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Effects of rate-limited mass transfer on modeling vapor intrusion with aerobic biodegradation Yiming Chen, Deyi Hou, Chunhui Lu, Jim C Spain, and Jian Luo Environ. Sci. Technol., Just Accepted Manuscript • DOI: 10.1021/acs.est.6b01840 • Publication Date (Web): 03 Aug 2016 Downloaded from http://pubs.acs.org on August 8, 2016

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1

Environmental Science & Technology

TOC Art

O2

O2

Biomass

Biomass

Air

Unsaturated Zone Contaminant vapor

Capillary Fringe Water Table Saturated Zone

Contaminant Equilibrium mass transfer

Contaminant Rate-limited mass transfer

1

ACS Paragon Plus Environment


1

Effects of rate-limited mass transfer on modeling vapor intrusion

2

with aerobic biodegradation

3

Yiming Chen†, Deyi Hou ‡, Chunhui Lu§, Jim C. Spain||, and Jian Luo*†

4

†

5

Atlanta, GA 30332-0355, USA

6

‡

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School of Civil and Environmental Engineering, Georgia Institute of Technology,

School of Environment, Tsinghua University, Beijing, China

7 8

§

9 10

||

State Key Laboratory of Hydrology-Water Resources and Hydraulic Engineering, Hohai University, Nanjing, China Center for Environmental Diagnostics & Bioremediation, University of West Florida, Pensacola, FL 32514-5751, USA

11 12 13

*

Corresponding author: J. Luo, E-mail: [email protected], Phone: (404) 385-6390, Fax: (404) 385-1131

14

15

1


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16


TOC Art

17

18

O2

O2

Biomass

Biomass

Air

Unsaturated Zone Contaminant vapor

Capillary Fringe Water Table Saturated Zone

Contaminant Equilibrium mass transfer

Contaminant Rate-limited mass transfer

2



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19

Abstract

20

Most of the models for simulating vapor intrusion accept the local equilibrium assumption for

21

multiphase concentration distributions, i.e., concentrations in solid, liquid and vapor phases are

22

in equilibrium. For simulating vapor transport with aerobic biodegradation controlled by

23

counter-diffusion processes, the local equilibrium assumption combined with dual-Monod

24

kinetics and biomass decay may yield near-instantaneous behavior at steady state. The present

25

research investigates how predicted concentration profiles and fluxes change as inter-phase mass

26

transfer resistances are increased for vapor intrusion with aerobic biodegradation. Our modeling

27

results indicate that the attenuation coefficients for cases with and without mass transfer

28

limitations can be significantly different by orders of magnitude. Rate-limited mass transfer may

29

lead to larger overlaps of contaminant vapor and oxygen concentrations, which cannot be

30

simulated by instantaneous reaction models with local equilibrium mass transfer. In addition, the

31

contaminant flux with rate-limited mass transfer is much smaller than that with local equilibrium

32

mass transfer, indicating that local equilibrium mass transfer assumption may significantly

33

overestimate the biodegradation rate and capacity for mitigating vapor intrusion through the

34

unsaturated zone. Our results indicate a strong research need for field tests to examine the

35

validity of local equilibrium mass transfer, a widely accepted assumption in modeling vapor

36

intrusion.

3


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37

Introduction

38

Volatile organic chemicals from subsurface sources can migrate through the unsaturated zone

39

and enter buildings as vapors, posing risks to indoor air quality and human health 1. Under

40

favorable conditions bacteria in the capillary fringe and unsaturated zone can substantially

41

mitigate contaminant transport by biodegrading organic vapors partitioning into soil moisture 2-8,

42

16,17

43

capacity in the capillary fringe and unsaturated zone and found compact and thin reactive zones

44

developed at oxic/anoxic interfaces with high microbial concentrations and low substrate

45

concentrations

46

approximated as an instantaneous reaction 4,9-13. Our recent modeling work of column studies of

47

chlorobenzene, 1,2-dichlorobenzene, and 1,4-dichlorobenzene characterized such thin reactive

48

zones at steady state with balanced contaminant and oxygen fluxes and theoretically proved the

49

validity of instantaneous reactions for describing biodegradation controlled by counter-diffusion

50

processes 14.

