P = {Pij, i,j = 1,2,3; pij = pji, i ^ j}, H = + ^ Aj/i

with f\, fi, and /3 being the right sides of Eqs. 7.29-7.31, respectively, and the elements of P being

Pn = (2/xx + HxxX) Xx - (2ax + crXxX) A2 + (2ex + £xxX) A3, P12 = (Ms + VxsX) Ai - (as + oXsX) A2 4- (es + ExsX) A3, P31 = (mp + VxpX) Ai - (aP + (TXpX) A2 + (ep + eXpX) A3, P22 = (mssAi - ass+ ess^s) X, P23 = (msp^I - asp\2 + 3)

The adjoint variables at a steady state are obtained from solution of Eq. 7.81, which in this case assume the form miAi — to2A2 + m3A3 = —vD, m4Aj - msA2 + m6A3 = 0, myXx - mgA2 -I- mgA3 = -D, mx~ D + Xfix, m2 = ct + Xux, m3 = e + Xex, m4 = m5 = D + m6 = Xss, m7 — X/xp, ms — Xap, 7719 = Xep — D. (7.99)

For the problem formulation described by Eqs. 7.29-7.31, depending on the nature of relations among the three rate processes, various bioprocesses can be classified into three types as [448, 449]: (I) bioprocesses where a and e are each related linearly to /i, (II) bioprocesses where /x, a and e are related by a single linear relation, and (III) bioprocesses where /i, a and e are not related linearly. For type I bioprocesses, the relations in Eq. 7.45 are applicable. In steady-state operations and forced periodic operations with variation in D alone, the state variables X, S, and P satisfy the stoichiometric relations in Eq. 7.50 (Sp = Sfo)- For type II bioprocesses, the three specific rates are related linearly as in Eq. 7.53. In steady-state operations and forced periodic operations with variation in D alone, the state variables X, S, and P satisfy the stoichiometric relation in Eq. 7.61 (Sf = Sfo)- The analysis of forced periodic operation is simplified considerably if the bioreactor state (X, S, P) in steady-state and forced periodic operations satisfies Eq. 7.50 or Eq. 7.61, as appropriate [448, 449]. For details of the analysis of the forced periodic operations of the three bioprocess types, the reader should refer to [448, 449].

Forced periodic operations of continuous cultures may in some situations extend the regions of the operating parameter space where non-washout solutions are admissible [322]. The extension of the regions of admissibility of the meaningful states of continuous cultures via forced periodic operations subject to weak variations in inputs can be investigated by applying the 7r-criterion to the washout steady state (X = P = 0, S = SF) when it is locally, asymptotically stable, the necessary and sufficient condition for which is that p(XF, SF, PF) < D [448]. It is established in [448] that weak periodic perturbations in D and Sf will allow for cell retention under conditions where such retention is not possible in steady-state continuous culture operation as long as the phase difference between the perturbations in the two inputs lies between 90° and 270°. For the performance index under consideration (Eq. 7.95), in view of the form of h in Eq. 7.97, Tin and i?22 (diagonal elements of R) are trivial and P and Q (Eq. 7.79) do not depend on w. For the three types of bioprocesses therefore, pn(w) and P22(w) are independent of w [448] .We import here numerical results from specific examples in [449].

Example 1.

This example pertains to type I bioprocesses (Eq. 7.45) with p, being dependent exclusively on S. For a locally, asymptotically stable steady state (ps > 0, uss < 0), improvement in bioprocess performance via periodic forcing in D or SF alone is not possible since pu < 0 and p22 < 0 for all to (0 < ui < 00) [448]. Superiority of forced periodic operations subject to simultaneous variations in D and Sf over steady-state operation at a locally, asymptotically stable non-trivial steady state is guaranteed at low and high frequencies (Figure 7.4). This observation is valid in the entire portion(s) of the Sp — D space where stable non-trivial steady states are admissible. The results in Figure 7.4 are for Monod kinetics [p = pmS/(Ks + S)\ with pm = 1.0 h~l, Ks = 0.05 gL'1, and w = 0.

Example 2.

This example pertains to type I bioprocesses with p being dependent on S, X, and P (Eq. 7.50), px and pp being negative. Case studies of these bioprocesses include fermentations producing alcohols [47, 239, 325, 326, 338, 395, 544]. For numerical illustration, p, is expressed as [326, 395]

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