development of characteristic equation for thermometer p m v subbarao professor mechanical...
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Development of Characteristic Equation for Thermometer
P M V SubbaraoProfessor
Mechanical Engineering Department
Clues for Design of a thermometer ….
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Common Thermometers
(1) An element
(2) Bobbin
(3) Sheath
(4) Connecting wires
Liquid in Glass
Thermometer
Thermocouple Thermometer
Resistance Thermometer
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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
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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
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System heat transfer coefficients in W/m2K @ 20 °C
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tfgs RRRU
1
dtTTUAinheat tfsbulb
Change in energy of thermometer: tftfbulb dTCV
tftfbulbtfsbulb TCVdtTTUA
sbulbtfbulbtf
tfbulb TUATUAdt
dTCV
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Thermal Resistance of Thermometer Bulb
Rg
The energy balance for mercury in the bulb:
igbulbtftfbulbtftf
tfbulb TAhTAhdt
dTCV .
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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
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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
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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
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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
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t
TtT stf exp1
bulb
tfbulb
UA
CV
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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
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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
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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
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22
max,
ssss s
tftf
22
max,1
sss s
tf
122
max,
ss
s stf
122
max,1
sstT s
tf
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t
tTe
TtT s
t
stf
1
22
max,
22max,
tan
sin11
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Ts,
max
- T
tf,m
ax
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Temperature: Standards & Insturuments
P M V SubbaraoProfessor
Mechanical Engineering Department
Selection of an Instrument for a Range….
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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.
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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
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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
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Accuracy Realization of Temperature Scale
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