Czichos H., Saito T., Smith L.E. (Eds.) Handbook of Metrology and Testing

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468 Part C Materials Properties Measurement

Adiabatic and some isoperibol instruments work

as heat-accumulation calorimeters, i. e. the heat to be

measured is detected via the temperature change of the

calorimeter. The calibration of the instrument is deter-

mined either by electrical calibration or by the use of

reference materials with well-known heat of solution,

reaction or adsorption.

A further method is to suppress the temperature

change by measuring the required compensating heat,

e.g. by means of electric energy (Joule heating or Peltier

cooling). In this case only a sensitive temperature sen-

sor (no calibration is required) with sufﬁcient long-term

stability is needed because the calorimeter operates in

the isothermal mode. The thermometer is used only for

the control of the compensation power, so that the tem-

perature remains constant.

A third method of heat measurement is the deter-

mination of heat exchanged between the reaction vessel

and the (isothermal) surroundings by means of heat ﬂux

sensors (Sect. 8.1.1). The vapor pressure of a mixture

depends on its composition. Heats of vaporization are

usually larger than heats of mixing. Therefore, the de-

termination and correction of the inﬂuence of the vapor

pressure change during the experiment is very impor-

tant. Measurement uncertainties of about 0.25% can be

achieved [8.30].

The advantages of continuously working ﬂow

calorimeters [8.31] are that the measurements can be

carried out in a shorter period of time and that less

material is needed. Measurements of heat of mixing in

a temperature range of 273–479 K and a pressure range

between 0.1MPaand40.5 MPa with relative uncertain-

ties of 0.5% being possible [8.32].

8.2.5 Combustion Calorimetry

The most common device for measuring the heat of

combustion or caloriﬁc value of a solid or liquid ma-

terial is the bomb calorimeter [8.33–36].

A sample is contained under a pressure (about

3 MPa) of pure oxygen in a pressure-tight stainless-

steel container (bomb, V ≈ 300 ml) and is burned under

standardized conditions. During recent years several

new micro-bomb calorimeters (V < 100 ml) have been

developed. All instruments are provided with an elec-

trical system for the ignition of the combustion process.

The determination of the heat of combustion is based

on the observed temperature rise, which is measured

with high-precision temperature sensors such as stand-

ard platinum resistance thermometers, thermistors or

quartz thermometers. Adiabatic, isoperibol or aneroid

instruments are commercially available. In most cases

a correction of the measured temperature increase is

needed to consider the heat exchange with the sur-

rounding. Furthermore, the calibration factor (energy

equivalent) of the calorimeter must be determined. For

that purpose a reference material with a well-known

heat of combustion is needed. Benzoic acid is the pre-

ferred material for that purpose. With well-designed

bomb calorimeters relative measurement uncertainties

of less than 0.01% can be achieved [8.37] with highly

puriﬁed materials containing C, H, and O. If other elem-

ents are present, the accuracy is limited by the extent

to which the stoichiometry of the combustion process

can be controlled and determined. Systematic devia-

tions can arise from incomplete combustion or from the

lack of a well-deﬁned ﬁnal state.

Calorimeters for the determination of the caloriﬁc

value of gases can be subdivided into three different

groups [8.38]:

•

combustion of the gas inside of a bomb calorimeter

(isochoric combustion)

•

combustion without a ﬂame on a catalyst (isobaric

combustion)

•

combustion of the gas in an open ﬂame of a gas

burner (isobaric combustion)

The ﬁrst two groups of methods are used only in rare

cases for very speciﬁc applications, e.g. investigations

on ﬂuorochlorohydrocarbons.

Calorimetric methods in the third group are based

on the combustion of the gas at constant pressure and

ﬂow rate (ﬂow calorimeter) with an open ﬂame. A heat

exchanger is used to transfer the combustion heat to

a heat-absorbing ﬂuid (air or water). The temperature

increase of the heat-absorbing ﬂuid is a measure of

the caloriﬁc value. For the calibration of these calorim-

eters various methods can be applied: the use of gases

with well-known caloriﬁc values, electrical calibration

techniques and methods based on the determination of

temperature increase, volume ﬂows and the knowledge

of the heat capacity of the heat absorbing ﬂuid. There

are several gas calorimeters commercially available,

e.g. the Junkers calorimeter, the Reinecke calorimeter,

the Thomas–Cambridge calorimeter, or the Cutler–

Hammer calorimeter. The typical relative measurement

uncertainty of these instruments is about 0.5%.

A high-precision constant-pressure gas-burning

calorimeter has been used for the determination of the

heat of combustion of methane [8.39]. The resulting

combined standard uncertainty was 0.21 kJ/mol which

corresponds to a relative uncertainty of 0.024%.

