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16.2 Theory of the Contact Method It is assumed that a solid body in contact with a surrounding gaseous medium, as shown in Figure 16.1, remains in a thermal steady-state.. While making

The non-contact, or pyrometric, methods are described in detail in Chapters 8 to 11 For

a rough estimation of surface temperatures, temperature indicators, described in Section 2.5 are also used The problem of measuring the internal temperatures of solid bodies, considered in Section 16.6, is also similar to surface temperature measurement.

16.2 Theory of the Contact Method

It is assumed that a solid body in contact with a surrounding gaseous medium, as shown in Figure 16.1, remains in a thermal steady-state A surfacial heat source is placed inside the solid body, whose true surface temperature, Ot, is higher than the ambient gas temperature,

Oa The surface temperature, Ot , is to be measured by a contact sensor, which is a bare thermocouple having a flat-cut measuring junction, in contact with the surface Figure 16.1(a) illustrates the original isotherms of the undisturbed thermal field When the contact sensor is introduced, the thermal field is deformed, as shown in Figure 16.1(b) Assume that the heat transfer between the investigated surface and the surrounding gaseous medium takes place through convection and conduction and that the isotherms in the gas are deformed in the vicinity of the sensor The corresponding temperature distribution in the direction normal to the surface is shown in Figures 16.1(c) and (d).

While making contact with the investigated surface, the sensor causes a more intense heat flow from the surface, resulting in a drop in the surface temperature from its original value, Ot, to a new temperature, 0', as shown in Figure 16.1(d) The temperature difference, Ad, = 0' - Ot , which is called the first partial error of the measurement, is caused by the deformation of the original temperature field Between the flat cut measuring

ISBNs: 0-471-86779-9 (Hardback); 0-470-84613-5 (Electronic)

ORIGINAL STATE STATE AFTER APPLYING CONTACT SENSOR

SOLID BODY q GAS SOLID BODY q GAS

A.

SENSOR

(c)~ (d) AA4 '% \ S-SENSOR SENSITIVE POINT('8T)

Wc-THERMAL CONTACT RESISTANCE S, > S 4r

junction of the thermocouple and the investigated surface there is always a thermal contact resistance, Wc , caused by a non-ideal contact The temperature drop across this contact resistance, A62 = 6" - 0' , is called the secondpartial error.

It is further assumed, as in all sensors, that there is also a sensitive point in a contact sensor determining the thermometer readings, OT From Figure 16.1(b) this point, S, in a contact thermocouple is placed at the distance, l' , from the investigated surface The temperature, OT,at the point S differs from 0" by a value A03 = OT - 0",called the third partial error, which depends on the sensor design All of the differences A?),, A02 and A63 are systematic errors of the contact method of surface temperature measurement

of a solid body in the thermal steady-state To determine their values the temperature field

of a solid body in contact with a sensor and the heat flux entering the sensor will be analysed In Sections 16.2.1 and 16.2.2, the temperature excess, O, over ambient will be used.

16.2.1 Disturbing temperature field

The investigated temperature field of a solid body in contact with a sensor, according to Kulakov and Makarov (1969), can be regarded as a superposition of two fields Firstly, there is the original temperature field in the body without the sensor, described by

Ob= f(x, r) and secondly, the disturbing temperature field Od = f(x, r) The disturbingtemperature field is caused by the disturbing heat flux density, resulting from the differencebetween the density, qT, of the heat flux conducted along the sensor and the density, qb, ofthe heat flux transferred from the body to its ambient surrounding This density, qd, ofdisturbing heat flux is given by

qd = qT - qb =f(x,r) (16.1)Using the semi-infinite body in Figure 16.2 as an example, gives an explanation of themanner of determining the disturbing flux and also shows the surfacial temperaturedistribution The medium value of disturbing temperature Od,m , shown in Figure 16 2(c), atthe contact surface between sensor and body permits determination of the first partial error,

061

The differential equation of heat conduction, describing the disturbing temperature field

in a semi-infinite cylindrical body, is

2 -y-~

d + a

2

zd + r

(a) BODY AND SENSOR (b) EOUIVALENT MODEL

II I

TEMPERATURE I l FIELD ON SURFACE I Bd.m

OF SOLID BODY

Ba(x=O)

Figure 16.2 Disturbing temperature field on the surface of a semi-infinite body, resulting from theapplication ofa contact temperature sensor

where A'b is the thermal conductivity of the body, and abis the heat transfer coefficient at the surface of the body.

Solving equation (16.2) with the boundary conditions of equation (16.3) gives:

To help with the practical use of equation(16.5) Kulakov and Makarov (1969) graphically display the values of the function F versus kXin Figure 16.3 and versus krin Figure 16 4 as well as the mean value Fm of the function F, at the contact surface between the sensor and the body, versus parameter B as in Figure 16.5.

F(r=O, B=0)

0,6 0,4 kx=RT Q2- II

F m (x=0 ; 0 < r -<RT)1,0 B= a'RT

0.6 0 0,2-I

0 0.5 1,0 1,5 2,0 B

Figure 16.5 Mean value, Fm,from equation (16.5) versus parameter, B, for x = 0, 0<r<_ _ RT

Figure 16.6 presents a case in which the disturbing heat flux density,qd,diffuses into an infinitely large plate of limited thickness, through a surface limited by a circle of radius RT

The differential equation of heat conduction, characterising the disturbing temperature field,

is the same as that given in equation (16.2) for the semi-infinite body with the boundary conditions :

Od = - gdRT °" (v + Bl )e[v(k, -k.)]+ (v - B2 )el-v(k, -k.)] jl (v)Io (kr dV

where k,,= x , kr = r , kl = lb , B1 = atRT andB2= azRT

As the functionHdepends upon several parameters it is difficult to display it graphically

in an universal way Figure 16 7 presents values of H, forkr = 0 andk, = 0, as a function ofrelative plate thickness klfor some chosen values ofB1 = Bz = B.This corresponds to a casewhen the heat transfer coefficients on both sides of the plate are the same (a1 = a2 = a).

Figure 16.7 permits the maximum value of disturbing temperature in the centre of thecontact surface to be found From the curves of H =f(kv, it follows that the disturbingtemperature values decrease with increasing plate thickness If the plate is sufficiently thick

it can be regarded as a semi-infinite body and equation (16.5) can be applied Kulakov andMakarov (1969) discuss some bodies with finite dimensions and other shapes

16.2.2 Heat flux entering the sensor

As follows from equation (16.1), determination of the density, qd, of the disturbing heatflux, requires knowledge of the density,qT, of the heat flux entering the sensor Thefour simple sensor models, given in Figure 16.8, will be considered under some simplifyingassumptions It will be assumed that the heat transfer coefficients have constant values andthat the temperature field in the cross-section of the rod or plate is uniform, with a uniformdensity of heat flux all over the front surface of the sensor as well In addition it is assumedthat the rods shown in Figure 16.8(b), Figure 16.8(c) and Figure 16.8(d) are infinitely long

B

7 B=10 k= Rb6

'B=10, BI =82 =B 5

4 B=103

(Cci=oc2=a)

0 3 B=10 B= a Rr ,.

2

~n 1

0 1 2 3 4 5 6 7 8 9 10 11

k i

-Figure 16.7 Function, H, from equation (16.7) versus relative plate thickness, kj, for different Bvalues

Also to be determined is the thermal resistance, WT, of the sensor, defined by Mackiewicz (1976a) in terms of, OT , the temperature of the surface of the sensor in contact with the investigated body as,

surface (1p << 2Rp ), and that Op I = Opt = 6T This corresponds to a thin disk, with a large value of thermal conductivity Xp The heat flux (DT is:

q'T =IrR, 2acAcRcOT (16.17) The heat flux density,qT,is then

IrRcand finally:

RC Following the definition of equation(16.9), the thermal resistance, WT, of the rod is given by:

Rcl= Rc2 = Rc > ac, = ace = ac ; OTi = OT2 =OT

and then the total heat flux, (DT,entering the sensor, is

and the thermal resistance of the sensor, following equation (16.9) will be:

Rcir 2ac Rc( jci + jc2 ) The heat flux density of each conductor is given by equation (16.18).

Disk thermocouple sensor: The simplifications for Figure 16.8(d), are the same as for the models of Figure 16.8(a) and 16 8(b) but with the assumption that Rc <<RP The total heat flux, (D,,,, entering the sensor then equals the sum of the heat fluxes of both conductors and of the disk (plate) :

(PT = [irR2a, +xR,P 2a~R, + Jz)JOT (16.22)The appropriate heat flux density in this case is:

gT =7rRz = aP+ Ri 2a~R, ( ~~, + ~~z) OT (16.23)

and the thermal resistance, WT, conforming with equation (16.9) is :

it[Reap +Rc 2a~R, (X,, + A2

16.2.3 Method errors and their reduction

The first partial error, A61 , agreeing with the definition from Section 16.2 is:

This error equals the medium value of the disturbing temperature Od,m, at the contactsurface between the sensor and the body For a semi-infinite body, from equation (16.5), theerror, A61 , is described by

0_<r_RT /~b x=0

0<r<RTThe density, qd, of the disturbing heat flux in equation (16.26), is calculated fromequation (16.1) as:

Determination of, qT , for the four sensor models is described in Section 16.2.2 Tosimplify the problem it is assumed that 6T = t9' (Wc = 0) The value of qb is calculatedfrom qb = abOT , while the value ofFm in equation (16.26) is found from Figure 16.5.Calculation of the error, A61 , is accomplished in a step by step iterative way orgraphically This will be explained in a numerical example To calculate these A61 errors,

a ready formula can also be used This is derived for a semi-infinite body by Mackiewicz(1976a), taking into consideration the thermal contact resistance, Wc, in the manner:

In a similar way, based on the theory of disturbing heat flux for an infinitely large plate,the error, A61 , can be found with the calculations based on equation (16.7) The appli-cation range ofthis method is limited to the case, when the heat transfer coefficients on bothplate surfaces are the same (al = a2 ) and the plate itself is composed ofonly one layer.There are different ways to reduce thefirst partial error A61

1 Error A01 can be reduced by increasing the contact surface area between the sensorand the investigated body In this way the heat flux density is diminished at the contactarea and thus the deformation of the original temperature field is also reduced One ofthe ways to achieve this is to apply an additional metal disk of high thermalconductivity as shown in Figure 16.9(a)

2 Reduction of the value of A01 can be obtained by decreasing the heat flux conductedfrom the measuring point along the sensor or its conductors For this purpose thethermocouple conductors should be as thin as possible and should initially be led for ashort distance, parallel to the investigated surface which is along the isotherms Anyinfluence of the conductor radius, RC, on the heat flux entering the sensor is seen fromthe relation ofequation (16.17)

3 Total elimination of the error, A61 , can be achieved by applying a thin disk sensor,made from a high conductivity material with the same emissivity as the investigatedsurface and fastened permanently to it In such a solution the heat flux densities qb and

qT are equal (qb = qT) as well as thermal resistances Wa = WT , (ab = ap ) For a diskfastened permanently to the surface the thermal contact resistance, between the sensorand the surface, is nearly zero (Wc~L, 0) From equations (16.26) and (16.28) it followsthat A61 = 0 in the case described above

The second partial error, A62 , is defined, in agreement with Figure 16.1, as:

SOLID BODY SOLID BODY q

Figure 16.9 Surface temperature measurement of a solid body by a disk thermocouple

This error results from the existence of a thermal contact resistance, W c, at the interface between the sensor and the body It can be regarded as a temperature drop across the resistance, W c, under the influence of heat flux, (DT, entering the sensor Thus, At9 2 is given by:

In the case ofa semi-infinite body, A0 2 is given by

1 +(1rRT b /F.)(WT + We ) where all symbols are as in equation(16.28)

It is rather difficult to calculate the thermal contact resistance, W c, which depends upon many factors such as the smoothness and cleanliness of the surface, the force with which the sensor is pressed to the investigated surface, the elasticities of the sensor and the surface materials, and so on Tye (1969) and Michalski (1978) show how it can be found experimentally using techniques similar to that illustrated in Figure16.10.Here, the thermal contact resistance between a copper rod and a steel surface is displayed against the contact force at different contact temperatures ec Based on these results Michalski (1978) has advised the application of a force of about30 Nfor copper-plate sensors of diameter8mm Reduction of the thermal contact resistance is possible by cleaning the contacting surfaces

to remove oxides, fats and other impurities followed by the application of a paste of high thermal conductivity (Tye, 1969; Michalski, 1978)

.,y =163°C

CONTACT FORCE , NIm'

Figure 16.10 Thermal contact resistance, Wc, of copper-brass contact versus contact force, P, at different temperatures

The third partial error, A63, is defined, conforming to Figure 16.1, as

The temperature distribution along the sensor must be known for it to be calculated However, for really small distances between the sensor sensitive point and its front surface, the heat flux from the side surface of the sensor along the length, P, can be neglected Thus the third partial error is given by:

where AT is the thermal conductivity ofthe sensor material.

To reduce the third partial error, A03, the distance P in Figure 16.1 should be kept as small as possible This can be achieved by using thin flat band-thermocouples, thin plates,

of well-conducting materials or thermocouples having nonjoined, pointed conductors, such

as in Figure 16.14(a), which can only be used on metallic surfaces.

Thermally compensated sensors A proper elimination of all partial errors could be achieved by applying thermally compensated sensors, which have additional heating by low power small heating elements (Michalski et al.,1991) Due to their complicated construction and measuring circuit, they are no longer produced.

Dynamic errors: In contact surface temperature measurements, especially when usinghand-held sensors, dynamic errors are also observed A theory of dynamic errors in contacttemperature measurements has not been properly developed so far (Mackiewicz, 1976b;Znichenko, 1969) These errors occur when readings are taken before the contact sensor hasreached a thermal steady state after making contact with the investigated surface, This errorcan be eliminated by a sufficiently long contact time before the readings are taken Inpractice, this time has to be below about 1 min, to prevent tiresome working for theoperator Also, the probability of ensuring the correct sensor position on the surfacedecreases with increasing time It is possible to reduce dynamic errors by applying a peak-picker device of the type described in Section 12.2.3

Numerical example

Calculate the method error of contact temperature measurement of a chromium-nickel steelsurface, using a disk thermocouple It is assumed that the thermal contact resistance, Wc, betweenthe sensor and investigated surface is null

Data: original surface temperature, Ot = 120 °C; ambient temperature, Oa = 20 °C,

body thermal conductivity, fb =10 W/m °C,

heat transfer coefficient at body's surface, ab = = 10 W/m2 °C

Sensor: type K thermocouple, with copper disk of:

Ip - 1 5 mm;Rp -7.5 mm, Ab = 372 W/m °C,

positive conductor:Rcl = 1 5 mm, Ac, = 13 W/m °C,

negative conductor: Rc2= 1 5 mm,Act= 58 W/m °C,

heat transfer coefficient on disk surface, ap = 10 W/m2 °C,

heat transfer coefficient on side surfaces of conductors, ac= 50 W/m2 °C

Assuming: A02 = 0 (Wc = 0) and A63 = 0 , the method error A O = Af91

Consequently : 6T =6t +Ar91 or OT = 13T-Aria = (OT-61)+Ana

Thus: qT= 127.3[(120 - 20) + A61] =12 730+127.3A6,

and finally: Az1 = (qT-12 730) 11273 ° C

2 Calculation of A61 =f(qT)

From equation (16.4): B = (10 x 7.5 x10-3)/15=5x10-'

From the diagram in Figure 16.5, for B = 5 x10-3, Fm = 0-9,

and thus, A61 =(qd x 7.5 x10-3 /15) x0.9 =0.45 x10-3 qd