Delay Sensitivity Analysis for Typical Reactor ... - ACS Publications

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Delay Sensitivity Analysis for Typical Reactor Systems with Flexibility Consideration Weiqing Huang,*,† Yu Qian,‡ Youyuan Shao,† Yongfu Qiu,† and Hongbo Fan† †

School of Chemistry and Environmental Engineering, Dongguan University of Technology, Dongguan, Guangdong 523808, PR China ‡ School of Chemical Engineering, South China University of Technology, Guangzhou 510640, PR China ABSTRACT: Chemical reactor systems are often complex dynamic time-delay systems that have to operate successfully in the presence of uncertainties. Under these circumstances, a concept of delay sensitivity and an integrated framework for delay sensitivity analysis of typical reactor systems with control and flexibility consideration is proposed in this work. In the proposed framework, two critical steps are used to push the dynamic system to satisfy the constrain requirement. In the first step, the Ziegler−Nichols method is combined with the Nonlinear Control Design (NCD) Package to optimize the control action. In the second step, the flexibility range is rectified based on the golden section method. The proposed strategy is investigated by two typical reactor systems with time delay. All the results demonstrate that the proposed framework may provide a powerful tool for delay sensitivity analysis of typical reactor systems with control and flexibility consideration.

1. INTRODUCTION Chemical plants are usually complex dynamic systems which need to be smoothly controlled during operation in the presence of uncertainties, especially when chemical reaction is included. These uncertainties generally correspond to variations in either external parameters, such as product demands and quality and quantity of feedstock, or in internal process parameters, such as heat/mass transfer coefficients and kinetic coefficient. Flexibility analysis with uncertainty consideration was originally proposed in the 1980s.1−6 Grossmann and Halemane proposed a decomposition strategy1 and optimization strategies2 for flexible chemical processes. Halemane and Grossmann3 used the max−min−max optimization problem to solve the flexibility test problem for process design under uncertainty. Swaney and Grossmann proposed a scalar flexibility index to quantify the maximum feasible uncertainty range and focused on the problem formulation4 and computational algorithms.5 Grossmann and Floudas6 proposed an active constraint strategy for flexibility analysis in a chemical process. Afterward, flexibility analysis problems, such as feasibility test, flexibility index, and two-stage optimization problems were discussed in the literature.7−12 Bansal et al.7 tried to solve the flexibility analysis and design problem for linear systems by using parametric programming. Ostrovsky et al.8 proposed a new algorithm name BB_ACTIVE which can avoid enumeration for computing process flexibility. Floudas et al.9 tried to solve the flexibility analysis problems for chemical systems by using a deterministic global optimization algorithm. Ostrovski and Volin10 tried to solve the flexibility evaluation problem by using new estimators of process constraints for chemical systems. Bansal et al.11 used a new parametric programming framework to unify the solution of various flexibility analysis and design optimization problems. Ostrovsky and Volin12 took into account fullness and accuracy of plant data at the operation stage and developed new formulations of the feasibility test and the two-stage optimization problems. Flexibility is the ability for © 2014 American Chemical Society

a particular system to maintain feasible operation over a range of uncertain conditions. The flexibility analysis problem generally consists of two tasks which are complementary to each other. The first task is known as the feasibility problem or flexibility test problem, which is to determine if a given design can feasibly operate over the range of uncertainty considered. The second task is known as the flexibility index problem and is usually tackled by establishing the maximum uncertainty parameter range over which the design can operate feasibly. However, some of these flexibility analysis theories were limited due to the assumption that the processes were operated at steady state within an uncertainty range, leading to a defective description of real systems. Furthermore, they did not take account of real-time control and thereby could not exactly describe the flexibility, especially when time delay was included in chemical systems. The effect of control on system flexibility was discussed in some literature. Dimitriadis and Pistikopoulos13 developed a new formulation for both the dynamic feasibility and dynamic flexibility analysis problem and presented a unified approach to solve those problems. Bahri et al.14 proposed a systematic approach for the synthesis of flexible and controllable plants, which provided a new quantitative measure for the flexibility and controllability of a design. Bansal et al.15 demonstrated how the design and control of processes described by large-scale, complex, mixed-integer dynamic models can be simultaneously optimized under parametric uncertainties. Sakizlis et al.16 developed a simultaneous process and control design methodology based on novel mixed integer dynamic optimization algorithms for the chemical system under uncertainty. Malcolm et al.17 applied dynamic flexibility analysis to integrate systems design and Received: Revised: Accepted: Published: 14721

March 19, 2014 August 25, 2014 August 28, 2014 August 28, 2014 dx.doi.org/10.1021/ie501164y | Ind. Eng. Chem. Res. 2014, 53, 14721−14734

Industrial & Engineering Chemistry Research

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solve the flexibility analysis problem of chemical reaction systems with time delay using a modified finite element collocation method.34 In the current work, the focus is delay sensitivity analysis for typical reactor systems with dynamic flexibility and control consideration. In the proposed strategy, two critical steps are used to push the dynamic system to satisfy the constrain requirement. In the first step, the Ziegler−Nichols method is combined with the Nonlinear Control Design (NCD) Package to optimize the control action. In the second step, the flexibility range is rectified based on the golden section method. The validity of the proposed strategy is investigated by two typical reactor systems with time delay. The proposed strategy can provide a simple and rigorous method for delay sensitivity analysis of typical reactor systems with dynamic flexibility and control consideration.

