A variable-stepsize Jacobian-free exponential integrator for simulating transport in heterogeneous porous media : application to wood drying

A Jacobian-free variable-stepsize method is developed for the numerical integration of the large, stiff systems of differential equations encountered when simulating transport in heterogeneous porous media. Our method utilises the exponential Rosenbrock-Euler method, which is explicit in nature and requires a matrix-vector product involving the exponential of the Jacobian matrix at each step of the integration process. These products can be approximated using Krylov subspace methods, which permit a large integration stepsize to be utilised without having to precondition the iterations. This means that our method is truly "Jacobian-free" - the Jacobian need never be formed or factored during the simulation. We assess the performance of the new algorithm for simulating the drying of softwood. Numerical experiments conducted for both low and high temperature drying demonstrates that the new approach outperforms (in terms of accuracy and efficiency) existing simulation codes that utilise the backward Euler method via a preconditioned Newton-Krylov strategy.

Exponential integrators and a dual-scale model for wood drying

For the timber industry, the ability to simulate the drying of wood is invaluable for manufacturing high quality wood products. Mathematically, however, modelling the drying of a wet porous material, such as wood, is a diffcult task due to its heterogeneous and anisotropic nature, and the complex geometry of the underlying pore structure. The well{ developed macroscopic modelling approach involves writing down classical conservation equations at a length scale where physical quantities (e.g., porosity) can be interpreted as averaged values over a small volume (typically containing hundreds or thousands of pores). This averaging procedure produces balance equations that resemble those of a continuum with the exception that effective coeffcients appear in their deffnitions. Exponential integrators are numerical schemes for initial value problems involving a system of ordinary differential equations. These methods differ from popular Newton{Krylov implicit methods (i.e., those based on the backward differentiation formulae (BDF)) in that they do not require the solution of a system of nonlinear equations at each time step but rather they require computation of matrix{vector products involving the exponential of the Jacobian matrix. Although originally appearing in the 1960s, exponential integrators have recently experienced a resurgence in interest due to a greater undertaking of research in Krylov subspace methods for matrix function approximation. One of the simplest examples of an exponential integrator is the exponential Euler method (EEM), which requires, at each time step, approximation of φ(A)b, where φ(z) = (ez - 1)/z, A E Rnxn and b E Rn. For drying in porous media, the most comprehensive macroscopic formulation is TransPore [Perre and Turner, Chem. Eng. J., 86: 117-131, 2002], which features three coupled, nonlinear partial differential equations. The focus of the first part of this thesis is the use of the exponential Euler method (EEM) for performing the time integration of the macroscopic set of equations featured in TransPore. In particular, a new variable{ stepsize algorithm for EEM is presented within a Krylov subspace framework, which allows control of the error during the integration process. The performance of the new algorithm highlights the great potential of exponential integrators not only for drying applications but across all disciplines of transport phenomena. For example, when applied to well{ known benchmark problems involving single{phase liquid ow in heterogeneous soils, the proposed algorithm requires half the number of function evaluations than that required for an equivalent (sophisticated) Newton{Krylov BDF implementation. Furthermore for all drying configurations tested, the new algorithm always produces, in less computational time, a solution of higher accuracy than the existing backward Euler module featured in TransPore. Some new results relating to Krylov subspace approximation of '(A)b are also developed in this thesis. Most notably, an alternative derivation of the approximation error estimate of Hochbruck, Lubich and Selhofer [SIAM J. Sci. Comput., 19(5): 1552{1574, 1998] is provided, which reveals why it performs well in the error control procedure. Two of the main drawbacks of the macroscopic approach outlined above include the effective coefficients must be supplied to the model, and it fails for some drying configurations, where typical dual{scale mechanisms occur. In the second part of this thesis, a new dual{scale approach for simulating wood drying is proposed that couples the porous medium (macroscale) with the underlying pore structure (microscale). The proposed model is applied to the convective drying of softwood at low temperatures and is valid in the so{called hygroscopic range, where hygroscopically held liquid water is present in the solid phase and water exits only as vapour in the pores. Coupling between scales is achieved by imposing the macroscopic gradient on the microscopic field using suitably defined periodic boundary conditions, which allows the macroscopic ux to be defined as an average of the microscopic ux over the unit cell. This formulation provides a first step for moving from the macroscopic formulation featured in TransPore to a comprehensive dual{scale formulation capable of addressing any drying configuration. Simulation results reported for a sample of spruce highlight the potential and flexibility of the new dual{scale approach. In particular, for a given unit cell configuration it is not necessary to supply the effective coefficients prior to each simulation.

A new control-volume finite-element scheme for heterogeneous porous media : application to the drying of softwood

An improved mesoscopic model is presented for simulating the drying of porous media. The aim of this model is to account for two scales simultaneously: the scale of the whole product and the scale of the heterogeneities of the porous medium. The innovation of this method is the utilization of a new mass-conservative scheme based on the Control-Volume Finite-Element (CV-FE) method that partitions the moisture content field over the individual sub-control volumes surrounding each node within the mesh. Although the new formulation has potential for application across a wide range of transport processes in heterogeneous porous media, the focus here is on applying the model to the drying of small sections of softwood consisting of several growth rings. The results conclude that, when compared to a previously published scheme, only the new mass-conservative formulation correctly captures the true moisture content evolution in the earlywood and latewood components of the growth rings during drying.

