1 TAME synthesis problem Tert-Amyl Methyl Ether (TAME) is an ...
1 TAME synthesis problem Tert-Amyl Methyl Ether (TAME) is an ...
1 TAME synthesis problem Tert-Amyl Methyl Ether (TAME) is an ...
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<strong>TAME</strong> <strong>synthes<strong>is</strong></strong> <strong>problem</strong><br />
<strong>Tert</strong>-<strong>Amyl</strong> <strong>Methyl</strong> <strong>Ether</strong> (<strong>TAME</strong>) <strong>is</strong> <strong>an</strong> oxygenated additive for green gasolines. Besides its use as <strong>an</strong><br />
oct<strong>an</strong>e enh<strong>an</strong>cer, it also improves the combustion of gasoline <strong>an</strong>d reduces the CO <strong>an</strong>d HC (<strong>an</strong>d, in a<br />
smaller extent, the NOx) automobile exhaust em<strong>is</strong>sions. Due to the environmental concerns related to<br />
those em<strong>is</strong>sions, th<strong>is</strong> <strong>an</strong>d other ethers (MTBE, ETBE, TAEE) have been lately studied intensively.<br />
<strong>TAME</strong> <strong>is</strong> currently catalytically produced in the liquid phase by the reaction of meth<strong>an</strong>ol (MeOH) <strong>an</strong>d<br />
the <strong>is</strong>oamylenes 2-methyl-1-butene (2M1B) <strong>an</strong>d 2-methyl-2-butene (2M2B). There are three<br />
simult<strong>an</strong>eous equilibrium reactions in the formation <strong>an</strong>d splitting of <strong>TAME</strong>: the two etherification<br />
reactions <strong>an</strong>d the <strong>is</strong>omerization between the <strong>is</strong>oamylenes:<br />
2 M1B + MeOH ⇔ <strong>TAME</strong><br />
(1)<br />
2 M 2B<br />
+ MeOH ⇔ <strong>TAME</strong><br />
(2)<br />
2M1B ⇔ 2M<br />
2B<br />
(3)<br />
These reactions are to be carried out in a plug flow reactor <strong>an</strong>d a membr<strong>an</strong>e reactor in which MeOH <strong>is</strong><br />
fed uniformly through the sides.<br />
For <strong>is</strong>othermal operation:<br />
a) Plot the concentration profiles for a 10 m 3 PFR.<br />
b) Vary the entering temperature, T0, <strong>an</strong>d plot the exit concentrations as a function of T0.<br />
For a reactor with heat exch<strong>an</strong>ge (U = 10 J.m -2 .s -1 .K -1 ):<br />
c) Plot the temperature <strong>an</strong>d concentration profiles for <strong>an</strong> entering temperature of 353 K<br />
d) Repeat (a) through (c) for a membr<strong>an</strong>e reactor.<br />
NOTE: To simplify the <strong>problem</strong>, consider the solution density <strong>an</strong>d heat capacity const<strong>an</strong>t (evaluate<br />
them for the entering temperature <strong>an</strong>d the feed mole fractions) <strong>an</strong>d consider the solution as ideal: ai = xi<br />
in the rate expressions.<br />
A new <strong>an</strong>d improved kinetic model, including the non-ideal character of the reacting solution, c<strong>an</strong> be<br />
found in “Number of Actives Sites in <strong>TAME</strong> Synthes<strong>is</strong>: Mech<strong>an</strong><strong>is</strong>m <strong>an</strong>d Kinetic Modeling”, M<strong>an</strong>uela<br />
V. Ferreira <strong>an</strong>d José M. Loureiro, Ind. Eng. Chem. Res. 43 (2004) 5156-5165.<br />
1
Additional information for solving the <strong>problem</strong><br />
• Thermodynamic equilibrium const<strong>an</strong>ts, activity based (Vilarinho Ferreira <strong>an</strong>d Loureiro, 2001)<br />
3<br />
⎛ 5.<br />
0166×<br />
10 ⎞<br />
Keq = exp ⎜<br />
− 10.<br />
839 ⎟<br />
1<br />
(I.1)<br />
⎝ T<br />
⎠<br />
3<br />
⎛ 3.<br />
7264×<br />
10 ⎞<br />
Keq = exp ⎜<br />
− 9.<br />
6367 ⎟<br />
2<br />
(I.2)<br />
⎝ T<br />
⎠<br />
Keq<br />
Keq = (I.3)<br />
1<br />
3<br />
Keq2<br />
with: T in Kelvin<br />
• Kinetic const<strong>an</strong>ts for the direct reactions (Kivir<strong>an</strong>ta-Pääkkönen et al., 1998)<br />