51

Unlike the instantaneous reaction in thin reactive zones, we also observed large concentration

52

overlaps of contaminants (cis-dichloroethene) and oxygen in column studies50, which were found

53

for other contaminants in field observations

54

reaction kinetics without the consideration of biomass may yield better modeling performance

55

than the model of instantaneous reactions 16,18-20. Such inconsistencies related to reaction kinetics

56

causes the uncertainties in understanding vapor intrusion processes and providing accurate

57

quantification of aerobic biodegradation for attenuating vapor threats. In fact, the instantaneous

58

reaction model implies an unrealistically high biodegradation capacity for the compact reactive

. Many lab experiments and field investigations have demonstrated high biodegradation

4,5,9,10

. Within the reactive zones, reactions occurred rapidly and might be

13,15,49

. For such observations, first-order or Monod

4



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59

zone by pushing the reactive zone towards the oxygen boundary to create larger oxygen fluxes,

60

leading to overly optimistic protective measures for risk assessments.

61

many limiting mechanisms that affect biodegradation, such as the biomass limitation in pore

62

space

63

geochemical conditions for bacterial growth 25, kinetic solute uptake by the biomass 26,27, etc.

64

In the present study, we investigate the effects of rate-limited mass transfer between vapor and

65

liquid phases as a limiting mechanism for biodegradation in the unsaturated zone. Such mass

66

transfer processes were widely considered for removing volatile organic contaminants in soil

67

vapor extraction systems,

68

porous media,

69

dual-porosity behavior.48 For simulating vapor intrusion in the unsaturated zone, multiphase

70

concentration distributions were included in most models, but the local equilibrium assumption

71

was applied,

72

equilibrium. The key objective of the present study is to extend the analysis of steady state vapor

73

transport with aerobic biodegradation to include rate-limited mass transfer for contaminants and

74

oxygen. Particularly from a modeling perspective we evaluate the effects of rate-limited mass

75

transfer on attenuation coefficients of vapor intrusion, concentration profiles and fluxes of

76

contaminants and oxygen. We also evaluate simplified reaction models such as first-order and

77

instantaneous reaction models, for describing vapor transport with biodegradation in the

78

unsaturated zone, and investigate whether a larger overlap of contaminants and oxygen and a

79

larger reactive zone can be explained by with rate-limited mass transfer.

80

Model

21-23

, the moisture content preferred by the bacteria

32,33

34-42

28-31

24

14

The model neglects

, optimal temperature and

in modeling multiphase transport of organic contaminants in

and in modeling solute transport in the unsaturated zone exhibiting

i.e., concentrations in solid, liquid and vapor phases are distributed in

5


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81

We conceptualize the contaminant transport as one-dimensional (vertical) in the unsaturated

82

zone with a steady-state soil moisture distribution. With an underlying contaminant source and

83

oxygen inputs from surface, the transport of electron donors (organic contaminants) and

84

acceptors (oxygen) is controlled by counter-diffusion processes in both vapor and liquid phases.

85

For simplicity, we assume that diffusion can be quantified by constant effective diffusion

86

coefficients

87

kinetic models are used to describe the interphase mass transfer between liquid and vapor phases

88

and between liquid and solid phases. Direct mass transfer between vapor and solid phases is

89

neglected.

90

The mass balance equations of individual phases at steady state are given by:

91

DDG

 L cDG  ∂ 2cDG  = 0 + k a D  cD − ∂z 2 H D  

(1a)

92

 L cAG  ∂ 2cAG  = 0 D + k A a cA − ∂z 2 H  A 

(1b)

93

DDL

 cDG  ∂ 2cDL  + k a − cDL  = θ L r D  2 ∂z  HD 

(1c)

94

 cAG ∂ 2 cAL L   = θ L FA r D + k a − c A A 2  ∂z  HA 

(1d)

95

Yr − bX = 0

(1e)

96

cDS =

34

. Aerobic biodgradation is assumed to occur only in the liquid phase. First-order

G A

L A

kD, f k D, b

cDL

(1f)

6



97

c AS =

k A, f k A, b

c AL

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(1g)

98

where z (L) is the vertical spatial coordinate; t (T) is the time; θ ( − ) is the effective porosity;

