MAM-An Aquivalence-based Dynamic Mass Balance Model for the Fate of Non-Volatile Organic Chemicals in the Agricultural Environment

A Multimedia Agricultural Model (MAM) for predicting the fate and transport of NonVolatile Organic Chemicals (NVOCs) in the agricultural environment was presented. It is an expanded and modified version of the three compartmental model introduced by Batiha and co-authors in 2007, which is an aquivalence-based level IV. MAM considered five environmental compartments to include the air, water, soil, sediment and vegetation. It calculates the complete steady-state mass budgets for the air, water and particulate organic carbon between the model compartments. MAM compartments were connected by advective and intermedia transport processes. Degradation can take place in every compartment. The mass balances for each of the compartments result in a system of five differential equations, solved numerically to yield estimates of concentrations, masses, transport fluxes and reaction rates as a function of time. All the equations required for MAM calculations were provided.


INTRODUCTION
It is becoming evident that vegetations play an important role in the environmental fate of many organic compounds [1][2][3][4] . It is clearly an important aspect of the agricultural environment since approximately 97% of the plantations land surface is covered by vegetation. Moreover, the total leave surface area may exceed by factors of up to 20 the ground area on which plants grows [5] . The total amount of cuticular material in temperate forests and agricultural plants communities may range from 180-1500 kg per hectare [6] . As a result of the high reactivity of some compounds on vegetation surface and the array of available deposition surfaces in vegetation canopies [7] , it has been suggested that vegetation could be an important sink for organic compounds [8] .
Many authors [9][10] have reported that the concentrations of organic chemicals in forest soils are higher than in agricultural soils. This phenomenon was explained by Horstmann et al. [11] that the elevated levels of organic chemicals present in forest soils compared to agricultural soils are due to atmospheric deposition processes. Atmospheric pollutants can be transferred to vegetation canopies either through wet or dry deposition [12] . Wet deposition, can be expected to be similar to vegetated and non-vegetated land. Hence, the elevated deposition to vegetations must arise from dry deposition. McLachlan et al. [13] have divided the dry deposition of organic chemicals to vegetations into three categories according to the volatility of the compound. For the relatively Non-Volatile Organic Chemicals (NVOCs), gaseous deposition is superseded in importance by dry-particle-bound deposition. Sehmel [14] sub-divided the dry-particle-bound deposition into three processes: particle sedimentation, particle impaction and diffusion [15] . Whereas, for NVOCs, the diffusion process from the surface compartments to the atmosphere is neglected [16] . Moreover, the dominant mechanisms that convey chemicals transport from vegetation to soil are also occurring under wet (i.e., canopy throughfall) and dry (i.e., litterfall) conditions [2] .
The water-sediment interface is an active layer, which typically consists of 95% particles and is often highly in organic nature. It may consists of deposited particles and fecal material from water column [17] . The main transport processes within water-sediment interface layer are deposition, resuspension and sediment burial. Therefore, the inclusion of vegetation and sediment compartments into multimedia models for NVOCs is an important issue. This study follows up the preliminary findings, reported by Batiha et al. [18] and constitute on recent attempts to include vegetation and sediment compartments into multimedia aquivalencebased models at the dynamic conditions. Herewith, we have described the Multimedia Agricultural Model (MAM) and provided all the necessary equations for the model feasibility calculation.

Model structure:
The main concepts of MAM have been presented in detail by Batiha et al. [18] . Hence, MAM is an aquivalence-based, dynamic (Level IV: unsteady-state, non-equilibrium with degradation, advection and inter-compartmental transfer) model. The MAM environment consists of up to five environmental compartments Fig. 1, namely, the Air (A), Water (W), Soil (S), Sediment (E) and Vegetation (V), which are believed to shape the agricultural environment. These compartments are linked by up to eight transport processes from i to j compartment (D ij , m 3 h −1 ). There are two possibilities through which NVOCs can enter the modeled environment: either by direct emission to compartment i (E i mol h −1 ) and/or by advection of air and water into the model domain (D AIN and D WIN , mol h −1 ). Once NVOCs is introduced to the environment, it can move amongst compartments and can be lost from each compartment i through various transformation processes (D Ri ), leaching from soil to ground water (D LS ) and sediment burial (D LE ). In addition, NVOCs can be lost by advection of air and water out of the model domain (D AUT and D WUT , m 3 h −1 ).

