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Effect of coolant flow rate on the dynamics of delayed recycle continuous stirred tank reactor
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Effect of coolant flow rate on the dynamics of delayed recycle continuous stirred tank reactor Balasubramanian Periyasamy∗, Mihir Bhalchandra Khanolkar School of Engineering and Physical Sciences, Heriot-Watt University, Dubai Campus, P.O. Box 294345 Dubai, United Arab Emirates
a r t i c l e
i n f o
Article history: Received 12 May 2016 Revised 26 October 2016 Accepted 1 March 2017 Available online xxx Keywords: Kinetics CSTR Delay differential equations Hopf bifurcation LMTD Coolant dynamics
a b s t r a c t Numerical bifurcation analysis of delayed recycle stream in a continuous stirred tank reactor was studied
1. Introduction For the past six decades, the steady state and dynamic characteristics of continuous stirred tank reactor [1–26] (CSTR) as well as reactor-separator-recycle system [27–43] were examined, and innumerable research articles were published on the bifurcation analysis of first-order irreversible exothermic reaction. In 1953, Van Heerden [1] observed the steady-state multiplicity for the firstorder irreversible exothermic reaction in a CSTR. Luss and Lapidus [3] presented a method for finding stability characteristics of the state variables in a CSTR. A detailed study on the nonlinear behavior of first-order irreversible exothermic reaction in a recycle CSTR was carried-out by Uppal et al. [5,6]. They presented the steadystate multiplicity in a parameter space by neglecting delay in the recycle stream. Kubieck et al. [9] studied the existence of isola, and mushroom regions for two inter-connected non-isothermal CSTRs. The chaotic behavior was observed by Mankin and Hudson [10] for two CSTRs connected in series with negligible delays in the recycle
∗
k2,1
using DDE-BIFTOOL. A first-order irreversible exothermic reaction s1 −−→ s2 was considered for analyzing effect of delay on the stability of the delayed recycle system. The non-isothermal CSTR was operated at both infinite and finite coolant inlet flowrates. A constant delay was considered in the recycle stream of CSTR for concentration of reactant, and temperature. DDE-BIFTOOL solver was used for finding dependency of delays on the bifurcation parameters, and its stability characteristics. The bifurcation parameters considered were: (i) fresh feed flowrate, (ii) coolant temperature, and (iii) coolant inlet flowrate. In the absence of delay, the system exhibits the region of dynamic instability for both infinite and finite coolant inlet flowrates. For infinite coolant flowrate, the region of dynamic instability on the reactor temperature was modified as a result of delay by varying either fresh feed flowrate or coolant temperature. The steady-state multiplicity was observed on the coolant temperature by varying fresh feed flowrate at finite coolant inlet flowrate. In the absence of delay, the fold and Hopf bifurcations were observed on the multiple steady-states of coolant temperature. The concentration and temperature delays do not alter substantially the dynamic characteristics of coolant temperature at Tc,in = 298 K. However, delays can alter the region of dynamic instability for coolant temperature when coolant inlet temperature is higher than 298 K. These are the main result of present work. © 2017 Taiwan Institute of Chemical Engineers. Published by Elsevier B.V. All rights reserved.
Corresponding author. E-mail address: [email protected] (B. Periyasamy).
stream. Subsequently, many researchers investigated the bifurcation analysis of CSTR by considering various reaction schemes [11– 23]. Singularity and bifurcation theory [14] was applied for finding the performance of first-order exothermic reaction in a CSTR. The chaotic behavior was observed for consecutive first-order irreversible reactions [16,17] in a CSTR through bifurcation analysis. The nonlinear bifurcation analysis of CSTR-separator-recycle system was investigated by various researchers [27–43]. For example, CSTR-distillation column [30,31], CSTR-flash-recycle system [34], and so on. Luyben [28] presented the snowball effect in a reactor-separator-recycle system and proposed a control scheme for avoiding snowball effect. Morud and Skogestad [29] observed the instability in a reactor as a result of recycle of mass and energy with large time constant. Singularity and bifurcation theory was applied for determining nonlinear behavior of CSTR-flash recycle system [34]. Bildea et al. [41] observed the large reactor volume prevents the accumulation and infinite recycle of reactant in a reactor-separator-recycle system. Steady-state multiplicity in polymerization reactions was presented by Kiss et al. [42,43]. Few researchers [44–50] investigated the stability analysis of delayed recycle CSTR. Lehman and coworkers [46–50] studied effect of delay on the stability of first-order exothermic reaction in a
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Please cite this article as: B. Periyasamy, M.B. Khanolkar, Effect of coolant flow rate on the dynamics of delayed recycle continuous stirred tank reactor, Journal of the Taiwan Institute of Chemical Engineers (2017), http://dx.doi.org/10.1016/j.jtice.2017.03.003
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Nomenclature A2,1 cp cp ,c cs1 , f cs1 cs2 E2,1 I J Jτ m n Q Qj r R si t T Tc Tc ,in Tf UA V Vj zi
pre-exponential factor, s−1 specific heat capacity of reaction mixture, J kg−1 K−1 specific heat capacity of coolant, J kg−1 K−1 initial concentration of reactant s1 in the feed, mol m−3 concentration of reactant s1 , mol m−3 concentration of product s2 , mol m−3 activation energy of reaction, J mol−1 identity matrix Jacobian matrix with respect to instantaneous state variables Jacobian matrix with respect to delayed state variables number of delays considered number of state variables volumetric flow rate, m3 s−1 volumetric flow rate of coolant, m3 s−1 recycle ratio (0 < r < 1) ideal gas constant, 8.314 J mol−1 K−1 symbol for reacting species i time, s temperature (K) coolant temperature (K) coolant inlet temperature (K) temperature (K) overall heat transfer coefficient, Js−1 K−1 volume of reactor, m3 volume of cooling jacket, m3 state variable of index i
Greek symbols ρ density of mixture, kg m−3 ρc density of coolant, kg m−3 ψ parameters τ delay, s (-H) heat of reaction, J mol−1 λ eigenvalue
CSTR with an assumption of infinite coolant flowrates. They found that a delay in the recycle stream does not alters the dynamic characteristics of the system. A detailed study on the dynamic behavior of first-order irreversible reaction in the delayed CSTRflash-recycle system was carried-out by Balasubramanian et al. [51–53]. A delay independent stability was determined analytically by controlling the fresh feed flowrate entering into an isothermal CSTR. Furthermore, a delay dependent stability was observed by controlling the effluent flowrate. For non-isothermal CSTR, the switching of stability characteristics from unstable to stable and vice-versa was recognized. Dynamic behavior of CSTR-mechanical separator-recycle system sustaining first-order irreversible exothermic reaction [54] was studied using DDE-BIFTOOL [55,56] with an assumption of infinite coolant flowrates. DDE-BIFTOOL [55,56] is a MATLAB based solver for numerical bifurcation analysis of delay differential equations. New isola regions were found at large delays. Furthermore, the switching of stability characteristics from unstable to stable state, and vice versa was observed for infinite coolant flowrate. Here, dimensionless parameters were considered in the bifurcation analysis. Uppal et al. [5,6] presented the bifurcation analysis of delayed recycle CSTR with negligible delays in the recycle stream for first-order irreversible exothermic reaction. Infinite coolant inlet flowrate was assumed for bifurcation analysis, and observed that
the system admits fold as well as Hopf bifurcations on the steadystate multiplicity. Likewise, Balasubramanian et al. [51–53] presented the effect of delay on the stability of first-order irreversible exothermic reaction in CSTR-flash-recycle system with an assumption of an infinite coolant inlet flowrate. A delay dependent stability was analytically proved by controlling the effluent flowrate using a flow controller. Previously, researchers studied the bifurcation analysis of first-order irreversible exothermic reaction in a CSTR with an assumption of an infinite coolant flowrate. That is, the exit and inlet coolant temperatures are alike for finding the heat of cooling in a reactor. In the earlier studies, the dynamics of coolant temperature was neglected. The coolant flow rate entering into a reactor must be finite in the realistic situation and reactor temperature of an exothermic reaction must be controlled at a desired value by manipulating the coolant inlet flowrate. Thus, the dynamics of coolant temperature in a reactor should be included in the bifurcation analysis. Various researchers used the dimensionless parameters for bifurcation analysis of delayed recycle systems and the realistic delay values were not included. Therefore, we decided to use realistic delays up to 180 seconds in the bifurcation analysis for better understanding of its effect on the process variables. Besides, this article describes a comprehensive numerical bifurcation analysis of first-order exothermic reaction in the CSTRmechanical separator-recycle system with delays up to 180 s using DDE-BIFTOOL for both infinite and finite coolant inlet flowrates. 2. Model description In the following, the mass and energy balance equations which describe the steady-state and dynamic performance of the CSTRmechanical separator-recycle system are presented. A first-order irreversible exothermic elementary reaction was considered for analyzing the steady- state and dynamic performance of the delayed recycle stream in a reactor. The stoichiometry of elementary reaction [57,58] can be represented as k2,1
s1 −−→ s2
(1)
In Eq. (1), the kinetic constant k2,1 indicates the formation of product s2 from reactant s1 by virtue of chemical reaction. According to law of mass action kinetics [57,58], the reaction rates for disappearance of reactant s1 is
rs1 = −k2,1 cs1
(2)
The constitute relationships considered for the nonlinearity of delayed recycle system are: (i) Arrhenius law of kinetics, and (ii) heat of cooling in a jacketed CSTR. According to Arrhenius law, the temperature dependency of kinetic constants can be represented as
E 2,1
k2,1 = A2,1 exp −
RT
(3)
Fig. 1 illustrates the schematic diagram of CSTR-mechanical separator recycle system. Here, the CSTR was connected with a mechanical separator for the separation of product s2 from unconverted reactant s1 . The latter was recycled back to the reactor. A constant delay was assumed in the recycle stream for both concentration of reactant, and temperature. The non-isothermal operation of the reactor was considered in the bifurcation analysis. The exothermic heat released by the reaction mixture in a jacketed CSTR during the reaction must be removed by the coolant flow in a jacket. The convective and conductive heat transfer between the cooling jacket and reactor should be included for finding the heat of cooling requirements. Moreover, the logarithmic mean temperature difference between the cooling medium and reaction mixture must be accounted in the heat of cooling calculations. Thus, the formula [59] for finding the heat of cooling in a
Please cite this article as: B. Periyasamy, M.B. Khanolkar, Effect of coolant flow rate on the dynamics of delayed recycle continuous stirred tank reactor, Journal of the Taiwan Institute of Chemical Engineers (2017), http://dx.doi.org/10.1016/j.jtice.2017.03.003
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Table 1 Parameters considered in the numerical bifurcation analysis using DDE-BIFTOOL.
Fig. 1. Schematic diagram of CSTR-mechanical separator-recycle system.
