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Open in new tab International Journal of Materials Sciences.
ISSN 0973-4589 Volume 3, Number 3 (2008), pp. 239–250
© Research India Publications
http://www.ripublication.com/ijoms.htm
Study of the Transfer Function of a Thermocouple
A. Hasseinea, A. Azzouz, H. Binousb and M.S. Remadnac
aDepartment of industrial chemistry, University of Biskra B.P.145 Biskra, Algérie
bDepartment of Chemical Engineering, INSAT, BP 676, 1080 Tunis, Tunisia
caDepartment of Civil Engineering, University of Biskra B.P.145 Biskra, Algérie
Corresponding author: hasseine@netcourrier.fr
Abstract
In the present paper, we studied the stationary and dynamic characteristics of a
thermocouple (Cu-Constantan) [1, 2, 3] in the case of natural convection of air or
liquid nitrogen. The outcome of this work was to highlight two thermoelectric
effects: the Seebeck and Peltier effects. We find that the majority of the
theoretical characteristics of the thermocouple (distribution of the temperature,
response time and transfer function) allow a judicious choice of the thermocouple
and make possible the correction of the measured temperature signals.
Keywords: Thermocouple, Seebeck effect, Peltier effect, Transfer function
Introduction
Thermocouples are widely used sensors, in industry or laboratory, to determine and
control temperature changes in a gas or a liquid. Although other techniques, such as the
ones based on optics, were recently developed, these sensors remain frequently used
because of their ease of employment and their low cost of implementation. In 1821,
German physicist Thomas Seebeck discovered that a current will flow in the presence of
a temperature gradient, as long as the circuit were made of dissimilar metals. This effect
is known as the Seebeck Effect [4]. In 1834, French watchmaker turned physicist, Jean
Charles Peltier, noted that if an electric current passes from one substance to another,
then heat would either be released or absorbed in the junction between the metals,
depending on the direction of current flow [4]. Heat is effectively pumped through the
circuit by charge carriers, and this is known as the Peltier effect [5]. What Peltier had
actually discovered, was that the Seebeck effect was completely reversible [5]. A Peltier
device is a thermoelectric device which converts both electrical current into a temperature
gradient, and a temperature gradient into electric current.
240
A. Hasseine et al
Governing equation of the thermocouple
Figure 1 shows the geometrical model used for the thermocouple made of copper and
constantan.
Figure 1: Thermocouple configuration
The current study assumes that the junction has the same diameter as wire; i.e. the
solder of the two wires is carried out with a great care and without filler.
Energy balance:
In the present study, we neglect both the heat transfer due to conversion of kinetic energy
of the fluid into calorific energy as well as the catalytic effects. In addition, we assume
that the sensitive element of the thermoelectric probe is compared to a cylinder and the
distribution of temperature is uniform along a cross-section of the thermocouple. The
governing equations are then given by:
1) For the copper section of the thermocouple
2
π
d
4
ρ
c
pw1
=
πλ
w
1
∂
‘
T
w
1
∂
‘
t
2
d
4
‘
2
∂
T
∂
2
‘
x
−
‘
(
TI
∂
α
w
1
∂
‘
T
)
‘
∂
T
∂
x
+
2
‘4
I
π
2
d
ω
1
[
(
+
αρ
‘
T
1
ω
1
0
]
)
−
‘
T
0
+
π
dh
ωω
1
1
(
Τ−Τ
ω
1
)’
‘
f
(1)
2) For the constantan section of the thermocouple
[
(
+
αρ
T
1
ρ
c
TI
(
+
=
−
)
2
2
2
‘
‘
‘
πλ
w
2
pw2
0
∂
α
w
2
∂
‘
T
∂
T
∂
x
I
‘4
π
2
d
ω
2
d
4
2
∂
T
∂
2
‘
x
∂
‘
T
w
1
∂
t
‘
π
d
4
]
)
‘
ω
2
−
‘
T
0
+
π
dh
ωω
2
2
(
Τ−Τ
‘
f
‘
ω
2
)
(2)
For the concerned materials, however, their Seebeck coefficients are very small, and
will be viewed as zero below [6]. Their thermal conductivity are assumed to be constant
in the concerned temperature range, while the electric resistivity is described by a linear
function with respect to the temperature.
We introduce the following non-dimensional definitions:
=
t
‘
at
d
2
d
;
x
;
=Τ
ω
‘=
x
d
Τ−Τ
‘
‘
ω
0
ΔΤ
‘
;
=Ι
1
λαρ
/
)
(
0
1
2
Ι
‘
1
d
with
Study of the Transfer Function of a Thermocouple
241
Τ−Τ=ΔΤ
‘
‘
c
‘
0
‘
The temperature,
0T , is taken equal to ambient temperature and
in the junction (i.e. at x=0) taken equal to the fluid’s temperature.
Next, we replace ‘
the following constants:
wT and ‘t with their expressions in Eqs. (1) and (2) and we also define
‘
CT is the temperature
=
β
1
4
h
d
ωω
1
1
ωλ
1
−
(2
I
4
π
2)
=Κ
1
⎡
⎢
⎢
⎣
4
h
d
ωω
1
1
ωλ
1
Τ
(
‘
f
Τ−
+
)0
(
4
π
2)
2
I
α
⎤
1
⎥
ΔΤ⎥
⎦
‘
β
2
=
4
h
d
ω
2
2
ω
λ
ω
2
−
I
2
(
2
)
4
π
Κ
2
=
⎡
⎢
⎣
4
h
ω
2
λω
d
2
ω
2
Τ
(
‘
f
Τ−
‘
0
)
+
(
4
π
2
)
2
I
α
1
ΔΤ
⎤
⎥
⎦
‘
The time-dependent governing equations in non-dimensional form become:
1) For the metal m1
δ
Τ
ω
δ
t
1
=
δ
2
Τ
δ
x
2) For the metal m2
ω
2
=
Τδ
δ
t
2
Τδ
δ
x
2
1
ω
2
−
β
1
Τ
ω
1
Κ+
1
ω
2
−
ΚΤβ
+
2
ω
2
2
(3)
(4)
1
,2
β,
β K1 and K2, which appear, in the equations (3) and (4),
It should be noted that
represent the inverse of the time-constant and the sensitivity of the thermocouple,
respectively. The heat transfer coefficient, h, is given by Kramer’s correlation [1] where
only natural convection is considered.