k<br />
k<br />
k<br />
1<br />
2<br />
3<br />
3<br />
10 ⎛ 76.<br />
8×<br />
10 ⎞<br />
= 3 . 2870 × 10 exp ⎜<br />
⎜−<br />
⎟<br />
(I.4)<br />
⎝ R T ⎠<br />
3<br />
13 ⎛ 99.<br />
7 × 10 ⎞<br />
= 3 . 9682 × 10 exp ⎜<br />
⎜−<br />
⎟<br />
(I.5)<br />
⎝ R T ⎠<br />
3<br />
10 ⎛ 81.<br />
7 × 10 ⎞<br />
= 7 . 4767 × 10 exp ⎜<br />
⎜−<br />
⎟<br />
(I.6)<br />
⎝ R T ⎠<br />
with: k in mol.<br />
T in Kelvin<br />
−1<br />
kg cat .s -1<br />
R = 8.314 J.mol -1 .K -1<br />
• Adsorption const<strong>an</strong>ts for each component, activity based (calculated <strong>an</strong>d adapted from Oktar et<br />
al., 1999)<br />
3<br />
⎛ 4.<br />
6825×<br />
10 ⎞<br />
K1B = exp ⎜<br />
− 10.<br />
157 ⎟ , 1B = 2M1B (I.7)<br />
⎝ T<br />
⎠<br />
3<br />
⎛ 3.<br />
4420 × 10 ⎞<br />
K = exp ⎜<br />
− 6.<br />
5849 ⎟<br />
2B<br />
, 2B = 2M2B (I.8)<br />
⎝ T<br />
⎠<br />
3<br />
⎛1.<br />
0014 × 10 ⎞<br />
K M = exp ⎜<br />
+ 4.<br />
7496 ⎟ , M = MeOH (I.9)<br />
⎝ T<br />
⎠<br />
3<br />
⎛ 2.<br />
3934 × 10 ⎞<br />
KT = exp ⎜<br />
− 3.<br />
5736 ⎟ , T = <strong>TAME</strong><br />
⎝ T<br />
⎠<br />
with: T in Kelvin<br />
(I.10)<br />
2
1B<br />
r<br />
2B<br />
• Heat of reaction (Vilarinho Ferreira <strong>an</strong>d Loureiro, 2001)<br />
R<br />
∆ H1 = - 41.708 kJ.mol -1<br />
R<br />
∆ H 2 = - 30.981 kJ.mol -1<br />
R<br />
∆ H 3 = - 10.727 kJ.mol -1<br />
• Bulk density <strong>an</strong>d bed porosity<br />
The PFR <strong>is</strong> filled with a macroreticular strong cation ion-exch<strong>an</strong>ge resin in hydrogen form<br />
(Amberlyst 15 Wet, Rohm & Haas).<br />
The bulk density <strong>is</strong>: ρ = 770 g / L<br />
The bed porosity <strong>is</strong>: ε = 0.4<br />
b<br />
• Rate equations for the formation of each component (Vilarinho Ferreira <strong>an</strong>d Loureiro, 2001)<br />
=<br />
=<br />
k<br />
1<br />
K<br />
2<br />
( 1 + K a + K a + K a + K a ) ( 1 + K a + K a + K a + K a )<br />
k<br />
2<br />
1B<br />
K<br />
M<br />
K<br />
1B<br />
1B<br />
a<br />
M<br />
2B<br />
a<br />
1B<br />
⎛ 1<br />
⎜<br />
⎝ Keq<br />
2B<br />
1<br />
M<br />
aT<br />
a a<br />
M<br />
M<br />
1B<br />
⎞<br />
−1<br />
⎟<br />
⎠<br />
T<br />
T<br />
+<br />
1B<br />
k<br />
3<br />
1B<br />
K<br />
1B<br />
a<br />
1B<br />
2B<br />
⎛ 1<br />
⎜<br />
⎝ Keq<br />
2B<br />
3<br />
a<br />
a<br />
2B<br />
1B<br />
M<br />
⎞<br />
−1<br />
⎟<br />
⎠<br />
2<br />
( 1 + K a + K a + K a + K a ) ( 1 + K a + K a + K a + K a )<br />
1B<br />
rM r1<br />
B r2<br />
B +<br />
T<br />
M<br />
K<br />
1B<br />
2B<br />
a<br />
M<br />
2B<br />
a<br />
1B<br />
⎛ 1<br />
⎜<br />
⎝ Keq<br />
2B<br />
2<br />
M<br />
aT<br />
a a<br />
M<br />
M<br />
2B<br />
⎞<br />
−1<br />
⎟<br />
⎠<br />
T<br />
T<br />
−<br />
1B<br />
k<br />
3<br />
1B<br />
K<br />
1B<br />
a<br />
1B<br />
2B<br />
⎛ 1<br />
⎜<br />
⎝ Keq<br />
2B<br />
3<br />
a<br />
a<br />
2B<br />
1B<br />
M<br />
M<br />
⎞<br />
−1<br />
⎟<br />
⎠<br />
M<br />
T<br />
T<br />
T<br />
T<br />
(I.11)<br />
(I.12)<br />
= (I.13)<br />
r = − r<br />
(I.14)<br />
M<br />
with: r in mol.<br />
−1<br />
kg cat .s -1<br />
ai st<strong>an</strong>ds for the activity of component i in the liquid phase, <strong>an</strong>d since we are considering the<br />
solution as ideal:<br />
a = x<br />
(I.15)<br />
i<br />
i<br />
where xi <strong>is</strong> the mole fraction of component i in the liquid phase.<br />
We c<strong>an</strong> also define the rate of each reaction:<br />
3
k<br />
K<br />
K<br />
a<br />
a<br />
1<br />
a<br />
⎞<br />
−1⎟<br />
( ) 2<br />
T<br />
1 M 1B<br />
M 1B<br />
⎜ Keq1<br />
aM<br />
a ⎟<br />
1B<br />
1 =<br />
⎝<br />
⎠<br />
(I.16)<br />
1 + K1B<br />
a1B<br />
+ K 2B<br />
a2<br />
B + K M aM<br />
+ KT<br />
aT<br />
r<br />
k<br />
K<br />
K<br />