99

c (ML

−3

) is the concentration; D (L 2 T

−1

) is the effective diffusion coefficients; the

100

superscripts 'G', 'L' and 'S' represent gas, liquid and solid phases, respectively; the subscripts 'D'

101

and 'A' represent electron donor and acceptor, respectively; H ( − ) is the Henry's law constant;

102

k DL a (T −1 ) and k AL a (T −1 ) are the overall rate constants based on the liquid-phase driving force

103

for rate-limited mass transfer between gas and liquid phase; a (L −1 ) is known as the specific

104

interfacial area; ρ b (ML −3 ) is the bulk density; k f (T −1 ) and k b (ML −3 T −1 ) are the forward and

105

backward rate coefficient of rate-limited mass transfer between liquid and solid phase; X (M bio

106

L −3 ) is the biomass concentration; FA ( − ) is the stoichiometric conversion factor; r (ML −3 T −1 )

107

−1 is the reaction rate; Y (M bio M donor ) is the yield coefficient; and b (T −1 ) is the rate coefficient of

108

biomass decay.

109

The reaction rate of aerobic biodegradation is described by the dual-Monod kinetic model with

110

biomass:

111

r=

112

where K A (ML −3 ) and K D (ML −3 ) are the half velocity constants; and

113

maximum specific growth rate.

114

Equations (1f) and (1g) indicate that the mass transfer between dissolved compounds and

115

sediment surface becomes equilibrium at steady state and does not affect the concentration

µ max c DL c AL X L L K D + cD K A + cA Y

(2)

µ max (T −1 ) is the

7


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116

distribution in the vapor and liquid phase described by equations (1a)-(1e). Thus, we only need

117

to discuss the mass transfer between the vapor and liquid phase at steady state.

118

For solving equations (1a) – (1g), the boundary conditions are defined as:

119

At the contaminant source, z = 0 ,

120

L L c DL = c D0 , c DG = H D c D0 , c AL = 0, c AG = 0

121

At the ground surface, z = L ,

122

cDL =

(3a)

c DG V ∂c G L L , E B B c DG − DDGη D = 0, c AL = c A0 , c AG = H A c A0 HD AB ∂z

(3b)

123

L G where cD0 is the contaminant concentration at the groundwater source, c A0 is the oxygen

124

concentration at the ground surface, EB is the indoor air exchange rate,

125

height of the enclosed space, and η is the fraction of surface area with permeable cracks. For

126

evaluating vapor intrusion, the boundary condition at the ground surface is associated with vapor

127

entry into buildings and the indoor mixing.32

128

Local Equilibrium Mass Transfer. For local equilibrium mass transfer between the vapor and

129

liquid phase, the model is written in terms of the total mass balance and reduced to:

130

DDG

131

2 L ∂ 2cAG L ∂ cA D + DA = θ L FA r 2 2 ∂z ∂z

132

2 L ∂ 2 cDG L ∂ cD + D = θ Lr D 2 2 ∂z ∂z

VB is known as the AB

(4a)

G A

(4b)

Note the total mass balance equations (4a) and (4b) are also valid for the rate-limited mass 8



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133

transfer model. The transport equations of individual phases in the kinetic model are replaced by

134

the local equilibrium relationships:

135

cDG = H D c DL , cAG = H A cAL

136

Substituting the local equilibrium relationships into equations (4a) and (4b) yields:

137

DDeff

∂ 2 cDL = θ Lr ∂z 2

(6a)

138

DAeff

∂ 2cAL = θ L FA r 2 ∂z

(6b)

139

where

140

DDeff = DDG H D + DDL , DAeff = DAG H A + DAL

141

For constant effective diffusion coefficients, the model can be solved analytically.14

142

Rate-limited Mass Transfer. For rate-limited mass transfer, the overall mass transfer

143

coefficient, k D a or k A a , according to the two-resistance theory of interface mass transfer,

144

27,29,44

145

constant, also known as the partition coefficient. Either the gas phase or the liquid phase may

146

control the rate of mass transfer, depending on the relative contributions of the individual

147

resistances. In the absence of bioreactions, i.e., r = 0 in equations (1c) and (1d), one can

148

conveniently verify that the concentrations in the vapor and liquid phase become equilibrium and

149

the transport is simplified to non-reactive transport with an effective diffusion coefficient. 34,42

150

For biodegradation, we assume that diffusion in the vapor phase dominates the diffusion

151

processes in the unsaturated zone and ignore diffusion in the liquid phase

(5)

(7)

depends on the mass transfer coefficients of individual phases and on the Henry's law

34

. To quantify the 9


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152

relative importance of diffusion and rate-limited mass transfer, we introduce the following

153

Sherwood numbers:

154

ShD =

155

where L (L) is the domain length. To quantify the relative reaction rates, we introduce the

156

Damkohler numbers:

157

Da D =

158

Thus, the transport equations in dimensionless coordinate, Z = z / L , can be simplified to:

159

 L c DG  ∂ 2 c DG  =0 + Sh D  cD − H D  ∂Z 2 

(10a)

160

 L c AG  ∂ 2 c AG  = 0 + ShA  c A − H ∂Z 2 A  

(10b)

161

 cG  c DL c AL θLX ShD  D − c DL  = Da D K D + c DL K A + c AL Y  HD 

(10c)

162

 c AG θ L FA X c DL c AL L    ShA  − c A  = Da A Y K D + c DL K A + c AL  HA 

(10d)

163

 cDL cAL b    X = 0 − L L  K D + cD K A + cA µ max 

(10f)

164

We define Sherwood numbers and Damkohler numbers based on the diffusion coefficients so

165

that it is more convenient to discuss their effects individually when changing the mass transfer

166

rate coefficient. One may also define the Damkohler numbers in terms of mass transfer rate

k DL aL2 k AL aL2 , Sh = A DDG DAG

µ max L2 DDG

, Da A =

(8)

µ max L2

(9)

DAG

10



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167

coefficients.

168

Base Model. To investigate the effects of rate-limited mass transfer and biodegradation on vapor

169

transport, we define a base model with typical geometry and transport parameters for vapor

170

intrusion in practical scenarios. 45 Table 1 lists the base model parameters. For simplicity, we

171

define the same diffusion coefficients and mass transfer rate constants for both volatile

172

contaminants and oxygen. The dimensionless parameters to quantify the relative importance of

173

gas-phase diffusion, rate-limited mass transfer and reaction rates are reduced to one Sherwood

174

number, Sh = ShD = ShA , and one Damkohler number, Da = DaD = DaA .

175

Table 1. Transport and reactive parameters for the base model Parameter

Symbol

Dimension of domain

L

Values 1 m 45

Effective diffusion coefficients in gas phase

DDG = DAG

0.2 m 2 /d 45

Effective diffusion coefficients in liquid phase

DDL = DAL

2 × 10 −5 m 2 /d 45

Overall liquid-phase mass transfer rate constants

k DL a = k AL a

various

Contaminant source concentration

L cD0

100 mg/L

Oxygen source concentration

L c A0

8 mg/L

Henry's law constant of oxygen

HA

30 46

Henry's law constant of volatile contaminant

HD

0.23 (benzene)46

Maximum specific growth rate

µ max

13.2 /d 47

Yield coefficient

Y

0.49 mg bio /mgdonor47

Rate coefficient of biomass decay

b

0.05 /d 47

Half velocity constants

KD = KA

0.1 mg/L 47

Stoichiometric conversion factor

FA

1

Indoor air exchange rate

EB

14/d 45 11


Page 13 of 27


Height of the enclosed space

VB / AB

2.5 m 45

Fraction of surface area with permeable cracks

η

0.001 45

176

Results and Discussion

177

Attenuation Coefficient. Figure 1 shows the attenuation coefficient decreases with the increase

178

of the Sherwood number or the mass transfer rate. At low mass transfer, Sh < 1 , vapor transport

179

can be approximated by conservative transport because biodegradation in the soil moisture has

180

almost no impact due to rate-limiting mass transfer. Following the Johnson-Ettinger model,34 the

181

attenuation coefficient for diffusion-dominant conservative transport cases can be evaluated by:

182

α=

ηDDG

(11)

V  ηD + E B  B  L  AB  G D

equation (11) yields α = 5.7 ×10 −6 for highly mass

183

For the given parameter values in Table,

184

transfer limited cases. For Sh > 10 4 , mass transfer can be approximated as an equilibrium

185

process, and the attenuation coefficient decreases by four orders of magnitude. That is, without

186

mass transfer limitations or with local equilibrium mass transfer assumption, biodegradation can

187

significantly mitigate contaminant vapor transport through the unsaturated zone. On the other

188

hand, if rate-limited mass transfer occurs, one may highly overestimate the effects of

189

biodegradation on preventing vapor transport based on the local equilibrium mass transfer

190

assumption, which is adopted in almost all vapor intrusion models.

12



Page 14 of 27

10 -5 10 -6

5.7 X 10

-6

α [-]

10 -7 10 -8 10 -9 10 -10 10 -11 10 -2

191

10 0

10 2

10 4

10 6

Sh [-]

192

Figure 1. Attenuation coefficient changes as a function of mass transfer rate. For this case,

193

Sh < 10 0 is mass transfer limited region, and Sh > 10 4 is equilibrium mass transfer region.

194

Concentration Distribution. Figure 2 shows the concentration distributions of electron donors

195

and acceptors in gas and liquid phases and the biomass at various mass transfer rate coefficients.

196

At low Sherwood number, Sh = 1 (Figure 2a), aerobic biodegradation is highly limited by mass

197

transfer and the effect of biodegradation on vapor transport becomes negligible. Thus, the vapor

198

transport becomes conservative and reactive transport in the liquid phase is controlled by liquid

199

phase diffusion and mass transfer. The microbial reaction simplifies to an instantaneous reaction

200

in a thin reactive zone in the liquid phase but with low biomass concentrations because of low

201

substrate fluxes introduced by liquid-phase diffusion and rate-limited mass transfer. At

202

intermediate Sherwood number, Sh = 100 (Figures 2b), the vapor concentrations gradually

203

decrease like first-order decay kinetics, the liquid-phase concentrations still follow instantaneous

204

reaction kinetics but the reactive zone starts to shrink with higher biomass concentrations. At

205

high Sherwood number, Sh = 10 4 (Figure 2c), vapor and liquid phase concentrations are at

206

equilibrium described by the Henry’s constant and both approach instantaneous reaction kinetics. 13


Page 15 of 27


207

As a result, the reactive zone shrinks to a compact, thin zone with very high biomass at the

208

equilibrium state. In summary, in terms of the contaminant concentration distribution in the

209

vapor phase, there are three scenarios: (1) conservative transport for low mass transfer rates; (2)

210

kinetic decay for intermediate mass transfer rates; and (3) instantaneous reaction for high mass

211

transfer rates. Concentration distributions in the vapor phase are drastically different between

212

cases with and without mass transfer limitations. The vapor concentrations of contaminants and

213

oxygen in mass transfer limited scenarios can have much larger overlaps (Figures 2a and 2b)

214

than those in the equilibrium case (Figure 2c).

215

We should notice that large concentration overlaps cannot be explained by low biodegradation

216

rates and local equilibrium partitioning without other limiting mechanisms. This is because local

217

equilibrium partitioning and large concentration overlaps in vapor phase imply large

218

concentration overlaps in liquid phase available for biodegradation. When both organic

219

compounds and oxygen are available, biomass will continue growing until either electron donors

220

or acceptors become limiting, leading to instantaneous reaction kinetics. On the other hand,

221

organic compounds and oxygen would not be expected to co-exist in the liquid phase at steady

222

state (Figure 2).

14


Page 16 of 27

1

50

(a) Sh = 1

0

c/c [-]

0.8

40

0.6

30

0.4

20

0.2

10

0

Biomass [mg/L]


0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1

(b) Sh = 100

0

c/c [-]

0.8

2000

0.6 0.4 cgD

0.2

cgA

clD

clA

X

0

0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

z/L [-]

(c) Sh = 10 4

0

c/c [-]

0.8

2

0.6 0.4

1

0.2 0

4

Biomass [mg/L]

×10

1

Biomass [mg/L]

z/L [-]

0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

z/L [-]

223 224

Figure 2. Concentration distributions in the liquid and vapor phase and biomass concentration

225

distributions at various mass transfer rate coefficients for the base model. Contaminant and

226

oxygen concentrations are normalized by the source concentration at boundaries. Red solid and

227

dashed lines are contaminant concentrations in the vapor and liquid phase, respectively; blue

228

solid and dashed lines are oxygen concentrations in the vapor and liquid phase, respectively; and

229

black lines are biomass concentrations. (a) Sh = 1, (b) Sh = 100, (c) Sh = 104.

