Correlations for mass transfer coefficients applicable to NAPL pool dissolution in subsurface formations C.V. Chrysikopoulos & T.-J. Kim Department of Civil and Environmental Engineering, University of California, Irvine, CA 92697, USA Abstract In this paper, we develop correlations describing the rate of interface mass transfer from single component nonaqueous phase liquid (NAPL) pools in saturated, homogeneous porous media. A three-dimensional contaminant transport model is employed to obtain overall mass transfer coefficients computed from concentration gradients at the NAPL-water interface. It is assumed that pool dissolution at the pool-water interface is fast enough so the dissolved concentration is only mass transfer limited, and the liquid phase concentration along the interface is constant and equal to the saturation concentration. Power-law correlations relate the overall Sherwood number to the appropriate overall Peclet numbers. The proposed relationships are fitted to numerically determined mass transfer coefficients and the correlation coefficients are determined by nonlinear least squares regression. 1 Introduction Groundwater contamination by nonaqueous phase liquids (NAPLs) originating from industrial and commercial activities currently is recognized as an important world-wide problem. Most of the NAPLs are organic solvents and petroleum hydrocarbons originating from leaking underground storage tanks, ruptured pipelines, surface spills, hazardous waste landfills, and disposal sites. As a NAPL is released into the subsurface environment, it infiltrates through the vadose zone leaving behind blobs or ganglia which are no longer connected to the main body of the organic liquid. Upon reaching the water table NAPLs lighter than water remain above the water
184 Computer Methods in Water Resources XII table in the form of a floating pool; NAPLs heavier than water continue to migrate downward until they encounter an impermeable layer where a flat pool starts to form. As groundwater flows past trapped ganglia or NAPL pools, a plume of dissolved hydrocarbons is created. The concentration of dissolved NAPLs in groundwater are primarily governed by inter phase mass-transfer processes that often are slow and rate-limited [16, 19]. There is a relatively large body of available literature on the migration of NAPLs and dissolution of residual blobs and lenses [1, 3, 8, 14, 19, 20], as well as pool dissolution [2, 5, 13, 15]. Furthermore, several theoretically and experimentally derived mass transfer relationships, expressed in terms of non-dimensional parameters, for residual NAPL blobs with simple geometry are presented in the literature [10, 17, 18]. Unfortunately, mass transfer correlations for NAPL pool dissolution in porous media have not been established yet. Mass transfer coefficients for NAPL pools are not easily determined with precision, because they may vary with location at the NAPL-water interface, and for unsteady-state flow conditions they exhibit temporal dependace [7, 11]. Concequently, mathematical models for contaminant transport originating from NAPL pool dissolution often employ an average and time invariant mass transfer coefficient applicable for the entire pool [6, 9]. In this paper, we develop overall mass transfer correlations for the rate of interface mass transfer from single component NAPL pools in saturated, homogeneous porous media. The correlations relate the dimensionless mass transfer coefficient, i.e., the Sherwood number with the appropriate Peclet numbers. 2 Mathematical Model The transient contaminant transport from a dissolving NAPL pool denser than water in a three-dimensional, homogeneous porous medium under steady-state uniform flow conditions, assuming that the dissolved organic is sorbing under local equilibrium conditions, is governed by the following dimensionless partial differential equation: ac i &c i &c i d*c dc the following dimensionless definitions were used C=-, X = ^ Y=T> Z=f' ^=T^' (2a,b,c,d,e) Cg tc t-c *-c ^c Pe - M± Pe -^ Pe -^ (2f * h) *&x n, *Zy, r^z ri ' V/M&»"/
Computer Methods in Water Resources XII 185..* ^"' = where c(t,x,y,z) is the liquid phase contaminant concentration; Cg is the aqueous concentration at the interface and for a pure organic liquid equals the liquid's aqueous saturation (solubility) concentration; D^,Dy,D^ are the longitudinal, transverse, and vertical hydrodynamic dispersion coefficients, respectively; Kd is the partition or distribution coefficient; ^ = (tx x iyy^ is the square root of the pool area and used as a characteristic length; R = 1 -f Kdp/9 is the dimensionless retardation factor for linear, reversible, instantaneous sorption; t is time; Ux is the average unidirectional interstitial fluid velocity; x, y, z are the spatial coordinates in the longitudinal, lateral, and vertical directions, respectively; 9 is the porosity of the porous medium; A is thefirst-orderdecay constant of the liquid phase concentration; A* is the first-order decay coefficient of the concentration sorbed onto the solid matrix; and p is the bulk density of the solid matrix. For the special case where A is equal to A*, the two decay terms in (1) are replaced by the term ARC. For a rectangular-shaped stagnant NAPL pool the appropriate dimensionless initial and boundary conditions are: =0, (3) )=0, (4) ) = 0, (5) y, o) = i x, y e %(,.), (6a).., C(T, X, y, oo) = 0, (7) where the dimensionless K^ is the domain defined by the rectangular poolwater interfacial area expressed as (8a,b) 2.1 Interface Mass Transfer For a uniform NAPL pool under local equilibrium conditions the interface mass flux is described by a single-resistance, linear-driving force model as follows [6] '^=fc(t.x.y)[c.-c(f,x.y,oo)], (9)
