BOOKCOMP, Inc. — John Wiley & Sons / Page 926 / 2nd Proofs / Heat Transfer Handbook / Bejan 926 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 [926], (14) Lines: 397 to 406 ——— -4.947pt PgVar ——— Normal Page PgEnds: T E X [926], (14) 22 22.5 23 23.5 24 T 0.2 0.4 0.6 0.8 1 1.2 ⌫ ⌫ G ⌫⌫, G Figure 12.4 Graphical representation of the observed probability distribution Γ (shown as black triangles) and the Gaussian distribution Γ G (straight line) from Table 12.3. thumb, when 0.1 < Γ χ 2 < 0.9 it cannot be said with confidence whether or not the expected distribution is followed. However, if Γ χ 2 < 0.1, the chi-square test shows that the distribution expected is not being followed. The other extreme range, when Γ χ 2 > 0.9, is very unlikely to happen naturally; therefore, this range is regarded as suspicious (Dally et al., 1993). (Note: It is important that the number of events of each measurement be at least five for the result to be statistically meaningful.) To bring into a better perspective the meaning of the chi-square test example considered in the preceding paragraph, the probability distribution Γ and Gaussian distribution Γ G observed, presented in Table 12.3, are shown graphically in Fig. 12.4. It is clear from Fig. 12.4 that the distribution Γ diverges from the Gaussian distribution Γ G at the extreme temperature values of T = 22°C and T = 24°C. 12.3 CALCULATION ERROR Once the uncertainty of a certain measured quantity is determined, the next compli- cation arises when a quantity is calculated from this measurement. The question is how the uncertainty of a measured quantity will affect the uncertainty of the quantity calculated. This question becomes even more difficult to answer when more than one measuring quantity is used for the calculation. The most common procedure used to evaluate the uncertainty of the quantity calculated in this case is based on the Kline and McClintock (1953) method. BOOKCOMP, Inc. — John Wiley & Sons / Page 927 / 2nd Proofs / Heat Transfer Handbook / Bejan CURVE FITTING 927 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 [927], (15) Lines: 406 to 441 ——— 8.72102pt PgVar ——— Normal Page * PgEnds: Eject [927], (15) When a certain quantity Q is calculated from a number n of measured quantities T , and each measured quantity has uncertainty ±U, the corresponding uncertainty of the calculated quantity U Q can be estimated from U Q = U 1 ∂Q ∂T 1 2 + U 2 ∂Q ∂T 2 2 +···+ U n ∂Q ∂T n 2 1/2 (12.15) The conceptual basis for eq. (12.15) is discussed by Coleman and Steel (1989). Equation (12.15) provides a good estimate of the uncertainty U Q when the odds of each measured quantity T i having uncertainty U i is the same, and when the measured quantities T 1 ,T 2 , ,T n are independent of each other (i.e., the error of one quantity does not correlate with the errors of the remaining quantities). If the errors correlate, eq. (12.15) should be modified as suggested by Coleman and Steel (1989). If the uncertainty of one measured quantity T i is more probable than the uncer- tainty of another variable, say T i+1 , eq. (12.15) must be corrected. In general, U Q = 1 Ω ω 1 U 1 ∂Q ∂T 1 2 + ω 2 U 2 ∂Q ∂T 2 2 +···+ ω n U n ∂Q ∂T n 2 1/2 (12.16) where ω i represents the odds of variable T i having uncertainty U i , and Ω, a represen- tative average of the odds, is defined as Ω = ω 2 1 + ω 2 2 +···+ω 2 n 1/2 n 1/2 (12.17) Further relevant information on uncertainty analysis was provided by Moffat (1982, 1985). 12.4 CURVE FITTING The most common curve-fitting technique is the least-squares technique. To apply this technique, the experimentalist must know (or guess) the form of the function for curve fitting a data set. Consider a certain data set of temperature values T i obtained when varying the heating power ˙q i . Considering the shape of the graph T versus ˙q, it could be assumed that T = f(a 1 ,a 2 , ,a p , ˙q), where a 1 ,a 2 , ,a p are all constants to be determined. The least squares method entails minimization of the cumulative square of the deviations between the measured value T i and the fitted value f(a 1 ,a 2 , ,a p , ˙q), namely, the minimization of D, where D = n i=1 T i − f(a 1 ,a 2 , ,a p , ˙q) 2 (12.18) BOOKCOMP, Inc. — John Wiley & Sons / Page 928 / 2nd Proofs / Heat Transfer Handbook / Bejan 928 