Self-heating in compost piles due to biological effects
Introduction
The phenomenon of spontaneous ignition due to internal heating in bulk solids such as coal, grain, hay, wool wastes, and other materials, can be described by thermal explosion theory (Bowes, 1984, Frank-Kamenetskii, 1969). In such models heat release is usually represented by a single Arrhenius reaction. However, in industrial processes involving large volumes of bulk organic materials there are two sources of heat-generation: a low-temperature process involving the growth and respiration of micro-organisms, such as aerobic mould-fungi and bacteria, and a high-temperature process due to oxidation of cellulosic materials. Examples of processes where biological heating is important include large-scale composting operations (Rynk, 2000), the storage of industrial waste fuel, such as municipal solid waste, and landfills. In these, for example in composting (Brinton et al., 1995), self-heating due to biological activity is a desired outcome. Elevated temperatures of the order may be found within a few months or even a few days (Hogland et al., 1996). Although it has been recognised for over 20 years that “biological heating may be an indispensable prelude to self-ignition” (Bowes 1984, p. 373), little information is available regarding the mechanism of fires when biological self-heating isinvolved—despite fires at landfills being common worldwide (Hudak, 2001).
The case when self-heating is entirely due to cellulosic oxidation is well known (Bowes, 1984, Frank-Kamenetskii, 1969). Here we consider a scenario when self-heating is entirely due to the biomass. This is a constructive step towards the analysis of a model containing heat-release due to both biological and chemical activity as it establishes regions in parameter space in which biological self-heating leads to elevated temperature raises—the main feature of interest in landfill sites. Furthermore, a model without chemical self-heating applies to environments under which anaerobic biodegradation predominates; anaerobic oxidation being a known technology to treat the organic component of municipal solid wastes. The resulting mathematical model also serves as a useful testing ground for a new method of carrying out bifurcational analysis of spatially distributed systems that has been developed by one of the authors. This is discussed in Section 4.2. An advantage of this method, over direct numerical integration, is that all solution branches are found, not just the stable branches.
Comprehensive mathematical models for the combustion of biomass have been published in recent years (Jand and Foscolo, 2005; Radmanesh et al., 2006, Wurzenberger et al., 2002). These models have been developed to optimise the thermal conversion of biomass in furnaces and to predict the composition of the product gas. Such models may include detailed consideration of the physics of the furnace (including hydrodynamics), chemical kinetics (pyrolysis chemistry and heterogeneous/homogeneous gas-phase reactions) and knowledge of physical data such as fuel density, thermodynamic, and transport properties. Thus knowledge of a large number of physical and kinetic parameters is required. Such detail is not warranted in the present case, where we extend classical thermal explosion theory to cover the case of biological self-heating. Our aim is to comprehensively identify the generic behaviour exhibited by a simple model, in particular identifying parameter regions where significant self-heating occurs, rather than to present detailed simulations showing the evolution of a compost pile in such regions.
In this paper we use a Galerkin method to obtain a semi-analytical approximation to the solution of a spatially distributed reactor model. This techniques yields a system of coupled non-linear ordinary differential equations, which can be analysed using standard methods. In this section we outline some alternative approaches that can be used to investigate the behaviour of spatially structured reactor models.
A naive approach to dealing with a distributed reactor model is to discretise it to produce a set of ordinary differential equations and to then use standard numerical bifurcation algorithms. For 1-d models good results can be obtained for discretisations using of the order of points. However, for 2-d and 3-d models the growth in nodal points makes it impractical to compute the Jacobian matrix and eigenvectors using such a direct approach (Balakotaiah and Khinast, 2000). For problems in one spatial dimension an efficient, and powerful technique, is to combine Liapunov–Schmidt reduction with shooting methods (Balakotaiah and Khinast, 2000, Subramanian and Balakotaiah, 1996). However, this method does not generalise to problems in two, or more, spatial dimensions. For problems in higher dimensions, Subramanian and Balakotaiah (1996) suggest that the model equations be averaged in the spatial directions in which the variables do not change rapidly.
