System-dependent behaviors of nanofluids for heat transfer: a particle-resolved computational study

Abstract

Nanofluids, the suspensions of metal or oxide nanoparticles in a fluid, have been extensively studied in the past decades for their potential in thermal enhancement in various heat transfer applications. However, controversial findings exist in the literature, and our knowledge and understanding of nanofluid thermal performances are still limited, considering the complicated microscopic mechanisms involved in nanofluids. In this article, unlike the one-phase or two-phase approaches typically adopted in nanofluid simulations, we present a particle-resolved numerical model with the nanoparticles and ambient fluid flows considered explicitly. Governing equations for the fluid flow, heat transfer, particle dynamics, and flow–particle interaction are solved simultaneously, and thermophysical properties of the nanoparticles and base fluids can be considered directly. Several simulations have been conducted for nanofluids of different particle volume fractions and under different thermal and flow conditions. Our results show that adding particles can improve the conductive heat transfer, benefiting from the high conductivity of the nanoparticle materials. However, the increased suspension viscosity can reduce the convection contribution on heat transfer; and the overall thermal performance could be deteriorated with the presence of nanoparticles. The results and analysis imply that nanofluids might be more appropriate for systems where the conduction part is more dominant over the convection part, such as in microchannels and porous materials. The model can also be extended to include other factors for future nanofluid studies.

1 Introduction

Over the past two decades, nanofluids, which are suspensions of nanoparticles (less than 100 nm in diameter) in base fluids, have gained increasing interest for various heat transfer applications, such as cooling of electronic equipment, solar collectors, car radiators, and various types of liquid-based heat exchangers [1,2,3,4,5,6,7,8]. In principle, the heat transfer enhancement for nanofluids is mainly due to the high thermal conductivity of nanoparticles [9,10,11]; however, other factors, including the size, shape, density, specific heat, and volume fraction of nanoparticles, are also important parameters that can affect the system performance [12,13,14,15]. Extensive analytical and experimental studies have been conducted to explore correlations between the effective thermophysical parameters of nanofluids and the nanoparticle and base fluid properties, and numerous models and formulas have been proposed for this purpose [16,17,18]. Considering the numerous parameters (such as the size, shape, distribution, density, specific heat, thermal conductivity, and volume fraction of nanoparticles; and the density, viscosity, specific heat, and thermal conductivity of the base fluid) and microscopic mechanisms (for example, the hydrodynamic interactions, the Brownian motion, the Soret effect, the van der Waals forces, and the electronic double layer interaction) involved in nanofluids [19, 20], it is difficult to perform exact theoretical analysis for such complicated systems. Instead, effective nanofluid properties are often developed using a theoretical-empirical approach: including the physical effect based on theoretical modeling with unknown coefficients, and obtaining the coefficient values by fitting the model to experimental measurements [3, 21,22,23,24]. The applicability of such correlations is usually limited to specified nanofluid systems [1, 22,23,24,25,26], and different or even contradictory predictions have been observed when using different correlations [27,28,29].

Typical measurements for nanofluid thermal conductivity are performed using a cylindrical configuration, with an electric heater in the center and the nanofluid surrounding it. The thermal conductivity is obtained from the electric power input and the measured temperatures [30,31,32,33,34,35]. These methods are called static or bulk measurements since there is no flow of the tested nanofluid in the device [30]. However, nanofluids are mainly utilized as working fluids in heat transfer applications with strong flow convection such as heat exchangers [36, 37]. To include the convection effect, researchers often adopt the flow-loop setup with the nanofluid circulating through [38,39,40]. In the test section, the nanofluid flows through a straight tube with constant heat flux or temperature maintained on the tube wall. Similar to the diverse findings in static measurements [20, 30], no general agreement can be established either for the thermal enhancement effect of nanofluids among these convective measurements using similar setups. For example, Heris et. al. [40] found that the heat transfer coefficient in the tube flow can be up to 20% higher than that calculated from the static nanofluid thermal conductivity, while Rea et al. [41] reported that the measured heat transfer coefficients of nanofluids agree well to the prediction from static thermal conductivity. On the other hand, Azari et al. [42] and Pak and Cho [38] observed that the heat transfer coefficients of nanofluids could be even smaller than that of the base fluid under the same flow rate or same Reynolds number condition; and Esfahani et al. [33] found that the thermal enhancements of nanofluids over the base fluid also depend on the heat flux even at the same Reynolds number. Moreover, Wen and Ding [39] observed that the improvement in heat transfer coefficient was more significant near the inlet of the tube and decreased gradually along the flow. The controversial research findings on the thermal performance of nanofluids in the literature [43,44,45] indicate that our current understanding and knowledge of the flow and thermal behaviors of nanofluids are far insufficient and more research is needed, in particular, to explore the underlying mechanisms. Actually, Mohamad [46] has pointed out that, in spite of the system complexity, basic physical principles should not be forgotten when discussing the nanofluid thermal improvement.

One unique feature of computer simulations is that more controllable virtual experiments can be conducted to investigate the individual effects of each parameter on the performance of a complex system, which might be difficult in real experiments. Although numerous computational studies have been published in the literature on the heat transfer performance of nanofluids in various flow situations [31, 47], these simulations apply traditional numerical methods for thermal flows in combination with various correlations for the effective viscosity and thermal conductivity for nanofluids. As a result, these studies are not helpful on advancing our understanding of the mechanisms behind the macroscopic nanofluid performances. In this direction, a few attempts using molecular dynamics simulations have been published recently. Wang and Jiang [48] found that there exists an interfacial layer of thickness $\sim$1 nm of liquid molecules from the nanoparticle surface, and the thermal conductivity in this layer could be few folds of that for the base fluid when the fluid–solid attraction is strong. Motlagh and Kalteh [49] studied the argon atom flow through a nanochannel, and they found the Nusselt number increased by 6% when four copper particles are added to the argon flow. Due to the large computational demand for such molecular dynamics simulations, these studies were conducted with much smaller systems compared to real nanofluid systems: the nanoparticle dimension was only $\sim$1 nm and there were only few nanoparticles considered: one in Ref. [48] and four in Ref. [49].

To have a more realistic representation of nanofluids, in this paper, we adopt the classical fluid mechanics and heat transfer theory for continuum substances, and model a nanofluid explicitly as a suspension of spherical particles dispersed in a fluid. Governing equations for the fluid flow, heat transfer, flow–particle interaction, and particle dynamics are solved simultaneously. With this particle-resolved approach, the three-dimensional (3D) nanoparticle geometry is represented exactly, and various thermophysical parameters for the nanoparticle materials and base fluids can be directly considered. The heat transfer performance of the fluid-particle system in then investigated by evaluating the heat flux through the system under different flow and thermal boundary conditions.

The remainder of this paper is organized as follows. In Sect. 2, we describe the governing equations involved in our model and the numerical methods for solving these equations. Then, in Sect. 3, simulations are conducted for the heat transfer of model nanofluids with different particle volume fractions and under different thermal and flow conditions, and their individual effects on heat transfer are examined. At last, Sect. 4 summarizes this research and also provides further discussions.

