Flow and mass transfer around a core-shell reservoir

I have developed an alternative numerical approach to study mass transfer from a stationary core-shell reservoir under channel flow conditions. I use the lattice Boltzmann method to compute both the solvent fluid flow and the diffusion and advection of the solute. I have investigated the impact of the flow by reporting mass transfer quantities such as the instantaneous solute concentration and the local Sherwood number at the surface of the reservoir. The flow is found to enhance the release of the encapsulated material, but it prevents the released material from reaching the channel walls.

Figure 1

Geometry of the problem for the flow part.

Figure 2

Geometry of the problem for the mass transfer part.

Figure 3

Evolution of flow and mass transfer around a stationary rigid core-shell reservoir. The fluid is initially at rest and it develops into a laminar steady flow. The core is initially loaded with a high concentration $c_0=10$ that creates a gradient with the rest of the domain, where the solute is initially absent. The concentration gradient triggers the release, from the core into the ambient fluid, until exhaustion of the internal solute and until it reaches equilibrium. Snapshots are taken at equal time intervals. Here $\mathrm{Pe}=50$.

Figure 4

Solute concentration profiles taken along the line $x=x_{\mathrm{c}}$ at different moments, for (a) stagnant fluid and (b) flowing fluid. The gray areas indicate the location of the shell. Under flow conditions, the released solute stays in the vicinity of the reservoir surface and the thickness of the boundary layer ${\delta }_{\mathrm{BL}}$ stays roughly constant.

Figure 5

Distribution of the solute as a function of the Péclet number $\mathrm{Pe}$, for the same fully developed flow pattern around the reservoir, when $80\%$ of the initially encapsulated amount is released. Here $\mathrm{Pe}$ is increased by decreasing the value of the diffusion coefficient in the fluid $D$ while holding all the other parameters constant. The concentration boundary layer gets thinner and extends farther downstream when increasing $\mathrm{Pe}$.

Figure 6

(a) Angular position $\theta$, (b) instantaneous solute concentration at the surface of the reservoir $c(R,\theta ,t)$, (c) instantaneous local mass flux across the shell $J(\theta ,t)$, and (d) instantaneous local Sherwood number $\mathrm{Sh}\left(\theta \right)$, taken at three different moments. All these quantities are not constant and are not uniform at the surface of the reservoir.

Figure 7

Instantaneous total Sherwood number ${\mathrm{Sh}}_{\mathrm{T}}\left(t\right)$ for a stagnant fluid (solid red line) and for a flowing fluid with Péclet number $\mathrm{Pe}=50$ (dashed blue line). The overall mass transfer from the core-shell reservoir is enhanced by the flow.