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13-1
Solutions for Chapter 13 – Distributions of Residence
Times for Chemical Reactors
P13-1 No solution will be given.
P13-2 (a)
The area of a triangle (h=0.044, b=5) can approximate the area of the tail :0.11
P13-2 (b)
P13-2 (c)
0t<
"
p
%
13-3
P13-2 (d)
X=0.75
For a PFR first order reaction:
( ) 39.14ln
1
1
ln ==
!
"
#
$
%
&
'
=X
Da
where
!
kDa =
Da
Da
Da
Da
Da
Da
+
!
#
$
&
+
+
!
"
#
$
%
&
=
exp4
2
exp
75.0
where
!
kDa =
P13-2 (e)
For a PFR, τ=5.15min, first order, liquid phase, irreversible reaction with k=0.1min-1.
402.01 =!=!
"
k
eX
13-4
P13-2 (f)
Decrease of 10
°
C in temperature
See Polymath program P13-2-f.pol
POLYMATH Results
Calculated values of the DEQ variables
Variable initial value minimal value maximal value final value
t 0 0 2.0E+04 2.0E+04
Xbar 0 0 0.6023837 0.6023837
ODE Report (RKF45)
Differential equations as entered by the user
Explicit equations as entered by the user
[1] k = .00493*exp(13300/1.9872*(1/323.15-1/313.15))
13-5
Decrease in reaction order from 2nd to pseudo 1st
See Polymath program P13-2-f-2.pol
POLYMATH Results
Calculated values of the DEQ variables
Variable initial value minimal value maximal value final value
t 0 0 2.0E+04 2.0E+04
ODE Report (RKF45)
Differential equations as entered by the user
[1] d(Xbar)/d(t) = X*E
Explicit equations as entered by the user
[1] k = 0.004
The decrease in reaction order from 2nd to pseudo 1st has the effect of increasing the exit
conversion by 20%. The smaller the dependency of the rate on CA means that when CA is below 1
13-6
POLYMATH Results
Calculated values of the DEQ variables
Variable initial value minimal value maximal value final value
t 0 0 2.0E+04 2.0E+04
Xbar 0 0 0.999375 0.999375
ODE Report (STIFF)
Differential equations as entered by the user
[1] d(Xbar)/d(t) = X*E
[2] d(X)/d(t) = k*(1-X)
Explicit equations as entered by the user
The mean conversion Xbar, the integral, is estimated to be 99.9%. The reaction is adiabatic and
exothermic as the temperature increases to a maximum of 1373.15 K once the batch conversion
P13-2 (g)
For a PFR, τ=40min, second order, liquid phase, irreversible reaction with k=0.01 dm3 /mol⋅min-1.
13-7
POLYMATH Results
Calculated values of the DEQ variables
Variable initial value minimal value maximal value final value
z 0 0 200 200
x 0 0 0.5938635 0.5632738
ODE Report (RKF45)
Differential equations as entered by the user
[1] d(x)/d(z) = -(ra/cao+E/(1-F)*x)
Explicit equations as entered by the user
[1] cao = 8
[2] k = .01
13-8
XMM
Xseg
XPFR
XCSTR
56%
61%
76%
58%
P13-2 (h)
Liquid phase, first order, Maximum Mixedness model
Rate Law:
A1A Ckr =!
where
-1
10.08minkk == Ao
C
CA=CAo 1"X
( )
POLYMATH Results
Calculated values of the DEQ variables
Variable initial value minimal value maximal value final value
z 0 0 200 200
x 0 0 0.7829342 0.7463946
cao 8 8 8 8
13-9
ODE Report (RKF45)
Differential equations as entered by the user
[1] d(x)/d(z) = -(ra/cao+E/(1-F)*x)
Explicit equations as entered by the user
[1] cao = 8
[2] k = 0.08
At z = 200, i.e. λ = 0 (exit), conversion X = 75 %.
The decrease in reaction order from 2nd to 1st has the effect of increasing the exit conversion by
19%. Once the concentration of A drops below 1 mol/dm3 then the rate of consumption of A does
not fall as rapidly (as the 2nd order reaction) and hence resulting in a larger conversion.
Liquid phase, third order, Maximum Mixedness model
Rate Law:
3
AA Ckr =!
( )
X1CC AoA !=
13-10
POLYMATH Results
Calculated values of the DEQ variables
Variable initial value minimal value maximal value final value
z 0 0 200 200
x 0 0 0.4867311 0.4614308
ODE Report (RKF45)
Differential equations as entered by the user
[1] d(x)/d(z) = -(-k*(1-x)^3+E/(1-F)*x)
Explicit equations as entered by the user
[1] cao = 8
[2] k = 0.08
[3] lam = 200-z
At z = 200, i.e. λ = 0 (exit), conversion X = 46.1 %.
Liquid phase, half order, Maximum Mixedness model
Rate Law:
2/1
AA Ck'r =!
( )
X1CC AoA !=
13-11
See Polymath program P13-2-h-3.pol
POLYMATH Results
Calculated values of the DEQ variables
Variable initial value minimal value maximal value final value
z 0 0 200 200
x 0 0 0.9334778 0.9038179
ODE Report (RKF45)
Differential equations as entered by the user
[1] d(x)/d(z) = -(-k*(1-x)^(.5)+E/(1-F)*x)
Explicit equations as entered by the user
[1] cao = 8
[2] k = 0.08
At z = 200, i.e. λ = 0 (exit), conversion X = 90 %.