Parameter
Symbol
Value
Unit
Volumetric flowrate Reactor volume Initial concentration of reactant Pre-exponential factor Activation energy Gas constant Feed temperature Heat of reaction Density of reaction mixture Specific heat of reaction mixture Overall heat transfer coefficient Volume of cooling jacket Density of coolant Specific heat of coolant
Q V c s1 , f A2,1 E2,1 R Tf (−H)
1.67 × 10−3 0.1 10 0 0 1.2 ×109 72,747.5 8.314 350 50,0 0 0 10 0 0 239 833.33 0.015 10 0 0 4183
m3 s−1 m3 mol m−3 s−1 J mol−1 J mol−1 K−1 K J mol−1 kg m−3 J kg− 1 K−1 J s− 1 K−1 m3 kg m−3 J kg− 1 K−1
ρ
cp UA Vj
ρc
cp,c
Various researchers [5,6,46–50,51–54] assumed the coolant inlet flowrate was infinite for finding the heat of cooling requirements and Eq. (4a) becomes
In the aforementioned model description, the asymptotic values of recycle ratio r represent the total recycle, and zero recycle of unconverted reactant from the mechanical separator to the CSTR, respectively, at r = 0 and 1. A reactor may not be operated under asymptotic recycle ratios. Thus, the recycle ratio r must be within the range of 0 and 1 for the feasible conversion of reactant.
Qcooling = UA(T − Tc )
3. Numerical bifurcation analysis
non-isothermal CSTR is
Qcooling =
UA(Tc − Tc,in ) ln
(4a)
T −Tc,in T −Tc
(4b)
However, Eq. (4b) may not be applicable as a result of finite coolant inlet flowrate entering into a reactor, and is being controlled using a flow controller. In this work, both formulas were used for heat of cooling calculations. The mole balance equation for the disappearance of reactant in the CSTR is
Q dcs1 (t ) rQ (1 − r )Q = cs + cs1 (t − τ ) − cs1 (t ) dt V 1, f V V −E2,1 −A2,1 exp cs1 (t ) RT (t )
(5)
The energy balance equations for the CSTR and cooling jacket are
Q dT (t ) rQ (1 − r )Q = T + T (t − τ ) − T (t ) dt V f V V Qcooling −E2,1 (−H ) + A exp cs1 (t ) − ρ c p 2,1 RT (t ) V ρ cp Qcooling Qj dTc (t ) = [Tc,in − Tc (t )] + dt Vj V j ρc c p,c
(6)
(7)
At steady states, Eqs. (5)−(7) become,
0=
rQ −E2,1 cs1, f − cs1 ,s − A2,1 exp cs1,s V RTs
0=
(−H ) Qcooling rQ −E2,1 T f − Ts + A exp cs1 ,s − V ρ c p 2,1 RTs V ρ cp
0=
Qcooling Qj [Tc,in − Tc,s ] + Vj V j ρc c p,c
(8)
(9) (10)
In the absence of delay, the following ordinary differential equations describe the time-dependent behavior of the system.
d cs1 rQ −E2,1 = cs1, f − cs1 − A2,1 exp cs1 dt V RT
(11)
(−H ) Qcooling dT rQ −E2,1 = Tf − T + A exp cs1 − dt V ρ c p 2,1 RT V ρ cp
(12)
Qcooling Qj dTc = [Tc,in − Tc ] + dt Vj V j ρc c p,c
(13)
A general form of system of multivariable delay differential equations with constant delays [52,53,55,56] is
dzi (t ) = f zi (t ), zi t − τ j , ψ zi (t ) = zi,0 for − τ j < t < 0 dt
(14)
In Eq. (14), i and j vary from 1 to n and 1 to m, respectively. The characteristic equation which was used for finding the real parts of roots and further analysis of stability of the steady-states of delay differential equations [55,56] is
m
Jτ j exp −λτ j = 0
λI − J −
j=1
(15)
Bifurcations occur whenever the roots move through the imaginary axis as one or more system parameters are varied. Fold bifurcation does not occur as a result of delayed recycle system cannot be destabilized by a real λ crossing the imaginary axis. Therefore, delays admit Hopf bifurcation in the CSTR-mechanical separatorrecycle system as a result of λ is purely imaginary. DDE-BIFTOOL [55,56] is a collection of MATLAB based solver for determining numerical bifurcation analysis of delay differential equations. This solver implements the continuation, and stability analysis of steady state solutions as well as fold and Hopf bifurcations. Furthermore, it allows switching from the latter to an emanating branch of periodic solutions. In this work, DDE-BIFTOOL was used for finding the steady-state profiles, real parts of the roots of characteristic equation, and dependency of delays on the bifurcation parameters. Eqs. (5)−(7) were numerically solved using MATLAB based dde23 solver [60] for finding the dynamic characteristics of the state variables. The bifurcation diagrams for the state variables were obtained using DDE-BIFTOOL. In the absence of delay, Eqs. (11)−(13) were numerically solved using MATLAB based ode45 solver for finding time-dependent behavior of the state variables. 4. Results and discussion The steady-state and dynamic characteristics of CSTRmechanical separator-recycle system sustaining first-order irreversible exothermic reaction were examined using DDE-BIFTOOL.
Please cite this article as: B. Periyasamy, M.B. Khanolkar, Effect of coolant flow rate on the dynamics of delayed recycle continuous stirred tank reactor, Journal of the Taiwan Institute of Chemical Engineers (2017), http://dx.doi.org/10.1016/j.jtice.2017.03.003
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a
b
c
d
e
Fig. 2. Dependency of delays on rQ at fixed Tc . The unit for rQ is m3 s−1 . (a) Tc = 310 K, (b) Tc = 315 K, (c) Tc = 316 K, (d) Tc = 320 K, and (e) Tc = 326 K.
a
b
Fig. 3. Bifurcation diagrams of the system at Tc = 320 K. Solid line: stable steady-states, and dashed line: unstable steady states (Hopf bifurcation). (a) Reactant concentration, and (b) temperature.
Please cite this article as: B. Periyasamy, M.B. Khanolkar, Effect of coolant flow rate on the dynamics of delayed recycle continuous stirred tank reactor, Journal of the Taiwan Institute of Chemical Engineers (2017), http://dx.doi.org/10.1016/j.jtice.2017.03.003
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5
b
c
d
e
f
Fig. 4. Real parts of roots of characteristic equation versus rQ for various delays at Tc = 320 K (Hopf bifurcation regions). (a) τ = 0 s, (b) τ = 10 s, (c) τ = 60 s, (d) τ = 90 s, (e) τ = 110 s, and (f) τ = 180 s. The unit for rQ is m3 s−1 .
In the following, numerical bifurcation analysis of delayed recycle CSTR is presented by varying three bifurcation parameters. Table 1 illustrates the system parameters [61–64] used in the bifurcation analysis. Here, the volume of cooling jacket, and coolant inlet flow rate were assumed for finding the steady-state multiplicity on the coolant temperature. 4.1. Infinite coolant inlet flowrate For the first instance, the fresh feed flowrate rQ was considered as a bifurcation parameter for stability analysis. The effect of delay on the steady-state and dynamic characteristics of the nonisothermal CSTR was studied by assuming infinite coolant inlet flowrate. Eqs. (4b), (5) and (6) were used in DDE-BIFTOOL for finding the appreciable changes on the region of dynamic instability
as a result of delay by varying rQ at a fixed coolant temperature. Here, the coolant dynamics was neglected in the bifurcation analysis. The dependency of delays on rQ at Tc = 310, 315, 316, 320 and 326 K is depicted in Fig. 2. For Tc < 315 K, a delay does not affect significantly on the dynamic characteristics of the delayed recycle CSTR. A delay can alter substantially the region of dynamic characteristics when Tc ≥ 315K. In particular, the system changes the region of dynamic instability into the stable steady-states, and viceversa (vide Fig. 2c−e) as a result of delay in the recycle stream. Here, the stable operation of the reactor is possible with small delays if the system is dynamically unstable in the absence of delay. Furthermore, Tc = 320 K was selected for finding better insights on the stability characteristics of the delayed recycle system in the absence of coolant dynamics.
Please cite this article as: B. Periyasamy, M.B. Khanolkar, Effect of coolant flow rate on the dynamics of delayed recycle continuous stirred tank reactor, Journal of the Taiwan Institute of Chemical Engineers (2017), http://dx.doi.org/10.1016/j.jtice.2017.03.003
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a
c
b
d
Fig. 5. Bifurcation diagrams of the system at fixed rQ. Solid line: stable steady-states, and dashed line: unstable steady states (Hopf bifurcation). (a and b) rQ = 6.68 × 10−4 m3 s−1 , and (c and d) rQ = 1.002 × 10−3 m3 s−1 .
b
a
Fig. 6. Dependency of delays on Tc at fixed rQ. (a) rQ = 6.68 × 10−4 m3 s−1 , and (b) rQ = 1.002 × 10−3 m3 s−1 .
Fig. 3 illustrates the bifurcation diagrams for the concentration of reactant, and reactor temperature by varying rQ at Tc = 320 K. In the absence of delay, the system exhibits Hopf bifurcation over the range of bifurcation parameter 5.71 × 10 − 4 ≤ rQ ≤ 1.13 × 10 − 3 m3 s−1 . Eqs. (4b), (11) and (12) were numerically solved using ode45 solver available in MATLAB for finding the dynamic characteristics of the delayed recycle system. The dynamic simulation plots for the concentration of reactant and reactor temperature at rQ = 7.15 × 10−4 , 1.0 × 10−3 , and 1.12 × 10−3 m3 s−1 are presented in Fig. 1S (vide supporting information). At Tc = 320 K, the sustained oscillations of reactant concentration, and temperature were observed. For better insights on the regions of Hopf bifurcation, the dependency of the real parts of roots of characteristic equation on rQ is depicted in Fig. 4 for various delays. Here, the values are positive over the range of rQ at the delays of 10, 60, 110 and 180 s. These stability characteristics indicate that the system exhibits sustained oscillations on the Hopf bifurcation regions. The real parts of roots of characteristic equation shown in Fig. 4 can be compared with Fig. 2d for finding the stability characteristics of the boundary.
