## 2 Materials and Methods
### Modeling Methods  
A model of monodisperse hydrocarbon particles of a single hydrocarbon component slowly dissolving into a bulk aqueous phase was developed and run with various conditions. The processes modeled were dissolution of the non-aqueous phase liquid (NAPL) into the aqueous phase, and subsequent biodegradation of the dissolved hydrocarbon component. The dissolution was governed by hydrocarbon solubility and mass transport into the bulk phase, evaluated as diffusion-only transport or advection transport. The model consisted of ordinary differential equations (ODEs) representing the mass balance of the hydrocarbon component in the NAPL phase and dissolved in the bulk phase. The solution of the two ODEs resulted in the estimation over time of the complete dissolution of the hydrocarbon component in the aqueous phase, and subsequent hydrocarbon biodegradation. The rate of dissolution was governed by the mass transfer, which is directly proportional to the surface area, concentration difference, and mass transfer coefficient. The following equation represents the mathematical form of the dissolution process:  
\[F=A_{S}K_{c}(C_{Sxi}-C_{Sx}) \tag{4}\]  
where \(F\) is the mass flux, \(A_{S}\) is the interfacial area, \(K_{c}\) is the mass transfer coefficient, \(C_{Sxi}\) is the hydrocarbon component concentration at the particle/bulk phase interface, and \(C_{Sx}\) is the bulk phase component concentration. The biodegradation of the dissolved form of the hydrocarbon component was represented as a first-order reaction, dependent only on the component concentration (oxygen required for aerobic biodegradation assumed not to be limiting). Thus, the following equations represent the dissolution and aerobic biodegradation phenomena. \[\frac{dM_{Sx}}{dt}=-K_{c}\times(C_{Sxi}-C_{Sx})\times A_{s}\times N_{p} \tag{5}\]  
\[\frac{dC_{Sx}}{dt}=K_{C}\times(C_{Sxi}-C_{Sx})\times A_{S}\times N_{p}-K_{r} \times C_{Sx} \tag{6}\]  
where \(M_{Sx}\) is the total mass of the NAPL phase, \(N_{P}\) is the number of NAPL particles, and \(K_{r}\) is the biodegradation rate constant for the specific hydrocarbon component. The term that requires further definition in these two equations, in order for the ODEs to be solved using numerical integration techniques, is the product of the particle surface area times the number of particles \(A_{S}\times N_{P}\). For this purpose, the initial diameter of the monodisperse particles was selected as an initial parameter, along with the initial hydrocarbon component mass in the NAPL phase. With these two values, the \(As\)\(\times N_{P}\) term was replaced according to the following equations, derived from the formulas for volume and surface area of a sphere (detailed derivation for Equation (7) is given in the supplementary material). The constant B defined in Equation (8) below is a function only of the parameter values specified at the outset of the problem, namely the monodisperse particle size and the initial hydrocarbon mass. \[A_{S}\star N_{p}=6\times\frac{M_{Sxi}^{\frac{1}{3}}}{D_{Sxi}\star\rho_{Sx}} \times M_{Sx}^{\frac{3}{3}} \tag{7}\]  
\[\text{Assuming constant, B}=\left(6\times\frac{M_{Sxi}^{\frac{1}{3}}}{D_{Sxi} \star\rho_{Sx}}\right) \tag{8}\]  
\[A_{surface}\times N_{p}=B\times M_{Sx}^{\frac{3}{3}} \tag{9}\]  
where \(M_{Sx}\) is the initial value of the total mass of the NAPL phase (initial value of the \(M_{Sx}\) parameter), \(\rho_{Sx}\) is the hydrocarbon density, and \(D_{Sxi}\) is the initial particle size.