Modeling of a Floc-Forming Fluid Using a Reversible Network Model

Transcription

1 ORIGINAL PAPER Journal of Textile Engineering (2014), Vol.60, No.4, The Textile Machinery Society of Japan Modeling of a Floc-Forming Fluid Using a Reversible Network Model Ya m a m o t o Takehiro a, * a Division of Mechanical Engineering, Graduate School of Engineering, Osaka University, 2-1 Yamadaoka, Suita, Osaka , Japan Received 23 April 2014, accepted for publication 24 June 2014 Abstract A reversible network model was applied to a model for a floc-forming fluid such as a pulp suspension. The reversible network model approximately represents the evolution of polymer network structures using non-interacting elastic dumbbells that have two states, active and dangling. The former corresponds to a network segment whose both ends connect to the network, while the latter corresponds to a network segment that its one end dissociates from the network. The transition between the active and the dangling states is stochastically simulated according to probability functions. In the present study, the repetition of forming and breakup of flocs in a flow was modeled by the reversible network model, and the model was examined by evaluating rheology of the model fluid and by simulating its Poiseuille flow in a circular tube. The predictions confirmed that the present model was appropriate for a model of floc-forming fluids. Key Words : Floc, Flocculation, Reversible network model, Numerical simulation 1. Introduction In flows of concentrated or semi dilute suspensions, flocs of dispersed particles are often formed. Flocs of fibers can be observed in flows of pulp fiber suspensions and nano-fiber suspensions [1,2]. For example, the flocculation occurs in the forming process of fiber-reinforced composites and in pulp suspension flows in the manufacturing process of paper. Floc is a relatively weak aggregation, and hence it is relatively easily broken up by flow and is also reformed owing to a flow-induced reaggregation. The repetition of forming and breakup of flocs originates complicated flow phenomena of floc-forming fluids. Furthermore the inhomogeneity due to the concentration distribution of dispersed materials significantly affects the flow of suspensions [1,2]. It is therefore significant to analyze the flow behavior of floc-forming fluids considering the effect of the formation and breakup of flocs in a flow. Computational simulation is a powerful tool for the analysis of complex fluids such as polymeric liquids, suspensions, and flocforming fluid. The numerical flow analysis based on continuum mechanics is a widely used approach, while the simulation of developing mesoscale structures of complex fluids under flow is being developed as a novel approach to the behavior of complex fluids. Appropriate constitutive equation is necessary for the continuum approach, whereas the constitutive model for flocforming fluids has not been established yet. Constitutive equations for viscoplastic fluids such as the Bingham model and the Herschel- Bulkley model are sometimes applied to the flow analysis of concentrated suspensions [2,3]. However they are not adequate for modeling the forming-breakup process of flocs. On the other hand, many researchers have performed direct simulations of the behavior of individual fibers under flow considering their interactions, contact, and entanglement [4-13]. Although this approach provides detailed information on the flow-induced structure of flocs, it requires much expensive computational costs. Therefore the application of the direct simulation was restricted to relatively simple problems such as simple shear flow. In the present study, we applied a reversible network model proposed by Hernández Cifre et al. [14], to describe floc-forming fluids. This model has been developed for describing the behavior of associative polymers. In this model, a polymer network is approximately represented by a set of numerous non-interacting elastic dumbbells and the deformation of network topology is not directly computed. Consequently the computational cost can be kept relatively low as compared to other direct simulations. The dumbbell consists of two beads connected by a spring and takes two conditions, active and dangling: Both ends of an active dumbbell belong to a polymer network and hence the beads of active * Corresponding author: take@mech.eng.osaka-u.ac.jp, Tel : , Fax:

2 Yamamoto Takehiro dumbbell are denoted by node beads. A dangling dumbbell has a node bead and a free bead, which represents the end of dumbbell that is not linked to a network as shown in Fig. 1. The resistance coefficient of a bead is different between node and free beads. The resistance coefficient ratio Z defined by Z = ς n / ς f is set to be larger than unity to describe the effect of entanglement of polymers. Here ς n and ς f denote the resistance coefficient for node beads and that for free beads, respectively. The transition between the two states, the association/dissociation process, is stochastically represented using probability functions. The behavior of floc can be characterized by the formation and breakup of entanglement of fibers, which is similar to the association/dissociation process of associative polymers, and hence the reversible network model is probably applicable for a model of floc-forming fluid. Stochastic behaviors of fiber flocs due to their non-uniform structures can be considered using probability functions. In addition, a random term in the probability differential equation, which describes the effect of Brownian motions of the reversible network model, can represent the effects of randomness due to the dispersion of fiber properties such as size and stiffness. In the present study therefore we examine the applicability of the reversible network model as a model of froc-forming fluids. The numerical simulation of flows in an axisymmetric tube was performed by coupling the simulation of floc behavior using the reversible network model and the macro flow simulation. 2. Basic equations 2.1 Reversible network model In the reversible network model, the behavior of a polymer network is represented by the orientation of non-interacting dumbbells and their status, active or dangling. The orientation of numerous numbers of dumbbells within a fluid element considered is computed and its ensemble average is used to obtain macroscopic variables such as a stress tensor. The flow field within a fluid element is assumed to be homogeneous and the motion of a dumbbell is represented by the motion of the end-to-end vector Q of the dumbbell (Fig. 2). The transition between the active and the dangling states is statistically simulated using probability functions. We considered the end-to-end vector Q of an elastic dumbbell that consists of two beads i and j with friction factors of ς i and ς j connected by a Hookean spring, whose spring coefficient is H. The time evolution equation of Q is described by [15], (1) where κ is the velocity gradient tensor, k B is the Boltzmann constant. T is the absolute temperature. t and Δt denote time and time interval, respectively.δw * is a vector of a Wiener process with zero average and Δt variance. In the reversible network model, ς i = ς j = ς n for active dumbbells and ς i = ς n and ς j = ς f for dangling dumbbells. Consequently Eq (1) is rewritten as for active dumbbells, and for dangling dumbbells. Here Q a and Q d indicate the end-to-end vector for the active and the dangling dumbbells, respectively. The friction coefficient ratio Z = ς n / ς f is set to be larger than unity. The spring is assumed to be Hookean, and hence the spring force F i is expressed by F i =HQ i (i=a or d) for each dumbbell. The extra stress tensor τ is calculated using a Kramers expression of the stress tensor: [15, 16] τ = nk B TI+2η s D+ n a F a Q a a + n d F d Q d d = nk B TI+2η s D+ n a H Q a Q a a + n d H Q d Q d d', (4) where n is the number density of dumbbells, I is the unit tensor, D is the rate-of-deformation tensor, and n i (i=a or d) indicates the number of active (a) or dangling (d) dumbbells. The brackets a and d mean the ensemble average of active and dangling dumbbells. The transition between the active and dangling states of dumbbells is simulated at each time step. The transition occurs according to a probability functions P d and P a for the dissociation and association processes, respectively. We applied probability functions proposed by Hernández Cifre et al. [14] for associative polymers: P d =1 exp[ β 1 exp(β 2 Q 2 a)δt], (5) P a =1 exp[ (α 0 +α 1 Q d )Δt]. (6) (2) (3) 62