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 [928], (16) Lines: 441 to 498 ——— 7.84288pt PgVar ——— Short Page * PgEnds: Eject [928], (16) The minimization process then yields the best values for a 1 ,a 2 , , and a p ,by imposing ∂D ∂a 1 = ∂D ∂a 2 =···= ∂D ∂a p = 0 (12.19) Each term of eq. (12.19) yields an algebraic equation involving a 1 ,a 2 , , and a p , in the form ∂D ∂a j = n i=1 2 T i − f(a 1 ,a 2 , ,a p , ˙q) − ∂f ∂a j = 0 (12.20) where j goes from 1 to p. There are p equation and p unknown. The solution of this system of algebraic equations leads to a 1 ,a 2 , , and a p values that minimize the cumulative square of the deviations. As an illustration of the applicability of the least squares fitting method, consider the temperature data set presented in Fig. 12.5. From the data distribution shown by the triangle symbols, the fitting polynomial function f = T = a 1 ˙q + a 2 ˙q 2 (12.21) could be proposed. The minimization equations in this case are ∂D ∂a 1 = n i=1 2 T i − a 1 ˙q i + a 2 ˙q 2 i ( −˙q i ) = 0 (12.22) ∂D ∂a 2 = n i=1 2 T i − a 1 ˙q i + a 2 ˙q 2 i −˙q 2 i = 0 (12.23) When written in terms of coefficients a 1 and a 2 , these two equations read, respectively, a 1 n i=1 ˙q 2 i = n i=1 T i ˙q i − a 2 n i=1 ˙q 3 i (12.24) a 2 n i=1 ˙q 4 i = n i=1 T i ˙q 2 i − a 1 n i=1 ˙q 3 i (12.25) Using eq. (12.25), eq. (12.24) can be rewritten in terms of a 1 only: a 1 = n i=1 ˙q 4 i n i=1 T i ˙q i − n i=1 T i ˙q 2 i n i=1 ˙q 3 i n i=1 ˙q 2 i n i=1 ˙q 4 i − n i=1 ˙q 3 i n i=1 ˙q 3 i (12.26) BOOKCOMP, Inc. — John Wiley & Sons / Page 929 / 2nd Proofs / Heat Transfer Handbook / Bejan CURVE FITTING 929 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 [929], (17) Lines: 498 to 526 ——— 0.29117pt PgVar ——— Short Page PgEnds: T E X [929], (17) Now, using eq. (12.26), eq. (12.25) can be rewritten in terms of a 2 only: a 2 = n i=1 ˙q 2 i n i=1 T i ˙q 2 i − n i=1 T i ˙q i n i=1 ˙q 3 i n i=1 ˙q 2 i n i=1 ˙q 4 i − n i=1 ˙q 3 i n i=1 ˙q 3 i (12.27) Once a 1 and a 2 are determined, the standard error (representative deviation) of the curve fit from the measured quantities T i can be estimated using (Figliola and Beasley, 1995) ε = n i=1 T i − f(a 1 ,a 2 , ,a p , ˙q) 2 n − p 1/2 (12.28) The appearance of n−p in the denominator of eq. (12.28) instead of n results from the use of the coefficients a 1 ,a 2 , ,and a p for determining f . Hence, p degrees of freedom have been removed from the system of data. There are situations in which it is important to estimate the uncertainties of the coefficients a 1 and a 2 . One common example is determination of the permeability K 024 6810 q . 0 10 20 40 30 50 70 60 T (°C) Figure 12.5 Graphical representation of experimental data (triangles) and curve-fit T = a 1 ˙q +a 2 ˙q 2 . The least squares fit yields a 1 = 2.0646,a 2 = 0.48645, with χ 2 = 8.3096 (Γ χ 2 = 0.5) and fitting error ε = 0.9608°C. BOOKCOMP, Inc. — John Wiley & Sons / Page 930 / 2nd Proofs / Heat Transfer Handbook / Bejan 930 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 [930], (18) Lines: 526 to 558 ——— 6.93274pt PgVar ——— Normal Page PgEnds: T E X [930], (18) and the form coefficient C of porous materials from experimentally measured cross- section-averaged fluid speed ν and pressure-drop ∆P data using the equation ∆P L = µ K ν + ρCν 2 (12.29) Comparing eq. (12.29) and eq. (12.21) it is observed that ∆P/Lplays the role of f, ν of ˙q,µ/K of a 1 , and ρC of a 2 . The coefficients a 1 and a 2 (and consequently, K and C) can be determined from eqs. (12.26) and (12.27), respectively. The issue at hand is very simple: how to determine the uncertainties of the resulting coefficients a 1 and a 2 (or K and C)? This question was answered by Antohe et al. (1997). From the point of view of uncertainty analysis, the variables T i and ˙q i can be considered independent. Hence, using the Kline–McClintock method, eq. (12.15), the uncertainties can be estimated from U a1 = n i=1 U ˙q i ∂a 1 ∂ ˙q i 2 + n i=1 U T i ∂a 1 ∂T i 2 1/2 (12.30) U a2 = n i=1 U ˙q i ∂a 2 ∂ ˙q i 2 + n i=1 U T i ∂a 2 ∂T i 2 1/2 (12.31) where U ˙q and U T are, respectively, the uncertainties of each experimental data pair ˙q i and T i . This methodology for estimating the uncertainty of coefficients obtained from curve-fitting experimental data is general. The fundamental step is to utilize, in the Kline–McClintock