Ž .Journal of Contaminant Hydrology 33 1998 133–148

Assessment of nonequilibrium in gas–water masstransfer during advective gas-phase transport

in soils

Ulrich Fischer a,), Christoph Hinz b, Rainer Schulin c,Fritz Stauffer d

a Department of Chemical Engineering, Swiss Federal Institute of Technology, ETH Zentrum, CH 8092Zurich, Switzerland

b Institute of Soil Science and Forest Nutrition, UniÕersity of Gottingen, Busgenweg 2, D-37077 Gottingen,¨ ¨ ¨Germany

c Institute of Terrestrial Ecology, Swiss Federal Institute of Technology ETH, Grabenstrasse 3r11a, CH 8952Schlieren, Switzerland

d Institute of Hydromechanics and Water Resources Management, Swiss Federal Institute of Technology, ETHHoenggerberg, CH 8093 Zurich, Switzerland

Abstract

In soils advective gas flow may be due to natural processes, e.g., water-table fluctuations, orŽ .may be a result of remediation techniques, such as soil vapor extraction SVE , which are used to

Ž .remove volatile organic compounds VOCs from the vadose zone. Recently, multicomponentexperiments as well as numerical simulations provided evidence that the efficiency of SVEoperations can be limited by slow diffusion of VOCs from interparticle water into the gas phase.Using a first-order kinetics approach, we show that the degree of local nonequilibrium between the

Ž .two phases and, thus, of the validity of the local equilibrium assumption LEA can becharacterized by the prefix denominator, P , for gas–water mass transfer which is derived by theD

Ž . wSKIT Separation of the Kinetically Influenced Term procedure of Bahr and Rubin Bahr, J.M.,Rubin, J., 1987. Direct comparison of kinetic and local equilibrium formulations for solute

Ž . xtransport affected by surface reactions. Water Resour. Res. 23 3 , 438–452 . Values of the prefixdenominator smaller than 1 indicated nonequilibrium between the two phases while P -values ofD

10 or larger indicated local equilibrium between aqueous and gaseous VOC concentrations. These

) Corresponding author. Fax: q41 1 632 11 89; e-mail: ufischer@tech.chem.ethz.ch

0169-7722r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved.Ž .PII S0169-7722 98 00068-0

( )U. Fischer et al.rJournal of Contaminant Hydrology 33 1998 133–148134

Ž .findings can be used to optimize field applications of soil vapor extraction SVE . q 1998 ElsevierScience B.V. All rights reserved.

Keywords: Gas advection; Mass transfer; Soil vapor extraction; Volatile organic compounds

1. Introduction

In soils, advective gas flow may be due to changes in the barometric pressure or tolocal turbulence when air is passing over the soil surface. Infiltration as well as watertable fluctuations may also cause advective gas flow in the unsaturated zone. By

Ž .applying remediation techniques such as soil vapor extraction SVE , advective gas flowŽ .is created artificially to remove volatile organic compounds VOCs . Design andŽ .operation of SVE systems is discussed in detail by Johnson et al. 1990 and Wilson and

Ž .Clarke 1994 . Usually, SVE is quite effective during the early stages of operation. Inlater stages, however, the removal of contaminants often decreases due to low masstransfer rates which are almost independent of the pumping rate. Decreasing efficiencyof SVE operations can be the result of mass transfer limitations from a nonaqueous

Ž .phase liquid NAPL , the soil water, or the solid phase into the mobile gas phase.Ž .Fischer et al. 1996 conducted multicomponent SVE experiments in a sand tank at

different water saturation showing that rate-limited mass transfer and, thus, localnonequilibrium may be caused solely by diffusion in the interparticle water. The

Ž .deviations of the gas concentrations from the local equilibrium assumption LEA couldbe described by means of a first-order kinetics approach using calibrated mass-transfercoefficients.

