Formation of Thermal Wave Trains in Loop Reactors: Stability Limits

Article pubs.acs.org/IECR

Formation of Thermal Wave Trains in Loop Reactors: Stability Limits and Spatiotemporal Structure for Reversible Reactions Pietro Altimari,†,* Erasmo Mancusi,‡,∥ and Silvestro Crescitelli§ †

Dipartimento di Chimica, Università “Sapienza” di Roma, Piazzale Aldo Moro 5, 00185 Roma, Italia Facoltà d’Ingegneria, Università del Sannio, Piazza Roma, 82100, Benevento, Italia ∥ Universidade Federal de Santa Catarina, Departamento de Engenharia Quı ́mica e de Alimentos 88040-970, Florianópolis, SC, Brazil § Dipartimento d’Ingegneria Chimica, Università “Federico II”, Piazzale Tecchio 80, I-80125 Napoli, Italia ‡

ABSTRACT: Networks of fixed bed reactors with periodically switched inlet and outlet sections are studied with reference to equilibrium limited reactions. The methanol synthesis is selected as representative reaction example and the mechanisms governing the emergence of periodic regimes corresponding to trains of traveling thermal waves are analyzed. Analytical approximations, accounting for the influence of the implemented switching strategy and in satisfactory agreement with numerical simulation, are derived for the switch time stability limits in terms of temperature fronts velocities. Large and sudden variations in the switch time stability range of any wave train solution are demonstrated to occur when the sum of the velocities of declining temperature fronts exceeds the sum of the velocities of rising temperature fronts. The illustrated results provide indications on how to design and operate the network so as to generate thermal wave trains with desired number of waves and stability limits.

1. INTRODUCTION The mechanisms governing the emergence of spatiotemporally varying patterns as traveling fronts and periodic waves in fixed bed reactors have been over the past decades the subject of numerous theoretical and experimental studies.1−6 Only a few studies have however focused on the development of techniques constructively exploiting results achieved in this field to design sustainable reactor solutions. Fixed bed reactors are commonly operated in stationary regimes and reactor parameters are selected to ensure robust stability margins from the emergence of dynamic regimes. Recent studies have however shown that the performance of chemical processes can significantly be improved by purposefully controlling, rather than suppressing, spatiotemporally varying patterns.7,8 The main idea is to steer by weak control impulses the evolution of traveling waves and/or fronts to optimize the spatiotemporal evolution of the state variables.9 Theoretical and experimental studies have been reported demonstrating the effectiveness of this approach to enhance the selectivity and/or the yield of chemical reactions on catalytic surfaces.7−10 In chemical reactor engineering, a promising route to the constructive application of these ideas is offered by networks of N fixed bed reactors with stepwise periodically shifted inlet and outlet ports.11,12 In this system, commonly referred to as loop reactor, the emergence of high conversion regimes is governed by the interaction between the periodic shift of inlet and outlet ports and the motion of traveling temperature fronts. The periodic shift of inlet and outlet ports can, for example, be exploited at low adiabatic temperature rise and feed temperature values to trap within the bed a reaction front traveling in the flow direction, thus ruling out the need of an external heat exchanger to preheat fresh reactants. This has motivated the extensive application of the loop reactor in the purification of gaseous streams with low concentration of volatile organic compounds.13−18 Besides enforcing autothermal operation, the interaction between traveling temperature fronts and shifted inlet and outlet © 2012 American Chemical Society

sections has also been proved to produce temperature and concentration patterns ensuring large conversion values in reversible exothermic reactions.19−21 The main advantage resulting from application of the loop reactor is, for this class of reactions, the emergence of thermal waves delimited by rising and declining temperature fronts. These temperature patterns reproduce the interstage cooling effect of multistage fixed bed reactors and thus enable to overcome thermodynamic equilibrium limitations.22 The potential of the loop reactor technology to enhance performance with respect to the stationary operation mode has been also demonstrated for the selective reduction of nitrogen oxides23 and the production of synthesis gas through the coupling of partial oxidation and steam reforming.24 Typically, the loop reactor is operated in periodic regimes with period T equal to the time interval needed to recover the initial feeding sequence. Such a regime is attained with inlet and outlet ports periodically shifted in the flow direction of ns = 1 reactors and constant time interval τ between two successive ports rotations. T-periodic regimes corresponding to a single rising temperature front followed by a declining one arise with this operating strategy at ratios σ = Vsw/Vth of the switching velocity Vsw = ns/τ to the purely thermal front velocity Vth around unity.25 This temperature pattern exhibits the structure of a single thermal wave traveling in the flow direction and is stable in a τ interval [τa,τb] delimited by extinction. The possibility to robustly operate the network under these conditions for the case of an irreversible exothermic reaction has been recently demonstrated by Madai and Sheintuch.26 No high conversion regime can be sustained at τ > τb while many dynamic regimes can be found at 0 < τ < τa. Multiperiodic, quasiperiodic, and Received: Revised: Accepted: Published: 9609

December 21, 2011 May 25, 2012 June 22, 2012 June 22, 2012 dx.doi.org/10.1021/ie2030008 | Ind. Eng. Chem. Res. 2012, 51, 9609−9619