model, the curve-fitting equations (e.g., least squares) to compute the local derivatives of the coefficients being considered. In most cases, it is not known if a certain curve-fit equation is valid or not. Hence, it is very instructive to question the validation of the curve-fitting equation. A very useful procedure for validating (or not) a certain model (or curve-fit equation) was proposed by Davis et al. (1992). The first step in the validation procedure is to calculate the residual δ i , defined as the deviation of the analytical value f(˙q i ) (or the curve-fit value) from the corresponding experimental value T i ; that is, δ i ( ˙q i ) = T i − f(˙q i ) (12.32) A parameter called semivariance, defined as ϕ(∆ ˙q) = 1 2N(∆ ˙q) n(∆ ˙q) j=1 δ j ( ˙q j + ∆ ˙q) − δ j ( ˙q j ) 2 (12.33) can be used to estimate the appropriateness of the curve-fitting equation. The semi- variance ϕ is a function of the lag distance ∆ ˙q between two experimental points and of the number of experimental points (or pair of points) N that are ∆ ˙q units apart. If the resulting semivariance tends to a finite value as the lag distance ∆ ˙q increases, BOOKCOMP, Inc. — John Wiley & Sons / Page 931 / 2nd Proofs / Heat Transfer Handbook / Bejan EQUIPMENT 931 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 [931], (19) Lines: 558 to 593 ——— 0.99008pt PgVar ——— Normal Page PgEnds: T E X [931], (19) the residuals do not correlate and the model (or curve fit) is valid. Otherwise, the residuals correlate and the theoretical model (or curve fit) is invalid. This procedure was tested successfully in a fluid mechanics application by Lage et al. (1997). 12.5 EQUIPMENT 12.5.1 Glass Thermometers The oldest and most common human-made temperature-measuring device is the glass (fluid) thermometer. The basic principle of operation is the volumetric change of the working fluid (e.g., air, water, mercury) with temperature, commonly represented by the coefficient of volumetric thermal expansion β, defined as β = 1 V dV dT (12.34) For compressible fluids, β must be measured at a constant pressure. The coef- ficient of volumetric thermal expansion varies with temperature, primarily for liq- uids and solids. Gases present similar volumetric coefficient of thermal expansion at atmospheric pressure. Some average values, from −20 to 100°C, are presented in Table 12.5. Glass thermometers are commercially available for measuring temperatures from −80°C to approximately 600°C. They are designed with a bulb where most of the working fluid is stored, connected to a capillary to accommodate the increase in fluid volume. Mercury-filled thermometers, also called clinical thermometers, are the most com- mon glass thermometers, available in several sizes and designed for various ranges of operation. Mercury is advantageous as a glass thermometer working fluid, as it has a coefficient of volumetric thermal expansion almost independent of temperature. From −10 to 330°C, for instance, the coefficient of volumetric thermal expansion of mercury is almost constant and equal to 1.8 × 10 −4 K −1 . Mercury thermometers are suitable for measuring temperatures from −40 to 550°C. TABLE 12.5 Average Value of Coefficient of Volumetric Thermal Expansion β for Several Gases from −20 to 100°C Gas β (×10 −3 K −1 ) Air 3.6 NH 3 3.8 CO 3.7 CO 2 3.7 H 2 3.7 NO 3.7 BOOKCOMP, Inc. — John Wiley & Sons / Page 932 / 2nd Proofs / Heat Transfer Handbook / Bejan 932 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 [932], (20) Lines: 593 to 610 ——— 0.062pt PgVar ——— Normal Page PgEnds: T E X [932], (20) Assuming β as a value independent of the temperature, eq. (12.34) can be inte- grated to yield V 2 = V 1 e β(T 2 −T 1 ) (12.35) with T 1 and T 2 being two distinct temperatures at which the fluid volume is V 1 and V 2 , respectively. Because β is generally a very small number, the exponential function of eq. (12.35) is very weak. In fact, the volume increment versus temperature variation for a mercury thermometer is almost linear, as can be seen in Fig. 12.6. This aspect facilitates calibration of the thermometer. In the case of mercury, a variation of 1.0°C in temperature causes the volume to change by 0.018%. Hence, a volume of 8 mm 3 of mercury will expand to 8.00144 mm 3 . When this expansion is accommodated by a 0.1-mm-diameter capillary, the