2 Governing equations, models and methods

2.1 Governing equations and lattice Boltzmann method

The incompressible flow and thermal fields in a nanofluid suspension are governed by the following continuity equation (Eq. 1), momentum equation (Eq. 2), and energy equation (Eq. 3):

$$\begin{aligned}&\nabla \cdot \mathbf{u}=0, \end{aligned}$$

(1)

$$\begin{aligned}&\rho \frac{\partial \mathbf{u}}{\partial t}+ \rho \mathbf{u} \cdot \nabla \mathbf{u}=-\nabla P + \mu \nabla ^2\mathbf{u}, \end{aligned}$$

(2)

$$\begin{aligned}&\frac{\partial (\rho c_pT)}{\partial t}+ \mathbf{u} \cdot \nabla (\rho c_p T) = \nabla \cdot (k\nabla T). \end{aligned}$$

(3)

In these equations, $\rho$ is the fluid density, $\mathbf{u}$ is the flow velocity, P is the pressure, $\mu$ is the fluid dynamic viscosity, $c_p$ is the specific heat, T is the temperature, k is the thermal conductivity, and t is the time. In this study, we use the smooth particle method (SPM), which adopts a density field to identify the solid and fluid phases [50, 51], and therefore, we can solve the above equations in a uniform way with no special treatments for the fluid–solid interfaces. More detailed descriptions of SPM will be presented in Sect. 2.2.

To solve these equations numerically, we employ the lattice Boltzmann method (LBM) with two sets of density distribution functions: $f_i(\mathbf{x},t)$ for the fluid flow and $g_i(\mathbf{x},t)$ for the temperature field. Here, the subscript i denotes the lattice direction and $\mathbf{x}$ represents the coordinates of a lattice node in the discretized computational domain. The evolution equations for these density distribution functions are the lattice Boltzmann equations with single-relaxation-time collision terms as [51,52,53]:

$$\begin{aligned}&f_i(\mathbf{x} +\mathbf{e}_i \Delta t, t+\Delta t)-f_i(\mathbf{x},t)\nonumber \\&\quad =-\frac{f_i(\mathbf{x},t)-f_i^{eq}(\mathbf{x},t)}{\tau _f} + F_i\Delta t, \end{aligned}$$

(4)

$$\begin{aligned}&g_i(\mathbf{x} +\mathbf{e}_i\Delta t, t+\Delta t)-g_i(\mathbf{x},t)\nonumber \\&\quad =-\frac{g_i(\mathbf{x},t)-g_i^{eq}(\mathbf{x},t)}{\tau _g} + G_i\Delta t, \end{aligned}$$

(5)

where $\Delta t$ is the time step and $\mathbf {e}_i$ is the lattice velocity. In this study, there is no heat source considered in the nanofluids and thus we have the source term $G_i=0$ inside the computation domain; however, different thermal situations can be established by imposing appropriate boundary conditions on the domain boundaries (see Sect. 3.1). The force term $F_i$ is related to the body force density on the fluid, and it will be discussed below in Sect. 3.1 as well.

The equilibrium density distributions $f_i^{eq}$ and $g_i^{eq}$ are given by [50, 52, 53]

$$\begin{aligned} f_{i}^{{eq}} = \omega _{i} \rho \left( {1 + \frac{{{\mathbf {e}}_{i} \cdot {\mathbf {u}}}}{{c_{s}^{2} }} + \frac{{({\mathbf {e}}_{i} \cdot {\mathbf {u}})^{2} }}{{2c_{s}^{4} }} - \frac{{{\mathbf {u}}^{{\mathbf {2}}} }}{{2c_{s}^{2} }}} \right) ~, \end{aligned}$$

(6)

$$\begin{aligned} g_i^{eq}= {\left\{ \begin{array}{ll} T\left[ \rho c_p-(1-\omega _0)\left( \rho c_p \right) _0\right] , &{} i=0 \\ \omega _i T \left[ \left( \rho c_p\right) _0 + \rho c_p \frac{\mathbf{e}_i\cdot \mathbf{u}}{c_{s}^2}\right] , &{} i\ne 0 . \end{array}\right. } \end{aligned}$$

(7)

Here, $(\rho c_p)_0$ is the reference heat capacity, and its value is to be specified in Sect. 2.2. The values for the lattice velocity $\mathbf{e}_i$, the lattice weight factor $\omega _i$, and the lattice sound speed $c_s$ depend on the lattice structure employed in a simulation. In this work, two different lattice models are utilized: D3Q19 (three dimensions and 19 lattice velocities) for the flow field, and D3Q7 (three dimensions and seven lattice velocities) for the temperature field. The reason for using the D3Q7 lattice model for solving the energy equation Eq. (3) is that it has a better computational efficiency (by using less lattice velocities) without sacrificing the numerical accuracy [54, 55]. The lattice parameters for them are [53, 56]

$$\begin{aligned}&\left( \begin{array}{c} \mathbf{e}_0 \\ \mathbf{e}_1 \\ \mathbf{e}_2 \\ \vdots \\ \mathbf{e}_{18}\\ \end{array} \right) ^T = \left[ \begin{array}{ccccccccccccccccccc} 0 &{} 0 &{} 1&{} 0&{}-1&{} 0&{} 0&{} 1&{} 1&{}-1&{}-1&{} 0&{} 1&{} 0&{}-1&{} 0&{} 1&{} 0&{}-1 \\ 0&{} 1&{} 0&{}-1&{} 0&{} 0&{} 0&{} 1&{}-1&{}-1&{} 1&{} 1&{} 0&{}-1&{} 0&{} 1&{} 0&{}-1&{} 0 \\ 0&{} 0&{} 0&{} 0&{} 0&{} 1&{}-1&{} 0&{} 0&{} 0&{} 0&{} 1&{} 1&{}1&{}1&{}-1&{}-1&{}-1&{}-1 \end{array} \right] \frac{\Delta x}{\Delta t}, \end{aligned}$$

(8)

$$\begin{aligned}&\left. \begin{array}{c} \omega _0=1/3,. \omega _{1\sim 6}=1/18,.\omega _{7\sim 18}=1/36,.c_s=(1/\sqrt{3})\Delta x/\Delta t.~\text{ for } \text{ D3Q19 };\\ \omega _0=1/4,. \omega _{1\sim 6}=1/8,.c_s=(1/2)\Delta x/\Delta t.~\text{ for } \text{ D3Q7 }. \end{array} \right. \end{aligned}$$

(9)

The relaxation times, $\tau _f$ for flow and $\tau _g$ for temperature, which control how fast the distributions $f_i$ and $g_i$ decay toward their equilibrium counterparts $f_i^{eq}$ and $g_i^{eq}$ [57, 58], are related to the macroscopic fluid viscosity $\mu$ and thermal conductivity k, respectively, as:

$$\begin{aligned} \tau _f=\frac{\nu }{c_s^2 \Delta t}+\frac{1}{2}, \tau _g=\frac{k}{(\rho c_p)_0 c_s^2 \Delta t}+\frac{1}{2}. \end{aligned}$$

(10)

In these equations, $\Delta x$ is the lattice grid resolution, i.e., the distance between two adjacent lattice nodes in the discretized space along the coordinate directions. Please note that only the first seven lattice velocities in Eq. 8 are used in the D3Q7 model for the thermal field.

