Resonance Response of Reverse Flow Reactors: A Numerical

May 29, 2015 - ... the resonance response is attributed to the breakdown of temporal and ... to avoid reactor extinction, and so does avoiding thermal...
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Resonance Response of Reverse Flow Reactors: A Numerical Simulation Yingying Zhu, Geng Chen,* Xinbao Li, and Guohua Yang Faculty of Maritime and Transportation, Ningbo University, Ningbo, Zhejiang 315211, China ABSTRACT: The effect of periodic time-varying inlet parameters on reverse flow reactor behavior is simulated using a onedimensional heterogeneous model with methane as the model reactant. When the flow-reversal cycle is an odd multiple of the inlet concentration variation cycle, a resonance response occurs. Such reactor behavior will occur with various values of superficial inlet velocity, mean inlet methane concentration, flow-reversal time and inert-to-catalyst ratios during this simulation work. A synchronization of the two cycles will cause a sharp increase in the variation of maximum temperature, which may impair reactor stability. Simulation results reveal that the resonance response is attributed to the breakdown of temporal and spatial symmetry in the temperature field of reactor. When a resonance response occurs, adjusting the flow-reversal time is a feasible method to maintain autothermal operation of the reactor. Different from commonly used control strategies both increasing and decreasing flow-reversal time can be used to avoid reactor extinction, and so does avoiding thermal runaway. Simulation work of introducing periodic variation of inlet gas velocity is performed, and a similar resonance response behavior is observed. Beld,14,15 Salomons16 and Snyder17 have shown that a RFR can overcome a sudden decrease in inlet concentration and recover to its former PSS several flow-reversal cycles after the feed has been restored. Such reactor behavior is most attributed to the large thermal inertial of the solid packing. If heat is removed from a RFR, e.g., by reactor cooling or heat withdraw, asymmetric and quasi-periodic states will occur.18,19 Such undesired complex dynamic behaviors that may lead to excessive temperature rise in the reactor is caused by the failure of bed temperature response to fast switching flow directions. During the actual process, the concentration of pollutants often varies with time in a stochastic manner.3,20−22 Several control strategies23−27 have been put forward to deal with this issue and have been proved to be quite effective and with efficiency. However, applying the RFR to a real process still necessitates a better understanding of reactor behavior with inlet condition variation. Numerical investigation carried out by Cittadini28 reveals that when periodic inlet concentration or flow rate was introduced, it would cause serious instability problems at a particular ratio between the feeding and flowreversal cycles. Our previous work shows that the interaction of inlet combustible concentration cycle and flow-reversal cycle would impair reactor stability and cause reactor extinction.29 A RFR is categorized into fixed bed reactor and shares some common characteristics of this kind of reactor. A fixed bed reactor at steady state operation can be regarded as a dynamical periodic system, which may have a natural frequency of its own. Capsaskis30 studied experimentally the subharmonic response of the heterogeneous catalytic combustion of CO/C3H6 in a fixed bed reactor with a square-wave feed input of C3H6. Na-

1. INTRODUCTION Reverse flow reactors (RFRs), a type of forced unsteady state catalytic reactor, have proved to be one of the effective solutions for waste gas decontamination, such as SO 2 oxidation,1−3 decontamination of a diluted VOC gas stream4−7 and the selective reduction of NOx.8−10 A RFR is a catalytic fixed bed reactor, in which the feed flow direction is reversed periodically. When exothermic reactions are conducted in the reactor, a temperature wave is generated that creeps along with the gas flow. The catalyst at either end acts as a heat regenerator, while the released heat is trapped in the reactor, which maintains the temperature plateau within the reactor and sustains the catalytic reaction even with weak exothermic effects. So, the catalyst packing at the two ends is normally substituted by inert materials for economic reasons. A determination of the steady states of such a system and the construction of a corresponding control strategies are of primary interest in chemical engineering because only steady state or pseudosteady state (PSS) operation has a realistic meaning. However, the RFR behaves in a rather sophisticated manner because of its inherent nonstationary character derived from forced unsteady operation. A number of efforts have been made to establish the dependence of reactor performance on many design and operating parameters, such as reactor length, gas flow rate, flow-reversal time, inlet concentration, wall effects and reaction kinetics. A number of mathematical models and simulation methods have been developed for the dynamic calculation of PSS and the sensitivity analysis over an extended range of parameters. Detailed reviews were provided by Matros11 and Kolios.12 A comprehensive survey on RFR ́ 4,13 performance and model simplification was given by Marin. A change in operating conditions can lead to significant qualitative changes in the system response, or even endanger reactor stability (by causing reactor extinction or thermal runaway). The effect of a temporary significant step change in the inlet concentration on reactor behavior is the most commonly discussed issue in this area. The works of van de © 2015 American Chemical Society

