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Mariusz NIKLAS, Dariusz ASENDRYCH*

Modelling of fluid flow with complex rheology

Abstract. The paper concerns numerical modelling of the paper pulp flow of a non-Newtonian fluid with complex rheological properties. The work is aimed at optimisation of the performance of a hydropulper used for the recovered paper defibration in paper recycling technology.

The phenomenon under consideration reveals high degree of complexity and its proper modelling requires to solve a number of particular problems related to duality of flow regimes (coexistence of turbulent and laminar regimes governed by different constitutive models), complicated geometry, presence of moving elements of machine, unsteady nature of phenomena and complex material properties of medium.

So high degree of complexity enforced the necessity of the use of commercial code Fluent as a computational tool. Its effective use was however possible after code extension with additional procedures (the so called UDF - the user defined functions) allowing to meet the requirements of the rheology.

1. Introduction

The still growing demand for high quality paper products as well as economical impacts cause that the interest in paper recycling is increasing. Nowadays more than 50% of world paper production comes from recovered paper [2]. It shows that the optimisation of this process may bring huge reduction in costs related to running production lines. Among the numerous stages of paper recycling technology the pulping process (aimed at defibration of recovered paper and making it as a homogenous paper pulp) carried out with the use of a hydropulper is recognised as the most energy-consuming [2, 3, 8]. The quality of pulping process is evaluated taking into account two opposite criterions: the defibration degree and the defragmentation of undesired pulp contaminants.

As the result of high quality pulping one should obtain the medium with perfectly separated cellulose fibres and with possibly the biggest contaminant particles. The latter requirements makes the pulp impurities easy removable in the farther stages of recycling. So it is necessary to find the optimal set of parameters allowing to reconcile the inconsistent process requirements. Optimised pulping process should lead to smaller demand for chemicals, reduced energy consumption and shorter running time.

* Politechnika Częstochowska, Instytut Maszyn Cieplnych, Al. Armii Krajowej 21, 42-200 Częstochowa

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The paper pulp flow is a very complex and phenomenon which makes it difficult for experimental analysis. The pulp is an opaque multiphase (as it consists of water, cellulose fibres, chemicals, inks, “sticky” particles coming from commonly used adhesive products – labels, tapes, stickers etc.) fluid with complex rheological properties. So there is no simple manner to measure the flow field inside working hydropulper and to have an insight into the physics of the process. As a consequence the numerical simulation of the flow gains special importance as it can easily overcome the problems encountered in experiment.

2. Modelling of paper pulp flow

The modelling of the cellulose fibres suspension requires at first to make an assumption about the way the discrete phase is treated. The approach regarded fibres as single objects carried by water would lead to the model which should take into account thousands of fibres and their mutual interactions resulting in extremely time-consuming computations. In order to make simulation times realistic the simplified assumption was applied about the pulp as a continuum. With this assumption the phase and interfacial interactions are neglected and the suspension is treated as a homogeneous medium with apparent viscosity [7]. Besides the above discussed specific features of the paper pulp flow one should also mention the duality of flow regimes observed in the hydropulper. Due to high viscosity of the medium the laminar flow regime dominates in the entire machine. In the neighbourhood of a rotor flow may become turbulent so the relevant constitutive model (either than that applied for the laminar flow regime) must be there employed. Although that region is of relatively small extent the proper modelling should take into account the turbulent character of the flow.

Main feature which differs paper pulp from Newtonian fluids in laminar regime is a nonlinear character of shear stress τ variation versus shear rate γ [4, 9]. Moreover, the τ=f(γ) distribution strongly depends on fibres consistency defined as a ratio of fibres mass to total suspension mass. For the pulp flow with relatively low consistency (below 1%) the resistance of the flow is adequately small and fluid behaviour does not deviate from Newtonian manner. Growth of the consistency makes the fibres interactions noticeably higher and fluid starts to behave as a non-Newtonian medium. In a case of sufficiently high consistency pulp starts to flow only if the shear stress exceeds the so-called yield stress τo (see fig.1). If the stress