The force term $F_i$ in Eq. 4 is given by [51, 59]

$$\begin{aligned} F_i=\omega _i \left( 1-\frac{1}{2\tau _f}\right) \left( \frac{\mathbf{e}_i-\mathbf{u}}{c_s^2}+\frac{\mathbf{e}_i \cdot \mathbf{u}}{c_s^4} \mathbf{e}_i\right) \cdot (\mathbf{f}_e+ \mathbf{f}_b), \end{aligned}$$

(11)

where $\mathbf{f}_e$ is the external force (the gravity, for example) and $\mathbf{f}_b$ the solid–fluid interaction force in SPM, which will be defined in the next section. The macroscopic properties, including the density, velocity, and temperature, can be obtained from the density distribution functions $f_i$ and $g_i$, respectively, as follows [51, 59]:

$$\begin{aligned}&\rho =\sum \limits _{i}f_i ,~\mathbf{u}=\frac{1}{\rho } \sum \limits _{i} f_i\mathbf{e}_i +\frac{\mathbf{f}_e+\mathbf{f}_b}{2\rho }\Delta t,\nonumber \\&T=\frac{1}{\rho c_p}\sum \limits _{i}g_i. \end{aligned}$$

(12)

Applying the multiscale expansion analysis to Eqs. 4-11, the macroscopic governing equations Eqs. 1-3 can be recovered [52, 53].

2.2 Smoothed profile method for particulate flows

Our simulations contain solid particles moving with the surrounding flow. To incorporate the dynamic flow–particle interaction, we adopt the smoothed profile method (SPM), which treats the solid phase in a similar fashion as the fluid phase and therefore the governing equations can be solved in a unified way [50, 51, 60]. The interface between the solid particle and the fluid is represented by a thin layer with a thickness $\epsilon$, which is set as $\epsilon =2~\Delta x$ in this work [50, 51, 60]. Please note that this interfacial layer is only for computational convenience, and it is not relevant to any physical aspects, for example, the nanolayer on nanoparticle surfaces [61,62,63]. A density field function $\Phi (\mathbf{x},t)$ is adopted to indicate the material phase in the simulation domain: $\Phi =1$ in the solid phase and $\Phi =0$ in the fluid phase; and its value varies smoothly from 0 and 1 across the interface. Different options for the $\Phi$ formulation are available; however, they usually do not affect the overall simulation results [50, 60, 64]. In this study, we follow Refs. [50, 51] and use the following transition function for the density field $\Phi$:

$$\begin{aligned}&\Phi (\mathbf{x},t)=\sum \limits _{i=1}^{N_p}\Phi _i (\mathbf{x},t) , \nonumber \\&\Phi _i (\mathbf{x},t)= {\left\{ \begin{array}{ll} 0, &{} d_i<-\epsilon /2 \\ \frac{1}{2}\left[ \sin (\frac{\pi d_i}{\epsilon })+1 \right] , &{} |d_i|\le \epsilon /2 ;\\ 1, &{} d_i>\epsilon /2 \end{array}\right. } \end{aligned}$$

(13)

where $\Phi _i$ is the density field function due to the i-th particle and $N_p$ is the total number of particles. $d_i=R_i-|\mathbf{x}-\mathbf{X}_i|$ is the distance of location $\mathbf{x}$ to the surface of the i-th particle with radius $R_i$ and center location $\mathbf{X}_i$. Using the density field function $\Phi$, the particle velocity field $\mathbf{u}_p(\mathbf{x},t)$ is defined as [50, 51]

$$\begin{aligned} \Phi (\mathbf{x},t)\mathbf{u}_p(\mathbf{x},t)= & {} \sum \limits _{i=1}^{N_p}\Phi _i (\mathbf{x},t) \left\{ \mathbf{V}_i(t)+ \varvec{\Omega _i}(t) \right. \nonumber \\&\times \left. \left[ \mathbf{x}-\mathbf{X}_i(t) \right] \right\} , \end{aligned}$$

(14)

where $\mathbf{V}_i$ and ${\varvec{\Omega }}_i$ are the translational and angular velocities of the i-th particle, respectively. Following Ref. [65], the time splitting method is utilized in our SPM calculation with an intermediate velocity field $\mathbf{u}^* (\mathbf{x},t)$ being calculated without involving the solid–fluid interaction as follows:

$$\begin{aligned} \mathbf{u}^*= \frac{1}{\rho } \sum _i f_i \mathbf{e}_i + \frac{\mathbf{f}_e}{2\rho }\Delta t. \end{aligned}$$

(15)

The solid–fluid interaction force $\mathbf{f}_b$ is then obtained according to the velocity difference:

$$\begin{aligned} \mathbf{f}_b= \rho \frac{\Phi \left( \mathbf{u}_p - \mathbf{u}^*\right) }{\Delta t} ; \end{aligned}$$

(16)

and this force is then used to calculate the force term $F_i$ in the lattice Boltzmann equation Eq. 4 via Eq. 11.

The density field function $\Phi$ also provides a convenient way to describe the material property variations for the temperature field solution, namely

$$\begin{aligned} \left( \rho c_p\right) (\mathbf{x})=(\rho c_p)_f+\left[ (\rho c_p)_s-(\rho c_p)_f\right] \Phi (\mathbf{x}) .~ \end{aligned}$$

(17)

for the volumetric heat capacity $\rho c_p$ and

$$\begin{aligned} k(\mathbf{x})=k_f+(k_s-k_f)\Phi (\mathbf{x}) \end{aligned}$$

(18)

for the thermal conductivity k, where subscripts f and s denote fluid and solid phases, respectively. According to Ref. [66], for a system consisting two phases of different thermal properties, the reference volumetric heat capacity $(\rho c_p)_0$ in Eq. 7 can be set as

$$\begin{aligned} (\rho c_p)_0=\frac{2(\rho c_p)_s (\rho c_p)_f}{(\rho c_p)_s+ (\rho c_p)_f} . \end{aligned}$$

(19)

2.3 Particle dynamics and interactions

To update the particle position $\mathbf{X}_i$, velocity $\mathbf{V}_i$, and angular velocity ${\varvec{\Omega }}_i$, the external forces and torques are needed in addition to the particle mass $m_i$ and moment of inertia $I_i$. The hydrodynamic force $\mathbf{F}_i^H$ and its torque $\mathbf{M}_i^H$ due to the flow–particle interaction can be obtained by summarizing the interaction $\mathbf{f}_b$ in Eq. 16 as [65]:

$$\begin{aligned}&\mathbf{F}_i^H =\int _{\Gamma _i}{} \mathbf{f}_b d\Gamma _i ; \end{aligned}$$

(20)

$$\begin{aligned}&\mathbf{M}_i^H =\int _{\Gamma _i}\left( \mathbf{x}-\mathbf{X}_i\right) \times \mathbf{f}_bd\Gamma _i . \end{aligned}$$

(21)

In the above equations, $\Gamma _i$ is the volume of the i-th particle.

In addition, to prevent a particle from overlapping with another particles or boundary walls, particle–particle and particle–wall collisions are considered via repulsive forces on the particles [50, 67]. The collision force $\mathbf{F}_{ij}^r$ between particles i and j is expressed as

$$\begin{aligned} {\mathbf {F}}_{{ij}}^{r} = \left\{ {\begin{array}{*{20}l} 0, &{} {|{\mathbf {X}}_{i} - {\mathbf {X}}_{j} | > R_{i} + R_{j} + \zeta } \\ {2.4\varepsilon \left[ {2\left( {\frac{{R_{i} + R_{j} }}{{|{\mathbf {X}}_{i} - {\mathbf {X}}_{j} |}}} \right) ^{{14}} - \left( {\frac{{R_{i} + R_{j} }}{{|{\mathbf {X}}_{i} - {\mathbf {X}}_{j} |}}} \right) ^{8} } \right] \frac{{{\mathbf {X}}_{{\mathbf {i}}} - {\mathbf {X}}_{{\mathbf {j}}} }}{{(R_{i} + R_{j} )^{2} }}}, &{} {|{\mathbf {X}}_{i} - {\mathbf {X}}_{j} | \le R_{i} + R_{j} + \zeta } \\ \end{array} } \right. \end{aligned}$$

