(1.) Temperature Measurements

TEMPERATURE MEASUREMENTS

I Dynamic Response and Dynamic Error

Instruments seldom respond instantaneously to changes in the measured

variables rather they show some delay in response The response

generally starts quickly and then becomes gradual in reaching a steady-

state value The time lapse when a system is exposed to a change until

the system starts to response is the lag time (or dead time) There are

several factors that influence the speed of response of a temperature

sensor such as

bull The mass of the shield of the temperature sensor

bull The insulating air space between the sensor and the protecting well

CHE 215 page of 21 H Muhamad 1

bull The velocity of the moving medium being measured

bull The type of medium being measured

When the temperature sensor (or any other sensorinstrument) is exposed

to a step change in the environment of which it is sensing the

temperature the temperature readout of the sensor is lower than the

actual temperature This difference is known as the dynamic error of the

sensor as shown in Figure1 and 2 below (for first-order response

instrument)

00

200

400

600

800

1000

1200

0 02 04 06 08 1 12 14 16 18 2Time (sec)

Tem

pera

ture

(C)

T-dynamic

Step change (ideal T response)

actual T response

dynamic errror

dynamic error

Figure 1 Temperature response to a step change


0

02

04

06

08

1

12

0 02 04 06 08 1 12 14 16 18 2

Time (sec)

Nor

mal

ized

dyn

amic

err

or (X

tXo)

Dynamic error

time constant = 025 s

XtXo = 0368

Figure 2 Dynamic errors

As an example shown in Figures 1 and 2 a temperature sensor is in an

environment at temperature T1 of 20oC The temperature of the

environment is assumed to have a surge in temperature to T2 of 100oC

(ideal step change) Therefore the initial (maximum) dynamic error Xo =

(T2-T1) = (100- 20) = 80oC The dynamic error of the temperature

reading is defined as

Xt = T2 -Td

Where Td is the actual temperature response (or temperature reading) of

the temperature sensor at a given time t after it was exposed to the step

change in temperature


As shown in Figure 2 the dynamic error of the temperature sensor shows

an exponential decay trend Therefore the rate of change of the dynamic

error can be written as

XkdtdX

minus= (1)

Integration of Equation (1) for time t = 0 to some time t gives

tkXX

o

t minus=⎟⎟⎠

⎞⎜⎜⎝

⎛ln (2)

kt

o

t eXX minus=⎟⎟

⎠

⎞⎜⎜⎝

⎛ (3)

When t is equal to the time constant tc of the sensor which is defined as

tc=1k then

36801 ==⎟⎟⎠

⎞⎜⎜⎝

⎛ minuseXX

o

t (4)

Therefore as indicated by Equation (4) when the elapsed time is equal

to the time constant the response of the sensor has covered 632 of the

step change in the temperature as shown below At t = tc

Td = T2 ndashXtc = T2 ndash 0368 Xo

Td = T2 ndash 0368(T2-T1)

Td ndashT1 = T2 ndash 0368T2 + 0368T1 ndash T1

(Td ndashT1) = 0632 (T2 ndash T1)


(Td ndashT1) is 632 of the overall step change (T2 ndash T1)

A first-order-response instrument shows a quick response initially the

response then slows down considerably to the point of achieving a

steady-state measurement The bare-wire thermocouple usually exhibits

a first-order response to a step change in temperature of the environment

it is in From the energy balance for the sensor and the surrounding the

first-order temperature response can be described by the following

differential equation

FTTtdTd

=+sdotτ (5)

where τ is the time constant of the temperature sensor T is the dynamic

temperature of the sensor TF is the step change of temperature and t is

time ( ) ( )( )τexp100 tTTTT F minusminussdotminus+=

However for a shielded temperature sensor with a low thermal

conductivity shield the temperature response could exhibit a second-

order temperature response as described by

( ) FSWSW TTdtdT

tdTd

=+++sdotsdot ττττ 2

2 (6)


where τW and τS are the time constant of the sensor itself and the shield

of the sensor respectively The ratio of (τW + τS)(2radic(τW τS)) is known

as the damping ratio For a large damping ratio (greater than 10) the

profile is similar to that of the first order system) On the other hand for

smaller damping ratio the profile is oscillating

( ) ( ) ( )⎟⎟⎠

⎞⎜⎜⎝

⎛minus

minusminusminus+minus+=

sw

wwssF

ttTTTTττ

ττττ expexp100

II Temperature Measurements

1 Thermometer

Common liquid used

bull Alcohol for temperature range from 50 ndash 200oC

bull Xylene for temperature range from -40 to 400oC

bull Hg for temperature range from -37 to 350oC (Hg freezes at -378oC)

