Conjugate heat and mass transfer in the lattice Boltzmann equation method

An interface treatment for conjugate heat and mass transfer in the lattice Boltzmann equation method is proposed based on our previously proposed second-order accurate Dirichlet and Neumann boundary schemes. The continuity of temperature (concentration) and its flux at the interface for heat (mass) transfer is intrinsically satisfied without iterative computations, and the interfacial temperature (concentration) and their fluxes are conveniently obtained from the microscopic distribution functions without finite-difference calculations. The present treatment takes into account the local geometry of the interface so that it can be directly applied to curved interface problems such as conjugate heat and mass transfer in porous media. For straight interfaces or curved interfaces with no tangential gradient, the coupling between the interfacial fluxes along the discrete lattice velocity directions is eliminated and thus the proposed interface schemes can be greatly simplified. Several numerical tests are conducted to verify the applicability and accuracy of the proposed conjugate interface treatment, including (i) steady convection-diffusion in a channel containing two different fluids, (ii) unsteady convection-diffusion in the channel, (iii) steady heat conduction inside a circular domain with two different solid materials, and (iv) unsteady mass transfer from a spherical droplet in an extensional creeping flow. The accuracy and order of convergence of the simulated interior temperature (concentration) field, the interfacial temperature (concentration), and heat (mass) flux are examined in detail and compared with those obtained from the “half-lattice division” treatment in the literature. The present analysis and numerical results show that the half-lattice division scheme is second-order accurate only when the interface is fixed at the center of the lattice links, while the present treatment preserves second-order accuracy for arbitrary link fractions. For curved interfaces, the present treatment yields second-order accurate interior and interfacial temperatures (concentrations) and first-order accurate interfacial heat (mass) flux. An increase of order of convergence by one degree is obtained for each of these three quantities compared with the half-lattice division scheme. The surface-averaged Sherwood numbers computed in test (iv) agree well with published results.

Figure 1

Illustration of the local geometry of an interface in the lattice (solid circles, lattice nodes in Domain 1; solid squares, interface nodes; open circles, lattice nodes in Domain 2).

Figure 2

Schematic layout of the square lattice on a 2D channel containing two fluids in Domain 1 (0 ⩽ $y$ ⩽ $h$) and Domain 2 ($h$ ⩽ $y$ ⩽ $H$).

Figure 3

Regions of stability and instability in the LBE computation for steady convection-diffusion in the channel for various fluid diffusivity ratios κ = $D_2/D_1$ using different relaxation coefficient ${\tau }_1$ values (a) on log-linear scale and (b) on log-log scale.

Figure 4

Contours of ϕ for steady convection-diffusion in the channel at Pe = 20 with simulation parameters $H$ = 2$h$ = 64δ$y$, Δ = 0.5, κ = 10, ${\tau }_1$ = 0.55, and ${\tau }_2$ = 1.0.

Figure 5

Profiles of (a) $\varphi (x,y)$ and (b) $D_i\partial {\varphi }_i/\partial y$ along vertical lines in the channel at Pe = 20 with $H$ = 2$h$ = 64δ$y$, κ = 10, ${\tau }_1$ = 0.55, and ${\tau }_2$ = 1.0.

Figure 6

Profiles of ${\varphi }_{\mathrm{int}}(x,y)$ and ${(D\partial \varphi /\partial y)}_{\mathrm{int}}$ at the interface ($y$ = $h$) in the channel with the same parameters as in Fig. 5.

Figure 7

Relative $L_2$ norm error, $E_2$, for the interior field of ϕ versus the grid resolution, 1/$H$, for steady convection-diffusion in the channel using the present interface schemes with (a) Δ = 0.50, 0.25, and 0.75, and (b) Δ = 0.50, 0.01, and 0.99.

Figure 8

Relative $L_2$ norm error, $E_{2\_\mathrm{tint}}$, for the interfacial value of ϕ versus 1/$H$ for steady convection-diffusion in the channel using the present interface schemes with (a) Δ = 0.50, 0.25, and 0.75, and (b) Δ = 0.50, 0.01, and 0.99.

