cItAfTER7:Laplace-Domain Dynamics 241- Time domainLaplacedomainFIGURE 7.6Deadtime transfer function.Let us see what happens when we Laplace-transform a function h,-o, that has been delayed by a deadtime. Laplace transformation is defined in Eq. (7.5 1).af,,,1 = oY(@I dt = F(s) I(7.5 1)The variable t in this equation is just a ummy variable�of integration. It is integrated out, leaving a function of only s. Thus, we can write Eq. (7.5 1) in a completely equivalent mathematical form: (7.52) where y is now the dummy variable of integration. Now let y =F(s) = I m fit-&SO--D) d(t - 0) = p f0t- D. (7.53)m fit-D,e-� dt0F(,) = eD�Lf&dTherefore,%ht-o)l = e-DsFcs)(7.54)Thus, time delay or deadtime in the time domain is equivalent to multiplication by eeDS in the Laplace domain. If the input into the deadtime element is ~(~1 and the output of the deadtime element is y(+ then u and y are related byY(f) = q-D)And in the Laplace domain, Y(,) = e -Ds 4) (7.55)Thus, the transfer function between output and input variables for a pure deadtime process is epDS, as sketched in Fig. 7.6.7.5EXAMPLES Now we are ready to apply all these Laplace transformation techniques to some typical chemical engineering processes.
