg/kg

eq. (7)

kg/kg

eq. (8)

(𝜌𝑓𝑢𝑠𝑑𝑝) (µ𝑓(1 ― 𝜀𝑝))

1

37

1.1

nm N/m3 1 kg/(m3.s)

21

Table S1 298

J/(m3.s) K K

0.856

m/s

0.57

cm3/g

eq. (21) eq. (14), 34 boundary condition boundary condition

10.00

µ𝑓 (ρ𝑓𝐷𝐴𝐵,𝑖)


34

measured 36

30

eq. (18) 38

eq. (6)

22


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10

molar volume of VOC adsorbed molar volume of water vapor adsorbed average micropore width of BAC polarizability empirical correction factor for mass diffusion terms Forchheimer’s drag coefficient bulk bed porosity

𝑀𝐴,1/𝜌𝑣

cm3/mol

𝑀𝐴,2/𝜌𝑤

cm3/mol

1.02

nm

Table S1 20

cm3 x 10-24 eq. (22), 34 38 1

𝜌𝑓(𝐶𝐹 𝜅)

kg/m4

25

)

25

𝑉𝑝𝑜𝑟𝑒𝜌𝑝

𝜿

particle porosity bed porosity as a function of radial distance from the center bed permeability

(𝜀𝑟3𝑑𝑝2) (150(1 ― 𝜀𝑟)2)

m2

25

µ𝒇

air viscosity

temperature dependent

Pa.s

𝝆𝒃 𝝆𝒇

bulk bed density air density

595 temperature dependent

kg/m3 kg/m3

𝝆𝒑 𝝈𝒊 𝝉𝒑 𝝊𝑨,𝒊

BAC particle density surface tension BAC particle tortuosity atomic diffusion volume of adsorbate atomic diffusion volume of air surface to pore diffusion flux ratio

𝜌𝑏 (1 ― 𝜀 ) 𝑏 1 𝜀 0.5 𝑝 Table S1

kg/m3 mN/m 1 1

COMSOL material database measured COMSOL material database

20.1

1

40

Table 2;

1

41

Table 2;

1

𝒗𝒗 𝒗𝒘 𝒘𝒎𝒊𝒄 α𝒊 𝜶𝟎

𝜷𝒇 𝜺𝒃 𝜺𝒑 𝜺𝒓

𝝊𝑩 𝝋𝒊

0.379 +

(0.078 ((𝐷

𝜀𝑏(1 + ((1 ― 𝜀𝑏)

𝑏

𝑑𝑝) ― 1.8)1

22 1 1 ∗ (𝑅 ― 𝑟)) 25 𝜀𝑏) ∗ 𝑒𝑥𝑝(( ―6 𝑑𝑝)

(𝜌𝑏𝐷𝑠,𝑖𝑉𝑜𝑣, 𝑚𝑎𝑥𝜌𝑣) (𝐷𝑒𝑓𝑓,𝑖𝐶𝑜,𝑖)

(𝜌𝑏𝐷𝑠,𝑖𝑉𝑜𝑤, 𝑚𝑎𝑥𝜌𝑤) (𝐷𝑒𝑓𝑓,𝑖𝐶𝑜,𝑖)

Indices 𝒊 N/A

22

component 1 (VOC); component 2 (water vapor) not applicable


)

22

eq. (22), 34 39

eq. (5), 34

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97

Governing Transport Phenomena

98

The model developed here is an extension of the one developed by Tefera et al. for adsorption

99

of single and mixtures of VOCs from a dry stream in a fixed-bed adsorber.21, 22

100

The model accounts for adsorbate mass balance in the gas phase and adsorbed phase as well as

101

the heat and momentum balance across the fixed bed adsorber. These transport phenomena

102

are described by partial differential equations (PDEs), ordinary differential equations, and

103

algebraic equations, as depicted in the following subsections.

104 105

Gas-Phase Mass Balance

106

Adsorbate mass transport in the gas phase of a fixed-bed adsorber is governed by dispersion

107

and convection, and is described as: 42, 43

108

―∇(𝐷𝑖 ∗ ∇𝐶𝑖) + (𝑢 ∗ ∇𝐶𝑖) +

109

where 𝐷𝑖 is the symmetric mass dispersion tensor:

110

𝐷𝑟,𝑖 0 𝐷𝑖 = 0 𝐷 𝑎𝑥,𝑖

111

The radial (𝐷𝑟,𝑖) and axial (𝐷𝑎𝑥,𝑖) dispersion coefficients are described in equations (3) and (4)

