## 2 Materials and Methods
### Data Reduction Equations  
Data reduction details for analyzing the experimental data are provided below. The experimental data covered the wide range of Reynolds number (from 5500 to 14,500). Heat-transfer and fluid-flow characteristics were expressed as non-dimensional parameters, i.e., Nusselt number (\(Nu\)) and friction  
Figure 3: Schematic diagram of double-sided delta-winglet vortex generators. Figure 2: Schematic diagram of the enhanced heat-transfer test section with measurement points for registering temperature and pressure. factor (_f_), respectively. When the steady state condition was achieved, temperature and pressure readings were obtained. For each variation in the test condition at least 18 data points were collected. Certain properties of water were characterized according to the thermo-physical properties of water at the bulk mean temperature. The heat-transfer rate of the cold water on the outer side of the tube (\(Q_{c}\)) was given as  
\[Q_{c}=\dot{m_{c}}c_{p,c}(T_{c,\rho}-T_{c,i})=h_{\rho}A_{o}(\overline{T}_{w, \rho}-T_{b,c}). \tag{1}\]  
\(T_{w,o}\) referred to the mean outer wall temperature of the inner tube and was calculated as  
\[\overline{T}_{w,\rho}=\frac{\sum T_{w,\rho}}{10}. \tag{2}\]  
\(T_{b,c}\) referred to the bulk cold water temperature and was calculated as  
\[T_{b,c}=\frac{T_{c,i}+T_{c,\rho}}{2}. \tag{3}\]  
The heat-transfer rate of the hot water in the inner tube side (\(Q_{h}\)) was evaluated as  
\[Q_{h}=\dot{m_{h}}c_{p,h}(T_{h,i}-T_{h,\rho})=U_{i}A_{i}\Delta T_{LMTD}. \tag{4}\]  
\(\Delta T_{LMTD}\) referred to the logarithmic mean temperature difference and was calculated according to the following equation:  
\[\Delta T_{LMTD}=\frac{(T_{h,i}-T_{c,\rho})-(T_{h,\rho}-T_{c,i})}{\ln{((T_{h,i} -T_{c,\rho})/(T_{h,\rho}-T_{c,i}))}}. \tag{5}\]  
To minimize the surrounding heat loss, the difference between \(Q_{c}\) and \(Q_{h}\) was expressed as \(Q_{loss}\):  
\[Q_{loss}=Q_{h}-Q_{c}. \tag{6}\]  
A negligible \(Q_{loss}\) of 4.72% was reported. The average heat transfer between the cold and hot fluid (\(Q_{ave}\)) was estimated according to the following equation:  
\[Q_{ave}=\frac{Q_{h}+Q_{c}}{2}. \tag{7}\]  
Based on the \(Q_{ave}\) value obtained from Equation (7), the overall heat-transfer coefficient of the inner tube (\(U_{i}\)) was expressed as  
\[Q_{ave}=U_{i}A_{i}\Delta T_{LMTD}. \tag{8}\]  
Therefore, the convective heat-transfer coefficient in the inner tube (\(h_{l}\)) was calculated using the following equation:  
Figure 2:0  
The average Nusselt number (\(Nu\)) based on the \(h_{i}\) was calculated using the following equation:  
Figure 2:1  
The inner diameter of the inner tube (\(d_{i}\)) and thermo-physical properties of water at the bulk mean temperature of the inner tube (\(T_{b}\)) were used to calculate the Reynolds number:  
Figure 2:2Given the representation of fluid flow as the friction factor (_f_), the friction factor for a fully developed turbulent flow inside the tube was defined as  
Figure 2:3  
To evaluate the effect of using DWTs and LS, a comparison of the enhanced heat transfer and friction factor improvement among DWTs, LS and plain tube under identical pumping power [18,19] is also performed. This comparison is denoted as the thermal performance factor of the system (_\(\eta\)_) and is expressed as:  
Figure 2:4  
Uncertainty analyses of non-dimensional parameters was performed based on ANSI/ASME (American National Standards Institute/American Society of Mechanical Engineers) [20] for determining errors in the measurement data. The uncertainty values for the Nusselt number, friction factor, and Reynolds number were \(\pm\)7.15%, \(\pm\)5.53%, and \(\pm\)6.62%, respectively.