Hamelers (1993) applied to a composting particle, the diffusional processes modelled in the mud-water interface by Harremoës (1978). Hamelers‟ particle was assumed to be a matrix of insoluble particles, inert matter and water-filled pores. All activity was assumed to occur in the water phase. Hydrolysis and fermentation reactions in the anaerobic core generated soluble substrates which then diffused to the aerobic surface of the particle and were composted, this activity inducing “the development of other
gradients as of biomass and substrate concentration” (Hamelers, 1993, p. 39). The rate of conversion of the polymeric substrate to monomeric substrate (insoluble to soluble), was modelled using a Monod type function while aerobic composting was modelled using biomass concentrations and the well known biomass yield per unit of substrate consumed. This framework resulted in five differential equations and one equality relationship which needed to be solved numerically (Hamelers, 1993). In his subsequent thesis (Hamelers, 2001) he took a mathematical approach to the model which included: identifiability analysis, and an analytical approximate solution for a single particle. Later chapters extended the model to several particle sizes, called the distributed OUR model. A later co-authored paper (Hamelers & Richard, 2001) extended the approach, first developed for his distributed OUR model, by using matric potential to determine particle size as
influenced by moisture content.
Hamelers‟ work contains the inherent assumption, arising from diffusion laws, that the volumetric oxygen uptake rate is constant throughout the aerobic zone. With this assumption the surface flux approach used by Hamelers is valid as any oxygen passing through the surface of the particle will be equally as likely to be consumed in any part of the aerobic zone. However, if this assumption is not true then a different diffusion law solution is needed. Such a solution has already been derived by Stępniewski (Gliński & Stępniewski, 1985), to investigate the oxygen profile in soil layers. Stępniewski‟s numerical model allowed for both different oxygen uptake rates and different diffusion coefficients in any number of horizontal layers.
Also implicit in Hamelers‟ work is recognition of the limit of mathematical knowledge for complex systems. In order to solve the equations for a variable particle size, Hamelers needed to make mathematical compromises by „lumping parameters‟. It was shown that this resulted in a 0.7 to 2.1% (Hamelers, 2001, p. 144) compromise in the accuracy of the solution. This raises the question as to what compromised accuracy would be needed to
solve the equations for variable temperature, such as the change from ambient
temperatures to the thermophilic temperature range met in large scale composting, or the diurnal/seasonal variation in temperatures experienced by a smaller compost pile. Changing temperature, and its effect on the rate constant, is surely a significant parameter to be incorporated into composting models.
Over-parameterization is a common result of this modelling approach (Haag et al., 2005; Hamelers, 2001); Hamelers needed to use identifiable combined parameters to overcome the identifiability problem (Hamelers, 2004).
In assuming that composting occurred in the aerobic part of the particle and that substrate diffused from the core of the particle, Hamelers has also assumed that the limit of oxygen penetration is constant as this is implicit in his assumption of constant oxygen uptake rate. When this assumption is relaxed it is readily apparent that the diffusion law solution
becomes one of a moving boundary problem, where the oxygen penetration depth is continuously increasing. The moving boundary problem has been solved for diffusion laws (Crank & Gupta, 1972) and the same mathematical solution also applies to heat transfer into a phase-change material (Crank, 1957), and in biology (Britton, 1986) where it is called a travelling wave, but not yet applied to composting.
Hamelers‟ (1993, 2001) work, while being a major step forward in compost understanding falls short of a full synecological approach as the assumptions implicit in the diffusion law solutions that are used are not all valid. In particular:
Oxygen uptake rate (VOR) is not constant as assumed for most diffusion law solutions. VOR changes:
o over time;
o with distance from the particle surface;
o with temperature.
A moving boundary solution is required.
In addition, a full synecological solution would give analytical space to microbial complexity, in particular:
the existence of alternative electron acceptors;
spatial heterogeneity of the biomass in response to the substrate and electron acceptor distributions.
Chapter 3
3 THEORY
The need for synecological approaches in which microbial interactions with their environment become explicit can be filled, in part, by applying diffusion laws to the dominant electron acceptor. A framework for understanding this complexity is needed and diffusion law solutions based on oxygen distribution are a good start to this
framework.
For a composting particle, organic matter is converted to biomass or humus, or undergoes a change of state (evolving mainly H2O, CO2). In the process, oxygen is consumed by the
microbial biomass resulting in a concentration gradient between the interstitial air and the inside of the particle. The process is known as composting and the organic matter is said to be degraded. Over time the amount of organic matter decreases, resulting in lower composting rates, lower volumetric oxygen consumption rates and greater oxygen penetration depths. Thus, if a composting particle is sufficiently large so that oxygen is initially unable to reach the core of the particle then over time, as the outer layers degrade, oxygen will penetrate further into the particle – eventually reaching the core. If
significant time is required before oxygen reaches the core of the particle, then the concentration of the organic matter in the outer layers will differ significantly from the concentration of the organic matter in the inner layers; spatial variation will arise. The theoretical basis for determining this variation is presented here.
This chapter is in 6 parts:
1) The thermodynamic perspective of microbial kinetics and its applicability to composting is discussed in section 3.1.
2) The limits of diffusion law solutions as they can be applied to composting are discussed in section 3.2.
3) The characteristics of microbial kinetics, in particular that oxygen is not the only electron acceptor, which further constrains the application of diffusion laws, is discussed in section 3.3.
4) The logic of the emergence of spatial variation in substrate concentrations at sub- particle scales is argued in section 3.4.
5) A novel analysis framework (called micro-environment analysis or MEA) which links the limits of diffusion law solutions, microbial kinetic characteristics and the spatial variation in substrate concentration is proposed in section 3.5.
6) The mathematical framework of micro-environment analysis is presented in section 3.6.