Received: Revised: Accepted: Published: 5885

October 24, 2014 January 27, 2015 May 19, 2015 May 29, 2015 DOI: 10.1021/ie504203v Ind. Eng. Chem. Res. 2015, 54, 5885−5893

Article

Industrial & Engineering Chemistry Research Ranong31 reported on an amplification phenomenon of periodic inlet temperature disturbances in the fixed bed when the disturbance frequency reached a certain value. According to Yakhnin’s research,32−34 such an inlet temperature amplification is attributed to a decoupling of the heat and mass waves through the reactor bed, and is referred to as the differential flow instability phenomenon. According to those works, it is reasonable to deduce that a RFR is possessed of similar characteristics too. In this work, the interaction of a time-varying inlet parameter and reactor behavior is investigated through numerical simulation. The catalytic oxidation of methane, an irreversible exothermal reaction, was chosen as a model reaction. The following simulation work focuses mainly on the effects of a sine waveform variation of inlet concentration and gas velocity on RFR behavior with different operating conditions and reactor setups. The reason why choosing sin waveform is that sin wave is simple but fundamental and discussion on effects of such inlet variation types could reveal some intrinsic nature of RFR. It is also a stepping stone for further study on effects of other types of periodic inlet variations because every periodic waveform can be represented by a series of sin waves using Fourier series expansion. Discussions of feasible control strategies for stable operation are included. Simulation work regarding effects of inlet gas velocity variation is also performed.

Table 1. Parameters Used in the Simulation inlet gas temperature reactor bed length, L bed void fraction, ε catalyst pellet diameter, dp,c catalyst density, cp,s catalyst thermal conductivity, λs activation energy, Ea pre-exponential factor, k∞ heat of combustion, ΔH inert pellet diameter, dp,I inert pellet density, ρI inert pellet thermal conductivity, λs,I

∂Tg ∂t

∂t

=−

u0ρg,0 ∂Tg ερg ∂x

+

keff,g ∂ 2Tg c p,gρg ∂x 2

∂t

=−

ερg

∂x

+ Deff

∂ 2Cg ∂x

2

av kg(Cg − Cs) ε

(4)

kga v (Cg − Cs) = (1 − ε)( −RA )V he

(5)

and in the inert sections: kga v (Cg − Cs) = 0

(6)

Boundary conditions: t > 0,

Deff

∂Cg ∂x

Z = 0: −

keff,g ε ∂Tg u0c p,gρg,0 ∂x

= (Cg − Cg,0)

= T0 − Tg (7)

u0 ε

(8)

∂Ts =0 ∂x t > 0,

(9)

Z = L:

∂Cg ∂x

=

∂Tg ∂T ∂Cs = s =0 = ∂x ∂x ∂x

(10)

Initial conditions: t = 0: Tg = Ts = T0 ,

Cg = Cs = 0

(11)

The dynamic simulation of the reactor behavior is conducted by a self-developed FORTRAN code. The above set of partial differential equations is discretized only in spatial dimension by means of method of lines. The resulting ordinary differential equations are solved by the FORTRAN routine VODE, which is developed by LLNL (Lawrence Livermore National Laboratory, University of California). The flow-reversal operation of the reactor is realized by inverting the temperature and concentration fields at the end of each half cycle. The simulation results are in coincidence with experimental results as shown in previous work,29 which also reveals the accuracy of this method.

ha v (Tg − Ts) − εc p,gρg

−

( −RA )V he = ηk∞exp( −Ea /(RTs))Cs

The mass balance in the catalyst section:

The mass balance in gas phase: u0ρg,0 ∂Cg

(3)

where

(1)