Fig. 1. Paper pulp flow characteristic according to [1]

Laminar flow

Turbulent flowτcr

τo

τ

γ γcr

AB

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level is too low the pulp is stagnant and behaves like a solid body thanks to the links between fibres creating a network. Figure 1 shows typical flow characteristic, i.e. dependence of shear stress on shear rate γ for paper fibres suspension [1]. Two different areas of flow were distinguished by the authors:

- laminar - limited by shear rate γcr for which paper pulp behaves as a non Newtonian fluid,

- turbulent (for shear rates γ>γcr), where the fluid can be assumed as Newtonian medium.

The authors [1] have determined the transition point (A in fig.1) corresponding to γcr by making the assumption that the stress starts to increase (above τcr level) due to the origin of turbulence. Such a hypothesis cannot be, however, accepted as the transition point lies within the range of decreasing pulp viscosity. For the paper pulp flow such a variation of viscosity can be explained with the qualitative change of an internal structure of a suspension, i.e. the joints of a fibre network are gradually weakened by increasing strain and finally the flow of the separate (not interacting) fibres is achieved. If such a hypothesis can be accepted the “transition” region should be followed by the Newtonian fluid characteristics, i.e. revealing constant viscosity. So for the purpose of current study the transition point location (point B in fig. 1) was determined by the tangential to the τ=f(γ) distribution (as indicated in figure 1). So the excessive stress growth (above Newtonian character –shown as a dashed area) is caused by turbulence.

In general the numerical model of paper pulp flow must include: - adequate rheological model in laminar regime, - the mechanism for switching between two kinds of constitutive models (laminar-

rheological and turbulent - Newtonian) according to the local stress level. As a computational tool for the simulation a commercial code Fluent was used. As described in following sections some necessary code modifications were introduced to make it capable to deal with current flow case.

2.1. Modelling of non-Newtonian fluid flow for laminar regime

Fluent as a commercially available software has a number of limitations prohibiting its application to certain flow cases among which a paper pulp flow can be mentioned. For the medium under considerations Fluent does not provide in its database the rheological model being capable to adequately describe the paper pulp behaviour. In order to overcome that limitation the UDF procedure was created with an adequate rheological model of the paper pulp flow (see section 3.1)

According to [1] the implementation of non-Newtonian model is not so obvious as it could be expected. The problem arises when the two or three-dimensional flow configuration is considered and the stress tensor is expressed as a function of second order velocity deformation tensor Dij (being the shear rate equivalent) according to the following relationship

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( ) ijijij DDη=τ (1)

Molecular viscosity η (dependent on the local state of flow field and as a consequence being one of the parameters to be found during iterative procedure) has to be estimated on the basis of second invariant of shear stress tensor IID. Such a procedure including the convergence monitoring of viscosity residuum (for details see [6]) was implemented in source code and successfully validated.

2.2. Modelling of flow with coexisting constitutive models

As it was mentioned in the previous section the process of pulp mixing in hydropulper reveals the existence of the different states of motion i.e.:

- laminar with the possibility to occur the stagnation areas (especially for high consistency),

- turbulent taking place mainly in the close neighbourhood of a rotor. Numerical model of pulping process built with assumption that the entire pulp volume behaves in laminar manner might only be appropriate for high consistency pulp (being beyond the scope of current work) for which turbulent flow appears in region of very small extent and its impact on the entire flow is negligible. The model to be constructed should, however, be capable to simulate the flow also for low and intermediate consistencies. In order to provide the possibility of the coexistence of two constitutive models the relevant UDF procedure was created enabling to control the way Fluent solves the flow. Taking into account the critical stress level the solver was switched to the appropriate system of equations. The detailed description of the algorithm applied along with some validation tests can be found in [6].