For example, the system exhibits stable steady-states at τ = 90 s in Fig. 2d, and the corresponding real parts of the roots of characteristic equation are negative over the range of rQ as illustrated in Fig. 4d. This result indicates that the critical delay boundary depicted in Fig. 2d is consistent with the stability characteristics of the plots shown in Fig. 4d. For τ = 240, and 300 s, the real parts of roots of characteristic equation on rQ plots are shown in Fig. 2S (vide supporting information). For these delays, the dynamic characteristics are unaltered, and are alike as that of no delay context. The dynamic simulation plots obtained for various delays (20 and 160 s) by numerically solving Eqs (4b), (5), and (6) at rQ = 1 × 10−3 m3 s−1 , and Tc = 320 K are shown in Fig. 3S (vide, supporting information). Here, the sustained oscillation of concentration of reactant, and reactor temperature are modified into the state of oscillations. For the second case, the fresh feed flowrates were fixed at 6.68 × 10−4 , 1.002 × 10−3 , and 1.336 × 10−3 m3 s−1 for 60%, 40%, and 20% recycle, respectively. The coolant temperature was considered as a bifurcation parameter over the interval between 300 and 350 K. The bifurcation diagrams obtained by numerically solv-
Please cite this article as: B. Periyasamy, M.B. Khanolkar, Effect of coolant flow rate on the dynamics of delayed recycle continuous stirred tank reactor, Journal of the Taiwan Institute of Chemical Engineers (2017), http://dx.doi.org/10.1016/j.jtice.2017.03.003
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7
b
d c
Fig. 7. Bifurcation diagrams of the system at Tc ,in = 308 K, τ = 0 s, and Qj = 4.425 × 10−4 m3 s−1 . Solid line: stable steady-states, and dashed line: unstable steady states (Hopf bifurcation).(a) Concentration of reactant, (b) reactor temperature, and (c) coolant temperature. (d) Real parts of roots of characteristic equation plot by varying rQ.
ing Eqs. (4b), (5) and (6) for concentration of reactant, and temperature are depicted in Fig. 5. In the absence of delay, the system exhibits Hopf bifurcation over the range of 316 ≤ Tc ≤ 327K, and 310 ≤ Tc ≤ 323K for 60%, and 40% recycle, respectively. The dependency of delays on Tc for 60% and 40% recycle are illustrated in Fig. 6. Here, the system alters substantially the region of dynamic characteristics from unstable to stable state and vice-versa at 60% recycle. The system exhibits a small stable region on the righthand side of delay plot at 40% recycle as shown in Fig. 6b. For 20% recycle, an incomplete region of Hopf bifurcation was observed for various delays (10 and180 s). The real parts of roots of characteristic equation on Tc plots are depicted in Fig. 4S (vide supporting information). For small recycle, delays do not alter significantly the region of dynamic characteristics. Besides, it was observed that the region of dynamic instability shifted towards low coolant temperature range. 4.2. Finite coolant inlet flowrate In the following, the bifurcation analysis of the delayed recycle system in the presence of coolant dynamics is described. The coolant inlet flowrate Qj , and temperature Tc,in considered for the bifurcation analysis are 4.425 × 10−4 m3 s−1 , and 308 K, respectively. Eqs. (4a), (5)−(7) were used in DDE-BIFTOOL for finding the delaydependent stability characteristics. The bifurcation diagrams ob-
tained for the concentration of reactant, reactor and coolant temperatures by varying rQ at fixed Qj and Tc,in are shown in Fig. 7. In the absence of delay, the system exhibits the region of dynamic instability for 7.22 × 10 − 4 ≤ rQ ≤ 1.08 × 10 − 3 m3 s−1 as illustrated in Fig. 7d, and multiple steady-states are not observed at Tc,in = 308 K. The dynamic simulation plots are presented in Fig. 5S (vide supporting information) at rQ = 8.95 × 10−4 m3 s−1 . The dynamic simulation results reveal that the sustained oscillation of coolant temperature in a reactor besides other two state variables. Thus, the dynamics of coolant temperature must be included in bifurcation analysis of the delayed recycle system. The dependency of delays on rQ at fixed Qj and Tc,in is depicted in Fig. 8. Here, the delays on concentration of reactant, and reactor temperature do not alter substantially the region of dynamic characteristics. However, a delay induces a small stable region on the right-hand side of Hopf bifurcation boundary. Afterwards, an ambient coolant inlet temperature (Tc,in = 298 K) was considered for analyzing the stability characteristics of the delayed recycle system. Fig. 9 illustrates the bifurcation diagrams for the concentration of reactant, reactor and coolant temperatures by varying rQ at Qj = 4.425 × 10−4 m3 s−1 . In the absence of delay, the system admits multiple steady-states for all three state variables. In particular, the bifurcation analysis reveals that the coolant temperature of the system exhibits steady-state multiplicity. This phenomenon was not observed for an infinite coolant inlet flowrate
Please cite this article as: B. Periyasamy, M.B. Khanolkar, Effect of coolant flow rate on the dynamics of delayed recycle continuous stirred tank reactor, Journal of the Taiwan Institute of Chemical Engineers (2017), http://dx.doi.org/10.1016/j.jtice.2017.03.003
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b
Fig. 8. (a) Dependency of delays on rQ at fixed Qj and Tc,in , and (b) Real parts of roots of characteristic equation versus rQ plot. τ = 130 s, Qj = 4.425 × 10−4 m3 s−1 , and Tc,in = 308 K. The unit of rQ is m3 s−1 .
a
c
b
d
Fig. 9. (a−c) Bifurcation diagrams obtained using DDE-BIFTOOL at Qj = 4.425 × 10− 4 m3 s−1 , and Tc, in = 298 K. (d) Real parts of roots of characteristic equation versus rQ plot. (a) Concentration of reactant, (b) reactor temperature, and (c) coolant temperature. The unit of rQ is m3 s−1 .
assumption with the parameters considered. The stability of the steady-states of state variables is depicted in Fig. 9d. Fold bifurcation persists on the middle portion, and an incomplete region of Hopf bifurcation can be found on the lower and upper portions of the profile. For Tc,in = 308 K, the system exhibits the region of dynamic instability in the absence of delay (vide Fig. 8a) for the state variables such as concentration of reactant, reactor and coolant temperatures. The concentration and temperature delays in the recycle stream of the reactor do not alter the region of dynamic instability for the range of small delay 0 < τ < 15 s as shown in Fig. 8a.
For τ > 15 s, the delay alters the right Hopf bifurcation boundary of the system considered. Another Hopf bifurcation region arises near the right Hopf bifurcation boundary when τ > 120 s. Here, the stable operation of the system is possible for the set of parameters. For example, the coolant temperature of the CSTR is stable at rQ = 1 m3 s−1 and 90 s < τ < 135 s. That is, a delay in the recycle stream changes sustained oscillations of the CSTR and coolant temperatures to stable steady-states. The oscillations in the coolant temperature affects the yield of product in a reactor for an exothermic reaction. Ultimately, it increases the possibilities of rise in the reactor temperature as well as the formation of undesired prod-
Please cite this article as: B. Periyasamy, M.B. Khanolkar, Effect of coolant flow rate on the dynamics of delayed recycle continuous stirred tank reactor, Journal of the Taiwan Institute of Chemical Engineers (2017), http://dx.doi.org/10.1016/j.jtice.2017.03.003
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a
b
c
d
e
f
9
Fig. 10. (a, c and e) Steady-state coolant temperature as a function of coolant inlet flowrate. (b, d and f) Real parts of roots of characteristic equation versus Qj plots. rQ = 1.12 × 10−3 m3 s−1 , and τ = 10 s. (a and b) Tc,in = 293 K, (c and d) Tc,in = 298 K, and (e and f) Tc,in = 302 K. The unit of Qj is m3 s−1 .
ucts. Thus, the delayed recycle stream can able to circumvent these problems. For Tc,in = 298 K, the coolant temperature exhibits multiple steady-states by varying rQ as shown in Fig. 9c. Here, the system does not exhibit a full region of dynamic instability as depicted in Fig. 9d and delay do not alter the dynamic characteristics significantly. Likewise, the bifurcation diagrams of coolant temperature on Qj instead of rQ are illustrated in Fig. 10a, b and c at Tc,in = 293, 298, and 302 K, respectively. In the absence of delay, the system admits the multiple steady-states for the state variables considered Fold and Hopf bifurcations were observed on the multiple steady-states of the state variables. For all three coolant temperatures, the incomplete regions of Hopf bifurcation were found on the upper and lower portion of the profiles. However, a broader Hopf bifurcation
region was observed on the steady-state profile at Tc,in = 302 K as illustrated in Fig. 10e and f. A region of fold bifurcation was found on the middle portion of the curve and was not induced by delay. Here, a delay on the recycle stream of the CSTR do not alter substantially the dynamic characteristics as a result of steady-state multiplicity on the coolant temperature. In this work, the effect of delay in the recycle stream of CSTR was analyzed using DDE-BIFTOOL for both infinite and finite coolant inlet flowrates at 0 ≤ τ ≤ 180 s. In the absence of coolant dynamics, the concentration and temperature delays in the recycle stream substantially alter the dynamic characteristics (vide Figs. 2 and 6) with the set of parameters used. However, the system exhibits different behaviors for the finite coolant flowrates. The coolant temperature in the delayed recycle CSTR exhibits
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sustained oscillations in addition to the oscillations in the other two state variables. These oscillations can be either minimized or circumvented to a stable operation (vide Fig. 8) for certain concentration and temperature delays in the recycle stream when Tc,in > 298 K. However, a delay does not alter the stability characteristics of the coolant temperature in the jacketed CSTR at Tc,in = 298 K. Lehman et al. [50] analytically proved that a delay in the recycle stream of CSTR does not alter the dynamic characteristics of the system for infinite coolant flowrate. In the present work, the logarithmic mean temperature difference was considered for finding dependency of delays on the coolant temperature using DDE-BIFTOOL. Besides, the bifurcation analysis results presented at Tc,in = 298 K are consistent with the results of Lehman et al. [50]. 5. Conclusion A comprehensive bifurcation analysis of first-order irreversible exothermic elementary reaction in a CSTR-mechanical separatorrecycle system was presented using DDE-BIFTOOL. For infinite coolant inlet flowrate, the system changes the region of Hopf bifurcation as a result of delay by varying either fresh feed flowrate or coolant temperature. In particular, the stable steady-states were observed on the parameter space for some delays. On contrary, fold and Hopf bifurcations exist for finite coolant inlet flowrate entering into the CSTR. Steady-state multiplicity was observed on the coolant temperature besides the reactor temperature and concentration of reactant. Delays do not induce fold bifurcation on the system considered. The heat of cooling based on logarithmic mean temperature difference between coolant and reaction mixture is not influenced significantly by temperature delays in the recycle stream. Thus, delays do not alter substantially the dynamic characteristics at finite coolant inlet flowrate into the CSTR. Besides, the bifurcation analysis results presented here provide guidelines for analyzing effect of delay on the dynamic characteristics of fluid catalytic cracking unit, CSTRs connected in series, and other process engineering systems. The input delays on fresh feed as well as coolant, and effect of flow controller for controlling coolant inlet flowrate based on coolant and reactor temperatures are not included here. These are beyond the scope of present work. Supplementary materials Supplementary material associated with this article can be found, in the online version, at doi:10.1016/j.jtice.2017.03.003. References [1] Van Heerden C. Autothermic processes. Ind Eng Chem 1953;45:1242–7. [2] Aris R, Amundson N. An analysis of chemical reactor stability and control-I. Chem Eng Sci 1958;7:121–31. [3] Luss R, Lapidus L. An averaging technique for stability analysis. Chem Eng Sci 1966;21:159–81. [4] Douglas JM. The use of optimization theory to design simple multivariable control systems. Chem Eng Sci 1966;21:519–32. [5] Uppal A, Ray WH, Poore AB. On the dynamic behavior of continuous stirred tank reactors. Chem. Eng. Sci 1974;29:967–85. [6] Uppal A, Ray WH, Poore AB. The classification of the dynamic behavior of continuous stirred tank reactors-influence of reactor residence time. Chem Eng Sci 1976;31:205–14. [7] Merzhanov GA, Abramov GV. Thermal regimes of exothermic processes in continuous stirred tank reactors. Chem Eng Sci 1977;32:475–81. [8] Vaganov DA, Samoilenko NG, Abramov VG. Periodic regimes of continuous stirred tank reactors. Chem Eng Sci 1978;33:1133–40. [9] Kubicek M, Hofmann H, Hlavacek V, Sinkule J. Multiplicity and stability in a sequence of two nonadiabatic nonisothermal CSTR. Chem Eng Sci 1980;35:987–96. [10] Mankin JC, Hudson JL. The dynamics of coupled nonisothermal continuous stirred tank reactors. Chem Eng Sci 1986;41:2651–61. [11] Wirges HP. Experimental study of self-sustained oscillations in a stirred tank reactor. Chem Eng Sci 1980;33:2141–6. [12] Zhabotinsky AM, Rovinsky AB. Mechanism and nonlinear dynamics of an oscillating chemical reaction. J Stat Phy 1987;48:959–75.
