BOOKCOMP, Inc. — John Wiley & Sons / Page 936 / 2nd Proofs / Heat Transfer Handbook / Bejan 936 EXPERIMENTAL METHODS 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [936], (24) Lines: 722 to 759 ——— -0.73187pt PgVar ——— Normal Page * PgEnds: Eject [936], (24) Figure 12.8 Seebeck effect (basic thermocouple). A difference in electrical potential across the thermal leads of two dissimilar but connected conductors is generated when a temperature gradient is imposed along each conductor. The ratio of electrical potential gradient to thermal potential gradient along each conductor is dependent on the local temperature of the conductor. where ε s1 and ε s2 are the thermoelectrical characteristics of each conductor (total thermoelectric power, equal to the Seebeck coefficient). From eq. (12.47), ε s1 must differ from ε s2 for the potential generated by the two conductors to be nonzero. That is, the two conductors must be different. Observe also that the sensitivity of each conductor is proportional to ε s . A very high ε s value means that a small temperature variation will generate a large electric potential difference. Figure 12.8 is the most fundamental representation of a thermocouple. By mea- suring the voltage ∆E across terminals 1 and 2 and the temperature T 0 , the junction temperature T c can be found once the Seebeck coefficient of each conductor in the thermocouple is known. It is very important to recognize that the length of each con- ductor is irrelevant for determining the junction temperature T c [see eq. (12.47)]. Moreover, the manner in which the temperature varies along the conductors is also irrelevant for determining the junction temperature T c . With these two observations in mind, the effect of attaching a third conductor at the extremities of conductor 1 and 2 as shown in Fig. 12.9 can be predicted. Using eq. (12.44) for conductor 1, E c − E 1 = T c T 0 ε s1 (T ) dT (12.48) and for the leg of conductor 3 connected to conductor 1, E 1 − E 3 = T 0 Ta ε s3 (T ) dT (12.49) Addition of eq. (12.48) to eq. (12.49) gives E c − E 3 = T c T 0 [ ε s1 (T ) ] dT + T 0 Ta [ ε s3 (T ) ] dT (12.50) BOOKCOMP, Inc. — John Wiley & Sons / Page 937 / 2nd Proofs / Heat Transfer Handbook / Bejan EQUIPMENT 937 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [937], (25) Lines: 759 to 815 ——— -0.84583pt PgVar ——— Normal Page PgEnds: T E X [937], (25) Figure 12.9 Third conductor attached to the leads of the two base thermocouple wires (con- ductors 1 and 2). Doing the same for conductor 2, E c − E 2 = Tc T 0 ε s2 (T ) dT (12.51) and for the leg of conductor 3 connected to conductor 2, we have E 2 − E 4 = T 0 Ta ε s3 (T ) dT (12.52) Addition of eq. (12.51) to eq. (12.52) yields E c − E 4 = T c T 0 [ ε s2 (T ) ] dT + T 0 Ta [ ε s3 (T ) ] dT (12.53) Finally, subtraction of eq. (12.50) from eq. (12.53) provides E 4 − E 3 = T c T 0 [ ε s1 (T ) − ε s2 (T ) ] dT (12.54) A comparison of eq. (12.54) with eq. (12.47) shows that the potential difference (E 4 −E 3 ) is identical to (E 2 −E 1 ). Therefore, the inclusion of a third homogeneous conductor is irrelevant to the determination of T c using the thermocouple configura- tion shown in Fig. 12.9. Existing standard pairs of conductors used as thermocouples have a wide range of temperature application, being commercially available for measuring temperatures BOOKCOMP, Inc. — John Wiley & Sons / Page 938 / 2nd Proofs / Heat Transfer Handbook / Bejan 938 EXPERIMENTAL METHODS 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [938], (26) Lines: 815 to 849 ——— 0.97804pt PgVar ——— Normal Page PgEnds: T E X [938], (26) TABLE 12.6 Most Common Standard Thermocouples a Range, T min to T max S (mV/°C) Thermocouple Wires Type (°C) T (°C) Chromel (nickel–chromiun) E −260 to 980 0.07313 Constantan (copper–nickel) 300 to 1000 Iron J −180 to 870 0.05451 Constantan (copper-nickel) 0 to 760 Chromel (nickel-chromiun) K −260 to 1370 0.04079 Alumel (nickel–aluminum) 0 to 1370 Tungsten G 15 to 2800 0.01474 Tungsten and 26% rhenium 800 to 2300 a Thermocouple type is a letter designation for certain thermocouples. Range refers to the applicability range and sensitivity S for specific temperature ranges with a reference junction temperature at 0°C included. from approximately −270°C to approximately 2300°C. The most common are listed in Table 