Analytical solutions for a soil vapor extraction model that incorporates gas phase dispersion and molecular diffusion

Abstract

To greatly simplify their solution, the equations describing radial advective/dispersive transport to an extraction well in a porous medium typically neglect molecular diffusion. While this simplification is appropriate to simulate transport in the saturated zone, it can result in significant errors when modeling gas phase transport in the vadose zone, as might be applied when simulating a soil vapor extraction (SVE) system to remediate vadose zone contamination. A new analytical solution for the equations describing radial gas phase transport of a sorbing contaminant to an extraction well is presented. The equations model advection, dispersion (including both mechanical dispersion and molecular diffusion), and rate-limited mass transfer of dissolved, separate phase, and sorbed contaminants into the gas phase. The model equations are analytically solved by using the Laplace transform with respect to time. The solutions are represented by confluent hypergeometric functions in the Laplace domain. The Laplace domain solutions are then evaluated using a numerical Laplace inversion algorithm. The solutions can be used to simulate the spatial distribution and the temporal evolution of contaminant concentrations during operation of a soil vapor extraction well. Results of model simulations show that the effect of gas phase molecular diffusion upon concentrations at the extraction well is relatively small, although the effect upon the distribution of concentrations in space is significant. This study provides a tool that can be useful in designing SVE remediation strategies, as well as verifying numerical models used to simulate SVE system performance.

1. Introduction

Soil vapor extraction (SVE) is a physical treatment process for in situ remediation of volatile organic contaminants (VOCs) in the vadose zone. The process involves inducing air flow in unsaturated soil, thereby enhancing the in situ volatilization of the VOC from dissolved and separate (i.e., non-aqueous phase liquid (NAPL)) phases, accompanied by transfer of the contaminants from the sorbed phase into the gas phase. The gas containing the contaminants is then pumped out via an extraction well. The engineering performance of an SVE system is governed by the processes affecting VOC fate and transport (e.g., advection, dispersion, inter-phase mass transfer and decay) (Gierke et al., 1992, Travis and Macinnis, 1992, Cho et al., 1993, Armstrong et al., 1994, Popovicova and Brusseau, 1998, Lorden et al., 1998). Conventionally, SVE models simulate VOC transport in a convergent gas flow field (e.g., Goltz and Oxley, 1994). Because of the dependence of the gas flow velocity on radial distance, the dispersion coefficient (D) is also modeled as a function of radial distance, and may be represented as D = a_Lv(r) + D_m, where a_L is the longitudinal dispersivity, v(r) is the velocity of gas flow as a function of the radial distance (r), and D_m is the effective molecular diffusion coefficient in the vadose zone. To analytically solve the radial advective/dispersive transport equation, some simplifications are usually implemented. In particular, a number of models have been presented to describe transport in both the saturated and unsaturated zone that neglect molecular diffusion (D_m ≡ 0) (Chen, 1985, Chen, 1987, Moench, 1989, Moench, 1995, Chen et al., 2002, Chen et al., 2003, Huang et al., 2010, Wang and Zhan, 2013, Goltz and Oxley, 1991, Goltz and Oxley, 1994, Huang and Goltz, 1999, Lai et al., 2016). Chen et al. (2007) developed an approximate analytical model that incorporated both dispersion and molecular diffusion to simulate SVE. The Chen et al. (2007) model approximated the exact solution by linearly coupling and weighting the diffusion- and dispersion-dominated analytical solutions, using a composite parameter (a Peclet number that characterized the combination of diffusive and dispersive processes). Recently, a more complete model that incorporates both scale-dependent dispersivity and molecular diffusion was presented by Haddad et al. (2015). In the Haddad et al. (2015) model the dispersion coefficient was represented as (D = (ar + b)v(r) + D_m), where a and b are constants. The Haddad et al. (2015) model describes divergent flow of a conservative tracer from an injection well coupled with diffusive mass transfer of tracer into fractures.

In this study, a new analytical solution is developed that may be used to simulate contaminant transport in response to operation of a soil vapor extraction system. The model accounts for the following processes that affect VOC fate and transport in the vadose zone: (1) radial gas phase advection, (2) mechanical dispersion and effective molecular diffusion, (3) first-order degradation in gas, dissolved, and sorbed phases, and (4) rate-limited mass transfer between the gas phase and the dissolved/sorbed/NAPL phases. The model assumes the vadose zone is initially contaminated and the contaminated zone can be represented by a cylinder. The solution, which incorporates processes such as molecular diffusion and NAPL desaturation that were not considered in earlier models, can be employed to better understand how SVE performance is affected by various process parameter values.

