16 Temperature Measurement of Solid Bodies by Contact Method 16 .1 Introduction One of the most frequently encountered problems in temperature measurement is measuring the temperature of solid bodies on their surfaces in contact with a surrounding gas or liquid . This may be solved by either contact or non-contact methods . Semi-contact or quasi-contact methods may also be used . Roeser and Mueller (1930) have previously discussed the problem . 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, O t , 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 Temperature Measurement Second Edition L. Michalski, K. Eckersdorf, J. Kucharski, J. McGhee Copyright © 2001 John Wiley & Sons Ltd ISBNs: 0-471-86779-9 (Hardback); 0-470-84613-5 (Electronic) 334 TEMPERATURE MEASUREMENT OF SOLID BODIES BYCONTACT METHOD ORIGINAL STATE STATE AFTER APPLYING CONTACT SENSOR SOLID BODY q GAS SOLID BODY q GAS (a) (b) A . SENSOR I (c)~ (d) A A4 '% \ S-SENSOR SENSITIVE POINT ('8 T ) W c -THERMAL CONTACT RESISTANCE .S, > .S 4 r ' 9 'T "o A . t ~ l Figure 16 .1 Surface temperature measurement of a solid body by a contact sensor . The isotherms and heat flux density lines, q, for the undisturbed conditions without the sensor are given in (a) with the corresponding temperature distribution in the direction normal to the surface shown in (c) . When the sensor is introduced the isotherms and heat flux density become distorted as in (b) and the temperature distribution becomes that in (d) junction of the thermocouple and the investigated surface there is always a thermal contact resistance, W c , caused by a non-ideal contact . The temperature drop across this contact resistance, A6 2 = 6" - 0' , is called the second partial error . It is further assumed, as in all sensors, that there is also a sensitive point in a contact sensor determining the thermometer readings, O T . 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 A0 3 = OT - 0", called the third partial error, which depends on the sensor design . All of the differences A?),, A0 2 and A6 3 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 THEORY OF CONTACT METHOD 335 O b = f (x, r) and secondly, the disturbing temperature field Od = f (x, r) . The disturbing temperature field is caused by the disturbing heat flux density, resulting from the difference between the density, q T , of the heat flux conducted along the sensor and the density, q b , of the heat flux transferred from the body to its ambient surrounding . This density, qd , of disturbing 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 the mannerof determining the disturbing flux and also shows the surfacial temperature distribution . The medium value of disturbing temperature Od,m , shown in Figure 16 .2(c), at the contact surface between sensor and body permits determination of the first partial error, 06 1 . The differential equation of heat conduction, describing the disturbing temperature field in a semi-infinite cylindrical body, is 2 -y-~ d + a 2 °d = o (16 .2) z d + r ~ with the boundary conditions : j dE)d - ~ Od]x- -0 _- ~ r<<-RT (16 .3a) L ax Od] x _0< °° ; Od]x -0 ; Od], =0 (16 .3b) (a) BODY AND SENSOR (b) EOUIVALENT MODEL 2R SENSOR GAS ZRT a qT qo qT - qb a qb _ //Tl X'b 1' XIl -ql I' I SOLID x BODY x II I (c) DISTURBING r TEMPERATURE I l FIELD ON SURFACE I B d .m OF SOLID BODY Ba (x=O) Figure 16 .2 Disturbing temperature field on the surface of a semi-infinite body, resulting from the application of a contact temperature sensor 336 TEMPERATURE MEASUREMENT OF SOLID BODIES BY CONTACT METHOD where A'b is the thermal conductivity of the body, and a b is the heat transfer coefficient at the surface of the body . Solving equation (16 .2) with the boundary conditions of equation (16 .3) gives : od = _ gdRT - e-kxyh(y)IO(kry),, v (16 .4) v+B where v is the variable of integration, k,, = x ; k r = r ; B = aRT and I D (v) and R T R T k b I, (krv) are Bessel functions of the first kind and of order zero and first, respectively . From equation (16 .4), let the value of the integral, a function of k,,, kr and B, be described by F(k X ,k r ,B), so that equation (16 .4) becomes : od =-gT F(kXkr,B) (16 .5) ~ To help with the practical use of equation(16 .5) Kulakov and Makarov (1969) graphically display the values of the function F versus k X in Figure 16 .3 and versus k r in Figure 16 .4 as well as the mean value F m 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) 1,0 B= a R T p,8 ~n 0,6 kx= R 0,4 T Q2- II 0 1 1 , . ~ I ~0 1 2 3 4 5 kx i I 11 5 6 7 8 9 10 Figure 16 .3 Function, F, from