An Immediate Formula for the Radius of Curvature of A Bimetallic Strip G. D. Angel School of Engineering and Technology, University of Hertfordshire G. Haritos School of Engineering and Technology, University of Hertfordshire Abstract An alternative formula has been derived to enable a close prediction of the radius of curvature of a thin bimetallic strip that at initial ambient temperature, is both flat and straight, but at above ambient temperature, forms into an arc of a circle. The formula enables the evaluation of the radius of curvature of the strip as a function of heating or cooling. A formula for calculating the radius of curvature of a bimetallic strip already exists, and was produced by Timoshenko in his paper on Bimetal Thermostats. The formula by Timoshenko has been vigorously proven, tried and tested and accepted in countless papers and journals since its original publication. The formula introduced by this work, very closely approximates to the Timoshenko formula for equal thicknesses of the two mating metals within the bimetallic. The drawbacks of the Timoshenko formula are that it is both complex and unwieldy to use, and requires some form of electronic spreadsheet to enable its evaluation. The formula put forward here is both simple and quick to use, making it more immediate. For the correlation of the new formula, Timoshenko’s formula is used as a datum, or benchmark. From the simulation a good overall correlation was shown to exist between the Timoshenko generated values and the values generated by the new formula put forward here. Key words: Timoshenko, bimetallic strip, radius of curvature, formula, thin. 1. Introduction The original Timoshenko [1] bimetallic bending formula was published in 1925 and since then it has been applied in by multitudes of engineers and scientist and referred to in many papers such as by Krulevitch [2], Prasad [3] and books Kanthal [4]. Whilst it has been proven and accepted to be the formula to evaluate the hot radius of curvature of an initially flat bimetallic strip, it is an unwieldy and a complex formula to evaluate. This work introduces a new simpler, quicker method of evaluating the radius of curvature of a bimetallic strip from an initially flat ambient condition that has been uniformly heated. When a bimetallic strip is uniformly heated along its entire length, it will bend or deform into an arc of a circle with a radius of curvature, the value of which, is dependent on the geometry and metal components making up the strip. As will be seen later on, the nature of the bend as a function of temperature change from ambient is characteristically asymptotic. The new formula introduced here, closely approximates to the Timoshenko formula with the exception of accommodating the change in the thicknesses of the two mating metals making up the bimetallic strip. It is important to note that in the majority of applications of bimetallic strip, the ratio of the thickness of the two constitute metals is normally one to one, i.e. of equal thickness. This comes about due to the way that the bimetallic strip is manufactured. The dominant method in the mass production of commercially available bimetallic strip involves either hot or cold rolling the two separate metals under intense pressures to produce interstitial bonding of the atoms at the bi- material interface Uhlig [5]. Under such conditions, it is expensive, because of setup costs, to make special separate metal thicknesses unless specifically required. Moreover, there is no data to support that different thicknesses of the bimetals in the strip would have any performance benefit over equal thickness bimetal strip. Cladding of metals Haga[6] is used to provide a product with a less expensive base metal that benefit from a thinner skin for decorative and or surface protection purposes, although clad metals are technically bimetallic metals, clad bimetallic strip is not being used for its functional bending qualities. Therefore the need to cater for separate material thicknesses is not required for most applications where the bimetallic bending qualities of a bimetallic strip are being exploited. 1312 International Journal of Engineering Research & Technology (IJERT) Vol. 2 Issue 12, December - 2013 ISSN: 2278-0181 www.ijert.org IJERTV2IS120106
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An Immediate Formula for the Radius of Curvature of A Bimetallic Strip
G. D. Angel
School of Engineering and Technology,
University of Hertfordshire
G. Haritos
School of Engineering and Technology,
University of Hertfordshire
Abstract An alternative formula has been derived to enable a
close prediction of the radius of curvature of a thin
bimetallic strip that at initial ambient temperature, is
both flat and straight, but at above ambient
temperature, forms into an arc of a circle. The formula
enables the evaluation of the radius of curvature of the
strip as a function of heating or cooling. A formula for
calculating the radius of curvature of a bimetallic strip
already exists, and was produced by Timoshenko in his
paper on Bimetal Thermostats. The formula by
Timoshenko has been vigorously proven, tried and
tested and accepted in countless papers and journals
since its original publication. The formula introduced
by this work, very closely approximates to the
Timoshenko formula for equal thicknesses of the two
mating metals within the bimetallic. The drawbacks of
the Timoshenko formula are that it is both complex and
unwieldy to use, and requires some form of electronic
spreadsheet to enable its evaluation. The formula put
forward here is both simple and quick to use, making it
more immediate. For the correlation of the new
formula, Timoshenko’s formula is used as a datum, or
benchmark. From the simulation a good overall
correlation was shown to exist between the Timoshenko
generated values and the values generated by the new
formula put forward here. Key words: Timoshenko, bimetallic strip, radius of
curvature, formula, thin.