Cautions

bull Do not immerse the whole stem of the thermometer in the

environment where the temperature is measured Only place the bulb

of the thermometer in the environment so to avoid error due to

different thermal expansion coefficients of liquid and glass

bull Allow the liquid level in the thermometer stabilized before taking the

temperature reading


bull Precision of liquid filled thermometer is frac12 of the smallest division on

the scale Highly accurate Hg thermometer from NIST (National

Institute of Standards and Testing) could measure to plusmn 005oC

2 Resistance temperature detector (RTD)

RTD is basically an electrical wire typically Pt Ag Ni Cu etc which

is used to sense the temperature change based on the principle of the

temperature dependency of the wire resistance as shown below

( )[ ]11 1 TTRR minus+= α (7)

Where R and R1 is the resistances of the wire at T and T1 respectively

and the linear temperature coefficient of the wire resistance which is

defined as below

( )( )121

12TTR

RRminus

minus=α (8)


Where R2 is the wire resistance at T2

However the relationship between the wire resistance and temperature

is usually non-linear The relationship can then be expressed as

[ ]21 bTaTRR o ++= (9)

Where Ro is the wire resistance at the reference temperature of zero oC

An RTD sensor can be integrated into a Wheatstone bridge circuit to

convert the RTD resistance change with temperature to a voltage output

The voltage output can be calibrated to give readings of temperature

a) No lead wire

1 At the calibrated temperature

Ro Rx Rx = R0

Vs Va = Ro(Ro+Ro)middotVs = Vs2

Ro Ro Vb = Ro(Ro+Ro)middotVs = Vs2

Vab = Va ndash Vb = 0


2 At some other temperature

Rx = Ro +ΔR

Va = Ro(Ro+Ro)middotVs = Vs2

Vb = Ro(Ro+Rx)middotVs = Ro(Ro + Ro + ΔR)middotVs = Ro( 2Ro + ΔR)middotVs

Vab= Vs2 - Ro( 2Ro + ΔR)middotVs = Vs[12 ndash Ro(2Ro + ΔR)]

Vab= Vs[(2Ro + ΔR-2Ro)(2(2Ro+ ΔR))]= Vs[(ΔRRo)(2(2+ ΔRRo))]

o

oSR2R1

RR4

VVabΔ+Δ

sdot=

Since ΔR ltlt Ro ΔR2Ro ltlt 1 Vab can be simplified as

o

SR

R4

VVab Δsdot= (10)

b) With lead wire

(in ldquoExperimental Methods for Engineersrdquo J Holman)


RL 1 At the calibrated temperature

Ro Rx (RL + Rx) = Ro




b) At some other temperature

(RL + Rx) = Ro +ΔRx+ ΔRL for the RTD wire and the lead wire


Vb = Ro(Ro + Ro + ΔRx+ ΔRL)middotVs

Vab= Vs2 - Ro(2 Ro + ΔRx+ ΔRL)middotVs

Vab= Vs[(2Ro + ΔRx+ ΔRL -2Ro)(2(2Ro+ ΔRx+ ΔRL))]

Vab = (Vs4) [((ΔRx+ ΔRL )Ro) (1+ ΔRx 2Ro + ΔRL 2Ro)]

Since ΔRx and ΔRL ltlt Ro ΔRx2Ro ltlt 1 and ΔRL2Ro ltlt 1 the

equation for Vab can be simplified as

( )o

LSR

RRx4

VVab Δ+Δsdot= (11)


