Preferred citation: D. Radoslavova, J.C. Roux and J. Silvy. Hydrodynamic modelling of the behaviour of the pulp suspensions during beating and its application to optimising the refi ning process. In The Fundametals of Papermaking Materials, Trans. of the XIth Fund. Res. Symp. Cambridge, 1997, (C.F. Baker, ed.), pp 607–639, FRC, Manchester, 2018. DOI: 10.15376/frc.1997.1.607. HYDRODYNAMIC MODELLING OF THE BEHAVIOUR OF THE PULP SUSPENSIONS DURING BEATING AND ITS APPLICATION TO OPTIMISING THE REFINING PROCESS Authors: D Radoslavova, J C Roux, J Silvy* UMR Génie des Procédés Papeties 5518 CNRS EFPG-INPG Domaine Universitaire, BP 65, 461 rue de la Papeterie 38402 Saint-Martin d'Héres, Cedex FRANCE * Presentaddress : Universidade da Beira Interior Rua Marques d'Avila e Bolama 6200 Covilhà PORTUGAL 608 INTRODUCTION The theoretical approach proposed in this research paper aims to relate the energy efficiency coefficient ofthe refiner to the rheological pulp properties . The theory is then used to predict the observed energy consumption as a function of the desired paper properties . Various aspects of the beating process have been previously discussed in review articles [1],[2] and this discussion will not be repeated in this paper. Even in the sixties, the major characteristics of the beating as a dynamic process, were pointed out very clearly. Halme [3], in an interesting survey , mentioned that the pressure in the beating zone must be combined with the relative motion of the surfaces to produce acceptable beating effects on the fibres. It was also clear that the beating effect of the turbulence alone was negligible. As a matter of conclusion, for future works on the beating process, Halme [3] recommended a better characterisation of the behaviour of stock in the refining zone. 25 years latter, Page [2] pointed out the necessary knowledge of the stresses that fibres can experience in the beating zone and their answer to these stresses, this last remark implies 'the development of a mechanistic theory of the beating'. Nowadays, we have not yet directly developed this approach but an other general one. The hydrodynamic aspect of the beating process is considered with a pulp apparent viscosity of the suspension in the beating zone. Various studies of flow in the refiners have been reported in the literature. Frazier [4] tried to use the hydrodynamic theory to analyse the process in high-consistency refining . The fact that the pulp apparent viscosity is unknown, was a major drawback and led to a limited success. Rance [5] and Steenberg [6] said earlier that we can develop the modelling of the beating through the lubrication theory but the beating process, in itself, modifies the pulp lubricant film - that is not the case in tribology. The previous authors have drawn the difficulties of such a research but have not solved the problem. How to do that is the purpose ofthe following paragraphs. We shall prove in this paper that it can be a fruitful approach . But, several conditions must be required before: 1 . The flow of the pulp suspension in the beating zone is laminar, 2. The value of the pulp apparent viscosity is known, at any time of the beating process. 609 APPLICATION OF LUBRICATION THEORY TO THE REFINING ANALYSIS By considering the refining machine as a bearing in which the pulp acts as a lubricant, it is possible to establish several relationships between the key refining parameters . The development starts with the equation ofmotion: the momentum transfer balance, written for the refining zone and for an incompressible fluid flow [7]: P V V) P-~ - Vp - ~V_ . -C + Pk This equation contains different forces expressed per unit volume: " the inertial forces are expressed from the derivatives of the velocity of the flow and a regroup an accumulation term and a connective term: p i + (V-. D)- pv at " the pressure forces can be deduced from the gradient ~p " the viscous forces are included in the divergence of the stress tensor T (spatial derivatives ofthe shear stresses according to the chosen co-ordinates) " the gravity forces . The shear stress tensor implies the knowledge of the rheological pulp behaviour in the refining zone. But, even for a non-Newtonian behaviour, it is always possible to consider a pulp apparent viscosity [7]. This parameter seems difficult to obtain owing to the complexity of the flow and to the pronounced modifications during the beating process. We obviously know that it affects the fibre length distribution, the hydration and the fibrillation and. brings to change in the fibre flexibility. The Theological equations for particulate suspensions, already published in the literature [8] indicate that changes in the fibre characteristics (length, diameter) can lead to modifying the pulp apparent viscosity. In other hand, the presence of inertial terms introduces a set of supplementary parameters : the derivatives of the speed of the flow. Time sampling of the problem The pulp properties change continuously during refining but if we consider a small period