Temperature Measurement

University of South Australia, Page 1 School of Pharmacy and Medical Sciences Process Instrumentation, IL2 Lecturer: Dr. Alexander Badalyan

3. Instrumentation for temperature measurements Temperature is one of the major physical parameter, which characterises the condition of substances involved in processes. In order to measure this parameter we need to choose an appropriate temperature scale and the unit of temperature. 3.1. Temperature scales, temperature units Anders Celsius, the Swedish astronomer, devised a scale for measuring temperature, which later was named after his name. This scale has the symbol °C. The UCelsius scale U was based on two fixed and easily reproducible points:

• the ice point, ie the temperature of a mixture of ice and water in equilibrium with saturated air at a pressure 101325 Pa. This temperature was numbered 0 °C;

• the steam point, ie the temperature of the water and steam in equilibrium at a pressure

101325 Pa. This temperature was numbered 100 °C. Later in 1954 this scale was redefined and was based on:

• a single fixed point - the triple point of water. This is the temperature at which solid, liquid and vapour phases of water exist together in equilibrium. The temperature of the triple point of water has the value of 0.01 °C; • the ideal-gas temperature scale. On this scale the steam point was experimentally found to be equal to 100.00 °C.

The Uthermodynamic scale U of temperature (or the Uabsolute scale U) was derived from the second law of thermodynamics. This scale is independent of any thermometric substance. The relation between the absolute scale and the Celsius scale is as follows: 15.273+=ϑT , (3.1) where: T - temperature in the absolute scale, K ; ϑ - temperature in the Celsius scale, Cο . The unit for the absolute scale is K - Kelvin, named after Lord Kelvin (William Thomson).

16.2731,1 =K of the temperature at the triple point of water.

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However, for practical purposes an UInternational Practical Temperature ScaleU, 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. Table 3.1 presents the primary fixed-point temperatures for IPTS-68. (Van Wylen G.J., Sonntag R.E. Fundamentals of Classical Thermodynamics, 1980). Table 3.1. Fixed points and corresponding temperatures for IPTS-68.

NN Fixed points 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

In this table: s - solid; l - liquid; v - vapour. The International Committee of Weights and Measures (CIPM) adopted a new International Temperature Scale (ITS-90) at its meeting in September 1989. It became the official international temperature scale on January 1, 1990. Table 3.2 shows fixed point for ITS-90. (Burns G.W., Scroger M.G., Strouse G.F., Croarkin M.C., Guthrie W.F. Temperature-Electromotive Force Reference Functions and Tables for the Letter-Designated Thermocouple Types Based on the ITS-90). Temperatures in ITS-90 are in closer agreement with thermodynamic values when compared with IPTS-68. The increased numbers of temperature subranges makes ITS-90 more flexible. There are certain differences between ITS-90 and IPTS-68 (similar fixed points are shown in italics in these tables).


Table 3.2. Fixed points and corresponding temperatures for ITS-90.

N

Fixed points 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

3.2. Liquid-in-glass thermometers These thermometers are used for temperature measurements from -200 to 750 °C. They are contact-type thermometers. Fig. 3.1 shows the principle of their design.

1 2 435

Figure 3.1. Liquid-in-glass thermometer


This thermometer consists of a glass bulb 1, which is connected with a glass capillary tube 2. A scale 3 in degrees of Celsius or Fahrenheit is placed behind the capillary tube. The bulb, the capillary tube and the scale are placed in a glass tube 4 to protect them against the damage. A thermometric liquid 5 fills the bulb and a part of the capillary tube. The operational principle of these thermometers is based on the difference between the volume expansion of liquids and glass with temperature. The relationship that governs the operation of this device is )*1(*

0TVV TT Δ+= β (3.2)

where, TV - volume of liquid at temperature T , 3m ;

0TV - volume of liquid at temperature 0T , 3m ;

0TTT −=Δ - difference of temperatures, K ;

β - volumetric thermal expansion coefficient, K1

.

The volumetric thermal expansion coefficient of glass is much less than that of liquids. The variation of temperature (up and down) of the bulb causes liquid in the system to expand or decrease its volume, respectively. As a result of such changes (the internal volume of the glass bulb and the glass capillary varies negligible), the length of the liquid column in the capillary tube goes up or down proportionally to the variation of temperature. The type of thermometric liquid depends on the lower and upper limits of the measuring temperature range. Table 3.3 presents the most common types of liquids used in these types of thermometer. Table 3.3. Types of thermometric liquids.

Liquid Temperature range, °C From To

Mercury -35 750 Toluene -90 200 Ethanol -80 70

Kerosene -60 300 Petroleum Ether -120 25

Pentane -200 20

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Among these liquids Umercury U is the most widely used, because:

• mercury is easy obtainable with high chemical purity; • mercury does not wet glass (this increases the accuracy of measurement/ reading); • mercury remains in liquid state in a wide temperature range.

Among Udisadvantages Uinherent to mercury-in-glass thermometers we can mention the following:

• mercury is a poisonous element, which affects the central and peripheral nervous system, its vapour is the most toxic; • small volumetric thermal expansion coefficient for mercury, therefore, mercury is used in thermometers with capillaries of small internal diameter;

The solidifying point of mercury, ie 38 °C, limits the lowest temperature that can be measured by mercury-in-glass thermometers. The upper temperature is determined by the temperature at which glass still retains its solid properties. This temperature is equal about 600 °C for glass, and about 750 °C for silicon glass. When air above mercury in the capillary is removed, a mercury-in-glass thermometer can be used at temperatures below 300 °C, because the boiling temperature of mercury at atmospheric pressure is equal 356.9 °C. In order to increase this temperature range it is necessary to increase the boiling temperature of mercury (saturation temperature). This can be achieved by increasing pressure in the capillary. Usually, the space above mercury in the capillary is filled by inert gas (such as nitrogen, argon) under pressure. Liquid-in-glass thermometers with organic thermometric liquids are used for temperature measurements from -200 to 200 °C. One Uadvantage Uof these thermometers is:

• a higher volume thermal expansion coefficient comparing with that for mercury (six times higher in average).

