Computer simulation of reactive extrusion processes for free radical polymerization.INTRODUCTION In the reactive extrusion process for polymerization polymerization Any process in which monomers combine chemically to produce a polymer. The monomer molecules—which in the polymer usually number from at least 100 to many thousands—may or may not all be the same. , the phenomena such as the chemical reaction, the evolutions of material structures, and physicochemical properties, the fluid flow and the forced convection heat transfer coexist and interact with each other [1-6]. To study the complex process, some experiments were carried out and the mathematical models were developed on the basis of experimental data, and then the numerical simulations were performed. Siadat et al. reported the mathematical models of the reactive extrusion process for polycondensation [7]. Hyun and Kim made an engineering analysis of the reactive extrusion process of thermoplastic A polymer material that turns to liquid when heated and becomes solid when cooled. There are more than 40 types of thermoplastics, including acrylic, polypropylene, polycarbonate and polyethylene. polyurethane in a single screw extruder via numerical simulation [8]. Janssen and coworker co·work·er or co-work·er n. One who works with another; a fellow worker. designed a mixing model for multicomponent reactions in counter-rotating twin screw extruders, applied it to the polymerization of urethanes [9], and carried out the numerical simulation of the reactive extrusion process for the free radical polymerization Radical polymerization is a type of polymerization in which the reactive center of a polymer chain consists of a radical. The polymerization reaction is initiated by three classes of free-radical initiators: n. A negatively charged ion, especially the ion that migrates to an anode in electrolysis. [From Greek, neuter present participle of anienai, to go up : ana-, ana- polymerizations of nylon 6 and polystyrene in closely intermeshing corotating twin screw extruders, on the basis of the models of the ideal reactor types "cascade of continuous stirred tank reactors" and "pipe reactor" [13, 14]. Kim and White performed an engineering analysis of the increase in the molecular weight and the shear-induced reduction of the molecular weight after the polymerization of [epsilon]-caprolactone during the reactive extrusion process [15]. Gimenez et al. carried out the simulation of the bulk polymerization of [epsilon]-caprolactone in corotating twin screw extruders [16-18]. Vergnes et al. proposed a computer software for polymer flows in corotating twin screw extruders on the basis of a ID approximated approach [19-21]. Zhu and Jaluria carried out the numerical simulation of the flow of chemically reactive non-Newtonian fluids in a fully intermeshing corotating twin screw extruder [22]. Fukuoka built a computer simulation of the flow field in a self-wiping corotating twin screw extruder on the basis of the reaction kinetics and rheological models [23]. To optimize reactive processing conditions and then design macromolecular mac·ro·mol·e·cule n. A very large molecule, such as a polymer or protein, consisting of many smaller structural units linked together. Also called supermolecule. materials according to according to prep. 1. As stated or indicated by; on the authority of: according to historians. 2. In keeping with: according to instructions. 3. requirements, a new semi-implicit iterative algorithm is proposed to deal with the complicated relationships among the various variables. Then the computer simulation of the reactive extrusion process for free radical polymerization is carried out, and the evolutions of the monomer concentration, the initiator concentration, the average molecular weight, and the fluid viscosity are shown and compared with the experimental results. MATHEMATICAL MODEL OF COROTATING TWIN SCREW EXTRUDERS Usually, the free radical polymerization of monomer takes place in a closely intermeshing corotating twin screw extruder, which often consists of several forward conveying screw elements, several reverse conveying screw elements, several kneading elements, and a die. The fluid flows forward by means of both the drag effect of the screws and the pressure effect [24-27]. Therefore, the space of fluid flow can be equivalently considered as a long axisymmetrical space, in which the effect of the axial motion of the solid wall on fluid flow is equivalent to the integrated effect of the screws and barrels on fluid flow. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII ASCII or American Standard Code for Information Interchange, a set of codes used to represent letters, numbers, a few symbols, and control characters. Originally designed for teletype operations, it has found wide application in computers. ] (2) where [C.sub.1] denotes the centerline cen·ter·line n. 1. A line that bisects something into equal parts. 2. A painted line running along the center of a road or highway that divides it into two sections for traffic moving in opposite directions, or, in the case of distance of the screws. The area [A.sub.s] of the cross section can be expressed as [A.sub.s] = 2 [[integral].sub.[[W.sub.b]/2].sup.[[W.sub.s]/2]] {[R.sub.s][1 + cos[[2[pi](x - [[W.sub.b]/2])]/[[T.sub.s] cos [bar.[phi]]]]]. - [square root of ([C.sub.L.sup.2] - [R.sub.s.sup.2][sin.sup.2] [[2[pi](x - [[W.sub.b]/2])]/[[T.sub.s] cos [bar.[phi]]]])] + [[delta].sub.f]}dx + (2[R.sub.s] - [C.sub.L] + [[delta].sub.f]) [W.sub.b] + [[delta].sub.f] [W.sub.b]. (3) If the number of thread starts is denoted as n, the equivalent radius [R.sub.e] of the axisymmetrical model is obtained as [R.sub.e] = [square root of ([(2n - 1)[A.sub.s]]/[pi])]. (4) SEMI-IMPLICIT ITERATIVE ALGORITHM To deal with the complicated relationships among the variables, such as reaction rate, average molecular weight, fluid viscosity, pressure, temperature, and flow velocity In fluid dynamics the flow velocity, or velocity field, of a fluid is a vector field which is used to mathematically describe the motion of the fluid. Definition The flow velocity of a fluid is a vector field If the axial length of the screw is denoted as L, the length [L.sub.m] of the axisymmetrical model can be calculated [28]. [L.sub.m] = [L/[T.sub.s]] [[T.sub.s]/[sin [bar.[phi]]]] = L/[sin[arc tan [[T.sub.s]/[[pi]([R.sub.s] + [R.sub.b])]]]]. (1) Where [T.sub.s] denotes the lead of screws, [bar.[phi]] the average helix angle, [R.sub.s] the radius of the excircle, and [R.sub.b] the radius of the radical circle. As shown in Fig. 1, the cross section of the screw channel is that surrounded by eight curves, which are denoted as IJ, JK, KL, LM, MN, NS, ST, and TI. According to the geometry of normal screws, the channel depth h(x) can be obtained [28] NUMERICAL CALCULATION OF REACTION EXTENT Calculation of Monomer Concentration A rigorous kinetic treatment of free radical polymerization mechanisms leads to immense complexities, hence it is necessary to introduce some simplifying assumptions and approximations: (1) chain transfer reactions are omitted from the polymerization mechanism; (2) the reactivity of the propagating radicals is independent of the size or degree of polymerization The degree of polymerization, or DP, is the number of repeat units in an average polymer chain at time t in a polymerization reaction [1]. The length is in monomer units. The degree of polymerization is a measure of molecular weight. of the radicals; (3) the kinetic chain lengths are very large; (4) all the free radicals present in the system are at steady-state concentrations [29]. Hence, initiated by initiators, the chain propagation Chain propagation is a process in which a reactive intermediate is continuously regenerated during the course of a chemical reaction. In polymerization reaction, the reactive end-groups of a polymer chain react in each propagation step with a new monomer molecule transferring the rate Vp can be obtained [FIGURE 1 OMITTED] [FIGURE 2 OMITTED] [V.sub.P] = [k.sub.p] ([f[k.sub.d]]/[k.sub.t])[.sup.1/2] [c.sub.ini.sup.1/2][c.sub.mono] (5) where [k.sub.d], [k.sub.p], [k.sub.t] denote the rate constants for initiator decomposition, chain propagation, and chain termination For the DNA sequencing method, see . Chain termination is any chemical reaction leading to the destruction of a reactive intermediate in a chain propagation step in the course of a polymerization, effectively bringing it to a halt. , respectively. f denotes the efficiency of initiation, [c.sub.ini] the initiator concentration, and [c.sub.mono] the monomer concentration. If the rate constants are written according to the Arrhenius formulation and f is considered to be independent of temperature T, Eq. 6 follows [V.sub.p] = [A.sub.p] ([A.sub.d]/[A.sub.t])[.sup.1/2] [f.sup.1/2][c.sub.ini.sup.1/2][e.sup.