An alternatively averaging method uses the Liapunov–Schmidt technique to perform spatial homogenization over small scales. Averaging is done over the local (transverse) dimensions. This leads to a series solution in a small parameter which is the ratio of local diffusion time to convection time of the system. This method has been applied to convection–diffusion–reaction (CDR) equations to obtain low-dimensional models. The first-order term in the series expansion parameter is sufficient to retain all the qualitative features of the CDR model (Chakraborty and Balakotaiah, 2002a). In the low-dimensional models the concentration of a chemical species is not given by a single equation but by a system of differential–algebraic equations. The differential equation describes the variation of the “cup-mixing” concentration with the residence time whilst the algebraic equation(s) captures mixing on local scales. Typically, one algebraic equation is used to model micromixing. (For non-isothermal systems there are multiple temperature equations).
This method has been applied to thermal and solutal dispersion (Balakotaiah and Chang, 2003), laminar flow in isothermal tubular reactors with homogeneous reactions (Chakraborty and Balakotaiah, 2002a, Chakraborty and Balakotaiah, 2002b, Chakraborty and Balakotaiah, 2003), to a variety of isothermal wall-catalyzed and coupled homogeneous–heterogeneous reacting flows (Balakotaiah and Chakraborty, 2003) and to non-isothermal reactor systems (Chakraborty and Balakotaiah, 2004). The methodology developed in these papers is limited to CDR models with Neumann (zero flux), Robin or periodic boundary conditions. Furthermore, the averaged models exist only when the local diffusion time is much smaller than the convective and characteristic reaction times.
Section snippets
Temperature dependence of the reaction rate
The basis of our model is that developed by Frank-Kamenetskii (FK) to explain the phenomenon of thermal explosions in spatially distributed systems (Bowes, 1984, Frank-Kamenetskii, 1969). However, the temperature dependence of the reaction rate is not given by an Arrhenius expression but is instead parameterised in the form:The formulation of Eq. (1) encapsulates that activation and inactivation processes occur over different temperature ranges. At low
Semi-analytical theory
The Galerkin method is used to obtain semi-analytical solutions for the compost problem in the 1- and 2d-dimensional slab geometries. This involves approximating the spatial structure of the temperature profile in the compost heap using trial functions that satisfy the boundary conditions. The semi-analytical model is then obtained by averaging the governing energy equation. As the expression for biological self-heating cannot be integrated explicitly, the semi-analytical model is given by a
Results
As stated above we make the standard simplifying assumption, the pre-exponential approximation, that . The biomass FK parameter includes the size of the compost heap and is a controllable parameter; it is taken as the primary bifurcation parameter. For a particular type of biomass the reduced inhibition activation energy and the reduced deactivation rate are fixed and these are therefore viewed as unfolding parameters.
In Section 4.1 we use the continuation and bifurcation
Discussion
When heat-release due to cellulosic oxidation is incorporated into the model the energy balance Eq. (1) becomesin which is a FK parameter for chemical heating and is a reduced activation energy for chemical heating. The dimensionless grouping contains the relevant kinetic parameters for chemical heating. It is defined in an analogous way to the standard approach (Bowes, 1984, Frank-Kamenetskii, 1969) except that the reference activation
Conclusions
In this paper we have investigated a model for biological self-heating in spatially distributed compost heaps. After making the pre-exponential approximation the model only contains three parameters. Consequently it is possible to thoroughly investigate the generic behaviour of the model and to identify parameter regions in which biological self-heating can give rise to elevated temperatures, which is the feature of practical interest. A novel semi-analytical method was used to generate highly
Notation
pre-exponential factor for biomass growth, pre-exponential factor for deactivation of biomass, dimensionless the biomass density, Representative value for the biomass density, the maximum biomass density, specific heat of the compost pile, activation energy for biomass growth, activation energy for the deactivation of biomass, activation energy for cellulose oxidation, thermal conductivity, the half length of the compost
Acknowledgements
This work is supported by a grant from Sultan Qaboos University (IG/SCI/DOMS/04/05). M.I.N. thanks the Department of Mathematics and Statistics at Sultan Qaboos University for their hospitality and support during his visit.
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