temperature change will cause the mercury column in the capillary to rise by 0.18 mm. Observe that the sensitivity of this thermometer is 0.18 mm/°C. Increasing (or decreasing) the initial volume of mercury in the thermometer and decreasing (or increasing) the diameter of the capillary increases (or decreases) the sensitivity of the thermometer. For instance, using an initial volume equal to 10 mm 3 of mercury Figure 12.6 Volume variation versus temperature change for mercury. The straight line indicates the result of the least squares curve fitting the points of the graph with the linear function (V 2 /V 1 ) − 1 =−9.4885 × 10 −4 − 1.9005 × 10 −4 (T 2 − T 1 ). BOOKCOMP, Inc. — John Wiley & Sons / Page 933 / 2nd Proofs / Heat Transfer Handbook / Bejan EQUIPMENT 933 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 [933], (21) Lines: 610 to 646 ——— 0.85413pt PgVar ——— Normal Page * PgEnds: Eject [933], (21) and a capillary with diameter 0.05 mm leads to a thermometer with sensitivity equal to 0.92 mm/°C. 12.5.2 Thermocouples Thermocouples are the most important components of thermoelectric systems func- tioning as electric thermometers. These thermometers operate based on the natural generation of electric potential through distinct but homogeneous electrical conduc- tors with terminals exposed to different temperatures—a thermoelectric phenomenon. The fundamental principle behind the operation of thermocouples is the cross- coupling between fluxes (heat and electricity) and driving forces (temperature and electric potential). A sound theoretical foundation for the study and design of thermo- electric systems, based on irreversible thermodynamics, is available in Bejan (1988). In this section, a brief review of the most important concepts is offered. Two fundamental empirical effects are important for understanding thermal– electric interactions: the Peltier effect (named after the French physicist Jean Peltier, who discovered the phenomenon in 1834) and Seebeck effect (discovered by the Ger- man physicist Thomas Seebeck in 1826) which is the basis for the thermocouple operation. The Peltier effect, observed experimentally, is the natural generation of a heat power flux of ˙q (in Watts per square meter) along a conductor by a finite electric potential gradient, even when the conductor is maintained at a uniform temperature (which brings to light one of the limitations of Fourier’s law). A simple mathematical representation of this statement is ˙q =−α 1 1 T dE dx dT /dx=0 (12.36) where α 1 is a proportionality parameter, T the temperature of the conductor (in Kelvin), E the electrostatic potential (in volts), and x is the coordinate measured along the electric current direction (in meters). It is known that a potential gradient along a conductor generates an electric current flux I (in amperes per meter-square) through the conductor. Therefore, I =−α 2 1 T dE dx dT /dx=0 (12.37) where α 2 is another proportionality parameter. By combining eqs. (12.36) and (12.37), the Peltier coefficient π(T ), measured in volts, can be defined as π(T ) = ˙q I = α 1 α 2 (12.38) Equation (12.38) can be rewritten by introducing ε s (T ), the absolute thermoelec- tric power of the conductor (in volts per Kelvin), also known as the absolute Seebeck coefficient, to obtain BOOKCOMP, Inc. — John Wiley & Sons / Page 934 / 2nd Proofs / Heat Transfer Handbook / Bejan 934 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 [934], (22) Lines: 646 to 680 ——— 0.427pt PgVar ——— Normal Page PgEnds: T E X [934], (22) ˙q = π(T )I = T ε s (T )I (12.39) Equation (12.39) shows the existing relationship between electric current flux and generated heat power flux along an isothermal conductor more directly. This equation also presents the relationship between the Peltier π(T ) and Seebeck ε s (T ) coefficients. The phenomenon responsible for the Peltier effect can be understood more clearly when considering two dissimilar thermoelectric conductors connected and main- tained at the same (uniform) temperature, as shown in Fig. 12.7. When a finite electric potential gradient is imposed along the conductors, a certain finite amount of electric current flux I can be transported through the conductors and across the junction. The discontinuity in the absolute thermoelectric power ε s (T ) across the