Dynamics of Thermometer

8/4/2019 Dynamics of Thermometer

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EXPERIMENT -2

STUDYING THE DYNAMICS OF THERMOMETER WITH AND WITHOUT

THERMOWELL

1. Objective

To determine the time constant of first order and second order kinematics of

thermometer.

2. Aim

To study the dynamics of thermometer with and without thermowell.

3. Theory

A simple mercury thermometer is a first order system under following assumptions:3.1 All the resistance to heat transfer resides in the film surrounding the bulb.

3.2 All the thermal capacity is in the mercury.

3.3 The glass wall containing the mercury does not expand or contract during the

transcient response.

The transfer function of the system is as follows

Y(s)/X(s) = 1/( ) (1)

Where = time constant

= mC/hA (2)

The time response of first order system is given by

Y(t) =0 t<0 (3)

Y(t) = A(1 - ()) t>0 (4)

The introduction of thermowell leads to second order system due to additional

resistance and capacitance of the coil. The transfer function is of the form

Y(s)/X(s) = 1/( +2 ) (5)

The system is overdamped with the values of and calculated using methods such as

slope intercept method, method of moments, method of Harriot.



The time response of first order system is given by

Y(t) = 0 t<0 (6)

Y(t) = A(1 - ()) t>0 (7)

4. Procedure

Steady state temperature of the thermometer was noted before switching on the

hot air blower.

Hot air blower was switched on and fall in temperature with respect to change in

time was noted till the steady state value was attained.

Then the air blower was switched off and the fall in temperature with respect to

time was noted till steady state was attained.

Above steps were repeated with the introduction of thermowell.

.

5. Calculation Without Thermowell

5.1 During Heating

T = A (1 - ())

T is deviation variable that is actual temperature reading of thermometer

A = 61-30 = 31°C

T/A = (1 -

()

)

() = 1-T/A

t/ = -ln(1-T/A)

so a plot of –ln(1-T/A) vs t gives the slope 1/



Table 1 Summary of calculation of thermometer without thermowell during heating

Temperature(°C) Time (s)

T Deviation

variable(°C) 1-T/A -ln(1-T/A)

30 0 0 1 0

31 4 1 0.967742 0.03279

32 6 2 0.935484 0.066691

35 11 5 0.83871 0.175891

36 13 6 0.806452 0.215111

38 16 8 0.741935 0.298493

41 23 11 0.645161 0.438255

44 27 14 0.548387 0.600774

45 29 15 0.516129 0.661398

46 31 16 0.483871 0.725937

47 33 17 0.451613 0.79493

48 36 18 0.419355 0.869038

49 39 19 0.387097 0.949081

50 43 20 0.354839 1.036092

51 45 21 0.322581 1.131402

52 49 22 0.290323 1.236763

53 53 23 0.258065 1.354546

54 57 24 0.225806 1.488077

55 64 25 0.193548 1.642228

56 70 26 0.16129 1.824549

57 77 27 0.129032 2.047693

58 90 28 0.096774 2.335375

59 109 29 0.064516 2.74084

60 136 30 0.032258 3.433987

61 178 31 0



Fig.1 Variation of –ln(1-T/A) vs time for thermometer without thermowell during heating

From Fig. 1 By linear regression of data we get

1/ =0.02644

= 37.82 s

0

0.5

1

1.5

2

2.5

3

3.5

4

0 50 100 150 200

- l n ( 1 - T

/ A

)

Time (s)



5.2 During Cooling

A = 31°C

Table 2 Summary of calculation of thermometer without thermowell during cooling

Temperature(°C)

Time

(s)

T Deviation

variable(°C) 1-T/A ln(1-T/A)

61 0 0 1 0

60 14 1 0.967742 0.03279

59 17 2 0.935484 0.066691

58 21 3 0.903226 0.101783

57 25 4 0.870968 0.13815

56 29 5 0.83871 0.175891

55 33 6 0.806452 0.21511154 38 7 0.774194 0.255933

53 43 8 0.741935 0.298493

52 49 9 0.709677 0.342945

51 56 10 0.677419 0.389465

50 62 11 0.645161 0.438255

49 70 12 0.612903 0.489548

48 77 13 0.580645 0.543615

47 88 14 0.548387 0.600774

46 98 15 0.516129 0.661398

45 108 16 0.483871 0.725937

44 120 17 0.451613 0.79493

43 137 18 0.419355 0.869038

42 156 19 0.387097 0.949081

41 171 20 0.354839 1.036092

40 182 21 0.322581 1.131402

39 198 22 0.290323 1.236763

38 231 23 0.258065 1.354546

37 249 24 0.225806 1.488077

36 279 25 0.193548 1.642228

35 333 26 0.16129 1.824549

34 399 27 0.129032 2.047693

33 481 28 0.096774 2.335375

32 580 29 0.064516 2.74084

31 750 30 0.032258 3.433987

30 1155 31 0 0



Fig.2 Variation of –ln(1-T/A) vs time for thermometer without thermowell during cooling

From Fig. 2

1/ = 0.004698

= 212.8565 s

6. Calculation With Thermowell

Method of moments is used for the calculation of and

Curves of (1-Y/A) vs t and t(1-Y/A) vs t are drawn and

∫ (1-Y/A)dt =

∫ t(1-Y/A)dt = 12

+ 22

+ 1 2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 50 100 150 200 250

- l n

( 1 - T / A

)

time (s)



