1、Liping Liu is an assistant professor in A. Leon Linton Department of Mechanical Engineering, Lawrence Technological University, Southfield, MI. Young-Gil Park is an assistant professor in the Department of Mechanical Engineering, University of Texas Rio Grande Valley, Edinburg, TX. Anthony M. Jacobi
2、 is a professor in the Department of Mechanical Science and Engineering, University of Illinois, Urbana, IL. Minimizing Data Reduction Uncertainty during Heat-Transfer Equipment Testing Liping Liu, PhD Young-Gil Park, PhD Anthony M. Jacobi, PhD ABSTRACT It is a widely adopted practiceeven adopted in
3、 ASHRAE/ANSI/ARI engineering standardsto use the arithmetic mean of two heat transfer measurements for the evaluation of heat-exchanger performance. However, this approach does not generally lead to a minimized experimental uncertainty because uncertainties of redundant measurements can vary conside
4、rably depending on the experimental techniques and the test conditions. Moreover, based on this approach it is preferred to discard information when its uncertainty exceeds some limit. It is proposed in this paper that Qaveshould be calculated based on a form of weighted-linear average, with weighti
5、ng factors depending on the individual uncertainties in Qhand Qc. Heat-transfer rate which has larger uncertainty will be weighed less in the average, and the other one with smaller uncertainty will be weighed more accordingly. Implementing this new methodology will minimize the uncertainty in heat-
6、transfer coefficient and Colburn j factors, which will consequently provide more accurate data for use in the development of correlations or for performance comparison purposes. Through analysis of experimental data with different uncertainties, the benefit of weighted average method was demonstrate
7、d. The results showed that the weighted averaging method recuded the average relative uncertainty in j factors from 11% to 10.3% for dry condition data, and from 21.8% to 13.1% for wet condition data. The benefit was more pronounced as the air-side Reynolds number increased. Because the air-side unc
8、ertainty is usually much higher under wet operating conditions, the weighted average method is highly recommended for data reduction with dehumidifying conditions. INTRODUCTION The accuracy of experimental results has always concerned engineers and scientists. The uncertainty of each parameter is de
9、sired to be minimized because these uncertainties will propagate in the data reduction process. In heat-transfer equipment testing, there are usually two independent measurements of heat-transfer rate in the hot and cold stream respectively (Qh and Qc). It is well accepted that an arithmetic mean nu
10、llnullnullnullnullnullnullnullnullnullnullnull/2 should be employed to acquire the average heat-transfer rate during the data reduction (Wang, et al., 2000; Pirompugd, et al., 2006; Azar, et al., 2014; Longo, et al., 2000; Ray, et al., 2014; Yang, et al., 2014). This is a widely adopted practice, in
11、cluding ASHRAE/ANSI/ARI engineering standards (ANSI/ASME PTC 19.1, 2006; ANSI/ASHRAE Standard 33, 2000; ARI Standard 410, 2001). However, because uncertainties of Qh and Qc can vary considerably depending on the experimental techniques and the test conditions, these prevalent practices do not always
12、 lead to reduced experimental uncertainty. Sometimes, when the uncertainty of one measurement is much larger than the other, using the data with smaller Associate Member ASHRAE Associate Member ASHRAE Fellow ASHRAEuncertainty and discarding the other one can be better than using the arithmetic mean.
13、 Still, discarding a measurement may not be the best decision no matter how large its uncertainty is. It is more sensible to use a weighted averaging method as proposed by (Park, et al., 2010). The Qave combined from two independent measurements should be calculated based on a form of weighted-linea
14、r average, with weighting factors depending on the individual uncertainties in Qh and Qc. Heat-transfer rate which has higher uncertainty will be weighed less in the average, and the other one with lower uncertainty will be weighed more heavily. The weighted-linear averaging method can be applied to
15、 two or more redundant measurements, of which the individual contributions to the average result are determined based on their uncertainties. Implementing this new methodology will minimize the uncertainty in heat-transfer coefficient and Colburn j factors, which will consequently provide more accur
16、ate data for use in the development of correlations or for performance comparison purposes. In this paper, the weighted-linear averaging method is applied to some experimental data obtained from heat-exchanger testing. The proposed method and the conventional method will be compared and conditions w
17、here the choice of method has a significant impact will be identified. The averaging method for minimal uncertainty improve the readability of data by reducing the size of error bars. Furthermore, it can also assist in identifying data trends. Through the analysis of heat-exchanger data with differe
18、nt uncertainties and energy balances, the impact of using this new approach will be thoroughly investigated. Recommendations will be made for the employment of this method, which will help enhance the veracity of heat-transfer performance evaluation. UNCERTAINTY MINIMIZATION: WEIGHTED-LINEAR AVERAGE
19、 Consider two independent measurements on heat-transfer rate from the hot and cold streams in an air-to-coolant heat exchanger. The conventional way is to use the arithmetic mean in order to combine the redundant data, as shown in Eqn. (1). nullnullnullnullnullnullnullnullnullnullnull(1)If the absol
20、ute uncertainties of Q1 and Q2 are u1 and u2, respectively, the combined uncertainty will be (Taylor nullnullnullnullnullnullnullnullnullnullnullnullnullnullnullnullnullnullnullnullnull; nullnullnullnullnullnullnullnullnullnullnullnullnullnullnullnullnullnullnullnullnull(7) The combined uncertainty
21、can also be acquired as nullnullnullnullnull nullnullnullnullnullnullnullnullnullnullnullnullnullnullnullnullnullnullnullnullnullnullnullnull(8) IMPACT ON EXPERIMENTAL RESULTSExperimental data from an air-to-coolant heat exchanger testing was analyzed using both the arithmetic mean and weighted-line
22、ar average methods. The test sample was a plain-fin round-tube heat exchanger with a fin spacing of 5.3 mm (0.2 in.). The schematic of heat exchanger is shown in Figure 3 and more details can be found in (Liu however, it can be seen that as the Reynolds number increases, the uncertainty of air-side
23、heat-transfer rate becomes larger, differing from the coolant-side uncertainty. Therefore, the advantage of using weighted-linear average method is more pronounced. In the overall range of Reynolds number tested, using the weighted method reduced the average relative uncertainties in j factors from
24、11% to 10.3%. 00.0010.0020.0030.0040.0050.0060.0070.0081500 2000 2500 3000 3500 4000arithmetic meanweighted-linear averageColburn j-factor(-)Air-Side Reynolds Number, Redh(-)Figure 4 Comparison of dry condition Colburn j-factor results from arithmetic mean and weighted-linear average methods 00.020.
