Tugas Matakuliah Instrumentasi dan Pengukuran

Dwindra W M (23814003)

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

Thus

10

)60(exp1795,2795,4)( ttV

second

Vol

tage

(mV

)

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)(

)()()(

b. -increase frequency cut-off of LPF > 0,063Hz in order to let frequency of 4th sine wave of the input.-Reduce time constant of the system, in order to stable the gain at 1 and reduce angle phase

)()(0)( tTtTtE M

0

011

1

111

1)(

1)(

2222

2222

jG

jG

00)(tan0

)(tan01

1