RESULTS AND DISCUSSION
NVOCs transfer fluxes: Chemical movement in the environment is strongly associated with the movement of air, water and organic matter [19] . NVOCs advective inter-compartmental transfer fluxes are calculated as the product of a flux of a carrier phase (F) in units of m 3 h −1 , namely, air (AF), water (WF) and particulate organic carbon (PF) and a NVOC concentration (C, mol m −3 ) in similar phase. Solving the NVOC mass balance thus requires the constructions of mass balance for air, water and Particulate Organic Carbon (POC) within the modeled system. These fluxes were assumed to be constant in time. In the following, details are provided for the construction of mass balances for air, water and POC.
Mass balance of air: As has been discussed by Batiha et al. [18] , the mass balance for air is calculated as the volume of atmospheric compartment divided by the atmospheric residence time.
Mass balance for water: The water mass balance consistent set of up to 11 inter-compartmental water fluxes, WF in m 3 h −1 , between the modeled compartments Fig. 2. The water flux from air to the surface compartments by precipitation, WF AX (Eq. 1, 5 and 8 in Table 1), is estimated from the rain rate (U R , m h −1 ) and the compartment surface area (A X , m 2 ). Evaporation loss from the surface compartments, WF XA (Eq. 2, 6 and 9 in Table 1  Evaporation from vegetation to air WF VA = φ V (WF AV +WF SV ) 3.
Throughfall from vegetation to soil Rainsplash from soil to vegetation WF SV = φ RS WF AS 5.
Precipitation from air to soil WF AS = U R A S 6.
Evaporation from soil to air WF SA = φ S (WF AS +WF VS ) 7.
Runoff and leaching from soil to water Precipitation from air to water WF AW = U R A W 9.
Evaporation from water to air WF WA = φ W (WF AW +WF SW ) 10.
Water flow from water to the outside Water flow from the outside to water POC sedimentations from water to sediment PF WE = υ sed A W C pw /P OC 17.
POC resuspension to water from sediment PF EW = φ res υ sed A W C PW /P OC 18.
POC burial in sediment POC mineralization in sediment PF Emin = (PF WE -PF EW ) φ Emin Note: U R is the rainfall rate, m 3 h −1 , A X is the surface area of compartment x, m 2 , φ X is the fraction of precipitation to compartment x that evaporates from that compartment, φ RS is the fraction of raindrop water reflected after reaching the soil surface to vegetation, f UT is the factor by which water is increased by inflow from the outside flux, ν SE , ν OE and ν SS are the volume fraction of solids in sediment, of organic carbon in sediment solids and of solids in soil, respectively, ρ OC is the density of organic carbon, g m −3 , P W is the primary productivity in water, g carbon/m 2 a, φ Wmin and φ Emin are the fraction of net POC input to water and sediment compartments, respectively, that is mineralized, C PIN is the concentration of POC in the water flow from the outside, g m −3 , υ sed and υ bur are the POC sedimentation and burial velocities in sediment, respectively, φ res is the fraction of POC deposition that is resuspended  Table 1), is derived as the balance of the remaining fraction of precipitation to vegetation that fall to soil from evaporation loss. The water flux from the soil to vegetation by rainsplash, WF SV (Eq. 4 in Table 1), is estimated by employing fractions φ RS of the raindrop water reflected after reaching the soil surface to vegetation. The run-off and leaching flux from soil to the water compartment, WF SW (Eq. 7 in Table 1), is derived as the balance of the remaining fraction of precipitation plus water throughfall from the vegetation to soil obtained from evaporation plus rainsplash losses. The net flow between water compartment and the outside, WF UT and WF IN (Eq. 10 and 11 in Table 1), is estimated employing the factor ( f UT ) by which water is increased by inflow from the outside flux.

