Page 1 Consider the unconstrained optimization of a CSTR with volume V. A ? B r = k[A] q [A] 0 [A] [B] V The goal is to maximize F B with respect to changes in the volumetric flow rate, q. F B = q[B] Steady state material balances on species A and B give: 0 = F A0 - F A - rV = q ([A] 0 - [A] ) - kV[A] 0 = F B0 - F B + rV = -q[B] + kV[A] Hence, [B] = k[A] (V / q ) and F B = rV = k[A]V ; thus production of B is maximized when [A] takes its maximum value, which is [A] 0 . Continuing with the material balances, we find: [A] = [A] 0 = [A] 0 1 + (kV / q ) 1 + k t When Da = k t << 1, [A] goes to [A] 0 . F B = rV = kV[A] = kV[A] o = kV[A] o 1 + k t 1 + kV / q o lim F B = lim ? ? ? kV[A] ? ? ? = kV[A] 0 q ? 8 q ? 8 ? 1 + kV / q ? Unfortunately, in the limiting case of infinite flow rate, the concentration of B in the output solution is vanishingly small: lim[B] = lim (k[A] (V / q ) ) = lim ? ? ? k [A] 0 (V / q ) ? ? ? = 0 . q ? 8 q ? 8 q ? 8 ? 1 + (kV / q ) ? Cite as: William Green, Jr., and K. Dane Wittrup, course materials for 10.37 Chemical and Biological Reaction Engineering, Spring 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].Read More

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