Better Standards, Better Life 1
June, 2014
Dr. Yong-Gyoo Kim
Center for Thermometry,
Division of Physical Metrology
Korea Research Institute of Standards and Science
Thermocouple thermometry
Better Standards, Better Life 2
Part I. Heat and Temperature
1. Unit and Scale of Temperature
2. Heat Transfer and Thermal Properties
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1. Unit and scale of Temperature
Lord Kelvin Ludwig Boltzmann
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1.1 What is temperature ?
Temperature Degree of hotness or coldness of objects
intuitively
Exactly, level of thermal energy (kT) k : Boltzmann constant
Kinetic energy due to a microscopic motion of particles, energy due to phonon ½ mv2 = ½ kT
Phonon: quantum mechanical description of an elementary vibrational motion
Heat is caused from the vibration of particles
Temperature is a quantitative description of this vibrational motion.
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Temperature Kelvin Degrees Celsius
Coldest temperature achieved 100 pK −273.149 999 999 900 °C
Triple point of water 273.16 K 0.01 °C
Sun’s visible surface 5,778 K 5,505 °C
Lightening bolt’s channel 28 kK 28,000 °C
Sun’s core 16 MK 16,000,000 °C
Thermonuclear weapon peak temperature 350 MK 350,000,000 °C
Core of a high-mass star on its last day 3 GK 3,000,000,000 °C
CERN’s proton vs nucleus collisions 10 TK 10 ×1012 °C
Universe 5.391×10−44 s after Big Bang 1.417×1032 K 1.417×1032 °C
Some examples of temperature
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The first thermometer
In 1594, G. Galilei A type of gas thermometer
Volume change with temperature
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1.2 Unit of temperature
International unit: K or oC Absolute thermodynamic scale : Kelvin, K degree Celsius (oC) Common and practical unit 1 oC = 1 K K = 273.15 + oC
Non SI unit Fahrenheit scale: oF In Europe and USA oF = oC x 9/5 + 32
Rankine scale: oR Chemical engineering in USA oR = K x 9/5
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degree Celsius (oC)
A. Celsius (Sweden ) in 1742
Ice point : 100 oC
Water boiling point : 0 oC
Divided into 100 parts
In 1744, M. Strømer
After Celsius passed away
Reverse the two fixed point value
Ice point :0 oC, boiling point : 100 oC
Currently used with Kelvin internationally
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Kelvin, K
For a reversible Carnot engine, Heat efficiency is proportional to temperature
Absolute temperature ,T, is defined as above. For ideal gases, PV = constant x T At T = 0, volume of gas should be zero.
For real gases, extrapolated volume goes to zero (T = 0 K).
1
2
1
2
TT
=
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Volume vs Temperature (Charles’ Law)
T[K]
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1.3 Temperature scale
Before 1954, Celsius scale used Ice point and boiling point of water
After 1954, thermodynamic scale used Triple point of water (TPW) was defined as definition of
temperature scale TPW 273.16 K
Definition of temperature scale The unit of thermodynamic temperature, the kelvin,
symbol K, is defined as the fraction 1/273.16 of the thermodynamic temperature of the triple point of water
≡
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Triple point cell of Water
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History of temperature scale
ITS-27 Boiling point of oxygen (-182.970 oC) ~ radiation thermometer range 6 fixed-points
ITS-48 Change of Ag fixed-point value Change of interpolation equation of pyrometer Include several secondary fixed-points
IPTS-68 Down to TPW of H2 (13.81 K) 11 fixed-points Change of interpolation equation of standard thermometer
ITS-90 Current temperature scale
PLTS-2000 Down to 0.9 mK
Mise en Pratique of Kelvin (MeP-K) Current definition of Kelvin
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Classification of thermometers
Primary vs. Secondary Primary : Clear equation of state
Measurement of thermodynamic temperature Secondary : Temperature-dependent parameter (voltage, current, length, etc)
Resistance thermometer, thermocouple, etc.
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Standard vs. Industrial thermometer Standard : Realization of temperature scale Gas thermometer Vapor pressure thermometer Standard platinum resistance thermometer Pyrometer
Industrial : Except for the standards Thermocouple, Glass thermometer, Thermistor, etc
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2. Heat Transfer and Thermal Properties Temperature measurement is just measurement of the
heat exchange between thermometer and system.