Part C 8.2

Thermal Properties 8.3 Thermal Expansion and Thermomechanical Analysis 469

8.3 Thermal Expansion and Thermomechanical Analysis

Thermomechanical analysis (TMA) measures the

change in dimension (deformation) of a sample under

constant (static force) compressive, tensile, or ﬂexural

loads as the temperature of the sample is changed. The

special case with negligible force is called dilatometry

or thermodilatometry. Typical applications of TMA are

the determination of the coefﬁcient of thermal expan-

sion, glass-transition temperatures, softening or shrink-

age behavior or the investigation of changes in dimen-

sion caused by sintering or chemical reactions. There

are several instruments commercially available covering

the temperature range between −260

◦

C and 2400

◦

In dynamic mechanical analysis DMA a dynamic

force is applied to the sample and the resulting dis-

placement is measured as a function of temperature,

frequency or time. A special case is the application of an

oscillating force. Dynamic mechanical analysis is used

for the determination of the modulus of elasticity (stiff-

ness), viscous modulus and damping coefﬁcient (tan δ)

of materials (for details see Chap. 7). In addition, DMA

measurements give insight into the temperature and fre-

quency dependence of molecular mobility and can be

used for the determination of glass-transition tempera-

tures.

Thermal expansion is the change of the length

of a specimen ΔL as a function of the temperature

change ΔT

ΔL = L −L

= L

αΔT . (8.17)

This behavior is described by the coefﬁcient of (linear)

thermal expansion (CTE)

α =α

(8.18)

and is a function of temperature α = α(T ). The ref-

erence length L

of the specimen is generally given

either at a temperature of 0

◦

Corat20

◦

C. In crystalline

anisotropic solids the coefﬁcient of thermal expansion

is direction dependent and may have up to six different

contributions.

The change of the volume of a body as a function of

temperature is described by the volumetric coefﬁcient

of thermal expansion

α =α

. (8.19)

In most cases the volumetric coefﬁcient of thermal ex-

pansion can be determined from the linear coefﬁcient of

thermal expansion with sufﬁcient accuracy by means of

≈3α

There are several different descriptions of thermal

expansion in use. In the deﬁnition according to (8.18)

α is often called the physical or differential coefﬁcient

of thermal expansion. Furthermore, in some cases a so

called mean or technical coefﬁcient of thermal expan-

sion according to

L(T

) −L(T

)

−T

ΔL

ΔT

(8.20)

is used. The third kind is the speciﬁcation of a relative

length change ΔL/L

at a (mean) temperature T .

The coefﬁcient of thermal expansion of a sample is

usually determined according to its deﬁnition (8.18)by

the measurement of the reference (initial) length L

the temperature T

, and the change of the length ΔL as

a result of a temperature change ΔT at different (mean)

temperatures.

The shape of the samples is typically a cylindrical

rod with a diameter in the rage 5–10 mm and a length

in the range 25–50 mm. Depending on the temperature

range of interest, cryostats, liquid baths, multizone or

heat pipe furnaces with excellent temperature stability

and homogeneity are used. These are operated either by

stepwise heating or by scanning at low rates. Thermo-

couples or radiation thermometers are mostly used for

the temperature measurements. More information can

be found in [8.40].

8.3.1 Optical Methods

For measurements of length changes with highest ac-

curacy, e.g., for the certiﬁcation of reference materials,

for materials with very low coefﬁcients of thermal ex-

pansion (Zerodur, Invar) and for cases when only small

samples are available, optical methods are chosen. Rel-

ative measurement uncertainties of the certiﬁed values

of α

of high-purity synthetic Al

of 0.18–1% were

achieved in a temperature range between −180

◦

Cand

+400

◦

C[8.41].

The optical methods can be divided into three main

types. The ﬁrst one is based on the creation of an image

of a sample and the determination of the spatial move-

ment of the ends or other marks along the length. In

this case the optical path is perpendicular to the dis-

placement direction. The image is formed either by

background illumination to give a silhouette effect or

by the radiant light emitted from the specimen itself.

These techniques are known as optical imaging, optical

comparator or twin telemicroscopy.

Part C 8.3

470 Part C Materials Properties Measurement

The second type is based on interferometry. Here the

displacement is determined by the measurement of the

path difference of beams reﬂected from opposite sur-

faces of the sample. Because the refractive index of air

or inert gases is not known with sufﬁcient accuracy most

measurements are carried out in vacuum.

In the third method speckle interferometry is used

to determine the displacement by means of changes in

the interference pattern on the surface of the sample.

There are further specialized techniques which do

not fall into the above categories. Details can be found

in [8.42].

For the certiﬁcation of reference materials instru-

ments based on Fizeau and Michelson interferometers

have been used [8.43, 44]. At temperatures above

800 K several problems lead to a decrease in the accu-

racy of this method. Therefore, instruments based on

optical heterodyne interferometers for absolute meas-

urements at temperatures of 300–1300 K [8.45]and

1300–2000 K [8.46] have been developed. For meas-

urements with these instruments combined standard

uncertainties of 1.1×10

−8

−1

[0.26% for 100(Δα/α)]

at 900 K and of 1.3% in the range 1300–2000 K have

been claimed.

The highest accuracies are achieved around room

temperature. Measurement results on ceramic and steel

gauge blocks with uncertainties of 1 × 10

−9

−1

a temperature range between −10

◦

C and 60

◦

Chave

been published [8.47].

