**1. Introduction**

Heat and mass transfer between multi-phases or different materials with interfacial conjugate conditions is frequently encountered in fundamental sciences and numerous engineering applications involving fluid dynamics, thermal transport, materials sciences, and chemical reactions. Examples are cooling of turbine blades, heat exchangers and electronic devices, thermal insulation on heat pipes and chemical reactors, heat conduction in composite materials, and heat and mass transfer between solid particles and their surrounding fluids [1–8], to name a few. The most well-known conjugate conditions include the continuity of both the temperature (concentration) and the heat (mass) flux at the interface. Other conjugate conditions, such as with temperature (concentration) jumps and/or flux discontinuities [9], and Henry's law relationship [10], are also noticed at fluid-solid interfaces or interfaces of two solids or fluids of different thermal (mass diffusion) properties.

The non-smoothness or discontinuities/jumps in the physical or transport properties, and consequently in the distribution of the temperature (concentration) field across the interface, pose a great challenge to any numerical method applied to solve the interface problems. Development of accurate and efficient numerical schemes to treat the interface conditions has attracted much attention in the literature, such as the immersed boundary method (IBM) [11, 12], the immersed interface method (IIM) [13, 14], the ghost fluid method (GFM) [15, 16], the sharp interface Cartesian grid method [17, 18], and the matched interface and boundary (MIB) method [19]. Most of these methods are formulated in the finite-difference, finite-volume or finite-element frameworks.

for steady cases. Improvement of this HLD based scheme was conducted by several groups (see a review in Ref. [23]) for unsteady cases. Importantly, with the interface treated as a shared boundary between the adjacent domains, the boundary conditions by Li et al. [21] were applied to interface conditions and particular interface schemes were proposed and verified in [23] for standard conjugate conditions, and in [9] for conjugate heat and mass transfer with interfacial jump conditions. This idea of developing analytical relationships for the DFs to satisfy the conjugate conditions was also extended to handle general interface conditions in [32, 33]. In all the previous schemes in [9, 23, 32, 33], the local geometry was taken into account and the second-order accuracy of the LBM solution can be preserved. There is another category of interface schemes that has attracted interest in the LBM community. In those schemes, additional source terms [34], alternative LBE formulations [35, 36], or modified equilibrium DFs [37, 38], were proposed to handle the conjugate conditions. The main motivation for those schemes is to avoid the consideration of the interface geometry or topology, which can be a challenge in complex systems such as porous media. As pointed out in [33], however, these schemes usually suffer from degraded numerical accuracy and/or convergence

*Interface Treatment for Conjugate Conditions in the Lattice Boltzmann Method…*

*DOI: http://dx.doi.org/10.5772/intechopen.86252*

orders. This perspective will be illustrated in detail in this chapter.

**2. Conjugate conditions in heat and mass transfer**

**<sup>n</sup>** � ð Þ *Dm*∇*<sup>ϕ</sup>* <sup>þ</sup> **<sup>u</sup>***<sup>ϕ</sup> <sup>f</sup>* <sup>¼</sup> **<sup>n</sup>** � ð Þ *Dm*∇*<sup>ϕ</sup>* <sup>þ</sup> **<sup>u</sup>***<sup>ϕ</sup> <sup>s</sup>* <sup>þ</sup> **<sup>n</sup>** � **<sup>q</sup>***<sup>C</sup>*

interfacial scalar values and their fluxes as in [32, 33].

**3. Lattice Boltzmann model for the general CDE**

written as a general convection diffusion equation as:

*<sup>f</sup>* <sup>¼</sup> **<sup>n</sup>** � *<sup>k</sup>*∇*<sup>ϕ</sup>* <sup>þ</sup> *<sup>ρ</sup>cp***u***<sup>ϕ</sup>*

mass transfer perspective, can be defined as:

**q**jump the jump conditions at the interface.

outlook are given in Section 6.