Various concepts have been used to assess the validity of the local equilibriumŽ . Ž .assumption. Valocchi 1985 and Parker and Valocchi 1986 based their analysis on the

comparison of temporal moments of solute breakthrough curves. Jennings and KirknerŽ .1984 expressed the validity of the LEA in terms of the Damkohler I number¨Ž . Ž .Damkohler, 1936 . Bahr and Rubin 1987 developed the procedure of separation of the¨

Ž .kinetically influenced term SKIT which is based on direct comparison of the mathe-matical formulations of equilibrium and kinetics model. By means of SKIT, the prefixdenominator, P , is derived which provides an alternative measure for nonequilibrium.D

Ž .Bahr and Rubin 1987 derived the prefix denominator for surface reactions. BahrŽ .1990 extended this approach to homogeneous and heterogeneous classical reactions.

The first objective of this paper is to derive the prefix denominator for gas–watermass transfer by applying the SKIT procedure to the equilibrium and kinetics model of

Ž . Ž .Armstrong et al. 1994 which was used by Fischer et al. 1996 to model gasconcentrations obtained from their SVE experiments. Furthermore, a sensitivity analysisof the kinetics model as a function of the prefix denominator is conducted, and for 32

Ž .data sets obtained by Fischer et al. 1996 the values of the prefix denominator as wellas the corresponding Damkohler I number are calculated. From the comparison of¨experimental data and numerical simulations, criteria are obtained that indicate equilib-rium or nonequilibrium gas–water mass transfer. We will discuss how these criteria canbe used to optimize field applications of soil vapor extraction.

( )U. Fischer et al.rJournal of Contaminant Hydrology 33 1998 133–148 135

2. Theory

Ž .Following the approach of Armstrong et al. 1994 , mass transport in the gas phaseunder local equilibrium between gaseous and aqueous concentrations of a compound canbe described by the following advection–dispersion equation:

E C E C E C E E C E Cg w s g gu qu qr s u D yu Õ 'L C ijsx , zŽ .g w b g i j g i gž /E t E t E t E x E x E xi j i

1Ž .

w x w y3 x w xwhere u is volumetric gas content y , C is gas concentration M L , t is time T ,g gw xu is volumetric content of the stationary water phase y , C is the aqueousw w

w y3 x w y3 xconcentration M L , r is the soil bulk density M L , C is the sorbedb sw y1 x w xconcentration M M , the x denote the spatial coordinates L , D are components ofi i j

w 2 y1 x w y1 x Ž .the dispersion tensor L T , the Õ are velocity components L T , and L C is thei gŽ .differential operator denoting the advective and dispersive terms. Eq. 1 can be written

as

E CgRu sL C 2Ž .Ž .g gE t

w x Ž .in which R y is the retardation factor for gas-phase transport Armstrong et al., 1994 :

u r Kw b dRs1q q 3Ž .

u H u Hg c g c

where H is the compound’s Henry’s law constant, and K is the solid–waterc dw 3 y1 xdistribution coefficient L M . This model is hereafter called the equilibrium model.

For local nonequilibrium between the soil water and the gas phase, Armstrong et al.Ž .1994 presented a first-order kinetics model in which transport in the gas phase isdescribed as

E Cgu qu l C yH C sL C 4Ž .Ž . Ž .g g gw g c w gE t

w y1 xwhere l is the gas–water mass-transfer coefficient T . Assuming equilibriumgwŽ .sorption the storage equation for the water–solids phase is Armstrong et al., 1994 :

E Cwu R su l C yH C 5Ž .Ž .w s g gw g c wE t

w xwhere R is the water–solid retardation factor y :s

r Kb dR s1q 6Ž .s

uw

Hereafter this model is referred to as the kinetics model.Ž .The SKIT Separation of the Kinetically Influenced Term procedure introduced byŽ .Bahr and Rubin 1987 consists in deriving equivalent dimensionless formulations of the

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equilibrium model and the kinetics model and identifying the additional term in thekinetics model which is called the kinetically influenced term. The dimensionless formof the equilibrium model is:

E CX 1 E E CXE CX 1g g g X Xs u Pe yV ' L C 7Ž .Ž .g i j i gž /E T u E X E X E X ug i j i g

Ž . XŽ X .in which, in analogy to L C , L C denotes the dimensionless differential operatorg g

representing the advective and dispersive terms. The dimensionless parameters aredefined as follows:

CgXC s 8aŽ .g Cg ,0

Õ txTs 8bŽ .