POC mass balance:
The POC fluxes illustrated in Eq. 12-15 in Table 1, has been discussed in detail by Batiha et al. [18] . Only the calculation of the new included POC fluxes over the sediment compartment needed for the construction of POC mass balance is discussed in this study Fig. 3 Table 1, respectively). Whereas, the user-defined parameters are the POC sedimentation and burial velocities in sediment, υ sed and υ bur , respectively, the fraction of POC deposition that is resuspended (φ res ), the density of organic carbon in g m −3 (ρ OC ), the fraction of net POC input to sediment compartment (φ Emin ), the volume fraction of solids in sediment (ν SE ) and the volume fraction of organic carbon in sediment solids (ν OE ).  The compartments volumes are constant, therefore, the equations can be rewritten as partial differential equations:

Description of chemical fate:
Substituting the differentials by finite differences: Z-values are independent of the aquivalences and can be calculated for time t+∆t. The time course of the aquivalences in all compartments can thus be calculated iteratively using: Estimating aquivalences in all MAM compartments as a function of time, the intermedia fluxes between MAM compartments, reaction rate, chemical residence time and chemical concentration in each compartment can be deduced. In the following context, the details of how the various Z-and D-values can be calculated in the MAM are provided.
Description of phase partitioning: Equilibrium phase partitioning in MAM as is typical for aquivalence-based models are expressed in terms of Z-values or the compartment partitioning capacities. The Z-values for sediment and vegetation compartments (Z E and Z V ) are described by Eq. 1 and 2, respectively: where Z W is the partitioning capacity of water compartment, ρ E is the sediment density and K EW and  Mass balance equations Air d(V A Z BA q A )/dt = (E A +D AIN ) -(D AUT +D RA +D AV +D AS +D AW ) q A Water d(V W Z BW q W )/dt = (E W +D WIN +D SW q S +D EW q A +D EW q E ) -(D WUT +D WE +D RW ) q W Soil d(V S Z BS q S )/dt = (E S +D AS q A +D VS q V ) -(D SV +D SW +D RS +D LS ) q S Sediment d(V E Z BE q E )/dt = (D EW q W )-(D EW +D RE +D LE )q E Vegetation d(V V Z BV q V )/dt = (E V +D SV q S +D AV q A ) -(D VS +D RV )q V K CW are the sediment-and cuticle-water partition coefficients, respectively. The K CW is the parameter describing the tendency of environmental chemicals to accumulate in the cuticle phase in an ecosystem as well as on a single leaf or fruit scale. Based on the available data, one of the following quantitative structureactivity relationship models can be used to calculate the K CW [20][21] : Where K OW is the octanol-water partition coefficient, S the water solubility, 3 χ v the valence third-order molecular connectivity index that calculated from the non-hydrogen part of the molecule and OH aliph N the number of aliphatic hydroxy groups.
The bulk Z-value for sediment and vegetation compartments (Z BE and Z BV ) is calculated by Eq. 6 and 7, respectively: where ν E is the volume fraction of solids in sediment, ν OE is the volume fraction of organic carbon in sediment solids, Z P is the Z-value for particulate organic carbon and ν AV , ν WV and ν CV are the volume fraction of air, water and cuticle in vegetation, respectively.
Description of chemical fate processes: Transport, loss and transformation processes in aquivalence-based models are described with the help of D-values in units of m 3 h −1 . The present model adds the D-values for sediment and vegetation compartments to those Dvalues previously described by Batiha et al. [18] for air, water and soil. The additional estimated D-values are drawn upon on recent modeling published papers [2,19,[22][23] . The transport and transformation processes from and to sediment and vegetation compartments are discussed below and D-values are listed in Table 3.
As MAM is intended to be applicable for NVOCs, the diffusion process from plant surface to air was neglected. Advective processes from air to vegetation are the wet and dry particle deposition. The D-value for wet deposition of particles to vegetation is given by Eq. 3 in Table 3. In this equation, the wet deposition interception fraction (If W ) depends on the leaf area index (LAI) in units of m 2 leaves per m 2 land, the rainfall rate (U R , m h −1 ) and the canopy water storage capacity (S bl ), and can be calculated from Müller and Pröhl [24] as follows: The D-value for dry particle deposition to vegetation is given by Eq. 4 in Table 3. The dry deposition interception fraction (If d ) is calculated from plant biomass as described by Whicker and Kirchner [25] , where bio inv is the plant dry mass inventory (kg dry mass m −2 ). Summation of D-values for wet and dry particle deposition (Eq. 3 and 4 in Table 3) give the overall D-value for air to vegetation.