a<br />
a<br />
⎛<br />
⎜<br />
1<br />
a<br />
⎞<br />
−1⎟<br />
( ) 2<br />
T<br />
2 M 2B<br />
M 1B<br />
⎜ Keq2<br />
aM<br />
a ⎟<br />
2B<br />
2 =<br />
⎝<br />
⎠<br />
(I.17)<br />
1 + K1B<br />
a1B<br />
+ K 2B<br />
a2<br />
B + K M aM<br />
+ KT<br />
aT<br />
r<br />
r<br />
3<br />
=<br />
1 + K<br />
1B<br />
k<br />
a<br />
3<br />
1B<br />
K<br />
1B<br />
a<br />
+ K<br />
1B<br />
2B<br />
2B<br />
⎛<br />
⎜<br />
⎛ 1 a<br />
⎜<br />
⎝ Keq3<br />
a<br />
a + K<br />
• Heat capacity of each component<br />
2B<br />
1B<br />
M<br />
⎞<br />
−1<br />
⎟<br />
⎠<br />
a +<br />
M<br />
K<br />
T<br />
a<br />
T<br />
(I.18)<br />
Cpi 2<br />
3<br />
= ai<br />
+ bi<br />
T + ci<br />
T + di<br />
T<br />
(I.19)<br />
with: Cp in kJ.mol -1 .K -1<br />
T in Kelvin<br />
component i 10 ai 10 4 bi 10 7 ci 10 10 di<br />
MeOH a<br />
2M1B b<br />
2M2B b<br />
<strong>TAME</strong> c<br />
0.077 1.62 2.06 2.87<br />
1.27 -0.609 5.08 1.69<br />
1.33 -1.48 7.51 -0.882<br />
1.73 2.29 -6.00 20.0<br />
a Zh<strong>an</strong>g <strong>an</strong>d Datta, 1995; b Kitchaiya <strong>an</strong>d Datta, 1995; c Estimated by the M<strong>is</strong>senard<br />
method (Reid et al., 1987)<br />
To simplify the <strong>problem</strong> we are going to consider the heat capacity of each component const<strong>an</strong>t<br />
<strong>an</strong>d equal to its value at the entry conditions: T = T0.<br />
The solution heat capacity will also be considered const<strong>an</strong>t <strong>an</strong>d equal to its value at the entry<br />
conditions:<br />
4<br />
∑<br />
i=<br />
1<br />
IN<br />
Cp = Cpi<br />
xi<br />
(I.20)<br />
where<br />
IN<br />
x i <strong>is</strong> the feed mole fraction of component i.<br />
4
• Density of each component (Perry <strong>an</strong>d Green, 1997)<br />
C1,<br />
i M i<br />
ρ (I.21)<br />
i<br />
= ⎡<br />
C ⎤<br />
⎢ ⎛ ⎞ 4,<br />
i<br />
⎜ T ⎟ ⎥<br />
⎢1<br />
+ 1 −<br />
⎜ ⎟ ⎥<br />
⎢<br />
⎣<br />
⎝<br />
C3,<br />
i ⎠ ⎥<br />
⎦<br />
C<br />
2,<br />
i<br />
with ρ in g.L -1<br />
T in Kelvin<br />
M in g.mol -1<br />
component i Mi C1,i C2,i C3,i C4,i<br />
MeOH<br />
2M1B<br />
2M2B<br />
<strong>TAME</strong><br />
32.042 2.288 0.2685 512.64 0.2453<br />
70.135 0.91619 0.26752 465 0.28164<br />
70.135 0.93322 0.27251 471 0.26031<br />
102.177 * * * *<br />
* as there <strong>is</strong> no data available for <strong>TAME</strong>, we will consider its density const<strong>an</strong>t <strong>an</strong>d equal to its<br />
value at 293 K: ρ = 770 g / L<br />
<strong>TAME</strong><br />
To simplify the <strong>problem</strong> we are going to consider the heat capacity of each component const<strong>an</strong>t<br />
<strong>an</strong>d equal to its value at the entry conditions: T = T0.<br />
The solution density will also be considered const<strong>an</strong>t <strong>an</strong>d equal to its value at the entry<br />
conditions:<br />
4<br />
∑<br />
i=<br />
1<br />
IN<br />
ρ = ρ<br />
(I.22)<br />
i xi<br />
5
Starting solving the <strong>problem</strong><br />
Writing mass <strong>an</strong>d energy bal<strong>an</strong>ces<br />
In the figure <strong>is</strong> a representation of a PFR filled with a catalyst.<br />
Fig. 1:Representaion of a PFR filled with catalyst.<br />
A. Steady state mass bal<strong>an</strong>ce<br />
For the steady state mass bal<strong>an</strong>ce of component i, in the volume element of length dz, we c<strong>an</strong> write:<br />
Total Flux IN = Total Flux OUT − What <strong>is</strong> formed by Chemical Re action<br />
(A.1)<br />
z<br />
z+<br />
dz<br />
If ε <strong>is</strong> the bed porosity, the area, A, available for the catalyst particles <strong>is</strong> [(1-ε) A] <strong>an</strong>d the area available<br />
for the fluid <strong>is</strong> [ε A]. Equation (A.1) becomes:<br />