RLx

XiX s 8cŽ .i Lx

Di jPe s 8dŽ .i j L Õx x

ÕiV s 8eŽ .i

Õx

X wwhere C is relative gas concentration, C is the gas concentration at time zero Mg g,0y3 x w y1 xL , T is dimensionless time, Õ is the gas velocity component in x-direction L T ,x

w xL is a reference length in x-direction L , the X are relative distances in x-andx i

z-direction normalized by L , Pe denotes a tensor, whose components are the inversex i j

Peclet numbers, and the V are relative velocities normalized by Õ . In order to derivei xŽ .the corresponding formulation representing the kinetics model, Eq. 4 is rearranged to

express C asw

1 E C 1 1gC s y L C q C 9Ž .Ž .w g g

l H E t u Hgw c g c

Its time derivative

E C 1 E E C 1 1 E Cw g gs y L C q 10Ž .Ž .gE t l H E t E t u H E tgw c g c

Ž .is then substituted into Eq. 1 yielding after some rearrangement the SKIT formulationof the kinetics model:

E C 1 1 E E C 1g gs L C y y L C 11Ž .Ž . Ž .g gH uE t u R E t E t uc gg g

l Rgwuw

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which in dimensionless form is:X X

E C 1 1 E 1 E C 1g gX X X Xs L C y y L C 12Ž .Ž . Ž .g gH uE T u E T R E T uc gg gv R

uw

Ž .where v is the Damkohler I number Damkohler, 1936 given as¨ ¨l Lgw x

vs 13Ž .Õx

Ž . Ž .The second term on the right hand side of Eq. 12 does not appear in Eq. 7 and, thus,represents the kinetically influenced term. The reciprocal of the factor in front of thedimensionless time derivative of the kinetically influenced term is the prefix denomina-tor, P , for gas–water mass transfer in the kinetics model:D

H uc gP sv R 14Ž .D

uw

Ž .Assuming that sorption to the solid phase is negligible Rs1 , this result is similar toŽ .that obtained by Bahr and Rubin 1987 for first-order sorption kinetics and one-dimen-

Ž .sional transport. On the basis of numerical simulations Bahr and Rubin 1987 con-cluded that the magnitude of the kinetically influenced term as a whole is controlled bythe magnitude of the prefix factor with denominator P and that this parameter,D

therefore, provides a measure for the degree of nonequilibrium and the validity of thelocal equilibrium assumption.

3. Results and discussion

3.1. Summary of soil Õapor extraction experiments and numerical simulations

Ž .The soil vapor extraction experiments of Fischer et al. 1996 were carried out in aŽ .tank Fig. 1 using quartz sand as a model soil. The four compounds 1,1,1-trichloro-

Fig. 1. Experimental set-up of soil vapor extraction experiments and positions of sampling ports.

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Table 1Characteristics of soil vapor extraction experiments V1 and V2

y1w x w x w xExperiment Water saturation y Gas pressure head difference cm Gas velocity cm s

Minimum Maximum Minimum Maximumy3 y2V1 0.07 0.30 11 9.74=10 1.42=10y3 y2V2 0.08 0.54 14 2.75=10 1.80=10

Ž . Ž . Ž .ethane 1,1,1-TCA , 1,1,2-trichloroethane 1,1,2-TCA , trichloroethylene TCE , andŽ . Ž .perchloroethylene PCE with Henry’s law constants Dilling, 1977; Fischer et al., 1996

of 0.73, 0.04, 0.45, and 0.69, respectively, were assayed in multicomponent experi-ments. Gas samples were taken at different locations of the tank and the concentrationsof the four compounds in these samples were determined by gas chromatography. Thecompounds did not adsorb onto the quartz sand. The data obtained during two soil

Ž .venting experiments V1, V2 performed at low water saturation are used in this study.Relevant characteristics and parameters of these experiments are summarized in Table 1.More details about the experimental procedures applied in the tank venting experiments

Ž .are given by Fischer et al. 1996 .Ž .Relative concentrations measured by Fischer et al. 1996 at each location for

1,1,1-TCA, TCE, and PCE during experiments V1 and V2 are quite similar, but differconsiderably from those observed for 1,1,2-TCA. Concentrations measured at differentsampling ports of the sand tank during one experiment show a strong effect of water

Ž .Fig. 2. Relative gas concentration distributions of 1,1,1-trichloroethane 1,1,1-TCA , 1,1,2-trichloroethaneŽ . Ž . Ž .1,1,2-TCA , trichloroethylene TCE , and perchloroethylene PCE recorded at four sampling ports during

Ž .soil vapor extraction experiments V1 and V2 plotted as a function of dimensionless time as defined in Eq. 5Ž .symbols not further defined .

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Ž .Fig. 3. Comparison of relative gas concentrations measured for 1,1,1-trichloroethane 1,1,1-TCA andŽ .1,1,2-trichloroethane 1,1,2-TCA at sampling port SP 13 during experiment V1 and simulations obtained with

Ž . Ž . Ž y6 y1.the equilibrium EQ model and the kinetics KIN model l s6=10 s .gw

saturation on the gas phase concentration of the VOCs. While overall relative concentra-tions obtained in each experiment differ considerably, they are all characterized by asteep initial decline and a long tail. When these curves are scaled to dimensionless time

Ž .as defined in Eq. 8b very similar curves are obtained as shown in Fig. 2.The hypothesis of local nonequilibrium between gaseous and aqueous VOCs as

indicated by the tailing of the gas concentration distributions is supported by numericalsimulations using the equilibrium and the kinetics model. The equilibrium modelmatches the initial steep decline in concentration observed for 1,1,1-TCA, TCE, and

Ž .PCE but fails to describe the tailing thereafter Fig. 3 . The kinetics model, on the otherhand, describes the experimental data in general well when values of the mass-transfer

Ž .coefficient are optimized to match the tailing Fig. 3 . The small Henry’s law constantfor 1,1,2-TCA resulted in much lower dimensionless times as compared to the otherthree compounds. Therefore the last measurements for this compound had not yetreached the stage of slow mass removal and could be successfully modeled with theequilibrium model. Because the VOC did not adsorb onto the quartz sand, Fischer et al.Ž .1996 concluded that diffusion within the interparticle water was the rate-limiting stepduring the extraction process.

3.2. EÕaluation of the prefix denominator as an indicator of nonequilibrium in gas–watermass transfer-model sensitiÕity

Fig. 4 shows a comparison of simulated 1,1,1-TCA concentrations obtained from theequilibrium model and the kinetics model. Water saturation is 0.22 and gas velocity is1.17=10y2 cm sy1. In the kinetics model, the value of the gas–water mass-transfer

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Fig. 4. Comparison of simulations obtained with the equilibrium model and the kinetics model. The kineticssimulations represent different values of the prefix denominator, P , and were obtained by varying the valueD

Ž .of the gas–water mass-transfer coefficient for a given combination of compound 1,1,1-TCA , water saturationŽ . Ž y2 y1.0.22 , and gas velocity 1.17=10 cm s .

coefficient, l , is varied such that values of the prefix denominator, P , range fromgw D

0.01 to 10. The kinetics simulation for P s10 is quite close to the equilibriumDŽ .simulation Fig. 4 and larger values of the prefix denominator yield concentrations even

Fig. 5. Relative water concentrations plotted as a function of the prefix denominator, P . Water concentrationsD

were calculated for a simulation period of 25 h and the same combination of parameters as investigated in Fig.4.

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closer to the equilibrium model results. The kinetics simulation for P s1 shows someDŽ .deviation from the equilibrium simulation but no tailing behavior at small times Fig. 4 .

Tailing of gas concentrations appears at a P -value of 0.1 and is quite pronounced forD

P s0.01.D

For the same combination of parameters as investigated in Fig. 4, relative waterconcentration at 25 h is plotted in Fig. 5 as a function of P . While similarD

concentrations are obtained for P -values of either smaller than 0.1, or larger than 10, aD

steep transition appears between these two values. This result means that data that arematched by a kinetics simulation with a prefix denominator of 10 or larger should bedescribed quite well by the equilibrium model. The equilibrium model may adequatelydescribe some cases with P between 1 and 10, depending on the specific problem andD

goals of the simulation.