[13] Aris R, Gray P, Scott SK. Modelling cubic autocatalysis by successive bimolecular steps. Chem Eng Sci 1988;43:207–11. [14] Balakotaiah V, Luss D. Structure of the steady-state solutions of lumped-parameter chemically reacting systems. Chem Eng Sci 1982;37:1611–23. [15] Balakotaiah V, Luss D. Multiplicity features of reacting systems dependence of the steady-states of a CSTR on the residence time. Chem Eng Sci 1983;38:1709–21. [16] Jorgensen DV, Aris R. On the dynamics of a stirred tank with consecutive reactions. Chem Eng Sci 1983;38:45–53. [17] Doedel JE, Heinemann FR. Numerical computation of periodic solution branches and oscillatory dynamics of the stirred tank reactor with A→B→C reactions. Chem Eng Sci 1983;38:1493–9. [18] Nelson MI, Sidhu HS. Bifurcation phenomena for an oxidation reaction in a continuously stirred tank reactor I. adiabatic operation. J Math Chem 2002;31:155–86. [19] Nelson MI, Sidhu HS. Bifurcation phenomena for an oxidation reaction in a continuously stirred tank reactor II. diabatic operation. Aust N Z Ind Appl Math J 2004;45:303–26. [20] Lee JH, Lim HC. Multiplicity and stability analysis of continuous pre-fermentation of cheese culture. Proc Biochem 1999;34:467–75. [21] Sanyasi Rao VN, Pushpavanam S. Effect of noise on the behavior of a zeroth-order reaction in a continuous stirred tank reactor. Ind Eng Chem Res 20 0 0;39:926–32. [22] Balakrishna A, Schmidt LD, Aris R. Pt-catalyzed combustion of CH4 -C3 H8 mixtures. Chem Eng Sci 1994;49:11–18. [23] Li R, Li H. Isolas, mushrooms and other forms of multistability in isothermal bimolecular reacting systems. Chem Eng Sci 1989;44:2995–3000. [24] Balakotaiah V, Luss D. Global analysis of the multiplicity features of multi-reaction lumped-parameter systems. Chem Eng Sci 1984;39:865–81. [25] Tsiligiannis CA, Lyberatost G. A linear algebraic approach to steady-state bifurcation of chemical reaction systems. Chem Eng Sci 1987;42:535–41. [26] Berezowski M, Burghardt A. A generalized analytical method for determining multiplicity features in chemical reactors with recycle. Chem Eng Sci 1989;44:2927–33. [27] Luyben ML, Floudas CA. Analyzing the interaction of design and control-2. reactor-separator-recycle system. Comp Chem Eng 1994;18:971–94. [28] Luyben WL. Snowball effects in reactor/separator process with recycle. Ind Eng Chem Res 1994;33:299–305. [29] Morud J, Skogestad S. Effects of recycle on dynamics and control of chemical processing plants. Comp Chem Eng 1994;18:S529–34. [30] Yi CK, Luyben WL. Design and control of coupled reactor/column systems-part 1. A binary coupled reactor/rectifier system. Comp Chem Eng 1997;21:25–46. [31] Yi CK, Luyben WL. Design and control of coupled reactor/column systems-part 2. A binary coupled reactor/rectifier system. Comp Chem Eng 1997;21:47–67. [32] Yi CK, Luyben WL. Design and control of coupled reactor/column systems-part 3. A binary coupled reactor/rectifier system. Comp Chem Eng 1997;21:69–86. [33] Read NK, Ray WH. Application of nonlinear dynamic analysis in the identification and control of nonlinear systems I. Simple dynamics. J Proc Cont 1998;8:1–15. [34] Pushpavanam S, Kienle A. Nonlinear behaviour of an ideal reactor separator network with mass recycle. Chem Eng Sci 2001;56:2837–49. [35] Larson T, Govatsmark SM, Skogestad S, Yu CC. Control structure selection for reactor separator and recycle processes. Ind Eng Chem Res 2003;42:1225–34. [36] Seki H, Naka Y. A hierarchical controller design for a reactor/separator system with recycle. Ind Eng Chem Res 2006;45:6518–24. [37] Xu S, Bao J. Distributed control of plant wide chemical processes. J Proc Cont 2009;19:1671–87. [38] Jogwar SV, Baldea M, Daoutidis P. Dynamics and control of process networks with large energy recycle. Ind Eng Chem Res 2009;48:6087–97. [39] Liu J, Munoz de la Pena D, Christofides DP. Distributed model predictive control of nonlinear process systems. AIChE J 2009;55:1171–84. [40] Antoniades C, Christofides DP. Feedback control of nonlinear differential difference equation systems. Chem Eng Sci 1999;54:5677–709. [41] Bildea CS, Dimian AC, Iedema PD. Nonlinear behavior of reactor-separator-recycle systems. Comp Chem Eng 20 0 0;24:209–15. [42] Kiss AA, Bildea CS, Dimian AC, Iedema PD. State multiplicity in CSTR-separator-recycle polymerization systems. Chem Eng Sci 2002;57:535–46. [43] Kiss AA, Bildea CS, Dimian AC, Iedema PD. State multiplicity in PFR-separator-recycle polymerization systems. Chem Eng Sci 2003;58:2973–84. [44] Hellinckx L, Grootjans J, Van Den Bosch B. Stability analysis of a reactor with a retarded recycle. Chem Eng J 1972;3:304–7. [45] Denn MM, Lavie R. Dynamics of plants with recycle. Chem Eng J 1982;24:55–9. [46] Lehman B, Verriest E. Stability of a continuous stirred reactor with delay in the recycle stream. In: Proceedings of the 30th conference on decision and control, England; 1991. [47] Lehman B. Stability of chemical reactions in a CSTR with delayed recycle stream. In: Proceedings of the American Control conference, Maryland; 1994. [48] Lehman B, Bentsman J, Verduyn S, Lunel V IE. Vibrational control of nonlinear time lag systems with arbitrarily large but bounded delay averaging theory stabilizability and transient behavior. In: Conference on decision and control, Tucson Arizona; 1992. [49] Lehman B, Widjaya I, Shujaee K. Application of vibrational control to chemical reactions a CSTR with delayed recycle stream. In: Proceedings of the American control conference, Maryland; 1994. [50] Lehman B, Widjaya I, Shujaee K. Vibrational control of chemical reactions in a CSTR with delayed recycle stream. J Math Anal Appl 1995;193:28–59.
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ARTICLE IN PRESS
[m5G;March 22, 2017;5:57]
B. Periyasamy, M.B. Khanolkar / Journal of the Taiwan Institute of Chemical Engineers 000 (2017) 1–11 [51] Balasubramanian P, Kosuri MR, Pushpavanam S, Kienle A. Effect of delay on the stability of a coupled reactor separator system. Ind Eng Chem Res 2003;42:3758–64. [52] Balasubramanian P, Pushpavanam S, Kienle A, Balaraman KS. Effect of delay on the stability of a coupled reactor flash system sustaining an elementary non-isothermal reaction. Ind Eng Chem Res 2005;44:3619–25. [53] Balasubramanian P. Numerical bifurcation analysis of delay in a coupled reactor separator system. Ind Eng Chem Res 2008;47:3920–9. [54] Gangadhar NR, Balasubramanian P. Numerical bifurcation analysis of delayed recycle stream in a continuously stirred tank reactor. In: Proceedings of the international conference on modeling, optimization and computing ICMOC 2010, 1298; 2010. p. 192–7. [55] Engelborghs K, Luzyanina T, Roose D. Numerical bifurcation analysis of delay differential equations. J Comput Appl Math 20 0 0;125:265–75. [56] Engelborghs K. A Matlab package for bifurcation analysis of delay differential equations. DDE-BIFTOOL: technical report TW-305 department of computer science 20 0 0. [57] Bhusare VH, Balasubramanian P. A new paradigm in kinetic modeling of complex reaction systems. Int Rev Model Simul 2010;3:1145–52.
11
[58] Bhusare VH, Balasubramanian P. A general kinetic modelling methodology for first order irreversible reaction systems. Research bulletin of australian institute of high energetic Materials, 1. QLD, Australia; 2011. ISBN: 978-0-9806811-9-2 p. 85–105. [59] Molnár A, Markoš J, Jelemenský Ľ. Safety analysis of CSTR towards changes in operating conditions. J Loss Preven Proc Ind 2003;16:373–80. [60] Shampine L.F., Thompson S. Solving DDE with MATLAB. Mathematics Department, Southern Methodist University, Dallas, TX. [61] Armaoua A, Ataei A. Piece-wise constant predictive feedback control of nonlinear systems. J Proc Contr 2014;24:326–35. [62] Hahn J, Mönnigmann M, Marquardt W. On the use of bifurcation analysis for robust controller tuning for nonlinear systems. J Proc Contr 2008;18:408–20. [63] Jouili K, Jerbi H, Braiek NB. An advanced fuzzy logic gain scheduling trajectory control for nonlinear systems. J Proc Contr 2010;20:426–40. [64] Di Ciccio MP, Bottini M, Pepe P, Foscolo PU. Observer-based nonlinear control law for a continuous stirred tank reactor with recycle. Chem Eng Sci 2011;66:4780–97.
