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17 Temperature Measurement of Fluids 17 .1 Low Velocity Gas 17 .1 .1 Contact sensors Assume that a gas, with a temperature, T ., flows in a tube with a flow-velocity below about 20 m/s, as shown in Figure 17 .1(a) . In the tube there is a sheathed sensor, whose sensitive part, at temperature, T T , is placed in the sheath bottom . This case corresponds to a thermocouple with the measuring junction welded into the sheath bottom . Also let the temperature of the internal surface of the tube wall be T W , while the length to diameter ratio of the sensor sheath is assumed to be so large that heat transfer through the sheath bottom can be neglected . It is further assumed that the gas temperature, T g , is higher than the temperature of the tube wall, T, Consider a cylindrical sheath element of length dx, placed at a distance, x, from the sheath end as represented in Figure 17 .1(b) . In the thermal steady- state it is apparent that the heat balance must take account of all conduction, convection and radiation effects . (a) ARRANGEMENT (b) SHEATH T 9 > TS D - d " TUBE ELEMENT T= '` T' v 0~ +d0, GAS T9 EJ " d O p c d0~ -~ ' `SENSOR TT SHEATH HEAT FLUX : 0k - CONVECTION O r - RADIATION 0, - CONDUCTION Figure 17 .1 Gas temperature measurement in a tube by a sheathed sensor 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) 362 TEMPERATURE MEASUREMENT OF FLUIDS The convection heat flux from the gas to an element of length dx is : d'Dk = a k 'rDdx(T g - T x ) (17 .1) where a k is the convective heat transfer coefficient, T g is the gas temperature, and T x is the sheath temperature at a distance x from its end . Assuming that the gas transmissivity equals unity, the heat flux dissipated from this element by radiation to the tube wall is given by : d(')r=ETCo7SDdx (100) 4- ( 00)4 (17 .2) where E T is the emissivity of the sheath surface, C o is the radiation constant of a black body and T,, is the temperature of the tube wall and the other symbols are as in equation (17 .1) . Equation (17 .2) is valid under the assumption that the sheath surface is many times smaller than that of the tube as considered in equation (8 .21a), while equation (17 .1) is valid for a k =constant . The conductive heat flux along the sensor is described by the relation : dam, =- .1,A d2T dx (17 .3) where A is the thermal conductivity of the sheath material, A = )rD 2 nd 2 4 4 and T x is the temperature at a distance x from the sheath end . A heat balance equation of element dx is then : d(D k = d0 c + d(D r (17 .4) Substituting the corresponding values from (17 .1), (17 .2) and (17 .3) into (17 .4) yields : dx2 ~1A+ak~cD(Tg-Tx)-ETC070 100 100 =d (17 .5) d2T x _( Tx )4 ( Tw )4 This nonlinear differential equation, which cannot be solved by direct integration, can be solved in some cases by a graphical method or by introducing some application specific simplifications . In most cases, similarly to the convective heat transfer coefficient, a k , an LOW VELOCITY GAS 363 equivalent heat transfer coefficient by radiation, a r , with a constant value is introduced . Equation (17 .5) then becomes : z ~2 AA +ak7rD(T g - T X ) - a r ; rD(T,, - Tw)=0 (17 .6) Making the substitution n (ak +a r )7rD = AA this equation has the solution : T g -T w a k _ +a r cosh nl (17 .7) TX - Tw a k cosh nl-1 Considering that the sensitive point of the sensor is mostly located at its end, where x = 0, equation (17 .7) is simplified to : Tg - Tw _ ak + ar cosh nl (17 .8) T T -T w a k cosh nl -cosh nx or after some transformations : ATc,r = TT _ Tg (Tw _ T g )Cl - a k (cosh _ nl -1) (17 .9) - (a k +a r )coshnl where AT, ,,r is the overall error resulting from heat conduction along the sensor and from radiant heat exchange . If radiant heat transfer can be neglected, so that a r = 0 , equation (17 .9) becomes : OT C =T T -T g = T w-Tg (17 .10) cosh ml where AT E is the error resulting from heat conduction along the sensor and k ~D m= a N AA If the sensor is sufficiently long so that any error resulting from heat conduction along the sensor could be neglected, or if the product, .l .A, is sufficiently small and substituting T x = T T , equation (17 .5) is simplified to : 4 a k 'rD(T g -TT)-ETC,,7D 100)-(00)4 0 (17 .11) [( L- 364 TEMPERATURE MEASUREMENT OF FLUIDS From equation (17 .11), the measuring error, AT, resulting from radiant heat transfer, is expressed by : AT, = T T g = ETC° T"' 4 TT 4 r T - ak (100) - (100) (17 .12) The simplest way of solving this equation is a graphical one, as shown in Figure 17 .2, where