junction yields a net amount of heat released (or absorbed) at the junction, as shown in Fig. 12.7. This heat power flux, q c , generated (or depleted) by the electric current passing through the junction, is the difference between the power flux generated along conductor 1, ˙q 1 , and the power flux generated along conductor 2, ˙q 2 . Expressions for q 1 and q 2 can be written using eq. (12.39): ˙q 1 = π 1 (T c )I = T c ε s1 (T c )I (12.40) ˙q 2 = π 2 (T c )I = T c ε s2 (T c )I (12.41) where T c is the junction temperature and ε s1 (T ) and ε s2 (T ) are the absolute thermo- electric power of each conductor. Assuming ˙q 1 greater than ˙q 2 (which is equivalent to saying that conductor 1 is more sensitive than conductor 2 to the generation of heat power flux when transport- ing electric current, and subtracting eq. (12.41) from eq. (12.40) yields ˙q c =˙q 1 −˙q 2 = T c [ ε s1 (T c ) − ε s2 (T c ) ] I (12.42) Figure 12.7 Peltier effect. It is necessary to remove (or add) a certain amount of heat at the junction to maintain two dissimilar thermoelectric conductors at the same uniform temperature when passing a current through them. The ratio of heat power flux ˙q to electric current flux I is dependent on the temperature T c at which the conductors and the connection are maintained. BOOKCOMP, Inc. — John Wiley & Sons / Page 935 / 2nd Proofs / Heat Transfer Handbook / Bejan EQUIPMENT 935 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 [935], (23) Lines: 680 to 722 ——— 1.55019pt PgVar ——— Normal Page * PgEnds: Eject [935], (23) Observe that the configuration shown in Fig. 12.7 is suitable for evaluating the junction temperature T c by measuring the heat power flux q c and electric current flux I when the Seebeck coefficient of the two conductors ε s1 (T ) and ε s2 (T ) are known. Unfortunately, the measuring of the heat power flux q c is very difficult in practice. An alternative for measuring the junction temperature T c is based on the Seebeck effect. This effect evolves from experimental observations that suggest the existence of a finite electromotive force when a temperature gradient is imposed along a con- ductor, even though no electric current passes through the conductor. The mathemat- ical representation of the Seebeck effect is dE dx I =0 = ε s (T ) dT dx I =0 (12.43) It can be shown from eq. (12.43) that the absolute Seebeck coefficient ε s (T ) represents the variation of electrostatic potential E with temperature T when zero electric current passes through the conductor, ε s (T ) = dE dT I =0 (12.44) Observe how the Peltier and Seebeck effects are related. The Peltier effect, eq. (12.36), represents the production of heat power flux by an electric potential gradient along an isothermal conductor (zero temperature gradient). The Seebeck effect, eq. (12.43), represents the creation of an electric potential gradient by a temperature gra- dient when no current flows through the conductor. Moreover, the Peltier coefficient π(T ), eq. (12.38), is defined as the ratio of thermal flux to electric flux, while the Seebeck coefficient, eq. (12.43), is the ratio of electric potential gradient to thermal potential gradient. Consider now the sketch shown in Fig. 12.8. Using eq. (12.40), the difference in electric potential along conductor 1 will be E c − E 1 = T c T 0 ε s1 (T ) dT (12.45) and along conductor 2, E c − E 2 = T c T 0 ε s2 (T ) dT (12.46) Subtraction of eq. (12.45) from eq. (12.46) provides the electric potential differ- ence [electromotive force (EMF)] ∆E = E 2 −E 1 generated across two homogeneous conductors connected at one extremity maintained at temperature T c , with the other extremities maintained at temperature T 0 ,as ∆E = E 2 − E 1 = T c T 0 [ ε s1 (T ) −ε s2 (T ) ] dT (12.47) . ˙q units apart. If the resulting semivariance tends to a finite value as the lag distance ∆ ˙q increases, BOOKCOMP, Inc. — John Wiley & Sons / Page 931 / 2nd Proofs / Heat Transfer Handbook. McClintock (1953) method. BOOKCOMP, Inc. — John Wiley & Sons / Page 927 / 2nd Proofs / Heat Transfer Handbook / Bejan CURVE FITTING 927 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 [927],. ,a p , ˙q) 2 (12.18) BOOKCOMP, Inc. — John Wiley & Sons / Page 928 / 2nd Proofs / Heat Transfer Handbook / Bejan 928 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 [928],