2. Model equations

The conceptual model of SVE in a contaminated vadose zone is depicted in Fig. 1. In the vadose zone, voids are filled with air, water, and NAPL. Air containing volatilized contaminant is extracted through a fully screened well, while dissolved and NAPL phase contaminant volatilize into the gas phase. To mathematically describe the conceptual model, the following assumptions are made: 1) the vapor extraction well induces a steady, radially convergent gas flow field; 2) water and NAPL phases are immobile; 3) transport of volatilized contaminant in the vadose zone is governed by advection, dispersion and molecular diffusion; 4) contaminant mass transfer between phases can be described by first-order kinetics; 5) first-order decay of contaminant occurs in the dissolved, sorbed and volatilized phases; and, 6) saturation of NAPL is low, so that NAPL desaturation does not significantly change gas and water phase saturations, which are assumed constant.

Fig. 1.

Conceptual model for the soil vapor extraction.

Based on these assumptions, we modify Brusseau’s (1991) one-dimensional formulation and Goltz and Oxley, 1991, Goltz and Oxley, 1994 radial flow model, to obtain the following equations describing the SVE process:

$$\theta s_g\frac{\partial C_g}{\partial t}=\frac{1}{r}\frac{\partial }{\partial r}[(a_{L}q+r{D}_m)\frac{\partial C_g}{\partial r}+q{C}_g]+{\alpha }_{gw}(C_w-k_H{C}_g)+{\alpha }_{sg}(k_A{C}_s-k_H{C}_g)-\theta s_g{\lambda }_g{C}_g+R_E$$ (1)

$$\theta s_w\frac{\partial C_w}{\partial t}={\alpha }_{gw}(k_H{C}_g-C_w)+{\alpha }_{ws}(k_A{C}_s-C_w)-\theta s_w{\lambda }_w{C}_w+R_D$$ (2)

$${\rho }_b\frac{\partial C_s}{\partial t}={\alpha }_{ws}(C_w-k_A{C}_s)+{\alpha }_{sg}(k_H{C}_g-k_A{C}_s)-{\rho }_b{\lambda }_s{C}_s$$ (3)

Eq. (1) describes the contaminant transport in the gas phase in the vadose zone; Eqs. (2), (3) describe the interphase mass transfer. The subscripts g, w, s respectively refer to gas, water and solid phases; the subscripts gw, ws and sg respectively refer to interphase gas-water, water-solid and solid-gas; C is the contaminant concentration in gas, water or solid phase; θ is the porosity; s_g and s_w are respectively the gas and water saturation; ρ_b is the bulk density; q = Q_w/(2πH), where Q_w is the well pumping rate and H is the thickness of the vadose zone; a_L is the longitudinal dispersivity in the gas phase; D_m is the effective molecular diffusion coefficient of contaminant in the gas phase; λ is the first-order decay constant of the contaminant, α is the non-equilibrium first-order mass transfer rate coefficient; k_H is the dimensionless Henry coefficient, expressed so C_w = k_HC_g; k_A is the adsorption partitioning constant, expressed so C_w = k_AC_s at equilibrium; R_E is the volatilization rate of NAPL; R_D is the dissolution rate of NAPL; t is time; r is the radial coordinate.

The volatilization and dissolution of NAPL can be described by Eqs. (4a), (4b) respectively, which assume first-order mass transfer processes in accordance with Raoult’s Law:

$$R_E=s_n^{\beta }{k}_E(C_{equ}-C_g)$$ (4a)

$$R_D=s_n^{\beta }{k}_D(C_{sol}-C_w)$$ (4b)

where s_n is the saturation of NAPL; k_E is the volatilization rate coefficient of NAPL; C_equ is the equilibrium vapor concentration of NAPL in the gas phase; k_D is the dissolution rate coefficient of NAPL; C_sol is the solubility of NAPL in the water phase; and β is an empirical parameter characterizing the morphology of NAPL. To find an analytical solution for the SVE model, Eqs. (4a), (4b) need to be linearized. Using the Taylor series expansion, neglecting the quadratic and higher order terms, Eqs. (4a), (4b) may be approximated as:

$$R_E=k_E{s}_{nx}^{\beta }(C_{equ}-C_g)+k_E\beta s_{nx}^{\beta -1}(C_{equ}-C_{gx})(s_n-s_{nx})$$ (5a)

$$R_D=k_D{s}_{nx}^{\beta }(C_{sol}-C_w)+k_D\beta s_{nx}^{\beta -1}(C_{sol}-C_{wx})(s_n-s_{nx})$$ (5b)

