**2. Materials and Methods**

The mathematical model can be summarized in the following governing and constitutive equations:

Continuity equation

$$\frac{\partial}{\partial t} \int\_{V} \rho \mathbf{d}V + \int\_{s} \rho v \cdot \mathbf{d}s = 0\tag{1}$$

where *ρ* is density and *v* is the velocity vector.

Momentum equation

$$\frac{d}{dt} \int\_{V} \rho \mathbf{v} dV + \int\_{s} \rho \mathbf{v} \mathbf{v} \cdot d\mathbf{s} = \int\_{s} \mathbf{T} \cdot d\mathbf{s} + \int\_{V} f\_{b} dV \tag{2}$$

where *<sup>T</sup>* the Cauchy stress tensor and *<sup>f</sup><sup>b</sup>* is the resultant body force.

Thermal energy equation

$$\frac{\mathbf{d}}{\mathbf{d}t} \int\_{V} \rho \mathbf{c}\_{v} T dV \, + \int\_{\mathbf{s}} \rho \mathbf{c}\_{p} T \mathbf{v} \cdot d\mathbf{s} = - \int\_{\mathbf{s}} \mathbf{q} \cdot d\mathbf{s} \, + \int\_{V} (T : \mathbf{grad} v) dV \tag{3}$$

where *cp* is the specific heat, *T* is the temperature and *q* is the heat flux vector.

Stoke's law

$$T = 2\mu \dot{D} - \frac{2}{3} \mu div \upsilon I - PI \tag{4}$$

where .

$$\dot{D} = \frac{1}{2} [\text{grad}\,\mathbf{v} + (\text{grad}\,\mathbf{v})^{\text{T}}] \tag{5}$$

is the rate of strain tensor, *μ* is the dynamic viscosity, *p* is the pressure and *I* is the unit tensor.

Fourier's law

$$q = -k \,\mathrm{grad}\,T\,\tag{6}$$

where *k* is thermal conductivity.

Thermophysical properties of ionic liquid and ionanofluids.

Relationships for thermophysical properties (density, thermal conductivity, viscosity and specific heat) of ionanofluids [C4mpyrr][NTf2] with the Al2O3 nanoparticles used in this study are obtained by curve-fitting the experimental results from the literature [22] and are given in the Table 1. Additionally, relationships for certain properties, such as density, were already given by Titan [22] and are used as such.

**Table 1.** Equations used for predicting the thermophysical properties for different weight percent wt%.


The curve-fits are applied to the experimental values of the thermophysical properties obtained by Titan [22], and the results are presented in Figure 1.

**Figure 1.** Comparison of the predicted values for thermophysical properties using equations from Table 1 with measured data obtained by [22].

As it can be seen from Figure 1, the temperature variations of density (Figure 1a), viscosity (Figure 1b), heat capacity (Figure 1c) and ratio of thermal conductivity of base liquid and corresponding ionanofluids (Figure 1d) were compared with experimental results obtained by Titan [22]. The comparison was made for the base liquid as well as for ionanofluids with concentrations of 0.5, 1.0 and 2.5 wt% (weight percent). The square of correlation factor (*R*2) for density was above 0.95, for viscosity above 0.99, for heat capacity above 0.98 and for the ratio of thermal conductivity above 0.8, regardless of weight percent.

The numerical method employed for modeling the flow and heat transfer of ionic liquid and ionanofluids was the finite volume method. The methodology closely follows the one presented in [23,24].

#### **3. Results**

The steady-state flow of ionic liquid [C4mpyrr][NTf2] and ionanofluids with Al2O3 nanoparticles through the horizontal straight tube of 1.75 m length and 0.014 m diameter was analyzed with convective heat transfer included. The geometry of the problem can be seen from Figure 2. The case study was analyzed for initial temperatures *Tin*<sup>1</sup> = 293 K, *Tin*<sup>2</sup> = 303 K and *Tin*<sup>3</sup> = 335 K, for two Reynolds number values (100 and 512), and for three values of weight percentage (0.5 wt%, 1.0 wt% and 2.5 wt%). Additionally, the heat transfer characteristics of pure ionic liquid was also analyzed. The wall

heat flux was constant *<sup>q</sup>* = 13 kW·m−2. At the outlet, the pressure was set to 0 Pa. For the purpose of set up, the inlet boundary conditions uniform velocity profile was used.

**Figure 2.** Geometry and boundary conditions.

The effects of natural convection were neglected. Due to computational efficiency, fluid flow was analyzed through a part of the tube in the shape of a longitudinal wedge with an angle of 5◦ as it can be seen from Figure 3.