∂Cg

keff,s ∂ 2Ts ha v (Ts − Tg) − c p,sρs ∂x 2 (1 − ε)c p,sρs ( −RA )V he ΔH c p,sρs

+

2. MATHEMATICAL MODELING Many types of RFR modes with different complexity have been proposed in the literature. A heterogeneous one-dimensional unstationary model taking plug flow assumption is used. Such a model is considered to be a good balance between accuracy and complexity4,5,35 and is chosen in this work for such reason. An unstationary model can give out the evolution of temperature and reactant concentration distribution under unsteady state conditions, which meets the requirements of this work. The model considers the gas phase and solid phase as different phases due to the highly exothermic reaction on catalytic surface. The model equations derived from energy and mass conservation for the two phases together with initial and boundary conditions are listed in eqs 1−11. Because oxygen is excessive during reaction, methane catalytic oxidation reaction is considered as first-order. Homogeneous combustion is ignored as the reaction temperature in this simulation is relatively low. Taking the fast reaction assumption, combustible accumulation on catalytic surface is ignored. The interparticle mass transfer is incorporated by accounting the effectiveness factor (Thiele modulus). The main physical properties used in the numerical simulation are listed in Table 1. The energy balance in gas phase: ∂Tg

=−

298 K 1.0 m 0.405 4 mm 1240 kg/m3 0.73 W/(m·K) 95244 J/mol 2.0 × 108 s−1 802358 J/mol 6 mm 1765 kg/m3 1.13 W/(m·K)

(2)

The energy balance in solid phase: 5886

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3. RESULTS AND DISCUSSION Due to the intrinsic dynamic nature, comparison of different RFR behaviors should be performed under a consistent criteria. The PSS that a reactor can achieve with the same operating conditions, initial conditions and surroundings is commonly used.36 In this work, such PSS is chosen as a baseline. Because the thermal history (e.g., the presence of an electric preheater) will affect the temperature field achieved in PSS,16 it is of importance to maintain the same initial conditions or at least a similar one for a series of comparative experiments. During model simulation, the initial conditions and all the other parameters are kept unchanged during each operation to achieve a reproducible PSS. When a PSS is attained and maintained for several full flow-reversal cycles, different types of periodic inlet variation are introduced and the simulation continues until the reactor reaches a new PSS. The interrelation between the inlet concentration variation cycle and flow-reversal cycle is illustrated in Figure 1. The inlet

Figure 1. Interrelation between the periodic inlet concentration variation cycle and flow-reversal cycle.

concentration varies in sinusoidal waveform. As shown in Figure 1, the average inlet concentration (Cave) denotes timeaveraged concentration over a full flow-reversal cycle, which also equals to the inlet concentration (C0) when periodic inlet variation is not introduced. The net energy imported to the reactor is the product of mean combustible concentration and conversion rate. It must be equal to the mean gas temperature rise between inlet and outlet if the reactor is adiabatic. Thus, maintaining Cave unchanged is to maintain the gross energy input consistent under various operation conditions. The initial phase difference of the two cycles are equal, which means inlet concentration is increased from average value to the maximum when gas flow direction is just switched from backward to forward. The amplitude of the variation is defined as A. In Figure 1, positive values of gas flow velocity refer to the forward flow cycle and negative values refer to the reversed cycle. In this work, the total reactor length is set to 1.0 m, whereas a portion of the catalytic section is varied in different cases. Figure 2 presents the maximum temperature variation as a function of tfeed/tc for different superficial gas velocities, cycle times and catalyst proportions. Here, tc is switch time whereas tfeed is feed variation cycle time. In all cases, the reactor is operated at a relatively high thermal level, with the inlet methane concentration being well above the extinction limit, which is determined from trial-and-error tests and simplified model calculations. The autothermal operation can be

Figure 2. Maximum temperature as a function of inlet concentration variation cycle to flow-reversal cycle ratios (tfeed/tc) over a various operating conditions, for all cases: Cave = 1000 ppmv (ΔTad = 32.5 K), A = 0.5.

maintained if the hot reaction zone is kept inside the reactor and a relatively high conversion rate is guaranteed. The reactor 5887