control. Ricardez-Sandoval et al.18 proposed a methodology for the simultaneous design and control of large-scale systems under process parameter uncertainty. Zhou et al.19 took into account initial conditions to improve the system flexibility. In recent years, there are more research works focused on the integrated analysis of flexibility, controllability, and stability for chemical systems under uncertainties. Ricardez-Sandoval20 presented a new methodology for the simultaneous design and control of systems under uncertainty which was applied to a continuous stirred tank reactor (CSTR) process. Escobar et al.21 proposed a computational framework based on two-stage strategy for the synthesis of flexible and controllable heat exchanger networks under uncertainties. Trainor et al.22 presented a new methodology for the optimal process and control design of dynamic systems under uncertainty. Bahakim and Ricardez-Sandoval23 proposed a stochastic-based simultaneous design and control methodology for chemical processes under uncertainty. Apparently, control strategy and critical influencing factors need to be considered in flexibility analysis for dynamic systems. Time delay occurs frequently in control system design which is very important for dynamic flexibility analysis of chemical systems. In the chemical reaction process, time delays often take place in the mass transmission in serial reactors, recycles, and intermediate storage tanks or in the heat transmission and measurement. Time delay was often ignored in previous research on flexibility analysis of chemical reaction systems which, however, has a significant impact on robust stability and stabilization of the entire process system during the operation.24 Control of time-delay systems is a subject of both practical and theoretical importance which has received much attention in the past years, which focuses on the design of effective feedback controller and the problem of robust H∞ control for time-delay systems. Xu and Chen25 discussed the robust H∞ control for uncertain discrete-time systems with time-varying delays. Xu et al.26 also discussed the robust H∞ control for uncertain discrete stochastic time-delay systems. Yang et al.27 discussed the robust H∞ control for network systems with random communication delays. Uma et al.28 used a modified Smith predictor to enhance control of the integrating cascade process with time delay. Xu and Bao29 discussed the control strategy for chemical processes with uncertain time delays. The effect of delay on reactor systems was also investigated in several literatures. Antoniades and Christofides30 discussed the feedback control for nonlinear differential difference equation systems, and chemical reactor systems with time delay are investigated in the case studies. Balasubramanian et al.31 discussed the effect of delay on the stability of coupled reactor−separator systems. Balasubramanian32 also discussed the numerical bifurcation analysis of delay in a coupled reactor−separator system. Di Ciccio et al.33 discussed nonlinear control strategy for a continuous stirred tank reactor with recycle. On the other hand, most chemical reaction processes are strongly exothermic, and therefore it is important to consider the critical influencing factors and control strategy during the operation to guarantee the safety. The neglect of time delays might give rise to defective control and inaccurate flexibility evaluation. It is meaningful to investigate the effect of time delay on the flexibility for chemical reactor systems. The main difficulty in this work is to explore a proper strategy/method which can evaluate the delay impact with control and flexibility consideration for chemical reactor systems. In our previous work, the researchers try to

2. PROBLEM DESCRIPTION Two examples without/with time delay consideration are investigated to show the effect of time delay on dynamic systems under uncertainty. The first case is a third-order system whose transfer function is given as G (s ) =

1.5 50s 3 + θ2s 2 + θ1s + 1

(1)

To show the impact of time delay on the dynamic systems under uncertainties, a time delay parameter (1.6 time unit) was introduced to the third-order system, and the corresponding transfer function is changed to G (s ) =

1.5 e−1.6s 50s + θ2s 2 + θ1s + 1 3

(2)

where θ1 and θ2 are uncertain parameters. For the sake of operation stability and somewhat dynamic response, such a system is subject to several constraints: t ∈ [0, 10], y ∈ [−0.01, 1.20]; t ∈ [10, 30], y ∈ [0.9, 1.20]; and t ∈ [30, 100], y ∈ [0.99, 1.01]. Meanwhile, the ranges of uncertainty parameters are given as θ1 ∈ [1.5, 6], θ1N = 3, and θ2 ∈ [40, 50], θ2N = 45. The step response is used to investigate the entire system controlled by a proportional−integral−derivative (PID) controller which is implemented with the nonlinear control design (NCD) package in MATLAB/SIMULINK. The results are shown in Figure 1, where the dashed line represents the process constraints and the solid curve denotes the system response output. As shown in Figure 1, when time delay is ignored, the system output can be maintained within the constraint area in the whole time horizon. But, as is apparent, the constraints are exceeded numbers of times under the given uncertainties when time delay is considered. The simulation results also demonstrates that the dynamic flexibility of the third-order system with time delay consideration cannot be guaranteed under the given uncertainties. The second example is a second-order system whose transfer function is given as G (s ) =

1.5 θ2s 2 + θ1s + 1

(3)

To show the impact of time delay on the second-order systems under uncertainties, a time delay parameter (1.6 time unit) was introduced to the system, and the corresponding transfer function is changed to 14722

dx.doi.org/10.1021/ie501164y | Ind. Eng. Chem. Res. 2014, 53, 14721−14734


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Figure 1. Simulation results of the third-order system without/with delay: (a) without time delay and (b) with time delay.

G (s ) =

1.5 e−1.6s θ2s 2 + θ1s + 1

Figure 2. Simulation results of the second-order system without/with delay: (a) without delay and (b) with delay.

As the simulation results show by the above examples, it is interesting that time delay has different impact on different dynamic systems under the same uncertainties and lag time. Since time delay occurs frequently in most dynamic chemical processes with control system, it should be taken into account when carrying out the dynamic flexibility analysis of chemical systems. In order to demonstrate the effect of time delay on dynamic flexibility for dynamic systems with control consideration, a concept of delay sensitivity is proposed in this work, and the delay sensitivity index is define as

(4)

where θ1 and θ2 are uncertain parameters. For the sake of operation stability and somewhat dynamic response, such a system is subject to several constraints: t ∈ [0, 10], y ∈ [−0.01, 1.20]; t ∈ [10, 30], y ∈ [0.9, 1.20]; and t ∈ [30, 100], y∈[0.99, 1.01]. Meanwhile, the ranges of uncertainty parameters are given as θ1 ∈ [1.5, 6], θ1N = 3, and θ2 ∈ [40, 50], θ2N = 45. The step response is used to investigate the entire system controlled by a PID controller which is implemented with the nonlinear control design (NCD) package in MATLAB/SIMULINK. The results are shown in Figure 2, where the dashed line represents the process constraints and the solid curve denotes the system response output. As shown in Figure 2, when time delay is ignored, the system output can be maintained within the constraint area in the whole time horizon. And the system output still can be maintained within the process constraint area under the given uncertainties when time delay (1.6 time unit) is considered, which means that the dynamic flexibility of the second-order system with time delay consideration still can be guaranteed under the given uncertainties.

