Attainable Region
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Transcript of Attainable Region
Attainable Region
S,S&L Chapt. 6
Attainable Region
• Graphical method that is used to determine the entire space feasible concentrations
• Useful for identifying reactor configurations that will yield the optimal products
Procedure
Step 1: Construct a trajectory for a PFR from the feed point, continuing to complete conversion or chemical equilibrium
Step 2: When the PFR bounds a convex region, this constitutes a candidate AR. The procedure terminates if the rate vectors outside the candidate AR do not point back into it.
Step 3: The PFR trajectory is expanded by linear arcs, representing mixing between the PFR effluent and the feed stream, extending the candidate AR.
Step 4: Construct a CSTR trajectory to see if the AR can be extended. Place linear arcs, which represent mixing, on the CSTR trajectory to ensure the trajectory remains convex.
Step 5: A PFR trajectory is drawn from the position where the mixing line meets the CSTR trajectory. If the PFR trajectory is convex, it extends the previous AR to form a expanded AR. Then return to step 2. Otherwise, repeat the procedure from Step 3.
Example
BBAB
ABAA
k
kk
k
CkCkCkdt
dC
CkCkCkdt
dC
DA
CdesiredBA
321
2421
4
31
2
2
)(
Reactions
Rate Equations
Step 1
Begin by constructing a trajectory for a PFR from the feed point, continuing to the complete conversion of A or chemical equilibrium
• Solve the PFR design equations numerically– Use the feed conditions as initial conditions to
the o.d.e.– Adjust integration range, (residence time),
until complete conversion or to equilibrium
PFR Design Equations
BBAB
ABAA
CkCkCkd
dC
CkCkCkd
dC
321
2421
x
AA r
dxFV
0
0
Solve Numerically
Solve Numerically
Step 2
Attainable Region
0
0.00005
0.0001
0.00015
0 0.2 0.4 0.6 0.8 1
Ca, kmol/m3
Cb
, km
ol/
m3
PFR
Plot the PFR trajectory from the previous results. Check to see if rate vectors outside AR point back into it (e.g. Look for non-convex regions on the curve)
Step 3Attainable Region
0
0.00002
0.00004
0.00006
0.00008
0.0001
0.00012
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
C a, kmol / m3
P FR
Expand the AR as much as possible with straight arcs that represent mixing of reactor effluent and feed stream
Interpreting points on mixing lineAttainable Region
0
0.00002
0.00004
0.00006
0.00008
0.0001
0.00012
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1
C a, kmol / m3
P FR
Desired operating point
PFR CA=0. 2187CB=0.00011
CA=0.72CB=0.0004
CA=1CB=0
(1-)
Mixing of StreamsReactant Bypass
21 )1( ccc α =fraction of mixture of stream 1in the mixed stream
)1(00011.0000004.0
)1(2187.0172.0
B
A
C
C
Feed mixing fraction: = 0. 64
Step 4
If a mixing arc extends the attainable region on a PFR trajectory, check to see if a CSTR trajectory can extend the attainable region
For CSTR trajectories that extend the attainable region, add mixing arcs to concave regions to ensure the attainable region remains convex
• Solve CSTR multiple NLE numerically– Vary until all feed is consumed or equilibrium is
reached
CSTR Design Equations
)(
)(
321
2421
BBAB
ABAAAo
CkCkCkC
CkCkCkCC
A
A
r
xFV
0
Solve numerically at various until complete conversion or equilibrium
is achieved
Plot extensions to attainable region
Attainable Region
0
0.00002
0.00004
0.00006
0.00008
0.0001
0.00012
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 1.05
C a, kmol / m3
CST R
P FR
mixing arc
0.38
i.c. for step 5
Step 5
A PFR trajectory is drawn from the position where the mixing line meets the CSTR trajectory. If this PFR trajectory is convex, it extends the previous AR to form an expanded candidate AR. Then return to Step 2. Otherwise repeat Step 3
Solve PFR equations with modified initial conditions
New feed point
Vary integration range
Attainable Region
Attainable Region
0
0.00002
0.00004
0.00006
0.00008
0.0001
0.00012
0.00014
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 1.05
C a, kmol / m3
CST R
P FR
P FR2
Keep track of feed points
• Initial feed point occurs at far right on AR
• Mixing lines connect two feed points
• Connect reactors and mixers with feed points to get network
Reactor configuration for highest selectivity
CSTR
PFR
CA=1CB=0
CA=0.38CB=0.0001
CA=0.185CB=0.000124
Reactor series occur when multiple feed points exist
Go back to calculations for optimal reactor sizing
0v
V
Other factors to consider
• Annualized, operating, and capital costs might favor designs that don’t give the highest selectivity
• If objective function (e.g. $ = f{CA} + f{CB}) can be expressed in terms of the axis variable, a family of objective contours can be plotted on top of the AR– The point where a contour becomes tangent to the AR is the
optimum
• Temperature effects– Changing temperature will change the AR– Need energy balance for non-isothermal reactions
• Make sure to keep track of temperature
Profit ($) = 15000*CB-15*CA2
Attainable Region
-0.00001
0.00001
0.00003
0.00005
0.00007
0.00009
0.00011
0.00013
0.00015
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 1.05
C a, kmol / m3
CSTR
PFR
PFR2
$=0.9
$=2
$=1.5
Optimal point not at highest selectivity
Conclusions
• Need to know feed conditions• AR graphical method is 2-D and limited to 2
independent species• Systems with rate expressions involving more
than 2 species need to be reduced– Atom balances are used to reduce independent species– Independent species = #molecular species - #atomic
species• If independent species < 2, AR can be used by Principle of
Reaction Invariants