- Email: [email protected]
Three-dimensional simulation of the rolling of a saturated fibro-porous media
Finite Elements in Analysis and Design 42 (2005) 90 – 104 www.elsevier.com/locate/finel
Three-dimensional simulation of the rolling of a saturated fibro-porous media J.G. Loughrana , Arasu Kannapiranb,∗ a School of Engineering, James Cook University, Townsville 4811, Australia b School of Engineering, The University of Queensland, St. Lucia 4072, Australia
Received 19 March 2004; accepted 25 May 2005 Available online 15 August 2005
Abstract This paper describes recent advances made in computational modelling of the sugar cane liquid extraction process. The saturated fibro-porous material is rolled between circumferentially grooved rolls, which enhance frictional grip and provide a low-resistance path for liquid flow during the extraction process. Previously reported two-dimensional (2D) computational models, account for the large deformation of the porous material by solving the fully coupled governing fibre stress and fluid-flow equations using finite element techniques. While the 2D simulations provide much insight into the overarching cause-effect relationships, predictions of mechanical quantities such as roll separating force and particularly torque as a function of roll speed and degree of compression are not satisfactory for industrial use. It is considered that the unsatisfactory response in roll torque prediction may be due to the stress levels that exist between the groove tips and roots which have been largely neglected in the geometrically simplified 2D model. This paper gives results for both two- and three-dimensional finite element models and highlights their strengths and weaknesses in predicting key milling parameters. 䉷 2005 Elsevier B.V. All rights reserved. Keywords: Porous material; Constitutive laws; Plane strain model; Grooving effects; 3D modelling
1. Introduction Many industrial processes rely on mechanical dewatering as the primary technique for separating liquid from saturated solids. Predictive models for these processes have been largely empirical due to ∗ Corresponding author. Tel.: +61 7 33652969; fax: +61 7 33654799.
E-mail address: [email protected] (A. Kannapiran). 0168-874X/$ - see front matter 䉷 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.finel.2005.05.006
J.G. Loughran, A. Kannapiran / Finite Elements in Analysis and Design 42 (2005) 90 – 104
91
Feed chute
Under feed roll
Pressure feed roll
Pressure feed chute
Pressure feed roll Top roll
Feed roll Delivery roll
Return chute
Fig. 1. Schematic of a six-roll crushing unit.
complex and often highly non-linear constitutive laws and in the case of rolling, evolution of boundary conditions. Nevertheless, over the last decade or so, computational methods in solid mechanics have matured considerably and when coupled with the marked advance in computing hardware, permit the numerical solution of complex industrial problems. Expression of juice from porous sugar cane is carried out in the initial stages of the raw sugar production process. The sugar cane undergoes comminution by heavy-duty swing hammer shredders before it is presented for rolling. The material is fibrous with particle sizes ranging from typically 1 mm to damaged fibro-vascular bundles up to 60 mm in length and 2–3 mm in diameter. The fibre material is hygroscopic and liquid is attached to all particles with varying propensity. Although Kauppila and Barnes [1] have shown that the liquid is more easily removed from the smaller particles, there is no factory process for segregation. The material is fed directly to a tandem of six-roll crushing units, which separate the liquid from the fibre under high-pressure rolling. Fig. 1 shows a schematic of a typical six-roll crushing unit. The rolls are circumferentially grooved to enhance feeding and to offer a low-resistance path for liquid escape from the compressed blanket. During the late 1950s, a considerable research effort was begun on understanding the fundamentals of sugar cane crushing [2–6]. Bullock [2] initiated the research with basic experiments on an experimental milling facility. That research was extended by Murry [3] and others [3–6] who consolidated Bullock’s experimental responses and began investigations into fundamental modelling of the porous media mechanics problem. The empirical responses are still used widely today by practitioners in the field. The porous media models were largely one-dimensional and were built around Darcy’s Law. No serious attempt was made to include the constitutive characteristics of the fibrous skeleton, and therefore the roll load and torque were estimated solely on the pressure effects of juice. In an effort to improve efficiency and reduce factory costs, there has been a research push, over the last decade, to apply modern computational and constitutive mechanics to the sugar cane crushing process.
J.G. Loughran, A. Kannapiran / Finite Elements in Analysis and Design 42 (2005) 90 – 104
92
Owen et al. [7] applied computational porous media mechanics to sugar cane crushing by treating the prepared cane as a fully coupled unsaturated/saturated two-phase problem. Prepared cane was considered a continuous isotropic porous medium from a macroscopic and statistical viewpoint, with fluid and solid phases forming overlapping regions. The governing equations were written for the porous medium assuming that an elemental volume [8] was large when compared to pores but small compared to the overall extent of the domain. For all practical purpose, the gas flow that may be present at atmospheric pressure was neglected. Some useful prediction trends were developed using a simplified non-linear constitutive law for the fibrous skeleton. deSouza et al. [9] extended the earlier work of Owen by employing CamClay and crushable foam constitutive laws for the fibre skeleton. The numerical procedures for rolling with evolving boundary conditions and adaptive meshing were also discussed. Adam [10] extended Owen’s model to include a Capped Drucker–Prager constitutive law with linear or porous elasticity. Adam and Loughran [11] report on parameter estimation procedures for Owen type models. Plaza and Kent [12] and Plaza et al. [13] have also reported experiments on characterizing the properties of the fibre skeleton subjected to normal shear loads. Adam provided computational solutions for internal and external parameters associated mostly with two-roll geometries and light to moderate nip compactions. For example, fibre velocities, strain, stress, liquid velocities, plasticity and elastic recovery could be predicted as well as roll load and torque. At the time it was reported that model validation was a critical issue. Adam was relying on 1950–1960 two-roll mill data for comparison with predictions. Recent studies by Loughran and Kannapiran [14] focused on exercising Adam’s numerical model and making direct comparisons with data obtained from modern day two-roll mill experiments. The cane used in the mill experiments has also passed through a series of basic tests (following Loughran and Adam [15]) to characterize its properties for computational models. Loughran and Kannapiran [14] found that the two-dimensional (2D) plane strain models used by previous researchers oversimplified the complex skeletal deformation response in the vicinity of the grooved circumferential rolls and led to poor predictions of roll torque. This paper describes the modelling approach adopted by Loughran and Kannapiran [14] and gives results for two- and three-dimensional (3D) simulations.
2. Modelling assumptions and approach The fundamental equations governing deformable porous media [7] can be simplified by noting the following assumptions: 1. 2. 3. 4. 5. 6.
solid and liquid phases are incompressible; the gas pressure in the medium is nominally zero; for quasi-static analysis, solid and liquid accelerations are negligible; gravitational effects are negligible; Darcy’s law is valid; Terzaghi’s principle of effective stress is valid. Hence LT = 0
is the equilibrium equation and L, are the differential operator and total stress vector respectively.
