Introduction

The major challenge of different industries with high heat flux, including heat exchanger, nuclear reactor, fuel cell and electronic, is enhancement of the efficiency, and miniaturization, cost and materials in construction. The miniature channels, vortex generators, pin fins and nanofluids are passive techniques of increasing the heat transfer, each of which separately has a significant effect on the thermal performance of the system. Thus, combining these techniques, the influence on performance of system will increase. Among the advantages of small channels are the high ratio of heat transfer area to fluid volume, low weight and materials and thus high heat transfer coefficient [1]. The nanofluids with higher thermal conductivity than pure water and also their Brownian motion and also the vortex generators and pin fins by increasing the total heat transfer surface and the creation of vortices and mixing inside flow field improve the thermal efficiency. Many numerical and experimental studies have been carried out on the nanofluids and or vortex generators [2,3,4,5,6,7,8]. But the number of studies on the use of vortex generator inside miniature channel with nanofluid as coolant fluid is limited. In general, in most studies, the shape of vortex generators is triangular and rectangular, and the coolant fluid is pure water. The more research is needed to comprehensive understand of the thermal and frictional performance of the different coolant fluids and also efficient design of the cooling miniature channels. Liu et al. [7] experimentally investigated the heat transfer and pressure drop of pure water in rectangular vortex generator micro-channel. They studied the number and various angles of vortex generators inside micro-channel. They observed that the heat transfer increased up to 9–21%, while pressure drop increased up to 34–83% for laminar flow regime. Ebrahimi et al. [6] numerically studied the thermal and frictional performance of pure water in rectangular vortex generator micro-channel. They investigated the various angles of vortex generators inside micro-channel. They reported that the heat transfer improved about 2–25%, while the friction factor increased about 4–30% with vortex generators inside channel. Datta et al. [9] numerically investigated the thermal and frictional performance of water in rectangular vortex generator micro-channel. The highest overall performance reported for angle of 30° of vortex generators inside micro-channel. Goodarzi et al. [10, 11] experimentally investigated the heat transfer of MWCNT-based nanofluid and nitrogen-doped graphene nanofluid. Ebrahimi et al. [12] numerically studied the thermal and frictional performance of CuO–water and Al2O3–water nanofluids in vortex generator micro-channel. They reported that the heat transfer improved about 2.29–30.63% for Al2O3–water and 9.44–53.06% for CuO–water. Also, the pressure drop increased about 3.49–16.85 and 6.5–17.7% for Al2O3–water and CuO–water nanofluids, respectively. Nasiri et al. [13] numerically investigated the heat transfer and flow of nanofluid around a cylinder by smoothed particle hydrodynamics method. Their results showed that the heat transfer improved using nanofluid flow comparing to base fluids. Heydari et al. [14] numerically studied the effect of triangular ribs attack angle in micro-channel. The working fluid was water/Ag nanofluid. They reported that using higher angle of triangular rib, the heat transfer significantly improved.

In most studies, the cylindrical pin fins and water flow as working fluid have been used inside channel [15,16,17,18,19]. All these studies reported increasing the heat transfer inside micro/mini-channel due to using the pin fins. Recently, Sadrabadi Haghighi et al. [20] experimentally investigated the thermal performance of plate cubic pin fin and plate fin heat sinks. They observed the higher thermal performance and lower thermal resistance for plate cubic pin fin heat sink than plate fin heat sink. Some studies have used nanofluids as coolant fluid in circular and non-circular channel [21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36]. In numerical research, the nanofluid flow simulated only as single phase [21, 23, 25]. Muhammad Ali and Arshad [26] experimentally studied the thermal performance inside inline and staggered square pin fin mini-channel. The working fluid was the TiO2–H2O nanofluid. They reported that the staggered pin fin mini-channel had the higher heat transfer than inline pin fin mini-channel. The higher heat transfer reported for TiO2–H2O nanofluid compared to pure water. Tullius et al. [37] numerically investigated the thermal and frictional performance of pure water in different pin fin mini-channel. The considered cross-sectional shapes of pin fin were the ellipse, square, triangular, hexagonal, diamond and circular. Their results show that the triangular and square pin fins had the highest thermal performance and pressure drop, respectively. Also, the circular and ellipse pin fins had the least performance.