112

respectively.38, 44

113

𝐷𝑟,𝑖 = 𝛼0 +

114

𝐷𝑎𝑥,𝑖 = 𝛼0 +

115

where 𝐷𝐴𝐵,𝑖 is the molecular diffusivity of the ith component described as:40

|

(

(

𝐷𝐴𝐵,𝑖 =

∂𝑡

+

( )∗𝑆 1 ― 𝜀𝑝 𝜀𝑝

=0 𝑚,𝑖

(1)

|

(2)

𝑆𝑐𝑖𝑅𝑒𝑝 𝐷𝐴𝐵,𝑖

)

8

(3)

𝜀𝑏

𝑆𝑐𝑖𝑅𝑒𝑝 𝐷𝐴𝐵,𝑖

)

2

(

𝑃((∑𝜐)𝐴,𝑖

0.33

(4)

𝜀𝑏

0.0101325 ∗ 10 ―3𝑇1.75

116

∂𝐶𝑖

― (∑𝜐)𝐵

𝑀𝐴,𝑖 + 𝑀𝐵 𝑀𝐴,𝑖𝑀𝐵

)

(5)

0.33 2

)



Page 12 of 38

12

117

The mass sink (𝑆𝑚,𝑖) of the gas phase is represented by the linear driving force (LDF) model

118

given as:45

119

𝑆𝑚,𝑖 = 𝑘𝑜𝑣,𝑖(𝐶𝑠𝑒,𝑖 ― 𝐶𝑠,𝑖)

120

The LDF model has similar accuracy, but is less time-consuming and complex than other

121

computationally demanding models that rely on individual particle mass transport.45

122

𝐶𝑠𝑒,𝑖, the equilibrium adsorbed-phase concentration, is obtained from the multicomponent

123

competitive adsorption isotherm (equations (7) and (8)).

124

𝐶𝑠𝑒,1 = 𝜌𝑏(𝑞𝑣 𝑜𝑟 𝑞𝑚𝑣)

(7)

125

𝐶𝑠𝑒,2 = 𝜌𝑏(𝑞𝑤 𝑜𝑟 𝑞𝑚𝑤)

(8)

126

Here, the LDF-based gas-phase mass sink acts as the source for the adsorbed phase.45

127 128

VOC-Water Vapor Multicomponent Adsorption Isotherm Formulation

129

An accurate prediction of the multicomponent adsorption isotherms serves as a good basis to

130

describe the competitive adsorption dynamics between water vapor and VOC.43 A

131

thermodynamically consistent, potential theory-based Manes method was applied to predict

132

the competitive multicomponent adsorption equilibrium between VOC and water vapor. The

133

method was originally developed for water-immiscible organics, and in this study it was

134

extended to polar VOCs by introducing a Raoult’s law-like equation. The Manes model (i.e. the

135

equations in the Manes method) was numerically solved using MATLAB. It requires only the

136

single-component adsorption isotherms of the studied VOCs (2-propanol, acetone, n-butanol,

137

toluene, 1,2,4- trimethylbenzene (TMB)) and water vapor as inputs. The modified Dubinin-

138

Radushkevich (MDR) and the Qi-Hay-Rood (QHR) equations were selected to represent the

139

single-component adsorption isotherm of VOC and water vapor, respectively. The equilibria

(6)


Page 13 of 38


13

140

predictions using this method were encouraging and therefore, were applied here to model

141

competitive adsorption kinetics between VOC and water vapor. Details on the equilibria

142

formulation and the extended Manes method are provided elsewhere.26

143 144

Adsorbed-Phase Mass Balance

145

The diffusive adsorbate transport in the adsorbed phase is characterized using the LDF model as

146

mentioned above, and is given as:42, 45

147

∂𝐶𝑠,𝑖 ∂𝑡

(9)

= 𝑘𝑜𝑣,𝑖(𝐶𝑠𝑒,𝑖 ― 𝐶𝑠,𝑖) = 𝑆𝑚,𝑖

148

The LDF overall mass transfer coefficient (𝑘𝑜𝑣,𝑖) considers both the internal (1/𝑘𝑖𝑛,𝑖) and

149

external (1/𝑘𝑒𝑥,𝑖) mass transfer resistances described as:38, 46

150

𝑘𝑜𝑣,𝑖

151

where

152

𝑘𝑒𝑥,𝑖 =

153

𝑘𝑖𝑛,𝑖 =

154

The internal mass transfer coefficient is controlled by macropore molecular diffusion.38 The

155

external mass transfer resistance applied here can be used for any type of fixed-bed dynamics

156

and configuration. 46 Applicability of the external mass transfer resistance depends on the pore

