Modeling and optimization of the cyclic steady state operation of adsorptive reactors

Mamoon Hussainy*,David W.Agar

Technical University of Dortmund,Faculty of Bio-and Chemical Engineering,Laboratory for Chemical Reaction Engineering,Emil-Figge-Stra?e 66,D-44227 Dortmund,Germany

Keywords:Adsorptive reactors Multifunctional reactors Cyclic steady state Claus reaction Deacon reaction

A B S T R A C T Adsorptive reactors(AR),in which an adsorptive functionality is incorporated into the catalytic reactors,offer enhanced performance over their conventional counterparts due to the effective manipulation of concentration and temperature profiles.The operation of these attractive reactors is,however,inherently unsteady state,complicating the design and operation of such sorption-enhanced processes.In order to capture,comprehend and capitalize upon the rich dynamic texture of adsorptive reactors,it is necessary to employ cyclic steady state algorithms describing the entire reaction-adsorption/desorption cycle.The stability of this cyclic steady state is of great importance for the design and operation of adsorptive reactors.In this paper,the cyclic steady state of previously proposed novel adsorptive reactor designs has been calculated and then optimized to give maximum space–time yields.The results obtained revealed unambiguously that an improvement potential of up to multifold level could be attained under the optimized cyclic steady state conditions.This additional improvement resulted from the reduction of the regeneration time well below the reaction-adsorption time,which means,in turn,more space–time yield.

1.Introduction

Cyclic fixed bed processes are operated,in most cases,by repeating a sequence of predefined steps continuously until these cycles reach,or asymptotically approach,a condition,at which any state of the bed at any point in the cycle is identical to the previous cycle.This condition is known as a periodic state or cyclic steady state.Determining and optimizing the cyclic steady state receive significant technological interest for developing new cyclic processes or improving already existing ones.Pressure swing adsorption(PSA),temperature swing adsorption(TSA),and reverse flow reactors are common examples of such processes.With no exception to the above mentioned inherently cyclic nature,adsorptive reactors,or sorption enhanced processes as also widely known in the literature,represent a promising and upcoming technology for several important and equilibrium-limited processes applied in industry.The cyclic steady state behavior and process cycle design of adsorptive reactors have been extensively studied in the literature.For instance,the working group of Rodriguesin Portugal designed three-,four-,and five-step cycle operation of adsorptive reactors,the former for dehydrogenation of ethane to ethylene[1]and the latter ones for steam-methane reforming[2,3].For all developed cycle designs,the periodic state was determined,for which an improved performance over the conventional processes was reported.A similar conclusion has been also made by Rawadieh and Gomes[4]for a five step cyclic adsorptive steam reforming reactor.With a five cyclic step design including reaction-adsorption,depressurization,purge I and II,and pressurization,Hufton et al.[5]reported about 200%process improvement over the conventional process.

Mathematical modeling and optimization offer an economical tool for investigating a wide range of process alternatives and configurations.Nevertheless,improved calculating algorithms and optimizing routines are necessary to accelerate the solution process over the extremely time-consuming conventional techniques.For example,cyclic steady states are typically determined by the so-called successive substitution method,in which the process is simulated one cycle after another until convergence to the cyclic state is reached.This method is computationally infeasible for complex systems.Several approaches to speed up the convergence and subsequently reach the solution have been successfully proposed and implemented.Gorbach et al.[6]introduced a general method to derive shortcut models for fast cyclic processes.Applying this methodology to a set of transient process equations converts them to a stationery reduced set of equations that can then be solved faster.Following this idea,two algorithms were introduced by Kolios et al.[7]:on the one hand,the perturbation algorithm which its convergence is comparable to New ton's method,and the dual-grid algorithm,on the other hand,which provides much faster solution process and facilitates efficient analysis and design of the system.In their study on cyclic reverse flow reactors,Salinger and Eigenberger[8]introduced a methodology to transform the initial-value problem into a steady state boundary-value problem in space and time.They reported the ability of their methodology to clearly identify the multiplicity region throughout the parameter space without varying the parameter values and initial conditions as with the successive substitution method.For complex systems with quite a slow solution process,fast simple methods for the direct determination[9]of cyclic steady states will be helpful.Two types of direct calculation methods have been proposed.One is a full discretization method[10,11]in which the model equations are temporally and spatially discretized using either a finite difference method or a finite element method.To obtain accurate results,large discretization dimensions in both the temporal and spatial domains are often required,resulting in a very large set of nonlinear algebraic equations[12].The solution is then enforced towards cyclic steady state by imposing cyclic constraints to set bed conditions at the end and the beginning of each simulated cycle as identical.The second method for direct determination of periodic states is the direct substitution,in which,and unlike the first method,the nonlinear governing partial differential equations are only discretized in the spatial domain resulting in a set of ordinary differential equations that are integrated over time[12–14].Recently,Munera-Parra et al.[15]studied the cyclic behavior of the adsorptive retro shift water-gas reactor using both the direct determination methods mentioned above.They found that direct substitution method gave sufficient resolution accuracy,within reasonable computer-time,to capture even sharp fronts expected for such system with highly dynamic nature.

