Hydrothermal performance of a turbulent nanofluid with different nanoparticle shapes in a duct fitted with various configurations of coiled-wire inserts

Abstract

This paper examines the turbulent hydrothermal performance of boehmite/water–ethylene glycol $(\upgamma -\mathrm{AlO}(\mathrm{OH})/{\mathrm{H}}_{2}\mathrm{O}-\mathrm{EG})$ nanofluid flowing through a square duct fitted with various coiled-wire inserts (CWIs) using the finite volume method. The turbulent flow of $\upgamma -\mathrm{AlO}(\mathrm{OH})/{\mathrm{H}}_{2}\mathrm{O}-\mathrm{EG}$ nanofluid is modeled using single-phase and $k-\varepsilon$ model. A parametric study is carried out on the effect of Reynolds number ($5.0\times {10}^{3}\le \mathrm{Re}\le 4.0\times {10}^{4}$), the geometry of wire (circular, triangular, square, square-diamond, hexagon, octagon, and decagon), nanoparticle volume ratio ($0\le \varphi \le 4\%$), and nanoparticle shapes (blade, brick, cylinder, platelet, and oblate-spheroid) on hydrodynamic and convective heat transfer performance (CHTP). The results showed that the combination between CWI and nanofluid enhances hydrothermal performance. For instance, among the geometries of CWI considered at $\mathrm{Re}=5.0\times {10}^{3}$, the square CWI has the highest normalized ${\mathrm{Nu}}^{\mathrm{G}}$ (referencing empty channel) of 2.58, while the decagon has the lowest value of 1.78. Furthermore, regarding the nanoparticle shapes, the platelet shape has a maximum normalized ${\mathrm{Nu}}^{\mathrm{N}}$ (referencing base fluid) of 1.53, while the oblate-spheroid has a minimum value of 0.93. Lastly, in terms of application, square and octagon wire-fitted channels are better than empty channel at low $\mathrm{Re}$, as the values of their hydrothermal performance evaluation criteria are greater than unity.

Introduction

Enhancement of convective heat transfer performance (CHTP) is indispensable toward achieving optimization and miniaturization of the thermal system. Generally, CHTP in a thermal system can be enhanced using active, passive, and compound methods. The active methods require external power for its operation, while the passive method, which is principally a modification of geometry or fluid properties (like the usage of nanofluid), does not require external power for its operation [1,2,3,4]. Last decades, many authors investigated various techniques of heat transfer enhancement such as corrugated tubes [5,6,7], twisted-tape inserts [8, 9], dimpled tubes [10], conical tubes [11], coil inserts [12, 13], nanofluids [14, 15], conical inserts [16], fins [17], and fluid injection [18, 19]. The physics behind the enhancement of CHTP due to geometry modification lies in the fact that the thinner the thickness of the thermal boundary layer (${\delta }_{\mathrm{th}}$), the better the rate of CHTP. Therefore, the geometry of the thermal system is modified such that the ${\delta }_{\mathrm{th}}$ of the fluid reduces along the flow direction [20,21,22,23].

The literature survey showed that a channel fitted with CWI is preferable among various geometry modifications examined so far because of its improved heat transfer performance and ease of fabrication and maintenance. Furthermore, it has a variety of industrial applications. Admane and Patil [24] experimentally investigated turbulent CHTP of distilled water flowing through a heat exchanger fitted with CWI. The study examined the effect of different pitch ratios for the insert on the Nusselt number $(\mathrm{Nu})$ and friction factor $\left(\mathrm{fr}\right)$ at Re between 4000 and 14,000. They concluded that the presence of CWI and decrease in its pitch enhances thermal performance. Shoji et al. [25] performed experiments to examine the laminar CHTP of water flowing through a pipe with a copper-made CWI. The work examined the influence of the length of the CWI on $\mathrm{dP}$ and $\mathrm{Nu}$. The authors reported that at low $\mathrm{Re}$, the length of the CWI is not sensitive to $\mathrm{Nu}$. However, at high $\mathrm{Re}$, the $\mathrm{Nu}$ increases as the wire’s length increases. Naphon [26] carried out an experimental study on the hydrothermal performance of water flowing in a concentric pipe fitted with a bended CWI. The study focused on the effect of $\mathrm{Re}$ and pitch of the inserted turbulator on $\mathrm{Nu}$ and $\mathrm{dP}$. The researcher posited that the insert had significant role on the enhancement of CHTP. Furthermore, an increase in the pitch of the iron-made CWI reduces both $\mathrm{Nu}$ and $\mathrm{dP}$. Finally, a correlation for $\mathrm{Nu}$ as a function of pitch and $\mathrm{Re}$ was introduced. Keklikcioglu and Ozceyhan [12] conducted experiments to investigate the effect of $\mathrm{Re}$, pitch-to-diameter ratio (P/D) of equilateral triangular cross-section CWI, and clearance (distance between the WCI and the inner wall of the pipe) on the average $\mathrm{Nu}$ and $\mathrm{fr}$ of water flowing inside circular pipe. The authors reported an optimum configuration with thermal performance evaluation criterion (PEC) of 1.8. Furthermore, they reported that decrease in P/D of the CWI and clearance distance enhances both $\mathrm{Nu}$ and $\mathrm{fr}$. Abbaspour et al. [27] compared turbulent CHTP of water flowing inside a straight tube fitted with conical rings and CWI with the empty straight tube. The numerical study considered the effect of coil diameter and pitch on $\mathrm{Nu}$ and $\mathrm{fr}$. The authors concluded that increase in diameter of the coil by 300% enhances $\mathrm{Nu}$ by 131%, while 37.5% decrease in value of the pitch increases the $\mathrm{Nu}$ by 143%. Abedin and Shakar [28] performed an experimental analysis to investigate the effect of the pitch of the CWI with corresponding helix angles on $\mathrm{Nu}$ and $\mathrm{dP}$ for a fully developed turbulent airflow through a pipe. The author reported that a low $\mathrm{Re}$, a maximum ratio of 2 and 4, was reached against $\mathrm{Nu}$ and $\mathrm{dP},$ respectively, while the maximum thermal enhancement factor was 1.25. Wahidifar and Kohrom [29] conducted experiments to study the thermal behavior and the pressure drop of water flowing through a horizontal heat exchanger fitted with a coiled-wire and rings inserts. The study focused on the effect of the pitch-to-coil diameter ratio and the wire-to-coil diameter ratio on the PEC. They presented the optimum configuration, which has a PEC value of 1.28 at $\mathrm{Re}$ = 10,000. Ali et al. [30] also studied experimentally the effect of the geometric parameters (ratio of pitch-to-coil diameter and ratio of wire diameter-to-coil diameter) on CHTP of air flowing through a copper tube fitted with helical-coil insert across a range of $\mathrm{Re}$ $\left(14200\le \mathrm{Re}\le 49800\right)$. The authors reported 82% increment in $\mathrm{Nu}$.

