Development of Characteristic Equation for Thermometer

P M V SubbaraoProfessor

Mechanical Engineering Department

Clues for Design of a thermometer ….

Common Thermometers

(1) An element

(2) Bobbin

(3) Sheath

(4) Connecting wires

Liquid in Glass

Thermometer

Thermocouple Thermometer

Resistance Thermometer

Development of Characteristic Equation for a Thermometer

Conservation of Energy during a time dt

Heat in – heat out = Change in energy of thermometer

System theof eTemperatur ousInstantanetTs

tTs tTtf

fluid ric thermometof eTemperatur ousInstantanetTtf

Thermal Resistance of Thermometer Bulb

Rg

Rs Rg Rd

Ts(t) Ttf(t)

Thermometric fluidAssume no losses from the stem.

Heat in = Change in energy of thermometer

System heat transfer coefficients in W/m2K @ 20 °C

tfgs RRRU

1

dtTTUAinheat tfsbulb

Change in energy of thermometer: tftfbulb dTCV

tftfbulbtfsbulb TCVdtTTUA

sbulbtfbulbtf

tfbulb TUATUAdt

dTCV

Thermal Resistance of Thermometer Bulb

Rg

The energy balance for mercury in the bulb:

igbulbtftfbulbtftf

tfbulb TAhTAhdt

dTCV .

The energy balance for the glass wall has both inflow and outflow of heat:

)()( ,,,

tfigbulbtfogsbulbsavgg

gg TTAhTTAhdt

dTCm

)()( tfgbulbtfggsbulbgsg

gg TTAUTTAUdt

dTCm

gbulbtfgtfbulbtfgtf

tfbulb TAUTAUdt

dTCV

Substitution of relevant expressions and after manipulations we get:

stftftf DTCT

dt

dTB

dt

TdA

2

2

)()( tfgbulbtfggsbulbgsg

gg TTAUTTAUdt

dTCm

tftf

bulbtfg

tfbulbg T

dt

dT

AU

CVT

stfntf

ntf TT

dt

dT

dt

Td 2

2

2

2

stftf

bulb

tfbulb TTdt

dT

UA

CV

bulb

tfbulb

UA

CV Define Time constant

stftf TT

dt

dT

s

Tsss s

tftf

s

Tss s

tf 1 1ss

Ts s

tf

Step Response of Thermometers : First Order Model

1s

T

s

Ts ss

tf

1

s

T

s

Ts ss

tf

1

11

ss

Ts stf

1

111

ss

TtT stf

t

TtT stf exp1

t

TtT stf exp1

bulb

tfbulb

UA

CV

Step Response of Thermometers : First Order Model

stfntf

ntf TT

dt

dT

dt

Td 2

2

2

2

22

2

2 nn

nstf sss

TST

Transient response of Liquid-in-Glass Thermometer

0

2

4

6

8

10

0 100 200 300 400 500 600

Time, s

Tem

pera

ture

(in

devi

atio

n fo

rm),

C

Response of Thermometers: Periodic Loading

• If the input is a sine-wave, the output response is quite different;

• but again, it will be found that there is a general solution for all situations of this kind.

stftf

bulb

tfbulb TTdt

dT

UA

CV

22

max,max

s

stSinTT ssss

22

max,

ssss s

tftf

22

max,1

sss s

tf

122

max,

ss

s stf

122

max,1

sstT s

tf

t

tTe

TtT s

t

stf

1

22

max,

22max,

tan

sin11

Ts,

max

- T

tf,m

ax

Temperature: Standards & Insturuments

P M V SubbaraoProfessor

Mechanical Engineering Department

Selection of an Instrument for a Range….

International Practice

• The thermodynamic scale of temperature (or the absolute scale) was derived from the second law of thermodynamics.

• This scale is independent of any thermometric substance. • The unit for the absolute scale is K - Kelvin, named after Lord

Kelvin (William Thomson). • 1K= 1/273.16 th of the temperature at the triple point of water.• For practical purposes an International Practical Temperature

Scale, IPTS-68 (adopted in 1968) has been used for 22 years. • This scale was based on a number of fixed and easily

reproducible points. • Definite numerical values of temperature were assigned to these

points. • For interpolation purposes between the fixed points, specified

formulas relating temperature to the readings on certain temperature-measuring instruments were used.

NN Fixed points : ipts1968 Temperature, C

1. Triple point of equilibrium-hydrogen (s+l+v) -259.34

2. Boiling point of equilibrium hydrogen (l+v) at 33.33 kPa -256.108

3. Normal boiling point of equilibrium hydrogen at 101325 Pa

-252.87

4. Normal boiling point of neon -246.048

5. Triple point of oxygen -218.789

6. Normal boiling point of oxygen -182.962

7. Triple point of water 0.01

8. Normal boiling point of water 100

9. Normal freezing point of zinc (s+l) at 101325 Pa 419.58

10. Normal freezing point of silver (s+l) at 101325 Pa 961.93

11. Normal freezing point of gold (s+l) at 101325 Pa 1064.43

N Fixed points:its1990 Temperature, C

1. Normal boiling point of helium -270.15 to --268.15

2. Triple point of equilibrium-hydrogen (s+l+v) 259.3467

3. Boiling point of equilibrium hydrogen (l+v) at 33.33 kPa -256.15

4. Normal boiling point of equilibrium hydrogen at 101325 Pa -252.85

5. Triple point of neon -248.5939

6. Triple point of oxygen -218.7916

7. Triple point of argon -189.3442

8. Triple point of mercury -38.8344

9. Triple point of water 0.01

10. Melting point of gallium 29.7646

11. Normal freezing point of indium (s+l) at 101325 Pa 156.5985

12. Normal freezing point of tin (s+l) at 101325 Pa 231.928

13. Normal freezing point of zinc (s+l) at 101325 Pa 419.527

14. Normal freezing point of aluminium (s+l) at 101325 Pa 660.323

15. Normal freezing point of silver (s+l) at 101325 Pa 961.78

16. Normal freezing point of gold (s+l) at 101325 Pa 1064.18

17. Normal freezing point of copper (s+l) at 101325 Pa 1084.62

Accuracy Realization of Temperature Scale