where C_gx, C_wx and s_nx are, respectively, reference values of C_g, C_w and s_n. We select these reference values so that β(C_equ − C_gx) = C_equ, β(C_sol − C_wx) = C_sol and s_nx = s_n0 (initial value of NAPL saturation) to guarantee that when the system is at equilibrium, the NAPL volatilization and dissolution rates go to zero. Using Eqs. (5a), (5b), the NAPL desaturation equation then can be written as:

$$\theta {\rho }_n\frac{\partial s_n}{\partial t}=-(R_E-R_D)=-k_E{s}_{n0}^{\beta -1}\left(C_{equ}{s}_n-{s_{n0}C}_g\right)-k_D{s}_{n0}^{\beta -1}(C_{sol}{s}_n-{s_{n0}C}_w)$$ (6)

where ρ_n is the density of NAPL. The validity of this linearized NAPL desaturation model (6) to approximate the original non-linear Eqs. (4a), (4b) is investigated in Section 4, Model Solution Verification. Note that neglecting the volume of NAPL in the pore space implies s_g = 1 − s_w − s_n ≈ 1 − s_w.

Initial conditions are given by Eqs. (7a), (7b), (7c), (7d), which assume all phases are initially at equilibrium.

$$C_g=C_{equ},t=0$$ (7a)

$$C_w=C_{sol},t=0$$ (7b)

$$C_s=\frac{1}{k_A}{C}_{sol},t=0$$ (7c)

$$s_n=s_{n0},t=0$$ (7d)

Boundary conditions are defined by Eqs. (8a), (8b). At the extraction well:

$$\frac{\partial C_s}{\partial r}=0,r=r_w$$ (8a)

where r_w is the radius of the extraction well. At the edge of the contamination zone, the contaminant inward flux is assumed to be zero:

$$(a_{L}q+r{D}_m)\frac{\partial C_s}{\partial r}+q{C}_g=0,r=r_b$$ (8b)

where r_b is the radius of the contaminated zone.

3. Model solutions

Taking the Laplace transform to Eqs. (1), (2), (3) and (6) with respect to time t subject to the initial conditions (7) and boundary conditions (8), we may obtain the solution in the Laplace domain (see Appendix A for detailed derivation):

$${\overline{C}}_g=\frac{\mu }{\phi }[1-e^{(r_b-r)h}\frac{U(a,b,z)G_M(a,b,z_w)-M(a,b,z)G_U(a,b,z_w)}{G_M(a,b,z_w)H_U(a,b,z_b)-G_U(a,b,z_w)H_M(a,b,z_b)}]$$ (9)

where M(a, b, z) and U(a, b, z) are the confluent hypergeometric functions of the first and second kind, respectively. Notation for other intermediate variables included in (9) can be found in Appendix A

The rate at which mass is extracted at the extraction well, r_mass, is calculated as:

$$r_{mass}=Q_w{C}_g(r_w,t)$$ (10)

At a given time, t, the cumulative mass that has been extracted ($A_{mass}^t$) is calculated as:

$$A_{mass}^t=Q_w{\int }_0^t{C}_g(r_w,t)dt$$ (11)

In the Laplace domain,

$${\overline{r}}_{mass}=Q_w{\overline{C}}_g(r_w,p)$$ (12)

$${\overline{A}}_{mass}^p=\frac{Q_w}{p}{\overline{C}}_g(r_w,p)$$ (13)

where p is the Laplace transform variable.

4. Model solution verification

Numerical Laplace inverse transformation can be used to evaluate the solutions. One algorithm that is described by de Hoog et al. (1982) and has been implemented in MATLAB by Hollenbeck (1998) is employed in this study. Moench (1991) demonstrated that this algorithm could accurately and rapidly invert the Laplace domain solution for convergent radial dispersive transport “…for all Peclet numbers (i.e., dispersivities) of practical interest, and beyond.” The confluent hypergeometric function of the first kind, M(a, b, z), is evaluated using a program developed by Nardin et al. (1992). The confluent hypergeometric function of the second kind U(a, b, z) is evaluated using a formula by the integral representation (Abramowitz and Stegun, 1970):

$$U(a,b,z)=\frac{z^{-a}}{\Gamma (a)}{\int }_0^{\mathrm{\infty }}{e}^{-t}{t}^{a-1}(1+t/z{)}^{b-a-1}dt$$ (14)

when $\mathcal{R}(a)>0$ and $\mathcal{R}(z)>0$ (the real parts of a and z are positive), or evaluated using a definition formula (Abramowitz and Stegun, 1970):

$$U(a,b,z)=\frac{\pi }{\mathit{sin}(\pi b)}[\frac{M(a,b,z)}{\Gamma (1+a-b)\Gamma (b)}-z^{1-b}\frac{M(1+a-b,2-b,z)}{\Gamma (a)\Gamma (2-b)}]$$ (15)

when $\mathcal{R}(a)$ and $\mathcal{R}(z)$ are non-positive; in addition, the asymptotic expansion is applicable for |z| large (Abramowitz and Stegun, 1970):

$$U(a,b,z)=z^{-b}\left\{\sum _{n-0}^{R-1}\frac{{(a)}_n{(1+a-b)}_n}{n!}{(-z)}^{-n}+O({\left|z\right|}^{-R})\right\}$$ (16)

where ${(a)}_n=a(z+1)(a+2)\dots (z+n-1),{(a)}_0=1$, R is the number of terms of the sequence.