(1)
J.G. Loughran, A. Kannapiran / Finite Elements in Analysis and Design 42 (2005) 90 – 104
93
The equilibrium equation of the liquid phase is related through Darcy’s law by ˙ = ∇Pl , k −1
(2)
where k is the Darcy coefficient matrix of permeability, ˙ is the superficial velocity of the pore fluid relative to the fibre and Pl is the liquid pore pressure. The liquid flow continuity equation is given by applying mass conservation principles, ˙ = mT ˙, −∇ T where
∇T =
j
,
j
(3)
,
j
jx jy jz
,
is the total strain vector and mT = [1, 1, 1, 0, 0, 0].
Following Terzaghi [16] the principle of effective stress links the effective stress e to the total stress
, through
= e − mP l .
(4)
The effective stress e is then related to the strain displacement d in incremental form by d e = D T d
(5)
d = LT du.
(6)
and
DT is the tangent stiffness matrix which is dependent on the level of effective stress and the total strain of the skeleton and du represents the incremental displacement vector. The above governing equations represent a fully coupled highly non-linear set of partial differential equations, which undergo spatial and temporal discretization in the Lagrangian model prior to solution [17]. For this coupled application it is noted that the stability of the finite element solution is enhanced if different interpolation/integration orders are used for the liquid pressure and velocity fields. Either a full Newmark scheme or reduced rank formulation can be used for time discretization. deSouza et al. [9] noted no practical difference in computational output. The resulting governing equations are then solved with a Newmark scheme. 3. Constitutive models for crushing The specification of constitutive models for crushing presents a significant challenge. The material enters a vertical chute under gravity in a loosely packed condition. Here the individual particles in the fibrous skeleton are approximately randomised and the collection of material is largely unsaturated. Under light loading conditions (for example, as material enters the first nip), fibres align orthogonal to the direction of maximum principal stress and saturation may occur. Saturation definitely occurs if the level of nip compression is moderate. This suggests that the material might develop some degree of anisotropy during the compression process. Constitutive models are required for the liquid phase, interaction of
J.G. Loughran, A. Kannapiran / Finite Elements in Analysis and Design 42 (2005) 90 – 104
94
the skeleton against roll boundaries or other rigid surfaces and large strain deformation response of the skeleton. Following previous researchers [18,19], the liquid phase is modelled with Darcy’s Law and the contact behaviour by a suitable friction law. Adam [10] developed an empirical friction law for plane strain modelling of grooved circumferential rolls which accounted for surface speed, groove angle, the effective normal stress and the degree of material fineness (in an average sense). For simplicity, we adopt a constant Coulomb friction of 0.5 between the fibrous skeleton and roll interface in this paper. The model of the fibrous skeleton must reproduce the elasto-plastic response of the material across a wide range of nip loadings and compression speeds. This is an onerous task for such a complex material. Experimental observations [10,12,13,20] show that the material: • • • • • • •
Is highly compressible and yields under hydrostatic pressure. Exhibits large elastic and plastic volumetric strains. Has a non-linear plastic strain hardening relation. Has uniaxial plastic strains co-directional with applied stress. Exhibits anisotropic behaviour due to layering of fibres. Shows no shear failure under uniaxial or triaxial tests. A marked shear failure when loaded in a typical geo-mechanics soil box experiment.
To date researchers have opted for critical state isotropic constitutive models due to the “ease” of obtaining experimental property data compared with the inherent complexity of anisotropic models and their respective data sets. Following Adam [10] we define the material response through an isotropic linear elastic law and Capped pl Drucker–Prager plasticity law. The flow rule of incremental plastic volumetric strain v versus hydrostatic yield stress pc for a typical tested cane takes the form pl
pc = 0.0346 e4.1321v .
(7)
Unlike the Cam-Clay models [21], the Drucker–Prager model has a yield surface which includes a shear failure surface, and a cap that intersects the equivalent pressure stress axis p (p = −1/3 trace ). The cap yield surface has an elliptical shape with constant eccentricity in the p–t plane (Fig. 2). The cap surface hardens or softens as a function of plastic volumetric strain. Associated plastic flow is assumed on the cap and non-associated flow in the cone and transition regions. The cone and cap regions are expressed in mathematical form as t = p tan + d
(8)
and t2 = 1 + −
2
( + pa tan )2 −
p − pa R
2 ,
(9)
respectively, where is the material’s angle of friction, d the cohesion for the material, t the deviatoric stress measure, R the material parameter which controls the shape of cap, a small number (typically 0.01–0.05) defining a smooth transition region between cone and cap, and pa the evolution parameter which describes the size of the yield surface.
J.G. Loughran, A. Kannapiran / Finite Elements in Analysis and Design 42 (2005) 90 – 104
95
Transition
t Similar ellipses Shear failure
α (d+pa tan β )
Cap
d+ pa tanβ
Cone
β
d pa
R(d+p a tanβ)
pb
p
Fig. 2. Yield surface for capped Drucker–Prager plasticity model.
For the associated flow rule in which the failure envelope grows uniformly, the plastic strain hardening relation for the material is developed using inverse-calibration from finite element simulation of a confined uniaxial test cell [15]. In undertaking this calibration, a somewhat arbitrary but high value of the critical state slope M is assumed (typically 3.5–3.8) [10]. This limits lateral expansion of the material and mimics the inherent fibre locking, which occurs at a micro level.
4. Application This section gives simulation results for 2D plane strain and 3D models of a single nip. The objective is to demonstrate strengths and weaknesses of the 2D approach and the need for a more general 3D solution which accounts for the geometric complexity of the circumferential grooves. To date, models that attempt to solve the full geometry of a crushing unit have not been adequately solved. 4.1. Two-roll modelling The finite element simulation of a single nip was carried out utilising the symmetry at half groove depth (Fig. 3) and modelling the rolls as circular 2D rigid surfaces. The z-direction corresponds to the axial direction of the roll where the groove profiles are placed. The blanket is modelled as fully saturated both initially and throughout the crushing process. The material behaviour under stress is considered to be isotropic. The frictional behaviour between rolls and blanket is specified by a constant contact friction coefficient of 0.5 for simplicity. The permeability data were identical to that used by Loughran and Adam [15] where for void ratios above 12, the permeability was increased by 100 times to account for partial saturation, and for the void ratios 5.2 < e < 8.2, the permeability was increased by a factor of 4 to account for seepage induced consolidation (Fig. 4). Because the 2D model simplifies the groove detail by a rigid surface at a mean radius, particular attention was given to estimation of the strain-hardening curve (7). Following Adam [10] the strainhardening curve was determined through reverse finite element modelling of prepared cane compressed under quasi-static uniaxial conditions between grooved surfaces. The grooves were identical to those
96
J.G. Loughran, A. Kannapiran / Finite Elements in Analysis and Design 42 (2005) 90 – 104
Axial plane D D= r ete iam
y
Contact angle α
x
1/2 Feed height (h α / 2)
Plane of symmetry
Cane
1/2 Work opening (w o / 2)
Fig. 3. Two-dimensional representation of plane strain model for a pair of symmetric rolls.