φ = φ (d , x , z , θ , τ )

(5)

where d is the vector of design variables, x is the vector of state variables, z is the vector of control variables, θ is the vector of uncertainty parameters, and τ is the lag time. The detail relation between the delay sensitivity index φ and the function φ(d, x, z, θ, τ) is very difficult to figure out, so the delay sensitivity index is defined as φ = S1/S to evaluate the impact of time delay semiquantitatively for dynamic systems. S denotes the process constraint area which is composed by the lower constraint and the upper constraint, S1 denotes the exceed area by system response curve which was not maintained within the process 14723



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constraint area. If the value of φ ≤ 0, which means the dynamic flexibility of the dynamic system still can be guaranteed under the given lag time and uncertainties, the delay sensitivity of the dynamic system is low. If the value of φ > 0, which means the dynamic flexibility of the dynamic system cannot be guaranteed under the effect of the lag time, the operation safety and stability may not be guaranteed either. And the higher the value of φ is, the more sensitive the dynamic system is. For example, the delay sensitivity index of the second-order system is φ = S1/ S = 0, which means the dynamic flexibility of the second-order system still can be guaranteed under the effect of the lag time (1.6 time unit). the delay sensitivity index of the third-order system is φ = S1/S ≈ 0.17 > 0, which means the dynamic flexibility of the third-order system cannot be guaranteed under the effect of the lag time (1.6 time unit), and the operation safety and stability of the dynamic system might not be guaranteed either. So, when time delay shows strong impact to the dynamic systems, dynamic flexibility analysis with control consideration should do some correction or improvement to guarantee the operation safety and stability. In this work, the “delay sensitivity” is used to evaluate how time delay will affect the flexibility that represents the feasible uncertainty range with control consideration. The efficacy of the control strategy is also evaluated during the flexibility analysis with delay consideration. Flexibility of chemical reactor systems varies between situations with time delay and those without time delay. It is affected by the control strategy too. So it is very important to figure out the effect of time delay to the flexibility of reactor systems with control consideration. In summary, the most important standard to assess the delay sensitivity would be the flexibility.

χ (d) = max

x(0) = x 0

g(d, x(t ), z(t ), θ(t ), t ) ≤ 0 L

max

g (d, x(t ), z(t ), θ(t ), t ) x(0) = x 0

s.t . h(d, ẋ(t ), x(t ), z(t ), θ(t ), t ) = 0; L

U

θ(t ) ∈ T(t ) = {θ(t )|θ ≤ θ(t ) ≤ θ } z(t ) ∈ Z(t ) = {z(t )|z L ≤ z(t ) ≤ zU }

(DFT)

where J is the index set for the feasibility constraints. If χ(d) ≤ 0, the system is dynamically feasible in T(t). Otherwise, there exists at least one possible profile, θ(t), for which no corresponding control action, Z(t), can be found to keep the system feasible at all times within the horizon considered. And dynamic flexibility index evaluation problem (DFI) of the process system without time delay can be formulated as follows: DF(d) =

max

δ

δ , z , t ∈ [0, H ]

s.t . χ (d) = max

min

max

g (d, x(t ), z(t ), θ(t ), t ) ≤ 0

θ(t ) ∈ T(t ) z(t ) ∈ Z(t ) j ∈ J, t ∈ [0, H ] j

h(d, ẋ(t ), x(t ), z(t ), θ(t ), t ) = 0;

x(0) = x 0

θ(t ) ∈ T(δ , t ) = {θ(t )|θ N (t ) − δ Δθ−(t ) ≤ θ(t ) ≤ θ N (t ) + δ Δθ+(t )} δ≥0 z(t ) ∈ Z(t ) = {z(t )|z L ≤ z(t ) ≤ zU }

(DFI)

where DF(d) is the dynamic flexibility index, δ is the flexibility index variables. Time delays may have significant impact on the operation flexibility of chemical reaction systems.34 Dynamic models of chemical reaction systems with time delay can be described as differential difference equation (DDE) systems:

3. METHODOLOGY Dynamic systems without time delay can be described mathematically as the ordinary differential equations (ODE) and algebraic equations: h(d, ẋ(t ), x(t ), z(t ), θ(t ), t ) = 0;

min

θ(t ) ∈ T(t ) z(t ) ∈ Z(t ) j ∈ J, t ∈ [0, H ] j

h(d, ẋ(t ), x(t ), x(t − τ ), z(t ), θ(t ), t ) = 0;

x(0) = x 0 (10)

g(d, x(t ), x(t − τ ), z(t ), θ(t ), t ) ≤ 0

(11)

θ(t ) ∈ T(t ) = {θ(t )|θ L ≤ θ(t ) ≤ θU }

(12)

z(t ) ∈ Z(t ) = {z(t )|z L ≤ z(t ) ≤ zU }

(13)

(6) (7)

U

θ(t ) ∈ T(t ) = {θ(t )|θ ≤ θ(t ) ≤ θ }

(8)

z(t ) ∈ Z(t ) = {z(t )|z L ≤ z(t ) ≤ zU }

(9)

Correspondingly, the dynamic flexibility test problem for dynamic systems with time delay can be formulated as χ (d) = max

min

max

g (d, x(t ), x(t − τ ), z(t ), θ(t ), t )

θ(t ) ∈ T(t ) z(t ) ∈ Z(t ) j ∈ J, t ∈ [0, H ] j

where dim {h} = dim {x}, h denotes the vector of equations (such as mass and energy balances or equilibrium relation), g denotes the vector of inequalities (such as product quality specifications or physical operation limits), t is the time factor, d is the vector of design variables, x is the vector of state variables and x(0) denotes the initial value of x, z is the vector of control variables, θ is the vector of uncertainty parameters, θL and θU represent the lower bound and upper bound of the uncertainty vector θ, and T(t) represents the space of uncertainty parameters. According to the research13 by Dimitriadis and Pistikopoulos, the dynamic flexibility test problem (DFT) of the process system without time delay can be reduced to the following max−min−max optimal problem:

s.t . h(d, ẋ(t ), x(t ), x(t − τ ), z(t ), θ(t ), t ) = 0;

x(0) = x 0

θ(t ) ∈ T(t ) = {θ(t )|θ L ≤ θ(t ) ≤ θU } z(t ) ∈ Z(t ) = {z(t )|z L ≤ z(t ) ≤ zU }

(DFT1)

If χ(d) ≤ 0, the system is dynamically feasible in T(t). Otherwise, there exists at least one possible profile, θ(t), for which no corresponding control action, Z(t), can be found to keep the system feasible at all times within the horizon considered under the effect of time delay. Accordingly, the dynamic flexibility index evaluation problem (DFI) of the process system with time delay can be formulated as follows: 14724


