Step changes coefficient

In principle, the step-response coefficients can be determined from the output response to a step change in the input. A typical response to a unit step change in input u is shown in Fig. 8-43. The step response coefficients are simply the values of the output variable at the samphng instants, after the initial value y(0) has been subtracted. Theoretically, they can be determined from a single-step response, but, in practice, a number of bump tests are required to compensate for unanticipated disturbances, process nonhnearities, and noisy measurements. 

Gal-Or and Hoelscher (G5) have recently developed a fast and simple transient-response method for the measurement of concentration and volumetric mass-transfer coefficients in gas-liquid dispersions. The method involves the use of a transient response to a step change in the composition of the feed gas. The resulting change in the composition of the liquid phase of the dispersion is measured by means of a Clark electrode, which permits the rapid and accurate analysis of oxygen or carbon dioxide concentrations in a gas, in blood, or in any liquid mixture. 

Step 1 Begin balancing equations by trying suitable coefficients that will give the same number of atoms of each element on both sides of the equation. Remember to change coefficients, not subscripts. 

A theoretical situation in which the system oscillates in response to a step change in the input value and the amplitude of the oscillations does not diminish with time the damping coefficient is 0. 

Dynamic matrix control uses time-domain step-response models (called convolution models). As sketched in Fig. 8.18, the response (x) of a process to a unit step change in the input (Ami = ) made at time equal zero can be described by the values of x at discrete points in time (the fc, s shown on the figure). At r nTJ, the value of X is h r,. If Affii is not equal to one, the value of x at f = n7 is b j Aibi, The complete response can be described using a finite number (NP) values of b coefficients. NP is typically chosen such that the response has reached 90 to 95 percent of its final value. 

The composition of the hydrolysate will affect growth rate, specific substrate uptake rates, and yield coefficients in a not easily predictable way (14). It is thus not possible to use a predetermined feed profile, but instead a closed-loop procedure should be used. One approach is to maximize the productivity at every point in time, and this can relatively easily be implemented in a closed-loop control. This approach was used by Taherzadeh et al. (14), who applied step changes in the feed rate and gradually increased the feed rate as long as there was a relative response in CER corresponding to at least include 50% of the relative increase in feed rate. The major drawbacks of this control were that the increase in feed rate was fixed and that the CER-response ratio was somewhat sensitive to measurement errors. 

If the system is to be fully decoupled in the steady-state, it is sufficient that c and m be equal after completion of a step change. This means that mj and m2 may be substituted for c and 2 and, therefore, the elements g -s of relation 8 define Gm in Figure 1. These elements are functions of the coefficients in the performance equation 5. 

Low-pass filter constant Vector of regression coefficients Magnitude of step change... 

The coefficient of friction at each PV step is continuously monitored. Because the interface temperature keeps changing during the test at each PV step, the coefficients of friction therefore are recorded during the wear test. The values of the coefficient of friction tend to decrease as the steps increase [1]. 

The coefficient representing axial dispersion, E, is measured using a tracer by pulse, sinusoidal, or step-change residence time distribution tests, or by measuring backflow. Sometimes the phenomenon is represented by a cefl model, in which the number of well-mixed cells fits dispersion. The coefficient representing radial dispersion, is determined by measuring the radial spread of a tracer from the centerline toward the wall. 

A quasi steady-state solution for the tracer distribution in a soilpolutnn has been developed for the inlet boundary concentration being a constant plus a Sinusoidal component. Then an unsteady state solution for tracer distribution a soil column was developed for the same inlet boundary condition as above. The unsteady-state tracer concentration distribution applies to the section of a soil column that still remembers the initial condition. The two solutions may be applicable to those planning experiments to measure parameters such as the dispersion coefficient from tracer tests. A sinusoidal loading of tracer at the inlet boundary may enable one to obtain repeated data traces at the column outlet as part of an extended experiment. Continued collection of tracer concentration vs. time data at the column outlet over a number of periods would enable one to collect data from repeated experiments, for each period of the sine wave would represent another experiment. This should enable one to obtain more replicates of data to improve statistical estimates of the dispersion coefficient than could be obtained by experimental methods that use a slug loading or a step change of concentration at the column inleL"... 

---