(22)

where $\varepsilon =\left( \frac{2R_iR_j}{R_i+R_j}\right) ^2$. $\zeta$ is the interaction threshold and it is set as $\zeta =\Delta x$ [50, 67] in this work. Similarly, the repulsion between the i-th particle at position $\mathbf{X}_i$ of radius $R_i$ and the wall at position $\mathbf{X}_w$ is given as follows [50, 67]

$$\begin{aligned} {\mathbf {F}}_{{iw}}^{r} = \left\{ {\begin{array}{*{20}l} 0, &{} {|{\mathbf {X}}_{i} - {\mathbf {X}}_{j} | > R_{i} + \zeta } \\ {2.4\varepsilon \left[ {2\left( {\frac{{R_{i} }}{{|{\mathbf {X}}_{i} - {\mathbf {X}}_{w} |}}} \right) ^{{14}} - \left( {\frac{{R_{i} }}{{|{\mathbf {X}}_{i} - {\mathbf {X}}_{w} |}}} \right) ^{8} } \right] \frac{{{\mathbf {X}}_{{\mathbf {i}}} - {\mathbf {X}}_{{\mathbf {w}}} }}{{(R_{i} )^{2} }}}, &{} {|{\mathbf {X}}_{i} - {\mathbf {X}}_{j} | \le R_{i} + \zeta } \\ \end{array} } \right. \end{aligned}$$

(23)

The particle velocity $\mathbf{V}_i$ can be updated based on the total force exerted on this particle via Newton’s second law:

$$\begin{aligned}&\mathbf{V}_i (t+\Delta t)= \mathbf{V}_i (t)\nonumber \\&\quad + \frac{\Delta t}{m_i} \left[ \mathbf{F}_i^H+ \sum _j\mathbf{F}_{ij}^r+\mathbf{F}_{iw}^r+\mathbf{F}_{i}^{ext}\right] , \end{aligned}$$

(24)

where $\mathbf{F}_i^{ext}$ represents the external force, such as the gravitational force, applied on the particle. The particle center position $\mathbf{X}_i$ can then be obtained using the average velocity at t and $t+\Delta t$:

$$\begin{aligned} \mathbf{X}_i (t+\Delta t)= \mathbf{X}_i (t)+ \frac{\mathbf{V}_i(t)+\mathbf{V}_i (t+\Delta t)}{2} \Delta t . \end{aligned}$$

(25)

The angular velocity ${\varvec{\Omega }}_i$ is calculated in a similar way as in Eq. 24 from the hydrodynamic torque exerted on the particle:

$$\begin{aligned} {\varvec{\Omega }}_i (t+\Delta t)= {\varvec{\Omega }}_i (t)+ \frac{\mathbf{M}_i^H\Delta t}{I_i} . \end{aligned}$$

(26)

Please note that the particle–particle and particle–wall interaction forces in Eqs. 22 and 23, as well as the gravitational force on spherical particles of uniform materials in this study, pass through the particle centers, and therefore, they have no impact on the particle rotation.

Fig. 1

Schematic representations of the simulation setup employed in this study, with the base fluid filling the space between the top and bottom walls and twelve spherical particles uniformly suspended in the fluid. More details are provided in the text

3 Simulations and results

The above described numerical models have been validated by several benchmark tests, including the particle sedimentation process and the flow and thermal fields for a particle–channel setup using different reference frames. The results are compared to previous publications and the agreement is satisfactory. These tests show that all the key physical mechanisms involved, including the fluid flow, thermal field, particle-flow interaction, and particle dynamics, have been accurately considered in our methods and programs. Please refer to the Appendix section for details.

3.1 Simulation setup and boundary conditions

In this study, we simulate the flow and heat transfer in a particulate flow system, which consists of twelve spherical particles suspending in the ambient fluid. This serves as a simplified numerical representation of nanofluids with nanoparticles approximately equally spaced. Using more particles is desirable for a more realistic nanofluid model; however, a larger simulation domain and longer calculation time will be required and the computational efficiency could be a concern to our in-house program. Figure 1a illustrates the 3D computational system utilized in this study. The computational domain has a length $L=220$, a width $W=55$, and a height $H=165$, all in the lattice unit $\Delta x$. The initial configuration of the twelve particles in the simulation domain is shown in Fig. 1b. Although the nanoparticle distribution is in general heterogeneous due to the particle aggregation, microscopic convection, Brownian motion, and thermophoretic diffusion [19, 20, 61], here we assume that the spatial distribution of nanoparticles is uniform for simplicity. A larger computation domain with more nanoparticles as well as more detailed descriptions of the underlying interaction mechanisms is necessary for a more realistic modeling of the nanoparticle spatial distribution. Two values for the particle radius R are considered: $R=10.0$ for a particle volume fraction of $\phi = 2.5\%$, and $R=12.6$ for $\phi =5\%$. In this work, with the domain dimensions and particle number held the same, it is not easy to adjust the particle size and volume fraction independently. While the volume fraction $\phi$ can be increased by simply increasing the particle number, it would be difficult to set up an approximately uniform configuration of the particles in the simulation domain. Nevertheless, we refer the system with $R=12.6$ and $\phi =5\%$ as the high-volume fraction case for convenience and clarity. The solid–fluid property ratios are set as $\rho _s/\rho _f=4$, $k_s/k_f=50$, and $(\rho c_p)_s/(\rho c_p)_f=0.8$. All these parameters are in the general ranges of typical nanofluids in the literature [12, 20, 68]. The Prandtl number for the ambient fluid is set at $Pr_f=7$. The properties for the fluid are: $k_f=0.025$, $\rho _f=1$, $\mu _f=0.175$, and $c_{pf}=1$, all in the non-dimensional simulation units.

With the no-slip boundary condition on the top and bottom, a channel flow can be generated by applying a body force $G_x$ in the x direction, i.e., $\mathbf{f}_e=G_x \mathbf{x}$ in Eq. 11. When no particles exist, the body force $G_x$ is related to the maximum fluid velocity $U_c$ at channel centerline via the Poiseuille law as

$$\begin{aligned} G_x=\frac{8\mu _f U_c}{H^2} . \end{aligned}$$

(27)

Different flow strengths can be generated by applying different $G_x$ values to examine the convection effect on system heat transfer. Two scenarios are considered for the thermal boundary conditions at the top and bottom walls in this study: (a) Constant wall temperatures:

$$\begin{aligned} T_{top}=T_H,~ T_{btm}=T_L; \end{aligned}$$

(28)

and (b) mixed isothermal-adiabatic conditions given by

$$\begin{aligned} {\left\{ \begin{array}{ll} T_{top}=T_{btm}=T_H, &{} ~0\le x<\frac{L}{4}; \\ \left( \frac{\partial T}{\partial z}\right) _{top}= \left( \frac{\partial T}{\partial z}\right) _{btm}=0, &{} ~\frac{L}{4}\le x<\frac{L}{2};\\ T_{top}=T_{btm}=T_L, &{} ~\frac{L}{2}\le x<\frac{3L}{4}; \\ \left( \frac{\partial T}{\partial z}\right) _{top}= \left( \frac{\partial T}{\partial z}\right) _{btm}=0, &{} ~\frac{3L}{4}\le x<L \end{array}\right. } \end{aligned}$$

(29)

Here, the subscripts top and btm denote the top ($z=H$) and bottom ($z=0$) domain boundaries, respectively. For simplicity, we use $T_H=1$ and $T_L=0$ in our calculations. Under these thermal boundary conditions, the overall heat transfer in the system will be either in vertical or horizontal direction (See Figs. 2 and 3). In addition, the periodic boundary condition is applied on other sides of the simulation box, for both the fluid flow and the temperature field.