the curves q k =f l (TT ) and q r = f 2 (T T ) are drawn . The convective heat flux density from the gas to the sensor surface is q k = a k (T g - T T ) while the radiant heat flux density, q r , from the sensor surface to the tube wall is : qr = ETCo 0J4 -~ 00)4 In the thermal steady-state both these quantities are equal so that : qk = 9r (17 .13) The point where both curves intersect in Figure 17 .2, indicates the sensor temperature, T T , and the radiant measuring error, AT r . Equations (17 .9), (17 .10) and (17 .12) enable calculation of the measuring errors and the introduction of any necessary corrections in particular cases . This latter possibility is rarely used . Mostly based on these equations, conclusions may be drawn for the design of sensors, so that indication errors can be kept as small as possible . Sometimes more complex models of a temperature sensor in a tube, through which gas flows, are also applied . In these models described by Blumroder (1981), Haas (1969) and Rudolphi (1969), the heat exchange between the part of the sensor sheath and sensor head, protruding from the wall into the environment, is considered . v TUBEWALL TEMPERATURE T =const GAS TEMPERATURE T 9 =const r r o l a w Y T 9 0 T T i AT r TEMPERATURE Figure 17 .2 Graphical method of determining the error, AT r , of equation (17 .2) LOW VELOCITY GAS 365 Error analysis of gas temperature measurement permits the practical establishment of the dependence of errors upon the sensor insertion depth into the medium or to establish the dependence of errors upon the values of the heat transfer coefficients between the sensor sheath and the gas . 17 .1 .2 Methods of reducing errors in contact measurements From the preceding section, it follows that reduction of the measuring errors requires : " increased heat flux, (Dk, gained by convection, " decreased heat flux, (D C , lost by conduction, " decreased heat flux, (D, ., lost by radiation . Increase of heat flux, Ok, by convection can be achieved by : 1 . Increase of the surface of the convective heat transfer by using a finned sensor as shown in Figure 17 .3 . This means that the fins should be made of a material with a high thermal conductivity and low emissivity, to reduce the radiant heat losses . 2 . Increase of the convective heat transfer coefficient by applying high gas velocity in the tube using a sensor sheath with as small a diameter as possible and by placing the sensor at an angle of about n/2 relative to the direction of the gas flow . 3 . Applying suction thermometers, in which the gas velocity is increased only in the surrounding of the sensor . This method is especially appropriate in those cases where an increase of the gas velocity in the tube is not possible . A suction thermometer, also called aspiration thermometer, was first proposed by R . Assmann (1892) . Its operating principle, shown in Figure 17 .4 (Wenzel and Schulze, 1926), has the end of a suction tube with a thermocouple inserted into a pipe-line or gas-filled enclosure, in which the gas temperature is to be measured . The compressed air produces suction of the gas in the nozzle . In this way, the gas, whose temperature is to be measured, streams past the thermocouple at high velocity . Thermocouple readings, which are a function of gas velocity, have the dependence on rate of gas flow shown in Figure 17 .5 for a gas temperature of 100 °C . These thermometer readings are practically independent of the position of the measuring junction in the suction tube . Industrial suction thermometers are often equipped with one or a number of concentric radiation shields to reduce the radiant heat exchange . Suction thermometers, which can be used up to about 1600 °C, are often water or air cooled, using MI thermocouples as temperature sensors . More TRANSVERSE FINS LONGITUDINAL FINS I Figure 17 .3 Finned sensors 366 TEMPERATURE MEASUREMENT OF FLUIDS COMPRESSED AIR MEASURING 1' NOZZLE OV t " 1 " . 1 1 1 v THERMOCOUPLE JUNCTION 1 SUCKED GAS Ei 1 1 0 1 40 50 Z RATE 1 1 Figure 17 .4 Suction thermometer by Wenzel Figure 17 .5 Thermocouple readings of the and Schulze (1926) thermometer from Figure 17 .4, versus gas flow detailed information rate at 'g g = 100 - C " " found " publications " " 4 "" " 41) and Ribaud et al . (1959) . It is also important to note that they can only be with the rate of gas flow in the pipe-line itself . Decrease used when the quantity of gas sucked by the thermometer is negligibly small compared I q )C9 1 conduction 1 ' obtained 1 Application " / I cross-sectional - . " " from materials with low thermal conductivity . Figure 17 .6 presents some ways of installing rather long sensors in pipe-lines . It is advisable that the ratio of the sensor length, 1, to its diameter, I should " " I 2 : 6 to 10 in flowing gas or Ild 2! 