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ANNUAL TRANSACTIONS OF THE NORDIC SOCIETY, VOL. 13, 2005

Interaction of Non-Newtonian 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 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 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

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 . 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 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 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. turbulent vortices and thereby dampens velocity fluctuation in the cross flow Mixing tank: direction and amplifies fluctuation in the Bleached softwood kraft pulp main flow direction. To take this feature into suspensions of 2-10% consistency, mixed account, we must resort to more enriched with a rotating disc type mixer in a models. container, were rheologically measured at the laboratory of the Energy and Process model: Engineering Institute, TUT. The container In the Reynolds stress model (RSM), was designed so as to prevent the walls from additional transport equations are needed for affecting the shear stress on the rotating disc each velocity fluctuation correlation and hence its torque resistance. The component of the dyadic product v’v’ equipment was run by an electromotor, (Reynolds stress component). In full 3-D mounted on bearings, and torque resistance simulation, this means six new equations, was measured on its axle. since the Reynolds stress tensor (Q) is The shear stress on the edge of the rotating symmetric. With the earlier definition of the disc may be calculated from the measured 6 velocity fluctuation correlation tensor (Q), torque values , and we can construct either a we can write the transport equations for the shear rate-shear stress or a shear rate- Reynolds stress components as apparent viscosity relation to describe the non-Newtonian behavior of paper pulp or T fiber suspensions (see Fig.1). (DQ/Dt) = − Qv' + p'[]()v' + ()v' + T T []µ eff Q − []Q ()v + Q v + T p'[]v'+()v' − 2µ eff ()v'v' (19)

180 The measured data used here was adopted from an article by Hämström et al.9, which examined the pipe flow of dilute fiber suspensions (0.67%, 1.27%, 2.47% and 3.41%) of bleached softwood kraft pulp. Measurements were made in a 100mm straight pipe by varying the suspension's flow rate. The mean velocity in the pipe was determined from the flow rate and used as reference in pressure drop calculations. Figure 1. Results of torque measurements These measurements were reanalyzed with rotating disc in container for water and because of their good documentation and for bleached softwood kraft pulp convenience. suspensions as shear rate-apparent viscosity

data. Converging channel:

Flow of a non-Newtonian fluid through Pipe flow: the converging channel is relevant In terms of rheological measurements, especially to the paper making industry, pipe flow is very interesting because it is because it resembles flow through the slice viscometric flow, which can be solved channel in the paper machine headbox. Such analytically by derivation. The solution can flows are usually turbulent in the core be applied to characterize material region. However, with viscous fluids some properties of fluids with a simple material signs of re-laminarization may appear in the function such as the Power-law model. If we . To analyze the role of adopt the notion that pressure drop in pipe turbulence in the headbox and its effects on flow is equal to the shear stress on the wall, produced paper quality, some measurements both multiplied by appropriate surface areas, must be made inside the channel. and if we assume that the shear stress of In this study, we did not measure flow in paper pulp or fiber suspensions on the wall a converging channel ourselves but analyzed can be expressed by the Power-law model, the measurements by Parsheh et al.10 to then pressure drop in a laminar case can be 2, 7 verify our computational results. The written as measurements were made with air, therefore

n n (n+1) they are not exactly relevant for fiber dp/dx = -2KV (3 + 1/n) / R , (21) suspension flow, but they were the best ones

8 in hands at the moment of this study. and in a turbulent case

2 Backward-facing step: dp/dx = 4fV / 2D (22) The flow of a non-Newtonian fluid

through a channel with a backward-facing where the friction factor (f) can be step is important in the paper making determined by Blasius's type of equation, by industry, because the step is often used to using 1/8-law for the velocity profile, as /6/ create turbulence. In such applications, the

2/9 flow always separates from the wall and 4f = 0.0964 / Rec (23) then re-attaches. The separation is necessary

n (2-n) (n-1) to produce turbulence, and highly Here Rec = D V / (8 K’) is the fluctuating velocities appear in the mixing modified , and K’ = K [(3n n layer between the main flow stream and the + 1) / (4n)] is the modified Power-law backflow behind the step. consistency.

181 In this study, we did not measure flow in non-Newtonian effects, such as yield stress, a channel with a backward-facing step but but only shear thinning. Compared with analyzed the measurements by Piirto et al.11 measurements, those predicted by the to verify our computations. They used a Power-law model were more than 100% in Particle Image Velocimetry (PIV) in their error. The Bird-Carreau model, too, failed to measurements and aimed to locate the predict accurate torque resistances at higher highly turbulent flow areas and to measure consistencies (C > 6%), the reason possibly their velocity fluctuation components. From being that the model does not actually the velocity fluctuations, they could describe any yield stress effects, but defines calculate the kinetic energy and dissipation explicitly material's zero and infinite rate of the turbulence by using image viscosity values. Though in terms of processing methods. numerical simulation, the material models with yield stress behavior (Bingham plastic NUMERICAL SIMULATIONS and Herschel-Bulkley models) were the All the cases were numerically simulated most difficult to handle, they described with commercial codes either in a laminar physically most accurately the behavior of case with the finite-element-method-based pulp suspensions in rotational flow. Their POLYFLOW, versions 3.5.4 to 3.8.0, or in a predicted results fell within 10-20% of those turbulent case with the control-volume- obtained with measurements for torque method-based FLUENT, versions 4.3.2 to resistance. Fig. 2 shows the torque 5.5.14. POLYFLOW was chosen for resistances of the disc, calculated based on laminar simulations, because experiments numerical predictions. showed that paper pulp and fiber suspensions displayed non-Newtonian and viscoelastic properties. FLUENT was used to simulate turbulent flow, because the program can exploit non-Newtonian fluid models for such simulation.