12.6, together with their temperature range of applicability and sensitivity. Keep in mind when selecting a thermocouple that not only is the temperature range important but also the conditions under which the thermocouple will be operating. When two thermocouples cover approximately the same temperature range (such as thermocouples type E and J) the one that presents the largest sensitivity S tends, in general, to yield more accurate readings. However, they tend to be more expensive and require more expensive equipment for data acquisition. Figure 12.10 is presented to assist in the verification of the linearity of the voltage as a function of temperature for each thermocouple. The straight lines are the best linear curve fit for each thermocouple within the temperature range shown. Observe that the agreement between curve-fit and real values is quite good in most cases. If the measuring temperature range is small, the voltage versus temperature sig- nal can be well approximated by a linear function. With this in mind, and eq. (12.47), Moffat (1990) devised a method for visualizing the temperature versus voltage rela- tionship of any thermocouple arrangement. For instance, consider the simple thermocouple configurations shown in Fig. 12.11. The first two thermocouples have wires A–B (left) and C–B (center), connected at the same temperature T 2 and with terminals at the same temperature T 1 . Each thermocouple will generate a certain electromotive force, which will be ∆E AB = E A − E B and ∆E CB = E C − E B , respectively. If the electric potential ∆E of each thermocouple is linear with the connection temperature T 2 , and the linearity function depends only on the thermocouple material, each thermocouple will follow a different voltage versus temperature line, as shown in Fig. 12.10. Hence, we can write ∆E AB = ε sAB (T 2 −T 1 ) and ∆E CB = ε sCB (T 2 −T 1 ). A third thermocouple, made of wires A–C, also shown in Fig. 12.11 (right), can now be considered. This third thermocouple yields the resulting effect of combining BOOKCOMP, Inc. — John Wiley & Sons / Page 939 / 2nd Proofs / Heat Transfer Handbook / Bejan EQUIPMENT 939 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [939], (27) Lines: 849 to 849 ——— * 88.854pt PgVar ——— Normal Page PgEnds: T E X [939], (27) Figure 12.10 Electromotive force generated, ∆E, as a function of junction temperature T for thermocouple types E, J, K, and G. The reference temperature is equal to 0°C in all cases. The straight lines represent a linear curve fit of the results for each thermocouple. Values for subzero temperatures are not shown. Figure 12.11 Two thermocouple arrangements, wires A–B (left), C–B (center), and A–C (right) connected at the same temperature T 2 and with terminals at the same temperature T 1 . BOOKCOMP, Inc. — John Wiley & Sons / Page 940 / 2nd Proofs / Heat Transfer Handbook / Bejan 940 EXPERIMENTAL METHODS 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [940], (28) Lines: 849 to 864 ——— 0.067pt PgVar ——— Normal Page PgEnds: T E X [940], (28) E 3 T c T c T h T h T E 2 3 1 E 1 ⌬E 3 1 2 2 A C Figure 12.12 Thermocouple connection (top) and corresponding resulting electromotive force ∆E = E 3 − E 1 as a function of temperature (bottom). the first two because we can write E A −E c = (E A −E B ) −(E C −E B ). In terms of the temperature variation, we can write ∆E AC = ∆E AB − ∆E CB = ε sAB (T 2 − T 1 ) − ε sCB (T 2 − T 1 ) This final result is shown in Fig. 12.12. The graph E versus T presents the curves for each original thermocouple (i.e., line 1–2 in Fig. 12.12 corresponds to curve ∆E AB of thermocouple A–B, and line 2–3 corresponds to curve −∆E CB of ther- mocouple C–B), with arrows linking each connection. Although the final result does not seem to depend on the thermocouple wire B, it is known that the inclination of each curve ∆E versus T (curves 1–2 and 2–3) depends on the material of wire B. If we obtain ∆E versus T curves, such as those of Fig. 12.10, for all thermocouple wires when connected to the same base wire B, the curves can be used to find the voltage of any thermocouple configuration. This concept is utilized in analyzing the four-thermocouple thermopile shown in Fig. 12.13. The resulting ∆E versus T diagram indicates that the final voltage of the thermopile, ∆E 1−9 = E 9 − E 1 , is four times the voltage of a single thermocouple, ∆E 1−3 = E 3 − E 1 . Thermopiles are useful for amplifying the voltage signal. 