3.3. Application of the prefix denominator as an indicator of nonequilibrium ingas–water mass transfer

In this section the prefix denominator is applied to the 32 data sets obtained byŽ .Fischer et al. 1996 during their experiments V1 and V2 in order to assess the degree of

nonequilibrium at the end of these experiments. Using the optimized values of themass-transfer coefficient, l , given in Table 2, the corresponding values of thegw

Damkohler I number, v, and the prefix denominator, P , are calculated according to¨ DŽ . Ž .Eqs. 13 and 14 , respectively. Linear gas velocities in x-direction, Õ , are approxi-x

mated by average linear velocities, Õ, assuming that in the experiments the differencesbetween these two quantities were negligible. Average linear gas velocities, Õ, arecalculated as

q 1 D pÕs syk k 15Ž .rg

u u m D xg g g

w y1 xwhere q is specific discharge L T , D p is the difference in the applied vacuumw y1 y2 xpressures M L T , and D x is the distance between the two wells of the sand tank

w x Ž .L . Bahr and Rubin 1987 pointed out that the choice of the reference length Lx

selected for the calculation of P to a certain extent depends on the problem and thatD

for predicting concentration histories in a natural system the distance of the observationpoint from the location of solute input would be an appropriate choice of the referencelength. In analogy, in the present study L is chosen as the distance between eachx

sampling port and the air injection well of the sand tank.The calculated values of v and P are given in Table 2. The largest are forD

1,1,2-TCA and range from 2 to 5. The corresponding data are described well by theŽ .equilibrium model see Fig. 3 for comparison . Experimental data corresponding to PD

smaller than 0.4 can not be described correctly by the equilibrium model at all times.Ž . Ž . Ž .Jennings and Kirkner 1984 , Valocchi 1985 , and Bahr and Rubin 1987 investigated

the validity of the LEA for modeling sorption and complexation processes. Theseauthors concluded that the LEA is applicable if the Damkohler I number exceeds 100.¨

Ž .Jennings and Kirkner 1984 obtained reasonably good approximations of the LEA for

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Table 2Values of the Damkohler I number, v, and the prefix denominator, P , calculated for the gas concentration distributions obtained for four compounds at four¨ D

Ž .sampling ports, SP, during soil vapor extraction experiments V1 and V2 see Table 1y1 y1w x w x w x w x w x w x w x w xExperiment SP S y Õ cm s Compound H y l s v y P y L m F yw c gw D x ,crit

y2 y6 y2 y2V1 13 0.22 1.17=10 1,1,1-TCA 0.73 6=10 2.1=10 7.6=10 54 131y2 y3 0 0V1 13 0.22 1.17=10 1,1,2-TCA 0.04 1=10 3.5=10 4.0=10 1.0 2.5y2 y5 y2 y2V1 13 0.22 1.17=10 TCE 0.45 1=10 3.5=10 9.2=10 45 109y2 y6 y2 y2V1 13 0.22 1.17=10 PCE 0.69 5=10 2.1=10 7.3=10 56 137y2 y6 y2 y2V1 23 0.12 1.35=10 1,1,1-TCA 0.73 6=10 1.2=10 7.8=10 53 129y2 y3 0 0V1 23 0.12 1.35=10 1,1,2-TCA 0.04 1=10 3.1=10 4.0=10 1.0 2.5y2 y5 y2 y1V1 23 0.12 1.35=10 TCE 0.45 1=10 2.4=10 1.1=10 39 95y2 y6 y2 y2V1 23 0.12 1.35=10 PCE 0.69 5=10 1.2=10 7.4=10 56 135y2 y6 y3 y2V1 33 0.09 1.41=10 1,1,1-TCA 0.73 6=10 5.9=10 4.9=10 84 203y2 y3 0 0V1 33 0.09 1.41=10 1,1,2-TCA 0.04 1=10 2.9=10 4.1=10 1.0 2.4y2 y5 y2 y2V1 33 0.09 1.41=10 TCE 0.45 1=10 1.2=10 6.5=10 63 153y2 y6 y3 y2V1 33 0.09 1.41=10 PCE 0.69 5=10 5.9=10 4.7=10 88 214