230

Fluxes. Figure 3 shows how the inflow, outflow and biodegraded contaminant fluxes vary with

231

the mass transfer rate. The flux is normalized by the reacted flux with local equilibrium mass

232

transfer, which is also the maximum vapor flux that can be biodegraded by an instantaneous

233

reaction: 12-14

15


Page 17 of 27

234


FLUX max =

G DDeff c D0 +

DAeff FA

G c A0

(12)

L

235

At low Sherwood number, Sh < 1 , the inflow flux is almost equal to the outflow flux, implying

236

low reaction rates and conservative transport of contaminant vapor, i.e., vapor transport is not

237

affected by biodegradation because of rate-limited mass transfer. At intermediate Sherwood

238

number, 1 < Sh ≤ 100 , inflow and biodegraded fluxes increase and outflow flux decreases with

239

the mass transfer, indicating fast mass transfer enhances the contaminant flux into the domain

240

and stimulates bioreactions in the liquid phase. At large Sherwood number, Sh > 100 , the

241

inflow and biodegraded fluxes are equal and balanced and the outflow flux drops to zero,

242

representing the situation where all inflow contaminants are degraded.

243

If the reacted flux, i.e., the difference between the inflow and outflow fluxes, is 95% of the

244

reacted flux of an instantaneous reaction (the dashed line shown in Figure 3), we may

245

approximate the kinetic reaction as an instantaneous reaction or the rate-limited mass transfer as

246

local equilibrium mass transfer. For the base model, the Sherwood number needs to reach 2000

247

to support an assumption of instantaneous reaction with local equilibrium mass transfer. For the

248

parameters listed in Table 1, the Sherwood number of 2000 implies the mass transfer rate

249

coefficient of 400 /d, a value much larger than many reported mass transfer rate coefficients for

250

volatile organic chemicals

251

addition, we notice that local equilibrium mass transfer and an instantaneous reaction are not

252

required for the outflow fluxes to be negligible (Figure 3). At intermediate Sherwood number,

253

such as Sh = 100 , almost all inflow fluxes of contaminants are reacted, but the reacted flux is

254

only 50% of the maximum reaction flux for an instantaneous reaction. Thus, zero outflow flux

255

does not necessarily imply local equilibrium mass transfer and the approximation of

28,29

. Thus, mass transfer limitation could occur in vapor intrusion. In

16



Page 18 of 27

256

instantaneous reaction kinetics may highly overestimate the reaction flux and the biodegradation

257

capacity.

1 Inflow Outflow Reacted

Ratio of flux [-]

0.8 0.6 0.4 0.2 0 10-2

10-1

100

101

102

103

104

105

106

Sh [-]

258 259

Figure 3. Inflow, outflow and reacted fluxes of contaminants as a function of mass transfer rate.

260

All fluxes are normalized by the reacted flux with local equilibrium mass transfer. The dashed

261

line indicates the Sherwood number for 0.95 normalized flux.

262

Location of Reactive Zone. The location with the maximum biomass is representative of the

263

location of the reactive zone. For local equilibrium mass transfer, it can be conveniently

264

approximated by the balanced diffusive mass fluxes from contaminant and oxygen sources

265

weighted by the stoichiometric factor:12-14

266

z L

= X = X max

L c D0 L c D0 +

DAeff FA DDeff

(13)

L c A0

17


Page 19 of 27


267

However, equation (13) is not valid when rate-limited mass transfer occurs. Figure 4 shows the

268

concentration distributions and the reactive zones for halved diffusion coefficients of oxygen.

