Detection of Safe Operating Regions - American Chemical Society

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Ind. Eng. Chem. Res. 2010, 49, 658–668

Detection of Safe Operating Regions: A Novel Dynamic Process Simulator Based Predictive Alarm Management Approach Tama´s Varga,* Ferenc Szeifert, and Ja´nos Abonyi Department of Process Engineering, UniVersity of Pannonia, P.O. Box 158, Veszpre´m 8201, Hungary

The operation of complex production processes is one of the most important research and development problems in process engineering. A safety instrumented system (SIS) performs specified functions to achieve or maintain a safe state of the process when unacceptable or dangerous process conditions are detected. A logic solver is required to receive the sensor input signal(s), to make appropriate decisions based on the nature of the signal(s), and to change its outputs according to user-defined logic. The change of the logic solver output(s) results in the final element(s) taking action on the process (e.g., closing a valve) to bring it back to a safe state. Alarm management is a powerful tool to support the operators’ work to control the process in safe operating regions and to detect process malfunctions. Predictive alarm management (PAM) systems should be able not only to detect a dangerous situation early enough, but also to give advice to process operators which safety action (or safety element(s)) must be applied. The aim of this paper is to develop a novel methodology to support the operators how to make necessary adjustments in operating variables at the proper time. The essential of the proposed methodology is the simulation of the effect of safety elements over a prediction horizon. Since different manipulations have different time demand to avoid the evolution of the unsafe situation (safety time), the process operators should know which safety action(s) should be taken at a given time. For this purpose a method for model based predictive stability analysis has been worked out based on Lyapunov’s stability analysis of simulated state trajectories. The proposed algorithm can be applied to explore the stable and unstable operating regimes of a process (set of safe states), information that can be used for PAM. The developed methodology has been applied to two industrial benchmark problems related to the thermal runaway. 1. Introduction In the rush to exploit the advantages in computer-based automation, many companies in the process industry have overlooked one of the most important elements in their business value chainsthe plant operator. The plant operatorsthe forgotten knowledge workersis on the frontline of real-time operations, making decisions that directly impact plant safety, reliability, profitability, and ultimately shareholder value. Operators like other knowledge workers analyze information, diagnose situations, predict outcomes, and take action to deliver value. Optimal operating conditions of production processes are getting closer to physical constraints. Therefore, the development of knowledge based expert systems is more and more important to support operators for supporting operators to keep operation conditions in this narrow range. Beside this requirement, it is necessary that an expert system is able to detect failures, discover the sources of faults, and forecast the false operations to prevent from development of production breakdowns.1-3 A process alarm is a mechanism to inform the operator about the development of an abnormal process condition for which an operator action is required. The operator is alerted in order to prevent or mitigate process abnormalities and equipment malfunctions. A poorly functioning alarm system is often noted as a contributing factor to the seriousness of upsets, incidents, and major accidents. Significant alarm system improvement is needed in most industries that utilizes computer based distributed control systems; it is a massively common and serious problem. Most companies have realized that they need to thoroughly investigate and understand their alarm system performance. The safe state is a state of the process operation where the hazardous event cannot occur. The set of safe states defines safe * To whom correspondence should be addressed. Tel.: +36 88 624447. Fax: +36 88 624171. E-mail: [email protected].

operating regions. A logic solver is required to receive the sensor input signal(s), to make appropriate decisions based on the nature of the signal(s), and to change its outputs according to user-defined logic. Beside the change of the logic solver output(s) result(s) in the final element(s) taking action on the process (e.g., closing a valve) to bring (back) it to a safe state. Defense against a possible abnormal situation can have one or more independent protection layers. To analyze the number of protection layers the method called layer of protection analysis (LOPA) can be applied.6 The possible protection layers have a hierarchy in which the design of the process and the basic process control systems is the basic levels at the bottom. Hence, the design of a very reliable and controllable process is crucial. Hence, the development of tools to support the design of a reliable process is a really important topic. Alarm management can be applied to extract the necessary information about the crucial parts of a process for designing or improving the safety system of the process. A proper alarm management results in improved safety, reliability, and overprofitability of the process.4 Alarm management is a fast growing, high profile topic in the process industry. The BP Upstream Technology Group proposed a five-level scale using the following nomenclature: overloaded, reactive, stable, robust, and predictive levels.5 Technology that can achieve the predictive performance level is still experimental and “bleeding edge”. In ideal case, predictive performance will involve the following kinds of techniques: • Early fault detection: Early detection of process deviation from its normal operation or breakdown of a process device by monitoring a set of process variables. Deviations can be detected even when each process variable is within their operating and alarm limits.