157

Biot number (𝐵𝑝,𝑖). If 𝐵𝑝,𝑖 is larger than unity, then the effect of external mass transfer

158

resistance is negligible and the overall mass transfer is governed by internal or intraparticle

159

diffusion.47

1

𝑑𝑝

1

(10)

= 𝑘𝑒𝑥,𝑖 + 𝑘𝑖𝑛,𝑖

[1 + 1.5(1 ― 𝜀𝑏)]𝐷𝐴𝐵,𝑖 𝑑𝑝

1/3 (2 + 0.644𝑅𝑒1/2 𝑝 𝑆𝑐𝑖 )

60𝜀𝑝𝐶𝑜,𝑖𝐷𝑒𝑓𝑓,𝑖

(11) (12)

τ𝑝𝐶𝑠𝑜,𝑖𝑑𝑝2



Page 14 of 38

14

160

The effective diffusion coefficient (𝐷𝑒𝑓𝑓,𝑖 ) comprises surface (𝐷𝑠,𝑖 ) and pore (𝐷𝑝,𝑖 ) diffusion;

161

and is written as:29, 48, 49

162

𝐷𝑒𝑓𝑓,𝑖 = 𝐷𝑝,𝑖 +

163

𝐷𝑠,𝑖 = 𝐷𝑠𝑜𝑒𝑥𝑝

164

𝐷𝑝,𝑖 = 𝐷𝐴𝐵,𝑖

165

Surface diffusion can be neglected from the effective diffusion resistance if the surface to pore

166

diffusion flux ratio (𝜑𝑖) is found to be less than unity; otherwise pore diffusion can be

167

neglected.41 Here, pore diffusion consists of only molecular diffusion.

(

∂𝐶𝑠,𝑖

(13)

∂𝐶𝑖 𝐷𝑠,𝑖

―5.38𝑇𝑏,𝑖

)

𝑇

(14) (15)

168 169

Momentum Balance

170

The momentum balance equation adopted here accounts for Darcy and Brinkman viscous

171

terms, Navier-Stokes’ convective term, and Forchheimer’s inertial term.25 This model has been

172

previously applied with success, and is described as:21, 22

173

(( ) + (𝑢 ∗ ∇) ) = ―∇𝑃 + ∇𝐽 ― 𝑆 + 𝐹

𝜌𝑓

∂𝑢

𝑢

𝜀𝑟

∂𝑡

𝜀𝑟

(16)

174

The shear stress (J) is defined in terms of gas viscosity (µf):

175

𝐽 = (𝜇𝑓𝜀𝑟 (∇𝑢 + (∇𝑢)′) ―

176

Momentum dissipation of the gas flow across the fixed-bed adsorber is represented by Darcy’s

177

friction loss factor, Forchheimer’s inertial term, and a sink term due to the adsorption of VOC

178

and/or water vapor (equation (18)).

179

𝑆=

1

(

𝜇𝑓 𝜅

(

1

(

2

(

2 3

))

∗ (∇𝑢)

))𝑢

∂𝐶𝑠,𝑖

+ 𝛽𝑓|𝑢| + 𝜀𝑟 ∑𝑖 = 1

∂𝑡

(17)

(18)


Page 15 of 38


15

180

The continuity equation given below accounts for the compressibility of the gas flow in the

181

fixed-bed adsorber and the sink due to VOC and/or water vapor adsorption:

182

∂(𝜀𝑟𝜌𝑓) ∂𝑡

2

∂𝐶𝑠,𝑖

+∇ ∗ (𝜌𝑓𝑢) = ∑𝑖 = 1

(19)

∂𝑡

183

Energy Balance

184

The main assumptions for formulating the energy balance equation across the fixed-bed

185

adsorber are: local thermal equilibrium between the solid adsorbent and the gas, and negligible

186

pressure work and viscous heat dissipation. The convection-diffusion-based heat transfer

187

equation was previously validated for single and multicomponent adsorption systems.21, 22

188

𝐶𝑒𝑓𝑓 ∂𝑡 + 𝐶𝑝,𝑓𝜌𝑓𝑢 ∗ ∇𝑇 ― ∇(𝑘𝑒𝑓𝑓∇𝑇) = ∑𝑖 = 1𝑆ℎ,𝑖

189

The domain heat source is the heat of adsorption of the ith component, and is described using

190

equation (21) for mixtures involving non-polar adsorbates and water vapor:

191

𝑆ℎ,𝑖 = ( ― ∆𝐻𝑎𝑑,𝑖)

192

where the heat of adsorption (∆𝐻𝑎𝑑,𝑖) is dependent on the properties of the adsorbate and the

193

adsorbent:50

194

―∆𝐻𝑎𝑑,𝑖 = 103.2 + 1.16α𝑖 +0.76∆𝐻𝑣𝑎𝑝,𝑖 ―3.87(𝐼𝑃𝑖) ―0.7𝜎𝑖 ―26.1𝑤𝑚𝑖𝑐

195

For mixtures involving polar adsorbates and water vapor, heat of dissolution (∆𝐻𝑠𝑜𝑙,𝑖) is also

196

incorporated (equation (23)).