The dominant transport mechanisms of NVOCs from vegetation to soil are also occurring under wet (canopy throughfall) and dry (litterfall) conditions. Litterfall is the process of vegetation foliage falling to the ground as a result of shedding of leaves and vegetation death and controlled by a first order rate constant (k LF ). The k LF is assumed by Bennett et al. [22] to be equal to one divided by the length of the growing season. The D-value for this process is given by Eq. 6 in Table 3. Canopy throughfall (Eq. 7 in Table 3) is the process of washing out of contaminant sorbed onto vegetations by precipitation, which is calculated as the product of the water flux from vegetation to soil (WF VS , m 3 h −1 ) and the vegetation bulk Z-value (Z BV ). Summation of D-values for canopy throughfall and litterfall yield the total D-value for the transport mechanisms from the vegetation to the soil.  Litterfall Canopy Degradation in vegetation Degradation in sediment D RE = V E k RE Z E Notes: A V and A W are the surface area of vegetation and water compartments, m 2 , If w and If d are the canopy wet and dry interception fraction, U Q is the dry deposition velocity, m h −1 ν Q , ν OE and ν E are the volume fraction of aerosol in air, of organic carbon in sediment and of solids in sediment, respectively, V V and V E are the volume of vegetation and sediment compartments, m 3 , k LF is the litterfall rate constant, h −1 , k EW is the mass transfer coefficient for diffusion across benthic boundary layer, m h −1 , h E is the depth of sediment compartment, m, B W is the molecular diffusivity in water, m 2 h −1 , B WE is the bioirrigation diffusivity of sediment pore water, m 2 h −1 , B PE is the bioturbation diffusivity of sediment solids, m 2 h −1 and k RV and k RE are the vegetation and sediment degradation rate constants, h −1 Rainsplash, which is the only transport mechanism from soil to vegetation considered, is based on the concept that once lifted off by the raindrop impact, the particle-sorbed contaminant in soil entrained into the splash droplets travel to vegetation. The D-value for this mechanism (Eq. 8 in Table 3) is calculated as a product of the water flux from soil to vegetation (WF SV , m 3 h −1 ) and the soil bulk Z-value (Z BV ).
The processes contribute the exchange of contaminants across the water-sediment interface as described by Wania et al. [19] are two advective and four different diffusive processes. The advective processes are physical sedimentation or deposition and resuspension of contaminants sorbed to POC (Eq. 12 and 13 in Table 3), described using the POC transport rates PF WE and PF EW in m 3 h −1 derived in Table 1. The overall diffusive process, which may act in either direction, is described as follows. On the waterside there is a resistance to transfer through the stagnant benthic boundary layer above the sediment surface, quantified using a benthic boundary layer mass transfer coefficient, k EW in m h −1 . Secondly, there are three parallel diffusive processes on the sediment side of the water-sediment interface. The first is the molecular diffusion through water-filled pore spaces in the sediment, quantified by the diffusivity in water B W , corrected for the sediment porosity. Bioturbation of sediment solids and sediment pore water is treated as two pseudo-diffusive processes invoking two equivalent bioturbation diffusivities, B PE and B WE in m 2 h −1 , respectively. In all three cases the diffusion path length is calculated as the log mean depth of the sediment compartment (h E , m). The final equation for diffusive water-sediment exchange is given by Eq. 11 in Table 3. Eq. 9 and 10 in this Table show the Dvalues describing the exchange of the contaminants across the water-sediment interface.
Finally, the D-value for degradation processes in the vegetation and sediment compartments may be quantified as the product of compartments volume (V V and V E , m 3 ), the compartment Z-value and the degradation rate constants (k RV and k RE , h −1 ) as given by Eq. 15 and 16 in Table 3.

CONCLUSION
A comprehensive multimedia agricultural model, MAM, has been developed that described the processes of NVOCs transport and transformation in the agroecosystem. MAM is a Level IV aquivalence-based model, an improved and expanded version of the model that has been described by Batiha et al. [18] . Major differences from the earlier model were the increased number of environmental compartments (sediment and vegetation) which together are believed to shape the agro-ecosystem. Therefore, MAM has utilized to treat five environmental compartments, namely, the air, water, soil, sediment and vegetation. MAM constructed complete steady-state mass balance for air, water and POC. Even though the rates of primary production and POC mineralization are not required for the contaminant mass balance, they were included in MAM calculation to complete and assure the consistency of the mass balance.