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2.1 Heat transfer
3 transfer mode Conduction, Convection, Radiation
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A. Conduction
transfer of thermal energy between regions of matter due to a temperature gradient In solid, liquid, gas, and plasma Vibration of solid atoms, collision or diffusion of liquid (gas) atoms
(molecules, radicals) Heat flow rate (watt, W)
k : thermal conductivity ( ) A : area ∆T : temperature gradient ∆x : distance
xTkAQ
∆∆
=ƥ
KW/m ⋅
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Thermal conductivity, k, of various materials
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Thermal conductivity, k, of various materials
Material Thermal conductivity W/(m·K) Material Thermal conductivity
W/(m·K)
Silica Aerogel 0.004 - 0.04 Thermal epoxy 1 - 7
Air 0.025 Glass 1.1
Wood 0.04 - 0.4 Soil 1.5
Hollow Fill Fibre Insulation 0.042 Concrete, stone 1.7
Alcohols and oils 0.1 - 0.21 Ice 2
Polypropylene 0.25 Sandstone 2.4
Mineral oil 0.138 Stainless steel 12.11 ~ 45.0
Rubber 0.16 Lead 35.3
LPG 0.23 - 0.26 Aluminum 237 (pure)120—180 (alloys)
Cement, Portland 0.29 Gold 318
Epoxy (silica-filled) 0.30 Copper 401
Epoxy (unfilled) 0.12 - 0.177 Silver 429
Water (liquid) 0.6 Diamond 900 - 2320
Thermal grease 0.7 - 3 Graphene (4840±440) - (5300±480)
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B. Convection
transfer of heat from one place to another by the physical movement of fluids In liquids, gases Natural (free) convection Forced convection by fan, pump, etc
Heat flow rate (watt, W) between solid and fluid
h : convection heat transfer ( W/m2K) A : area Ts : surface temperature of solid Tenvironm: temperature of fluid far from the surface
)( environms TThAQ −=∆•
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Some examples of convection
Mantle convection
Atmospheric convection Foehn phenomenon
Heat sink radiator
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Convection heat transfer coefficient, h, of some phases
Process h [Wm-2 K-1]
Free Convection Gases 2 ~ 25
Liquids 50 ~ 1 000
Forced Convection Gases 25 ~ 250
Liquids 50 ~ 20 000
Convection with phase change
Boiling or condensation 2 500 ~ 100 000
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C. Radiation
Energy in the form of electromagnetic waves with continuous spectrum
All matter higher than absolute zero temperature
Sunlight is thermal radiation generated by the hot plasma of the Sun.
Stefan-Boltzmann law
E : energy radiated per unit area (W/m2)
ε : emissivity
σ : Stefan-Boltzmann constant
T: Absolute temperature
4TE σε=
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2.2 Thermal properties
Three important properties Thermal conductivity, heat capacity, latent heat
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A. Heat capacity (C)
Heat (q) required to raise the temperature by 1 K C = q/(T2-T1)
Higher C, larger q ⇒ need more time to stabilize
For fast measurement, sensor having higher k and smaller C required.
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Heat capacity of some materials
Material Per unit mass
JK-1kg-1 Per unit volume
JK-1cm-3
Copper Gold Silver Aluminum Zinc Tin Mercury Stainless steel Silicon water Ice Pyrex Alumina (Al2O3)
Magnesia (MgO) Polystyrene Silicone oil
385 129 235 903 389 227
139.3 480 712
4 179 2 040 835 800 960
1 300 1 548
3.44 2.49 2.47 2.44 2.78 1.66 1.88 3.83 1.65 4.17 1.88 1.85 3.04 3.46 1.37 1.45
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B. Latent heat
Heat associated with phase change solid ⇔ liquid ⇔ gas Fixed-points use latent heat of materials
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Simple melting and freezing of pure solid
For melting, Qin = latent heat For freezing, Qout = latent heat Basically, Tmelting = Tfreezing
But little difference due to impurities, solid structure During freezing, supercooling and recovery observed
Require more heat for solidification at initial to overcome the surface energy between solid and liquid
There is a critical size for nucleation of solid
time
tem
pera
ture
Melting
melting plateau
time
Freezing
freezing plateau
supercooling and recovery
tem
pera
ture
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Nucleation in pure metal
Gibbs free energy change, ∆G
Critical value to solidify
Liquid Liquid
Solid
freezing
Energy G1 G2 = G1 + ∆G
SLv
SLSLvs
rGr
AGVG
γππ
γ
23 434
+∆−=
+∆−=∆
S,L: solid, liquid V: volume A : surface area γ : surface energy r : radius of solid Lv : latent heat ∆T : undercooling
m
v
Sv
Lvv
TTL
GGG∆
=
−=∆
TLTr
v
mSL
∆
=
12* γ22
23*
)(1
316
TLTG
v
mSL
∆
=∆
πγ
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2.3 Errors in Use of Thermometers
Thermometer indicates the temperature of a system.
In many cases, perfect measurement is not possible. No thermal equilibrium between thermometer and system Insertion of thermometer disturbs the equilibrium.
Possible and common errors Immersion errors Heat capacity errors Settling response errors Lag errors with steadily changing temperature Radiation errors
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A. Immersion errors
Thermometer stem, sheath and lead wires are acting as path for a continuous heat flow.