-[[[E.sub.p] - [[E.sub.t]/2] + [[E.sub.d]/2]]/RT]] [c.sub.mono] (6) where [A.sub.d], [A.sub.p], [A.sub.t] denote the frequency factors for initiator decomposition, chain propagation and chain termination, respectively. [E.sub.d], [E.sub.p], [E.sub.t] denote the activation energies for initiator decomposition, chain propagation, and chain termination, respectively. R denotes the general gas constant. For a first-order chain propagation reaction, the monomer concentration can be obtained [c.sub.mono] = [c.sub.mono,0] [e.sup.-[k.sub.0][c.sub.ini.sup.1/2][te.sup.-[[E.sub.a]/RT]]] (7) where [c.sub.mono,0] denotes the initial monomer concentration, t the reaction time, [k.sub.0] the total frequency factor, and [E.sub.a] the total activation energy activation energy, in chemistry, minimum energy needed to cause a chemical reaction. A chemical reaction between two substances occurs only when an atom, ion, or molecule of one collides with an atom, ion, or molecule of the other. , namely, [k.sub.0] = [A.sub.p]([A.sub.d]/[A.sub.t])[.sup.1/2] [f.sup.1/2] (8) [E.sub.a] = [E.sub.p] - [[E.sub.t]/2] + [[E.sub.d]/2]. (9) Numerical Calculation of Monomer Conversion In the process of free radical polymerization, the monomer conversion X can be expressed as [29, 30] X = 1 - [e.sup.-[k.sub.0][c.sub.ini.sup.1/2][te.sup.-[[E.sub.a]/RT]]]. (10) For an isothermal process Isothermal process A thermodynamic process which occurs with a heat addition or removal rate just adequate to maintain constant temperature. The change in the internal energy per mole U , if the initiator concentration is known, the monomer conversion can be calculated directly with Eq. 10. But in practical processes, both the reaction temperature and the initiator concentration are variable, therefore the monomer conversion can not be calculated directly via Eq. 10. To solve this problem, the incremental theory is introduced. Assuming that X(I,J) denotes the monomer conversion on the Jth space point in the Ith time step, X(I - 1,J) the monomer conversion on the Jth space point in the (I - 1)th time step, and [DELTA]X(I,J) the increment of the monomer conversion on the Jth space point in the Ith time step, the following equation can be obtained [[partial derivative]X/[partial derivative]t][|.sub.I,J] = [[DELTA]X(I,J)/[DELTA]t] + O([DELTA]t) = [[X(I,J) - X(I - 1,J)]/[DELTA]t] + O([DELTA]t) (11) where O([DELTA]t) denotes the truncation error Noun 1. truncation error - (mathematics) a miscalculation that results from cutting off a numerical calculation before it is finished miscalculation, misestimation, misreckoning - a mistake in calculating for the finite-difference equation, [DELTA]t the time step. The numerical computation expression of the increment of the monomer conversion can be obtained [DELTA]X(I,J) = [[k.sub.0][c.sub.ini.sup.1/2][e.sup.-[[E.sub.a]/RT]][DELTA]t[1 - X(I - 1,J)]]/[1 + [k.sub.0][c.sub.ini.sup.1/2][e.sup.-[[E.sub.a]/RT]][DELTA]t]. (12) Therefore, the numerical computation expression of the monomer conversion can be derived X(I,J) = [X(I - 1,J) + [k.sub.0][c.sub.ini.sup.1/2][e.sup.-[[E.sub.a]/RT]][DELTA]t]/[1 + [k.sub.0][c.sub.ini.sup.1/2][e.sup.-[[E.sub.a]/RT]][DELTA]t]. (13) The initial monomer conversion X(0,J) is zero and the initiator concentration can be obtained according to the semi-implicit iterative algorithm, therefore the monomer conversion in any time step can be calculated. Numerical Calculation of Initiator Concentration For a first-order decomposition reaction Noun 1. decomposition reaction - (chemistry) separation of a substance into two or more substances that may differ from each other and from the original substance chemical decomposition reaction, decomposition of initiator, similar to the numerical calculation of monomer conversion, the numerical computation expression of initiator concentration can be obtained [c.sub.ini](I,J) = [[c.sub.ini](I - 1,J)]/[1 + [k.sub.d][DELTA]t] (14) where [c.sub.ini] (I,J) denotes the initiator concentration on the Jth space point in the Ith time step, [c.sub.ini] (I - 1,J) the initiator concentration on the Jth space point in the (I - 1)th time step. Because the initial initiator concentration [c.sub.ini] (0,J) is known, the initiator concentration in any time step can be calculated. NUMERICAL CALCULATION OF AVERAGE MOLECULAR WEIGHT Hierarchies of Average Molecular Weight To carry out the numerical calculation of the average molecular weight in reactive extrusion processes by means of the semi-implicit iterative algorithm, the average molecular weight is considered to be those in three hierarchies: (1) the average molecular weight in the first hierarchy, [bar.M] (I,J), is that of the polymer chains produced on node (I,J); (2) the average molecular weight in the second hierarchy, [bar.M.sub.e] (I,J), is that of the total polymer chains lying on node (I,J), with the cumulative effect of polymer chains taken into account; (3) the average molecular weight in the third hierarchy, [bar.M.sub.ee] (I,J), is that of the fluid lying on node (I,J), with both reactant and production taken into account. Numerical Calculation of [bar.M] (I,J) When the chains are terminated via both combination and disproportionation Disproportionation or dismutation is used to describe two particular types of chemical reaction:[1]
[bar.M.sub.n](I,J) = [[M.sub.mono]([k.sub.t])[.sup.1/2][k.sub.p][c.sub.mono](I,J)]/[([k.sub.t] + [k".sub.t])[f[k.sub.d][c.sub.ini](I,J)][.sup.1/2]] (15) where [M.sub.mono] denotes the molecular weight of monomer, and [k.sub.t]" the rate constant for the disproportionation reaction. The weight-average molecular weight weight-average molecular weight: see molecular weight. [bar.M.sub.w] (I,J) can be calculated as [bar.M.sub.w] (I,J) = [[[gamma][k.sub.p][c.sub.mono](I,J)[M.sub.mono]]/[[f[k.sub.d][k".sub.t][c.sub.ini](I,J)][.sup.1/2]]] + [[1.5(1 - [gamma])[k.sub.p][c.sub.mono](I,J)[M.sub.mono]]/[[f[k.sub.d][k'.sub.t][c.sub.ini](I,J)][.sup.1/2]]] (16) where [gamma] denotes the fraction of termination by disproportionation, and [k.sub.t]" the rate constant for the combination reaction. Numerical Calculation of [bar.M.sub.e] (I,J) If [bar.x.sub.en] (I,J) denotes the number-average degree of polymerization of the total polymer chains lying on node (I,J), Eq. 17 can be obtained as [bar.x.sub.en](I,J) = [[1.summation over (i=1)] [[bar.x.sub.n](i,J)N(i,J)]]/[[1.summation over (i=1)]N(i,J)] (17) where N (i,J) denotes the concentration of the polymer chains, whose number-average degree of polymerization is [bar.x.sub.n] (i,J), produced on node (i,J). According to the incremental theory, Eq. 18 can be obtained N(i,J) = [[DELTA]X(i,J)[c.sub.mono,0]]/[[bar.x.sub.n](i,J)]. (18) Therefore, the numerical computation expression of number-average molecular weight, [bar.M.sub.en] (I,J), in the second hierarchy can be obtained [bar.M.sub.en](I,J) = [[M.sub.mono]X(I,J)]/[[1.summation over (i=1)] [[[DELTA]X(i,J)]/[[bar.x.sub.n](i,J)]]]. (19) Similar to the above-mentioned calculation, the numerical computation expression of weight-average molecular weight, [bar.M.sub.ew] (I,J), can be obtained as [bar.M.sub.ew](I,J) = [[M.sub.mono][1.summation over (i=1)] [bar.x.sub.n](i,J)[DELTA]X(i,J)]/X(I,J). (20) Numerical Calculation of [bar.M.sub.ee] (I,J) The numerical computation expression of weight-average molecular weight in the third hierarchy, [bar.M.sub.eew] (I,J), can be calculated [13] [bar.M.sub.eew] (I,J) = [M.sub.mono]X(I,J)[bar.x.sub.ew](I,J) + [M.sub.mono][1 - X(I,J)] = [M.sub.mono][1.summation over (i=1)][bar.x.sub.n](i,J)[DELTA]X(i,J) + [M.sub.mono][1 - X(I,J)]. (21) CONSTRUCTION OF CHEMORHEOLOGICAL MODELS To describe the rheological property of the steady-state flow of incompressible in·com·press·i·ble adj. Impossible to compress; resisting compression: mounds of incompressible garbage. in non-Newtonian fluids, the power-law constitutive equation In structural analysis, constitutive relations connect applied stresses or forces to strains or deformations. The constitutive relations for linear materials are linear, and termed Hooke's law. is adopted in this paper [19, 31] [eta] = [tau]/[bar.[gamma]] = a/[(1 + b[dot.