An improved scaling procedure for analysis and simplification of process models
Introduction
Scaling analysis is a systematic approach that can be used to identify phenomena occurring at various scales. This information can be used to simplify a given set of equations by neglecting phenomena, which occur at scales that are different from the scale of interest. In this approach, a given set of equations is made dimensionless, resulting in several dimensionless groups of varying magnitudes. These dimensionless groups represent the relative effects of phenomena or mechanisms and therefore help identify dominant phenomena/mechanisms in the scale of interest. A number of authors have used scaling analysis for model simplification and identification. For example, Dahl et al. (2004) have used scaling analysis to get insights into the behavior of fluid aerosol reactor without performing actual simulations. Kopaygorodsky et al. (2004) have used scaling analysis to identify key differences between the modeling assumptions for conventional pressure swing adsorption and ultra-rapid pressure swing adsorption. Kaisare et al. (2005) have used scaling analysis to identify phenomena occurring at varying scales in a reverse flow reactor. Balaji et al. (2008) have used scaling analysis for reverse flow reactor and have shown ways of simplifying the model equations. Rao et al. (2010) have used scaling analysis for pulsed pressure swing adsorber to identify useful correlations in terms of dimensionless numbers. Rezvanpour et al. (2012) have studied electro-hydrodynamic atomization process using scaling analysis to simply the model and to find a correlation relating efficiency with a single dimensionless number involving the parameters of the process. Baldea and Daoutidis (2007) have used scaling analysis for auto-thermal reactors to identify a non-stiff model by separating fast and slow time scales. Krantz (2007) have described the method of scaling analysis in a book for various transport and reaction process.
There are two important gaps in all these works that use scaling analysis. In the scaling methodology described by Krantz (2007), all dependent and independent variables in the equations are made dimensionless by choosing appropriate scale(s) and reference factor(s). This results in a minimum parametric representation of the model equations. Thus the solution of these equations can be expressed in terms of dimensionless groups. The form of these dimensionless groups and the methods used for obtaining scales usually involve trial and error methods. In Krantz et al. (2012), it is mentioned that one has to know the controlling mechanism while forming a unique dimensionless equation which varies in the order of 1. Identifying this controlling mechanism is not obvious and in most cases this usually involves a trial and error process. Further, in all these works on scaling analysis, the scaling of nonlinear terms in the equations is addressed in an empirical manner. The scale for nonlinear terms are usually taken to be some characteristic maximum (Balaji et al., 2008, Krantz, 2007), but obtaining this maximum is not obvious without simulating the corresponding equations. We address these gaps in the literature by: (i) proposing an approach that avoids the trial-and-error method for deriving scales, and (ii) we focus on nonlinear terms and suggest a systematic way to obtain appropriate scales for these terms. The proposed method for scaling analysis is general and straight-forward to apply to any given set of equations. The proposed method is described in several steps and explained though examples of varying complexity. We apply the techniques developed in this paper and calculate scale and reference values for a 1D model of water gas shift (WGS) reactor system, which involves complex nonlinear terms and differential algebraic equations. The obtained scale and reference values are shown to be appropriate in making the corresponding dimensionless variables to vary in the order of 1. Analysis of obtained scale and reference values through relevant dimensionless groups results in a simplified model. The performance of the simplified model based on these scales is evaluated by comparing the simulation results with a detailed model and bench-marking the respective computational performances.
Section snippets
Model simplification using current method of scaling analysis
Systematic scaling analysis of model equations can identify phenomena with varying importance thereby providing a rational approach for model simplification through elimination of terms and elimination of equations with minimal impact on the simulation results. Scaling analysis involves identifying appropriate scale and reference values to make the entire dependent and independent variables in a model to be dimensionless and vary in the order of 1, i.e. these dimensionless variables vary from
Proposed improvements in scaling analysis
In the scaling analysis described in the literature, we see that handling of nonlinear terms during scaling analysis is not clear and not general. Also during scaling analysis, one need to follow a trial and error procedure during selection correct form of dimensionless equation and during calculation of scales. In this paper we propose an systematic approach where scaling analysis is general and intuitive to any given model equations. This new approach is explained through following steps
Step
Examples of varying complexity to illustrate the proposed idea for scaling analysis
In this section, we illustrate the proposed idea using various examples of different complexity. The first example consider model equations with single independent variable and the second example consider equations with two independent variables. In both the examples, we use Dirichlet type boundary conditions and in third example, we use Neumann type boundary condition. Finally, we consider an example involving DAEs with two independent variables and involving Neumann type boundary conditions.
Numerical calculation of scale and reference values for the given parameters for WGS reactor example
In this section, scale and reference values are calculated using given parameters and boundary conditions for the WGS reactor model (step 5 in the proposed scaling analysis). The parameters and boundary condition values for WGS reactor model are given in Table 1 and these are introduced into the above algebraic equation(Eqs. (88)–(100) and (102)) to calculate the appropriate values. The reference values are obtained from the corresponding boundary condition values, based on Eq. (101). The
Verification of calculated scale and reference values using simulation for WGS reactor example
In this section, calculated scale and reference values are verified to make the corresponding dimensionless variables to vary in the order of 1. This is to show that proposed method of scaling analysis results in scale and reference values that are appropriate in representing the model equation and hence can be used for further analysis such as model simplification. For this verification, first actual dimensional values of variables are obtained through simulation of WGS reactor model (Eqs. (70)
Simplification of the model based on the dimensionless groups – WGS reactor example
This section focuses on application of scaling analysis in simplifying the model equations. We analyze the model equations of WGS reactor, using dimensionless groups formed from the calculated scale and reference values. Each term in a model equation represent some phenomena and ratio of two such phenomenon form a dimensionless group. Once scale and reference values are calculated, value of these phenomenon can easily be calculated and from which dominant phenomenon (phenomenon with higher
Simulation and comparison of the detailed and simplified models
The simplified model obtained through scaling analysis is tested by comparing the simulation results with that generated from the detailed model. The percentage change in the average values of the variables between detailed and simplified model and the computational load are considered as metrics for comparison. These metrics represents the accuracy of the simplified model and its computational efficiency compared to the corresponding detailed model. For example, the average difference in the
Conclusion
Scaling analysis has been systematically applied to model equations in obtaining a simplified model. In this analysis, two important difficulties, one in handling nonlinear terms and another in performing trial and error procedure, are addressed. The proposed improvements in the method of scaling analysis results in straight-forward handling of nonlinear terms and avoids trial and error procedure by solving a set of algebraic equations to obtain the scale and reference values. This method is
Acknowledgement
The authors gratefully acknowledge support from DOE through grant no. DE-FE0005749 titled “Model-Based Sensor Placement for Component Condition Monitoring and Fault Diagnosis in Fossil Energy Systems”.
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