Comparative study of two-phase flow approaches for modeling conjugate heat transfer in a three-dimensional wavy microchannel-heat sink with CuO-water nanofluid

Abstract

This study aims to numerically investigate the conjugate heat transfer (CHT) and the changes in the hydraulic and thermal terms of laminar flow (300 ≤ Re ≤ 800) in a 3-D wavy microchannel heat sink (MCHS). The obtained results from different nanofluid flow simulated models will be compared. The numerical simulation approaches used include single-phase, mixture, Eulerian, DPM, and DDPM. The effect of adding 1%, 3%, and 6% concentration of CuO nanoparticles (100 nm) in deionized water as the base fluid is examined on various parameters such as local and axial temperature, heat transfer convective coefficient, friction factor, Nusselt number (Nu), velocity distribution, and thermal and hydraulic boundary layer parameters. The substrate surface, due to its larger area surface and the amplitude of the sinusoidal wave, has a greater impact on the nanofluid flow as a channel bottom conjugate surface compared to others. The results obtained from the different models show close agreement with each other for various cases. For example, at Re = 304, the average Nu and friction factor for pure water are approximately 10.43 and 0.0260, respectively. For water with 0.01 concentration of CuO nanoparticles, Nu is found to be in the range of 10.43–10.78, and the friction factor ranges from 0.260 to 0.290, depending on the approach used for simulation. This shows a 2% increase in Nusselt number compared to a 0.05% increase in the friction factor. Moreover, these values increase by approximately 5.7% and 11.3% on average with an increase in CuO nanoparticle concentration to 0.06. Additionally, the DPM and DDPM models show a 15% difference in the scattering of CuO nanoparticles compared to the Eulerian-Eulerian models. The single-phase model presented in this study yields similar results to the multiphase approaches. Furthermore, the PEC parameter increases by about 1.9% with the addition of CuO nanoparticles at a concentration of 0.06.

Keywords

CFD simulation nanofluid flow Eulerian-Lagrangian approach Eulerian-Eulerian approach wavy microchannel heat sink

Introduction

The heat transfer enhancement issue has received basic attention in thermal applications. Several approaches have been examined to enhance heat transfer and get more efficient/effective heat exchanger devices in different engineering applications in recent years. The active and the passive methods in heating and especially in cooling are two common methods of heat exchange.¹ In the active method, the heat transfer system uses the energy, whereas the no energy is used in passive method. Various passive methods have been suggested in the literature to increase the performance of the heat transfer mechanism, for example using extensive surfaces, channels with non-circular cross-sections, vortex generators, surface materials, microchannels, additives in liquids, etc.^2,3 Because of the more conductive heat transfer coefficient of metals, adding their particles to the base fluid has increased the thermal conductivity of the fluid. Furthermore, one of the attractive means for thermal control and convective heat transfer is the microchannel.⁴ Also, using wavy surfaces in the microchannels as a passive method has been used to enhance the heat transfer mechanism in several studies.^5,6 As the scale of the devices becomes smaller, more investigations need to study the various aspects of the heat and flow distribution of the microchannel. Unsteady conjugate heat transfer (CHT) of a thermally developing flow in a microtube is studied by Şen and Darici.⁷ They observed the fluid temperature increases by increasing in Brinkman number in the upstream region. Borah et al.⁸ considered the conjugate force convective of the microchannel under the sinusoidal heat flux distribution with a phase difference between two walls. They claimed that the CHT depends on both the wall thickness and the thermal conductivity of the wall. In a numerical investigation of the MCHS by Dehghan et al.,⁹ a 33% growth in the convective heat transfer in a converging flow channel more than the straight one in the case of 2% Al₂O₃/water work fluid was reported. Rostami et al.¹⁰ in another numerical research via discrete phase model show in the case of water/cu in 100nm and 2% concentration the Nusselt number of the conjugate surfaces rises by 33.2% in compare with water through a wavy MCHS. The heat transfer and fluid flow of the MCHS with the secondary flow was modeled by Alsaiari et al.¹¹ Abdollahi et al.¹² simulated a microchannel with longitudinal fins under a constant heat flux. They show the SiO₂ in 2% and 30 nm had more performance in comparison to CuO, ZnO, and Al₂O₃. The MCHS performance under different condition were studied by Ambreen et al.^13–15 They studied numerically the thermohydraulic parameters of the 15 nm spherical shape of Al₂O₃-Cu/water in 1% of concentration hybrid nanofluid via the DPM approach through the micro diamond, circular, and elliptical fin heat sink. Their obtained results show a 25.14%, 19.65%, and 24% enhancement in mean Nuseelt number for diamond, circular, and elliptical fins, respectively. Saadoon et al.¹⁶ investigated the impacts of the amplitude of the wavy MCHS and the volume fraction of CuO. Al₂O₃, TiO₂, Ag, and Fe₃O₄ in 2.5%, 5%, and 7.5% in laminar flow. Sadripour¹⁷ presented the 1.315 increases in the PEC parameter in the case of 4% of SiO₂/water nanofluid through a corrugated channel. Ali et al.¹⁸ considered numerically the thermal behavior of the Al₂O₃ nanofluid pass along MCHS with different shapes of fin in it.

Amiri et al.¹⁹ in their recent paper studied thermal management enhancement by water/Ag nanofluid with sinusoidal and absolute sinusoidal wavy MCHS walls. They reported by utilizing unnecessary sinusoidal periods wall the heat transfer and thermal efficiency reduced, while employing only one absolute sinusoidal wavelength enhances thermal efficiency by 3.35%.

As a passive method in enhancing heat transfer in electronic devices, MCHS used an efficient solution to reduce the undesirable heat flux in the thermal devices. Wavy designs in microchannel are the two main varieties that have been the focus of extensive research in this field.

Therefore, due to earlier mentions, it has been tried to study a comprehensive numerical approach to survey the heat and flow characteristics (conjugate heat flux, local and axial temperature, heat transfer convective coefficient, friction factor, Nusselt number, velocity distribution, and thermal and hydraulic boundary layers) of the CuO-water as the nanofluid in a 3-D wavy MCHS via the single-phase, Eulerian-Eulerian (mixture and Euler multiphase models), and Eulerian-Lagrangian (DPM and DDPM models) approaches to compare the obtained results to each other and also study the behavior of the nanofluid flow behavior versus the base fluid. To study the effect of adding CuO nanoparticle to the pure water from the point of the thermohydraulic, the PEC term in all approaches were computed for all studied simulation cases, also.

Mathematical formulation

Supposing a three-dimensional fluid and solid CHT problem, the physical properties of the fluid flow and solid are independent of temperature if the flow is laminar. Then, the governing equations for the nanofluid flow and solid parts could be written as follow in each CFD model.²⁰ Moreover, the following assumptions have been considered to simplify the heat and flow modeling in the computational domain:

- Model is considered to be steady-state.

- The flow is assumed to be laminar and incompressible.

- Physical parameters of solids and fluids are constant.

- The viscous dissipation is ignored (Tables 1 –3).

- The natural convective and radiative heat transfers from the MCHS are ignored.

Table 1.

The Governing equations in the presented methods (for subsequent equations without explicit references, the preceding reference is implied).