12.5.3 Resistance Temperature Detectors Resistance temperature detectors (RTDs) are devices designed for obtaining temper- ature values from measured changes in the resistance of an element. Many types of BOOKCOMP, Inc. — John Wiley & Sons / Page 941 / 2nd Proofs / Heat Transfer Handbook / Bejan EQUIPMENT 941 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [941], (29) Lines: 864 to 880 ——— -2.903pt PgVar ——— Normal Page PgEnds: T E X [941], (29) E 9 E E 7 E 8 T c T c T h T h T E 5 E 6 E 3 E 4 E 1 ⌬E 9 7 8 5 6 3 4 1 2 E 2 Figure 12.13 Four-thermocouple thermopile (top) and corresponding voltage versus temper- ature graph (bottom). materials, such as gold, iron, mercury, nickel, and tungsten, can be used as resistive elements. As the temperature of the element changes, so does the resistance. Experi- mental correlations to obtain the temperature of the element by measuring the element resistance, are generally written as R = R 0 1 + c 1 (T − T 0 ) + c 2 (T − T 0 ) 2 (12.55) where R is the resistance of the element measured at temperature T,R 0 the resistance at T = T 0 , and c 1 and c 2 are two experimentally determined correlation constants. For small temperature variation [i.e., small (T −T 0 )], eq. (12.55) can be simplified to R = R 0 [ 1 + c 1 (T − T 0 ) ] (12.56) where c 1 , the temperature coefficient of resistance, can vary from 0.001 to 0.007°C −1 , depending on the material used for the resistance element. For most materials, the electrical resistance increases with the temperature of the element (i.e., c 1 is positive). However, there are semiconductors that present opposite BOOKCOMP, Inc. — John Wiley & Sons / Page 942 / 2nd Proofs / Heat Transfer Handbook / Bejan 942 EXPERIMENTAL METHODS 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [942], (30) Lines: 880 to 903 ——— 0.09003pt PgVar ——— Custom Page (9.0pt) PgEnds: T E X [942], (30) behavior. RTDs using semiconductors are called thermistors, and their resistance varies exponentially with temperature, R = R 0 exp α 1 T − 1 T 0 (12.57) where α is an experimentally determined constant. The resistance of thermistors not only decreases with temperature, specific and their resistance is much higher than that of common metals. For instance, the specific resistance of copper is approximately 0.17 µΩ · m, and for semiconductors used in thermistors is approximately 10 7 µΩ · m. Some of the main advantages in using thermistors are related directly to their high resistivity: (1) small resistance effect of connecting leads (which is an important issue in most RTDs), and (2) reduced elec- tric current for measuring resistance, and consequently, reduced self-heating. Other advantages are the very high sensitivity of the device [because of the exponential characteristics of eq. (12.57)] and very good accuracy. The accuracy of RTDs is about ±0.0025°C, whereas the accuracy of thermistors can be as high as ±0.005°C which is much better than that of thermocouples. Moreover, the rapid time response to changes in temperature is also important. However, semiconductors tend to deteriorate rather rapidly, and thermistors are generally limited in application to benign environments at temperatures below 300°C. 12.5.4 Liquid Crystals Liquid crystals, discovered in 1888 by Friedrich Reinitzer (an Austrian botanist), are now an integral part of the measuring arsenal available to the thermal engineer. They are suitable for temperature measurements because the molecular structure of the material changes with temperature, and by doing so, the optical characteristics (light diffraction) of the material changes as well. Liquid crystals are ideal for surface temperature measurements and for volumet- ric clear liquid temperature measurement. Two particular types of liquid crystals, the cholesteric and the nematic, are used for thermal measurement. At present, liquid crystals are available in encapsulated form (microcapsules) that can be used as paint (thin film) for measuring surface temperature, or mixed with clear fluids (e.g., air, water) for volumetric temperature measurements. The microencapsulation of liquid crystal is extremely important for minimizing the effect of shear on the molecular