( ε ) ϕ ( ε A)<br />
ϕ − ρ r Adz<br />
A i,<br />
z = i,<br />
z+<br />
dz b i<br />
(A.2)<br />
where ϕi <strong>is</strong> the molar flux of component i, ρb <strong>is</strong> the bulk density <strong>an</strong>d ri <strong>is</strong> the rate of formation of<br />
component i.<br />
Re-writing equation (A.2):<br />
ϕ<br />
dϕ<br />
dz<br />
i,<br />
z+<br />
dz<br />
− ϕ i,<br />
z ρ b<br />
− ri<br />
= 0<br />
dz ε<br />
ρ<br />
i b<br />
− ri<br />
ε<br />
= 0<br />
IN<br />
Ci<br />
Considering a plug flow with axial d<strong>is</strong>persion, the molar flux <strong>is</strong> given by:<br />
z<br />
dz<br />
z+dz<br />
z = 0 z = L<br />
A<br />
OUT<br />
Ci<br />
(A.3)<br />
(A.4)<br />
6
dCi<br />
ϕ i = ui<br />
Ci<br />
− Dax<br />
(A.5)<br />
dz<br />
where ui <strong>is</strong> the interstitial velocity of the fluid, Ci <strong>is</strong> the concentration of component i in the fluid <strong>an</strong>d<br />
Dax <strong>is</strong> the axial d<strong>is</strong>persion coefficient.<br />
u<br />
i<br />
dC<br />
dz<br />
d<br />
2<br />
i<br />
− Dax<br />
i<br />
2 −<br />
b<br />
ri<br />
dz<br />
C<br />
ρ<br />
ε<br />
= 0<br />
Normalizing some of the variables:<br />
IN<br />
C i <strong>is</strong> the feed concentration of component i),<br />
ui<br />
C<br />
L<br />
IN<br />
total<br />
dividing by<br />
df<br />
i<br />
dX<br />
df i<br />
dX<br />
u<br />
D<br />
d<br />
2<br />
−<br />
ax<br />
2<br />
IN<br />
Ctotal<br />
i<br />
2 −<br />
b<br />
ri<br />
L<br />
i IN<br />
Ctotal<br />
L<br />
1 d<br />
Pe<br />
f<br />
dX<br />
2<br />
i b<br />
− − τ r<br />
2 IN i<br />
dX ε Ctotal<br />
f<br />
ρ<br />
ε<br />
(A.6)<br />
z<br />
Ci<br />
IN<br />
IN<br />
X = (L <strong>is</strong> the PFR length), f = ( C IN total ∑C<br />
i where<br />
L<br />
= 0<br />
, equation (A.7) becomes:<br />
ρ<br />
= 0<br />
where Pe <strong>is</strong> the dimensionless Peclet number given by<br />
ρ b<br />
Grouping the term τ r IN i<br />
ε C<br />
component i becomes:<br />
df<br />
i<br />
dX<br />
total<br />
1 d f<br />
Pe dX<br />
2<br />
i<br />
− − ℜ 2 i<br />
ax<br />
i<br />
Ctotal<br />
= i<br />
ui<br />
L<br />
, <strong>an</strong>d τ <strong>is</strong> the space-time given by<br />
D<br />
L<br />
.<br />
ui<br />
(A.7)<br />
(A.8)<br />
as a reaction term represented by ℜ i , the steady state mass bal<strong>an</strong>ce of<br />
= 0<br />
⎛ 1 ⎞<br />
In the limiting case of absence of d<strong>is</strong>persion ⎜ →0 ⎟ , equation (A.9) becomes:<br />
⎝ Pe ⎠<br />
df i<br />
dX<br />
− ℜi<br />
= 0<br />
(A.9)<br />
(A.10)<br />
7
B. Steady state energy bal<strong>an</strong>ce<br />
For the steady state energy bal<strong>an</strong>ce in the volume element of length dz, considering no d<strong>is</strong>persion, we<br />
c<strong>an</strong> write:<br />
heat produced by<br />
Total Flux IN + = Total Flux OUT<br />
z<br />
z+<br />
dz<br />
Chemical Re action<br />
M reactions<br />
h<br />
R<br />
h<br />
( ε A) ϕ z + ρ b ∑(<br />
∆H<br />
j rj<br />
) Adz<br />
= ( ε A)<br />
ϕ z+<br />
dz + Alat<br />
U ( T −Tw<br />
)<br />
j=<br />
1<br />
where ϕ h <strong>is</strong> the heat flux,<br />
R<br />
j<br />
heat losses through<br />
+<br />
reactor walls<br />
(B.1)<br />
(B.2)<br />
∆ H <strong>is</strong> the heat of reaction j, rj <strong>is</strong> the rate of reaction j, Alat <strong>is</strong> the lateral area<br />
of the volume element, U <strong>is</strong> the overall heat-tr<strong>an</strong>sfer coefficient, T <strong>is</strong> the reactor temperature <strong>an</strong>d Tw <strong>is</strong><br />
the reactor wall temperature.<br />
Re-writing equation (B.2):<br />
M reactions<br />
b ∑<br />
j=<br />
1<br />
R Alat<br />
( ∆H<br />
r ) + U ( T −T<br />
) = 0<br />
h h<br />
ϕ z+<br />
dz −ϕ<br />
x ε − ρ<br />
j j<br />
w<br />
