Box models

Types of distribution models

There is a number of distribution models types that differ in focus, size, number of environmental components included, and purpose. First there are models describing some specific processes air transport, wet atmospherical deposition (rain, snow, fog etc.), or elution from soil. Some models provide general description of the environmental fate and are employed for comparison of different chemicals fate. And finally there are models that focus on selected region of interest and differ in a degree of spatial resolution, size of modelled area, and complexity of processes included.

Examples of regional models (Wania et al., 2000)

Further there are models dealing with pollutants uptake by living organisms, eventually food nets.

Box models may contain various subsets of basic models that describe individual distribution processes. It could be for example distribution of pollutant content among various phases in atmosphere (pollutants may be present in vapour phase or bound to atmospherical particles such as dust), equilibrium distribution among various environmental components (distribution coefficient and regression relationships play role in this case), or degradation of the pollutant.

Model of a substance transport over the air-water interface

CalTOX, A Multimedia Total-Exposure Model for Hazardous-Waste Sites Part II: The Dynamic Multimedia Transport and Transformation Model, A technical report, The Office of Scientific Affairs, Department of Toxic Substances Control, California Environmental Protection Agency Sacramento, California, December 1993

Models may vary in the space size that they present. The smallest models may simulate individual organisms such as fish or plants.

Model example of plant, (Mackay et al., 1994)

Ian T. Cousins, Donald Mackay, Strategies for including vegetation compartments in multimedia models, Chemosphere 44 (2001), 643-653

Basl4- Model example of earthworm, (Webster et al., 2007)

Lauren Hughes, Eva Webster, Don Mackay, James Armitage, Frank Gobas, Development and Application of Models of Chemical Fate in Canada, Modelling the Fate of Substances in Sludge-Amended Soils Report to Environment Canada, CEMN Report No. 200502

Progress of a box model from the simplest to global (Wania, 1999)

Frank Wania, Diferencies, Similarities, and complementarity of various approaches to modelling persistent organic pollutant distribution in the environment. WHO/EMEP/UNEP Workshop on modelling of atmospheric transport and deposition of persistent organic pollutants and heavy metals, Geneva, Switzerland, 16-19 November 1999.

The highest stage is incorporation of more models to one complex. These may consist of many tens, hundreds and thousands of unified units. Such models are often used for the simulation of atmosphere motion.

Mathematical description of a box model

Mathematical solution of an environmental model can be based on several approaches that define chemical equilibrium, e.g. chemical potential, activity, or fugacity.

The fugacity approach was introduced by G. N. Lewis in 1901 as a thermodynamic criterion of equilibrium, which is more suitable than chemical potential, because fugacity is linearly dependent on concentration. Its suitability for the environmental equilibriums solution has come out during last decades, as allowed for an easy expression of mass transport velocity over the interface (both diffusive and advective). The transport velocity is proportional to fugacity difference and velocity equations of chemical transformations can also be transferred to fugacity form. Fugacity may be defined as "escape tendency", whose unit is Pa (Pascal). It also characterizes partial pressure of the compound in a given phase. In contrast to chemical potential, its absolute value is findable, because it is equal to low values of partial pressure at ideal conditions.

Relationship between fugacity (f) and real concentration (C) is expressed by fugacity capacitance (Z) defined by Mackay in 1979. It is a reference constant linearly correlated with concentration.

C = Z * f

Its unit is (mol m^-3 Pa^-1). Final value depends on physical and chemical properties of the chemical compound as well as on properties of the environment. The dependence on concentration and pressure can be neglected at common conditions and pollutants environmental levels.

Fugacity capacitance calculations are analogous to thermal capacitance. If the environment has high fugacity capacitance, the value of "partial pressure" fugacity will be low even at relatively high concentration and the compound will tend to escape given the environment to the phase with lower fugacity capacitance. Thus, the fugacity capacitance expresses affinity of a specific compound to real phase.

The fugacity capacitance calculation is based on the computation of its absolute value for air and is derived by means of appropriate equilibrium distribution coefficients for other environmental components. Fugacity has a meaning of partial pressure, i.e.:

P * V = n * R * T

where the substance concentration in a given component is expressed as c = n / V; i.e. c = p / R * T. Therefore, fugacity capacitance for air is equal to 1 / R * T at ideal conditions for all dissolved substances. R is universal gas constant, n amount of substance, T thermodynamic temperature and p partial pressure.

For water and other environmental phases the fugacity capacitance may be derived by means of distribution coefficients:

K_aw = Z_a / Z_w = H / R * T = ps / cs * R * T
Z_w = Z_a * R * T / H = 1 / H = cs / ps

The box model mathematical solution itself is based on the mass conservation law. Amount of chemical substance entering the given compartment must also go out from it. Thus, the total mass balance of the box must be zero. We may therefore write a set of balance equations for the box that describe compound migration between compartments. Assuming that the time change in concentration in compartments is zero, equations in the set are then linear. If the concentrations changes in time, then the equations are differential. Results of the equations set solution is fugacity values in individual environmental compartments or their dependence on time, respectively. Pollutant concentration may be calculated by means of appropriate fugacity capacitance values.

Solution procedure of the box model

As mentioned above, the model solution lies in a set of balance equations for individual compartments and its solution. For this purpose it is necessary to perform preliminary calculations concerning interactions of individual environmental compartments with chemical substances and describe dynamics of transport processes.