Safaei et al. [38] numerically studied the mixed convection heat transfer in inclined ribbed micro-channel. The working fluid was the Cu–water nanofluid. They investigated the effect of inclination angle, rib shape, volume fraction of nanoparticles and shear and buoyancy forces on thermal and hydraulic performance. They reported that the heat transfer enhanced by increasing the volume fraction of nanoparticles and inclination angle. Abbasian Arani et al. [39] numerically investigated the heat transfer and flow of SWCNT–water nanofluid in double layered micro-channel heat sink. They observed that the overall performance improved by increasing the concentration of nanoparticles. Also, the thermal resistance reduced by decreasing the truncated lengths.

In the present study, the effect of different angles of wavy vortex generators in cooling miniature channel on the thermal and frictional performance of the 0.5 vol% single-walled carbon nanotube (SWCNT)–H2O nanofluid and also the performance of 1 vol% alumina (Al2O3)–H2O nanofluid in triangular longitudinal pin fin miniature channel are numerically investigated. For validation of numerical modeling, the experimental study is conducted for pure water inside wavy vortex generator miniature channel and triangular pin fin miniature channel.

Experimental

Experimental setup

The test loop consists of a tank, a centrifuge pump, a flow line, a bypass flow line, test section, a DC power supply and cooling unit. The coolant fluid flows inside the main line and bypass line by a ball valve and a needle valve, respectively. The cooling unit includes a plate heat exchanger and also a constant temperature bath. The thermal and frictional performances are estimated by the recorded fluid and the hot wall temperatures and also pressure drop of fluid, respectively. Two Pt-100 thermocouples with the accuracy of ± 0.1 °C are used to measure the inlet and outlet temperatures of coolant fluid and also the hot wall temperature are measured by three Pt-100 thermocouples. The two pressure transmitters (model: Sensys, PTCH0001BCIA) with accuracy of ± 1 Pa are used to measure the inlet and outlet pressures of coolant fluid. The mass flow rate is measured by a digital balance with accuracy of 0.001 g, a digital timer and two solenoid valves in wavy vortex generator miniature channel. An ultrasonic flow meter (Flownetix® 100series™) is used to measure the volumetric flow rate in triangular pin fin miniature channel. To investigate the thermal and frictional performance of water inside wavy vortex generator miniature channel and triangular longitudinal pin fin miniature channel, the two different test sections [i.e., type (1) for wavy vortex generator and type (2) for triangular pin fin] are constructed and tested. The test sections are made by the CNC device. The test section of type (1) is an aluminum block that consists of two inlet and outlet manifolds, a wavy vortex generator miniature channel and ten cartridge heaters. The heaters which are put in the bottom of the block are parallel to each other as located in two rows with equal distances. The test section of type (2) consists of three steel sheets (i.e., top, middle and bottom), two cylindrical inlet and outlet manifolds and a plate heater. The miniature channel is created on two sheets in two upper and lower halves (i.e., top and middle sheets). The triangular pin fins are inserted in lower half of miniature channel with equal distance. Figure 1 shows the designed experimental setup, test sections and their dimensions.

Fig. 1
figure 1

The designed experimental setup and test sections

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Data reduction and experimental uncertainties

The inlet velocity of fluid is calculated based to Reynolds number (Re):

$$Re = \frac{{\rho u_{\text{in}} D_{\text{h}} }}{\mu }$$
(1)

and the mass flow rate of fluid is computed as follows:

$$\dot{m} = \rho u_{\text{in}} A_{\text{c}}$$
(2)

The total heat absorbed by coolant fluid can be estimated as follows:

$$Q = \dot{m}C_{\text{p}} (T_{{{\text{f}},{\text{out}}}} - T_{{{\text{f}},{\text{in}}}} )$$
(3)

Using Newton’s law of cooling, the average convective heat transfer coefficient (h) is applied to evaluate the thermal performance with the total heat transfer area (At), the average of bulk fluid temperatures (Tf), and the average of hot wall temperatures (Tavg,W):

$$h = \frac{Q}{{A_{\text{t}} (T_{\text{W,Avg}} - T_{\text{f}} )}}$$
(4)
$$T_{\text{f}} = \frac{{T_{\text{in}} + T_{\text{out}} }}{2}$$
(5)

The hydraulic diameter of channel (Dh) based on the channel height (Hch), channel width (Wch) and the average Nusselt number (Nu) is defined as follows:

$$D_{\text{h}} = \frac{{2 \times H_{\text{ch}} \times W_{\text{ch}} }}{{H_{\text{ch}} + W_{\text{ch}} }}$$
(6)
$$Nu = \frac{{hD_{\text{h}} }}{{k_{\text{f}} }}$$
(7)