ΔRL is not known not calibrated and uncontrollable since it is exposed

to the outside environment that is subject to sporadic changes This is a

source of error in the temperature measurement by RTD

c) Lead wire compensation



Ro Rx (RL + Rx) = Ro (RL + RD) = Ro


Ro RL Vb = Ro(Ro+Ro)middotVs = Vs2

RD Vab = Va ndash Vb = 0




(RL + RD) = Ro + ΔRL for the dummy wire


Vb = (Ro+ΔRL) [(Ro+ΔRL) + (Ro + ΔRx+ ΔRL)]middotVs

Vab= Vs2 - (Ro+ΔRL)(2 Ro + ΔRx+ 2ΔRL)middotVs

Vab= Vs[(2Ro + ΔRx+ 2ΔRL -2Ro- 2ΔRL)(2(2Ro+ ΔRx+ 2ΔRL))]

Vab = (Vs4) [ΔRx (Ro + ΔRx 2 + ΔRL)]

Vab = (Vs4) [ΔRxRo (1 + ΔRx 2Ro + ΔRLRo)]

Since ΔRx and ΔRL ltlt Ro ΔRx2Ro ltlt 1 and ΔRLRo ltlt 1 the


o

SRRx

4VVab Δ

sdot= (12)

Equation (12) has the same form as that of Equation (10) where the

uncertain ΔRL is eliminated


3 Thermistors

Thermistors are semiconductor sensors whose resistances decrease with

temperature such as FeO NiO MnO hellip The relationship of the

thermistor resistance and temperature can be written as below

⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minusβ

= oT1

T1

o eRR (13)

where Ro is the resistance of the thermistor at the ref temperature To


Generally thermistors are more sensitive than RTD They can have a

precision of plusmn 001oC as compared with plusmn 01oC for RTD In addition

the error due to lead wire is insignificant since the resistance of the

thermistor is very high compared to that of the lead wire However the

suitable temperature range for thermistors is narrower from -100 to

400oC while RTD can cover from -100 to 1000 oC

4 Thermocouple

A thermocouple consists of two dissimilar metal wires twisted together

to form a junction that can act as a temperature sensor For example

copper wire and constantan wire (55 copper plus 45 nickel)

bull Seebeck effect

The use of a thermocouple as a temperature sensor is based on the

principle of the Seebeck effect The Seebeck effect states that an emf is

generated in the thermocouple when the junction is heated or cooled

Junction copper A

Constantan B


VAB is finite when the junction is at a temperature different from that at

the terminals A and B

bull Law of intermediate metals

Junction copper A

iron

Constantan B

When another metal (iron) is added to the thermocouple circuit as shown

in the above sketch two more junctions are formed at points A and B

As long as the junctions at A and B are kept at the same temperature the

emf generated by the main junction isnrsquot altered ie the temperature

reading doesnrsquot change

bull Law of intermediate temperatures

hot junction copper cold junction

E

constantan


For the same thermocouple ie the same two dissimilar metal junctions

the emf generated at different temperatures are additive

E3 = E1 + E2

Where E1 is the emf for the cold junction at 0oC and the hot junction at

50 oC E2 is the emf for the cold junction at 50oC and hot junction at 100

oC E3 is the emf for the cold junction at 0oC and hot junction at 100 oC

Precautions for use of TC

bull Extension wire should have similar properties as those of the TC

otherwise the two junctions of the extension wires and the TC wires

should be kept at the same temperature

bull Extension wire should be thick (having a low resistance) so to

minimize the error associated with the wire heating since the wire

resistance R = ρLS where ρ is the resistivity of the wire L is the

length and S is the cross-sectional area of the wire

5 Correction for temperature measurement under high-velocity

For temperature measurements of a flow at high speed usually gas flow

the effect of the stagnation point at the tip of the temperature sensor on


the reading could be significant At the stagnation point the stagnation

temperature Ts is

cp

S gCu

TT2

2α+= infin (14)

Where Tinfin is the temperature of the bulk gas uinfin is the velocity of gas the

flow Cp is the specific heat of the gas

The temperature registered by the sensor (the measured temperature) is

usually different than the stagnation temperature due to the effect of the

shield intruding in the flow path The measured temperature is called the

recovery temperature It is strongly dependent on the configuration of

the sensor and the shield The relative difference between the measured

temperature by the sensor Tr and the stagnation temperature TS is

known as the recovery factor r

infinminusinfinminus

=TTTTr

s

r (15)

