## 3.8.1. Multicompartment modeling

Authors: Dik van de Meent and Michael Matthies

Reviewer: John Parsons

Learning objectives:

You should be able to

- explain what a mass balance equation is
- describe how mass balance equations are used in multimedia fate modeling
- explain the concepts of thermodynamic equilibrium and steady state

- give some examples of the use of multimedia mass balance modeling

Keywords: mass balance equation, environmental fate model

The mass balance equation

Multicompartment (or multimedia) mass balance modeling starts from the universal conservation principle, formulated as balance equation. The governing principle is that the rate of change (of any entity, in any system) equals the difference between the sum of all inputs (of that entity) to the system and the sum of all outputs from it. Environmental modelers use the balance equation to predict exposure concentrations of chemicals in the environment by deduction from knowledge of the rates of input- and output processes, which can be understood easiest from considering the mass balance equation for one single environmental compartment (Figure 1):

(eq. 1)

where dmi,j/dt represents the change of mass of chemical i in compartment j (kg) over time (s), and inputi,j and outputi,j denote the rates of input and output of chemical to and from compartment j, respectively.

Figure 1. The mass of a chemical in a lake is like the mass of water in a leaking bucket: both can be described with the universal (mass) balance equation, which says differences in inputs and outputs make amounts held up in systems change: .

One compartment model

In multimedia mass balance modeling, mass balance equations (of the type shown in equation 1) are formulated for each environmental compartment. Outflows of chemical from the compartments are often proportional to the amounts of chemical present in the compartments, while external inputs (emissions) may often be assumed constant. In such cases, i.e. when first-order kinetics apply (see section 3.3 on Environmental fate of chemicals), mass balance equations take the form of equation 1 in section 3.3. For one compartment (e.g. a lake, as in Figure 1) only:

(eq. 2)

in which dm/dt (kg.s-1) is the rate of change of the mass (kg) of chemical in the lake, I (kg.s-1) is the (constant) emission rate, and the product k.m (kg.s-1) denotes the first-order loss rate of the chemical from the lake. It is obvious that eventually a steady state must develop, in which the mass of chemical in the lake reaches a predictable maximum

(eq. 2a)

A very intuitive result: mass (thus concentration) at steady state is proportional to the rate of emission (twice the emission yields twice the mass or concentration); steady-state mass is inversely proportional to the rate (constant) of degradation (more readily degrading chemicals reach lower steady-state masses). It can be demonstrated mathematically (not here) that the general (transient) solution of equation 2 exists and can be found (Figure 2):

Figure 2. For one compartment only, in case loss processes obey first-order kinetics and emissions are constant (i.e. not varying with time), the mass of chemical is expected to increase exponentially from its initial mass m0 to its steady-state level , which will be reached at ininite time . After Van de Meent et al. (2011).

(eq. 3)

When the input rate (emission) is constant, i.e. that it does not vary with time, and is independent of the mass of chemical present, the mass of chemical in the systems is expected to increase exponentially, from its initial value at , to a steady level at . According to equation 3, a final mass level equal to is to be expected.

Multi-compartment model

The prefix ‘multi’ indicates that generally (many) more than one environmental compartment is considered. The Unit World (see below) contains air, water, biota, sediment and soil; more advanced global modeling systems may use hundreds of compartments. The case of three compartments (typically one air, one water, one soil) is schematically worked out in Figure 3.

Figure 3. Three-compartment mass balance model. Arrows represent mass flows of chemical to and from compartments. Losses from source compartments (negative signs) become gains to receiving compartments (positive sign). The model consists of three differential mass balance equations, with (nine) known rate constants ki,j (for flows out of source compartments i, into receiving compartments j, in s-1), and (three) unknown masses mi (kg). From Van de Meent et al. (2011), with permission.