Increased by 100 times for partial saturation
Permeability
Increased by a factor of 4 for seepage-induced consolidation
Compression ratio 1.0 < C < 1.5
Compression ratio C < 1.0
Apparent permeability from experiments
Void ratio e
Fig. 4. Permeability response accounting for partial saturation and seepage flow [15].
specified in the two-roll mill. The boundary conditions on the roll surface of the 2D model are assumed free draining. As the material passes through the nip, reduction in volume and expression of juice causes a reduction in void ratio as seen in Fig. 5. The maximum pore pressure occurs on the horizontal symmetry line of the blanket near minimum opening (Fig. 6). At the nip exit, elastic recovery of the material causes negative pore pressure. This is attributed to the full saturation assumption and free surface at the roll boundary. The effect of elastic recovery on roll loads and torques is small (< 5%). The juice velocity vectors relative to the fibre are shown in Fig. 7. The forward juice flow at the roll boundary near the contact point of the roll, and backward juice flow in the entry region are typical characteristics of juice flow behaviour. The fibre velocities follow the direction of the blanket and in general are unidirectional as shown in Fig. 8.
J.G. Loughran, A. Kannapiran / Finite Elements in Analysis and Design 42 (2005) 90 – 104
97
Fig. 5. Contour plot of void ratio (Nip void ratio = 5.15, S = 0.15 m/s).
Fig. 6. Contour plot of pore pressure (Nip void ratio = 5.15, S = 0.15 m/s).
Fig. 7. Juice velocity vectors at nip void ratio = 5.15 (Maximum velocity = 4.4 cm/s).
Fig. 8. Fibre velocity vectors at nip void ratio = 5.15 (Maximum velocity = 45.5 cm/s).
The experimental observations and the simulation results of roll load indicate the roll load is proportional to decreasing void ratio and approximately independent of roll surface speed [14]. Fig. 9 shows a plot of roll load versus void ratio at two speeds. The predicted trend and level of response for roll load is
98
J.G. Loughran, A. Kannapiran / Finite Elements in Analysis and Design 42 (2005) 90 – 104 250 S = 0.15 m/s (Experiment) S = 0.30 m/s (Experiment)
200
S = 0.15 m/s (Prediction)
Roll load kN
S = 0.30 m/s (Prediction)
150 100
50 0 6
5
4
3
2
1
Void ratio e
Fig. 9. Comparative response of roll load to nip void ratio.
25 S = 0.15 m/s (Experiment) S = 0.30 m/s (Experiment)
Roll torque kN m
20
S = 0.15 m/s (Prediction) S = 0.30 m/s (Prediction)
15
10 5
0 6
5
4
3
2
1
Void ratio e
Fig. 10. Comparative response of roll torque to nip void ratio.
in reasonable agreement. Fig. 10 shows the comparative response for roll torque. Here, the trends are roughly consistent with experiment, but the level of response is unsatisfactory. The poor response for torque is attributed to over simplification of the groove detail with 2D modelling. It is known that the fibrous material packs tightly against the groove tip and loosely at the root of the groove. Since roll torque depends strongly on contact behaviour and normal stress, the local compaction behaviour will be crucial to the development of plausible predictions. 4.2. Three-dimensional simulation The experimental data used for verification purposes in Section 4.1 had negative set openings. Numerical modelling with negative set openings in three-dimensions produces extensive deformation around the apex of the teeth. This results in severe mesh distortion and convergence of solution is severely affected. Adaptive meshing would be essential to solve such problems. Here, we opt to increase the set opening while maintaining the same theoretical nip void ratio (based on volumetric theory [3,4]). This was achieved
J.G. Loughran, A. Kannapiran / Finite Elements in Analysis and Design 42 (2005) 90 – 104
99
Fig. 11. Blanket in three-dimensions.
Fig. 12. Contour plot of vertical stress at nip void ratio = 2.10.
by increasing the initial blanket height while maintaining the same initial void ratio. In practical terms, the crushing capacity of the system was increased. Input property parameters to the numerical model were identical to the plane strain simulation except for the strain hardening relation. In this case, the strain hardening relation was estimated through reverse analysis of prepared cane compressed in a confined uniaxial test between flat porous platens. Fig. 11 shows the deformed mesh for a typical 3D roll groove accounting for existing non-symmetry between top and bottom rolls. The rolls are not shown for clarity. The compression of the top roll groove tip into the blanket is clearly visible. The vertical compressive stress is shown in Fig. 12 for an average nip void ratio of 2.10. The corresponding spatial void ratio is shown in Fig. 13. Penetration of fibrous material into the groove region is evident, particularly in close proximity to the nip region. Vector plots of liquid velocity and fibre are shown in Figs. 14 and 15, respectively. Liquid flows relatively freely at the entry region of the rolls where the skeleton is less compacted. There is also evidence that the fibre compaction is largely unidirectional.
100
J.G. Loughran, A. Kannapiran / Finite Elements in Analysis and Design 42 (2005) 90 – 104
Fig. 13. Void ratio contours at nip void ratio = 2.10.
Fig. 14. Juice velocity vectors at nip void ratio = 2.10 (Maximum velocity = 28.12 cm/s).
Results from controlled uniaxial laboratory experiments [10,22] suggest that bagasse does not penetrate to the base of a groove under typical milling conditions (Fig. 16). The groove penetration is a function of groove geometries and speed of compression. Defining compression ratio as the ratio of no-void volume of cane before compression, to the volume of cane at the particular location under investigation, the actual compression ratio Cc in the grooves, is higher than the compression ratio Co based on mill settings. This has been quantitatively reported by Leitch et al. [22] as Cc = 1.349Co − 0.158
(10)
for 35◦ grooves across all platen speeds. The 3D simulation result of roll load is compared with the experimental data of corrected compression ratio in Fig. 17. The experimental data were averaged at each void ratio. Void ratio e is based on
J.G. Loughran, A. Kannapiran / Finite Elements in Analysis and Design 42 (2005) 90 – 104
101
Fig. 15. Fibre velocity vectors at nip void ratio = 2.10 (Maximum velocity = 27.42 cm/s).
Fig. 16. Groove penetration and variation of local stresses [20].
Leitch et al.’s interpretation of compression ratio C, i.e. e=
1 1 (1 − f /j ) + ( / ) − 1, C fC f j
(11)
where f and j are the densities of the fibre and juice, respectively, and f is the fibre content. As the 3D simulation was carried out with increased set openings but with the same nip void ratio, the effect of increasing (wo /D) is to increase crushing rate and corresponding increase roll torque [3]. It was shown theoretically by Murry [3] that torque load number N is proportional to the square root of the work opening to diameter ratio (wo /D) and a function of the mill compression ratio Co . This function
J.G. Loughran, A. Kannapiran / Finite Elements in Analysis and Design 42 (2005) 90 – 104
102
25 0 S = 0.15 m/s (Experiment) S = 0.30 m/s (Experiment)
Roll load kN
20 0
S = 0.15 m/s (Prediction- 3D)
15 0 10 0 50 0 5
4
3 Void ratio e
2
1
Fig. 17. Roll load from 3D simulation.