Physical and mechanical properties of thermally modified wood from E. grandis

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Decay resistance of thermally-modified Eucalyptus grandis wood at 140 degrees C, 160 degrees C, 180 degrees C, 200 degrees C and 220 degrees C

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Behavior of the brown-rot fungus Gloeophyllum trabeum on thermally-modified Eucalyptus grandis wood

In this study, we aimed evaluate the behavior of the brown-rot fungus Gloeophylum trabeum and white-rot fungus Pycnoporus sanguineus on thermally-modified Eucalyptus grandis wood. To this end, boards from five-year-eleven-month-old E. grandis trees, taken from the Duratex-SA company stock, were thermally-modified between 180 ºC and 220 ºC in the Laboratory of Wood Drying and Preservation at Universidade Estadual Paulista - UNESP, Botucatu, Sao Paulo state Brazil. Samples of each treatment were tested according to the ASTM D-2017 (2008) technical norm. The accelerated decay caused by the brown-rot fungus G. trabeum was compared with the decay caused by the white-rot fungus P. sanguineus, studied by Calonego et al. (2010). The results showed that (1) brown-rot fungus caused greater decay than white-rot fungus; and (2) the increase in temperature from 180 to 220 ºC caused reductions between 28.2% and 70.0% in the weight loss of E. grandis samples incubated with G. trabeum.

Krylov subspace approximations for the exponential Euler method: Error estimates and the harmonic Ritz approximant

We study Krylov subspace methods for approximating the matrix-function vector product φ(tA)b where φ(z) = [exp(z) - 1]/z. This product arises in the numerical integration of large stiff systems of differential equations by the Exponential Euler Method, where A is the Jacobian matrix of the system. Recently, this method has found application in the simulation of transport phenomena in porous media within mathematical models of wood drying and groundwater flow. We develop an a posteriori upper bound on the Krylov subspace approximation error and provide a new interpretation of a previously published error estimate. This leads to an alternative Krylov approximation to φ(tA)b, the so-called Harmonic Ritz approximant, which we find does not exhibit oscillatory behaviour of the residual error.

Physical and chemical changes in juvenile and mature woods of Pinus elliottii var. elliottii by thermal modification

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Estimativa do tempo de vaporização das toras e sua implicação no desdobro e na secagem da madeira de Eucalyptus grandis

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Effect of thermal modification on the physical properties of juvenile and mature woods of Eucalyptus grandis

This study aimed to evaluate the effect of thermal treatment on the physical properties of juvenile and mature woods of Eucalyptus grandis. Boards were taken from 30-year-old E. grandis trees. The boards were thermally modified at 180 °C in the Laboratory of Wood Drying and Preservation at UNESP, Botucatu, Sao Paulo state, Brazil. The results showed that thermal modification caused: (1) decrease of 6.8% in the density at 0% equilibrium moisture content of mature wood; (2) significant decreases of 14.7% and 35.6% in the maximum volumetric swellings of juvenile and mature woods, respectively; (3) significant decreases of 13.7% and 21.3% in the equilibrium moisture content of juvenile and mature woods, respectively. The influence of thermal modification in juvenile wood was lower than in mature wood and caused greater uniformity in the physical variations between these types of wood in E. grandis.

Dimensional change behavior of caribbean pine using an environmental scanning electron microscope

Moisture transport and dimensional change during wood drying or wetting processes were analyzed based on pictures from an environmental scanning electron microscope (ESEM). This provides quantitative relationships between dimensional changes of total area, cell wall, and lumen, and moisture content for earlywood and latewood. Earlywood and latewood behave similarly but show some quantitative differences. The overall outcome for sections containing both kinds of wood seems to be dominated by the latewood behavior. The observed strain behavior of wood during drying is anisotropic in ways that are inconsistent with explanations solely related to microfibril orientation or earlywood/latewood interactions and more likely may be influenced by ray tracheids.

Characterisation of wood properties and transverse anatomy for vacuum drying modelling of commercially important Australian hardwood species.

Characterisation and investigation of a number of key wood properties, critical for further modelling work, has been achieved. The key results were: • Morphological characterisation, in terms of fibre cell wall thickness and porosity, was completed. A clear difference in fibre porosity, size, wall thickness and orientation was evident between species. Results were consistent with published data for other species. • Viscoelastic properties of wood were shown to differ greatly between species and in the radial and tangential directions, largely due to anatomical and chemical variations. Consistent with published data, the radial direction shows higher stiffness, internal friction and glass transition temperature than the tangential directions. The loss of stiffness over the measured temperature range was greater in the tangential direction than the radial direction. Due to time dependant molecular relaxation, the storage modulus and glass transition temperature decreased with decreasing test frequency, approaching an asymptotic limit. Thus the viscoelastic properties measured at lower frequencies are more representative of static material. • Dynamic interactions between relative humidity, moisture content and shrinkage of four Australian hardwood timbers can be accurately monitored on micro-samples using a specialised experimental device developed by AgroParisTech – ENGREF. The device generated shrinkage data that varied between species but were consistent (repeatable) within a species. Collapse shrinkage was clearly evident with this method for Eucalyptus obliqua, but not with other species, consistent with industrial seasoning experience. To characterise the wood-water relations of this species, free of collapse, thinner sample sections (in the R-T plane) should be used.

Wood veneer : log selection, cutting, and drying /

Issued Jan. 1978.