Continuity	Momentum	Energy	Volume fraction
Single-phase
$\nabla . (ρ_{nf} V_{m}) = 0$ ²¹	$\nabla . (ρ_{nf} V_{m} V_{m}) = - \nabla P + \nabla . (μ_{nf} \nabla V_{m})$ ²²	$\nabla . (ρ_{nf} C V_{m} T) = \nabla . (k_{nf} \nabla T)$ ²³
$ρ_{m} = (1 - ϕ) ρ_{p} + ϕ ρ_{s}$	$Log (μ) = A e^{- BT}$ ²⁴	$C_{p, m} = \frac{(1 - ϕ) {(ρ C)}_{p} + ϕ {(ρ C)}_{s}}{(1 - ϕ) ρ_{p} + ϕ ρ_{s}}$
	$A = 1.8375 (ϕ)^{2} - 29.643 (ϕ) + 165.56$	$k_{m} = k_{static} + k_{Brownian}$ ²⁵
	$B = 4 \times 10^{- 6} (ϕ)^{2} - 0.001 (ϕ) + 0.0186$	$\frac{k_{static}}{k_{p}} = \frac{k_{s} + 2 k_{p} + 2 (k_{s} - k_{p}) ϕ}{k_{s} + 2 k_{p} - (k_{s} - k_{p}) ϕ}$
		$k_{Brownian} = 5 \times 10^{4} β ϕ ρ_{p} c_{p, p} \sqrt{\frac{k_{B} T}{ρ_{s} d_{s}}} f (T, ϕ)$
		$\begin{array}{l} f (T, ϕ) = (- 134.63 + 1722.3 ϕ) \\ + (0.4705 - 6.04 ϕ) T \end{array}$
		$β = 0.0011 (100 ϕ)^{- 0.7272}$ for $ϕ > 1 %$
Mixture
$\nabla . (ρ_{m} V_{m}) = 0$ ²⁶	$\nabla . (ρ {\vec{V}}_{m} {\vec{V}}_{m}) = - \nabla P + \nabla . (μ \nabla {\vec{V}}_{m}) + \nabla . (\sum_{k = 1}^{n} ϕ_{k} ρ_{k} {\vec{V}}_{d r, k} {\vec{V}}_{d r, k})$ ²⁷	$\nabla . \sum_{k 1}^{n} (ϕ_{k} {\vec{V}}_{k} (ρ_{k} H_{k} + P)) = \nabla . (k \nabla T)$ ²⁸	$\nabla . (ϕ_{p} ρ_{p} {\vec{V}}_{m}) = - \nabla . (ϕ_{p} ρ_{p} {\vec{V}}_{dr, p})$ ²⁹
${\vec{V}}_{m} = \frac{Σ_{k = 1}^{n} ϕ_{k} ρ_{k} {\vec{V}}_{k}}{ρ_{m}}$ ³⁰	$μ_{m} = \sum_{k = 1}^{n} ϕ_{k} μ_{k}$	$H_{k} = h_{k} - \frac{P}{ρ_{k}} + \frac{v_{k}^{2}}{2}$	${\vec{V}}_{dr, p} = {\vec{V}}_{p, f} - \sum_{k = 1}^{n} \frac{ϕ_{k} p_{k}}{ρ_{m}} {\vec{V}}_{pk}$
$ρ_{m} = \sum_{k = 1}^{n} ϕ_{k} ρ_{k}$	${\vec{V}}_{dr, k} = {\vec{V}}_{pk} - {\vec{V}}_{m}$ ³¹		${\vec{V}}_{p, f} = {\vec{V}}_{p} - {\vec{V}}_{f}$
	${\vec{V}}_{pk} = \frac{p_{p} d_{p}^{2}}{18 μ_{m} f_{drg}} \frac{(ρ_{p} - ρ_{m})}{ρ_{p}} \vec{a}$
	$f_{drag} = {\begin{matrix} 1 + 0.15 {Re}_{p}^{0.687} R e_{p} \leq 1000 \\ 0.0183 R e_{p} R e_{p} > 1000 \end{matrix}$
	$\vec{a} = \vec{g} - ({\vec{V}}_{m} . \nabla) {\vec{V}}_{m}$
	$R e_{p} = \frac{V_{m} d_{p}}{υ_{m}}$
Eulerian
$\nabla . (ϕ_{k} ρ_{k} {\vec{V}}_{k}) = 0$ ³²	$\begin{array}{l} \nabla \cdot (ϕ_{k} ρ_{k} {\vec{V}}_{k} {\vec{V}}_{k}) = - ϕ_{k} \nabla p + \nabla \cdot (ϕ_{k} μ_{k} \nabla {\vec{V}}_{k}) \\ + ϕ_{k} ρ_{k} \vec{g} + {\vec{R}}_{p k} + {\vec{F}}_{l i f t, k} \end{array}$ ³³	$\begin{array}{l} \nabla \cdot (ϕ_{k} ρ_{k} V_{k} H_{h}) = - \nabla . (k_{k} \nabla T_{k}) \\ - τ_{q} : \nabla V_{k} + Q_{p k} \end{array}$ ³⁴	$\forall_{k} = \int_{\forall} ϕ_{k} d \forall$ ³⁵
${\vec{V}}_{k} = \int_{v} ϕ_{k} dV$	${\vec{R}}_{pk} = K_{pk} ({\vec{V}}_{k} - {\vec{V}}_{p})$ ¹⁵	$Q_{pk} = h_{V} (T_{p} - T_{k})$	$\sum_{k = 1}^{n} ϕ_{k} = 1$
	$K_{pk} = \frac{ϕ_{p} ϕ_{k} ρ_{k} C_{D} R e_{k}}{24 τ_{k}}$	$h_{V} = \frac{6 k_{p} ϕ_{k} ϕ_{p} N u_{k}}{d_{k}^{2}}$
	$τ_{k} = \frac{ρ_{k} d_{k}^{2}}{18 μ_{p}}$	$N u_{k} = 2 + 0.6 {Re}_{k}^{0.5} \Pr_{p}^{1 / 3}$
	$C_{D} = {\begin{matrix} \frac{24 (1 + 0.15 {Re}_{k}^{0.687})}{R e_{k}} Re \leq 1000 \\ 0.44 Re > 1000 \end{matrix}$	$P r_{p} = \frac{C_{p, p} μ_{p}}{K_{p}}$
	$R e_{k} = \frac{ρ_{p} \| {\vec{V}}_{k} - {\vec{V}}_{p} \| d_{k}}{μ_{p}}$
	$F_{lift, k} = 0.5 ϕ_{k} ρ_{p} \frac{D ({\vec{V}}_{p} - {\vec{V}}_{k})}{Dt}$
DPM
$\nabla . (ρ \vec{V}) = 0$ ¹³	$\begin{array}{l} \nabla (ρ \vec{V} \vec{V}) = - \nabla P + \nabla . (μ \nabla \vec{V}) \\ + \nabla . (μ_{T} \nabla \vec{V}) + ϕ ρ \vec{g} + {\vec{S}}_{u} \end{array}$	$\begin{array}{l} \nabla (ρ \vec{V} C_{p} T) = \nabla . {(K + \frac{C_{p} μ_{T}}{P r_{T}}) \nabla T} - (\nabla . P \vec{V}) \\ + ϕ ρ \vec{g} + S_{E} \end{array}$ ³⁶	$\sum_{k = 1}^{n} ϕ_{k} = 1$
	${\vec{S}}_{u} = \frac{d {\vec{V}}_{p}}{dt} m_{p}$ ³⁷	$S_{E} = \frac{d T_{P}}{dt} m_{p} C_{p}$ ³⁸	$ϕ_{k} = \frac{1}{Δ \forall_{cell}} \sum_{i}^{n} \frac{Q_{i} . t}{ρ_{i}}$
	$\begin{array}{l} m_{p} \frac{d {\vec{V}}_{p}}{d t} = \sum \vec{F} = {\vec{F}}_{d r a g} + {\vec{F}}_{g r a ν i t y} + {\vec{F}}_{B r o w n i n} \\ + {\vec{F}}_{t h e r o p h o r e s i s} + {\vec{F}}_{l i f t} + {\vec{F}}_{v i r t u a l} + {\vec{F}}_{p r e s s u r e} \end{array}$	$m_{p} C_{p} \frac{d T_{p}}{dt} = ℏ A_{p} (T_{f} - T_{p})$ ³⁹	$n_{p} = \frac{{\overset{\cdot}{m}}_{parcel} Δ t}{m_{p}}$
	${\vec{F}}_{drag} = \frac{18 μ_{f}}{d_{p}^{2} ρ_{p} C_{c}}$ ⁴⁰	$Nu = \frac{h d_{p}}{K_{f}} = 2 + 0.6 {Re}_{P}^{0.5} \Pr_{f}^{1 / 3}$ ⁴¹
	$C_{c} = 1 + \frac{2 λ}{d_{p}} (1.257 + 0.4 e^{(\frac{1.1 dp}{2 λ})})$
	${\vec{F}}_{gravity} = \frac{\vec{g} (ρ_{p} - ρ_{f})}{ρ_{p}}$ ⁴²
	${\vec{F}}_{Brownian} = \frac{216 V K_{B} T}{π^{2} ρ_{f}^{5} C_{C} {(\frac{ρ_{p}}{ρ_{f}})}^{2}}$ ¹⁴
	${\vec{F}}_{therophoresis} = \frac{7.02 π d_{p} μ_{f}^{2} (\frac{k_{f}}{k_{p}} + 2.18 K_{n})}{m_{p} ρ_{f} T (1 + 3.42 K_{n}) (1 + 2 \frac{k_{f}}{k_{p}} + 4.36 K_{n})} \times \nabla T$ ⁴³
	${\vec{F}}_{lift} = \frac{5.188 v_{f}^{0.5} ρ_{f} d_{ij} (μ_{f} - μ_{p})}{ρ_{p} d_{p} {(d_{ik} d_{ki})}^{0.25}}$ ⁴⁴
	${\vec{F}}_{virtual} = 0.5 \frac{p_{f}}{ρ_{p}} \frac{d (μ_{f} - μ_{p})}{dt}$ ⁴⁵
	${\vec{F}}_{pressure} = (\frac{ρ_{f}}{ρ_{p}}) μ_{p} \frac{d μ_{f}}{dx}$ ¹⁴
	$K_{n} = \frac{2 λ}{d_{p}}$
DDPM
$\nabla . (ϕ_{P} ρ_{P} {\vec{V}}_{P}) = 0$ ⁴⁶	$\begin{array}{l} \nabla (ϕ_{P} ρ_{P} {\vec{V}}_{P} {\vec{V}}_{P}) = - ϕ_{P} \nabla P_{P} + \nabla . (ϕ_{P} μ_{P} \nabla {\vec{V}}_{P}) \\ + \nabla . (ϕ_{P} μ_{T} \nabla {\vec{V}}_{P}) + ϕ_{P} ρ \vec{g} + K_{f p} ({\vec{V}}_{f} - {\vec{V}}_{p}) \end{array}$ ⁴⁷	$\begin{array}{l} \nabla (ρ \vec{V} C_{p} T) = \nabla . {(K + \frac{C_{p} μ_{T}}{P r_{T}}) \nabla T} \\ - (\nabla . P \vec{V}) + ϕ ρ \vec{g} + S_{E} \end{array}$ ⁴⁸	$\sum_{k = 1}^{n} ϕ_{k} = 1$ ⁴⁹
	$m_{p} \frac{d {\vec{V}}_{P}}{dt} = \sum^{\vec{F}} + {\vec{F}}_{interact}$ ⁴⁹	$S_{E} = \frac{d T_{P}}{dt} m_{p} C_{p}$	$ϕ_{k} = \frac{1}{Δ \forall_{cell}} \sum_{i}^{n} \frac{Q_{i} \cdot Δ t}{ρ_{i}}$
	${\vec{F}}_{interact} = - \frac{1}{ρ_{p}} (\nabla . {\vec{τ}}_{p})$	$m_{p} C_{p} \frac{d T_{p}}{dt} = ℏ A_{p} (T_{f} - T_{p})$	$n_{p} = \frac{{\overset{\cdot}{m}}_{parcel} Δ t}{m_{p}}$
	${\vec{τ}}_{p} = - p_{p} + ϕ_{p} μ_{p} \nabla {\vec{V}}_{P} + ϕ_{p} (λ_{p} - \frac{2}{3} μ_{p}) \nabla {\vec{V}}_{P}$ ⁵⁰
	$F_{D} = \frac{3 μ C_{d} R e_{p}}{4 ρ_{p} d_{p}^{2}}$
	$C_{d} = \frac{24}{ϕ_{f} R e_{p}} [1 + 0.15 (ϕ_{f} R e_{p})^{0.687}] ϕ_{f}^{- 3.65}$ ⁵¹
	$R e_{p} = \frac{ϕ_{f} ρ_{f} \| {\vec{V}}_{f} - {\vec{V}}_{p} \| d_{p}}{μ_{f}}$

Table 2.

Thermophysical properties.

Material	$ρ (\frac{Kg}{m^{3}})$	$K (\frac{W}{m . K})$	$C_{P} (\frac{J}{Kg . K})$	$μ (Pa . s)$	$d_{P} (nm)$
Water	994	0.615	4179	0.0007	-
Copper oxide	8933	381	400	-	100

Table 3.

The applied boundary condition of the MCHS simulation.

Boundary	Type of boundary condition	Eqs.
Inlet	$u = u_{inlet}$ (Uniform velocity) $v = w = 0$ $T_{in} = 300 K$ (Uniformtemperature)	(14)(15)
MCHS bottom wall	$q ″ = 50 W / c m^{2}$ (Uniform heat flux)
MCHS side walls	Periodic boundaries
MCHS up walls	$\partial T / \partial z = 0$	(16)
Conjugate surfaces	No slip velocityNo jump temperature $u = v = w = 0$ $T_{s} = T_{nf}$ $\| K_{s} \frac{\partial T_{s}}{\partial N} \| = \| K_{nf} \frac{\partial T_{nf}}{\partial N} \|$	(17)(18)
Outlet	$\frac{\partial u}{\partial x} = \frac{\partial v}{\partial x} = \frac{\partial w}{\partial x} = \frac{\partial T}{\partial x} = 0$	(19)

It is also needed to mention that the energy equation in the solid part is $K_{s} \nabla^{2} T = 0$ .⁵² According to the definition of the convection heat transfer coefficient and consequently the Nusselt number,^8,53 it can be claimed that both of those changes are similar to the MCHS. Since the nanofluid flow is exposed to the conjugate surfaces and the heat flux from the substrate and sides of MCHS the average of the temperature and heat flux on the fluid side is used. In addition, to calculate the nanofluid mean temperature ( $T_{m}$ ) that has been extracted from the centerline of the channel. It is necessary to note that the mean temperature of the multiphase model is calculated by equation (1) (refer to Yao et al.⁵⁴); In this paper, two phases of nanofluid flow were studied where the integration is done on cross-sections of each channel, and along its path through the microchannel. Also, the heat transfer rate and consequently the convective heat flux on the conjugate surfaces could be computed by equation (2). In this study, the $p$ index denoted the primary phase (water), and the $s$ index denoted the secondary phase (CuO nanoparticle).

T_{m} = \frac{\sum_{i = 1}^{k} (\int \int ρ_{i} u_{i} c_{pi} T_{i} dA)}{\sum_{i = 1}^{k} (\int \int ρ_{i} u_{i} c_{pi} dA)}

(1)

q_{conj} = \sum_{i = 1}^{k} (\int \int ϕ_{i} K_{i} {\frac{\partial T_{i}}{\partial n} |}_{conj} dA)

(2)

Where the $n$ is the normal vector on channel surfaces which can be in the Y and Z direction. Meanwhile, in equation (1) the effect of the nanoparticle concentration is considered in the $ρ$ parameter. To calculate the mean velocity of the nanofluid flow, the below correlation, was used to consider both base fluid (the primary phase) and nanoparticles (the secondary phase) velocity in multi-phase models.

V_{m} = \frac{\sum_{i = 1}^{k} (\int \int ρ_{i} u_{i} dA)}{\sum_{i = 1}^{k} (\int \int ρ_{i} dA)}

(3)

Geometry model

The wavy microchannel heat sink has been deemed as one of the several methods employed for increasing heat transfer efficiency. Among the various waveforms of the channel, the sinusoidal type has been used in different studies because of its geometry. The wavelength of the sinusoidal waveforms and its amplitude are playing important roles to conduct the flow and the governing regime.

