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Numerical simulation of micromixing effect on the reactive flow in a co-rotating twin screw extruder
Numerical simulation of micromixing effect on the reactive flow in a corotating twin screw extruder Hao Tang, Yuan Zong, Ling Zhao PII: DOI: Reference:
S1004-9541(16)30038-6 doi: 10.1016/j.cjche.2016.04.040 CJCHE 542
To appear in: Received date: Revised date: Accepted date:
20 January 2016 5 April 2016 27 April 2016
Please cite this article as: Hao Tang, Yuan Zong, Ling Zhao, Numerical simulation of micromixing effect on the reactive flow in a co-rotating twin screw extruder, (2016), doi: 10.1016/j.cjche.2016.04.040
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ACCEPTED MANUSCRIPT Numerical simulation of micromixing effect on the reactive flow in a corotating twin screw extruder* TANG Hao (唐豪), ZONG Yuan (宗原), and ZHAO Ling (赵玲)**
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State Key Laboratory of Chemical Engineering, East China University of Science and Technology, Shanghai 200237, China
Abstract To control the multicomponent reactions in extrusion, reactive-mixing flow in a co-rotating twin screw extruder was
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numerically studied in the present paper. Effects of initial species distribution, rotating speed and flow rate on a competitive-
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parallel reaction were investigated and the relationship between mixing and reactions were discussed from the view of chemical reaction engineering. The simulation results show the studied operational parameters, which determine residence time distribution, earliness of mixing and segregation degree of reactive-mixing flows, affect the local species concentration and reaction time and hence have significant influences on the reaction extent. Orthogonal test was adopted to clarify the significance of operational
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parameters. The analysis shows initial species distribution and flow rate are the most important factors in the control of reaction extent, and effect of rotating speed is conditional depending on the micro-mixing status of the fluid.
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Keywords Multicomponent reaction, Mixing, Numerical simulation, Extrusion
Received 0000-00-00, accepted 0000-00-00. * Supported by National Program on Key Basic Research Project (No. 2011CB606100) and the National Natural Science Foundation of China (Grant No.21406059). ** To whom correspondence should be addressed. E-mail address: [email protected] (L. Zhao)
ACCEPTED MANUSCRIPT 1 INTRODUCTION The co-rotating twin-screw extruder (CoTSE) is very attractive reactor for reactive extrusion (REX) due to
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its self-wiping performance and modular design characteristic. The reactions in a CoTSE can be classified into two categories: single component reaction, such as bulk polymerization, and multicomponent reaction, such as
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synthesis of urethanes, co-polymerizations and grafting reactions, etc. For the former, studies were mainly
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focused on the macromixing and temperature in the extruder[1, 2]. For the latter, in most cases, multicomponent reactions take place on the interface between striation structure and its area is increased with deforming and reorientation during laminar mixing[3]. Once the mixing time has the equivalent magnitude with reaction times or longer than that, incomplete local mixing of the reactants will significantly affects the course of reaction or
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even makes the conversion independent on Damköhler number[4, 5]. Therefore, detailed information of the mixing phenomena is important for the control of reaction in REX process.
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From a view of hydrodynamics, the essence of the multicomponent reactive extrusion is a combination of mass and momentum transfer in laminar regime. Like other reaction process, mixing between reactants is the
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precondition for the occurrence of reaction. To reduce the computing load, traditional approaches simplify the
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geometry of extruder into ideal chemical reactor models based on chemical reaction engineering theory, and employ the residence time distribution (RTD) as a measure of axial mixing performance. While this measure is not very useful for the laminar mixing of viscous fluids because it provides little information on transverse mixing[6], which is of more interest for the micromixing of reactive species. Since reaction occurs at the interface between striations, pioneering micromixing work of Ottino proposed a lamellar mixing model with alternating lamellar structure in the modeling of solution for concentrations field with multicomponent diffusion and chemical reaction[7, 8]. In this model, the striation thickness and average deformation rate were utilized as a measure of mixing state[9-11]. This approach is proved feasible in the analysis of relationship between molecular weight distribution and micromixing in a CoTSE[12]. However, the lamellar micromixing model simplifies the mixing as one-dimensional convection and diffusion problem because of the complexity in describing flow dynamics in the screw channel[13]. With the development of computational capabilities, approaches based on computational fluid dynamics (CFD) theory allows to solve these reactive-mixing problems in a three-dimensional way[14]. Thus, more realistic flow
ACCEPTED MANUSCRIPT patterns can be obtained[15]. Various simulations have been reported for the reactive extrusion flow in recent decades, such as peroxide-initiated degradation of polypropylene and polymerization of
-caprolactone in
CoTSE[16-19]. While most of these investigations were only concerned with macromixing and research on
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micromixing was rarely reported. It is reported the inadequate mixing between the polymers and initiators can significantly influence on the molecular weight distribution (MWD)[20], which was ubiquitous in
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multicomponent reactions[2]. Consequently, micromixing behavior should also be taken into consideration in the
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three dimensional simulation of REX.
Generally, there lies three decisive factors for the reactive-mixing process, i.e., macromixing state, which can be measured by RTD, degree of segregation, and earliness/lateness of mixing[21]. In the present paper, numerical simulation with three-dimensional mixing analysis was performed for the REX process. Effect of
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operational parameters, including initial species distribution, screw rotating speed and specific throughput, on product quality were numerically investigated and their relationships with these three factors were discussed.
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The aim of this study is to get a grasp on the REX process from the view of chemical reaction engineering and
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find the decisive factors in the controlling of product quality.
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2 NUMERICAL SIMULATION 2.1 Model reactions
Competitive-parallel reaction has been widely used in the investigation of micromixing in reactor[5, 11, 22]. In order to obtain realistic results in extruders, the competitive-parallel polymerization of polyurethane is employed in this work. The modeling reactions are: k1
(a)
A B C k2
(b) A D E where species A is –R-OH (1,4-butane diol, BDO), species B is –R’-NCO, i.e., the prepolymer of dicyclohexylmethane 4,4’-diisocyanate (H12MDI) and polyester (Mn= 1000 g·mol-1), species D is –R”-NCO, i.e., the prepolymer of 4,4’-diphenyl methane diisocyanate (MDI) and polyester (Mn= 1000 g·mol-1)[23, 24]. k1 and k2 are the rate constants of reaction (a) and (b), respectively. Both reactions are second-order reactions with suitable catalysts. According to k=5.45×109[Cat]exp(-Ea/RT), where [Cat]= 4.75×10-4 mol·L-1 and Ea=48.5 kJ·mol-1
ACCEPTED MANUSCRIPT (Ref[23]), the reaction constant is in the range of 10-1~10-2 L·mol-1·s-1. Thus, k1= 0.16 L·mol-1·s-1 and k2= 0.016 L·mol-1·s-1 are set in the model.
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2.2 Simulation methods
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In practical process, the barrel temperature is usually kept at a constant to ensure stable extrusion. By this way, the temperature rise in reactive extrusion becomes small and its effect on the process can be neglected.
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Thus, incompressible non-Newtonian flow, pseudo-steady state and isothermal condition are assumed for the numerical simulation. The governing equations are as follows:
Navier-Stokes equation:
U 0
(1)
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Continuity equation:
(2)
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p UU
[(U ) (U )T ]
(3)
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where p is the pressure in extruder, U is the local velocity. η is the viscosity of system. The constitutive equation
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of reaction system is regressed by literature data[23]:
0 [1 ( ) a ](1 n ')/ a
(4)
where a 1.24 is the Yasuda index, n ' 0.42 is the power law index, 0.014 s is the relaxation time constant and
is the shear rate.
The molecular weight (MW) of prepolymer is 7.2 kg·mol-1, at that range, the zero-shear viscosity of reaction system is[25]:
0 1.24 104 MW3.45
(5)
In practical polymerization, zero-shear viscosity is yielding 0.1~105 Pa·s. During the simulation in this work, it ranges from 0.1~10 Pa·s. That makes Reynolds number ranges from 0.02~2. The mass balance of species i gives the following convection-diffusion equation: ( i U ) ( Di i ) Ri
(6)
ACCEPTED MANUSCRIPT where ρ is the density of mixture, ωi and Di are the mass fraction and the molecular diffusivity for species i in the mixture, and Ri represents the net mass rate of reaction(s) for species i, which could be estimated as follows: j
Ri M i ij rj
(7)
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j 1
where Mi is the mole mass of species i, υij is the stoichiometric coefficient of the i-th species in the j-th reaction.
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For reactants, υij<0, for products, υij>0, and rj is the molar reaction rate of reaction j.
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Divide both sides of Eq.(6) by Mi, the laminar diffusion equation of species concentration can be reformed into:
j
(ci U ) ( Di ci ) ij rj
(8)
j 1
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By solving the above equations, the characteristics of the flow regime and species transport can be obtained. The
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dimension of ci in Eq.(8) is [mol·m-3], and we convert it into [mol·L-1] for convenience.
2.3 Simulation details
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Figure 1 illustrates the geometrical features of three-dimensional model. The origin of coordinates is at the center of one of the screw inlets, and the extrusion direction is along the negative Z-axis. Screw elements are
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right-handed with 40 mm pitch and 60 mm length. For laminar flow, the length of develop section is Ldevelop=0.05Re*D≈4 mm[26]. Thus, two 10 mm long flow domains without screws are added before the inlet and after the outlet separately to ensure the flow is fully developed. In the present simulation, two feeding streams are introduced into extruder through inlet 1 and inlet 2 separately. Two planes are chosen as sample planes, i.e., plane y/D= 0.38 in the middle of the screw chamber and plane |Z|/L= 0.75 approaching the outlet, where D= 42 mm is the barrel diameter and L= 80 mm is the axial length of flow domain. Besides, a sample line from (0, 16 mm, 0) to (0, 16 mm, -80 mm) is created to investigate the local properties along the axial distance.