UDisadvantage Uof thermometers with organic liquids is:

• these liquids wet glass, therefore, in order to increase the accuracy of measurement/reading, glass capillaries with bigger internal diameters (up to 1 mm) are used.

UAdvantages Uof liquid-in-glass thermometers are as follows:

• they are simple in design;

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• they are relatively highly accurate in temperature measurement.

There are several Udisadvantages Uinherent to liquid-in-glass thermometers

• they are fragile;

• it is difficult to perform readings due to low visibility of the scale;

• they are not capable of distance transmission of a measuring signal, therefore, they are used as locally placed devices;

• impossibility to repair;

• high values of time lag;

• low visibility of mercury in the capillary. 3.2.1. Dynamic characteristic of liquid-in-glass thermometer (thermal capacitance of glass

wall is not included) The heat energy balance for mercury in the bulb:

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

−

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

=

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

periodtimesystemtheofout

heatofflow

periodtimesystemthein

heatofflow

periodtimesystemawithin

heatofonaccumulati

____

__

___

__

___

__

(3.3)

Thermal capacitance of the glass walls is neglected.

tQ

tQ

tQ mmm

outinaccum

Δ

Δ−

Δ

Δ=

Δ

Δ. , (3.4) ⇒ 0=m

outQ , (3.5)

tQ

tQ mm

inaccum

Δ

Δ=

Δ

Δ. , (3.6) ⇒

t

TcM

tQ m

mm

mpaccum

Δ

Δ=

Δ

Δ. , (3.7)

mf

mflm

RTT

tQ

in

,

)( −=

Δ

Δ, (3.8) ⇒

mf

mflmm R

TTt

TC

,

)( −=

ΔΔ

, (3.9)


flmm

mmf TTt

TCR =+

ΔΔ

, , (3.10)

Let Δt→0, then we can get a first order differential equation:

flmm

mmf TTdt

dTCR =+, , (3.11)

Explanations of variables used in the above equations is given below:

mC - thermal capacitance of mercury, KJ

;

mp

c - specific heat of mercury, Kkg

J*

;

mM - mass of mercury, kg ; maccum

Q.

Δ - amount of heat energy accumulated by mercury during a period of time tΔ , J ; min

QΔ - amount of heat energy transferred to mercury during a period of time tΔ , J ; moutQΔ - outflow of heat energy from mercury during a period of time tΔ , J ;

mfR , - thermal resistance between mercury and outside fluid, WK

;

tΔ - period of time, s ; mT - temperature of mercury, K ;

tTm

ΔΔ

- rate of change of temperature of mercury, sK

;

dtdTm - instantaneous rate of change of temperature of mercury,

sK

;

flT - temperature of the fluid outside the bulb, K .

mggg

g

flgmf hAkA

xhA

R 11, ++= , (3.12)

gA - heat transfer surface area, 2m ;


mfl hh , - film coefficients of fluid and mercury, respectively, Km

W

*2;

gk - thermal conductivity of glass, Km

W*

;

gx - thickness of glass wall, m . Differential equation with variables in deviation form:

flmm

mmf TTdt

dTCR ''

', =+ , (3.13)

{ }flmm

mmf TLTdt

dTCRL ''

', =

⎭⎬⎫

⎩⎨⎧ + , (3.14)

Let: AT fl =' - step change. Then we have:

sAsTsTsCR mmmmf =+ )()( ''

, (3.15) ⇒ sAsTsCR mmmf =+ )()1( '

, (3.16)

)1()(

,

'

+=

sCRsAsT

mmfm (3.17)

Use inverse Laplace transform:

{ } ⎟⎟⎠

⎞⎜⎜⎝

⎛−=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛−=

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

+==

−−−− τ

tCR

t

mmfmm eAeA

sCRsALsTLT mmf 1*1*

)1()(' ,

,

1'1 (3.18)

where,

mmf CR ,=τ - time constant, s . Let: CA ο10= ; WKR mf ,131, = ;

KJCm ,56.0= .

Figure 3.2 shows a dynamic response of this thermometer to a step change in temperature.


Dynamic Response of a Liquid-in-Glass Thermometer to a Step Change in Temperature

0

2

4

6

8

10

0 50 100 150 200 250 300 350 400

Time, s

Tem

pera

ture

(in

devi

atio

n fo

rm),

C

Figure 3.2. Dynamic response of liquid-in-glass thermometer to a step change in temperature. From equation (3.14) we can get:

)()()( ''', sTsTsTsCR flmmmmf =+ , (3.19) ⇒ )()()1( '' sTsTs flm =+τ , (3.20)

Using a block diagram in Figure 3.3 we can get the following expression for a transfer function:

T h e rm o m e te r

)(' sf

in p u t o u tp u t

)(' sf)(' sy

)(' sT m

G (s ))(' sy

)(' sT fl )(' sT m)(' sT fl

Figure 3.3. Block diagram of a thermometer.