10.1021/ie9005222  2010 American Chemical Society Published on Web 11/25/2009

Ind. Eng. Chem. Res., Vol. 49, No. 2, 2010

• Early fault diagnosis and advice: Specific actions are advised to the operator to avoid the upset from occurring. • Procedural automation: Automation of standard and emergency operating procedures associated with both normal plant transitions (e.g., startup, shutdown, product change, and so forth) as well as critical corrective actions to recurring disturbances. (Procedural automation allows both online sequence execution and monitoring of steps in a procedure as well as collaboration between field and control room board operators.) • Extensive uses of operator support systems involving pattern recognition, adaptive graphics, artificial intelligence, and other new experimental methodologies. There are many other tools which can be used to analyze the process and to determine the reason of the development of unsafe situations and the possible ways to avoid them. A safetybarrier diagram (SBD) can be applied to summarize and show how safety barriers prevent the propagation of initiating events or conditions into accidents.7 The diagram presents all the possible scenarios to develop the accident. Each scenario is a sort of barrier sequence, and if a safety barrier is successful, the scenario stops at that barrier; if it fails, the diagram shows the next barrier until the accident occurs when all barriers have failed. Event tree (ET) and fault tree (FT) analyses can also be applied, but the results of these techniques are a bit more complicated to understand since the generated trees are built from more elements than SBDs.8 Applying ET and FT, a cause-consequence diagram can be developed which is similar to SBD but is applicable to considering some dynamic aspects of safety analysis.9 All these tools have limitations since additional knowledge is needed to characterize the boundary of safe operation and to determine the safety actions for every scenario. Predictive alarm management (PAM) is an important tool since the operator’s ability to respond to an alarm in a timely fashion determines the degree of success in preventing loss. The consequences of an uncorrected alarm generally worsens with the passage of time. During an abnormal condition, the board operator is confronted with making decisions on numerous tasks that must be performed in an appropriate sequence. The timing and the order of executing these tasks determine the outcome of the operator’s effort. For example, if two process variables are deviating from normal values and can potentially cause the same significant loss, the operator must quickly decide which variable to address first. In such a case, the operator must take an action to address the variable that is more volatile or can reach the point of loss in the shortest time. Therefore, the shorter the time available to respond, the higher the priority of the alarm will be, assuming that equal consequences can be caused. PAM systems should be able not only to detect a dangerous situation early enough, but also to give advice to process operators as to which safety action (or safety element(s)) must be applied. Hazardous operation cannot usually be avoided with any of the possible manipulations within the time of detection. In the case of a reactor runaway, there is a last controllable operating point at each manipulation before the reactor becomes unstable; see Figure 1. However, different safety actions have different time demands (safety time) before an emergency situation evolves, so process operators should know which manipulation to use and when it must be used. This process safety time means the period of time in which the process will move from a safe operating condition to a dangerous condition. To support process operators, it is not enough to detect failures and information should be given about necessary manipulations which must be applied to avoid any unsafe operation.

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Figure 1. Avoiding unsafe operation with a manipulation in the appropriate time.