Mixing tank: We simulated of pulp Figure 2. Torque resistance of disc: a) suspensions with POLYFLOW, version laminar flow (N = 100 rpm) with different 3.5.4, near a rotating disc in a mixing tank, fluid models as function of fiber and analyzed and compared the results with consistency. the experiments discussed earlier in previous report2. In the simulations, we used We simulated turbulent flow of pulp some generalized Newtonian fluid models suspensions in a mixing tank with FLUENT, (constant viscosity, Power-law, Bingham version 5.5.14. We used Power-law fluid plastic, Herschel-Bulkley, and Bird-Carreau model, with the same material parameters as models). Material functions were in the laminar series, because the model was determined based on measurements made in presently the only non-Newtonian fluid our laboratory at TUT. model available in FLUENT to simulate All the fluid models we used predicted turbulent flow. Both the k-e and RSM increased torque resistance as a function of turbulence models were tested in simulation the fiber consistency of the paper pulp with two different boundary treatment suspensions, and the increase was almost schemes, standard wall functions and two- exponential. However, the Power-law model layer zones, respectively. Fig. 3 shows the could not produce accurate torque values, disc's torque resistances in turbulent water because it cannot take into account complex flow with different turbulence models.

182 loss was determined by using the Power-law fluid model as a material function to define the friction factor of pulp suspension in pipe flow. The calculations fully agreed with the measurements. By using the 1/8-law for the velocity profile instead of the normal 1/7- law for Newtonian fluids to determine the friction factor for turbulent pipe flow of non-Newtonian fluid, we could predict the Figure 3. Torque resistance of disc: pressure loss very well. Measured and turbulent flow of water as function of calculated results are compared in Fig. 5. rotational speed of disc, calculated based on numerical predictions and compared with measured data.

Pipe flow: Laminar paper pulp suspension flow in a pipe was numerically simulated with POLYFLOW, by constructing a 2-D computational domain of a pipe cross- section, through which a volumetric flow rate was imposed. A slip friction boundary Figure 5. Comparison of measured and condition was introduced on the pipe wall, calculated pressure losses in pulp and slip factor values were determined by suspensions in turbulent pipe flow. analyzing2 the measurements of Hemström et al.9. Converging channel: When the measurements were analyzed, In numerical simulation of converging the same trend was observed in each case: channel flow with FLUENT, version 5.3, we slip factor values seemed to exhibit a kind of used the non-Newtonian Power-law fluid power-law relation with fluid velocity. The model as a material function and studied in results of simulated pressure loss curves series the effects on turbulence quantities of with slip boundary condition are shown in non-constant molecular viscosity and level Fig. 4. of viscosity. The objective was to study the effects of shear thinning by varying the exponent (n) value rather than to try to correctly model fiber suspension as non- Newtonian material. The effects were significant. Signs of re-laminarization near the walls in the boundary layer can be seen in the velocity profiles in Fig. 6 for water and certain non-Newtonian fluids as a function Figure 4. Results of simulated pressure loss of the shear index. The consistency of the 2 in paper pulp suspension flow in pipe . Power-law model (K = 100 * v) was kept constant throughout the simulations, which In this study, we did not simulate gave us the variation of acceleration numerically turbulent flow in a pipe but parameter10 from 4.89 10-7 to 4.89 10-5, compared measurements with theoretical which should indicate the evident change calculations in equations, where the pressure from turbulent to laminar boundary layer.