)1(1

)(

)()(')(')(

'

'

+===

ssT

sTsfsysG

fl

m

τ. (3.21)

3.2.2. Dynamic characteristic of liquid-in-glass thermometer (thermal capacitance of glass

wall is included) The heat energy balance:

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

−

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

=

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

periodtimesystemtheofout

heatofflow

periodtimesystemthein

heatofflow

periodtimesystemawithin

heatofonaccumulati

____

__

___

__

___

__

, (3.22)

Thermal capacitance of the glass walls is included. a). The heat energy balance for mercury in the bulb:

tQ

tQ

tQ mmm

outinaccum

Δ

Δ−

Δ

Δ=

Δ

Δ. , (3.23) ⇒ 0=m

outQ , (3.24)

tQ

tQ mm

inaccum

Δ

Δ=

Δ

Δ. (3.25) ⇒

t

TcM

tQ m

mm

mpaccum

Δ

Δ=

Δ

Δ. (3.26)


Mercury

Glass

Fliud (measuringmedia)

gT

mT

R fl g,Rg m,

flT

Figure 3.4. Liquid-in-glass thermometer.

mg

mgm

RTT

tQ

in

,

)( −=

Δ

Δ (3.27) ⇒

mg

mgmm R

TTt

TC

,

)( −=

ΔΔ

(3.28)

gmm

mmg TTt

TCR =+

ΔΔ

, (3.29) ⇒ Let Δt→0

We get the first order differential equation:

gmm

mmg TTdt

dTCR =+, (3.30)

b). The heat energy balance for the glass wall Uhas both inflow and outflow of heat U:


t

Q

t

Q

t

Q gggoutinaccum

Δ

Δ−

Δ

Δ=

Δ

Δ. , (3.31) ⇒

tTC

t

TcM

t

Q gggg

gg

paccum

Δ

Δ=

Δ

Δ=

Δ

Δ. , (3.32)

gfl

gflg

RTT

t

Qin

,

)( −=

Δ

Δ, (3.33) ⇒

mg

mggout

RTT

tQ

,

)( −=

ΔΔ

, (3.34)

mg

mg

gfl

gflgg R

TTR

TTt

TC

,,

)()( −−

−=

Δ

Δ. (3.35) Let Δt→0, then

mg

mg

gfl

gflgg R

TTR

TTdt

dTC

,,

)()( −−

−= . (3.36)

Substitute (3.30) into (3.36) and after manipulations we get:

( ) flmm

mgflmmgggflm

mmgggfl TTdt

dTCRCRCR

dtTd

CRCR =++++ ,,,2

2

,, , (3.37)

or

( ) flmm

mgflm TT

dtdTCR

dtTd

=++++ ,212

2

21 ττττ . (3.38)

Equation (3.38) is a second-order differential equation.

gC - thermal capacitance of glass bulb, KJ

;

gp

c - specific heat of glass, Kkg

J*

;

gM - mass of glass bulb, kg ; gaccum

Q.

Δ - amount of heat energy accumulated by glass bulb during a period of time tΔ ,

J ; gin

QΔ - amount of heat energy transferred to glass bulb from fluid during a period of

time tΔ , J ; goutQΔ - outflow of heat energy from glass bulb during a period of time tΔ , J ;


gfR , - thermal resistance of fluid film and glass wall, WK

;

mgR , - thermal resistance of glass and mercury film, WK

;

gT - temperature of glass, K .

gg

g

fggf kA

xhA

R +=1

, , (3.39) gg

g

mgmg kA

xhA

R +=1

, , (3.40)

where,

gA - heat transfer surface area, 2m ;

mf hh , - film coefficients of fluid and mercury, respectively, Km

W*2 ;

gk - thermal conductivity of glass, Km

W*

;

gx - thickness of glass wall, m . Transfer function is as follows:

1)(1

)(

)()(')(')(

2,1212

21'

'

++++===

sssT

sTsfsysG

fl

m

τττττ. (3.41)

where, mgfl CR ,2,1 =τ . Let, 2

21 τττ = , (3.42) and ζττττ 22,121 =++ . (3.43)

Then, 12

)(')( 22

'

++=

ss

sTsT fl

m ζττ. (3.44)

Let: AT fl =' - step change, and 1=ζ . Then we have:


222'

)1()12()(

+=

++=

ssA

sssAsTm τττ

. (3.45)

Use inverse Laplace transform:

{ } ⎟⎟⎠

⎞⎜⎜⎝

⎛ +−=

⎭⎬⎫

⎩⎨⎧

+==

−−− τ

ττ

τ

t

mm etAssALsTLT 1*

)1()(' 2

1'1 (3.46)

Let: CA ο10= and s,85=τ . Then we can plot a transient response of this thermometer to a step change in the input variable (see Figure 3.5).

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

Figure 3.5. Transient response of liquid-in-glass thermometer.


3.3. Filled thermal systems

Another class of thermometers that utilise the principle of expansion of substances with temperature is called Ufilled thermal U systems. Depending on the phase of the substance, which fills these devices, these systems are sub-categorised into Ugas- U, Uliquid- U and Uvapour-filledU systems.

UGas-filledU systems are based on a basic law of gases. If a gas is kept in a metallic bulb (or a container) at a constant volume, then if the temperature varies, so does the pressure according to the relationship )(** 12112 TTPPP −=− β , (3.47) where:

1P and 1T - absolute pressure ( Pa ) and temperature ( K ) at state 1;

2P and 2T - absolute pressure ( Pa ) and temperature ( K ) at state 2; β - thermal coefficient of pressure, equal to the volumetric thermal expansion

coefficient, 1−K .

1

34

2

1

3

5

2

4

Figure 3.6. Gas- or liquid-filled thermometer. Figure 3.7. Vapour-pressure system.

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Figure 3.6 schematically shows the design of a gas-filled thermometer. Gas (nitrogen or helium) 1 fills the thermal bulb 2, capillary tube 3 and Bourdon tube of a manometer 4. The thermal bulb (usually made of a stainless steel) is immersed in the measuring media. Variation of its temperature causes change in pressure of the gas in the system. The manometer measures this variation of pressure. The scale of the manometer is graduated in °C, but not in Pa. The length of the capillary tube (usually made of a stainless steel) varies from 0.6 to 60 m. UThe accuracy of measurements for these thermometers is greatly influenced by variation of ambient temperatureU (since it can change the pressure of a gas in the system). Two methods are used to reduce this effect:

• a thermal bimetallic temperature compensator is used in the manometer; • an internal volume of the thermal bulb should be greater than that of the capillary tube,

the ratio c

b

VV

(where bV and cV are volumes of the thermal bulb and of the capillary,

respectively) may vary from 40 to 60; this can be achieved by reducing the internal diameter of the capillary tube or increasing the internal volume of the thermal bulb. The longer the capillary tube, the bigger the thermal bulb should be.

Therefore, gas-filled thermometers are not widely used in practice. Depending on the measured temperature range, the system may be filled with a gas under pressure higher than atmospheric. That is why variations in atmospheric pressure have no effect on the indications of gas-filled thermometers. Gas-filled thermometers have several Uadvantages U:

• they have the widest temperature range of all filled systems; • as follows from the equation (3.47) these thermometers have uniform scales; • they have the longest capillary length compared with other filled systems.

These thermometers are usually used for temperature measurement in the range from -200 to 600 °C. ULiquid-filledU systems have similar design with gas-filled thermometers (see Fig. 3.6). Organosilicone liquids, propanol and mercury are used as thermometric liquids, which fill the entire system. Since the total volume of the thermal system is constant, then variation of temperature of the media, where the thermal bulb is immersed, causes variation in the pressure of

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the thermometric liquid. This variation in pressure is proportional to the variation of temperature. Therefore, scales of liquid-filled thermometers are uniform. Several factors influence the accuracy during temperature measurements, namely:

• variation in ambient temperature; • variation in pressure head; • variation in atmospheric pressure.

In order to compensate the influence of variation of an ambient temperature it is necessary to increase the Uratio internal volume of the thermal bulb/internal volume of the capillary tubeU, and employ thermal bimetallic compensators (see gas-filled thermometers). The error due to variation of an ambient temperature is bigger in the case of liquid-filled systems, compared to gas-filled systems. Therefore, the capillary length for liquid-filled systems can not exceed 10 m. When the thermal bulb is placed below or above the manometer, results of such temperature measurements will not be correct. This is because of different pressure head of the liquid column compared with the case when this thermometer was calibrated (the manometer and the thermal bulb were placed on the same level). In this case the error can be eliminated by zero correction of manometer. The ultimate elevation distance between the thermal bulb and the manometer are given in the calibration certificate supplied with the liquid-filled thermometer. To reduce influence of variation of atmospheric pressure, the system is filled with liquid under pressure from 0.5 to 2.0 MPa. Here are the Uadvantages Uof liquid-filled thermometers:

• small time lag; • small dimensions of thermal bulb.

These thermometers are used for temperature measurement in the range from -150 to 300 °C. UVapour/pressure U systems (see Fig. 3.7) are filled by 2/3 of the volume of the thermal bulb 1 by liquid 2 which has a low boiling temperature, for example, freon (refrigerant), propylene, acetone, ethylbenzene, methyl chloride, etc. Another (upper) part of the thermal bulb and the capillary tube 3 is occupied by saturated vapour 4 of this liquid. Vapour pressure depends only on the temperature of saturated liquid in the thermal bulb, and therefore, does not depend on the variation of the ambient temperature (this is an Uadvantage U). Relationship between saturation pressure and temperature for liquids is non-linear (see Fig. 3.8). Hence, the scales of these

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thermometers are non-uniform, with more widely spaced increments at high temperatures. The length of the capillary tube usually does not exceed 25 m. UDisadvantages U of vapour-pressure thermometers are as follows:

• narrow temperature range, from -50 to 300 °C; • slow response time (time lag) of about 20 seconds; • non-uniformity of the temperature scale.

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

0 20 40 60 80 100Temperature, C

Vapo

ur (s

atur

atio

n)

pres

sure

, MPa

Methyl chloride

Figure 3.8. Saturation (vapour pressure) curve for methyl chloride. 3.4. Thermocouples


Seebeck in 1821 discovered that Uthermal electromotive force (t.e.m.f.) is generated in a closed circuit of two wires made of dissimilar metals if two junction are at different temperaturesU. One junction is inserted into a measuring media, and it is called a Uhot U or Umeasuring junction U. Another one, called a UcoldU or Ureference junctionU, is kept either at 0 °C or at ambient temperature and is connected to a measuring instrument (millivoltmeter). The electronic explanation of this phenomenon is as follows:

the density of conduction electrons in two dissimilar metals is different. So, in the case when metals are brought into contact (welded together), the free (or conduction) electrons will flow from the metal with high their density to the metal with low density of the conduction electrons. As the result of this drift, a potential difference is produced in the boundary between these two metals. This potential difference will stop the flow of electrons. Since the metals are different, so they will differently respond to temperature variations. In other words, the variation of temperature will change the density and velocities of free electrons in two metals differently. This will cause the change in the magnitude of the thermal electromotive force.

Figure 3.9 schematically shows a thermocouple and a measuring instrument.

1 2 5

3

8

6

7

+

-

+

-

4

Figure 3.9. Thermocouple and measuring instrument. 1 - hot junction; 2 - metal A; 3 - metal B; 4 - connection head; 5 - extension wires; 6, 7 - positive and negative terminals, respectively, of a measuring instrument; 8 - measuring instrument. T.e.m.f. is proportional to the difference of temperatures between the two junctions. All tables, correlated t.e.m.f. of thermocouple (measured in mV) and temperature, are developed when the

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temperature of a cold junction is equal to 0 °C. T.e.m.f. is the function of temperature difference between the hot and the cold junctions: )( 0ϑϑ −= fEAB , (3.48) where:

ABE - t.e.m.f. developed by a thermocouple, mV ;

ϑ and 0ϑ - temperatures of the hot and the cold junctions of a thermocouple, Cο . If the temperature of the cold junction is kept constant (say at Cο,0 ), then t.e.m.f. is proportional to the temperature of the hot junction (the measuring temperature), ie )(tfEAB = . (3.49) In reality, in industrial environment, however, it is not possible (or is not convenient) to keep the temperature of the cold junction at Cο,0 . Therefore, to evaluate the actual t.e.m.f. and, finally, the actual measuring temperature, we should introduce a correction. A final equation has the following form:

),(),(),( 0'0

'00 ϑϑϑϑϑϑ ABABAB EEE += (3.50)

where:

),( 0ϑϑABE - t.e.m.f. developed by a thermocouple when the temperature of the hot junction is equal to ϑ and the temperature of the cold junction is equal to

Cο,00 =ϑ , mV ;

),( '0ϑϑABE - t.e.m.f. developed by a thermocouple when the temperature of the hot

junction is equal to ϑ and the temperature of the cold junction is equal to '0ϑ (different from Cο,0 ) – this t.e.m.f. is measured by a millivoltmeter, mV ;

),( 0'0 ϑϑABE - t.e.m.f. developed by a thermocouple when the temperature of the hot

junction is equal to '0ϑ and the temperature of the cold junction is equal to

Cο,00 =ϑ , mV . There are various types of thermocouples:

• UPlatinumU and UPlatinum - 10% RhodiumU ( Utype SU) from -50 to 1765 °C;

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• UPlatinum - 6% RhodiumU and UPlatinum - 30% RhodiumU ( Utype BU) from 0 to 1820 °C; • UNickel - ChromiumU and UNickel - AluminiumU (Chromel-Alumel, Utype KU) from -270 to 1370 °C; • UIron U and UCopper - NickelU (Iron - Constantan, Utype J U) from -210 to 1200 °C; • UCopper U and UCopper - NickelU (Copper - Constantan, Utype TU) from -270 to 400 °C; • UNickel - ChromiumU and UCopper - NickelU (Chromel - Constantan, Utype EU) from -270 to 1000 °C.

Figure 3.10 presents experimental curves thermal electromotive force vs temperature for several types of thermocouples.

-10

0

10

20

30

40

50

60

-300 0 300 600 900 1200 1500 1800

Temperature, C

Ther

mal

ele

ctro

mot

ive

forc

e, m

V

Type K

Type JType T

Type S

Figure 3.10. Experimental curves Uthermal electromotive forceU vs Utemperature U. Requirements imposed to the properties of metals used as electrodes for thermocouples are as follows:

• reproducibility of material, ie possibility of obtaining of metal wires with the same properties;

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• resistance of metal electrodes should be small and have a weak relationship vs temperature; • stability of a static characteristic )(ϑfEAB = , ie recovery of properties after measurements; • high sensitivity; • correlation )(ϑfEAB = should be close to linear relationship as much as possible.

The highest sensitivity has thermocouple of Type J (Iron - Constantan): C

mVS ο,055.0= .

3.5. Resistance temperature detectors The principle of resistance temperature detectors (RTD) is based on the variation of electrical resistance of metals with temperature. For this purpose several metals are used, namely, platinum, copper, nickel. When temperature increases the resistance of these metals increases. Temperature function of resistance for metals in a narrow temperature interval can be expressed by a relationship: )1(0 αϑϑ += RR , (3.51) where: ϑR and 0R - are the values of electrical resistance of a metal conductor at temperatures ϑ and

Cο,0 , respectively, Ohm ;

α - thermal coefficient of electrical resistance, Cο1

.

For metals this coefficient is positive. Fig. 3.11 shows relationship between resistance of platinum and copper RTD and temperature. Platinum RTDs are used for temperature measurements from -220 to 850 oC (they are used as reference RTDs, as well), copper RTD - from -50 to 150 oC, and Nickel RTD - from -215 to 320 oC.

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0

50

100

150

200

250

300

350

400

-300 -100 100 300 500 700 900

t, oC

R, O

hm

Platinum RTD

Copper RTD

Figure 3.11. Resistance vs temperature for platinum and copper RTD. Fig. 3.12 shows the assembly of RTDs. Sensitive elements of RTDs are made of a thin wire 1 with outside diameter equal to 0.025 mm (platinum RTD) and 0.1 mm (copper RTD) double wounded (non-inductive) on a micaceous or porcelain stem 2. For mechanical strength the sensitive element is placed in the ceramic insulator tube 3 filled by extremely fine granular powder; extension wires are placed in the ceramic insulator 4, and entire assembly is covered by a protective sheath of stainless steel 5. The space between the sheath and ceramic insulator is filled by ceramic packing powder 6. To avoid contact of sensitive element with environment, sensitive assembly is protected by high-temperature hermetic seal 7. The contact between the wire of the sensitive element and the ceramic encapsulation permits a rapid speed of response.