The computational domain has been shown in Figure 1 which is experimentally used by Sui et al.⁵⁵ The working fluid is pure water. The width and depth of the investigated microchannel are Sc = 207 µm and H = 406 µm, respectively. The waveform part of the channel follows the $y = ACos (\frac{2 π x}{L})$ equation, where A and L indicate the amplitude and wavelength of the wave, respectively. Thus, the total length of the channel with 10 waves is 25 mm.

Figure 1.

The computational geometry. (a) Copper block contains 60 wavy microchannel. (b) Conjugate domain along the microchannel. (c) Conjugate domain across the microchannel. (d) Cross-section of the microchannel sample.

According to the standard procedure, the heat ( $q$ ) received by the coolant is calculated as:

q = ρ C_{P} Q ({T_{m}}_{out} - {T_{m}}_{in})

(4)

where Q denotes the flow rate, ${T_{m}}_{in}$ and ${T_{m}}_{out}$ denote the inlet and outlet nanofluid temperatures, respectively; $ρ$ is the fluid density and $C_{P}$ is the specific heat.

To calculate the temperature of the microchannel wall, 1-D heat conduction is considered as below:

T_{w} = (0.25 T_{1} + 0.5 T_{2} + 0.25 T_{3}) - \frac{S q^{″}}{k_{Cu}}

(5)

In equation (5), the temperatures $T_{1}$ , $T_{2}$ , and $T_{3}$ illustrate the temperature of the wall channel at the distance of 3, 12.5, and 22 mm from the inlet. The distance $S$ (the distance of the thermocouple installed in the copper test of the experimental research⁵⁵), is presented in Figure 1(d). The term $k_{Cu}$ which denotes the thermal conductivity of copper is $400 (\frac{W}{mk})$ and the heat flux is maintained at the base surface is $50 (\frac{W}{c m^{2}})$ . The average heat transfer coefficient is evaluated from Yao et al.⁵⁴:

h = \frac{q}{N (A_{cb} + 2 A_{cs} η) (T_{w} - T_{m})}

(6)

where N = 60 in this simulation, and $T_{m}$ calculated as below:

T_{m} = \frac{T_{in} + T_{out}}{2}

(7)

The term $η$ denotes the efficiency of fin assuming an adiabatic tip condition:

η = \frac{\tanh (mH)}{(mH)}

(8)

where

m^{2} = \frac{2 h}{K S_{w}}

(9)

The fin efficiency is evaluated using an iterative approach as described in Yao et al.⁵⁴ Then, the average Nusselt number is evaluated as:

Nu = \frac{h . D_{h}}{K}

(10)

where $D_{h}$ is the hydraulic diameter. In this paper, the Re is defined by:

Re = \frac{ρ . V . D_{h}}{μ}

(11)

where $μ$ and $V$ are denoted the nanofluid viscosity and average velocity, respectively. Finally, the friction factor is defined as:

f = - \frac{2 (\frac{dp}{dx}) D_{h}}{ρ V^{2}}

(12)

where the pressure gradient is calculated along the microchannel length.

To calculate the fin efficiency ( $η$ ) in the correlation of the heat transfer coefficient (equation (6)), it is needed to use the iterative method. For this purpose, in the first place, the fin efficiency would be considered one (that means 100%), and the amount of $h$ was calculated by (equation (6)). Next, the fin efficiency was calculated by using a new amount for $h$ in the previous stage (equations (8) and (9)) then the new $h$ was calculated with the updated value of $η$ . In this step, the difference between the obtained $h$ in the two previous steps was computed. If this amount is acceptable, for example, less than $10^{- 4}$ , the iteration process will be stopped.

Moreover, to evaluate the overall performance of the MCHS by adding CuO nanoparticles to the pure water and study its hydrothermal performance in simulated numerical methods, the performance evaluation criteria (PEC) is used. The PEC parameter is defined as follows⁵⁶:

PEC = (Nu / N u_{bf}) / (f / f_{bf})^{1 / 3}

(13)

where $N u_{bf}$ and $f_{bf}$ return to the base fluid parameters, and $Nu$ and $f$ return to each simulated numerical model, respectively. In this paper, 3D CHT considered both convection and conduction in the wavy MCHS and the copper substrate.

In general, the geometry has been divided to two parts; the MCHS solid body and the flow channel which means these two parts exchange the heat transfer via conjugate surfaces. In such a way the produced heat flux in the solid part of the MCHS exchange in the conjugate surfaces by the developed fluid flow enters to the channel with uniform and constant values in velocity and temperature.

The governing equations have been discretized by the finite volume method (FVM) via Ansys Fluent package. It is needed to note; the SIMPLE scheme is employed to solve the pressure-velocity coupling. while the PRESTO scheme is applied to discretize the pressure, the second-order upwind is selected for the spatial discretization of momentum and energy equations. The continuity, momentum, and energy residuals are set to 10⁻⁶, 10⁻⁶, and 10⁻¹⁰, respectively.

More cell elements are needed to obtain reasonable results in Eulerian-Lagrangian approaches in comparison with the Eulerian-Eulerian approaches and also in contrast to the single-phase methods. To consider the unique mesh system for the simulation of all approaches, it is tried to obtain results from the DPM method to be the criteria of this evaluation. To examine the effect of the mesh grid size on the numerical results, different uniform cell systems in both the solid and the fluid part of the microchannel have been implemented (see Figure 2(a)). For this purpose, the Nusselt number has been investigated which has been depicted in Figure 2(b). Therefore, the mesh system with the 582,250 cells has been selected by less than a 0.17% difference in the both friction factor and Nusselt number. Meanwhile, despite the Nusselt number which is an integral variable, the temperature profile at the middle section of the outlet boundary in the height channel direction was investigated to be sure of the chosen mesh grid systems (see Figure 2(b)).

Figure 2.

(a) Structured mesh system in all domains. (b) Changes in the Nusselt number based on the cells number.(c) Temperature distribution in the x = L cross section.

All of the mentioned approaches for nanofluid simulation have been validated based on the two various experimental studies. The former was provided by Sui et al.,⁵⁵ shows that the flow friction and heat transfer of deionized water in 3-D wavy microchannels experimentally is more important from this point of view which has been used in this research to investigate the effect of the CuO as the nanoparticle in the base fluid in this geometry as the passive methods. In addition, the latter considered the convective heat transfer in nanofluid in a circle tube under a constant heat flux which is done by Anoop et al.⁵⁷ Eulerian-Eulerian model treats both the base fluid and nanoparticles as continuous phases. The governing equations for each phase are solved simultaneously to determine their respective velocities, pressures, and concentrations. The interactions between the two phases are accounted for by incorporating additional terms in the equations. This model is suitable for studying highly loaded nanofluids containing a high volume fraction of nanoparticles. In Eulerian-Lagrangian model, the base fluid is considered as a continuous phase using the Eulerian method, while the nanoparticles are treated as discrete Lagrangian particles. The base fluid is modeled using the Navier-Stokes equations, while the particles’ motion is determined by solving the motion equations for each particle. This model is appropriate for dilute nanofluids with low particle volume fractions. It seems that the Euler, single phase, and mixture models have the least deviations from the laboratory results. The main difference between the Euler two-phase and the mixture model lies in how they treat the dispersed nanoparticles. In the Euler two-phase model, the nanoparticles are considered as a separate phase, while in the mixture model, they are treated as part of the fluid mixture. Each model has its advantages and limitations, and the choice depends on the specific characteristics of the nanofluid being simulated and the desired level of detail in the analysis. In the single-phase model, the properties of the nanofluid, such as density, viscosity, and thermal conductivity, are calculated as effective properties based on the nanoparticles volume fraction in the mixture. The model also considers the energy transfer between the nanoparticles and the base fluid, taking into account factors such as thermophoresis, Brownian motion, and convection compared to two-phase models, which treat the nanoparticles and base fluid as separate phases, the single-phase model is more convenience and computationally less demanding. It provides an efficient way to predict the overall behavior of nanofluids, such as their heat transfer performance, without the need for detailed information about the individual phases. However, one limitation of the single-phase model is that it does not capture the spatial distribution and interaction of nanoparticles within the fluid. It assumes a homogeneous distribution, neglecting any concentration gradients or clustering effects that may occur in practice. This limitation can make the single-phase model less accurate for certain applications, such as predicting the behavior of nanofluids in complex flow patterns or near solid surfaces. In contrast, two-phase models, such as the Eulerian-Eulerian and Eulerian-Lagrangian methods, explicitly consider the separate phases of the nanofluid. These models can capture the spatial distribution and movement of nanoparticles, which may be important in situations where particle clustering or sedimentation occurs. However, two-phase models are more computationally intensive and require more detailed input data, such as the particle size distribution and interphase forces. To be sure from the numerical methods, obtained average Nusselt number in different Res at the 0.698 m section has been compared by a maximum difference of about 9% in the DDPM and the single-phase models (see Figure 3). Overall, the choice between single-phase and two-phase models depends on the specific application and the level of accuracy required. The single-phase model provides a simplified representation of nanofluid behavior, suitable for many engineering applications, while two-phase models offer a more detailed and accurate description but at a higher computational cost.

