Temperature - I
- Temperature Scales
- Step Response of first order system
- RTD
Temperature Scales
• Statistical mechanics relates temperature to the mean kinetic energies
of molecules, which depends on mass, length, and time. However, since
these energies can not be measured directly (or at least not easily) at
present, independent temperature standards are required.
• Several temperature scales have been introduced in the past, based on
some easily reproducible states, such as the freezing and boiling points
of water (these points are not used anymore).
• In early 1700’s, Fahrenheit produced an accurate and repeatable
mercury thermometer. Using salt and water, he managed to reproduce
what is now known as 0F and chose this as the low end of his
temperature scale. He chose the blood temperature of a healthy human
(96F) as the high end.
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Temperature Scales

Around 1742, Celsius proposed that the melting point of ice and the
boiling point of water be used for the two benchmarks. Originally, he
chose the boiling point as 0C and the melting point as 100C, at 1 atm.
Later the end points were reversed.

In 1854, Lord Kelvin developed a universal thermodynamic scale based
on the coefficient of expansion of an ideal gas. He established the
concept of absolute zero (0K) and his scale remains the standard for
modern thermometry.

The Rankine scale is simply the Fahrenheit equivalent of the Kelvin
scale (R).
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Temperature Scales
Comparison of the four temperature scales (approximate values)
Fahrenheit F
Boiling point
of water (1atm)
Celsius C
100
212
180
Rankine R
Kelvin K
373
672
100
100
180
32
0
492
273
-460
-273
0
0
Freezing/melting
point of water
(1atm)
Absolute zero
K   C  27315
. ,

R   F  459.67,

F
9 
 C  32.0
5
• We must rely on temperatures established by physical phenomena
which can easily be observed and consistent in nature. The International
Practical Temperature Scale (IPTS) is based on such phenomena.
Revised in 1990, it established seventeen reference temperatures,
which covers the temperature range from -270.15C to 1084.62C. Note
that at present there is no established reference temperatures outside
this range.
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Temperature Scales
Fixed Point
No.
1
Material
State
Temperature (c)
He
Vapor
-270.15 to -268.15
6
O2
Triple Point
-218.7916
8
Hg
Triple Point
-38.8344
9
H 2O
Triple Point
0.01
10
Ga
Melting
27.7646
11
In
Freezing
156.5985
14
Al
Freezing
660.323
15
Ag
Freezing
961.78
17
Cu
Freezing
1084.62
• Since we have only these fixed temperatures to use as references, we
need to use instruments to interpolate between them. We shall discuss
Resistance Temperature detectors (RTD), thermistors, IC sensors,
thermocouples, and pyrometers. Before we go into these topics, we
discuss briefly the speed of response of temperature sensors.
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Step Response of a First Order System
• When a thermometer is placed in a hot liquid, the thermometer
undergoes a step change but the reading does not reach the steadystate value immediately. This is due to the heat-transfer which takes
place between the liquid and thermometer (a dynamic characteristic). All
other sensors we have seen in this course so far are assumed to have a
very fast response; their output responds almost immediately when an
input is applied or changed, so that the measurement reading can be
taken right away.
Temperature
Original temp.
Liquid temp.
Thermometer
Thermometer
reading
Time
Hot liquid
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Step Response of a First Order System
• A convenient and widely used index for showing how fast a sensor can
reach its steady-state is the time constant . This is the time it takes for
the response to reach about 63% of the total response, when the sensor
undergoes a step change in its input. In practice, if the thermometer
reaches 99% of the liquid temperature, it can be considered to have
reached the steady-state value. This occurs in about t=5.
• [The actual step response of a thermometer may not show such a
smooth curve, as you will find out in the lab. The step response of an
ideal first-order system is known to be given by
 t

TTh  TL 1  e  


Temperature
1
•
63.2%
TL
95%
98%
99%
Final value
TTh
• which is shown in the figure]

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2
3
4
Time
5
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6
Step Response of a First Order System
Ex. A thermometer with the time constant of 40 seconds is used to measure
your fever. For 1% accuracy, how long do you have to wait?
• Since =40s, for 1% accuracy, you have to wait for about 5=200s; i.e.,
3min. 20s. If a digital thermometer with =10s is used, you need to wait
for only 50s.
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RTD
• The fact that resistivity of metal depends on temperature is the basis of
Resistance Temperature Detector (RTD). Platinum, nickel, and copper
are the commonly used material for this sensor. Some are like
resistance-type strain gages, but more sensitive to temperature than to
strain. The resistance R at temperature T can be approximated by
R  R0 1  aT  T0 
• where R0 is the nominal resistance at temperature To and a is the
temperature coefficient. The temperature coefficient of platinum wire is
a=0.00385/C. For a 100 platinum RTD, this corresponds to the
sensitivity of 0.385/C at 0C. By measuring the resistance of a RTD,
the temperature can be inferred. To measure the resistance, the bridge
circuit with one variable resistor and one RTD can be used. The
resistance of the variable resistor is adjusted such that the bridge is
balanced.
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RTD
Ex. Consider the bridge circuit shown below, where is the RTD with the
nominal resistance of 25 at 0C and is used to measure the constant
temperature T. The temperature coefficient of the RTD is 0.003925/C. If
the output voltage EAC is 0 when R2 is 37.36, find the temperature T in
C.
R1 
+
R2
25
E ex
A
-
R4 
25
C
R3 (RTD)
The bridge balance equation is R1R3=R2R4, which leads to
R3 
R2 R4
 37.36
R1
(If R1=R4, then R3=R2 for balance.) Solving R  R0 1  aT  T0  for T yields
T

