**Direct Numerical Simulation on
Single-Droplet Flow with Mass Transfer**

by Rho-Taek Jung, Toru Sato

The University of Tokyo, Environmental and Ocean Engineering
Department, Bunkyou-ku, Tokyo, 133-5868, JAPAN

** **

The sequestration
of carbon dioxide (CO_{2}), one of the so-called greenhouse gases, in
the ocean is a promising method to reduce CO_{2} in the atmosphere
[1,2]. The CO_{2} extracted from thermal power plants on land is liquefied
(that is, LCO_{2}) and transported by ships, and injected in the form
of droplets at the depth of 1000〜2000m in the deep ocean. Emitted LCO_{2} is regarded to form a
plume of rising droplets, which entrains surrounding seawater. The LCO_{2}
droplets are dissolved out during the rise and water of the large concentration
of CO_{2} is expected to peel out of the rising plume, to sink as a
density current and to intrude into the surrounding stratified ocean the
density of which is equal to that of the CO_{2} rich water. A prior
proposition on the CO_{2} ocean sequestration is how to realize the
slow dissolution of CO_{2} into seawater and, consequently, the large
dilution of dissolved CO_{2} near the injection site in order to
minimize biological impacts in the deep ocean.

To accomplish both slow dissolution and large dilution of CO_{2},
it is very important to make the droplet flow to be a laminar, and to control
initial droplet size that determines its rise velocity and the dissolution rate
of CO_{2} from it.

If there is a mass transfer, the flow is divided into three
characteristic regions depending on the Schmidt number; that is, the Schmidt
number of O(1), Sc<<1 and Sc>>1. At the regime where the Schmidt
number is much greater than unity, mass diffusion proceeds at a slower speed
than momentum diffusion, and the mass boundary layer become much thinner than
the momentum boundary layer. There are some experimental studies of the high
mass transfer over a sphere [3,4]. Experiments on three-dimensional mass
transfer needs high experimental technique, while the numerical simulation may
be appropriate method for understanding its mechanism, as the power of
calculation speed and capacity of hardware grades up incredibly.

The aim of this article is to develop a CFD code for two-phase flows
with unidirectional dissolution from a dispersed phase into a continuous phase.
Direct CFD codes of front-capturing type for bubble/droplet flows have been
developed by a number of researchers[5,6,7]. The interface is expressed by a
variety of scalar-function methods, e.g. volume of fluid (VOF), marker-density
function (MDF), VOF in micro-cells or the level-set methods. On the other hand,
Matsumoto *et al.* [8] applied a
front-tracking method by the boundary-fitted coordinates to single-bubble
flows. Although it seems that the front-tracking type gives more accurate
representation of the interface shape than the front-capturing type because it
tracks the interface in the geometrically direct way, grid skew may cause
numerical inaccuracy when the deformation of the interface becomes large.
Tryggvason *et al.* [9] developed the
explicit front-tracking mesh for bubble/droplet flows, in which the interface
is expressed by the moving mesh in the orthogonal grids which never move. This
seems to concur the grid skew problem.

Here we have selected the
MDF method because the front-capturing method with volume fraction has more
flexibility in coping with large curvature, coalescence, pinch-off and so on
than the front-tracking type. In the numerical procedure of the MDF, steep
shock surfaces are transferred, so that the present method adopts the total variation-diminishing
(TVD) scheme to suppress artificial oscillations.

Since it is still not
cheap to treat a number of droplets by direct simulations with current
computing facilities, we only focus on the movement of a single droplet. For
the same reason, the thickness of mass boundary layer is assumed to have the
same order of magnitude of that of momentum boundary layer in this study.
Accordingly, high Schmidt number problem such as the LCO_{2}-seawater
systems, _{} of which is over
500, is difficult to handle by the MDF.

On the other hand, we are underway on the high Schmidt number problem
by the front-tracking method. At first, we have carried out a simulation on
flow over a solid sphere with mass transfer from the wall constructed by
unstructured mesh. There are very thin layers in the mass boundary for solving
high Schmidt number problem.

Moreover, the formation of CO_{2} clathrate hydrate is left for
the future challenge.

The problems we tackle in this study are (1) the movement of the
interface between a droplet and the liquid continuous phase, (2) the
dissolution of mass from the droplet to the continuous phase through the
interface for low _{}, (3) the transfer of dissolved mass on the solid sphere for
high _{}. This article gives an outline the present CFD method (ref.
[10]) in detail and leads to the results from case studies done for its
validation.

(a) Re=990 (b)
Re=1131

Fig.1 3D images of a rising droplet: (a) Dissolved Sodium Fluorescein
into the Water (b) Simulated Iso-Surface of Laplacian of Pressure (DP=1.0)

Fig.4
Mass transfer group for a sphere at high Schmidt number. Lines of Eq.10 and Eq.11
are recommend by Clift and Eq.9 by Ranz-Marshall. Dots are numerical results

Fig.5 Isometric view of solid
sphere on concentration contour and a streamline. An equatorial plane cutting
through the interior of hybrid mesh comprised 799 wall boundary nodes and 1594
triangular prisms. (shown in small box for detailed concentration contour at
z-y plan)