Industrial & Engineering Chemistry Research DF(d) =

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In this work, Problem DFT2 is used to test if a particular system can feasibly operate over the range of uncertainty considered under the effect of time delay. And Problem DFI2 is used to find the maximum uncertainty range over which the system can operate feasibly under the effect of time delay. With the above assumption of DFI2 and DFT2, when delay time τ = 0, Problems DFI2 and DFT2 will become flexibility analysis problems with given uncertainty profiles in which time delay is ignored, and more details can be traced in the paper13 researched by Dimitriadis and Pistikopoulos. In this work, to assess the delay sensitivity for typical reactor systems with flexibility and control consideration, the reactor systems modeled by ODE/DDE are linearized by the Taylor series spreader formula and finally changed to the transform function form using a Laplace transform. An integrated framework for delay sensitivity analysis of typical reactor systems with flexibility and control consideration is proposed in this work, which is described in Figure 3. As shown in Figure 3,

δ

max

δ , z , t ∈ [0, H ]

s.t . χ (d) = max

min

max

g (d, x(t ), x(t − τ ), z(t ), θ(t ), t )

θ(t ) ∈ T(t ) z(t ) ∈ Z(t ) j ∈ J, t ∈ [0, H ] j

≤0 h(d, ẋ(t ), x(t ), x(t − τ ), z(t ), θ(t ), t ) = 0;

x(0) = x 0

θ(t ) ∈ T(δ , t ) = {θ(t )|θ N (t ) − δ Δθ−(t ) ≤ θ(t ) ≤ θ N (t ) + δ Δθ+(t )} δ≥0 z(t ) ∈ Z(t ) = {z(t )|z L ≤ z(t ) ≤ zU }

(DFI1)

The dynamic flexibility index evaluation problem can be simplified by assuming that the direction, Δθc, from the nominal profile, θN, on which the critical point that limits flexibility lies is either known or is assumed to be one of the vertex directions of the uncertainty space, and then the dynamic flexibility index evaluation problem with time delay consideration can be reduced to a dynamic optimization problem of the following form: DF(d) =

max

δ

δ , z , t ∈ [0, H ]


x(0) = x 0

g(d, x(t ), x(t − τ ), z(t ), θ(t ), t ) ≤ 0 θ(t ) = θ N (t ) + δ Δθ c(t ) δ≥0 z(t ) ∈ Z(t ) = {z(t )|z L ≤ z(t ) ≤ zU }

(DFI2)

The dynamic flexibility index DF(d) represents the largest scaled deviation of the uncertainty parameter profile that the system can tolerate while remaining feasible within the horizon considered. According to the assumption, when the critical direction, Δθc, from the nominal profile, θN, on which the critical point that limits flexibility lies is known, the maximum δ can be searched on the critical direction. Because the critical point represents the worst condition, the searching result of δ will be the solution of problem DFI2. Correspondingly, The dynamic flexibility test problem (DFT1) can also be greatly simplified by assuming that the exact profile of θ(t) is given or the critical point of the uncertainty space that limits flexibility is known or can be found. Under this assumption the dynamic flexibility test problem for dynamic systems with time delay can be formulated as χ (d) =

min

u , z(t ) ∈ Z(t ), t ∈ [0, H ]

Figure 3. Delay sensitivity analysis framework for dynamic system with control and flexibility consideration.

system model and constraints are demonstrated in Step 1, and the original expected feasible uncertainty range should be determined in Step 2. After system modeling and original flexibility analysis, delay sensitivity analysis and dynamic response analysis should be done in Step 3. The control strategy used in Step 3 for a particular system with time delay is the same as the situation without time delay in the first round, so the effect of time delay on the flexibility can be evaluated. If system constraints can be satisfied completely in Step 3, this demonstrates that the system can operate feasibly over the original expected uncertainty range. If system constraints cannot be satisfied completely in Step 3, Step 4 should be carried out. Another control strategy using the Ziegler−Nichols method combined with the Nonlinear Control Design (NCD) Package is carried out to optimize the control action and test if the control action can be improved, and then Step 3 should be

u


x(0) = x 0

g(d, x(t ), x(t − τ ), z(t ), θ(t ), t ) ≤ u θ(t ) = θ c(t ) z(t ) ∈ Z(t ) = {z(t )|z L ≤ z(t ) ≤ zU }

(DFT2)

If χ(d) ≤ 0 when the uncertainty θ is at the critical point θc which represents the worst condition, then the system is dynamically feasible in T(t). Otherwise, there exists at least one possible profile, θ(t), for which no corresponding control action, Z(t), can be found to keep the system feasible at all times within the horizon considered under the effect of time delay. 14725



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carried out as the second round to test the delay sensitivity and dynamic response analysis. If system constraints and requirements can be satisfied completely after control improvement, then the whole delay analysis can be finished. But if system constraints and requirements still cannot be satisfied completely after control improvement, which means the system cannot operate feasibly over the whole uncertainty range by using the proposed control strategy, then Step 5 should be carried out to find the maximum feasible uncertainty range. Still, after flexibility correction in Step 5, a new feasible uncertainty range can be determined in Step 2, and delay sensitivity and dynamic response can be tested in Step 3 as the third round. After the testing by Step 3, the maximum feasible uncertainty range can be determined and the whole analysis procedure can be finished. The experimental results of the integrated strategy on two illustrative cases are reported in the next section, and it should be noticed that just single delay, especially the heat transfer delay, is considered in the case studies in this work.

where k = k 0e(−E / RT )

The reaction rate is assumed to be of first-order, where k0 and E denote the pre-exponential constant and activation energy of the reaction and T and CA denote the temperature and concentration of species A in the reactor. F is the flow rate of the inlet stream to the reactor and consists of a fresh feed of pure A. U represents the coefficient of heat transfer, and HR represents the reaction heat. TJ denotes the jacket temperature, and the system is under the assumptions of constant volume of the reacting liquid, V, constant density, ρ, and heat capacity, Cp, of the reacting liquid. The system parameters are listed in Table 1. For these values the corresponding steady state is CAs = 3.92 Table 1. Parameters of CSTR System

4. CASE STUDIES To demonstrate the efficacy of the proposed integrated strategy/framework for delay sensitivity assessment with flexibility and control consideration, two typical reactor system examples are investigated. Case A: Analysis of a Nonisothermal CSTR Operation. Figure 4 shows a jacketed continuously stirred tank reactor