050

100150200

0 40 80 120 160 200

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ab

c

1 52 3 6 74

Figure 3.12. RTD assembly. For measurements of resistance of RTDs several methods are used. Among them the most widely used is a method employing a UWheatstone bridge U (Figure 3.13). In this case RTD is connected to the bridge by two connecting cables (conductors). The bridge is powered by direct current power supply in the points ""a and ""b . RTD is immersed in the media, which temperature to be measured. When this bridge is in balance, then there is no voltage between points ""c and ""d , and zero-indicator (ZI) shows no current. For this condition we can write the following equation: src RRRRR *)2(* 21 =+ϑ , (3.52) or

csr RRRR

R 2*1

2 −=ϑ , (3.53)

where:

1R and 2R - electrical resistance of two invariable resistors, usually, 21 RR = , Ohm ; ϑR - electrical resistance of RTD, Ohm ;

cR - electrical resistance of a connecting cable, Ohm ;

srR - electrical resistance of a slide (variable) resistor, Ohm .

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Rt

ZI

Rsr R1

R2

RcRc

Rt

a bd

c

U

ZI

Rsr R1

R2

RcRc

a bd

c

U

Rc

Figure 3.13. Two-conductor connecting. Figure 3.14. Three-conductor connecting. As follows from these equations, the certain position of a slide of the variable resistor (ie, certain value of srR ) corresponds to each value of a measuring temperature (ie, for each value of ϑR ) at any balance condition of the bridge. Therefore, the scale of the bridge may be calibrated in degrees Celsius. For Ua two-conductor connecting scheme U (see Fig. 3.13), variation of an ambient temperature will effect values of an electrical resistance of connecting cables cR and, therefore, the results of measurements will be erroneous. With all resistance temperature detectors Ua three-conductor connection scheme U is recommended (see Fig. 3.14). One conductor is common to both sides of the bridge, while other two connect the RTD to each side of the bridge. Any change in the cable temperature (as the result of variations in ambient temperature) will be cancelled because the resistance of both sides of the bridge change by the same value (providing three connecting cables are at the same temperature).

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A Ufour-conductor connection scheme U is used when very accurate measurements of temperature are required, up to ±0.01 °C of accuracy. 3.5.1. Derivation of a balanced condition for a Wheatstone Bridge Below I give you the derivation of the balanced condition for a Wheatstone bridge. Figure 3.14a shows a circuit of five electrical resistors connected to form the Wheatstone bridge.

fe

d

c

b

a + -

R1 R2

R3 R4

R5

I1 I1-I5

I3+I5I3

I5

I5

I5

U

g

Figure 3.14a. Wheatstone bridge. Here we use two Kirchoff’s Laws. The first Kirchoff’s law says: Uthe total current flowing into any junction is equal to the total current flowing out of this junctionU. The second Kirchoff’s law states: Uthe total change in potential around any closed circuit loop is equal to zero. For the second Kirchoff’s law we should be sure to include the sign of potential energy correctly: the potential decreases around a resistor in the direction of current flow and increases in the direction opposite to current flow; potential increases from negative to positive terminals of the battery.


Below I give you two different methods for the derivation of the balanced condition for a Wheatstone bridge. Now, let’s consider several closed circuit loops and apply the above law to them. Ua. First method 1. Loop efcbae −−−−− : 0)( 25111 =−−+ URIIRI (3.54) 2. Loop adgba −−−− 0335511 =−+ RIRIRI (3.55) 3. Loop bgdcb −−−− ( ) ( ) 055453251 =−+−− RIRIIRII (3.56) Here, U is a dc voltage, and 0≠U . So, we have three equations with 3 unknowns - 1I ; 3I and 5I .

URIRIRI =−+ 252111 (3.57)

0335511 =−+ RIRIRI (3.57)

05545432521 =−−−− RIRIRIRIRI (3.59) Now we combine terms with 1I ; 3I and 5I . ( ) UIRIIRR =−++ 523121 *0 (3.60)

0553311 =+− IRIRIR (3.61)

( ) 055423412 =++−− IRRRIRIR (3.62)


Now we evaluate the determinant of the matrix developed using this set of equations:

( )

( )

)(

))((*0

)()(

)(

0

32412

525421

54542321

54242

531

221

RRRRR

RRRRRR

RRRRRRRR

RRRRRRRRRRR

+−−

−−++−−

−++++=

=++−−

−−+

=Δ

(3.63)

In the case of the Wheatstone bridge the resistance 5R is substituted by an ampmeter. Ampmeters have a negligibly small resistance, so we can write that 05 =R . With this condition we can re-write (3.63) as follows:

( )

421432431321

4213224323

22431321

41322433221 )()(

RRRRRRRRRRRR

RRRRRRRRRRRRRRRR

RRRRRRRRRRR

+++=

=+−+++=

=−−++=Δ

(3.64)

Because all 0>iR we have that 0>Δ and 0≠Δ . For current 5I we have:


( )

)()(

)()((

)0*0*(*0)0*0*)((

00

0

41323241

3241

214321

42

31

21

5

RRRRURRRRU

RRRRU

RRRRRR

RRRR

URR

I

−=+−=

=−−−+

+−−+−+=

=−−

+=Δ

(3.65)

Now we can evaluate the value of the current, which flows through an ammeter as follows:

Δ−

=Δ

Δ=

)( 41325

5 RRRRUI I . (3.66)

We know that a balanced condition is when current 05 =I , and because 0>Δ , and 0≠Δ and