186 Computer Methods in Water Resources XII where D? = T>/r is the effective aqueous diffusion coefficient in the porous medium (where T> is the molecular diffusion coefficient; and r > 1 is the tortuosity coefficient); k(t,x,y) is the local mass transfer coefficient dependent on time and location on the NAPL-water interface; c(,x,?/, oo) ~ 0 represents the bulk contaminant concentration outside the boundary layer, which, for the present study, is considered equal to zero. The relationship (9) implies that at the interface there is no fluid motion and mass transfer occurs only by effective molecular diffusion. The local mass transfer coefficient is physically proportional to the gradient of the contaminant concentration at the NAPL-water interface, and is obtained from (9) as follows,,. Cg-c(2,z,2/, oo) '* >. do) The time required for complete pool dissolution is much longer than the contact time between the pool and the flowing groundwater [13]. Therefore, in this work local mass transfer coefficients are estimated at steady-state conditions. The time invariant local mass transfer coefficient is denoted by k(x,y). Because the local mass transfer coefficient at a specific location is usually not an easy parameter to determine with precision, in mathematical models of contaminant transport k(x, y) is often replaced by a time invariant overall (average) mass transfer coefficient &*, for the entire pool expressed as [11] where A is the surface area of the NAPL pool, and cpa is a differential surface area. In this work, A:* is numerically calculated by simply dividing the sum of the estimated time independent local mass transfer coefficients at the nodal points of the entire pool surface by the total number of nodal points used. It should be noted that the relationship (10) for the local mass transfer coefficient can also be expressed as (11) The dimensionless form of the local mass transfer coefficient, i.e., local Sherwood number, is given by SMTXY] c_.., Sh(T,X,Y)- -- ^. (13) Corresponding to the time invariant local mass transfer coefficientfc(x,y), the time invariant local Sherwood number is denoted by Sh(X,Y). The overall Sherwood number is defined as. (14)
Computer Methods in Water Resources XII 187 Figure 1. Schemetic illustrations of the three-dimensional numerical domains showing the rectangular DNAPL pool and the appropriate boundary conditions. 2.2 Numerical Solution The three-dimensional mathematical model presented by equation (21) is solved numerically by an alternating direction implicit (ADI), finitedifference scheme. The ADI algorithm is unconditionally stable and leads to a set of algebraic equations that form of a tridiagonal matrix which is solved by the highly efficient Thomas algorithm. Numerical simulations are performed to estimate profiles of dissolved concentration for various interstitial velocities, pool dimensions and pool geometries. The numerical domain as well as the corresponding boundary conditions used in this work are schematically illustrated in Figure 1. A zero flux boundary condition is applied to all outer boundaries of the numerical domain, and a constant boundary condition (6a) is applied at the poolwater interface. It should be noted that the numerical codes developed were consistently run until the simulated dissolved concentration profiles become time independent, because in this study the local mass transfer coefficients are calculated at steady-state conditions. 3 Development of Mass Transfer Correlations A total of 121 different rectangular pools with dimensions x*ty in the range from 5.0 mx5.0 m to 10.0 mxlo.o m are examined in this work. Overall Sherwood numbers are obtained from numerical simulations, varying the
188 Computer Methods in Water Resources XII Sh 12.5 2.5 3.0 3.5 4.0 4.5 5.0 Pe, Sh 10.4 9.7 9.0 25.0 30.0 35.0 40.0 45.0 50.0 Pe Figure 2. Computed (open circles) and predicted (solid lines) overall Sherwood numbers as a function of (a) Pe* and (b) Pey. Peclet numbers, pool size and pool shape. The general form of mass transfer relationships for single component NAPL pools is formulated through a dimensional analysis using the Buckingham Pi theorem which states that the smallest number of dimensionless groups associated with a system is determined by the difference between the number of variables and the number of basic dimensions of the variables [4, 22]. The mass transfer correlation in the form of power-law model
Computer Methods in Water Resources XII 189 of the two-directional Peclet numbers with a leading coefficient is developed for rectangular pools under the uniform flow condition in a concentration boundary layer: Sh = pipgp^, (15) where /3i, /%, and /% are empirical parameters to be determined from appropriate data. The nonlinear least squares regression routine RNLIN [12] is employed to estimate the parameters, f3\, 02, and /%, by fitting the nonlinear power law correlation (15) to 484 overall Sherwood numbers computed for 121 different pool dimensions and four different hydrodynamic conditions. The corresponding Pe* and Pey to simulation conditions range from 2.5 to 5.0 and 25.0 to 50.0, respectively. The resulting overall mass transfer correlation for rectangular pools is: ]%=1.587^^P^. (16) Figure 2 compares the Sherwood numbers predicted by (14) (open circles) to those computed from equation (16) (solid lines) as a function of Pe* (Fig. 2a) and Pey (Fig. 2b). 4 Summary Overall mass transfer correlations for the prediction of interface mass transfer coefficients associated with the dissolution of rectangular NAPL pools in saturated porous media are developed. A nondimensionalized advectivedispersion equation is solved using an ADI finite difference numerical scheme in order to estimate overall mass transfer coefficients. The transport model assumes that the aqueous phase concentration of the dissolved solute adjacent to the source is considered to be equal to the solubility limit. The overall mass transfer correlations developed, relate dimensionless overall mass transfer coefficients (Sherwood numbers) to overall Peclet numbers. The proposed power-law correlations are calibrated with overall mass transfer coefficients obtained from numerical simulations. Nonlinear least squares regression is used for the estimation of the leading and exponential coefficients of the correlations. The correlations presented here can be used to predict the mass transfer coefficients of single component NAPL pool dissolution in groundwater under equilibrium conditions. Acknowledgements. This work was sponsored by the U.S. Environmental Protection Agency, under award R-823579-01-0. However, the manuscript has not been subjected to the Agency's peer and administrative review and therefore does not necessarily reflect the views of the Agency and no official endorsement should be inferred.
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