Physically Consistent Implementation of the Mixture Model for Modelling Nanofluid Conjugate Heat Transfer in Minichannel Heat Sinks

Abstract

As much as two-phase mixture models resolve more physics than single-phase homogeneous models, their inconsistent heat transfer predictions have limited their use in modelling nanofluid cooled minichannel heat sinks. This work investigates, addresses, and solves this key shortcoming, enabling reliable physically sound predictions of minichannel nanoflows, using the two-phase mixture model. It does so by applying the single-phase and the two-phase mixture model to a nine-passages rectangular minichannel, 3 mm deep and 1 mm wide, cooled by a 1% by volume suspension of Al₂O₃ nanoparticles in water, over the Reynolds number range 92 to 455. By varying the volume fraction ${\alpha }_{nf}$ of the second phase between 2% and 50%, under a constant heat flux of 16.67 W/cm² and 30 Celsius coolant inflow, it is shown that the two-phase mixture model predicts heat transfer coefficient, pressure loss, friction factor, exergy destruction rate, exergy expenditure rate, and second law efficiency values converging to the single-phase model ones at increasing ${\alpha }_{nf}$. A two-phase mixture model defined with 1% second phase volume fraction and 100% nanoparticles volume fraction in the second phase breaks the Newtonian fluid assumption within the model and produces outlier predictions. By avoiding this unphysical regime, the two-phase mixture model matched experimental measurements of average heat transfer coefficient to within 1.76%. This has opened the way for using the two-phase mixture model with confidence to assess and resolve uneven nanoparticle dispersion effects and increase the thermal and mass transport performance of minichannels.

1. Introduction

The performance growth and size reduction of micro-electronic devices, such as computer cores, are currently limited by the ability to dissipate the heat produced by the supplied electrical power, through a conventional air-cooled heat sink. There is therefore a sustained research interest in water-based liquid coolants for mini/microchannels cooling systems, owing to the higher specific heat capacity of water compared to air. Water-based suspensions of nanoparticles, referred to as nanofluids, have an even better cooling potential. Compared to the base fluid (water), the nanofluids can extract more heat for the same volume flow rate. Many studies have investigated nanofluids by way of experiment and/or simulation. As the tracking of individual nanoparticles by a Eulerian-Lagrangian approach remains computationally expensive, nanofluid numerical computations use either a single-phase or a two-phase approach.

In the single-phase approach, the suspension of nanoparticles in a base fluid is assumed to be homogeneous and the nanoparticles are assumed to be in thermal equilibrium with the base fluid [1]. The nanofluid is replaced by a single fluid of equivalent thermophysical properties to the ones of the suspension, determined from experiment or by a physical model. The accuracy to which the thermophysical properties of the single fluid are estimated is very important for this type of flow simulation. Santra, et al. [2], Koo and Kleinstreuer [3], and Xuan and Roetzel [4] used this single-phase approach, which is relatively computationally inexpensive and has been shown to reproduce flow predictions close to experimental measurements.

The single-phase approach is popular for modelling the nanofluid flow for electronic core heat sinks. For example, Bhattacharya, et al. [5] examined the laminar forced convective heat transfer of Al₂O₃ nanofluid in a microchannel heat sink of a rectangular cross-section and found the addition of nanoparticles to be more beneficial to cooling at a lower Reynolds number. Nebbati and Kadja [6] and Ali, et al. [7,8] modelled Al₂O₃ nanofluid in a microchannel heat sink and showed that the nanofluids can increase the Nusselt number and reduce the heat sink contact surface temperature compared to water, at an acceptable through-flow pressure drop penalty.

In the two-phase approach, nanoparticles and their base flow are split in two separate phases. These phases move and interact according to a volume of fluid model (VOF) [9], a mixture model [10], an Eulerian model [11], or a Discrete Phase model [12]. Each model is distinct in the way it manages the mass, momentum, and energy conservation of each phase. The mixture model solves one momentum and one energy equation for the mixture, one volume fraction equation, and individual continuity equations for each phase. The phase velocities are then obtained by empirical correlations [13,14]. By this, a greater insight into the role of each phase on heat transfer is obtained compared to a single-phase approach [15].

Mixture models of microchannel flows have shown heat transfer enhancement with increasing nanoparticle concentration, in Esmaeilnejad, et al. [16], in line with previous findings by Bianco, et al. [17], secondary flow generated by helical inserts, in Narrein, et al. [18], and nanoparticle size effect on heat transport, in Mirmasoumi and Behzadmehr [19] and in Akbarinia and Laur [20].

Comparisons between single-phase and two-phase mixture model predictions are shown in Akbari, et al. [21], who report the two-phase model in good agreement in Nusselt number with a CuO nanofluid experiment, and in Sidik, et al. [22], where the laminar and turbulent forced convection by the two-phase model was in closer agreement with experiments by Chein and Chuang [23]. Ambreen and Kim [24] modelled a microchannel cooled by Al₂O₃ nanofluid, showing that single-phase and two-phase mixture models produced a maximum Nusselt number deviation of 11% and 3.2% compared to experiment, respectively. Kumar and Sarkar [25] modelled the laminar forced convection of Al₂O₃ nanofluids in a minichannel heat sink. The single-phase and mixture models predicted average Nusselt numbers 22.57% and 13.21% different than in experiment, respectively.

The main alternatives to the mixture model for the two-phase approach are compared in Moraveji and Ardehali [26], where a steady, laminar microchannel flow simulation with Al₂O₃ nanofluid uses the single-phase, VOF, mixture, and Eulerian methods. The two-phase model predictions are in closer agreement to experiment than the single-phase model predictions. Naphon and Nakharintr [27] reported the same finding using TiO₂ nanofluids in a minichannel heat sink and so did Moraveji and Ardehali [26] by way of applying Al₂O₃ nanofluid in a liquid block. Kurowski, et al. [28] made use of the single-phase, Eulerian–Lagrangian, and mixture models to simulate nanofluid flow inside a minichannel and documented a consistent flow and heat transfer behavior among the three models. Haghshenas Fard, et al. [29] analyzed the 0.2% CuO nanofluid heat transfer inside a tube and found the average relative error between the experiment and single-phase and two-phase model predictions to be 16% and 8%, respectively, with the best performing two-phase model. Kamyar, et al. [30] studied the forced convection of a nanofluid in a horizontal tube using single-phase, mixture, and Eulerian two-phase models, showing that the mixture model is more accurate than the other two models.

Whereas the reduced assumptions of the two-phase mixture model compared with the single-phase approach can provide higher fidelity predictions, some authors reported over-estimations of the heat transfer with the two-phase mixture model compared to the single-phase approach [31]. This generates lack of confidence in the application of the two-phase mixture simulations for modelling microchannel and minichannel heat sink flows.

The main aim of this work is to review the implementation of the two-phase mixture model to microchannel and mini-channel heat exchangers and show that improved heat prediction performance with respect to a single-phase approach can be achieved when appropriate physical modelling assumptions are made. This provides a significant contribution to the scientific community by identifying how the two-phase mixture model can be used with confidence and exploit its greater fidelity to the physical flow. The main expected outcome is the wider adoption of this higher fidelity model, to provide significant energy savings, longevity, and cost reduction to microprocessor-aided technology, now ubiquitous across all industrial sectors.