1. Introduction The original Timoshenko [1] bimetallic bending
formula was published in 1925 and since then it has
been applied in by multitudes of engineers and scientist
and referred to in many papers such as by Krulevitch
[2], Prasad [3] and books Kanthal [4]. Whilst it has
been proven and accepted to be the formula to evaluate
the hot radius of curvature of an initially flat bimetallic
strip, it is an unwieldy and a complex formula to
evaluate. This work introduces a new simpler, quicker
method of evaluating the radius of curvature of a
bimetallic strip from an initially flat ambient condition
that has been uniformly heated. When a bimetallic strip
is uniformly heated along its entire length, it will bend
or deform into an arc of a circle with a radius of
curvature, the value of which, is dependent on the
geometry and metal components making up the strip.
As will be seen later on, the nature of the bend as a
function of temperature change from ambient is
characteristically asymptotic. The new formula
introduced here, closely approximates to the
Timoshenko formula with the exception of
accommodating the change in the thicknesses of the
two mating metals making up the bimetallic strip. It is
important to note that in the majority of applications of
bimetallic strip, the ratio of the thickness of the two
constitute metals is normally one to one, i.e. of equal
thickness. This comes about due to the way that the
bimetallic strip is manufactured. The dominant method
in the mass production of commercially available
bimetallic strip involves either hot or cold rolling the
two separate metals under intense pressures to produce
interstitial bonding of the atoms at the bi- material
interface Uhlig [5]. Under such conditions, it is
expensive, because of setup costs, to make special
separate metal thicknesses unless specifically required.
Moreover, there is no data to support that different
thicknesses of the bimetals in the strip would have any
performance benefit over equal thickness bimetal strip.
Cladding of metals Haga[6] is used to provide a
product with a less expensive base metal that benefit
from a thinner skin for decorative and or surface
protection purposes, although clad metals are
technically bimetallic metals, clad bimetallic strip is not
being used for its functional bending qualities.
Therefore the need to cater for separate material
thicknesses is not required for most applications where
the bimetallic bending qualities of a bimetallic strip are
being exploited.
1312
International Journal of Engineering Research & Technology (IJERT)
Vol. 2 Issue 12, December - 2013
IJERT
IJERT
ISSN: 2278-0181
www.ijert.orgIJERTV2IS120106
2. Timoshenko Formula From the Timoshenko [1], the radius of curvature
of a bimetallic strip is given by:
eqn.(1)
Where ρ is the radius of curvature,
total thickness of the strip.
are the individual material thicknesses.
is the ratio of thicknesses.
is the ratio of the Young’s Moduli.
are the hot and cold temperatures states.
, is the linear Modulus of the two materials.
& are the coefficients of linear thermal
expansion for the two metals.
is assumed to be numerically larger than
Fig.1 shows a bimetallic strip in two states of
heating, at state 1, at ambient temperature, the strip
will be flat with no discernible radius of curvature
R. At state 2, uniformly distributed heating will
cause the strip to form into a radius of curvature.
Note that has a numerically higher coefficient
of linear thermal expansion and thus naturally
wants to extend further than the side with The
differences, leads to internal stresses, forces and
moments at the material interface, resulting in the
bending as shown at state 2.
Fig.1 Bimetallic strip in two states of
heating
3. Derivation of the Approximation
Formula The derivation is based upon the amalgamation of
two well established formulae, with the addition of
new correction relationships that are a combination
of the ratios, sums, and quotients of the coefficient
of linear expansion of the metals.
Where possible, the nomenclature employed in the
Timoshenko formula will be used in the new
formula.
It is commonly known that the internal force
developed within a metal bar by heating or cooling,
can be written as follows:
eqn.(2)
Where F is the force (N).
α is the coefficient of linear expansion of
the metal ( .
ΔT is the temperature change of the metal
from ambient (K).
A is the cross-sectional area of the bar(
).