The analytical solutions developed here were verified by a numerical solution using time integration and spatial finite-difference methods. Implementing a numerical solution, the spatial discretization transforms the partial differential Eqs. (1), (2), (3) and (6) into an ordinary differential equation system, which then is solved by the Runge-Kutta method for the time integration. Concentration distributions along the radial direction at four times, calculated using the analytical and numerical methods and Table 1 parameter values, are depicted in Fig. 2. The excellent match gives confidence in the correctness of the analytical solution.

Table 1.

Base line parameter values used for simulations.

Parameter	θ	sw	ρb (kg/L)	αgw (1/day)	αws (1/day)	αsg (1/day)	kH	kA (kg/L)
Value	0.35	0.1	1.76	0.5	0.06	0.03	2.619	0.25
Parameter	λg (1/day)	λw (1/day)	λs (1/day)	Qw (m3/day)	aL (m)	Dm (m2/day)	H (m)	rw (m)
Value	0.0002	0.0003	0.0004	200	0.04	0.25	3	0.2
Parameter	rb (m)	kE (1/day)	kD (1/day)	sn0	β	Cequ (mg/L)	Csol (mg/L)	ρn (mg/L)
Value	50	0.3	0.1	0.02	0.746	420	1100	1.46 × 106

Fig. 2.

Concentration distributions calculated analytically and numerically at four times for the Table 1 parameter values.

In addition, the validity of linearization of the NAPL desaturation model is verified by comparing the analytical solution that uses the linearized NAPL desaturation model (Eqs. (5a), (5b)) and the numerical solution that uses the nonlinear NAPL desaturation model (Eqs. (4a), (4b)). Fig. 3 shows that the linearization of the NAPL desaturation model adequately approximates the process of NAPL dissolution and evaporation for the system where the initial NAPL saturation is low (s_n0 < 0.02) and when the NAPL morphology parameter, β, is unity. Here, in order to evaluate the accuracy of the linearized NAPL desaturation model (Eqs. (1), (2), (3) and (6)) we are assuming that the Runge-Kutta numerical approximation of the nonlinear model (Eqs. (1), (2), (3), (4a), (4b)) is accurate. Given that the Runge-Kutta method is fourth order, and considering the agreement between numerical and analytical results shown in Fig. 2, this assumption is reasonable. Further investigations of the effect of initial NAPL saturation (s_n0) on the validity of linearizing the NAPL desaturation model are conducted by calculating the root-mean-square deviation (RMSD) between the two solutions:

$$RMSD(C_g)={\left[\frac{1}{N}\sum _{i=1}^N{\left(C_g^L(r_i)-C_g^{nL}(r_i)\right)}^2\right]}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$$ (17)

where $C_g^L$ is the NAPL concentration calculated using the analytical solution with the linear NAPL desaturation model; $C_g^{nL}$ is the NAPL concentration calculated using the numerical solution with the nonlinear NAPL desaturation model; r_i is the radial coordinate specifying the location of the concentration sample; and N is the total number of samples. Fig. 4 shows the RMSD comparing the numerical and analytical solutions; the lower the initial NAPL saturation (s_n0), the better the approximation of linearizing NAPL desaturation, while the linearization improves as β, the model parameter characterizing NAPL morphology, increases.

Fig. 3.

Concentration distributions along the radial direction comparing the analytical solution using the linearized NAPL desaturation model (Eqs. (5a), (5b)) with the numerical solution that uses the nonlinear NAPL desaturation model (Eqs. (4a), (4b)) at four initial NAPL saturations (s_n0 = 0.001, 0.005, 0.02 and 0.05 for Figs. 3a–d, respectively) with β = 1.0 and other Table 1 parameter values.

Fig. 4.

Root-Mean-Square Deviation (RMSD) for the NAPL concentrations calculated using the linear and nonlinear NAPL desaturation models for varying values of initial NAPL saturation (s_no) and the NAPL morphology parameter (β).