8.3.2 Push Rod Dilatometry

The most common method of measuring thermal ex-

pansion is push rod dilatometry. Several instruments

are commercially available for the temperature range

between −260

◦

C and 2800

◦

A sample is heated in a furnace or other temper-

ature-controlled environment and the displacement of

the ends is mechanically transmitted to a displacement

sensor (e.g. a linear variable differential transformer,

(LVDT)) by means of push rods. There are several pos-

sible arrangements of the push rods. The ﬁrst is the

parallel arrangement of two push rods, the double push

rod system. This arrangement can be modiﬁed by simul-

taneous use of sample and reference samples, known as

the differential method. A further arrangement is for one

of the rods to be in the form of a closed tube. The sam-

ple and the other push rod are located along the central

axis of the tube. In this case the sample is clamped be-

tween the central push rod and the closed end of the

other tube-shaped push rod.

The displacement sensor is maintained in a con-

trolled-temperature environment close to room tem-

perature. Therefore, the most critical part is the push

rod, which transmits the expansion signal from the

sample to the displacement sensor. The resulting tem-

perature difference between opposite ends of push rods

can be more than 2000 K. Therefore, the homogeneity

of the temperature ﬁeld in the furnace, the repeatabil-

ity of a temperature program and the material used for

the push rods are of critical importance for high-quality

measurements.

The main criterion for the choice of a material used

for push rods is a low, reproducible, and accurately

known coefﬁcient of thermal expansion. At low tem-

peratures vitreous silica is the preferred material. There

are very different recommendations for the upper tem-

perature for the use of vitreous silica for this purpose,

ranging from the temperature of the α–β transition of

about 550

◦

C to 1000

◦

C as the maximum tempera-

ture to avoid devitriﬁcation, i. e. the transition from the

glassy state to a crystalline structure. For temperatures

up to 1600

◦

C push rods are made from silica, either

in single-crystal form (sapphire) or sintered (polycrys-

talline) state, at higher temperatures rods consisting of

isotropic graphite are used.

Dilatometers are often operated in a dynamic mode

by temperature scanning with a constant rate of typ-

ically less than 5 K/min. Depending on the method

of construction more than one measurement may be

necessary to determine the correction for the ther-

mal expansion of the push rods and to calibrate both

the displacement- and the temperature sensor. For this

purpose certiﬁed reference materials are necessary.

High-purity materials such as silicon, tungsten, plat-

inum, copper, aluminum, sapphire or vitreous silica

are mostly used for displacement calibration. The cal-

ibration of the temperature sensor is carried out by

the measurement of the dimensional change of a high-

purity metal during melting.

With careful work relative measurement uncertain-

ties for the CTE of 2% at temperatures between room

temperature and 973

◦

C have been achieved [8.48].

8.3.3 Thermomechanical Analysis

The basic difference between thermomechanical analy-

sis (TMA) and dilatometry is that for TMA a load (static

force) is applied to the sample. As a consequence in-

struments for TMA are operated with vertical sample

orientation. This is different from most dilatometers,

where horizontal sample and furnace orientation is cho-

Part C 8.3

Thermal Properties 8.5 Temperature Sensors 471

sen as this gives better temperature uniformity. There

are several modes of force control and sample con-

ﬁguration for TMA. At negligible load the thermal

expansion measurements are carried out in a very sim-

ilar manner to dilatometry. In the compression mode

a rod with well-known cross section or geometry

presses with a known force on the sample and the

compression, penetration or bending is determined as

a function of force, temperature or time. From these

kinds of measurements elastic modulus, creep or cure

behavior, softening temperatures or phase transition are

determined. This is also of advantage for the inves-

tigation of transitions in thin ﬁlms such as polymer

coatings or lacquers. Further applications are measure-

ments in tension mode and investigations of the viscous

properties or gelation of ﬂuids. There are instruments

commercially available for a temperature range between

−160

◦

C and 2400

◦

8.4 Thermogravimetry

Thermogravimetry (TG) is a method of thermal analysis

in which the mass of a sample is measured as a func-

tion of temperature whilst the sample is subjected to

a controlled temperature program. In many cases the

reaction products are analyzed by supplementary inves-

tigations. As for most other methods of thermal analysis

the typical temperature program is a linear scan, i. e.

the temperature changes linearly in time. During re-

cent years so called controlled-rate thermal analysis

has been increasingly used. This means that the scan-

ning rate can vary as a function of time depending on

the magnitude of the measured quantity; for example,

the scanning rate of a thermogravimetric instrument is

controlled as a function of the measured mass change

of the sample or the amount of a speciﬁc evolved

gas.

Typical applications are the evaluation of the ther-

mal decomposition kinetics of materials such as rubbers

or polymers, the investigation of processes such as sin-

tering, drying, oxidation or reduction. There are several

instruments for investigations in a temperature range

between −150

◦

C and 2400

◦

C commercially available.

Typical sample masses are in the range 5–25 mg. Mass

changes can be detected with resolutions in the range

0.1–10μg. Most TG instruments can be combined

with calorimetric detection systems for simultaneous

differential thermal analysis or differential scanning

calorimetry. Thermogravimetry is performed in a con-

trolled atmosphere including oxygen, nitrogen, helium

or argon with adjustable ﬂow rates.