**<sup>n</sup>** � *<sup>k</sup>*∇*<sup>ϕ</sup>* <sup>þ</sup> *<sup>ρ</sup>cp***u***<sup>ϕ</sup>*

**67**

The rest of this chapter is organized as follows. In Section 2, the different types of conjugate conditions are presented. The LB model for the general CDE is briefly described in Section 3. In Section 4, the representative interface schemes are summarized. Discussion on the numerical accuracy of the selected interface schemes is provided in Section 5 with representative numerical examples. Conclusions and

In order to define the conjugate conditions, we begin by defining two domains 1 and 2, as shown in **Figure 1**. The conjugate interface conditions, from a heat and

where **n** represents the normal direction, *ϕ* the macroscopic scalar variable of interest (temperature or concentration), *k* the thermal conductivity, *ρ* the density, *cp* the heat capacity, **u** the velocity vector, *Dm* the mass diffusivity, and *ϕ*jump and

Some examples of jump conditions can be found in cases such as concentration jumps (Henry's law [10]) or temperature jumps at the interface [9]. Eqs. (1a)–(1c) reduce to the standard conjugate conditions in [23] with no jumps and zero normal velocity; they can also be extended to yield two general relationships between

The governing heat and mass transfer equation within each domain can be

*<sup>s</sup>* <sup>þ</sup> **<sup>n</sup>** � **<sup>q</sup>***<sup>T</sup>*

*Φdf* ¼ *Φds* þ *ϕ*jump (1a)

jump in heat transfer*,* (1b)

jump in mass transfer*,* (1c)

When applying those traditional numerical methods, a popular approach to implement the conjugate conditions is to employ iterative schemes, in which a Dirichlet interface condition is imposed for one phase or material and a Neumann interface condition for the other. The heat and mass transfer in each phase is separately solved, and the continuity or prescribed jump condition at the interface could be satisfied after multiple iterations. For conjugate transport with complex interface geometry, the iterative schemes would become difficult to implement and they normally necessitate a considerable amount of computational effort.

The lattice Boltzmann method (LBM), which has emerged as an attractive alternative numerical method for modeling fluid flows and heat mass transfer (see [20–22] and Refs. therein), has been demonstrated to be an effective and efficient numerical approach for conjugate interface conditions in tandem with the convection diffusion equation (CDE) [9, 23]. In this chapter, we present a critical review of the various interface schemes proposed in the literature, with a focus on the comparison of numerical accuracy.

The well-known features of the LBM method include its explicit algorithm, ease in implementation, capability to treat complex geometry, and compatibility with parallel computing [20, 21]. Boundary condition treatment is essential to the integrity of LBM since the kinetic theory-based method deals directly with the microscopic distribution functions (DFs) rather than the macroscopic conservation equations. Earlier LB models treat the collision effects with a single-relaxation-time (SRT) approximation, commonly referred to as the Bhatnagar-Gross-Krook (BGK) model [24–26]. However, the SRT model is limited such that it can only describe isotropic diffusion [20]. In recent years, models such as the two-relaxation-time (TRT) [27, 28] and multiple-relaxation-time (MRT) [20, 29, 30] LB models have been proposed that can handle anisotropic diffusion. Representative LB models proposed in the literature include the general BGK model by Shi and Guo [31] for the nonlinear CDE, the D3Q7/D2Q5 MRT models by Yoshida and Nagaoka [20] for the general convection anisotropic diffusion equation, and the D1Q3/D2Q9/D3Q19 MRT models by Chai and Zhao [30] for the general nonlinear convection anisotropic diffusion equation, to name a few. The MRT models have improved numerical accuracy and stability compared to the SRT models [20, 28, 30]. The D3Q7/ D2Q5 model proposed in [20] is used for this review, as it preserves second order spatial accuracy when recovering the general CDE following an asymptotic analysis. Based on the D3Q7/D2Q5 LB models, Li et al. [21] proposed second-order accurate boundary treatments for both the Dirichlet and Neumann conditions; they have also established a general framework for heat and mass transfer simulations with direct extension to curved boundary situations. In their framework, explicit analytical expressions were developed to relate the macroscopic quantities, such as boundary temperature (concentration) and their fluxes, and interior temperature (concentration) gradients, to the microscopic DFs in the LB model.

