Dynamic response model

When operating conditions were changed, transient phenomena were sometimes observed that first move in one direction and in the reverse direction on going to the final steady state. To study these transients and to design an improved control strategy for the unit, a dynamic response model was needed. With the inclusion of the fast coke in the model, it became possible to extend the steady-state model to obtain useful dynamic response results by the addition of time-dependent accumulation terms (Weekman et al., 1967). 

Baratta, R. and Solomonow, M., The dynamic response model of nine different skeletal muscles, IEEE Trans. Biomed. Eng. BME-36 243-251,1989. 

Site Response 1-D Time Domain Analyses, Fig. 1 Lumped mass discretization for 1-D dynamic response model (Matasovic 1993)... 

Dyna.micPerforma.nce, Most models do not attempt to separate the equiUbrium behavior from the mass-transfer behavior. Rather they treat adsorption as one dynamic process with an overall dynamic response of the adsorbent bed to the feed stream. Although numerical solutions can be attempted for the rigorous partial differential equations, simplifying assumptions are often made to yield more manageable calculating techniques. 

Mathematically speaking, a process simulation model consists of a set of variables (stream flows, stream conditions and compositions, conditions of process equipment, etc) that can be equalities and inequalities. Simulation of steady-state processes assume that the values of all the variables are independent of time a mathematical model results in a set of algebraic equations. If, on the other hand, many of the variables were to be time dependent (m the case of simulation of batch processes, shutdowns and startups of plants, dynamic response to disturbances in a plant, etc), then the mathematical model would consist of a set of differential equations or a mixed set of differential and algebraic equations. 

As an alternative to deriving Eq. (8-2) from a dynamic mass balance, one could simply postulate a first-order differential equation to be valid (empirical modeling). Then it would be necessary to estimate values for T and K so that the postulated model described the reactor s dynamic response. The advantage of the physical model over the empirical model is that the physical model gives insight into how reactor parameters affec t the v ues of T, and which in turn affects the dynamic response of the reac tor. 

Feedforward Control If the process exhibits slow dynamic response and disturbances are frequent, then the apphcation of feedforward control may be advantageous. Feedforward (FF) control differs from feedback (FB) control in that the primary disturbance or load (L) is measured via a sensor and the manipulated variable (m) is adjusted so that deviations in the controlled variable from the set point are minimized or eliminated (see Fig. 8-29). By taking control action based on measured disturbances rather than controlled variable error, the controller can reject disturbances before they affec t the controlled variable c. In order to determine the appropriate settings for the manipulated variable, one must develop mathematical models that relate ... 

A key featui-e of MPC is that a dynamic model of the pi ocess is used to pi-edict futui e values of the contmlled outputs. Thei-e is considei--able flexibihty concei-ning the choice of the dynamic model. Fof example, a physical model based on fifst principles (e.g., mass and energy balances) or an empirical model coiild be selected. Also, the empirical model could be a linear model (e.g., transfer function, step response model, or state space model) or a nonhnear model (e.g., neural net model). However, most industrial applications of MPC have relied on linear empirical models, which may include simple nonlinear transformations of process variables. 

Although dynamic responses of microbial systems are poorly understood, models with some basic features and some empirical features have been found to correlate with actual data fairly well. Real fermentations take days to run, but many variables can be tried in a few minutes using computer simulation. Optimization of fermentation with models and reaf-time dynamic control is in its early infancy however, bases for such work are advancing steadily. The foundations for all such studies are accurate material Balances. 

Many HVAC system engineering problems focus on the operation and the control of the system. In many cases, the optimization of the system s control and operation is the objective of the simulation. Therefore, the appropriate modeling of the controllers and the selected control strategies are of crucial importance in the simulation. Once the system is correctly set up, the use of simulation tools is very helpful when dealing with such problems. Dynamic system operation is often approximated by series of quasi-steady-state operating conditions, provided that the time step of the simulation is large compared to the dynamic response time of the HVAC equipment. However, for dynamic systems and plant simulation and, most important, for the realistic simulation... 

The time that a molecule spends in a reactive system will affect its probability of reacting and the measurement, interpretation, and modeling of residence time distributions are important aspects of chemical reaction engineering. Part of the inspiration for residence time theory came from the black box analysis techniques used by electrical engineers to study circuits. These are stimulus-response or input-output methods where a system is disturbed and its response to the disturbance is measured. The measured response, when properly interpreted, is used to predict the response of the system to other inputs. For residence time measurements, an inert tracer is injected at the inlet to the reactor, and the tracer concentration is measured at the outlet. The injection is carried out in a standardized way to allow easy interpretation of the results, which can then be used to make predictions. Predictions include the dynamic response of the system to arbitrary tracer inputs. More important, however, are the predictions of the steady-state yield of reactions in continuous-flow systems. All this can be done without opening the black box. 