(B.3)<br />
dz<br />
A dz<br />
The heat flux c<strong>an</strong> be given by:<br />
h<br />
ϕ = ρ CpT<br />
(B.4)<br />
u i<br />
where ρ <strong>is</strong> the solution density <strong>an</strong>d Cp <strong>is</strong> the solution heat capacity.<br />
If R0 <strong>is</strong> the reactor radius, the lateral area of the volume element of length dz <strong>is</strong>:<br />
= 2π R dz<br />
(B.5)<br />
A lat 0<br />
<strong>an</strong>d its sectional area <strong>is</strong>:<br />
A = π R<br />
(B.6)<br />
2<br />
0<br />
leading to:<br />
Alat 2<br />
= (B.7)<br />
Adz<br />
R<br />
0<br />
Substituting into equation (B.3):<br />
8
− b ∑<br />
j=<br />
1<br />
R 2<br />
( ∆H<br />
r ) + U ( T −T<br />
) = 0<br />
M reactions<br />
dT<br />
ε ui ρ Cp ρ<br />
j j<br />
w<br />
(B.8)<br />
dz<br />
R<br />
Normalizing the space variable:<br />
rearr<strong>an</strong>ging equation (B.8):<br />
M reactions<br />
b<br />
− ∑ ρ Cp j=<br />
1<br />
0<br />
z<br />
X = , remembering the space-time definition<br />
⎛ ⎞<br />
⎜τ<br />
= L ⎟ <strong>an</strong>d<br />
L<br />
⎝ ui<br />
⎠<br />
R 2U<br />
τ<br />
( ∆H<br />
r ) + ( T −T<br />
) = 0<br />
dT ρ τ<br />
ε j j<br />
w<br />
(B.9)<br />
dX<br />
ρ Cp R<br />
The term<br />
2U<br />
τ<br />
ρ Cp R<br />
0<br />
0<br />
<strong>is</strong> dimensionless <strong>an</strong>d <strong>is</strong> referred as NTU (number of heat tr<strong>an</strong>sfer units).<br />
Finally, the steady state energy bal<strong>an</strong>ce becomes:<br />
M reactions<br />
b<br />
− ∑ ρ Cp j=<br />
1<br />
R ( ∆H<br />
r ) + NTU ( T −T<br />
) = 0<br />
dT ρ τ<br />
ε j j<br />
w<br />
(B.10)<br />
dX<br />
C. Boundary Conditions<br />
In the absence of d<strong>is</strong>persion, we only need one boundary condition:<br />
IN ⎧ Ci<br />
⎪ fi<br />
=<br />
X = 0 ⇒<br />
IN<br />
⎨ C<br />
(C.1)<br />
total<br />
⎪<br />
⎩ T = T0<br />
where T0 <strong>is</strong> the initial reactor temperature.<br />
9
Algorithm to solve the <strong>problem</strong><br />
We have a non-linear system of Ordinary Differential Equations (ODE) to solve:<br />
⎧ 1B<br />
⎪ − ℜ1B<br />
= 0<br />
⎪<br />
2B<br />
⎪ − ℜ2<br />
B = 0<br />
⎪<br />
⎪ M<br />
⎨ − ℜ M = 0<br />
⎪<br />
⎪<br />
dfT<br />
− ℜT<br />
= 0<br />
⎪ dX<br />
3<br />
⎪ dT b<br />
⎪ − ∑<br />
⎩ dX Cp j=<br />
dX<br />
df<br />
dX<br />
df<br />
dX<br />
df<br />
ρ τ<br />
ε<br />
ρ<br />
1<br />
R ( ∆H<br />
r ) + NTU ( T −T<br />
)<br />
j<br />
j<br />
w<br />
= 0<br />
The program we developed uses subroutine DDASSL (Bren<strong>an</strong> et al., 1989) to solve th<strong>is</strong> system. Th<strong>is</strong><br />
code solves a system of differential/algebraic equations of the form delta(t, y, yprime)=0, with<br />
delta(i) = yprime(i) – y(i), using the backward differentiation formulas of orders one trough five. t <strong>is</strong><br />
the current value of the independent variable (in our case t = X), y <strong>is</strong> the array that contains the solution<br />
components at t (in our case we have: y(i) = f(i), i=1,4 <strong>an</strong>d y(5) = T) <strong>an</strong>d yprime <strong>is</strong> the array that<br />
contains the derivatives of the solution components at t.<br />
The program solves the system from t to tout <strong>an</strong>d it <strong>is</strong> easy to continue the solution to get results at<br />
additional tout. In our case, we are going to get results at different values of X, between 0 <strong>an</strong>d 1.<br />
Th<strong>is</strong> <strong>problem</strong> <strong>is</strong> rather complex because most of the other variables depend on yi: the kinetic, adsorption<br />
<strong>an</strong>d thermodynamic equilibrium “const<strong>an</strong>ts” depend on T (y5), the reaction rate <strong>an</strong>d the components rate<br />
