BOOKCOMP, Inc. — John Wiley & Sons / Page 916 / 2nd Proofs / Heat Transfer Handbook / Bejan 916 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 [916], (4) Lines: 113 to 140 ——— -0.903pt PgVar ——— Normal Page PgEnds: T E X [916], (4) 0 5 10 15 20 10 15 20 25 30 35 40 ⌬E (mV) T (°C) Figure 12.1 Voltage ∆E versus temperature T measurements from a thermocouple system exhibiting hysteresis. When measuring an increase in T (continuous line), the sensitivity of the instrument is uniform and equal to 1.5 mV/°C. When measuring a decrease in T (dashed line), the sensitivity is nonuniform and equal to 1.5(2.33 − 0.133T + 0.033T 2 ) mV/°C. 12.1.3 Calibration All measuring instruments must be calibrated for determining their accuracy a. The accuracy quantifies an instrument’s ability to measure a known value or standard. Standards define units of measurement and are essential in building measuring scales. Standards for measuring temperature and heat flow using different scales are main- tained in the United States by the National Institute of Standards and Technology (NIST). The International Temperature Scale of 1990 (ITS-90) is defined with several fixed temperature points, or standards, listed in Table 12.1. This scale evolved from the pioneering work of Lord Kelvin, who in 1884 proposed an absolute temperature scale, the Kelvin scale, for measuring temperatures. The Kelvin temperature scale (with only positive values) is an absolute scale because the scale covers the entire range of possible temperatures, as established by the second law of thermodynamics. When calibrating a thermometer, for instance, the experimentalist compares the measured temperature value at a certain state to the standard temperature value set BOOKCOMP, Inc. — John Wiley & Sons / Page 917 / 2nd Proofs / Heat Transfer Handbook / Bejan FUNDAMENTALS 917 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 [917], (5) Lines: 140 to 155 ——— -0.93994pt PgVar ——— Normal Page PgEnds: T E X [917], (5) TABLE 12.1 Some Fixed Temperature Points Used as Standards for Defining the ITS-90 State T(K) Triple equilibrium state of hydrogen 13.8033 Triple equilibrium state of neon 24.5561 Triple equilibrium state of oxygen 54.3584 Triple equilibrium state of argon 83.8058 Triple equilibrium state of water 273.16 Solid–liquid equilibrium of galium at 1 atm 302.9146 Solid–liquid equilibrium of tin at 1 atm 505.078 Solid–liquid equilibrium of zinc at 1 atm 692.677 Solid–liquid equilibrium of silver at 1 atm 1234.93 Solid–liquid equilibrium of gold at 1 atm 1337.33 Solid–liquid equilibrium of copper at 1 atm 1357.77 Source: Holman (2001). forth by the ITS-90 for that particular state. When the discrepancy (or accuracy)is uniform along the entire scale, the instrument can be calibrated simply by moving the origin of the scale so that the new values measured by the instrument agree with the standard values. Often, standards provide a physical meaning to the values measured by the scale as is the case of the temperature measured with the Kelvin absolute temperature scale. Other temperature scales commonly used are the Celsius (°C) and Fahrenheit (°F). Neither is an absolute scale, as they both allow for negative temperature values. Al- though having different standards, a thermocouple system calibrated against the Cel- sius scale will have the same sensitivity as a thermocouple system calibrated against the Kelvin scale. That is, the thermocouple voltage variation per degree Celsius is the same as the voltage variation per degree Kelvin. Therefore, the Kelvin and Celsius scales are related for yielding the same sensitivity (the same variation per degree). The absolute Rankine scale (°R) provides the same sensitivity as the Fahrenheit scale. The conversions from one temperature scale to another can be performed using K = 273.15 + °C (12.1) °R = 459.67 + °F (12.2) °F = 32 + 9 5 °C (12.3) 12.1.4 Readability A measuring scale is based on standard values chosen to represent the measurement at certain states. It is possible to divide the space between standard values, mark- ing equal increments along the instrument read out to obtain a better estimate of measurements for a state different from the standard state. The number of divisions in BOOKCOMP, Inc. — John Wiley & Sons / Page 918 / 2nd Proofs / Heat Transfer Handbook / Bejan 918 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 [918], (6) Lines: 155 to 168 ——— 5.7pt PgVar ——— Normal Page PgEnds: T E X [918], (6) the readout of an instrument determines the readability of the instrument. For exam- ple, if a thermometer using the Kelvin scale is build with 50 equal increments between the triple equilibrium state of oxygen (54.3584 K) and the triple equilibrium state of water (273.16 K), each increment of this thermometer (referred to as the least count L c of the thermometer) would be L c = (273.16 K − 54.3584 K)/50 = 4.3726 K. If instead of 50, 150 equal increments were employed, the least count of the thermome- ter would be L c = 1.4587 K. The readability of an instrument, or the least count, determines how close the real value can actually be read from the instrument scale. For instance, in an attempt to measure the temperature T at the triple equilibrium state of argon (83.8058 K) using a thermometer calibrated for the triple equilibrium state of oxygen (54.3584 K) and with least count equal to 4.3726 K, the thermometer would indicate a value between the sixth (80.594 K) and seventh (84.967 K) marks on the scale. The measured value in this case could be any value between 80.594 and 84.976 K. The certainty of the measured value would be assured only to within the least count of the instrument, in this case 4.3726 K. One common representation for the measured value in this case is based on the arithmetic average of the two bounding values and half the least count, i.e., T ={ [ (80.594 K + 84.976 K)/2 ] ± (4.3726 K/2)}. The resulting value, T = 82.785 ± 2.186 K, covers the entire range from 80.594 to 84.976 K. However, the use of the same thermometer, but now with least count equal to 1.4587 K, the same measurement would fall between the twentieth (83.532 K) and twenty-first (84.991 K) marks, and the certainty would be to within 1.4587 K. The measured value in this case could be T = 84.261 ± 0.73 K, which is closer to the precise value 83.8058 K than the value obtained by the thermometer with the least count 4.3726 K. Even without knowing the precise value being measured, it can be stated that the certainty of the second reading (±0.73 K) is much better than the certainty of the first reading (±2.186 K). This conclusion is a direct consequence of the least count being smaller in the second thermometer than in the first thermometer. When the state at which a value is being measured coincides with a standard state, the error of the instrument can always be checked by comparing the measurement with the standard value. However, the standards are values set at some prescribed (discrete) states. If the state at which a value is to be measured does not coincide with any of the standard states, the experimentalist cannot be certain of the accuracy of the measurement, even if the instrument has been calibrated. 12.2 MEASUREMENT ERROR 12.2.1 Uncertainty: Bias and Precision Errors In the example described in Section 12.1, the error between the measured and real values could not be quantified precisely because the measured value was not a stan- dard value. However, the error between measured and exact values could be estimated to be no more than one-half the least count of the instrument. Although uncertain, the measurement error in this case can be estimated or determined (deterministic as BOOKCOMP, Inc. — John Wiley & Sons / Page 919 / 2nd Proofs / Heat Transfer Handbook / Bejan MEASUREMENT ERROR 919 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 [919], (7) Lines: 168 to 188 ——— 0.0pt PgVar ——— Normal Page PgEnds: T E X [919], (7) opposed to random). Observe that this error is independent of the number of times the measurement is taken, or equivalently, independent of the sample size. This error (related to the least count) is then said to be a systematic error, contributing to the bias error B of the measurement. The bias error remains constant if the measurement is performed under the same environmental conditions (under which condition, for instance, the least count of the measuring device would not change). All errors that are known to exist (deterministic) but that have not been eliminated (such as accuracy, hysteresis, and linearity) should be included in the bias error of the measurement. The bias error is one of the components of the overall measurement error, called the uncertainty U of the measurement. The uncertainty of a measurement is affected not only by the bias error, but also by the precision σ p of the instrument. In general, the uncertainty of a measurement can be obtained from the bias and precision errors (Wheeler and Ganji, 1996) as U = B 2 + σ 2 p 1/2 (12.4) The precision σ p of an instrument is a measure of the repeatability of the mea- surement. Most instruments, especially when used in the field, are affected by un- controllable effects such as ambient temperature, humidity, and pressure. Even when trying to calibrate an instrument by measuring a certain standard, the response of the instrument might not be the same when the measurement is repeated several times. Assume, for example, that it is required to calibrate a mercury thermometer using the triple point of water. The thermometer is placed in contact with a mixture of solid (ice), liquid, and gas (vapor) water in thermal equilibrium. Once the thermometer reaches thermal equilibrium with the mixture, the location of the mercury meniscus in the thermometer is marked. In repeated