Heat absorbed by thermometer Heat lost from thermometer
Temperature profile along thermometer stem
A simple model for error
)exp()(eff
sysambm DLKTTT −
−=∆
• Tsys : Temperature of system • Tamb : Temperature of ambient • L : depth of immersion of sensor • Deff: effective diameter of thermometer • K : constant (~ 1)
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Relative immersion error
For 1 % accuracy to 5 diameters + sensor
length For 0.01 % accuracy
to 10 diameters + sensor length
For 0.0001 % accuracy, to 15 diameters + sensor
length Highest level using fixed-
point measurement
Immersion (diameter)
For K = 1
Rel
ativ
e te
mpe
ratu
re e
rror
(%)
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B. Heat capacity errors
If a thermometer is immersed into system, the temperature of system should be changed to lower or higher temperature.
This is due to heat flow to (from) thermometer from (to) system. In adiabatic condition (there is no heat flow !),
)( sTT
CCCTT ini
ts
tsmeasured −
++=
system
Ts, Cs
Tini, Ct
Tmeasured
equilibrium
• Tmeasured : Measured temperature • Ts : Temperature of system • Tini : Initial temperature of thermometer • Cs, Ct : heat capacity of system and thermometer
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C. Settling response error
In practical cases, for control system, it will take time to replace heat change of thermometer.
If insufficient time is allowed, error due to response will occur.
Time
Tem
pera
ture
1/e
−−=∆
0s exp)(
ττTTT inim
• ∆Tm : temperature error • Ts : Initial temperature of system • Tini : Initial temperature of thermometer • τ : time • τ0 : 1/e time constant
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Relative error
For 1 % accuracy industrial Wait 5 τ0
For 0.01 % accuracy laboratory Wait 10 τ0
For 0.0001 % accuracy, Standard lab Wait 15 τ0
Time (units of τ0)
Perc
enta
ge re
lativ
e te
mpe
ratu
re e
rror
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Part II. Thermocouple Thermometry
1. Thermoelectric Effects
2. Type and Features of Thermocouples
3. Measurement Systems
4. Errors and Usage
5. Calibration and Uncertainty
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1. Thermoelectric Effects Seebeck effect
Temperature difference ⇒ Electric power Peltier effect
Electric power ⇒ Temperature difference Thomson effect
Combine of above two effects
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1.1 Seebeck effects
In 1821, T.J.Seebeck found
Hot Cold current
A
B
A and B are two different metallic wires.
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Basic principle
For a single conductor under the temperature gradient Th side is in higher thermal energy state than Tc. So, electrons in Th side move toward Tc side. Then, number of electrons in Tc side is larger than those in Th
side. It means a non-equilibrium state. Finally, there is a driving force for the movement of electrons
in Tc side toward Th side. This is the thermoelectric voltage, in other words, Seebeck
voltage.
Th Tc
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Temperature coefficient
Seebeck coefficient, S (voltage per oC) E depends on the material and temperature gradient.
Th Tc E
tES
t ∆=
=∆ 0lim
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For a specific material Temperature is an intrinsic property, so it is continuous.
Th Tc E
∫= dttSE )(
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Thermocouple itself is….batteries !
A series of batteries which is operated using temperature gradient.
Tem
pera
ture
Position
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Circuit analysis
Two different conducting wires Single conductor can not be used as a thermocouple.
STdtSS
dtSdtS
EEE
T
TBA
T
TB
T
TA
BAnet
∆⋅∆=−=
+=
+=
∫
∫∫1
0
0
1
1
0
)(
T1 T0 Enet
EA
EB
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Same wires are connected to the circuit.
STdtSS
dtSdtSdtSdtS
EEEEE
T
TBA
T
TC
T
TB
T
TA
T
TC
CBACnet
∆⋅∆=−=
+++=
+++=
∫
∫∫∫∫2
1
0
1
1
2
2
1
1
0
)(
If T1 = zero ?
T2 T0 Enet
EA
EB
T1
EC
EC
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Some part is damaged.
∫∫∫
∫∫∫∫∫∫
−+−+−=
+++++=
++=
3
2
2
1
1
0
0
1
1
2
2
3
3
2
2
1
1
0
)()()( '
'
'
T
TBA
T
TBA
T
TBA
T
TB
T
TB
T
TB
T
TA
T
TA
T
TA
BAAnet
dtSSdtSSdtSS
dtSdtSdtSdtSdtSdtS
EEEE
If T1 = T2 ?
EB
T3 T0 Enet
EA
T1
EA’
T2
Let SA’ = SA + ∆S
∫∫
∫∫∫
∆+−=
−+−∆++−=
2
1
3
0
3
2
2
1
1
0
)()(
)()()(
T
TAB
T
TBA
T
TBA
T
TBABA
T
TBAnet
dtSdtSS
dtSSdtSSSdtSSE
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Ideal vs Practical
For ideal and homogeneous wires E only depends on the temperature gradient and Seebeck
coefficients.
For practical thermocouples E = f(t, S, x)
t : temperature S : Seebeck coefficient x : position
Due to the thermoelectric inhomogeneity Epractical = Ehomo + Einhomo
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Thermoelectric inhomogeneity
Caused from the factors affecting the Fermi energy level of material Composition change
Oxidation or evaporation
Structural change
Defects
Plastic or Elastic deformation
Produced in the local area
No temperature gradient, No effects !
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Calibration uncertainty by KRISS, Type S
Type S
TC
0 2 4 6 8 10 12 14 16 18 20 22
Unc
ertIa
inty
/o C
0.1
0.2
0.3
0.4
0.5
0.6Ref TCOther factorsInhomogeneityCombined
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Calibration uncertainty by KRISS, Type R
Type R
TC
0 1 2 3 4
Unc
erta
inty
/o C
0.0
0.1
0.2
0.3
0.4
0.5
0.6
Ref TCInhomogeneityOther factorsCombined
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Type B
TC
0 1 2 3
Unc
erta
inty
/o C
0
1
2
3
4
5
6Ref TCInhomogeneityOther factorsCombined
Calibration uncertainty by KRISS, Type S
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Change of inhomogeneity with temperature
Type S / R t/c
Little dependence on temperature
within ± 0.01 %
2002 Temperature Symposium by M. Ballico
Possible to estimate inhomogeneity at whole
temperature range.
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KRISS Inhomogeneity Test System
Vertical Liquid bath Operation temperature : 180
oC ~ 200 oC Maximum immersion depth of
43 cm Stepping motor driven by
computer Bath temperature is monitored
by F250 thermometer. Emf is measured using
Keithley 2000 DMM. Ice point as reference
temperature TCs are protected using quartz
tubes.
Motor
TCs
Bath
DMM
Thermometer
Motor controller
Control PC
Ice bath
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Example 1, Type S thermocouple Type S
Immersion depth /mm
0 100 200 300 400 500
Emf d
iffer
ence
(Em -
E 180
o C) /
µV
-140
-120
-100
-80
-60
-40
-20
0
0 100 200 300 400-10
-8
-6
-4
-2
0
∆E = 1.8 µV
%159.0
%1006.1423.1273
8.1
%100 ityInhomogeneC25C180 oo
=
×−
=
×−
∆=
EEE
Uncertainty at 1100 oC due to inhomogeneity (Rectangular distribution)
C0.42μV94.4
32/)5.1075600159.0(
o=
=
×=
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Type B
Immersion depth /mm
0 100 200 300 400 500
Emf d
iffer
ence
(Em -
E 180
o C) /
µV
-14
-12
-10
-8
-6
-4
-2
∆E = 2.2 µV
%56.1
%1003.05.140
2.2
%100 ityInhomogeneC25C180 oo
=
×+
=
×−
∆=
EEE
Example 2, Type B thermocouple
C93.3μV5.45
32/)1.100990156.0(
o=
=
×=Uncertainty at 1500 oC due to inhomogeneity
(Rectangular distribution)
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Dependency of inhomogeneity in KRISS
Type Combined Uncertainty
/oC Uncertainty from
Inhomogeneity /oC Dependency
B 3.45 2.87 83 %
B 3.91 3.42 86 %
S 0.47 0.44 93 %
S 0.40 0.36 90 %
S 0.49 0.46 93 %
S 0.53 0.50 94 %
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2. Type and Features of thermocouples Letter-designated Noble metal thermocouples Base metal thermocouples
Non letter-designated Pure metal thermocouples W/Re thermocouples Others
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2.1 Thermocouple Types
Temperature (oC)
-250 0 250 500 750 1000 1250 1500 1750 2000
EMF
(mV)
-20
0
20
40
60
80
100
E
J
T
KN
RS B
Temperature vs. Emf relation of commercial thermocouples
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2.2 Features of Thermocouples
Type Composition
Properties + -
S R B
Pt90Rh10
Pt87Rh13
Pt70Rh30
Pt Pt Pt94Rh6
Most stable in air. S and R can be used to 1450 oC, B is to 1700 oC. Very sensitive to the metallic impurities. Should not be used in vacuum without protecting tube.
K Ni90Cr10
(Chromel) Ni95(Mn,Al,Si)5
(Alumel)
Most well-known thermocouple. In air or inert atmosphere. Severe degradation at high temperature
N Ni84.5Cr14.2Si1.3
(Nicrosil) Ni95.5(Si,Mg)4.5
(Nisil)
Alternative to the type K Enhance the high temperature stability Newest developed one
T J E
Cu Fe Ni90Cr10
Cu55Ni45
(Constantan)
T is usually used in low temperature range, below 0 oC. J and E are rarely used.