A mass spectrometer (MS)oraFourier-transform

infrared (FTIR) spectrometer can be coupled to most

TG instruments for continuous, online identiﬁcation

and analysis of the evolved gases during heating of the

sample. To avoid a separation or loss of evolved gases

by condensation these are routed to the MS via a heated

capillary or a system of oriﬁces held at the same tem-

perature as the sample (Skimmer coupling).

A recent extension of TG systems is pulse ther-

mal analysis [8.49]. This is based on the injection of

a speciﬁc amount of liquid or gaseous reactant into the

inert carrier gas stream and monitoring of the changes

in the mass, enthalpy, and/or gas composition. By this

method gas–solid reactions, adsorption or catalytic pro-

cesses can be studied. A further application is the direct

calibration of the mass spectrometer or FTIR spectrom-

eter by injecting a known amount of substance into the

inert carrier gas stream and relating it to the spectrom-

eter signal. For more information the reader is referred

to [8.50].

8.5 Temperature Sensors

Temperature sensors are needed for the measurement

of thermal properties of materials, of course, but they

are used in a much larger ﬁeld of applications, since

temperature is the quantity measured most frequently in

science, industry, and daily life. In the following chapter

a selected number of temperature sensors of sufﬁcient

reliability for scientiﬁc and industrial measurements is

described as well as the temperature scale against which

they should be calibrated.

8.5.1 Temperature and Temperature Scale

Temperature characterizes the thermal state of matter

independently of the nature of the substance. It is an

intensive quantity, not depending on the addition or re-

duction of the amount of matter concerned, but changed

by supplying or removing heat or mechanical work.

Classically, the deﬁnition of temperature is derived from

the thermodynamic description of a Carnot engine re-

Part C 8.5

472 Part C Materials Properties Measurement

versibly driven in a thermodynamic cycle, the efﬁciency

of which depends only on temperature. The ratio of the

heat Q

fed to the engine isothermally at high temper-

ature to the heat Q

removed at low temperature to get

back mechanical work was found to be identical to the

ratio of the temperatures T

and T

at both isothermal

parts of the cycle by William Thomson, the later Lord

Kelvin:

= T

. (8.21)

From this deﬁnition of temperature he could derive the

equation of state for an ideal gas to be

pV = const T . (8.22)

This states that the product of the gas pressure p and

the volume V ﬁlled with the gas is proportional to the

absolute temperature. The constant in (8.22) turned out

to be equal to nR, the product of the number n of moles

involved and the gas constant R.

By statistical methods it was shown during the sec-

ond half of the 19th century that the average kinetic

energy of a molecule of an ideal gas amounts to

pV = const





, (8.23)

Table 8.3 Primary thermometers and the fundamental relations on which they are based

Thermometer Fundamental relation

Constant-volume gas thermometer (ideal gas)

pressure p, volume V , number of moles n, molar gas constant R,

temperature T

pV = nRT

Acoustic gas thermometer (ideal gas)

speed of sound c

, ratio γ of speciﬁc heat capacities

at constant pressure and constant volume, gas constant R,

temperature T ,molarmassM

=(γ RT/M)

1/2

Dielectric-constant gas thermometer (ideal gas)

pressure p, Boltzmann constant k

, temperature T ,

dielectric constant ε, electric constant ε

static electric dipole polarisability of

a gas atom α

p =k

T(ε −ε

)/α

(Clausius–Mosotti equation)

Total radiation thermometer

total radiance L, Boltzmann constant k

, temperature T , speed of

light in vacuum c, Planck constant h

L =



2π





15c



(Stefan–Boltzmann law)

Spectral band radiation thermometer

spectral radiance L

, Planck constant h, frequency of light in

vacuum ν, speed of light in vacuum c, Boltzmann constant k

temperature T

= 2hν

[exp(hν/k

T ) − 1]

−1

(Planck’s Law)

Noise thermometer

mean square noise voltage





, Boltzmann constant k

, temperature

T, resistance R, bandwidth Δ f





=4k

TRΔ f

(Nyquist formula)

a fact, which demonstrates that temperature is a measure

of the averaged internal energy of the gas molecules,

which are ascribed a mass m and a velocity v.

An apparatus built to determine the pressure and

the volume of a known number of moles of an ideal

gas can be used to measure the temperature of the

gas according to (8.22). Such a gas thermometer is

called a primary thermometer, since there is no need

to calibrate it against another thermometer and it is

based on a fundamental physical law including the

thermodynamic temperature. There are several other

primary thermometers based on different fundamental

laws listed in Table 8.3.

Since these primary thermometers are usually com-

plicated and difﬁcult to handle, a temperature scale was

introduced for practical purpose and reﬁned over the

years. Such a scale consists of temperature ﬁxed points,

the temperature values of which are determined with

great effort by comparison with primary thermometers,

and of interpolating instruments, which are calibrated

at those ﬁxed points and deﬁne the temperature in be-

tween. After some early attempts such a scale was

adopted in 1927 following a suggestion by Callendar

Part C 8.5

Thermal Properties 8.5 Temperature Sensors 473

from 1899. The actual revision of the temperature scale

dates from 1990. It is called the International Temper-

ature Scale of 1990, ITS-90. It is based on 14 ﬁxed

points from the triple point of hydrogen (13.8033 K) to

the freezing point of copper (1357.77 K) and the vapor

pressure of the stable helium isotopes below.