[gamma]]).sup.c] (22) where [eta] denotes the apparent viscosity, [tau] the shear stress shear stress n. See shear. shear stress A form of stress that subjects an object to which force is applied to skew, tending to cause shear strain. , and [dot.[gamma]] the shear rate Shear rate is a measure of the rate of shear deformation:  For the simple shear case, it is just a gradient of velocity in a flowing material. . a, b, and c are three parameters. When [dot.[gamma]] = 0 and a = [[eta].sub.0]. To describe the relationship among zero shear viscosity [[eta].sub.0] of the fluid, mass concentration [c.sub.M] and weight-average molecular weight [bar.M.sub.ew] of the polymer chains, the formulation is adopted [32] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (23) Where [M.sub.c] denotes the critical molecular weight for entanglement effects in viscosity, [K.sub.1] and [K.sub.2] are constants which are related to temperature and molecular structure. To describe the temperature dependence of friction coefficient, the thermally activated Arrhenius equation The Arrhenius equation is a simple, but remarkably accurate, formula for the temperature dependence of a chemical reaction rate, more correctly, of a rate coefficient, as this coefficient includes all magnitudes that affect reaction rate except for concentration. is adopted [32] [[eta].sub.0](T) = K[e.sup.[[E.sub.[eta]]/RT]]. (24) Where K denotes the material constant, [E.sub.[eta]] the activation energy for fluid flow. Because closely intermeshing corotating twin screw extruders are considered as axisymmetrical models, the shear rate in the cylindrical coordinate system The cylindrical coordinate system is a three-dimensional coordinate system which essentially extends circular polar coordinates by adding a third coordinate (usually denoted can be obtained [33] [dot.[gamma].sub.rz] = [dot.[gamma].sub.zr] = [[[partial derivative][u.sub.r]]/[[partial derivative].sub.z]] + [[[partial derivative][u.sub.z]]/[partial derivative]r] (25) where [u.sub.r] and [u.sub.z] denote the r and z components of velocity. PROCESS OF SEMI-IMPLICIT ITERATIVE COMPUTATION According to the numerical computation expressions mentioned in the sections above, the semi-implicit iterative computation can be carried out. The detailed computation process is presented as follows. 1. On the basis of the temperature and the weight-average molecular weight in the ith iterative step, the zero shear viscosity in the (i + 1)th iterative step is calculated via Eqs. 23 and 24. 2. Based on the flow velocity in the ith iterative step, the shear rate in the (i + 1)th iterative step is calculated via Eq. 25. 3. The apparent viscosity in the (i + 1)th iterative step is calculated via Eq. 22. 4. The flow velocity in the (i + 1)th iterative step is calculated by means of the governing equations of fluid dynamics fluid dynamics n. (used with a sing. verb) The branch of applied science that is concerned with the movement of gases and liquids. [33, 34]. 5. The monomer conversion in the (i + 1)th iterative step is calculated via Eq. 13. 6. The weight-average molecular weight of the fluid in the (i + 1)th iterative step is calculated via Eq. 21. 7. Analogously repeating the above mentioned procedures, the variables in the (i + 2)th iterative step can be calculated. EXAMPLE AND VERIFICATION Initial and Boundary Conditions of an Example To validate the simulated results obtained on the basis of the three-dimensional equivalent model of a whole extruder and the semi-implicit iterative algorithm, the reactive extrusion process for the homopolymerization of n-butylmethacrylate (BMA BMA British Medical Association. ) in a self-wiping corotating twin screw extruder was investigated, and the simulated results were compared with the corresponding experimental results in Jongbloed's paper [35]. The input key data of the example are shown in Table 1 [1, 2, 10, 29, 32, 35]. The number of the nodes in the axial direction of the extruder is 8807, and the number of the nodes in the radial direction of the extruder is 119, therefore there are 1,048,033 nodes in the geometrical model. Consequently, the average dimensions of a grid can be obtained according to the dimensions of the model. According to the