performance of the same sequence of events with the same thermometer, the chances are that the location of mercury meniscus at equilibrium would not be the same as the location of the first measurement. The discrepancy gives a measure of the precision of the thermometer. The precision of an instrument can be accessed by performing repeated measure- ments of the same state (not necessarily a standard state). Hence, to access the pre- cision of an instrument, multiple measurements of the same state must be taken (one single measurement will never allow for the determination of the precision of the measuring instrument). Keep in mind that precision and accuracy are very different concepts. The accu- racy of a measurement is a deterministic error that can be eliminated by calibrating the instrument, hence it affects the bias error. The precision of the instrument, intrinsic to the instrument itself, is affected in most cases by the environment. This precision error cannot be eliminated unless the instrument is rebuilt. [Observe that even if the instrument is used in the same controlled environment (temperature, pressure, hu- midity, etc.), the deterioration of the instrument in time will affect the precision.] Precision is a random error contributing to the uncertainty of the measurement. Although impossible to predict precisely, the precision of an instrument can often be estimated with reasonable confidence via statistical analysis. BOOKCOMP, Inc. — John Wiley & Sons / Page 920 / 2nd Proofs / Heat Transfer Handbook / Bejan 920 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 [920], (8) Lines: 188 to 233 ——— -1.94084pt PgVar ——— Short Page PgEnds: T E X [920], (8) Finally, it is important to point out that the readability error can influence not only the bias error (because of the least count) but also the precision error of the instrument (because it can change the resolution, or least count). For instance, when the environmental conditions change, the least count of the instrument might change as well due to the expansion–contraction of the scale. This contribution to the error is random, or nondeterministic. Therefore, this readability error adds to the precision error of the instrument. 12.2.2 Mean and Deviation When n measurements T i are taken of the same quantity, a single representative measured quantity can be found by calculating the arithmetic mean T of all values, T = 1 n n i=1 T i (12.5) The deviation d i from the mean T of each measurement T i is defined as d i = T i − T (12.6) The standard deviation σ can be used to quantify the spreading of the measured values with respect to the mean: σ = 1 n n i=1 (T i − T) 2 1/2 (12.7) The standard deviation, eq. (12.7), is representative of the precision σ p of the instru- ment when the number of measurements is large (n>20). Otherwise, σ should be multiplied by [ n/(n − 1) ] 1/2 . Notice that the standard deviation σ has the dimension(s) of the measured quan- tity T . Moreover, the standard deviation of the mean value T , namely, σ T , can be estimated from σ T = σ n 1/2 (12.8) where n is larger than 10. The standard deviation of a set of measurements is an important parameter to represent the probability of a certain measurement to fall within a certain interval. Although similar, this concept differs from the readability concept discussed earlier. Recall that every measurement has a certain readability error associated with it and this error is proportional to the least count of the measuring device. The precision of the instrument (or measurement) is an additional uncertainty, as it relates to a series of measurements taken of the same state (or quantity). BOOKCOMP, Inc. — John Wiley & Sons / Page 921 / 2nd Proofs / Heat Transfer Handbook / Bejan MEASUREMENT ERROR 921 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 [921], (9) Lines: 233 to 252 ——— 0.68909pt PgVar ——— Short Page PgEnds: T E X [921], (9) 12.2.3 Error Distribution When several measurements of a quantity T are taken, and all of them are affected by small random errors equally likely to be positive or negative, the probability Γ G (T ) of one of the measurements falling within T and T +dT is given by the Gaussian or normal error distribution (also known as random distribution), Γ G (T ) = 1 σ(2π) 1/2 e −(T −T) 2 /2σ 2 (12.9) Figure 12.2 presents some normalized Gaussian distribution results ∞ −∞ Γ G (T ) dT = 1.0 from simulated temperature measurements. As expected, the most probable measurement is the one coinciding with the mean (average) value T . Moreover, when the standard deviation σ is large, the spreading of the measurements is large as well. Hence, the probability of any measurement falling far from T increases with σ, as predicted by Γ G (T ). Observe from eq. (12.9) that the probability of a measurement yielding the average value T equals 1/ σ(2π) 1/2 . In the cases presented in Fig. (12.2), the probabilities are Γ G (T)= 0.997 when σ = 0.4 and Γ G (T)= 0.665 when σ = 0.6. Evidently, Γ G (T)will be greater than unity when σ is smaller than (2π) −1/2 ,or equivalently, when σ < 0.399. Because of the relation with σ, the value of Γ G (T)is also a measure of the precision σ p of the instrument. 20 21 22 23 24 0 0.2 0.4 0.6 0.8 1 ⌫ G ()T T = 0.4 = 0.6 Figure 12.2 Simulated Gaussian distribution of temperature with standard deviation 0.4 and 0.6, and average temperature T = 22°C. BOOKCOMP, Inc. — John Wiley & Sons / Page 922 / 2nd Proofs / Heat Transfer Handbook / Bejan 922 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 [922], (10) Lines: 252 to 284 ——— 0.01611pt PgVar ——— Short Page PgEnds: T E X [922], (10) A more interesting and practical quantity is the probability that a certain measure- ment will fall within a certain distance (deviation) ∆T from the mean value T . This deviation probability is given by Γ d (∆ζ): Γ d (∆ζ) = 2 π 1/2 ∆ζ 0 e −ζ 2 /2 dζ (12.10) where ζ = (T − T)/σ and ∆ζ = ∆T/σ. It can be shown that the equation for Γ d (∆ζ) can be written in terms of the error function: Γ d (∆ζ) = erf ∆ζ 2 1/2 (12.11) where erf(x) = 2 π 1/2 x 0 e −m 2 dm (12.12) Results from eq. (12.11) are presented in Fig. 12.3. In practice, when trying to find the probability of a measurement falling within a distance ∆T from the mean value, it is only necessary to calculate ∆ζ = ∆T/σ and then find the corresponding Γ d (∆ζ) value in Fig. 12.3. Figure 12.3 Graphical representation of eq. (12.11). BOOKCOMP, Inc. — John Wiley & Sons / Page 923 / 2nd Proofs / Heat Transfer Handbook / Bejan MEASUREMENT ERROR 923 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 [923], (11) Lines: 284 to 322 ——— 1.07602pt PgVar ——— Short Page PgEnds: T E X [923], (11) Consider, for instance, the case of a series of temperature measurements yielding a standard deviation σ = 0.4. The probability of a measurement falling within ∆T = 0.4 unit from the mean value is approximately 0.68. This value is the Γ d (∆ζ) value, obtained from Fig. 12.3, for ∆ζ = ∆T/σ = 1.0. It is worth noting from Fig. 12.3 that the probability that a measurement is within 3σ units (or greater) of the mean value is precisely Γ d (3) = 0.997, or approximately unity. Recall that these results are valid when the distribution of measurements follows a Gaussian distribution, which is generally valid when several measurements of the same quantity are taken. Another interesting aspect related to Γ d (∆ζ) is now considered. If it can be con- cluded from a series of measurements that the probability of any measurement falling within a ∆T interval from the mean is given by Γ d (∆T/σ), it can be stated that this same probability is the probability of the mean value falling within the same interval. When related to the mean, the probability Γ d (∆ζ) is called the confidence level, and ∆ζ is called the confidence interval. In the example described in the preceding para- graph, the measurements present a confidence level of 68% for a confidence interval of 1.0—that is, the probability that the mean value T would fall within an interval of ±1σ units is 68%! Multiple measurements of the same quantity are very often written is terms of the mean value and the standard deviation representing the precision error, such as T ±σ. According to the previous assessment, this representation of the mean value (using a confidence interval equal to unity) has a confidence level of 68% if the probability of the measurements follows a Gaussian distribution. The number of measurements necessary to achieve a certain confidence level can be determined from n = σ p ∆ζ σ 2 (12.13) As an example, consider the measurement of temperature with a device having preci- sion σ p =±0.4°C. If the interest is in achieving a confidence interval ∆ζ = ∆T/σ = 2 (equivalent to a confidence level of 95.4%; see Fig. 12.3) for a standard deviation of ±0.2°C, at least 16 measurements would be required. 12.2.4 Chauvenet’s Criterion and the Chi-Square Test Frequently, repeated measurements of the same quantity might yield results that are far off the mean value. A criterion commonly used to identify unacceptable measurement values is Chauvenet’s criterion. This criterion is based on the fact that the probability of obtaining a certain deviation from the mean by any measurement should not be smaller than 1/(2n), where n is the total number of measurements. This probability is equivalent to [ 1 − Γ d (∆T/σ) ] . Therefore, Fig. 12.3 can be employed to determine the maximum deviation ∆T = σ∆ζ for any number of readings. Table 12.2 presents the corresponding maximum ∆ζ acceptable for several numbers of measurements. BOOKCOMP, Inc. — John Wiley & Sons / Page 924 / 2nd Proofs / Heat Transfer Handbook / Bejan 924 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 [924], (12) Lines: 322 to 360 ——— -1.23857pt PgVar ——— Normal Page PgEnds: T E X [924], (12) TABLE 12.2 Values of Maximum Deviation ∆T max from the Mean for Application of Chauvenet’s Criterion n a ∆T max /σ n a ∆T max /σ 2 1.28 9 1.91 3 1.38 10 1.96 4 1.54 20 2.24 5 1.65 40 2.52 6 1.73 60 2.64 7 1.80 80 2.73 8 1.85 100 2.82 a Number of measurements from which the standard deviation σ is calculated. Notice that the criterion is applied only once to a set of measurements data, by comparing the deviation from the mean of each measurement with the maximum deviation ∆T obtained from Table 12.2 once the standard deviation σ of the data set is calculated. The application of Chauvenet’s criterion on a data set allows not only consistent discard of questionable data, but also recalculation of the standard deviation from the remaining data. This new standard deviation is believed to be a more accurate representation of the average deviation of the experimental data from the mean value. It is important to verify how close a certain data set follows a certain distribution before relying too heavily on statistical analysis. A common test for evaluating the closeness of the data to a certain distribution is the chi-square test. This test is based on χ 2 , defined as the relative deviation of the value observed from the value expected in a set of n measurements. If the value in question is the probability of a certain measurement, χ 2 = n v i=1 (Γ i − Γ Gi ) 2 Γ Gi (12.14) where n v is the number of different values measured, Γ i the observed probability of a certain measurement, and Γ Gi the expected Gaussian distribution for the particular measurement. To illustrate the use of the chi-square test, consider the temperature data set shown in Table 12.3, where n v = 11. The mean temperature of the data set is T = 23°C and the standard deviation is σ = 0.724°C. To verify if the observed probability distribution of the data follows a Gaussian distribution, an estimate Γ of each measurement is made by dividing the number of times the measurement is observed in the data set (number of events) by the total number of measurements (equal to 122). To normalize the results for comparison with the Gaussian distribution, the probability was divided by the increment in tem- perature, 0.2°C, and is presented in Table 12.3 as Γ. If this data set follows a Gaussian distribution, the probability of each measurement must be given by eq. (12.9), with BOOKCOMP, Inc. — John Wiley & Sons / Page 925 / 2nd Proofs / Heat Transfer Handbook / Bejan MEASUREMENT ERROR 925 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 [925], (13) Lines: 360 to 397 ——— -1.71884pt PgVar ——— Normal Page PgEnds: T E X [925], (13) TABLE 12.3 Data Set of Measured Temperature T with Number of Events for Each Measurement, Observed Probability Γ, and Gaussian Probability Γ G a T(°C) Events Γ (×10 −1 ) Γ G (×10 −1 ) (Γ − Γ G ) 2 Γ G 22.0 19 7.79 2.10 1.5500 22.2 12 4.92 2.96 0.1290 22.4 6 2.46 3.88 0.0520 22.6 8 3.28 4.71 0.0434 22.8 12 4.92 5.29 0.0026 23.0 13 5.33 5.51 0.0006 23.2 6 2.46 5.32 0.1540 23.4 9 3.69 4.75 0.0239 23.6 5 2.05 3.94 0.0906 23.8 6 2.46 3.02 0.0105 24.0 26 10.70 2.15 3.3700 a The final column shows values to be used in calculating χ 2 from eq. (12.14). results shown in Table 12.3. Using both observed and predicted probability, the value of χ 2 using eq. (12.14), is obtained as χ 2 = 5.42. The chi-square probability Γ χ 2 , indicating how well the distribution observed follows the Gaussian distribution, is obtained from Table 12.4 once the number of degrees of freedom N is found. Observe that N = n v −p, where p is the number of imposed conditions in the data set. From Table 12.3 it is found that the number of conditions imposed equals unity (the only condition imposed is that the total number of measurements be equal to 122), hence N = 10. From Table 12.4, Γ χ 2 = 0.856 is then obtained. As a rule of TABLE 12.4 Measurement Data Obtained with Various Degrees of Freedom N a Γ χ 2 N 0.99 0.9 0.5 0.1 0.01 1 0.000157 0.0158 0.455 2.71 6.63 2 0.0201 0.211 1.39 4.61 9.21 3 0.115 0.584 2.37 6.25 11.3 4 0.297 1.06 3.36 7.78 13.3 5 0.554 1.61 4.35 9.24 15.1 6 0.872 2.20 5.35 10.6 16.8 7 1.24 2.83 6.35 12.0 18.5 8 1.65 3.49 7.34 13.4 20.1 9 2.09 4.17 8.34 14.7 21.7 10 2.56 4.87 9.34 16.0 23.2 Source: Holman (2001). a Presents the natural probability Γ χ 2 of chi-square being higher than the value calculated for χ 2 . . BOOKCOMP, Inc. — John Wiley & Sons / Page 916 / 2nd Proofs / Heat Transfer Handbook / Bejan 916 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 [916],. standard temperature value set BOOKCOMP, Inc. — John Wiley & Sons / Page 917 / 2nd Proofs / Heat Transfer Handbook / Bejan FUNDAMENTALS 917 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 [917],. number of divisions in BOOKCOMP, Inc. — John Wiley & Sons / Page 918 / 2nd Proofs / Heat Transfer Handbook / Bejan 918 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 [918],