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Interaction of Non-Newtonian Fluid Dynamics and Turbulence on the Behavior of Pulp Suspension Flows

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Interaction of Non-Newtonian Fluid Dynamics and Turbulence on the Behavior of Pulp Suspension Flows ANNUAL TRANSACTIONS OF THE NORDIC RHEOLOGY SOCIETY, VOL. 13, 2005 Interaction of Non-Newtonian Fluid Dynamics and Turbulence on the Behavior of Pulp Suspension Flows Juha-Pekka T. Huhtanen, and Reijo J. Karvinen Tampere University of Technology, Tampere, Finland. ABSTRACT forces, determines pressure production of a This research paper deals with the hydraulic machine. Flow of dilute water- interaction of non-Newtonian fluid fiber suspensions (C < 1%) in a converging dynamics and turbulence in paper making channel is similar to the flow in a paper flows. Applications are involved with highly machine head-box, and the flow conditions non-Newtonian wood fiber and paper pulp there determine the quality of produced suspensions in laminar and turbulent flows. paper. Flow of dilute water-fiber The flow is characterized by using the suspensions over a backward facing step is continuum approach. The goal of the study present in many unit processes in paper is to examine the possibilities of using non- production, especially in the wet end of Newtonian fluid models for simulation of paper machine, and is an important mixing these flows. element for fiber suspension flows, because it is very effective turbulence generator. INTRODUCTION The theories of non-Newtonian fluid BASIC THEORY dynamics and rheology have for decades Basics of continuum mechanics and the been successfully applied in polymer generalized Newtonian fluid models used in processing, but in the papermaking industry numerical simulations in our study are given it is only now taking its first steps. Attempts below. In addition, the turbulence models have been made to apply non-Newtonian used in this study are also expressed. fluid dynamics to laminar flows of paper pulp suspensions1, 2, 3, but hardly at all to 1. Continuum mechanics turbulent flow4, 5, 6. For an incompressible fluid, the In the study, four distinct flow situations continuity equation can be written as are examined. A pipe flow of dilute pulp suspensions (C < 4%), which is an v = 0 . (1) important pulp transport application from one single process to another, and the The conservation law of linear pressure losses during the transport are momentum can be written in general form as directly related to pumping costs. Flow of follows: concentrated pulp suspensions (up to 10%) in a mixing tank is important for all (Dv/Dt) = T + f , (2) rotational devices used in papermaking (refiners, mixers, pumps, etc.), because it where T is the total stress tensor, i.e., takes into account the centrifugal force Cauchy's stress tensor, and f is the body component, which, together with friction 177 force vector. The first term on the left hand = K(2(D : D))((n-1)/2) D , (6) side of the equation is the inertial term with the total derivative of the velocity vector: where n is the Power-law index or shear index, and K is the consistency factor. Dv/Dt = v/t + v v . (3) Bingham plastic fluid: The total stress tensor may be divided Some fluids may exhibit solid type into spherical and deviatoric components as behavior at rest, even a solid body, and follows: hence need finite start-up stress, often called yield stress (ty), before they yield and start T = -p + , (4) to flow. Such behavior can be described by the Bingham plastic fluid model in its where is the unit tensor and p = -(tr T)/3 general form of the hydrostatic pressure in the matter. 2 = 2(y/γ + p) D , ½( : ) y , (7) 2. Generalized Newtonian fluid models 2 In this study, we used only generalized D = 0 , ½( : ) y , (8) Newtonian models. The term generalized Newtonian fluid is used here to describe where p is the so-called plastic viscosity. behavior, which may be shear thinning or shear thickening, depending on the shear Herschel-Bulkley fluid: index value. Also start-up stress, often To combine the advantages of the Power- called yield stress, behavior can be law and the Bingham plastic fluid models, described by some of these models. we can use the Herschel-Bulkley equation to Moreover, the constant viscosity fluid model describe the rheological behavior of a fluid may also be subsumed within this category. (n-1) 2 The following fluid models were used = 2(y/γ + Kγ ) D , ½( : ) y , (9) here: 2 D = 0 , ½( : ) y , (10) Newtonian fluid: The behavior of a Newtonian fluid may The model describes both yield stress be described with the aid of a rheological behavior and shear thinning or thickening model, which connects the "extra-stress" effects. tensor to the rate of deformation tensor. For an incompressible fluid, the relation can be Bird-Carreau fluid: written as Perhaps the most convenient of the generalized Newtonian models is the Bird- = 2 D (5) Carreau fluid model Power-law fluid: (n−1)/2 = 2µ D + µ − µ 1+ 2 2 D : D The shear thinning or shear thickening ()0 [ t ] () behavior of materials may