30 S = 0.15 m/s (Experiment) S = 0.30 m/s (Experiment)
25 Roll torque kN m
S = 0.15 m/s (Prediction- 3D)
20 15 10 5 0 5
4
3 Void ratio e
2
1
Fig. 18. Roll torque from 3D simulation.
was found empirically from experimental mill results as 3/4 wo , N = Co D
(12)
where depends on the preparation and roll diameter. The torque-load number N is defined by N=
Ttot RD
(13)
and Ttot is the total torque for the rolls and R is the roll load. Roll load is simply a function of the average nip compression ratio, and hence is unaffected by increased capacity. The groove penetration and (wo /D) effects are applied to the torque values of the measured experimental data from Eqs. (12) and (13), respectively. The manipulated experimental data are compared with the simulation result in Fig. 18 for roll surface speed of 0.15 m/s. It is evident that the predicted values are still lower than the experimental values, and this may be due to the assumption of porous boundary conditions
J.G. Loughran, A. Kannapiran / Finite Elements in Analysis and Design 42 (2005) 90 – 104
103
on the roll surface. The actual pore pressure distribution on the roll is not known a prior and will evolve during the compression of the skeleton into the groove surface. Consequently the zero pore pressure boundary condition used in the simulation will result in an underestimation of the spatial pore pressure within the blanket and the evolution of effective stress at the interfaces. Torque prediction is likely to be significantly affected by the boundary conditions. The effect of mesh density on the predicted response was studied using different number of elements, namely at 600, 1200 and 2400 for a nip void ratio of 3.15. The steady state values of roll load or roll torque were not affected seriously by these mesh discretizations. However, this study could not be conducted at lower void ratios due to convergence problems associated with severe element mesh distortion. Further work with 3D simulation requires adaptive mesh refinement strategies. It may be noted that the experimental mill had side plates at the minimum opening area. In the numerical simulation, this end effect was not considered. The end boundary condition introduces additional frictional torque in the numerical simulation. The prediction in roll torque values is expected to improve further, if this end effect is accounted for. 5. Conclusions The simulation of the dewatering of a saturated fibro-porous media (prepared sugar cane) during rolling is described. Both 2D and 3D models have been used to predict evolution of strain, stress, pore pressure and mechanical quantities (roll load and torque). The finite element model is two-phase and highly nonlinear. While the outputs of the simulation sheds significant light on the internal mechanisms occurring during the dewatering process, with reasonable predicted trends, more work is required to capture the local effects which occur at the interface between a grooved surface and the saturated fibre. Acknowledgements The support of the SRDC under projects JCU4S, JCO11 and JCUO20 are gratefully acknowledged. References [1] D.J. Kauppila, G. Barnes, An experimental study into the extraction performance of sieved portions of prepared cane, in: Proceedings of the Australian Society Sugar Cane Technology, 2003, Paper M43. [2] K.J. Bullock, An investigation into the crushing and physical properties of sugar cane and bagasse, School of Engineering, The University of Queensland, 1957. [3] C.R. Murry, A theoretical and experimental investigation into the mechanics of crushing prepared sugar cane, School of Engineering, The University of Queensland, 1961. [4] J.E. Holt, The prediction of roll loads in the crushing of prepared sugar cane, School of Engineering, The University of Queensland, 1963. [5] T.J. Solomon, Theoretical and experimental studies in the mechanics of crushing sugar cane, School of Engineering, The University of Queensland, 1967. [6] G.E. Russell, An investigation of the extraction performance of sugar cane crushing trains, School of Engineering, The University of Queensland, 1968. [7] D.R.J. Owen, S.Y. Zhao, J.G. Loughran, An overview of crushing theory investigations at Swansea, Part 1 & Part 2, in: Proceedings of the Australian Society Sugar Cane Technology, 1994, pp. 264–277.
104
J.G. Loughran, A. Kannapiran / Finite Elements in Analysis and Design 42 (2005) 90 – 104
[8] D.R.J. Owen, S.Y. Zhao, J.G. Loughran, Application of porous-media mechanics to the numerical-simulation of the rolling of sugar-cane, Eng. Comput. 12 (3) (1995) 281–302. [9] E.A. de Souza Neta, S.Y. Zhao, D.R.J. Owen, J.G. Loughran, Finite element simulation of the rolling and extrusion of multi-phase materials, in: Proceedings of the Fifth International Conference on Computational Plasticity, Barcelona, 1997. [10] C.J. Adam, Application of computational porous media mechanics to the rolling of prepared cane, School of Engineering, James Cook University, 1997. [11] C.J.Adam, J.G. Loughran,Application of computational mechanics to rolling of a saturated porous material, in: Proceedings of the Sixth International Conference on Numerical Methods in Industrial Forming Processes, Enschede, The Netherlands, 1998. [12] F. Plaza, G.A. Kent, Using soil shear tests to investigate mill feeding, in: Proceedings of the Australian Society Sugar Cane Technology, 1997, pp. 300–339. [13] F. Plaza, G.A. Kent, J.M. Kirby, Modelling the compression, shear and volume behaviour of final bagasse, in: Proceedings of the Australian Society Sugar Cane Technology, 2001, pp. 428–436. [14] J.G. Loughran, A. Kannapiran, Finite element modelling of the crushing of prepared cane and bagasse, in: Proceedings of the Australian Society Sugar Cane Technology, 2002, Paper 47. [15] J.G. Loughran, C.J. Adam, Properties of prepared cane for computational crushing models, in: Proceedings of the Australian Society Sugar Cane Technology, 1998, pp. 307–312. [16] K.V. Terzaghi, Theoretical Soil Mechanics, Wiley, Chapman and Hall, 1943. [17] S.Y. Zhao, Finite element solution of porous materials with application to the rolling of prepared cane, University college of Swansea, 1993. [18] C.R. Murry, J.E. Holt, The Mechanics of Crushing Sugar Cane, Elsevier, Amsterdam, 1967. [19] J.G. Loughran, Mathematical and experimental modelling of the crushing of prepared sugar cane, School of Engineering, The University of Queensland, 1990. [20] C.J. Leitch, An experimental investigation into the constitutive behaviour of prepared sugar cane, School of Engineering, James Cook University, 1996. [21] A. Schofield, P. Wroth, Critical State Soil Mechanics, McGraw-Hill, New York, 1968. [22] C.J. Leitch, C.J. Adam, J.G. Loughran, On the capacity of prepared cane to penetrate grooved surfaces, in: Proceedings of the Australian Society Sugar Cane Technology, 1997, pp. 341–349.