25、040.060.080.10.120.140.161500 2000 2500 3000 3500 4000arithmetic meanweighted-linear averagej-RelativeUncertainty (-)Air-Side Reynolds Number, Redh(-)Figure 5 Comparison of relative uncertainties in dry j factors from arithmetic mean and weighted-linear average methods One may notice in Figure 5 tha
26、t for some cases the relative uncertainty in j factor was actually slightly higher when the weighted average method was used. It is due to the introduction of additional equations and parameters (Eqns. (3) and (4) during the error propagation analysis conducted by the Engineering Equation Solver (EE
27、S) software. However, the impact is usually very small. It should also be noted that the weighted-linear averaging method is targeted at minimizing the absolute uncertainty in Qave. Therefore, when the value of Qavewas shifted a little lower due to change of weighting factors, i.e. the smaller one f
28、rom Qhand Qcwas weighed more heavily during the averaging, it may also be a reason to cause a slightly higher relative uncertainty. When the heat exchanger is being operated under wet-surface conditions, the fin surface temperature is lower than the dew point of circulating air and therefore water c
29、ondenses on the air-side surface of the heat exchanger. Due to the existence of a latent load the heat-transfer rate on the air side needs to be calculated using enthalpy difference instead of temperature difference (replacing Eqn. (9). ,h a ai aoQmi i (12) Instead of the LMTD method, a log-mean-ent
30、halpy-difference method (LMED) was employed to solve for the air-side heat-transfer coefficient (details shown in (Liu & Jacobi, 2014). The wet j-factor results acquired through two different averaging methods are presented in Figure 6. It can be seen that the error bars from the weighted-linear ave
31、rage method is considerably smaller compared to the ones from arithmetic mean method. This is because under wet test conditions, Qhwas calculated based on enthalpy difference, which was evaluated based on dew-point measurements from air upstream and downstream. The uncertainties in dew point and ent
32、halpy are usually much higher comparing to those in temperature measurements. As a result, the weighted average method helped reduce the combined uncertainty by giving Qha smaller weighting factor and Qcwhich has lower uncertainty plays a more important role in the average. This is more clearly show
33、n in Figure 7 by showing the relative uncertainty in j factors from both methods. In the overall range of Reynolds numbers tested, using the weighted-linear averaging method reduced the average relative uncertainty in j factors from 21.8% to 13.1%. It is also obvious that as the Reynolds number incr
34、eased, the uncertainty in air-side heat-transfer rate increased rapidly. Therefore, the advantage of performing a weighted-linear average is more and more significant. 00.0020.0040.0060.0080.010.0121500 2000 2500 3000 3500 4000arithmetic meanweighted-linear averageColburn j-factor(-)Air-Side Reynold
35、s Number, Redh(-)Figure 6 Comparison of wet condition Colburn j-factor results from arithmetic mean and weighted-linear average methods 00.050.10.150.20.250.30.351500 2000 2500 3000 3500 4000arithmetic meanweighted-linear averagej-RelativeUncertainty (-)Air-Side Reynolds Number, Redh(-)Figure 7 Comp
36、arison of relative uncertainties in wet j factors from arithmetic mean and weighted-linear average methods CONCLUSIONIn this study, a weighted-linear averaging method was applied to experimental data reduction for a heat exchanger tested under both dry and wet surface conditions. The proposed method
37、 and the conventional arithmetic mean method were compared and conditions where the choice of method had a significant impact were identified. The results showed that the weighted averaging method recuded the average relative uncertainty in j factors from 11% to 10.3% for dry condition data, and fro
38、m 21.8% to 13.1% for wet condition data. The benefit was more pronounced as the air-side Reynolds number increased. It was demonstrated that when the uncertainties in heat-transfer rates from both sides are comparable, it didnt matter which method to be employed. However, as the uncertainties differ
39、ed from each side, the new method palyed an important role in minimizing the combined uncertainty. Because the air-side uncertainty is often much higher under wet operating conditions, the weighted average method is highly recommended for data reduction with dehumidifying conditions. The averaging m
40、ethod for minimal uncertainty improves the readability of data by reducing the size of error bars. Furthermore, it can also assist in identifying data trends because heat-exchanger performance data often have a strong parametric dependence (e.g. on mass flow rate). NOMENCLATURE Cp= specific heat G =
41、 mass flux at minimum flow area h = heat-transfer coefficient i = enthalpy j = Colburn j-factor m = mass flow rate Nu = Nusselt number Pr = Prandtl number Q = heat-transfer rate Re = Reynolds number T = temperature u = absolute uncertainty = weighting factor = uncertainty ratio Subscripts a = air av
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