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R  R0 1  aT0 
R0 a
37.36  25(1  0.003925  0)
 126.0 C
25  0.003925
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RTD
• Three-Wire-Bridge
The resistance of most common RTD is around 100 at 0C (10 for
platinum), while the measurement wires leading to the RTD sensor may
be tens of ohms. The following three-wire-bridge is better than the usual
two-wire bridge to solve this problem. (This bridge is useful also for
strain gages.)
[Not discussed this year.]
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10
Temperature - II
- Thermistors
- Thermocouples
Thermistors
• Similarly to RTDs, the thermistor is also a temperature sensitive resistor.
Typical sensitivity characteristics of RTDs, thermistors, and
thermocouples are compared below:
R  (RTD & Thermistor)
E Volts (Thermocouple)
Thermistor
Most sensitive (no bridge circuit required).
Nonlinear (linearization circuit desired).
Negative slope (self-destructive).
Fragile. Small temperature range. R  R0 e
1 1

 T T0
 




RTD
Most linear, bridge circuit required.
R  R0 1  1 (T  T0 )
Thermocouple
Widest temperature range. Rugged.
Nonlinear.
Temperature T
E   1 (T  T0 )   2 (T  T0 )2   3 (T  T0 )3 
where  1   2 and  1   3 .
Thermocouple tables are used.
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Thermistors
•
Typical thermistors have the resistance values as follows:
Type
Bead
coated)
Disk
(glass-
R at 0C
R at 25C
8.8 k
3.1 k
1.3 k
300 C
283
100
40.7
127 C
• When current flows through a
thermistor, heat is generated,
reducing its resistance. This will
increase current and, thus, heat.
This goes on until the thermistor
self-destruct.
Therefore,
an
additional resistor is always placed
in series with the thermistor. [A
thermistor is usually used in the
voltage divider, which solves the
self-destruction problem.]
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R at 50C
Maximum
RT
I
+
E ex
  as RT  0,
RT
i.e., as T increases.
E ex
I
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Thermocouples(TC)
• Seebeck Effect
• If two different metals (electrical conductors) come in contact with each
other at one end, a voltage (in mV) is developed at the other open end.
This is the so called Seebeck effect. A junction may be created by
welding, soldering, or twisting the wires around one another, so that a
good electrical contact is made. The amplitude of this voltage depends
on
- the metals involved, and
- the absolute temperature (K) at the junction.
Metal A
Junction
+
Seebeck Voltage
Metal B
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Thermocouples(TC)
• With no loading effect (i.e., open circuit or a meter with an infinitely high
resistance is connected, so that there is no current flow), the voltage
developed by a pair of different metals is due only to the Seebeck effect.
The thermocouple (TC) is a utilization of this thermoelectric
phenomenon.
• [When there is current flow, the Peltier and Thomson Effects are also
involved. In this course, it is assumed that current flow is small and
these two effects are negligible.]
• Any pair of metals produces the Seebeck voltage, but certain alloys and
combinations have been adopted as standard, such as those shown in
the next table.
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Thermocouples(TC)
Type
+ Lead
- Lead
Temp. range
(C)
0 to 750
Bias error
Comments
J
Iron
(Fe)
Constantan
(Cu-Ni)
K
Chromel
(Ni-Cr)
Copper
(Cu)
Chromel
Alumel
(Ni-Al)
Constantan
-200 to 1250
2.2C or 0.75% High temperature
-200 to 350
Constantan
-200 to 900
1.0C or 0.75% Moist environment
Low temperature
1.7C or 0.5% Highest sensitivity
T
E
2.2C or 0.75% Non-oxidizing
environment
Measuring Thermocouple Voltage (Ice bath Compensation)
We can not measure the Seeback voltage directly, because we must first
connect a voltmeter to the thermocouple. Even if the meter does not
cause the loading effect, there is one issue which we must to consider;
the voltmeter leads themselves create a new thermocouple. In the
special case where one of the TC wire is the same as the meter wire
(Cu), only one of the new junctions is a thermocouple junction.
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Thermocouples (TC)
• The other is a Cu-Cu junction and creates no voltage. Thus, the
voltmeter reading E will be related to the temperature difference
between Junctions 1 and 2.
Type-T Thermocouple
Cu
Cu
E
Cu
T1
J1
-E
C
C
Cu
J2
hot
-
E  E1  E2
+
Eref
Cu
T2
Circuit in Cu
+
E2
+E1
Meter with high
resistance (open
circuit)
This thermocouple circuit can be redrawn as
Metal A
Hot Junction
+
-
Metal A
+
+
-
-
Reference Junction
Metal B
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Thermocouples (TC)
where Metal A is copper and Metal B is constantan. This figure shows
more clearly that a basic thermocouple circuit has two junctions. One of
them is called the reference junction and placed at a known, constant
reference temperature. The other is called the hot junction and becomes
a measuring junction. Since thermocouples are temperature difference
measuring device, the reference junction temperature must be known.
One way to solve this problem is to place the reference junction
physically into an ice bath, forcing Tref=0.
• Thermocouple tables which relate the hot junction temperature Thot to the
measured voltage E when the reference junction temperature is 0C, are
available for the standard thermocouples
Cu
Thot
-
Ehot
C
Ice Bath
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+
E2
+ E1
-
E  E1  E2
E+ref
Cu
Tref  0 C
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