of formation depend on yi (fi (i=1,4) <strong>an</strong>d T), as the activity coefficients.<br />
In Figure 2 <strong>is</strong> the algorithm to solve our <strong>problem</strong>.<br />
(1)<br />
10
Read data from file<br />
T0, Tw IN<br />
C i<br />
Q (operation flow)<br />
Input const<strong>an</strong>ts values<br />
ε, ρb, U, R 0,<br />
V (reactor volume)<br />
∆ H<br />
R<br />
j<br />
Calculate const<strong>an</strong>ts<br />
ε V<br />
τ =<br />
Q<br />
C C<br />
IN<br />
total = ∑<br />
i<br />
IN<br />
i<br />
IN<br />
0 Ci<br />
f i = IN<br />
Ctotal<br />
Initialize the variables<br />
t = X = 0<br />
yi (i=1,4) = f<br />
y5 = T 0<br />
0<br />
i<br />
Solve system of ODEs<br />
Call<br />
subroutine DASSL<br />
New values<br />
of<br />
yi<br />
Cp<br />
ρ<br />
NTU<br />
tout = X =<br />
X + ∆X<br />
No<br />
tout = X = 1 ?<br />
Fig. 2: Algorithm to solve the <strong>problem</strong>.<br />
Yes<br />
STOP<br />
11
Some results<br />
For both cases, <strong>is</strong>othermal <strong>an</strong>d non-<strong>is</strong>othermal, the feed concentrations were the same:<br />
C<br />
C<br />
C<br />
C<br />
IN<br />
2M<br />
1B<br />
IN<br />
2M<br />
2B<br />
IN<br />
MeOH<br />
IN<br />
<strong>TAME</strong><br />
=<br />
=<br />
=<br />
= 0<br />
3.<br />
33<br />
3.<br />
33<br />
6.<br />
66<br />
mol / L<br />
mol / L<br />
mol / L<br />
These concentrations lead to a feed mole ratio MeOH/<strong>is</strong>oamylenes, RM/IA, of 1.0.<br />
a) Figures R.1, R.2 <strong>an</strong>d R.3 represent the concentration profiles for a 10 m 3 <strong>is</strong>othermal PFR, operating<br />
at 323 K, 343 K <strong>an</strong>d 363 K, <strong>an</strong>d a flow of 40 L/min.<br />
C (mol/L)<br />
7<br />
6<br />
5<br />
4<br />
3<br />
2<br />
1<br />
0<br />
Isothermal: T = 323 K<br />
RM/IA = 1, Q = 40 L/min<br />
0 0.2 0.4 0.6 0.8 1<br />
X<br />
2M1B 2M2B MeOH <strong>TAME</strong><br />
Fig. R.1: Concentration profiles for a 10 m 3 <strong>is</strong>othermal PFR operating at 323 K.<br />
12
C (mol/L)<br />
Fig. R.2: Concentration profiles for a 10 m 3 <strong>is</strong>othermal PFR operating at 343 K.<br />
C (mol/L)<br />
7<br />
6<br />
5<br />
4<br />
3<br />
2<br />
1<br />
0<br />
7<br />
6<br />
5<br />
4<br />
3<br />
2<br />
1<br />
0<br />
Isothermal: T = 343 K<br />
RM/IA = 1, Q = 40 L/min<br />
0 0.2 0.4 0.6 0.8 1<br />
X<br />
2M1B 2M2B MeOH <strong>TAME</strong><br />
Isothermal: T = 363 K<br />
RM/IA = 1, Q = 40 L/min<br />
0 0.2 0.4 0.6 0.8 1<br />
X<br />
2M1B 2M2B MeOH <strong>TAME</strong><br />
Fig. R.3: Concentration profiles for a 10 m 3 <strong>is</strong>othermal PFR operating at 363 K<br />
As the temperature increases, the reactions are faster, favoring <strong>TAME</strong> production, but the chemical<br />
equilibrium <strong>is</strong> moved to the opposite direction: for <strong>an</strong> operating temperature of 363 K, the equilibrium<br />
concentration of <strong>TAME</strong> <strong>is</strong> reached faster th<strong>an</strong> for 343 K, but its value <strong>is</strong> lower.<br />
13
) Figure R.4 represents the exit concentrations as a function of the entering temperature, T0, for a 10<br />
m 3 <strong>is</strong>othermal PFR operating with a flow of 40 L/min.<br />
C OUT (mol/L)<br />
5<br />
4<br />
3<br />
2<br />
1<br />
0<br />
Isothermal operation<br />
RM/IA = 1.0, Q = 40 L/min<br />
323 333 343 353 363<br />
T0 (K)<br />
2M1B 2M2B MeOH <strong>TAME</strong><br />
Fig. R.4: Exit concentrations as function of T0, for a 10 m 3 <strong>is</strong>othermal PFR<br />
Figure R.4 shows that there <strong>is</strong> <strong>an</strong> optimum operating temperature around 333 K that leads to a<br />
maximum exit concentration for <strong>TAME</strong>. It <strong>is</strong> due to the fact that we have a system with competition<br />
