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When both pn and P22 are non-negative, on the family of curves 5 J = c (c is an arbitrary constant), an increase (a decrease) in r2 is accompanied by a decrease (an increase) in ri [Figure 7.3(b)], since dn {fn + p22r2) y '

There therefore are infinite sets of ri and r2 which lead to the same value of 8J for a particular Js when min(pn, p22) > 0, max(pn, p22) > 0 and / > 0. An optimum amplitude ratio, r\/r2, is as a result not admissible.

The contribution of the off-diagonal elements of II, pjk {j k, j, k = 1, 2 in the present case), to 5J is more significant than that of the diagonal elements, pjj (j = 1, 2) [449, 450, 571, 572, 573]. In the case study that follows, the forcing frequencies of inputs subject to periodic variation are therefore considered to be equal.

7.3.2 Case Study - Forced Periodic Operations

A unified analysis of optimality of forced periodic operation of continuous cultures producing a wide range of products and subject to periodic variation in dilution rate and/or feed concentration of the limiting substrate has been reported by Parulekar [448, 449]. It was established that very low frequency periodic operations around the optimal steady state, where admissible, are non-optimal. Conditions for properness of periodic control and expressions for frequency ranges where periodic control is proper and optimum cycling frequency were obtained analytically. It was established that subjecting a bioprocess to simultaneous periodic variations in dilution rate and substrate feed concentration does always lead to improved performance, at least at high frequencies [449].

Problem Formulation

The dynamics of continuous bioprocesses of interest (continuous pure cultures) is considered to be described adequately by the conservation equations for cell mass (biomass), limiting substrate and the desired non-biomass product. The mass balances for cell- and product-free feed are provided in Eqs. 7.29-7.31 with F/V being referred to as the dilution rate (D) for continuous culture. In situations where the desired product is excreted to a large extent and is subject to degradation in the abiotic phase, e can be expressed in terms of the cell mass-specific product synthesis rate (eo) and the volume-specific product degradation rate (Rd) as e = £0 — Rd/X• It is of interest to maximize the performance index of the type [448, 449]

with w (w > 0) being the cost of limiting substrate relative to the price of the desired product. The term involving the coefficient v in Eq. 7.95 accounts for the difference between the price of those products (other than the desired product) whose formation is associated with cell growth and the cost of separation of cell mass from the desired product relative to the price of the desired product. The objective function in Eq. 7.95 is therefore appropriate for optimizing operation of continuous bioprocesses that generate growth-associated and non-growth associated products and incorporates costs associated with separation of the desired product from cell mass. In both steady-state and periodic operations of continuous cultures, the input variable space is defined by the inequality constraints

with D* being the dilution rate beyond which retention of cells is not possible in a steady state continuous culture and Sp the maximum permissible concentration of the limiting substrate in the bioreactor feed (usually decided by solubility limits of the substrate in the feed medium).

Since the performance index considered here (Eq. 7.95) is non-positive for steady-state operations at D — 0, D — D* or SF = 0 (P = X = 0 for D > D* or SF =0), the optimal steady state solutions cannot lie on the boundaries D — 0, D = D* and SF = 0 of the control variable space. The optimal steady state solutions may therefore lie strictly in the interior of the control variable space (defined by the inequality constraints in Eq. 7.96) or on the boundary Sp = Sp.

The expressions for the scalars and vectors involved in evaluation of n(w) have the form h = D[P + vX - wSp]

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