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Non Letter-designated
Type Composition
Properties + -
Au/Pt Pt/Pd Au/Pd
Au Pt Au
Pt Pd Pd
Very high purity wires (5N up) Strong resistance to the composition change Most stable and accurate thermocouples
W3Re/W25Re W97Re3 W75Re25 Up to 2300 oC Specially designed ones for very high temperature. Only for inert, vacuum or hydrogen atmosphere W5Re/W26Re W95Re5 W74Re26
More than several hundreds of thermocouples
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3. Measurement Systems
Thermocouple probe Reference junction Insulator and protecting tube Compensating wire
Readout
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3.1 Thermocouple Probe
Sheath vs. Unsheathed Quick connector
Reference thermocouple
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Color codes
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3.2 Reference junction
Silicon oil
J Cold junction
Stability of ± 5 mK
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3.3 Protecting tube
Temperature
Strength
Surrounding atmosphere
Material composition of protecting tube
Chemical composition of environment
For base metal TCs, metallic tube
For noble metal TCs, ceramic tube
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Some insulators
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3.4 Compensating wire
Economic point of view For noble metal thermocouples
Composition Type
+ - Temperature error
B S, R
K J E T N
Copper Copper Chromel Iron Chromel Copper Nicrosil
Copper Copper Alloy Alumel Constantan Constantan Constantan Nisil
0 ~100 0 ~ 200 0 ~ 200 0 ~ 200 0 ~ 200
-60 ~ 100 0 ~ 200
0 ± 5.0 ± 2.2 ± 2.2 ± 1.7 ± 1.0 ± 2.2
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3.5 Readouts
Digital volt(multi)meters (0.1 µV resolution) Digital indicators (with internal reference
Junction)
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4. Errors and Usage
Error tree of thermocouple
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4.1 Composition change at high temperature
•Type S, 1400 oC, 1440 h
•SEM image
Negative : 95% Pt + 5% Rh
(originally 100 % Pt)
Negative leg
Positive leg
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4.2 Thermocouples in different atmosphere
Type Gas Time /h Temperature change Result
Argon 10,000 - 19 ~ -28 K Completed Vacuum 1,610 - 1.5 K Failed R
Air 10,000 - 4.9 K Completed Argon 10,000 - 7 ~ - 18 K Completed
B Vacuum 2,780 - 3.2 K Failed
Test temperature: 1600 K
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4.3 Effects of microstructual change of type K
•Chromel : Short-range ordering of Cr between 200 oC ~ 600 oC
•Increase of emf due to short-range ordering
Chromel
Alumel
Net emf
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4.4 Effects of protecting materials
•Thermocouples in various protecting tubes at about 1000 oC
•Type S in Pt10Rh, Pt20Rh, Inconel, S.steel
•Type B in Inconel and S.steel
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4.5 Annealing treatments
•Standard annealing -1450 oC, 1 h -1100 oC, 10 h
•Emfannealed-Emfas-received
• ~ 2 oC difference
Temperature /oC0 200 400 600 800 1000 1200
Tem
p. D
iff /K
0
1
2
3
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4.6 Inhomogeneity of type K and N 1/4", IN Sheath, Type K, Grounded junction
Immersion depth /mm
0 100 200 300 400 500
Tem
pera
ture
/o C
130
140
150
160
170
180
190
1000 h600 h300 hAs-received
48 oC
1/4", IN Sheath, Type N, Grounded junction
Immersion depth /mm
0 100 200 300 400 500
Tem
pera
ture
/o C
172
174
176
178
180
182
1000 h600 h300 hAs-received
• 1000 oC annealing treatment with time • Type K: inhomogeneous temperature variation of about 48 oC • Type N: very small change ( ~ 1 oC)
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5. Calibration and Uncertainty
Calibration method Procedures of comparison calibration Uncertainty of calibration
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5.1 Calibration method
Fixed-point method At the defining fixed-points by ITS-90
Comparison method Compare with the standard (reference)
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5.1.1 Fixed-point method
Most accurate method
Pd
TC wire
PtPd thermocouple at Cu Point (1084.62 oC)
Time /h.min 15:40 16:00 16:20 16:40
Emf /
µV
13242
13243
13244
13245
13246
13247
Standard deviation: ± 0.02 µV (=1 mK)
0.05 oC
Freezing cell Typical freezing curve
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Calibration uncertainty of thermocouple
Temperature/oC Expanded uncertainty (k=2)
Au/Pt Pt/Pd S, R, B
231.928 (Sn) 0.02 oC 0.03 oC 0.04 oC
419.527 (Zn) 0.02 oC 0.03 oC 0.05 oC
660.323 (Al) 0.03 oC 0.03 oC 0.06 oC
961.78 (Ag) 0.03 oC 0.04 oC 0.08 oC
1084.62 (Cu) - 0.05 oC 0.09 oC
1154 (Fe/C) - 0.15 oC 0.2 oC
1324 (Co/C) - 0.65 oC 0.65 oC
1554.8 (Pd) - - 0.8 oC
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Most common method Compare with Reference thermometer in
furnace or liquid-bath PRT s Thermocouples Liquid-in glass thermometers Bi-metal etc.
