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