As stated earlier, another means of enhancing CHTP is by replacing the conventional heat transfer fluid with a nanofluid. Nanofluid is a type of improved heat transfer fluid obtained by dispersing small amounts of nanoparticles of higher thermal conductivity (SiO₂, Al₂O₃, TiO_2, and Fe₂O₃) in base fluid (water and ethylene glycol). These resultant fluids have better thermophysical properties (thermal conductivity) than the base fluid [31,32,33]. Furthermore, it has been shown by various researchers that the nanoparticle’s shape has an influence on its CHTP [34, 35]. For instance, Benkhedda et al. [36] numerically examined the influence of nanoparticle shapes (platelet, cylindrical, blade, and spherical) of TiO₂/H₂O nanofluid on hydrodynamic and CHTP. They revealed that among the four nanoparticle shapes considered, the blade nanoparticle has the highest $\mathrm{Nu}$ while spherical shape has the lowest. Experiments by Jeong et al. [37] were performed to examine the influence of nanoparticle shapes (nearly rectangular and spherical) on the viscosity and thermal conductivity of ZnO/H₂O. The results mentioned that the nearly rectangular nanoparticle has a higher viscosity and thermal conductivity than nanoparticles of spherical shape. In another study, an experimental analysis of the effect of the nanoparticle shape of ZnO/H₂O and SiO₂/H₂O on CHTP and energetic performance was conducted by Ferrouillat et al. [38]. The authors concluded that the shape of a nanoparticle affects both $\mathrm{Nu}$ and $\mathrm{fr}$. A detailed experimental work by Maheshwary et al. [39] investigated the impact of nanoparticle’s concentration, shape (cubic, rod, and spherical), and size on thermophysical properties of TiO₂/H₂O nanofluid and its CHTP. The researchers disclosed that nanoparticle shapes enhanced the thermal conductivity by 5.62% over the base fluid. Furthermore, the cubic nanoparticle has the highest pumping power among the aforementioned shapes, while the spherical nanoparticle has the lowest.

Based on this review, the following conclusion can be drawn:

• 1. There are a lot of existing works on CHTP due to CWI. However, the majority of these works only examined the traditional CWI of a circular shape [40].

• 2. The effect of nanoparticle shapes on the flow characteristics and the thermo-hydraulic performance has been studied by many researchers. However, there is no work in this literature review that has examined the nanoparticle shapes' impact (blade, brick, cylinder, oblate-spheroid, and platelet) on the hydrodynamic and CHTP of a nanofluid flow in a square channel fitted with CWI having various geometries (circular, triangular, square, square-diamond, hexagon, octagon, and decagon).

As a result of these aforementioned research gaps, this study aims to investigate the turbulent thermo-hydraulic performance of $\upgamma -\mathrm{AlO}(\mathrm{OH})/{\mathrm{H}}_{2}\mathrm{O}-\mathrm{EG}$ nanofluid, flowing through a square duct fitted with CWI having various geometries by considering the effect of $\mathrm{Re}$, nanoparticle shapes (blade, brick, cylinder, oblate-spheroid, and platelet), and shapes of the wire-coil (circular, triangular, square, square-diamond, hexagon, octagon, and decagon). The boehmite/water–ethylene glycol nanofluid has been chosen because its nanoparticles exist in various phases (shapes).

Materials and methods

Problem description

This study investigates the turbulent thermal–hydraulic performance of $\upgamma -\mathrm{AlO}(\mathrm{OH})/{\mathrm{H}}_{2}\mathrm{O}-\mathrm{EG}$ nanofluid flowing in a square duct with and without CWI and subjected to constant heat flux, as shown in Fig. 1. The square duct has a cross-section area of 0.1 × 0.1 m² and 1-m long. The CWIs are of various geometries (circular, triangular, square, square-diamond, hexagon, octagon, and decagon), as shown in Fig. 2. The dimensions for the CWIs are mentioned in Table 1 and illustrated in Fig. 1b. Also, the boehmite/water–ethylene glycol nanofluid is prepared from nanoparticles of various shapes (blade, brick, cylinder, oblate-spheroid, and platelet), as shown in Fig. 3.

Fig. 1

a Schematic of the mathematical model with applied boundary conditions. b Geometrical parameters of duct and CWI

Fig. 2

Configuration of the square channel with CWI having various geometries. a Empty channel, b circular CW, c triangular CW, d square CW, e square-diamond CW, f hexagon CW, g octagon CW, and h decagon CW

Table 1 Dimensions for the CWIs

Fig. 3

Schematic for nanoparticle shapes used in the nanofluid [34]

In this study, the nanoparticles are assumed to be homogenously mixed with the base fluid consisting of distilled water and ethylene glycol mixed together at an equal ratio (50:50). As a result, a single-phase model is used to model the nanofluid. The other assumptions regarding this model are available in the previous works [5, 11].

Governing equations

Considering the aforementioned system, a 3D model has been established to simulate turbulent flow through the different channel configurations under steady-state conditions. The control volume is governed by the following equations (continuity, momentum, energy, turbulence kinetic energy, dissipation rate, and turbulence kinetic energy (TKE)), which are expressed as follows [2, 11]:

Continuity equation:

$$\frac{\partial }{{\partial x_\text{i} }}\left( {\rho_{{{\text{nf}}}} u_\text{i} } \right) = 0$$

(1)

Momentum equation:

$$\frac{\partial }{{\partial x_\text{i} }}\left( {\rho_{{{\text{nf}}}} u_\text{i} u_\text{j} } \right) = \frac{\partial }{{\partial x_\text{j} }}\left[ {\mu_{{{\text{nf}}}} \left( {\frac{{\partial u_\text{i} }}{{\partial x_\text{j} }} + \frac{{\partial u_\text{j} }}{{\partial x_\text{i} }} - \frac{2}{3}\delta_\text{ij} \frac{{\partial u_\text{i} }}{{\partial x_\text{j} }}} \right)} \right] + \frac{\partial }{{\partial x_\text{j} }}\left( { - \rho \mathop u\limits^{^{\prime}}_\text{j} \mathop u\limits^{^{\prime}}_\text{i} } \right) - \frac{\partial P}{{\partial x_\text{i} }}$$

(2)

Energy equation:

$$\frac{\partial }{{\partial x_\text{i} }}\left[ {u_\text{i} \left( {\rho E + P} \right)} \right] = \frac{\partial }{{\partial x_\text{j} }}\left[ {\left( {\lambda_{{{\text{nf}}}} + \frac{{c_\text{p} \mu_\text{t} }}{{Pr_\text{t} }}} \right)\frac{\partial T}{{\partial x_\text{j} }} + u_\text{i} \left( {\tau_\text{ij} } \right)_{{{\text{eff}}}} } \right] = 0$$

(3)