0≠U , we can give an expression for the balanced condition for the Wheatstone bridge as follows:

04132 =− RRRR , (3.67) or, finally,

3241 RRRR = , (3.68) b. Second method When the bridge is balanced, ie 05 =I , and current 1. Loop efcbae −−−−− : 0)( 25111 =−−+ URIIRI (3.69) or, URIRI =+ 2111 . (3.70) 2. Loop efcdae −−−−−


0)( 45333 =−++ URIIRI (3.71) or, URIRI =+ 4333 . (3.72) Assume point f is at earth potential, then potential at point e is equal to UU a = . Potential at point b is equal to 1111 RIURIUU ab −=−= . (3.73) Potential at point d is equal to 3333 RIURIUU ad −=−= . (3.74) Potential difference between points b and d is equal to )()( 3311 RIURIUU bd −−−= . (3.75) Using equations (3.70) and (3.72) we can get:

21

1 RRUI+

= (3.76)

and

43

3 RRUI+

= . (3.77)

Substitute equations (3.76) and (3.77) into equation (3.75):

⎟⎟⎠

⎞⎜⎜⎝

⎛+

−−⎟⎟⎠

⎞⎜⎜⎝

⎛+

−= 343

121

RRR

UURRR

UUU bd , (3.78)

or,

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+

−−⎟⎟⎠

⎞⎜⎜⎝

⎛+

−= 343

121

1111 RRR

RRR

UU bd , (3.79)


or,

⎟⎟⎠

⎞⎜⎜⎝

⎛+

−+

=21

1

43

3

RRR

RRR

UU bd . (3.80)

Since 03 ≠R and 01 ≠R , then

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

+−

+=

1

2

3

4 1

1

1

1

RR

RRUUbd . (3.81)

We know for balanced condition 05 =I , then voltage 0=bdU . Because 0≠U , then

01

1

1

1

1

2

3

4=

+−

+RR

RR

, (3.82)

or,

1

2

3

4 11RR

RR

+=+ , (3.83)

or,

1

2

3

4

RR

RR

= , (3.84)

So, finally, we get

3241 RRRR = . (3.85) 3.6. Thermistors


If semiconductors or heat-treated metallic oxides (oxides of cobalt, copper, iron, tin, titanium, etc.) are used as the materials for producing temperature sensitive elements, then these temperature transducers are called Uthermistors U (the name is derived from the term of ‘thermally sensitive resistor’). These oxides are compressed into the desired shape from the specially formulated powder. After that, the oxides are heat-treated to recrystallise them. As the result of this treatment the ceramic body becomes dense. The leadwires are then attached to this sensor for maintaining electrical contact. The following relationship applies to most thermistors:

⎟⎟⎠

⎞⎜⎜⎝

⎛−

= 0

11*

0 * TTB

t eRR (3.86) where,

0TR - resistance of thermistor at reference temperature KT ,0 , Ohm ;

TR - resistance of thermistor at temperature KT , , Ohm ; B - constant over temperature range, depends on manufacturing process and

construction characteristics, K1

.

Fig. 3.15 shows relationship between temperature and resistance for a thermistor. Thermistors have negative thermal coefficient of electrical resistance. It means that when temperature increases the electrical resistance of thermistor decreases. They have greater resistance change (this is an advantage) compared with RTD in a given temperature range. For example, if we compare what change in resistance will be caused by variation of temperature in 1 °C for Platinum and Copper RTD (see Fig. 3.11) and for thermistor (see Fig. 3.15) in the temperature range from 273.15 to 423.15 K (ie, from 0 to 150 °C), we will obtain the following values:

• for platinum RTD - C

Ohmο

,38.0 ;

• for copper RTD - C

Ohmο

,04.0 ;

• for thermistor - C

Ohmο

,65.0 .

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430

460

490

520

550

150 200 250 300 350 400 450 500 550

Temperature, K

Res

ista

nce,

Ohm

Thermistor

Figure 3.15. Thermistor resistance vs temperature curve.

Wheatstone bridge and resistance measuring constant current circuits, similar to that used in the case of RTDs, are used for resistance measurement of thermistors (see Fig. 3.14). Despite their high sensitivity, thermistors have a worse accuracy and repeatability (this is the disadvantage) comparing with metallic RTDs. Since the resistance vs temperature function for thermistors is non-linear (although, some modern thermistors have a nearly linear relationship of temperature vs resistance), it is necessary to use prelinearisation circuits before interacting with related system instrumentation. In addition, due to the negative thermal coefficient of electrical resistance an inversion of the signal to positive form is required when interfacing with some analog or digital instrumentation. Therefore, thermistors are not widely used in process instrumentation field, at least at present. However, they have been well accepted in the food transportation industry, because they are small, portable and convenient. Another field of their growing application are heating and air-conditioning systems, where thermistors are used for checking the temperature in flow and return pipes.

All the discussed above instrumentation for temperature measurement (see from 3.2 to 3.6) refers to Ucontact-typeU devices, because their sensitive elements are immersed in the measuring media. When dealing with temperatures above 1500 °C, contact-type temperature measuring devices are not applicable, because irreversible changes occur in metals which form their sensitive elements. It is possible to perform Unon-contact U measurement of temperature by optoelectronic transducers.

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3.7. Optical and radiation pyrometers UPyrometerU is a device which uses the relationship between the electromagnetic radiation emitted by a body and the temperature of this body. In order to better understand the phenomenon which forms the basis of pyrometry, it is useful to explain the concept of the Ublackbody U, and the differences between it and real objects. The term Ublackbody U is ideal, and designates a body which Uradiates more electromagnetic energy for all wavelengths intervals than any other body of the same area and at the same temperature, and absorbs all the radiation it interceptsU. Fig. 3.16 presents one of the classical blackbody model.