For the thermal flow system considered here, both the thermal conduction and flow convection contribute to the total heat transfer in the system. To examine the conduction and convection effects individually, the conductive heat flux $\mathbf{q}^k$ and the convective heat flux $\mathbf{q}^v$ are calculated for different flow and thermal situations:

$$\begin{aligned} \mathbf{q}^k=-k\nabla T,~\mathbf{q}^v=\rho c_p T \mathbf{u}. \end{aligned}$$

(30)

In the heat flux calculation, the local values for k and $\rho c_p$ are obtained via Eqs. 18 and 17, respectively, and the temperature gradient $\nabla T$ is estimated via the centered finite difference approximation.

Fig. 2

The temperature distributions at the vertical middle plane $y=W/2$ for Cases V0 (top row) and V10 (bottom row) with different particle volume fractions: $\phi =0$ (pure fluid, left column), 2.5% (middle column), and 5% (right column). Also the black circles represent the particles and the fluid flow is indicated by gray arrows for Case V10 in the bottom row

Table 1 Heat flux ${{{\bar{q}}}}_z$ across the system under the vertical thermal gradient with different particle volume fractions and flow conditions. The common exponential part $10^{-4}$ for all ${\bar{q}}_z$ values is omitted for clarity

3.2 Heat transfer under vertical thermal gradient

We first look at the heat transfer performance of the system under the vertical thermal gradient generated from the boundary condition Eq. 28. Three particle volume fractions, $\phi =0$ (pure fluid), 2.5%, and 5%, and two flow situations, Case V0 with no fluid flow (i.e., $U_c=0$) and Case V10 with $U_c=0.01$, are considered. The corresponding Reynolds number is $Re=\rho _f U_c H/\mu _f =9.43$. This is relatively low compared to the Re values in typical nanofluid applications; however, larger simulation domains and smaller time steps will be required for higher Re values and the simulation time will be long for our in-house computer code. Please note that the $U_c=0.01$ value is only used to find the corresponding body force $G_x$ from Eq. 27, and this same $G_x$ value is adopted for all V10 cases of different particle volume fractions.

For all V0 cases where there is no fluid flow and particle motion and the V10 case of pure fluid, steady states are quickly established. For V10 cases of $\phi =2.5\%$ and 5%, due to the dynamic motion of particles, only quasi-steady states are observed. The steady and quasi-steady temperature and flow distributions in the vertical middle plane $y=W/2$ are displayed in Fig. 2. The mean flow velocity in the horizontal middle plane $z=H/2$, ${{\bar{u}}}_x=\iint _{z=H/2}u_x dx dy /LW$, is 9.99$\times 10^{-3}$ for the Case V10 with $\phi =0$. The relative difference from the expected value $U_c=0.01$ is 0.1%, and this tiny difference is attributed to the inevitable numerical errors. The center velocity ${{\bar{u}}}_x$ values for other two V10 systems are $9.53\pm 0.02\times 10^{-3}$ (mean±variation) for $\phi =2.5\%$ and $9.06\pm 0.01\times 10^{-3}$ for $\phi =5\%$. The reduced ${{\bar{u}}}_x$ values are expected since the presence of particles increases the apparent viscosity of a fluid suspension. Also the small oscillation magnitudes indicate quasi-steady states have established. The effective viscosity of the particle suspension can be obtained from the center velocity as $\mu _{eff}/\mu _f=U_c /{{\bar{u}}}_x=1.0526$ for $\phi =2.5\%$ and 1.0989 for $\phi =5\%$. For a comparison, the effective viscosity ratio calculated from the Einstein formula [69]

$$\begin{aligned} \frac{\mu _{eff}^E}{\mu _f}=1+2.5 \phi \end{aligned}$$

(31)

is 1.0625 for $\phi =2.5\%$ and 1.125 for $\phi =5\%$.

The temperature distributions in Fig. 2 overall appear very similar, except the local distortions around particles caused by the different solid and fluid thermal conductivities. With a high temperature on top and a low temperature at bottom, heat flows downward through the fluid-particle suspension. The mean conductive heat flux ${{\bar{q}}}_z^k$ in the horizontal middle plane $z=H/2$ is calculated according to Eq. 34, and the calculated results for different particle volume fractions, $\phi =0$ (pure fluid), 2.5% and 5%, are listed in Table 1. On the other hand, the convective heat flux in the vertical direction $q_z^v$ is either exactly zero for steady cases or negligible for quasi-steady cases (since the flow mainly in the x direction). This is confirmed in our calculations: The magnitude of ${{\bar{q}}}_z^v$ (mean value in the horizontal middle plane) is $\sim 10^{-9}$ for Cases V10 of $\phi =2.5\%$ and 5%, which is orders smaller than the conductive heat flux ${{\bar{q}}}_z^k$. Also for ${{\bar{q}}}_z^k$, the differences between Cases V0 and V10 are tiny for both the $\phi =2.5\%$ and 5% systems, suggesting that the horizontal fluid flow in Case V10 has no apparent influence on the heat transfer behavior in this vertical thermal gradient dominated system. This is consistent to the convective term $\mathbf{u} \cdot \nabla (\rho c_p T)$ in the energy equation Eq. 3: With the thermal gradient mainly in the vertical direction and the flow mainly in the horizontal direction (see Fig. 2, bottom row), the convection effect becomes negligible in such situations.

As for the particle volume fraction effect on heat transfer, we take the heat flux values for Cases V0 in Table 1 for simplicity. We see the mean heat flux ${{\bar{q}}}_z^k$ increases 9.11% for $\phi =2.5\%$ and 18.20% for $\phi =5\%$. As in typical nanofluid literature, the increase in ${{\bar{q}}}_z^k$ can be interpreted as an improvement in the effective thermal conductivity $k_{eff}$, i.e., $k_{eff}/k_f=1.0911$ for $\phi =2.5\%$ and $k_{eff}/k_f=1.1820$ for $\phi =5\%$. In comparison, we also calculate the effective conductivity $k^M_{eff}$ from the well-known Maxwell model [46]

$$\begin{aligned} k^M_{eff}=\frac{k_s+2k_f+2(k_s-k_f)\phi }{k_s+2k_f-(k_s-k_f)\phi }k_f \end{aligned}$$

(32)

which has been adopted in many studies for nanofluids and porous thermal flows [2, 70]. The calculated effective conductivity from Eq. 32 is $k^M_{eff}/k_f=1.0728$ for $\phi =2.5\%$ and $k^M_{eff}/k_f=1.1484$ for $\phi =5\%$. These values are slightly smaller than those obtained from our direct simulation. One should not expect a perfect agreement from the highly idealized Einstein and Maxwell models, and larger deviations between experimental and numerical studies and these theoretical models have also been reported in the literature [23, 32, 71].