Here ${u}_\text{i} and {u}_\text{j}$ are the nanofluid velocities in i and j direction, respectively. $P$, $\rho$, ${c}_\text{p}$, and $T$ are the local nanofluid pressure, density, constant pressure specific heat, and temperature, respectively. $E={c}_\text{p}T-\frac{P}{\rho }+\frac{{u}^{2}}{2}$ represents the total energy, and $\rho \mathop u\limits^{^{\prime}}_{j} \mathop u\limits^{^{\prime}}_{i}$ is the Reynolds stress. ${\mathrm{Pr}}_{\text{t}}$ announces the turbulent Prandtl number, while the deviatoric tensor ${\left({\tau }_\text{ij}\right)}_{\mathrm{eff}}$ is given as follows:

$$\left( {\tau_\text{ij} } \right)_{{{\text{eff}}}} = \mu_{{{\text{eff}}}} \left[ {\left( {\frac{{\partial u_\text{j} }}{{\partial x_\text{i} }} + \frac{{\partial u_\text{i} }}{{\partial x_\text{j} }}} \right) - \frac{2}{3}\delta_\text{ij} \frac{{\partial u_\text{i} }}{{\partial x_\text{j} }}} \right]$$

(4)

where $\mu_{{{\text{eff}}}}$ is the effective molecular viscosity and is denoted as follows:

$$\mu_{{{\text{eff}}}} = \mu_{{{\text{nf}}}} + \mu_\text{t}$$

(5)

In the previous equation, ${\mu }_{\mathrm{nf}}$ and ${\mu}_\text{t}$ represent the dynamic and turbulent viscosities. The realizable $k-\varepsilon$ turbulent model has been chosen based on other similar previous studies [41,42,43]. In addition, its superior prediction performance for the fluid flow involves recirculation, separation, rotation, and boundary layers under strong adverse pressure gradients [44]. The governing equation for the rate of production of turbulent kinetic energy $\left(k\right)$ and the rate of dissipation $\left(\varepsilon \right)$ are expressed in Eqs. (6 and 7), respectively [45].

$$\frac{{\partial \left( {\rho_{{{\text{nf}}}} ku_\text{i} } \right)}}{{\partial x_\text{i} }} = \frac{\partial }{{\partial x_\text{i} }}\left[ {\left( {\mu_{{{\text{nf}}}} + \frac{{\mu_\text{t} }}{{\sigma_\text{k} }}} \right)\frac{\partial k}{{\partial x_\text{i} }}} \right] + G_\text{k} - \rho_{{{\text{nf}}}} \varepsilon$$

(6)

$$\frac{{\partial \left( {\rho_{{{\text{nf}}}} \varepsilon u_\text{i} } \right)}}{{\partial x_\text{i} }} = \frac{\partial }{{\partial x_\text{i} }}\left[ {\left( {\mu_{{{\text{nf}}}} + \frac{{\mu_\text{t} }}{{\sigma_{\smallint \varepsilon } }}} \right)\frac{\partial \varepsilon }{{\partial x_\text{i} }}} \right] + \rho_{{{\text{nf}}}} C_{1\varepsilon } S _{{}} \varepsilon - C_{2\varepsilon } \frac{{\varepsilon^{2} \rho_{{{\text{nf}}}} }}{{k + \sqrt {v\varepsilon } }}$$

(7)

where ${G}_\text{k}$ is the turbulence energy generation due to mean velocity gradients and is defined as follows:

$$G_\text{k} = \mu_\text{t} S^{2}$$

(8)

With

$$\mu_\text{t} = \rho_{{{\text{nf}}}} C_{\upmu } \frac{{k^{2} }}{\varepsilon }$$

(9)

$$S = \sqrt {2S_\text{ij} S_\text{ij} }$$

(10)

$$S_\text{ij} = 0.5\left( {\frac{{\partial u_\text{i} }}{{\partial x_\text{j} }} + \frac{{\partial u_\text{j} }}{{\partial x_\text{i} }}} \right)$$

(11)

$$C_{1\varepsilon } = {\text{max}}\left[ {0.43,\frac{\eta }{\eta + 5}} \right]$$

(12)

$$\eta = S\frac{k}{\varepsilon }$$

(13)

The model constants (${C}_{2\upepsilon }{, {C}_{\upmu },\sigma }_\text{k}, and {\sigma }_{\upepsilon })$ are provided in Table 2. Also, the flow conditions for the turbulent nanofluid are afforded in Table 3.

Table 2 Realizable $k-\varepsilon$ model constants [46]

Table 3 Summarize of boundary conditions

Nanofluid description

As stated earlier, the $\upgamma -\mathrm{AlO}(\mathrm{OH})/{\mathrm{H}}_{2}\mathrm{O}-\mathrm{EG}$ nanofluid has been prepared from boehmite ($\upgamma -\mathrm{AlO}(\mathrm{OH}))$ nanoparticles of various shapes (blade, brick, cylinder, oblate-spheroid, and platelet). The nanofluid's thermophysical properties (thermal conductivity and viscosity) largely depend on their shapes. Thus, the thermal conductivity $\left({\lambda }_{\mathrm{nf}}\right)$ of nanofluid prepared from blade, brick, cylinder, and platelet nanoparticles is estimated using the following formula [47].

$$\frac{{\lambda_{{{\text{nf}}}} }}{{\lambda_\text{f} }} = 1 + \left( {C_\text{k}^{{{\text{shape}}}} + C_\text{k}^{{{\text{surface}}}} } \right)\varphi = 1 + C_\text{k} \varphi$$

(14)

While the corresponding viscosities (${\mu }_{\mathrm{nf}}$) are estimated using

$$\frac{{\mu_{{{\text{nf}}}} }}{{\mu_\text{f} }} = \left( {1 + A_{1} \varphi + A_{2} \varphi^{2} } \right)$$

(15)

where the $\varphi$, $\mathrm{nf}$, and $f$ represent the nanoparticle volume ratio ($\mathrm{VR}$), the nanofluid, and the ${\mathrm{H}}_{2}\mathrm{O}-\mathrm{EG}$ (50:50) base fluid, respectively. The aspect ratio of these nanoparticles and the values for their ${C}_{\text{k}}^{\mathrm{shape}}$, ${C}_{\text{k}}^{\mathrm{surface}}$, and ${C}_{\text{k}}$ are listed in Table 4. The factors ${A}_{1}$ and ${A}_{2}$ are given in Table 5.

Table 4 Aspect ratio and values for essential factors employed in Eq. (14) to evaluate thermal conductivity [48]

Table 5 Values for parameters used in Eq. (15) to estimate viscosity [48]

For oblate-spheroid nanoparticles, the thermal conductivity (${\lambda }_{\mathrm{nf}}$) and viscosity (${\mu }_{\mathrm{nf}}$) are estimated as follows [49]:

$$\frac{{\lambda_{{{\text{nf}}}} }}{{\lambda_\text{f} }} = \left( {\frac{{\left( {\lambda_{{{\text{np}}}} + \left( {n - 1} \right)\lambda_\text{f} - \left( {n - 1} \right)(\lambda_\text{f} - \lambda_{{{\text{np}}}} } \right)\varphi }}{{\left( {\lambda_{{{\text{np}}}} + \left( {n - 1} \right)\lambda_\text{f} + (\lambda_\text{f} - \lambda_{{{\text{np}}}} } \right)\varphi }}} \right)$$

(16)

$$\frac{{\mu_{{{\text{nf}}}} }}{{\mu_\text{f} }} = \left( {1 - \frac{\varphi }{{\varphi_\text{m} }}} \right)^{ - 2}$$

(17)

where subscript $\mathrm{np}$ denotes the boehmite nanoparticles, and $n=\frac{3}{\Psi }$ refers to the shape factor. The values of $\Psi$ and ${\varphi }_\text{m}$ for the oblate-spheroid nanoparticles are presented in Table 6. The value of ${\varphi }_\text{m}$ which is empirically found depends on the geometric property $\delta =\frac{c}{a}$, as represented in Fig. 3. In addition, the thermophysical properties of $\upgamma -\mathrm{AlO}(\mathrm{OH})$ nanoparticles and water–ethylene glycol (50:50) are provided in Table 7.