The first work that explicitly addressed the fluid-solid interface condition in LBM was conducted by Wang et al. [6]. They proposed a simple "half lattice division" (HLD) treatment in which no special treatment is required and the temperature and flux continuity condition at the interface was automatically satisfied

#### *Interface Treatment for Conjugate Conditions in the Lattice Boltzmann Method… DOI: http://dx.doi.org/10.5772/intechopen.86252*

for steady cases. Improvement of this HLD based scheme was conducted by several groups (see a review in Ref. [23]) for unsteady cases. Importantly, with the interface treated as a shared boundary between the adjacent domains, the boundary conditions by Li et al. [21] were applied to interface conditions and particular interface schemes were proposed and verified in [23] for standard conjugate conditions, and in [9] for conjugate heat and mass transfer with interfacial jump conditions. This idea of developing analytical relationships for the DFs to satisfy the conjugate conditions was also extended to handle general interface conditions in [32, 33]. In all the previous schemes in [9, 23, 32, 33], the local geometry was taken into account and the second-order accuracy of the LBM solution can be preserved.

There is another category of interface schemes that has attracted interest in the LBM community. In those schemes, additional source terms [34], alternative LBE formulations [35, 36], or modified equilibrium DFs [37, 38], were proposed to handle the conjugate conditions. The main motivation for those schemes is to avoid the consideration of the interface geometry or topology, which can be a challenge in complex systems such as porous media. As pointed out in [33], however, these schemes usually suffer from degraded numerical accuracy and/or convergence orders. This perspective will be illustrated in detail in this chapter.

The rest of this chapter is organized as follows. In Section 2, the different types of conjugate conditions are presented. The LB model for the general CDE is briefly described in Section 3. In Section 4, the representative interface schemes are summarized. Discussion on the numerical accuracy of the selected interface schemes is provided in Section 5 with representative numerical examples. Conclusions and outlook are given in Section 6.

### **2. Conjugate conditions in heat and mass transfer**

In order to define the conjugate conditions, we begin by defining two domains 1 and 2, as shown in **Figure 1**. The conjugate interface conditions, from a heat and mass transfer perspective, can be defined as:

$$
\Phi\_{df} = \Phi\_{ds} + \phi\_{\text{jump}} \tag{1a}
$$

$$\mathbf{n} \cdot \left( k \nabla \phi + \rho c\_p \mathbf{u} \phi \right)\_f = \mathbf{n} \cdot \left( k \nabla \phi + \rho c\_p \mathbf{u} \phi \right)\_s + \mathbf{n} \cdot \mathbf{q}\_{\text{jump}}^T \text{ in heat transfer}, \qquad \text{(1b)}$$

$$(\mathbf{n} \cdot (D\_m \nabla \phi + \mathbf{u}\phi)\_{\!\!\!/} = \mathbf{n} \cdot (D\_m \nabla \phi + \mathbf{u}\phi)\_{\!\!\!/} + \mathbf{n} \cdot \mathbf{q}\_{\!\!\!/}^{\text{C}} \text{ in mass transfer}, \qquad \text{(1c)}$$

where **n** represents the normal direction, *ϕ* the macroscopic scalar variable of interest (temperature or concentration), *k* the thermal conductivity, *ρ* the density, *cp* the heat capacity, **u** the velocity vector, *Dm* the mass diffusivity, and *ϕ*jump and **q**jump the jump conditions at the interface.

Some examples of jump conditions can be found in cases such as concentration jumps (Henry's law [10]) or temperature jumps at the interface [9]. Eqs. (1a)–(1c) reduce to the standard conjugate conditions in [23] with no jumps and zero normal velocity; they can also be extended to yield two general relationships between interfacial scalar values and their fluxes as in [32, 33].