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Effect of coolant flow rate on the dynamics of delayed recycle continuous stirred tank reactor Balasubramanian Periyasamy∗, Mihir Bhalchandra Khanolkar School of Engineering and Physical Sciences, Heriot-Watt University, Dubai Campus, P.O. Box 294345 Dubai, United Arab Emirates
a r t i c l e
i n f o
Article history: Received 12 May 2016 Revised 26 October 2016 Accepted 1 March 2017 Available online xxx Keywords: Kinetics CSTR Delay differential equations Hopf bifurcation LMTD Coolant dynamics
a b s t r a c t Numerical bifurcation analysis of delayed recycle stream in a continuous stirred tank reactor was studied
1. Introduction For the past six decades, the steady state and dynamic characteristics of continuous stirred tank reactor [1–26] (CSTR) as well as reactor-separator-recycle system [27–43] were examined, and innumerable research articles were published on the bifurcation analysis of first-order irreversible exothermic reaction. In 1953, Van Heerden [1] observed the steady-state multiplicity for the firstorder irreversible exothermic reaction in a CSTR. Luss and Lapidus [3] presented a method for finding stability characteristics of the state variables in a CSTR. A detailed study on the nonlinear behavior of first-order irreversible exothermic reaction in a recycle CSTR was carried-out by Uppal et al. [5,6]. They presented the steadystate multiplicity in a parameter space by neglecting delay in the recycle stream. Kubieck et al. [9] studied the existence of isola, and mushroom regions for two inter-connected non-isothermal CSTRs. The chaotic behavior was observed by Mankin and Hudson [10] for two CSTRs connected in series with negligible delays in the recycle
∗
k2,1
using DDE-BIFTOOL. A first-order irreversible exothermic reaction s1 −−→ s2 was considered for analyzing effect of delay on the stability of the delayed recycle system. The non-isothermal CSTR was operated at both infinite and finite coolant inlet flowrates. A constant delay was considered in the recycle stream of CSTR for concentration of reactant, and temperature. DDE-BIFTOOL solver was used for finding dependency of delays on the bifurcation parameters, and its stability characteristics. The bifurcation parameters considered were: (i) fresh feed flowrate, (ii) coolant temperature, and (iii) coolant inlet flowrate. In the absence of delay, the system exhibits the region of dynamic instability for both infinite and finite coolant inlet flowrates. For infinite coolant flowrate, the region of dynamic instability on the reactor temperature was modified as a result of delay by varying either fresh feed flowrate or coolant temperature. The steady-state multiplicity was observed on the coolant temperature by varying fresh feed flowrate at finite coolant inlet flowrate. In the absence of delay, the fold and Hopf bifurcations were observed on the multiple steady-states of coolant temperature. The concentration and temperature delays do not alter substantially the dynamic characteristics of coolant temperature at Tc,in = 298 K. However, delays can alter the region of dynamic instability for coolant temperature when coolant inlet temperature is higher than 298 K. These are the main result of present work. © 2017 Taiwan Institute of Chemical Engineers. Published by Elsevier B.V. All rights reserved.
Corresponding author. E-mail address: [email protected] (B. Periyasamy).
stream. Subsequently, many researchers investigated the bifurcation analysis of CSTR by considering various reaction schemes [11– 23]. Singularity and bifurcation theory [14] was applied for finding the performance of first-order exothermic reaction in a CSTR. The chaotic behavior was observed for consecutive first-order irreversible reactions [16,17] in a CSTR through bifurcation analysis. The nonlinear bifurcation analysis of CSTR-separator-recycle system was investigated by various researchers [27–43]. For example, CSTR-distillation column [30,31], CSTR-flash-recycle system [34], and so on. Luyben [28] presented the snowball effect in a reactor-separator-recycle system and proposed a control scheme for avoiding snowball effect. Morud and Skogestad [29] observed the instability in a reactor as a result of recycle of mass and energy with large time constant. Singularity and bifurcation theory was applied for determining nonlinear behavior of CSTR-flash recycle system [34]. Bildea et al. [41] observed the large reactor volume prevents the accumulation and infinite recycle of reactant in a reactor-separator-recycle system. Steady-state multiplicity in polymerization reactions was presented by Kiss et al. [42,43]. Few researchers [44–50] investigated the stability analysis of delayed recycle CSTR. Lehman and coworkers [46–50] studied effect of delay on the stability of first-order exothermic reaction in a
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Nomenclature A2,1 cp cp ,c cs1 , f cs1 cs2 E2,1 I J Jτ m n Q Qj r R si t T Tc Tc ,in Tf UA V Vj zi
pre-exponential factor, s−1 specific heat capacity of reaction mixture, J kg−1 K−1 specific heat capacity of coolant, J kg−1 K−1 initial concentration of reactant s1 in the feed, mol m−3 concentration of reactant s1 , mol m−3 concentration of product s2 , mol m−3 activation energy of reaction, J mol−1 identity matrix Jacobian matrix with respect to instantaneous state variables Jacobian matrix with respect to delayed state variables number of delays considered number of state variables volumetric flow rate, m3 s−1 volumetric flow rate of coolant, m3 s−1 recycle ratio (0 < r < 1) ideal gas constant, 8.314 J mol−1 K−1 symbol for reacting species i time, s temperature (K) coolant temperature (K) coolant inlet temperature (K) temperature (K) overall heat transfer coefficient, Js−1 K−1 volume of reactor, m3 volume of cooling jacket, m3 state variable of index i
Greek symbols ρ density of mixture, kg m−3 ρc density of coolant, kg m−3 ψ parameters τ delay, s (-H) heat of reaction, J mol−1 λ eigenvalue
CSTR with an assumption of infinite coolant flowrates. They found that a delay in the recycle stream does not alters the dynamic characteristics of the system. A detailed study on the dynamic behavior of first-order irreversible reaction in the delayed CSTRflash-recycle system was carried-out by Balasubramanian et al. [51–53]. A delay independent stability was determined analytically by controlling the fresh feed flowrate entering into an isothermal CSTR. Furthermore, a delay dependent stability was observed by controlling the effluent flowrate. For non-isothermal CSTR, the switching of stability characteristics from unstable to stable and vice-versa was recognized. Dynamic behavior of CSTR-mechanical separator-recycle system sustaining first-order irreversible exothermic reaction [54] was studied using DDE-BIFTOOL [55,56] with an assumption of infinite coolant flowrates. DDE-BIFTOOL [55,56] is a MATLAB based solver for numerical bifurcation analysis of delay differential equations. New isola regions were found at large delays. Furthermore, the switching of stability characteristics from unstable to stable state, and vice versa was observed for infinite coolant flowrate. Here, dimensionless parameters were considered in the bifurcation analysis. Uppal et al. [5,6] presented the bifurcation analysis of delayed recycle CSTR with negligible delays in the recycle stream for first-order irreversible exothermic reaction. Infinite coolant inlet flowrate was assumed for bifurcation analysis, and observed that
the system admits fold as well as Hopf bifurcations on the steadystate multiplicity. Likewise, Balasubramanian et al. [51–53] presented the effect of delay on the stability of first-order irreversible exothermic reaction in CSTR-flash-recycle system with an assumption of an infinite coolant inlet flowrate. A delay dependent stability was analytically proved by controlling the effluent flowrate using a flow controller. Previously, researchers studied the bifurcation analysis of first-order irreversible exothermic reaction in a CSTR with an assumption of an infinite coolant flowrate. That is, the exit and inlet coolant temperatures are alike for finding the heat of cooling in a reactor. In the earlier studies, the dynamics of coolant temperature was neglected. The coolant flow rate entering into a reactor must be finite in the realistic situation and reactor temperature of an exothermic reaction must be controlled at a desired value by manipulating the coolant inlet flowrate. Thus, the dynamics of coolant temperature in a reactor should be included in the bifurcation analysis. Various researchers used the dimensionless parameters for bifurcation analysis of delayed recycle systems and the realistic delay values were not included. Therefore, we decided to use realistic delays up to 180 seconds in the bifurcation analysis for better understanding of its effect on the process variables. Besides, this article describes a comprehensive numerical bifurcation analysis of first-order exothermic reaction in the CSTRmechanical separator-recycle system with delays up to 180 s using DDE-BIFTOOL for both infinite and finite coolant inlet flowrates. 2. Model description In the following, the mass and energy balance equations which describe the steady-state and dynamic performance of the CSTRmechanical separator-recycle system are presented. A first-order irreversible exothermic elementary reaction was considered for analyzing the steady- state and dynamic performance of the delayed recycle stream in a reactor. The stoichiometry of elementary reaction [57,58] can be represented as k2,1
s1 −−→ s2
(1)
In Eq. (1), the kinetic constant k2,1 indicates the formation of product s2 from reactant s1 by virtue of chemical reaction. According to law of mass action kinetics [57,58], the reaction rates for disappearance of reactant s1 is
rs1 = −k2,1 cs1
(2)
The constitute relationships considered for the nonlinearity of delayed recycle system are: (i) Arrhenius law of kinetics, and (ii) heat of cooling in a jacketed CSTR. According to Arrhenius law, the temperature dependency of kinetic constants can be represented as
E 2,1
k2,1 = A2,1 exp −
RT
(3)
Fig. 1 illustrates the schematic diagram of CSTR-mechanical separator recycle system. Here, the CSTR was connected with a mechanical separator for the separation of product s2 from unconverted reactant s1 . The latter was recycled back to the reactor. A constant delay was assumed in the recycle stream for both concentration of reactant, and temperature. The non-isothermal operation of the reactor was considered in the bifurcation analysis. The exothermic heat released by the reaction mixture in a jacketed CSTR during the reaction must be removed by the coolant flow in a jacket. The convective and conductive heat transfer between the cooling jacket and reactor should be included for finding the heat of cooling requirements. Moreover, the logarithmic mean temperature difference between the cooling medium and reaction mixture must be accounted in the heat of cooling calculations. Thus, the formula [59] for finding the heat of cooling in a
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Table 1 Parameters considered in the numerical bifurcation analysis using DDE-BIFTOOL.
Fig. 1. Schematic diagram of CSTR-mechanical separator-recycle system.
Parameter
Symbol
Value
Unit
Volumetric flowrate Reactor volume Initial concentration of reactant Pre-exponential factor Activation energy Gas constant Feed temperature Heat of reaction Density of reaction mixture Specific heat of reaction mixture Overall heat transfer coefficient Volume of cooling jacket Density of coolant Specific heat of coolant
Q V c s1 , f A2,1 E2,1 R Tf (−H)
1.67 × 10−3 0.1 10 0 0 1.2 ×109 72,747.5 8.314 350 50,0 0 0 10 0 0 239 833.33 0.015 10 0 0 4183
m3 s−1 m3 mol m−3 s−1 J mol−1 J mol−1 K−1 K J mol−1 kg m−3 J kg− 1 K−1 J s− 1 K−1 m3 kg m−3 J kg− 1 K−1
ρ
cp UA Vj
ρc
cp,c
Various researchers [5,6,46–50,51–54] assumed the coolant inlet flowrate was infinite for finding the heat of cooling requirements and Eq. (4a) becomes
In the aforementioned model description, the asymptotic values of recycle ratio r represent the total recycle, and zero recycle of unconverted reactant from the mechanical separator to the CSTR, respectively, at r = 0 and 1. A reactor may not be operated under asymptotic recycle ratios. Thus, the recycle ratio r must be within the range of 0 and 1 for the feasible conversion of reactant.