To introduce the proposed methodology for PAM, a very important problem in process industry was chosen. This problem is referred to as reactor runaway. It means a sudden and significant change in process variables (such as the reactor temperature) that is a serious problem in many chemical industrial technologies like oxidation processes and polymerization. In case a highly exothermic reaction thermal runaway occurs when the reaction rate increases due to an increase in temperature, causing a further increase in temperature and hence a further increase in the reaction rate while the reactants are depleted. Since the correlation between the temperature and the reaction rate is exponential, the reaction rate is acceleratedly increased. The essential of the proposed methodology is the simulation of the effect of safety elements over a prediction horizon. Since different manipulations have different time demands to avoid the evolution of an unsafe situation (safety time), the process operators should know which safety action(s) should be taken at a given time. For this purpose, a method for model based predictive stability analysis has been worked out based on Lyapunov’s stability analysis of simulated state trajectories. The proposed algorithm can be applied to explore the stable and unstable operating regimes of a process (set of safe states), which information can be used for PAM. The developed methodology has been applied to two industrial benchmark problems related to the thermal runaway. The possible (thermal) runaway of reactors should be forecasted until the time point when the runaway can be avoided by the optimal control of the process. This task requires predictive stability analysis (for PAM). For this purpose in this work, a model based technique is worked out, where the Lyapunov’s indirect stability analysis of the state variables along simulated trajectories are used to detect the boundary of the controllable region of the process. In this paper, the term controllability means that at a given state of the process it is possible to find a future control trajectory to maintain safe operation and to avoid development of reactor runaway, i.e. a command signal is needed to keep the state variables of the system in the stable region. A not controllable state means that there is no safety action which is able to control the state. The boundary of controllability represents critical states of the reactor. When state-variables cross these boundaries during the operation, a process operator cannot do anything to avoid the development of reactor runaway. Hence, the aim of this work is to develop a tool to explore the safe regions of operations to keep process variables in controllable regions of the process. To introduce the proposed approach of controllability, a wellstirred fed-batch reactor and a heterocatalytic tube reactor are

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analyzed. The paper is organized as follows. After the introduction, the applicability of stability analysis in runaway detection is explained in section 2, Section 2.3 introduces the proposed algorithm which can be applied to determine boundaries of controllability region of complex process systems. Section 3 briefly describes reactors used in the case studies, while sections 3.1.2 and 3.2.2 summarize the obtained results. Finally, section 4 summarizes the most important conclusions. 2. Stability Analysis Based Forecast of Runaway Reactor runaway causes a serious problem in many chemical industrial technologies and could often occur where highly exothermic reaction(s) takes place. This phenomenon has contributed to the most serious industrial chemical accidents, most notably the 1984 explosion of a Union Carbide plant in Bhopal, India,10 that produced methyl isocyanate and the disaster in Seveso, Italy, in 1976 which gave rise to standardized industrial safety regulations.11 Thermal runaway is also a concern in hydrocracking and other oil refinery processes. Detection of runaway has two important aspects. On the one hand, runaway forecasting has a safety aspect since it is important for avoiding damaging construction materials or in the worst case the explosion of reactor. On the other hand, it has a technological aspect, since forecasting the runaway is important for avoiding the development of hot spots in a catalytic bed, which speed up the aging of catalyst, or it is important for decreasing the quantity of of byproducts, e.g. in the synthesis of 2-octanone from 2-octanol by oxidation, a huge amount of different carboxylic acids are formed when runaway occurs.12-14 2.1. Classical Approaches for Runaway Detection. Advanced process control (APC) systems should be able to forecast the effect of the variation in operating condition which can decrease production costs and increase the safety of production.15 Probably for this reason, process stability is given by two-thirds of the APC users as a main profit factor but only by less than 50% of the APC suppliers. Process stability ensures that the product meets customer specifications consistently and that operations run smoothly. The first step in the development of safety elements of APC that can be used to detect runaway is the generation of a reliable runaway criterion. Many detection techniques are introduced in literature which can be classified as data16-20 or model based analysis.21-24 However, these techniques can only be applied as a part of a support system to detect runaway becaues a more detailed analysis of the process is needed to inform the operator about the necessary manipulation. To apply a data based criterion, it is necessary to have some measured data that makes it impossible to apply a data based criterion to forecast the development of runaway. Other problems with data based methods are found in measurement conditions, e.g. measurement noise can result false detections. Model-based criteria require parametric sensitivity and/or stability analysis; therefore, it is necessary to have exact process model with correct model parameters for the application of these kinds of criteria. 2.2. Stability Analysis Based Runaway Detection. In advanced model based, techniques reactor runaway is detected through the stability analysis of the process model, e.g. by using Lyapunov’s indirect method.25 If all the solutions of a dynamical system which starts near a specific point stay close to that point, the specific point is Lyapunov stable. If the solutions are converging to the specific point, the point is called asymptotically stable.

