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61
CHAPTER 6 SOLUTIONS
6.1
This exercise is solved using HYSYS.Plant. The HYSYS PFD below indicates how
the simulation is put together. Note that each of the three PFRs is defined as a
diabatic bed with uniform heat transfer rate. The optimizer is set up to manipulate the
effluent temperatures in each bed, to maximize the ammonia effluent composition.
To maximize the conversion in the reactor, the following nonlinear program (NLP) of
the type discussed in Chapter 18, is formulated:
w. r. t.
123
max
,,
TTT
ξ
(1)
Subject to (s. t.)
( )
0
=
x
f
(2)
1
270
450
T
≤≤
o
C
(3)
2
270
450
T
o
C
(4)
3
270
450
T
o
C
(5)
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In the above, Eq. (2) refers to the kinetics and material and energy balances for the
converter, and Eqs. (3) – (5) define upper and lower limits for the effluent
temperatures of each bed.
The NLP in Eqs. (1)–(5) is solved efficiently using successive quadratic programming
(SQP) in HYSYS.Plant, described in Chapter 18. The final ammonia composition in
the converter effluent is 24 mol %, obtained with optimal effluent temperatures of
358, 320 and 272
o
C, respectively, with
Q
1
= 57,850 kW,
Q
2
= 61,950 kW, and
Q
3
=
40,740 kW.
0
5
10
15
20
25
250
300
350
400
450
Temperature [
o
C]
NH
3
Conc. [mol %]
First Bed
Second Bed
Third Bed
The above graph shows the coldshot profile obtained in Example 6.3, with an
ammonia effluent composition of 15.9 mol %. The design involving uniform heat
transfer in each bed of 2 m length is also shown in the figure, demonstrating that this
more precise temperature control means that the reactor temperaturecomposition
trajectory more closely matches the optimal trajectory, indicated as the dashed red
line. This solution can be reproduce using the file Exer_6_1.hsc.
63
6.2
The attainable region is constructed as follows:
Step 1:
Begin by constructing a trajectory for a PFR from the feed point,
continuing to the complete conversion of A or chemical equilibrium.
In this
case, the PFR trajectory is computed by solving simultaneously the kinetic
equations for A and C:
2
12
3
A
AA
dC
kk
C k
C
d
τ
=− −
−
−
(1)
2
C
A
dC
kC
d
=
(2)
where
τ
is the PFR residence time. The trajectory in
C
A
–C
C
space is plotted
in Fig. 1 as curve ABC.
Step 2:
When the PFR trajectory bounds a convex region, this constitutes a
candidate attainable region. When the rate vectors at concentrations
outside of the candidate AR do not point back into it, the current limits are
the boundary of the AR and the procedure terminates.
In Fig.1, the PFR
trajectory is not convex, so proceed to the next step.
Step 3:
The PFR trajectory is expanded by linear arcs, representing mixing between
the PFR effluent and the feed stream, extending the candidate attainable
region.
Here, a linear arc AB is introduced to form a convex hull, from the
feed point A to a point B, tangent to the PFR trajectory, as shown in Fig.1.
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This note was uploaded on 04/10/2008 for the course CHE 233W taught by Professor Debelak during the Spring '08 term at Vanderbilt.
 Spring '08
 Debelak

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