structure of the material, which can induce changes in optical properties as the tem- perature does. When temperature of an illuminated liquid crystal increases, the color of the crystal changes in bands. For instance, the crystal might seem yellow at temperatures from 31 to 34°C. The temperature range within which a single color is observed is called the ambiguity band. Liquid crystals are available in several different bands. For quantitative measurement, liquid crystals must be calibrated against known temperatures, and this sometimes represents a nuisance. They can be calibrated in terms of color bands (ambiguity bands) or in terms of wavelength (chromaticity). The latter is more involved and more precise for providing continuous calibration BOOKCOMP, Inc. — John Wiley & Sons / Page 943 / 2nd Proofs / Heat Transfer Handbook / Bejan EQUIPMENT 943 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [943], (31) Lines: 903 to 926 ——— 3.16003pt PgVar ——— Custom Page (9.0pt) PgEnds: T E X [943], (31) (calibration based on wavelength values instead of a single color). The time of re- sponse of liquid crystals is good, generally faster than 10 −3 s. The microcapsules are also very close to being neutrally buoyant in water (specific gravity equal to 1.02), reducing the interference with the flow field. 12.5.5 Pyrometers Pyrometers are devices based on radiation heattransfer commonly used formeasuring the temperature of hot gases, above 500°C. However, their absolute accuracy is generally poor, about ±10°C. They can be considered a subgroup of optical devices, as they require a line of sight for performing the measurement. The most common are specific to the radiation absorption technique, being particularly useful for steady or unsteady measurement of gas temperature. Other existing devices are based on Rayleigh radiation scattering, spontaneous Raman scattering, laser-induced fluorescence, and spectral radiation emission. Some provide temporal resolution of more than 10 −10 s. When using the laser as probe, the spatial resolution can be made extremely small, less than 10 −9 m. These systems tend to be complex and very expensive, especially because of the base equipment. Some of these devices, such as laser-induced fluorescence, can be designed to provide concur- rent multipoint measurement. Some require seeding the gas, which can be detrimental in certain circumstances for making the measuring process more intrusive. 12.5.6 Heat Flow Meters Devices for measuring local heat flow based on measurements of the time evolution of local temperature are generally called the bead calorimeter or slug calorimeter. Here the preference is to call them sensible capacitors, which is more descriptive of the two main characteristics of the device: (1) it stores energy as a capacitor, and (2) energy is stored in the form of sensible heat. When measuring temperature, the measuring device (e.g., a thermocouple) is to have very small thermal inertia (or heat capacity) to achieve thermal equilibrium (the same temperature) as fast as possible with the point where the temperature is being measured. In contrast, the measurement of heat flow by sensible capacitors requires large thermal inertia, so the time for thermal equilibrium is longer and measurable. Thermal inertia can be adjusted by changing the mass m or the specific heat c of the material within the capacitor. The heat flow ˙q can be calculated using the first law of thermodynamics for a solid system, ˙q = mc dT dt (12.58) as long as the time evolution of the temperature dT /dt is measurable. It is important that the temperature within the device be as uniform as possible, to satisfy an implicit assumption of eq. (12.58). Consequently, materials with very high thermal conduc- tivity (e.g., deoxidized copper) designed for a very small volume/surface area ratio are preferable (e.g., spherical shapes). BOOKCOMP, Inc. — John Wiley & Sons / Page 944 / 2nd Proofs / Heat Transfer Handbook / Bejan 944 EXPERIMENTAL METHODS 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [944], (32) Lines: 926 to 984 ——— 0.36452pt PgVar ——— Normal Page PgEnds: T E X [944], (32) Some disadvantages of sensible capacitors include inaccuracy in obtaining dT /dt from the temperature–time evolution measurement, the need to cool the device to a temperature