(a) (b) Figure 1 Geometrical features of simulated screw (a) constrains of screws (b) three-dimensional model
ACCEPTED MANUSCRIPT Both inlets have the same flow rate. Different species concentrations are given for both inlets in order to investigate the reaction extent in accompany with convection-diffusion process. In practical process, the initial concentration of diol or diisocyanate is 3.12 mol·L-1)[23-25]. Thus, the concentrations of -OH and -NCO are set
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as 2 mol·L-1 with consideration of perpolymerization before reactive extrusion. The feeding streams are shown in Table 1. In this way, generation of a lamellar structure for species concentrations can be displayed during
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extrusion.
Reactant concentrations imposed on two inlets
cA1
Inlet 1 cB1
① ② ③ ④
2 2 1 2
0 0 1 2
cD1 cA2 / mol·L-1 0 0 2 0 1 1 2 0
Inlet 2 cB2
cD2
2 2 1 0
2 0 1 0
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Feeding condition
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Table 1
No-slip condition is imposed on the screws and barrel surfaces. The barrel is static and the screws are set as
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moving wall at certain rotating speed. The simulation work is performed on commercial CFD platform POLYFLOW based on finite element method. To eliminate the possible negative volume meshes arising due to
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screw rotating, the mesh superposition technique (MST) was implemented: fluid mesh elements obey the conservation equations, while the mesh elements occupied by moving screws are forced to rotate as rigid
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bodies[27]. Thus, three-dimensional finite element meshes are constructed covering moving parts and flow domain (Fig.2), the flow domain is discretized into 48 k hexahedral elements and each screw is divided into 40 k hexahedral elements. All the properties including velocity, species, pressure and viscosity, etc. are coupled with each other in the simulation to ensure convergence. The convergence of calculation is achieved until the residuals are less than 0.0001. Typical CPU time for an FEM task on a workstation with 8 processors is around 1.6×104 s. The mixing task is implemented with particle tracking analysis. The trajectories of 5000 marker particles are calculated per case in order to ensure the accuracy of the statistical results[28, 29]. All these particles are volume less, massless and independent from each other[30]. Initially, these particles are randomly released at the inlet plane. The particles released at inlet 1 and inlet 2 can be distinguished with different colors, seen in Figure 3. In the computational code, these particles are assigned as "1" and "0", respectively. Then their trajectories are calculated from the flow field. By this way, the material deformation can be estimated via the displacements of particle pairs, and the RTD can be obtained by the statistics of particle lifetime at outlet plane. Typical CPU time for a mixing task on a workstation with 8 processors is 1.4×103 s.
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(a) (b) Figure 2 Finite element meshes of (a) screws and (b) flow domain
2.4 Characterization of micromixing
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Figure 3 Initial distribution of particles
Convection, diffusion and reaction determines the effect of mixing on the course of chemical reaction[6].
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Thus, the characteristic reaction time and mixing time need to be evaluated. The characteristic reaction time is defined as[10]:
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tR j 1/ k j ci 0 ( j 1, 2)
(9)
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where ci 0 1 mol·L-1 is the initial average concentration of species i. For the main reaction, tR 6.25 s , and for 1
the side reaction, tR 62.5 s . 2
The characteristic micromixing time with diffusion mechanism is defined as[10]: tM arc sinh(0.76 s02 / D) / (2 )
(10)
where α is the deformation rate of striation and s0 is the initial striation thickness. The micromixing time in the present simulation of the order of magnitude 1~10 s. Apparently, the micromixing time has similar magnitude to the characteristic main reaction time, indicating the significance of spatial mixing performance on the reactions. When multiple reactions take place between two reactant fluids and once these reactions proceed to a considerable extent before homogeneity is attained, product distribution will be affected by the segregation status of the flow[31]. Here we adopt segregation scale and segregation intensity to characterize the mixing flow. The former measures the region size of homogeneous concentration and the latter reflects the spatial uniformity of the concentration[32, 33]. Reduction of segregation scale and segregation intensity represent the
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Ls R( r )d r
(11)
0
where cP c"j cP
M
j 1
M c2 NP
P 1
2
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c2
(cP cP )
(12)
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R r
' j
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c
NP 1
(13)
R(|r|) is the correlation coefficient between concentration of pairs of particles separated by |r| and M=NP(NP-1)/2
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is the number of pairs, where NP is the number of particles, and σ2c is the sample variance. The c 'j and c"j denote the concentrations of both particles in the j-th pair. cp is the concentration of a certain particle which equals to either 0 or 1 and remains constant since the diffusion vanishes.
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The segregation intensity Iseg is the standard deviation of the concentration around its mean, and is defined
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as follow[33, 34]:
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I seg
x2 02
N
(x j x j ) j 1
N 1
2
02
(14)
where σ20 is the variance of a completely segregated system and σ2x is the measured variance. xj denotes the concentration of j-th element in the extruder and N is the number of elements. The concentration of these elements will be homogenized due to convection and diffusion effect. Another important factor in micromixing is the earliness/lateness of mixing, i.e., whether fluid mixes early or late as it flows through the vessel. For multicomponent reaction, the mixing of reactant is the prerequisite for reaction, which is relate to the premixed condition. In the simulation, the earliness/lateness of mixing is controlled by the reactant concentrations imposed on two inlets (seen in Table 1).
2.5 Characterization of reaction In order to quantify the reaction extent, firstly, the area-averaged features of i-th species on a plane is defined as[19]:
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N
ci
c Aj
j 1 ij N j 1
(15)
Aj
where Aj and cij are the area and the concentration of species i at cell j.
cC cC 0 cA0 c A
cC
(16)
c A0 c A
(17)
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where, cA0 1 mol·L-1 and cC 0 0 .
2.6 Evaluation of simulation results
c A0
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c A0 c A
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x
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Conversion x and selectivity of species A can be calculated by following equations:
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The evaluation of simulation results includes two aspects: verification and validation[36]. Verification evaluates the uncertainty in numerical simulation. Here, mesh independency was checked. Validation
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means comparison between simulation and experiment. To prove the versatility of numerical methods, RTD and species distribution were compared with different sizes of extruders.
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2.6.1 Verification
In the simulation, mesh size is extremely important to ensure the accuracy of species transportation. Fig.4 shows the meshes at a cross section normal to the extrusion direction with three mesh sizes in mesh independency check. The total elements quantities are 41 k, 88 k and 190 k for scheme a, b and c, respectively.
Figure 4 Meshes at a cross section normal to the extrusion direction.
Feeding ① with Q/n=7000/30 were employed in mesh check, where the reactants are initial segregated in different inlets, as seen in Table 1. Fig.5a illustrates the velocity on the sample line and Fig.5b shows the evolution of average conversion along axial distance with three mesh sizes. The curves of mesh (b) and (c) are
ACCEPTED MANUSCRIPT very close in both figures, while perceptible difference could be recognized between results from mesh (a) and others. With compromising consideration of computational load and accuracy, mesh size (b) is chosen as an
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optimized mesh for the simulation work.
(a)
(b)
Figure 5 Mesh independency check (a) velocity distribution on sample line 1. (b) average conversion evolution along the axial distance
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2.6.2 Validation
In order to test the capabilities and accuracy of numerical methods, RTD of a flow through KD2 structure
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(two pairs of 60/4/32 kneading discs with D=35 mm) is simulated and compared with that in literature[30], as shown in Fig.6a. In Zhang’s experiment[30, 37], the test elements were installed in the mixing section of
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extruder and two probes were located at the upstream and downstream. Anthracene was chosen as the tracer and the in-line fluorescent light measuring system was developed to measure the local RTD at probe locations. Then
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the RTD between them was calculated by a deconvolution method[38]. The experimental material is polystyrene and its constitutive equation can be represented by Carreau model[30]:
2814[1 (0.148 )2 ](0.2781)/2
-6
3
(18)
-1
The flow rate is 3.06×10 m ·s and rotating speed is 120 rpm in the simulation according to literature[30]. A good agreement is found in the comparison of both work (Fig.6b). And the relative error is less than 5%.
(a) (b) Figure 6 Validation of simulation on RTD (a) Elements geometry (b) RTD curves
Besides, distribution of species is also validated by comparing the deformation of striation between the simulation work and the experiment work of Kalyon[39]. The geometry of screw elements are full flight
ACCEPTED MANUSCRIPT elements, diameter and pitch in 50.8 mm, and the operational conditions are identical to literature[39] in the simulation. Fig.7a shows the initial position of striation. Fig.7b and 7c illustrate simulated and experimental shapes of striation after half revolution of screws, respectively. The agreement between those shapes seems fine.
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The above validations indicate that the employed numerical method is feasible and the parameters defined in
(b) Figure 7 Spatial distribution of tracer in extruder after 1/2 revolution (a) initial position (b) simulated result (c) experimental result
(c)
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numerical analysis are reasonable.
3 RESULTS AND DISCUSSION
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3.1 Flow pattern in CoTSE
In twin screw extruders, the distribution of fluid elements is improved due to relative movement between
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both screws and barrel, which have significant influence on flow behavior. In order to trace the transfer movement of species, species A is introduced into inlet 1 with a concentration of 2 mol·L-1. The screw rotating
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speed is 30 rpm and both inlets have the same flow rate of 3500 mm3·s-1. Fig.8 gives the distribution evolution of species A at different axial locations. Obviously, it can be seen that the flow is transferred from one screw to the other because the other screw picks up the material and drags it away from intermeshing region. As a result, species A is delivered in a “
” pattern with helical flow and new material interfaces are generated with each
screw revolution[40, 41]. With the development of material exchange and accumulation of the mixing effect, the distribution of concentration gradually becomes uniform along the extrusion direction as seen after plane |Z|/L= 0.75.