Qcooling = UA(T − Tc )
3. Numerical bifurcation analysis
non-isothermal CSTR is
Qcooling =
UA(Tc − Tc,in ) ln
(4a)
T −Tc,in T −Tc
(4b)
However, Eq. (4b) may not be applicable as a result of finite coolant inlet flowrate entering into a reactor, and is being controlled using a flow controller. In this work, both formulas were used for heat of cooling calculations. The mole balance equation for the disappearance of reactant in the CSTR is
Q dcs1 (t ) rQ (1 − r )Q = cs + cs1 (t − τ ) − cs1 (t ) dt V 1, f V V −E2,1 −A2,1 exp cs1 (t ) RT (t )
(5)
The energy balance equations for the CSTR and cooling jacket are
Q dT (t ) rQ (1 − r )Q = T + T (t − τ ) − T (t ) dt V f V V Qcooling −E2,1 (−H ) + A exp cs1 (t ) − ρ c p 2,1 RT (t ) V ρ cp Qcooling Qj dTc (t ) = [Tc,in − Tc (t )] + dt Vj V j ρc c p,c
(6)
(7)
At steady states, Eqs. (5)−(7) become,
0=
rQ −E2,1 cs1, f − cs1 ,s − A2,1 exp cs1,s V RTs
0=
(−H ) Qcooling rQ −E2,1 T f − Ts + A exp cs1 ,s − V ρ c p 2,1 RTs V ρ cp
0=
Qcooling Qj [Tc,in − Tc,s ] + Vj V j ρc c p,c
(8)
(9) (10)
In the absence of delay, the following ordinary differential equations describe the time-dependent behavior of the system.
d cs1 rQ −E2,1 = cs1, f − cs1 − A2,1 exp cs1 dt V RT
(11)
(−H ) Qcooling dT rQ −E2,1 = Tf − T + A exp cs1 − dt V ρ c p 2,1 RT V ρ cp
(12)
Qcooling Qj dTc = [Tc,in − Tc ] + dt Vj V j ρc c p,c
(13)
A general form of system of multivariable delay differential equations with constant delays [52,53,55,56] is
dzi (t ) = f zi (t ), zi t − τ j , ψ zi (t ) = zi,0 for − τ j < t < 0 dt
(14)
In Eq. (14), i and j vary from 1 to n and 1 to m, respectively. The characteristic equation which was used for finding the real parts of roots and further analysis of stability of the steady-states of delay differential equations [55,56] is
m
Jτ j exp −λτ j = 0
λI − J −
j=1
(15)
Bifurcations occur whenever the roots move through the imaginary axis as one or more system parameters are varied. Fold bifurcation does not occur as a result of delayed recycle system cannot be destabilized by a real λ crossing the imaginary axis. Therefore, delays admit Hopf bifurcation in the CSTR-mechanical separatorrecycle system as a result of λ is purely imaginary. DDE-BIFTOOL [55,56] is a collection of MATLAB based solver for determining numerical bifurcation analysis of delay differential equations. This solver implements the continuation, and stability analysis of steady state solutions as well as fold and Hopf bifurcations. Furthermore, it allows switching from the latter to an emanating branch of periodic solutions. In this work, DDE-BIFTOOL was used for finding the steady-state profiles, real parts of the roots of characteristic equation, and dependency of delays on the bifurcation parameters. Eqs. (5)−(7) were numerically solved using MATLAB based dde23 solver [60] for finding the dynamic characteristics of the state variables. The bifurcation diagrams for the state variables were obtained using DDE-BIFTOOL. In the absence of delay, Eqs. (11)−(13) were numerically solved using MATLAB based ode45 solver for finding time-dependent behavior of the state variables. 4. Results and discussion The steady-state and dynamic characteristics of CSTRmechanical separator-recycle system sustaining first-order irreversible exothermic reaction were examined using DDE-BIFTOOL.
Please cite this article as: B. Periyasamy, M.B. Khanolkar, Effect of coolant flow rate on the dynamics of delayed recycle continuous stirred tank reactor, Journal of the Taiwan Institute of Chemical Engineers (2017), http://dx.doi.org/10.1016/j.jtice.2017.03.003
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Fig. 2. Dependency of delays on rQ at fixed Tc . The unit for rQ is m3 s−1 . (a) Tc = 310 K, (b) Tc = 315 K, (c) Tc = 316 K, (d) Tc = 320 K, and (e) Tc = 326 K.
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Fig. 3. Bifurcation diagrams of the system at Tc = 320 K. Solid line: stable steady-states, and dashed line: unstable steady states (Hopf bifurcation). (a) Reactant concentration, and (b) temperature.
Please cite this article as: B. Periyasamy, M.B. Khanolkar, Effect of coolant flow rate on the dynamics of delayed recycle continuous stirred tank reactor, Journal of the Taiwan Institute of Chemical Engineers (2017), http://dx.doi.org/10.1016/j.jtice.2017.03.003
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Fig. 4. Real parts of roots of characteristic equation versus rQ for various delays at Tc = 320 K (Hopf bifurcation regions). (a) τ = 0 s, (b) τ = 10 s, (c) τ = 60 s, (d) τ = 90 s, (e) τ = 110 s, and (f) τ = 180 s. The unit for rQ is m3 s−1 .
In the following, numerical bifurcation analysis of delayed recycle CSTR is presented by varying three bifurcation parameters. Table 1 illustrates the system parameters [61–64] used in the bifurcation analysis. Here, the volume of cooling jacket, and coolant inlet flow rate were assumed for finding the steady-state multiplicity on the coolant temperature. 4.1. Infinite coolant inlet flowrate For the first instance, the fresh feed flowrate rQ was considered as a bifurcation parameter for stability analysis. The effect of delay on the steady-state and dynamic characteristics of the nonisothermal CSTR was studied by assuming infinite coolant inlet flowrate. Eqs. (4b), (5) and (6) were used in DDE-BIFTOOL for finding the appreciable changes on the region of dynamic instability
as a result of delay by varying rQ at a fixed coolant temperature. Here, the coolant dynamics was neglected in the bifurcation analysis. The dependency of delays on rQ at Tc = 310, 315, 316, 320 and 326 K is depicted in Fig. 2. For Tc < 315 K, a delay does not affect significantly on the dynamic characteristics of the delayed recycle CSTR. A delay can alter substantially the region of dynamic characteristics when Tc ≥ 315K. In particular, the system changes the region of dynamic instability into the stable steady-states, and viceversa (vide Fig. 2c−e) as a result of delay in the recycle stream. Here, the stable operation of the reactor is possible with small delays if the system is dynamically unstable in the absence of delay. Furthermore, Tc = 320 K was selected for finding better insights on the stability characteristics of the delayed recycle system in the absence of coolant dynamics.
Please cite this article as: B. Periyasamy, M.B. Khanolkar, Effect of coolant flow rate on the dynamics of delayed recycle continuous stirred tank reactor, Journal of the Taiwan Institute of Chemical Engineers (2017), http://dx.doi.org/10.1016/j.jtice.2017.03.003
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Fig. 5. Bifurcation diagrams of the system at fixed rQ. Solid line: stable steady-states, and dashed line: unstable steady states (Hopf bifurcation). (a and b) rQ = 6.68 × 10−4 m3 s−1 , and (c and d) rQ = 1.002 × 10−3 m3 s−1 .
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Fig. 6. Dependency of delays on Tc at fixed rQ. (a) rQ = 6.68 × 10−4 m3 s−1 , and (b) rQ = 1.002 × 10−3 m3 s−1 .
Fig. 3 illustrates the bifurcation diagrams for the concentration of reactant, and reactor temperature by varying rQ at Tc = 320 K. In the absence of delay, the system exhibits Hopf bifurcation over the range of bifurcation parameter 5.71 × 10 − 4 ≤ rQ ≤ 1.13 × 10 − 3 m3 s−1 . Eqs. (4b), (11) and (12) were numerically solved using ode45 solver available in MATLAB for finding the dynamic characteristics of the delayed recycle system. The dynamic simulation plots for the concentration of reactant and reactor temperature at rQ = 7.15 × 10−4 , 1.0 × 10−3 , and 1.12 × 10−3 m3 s−1 are presented in Fig. 1S (vide supporting information). At Tc = 320 K, the sustained oscillations of reactant concentration, and temperature were observed. For better insights on the regions of Hopf bifurcation, the dependency of the real parts of roots of characteristic equation on rQ is depicted in Fig. 4 for various delays. Here, the values are positive over the range of rQ at the delays of 10, 60, 110 and 180 s. These stability characteristics indicate that the system exhibits sustained oscillations on the Hopf bifurcation regions. The real parts of roots of characteristic equation shown in Fig. 4 can be compared with Fig. 2d for finding the stability characteristics of the boundary.