Industrial & Engineering Chemistry Research

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chaotic regimes have been indeed predicted besides T-periodic ones in this latter τ range.27 T-periodic regimes arising at 0 < τ < τa correspond to trains of traveling thermal waves and are again stable within τ intervals delimited by extinction.28 These solutions have attracted considerable interest since they produce in equilibrium limited processes average conversion values significantly larger than those found with single thermal waves at τ ∈ [τ a , τ b ].20,22 Unfortunately, T-periodic regimes corresponding to thermal wave trains have been detected with ns = 1 in narrow τ ranges and at low τ values. These conditions are unfeasible since they would require tight control of τ and frequent ports rotations, with the risk of damages to the external valves system. Mancusi et al.22 have however recently shown that the number and the τ stability interval of T-periodic regimes corresponding to thermal wave trains increase with ns. These results were however based on numerical simulation alone and no explanation was proposed for the mechanisms governing the emergence of thermal wave trains. A first analysis of these mechanisms was proposed by Altimari et al.29 On the basis of a geometrical approach, infinitely many domains of T-periodic regimes corresponding to thermal wave trains with different number of waves were proved to exist for any ns and N, and analytical approximations to the stability limits and the spatiotemporal pattern of these solutions were derived. The analysis was in this latter study simplified by considering a fast irreversible exothermic reaction. In such a case, fresh reactants are completely converted in a narrow reactor region giving rise to the formation of a unique reaction front traveling at constant velocity Vfr < Vth followed by only purely thermal fronts with velocity Vth. On the contrary, multiple reaction fronts characterized by different velocities simultaneously travel through the network as reversible reactions are considered.22 It is demonstrated in the present work that the latter aspect implies significant differences in the stability boundaries and in the spatiotemporal pattern of T-periodic regimes corresponding to thermal wave trains compared to the case of irreversible exothermic reactions. We here investigate the influence of ns and N on the stability and the structure of thermal wave trains for reversible exothermic reactions. The geometric analysis of the spatiotemporal temperature pattern presented in ref 29 is revisited to account for the emergence of multiple reaction fronts. Analytical approximations to the stability limits of T-periodic regimes corresponding to thermal wave trains are derived in terms of τ, ns, N, and reaction front velocities. Mechanisms undetected in the analysis of irreversible exothermic reactions are shown to govern the evolution of the stability boundaries and of the spatiotemporal pattern of T-periodic regimes corresponding to thermal wave trains as reactor parameters vary. The paper is structured as follows. In section 2, the mathematical model of the network and the examined class of operating strategies are described. In section 3, analytical approximations to the stability limits of thermal wave trains are derived and the mechanisms determining the spatiotemporal temperature and conversion patterns are described. Final remarks end the paper.

Figure 1. Schematic representation of the loop reactor. Inlet and outlet ports are stepwise periodically rotated in the flow direction of ns reactors.

switching velocity is defined as Vsw = ns/τ. ns and τ do not vary during network operation. A one-dimensional pseudohomogeneous model taking into account axial mass and energy dispersive transport is considered to describe the dynamics of each reactor unit. The methanol synthesis is selected as representative process. Particularly, it is assumed that the following reaction alone takes place within the network: r1

CO + 2H 2 ⇄ CH3OH r2

(1)

The simplified kinetic model derived in ref 21 is employed. Therefore, the dimensionless mass and energy balances for the ith reactor unit in the loop are written as follows: 2 ⎧ ∂θ ∂θ 1 ∂ θi ⎪ Le i + v i = + Br(xi , θi) ∂ξ Peh ∂ξ 2 ⎪ ∂t * ⎪ 2 ∂x ⎪ ∂xi 1 ∂ xi +v i = − r(xi , θi) ⎪ 2 ∂ * ∂ t Pe ξ ⎪ m ∂ξ ⎨ ⎪ ⎛ θγ̃ ⎞ ⎪ r(xi , θi) = Da exp⎜ i ⎟ ⎝ θi + γ ̃ ⎠ ⎪ ⎪ ⎛ 2 ⎞⎞⎞ ⎛ ⎛ ⎪ × ⎜1 − x ⎜⎜1 + ψ ⎜ (1 − μ)γ ̃ ⎟⎟⎟⎟ (for i = 1, 2, 3, 4) i ⎜ ⎟ ⎪ ⎝ θi + γ ̃ ⎠⎠⎠ ⎝ ⎝ ⎩

(2)

with the following definitions for dimensionless variables and parameters: μ=

tu T − T0 E2 A z ; ψ = 2 ; ξ = ; t* = 0 ; θ = γ ̃ ; E1 A1 L L T

x=1− Da = Pem =

( −ΔH )C0γ ̃ C E u ; γ̃ = ; vs = ;B= ; (ρc p)f T0 C0 RT u0

(ρc p)f Luo (ρc p)eff AL ; Peh = ; exp( −γ ); ̃ Le (ρc p)f ke uo (ρc p)f Luo Df

(3)

where x is the conversion, θ is the dimensionless temperature, t* is the dimensionless time, γ is dimensionless activation energy, μ is the ratio between the activation energies, ψ is the ratio between the pre-exponential factors of reactions 1 and 2, respectively, v is the dimensionless velocity, B is the dimensionless adiabatic temperature rise, Da is the Damkohler number, Le is the ratio of the solid to gas heat capacity, and Peh and Pem are the Peclet numbers for the energy and the mass balance, respectively. Dimensional variables are defined in the nomenclature section. Danckwerts boundary conditions are applied at the inlet and

2. MATHEMATICAL MODEL Networks of connected N identical fixed-bed reactors are considered (see Figure 1 for a representative scheme). Inlet and outlet ports are stepwise periodically rotated in the flow direction of ns reactor units. The period between two consecutive ports rotations is referred to as the switch time τ and the 9610

dx.doi.org/10.1021/ie2030008 | Ind. Eng. Chem. Res. 2012, 51, 9609−9619


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Figure 2. Temperature and conversion profiles observed during two successive cycles in T-periodic regime at τ = 90 with ns = 3 and N = 5 (solid and dotted lines correspond to the beginning and the end of the cycle respectively while vertical bold lines are used to denote the position of the network inlet and outlet ports).