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Figure 8 Distribution of species A along extrusion direction at n= 30 rpm
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3.2 Effect of feeding condition
For continuous process, the initial species distribution is determined by the feeding condition. In order to
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clarify the interaction between flow field and reactions, here the species are fed through different inlets, which can be regarded as completely separated condition[42-44]. Table 1 shows the initial species distributions of
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cases.
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various feedings. The flow rate is set as 3500 mm3·s-1 for each inlet and rotating speed equals to 30 rpm for all
Figure 9 illustrates the concentration distribution of main product (C) and side product (E) on plane y/D= 0.38 with various feedings. The results clearly illustrate the sensitivity of species C and E to the premixing status of their relevant reactants. It is noticed that the concentration of species E is improved in Fig.9b compared with that in Fig.9a, indicating prior mixing between species A and D is beneficial to the side reaction. The same observation can be made for the concentration distribution of species C and E in Fig.9c and Fig.9d compared with that in Fig.9a, which implies the initial homogeneous distribution of the reactants can simultaneously promote the main and side reactions to a certain extent.
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Figure 9 Distributions of species C (left) and E (right) on plane y/D= 0.38 with (a) feeding ① (b) feeding ② (c) feeding ③ (d) feeding ④
Table 2 provides reaction results on plane |Z|/L= 0.75. From the table, it can be seen that the lowest averaged conversion is found in the case with feeding condition ①, this is caused by the initial condition of species. In that condition, species A is completely separated from B and D, and the insufficient mixing between these reactants can decrease the reaction rate. Higher conversion and lowest selectivity under feeding condition ② is caused by the only premixing of species A with species D at the inlet, hence the side reaction is enhanced while the main reaction is suppressed. Better mixing of the reactants under feeding condition ③ and ④ leads to higher reaction conversions. And the highest local reactant concentrations in the case with feeding condition ④ are responsible for the highest conversion. Moreover, it is also observed that the selectivity of the cases with
ACCEPTED MANUSCRIPT feeding condition ①, ③ and ④ are very close. This can be attributed to the fact that the mixing of species B and D are identical in these cases. Table 2 Average concentrations, selectivity and conversion on plane |Z|/L= 0.75 / mol•L 0.436 0.069 0.395 0.212 0.634 0.099 0.696 0.117
x
0.864 0.651 0.865 0.856
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cE
-1
0.505 0.607 0.733 0.813
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① ② ③ ④
cC
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Feeding
Figure 10 gives the yield of species C (Φ) along the extrusion direction, which is the product of selectivity and conversion. As far as feeding condition ③ and ④ are concerned, earlier mixing between reactants results in a jump in the yield of species C, and then the growth is gradually weakened approaching the outlet since the
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reactant A is exhausted. The curves with feeding condition ① and ② are almost overlapped in the inlet section due to their low conversions. Downstream, the differences between the two curves become larger attributing to
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their distinguishing selectivity.
Figure 10 Effect of feeding condition on yield of species C
The above discussion reveals different micromixing behaviors of the fluids. Flow through the inlets at feeding condition ①, ② and ④, is often called “macrofluid”, where fluids are completely segregated. When identical flow through both inlets, like feeding condition ③, is called “microfluid”, and fluids will be in the state of maximum mixedness during being transported. These two groups of fluids restrict earliness/lateness of mixing, which is also a prominent issue for other reactors. The above observations confirms that earliness/lateness of mixing has a determined effect on the local concentration distribution, which can further affect the conversion and selectivity for these multicomponent reactions in CoTSE. It also provides an evidence that attentions should be paid to the feeding condition in order to achieve the desired reaction results in the REX process.
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3.3 Effect of rotating speed Figure 11 illustrates the product distributions at various screw rotating speeds with feeding condition ②.
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The flow rate is fixed at 3500 mm3·s-1 per inlet. With increasing rotating speed, the concentration of species C is
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while the side reaction has been suppressed on the contrary.
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improved while the concentration of species E is decreased, indicating the main reaction has been enhanced
Figure 11 Distributions of species C (left) and E (right) on plane y/D= 0.38 at (a) 20rpm (b) 30rpm (c) 40rpm
Figures 11a and 11b show the selectivity and conversion evolution along the axial direction at various rotating speeds. In order to illustrate the mixing effect on reaction, fully premixed case with feeding condition ③ and rotating speed of 30 rpm is provided for comparison. The reaction conversion and selectivity along the extrusion direction are found increasing with the increase of screw rotating speeds, indicating higher screw rotating speed is beneficial to the proceeding of reaction under the given feeding condition. Besides, apparent distinct can be noticed between premixed case and non-premixed cases. The selectivity for the premixed case keeps a high value in the all range, which proves the importance of reactant segregation on the reactions. Here define “product gap” to demonstrate the difference of yields between premixed and segregated feeding:
ACCEPTED MANUSCRIPT premix
(19)
where premix is the yield of the case that with premixed feeding. The yield difference along the extrusion direction is shown in Fig.12c. At the beginning of extrusion, the rise of target product in the microfluid is larger
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than that in the macrofluid due to the earliness of mixing between reactants. With the development of flow and
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accumulation of mixing effect, this difference reaches its maximum. Then, the gap between mixed and nonmixed cases declines with the consumption of reactants. The smaller rotating speed, the bigger gap is found.
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RTD is typically employed as a measure of a kinematic description of flow, to understand the axial mixing in the reactor. Fig.12d describes the local RTD at plane |Z|/L= 0.75 at different speeds. There is no obvious effect of rotating speed on the RTD profile shape. Moreover, the mean residence time is found slightly reduced,
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which is 11.58 s, 10.56 s and 9.64 s with the increasing of rotating speed. Change to the results of reaction extent, it is clearly insufficient to analyze the reactive-mixing flow only by RTD. Because RTD is only a good measure for macromixing and becomes invalid in the characterization of micromixing. Hence, statistical
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description of mixture is processed to analyze the degree of segregation in the extruder. Fig.12e shows the evolution of segregation scale along extrusion direction. The segregation scale presents a sharp decline in all
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curves near the entrance section, illustrating the region size with homogeneous concentration is decreased
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rapidly. It is also found the segregation scale oscillates in the remaining section, which may be caused by the stretching, folding and reorientation of the striations during extrusion[45]. Approaching the outlet, the segregation scale at higher speed is found smaller than that at lower speed. Fig.12f shows the evolution of segregation intensity along extrusion direction. All curves show a decreasing trend. The higher the screw rotating speed, the quicker the curve goes down. Lower value of segregation intensity is obtained at higher speed near the outlets. It shows that rotating speed is more relevant to micromixing rather than macromixing and higher rotating speed can achieve more uniform distribution of species in the extruder. The results also illustrate the scalar mixing analysis is a necessary supplement for the characterization of reactive-mixing in REX.
(a)
(b)
(c)
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(e) (f) Figure 12 Effect of rotating speed on (a) selectivity (b) conversion (c) yield difference (d) RTD (e) segregation scale and (f) segregation intensity
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(d)
3.4 Effect of specific throughput
It has been reported that the specific throughput, defined as the ratio between flow rate Q and rotating speed n, has significant influence on partial as well as overall residence time/revolution/volume distribution[37].
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However, not much is mentioned about its effect on reaction. Since the REX process is determined by a combination of mixing, reaction kinetics and residence time in the channel of fluid element, it is of interest to
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figure out the relationship between specific throughput and reactions. Figure 13 provides the species distribution of C and E at y/D= 0.38 at the same specific throughput with
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feeding condition ②. It can be seen that both the concentrations of species C and E in the whole extruder are
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reduced with increasing flow rate, which is due to the decrease of residence time.
ACCEPTED MANUSCRIPT Figure 13 Distributions of species C (left) and E (right) on plane y/D= 0.38 at specific throughput of (a) 3500/15 (b) 7000/30 (c) 10500/45
Figures 14a and 14b show the selectivity and conversion evolution along the extrusion direction. The result of
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conversion indicates that reactions are suppressed with increasing flow rate. However, the reaction selectivity is increased with increasing screw rotating speed. There may be two possible reasons behind this phenomenon: (1)
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The local micromixing status in extruder may not change at the same specific throughput, while the time to
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achieve the same micro-mixing status is shortened with higher flow rate. Thus, the side reaction is suppressed. (2) The micromixing of reactive flow is enhanced by the increasing rotating speed with identical specific throughput. The yield difference along the extrusion direction is shown in Fig.14c. All curves show a trend from increase to decline as except. Moreover, the yield difference is decreasing with higher flow rate. That is because
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shorter residence time leads to shorter reaction time, and hence the yields with premixed and segregated feeding are decreased.
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To confirm the true reason behind the improvement of reaction selectivity, here we analyze the effect of specific throughput on the mixing performance of flow firstly. With the increasing of flow rate and rotating
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speed while the specific throughput remains constant, the RTD curve shifts to the left and becomes narrower as
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expected, as seen in Fig.14d. The mean residence times t are 21.40 s, 10.56 s and 7.14 s with increasing flow rate. As the mixing performance is concerned, it is interesting to find the curves of segregation scale and segregation intensity are almost superimposed on a single master one (Fig.14e and 14f). Therefore, we can conclude that the same specific throughput bring about differences in the duration of mixing while identity in micro-mixing status. And it gives the answer that the improved selectivity is mainly attributed to the shortened reaction time rather than the mixing performance. Combined with the results in Section 3.3, it confirms that the mixing is closely dependent on the specific throughput instead of simply rotating speed, and the reaction extent is the consequence of both mixing and residence time.