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Industrial & Engineering Chemistry Research gradually reaches a new PSS within several flow-reversal cycles after introducing the inlet concentration variation. The methane conversion is 97% or higher in the cases selected. So, a wider range of tfeed/tc can be applied in the simulation, whereas reactor extinction will occur when Cave is low and tfeed/ tc is of certain values. When each point is determined in Figure 2, a PSS is achieved and maintained for over 10 full flowreversal cycles with the same operating conditions and initial conditions first. Then the inlet concentration varies periodically until the reactor reaches a new PSS, the evolution of temperature profile and maximum temperature in 30 successive full flow-reversal cycles is recorded. When introducing periodic inlet variations, a low frequency fluctuation in temperature fields may occur in some circumstances, which means the axial temperature profile of two neighboring cycles probably do not coincide with each another. As shown in Figure 2, all curves are similar in shape: the maximum temperature increases gradually with increasing tfeed/ tc, whereas several peaks emerge on these curves when tfeed/tc is 1/7, 1/5, 1/3 and close to 1.0. A maximum value is reached when tfeed/tc equals 1.0, whatever the operating parameters is in this simulation work. According to such simulation results, the flow-reversal time is defined as the characteristic response time of a RFR. If the cycle time of periodic inlet and the characteristic response time are in specific ration, a large temperature variation in the reactor can be expected. Such RFR behavior is defined as resonance behavior herein. With the periodic flow-reversal operation, a creeping heat front is generated in a RFR, which moves back and forth along the reactor. This is one of the major features of such kind of reactor. The velocity of the moving heat front, derived from a dynamic simulation without inlet concentration variation is presented in Figure 3. As shown in Figure 3, the shape of the

cycles are the same, the temporal symmetry of the RFR is maintained while axial temperature distribution or movement of heat front between two neighboring cycles are the same. If a RFR is considered as a dynamic system, into which a variation in inlet conditions introduces a periodic disturbance, the switching frequency of gas flow direction in the reactor is then taken to be the fundamental frequency of the dynamic system. As shown in Figure 3, the profiles of heat wave velocity are generally in rectangular waveform. When rewriting the expression of heat wave velocity into a Fourier series, only the odd harmonic components remain, as follows: n

u(t ) =

⎛

⎞ 2πt + θi⎟ ⎝ (2i − 1)tc ⎠

∑ Aicos⎜ i=1

(12)

Here, A is the Fourier coefficient and θ is the initial phase of each component, i = 1, 2, 3, ... n. According to harmonic analysis theory used in electrical networks, a dynamic system would resonance if its fundamental frequency is an integral multiple of the disturbance frequency. As shown in Figure 2, several peaks emerge for a tfeed/tc of 1/7, 1/5, 1/3 and close to 1.0. This implies that only the odd harmonic components remain in the RFR system discussed here. Larger fluctuation in maximum temperature implies the reactor system tends to be more thermally unstable. Thermal runaway will occur if maximum temperature in catalysts is exorbitant and endanger safety operation of the reactor. The maximum temperature variation reaches its peak value when tfeed/tc is 1.0, because the amplitude of the first harmonic component is of the maximum in all the harmonic components, as calculated in eq 12. In Figure 2, the curve should have peaked at tfeed/tc = 1/11, but the variation was too small and it overlapped the nearby peak at tfeed/tc = 1/7. A periodic inlet disturbance with higher amplitude (A, as defined in Figure 1) will cause a larger temperature variation in the reactor while cycle time and wave type are kept the same. The temporal and spatial symmetry breakdown of the reactor is thought to be the cause of the resonance response in the RFR. Development of axial temperature profile with periodic inlet concentration variation is shown in Figure 4. Here, t = 0 h is the time flow direction switches from backward to forward and inlet concentration variation starts. It takes about 20 full

Figure 3. Velocity of creeping heat front in the reactor within one full flow-reversal cycle, calculated from dynamic simulation.

plot of heat wave velocity versus time is a mirror image between two neighboring half cycles. The preheating of the cold inlet gas is improved with an increase in inert packing ratio with higher thermal inertia, and results in a higher reactor temperature level. Methane is oxidized almost completely soon after it reaches the catalyst section. Heat released by exothermal reaction tends to accumulate near the interface between inert and catalyst section. This slows down the creeping velocity of the heat front significantly immediately after the flow direction is switched. However, if the flow-reversal times of the two half