Calculation of the Jacobian matrix of the process model is the first step in the application of Lyapunov’s indirect method to investigate stability. The Jacobian matrix is the matrix of all first-order partial derivatives of the differential model equations with respect to the state variables of the process. On the basis of the eigenvalues of the matrix, the stability of the process can be determined. When all of the real parts of the eigenvalues are negative, then the process is stable, but if one of the real parts of eigenvalues is over zero, then the model is unstable at the investigated operating point. Lyapunov’s stability analysis is suitable for the detection of the development of runaway,26 and the algorithm based on this stability analysis can separate the cases where runaway occurs in the reactor or not. Although it is applicable to detect when the runaway has occurred, which is interesting from veiwpoint of the analysis of historical process data, it cannot be used in online process monitoring and control, where the operator is interested in the region of safe operation and the last time instant when the runaway could have been avoided by the control system. 2.3. Concept of Predictive Stability Analysis. For the predictive analysis of the process, not only is a detailed (accurate) process model needed but a process simulator that is able to estimate trajectories of process variables in the case of normal and abnormal operations is also needed. This simulator should also be able to model the dynamic behavior of the control system including its safety elements. This knowledge is extremely important since these elements of a control system determine and sometimes “expand” the region of safe operation. An algorithm has been worked out to detect the boundary of stability and controllability of the process in the feature space of the most important process variables. Such knowledge is extremely useful since it can be interpreted as (multivariate) constraints defined on the process variables that can be given to process operators, built into the control system, or into an algorithm used for the optimization of the operation. As Figure 2 illustrates, the first step of the proposed framework is the application of the classical Lyapunov’s stability analysis. If the algorithm cannot find at least one unstable state during the operation of a prediction horizon, then the simulation stops. Otherwise, it finds the last controllable state. The algorithm tests all the possible safety actions and determines which of them can be applied to avoid the development of the detected unstable state. The developed process model is applied to simulate the effect of the determined safety elements and to check which elements can be applied to avoid unstable operation. To check the effect of safety actions in avoiding the development of abnormal situations, the algorithm takes one step back in simulation time since the aim of this analysis is to find the last controllable state. Effects of all possible safety actions are simulated by solving the process model in that simulation time. When the current state is stable, then the analyzed safety element is labeled as it can be used to avoid the development of runaway and that state is labeled as the last controllable state. Otherwise, the algorithm takes another step back in simulation time until it finds a stable state. To speed up the search for the last controllable state, the classical secant method for solving univariate equations is applied in the algorithm. The time difference between the first unstable and last controllable states is calculated in case of every safety element which seems to be applicable in avoiding the development of unstable states. On the basis of this time


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Figure 2. Detection of the last controllable state of reactor.

Figure 3. Trajectories of state variables during normal reactor operation.

difference, the safety elements are ranked, and the best safety element, which requires the least time to avoid the development of runaway, is chosen. Beside the determination of the time requirement of safety elements, the introduced algorithm can be applied to explore the stable and unstable operating regimes of a process. Hence

during the operation of the process, it can be used as a PAM system. In this work, the developed algorithm is applied to determine the boundary of stable operation regions. To achieve this aim, the developed algorithm should analyze the model of safety system by several randomly generated inlet and initial conditions.

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Figure 4. Trajectories of state variables when runaway occurs.

Figure 5. Time instance when stability and/or controllability of the process is lost under given conditions as a function of the initial reactor volume and feeding volume of the reactant.

Figure 6. Stability regions in the function of initial reactor volume and the feeding volume of the reactant.

The result of the analysis is two hypersurfaces in state space which represent the boundaries of stability and controllability.

Figure 7. Controllability regions in the function of initial reactor volume and the feeding volume of the reactant.

After the characterization of unstable operating regimes, the results can be applied to design the inlet conditions of the reactor. These results are useful also during the operation because they can be applied to detect the development of runaway. In case the measured state variables are crossing the boundary of controllability, the first unstable state occurs soon and the operator must be warned to make some modifications in operating conditions. After crossing the boundary of stability, the reactor becomes unstable at that moment and the runaway cannot be avoided. Let us see the steps that are needed to apply the proposed tool in a PAM system. First all the knowledge about the analyzed system must be collected and arranged in a model. Applying classical alarm management techniques, the possible safety actions must be determined. To analyze the effect of possible manipulations, a review of the safety system is needed. In the next chapter, two applications of the developed algorithm are described. In both examples the algorithm was applied to determine the safe operating regimes. In the first example a fed-batch reactor with an exothermic reaction system


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engineering. In ref 13, the control of this process has been analyzed and the safety issues related to its operation were discussed. From the viewpoint of this paper, the most relevant work is published in ref 14, where the hazard and operability analysis (HAZOP) of this process has been developed based on the application of process simulator. 3.1.1. Introduction to the Mathematical Model of the Fed-Batch Reactor. A mathematical model of the system has been worked out by van Woezik and Westerterp13 to predict the dynamic behavior of the reactor; the simplified version of the reaction scheme is applied as follows: A + B f P + 2B P+BfX Figure 8. Time difference between stability and controllability regions in the function of initial reactor volume and feeding volume of reactant.