One of these ﬁxed points is the triple point of water,

the value of which is ﬁxed to 273.16 K, as the deﬁn-

ing point of the temperature unit kelvin. This deﬁnition

states: the kelvin is the fraction 1/273.16 of the thermo-

dynamic temperature of the triple point of water.

Interpolating instruments are the vapor pressure

thermometer with

He or

He as the working gas from

0.65 K to 5 K, which is part of the scale’s deﬁnition as

well, an interpolating gas thermometer with the same

gases up to 25 K and standard platinum resistance ther-

mometers (SPRTs) above. From the freezing point of

silver (1234.93 K) to the highest temperatures to be

reached the relative spectral radiance thermometer is

used, being calibrated at the Ag, Au or Cu ﬁxed point,

respectively. Table 8.4 includes the temperature ﬁxed

points and indicates the uncertainty ΔT

of the real-

ization of the ITS-90.

In 2000 the CIPM (Comité International des Poids

et Mesures) adopted an extrapolation of the tem-

perature scale to subkelvin temperatures. This scale,

called the Provisional Low Temperature Scale of 2000,

PLTS-2000, is based on magnetic and superﬂuid phase

Table 8.4 The deﬁning ﬁxed points of ITS-90 (after [8.51])

No. T

(K) t

(

◦

C) T

(mK) Substance State

1 3–5 −270.15 to −268.15 0.1

He and

He Vapor pressure

2 13.8033 −259.3467 0.1 e-H

Triple point

3 ≈17 ≈−256.15 0.2 e-H

Vapor pressure ≈33.3213 kPa

4 ≈20.3 ≈−252.85 0.2 e-H

Vapor pressure ≈101.292 kPa

5 24.5561 −248.5939 0.2 Ne Triple point

6 54.3584 −218.7916 0.1 O

Triple point

7 83.8058 −189.3442 0.1 Ar Triple point

8 234.3156 −38.8344 0.05 Hg Triple point

9 273.16 0.01 0.02 H

O Triple point

10 302.9146 29.7646 0.05 Ga Melting point

11 429.7485 156.5985 0.1 In Freezing point

12 505.078 231.928 0.1 Sn Freezing point

13 692.677 419.527 0.1 Zn Freezing point

14 933.473 660.323 0.3 Al Freezing point

15 1234.93 961.78 1–10 Ag Freezing point

16 1337.33 1064.18 10 Au Freezing point

17 1357.77 1084.62 15 Cu Freezing point

e-H

is hydrogen at the equilibrium concentration of the ortho and para molecular forms

transitions in

He and the minimum of its melting

pressure as ﬁxed points and the

He melting curve ther-

mometer as the interpolating instrument.

The national metrological systems, often divided

into legal and industrial branches, provide national tem-

perature standards and calibration facilities for science

and industry to trace back the readings of thermometers

of individual users. The lowest uncertainty is avail-

able from those national metrological institutes, which

compare their particular representations of ITS-90 from

time to time by means of so called key comparisons to

guarantee the uniformity of temperature measurements

throughout the world. The deﬁnition of temperature, the

development of temperature scales and the principles

of thermometry are extensively explained in the mono-

graph [8.52].

Very recently, efforts have been started by several

national metrological institutes to replace the depen-

dence of the kelvin on a material property, namely the

already mentioned triple point of water. Instead, it is

suggested that the kelvin should be based on a funda-

mental constant as is done with other SI units, which

are already related to a fundamental constant, such as

the meter to the speed of light in vacuum, or are in

the process of being related, like the kilogram. In the

case of the kelvin the appropriate constant would be the

Boltzmann constant k

. If its value is determined with

sufﬁcient accuracy – about one order of magnitude bet-

Part C 8.5

474 Part C Materials Properties Measurement

ter than now – and then ﬁxed by deﬁnition, temperature

could be traced back to the internal energy k

T,the

quantity to which it is proportional in the microscopic

view, as mentioned at the beginning of this section.

8.5.2 Use of Thermometers

Strictly speaking, a thermometer never measures the

temperature of the sample of interest, but always its

own. To get an indication of the sample’s temperature

its temperature and the temperature of the thermometer

should agree within the requested uncertainty of the

measurement. To meet this requirement within a ﬁnite

amount of time heat should easily be able to ﬂow be-

tween them. If the heat ﬂow has approached zero, so

called thermal equilibrium is reached between sample

and thermometer, and the thermometer is ready to in-

dicate the sample’s temperature. In the case of contact

thermometry the thermal resistance between the sample

and the thermometer should be low. Thermal conduc-

tivity should be high within the sample and within

the thermometer to provide thermal equilibrium within

both. A small heat capacity of the thermometer is advan-

tageous, since then only a small amount of heat is forced

to ﬂow in order to reach thermal equilibrium. Small

heat capacity, good thermal contact and large internal

thermal conductivity make the thermometer fast.