Three-dimensional simulation of the rolling of a saturated fibro-porous media J.G. Loughrana , Arasu Kannapiranb,∗ a School of Engineering, James Cook University, Townsville 4811, Australia b School of Engineering, The University of Queensland, St. Lucia 4072, Australia
Received 19 March 2004; accepted 25 May 2005 Available online 15 August 2005
Abstract This paper describes recent advances made in computational modelling of the sugar cane liquid extraction process. The saturated fibro-porous material is rolled between circumferentially grooved rolls, which enhance frictional grip and provide a low-resistance path for liquid flow during the extraction process. Previously reported two-dimensional (2D) computational models, account for the large deformation of the porous material by solving the fully coupled governing fibre stress and fluid-flow equations using finite element techniques. While the 2D simulations provide much insight into the overarching cause-effect relationships, predictions of mechanical quantities such as roll separating force and particularly torque as a function of roll speed and degree of compression are not satisfactory for industrial use. It is considered that the unsatisfactory response in roll torque prediction may be due to the stress levels that exist between the groove tips and roots which have been largely neglected in the geometrically simplified 2D model. This paper gives results for both two- and three-dimensional finite element models and highlights their strengths and weaknesses in predicting key milling parameters. 䉷 2005 Elsevier B.V. All rights reserved. Keywords: Porous material; Constitutive laws; Plane strain model; Grooving effects; 3D modelling
1. Introduction Many industrial processes rely on mechanical dewatering as the primary technique for separating liquid from saturated solids. Predictive models for these processes have been largely empirical due to ∗ Corresponding author. Tel.: +61 7 33652969; fax: +61 7 33654799.
E-mail address: [email protected] (A. Kannapiran). 0168-874X/$ - see front matter 䉷 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.finel.2005.05.006
J.G. Loughran, A. Kannapiran / Finite Elements in Analysis and Design 42 (2005) 90 – 104
91
Feed chute
Under feed roll
Pressure feed roll
Pressure feed chute
Pressure feed roll Top roll
Feed roll Delivery roll
Return chute
Fig. 1. Schematic of a six-roll crushing unit.
complex and often highly non-linear constitutive laws and in the case of rolling, evolution of boundary conditions. Nevertheless, over the last decade or so, computational methods in solid mechanics have matured considerably and when coupled with the marked advance in computing hardware, permit the numerical solution of complex industrial problems. Expression of juice from porous sugar cane is carried out in the initial stages of the raw sugar production process. The sugar cane undergoes comminution by heavy-duty swing hammer shredders before it is presented for rolling. The material is fibrous with particle sizes ranging from typically 1 mm to damaged fibro-vascular bundles up to 60 mm in length and 2–3 mm in diameter. The fibre material is hygroscopic and liquid is attached to all particles with varying propensity. Although Kauppila and Barnes [1] have shown that the liquid is more easily removed from the smaller particles, there is no factory process for segregation. The material is fed directly to a tandem of six-roll crushing units, which separate the liquid from the fibre under high-pressure rolling. Fig. 1 shows a schematic of a typical six-roll crushing unit. The rolls are circumferentially grooved to enhance feeding and to offer a low-resistance path for liquid escape from the compressed blanket. During the late 1950s, a considerable research effort was begun on understanding the fundamentals of sugar cane crushing [2–6]. Bullock [2] initiated the research with basic experiments on an experimental milling facility. That research was extended by Murry [3] and others [3–6] who consolidated Bullock’s experimental responses and began investigations into fundamental modelling of the porous media mechanics problem. The empirical responses are still used widely today by practitioners in the field. The porous media models were largely one-dimensional and were built around Darcy’s Law. No serious attempt was made to include the constitutive characteristics of the fibrous skeleton, and therefore the roll load and torque were estimated solely on the pressure effects of juice. In an effort to improve efficiency and reduce factory costs, there has been a research push, over the last decade, to apply modern computational and constitutive mechanics to the sugar cane crushing process.
J.G. Loughran, A. Kannapiran / Finite Elements in Analysis and Design 42 (2005) 90 – 104
92
Owen et al. [7] applied computational porous media mechanics to sugar cane crushing by treating the prepared cane as a fully coupled unsaturated/saturated two-phase problem. Prepared cane was considered a continuous isotropic porous medium from a macroscopic and statistical viewpoint, with fluid and solid phases forming overlapping regions. The governing equations were written for the porous medium assuming that an elemental volume [8] was large when compared to pores but small compared to the overall extent of the domain. For all practical purpose, the gas flow that may be present at atmospheric pressure was neglected. Some useful prediction trends were developed using a simplified non-linear constitutive law for the fibrous skeleton. deSouza et al. [9] extended the earlier work of Owen by employing CamClay and crushable foam constitutive laws for the fibre skeleton. The numerical procedures for rolling with evolving boundary conditions and adaptive meshing were also discussed. Adam [10] extended Owen’s model to include a Capped Drucker–Prager constitutive law with linear or porous elasticity. Adam and Loughran [11] report on parameter estimation procedures for Owen type models. Plaza and Kent [12] and Plaza et al. [13] have also reported experiments on characterizing the properties of the fibre skeleton subjected to normal shear loads. Adam provided computational solutions for internal and external parameters associated mostly with two-roll geometries and light to moderate nip compactions. For example, fibre velocities, strain, stress, liquid velocities, plasticity and elastic recovery could be predicted as well as roll load and torque. At the time it was reported that model validation was a critical issue. Adam was relying on 1950–1960 two-roll mill data for comparison with predictions. Recent studies by Loughran and Kannapiran [14] focused on exercising Adam’s numerical model and making direct comparisons with data obtained from modern day two-roll mill experiments. The cane used in the mill experiments has also passed through a series of basic tests (following Loughran and Adam [15]) to characterize its properties for computational models. Loughran and Kannapiran [14] found that the two-dimensional (2D) plane strain models used by previous researchers oversimplified the complex skeletal deformation response in the vicinity of the grooved circumferential rolls and led to poor predictions of roll torque. This paper describes the modelling approach adopted by Loughran and Kannapiran [14] and gives results for two- and three-dimensional (3D) simulations.
2. Modelling assumptions and approach The fundamental equations governing deformable porous media [7] can be simplified by noting the following assumptions: 1. 2. 3. 4. 5. 6.
solid and liquid phases are incompressible; the gas pressure in the medium is nominally zero; for quasi-static analysis, solid and liquid accelerations are negligible; gravitational effects are negligible; Darcy’s law is valid; Terzaghi’s principle of effective stress is valid. Hence LT = 0
is the equilibrium equation and L, are the differential operator and total stress vector respectively.