1. Calculation of properties of the environment, distribution coefficients and their temperature dependence for a specific chemical substance

Properties of individual environmental components necessary for the calculation are defined in input parameters. They include particularly information about size and volume of individual phases, mass density, matrix composition, and organic carbon content, and further velocity coefficients characterizing dynamic processes. These coefficients describe velocity of substance transport over the phase interface. They include e.g. diffusion coefficients, velocity constants, precipitation intensity, elution coefficient, or dry atmospherical deposition velocity (dust). Physical-chemical properties of assessed substance must be known.

2. Fugacity capacitance calculation

Fugacity capacitance defines compound affinity to the appropriate environmental component. It is dependent on properties of both the phase and individual compounds. Fugacity capacitance values are derived for both environmental subcomponents (aerosols) and complete phases that are their combination (air, soil). Derivation of fugacity capacitance proceeds by means of distribution coefficients, because they are mutually in constant ratio. Fugacity capacitance of air is fundamental and is equal to 1 / R * T, fugacity capacitance for all remaining components may be derived by means of specific distribution coefficients, such as Kaw (air-water), Kow (octanol-water), Koc (octanol-organic carbon).

Water fugacity capacitance calculation may serve as an example:
Z water = 1 / ( Kawt * 8.314 * T), where T is thermodynamic temperature and Kawt is distribution coefficient air-water for a specific compound at temperature T.

3. Calculation of transport coefficients

Use of D coefficients is one of the biggest advantages of fugacity models. Each D coefficient describes velocity of transport process over the phase interface per fugacity unit. It may therefore be matched with arrows illustrating transport processes in scheme of the model. Their advantage is additivity. It is therefore possible to define new processes and simply add them to the D coefficients group without necessity of intervention to the model structure or new solution of the equations set. This allows for an easy development of the model. If the individual D coefficient is multiplied by fugacity in appropriate medium, mass flow over the phase interface is obtained and their dependencies on environment properties may be studied separately. Setting of a D coefficient to zero deactivates the appropriate transport process without affecting model functionality. D coefficients are calculated from appropriate fugacity capacitance values, environmental compartments properties, and transport coefficients. The transport D coefficient consists of fugacity capacitance of the given compartment, phase interface area, over which the transport proceeds (or phase volume in case of degradation) and appropriate velocity coefficient.

As example follows pollutants amount transferred over the phase interface air-soil by means of rain.

Drain-soil = S * Ur * Z_water,

where S is soil area, Ur precipitation intensity (m hod^-1) and Z_water is fugacity capacitance of water.

Thus, pollutant molar flow over the phase interface air-water by means of precipitation will be:

F = f(air) * Drain-soil,

where F is the flow (mol hod^-1), f(air) is pollutant fugacity in air, and Drain-soil is appropriate transport D coefficient.

4. Solution of equations set

Principle of the solution consists in expressing mass balance equations for each phase containing fugacity values as unknown quantities. Solution of this linear equations set are appropriate fugacity values of individual phases. The Matlab software is the most frequently used for the solution. Pollutant mass flow over the phase interface (mol.h^-1) is expressed as multiple of D coefficient of the appropriate process and substance fugacity in appropriate environmental component.

Example of solution of a simple box model

Model environment:

• Two compartments
• Emisson flux E₁ to compartment no. 1
• Degradation of pollutants in both compartments is expressed by reaction rate coefficients k₁, k₂
• Two transport processes over interphase of compartments described by two "D" coefficients ( D₁₂ a D₂₁)

"D" coefficients:

E1	emission rate to compartment no. 1	mol / h
D12	D21 overall "D" coefficient of transport between comparments	mol / ( h * Pa )
Dr1	„D" coeff. degradation of pollutants in 1 compatments	mol / ( h * Pa )

Dr₁ = V₁ * Z₁ * k₁

V1	volume of first compartment	m3
Z1	fugacity capacity of first compartment	mol / ( m3 * Pa )
k1	first order reaction rate coefficient	h-1

D_1x - sum of all "out from first compartment" D coefficients

D_1x = D₁₂ + Dr₁

Dr₂ - D of degradation of pollutant in comparment no. 2

Dr₂ = V₂ * Z₂ * k₂

V2	volume of second compartment	m3
Z2	fugacity capacity of second compartment	mol / ( m3 * Pa )
k2	first order reaction rate coefficient	h-1

D_2x - sum of all "out from second compartment" D coefficients

D_2x = D₂₁ + D_r2

Set of balance equations:

First compartment input = output
E₁ + f₂ * D₂₁ = f₁ * D_1x

First compartment input = output
f₁ * D₁₂ = f₂ * D_2x

Solution:

f₁ = f₂ * D_2x / D₁₂
f₂ = E₁ / ( D_2x * D_1x / D₁₂ - D₂₁ )

Resulting concentrations have units (mol / m³):

C₁ = Z₁ * f₁

C₂ = Z₂ * f₂

A basic method using the box model is calculation of concentrations in individual environmental compartments based on emission to individual phases and concentrations in advective influxes. Properties changes of chemical compounds and the environment result in changes of distribution. Transport D coefficients may be used for independent research of interphase transport processes and their relationship to properties of the environment and compounds. Thus, use of the model is not limited to the box space only, but it may be applied to "one point", such as for study of pollutants volatilization from soil by observation of equilibrium changes between individual media. The model may be divided to individual parts, which can be used for interpretation of laboratory or field experiments. Parts of the model may therefore be also used for common calculations associated with establishing of equilibriums, their relationships and interactions.

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