13-41
Maximum Mixedness
( )
( )X
F
E
C
r
d
dX
Ao
A
!
!
!
“
+= 1
See Polymath program P13-9-b-2.pol
POLYMATH Results
Calculated values of the DEQ variables
Variable initial value minimal value maximal value final value
z 0 0 14 14
X 0 0 0.3536026 0.3536026
13-42
ODE Report (RKF45)
Differential equations as entered by the user
[1] d(X)/d(z) = -(ra/Cao+E/(1-F)*X)
[2] d(F)/d(z) = -E
Explicit equations as entered by the user
[1] Cbo = .0313
[2] k = 175
X=35.4%
P13-9 (c)
Exit time(t), internal age(α) and life expectancy
λ
P13-9 (d)
Adiabatic reaction
Segregation model
13-43
POLYMATH Results
Calculated values of the DEQ variables
Variable initial value minimal value maximal value final value
t 0 0 14 14
xbar 0 0 0.7585435 0.7585435
ODE Report (RKF45)
Differential equations as entered by the user
Explicit equations as entered by the user
[1] cbo = 0.0313
X
=76%
Maximum Mixedness Model
( )
( )X
F
E
C
r
d
dX
Ao
A
!
!
!
“
+= 1
POLYMATH Results
Calculated values of the DEQ variables
Variable initial value minimal value maximal value final value
z 0 0 14 14
X 0 0 0.7189248 0.7189248
F 0.9999 -0.1138842 0.9999 -0.1138842
ODE Report (RKF45)
Differential equations as entered by the user
[1] d(X)/d(z) = -(ra/Cao+E/(1-F)*X)
[2] d(F)/d(z) = -E
Explicit equations as entered by the user
[1] Cbo = .0313
[2] T = 320+150*X
gives X=72%
If the reaction is carried out adiabatically the conversions are more than doubled.
13-45
P13-10
Irreversible, first order, long tubular reactor, constant volume, isothermal
For
0.2865.0 =!=
“
kX
For laminar flow with negligible diffusion (LFR), the mean conversion is given by:
2/
2/
$$
We can apply the approximated solution due to Hilder:
( )
( ) 782.0
4
44
5.0
5.0
=
++
!++
=
DaeDa
DaeDa
XDa
Da
where Da=kτ=2
85.0782.0 =<= PFR
XX
P13-11 (a)
First Moment about the mean: by definition is always equal to zero.
13-46
P13-11 (b)
Second-order liquid-phase reaction Da= τkCAo=1.0,τ=2min and kCAo=0.5min-1.
CSTR
VrFF AAAo !=!
XFFF AoAAo =!
PFR
A
Ar
dV
dF !=
!
AAo r
dV
dX
F!=
Liquid-phase ε = 0 and integrating
!
“
#
$
%
&
‘
=X
X
kC Ao 1
1
(
or
5.0
1=
+
=Da
Da
X
LFR
13-47
Evaluate for Da=1,
451.0=X
See Polymath program P13-11-b.pol
CSTR
PFR
LFR
P13-12
( )2
1AA
A
ACK
kC
r
+
=!
The criteria
The following figure shows the reaction rate as function of the concentration.
.
0.01
13-48
The second derivative is initially negative (Xseg<XMM), then positive (Xseg>XMM). The flex point
is for CA=8mol/dm3 (Xseg=XMM).
P13-13 No solution will be given
P13-14 (a)
Liquid phase, Segregation Model, second order, non-ideal CSTR, adiabatic:
0
E(t)=IF (t<=1) THEN (t) ELSE ( IF (t>=2) THEN (0) ELSE (2-t))
For a batch globule:
AAo r
dt
dX
C!=
2
AA kCr =!