Table 6 Applied constants used to estimate thermophysical properties of oblate-spheroid nanoparticles in the present work [49]

Table 7 Thermophysical properties for boehmite nanoparticles and water–ethylene glycol base fluid at 298 K [35]

Furthermore, it has been shown that nanofluid’s density and specific heat capacity are not sensitive to the nanoparticle’s shape. Accordingly, the density and the constant pressure-specific heat capacity for the aforesaid shapes are estimated using Eqs. (18 and 19).

$$\rho_{\text{nf}} = \left( {1 - \varphi } \right)\rho_\text{f} + \varphi \rho_{\text{np}}$$

(18)

$${C}_{p_{\text{nf}} } = \frac {(1-\varphi) \rho_{\text{f}}C_{p_{\text{f}} }+ \varphi\rho_{\text {np}}C_{\text{p}_{\text {np}}} }{\rho_{\text {nf}}}$$

(19)

Performance indices

The inlet velocity is estimated from the prescribed $\mathrm{Re}$ as

$$u_{{{\text{in}}}} = \frac{{{\text{Re }}\mu_{{{\text{nf}}}} }}{{\rho_{{{\text{nf}}}} { }D_\text{h} }}$$

(20)

To address the thermal–hydraulic performance of the nanofluid, the dimensionless parameters, particularly, the friction factor and the surface average Nusselt number ($\mathrm{Nu}$), are used. Both are defined as follows:

$$f = \frac{{2D_\text{h} { }\Delta p}}{{\rho_{{{\text{nf}}}} { }u_{{{\text{mean}}}}^{2} L}}$$

(21)

$${\text{Nu}} = \frac{{q^{\prime\prime}{ }D_\text{h} }}{{\left( {T_{{{\text{wall}}}} - T_{{{\text{bulk}}}} } \right)\lambda_{{{\text{nf}}}} }}$$

(22)

where ${T}_{\mathrm{bulk}}$ is evaluated as follows:

$$T_{{{\text{bulk}}}} = 0.5\left( {T_{{{\text{in}}}} + T_{{{\text{out}}}} } \right)$$

(23)

The thermo-hydraulic performance evaluation criterion due to the presence of wire-coil insert is expressed in terms of (${\text{PEC}}_{{{\text{geo}}}}$), which is calculated as follows:

$${\text{PEC}}_{{{\text{geo}}}} = \frac{{\left( {\frac{{{\text{Nu}}_{{{\text{in}}}} }}{{{\text{Nu}}_{{{\text{em}}}} }}} \right)}}{{\left( {\frac{{{\text{fr}}_{{{\text{in}}}} }}{{{\text{fr}}_{{{\text{em}}}} }}} \right)^{1/3} }}$$

(24)

where subscripts (in and em) connote channels with CWI and empty channel, respectively.

The corresponding thermo-hydraulic performance evaluation criteria due to the usage of nanofluid is announced through (${\text{PEC}}_{{{\text{nf}}}}$), which is defined as follows:

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

(25)

where subscripts (nf and f) connote nanofluid and base fluid, respectively.

The normalized friction factor $\left({\mathrm{fr}}^{\mathrm{G}}\right)$ and the surface average Nusselt number $\left({\mathrm{Nu}}^{\mathrm{G}}\right)$ due to geometry modifications are also defined as follows:

$${\text{fr}}^{{\text{G}}} = \frac{{{\text{fr}}_{{{\text{in}}}} }}{{{\text{fr}}_{{{\text{em}}}} }}$$

(26)

$${\text{Nu}}^{{\text{G}}} = \frac{{{\text{Nu}}_{{{\text{in}}}} }}{{{\text{Nu}}_{{{\text{em}}}} }}$$

(27)

While the corresponding normalized friction factor $\left({\mathrm{fr}}^{\mathrm{N}}\right)$ and the surface average Nusselt number $\left({\mathrm{Nu}}^{\mathrm{N}}\right)$ due to the usage of nanofluid are delineated with

$${\text{fr}}^{{\text{N}}} = \frac{{{\text{fr}}_{{{\text{nf}}}} }}{{{\text{fr}}_\text{f} }}$$

(28)

$${\text{Nu}}^{{\text{N}}} = \frac{{{\text{Nu}}_{{{\text{nf}}}} }}{{{\text{Nu}}_\text{f} }}$$

(29)

Computational model development, grid sensitivity, and validation

The entire 3D domain for squared channels with and without CWIs has been generated based on the polyhedron finite mesh technique. Enhanced wall treatment technique was applied for near-wall modeling to satisfy the desirable value $\left({y}^{+}\approx 1\right)$, such that a small mesh size was used near the wall of the channel, and the surface of CWI, as shown in Fig. 4. This has been done in order to apprehend the boundary layer effect while a coarser mesh was used in the core fluid region. The governing equations (continuity, momentum, and energy) and the turbulence equations have been solved using the finite volume method through ANSYS Fluent® [50]. The imposed boundary conditions are shown in Fig. 1a and summarized in Table 3. Semi-implicit pressure link equation (SIMPLE algorithm) was employed for coupling pressure and velocity equations. The momentum, energy, turbulent kinetic energy generation $\left(k\right)$, and its rate of dissipation $\left(\varepsilon \right)$ equations were discretized using second-order upwind scheme, while the resulting algebraic equations were solved using the iterative method. The convergence criteria for all equations were set to 10^–6 such that

$$\left| {\frac{{\Gamma^\text{k + 1} - \Gamma^\text{k} }}{{\Gamma^\text{k + 1} }}} \right| \le 10^{ - 6}$$

(30)

where $k$ $and \Gamma$ denote iteration number and field variables, respectively. Grid sensitivity analysis of the model was performed by estimating the $\mathrm{Nu}$ and $\mathrm{fr}$ for distilled water flowing through a square channel provided with a circular CWI. The variation of $\mathrm{Nu}$ and $\mathrm{fr}$ against the number of grids is shown in Fig. 5a. In order to reduce computational time without compromising the accuracy of the results, the mesh configuration of ${5.137\times 10}^{6}$ was chosen and used for the entire models. Also, the results of the grid sensitivity test in terms of local Nusselt number and pressure for coarse, fine, and finest generated grid are shown in Fig. 5b–c. The reliability of this model was further validated by estimating the $\mathrm{Nu}$ and $\mathrm{fr}$ of distilled water in an empty square duct across a range of $\mathrm{Re}$. The obtained values for $\mathrm{Nu}$ were compared with existing empirical correlations (Appendix A) by Dittus–Boelter [51], Sieder and Tate [52], and Gnielinski [53], while the values of $\mathrm{fr}$ have been compared with Petukhov [54], Blasius [55], and Filonenko [56] correlations as shown in Fig. 6a–b. It could be seen that a good agreement exists between the estimated values and their corresponding correlations. For instance, the average percentage deviation of estimated $\mathrm{Nu}$ relative to Dittus–Boelter, Sieder and Tate, and Gnielinski correlations is 6.63, 1.57, and 5.40%, respectively. The average percentage deviation of $\mathrm{fr}$ estimated relative to Petukhov, Blasius, and Filonenko correlation is 3.28, 4.48, and 3.26%, respectively. For more robust of the model, another validation with experimental results of nanofluid flow in circular duct in terms of local Nusselt number Nu(x) investigated by Hafiz [57]. The average deviation between the present model and the experimental results is 7.12%, as shown in Fig. 6c. Consequently, the present model is reliable and can be used in the rest of cases.