For example, the system exhibits stable steady-states at τ = 90 s in Fig. 2d, and the corresponding real parts of the roots of characteristic equation are negative over the range of rQ as illustrated in Fig. 4d. This result indicates that the critical delay boundary depicted in Fig. 2d is consistent with the stability characteristics of the plots shown in Fig. 4d. For τ = 240, and 300 s, the real parts of roots of characteristic equation on rQ plots are shown in Fig. 2S (vide supporting information). For these delays, the dynamic characteristics are unaltered, and are alike as that of no delay context. The dynamic simulation plots obtained for various delays (20 and 160 s) by numerically solving Eqs (4b), (5), and (6) at rQ = 1 × 10−3 m3 s−1 , and Tc = 320 K are shown in Fig. 3S (vide, supporting information). Here, the sustained oscillation of concentration of reactant, and reactor temperature are modified into the state of oscillations. For the second case, the fresh feed flowrates were fixed at 6.68 × 10−4 , 1.002 × 10−3 , and 1.336 × 10−3 m3 s−1 for 60%, 40%, and 20% recycle, respectively. The coolant temperature was considered as a bifurcation parameter over the interval between 300 and 350 K. The bifurcation diagrams obtained by numerically solv-
Please cite this article as: B. Periyasamy, M.B. Khanolkar, Effect of coolant flow rate on the dynamics of delayed recycle continuous stirred tank reactor, Journal of the Taiwan Institute of Chemical Engineers (2017), http://dx.doi.org/10.1016/j.jtice.2017.03.003
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Fig. 7. Bifurcation diagrams of the system at Tc ,in = 308 K, τ = 0 s, and Qj = 4.425 × 10−4 m3 s−1 . Solid line: stable steady-states, and dashed line: unstable steady states (Hopf bifurcation).(a) Concentration of reactant, (b) reactor temperature, and (c) coolant temperature. (d) Real parts of roots of characteristic equation plot by varying rQ.
ing Eqs. (4b), (5) and (6) for concentration of reactant, and temperature are depicted in Fig. 5. In the absence of delay, the system exhibits Hopf bifurcation over the range of 316 ≤ Tc ≤ 327K, and 310 ≤ Tc ≤ 323K for 60%, and 40% recycle, respectively. The dependency of delays on Tc for 60% and 40% recycle are illustrated in Fig. 6. Here, the system alters substantially the region of dynamic characteristics from unstable to stable state and vice-versa at 60% recycle. The system exhibits a small stable region on the righthand side of delay plot at 40% recycle as shown in Fig. 6b. For 20% recycle, an incomplete region of Hopf bifurcation was observed for various delays (10 and180 s). The real parts of roots of characteristic equation on Tc plots are depicted in Fig. 4S (vide supporting information). For small recycle, delays do not alter significantly the region of dynamic characteristics. Besides, it was observed that the region of dynamic instability shifted towards low coolant temperature range. 4.2. Finite coolant inlet flowrate In the following, the bifurcation analysis of the delayed recycle system in the presence of coolant dynamics is described. The coolant inlet flowrate Qj , and temperature Tc,in considered for the bifurcation analysis are 4.425 × 10−4 m3 s−1 , and 308 K, respectively. Eqs. (4a), (5)−(7) were used in DDE-BIFTOOL for finding the delaydependent stability characteristics. The bifurcation diagrams ob-
tained for the concentration of reactant, reactor and coolant temperatures by varying rQ at fixed Qj and Tc,in are shown in Fig. 7. In the absence of delay, the system exhibits the region of dynamic instability for 7.22 × 10 − 4 ≤ rQ ≤ 1.08 × 10 − 3 m3 s−1 as illustrated in Fig. 7d, and multiple steady-states are not observed at Tc,in = 308 K. The dynamic simulation plots are presented in Fig. 5S (vide supporting information) at rQ = 8.95 × 10−4 m3 s−1 . The dynamic simulation results reveal that the sustained oscillation of coolant temperature in a reactor besides other two state variables. Thus, the dynamics of coolant temperature must be included in bifurcation analysis of the delayed recycle system. The dependency of delays on rQ at fixed Qj and Tc,in is depicted in Fig. 8. Here, the delays on concentration of reactant, and reactor temperature do not alter substantially the region of dynamic characteristics. However, a delay induces a small stable region on the right-hand side of Hopf bifurcation boundary. Afterwards, an ambient coolant inlet temperature (Tc,in = 298 K) was considered for analyzing the stability characteristics of the delayed recycle system. Fig. 9 illustrates the bifurcation diagrams for the concentration of reactant, reactor and coolant temperatures by varying rQ at Qj = 4.425 × 10−4 m3 s−1 . In the absence of delay, the system admits multiple steady-states for all three state variables. In particular, the bifurcation analysis reveals that the coolant temperature of the system exhibits steady-state multiplicity. This phenomenon was not observed for an infinite coolant inlet flowrate
Please cite this article as: B. Periyasamy, M.B. Khanolkar, Effect of coolant flow rate on the dynamics of delayed recycle continuous stirred tank reactor, Journal of the Taiwan Institute of Chemical Engineers (2017), http://dx.doi.org/10.1016/j.jtice.2017.03.003
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Fig. 8. (a) Dependency of delays on rQ at fixed Qj and Tc,in , and (b) Real parts of roots of characteristic equation versus rQ plot. τ = 130 s, Qj = 4.425 × 10−4 m3 s−1 , and Tc,in = 308 K. The unit of rQ is m3 s−1 .
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Fig. 9. (a−c) Bifurcation diagrams obtained using DDE-BIFTOOL at Qj = 4.425 × 10− 4 m3 s−1 , and Tc, in = 298 K. (d) Real parts of roots of characteristic equation versus rQ plot. (a) Concentration of reactant, (b) reactor temperature, and (c) coolant temperature. The unit of rQ is m3 s−1 .
assumption with the parameters considered. The stability of the steady-states of state variables is depicted in Fig. 9d. Fold bifurcation persists on the middle portion, and an incomplete region of Hopf bifurcation can be found on the lower and upper portions of the profile. For Tc,in = 308 K, the system exhibits the region of dynamic instability in the absence of delay (vide Fig. 8a) for the state variables such as concentration of reactant, reactor and coolant temperatures. The concentration and temperature delays in the recycle stream of the reactor do not alter the region of dynamic instability for the range of small delay 0 < τ < 15 s as shown in Fig. 8a.
For τ > 15 s, the delay alters the right Hopf bifurcation boundary of the system considered. Another Hopf bifurcation region arises near the right Hopf bifurcation boundary when τ > 120 s. Here, the stable operation of the system is possible for the set of parameters. For example, the coolant temperature of the CSTR is stable at rQ = 1 m3 s−1 and 90 s < τ < 135 s. That is, a delay in the recycle stream changes sustained oscillations of the CSTR and coolant temperatures to stable steady-states. The oscillations in the coolant temperature affects the yield of product in a reactor for an exothermic reaction. Ultimately, it increases the possibilities of rise in the reactor temperature as well as the formation of undesired prod-
Please cite this article as: B. Periyasamy, M.B. Khanolkar, Effect of coolant flow rate on the dynamics of delayed recycle continuous stirred tank reactor, Journal of the Taiwan Institute of Chemical Engineers (2017), http://dx.doi.org/10.1016/j.jtice.2017.03.003
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Fig. 10. (a, c and e) Steady-state coolant temperature as a function of coolant inlet flowrate. (b, d and f) Real parts of roots of characteristic equation versus Qj plots. rQ = 1.12 × 10−3 m3 s−1 , and τ = 10 s. (a and b) Tc,in = 293 K, (c and d) Tc,in = 298 K, and (e and f) Tc,in = 302 K. The unit of Qj is m3 s−1 .
ucts. Thus, the delayed recycle stream can able to circumvent these problems. For Tc,in = 298 K, the coolant temperature exhibits multiple steady-states by varying rQ as shown in Fig. 9c. Here, the system does not exhibit a full region of dynamic instability as depicted in Fig. 9d and delay do not alter the dynamic characteristics significantly. Likewise, the bifurcation diagrams of coolant temperature on Qj instead of rQ are illustrated in Fig. 10a, b and c at Tc,in = 293, 298, and 302 K, respectively. In the absence of delay, the system admits the multiple steady-states for the state variables considered Fold and Hopf bifurcations were observed on the multiple steady-states of the state variables. For all three coolant temperatures, the incomplete regions of Hopf bifurcation were found on the upper and lower portion of the profiles. However, a broader Hopf bifurcation
region was observed on the steady-state profile at Tc,in = 302 K as illustrated in Fig. 10e and f. A region of fold bifurcation was found on the middle portion of the curve and was not induced by delay. Here, a delay on the recycle stream of the CSTR do not alter substantially the dynamic characteristics as a result of steady-state multiplicity on the coolant temperature. In this work, the effect of delay in the recycle stream of CSTR was analyzed using DDE-BIFTOOL for both infinite and finite coolant inlet flowrates at 0 ≤ τ ≤ 180 s. In the absence of coolant dynamics, the concentration and temperature delays in the recycle stream substantially alter the dynamic characteristics (vide Figs. 2 and 6) with the set of parameters used. However, the system exhibits different behaviors for the finite coolant flowrates. The coolant temperature in the delayed recycle CSTR exhibits
Please cite this article as: B. Periyasamy, M.B. Khanolkar, Effect of coolant flow rate on the dynamics of delayed recycle continuous stirred tank reactor, Journal of the Taiwan Institute of Chemical Engineers (2017), http://dx.doi.org/10.1016/j.jtice.2017.03.003
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sustained oscillations in addition to the oscillations in the other two state variables. These oscillations can be either minimized or circumvented to a stable operation (vide Fig. 8) for certain concentration and temperature delays in the recycle stream when Tc,in > 298 K. However, a delay does not alter the stability characteristics of the coolant temperature in the jacketed CSTR at Tc,in = 298 K. Lehman et al. [50] analytically proved that a delay in the recycle stream of CSTR does not alter the dynamic characteristics of the system for infinite coolant flowrate. In the present work, the logarithmic mean temperature difference was considered for finding dependency of delays on the coolant temperature using DDE-BIFTOOL. Besides, the bifurcation analysis results presented at Tc,in = 298 K are consistent with the results of Lehman et al. [50]. 5. Conclusion A comprehensive bifurcation analysis of first-order irreversible exothermic elementary reaction in a CSTR-mechanical separatorrecycle system was presented using DDE-BIFTOOL. For infinite coolant inlet flowrate, the system changes the region of Hopf bifurcation as a result of delay by varying either fresh feed flowrate or coolant temperature. In particular, the stable steady-states were observed on the parameter space for some delays. On contrary, fold and Hopf bifurcations exist for finite coolant inlet flowrate entering into the CSTR. Steady-state multiplicity was observed on the coolant temperature besides the reactor temperature and concentration of reactant. Delays do not induce fold bifurcation on the system considered. The heat of cooling based on logarithmic mean temperature difference between coolant and reaction mixture is not influenced significantly by temperature delays in the recycle stream. Thus, delays do not alter substantially the dynamic characteristics at finite coolant inlet flowrate into the CSTR. Besides, the bifurcation analysis results presented here provide guidelines for analyzing effect of delay on the dynamic characteristics of fluid catalytic cracking unit, CSTRs connected in series, and other process engineering systems. The input delays on fresh feed as well as coolant, and effect of flow controller for controlling coolant inlet flowrate based on coolant and reactor temperatures are not included here. These are beyond the scope of present work. Supplementary materials Supplementary material associated with this article can be found, in the online version, at doi:10.1016/j.jtice.2017.03.003. References [1] Van Heerden C. Autothermic processes. Ind Eng Chem 1953;45:1242–7. [2] Aris R, Amundson N. An analysis of chemical reactor stability and control-I. Chem Eng Sci 1958;7:121–31. [3] Luss R, Lapidus L. An averaging technique for stability analysis. Chem Eng Sci 1966;21:159–81. [4] Douglas JM. The use of optimization theory to design simple multivariable control systems. Chem Eng Sci 1966;21:519–32. [5] Uppal A, Ray WH, Poore AB. On the dynamic behavior of continuous stirred tank reactors. Chem. Eng. Sci 1974;29:967–85. [6] Uppal A, Ray WH, Poore AB. The classification of the dynamic behavior of continuous stirred tank reactors-influence of reactor residence time. Chem Eng Sci 1976;31:205–14. [7] Merzhanov GA, Abramov GV. Thermal regimes of exothermic processes in continuous stirred tank reactors. Chem Eng Sci 1977;32:475–81. [8] Vaganov DA, Samoilenko NG, Abramov VG. Periodic regimes of continuous stirred tank reactors. Chem Eng Sci 1978;33:1133–40. [9] Kubicek M, Hofmann H, Hlavacek V, Sinkule J. Multiplicity and stability in a sequence of two nonadiabatic nonisothermal CSTR. Chem Eng Sci 1980;35:987–96. [10] Mankin JC, Hudson JL. The dynamics of coupled nonisothermal continuous stirred tank reactors. Chem Eng Sci 1986;41:2651–61. [11] Wirges HP. Experimental study of self-sustained oscillations in a stirred tank reactor. Chem Eng Sci 1980;33:2141–6. [12] Zhabotinsky AM, Rovinsky AB. Mechanism and nonlinear dynamics of an oscillating chemical reaction. J Stat Phy 1987;48:959–75.