The frictional performance is estimated by the fanning friction factor (f) based on the pressure drop of coolant fluid (ΔP) as follows:

$$f = \frac{{\Delta PD_{\text{h}} }}{{2L_{\text{ch}} \rho u_{\text{in}}^{2} }}$$
(8)

The considered performance evaluation criteria (PEC) for evaluating the overall performance of different geometries of the miniature channel are [40]:

$${\text{PEC}} = \frac{{{{Nu_{\text{i}} } \mathord{\left/ {\vphantom {{Nu_{\text{i}} } {Nu_{\text{o}} }}} \right. \kern-0pt} {Nu_{\text{o}} }}}}{{({{f_{\text{i}} } \mathord{\left/ {\vphantom {{f_{\text{i}} } {f_{\text{o}} }}} \right. \kern-0pt} {f_{\text{o}} }})^{1/3} }}$$
(9)

where Nui and fo are the Nu number and f factor of reference model, respectively. In the triangular pin fin, the performance of water inside miniature channel is considered as reference. In the wavy vortex generator miniature channel, the reference model is the performance of water inside miniature channel with vortex generator including of angle of 0°, wave amplitude of 1 mm, wavelength of 10 mm and the number of 6. The local skin friction coefficient (Cf) is:

$$C_{\text{f}} = \frac{{\tau_{\text{W}} }}{{1/2\rho U_{{}}^{2} }}$$
(10)

where U is the averaged velocity of coolant fluid and τW is the wall shear stress. The experimental uncertainties are calculated using the Kline and McClintock method [41] and the following equation,

$$\frac{\delta P}{P} = \left[ {\left( {n_{1} \frac{{\delta x_{1} }}{{x_{1} }}} \right)^{2} + \left( {n_{2} \frac{{\delta x_{2} }}{{x_{2} }}} \right)^{2} + \left( {n_{3} \frac{{\delta x_{3} }}{{x_{3} }}} \right)^{2} + \cdots } \right]^{1/2}$$
(11)

where δP and δx are the uncertainties of the dependent and independent parameters, respectively, and the ni is power of independent parameters. The uncertainty equations for the heat transfer coefficient and friction factor are:

$$\frac{\delta h}{h} = \left[ {\left( {\frac{{\delta \dot{m}}}{{\dot{m}}}} \right)^{2} + \left( {\frac{{\delta c_{\text{p}} }}{{c_{\text{p}} }}} \right)^{2} + \left( {\frac{{\delta (T_{\text{f,out}} - T_{\text{f,in}} )}}{{(T_{\text{f,out}} - T_{\text{f,in}} )}}} \right)^{2} + \left( {\frac{{\delta (T_{\text{W,avg}} - T_{\text{f}} )}}{{(T_{\text{W,avg}} - T_{\text{f}} )}}} \right)^{2} + \left( {\frac{{\delta A_{\text{t}} }}{{A_{\text{t}} }}} \right)^{2} } \right]^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}}}$$
(12)
$$\frac{\delta f}{f} = \left[ {\left( {\frac{\delta \Delta P}{\Delta P}} \right)^{2} + \left( {\frac{{\delta D_{\text{h}} }}{{D_{\text{h}} }}} \right)^{2} + \left( {\frac{{\delta L_{\text{ch}} }}{{L_{\text{ch}} }}} \right)^{2} + \left( {\frac{\delta \rho }{\rho }} \right)^{2} + \left( {2\frac{{\delta u_{\text{in}} }}{{u_{\text{in}} }}} \right)^{2} } \right]^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}}}$$
(13)

Table 1 represents the uncertainty of the experimentally obtained parameters. The maximum uncertainty for the f factor and the h is about 5.8% and 4.15%, respectively.

Table 1 Uncertainty of the experimental measured parameters
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Numerical section

Geometries and computational domains

In Fig. 2a, b, the computational domains, the geometric parameters of the triangular pin fin miniature channel and wavy vortex generator miniature channel and also their dimensions given in Table 2 are presented. The channel width (Wch) of 10 mm, channel height (Hch) of 10 mm and channel length (Lch) of 150 mm are considered for triangular pin fin miniature channel. Three triangular longitudinal pin fins are inserted at the top of the miniature channel. The space between the pin fins is constant (i.e., 37.5 mm). The triangular pin fins length (Lf) is equal to the channel width (i.e., 10 mm). The height of pin fins (Hf) is half the channel height (i.e., 5 mm). The width of pin fins (Wf) is 6 mm. The base height (Hb) in triangular pin fin miniature channel is 3 mm. Also, the dimensions of miniature channel in geometry of the wavy vortex generator miniature channel are 20 mm × 1 mm × 100 mm (width × height × length). The spacing between wavy vortex generators is the same, and their number is 4. The wavelength (Lf), height (Hf) and width (Wf) of wavy vortex generators is 15 mm, 1 mm and 1 mm, respectively. The wave amplitude (A) of wavy vortex generator is 1 mm. The angle of wavy vortex generator is the angle of axis of a wavy vortex generator with x-axis. For improvement of the thermal performance of the miniature channel, the different angles of wavy vortex generators have been investigated. In Fig. 2c, the different angles of wavy vortex generators are shown. The angles of 0°, 60° and − 60° are considered. The base height (Hb) in wavy vortex generator miniature channel is 0.5 mm.

Fig. 2
figure 2

Computational domains, geometric parameters and boundary conditions of a triangular longitudinal pin fin miniature channel and b wavy vortex generator miniature channel and c different angles of wavy vortex generator

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Table 2 Geometric parameters and their dimensions (mm)
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Thermophysical properties

The present study, the working fluids are composed of the pure water, 1 vol% Al2O3–H2O nanofluid and 0.5 vol% SWCNT–H2O nanofluid. Their thermophysical properties in T = 300 K are presented in Table 3 [42, 43]. The density and specific heat of coolant nanofluids [44, 45], the thermal conductivity of SWCNT–H2O nanofluid according to Hamilton and Crosser model [46] and also the thermal conductivity of Al2O3–H2O nanofluid according to Maxwell–Garnett’s model [47] are estimated using the following equations:

$$\rho_{\text{m}} = (1 - \varphi )\rho_{\text{f}} + \varphi \rho_{\text{p}}$$
(14)
$$(\rho C_{\text{p}} )_{\text{m}} = (1 - \varphi )(\rho C_{\text{p}} )_{\text{f}} + \varphi (\rho C_{\text{p}} )_{\text{p}}$$
(15)
$${{k_{\text{m}} } \mathord{\left/ {\vphantom {{k_{\text{m}} } {k_{\text{f}} }}} \right. \kern-0pt} {k_{\text{f}} }} = \frac{{k_{\text{p}} (1 + 5\varphi ) + 5k_{\text{f}} (1 - \varphi )}}{{k_{\text{p}} (1 - \varphi ) + k_{\text{f}} (5 + \varphi )}},\quad n = 6$$
(16)
$$k_{\text{m}} = \frac{{k_{\text{p}} + (n - 1)k_{\text{f}} - (n - 1)\varphi (k_{\text{f}} - k_{\text{p}} )}}{{k_{\text{p}} + (n - 1)k_{\text{f}} + \varphi (k_{\text{f}} - k_{\text{p}} )}}k_{\text{f}} ,\quad n = 3$$
(17)

The viscosity of the SWCNT–H2O nanofluid and the viscosity of Al2O3–H2O nanofluid are calculated by the Baboo et al. model [48] and the Brinkman model [49], respectively:

Table 3 The thermophysical properties of H2O, SWCNT and Al2O3 nanoparticles at T = 300 K
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$$\frac{{\mu_{\text{m}} }}{{\mu_{\text{f}} }} = 1 + a\varphi + b\varphi^{2} ,\quad a = - 0.50437,\quad b = 1.744$$
(18)
$$\mu_{\text{m}} = \frac{1}{{(1 - \varphi )^{2.5} \,}}\mu_{\text{f}}$$
(19)

Boundary conditions, assumptions and governing equations

The computational domains consist of the two inlet and outlet extended regions and the miniature channel. The velocity inlet and outflow conditions are applied at the inlet and outlet of miniature channel, respectively. The constant heat flux condition is used at the bottom wall of channel. For the left and right side surfaces, the symmetry condition is applied. Also, for top wall of channel and the top, bottom, left and right walls of two extended regions, the no-slip and adiabatic condition is used. The following assumptions for nanofluids flow are considered:

  • laminar flow, Newtonian, steady state, incompressible, two-phase and three dimensional.

The free convection and radiation heat transfer have been considered to be negligible. The nanofluid flow has been simulated by the mixture two-phase model. According to the boundary conditions and mentioned assumptions, the governing equations are as follows [50]:

  • Continuity equation using the nanofluid mixture density (ρm) and mass average velocity (\(\vec{u}_{\text{m}}\)) is:

    $$\nabla .(\rho_{\text{m}} \vec{u}_{\text{m}} ) = 0$$
    (20)
    $$\vec{u}_{\text{m}} = \frac{{\sum\nolimits_{k = 1}^{n} {\varphi_{\text{k}} \rho_{\text{k}} \vec{u}_{\text{k}} \,} }}{{\rho_{\text{m}} }}$$
    (21)

    where \(\vec{u}_{k}\) is the velocity of phase k (primary phase: base fluid and secondary phase: nanoparticles) and n is the number of phases.

  • Momentum equation using the drift velocity of phase k (\(\vec{u}_{dr,k}\)):

    $$\nabla .(\rho_{\text{m}} .\vec{u}_{\text{m}} .\vec{u}_{\text{m}} ) = - \nabla P + \nabla .(\mu_{\text{m}} \nabla \vec{u}_{\text{m}} ) + \nabla .\left( {\sum\limits_{k = 1}^{n} {\varphi_{\text{k}} \rho_{\text{k}} } \vec{u}_{{{\text{dr}},{\text{k}}}} \,\vec{u}_{{{\text{dr}},{\text{k}}}} } \right)$$
    (22)
    $$\vec{u}_{\text{{dr}},{\text{k}}} = \vec{u}_{\text{k}} - \vec{u}_{\text{m}}$$
    (23)

  • Energy equation:

    $$\nabla .\sum\limits_{k = 1}^{n} {(\varphi_{\text{k}} \rho_{\text{k}} \vec{u}_{\text{k}} C_{\text{pk}} T)} = \nabla .(k_{\text{m}} \nabla T)$$
    (24)

  • Volume fraction equation:

    $$\nabla .(\varphi_{\text{p}} \rho_{\text{p}} \vec{u}_{\text{m}} ) = - \nabla .(\varphi_{\text{p}} \rho_{\text{p}} \vec{u}_{\text{dr,p}} )$$
    (25)

The slip velocity is the secondary phase velocity than the primary phase velocity:

$$\vec{u}_{\text{pf}} = \vec{u}_{\text{p}} - \vec{u}_{\text{f}}$$
(26)

The secondary phase drift velocity is:

$$\vec{u}_{\text{dr,p}} = \vec{u}_{\text{pf}} - \sum\limits_{k = 1}^{n} {\frac{{\varphi_{k} \rho_{k} }}{{\rho_{\text{m}} }}} \vec{u}_{{{\text{f}}k}}$$
(27)

The correlation of Manninen et al. [50] is applied for the slip velocity:

$$\vec{u}_{\text{pf}} = \frac{{\rho_{\text{p}} d_{\text{p}}^{2} }}{{18\mu_{\text{f}} f_{\text{drag}} }}\frac{{(\rho_{\text{p}} - \rho_{\text{m}} )}}{{\rho_{\text{p}} }}\vec{a}_{\text{p}}$$
(28)

The drag coefficient is estimated using correlation of Schiller and Naumann [51]:

$$f_{\text{drag}} = \left\{ {\begin{array}{*{20}l} {1 + 0.15\,Re_{\text{p}}^{0.687} } \hfill & {Re_{\text{p}} \le 1000} \hfill \\ {0.0183\,Re_{\text{p}} } \hfill & {Re_{\text{p}} \succ 1000} \hfill \\ \end{array} } \right.$$
(29)

where the Re number and the nanoparticles acceleration based to the gravitational acceleration (g) are defined as follows:

$$Re_{\text{p}} = \frac{{\rho_{\text{m}} u_{\text{m}} d_{\text{p}} }}{{\mu_{\text{m}} }}$$
(30)
$$\vec{a}_{\text{p}} = \vec{g} - (\vec{u}_{\text{m}} .\nabla )\vec{u}_{\text{m}}$$
(31)

Numerical method

The finite volume method is used to solve the governing equations. The second-order upwind scheme is applied for discretizing the momentum and energy equations, and the SIMPLE algorithm is used for coupling the velocity and pressure. The convergence criteria or residuals of 10−5 for the continuity, momentum and volume fraction and that of 10−8 for energy equation are considered.

Results and discussion

Grid independence and validation procedure

The non-uniform grids with structured and hexagonal meshes are used for triangular pin fin miniature channel and wavy vortex generator miniature channel. To check the independence of the numerical results to the mesh number, the grid independence studies are performed. For triangular pin fin at Re = 1200, the h is calculated for five grids with total number of 697,900, 968,240, 1,183,140, 1,370,250 and 1,574,500. The grid number of 1,370,250 with mean deviation 1.88% is applied for the modeling. For wavy vortex generator miniature channel with wave amplitude of 1 mm, angle of 0° and wavelength of 15 mm at Re = 800, the h and ΔP are computed for seven grids with total number of 388,248, 610,304, 903,528, 1,293,732, 1,604,541, 1,993,920 and 2,208,192. The grid of 1,993,920 with mean deviation of 0.72% for the h and 0.03% for the ΔP is adopted for present simulation.

To validate of numerical simulation, the experiments for water flow inside triangular longitudinal pin fin miniature channel and the wavy vortex generator miniature channel are conducted. The heater power of 170 W is used on bottom wall of triangular pin fin and 252 W for wavy vortex generator miniature channel. For both geometries, the numerical results and experimental data have been compared and also an acceptable agreement has been observed (Fig. 3a, b). The mean deviations of 4% and 6.17% have been obtained for the h in triangular pin fin miniature channel and the wavy vortex generator miniature channel, respectively.

Fig. 3
figure 3

The comparison between calculated numerical results and recorded experimental data a triangular longitudinal pin fin miniature channel and b wavy vortex generator miniature channel

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Triangular longitudinal pin fin miniature channel

Figure 4a shows the variation of h versus Re number for pure water inside miniature channel, 1 vol% Al2O3–H2O nanofluid inside miniature channel and also 1 vol% Al2O3–H2O nanofluid inside triangular longitudinal pin fin miniature channel. In all Re numbers, the h is increased by increasing the Re number. For example, in miniature channel, with increasing the Re = 300–1500, the h of pure water is increased about 86%. By adding nanoparticles of 1 vol% Al2O3 into the water inside miniature channel, the h is increased about 39.5% in all Re numbers. Also, by inserting of three triangular longitudinal pin fins in miniature channel and also using the Al2O3–H2O nanofluid, the h is significantly improved about 82% compared to Al2O3–H2O nanofluid in miniature channel and about 152% compared to pure water in miniature channel over all Re numbers. The adding of Al2O3 nanoparticles into the pure water leads to improve the thermal performance due to higher thermal conductivity. Also, inserting the pin fins in the channel affect the fluid behavior around and after from them. The phenomena such as the separation in fluid layers, deflection, recirculation and reattachment occur in the flow field that lead to creation the vortices and the mixing within fluid and more heat dispersal between fluid layers; thus, the heat exchange is increased. As shown in Fig. 4, with increasing the Re number, the effect of geometry of miniature channel or embedding of pin fins on the thermal performance is greater. As, in Re numbers below 600, the increase rate of h for Al2O3–H2O nanofluid inside triangular longitudinal pin fin miniature channel is about 54.5% than that of in miniature channel. But, with increasing the Re number gradually, the increase rate is significantly increased; about 105.6%. At higher flow rates, the vortices