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Figure 6. Effect of shear index on velocity Figure 8. Effects of shear index and profiles in converging channel flow. consistency values [( ) = , ( o ) = 10 Simulations of water and Power-law fluids v * v, ( * ) K = 10 * v & n = 0.8, ( ) K = 10 with K = 100 * v. * v & n = 0.7, ( + ) K = 10 * v & n = 0.5] In a flow domain where the cross section on velocity profile in channel flow after is not constant, as in a contracting channel, backward facing step. turbulence is non-isotropic. In such a case, Our simulations agreed quite well with the RSM turbulence model must be used in 12 13 order to describe correctly the effects on earlier measurements , and simulations . turbulence quantities of the main strain rate The dimensionless mean flow and field through stretching. The level of turbulence quantity profiles, predicted by velocity fluctuation in the flow direction is different turbulence models, agreed, in fact, excellently with earlier DNS simulations by almost an order of magnitude higher than in 13 the cross direction at the end of the channel. Le et al. . This is encouraging, because Le The results of the velocity and turbulence et al. analyzed a great number of well- quantity profiles agree qualitatively with the established measurements. However, measurements by Parsheh et al.10. comparison with measurements made at TUT by Piirto et al.11 indicated some differences, especially in the core flow area. Measurements by Piirto et al. showed too flat velocity profiles and hence too low velocity fluctuation and turbulence kinetic energy. Yet our prediction of the trends coincided with the trends of their measurement, even when fibers were added to the flow. Figure 7. Development of scaled rms values of velocity fluctuations (u', v' and w') along channel axis for water flow.

Backward-facing step: We simulated flow over the backward- facing step numerically with FLUENT, version 5.3, and used both the k- and RSM turbulence models with standard wall functions or a two-layer wall to compare the Figure 9. Effect of shear index on location accuracy of the two schemes in the wall of re-attachment point (XR / H) after layer. The Power-law model was used to backward-facing step. simulate the shear-thinning behavior of the material.

184 The shear index had a marked effect on the 2. Huhtanen, J-P. (1998), Non-Newtonian location of the re-attachment point after the Flows in Paper Making. Licentiate Thesis, backward-facing step. As the index Tampere University of Technology, Energy decreased, so did also the local molecular and Process Engineering. viscosity near the walls, and the location of the re-attachment point moved further 3. Swerin, A. (1995), Flocculation and fiber downstream, as shown in Fig. 9 above. network strength in papermaking suspensions flocculated by retention aid DISCUSSION AND CONCLUSIONS systems. Doctoral Thesis, Royal Institute of This study examined the behavior of Technology, Department of Pulp and Paper paper pulp and wood fiber suspension flows Chemistry and Technology, Division of in the paper making industry with focus on Paper Technology, Stockholm. continuum mechanics and turbulence. Both the above materials are concentrated 4. Wikstömin, T. (2002), Flow and suspensions, and therefore their response to Rheology of Pulp Suspensions at Medium a shear stress or shear rate field was Consistency. Doctoral Thesis, Chalmers analyzed in terms of rheology. University of Technology, Department of Continuum mechanics and non- Chemical Engineering Design. Newtonian fluid dynamics combined with turbulence theory turned out to be a suitable 5. Hammarström, D. (2004), A Model for theoretical framework to simulate flows in Simulation of Fiber Suspension Flows. the paper making processes. Though we Licentiate thesis, Technical Reports from have to keep in mind the limitations of KTH Mechanics, Royal Institute of continuum models, numerical simulation Technology, Stockholm, Sweden. does offer obvious advantages in flow conditions such as these, where accurate 6. Huhtanen, J-P. (2004), Modeling of Fiber measurements are tedious and time- Suspension Flows in Refiner and Other consuming and sometimes even impossible. Papermaking Processes by Combining Non- Simulation of concentrated suspensions in Newtonian Fluid Dynamics and Turbulence. turbulent flow is practically impossible by Doctoral Thesis, Tampere University of means other than homogenous models, such Technology, Energy and Process as non-Newtonian fluid models. But Engineering. together with proper turbulence models and wall treatment schemes, these fluid models 7. Tanner, R. I. (1985), Engineering yield quite reliable results compared to flow Rheology. The Oxford Engineering Science measurements. Therefore, the combinatory Series. Claredon Press, Oxford. approach of non-Newtonian fluid dynamics and turbulence theories can be considered 8. Tennekes, H., Lumley, J. L. (1972), A successful and one of the most interesting First Course in Turbulence. MIT Press, future fields of modern flow simulation. Cambridge, Massachusetts.

REFERENCES 9. Hemström, G., Möller, K., Norman, B. 1. Wikström, T. (1998), Flow of Pulp (1976), Boundary Layer Studies in Pulp Suspensions at Low Shear Rates. Licentiate Suspension Flow. TAPPI Journal, 59, 115- Thesis, Chalmers University of Technology, 118. Department of Chemical Engineering Design.

185 10. Parsheh, M. (2001), Flow in Contractions with Application to Headbox. Doctoral Thesis, Royal Institute of Technology, Stockholm.

11. Piirto, M. (2003), Characterization and Control of Turbulence with Particle Image Velocimetry in a Backward-Facing Step Flow. Doctoral Thesis, Tampere University of Technology, Energy and Process Engineering.

12. Jovic, S., Driver, D. M. (1994), Backward-facing step measurement at low Reynolds number, Reh = 5000. NASA Tech. Mem. 108807.

13. Le, H., Moin, P., Kim, J. (1997), Direct Numerical Simulation of Turbulent Flow Over Backward-facing Step. J. Fluid Mech., 330, 349-374.

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