18th European Symposium on Computer Aided Process Engineering – ESCAPE 18
Bertrand Braunschweig and Xavier Joulia (Editors)
© 2008 Elsevier B.V./Ltd. All rights reserved.
Integrated design and control of processes coupling
exothermic and endothermic reactions
Pietro Altimaria and Costin Sorin Bildeab
a
b
Delft University of Technology, Julianalaan 136, 2628BL, Delft, The Netherlands
University “Politehnica” of Bucharest, 1-7 Polizu, 011061 - Bucharest, Romania
Abstract
The nonlinear behavior of plantwide systems, where the endothermic first-order
reaction A → R+Q and the exothermic second-order reaction B+Q → P
simultaneously take place, is investigated. As the physical properties of the species
involved vary, possible flowsheets are identified and feasible control strategies are
suggested. Bifurcation analysis is performed by varying the set-point variables. Steady
state multiplicity is detected leading to complex behavior. Implications on plantwide
control are discussed and guidelines are provided which enable to select values of setpoint variables in a way that guarantees feasible operation. Singularity theory is
exploited to investigate the influence of reactor design parameters on plant operability
and complex multiplicity patterns with a maximum of five coexisting steady states and
isolated solution branches are predicted.
Keywords: Reaction coupling; Recycle systems; Nonlinear dynamics; Design; Control.
1. Introduction
As result of the increasing demand for sustainable process solutions, several efforts
have been carried out, over the past decades, towards the development of effective
process intensification technologies. In this framework, coupling endothermic and
exothermic reactions in a single reactor unit has been extensively recognized to offer
several advantages [1]. It guarantees in general increased thermal efficiency and, for
reversible chemical reactions, can lead to increased reactor equilibrium conversion and
increased reaction rate due to equilibrium displacement. As a result, energy savings and
reduced reactor size are achieved. Some examples demonstrating the effectiveness of
this idea are in situ hydrogen combustion in oxidative dehydrogenation [2], coupling
propane combustion and endothermic thermal cracking of propane to ethylene and
propylene [3], and coupling methane steam reforming with catalytic oxidation of
methane in partial oxidation reactors [4].
While enhancing reactor performances, performing endothermic and exothermic
chemical reactions in a single unit may lead to increased complexity of the external
plant, making necessary additional separation and recycles. In practice, energy savings
and reduced reactor investments must outweigh the cost of required additional units.
Furthermore, operational and control difficulties arising from a more complex behavior
should be taken into account to assess feasibility of this operation mode. To this regard,
we remark that introducing recycle streams is recognized to be source of nonlinear
phenomena [5, 6]. However, despite the great interest towards the development of
intensified plant solutions, the behavior of reactor separation recycle systems coupling
exothermic and endothermic chemical reactions has not been studied.
In this work, we investigate the steady state behavior of Reactor-Separation-Recycle
(RSR) systems simultaneously carrying on the following reactions:
2
P. Altimari and C. S. Bildea
A → R+Q, endothermic, first order
B+Q → P, exothermic, second order
This reaction scheme has been chosen as representative of a large class of chemical
processes, where the heat necessary to perform a desired endothermic reaction is
provided by conversion of a byproduct Q in a secondary exothermic reaction.
The paper is structured as follows. Flowsheets which are possible, depending on the
physical properties of the species involved, are analyzed. Then, model equations are
presented in dimensionless form. Finally, the steady state behavior of the plantwide
system is described and operational difficulties are addressed. Final remarks end the
paper.
2. Conceptual design and plantwide control
We consider a RSR where the reactants A and B are fed to the plant whereas the amount
of Q required to perform the exothermic reaction is provided by the endothermic one.
Unconverted reactants A, B, Q are totally recovered and recycled to the reactor, while
products P and R are removed from the plant. Under these assumptions, mole balance
equations governing the steady state behavior of the RSR system can be written as
follows:
f A,1  f A,0   A  0
(1)
f B ,1  f B ,0   B  0
(2)
f Q ,1   Q  0
(3)
where fK,0 and fK,1 (K = A, B, Q) denote the dimensionless molar flow rates of the
species K at the plant inlet and reactor inlet, respectively. A B, Q are functions
describing the dependence of recycle flow rates on reactor-inlet flow rates, reactor
parameters and separation performance. Five unknown lumped variables are to be
determined from three equations. Hence, two consistent specifications must be provided
to solve the balance equations.
In practice, these specifications can be achieved by flow or concentration control loops.
However, since concentration measurements are expensive and unreliable, controlling
flow rates is, in general, the preferred option. For this reason, we focus on control
strategies which can be applied without using concentration measurements.
The number and the components of the recycle streams are determined by the physical
properties of the species involved. Based on this, the set of possible flowsheets of the
RSR system is structured according to the classes described in Fig.1. Based on the
analysis of the flowsheet structures presented in Fig.1, eight admissible control
strategies have been identified. While the steady state behavior of the RSR system was
analyzed for all these control structures, we present in this paper only results concerning
a control strategy fixing the flow rates of the endothermic reactant A and exothermic
reactant B at the reactor inlet. This strategy gives rise to the following additional
equations:
f A,1  m ; f B ,1  p
(4)
Integrated design and control of processes coupling exothermic and endothermic
reactions
3
where m and p are the set-points of the flow rate control loops. This is, in general, the
recommended strategy when coping with RSR systems because it removes the effect of
recycling reactants. As a result, the system is expected to behave as a plug flow reactor
followed by a separation section, which exhibits a unique steady state. We show that,
for the reaction scheme here considered, a complex behavior still occurs.
(a)
(b)
A
A,B
A,B
P,R
Reactor
Separation
A,B
P,R
Reactor
Separation
B
Q
Q
(c)
(d)
B,Q
A,Q
A,B
P,R
Reactor
A,B
Separation
A
P,R
Reactor
Separation
B
Figure 1 Classification of the flowsheets of the RSR system which are possible as the physical
properties of the species involved varies.
3. The mathematical model
In this section, we present the mathematical model governing the steady state behavior
of the RSR system. The reactor-inlet temperature is fixed by feedback control of a heat
exchanger placed upstream of the reactor. A plug flow model is used to describe the
steady state behavior of the reactor. Arrhenius dependence of the kinetic constants on
temperature is assumed. Mass and energy balance equations describing the steady state
behavior of the RSR system write as follows:
df A
   f A
  Da exp 