(a)
(b)
(c)
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(e) (f) Figure 14 Effect of specific throughput on (a) selectivity (b) conversion (c) yield difference (d) RTD (e) segregation scale and (f) segregation intensity
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(d)
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3.5 Reactive-mixing flow in extruder
To compare the significance of the effects of feeding, rotating speed and flow rate on reactions, an orthogonal test is performed. Each factor has three levels and the interaction of feeding condition and rotating
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speed is considered (Table 3). The results of orthogonal test are listed in Table 4. Columns 6 and 7 show level errors, the deviations are quite small. While, the same outcomes are found in columns 3 and 4, indicating the
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interaction between feeding condition and rotating speed can be neglected, column 3, 4, 6 and 7 could be regard as level error columns. It can be concluded that sequence of significance of these factors on the reaction yield is
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feeding condition > flow rate > rotating speed, based on the range and sum of square of deviations analysis. The F examination tells both the differences of flow rate and feeding condition are highly significant (P<0.01), and
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the difference of rotating speed is also remarkable (P<0.1).
Table 3 Factors and levels in orthogonal test Factors Level 1 Level 2 Feeding condition A ① ② Rotating speed (rpm) B 15 30 Flow rate (mm3·s-1) C 3500 7000
Level 3 ③ 45 10500
Note: the interaction of A and B is also considered
Table 4 The results of orthogonal test Test number 1 2 3 4 5 6 7 8 9 10
Factors and levels A B 1 2 1 1 1 2 1 3 2 1 2 2 2 3 3 1 3 2 3 3 1 1
Yield A 3 1 2 3 1 2 3 2 3 1 3
×
B 4 1 2 3 2 3 1 1 2 3 3
C 5 1 2 3 2 3 1 3 1 2 2
6 1 2 3 3 1 2 2 3 1 2
7 1 2 3 3 1 2 3 1 2 1
0.552 0.436 0.394 0.265 0.355 0.701 0.547 0.722 0.619 0.271
ACCEPTED MANUSCRIPT 1 2 2 3 1 3 1 2 0.473 0.518 0.505 0.045 0.007
1 2 3 1 2 2 3 1 0.524 0.500 0.473 0.051 0.008
3 1 1 2 3 3 1 2 0.642 0.434 0.421 0.221 0.185 24.68
3 1 3 1 2 1 2 3 0.524 0.505 0.467 0.057 0.010
2 3 2 3 1 2 3 1 0.479 0.518 0.500 0.039 0.005
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2 3 1 2 3 1 2 3 0.443 0.493 0.561 0.103 0.042 5.60
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1 1 2 2 2 3 3 3 0.443 0.424 0.629 0.205 0.154 20.53
0.327 0.678 0.476 0.395 0.353 0.547 0.722 0.619
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11 12 13 14 15 16 17 18 K1 K2 K3 R S F
Note: R and S represent range and sum of square of deviations, respectively, and F is the value of F examination
It should be pointed out screw rotating speed may have different effects on reaction according to the micro-
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mixing behavior of the fluid. For macrofluid, increasing rotating speed enhance the micro-mixing, which can compensate the shortening of residence time and results in a higher reaction extent; for microfluid, increasing rotating speed only reduce the reaction time, leading to lower reaction extent. The orthogonal test proves when
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the screw rotating speed is low, microfluid status is more beneficial for the reaction because macrofluid takes
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long time for micro-mixing. When the screw rotating speed is extremely increased, higher than the studied speed, the micro-mixing time may be greatly shortened and the significance of rotating speed can be
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strengthened. However, in order to ensure enough reaction time, lower rotating speed is more reliable. In brief, the highly significant differences of feeding condition and flow rate on reaction yield can be attributed to their relationship with earliness of mixing and residence time, respectively. The effect of rotating speed relies on the micro-mixing status of the fluid.
4 CONCLUSIONS Numerical modeling of reactive extrusion systems is of great interest since it provides an accessible way to choose favorable operational conditions from a practical point of view. In the present paper, effects of initial species distribution, screw rotating speed and specific throughput on the multicomponent reaction were numerically investigated. In accordance with the above discussions and analysis about their effects and interaction, the following conclusions are drawn. The results show that RTD alone is inadequate to characterize the mixing status in REX process. Micromixing status of fluids, i.e., microfluid and macrofluid, which controls earliness/lateness of mixing, has the
ACCEPTED MANUSCRIPT highest priority in affecting the flow field and the product quality. The analysis of segregation degree, including segregation scale and segregation intensity, shows at the same flow rate, effective mixing resulted from higher screw rotating speed can compensate the shortening of residence
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time and promote the proceeding of reactions with macrofluid feeding. On the contrary, prominent effect of increasing screw rotating speed decreases the reaction time with microfluid feeding.
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The relationship between flow rate and rotating speed is identified by specific throughput. The mixing
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performance keeps unchanged at the same specific throughput, while higher flow rate and rotating speed lead to significant decrease of products due to the reduction of reaction time.
In order to discriminate the significance of these operational conditions on the reaction, orthogonal test is performed and the results confirmed the initial species distribution and flow rate play decisive roles in the yield
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controlling.
Aj
area of cell j
a
Yasuda index
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NOMENCLATURE
concentration of species i at inlet 1 and 2, mol·L-1
c 'j , c"j
concentrations (volume fractions) of particles in the j-th pair
cp
concentrations (volume fractions) of particles
D
barrel diameter, mm
Di
molecular diffusivity, m2·s-1
Iseg
segregation intensity
kj
second order reaction rate constants, L·mol-1·s-1
L
axial length of flow domain, mm
Ls
segregation scale, mm
M
number of particle pairs
Mi
mole mass of species i, kg·mol-1
N
number of elements
NP
number of particles
n
rotating speed, rpm
n’
power-law index
p
pressure, Pa
Q
flow rate, mm3·s-1
Ri
net mass rate of species i, kg·m-3·s-1
R(|r|)
correlation coefficient between concentration of pairs of points separated by |r|
rj
molar reaction rate of reaction j, mol·m-3·s-1
s0
initial striation thickness, m
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ci1, ci2
mean residence time, s
tRi
characteristic reaction time of reaction j, s
tM
micro-mixing time with diffusion mechanism, s
U
velocity vector, m·s-1
V
volume of flow domain in an extruder, mm3
x
conversion of species A
xj
concentration (volume fraction) of the j-th element in extruder
α
deformation rate of striations, s-1
β
selectivity
η
viscosity, Pa·s
ρ
density, kg·m-3
σ20
variance of a completely segregated system
σ
variance in calculation of segregation scale
2 c
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t
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ACCEPTED MANUSCRIPT
variance in calculation of segregation intensity
υij
stoichiometric coefficient of the i-th species in the j-th reaction
τ
relaxation time index, s
ωi
mass fraction
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σ2x
1 Berzin F., Tara A., Tighzert L., Vergnes B., Importance of coupling between specific energy and viscosity in the
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modeling of twin screw extrusion of starchy products, Polym. Eng. Sci., 50 (9) (2010) 1758-1766. 2 Ganzeveld K. J., Janssen L. P. B. M., Role of mixing and rheology in reactive extrusion, Ind. Eng. Chem. Res., 33
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ACCEPTED MANUSCRIPT 11 Rożeń A., Bakker R. A., Bałdyga J., Application of an integral method to modelling of laminar micromixing, Chem. Eng. J., 84 (3) (2001) 413-428. 12 Iedema P. D., Remerie K., van der Ham M., Biemond E., Tacx J., Controlled peroxide-induced degradation of polypropylene in a twin-screw extruder: Change of molecular weight distribution under conditions controlled by
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14 L. Marchisio D., A. Barresi A., CFD simulation of mixing and reaction: The relevance of the micro-mixing model,
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17 Ortiz-Rodriguez E., Tzoganakis C., A 3d simulation analysis of reactive flow in screw elements of closely intermeshing twin screw extruders, Int. Polym. Proc., 27 (4) (2012) 442-451. 18 Zhu L. J., Narh K. A., Hyun K. S., Investigation of mixing mechanisms and energy balance in reactive extrusion
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Tech., 24 (3) (2005) 183-193.
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reactor, Chem. Eng. Sci., 56 (23) (2001) 6589-6603. 21 Smit D. J., Paterson W. R., Hounslow M. J., Aggregation and gelation—II. Mixing effects in continuous flow vessels, Chem. Eng. Sci., 49 (18) (1994) 3147-3167. 22 Han Y., Wang J. J., Gu X. P., Feng L. F., Numerical simulation on micromixing of viscous fluids in a stirred-tank reactor, Chem. Eng. Sci., 74 (2012) 9-17. 23 Cassagnau P., Nietsch T., Michel A., Bulk and dispersed phase polymerization of urethane in twin screw extruders, Int. Polym. Proc., 14 (2) (1999) 144-151. 24 Cassagnau P., Mélis F., Michel A., Correlation of linear viscoelastic behavior and molecular weight evolution in the bulk urethane polymerization, J. Appl. Polym. Sci., 65 (12) (1997) 2395-2406. 25 Cassagnau P., Nietsch T., Bert M., Michel A., Reactive blending by in situ polymerization of the dispersed phase, Polymer, 40 (1998) 131-138. 26 Hunger M., Hüsken G., Brouwers H. J. H., Photocatalytic degradation of air pollutants — from modeling to large scale application, Cem. Concr. Res., 40 (2) (2010) 313-320. 27 Vyakaranam K. V., Ashokan B. K., Kokini J. L., Evaluation of effect of paddle element stagger angle on the local velocity profiles in a twin-screw continuous mixer with viscous flow using finite element method simulations, J. Food Eng., 108 (4) (2012) 585-599.
ACCEPTED MANUSCRIPT 28 Yang K. X., Xin C. L., Yu D. Q., Yan B. R., Pang J. J., He Y. D., Numerical simulation and experimental study of pressure and residence time distribution of triple-screw extruder, Polym. Eng. Sci., 55 (1) (2015) 156-162. 29 Hirata K., Ishida H., Hiragohri M., Nakayama Y., Kajiwara T., Effectiveness of a backward mixing screw element for glass fiber dispersion in a twin-screw extruder, Polym. Eng. Sci., 54 (9) (2014) 2005-2012.