[13] Aris R, Gray P, Scott SK. Modelling cubic autocatalysis by successive bimolecular steps. Chem Eng Sci 1988;43:207–11. [14] Balakotaiah V, Luss D. Structure of the steady-state solutions of lumped-parameter chemically reacting systems. Chem Eng Sci 1982;37:1611–23. [15] Balakotaiah V, Luss D. Multiplicity features of reacting systems dependence of the steady-states of a CSTR on the residence time. Chem Eng Sci 1983;38:1709–21. [16] Jorgensen DV, Aris R. On the dynamics of a stirred tank with consecutive reactions. Chem Eng Sci 1983;38:45–53. [17] Doedel JE, Heinemann FR. Numerical computation of periodic solution branches and oscillatory dynamics of the stirred tank reactor with A→B→C reactions. Chem Eng Sci 1983;38:1493–9. [18] Nelson MI, Sidhu HS. Bifurcation phenomena for an oxidation reaction in a continuously stirred tank reactor I. adiabatic operation. J Math Chem 2002;31:155–86. [19] Nelson MI, Sidhu HS. Bifurcation phenomena for an oxidation reaction in a continuously stirred tank reactor II. diabatic operation. Aust N Z Ind Appl Math J 2004;45:303–26. [20] Lee JH, Lim HC. Multiplicity and stability analysis of continuous pre-fermentation of cheese culture. Proc Biochem 1999;34:467–75. [21] Sanyasi Rao VN, Pushpavanam S. Effect of noise on the behavior of a zeroth-order reaction in a continuous stirred tank reactor. Ind Eng Chem Res 20 0 0;39:926–32. [22] Balakrishna A, Schmidt LD, Aris R. Pt-catalyzed combustion of CH4 -C3 H8 mixtures. Chem Eng Sci 1994;49:11–18. [23] Li R, Li H. Isolas, mushrooms and other forms of multistability in isothermal bimolecular reacting systems. Chem Eng Sci 1989;44:2995–3000. [24] Balakotaiah V, Luss D. Global analysis of the multiplicity features of multi-reaction lumped-parameter systems. Chem Eng Sci 1984;39:865–81. [25] Tsiligiannis CA, Lyberatost G. A linear algebraic approach to steady-state bifurcation of chemical reaction systems. Chem Eng Sci 1987;42:535–41. [26] Berezowski M, Burghardt A. A generalized analytical method for determining multiplicity features in chemical reactors with recycle. Chem Eng Sci 1989;44:2927–33. [27] Luyben ML, Floudas CA. Analyzing the interaction of design and control-2. reactor-separator-recycle system. Comp Chem Eng 1994;18:971–94. [28] Luyben WL. Snowball effects in reactor/separator process with recycle. Ind Eng Chem Res 1994;33:299–305. [29] Morud J, Skogestad S. Effects of recycle on dynamics and control of chemical processing plants. Comp Chem Eng 1994;18:S529–34. [30] Yi CK, Luyben WL. Design and control of coupled reactor/column systems-part 1. A binary coupled reactor/rectifier system. Comp Chem Eng 1997;21:25–46. [31] Yi CK, Luyben WL. Design and control of coupled reactor/column systems-part 2. A binary coupled reactor/rectifier system. Comp Chem Eng 1997;21:47–67. [32] Yi CK, Luyben WL. Design and control of coupled reactor/column systems-part 3. A binary coupled reactor/rectifier system. Comp Chem Eng 1997;21:69–86. [33] Read NK, Ray WH. Application of nonlinear dynamic analysis in the identification and control of nonlinear systems I. Simple dynamics. J Proc Cont 1998;8:1–15. [34] Pushpavanam S, Kienle A. Nonlinear behaviour of an ideal reactor separator network with mass recycle. Chem Eng Sci 2001;56:2837–49. [35] Larson T, Govatsmark SM, Skogestad S, Yu CC. Control structure selection for reactor separator and recycle processes. Ind Eng Chem Res 2003;42:1225–34. [36] Seki H, Naka Y. A hierarchical controller design for a reactor/separator system with recycle. Ind Eng Chem Res 2006;45:6518–24. [37] Xu S, Bao J. Distributed control of plant wide chemical processes. J Proc Cont 2009;19:1671–87. [38] Jogwar SV, Baldea M, Daoutidis P. Dynamics and control of process networks with large energy recycle. Ind Eng Chem Res 2009;48:6087–97. [39] Liu J, Munoz de la Pena D, Christofides DP. Distributed model predictive control of nonlinear process systems. AIChE J 2009;55:1171–84. [40] Antoniades C, Christofides DP. Feedback control of nonlinear differential difference equation systems. Chem Eng Sci 1999;54:5677–709. [41] Bildea CS, Dimian AC, Iedema PD. Nonlinear behavior of reactor-separator-recycle systems. Comp Chem Eng 20 0 0;24:209–15. [42] Kiss AA, Bildea CS, Dimian AC, Iedema PD. State multiplicity in CSTR-separator-recycle polymerization systems. Chem Eng Sci 2002;57:535–46. [43] Kiss AA, Bildea CS, Dimian AC, Iedema PD. State multiplicity in PFR-separator-recycle polymerization systems. Chem Eng Sci 2003;58:2973–84. [44] Hellinckx L, Grootjans J, Van Den Bosch B. Stability analysis of a reactor with a retarded recycle. Chem Eng J 1972;3:304–7. [45] Denn MM, Lavie R. Dynamics of plants with recycle. Chem Eng J 1982;24:55–9. [46] Lehman B, Verriest E. Stability of a continuous stirred reactor with delay in the recycle stream. In: Proceedings of the 30th conference on decision and control, England; 1991. [47] Lehman B. Stability of chemical reactions in a CSTR with delayed recycle stream. In: Proceedings of the American Control conference, Maryland; 1994. [48] Lehman B, Bentsman J, Verduyn S, Lunel V IE. Vibrational control of nonlinear time lag systems with arbitrarily large but bounded delay averaging theory stabilizability and transient behavior. In: Conference on decision and control, Tucson Arizona; 1992. [49] Lehman B, Widjaya I, Shujaee K. Application of vibrational control to chemical reactions a CSTR with delayed recycle stream. In: Proceedings of the American control conference, Maryland; 1994. [50] Lehman B, Widjaya I, Shujaee K. Vibrational control of chemical reactions in a CSTR with delayed recycle stream. J Math Anal Appl 1995;193:28–59.
Please cite this article as: B. Periyasamy, M.B. Khanolkar, Effect of coolant flow rate on the dynamics of delayed recycle continuous stirred tank reactor, Journal of the Taiwan Institute of Chemical Engineers (2017), http://dx.doi.org/10.1016/j.jtice.2017.03.003
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ARTICLE IN PRESS
[m5G;March 22, 2017;5:57]
B. Periyasamy, M.B. Khanolkar / Journal of the Taiwan Institute of Chemical Engineers 000 (2017) 1–11 [51] Balasubramanian P, Kosuri MR, Pushpavanam S, Kienle A. Effect of delay on the stability of a coupled reactor separator system. Ind Eng Chem Res 2003;42:3758–64. [52] Balasubramanian P, Pushpavanam S, Kienle A, Balaraman KS. Effect of delay on the stability of a coupled reactor flash system sustaining an elementary non-isothermal reaction. Ind Eng Chem Res 2005;44:3619–25. [53] Balasubramanian P. Numerical bifurcation analysis of delay in a coupled reactor separator system. Ind Eng Chem Res 2008;47:3920–9. [54] Gangadhar NR, Balasubramanian P. Numerical bifurcation analysis of delayed recycle stream in a continuously stirred tank reactor. In: Proceedings of the international conference on modeling, optimization and computing ICMOC 2010, 1298; 2010. p. 192–7. [55] Engelborghs K, Luzyanina T, Roose D. Numerical bifurcation analysis of delay differential equations. J Comput Appl Math 20 0 0;125:265–75. [56] Engelborghs K. A Matlab package for bifurcation analysis of delay differential equations. DDE-BIFTOOL: technical report TW-305 department of computer science 20 0 0. [57] Bhusare VH, Balasubramanian P. A new paradigm in kinetic modeling of complex reaction systems. Int Rev Model Simul 2010;3:1145–52.
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[58] Bhusare VH, Balasubramanian P. A general kinetic modelling methodology for first order irreversible reaction systems. Research bulletin of australian institute of high energetic Materials, 1. QLD, Australia; 2011. ISBN: 978-0-9806811-9-2 p. 85–105. [59] Molnár A, Markoš J, Jelemenský Ľ. Safety analysis of CSTR towards changes in operating conditions. J Loss Preven Proc Ind 2003;16:373–80. [60] Shampine L.F., Thompson S. Solving DDE with MATLAB. Mathematics Department, Southern Methodist University, Dallas, TX. [61] Armaoua A, Ataei A. Piece-wise constant predictive feedback control of nonlinear systems. J Proc Contr 2014;24:326–35. [62] Hahn J, Mönnigmann M, Marquardt W. On the use of bifurcation analysis for robust controller tuning for nonlinear systems. J Proc Contr 2008;18:408–20. [63] Jouili K, Jerbi H, Braiek NB. An advanced fuzzy logic gain scheduling trajectory control for nonlinear systems. J Proc Contr 2010;20:426–40. [64] Di Ciccio MP, Bottini M, Pepe P, Foscolo PU. Observer-based nonlinear control law for a continuous stirred tank reactor with recycle. Chem Eng Sci 2011;66:4780–97.
Please cite this article as: B. Periyasamy, M.B. Khanolkar, Effect of coolant flow rate on the dynamics of delayed recycle continuous stirred tank reactor, Journal of the Taiwan Institute of Chemical Engineers (2017), http://dx.doi.org/10.1016/j.jtice.2017.03.003