To reach this aim, this paper provides the following content: Section 2 Methodology states the two-phase mixture model formulation used in this work, which is essential in a paper aimed at providing an archival reference on how to implement such a model. Therein, Section 2.9 validates the implementation against experimental results recorded by Kumar and Sarkar [25] in a minichannel cooled by Al₂O₃ nanofluid. Section 3 shows a key aspect for obtaining physically consistent and reliable heat transfer model predictions from the minichannel two-phase mixture model, namely, to define two fluid phases, the first being water and the second being a suspension of Al₂O₃ nanoparticles in water, modelled as a fluid of equivalent properties. Model limitations with respect to the nanoparticle volume fraction in the second phase and with respect to the size of the nanoparticles compared to that of the ‘bubbles’ of the second phase are documented.

Section 3.4 explores another important aspect of minichannel heat sinks, from the second law of thermodynamics perspective. The heat sink heat extraction process involves the irreversible conversion of higher-grade input heat into lower grade output heat, through thermal resistances, and of coolant pumping power into heat, by friction. This can be assessed by means of entropy generation [32,33,34] and exergy [35] and by different approaches based on the second law of thermodynamics [36,37,38,39,40]. Khaleduzzaman, et al. [38] and Bahiraei, et al. [40] reported an increasing frictional entropy production and a decreasing thermal entropy production with increasing Reynolds number or flow velocity. In a TiO₂ nanofluid heat sink model of Khaleduzzaman, et al. [38], the outlet exergy increased with the Reynolds number. Bahiraei, et al. [40] showed these trends are nanoparticle volume fraction invariant. The current investigation aims to confirm these findings using the physically consistent two-phase mixture model implementation detailed in this work, for a constant minichannel heat flux input of 16.67 W/cm², 1% Al₂O₃ nanofluid, over the Reynolds number range 91 to 455.

2. Methodology

2.1. Mathematical Formulation

A three-dimensional numerical analysis of the laminar nanofluid forced convection in the minichannel of Figure 1 is performed. Figure 1 outlines the geometry and the computational domain. A constant heat flux is applied at the bottom plate of the heat sink. The conjugate heat transfer through the aluminum walls and through the coolant flow is evaluated numerically by means of coupling the Laplace equation,

$${\nabla }^2{T}_s=0$$

(1)

which models the distribution of temperature $T_s$ through the solid walls, with the time-averaged Navier-Stokes equations, which model the coolant [8]. The coolant is represented as a Newtonian fluid using either a single-phase approach or the two-phase mixture model approach.

2.1.1. Single-Phase Approach

By the single-phase approach, the nanofluid is modelled as a homogeneous Newtonian fluid. The continuity, momentum, and energy equations are solved numerically in line with the assumptions shown below:

• The flow is three dimensional, laminar, Newtonian, steady, and incompressible.
• The flow is steady, temperature dependent, and the flow velocity is uniform at the inlet.
• The heat sink radiative heat transfer is negligible compared to that by conduction and convection.

The fluid flow governing equations are: [24]:

$$\nabla ·\left({\rho }_{nf}{\overrightarrow{u}}_{nf}\right)=0$$

(2)

$$\nabla ·\left({\rho }_{nf}{\overrightarrow{u}}_i{\overrightarrow{u}}_i\right)=-\nabla p_{nf}+{\mu }_{nf}{\nabla }^2{\overrightarrow{u}}_{nf}+{\rho }_{nf}\overrightarrow{g}$$

(3)

$${\rho }_{nf}{\overrightarrow{u}}_{nf}·\nabla T_f=\nabla ·\left(k_{nf} \nabla T_f\right)$$

(4)

where ${\rho }_{nf}, {\mu }_{nf}, k_{nf}, {\overrightarrow{u}}_{nf}, p_{nf},\overrightarrow{g}, \mathrm{and} T_f$ are the nanofluid density, nanofluid molecular viscosity, nanofluid thermal conductivity, the velocity vector, the absolute pressure, the gravitational acceleration in the negative $x_2$ direction, and the absolute temperature of the nanofluid, respectively.

2.1.2. Multiphase Mixture Model

This model represents the flow by two phases, the first phase is water, and the second phase is a homogeneous Newtonian fluid, the properties of which are equivalent to those of a suspension of nanoparticles in water. In this second phase, the coupling between the motion of the nanoparticles and the fluid motion surrounding them is assumed to be strong, so that the nanoparticles move with their neighboring fluid in the second phase. Each phase has its own velocity vector field. The first and second phases are immiscible and the second phase is modelled as spherical bubbles dispersed in the first phase continuum, in a bubbly flow regime [41]. The portion of space occupied by the second phase bubbles at any spatial location is defined by the volume fraction. The first phase moves the second phase by way of a drag force and the second phase affects the first phase because of a reduction in the first phase mean momentum. The coolant flow is modelled as a two-phase mixture based on the following assumptions:

• The flow is steady, laminar, incompressible, as well as three-dimensional.
• The second phase comprises spherical bubbles of prescribed uniform size [42].
• All of the phases share a single pressure [22].
• The working fluid thermal properties are volume fraction dependent [22].
• Gravitational effects are modelled, and radiative heat transfer is disregarded [14].
• The volume fraction of nanoparticles in the mixture is sufficiently low so that collision and heat conduction by contact between particles are negligible [42].
• The diameter of the nanoparticles is smaller than the diameter of the bubbles so that the nanoparticles are uniformly dispersed in the second phase [19], so that the volume fraction $\mathsf{\Phi }$ of the nanoparticles in the second phase has one average value and that the spherical shape of the bubbles is unaffected by the presence of the nanoparticles in it.
• The diameter of the bubbles is smaller than the minichannel hydraulic diameter so that a local equilibrium is established over short spatial length scales [42].

The continuity equation in relation to the mixture is:

$$\nabla ·\left({\rho }_m{\overrightarrow{u}}_m\right)=0$$

(5)

where ${\overrightarrow{u}}_m$ is the mass-averaged mixture velocity

$${\overrightarrow{u}}_m=\frac{{\alpha }_{bf}{\rho }_{bf}{\overrightarrow{u}}_{bf}+{\alpha }_{nf}{\rho }_{nf}{\overrightarrow{u}}_{nf}}{{\rho }_m}$$

(6)

and ${\rho }_m$ is the mixture density

$${\rho }_m={\alpha }_{bf}{\rho }_{bf}+{\alpha }_{nf}{\rho }_{nf}$$

(7)

The momentum equation for the mixture can be shown as:

$$\begin{gathered} \nabla ·\left({\rho }_m{\overrightarrow{u}}_m{\overrightarrow{u}}_m\right)=-\nabla p+{\mu }_m{\nabla }^2{\overrightarrow{u}}_m+{\rho }_m\overrightarrow{g}+ \\ \nabla ·\left({\alpha }_{bf}{\rho }_{bf}{\overrightarrow{u}}_{dr,bf}{\overrightarrow{u}}_{dr,bf}+{\alpha }_{nf}{\rho }_{nf}{\overrightarrow{u}}_{dr,nf}{\overrightarrow{u}}_{dr,nf}\right) \end{gathered}$$

(8)

The molecular viscosity of the mixture ${\mu }_m$ is

$${\mu }_m={\alpha }_{bf}{\mu }_{bf}+{\alpha }_{nf}{\mu }_{nf}$$

(9)

where ${\mu }_{bf}$ and ${\mu }_{nf}$ are the molecular viscosities of the first and of the second phases, respectively.