Figure 9. Simplified scheme of the investigated tube reactor.

is analyzed while in the other example is heterocatalytic tube reactor also with an exothermic reaction. 3. Application of the Developed Algorithm in Industrial Problems To illustrate the applicability of the previously introduced approach in characterizing the boundaries of reactor controllability, the operation of a well-stirred fed-batch reactor and a heterocatalytic tube reactor with jacket cooling are analyzed. In the fed-batch reactor 2-octanone is produced from 2-octanol while in the other example a highly exothermic reaction takes place. Both studied reaction systems apply some catalysts. In the first example, runaway causes a significant decrease in the amount of the produced 2-octanol while in the other reactor the runaway must be avoided to reduce the speed of the catalyst aging. 3.1. Safe Operation of a Fed-Batch Reactor for the Production of 2-Octanone. This process can be considered as an advanced benchmark problem in the field of process

where A is 2-octanol, B is the nitrosonium ion, P is 2-octanone, and X represents all the byproducts. In the following part, just a brief introduction will be given about this model; a very detailed description can be found in refs 12 and 13. The main product, 2-octanone is produced in a two-phase reaction system in a fed-batch reactor from 2-octanol. 2-Octanol is continuously fed into the organic phase, which does not exist before the beginning of feeding. Hence, the organic phase is the dispersed one while the aqueous nitric acid phase where all the reaction steps take place is the continuous phase. Two phases are coupled by mass and heat transfer processes. On the basis of Castellan et al.’s work27 which shows that the oxidation processes in room temperature mostly take place according to an ionic mechanism, van Woezik and Westerterp13 determined a reaction kinetic describing the method of oxidation of 2-octanol to form 2-octanone in nitric acid, which is proved to be feasible. In the first step of this reaction mechanism, the nitrosonium ion forms. This reaction has a very long induction time, which can be shortened by adding a small amount of initiator like NaNO2 to generate the necessary nitrous acid faster. The nitrosonium ion forms only in the aqueous phase. It is followed by the oxidation of 2-octanol forming 2-octanone and the further oxidation producing different carboxylic acids. The mathematical description of the fed-batch reactor can be found in the Appendix, while the numerical description of the applied stability analysis of the model can be found at the following Web site: http://www.fmt.vein.hu/softcomp/. 3.1.2. Exploring the Unstable Regions in a Fed-Batch Reactor Example. To illustrate the complex dynamic behavior of this process, the following simple simulation study has been performed. Trajectories of state variables in Figure 3 show the normal operation of the reactor when 2-octanone is the main product and the total quantity of byproducts is low. A small, 3 K variation in the inlet temperature of the jacket results in a reactor runaway (see Figure 4). In this case, the byproducts are generated mainly during the operation while the conversion of 2-octanol significantly decreases at the end of the operation. In Figures 3 and 4, the gray areas represent the instability region of the process based on Lyapunov’s stability analysis. These areas have been determined by using Lyapunov indirect stability analysis. It can be seen that in a small time interval the reactor becomes unstable in the case of normal reactor operation. This region does not represent thermal instability, which is a synonym of reactor runaway, but it can be originated from the autocatalytic reaction scheme. In order to present the difference between the stability and the proposed controllability region, a detailed simulation experiment has been performed where the initial volume of the aqueous and the organic phase are varied at a given constant inlet flow rate. At these scenarios, the stability of the process is

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Figure 10. Model structure of the tube reactor.