Fig. 3

The temperature distributions at the vertical middle plane $y=W/2$ for Cases H0 (top row), H2 (middle row), and H10 (bottom row) with different particle volume fractions: $\phi =0$ (pure fluid, left column), 2.5% (middle column), and 5% (right column). Also the black circles represent the particles and the fluid flow is indicated by gray arrows for Cases H2 and H10. Please note that different scales are adopted when plotting the velocity arrows for Cases H2 and H10

3.3 Heat transfer under horizontal thermal gradient

For the V10 cases in the previous section, the flow and the temperature gradient are perpendicular to each other, and our results show that the flow convection has no apparent influence on the overall heat transfer behavior. However, such situations with perpendicular thermal-flow directions are rare in practical heat transfer processes, and the flow convection often plays a significant role in the overall thermal performance. To investigate the convection effect, we adopt the mixed boundary condition in Eq. 29. Four different flow strengths with $U_c=0$ (Case H0), 0.001 (Case H1), 0.002 (Case H2), and 0.01 (Case H10), plus three different particle volume fraction values ($\phi =0$, 2.5%, and 5%), are considered in this section.

Figure 3 shows the thermal and flow fields in the vertical middle plane $y=W/2$ for Cases H0, H2 and H10. The H1 results are similar to those from H2 and they are thus not displayed. The high- and low-temperature regions near the channel walls are consistent with specified boundary temperatures; and the isotherm lines intersect perpendicularly with the boundary walls between the hot and cold spots as expected from the adiabatic thermal condition there. These observations indicate that the thermal condition in Eq. 29 has been correctly implemented in our simulations. Without fluid flow in Case H0, the temperature distributions are similar among the $\phi =0$, 2.5% and 5% systems, except the local distortions around the particles. Also the isothermal patterns are symmetric about $x=L/8$ and 5L/8, the center positions of the hot and cold sections on the walls (see Eq. 29). This is reasonable considering the particular system configuration, including the periodic inlet/out boundary condition, the uniform particle spacing, and the specific thermal condition on the top and bottom walls. When fluid flow is introduced in Cases H2 and H10, the flow current drifts the isotherm patterns toward the downstream direction and also depresses the hot and cold regions near the walls; and such influences are more profound in Case H10 than in Case H2 due to the higher flow velocity. Other than the local distortions mentioned before, the effects of particle volume fraction $\phi$ are not obvious in the temperature distributions in Fig. 3.

Fig. 4

Comparisons of the convective heat flux ${{\bar{q}}}_x^v$ (a), the conductive heat flux ${{\bar{q}}}_x^k$ (b), and total heat flux ${{\bar{q}}}_x^k$+${{\bar{q}}}_x^v$ (c) under the horizontal thermal gradient with different flow strengths (Cases H0, H1, H2, and H10) and particle volume fractions ($\phi =0$, 2.5 and 5%)

To characterize the heat transfer behavior more quantitatively, we calculate the mean values (averaged over the $W\times H$ vertical cross section) of conductive and convective heat fluxes in the x direction at $x=3L/8$, the center location between the hot and cold sections on the walls. Figure 4 compares these calculated heat fluxes values with different flow situations (Cases H0, H1, H2, and H10) and particle volume fractions ($\phi =0$, 2.5%, and 5%). Clearly in Fig. 4a, the convective heat flux ${{\bar{q}}}_x^v$ is zero for Case H0 with no flow, and it increases approximately linearly with the $U_c$ values adopted in Cases H1, H2, and H10. An exact linear relationship should not be expected since the temperature distribution has been altered by the flow field (Fig. 3). On the other hand, the conductive heat flux ${{\bar{q}}}_x^k$ in Fig. 4b decreases with the flow strength. This trend can be qualitatively explained by looking at the changes in isotherm patters in Fig. 3. As the flow increases from Cases H0 to H2 and then H10, the influence of the hot and cold sections on the boundary walls shrinks to the closer vicinity to the walls. This results in a larger region in the channel center of small temperature variations. Since the conductive heat flux is associated with the spatial temperature gradient, the large central region of small temperature variations contributes much less to the mean conductive heat flux in Case H10 compared to Case H0. The relative significance of the convection and diffusion effects in transport processes can be expressed by the Peclet number $Pe=2U_c R/\alpha$, where $\alpha =k/\rho c_p$ is the thermal diffusivity [72]. Using the pure fluid properties and $R=10$ for simplicity, we have $Pe=0.8$, 1.6, and 8 for flow situations H1, H2, and H10, respectively. These numbers in general agree with the heat flux magnitudes in Fig. 4a and b. Actually, the opposite trends in ${{\bar{q}}}_x^v$ (increasing with flow) and ${{\bar{q}}}_x^k$ (decreasing with flow) make the conductive contribution further less important, and the convection part ${{\bar{q}}}_x^v$ dominates the total heat flux ${{\bar{q}}}_x^v+{{\bar{q}}}_x^k$ in Cases H2 and H10 (Fig. 4c).

Table 2 Effective nanofluid properties and non-dimensional parameters for the H10 system with different particle volume fraction $\phi$

Next we look at the effect of the particle volume fraction $\phi$ on the heat fluxes. In Fig. 4b, the conductive flux ${{\bar{q}}}_x^k$ increases with $\phi$ for all flow situations. The improvement percentages over the pure fluid values are in the ranges of $5.0\sim 10.1\%$ for $\phi =2.5\%$ and $12.9\sim 23.3\%$ for $\phi =5\%$, which are also similar to those observed in the vertical thermal gradient systems and those calculated from the Maxwell equation in Sect. 3.2. However, the presence of particles brings the convective flux ${{\bar{q}}}_x^v$ down slightly. This is mainly due to the reduced flow velocity for $\phi =2.5\%$ and 5%: for example, in Case H2, the centerline velocity is 1.99$\times 10^{-3}$ for $\phi =0$ (0.5% different from the input parameter $U_c=2\times 10^{-3}$), 1.92$\times 10^{-3}$ for $\phi =2.5\%$, and 1.84$\times 10^{-3}$ for $\phi =5\%$. For the parameters used in this research, the effective heat capacity $(\rho c_p)_{eff}$=$(1-\phi )(\rho c_p)_f$ +$\phi (\rho c_p)_s$ [12, 68, 73] is 0.995 for $\phi =2.5\%$ and 0.99 for $\phi =5\%$, which are very close to that for the pure fluid $(\rho c_p)_f=1$. The slightly reduced heat capacity of the particle suspension may also be responsible for the ${{\bar{q}}}_x^v$ decrease with $\phi$; however, this should be less significant compared with the effect from the reduced velocity. When looking at the combined heat flux ${{\bar{q}}}_x^k+{{\bar{q}}}_x^v$ in Fig. 4c, we find that the suppressing $\phi$ effect on ${{\bar{q}}}_x^v$ overcomes its enhancing effect on ${\bar{q}}_x^k$, and consequently the total heat flux ${{\bar{q}}}_x^k + {\bar{q}}_x^v$ decreases slightly with the particle volume fraction $\phi$, except the Case H0 when there is no fluid flow. These results and discussions show that, for the systems studied here, the benefit of adding high-conductivity particles is limited to the conductive heat transfer, but the increased hydrodynamic resistance inhibits the convection counterpart; and the overall heat transfer performance is deteriorated. This apparently counter-intuitive observations suggest that adding nanoparticles to a base fluid may not necessarily enhance the thermal performance of heat transfer systems, especially for those where the convection plays a dominant role. Similar findings have been reported in previous studies using different research methods [38, 42, 45, 46].