outlet of each reactor and are modified after each cycle to take into account the permutation of the network feeding sequence. In particular, dimensionless boundary conditions can be written as follows: 1 ∂xi Pem ∂ξ

spatiotemporal structure of thermal wave trains is examined in subsection 3.4. 3.1. Formation of Thermal Wave Trains. The periodic shift of inlet and outlet ports enables trapping thermal fronts traveling in the flow direction within the loop reactor and thus determines the emergence of traveling thermal waves. As representative example, the temperature and conversion profiles observed under T-periodic regime at τ = 90 with N = 5 and ns = 3 during two successive cycles are displayed in Figure 2. Two rising and one declining temperature front traveling with dimensionless velocities V1, V3, and V2, respectively, are found. The temperature profile exhibits therefore the structure of a single thermal wave followed by a downstream rising temperature front. Reaction takes place over narrow regions around the three temperature fronts. A first conversion step Δx1 occurs around the upstream rising temperature front as fresh reactants reach the required temperature. Temperature and conversion become constant as thermodynamic equilibrium conditions are approached. The temperature reduction around the declining temperature front determines then a displacement of the equilibrium conditions leading to a second conversion step Δx2. Finally, the downstream rising temperature front produces a further increase in the reaction rate giving a third conversion step Δx3. Therefore, a three steps conversion pattern is found. The maximum temperature is determined by thermodynamic equilibrium and does not vary appreciably during the cycle. The structure of the temperature and conversion profiles changes on the contrary because of the relative motion between reaction fronts. The velocities V1, V2, and V3 exhibit indeed different values. Any velocity Vi mainly depends on the dimensionless adiabatic temperature rise B, on the dimensionless feed temperature θin,32 and on the conversion step Δxi, and approaches the velocity of a purely thermal front Vth = 1/Le in the limit Δxi → 0. This latter condition is achieved in the limit of zero adiabatic temperature rise, that is B → 0. It is worth remarking that the velocities of rising and declining temperature fronts are always lower and larger than Vth, respectively. With reference to the pattern displayed in Figure 1, it is, for example, found that V1 < Vth and V3 < Vth while V2 > Vth. This difference can be explained by noting that the displacement of a rising temperature front has the effect of cooling down the upstream region, while the motion of a declining temperature front warms up the downstream region.

= −[1 − fi (t )]x in − fi (t )xi − 1(i − 1, t ) 0

+ xi(i − 1, t ) 1 ∂θi Peh ∂ξ

= −[1 − fi (t )]θin − fi (t )θi − 1(i − 1, t ) 0

+ θi(i − 1, t ) ∂xi ∂ξ

= 1

∂θi ∂ξ

=0 1

(4)

with f i(t) = g(t − mod((i − 1)ns, N)τ) with mod(*, ·) denoting the standard modulo function30 and g(t) describing piecewise constant periodic function (see ref 29 for a description). Results presented in this paper are obtained otherwise specified with the same parameter values adopted in ref 22. These and all the parameter values considered in the study are selected to ensure that the only high conversion solution found in stationary operation of the network, that is with constant feeding sequence, is a reaction front traveling in the flow direction. Under these conditions, no high conversion regime can be found if the network is operated in stationary operation. Model order reduction of the infinite dimensional PDE system (eqs 2−4) has been achieved through finite difference approximation of the spatial derivatives. The resulting set of ordinary differential equations has been numerically integrated by the Fortran routine DLSODES.31

3. THERMAL WAVE TRAINS In this section, the influence of ns and N on the stability and the structure of thermal wave trains is analyzed. A qualitative description of the mechanisms responsible for the emergence of thermal wave trains is initially reported in subsection 3.1. Then, analytical approximations to the stability limits of thermal wave trains are derived in subsections 3.2 and 3.3. Finally, the 9611



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Figure 3. Temperature and conversion profiles observed during two successive cycles in T-periodic regime at τ = 40 with ns = 3 and N = 5 (solid and dotted lines correspond to the beginning and the end of the cycle respectively).

T-periodic regime at τ = 40 with N = 5 and ns = 3 during two successive cycles are displayed in Figure.3. Unlike Figure 2, switching is here performed before the temperature fronts with position indexes 2 and 3 reach the network outlet port. This determines an increase in the number of temperature fronts simultaneously traveling through the network. In particular, a train of three thermal waves followed by a rising temperature front is observed. Again, reaction takes place over narrow regions around the fronts, and hence, seven distinct conversion steps are found. It is important to remark that the conversion step achieved around the ith reaction front and, consequently, the heat locally generated, decrease in general with the position index i, that is, when reaction fronts closer to the network outlet port are considered. The velocity of a rising temperature front is therefore greater than the velocity of any upstream rising temperature front while the velocity of a declining temperature front is lower than the velocity of any upstream declining temperature front. Since rising and declining temperature fronts are associated to odd and even position indexes respectively, this implies that V2i ⩾ V2i+2 ⩾ Vth and V2i+1 ⩽ V2i+3 ⩽ Vth for any position index i, the equalities being valid in the absence of reaction. 3.2. Stability Limits of Thermal Wave Trains. The displacement of temperature fronts shift hot bed regions in the flow direction. Ensuring that fresh reactants are fed immediately after switching to a hot section requires therefore that the rotation of the network inlet and outlet ports is synchronized with the motion of temperature fronts. Indications on how to select the switch time τ so as to enforce such synchronization can be derived by geometric analysis of the network spatiotemporal temperature pattern. To illustrate this analysis, we display in Figure 4 the spatiotemporal temperature patterns of the periodic regimes arising with ns = 3 and N = 5 at τ = 90 and at τ = 40. The time t* is scaled in Figures 4a and b by τ and, hence, each period τ here corresponds to a unit interval of variation of t*/τ. The position of inlet and outlet ports during each cycle is also indicated by vertical bold segments. Straight lines separating adjacent cold and hot regions define in Figure 4 the actual axial position of temperature fronts and will be referred to as spatiotemporal paths followed by the temperature fronts. The slope of the any of such path is equal to 1/Vτ, with V as the