E is the Youngs modulus of the material of
the bar ( ).
Eqn.(2) can be re-written in terms of the stress ,
since
, thus the internal stress due to heating:
eqn.(3)
By substitution of where y is assumed to be
the distance from bi- metal interface to the outer
edge, this is also equal to half the total thickness of
the bimetallic strip.
From the well-known simple bending equation
substituting and re-arranging using the
first two terms of the simple bending equation, thus
;
eqn.(4)
Where:
R is the radius of curvature of the bimetallic strip
to the bimetallic joint center line (mm).
t is the total thickness of the bimetallic
strip(mm).
with the average coefficient of
linear expansion of both metals ( ).
Thus eqn.(5)
This is a first order estimate of the radius of
curvature of the strip as a function of change in
temperature that approximates to the Timoshenko
formula, see Fig.2.
The simple derivation resulting in eqn.5 and shown
in Fig.2, provides a rough or first order estimate of
the radius of curvature as a function of temperature
change in the strip. It should be noted that although
eqn.5 is an approximation to the Timoshenko
formula, the accuracy of eqn.5 tends to improve as
the temperature increases, or as the expression
becomes more asymptotic. Furthermore, the nature
the first order derivation, closes resembles the
Timoshenko formula and follows a similar
trajectory, that of an asymptotic curve.
1313
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Vol. 2 Issue 12, December - 2013
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ISSN: 2278-0181
www.ijert.orgIJERTV2IS120106
Fig. 2 Comparison of Timoshenko ρ vs. R
Using a similar approach to Timoshenko where the
ratios of the thicknesses of the metals and the sum
of the thicknesses in the metals, play an influential
part of the derivation. In the derivation put forward
here, the ratios of the coefficients of linear
expansions, sum and differences are used as a
correction factor with similar effect that ultimately
modifies the first order expression eqn.5, to a close
approximation of the Timoshenko formula. The
rationale for using the coefficients of linear
expansions to correct the first order curve, is based
upon the fact that in this derivation, the thicknesses
of the two constitute metals are assumed equal for
reasons explained earlier.
From Fig.2, it can be seen that the radius of
curvature of the approximate curve R, is slightly
larger numerically, than the Timoshenko line. Thus
a possible correction factor needs to multiply R by
a number slightly less than unity. Introducing a
proportional correction factor reduces the value of
R to a very close approximation of Timoshenko ρ.
Thus letting and this provides
an initial lowering of R.
and also letting and
be the proportional difference in the coefficients of
linear thermal expansions, then the proportional
correction influence on R is:
=
eqn.(6)
Eqn.6 is a very effective modifier when multiplied
by the eqn.5 see Fig. 3 for the very close
approximation of .
At the same time, the correction must take into
account the rapid change of curvature in the lower
temperature range of 20°C to 80°C. A further
correction factor is achieved if the ratios of the
sums and differences are also considered, thus
letting:
and and combining the ratios.
Thus that expands to
Combining and re-arranging and reducing, thus:
eqn.(7)
Fig.3 showing effects of the correction coefficients
Therefore adding derived components, to
eqn.5, enhances the accuracy of eqn.5 to that of
Timoshenko. Thus the radius of curvature R, as a
function of temperature is given by:
eqn.(8)
From eqn.8 it can be seen that in this derivation,
there is no requirement to know E, the Young’s
modulus of the two separate metals of the
bimetallic strip.
Also eqn.8 can be also be expressed directly as:
eqn.(9)
Where t is total thickness of the bimetallic strip
(m).
is the change in temperature of the strip from
ambient (°C ).
R is the radius of curvature of the strip (m).
is the correction factors multiplied by
(°C ).
Eqn.10 was used to generate data in the simulation.
comparison to eqn.1 by Timoshenko.
4. Simulation Data For the generation of comparison data, parameters
& were varied in the simulation; = 0.4, 0.8,
1.2, 1.6, 2, 4, 6, 8 and 10 mm, being the total
thickness of strip. It should be noted that in most
applications of bimetallic strip, the total thickness
is usually quite thin, up to 1mm thick for switching
applications [4].
1314
International Journal of Engineering Research & Technology (IJERT)
Vol. 2 Issue 12, December - 2013
IJERT
IJERT
ISSN: 2278-0181
www.ijert.orgIJERTV2IS120106
5. Simulation Data For the proof of the correlation between the new
formula proposed in this paper, and Timoshenko’s
original formula, two separate simulations were
performed.
The first simulation was based around a bespoke
Bimetallic strip SBC206-1 from Shivalik [7] ;
100mm long x 5mm wide x 0.4mm thick , these
were the starting values of the first simulation set.
For simulation set 1, the following data was
assumed;
The strip thickness t, was varied from 0.4mm thick
to 10mm thick.
= 213 ( ) ; Young’s modulus of Steel side
of the bimetallic strip.
= 145 ( ) ; Young’s modulus of Invar 36
side of the bimetallic strip.
= ( ) coefficient of linear thermal
expansion for Steel side of the strip.
= ( ) coefficient of linear thermal
expansion for Invar 36 side of the strip.
both equal, total thickness of
the strip.
= (20°C) assumed ambient temperature constant
throughout the simulation.
Input variable temperature (°C).
change in temperature , (°C).
= radius of curvature evaluated by eqn.(1)
Timoshenko formula (m).
R = radius of curvature evaluated by eqn.(10)
proposed new formula (m).
For the second simulation, a combination of
different materials within the bimetallic strip and
also a variety of thicknesses of strip were used. It
should be noted that the material combinations put
forward in the second simulation may not be
practical for the manufacture of bimetallic strip by
modern mass production methods of cold pressure
rolling, but they can be produced by other, older
fabrication methods such as by riveting the two
metals together.
The simulation data in set 2 have been included in a
random fashion to test the robustness of the new
formula.
The second simulation data set is as shown in Table
1. Note that Invar 36 is used as the common mating
material since it possesses a very low coefficient of
linear expansion as compared with all other
engineering materials.
As per simulation 1 data set, the ambient
temperature is assumed to be = 20 °C.
Ma
teri
al M
ixS
peci
fica
ton
Thic
kness
es
Co
eff
icie
nt
of
Lin
ea
r E
xpa
nsi
on
Yo
ung
s M
odulu
s R
efe
rence
Inva
r 36
B
388
-06
20
060
.254
mm
1.4
5 x
10
-6m
/m
K1
37 -
145
Gp
a [8
]
Alu
min
ium
EN
AW
10
50A
H1
40
.254
mm
23.
5 x
10
-6m
/m
K6
9 G
Pa
[9]
Inva
r 36
B
388
-06
20
060
.3m
m
1.4
5 x
10
-6m
/m
K1
37 -
145
Gp
a [8
]
Nic
kel
Silv
erB
S2
870
NS
103
0.3
mm
1
3 x
10
-6m
/m
K1
18 G
Pa
[10]
Inva
r 36
B
388
-06
20
060
.65m
m
1.4
5 x
10
-6m
/m
K1
37 -
145
Gp
a [8
]
Bra
ssB
S 2
870
CZ
108
0.6
5mm
1
8.7 x
10
-6m
/m
K1
11G
Pa
[11]
Inva
r 36
B
388
-06
20
060
.8m
m
1.4
5 x
10
-6m
/m
K1
37 -
145
Gp
a [8
]
Mild
Ste
elB
S E
N1
A0
.8m
m
11
x 10
-6m
/m
K1
95 G
Pa
[12]
Inva
r 36
B
388
-06
20
061
.3m
m
1.4
5 x
10
-6m
/m
K1
37 -
145
Gp
a [8
]
Co
pper
B
S 2
879
C1
061
.3m
m
16.
6 x
10
-6m
/m
K1
17 G
Pa
[13]
Inva
r 36
B
388
-06
20
060
.4m
m
1.4
5 x
10
-6m
/m
K1
37 -
145
Gp
a [8
]
Sta
inle
ss S
teel
AIS
I B
S 3
04
0.4
mm
1
7.3 x
10
-6m
/m
K2
13G
pa
[14]
Table 1 Simulation set 2
6. Simulation Results
The two formulae of eqn.1 and eqn.10 were
used to generate data values of and R
respectively for both simulation sets. The
radii of curvature for simulation set 1 were
plotted against the change of temperature for
each thickness of bimetallic strip, see Fig.4
1315
International Journal of Engineering Research & Technology (IJERT)
Vol. 2 Issue 12, December - 2013
IJERT
IJERT
ISSN: 2278-0181
www.ijert.orgIJERTV2IS120106
Fig.4 Timoshenko Comparison range 0.4mm to
10mm
For simulation set 2, see Fig.’s 5,6,7,8,9,10.
Fig.5 Timoshenko Comparison Invar vs.
Aluminium 0.508mm thick
Fig.6 Timoshenko Comparison Invar vs. Nickel
silver 0.6mm thick
Fig.7 Timoshenko Comparison Invar vs. Brass
1.3mm thick
Fig.8 Timoshenko Comparison Invar vs. Mild
Steel 1.6mm thick
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ISSN: 2278-0181
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Fig.9 Timoshenko Comparison Invar vs. Copper
2.6 mm thick
Fig.10 Timoshenko Comparison Invar vs. St.
Steel 0.8 mm thick
From Fig. 4 it is evident that for the 153 overall
data points from simulation set 1, yielding nine
different thicknesses of the strip, an excellent
correlation between the two formulae has resulted.
From simulation set 2, comparing the six different
materials types and thickness combinations
yielding 66 data points, a very good overall
correlation between the new formula and
Timoshenko was shown to exist.
7. Discussion of results From the 9 data tables generated from simulation 1,
the overall average error was 0.64%.
The break down in average error for each thickness
was as follows:
0.4mm thick = 0.9927 average 0.73%.
0.8mm thick = 0.9938 average 0.62%.
1.2mm thick = 0.9949 average 0.51%.
1.6mm thick = 0.9936 average 0.64%.
2.0 mm thick = 0.9912 average 0.88%.
4.0mm thick = 0.9938 average 0.62%.
6.0mm thick = 0.9960 average 0.40%
8.0mm thick = 0.9940 average 0.60%
10mm thick = 0.9940 average 0.60%
From the breakdown of average error it is evident
that the error fluctuates slightly as a function of the
thickness of the strip. The maximum fluctuation of
error lies between the 2mm thick and 6mm thick
test strips, was only 0.48%.
From simulation set 2, the error breakdown was as
follows:
0.508mm thick = 0.899 average 1.01% Invar vs.
Aluminium
0.6mm thick = 1.054 average 0.54% Invar vs.
Nickel Silver
1.3mm thick = 0.925 average 0.75% Invar vs. Brass
1.6mm thick = 1.158 average1.58% Invar vs. Mild
Steel
2.6mm thick = 0.956 average 0.4% Invar vs.
Copper
0.8mm thick = 0.948 average 0.52% Invar vs. St.
Steel
The maximum fluctuation of error of the function
was 1.18% which occurred between the 1.6mm and
2.6mm simulation data. It should be noted that the
second test was simultaneously subjecting the
formula to all changes of the data, i.e. different
thicknesses, different Young’s modulus, and
different coefficients of linear expansions.
The average error in simulation set 2 was 0.8% and
the maximum fluctuation error was 1.18%.
From simulation set 1 the average error was
0.622% and the maximum fluctuation error was
0.48%.
The derivation has shown, and the correlation of
the new formula to the Timoshenko formula has
proven, that the values of Young’s Modulus for
each metal within the bimetallic strip are not
required in the evaluation of the new formula. This
is very useful since it is not always quick and easy
to find the Young’s modulus of the metals, and the
value as used in the Timoshenko formula, takes the
average of both Young’s moduli which can only be
an approximation at best. It should also be noted
that this work assumes that Young’s modulus for a
bimetallic strip is the average of the two constitute
metals making up the strip, as per the original
Timoshenko formula.
8. Conclusions The results prove an acceptable overall maximum
error of 1.18%, and an overall average error of
0.64%. Thus it has been demonstrated that the
formula put forward here can be a useful, quick,
easier alternative to Timoshenko’s radius of
curvature formula for close estimates of the radius
of curvature as a function of temperature change.
Furthermore, it has been shown that the new
formula works without the requirement of first
knowing the Young’s moduli of the two metals
within the bimetallic strip. Most usefully, the
formula presented in this work can be evaluated
without the need of an electronic spread sheet or
program as is required with the more complex
Timoshenko’s formula, but can be easily used on a
hand held calculator at a fraction of the time.
1317
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Vol. 2 Issue 12, December - 2013
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ISSN: 2278-0181
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9. References [1] Timoshenko, S., Analysis of Bi-metal
Thermostats. JOSA, 1925. 11(3): p. 233-255.
[2] Krulevitch P, J.G.C., Curvature of a Cantilever
Beam Subjected to an Equi-Biaxial Bending
Moment, in Materials Research Society
Conference. 1998.
[3] Prasad, K., Principle and Properties of
Thermoststat. Journal of Materials, 1993.
[4] Kanthal, Kanthal Thermostatic Bimetal
Handbook. 2008, Box 502, SE-734 27
Hallstahammar, Sweden: Kanthal. 130.
[5] Uhlig, W., et al, Thermostatic Metal,
Manufacture and Application. 2nd, revised ed.
2007, Hammerplatz 1, D-08280,Aue/Sachsen:
Auerhammer Metallwerk GMBH. 198.
[6] Haga, T., Clad strip casting by a twin roll
caster. World Academy of Materials and
Manufacturing Engineering, 2009. 37(2): p.
117-124.
[7] Shivalik,S.Bimetallic
strip supplier. 2013; Available
from:
http://www.shivalikbimetals.com.
Accessed online Sept.2103.
[8 http://www.nickel- alloys.net
/invar_nickel_iron_alloy.html
Physical_properties,
WWW accessed Nov 2013.
[9]http://www.smithmetal.com
/downloads/1050A.pdf,
WWW accessed Nov 2013.
[10]
http://www.cuivre.org/contenu/docs/doc/pdf/CuNi
Zn/CuNi10Zn27.pdf
WWW accessed Nov 2013.
[11]http://www.copperinfo.co.uk/alloys/brass,
WWW accessed Nov 2013.
[12]http://www.matweb.com/ mild steel,
WWW accessed Nov 2013.
[13]
http://www.aalco.co.uk/datasheets/Copper~Brass~
Bronze_CW024A-C106_122.ashx
WWW accessed Nov 2013.
[14]Munday. A,J, and Farrar .R.A An Engineering
Data book , MacMillan, London,
ISBN 0 333 25829 0
11. Appendix Data tables Simulation set 1
m m
Tc(°C) Th(°C) ΔT Rt Rg Rt/Rg
20 30 10 1.525 1.535 0.993
20 40 20 0.763 0.767 0.995
20 50 30 0.508 0.512 0.992
20 60 40 0.381 0.383 0.995
20 90 70 0.218 0.219 0.995
20 120 100 0.152 0.153 0.993
20 150 130 0.117 0.118 0.992
20 180 160 0.095 0.096 0.993
20 210 190 0.080 0.081 0.994
20 240 220 0.069 0.070 0.986
20 270 250 0.061 0.061 0.993
20 300 280 0.054 0.055 0.982
20 330 310 0.049 0.050 0.994
20 370 350 0.044 0.044 0.995
20 400 380 0.040 0.040 0.993
20 430 410 0.037 0.037 0.995
20 470 450 0.034 0.034 0.997
0.993
t2+t1=0.4mm
m m
Tc(°C) Th(°C) ΔT Rt Rg Rt/Rg
20 30 10 3.049 3.069 0.993
20 40 20 1.525 1.534 0.994
20 50 30 1.016 1.023 0.993
20 60 40 0.762 0.767 0.993
20 90 70 0.436 0.438 0.995
20 120 100 0.305 0.306 0.997
20 150 130 0.234 0.236 0.992
20 180 160 0.191 0.192 0.995
20 210 190 0.160 0.161 0.994
20 240 220 0.139 0.139 1.000
20 270 250 0.122 0.123 0.992
20 300 280 0.109 0.109 1.000
20 330 310 0.099 0.099 1.000
20 370 350 0.087 0.088 0.989
20 400 380 0.080 0.081 0.988
20 430 410 0.074 0.075 0.987
20 470 450 0.068 0.068 0.994
0.994
t2+t1=0.8mm
m m
Tc(°C) Th(°C) ΔT Rt Rg Rt/Rg
20 30 10 4.575 4.604 0.994
20 40 20 2.287 2.302 0.993
20 50 30 1.525 1.535 0.993
20 60 40 1.144 1.151 0.994
20 90 70 0.653 0.657 0.994
20 120 100 0.457 0.460 0.993
20 150 130 0.352 0.354 0.994
20 180 160 0.286 0.288 0.993
20 210 190 0.241 0.242 0.996
20 240 220 0.208 0.209 0.995
20 270 250 0.183 0.184 0.995