fit, a mean value ofthe suspension apparent viscosity can be assumed. Then, if we sample 61 0 the refining process, during each elementary period, a continuous stationary flow is supposed to establish in the refining confined zone. The initial equation can be written under the following form: pv-Vp-V.i+pg=0 (2) Rheological pulp behaviour during an elementary period The apparent viscosity is, by definition, the non-Newtonian effective viscosity [7]. In an elementary period of time ( ti , t; + At ), the pulp apparent viscosity is coincident with the dynamic viscosity of the equivalent Newtonian fluid ~t a but [t,, is a function of time during the whole refining process. This point is illustrated on the figure 1 in order to reconstruct the pulp apparent viscosity function from the different discrete numerical values ~i, . Figure 1 . Reconstruction ofthe pulp apparent viscosity function vs. time during the refining process. This last statement is very important in the proposed analysis . It simply means that the equation (2) is a set of Navier-Stockes equations for each period of time during the whole refining process. p _~t.V2~ +Pk = 6 for4,, = ~t,, , ( t, tj + At (3) Estimation of the Reynolds number In order to determine the nature of the flow in the refining confined zone, the Reynolds number must be calculated. In other hand, this number characterises the momentum flux transferred by convection versus those transferred by viscous diffusion, taking into account the viscosity The Reynolds number is defined by the following equation: Re = -P(DU P. (4) where p is the pulp density kg. M-3 U is the mean velocity of the flow M . S -' g,, is the pulp apparent viscosity Pa. S (D is the hydraulic equivalent diameter of the gap clearance M For example, if we consider a refiner gap equal to 300 pm, then (D = 6.10-'m , p can be chosen as the density of the dry cellulose p = 1,54.103kg. m-' . A mean value of the velocity is U = 12m.s-' . Then, according to Batchelor equation [9], an apparent viscosity of 20 mPa.s is retained for the calculation. The Reynolds number can be calculated as: 154.1 0' .12.6.10-' = Re - 554 < 1000 20.10-' - 61 2 (5) Even if the numerical values above are slightly different, we must conclude that the nature of the flow is laminar in the refining zone. The inertial term is equal to 0 owing to the orthogonality of the flow ((v . V)v = 0 ). Thus, the equation (3) can be written in the following form ifwe neglect the gravity forces _VP ~t + aV2v + Pg = 0 (6) Considering the above assumptions, the gravity forces will be neglected then the modelling of the refining process will be based upon this last equation: -Vp + ~tUV2. = 0 (7) It is a single form of the Navier-Stockes equation but the resolution requires first the selection of the most appropriate co-ordinate system [7]. In other terms, the resolution depends on the refining geometry. This knowledge then allows supplementary simplifications of the model to be performed. After the model resolution, the laminar nature ofthe flow will be validated a-posteriori. THE MODEL RESOLUTION The following developments are performed for a laboratory Voith Hollander beater chosen for its simple geometry, described on figure 2. In order to obtain the input parameters of the model , the refiner is equipped with sensors. These allow measurements of the vertical displacement of the cylinder, the loading force, the shaft torque, the rotation speed and the temperature of the pulp suspension . An automatic control system is developed to run in different beating modes: " maintaining a constant vertical displacement of the cylinder (i.e. constant set of gap clearance) " maintaining a constant loading force (normal beating mode for this refiner) " maintaining a constant net power " maintaining a constant torque (if the rotation speed is also constant, the running mode keeps*the total power constant). 61 3 Figure 2. Scheme of the Voith Hollander beater . Owing to the refining geometry, the cylindrical co-ordinate system is chosen according to the figure 3. First of all, the two bars of the cylinder in the converging zone are considered (see figure 6) but we assume that the bedplate is smooth with no bars inside . This last assumption will be validated afterwards. The flow according to the z-direction is not taken into account in this analysis. The problem to be solved remains bidimensionnal (0, l, í9 ) . In the lubrication theory, the thickness of the wedge zone is very small compared to the other characteristic dimensions (radius of the cylinder, length of the bedplate,...). The complementary following assumptions can be reasonably considered : no radial flow yr = 0 no radial pressure gradient aP = 0 ar The vectorial equation (7) finally leads to only one scalar equation 61 4 _I dp(8) - _a _(I _a ))) ~u r d8 ar r ar ~rve (g) r radial co-ordinate in the gap clearance m v angular co-ordinate ofthe speed ofthe flow m.s""` It is easier to manipulate dimensionless formulations so several reference parameters must be chosen accordingly.