5. Analysis of model solutions

To characterize the soil vapor extraction process, we evaluate the model solutions and investigate the impact of model parameters on performance. The base line parameter values used for simulations are listed in Table 1. Fig. 5a depicts the gas phase concentration breakthrough at the extraction well for three effective molecular diffusion coefficients (D_m = 0.25, 0.125, and 0.0 (m²/day)), with D_m = 0.0 m²/day corresponding to the conventional model scenario that only takes mechanical dispersion into account. Fig. 5b depicts the gas phase concentration profiles in space at three times for D_m = 0.25 and 0.0 m²/day. As is apparent, there is an impact when molecular diffusion is considered, with the impact more apparent when considering the concentration profiles in space. This may be explained by the fact that at the extraction well, where the breakthrough concentrations are measured, the a_Lq term in Eq. (1) is significantly larger than the rD_m term, so the impact of molecular diffusion is small at the well. Fig. 6 shows VOC gas phase concentration breakthrough at the extraction well using different interphase mass transfer rate coefficients: (a) for the gas-water interphase; (b) for the water-solid interphase; (c) for the solid-gas interphase; and (d) for the gas-water interphase while there is no trapped NAPL (s_n0 = 0). Notice from Figs. 6a–c, that the value of the mass transfer coefficient has no significant effect on the breakthrough concentration at the extraction well, particularly when considering the difference in long time concentrations. However, Fig. 6d shows that when there is no trapped NAPL, as would be the case nearing the conclusion of a remediation effort, the value of the mass transfer coefficient has a significant impact on breakthrough concentrations. As has been shown in other studies of the impact of mass transfer limitations on SVE effectiveness (e.g., Goltz and Oxley, 1994) when interphase mass transfer is rate-limited (see the α_gw = 0.001/day simulation in Fig. 6d), low-level concentration “tails” persist. This is in contrast to when mass transfer is “fast” (the breakthrough curve in Fig. 6d for the α_gw = 0.5/day simulation) where concentrations more rapidly decrease at long times. Fig. 7 shows VOC concentration breakthrough at the extraction well for varying NAPL volatilization (k_E) and dissolution (k_D) rate coefficients. As would be expected, at smaller k_E and k_D values, less mass per time is entering the gas stream from the NAPL source, so gas phase concentrations at the extraction well are lower at early times while there are longer breakthrough tails at later times. Fig. 8a shows the impact of initial NAPL saturation on cumulative extracted mass. For s_n0 = 0 (no separate phase NAPL), the extracted mass quickly approaches a constant value (steady state) equal to the mass initially stored in the soil, water, and air in the vadose zone, while for s_n0 > 0 the extracted mass increases to a higher level before approaching a steady state. Note that the steady state means there is no more mass to be extracted from the vadose zone; the clean-up time for s_n0 > 0 is much longer than that for s_n0 = 0, indicating the persistence of contamination when separate phase NAPL exists. Fig. 8b compares the impact of the contaminated area radius on cumulative extracted mass. Of course, more mass is extracted from the larger area and therefore the clean-up time is greater for the larger area source. The agreement between total mass extracted and initial mass gives us further confidence in the accuracy of the analytical solution presented here.

Fig. 5.

VOC gas phase concentration (a) breakthrough at the extraction well, and (b) profiles in space using different effective molecular diffusion coefficients, with D_m = 0.0 (m²/day) corresponding to the conventional model solution that ignores the molecular diffusion coefficient.

Fig. 6.

VOC concentration breakthrough at the extraction well for varying interphase mass transfer coefficients, (a) varying gas-water mass transfer coefficients; (b) varying water-solid mass transfer coefficients; (c) varying solid-gas mass transfer coefficients; (d) varying gas-water mass transfer coefficients for no trapped NAPL (s_n0 = 0). Note Fig. 4d is a semi-log plot.

Fig. 7.

VOC concentration breakthrough at the extraction well while varying (a) vaporization and (b) dissolution rate constants.

Fig. 8.

Cumulative contaminant mass extracted vs time for varying (a) initial NAPL saturations and (b) contaminated area radii. Table 1 base line parameter values but with decay constants in all phases set to zero.

Fig. 9 shows the impact of gas extraction rate on the concentration breakthrough; increasing the gas extraction rate decreases the concentration at the extraction well at any given time. The time required for an SVE system at a given extraction rate to attain a specific remediation target (concentration goal) can be estimated from the model solution. The figure shows the obvious result that the lower the extraction rate, which induces slower gas phase velocities, the longer time required for the SVE system to attain a gas phase concentration goal.

Fig. 9.

VOC gas phase concentrations at the extraction well vs time for different extraction rates (Q_w).