3 Journal of Textile Engineering (2014), Vol.60, No.4, These functions describe a probability that the transition occurs within time interval Δt. α 0, α 1, β 1, and β 2 are model parameters, and Q i (i=a or d) is the dumbbell length. In the present paper, we omit the detail of the derivation of these functions and explain their features briefly (see [14] for details). The probability P d is derived considering the force balance between the associative force and the elastic spring force, and the dissociation occurs more frequently when the spring force was larger. β 1, and β 2 are related to a thermal vibration and the spring strength, respectively. The association probability P a was derived by assuming the probability linearly depends on the dumbbell length, which was suggested by the Brownian dynamics simulations by van den Brule and Hoogerbrugge [17]. α 0 is associated with a thermal diffusion of the dangling bead and α 1 is related to the space explored by the reactive motion of the extended dangling segment. Although these probability functions were derived for associative polymers, they can phenomenologically simulate the forming and breakup of flocs. The present study does not treat the improvement of the probability functions, while they should be constructed based on the dynamics of flocs to improve the model in future. 2.2 Governing equation for flows In the present study, isothermal incompressible flows are considered. Consequently, the governing equations are described by the equation of continuity (7) and the equation of motion (8). u = 0, (7), (8) where u is the velocity vector, p is the isotropic pressure, τ is the extra stress tensor, and ρ is the fluid density. D/Dt denotes the material derivative. Here the effect of external force such as the gravity is neglected. The extra stress tensor τ is calculated from Eq (4). In the present study, we considered Poiseuille flows in a circular tube with radius R illustrated in Fig. 3. The flow was assumed to be fully developed in the flow direction (z) and the average velocity V was fixed. As for the boundary condition, the non-slip condition was adopted on the tube wall, and the axisymmetric condition was considered at the channel center. When the inertial effect is ignored, the governing equations for the Poiseuille flow are written in the cylindrical coordinate system (r, θ, z) as follows: u r = 0, u z =u z (r), (9), (10) where u r, u z, and τ rz indicate the r-component of u, the z-component of u, and the rz-component of τ, respectively. When the extra stress tensor τ is expressed as τ =2η s D+τ p, Eq (10) is rewritten as The mean velocity V can be expressed as (11) (12) From Eqs (11) and (12), one obtains the following equation: (13) The development of flow field can be computed by the following procedure: (a) A Newtonian velocity profile and random dumbbell orientation are given as the initial condition. The numbers of active dumbbells and dangling dumbbells are set to be equal. (b) Λ is calculated from Eq (13). (c) The velocity gradient u z / r is calculated from Eq (11). (d) The end-to-end vector of the dumbbells at each computational point is simulated for the velocity gradient calculated in (c). (e) The transition of active/dangling states is simulated according to the probability functions (5) and (6). When a number created by a uniform random number generator is less than the corresponding probability (5) or (6), the transition occurs. p (f) Shear stress τ r z is calculated by Eq (4) using the results in (d) and (e). (g) Repeat the procedures (b)-(f) until the flow reaches a steady state. 2.3 Non-dimensional form We rewrite the basic equations in a non-dimensionalized form using the following scaling rule: t=λ H t *, r=rr *, u=vu *, p=(nk B T)p *, τ =(nk B T) τ *, and D=(V/R)D *, α 0 = α * 0 / λ H, β 1 = β * 1 / λ H, β 2 = Hβ * 2 / (k B T), where r is the position vector and λ H = ζ f /(4H). Here the variables with an asterisk denote dimensionless variables. The numerical simulation 63

4 Yamamoto Takehiro was performed based on the governing equations in the nondimensional form. The evolution equations of Q* in non-dimensional form are expressed as where the Deborah number De is defined by, (14), (15). (16) As for the governing equations of the flow field, Eqs (11) and (13) are respectively described in non-dimensional from as: 1, (17), (18) where the Weissenberg number We is defined by, (19) which expresses the ratio of elastic force due to the dumbbell orientation to viscous force. For simplicity, the asterisk indicating non-dimensional variables is omitted hereafter. 3. Numerical scheme The computational domain was uniformly divided into 200 segments and the differential in the governing equations was approximated by the forward differential scheme. The number of dumbbells N d = n a +n d at each computational point was set to The time increment Δt is We performed preliminary computations to determine the values of both N d and Δt such that the numerical results are independent of N d or Δt. Normal random numbers utilized for the computation of the Wiener process and for the active/dangling transition were generated by Box-Muller s method [18]. 4. Results and discussion 4.1 Rheology Firstly we evaluated rheological properties of the model fluid. The flow behavior of the reversible network model under shear can be simulated by computing the behavior of Q a and Q d for the velocity gradient tensor where γ is shear rate., (20) Shear properties of the model fluid can be evaluated using the stress tensor obtained. Figure 4 shows the shear viscosity of the model fluids. The model parameter applied are as follows: α 0 =0.83, α 1 =0.17 β 1 =0.333, and β 2 = , which were determined considering the model by Hernández Cifre et al. [14]. The resistance coefficient ratio Z was varied as Z=1, 2, 5, 10, 50, and 100. The number of dumbbells is We performed preliminary computations to determine the number of dumbbells, as the numerical results are independent of it. The shear viscosity was evaluated by the average for shear strain of 5 after the flow reaches a steady state. The shear viscosity shows shear-thinning property, which is a typical property of pulp fiber suspensions [2]. Zeroshear-viscosity is larger for larger Z and the shear-thinning appears at lower shear rate for larger Z because the viscous resistance of active dumbbells increases more and their relaxation time becomes longer as Z increases. Figure 5 shows the flow curve, the shear stress versus the shear rate, of the model fluid at We=1. For Newtonian fluids, the shear stress linearly depends on the shear rate γ, while the model fluid shows anomalous behavior. The shear stress linearly increases depending on the shear rate at small γ, a plateau region emerges at moderate shear rates, and then the shear stress increases again at large γ, while the shear-thinning behavior appears. The emergence of plateau region is due to the balance of the association rate and the dissociation rate. That is, the network structure varies only slightly and hence the shear stress remains unchanged. Chen et 64