An increase in the conversion step Δxi determines therefore for a rising temperature front a reduction in the velocity at which the upstream region is cooled, while it produces an increase in the velocity at which a declining temperature front warms up the downstream region. Since the velocity of each temperature front approaches Vth in the limit Δxi → 0, the velocities of rising and declining temperature fronts must necessarily result at Δxi > 0 lower and greater than Vth, respectively. In the following, Vi and Δxi will be used to denote the velocity and the conversion step corresponding to a reaction front following i − 1 upstream reaction fronts and the integer i will be referred to as position index of the front. It must be emphasized that the conversion step achieved around a reaction front is uniquely determined by its position index. Since any reaction front exhibits position index varying between two successive cycles, different conversion steps are therefore achieved around the same reaction front depending on the considered cycle. As a consequence, the same front will also move with different velocities during successive cycles. In Figure 2, for example, the front forming at the network inlet at t* = 0 moves with velocity V1 during the first cycle. The same front follows however two fronts and exhibits therefore velocity V3 during the successive cycle. It is apparent from Figure 2 that the feed stream is shifted at switching to a section covered by the upstream thermal wave. This ensures the formation of a new reaction front traveling with velocity V1 at the network inlet and therefore prevents reaction extinction. This condition identifies the fundamental mechanism responsible for the emergence of high conversion regimes in presence of reaction fronts traveling in the flow direction: reaction extinction can be prevented as far as that the feed stream is switched to a hot section before the upstream reaction front has reached the network outlet. From this perspective, the switching of the feed stream can be thought as realizing a tracking of thermal waves which would be otherwise swept from the network. On the basis of this idea, infinitely many ways to orchestrate the periodic shift of inlet and outlet ports can be conceived still enabling to prevent reaction extinction. One may particularly think to decrease τ leaving unchanged ns in the illustrated numerical experiment so that the feed stream can still reach at switching a hot region. As an example, the evolution of the temperature and conversion profiles predicted under 9612



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Figure 4. T-periodic regime corresponding traveling thermal waves predicted at τ = 90 (a, c) and τ = 40 (b, d) with ns = 3 and N = 4. (a, b) Spatiotemporal temperature patterns (blue and red correspond to cold and hot regions respectively while vertical bold lines denote the position of inlet and outlet ports). (c, d) Temperature profiles observed at t* = 0.5τ.

fronts and can make no longer possible the fulfillment of the energy balance as described above. Increasing τ can, for example, cause the point b to lie downstream to the network inlet section. Under these conditions, only part of the cold region ab is swept from the network and a progressive reduction in the enthalpy of the bed is observed. This eventually causes the achievement of an extinguished regime. The maximum τ ensuring the existence of spatiotemporal temperature patterns of the type described in Figure 4a is therefore τb =ns/V1. This τ value ensures that the point b exactly lies at switching at the network inlet section. It can analogously be verified that spatiotemporal temperature pattern of Figure 4a extinguishes at τ values lower than τa = ns/V2. In the latter case, the reaction front forming in a cannot reach the network outlet section within a single cycle still causing a progressive reduction in the enthalpy of the bed. It is important to note that the computed τ stability interval is valid provided that the time required for the two temperature fronts forming in a and b to reach each other is greater than the time needed for the same fronts to reach the network outlet port. This is equivalent to require that the temperature front forming in a takes less than the temperature front forming in b to reach the network outlet port. The time intervals needed for the temperature fronts forming in a and b to reach the network outlet port are respectively equal to Δta* = ns/V2 and Δtb* = (ns − V1τ)/ V3. Since the inequality must hold for any τ falling in the computed stability range, the lower switch time stability limit

velocity of the corresponding temperature front, and can be estimated to get a numerical approximation to V. Figure 4a shows that the displacement of the upstream temperature front forming at switching at the network inlet section determines during the following cycle the formation of a cold region ab. This cold region exhibits width equal to V1τ. Starting from the points a and b, two temperature fronts then form and move with velocities V2 and V3 shifting downstream the cold region ab. These two temperature fronts do not move parallel but approach each other because V2 > V3. Consequently, the cold region ab is shifted downstream and exhibits decreasing width. The two temperature fronts reach the network outlet section and thus extinguish before the width of the cold region ab becomes zero. In this way, the cold region ab is swept from the network. Physically, the reduction in the width of the cold region ab must be attributed to the heat generated by the exothermic reaction around the two temperature fronts forming in a and b while the shift of the cold region ab is determined by thermal convection. Overall, while the displacement of the upstream rising temperature front determines during a single cycle the formation of a cold region ab, the motion of the two downstream temperature fronts warms up a cold region of identical width. This fulfills the network energy balance over a single cycle. Since t* is scaled by τ, decreasing τ produces in Figure 4a a rigid counterclockwise rotation of the spatiotemporal paths followed by temperature 9613