between kinetics <strong>an</strong>d equilibrium, as we are going to see for the non-<strong>is</strong>othermal PFR<br />
c) We chose a reactor diameter of 1 m (for a volume of 10 m 3 it leads to a reactor length of 12.7 m<br />
approximately) <strong>an</strong>d for the wall temperature, we decided to use room temperature: 298 K. The entering<br />
temperature <strong>is</strong> 353 K.<br />
It <strong>is</strong> import<strong>an</strong>t to notice that the catalyst used in <strong>TAME</strong> production <strong>is</strong> a macroreticular strong cation ionexch<strong>an</strong>ge<br />
resin in hydrogen form (Amberlyst 15 Wet, Rohm & Haas) that has a maximum operating<br />
temperature of 393 K.<br />
Figure R.5 represents the concentration (a) <strong>an</strong>d temperature (b) profiles for <strong>an</strong> operating flow of 200<br />
L/min.<br />
14
(a)<br />
(b)<br />
C (mol/L)<br />
T (K)<br />
7<br />
6<br />
5<br />
4<br />
3<br />
2<br />
1<br />
0<br />
390<br />
380<br />
370<br />
360<br />
350<br />
Non-Isothernal: To = 353 K, Tw = 298 K<br />
RM/IA = 1.0, Q = 200 L/min<br />
0 0.2 0.4 0.6 0.8 1<br />
X<br />
2M1B 2M2B MeOH <strong>TAME</strong><br />
0 0.2 0.4 0.6 0.8 1<br />
X<br />
Fig. R.5: Non-<strong>is</strong>othermal PFR: (a) concentration profiles; (b) temperature profile.<br />
Figure R.5(a) shows the competition between the three reactions: first, 2M1B <strong>an</strong>d 2M2B react with<br />
MeOH to produce <strong>TAME</strong> <strong>an</strong>d the react<strong>an</strong>ts concentrations decrease <strong>an</strong>d <strong>TAME</strong> concentration<br />
increases; but then, although MeOH <strong>an</strong>d <strong>TAME</strong> concentrations are almost const<strong>an</strong>t, 2M1B <strong>is</strong> still<br />
decreasing <strong>an</strong>d 2M2B starts to increase: the third reaction, the <strong>is</strong>omerization, <strong>is</strong> now more evident.<br />
The temperature profile (Fig. R.5(b)) shows that the reactor seams to be almost adiabatic (no heat losses<br />
through the reactor walls) since the reactor temperature <strong>is</strong> always increasing. Th<strong>is</strong> close to adiabatic<br />
behavior was expected since the reactor diameter <strong>is</strong> rather large: 1 m. But the temperature reaches a<br />
value that <strong>is</strong> very close to the catalyst limit: remember that its maximum operating temperature <strong>is</strong> 393 K<br />
<strong>an</strong>d the reactor <strong>is</strong> reaching 385 K.<br />
15
To see what <strong>is</strong> the maximum temperature that the reactor reaches, we c<strong>an</strong> make it adiabatic setting the<br />
overall heat-tr<strong>an</strong>sfer coefficient equal to zero: U = 0. In Figure R.6 are the results for the adiabatic<br />
reactor.<br />
(a)<br />
C (mol/L)<br />
(b)<br />
T (K)<br />
7<br />
6<br />
5<br />
4<br />
3<br />
2<br />
1<br />
0<br />
390<br />
380<br />
370<br />
360<br />
350<br />
Adiabatic operation: To = 353 K<br />
RM/IA = 1.0, Q = 200 L/min<br />
0 0.2 0.4 0.6 0.8 1<br />
X<br />
2M1B 2M2B MeOH <strong>TAME</strong><br />
0 0.2 0.4 0.6 0.8 1<br />
X<br />
Fig. R.6: Adiabatic PFR: (a) concentration profiles; (b) temperature profile.<br />
The maximum temperature reached by the reactor <strong>is</strong> 388 K. Comparing Figures R.5 <strong>an</strong>d R.6 it <strong>is</strong> easy to<br />
see that the non-<strong>is</strong>othermal 1m diameter PFR c<strong>an</strong> be considered adiabatic: the concentration <strong>an</strong>d<br />
temperature profiles are almost the same.<br />
To improve the reactor production, i.e., to reach higher exit concentrations of <strong>TAME</strong>, we c<strong>an</strong> use a<br />
multi-tubular reactor. Lets think of a reactor composed of 4000 tubes (each one considered as a PFR)<br />
16