Figure 11. Simulated temperature profiles in the gas phase (a) and the result of stability analysis of the reactor model (b). Table 1. Simulated Temperature Profiles in Gas Phase

original modified

cA [mol/m3]

cB [mol/m3]

pG [bar]

TG [K]

TW [K]

52.8 0

18.1 0

1.80 1.25

320 290

320 290

analyzed and the time instant when the process became unstable was determined. When the process became unstable (circles), the last time-instance when the runaway could have been avoided (dots) was also determined by using the proposed algorithm. Figure 5 shows the result of this analysis. As can be seen in some cases, there is a significant difference among the time instance when the process loses its controllability and the time instance when it becomes unstable. This difference illustrates the necessity of controllability analysis given in this paper. If the trajectory of state variables crosses this boundary, then the reactor cannot be controlled anymore and reactor runaway develops soon. To illustrate the effect of the feeding profile on stability and controllability of the reactor, the flow rate of the reactant feeding has been varied on the range of 100-200 dm3/min, while the initial reactor volume has also been varied in the range of

1.2-1.8 m3. In Figure 5, the time instances when the process became unstable and uncontrollable are not shown, but a marker is given when these phenomena occurred or not. The effect of the cooling has been analyzed by varying the feeding of the coolant media into the jacket. The stability and controllability regions of the reactor have been explored by the proposed algorithm in the case of three different flowrates 100 (dots), 150 (circles), and 200 (crosses) dm3/s. The result of the analysis is shown in Figures 6 and 7. The differences between the regions of stability and controllability are plotted in the case of every investigated condition in Figure 8. As it can be seen in this figure, by increasing the inlet flow rate of the cooling media the reactor becomes instable later but the difference between the stability and controllability region is getting wider. Figure 8 is extremely important because it gives an insight to the process engineer and to the operator as to what kind of combinations of the initial conditions and feeding profile are allowed and what is the final time instant when the reaction must be stopped by the auxiliary functions of the control system. As it can be seen in Figure 8, a wide region of process variables


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Table 2. Results of Controllability Analysis

exists where the safety time is significantly increased. Hence, it is important to determine the boundary of controllability in a fed-batch reactor with a highly exothermic reaction system. On the basis of the proposed methodology for PAM, the safe operating regimes are characterized in a fed-batch reactor with a highly exothermic reaction system. The obtained results show that in some operating regimes the time necessity of the investigated safety action is more than the sample time. Hence, the proposed tool can be applied to detect the development of undesired states and support the process operator to keep the process from unstable situations. To check the developed methodology in the case of more complex systems, a tube reactor with fixed bed is analyzed as well. In the second example, a set of control actions are analyzed and rated. 3.2. Highly Exothermic Reaction in a Tube Reactor. The second example relates to the analysis of a tube reactor with a highly exothermic reaction. The studied vertically built up reactor contains a great number of tubes with catalyst as it is shown in Figure 9. The highly exothermic reaction occurs as the reactants rise up in tubes and pass through the fixed bed of catalyst particles. The generated heat is removed by the cooling media circulated around tubes in the reactor jacket. Only properties of inlet and outlet streams are measured continuously to monitor the reactor and check the safety of operation. Steadystate temperature profiles are collected every 3-6 months to check the activity of catalyst bed. To track the development of stability boundaries, a detailed dynamic model has been developed. The developed model of the reactor can be found in the Appendix, while the numerical description of the applied stability analysis of the model can be found at the Web site mentioned above. 3.2.1. Introduction to the Mathematical Model of the Tube Reactor. A one-dimensional two-phase dynamic model has been worked out to simulate the operation of investigated

reactor. The complexity of the model allows observation of how the boundaries of stability are developing in time along the catalyst bed. Dynamic behavior of tube reactor can be obtained by solving the mass and energy balances. The developed model calculates all the transport processes between each of the connected phases. The structure of the dynamic model can be seen in Figure 10 with notation of the connection at each hierarchy level. The dynamic model contains partial differential equations (PDEs) besides the ordinary and scalar expressions. All variables in PDEs correspond to the time and the position in the catalyst bed along the symmetric axis. In the model mass and heat transport between the solid and the gas phase and the chemical reaction in the solid phase are considered. Mathematical simulations were achieved by applying the finite differences method to solve model equations. An explanation of the model parameters is given in the Nomenclature, while a very detailed introduction of model equations can be found in the Appendix. 3.2.2. Exploring the Unstable Regions. The aim of this section is to investigate the tube reactor with a highly exothermic reaction when manipulations must be performed to keep the reactor in the controllable region. The feeding rate of the reagents is modified to generate different operating conditions. The algorithm tests all the considered safety elements in case of six different operating regimes. For this purpose, an algorithm has been developed based on the simulator of the reactor and Lyapunov’s indirect stability analysis. Stability analysis of the process model is used to detect unstable operating points. The algorithm has already been introduced in section 2.3. Development of temperature profiles in the case of reactor runaway can be seen in Figure 11a. As can be seen, the temperature maximum is moving in the opposite direction of the reagent flow. The result of the stability analysis can be seen in Figure 11b. The boundary of the unstable region is moving as it is expected, and it is moving toward to inlet spot. As can be seen in this

