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Part B
Convection Challenges and Energy Balances
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5
Wilson Plots and Measurement Accuracy
Jaime Sieres
University of Vigo
CONTENTS
5.1 Introduction
5.2 Wilson Plot Method
5.2.1 Estimation of HTC
5.2.2 Standard Errors of the HTC Parameters
5.2.3 Experimental Procedure and Limitations
5.3 Modified Wilson Plot Methods
5.3.1 Reynolds Exponent Limitation
5.3.2 Variable Property Effects or Fin Resistance Limitations
5.3.3 Constant Thermal Resistance Limitation
5.3.4 Experimental Procedure and Limitations
5.4 Weighted Fits
5.5 General Wilson Plot
5.6 Example Application
5.6.1 Data
5.6.2 HTC Calculation
5.6.2.1 Wilson Plot Method
5.6.2.2 Modified Wilson Plot Methods
5.6.2.3 Weighted Wilson Plot Method
5.6.2.4 General Wilson Plot Method
5.6.3 Discussion
References
5.1 Introduction
Convection is a heat transfer mechanism of utmost importance in many thermal engineering applications and equipment. The physics that describe convection heat transfer are complex, and most convection heat transfer problems are actually solved by virtue of the Newton law of cooling, which relies on the knowledge of a convection heat transfer coefficient (HTC). In order to obtain the HTC, carefully designed experiments are generally performed. Frequently, the desired outcome of this experimental work is a built-in correlation based on dimensional analysis.
The HTC is a local parameter; however, in heat exchange devices we are often interested on an average value for the HTC. According to Newton’s law of cooling, the heat transfer rate () exchanged by convection between a solid surface at temperature (Ts) and a fluid with temperature (Tf) is:
where A is the area of the surface and h is the average HTC for the entire surface.
Obtaining the HTC from the previous equation requires knowing the surface temperature (Ts). In many applications, it is not feasible to measure this temperature because the surface area is not accessible. In others, measuring this temperature might not be recommended since the use of a temperature sensor might affect the heat transfer rate or the fluid flow near the surface, which will also affect the measured value of the HTC. One alternative simple method to determine the HTC from experimental data is the Wilson plot method (Wilson 1915) or any of its subsequent modifications.
The Wilson plot method dates back from 1915 when Wilson proposed a technique to obtain HTC for turbulent flow inside a circular tube. The HTC correlation involved two unknown parameters that were determined by a linear regression analysis of the experimental data. Since then, multitude improvements and modifications of the original method have been proposed in order to improve its accuracy and applications. A literature review (Fernández-Seara et al. 2007) shows that most works are related to the application of the Wilson plot method in order to obtain HTC correlations that involve two parameters. However, multiple applications that involve three or more parameters have been studied through modifications of the original method. In these last cases, a simple linear regression analysis can no longer be used and iterative regression schemes are used to determine the HTC correlation parameters.
As the HTCs obtained from the Wilson method (or its subsequent modifications) are based on experimental data, they are sensitive to measurement errors. Then, good and accurate measured data with known uncertainties are needed for reliable correlations of HTCs.
Little attention has been given in the open literature to the effect of the uncertainty of the measured data in the determination of the HTC model parameters and their corresponding uncertainties (Khartabil and Christensen 1992, Uhía et al. 2013, Sieres and Campo 2018). As pointed out by van Rooyen et al. (2012), simple statistics such as the coefficient of determination of t...