12 to 15 in still gas . 2 . Laying the sensor from the measuring point, along isotherms, as shown in Figures Figure 17 .6(a) . Similar results can be obtained in arrangements like those in I i (a) IN ELBOW (b) OBLIQUELY (c) ALONG PIPELINE AXLE IbI (CI Figure 17 .6 Methods of installing temperature sensors in a pipe-line LOW VELOCITY GAS 367 3 . Thermally insulating or heating of the sensor head to increase its temperature in the manner shown in Figure 17 .7 . Decrease of heat flux, (Dr, lost by radiation, can be obtained by : 1 . Covering the sensor surface with materials such as gold, silver or platinum which have a low emissivity, ET . 2 . Application of radiation shields . This is the most popular method of reducing radiant heat exchange between sensor and surrounding walls . A shield of low emissivity, E,, is placed between the sensor and tube wall as shown in Figure 17 .8 . Assuming that the shield is sufficiently long, to be able to neglect the influence of its open ends on the sensor heat balance, and that the shield inner surface is many times larger than that of the sensor, the measurement error due to radiant heat exchange can be calculated from equation (17 .12) . In that relation the shield temperature T S is substituted instead of the wall temperature, T W . The relevant error, AT r,s , is given by : AT"s = TT -T 9 = ETC . (lO i O) 4 -(100)4 (17 .14) k The shield temperature, T s , is determined from the shield heat balance : O k,s = (Dr,s (17 .15) where (Dk,s is the heat flux gained by convection and (Dr,s is the heat flux lost by radiation . Heat flux 4)k,, is given by : (Dk,s =270 s l s a k's (T g -T S ) (17 .16) where ak ,s is the convectioe heat transfer coefficient of the shield and the other symbols are as in Figure 17 .8 . (a) INSULATED SENSOR HEAD (b) HEATED SENSORHEAD INSULATION ELECTRIC HEATING ELEMENT T s T s Figure 17 .7 Methods of reducing heat-flux conducted along the sensor 368 TEMPERATURE MEASUREMENT OF FLUIDS PIPELINE SENSOR IT Tw E EE 0 0 T y T r s E s SHIELD Figure 17 .8 Sensor with a radiation shield In equation (17 .16) double, internal and external shield surfaces, have been considered . The heat flux, (Dr,S, is described by : c l?r,s _9GDsls"Co (100)4-(100)4 (17 .17) [ 1 where e s is the shield surface emissivity, and the other symbols are as in Figure 17 .8 . Equation (17 .17), which is based on equation (8 .21 a), is valid under the assumption that the external shield surface is many times smaller than that of the surrounding surface of the tube wall . After some simplifications, the heat balance equation in steady-state, becomes : 2aks(Tg-Ts)=EsCO (100) 4- ( 00)4 (17 .18) The value of shield temperature, T s , can be found graphically as in Figure 17 .2 . Assuming that T g > Tw the shield temperature, T s , is always higher than that of the wall, T W . Comparing equations (17 .12) and (17 .14) it is clearly seen that error due to the radiant heat exchange between the sensor and its surroundings decreases due to the application of the shield . Analogous reasoning can be made for two, three and more shields . Each consecutive shield decreases the error resulting from the radiant heat exchange with a progressively smaller and smaller influence . According to King (1943) this error, OT r , ns , for n shields is given by the approximate relation : AT im = i T , (17 .19) A more detailed analysis of the influence of shields on the readings of a gas measuring thermometer is presented by Moffat (1952) . The distances between the shields should be large enough to enable a free gas flow . 3 . Application of heated radiation shields (Mullikin, 1941) . In this method a thermocouple is placed on the axis of a shield which is heated by an additional low-power heating LOW VELOCITY GAS 369 element . A second thermocouple, which measures the shield temperature, allows adjustment of the heating power until the readings of both thermocouples are the same . When this state is reached, the thermocouple which measures the gas temperature, does not exchange any energy with the surroundings by radiation so that its readings are correct . It is a rather time consuming method unless it is automated . The simultaneous fulfilment of all three conditionsfor reducing measuring errors may be realised with the use of a bare thermocouple of very small wire diameter . The diameter of the thermocouple wire and its surface area are prime influences which determine radiation heat exchange . As a thin gas film always exists around a wire, the convective heat exchange depends upon the wire diameter plus double film thickness . With wire diameter approaching zero the convective heat exchange is mainly determined by the double film thickness while the radiant heat exchange disappears . Thus, as only convective heat exchange remains, any radiant measuring errors, AT, disappear . At the same time, with wire diameter approaching zero any conductive measuring errors, AT, also disappear . In practice it is advisable to use bare or MI thermocouples which are as thin as possible . The lower diameter limit is imposed by the mechanical strength and the corrosion resistance of the wires . This method is suitable for both laboratory and industrial applications . The extrapolation method is another very precise method of gas temperature measurement . Gas temperature is simultaneously measured by a number of bare or MI thermocouples of different diameters . The results, which are displayed graphically as a function of the thermocouple diameters are extrapolated to the zero diameter . This value is then the true gas temperature . As an example, Figure 17 .9 presents results obtained while using bare Type K thermocouples to measure the temperature of hot air with velocity 8 m/s flowing through a tube having a wall temperature of 15 °C . The extrapolated value was 175 °C . Before the measurements it is very important to check the identity of the thermoelectric characteristics of the thermocouples used . MI thermocouples, as described in Section 3 .3 .3, which are produced with a diameter as thin as 0 .2 mm, are very convenient in the extrapolation method . -GAS TEMPERATURE EXTRAPOLATED VALUE 190 TYPE K THERMOCOUPLE 0 170 4 T 150 ~ w 15°C a E 130 - 110 ' - w r n r r ~ r 0 0,5 1 1,5 2 2,5 3 WIRE DIAMETER d , mm Figure 17 .9 Temperature indicated, 9T, by bare thermocouples in flowing air of velocity, v = 1 .8 m/s, versus wire diameter, d 370 TEMPERATURE MEASUREMENT OF FLUIDS An important source of errors occurs in the temperature measurement of flowing gas when a non-uniform distribution of gas temperature occurs across the tube section . To get readings approaching the average gas temperature a number of sensors is used as given in Figure 17 .10 . If thermocouples are used, they have to be connected in series . The measured thermal emf must then be converted into a temperature value . A correct average temperature can be obtained directly, using parallel connected thermocouples, provided all of the thermocouples used have precisely the same resistance . A merit of the parallel connection is that correct readings are obtained even in the case of a broken circuit in one of the measuring loops . For gas temperature measurement in the temperature range of from about 1000 to 3000 °C, only thermocouples of the metal group, Pt, Rh and Ir, whose properties have been described in Section 3 .4, are used . However, besides the changes in their characteristics, described before, these thermocouples can also act as catalysts in oxidisation and combustion processes . These phenomena result in additional heating of the thermocouples causing important additional errors . A detailed analysis of these phenomena, which can also occur in any other temperature sensors with protecting shields made of the previously mentioned metals, is presented in Ash and Grossmann (1972) and Thomas and Freeze (1972) . Many references and methods for the experimental detection of these phenomena are also given by these authors . As an example, in measuring the temperature of not completely burned exhaust gases of internal combustion engines or of gas turbines at about 1000 to 1500 °C, the measuring error can be as high as about 400 °C (Thomas and Freeze, 1972) . To prevent additional catalytic heating, thermocouples or their protective sheaths as pointed out by Kinzie (1973), have to be covered by a layer of BeO, ZrO, Si 2 0 3 , Cr 2 0 3 or other appropriate materials . Detailed information on methods of gas temperature measurement and many references can be found in papers by Baas and Mai (1972), Benson and Brundrett (1962), Moffat (1952), Mullikin (1941), Mullikin and Osborn (1941) and Torkelsson (1980) . T 2 9 T 1 T3 T 1 , T 2 , T 3 -THERMOCOUPLES + - TO MEASURING INSTRUMENT Figure 17 .10 Series thermocouple connection for measuring the average temperature of a flowing gas
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