7 Temperature measurements by radiation

The temperature of an object can be determined from the total thermal

radiation emitted from the object For an ideal body the total energy

emitted is proportional to the fourth power of the temperature


However most practical objects are non-ideal bodies and a correction

must be made to obtain the true temperature of the target This

correction is done by the emissivity of the non-ideal body as below

(16) 4TE σε=

Where E is the energy emitted by the object at the temperature T (in K

or oR) ɛ is the emissivity of the object (ɛ le 10) and σ is the Stefan-

Boltzmann constant

σ = 174x10-9 BTU(hft2oR4) or 5669x10-8 W(m2K4)

When an object is an ideal body the emissivity ɛ is equal to 10 and

Equation (16) is then known as the Stefan-Boltzmann equation


71 Optical Pyrometer

The optical pyrometer is simply a photometer that matches the energy

emitted from a reference source with the incoming one from the target

source As an example in the laboratory of this course the target source

is an incandescent lamp whose tungsten filament temperature can be

varied by adjusting the current through it (by adjusting a knob on the

optical pyrometer) When the brightness of the reference filament in the

optical pyrometer matches the brightness of the target filament being

measured the image of the reference filament blends into the colour of

the background of the viewing field in the optical pyrometer At this

point the intensity of the energy emitted from the target is equal to that

emitted from the reference and hence the temperature of the target (light

bulb) can be read from the display on a digital readout

2 Total Radiation Pyrometer

The total radiation pyrometer is the most commonly used pyrometer for

continuous non-contacting measurement of temperature Radiation (both

infrared and visible) passes through a lens and is then concentrated on a

heat-sensing element which produces an electric signal This sensing

element or detector may be a thermopile a thin thermistor flake or a


resistance thermometer Thermopiles are the most commonly used

Ambient temperature compensation may be required since the body of

the pyrometer is heated by the target radiation and the thermopile is

sensitive to the body heat

The current (or voltage) produced by the sensing detector is sent to a

potentiometer or a millivoltmeter where the current can be read The

total radiation pyrometer can be used to measure low temperature

However temperatures measured are also subject to errors

bull if the source is not an ideal body Error = (1 - ɛ14)

bull if the optical system gets dirty and adsorbs too much radiation

bull If part of the radiant energy is absorbed before it reaches the

instruments such as absorption by smoke and dust Gases such as

carbon dioxide sulphur dioxide and water vapour absorb infrared

radiation

The emissivity of a material (usually written ε or e) is the relative ability

of its surface to emit energy by radiation It is the ratio of energy

radiated by a particular material to energy radiated by a black body at



the same temperature A true black body would have an ε = 1 while any

real object would have ε lt 1 Emissivity is a dimensionless quantity

In general the duller and blacker a material is the closer its emissivity is

to 1 The more reflective a material is the lower its emissivity Highly

polished silver has an emissivity of about 002[1]

bull The velocity of the moving medium being measured

bull The type of medium being measured

When the temperature sensor (or any other sensorinstrument) is exposed

to a step change in the environment of which it is sensing the

temperature the temperature readout of the sensor is lower than the

actual temperature This difference is known as the dynamic error of the

sensor as shown in Figure1 and 2 below (for first-order response

instrument)

00

200

400

600

800

1000

1200

0 02 04 06 08 1 12 14 16 18 2Time (sec)

Tem

pera

ture

(C)

T-dynamic

Step change (ideal T response)

actual T response

dynamic errror

dynamic error

Figure 1 Temperature response to a step change


0

02

04

06

08

1

12

0 02 04 06 08 1 12 14 16 18 2

Time (sec)

Nor

mal

ized

dyn

amic

err

or (X

tXo)

Dynamic error


XtXo = 0368








Xt = T2 -Td








XkdtdX

minus= (1)


tkXX

o

t minus=⎟⎟⎠

⎞⎜⎜⎝

⎛ln (2)

kt

o

t eXX minus=⎟⎟

⎠

⎞⎜⎜⎝

⎛ (3)


tc=1k then

36801 ==⎟⎟⎠

⎞⎜⎜⎝

⎛ minuseXX

o

t (4)

















FTTtdTd

=+sdotτ (5)







( ) FSWSW TTdtdT

tdTd


2 (6)