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Figure 12. Avoiding the development of reactor runaway.

figure, the first unstable operating point is detected at 128 s, so the runaway of the reactor occurs in this operation and there is need for a safety action to prevent this situation. The proposed algorithm can be applied to determine the boundary of controllability of reactor at a given safety action. In this case study, five possible safety elements are considered and summarized in Table 1. In the first scenario, two of the possible five safety elements cannot be used to avoid the development of reactor runaway. The case when the applied safety element is not working is represented with an infinity sign in the last row in every scenario in Table 2. The concentration of reagent B is the perfect choice to prevent runaway in this region because 1 s is enough before the detection to decrease the concentration to 0 to keep the reactor in the controllable state. Further analysis of results shows that this safety element does not work in all of the investigated regimes (see scenarios III and V). There is one operating regime (scenario III) in which none of the investigated safety elements are capable of immediately avoiding runaway. The results highlighted that the concentration of reagent A and the inlet temperature of cooling media can be applied in all operating regimes to prevent the development of runaway. To validate the reliability of the results, the offered safety element has been tested by a tailored simulation experiment; see Figure 12. It is well demonstrated that the applied safety elements can bee applied to avoid runaway and to keep the reactor in safe operation regime, assuming that they are performed in proper time.

4. Conclusions and Future Work The operation of complex production processes is one of the most important research and development problems in process engineering. The identification of possible hazards by any kind of process hazard analysis techniques requires detailed knowledge about the technology. Unfortunately, at the application of these techniques, the time aspect of the operation and the dynamic behavior of the process are usually neglected. This paper showed that the possible (thermal) runaway of these reactors should be forecasted to avoid uncontrollability of the process. For this purpose, model based predictive stability analysis method has been worked out, where Lyapunov’s stability analysis is applied to detect the boundary of the controllable region of the analyzed processes. The methodology is based on the dynamical analysis of the system and the possible safety actions. On the basis of the proposed methodology, an algorithm has been worked out to detect the development of unsafe situations and to determine the necessary safety action which can be applied to avoid them. The algorithm has been applied to explore the stable and unstable operating regimes of two industrial benchmark problems with highly exothermic reactions. The characterized operating regions can easily be interpreted by the operators and process engineers and can be implemented in realtime process monitoring and/or online process optimization algorithms. The results of the introduced method for the analysis of process controllability can be evaluated with decision trees (DTs) and a novel tool can be developed, which can be applied to alarm the process operators when state variables of the process


leave the controllable region. Therefore, further simulation experiments should be constructed and evaluated to generate the necessary learning samples for DT induction.

• To calculate pressure drop along the packed bed, a modified Ergun equation is applied. VG

Acknowledgment The financial support from the TAMOP-4.2.2-08/1/2008-0018 (Livable environment and healthier people Bioinnovation and Green Technology research at the University of Pannonia) project is gratefully acknowledged.

VGFGcGp

∂(BGcGi ) ∂cGi + ) υiVSrG ∂t ∂x

Appendix

∂TW ∂TW + BWFWcW ) AGWjQGW p ∂t ∂x

(13)

;TG ) TG,in cGi ) cG,in i

(14)

x ) 0, Rates of reactions are calculated with the following equations: r1 ) (1 - εorg)keff,1mAcAorgcaq B r2 ) (1 - ε

org

keff,i )

(

exp -

EA,eff,i RTR

- mH0,eff,iH0

( )

(2)

where effective reaction rate constants depend on the temperature and the acidity of the aqueous phase, i.e. 0 keff,i

A modified Langmuir-Hinwhelwood reaction kinetics expression has been applied to calculate the overall rate of the process:

(1)

)keff,2mPcPorgcaq B

)

(3)

(11)

∂TG ∂TG + BGFGcGp ) VSrG(-∆Hr) - AGWjQGW ∂t ∂x (12)

VWFWcW p

Dynamic Model of a Fed-Batch Reactor.