The heat transfer characteristics of the ionic liquid and corresponding ionanofluids were investigated by analyzing the heat transfer coefficient and Nusselt number. Additionally, the temperature profile for each case is presented.

The heat transfer coefficient and Nusselt number were calculated using the following equations:

$$h\_{IL/NEIL} = \frac{q}{\left(T\_{wall} - T\_{in}\right)'} \left[\mathbf{W} \cdot \mathbf{m}^{-2} \cdot \mathbf{K}^{-1}\right] \tag{7}$$

$$\text{Nu}\_{\text{IL/NELL}} = h\_{\text{IL/NELL}} \frac{D}{k\_{\text{IL/NELL}}} \begin{bmatrix} - \end{bmatrix} \tag{8}$$

where

*q* ) <sup>W</sup>·m−<sup>2</sup> \* is heat flux through the shell of the tube given as the boundary condition *Twall* [K] is temperature of the tube shell

*Tin* [K] is initial (reference) temperature

*hIL*/*NEIL* <sup>W</sup>·m−2·K−<sup>1</sup> is the heat transfer coefficient of the ionic liquid (IL) or ionanofluids (NEIL)

*D* [m] is diameter of the tube

*kIL*/*NEIL* <sup>W</sup>·m−1·K−<sup>1</sup> is the thermal conductivity of the ionic liquid (IL) or ionanofluids (NEIL)

The validation of the presented model along with the grid refinement was done for the ionic liquid [C4mpyrr][NTf2] and the results were compared with results obtained by using the Shah's equation. The grid sensitivity study was performed for four different values of base cell sizes. The characteristics of the analyzed grids are given in Table 2.


**Table 2.** Grid characteristics for grid sensitivity study.

The results for each grid, and the results obtained by using Shah's equation, can be seen from Figure 4. It can be seen from Figure 4 that as the grid is systematically refined, the results approach the results obtained from the Shah's equations.

**Figure 4.** Grid sensitivity study and model validation.

Furthermore, one can conclude that for grid 3 (base cell size of 0.6 mm), the obtained results are grid independent since for further refinement the results remain the same. Hence, the numerical study was performed for grid 3, with a base cell size of 0.6 mm.

Over 40 numerical simulations were performed in order to compare temperature profiles and hence heat transfer characteristics (heat transfer coefficient and Nusselt number) for ionic liquid and ionanofluids with temperature-dependent thermophysical properties and with constant thermophysical properties. For better presentation of the findings, the results were compared in accordance with different perspectives, i.e., in accordance with the weight percent, Reynold numbers, initial temperatures, and/or in accordance with the temperature/dependency of thermophysical properties.

#### **4. Discussion**

Figure 5 presents the temperature distribution along the top wall, *Twall*, at *Re* = 100 and *Re* = 500 and 0.5 ≤ wt% ≤ 2.5 for fluids with constant thermophysical properties, whereas Figure 6 presents the temperature distribution along the top wall, *Twall*, at *Re* = 100 and *Re* = 500 and 0.5 ≤ wt% ≤ 2.5 for fluids with temperature-dependent thermophysical properties in accordance with Table 1.

**Figure 5.** Temperature profile for *Re* = 100 and *Re* = 512 for 0.5 ≤ wt% ≤ 2.5 and constant thermophysical properties.

**Figure 6.** *Cont*.

**Figure 6.** Temperature profile for *Re* = 100 and *Re* = 512 for 0.5 ≤ wt% ≤ 2.5 and temperature-dependent thermophysical properties.

It can be seen from the Figures 5 and 6 that the temperature at *x* = 0 m is same as the inlet temperature, and in the developing region (up to *x* = 0.4 m) the immediate increase is noticeable. Following the gradual linear increase in the developed region due to the constant heat flux applied to the wall surface, the temperature at the upper wall reaches the maximum value at the outlet (*x* = 1.75 m).

Furthermore, for both constant and temperature-dependent thermophysical properties, the increase in the weight percent of nanoparticles results in lower temperature profiles of the upper wall. One can conclude that the increase in the weight percent of nanoparticles results in more heat transfer from the wall to the fluid regardless of the temperature-dependency of the thermophysical properties. The same can be concluded for the influence of the Reynold's number; the increase in the Reynold's number results in an increase in heat transfer from the wall to the fluid for ionic liquid and ionanofluids for constant and temperature-dependent thermophysical properties.

To better understand the influence of the temperature-dependency of the thermophysical properties of the ionic liquid and ionanofluids on the temperature profile, Figure 7 presents the temperature distributions on the upper wall for *Re* = 100, inlet temperatures *T* = 293 K, *T* = 303 K and *T* = 335 K and wt% of 0, 0.5 and 1.0 for both constant and temperature-dependent thermophysical properties.