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τa = ns/V2 minimizing Δt*b is selected. Therefore, the computed stability limits are valid provided that ns/V2 ≥ ns (1 − V1/V2)/V3 which can be solved to give V1 + V3 ≥ V2, that is, the sum of the velocities of the two rising temperature fronts must be greater than the velocity of the declining temperature front. Since decreasing τ produces a rigid counterclockwise rotation of the spatiotemporal path associated to any temperature front, the two temperature fronts formed in a and b can reach the network outlet section after a number of cycles greater than one as τ is decreased. In this way, the cold region ab can still be swept away from the network, thus enabling to fulfill the network energy balance. This situation corresponds, for example, to the emergence of the spatiotemporal temperature pattern displayed in Figure 4b. The time needed for the two temperature fronts forming in a and b to reach the network outlet section ranges between 2τ and 3τ. Since two temperature fronts form in a and b at each switching instant and take about three cycles to reach the network outlet section, the bed is crossed during each cycle by six temperature fronts. In addition, an upstream temperature front always forms at switching near the network inlet section. Therefore, a train of three thermal waves is found. The application of simple geometric rules still enables us to determine an approximation to the stability limits of the spatiotemporal temperature pattern displayed in Figure 4b. For this purpose, it is important to observe that the temperature fronts forming in a and b exhibit velocities varying between two successive cycles. The temperature front forming in a follows initially three temperature fronts and is therefore characterized by velocity V4. During the following two cycles, the same temperature front moves however downstream to a single and to five temperature fronts, respectively, and is therefore characterized by velocities V2 and V6, respectively. The temperature front arising in b moves with velocity V5, V3, and V7 during the first, the second, and the third cycle following its formation, respectively. The minimum switch time τ enabling the formation of the spatiotemporal temperature pattern displayed in Figure 4b must therefore fulfill the equality V2τ + V4τ + V6τ = 2. It can be analogously verified that the maximum switch time must fulfill the equality V1τ + V3τ + V5τ = 2. Again, these results are valid as far as the time interval required for the two temperature fronts forming in a and b to reach each other is greater than the time interval needed for the two temperature fronts to reach the network outlet section. It can be proved based on geometric arguments that this condition is fulfilled if (V1 + V3 + V5 + V7) ≥ (V2 + V4 + V6). The illustrated analysis can be extended to determine approximations to the stability limits of any spatiotemporal temperature pattern arising as τ is decreased and the two temperature fronts forming in a and b reach the network outlet section during the same cycle. Particularly, a generalization to any ns and N is possible. For this purpose, it is sufficient to note that the axial position of the network inlet section becomes q = mod(ns·k + q0, N) after k switches, q0 being its initial axial position, and does not change during the following cycle. Let us indeed assume that the time interval needed for the two temperature fronts forming in a and b to reach the network outlet section ranges between (p − 1)τ and pτ. Under these conditions, the network is crossed by p couples of temperature fronts formed in a and b at the previous p switching instants, and by an upstream temperature front formed near the network inlet section at the previous switching instant. This generates a train of p thermal waves. The schematic representation in Figure 5 shows

Figure 5. Geometric based derivation of the stability limits of thermal wave trains.

that the switch time stability limits of such a pattern can be expressed in terms of reaction front velocities as follows: mod(nsp , N ) mod(nsp , N ) ≤τ≤ p p−1 ∑i = 1 V2i ∑i = 0 V2i + 1

(5)

Inequalities 5 are derived by assuming that the velocities of temperature fronts are constant during a single cycle and change discontinuously between two successive cycles. Particularly, temperature fronts forming in a and b experience velocities V2i and V2i+1 (i = 1, ..., p), respectively. Moreover, the computed stability limits (5) are valid as far as the temperature front forming in a takes a time interval greater than the temperature front forming in b to reach the network outlet section. It can be demonstrated based on geometric arguments that this condition is fulfilled if and only if the sum of the velocities V2i of declining temperature fronts is lower than the sum of the velocities V2i+1 of rising temperature fronts, that is: p

p

∑ V2i ≤ ∑ V2i+ 1 i=1

i=0

(6)

Details concerning the derivation of such inequality are sketched in the Appendix at the end of the article. Since the velocity of a rising temperature front is in general lower than the velocity of a declining temperature front, that is, V2i+1 ⩽ V2i, inequality 6 can be fulfilled only because the number of rising temperature fronts is one unit greater than the number of declining temperature fronts. If the heat generated by the reaction around each temperature front increases, differences between the velocities of rising and declining temperature fronts also increase and can thus cause, as will be shown later, the violation of 6. When relationships describing the dependence of temperature front velocities on reactor parameters are available, inequalities 5 can be employed to trace approximate stability boundaries of T-periodic regimes corresponding to thermal wave trains. No example of such relationship has been however developed to date. An alternative approach is to compute, as previously shown, the velocity of any temperature front through evaluation of the slope of the associated spatiotemporal path. The evaluation of the temperature front velocities based on such an analysis makes 9614



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however necessary to preliminarily compute a T-periodic regime corresponding to a thermal wave train to trace its spatiotemporal temperature pattern. For this purpose, estimates of the switch time values sustaining the emergence of thermal wave trains must be derived before the velocities of temperature fronts can be computed. Since V2i ⩾ Vth and V2i+1 ⩽ Vth for any position index i, stability subdomains of thermal wave trains can be computed from inequalities 5 as follows: mod(nsp , N ) mod(nsp , N ) ≤τ≤ pVth V1 + (p − 1)Vth

(7)