(1)
J.G. Loughran, A. Kannapiran / Finite Elements in Analysis and Design 42 (2005) 90 – 104
93
The equilibrium equation of the liquid phase is related through Darcy’s law by ˙ = ∇Pl , k −1
(2)
where k is the Darcy coefficient matrix of permeability, ˙ is the superficial velocity of the pore fluid relative to the fibre and Pl is the liquid pore pressure. The liquid flow continuity equation is given by applying mass conservation principles, ˙ = mT ˙, −∇ T where
∇T =
j
,
j
(3)
,
j
jx jy jz
,
is the total strain vector and mT = [1, 1, 1, 0, 0, 0].
Following Terzaghi [16] the principle of effective stress links the effective stress e to the total stress
, through
= e − mP l .
(4)
The effective stress e is then related to the strain displacement d in incremental form by d e = D T d
(5)
d = LT du.
(6)
and
DT is the tangent stiffness matrix which is dependent on the level of effective stress and the total strain of the skeleton and du represents the incremental displacement vector. The above governing equations represent a fully coupled highly non-linear set of partial differential equations, which undergo spatial and temporal discretization in the Lagrangian model prior to solution [17]. For this coupled application it is noted that the stability of the finite element solution is enhanced if different interpolation/integration orders are used for the liquid pressure and velocity fields. Either a full Newmark scheme or reduced rank formulation can be used for time discretization. deSouza et al. [9] noted no practical difference in computational output. The resulting governing equations are then solved with a Newmark scheme. 3. Constitutive models for crushing The specification of constitutive models for crushing presents a significant challenge. The material enters a vertical chute under gravity in a loosely packed condition. Here the individual particles in the fibrous skeleton are approximately randomised and the collection of material is largely unsaturated. Under light loading conditions (for example, as material enters the first nip), fibres align orthogonal to the direction of maximum principal stress and saturation may occur. Saturation definitely occurs if the level of nip compression is moderate. This suggests that the material might develop some degree of anisotropy during the compression process. Constitutive models are required for the liquid phase, interaction of
J.G. Loughran, A. Kannapiran / Finite Elements in Analysis and Design 42 (2005) 90 – 104
94
the skeleton against roll boundaries or other rigid surfaces and large strain deformation response of the skeleton. Following previous researchers [18,19], the liquid phase is modelled with Darcy’s Law and the contact behaviour by a suitable friction law. Adam [10] developed an empirical friction law for plane strain modelling of grooved circumferential rolls which accounted for surface speed, groove angle, the effective normal stress and the degree of material fineness (in an average sense). For simplicity, we adopt a constant Coulomb friction of 0.5 between the fibrous skeleton and roll interface in this paper. The model of the fibrous skeleton must reproduce the elasto-plastic response of the material across a wide range of nip loadings and compression speeds. This is an onerous task for such a complex material. Experimental observations [10,12,13,20] show that the material: • • • • • • •
Is highly compressible and yields under hydrostatic pressure. Exhibits large elastic and plastic volumetric strains. Has a non-linear plastic strain hardening relation. Has uniaxial plastic strains co-directional with applied stress. Exhibits anisotropic behaviour due to layering of fibres. Shows no shear failure under uniaxial or triaxial tests. A marked shear failure when loaded in a typical geo-mechanics soil box experiment.
To date researchers have opted for critical state isotropic constitutive models due to the “ease” of obtaining experimental property data compared with the inherent complexity of anisotropic models and their respective data sets. Following Adam [10] we define the material response through an isotropic linear elastic law and Capped pl Drucker–Prager plasticity law. The flow rule of incremental plastic volumetric strain v versus hydrostatic yield stress pc for a typical tested cane takes the form pl
pc = 0.0346 e4.1321v .
(7)
Unlike the Cam-Clay models [21], the Drucker–Prager model has a yield surface which includes a shear failure surface, and a cap that intersects the equivalent pressure stress axis p (p = −1/3 trace ). The cap yield surface has an elliptical shape with constant eccentricity in the p–t plane (Fig. 2). The cap surface hardens or softens as a function of plastic volumetric strain. Associated plastic flow is assumed on the cap and non-associated flow in the cone and transition regions. The cone and cap regions are expressed in mathematical form as t = p tan + d
(8)
and t2 = 1 + −
2
( + pa tan )2 −
p − pa R
2 ,
(9)
respectively, where is the material’s angle of friction, d the cohesion for the material, t the deviatoric stress measure, R the material parameter which controls the shape of cap, a small number (typically 0.01–0.05) defining a smooth transition region between cone and cap, and pa the evolution parameter which describes the size of the yield surface.
J.G. Loughran, A. Kannapiran / Finite Elements in Analysis and Design 42 (2005) 90 – 104
95
Transition
t Similar ellipses Shear failure
α (d+pa tan β )
Cap
d+ pa tanβ
Cone
β
d pa
R(d+p a tanβ)
pb
p
Fig. 2. Yield surface for capped Drucker–Prager plasticity model.
For the associated flow rule in which the failure envelope grows uniformly, the plastic strain hardening relation for the material is developed using inverse-calibration from finite element simulation of a confined uniaxial test cell [15]. In undertaking this calibration, a somewhat arbitrary but high value of the critical state slope M is assumed (typically 3.5–3.8) [10]. This limits lateral expansion of the material and mimics the inherent fibre locking, which occurs at a micro level.
4. Application This section gives simulation results for 2D plane strain and 3D models of a single nip. The objective is to demonstrate strengths and weaknesses of the 2D approach and the need for a more general 3D solution which accounts for the geometric complexity of the circumferential grooves. To date, models that attempt to solve the full geometry of a crushing unit have not been adequately solved. 4.1. Two-roll modelling The finite element simulation of a single nip was carried out utilising the symmetry at half groove depth (Fig. 3) and modelling the rolls as circular 2D rigid surfaces. The z-direction corresponds to the axial direction of the roll where the groove profiles are placed. The blanket is modelled as fully saturated both initially and throughout the crushing process. The material behaviour under stress is considered to be isotropic. The frictional behaviour between rolls and blanket is specified by a constant contact friction coefficient of 0.5 for simplicity. The permeability data were identical to that used by Loughran and Adam [15] where for void ratios above 12, the permeability was increased by 100 times to account for partial saturation, and for the void ratios 5.2 < e < 8.2, the permeability was increased by a factor of 4 to account for seepage induced consolidation (Fig. 4). Because the 2D model simplifies the groove detail by a rigid surface at a mean radius, particular attention was given to estimation of the strain-hardening curve (7). Following Adam [10] the strainhardening curve was determined through reverse finite element modelling of prepared cane compressed under quasi-static uniaxial conditions between grooved surfaces. The grooves were identical to those
96
J.G. Loughran, A. Kannapiran / Finite Elements in Analysis and Design 42 (2005) 90 – 104
Axial plane D D= r ete iam
y
Contact angle α
x
1/2 Feed height (h α / 2)
Plane of symmetry
Cane
1/2 Work opening (w o / 2)
Fig. 3. Two-dimensional representation of plane strain model for a pair of symmetric rolls.