6. Conclusions

In this study, an analytical solution to equations describing single well soil vapor extraction of a contaminant assuming advection, mechanical dispersion and molecular diffusion, first order decay and rate-limited interphase mass transfer (between gas, water, solid, and NAPL phases), subject to a constant initial concentration distribution is derived. The model solution includes gas phase molecular diffusion, a process that was not included in previous derivations. The model solution was applied to estimate VOC concentration breakthrough at the extraction well, as well as cumulative extracted mass, for various values of contamination, engineered and natural parameters. Results of model simulations show that the effect of gas phase molecular diffusion upon concentrations at the extraction well is relatively small, although the effect on concentration distribution in space is significant. More significantly, the linearization of NAPL desaturation through dissolution and evaporation is shown to be accurate for the cases where the initial NAPL saturation is lower (s_n0 < 0.02) and the NAPL morphology parameter is larger (β > 1). This study provides a tool that can be useful in designing SVE remediation strategies, as well as verifying numerical models used to simulate SVE system performance.

Appendix. A: Derivation of analytical solutions

Taking the Laplace transform to Eqs. (1), (2), (3), (6) with respect to time t, we have:

$$r\theta s_g(p{\overline{C}}_g-C_{equ})=(a_{L}q-r{D}_m)\frac{{\partial }^2{\overline{C}}_g}{\partial r^2}+(q+D_m)\frac{\partial {\overline{C}}_g}{\partial r}+r{\alpha }_{gw}({\overline{C}}_w-k_H{\overline{C}}_g)+r{\alpha }_{sg}(k_A{\overline{C}}_s-k_H{\overline{C}}_g)-r\theta s_g{\lambda }_g{\overline{C}}_g+r{k}_E{s}_{n0}^{\beta -1}(C_{equ}{\overline{s}}_n-s_{n0}{\overline{C}}_g)$$ (A.1)

$$\theta s_w(p{\overline{C}}_w-C_{sol})={\alpha }_{gw}(k_H{\overline{C}}_g-{\overline{C}}_w)+{\alpha }_{ws}(k_A{\overline{C}}_s-{\overline{C}}_w)-\theta s_w{\lambda }_w{\overline{C}}_w+k_D{s}_{n0}^{\beta -1}(C_{sol}{\overline{s}}_n-s_{n0}{\overline{C}}_w)$$ (A.2)

$${\rho }_b\left(p{\overline{C}}_s-\frac{1}{k_A}{C}_{sol}\right)={\alpha }_{ws}({\overline{C}}_w-k_A{\overline{C}}_s)+{\alpha }_{sg}(k_H{\overline{C}}_g-{k_A\overline{C}}_s)-{\rho }_b{\lambda }_s{\overline{C}}_s$$ (A.3)

$$\theta {\rho }_n(p{s}_n-s_{n0})={-k}_E{s}_{n0}^{\beta -1}(C_{equ}{\overline{s}}_n-s_{n0)}{\overline{C}}_g){-k}_D{s}_{n0}^{\beta -1}(C_{sol}{\overline{s}}_n-s_{n0)}{\overline{C}}_w)$$ (A.4)

where the variables with the bar indicate the Laplace transform, p is the Laplace transform parameter.

Rearranging (A.2), (A.3), (A.4) yields:

$$\left(\begin{array}{ccc}{a}_{11}& {-\alpha }_{ws}{k}_A& {-k}_D{s}_{n0}^{\beta -1}{C}_{sol}\\ {-\alpha }_{ws}& a_{22}& 0\\ {-k}_D{s}_{n0}^{\beta }& 0& a_{33}\end{array}\right)\left(\begin{array}{c}{\overline{C}}_w\\ {\overline{C}}_s\\ {\overline{s}}_n\end{array}\right)=\left(\begin{array}{c}{\alpha }_{gw}{k}_H{\overline{C}}_g+\theta s_w{C}_{sol}\\ {\alpha }_{sg}{k}_H{\overline{C}}_g+({\rho }_b/k_A)C_{sol}\\ k_E{s}_{n0}^{\beta }{\overline{C}}_g+\theta {\rho }_n{s}_{n0}\end{array}\right)$$ (A.5)

$$a_{11}=\theta s_w(p+{\lambda }_w)+{\alpha }_{gw}+{\alpha }_{ws}+k_D{s}_{n0}^{\beta }$$ (A.6)

$$a_{22}={\rho }_b(p+{\lambda }_s)+k_A({\alpha }_{ws}+{\alpha }_{sg})$$ (A.7)

$$a_{33}=\theta {\rho }_{n}p+s_{n0}^{\beta -1}(k_E{C}_{equ}+k_D{C}_{sol})$$ (A.8)