The drift velocity of the second phase ${\overrightarrow{u}}_{dr,nf}$ is defined as the phase dispersion velocity relative to the mixture velocity:

$${\overrightarrow{u}}_{dr,nf}={\overrightarrow{u}}_{nf bf}-{\overrightarrow{u}}_m$$

(10)

$${\overrightarrow{u}}_{nf bf}=\frac{\left({\rho }_{nf}-{\rho }_m\right)d_{nf}^2}{18{\mu }_m{f}_{drag}}\overrightarrow{a}$$

(11)

where $d_{nf}$ is the uniform diameter of the second phase bubbles. The drag coefficient $f_D$ is evaluated according to Naumann and Schiller [43] as

$$f_D=\left\{\begin{array}{c}1+0.15R{e}_{nf}^{0.687} R{e}_p\le 1000\\ 0.0183R{e}_{nf} R{e}_p>1000\end{array}\right.$$

(12)

where $\overrightarrow{a}$ is the second phase bubble acceleration

$$\overrightarrow{a}=\overrightarrow{g}-\left({\overrightarrow{u}}_m·\nabla \right){\overrightarrow{u}}_m$$

(13)

Using a length scale analysis, Buongiorno [44] shows that the effect of gravity and of drag on nanoparticles is insignificant. Whereas nanoparticles have a size of order 1 nm, the second phase bubbles that model their presence in the two-phase mixture model have size of order 100 nm. The larger scale of these bubbles, compared with that of the nanoparticles, enables the estimation of gravitational and drag effects on the bubble motion, albeit the length scale analysis suggests these effects are small. This limits the application of this two-phase model to larger-sized bubbles, above 80 nm, as gravity and drag estimates become progressively smaller and numerically more difficult to account for in the flow force-momentum balance with smaller bubble sizes.

Due to their small size, the nanofluid particles are assumed to be in thermal equilibrium within each second phase bubble and the temperature field is assumed to be smooth across phase boundaries. This enables defining a single temperature field for the mixture that is governed by the following scalar transport equation:

$$\nabla ·\left({\alpha }_{bf}{\rho }_{bf}C{p}_{bf}{\overrightarrow{u}}_{bf}+{\alpha }_{nf}{\rho }_{nf}C{p}_{nf}{\overrightarrow{u}}_{nf}\right)T_f=\nabla ·\left(k_m \nabla T_f\right)$$

(14)

where $T_f$ is the mixture temperature, $C{p}_{bf}$ is the constant pressure specific heat capacity of the first phase, $C{p}_{nf}$ is the constant pressure specific heat capacity of the second phase, and $k_m$ is the mixture thermal conductivity

$$k_m={\alpha }_{bf}{k}_{bf}+{\alpha }_{nf}{k}_{nf}$$

(15)

where $k_{bf}$ is the thermal conductivity of the first phase and $k_{nf}$ is the thermal conductivity of the second phase.

The two-phase mixture model does not track the motion of individual particles and is therefore unable to model particle agglomeration by particle surface attraction, for instance by chemical bonding. In addition, it is unable to model particle sedimentation, which entails the mechanical locking of particles onto one another. However, by Equation (13), the accelerations due to flow path curvature and due to gravity on the second phase are accounted for. This, in principle, enables the model to represent phase stratification (but not particle sedimentation), due to gravity, and phase segregation (but not particle agglomeration), due to flow path curvature, both of which are not representable by the single-phase approach.

2.2. Nanofluid Model

The thermal and physical properties of water and Al₂O₃ nanoparticles are displayed in

Table 1. The thermo-physical properties of water at 20 °C [45] and of nanoparticles [46].

	$\mathit{\rho } \left(\mathbf{k}\mathbf{g}/{\mathbf{m}}^3\right)$	${\mathit{C}}_{\mathit{P}} \left(\mathbf{J}/\mathbf{k}\mathbf{g}\cdot \mathbf{K}\right)$	$\mathit{k} \left(\mathbf{W}/\mathbf{m}\cdot \mathbf{K}\right)$	$\mathit{\mu } \left(\mathbf{N}\cdot \mathbf{s}/{\mathbf{m}}^2\right)$	Diameter of Nanoparticles (nm)
Water	998.2	4183	0.603	0.001002	-
Al2O3	3970	765	40	-	15

. Two alternative approaches are used to define a suspension of Al₂O₃ nanoparticles in water at the nanoparticle module fraction $\varphi$ of 1%. In the single-phase approach, the nanofluid is simulated by one homogeneous fluid of equivalent thermal and physical properties. The two-phase approach uses water as the first phase and a homogeneous fluid of equivalent thermal and physical properties as the second phase.

In the two-phase approach, this work uses the seven combinations of second phase volume fraction ${\alpha }_{nf}$ and of second phase Al₂O₃ nanoparticle volume concentration $\mathsf{\Phi }$ listed in

Table 2. Combinations of second phase volume fraction and of second phase nanoparticle volume fraction giving the same $\varphi$ = 1% Al₂O₃ nanofluid mixture.

Second phase $\mathrm{volume} \mathrm{fraction} {\alpha }_{nf}$	0.5	0.2	0.1	0.08	0.06	0.04	0.02
Second phase Al2O3 nanoparticle $\mathrm{volume} \mathrm{fraction} \mathsf{\Phi }$	0.02	0.05	0.1	0.125	0.1667	0.25	0.5

. Each combination provides the same 1% Al₂O₃ nanoparticle volume fraction mixture $\varphi$.

The thermophysical properties of the equivalent homogeneous fluid are evaluated as follows:

The density of the equivalent homogeneous fluid is [4]:

$${\rho }_{nf}=\left(1-\varphi \right){\rho }_{bf}+\varphi {\rho }_{np}$$

(16)

The equivalent homogeneous fluid specific heat capacity is [4]:

$$C{p}_{nf}=\left(1-\varphi \right)C{p}_{bf}+\varphi C{p}_{np}$$

(17)

The thermal conductivity of the equivalent homogeneous fluid is [4]:

$$k_{nf}=k_{bf}\frac{k_{np}+2k_{bf}-2\varphi \left(k_{bf}-k_{np}\right)}{k_{np}+2k_{bf}+\varphi \left(k_{bf}-k_{np}\right)}$$

(18)

The effective dynamic viscosity of the equivalent homogeneous fluid is [8]:

$${\mu }_{nf}={\mu }_{bf}\left(123{\varphi }^2+7.3\varphi +1\right)$$

(19)

In the two-phase mixture model, $\mathsf{\Phi }$ from Table 2 replaces $\varphi$ in Equations (16)–(19) to evaluate the second phase thermophysical properties.

2.3. Boundary Conditions

To solve the governing equations in Section 2.1, Section 2.1.1 and Section 2.1.2, the following time-invariant conditions are imposed uniformly at each computational domain boundary:

1.	Minichannel inlet first and second phases	$u_1=u_{in}, u_2=u_3=0$	$\mathrm{for} \mathrm{fluid} T_f=T_{in}=303.15 \mathrm{K}$ $\mathrm{for} \mathrm{solid} \frac{\partial T_s}{\partial x_1}=0$
2.	Minichannel outlet first and second phases	$p_f=p_{out}=1 \mathrm{atm}$	$\mathrm{for} \mathrm{fluid} \frac{\partial T_f}{\partial x_1}=0$ $\mathrm{for} \mathrm{solid} \frac{\partial T_s}{\partial x_1}=0$
3.	The top wall		$\mathrm{solid} \frac{\partial T_s}{\partial x_2}=0$
4.	Right and left walls	(symmetry)	$\mathrm{solid} \frac{\partial T_s}{\partial x_3}=0$
5.	Base wall		$-k_s\frac{\partial T_s}{\partial x_2}={\dot{q}}_2=16.67 \mathrm{W}/{\mathrm{cm}}^2$
6.	Inner walls (fluid/solid surface)	$u_1=u_2=u_3=0$	$-k_s\frac{\partial T_s}{\partial x_n}=-k_f\frac{\partial T_f}{\partial x_n}$

where $n, k_f$ and $k_s$ are, respectively, the coordinate normal to the wall, the working fluid thermal conductivity, and the solid wall thermal conductivity. In the single-phase approach $k_f=k_{nf}$ and in the two-phase approach $k_f=k_m$.