( ) ( ) ( )⎟⎟⎠

⎞⎜⎜⎝

⎛minus


sw

wwssF

ttTTTTττ

ττττ expexp100


1 Thermometer

Common liquid used




Cautions






temperature reading









( )[ ]11 1 TTRR minus+= α (7)



defined as below

( )( )121

12TTR

RRminus

minus=α (8)





[ ]21 bTaTRR o ++= (9)





a) No lead wire


Ro Rx Rx = R0






Rx = Ro +ΔR





o

oSR2R1

RR4

VVabΔ+Δ

sdot=


o

SR

R4

VVab Δsdot= (10)

b) With lead wire

















( )o

LSR

RRx4

























o

SRRx

4VVab Δ

sdot= (12)




3 Thermistors




⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minusβ

= oT1

T1

o eRR (13)









4 Thermocouple




bull Seebeck effect




Junction copper A

Constantan B





Junction copper A

iron

Constantan B








E

constantan




E3 = E1 + E2

















temperature Ts is

cp

S gCu

TT2

2α+= infin (14)











=TTTTr

s

r (15)









(16) 4TE σε=



Boltzmann constant






































radiation











0

02

04

06

08

1

12

0 02 04 06 08 1 12 14 16 18 2

Time (sec)

Nor

mal

ized

dyn

amic

err

or (X

tXo)

Dynamic error


XtXo = 0368








Xt = T2 -Td








XkdtdX

minus= (1)


tkXX

o

t minus=⎟⎟⎠

⎞⎜⎜⎝

⎛ln (2)

kt

o

t eXX minus=⎟⎟

⎠

⎞⎜⎜⎝

⎛ (3)


tc=1k then

36801 ==⎟⎟⎠

⎞⎜⎜⎝

⎛ minuseXX

o

t (4)

















FTTtdTd

=+sdotτ (5)







( ) FSWSW TTdtdT

tdTd


2 (6)







( ) ( ) ( )⎟⎟⎠

⎞⎜⎜⎝

⎛minus


sw

wwssF

ttTTTTττ

ττττ expexp100


1 Thermometer

Common liquid used




Cautions






temperature reading









( )[ ]11 1 TTRR minus+= α (7)



defined as below

( )( )121

12TTR

RRminus

minus=α (8)





[ ]21 bTaTRR o ++= (9)





a) No lead wire


Ro Rx Rx = R0






Rx = Ro +ΔR





o

oSR2R1

RR4

VVabΔ+Δ

sdot=


o

SR

R4

VVab Δsdot= (10)

b) With lead wire

















( )o

LSR

RRx4

























o

SRRx

4VVab Δ

sdot= (12)




3 Thermistors




⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minusβ

= oT1

T1

o eRR (13)









4 Thermocouple




bull Seebeck effect




Junction copper A

Constantan B





Junction copper A

iron

Constantan B








E

constantan




E3 = E1 + E2

















temperature Ts is

cp

S gCu

TT2

2α+= infin (14)











=TTTTr

s

r (15)









(16) 4TE σε=



Boltzmann constant






































radiation














XkdtdX

minus= (1)


tkXX

o

t minus=⎟⎟⎠

⎞⎜⎜⎝

⎛ln (2)

kt

o

t eXX minus=⎟⎟

⎠

⎞⎜⎜⎝

⎛ (3)


tc=1k then

36801 ==⎟⎟⎠

⎞⎜⎜⎝

⎛ minuseXX

o

t (4)

















FTTtdTd

=+sdotτ (5)







( ) FSWSW TTdtdT

tdTd


2 (6)







( ) ( ) ( )⎟⎟⎠

⎞⎜⎜⎝

⎛minus


sw

wwssF

ttTTTTττ

ττττ expexp100


1 Thermometer

Common liquid used




Cautions






temperature reading









( )[ ]11 1 TTRR minus+= α (7)



defined as below

( )( )121

12TTR

RRminus

minus=α (8)





[ ]21 bTaTRR o ++= (9)





a) No lead wire


Ro Rx Rx = R0






Rx = Ro +ΔR





o

oSR2R1

RR4

VVabΔ+Δ

sdot=


o

SR

R4

VVab Δsdot= (10)

b) With lead wire

















( )o

LSR

RRx4

























o

SRRx

4VVab Δ

sdot= (12)