(CSTR) system, originally given by Luyben,35 that later was studied by Uygun et al.36 An irreversible reaction of the form A → B takes place in the reactor, where A is the reactant species and B is the product species. The reaction is exothermic, and a cooling jacket is used to remove heat from the reactor. According to the proposed framework, first the system modeling should be carried out in the Step 1, and the original expected feasible uncertainty range should be determined in Step 2. The nominal system model is given below

VρCp

parameters

value

reactor feed/output flow rate F (m3 min−1) reactor feed temperature T0 (K) inlet coolant temperature TJ0 (K) inlet concentration CA0 (kmol m−3) volume of liquid in reactor V (m3) total heat capacity Cp (kJ kg−1 K−1) activation energy E (kJ kmol−1) gas constant R (kJ kmol−1 K−1) heat of reaction HR (kJ kmol−1) pre-exponential constant k0 (min−1) density of liquid in reactor ρ (kg m−3) heat transfer coefficient U (kJ m−2 min−1 K−1) heat transfer area A (m2)

1.883 × 10−2 294.44 294.44 8.01 1.36 3.14 6.978 × 104 8.314 −6.978 × 104 1.18 × 109 800.95 51.105 23.23

kmol m−3, Ts = 333.33 K, and TJ = 330.33 K. In all the simulation runs, the process was initially assumed to be at the steady state. For operation safety and stability, the reactor temperature T should be controlled in a suitable way to prevent the possible runaway, so the reactor temperature T is treated as the critical state variable, and the jacket temperature TJ is treated as the manipulated variable. In fact, a control system with PID controller embedded in the closed-loop is considered to make the control action more suitable, where the PID controller parameters are tuned by the NCD Package. The NCD package is a blockset which can be used for tuning PID controller parameters in MATLAB/SIMULINK, and the user guideline of NCD blockset can be easily found at the relevant public Web site. The CSTR system model is linearized by Taylor series spreader formula, and eq 15 can be changed as the increment equation:

Figure 4. CSTR system in Case A.

dC V A = F(CA0 − CA ) − VkCA dt

(16)

VρCp

d(ΔT ) = CAsV ( −HR )Δk − UA(ΔT − ΔTJ) dt − FρCpΔT

(17)

where Δk =

(14)

∂k ∂T

ΔT = K r ΔT = CAs

⎛ E ⎞ E k exp⎜ − ⎟ΔT 2 0 RTs ⎝ RTs ⎠ (18)

dT = kCAV ( −HR ) − UA(T − TJ ) − FρCp(T − T0) dt

Finally eq 17 can be changed as the time domain equation form using Laplace transform which is given as

(15) 14726



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(UA + FρCp − CAV ( − HR )K r )T (s) + VρCpsT (s) = UATJ(s)

(19)

So the control relation between TJ(s) and T(s) can be given as a transform function: T (s ) UA = G (s ) = TJ(s) (UA + FρCp − CAV ( −HR )K r ) + VρCps =

θ1 θ2s + 1

(20)

UA (UA + FρCp − CAV ( −HR )K r )

(21)

where θ1 =

θ2 =

VρCp (UA + FρCp − CAV ( −HR )K r )

(22)

In this practical process, the coefficient of heat transfer varies due to the fouling in the tube wall of the heat exchanger with the running time, so the coefficient of heat transfer U is the uncertainty parameter of the system. If the value of uncertainty U changed, θ1 and θ2 will also change according to eq 21 and eq 22. Define y(s) = T(s) and u(s) = TJ(s), and the CSTR system model can be written as y(s) = G(s)u(s) =

θ1 u(s) θ2s + 1

(23)

In this CSTR system, because the uncertainty parameter is affected by the fouling in the tube wall of the heat exchanger, the uncertainty parameter must satisfy the process constraint of 0 ≤ U ≤ UN = 51.105 kJ m−2 min−1 K−1, and the critical point that limits flexibility lies on the decreased direction of the uncertainty U. The nominal profile and expected deviations of the uncertain parameter are UN = 51.105 kJ m−2 min−1 K−1, ΔU− = 30%UN. So the expected feasible range of the uncertainty is U ∈ [0.7UN, UN] kJ m−2 min−1 K−1. To evaluate the stability and output tracking capabilities of the CSTR system, a 5K change in the input value of jacket temperature TJ is imposed at time t = 0 s, and the output value of reactor temperature T is expected to track the temperature change (ΔT = 5K) quickly. The CSTR system is subject to the following constraints: t ∈ [0, 5], y ∈ [−0.01, 5.8]; t ∈ [5, 15], y ∈ [4.4, 5.8]; and t ∈ [15, 50], y ∈ [4.9, 5.1]. So far, Step 1 and Step 2 of the proposed framework are finished, and now Step 3 should be carried out to evaluate the delay sensitivity. To examine the dynamic flexibility test problem for the CSTR system without delay consideration, the system is simulated in MATLAB/ SIMULINK, where the PID controller is designed by the NCD Package. The step response results are shown in Figure 5 at the critical point U = 0.7UN, where the dashed line represents the process constraints and the solid curve denotes the system response output. The optimized PID controller parameters are (Kp = 1.7127; Ki = 0.22626; Kd = −0.1168) after tuning by NCD package based on the initial values (Kp = 1.1; Ki = 0.1; Kd = 0.01). As shown in Figure 5, when time delay is ignored, the system output can maintain within the constraint area completely in the whole time horizon after optimizing the control action. The simulation results also demonstrate that the dynamic flexibility of the CSTR system can be guaranteed

Figure 5. Process output and manipulated input profiles of the CSTR system without delay.

under the uncertainty U ∈ [0.7UN, UN] without time delay consideration. When heat transfer delay is considered in the CSTR system and the expected lag time is τ = 2 min the CSTR system model can be written as y(s) = G(s)u(s) =

θ1 e−2su(s) θ2s + 1

(24)

The simulation results with delay consideration for the CSTR system are shown in Figure 6 at the critical point U = 0.7UN, where the optimized PID controller parameters are Kp = 1.4434; Ki = 0.18401; and Kd = −0.01532 after tuning by NCD package based on the initial values Kp = 1.1; Ki = 0.1; and Kd = 0.01. As shown in Figure 6, the delay sensitivity index of the CSTR system is φ1 = S1/S ≈ 0.02 > 0. When time delay is considered, the system output cannot be maintained within the constraint area completely in the whole time horizon after optimizing the control action by the NCD package, where the initial PID controller parameters are the same as the situation without delay. The simulation results also demonstrate that the dynamic flexibility of the CSTR system cannot be guaranteed under the uncertainty U ∈ [0.7UN, UN] with time delay consideration (τ = 2 min) using the NCD package to optimize 14727