5 Journal of Textile Engineering (2014), Vol.60, No.4, al. [19] have been found similar phenomena in their experimental study. They reported stress jumps in a region between Newtonian regions at high and low shear rates in a flow curve of hardwood bleached kraft pulp fiber. They concluded that the stress jumps were attributed to the flocculation of pulp fiber suspensions. Tatsumi et al. [20] reported that suspensions of a cotton cellulose fiber and those of a microfibrillated cellulose fiber made from purified wood pulp showed plateau regions in their flow curves and stated that the stress at the plateau corresponds to the yield stress of the suspensions. The present model can describe the emergence of the plateau region in a flow curve. 4.2 Poiseuille flow Next we considered the Poiseuille flow in a circular tube. The model parameters applied are as follows: α 0 =0.83, α 1 =0.17 β 1 =0.333, β 2 = , De=60, We=2, 5, and 10, and Z=10, 50, and 100. The velocity profiles for We=2, 5, and 10 at De=60 and Z=50 are compared in Fig. 6. The effect of dumbbell orientation on the velocity field via the stress field is intensified as the Weissenberg number increases as is known from Eqs (17) and (4), and hence the profile deviates from the Newtonian profile at high Weissenberg numbers. Considering these results we decided to fix the value of We to 10 in the subsequent analysis. There are small fluctuations in the velocity profile; it is probably due to numerical noises in the computation of Q originating from the random term in Eq (1). We fixed the values of De and We to 60 and 10, respectively and varied the value of the resistance coefficient Z. The resistance coefficient ratio is relevant to the density of network structures, that is the strength of a floc: The strength is stronger for larger Z. Figure 7 shows the velocity profiles for Z=10, 50, and 100 at De=60 and We=10. The profile for Newtonian fluids is plotted by a gray line for comparison. As Z increases, the velocity profile becomes a non- Newtonian profile, which has a flat region near the centerline. This tendency is remarkably seen for Z =50 and 100 because the shearthinning property appears more remarkably for larger Z as shown in Fig. 4. Experimental observations for pulp flows in a circular tube [21] indicate that the velocity profile approaches a plug-like shape as the pulp concentration increases. In general, the orientation behavior of the dumbbells affects the flow field. However, in the Poiseuille flow under the present conditions, the dumbbells were oriented in the flow direction at a relatively high orientation degree at each value of Z except for a 65

6 Yamamoto Takehiro region near the centerline and showed little dependence on Z. Thus we focused on the dumbbell length. Figure 8 shows distributions of lengths of the active and the dangling dumbbells along the r-axis for Z=10, 50, and 100. The dumbbells tend to be highly stretched near the tube wall and it accelerates the transition between active and dangling states. In general, the dumbbells are highly stretched in a high shear rate region near the wall, and the dangling dumbbells are stretched longer. For Z=50 and 100, the length of the dangling dumbbells increases rapidly around r/r=0.5 because of a rapid change in the velocity gradient due to a plug-like velocity profile. Figure 9 shows the fraction of active dumbbells for Z=10, 50, and 100. The fraction n a / N d is close to unity in the entire region of the tube channel. It means that flocs are formed in the entire region. On the centerline, however, the fraction is smaller than the other region because the shear rate is zero and hence the dumbbell length becomes short: according to Eq (6), the association probability decreases when the dumbbell length is short. Near the tube wall, n a / N d decreases for Z=50 and 100 because of the dissociation process occurs more frequently at high-shear-rate regions, where dumbbells are strongly stretched. This tendency is due to the plug-like velocity profile, which shows rapid change in the velocity gradient near the wall. The change in the fraction of active dumbbells depending on the velocity gradient suggests that the repetition of forming and breakup of flocs can be represented by the reversible network model. Considering these results we can conclude that properties of flocforming fluids are represented by setting the value of the resistance coefficient ratio Z be sufficiently large. 5. Conclusion In the present study, we examined the reversible network model as a constitutive model for floc-forming fluids. This model describes both the association and the dissociation of polymer network segment by stochastic processes according to probability functions for the association and the dissociation processes. This 66