3 Thermistors




⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minusβ

= oT1

T1

o eRR (13)









4 Thermocouple




bull Seebeck effect




Junction copper A

Constantan B





Junction copper A

iron

Constantan B








E

constantan




E3 = E1 + E2

















temperature Ts is

cp

S gCu

TT2

2α+= infin (14)











=TTTTr

s

r (15)









(16) 4TE σε=



Boltzmann constant






































radiation



















FTTtdTd

=+sdotτ (5)







( ) FSWSW TTdtdT

tdTd


2 (6)







( ) ( ) ( )⎟⎟⎠

⎞⎜⎜⎝

⎛minus


sw

wwssF

ttTTTTττ

ττττ expexp100


1 Thermometer

Common liquid used




Cautions






temperature reading









( )[ ]11 1 TTRR minus+= α (7)



defined as below

( )( )121

12TTR

RRminus

minus=α (8)





[ ]21 bTaTRR o ++= (9)





a) No lead wire


Ro Rx Rx = R0






Rx = Ro +ΔR





o

oSR2R1

RR4

VVabΔ+Δ

sdot=


o

SR

R4

VVab Δsdot= (10)

b) With lead wire

















( )o

LSR

RRx4

























o

SRRx

4VVab Δ

sdot= (12)




3 Thermistors




⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minusβ

= oT1

T1

o eRR (13)









4 Thermocouple




bull Seebeck effect




Junction copper A

Constantan B





Junction copper A

iron

Constantan B








E

constantan




E3 = E1 + E2

















temperature Ts is

cp

S gCu

TT2

2α+= infin (14)











=TTTTr

s

r (15)









(16) 4TE σε=



Boltzmann constant






































radiation
















( ) ( ) ( )⎟⎟⎠

⎞⎜⎜⎝

⎛minus


sw

wwssF

ttTTTTττ

ττττ expexp100


1 Thermometer

Common liquid used




Cautions






temperature reading









( )[ ]11 1 TTRR minus+= α (7)



defined as below

( )( )121

12TTR

RRminus

minus=α (8)





[ ]21 bTaTRR o ++= (9)





a) No lead wire


Ro Rx Rx = R0






Rx = Ro +ΔR





o

oSR2R1

RR4

VVabΔ+Δ

sdot=


o

SR

R4

VVab Δsdot= (10)

b) With lead wire

















( )o

LSR

RRx4

























o

SRRx

4VVab Δ

sdot= (12)




3 Thermistors




⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minusβ

= oT1

T1

o eRR (13)









4 Thermocouple




bull Seebeck effect




Junction copper A

Constantan B





Junction copper A

iron

Constantan B








E

constantan




E3 = E1 + E2

















temperature Ts is

cp

S gCu

TT2

2α+= infin (14)











=TTTTr

s

r (15)









(16) 4TE σε=



Boltzmann constant






































radiation


















( )[ ]11 1 TTRR minus+= α (7)



defined as below

( )( )121

12TTR

RRminus

minus=α (8)





[ ]21 bTaTRR o ++= (9)





a) No lead wire


Ro Rx Rx = R0






Rx = Ro +ΔR





o

oSR2R1

RR4

VVabΔ+Δ

sdot=


o

SR

R4

VVab Δsdot= (10)

b) With lead wire

















( )o

LSR

RRx4

























o

SRRx

4VVab Δ

sdot= (12)




3 Thermistors




⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minusβ

= oT1

T1

o eRR (13)









4 Thermocouple




bull Seebeck effect




Junction copper A

Constantan B





Junction copper A

iron

Constantan B








E

constantan




E3 = E1 + E2

















temperature Ts is

cp

S gCu

TT2

2α+= infin (14)











=TTTTr

s

r (15)









(16) 4TE σε=



Boltzmann constant






































radiation














[ ]21 bTaTRR o ++= (9)





a) No lead wire


Ro Rx Rx = R0






Rx = Ro +ΔR





o

oSR2R1

RR4

VVabΔ+Δ

sdot=


o

SR

R4

VVab Δsdot= (10)

b) With lead wire

















( )o

LSR

RRx4

























o

SRRx

4VVab Δ

sdot= (12)