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controller parameters are optimized as Kp = 2.473; Ki = 0.618; and Kd = 2.473 at the critical point U = 0.7UN by using the Ziegler−Nichols tuning formula, and then the controller parameters are regulated by the NCD package. The step response results at the critical point U = 0.7UN are shown in Figure 7. The optimized PID controller parameters are Kp =

Figure 6. Process output and manipulated input profiles of the CSTR system using NCD packege to tune the PID controller parameters at uncertainty U = 0.7UN and delay τ = 2 min.

the control action. Now, Step 4 should be carried out to improve the control action by using a new control strategy with then returning back to Step 3 to test the delay sensitivity as the second round. For ensuring that the control action has been optimized properly, the Ziegler−Nichols method is combined with the NCD Package to optimize the control action. Equation 24 demonstrates that the CSTR system is a first-order lag plus delay (FOLPD) model, and the Ziegler−Nichols tuning formula is well-known and used a lot as a feasible and effective method for FOLPD models. The details about the Ziegler− Nichols tuning formula are shown in Table 2. The PID

Figure 7. Process output and manipulated input profiles of the CSTR system using Ziegler−Nichols method combined with NCD package to tune the PID controller parameters at uncertainty U = 0.7UN and delay τ = 2 min.

2.5293; Ki = 0.19053; and Kd = 2.3405 after tuning by the NCD package based on the initial values Kp = 2.473; Ki = 0.618; and Kd = 2.473. As shown in Figure 7, the delay sensitivity index of the CSTR system is φ2 = S1/S ≈ 0.01 > 0, and φ2 is lower than φ1, so the Ziegler−Nichols method embedded in the NCD package shows some advantage. But still the system output cannot be maintained within the constraint area completely in the whole time horizon, which means after trying to enhance the control for the CSTR system by combining the Ziegler− Nichols method with the NCD package, the dynamic flexibility of the CSTR system cannot be guaranteed under the uncertainty U ∈ [0.7UN, UN] with time delay consideration (τ = 2 min). Now, Step 5 should be carried out, and the

Table 2. Ziegler−Nichols Tuning Formula system model

G(s) =

θ1 e−Ls θ2s + 1

controller type

controller parameters (from step response)

PID

Kp = 1.2/a Ti = 2L

Lθ a= 1 θ2

Td = L/2

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flexibility range must be rectified to ensure the operation safety and stability. The flexibility correction step is based on the golden section method. After the searching, the solution of problem DFI2 is δmax = 0.77, and the feasible uncertainty range is U ∈ [0.77UN, UN]. The step response results at the critical point U = 0.77UN are shown in Figure 8 and Figure 9. The

Figure 9. Process output and manipulated input profiles of the CSTR system using Ziegler−Nichols method combined with NCD package to tune the PID controller parameters at uncertainty U = 0.77UN and delay τ = 2 min.

shows a strong regulation capability, and the solution of the flexibility index problem is the same; the main reason is the initial PID controller parameters can affect the regulation result by the NCD package, and the initial values (Kp = 1.1; Ki = 0.1; Kd = 0.01) may be a good choice for the situation with delay consideration in this CSTR system. The simulation results also demonstrate that the dynamic flexibility of the CSTR system with control consideration can be guaranteed under the uncertainty U ∈ [0.77UN, UN] and time delay τ = 2 min. So the delay sensitivity analysis of the CSTR system with dynamic flexibility and control consideration is solved effectively by the proposed strategy/framework. In summary, The CSTR system modeling and original feasible uncertainty range are determined follow Step 1 and Step 2 of the proposed framework. The system constraints can be satisfied completely when time delay is ignored, but the system constraints cannot be satisfied completely under the expected uncertainty range involving time delay if the same control strategy is used as the situation without time delay (delay sensitivity assessed directly in Step 3). So Step 4 should

Figure 8. Process output and manipulated input profiles of the CSTR system using NCD package to tune the PID controller parameters at uncertainty U = 0.77UN and delay τ = 2 min.

optimized PID controller parameters of Figure 8 are Kp = 1.4113; Ki = 0.20341; and Kd = 0.53475 after tuning by the NCD package based on the initial values (Kp = 1.1; Ki = 0.1; Kd = 0.01). And the optimized PID controller parameters of Figure 9 are Kp = 2.2564; Ki = 0.20004; and Kd = 2.3982 after tuning by Ziegler−Nichols method combined with NCD package, where the initial values are Kp = 2.245; Ki = 0.561; and Kd = 2.245 based on the Ziegler−Nichols formula. As shown in Figure 8 and Figure 9, when flexibility is rectified, the system output can be maintained within the constraint area completely in the whole time horizon with delay consideration. In this CSTR case, although the PID controller parameters are optimized based on different initial values, the NCD package 14729



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−HR (1 − λ)F dT F λF Tf − T + T+ kCA = dt V V V ρCp

be carried out by using a proper control strategy (combined Ziegler−Nichols method with the Nonlinear Control Design (NCD) Package) with delay consideration to optimize the control action. And then go back to Step 3 to assess the flexibility test problem (DFT2) with control consideration; the efficacy of the new control strategy is tested in this part. At last, the system still cannot satisfy the constraints completely, and the flexibility should be corrected to find the maximum feasible uncertainty range (DFI2) which is carried out in Step 5. The whole analyses above are following the steps in the proposed framework. Case B: Reactor−Separator System with Recycle Loop. Consider the process shown in Figure 10, which

−

UA (T − TJ) VρCp

(26)

Equation 26 can be reduced as −HR dT λF λF UA = Tf − T+ kCA − (T − TJ) dt V V ρCp VρCp (27)

The system parameters are listed in Table 3. For these values the corresponding steady state is CAs = 0.5 mol l−1, CAfs = 0.709 Table 3. Parameters of Reactor−Separator System parameters

value

Figure 10. Reactor−separator system with recycle in case B.