197

𝑆ℎ,𝑖 = ( ― ∆𝐻𝑎𝑑,𝑖 ― ∆𝐻𝑠𝑜𝑙,𝑖)

198

The effective volumetric heat capacity (𝐶𝑒𝑓𝑓) of the solid-gas system is calculated from:

199

𝐶𝑒𝑓𝑓 = (1 ― 𝜀𝑝)𝜌𝑝𝐶𝑝,𝑝 + 𝜀𝑝𝜌𝑓𝐶𝑝,𝑓

200

The effective thermal conductivity tensor (𝑘𝑒𝑓𝑓) is given as:

∂𝑇

2

(20)

𝑑𝐶𝑠,𝑖

(21)

𝑑𝑡

(22)

𝑑𝐶𝑠,𝑖

(23)

𝑑𝑡

(24)



Page 16 of 38

16

|

|

201

𝑘𝑟 0 𝑘𝑒𝑓𝑓 = 0 𝑘 𝑎𝑥

202

𝑘𝑟 and 𝑘𝑎𝑥 are the radial and axial effective thermal conductivities of the fixed-bed adsorber

203

respectively (equations (26) and (27)).29

204

𝑘𝑟 = 𝑘𝑏 + 8𝑃𝑒𝑜𝑘𝑓

205

𝑘𝑎𝑥 = 𝑘𝑏 + 2𝑃𝑒𝑜𝑘𝑓

206

where 𝑘𝑏 is the stagnant bed thermal conductivity, that is, the thermal conductivity of the

207

fixed-bed with stagnant gas.29

208

𝑘𝑏 = (1 ― 𝜀𝑝)𝑘𝑝 + 𝜀𝑝𝑘𝑓

(25)

1

(26)

1

(27)

(28)

209 210

Initial and Boundary Conditions

211

The initial and boundary conditions applied to the 2D mathematical model here are described

212

graphically in Figure 1. For mass transfer, a constant concentration boundary condition (𝐶𝑖 =

213

𝐶𝑜,𝑖; 𝐶𝑠,𝑖 = 𝐶𝑠𝑜,𝑖) and a flux boundary condition ( ―𝑛 ∗ (𝐷𝑖∇𝐶𝑖) = 0; ―𝑛 ∗ (𝐷𝑖∇𝐶𝑠,𝑖) = 0) are set

214

at the inlet (𝑍 = 𝐿) and the outlet (𝑍 = 0) of the fixed-bed adsorption tube, respectively. In

215

addition, zero flux was considered at the adsorber wall (𝑟 = 𝑅); with (𝐶𝑜,𝑖 = 0;𝐶𝑠𝑜,𝑖 = 0) during

216

initial conditions (𝑡 = 0). For momentum balance, normal inflow velocity boundary condition (

217

𝑢𝑠 = 0.856 𝑚/𝑠) is set at the inlet, atmospheric pressure (𝑃 = 101.325 𝑘𝑃𝑎) is set at the

218

outlet (𝑢 = 0 m/s; 𝑃 = 101.325 𝑘𝑃𝑎) at 𝑡 = 0, and a no slip boundary condition is applied to

219

the wall of the adsorption tube. For heat transfer, a constant temperature boundary condition (

220

𝑇 = 298 𝐾), a flux boundary condition ( ― 𝑛 ∗ (𝑘𝑒𝑓𝑓∇𝑇) = 0), (𝑇 = 298 𝐾), and a convective


Page 17 of 38


17

221

heat flux (𝑞𝑜 = ℎ ∗ (𝑇𝑤 ― 𝑇)) are specified at the inlet, outlet, initial conditions, and wall of the

222

fixed-bed adsorber, respectively.

