Applications for Parametric Sensitivity Analysis

Parametric Sensitivity Analysis (PSA) has various applications in chemical engineering, such as model discrimination, optimisation, control system design, parameter estimation, model simplification, process sensitivity and multiplicity and experimental design (Takamatsu et al. 1970; Rabitz et al. 1983, Ungureanu 1989; Curteanu and Ungureanu 1995, Varma et al. 1999; Kelkar and Ng 1998, 2000). The implementation of PSA for chemical reactors was first identified by Bilous and Amundson (1956). They defined that a chemical reactor operates in the parametric sensitivity region when, for given small variations of some of the input parameters o f the reactor, one or more of the outputs undergo large variations.

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Historically, most of research on PSA for chemical reactors is related to exothermic reactions and deals with the magnitude of the temperature peak, or hot spot, which almost inevitably develops. Alder and Enig (1964) studied reaction runaway in thermal explosion theory and their work represents one o f the early developments in parametric sensitivity approach. Criteria were developed to predict the ignition temperature. Froment and co-workers (Van Welsenaere and Froment 1970, Hosten and Froment 1986) introduced criteria for runaway in fixed bed tubular reactors based on geometrical properties o f the temperature profile along the reactor to predict critical values for operation variables. They showed that if there is a positive second derivative of temperature with respect to reactor length before the temperature reaches its maximum, then the approach to maximum temperature is more sudden than when the second derivative is negative. Thus, avoiding positive second derivative could mean avoiding runaway. McGreavy and Adderley (1973) developed a similar criterion based on a heterogeneous model of a fixed bed reactor. The authors underlined the importance of intraphase resistances, especially for cases having the effectiveness factor greater than unity, when the criteria for the quasi-homogeneous models can be misleading. Morbidelli and Varma (1982) provided a necessary and sufficient condition for reactor runaway based on the method of isoclines. For all positive-order exothermic reactions, using the full Arrhenius temperature dependence o f the reaction rate, critical values of the heat of reaction and heat transfer parameters beyond which runaway was encountered were derived. The authors concluded that runaway is more likely as the reaction order decreases, the reaction activation energy increases, or as the inlet temperature o f the reaction mixture increases.

Although the geometry-based criteria give a fundamentally correct description of thermal runaway, they do not give any measure o f runaway intensity. For this purpose

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sensitivity-based criteria can be employed. The normalised objective sensitivity is defined as the scaled derivative of the maximum temperature with respect to a certain reactor inlet condition or physico-chemical parameter. Morbidelli and Varma (1988) defined criticality as the situation where the normalised objective sensitivity of the temperature maximum to any o f the physico-chemical parameters of the model is a maximum. This criterion was utilised to study a variety o f reacting systems. Morbidelli and Varma (1986, 1987) identified parametrically sensitive regions for heterogeneous plug flow reactors. Chemburkar et al. (1986) showed for non-adiabatic CSTRs that if operating conditions are chosen so that to avoid the possibility o f parametric sensitivity then steady-state multiplicity is automatically avoided. Tjahjadi et al. (1987) applied the same sensitivity criterion to tubular polymerisation reactors to find design constraints for various operating parameters. Morbidelli and Varma (1989) studied tubular reactors where multiple reactions take place and analysed the connection between thermal runaway and runaway of yield and selectivity. Wu et al. (1998) demonstrated that if one uses reactant conversion instead of reactor axial co-ordinate as independent variable to identify the critical conditions, more conservative runaway boundaries are predicted.

In an attempt to reduce the sensitivity o f a catalytic reactor where an exothermic reaction takes place van der Vaart and van der Vaart (1991, 1992), introduced an endothermie reaction in the system. Using various parametric sensitivity criteria they calculated the proportion of the two catalysts to provide insensitive reactor operation.

Apart from run-away criteria, the application of PSA in chemical reactor study is related to quantitatively prediction of the effect of parameter uncertainty on the reactor model. Priestley and Agnew (1975) analysed the catalytic hydrochlorination o f acetylene to

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concept by the statistical theory as the sum of squares o f temperature deviations from its nominal value due to perturbations in the transport and kinetic parameters. They calculated further the fractional contribution to the total variance in the system performance produced by a single parameter. The results o f sensitivity calculations showed that the contribution of the heat transfer parameters to total variance of temperature is 53% followed by the kinetic parameters by 46 %, while the fluid-particle heat and mass transfer contribution is low, namely 1%. Skilivaniotis et al. (1988) examined the sensitivity behaviour of a fixed-bed heat exchanger showing that during experimental measurements the contours of the sensitivity in axial and radial direction can be used as a guide to locate the sensors at near optimal location. Ungureanu et al. (1994) employed a quasi-homogeneous two-dimensional mathematical model for a reactor where the endothermie ethyl-benzene dehydrogenation takes place. Parameters affecting radial heat transfer such as reactor radius had the greater impact on reactor sensitivity. Quina et al. (1999) studied the partial oxidation of methanol to formaldehyde in a fixed bed reactor with two distinct zones: at the entrance the catalyst is diluted with inert, followed by a region with pure catalyst. It was found that the system was particularly sensitive to the wall temperature and almost insensitive to mass transfer parameters.