Solving (A.5):

$${\overline{C}}_w=A_{wg}{\overline{C}}_g+A_{w0}$$ (A.9)

$${\overline{C}}_s=A_{sg}{\overline{C}}_g+A_{s0}$$ (A.10)

$${\overline{s}}_n=A_{ng}{\overline{C}}_g+A_{n0}$$ (A.11)

Where

$$A_{wg}=\frac{1}{\Delta }\left[a_{33}(a_{22}{\alpha }_{gw}+{\alpha }_{sg}{\alpha }_{ws}{k}_A)k_H+a_{22}{k}_D{k}_E{s}_{n0}^{2\beta -1}{C}_{sol}\right]$$ (A.12)

$$A_{w0}=\frac{1}{\Delta }\left[a_{33}(a_{22}{\theta s}_w+{\rho }_b{\alpha }_{ws})C_{sol}+a_{22}{k}_D{s}_{n0}^{\beta -1}\theta {\rho }_n{s}_{n0}{C}_{sol}\right]$$ (A.13)

$$A_{sg}=\frac{1}{\Delta }\left[a_{33}(a_{11}{\alpha }_{sg}+{\alpha }_{gw}{\alpha }_{ws})k_H+({\alpha }_{ws}{k}_E-{\alpha }_{sg}{k}_D{k}_H)k_D{s}_{n0}^{2\beta -1}{C}_{sol}\right]$$ (A.14)

$$A_{s0}=\frac{1}{\Delta }\left[a_{33}(a_{11}{\rho }_b/k_A+{\alpha }_{ws}\theta s_w)C_{sol}+({\alpha }_{ws}\theta {\rho }_n-k_D{s_{n0}^{\beta -1}C}_{sol}{\rho }_b/k_A)k_D{s}_{n0}^{\beta }{C}_{sol}\right]$$ (A.15)

$$A_{ng}=\frac{1}{\Delta }\left[a_{22}(a_{11}{k}_E+{\alpha }_{gw}{k}_D{k}_H)+{\alpha }_{ws}({\alpha }_{sg}{k}_D{k}_H-{\alpha }_{ws}{k}_E)k_{A}f\right]s_{n0}^{\beta }$$ (A.16)

$$A_{n0}=\frac{1}{\Delta }\left[a_{22}(a_{11}{\rho }_n+k_D{s}_{n0}^{\beta -1}{s}_w{C}_{sol})\theta s_{n0}+{\alpha }_{ws}(k_D{s}_{n0}^{\beta -1}{\rho }_b{C}_{sol}-{\alpha }_{ws}{k}_A\theta {\rho }_n)s_{n0}\right]$$ (A.17)

$$\Delta =a_{11}{a}_{22}{a}_{33}-a_{22}{k}_D^2{s}_{n0}^{2\beta -1}{C}_{sol}-a_{33}{\alpha }_{ws}^2{k}_A$$ (A.18)

Substituting (A.9), (A.10), (A.11) into (A.1), we get:

$$(a_{L}q+r{D}_m)\frac{{\partial }^2{\overline{C}}_g}{\partial r^2}+(q+D_m)\frac{\partial {\overline{C}}_g}{\partial r}-r\phi (p){\overline{C}}_g=-r\mu (p)$$ (A.19)

Where

$$\phi (p)=\theta s_g(p+{\lambda }_g)+{\alpha }_{gw}(k_H-A_{wg})+{\alpha }_{sg}(k_H-k_A{A}_{sg})+k_E{s}_{n0}^{\beta -1}(s_{n0}-C_{equ}{A}_{ng})$$ (A.20)

$$\mu (p)=\theta s_g{C}_{equ}+{\alpha }_{gw}{A}_{w0}+{\alpha }_{sg}{k}_A{A}_{s0}+k_E{s}_{n0}^{\beta -1}{C}_{equ}{A}_{n0}$$ (A.21)

The heterogeneous equation (A.19) has a specific solution:

$${\overline{C}}_g^{\mathrm{*}}=\frac{\mu (p)}{\phi (p)}$$ (A.22)

$$(a_{L}q+r{D}_m)\frac{{\partial }^2{\overline{C}}_g}{\partial r^2}+(q+D_m)\frac{\partial {\overline{C}}_g}{\partial r}-r\phi (p){\overline{C}}_g=0$$ (A.23)

Using the variables substitution:

$${\overline{C}}_g=e^{-rh}y(z)$$ (A.24)

$$z=2h(r+a_L\kappa )$$ (A.25)

$$h=\sqrt{\phi /D_m}$$ (A.26)

$$\kappa =q/D_m$$ (A.27)