223 224

Solution Method

225

The mass, momentum, and heat transfer across the fixed-bed adsorber were coupled and

226

simultaneously solved using COMSOL Multiphysics software (Version 4.3a). This simulation was

227

coupled with the interpolation-function-based MATLAB-coded Manes method. The

228

computation time was typically 2 to 3 hours on a computer with current generation Intel Core

229

i7 processor and 12 GB RAM. In COMSOL Multiphysics, the governing equations were solved

230

numerically using the finite element method. The software’s coefficient form PDE interfaces,

231

built-in momentum and energy interfaces were used respectively to represent the mass,

232

momentum, and heat transfer equations. A second-order element was used for concentration,

233

pressure, and temperature, while a third-order element was used for velocity to improve model

234

convergence and stability.51-53 Convergence of the model was validated by using systematic

235

mesh refinement to its geometry until grid-independent results were obtained. The meshing of

236

the 2D model’s geometry was finally optimized to 4,488 mesh elements, which showed a

237

relative deviation of only 0.7% from the solution obtained through fine meshing (25,964 mesh

238

elements) and reduced the computation time by 75%.

239 240

Model Validation Method

241

To validate the model, adsorption capacities, breakthrough concentrations, and bed

242

temperatures were measured from experiments and compared with the model results. For the



Page 18 of 38

18

243

nonzero data points, the experiment and model were compared through the mean relative

244

absolute error (MRAE).54

245

𝑀𝑅𝐴𝐸 = 𝑁∑1

246

where N is the number of data points.

247

The normalized root-mean-square error (NRMSE) was also used to evaluate the overall error

248

between the experimental and modeled values.22

1

(

𝑁 |𝑒𝑥𝑝𝑒𝑟𝑖𝑚𝑒𝑛𝑡𝑎𝑙 𝑣𝑎𝑙𝑢𝑒 ― 𝑚𝑜𝑑𝑒𝑙𝑒𝑑 𝑣𝑎𝑙𝑢𝑒| 𝑒𝑥𝑝𝑒𝑟𝑖𝑚𝑒𝑛𝑡𝑎𝑙 𝑣𝑎𝑙𝑢𝑒

1 𝑁 ∑ 𝑁 1 (𝑒𝑥𝑝𝑒𝑟𝑖𝑚𝑒𝑛𝑡𝑎𝑙

)

∗ 100%

𝑣𝑎𝑙𝑢𝑒 ― 𝑚𝑜𝑑𝑒𝑙𝑒𝑑 𝑣𝑎𝑙𝑢𝑒)2

(29)

(30)

249

𝑁𝑅𝑀𝑆𝐸 =

250

For measuring deviations in bed temperature results using MRAE and NRMSE, the temperature

251

data points were expressed in degree Celsius instead of Kelvin to avoid low relative error bias.

252 253

RESULTS AND DISCUSSION

254

Validation of Adsorption Breakthrough Profiles

255

Figure 2 shows the adsorption breakthrough profiles at 0%, 55%, and 95% RH levels and 298 K

256

for the selected VOCs. The pore Biot number (𝐵𝑝,𝑖) was larger than unity, and therefore the

257

overall mass transfer was governed by intraparticle diffusion (Table 2). Surface diffusion was

258

assumed to be negligible because the surface to pore diffusion flux ratio (𝜑𝑖) was found to be

259

less than unity (Table 2). Pore diffusion, specifically molecular diffusion, was thus considered to

260

be the effective diffusion mechanism to calculate the overall mass transfer resistance.

261

Water vapor adsorption breakthrough profiles at 55%/95% RH and 298 K are presented in

262

Figure 3. The model predicted the breakthrough curves with an overall MRAE of 8% and 11% at

263

55% and 95% RH respectively. Due to the nature of water vapor adsorption isotherm on

264

activated carbon (type V) (Figure S2), there is immediate breakthrough at both humidity levels.

𝑖𝑛𝑓𝑙𝑢𝑒𝑛𝑡 𝑠𝑡𝑟𝑒𝑎𝑚 𝑣𝑎𝑙𝑢𝑒

∗ 100%


Page 19 of 38


19

265

However, at 95% RH there is considerable adsorption of water for the first 70 minutes after

266

which water adsorption becomes negligible.



Page 20 of 38

20

1.00E+03

Acetone (ppmv)

2-propanol (ppmv)

1000

8.00E+02 6.00E+02 4.00E+02 2.00E+02

800 600 400 200 0

0.00E+00 0

100 Time (min)

0

200

(a)

600 400 200 0

268 269

800 600 400 200 0

0

267

1000 1,2,4-TMB (ppmv)

Toluene (ppmv)

n-butanol (ppmv)

800

100 (min) Time (c)

200

200

(b)