In this present work,the operation of the novel adsorptive reactor designs proposed by Hussainy and Agar[16]was extended to account for the role of regeneration step within the entire cycle.The cyclic steady state was first calculated using the direct distribution approach and then optimized to ensure the overall optimality of the proposed adsorptive reactor designs.As this paper is based on our previous work[16]and draw s heavily on it,we introduce in the following two sections a short summary of our findings and mathematical modeling,recommending at the same time to read our mentioned previous paper for best insight.

2.Novel Designs of Adsorptive Reactors

Several degrees of freedom are available in design and operation of adsorptive reactors.Among these,the potential of temperature profiling,in particular,to expedite the macrostructuring of adsorptive reactors for performance optimization was presented in[16].Two industrially relevant and equilibrium-limited reactions were taken as test cases:the Claus reaction for recovering sulfur from acid gas(React.1)and the Deacon reaction for the plant internal recycling of chlorine via HCl-oxidation(React.2).

In place of simple uniform functionality distributions for isothermal and adiabatic operation(benchmark cases;see Fig.1),two optimally macrostructured adsorptive reactorsw ere proposed for thesereactions:multilevel isothermal(Fig.2)and central isothermal(Fig.3)sandw ich designs.Each of these designs consists of three spatial segments;two reactive(20%of the fixed-bed)at both ends of the reactor and one adsorptive(80%of the fixed-bed)in betw een.The adsorptive segment is operated under isothermal conditions and the other two segments are operated either isothermally or adiabatically.Doing so,a substantial process performance has been reported for both reaction schemes under consideration.

Fig.1.The benchmark case for both Claus and Deacon reactions under both isothermal and adiabatic conditions:uniform functionality distribution.

Fig.2.Multilevel isothermal sandwich design.a)for Clausreaction;b)for Deacon reaction.

Fig.3.Central isothermal sandwich design for both Claus and Deacon reactions.

3.Mathematical Formulation

The one-dimensional,pseudo-heterogeneous and dynamic model previously derived to describe the behavior of isothermal and adiabatic fixed-bed adsorptive reactors was used further in this study unchanged except for the boundary conditions which have been changed to account for the countercurrent regeneration step considered here.

3.1.Governing equations and cycle description

The governing equations including the overall and component mass balances,the heat balance in case of adiabatic sections,and the momentum balance over the whole length of the adsorptive fixed-bed reactor along with the bed initial and boundary conditions are listed in Table 1.It is worth noting that the initial conditions mentioned in Table 1 represent the starting bed conditions before cycle 1 begins and not the initial states from cycle to cycle when calculating the cyclic steady state.

The values used in the simulations for feed operating parameters during the two steps of the designed cycle,which is described below,are given in Table2 for both considered reactions.As it will be explained in Section 4,the following operating parameters uf,ureg,and Tregin case of adiabatic design were chosen as design variables for the dynamic optimization of the cyclic steady state and thus their values mentioned in Table 2 are the values for the benchmark case,and for multilevel isothermal design as well in case of Treg.

