(368d) Nonlinear Analysis of Pfr – Separation – Recycle Systems Coupling Exothermic and Endothermic Reactions

Performing endothermic and exothermic chemical reactions in a single unit offers several advantages. 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 [1]. 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, 3] coupling propane combustion and endothermic thermal cracking of propane to ethylene and propylene [4], coupling methane steam reforming with catalytic oxidation of methane in partial oxidation reactors [5, 6].

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. For example, [7, 8] investigated the behavior of reactor separation recycle systems demonstrating the occurrence of nonlinear phenomena such as steady state multiplicity and dynamic bifurcations leading to autonomous periodic oscillations. The influence of these undesired phenomena on design and control is thoroughly discussed in [9]. However, in spite of these results, the behavior of reactor separation recycle systems where exothermic and endothermic chemical reactions simultaneously take place has not been studied.

In this work, we investigate the steady state behavior of a reactor separation recycle system coupling the following reactions:

,  endothermic, first order

, exothermic, second order

Figure 1. PFR ? Separation ? Recycle systems coupling exothermic and endothermic reactions; (a) flowsheet structure; (b) control structure fixing the flow rates of A feed and reactor inlet B.

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 reaction. Unconverted reactants A, B, Q are recovered and recycled to the reactor. Depending on the physical properties of the species involved, several flowsheets are possible. A typical flowsheet is presented in figure 1a.

We assume that heat feedback effects are removed by fixing the reactor inlet temperature by the use of a heat exchanger placed upstream the chemical reactor. The reactor is described by a plug flow model. In this way, the stand ? alone unit invariably exhibits a unique steady state solution regime. Therefore, our analysis reveals nonlinear phenomena which only can be attributed to the interaction between mass recycle and coupling endothermic and exothermic reactions.

Consistent flow rates/concentration specifications must be provided to solve steady mole balance equations. We show that, once separation performances are defined, the number of required flow rates/concentration specifications is, independently of the recycle structure, equal to the number of reactants fed to the system. These specifications define the control strategies.

Our work presents a thorough investigation of the nonlinear steady state behavior of the reactor separation recycle system, for several flowsheet structures and control strategies. Singularity theory is exploited to divide the space of the Damkholer number Da and the reactor dimensionless heat transfer coefficient b in regions characterized by qualitatively different steady state solution diagrams.

In this abstract, some results concerning the steady state behavior of the plug flow reactor separation recycle system are presented, for the flowsheet and control structure described in figure 1a and 1b respectively.

Hysteresis and isola varieties are found which divide the Da ? b space into six regions (figure 2), each of them characterized by a qualitatively different steady state solution diagram (figure 3).

                            

Figure 2. Classification of the steady state behavior. Hysteresis and isola varieties are described by dashed and continuous lines respectively. Bifurcation diagrams corresponding to regions (i) ? (vi) are displayed in figure 3.

In region (i), two steady state solution regimes coexist at low f_A,0values. As f_A,0is increased, a saddle node bifurcation point S₁ is predicted leading to the disappearing of feasible steady state solution regimes.

Crossing the isola variety and moving to region (ii), an isolated high conversion steady state solution branch arises at large f_A,0 values. The isolated branch is delimitated by the saddle node bifurcation points S₂ and S₃ and covers a narrow  f_A,0range.

As the hysteresis variety is crossed moving to region (iii), two saddle node bifurcation points S₄ and S₅ are detected delimitating a range where four steady state solution regimes coexist.

Crossing again the isola variety and moving to region (iv), the isolated steady state solution branch delimitated by the saddle node bifurcation points S₂ and S₃ vanishes and a sudden growth of the steady state reactor conversion is observed as the saddle node bifurcation point S₁ is approached.

Crossing the hysteresis variety and moving to region (v), two saddle node bifurcation points S₂ and S₃ appear delimitating a narrow f_A,0  range where four steady state solution regimes coexist.

As the isola variety is crossed and parameter values falling in region (vi) are selected, saddle node bifurcation points S₅ and S₃ vanish giving rise to an isolated high conversion steady state solution branch. The isolated branch is delimitated by the saddle node bifurcation points S₂ and S₄ and extends over a narrow f_A,0 range.

Figure 3. Qualitatively different steady state solution diagrams corresponding to regions (i) ? (vi) displayed in figure 2.

The previous results show that complex multiplicity patterns and unfeasible operating ranges can occur in reactor separation recycle system coupling exothermic and endothermic reactions. In this contribution, the analysis of the steady state behavior of the system is extended to several control strategies. In particular, the relationship between the nonlinear behavior of the system and plant control is detailed and design guidelines to improve plant controllability are provided. 

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