1000

1000

100 Time (min)

800 600 400 200 0

0

100 Time (min) (d)

200

0

100 Time (min) (e)

Figure 2. Comparison of experimental and modeled breakthrough curves of VOCs during competitive adsorption of water vapor, (a) 2-propanol, (b) acetone, (c) n-butanol, (d) toluene, and (e) 1,2,4-TMB on BAC at 298 K. ACS Paragon Plus Environment

200

Page 21 of 38


21

270 271

Table 2. Surface diffusion, pore diffusion, and external mass transfer coefficient of the selected adsorbates at 298 K adsorbate

𝑫𝒔,𝒊 (m2/s)

𝑫𝒑,𝒊 (m2/s)

𝝋𝒊 (–)

𝒌𝒆𝒙,𝒊 (m/s)

𝑩𝒑,𝒊 (–)

2-propanol

1.81E-11

1.06E-05

0.12

0.26

9.24

acetone

2.89E-11

1.09E-05

0.09

0.27

9.17

toluene

1.08E-11

8.26E-06

0.08

0.22

9.86

n-butanol

9.53E-12

9.22E-06

0.08

0.24

9.58

1,2,4-TMB

3.74E-12

7.02E-06

0.03

0.19

10.31

water

1.31E-11

2.56E-05

0.01

0.50

7.37

272 273

The model predicted the breakthrough curves with an overall MRAE of 11.8% and 17.2%, and

274

NRMSE of 5.5% and 8.4%, for dry (0% RH) and humid conditions (55% and 95% RH) respectively.

275

The MRAE analysis was conducted only for the nonzero data points. Numerical error, model

276

assumptions (negligible variation of flow properties in the angular direction of the tube,

277

negligible adsorption of the carrier gas (air), ideal gas behaviour, and symmetric flow

278

conditions), and/or experimental error in concentration measurements are possible

279

contributors to the deviations between the modeled and experimental breakthrough profiles.

280

At 95% RH, all the tested VOCs except 1,2,4-TMB experienced earlier breakthrough due to

281

competition from water vapor (Figure 2), owing to its increased affinity towards activated

282

carbon adsorption (Figure S2, Figure 3b). However, at 55% RH no changes in the VOC

283

breakthrough profiles were observed due to the significantly lower adsorption affinity of water

284

vapor compared to VOCs at the tested concentration (1000 ppmv for VOC and 55% RH for water

285

vapor) (Figure S2, Figure 3a).

286

As expected, at 95% RH polar VOCs such as 2-propanol and acetone yielded the highest

287

reduction in the bed service time (5% breakthrough time, that is the time when the outlet



Page 22 of 38

22

288

adsorbate concentration is 5% of the inlet) of 16.9% and 10.7% respectively (Figure 4). This was

289

followed by the non-polar VOCs in the order: n-butanol (9.6%), toluene (7.8%), and 1,2,4-TMB

290

(0.0%) (Figure 4). The high susceptibility of polar VOCs to the impact of RH during competitive

291

adsorption with VOCs was also reported in other studies where adsorption capacity for polar

292

VOCs such as acetone were reduced by up to 50% compared to about 40% for non-polar VOCs

293

such as benzene and toluene at 90% RH and 303 K on granular activated carbon (GAC) and inlet

294

concentrations ranging from 200 – 3500 ppmv.11, 55 The breakthrough curve results here show

295

good agreement between model and experiment, enabled by an effective combination of the

296

MATLAB code that applied the extended Manes method and the COMSOL Multiphysics

297

software that used the MATLAB results to solve the adsorption breakthrough profiles.

298

299 300 301

Figure 3. Comparison of experimental and modeled breakthrough curves of water vapor at (a) 55% RH, and (b) 95% RH on BAC at 298 K.


Page 23 of 38


23

302 303 304 305

Figure 4. Experimental breakthrough times of polar VOCs (acetone, 2-propanol) and non-polar VOCs (toluene, n-butanol, and 1,2,4-trimethylbenzene) during competitive adsorption with water vapor (0%, 55%, 95% RH) on BAC at 298 K.



Page 24 of 38

24

306

Absorbed-Phase Concentration Distribution

307

Figure 5 shows the development of the 2D adsorbed-phase concentration distribution of 2-

308

propanol during its competitive adsorption with water vapor at 298 K, and its comparison to dry

309

conditions at 0% RH. As observed earlier with the validations of breakthrough profile

310

predictions, water vapor has no effect on VOC adsorption at 55% RH (Figure 4). However, at

311

95% RH, the movement of the mass transfer zone (MTZ) is faster than at 0% or 55% RH. At 30

312

min after the start of adsorption, the MTZ at 95% RH is 35 mm from the bed top compared to

313

25 mm for 0%/55% RH. At the end of adsorption, the equilibrium adsorption capacity at 95% RH