A thermometer gives an indication of its tempera-

ture by measuring some other property that is somehow

dependent on temperature. With a constant-volume gas

thermometer the pressure is the quantity that measures

temperature, with a resistance thermometer it is the

electrical resistance of the sensor. Usually some en-

ergy is needed to read the thermometer on the display.

This energy is fed to the sensor and, in principle, due

to the permanent heat ﬂow, keeps the thermometer out

of equilibrium. Therefore, the measuring energy must

be reduced to a value that does not disturb the result.

Since thermal conductivity, thermal contact resistance

and heat capacity change with temperature, the measur-

ing energy must also be changed in most cases over the

temperature range.

In the following, we will consider some frequently

appearing errors in the use of thermometers, which are

discussed in [8.53] in great detail.

•

Electromagnetic interference

Since the energy ﬂowing between sample and ther-

mometer is not limited to the measuring energy,

other energy contributions coupled into the sys-

tem, such as heat leaks, must be kept under control

during the measurement. If electromagnetic energy

from radio and television broadcasts irradiates the

thermometer and is fed in by the wiring, or if

the electromagnetic radiation emitted from hot sur-

faces nearby is absorbed by the thermometer, the

measuring result can be signiﬁcantly affected. Elec-

tromagnetic shielding of the thermometric sensors

and electronics, ﬁltering circuits in the sensor wiring

and radiation shields are necessary tools when the

amount of irradiated external energy can not be

tolerated. In extreme situations the whole measure-

ment should be located in an electromagnetically

shielded room.

•

Immersion error

Immersion error is an issue for liquid-in-glass ther-

mometers and others that are usually not fully

immersed in the liquid to be measured. In this case,

heat is permanently ﬂowing through the stem of

the thermometer from the bath to the surrounding

air. This is a typical nonequilibrium situation inap-

propriate for an exact result. As a rule of thumb

a thermometer should be immersed into the bath

by more than ten diameters of the thermometer

(that means 40 mm beyond the sensing element with

a sensor diameter of 4 mm) to limit the error to

0.01%.

•

Heat capacity error

If the heat capacity of the thermometer is not neg-

ligible compared to the heat capacity of the sample

and the sample is more or less thermally isolated,

a certain amount of heat ﬂows between them when

the starting temperatures are different. After reach-

ing thermal equilibrium both temperatures have

changed to a common value somewhere between

their initial temperatures. This means that, if the

sample’s temperature was, e.g. much hotter than that

of the thermometer, the ﬁnally indicated tempera-

ture of the sample would be less than its value before

the measurement. This difference is called the heat

capacity error.

•

Time constant

Even if the sample’s heat capacity is large enough

to avoid the heat capacity error it may take a signif-

icant time to replace the heat in the sample that has

ﬂown into the thermometer while thermal equilib-

rium was established. Therefore, the reading of the

thermometer approaches its ﬁnal value following an

exponential trend with a time constant τ

. If insufﬁ-

cient time is given to the system, a settling response

error is arising. If the temperature of a sample is

changing at a certain rate, and the settling rate of

Part C 8.5

Thermal Properties 8.5 Temperature Sensors 475

the thermometer is too slow compared to the tem-

perature variation rate, a lag between the real and

the measured temperature will develop. This can be

avoided either by slowing down the rate of tempera-

ture change or by a constructive improvement of the

thermal coupling between the parts of the system.

•

Environmental irradiation

According to the Stefan–Boltzmann law (see Ta-

ble 8.3) all material surfaces emit electromagnetic

radiation proportional to T

. Therefore, all radiat-

ing sources in the environment should be removed,

if possible, or at least shielded, since thermometers

should not see surfaces at temperatures very differ-

ent from their own. In addition, thermometers show

signiﬁcantly different readings up to the order of

some kelvin depending on their surface emissivity.

This fact recommends proper shielding, as well.

To clarify the different inﬂuences on thermometers

during the measurement of temperature it is helpful to

use analogue electrical modeling. Since thermal con-

ductivity can be described in a similar way as its

electrical correspondent, one can derive analogue re-

lations. Heat Q corresponds to electrical charge Q

heat ﬂux q to electrical current I, temperature dif-

ference ΔT to voltage V, thermal resistance R

electrical resistance R

, and heat capacity c

to elec-

trical capacitance C. Thus, Ohm’s law can be applied in

analogy to heat ﬂow processes in the following form:

ΔT = T

−T

=qR

. (8.24)

If the heat ﬂow crosses several materials with different

thermal conductivities and thermal contact resistances

in between, the thermal resistances can be added like

electrical resistances in series. If there are several heat

ﬂows from different sources to one thermometer, such

as heat conduction from the sample and from the sur-

roundings and thermal radiation from other nearby

objects, the inverse of the thermal resistances can be

added again. Within this electrical analogue picture one

can state that a thermometer measures the correct sam-

ple temperature when the thermal resistance between

them goes to zero, whereas the thermal resistances to

other objects such as the surroundings tend to inﬁnity

by screening or shielding measures.