In particular it can be shown that the dynamic flocculation model of stress softening and hysteresis fulfils a plausibility criterion, important, e.g., for finite element (FE) apphcations. Accordingly, any deformation mode can be predicted based solely on uniaxial stress-strain measurements, which can be carried out relatively easily. From the simulations of stress-strain cycles at medium and large strain it can be concluded that the model of cluster breakdown and reaggregation for prestrained samples represents a fundamental micromechanical basis for the description of nonlinear viscoelasticity of filler-reinforced rubbers. Thereby, the mechanisms of energy storage and dissipation are traced back to the elastic response of tender but fragile filler clusters [24]. 

From careful measurements of such processs, it is possible to develop quantitative models to describe the molecular dynamical response to impulsive laser excitation. These enable the fundamental interaction of intense, ultrafast laser light with molecules to be understood from first... 

The theoretical approach described before dealt with the short-time dynamic response of the star molecules. However, in the case of completely labelled stars [148] it was found that the line shape of the Zimm model provides a good description of the NSE spectra not only in the short-time regime (t < 5 ns), but also on longer time scales. 

A quantitative model is proposed for the dynamic response, which is based on the hypothesis that the elastic part is due to thermal motions, while the viscous part is due to adsorption and desorption of chain segments and molecules. 

The studies described in the preceding two sections have identified several processes that affect the dynamic behavior of three-way catalysts. Further studies are required to identify all of the chemical and physical processes that influence the behavior of these catalysts under cycled air-fuel ratio conditions. The approaches used in future studies should include (1) direct measurement of dynamic responses, (2) mathematical analysis of experimental data, and (3) formulation and validation of mathematical models of dynamic converter operation. 

To define a feature extraction procedure it is necessary to consider that the output signal of a chemical sensor follows the variation of the concentration of gases at which it is exposed with a certain dynamics. The nontrivial handling of gas samples complicates the investigation of the dynamics of the sensor response. Generally, sensor response models based on the assumption of a very rapid concentration transition from two steady states results in exponential behaviour. 

The example simulation THERMFF illustrates this method of using a dynamic process model to develop a feedforward control strategy. At the desired setpoint the process will be at steady-state. Therefore the steady-state form of the model is used to make the feedforward calculations. This example involves a continuous tank reactor with exothermic reaction and jacket cooling. It is assumed here that variations of inlet concentration and inlet temperature will disturb the reactor operation. As shown in the example description, the steady state material balance is used to calculate the required response of flowrate and the steady state energy balance is used to calculate the required variation in jacket temperature. This feedforward strategy results in perfect control of the simulated process, but limitations required on the jacket temperature lead to imperfections in the control. 

The phase angle changes with frequency and this is shown in Figure 4.7. As the frequency increases the sample becomes more elastic. Thus the phase difference between the stress and the strain reduces. There is an important feature that we can obtain from the dynamic response of a viscoelastic model and that is the dynamic viscosity. In oscillatory flow there is an analogue to the viscosity measured in continuous shear flow. We can illustrate this by considering the relationship between the stress and the strain. This defines the complex modulus ... 

Our steadystate design calculations teU us nothing about what the dynamic response to the system will be. They teU us where we will start and where we will end up but not how we get there. This kind of information is what a study of the dynamics of the system will reveal. We will return to this system later in the book to derive a mathematical model of it and to determine its dynamic response quantitatively by simulation. 

Finally, we will neglect the dynamics of the condenser and the reboiler. In commercial-scale colunms, the dynamic response of these heat exchangers is usually much faster than the response of the column itself In some systems, however, the dynamics of this peripheral equipment are important and must be included in the model... 

You may wonder why we would ever be satisfied with anything less than a very accurate integration. The ODEs that make up the mathematical models of most practical chemical engineering systems usually represent a mixture of fast dynamics and slow dynamics. For example, in a distillation column the liquid flow or hydraulic dynamic response occurs fairly rapidly, of the order of a few seconds per tray. The composition dynamics, the rate of change of hquid mole fractions on the trays, are usually much slower—minutes or even hours for columns with many trays. Systems with this mixture of fast and slow ODEs are called stiff systems. 

---