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Response of a thermocouple to transient temperature changes in a metal to which it is attached
Int. J. Mech. Sci, Vol.33, No. 7, pp. 551 561,1991 Printedin Great Britain.
0021~7403/91 $3.00+.00 © 1991PergamonPresspie
RESPONSE OF A THERMOCOUPLE TO TRANSIENT TEMPERATURE C H A N G E S I N A M E T A L T O W H I C H IT IS ATTACHED T. SAWADAand N. NISHIWAKI Tokyo University of Agriculture and Technology, Nakacho, Koganeishi, Tokyo 184, Japan (Received 16 September 1987; and in revised form 3 M a y 1990)
Abstract--Relationships between the temperature at a thermocouple junction and the electromotive force under suddenly changing temperature, such as a step function, are discussed. The accuracy or the time response of temperature measurement of quickly heated metal by the thermocouple, each wire of which is separately attached to the surface of the metal in order to improve the response of the thermocouplc, is then discussed. It is demonstrated that the accuracy of temperature measurement can be estimated by a Fourier number defined in this paper and can be improved by selecting suitable combinations of the two kinds of thermocouple wirc and their diamctcrs.
NOTATION a c d e Fo h J M Q q R t T To
Tap V v W Wa Wb X z ~w~ 6 2 p 4,
thermal diffusivity (m2/s) specific heat capacity (J/kg K) diameter of thermocouple wire (m) voltage per Kelvin caused by electromotive force (emf) for a given pair of metals Fourier number apparent coefficient of heat transfer (kJ/mZsK) junction between metal and thermocouple wire metal measured with thermocouple rate of heat generation per unit volume (kJ/m3s) heat flow (kJ/s) thermal contact resistance (m2sK/kJ) time temperature reference temperature kept at a constant, e.g. room temperature apparent temperature calculated from the voltage measured in thermocouple circuit voltage generated in thermocouple circuit voltage per Kelvin generated between metals in one of which temperature rises like a step function thermocouple wire one wire used as thermocouple the other wire used as thermocouple standard metal for determining relative electromotive force, e.g. platinum distance between the points divided as shown in Fig. 3 (pWCW2w/PMCM2M)°'s gap between metal and thermocouple thermal conductivity (kJ/mhK) density (kg/m 3) rate of apparent temperature rise measured with thermocouple for real temperature of heated metal
1. I N T R O D U C T I O N
In plastic working processes, most of the work done reappears in the form of heat, whereby the temperature of the metal is quickly raised. Especially in the hot forging process, a die is subjected to the high temperature of the material, the heat evolution due to plastic deformation, and also to cyclically forced cooling within a short period. The surface of the die has such severe variations of temperature and stress, that abrasion and fatigue failures will increase. This influences the life of the die and punch. The worked material is also subject to cooling, distorts in shape, and introduces dimensional errors, especially in the rolling process of a sheet metal. In order to prevent or control such problems, the temperatures of the die and the material must be measured. 551
552
T. SAWADAand N. NISHIWAKI
The relationships between the deformation of metal and the temperature rise or the heat evolution have been investigated theoretically [1-3] and experimentally. So far in experimental studies, electric resistance thermometers [4], the structure of which is like a resistance wire strain gauge, a thermocouple [5-7] and an infrared ray thermometer [8] have been used in order to measure the temperature of plastically deforming metal. However, the temperature rises in the general plastic working process so quickly that it cannot always be measured easily and accurately with an electric resistance thermometer. This is because there is an adhesion film between the resistance wire and the metal surface and the contact area of the thermocouple junction cannot be made to touch perfectly the measured surface. It is also difficult to measure the temperature with the infrared ray thermometer because the measured temperature is not so high and the emissivity of the measured surface changes with the plastic deformation of the metal. A thermocouple is inexpensive and easily measures the temperature at a given location of the material but there is a comparatively large heat capacity at its junction and it cannot measure the temperature quickly enough. In order to improve the measurement accuracy and the response of the thermocouple, the following method is often used. Two different kinds of wires used as a thermocouple are separately attached to different but nearby locations on the measured surface so that each of them may be electrically connected through the measured metal. In general, the temperatures at these two locations are considered to be equal, but in fact they are not because the thermal properties of the wires and the measured metal are different from each other. This study deals with the accurate measurement of the temperatures of quickly heated metal under plastic working by using a thermocouple and presents a method of improving this accuracy by analysing the junction temperature between the measured metal and the thermocouple wires by introducing the electromotive force of the metal. 2. RELATIONSHIPS BETWEEN THE TEMPERATURE AT A THERMOCOUPLE JUNCTION AND THE ELECTROMOTIVE FORCE A thermocouple is generally used in an electric circuit such as that shown in Fig. l(a). The junction temperature T on a metal,surface can be obtained from the measured voltage V by the following equation: V = el(T-
To) + e 2 ( T - To)2 + e 3 ( T - To)3 + ' ' ".
(1)
Here, To is a reference temperature kept constant and el, e2, e3 and so on are constants determined by the properties of two different thermocouple wires Wa and Wb. When the temperature difference ( T - To) is not too large, Eqn (1) is approximately represented by the following equation: V = e w a w b ( T - To).
(2)
Here, ewawb represents the voltage per Kelvin caused by the electromotive force (emf) which is determined by the combination of the thermocouple wires Wa and Wb. For example, in the case of obtaining the value ec,ct of copper-constantan thermocouple wires, the electromotive forces ec,pl and ectpl of each wire against platinum have been already
(a)
(b)
~a
M
FIO. 1. Temperature measurement of metal by thermocouple:(a) Type I; (b) Type II.
Response of a thermocouple to transient temperature changes in metal
553
measured as shown in Table 1. Then the value e~ct is obtained by considering the direction of the current as follows: ecuct = ec~Pi - ectPi = 0.76 - ( - 3.51) ---- ecupl 4- epIct ----0.76
+ 3.51 = 4.27 (mY/100 K).
(3)
The thermocouple circuit shown in Fig. l(b) is often used as a method of temperature measurement of the metal in order to improve its measurement accuracy or its response. The thermocouple wires Wa and Wb are electrically joined through the measured metal M and located near each other. The junction temperature Two is not always equal to the junction temperature Twb, especially when the temperature of the heated metal M varies. The voltage measured in this circuit is represented as follows from Eqn (2) assuming the current direction is from junction Two to junction Twb: V = ew, x( Two - To) + e u x ( Twb -- Two) + ewbx( To -- Twb).
(4)
Here, X represents the standard metal for determining the relative electromotive force. The second term in Eqn (4) is represented by considering the current direction as (5)
etax( Twb -- Twa) = exM( Tw, -- Twb).
The law of intermediate temperature gives e x M ( T w a - - T w b ) 4- e x M ( T w b - - TO) = e x M ( T W a - - To).
(6)
Equation (4) becomes, using Eqns (5) and (6), V = ewax(Twa -
To) +
exM(Twa --
To) +
e w b x ( T o - - T w b ) + e x M ( T o - - Twb).
(7)
The law of intermediate metal is represented as ewox( Twa - To) + e x u ( Twa -- To) = ewoM( Twa -- To).
(8)
Equation (7) becomes, using Eqn (8), V -- e W a M ( T W a - -
To) +
eWbM(T 0 --
Twb ) = ewouATwa
-
ewbMATwb ,
(9)
where, ATwo = Twa - To and ATwb= T w b - - T o. In practice neither the temperature rise ATwo nor ATwb but rather the apparent temperature rise ATap of the metal M can be obtained from the measured electromotive force as follows: ATap = V / e w a w b = (ewoMATwo - - e w b M A T w b ) / e W a W b ,
(10)
where ATap = T a p - T 0. The value ATap is determined by the temperature rises at two junctions and the electromotive forces of the three kinds of metals W~, Wb and M.
TABLE 1. THERMALPROPERTIESOF METALS[10--13"1
Thermocouple Chromel Stainless Nichrome Constantan Alumei Mild steel Nickel Lead Brass Aluminum Copper Silver
,t/pc
pc,t
Symbol
x 10- 6 m2/s
kj2/m%K
Emf for platinum e mV/100 K
Cm St Nic Ct Alu Fe Ni Pb Br AI Cu Ag
3.75 4.44 4.53 6.03 8.19 20.4 22.8 23.6 33.7 93.9 112.5 170.3
51.2 58.5 67.2 81.0 107 260 351 51.4 363 554 1327 1031
2.81 -2.20 - 3.51 - 1.29 1.98 - 1.48 0.44 0.60 0.42 0.76 0.74
554
T. SAWADAand N. NISHIWAKI 3. M E A S U R E M E N T
ACCURACY
OF THERMOCOUPLE
3.1. Step change in temperature When the metal M is plastically deformed, the temperature rises quickly from To to TM. In such conditions that the temperature changes like a step function, the junction temperatures Tw° and Twb in Fig. l(b) are not always equal to TM at the beginning of the temperature rise. Here, both metal block M and thermocouple wire W can be considered as elastic half-spaces only at the instant that they are brought into contact with each other at the free surfaces. That is, it is assumed that heat flows one dimensionally through the boundary. Then, the junction temperature is obtained from the temperatures in both solids [9-1. Just after the temperature of the metal M rises by A TMlike a step function, the junction temperature rises A T j as follows:
ATj = ATM/(1 +
(11)
~WM),
(12)
o~wM = (pWCW2w/pMcM2M) °'5.
Here, ATj = Tj - T o, A T M = T M - To. Also p, c and 2 represent the density, the specific heat capacity and the thermal conductivity, respectively. The junction temperatures Tw, or T,,b between the metal M and the thermocouple wires Wa or Wb are obtained by changing subscripts from W to Wa or Wb in Eqns (11) and (12), respectively. The apparent temperature Tap or its rise AT, D is obtained from Eqns (10) and (11) as follows: q5 = ATap/ATM = (VWaM -- VWbM)/ewawb ,
(13) (14)
VWaM = ewaM/(1 + C~WaM), VWbM = ewbM/(1 + ~WbM)"
Here q~is the rate of apparent temperature rise ATap with respect to the real temperature rise ATM. The value VWM represents the voltage generated at each junction which is obtained from the combination of the measured metal M and the thermocouple wire W, or Wb, that is, their thermal properties ~ and electromotive forces e as shown in Table 1. The relationships between the values of vwM and eWM for many kinds of the combination of the measured metal M and the thermocouple wire W are shown in Fig. 2. The mark and the number indicate the nature of the measured metal and the thermocouple wire, respectively. Each point in this figure represents the temperature just after the measured metal is heated
i
i
i
i
i
3
4
!
,-3 O O
1 -5
-4
-3
-2
-1
I
I
I
l
4
=;
1
2
Metal
Thermocouple Wire W
M
k o b3
/
o ~.
Fe Cu
-2
Ct Ni Alu AI
rn
Br
®
Pb
5
Cu
V
AI
6
il
Ni
7
Fe Cm
~ 3
i
I 2 3 4
i
I
I
Fie. 2. Voltage vs electromotive force between thermocouple wire and measured metal (chain line and broken one represent the cases ~b > 1 and ~b < 0, respectively).
Response of a thermocouple to transient temperature changes in metal
555
suddenly in the manner of a step-function. The line VwM = ewu represents the ideal state of temperature measurement A Tap = A TM in Eqn (13) without error, when time has sufficiently elapsed after the sudden evolution of heat in the metal without other thermal effects. Accordingly, the difference between the line and each point parallel to the vertical axis represents the value (1 - ~b) of temperature and/or voltage after heat evolution of metal. This will be discussed in detail in the next section. The value ~b in Eqn (13) just after the heat evolution of the metal is obtained from the gradient of the line connecting two points with the mark showing the same metal, where two different kinds of the wire are affixed to the measured metal surface as shown in Fig. 1(b). For example, when mild steel and chromel wires or aluminum and copper wires are affixed to the nickel plate surface, the gradient of the chain line between n. 6 and n. 7 or the broken line between n. 4 and n. 5 is found to be 1.50 or - 0.24 as shown in Fig. 2, respectively. In these cases the accuracy of the temperature measurement is not so good. In the former case the measured temperature rise is 50% larger than the actual one, and in the latter case it is 124% less and drops although the actual temperature rises. The combination of mild steelchromel affixed to copper plate represents a gradient ~b = 1.04, which indicates the least error of measured temperature for all of the combinations. The gradient ~b in the thermocouple circuit, however, approaches 1.0 soon after heat evolution as mentioned above. 3.2. Heat produced at a constant rate in the metal The error in junction temperature is now discussed for the case where heat is produced uniformly in the metal at a constant rate. The junction temperature varies, as mentioned above, so that it should be obtained by computing heat conduction near the junction. The thermocouple wire and the metal near the junction are divided into meshes axisymmetrically arranged as shown in Fig. 3(a). The finite difference equation of heat conduction at the junction is n(d/2) 2 {(aZw/2)pwCw + (AZM/2)pMcM } ( T ~ x -- TI,j) = (ql + q2 + q3)At + n(d/2)2(AzM/2)QMAt,
where
(15)
qx = 2w{(T~,j+ l - T[,j)/Azw}n(d/2) 2, q2 = 2M{(T[,j-I -- TI,j)/AzM}n(d/2) 2, q3 = AM{(TI +x.j -- T[,j)/d} 2rt(d/2)(AZM/2 ),
where t, d, q, QM, i and j, respectively, represent time, the diameter of the wire, the heat flow, the heat production rate per unit volume and the location of the temperature in the wire and metal. The heat transfer and radiation at the surfaces of the wire and the metal are neglected in order to discuss the accuracy of the measured temperature only a short time after the beginning of heat production. The temperature rise ATs between a thermocouple wire and a metal was computed for a given heat rate QM by use of the thermal properties of metals in Table 1. The temperature
(a)
(h)
(c)
ri.i-t,k FIG. 3. Mesh divided for numerical analysis of heat conduction: (a) Model I; (b) Model II; (c) Model III.
556
T. SAWADAand N. NISHIWAKI
rise ATM due to heat produced in the metal is given as
ATM = (QM/pMCM)At.
(16)
The junction temperature A T s is computed by employing the heat conduction equations. Applying Eqn (13) to the junction between a wire and a metal, the value 4) is defined as
49 = ATs/ATM.
(17)
The value (1 - 49) represents the temperature error at the junction, that is, the departure from the rise in temperature ATM as mentioned in Section 3.1. It decreases with time as shown in Fig. 4, where the mark Fe-(Ct) represents the case where the thermocouple wire, constantan Ct, is affixed to the measured metal, mild steel Fe. The error (1 - 49) in Fe-(Ct) is smaller than that of Fe-(Fe), which is caused by a small dissipation of heat flow through the constantan wire due to the smaller thermal diffusivity 2/pc of this wire than that of the Fe wire. The values (1 - 49) of the measured metal, copper Cu, are smaller than those of the metal Fe, because heat flows faster from the other place in the metal to the junction due to the large 2~pc value of copper. The value (1 - 49) at t = 0 in Fig. 4 is equal to that shown in Fig. 2 when a thermocouple wire is affixed to a metal. The value at t > 0 is discussed in the next step. Putting AzM = d and Azw/AzM = [(2w/pWCw)(pMcM/2~)] °'5 = (2W/2~)/CtWM, Eqn (15) is t t _,,jT t.+.l = T~,j{1 - 2(0tWM + 3)Fo} + 2Fo(~twMTi,i+ ~ + Ti,j_ t
where,
+ 2~+~,~) + FoQMd2/2~,
(18)
Fo = (aM/d2){At/(1 + ~wM)},
(19)
aM = 2M/PMCM.
(20)
Therefore, the junction temperature rise ATs after heat production becomes a function of the Fourier number Fo, which includes the coefficient ~ of thermal properties. Equation (18) is numerically analysed and the results are shown in Fig. 5, where temperature error (1 - 49)(1 + ct)/~ [Eqn (11)] is plotted against F o in the range of ct = 0.25-2.27. Then it was found that the following equation can be approximately given in the range of Fo less than 10: (1 - 49)(1 + OtWM)/~WM= exp(Fo/x//3)~~erfc(Fo/x/3) °'55,
(21)
49 = 1 - {~w~t/(1 + CtWM)}exp(Fo/x/3)LIerfc(Fo/x//3)°55,
(22)
or The voltage V per 100 K temperature rise at the junction between the thermocouple wire W and the metal M at time t = 0 is given by Eqn (14). The voltage at t = tl is represented as
0.5
0.4
0.3
""''--. Fe-(Fe)
I
-
0.2
~
0.!
Cu-(Cu)
~.._
"
---___
",,,..~ct ) i
o
I
I
I
2 t
ms
FIG. 4. Effectof thermal diffusivityon temperature error at a junction (Model I).
Response of a thermocoupleto transient temperaturechangesin metal
557
1.0 Metal Wire " "
0.8
~ ,
0.6
A
....
s~,o.55
"l=tr~//3/ !
F e - Fe F e - Ct
•
Fe
o ®
Cu - Cu
•
Cu
-
-
Cu
Ct
0.2
0
i
i
i
t
2
t;
6
8
Fo
tO
FIG. 5. Relationshipsbetween Fourier number and temperatureerror at a junction.
follows using Eqn (22) and applying Eqns (13) and (2) to the case of contact between a wire and a metal: Vt=,, = e w ~ A T M [ 1 - {~wM/(1 + o~wM)}exp(aMtl/v/3d2(1
+ OtwM)) 1"1
x erfc((aMtl/,v/3d2(1 + aw~))°s5l.
(23)
At time t = Vt= ~o/ATM = vt= ~ = ewM.
(24)
This shows that the voltage V at each junction at t = 0 approaches the line with a unit gradient ~ = 1 parallel to the vertical axis as time increases as shown in Fig. 2. The voltage Vgenerated in the thermocouple wires A and B through a metal M is derived from Eqns (9) and (17) as V = (ev,.,,uck.lw,, - ewbMdP~wb)ATM.
(25)
Using Eqn (22) V = A T M [eve,,Wb -- ewaM {~W°M/(1 + ~W=M)}exp(Fo/v/3) H erfc(Fo/~/3) °'s5 + eWbM{OtWbM/(1 + o ~ w b ~ ) } e x p ( F o / ~ / 3 ) H e r f c ( F o / v / 3 ) ° ' s s ] .
(26)
4. NUMERICAL RESULTS Two types of thermocouples were analysed numerically to investigate the accuracy of measured temperature. Type I in Fig. l(a) shows the usual thermocouple circuit where a junction of thermocouple wires was affixed to a metal in order to measure the surface temperature. Type II represents a thermocouple circuit to improve the accuracy or the time response of the thermocouple. The discretization of the wire and metal is shown in Fig. 3 to analyse the distribution of temperature by a finite difference method such as Eqn (15). The dimension of each wire is chosen as d = AzM = 0.1 ram. 4.1. T y p e I I thermocouple
The results for Type II are shown in Fig. 6. The mark Cu-(Fe.Ct) represents the thermocouple combined with the measured metal copper Cu and mild steel Fe and constantan Ct wires. The error (1 -- ~) in the measured temperature Cu-(Fe.Ct) decreases with time. This error represents a value between the errors for Cu-(Fe) and Cu-(Ct). When the metal is mild steel Fe, the error is larger than when it is Cu as shown in Fig. 4. The error in the Fe-(Cu.Ct) system is smaller than that of the component Fe-(Cu) and Fe-(Ct) systems. This may be understood with reference to the values obtained at time zero (which
558
T. SAWADA and N. NISHIWAKI
0.7 t\\ 0.6 t \\
\
\
0,5
N\
" "--..
0.4
Fe-(Cu)
I
0.3 h\
i\
o. 2 t L " ~ e
-(ct )
J
_
[¢u-(Fe~Ct)
~-(-d~. . . . .
,. . . . . . 1
0
l
2 t
ms
FIG. 6. Temperature error at a junction and error in an electrically measured temperature with thermocouple composed of two junctions (Model I).
also represent the step change error). The Fe-(Ct) system has a low 0( value (0.558) and a high e value (5.49) whereas the Fe-(Cu) system has a high ~ value (2.26) and low e (1.22). Hence, from Eqn (13) it can be seen that this system will have a 4) value higher than either of the two junctions (that is, a lower error). Had the combination been such that a high ~ value couple had a high e, then the error of measurement would have been greater than either of the two couples and this would be an unsatisfactory combination. The method of reducing the measured temperature error is to make the difference larger between the two products of the electromotive force and the temperature rise A Tw or the value 4b,,, at two junctions as shown in Eqns (10) or (25). These values depend on the magnitude of the heat flow, so that the dissipation of heat flow through the thermocouple wires can be controlled by varying the diameter. Figure 7 shows results, where the diameters of the wires are 0.2 and 0.05 mm in copper Cu and constantan Ct, respectively. The temperature error (1 - 4)) in Fe-(Cu.Ct) decreases in a short time. This method contributes to the decrease of the temperature error with time, that is, improving the response time of the thermocouple by the proper combination of metals. Figure 8 shows the distribution of the measured temperature error around the junction caused by the dissipation of heat through the wire. The error curve does not increase more than the broken line in the case of Cu--(Cu). It does increase in the case of Fe--(Cu), but it is less than 1% of the metal temperature rise A TMaround the range of three and four times of the diameter d of the wire apart from the center of the junction (the origin of the axes), even at about 2.5 ms after heat evolution. To improve the accuracy, a hole is bored 4d in depth into the surface of the metal at the bottom of which the tip of the wire is affixed. The other part of the wire is assumed adiabatic. The error decreases by 1.4 and 3.6% at 0.5 and 2.5 ms in the case of Fe-(Cu) as shown in Fig. 6. Finally the factors which decrease the measured temperature error in a Type II thermocouple in descending order of significance are as follows: (1) to make the thermal diffusivity of the measured metal larger; (2) to make the thermal diffusivity of the thermocouple wire smaller; (3) to make the difference between the electromotive forces of the thermocouple wires larger [in Eqn (10)]; and (4) to bore a hole into the metal to insert the thermocouple wire. 4.2. Type I thermocouple It is supposed that the temperature error in Type I depends on the thermal diffusivity of the measured metal, as in the Type II case, where the dimension of the thermocouple wire in
Response of a thermocouple to transient temperature changes in metal
~,
559
r
0.08
0.8
0.06
.0.6
-.-.‘0
Fe-(CwCt)(l,-d) IO
t
20
ms
30
FIG. 7. Improvement
of the error in an electrically measured temperature by use of the thermocouple with different diameters of wire in the case of Fe-(CuCt) (Model I).
0.06 0.05 t
i
0
d
0
Fe-(Cu)
FIG. 8. Distribution of the measured temperature error, where the origin is the center of the wire.
Type I is 0.1 mm. In addition, the temperature measured by two wires W, and W, also depends on the electromotive forces of the wires and the metal and the temperatures at each contact surface. Therefore, it is assumed that the temperature error variations may be more complicated than for Type II. That is, three kinds of metal compose three electrical circuits through the divided contact points, the average value of two of which is calculated to estimate the measured voltage in Fig. 3(b). In addition, the thermal conduction from the metal to two wires is not as simple as in Type II. In Fig. 3(b) there are contact areas l-2 between the wires W, and W,, l-3 between the wire W, and the metal M, and l-4 between the wire W, and the metal M. Therefore, the temperature along each contact surface is calculated with due consideration of the contact area. The metals in Fig. 9 are copper and mild steel and the temperature errors shown are larger than those in Type II. This is because heat dissipates through two wires in Type I, the areas at one junction of which are about two times larger than those in Type II, even if the difference between rectangular wires and rounds ones is neglected. Both figures also show that the thermal diffusivity of the measured metal mainly influences the temperature error. There is, in general, a thermal contact resistance R between the wire and the metal such as a metal sheathed ceramic insulated thermocouple as shown in Fig. 3(c). The value of this depends on the surface roughness, the contact pressure and the metal properties. Here, it is assumed that there is an air gap 6 between the two flat planes of the metal and the thermocouple. The resistance R is converted to a coefficient of apparent heat transfer h( = l/R) to analyse the temperature error, and it has a value of 1 kJ/m’sK when 6 = 25 ,um and Aair= 0.090 kJ/mhK as shown in Fig. 10. The figure shows two results where
560
T. SAWADAand N. NISHlWAKI (a)
(b) i
0.5
0.5
O.4[
0.4
i
~ ~Fe-(Fe-Ct) "~ 0.3
0.3
I
I
0.2
0.2
~'X~,,,.._Cu-(Cxn.Ct)
O.l
~C,~._~._
,Cu-(Fe.Ct)
Cu-(Cu'Ct)
~
0.1
"-=:'-'-== i
0
I
ms
t
2
I
0
J
,
1
I
2 t
ms
FIG. 9. Effectof thermal diffusivityon temperature error (Model II, AzM= 0.1 mm): (a) copper; (b) mild steel.
1.0
i
i
h : ~ kJ/m2sK
0.8
.0.
0.6 ~
0~
~
!
0./* 0.2
St- (Cm'Ct) ----- St - (Cu" Ct ) 0.1 l
I
0.2
FIG. 10. Effectof thermal diffusivityon temperatureerror in the case of heat transferbetweenwires and metal surface(Model III, AzM= 0.1 mm).
thermocouples chromel-constantan (Cm'Ct) and copper-constantan (Cu'Ct) are both affixed to stainless steel St. The temperature error in the latter is larger than that in the former. This shows that large thermal diffusivity through a wire increases the error. The errors in Type I in Figs. 9 and 10 are, however, larger than those in Type II in Fig. 6. 5. CONCLUSIONS In this paper the behavior of the measured voltage in thermocouples composed of several kind of metals has been discussed and schematically shown for the case in which heat is quickly produced in measured metal. The accuracy and/or the response of the temperature measurement at the junction between the thermocouple and the metal were estimated by considering the electromotive force and thermal diffusivity of the thermocouplc wire and metal. It was shown to be influenced by a Fourier number modified by the coefficient of the thermal properties of the wire and the metal. In addition, the accuracy in two kinds of thermocouple, the usual Type I and Type II, the wires of which are affixed separately to the metal in order to improve the measurement accuracy, has been analysed. The following cases were found to be accurate if the thermal diffusivity of the measured metal is larger and
Response of a thermocouple to transient temperature changes in metal
561
that of the thermocouple is small when the diameter of the wire is 0.1 mm. Methods to improve the accuracy and the time response of temperature measurement by assembling metals with an appropriate thermal diffusivity and an electromotive force, were proposed. Metal under plastic deformation is subjected to the constraint of the deformation at the junction by the thermocouple wire, so that the error in the temperature measurement is influenced to some degree [14]. REFERENCES 1. J. F. W. BISHOP,An approximate method for determining the temperatures reached in steady motion problems of plane plastic strain. Quart. J. Mech. Appl. Math. 9, 236 (1956). 2. N. RESELOand S. KOBAYASHI,A coupled analysis of viscoplastic deformation and heat transfer--I. Theoretical consideration. Int. J. Mech. Sci. 22, 699 (1980). 3. J. H. ARGYRISand J. ST. DOLTSINIS,On the natural formulation and analysis of large deformation coupled thermomechanical problems. Comp. Meths. Appl. Mech. Engng 25, 195 (1981). 4. M. SOGIMOTOand K. SAITO,Thermal behavior in specimen during tensile testing. Preprint of Jap. Soc. Mech. Engng 774-2, 9 (1977). (In Japanese.) 5. A. R. E. SINGERand J. W. COAKHAM,Temperature changes occurring during the extrusion of aluminium, tin and lead. J. Inst. Metals 89, 177 (1960). 6. W. S. FARRENand G. I. TAYLOR,The heat developed during plastic extension of metals. Proc. Roy. Soc. London. A107, 422 (1925). 7. R. O. WILLIAMS,A defomation calorimeter. Rev. Scientific Inst. 31, 1336 (1960). 8. Y. KASUGA,T. JIMMA and Y. YAMAGUCHI,Analysis of temperature and membrane stress on welding and cooling in pipe producing processes of 30,* stainless steel. J. Jap. Soc. Tech. Plasticity 30, 371 (1989). (In Japanese.) 9. H. S, CARSLAWand J. C. JAEGER,Conduction of heat in solids, 89. Clarendon Press, Oxford (1959). 10. Ed. Japan Soc. Mech. Eng., JSME mechanical engineer's handbook, 11-7, JSME (1974). (In Japanese.) 11. Tokyo Astronomical Observatory, Scientific chronology, Phy. 54, Maruzen Co. (1980). (In Japanese.) 12. Ed. Japan Soc. Mech, Eng., Report on heat conduction, 293, JSME (1977). (In Japanese.) 13. Ed. Committee for Industrial Engineering, Temperature, 73, Nikkan Kogyo Shinbunsha (1965). (In Japanese.) 14. T. SAWADA,N. NISHIWAKIand T. KUWABARA,Error estimates for temperature measurement of plastically deformed solid with thermocouple. J. Jap. Soc. Tech. Plasticity 26, 848 (1985). (In Japanese.)
0021~7403/91 $3.00+.00 © 1991PergamonPresspie
RESPONSE OF A THERMOCOUPLE TO TRANSIENT TEMPERATURE C H A N G E S I N A M E T A L T O W H I C H IT IS ATTACHED T. SAWADAand N. NISHIWAKI Tokyo University of Agriculture and Technology, Nakacho, Koganeishi, Tokyo 184, Japan (Received 16 September 1987; and in revised form 3 M a y 1990)
Abstract--Relationships between the temperature at a thermocouple junction and the electromotive force under suddenly changing temperature, such as a step function, are discussed. The accuracy or the time response of temperature measurement of quickly heated metal by the thermocouple, each wire of which is separately attached to the surface of the metal in order to improve the response of the thermocouplc, is then discussed. It is demonstrated that the accuracy of temperature measurement can be estimated by a Fourier number defined in this paper and can be improved by selecting suitable combinations of the two kinds of thermocouple wirc and their diamctcrs.
NOTATION a c d e Fo h J M Q q R t T To
Tap V v W Wa Wb X z ~w~ 6 2 p 4,
thermal diffusivity (m2/s) specific heat capacity (J/kg K) diameter of thermocouple wire (m) voltage per Kelvin caused by electromotive force (emf) for a given pair of metals Fourier number apparent coefficient of heat transfer (kJ/mZsK) junction between metal and thermocouple wire metal measured with thermocouple rate of heat generation per unit volume (kJ/m3s) heat flow (kJ/s) thermal contact resistance (m2sK/kJ) time temperature reference temperature kept at a constant, e.g. room temperature apparent temperature calculated from the voltage measured in thermocouple circuit voltage generated in thermocouple circuit voltage per Kelvin generated between metals in one of which temperature rises like a step function thermocouple wire one wire used as thermocouple the other wire used as thermocouple standard metal for determining relative electromotive force, e.g. platinum distance between the points divided as shown in Fig. 3 (pWCW2w/PMCM2M)°'s gap between metal and thermocouple thermal conductivity (kJ/mhK) density (kg/m 3) rate of apparent temperature rise measured with thermocouple for real temperature of heated metal
1. I N T R O D U C T I O N
In plastic working processes, most of the work done reappears in the form of heat, whereby the temperature of the metal is quickly raised. Especially in the hot forging process, a die is subjected to the high temperature of the material, the heat evolution due to plastic deformation, and also to cyclically forced cooling within a short period. The surface of the die has such severe variations of temperature and stress, that abrasion and fatigue failures will increase. This influences the life of the die and punch. The worked material is also subject to cooling, distorts in shape, and introduces dimensional errors, especially in the rolling process of a sheet metal. In order to prevent or control such problems, the temperatures of the die and the material must be measured. 551
552
T. SAWADAand N. NISHIWAKI
The relationships between the deformation of metal and the temperature rise or the heat evolution have been investigated theoretically [1-3] and experimentally. So far in experimental studies, electric resistance thermometers [4], the structure of which is like a resistance wire strain gauge, a thermocouple [5-7] and an infrared ray thermometer [8] have been used in order to measure the temperature of plastically deforming metal. However, the temperature rises in the general plastic working process so quickly that it cannot always be measured easily and accurately with an electric resistance thermometer. This is because there is an adhesion film between the resistance wire and the metal surface and the contact area of the thermocouple junction cannot be made to touch perfectly the measured surface. It is also difficult to measure the temperature with the infrared ray thermometer because the measured temperature is not so high and the emissivity of the measured surface changes with the plastic deformation of the metal. A thermocouple is inexpensive and easily measures the temperature at a given location of the material but there is a comparatively large heat capacity at its junction and it cannot measure the temperature quickly enough. In order to improve the measurement accuracy and the response of the thermocouple, the following method is often used. Two different kinds of wires used as a thermocouple are separately attached to different but nearby locations on the measured surface so that each of them may be electrically connected through the measured metal. In general, the temperatures at these two locations are considered to be equal, but in fact they are not because the thermal properties of the wires and the measured metal are different from each other. This study deals with the accurate measurement of the temperatures of quickly heated metal under plastic working by using a thermocouple and presents a method of improving this accuracy by analysing the junction temperature between the measured metal and the thermocouple wires by introducing the electromotive force of the metal. 2. RELATIONSHIPS BETWEEN THE TEMPERATURE AT A THERMOCOUPLE JUNCTION AND THE ELECTROMOTIVE FORCE A thermocouple is generally used in an electric circuit such as that shown in Fig. l(a). The junction temperature T on a metal,surface can be obtained from the measured voltage V by the following equation: V = el(T-
To) + e 2 ( T - To)2 + e 3 ( T - To)3 + ' ' ".
(1)
Here, To is a reference temperature kept constant and el, e2, e3 and so on are constants determined by the properties of two different thermocouple wires Wa and Wb. When the temperature difference ( T - To) is not too large, Eqn (1) is approximately represented by the following equation: V = e w a w b ( T - To).
(2)
Here, ewawb represents the voltage per Kelvin caused by the electromotive force (emf) which is determined by the combination of the thermocouple wires Wa and Wb. For example, in the case of obtaining the value ec,ct of copper-constantan thermocouple wires, the electromotive forces ec,pl and ectpl of each wire against platinum have been already
(a)
(b)
~a
M
FIO. 1. Temperature measurement of metal by thermocouple:(a) Type I; (b) Type II.
Response of a thermocouple to transient temperature changes in metal
553
measured as shown in Table 1. Then the value e~ct is obtained by considering the direction of the current as follows: ecuct = ec~Pi - ectPi = 0.76 - ( - 3.51) ---- ecupl 4- epIct ----0.76
+ 3.51 = 4.27 (mY/100 K).
(3)
The thermocouple circuit shown in Fig. l(b) is often used as a method of temperature measurement of the metal in order to improve its measurement accuracy or its response. The thermocouple wires Wa and Wb are electrically joined through the measured metal M and located near each other. The junction temperature Two is not always equal to the junction temperature Twb, especially when the temperature of the heated metal M varies. The voltage measured in this circuit is represented as follows from Eqn (2) assuming the current direction is from junction Two to junction Twb: V = ew, x( Two - To) + e u x ( Twb -- Two) + ewbx( To -- Twb).
(4)
Here, X represents the standard metal for determining the relative electromotive force. The second term in Eqn (4) is represented by considering the current direction as (5)
etax( Twb -- Twa) = exM( Tw, -- Twb).
The law of intermediate temperature gives e x M ( T w a - - T w b ) 4- e x M ( T w b - - TO) = e x M ( T W a - - To).
(6)
Equation (4) becomes, using Eqns (5) and (6), V = ewax(Twa -
To) +
exM(Twa --
To) +
e w b x ( T o - - T w b ) + e x M ( T o - - Twb).
(7)
The law of intermediate metal is represented as ewox( Twa - To) + e x u ( Twa -- To) = ewoM( Twa -- To).
(8)
Equation (7) becomes, using Eqn (8), V -- e W a M ( T W a - -
To) +
eWbM(T 0 --
Twb ) = ewouATwa
-
ewbMATwb ,
(9)
where, ATwo = Twa - To and ATwb= T w b - - T o. In practice neither the temperature rise ATwo nor ATwb but rather the apparent temperature rise ATap of the metal M can be obtained from the measured electromotive force as follows: ATap = V / e w a w b = (ewoMATwo - - e w b M A T w b ) / e W a W b ,
(10)
where ATap = T a p - T 0. The value ATap is determined by the temperature rises at two junctions and the electromotive forces of the three kinds of metals W~, Wb and M.
TABLE 1. THERMALPROPERTIESOF METALS[10--13"1
Thermocouple Chromel Stainless Nichrome Constantan Alumei Mild steel Nickel Lead Brass Aluminum Copper Silver
,t/pc
pc,t
Symbol
x 10- 6 m2/s
kj2/m%K
Emf for platinum e mV/100 K
Cm St Nic Ct Alu Fe Ni Pb Br AI Cu Ag
3.75 4.44 4.53 6.03 8.19 20.4 22.8 23.6 33.7 93.9 112.5 170.3
51.2 58.5 67.2 81.0 107 260 351 51.4 363 554 1327 1031
2.81 -2.20 - 3.51 - 1.29 1.98 - 1.48 0.44 0.60 0.42 0.76 0.74
554
T. SAWADAand N. NISHIWAKI 3. M E A S U R E M E N T
ACCURACY
OF THERMOCOUPLE
3.1. Step change in temperature When the metal M is plastically deformed, the temperature rises quickly from To to TM. In such conditions that the temperature changes like a step function, the junction temperatures Tw° and Twb in Fig. l(b) are not always equal to TM at the beginning of the temperature rise. Here, both metal block M and thermocouple wire W can be considered as elastic half-spaces only at the instant that they are brought into contact with each other at the free surfaces. That is, it is assumed that heat flows one dimensionally through the boundary. Then, the junction temperature is obtained from the temperatures in both solids [9-1. Just after the temperature of the metal M rises by A TMlike a step function, the junction temperature rises A T j as follows:
ATj = ATM/(1 +
(11)
~WM),
(12)
o~wM = (pWCW2w/pMcM2M) °'5.
Here, ATj = Tj - T o, A T M = T M - To. Also p, c and 2 represent the density, the specific heat capacity and the thermal conductivity, respectively. The junction temperatures Tw, or T,,b between the metal M and the thermocouple wires Wa or Wb are obtained by changing subscripts from W to Wa or Wb in Eqns (11) and (12), respectively. The apparent temperature Tap or its rise AT, D is obtained from Eqns (10) and (11) as follows: q5 = ATap/ATM = (VWaM -- VWbM)/ewawb ,
(13) (14)
VWaM = ewaM/(1 + C~WaM), VWbM = ewbM/(1 + ~WbM)"
Here q~is the rate of apparent temperature rise ATap with respect to the real temperature rise ATM. The value VWM represents the voltage generated at each junction which is obtained from the combination of the measured metal M and the thermocouple wire W, or Wb, that is, their thermal properties ~ and electromotive forces e as shown in Table 1. The relationships between the values of vwM and eWM for many kinds of the combination of the measured metal M and the thermocouple wire W are shown in Fig. 2. The mark and the number indicate the nature of the measured metal and the thermocouple wire, respectively. Each point in this figure represents the temperature just after the measured metal is heated
i
i
i
i
i
3
4
!
,-3 O O
1 -5
-4
-3
-2
-1
I
I
I
l
4
=;
1
2
Metal
Thermocouple Wire W
M
k o b3
/
o ~.
Fe Cu
-2
Ct Ni Alu AI
rn
Br
®
Pb
5
Cu
V
AI
6
il
Ni
7
Fe Cm
~ 3
i
I 2 3 4
i
I
I
Fie. 2. Voltage vs electromotive force between thermocouple wire and measured metal (chain line and broken one represent the cases ~b > 1 and ~b < 0, respectively).
Response of a thermocouple to transient temperature changes in metal
555
suddenly in the manner of a step-function. The line VwM = ewu represents the ideal state of temperature measurement A Tap = A TM in Eqn (13) without error, when time has sufficiently elapsed after the sudden evolution of heat in the metal without other thermal effects. Accordingly, the difference between the line and each point parallel to the vertical axis represents the value (1 - ~b) of temperature and/or voltage after heat evolution of metal. This will be discussed in detail in the next section. The value ~b in Eqn (13) just after the heat evolution of the metal is obtained from the gradient of the line connecting two points with the mark showing the same metal, where two different kinds of the wire are affixed to the measured metal surface as shown in Fig. 1(b). For example, when mild steel and chromel wires or aluminum and copper wires are affixed to the nickel plate surface, the gradient of the chain line between n. 6 and n. 7 or the broken line between n. 4 and n. 5 is found to be 1.50 or - 0.24 as shown in Fig. 2, respectively. In these cases the accuracy of the temperature measurement is not so good. In the former case the measured temperature rise is 50% larger than the actual one, and in the latter case it is 124% less and drops although the actual temperature rises. The combination of mild steelchromel affixed to copper plate represents a gradient ~b = 1.04, which indicates the least error of measured temperature for all of the combinations. The gradient ~b in the thermocouple circuit, however, approaches 1.0 soon after heat evolution as mentioned above. 3.2. Heat produced at a constant rate in the metal The error in junction temperature is now discussed for the case where heat is produced uniformly in the metal at a constant rate. The junction temperature varies, as mentioned above, so that it should be obtained by computing heat conduction near the junction. The thermocouple wire and the metal near the junction are divided into meshes axisymmetrically arranged as shown in Fig. 3(a). The finite difference equation of heat conduction at the junction is n(d/2) 2 {(aZw/2)pwCw + (AZM/2)pMcM } ( T ~ x -- TI,j) = (ql + q2 + q3)At + n(d/2)2(AzM/2)QMAt,
where
(15)
qx = 2w{(T~,j+ l - T[,j)/Azw}n(d/2) 2, q2 = 2M{(T[,j-I -- TI,j)/AzM}n(d/2) 2, q3 = AM{(TI +x.j -- T[,j)/d} 2rt(d/2)(AZM/2 ),
where t, d, q, QM, i and j, respectively, represent time, the diameter of the wire, the heat flow, the heat production rate per unit volume and the location of the temperature in the wire and metal. The heat transfer and radiation at the surfaces of the wire and the metal are neglected in order to discuss the accuracy of the measured temperature only a short time after the beginning of heat production. The temperature rise ATs between a thermocouple wire and a metal was computed for a given heat rate QM by use of the thermal properties of metals in Table 1. The temperature
(a)
(h)
(c)
ri.i-t,k FIG. 3. Mesh divided for numerical analysis of heat conduction: (a) Model I; (b) Model II; (c) Model III.
556
T. SAWADAand N. NISHIWAKI
rise ATM due to heat produced in the metal is given as
ATM = (QM/pMCM)At.
(16)
The junction temperature A T s is computed by employing the heat conduction equations. Applying Eqn (13) to the junction between a wire and a metal, the value 4) is defined as
49 = ATs/ATM.
(17)
The value (1 - 49) represents the temperature error at the junction, that is, the departure from the rise in temperature ATM as mentioned in Section 3.1. It decreases with time as shown in Fig. 4, where the mark Fe-(Ct) represents the case where the thermocouple wire, constantan Ct, is affixed to the measured metal, mild steel Fe. The error (1 - 49) in Fe-(Ct) is smaller than that of Fe-(Fe), which is caused by a small dissipation of heat flow through the constantan wire due to the smaller thermal diffusivity 2/pc of this wire than that of the Fe wire. The values (1 - 49) of the measured metal, copper Cu, are smaller than those of the metal Fe, because heat flows faster from the other place in the metal to the junction due to the large 2~pc value of copper. The value (1 - 49) at t = 0 in Fig. 4 is equal to that shown in Fig. 2 when a thermocouple wire is affixed to a metal. The value at t > 0 is discussed in the next step. Putting AzM = d and Azw/AzM = [(2w/pWCw)(pMcM/2~)] °'5 = (2W/2~)/CtWM, Eqn (15) is t t _,,jT t.+.l = T~,j{1 - 2(0tWM + 3)Fo} + 2Fo(~twMTi,i+ ~ + Ti,j_ t
where,
+ 2~+~,~) + FoQMd2/2~,
(18)
Fo = (aM/d2){At/(1 + ~wM)},
(19)
aM = 2M/PMCM.
(20)
Therefore, the junction temperature rise ATs after heat production becomes a function of the Fourier number Fo, which includes the coefficient ~ of thermal properties. Equation (18) is numerically analysed and the results are shown in Fig. 5, where temperature error (1 - 49)(1 + ct)/~ [Eqn (11)] is plotted against F o in the range of ct = 0.25-2.27. Then it was found that the following equation can be approximately given in the range of Fo less than 10: (1 - 49)(1 + OtWM)/~WM= exp(Fo/x//3)~~erfc(Fo/x/3) °'55,
(21)
49 = 1 - {~w~t/(1 + CtWM)}exp(Fo/x/3)LIerfc(Fo/x//3)°55,
(22)
or The voltage V per 100 K temperature rise at the junction between the thermocouple wire W and the metal M at time t = 0 is given by Eqn (14). The voltage at t = tl is represented as
0.5
0.4
0.3
""''--. Fe-(Fe)
I
-
0.2
~
0.!
Cu-(Cu)
~.._
"
---___
",,,..~ct ) i
o
I
I
I
2 t
ms
FIG. 4. Effectof thermal diffusivityon temperature error at a junction (Model I).
Response of a thermocoupleto transient temperaturechangesin metal
557
1.0 Metal Wire " "
0.8
~ ,
0.6
A
....
s~,o.55
"l=tr~//3/ !
F e - Fe F e - Ct
•
Fe
o ®
Cu - Cu
•
Cu
-
-
Cu
Ct
0.2
0
i
i
i
t
2
t;
6
8
Fo
tO
FIG. 5. Relationshipsbetween Fourier number and temperatureerror at a junction.
follows using Eqn (22) and applying Eqns (13) and (2) to the case of contact between a wire and a metal: Vt=,, = e w ~ A T M [ 1 - {~wM/(1 + o~wM)}exp(aMtl/v/3d2(1
+ OtwM)) 1"1
x erfc((aMtl/,v/3d2(1 + aw~))°s5l.
(23)
At time t = Vt= ~o/ATM = vt= ~ = ewM.
(24)
This shows that the voltage V at each junction at t = 0 approaches the line with a unit gradient ~ = 1 parallel to the vertical axis as time increases as shown in Fig. 2. The voltage Vgenerated in the thermocouple wires A and B through a metal M is derived from Eqns (9) and (17) as V = (ev,.,,uck.lw,, - ewbMdP~wb)ATM.
(25)
Using Eqn (22) V = A T M [eve,,Wb -- ewaM {~W°M/(1 + ~W=M)}exp(Fo/v/3) H erfc(Fo/~/3) °'s5 + eWbM{OtWbM/(1 + o ~ w b ~ ) } e x p ( F o / ~ / 3 ) H e r f c ( F o / v / 3 ) ° ' s s ] .
(26)
4. NUMERICAL RESULTS Two types of thermocouples were analysed numerically to investigate the accuracy of measured temperature. Type I in Fig. l(a) shows the usual thermocouple circuit where a junction of thermocouple wires was affixed to a metal in order to measure the surface temperature. Type II represents a thermocouple circuit to improve the accuracy or the time response of the thermocouple. The discretization of the wire and metal is shown in Fig. 3 to analyse the distribution of temperature by a finite difference method such as Eqn (15). The dimension of each wire is chosen as d = AzM = 0.1 ram. 4.1. T y p e I I thermocouple
The results for Type II are shown in Fig. 6. The mark Cu-(Fe.Ct) represents the thermocouple combined with the measured metal copper Cu and mild steel Fe and constantan Ct wires. The error (1 -- ~) in the measured temperature Cu-(Fe.Ct) decreases with time. This error represents a value between the errors for Cu-(Fe) and Cu-(Ct). When the metal is mild steel Fe, the error is larger than when it is Cu as shown in Fig. 4. The error in the Fe-(Cu.Ct) system is smaller than that of the component Fe-(Cu) and Fe-(Ct) systems. This may be understood with reference to the values obtained at time zero (which
558
T. SAWADA and N. NISHIWAKI
0.7 t\\ 0.6 t \\
\
\
0,5
N\
" "--..
0.4
Fe-(Cu)
I
0.3 h\
i\
o. 2 t L " ~ e
-(ct )
J
_
[¢u-(Fe~Ct)
~-(-d~. . . . .
,. . . . . . 1
0
l
2 t
ms
FIG. 6. Temperature error at a junction and error in an electrically measured temperature with thermocouple composed of two junctions (Model I).
also represent the step change error). The Fe-(Ct) system has a low 0( value (0.558) and a high e value (5.49) whereas the Fe-(Cu) system has a high ~ value (2.26) and low e (1.22). Hence, from Eqn (13) it can be seen that this system will have a 4) value higher than either of the two junctions (that is, a lower error). Had the combination been such that a high ~ value couple had a high e, then the error of measurement would have been greater than either of the two couples and this would be an unsatisfactory combination. The method of reducing the measured temperature error is to make the difference larger between the two products of the electromotive force and the temperature rise A Tw or the value 4b,,, at two junctions as shown in Eqns (10) or (25). These values depend on the magnitude of the heat flow, so that the dissipation of heat flow through the thermocouple wires can be controlled by varying the diameter. Figure 7 shows results, where the diameters of the wires are 0.2 and 0.05 mm in copper Cu and constantan Ct, respectively. The temperature error (1 - 4)) in Fe-(Cu.Ct) decreases in a short time. This method contributes to the decrease of the temperature error with time, that is, improving the response time of the thermocouple by the proper combination of metals. Figure 8 shows the distribution of the measured temperature error around the junction caused by the dissipation of heat through the wire. The error curve does not increase more than the broken line in the case of Cu--(Cu). It does increase in the case of Fe--(Cu), but it is less than 1% of the metal temperature rise A TMaround the range of three and four times of the diameter d of the wire apart from the center of the junction (the origin of the axes), even at about 2.5 ms after heat evolution. To improve the accuracy, a hole is bored 4d in depth into the surface of the metal at the bottom of which the tip of the wire is affixed. The other part of the wire is assumed adiabatic. The error decreases by 1.4 and 3.6% at 0.5 and 2.5 ms in the case of Fe-(Cu) as shown in Fig. 6. Finally the factors which decrease the measured temperature error in a Type II thermocouple in descending order of significance are as follows: (1) to make the thermal diffusivity of the measured metal larger; (2) to make the thermal diffusivity of the thermocouple wire smaller; (3) to make the difference between the electromotive forces of the thermocouple wires larger [in Eqn (10)]; and (4) to bore a hole into the metal to insert the thermocouple wire. 4.2. Type I thermocouple It is supposed that the temperature error in Type I depends on the thermal diffusivity of the measured metal, as in the Type II case, where the dimension of the thermocouple wire in
Response of a thermocouple to transient temperature changes in metal
~,
559
r
0.08
0.8
0.06
.0.6
-.-.‘0
Fe-(CwCt)(l,-d) IO
t
20
ms
30
FIG. 7. Improvement
of the error in an electrically measured temperature by use of the thermocouple with different diameters of wire in the case of Fe-(CuCt) (Model I).
0.06 0.05 t
i
0
d
0
Fe-(Cu)
FIG. 8. Distribution of the measured temperature error, where the origin is the center of the wire.
Type I is 0.1 mm. In addition, the temperature measured by two wires W, and W, also depends on the electromotive forces of the wires and the metal and the temperatures at each contact surface. Therefore, it is assumed that the temperature error variations may be more complicated than for Type II. That is, three kinds of metal compose three electrical circuits through the divided contact points, the average value of two of which is calculated to estimate the measured voltage in Fig. 3(b). In addition, the thermal conduction from the metal to two wires is not as simple as in Type II. In Fig. 3(b) there are contact areas l-2 between the wires W, and W,, l-3 between the wire W, and the metal M, and l-4 between the wire W, and the metal M. Therefore, the temperature along each contact surface is calculated with due consideration of the contact area. The metals in Fig. 9 are copper and mild steel and the temperature errors shown are larger than those in Type II. This is because heat dissipates through two wires in Type I, the areas at one junction of which are about two times larger than those in Type II, even if the difference between rectangular wires and rounds ones is neglected. Both figures also show that the thermal diffusivity of the measured metal mainly influences the temperature error. There is, in general, a thermal contact resistance R between the wire and the metal such as a metal sheathed ceramic insulated thermocouple as shown in Fig. 3(c). The value of this depends on the surface roughness, the contact pressure and the metal properties. Here, it is assumed that there is an air gap 6 between the two flat planes of the metal and the thermocouple. The resistance R is converted to a coefficient of apparent heat transfer h( = l/R) to analyse the temperature error, and it has a value of 1 kJ/m’sK when 6 = 25 ,um and Aair= 0.090 kJ/mhK as shown in Fig. 10. The figure shows two results where
560
T. SAWADAand N. NISHlWAKI (a)
(b) i
0.5
0.5
O.4[
0.4
i
~ ~Fe-(Fe-Ct) "~ 0.3
0.3
I
I
0.2
0.2
~'X~,,,.._Cu-(Cxn.Ct)
O.l
~C,~._~._
,Cu-(Fe.Ct)
Cu-(Cu'Ct)
~
0.1
"-=:'-'-== i
0
I
ms
t
2
I
0
J
,
1
I
2 t
ms
FIG. 9. Effectof thermal diffusivityon temperature error (Model II, AzM= 0.1 mm): (a) copper; (b) mild steel.
1.0
i
i
h : ~ kJ/m2sK
0.8
.0.
0.6 ~
0~
~
!
0./* 0.2
St- (Cm'Ct) ----- St - (Cu" Ct ) 0.1 l
I
0.2
FIG. 10. Effectof thermal diffusivityon temperatureerror in the case of heat transferbetweenwires and metal surface(Model III, AzM= 0.1 mm).
thermocouples chromel-constantan (Cm'Ct) and copper-constantan (Cu'Ct) are both affixed to stainless steel St. The temperature error in the latter is larger than that in the former. This shows that large thermal diffusivity through a wire increases the error. The errors in Type I in Figs. 9 and 10 are, however, larger than those in Type II in Fig. 6. 5. CONCLUSIONS In this paper the behavior of the measured voltage in thermocouples composed of several kind of metals has been discussed and schematically shown for the case in which heat is quickly produced in measured metal. The accuracy and/or the response of the temperature measurement at the junction between the thermocouple and the metal were estimated by considering the electromotive force and thermal diffusivity of the thermocouplc wire and metal. It was shown to be influenced by a Fourier number modified by the coefficient of the thermal properties of the wire and the metal. In addition, the accuracy in two kinds of thermocouple, the usual Type I and Type II, the wires of which are affixed separately to the metal in order to improve the measurement accuracy, has been analysed. The following cases were found to be accurate if the thermal diffusivity of the measured metal is larger and
Response of a thermocouple to transient temperature changes in metal
561
that of the thermocouple is small when the diameter of the wire is 0.1 mm. Methods to improve the accuracy and the time response of temperature measurement by assembling metals with an appropriate thermal diffusivity and an electromotive force, were proposed. Metal under plastic deformation is subjected to the constraint of the deformation at the junction by the thermocouple wire, so that the error in the temperature measurement is influenced to some degree [14]. REFERENCES 1. J. F. W. BISHOP,An approximate method for determining the temperatures reached in steady motion problems of plane plastic strain. Quart. J. Mech. Appl. Math. 9, 236 (1956). 2. N. RESELOand S. KOBAYASHI,A coupled analysis of viscoplastic deformation and heat transfer--I. Theoretical consideration. Int. J. Mech. Sci. 22, 699 (1980). 3. J. H. ARGYRISand J. ST. DOLTSINIS,On the natural formulation and analysis of large deformation coupled thermomechanical problems. Comp. Meths. Appl. Mech. Engng 25, 195 (1981). 4. M. SOGIMOTOand K. SAITO,Thermal behavior in specimen during tensile testing. Preprint of Jap. Soc. Mech. Engng 774-2, 9 (1977). (In Japanese.) 5. A. R. E. SINGERand J. W. COAKHAM,Temperature changes occurring during the extrusion of aluminium, tin and lead. J. Inst. Metals 89, 177 (1960). 6. W. S. FARRENand G. I. TAYLOR,The heat developed during plastic extension of metals. Proc. Roy. Soc. London. A107, 422 (1925). 7. R. O. WILLIAMS,A defomation calorimeter. Rev. Scientific Inst. 31, 1336 (1960). 8. Y. KASUGA,T. JIMMA and Y. YAMAGUCHI,Analysis of temperature and membrane stress on welding and cooling in pipe producing processes of 30,* stainless steel. J. Jap. Soc. Tech. Plasticity 30, 371 (1989). (In Japanese.) 9. H. S, CARSLAWand J. C. JAEGER,Conduction of heat in solids, 89. Clarendon Press, Oxford (1959). 10. Ed. Japan Soc. Mech. Eng., JSME mechanical engineer's handbook, 11-7, JSME (1974). (In Japanese.) 11. Tokyo Astronomical Observatory, Scientific chronology, Phy. 54, Maruzen Co. (1980). (In Japanese.) 12. Ed. Japan Soc. Mech, Eng., Report on heat conduction, 293, JSME (1977). (In Japanese.) 13. Ed. Committee for Industrial Engineering, Temperature, 73, Nikkan Kogyo Shinbunsha (1965). (In Japanese.) 14. T. SAWADA,N. NISHIWAKIand T. KUWABARA,Error estimates for temperature measurement of plastically deformed solid with thermocouple. J. Jap. Soc. Tech. Plasticity 26, 848 (1985). (In Japanese.)