Feedback control system to control the composition of the output stream in a stirred tank blending process is shown in F

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Feedback control system to control the composition of the output
stream in a stirred tank blending process is shown in Figure 11.1,
page 176 of Textbook (as shown below). fig 11 Mass fraction x of
the output stream is the controlled variable, flow rate w 2 of the
input stream is the manipulated variable and mass fraction x 1 of
the other input stream is the disturbance variable. The following
data are available: Volume and density are constant. V= 3.2 m 3, ρ=
900 kg/m 3 The process is operating at steady state with w 1=500
kg/min, w 2= 300 kg/min, x 1= 0.4, x 2= 0.8 G m= K m = 16 mA/(mass
fraction), G v= K v = 20 kg/min mA The process transfer function G
p= X’(s)/W 2’(s) = K 1 /(τs+1) where τ = Vρ/w and K 1 =(1-x)/w The
transfer function relative to the disturbance variable G d =
X’(s)/X 1’(s) = K 2 /(τs+1) where K 2 = w 1/w A PI controller is
used with K c=3 and τ I = 1 min The set point for the exit mass
fraction x is set at the initial steady state value. (a) If the
disturbance variable x 1 is suddenly decreased to 0.2 from the
initial steady state value of 0.4, derive an expression for the
response of outlet composition x to this step change . (b)
Calculate the composition of the exit stream (x) 1 minutes after
the change. (c) Calculate the composition of the exit stream (x) 2
minutes after the change. (d) What is the composition x when
a new steady state is reached? (e) What is the
offset?