314

is 17.5% lower than that for 0% or 55% RH, as predicted by the underlying extended Manes

315

method.26 Such behavior at high relative humidity is due to competition between 2-propanol

316

and water molecules for the limited adsorption sites on the BAC. This competition results in

317

displacement of the VOC’s MTZ and reduction in its adsorption capacity. In general, the

318

reduction in breakthrough time is governed by the affinity, adsorption potential, and polarity of

319

both the VOC and water vapor at their given inlet concentrations. The difference between the

320

MTZ at 0%/55% and 95% RH is within 10 to 20 mm (9 to 17% of the entire bed length)

321

throughout the adsorption period until saturation, which in turn leads to an earlier bed-

322

saturation (160 min for 95% RH compared to 200 min for 0%/55% RH). The movement of the

323

MTZ for 2-propanol is consistent with its adsorption breakthrough profile measured at the

324

centre of the adsorber outlet (Figure 2). Furthermore, the similarity in the breakthrough curve

325

slopes during dry and humid conditions suggests that the mass transfer resistance of 2-

326

propanol did not have any notable impact during high relative humidity conditions; likely


Page 25 of 38


25

327

because of the small size and very low diffusion resistance of the water molecules relative to 2-

328

propanol (Table S1). The same finding is true for all the selected VOCs (Table S1).

329



Page 26 of 38

26

330 331 332

Figure 5. 2D adsorbed-phase concentration distribution of 2-propanol during competitive adsorption with water vapor on BAC at 298 K.


Page 27 of 38


27

333

Also, the variation of the adsorbed phase concentration in the radial direction reveals that the

334

bed becomes saturated near the wall earlier than at its centre due to wall channeling. This

335

observation is consistent with previous 2D adsorption modeling studies.21-24 A one-dimensional

336

simulation along the bed centre would overestimate the breakthrough time for the same

337

scenario.

338 339

Absorber Bed Temperature Distribution

340

Figure 6 depicts the experimental and modeled adsorber bed temperature profiles at 75 mm

341

from the bottom along the centerline of the bed (r = 0.0 m, z = 0.075 m) during competitive

342

adsorption for different VOC-water vapor systems. The model predictions were good with an

343

MRAE of 1.6% and 2.4% and NRMSE of 2.1% and 2.9% for dry and humid conditions,

344

respectively. While experimental error in measuring bed temperature could be a factor, the

345

most likely contributor to the error could be the competitive adsorption isotherm model’s

346

assumption of ideal adsorbed-phase for polar VOCs.26 The MRAE and NRMSE for polar VOCs

347

was also found to be higher at 2.6% and 3.4% respectively, compared to 1.9% and 2.5%

348

respectively for non-polar VOCs. For all the selected VOC-water vapor systems except 1,2,4-

349

TMB, the average bed temperature increased by up to 1 to 2K during competitive adsorption at

350

95% RH when compared to 0%/55% RH. Adsorption being an exothermic process, the

351

temperature rise can be attributed to the heat of adsorption and heat of dissolution (for polar

352

adsorbates in a gas mixture). The more polar VOCs, which exhibited RH impacts, also had the

353

highest average bed-temperature increase of up to 2K during competitive adsorption with

354

water vapor, whereas for non-polar VOCs, the increase was about 1K.



Page 28 of 38

28 355

356 357 358

Figure 6. Comparison of experimental and modeled BAC-bed temperature profiles at the centre of the reactor (r = 0.0 m, z = 0.075 m) during competition adsorption of water vapor with (a) 2-propanol, (b) acetone, (c) n-butanol, (d) toluene, (e) 1,2,4-TMB.


Page 29 of 38


29

359

Following bed temperature validations, 2D adsorber bed temperature distributions were

360

generated showing the evolution of heat transfer zones (HTZ) for systems with 2-propanol and

361

water vapor (Figure 7). Higher temperatures, as predicted earlier, are observed at 95% RH

362

compared to 0% and 55% RH. This leads to a difference in HTZ at 0%/55% RH and 95% RH of up

363

to 30 mm (26% of the entire bed length) throughout the adsorption period. Consequently, the

364

fixed-bed at 0%/55% RH reached thermal equilibrium with the adsorption temperature at 200

365

min, at least 40 min earlier than the fixed-bed at 95% RH. The 2D temperature distribution plot

366

corresponds well with the 2D adsorbed-phase concentration plot (Figure 5); albeit the HTZ had

367

a higher velocity than the MTZ, creating a difference of at least 10 mm at all times and

368

conditions during adsorption. It should also be noted that the temperature during adsorption

369

varied across the bed, and was higher at the center than at the periphery because of convective

370

heat transfer at the tube wall. In dry conditions, similar observations were made in adsorber

371

bed temperature and HTZ across the adsorber bed in previous modeling and experimental

372

works on activated carbon adsorption of VOCs such as benzene, toluene, acetone, ethanol,

373

pentane,22, 24, 36, 56 which gives confidence in the model for simulating heat transfer kinetics.