2.4. Numerical Solver

The three-dimensional computational domain of Figure 1 is discretized in finite volumes. The conjugate heat transfer problem is solved numerically using the commercial software FLUENT version 19.5, by ANSYS Inc., Canonsburg, PA, USA. The flow is defined either as constant property, incompressible, laminar, and single-phase or a constant property, incompressible, laminar, and two-phase mixture. The boundary conditions in Section 2.3 for the fluid domain are implemented by the velocity-inlet, pressure-outlet, symmetry, and wall boundary types in FLUENT 19.5. At the minichannel inlet, the velocity $(u_{in})$ is set uniform and normal to the boundary, the inflow temperature is 303.15 K and uniform and, in two-phase simulations, the second phase volume fraction is set according to Table 2. At the pressure outlet, a uniform gauge pressure of zero is applied. A steady heat flux input of 16.67 W/cm² is prescribed on the base wall and of 0 W/cm² on all other exterior walls. The reference absolute pressure is set to 1 bar. The single-phase approach uses constant values for ${\rho }_{nf},C{p}_{nf}, {\mu }_{nf} \mathrm{and} k_{nf}$ which are set at the start of the computation by evaluating Equations (16)–(19) with $\varphi$ = 1%. The two-phase mixture model uses constant values of ${\rho }_{bf},C{p}_{bf}, {\mu }_{bf} \mathrm{and} k_{bf}$ for the first phase and constant values for ${\rho }_{nf},C{p}_{nf}, {\mu }_{nf} \mathrm{and} k_{nf}$ for the second phase, which are also set at the start of the computation by evaluating Equations (16)–(19) with $\mathsf{\Phi }$ from Table 2. These constant properties are space and time invariant in both the single-phase and the two-phase simulations.

The double-precision steady pressure-based coupled flow solver of ANSYS FLUENT 19.5 is used with the PRESTO! scheme for pressure, with the second-order upwind method for the momentum and energy equations, and with the QUICK scheme for the volume fraction equation [47]. The hybrid initialization of ANSYS FLUENT 19.5 provides the interior flow conditions and the temperature distribution in the solid domain portion at the start of the computation. The implicit scheme iterates the solution with a Courant number limit of 40, an explicit relaxation factor of 0.5 for momentum and pressure, and under-relaxation factors of 0.1 and 0.4 for the slip velocity and the volume fraction, respectively. Multi-grid acceleration is implemented for the volume fraction with a flexible cycle, which is limited to two sweeps, 30 fine relaxation levels, and 50 coarse relaxation levels. The solution was deemed converged when the residuals for the continuity, momentum and energy equations fell below 1 × 10⁻⁶ [48]. The computational cost of each simulation was 30 min on a 3.2 GHz sixteen core shared memory computer cluster.

2.5. Geometry

The minichannel heat sink consists of nine parallel rectangular-shaped ducts with a length of 30 mm, a height of 4 mm, a channel width of 1 mm, a fin width of 1 mm, and a channel height of 3 mm. Only one duct is modelled to decrease the complexity and the computational cost. The temperature field is assumed to be symmetric between adjacent ducts, about the sectioned planes shown with dashed lines in Figure 1a.

Table 3. The thermo-physical properties of aluminum [49].

	$\mathit{\rho } \left(\mathbf{k}\mathbf{g}/{\mathbf{m}}^3\right)$	$\mathit{C}\mathit{p} \left(\mathbf{J}/\mathbf{k}\mathbf{g}\cdot \mathbf{K}\right)$	$\mathit{k} \left(\mathbf{W}/\mathbf{m}\cdot \mathbf{K}\right)$
Aluminum	2719	871	202.4

shows the thermo-physical properties of the solid domain made from aluminum. The Reynolds number referenced to the inlet conditions ranges from 91 to 455 and it is defined as:

$$R_e=\frac{{\rho }_{nf}{u}_{in}{D}_h}{{\mu }_{nf}}$$

(20)

where ${\rho }_{nf}$ is the density of the working fluid evaluated by Equation (16) with $\varphi$ = 1%, $u_{in}$ is the average fluid velocity at the minichannel inlet, and ${\mu }_{nf}$ is the dynamic viscosity of the working fluid evaluated by Equation (19) with $\varphi$ = 1%. The hydraulic diameter ($D_h$) is

$$D_h=\frac{4A}{P}=\frac{4ab}{2\left(a+b\right)}=\frac{2ab}{a+b}$$

(21)

where, P, a and b are, respectively, the perimeter, width, and thickness of the duct.

An auxiliary test case is defined to test the model limitations identified in the minichannel heat sink simulations, using a geometry that features greater flow complexity. To this end, a simulation is performed of the flow and heat transport over a microscale backwards-facing step, which is uniformly heated over its base, as shown by the two-dimensional schematic of Figure 2. The inlet channel height $h_u=0.4 \mathrm{mm}$, the step height $h_d=0.6 \mathrm{mm}$, the length of the narrow channel upstream of the step $L_u=100 \mathrm{mm}$, and the length of the wide channel downstream of the step $L_d=150 \mathrm{mm}$. The geometry is tested at a constant Reynolds number of 280, which is based on the inflow conditions and on the hydraulic diameter $D_h=\left(h_u+h_d\right)$. The nanofluid, made by a 1% by volume suspension of Al₂O₃ nanoparticles in water, flows through the conduit. A constant heat flux ${\dot{q}}_2=30$ kW/m² is applied to the bottom wall, as in Ahmed, et al. [50]. Further details are provided in Kherbeet, et al. [51] and Klazly and Bognar [52] where both measurements and numerical predictions are available for comparison purposes.

Considering the purpose of this auxiliary test case is to address the model limitations in the presence of greater flow complexity, this computation is limited to the flow domain, which removes the requirement of solving a conjugate heat transfer problem through the solid walls. To this end, the single-phase approach of Section 2.1.1 and the two-phase mixture model of Section 2.1.2 are used, together with the nanofluid model of Section 2.2. At the inlet, boundary condition 1 of Section 2.3 is used. At the outlet, boundary condition 2 of Section 2.3 is used. The bottom wall is modelled with the no-slip boundary condition and $-k_f\frac{\partial T_f}{\partial x_2}={\dot{q}}_2$. The remaining walls are modelled as no-slip and adiabatic, by which $u_1=u_2=0$ and $\frac{\partial T_f}{\partial x_n}=0$, where $n$ is the coordinate normal to the wall.