Each compartment can receive constant inputs (emissions, imports), and chemical can be exported from each compartment by degradation or advective outflow, as in the one-compartment model. In addition, chemical can be transported between compartments (simultaneous import-export). All mass flows are characterized by (pseudo) first-order rate constants (see section 3.3 on Environmental fate processes). The three mass balance equations eventually balance to zero at infinite time:

(eq. 4)

where the symbols denote mass in compartments i at steady state. Sets of n linear equations with n unknowns can be solved algebraically, by manually manipulating equations 4, until clean expressions for each of the three mi values are obtained, which inevitably becomes tedious as soon as more than two mass balance equations are to be solved – this did not withhold one of Prof. Mackay’s most famous PhD students from successfully solving a set of 14 equations! An easier way of solving sets of n linear equations with n unknowns is by means of linear algebra. Using linear algebraic vector-matrix calculus, the equations 4 can be rewritten into one linear-algebraic equation:

(eq. 5)

in which in which is the vector of masses in the three compartments, is the model matrix of known rate constants and is the vector of known emission rates:

The solution of equation 5 is

in which is the vector of masses at steady state and is the inverse of model matrix . The linear algebraic method of solving linear mass balance equations is easily carried with spreadsheet software (such as MS Excel, LibreOffice Calc or Google Sheets), which contain built-in array functions for inverting matrices and multiplying them by vectors.

Unit World modeling

In the late 1970s, pioneering environmental scientists at the USEPA Environmental Research Laboratory in Athens GA, recognized that the universal (mass) balance equation, applied to compartments of environmental media (air, water, biota, sediment, soil) could serve as a means to analyze and understand differences in environmental behavior and fate of chemicals. Their ‘evaluative Unit World Modeling’ (Baughman and Lassiter, 1978; Neely and Blau, 1985) was the start of what is now known as multimedia mass balance modeling. The Unit World concept was further developed and polished by Mackay and co-workers (Neely and Mackay, 1982; Mackay and Paterson, 1982; Mackay et al., 1985; Paterson and Mackay, 1985, 1989). In Unit World modeling, the environment is viewed of as a set of well-mixed chemical reactors, each representing one environmental medium (compartment), to and from which chemical flows, driven by ‘departure from equilibrium’ – this is chemical technology jargon for expressing the degree to which thermodynamic equilibrium properties such as ‘chemical potential’ or ‘fugacity’ differ (Figure 4). Mackay and co-workers used fugacity in mass balance modeling as the central state variable. Soon after publication of this ‘fugacity approach’ (Mackay, 1991), the term ‘fugacity model’ became widely used to name all models of the ‘Mackay-type’, which applied ‘Unit World mass balance modeling’, even though most of these models kept using the more traditional chemical mass as a state variable.

Figure 4. Unit World mass balance modeling as described by Mackay and co-workers. The environment is viewed of as a set of well-mixed chemical reactors (A), for which mass balance equations are formulated. Chemical flows from one environmental compartment to another, driven by ‘departure from equilibrium’, until a steady (= not changing) state has been reached. This may be understood by regarding the hydraulic equivalent of chemical mass flow (B). Figure redrawn after Mackay (1991).

Complexity levels

While conceptually simple (environmental fate is like a leaking bucket, in the sense that its steady-state water height is predictable from first-order kinetics), the dynamic character of mass balance modeling is often not so intuitive. The abstract mathematical perspective may best suit explain mass balance modeling, but this may not be practical for all students. In his book about multimedia mass balance modeling, Mackay chose to teach his students the intuitive approach, by means of his famous water tank analogy (Figure 4B).

According to this intuitive approach, mass balance modeling can be done at levels of increasing complexity, where the lowest, simplest, level that serves the purpose should be regarded as the most suitable. The least complex is level I assuming no input and output. A chemical can freely (i.e. without restriction) flow from one environmental compartment to another, until it reaches its state of lowest energy: the state of thermodynamic equilibrium. In this state, the chemical has equal chemical potential and fugacity in all environmental media. The system is at rest; in the hydraulic analogy, water has equal levels in all tanks. This is the lowest level of model complexity, because this model only requires knowledge of a few thermodynamic equilibrium constants, which can be reasoned from basic physical substance properties.

