Page 266 - Numerical Methods for Chemical Engineering

Problems 255

subject to the constraints that for each road segment,
[k+1]
z r − z r
[k]
≤ max max = 0.08 (5.181)
,
2 2
[k+1] [k] [k+1] [k]
x − x + y − y
Using this approach, propose a path for the road to follow.
5.B.4. We wish to produce C from A and B by the reaction network
A + B → C + D r R1 = k 1 (T )c A c B
C + B → S 1 + D r R2 = k 2 (T )c C c B
(5.182)
A + D → S 2 r R3 = k 3 (T )c A c D
A + B → S 3 r R4 = k 4 (T )c A c B
C + B → S 4 r R5 = k 5 (T )c C c B
A CSTR has an input stream with a velocity of 1 l/s containing species A and B in a carrier
solvent, such that
c A0 + c B0 < 2 M (5.183)

We have the following temperature-dependent rate constant data,
l l
k 1 (298 K) = 0.01 k 1 (310 K) = 0.02 k 2 (T ) = k 1 (T )
mol s mol s
l l
k 3 (298 K) = 0.001 k 3 (310 K) = 0.005 (5.184)
mol s mol s
l l
k 4 (298 K) = 0.001 k 4 (310 K) = 0.005 k 5 (T ) = k 4 (T )
mol s mol s
We wish to design the reactor (assumed operated isothermally) to maximize the concentra-
tion of C in the output stream. We vary the inlet concentrations c A0 and c B0 , the volume of
the reactor V, within the range
10 l ≤ V ≤ 10 000 l (5.185)

and the temperature T within the range

298 K ≤ T ≤ 335 K (5.186)
Propose an optimal steady-state CSTR design.
5.C.1. You wish to control the height h(t) of water in a cylindrical tank of diameter 50 cm
by varying the inlet volumetric flow rate υ 0 (t) in liters per second. There is an outlet hole
at the bottom of the tank of diameter 1 cm. Use Bernoulli’s equation to propose an ODE
model for h(t). Then, compute the optimal feed control law υ 0 (h), based on minimizing the
cost functional

t H '
1
[0] C U 2 2 2
F υ 0 (t); h = [υ 0 (s) − υ 0, set ] + [h(s) − h set ] ds + C H [h(t H ) − h set ]
2
0
(5.187)