If temperature measurements are performed in

a way that excludes all errors, most of which are men-

tioned above, the result is reliable only with the correct

calibration of the instrument. This means, that its in-

dicated result must be traced to the SI unit kelvin by

linking it to the international temperature scale. Further-

more, the uncertainty of the calibration caused by the

number of steps or the special way of this trace back

must be known or investigated, and the reliability of

the instrument over a longer period of time should be

proven. The last requirement can be met by repeated

calibrations from time to time, depending on the fre-

quency and intensity of the thermometer’s use.

8.5.3 Resistance Thermometers

The type of thermometers most frequently found in

scientiﬁc and technical equipment is the resistance ther-

mometer, since it is easily read by electronics and can

simply be used for temperature control cycles. The most

widespread type among the resistance thermometers is

the platinum resistance thermometer, which at the same

time is the interpolating instrument of the ITS-90 at the

highest level of accuracy. In the latter case it is called the

standard platinum resistance thermometer (SPRT)and

is used in different forms between the triple point of hy-

drogen at 13.8033 K and the freezing point of silver at

1234.93 K.

The electrical resistance of metallic samples comes

from the scattering of the conduction electrons at

the atoms or molecules of the solid lattice. There is

a temperature-dependent part of the resistance R

due

to lattice vibrations, which increases with increasing in-

ternal energy, and a temperature-independent part R

caused by impurities and by lattice defects. The total

sample resistance arises as the sum of both

R(T ) = R

(T )+ R

. (8.25)

This equation, in the literature called Mathiessens’s

rule, can be reformulated as

R(T ) = R(0

◦

C)(1+αt) , (8.26)

where α is the temperature coefﬁcient of resistance.

Fewer impurities or defects reduce R

or R at 0

◦

Cand

increase α. Therefore, the metals that should be used for

resistance thermometers are those that can be puriﬁed

to the highest values and effectively annealed. High-

purity platinum most closely obeys Mathiessen’s rule

with small deviations found by Callendar and later by

van Dusen. Finally, (8.26) is modiﬁed to the Callendar–

van Dusen equation

R(T ) = R(0

◦



1 + At +Bt

+C(t −100)t



(8.27)

with coefﬁcients of the order: A = 4×10

−3 ◦

−1

B =−6×10

−7 ◦

−2

,andC = 4×10

−12 ◦

−3

Part C 8.5

476 Part C Materials Properties Measurement

This ideal behavior of platinum thermometers is

only true, if no other effects inﬂuence its resistance.

Such disturbing effects to be avoided are strain, mech-

anical shock, vibration, pressure, humidity, and corro-

sion. Some of them, such as strain, are caused by tem-

perature changes in an uncontrolled manner. Others can

be reduced by careful handling. Further ones must be

minimized by using a sealed and shielded construction.

Standard platinum resistance thermometers for ITS-

90 realization are manufactured in three different types,

as listed in Table 8.5.

SPRTs are usually constructed of high-purity coiled

platinum wire within a sheath of glass or quartz em-

bedded in a platinum tube. This is a setup free of strain

and danger of contamination, but extremely sensitive to

shock, vibrations or other mechanical stress. Therefore,

SPRTs are, on one hand, temperature standards with

high stability and an uncertainty of less than 1 mK, but

on the other not suitable for rough industrial environ-

ments. To obtain a more robust device, the support of

the platinum wire should be improved. This inevitably

causes more strain during temperature variations and

increases the uncertainty to about 10 mK. A further dis-

advantage of all wire-made devices is the poor thermal

contact only made by the gas in the capsule and the

leads, resulting in time constants of several seconds.

As an alternative solution, thick-ﬁlm thermometers are

available, which can be glued to the sample of interest

and have improved thermal contact and time constant.

They mostly have a nominal resistance of 100 Ω and

are well suited as sensors for rough environments and

temperature control circuits, but they feel much more

strain and contamination effects and suffer from a larger

uncertainty of about 100 mK. All of these devices are

available commercially and should not be fabricated by

the user except for very special needs. Some of them are

showninFig.8.10.

For the electronics of resistance thermometers two

different concepts are in use: the potentiometric method

and the resistance bridge (Fig. 8.11). The ﬁrst is the

most common concept, especially since precise elec-

tronic devices appeared on the market. A reference

resistor is placed in series with the temperature sen-

sor and the voltage drop across each of the resistors

Table 8.5 Types of standard platinum resistance thermometers

Type of thermometer Temperature range Typical resistance

Capsule thermometers, 50–60 mm long 13.8–430 K 25.5 Ω

Long-stem thermometers, 450 mm long 84–933 K 25.5 Ω

High-temperature long-stem thermometers 0.01–962

◦

C 0.25 Ω

Fig. 8.10 (from left to right) three thick-ﬁlm Pt-100 re-

sistors; Pt-500 chip resistor prepared for SMD mounting;

Pt-100 resistor in ceramics; two NTC thermistors in glass;

Pt-100 resistor in glass; double Pt-100 resistors in glass;

long-stem SPRT (Pt-25); capsule SPRT (Pt-25)

is determined when a constant current is applied. The

voltage drop across the known reference resistor R

ref

yields the current I through the sensor. Using the

voltage drop V (T ) measured across the sensor, the re-

sistance R(T ) can be calculated using Ohm’s law again.