are intensified, and as a result the heat transfer rate is improved. The effect of coolant fluid type and geometry of cooling miniature channel on the Nu number is shown in Fig. 4b. The same trend with the h is also observed for the Nu number of pure water and Al2O3–H2O nanofluid in miniature channel and Al2O3–H2O nanofluid in triangular pin fin miniature channel. The value of Nu number of Al2O3–H2O nanofluid is enhanced about 35.6% compared to that of pure water in miniature channel. Also, using triangular pin fins, the thermal performance has improved significantly. The Nu number of Al2O3–H2O nanofluid is increased about 96.5% in triangular pin fin miniature channel. Also, the simultaneous use of two passive methods of the nanofluid and triangular pin fins has been led to an increase of about 166% of the Nu number than pure water in miniature channel. The reasons for improving the thermal performance are described above. Figure 5 shows the ΔP of the pure water and 1 vol% Al2O3–H2O nanofluid in miniature channel and Al2O3–H2O nanofluid in triangular longitudinal pin fin miniature channel. According to the results shown in the figure, the value of ΔP is increased with increasing the Re number, so that the increase rate at Re numbers above 600 is the highest. The nanoparticles due to their density and viscosity lead to increase pressure drop of the base fluid. The use of Al2O3–H2O nanofluid led to an increase in ΔP of about 14% compared to pure water inside the miniature channel. By inserting the pin fins in miniature channel, the ΔP of fluid in all Re numbers is increased. So that this enhancement rate for pin fins is much higher than that for the nanoparticles. The increasing of the ΔP of Al2O3–H2O nanofluid in triangular pin fin miniature channel is about 178% than that of inside miniature channel. The fluid collision with pin fins inside miniature channel leads to an increase in pressure drop of coolant fluid. Figure 6 shows the variation of Cf along the channel length for the pure water and Al2O3–H2O nanofluid in miniature channel and Al2O3–H2O nanofluid in triangular pin fin miniature channel at Re = 1200. In fact, the effect of coolant fluid type and geometry of channel on the frictional performance is investigated. By increasing the thickness of the boundary layer, the value of Cf along the channel length is reduced. The Al2O3–H2O nanofluid has higher Cf due to their density and viscosity higher than pure water, about 11.4%. By inserting triangular pin fins within the miniature channel, the trend of Cf variation along channel length is different and its amount is increased; on average about 178% for Al2O3–H2O nanofluid. The pin fins lead to the growth and destruction of the boundary layer along the length of the miniature channel. With the destruction of the boundary layer, the amount of Cf increases, and with the growth of the boundary layer, its amount decreases. The value of overall performance or PEC at Re = 300, 900 and 1500 for pure water and Al2O3–H2O nanofluid in miniature channel and Al2O3–H2O nanofluid inside triangular pin fin miniature channel is shown in Fig. 7. The water flow inside miniature channel is considered as reference model, and its performance is compared with that of Al2O3–H2O nanofluid and also Al2O3–H2O nanofluid inside triangular pin fin miniature channel. The highest overall performance is observed for Al2O3–H2O nanofluid in triangular pin fin miniature channel. As, the increase rate of PEC of the Al2O3–H2O nanofluid inside triangular pin fin miniature channel is about 83% compared to that of pure water inside miniature channel. Also, the PEC of Al2O3–H2O nanofluid than pure water is increased about 35%.