be described as (11) an exponential relationship between shear where 0 and are the first and second rate and shear stress. The simplest and often Newtonian viscosities at zero and infinite a very useful relationship is expressed by shear rates, respectively, and t is the the Power-law model, which may be written inverse of a cut-off shear rate in the as viscosity curve, often called the characteristic time of the fluid or time 178 constant. (More on rheological models can To evaluate the turbulent viscosity for be found in the literature7.) calculation, we need to determine the values of turbulent quantities or turbulence kinetic 3. Turbulence energy and the dissipation rate. And to do Non-Newtonian fluids are highly viscous this, we need a turbulence model to describe fluids with their apparent molecular turbulent flow behavior. In fact, the same viscosities exceeding at low shear rates equations as defined in previous section, thousands of times the viscosity of water. i.e., continuity and momentum equations, However, many non-Newtonian fluids are determine the transport phenomena in the also shear thinning, which means that more main flow field. Since we can write similar and more bonds between molecules or transport equations for the above turbulent particles in suspension are broken as the quantities, we can also determine the shear rate increases and the fluid's apparent turbulent viscosity at every single point in molecular viscosity decreases. The flow the flow field. field's inertial forces may now become Turbulent quantities in the flow field can greater than the viscous forces and make the be characterized in several different ways, flow turbulent. and different turbulence models are One way to approach the problem is to categorized based on of the number of combine the two theories, non-Newtonian equations needed to determine the fluid dynamics and turbulence, by exploiting quantities. In engineering applications, the the effective viscosity, which links the k- model is the best known. More enriched effects of molecular and turbulent models may be categorized under multi- viscosities as follows: equation models; their ability to describe turbulence quantities depends on the number eff = ap + t , (12) of additional equations employed. (More on turbulence as phenomenon and its models 8 where t is the turbulent dynamic viscosity, can be found in the literature .) defined in terms of the turbulence kinetic energy (k) and dissipation rate () k- model: When we use the k- model, we exploit 2 t = Ck / . (13) two additional scalar transport equations in addition to the continuity and momentum When the Reynolds decomposition is equations for isothermal flow, one for inserted in the momentum equations, Eq. turbulence kinetic energy and one for (3), the form of the equations remains, if the dissipation rate. Those equations can be new denotation is taken for the total stress written as follows: tensor. Taking into account both the laminar and turbulent contributions of motion, we (Dk/Dt) = (eff k) +Pk - , (15) can now write the new stress tensor for a turbulent flow field in tensorial notation as (D/Dt) = ((ap + (t/)) ) + (C1Pk - C2)/k , (16) T = -p + - Q , (14) where C1 = 1.44, C2 = 1.92, and = 1.3 where Q = v' v' is the velocity fluctuation are constants. correlation tensor defined by the velocity fluctuation vector in ( v'). The overhead bar Pk is the turbulence energy production denotes time averaging. term, which describes how turbulence takes its energy from the main flow field via 179 viscous dissipation. Since this happens where v’ is the velocity fluctuation vector. through deformation work in the main flow From the correlation of the velocity field, turbulence energy production is fluctuation components, we can determine determined by the rate of strain tensor (S), the turbulence kinetic energy field by or by the velocity vector gradient as follows k = ½ (tr Q) (20) 2 Pk = t S = t (D : v) (17) The transport equation for the turbulence This straining motion of the main flow dissipation rate is similar to that defined for field produces velocity fluctuations in the the k- model. velocity field and thereby feeds energy into the turbulent motion. Therefore, the EXPERIMENTS production of turbulence kinetic energy can For this study, a representative set of be defined also by these velocity measurements was prepared to characterize fluctuations as the rheological properties of paper pulp and fiber suspensions to view the subject as a Pk = -(Q : S) (18) whole. Measurements were also extended to the turbulent flow region to observe the The biggest problem with the k- model above suspensions in this situation. is that it assumes turbulence to be Furthermore, measurements by other homogenous and isotropic and cannot take researchers were analyzed and their into account non-isotropic turbulence. If the experimental studies consulted, especially streamlines are curved, the straining motion when the rheological properties of the pulp of the main flow field stretches also suspensions were analyzed.
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