d
 1  f
  G2,1  f Q f B
df B
  DaK 2,1 exp 
 2
d
 1  f

   G2,1  fQ f B
d Da 
   f A

 K 2,1 H 2,1 B exp 
 2  B exp 
   (   c ) 
d
f 
 1  f
 1  f

f Q  f Q , 1   f A ,1  f A    f B ,1  f B  ; f R   f A , 1  f A  ; f P   f B , 1  f B 
f  f A  f B  fQ  f P  f R
(5)
(6)
(7)
(8)
(9)
f A (0)  f A,1 ; f B (0)  f B,1 ;  (0)  0;
(10)
f A (1)   A  0 ; f B (1)   B  0 ; f Q (1)   Q  0
(11)
f A ,0   A  f A ,1  0 ; f B , 0   B  f B , 1  0 ;  Q  f Q ,1  0
(12)
4
P. Altimari and C. S. Bildea
The variables appearing in the mathematical model are the dimensionless axial
coordinate , the dimensionless flow rates f, the temperature  along the reactor, and the
dimensionless recycle flow rates K. The model parameters are the Damkohler number
Da, the ratio between the kinetic constants K2,1, the dimensionless adiabatic temperature
rise B, the ratio between the reactions enthalpies H2,1, the ratio between the activation
energies G2,1, the dimensionless heat transfer capacity , the dimensionless cooling
temperature c.
4. Influence of the set point variables m and p
Typical steady state solution regimes of the RSR system controlled by strategy (4) are
described in Fig.2 as m varies, at p = 2.5. The reactor conversion of A is employed as
variable representative of the state of the system. Four saddle-node bifurcation points
S1, S2, S3, and S4 are predicted.
1,0
S1
xA
p = 2.5
S2
0,5
S4
S3
0,0
0
1
2
3
m1
4
m
Figure 2 Steady state solution diagram of the RSR system controlled by strategy (4), with m as
bifurcation parameter. (Da = 2.5, K2,1 = 2.5, B = 0.3, 
 = 12.5, G2,1 = 2.5, H2,1 = 4.5, c =
0).
A unique steady state is found at low m values. As m is increased, three coexisting
steady states are detected between S1 and S2. In this range, two coexisting regimes
exhibit very close conversion values resulting barely distinguishable in Fig.2. At larger
m values, a unique steady state is detected in the range delimitated by S2 and S3. As m is
further increased, three coexisting steady states arise between S 3 and S4. After the
saddle-node bifurcation point S4, a unique steady state is detected at m values lower than
m1. Increasing m in this range, the reactor conversion becomes smaller and smaller and
approaches zero in the limit m →m1. No feasible steady states are detected at larger m
values.
It is remarkable to observe that feasible steady states are always found at m values lower
than m1. Hence, m1 can provide useful indications to select m values guaranteeing
feasible operation. Analytic prediction of m1 is possible by observing that the recycle of
Q becomes infinitely large as m → m1. To verify it, it is sufficient to note that the
reactor conversion of A approaches zero as m → m1. Since finite amounts of A and B are
fed to the reactor, this can be only due to infinite recycle of Q. As the recycle of Q
becomes infinitely large and reactor conversion approaches zero, concentrations and
temperature are constant along the reactor. Hence, the reactor behaves as a CSTR
Integrated design and control of processes coupling exothermic and endothermic
reactions
5
operated at the feed temperature. For this reason, the dependence of m1 on system
parameters can be predicted by replacing the plug-flow reactor model (5) - (9) by a
CSTR model and solving it jointly with the condition xA=0. This gives the following
inequality guaranteeing the existence of feasible steady states:
m
 K 2,1
p
(13)
where the equality holds at m = m1.
Based on (13), a limit reactor-inlet flow rate of B, p1 = m/K2,1, can be equivalently
determined. Once m is given, feasible steady states are guaranteed to exist for p > p1.
5. Influence of Da and 
In this section we investigate the influence of reactor parameters on plant operability.
With this objective, we analyze the evolution of the steady state solution diagram of the
controlled RSR system, with m as bifurcation parameter, as the Damkohler number Da
and the dimensionless heat transfer capacity  are varied.
When the RSR system is controlled by strategy (4), hysteresis, isola, and double limit
varieties are computed, dividing the Da -  space in twelve regions (Fig.3) characterized
by qualitatively different bifurcation diagrams (Fig.4).
12
Isola
(iii)
10
(vii)
Double limit
1
(v)
Hysteresis
(vi)
(iv)
8
0
2,1
2,2
2,3
(x)

