Fall 2004
ICE Topics: Process Control by Design
Problem Set 3: Calculating RGA and DC
10.492
For this problem set, I will accept submissions with one or two names.
(1) Beginning with linearized material and energy balances for the shower, expressed in physical
terms,
F = Fh + Fc
T=
Fhr Thr + Fcr Tcr ⎡ Thr
Fhr
F T +F T ⎤
(Th − Thr )
+⎢
− hr hr cr 2 cr ⎥ (Fh − Fhr ) +
Fhr + Fcr
Fhr + Fcr
(Fhr + Fcr ) ⎦
⎣ Fhr + Fcr
⎡ Tcr
F T +F T ⎤
Fcr
(Tc − Tcr )
+⎢
− hr hr cr 2 cr ⎥ (Fc − Fcr ) +
Fhr + Fcr
(Fhr + Fcr ) ⎦
⎣ Fhr + Fcr
derive an expression for each individual element λij of the RGA from the definition
λ ij ≡
gain of the x mj → y i loop with other loops open
gain of the x mj → y i loop with other loops perfect
∂y i
∂x mj
=
•
x
∂y
i
∂x mj
mk
≠x
mj
constant
y ≠ y constant
i
k
(please use the nomenclature of the Lecture Notes)
(2) Beginning with linearized material and energy balances for the shower, expressed in scaled
deviation terms,
F*' =
∆Fh *' ∆Fc *'
Fh +
Fc
∆F
∆F
⎡ T − Tr ⎤ ∆Fh *' ⎡ Tcr − Tr ⎤ ∆Fc *' Fhr ∆Th *' Fcr ∆Tc *'
T *' = ⎢ hr
Fh + ⎢
Fc +
Th +
Tc
⎥
⎥
Fr ∆T
Fr ∆T
⎣ Fr ⎦ ∆T
⎣ Fr ⎦ ∆T
derive the RGA matrix Λ from the matrix of gain coefficients Pm*.
revised 2004 Nov 9
Dr. Barry S. Johnston, Copyright 2004. 1
Fall 2004 ICE Topics: Process Control by Design
Problem Set 3: Calculating RGA and DC
10.492
( )
Λ = Pm* ⊗ Pm*
•
−1 T
(please use the nomenclature of the Lecture Notes)
(3) For each variable in the shower process, state your reference value and scaling (max-min)
limits. Be sure to include units. Then calculate numeric values for the RGA.
• Which pairing of variables do you recommend for control?
• Can you find other reference values that are reasonable, but would lead you to the
opposite conclusion?
(4) Calculate DC for various combinations of the disturbance variables. The disturbance
variables should be set at their minimum, zero, and maximum deviations. Express the results as
a plot of DC versus the angle of the disturbance vector, as shown in the Notes.
• (the example in the Notes presumed that each reference value was centered in its
operating range – your reference and scaling values may well be different, so that your
angles may not be uniformly spaced at 45°.)
revised 2004 Nov 9 Dr. Barry S. Johnston, Copyright 2004.
2