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Polym. Tech., 8 (4) (1988) 337-353.
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ACCEPTED MANUSCRIPT Graphic abstract Feeding condition and rotational speed determine the local concentrations of reactants by controlling the earliness/lateness of mixing and the degree of segregation, respectively. Besides, rotational speed and flow rate also have effect on the RTD and duration of the reaction. The combination of local concentration and time accumulation restricts the reaction extents of
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multicomponent reactions during extrusion.
S1004-9541(16)30038-6 doi: 10.1016/j.cjche.2016.04.040 CJCHE 542
To appear in: Received date: Revised date: Accepted date:
20 January 2016 5 April 2016 27 April 2016
Please cite this article as: Hao Tang, Yuan Zong, Ling Zhao, Numerical simulation of micromixing effect on the reactive flow in a co-rotating twin screw extruder, (2016), doi: 10.1016/j.cjche.2016.04.040
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ACCEPTED MANUSCRIPT Numerical simulation of micromixing effect on the reactive flow in a corotating twin screw extruder* TANG Hao (唐豪), ZONG Yuan (宗原), and ZHAO Ling (赵玲)**
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State Key Laboratory of Chemical Engineering, East China University of Science and Technology, Shanghai 200237, China
Abstract To control the multicomponent reactions in extrusion, reactive-mixing flow in a co-rotating twin screw extruder was
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numerically studied in the present paper. Effects of initial species distribution, rotating speed and flow rate on a competitive-
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parallel reaction were investigated and the relationship between mixing and reactions were discussed from the view of chemical reaction engineering. The simulation results show the studied operational parameters, which determine residence time distribution, earliness of mixing and segregation degree of reactive-mixing flows, affect the local species concentration and reaction time and hence have significant influences on the reaction extent. Orthogonal test was adopted to clarify the significance of operational
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parameters. The analysis shows initial species distribution and flow rate are the most important factors in the control of reaction extent, and effect of rotating speed is conditional depending on the micro-mixing status of the fluid.
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Keywords Multicomponent reaction, Mixing, Numerical simulation, Extrusion
Received 0000-00-00, accepted 0000-00-00. * Supported by National Program on Key Basic Research Project (No. 2011CB606100) and the National Natural Science Foundation of China (Grant No.21406059). ** To whom correspondence should be addressed. E-mail address: [email protected] (L. Zhao)
ACCEPTED MANUSCRIPT 1 INTRODUCTION The co-rotating twin-screw extruder (CoTSE) is very attractive reactor for reactive extrusion (REX) due to
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its self-wiping performance and modular design characteristic. The reactions in a CoTSE can be classified into two categories: single component reaction, such as bulk polymerization, and multicomponent reaction, such as
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synthesis of urethanes, co-polymerizations and grafting reactions, etc. For the former, studies were mainly
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focused on the macromixing and temperature in the extruder[1, 2]. For the latter, in most cases, multicomponent reactions take place on the interface between striation structure and its area is increased with deforming and reorientation during laminar mixing[3]. Once the mixing time has the equivalent magnitude with reaction times or longer than that, incomplete local mixing of the reactants will significantly affects the course of reaction or
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even makes the conversion independent on Damköhler number[4, 5]. Therefore, detailed information of the mixing phenomena is important for the control of reaction in REX process.
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From a view of hydrodynamics, the essence of the multicomponent reactive extrusion is a combination of mass and momentum transfer in laminar regime. Like other reaction process, mixing between reactants is the
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precondition for the occurrence of reaction. To reduce the computing load, traditional approaches simplify the
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geometry of extruder into ideal chemical reactor models based on chemical reaction engineering theory, and employ the residence time distribution (RTD) as a measure of axial mixing performance. While this measure is not very useful for the laminar mixing of viscous fluids because it provides little information on transverse mixing[6], which is of more interest for the micromixing of reactive species. Since reaction occurs at the interface between striations, pioneering micromixing work of Ottino proposed a lamellar mixing model with alternating lamellar structure in the modeling of solution for concentrations field with multicomponent diffusion and chemical reaction[7, 8]. In this model, the striation thickness and average deformation rate were utilized as a measure of mixing state[9-11]. This approach is proved feasible in the analysis of relationship between molecular weight distribution and micromixing in a CoTSE[12]. However, the lamellar micromixing model simplifies the mixing as one-dimensional convection and diffusion problem because of the complexity in describing flow dynamics in the screw channel[13]. With the development of computational capabilities, approaches based on computational fluid dynamics (CFD) theory allows to solve these reactive-mixing problems in a three-dimensional way[14]. Thus, more realistic flow
ACCEPTED MANUSCRIPT patterns can be obtained[15]. Various simulations have been reported for the reactive extrusion flow in recent decades, such as peroxide-initiated degradation of polypropylene and polymerization of
-caprolactone in
CoTSE[16-19]. While most of these investigations were only concerned with macromixing and research on
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micromixing was rarely reported. It is reported the inadequate mixing between the polymers and initiators can significantly influence on the molecular weight distribution (MWD)[20], which was ubiquitous in
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multicomponent reactions[2]. Consequently, micromixing behavior should also be taken into consideration in the
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three dimensional simulation of REX.
Generally, there lies three decisive factors for the reactive-mixing process, i.e., macromixing state, which can be measured by RTD, degree of segregation, and earliness/lateness of mixing[21]. In the present paper, numerical simulation with three-dimensional mixing analysis was performed for the REX process. Effect of
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operational parameters, including initial species distribution, screw rotating speed and specific throughput, on product quality were numerically investigated and their relationships with these three factors were discussed.
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The aim of this study is to get a grasp on the REX process from the view of chemical reaction engineering and
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find the decisive factors in the controlling of product quality.
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2 NUMERICAL SIMULATION 2.1 Model reactions
Competitive-parallel reaction has been widely used in the investigation of micromixing in reactor[5, 11, 22]. In order to obtain realistic results in extruders, the competitive-parallel polymerization of polyurethane is employed in this work. The modeling reactions are: k1
(a)
A B C k2
(b) A D E where species A is –R-OH (1,4-butane diol, BDO), species B is –R’-NCO, i.e., the prepolymer of dicyclohexylmethane 4,4’-diisocyanate (H12MDI) and polyester (Mn= 1000 g·mol-1), species D is –R”-NCO, i.e., the prepolymer of 4,4’-diphenyl methane diisocyanate (MDI) and polyester (Mn= 1000 g·mol-1)[23, 24]. k1 and k2 are the rate constants of reaction (a) and (b), respectively. Both reactions are second-order reactions with suitable catalysts. According to k=5.45×109[Cat]exp(-Ea/RT), where [Cat]= 4.75×10-4 mol·L-1 and Ea=48.5 kJ·mol-1
ACCEPTED MANUSCRIPT (Ref[23]), the reaction constant is in the range of 10-1~10-2 L·mol-1·s-1. Thus, k1= 0.16 L·mol-1·s-1 and k2= 0.016 L·mol-1·s-1 are set in the model.
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2.2 Simulation methods
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In practical process, the barrel temperature is usually kept at a constant to ensure stable extrusion. By this way, the temperature rise in reactive extrusion becomes small and its effect on the process can be neglected.
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Thus, incompressible non-Newtonian flow, pseudo-steady state and isothermal condition are assumed for the numerical simulation. The governing equations are as follows:
Navier-Stokes equation:
U 0
(1)
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Continuity equation:
(2)
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p UU
[(U ) (U )T ]
(3)
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where p is the pressure in extruder, U is the local velocity. η is the viscosity of system. The constitutive equation
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of reaction system is regressed by literature data[23]:
0 [1 ( ) a ](1 n ')/ a
(4)
where a 1.24 is the Yasuda index, n ' 0.42 is the power law index, 0.014 s is the relaxation time constant and
is the shear rate.
The molecular weight (MW) of prepolymer is 7.2 kg·mol-1, at that range, the zero-shear viscosity of reaction system is[25]:
0 1.24 104 MW3.45
(5)
In practical polymerization, zero-shear viscosity is yielding 0.1~105 Pa·s. During the simulation in this work, it ranges from 0.1~10 Pa·s. That makes Reynolds number ranges from 0.02~2. The mass balance of species i gives the following convection-diffusion equation: ( i U ) ( Di i ) Ri
(6)
ACCEPTED MANUSCRIPT where ρ is the density of mixture, ωi and Di are the mass fraction and the molecular diffusivity for species i in the mixture, and Ri represents the net mass rate of reaction(s) for species i, which could be estimated as follows: j
Ri M i ij rj
(7)
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j 1
where Mi is the mole mass of species i, υij is the stoichiometric coefficient of the i-th species in the j-th reaction.
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For reactants, υij<0, for products, υij>0, and rj is the molar reaction rate of reaction j.
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Divide both sides of Eq.(6) by Mi, the laminar diffusion equation of species concentration can be reformed into:
j
(ci U ) ( Di ci ) ij rj
(8)
j 1
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By solving the above equations, the characteristics of the flow regime and species transport can be obtained. The
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dimension of ci in Eq.(8) is [mol·m-3], and we convert it into [mol·L-1] for convenience.
2.3 Simulation details
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Figure 1 illustrates the geometrical features of three-dimensional model. The origin of coordinates is at the center of one of the screw inlets, and the extrusion direction is along the negative Z-axis. Screw elements are
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right-handed with 40 mm pitch and 60 mm length. For laminar flow, the length of develop section is Ldevelop=0.05Re*D≈4 mm[26]. Thus, two 10 mm long flow domains without screws are added before the inlet and after the outlet separately to ensure the flow is fully developed. In the present simulation, two feeding streams are introduced into extruder through inlet 1 and inlet 2 separately. Two planes are chosen as sample planes, i.e., plane y/D= 0.38 in the middle of the screw chamber and plane |Z|/L= 0.75 approaching the outlet, where D= 42 mm is the barrel diameter and L= 80 mm is the axial length of flow domain. Besides, a sample line from (0, 16 mm, 0) to (0, 16 mm, -80 mm) is created to investigate the local properties along the axial distance.