374 375



Page 30 of 38

30

376 377

Figure 7. 2D adsorber bed temperature distribution during competitive adsorption of 2-propanol with water vapor.


Page 31 of 38


31 378

Effect of Adsorption Temperature

379

To evaluate the effect of adsorbent bed temperature, the bed temperature heater set point was

380

increased from 298 K (25 °C) to 305 K (32 °C), while the dry/humid air stream entering the bed was

381

maintained at 298 K and 0%/55%/95% RH.

382 383

Figure 8 shows the experimental and modeled adsorption breakthrough profiles of 2-propanol at a

384

bed temperature of 298 K and 305 K at inlet RH of 0%, 55%, and 95%. Increasing the adsorber

385

temperature from 298 K to 305 K, reduces the RH from 95 and 55% to 60 and 35% respectively inside

386

the adsorber column. This effect offsets the impact of RH on the adsorber cycle duration and VOC

387

adsorption capacity. At 298 K, the 5% breakthrough time and VOC adsorption capacity decreased by

388

16.9% and 17.5% respectively at 95% RH compared to 0/55% RH. At 305 K, no deterioration in the 5% ACS Paragon Plus Environment


Page 32 of 38

32 389

breakthrough time and the VOC adsorption capacity was observed over the range of RH levels tested.

390

However, the VOC adsorption capacity decreased by 11% at 305 K compared to 298 K (both at

391

0%/55% RH). Therefore, optimization of bed temperature may allow optimization of the effect of

392

temperature on RH and competitive adsorption along with its effect on adsorption capacity and

393

breakthrough time. To accomplish this, the utility and stability of the model is crucial. The model

394

predicted the adsorption behavior of 2-propanol at 305 K (0%, 55%, 95% inlet RH) with an overall

395

MRAE of 12.4% and 7.1% for the breakthrough profiles and adsorption capacity respectively.

396

Furthermore at 298 K, the model predicted 2-propanol’s breakthrough profile and adsorption capacity

397

with an overall MRAE of 18.5% and 5.9% respectively. These results demonstrate the model’s

398

sensitivity to changes in important operational conditions. The model’s potential utility for optimizing

399

bed temperature could enable increases in adsorber service lifetime and decreases in overall fixed-

400

bed adsorber operational costs.


Page 33 of 38


33

401 402 403

Figure 8. Comparison of experimental and modeled breakthrough curves during competition adsorption of water vapor with 2-propanol on BAC at 298 K (25 °C) and 305 K (32 °C).

404 405

A mathematical model, consisting of VOC-water vapor competitive adsorption isotherms and

406

transport phenomena equations, was developed to study the effect of carrier gas relative humidity on

407

VOC adsorption onto activated carbon. The results obtained were encouraging, especially because the

408

model was able to predict the mass, heat, and momentum transfer during VOC-water vapor

409

competitive adsorption process with a reasonable accuracy, using independently determined

410

adsorbate and adsorbent properties, adsorber geometry and operating conditions. The model

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developed in this study will help the industry to optimize large-scale adsorber designs and operating



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conditions and minimize the negative impact of RH during adsorption of VOCs from gas streams,

413

leading to lower operational costs. The validated model may also reduce the cost of pilot-scale testing

414

required to optimize the effect of process parameters and variables.

415 416

ACKNOWLEDGEMENTS

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The authors would like to acknowledge financial support for this research from Ford Motor Company

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and the Natural Science and Engineering Research Council (NSERC) of Canada.

419

While this article is believed to contain correct information, Ford Motor Company (Ford) does not

420

expressly or impliedly warrant, nor assume any responsibility, for the accuracy, completeness, or

421

usefulness of any information, apparatus, product, or process disclosed, nor represent that its use

422

would not infringe the rights of third parties. Reference to any commercial product or process does

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not constitute its endorsement. This article does not provide financial, safety, medical, consumer

424

product, or public policy advice or recommendation. Readers should independently replicate all

425

experiments, calculations, and results. The views and opinions expressed are of the authors and do

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not necessarily reflect those of Ford. This disclaimer may not be removed, altered, superseded or

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modified without prior Ford permission.

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REFERENCES

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