experimental design work in Ref. 35, the evolution of T along the axial direction of the extruder was mathematically formulated as shown in Fig. 3 (the axial length "0.7 m" is the extruder length, but not the length of the equivalent reactor model, which is calculated via Eq. 1). [FIGURE 3 OMITTED] [FIGURE 4 OMITTED] Simulated Results at the Throughput of 25 g/min The evolutions of [c.sub.mono], [c.sub.ini], X, [bar.M.sub.w], [bar.M.sub.ew], [bar.M.sub.eew], and [eta] along the axial direction of the extruder are shown in Figs. 4-10 ("R = xx m" represents the nodes whose distance to the centerline is xx m), respectively. It can be seen that with the increase of the fluid flow length, [c.sub.mono], [c.sub.ini], and [bar.M.sub.ew] decrease; X and [bar.M.sub.eew] increase; [bar.M.sub.w] first decreases, and then increases. It can be seen from Eq. 16 that [bar.M.sub.w] is dependent on [c.sub.mono] and [c.sub.ini]. With the increase of the fluid flow length, the decreasing tendency of [c.sub.mono] is different from the one of [c.sub.ini], resulting in the complex change of [bar.M.sub.w]. [FIGURE 5 OMITTED] With the increase of the fluid flow length, the evolution of [eta] is complex. It can be seen from Eqs. 22-24. that the zero shear viscosity is the increasing function (Math.) a function whose value increases when that of the variable increases, and decreases when the latter is diminished; also called a monotonically increasing function ltname>. See also: Increase of the weight-average molecular weight and the decreasing function of the temperature, and the apparent viscosity is the increasing function of the zero shear viscosity and the decreasing function of the shear rate. When the fluid flow length varies from 0 m to 0.09 m, the increase of [bar.M.sub.eew] is rapid but T is changeless change·less adj. Unchanging; constant. Adj. 1. changeless - not subject or susceptible to change or variation in form or quality or nature; "the view of that time was that all species were immutable, created by God" , so [eta] increases rapidly. When the fluid flow length varies from 0.09 m to 0.16 m, T is rapidly increasing but [bar.M.sub.eew] is slowly increasing, and then [eta] promptly decreases. When the fluid flow length varies from 0.16 m to 0.70 m, T is changeless again but [bar.M.sub.eew] is very slowly increasing, so [eta] slowly increases. [FIGURE 6 OMITTED] On the nodes with the same axial length of screws but the different radial distance, the values of the variables are different. The reasons are presented as follows. According to the equivalent reactor model, the velocity of the fluid near the model wall is bigger than the one in the inner. Therefore, with the increase of the distance to the centerline, the residence time of the fluid decreases, and resulting in the above phenomena. [FIGURE 7 OMITTED] Verification To make the comparison between the simulated conversion and the experimental results, Fig. 4 in Ref. 35 is redrawn in Fig. 11 in this article. In the simulation, the flow speeds in the entrance of the geometrical model are, respectively, 0.00222 m/s, 0.00444 m/s, and 0.00666 m/s, corresponding to the throughputs of 25 g/min, 50 g/min, and 75 g/min, respectively. [FIGURE 8 OMITTED] It can be seen that the simulated results are bigger than the experimental results when the throughput is 25 g/min, that the difference between them becomes small when the throughput is 50 g/min, and that the former is less than the latter when the throughput is 75 g/min. The main reasons for the above phenomena are as follows. 1. In the simulation, it is assumed that both the monomer and the initiator are well mixed in every element of the equivalent model during the whole extrusion process. In the experiment, the effects of the leakage flow and shear flow Shear flow is:-