269

Compared with Figure 2, at fast mass transfer (Figure 4c), the reactive zone moves toward the

270

oxygen boundary according to equation (13); at small and intermediate mass transfer (Figurea 4a

271

and 4b) the location of reactive zone is almost unchanged. In fact, for cases with small Sherwood

272

numbers, the substrate bioavailability in the liquid phase is mainly determined by mass transfer

273

and the location of reactive zone can be evaluated by:

274

z L

= X = X max

G c D0 G c D0 +

k AL FA k DL

HD HA

(14)

G c A0

276

transfer-limited cases, respectively.

0

1 0.8 0.6 0.4 0.2 0

50 40 30 20 10 0

(a) Sh = 1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Biomass [mg/L]

Equations (13) and (14) define two bounding cases, gas phase diffusion-limited and mass

c/c [-]

275

1

(b) Sh = 100 cgD

0

0.1

0.2

0.3

0.4

0.5

0.6

cgA

0.7

clD

0.8

clA

0.9

0

c/c [-]

z/L [-] 1 0.8 0.6 0.4 0.2 0

2000 1500 1000 500 0

X

1 ×10 4

(c) Sh = 10 4

0

0.1

0.2

0.3

0.4

0.5

Biomass [mg/L]

1 0.8 0.6 0.4 0.2 0

2 1.5 1 0.5 0 0.6

0.7

0.8

0.9

Biomass [mg/L]

0

c/c [-]

z/L [-]

1

z/L [-]

277 18



Page 20 of 27

278

Figure 4. Concentration distributions and change of location of the maximum biomass for halved

279

diffusion coefficient of oxygen in the vapor phase, 0.1 m2/d. (a) Sh = 1, (b) Sh = 100, (c) Sh =

280

104.

281

Implication. Our analyses indicate that rate-limited mass transfer can influence the role of

282

aerobic biodegradation in mitigating vapor intrusion because the biodegradation rate in soil is

283

limited by mass transfer controlled bioavailability. When local equilibrium mass transfer is

284

assumed, the reaction at steady state approximated by an instantaneous reaction model can

285

highly overestimate the biodegradation rate and capacity and underestimate vapor intrusion

286

fluxes through the unsaturated zone. For the defined model with practical parameter values, the

287

attenuation coefficient can vary by four orders of magnitude. For more general findings, model

288

parameters can be varied to conduct uncertainty analyses to examine the effects of rate-limited

289

mass transfer under various field conditions. In addition, model simulations indicate that

290

rate-limited mass transfer can lead to larger overlaps of contaminant vapor and oxygen

291

concentrations, which cannot be explained by the instantaneous reaction model with local

292

equilibrium mass transfer. On the other hand, the biodegradation reaction often simplifies to an

293

instantaneous reaction in the liquid phase (see Figure 2) regardless of the bioavailability

294

controlled by diffusion in the vapor phase or mass transfer. By assuming complete degradation in

295

the liquid phase, equations (1a) and (1b) can be simplified to first-order reaction:

296

DDG

∂ 2 c DG c DG − k a =0 D HD ∂z 2

(15a)

297

DAG

∂ 2 cAG cAG − k a =0 A ∂z 2 HA

(15b)

19


Page 21 of 27


298

Equations (15a) and (15b) define the limiting case with the fastest degradation completely

299

controlled by rate-limited mass transfer. Thus, the effect of mass transfer limitations on steady

300

state concentration distributions could be accounted for by the first-order model with the reaction

301

rate coefficient determined by mass transfer and the biodegradation kinetics. Thus, models with

302

the consideration of rate-limited mass transfer can explain both the instantaneous and first-order

303

reaction kinetics.

304

As we describe in the introduction, there are many mechanisms that can constrain biodegradation

305

in vapor transport and could even have similar effects as rate-limited mass transfer on

306

concentration distributions, reaction flux and location based on our modeling. To determine the

307

occurrence of rate-limited mass transfer, concentrations in both vapor and liquid phase should be

308

measured to verify the chemical equilibrium relationship. It will be important to explore the

309

effects of other limiting mechanisms and the relative importance of various mechanisms in future

310

studies. In addition, for unsteady-state systems, such as the unsaturated zone with a fluctuating

311

water table, rate-limited mass transfer may also affect situations without aerobic biodegradation.

312

References:

313 314 315

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