Aspen Properties has been used to calculate the physical properties of the gas phase at each discrete point depending on the corresponding temperature,pressure,and component mole fraction in the gas phase.The reactions' kinetics and adsorption kinetics and isotherm used in the simulations can be found in Table 3[17-23].Detailed information about that and about the correlations from which the mass and heat axial dispersion coefficients(Daxand Λaxrespectively),and the heat transfer coefficient(h)were estimated can be reviewed in[16].

The cycle considered in this work consists of the following two steps(Fig.4 illustrates schematically the cycle design):

Step 1 Isothermal or adiabatic reaction and isothermal adsorption step.In

this step,the mixture of reactants carried by N2gas is fed into theadsorptive reactor at a temperature(Tf)and feed velocity(uf).The step is terminated when the calculated H2S-conversion reaches a value below 99.5%,which is determined by environmental legislation,in case of the Claus reaction and when the calculated HCl-conversion drops below 96%,set based on the extent of any subsequent downstream processing still required,for the Deacon reaction.The duration of this step is labeled as reaction-adsorption time(treac).

Table 1The governing equations with initial and boundary conditions

Table 2Feed conditions for both cycle steps

Table 3Reactions' kinetics and adsorption isotherm and kinetics expressions

Fig.4.Cycle design with switching conditions considered in this study.

Step 2 Countercurrent desorption(-cooling)step.Here a switch in the boundary conditions is performed to simulate the countercurrent desorption,and the cooling in case of the adiabatic operation in the reactive sections of the simulated adsorptive reactor,achieved by introducing an N2gas stream at a regeneration velocity(ureg).This step is terminated and a switch back to step 1 is activated when the average loading in the adsorption section reaches or drops below 10?5kmol·m?3and the average solid temperature reaches or drops below 250°C.The duration of step 2 is donated as regeneration time(treg).

3.2.Numerical approach

The cyclic steady state of the spatially segmented adsorptive reactor designs,with a total fixed-bed length of 1 m and 0.06 m inner diameter,was simulated and calculated using the shooting method based on the method of lines available in Aspen Custom Modeler(ACM).The governing equations were discretized in the spatial domain into fifty finite elements using the central finite difference method.This number of discretization points was chosen based on a grid independencetest in which a reduced computational effort without compromising with accuracy of the solution can be attained.

On contrary to the standard successive substitution method,the direct substitution approach was applied using New ton's method to obtain efficient estimates for the cycle initial conditions and thus achieve quadratic acceleration of the convergence to cyclic steady state.In addition to this,the dynamic nonlinear constrained optimization problem,which will be introduced in the following section,was solved employing the feasible path successive quadratic programming optimizer(FEASOPT)which is part of ACM.

4.Optimization Problem and Results

The adsorptive reactor designs shown in Section 2 were a result of optimized functionality structure and operating temperature during the reaction-adsorption stage.Yet,to verify the overall feasibility of these designs,considering the regeneration step in the entire cyclic operation is absolutely fundamental.Therefore,the cyclic operational optimality of those designs was sought by setting an optimization problem addressing the role of the regeneration step.

For transient and cyclic operations,such as adsorptive reactors,the space time yield(STY),which is defined as the molar production rate per unit volume of the reactor,is an appropriate key performance indicator[22,23].Hence,the objective function was to maximize the space time yield of the simulated adsorptive reactor by finding the optimal feed velocity in step 1 and the optimal regeneration velocity in step 2 of the considered cycle.Additionally,in case of the central isothermal design,the regeneration feed temperature was to be chosen by the optimizer.The upper and lower values of the optimizing variables were set according to technical or safety considerations.For instance,the upper limit for velocity was calculated according to the fluidization point.The upper limit of the regeneration temperature,on the other hand,was determined based on safety issues and to avoid damaging the zeolite particles,the lower limit was ascertained considering technical limitations by super-cooling the fixed-bed.The mathematical expression of the optimization problem is given below:

The optimized values of the decision operating variables are shown in Table 4.The temporal development and spatial development of the loading profiles over the whole cycle for the benchmark cases and sandwich designs are sketched in Fig.5 for the Claus reaction and in Fig.6 for the Deacon reaction.In comparison to the benchmark cases,the increased adsorbent's capacity and appropriately adjusted temperature profiles established by the suggested novel adsorptive reactor arrangements resulted in maximized productivities in the first stage of the cycle(see Figs.5a to f and 6a to f).Additionally,the earlydesorption phenomenon already occurring during the reaction adsorption cycle step at the entrance of the adsorptive reactor(Figs.5c and 6c)observed for the adiabatic benchmark cases,caused by the hot-spot formation at that location(Figs.5e and 6e),can be avoided by spatially segregating the adsorptive and reactive functionalities as suggested by the novel sandwich designs.This spatial segregation of involved functionalities reflects the fact of existence of three distinct regions in the adsorptive fixed bed reactor.In order to comprehensively analyze these regions,the approach to equilibrium parameter(χ)defined by Eq.(31)represents a direct measure to indicate the truly utilization of the functionalities at any location within the fixed bed and describes the potential driving force for a reversible reaction to take place.

Fig.5.Temporal and spatial loading and temperature profiles for Claus reaction during reaction-adsorption(ato f)and loading profiles during regeneration(g to j;the loading profile at t=0 here represents the profile at the end of the first cycle step)cycle steps:a)and g):benchmark isothermal,b)and h):multilevel isothermal sandwich design,c),e),and i):benchmark adiabatic,and d),f),and j):central isothermal sandwich design.

Fig.6.Temporal and spatial loading and temperature profiles for Deacon reaction during reaction-adsorption(ato f)and loading profiles during regeneration(g to j;the loading profile at t=0 here represents the profile at the end of the first cycle step)cycle steps:a)and g):benchmark isothermal,b)and h):multilevel isothermal sandwich design,c),e),and i):benchmark adiabatic,and d),f),and j):central isothermal sandwich design.

At χ =1,the reaction is at thermodynamic equilibrium.If χ ＞ 1,the forward reaction takes place,while the backward reaction occurs when χ＜1.

Fig.7 shows the variation of this parameter in space and time for the isothermal and adiabatic adsorptive Claus reactor.It can be clearly seen that roughly the first tenth of the fixed bed is always kinetically-controlled as the value of χ is always greater than one which suggests the non-efficiency of the adsorptive functionality in this region.In contrast,within nearly the next eight-tenths of the reactor,χ starts greater than one and then drops to the equilibrium value(one)as the adsorbent has been saturated and the equilibrium can thus no longer be moved towards the desired forward reaction.Therefore,the existence of any reactive functionality in this region would be limitedly beneficial.The last remaining part of the bed can be described as the former one except for that χ here never reaches the equilibrium value due to the fact that the adsorbent never reaches saturation until the process constraint is violated and thus this section is kinetically-controlled.The same tendency was reported by Lawrence et al.[24].

The optimized value for the regeneration velocity was the same in all cases and it is three times the corresponding value for the benchmark case.This value guarantees much faster regeneration process maintaining at the same time the allowed pressure drop in the bed and lies well below the fluidization value.Moreover,it assures avoiding the negatively affecting re-adsorption during the regeneration phase at the entrance of the adiabatically operated conventional adsorptive reactor(see Figs.5i and 6i).In contrast,the temperature of the regeneration feed does not differ from its value for the benchmark case,where higher values will lead to a slow cooling of the adiabatic reactive segments and thus longer regeneration time.In other words,the enhancement to faster desorption by higher regeneration feed temperature will not compensate the much longer cooling duration of the reactive sections in the central isothermal adsorptive reactor design.

Fig.7.Temporal and spatial profiles of the approach to equilibrium during the reaction adsorption cycle step for the conventional adsorptive Claus reactor for a)isothermal operation and b)adiabatic operation(the adsorbent volume fraction at any spatial location here is 43%).