consists of a reactor and a separator. This process was studied by Antoniades and Christofides30 as a typical reactor system focusing on the feedback control analysis. An irreversible reaction of the form A → B takes place in the reactor, where A is the reactant species and B is the product species. The reaction is exothermic, and a cooling jacket is used to remove heat from the reactor. The reaction rate is assumed to be of first-order and is given by r = kCA = k0 exp(−E/RT)CA, where k0 and E denote the pre-exponential constant and activation energy of the reaction and T and CA denote the temperature and concentration of species A in the reactor. The outlet of the reactor is fed to a separator where the unreacted species A is separated from the product B. The unreacted amount of species A is fed back to the reactor through a recycle loop, which allows increasing the overall conversion of the reaction and minimizing reactant wastes. According to the proposed framework, first the system modeling should be carried out in the Step 1, and the original expected feasible uncertainty range should be determined in Step 2. The inlet stream to the reactor consists of a fresh feed of pure A, at flow-rate λF, concentration CAf and temperature Tf, and of the recycle stream at flow rate (1−λ)F, concentration CA(t), and temperature T(t), where F is the total reactor flow rate, λ is the recirculation coefficient, and TJ denotes the jacket temperature. Under the assumptions of constant volume of the reacting liquid, V, constant density, ρ, heat capacity, Cp, of the reacting liquid, and negligible heat losses, a process dynamic model can be derived from mass and energy balances and consists of the following two nonlinear differential difference equations: dCA (1 − λ)F λF F CA − kCA = CAf − CA + dt V V V

−1

total reactor flow rate F (m s ) recirculation coefficient λ fresh feed temperature Tf (K) inlet coolant temperature TJ0 (K) fresh feed concentration CAf (kmol m−3) volume of liquid in reactor V (m3) heat capacity Cp (kJ kg−1 K−1) activation energy E (kJ kmol−1) gas constant R (kJ kmol−1 K−1) heat of reaction HR (kJ kmol−1) pre-exponential constant k0 (s−1) density of liquid in reactor ρ (kg m−3) heat transfer coefficient U (kJ m−2 s−1 K−1) heat transfer area A (m2) 3

1.3333 × 10−2 0.25 300 295 0.709 0.1 4.184 5.188 × 104 8.314 −2.092 × 105 1.0 × 107 1000 125.52 1

mol l−1, and Ts = 296.16 K. In all the simulation runs, the process was initially assumed to be at the steady state. For operation safety and stability, the reactor temperature T should be controlled in a suitable way to prevent the possible runaway, so the reactor temperature T is treated as the critical state variable, and the jacket temperature TJ is treated as the manipulated variable. In fact, a control system with PID controller embedded in the closed-loop is considered to make the control action more suitable, where the PID controller parameters are tuned by the NCD Package. The CSTR system model is linearized by Taylor series spreader formula, and eq 27 is changed as the following increment equation: −HR d(ΔT ) λF UA (ΔT − ΔTJ) CAsΔk − = − ΔT + dt V ρCp VρCp (28)

where Δk =

∂k ∂T

ΔT = K r ΔT = CAs

⎛ E ⎞ E k exp ⎜− ⎟ΔT 0 RTs 2 ⎝ RTs ⎠ (29)

Finally, eq 28 can changed to time domain equation form using Laplace transform which is given as −HR λF UA T (s ) sT (s) + T (s ) − CAsK rT (s) + ρCp V VρCp =

(25) 14730

UA ΔTJ(s) VρCp

(30)



Article

So the control relation between TJ(s) and T(s) can be given as a transform function: T (s ) = G (s ) TJ(s) =

UA (UA + λFρCp − CAsV ( −HR )K r ) + VρCps

=

θ1 θ2s + 1

(31)

where θ1 =

θ2 =

UA (UA + λFρCp − CAsV ( −HR )K r )

(32)

VρCp (UA + λFρCp − CAsV ( −HR )K r )

(33)

In the practical process, the coefficient of heat transfer varies due to the fouling in the tube wall of the heat exchanger with the running time, so the coefficient of heat transfer U is the uncertainty parameter of the system. If the value of uncertainty U changed, θ1 and θ2 will also change according to eq 32 and eq 33. Define y(s) = T(s) and u(s) = TJ(s), and the reactor− separator system model can be written as y(s) = G(s)u(s) =

θ1 u(s) θ2s + 1

(34)

In this reactor−separator system, because the uncertainty parameter is affected by the fouling in the tube wall of the heat exchanger, the uncertainty parameter must satisfy the process constraint of 0 ≤ U ≤ UN = 125.52 kJ m−2 s−1 K−1, and the critical point that limits flexibility lies on the decreased direction of uncertainty U. The nominal profile and expected deviations of the uncertain parameter are UN = 125.52 kJ m−2 s−1 k−1 and ΔU− = 41.84 kJ m−2 s−1 K−1. So the expected feasible range of the uncertainty is U ∈ [83.68, 125.52] kJ m−2 s−1 K−1. To evaluate the stability and output tracking capabilities of the reactor−separator system, a 5K change in the input value of jacket temperature TJ is imposed at time t = 0 s, and the output value of reactor temperature T is expected to track the temperature change (ΔT = 5K) quickly, so if the reactor temperature suddenly increases, it is hoped that the rector temperature can be pulled back to the safe state by controlling the jacket temperature. The reactor−separator system is subject to the following constraints: t ∈ [0, 5], y ∈ [−0.01, 5.8]; t ∈ [5, 15], y ∈ [4.4, 5.8]; and t ∈ [15, 50], y ∈ [4.9, 5.1]. So far, Step 1 and Step 2 of the proposed framework are finished, and Step 3 should be carried out to evaluate the delay sensitivity. To examine the dynamic flexibility test problem for the reactor− separator system without delay consideration, the system is simulated in MATLAB/SIMULINK, where the PID controller is designed by the NCD Package. The step response results are shown in Figure 11 at the critical point U = 83.68 kJ m−2 s−1 K−1, where the dashed line represents the process constraints and the solid curve denotes the system response output. The optimized PID controller parameters are Kp = 0.1; Ki = 0.2; and Kd = 1.1 after tuning by NCD package based on the initial values (Kp = 0.1; Ki = 0.2; and Kd = 1.1). As shown in Figure 11, when time delay is ignored, the system output can be maintained within the constraint area completely in the whole time horizon after optimizing the control action. The

Figure 11. Process output and manipulated input profiles of the reactor−separator system without delay.

