1
, Chapters 2-15 & Powerpoint Slides Covered
Chapter 2
2.1 The equation for conservation of mass with a constant density simplifies to
dV / dt q f q , where V is the volume2oHf liquid at any time. V(t) = BLh 2/2H. With a bit
dh2
of manipulation we then obtain q f q , which integrates for constant q f q
dt BL
2H
to h h
2 2
o
q qt t o , where ho = h(to). The height is then
BL f
2H
h(t) ho
2
qf qt to .
BL
2.2 This is hokey problem, but it is a good exercise in writing a mass balance.
M < Mo, dM/dt = W1; M ≥ Mo, dM/dt = W1 – k(M – Mo)
(a) M = W1t for M < Mo . Thus to = Mo/W1.
(b) The simplest way to integrate the equation for t > to (i.e., M > Mo) is to define a new
variable u = M – Mo – W1/k, which turns the equation into du/dt = –ku, which is
k t Mo
W W1
separable. u = –W /k at t = M /W . The solution is M (t) M 1 1 e
1 o 1 . A
o
k
W1
steady state M M o is reached as t → ∞
k
d (h h* ) K fb * Q(t)
2.3 (a) The equation for the height becomes (h h ) (1 K ff ) ,
dt A A
t
e K fbt / A
with a solution h(t) h* K fb/ A (1 K ff )Q()d in analogy to the development
A 0 e
leading to Eq. 2.9. If Kff = 1 the solution is h = h* for all time; i.e., perfect control. If Kff <
1 the steady-state offset is reduced to Q*(1 – Kff)/Kfb. There is no reason to use Kfb > 1,
which would change the sign of the steady-state offset.
t
e K fbt / A
(b) Now h(t) h *
K fb/ A
[Qu() (1 K ff)Q m()]d. Clearly the design
A 0 e
equation for Kff = 1 is identical the that in Section 2.6.3, with Q(t) replaced by Qu(t). A
conservative design for the feedback controller design for Kff < 1 would use the
maximum expected value of Qu + (1 – Kff)Qm in the development in Section 2.6.3.
(c) Let u = th – h*. The equation for the height then becomes
du
A K u K u()dQ(t) . If we differentiate this once with respect to t and
I
dt fb
0
t
d u()d u(t) we obtain d 2u dQ
recall that dt A 2 K fb
du
KI u . This is identical to
0 dt dt dt
the equation for a forced damped harmonic oscillator. If Q = constant the right-hand side
goes to zero, and we know that an unforced damped oscillator will decay to u = 0, or h =
h*.
2
, Chapters 2-15 & Powerpoint Slides Covered
Chapter 2
2.1 The equation for conservation of mass with a constant density simplifies to
dV / dt q f q , where V is the volume2oHf liquid at any time. V(t) = BLh 2/2H. With a bit
dh2
of manipulation we then obtain q f q , which integrates for constant q f q
dt BL
2H
to h h
2 2
o
q qt t o , where ho = h(to). The height is then
BL f
2H
h(t) ho
2
qf qt to .
BL
2.2 This is hokey problem, but it is a good exercise in writing a mass balance.
M < Mo, dM/dt = W1; M ≥ Mo, dM/dt = W1 – k(M – Mo)
(a) M = W1t for M < Mo . Thus to = Mo/W1.
(b) The simplest way to integrate the equation for t > to (i.e., M > Mo) is to define a new
variable u = M – Mo – W1/k, which turns the equation into du/dt = –ku, which is
k t Mo
W W1
separable. u = –W /k at t = M /W . The solution is M (t) M 1 1 e
1 o 1 . A
o
k
W1
steady state M M o is reached as t → ∞
k
d (h h* ) K fb * Q(t)
2.3 (a) The equation for the height becomes (h h ) (1 K ff ) ,
dt A A
t
e K fbt / A
with a solution h(t) h* K fb/ A (1 K ff )Q()d in analogy to the development
A 0 e
leading to Eq. 2.9. If Kff = 1 the solution is h = h* for all time; i.e., perfect control. If Kff <
1 the steady-state offset is reduced to Q*(1 – Kff)/Kfb. There is no reason to use Kfb > 1,
which would change the sign of the steady-state offset.
t
e K fbt / A
(b) Now h(t) h *
K fb/ A
[Qu() (1 K ff)Q m()]d. Clearly the design
A 0 e
equation for Kff = 1 is identical the that in Section 2.6.3, with Q(t) replaced by Qu(t). A
conservative design for the feedback controller design for Kff < 1 would use the
maximum expected value of Qu + (1 – Kff)Qm in the development in Section 2.6.3.
(c) Let u = th – h*. The equation for the height then becomes
du
A K u K u()dQ(t) . If we differentiate this once with respect to t and
I
dt fb
0
t
d u()d u(t) we obtain d 2u dQ
recall that dt A 2 K fb
du
KI u . This is identical to
0 dt dt dt
the equation for a forced damped harmonic oscillator. If Q = constant the right-hand side
goes to zero, and we know that an unforced damped oscillator will decay to u = 0, or h =
h*.
2