13-12
P13-2 (i)
Assymetric RTD:
See Polymath program P13-2-i-1.pol
POLYMATH Results
Calculated values of the DEQ variables
Variable initial value minimal value maximal value final value
t 0 0 2.52 2.52
ca 1 0.0228578 1 0.0228578
cb 1 0.2840909 1 0.2840909
T 350 350 350 350
k1 1 1 1 1
k2 1 1 1 1
E1 -0.004 -27.414373 0.958793 -27.414373
ODE Report (RKF45)
Differential equations as entered by the user
[1] d(ca)/d(t) = ra
[2] d(cb)/d(t) = rb
13-13
Explicit equations as entered by the user
[1] T = 350
[2] k1 = exp((5000/1.987)*(1/350-1/T))
If the temperature is raised, the conversion of A increases. The selectivity Sc/d increases with
temperature and Sd/e decreases with increasing temperature
POLYMATH Results
Calculated values of the DEQ variables
Variable initial value minimal value maximal value final value
z 0 0 6 6
ca 1 0.2660482 1 0.2660482
cb 1 0.5350642 1 0.5352659
ceo 0 0 0 0
lam 6 0 6 0
T 350 350 350 350
k2 1 1 1 1
k1 1 1 1 1
13-14
ODE Report (RKF45)
Differential equations as entered by the user
[1] d(ca)/d(z) = -(-ra+(ca-cao)*EF)
Explicit equations as entered by the user
[1] cbo = 1
[10] rc = k1*ca*cb
[11] k3 = exp((9000/1.987)*(1/350-1/T))
[12] E1 = 0.47219*lam^4-1.30733*lam^3+0.31723*lam^2+0.85688*lam+0.20909
[13] E2 = 3.83999*lam^6-58.16185*lam^5+366.2097*lam^4-1224.66963*lam^3+2289.84857*lam^2-
2265.62125*lam+925.46463
[14] E3 = 0.00410*lam^4-0.07593*lam^3+0.52276*lam^2-1.59457*lam+1.84445
P13-2 (j)
13-15
POLYMATH Results
Calculated values of the DEQ variables
Variable initial value minimal value maximal value final value
z 0 0 200 200
x 0 0 0.9628524 0.9579239
cao 8 8 8 8
ODE Report (RKF45)
Differential equations as entered by the user
[1] d(x)/d(z) = -(ra/cao+E/(1-F)*x)
Explicit equations as entered by the user
[1] cao = 8
[2] T = 320+150*x
Endothermic Reaction: E=45Kj/mol
13-16
See Polymath program P13-2-j-1.pol
POLYMATH Results
Calculated values of the DEQ variables
Variable initial value minimal value maximal value final value
z 0 0 200 200
x 0 0 0.3017158 0.2860515
ODE Report (RKF45)
Differential equations as entered by the user
[1] d(x)/d(z) = -(ra/cao+E/(1-F)*x)
Explicit equations as entered by the user
[1] cao = 8
[2] T = 320-100*x
13-17
P13-2 (k)
Base case:
ODE Report (RKF45)
Differential equations as entered by the user
[1] d(Ca)/d(t) = ra/vo
Explicit equations as entered by the user
[1] vo = 10
[2] k1 = 1
PFR
K1/K2=1
K1/K2=2
K1/K2=0.5
CA
0.284
0.080
0.284
See Polymath program P13-2-k-2.pol
POLYMATH Results
NLES Solution
Variable Value f(x) Ini Guess
ca 0.4424779 4.704E-10 1
cb 0.2466912 -3.531E-10 0
cc 0.3108309 0 0
NLES Report (safenewt)
Nonlinear equations
13-18
Explicit equations
[1] cao = 1
[2] tau = 1.26
CSTR
K1/K2=1
K1/K2=2
K1/K2=0.5
CA
0.443
0.284
0.443
See Polymath program P13-2-k-3.pol
POLYMATH Results
Calculated values of the DEQ variables
Variable initial value minimal value maximal value final value
t 0 0 2.52 2.52
ca 1 0.0804596 1 0.0804596
cb 0 0 0.3678466 0.2027582
ODE Report (RKF45)
Differential equations as entered by the user
[1] d(ca)/d(t) = ra
13-19
Explicit equations as entered by the user
[1] k1 = 1
[2] k2 = 1
Asymmetric RTD
Segregation
Model
K1/K2=1
K1/K2=2
K1/K2=0.5
POLYMATH Results
Calculated values of the DEQ variables
Variable initial value minimal value maximal value final value
t 0 0 7 7
ca 1 9.119E-04 1 9.119E-04
cb 0 0 0.3678325 0.0063832
cc 0 0 0.9927049 0.9927049
cabar 0 0 0.3879174 0.3879174
ODE Report (RKF45)
Differential equations as entered by the user
[1] d(ca)/d(t) = ra
[2] d(cb)/d(t) = rb
Explicit equations as entered by the user
[1] k1 = 1
[2] k2 = 1
[3] cao = 1
Bimodal RTD
Segregation
Model
K1/K2=1
K1/K2=2
K1/K2=0.5
CA
0.388
0.213
0.388
Asymmetric RTD
Maximum
Mixedness
K1/K2=1
K1/K2=2
K1/K2=0.5
CA
0.306
0.110
0.306
P13-2 (l-r) No solution will be given at this time.
P13-3
Equivalency Maximum Mixedness and Segregation model for first order reaction:
