Step input

Concenti ation The main special lands are Cg, that of the effluent from a vessel with impulse input of tracer = m/V,., the initial mean concentration resulting from impulse input of magnitude m C, that of the effluent from a vessel with a step input of magnitude Cy. 

The chief quantities based on tracer tests are summarized in Table 23-4. Effluent concentrations resulting from impulse and step inputs are designated Cg and C , respectively. The initial mean concentration resulting from an impulse of magnitude m into a vessel of volume is C = mfVr- The mean residence time is the ratio of the vessel volume to the volumetric flow rate, t = V fV or t = jo tCg dt/jo Cg dt. The reduced time is t = t/t. 

Figure 23-7 illustrates the responses of CSTRs and PFRs to impmse or step inputs of tracers. 

Ideal CSTR Witt a step input of magnitude Cy the unsteady material balance is... 

FIG. 23-7 Imp ulse and step inputs and responses. Typical, PFR and CSTR. (a) Experiment with impulse input of tracer, (h) Typical behavior area between ordinates at tg and ty equals the fraction of the tracer with residence time in that range, (c) Plug flow behavior all molecules have the same residence time, (d) Completely mixed vessel residence times range between zero and infinity, e) Experiment with step input of tracer initial concentration zero. (/) Typical behavior fraction with ages between and ty equals the difference between the ordinates, h — a. (g) Plug flow behavior zero response until t =t has elapsed, then constant concentration Cy. (h) Completely mixed behavior response begins at once, and ultimately reaches feed concentration. 

FIG. Tracer responses to n-stage continuous stirred tank batteries the Erlang model (a) impulse inputs, (h) step input. 

Generalized second-order system response to a unit step input... 

Consider a second-order system whose steady-state gain is K, undamped natural frequency is Wn and whose damping ratio is (, where C < 1 For a unit step input, the block diagram is as shown in Figure 3.18. From Figure 3.18... 

From equation (3.57) it ean be seen that when there is no damping, a step input will eause the system to oseillate eontinuously at Wn (rad/s). 

The generalized seeond-order system response to a unit step input is shown in Figure 3.19 for the eondition K = 1 (see also Appendix 1, sec ord.m). 

When a unity gain seeond-order system is subjeet to a unit step input, its transient response eontains a first overshoot of 77%, oeeurring after 32.5 ms has elapsed. Find... 

If a step input funetion of 10 units is applied to the system, find an expression for the time response. Assume zero initial eonditions. 

Fig. 6.22 Relationship between system type classification and the Nyquist diagram. For a step input,...

The compensated and uncompensated open-loop frequency response is shown in Figure 6.41. From this Figure the compensated gain margin is 12.5 dB, and the phase margin is 48°. In equation (6.117), K does not need to be adjusted, and can be set to unity. When responding to a step input, the steady-state error is now 4.6%. 

If the sampling time is one seeond and the system is subjeet to a unit step input funetion, determine the diserete time response. (N.B. normally, a zero-order hold would be ineluded, but, in the interest of simplieity, has been omitted.) Now... 

For the spring-mass-damper system given in Example 8.6, evaluate the transient response of the state variables to a unit step input using... 

In Example 6.4, when there was no model uneertainty, K for marginal stability was 8, and for a gain margin of 6dB, K was 4. In this example with model uneertainty, from equation (9.154) marginal stability oeeurs with K = 3.5, so this is the maximum value for robust stability. For robust performanee, equation (9.150) applies. For a speeifie step input let lV(s) = 1 /s now... 

For a perfeetiy mixed CSTR, it is assumed that the vessel eontents are perfeetiy homogeneous and have the same eomposition as the exit stream. Considering a step input into the CSTR, a maeroseopie material balanee gives... 

During plug flow, all material passes through the vessel without any mixing, and eaeh fluid element stays in the vessel for exaetly the same length of time. Eor a step input, the front or interfaee between the traeer and non-traeer fluids traverses down the vessel and exits at the other end in a time equal to the mean residenee time f. Therefore, the E(6) eurve is a step funetion and is expressed as... 

Missen et al. [15] have developed numerieal methods for the step input F(t). Figure 8-16 illustrates a typieal method of determining the area under C(t) versus t eurve. 

Step Input An input in which the concentration of tracer is changed to some constant value Cf at time zero and maintained at this level indefinitely. Eor example, CfU(t - a) represents a step of magnitude Cf beginning at t = a. The resulting effluent concentration is C. ... 

Title section for the second job step Input continues... 

Allowing a step input disturbance of magnitude Co, as shown in Fig. 2.23, the constants x and to are derived by drawing a tangent to the step response curve at the point of inflexion. 

Note that when k] = k.2 = k3 = 0, Cao = 1> Cbo = 0- the program solves the case of a step input of tracer solution, which can be used to generate the typical F-diagram for a single perfectly mixed tank. Compare this result with the analytical solution. 

Obtain the tracer response curve to a step input disturbance of tracer solution by setting k = 0. 

We can show that with a step input, the output is a ramp function. When we have an impulse input, the output will not return to the original steady state value, but accumulates whatever we have added. (Both items are exercises in the Review Problems.)... 

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