3 Thermistors




⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minusβ

= oT1

T1

o eRR (13)









4 Thermocouple




bull Seebeck effect




Junction copper A

Constantan B





Junction copper A

iron

Constantan B








E

constantan




E3 = E1 + E2

















temperature Ts is

cp

S gCu

TT2

2α+= infin (14)











=TTTTr

s

r (15)









(16) 4TE σε=



Boltzmann constant






































radiation












Rx = Ro +ΔR





o

oSR2R1

RR4

VVabΔ+Δ

sdot=


o

SR

R4

VVab Δsdot= (10)

b) With lead wire

















( )o

LSR

RRx4

























o

SRRx

4VVab Δ

sdot= (12)




3 Thermistors




⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minusβ

= oT1

T1

o eRR (13)









4 Thermocouple




bull Seebeck effect




Junction copper A

Constantan B





Junction copper A

iron

Constantan B








E

constantan




E3 = E1 + E2

















temperature Ts is

cp

S gCu

TT2

2α+= infin (14)











=TTTTr

s

r (15)









(16) 4TE σε=



Boltzmann constant






































radiation

























( )o

LSR

RRx4

























o

SRRx

4VVab Δ

sdot= (12)




3 Thermistors




⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minusβ

= oT1

T1

o eRR (13)









4 Thermocouple




bull Seebeck effect




Junction copper A

Constantan B





Junction copper A

iron

Constantan B








E

constantan




E3 = E1 + E2

















temperature Ts is

cp

S gCu

TT2

2α+= infin (14)











=TTTTr

s

r (15)









(16) 4TE σε=



Boltzmann constant






































radiation

































o

SRRx

4VVab Δ

sdot= (12)




3 Thermistors




⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minusβ

= oT1

T1

o eRR (13)









4 Thermocouple




bull Seebeck effect




Junction copper A

Constantan B





Junction copper A

iron

Constantan B








E

constantan




E3 = E1 + E2

















temperature Ts is

cp

S gCu

TT2

2α+= infin (14)











=TTTTr

s

r (15)









(16) 4TE σε=



Boltzmann constant






































radiation






















o

SRRx

4VVab Δ

sdot= (12)




3 Thermistors




⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minusβ

= oT1

T1

o eRR (13)









4 Thermocouple




bull Seebeck effect




Junction copper A

Constantan B





Junction copper A

iron

Constantan B








E

constantan




E3 = E1 + E2

















temperature Ts is

cp

S gCu

TT2

2α+= infin (14)











=TTTTr

s

r (15)









(16) 4TE σε=



Boltzmann constant






































radiation











3 Thermistors




⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minusβ

= oT1

T1

o eRR (13)









4 Thermocouple




bull Seebeck effect




Junction copper A

Constantan B





Junction copper A

iron

Constantan B








E

constantan




E3 = E1 + E2

















temperature Ts is

cp

S gCu

TT2

2α+= infin (14)











=TTTTr

s

r (15)









(16) 4TE σε=



Boltzmann constant






































radiation

















4 Thermocouple




bull Seebeck effect




Junction copper A

Constantan B





Junction copper A

iron

Constantan B








E

constantan




E3 = E1 + E2

















temperature Ts is

cp

S gCu

TT2

2α+= infin (14)











=TTTTr

s

r (15)









(16) 4TE σε=



Boltzmann constant






































radiation














Junction copper A

iron

Constantan B








E

constantan




E3 = E1 + E2

















temperature Ts is

cp

S gCu

TT2

2α+= infin (14)











=TTTTr

s

r (15)









(16) 4TE σε=



Boltzmann constant






































radiation













E3 = E1 + E2

















temperature Ts is

cp

S gCu

TT2

2α+= infin (14)











=TTTTr

s

r (15)









(16) 4TE σε=



Boltzmann constant






































radiation












temperature Ts is

cp

S gCu

TT2

2α+= infin (14)











=TTTTr

s

r (15)









(16) 4TE σε=



Boltzmann constant






































radiation














(16) 4TE σε=



Boltzmann constant






































radiation












































radiation
























radiation

















Documents

(1.) Temperature Measurements