Reduced-order flow model

Compared with the use of arbitrary grid interfaces in combination with reduced-order flow models, the porous medium approach allows one to deal with an even larger multitude of micro channels. Furthermore, for comparatively simple geometries with only a limited number of channels, it represents a simple way to provide qualitative estimates of the flow distribution. However, as a coarse-grained description it does not reach the level of accuracy as reduced-order models. Compared with the macromodel approach as propagated by Commenge et al, the porous medium approach has a broader scope of applicability and can also be applied when recirculation zones appear in the flow distribution chamber. However, the macromodel approach is computationally less expensive and can ideally be used for optimization studies. 

The above time-scale decomposition provides a transparent framework for the selection of manipulated inputs that can be used for control in the two time scales. Specifically, it establishes that output variables y1 need to be controlled in the fast time scale, using the large flow rates u1, while the control of the variables ys is to be considered in the slow time scale, using the variables us. Moreover, the reduced-order approximate models for the fast (Equation (3.11)) and slow (the state-space realization of Equation (3.16)) dynamics can serve as a basis for the synthesis of well-conditioned nonlinear controllers in each time scale. 

Thanks to the great computer power available nowadays, CN theory models are outperformed by reduced-order physical models, e.g. DC power flow in power grids. Nonetheless, power-flow models alone lack the capability of capturing time-varying interdependencies and discrete processes, e.g. cascading failures, therefore, it is advisable to integrate them into approaches such as ABM to represent discrete event-driven processes (Schlapfer et al., 2008 Nan et al., 2013). 

In summary, in the equilibrium-chemistry limit, the computational problem associated with turbulent reacting flows is greatly simplified by employing the presumed mixture-fraction PDF method. Indeed, because the chemical source term usually leads to a stiff system of ODEs (see (5.151)) that are solved off-line, the equilibrium-chemistry limit significantly reduces the computational load needed to solve a turbulent-reacting-flow problem. In a CFD code, a second-order transport model for inert scalars such as those discussed in Chapter 3 is utilized to find ( ) and and the equifibrium com-... 

The solution of Eq. (173) poses a rather formidable task in general. Thus the dispersed plug-flow model has not been as extensively studied as the axial-dispersed plug-flow model. Actually, if there are no initial radial gradients in C, the radial terms will be identically zero, and Eq. (173) will reduce to the simpler Eq. (167). Thus for a simple isothermal reactor, the dispersed plug flow model is not useful. Its greatest use is for either nonisothermal reactions with radial temperature gradients or tube wall catalysed reactions. Of course, if the reactants were not introduced uniformly across a plane the model could be used, but this would not be a common practice. Paneth and Herzfeld (P2) have used this model for a first order wall catalysed reaction. The boundary conditions used were the same as those discussed for tracer measurements for radial dispersion coefficients in Section II,C,3,b, except that at the wall. 

Typically, a non-linear system dynamic model is made up of individual lumped models of the components which at a minimum conserve mass and energy across the given component, but may also have a momentum equation if pressure drops must also be analyzed. For most dynamic problems of interest in hybrid studies, however, the momentum equation may be taken as quasi-steady (unless the solver requires the dynamic form to perform the numerical solution). Higher fidelity individual models or reduced order models (ROMs) can also be used, where the connection to the system model would be made at each subcomponent boundary. Since dynamic systems modeling is not as common as steady-state modeling, some discussion of modeling approaches will be given. There are two primary methods used to provide solutions for the pressure-flow dynamics of a system model. 

The axial dispersion plug flow model is used to determine the performance of a reactor with non-ideal flow. Consider a steady state reacting species A, under isothermal operation for a system at constant density Equation 8-121 reduces to a second order differential equation ... 

Finally, we analyzed the control implications of the presence of impurities in a process, concluding that the control of impurity levels must be addressed over an extended time horizon using the flow rate of the purge stream as a manipulated input. To close the impurity-levels loop, one should resort either to an appropriately tuned linear controller (e.g., a PI controller with long reset time) or to a (nonlinear) model-based controller that uses (an inverse of) the reduced-order model of the slow dynamics - as developed in this chapter - to compute the necessary control action. 

Each of the reduced-order models derived for the fast, intermediate, and slow dynamics (Equations (5.12), (5.21), and (5.28)) involves only one group of manipulated inputs, namely the large internal flow rates u1, the small flow rates us, and the purge flow rate up, respectively. Thus, control objectives in each of the... 

The terms lime, o(l/ )r 1(x, 0)lo1 (which, being based on Equation (6.7), represent differences between large internal energy flows), become indeterminate in the slow time scale. These terms do, however, remain finite, and constitute an additional set of algebraic (rather than differential) variables in the model of the slow dynamics. On defining z = lime, o(l/ )f (x, 0)u> the reduced-order representation of the slow dynamics becomes... 

Optimization of internal engine combustion in respect of fuel efficiency and pollutant minimization requires detailed insight in the microscopic processes in which complex chemical kinetics is coupled with transport phenomena. Due to the development of various pulsed high power laser sources, experimental possibilities have expanded quite dramatically in recent years. Laser spectroscopic techniques allow nonintrusive measurements with high temporal, spectral and spatial resolution. New in situ detection techniques with high sensitivity allow the measurement of multidimensional temperature and species distributions required for the validation of reactive flow modeling calculations. The validated models are then used to And optimal conditions for the various combustion parameters in order to reduce pollutant formation and fuel consumption. 

Progress has been made in both reduced-order modeling and model-based control of combustion dynamics. Advances in modeling were obtained by investigating shear-flow driven combustion instability. The authors of this chapter determined that shear-layer instability occurs when an absolutely unstable mode is present. This mode can be predicted and matched to experimental data. Also, it is shown that the temperature profile determines a transition to absolutely unstable operation. Parameters of the shear-flow modes together with a POD-based approach led to the derivation of a new reduced-order model that sheds light on the interactions between hydrodynamics, acoustics, and heat release. A RePOD... 

This example illustrates the complexity of even elementary second order equations when practical boundary conditions are applied. Fig. 2.2 illustrates the reactor composition profiles predicted for plug and dispersive flow models. Under quite realizable operating conditions, the effect of backmbcing (diffusion) is readily seen. Diffusion tends to reduce the effective number of stages for a packed column. Its effect can be reduced by using smaller particle sizes, since Klinkenberg and Sjenitzer (1956) have shown that the effective diffusion coefficient (Dg) varies as Vgdp, where dp is particle size. This also implies that Peclet number is practically independent of velocity Pe L/dp). 

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