**Figure 7.** *Cont*.

(**c**) 1.0 wt%

**Figure 7.** Temperature profiles comparison for ionic liquid and ionanofluids with and without constant thermophysical properties for *Re* = 100.

When analyzing the temperature profile on the upper wall for ionanofluids with and without constant properties for constant weight percent and Reynolds number, and with variable inlet temperatures, it can be concluded that the temperature profile is higher when the assumption of constant thermophysical properties is made, regardless of the weight percent or the inlet temperature. Therefore, it can be concluded that the temperaturedependent thermophysical properties of ionanofluids cause better heat transfer from the wall of the tube to the fluid.

Heat transfer performances of ionic liquid and ionanofluids for different weight percent were analyzed through the heat transfer coefficient (Equation (7)) and Nusselt number (Equation (8)). Figure 8 presents the heat transfer coefficient along the tube for ionic liquid, and ionanofluids of 0.5 wt%, 1.0 wt% and 2.5 wt%, and for Reynolds numbers of 100 and 512, for reference temperatures *T* = 293 K, *T* = 303 K and *T* = 335 K.

**Figure 8.** *Cont*.

**Figure 8.** Heat transfer coefficient for 0.5 ≤ wt% ≤ 2.5 and *Re* = 100 and *Re* = 512.

It is shown in Figure 8 that the increase in the Reynolds number, as well as the increase in the nanoparticle weight percent, causes an increase in the heat transfer coefficient, meaning that the heat transfer is higher for higher values of wt% and Re. The same conclusion can be made for each inlet temperature. Furthermore, it is noticeable that the heat transfer coefficient has the highest values at the inlet of the tube, leading to the exponential decrease in the developing region (up to *x* = 0.4 m). Following this, the values of the heat transfer coefficient gradually decrease in a linear manner when progressing towards the outlet of the tube where the heat transfer coefficient has the minimum value.

Moreover, in this study, the influence of the assumption of the constant thermophysical properties on the heat transfer performances for ionic liquid and ionanofluids was analyzed. Therefore, Figure 9 presents the Nusselt number values along the tube for both ionanofluids with temperature-dependent thermophysical properties and for ionanofluids with constant thermophysical properties. Analysis was undertaken at a constant weight percentage, so that the influence of the assumption could be analyzed for different inlet temperatures and Reynolds numbers.

**Figure 9.** *Cont*.

**Figure 9.** Nusselt number for ionanofluids with and without constant thermal properties for different temperature values and different Reynold numbers.

When analyzing Figure 9, one can conclude that the assumption of the constant thermophysical properties of both ionic liquid and ionanofluids has a great influence on the Nusselt number. The influence is more significant as the weight percent increases. The greatest difference between the results for constant and variable thermophysical properties is for the weight percentage of 2.5%, where significant divergence is noticeable for the Nusselt number, both for *Re* = 100 and *Re* = 512, as well as for each inlet temperature. When analyzing curves for the temperature-dependent thermophysical properties (Figure 1) it can be seen that, for wt% 2.5, the viscosity exponentially decreases, resulting in a significant difference in the viscosity values for different temperatures. The Nusselt number values for constant thermophysical properties for *Re* = 512 and *T* = 293 K correspond to the Nusselt number values obtained by Chereches et al. [17].

For better understanding of the heat transfer performances of ionic liquids and ionanofluids, the heat transfer coefficient was analyzed and compared in accordance of weight percent for ionic liquid and ionanofluids, with and without constant properties for *Re* = 100 (Figure 10) and *Re* = 512 (Figure 11).

**Figure 10.** *Cont*.

**Figure 10.** Heat transfer coefficient of ionic liquid and ionanofluids with and without constant thermal properties for 0.5 ≤ wt% ≤ 2.5 and *Re* = 100.

**Figure 11.** Heat transfer coefficient of ionic liquid and ionanofluids with and without constant thermal properties for 0.5 ≤ wt% ≤ 2.5 and *Re* = 512.

The heat transfer coefficient increases as the weight percent of ionanofluids increase, and it has the lowest value for ionic liquid for both *Re* = 100 and *Re* = 512, regardless of inlet temperature. Furthermore, it is noticeable from Figures 10 and 11 that the assumption of the constant thermophysical properties has the greatest influence on the heat transfer performances of the observed ionanofluids. The greatest influence is for those with a weight percent of 2.5%. It can also be seen from Figures 10 and 11 that for ionic liquids and ionanofluids with constant thermophysical properties, the heat transfer coefficient is higher or lower (depending on the weight percent and inlet temperature), which can mislead a conclusion that the heat transfer performances are better or worse than they really are.
