Process Dynamics and Control Solutions
February 6, 2017 | Author: ciotti6209 | Category: N/A
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7.1
In the absence of more accurate data, use a first-order transfer function as T '( s ) Ke −θs = Qi '( s ) τs + 1 o T (∞) − T (0) (124.7 − 120) F = = 0.118 ∆qi 540 − 500 gal/min θ = 3:09 am – 3:05 am = 4 min
K=
Assuming that the operator logs a 99% complete system response as “no change after 3:34 am”, 5 time constants elapse between 3:09 and 3:34 am. 5τ = 3:34 min − 3:09 min = 25 min τ = 25/5 min = 5 min Therefore, T '( s ) 0.188e−4 s = Qi '( s ) 5s + 1 To obtain a better estimate of the transfer function, the operator should log more data between the first change in T and the new steady state.
7.2 h(5.0) − h(0) (6.52 − 5.50) min = = 0.336 2 ∆qi 30.4 × 0.1 ft Output at 63.2% of the total change Process gain, K = a)
= 5.50 + 0.632(6.52-5.50) = 6.145 ft Interpolating between h = 6.07 ft
and
h = 6.18 ft
Solution Manual for Process Dynamics and Control, 2nd edition, Copyright © 2004 by Dale E. Seborg, Thomas F. Edgar and Duncan A. Mellichamp.
7-1
τ = 0.6 +
(0.8 − 0.6) (6.145 − 6.07) min = 0.74 min (6.18 − 6.07)
b)
dh h(0.2) − h(0) 5.75 − 5.50 ft ft ≈ = = 1.25 dt t = 0 0.2 − 0 0.2 min min Using Eq. 7-15, τ=
c)
KM 0.347 × (30.4 × 0.1) = = 0.84 min 1.25 dh dt t =0
h(t i ) − h(0) The slope of the linear fit between ti and z i ≡ ln 1 − gives an h ( ∞ ) − h ( 0) approximation of (-1/τ) according to Eq. 7-13. Using h(∞) = h(5.0) = 6 .52, the values of zi are ti 0.0 0.2 0.4 0.6 0.8 1.0 1.2
zi 0.00 -0.28 -0.55 -0.82 -1.10 -1.37 -1.63
ti 1.4 1.6 1.8 2.0 3.0 4.0 5.0
zi -1.92 -2.14 -2.43 -2.68 -3.93 -4.62 -∞
Then the slope of the best-fit line, using Eq. 7-6 is
1 13Stz − St S z slope = − = 2 τ 13Stt − ( St )
(1)
where the datum at ti = 5.0 has been ignored. Using definitions, St = 18.0 S z = −23.5
Stt = 40.4 Stz = −51.1
Substituting in (1), 1 − = −1.213 τ
τ = 0.82 min
7-2
d) 6.8
6.6
6.4
6.2
6 Experimental data Model a) Model b) Model c)
5.8
5.6
5.4
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Figure S7.2. Comparison between models a), b) and c) for step response.
7.3
a)
T1′( s ) K1 = Q′( s ) τ1s + 1
T2′( s ) K2 = T1′( s ) τ2 s + 1
T2′( s ) K1 K 2 K1 K 2 e −τ2 s = ≈ Q′( s ) (τ1s + 1)(τ2 s + 1) (τ1s + 1)
(1)
where the approximation follows from Eq. 6-58 and the fact that τ1>τ2 as revealed by an inspection of the data. T1 (50) − T1 (0) 18.0 − 10.0 = = 2.667 ∆q 85 − 82 T (50) − T2 (0) 26.0 − 20.0 K2 = 2 = = 0.75 T1 (50) − T1 (0) 18.0 − 10.0
K1 =
Let z1, z2 be the natural log of the fraction incomplete response for T1,T2, respectively. Then,
7-3
T (50) − T1 (t ) 18 − T1 (t ) = ln z1 (t ) = ln 1 8 T1 (50) − T1 (0) T (50) − T2 (t ) 26 − T2 (t ) z2 (t ) = ln 2 = ln 6 T2 (50) − T2 (0) A graph of z1 and z2 versus t is shown below. The slope of z1 versus t line is –0.333 ; hence (1/-τ1)=-0.333 and τ1=3.0 From the best-fit line for z2 versus t, the projection intersects z2 = 0 at t≈1.15. Hence τ2 =1.15. T1 ' ( s ) 2.667 = Q ' ( s ) 3s + 1 T2 ' ( s ) 0.75 = T1 ' ( s ) 1.15s + 1
(2) (3)
0.0 -1.0
0
5
10
15
20
-2.0 z 1,z 2
-3.0 -4.0 -5.0 -6.0 -7.0 -8.0 time,t
Figure S7.3a. z1 and z2 versus t
By means of Simulink-MATLAB, the following simulations are obtained 28
26
24
22
20
T1 , T2
b)
18
16
T1 T2 T1 (experimental) T2 (experimental)
14
12
10
0
2
4
6
8
10
12
14
16
18
20
22
time
Figure S7.3b. Comparison of experimental data and models for step change
7-4
7.4 Y (s) = G( s) X ( s) =
2 1.5 × (5s + 1)(3s + 1)( s + 1) s
Taking the inverse Laplace transform y (t ) = -75/8*exp(-1/5*t)+27/4*exp(-1/3*t)-3/8*exp(-t)+3
a)
Fraction incomplete response y (t ) z (t ) = ln 1 − 3 0.0 -1.0 0
10
20
30
40
50
-2.0
z(t)
-3.0 -4.0 -5.0 -6.0 -7.0 z(t) = -0.1791 t + 0.5734
-8.0 -9.0
time,t
Figure S7.4a. Fraction incomplete response; linear regression
From the graph, slope = -0.179 and intercept ≈ 3.2 Hence, -1/τ = -0.179 and τ = 5.6 θ = 3.2 G (s) = b)
2e −3.2 s 5.6 s + 1
In order to use Smith’s method, find t20 and t60 y(t20)= 0.2 × 3 =0.6 y(t60)= 0.6 × 3 =1.8 Using either Eq. 1 or the plot of this equation, t20 = 4.2 , t60 = 9.0 Using Fig. 7.7 for t20/ t60 = 0.47 ζ= 0.65 , t60/τ= 1.75, and τ = 5.14
7-5
(1)
G (s) ≈
2 26.4 s + 6.68s + 1 2
The models are compared in the following graph: 2.5
2
1.5
y(t)
Third-order model First order model Second order model 1
0.5
0
0
5
10
15
20
25
30
35
40
time,t
Figure S7.4b. Comparison of three models for step input
7.5 The integrator plus time delay model is K G(s) e −θs s In the time domain, y(t) = 0 t
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