(a) (b) Figure 1 Geometrical features of simulated screw (a) constrains of screws (b) three-dimensional model
ACCEPTED MANUSCRIPT Both inlets have the same flow rate. Different species concentrations are given for both inlets in order to investigate the reaction extent in accompany with convection-diffusion process. In practical process, the initial concentration of diol or diisocyanate is 3.12 mol·L-1)[23-25]. Thus, the concentrations of -OH and -NCO are set
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as 2 mol·L-1 with consideration of perpolymerization before reactive extrusion. The feeding streams are shown in Table 1. In this way, generation of a lamellar structure for species concentrations can be displayed during
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extrusion.
Reactant concentrations imposed on two inlets
cA1
Inlet 1 cB1
① ② ③ ④
2 2 1 2
0 0 1 2
cD1 cA2 / mol·L-1 0 0 2 0 1 1 2 0
Inlet 2 cB2
cD2
2 2 1 0
2 0 1 0
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Feeding condition
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Table 1
No-slip condition is imposed on the screws and barrel surfaces. The barrel is static and the screws are set as
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moving wall at certain rotating speed. The simulation work is performed on commercial CFD platform POLYFLOW based on finite element method. To eliminate the possible negative volume meshes arising due to
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screw rotating, the mesh superposition technique (MST) was implemented: fluid mesh elements obey the conservation equations, while the mesh elements occupied by moving screws are forced to rotate as rigid
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bodies[27]. Thus, three-dimensional finite element meshes are constructed covering moving parts and flow domain (Fig.2), the flow domain is discretized into 48 k hexahedral elements and each screw is divided into 40 k hexahedral elements. All the properties including velocity, species, pressure and viscosity, etc. are coupled with each other in the simulation to ensure convergence. The convergence of calculation is achieved until the residuals are less than 0.0001. Typical CPU time for an FEM task on a workstation with 8 processors is around 1.6×104 s. The mixing task is implemented with particle tracking analysis. The trajectories of 5000 marker particles are calculated per case in order to ensure the accuracy of the statistical results[28, 29]. All these particles are volume less, massless and independent from each other[30]. Initially, these particles are randomly released at the inlet plane. The particles released at inlet 1 and inlet 2 can be distinguished with different colors, seen in Figure 3. In the computational code, these particles are assigned as "1" and "0", respectively. Then their trajectories are calculated from the flow field. By this way, the material deformation can be estimated via the displacements of particle pairs, and the RTD can be obtained by the statistics of particle lifetime at outlet plane. Typical CPU time for a mixing task on a workstation with 8 processors is 1.4×103 s.
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(a) (b) Figure 2 Finite element meshes of (a) screws and (b) flow domain
2.4 Characterization of micromixing
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Figure 3 Initial distribution of particles
Convection, diffusion and reaction determines the effect of mixing on the course of chemical reaction[6].
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Thus, the characteristic reaction time and mixing time need to be evaluated. The characteristic reaction time is defined as[10]:
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tR j 1/ k j ci 0 ( j 1, 2)
(9)
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where ci 0 1 mol·L-1 is the initial average concentration of species i. For the main reaction, tR 6.25 s , and for 1
the side reaction, tR 62.5 s . 2
The characteristic micromixing time with diffusion mechanism is defined as[10]: tM arc sinh(0.76 s02 / D) / (2 )
(10)
where α is the deformation rate of striation and s0 is the initial striation thickness. The micromixing time in the present simulation of the order of magnitude 1~10 s. Apparently, the micromixing time has similar magnitude to the characteristic main reaction time, indicating the significance of spatial mixing performance on the reactions. When multiple reactions take place between two reactant fluids and once these reactions proceed to a considerable extent before homogeneity is attained, product distribution will be affected by the segregation status of the flow[31]. Here we adopt segregation scale and segregation intensity to characterize the mixing flow. The former measures the region size of homogeneous concentration and the latter reflects the spatial uniformity of the concentration[32, 33]. Reduction of segregation scale and segregation intensity represent the
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Ls R( r )d r
(11)
0
where cP c"j cP
M
j 1
M c2 NP
P 1
2
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c2
(cP cP )
(12)
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R r
' j
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c
NP 1
(13)
R(|r|) is the correlation coefficient between concentration of pairs of particles separated by |r| and M=NP(NP-1)/2
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is the number of pairs, where NP is the number of particles, and σ2c is the sample variance. The c 'j and c"j denote the concentrations of both particles in the j-th pair. cp is the concentration of a certain particle which equals to either 0 or 1 and remains constant since the diffusion vanishes.
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The segregation intensity Iseg is the standard deviation of the concentration around its mean, and is defined
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as follow[33, 34]:
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I seg
x2 02
N
(x j x j ) j 1
N 1
2
02
(14)
where σ20 is the variance of a completely segregated system and σ2x is the measured variance. xj denotes the concentration of j-th element in the extruder and N is the number of elements. The concentration of these elements will be homogenized due to convection and diffusion effect. Another important factor in micromixing is the earliness/lateness of mixing, i.e., whether fluid mixes early or late as it flows through the vessel. For multicomponent reaction, the mixing of reactant is the prerequisite for reaction, which is relate to the premixed condition. In the simulation, the earliness/lateness of mixing is controlled by the reactant concentrations imposed on two inlets (seen in Table 1).
2.5 Characterization of reaction In order to quantify the reaction extent, firstly, the area-averaged features of i-th species on a plane is defined as[19]:
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N
ci
c Aj
j 1 ij N j 1
(15)
Aj
where Aj and cij are the area and the concentration of species i at cell j.
cC cC 0 cA0 c A
cC
(16)
c A0 c A
(17)
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where, cA0 1 mol·L-1 and cC 0 0 .
2.6 Evaluation of simulation results
c A0
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c A0 c A
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x
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Conversion x and selectivity of species A can be calculated by following equations:
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The evaluation of simulation results includes two aspects: verification and validation[36]. Verification evaluates the uncertainty in numerical simulation. Here, mesh independency was checked. Validation
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means comparison between simulation and experiment. To prove the versatility of numerical methods, RTD and species distribution were compared with different sizes of extruders.
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2.6.1 Verification
In the simulation, mesh size is extremely important to ensure the accuracy of species transportation. Fig.4 shows the meshes at a cross section normal to the extrusion direction with three mesh sizes in mesh independency check. The total elements quantities are 41 k, 88 k and 190 k for scheme a, b and c, respectively.
Figure 4 Meshes at a cross section normal to the extrusion direction.
Feeding ① with Q/n=7000/30 were employed in mesh check, where the reactants are initial segregated in different inlets, as seen in Table 1. Fig.5a illustrates the velocity on the sample line and Fig.5b shows the evolution of average conversion along axial distance with three mesh sizes. The curves of mesh (b) and (c) are
ACCEPTED MANUSCRIPT very close in both figures, while perceptible difference could be recognized between results from mesh (a) and others. With compromising consideration of computational load and accuracy, mesh size (b) is chosen as an
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optimized mesh for the simulation work.
(a)
(b)
Figure 5 Mesh independency check (a) velocity distribution on sample line 1. (b) average conversion evolution along the axial distance
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2.6.2 Validation
In order to test the capabilities and accuracy of numerical methods, RTD of a flow through KD2 structure
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(two pairs of 60/4/32 kneading discs with D=35 mm) is simulated and compared with that in literature[30], as shown in Fig.6a. In Zhang’s experiment[30, 37], the test elements were installed in the mixing section of
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extruder and two probes were located at the upstream and downstream. Anthracene was chosen as the tracer and the in-line fluorescent light measuring system was developed to measure the local RTD at probe locations. Then
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the RTD between them was calculated by a deconvolution method[38]. The experimental material is polystyrene and its constitutive equation can be represented by Carreau model[30]:
2814[1 (0.148 )2 ](0.2781)/2
-6
3
(18)
-1
The flow rate is 3.06×10 m ·s and rotating speed is 120 rpm in the simulation according to literature[30]. A good agreement is found in the comparison of both work (Fig.6b). And the relative error is less than 5%.
(a) (b) Figure 6 Validation of simulation on RTD (a) Elements geometry (b) RTD curves
Besides, distribution of species is also validated by comparing the deformation of striation between the simulation work and the experiment work of Kalyon[39]. The geometry of screw elements are full flight
ACCEPTED MANUSCRIPT elements, diameter and pitch in 50.8 mm, and the operational conditions are identical to literature[39] in the simulation. Fig.7a shows the initial position of striation. Fig.7b and 7c illustrate simulated and experimental shapes of striation after half revolution of screws, respectively. The agreement between those shapes seems fine.
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The above validations indicate that the employed numerical method is feasible and the parameters defined in
(b) Figure 7 Spatial distribution of tracer in extruder after 1/2 revolution (a) initial position (b) simulated result (c) experimental result
(c)
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(a)
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numerical analysis are reasonable.
3 RESULTS AND DISCUSSION
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3.1 Flow pattern in CoTSE
In twin screw extruders, the distribution of fluid elements is improved due to relative movement between
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both screws and barrel, which have significant influence on flow behavior. In order to trace the transfer movement of species, species A is introduced into inlet 1 with a concentration of 2 mol·L-1. The screw rotating
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speed is 30 rpm and both inlets have the same flow rate of 3500 mm3·s-1. Fig.8 gives the distribution evolution of species A at different axial locations. Obviously, it can be seen that the flow is transferred from one screw to the other because the other screw picks up the material and drags it away from intermeshing region. As a result, species A is delivered in a “
” pattern with helical flow and new material interfaces are generated with each
screw revolution[40, 41]. With the development of material exchange and accumulation of the mixing effect, the distribution of concentration gradually becomes uniform along the extrusion direction as seen after plane |Z|/L= 0.75.