3. Experimental trials

The numerical simulations of the paper pulp flow in the pulping machine require the knowledge about the detailed material properties of the medium. As the literature data are very poor it was decided to undertake the experimental trials and to determine the stress-strain characteristics for the commonly used recovered paper composed of newspapers. The experiment was performed in collaboration with Pulp and Paper Research Institute in Łódź and Institute of Chemical Engineering, Technical University of Łódź. Besides rheological pulp properties the experiment was additionally aimed at the flow field studies with the special attention laid on the shape and velocity distribution at the free surface. As the pulp is a non-transparent medium no additional insight into the flow field was possible.

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3.1. Rheological properties of paper pulp

The measurements were performed with the use of pipe rheometer (with pipe diameter of 40mm and 50mm) for the pulps of two consistencies i.e. 3% and 4% (corresponding to the low-consistency pulping process). Due to technical limitations of available measuring equipment it was not possible to reach the high-consistency regime, i.e. exceeding 7%. It is planned, however, in future works to continue the experimental trials and to cover completely the consistency range applied in industrial applications.

As a second disadvantage of the rheometer used in the experiment its limited shear rate range should be noted allowing to obtain flow characteristics up to γ ≈ 200s-1. As for the pulper under interest the expected values of velocity gradients were noticeably higher the available experimental data had to be complemented. It was decided to use for that purpose the literature data [5] devoted to similar cellulose fibre type and matching the pulp consistency range of own studies.

The results of the experimental trials together with corresponding source data are shown in figure 2. As a first remark the similarity of presented curves to the model of Bakker [1] should be stated. As it can be seen the distributions τ=f(γ) for both pulp consistencies c reveal strong non-Newtonian character with qualitative similarity. Both stress-strain curves start from non-zero stress value (yield stress) being a typical feature of the Bingham-type fluids. For low shear rates (approximately up to 50s-1) stress is growing relatively quickly and then the distributions τ(γ) reveal the plateaus (with hardly noticeable local maximum for pulp consistency c=4%). That is the range for which the strong reduction of viscosity is observed. According to the hypothesis made in section 2 of current paper the transition to turbulence for pulp consistency c=3% takes place for γ≈210s-1 while for c=4% the transition point lies outside the available data range.

3.2. The characteristics of a flow in an industrial pulper

As it was mentioned before the main aim of the experiment was to deliver the necessary information about the paper pulp material properties. But as these studies had to be preceded by the pulping trials (needed to produce the pulp) it was decided to use that opportunity to get some information about the flow which could be later employed for the validation of a numerical model. The pulping trials were conducted

Fig. 2. Flow characteristics of paper pulp

0

20

40

60

80

100

0 100 200 300

c=3%, own datac=3%, [5]c=4%, own datac=4%, [5]

τ [N/m2]

γ [s-1]

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at the Pulp and Paper Research Institute in Łódź with the use of a laboratory drum pulper. Its main dimensions are as follows:

- internal diameter - 0.9 m, - hydropulper height - 0.7 m, - rotor diameter - 0.27 m, - working capacity - 200 litres.

The general view of pulper’s geometry is shown in figure 4. The measurements concerned the velocity field and the shape of free surface for

three consistencies of paper pulp (3%, 4% and 7%) and for two rotational speeds of the rotor (420rpm and 620rpm). Additionally measurements for water were performed in order to compare the paper pulp flow to Newtonian fluid behaviour.

Figure 3 shows the shape of pulp free surface for the rotor speed 420rpm. The influence of consistency on the flow is clearly visible. For the lowest pulp consistency (c=3%) the influence of rotor is the strongest one and the free surface shape is similar to the shape of water flow. Growth of the consistency leads to increased viscosity and in turn to weaker impact of the rotor on the upper region of the flow which is especially visible for the pulp consistency c=7%. It is also worth to mention that for the highest pulp consistency the free surface shape deviates noticeably from remaining cases and reveals local maximum in certain radial position accompanied by free surface drop towards the wall. Such a behaviour could be explained by the locally reduced pulp consistency in near-wall region where a kind of water film at the wall is produced.

