*2.3. Data Processing*

In the current study, primary data were collected from an experimental setup and handled using very well-known procedures, as described in earlier studies [27]. The present laboratory analysis focused on evaluation of the heat transfer enhancement and hydrodynamic effectiveness under the condition of fully developed turbulent flow. The approximate heat flux, heat transfer coefficient, average Nusselt number, friction factor, Reynolds number, and Prandtl number; are presented as follows:



$$\text{Nusselt number (Nu)}\tag{3}$$

$$\begin{array}{ll}\text{Friction factor } \langle f \rangle & \frac{\Delta P}{\left(\frac{l}{\Delta}\right) \left(\frac{\sigma^2}{2}\right)} \\\end{array} \tag{4}$$

$$\text{Reynolds Number (Re)}\tag{5}$$

$$\begin{array}{cccc} \frac{4\text{ }\dot{\text{m}}}{\pi \text{D}\_h \mu} & \end{array} \tag{5}$$

$$\text{Prandtl number } (Pr) \tag{6}$$

In this regard, *T<sup>w</sup>* = <sup>∑</sup> *<sup>T</sup>* 5 . (*T<sup>w</sup>* = average wall surface temp.), *T<sup>b</sup>* = *To*−*T<sup>i</sup>* 2 . *D<sup>h</sup>* = 4*Ac P* , *A<sup>c</sup>* = cross-section area of square pipe, while *P* is the wetted perimeter.

The Gnielinski [28] relationship is justifiable, especially for the single-phase fluids flowing:

$$Nu = \frac{\left(\frac{f}{8}\right)(Re - 1000)Pr}{1 + 12.7\left(\frac{f}{8}\right)^{0.5}\left(Pr^{2/3} - 1\right)}\left[1 + \left(\frac{d}{L}\right)^{2/3}\right]\left(\frac{Pr\_m}{Pr\_w}\right)^{0.11} \tag{7}$$

where, *Pr<sup>m</sup>* = the bulk temperature-related Prandtl number and *Pr<sup>w</sup>* = wall temperaturerelated Prandtl number. The Gnielinski correlation remains valid in the range of 3000 < Re < <sup>5</sup> <sup>×</sup> <sup>10</sup><sup>6</sup> and 0.5 < *Pr* < 2000.

The Colebrook equation [29] is applicable, based upon Re-number, in order to identify the friction factor of a fully developed turbulent flow using Equation (8).

$$\frac{1}{\sqrt{f}} = -2.0 \log \left( \frac{\varepsilon/D}{3.7} + \frac{2.51}{\text{Re}\sqrt{f}} \right) \tag{8}$$

Petukhov's equation [30] of a fully developed turbulent flow is as shown in Equation (9):

$$Nu = \frac{\left(\frac{f}{8}\right) RePr}{1.07 + 12.7\left(\frac{f}{8}\right)^{0.5}\left(Pr^{2/3} - 1\right)}\tag{9}$$

Here, the formula is applicable for the requirements of 3000 < *Re* < 5 <sup>×</sup> <sup>10</sup><sup>6</sup> and 0.5 < *Pr* < 2000.

The values of the Darcy friction factor were determined from the approximate pressure loss along the heated square pipe. The Blasius and Petukhov correlations were employed for the validation of the results obtained for the base fluid [31]:

Petukhov [30]:

$$f = \left(0.79\ln(Re) - 1.64\right)^{-2} \tag{10}$$

Blasius [32]:

$$f = \frac{0.316}{Re^{0.25}}\tag{11}$$

A performance index (*PI*) indicates an appropriate parameter to define various velocity and temperature ranges usable by various nanofluids [33]:

$$PI = \frac{h\_{\text{nf}} / h\_{bf}}{\Delta P\_{\text{nf}} / \Delta P\_{bf}} = \frac{R\_h}{R\_{\Delta P}} \tag{12}$$

where (*R<sup>h</sup>* ) is the ratio between nanofluids heat transfer and DW heat transfer, while (*R*∆*P*) is the ratio between nanofluids pressure drop and DW pressure drop. An energysaving indicator within the turbulent flow region calculated the pumping power using Equation (13).

$$\frac{\mathcal{W}\_{nf}}{\mathcal{W}\_{bf}} = \left(\frac{\mu\_{nf}}{\mu\_{bf}}\right)^{0.25} \left(\frac{\rho\_{bf}}{\rho\_{nf}}\right)^{2} \tag{13}$$

where *<sup>W</sup>n f* is the nanofluids' pumping power and (*Wb f*) is the DW pumping power.

The overall performance was evaluated (in terms of the thermal and hydraulic performances) using a performance evaluation criterion (PEC), which depicts the ratio of the heat performance to the nanofluids compared to DW. The formula of the PEC was expressed as [34]:

$$PEC = \frac{\text{Nu}\_{\text{nf}} / \text{Nu}\_{\text{bf}}}{\left(f\_{\text{nf}} / f\_{\text{bf}}\right)^{1/3}} \tag{14}$$

Table 1 presents and outlines the range of uncertainties [35].

**Table 1.** Uncertainty ranges for heat transfer and fluid flow variables.

