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

Downloaded By: [HEAL-L**in**k Consortium] At: 12:27 29 July 2008 4 C. KIPARISSIDES et al. To simplify the above mathematical description **of** the process, the follow**in**g assumptions are made: (i) the density **of** the reaction mixture rema**in**s constant, (ii) transfer reactions are not significant, (iii) term**in**ation occurs by disproportionation only, and (iv) the quasi-steady state approximation (QSSA) for the live radicals is applicable. Accord**in**gly, Eqs. (I), (2), (6) and (8) are written: Note that pI can be expressed **in** terms **of** the monomer conversion and the **in**itial monomer concentration. pI=pl+AI=MO-M=M# (15) The number- and **weight**-average **molecular** **weight**s are given by where MW represents the **molecular** **weight** **of** MMA. The k**in**etic parameters appear**in**g **in** Eqs. (11)-(14) are the rate constant for **in**itiator decomposition, kd, and the ratio **of** the square **of** the propagation rate constant, k,,, over the term**in**ation 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 fitt**in**g the model Eqs. (11)-(18) to experimental conversion and **molecular** **weight** measurements obta**in**ed by Balke (1972). LINEAR QUADRATIC FEEDBACK CONTROL Our **control** objective is to design a l**in**ear feedback **control**ler, **in** order to ma**in**ta**in** 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 **in**itiator concentration. Assum**in**g that the state and **control** variables deviate slightly from their desired values, Eqs. (12)-(14) can be l**in**earized by a Taylor series expansion **of** the nonl**in**ear terms to obta**in** a l**in**earized state-space model **of** the form,

Downloaded By: [HEAL-L**in**k 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 represent**in**g the deviation **of** temperature from its nom**in**al 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 obta**in**ed by l**in**earization **of** the nonl**in**ear model equations around some specified nom**in**al trajectories (see Table I). S**in**ce the **reactor** is to be **control**led by a digital computer, it is more convenient to write Eq. (19) **in**to 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 **in**terval. The l**in**earized model was discretized based on a sampl**in**g time **of** five m**in**utes, which is the expected time for the on-l**in**e measurement **of** **molecular** **weight** distribution (Ponnuswamy, 1984). The design **of** constra**in**ed feedback **control**lers for l**in**ear, discrete state-space systems is a well known problem (Astrom and Wittenmark, 1984; Frankl**in** and Powell, 1981). For a l**in**ear plant Eq. (21), the optimal feedback **control** law

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