Fig. 4

Mesh generation for channel fitted with circular CWI

Fig. 5

a Results of grid sensitivity test in terms of average Nusselt number and friction factor (The selected mesh size is marked with the blue elliptic shape). b Results of grid sensitivity test in terms of local Nusselt number for coarse, fine, and finest generated grids. c Results of grid sensitivity test in terms of local pressure for coarse, fine, and finest generated grids

Fig. 6

a Validation of the present model against experimental correlations in terms of Nu. b Validation of present model against experimental correlations in terms of f. c Validation of present model against experimental results by Hafiz [57] in terms of local Nusselt number Nu(x)

Results and discussion

This section provides and discusses simulation results for the turbulent thermo-hydraulic performance of $\upgamma -\mathrm{AlO}(\mathrm{OH})/{\mathrm{H}}_{2}\mathrm{O}-\mathrm{EG}$ nanofluid, prepared from previously described nanoparticles and flowing through a square duct with various CWIs. In summary, there are an empty channel and seven other designs, all simulated at six values of Re, for five different nanofluids (each with $\mathrm{VR}=2\mathrm{ and }4\mathrm{\%}$) and a base fluid ($\mathrm{VR}=0\mathrm{\%}$) for comparison; which means 528 simulated cases. Sect. "Hydrodynamic performance" examines the effect of nanoparticle shapes, nanoparticle volume ratio ($0\le \mathrm{\varphi }\le 4\mathrm{\%}$), $\mathrm{Re}$ ($5.0\times {10}^{3}\le \mathrm{Re}\le 4.0\times {10}^{4})$, and geometries of CWI on hydrodynamics. Sect. "Heat transfer performance" examines the effect of the aforementioned terms on heat transfer performance, while Sect. "Thermo-hydraulic performance" discusses the thermo-hydraulic performance criterion.

Hydrodynamic performance

Figure 7a shows the contour of the planar view of TKE for nanofluid flow with platelet nanoparticles in the duct fitted with various CWIs at $\mathrm{Re}=5.0\times {10}^{3} -\mathrm{ \varphi }=4\mathrm{\%}$. This term is essential to investigate hydrodynamic and heat transfer performance in a thermal system as it measures the intensity at which fluid at the viscous sublayer is being mixed with fluid at the core fluid zone. From the figure, it is evident that TKE is higher in ducts fitted with CWIs than in empty ducts. Furthermore, high intensity of TKE is noticed near the CWI surface. This result is expected as the presence of CWI in the duct induces secondary flow in the core region, resulting in high-flow velocity fluctuations. The high intensity noticed near the wall of CWI is attributed to the high disturbance of the radial and tangential velocities of the fluid near the CWI by the vortex of the secondary flow. This leads to high fluctuation of the velocity near the wire’s surface. A higher pressure drop accompanies this process. It should be noticed that the high TKE observed in channel designs contributes to the improvement of CHTP by increasing the rate of mixing between hot fluid zones near the wall and cold fluid zones, which also decreases the thickness of the thermal boundary layer (${\delta }_{\mathrm{th}}$). Figure 7b reveals the distribution of TKE in nanofluid prepared from various nanoparticle shapes flowing in the duct fitted with the square CWI of various nanoparticle shapes at $\mathrm{Re}=5.0\times {10}^{3}-\mathrm{ \varphi }=4\mathrm{\%}$. A significant value of TKE is announced near the wall of the ducts which is attributed to the previously indicated velocity fluctuation. Furthermore, it is shown that TKE is higher in nanofluid than base fluid except in oblate-spheroid. Additionally, it can be seen that the nanofluid prepared from platelet nanoparticles has the highest TKE among the nanofluids considered. This high TKE is due to the relatively high viscosity of the platelet nanofluid. It shall be emphasized that high viscosity will be counterbalanced with the high inlet velocity since we operate at constant $\mathrm{Re}$. Later on, the reason for the high viscosity of nanofluid made from platelet nanoparticles will be explained. Figure 8a depicts the variation of $\mathrm{fr}$ for all nanofluids against $\mathrm{Re}$ for aforesaid channel configurations. It is clear that $\mathrm{fr}$ decreases as $\mathrm{Re}$ increases for multiple duct configurations considered. Whereas shown, across the range of $\mathrm{Re}$ considered $\left(5.0\times {10}^{3}\le \mathrm{Re}\le 4.0\times {10}^{4}\right)$, the values of $\mathrm{fr}$ for channels fitted with circular, triangular, square, square-diamond, hexagon, octagon, and decagon wires decrease by 38.52, 42.02, 38.82, 44.41, 38.62, 37.15, and 40.23%, respectively. While the corresponding percentage reduction for empty ducts is 48.54%. This result is expected as an increase in $\mathrm{Re}$ in the turbulent flow regime reduces the thickness of the viscous sublayer. Figure 8b shows the variation of ${\mathrm{fr}}^{\mathrm{G}}$ against$\mathrm{Re}$. As represented before, it could be seen that the empty square duct has the lowest $\mathrm{fr}$ compared with ducts with CWI. For instance, at $\mathrm{Re}=1.5\times {10}^{4},$ the values of ${\mathrm{fr}}^{\mathrm{G}}$ in channels fitted with circular, triangular, square, square-diamond, hexagon, octagon, and decagon wire are 11.28, 11.57, 10.52, 11.70, 12.01, 11.28, and 11.93, respectively.