lower then the temperature where the heat flow is to be measured before measurements can be taken, and the intrusiveness of the device (for absorbing energy for measurement, the device disturbs the state or temperature of the region where a measurement is to be taken, which affects the existing heat flow). A different family of heat flow gauges, called diffusion meters, operates based on Fourier’s law of conduction heat transfer for steady state, ˙q = kA ∆T ∆x (12.59) where k is the thermal conductivity of the material used for measurement, A the cross- sectional area of the material through which heat flows, and ∆T is the temperature difference measured across a distance ∆x within the device. The geometry of the meter is extremely important for the proper functioning of dif- fusion meters. It is necessary that the heat flow within the device be unidirectional for eq. (12.59) to be valid. Thin layers (films) of uniform and isotropic low-conductivity material help prevent multidimensional effects and provide a consistent frame for cal- culating the heat flow from eq. (12.5). More involve designs with different geometry are also common [e.g., the Gardon gauge, described by Moffat (1990)]. Heat fluxes as high as 0.6 MW/m 2 can be measured with this device. Some disadvantages of diffusion meters include difficulty in obtaining unidirec- tional heat flow, the need for protective layers around the measuring elements (which increase the thermal inertia of the element, and as a consequence, is more disturbing to the state of the measuring region), and the need for steady-state measurement [eq. (12.59) is inappropriate if the heat flow varies in time]. NOMENCLATURE Roman Letter Symbols a accuracy A area, m 2 B bias error, dimensionless c specific heat, J/kg · K C form coefficient, m −1 d i deviation of measurement i, dimensionless E electrostatic potential, V E c connection electrostatic potential, V ∆E EMF, voltage, V f function, dimensionless I electric current, A I electric current flux, A/m 2 k thermal conductivity, W/m · K BOOKCOMP, Inc. — John Wiley & Sons / Page 945 / 2nd Proofs / Heat Transfer Handbook / Bejan NOMENCLATURE 945 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [945], (33) Lines: 984 to 1011 ——— -0.06432pt PgVar ——— Normal Page * PgEnds: PageBreak [945], (33) K permeability, m 2 L length, m L c least count, dimensionless m mass, kg n number of measurements, dimensionless n v number of measures, dimensionless N number of degrees of freedom, dimensionless p number of imposed conditions, dimensionless P pressure, Pa ˙q heat power, W ˙q heat power flux, W/m 2 Q quantity calculated from T i , dimensionless R electric resistance, Ω R 0 reference resistance, Ω S sensitivity, dimensionless t time, s T temperature, K T c connection temperature, K T i measurement i, dimensionless T mean value of measurements T i , dimensionless U uncertainty, dimensionless v fluid speed, m/s V volume, m 3 x coordinate, m Greek Letter Symbols α proportionality/sensitivity parameter, dimensionless β volumetric coefficient of thermal expansion, K −1 Γ observed probability, dimensionless Γ d deviation probability, confidence level, dimensionless Γ G Gaussian probability, dimensionless Γ χ 2 χ 2 probability, dimensionless δ i residual ε standard error of curve fitting, dimensionless ε s Seebeck coefficient, dimensionless µ dynamic viscosity, dimensionless π Peltier coefficient, dimensionless ρ fluid density, kg/m 3 σ standard deviation, dimensionless σ p precision, dimensionless ϕ semivariance, dimensionless χ 2 chi-square, dimensionless ω odds of variable T i having uncertainty U i , dimensionless Ω average of odds, dimensionless ∆ζ confidence interval, dimensionless . taken, which affects the existing heat flow). A different family of heat flow gauges, called diffusion meters, operates based on Fourier’s law of conduction heat transfer for steady state, ˙q = kA ∆T ∆x (12.59) where. + T 0 Ta [ ε s3 (T ) ] dT (12.50) BOOKCOMP, Inc. — John Wiley & Sons / Page 937 / 2nd Proofs / Heat Transfer Handbook / Bejan EQUIPMENT 937 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [937],. measuring temperatures BOOKCOMP, Inc. — John Wiley & Sons / Page 938 / 2nd Proofs / Heat Transfer Handbook / Bejan 938 EXPERIMENTAL METHODS 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [938],