(xi)
6
(i)
4
(ii)
(ix)
(vii)
2
(xii)
(viii)
(iii)
(iii) - (vi)
0
1
2
3
4
Da
Figure 3 Classification of the steady state behaviors of the RSR system controlled by strategy (4).
Bifurcation diagrams corresponding to regions (i) - (xii) are displayed in Figure 4 (K2,1 = 2.5, B =
0.3,  = 12.5, G2,1 = 2.5, H2,1 = 4.5, p = 1.5, c = 0).
As reactor parameters vary, complex multiplicity patterns with a maximum of five
coexisting steady states and separated solution branches are observed (Fig.4). Parameter
values belonging to region (i) should be, in principle, recommended, because they
guarantee a unique steady state leading to safe operation. However, it must be noted
that, in this region, reactor conversion is low resulting in large recycles. To explain it, it
is sufficient to observe that region (i) mainly extends at low Damkholer values and high
heat transfer capacities. For this reason, it might be advantageous to select reactor
parameter values in different regions covering a large area of the parameter plane Da - 
(for example region (ii) or (x)). Although steady state multiplicity is invariably
experienced in regions (ii)-(xii), the analysis provided in sec.(3) can enable the selection
of m and p in a way that guarantees safe operation.
6
P. Altimari and C. S. Bildea
1.0
S1
(i)
S1
(ii)
S1
S2
S2
S3
xA
S3
S2
S2
0.5
S2
S4
S4
S3
S3
(iii)
(iv)
0.0
S1
1.0
S1
S2
S1-S2
S1
S2
S6
xA
S5
0.5
S5
S6
S3
S2
S6
S6
S3
S2
S6
S2
S5
S4
S5
S3
(vi)
S4
S4
S4
S3
(v)
S1
S5
S3
(vii)
(viii)
0.0
1.0
S1-S2
S1
S1
S1
S1-S2
S4
S5
S6
S3
S2
xA
xA
S2
0.5
S4
S2
S3
S4
(x)
(ix)
S3
(xi)
(xii)
0.0
0
1
2
3
m
4
5 0
1
2
3
m
4
5 0
1
2
3
m
4
5 0
1
2
3
4
5
m
Figure 4. Qualitatively different steady state solution diagrams of the RSR system controlled by
strategy (4) corresponding to regions (i) - (xii) displayed in Fig.3.
6. Conclusions
The steady state behavior of RSR systems, where the first order endothermic reaction A
→ R + Q and the second order exothermic reaction B + Q → P simultaneously take
place, was addressed. Flowsheets which are possible, as the physical properties of the
species involved vary, were classified. In this contribution, results concerning the
behavior of the system where the flow rates of the reactants A and B at the reactor inlet
are on flow control were presented.
Bifurcation analysis was performed as the set points of the flow control loops are varied
and feasibility boundaries for the reactor-inlet flow rates of A and B were analytically
predicted. The influence of reactor model parameters on the steady state behavior of the
RSR system was also addressed. Singularity theory was exploited to divide the
Damkholer number – reactor dimensionless heat transfer capacity space in regions
characterized by qualitatively different bifurcation diagrams. Isola, double limit and
cusp varieties were computed and twelve steady state solution diagrams exhibiting a
maximum of five coexisting solution regimes and separated branches were presented.
Our analysis shows that RSR systems coupling exothermic and endothermic reactions
exhibit complex steady state behavior. The reaction scheme considered in this work
involves three reactants, while the plantwide control structure allows fixing only two
degrees of freedom. Since it is impossible to completely remove the effect of material
recycle, no control structure allows to rule out the possibility of state multiplicity.
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