13-49
0502/1002/1 =!=!=“papb CCCp
T=300+150X
POLYMATH Results
Calculated values of the DEQ variables
Variable initial value minimal value maximal value final value
t 0 0 3 3
ODE Report (RKF45)
Differential equations as entered by the user
[1] d(xbar)/d(t) = E*x
[2] d(x)/d(t) = k*cao*((1-x)^2)
Explicit equations as entered by the user
[1] cao = 2
P13-14 (b)
Parallel reactions, isothermal, segregation model
Batch globules
13-50
BACAC
cCCkrr
dt
dC
22 =!==
Exit concentrations
E(t)=IF (t<=1) THEN (t) ELSE ( IF (t>=2) THEN (0) ELSE (2-t))
P13-15
Reactor: fluidised CSTR (V=1m3; F=10dm3/s, CC
0=2 Kmol/m3)
τ = 1000dm3s/10dm3 = 100s
The system of complex reactions for the Kentucky coal n.9 is given by
P13-15 (a)
Segregation Model
See Polymath program P13-15-a.pol
POLYMATH Results
Calculated values of the DEQ variables
Variable initial value minimal value maximal value final value
t 0 0 20 20
ca 0 0 900.42918 900.42918
cp 0 0 689.61265 689.61265
ODE Report (RKF45)
Differential equations as entered by the user
[1] d(ca)/d(t) = ra
Explicit equations as entered by the user
[1] k1 = 0.012
[2] k2 = 0.046
P13-15 (b)
Maximum Mixedness model
See Polymath program P13-15-b.pol
POLYMATH Results
Calculated values of the DEQ variables
Variable initial value minimal value maximal value final value
z 0 0 20 20
Ca 0 0 875.11273 875.11273
rc 0 -47.836902 0 -47.836902
ra 92 15.190162 92 15.190162
rp 24 24 37.327161 34.677371
tau 1.667 1.667 1.667 1.667
ro 0 0 35.004509 35.004509
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] k1 = 0.012
[2] k2 = 0.046
[3] k3 = 0.020
13-53
P13-15 (c)
The selectivities are reported in the following table:
P13-15 (d)
Normal Distribution with τ = 5min and σ = 3min
POLYMATH Results
Calculated values of the DEQ variables
Variable initial value minimal value maximal value final value
t 0 0 2 2
ca 0 0 171.75027 171.75027
cp 0 0 52.879376 52.879376
cc 2000 1991.2989 2000 1991.2989
k1 0.012 0.012 0.012 0.012
k2 0.046 0.046 0.046 0.046
k3 0.02 0.02 0.02 0.02
k4 0.034 0.034 0.034 0.034
ODE Report (RKF45)
Differential equations as entered by the user
[1] d(ca)/d(t) = ra
[2] d(cp)/d(t) = rp
13-54
Explicit equations as entered by the user
[1] k1 = 0.012
[2] k2 = 0.046
[3] k3 = 0.020
Maximum Mixedness Model
See Polymath program P13-15-d-2.pol
POLYMATH Results
POLYMATH Report 08-25-2005, Rev5.1.233
Calculated values of the DEQ variables
Variable initial value minimal value maximal value final value
z 0 0 2 2
Ca 0 0 120.77791 120.77791
Cp 0 0 35.906087 35.906087
rc 0 -5.986657 0 -5.986657
ra 92 83.563477 92 83.563477
rp 24 24 27.331698 27.331698
lam 2 0 2 0
ro 0 0 4.8311165 4.8311165
ODE Report (RKF45)
Differential equations as entered by the user
[1] d(Ca)/d(z) = -(-ra+(Ca-Cao)*EF)
13-55
Explicit equations as entered by the user
[1] k1 = 0.012
[2] k2 = 0.046
[3] k3 = 0.020
[4] k4 = 0.034
[5] k5 = 0.04
P13-16
Multiple parallel reactions, isothermal
P13-16 (a)
Segregation model
3/2
42
2
13
2
ACBABA
ACCkCCkCCk
dt
dC !!!=
2
32
2
175.025.1 CBBABA
BCCkCCkCCk
dt
dC !!!=
13-56
D
E
F
See Polymath program P13-16-a.pol
Exit concentrations: Selectivities:
3
/026.0 dmmolC A=
00196.0=
CD
S
P13-16 (b)
Maximum Mixedness model
E(t)=0.0279693-0.0008527λ+1.2778e-5λ2-1.0661e-7 λ3+4.5747e-10 λ4-7.73108e-13 λ5
And λ=tf-z where z=t, tf=200min (extent of E(t)).
and so on for other species.
( ) ( )
!
!
E
dz
dF “=
gives F(λ)
See Polymath program P13-16-b.pol
Exit concentrations: Selectivities:
3
/028.0 dmmolC A=
0009.0=
CD
S
13-57
P13-16 (c)
Ideal CSTR
tm=τ and
( )
!
!
/t
e
tE
“
=
Mol balances:
and so on for the other species.
See Polymath program P13-16-c.pol
Exit concentrations: Selectivities:
3
/050.0 dmmolC A=
068.0=
CD
S
Ideal PFR
tm=τ and RTD function
( ) ( )
!“
#=ttE
( )
A
Ar
dV
dF !=
Where
3/2
42
2
13
2
ACBABAA CCkCCkCCkr !!!=
and so on for other species.