equation (16 .5) versus parameter, kX , for r = 0, B = 0 F(x =0, B=01 1,0 F m (0 ~ r < RTI B-_ 1Rr 0,8- 0,6- k,= RT 0,4 I 0,2 I II I 1 -~0~ 1 2 3 4 5 ~k, 11 5 6 7 8 9 10 Figure 16 .4 Function, F, from equation (16 .5) versus parameter, k r , for x = 0, B= 0 THEORY OF CONTACT METHOD 337 F m (x=0 ; 0 < r -<R T ) 1,0 B= a'RT 0,8 ~b 0 .6 0 .4- 0,2-I 0 0 .5 1,0 1,5 2,0 B Figure 16 .5 Mean value, F m , 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, q d , diffuses into an infinitely large plate of limited thickness, through a surface limited by a circle of radius R T 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 : d9d Xb Od Jx- __ _ _ 9d 0 ' 1 13 lr<-RT C aod _ od 0 (16 .6) ~x-!b Od Jr-O < °° ; Od Jr - 0 The solution of equation (16 .2) for the boundary conditions (16 .6) is : Od = - gdRT °" (v + Bl )e [v(k, -k .)] + (v - B2 )el -v (k, -k .)] jl (v)Io (kr dV Ab 0 (v + Bj)(v + B2 )ek)v - (v - Bl)(v - B2 )e-k`v ) (16 .x) q~ T H(kx,kr,kl,Bl,B2) b (a) PLATE AND SENSOR (b) EQUIVALENT MODEL 2R SENSOR 2R a, }qd qT - qb °b, q b qT - qe r j i ,Zy / Ab ` PLATE qd -DISTURBING HEAT x FLUX x Figure 16 .6 Infinitely large plate with a surface temperature sensor 338 TEMPERATURE MEASUREMENT OF SOLID BODIES BYCONTACT METHOD where k,, = x , k r = r , k l = lb , B 1 = atRT and B 2 = azRT . RT RT R T 4 Ab To calculate the value of the first partial error, At9 1 , which equals the medium value of the disturbing temperature, O d,m , at the contact surface between the body and sensor, the function H from equation (16 .7) must be known for k, varying in the range 0<kr <_1(0<_rSRT) and for k x =0 . As the function H depends upon several parameters it is difficult to display it graphically in an universal way . Figure 16 .7 presents values of H, for k r = 0 and k, = 0, as a function of relative plate thickness k l for some chosen values of B 1 = B z = B . This corresponds to a case when the heat transfer coefficients on both sides of the plate are the same (a1 = a 2 = a) . Figure 16 .7 permits the maximum value of disturbing temperature in the centre of the contact surface to be found . From the curves of H = f(kv, it follows that the disturbing temperature values decreasewith 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 and Makarov (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, q d , of the disturbing heat flux, requires knowledge of the density, q T , of the heat flux entering the sensor . The four simple sensor models, given in Figure 16 .8, will be considered under some simplifying assumptions . It will be assumed that the heat transfer coefficients have constant values and that the temperature field in the cross-section of the rod or plate is uniform, with a uniform density of heat flux all over the front surface of the sensor as well . In addition it is assumed that 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= R b 6 'B=10, BI =82 =B 5 4 B=103 (Cci=oc2=a) Y -2 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 B values THEORY OF CONTACT METHOD 339 x (a) DISK x~ GAS (b) ROD 2R , 2RP 1% 2 a, P 7- JT X P x r `qT ~ SOLID BODY qT OT 4T r OT ~Pi (c) THERMOCOUPLE SENSOR (d) DISK THERMOCOUPLE SENSOR x x 2R, i 2R 12 2R, , 2Rc2 oc c am ac2 ' as X,2 '% P 2 '~C1 Xc2 a : r 77/71=/ 1-777 r Y , qci ~'l q,2 0 ,2 ' q T 2 R P ~P1 l J 0 T = 0 c1 +0 ,2 T Figure 16 .8 Sensor models for evaluating the heat flux, (DT, entering the sensor with the associated heat flux density, qT For each model, both the heat flux, (DT, entering the sensor, as well as the heat flux density, q T , at the sensor surface, A T , in contact with the investigated body, will be determined . These two are related by : qT = AT (16 .8) T Also to be determined is the thermal resistance, W T , of the sensor, defined by Mackiewicz (1976a) in terms of, O T , the temperature of the surface of the sensor in contact with the investigated body as, W T = ~T (16 .9) T Disk sensor : To simplify the problem of the disk, shown in Figure 16 .8(a), it is assumed that the disk (plate) only transfers heat to the environment from its upper 340 TEMPERATURE MEASUREMENT OF SOLID BODIES BYCONTACT METHOD surface (1 p << 2R p ), and that Op I = Opt = 6T . This corresponds to a thin disk, with a large value of thermal conductivity X p . The heat flux (DT is : (DT =KRpapWT -6a)=IrRpapOT (16 .10) with the heat flux density given by : 9T = icR2 -apOT (16 .11) P The thermal resistance of the disk, following equation (16 .9), is : W T = 1 (16 .12) 7rR 2 a p Rod sensor : The differential equation describing the temperature distribution along the rod (conductor) of the sensor in Figure 16 .8(b) is : d 2 O c (x) - 2ac O c (x)=0 (16 .13) dx 2 ACR, with the boundary conditions : Oc(x)],=o=OT ; Oc(x)]x =0 (16 .14) The solution of equation (16 .13) for the boundary conditions in equation (16 .14) is : Oc(x)=0T exp - F27, x (16 .15) ( Ac R c ) The heat flux, (DT, entering the sensor equals the total heat flux transferred to the environment from the side surface of the rod in accordance with : (DT = f ~21rR c a c O c (x)dx (16 .16) 0 Substituting the expression for Oc(x) from equation (16 .15) into equation (16 .16), yields : THEORY OF CONTACT METHOD 341 q'T = IrR, 2a c A c R c O T (16 .17) The heat flux density, q T , is then q T = z (16 .18a) IrR c and finally : q T - 2a ° A ° O T (16 .18b) R C Following the definition of equation(16 .9), the thermal resistance, W T , of the rod is given by : WT 7rR c 2acX~Rc (16 .19) Double conductor thermocouple sensor : Figure 16 .8(c) shows that this sensor may be regarded as two independent rods from the model in Figure 16 .8(b) . In most cases it can be assumed that : Rcl = Rc2 = R c > ac, = ace = ac ; OTi = OT2 = OT and then the total heat flux, (DT, entering the sensor, is (DT = R c ir 2acRc (XI + jc2 )E)T (16 .20) and the thermal resistance of the sensor, following equation (16 .9) will be : W T = I (16 .21) R c ir 2a c R c ( 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 R c << 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) : 342 TEMPERATURE MEASUREMENT OF SOLID BODIES BY CONTACT METHOD (PT = [irR 2 a, +xR, P 2a~R, + Jz ) J OT (16 .22) The appropriate heat flux density in this case is : gT = 7rRz = a P + Ri 2a~R, ( ~~, + ~~ z) O T (16 .23) P P and the thermal resistance, W T , conforming with equation (16 .9) is : W T = 1 (16 .24) it[Reap +R c 2a~R, (X,, + A2 16 .2 .3 Method errors and their reduction The first partial error, A6 1 , agreeing with the definition from Section 16 .2 is : A61 = 6'- 6t (16 .25) This error equals the medium value of the disturbing temperature Od,m, at the contact surface between the sensorand the body . For a semi-infinite body, from equation (16 .5), the error, A6 1 , is described by 00 1 = Od .m]x=0 gdRT Fn ~ (16 .26) 0_<r_R T /~b x=0 0 <r<RT The density, q d , of the disturbing heat flux in equation (16 .26), is calculated from equation (16 .1) as : qd = qT- qb (16 .27) Determination of, q T , for the four sensor models is described in Section 16 .2 .2 . To simplify the problem it is assumed that 6 T = t9' (W c = 0) . The value of qb is calculated from qb = abO T , while the value of F m in equation (16 .26) is found from Figure 16 .5 . Calculation of the error, A6 1 , is accomplished in a step by step iterative way or graphically . This will be explained in a numerical example . To calculate these A6 1 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, W c , in the manner : [...]... partial errors 16.3 Sensors for Surface Temperature Measurement 16.3.1 Portable contact sensors Portable contact sensors, or probes, pressed to the investigated surface, give good readings, known as spot readings, when a thermal steady-state is reached Many such thermometers have exchangeable measuring tips, adapted to the condition, shape and material of the surface All of the sensor tips used should... errors in thermal contact resistance measurements J Heat Transfer, 97(5), 305-307 Tye, R.P (1969) Thermal Conductivity, Academic Press, London Wilcox, S.J and Rohsenow, W.M (1970) Film condensation of potassium using copper block for precise wall temperature measurement J Heat Transfer, 92(8), 359-371 Yarishev, N.A and Minin, O.W (1969) Extrapolation method of measurement of temperature and heat-flux . 2R p ), and that Op I = Opt = 6T . This corresponds to a thin disk, with a large value of thermal conductivity X p . The heat flux (DT is : (DT =KRpapWT -6a)=IrRpapOT (16 .10) with the heat flux density given by : 9T = icR2 -apOT (16 .11) P The thermal resistance of the disk, following equation (16 .9), is : W T = 1 (16 .12) 7rR 2 a p Rod sensor : The differential equation describing the temperature distribution along the rod (conductor) of the sensor in Figure 16 .8(b) is : d 2 O c (x) - 2ac O c (x)=0 (16 .13) dx 2 ACR, with the boundary conditions : Oc(x)],=o=OT ; Oc(x)]x. contact sensors, or probes, pressed to the investigated surface, give good readings, known as spot readings, when a thermal steady-state is reached . Many such thermometers have exchangeable measuring tips, adapted to the condition, shape and material of the surface . All of the sensor tips used should fit the