Eq. (A.23) becomes

$$z\frac{{\partial }^{2}y}{d{z}^2}+(b-z)\frac{\partial y}{\partial z}-ay=0$$ (A.28)

Where

$$a=\frac{1}{2}(1+\kappa )-\frac{1}{2}{a}_L\kappa h$$ (A.29)

$$b=1+\kappa$$ (A.30)

Eq. (A.28) is the confluent hypergeometric equation and has the general solutions (Abramowitz and Stegun, 1970),

$$y=AM(a,b,z)+BU(a,b,z)$$ (A.31)

here, A and B are the constants of integration, M(a,b,z) and U(a,b,z) are the confluent hypergeometric functions of the first and second kind.

Substituting (A.31) into (A.24), and noting (A.22), we may finally have the general solution of Eq. (A.19):

$${\overline{C}}_g=A{e}^{-rh}M(a,b,z)+B{e}^{-rh}U(a,b,z)+{\overline{C}}_g^{\mathrm{*}}$$ (A.32)

Appling the boundary conditions (8a), (8b), A and B can be determined, and the solution can be expressed as:

$${\overline{C}}_g=\frac{\mu }{\phi }\left[1-e^{(r_b-r)h}\frac{U(a,b,z)G_M(a,b,z_w)-M(a,b,z)G_U(a,b,z_w)}{G_M(a,b,z_w)H_U(a,b,z_b)-G_U(a,b,z_w)H_M(a,b,z_b)}\right]$$ (A.33)

Where

$$G_U(a,b,z_w)=-U(a,b,z_w)+2U\mathrm{\prime }(a,b,z_w)$$ (A.34)

$$G_M(a,b,z_w)=-M(a,b,z_w)+2M\mathrm{\prime }(a,b,z_w)$$ (A.35)

$$H_U(a,b,z_b)=(1-\gamma )U(a,b,z_b)+2\gamma U\mathrm{\prime }(a,b,z_b)$$ (A.36)

$$H_M(a,b,z_b)=(1-\gamma )M(a,b,z_b)+2\gamma M\mathrm{\prime }(a,b,z_b)$$ (A.37)

$$z_w=2h(r_w+a_L\kappa )$$ (A.38)

$$z_b=2h(r_b+a_L\kappa )$$ (A.39)

$$\gamma =z_b/(2\kappa )$$ (A.40)

Note that if the molecular diffusion coefficient D_m is set to zero, as has been done in earlier solutions, the governing Eq. (A.19) would become:

$$a_{L}q\frac{{\partial }^2{\overline{C}}_g}{\partial r^2}+q\frac{\partial {\overline{C}}_g}{\partial r}-r\phi (p){\overline{C}}_g=r\mu (p)$$ (A.41)

Applying the method used in Goltz and Oxley (1991), the solution of Eq. (A.41) subject to the boundary conditions (8a), (8b) can be written as:

$${\overline{C}}_g=\frac{\mu }{\phi }\left[1-e^{(r_b-r)/2a_L}\frac{Bi(x)G_A(,x_w)-Ai(x)G_B(x_w)}{G_A(x_w)H_B(x_b)-G_B(x_w)H_A(x_b)}\right]$$ (A.42)

$$G_A(x_w)=-\frac{1}{2}Ai(x_w)+\gamma \mathrm{\prime }Ai\mathrm{\prime }(x_w)$$ (A.43)

$$G_B(x_w)=-\frac{1}{2}Bi(x_w)+\gamma \mathrm{\prime }Bi\mathrm{\prime }(x_w)$$ (A.44)

$$H_A(x_b)=\frac{1}{2}Ai(x_b)+\gamma \mathrm{\prime }Ai\mathrm{\prime }(x_b)$$ (A.45)

$$H_B(x_b)=\frac{1}{2}Bi(x_b)+\gamma \mathrm{\prime }Bi\mathrm{\prime }(x_b)$$ (A.46)

$$x={(a_{L}q)}^{-1/3}(\frac{q}{4a_L}+\phi r){\phi }^{-2/3}$$ (A.47)

$$x_w={(a_{L}q)}^{-1/3}(\frac{q}{4a_L}+\phi r_w){\phi }^{-2/3}$$ (A.48)

$$x_b={(a_{L}q)}^{-1/3}(\frac{q}{4a_L}+\phi r_b){\phi }^{-2/3}$$ (A.49)

$$\gamma \mathrm{\prime }={a_L}^{2/3}{(\phi /q)}^{1/3}$$ (A.50)

where Ai(⋅) and Bi(⋅) are the Airy functions, Ai′(⋅) and Bi′(⋅) are the derivative of the Airy functions.