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7-41
C
g
dC r
dt =
P7-18 (c)
Two CSTR’s
For 1st CSTR,
POLYMATH Results
NLES Solution
Variable Value f(x) Ini Guess
Cc 4.3333333 9.878E-12 4
Cs 1.3333333 1.976E-11 5
NLES Report (safenewt)
Explicit equations
[1] umax = 0.8
[8] V = 5000
CC1 = 4.33 g/dm3 X = 0.867
CS1 = 1.33 g/dm3 CP1 = YP/CCC1 =0.866 g/dm3
POLYMATH Results
NLES Solution
Variable Value f(x) Ini Guess
Cc 4.9334151 3.004E-10 4
Cs 0.1261699 6.008E-10 5
umax 0.8
NLES Report (safenewt)
Nonlinear equations
[1] f(Cc) = D*(Cc-Cc1)-rg = 0
[2] f(Cs) = D*(Cs1-Cs)+rs = 0
Explicit equations
[1] umax = 0.8
CC2 = 4.933 g/dm3 X = 0.987
CS2 = 1.26 g/dm3 CP1 = YP/CCC1 =0.9866 g/dm3
P7-18 (d)
For washout dilution rate, CC = 0
P7-18 (e)
For batch reactor,
See Polymath program P7-18-e.pol.
POLYMATH Results
Calculated values of the DEQ variables
Variable initial value minimal value maximal value final value
t 0 0 6 6
ODE Report (RKF45)
Differential equations as entered by the user
[1] d(Cc)/d(t) = rg
[2] d(Cs)/d(t) = rs
Explicit equations as entered by the user
P7-18 (g) Individualized solution
P7-18 (f) Individualized solution
7-44
P7-19
Given constants:
1
max / /
3 3
.5 , 2.0 , .2 , .1 , 50 , .2 , .3 , .1
s SO X S P S
g g g g g g
hr K C Y Y f
dm g ghr dm g g
µ! "
#
= = = = = = = =
Balance on cells
Balance on product ( lactic acid)
/
1
0 ( )
P
P C
P S
dC r C
dt Y
!µ"
= = #+ +
(3)
Rate of production = rP
Recycled cells
0.1 CC
7-45
fD =
µ
=
µ
max S
KS+S
S=fDKS
µ
max "fD
Plot rP versus D or
( ) 0
P
d r
dD =
to obtain the optimum D for maximizing (rP) the rate of lactic acid
production by utilizing the constants mentioned above.
Recycle CSTR
10
12
14
DP
Cell conc
P7-19 (a,b) Individualized solutions.
7-46
P7-20 (a)
X1 + S → More X1 + P1
X2 + X1 → More X2 + P2
For CSTR,
POLYMATH Results
Calculated values of the DEQ variables
Variable initial value minimal value maximal value final value
t 0 0 1 1
Cs 10 1.2366496 10 1.2366496
Cx1 25 25 25.791753 25.456756
ODE Report (RKF45)
Differential equations as entered by the user
[1] d(Cs)/d(t) = D*(Cso-Cs)-Ysx1*rgx1
Explicit equations as entered by the user
[1] Km1 = 10
7-47
P7-20 (b) When we increase D, CS increases, CX1 decreases, and CX2 has very little decrease.
P7-21 (a) and (b)
P7-21 (c)
P7-21 (d) Production starts at the end of the exponential (for both runs)
P7-21 (e)
P7-22 (a)
P7-22 (c)
P7-22 (d)
POLYMATH Results
Calculated values of the DEQ variables
Variable initial value minimal value maximal value final value
t 0 0 2 2
Cc 0.5 0.5 2.9 2.9
ODE Report (RKF45)
Differential equations as entered by the user
Explicit equations as entered by the user
[1] Ki = 50
7-51
P7-22 (e)
For semi-batch reactor,
POLYMATH Results
Calculated values of the DEQ variables
Variable initial value minimal value maximal value final value
t 0 0 4.5 4.5
Cc 0.5 0.2329971 2.2593341 2.2593341
Cs 2 0.8327919 24.016878 0.8327919
ODE Report (RKF45)
Differential equations as entered by the user
[1] d(Cc)/d(t) = rg-vo*Cc/V
[2] d(Cs)/d(t) = -Ysc*rg+vo*(Csin-Cs)/V
Explicit equations as entered by the user
[1] Ki = 50
[2] vo = 50
7-52
P7-22 (f) Individualized solution
P7-22 (g) Individualized solution
P7-23 (a)
P7-23 (b)
Answer correct, typos in Equation to get answer.
Rearranging Equation (7-89)
P7-23 (c)
P7-23 (d)
Vary CC0 and CS0 in Part (c) and describe what you find.
7-54
7-55
P7-24 No solution will be given.
P7-25 No solution will be given
CDP7-B
7-56
CDP7-C
CDP7-C (a)
Assumptions
7-57
CDP7-C (b)
To increase the growth rate, you could:
CDP7-C (c)
RC=yO2RO2
From equation (4), have:
CDP7-C (d)
Assumptions:
