The startup procedure for a batchmode operating reactor incl

The start-up procedure for a batch-mode operating reactor includes a heating step where the reactor temperature is gradually heated to the nominal operating temperature of 75 degree C. The desired temperature profile T(t) is shown in the drawing. What is the Laplace transform T(s) for this T(t)?

Solution

equation of line in the interval 20 and 30 min is given by

(y-0)/(x-20)=(0-75)/(20-30)

y=(x-20)*7.5

Thus T(t)=7.5*(t-20)[u(t-30)-u(t-20)] +75 u(t-30)

Now taking laplace transform

T(s)= 7.5e^-20s (e^-10s/s²+e^-10s*10/s+1/s²) +75 e^-30s/s