2.6. Data Reduction

The average convection heat transfer coefficient $\left(h\right)$ of the minichannel heat sink is calculated as [25]:

$$h=\frac{\dot{Q}}{A_b\left(T_s-T_{mean}\right)}$$

(22)

$$\dot{Q}={\dot{q}}_{2}L{x}_{2}L{x}_3$$

(23)

where $\dot{Q}$ is the heat transfer rate, $A_b$ is the area of the minichannel heat sink base, which is determined by Equation (24) [25], $T_s$ is the average bottom wall temperature of the minichannel heat sink, and $T_{mean}$ is the mean fluid temperature, which is determined from Equation (25).

$$A_b=L{x}_{2}N\left(W_{ch}+W_{fin}\right)$$

(24)

where, $L_{x2}$ is the channel length, $N$ is number of ducts, $W_{ch}$ is the channel width, and $W_{fin}$ is the width of the fin, as shown in Figure 1.

$$T_{mean}=\frac{T_f{}_{in}+T_f{}_{out}}{2}$$

(25)

where $T_f{}_{in}$ and $T_f{}_{out}$ are, respectively, the inlet and the outlet fluid temperatures of the minichannel heat sink.

The friction factor $f$ is [53]:

$$f=\frac{2D_h\Delta p}{{\rho }_{nf}{u}_{in}^{2}L{x}_1}$$

(26)

where ${\rho }_{nf}$ is the density of the working fluid evaluated by Equation (16) with $\varphi$ = 1% and $\mathsf{\Delta }p$ is the pressure drop of the working fluid between the inlet and the outlet [54]:

$$\Delta p=p_{in}-p_{out}$$

(27)

2.7. The Second Law Efficiency Analysis of the Minichannel Heat Sink Flow

The second law efficiency evaluates the quality of the thermodynamic process performed by a heat sink. It does so by considering the irreversibility of the cooling process. The primary sources of irreversibility are the temperature drop in the heat that is transported and the friction losses in the working fluid. One method to determine the magnitudes of these irreversibilities is evaluating the steady flow exergy balance [55], using the inlet flow temperature and the outflow pressure of the working fluid as the reference dead state. Figure 3 shows the control-volume two-dimensional schematic of a heat sink with a steady through flow and a constant heat flux from the bottom. The exergy destruction rate $({\dot{E}}_{xd})$ from the exergy balance through the dashed line control volume is [7]:

$${\dot{E}}_{xd}=-{{\displaystyle \iint }}_{base}^{}\left(1-\frac{T_r}{T}\right) {\dot{q}}_i{n}_i dA-{{\displaystyle \iint }}_{inlet}^{}{\rho }_{nf}\psi u_i{n}_i dA-{{\displaystyle \iint }}_{outlet}^{}{\rho }_{nf}\psi u_i{n}_i dA$$

(28)

where $T_r$ is the dead state reference temperature, which is equal to the inlet temperature $T_{in}=303.15$ K, $T$ is the temperature at the control volume surface, ${\dot{q}}_i$ is the heat flux vector, which is defined positive outwards from the control volume, $n_i$ is the control volume boundary normal unit vector, which is positive outwards, $u_i$ is the working fluid velocity, and $\psi$ is the flow exergy. The latter is defined as

$$\psi =\left(h_0-h_{0r}\right)-T_r\left(s-s_r\right)$$

(29)

where $h_0$, $h_{0r},$ $s$, and $s_r$ are the specific stagnation enthalpy, the specific stagnation enthalpy at $T_r$, the specific entropy, and the specific entropy at $T_r$, respectively. The contribution to the flow exergy owing to gravity is disregarded, as the inlet and the outlet of the minichannel are horizontally aligned [8]. The constant property incompressible flow assumption in the single-phase and two-phase mixture models enables restating Equation (29) as

$$\psi =C{p}_{nf} \left(T_0+\frac{u_i{}^2}{2C{p}_{nf}}-T_{0r}-\frac{u_i{}^2}{2C{p}_{nf}}-T_r\ln \left(\frac{T}{T_r}\right)\right)$$

(30)

where $C{p}_{nf}$ is evaluated from Equation (17) using $\varphi$ = 1% for all simulations.

The rate of exergy expended $({\dot{E}}_x$) through the minichannel cooling process is the sum of the total reversible work rate potential of the heat supplied through the base, as identified in Liu, et al. [35], plus the reversible work rate potential of the coolant supplied to the inlet, as identified in Ali, et al. [7]. This gives

$${\dot{E}}_x=-{\iint }_{base}^{}\left(1-\frac{T_r}{T}\right) {\dot{q}}_i{n}_i dA-{\iint }_{inlet}^{}{\rho }_{nf}\psi u_i{n}_i dA$$

(31)

In Equations (28) and (31), ${\rho }_{nf}$ is the density of the working fluid evaluated by Equation (16) with $\varphi$ = 1%.

By combining the exergy destruction rate and the rate of exergy expended, the second law efficiency of the minichannel heat sink is determined as [8].

$${\eta }_{II}=\left(1- \frac{{\dot{E}}_{xd}}{{\dot{E}}_x}\right)$$

(32)

2.8. Numerical Mesh

In this work, the rectangular prism shape of the minichannel enables the use of a topologically simple hexahedral computational mesh. Such mesh can provide a better computational accuracy and an improved convergence speed over tetrahedral and/or mixed element meshes in simple geometries [56]. The sensitivity of the predictions to the level of spatial mesh refinement was determined by comparing the numerical results from six computational meshes ranging from 0.24 million cells to 1.74 million cells. Predictions were obtained of the average minichannel bottom wall temperature and of the inlet to outlet pressure drop, at the coolant Reynolds number of 274, coolant inlet temperature of 303.15 K, base heat flux input of 16.76 W/cm², and with a suspension of 1% Al₂O₃ nanoparticles in water. The single-phase approach and the two-phase mixture model were both used, the latter with a combination of second phase volume fraction ${\alpha }_{nf}=10\%$ and of second phase nanoparticle volume fraction $\mathsf{\Phi }=$10%, and with $\left({\alpha }_{nf}=1\%,\mathsf{\Phi }=100\%\right)$. Figure 4a,b indicates differences among the models in the average minichannel bottom wall temperature and in the pressure drop, but for each model the predictions become substantially mesh independent above 1.12 million cells. Consequently, the mesh with 1.12 million cells is used throughout the remainder of this investigation. Figure 5 shows the three-dimensional structure of this mesh at the minichannel inlet.

Figure 6 shows an example of the convergence history with the two-phase mixture model. In this run, the minichannel through flow at a Reynolds number of 455 is simulated with a second phase volume fraction ${\alpha }_{nf}$ of 4% and with $\mathsf{\Phi }=25\%$. The residuals for continuity, x-velocity, y-velocity, z-velocity, and energy equations reduce monotonically below 10⁻⁶ in less than 100 iterations. This indicates an appropriate convergence of the results for the purpose of comparing them against corresponding ones from the single-phase approach and other two-phase mixture model runs.

2.9. Validation

To assess the numerical models, Figure 7a compares the average heat transfer coefficient of the minichannel heat sink predicted by the single-phase approach using water as the working fluid against experimental results recorded by Kumar and Sarkar [25]. The single-phase approach predictions are shown to follow reasonably well the experimental results, with a 16.7% maximum discrepancy at the Reynolds number of 92. In addition, Figure 7b compares the friction factor from plain water flowing through the minichannel single-phase simulation against the friction factor correlation for laminar square duct flows, the Darcy-Weisbach friction factor $f=62.2 R{e}^{-1}$ [55]. As this friction factor correlation applies to a fully developed laminar duct flow, $f$ was estimated from the pressure drop between the end of the hydrodynamic entrance region and the minichannel outlet. The hydrodynamic entrance region length $L_h\cong 0.05Re{D}_h$ [55]. There is good agreement in trend and values between the numerical predictions and the empirical correlation, which both follow the ($1/Re$) proportionality.