The lower bound of this interval is known since 1/Vth coincides with the ratio Le of the solid to gas heat capacity. The upper bound requires on the other hand an estimate of the only velocity V1 of the upstream reaction front velocity. This can be derived by numerical simulation of the network in stationary operation starting from a constant initial temperature profile. It is worth noting that the stability limits 7 are identical to those found for fast irreversible exothermic reactions29 and can be derived based on the illustrated geometric analysis by assuming that the upstream temperature front moves with velocity V1 while all the other temperature fronts are characterized by velocity Vth. 3.3. Comparison between Numerical and Analytical Predictions: Merging of Temperature Fronts. The limits of the range eq 7 fall at low adiabatic temperature rise values not far from the stability limits predicted by eq 5. The largest fraction of the reactor conversion is indeed achieved under these conditions around the upstream temperature front characterized by velocity V1 while small conversion steps are realized around the downstream temperature fronts. Moreover, conversion steps are lower around temperature fronts closer to the network outlet port. Therefore, Vth represents at low adiabatic temperature rise a good approximation to the velocities of temperature fronts following the upstream temperature front exhibiting velocity V1. This approximation corresponds to assume that the spatiotemporal paths associated to the temperature fronts forming in a and b move parallel and, hence, that a cold region ab is warmed up only due to the convective transport of heat. As the adiabatic temperature rise increases, a larger conversion step is however achieved around any reaction front. Consequently, deviations between Vth and reaction front velocities become larger and the fraction of the switch time interval 5 covered by 7 becomes progressively lower. This is confirmed by Figure 6 where the τ stability limits computed by numerical simulation are compared to those predicted with inequalities 5 and 7. Deviations between the predictions of inequalities 5 and 7 become larger as the adiabatic temperature rise is increased. It is worth noting that inequalities 7 provide, in contrast to inequalities 5, a conservative approximation of the stability limits. Inequalities 7 always give τ values falling within the actual stability range while the upper limit is slightly overestimated by inequalities 5. Identical behavior has been found for any considered solution. The upper limit predicted by 5 cannot therefore be directly used for design purposes but should be refined by numerical simulation or appropriately reduced to prevent overestimation. In any case, τ values provided by inequality 7 can be used without further analysis. Figure 6 shows that the estimate of the stability limits based on inequalities 5 becomes unsatisfactory at large B values. Particularly, the lower and the upper stability limits described in 5 loose validity as B is increased and the values B1 and B2 are crossed respectively. The achievement of these adiabatic

Figure 6. Comparison between the τ−B stability boundaries computed by numerical simulation (solid line) and by geometric analysis (dashed and dotted lines) of the T-periodic regime corresponding to a train of three thermal waves arising with ns = 3, N = 5 (dashed and dotted lines correspond to the predictions of inequalities 5 and 7, respectively).

temperature rise values corresponds to a modification in the physical mechanism governing the emergence of thermal wave trains. To illustrate this modification, the train of two thermal waves arising with N = 5 and ns = 3 at τ = 40 is again considered in Figure 7. The spatiotemporal temperature pattern computed near the lower τ stability limit predicted by numerical simulation at adiabatic temperature rise B > B1 is analyzed in Figure 7a. Inequality 6 is apparently no longer fulfilled. The two temperature fronts forming in a and b reach indeed each other and thus merge before reaching the network outlet section. The cold region ab is warmed up under these conditions because of the heat generated around the two temperature fronts. If τ values close to the upper stability limit computed by numerical simulation and adiabatic temperature rise values greater than B2 are selected, the two temperature fronts forming in a and b are analogously found to merge before reaching the network outlet port (Figure 7b). At adiabatic temperature rise B ∈ [B1, B2], inequality 6 is fulfilled only over a fraction of the τ interval 5 close to its upper bound. This fraction decreases with B and becomes zero as the value B2 is crossed. The lower τ stability limit can no longer be computed based on inequalities 5 when B ∈ [B1, B2]. On the contrary, the upper τ stability limits is still described by 5 in such B interval. Inequalities 5 fail to predict both the lower and the upper τ stability limit when B > B2. The lower and the upper τ stability limits fall under these conditions outside the interval 5. The temperature fronts forming in a and b reach the network outlet section and, at the same time, meet each other after pτ and (p − 1)τ when B = B1 and B = B2, respectively. Therefore, the following equalities 8 and 9 are fulfilled at B1 and B2, respectively: p

p

∑ V2i = ∑ V2i+ 1 i=1

i=0

p−1

p−1

∑ V2i = ∑ V2i+ 1 i=1

i=0

(8)

(9)

It must be noted that equality 8 represents the limit form of inequality 6. Equalities 8 and 9 can be used to predict B1 and B2 when the evolution of any Vi on B is determined. 9615



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Figure 7. Merging of consecutive temperature fronts at adiabatic temperature values B < B1. (a) Spatiotemporal temperature pattern of the T-periodic regime corresponding to a train of three thermal waves arising with ns = 3, N = 5 at B = 46.5 and τ = 30.5. (b) Spatiotemporal temperature pattern of the T-periodic regime corresponding to a train of three thermal waves arising with ns = 3, N = 5 at B = 70 and τ = 57.