5.1.2 Comparison calibration
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Calibration uncertainty by comparison calibration
Sensors Temperature range Expanded uncertainty (k=2)
S, R, B 0 oC ~ 1100 oC 0.2 oC
1100 oC ~ 1554 oC 2.0 oC
IPRT
-80 oC ~ 0 oC 16 mK
0 oC ~ 250 oC 16 mK
250 oC ~ 550 oC 20 mK
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Type of calibration
Single point At a specific temperature
Temperature vs. temperature (or emf)
Multiple points Point to point calculation
Make a result at each calibration temperature Continuous calculation
Make a result with finite temperature interval Normally, interpolation used Above the maximum point, extrapolation applied
It is very general to calculate the deviation emf. It is used to minimize the interpolation error. Deviation emf = Measured emf – Standard emf Standard emf is from the reference table (ASTM E230, IEC 584-1)
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5.2 Procedures of comparison calibration
Inspect the test item
Place the reference standard and test item into the temperature enclosure
Set the temperature enclosure to specified temperature
Confirm the temperature stable
Read the indications from reference and test item
Repeat
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• Step 1 : Visual inspection
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• Step 2 : Dismentle
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• Step 3 : Cleaning
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• Step 4 : First heat treatment
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• Step 5 : Wire insertion
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• Step 6 : Re-assemble
Better Standards, Better Life 90
• Step 7 : Making hot junction
Better Standards, Better Life 91
• Step 8 : Second heat treatment
Better Standards, Better Life 92
• Step 9 : Ice point preparation
Better Standards, Better Life 93
• Step 10 : Installing pyrex tubes for ice point
Better Standards, Better Life 94
Weld the hot junctions Or, use the temperature
equalizing block made of Pt Install at the uniform
temperature zone
Welding
Pt block
Installation of thermocouples
Better Standards, Better Life 95
From low to high temperature Measure if temperature change is below (0.05 ~ 0.1)
oC/min Repeat 10 times at each point About interval of 100 oC ~ 200 oC is suitable.
Measurement procedures
Better Standards, Better Life 96
∆E = (Ex – Es)+ ∆Efit
∆E : deviation emf Ex : DUT emf Er : standard emf ∆Efit :interpolation fit
correction
Example of interpolation fit of the deviation emf
Plot the deviation emf with temperature
Temperature /oC
200 400 600 800 1000 1200
Dev
iatio
n E
mf /
µV
0
2
4
6
8
10
12
∆Ei = Ex - Er
∆Efit
∆E(=a0+a1t+a2t2+a3t3)
Better Standards, Better Life 97
E’ = E + ∆E
= (c0 + c1t + ….+ c8t8) + a0+a1t+a2t2+a3t3
= ( c0+a0) + (c1+a1)t + (c2+a2)t2 + (c3+a3)t3
+ c4t4 + …. + c8t8
Making a new reference function
Better Standards, Better Life 98
5.3 Uncertainty of calibration
Comparison with reference thermocouple Normally Pt/Rh-based noble thermocouples are
used as reference thermometer. Au/Pt or Pt/Pd can be used as higher grade
standards.
Better Standards, Better Life 99
5.3.1 Measurement equation
Ed = (Ex – Es) + ∆Efit (1)
Ed : Deviation emf /µV Ex : Measured emf of DUT /µV Es : Standard emf of DUT /µV ∆Efit : Correction due to interpolation /µV
Better Standards, Better Life 100
Measured emf of DUT
Ex = Em,x + ∆Eex + ∆Ert,x (2)
Em,x : Read emf of DUT ∆Eex : Correction due to compensation(extension) wire ∆Ert,x : Correction due to room temperature
compensation
Better Standards, Better Life 101
Standard emf of DUT
Es = f(tr) (3)
tr = g(Em,r + ∆Ert,r) (4)
tr : Reference temperature from the certificate of reference thermocouple /oC
f(tr) : Standard emf from the IEC table Em,r : Measured emf of reference thermocouple ∆Ert,r : Correction due to room temperature compensation
Better Standards, Better Life 102
Combined Standard Uncertainty
(5)
(6)
(7)
(8)
α : Coefficient of voltage with temperature, µV/oC β : Reverse of α, oC/µV
)()()()( 222fitsxdc EuEuEuEu ∆++=
)()()()( ,22
,2
xrtexxmx EuEuEuEu ∆+∆+=
)()()( 222rs tufuEu α+=
)()()()( ,22
,222
rrtrmr EuEugutu ∆++= ββ
Better Standards, Better Life 103
5.3.2 Example of Comparison
Reference(REF): type R, DUT: type S Calibration at 8 points (230 oC, 500 oC, 600 oC, 700 oC, 800 oC, 890 oC, 980
oC, 1060 oC) 10 measurements at each point Calibration uncertainty and long term stability of REF are 0.4 oC (k =2) and ±
0.1 oC. Calibration uncertainty and long term stability of DVM(Res. 0.1 µV) are 10
µV/V (k = 2) and ± 20 µV/V. The accuracy of selector switch is ± 0.5 µV, and extension wires are not used. Two thermocouples are welded at their measuring junction. The stability of ice point (water and ice mixture) is ± 0.05 oC. The inhomogeneity of DUT at 180 oC is ± 0.078 %. The maximum deviation of interpolation is 0.15 µV. The combined standard uncertainties are calculated at each point. In this case, we calculate the uncertainty at 1060 oC.