20 300 280 0.163 0.164 0.994
20 330 310 0.147 0.148 0.993
20 370 350 0.131 0.131 1
20 400 380 0.120 0.121 0.992
20 430 410 0.112 0.112 1
20 470 450 0.102 0.102 1
0.995
t2+t1=1.2mm
m m
Tc(°C) Th(°C) ΔT Rt Rg Rt/Rg
20 30 10 6.100 6.140 0.993
20 40 20 3.050 3.070 0.993
20 50 30 2.033 2.050 0.992
20 60 40 1.525 1.534 0.994
20 90 70 0.871 0.877 0.993
20 120 100 0.610 0.614 0.993
20 150 130 0.469 0.472 0.994
20 180 160 0.381 0.384 0.992
20 210 190 0.321 0.323 0.994
20 240 220 0.277 0.279 0.993
20 270 250 0.244 0.245 0.996
20 300 280 0.218 0.219 0.995
20 330 310 0.197 0.198 0.995
20 370 350 0.174 0.175 0.994
20 400 380 0.160 0.161 0.994
20 430 410 0.149 0.150 0.993
20 470 450 0.135 0.136 0.993
0.994
t2+t1=1.6mm
m m
Tc(°C) Th(°C) ΔT Rt Rg Rt/Rg
20 30 10 7.625 7.674 0.994
20 40 20 3.812 3.837 0.993
20 50 30 2.542 2.558 0.994
20 60 40 1.906 1.918 0.994
20 90 70 1.089 1.096 0.994
20 120 100 0.762 0.767 0.993
20 150 130 0.586 0.590 0.993
20 180 160 0.476 0.479 0.994
20 210 190 0.401 0.404 0.993
20 240 220 0.346 0.348 0.994
20 270 250 0.305 0.307 0.993
20 300 280 0.272 0.274 0.993
20 330 310 0.246 0.247 0.996
20 370 350 0.218 0.219 0.995
20 400 380 0.201 0.212 0.948
20 430 410 0.186 0.187 0.995
20 470 450 0.169 0.170 0.994
0.991
t2+t1=2.0mm
m m
Tc(°C) Th(°C) ΔT Rt Rg Rt/Rg
20 30 10 15.250 15.350 0.993
20 40 20 7.625 7.674 0.994
20 50 30 5.083 5.116 0.994
20 60 40 3.812 3.840 0.993
20 90 70 2.178 2.192 0.994
20 120 100 1.525 1.539 0.991
20 150 130 1.173 1.181 0.993
20 180 160 0.953 0.959 0.994
20 210 190 0.803 0.807 0.995
20 240 220 0.693 0.697 0.994
20 270 250 0.610 0.614 0.993
20 300 280 0.546 0.548 0.996
20 330 310 0.492 0.495 0.994
20 370 350 0.436 0.438 0.995
20 400 380 0.401 0.404 0.993
20 430 410 0.372 0.374 0.995
20 470 450 0.339 0.341 0.994
0.994
t2+t1=4.0mm
1318
International Journal of Engineering Research & Technology (IJERT)
Vol. 2 Issue 12, December - 2013
IJERT
IJERT
ISSN: 2278-0181
www.ijert.orgIJERTV2IS120106
m m
Tc(°C) Th(°C) ΔT ρ R ρ/R
20 30 10 22.875 23.024 0.994
20 40 20 11.437 11.512 0.993
20 50 30 7.625 7.674 0.994
20 60 40 5.718 5.755 0.994
20 90 70 3.268 3.289 0.994
20 120 100 2.287 2.302 0.993
20 150 130 1.759 1.771 0.993
20 180 160 1.429 1.438 0.994
20 210 190 1.204 1.211 0.994
20 240 220 1.039 1.046 0.993
20 270 250 0.915 0.920 0.995
20 300 280 0.817 0.822 0.994
20 330 310 0.738 0.742 0.995
20 370 350 0.653 0.657 0.994
20 400 380 0.602 0.606 0.993
20 430 410 0.577 0.562 1.027
20 470 450 0.508 0.511 0.994
0.996
t2+t1=6.0mm
m m
Tc(°C) Th(°C) ΔT ρ R ρ/R
20 30 10 30.500 30.698 0.994
20 40 20 15.250 15.349 0.994
20 50 30 10.166 10.232 0.994
20 60 40 7.624 7.674 0.993
20 90 70 4.357 4.385 0.994
20 120 100 3.049 3.069 0.993
20 150 130 2.346 2.361 0.994
