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control of molecular weight in a batch polymerization reactor using ...

control of molecular weight in a batch polymerization reactor using ...

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Downloaded By: [HEAL-Link Consortium] At: 12:27 29 July 2008 4 C. KIPARISSIDES et al. To simplify the above mathematical description of the process, the following assumptions are made: (i) the density of the reaction mixture remains constant, (ii) transfer reactions are not significant, (iii) termination occurs by disproportionation only, and (iv) the quasi-steady state approximation (QSSA) for the live radicals is applicable. Accordingly, Eqs. (I), (2), (6) and (8) are written: Note that pI can be expressed in terms of the monomer conversion and the initial monomer concentration. pI=pl+AI=MO-M=M# (15) The number- and weight-average molecular weights are given by where MW represents the molecular weight of MMA. The kinetic parameters appearing in Eqs. (11)-(14) are the rate constant for initiator decomposition, kd, and the ratio of the square of the propagation rate constant, k,,, over the termination rate constant, k,, (k, = kilk,). Experimental data on free-radical bulk polymerization of MMA show that k, varies with conversion and temperature due to the appearance of gel effect. Thomas and Kiparissides (1984) developed an empirical model to express the variation of k, with temperature and conversion. k, = klog(x, T) = k,, exp(Clx3 + C2x2 + Cg) (17) where kl0 is the value of k, at zero conversion and the parameters C1, C2 and C3 are assumed to be functions of temperature. ci = Cil/~ + c,,; i = I, 2, 3 (18) The numerical values of C, parameters were estimated by fitting the model Eqs. (11)-(18) to experimental conversion and molecular weight measurements obtained by Balke (1972). LINEAR QUADRATIC FEEDBACK CONTROL Our control objective is to design a linear feedback controller, in order to maintain the state variables x, p0 and p, along some specified desired trajectories, x,, p, and p, despite the presence of process disturbances in the total initiator concentration. Assuming that the state and control variables deviate slightly from their desired values, Eqs. (12)-(14) can be linearized by a Taylor series expansion of the nonlinear terms to obtain a linearized state-space model of the form,

Downloaded By: [HEAL-Link Consortium] At: 12:27 29 July 2008 POLYMERIZATION REACTOR CONTROL 5 TABLE I Elements of A, B and D matrices in Eq. (19) all = (k1,g,l,)ln[-1 + 1.5CI,x~(1 -x,) + C,x,(l -x,)] a,, = 2k,,gSM~(1 -I,)[-2 + 3Cl,xf(l -x,) + 2C,x,(l -x,)] b, = (k1,g,l,)ln(l -xs)(0.5/T~)[E,lR b2 = 'WdrI,EdI(RT:) - Cl1xS - C+:- C,,] b3=2k1,gSMg(1 -x,)~(~IT:)[E,IR - Cllx: - CZ2x: - C,,] 4 = ( k1,~~lJ'~(1- *,)I2 4 = 2Fdr where Y is a (3 x 1) state vector U is the control variable representing the deviation of temperature from its nominal value, (T - T,), and V = (I - I,) represents the disturbance variable. A, B and D are (3 x 3), (3 x 1) and (3 x 1) time dependent system matrices obtained by linearization of the nonlinear model equations around some specified nominal trajectories (see Table I). Since the reactor is to be controlled by a digital computer, it is more convenient to write Eq. (19) into an equivalent discrete state-space form (Astrom and Wittenmark, 1984). cu,i, bi and d, are the elements of A, B and D matrices, respectively, and s is the discrete time interval. The linearized model was discretized based on a sampling time of five minutes, which is the expected time for the on-line measurement of molecular weight distribution (Ponnuswamy, 1984). The design of constrained feedback controllers for linear, discrete state-space systems is a well known problem (Astrom and Wittenmark, 1984; Franklin and Powell, 1981). For a linear plant Eq. (21), the optimal feedback control law

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