Figure 4. Development of axial temperature profile with periodic inlet concentration variation, u0 = 0.1 m/s, Cave = 1500 ppmv, tfeed = tc = 1800 s, A = 1.0, Lcatal = 0.7 m. 5888

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Industrial & Engineering Chemistry Research

Figure 5. Reactor extinction because of resonance response for low inlet methane concentration and the feasibility of changing the flow-reversal time to avoid extinction, u0 = 0.1 m/s, Cave = 545 ppmv, tc = 1800 s, A = 1.0, Lcatal = 0.7 m. The point at which a periodic concentration variation is introduced is denoted as the starting point (t = 0).

flow-reversal cycles to establish a new PSS at given tfeed/tc. The temperature plateau gradually shifts to one side of the reactor and shrinks during such process. The maximum temperature in the reactor increases from 750 to 840 K and moves aside. Because the time-averaged methane concentration is well above Cave in the forward flow cycle (the flow direction is from left to right in Figure 4), heat released by methane catalytic combustion tends to accumulate in a smaller portion at the left side of the reactor in forward flow cycle. Because the maximum temperature is below 900 K and heat wave is moving back and forth within one flow-reversal cycle, the effect of catalysts deactivation is ignored. After flow reversal, the timeaveraged methane concentration is all below Cave, the right portion of the reactor is cooled by inlet gas flow rather than heated up by methane combustion in the backward flow cycle (the flow direction is from right to left in Figure 4). Due to the shrinking of temperature plateau and shifting of reaction zone, the reactor tends to be thermally unstable. This is more likely to cause reactor extinction when the inlet concentration is low because of the shrinking temperature plateau, while thermal runaway most likely occurs when the inlet concentration is high

because of large amount of heat accumulation in a relatively small portion of the reactor. Figure 5 provides an example of reactor extinction because of the resonance response encountered during dynamic simulation. Bed temperature at the center of the reactor is used as ordinate while operating time as abscissa. The minimum methane concentration required to maintain autothermal operation under such an operating condition is about 512 ppmv. As shown in Figure 5a, reactor extinction occurs when tfeed equals tc and Cave is 545 ppmv, but autothermal operation can be maintained with the same Cave when tfeed does not equal to tc (e.g., tfeed = 1700 s as shown in Figure 5b). It usually takes several tens of full flow-reversal cycles for a RFR to extinct under adiabatic conditions. Such a process caused by resonance response is also rather long, as shown in Figure 5a. Further dynamic simulation also reveals that the extinction process will be accelerated by the resonance response if the Cave is lower than the extinction limit. The flow-reversal time is an important operating variable for a RFR control system. With a given reactor length, decreasing flow-reversal time means more heat is trapped inside the 5889

DOI: 10.1021/ie504203v Ind. Eng. Chem. Res. 2015, 54, 5885−5893

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Industrial & Engineering Chemistry Research

Figure 6. Thermal runaway because of resonance response for high inlet methane concentration and feasibility of changing flow-reversal time to avoid thermal runaway, u0 = 0.1 m/s, Cave = 2000 ppmv, tc = 1800 s, A = 0.6, L = 1.0 m, Lcatal = 0.7 m. The point at which a periodic concentration variation is introduced is denoted as the starting point (t = 0).

when tc equals 2100 s, which is slightly greater than the value used in this case. The discussion of thermal runaway of fixed bed reactors is rather empirical but is of practical interest. In this work, a sharp increase in maximum temperature over 900 K is considered to be thermal runaway. Catalyst deactivation or damage to equipment becomes possible at such a high temperature. The defined runaway phenomenon is observed when the inlet methane concentration is high and the inlet variation cycle time is equal to the flow-reversal cycle time, as illustrated in Figure 6a. Temperature at the center of the reactor is used as ordinate while operating time as abscissa, the same as in Figure 5. A similar thermal runaway phenomenon and dramatic fluctuation of maximum temperature can be observed when the methane concentration is over 2500 ppmv for a steady inlet feed while other operating parameters are the same as given in Figure 5a. A large amount of heat will accumulate in a small portion of the reactor, form local hot spots, and eventually lead to a thermal runaway when the inlet concentration variation cycle and flowreversal cycles are synchronized. Even when tfeed is close to tc, wide fluctuations of bed temperature will occur. This behavior

