r ie r2e

The complex exponential notation used here simplifies the analysis [549, 632, 634], Let Pij{uS) {i, j — 1,2) denote the individual elements of n(w). Then Eq. 7.84 can be deduced to have the form [p2i(w) = pi2(w)]

2ti tJO

In what follows, we examine the forms Eq. 7.85 reduces to when the number of inputs subject to periodic variation is 1 or 2.

Periodic Variation in Single Input

In bioreactor operations involving periodic variation in only one feed parameter (ri or r2 is positive), Eq. 7.85 reduces to 5J = pn(u})rf/2 (i = 1 or 2). It follows then that for superiority of forced periodic operation vis-a-vis steady-state operation, Pu(uj) (i = 1 or 2) must be positive for some iv.

Periodic Variations in Two Inputs

In bioreactor operations involving periodic variations in two feed parameters (i.e., rj ^ 0 or r2 ^ 0), let the frequencies of variations in Ui and u2 be the same or different with = 2wu>kt + <t>k {k — 1, 2, <j>i — 0).

Unequal forcing frequencies.

When lj\ ^ oj2, let the maximum of u>\ and ui2 be an integral multiple of the minimum of ui and u>2, i.e., max(ti>i, W2)/min(u;1, u>2) = n (n > 1, n an integer). Then

61 = 2nniu)t, 02 = 2-Kn2ujt + <p, nj = lj\/uj >1, n2= w2/w > 1, u! = min(o'i, u>2), r = max(ri, t2), ujjtj = u>t — 1, j = 1, 2 (7.86)

where m and n2 are integers.

In this case, Eq. 7.85 reduces to 5J=\ [piiMr? + P22(w2)r£] , Tj = 0 if Pjj{uj3) < 0, j = 1, 2. (7.87)

It is evident from Eq. 7.87 that the interaction between the control variables ui and U'2 vanishes when u>i ^ uio. Simultaneous periodic variations in u\ and u2 may provide improvement in process performance vis-a-vis periodic variations in u\ or u2 alone for those intervals of lo where both p\i and p-22 are positive. For a particular r) (= r2/VL). the optimum frequency (ujq) then is the frequency at which (pi \ + P22V2) maximized.

Equal forcing frequencies.

= \ [Pn(u)r{ + pnMrl + 2/nr2] , f(<t>, to) = \Re{p21) cos(^) + im(p21) sin(<£)] . (7.88)

The third term on the right side of Eq. 7.88 represents the interaction between the control variables u\ and w2. A positive effect of interaction between the two control variables in forced periodic operation involving perturbations in both u\ and i/,2 requires that / be positive. Maximization of 5J for a particular steady state requires that / be maximized. Since

\ I I /I \ Im{p2i) Re(p2i) , , f(<P, w) = P21 COS ((j) - X), \ = , x = \P21 , sm(x) cos(x) |p2l|= [{/2e(p2i)}2 + {/m(p21)}2]1/2, fmax = IP211 j (7.89)

a maximum in / (/max) occurs when cp — X- Positivity of / requires that <p lie in the interval — 7t/2 < ((¡> — x) < ^Where possible, optimum ratios of the amplitudes of weak perturbations in «1 and u2 which lead to maximization of 5J are identified next.

Fixed r2, variable 7*1.

The necessary and sufficient condition for occurrence of a local maximum in 5J and its value are

W = (p22 - —) at 77 = — = -J- if pn < 0. (7.90) 2 V Pn/ r2 p 11

The upper bound on 6J in Eq. 7.90 corresponds to / = \p2\\ (defined in Eq. 7.89). The necessary condition for a positive maximum in 5J is that |P2i|2 > P11P22.

If weak perturbations in u2 alone lead to improved process performance (p22 > 0), then it is evident from Eq. 7.90 that simultaneous variations in «1 and u2 lead to further improvement in the process performance. When P22 > 0, forced periodic operation involving variations in u\ and u2 is superior to steady-state operation only in the range 0 < rj < rj2 [tj — ri/r2, Figure 7.3(a)], with r]2 being

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