simulation results also demonstrate that the dynamic flexibility of the reactor−separator system can be guaranteed under the uncertainty U ∈ [83.68, 125.52] kJ m−2 s−1 K−1 without time delay consideration. When heat transfer delay is considered in the reactor− separator system and the expected lag time is τ = 10 s, the reactor−separator system model can be written as y(s) = G(s)u(s) =

θ1 e−10su(s) θ2s + 1

(35)

Equation 35 is also a first-order lag plus delay (FOLPD) model. For testing the efficiency of Ziegler−Nichols method combined with the NCD package, the simulation results with delay consideration for the reactor system are shown in Figure 12 and Figure 13 at the critical point U = 83.68 kJ m−2 s−1 K−1, where the optimized PID controller parameters in Figure 12 are Kp = 0.104; Ki = 0.201; and Kd = 1.1 after tuning by NCD package based on the initial values (Kp = 0.1; Ki = 0.2; and Kd = 1.1), and the optimized PID controller parameters in Figure 13 are Kp = 0.68916; Ki = 0.0680; and Kd = 2.8875 after tuning by NCD package based on the initial values (Kp = 0.6; Ki = 14731



Article

Figure 13. Process output and manipulated input profiles of the reactor−separator system using Ziegler−Nichols method combined with NCD to tune the PID controller parameters at uncertainty U = 83.68 kJ m−2 s−1 K−1 and delay τ = 10 s.

Figure 12. Process output and manipulated input profiles of the reactor−separator system using the NCD package to tune the PID controller parameters at uncertainty U = 83.68 kJ m−2 s−1 K−1 and delay τ = 10 s.

0.03;and Kd = 3.0) which are calculated by the Ziegler−Nichols formula. As shown in Figure 12, the delay sensitivity index of the reactor−separator system is φ3 = S1/S ≈ 2.33 > 0. When time delay is considered, the system output cannot be maintained within the constraint area completely in the whole time horizon after optimizing the control action by the NCD package. But as shown in Figure 13, the delay sensitivity index of the reactor−separator system is φ4 = S1/S = 0. When time delay is considered, the system output can be maintained within the constraint area completely in the whole time horizon after optimizing the control action by Ziegler−Nichols method combined with NCD package. In this reactor−separator system, initial controller parameters shows great impact on the reactor−separator system. The initial values (Kp = 0.1; Ki = 0.2; and Kd = 1.1) are a bad choice for the situation with delay consideration, and the Ziegler−Nichols method combined with NCD package shows great advantage and efficiency. The simulation results also demonstrate that the dynamic flexibility of the reactor−separator system can be guaranteed under the uncertainty U ∈ [83.68, 125.52] kJ m−2 s−1 K−1 with time delay

consideration (τ = 10 s) using Ziegler−Nichols combined with NCD package to optimize the control action. So the delay sensitivity analysis of the reactor−separator system with dynamic flexibility and control consideration are solved effectively by the proposed strategy/framework. In summary, the reactor−separator system modeling and original feasible uncertainty range are determined to follow the Step 1 and Step 2 of the proposed framework. The system constraints can be satisfied completely when time delay is ignored, but the system constraints cannot be satisfied completely under the expected uncertainty range involving time delay if the same control strategy is used as the situation without time delay (delay sensitivity assessed directly in Step 3). So Step 4 should be carried out by using a proper control strategy (combined Ziegler−Nichols method with the Nonlinear Control Design (NCD) Package) with delay consideration to optimize the control action. And then go back Step 3 to assess the flexibility test problem (DFT2) with control consideration. The system can satisfy the constraints completely after control improvement, so there is no need to 14732



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go to Step 5. The whole analyses above are following the steps in the proposed framework. The above two reactor system cases are modeled as firstorder systems. However, in the case when the Ziegler−Nichols method is not suitable for calculating the initial value of the PID controller parameters for some higher order complex systems, other suitable methods could be found to combine with the NCD package to test if the control efficiency can be improved in Step 4 of the proposed framework.

5. CONCLUSIONS A delay sensitivity analysis framework for chemical reactor systems with control and flexibility consideration is proposed in this work. In the proposed framework, the Ziegler−Nichols method is combined with the Nonlinear Control Design (NCD) Package to optimize the control action. Flexibility of chemical reactor systems varies between situations with time delay and those without time delay. It is affected by the control strategy too. So it is very important to figure out the effect of delay to the flexibility of reactor systems with control consideration. All the results demonstrate that the proposed method may provide a powerful tool and framework for studying the delay sensitivity analysis of typical chemical reactor systems with control and flexibility consideration. It should be noticed that just single delay and uncertainty, especially the heat transfer delay and uncertainty, is considered in the case studies in this work. The reactor system simulation results will be different when the value of delay is changed, and the constraints of the reactor system might not be satisfied completely if the delay time is too large even when the uncertainty is at the optimal point. The simulation results will also be different if a more advanced/suitable controller can be found or adopted to the reactor systems. And there is still further work that should be done to make the finding more suitable for real chemical plants, especially when the plants are nonlinear systems. However, it would be interesting to investigate the further application of the proposed framework for delay sensitivity analysis of chemical reactor systems with control and flexibility consideration to deal with the above situations especially when more delays and uncertainties are considered.

■

■

G = transfer function g = vector of inequalities HR = the heat of reaction h = vector of equations L = lag time k = reaction rate constant R = gas constant t = time T = temperature u = process input x = vector of state variables y = process output z = vector of control variables ZL = minimum control ranges ZU = maximum control ranges δ = flexibility index variables θ = vector of uncertain parameters θL = lower bounds of uncertain parameter θN = nominal profile of uncertain parameter θU = upper bounds of uncertain parameter φ = delay sensitivity index τ = time delay

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AUTHOR INFORMATION

Corresponding Author

*Tel./Fax: +86-769-22861232. E-mail: [email protected]. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS This work was supported by the National Natural Science Foundation of China (21136003), the National Sci. Tech Supporting Program (2012BAK13B02), the Guangdong Province Team Project (S2011030001366), the research fund of The Guangdong Provincial Key Laboratory of Green Chemical Product Technology (GC201204) and the National Natural Science Foundation of China (51176036).

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NOMENCLATURE C = concentration d = vector of design variables DF(d) = dynamic flexibility index E = activation energy 14733



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