6.1 During heating

Table 2 Summary of calculation of thermometer with thermowell during heating

Time(s) Temperature (°C) Y(deviation variable) (°C) (1-Y/A) t(1-Y/A)

0 30 0 1 0

17 31 1 0.96875 16.46875

30 32 2 0.9375 28.125

41 33 3 0.90625 37.15625

50 34 4 0.875 43.75

60 35 5 0.84375 50.625

67 36 6 0.8125 54.4375

76 37 7 0.78125 59.375

86 38 8 0.75 64.5

93 39 9 0.71875 66.84375102 40 10 0.6875 70.125

111 41 11 0.65625 72.84375

122 42 12 0.625 76.25

134 43 13 0.59375 79.5625

143 44 14 0.5625 80.4375

153 45 15 0.53125 81.28125

163 46 16 0.5 81.5

174 47 17 0.46875 81.5625

188 48 18 0.4375 82.25

203 49 19 0.40625 82.46875

209 50 20 0.375 78.375

231 51 21 0.34375 79.40625

253 52 22 0.3125 79.0625

270 53 23 0.28125 75.9375

298 54 24 0.25 74.5

325 55 25 0.21875 71.09375

354 56 26 0.1875 66.375

395 57 27 0.15625 61.71875

440 58 28 0.125 55

507 59 29 0.09375 47.53125

585 60 30 0.0625 36.5625

680 61 31 0.03125 21.25

918 62 32 0 0



Fig.3 Variation of (1-Y/A) vs time for thermometer with thermowell during heating

Fig. 4 Variation of t(1-Y/A) vs time for thermometer with thermowell during heating

From Fig. 3

∫ (1-Y/A)dt = = 219.3438

From Fig. 4

∫ t(1-Y/A)dt = 12

+ 22

+ 1 2= 40804.94

0

0.2

0.4

0.6

0.8

1

1.2

0 100 200 300 400 500 600 700 800 900 1000

1 - Y

/ A

time (s)

0

10

20

30

40

50

60

70

80

90

0 200 400 600 800 1000

t ( 1 - Y

/ A )

time (s)



Solving for and gives

= 40.96 s

= 178.3838 s

6.2 During Cooling

Table 2 Summary of calculation of thermometer with thermowell during cooling

Time (s) Temperature (°C)

Y deviation

variable(°C) (1-Y/A) t(1-Y/A)

0 61 0 1 0

56 60 1 0.966667 54.13333

85 59 2 0.933333 79.33333

114 58 3 0.9 102.6

143 57 4 0.866667 123.9333173 56 5 0.833333 144.1667

204 55 6 0.8 163.2

236 54 7 0.766667 180.9333

266 53 8 0.733333 195.0667

304 52 9 0.7 212.8

344 51 10 0.666667 229.3333

379 50 11 0.633333 240.0333

416 49 12 0.6 249.6

467 48 13 0.566667 264.6333

515 47 14 0.533333 274.6667563 46 15 0.5 281.5

607 45 16 0.466667 283.2667

660 44 17 0.433333 286

715 43 18 0.4 286

777 42 19 0.366667 284.9

836 41 20 0.333333 278.6667

902 40 21 0.3 270.6

988 39 22 0.266667 263.4667

1078 38 23 0.233333 251.5333

1184 37 24 0.2 236.8

1290 36 25 0.166667 215

1430 35 26 0.133333 190.6667

1596 34 27 0.1 159.6

1775 33 28 0.066667 118.3333

2024 32 29 0.033333 67.46667

2421 31 30 0 0



Fig. 5 Variation of (1-Y/A) vs time for thermometer with thermowell during cooling

Fig. 6 Variation of t(1-Y/A) vs time for thermometer with thermowell during cooling

From Fig. 5

∫ (1-Y/A)dt = = 711.25

From Fig. 6

∫ t(1-Y/A)dt = 12

+ 22

+ 1 2= 413098.4

0

0.2

0.4

0.6

0.8

1

1.2

0 500 1000 1500 2000 2500 3000

1 - Y

/ A

Time (s)

0

50

100

150

200

250

300

350

0 500 1000 1500 2000 2500 3000

t ( 1

- Y / A )

Time (s)



Solving for and gives

= 172.03 s

= 539.176 s

7. Result

Temperature vs. time graphs were plotted for first and second order dynamics

The time constants for thermometer were found to be

1) Without Thermowell

During Heating = 37.82 s

During Cooling = 212.8565 s

Average value is (37.82+212.8565)/2 = 125.33825 s

2) With Thermowell

During Heating = 40.96 s

= 178.3838 s

During Cooling = 172.03 s

= 539.176 s

Average value of is (40.96+172.03)/2 = 106.495 s

Average value of is (178.3838+539.176)/2 = 358.7799 s

8. Discussion

Higher during cooling is due to the reason that it is not strictly a step input. Air inside

the tube does not drop instantaneously to final steady state value when blower is

switched off.

9. Conclusion

Dynamics of thermometer is studied with and without thermowell. The study of the

dynamic response of the thermometer has shown that the time constant of the particular



system without thermowell is about 125.33825 s which is fairly large. The study of the

dynamic response of the thermometer has shown that the time constant of the particular

system with thermowell are = 106.495 s and =358.7799 s which are fairly large.

10. Precautions

There may be human error.

There may be erroneous reading because of parallax.

11. Reference

[1] Lab Manual of study of dynamics of thermometer with and without thermowell.

Documents

Dynamics of Thermometer