V1 43 0.07 1.41=10y2 1,1,1-TCA 0.73 6=10y6 5.9=10y3 6.3=10y2 66 160y2 y3 0 0V1 43 0.07 1.41=10 1,1,2-TCA 0.04 1=10 2.9=10 4.4=10 0.9 2.2y2 y5 y3 y2V1 43 0.07 1.41=10 TCE 0.45 1=10 8.8=10 6.1=10 67 163y2 y6 y3 y2V1 43 0.07 1.41=10 PCE 0.69 5=10 5.9=10 5.9=10 69 168y3 y5 y2 y2V2 13 0.38 9.14=10 1,1,1-TCA 0.73 2=10 1.8=10 4.0=10 104 253y3 y4 0 0V2 13 0.38 9.14=10 1,1,2-TCA 0.04 8=10 3.6=10 3.9=10 1.1 2.6

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y3 y5 y1 y1V2 13 0.38 9.14=10 TCE 0.45 3=10 2.7=10 4.7=10 8.8 21y3 y6 y2 y2V2 13 0.38 9.14=10 PCE 0.69 8=10 4.5=10 9.6=10 43 104y2 y5 y3 y2V2 23 0.17 1.62=10 1,1,1-TCA 0.73 2=10 5.1=10 2.3=10 177 429y2 y4 0 0V2 23 0.17 1.62=10 1,1,2-TCA 0.04 8=10 2.0=10 2.4=10 1.7 4.1y2 y5 y2 y2V2 23 0.17 1.62=10 TCE 0.45 3=10 1.0=10 3.3=10 126 306y2 y6 y2 y2V2 23 0.17 1.62=10 PCE 0.69 8=10 1.5=10 6.7=10 62 149y2 y5 y3 y2V2 33 0.11 1.76=10 1,1,1-TCA 0.73 2=10 4.7=10 3.2=10 128 309y2 y4 0 0V2 33 0.11 1.76=10 1,1,2-TCA 0.04 8=10 1.9=10 2.5=10 1.7 4.0y2 y5 y3 y2V2 33 0.11 1.76=10 TCE 0.45 3=10 9.4=10 4.4=10 95 230y2 y6 y3 y2V2 33 0.11 1.76=10 PCE 0.69 8=10 4.7=10 3.1=10 134 324y2 y5 y3 y2V2 43 0.08 1.78=10 1,1,1-TCA 0.73 2=10 4.6=10 4.4=10 95 230y2 y4 0 0V2 43 0.08 1.78=10 1,1,2-TCA 0.04 8=10 1.9=10 2.7=10 1.5 3.7y2 y5 y3 y2V2 43 0.08 1.78=10 TCE 0.45 3=10 9.3=10 5.7=10 72 175

V2 43 0.08 1.78=10y2 PCE 0.69 8=10y6 4.6=10y3 4.1=10y2 100 242

The values of v and P were calculated using the values of water saturation, S , and gas velocity, Õ, at the sampling port under consideration, the Henry’s lawD w

constant of the compound, H , and the values of the calibrated gas–water mass-transfer coefficients, l . Assuming that a P -value of at least 10 indicatesc gw D,crit

equilibrium gas–water mass transfer, the values of L give the minimum distances needed to reach equilibrium conditions and the values of F are the ratios ofx ,crit

L rL s Õr Õ where Õ is the maximum gas velocity to be established which still results in equilibrium gas–water mass transfer.x ,crit x crit crit

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values of v as low as 10. For values below 1 the LEA simulation led to large deviationsfrom the experimental data at late times. Thus, the results found in the present study

Ž .agree quite well with the findings of Jennings and Kirkner 1984 .Ž .Bahr and Rubin 1987 reported that equilibrium transport may be assumed for values

of the dimensionless prefix denominator, P , on the order of 100 or greater. In thisD

study, the values of P for which transport has been found to be close to localD

equilibrium conditions are at least one order of magnitude smaller and suggest thatalready for P -values of 5 to 10 the LEA may be a good approximation to describeD

gas–water mass transfer.Compared to the Damkohler I number, using the prefix denominator as an indicator¨

for nonequilibrium has the advantage that it includes capacity factors which representŽ .the varying physical conditions of a soil gas and water saturation and the physico-