The two-phase mixture model was run using water for both phases and produced the same predictions for the average heat transfer coefficient and for the friction factor as the single-phase approach, over the range $92\le Re\le 455$.

3. Results and Discussion

In this section, the influence of Al₂O₃ nanofluid on the thermal and hydraulic characteristics of a minichannel heat sink has been evaluated using both the single-phase approach and the two-phase mixture model.

3.1. Effect of the Volume Fraction of the Second Phase ${\alpha }_{nf}$ on the Thermal Characteristics

Figure 8 shows the average heat transfer coefficient obtained with single-phase approach and with the two-phase mixture model. Different combinations of phase two volume fraction ${\alpha }_{nf}$ and of phase two Al₂O₃ nanoparticle volume fraction $\mathsf{\Phi }$, as defined in Table 2, provide the same 1% Al₂O₃ nanoparticle volume fraction in the mixture. The results are compared with the experiments by Kumar and Sarkar [25], over the Reynolds number range 92 to 455. Overall, the average heat transfer coefficient in the minichannel heat sink increases with the Reynolds number, due to the higher velocity providing a stronger heat convection. This improvement in $h$ is similar to that reported in the literature [57,58]. The mixture model with $\left({\alpha }_{nf}=1\%,\mathsf{\Phi }=100\%\right)$ is shown to overpredict the average heat transfer coefficient when compared to the experiments and to any of the other models, due to the overestimation of the phase interaction between nanoparticles and the base fluid, consistent with the results by Akbari, et al. [11], Moraveji and Ardehali [26], and Mojarrad, et al. [31]. On the contrary, the average heat transfer coefficient from the single-phase approach and from the two-phase mixture model with $\left({\alpha }_{nf}\ge 2\%,\mathsf{\Phi }\le 50\%\right)$ is underestimated, particularly at lower Reynolds numbers. This is in line with the results obtained for water reported in Section 2.9, which show a similar underestimation at low Reynolds numbers.

It is observed that the predictions from the single-phase approach and from the two-phase mixture with $\left({\alpha }_{nf}\ge 2\%,\mathsf{\Phi }\le 50\%\right)$ essentially overlap. The close agreement between the two-phase mixture model at $\left({\alpha }_{nf}=50\%,\mathsf{\Phi }=2\%\right)$ and the single-phase approach at $\varphi =1\%$ shows that the two-phase mixture model predicts the same heat transport as the nanoparticles approach a uniform distribution in the flow, which is what the single-phase approach simulates. The convergence between the two models in their heat transport prediction is an important outcome of this investigation. It shows that the reservations in the application of the two-phase mixture model to minichannel heat transport problems can be overcome by a judicious model use, as further elaborated later in this paper.

Figure 8 shows that the average heat transfer coefficient predicted by the two-phase mixture model is substantially invariant over a range of ${\alpha }_{nf}$ and $\mathsf{\Phi }$ combinations. This is likely to result from the simple geometry of the channel, which promotes the nanoparticle dispersion towards a uniform distribution. The absence of significant flow curvature to drive phase segregation indicates that the mass transport mechanism in this minichannel is unlikely to form areas of high nanoparticle concentration, which the two-phase mixture model is designed to resolve.

3.2. Effect of the Volume Fraction of the Second Phase ${\alpha }_{nf}$ on the Hydraulic Characteristics

Figure 9 shows the pressure drop along the minichannel predicted by the single-phase approach and by the two-phase mixture model at the conditions defined in Table 2. The results are compared to the experiments by Kumar and Sarkar [25] in the Reynolds number range 92 to 455 at the same Al₂O₃ nanofluid volume fraction $\varphi =1\%$. The pressure drop in the minichannel heat sink is shown to increase with the increasing Reynolds number. The single-phase approach and the two-phase mixture model with $\left({\alpha }_{nf}\ge 6\%,\mathsf{\Phi }\le 16.67\%\right)$ underpredict the pressure drop compared to the experiments, and so does the $\left({\alpha }_{nf}=1\%,\mathsf{\Phi }=100\%\right)$ two-phase mixture model, as previously reported by Akbari, et al. [21] and Ambreen and Kim [24]. In the two-phase mixture model with $\left({\alpha }_{nf}=1\%,\mathsf{\Phi }=100\%\right)$, the second phase is made up entirely of nanoparticles, yet the phase is modelled as a Newtonian fluid, with a finite molecular viscosity. This is an unphysical implementation of the two-phase mixture model, as a volume of nanoparticles packed at 100% volume fraction would respond to stress by a strain rather than by a strain rate. As such, Equation (19) appears to be used beyond its range of applicability. Figure 9 shows that the $\left({\alpha }_{nf}=1\%,\mathsf{\Phi }=100\%\right)$ prediction is an outlier. Specifically, Figure 9 shows a monotonic increment of the pressure drop predicted by the two-phase mixture model with decreasing ${\alpha }_{nf}$, down to ${\alpha }_{nf}$ = 2%, followed by a sharp drop in pressure loss, as ${\alpha }_{nf}$ is reduced from 2% to 1%, across the whole Reynolds number range 92 to 455. This behaviour typically suggests an inconsistency in the use of the physical model, which this work has identified in the definition of the second phase viscosity.

Concerning the two-phase mixture model with ${\alpha }_{nf}\ge 2\%$, Figure 9 shows that increasing ${\alpha }_{nf}$ produces pressure drop predictions that monotonically reduce at increasing ${\alpha }_{nf}$ and that incrementally approach the pressure drop predicted by the single-phase simulation. The closest two-phase mixture model prediction to that from the single-phase approach is that with $\left({\alpha }_{nf}=50\%,\mathsf{\Phi }=2\%\right)$, which most closely represents the homogeneous mixing of the single-phase approach among the two-phase mixture simulations. This convergence in predictions provides further confidence in using the two-phase mixture model to perform minichannel flow predictions, provided its modelling assumptions are upheld.

Figure 9 shows that, unlike the average heat transfer coefficient, the pressure drop depends on the combination of the phase two volume fraction ${\alpha }_{nf}$ and of the phase two Al₂O₃ nanoparticle volume fraction $\mathsf{\Phi }$. As the phase two volume fraction decreases below 50%, the pressure drop approaches the experimental results. For ${\alpha }_{nf}$ = 4%, the mixture corresponds to the case with the highest $\mathsf{\Phi }$ for which the homogenous model used within a single bubble is valid. For ${\alpha }_{nf}$ = 2%, the volume fraction of nanoparticles within a bubble is above 30%, beyond the range of applicability of the homogenous model [4], and this is reflected by a steep increase in pressure drop compared to the ${\alpha }_{nf}=4\%$ case. These results confirm the challenge of modelling nanofluids. As the homogeneous mixture relation between viscosity and nanoparticle volume fraction is not linear, Equation (19), the overall pressure drop is sensitive to the second phase nanoparticle volume fraction $\mathsf{\Phi }$ in the two-phase mixture model.

This means, in principle, that it is possible to cluster nanoparticles to obtain a desired pressure drop independently from the average heat transfer coefficient. This may be obtained by adding a coagulant to the nanofluid to promote flocculation or by adding a surfactant to promote nanofluid particle dispersion. Surfactants appear to be of greater potential benefit to a minichannel heat sink than coagulants, since, by promoting the nanoparticle dispersion, the condition of a homogeneous mixture of the single-phase model could be induced, which Figure 9 shows being that associated with the smallest pressure drop, among the physically valid models.

