It is readily apparent that if a composting particle has an anaerobic core for a period of time, then variation in substrate concentration must arise; the outer layers will be more degraded than the inner layers. A useful analogy for this variation is that of the layers of an onion, where each layer differs from its neighbour in its degree of degradation. The task has been to derive a framework for understanding this variation. As oxygen presence
is the main source of this variation then analysing oxygen distribution, which is in turn a consequence of diffusion laws, bases the analysis on the laws of physics.
To derive micro-environment analysis, the constraints of diffusion laws and their currently available solutions were explored to determine oxygen penetration distance in order to determine a micro-environment volume. Diffusion theory was then meshed with microbial reality in which alternative electron acceptors and several rate constants are known to exist. Micro-environment analysis however only accommodates a single solute diffusion coefficient16, in this case the diffusion of oxygen in water, and it is
mathematically necessary for the diffusion to occur from a surface. Hence, a particle surface is defined as the „analytical‟ boundary in this analysis. This analytical boundary also coincides with a particle being: 1) a coherent element of a composting pile; 2) separate from the gas phase in the pile; 3) has a surface which is approximately spherical. With the particle as the analytical element, then particle geometry effects also need to form part of the solution, where inner parts of a particle contain proportionately less compost volume than surface parts.
To achieve optimum knowledge, no single aspect of the theory dominated other aspects in the final solution. In particular, there exists a level of precision for any single aspect at which further precision will be swamped by the „noise‟ in the system. Almost all aspects needed to be compromised:
Diffusion theory – the derivation used only zero-order solutions, yet it is known that first-order solutions apply at the oxygen penetration limit.
Steady-state, non-diffusible substrate diffusion law solutions are used, even though evidence is presented here that substrate diffusion does occur. However, the simpler calculations of these solutions help clarify the analysis necessary to identify the spatial effects (micro-environments). The more complex diffusible substrate solutions, while not used here, are entirely compatible with micro-environment analysis. Indeed, comparing the non-diffusible solutions with diffusible solutions enables identification of those elements of the time course attributable to substrate diffusion (plus the role of other, non-oxygen electron acceptors).
First-order microbial kinetics with no interactions within or between micro- environments is assumed. Clearly there are interactions within and between micro-environments and these could be in the form of pH, toxicity effects, substrate
16 Other diffusion coefficients may exist in either of two forms:
Where the second diffusion coefficient is consequent on the solute‟s diffusion coefficient (that is
concentration differences in substrate that arise from the solute‟s diffusion). These have solutions in diffusion laws, but are not used here.
Where the concentration gradients arise from processes independent of the solute. For example,
breakdown products from anaerobic degradation would require a different diffusion law formulation for understanding.
diffusion (both diffusion of the solid substrate and movement of gaseous breakdown products).
Anaerobic contribution to composting is negligible, yet it is known that low pH arising, in part, from anaerobic degradation impacts on the composting rate.
Particle size and shape – the geometry is based on a spherical particle which would be unlikely to occur in a composting pile, adjustments for non-sphericity could be made but these are ignored. Similarly knowing particle size with any degree of precision would be unlikely without a laborious measurement process.
Micro-environment thickness is determined using planar geometry, whereas spherical geometry should be used. This compromise was possible as micro- environment thickness is small (<10 μm for time interval of 3 hours) compared to a typical particle radius. The computational advantages of being able to use each micro-environment calculation across a range of particle sizes outweighs any error in calculating micro-environment thickness using planar geometry.
The compromises noted above could be placed in either of two categories:
1) Those required only for the formation of micro-environments, particularly the zero- order diffusion law solution.
2) Those which could be used without compromising the organisational form, such as diffusible substrate solutions to diffusion law and different microbial kinetics. These would, however, increase the computational difficulty, the benefit of the increased analytical precision needs to be balanced with this increased
computational difficulty.
For 1) above, the zero-order solution to diffusion law compromise only needs to exist at that instant in time that a micro-environment is formed. This arises as the formation of micro-environments is integral with the advance of the oxygen penetration boundary, and hence they increment with the oxygen front. Beyond this instant of formation, the micro- environment becomes purely a volume of compost containing substrate with a
concentration known to very high precision. As such its allegiance to zero-order oxygen diffusion solutions, necessary for its formation, ceases and any diffusion law solution can be used on this volume of compost (even ones which do not have an oxygen penetration limit). The micro-environment undergoes a transition from a compromised logic system to a high quality analysis tool.
Beyond the instant of the formation of a micro-environment, the parameters contained within the micro-environment space can be used in any computational form that is desired. For most of the compromises above, it is not the micro-environment which is compromised (as in 1) above), but the integrity of the model being used. For example, one may debate the adequacy of first-order microbial kinetics, but as any kinetics system can be used in
micro-environment analysis, the debate has no impact on the validity of the micro- environment itself.
By contrast, using a diffusible substrate solution of diffusion laws would impact
significantly on the time course of the oxygen penetration depth, as substrate will diffuse into the micro-environment. This would need additional considerations as the micro- environments formed may be smaller in thickness and have a different time course, but the micro-environment organisational form remains unaffected.
Micro-environment analysis accommodates the spatial variation of anaerobic/ aerobic rate constants, where aerobic rate constants occur in the outer parts of a particle and anaerobic rate constants occur in the core. However, with complex substrates, rate constants will be an average of all the rate constants of all the compounds being degraded. The difficulty in reliably modelling this complexity implies a case for using a simple kinetics and making explicit the spatial variation in substrate concentration that must arise due to the difference between aerobic and anaerobic rate constants. This then provides a basis on which increased computational complexity can sit.