Fig. 7

a Contour of local TKE for nanofluid flow with platelet nanoparticles in the duct fitted with various CWIs at Re = 5000–$\varphi =4\%$. b Contour of local TKE in the duct fitted with square CWI of various nanoparticle shapes at Re = 5000–$\varphi =4\%$

Fig. 8

a Variation of global friction factor against Re for empty duct and all other configurations of fitted CWIs ($\varphi =4\%$). b Variation of normalized friction factor against Re for the various CWIs compared to the empty one ($\varphi =4\%$)

The relatively high values of ${\mathrm{fr}}^{\mathrm{G}}$ observed for the duct with CWI are attributed to the following factors: (i) high shear stress and pressure drag created by the presence of the CWI, (ii) constriction of the flow passage due to the presence of CWI, (iii) the adverse pressure drop (APG) which result in flow recirculation, and (iv) swirl flow induced by the interaction between the flow and the coil, boundary layer separation, and reattachment. These factors contributed to the increase in pressure drop and consequently $\mathrm{fr}$ in the duct with CWI. For instance, the swirl flow increases the flow turbulence, the constriction of the flow passage increases the flow’s velocity, and the accelerated fluid increases the flow resistance. Lastly, the APG increases the pressure drop within the channel. The implication of high ${\mathrm{fr}}^{\mathrm{G}}$ is that high pumping power will be required in the operating channel with CWI. Also, it is explicit that among various CWIs examined, duct with square CWI has relatively lowest ${\mathrm{fr}}^{\mathrm{G}},$ while duct with hexagonal one has the highest value. The low value observed for duct fitted with square CWI is associated with its geometry similarity with the outer duct. Lastly, the values of ${\mathrm{fr}}^{\mathrm{G}}$ increase as $\mathrm{Re}$ increases for all channel designs. For example, at $\mathrm{Re}=5.0\times {10}^{3}$, the values of ${\mathrm{fr}}^{\mathrm{G}}$ are 10.27, 10.84, 9.42, 11.14, 10.91, 10.20, and 10.97 in circular, triangular, square, square-diamond, hexagon, octagon, and decagon wire shapes, respectively. While the corresponding values at $\mathrm{Re}=4.0\times {10}^{4}$ are 12.27, 12.21, 11.57, 12.03, 13.01, 12.46, and 12.74, respectively. This high value of ${\mathrm{fr}}^{\mathrm{G}}$ is attributed to an abnormal rise in shear stress, pressure drag, swirl flow, and APG as $\mathrm{Re}$ increases.

Heat transfer performance

The temperature contours shown in Fig. 9 illustrate the effect of using platelet nanofluid flow (VR = 4%) within our previously described channels, by presenting the temperature distribution along the axial flow direction. It could be seen that the presence of the CWI undermined the development of the thermal boundary layer (TBL), which leads to enhancement in CHTP. Also, we noticed that temperature distribution is relatively uniform in the channel with CWI compared with empty one. This result is attributed to high turbulence induced by the presence of CWI in the channel. The high turbulence increases the rate of mixing rate of hot fluid near the wall with cold fluid at the core region. This reduces the temperature gradient of the fluid. Furthermore, it was noticed that the thickness of the thermal boundary layer $({\delta }_{th})$ in base fluid (water–ethylene glycol) is bigger compared with $\upgamma -\mathrm{AlO}(\mathrm{OH})/{\mathrm{H}}_{2}\mathrm{O}-\mathrm{EG}$ nanofluid. This implies that the nanofluid enhances CHTP compared with base fluid. Also, it shall be noted that the thinner the ${\delta }_{th}$ around a hot solid object the higher the rate of CHTP to the surrounding fluid.

Fig. 9

Lateral views of temperature distribution for platelet nanofluid flow ($\mathrm{Re }= 5000 -\mathrm{\varphi }=4\mathrm{\%}$) in various channel designs

Figure 10a shows the variation of $\mathrm{Nu}$ for $\upgamma -\mathrm{AlO}(\mathrm{OH})/{\mathrm{H}}_{2}\mathrm{O}-\mathrm{EG}$ nanofluid in empty and other seven designed channels against $\mathrm{Re}$. It is observed that wire-fitted channels always outperform the empty duct in terms of $\mathrm{Nu}$ at the same $\mathrm{Re}$. For instance, at $\mathrm{Re}=1.5\times {10}^{4},$ the values of $Nu$ for blade nanofluid ($\mathrm{VR}=2\mathrm{\%})$ flowing inside the channel fitted with CWI (circular, triangular, square, square-diamond, hexagon, octagon, and decagon) are 295.82, 312.35, 323.08, 310.71, 306.48, 313.32, and 291.67, respectively. The corresponding value of $\mathrm{Nu}$ in the empty one is 201.26. This remarkable thermal enhancement is associated with the existence of CWI which (i) interrupts the development of the thermal boundary layer, (ii) produces swirl flow that induces tangential velocity component, which consequently enhances the rate of mixing of hot fluid near the wall with the accelerated core fluid [12, 26], (iii) reduces the cross section of the flow field which leads to high advection heat transfer that eventually enhances CHTP, (iv) produces APG which induces secondary flow, and (v) produces longitudinal vortices which also enhances the mixing rate. Also, it is obvious that an increase in $\mathrm{Re}$ enhances $\mathrm{Nu}.$ For example, across the range of $\mathrm{Re}$ considered $\left(5.0\times {10}^{3}\le \mathrm{Re}\le 4.0\times {10}^{4}\right)$, the values of $\mathrm{Nu}$ for blade nanofluid ($\mathrm{VR}=2\mathrm{\%})$ for the same aforementioned channel designs increase by factor of 4.02, 4.02, 2.61, 4.57, 3.73, 3.42, and 4.43, respectively. The corresponding factor in empty duct is 5.74. The increment is attributed to decrease in ${\delta }_{th}$ as $\mathrm{Re}$ increases, $\left({\delta }_{\mathrm{th}}\propto \frac{1}{{\mathrm{Re}}^{0.2}}\right)$[32]. The variation of ${\mathrm{Nu}}^{\mathrm{G}}$ against $\mathrm{Re}$ for the various channel designs is depicted in Fig. 10b. As demonstrated in the figure, the enhancement of $\mathrm{Nu}$ archived through CWI, compared to an empty channel, diminishes as $\mathrm{Re}$ increases. For instance, when the previously mentioned channel configurations are operating at $\mathrm{Re}=5.0\times {10}^{3}$, the values of ${\mathrm{Nu}}^{\mathrm{G}}$ for blade nanofluid ($\mathrm{VR}=2\mathrm{\%})$ are 1.94, 1.97, 2.64, 1.84, 2.09, 2.24, and 1.79, respectively. While that, at $\mathrm{Re}=4.0\times {10}^{4}$, the corresponding values are 1.36, 1.38, 1.20, 1.47, 1.36, 1.33, and 1.39, respectively. This result shows that as $\mathrm{Re}$ increases, the effect of formation factors (reduction in thickness of TBL, swirl flow, contraction of the flow passage, turbulence, and secondary flow) that contribute to the enhancement of CHTP in the channel with CWI began to wane. In other words, the high-flow rate mainly dominates the thermal enhancement over the other thermal improvement parameters. Back to Fig. 10b, it is noted that at low $\mathrm{Re}$ $\left(\mathrm{Re}=5\times {10}^{3}\right)$, duct with square CWI configuration has the highest value for ${\mathrm{Nu}}^{\mathrm{G}}$, while the square-diamond wire-fitted channel has the lowest. However, the opposite is the case at high $\mathrm{Re}$ $\left(\mathrm{Re}=4.0\times {10}^{4}\right)$. Hence, provided that $\mathrm{dP}$ is not an issue, duct with square CWI is advantageous at low $\mathrm{Re},$ while at a high-flow rate switching to a square-diamond design is better. On the other hand, the impact of boehmite-to-water–ethylene glycol $\mathrm{VRs}$ on the thermal performance of different nanoparticle shapes used in the present exploration is displayed in Fig. 11. Figure 11a–b better illustrates the variation of ${\lambda }_{\mathrm{nf}}$ and ${\mu }_{\mathrm{nf}}$ for $\upgamma -\mathrm{AlO}(\mathrm{OH})/{\mathrm{H}}_{2}\mathrm{O}-\mathrm{EG}$ nanofluids with nanoparticle $\mathrm{VRs}$. It is evident that the augmentation of nanoparticle VR elevates the magnitudes of the two quantities. For instance, across the range of nanoparticle $\mathrm{VR}$ examined $\left(0.0\le \varphi \le 0.04\right)$, the ${\lambda }_{\mathrm{nf}}$ prepared from platelet, blade, cylinder, brick, and oblate-spheroid nanoparticles increase by a factor of 1.10, 1.11, 1.16, 1.13, and 1.24, respectively. The corresponding increase in ${\mu }_{\mathrm{nf}}$ is 3.46, 1.78, 2.99, 1.83, and 1.16, respectively. Moreover, it is perceived that nanofluid prepared from oblate-spheroid nanoparticles has the highest ${\lambda }_{\mathrm{nf}}$, while platelet nanofluid has the lowest. Regarding ${\mu }_{\mathrm{nf}}$, the opposite is achieved. The high ${\mu }_{\mathrm{nf}}$ of nanofluid with platelet nanoparticles is attributed to its relatively small size compared to other shapes considered. This small size allows more nanoparticles to be packed into the base fluid, which invariably improves ${\mu }_{\mathrm{nf}}$. The variation of Prandtl number $\left(\mathrm{Pr}\right)$ for nanofluids prepared from various nanoparticle shapes against their volume concentrations is demonstrated in Fig. 11c. It is clear that platelet nanofluid has a maximum value of $\mathrm{Pr},$ while nanofluid prepared from oblate-spheroid nanoparticles has the minimum. This high $\mathrm{Pr}$ of platelet shape is associated with its relatively high ${\mu }_{\mathrm{nf}}$ and low ${\lambda }_{\mathrm{nf}}$, whereas all nanofluids have the same density and specific heat capacity.