Increased by 100 times for partial saturation
Permeability
Increased by a factor of 4 for seepage-induced consolidation
Compression ratio 1.0 < C < 1.5
Compression ratio C < 1.0
Apparent permeability from experiments
Void ratio e
Fig. 4. Permeability response accounting for partial saturation and seepage flow [15].
specified in the two-roll mill. The boundary conditions on the roll surface of the 2D model are assumed free draining. As the material passes through the nip, reduction in volume and expression of juice causes a reduction in void ratio as seen in Fig. 5. The maximum pore pressure occurs on the horizontal symmetry line of the blanket near minimum opening (Fig. 6). At the nip exit, elastic recovery of the material causes negative pore pressure. This is attributed to the full saturation assumption and free surface at the roll boundary. The effect of elastic recovery on roll loads and torques is small (< 5%). The juice velocity vectors relative to the fibre are shown in Fig. 7. The forward juice flow at the roll boundary near the contact point of the roll, and backward juice flow in the entry region are typical characteristics of juice flow behaviour. The fibre velocities follow the direction of the blanket and in general are unidirectional as shown in Fig. 8.
J.G. Loughran, A. Kannapiran / Finite Elements in Analysis and Design 42 (2005) 90 – 104
97
Fig. 5. Contour plot of void ratio (Nip void ratio = 5.15, S = 0.15 m/s).
Fig. 6. Contour plot of pore pressure (Nip void ratio = 5.15, S = 0.15 m/s).
Fig. 7. Juice velocity vectors at nip void ratio = 5.15 (Maximum velocity = 4.4 cm/s).
Fig. 8. Fibre velocity vectors at nip void ratio = 5.15 (Maximum velocity = 45.5 cm/s).
The experimental observations and the simulation results of roll load indicate the roll load is proportional to decreasing void ratio and approximately independent of roll surface speed [14]. Fig. 9 shows a plot of roll load versus void ratio at two speeds. The predicted trend and level of response for roll load is
98
J.G. Loughran, A. Kannapiran / Finite Elements in Analysis and Design 42 (2005) 90 – 104 250 S = 0.15 m/s (Experiment) S = 0.30 m/s (Experiment)
200
S = 0.15 m/s (Prediction)
Roll load kN
S = 0.30 m/s (Prediction)
150 100
50 0 6
5
4
3
2
1
Void ratio e
Fig. 9. Comparative response of roll load to nip void ratio.
25 S = 0.15 m/s (Experiment) S = 0.30 m/s (Experiment)
Roll torque kN m
20
S = 0.15 m/s (Prediction) S = 0.30 m/s (Prediction)
15
10 5
0 6
5
4
3
2
1
Void ratio e
Fig. 10. Comparative response of roll torque to nip void ratio.
in reasonable agreement. Fig. 10 shows the comparative response for roll torque. Here, the trends are roughly consistent with experiment, but the level of response is unsatisfactory. The poor response for torque is attributed to over simplification of the groove detail with 2D modelling. It is known that the fibrous material packs tightly against the groove tip and loosely at the root of the groove. Since roll torque depends strongly on contact behaviour and normal stress, the local compaction behaviour will be crucial to the development of plausible predictions. 4.2. Three-dimensional simulation The experimental data used for verification purposes in Section 4.1 had negative set openings. Numerical modelling with negative set openings in three-dimensions produces extensive deformation around the apex of the teeth. This results in severe mesh distortion and convergence of solution is severely affected. Adaptive meshing would be essential to solve such problems. Here, we opt to increase the set opening while maintaining the same theoretical nip void ratio (based on volumetric theory [3,4]). This was achieved
J.G. Loughran, A. Kannapiran / Finite Elements in Analysis and Design 42 (2005) 90 – 104
99
Fig. 11. Blanket in three-dimensions.
Fig. 12. Contour plot of vertical stress at nip void ratio = 2.10.
by increasing the initial blanket height while maintaining the same initial void ratio. In practical terms, the crushing capacity of the system was increased. Input property parameters to the numerical model were identical to the plane strain simulation except for the strain hardening relation. In this case, the strain hardening relation was estimated through reverse analysis of prepared cane compressed in a confined uniaxial test between flat porous platens. Fig. 11 shows the deformed mesh for a typical 3D roll groove accounting for existing non-symmetry between top and bottom rolls. The rolls are not shown for clarity. The compression of the top roll groove tip into the blanket is clearly visible. The vertical compressive stress is shown in Fig. 12 for an average nip void ratio of 2.10. The corresponding spatial void ratio is shown in Fig. 13. Penetration of fibrous material into the groove region is evident, particularly in close proximity to the nip region. Vector plots of liquid velocity and fibre are shown in Figs. 14 and 15, respectively. Liquid flows relatively freely at the entry region of the rolls where the skeleton is less compacted. There is also evidence that the fibre compaction is largely unidirectional.
100
J.G. Loughran, A. Kannapiran / Finite Elements in Analysis and Design 42 (2005) 90 – 104
Fig. 13. Void ratio contours at nip void ratio = 2.10.
Fig. 14. Juice velocity vectors at nip void ratio = 2.10 (Maximum velocity = 28.12 cm/s).
Results from controlled uniaxial laboratory experiments [10,22] suggest that bagasse does not penetrate to the base of a groove under typical milling conditions (Fig. 16). The groove penetration is a function of groove geometries and speed of compression. Defining compression ratio as the ratio of no-void volume of cane before compression, to the volume of cane at the particular location under investigation, the actual compression ratio Cc in the grooves, is higher than the compression ratio Co based on mill settings. This has been quantitatively reported by Leitch et al. [22] as Cc = 1.349Co − 0.158
(10)
for 35◦ grooves across all platen speeds. The 3D simulation result of roll load is compared with the experimental data of corrected compression ratio in Fig. 17. The experimental data were averaged at each void ratio. Void ratio e is based on
J.G. Loughran, A. Kannapiran / Finite Elements in Analysis and Design 42 (2005) 90 – 104
101
Fig. 15. Fibre velocity vectors at nip void ratio = 2.10 (Maximum velocity = 27.42 cm/s).
Fig. 16. Groove penetration and variation of local stresses [20].
Leitch et al.’s interpretation of compression ratio C, i.e. e=
1 1 (1 − f /j ) + ( / ) − 1, C fC f j
(11)
where f and j are the densities of the fibre and juice, respectively, and f is the fibre content. As the 3D simulation was carried out with increased set openings but with the same nip void ratio, the effect of increasing (wo /D) is to increase crushing rate and corresponding increase roll torque [3]. It was shown theoretically by Murry [3] that torque load number N is proportional to the square root of the work opening to diameter ratio (wo /D) and a function of the mill compression ratio Co . This function
J.G. Loughran, A. Kannapiran / Finite Elements in Analysis and Design 42 (2005) 90 – 104
102
25 0 S = 0.15 m/s (Experiment) S = 0.30 m/s (Experiment)
Roll load kN
20 0
S = 0.15 m/s (Prediction- 3D)
15 0 10 0 50 0 5
4
3 Void ratio e
2
1
Fig. 17. Roll load from 3D simulation.