7 Journal of Textile Engineering (2014), Vol.60, No.4, approach can be utilized for the forming and breakup behavior of flocs. We examined the applicability of this model to a constitutive model for floc-forming fluids by evaluating rheological properties of this model and by simulating a Poiseuille flow in a circular tube. The reversible network model can describe typical rheological properties of floc-forming liquids such as pulp suspensions, and can qualitatively express the flow behavior of floc-forming fluids in the Poiseuille flow. Consequently this model is applicable as a constitutive model for floc-forming fluids. However, the probability function should be developed considering dynamics of flocs in a flow, and it is a future challenge. For example, the development of the probability function including the effect of the yield stress of pulp fiber suspensions reported in experimental studies [3] will be a first step of the further study. References [1] Cui H, Grace JR (2007) Int J Multiphase Flow, 33, [2] Derakhshandeh B, Kerekes RJ, Hatzikiriakos SG, Bennington CPJ (2011) Chem Eng Sci, 66, [3] Derakhshandeh B, Hatzikiriakos SG, Bennington CPJ (2010) J Rheol, 54, [4] Sundararajakumar RR, Koch DL (1997) J Non-Newtonian Fluid Mech, 73, [5] Fan X, Phan-Thien N, Zheng R (1998) J Non-Newtonian Fluid Mech, 74, [6] Servais C, Luciani A, Ma nson J-AE (2002) J Non-Newtonian Fluid Mech, 104, [7] Switzer III LH, Klingenberg DJ (2003) J Rheol, 47, [8] Ausias G, Fan XJ, Tanner RI (2006) J Non-Newtonian Fluid Mech, 135, [9] Fe rec J, Ausias G, Heuzey MC, Carreau PJ (2009) J Rheol, 53, [10] Lindstro m SB, Uesaka T (2009) Phys Fluids, 21, [11] Yamanoi M, Maia JM (2010) J Non-Newtonian Fluid Mech, 165, [12] Yamanoi M, Maia J, Kwak T-S (2010) J Non-Newtonian Fluid Mech, 165, [13] Yamanoi M, Maia JM (2010) J Non-Newtonian Fluid Mech, 165, [14] Hernández Cifre JG, Barenbrug ThMAOM, Schieber JD, van den Brule BHAA (2003) J Non-Newtonian Fluid Mech, 113, [15] Bird RB, Curtiss CF, Armstrong RC, Hassager O (1987) Dynamics of Polymeric Liquids, Vol.2, John Wiley & Sons, New York [16] Öttinger HC (1996) Stochastic Processes in Polymeric Fluids, Springer, Berlin [17] Van den Brule BHAA, Hoogerbrugge PJ (1995) J Non- Newtonian Fluid Mech, 60, [18] Press WH, Teukolsky SA, Vetterling WT, Flannery BO (2007) Numerical Recipes: The Art of Scientific Computing, 3rd Ed, Cambridge Univ Press, Cambridge [19] Chen B, Tatsumi D, Matsumoto T (2002) Nihon Reoroji Gakkaishi, 30, [20] Tatsumi D, Ishioka S, Matsumoto T (2002) Nihon Reoroji Gakkaishi, 30, [21] Ogawa K, Yoshikawa S, Suguro A, Ikeda J, Ogawa H (1990) J Chem Eng Japan, 23,

8