Figure 3.

Validation of the current numerical solutions: (a) friction factor and average Nusselt number ofSui et al.⁵⁵ (b) Average Nusselt number obtained via different methods with experimental in Anoop et al.⁵⁷

Results and discussion

The numerical simulation results are extracted for CuO-water nanofluid in different concentrations as the working fluid and a comparison is made among the mentioned models and also the experimental data available in the literature.⁵⁵ The effect of adding nanoparticles in the heat and fluid flow parameters and the CHT has been noticed.

It is necessary to note that in the interface between the solid surface and its adjacent fluid, these two conditions must be confirmed. The condition first is that due to the principle of conservation of energy, the constant heat flux must be transferred on the conjugate surfaces. The second one is that the conjugate surfaces must be in thermal equilibrium with their adjacent fluid flow.

Figure 4 shows the transferred heat flux from the conjugate surfaces to the nanofluid flow which is located in a positive direction. While this amount for the solid is in the reverse direction. The substrate surface transferred about 218 $\frac{kW}{m^{2}}$ from the solid part of the MCHS to the nanofluid, which is about 180 $\frac{kW}{m^{2}}$ more than the sides surface along the channel at Re = 589 at 0.01 CuO nanoparticle concentration via the DDPM approach. The average value of the CHT through the channel obtained is 190 $\frac{kW}{m^{2}}$ which is less than this amount for the substrate surface. Also, it is observed that more heat is transferred by increasing the nanoparticle concentration in the same Re. The changes in the nanofluid temperature adjacent to the substrate surface are presented for all of the CFD approaches in the case of Re = 794, ϕ = 6% (see Figure 5). Meanwhile, the changes in the temperature for the substrate surface are similar to the Nanofluid flow side, which means their changes match perfectly. The nanofluid temperature distribution in the Eulerian-Eulerian models experienced similar trends in comparison to the Eulerian-Lagrangian approaches. In addition, the obtained results from the presented single-phase model had acceptable conformity with the Eulerian-Eulerian models. Although, this trend is different in DPM and DDPM models even with each other because of the particle distributions in the channel cross-section. Due to the shape of the channel, the temperature distribution in each section along every wavelength of the channel obtained is different. In detail, at the center of the $x = 5 λ$ section in the MCHS, the nanofluid temperature is 303.64, 303.7, 303.724, 302.922, and 304.198 K for single-phase, mixture, Euler, DPM and DDPM approaches, respectively. Also, a more temperature gradient in the hot surface of the section is observed. The channel wave shape conducted the nanofluid flow to the channel wall sides and caused more heat transfer which led to a more temperature gradient. While all of the presented models depict the approximately same temperature gradient changes adjacent to the substrate channel surface, the Eulerian-Eulerian models show the same changes at the adiabatic to channel surface in comparison to the Eulerian-Lagrangian models. The present status of computational fluid dynamics-discrete phase method (CFD-DPM) simulations in process engineering is characterized by notable progress and an expanding array of applications. This approach showcases its capabilities across diverse engineering domains such as transportation, erosion, spray, comminution, separation, mixing, tracking, and thermochemical conversion. It offers valuable insights into the intricate interplay between fluid and discrete phases, resulting in enhanced process optimization, design, product quality control, safety and risk assessment, as well as energy efficiency.^58–60

Figure 4.

Changes in the conjugate heat flux along the channel in the DDPM approach. (a) $Re = 589, ϕ = 0.01$ .(b) $Re = 494, ϕ = 0.03$ .

Figure 5.

Changes in the nanofluid temperature in the case of $Re = 794, ϕ = 0.06$ in all approaches: (a) axial temperature adjacent to the substrate surface; (b) temperature distribution in the channel width (at x = 5 $λ$ in the Y-direction); (c) temperature distribution in the channel height (at x = 5 $λ$ in the Z-direction).

As is seen from Figure 4 the variations in the trend of the conjugate heat flux in the substrate surface are less than the sides, while its average amount is more than those (217.6 kW/m²). Although the changes in the heat transfer coefficient depend on the shape of the wavy channel (A, L) and its amount fluctuations along the channel in all of the presented CFD approaches the same trends were observed. Of course, this trend has differences in the DPM approach in comparison with others. Since the wall of the microchannels is wavy, causes the thermal resistance of the conduction to be different compared to the heat dissipation during the microchannel, and the heat flux profile on the surface of the microchannel is different. Meanwhile, the amount of axial heat transfer coefficient is almost the same in these CFD simulations. For example, the heat transfer coefficient at Re = 589 with the 0.03 CuO nanoparticle concentration is 37.885, 37.188, 36.933, 37.175, and 37.675 $\frac{kW}{m^{2}}$ for the single-phase, mixture, Euler, DPM, and DDPM approaches, respectively. The obtained data for pure water in this case is 35.662 $\frac{kW}{m^{2}}$ , about a 6% increment in $h$ by adding 0.03 CuO nanoparticle concentration. In addition, the axial heat transfer coefficient increased slightly by increasing nanoparticle concentrations. For example, in the above case, the value of $h$ increased approximately 2.62%, 7.086%, 7.44%, 4.61%, and 9.18% by increasing CuO concentration ( $ϕ$ ) from 1% to 6% for the mentioned models, respectively.

As mentioned before, the Nusselt number changes as similar to the convection heat transfer coefficient, consequently the maximum Nusselt number occurs at the channel entrance due to the wave-shaped flow path fluctuating along the channel. Due to the growth of the thermal boundary layer on the conjugate surfaces and the tiny size of the cross-section, nanofluid flow is developed. Also, due to the rise in the bulk temperature of the fluid, the $Nu$ decreased along the channel. In the present study, the obtained Nusselt number at Re = 589 with the 0.03 CuO nanoparticle concentration is 16.87, 16.58, 16.46, 16.57, and 16.80 for the single-phase, mixture, Euler, DPM, and DDPM approaches, respectively. While this amount obtained 15.90 for the pure water (about 5.3% increases in the Nusselt number by adding 0.03 CuO concentration to pure water in the same Re). The axial Nusselt number increased with an increment in both nanoparticle concentration and Re. For example, in the above case, the axial Nusselt number grows around 48%, 51%, 50%, 50%, and 52% by increasing in Re from 304 to 794, respectively. While, the axial Nusselt number increased exactly with that same percent in the heat transfer coefficient changes by adding CuO nanoparticle concentration ( $ϕ$ ) from 1% to 6% for the mentioned models, respectively.

Besides, the variations in the average Nusselt number based on the Re and nanoparticle concentration are observed in Figure 6. Increasing the Re increases the Nusselt number. In the Re = 304, the obtained 11.19 Nusselt number reaches 20.97 for the Re = 794 by 0.06 CuO nanoparticle concentration in the Eulerian model. While, this amount in the DPM model obtained 11.13–21.60, respectively. Although, all of the presented models show similar trends, about a 5% difference is observed in the DDPM approach (11.66–21.85, respectively). Meanwhile, there is a margin (approximately 4%) of obtained Nusselt number in the DDPM model shown in the 0.06 of CuO concentration, the average Nusselt number extracted is a little more in different concentrations, for example, Nusselt number of 11.66 in DDPM approach in contrast with 11.19 in Eulerian model (see Figure 6(b)). Firstly, the addition of nanoparticles rises the effective thermal conductivity of the fluid. Nanoparticles have a high surface area-to-volume ratio, which enhances the interaction between the nanoparticles and the surrounding fluid molecules. This increased interaction leads to improved thermal conductivity, resulting in enhanced heat transfer. Secondly, the presence of nanoparticles alters the fluid viscosity. Most nanoparticles have a higher viscosity compared to the base fluid. This increased viscosity affects the flow behavior of the fluid, leading to changes in convective heat transfer. The flow resistance in the presence of nanoparticles may increase, reducing the overall heat transfer rate. Moreover, the addition of nanoparticles can augment the convective heat transfer coefficient. Due to their small size and the fractal nature of their surface, nanoparticles have a higher probability of being involved in Brownian motion. This increased motion leads to stronger mixing and improved heat transfer at the fluid-solid interface. Furthermore, the presence of nanoparticles can also affect the thickness of the thermal boundary layer. The interaction between nanoparticles and the fluid molecules can result in changes in the boundary layer formation and growth. This alteration in the boundary layer affects the overall heat transfer characteristics of the system. According to the differences in the two-phase and single-phase models for nanofluid modeling, the amount of changes in different models is different.

Figure 6.

Changes in average Nusselt number (a) in different approaches for $ϕ = 0.06$ , (b) in different CuO nanoparticle concentrations for $Re = 304$ .

Abugnah et al.⁶¹ reported that the predictions from three two-phase models – Volume of Fluid (VOF), Eulerian Mixture Model (EMM), and Eulerian -Eulerian Model (EEM) – exhibit significant similarity in the average Nusselt number, showing a 24% increase compared to the single-phase model (SPM).

Figure 7 presents the temperature distribution across the solid and nanofluid parts of MCHS for Eulerian-Eulerian and Eulerian-Lagrangian methods in the case of Re = 702 with 0.06 CuO nanoparticle concentration, in this study. As it is observed, the temperature distribution decreased from about 311 K (310.283, 310.717, 310.358, 310.282, and 310.161 K for the single-phase, mixture, Euler, DPM, DDPM approaches, respectively) for the Solid part on the bottom surface of MCHS, the boundary of applying heat flux, to about 309 K (309.157, 309.592, 309.233, 309.157, 309.035 K for the single-phase, mixture, Euler, DPM, DDPM approaches, respectively) close the adiabatic surface at the top solid boundary. This change for the nanofluid flow part is from about 309 K (309.691, 310.125, 309.765, 309.690, 309.569 K for the single-phase, mixture, Euler, DPM, DDPM approaches, respectively) at the conjugate surfaces to about 304 K (304.672, 304.778, 304.655, 304.484, 304.349 K for the single-phase, mixture, Euler, DPM, DDPM approaches, respectively) at the top surface. Of course, these trends for the Eulerian model are less than the others. The isothermal lines are shown by dash lines in these contours. While the thermal boundary layer is depicted by a solid line across the channel. As is seen in Figure 7, there is no thermal boundary layer formed for the adiabatic surface because there is no temperature gradient in the top boundary. The thin thermal boundary layer of the substrate conjugate surface can be justified by the more heat flux on the channel substrate. Consequently, the thickness of the thermal boundary layer is less and the flow will be developed at less distance from the substrate in the Z-direction. Also, the wave-shaped channel drives the flow to the special areas of the cross-section (maximum amplitude). Also, it is seen that there are more streamlines in the adjacent of these areas. Additionally, the thickness of the thermal boundary layer is thin in these areas.

Figure 7.

Contours of temperature (K) in the solid and fluid part in the case of $Re = 702, ϕ = 0.06$ for (a) single-phase,(b) mixture, (c) Eulerian, (d) DPM, (e) DDPM approaches in the Y-Z plane at x/L = 0.5 cross-section.

Changes in friction factor based on Re are observed in Figure 8. In the 0.06 of CuO concentration. Almost in all approaches, it is shown that a considerable decrease in friction factor occurs by increasing Re, which this reduction is greater in the low Res. For example, the friction factor value decreased by around 4.5% and 1.3% by increasing in Re from 304 to 397 and 702 to 794, respectively. Meanwhile, in DPM approach the reduction trend was different in some of the cases. In the case of Re 589, the Re was unjustifiable in comparison to the other methods. In addition, it is necessary to note that these changes in friction factor parameter versus the Reynolds and CuO nanoparticle concentration along the channel and in different cross-sections were similar to the other cases. Also, a significant difference was observed (about 6.8% and 3.05% in comparison to the other methods despite of DPM model). In detail, the friction factor for Re = 304 obtained about 0.2606, 0.3281, 0.3327, 0.2876, 0.3159 reduced to about 0.1456, 0.1856, 0.1844, 0.1763, 0.1767 in the case of Re = 794 for the 0.06 CuO nanoparticle concentration via single-phase, mixture, Euler, DPM, DDPM approaches, respectively. Moreover, by adding more nanoparticles in the working fluid, more friction factor is extracted. As is seen in Figure 8(b), the single-phase and DPM models presented different trends in contrast to the other models. The friction factor in the single-phase model was independent of the nanoparticle concentration. While the obtained value in the DPM model was different, the other models lead to closer results. For example, at Re = 589 in 0.01 of concentration, the friction factor extracted 0.1693, 0.1778, 0.1801, 0.1668, 0.1776 in contrast to 0.1693, 0.2160, 0.2187, 0.1846, 0.2103 for 0.03 of concentration for the single-phase, mixture, Euler, DPM, and DDPM approaches, respectively. The variations in different models differ in terms of the changes in nanofluid modeling, as indicated by the disparities between the single-phase and two-phase models.

Figure 8.

Changes in the friction factor (a) in different approaches for $ϕ = 0.06$ , (b) in different CuO nanoparticle concentrations for $Re = 589$ .

Due to the presented approaches in this paper to simulate the nanofluid flow, various nanofluid velocity profiles were extracted in the both Y and Z direction in different cross sections. The velocity distribution in the 12.5 mm (at $5 λ$ along the channel) at Re = 794 with 6% CuO nanoparticle concentration via different methods observed in Figure 9. It is necessary to note that because of the channel’s wavy shape, the velocity distribution can be different in the other cross sections, especially the changes in velocity in the Y direction. For example, as is seen in the $5 λ$ cross-section, the velocity behavior in the Z direction has fewer changes in comparison to the Y direction. Also, the velocity gradient in the Z direction was undergone a minor difference in contrast to the Z direction (around 1.37% changes). While, due to the channel’s wavy shape and just in the $5 λ$ section, a more velocity gradient has been created adjacent to the outer curved wall of the channel ( $y = 241.5 μ m$ ) which can be because of the more compressed streamlines near this wall in comparison to the inner curved wall of the channel ( $y = 34.5 μ m$ ) with around 59.49% changes (see Figure 9(b)). The velocity distribution in the Z direction changes around the average nanofluid velocity, about 2.2916 m/s, at this cross-section. Whereas, the average velocity of the nanofluid flow along the Y direction extracted about 1.8912 m/s, which observed more changes in Figure 9(a) because of the channel wave amplitude in this direction. The obtained changes in the velocity distribution in the Eulerian-Eulerian methods are almost in a similar trend. Meanwhile, the presented single-phase model shows less difference in the value and its changes to the Eulerian-Eulerian models. The reason for the difference in velocity distribution of nanofluid in the cross-sectional area of a microchannel between the single-phase model and the two-phase model can be explained as follows: In the single-phase model, the nanofluid is considered a homogeneous fluid with uniform properties throughout the microchannel. The velocity profile is assumed to be relatively uniform due to the absence of any phase change or interaction between different phases. Therefore, the velocity profile in the microchannel is expected to be relatively smooth and consistent. On the other hand, in the two-phase model, the nanofluid is treated as a mixture of two phases: the liquid phase and the dispersed nanoparticles. This model considers the interactions between the liquid phase and the nanoparticles, such as Brownian motion, particle-particle, and particle-liquid interactions. These interactions can lead to the formation of clusters or agglomerates of nanoparticles, causing local variations in the concentration and velocity distribution of the nanofluid within the microchannel. These variations may result in non-uniform velocity profiles with higher velocities in certain regions and lower velocities in others. Therefore, the difference in the velocity distribution of nanofluid between the single-phase and two-phase models can be attributed to the consideration of phase interactions and the formation of nanoparticle clusters in the two-phase model. While, the obtained results for the Eulerian-Lagrangian methods (DPM and DDPM models) experienced different trends around the same average velocity (see Figure 9). In detail, the average nanofluid velocity at Re = 794 with 0.06 CuO concentration at the $5 λ$ cross section in the Y direction extracted 1.9347, 1.8782, 1.9468, 1.6988, and 2.0105 m/s in comparison to 2.2364, 2.2226, 2.2300, 2.4346, and 2.4006 m/s for the average velocity in the Z-direction for the single-phase, mixture, Euler, DPM, and DDPM approaches, respectively.

Figure 9.

Changes in the nanofluid velocity distribution in the case of $Re = 794, ϕ = 0.06$ for all approaches at x = 5 $λ$ cross-section: (a) in the channel width at Y direction; (b) in the channel height at Z direction.

The velocity contours for Re = 702 with 0.06 CuO nanoparticle concentration in the single-phase, mixture, Eulerian, DPM, and DDPM models across the $5 λ$ channel section is shown in Figure 10. The iso surface velocity is shown with the dashed line while the hydraulic boundary layer is shown by the solid line in Figure 10. As it is seen from the velocity contours, the single-phase and the Eulerian-Eulerian models have more streamlines in the almost middle of the channel cross-section (Besides the small wakes formed in the top and bottom section corners). The hydraulic boundary layer was thinner in the outer wall of the channel in contrast to the outer wall of the channel and as it was mentioned before via the more velocity gradient in its adjacent (see Figure 10). Due to the wavelength and amplitude along the channel, velocity distribution experienced more changes in the side walls in comparison to the top and bottom walls. The formed hydraulic boundary layer is approximately similar in the single-phase and Eulerian-Eulerian approaches, while there is a difference between the hydraulic boundary layers of DPM and DDPM models. The created streamlines in DDPM models show more secondary wakes in particle adjacent to the channel bottom wall. The value of the hydraulic boundary layer for this case was recorded at about 1.4557, 1.6097, 1.6012, 1.6108, and 1.6116 m/s via the single-phase, mixture, Eulerian, DPM, and DDPM approaches, respectively. Also, for all the presented models, the least velocity was zero adjacent to the channel walls because of the no-slip boundary conditions. Whereas its maximum value obtained was 2.7281, 2.9476, 2.8173, 2.9299, and 2.6208 m/s, respectively.

Figure 10.

Contours of the velocity (m/s) distribution in the case of $Re = 702, ϕ = 0.06$ for (a) single-phase, (b) mixture,(c) Eulerian, (d) DPM, (e) DDPM approaches in the Y-Z plane at x/L = 0.5 cross-section.

The CuO nanoparticles distribution in the wavy channel cross-section is another important parameter in this study. Figure 11 shows the particle scattering of the 0.06 CuO concentration at Re = 794 in the width and height of the middle cross-section of the channel. There is no access to the nanoparticles scattering in the single-phase model. Also, obtained results present the changes in the nanoparticle distribution via Eulerian-Eulerian approaches (mixture, Euler) so close. Whereas there were small changes in the micro-scale in the Eulerian model which are shown in Figure 11. On the other hand, the CuO nanoparticles concentration scattered in the wide range via Eulerian-Lagrangian approaches (DPM and DDPM). The obtained data shows these changes from 0 to 0.1224 via the DPM model in comparison to 0 to 0.2206 via the DDPM model in the width of the section. In this section, the particles dispersed around 0.06198 and 0.06618 in the DPM and DDPM models, respectively, by about a 9.34% difference relative to the particles in the inlet boundary. The distribution of nanoparticles in microchannels for nanofluid flow using two-phase models can be attributed to several factors. One major factor is the interplay between fluid flow and particle dynamics. In microchannels, fluid flow is significantly influenced by the channel geometry and the flow rate. As the fluid flows through the microchannel, it experiences various flow regimes such as laminar flow, transition flow, or turbulent flow. These different flow regimes affect the distribution of nanoparticles due to their varying velocities and shear forces acting on them. Additionally, the properties of nanoparticles themselves, such as size, shape, and surface charge, play a crucial role in their distribution. Nanoparticles tend to aggregate or disperse depending on their surface properties and interaction forces with the surrounding fluid. These interactions can cause agglomeration or dispersion of nanoparticles, leading to uneven distribution along the microchannel. Moreover, the presence of other forces, such as Brownian motion and thermophoresis, also affects the distribution of nanoparticles. Brownian motion causes random movements of nanoparticles due to collisions with fluid molecules, leading to diffusion-like behavior. Thermophoresis, on the other hand, is the motion of particles induced by temperature gradients, which can cause nanoparticles to migrate toward cooler or hotter regions within the microchannel. Furthermore, the boundary conditions and inlet configurations of the microchannel can also influence the distribution of nanoparticles. For example, the use of different surface coatings or functionalization can alter the interaction between nanoparticles and the channel walls, affecting their distribution throughout the microchannel. In conclusion, the distribution of nanoparticles in microchannels for nanofluid flow is influenced by various factors, including fluid flow characteristics, nanoparticle properties, interparticle interactions, external forces, and boundary conditions. Understanding and modeling these factors using two-phase models can provide insights into the distribution behavior, allowing for better control and optimization of nanofluid flow in microchannels.

Figure 11.

CuO nanoparticles Scattering in the case of $Re = 794, ϕ = 0.06$ for all approaches at x = 5 $λ$ cross-section (a) in the channel width at Y direction, (b) in the channel height at Z direction.

Furthermore, for a comprehensive comparison between the MCHS simulated approaches, the obtained results at Re = 589 are presented for different CuO nanoparticle concentrations in Tables 4 and 5. The extracted results from the presented single-phase model give close results in comparison to multiphase models, be mindful that there are no results from the nanoparticle concentration. The parameter of $τ_{c - ave}$ denotes the average wall shear stress on the conjugate surfaces of the wavy channel. Although that is expected its value grows by adding nanoparticles and increasing its concentration, the value of $τ_{c - ave}$ decreased even its amount for the pure water in some simulations in this study. In the DPM model, the average wall shear stress is reduced to 41.043 Pa while this amount was extracted at 48.174 Pa for pure water, for example. Also, this issue was observed in some other parameters like the MCHS efficiency and the absolute pressure gradient. The friction factor parameter gets more by increasing the nanoparticle concentration in multiphase models while its obtained value was the same for the single-phase model with different concentrations. The parameter of $φ_{ave}$ refers to the nanoparticle concentration at the outlet boundary of the MCHS. The Eulerian-Eulerian approaches presented uniform nanoparticle distribution in different cross sections like the outlet MCHS boundary in contrast to the Eulerian-Lagrangian approaches. It is also necessary to note that the geometry problem affects the nanoparticle concentration, for example in the same conditions uniform nanoparticle concentration is obtained for the circular tube via DPM and DDPM approaches. Due to the equation (4), the gained heat flux of the nanofluid flow from the $50 \frac{w}{c m^{2}}$ applied heat flux on the bottom wall of the MCHS listed in the ${q ″}_{coolant}'$ column of Table 5. As this is expected the ${q ″}_{coolant}$ value increased by adding CuO nanoparticles and its concentration increment to 6% to the maximum amount of 12.78% for the case of DPM model. $T_{w - ave}$ and $T_{c - ave}$ (K) denote the average temperature of the MCHS bottom wall and conjugate bottom surface, respectively. In each case, it is observed that the temperature reduced from the substrate distance which means $T_{c - ave} < T_{w - ave}$ , and causes about $16.83 \frac{K}{cm}$ temperature gradient in the substrate thickness for the 6% CuO nanoparticle concentration compared with about $5.21 \frac{K}{cm}$ for the pure water. Also, the parameter of the temperature rise defines the difference between the outlet and the inlet temperature of the nanofluid flow showing around 3.93%, 7.53%, and 11.50% increases on average by adding the 1%, 3%, and 6% CuO nanoparticle concentration. In all nanofluids simulation models, the heat transfer coefficient and consequently Nusselt number increased by adding nanoparticles in the base fluid while the obtained results were less than in their values in contrast to deionized water for the mixture model. The average value of Nusselt grows by about 3.47%, 4.95%, and 7.01% by increasing CuO nanoparticles in 1%, 3%, and 6% concentration.