Figure 9

Relative $L_2$ norm error, $E_{2\_\mathrm{qint}}$, for the interfacial flux versus 1/$H$ for steady convection-diffusion in the channel using the present interface schemes with (a) Δ = 0.50, 0.25, and 0.75, and (b) Δ = 0.50, 0.01, and 0.99.

Figure 10

Comparison of $E_2$ obtained using the present interface Scheme 2 with that from the HLD scheme for steady convection-diffusion in the channel.

Figure 11

$E_{2\_\mathrm{tint}}$ versus 1/$H$ obtained using the HLD scheme for steady convection-diffusion in the channel with (a) Δ = 0.25 and (b) Δ = 0.75.

Figure 12

$E_{2\_\mathrm{qint}}$ versus 1/$H$ obtained using the HLD scheme for steady convection-diffusion in the channel with (a) Δ = 0.25 and (b) Δ = 0.75.

Figure 13

Time-averaged $L_2$ norm error, $E_2$, versus 1/$H$ for unsteady convection-diffusion in the channel using the present interface Scheme 2.

Figure 14

Time-averaged $L_2$ norm error, $E_{2\_\mathrm{tint}}$, versus 1/$H$ for unsteady convection-diffusion in the channel using the present interface Scheme 2.

Figure 15

Time-averaged $L_2$ norm error, $E_{2\_\mathrm{qint}}$, versus 1/$H$ for unsteady convection-diffusion in the channel using the present interface Scheme 2.

Figure 16

Schematic depiction of the lattice in a 2D circular domain with a curved interface and a curved outer boundary.

Figure 17

Profiles of (a) $\varphi (r,\phi )$ and (b) $q_r=-D_i\partial {\varphi }_i/\partial r$ along the lines with constant ϕ = 0 and ϕ = π/4, respectively, in the circular domain with $R_2$ = 2$R_1$ = 32.5δ$x$, κ = 10, ${\tau }_1$ = 0.55, ${\tau }_2$ = 1.0, and $n=4$.

Figure 18

Profiles of (a) ${\varphi }_{\mathrm{int}}(r,\phi )$ and (b) $q_{\mathrm{int}}=-{(D\partial \varphi /\partial x)}_{\mathrm{int}}$ or $-{(D\partial \varphi /\partial y)}_{\mathrm{int}}$ at the interface in the circular domain with the same parameters as in Fig. 17.

Figure 19

Comparison of $E_2$ obtained using the present interface schemes with that from the HLD scheme for steady heat conduction in the circular domain with a curved interface.

Figure 20

Comparison of $E_{2\_\mathrm{tint}}$ obtained using the present interface schemes with that from the HLD scheme for steady heat conduction in the circular domain with a curved interface.

Figure 21

Comparison of $E_{2\_\mathrm{qint}}$ obtained using the present interface schemes with that from the HLD scheme for steady heat conduction in the circular domain with a curved interface.

Figure 22

Schematic depiction of the computational domain and the square lattice layout for conjugate mass transfer across the interface of a spherical droplet immersed in an extensional creeping flow.

Figure 23

Dimensionless concentration contours for mass transfer from the droplet in the extensional creeping flow at Fo${}_1$ = 5$\times$10${}^{-2}$ and (a) Pe${}_1$ = 1, $D_2/D_1$ = 1; (b) Pe${}_1$ = 100, $D_2/D_1$ = 1; (c) Pe${}_1$ = 500, $D_2/D_1$ = 1; (d) Pe${}_1$ = 100, $D_2/D_1$ = 0.1; and (e) Pe${}_1$ = 100, $D_2/D_1$ = 10.

Figure 24

Profiles of ${\varphi }^{*}(r,z)$ and $q_r=-D_i\partial {\varphi }_i^{*}/\partial r$ along the horizontal line at $z$ = 0 for Cases (d) and (e) in Fig. 23.