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Differ bv the steadv-state values of all variables. 7. 55) ~~AO(,, = Kf where K is a constant. C,, is initially xro. Rearranging Eq. 5 I) gives *. tllIL!! E, tl! 56) The solution, according to Eq. 57) The integral in Eq. 57) can be looked up in mathematics tables or found by integrating by parts. Let 11 = t and dv = &11 dt du = dt and v = ~oe”r~l Then Since j udv = uv - j vdu, I ter’T~, dt = TOtel’T(~ - T,e’lT~~ dt I = T(,&“T” - (&pIJ Therefore Eq. 5 Ramp response of a first-order system. The linal solulioli is The ramp response is skc~chctl in f;ig.
Kc=4 : . _ _ -:- - - _ _ _ _ :~------:-------~--- - . . . . . . . . . . . . . . . . . . . . . . ~................ 5 I I . . . . . . . . . . . . . . . . . ~.................. 01 hr in the measurement of the temperature 7’, . The temperature controller sees a PV signal that is delayed by D hours. The effect this has on control is illustrated in Fig. 13. 01 hr because of the deadtime. Comparing Figs. 13 shows clearly that the dynamic performance with deadtime is worse.
7 ft. So at the new steady-state conditions, the value of SP is not equal to PV in Eqs. 8). ” This example illustrates that a proportional controller does not give zero steady-state error. For the control of levels in surge tanks we normally are not concerned about holding a constant level, so offset is not a problem. But for many control loops, we do want to drive the PV back to the SP value. This is accomplished by adding “integral” or “reset” action to the controller. The second type of level controller is a “proportional-integral” (PI) controller, in which the CO signal varies with both the PV signal and time integral of “error” (the difference between the SP and PV signals).
Chemical Engineering. Essentials of Process Control by Dupont