Fig. 10

a Variation of Nusselt number against Re for empty duct and all other configurations of fitted CWIs ($\varphi =4\%$). b Variation of normalized Nusselt number against Re for the various CWIs compared to the empty one ($\varphi =4\%$)

Fig. 11

a Variation of thermal conductivity, b Dynamic viscosity, c Prandtl number, and d Normalized average Nusselt number for empty channel at $\mathrm{Re}=5\times {10}^{3}$ for various nanofluids shapes against nanoparticle volume ratios

The influence of nanoparticle shapes on ${\mathrm{Nu}}^{\mathrm{N}}$ is shown in Fig. 11d. It is explicit that the ${\mathrm{Nu}}^{\mathrm{N}}$ is maximum for platelet nanofluid and minimum for oblate-spheroid nanofluid. For instance, across the range of $\mathrm{VR}$ considered $\left(0\le \varphi \le 4\%\right)$ and while the empty channel is operating at $\mathrm{Re}=5.0\times {10}^{3}$, the percentage enhancements of ${\mathrm{Nu}}^{\mathrm{N}}$ for platelet, blade, cylinder, brick, and oblate-spheroid nanofluids are 35.37, 11.96, 28.19, 12.12, and −4.66%, respectively. The high ${\mathrm{Nu}}^{\mathrm{N}}$ achieved for nanofluid prepared from platelet nanoparticles is attributed to its high $\mathrm{Pr}$, and the increment is attributed to increase in ${\mu }_{\mathrm{nf}}$ and ${\lambda }_{\mathrm{nf}}$ as $\varphi$ increases. It shall be noted that increase in the former leads to high inlet velocity since we are operating at constant $\mathrm{Re}$, and an increase in the latter leads to high heat transfer performance.

Thermo-hydraulic performance

Figure 12a maps the variation of ${\mathrm{PEC}}_{\mathrm{geo}}$ of $\upgamma -\mathrm{AlO}(\mathrm{OH})/{\mathrm{H}}_{2}\mathrm{O}-\mathrm{EG}$ nanofluid prepared from blade nanoparticles $(\varphi =4\%)$ against $\mathrm{Re}$ for all simulated duct configurations. Driven by exceptional thermal and hydrodynamic performance, the channels with square and octagon CWIs have a value of ${\mathrm{PEC}}_{\mathrm{geo}}$ greater than unity at low $\mathrm{Re}$ ($\mathrm{Re}=5.0\times {10}^{3})$, while for other configurations, the values are lower than unity across the entire $\mathrm{Re}$ range $\left(5.0\times {10}^{3}\le \mathrm{Re}\le 4.0\times {10}^{4}\right).$ This implies that if it is taken into consideration enhancement in convective heat transfer and $\mathrm{dP}$ due to existence of turbulators, only square and octagon wire-fitted channels have more favorable energy performance, and they allow for operating at a low-flow rate ($\mathrm{Re}=5.0\times {10}^{3})$ while surpassing the performance of an empty channel. Beyond this $\mathrm{Re}$, there is no enhancement in the overall thermal–hydraulic performance for all channel designs. This is attributed to the fact that at $\mathrm{Re}=5.0\times {10}^{3}$, enhancement in average surface $\mathrm{Nu}$ obtained for both configurations (square and octagon CWIs) is accompanied by a moderate increase in $\mathrm{dP}$ while the enhancement in $\mathrm{Nu}$ observed for other designs at higher $\mathrm{Re}$ $\left(\mathrm{Re}>5.0\times {10}^{3}\right)$ is accompanied with high increase in $\mathrm{dP}$. Of course, this remark assumes that hydrodynamic performance is as important as CHTP. Otherwise, significant improvements in CHTP are easily achieved with various channel designs used in the present work. Also, it is notable that as $\mathrm{Re}$ increases the values of ${\mathrm{PEC}}_{\mathrm{geo}}$ decreases. For instance, while using cylinder nanofluid ($\mathrm{VR}=2\mathrm{\%})$ across the aforementioned entire range of $\mathrm{Re}$, the values of ${\mathrm{PEC}}_{\mathrm{geo}}$ for ducts provided with circular, triangular, square, square-diamond, hexagon, octagon, and decagon CWIs decrease by 34.59, 33.16, 57.92, 22.75, 39.22, 44.75, and 26.96%, respectively. This result is expected as ${\mathrm{Nu}}^{\mathrm{G}}$ decreases as $\mathrm{Re}$ increases while ${\mathrm{fr}}^{\mathrm{G}}$ increases as $\mathrm{Re}$ increases. To examine the influence of nanoparticle shapes on the overall energetic performance, Fig. 12b shows the variation in ${\mathrm{PEC}}_{\mathrm{nf}}.$ It can be seen that while the empty channel is operating at $\mathrm{Re}=5.0\times {10}^{3}$, nanofluid prepared from platelet, cylinder, brick, and blade nanoparticles has values of ${\mathrm{PEC}}_{\mathrm{nf}}$ that is greater than unity. This implies that the usage of these nanofluids is proven to be an energy-efficient solution compared with base fluid (water–ethylene glycol). Furthermore $,$ according to the previously mentioned operating condition and across the range of nanoparticle $\mathrm{VR}$ considered $\left(0\le \varphi \le 4\%\right)$, the percentage improvements of ${\mathrm{PEC}}_{\mathrm{nf}}$ for platelet, blade, cylinder, brick, and oblate-spheroid boehmite/water–ethylene glycol are 35.37, 11.96, 28.19, 12.12, and − 4.66%, respectively.

