Attainable Region S,S&L Chapt. 7

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 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

Solve Numerically Runge-Kutta

Solve Numerically

Step 2 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. Tangent line passing (1,0)) Desired

Step 3 Expand the AR as much as possible with straight arcs that represent mixing of reactor effluent and feed stream PFR  (1-  )

Interpreting points on mixing line Larger Attainable Region PFR C A = C B = C A =0.72 C B = C A =1 C B =0  (1-  ) PFR C A =1 C B =0 (1-  )

Mixing of Streams Reactant Bypass Vector Equation, i component is C A, j component is C B α =fraction of mixture of stream 1in the mixed stream 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

Solve numerically at various  until complete conversion or equilibrium is achieved

CSTR Extends Attainable Region CSTR

Plot extends attainable region i.c. for step 5 CSTR C A =1 C B =0  (1-  ) Enlarges Attainable Region

Possible Configuration at this point CSTR C A =1 C B =0  β PFR  β = 0  β = 0 1-α-β 0.38 β = 1 α = 0

But you can do better than even this! Add PFR after mix point

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

Max. Attainable Region CSTR  (1-  ) PRF 0.38

Keep track of feed points Initial feed point occurs at far right on AR Mixing lines connect two points Connect reactors and mixers with feed points to get network

Reactor configuration for highest selectivity CSTR C A =1 C B =0 C A =0.38 C B = C A =0.185 C B = PFR

Go back to calculations for optimal reactor sizing

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{C A } + f{C B }) 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*C B -15*C A 2 Optimal point not at highest selectivity PFR CSTR

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