All experimental trials were also utilised to visualise the flow by the video camera. It is planned to use the available movies to recover the velocity field at the free surface and to provide the data for the validation of numerical simulation.

4. Pulp flow model

4.1. Geometry

Figure 4 shows the general view of hydropulper geometry following the existing pulper available at Pulp and Paper Research Institute in Łódź. The hydropulper consists of mixing vat (fig. 4a) and rotor (fig. 4b). In order to support the defibring process pulper is equipment with 12 stator blades and 24 rotor profiles (consisting of 2 sets – internal and external ones with 8 and 16 profiles, respectively). Complicated

0

0,1

0,2

0,3

0,4

0,5

-1 -0,8 -0,6 -0,4 -0,2 0 0,2 0,4 0,6 0,8 1

r/R

h [

m]

water norm

C=3% norm

c=4% norm

c=7% norm

Fig. 3. The dependence of free surface shape on pulp consistency for rotor velocity

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geometry of the machine enforced the necessity of dividing the whole volume into a number of sub volumes. Additionally because of the presence of moving elements (rotor) the concept of sliding mesh was used.

The geometry and computational mesh were created with the Gambit generator ran by the previously written scripts, allowing for full documentation and parameterisation of the process. It should be noted that the simulation was conducted for the model to scale 5:1 in order to reduce the computational effort. Although the scale effects can noticeably influence the solution, the main aim of the present work was to test the model of the pulping in reliable time. It is, however, planned to perform real-scale computations as one of the next simulation steps.

4.2. Numerical method

The simulation of the flow was performed by using Fluent version 6.2.16. For the computations Volume of Fluid method (a 2-phase flow model) was employed. The equations were discretized using first order approximation schemes. The segregated solver was applied with the Body Force Weighted pressure correction method. The solution was regarded as converged when the level of residuals in given time step was lower than 10-3. The grid quality was checked by performing grid independence tests of the flow field. As a result grid consisting of approximately 150 000 volume elements was accepted.

5. Simulations results

The preliminary simulation was performed for the rotor speed 620rpm and for pulp consistency c=3%. All results presented below describe the flow field 3 seconds after the start-up of the machine which corresponds to 30 revolutions.

Figure 5 shows the shape of a free surface in selected plane crossing the symmetry axis of the pulper. It should be noted that thanks to high viscosity the shape is independent of the circumferential position of a control plane. As it can be easily found from the data presented in the fig. 5 the simulation leads to much weaker free surface deviation from the horizontal plane than it was measured during the experiment. Even for lower rotational speed (420 rpm) the free surface reveals noticeably greater shape variations (see fig. 3) when compared to simulation. It is

Fig. 4. General view of hydropulper geometry

b x z

h

a

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believed, however, that such significant discrepancies are caused by differences in model formulation, mainly due to scale effects (computations performed to scale 5:1). As an another important source of inaccuracies, the presence of a water film in close vicinity of a wall (not implemented in a numerical model) can be mentioned. Its impact on the flow would be to reduce the shear stresses at pulper walls and in turn to increase inertial effects.

The analysis of velocity field (see fig. 6 with a contour map and fig. 7 with vectorial representation) indicates strong variations of velocity magnitude. It is clearly visible that the region of high velocities is of very limited extent in the close neighbourhood of a rotor.

Fig. 6. Velocity magnitude Fig. 7. Vectorial representation of velocity

Figure 8a shows the velocity magnitude at a horizontal plane located 2cm above the

rotor and crossing the stator blades. As it is seen the pulp is flowing in entire cross section except of the stator cavities where small stagnation regions can be identified. In figure 8a these regions (with velocities less than 10-4m/s) located in corners of cavities are marked by white lines. It should be remarked that no flow unsteadiness is observed in the flow due to rotor motion. Moreover, the maximum velocity in analysed cross section is equal to 0.4 m/s while the linear speed of rotor profile tips exceeds 3m/s. These observations show the strong influence of high viscosity on the flow field inhibiting the transfer processes. As the pulp is highly non-Newtonian fluid it is worth to have a look at the corresponding viscosity distribution shown in figure 8b. Huge viscosity variations should be noticed ranging from 0.1 Pa·s for the near-rotor region up to 200 Pa·s for the stagnation zones.