with a diameter of 1’’ each. To have a total reactor volume of 10 m 3 , each tube has a length of 5 m. In<br />
order to compare the results of the multi-tubular reactor with the ones obtained with the 1 m diameter<br />
PFR (Fig. R.5), we have to choose similar operating conditions: to have a total operating flow of 200<br />
L/min, the equivalent flow in each tube <strong>is</strong> 0.05 L/min; since the reactor <strong>is</strong> now really cooled, we will<br />
choose the “best” temperature for the cooling fluid (333 K, as seen with the <strong>is</strong>othermal behavior runs).<br />
Figure R.7 shows the results for one of th<strong>is</strong> tubes operating in the above conditions.<br />
(a)<br />
C (mol/L)<br />
(b)<br />
T (K)<br />
7<br />
6<br />
5<br />
4<br />
3<br />
2<br />
1<br />
0<br />
360<br />
358<br />
356<br />
354<br />
352<br />
350<br />
Non-Isothermal Multi-tubular reactor<br />
To = 353 K, Tw = 333 K, Qtotal = 200 L/min<br />
0 0.2 0.4 0.6 0.8 1<br />
X<br />
2M1B 2M2B MeOH <strong>TAME</strong><br />
0 0.2 0.4 0.6 0.8 1<br />
X<br />
Fig. R.7: Multi-tubular reactor: (a) concentration profiles; (b) temperature profile.<br />
The maximum temperature reached <strong>is</strong> 358 K - in th<strong>is</strong> case there are no <strong>problem</strong>s with the catalyst - <strong>an</strong>d<br />
the exit concentration of <strong>TAME</strong> <strong>is</strong> higher: 2.603 mol/L against 1.862 mol/L for the PFR in Figure R.5,<br />
which represents <strong>an</strong> increase of 28.5 %.<br />
17
References<br />
Bren<strong>an</strong>, K., Campbell, S. <strong>an</strong>d L. Petzold, “Numerical Solution of Initial-Value Problems in<br />
Differencial-Algebraic Equations, Elsevier, N.Y. (1989).<br />
Kitchaiya, P. <strong>an</strong>d R. Datta, “<strong>Ether</strong>s from Eth<strong>an</strong>ol. 2. Reaction Equilibria of Simult<strong>an</strong>eous tert-amyl<br />
Ethyl <strong>Ether</strong> Synthes<strong>is</strong> <strong>an</strong>d Isoamylene Isomerization”, Ind. Eng. Chem. Res. 34 (1995) 1092-1101.<br />
Kivir<strong>an</strong>ta-Pääkkönen, P.K., Struckm<strong>an</strong>, L.K. <strong>an</strong>d A.O.I. Krause, “Compar<strong>is</strong>on of the Various Kinetic<br />
Models of <strong>TAME</strong> Formation by Simulation <strong>an</strong>d Parameter Estimation”, Chem. Eng. Technol. 21 (1998)<br />
321-326.<br />
Oktar, N., Mürtezaoglu, K., Dogu, T. <strong>an</strong>d Gülsen Dogu, “Dynamic Analys<strong>is</strong> of Adsorption Equilibrium<br />
<strong>an</strong>d Rate Parameters of React<strong>an</strong>ts <strong>an</strong>d Products in MTBE, ETBE <strong>an</strong>d <strong>TAME</strong> Production”, C<strong>an</strong>.J.<br />
Chem. Eng. 77 (1999) 406-412.<br />
Perry, R.H. <strong>an</strong>d D<strong>an</strong> W. Green, “Perry’s Chemical Engineers’ H<strong>an</strong>dbook”, 7 th edition, McGraw-Hill<br />
(1997).<br />
Reid, R.C., Prausnitz, J.M. <strong>an</strong>d Thomas K. Sherwood, “The Properties of Gases <strong>an</strong>d Liquids”, 3 rd<br />
edition, McGraw-Hill (1977).<br />
Reid, R.C., Prausnitz, J.M. <strong>an</strong>d B.E. Poling, “The Properties of Gases <strong>an</strong>d Liquids”, 4 th edition,<br />
McGraw-Hill (1987).<br />
Vilarinho Ferreira, M.M. <strong>an</strong>d J.M. Loureiro, “Synthes<strong>is</strong> of <strong>TAME</strong>: Kinetics in Batch Reactor <strong>an</strong>d<br />
Thermodynamic Study”, presented on the “3 rd Europe<strong>an</strong> Congress of Chemical Engineering, ECCE3”,<br />
Nuremberg, Germ<strong>an</strong>y (June 2001).<br />
Zh<strong>an</strong>g, T. <strong>an</strong>d R. Datta, “Integral Analys<strong>is</strong> of <strong>Methyl</strong> tert-Butyl <strong>Ether</strong> Synthes<strong>is</strong> Kinetics”, Ind. Eng.<br />
Chem. Res. 34 (1995) 730-740.<br />
18