Increasing B over B2 determines a reduction in the velocities of rising temperature fronts and an increase in the velocities of declining temperature fronts. This decreases the time needed for the two temperature fronts forming in a and b to merge. Even though not shown in Figure 6, further discontinuous increases in the stability interval of thermal wave trains, analogous to those occurring at B1 and B2, are therefore observed as the time needed for the temperature fronts forming in a and b to merge becomes again lower than the time needed for the two temperature fronts to reach the network inlet section. The τ stability interval of the spatiotemporal pattern shown in Figure 6 is, for example, again discontinuously enlarged when the two fronts forming in a and b merge within less than τ. It is important to note that while the merging of two consecutive temperature fronts enlarges the stability window of thermal wave trains, it causes at the same time a reduction in the average conversion. The disappearance of two temperature fronts is indeed accompanied by the suppression of two conversion steps. It must also be emphasized that, according to inequality 6, the boundaries of the reactor parameter space delimiting the region characterized by the merging of temperature fronts only depend on the number of thermal waves p. The validity of this theoretical prediction is confirmed by the results of numerical simulation. We have, for example, found that B1 ∼ 43 and B2 ∼ 60 for the train of three thermal waves computed with ns = 4 and N = 5. These B values are very close to the values B1 ∼ 44.5 and B2 ∼62 found for the train of three thermal waves computed with ns = 3 and N = 5 and examined in Figures 4, 6, and 7. B1 and B2 values

Again, approximations to the stability limits of thermal wave trains can be derived when B > B1 through geometrical analysis of the network spatiotemporal temperature pattern. Decreasing τ below the lower stability limit predicted by 5 at B > B1 produces a proportional counterclockwise rotation of the spatiotemporal path associated to any temperature front. The considered thermal wave train therefore extinguishes as the spatiotemporal path associated to the temperature front forming in a reaches the network inlet section. The same arguments can be followed to compute the upper switch time stability limit when B > B2. The lower switch time stability limit of the spatiotemporal pattern shown in Figure 7 is, for example, achieved at B > B1 when the reaction front formed in a reaches the network inlet section in one cycle and after covering a single reactor. This gives τ = 1/V4. Analogously, it can be verified that the maximum switch time sustaining the same spatiotemporal pattern is achieved at B > B2 when the reaction front forming in b reaches the network outlet section in one cycle and after covering three reactors. This corresponds to about τ = 3/(V1 + V5). This latter estimate is based on the assumption that the two temperature fronts forming in a and b travel during the first cycle following their formation with velocities V4 and V5. It can therefore be further refined by taking into account that, as apparent from Figure 7b, the velocities of the two fronts forming in a and b vary during the initial cycle from V4 to V2 and from V5 to V3, respectively, because of the merging of temperature fronts. The description of how to get such refinement entails a simple but lengthy application of geometric rules and is therefore omitted for sake of clarity. 9616



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Figure 8. Geometric derivation of the spatiotemporal structure of thermal wave trains. The structure of a train of p thermal waves can be derived at any instant of the cycle by tracing the spatiotemporal paths of the temperature fronts forming in a and b at the previous p switching instant and of the temperature front forming at the network inlet at the previous switching instant.

reported above have been derived from the B−τ stability boundaries traced by numerical simulation. We note however that good agreement is found between B1 and B2 values computed numerically and those predicted based on equalities 8 and 9. In summary, approximations to the τ stability limits of a T-periodic regime corresponding to a thermal wave train can be computed through the following procedure: (1) Compute an estimate for V1 through numerical simulation of the network in stationary operation. (2) Determine based on inequalities 7 a coarse estimate of the τ stability limits. (3) Compute through analysis of the spatiotemporal temperature pattern the unknown reaction fronts velocities values Vi. (4) If inequality 6 is fulfilled with the computed Vi, then use eq 5 to get a refined approximation of the τ stability limits; otherwise, proceed through geometric inspection of the spatiotemporal temperature pattern as illustrated above. 3.4. Spatiotemporal Structure of Thermal Wave Trains. The structure of a thermal wave train is uniquely determined by the number and the axial positions of temperature fronts traveling through the network. The number of temperature fronts is determined at B < B1 by the number of cycles needed for the temperature fronts forming in a and b to reach the network outlet section. If this time interval ranges between (p − 1)τ and pτ, 2p + 1 temperature fronts are found to travel during any cycle through the network. The upstream temperature front is formed at the previous switching instant at the network inlet section, while the downstream 2p temperature fronts form at the previous p switching instants in a and b. The structure of a thermal wave train can therefore be computed at any instant by tracing the

spatiotemporal paths of these temperature fronts. A schematic representation illustrating the application of these ideas is reported in Figure 8. The structure of the train of three thermal waves arising with ns = 3 and N = 5 at τ = 40 is predicted. Note that the spatiotemporal paths followed by temperature fronts can be readily traced once the temperature fronts velocities are known. The illustrated procedure can also be implemented to predict the structure of thermal wave trains when B > B1. In this latter case, the spatiotemporal paths of temperature fronts forming in a and b will collide before reaching the network outlet port. It is important to note in this context that while the stability limits of thermal wave trains with number p of waves are described by inequalities 5, τ values fulfilling inequalities 5 may give rise, for certain p values, to solutions with number of waves lower than p. The mechanism responsible for this effect is described in Figure 9. The case N = 4 is here considered as representative example. At τ values fulfilling 5 with p = 3 and ns = 1, temperature fronts forming in a and b reach the network outlet section between zero and one cycle. Accordingly, a single thermal wave, instead of a train of three thermal waves, is generated. The same happens if τ values fulfilling 5 with p = 2 or p = 3 are selected. It can be verified that the selection of ns = 3 ensures on the contrary the emergence of thermal wave trains with number of waves p ranging between 1 and 3. If a prescribed number p of thermal waves is to be generated, ns and N must therefore be appropriately selected so as to ensure that, at τ values fulfilling 5, temperature fronts forming in a and b can reach the network outlet section within a time interval ranging between (p − 1)τ and pτ. The fulfillment of this condition can be rapidly checked by graphical inspection as shown in Figure 9. 9617



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coexist with T-periodic regimes. A systematic study of the stability limits and the spatiotemporal pattern of multiperiodic regimes goes however beyond the scope of the present article and will therefore be the subject of a future study.