[FIGURE 9 OMITTED] 2. In the simulation, the approximation on the steady-state concentrations of all the free radicals present in the system is adopted. But in the experiment, the gel effect often takes place at intermediate or high degrees of polymerization, and then results in the autoacceleration of the rate of the polymerization [38, 39]. [FIGURE 10 OMITTED] SUMMARY A new semi-implicit iterative algorithm has been proposed and used to deal with the complicated relationships among the variables, such as fluid flow velocity, pressure, macromolecular weight, and viscosity. The numerical computation expressions of the monomer conversion, the initiator concentration and the average molecular weight have been deduced, and then the finite volume simulation of the reactive extrusion process for free radical polymerization has been carried out. The simulated results and the experimental results of an example are nearly in agreement. Mixing elements are always present in twin screw extrusion, and the fill factor is not always equal to one, so the more precise three-dimensional reactor model should be built by means of computational fluid dynamics Computational fluid dynamics The numerical approximation to the solution of mathematical models of fluid flow and heat transfer. Computational fluid dynamics is one of the tools (in addition to experimental and theoretical methods) available to solve . Eq. 5 is deduced from low monomer conversion. At high monomer conversion phase, the polymerization kinetic equation should be more complex and built by means of dynamics of diffusion. [FIGURE 11 OMITTED] REFERENCES 1. A.J. Van Der Goot and L.P.B.M. Janssen, Adv. Polym. Tech., 16, 85 (1997). 2. L.P.B.M. Janssen, Polym. Eng. Sci., 38, 2010 (1998). 3. H.A. Jongbloed, R.K.S. Mulder, and L.P.B.M. Janssen, Polym. Eng. Sci., 35, 587 (1995). 4. B.J. Kim and J.L. White, J. Appl. Polym. Sci., 88, 1429 (2003). 5. L.X. Si, A.N. Zheng, H.B. Yang, R.Y. Guo, Z.N. Zhu, and Y.M. Zhang, J. Appl. Polym. Sci., 85, 2130 (2002). 6. W. Michaeli, A. Grefenstein, and W. Frings, Adv. Polym. Tech., 12, 25 (1993). 7. B. Siadat, M. Malone, and S. Middleman mid·dle·man n. 1. A trader who buys from producers and sells to retailers or consumers. 2. An intermediary; a go-between. , Polym. Eng. Sci., 19,787 (1979). 8. M.E. Hyun and S.C. Kim, Polym. Eng. Sci., 28, 743 (1988). 9. K.J. Ganzeveld and L.P.B.M. Janssen, Polym. Eng. Sci., 32, 457 (1992). 10. K.J. Ganzeveld, J.E. Capel, D.J. Vanderwal, and L.P.B.M. Janssen, Chem. Eng. Sci., 49, 1639 (1994). 11. R.A. de Graaf, M. Rohde, and L.P.B.M. Janssen, Chem. Eng. Sci., 52, 4345 (1997). 12. L.P.B.M. Janssen, P.F. Rozendal, H.W. Hoogstraten, and M. Cioffi, Int. Polym. Proc., 16, 263 (2001). 13. W. Michaeli and A. Grefenstein, Polym. Eng. Sci., 35, 1485 (1995). 14. W. Michaeli and A. Grefenstein, Adv. Polym. Tech., 14, 263 (1995). 15. B.J. Kim and J.L. White, J. Appl. Polym. Sci., 94, 1007 (2004). 16. J. Gimenez, M. Boudris, P. Cassagnau, and A. Michel, Polym. React. Eng., 8, 135 (2000). 17. J. Gimenez, M. Boudris, P. Cassagnau, and A. Michel, Int. Polym. Proc., 15, 20 (2000). 18. A. Poulesquen, B. Vergnes, P. Cassagnau, J. Gimenez, and A. Michel, Int. Polym. Proc., 16, 31 (2001). 19. B. Vergnes, V.G. Della, and L. Delamare, Polym. Eng. Sci., 38, 1781 (1998). 20. B. Vergnes, M. Vincent, Y. Demay, T. Coupez, N. Billon bil·lon n. 1. An alloy of gold or silver with a greater proportion of another metal, such as copper, used in making coins. 2. An alloy of silver with a high percentage of copper, used in making medals and tokens. , and J. F. Agassant, Can. J. Chem. Eng., 80, 1143 (2002). 21. F. Berzin and B. Vergnes, Int. Polym. Proc., 13, 13 (1998). 22. W. Zhu and Y. Jaluria, Polym. Eng. Sci., 42, 2120 (2002). 23. T. Fukuoka, Polym. Eng. Sci., 40, 2524 (2000). 24. S. David, T. Costas, and A.D. Thomas, Adv. Polym. Tech., 19, 22 (2000). 25. D. Goffart, D.W. Van, E.M. Klomp, H.W. Hoogstraten, L.P.B.M. Janssen, L. Breysse, and Y. Trolez, Polym. Eng. Sci., 36, 901 (1996). 26. D.W. Van, D. Goffart, E.M. Klomp, H.W. Hoogstraten, and L.P.B.M. Janssen, Polym. Eng. Sci., 36, 912 (1996). 27. D.M. Kalyon, A. Lawal, R. Yazici, P. Yaras, and S. Railkar, Polym. Eng. Sci., 39, 1139 (1999). 28. X.Z. Geng, Twin Screw Extruders & Its Application, China Light Industry Press, Beijing (2003). 29. H.R. Allcock, F.W. Lampe, and J.E. Mark. Contemporary Polymer Chemistry Polymer chemistry or macromolecular chemistry is a multidisciplinary science that deals with the chemical synthesis and chemical properties of polymers or macromolecules. , Science Press and Pearson Education Pearson Education is an international publisher of textbooks and other educational material, such as multimedia learning tools. Pearson Education is part of Pearson PLC. It is headquartered in Upper Saddle River, New Jersey. North Asia North Asia or Northern Asia is a subregion of Asia. The most common definition of the term is;