At last, we also calculate the relevant non-dimensional parameters, including the Prandtl number Pr, Reynolds number Re, and Nusselt number $N\!u$ for the H10 systems. Here, the Nusselt number is defined as

$$\begin{aligned} N\!u=\frac{{\bar{q}}_x}{k_{eff}\left( \frac{T_H-T_L}{L/2}\right) } . \end{aligned}$$

(33)

Three Nusselt number values, $N\!u_v$, $N\!u_k$, and $N\!u_{v+k}$, can be obtained by using the respective heat flux values in Fig. 4 for the numerator term ${{\bar{q}}}_x$ in this equation. These calculated non-dimensional values, as well as other effective properties of the nanofluids, are listed in Table 2. It is not surprising to see the big difference between $N\!u_v$ and $N\!u_k$, since they are directly related the convective (${{\bar{q}}}_x^v$) and conductive (${\bar{q}}_x^k$) heat fluxes discussed above. The decreasing Prandtl number with $\phi$ suggests, in general, that the thermal benefit (represented by the thermal diffusivity $\alpha$) from the high conductivity particles surpasses the increase in hydrodynamic resistance (represented by the kinematic viscosity $\nu$). The slight decrease in the Reynolds number with the increase in $\phi$ comes from two aspects: the increase in the effective viscosity $\mu _{eff}$ and the consequential reduced flow velocity under the same driving force $G_x$. The convective Nusselt number $N\!u_v$, as we discussed above, is directly related to the flow strength and thus $N\!u_v$ is decreasing with $\phi$ as well. On the other hand, the conductive Nusselt number $N\!u_k$ is approximately constant with slight variations with $\phi$. This less dramatic and non-monotonic response in $N\!u_k$ can be explained based on the Nusselt number definition in Eq. (33): both the conductive heat flux ${{\bar{q}}}_x^k$ (Fig. 4b) and the effective thermal conductivity $k_{eff}$ (Sect. 3.2) increase with $\phi$. Nevertheless, the small and nearly unchanged conductive Nusselt number $N\!u_k$ suggests that the high conductivity particles do not help much in improving the overall thermal performance of the system.

4 Conclusion

In this study, we model nanofluids as suspensions of spherical particles in a base fluid, and the fluid and particle properties can be considered explicitly based on physical material parameters of nanofluids. The governing equations for the fluid flow, heat transfer, particle dynamics and flow–particle interaction are solved simultaneously by integrating LBM and SPM. Compared to the single- or two-phase continuum methods commonly used for nanofluid simulations, our approach can incorporate the microscopic nanoparticle parameters explicitly. Therefore, our method could be useful in future fundamental studies of the underlying microscopic mechanisms for the thermal enhancement of nanofluids. This particle-resolved model is then adopted to study the heat transfer performances of suspension systems. Two thermal situations, one with the overall thermal gradient perpendicular to the flow stream and one with the thermal gradient along the flow direction, are considered to study the nanoparticle thermal enhancement effects on conduction-dominant and convection-dominant heat transfer systems. The effect of the particle volume fraction is also examined. Our results show that adding high-conductivity solid particles to the base fluid can enhance the conductive heat transfer, mainly from the elevated effective thermal conductivity of the nanofluid suspensions. However, the presence of these particles elevates the effective fluid viscosity and thus the hydrodynamic resistance; as a result, the convective heat transfer is affected negatively. The overall thermal performance may even be deteriorated in situations where convection dominates the heat transfer process. These results and analysis imply that nanofluids might be more suitable for heat transfer enhancement in microchannels or porous materials where the flow velocity is small.

We acknowledge that the nanofluid system adopted in this work is relatively simple and it could be improved in future studies. Only twelve spherical particles are considered and the Reynolds number is relatively low in this study. In the future, for a more realistic representation of nanofluid systems and thus more reliable results, it would be desirable to conduct the simulations in a larger domain of more nanoparticles and with higher Reynolds numbers, using more efficient computer programs. The thermal boundary conditions are set to generate thermal gradients primarily perpendicular or along the flow direction such that the convective and conductive contributions on overall heat transfer can be conveniently accessed. However, flow and thermal situations in realistic thermal systems can be dramatically different. Furthermore, many factors, such as the segregations and agglomeration, Brownian motion, particle wettability, thermophoresis, fluid pH value, and natural convection, may also play critical roles in nanofluid behaviors however they are not considered in this research. Nevertheless, unlike most existing numerical calculations relying on empirical correlations of effective viscosity and thermal conductivity, our model and simulations use the fluid and particle thermophysical properties directly with no empirical relations involved. The major mechanisms in nanofluids, namely the fluid mechanics, heat transfer, particle dynamics, and flow–particle interaction, are all incorporated via well established numerical methods. The information revealed in this work, therefore, could be beneficial for better understanding the complexity of nanofluid systems.

Appendix: algorithm and program validations

To validate the numerical methods utilized in this research, several simulations are conducted in these sections, and the results are compared with those found in the literature. The good agreement confirms that the fluid flow, flow–particle interaction, and heat transfer have been accurately calculated in our programs.

1.1 Sedimentation process of single particle

Fig. 5

The center position $Z_c$ (a) and velocity $V_z$ (b) of the particle during the sedimentation process. Results from this work (black lines) are compared to those from previous studies (symbols) for the identical system

Our first test considers the sedimentation process of a two-dimensional (2D) circular particle in a viscous fluid under gravity. Following Refs. [67, 74], the fluid is confined in a rectangular box of 2 cm width and 6 cm height, in the horizontal (x) and vertical (z) directions, respectively. A particle of radius $R=0.125$ cm and density $\rho _s=1.25$ g/cm$^3$ is initially set with its center at position ($X_0=1$ cm, $Z_0=4$ cm). The fluid density and viscosity are $\rho _f=1.0$ g/cm$^3$ and $\mu =0.1$ g/(cm$\cdot$s), respectively. The gravitational acceleration is $g=980$ cm/s$^2$, in the negative z direction. Since our programs are developed based on the 3D D3Q19 lattice structure, we consider a thickness of 0.05 cm of the above 2D system. The lattice resolution is $\Delta x=0.01$ cm [67]. Figure 5 displays our results of the particle center position and velocity with time during the sedimentation process (black lines). For the left-right symmetry of the system, the particle simply falls downward (i.e., the negative z direction) along the system centerline, and there is no particle motion in the x direction or particle rotation. After being released, the particle falls downward from its initial position $Z_c=4$ cm, until it reaches the bottom and stops at $t=0.8$ s (Fig. 5a). The particle velocity $V_z$ is also shown in Fig. 5b for a better illustration of the particle sedimentation process. The falling velocity increases at the beginning of the process, since the net gravity force (after canceling the buoyant force) overcomes the weak viscous resistant force due to the slow particle motion at the early stage. The gradually increasing velocity reaches a plateau of $\sim 5.2$ cm/s at $t=0.5$ s, suggesting that the driving gravity force is balanced with the viscous force at this particular velocity. The particle keeps this falling velocity for about 0.25 s, till it approaches the bottom wall and finally stops completely at $t=0.8$ s. Also shown in Fig. 5 are the results from previous studies (symbols) [67, 75, 76] of the identical system, and a good agreement can be observed among them. The small differences between our present and previous results after the particle reaches the bottom could be attributed to the different particle–wall collision rules utilized in these studies. The tested system appears simple; however, it involves the LBM solution of the flow field, the SPM treatment for the flow–particle interaction, the particle dynamics, and the particle–wall collision. The good agreement suggests our programs have considered these mechanisms correctly and accurately.

