10217

# Mass Transfer with Reaction in a Spherical Drop

Consider a spherical drop of nonreacting fluid . Species absorbs and diffuses into the interior of the drop and then reacts with the liquid. Assume that the kinetics for this reaction are , where is the reaction order.
The concentration of species is taken to be a function of only. Here, is the radial position, is the drop's radius, is the diffusion coefficient, is the convective heat transfer coefficient, is the concentration of species , is the reaction rate constant, and is the reaction order. The boundary value problem is (BC means boundary condition, IC means initial condition):
, ,
IC: ,
BC1: ,
BC2: at .
BC2 defines the mass transfer flux across the interface of the drop and is the far field concentration.
Dimensionless Form
For the analysis we define the following dimensionless variables:
, , ,
where, , , and are the dimensionless concentration, radial position, and time, respectively.
In dimensionless form the PDE becomes:
, (1)
IC: ,
BC1: ,
BC2: at .
The problem then depends on two dimensionless quantities: (i.e. the Sherwood number for mass transfer) and (i.e. the Damköhler number). represents the relative importance of convective mass transport to diffusional transport. When convective transport dominates diffusional transport (i.e. ), the concentration gradients in the drop are negligible. For small Sherwood numbers, the concentration in the drop is governed by diffusional processes. When , one recovers the case when there is no reaction.
Numerical Analysis
We solve the dimensionless form of the problem using the method of lines. Note the problem is linear and the only issue we need to address is the apparent singularity at . Evaluating the limiting behavior at shows that the appropriate equations are
, for ,
, at .
Finite Difference Operators for Spatial Derivatives
The central difference template for the spatial derivatives of our PDE at the node is
, ,
, ,
where we have defined the node spacing by: .
For our problem, we subdivide the drop radius into a grid of intervals such that the grid coordinates are , where . With this definition it follows that and .
The PDE can then be expressed as:
(2)
, (3)
Thus we have equations in unknowns. At we have the following BC:
.
Approximating the spatial derivative with a central difference gives
.
Solving for gives:
, (4)
while at we have
;
a central difference formula gives
, (5)
where is evaluated at ξ =ξ1-Δξ=-Δξ. Thus equations (4) and (5) allow us to eliminate the fictitious nodes from equations (2) and (3) to give equations in terms of unknown values.
This Demonstration plots the concentration profile versus the radial position, , for several values of the dimensionless time, . You can vary the Sherwood number, the reaction order (e.g. is a first-order reaction), and the Damköhler number (e.g. or no chemical reaction, see the last snapshot). For large values of , the boundary condition BC2 becomes .

### SNAPSHOTS

 Share: Embed Interactive Demonstration New! Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details » Download Demonstration as CDF » Download Author Code »(preview ») Files require Wolfram CDF Player or Mathematica.

#### Related Topics

 RELATED RESOURCES
 The #1 tool for creating Demonstrations and anything technical. Explore anything with the first computational knowledge engine. The web's most extensive mathematics resource. An app for every course—right in the palm of your hand. Read our views on math,science, and technology. The format that makes Demonstrations (and any information) easy to share and interact with. Programs & resources for educators, schools & students. Join the initiative for modernizing math education. Walk through homework problems one step at a time, with hints to help along the way. Unlimited random practice problems and answers with built-in Step-by-step solutions. Practice online or make a printable study sheet. Knowledge-based programming for everyone.