Better Standards, Better Life 104
Results of calibration
Cal. point (℃)
Reference thermocouple DUT Standard emf from IEC table
/㎶
Deviation emf /㎶ Average
/㎶ Temperature
/℃
Standard deviation
/㎶
Average /㎶
Standard deviation
/㎶
230 1 887.30 246.25 0.21 1 840.62 0.18 1839.26 1.36
500 4 465.73 499.37 0.23 4 230.18 0.27 4225.86 4.32
600 5 578.02 599.26 0.31 5 235.63 0.39 5229.98 5.65
700 6 745.01 699.80 0.24 6 279.13 0.24 6271.93 7.20
800 7 955.13 799.91 0.35 7 351.66 0.36 7342.8 8.86
890 9 217.93 900.40 0.42 8 462.70 0.40 8452.49 10.21
980 10 258.31 980.54 0.46 9 372.94 0.45 9361.89 11.05
1 060 11 328.53 1 060.84 0.58 10 304.97 0.69 10293.77 11.20
Better Standards, Better Life 105
5.3.3 Standard uncertainty of DUT, u(Ex)
Standard uncertainty of emf measurement, u(Em,x)
u(EVD,x): Resolution of DVM u(EVC,x): Calibration uncertainty of DVM u(EVL,x): Long-term stability of DVM u(ERe,x): Repeatability of measurement u(ESC,x): Accuracy of selector switch u(EEN,x): Electrical noise u(EIH): Inhomogeneity of DUT u(thj): Tip temperature difference due to temperature gradient
)()()()(
)()()()()(222
,2
,2
Re,2
,2
,2
,2
,2
hjIHxENxSC
xxVLxVCxVDxm
tuEuEuEu
EuEuEuEuEu
α++++
+++=
Better Standards, Better Life 106
Standard uncertainty of compensation wire, u(∆Eex) In case of using compensating or extension wires Correction through the measurement of
temperature of connecting point/ Or apply the specification Direct measurement : - 2.3 µV ± 0.3 µV u(∆Eex) = 0.15 µV
Using the specification
In this case, it is not used.
35)( CEu
o
ex ×=∆ α
Better Standards, Better Life 107
Standard uncertainty of room temperature compensation, u(∆Ert,x)
Factor due to the change of reference point
Usually, use the ice temperature (0 oC)
Apply the rectangular distribution
In this case, it is ± 0.05 oC.
C
CEu
o
o
xrt
029.03
05.0)( ,
=
=∆V
CVCyu ooi
µµ
16.0/40.5029.0)(
=×=
Apply the sensitivity coefficient of type S at 0 oC
Better Standards, Better Life 108
5.3.4 Standard uncertainty of REF, u(Es)
Standard uncertainty of IEC table, u(f) Internationally approved value, so it is zero.
Uncertainty reference temperature, u(tr) From the calibration certificate of reference
thermocouple tr = g(Em,r + ∆Ert,r)
Better Standards, Better Life 109
5.3.4.1 Uncertainty of reference temperature, u(g)
Certificate of reference thermocouple
Normal distribution
In this case, it is 0.4 oC (k = 2). Cgu o2.0)( =
Better Standards, Better Life 110
5.3.4.2 Uncertainty of reference temperature, u(tr)
Standard uncertainty of emf measurement, u(Em,r)
u(EVD,r): Resolution of DVM u(EVC,r): Calibration uncertainty of DVM u(EVL,r): Long-term stability of DVM u(ERe,r): Repeatability of measurement u(ESC,r): Accuracy of selector switch u(EEN,r): Electrical noise u(tLS): Long-term stability of reference thermocouple
)()()(
)()()()()(22
,2
,2
Re,2
,2
,2
,2
,2
LSrENrSC
rrVLrVCrVDrm
tuEuEu
EuEuEuEuEu
α+++
+++=
Better Standards, Better Life 111
Room temperature compensation, u(∆Ert,r)
In this case, it is ± 0.05 oC.
C
Ctu
o
o
hj
029.03
05.0)(
=
=
CCVCVCyu
o
o
oo
i
011.0/48.13/29.5029.0)(
=
×=µµ
Coefficients at ice point and 1060 oC are different each other.