The effect of the second phase ${\alpha }_{nf}$ on the hydraulic characteristics is consistent over the Reynolds number range 92 to 455 and Figure 10 enables the appreciation this consistency by providing a larger spread among the friction factor curves at the low end of the Reynolds number range compared to Figure 9. The friction factor prediction that follows most closely the experimental results is that from the two-phase mixture model with $\left({\alpha }_{nf}=4\%,\mathsf{\Phi }=25\%\right)$ that under-predicts the measured friction factor by 4%. The single-phase model under-predicts the measured friction factor by 16%, and the two-phase model with $\left({\alpha }_{nf}=1\%,\mathsf{\Phi }=100\%\right)$ under-predicts the measured friction factor by 20%.

3.3. Effect of the Diameter Size of the Second Phase Nanofluid Bubbles on Thermal and Hydraulic Characteristics

Figure 11 presents a parametric study of the minichannel average bottom wall temperature and of its pressure drop using the two-phase mixture model with (${\alpha }_{nf}=20\%,\mathsf{\Phi }=5\%)$ and $\left({\alpha }_{nf}=4\%,\mathsf{\Phi }=25\%\right)$, in which the second phase nanofluid bubble diameter is varied over the range $85 \mathrm{nm}\le d_{nf} \le 150 \mathrm{nm}$. The average bottom wall temperature and the pressure drop are shown to be substantially independent from $d_{nf}$ over the range 85 nm to 150 nm. The two-phase mixture model with $\left({\alpha }_{nf}=4\%,\mathsf{\Phi }=25\%\right)$ did not meet the residuals convergence threshold of 1 × 10⁻⁶ when using a second phase nanofluid diameter in the range $137 \mathrm{nm}\le d_{nf} \le 150 \mathrm{nm}$. These less converged results are shown in Figure 11 by the open symbols. While not meeting the 10⁻⁶ residuals convergence criterion, these less converged results support the substantial independence of the two-phase mixture model predictions from the second phase bubble diameter $d_{nf}$, over the range $85 \mathrm{nm}\le d_{nf} \le 150 \mathrm{nm}$. All other two-phase mixture model simulations presented in this paper use a fixed value of $d_{nf}=98 \mathrm{nm}$, which is within this range.

3.4. The Second Law Efficiency of the Minichannel Heat Sink

Figure 12 displays the exergy destruction rate ${\dot{E}}_{xd}$ across the minichannel heat sink over the Reynolds number range 92 to 455 at a Al₂O₃ nanoparticle volume fraction $\varphi =1\%$. All models show a reduction in ${\dot{E}}_{xd}$ with increasing Reynolds number, which is due to the reduction in the minichannel bottom surface temperature at increasing Reynolds numbers [7,8]. In fact, reducing $T$ decreases $\left(1-\frac{T_r}{T}\right)$ in the dominant first term of Equation (28) and hence it lowers ${\dot{E}}_{xd}$. The exergy inflow transported by the nanofluid is zero since the dead state temperature $T_r$ is equal to the inlet temperature $T_{in}$. The minimum exergy destruction rate ${\dot{E}}_{xd}$ is achieved at the highest Reynolds number, at which the minichannel average bottom surface temperature is lowest, owing to the average heat transfer coefficient being highest, as shown in Figure 8. The ${\dot{E}}_{xd}$ predictions from the single-phase approach and from the two-phase mixture model with $\left({\alpha }_{nf}\ge 2\%,\mathsf{\Phi }\le 50\%\right)$ essentially overlap, confirming the good agreement among the models in predicting the minichannel thermal performance reported in Figure 8. The two-phase mixture model with $\left({\alpha }_{nf}=1\%,\mathsf{\Phi }=100\%\right)$ produces predictions discordant with this data collapse. This indicates that the two-phase mixture model is improperly applied to a second phase made up solely of nanoparticles, as discussed in Section 3.2. This agrees with the finding in Figure 8, where the heat transfer coefficient of the mixture model with $\left({\alpha }_{nf}=1\%,\mathsf{\Phi }=100\%\right)$ is reported as being well above that from all other models, as well as above the experimental measurements.

Figure 13 displays the rate of exergy expended ${\dot{E}}_x$ in the minichannel heat sink, predicted by the single-phase approach and by the two-phase mixture model, over the Reynolds number range 92 to 455, at a Al₂O₃ nanoparticle volume fraction $\varphi =1\%$. The only contribution of the rate of exergy expended ${\dot{E}}_x$ is the heat transfer through the heated base, since the dead state reference temperature $T_r$ is equal to the inlet temperature T so that $\psi =0$ across the inlet. Figure 13 shows the same decreasing trend in ${\dot{E}}_x$ with increasing Reynolds number as for ${\dot{E}}_{xd}$ in Figure 12. There is good agreement among the predictions from the single-phase approach and from the two-phase mixture model with $\left({\alpha }_{nf}\ge 2\%,\mathsf{\Phi }\le 50\%\right)$, which indicate that the lowest rate of exergy expended occurs at the highest Reynolds number. The two-phase mixture model with $\left({\alpha }_{nf}=1\%,\mathsf{\Phi }=100\%\right)$ gives lower exergy destruction rate predictions that are lower than the ones shown in the data collapse with $\left({\alpha }_{nf}\ge 2\%,\mathsf{\Phi }\le 50\%\right)$. This supports the $\left({\alpha }_{nf}=1\%,\mathsf{\Phi }=100\%\right)$ combination not being an appropriate two-phase mixture model setting, as discussed in the context of Figure 8.

Figure 14 shows the second law efficiency ${\eta }_{II}$. of the minichannel heat sink, estimated using the single-phase approach and the two-phase mixture model, at Reynolds numbers between 92 and 455, at a Al₂O₃ nanoparticle volume fraction $\varphi =1\%$. The second law efficiency is evaluated by combining ${\dot{E}}_{xd}$ and ${\dot{E}}_x$ in Equation (32).

The main trend is a decrement in the second law efficiency with increasing Reynolds number. This trend differs from that reported in Ali, et al. [7] for the flow through a rectangular microchannel at similar Reynolds numbers, which displayed a minimum and involved higher numerical values of second law efficiency. The cause of this trend change is unclear. One possible explanation is that the length scale difference between minichannels and microchannels causes a difference in the heat conduction resistance across the thicker minichannel walls. This may cause the second law efficiency minimum to occur at a higher Reynolds number in a minichannel than in a microchannel.

The two-phase mixture model with $\left({\alpha }_{nf}=1\%,\mathsf{\Phi }=100\%\right)$ predicts a second law efficiency 15.15% higher than that from the single-phase approach at $Re$ = 93 and 1.79% higher at $Re$ = 455. Using $\left({\alpha }_{nf}\ge 2\%,\mathsf{\Phi }\le 50\%\right)$ this discrepancy reduces to a maximum of 0.0314% and of 0.0115% at Reynolds numbers 92 and 455, respectively. This reaffirms the $\left({\alpha }_{nf}=1\%,\mathsf{\Phi }=100\%\right)$ combination not being an appropriate two-phase model setting.

Upon close inspection, Figure 14 shows that that two-phase mixture model predictions with $\left({\alpha }_{nf}=2\%,\mathsf{\Phi }=50\%\right)$ are second farthest from the single-phase approach prediction and that the two-phase mixture model ${\eta }_{II}$ predictions display a monotonic and progressive approach to the ${\eta }_{II}$ predicted by the single-phase approach, with increasing ${\alpha }_{nf}$. This behaviour is mainly driven by the flow exergy term in the definition of ${\dot{E}}_x$ and ${\dot{E}}_{xd}$, which includes the exergy change due to friction over the channel walls, which is reflected in the inlet to outlet pressure drop in Figure 9. As the contribution from the flow exergy term is relatively small compared to that generated by the heat flux through the minichannel base, its effect on ${\eta }_{II}$ is minor, as shown in Figure 14.