Ž .chemical properties Henry’s law constant, solid–water distribution coefficient of thecompounds. In Fig. 6 the values of the prefix denominator calculated for the 32 data setsshown in Fig. 2 are plotted as a function of dimensionless time at the end of theexperiments, T . The values of P tend to decrease with dimensionless time, approach-E D

ing a limiting value of about 10y2 . This is due to the fact that the best-fit mass transfercoefficients tend to decrease for increasing dimensionless time at the end of theexperiments, thus, suggesting that the assumption of first-order kinetics may not be

Ž .entirely valid as previously discussed in more detail by Fischer et al. 1996 . Thetransition of P -values signifying local equilibrium conditions to those values signifyingD

Ž .nonequilibrium transport appears at the same dimensionless time Ts2 at which thescaled concentrations show a sharp transition from a steep decline to a gradual approachŽ . Ž .tailing to a nearly constant value at later times see Figs. 2 and 3 for comparison .

Fig. 6. Degree of nonequilibrium at the end of experiments V1 and V2 expressed in terms of the prefixdenominator, P , and plotted as a function of the corresponding dimensionless time, T , for 1,1,1-trichloro-D E

Ž . Ž . Ž . Ž .ethane 1,1,1-TCA , 1,1,2-trichloroethane 1,1,2-TCA , trichloroethylene TCE , and perchloroethylene PCE .P -values signifying conditions close to local equilibrium are represented by filled symbols while those valuesD

signifying nonequilibrium transport are shown by open symbols.

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The findings presented here are useful in determining physical constraints for theŽ .acceptance of the LEA in field applications of soil vapor extraction. Inserting Eq. 13

Ž .into Eq. 14 and given a critical value of the prefix denominator, P , equations forDcrit

calculating a critical distance, L , or a critical gas velocity, Õ , are obtained. Thex ,crit crit

critical distance in this sense is the minimum distance which is needed to reachequilibrium conditions for a given combination of compound, water saturation, gasvelocity, and gas–water mass-transfer coefficient:

P ÕuD ,crit wL s 16Ž .x ,crit

l RH ugw c g

For a given length L the critical gas velocity represents the maximum velocity to bex

established which still results in gas–water equilibrium at the extraction well:

l L RH ugw x c gÕ s 17Ž .crit P uD ,crit w y

Calculating values of L and Õ we choose the value of the prefix denominator tox ,crit crit

be 10, since according to our findings for a P -value of this magnitude or larger,D

gas–water mass transfer should be described quite well by means of the local equilib-rium assumption.

Ž .In the experiments of Fischer et al. 1996 , the gas velocity would have had to beŽ .decreased by factors of up to 429 see values of F in Table 2 in order to establish

equilibrium conditions at the end of the experiment for the compounds 1,1,1-TCA, TCE,and PCE. For these three compounds and the gas velocities actually applied in theexperiments, which were on the order of those in typical field applications of soil vaporextraction, equilibrium conditions would have been reached at distances of 8.8 to 177 mŽ .Table 2 compared to the distance of 0.41 m at which the sampling ports investigatedhere were installed from the air entry well of the sand tank.

If the gas–water mass-transfer coefficient can be estimated from soil properties andŽ .physico-chemical data e.g., Brusseau, 1991 or from curve-fitting of effluent concentra-

tions at extraction wells, then the calculation of critical length and critical velocity canbe used to optimize field applications of soil vapor extraction. Mass transfer limitationsduring SVE lead to low effluent concentrations resulting in cost-ineffective ventingoperations because large volumes of diluted soil gas are extracted which have to bepurified. The question whether low contaminant removal rates signify a strongly reducedcontaminant mass in the soil or indicate mass transfer limitations can be answered bymeans of the prefix denominator. Energy and money may be saved by reducing the flowrate or by pulsed pumping. For the same volume of extracted gas, a low pumping rate isadvantageous above pulsed pumping because it maintains a high concentration gradientbetween the phases. The lowest pumping rate which still results in the highest possibleloading rate at the extraction well can be calculated according to Õ . With regard to thecrit

installation of air-entry or extraction wells, the spacing between wells can be adjustedŽ .according to L if the minimal pumping rate is fixed due to technical specificationsx ,crit

of the pumping unit and the gas permeability of the soil.