Table 4.

Hydraulically obtained results of the MCHS at Re = 589.

CFD approach	$ϕ$	$Q$ (cm³/s)	N	$V_{out}$ (m/s)	$τ_{c - ave}$ (Pa)	$η_{HSMC}$	${- \frac{dp}{dx} \|}_{c - ave}$ (kPa/m)	$f$	$φ_{ave}$
Pure water	—	0.1272	60	1.521	48.174	0.9521	702.76	0.16926	—
Single-phase	0.01	0.1238	60	1.479	48.139	0.9500	701.68	0.16926	—
	0.03	0.1184	60	1.414	48.525	0.9494	707.30	0.16926	—
	0.06	0.1129	60	1.349	50.234	0.9487	732.21	0.16926	—
Mixture	0.01	0.1238	60	1.481	47.887	0.9565	699.58	0.17779	0.010
	0.03	0.1184	60	1.416	47.725	0.9551	698.25	0.19395	0.030
	0.06	0.1129	60	1.351	48.524	0.9529	707.25	0.21605	0.060
Eulerian	0.01	0.1238	60	1.478	47.804	0.9512	708.85	0.18015	0.010000117
	0.03	0.1184	60	1.414	47.656	0.9505	705.94	0.19609	0.030000290
	0.06	0.1129	60	1.348	48.302	0.9475	715.89	0.21869	0.060000596
DPM	0.01	0.1238	60	1.414	42.044	0.9516	656.54	0.16686	0.012183217
	0.03	0.1184	60	1.415	44.383	0.9502	709.80	0.19716	0.030716262
	0.06	0.1129	60	1.349	41.043	0.9494	604.54	0.18467	0.058114327
DDPM	0.01	0.1238	60	1.480	47.173	0.9512	698.72	0.17757	0.010298280
	0.03	0.1184	60	1.416	46.449	0.9496	687.58	0.19099	0.035051507
	0.06	0.1129	60	1.345	46.634	0.9466	688.58	0.21034	0.057831896

${\frac{dp}{dx} |}_{c - ave}$ is the absolute average pressure gradient on the conjugate surfaces of the wavy channel.

Table 5.

Thermally obtained results of the MCHS at Re = 589.

CFD approach	$ϕ$	${q ″}_{coolant}$ (w/m²)	$T_{w - ave}$ (°k)	$T_{c - ave}$ (°k)	$T_{rise}$ (°k)	$η_{MCHS}$	$h$ (w/m²°k)	Nu
Pure water	—	8824.0	310.932	310.670	10.020	0.9521	35,662.4	15.90
Single-phase	0.01	8939.6	310.959	310.703	10.448	0.9500	37,367.5	16.66
	0.03	8904.8	311.103	310.846	10.922	0.9494	37,855.6	16.88
	0.06	8868.5	311.283	311.024	11.473	0.9487	38,372.2	17.11
Mixture	0.01	8911.5	311.665	310.821	10.322	0.9565	32,259.6	14.38
	0.03	8947.3	311.801	310.957	10.918	0.9551	33,354.5	14.87
	0.06	8969.0	311.821	310.974	11.377	0.9529	35,072.0	15.64
Eulerian	0.01	9028.6	311.738	310.894	10.532	0.9512	36,410.8	16.23
	0.03	8970.4	311.828	310.983	10.939	0.9505	36,933.9	16.47
	0.06	9034.0	311.845	310.997	11.553	0.9475	39,336.8	17.54
DPM	0.01	9369.6	312.072	311.230	10.640	0.9516	36,083.6	16.09
	0.03	9452.3	311.981	311.136	10.734	0.9502	37,175.2	16.57
	0.06	10,117.0	312.703	311.858	11.488	0.9494	37,825.2	16.86
DDPM	0.01	9093.0	311.730	310.886	10.326	0.9512	36,426.1	16.24
	0.03	9331.0	311.842	310.997	10.596	0.9496	37,675.1	16.80
	0.06	9429.2	311.604	310.758	10.707	0.9466	40,108.9	17.88

In summary, due to the extracted friction factor in each simulated model, unexpected changes in the PEC are observed. For example, the PEC after an increase in Re = 397 decreased by about 3% in 494 of Re with 3% of CuO concentration in the DDPM approach. This trend is observed in the DPM approach by increasing the CuO concentration to 6% in Re = 702, also (see Figure 12). Due to the obtained results of the friction factor in the single-phase approach in low difference with pure water work fluid relative to the presented multiphase model, the PEC parameter extracted in higher amount. By increasing CuO concentration to 0.06 the PEC parameter increased about 1.9% on average despite in DPM approach which decreased by 2.7%. It returns to the more percentage growth in $f$ relative to $Nu$ . In this case, $f$ rises by about 21% while $Nu$ grows by 9%. In addition. the PEC experienced different trends by increasing in $Re$ .

Figure 12.

The PEC changes in all approaches (a) based on the CuO concentration in $Re = 702$ ; (b) based on the Re.

Conclusion

The CHT model in a 3-D wavy microchannel by nanofluid as a working fluid to analyze the influence of nanoparticles on thermo-hydraulic features in a laminar flow by different models including single-phase, Mixture, Eulerian, DPM, and DDPM models are studied. In this research, the following important results are obtained:

The obtained data from the presented single-phase model depicted the acceptable data in thermal parameters, while in the flow parameters, there are differences between them.

The changes in the thermal gradient at each section along the channel can lead to oscillations in the trend of the Nusselt number.

The amount of heat flux on the substrate conjugate surface is more than the others (305 kW/m² in comparison with 130 kW/m²), while its fluctuations domain is less.

Decreasing the temperature of conjugate surfaces along the wavy channel increases $h$ and $Nu$ .

The distribution of the temperature in the solid and fluid parts of the MCHS is presented (The acceptable match in each conjugate surface was observed).

The amount of $h$ and consequently $Nu$ increased by increasing $Re$ (Almost 100% from Re = 304–794 in 1% concentration in the DPM approach).

There were fewer changes in the CuO nanoparticles scattering in the Eulerian-Eulerian models in comparison to the Eulerian-Lagrangian models.

There are more differences (about 15%) between the particle’s concentrations in DPM and DDPM models in comparison to the mixture, and Eulerian models.

In most cases simulated in this study, more changes have been observed for Eulerian-Lagrangian (DPM and DDPM) approaches. However, the average parameters of the obtained results are almost the same

Footnotes

Appendix

Handling Editor: Sharmili Pandian

Author contributions

All authors contributed to the study conception and design. Material preparation, data collection, and analysis were performed by Farshid Marzban, Mahshid Marzban, Kazem Mohammadzadeh, and Somayeh Davoodabadi Farahani. The first draft of the manuscript was written by Kazem Mohammadzadeh and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Kazem Mohammadzadeh

Data Availability Statement

No data associated in the manuscript.

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