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Figure 8 Distribution of species A along extrusion direction at n= 30 rpm
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3.2 Effect of feeding condition
For continuous process, the initial species distribution is determined by the feeding condition. In order to
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clarify the interaction between flow field and reactions, here the species are fed through different inlets, which can be regarded as completely separated condition[42-44]. Table 1 shows the initial species distributions of
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cases.
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various feedings. The flow rate is set as 3500 mm3·s-1 for each inlet and rotating speed equals to 30 rpm for all
Figure 9 illustrates the concentration distribution of main product (C) and side product (E) on plane y/D= 0.38 with various feedings. The results clearly illustrate the sensitivity of species C and E to the premixing status of their relevant reactants. It is noticed that the concentration of species E is improved in Fig.9b compared with that in Fig.9a, indicating prior mixing between species A and D is beneficial to the side reaction. The same observation can be made for the concentration distribution of species C and E in Fig.9c and Fig.9d compared with that in Fig.9a, which implies the initial homogeneous distribution of the reactants can simultaneously promote the main and side reactions to a certain extent.
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Figure 9 Distributions of species C (left) and E (right) on plane y/D= 0.38 with (a) feeding ① (b) feeding ② (c) feeding ③ (d) feeding ④
Table 2 provides reaction results on plane |Z|/L= 0.75. From the table, it can be seen that the lowest averaged conversion is found in the case with feeding condition ①, this is caused by the initial condition of species. In that condition, species A is completely separated from B and D, and the insufficient mixing between these reactants can decrease the reaction rate. Higher conversion and lowest selectivity under feeding condition ② is caused by the only premixing of species A with species D at the inlet, hence the side reaction is enhanced while the main reaction is suppressed. Better mixing of the reactants under feeding condition ③ and ④ leads to higher reaction conversions. And the highest local reactant concentrations in the case with feeding condition ④ are responsible for the highest conversion. Moreover, it is also observed that the selectivity of the cases with
ACCEPTED MANUSCRIPT feeding condition ①, ③ and ④ are very close. This can be attributed to the fact that the mixing of species B and D are identical in these cases. Table 2 Average concentrations, selectivity and conversion on plane |Z|/L= 0.75 / mol•L 0.436 0.069 0.395 0.212 0.634 0.099 0.696 0.117
x
0.864 0.651 0.865 0.856
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cE
-1
0.505 0.607 0.733 0.813
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① ② ③ ④
cC
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Feeding
Figure 10 gives the yield of species C (Φ) along the extrusion direction, which is the product of selectivity and conversion. As far as feeding condition ③ and ④ are concerned, earlier mixing between reactants results in a jump in the yield of species C, and then the growth is gradually weakened approaching the outlet since the
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reactant A is exhausted. The curves with feeding condition ① and ② are almost overlapped in the inlet section due to their low conversions. Downstream, the differences between the two curves become larger attributing to
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their distinguishing selectivity.
Figure 10 Effect of feeding condition on yield of species C
The above discussion reveals different micromixing behaviors of the fluids. Flow through the inlets at feeding condition ①, ② and ④, is often called “macrofluid”, where fluids are completely segregated. When identical flow through both inlets, like feeding condition ③, is called “microfluid”, and fluids will be in the state of maximum mixedness during being transported. These two groups of fluids restrict earliness/lateness of mixing, which is also a prominent issue for other reactors. The above observations confirms that earliness/lateness of mixing has a determined effect on the local concentration distribution, which can further affect the conversion and selectivity for these multicomponent reactions in CoTSE. It also provides an evidence that attentions should be paid to the feeding condition in order to achieve the desired reaction results in the REX process.
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3.3 Effect of rotating speed Figure 11 illustrates the product distributions at various screw rotating speeds with feeding condition ②.
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The flow rate is fixed at 3500 mm3·s-1 per inlet. With increasing rotating speed, the concentration of species C is
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while the side reaction has been suppressed on the contrary.
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improved while the concentration of species E is decreased, indicating the main reaction has been enhanced
Figure 11 Distributions of species C (left) and E (right) on plane y/D= 0.38 at (a) 20rpm (b) 30rpm (c) 40rpm
Figures 11a and 11b show the selectivity and conversion evolution along the axial direction at various rotating speeds. In order to illustrate the mixing effect on reaction, fully premixed case with feeding condition ③ and rotating speed of 30 rpm is provided for comparison. The reaction conversion and selectivity along the extrusion direction are found increasing with the increase of screw rotating speeds, indicating higher screw rotating speed is beneficial to the proceeding of reaction under the given feeding condition. Besides, apparent distinct can be noticed between premixed case and non-premixed cases. The selectivity for the premixed case keeps a high value in the all range, which proves the importance of reactant segregation on the reactions. Here define “product gap” to demonstrate the difference of yields between premixed and segregated feeding:
ACCEPTED MANUSCRIPT premix
(19)
where premix is the yield of the case that with premixed feeding. The yield difference along the extrusion direction is shown in Fig.12c. At the beginning of extrusion, the rise of target product in the microfluid is larger
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than that in the macrofluid due to the earliness of mixing between reactants. With the development of flow and
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accumulation of mixing effect, this difference reaches its maximum. Then, the gap between mixed and nonmixed cases declines with the consumption of reactants. The smaller rotating speed, the bigger gap is found.
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RTD is typically employed as a measure of a kinematic description of flow, to understand the axial mixing in the reactor. Fig.12d describes the local RTD at plane |Z|/L= 0.75 at different speeds. There is no obvious effect of rotating speed on the RTD profile shape. Moreover, the mean residence time is found slightly reduced,
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which is 11.58 s, 10.56 s and 9.64 s with the increasing of rotating speed. Change to the results of reaction extent, it is clearly insufficient to analyze the reactive-mixing flow only by RTD. Because RTD is only a good measure for macromixing and becomes invalid in the characterization of micromixing. Hence, statistical
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description of mixture is processed to analyze the degree of segregation in the extruder. Fig.12e shows the evolution of segregation scale along extrusion direction. The segregation scale presents a sharp decline in all
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curves near the entrance section, illustrating the region size with homogeneous concentration is decreased
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rapidly. It is also found the segregation scale oscillates in the remaining section, which may be caused by the stretching, folding and reorientation of the striations during extrusion[45]. Approaching the outlet, the segregation scale at higher speed is found smaller than that at lower speed. Fig.12f shows the evolution of segregation intensity along extrusion direction. All curves show a decreasing trend. The higher the screw rotating speed, the quicker the curve goes down. Lower value of segregation intensity is obtained at higher speed near the outlets. It shows that rotating speed is more relevant to micromixing rather than macromixing and higher rotating speed can achieve more uniform distribution of species in the extruder. The results also illustrate the scalar mixing analysis is a necessary supplement for the characterization of reactive-mixing in REX.
(a)
(b)
(c)
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(e) (f) Figure 12 Effect of rotating speed on (a) selectivity (b) conversion (c) yield difference (d) RTD (e) segregation scale and (f) segregation intensity
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(d)
3.4 Effect of specific throughput
It has been reported that the specific throughput, defined as the ratio between flow rate Q and rotating speed n, has significant influence on partial as well as overall residence time/revolution/volume distribution[37].
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However, not much is mentioned about its effect on reaction. Since the REX process is determined by a combination of mixing, reaction kinetics and residence time in the channel of fluid element, it is of interest to
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figure out the relationship between specific throughput and reactions. Figure 13 provides the species distribution of C and E at y/D= 0.38 at the same specific throughput with
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feeding condition ②. It can be seen that both the concentrations of species C and E in the whole extruder are
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reduced with increasing flow rate, which is due to the decrease of residence time.
ACCEPTED MANUSCRIPT Figure 13 Distributions of species C (left) and E (right) on plane y/D= 0.38 at specific throughput of (a) 3500/15 (b) 7000/30 (c) 10500/45
Figures 14a and 14b show the selectivity and conversion evolution along the extrusion direction. The result of
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conversion indicates that reactions are suppressed with increasing flow rate. However, the reaction selectivity is increased with increasing screw rotating speed. There may be two possible reasons behind this phenomenon: (1)
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The local micromixing status in extruder may not change at the same specific throughput, while the time to
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achieve the same micro-mixing status is shortened with higher flow rate. Thus, the side reaction is suppressed. (2) The micromixing of reactive flow is enhanced by the increasing rotating speed with identical specific throughput. The yield difference along the extrusion direction is shown in Fig.14c. All curves show a trend from increase to decline as except. Moreover, the yield difference is decreasing with higher flow rate. That is because
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shorter residence time leads to shorter reaction time, and hence the yields with premixed and segregated feeding are decreased.
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To confirm the true reason behind the improvement of reaction selectivity, here we analyze the effect of specific throughput on the mixing performance of flow firstly. With the increasing of flow rate and rotating
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speed while the specific throughput remains constant, the RTD curve shifts to the left and becomes narrower as
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expected, as seen in Fig.14d. The mean residence times t are 21.40 s, 10.56 s and 7.14 s with increasing flow rate. As the mixing performance is concerned, it is interesting to find the curves of segregation scale and segregation intensity are almost superimposed on a single master one (Fig.14e and 14f). Therefore, we can conclude that the same specific throughput bring about differences in the duration of mixing while identity in micro-mixing status. And it gives the answer that the improved selectivity is mainly attributed to the shortened reaction time rather than the mixing performance. Combined with the results in Section 3.3, it confirms that the mixing is closely dependent on the specific throughput instead of simply rotating speed, and the reaction extent is the consequence of both mixing and residence time.