The more complex modeling level III describes an environment in which flow of chemical between compartments experiences flow resistance, so that a steady state of balance between outputs and inputs is reached only at the cost of permanent ‘departure from equilibrium’. Degradation in all compartments and advective flows, e.g. rain fall or wind and water currents, are also considered. The steady state of level III is one in which fugacities of chemical in the compartments are unequal (no thermodynamic equilibrium); in the hydraulic analogy, water in the tanks rest at different heights. Naturally, solving modeling level III requires detailed knowledge of the inputs (into which compartment(s) is the chemical emitted?), the outputs (at what rates is the chemical degraded in the various compartments?) and the transfer resistances (how rapid or slow is the mass transfer between the various compartments?). Level III modelers are rewarded for this by obtaining more realistic model results.

The fourth complex level of multimedia mass balance modeling (level IV, not shown in Figure 4B) produces transient (time dependent) solutions. Model simulations start (t = 0) with zero chemical (m = 0; empty water tanks). Compartments (tanks), fill up gradually until the system comes to a steady state, in which generally one or more compartments depart from equilibrium, as in level III modeling. Level IV is the most realistic representation of environmental fate of chemicals, but requires most detailed knowledge of mass flows and mass transfer resistances. Moreover, time-varying states are least easy to interpret and not always most informative of chemical fate. The most important piece of information to be gained from level IV modeling is the indication of time to steady state: how long does it take to clear the environment from persistent chemicals that are no longer used?

Mackay describes an intermediate level of complexity (level II), in which outputs (degradation, advective outflows) balance inputs (as in level III), and chemical is allowed to freely flow between compartments (as in level I). A steady state develops in level II and there is thermodynamic equilibrium at all times. Modeling at level II does not require knowledge of mass transfer resistances (other than that resistances are negligible!), but degradation and outflow rates increase the model complexity compared to that of level I. In many situations, level II modeling yields surprisingly realistic results.

Use of multimedia mass balance models

Soon after publication of the first use of ‘evaluative Unit World modeling’ (Mackay and Paterson, 1982), specific applications of the ‘Mackay approach’ to multimedia mass balance modeling started to appear. The Mackay group published several models for the evaluation of chemicals in Canada, of which ChemCAN (Mackay et al., 1995) is known best. Even before ChemCAN, the Californian model CalTOX (Mckone, 1993) and the Dutch model SimpleBox (Van de Meent, 1993) came out, followed by publication of the model HAZCHEM by the European Centre for Ecotoxicology and Toxicology of Chemicals (ECETOC, 1994) and the German Umwelt Bundesamt’s model ELPOS (Beyer and Matthies, 2002). Essentially, all these models serve the very same purpose as the original Unit World model, namely providing standardized modeling platforms for evaluating the possible environmental risks from societal use of chemical substances.

Multimedia mass balance models became essential tools in regulatory environmental decision making about chemical substances. In Europe, chemical substances can be registered for marketing under the REACH regulation only when it is demonstrated that the chemical can be used safely. Multimedia mass balance modeling with SimpleBox (Hollander et al., 2014) and SimpleTreat (Struijs et al, 2016) plays an important role in registration.

While early multimedia mass balance models all followed in the footsteps of Mackay’s Unit World concept (taking the steady-state approach and using one compartment per environmental medium), later models became larger and spatially and temporally explicit, and were used for in-depth analysis of chemical fate.

In the late 1990s, Wania and co-workers developed a Global Distribution Model for Persistent Organic Pollutants (GloboPOP). They used their global multimedia mass balance model to explore the so-called cold condensation effect, by which they explained the occurrence of relatively large amounts of persistent organic chemicals in the Arctic, where no one had ever used them (Wania, 1999). Scheringer and co-workers used their CliMoChem model to investigate long-range transport of persistent chemicals into Alpine regions (Scheringer, 1996; Wegmann et al., 2005). MacLeod and co-workers (Toose et al., 2004) constructed a global multimedia mass balance model (BETR World) to study long-range, global transport of pollutants.

References

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