30 S = 0.15 m/s (Experiment) S = 0.30 m/s (Experiment)
25 Roll torque kN m
S = 0.15 m/s (Prediction- 3D)
20 15 10 5 0 5
4
3 Void ratio e
2
1
Fig. 18. Roll torque from 3D simulation.
was found empirically from experimental mill results as 3/4 wo , N = Co D
(12)
where depends on the preparation and roll diameter. The torque-load number N is defined by N=
Ttot RD
(13)
and Ttot is the total torque for the rolls and R is the roll load. Roll load is simply a function of the average nip compression ratio, and hence is unaffected by increased capacity. The groove penetration and (wo /D) effects are applied to the torque values of the measured experimental data from Eqs. (12) and (13), respectively. The manipulated experimental data are compared with the simulation result in Fig. 18 for roll surface speed of 0.15 m/s. It is evident that the predicted values are still lower than the experimental values, and this may be due to the assumption of porous boundary conditions
J.G. Loughran, A. Kannapiran / Finite Elements in Analysis and Design 42 (2005) 90 – 104
103
on the roll surface. The actual pore pressure distribution on the roll is not known a prior and will evolve during the compression of the skeleton into the groove surface. Consequently the zero pore pressure boundary condition used in the simulation will result in an underestimation of the spatial pore pressure within the blanket and the evolution of effective stress at the interfaces. Torque prediction is likely to be significantly affected by the boundary conditions. The effect of mesh density on the predicted response was studied using different number of elements, namely at 600, 1200 and 2400 for a nip void ratio of 3.15. The steady state values of roll load or roll torque were not affected seriously by these mesh discretizations. However, this study could not be conducted at lower void ratios due to convergence problems associated with severe element mesh distortion. Further work with 3D simulation requires adaptive mesh refinement strategies. It may be noted that the experimental mill had side plates at the minimum opening area. In the numerical simulation, this end effect was not considered. The end boundary condition introduces additional frictional torque in the numerical simulation. The prediction in roll torque values is expected to improve further, if this end effect is accounted for. 5. Conclusions The simulation of the dewatering of a saturated fibro-porous media (prepared sugar cane) during rolling is described. Both 2D and 3D models have been used to predict evolution of strain, stress, pore pressure and mechanical quantities (roll load and torque). The finite element model is two-phase and highly nonlinear. While the outputs of the simulation sheds significant light on the internal mechanisms occurring during the dewatering process, with reasonable predicted trends, more work is required to capture the local effects which occur at the interface between a grooved surface and the saturated fibre. Acknowledgements The support of the SRDC under projects JCU4S, JCO11 and JCUO20 are gratefully acknowledged. References [1] D.J. Kauppila, G. Barnes, An experimental study into the extraction performance of sieved portions of prepared cane, in: Proceedings of the Australian Society Sugar Cane Technology, 2003, Paper M43. [2] K.J. Bullock, An investigation into the crushing and physical properties of sugar cane and bagasse, School of Engineering, The University of Queensland, 1957. [3] C.R. Murry, A theoretical and experimental investigation into the mechanics of crushing prepared sugar cane, School of Engineering, The University of Queensland, 1961. [4] J.E. Holt, The prediction of roll loads in the crushing of prepared sugar cane, School of Engineering, The University of Queensland, 1963. [5] T.J. Solomon, Theoretical and experimental studies in the mechanics of crushing sugar cane, School of Engineering, The University of Queensland, 1967. [6] G.E. Russell, An investigation of the extraction performance of sugar cane crushing trains, School of Engineering, The University of Queensland, 1968. [7] D.R.J. Owen, S.Y. Zhao, J.G. Loughran, An overview of crushing theory investigations at Swansea, Part 1 & Part 2, in: Proceedings of the Australian Society Sugar Cane Technology, 1994, pp. 264–277.
104
J.G. Loughran, A. Kannapiran / Finite Elements in Analysis and Design 42 (2005) 90 – 104
[8] D.R.J. Owen, S.Y. Zhao, J.G. Loughran, Application of porous-media mechanics to the numerical-simulation of the rolling of sugar-cane, Eng. Comput. 12 (3) (1995) 281–302. [9] E.A. de Souza Neta, S.Y. Zhao, D.R.J. Owen, J.G. Loughran, Finite element simulation of the rolling and extrusion of multi-phase materials, in: Proceedings of the Fifth International Conference on Computational Plasticity, Barcelona, 1997. [10] C.J. Adam, Application of computational porous media mechanics to the rolling of prepared cane, School of Engineering, James Cook University, 1997. [11] C.J.Adam, J.G. Loughran,Application of computational mechanics to rolling of a saturated porous material, in: Proceedings of the Sixth International Conference on Numerical Methods in Industrial Forming Processes, Enschede, The Netherlands, 1998. [12] F. Plaza, G.A. Kent, Using soil shear tests to investigate mill feeding, in: Proceedings of the Australian Society Sugar Cane Technology, 1997, pp. 300–339. [13] F. Plaza, G.A. Kent, J.M. Kirby, Modelling the compression, shear and volume behaviour of final bagasse, in: Proceedings of the Australian Society Sugar Cane Technology, 2001, pp. 428–436. [14] J.G. Loughran, A. Kannapiran, Finite element modelling of the crushing of prepared cane and bagasse, in: Proceedings of the Australian Society Sugar Cane Technology, 2002, Paper 47. [15] J.G. Loughran, C.J. Adam, Properties of prepared cane for computational crushing models, in: Proceedings of the Australian Society Sugar Cane Technology, 1998, pp. 307–312. [16] K.V. Terzaghi, Theoretical Soil Mechanics, Wiley, Chapman and Hall, 1943. [17] S.Y. Zhao, Finite element solution of porous materials with application to the rolling of prepared cane, University college of Swansea, 1993. [18] C.R. Murry, J.E. Holt, The Mechanics of Crushing Sugar Cane, Elsevier, Amsterdam, 1967. [19] J.G. Loughran, Mathematical and experimental modelling of the crushing of prepared sugar cane, School of Engineering, The University of Queensland, 1990. [20] C.J. Leitch, An experimental investigation into the constitutive behaviour of prepared sugar cane, School of Engineering, James Cook University, 1996. [21] A. Schofield, P. Wroth, Critical State Soil Mechanics, McGraw-Hill, New York, 1968. [22] C.J. Leitch, C.J. Adam, J.G. Loughran, On the capacity of prepared cane to penetrate grooved surfaces, in: Proceedings of the Australian Society Sugar Cane Technology, 1997, pp. 341–349.