(a)
(b)
(c)
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(e) (f) Figure 14 Effect of specific throughput on (a) selectivity (b) conversion (c) yield difference (d) RTD (e) segregation scale and (f) segregation intensity
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(d)
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3.5 Reactive-mixing flow in extruder
To compare the significance of the effects of feeding, rotating speed and flow rate on reactions, an orthogonal test is performed. Each factor has three levels and the interaction of feeding condition and rotating
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speed is considered (Table 3). The results of orthogonal test are listed in Table 4. Columns 6 and 7 show level errors, the deviations are quite small. While, the same outcomes are found in columns 3 and 4, indicating the
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interaction between feeding condition and rotating speed can be neglected, column 3, 4, 6 and 7 could be regard as level error columns. It can be concluded that sequence of significance of these factors on the reaction yield is
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feeding condition > flow rate > rotating speed, based on the range and sum of square of deviations analysis. The F examination tells both the differences of flow rate and feeding condition are highly significant (P<0.01), and
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the difference of rotating speed is also remarkable (P<0.1).
Table 3 Factors and levels in orthogonal test Factors Level 1 Level 2 Feeding condition A ① ② Rotating speed (rpm) B 15 30 Flow rate (mm3·s-1) C 3500 7000
Level 3 ③ 45 10500
Note: the interaction of A and B is also considered
Table 4 The results of orthogonal test Test number 1 2 3 4 5 6 7 8 9 10
Factors and levels A B 1 2 1 1 1 2 1 3 2 1 2 2 2 3 3 1 3 2 3 3 1 1
Yield A 3 1 2 3 1 2 3 2 3 1 3
×
B 4 1 2 3 2 3 1 1 2 3 3
C 5 1 2 3 2 3 1 3 1 2 2
6 1 2 3 3 1 2 2 3 1 2
7 1 2 3 3 1 2 3 1 2 1
0.552 0.436 0.394 0.265 0.355 0.701 0.547 0.722 0.619 0.271
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1 2 3 1 2 2 3 1 0.524 0.500 0.473 0.051 0.008
3 1 1 2 3 3 1 2 0.642 0.434 0.421 0.221 0.185 24.68
3 1 3 1 2 1 2 3 0.524 0.505 0.467 0.057 0.010
2 3 2 3 1 2 3 1 0.479 0.518 0.500 0.039 0.005
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2 3 1 2 3 1 2 3 0.443 0.493 0.561 0.103 0.042 5.60
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1 1 2 2 2 3 3 3 0.443 0.424 0.629 0.205 0.154 20.53
0.327 0.678 0.476 0.395 0.353 0.547 0.722 0.619
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11 12 13 14 15 16 17 18 K1 K2 K3 R S F
Note: R and S represent range and sum of square of deviations, respectively, and F is the value of F examination
It should be pointed out screw rotating speed may have different effects on reaction according to the micro-
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mixing behavior of the fluid. For macrofluid, increasing rotating speed enhance the micro-mixing, which can compensate the shortening of residence time and results in a higher reaction extent; for microfluid, increasing rotating speed only reduce the reaction time, leading to lower reaction extent. The orthogonal test proves when
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the screw rotating speed is low, microfluid status is more beneficial for the reaction because macrofluid takes
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long time for micro-mixing. When the screw rotating speed is extremely increased, higher than the studied speed, the micro-mixing time may be greatly shortened and the significance of rotating speed can be
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strengthened. However, in order to ensure enough reaction time, lower rotating speed is more reliable. In brief, the highly significant differences of feeding condition and flow rate on reaction yield can be attributed to their relationship with earliness of mixing and residence time, respectively. The effect of rotating speed relies on the micro-mixing status of the fluid.
4 CONCLUSIONS Numerical modeling of reactive extrusion systems is of great interest since it provides an accessible way to choose favorable operational conditions from a practical point of view. In the present paper, effects of initial species distribution, screw rotating speed and specific throughput on the multicomponent reaction were numerically investigated. In accordance with the above discussions and analysis about their effects and interaction, the following conclusions are drawn. The results show that RTD alone is inadequate to characterize the mixing status in REX process. Micromixing status of fluids, i.e., microfluid and macrofluid, which controls earliness/lateness of mixing, has the
ACCEPTED MANUSCRIPT highest priority in affecting the flow field and the product quality. The analysis of segregation degree, including segregation scale and segregation intensity, shows at the same flow rate, effective mixing resulted from higher screw rotating speed can compensate the shortening of residence
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time and promote the proceeding of reactions with macrofluid feeding. On the contrary, prominent effect of increasing screw rotating speed decreases the reaction time with microfluid feeding.
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The relationship between flow rate and rotating speed is identified by specific throughput. The mixing
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performance keeps unchanged at the same specific throughput, while higher flow rate and rotating speed lead to significant decrease of products due to the reduction of reaction time.
In order to discriminate the significance of these operational conditions on the reaction, orthogonal test is performed and the results confirmed the initial species distribution and flow rate play decisive roles in the yield
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controlling.
Aj
area of cell j
a
Yasuda index
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NOMENCLATURE
concentration of species i at inlet 1 and 2, mol·L-1
c 'j , c"j
concentrations (volume fractions) of particles in the j-th pair
cp
concentrations (volume fractions) of particles
D
barrel diameter, mm
Di
molecular diffusivity, m2·s-1
Iseg
segregation intensity
kj
second order reaction rate constants, L·mol-1·s-1
L
axial length of flow domain, mm
Ls
segregation scale, mm
M
number of particle pairs
Mi
mole mass of species i, kg·mol-1
N
number of elements
NP
number of particles
n
rotating speed, rpm
n’
power-law index
p
pressure, Pa
Q
flow rate, mm3·s-1
Ri
net mass rate of species i, kg·m-3·s-1
R(|r|)
correlation coefficient between concentration of pairs of points separated by |r|
rj
molar reaction rate of reaction j, mol·m-3·s-1
s0
initial striation thickness, m
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ci1, ci2
mean residence time, s
tRi
characteristic reaction time of reaction j, s
tM
micro-mixing time with diffusion mechanism, s
U
velocity vector, m·s-1
V
volume of flow domain in an extruder, mm3
x
conversion of species A
xj
concentration (volume fraction) of the j-th element in extruder
α
deformation rate of striations, s-1
β
selectivity
η
viscosity, Pa·s
ρ
density, kg·m-3
σ20
variance of a completely segregated system
σ
variance in calculation of segregation scale
2 c
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t
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variance in calculation of segregation intensity
υij
stoichiometric coefficient of the i-th species in the j-th reaction
τ
relaxation time index, s
ωi
mass fraction
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σ2x
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modeling of twin screw extrusion of starchy products, Polym. Eng. Sci., 50 (9) (2010) 1758-1766. 2 Ganzeveld K. J., Janssen L. P. B. M., Role of mixing and rheology in reactive extrusion, Ind. Eng. Chem. Res., 33
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(10) (1994) 2398-2403.
3 Ganzeveld K. J., Janssen L. P. B. M., A mixing model for multicomponent reactions in twin screw extruders applied to the polymerization of urethanes, Polym. Eng. Sci., 32 (7) (1992) 457-466. 4 Suresh A., Chakraborty S., Kargupta K., Ganguly S., Low-dimensional models for describing mixing effects in reactive extrusion of polypropylene, Chem. Eng. Sci., 63 (14) (2008) 3788-3801. 5 Rożeń A., Bakker R. A., Bałdyga J., Effect of operating parameters and screw geometry on micromixing in a corotating twin-screw extruder, Chem. Eng. Res. Des., 79 (8) (2001) 938-942. 6 Ottino J. M., Chella R., Laminar mixing of polymeric liquids; a brief review and recent theoretical developments, Polym. Eng. Sci., 23 (7) (1983) 357-379. 7 Ottino J. M., Lamellar mixing models for structured chemical reactions and their relationship to statistical models; macro- and micromixing and the problem of averages, Chem. Eng. Sci., 35 (6) (1980) 1377-1381. 8 Ottino J. M., Ranz W. E., Macosko C. W., A framework for description of mechanical mixing of fluids, AlChE J., 27 (4) (1981) 565-577. 9 Wu J. C., Micromixing of a polymer melt in twin screw extruders with kneading disks, Ph. D. Thesis, University of Delaware, Newark, 1994. 10 Bałldyga J., Rozeń A., Mostert F., A model of laminar micromixing with application to parallel chemical reactions, Chem. Eng. J., 69 (1) (1998) 7-20.
ACCEPTED MANUSCRIPT 11 Rożeń A., Bakker R. A., Bałdyga J., Application of an integral method to modelling of laminar micromixing, Chem. Eng. J., 84 (3) (2001) 413-428. 12 Iedema P. D., Remerie K., van der Ham M., Biemond E., Tacx J., Controlled peroxide-induced degradation of polypropylene in a twin-screw extruder: Change of molecular weight distribution under conditions controlled by
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micromixing, Chem. Eng. Sci., 66 (22) (2011) 5474-5486.
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dimensional flow, Chem. Eng. J., 71 (1) (1998) 49-56.
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14 L. Marchisio D., A. Barresi A., CFD simulation of mixing and reaction: The relevance of the micro-mixing model,
15 Xu X. M., Zhao G. Q., Qin S. X., Wang W., Numerical simulation of viscoelastic extrudate swell through elliptical ring die, Chin. J. Chem. Eng., 19 (1) (2011) 10-17.
16 Strutt D., Tzoganakis C., Duever T. A., Mixing analysis of reactive polymer flow in conveying elements of a corotating twin screw extruder, Adv. Polym. Tech., 19 (1) (2000) 22-33.
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ACCEPTED MANUSCRIPT Graphic abstract Feeding condition and rotational speed determine the local concentrations of reactants by controlling the earliness/lateness of mixing and the degree of segregation, respectively. Besides, rotational speed and flow rate also have effect on the RTD and duration of the reaction. The combination of local concentration and time accumulation restricts the reaction extents of
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multicomponent reactions during extrusion.