The obtained values of the objective function under the optimized cyclic steady state conditions of the novel adsorptive reactor designs are starkly juxtaposed with those of the conventional adsorptive counterparts for the Claus reaction in Fig.8 and for the Deacon reaction in Fig.9.It is obvious that the multilevel isothermal sandwich design raises the space time yield by c.700%for the Claus reaction and thirty- five fold for the Deacon reaction,while the central isothermal sandwich design magnifies the space time yield by about 650%for the Claus and by roughly a factor of 18.5 for the Deacon process.The superior improvement reported here is coming from the extra prolonged production time(i.e.the reaction adsorption time)achieved by macrostructuring and temperature profiling and the much more efficient regeneration process(short regeneration time well below the production time)established by setting the optimal feed and regeneration velocities.

Fig.8.Obtained values of STY under cyclic steady state conditions for conventional and novel adsorptive Claus reactor designs.

Fig.9.Obtained values of STY under cyclic steady state conditions for conventional and novel adsorptive Deacon reactor designs.

Although the presented results reflect the extreme attractiveness of adsorptive reactors as an up-and-coming technology,one should not overlook certain technical challenges that are still to be solved to successfully commercialize this concept.These challenges include,in the first place,the sulfur condensation by implementing the adsorptive Claus reactor concept and developing an appropriate adsorbent that can be used under the highly corrosive conditions of the Deacon reaction.

5.Conclusions

In this paper,we carried out numerical simulations and dynamic optimizations using the direct substitution approach based on the methods of line in Aspen Custom Modeler to calculate and optimize the cyclic steady state of the novel adsorptive Claus and Deacon reactor designs proposed in our previous work.The overall feasibility of these novel designs(the multilevel isothermal and central isothermal sandwich designs)for the cyclic operation was demonstrated.The efficient regeneration process proposed here eliminates possible problems associated with other reactor configurations,such as fluidized reactors.

So far,we verified the extremely high potential for process improvements of adsorptive reactors only by two degrees of freedom,the spatial temperature profiling,and macrostructuring.Our next works will focus on utilizing other available design and operational alternatives,for instance,the side-stream feeds,and temporal profiling of operating parameters.

Acknowledgment

The authors are grateful to the German research council(Deutsche Forschungsgemeinschaft)for their financial support to the project:AG 26/18-1.

Nomenclature

C Concentration,kmol·m?3

D Mass dispersion coefficient,m2·s?1

d Diameter of the reactor,m

dmMolecular diffusivity,m2·s?1

h Heat transfer coefficient,W·m?2·K?1

k1,ClausForward rate constant for Claus reaction,kmol·kg?1·s?1·Pa?1.25

k1,DeaconForward rate constant for Deacon reaction,kmol·kg?1·s?1·Pa?1.5

k2,ClausBackward rate constant for Claus reaction,

kmol·kg?1·s?1·Pa?1

L Reactor length,m

P Pressure,Pa

Pr Prandtl number

Re Reynolds number

r Rate of reaction or adsorption,kmol·kg?1·s?1

rcZeolite crystal radius,m

Sc Schmidt number

T Temperature,K

t Time,s

u Superficial velocity of the gas phase,m·s?1

X Conversion

x Axial coordinate,m

α Adsorbent volume fraction

β Ratio of area to volume of a solid particle,m2·m?3

ΔH Enthalpy change,J·mol?1

ε Bed void fraction

κ Equilibrium constant

λ Heat conductivity,W·m?1·K?1

Λ Heat dispersion coefficient,W·m?1·K?1

μ Viscosity,Pa·s

ν Stoichiometric coefficient

ρ Density,kg·m?3

τ Cycle time,s

χ Approach to equilibriumSubscripts

A Adsorption

ads Adsorbent

avg Average

ax Axial

cat Catalyst

DP Desired product;S8for Claus reaction and Cl2for Deacon reaction

eq Equilibrium

f Feed

g Gas phase

i Species i in the gas phase

j Segment number

min Minimum

p Particle

R Reaction

reac Reaction-adsorption phase

reg Regeneration

s Solid phase