Flow Model Parameter Estimation

With multiparameter flow models, the accurate estimation of the parameters can be far from a trivial task. The basic problem is, of course, similar to those considered in Chapters 1 and 2 for kinetic rate coefficients, but since many flow models are partial differential equations, the problems are more severe. The mixing of tracer concentrations is inherently a linear process, and if other diffusion and dispersion steps are also linear, the governing differential equations will then be linear (although the parameters may appear in nonlinear ways), and the methods of systems engineering can be useful. We will only give a brief outline here, focusing on a few of the special problems involved for flow models. An excellent reference to many useful techniques is Seinfeld and Lapidus [49]. 

Let us first discuss the choice of input signals. The practical aspects of type of tracer for various situations is surveyed in Wen and Fan [2]. Also see Hougen [95] for an extensive discussion. The advantages and disadvantages of various types of signals are as follows. 

A pulse input is often best in principle, since the output is directly the impulse response of the system also, the model response is often simplest in this case. In addition, the test only causes a brief disturbance to the process. The main disadvantage is that it is difficult to experimentally generate a perfect impulse input, and especially, small deviations in the tail can cause important deviations from ideal behavior. Finally, in principle, all frequency modes of the system are stimulated, and this may not be optimum. 

Z A step input is relatively easy to experimentally generate, and the result is simply related to the impulse response. However, a long time of input stimulation is required to achieve the final lined out response a related problem is that it may be difficult to accurately determine the final value, and thus the normalization factor. 

let us discuss the various techniques that have been used for the actual parameter estimation, describing their strong and weak points. For the axial dispersion model, this has been done by Boxkcs and Hofmann [96], and illustrates the typical problems involved. 

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In this chapter we present an individual-based population model (Metapopulation model for Assessing Spatial and Temporal Effects of Pesticides [MASTEP]). M ASTEP describes the effects on, and recovery of, populations of the water louse Asellus aqua-ticus following exposure to a fast-acting, nonpersistent insecticide caused by spray drift for pond, ditch, and stream scenarios. The model used the spatial and temporal distribution of the exposure in different treatment conditions as an input parameter. A dose-response relation derived from a hypothetical mesocosm study was used to link the exposure with the effects. The modeled landscape was represented as a lattice of 1 x 1 m cells. The model included processes of mortality of A. aquaticus, life history, random walk between cells, density-dependent population regulation, and in the case of the stream scenario, medium-distance drift of A. aquaticus due to flow. All parameter estimates were based on the results of a thorough review of published information on the ecology of A. aquaticus and expert judgment. 

The vertices are connected with hues indicating information flow. Measurements from the plant flow to plant data, where raw measurements are converted to typical engineering units. The plant data information flows via reconciliation, rec tification, and interpretation to the plant model. The results of the model (i.e., troubleshooting, model building, or parameter estimation) are then used to improve plant operation through remedial action, control, and design. 

Parameter Estimation Relational and physical models require adjustable parameters to match the predicted output (e.g., distillate composition, tower profiles, and reactor conversions) to the operating specifications (e.g., distillation material and energy balance) and the unit input, feed compositions, conditions, and flows. The physical-model adjustable parameters bear a loose tie to theory with the limitations discussed in previous sections. The relational models have no tie to theory or the internal equipment processes. The purpose of this interpretation procedure is to develop estimates for these parameters. It is these parameters hnked with the model that provide a mathematical representation of the unit that can be used in fault detection, control, and design. 

Observe that the axial dispersion model provides a lower and thus more conservative estimate of conversion than does the piston flow model given the same values for the input parameters. There is a more subtle possibility. The model may show that it is possible to operate with less conservative values for some parameters—e.g., higher values for Tin and T aii— without provoking adverse side reactions. 

In Fig. 1, a comparison can be observed for the prediction by the honeycomb reactor model developed with the parameters directly obtained from the kinetic study over the packed-bed flow reactor [6] and from the extruded honeycomb reactor for the 10 and 100 CPSI honeycomb reactors. The model with both parameters well describes the performance of both reactors although the parameters estimated from the honeycomb reactor more closely predict the experiment data than the parameters estimated from the kinetic study over the packed-bed reactor. The model with the parameters from the packed-bed reactor predicts slightly higher conversion of NO and lower emission of NHj as the reaction temperature decreases. The discrepancy also varies with respect to the reactor space velocity. 