3.5. Thermal and Hydraulic Characteristics on a Horizontal Microscale Backward-Facing Step

Figure 15 shows the velocity profile across the horizontal microscale backwards-facing step at the streamwise location where the channel undergoes the sudden enlargement, that is, at the step edge. These numerical predictions are obtained with the single-phase approach and with the two-phase mixture model. Different combinations of phase two volume fraction and of phase two Al₂O₃ nanoparticle volume fraction, as defined in Table 2, provide the same 1% Al₂O₃ nanoparticle volume fraction in the mixture. The results are compared to corresponding numerical predictions by Klazly and Bognar [52] at the Reynolds number of 280.

The length to height ratio $L_u/h_u=250$ is much greater than the entry length to height ratio of 3.41 of a laminar channel flow [59]. This establishes a fully developed laminar channel flow, with its characteristic parabolic velocity profile in $x_2$. The simulation is two-dimensional and therefore the flow is uniform spanwise. The predictions from all the combinations of phase two volume fraction in Table 2 as well as from the single-phase approach collapse on essentially a single, parabolic-shaped curve. The velocity profile is substantially symmetric about $x_2$ = 0, suggesting the absence of any significant flow stratification effect, due to the relatively small difference in density between the first and the second phases in the two-phase mixture model. The numerical results from Klazly and Bognar [52] follow the same trend and their scatter about the parabolic-shaped data collapse from the current models is driven by the digitization process of the reference data from the article print.

As shown by the flow visualizations in Klazly and Bognar [52], the flow separates at the backwards-facing step edge, creating a laminar shear layer that reattaches on the floor, about six step heights downstream. The streamline connecting the separation point and the reattachment point defines a clockwise flow recirculation region, which is relatively segregated from the inlet cold coolant feed running over the top of the step. This creates a region of elevated temperature, in which the more modest wall-normal thermal gradient as well as the more modest convective heat transport away from the wall produces low local values of Nusselt number.

This effect is shown in Figure 16, where the predictions of the local Nusselt number are plotted along the bottom heated wall. The results are compared to the experiments by Kherbeet, et al. [51] and to numerical results by Klazly and Bognar [52] at the same Reynolds number of 280 as in Figure 15. Just downstream of the backwards-facing step face, which is located at $x_1=0.1 \mathrm{m}$, the local value of the Nusselt number is relatively low and it increases in $x_1$ over the $x_1$ range from 0.1 m to 0.104 m. Just downstream of the reattachment point, the cold flow from the top of the step meets the heated wall, creating both a strong wall-normal thermal gradient and the means for transporting heat away from the wall, by the increase in coolant specific internal energy. This produces the observed local heat transfer maximum. Further downstream, the temperature increment of the coolant due to the heat that it has picked up reduces its temperature difference with the wall and therefore its ability to extract further heat. This produces a monotonic decay in the local Nusselt number all the way to the exit of the duct. Figure 16 shows good agreement among the single-phase model and the mixture models with a second phase nanoparticle volume fraction less or equal to 50%. These predictions are consistent with the reference measurements reported in Kherbeet, et al. [51]. The mixture model prediction with a second phase nanoparticle volume fraction of 100%, that is, where the second phase is entirely made up of nanoparticles, over-predicts the local Nusselt number and is a clear outlier. As with the rectangular minichannel, the microscale backwards-facing step is inappropriately modelled by the two-phase mixture model in which the second phase is not a fluid.

4. Conclusions

This work has identified, assessed, and solved the root cause of the research community’s lack of confidence in two-phase mixture models for predicting the performance of a nanofluid cooled minichannel heat sink, compared to the more widely adopted and simpler single-phase approach.

It has done so by implementing both single-phase and two-phase mixture models for the forced convection in a minichannel heat sink of rectangular cross-section, cooled by water and Al₂O₃ nanoparticles, at a nanoparticle volume fraction of $\varphi$ =1%, in the Reynolds number range 92 to 455. The key to a successful two-phase mixture model implementation is using water as one phase and, for the second phase, a Newtonian fluid of equivalent thermal and physical properties of a suspension of nanoparticles in water. The second phase nanoparticle volume fraction must be less than 100% to provide a physical fluid, rather than a solid, for the second phase. The same conclusion is arrived at by modelling the flow and heat transport through a microscale backwards-facing step, with a uniformly heated bottom wall. The flow separation and recirculation near the step shows that, in the presence of greater flow complexity, the two-phase mixture model with a 100% phase two volume fraction of nanoparticles is still at variance with the other models. Specifically, it over-predicts the local Nusselt number peak by 35%.

Various combinations of second phase volume fraction ${\alpha }_{nf}$ and second phase nanoparticle volume fraction $\mathsf{\Phi }$ were used in the two-phase mixture model to represent the same 1% Al₂O₃ nanoparticle water mixture. For all thermal and flow performance quantities considered, increasing the second phase volume fraction above 2% in the minichannel heat sink provided a monotonic and incremental approach to the predictions from the single-phase model. This clearly shows that the two-phase mixture model converges towards the single-phase predictions for a homogeneous nanofluid.

The minichannel mean heat transfer coefficient predictions and, to a good extent, the rates of exergy expended and destroyed, as well as the second law efficiency, were found to be relatively insensitive to the second phase volume fraction, within the range ${\alpha }_{nf}$ for which the model remains physically sound. The agreement with experiment in the mean heat transfer coefficient was within 2.5% and 1.76% at $Re$ = 455 with the single-phase approach and with the two-phase mixture model $\left({\alpha }_{nf}\ge 2\%,\mathsf{\Phi }\le 50\%\right)$, respectively. On the other hand, the minichannel friction factor and pressure drop display variations with appreciable ${\alpha }_{nf}$ dependence. The agreement with experiment varies from −2% at ${\alpha }_{nf}$ = 4% to 14.7% at ${\alpha }_{nf}$= 50%. The latter is similar to the 15.4% agreement with experiment that is obtained using the single-phase approach.

These results, albeit limited in applicability by the minichannel simple geometry used in this study, have important practical implications for the progress of heat sink technology. It is shown that in the rectangular minichannel at the tested flow conditions, the lack of significant flow curvature to drive phase segregation develops a condition close to that of a homogeneous mixture. This is evidenced by the average heat transfer coefficient predicted by the two-phase mixture model being substantially invariant over a range of ${\alpha }_{nf}$ and $\mathsf{\Phi }$ combinations. This provides an important performance feature to rectangular minichannel heat sinks required to work within tight temperature tolerances. As the current model resolves neither sedimentation nor agglomeration nor other direct particle-particle interaction, the lack of uniformity in the nanofluid due to these processes was not modelled. Therefore, heat sinks featuring such phenomena may well not have this performance feature.

Furthermore, this work shows that a well-dispersed, homogeneous nanofluid suspension provides the best choice for minimizing the channel pressure loss and therefore the expended pumping power.

Finally, this paper has provided the protocol for using the two-phase mixture model within appropriate volume fraction bounds. This higher fidelity method, unconstrained by the uniform particle distribution assumption of the single-phase approach, can now be used with confidence to explore more complex minichannel geometries and resolve localized concentrations of nanoparticles, exploring their effect on the thermal and mass transport characteristics.