Fig. 5. Free surface of the pulp flow

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Fig. 8. Velocity magnitude (a) and molecular viscosity (b) distributions at horizontal plane h=2cm

As it was described in section 2 the simulation was performed with the use of two

constitutive models (laminar – rheological and turbulent – Newtonian) which were alternatively applied depending on the shear rate. Figure 9 shows the border between both domains for particular circumferential position of a control plane. As it can be easily found the extent of the region, for which turbulent model had to be applied, is relatively small. Its contour is strongly affected by the rotor profiles which are responsible for the shear production. Regardless the limited extent of that region the application of constitutive models duality seems to be necessary if the behaviour of the pulp flow has to be recovered in realistic way.

Fig. 9. The border between laminar and turbulent flow regions

Conclusions

The paper presents the preliminary results of the paper pulp flow modelling in a hydropulper. The numerical model applied for the simulation allows to capture the dual nature of the flow revealing coexistence of laminar and turbulent regimes described by either rheological or Newtonian behaviour, respectively. Fluent commercial software used as a numerical tool was equipped with own-made

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procedures enabling for the control of its functioning in particular for the implementation of the adequate rheological material properties and the coexistence of constitutive models. The results of the simulation provided the insight into the physics of the flow in a pulper which is in agreement with the industrial experience. The regions of the stagnant fluid as well as of intensive transport processes were identified allowing for the evaluation of the defibring. Rough comparison with the experimental trials devoted to the free surface shape revealed great discrepancies which can be mainly explained by scale effects (the simulation was conducted for the pulper model to scale 5:1). It is also believed that the physical model of the flow can be improved by the introduction of a thin water film to near-wall region influencing the flow resistance.

Summing up, the preliminary results of the simulation allow to evaluate the applied model as a reliable tool for the prediction of the paper pulp behaviour and for the optimisation of the defibring process in a pulper.

Acknowledgments

The work was supported by Ministry of Education and Science under the grant number 4 T07A 027 27 and by statutory funds of Institute of Thermal Machinery under the number BS-1-103-301/2004/P.

Literature

[1] Bakker A., Fasano, J.B., The flow pattern in an industrial paper pulp chest with a side entering impeller, The Online CFM http://www.bakker.org/cfmbook/pprpulp.pdf

[2] Bennington C.P.J., Smith J.D., Sui O.S., Wang M-H., Characterisation of repulper operation for newsprint deinking, TAPPI Pulping Conference, Montreal, QC, October 25-29 (1998) pp. 1083-1095.

[3] Bennington C.P.J., Kerekes R.J., Grace J.R., Mixing in pulp bleaching, Journal of Pulp and Paper Science, vol 15, no. 5, September 1989

[4] Ferguson J., Kembłowski Z., Reologia stosowana płynów, Wyd. Marcus Łódź 1995 [5] Kembłowski Z., Giza R., Opory lokalne przy przepływie zawiesin włóknistych przez przewody,

Przegląd papierniczy – 1971, str. 332-337 [6] Niklas M., Asendrych D., Modelowanie procesu rozwłókniania masy papierniczej w hydropulperze,

IV Warsztaty Modelowanie przepływów wielofazowych w układach termochemicznych – Stawiska 2004

[7] Radoslavova D., Silvy J., Roux J. C., Wang M-H., The concept of apparent viscosity of pulp for beating analysis and the development of the paper properties, 1996 Papermakers Conference, Philadelphia

[8] Tyralski T., Grace J.R., Maszyny do przygotowania masy papierniczej i reologia zawiesiny włóknistej, Wydawnictwo Naukowe Politechniki Łódzkiej, Łódź 1980

[9] Wilczyński K., Reologia w przetwórstwie tworzyw sztucznych, WNT Warszawa 2001