30. Y.X. Jia, S. Sun, S.X. Xue, L.L. Liu, and G.Q. Zhao, Polym. Plast. Technol. Eng., 42, 883 (2003). 31. A.J. Van Der Goot, R. Hettema, and L.P.B.M. Janssen, Polym. Eng. Sci., 37, 511 (1997). 32. Q.Y. Wu and J.A. Wu, Polymer Rheology, Higher Education Press, Beijing (2002). 33. J.D. Anderson, Computational Fluid Dynamics: The Basics With Applications, Tsinghua University Press and McGraw-Hill Beijing Office, Beijing (2002). 34. H.K. Versteeg and W. Malalasekera, An Introduction to Computational Fluid Dynamics: The Finite Volume Method The finite volume method is a method for representing and evaluating partial differential equations as algebraic equations. Similar to the finite difference method, values are calculated at discrete places on a meshed geometry. , World Publishing Corporation, Beijing (2000). 35. H.A. Jongbloed, J.A. Kiewiet, J.H.V. Dijk, and L.P.B.M. Janssen, Polym. Eng. Sci., 35, 1569 (1995). 36. E.J. Troelstra, L.L. Van Dierendonck, and L.P.B.M. Janssen, Polym. Eng. Sci., 42, 240 (2002). 37. K.J. Ganzeveld and L.P.B.M. Janssen, Ind. Eng. Chem. Res., 33, 2398 (1994). 38. M. Cioffi, A.C. Hoffmann, and L.P.B.M. Janssen, Chem. Eng. Sci., 56, 911 (2001). 39. M. Cioffi, K.J. Ganzeveld, A.C. Hoffmann, and L.P.B.M. Janssen, Polym. Eng. Sci., 44, 179 (2004). Yuxi Jia, (1,2) Guofang Zhang, (1) Lili Wu, (1) Sheng sheng (Chinese; “sage” or “saint”) In Chinese belief, a mortal who attains extraordinary or supernatural powers by self-cultivation and serves as a model for others. Confucius used the term to refer to exemplary rulers of the past. Sun, (1) Guoqun Zhao, (1) Lijia An (2) (1) School of Materials Science and Engineering Materials science and engineering A multidisciplinary field concerned with the generation and application of knowledge relating to the composition, structure, and processing of materials to their properties and uses. , Shandong University, Jinan 250061, People's Republic of China (2) Chinese Academy of Sciences The Chinese Academy of Sciences (CAS) (Simplified Chinese: 中国科学院; Pinyin: Zhōngguó Kēxuéyuàn), formerly known as Academia Sinica , Changchun Institute of Applied Chemistry, State Key Laboratory of Polymer Physics and Chemistry, Changchun 130022, People's Republic of China Correspondence to: Y. Jia; e-mail: jia_yuxi@sdu.edu.cn or L. An; e-mail: ljan@ciac.jl.cn Contract grant sponsor: National Natural Science Foundation of China; contract grant numbers: 50425517, 50573079, 50390096, 50340420392. Contract grant sponsor: Special Funds for Major State Basic Research Projects; contract grant number: 2003CB615601. TABLE 1. The input key data of the example Refs. 1, 2, 10, 29, 32, and 35. Parameters Numerical values Nominal diameter of screws 50 mm Centerline distance of screws 45 mm Slenderness ratio of screws 14 Number of thread starts 2 Lead of screws 52 mm Molecular weight of monomer 142 g [mol.sup.-1] Initial concentration of 6183.1 mol [m.sup.-3] monomer Initial concentration of 70.0 mol [m.sup.-3] initiator Molecular weight of initiator 208.4 g [mol.sup.-1] Efficiency of initiation 0.955 Frequency factor for 1.584082 x [10.sup.16] [s.sup.-1] initiator decomposition reaction Frequency factor for chain 5.1 x [10.sup.3] [m.sup.3] propagation reaction (mol s)[.sup.-1] Frequency factor for chain 7.0 x [10.sup.5] [m.sup.3] termination reaction (mol s)[.sup.-1] Activation energy for 1.42818 x [10.sup.5] J [mol.sup.-1] initiator decomposition reaction Activation energy for chain 2.64 x [10.sup.4] J [mol.sup.-1] propagation reaction Activation energy for chain 1.17 x [10.sup.4] J [mol.sup.-1] termination reaction The parameter b in power-law 0.2649 s equation The parameter c in power-law 0.7899 equation Critical molecular weight for 27500 g [mol.sup.-1] entanglement effects Activation energy for fluid 1.76 x [10.sup.5] J [mol.sup.-1] flow Flow speed on the wall of the 0.00333 m/s geometrical model Flow speed in the entrance of 0.00222 m/s the geometrical model |
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