Fig. 12

a Variation of ${\mathrm{PEC}}_{\mathrm{geo}}$ for blade nanofluid ($\varphi$ = 4%) against Re. b Variation of ${\mathrm{PEC}}_{\mathrm{nf}}$ for various nanofluids shape against nanoparticle volume ratios while empty duct at Re = 5000

Conclusions

This study examined the turbulent heat transfer performance of nanofluid [($\upgamma -\mathrm{AlO}(\mathrm{OH}))$/water–ethylene glycol (50:50)] prepared from various nanoparticle shapes (blade, brick, cylinder, oblate-spheroid, and platelet), which flows through a square duct fitted with CWI of various geometries (circular, triangular, square, square-diamond, hexagon, octagon, and decagon). The channel is subject to a uniform external heat flux, a well-validated computational model has been developed to capture the hydrothermal characteristics in the duct when changing the fitted CWI configurations, nanoparticle’s shape, and volume ratio $(\mathrm{VR})$. Some of the main findings of this study are as follows:

• 1. The presence of CWI in the channel enhances CHTP. When blade nanofluid $(\varphi =2\%)$ is at a minimum flow rate $(\mathrm{Re}=5.0\times {10}^{3})$, the normalized $\left({\mathrm{Nu}}^{\mathrm{G}}\right)$ for duct provided with CWI (circular, triangular, square, square-diamond, hexagon, octagon, and decagon) is 1.94, 1.97, 2.64, 1.84, 2.09, 2.24, and 1.79, respectively.

• 2. The increase in $\mathrm{Re}$ enhances CHTP for all channel configurations considered. Across the range of $\mathrm{Re}$ considered $\left(5.0\times {10}^{3}\le \mathrm{Re}\le 4.0\times {10}^{4}\right)$, the $\mathrm{Nu}$ for blade nanofluid ($\varphi =2\%)$ flowing through the same aforementioned channel designs increases by a factor of 4.02, 4.02, 2.61, 4.57, 3.73, 3.42, and 4.43, respectively.

• 3. Among the various shapes of nanoparticles considered, it was found that nanofluid prepared from platelet nanoparticles has the best thermal performance while oblate-spheroid nanofluid has the minimum $\mathrm{Nu}$. At the lowest $\mathrm{Re}$, the $\mathrm{Nu}$ for platelet, blade, cylinder, brick, and oblate-spheroid boehmite/water–ethylene glycol nanofluids flowing through the empty channel is 89.31, 79.62, 83.77, 77.27, and 73.26, respectively, for $\varphi =2\%$.

• 4. The wire-fitted channels have higher $\mathrm{fr}$ compared with the empty channel. At $\mathrm{Re}=5.0\times {10}^{3},$ the normalized ${(\mathrm{fr}}^{\mathrm{G}})$ assigns to 10.27, 10.84, 9.42, 11.14, 10.91, 10.20, and 10.97, respectively, while using blade nanofluid ($\varphi =2\%)$ for the same design order as described above.

• 5. The channels with square and octagon CW configurations are better than empty channel at low $\mathrm{Re}$ (${\mathrm{PEC}}_{\mathrm{geo}}>1)$. However, at high $\mathrm{Re}$, neither of them perform very well (${\mathrm{PEC}}_{\mathrm{geo}}<1)$.

• 6. The increase in $\mathrm{Re}$ decreases ${\mathrm{PEC}}_{\mathrm{geo}}$. Across the aforesaid range of $\mathrm{Re}$, the ${\mathrm{PEC}}_{\mathrm{geo}}$ for channels provided with circular, triangular, square, square-diamond, hexagon, octagon, and decagon CWI decreases by 34.59, 33.16, 57.92, 22.75, 39.22, 44.75, and 26.96%, respectively, while using cylinder nanofluid ($\varphi =2\%)$.

• 7. In terms of ${\mathrm{PEC}}_{\mathrm{nf}}$, all nanofluids prepared from various nanoparticle shapes are better compared with base fluid (boehmite/water–ethylene glycol (50:50)), except the one prepared from oblate-spheroid nanoparticles.

• 8. The increase in nanoparticle $\varphi$ enhances ${\mathrm{PEC}}_{\mathrm{nf}}$. When the empty channel is operating at $\mathrm{Re}=5.0\times {10}^{3}$, the ${\mathrm{PEC}}_{\mathrm{nf}}$ for platelet, blade, cylinder, brick, and oblate-spheroid boehmite/water–ethylene glycol nanofluids increases by 35.37, 11.96, 28.19, 12.12, and − 4.66%, respectively.

Appendices

Appendix

Appendix A: Correlations used for model validation

Dittus–Boelter [51]:

$${\text{Nu}} = 0.023 {\text{Re}}^{0.8} {\text{Pr}}^{0.4} ;\;0.6 \le {\text{Pr}} \le 160;\;{\text{Re}} \ge 10^{4}$$

(31)

Sieder and Tate [52]:

$${\text{Nu}} = 0.027 {\text{Re}}^{0.8} {\text{Pr}}^{0.33} \left( {\frac{\mu }{{\mu_\text{w} }}} \right)^{0.14} ;\;0.7 \le {\text{Pr}} \le 16700;\;{\text{Re}} \ge 10^{4}$$

(32)

Gnielinski [53]:

$${\text{Nu}} = 0.012 \left( {{\text{Re}}^{0.87} - 280} \right){\text{Pr}}^{0.4} ;\;1.5 \le {\text{Pr}} \le 500;\;3000 \le {\text{Re}} \le 10^{5}$$

(33)

Petukhov [54]:

$$f = \left( {0.79 {\text{ln}} {\text{Re}} - 1.64} \right)^{ - 2} ;\;3,000 \le {\text{Re}} \le 5 \times 10^{6}$$

(34)

Blasius [55]:

$$f = 0.316 {\text{Re}}^{ - 0.25} ;\;3000 \le {\text{Re}} \le 2 \times 10^{4}$$

(35)

$$f = 0.184 {\text{Re}}^{ - 0.25} ;\;{\text{Re}} \ge 2 \times 10^{4}$$

(36)

Filonenko [56]:

$$f = \left( {1.82 {\text{log}} {\text{Re}} - 1.64} \right)^{ - 2} ;\;2300 < {\text{Re}} < 10^{6}$$

(37)