tugas matakuliah instrumentasi dan pengukuran.ppt
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Problem 4.8• A thermocouple is used to measure the temperature inside a
vessel, which is part of a high-speed batch process. At time t = 0, with the vessel at an initial temperature of 50 °C, the vessel is instantaneously filled with gas at 150°C. One minute later, instantaneously the gas is removed and the vessel is filled with liquid at 50°C. The thermocouple can be regarded as having linear steady-state characteristics and first-order dynamics.
• (a) Use the data given below to sketch a graph of how the thermocouple e.m.f. changes with time. The axes of the graph should have suitable scales and the answer should include supporting numerical calculations.
• (b) Comment on whether the thermocouple is suitable for this application.
• (c) What modifications should be made?
Ans
a. First condition(with unit step 150 oC of gas temperature with specify time of 60s)
16
23
12
12
11
2
1040/
10
0.1
2.0
2.0
105
CVxCV
mA
CWmU
CWmU
CJkgC
KgxM
o
ol
og
o
sx
xxAU
MC
sx
xxAU
MC
gl
gg
10101
2.0105
50102.0
2.0105
3
2
3
2
mVxxTTxVV
CTTT
mVxxTxTVV
CT
mVxxTxTVV
CT
o
o
oo
41001040/
100
61501040/
150
2501040/
50
6
01
611
1
600
In this case
At t=60s, we find
Second condition(with unit step 50 C of liquid temperature)
Let the result of V(60) first condition is the value of emf
Let
as response of first-order element with unit step. thus we got first order equation
ttF o exp1)(
tVVtV o
exp1)(
50exp142)( ttV
mVxV 795,46988,0425060exp142)60(
mVmVVVVmVxxTxTVV
CT
mVVo
795,2795,422501040/
50
758,4
01
611
1
0
b. Not suitablec. Change the value of smaller
which C and U is constant
-Change mass of thermocouple (M) smaller -Change surface area of thermocouple (A) greater-add specify time value (t) more than 5
AUMC
gg
g
g
Problem 4.9A temperature measurement system for a gas reactor consists of linear elements and has an overall steady-state sensitivity of unity. The temperature sensor has a time constant of 5.0s; an ideal low-pass filter with a cut-off frequency of 0.05 Hz is also present. The input temperature signal is periodic with period 63 s and can be approximated by the Fourier series:
where ω0 is the angular frequency of the fundamental component.(a) Calculate expressions for the time response of:
(i) the system output signal(ii) the system dynamic error.
(b) Explain what modifications are necessary to the system to minimize the dynamic error in (a).
NoteAn ideal low-pass filter has a gain of one and zero phase shift up to the cut-off frequency. The gain is zero above the cut-off frequency.
)4sin413sin
312sin
21(sin10)( tttttT oooo
Solv
i. Open loop
n
noon tnjnGItO1
)sin()()(
HzfT
sT
s
c
o
05,0
1,0263
5
)5,261,0sin(8944,01010
015,021,0
22
56,26)51,0(tan)(tan
8944,051,01
1
1
1)(
11
1111
11
11
22221
o
o
txOI
Hzff
x
jG
HzfLPF
c 05,0s
lethermocoup
511
)(tTM)(tT
)452,0sin(7170,0210
210
031,022,0
22
45)51,02(tan)2(tan
7071,0)51,02(1
1)2(1
1)2(
22
1
2222
112
22
o
oo
o
tOI
Hzff
xx
xxjG
)3,563,0sin(55,0310
310
047,023,0
22
3,56)51,03(tan)3(tan
554,0)51,03(1
1
)3(1
1)3(
33
3333
113
22
o
oo
o
tOI
Hzff
xx
xxjG
Because frequency upper frequency cut-off of low pass filter, thus
41
063,024,0
22
43,63)51,04(tan)4(tan
447,0)51,04(1
1)4(1
1)4(
4
4444
114
22
I
Hzff
xx
xxjG
oo
o
)3,563,0sin(
355,0)5,262,0sin(
2707,0)5,261,0(sin(894,010)( ooo
M ttttT
04 O
ii. Dynamic Error
or)()()( tItOtE
ttt
tttttE
tttt
ttttE
tTtTtE
o
oo
oooo
ooo
M
4,0sin4103,0sin)3,563,0sin(55,0
310
)2,0sin()5,262,0sin(707,0210)1,0sin()5,261,0(sin(894,010)(
)4sin413sin
312sin
21(sin10
)3,563,0sin(355,0)5,262,0sin(
2707,0)5,261,0(sin(894,010)(
)()()(
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