1.2 Sedimentation process of two circular particles

Fig. 6

The initial particle positions (a) and vorticity contours (b-e) at four time instants during the two-particle sedimentation process

To validate the particle–particle interaction model, next we simulate the two-particle system as shown in Fig. 6a. For a direct comparison, we adopt the same system parameters as in Refs. [67, 76]. The 2D system consists of two circular particles, P1 and P2, in a fluid-filled channel of 2 cm width and 8 cm length. The two particles are identical, with radius $R_1=R_2=0.1$ cm and density $\rho _s=1.01$ g/cm$^3$. Their center locations are set at ($X_{1,0}=0.999$ cm, $Z_{1,0}=7.2$ cm), and ($X_{2,0}=1$ cm, $Z_{2,0}=6.8$ cm) [67]. The density and viscosity of the fluid inside the box are $\rho _f=1.0$ g/cm$^3$ and $\mu =0.01$ g/(cm$\cdot$s), respectively; and the gravitational acceleration is $g=980$ cm/s$^2$ downward [76]. Again a thickness of 0.05 cm of the 2D system is adopted for our 3D program and the lattice resolution is $\Delta x=0.01$ cm [67].

The particles undergo the so-called drafting–kissing–tumbling (DKT) process [77, 78], which can be seen in the instantaneous vorticity plots in Fig. 6. In addition, Fig. 7 shows the particle trajectories during the sedimentation process. After being released, the two particles P1 and P2 fall at approximately the same velocity and an approximately constant distance is maintained for about 1 s. Since the leading particle P2 creates a wake of low pressure behind as it falls down, this allows the trailing particle P1 to fall faster. This is the drafting period. The trailing particle P1 gradually catches up with the leading particle P2, and the two particles become in contact at $t=1.5$ s—the kissing moment. The engagement lasts for about 1 s; during this period the interaction force pushes the particles slightly away from the channel centerline, and also the relative vertical positions are switched: Particle P2 moves ahead particle P1 at $t=2.5$ s, and this the so-called tumbling process. After that, the two particles become separate. Since particle P1 is falling slower along the right wall than particle P2 around the channel centerline, the distance between them keeps increasing with time. The flow patterns in Fig. 6 are nearly identical to those in previous publications [67, 76], and the particle trajectories in Fig. 7 also show good agreement with those in the literature [50, 67, 76]. It should be noted that the tumbling process is made up of a series of unstable configurations, and one should not expect to see identical results among different studies [50, 67].

Fig. 7

The center positions ($X_1$ and $X_2$ in a, and $Z_1$ and $Z_2$ in b) of particles P1 (blue) and P2 (red) during the sedimentation process. Results from this work (lines) are compared to those from previous studies (symbols) for the identical system

1.3 Heat transfer with spherical particle moving in a channel

Fig. 8

The simulated flow fields for the two particle–channel setups: a1 for Case A, and (a2) and (a3) for two representative moments of Case B. The velocity profiles from (a1)-(a3), with the different reference frames considered, across the channel are compared in (b1) at $x=X_c$ and (b2) at $x=X_c+L/2$. These two x locations are also indicated as dashed lines in (a1)-(a3) when visible

Heat transfer is not involved and the systems are 2D in the previous simulations. To address these concerns, in this part we consider the flow and thermal fields for a 3D spherical particle moving in a flat channel. The computational domain has a length $L=101$, a width $W=81$, and a height $H=81$, all in lattice unit $\Delta x$. A spherical particle of radius $R=20$ is initially located at the center of the domain. Constant temperature boundary conditions are applied as $T_{top}=1$ on the top and $T_{btm}=0$ at the bottom. Periodic boundary conditions, for both the fluid flow and the thermal field, are applied on the side surfaces of the simulation box. The fluid and solid heat capacities are set as the same $(\rho c_p)_f= (\rho c_p)_s=1$ and the thermal conductivity ratio is $k_s/k_f=5$. Two cases are considered for this system: Case A: The top and bottom walls are moving at $U_w=-U_0$ in the opposite x direction while the particle is fixed at the domain center; and Case B: The particle is moving at $U_p=U_0$ in the x direction along the channel centerline with the top and bottom walls at rest. $U_0$=0.001 in our next simulations. These two simulation setups actually represent a same physical system, and they are only different in the reference frames: the reference frame is fixed on the sphere in Case A; however, it is fixed with the top and bottom walls in Case B. Accordingly, a correct computer program should yield identical results for these two cases after the difference in reference frames being considered.

Fig. 9

The simulated temperature fields for the two particle–channel setups: a1 for Case A, and (a2) and (a3) for two representative moments of Case B. The temperature profiles from (a1)-(a3) across the channel are compared in (b1) at $x=X_c-L/4$, (b2) at $x=X_c$, and (b3) at $x=X_c+L/2$ These three x locations are also indicated as dashed lines in (a1)-(a3) when visible

A steady state can be quickly established for Case A, and the flow and thermal distributions exhibit periodic variations in time in Case B as the particle passing through the channel repeatedly. In our next analysis, we choose the steady state of Case A and two representative instants of Case B: one when the particle locates at the domain center and one as it crosses the left-right boundary. Figures 8 and 9 collect the simulation results of flow velocity and temperature distribution from these two setups. The absolute velocity vectors appear quite different in Figs. 8a1–a3, and this is due to the different reference frames adopted in these two setups. In Figs. 8b1, b2, we compare the profiles of the calculated velocity in Case A to those in Case B (after subtracting the particle velocity $U_0$) along two vertical lines: one passes through the particle center $X_c$ and one locates at $X_c+L/2$. As anticipated, the profiles from the two setups and at the two instants from Case B all agree to each other excellently. Similarly, the temperature distributions and profiles from Cases A and B are displayed in Fig. 9. The reference frame velocity has no influence on the thermal field, and thus no difference is observed in the isotherms and temperature profiles in Fig. 9. The wider gaps between adjacent isotherm lines inside the particle are due to the higher thermal conductivity in the solid domain (i.e., $k_s/k_f=5$), and the continuity in isotherms and temperature profiles across the solid–fluid interface suggests that our model has considered the interfacial conditions correctly.

Table 3 Comparisons of the calculated wall shear force $F_w$, particle drag force $F_p$, and mean heat flux ${{\bar{q}}}_z$ from Cases A and B. The force values have been normalized with $F_0=2\mu _f L W U_0 /H$ and the heat flux values have been normalized by $q_0=k_f (T_{top}-T_{btm})/H$

For a more quantitative verification, we also calculate the shear force on the bottom wall $F_w$ and the drag force on the particle $F_p$ using the momentum–exchange method in LBM [79], and the mean heat flux ${{\bar{q}}_z}$ across the horizontal middle plane $z=H/2$

$$\begin{aligned} {{\bar{q}}}_z=\frac{1}{LW}\int \int \left( -k\frac{\partial T}{\partial z}\right) _{z=H/2}dxdy. \end{aligned}$$

(34)

These values are further normalized with respect to $F_0=2\mu _f L W U_0 /H$ (the shear force on the channel wall when the spherical particle is replaced with a thin plate at $z=H/2$) and $q_0=k_f (T_{top}-T_{btm})/H$ (the heat flux through the fluid when the spherical particle is removed), respectively. Also, considering the mechanical balance, we should have $F_p/F_w=2$ for this system. Table 3 collects these values for both Cases A and B, and we find the relative differences between them are negligible. The $F_p/F_w=2$ relation is also nearly perfectly reproduced. The ratio ${{\bar{q}}}_z/q_0$ is slightly larger than unity, suggesting that the existence of the high-conductivity particle enhances the heat transfer in this system. The tiny relative differences in the last column between results from Cases A and B further confirm the nearly identical flow and temperature profiles in Figs. 8 and 9.