Light

Figure 3.16. A classical blackbody model. The temperature of the blackbody determines the nature and extent of such radiation. UStefan-Boltzmann’s law Usays, that

Uthe blackbody with a finite absolute temperature (T ) emits radiant (ie, in all directions) electromagnetic radiation (EMR) per unit area of this blackbody and per second with intensity which is proportional to 4T , according to an equation:

4*TET σ= , (3.87) where,

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TE - total EM energy emitted by the blackbody in all directions per unit area ( 2,1 m )

and per unit time ( s,1 ), 2mW

;

σ - Stefan-Boltzmann’s constant, equal to 428

*,10*67051.5

KmW− ;

T - an absolute temperature of the blackbody, K . Fig. 3.17 shows the relationship between EMR emitted by a perfect blackbody as a function of temperature. The area under these curves is equal to the total energy (emitted by a black body) per second per unit area. This body at low temperatures emits EMR in the region of long wavelengths. This region spreads from far-infrared to microwave region (5 μm < λ < 100 μm, where λ is the wavelength in μm, 10-6 m). With increasing the blackbody temperature, the emission peaks move into the region of shorter wavelengths. At very high temperatures the blackbody emits in the near visible wavelengths region. Visible region corresponds to the wavelengths from 0.7 μm (red) through 0.62 μm (orange), 0.58 μm (yellow), 0.53 μm (green), 0.47 μm (blue) to 0.42 μm (violet). UReal objects emit and absorb less EMR than blackbodies, and this difference is dependent on the wavelengthU, so nonblackbodies can not exactly follow relationship shown in Fig. 3.17. For this purposes corrections should be used, otherwise, the apparent temperature will be lower that the actual temperature. Also, it is necessary to take into account the loss of emitted radiation when it passes through the media between the emitting body and a measuring instrument.

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0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 1 2 3 4 5 6

Wavelength, microns

EM re

diat

ion

ener

gy d

ensi

ty(r

elat

ive

scal

e)2000 K1600 K1200 K800 K500 K

Figure 3.17. EM radiation emitted by the blackbody at various temperatures. There are two types of pyrometers: Uoptical U (monochromatic or narrowband) and UradiationU (total radiation or broadband) pyrometers. The last devices originally were called radiation pyrometers, then radiation thermometers, and more recently infrared thermometers. However, the first their name (radiation pyrometers) is still widely used at present. These devices have high accuracy of ±0.01 °C as a standard instruments, and from ±0.5 to ±1% for industrial purposes. a). UOptical pyrometersU, sometimes referred to as Ubrightness thermometersU, generally involve wavelengths only in the visible part of the spectrum. When the temperature of the body increases, so does the intensity at any particular wavelength. If two bodies have the same temperature, then intensities of those two objects are equal. In this type of a pyrometer the intensity of a certain wavelength of a heated body is compared with that of a heated platinum filament of a lamp (see Fig. 3.18).

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11

1 2 3

4

5 6 7 8

10

9

Figure 3.18. An optical pyrometer. An object 1 which temperature is to be measured, emits electromagnetic radiation with intensity proportional to its absolute temperature. This radiation passes through lens 2 and red optical filter 3. Optical filter picks out only the desired wavelength - red. Then radiation focuses on the platinum filament of a lamp 4, and passes through another filter 5, lens 6, viewing system 7. The viewer 8 sees the platinum filament superimposed on an image of the object 1. When the temperature of the filament is low comparing with that of the object, the viewer sees the filament as a dark line on the bright background image of the object. The lamp 4 is connected in series with an electrical battery 9, a variable resistor 10 and an ampermeter 11. By reducing the resistance of the resistor an electrical current passing through the filament increases. So does the temperature of the filament and its brightness. For a certain value of an electrical current (corresponded to a certain value of an object temperature), the brightness of the platinum filament will match the brightness of the object 1. At this setting the viewer cannot distinguish between the image of the object and the filament. At this time the measurement of temperature is performed. The scale of the ampermeter is calibrated in the units of temperature. The lower temperature limit for optical pyrometers is determined by the temperature at which objects become visible in red (about 225 °C). However, there are devices which are able to measure even lower temperatures down to -50 °C. The upper limit varies from 600 to 3000 °C, and is limited by the melting point of the platinum filament. An accuracy is typically varied from ±5 to ±10 K.

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b). URadiation pyrometersU, being very simple and cheap, use an exponential relationship between a total emitted EMR energy and given temperature. In radiation pyrometers (see Fig. 3.19) EMR energy emitted at infrared (2.5 < λ < 20 μm) to visible wavelengths (0.42 < λ < 0.7 μm) from an object 1 is focused by a spherical reflector 2 on a series of micro-thermocouples attached to a blackened platinum disc 3. The radiation is absorbed by the disc, which temperature is increased, so does thermal electromotive force U developed by the series of thermocouples. This thermal electromotive force is proportional to the temperature of hot junctions of thermocouples, and, finally, to the temperature of the object 1. The advantage of these pyrometers is that their operation slightly depends on the wavelength.

1 2

3

U

Figure 3.19. A total radiation pyrometer The lower limits for radiation pyrometers vary from 0 to 600 °C, the upper limits vary from 1000 to 1900 °C. The accuracy varies from ±0.5 to ±5 K, depending on cost. They are widely used for temperature measurements in metal production facilities, glass industries, semiconductor processes, etc.

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Temperature Measurement

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