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APPENDIX: DERIVATION OF INEQUALITY 6 The switch time stability interval 5 is valid provided that, for any switch time falling in such interval, the time needed for the temperature fronts forming in a and b to reach the network outlet port is greater than the time needed for the two fronts to reach each other. The fulfillment of this constraint can be imposed by requiring that the temperature front forming in a takes more than the one forming in b to reach the network outlet port. On the basis of Figure 5, this gives p−1

(p − 1)τ + Figure 9. Schematic representation illustrating the impossibility for temperature fronts forming in a and b to reach the network outlet port after any number of switches. (a) With ns = 1 and N = 4, it is impossible for the two temperature fronts to reach the network outlet port after 1, 2, and 3 switches. (b) Having ns = 2 and N = 4 allow reaching the network outlet port after any number of switches ranging between 1 and 3.

mod(nsp , N ) − (∑i = 1 V2i)τ V2p p−1

≥ (p − 1)τ +

mod(nsp , N ) − (∑i = 0 V2i + 1)τ V2p + 1

(11)

The left- and right-hand side of inequality 11 are the time intervals needed for the temperature front forming in a and b to reach the network outlet section respectively. To ensure that inequality 11 is valid for any τ falling in the interval 5, τ must be set equal to the lower limit of this interval. Accordingly, inequality 11 is recast in the following form:

4. CONCLUSIONS Thermal pattern formation mechanisms of the loop reactor were analyzed with reference to a specific simple equilibrium limited reaction. Parameters values sustaining the formation of temperature fronts traveling at constant velocities were selected and the influence of ns and N on the stability and the spatiotemporal pattern of high conversion T-periodic regimes corresponding to thermal wave trains was investigated. These periodic regimes can be obtained when the feed stream is switched to the hot section before the upstream thermal front does reach the network outlet. This requires synchronization between the motion of thermal fronts and the shifting of inlet and outlet ports. A geometric analysis of the spatiotemporal temperature pattern was presented to determine the switch time values enforcing such synchronization. Analytical approximations were derived for the dependence of the stability limits of thermal wave trains on ns, N, and on reaction fronts velocities. Moreover, it was demonstrated that two different mechanisms govern the emergence of thermal wave trains as reaction front velocities vary: (1) At reaction front velocities values close to the purely thermal front velocity, the complete cooling of the bed is prevented because of the heat transferred by declining temperature fronts to cold regions generated by the displacement of rising temperature fronts. (2) With large differences between reaction fronts velocities and the purely thermal front velocity, cold regions generated by the displacement of rising temperature fronts are warmed up because of the heat generated by the exothermic reaction around successive rising and declining temperature fronts. Transition to the latter mechanism is accompanied by the merging of two consecutive temperature fronts and determines a discontinuous increase in the width of the switch time stability interval of thermal wave trains. Finally, a procedure to determine the structure of thermal wave trains was discussed. It is worth concluding by remarking that the illustrated study did not consider multiperiodic regimes, that is, with period multiple of T. Results of numerical simulation and bifurcation analysis recently reported in ref 33 have shown that these latter regimes can also give rise to thermal wave trains and always

p−1

mod(nsp , N )(∑i = 1 V2i) mod(nsp , N ) − p V2p (∑i = 1 V2i)V2p p−1

mod(nsp , N )(∑i = 0 V2i + 1) mod(nsp , N ) ≥ − p V2p + 1 (∑i = 1 V2i)V2p + 1

(12)

which can be solved to give p

p−1

(∑ V2i)V2p + 1 − ( ∑ V2i)V2p + 1 i=1

i=1 p

p−1

≥ (∑ V2i)V2p − ( ∑ V2i + 1)V2p i=1

i=0

(13)

After manipulating the sums in 13, inequality 6 is finally obtained.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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NOMENCLATURE A = Arrhenius constant B = dimensionless adiabatic temperature rise cp = specific heat capacity C = concentration Da = Damköhler number Df = mass axial dispersion coefficient E = activation energy f(t) = forcing function g(t) = piecewise constant function defined by eq 5 ke = solid phase axial heat conductivity L = length of a single reactor unit dx.doi.org/10.1021/ie2030008 | Ind. Eng. Chem. Res. 2012, 51, 9609−9619


Article

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L0 = network length ns = number of reactors periodically jumped by inlet and outlet sections N = number of reactors composing the network p = number of thermal waves Pe = Peclet number R = gas constant r = reaction rate t = time t* = dimensionless time T = temperature u = gas rate v = dimensionless gas rate Vi = dimensionless velocity of the i-th temperature front following the network inlet Vsw = dimensionless switching velocity Vth = dimensionless purely thermal front velocity x = conversion z = axial coordinate Greek Letters

γ = dimensionless activation energy ΔH = heat of reaction Δx = conversion step ε = reactor void fraction θ = dimensionless temperature μ = ratio between activation energies ξ = dimensionless axial coordinate ρ = density τ = dimensionless switch time ψ = ratio between Arrhenius constants

Subscripts and Superscripts

0 = reference conditions f = fluid h = enthalpy in = inlet m = mass

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REFERENCES

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Formation of Thermal Wave Trains in Loop Reactors: Stability Limits

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