If we take dw1 = dw2 = d, the boundary and initial conditions, associated to Eqs. (3) and
(4), are the following:
1) at
x
−=
;
T
−ω
(
1
t
),
=
T
0
l
2
d
l
d
2
l
2
d
l
2
d
at
x
=
;
T
(
ω
2
t
),
=
T
0
242
A. Hasseine et al
2) at
0=x
temperature continuity writes :
Τ
ω
1
t
),0(
Τ=
ω
2
t
),0(
3) at
0=x
both fluxes are set to be equal:
λ
1
S
1
tx
),
Τ
d
(
ω
1
dt
=
0
x
=
λ
2
S
2
tx
),
Τ
d
(
ω
2
dt
=
0
x
4) at t=0 initially the wires are at a specified temperature
Steady-state solution of the equation of the thermocouple
The solution of balance equations at steady-state, equations (3) and (4), give the
temperature profile in the wires.
a) Steady-state temperature in the copper section
Τ
ω
1
(
x
)0,
=
−
⎡
⎢
⎣
k
01
β
01
sinh(
1
2
β
02
l
d
2
−
)
k
02
β
02
α
sinh(
1
2
β
01
)
cosh(
β
2/1
01
x
)
cosh(
β
2/1
01
)
sinh(
β
2/1
02
+
α
)
cosh(
β
2/1
02
)
sinh(
β
2/1
01
l
d
2
l
d
2
⎤
⎥
⎦
2
l
d
l
d
2
l
d
2
)
⎤
⎥⎦
⎡
⎢⎣
−
⎡
⎢⎣
l
2
d
k
02
β
02
⎡
⎢⎣
l
d
2
α
⎡
⎢
⎣
k
01
β
01
cosh(
β
2/1
02
l
2
d
−
)
k
02
β
02
+
cosh(
β
2/1
01
)
sinh(
β
2/1
01
x
)
cosh(
β
2/1
01
)
sinh(
β
2/1
02
+
α
)
cosh(
β
2/1
02
)
sinh(
β
2/1
01
l
2
d
⎤
⎥
⎦
l
2
d
(5)
+
k
01
β
01
l
2
d
)
⎤
⎥⎦
b) Steady-state temperature in the constantan section
Τ
(
x
)0,
ω
2
=
cosh(
β
2/1
01
)
sinh(
β
2/1
02
x
)
+
α
sinh(
β
2/1
01
)
cosh(
β
2/1
02
x
)
cosh(
β
2/1
01
)
sinh(
β
2/1
02
+
α
)
cosh(
β
2/1
02
)
sinh(
β
2/1
01
l
2
d
l
2
d
l
2
d
l
2
d
⎤
⎥⎦
⎤
⎥⎦
l
2
d
)
−
−
k
01
β
01
sinh(
β
2/1
02
(
−
x
))
l
d
2
cosh(
β
2/1
01
)
sinh(
β
2/1
02
+
α
)
cosh(
β
2/1
01
)
cosh(
β
2/1
02
l
d
2
(6)
+
k
02
β
02
l
d
2
)
⎤
⎥⎦
l
2
d
l
2
d
l
d
2
⎡
⎢⎣
⎡
⎢⎣
With
α
=
ω
λ β
2
λ β
ω
1
0 2
0 1
1 / 2
1 / 2
Study of the Transfer Function of a Thermocouple
243
Study of the dynamic response of the thermocouple
The response time of the thermocouple is a very important characteristic. The
thermocouple is generally regarded as a sensor controlled by a first order differential
equation. The traditional process used for the determination of the time-constant of the
thermocouple is a unit step; for this type of sensor the steady-state is often reached after
approximately five time-constants.
Let us write the equations (3) and (4) when conduction along wires is neglected:
1) For the metal m1
We proceed by carrying out the following change of functions:
θ β=
2
θ β=
Tw k
1
1
1
Tw
2
−
−
k
2
2
and
1
are respectively the time-constants of the copper and
After taking the Laplace transform of each term of equations (10) and (11), one
obtains the solutions in the field time; as follows:
2T are the time distributions of temperatures in the two wires.
∂
T
∂
1
ω β
+
1
t
T
ω
1
−
k
1
=
0
2) For the metal m2
∂
T
∂
2
ω
t
+
β
2
T
ω
2
−
k
2
=
0
θ
∂
1
∂
t
θ
∂
2
∂
t
Eqs. (7) and (8) become:
τ
1
+
θ
1
=
0
τ
2
+
θ
2
=
0
τ =
1
Where
1
β
1
constantan wires.
and
τ =
2
1
β
2
( )
T t
1
= −
−
t
τ
1
e
+
k
1
β
1
k
1
β
1
T t
( )
2
= −
t
−
τ
2
e
+
k
2
β
2
k
2
β
2
1T ,
(7)
(8)
(9)
(10)
(11)
(12)
(13)