reactor, which is a feasible way to maintain autothermal operation when inlet combustible concentration decreased or superficial gas velocity increases. So when operated near the extinction limit and with an inlet perturbation, a commonly used control strategy is to reduce the flow-reversal time to increase the thermal stability of the reactor, as shown in Figure 5c. Because reactor extinction is caused by a resonance response, maintaining autothermal operation by adjusting the flow-reversal time to avoid synchronization of the inlet variation cycle and flow-reversal cycle is also an option, as illustrated in Figure 5d. Autothermal operation can also be maintained after tc is increased to 1900 s, after introducing a variation in inlet concentration for 30 full flow-reversal cycles. However, a too high tc will also lead to reactor extinction. This is attributed to a higher extinction limit and wider moving range of heat front as tc increases. For example, under the same operating conditions as shown in Figure 5d, if tc is further increased to 2100 s, reactor extinction will finally occur whenever a change is made in tc. This is because the minimum required methane concentration is about 550 ppmv (with stable inlet feed) 5890

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composite wave types can always be written in Fourier series. Large difference of total combustible input between the two half flow cycles will lead to breakdown of temporal and spatial symmetry of a RFR. Thus, it will cause resonance response in the reactor. Variation in the inlet gas temperature or superficial gas velocity can also be regarded as transient (dynamic) perturbations if a RFR is treated as a dynamic system. Therefore, resonance response can be expected when introducing those periodic inlet variations. However, perturbations in inlet temperature is damped and reactor stability is hardly affected during simulation due to the large thermal inertia of catalyst or inert packing at the two ends. The influence of periodic variation in superficial gas velocity on the maximum temperature is similar to that of inlet concentration variation, as shown in Figure 8. Inlet concentration is constant

can be regarded as a result of the resonance response when the reactor is at a relatively high thermal level, causing reactor instability. Compared with the case discussed in Figure 5, it is reasonable to deduce that both increasing and shortening the flow-reversal time are feasible ways to avoid thermal runaway. Such simulation work has been conducted with results as shown in Figure 6b−d. Increasing the flow-reversal cycle time is a common method to decrease the maximum temperature when the reactor thermal level is high. This also yields satisfying results by avoiding thermal runaway if a resonance response occurs. However, as shown in Figure 6b, the bed temperature starts dropping before going through an inertial increasing stage of several cycles with an increased tc. Such an inertial increasing stage may endanger the safe operation of the reactor or cause catalyst thermal deactivation if the runaway process is severe. This stage can be shortened by changing tc to a higher value. As shown in Figure 6b,c, the bed temperature requires about 10 cycles to drop below 900 K when tc is changed to 1900 s, but only requires about 5 cycles when tc is changed to the higher value of 2100 s. Figure 7 gives a graphic explanation of the

Figure 8. Maximum temperature as a function of periodic variation in superficial gas velocity cycle to flow-reversal cycle ratios (tfeed/tc) over a various operating conditions and reactor setups. For all cases: Cave = 1000 ppmv (ΔTad = 32.5 K), u = 0.1 m/s, tc = 1800 s, Lcatal = 0.7 m, A = 0.02.

Figure 7. Cause of temporary maximum temperature increase in Figure 6b,c. Portion marked with a thick line represents the hot portion of the reactor still being fed with high methane concentration.