Fig. 4
figure 4

The variation of a heat transfer coefficient and b Nusselt number versus Reynolds number for pure water in miniature channel and Al2O3–H2O nanofluid in miniature channel and triangular pin fin miniature channel

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Fig. 5
figure 5

The variation of pressure drop versus Reynolds number for pure water in miniature channel and Al2O3–H2O nanofluid in miniature channel and triangular pin fin miniature channel

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Fig. 6
figure 6

The variation of local skin friction coefficient along the channel length for a pure water in miniature channel, b Al2O3–H2O nanofluid in miniature channel, and c Al2O3–H2O nanofluid in triangular pin fin miniature channel at Re = 1200

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Fig. 7
figure 7

The variation of overall performance versus Reynolds number for pure water in miniature channel and Al2O3–H2O nanofluid in miniature channel and triangular pin fin miniature channel

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Wavy vortex generator miniature channel

The effect of angle of 0°, − 60° and 60° of wavy vortex generator with wave amplitude of 1 mm and wavelength of 15 mm inside miniature channel on the h, ΔP and Nu number of the pure water and 0.5 vol% SWCNT–H2O nanofluid at Re = 600 is shown in Fig. 8a–c. The distance from the beginning of the channel and also the spacing between wavy vortex generators is 5 mm. In all different angles, the heat transfer and pressure drop are increased by adding the SWCNT nanoparticles to pure water, as the h and ΔP is increased about 197% and 4.2% for SWCNT–H2O nanofluid than that of pure water, respectively. As can be seen in Fig. 8a, b, the effect of adding nanoparticles to pure water on the thermal performance is more than its effect on the frictional performance of base fluid. By changing the angle of wavy vortex generator of 0°, the thermal and frictional performance of the pure water and also nanofluid is varied. For both the pure water and SWCNT–H2O nanofluid, the highest thermal and frictional performance is observed for angle of 60° and then for angle of − 60°. By changing the angle of 0° to 60°, the h is increased about 25.1% for SWCNT–H2O nanofluid and about 45.5% for pure water. Also, the ΔP is increased about 119.5% for the pure water and about 118% for the SWCNT–H2O nanofluid. By changing of angle of 60° to − 60°, the less effect on the h and ΔP is observed; about 10.5% for the h of the pure water and SWCNT–H2O nanofluid and about 1.03% for the ΔP of the pure water and 0.7% for SWCNT–H2O nanofluid. In the angle of 0°, near the walls where the changes are intense, the more of the streamlines are parallel. By changing the angle to 60°, the phenomena such as separation, deflection, recirculation and reattachment occur in flow field. Thus, vortices are created which lead to the mixing of fluid layers and destruction of the boundary layer. Hence, the heat transfer is increased and the thermal performance is improved. As shown in Fig. 8c, at all angles of the wavy vortex generators, the higher Nu numbers are observed for SWCNT–H2O nanofluid than pure water; for example about 63% at α = 60°. By changing of angle from 0° to 60° or − 60°, the thermal performance is improved in miniature channel, for example, by changing the α = 0° to 60° about 25% for nanofluid and 45.6% for pure water. For both coolant fluids, the order of enhancement rate is 60° > − 60° > 0°. The reasons for improving the thermal performance similar to that of the heat transfer coefficient are described above. The variation of Cf for SWCNT–H2O nanofluid along the length of the miniature channel at different angles of wavy vortex generators is shown in Fig. 9. Generally, by inserting the wavy vortex generators within the miniature channel, they lead to the growth and destruction of the boundary layer due to the creation of vortices in the corrugation troughs and behind them. As a result, with the thinning of the boundary layer, the amount of Cf is increased and with its growth, the Cf is decreased. As shown in Fig. 9, the trend of Cf variation along the channel length is different according to the angle of rotation of the wavy vortex generators inside the flow field. The effect of different angles of wavy vortex generators with wave amplitude of 1 mm and wavelength of 15 mm on the overall performance or PEC of the SWCNT–H2O nanofluid is shown in Fig. 10. According to figure, the angle of 60° of wavy vortex generator has highest overall performance in all Re numbers. Also, the angle of − 60° has lowest amount of the PEC. At Re numbers less than 600, the gradient of overall performance with Re number for all angles is more than Re numbers above 600. By changing the angle from 0° to 60°, the overall performance is increased about 12.2% for 0.5 vol% SWCNT–H2O nanofluid.

Fig. 8
figure 8

The variation of heat transfer coefficient and pressure drop of a pure water and b 0.5 vol% SWCNT–H2O nanofluid and c Nusselt number at different angles of wavy vortex generator with wavelength of 15 mm and wave amplitude of 1 mm inside miniature channel for at Re = 600

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Fig. 9
figure 9

The variation of local skin friction coefficient for 0.5 vol% SWCNT–H2O nanofluid at different angles of wavy vortex generator with wavelength of 15 mm and wave amplitude of 1 mm inside miniature channel at Re = 600

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Fig. 10
figure 10

The variation of overall performance in different angles of wavy vortex generator with wavelength of 15 mm and wave amplitude of 1 mm inside miniature channel for 0.5 vol% SWCNT–H2O nanofluid

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Conclusions

The most important findings of the present study are:

  • At all Re numbers, the h has been significantly improved by inserting the triangular longitudinal pin fins in miniature channel and also using the Al2O3–H2O nanofluid.

  • The amount of Cf along channel length has been increased by using the triangular pin fins within the miniature channel.

  • The highest overall performance has been observed for Al2O3–H2O nanofluid inside triangular longitudinal pin fin compared to that for Al2O3–H2O nanofluid and pure water inside miniature channel.

  • At all angles of wavy vortex generators, the thermal and frictional performances have been increased by adding the SWCNT nanoparticles. But, the effect of nanoparticles on the frictional performance was negligible.

  • By changing the angle of 0°, the thermal and frictional performances have been increased. Hence, the angle of wavy vortex generators has the significant effect on the heat transfer and pressure drop of the pure water and SWCNT–H2O nanofluid.

  • For both the pure water and SWCNT–H2O nanofluid, the highest and lowest amount of the h and ΔP has been observed for angle of 60° and 0°, respectively.

  • The highest and least amount of overall performance of SWCNT–H2O nanofluid has been observed for angles of 60° and − 60°, respectively.