Apply the coefficient of type R thermocouple at each temperature
Note
Better Standards, Better Life 112
5.3.5 Interpolation, u(∆Efit)
Uncertainty due to curve fit to get a deviation function Apply the maximum difference between the
measured and fitted value Apply the rectangular distribution
Apply the maximum difference as the half width of distribution
In this case, it is 0.15 µV. VVEu fit µµ 09.0
315.0)( ==∆
Better Standards, Better Life 113
5.3.6 Combined Standard Uncertainty – uc(Ed)
V
V
EuEuEuEu fitsxdc
µµ
33.509.056.267.4
)()()()(222
222
=++=
∆++=
VV
EuEuEuEu xrtexxmx
µµ
67.416.00.067.4
)()()()(222
,22
,2
=++=
∆+∆+=
VV
tufuEu rs
µµ
α
56.2218.073.110
)()()(22
222
=×+=
+=
CC
EuEugutu
o
o
rrtrmr
218.0029.0392.001.1074.02.0
)()()()(22222
,22
,222
=
×+×+=
∆++= ββ
Note ! Sensitivity of DUT
Better Standards, Better Life 114
5.3.7 Uncertainty budget – u(Ex)
Quantity Xi
Estimated xi
Std. uncertainty
u(xi)
Sensitivity coefficient
ci
Uncertainty ui(y)
Distribution Deg.
Freedom
Ex 10304.97 ㎶ 4.676 ㎶ 1 4.67 ㎶ Normal ∞
Em,x 10304.97 ㎶ 4.67 ㎶ 1 4.67 ㎶ Normal ∞
EVD,x - 0.03 ㎶ 1 0.03 ㎶ Rectang. ∞
EVC,x - 0.05 ㎶ 1 0.05 ㎶ Normal ∞
EVL,x - 0.12 ㎶ 1 0.12 ㎶ Rectang. ∞
ERe,x - 0.22 ㎶ 1 0.22 ㎶ Normal 9
ESC,x - 0.58 ㎶ 1 0.58 ㎶ Rectang. ∞
EEN,x - 0.58 ㎶ 1 0.58 ㎶ Rectang. ∞
EIH - 4.58 ㎶ 1 4.58 ㎶ Rectang. ∞
thj - 0.058 ℃ 11.73 ㎶/℃ 0.68 ㎶ Rectang. ∞
∆Eex 0 ㎶ 0 ㎶ 1 0 ㎶ Rectang. ∞
∆Ert,x 0 ㎶ 0.029 ℃ 5.40 ㎶/℃ 0.16 ㎶ Rectang. ∞
Better Standards, Better Life 115
Uncertainty budget – u(Es), u(∆Efit) Quantity
Xi
Estimated xi
Std. uncertainty
u(xi)
Sensitivity coefficient
ci
Uncertainty ui(y)
Distribution Deg. Freedom
Es 10 293.77 ㎶ 2.56 ㎶ 1 2.56 ㎶ Normal ∞
f(tr) - 0 1 0 - ∞
tr - 0.218 oC 11.73 ㎶/℃ 2.56 ㎶ Normal ∞
g(tr) - 0.2 oC 1 0.2 oC Normal ∞
Em,r - 1.16 ㎶ 0.074 oC/ ㎶ 0.086 oC Normal ∞
EVD,r - 0.03 ㎶ 1 0.03 ㎶ Rectang. ∞
EVC,r - 0.06 ㎶ 1 0.06 ㎶ Normal ∞
EVL,r - 0.13 ㎶ 1 0.13 ㎶ Rectang. ∞
ERe,r - 0.18 ㎶ 1 0.18 ㎶ Normal 9
ESC,r - 0.58 ㎶ 1 0.58 ㎶ Rectang. ∞
EEN,r - 0.58 ㎶ 1 0.58 ㎶ Rectang. ∞
tLS - 0.058 ℃ 13.48 ㎶/℃ 0.78 ㎶ Rectang. ∞
∆Ert,r 0 ㎶ 0.029 ℃ 0.392 0.011 oC Rectang. ∞
∆Efit 0.14 ㎶ 0.09 ㎶ 1 0.09 ㎶ Rectang. ∞
Better Standards, Better Life 116
5.3.8 Uncertainty with temperature
Calibration point /℃
Combined standard uncertainty /㎶
Expanded uncertainty (k = 2)
/㎶
Expanded uncertainty (k = 2)
/℃
246.25 2.55 5.10 0.58
499.37 3.19 6.38 0.64
599.26 3.51 7.02 0.69
699.80 3.87 7.74 0.73
799.91 4.27 8.54 0.79
900.40 4.70 9.40 0.84
980.54 5.07 10.14 0.88
1 060.84 5.33 10.66 0.91
Better Standards, Better Life 117
5.3.9 Report of uncertainty
Choose the maximum uncertainty among the calculated
values
Expanded uncertainty is determined with k = 2.
In this case, U95 = 0.9 oC
Better Standards, Better Life 118
Further reading T.D. McGee, “Principles and methods of temperature measurement”,
1988, John Wiley & Sons J.V.Nicholas and D.R.white, “Traceable Temperatures”, 2nd edition,
2001, John Wiley & Sons “Manual on the use of thermocouples in temperature measurement”, 4th
edition, 1993, ASTM