The kinetic parameters estimated by the experimental data obtained frmn the honeycomb reactor along with the packed bed flow reactor as listed in Table 1 reveal that all the kinetic parameters estimated from both reactors are similar to each other. This indicates that the honeycomb reactor model developed in the present study can directly employ intrinsic kinetic parameters estimated from the kinetic study over the packed-bed flow reactor. It will significantly reduce the efibrt for predicting the performance of monolith and estimating the parameters for the design of the commercial SCR reactor along with the reaction kinetics. 

Fig. 1. Prediction of the model for 10 and 100 CPSI honeycomb reactors extruded with the ViOs/sulfated Xi02 catalyst. (—, prediction with the parameters estimated from the experimental data over a packed-bed flow reactor —, prediction with the parameters estimated from the experimental data over a honeycomb reactor).

A pilot scale UASB reactor was simulated by the dispersed plug flow model with Monod kinetic parameters for the hypothetical influent composition for the three VPA ccmiponents. As a result, the COD removal efflciency for the propionic acid is smallest because its decomposition rate is cptite slow compared with other substrate components their COD removal eflSciencies are in order as, acetic acid 0.765 > butyric acid 0.705 > propionic acid 0.138. And the estimated value of the total COD removal efficiency is 0.561. This means that flie inclusion of large amount of propionic acid will lead to a significant reduction in the total VFA removal efficiency. 

We also use a linearized covariance analysis [34, 36] to evaluate the accuracy of estimates and take the measurement errors to be normally distributed with a zero mean and covariance matrix Assuming that the mathematical model is correct and that our selected partitions can represent the true multiphase flow functions, the mean of the error in the estimates is zero and the parameter covariance matrix of the errors in the parameter estimates is ... 

Important issues in groundwater model validation are the estimation of the aquifer physical properties, the estimation of the pollutant diffusion and decay coefficient. The aquifer properties are obtained via flow model calibration (i.e., parameter estimation see Bear, 20), and by employing various mathematical techniques such as kriging. The other parameters are obtained by comparing model output (i.e., predicted concentrations) to field measurements a quite difficult task, because clear contaminant plume shapes do not always exist in real life. 

PBPK and classical pharmacokinetic models both have valid applications in lead risk assessment. Both approaches can incorporate capacity-limited or nonlinear kinetic behavior in parameter estimates. An advantage of classical pharmacokinetic models is that, because the kinetic characteristics of the compartments of which they are composed are not constrained, a best possible fit to empirical data can be arrived at by varying the values of the parameters (O Flaherty 1987). However, such models are not readily extrapolated to other species because the parameters do not have precise physiological correlates. Compartmental models developed to date also do not simulate changes in bone metabolism, tissue volumes, blood flow rates, and enzyme activities associated with pregnancy, adverse nutritional states, aging, or osteoporotic diseases. Therefore, extrapolation of classical compartmental model simulations... 

Some recent applications have benefited from advances in computing and computational techniques. Steady-state simulation is being used off-line for process analysis, design, and retrofit process simulators can model flow sheets with up to about a million equations by employing nested procedures. Other applications have resulted in great economic benefits these include on-line real-time optimization models for data reconciliation and parameter estimation followed by optimal adjustment of operating conditions. Models of up to 500,000 variables have been used on a refinery-wide basis. 

The formulation described above provides a useful framework for treating feedback control of combustion instability. However, direct application of the model to practical problems must be exercised with caution due to uncertainties associated with system parameters such as and Eni in Eq. (22.12), and time delays and spatial distribution parameters bk in Eq. (22.13). The intrinsic complexities in combustor flows prohibit precise estimates of those parameters without considerable errors, except for some simple well-defined configurations. Furthermore, the model may not accommodate all the essential processes involved because of the physical assumptions and mathematical approximations employed. These model and parameter uncertainties must be carefully treated in the development of a robust controller. To this end, the system dynamics equations, Eqs. (22.12)-(22.14), are extended to include uncertainties, and can be represented with the following state-space model ... 

The results confirm that the adsorption of ammonia is very fast and that ammonia is strongly adsorbed on the catalyst surface. The data were analyzed by a dynamic isothermal plug flow reactor model and estimates of the relevant kinetic parameters were obtained by global nonlinear regression over the entire set of runs. The influences of both intra-particle and external mass transfer limitations were estimated to be negligible, on the basis of theoretical diagnostic criteria. 

We clearly cannot cover all aspects of physical chemistry here. However, we attempt to provide enough theoretical background for the reader with training in mechanical or chemical eningeering to understand what is needed to develop and analyze chemically reacting flow models. This includes understanding the chemistry input parameters that someone else has determined and, more important, being able to estimate parameters that are needed to do a simulation but simply do not exist in the literature. 

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