667

rG ) k0 exp -

EA

(pAG)nA(pGB )nB -

pGC K(pAG)1-nA(pGB )1-nB

1 + (pGC )nC

RTG

(15) Pressure drop through the catalyst bed is calculated by the Ergun equation:

where the H0 represents the Hammett-acidity function. To describe the concentration trajectories during the operation, the following ordinary differential equations must be solved:

FG(BG)2 1 - ε 1-ε dpG ) -2fc 1.75 + 150 dx Re dpA2 ε3

d(VrcAorg) ) BR,incA,in - r1VR dt

(4)

d(Vrcaq B) ) VR(r1 - r2) dt

(5)

All the missing model parameters were identified comparing measured and calculated temperature profiles. The simulators have been performed in MATLAB and can be downloaded from http://fmt.uni-pannon.hu/softcomp/. Every operation and reactor parameter which characterizes the industrial reactor has been modified.

d(VrcPorg) ) VR(r1 - r2) dt

(6)

d(VrcXorg) ) VRr2 dt

(7)

d(VrcNaq) ) -VR(r1 + r2) (8) dt To describe the change in temperature of reactor and jacket, the following equations must be solved: dTR 1 ) (Qr + QRin - Qcool - Qloss) R dt HC

(9)

1 dTC ) (QCin + Qcool) dt HCC

(10)

Dynamic Model of a Tube Reactor. The elements inside the reactor, i.e. the catalyst bed and the flowing gas, are considered as a quasi-single phase, and a summarized reaction kinetic expression is applied to calculate the reaction rate. Assumptions which have been considered in this model are summarized as follows: • Only the gas phase is considered in tubes. • Reaction takes place in the gas phase. • To calculate the rate of reaction, Langmuir-Hinselwood kinetics is modified with a term which describes the reaction equilibrium. • The temperatures of the gas and solid phases are equal.

(

)

(16)

Nomenclature A ) cross section of tubes [m2] A- ) transport area [m2] B- ) volumetric velocity [m3 · s-1] ci- ) concentration, where i ) {A; B; C; X} [m3 · s-1] cp- ) heat capacity of the phase [J · mol-1 · K-1] dp ) mean diameter of catalyst particles [m] EA ) activation energy [J · mol-1] GS jM,i ) mass transfer flux density, where i ) {A; B; C} [mol · s-1 · m-2] jQ- ) heat transfer flux density [J · s-1 · m-2] k- ) reaction rate coefficient ) k0 exp(EA/RTG) [m3 · s-1 · mol-1] k0 ) pre-exponential factor [m3 · s-1 · mol-1] Kad,i ) equilibrium constant of the adsorption processes [m3 · mol-1] Kreac ) equilibrium constant of the reaction ) exp(16.14 - 102845/ RTG) [-] L ) length of reactor [m] p- ) pressure [Pa] r- ) reaction rate [mol · s-1 · m-3] R ) gas constant ) 8.314 [J · mol-1 · K-1] Re- ) Reynolds number ) BGFGd/AεGµG [-] t ) time [s] T- ) temperature [K] x ) position in the reactor [m] V- ) volume [m3] Greek Letters β- ) mass transfer coefficient [m4 · mol-1 · s-1] ∆ H- ) heat of mass transport processes and the reaction [J · mol-1]

668


ε- ) void fraction [-] µ- ) dynamic viscosity [Pa · s] νi ) stoichiometric coefficient, where i ) {A; B; C} [-] F- ) density [kg · m-3] Superscripts R ) reactor inside C ) reactor jacket org ) organic phase aq ) aqueous phase G ) gas phase GS ) transport from the gas to the solid phase GW ) transport from the gas to cooling media S ) solid phase W ) cooling media Subscripts eff ) effective ad ) adsorption cool ) heat flux between reactor inside and jacket in ) reactor inlet loss ) heat loss corr ) correction reacd ) reaction in the detailed process reacd ) reaction in the detailed process

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ReceiVed for reView April 1, 2009 ReVised manuscript receiVed October 26, 2009 Accepted November 10, 2009 IE9005222

Detection of Safe Operating Regions - American Chemical Society

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