First order dynamics

Gas solenoid valve This is assumed to have first-order dynamics of the form... 

A mercury thermometer having first-order dynamics with a time constant of 60 s is placed in a bath at 308 K (35°C). After the thermometer reaches a steady state it is suddenly placed in a bath at 313 K (40° C) at / = 0 and left there for 60 s, after which it is immediately returned to the bath at 308 K (35°C). 

At / = 60 s, a further negative step change is imposed of 3.16 degK. The thermometer will respond immediately to this as it has only first order dynamics. 

There is a first-order dynamic lag of t minutes between a change in the signal to the steam valve and vapor boilup. The low base-level override controller pinches the reboiler steam valve over the lower 25 percent of the level transmitter span. 

Now, let us consider a system where an achiral molecule (A) and a chiral molecule (C) have a fixed mutual orientation. An electronic transition of the achiral molecule from the ground state z(0> to the excited state Aa, higher in energy by E0a, has a zero-order (non-perturbed) electric dipole moment po0 and an orthogonal magnetic dipole moment ma0. These moments are increased in the molecular pair (A -C) by first-order dynamic coupling as ... 

Computed fits of experimental signals probed at different wavelengths allow for the careful investigation of ultrafast electronic pathways (Fig.2). The transient signal at 400 nm is assigned to a very short-lived CTTS state of aqueous hydroxyl ions (OH), . This excited state is instantaneously populated, typically in less than 50 fs and follows a pseudo first order dynamics with a frequency rate of 5 x 1012 s. Semi-quantum MD simulations emphasize that transient excited CTTS states play a crucial role in photoinduced electron transfers [4-6]. 

The particle then approximately behaves as a first order dynamic system of time constant t with respect to mass transfer. The problem, when molecular diffusion is retained as the mixing mechanism... 

Systems with first-order dynamic behavior are not the only ones encountered in a chemical process. An output may change, under the influence of an input, in a drastically different way from that of a first-order system, following higher-order dynamics. In this chapter we analyze (1) the physical origin of systems with second-order dynamics, and (2) their dynamic characteristics. The analysis of systems with higher than second-order dynamics is left for Chapter 12. 

Example 11.4 demonstrates very clearly how the simple first-order dynamic behavior of a tank can change to that of a second-order when a proportional-integral controller is added to the process. Also, it indicates that the control parameters Kc and r can have a very profound effect on the dynamic behavior of the system, which can range from an underdamped to an overdamped response. 

Show that the concentration cA of reactant A in an isothermal continuous stirred tank reactor exhibits first-order dynamics to changes in the inlet composition, cA/. The reaction is irreversible, A - B, and has first-order kinetics (i.e., r = kcA). Furthermore (a) identify the time constant and static gain for the system, (b) derive the transfer function between cA and cA (c) draw the corresponding block diagram, and (d) sketch the qualitative response of cA to a unit pulse change in cAj. The reactor has a volume V, and the inlet and outlet flow rates are equal to F. 

Control valve. Assume first-order dynamics ... 

Discuss a system that stores momentum and exhibits first-order dynamics. 

Show that the concentration cA of reactant A in the reacting mixture exhibits first-order dynamic behavior with respect to the initial concen-... 

Let the measuring device and the control valve (final control element) have first-order dynamics ... 

Figure 4.4 shows the open-loop response over three hours for the following scenario after 0.5 h the feed temperature increases to 308.15 K for one hour, then for 0.5 h the total flow rises to 2000 kmol/h, followed by feed reset to 1870 kmol/hr, and finally feed temperature reset to 308.15 K. It can be observed that the flash temperature follows closely the disturbances with apparently first-order dynamics. The pressure is not affected by the disturbance in temperature, and only slightly by the feed flow. 

In this exercise, we will evaluate the controllability of a CSTR with heating jacket (Fig. 12.10). The reaction y4 —> 5 is first-order, irreversible and moderate exothermic. Since the heat of reaction is not enough to achieve a temperature that gives high conversion, heat is provided by pressurised hot water (inlet temperature 383 K). Temperature measurements follow a first order dynamics with a time constant of 60 s. Valve dynamics is represented by first order elements with a time constant of 30 s. Study the controllability property of the SISO loop keeping the reactor temperature at set-point by manipulating the hot-water flow rate. Disturbances in reactor inlet temperature and reactor inlet concentration are expected. 

The model also includes dynamical synapses based upon first order dynamics. In this case, currents generated by a synapse in response to a spike train is through an exponential decay over multiple spike inputs occurring at times (fi, tx,.. . t ) to... 

The peripheral (arterial) chemoreceptors response to changes in PaC02 maybe modeled by linear first-order dynamics [BeUville et al., 1979]... 

The aggregate then behaves approximately as a first-order dynamic system of time constant tdiffwith respect to mass transfer. The choice of the characteristic dimension l=2R depends on the kind of microstructure which is considered to ecist when molecular diffusion becomes controlling. In complex real flow, the shape of the structures is of course impossible to define because of the multiple laminar vortices which deform the structures along the three dimensions of space. The previous relation of the shape factor, however, enables one to give an evaluation of the mixing time by simple diffusion. 

There are a wide spectrum of AC-machine models ranging from the previously presented full-order, over reduced order, down to static or zero-order models, each of them being used according to the system analysis aims. The order reduction is largely based on heuristical or engineering considerations of experimental data as related to the full-order models, see [26], for instance. In the sequel only two of such models will be presented for the induction motor with short-circuited rotor. The first is a nonlinear static one, and the second is the former augmented with a first-order dynamics. For more details on the subject of this section, see [9, chapters 3 and 10]. 

The first-order dynamics of this type of condenser make it much easier to control than the condenser with once-through coolant. 

From equation (17.6) we can see that open-loop pressure dynamics are essentially first order. Since in most cases the inert gas bleed and the vent flow are fiiirly small, the valve gains, SwjgIBBc and dwJdBc, tend to be small. Together with the first-order dynamics, this commonly leads to large controller gains (small proporticmal bands) and control valve saturation fcM" fairly small disturbances. 

A heater for a semiconductor wafer has first-order dynamics, that is, the transfer function relating changes in temperature T to changes in the heater input power level P is... 

In Eq. 6-3 the standard first-order dynamics have been modified by the addition of the du/dt term multiphed by a time constant t. The corresponding transfer function is... 

Any measurement transducer output contains some dynamic error an estimate of the error can be calculated if transducer time constant t and the maximum expected rate of change of the measured variable are known. For a ramp input, x(t) = at, and a first-order dynamic model (see Eq. 9-15), the transducer output y is related to x by ... 

The ramp response y(t) of a first-order system was obtained in Eqs. 5-19 through 5-21. The maximum deviation between input and output is at (obtained when t j), as shown in Fig. 5.5. Hence, as a general result, we can say that the maximum dynamic error that can occur for any instrument mth first-order dynamics is... 

Equations 11-37 to 11-40 indicate that the closed-loop process has first-order dynamics with a time constant ti that is smaller than the process time constant t. We assume here that Kql > Oj otherwise, the control system would not function properly, as will be apparent from the stability analysis later in this chapter. Because ti < t, the feedback controller enables the controlled process to respond more quickly than the uncontrolled process. 

Suppose the IMA noise model is to be employed in a minimum variance controller for a process model with gain K that has no dynamics. It can be shown theoretically that for this simple case, the minimum variance controller has the same attributes as the IMC controller. Namely, the controller is the inverse of the process gain, and the IMC filter F is a first-order filter (MacGregor, 1988 Ogunnaike and Ray, 1994). A similar analysis can be performed for the case when the process model has first-order dynamics and the distur-... 

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