temporary increase in maximum temperature that occurs in Figure 6b. The hot portion of the reactor (at one side of the reactor where a higher methane concentration is fed during the half flow cycle, as shown in Figure 4) is still fed with methane of high concentration immediately after tc is increased. This part of the reactor could not be cooled sufficiently until the methane feed between the two half cycles is comparable, and this requires several full flow-reversal cycles. On the basis of the above discussion, shortening tc seems to be a more feasible way to avoid the thermal runaway caused by the resonance response, because the inertial increasing stage of bed temperature can be eliminated, as shown in Figure 6d. However, tc should be chosen with caution, otherwise a continuous increase in reactor thermal level or even runaway will occur for low tc as more heat is trapped in the reactor in such case. Further simulation results reveal that if tc is reduced to below 1200 s, the maximum temperature will drop first in a similar way as shown in Figure 6d, and then start to increase very slowly soon after. In this work, only sine wave inlet variation is discussed. If periodic inlet variation is introduced in other waveforms, e.g., rectangular wave or sawtoothed wave, similar reactor behavior can be expected. Because mathematical expression of such

when the gas superficial velocity is changed periodically. Several peaks with a smaller variation of maximum temperature on the curves are identified when tfeed/tc is 1/7, 1/5, 1/3 and close to 1.0. Further simulation work shows that the reactor extinction occurs when tfeed/tc is 1.0 if the amplitude of periodic variation (A) is increased to 0.05. If A is gradually increased, reactor extinction occurs when tfeed/tc is 1/3, 1/5, 1/7, one after another.

4. CONCLUSION The effects of periodic inlet parameter variation, including concentration and superficial gas velocity, on RFR behavior are discussed through a numerical study. Simulation results reveal that a resonance response will occur when the flow-reversal cycle time is an odd multiple of the inlet variation cycle time. Such reactor behavior will occur at various values of superficial inlet velocity, mean inlet methane concentration, flow-reversal time and inert-to-catalyst ratios under the given conditions applied in this work. When the two cycle times are equal, sharp increase in variation of maximum temperature is observed. If a RFR is regarded as a dynamic periodic system, the flow-reversal time is said to be a characteristic response time of the system. 5891

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Industrial & Engineering Chemistry Research By applying harmonic analysis, a mathematical expression of the creeping heat wave velocity versus time can be written in Fourier series by retaining the odd harmonic components. This is given as an explanation for the observed resonance response phenomenon. A synchronization of the periodic inlet parameter variation and flow-reversal cycles leads to temporal and spatial symmetry breakdown in reactor. It may impair the reactor stability and lead to extinction or thermal runaway when the reactor is operated near the operable limits. A discussion of some basic control strategies is provided. The basic idea is desynchronizing the two cycles to avoid the resonance response. Besides commonly used control methods, increasing flow-reversal time when reactor extinction caused by resonance behavior occurs is also a feasible way to maintain autothermal operation of the reactor. And reactor thermal runaway can be avoided when resonance response occurs by decreasing flow-reversal time. However, the flow-reversal time should be adjusted with caution, otherwise the reactor stability may be compromised as well.

■

■

g = gas phase out = outlet conditions s = solid phase

REFERENCES

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AUTHOR INFORMATION

Corresponding Author

*G. Chen. Tel.: +86 0574 87609539. Fax: +86 0574 87605311. E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS Financial support by the National Natural Science Foundation of China (grant No. 51406094), Zhejiang Provincial Natural Science Foundation of China (grant No. LQ13E060003), Ningbo Natural Science Foundation (grant No. 2013A610050) and K. C. Wong Magna Fund in Ningbo University are gratefully acknowledged.

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NOMENCLATURE av = surface-gas volume ratio, m2/m3 C = molar concentration, mol/m3 cp = heat capacity, J/(Kg·K) Deff = axial mass dispersion coefficient m2/s Ea = activation energy, J/mol ΔH = heat of combustion, J/mol h = gas−solid heat transmission coefficient, W/(m2·K) k∞ = pre-exponential factor, 1/s kg = gas−solid mass transmission coefficient, m/s keff = axial heat dispersion coefficient W/(m·K) L = reactor length, m R = ideal gas constant, J/(mol·K) T = temperature, K ΔTad = adiabatic temperature rise, K t = time, s tc = flow-reversal cycle time, s tfeed = feed variation cycle time, s u = gas velocity, m/s x = axial position along the reactor, m

Greek Letters

ε = bed void fraction η = internal mass transfer effectiveness factor ρ = density, kg/m3

Subscripts

0 = inlet conditions 5892

DOI: 10.1021/ie504203v Ind. Eng. Chem. Res. 2015, 54, 5885−5893

Article

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DOI: 10.1021/ie504203v Ind. Eng. Chem. Res. 2015, 54, 5885−5893