The use of bridge circuits is an alternative method.

These circuits rely on the principle of the Wheatstone

bridge. The design consists of two parallel branches of

two resistors in series, e.g. R

and the sensor R(T )in

one branch and R

and R

in the other. In the balanced

bridge mode, the resistor R

is adjusted so that the volt-

age between the centers of the two branches is zero.

This adjustment has to be done for each temperature T.

R(T )isthengivenby

R(T ) = R





. (8.28)

If this balancing is performed at only one temperature,

say T

, the bridge output V

−V

differs from zero at

other temperatures and is proportional to the measured

temperature in the linear range of the bridge around T

When measuring the temperature sensor’s resis-

tance one has to consider that there always is a lead

resistance in series. The best method to avoid this error

is to measure with a four-lead arrangement, where the

Part C 8.5

Thermal Properties 8.5 Temperature Sensors 477

two voltage probe leads are ﬁxed as close to the sensor

as possible. If the voltmeter’s internal resistance is close

to inﬁnity, only a negligible current ﬂows through the

voltage leads and their voltage drop can be neglected. If

four leads are too expensive and less accuracy is suf-

ﬁcient, a two-lead arrangement can be used. For the

lead wires, materials with a low temperature coefﬁcient

of electrical resistance are preferred. Furthermore, the

wiring should be performed such that the temperature

along these leads will not change during the measure-

ment to keep their resistance constant. There is also

a third method, which is a sort of compromise in ex-

pense and accuracy between both methods described:

the three-lead concept. Here only one voltage lead is

connected close to the sensor. Thus, one can measure

the voltage drop across the sensor and one current lead

in series and subtract the voltage drop of the other cur-

rent lead measured separately.

There are further sources of errors not dealt with

up to now, the thermoelectric effect at connections of

different materials along the leads and the input off-

set voltage of the voltmeter. By reversing the direction

of the measuring current those errors can be averaged

out. If this is done systematically by applying alternat-

ing current for the measurement, one can remove these

errors quantitatively. In addition, the resolution of the

measurement can be improved by the application of

lock-in techniques.

Electromagnetic interference (EMI) can be a prob-

lem in particular with alternating current (AC) mea-

surements, if the ﬁltering of, e.g., the power lines is

not adequate in the electronics. In addition, the sensor

leads should be protected from extra currents induced

by changing external magnetic ﬁelds. This could be

achieved using the twisted-pair design or by coaxial ca-

ble to minimize the loop area between the lead wires.

The heating of the sensor by captured radio and tele-

vision frequencies should be prevented by appropriate

shielding of the whole measuring arrangement. Other

errors are primarily due to the handling of the measure-

ment and the methods of avoiding these are described

in the previous section.

A different type of resistance thermometer is the

rhodium–iron thermometer (0.5% Fe in Rh). Its mer-

its are at the low temperature range between 0.5K and

30 K, where it is more sensitive than the platinum ther-

mometer, but it is still useful up to room temperature.

Besides the wire-type device with a resistance in the

range 20–50 Ω, thick-ﬁlm versions are commercially

available. For very low temperatures different resistance

materials are used and described in the last section.

a) b)

ref

I =

ref

R(T) V(T) = R(T)×I

R(T)

–V

= 0

Fig. 8.11a,b Resistance measurement circuits (a) potentiometric

type. (b) bridge type

There are also resistance thermometers made from

semiconducting materials, known as negative temper-

ature coefﬁcient (NTC) or positive temperature coefﬁ-

cient (PTC) thermistors. The temperature dependence

of the resistance of NTCs obeys an exponential law

R(T ) = A exp(B/T ) . (8.29)

Compared to platinum resistance thermometers, NTCs

show more than 100 times the sensitivity (3–6%/

◦

C) in

the temperature range from −100

◦

Cto+150

◦

Caswell

as being smaller and faster. Two such devices are shown

in Fig. 8.10. However, they are signiﬁcantly less stable

than PRTs and their temperature resistance relation is

extremely nonlinear (8.29) and difﬁcult to ﬁt accurately

over a wide range. They are ideal for use in control

circuits and for differential temperature measurements.

8.5.4 Liquid-in-Glass Thermometers

A liquid-in-glass thermometer, the oldest type of ther-

mometer, is based on the thermal expansion of a liquid

as a function of temperature. It consists of a glass bulb

connected to a capillary containing the sensing liquid

– in most cases mercury or an organic liquid. Attached

to it is a scale scratched on a ﬂat glass strip along the

capillary or on the capillary itself. The bulb, capillary,

and scale are placed in a slightly wider glass tube as

a protective container.

In spite of the image of simplicity that liquid-

in-glass thermometers have obtained, they are more

complicated to understand and to handle than is com-

monly recognized. Such a thermometer is a system

with strong interference between its components. The

liquid is the temperature sensor and its indicator at

the same time, which means that the volume of the

Part C 8.5