( )U. Fischer et al.rJournal of Contaminant Hydrology 33 1998 133–148146

Following the procedure described in this paper, the prefix denominator may also bederived for water–solid mass transfer if this process is rate-limiting. Thus, for situationsin which sorption can not be described with the local equilibrium assumption, masstransfer from the aqueous and solid phases to the gas phase is characterized either by theprefix denominator for water–solid mass transfer or by two parameters assessing thedegree of local nonequilibrium for water–solid as well as gas–water mass transfer. Thelower value of these two parameters then indicates the overall rate-limiting step inadvective gas-phase transport in soils.

4. Conclusions

The prefix denominator, P , as derived by the SKIT procedure can be used as anD

indicator of nonequilibrium in gas–water mass transfer during advective gas-phasetransport in soils. As compared to the Damkohler I number, v, the prefix denominator¨includes additional capacity factors. Local equilibrium between gaseous and aqueousVOC concentrations may be assumed for values of the prefix denominator of 10 orlarger. This finding can be used to optimize well spacing and pumping rates in fieldapplications of soil vapor extraction. The analysis of local nonequilibrium in gas-phasetransport using the prefix denominator may be extended to situations where water–solidmass transfer is also a rate-limited process.

5. Notation

Latin symbolsw y3 xC Concentration in the gas phase M Lg

X w xC Relative concentration in the gas phase ygw y3 xC Concentration in the aqueous phase M Lw

w y3 xC Gas-phase concentration at time zero M Lg,0w y1 xC Sorbed concentration M Ms

w 2 y1 xD Components of dispersion tensor L Ti jw xH Henry’s law constant yc

w 3 y1 xK Solid–water distribution coefficient L Mdw 2 xk Intrinsic permeability L

w xk Relative gas permeability yrgw xL Reference length in x-direction Lx

w xL Critical length in x-direction Lx ,critŽ .L C Differential operator denoting the advective and dispersive termsg

w y3 y1 xin the advection–dispersion M L TXŽ X .L C Dimensionless differential operator denoting the advective andg

w xdispersive terms in the advection–dispersion equation yD p Difference between the gas pressures at the injection and the

w y1 y2 xextraction well M L Tw xP Prefix denominator derived by the SKIT procedure yD

( )U. Fischer et al.rJournal of Contaminant Hydrology 33 1998 133–148 147

w xP Critical prefix denominator yD,critw xPe Inverse Peclet numbers yi j

w y1 xq Specific discharge L Tw xR Retardation factor for gas-phase transport y

w xR Water–solid retardation factor ysw xt Time T

w xT Dimensionless time yw xT Dimensionless time at end of experiment yE

w y1 xÕ Average gas linear velocity L Tw y1 xÕ Critical average linear gas velocity L Tcrit

w y1 xÕ Gas velocity components L Tiw xV Relative gas velocities in i-direction yi

w xD x Distance between the two wells of the sand tank Lw xx Spatial coordinates Li

w xX Relative distance in i-direction yi

Greek symbolsl Gas–water mass transfer coefficient in the first-order kineticsgw

w y1 xmodel Tw y1 y1 xm Dynamic gas viscosity M L Tg

w y3 xr Soil bulk density M Lbw xu Volumetric gas content yg

w xu Volumetric water content yww xv Damkohler I number y¨

AbbreÕiationsEQ EquilibriumKIN KineticsLEA Local equilibrium assumption

Ž .PCE Perchloroethylene tetrachloroethyleneSKIT Separation of the kinetically influenced termSP Sampling portSVE Soil vapor extraction1,1,1-TCA 1,1,1-trichloroethane1,1,2-TCA 1,1,2-trichloroethaneTCE TrichloroethyleneV1, V2 Designation of soil vapor extraction experimentVOC Volatile organic compound

Acknowledgements

We would like to thank R. Haggerty and two anonymous reviewers as well as D.J.Goode for their valuable comments and suggestions on the manuscript. Financial supportfor the second author was provided by the Board of the Swiss Federal Institutes ofTechnology. This work was conducted at Swiss Federal Institute of Technology ETH.

( )U. Fischer et al.rJournal of Contaminant Hydrology 33 1998 133–148148

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