20 180 160 1.910 1.918 0.996
20 210 190 1.605 1.615 0.994
20 240 220 1.386 1.395 0.994
20 270 250 1.220 1.227 0.994
20 300 280 1.089 1.096 0.994
20 330 310 0.984 0.990 0.994
20 370 350 0.871 0.877 0.993
20 400 380 0.803 0.807 0.995
20 430 410 0.744 0.748 0.995
20 470 450 0.678 0.682 0.994
0.994
t2+t1= 8.0mm
m m
Tc(°C) Th(°C) ΔT ρ R ρ/R
20 30 10 38.125 38.372 0.994
20 40 20 19.062 19.186 0.994
20 50 30 12.708 12.790 0.994
20 60 40 9.531 9.593 0.994
20 90 70 5.446 5.481 0.994
20 120 100 3.812 3.837 0.993
20 150 130 2.932 2.951 0.994
20 180 160 2.383 2.398 0.994
20 210 190 2.006 2.019 0.994
20 240 220 1.733 1.744 0.994
20 270 250 1.525 1.534 0.994
20 300 280 1.362 1.370 0.994
20 330 310 1.229 1.237 0.994
20 370 350 1.089 1.096 0.994
20 400 380 1.003 1.009 0.994
20 430 410 0.929 0.935 0.994
20 470 450 0.847 0.852 0.994
0.994
t2+t1= 10.0mm
Simulation Set 2; 0.5mm Simulation Set 2; 0.6mm
Invar 36 vs. Aluminium Invar 36 vs. Nickel silver
T ρ R ρ/R30 1.594 1.78 0.896
50 0.532 0.593 0.897
75 0.29 0.323 0.898
100 0.199 0.222 0.896
150 0.123 0.136 0.904
200 0.088 0.0989 0.89
250 0.07 0.077 0.909
300 0.057 0.063 0.905
350 0.048 0.0539 0.891
400 0.042 0.047 0.894
450 0.0371 0.041 0.905
0.899
T ρ R ρ/R30 3.472 3.3 1.052
50 1.157 1.1 1.052
75 0.631 0.6 1.052
100 0.434 0.41 1.059
150 0.267 0.252 1.06
200 0.193 0.182 1.06
250 0.151 0.143 1.056
300 0.124 0.12 1.033
350 0.105 0.1 1.05
400 0.091 0.086 1.058
450 0.081 0.076 1.066
1.054
Simulation Set 2; 1.3mm Simulation Set 2; 1.6mm
Invar 36 vs. Brass Invar 36 vs. Mild Steel
T ρ R ρ/R30 5.046 5.489 0.919
50 1.68 1.829 0.919
75 0.917 0.998 0.919
100 0.631 0.686 0.92
150 0.4 0.422 0.948
200 0.28 0.304 0.921
250 0.219 0.238 0.92
300 0.18 0.196 0.918
350 0.153 0.166 0.922
400 0.133 0.144 0.924
450 0.12 0.127 0.945
0.925
T ρ R ρ/R30 11.23 9.7 1.158
50 3.743 3.23 1.159
75 2.042 1.76 1.16
100 1.4 1.21 1.157
150 0.864 0.745 1.16
200 0.623 0.538 1.158
250 0.488 0.421 1.159
300 0.401 0.346 1.159
350 0.34 0.294 1.156
400 0.295 0.255 1.157
450 0.261 0.225 1.16
1.158
Simulation Set 2; 2.6mm Simulation Set 2; 0.8mm
Invar 36 vs. Copper Invar 36 vs. St. Steel
T ρ R ρ/R30 11.474 11.997 0.956
50 3.825 3.999 0.956
75 2.086 2.181 0.956
100 1.434 1.499 0.957
150 0.883 0.923 0.957
200 0.637 0.666 0.956
250 0.498 0.522 0.954
300 0.409 0.428 0.956
350 0.347 0.363 0.956
400 0.302 0.315 0.959
450 0.267 0.279 0.957
0.956
T ρ R ρ/R30 3.396 3.581 0.948
50 1.132 1.194 0.948
75 0.617 0.651 0.948
100 0.424 0.447 0.949
150 0.261 0.275 0.949
200 0.189 0.199 0.95
250 0.147 0.155 0.948
300 0.121 0.127 0.953
350 0.103 0.108 0.954
400 0.089 0.094 0.947
450 0.078 0.083 0.94
0.948
1319
International Journal of Engineering Research & Technology (IJERT)