INAUGURAL–DISSERTATION zur Erlangung der Doktorwürde der ...

46 3. Numerical Methods
3.1 Finite Difference Method for Bi-component
Droplet Evaporation and Solid Layer
Formation
The partial differential equation, Eq. (2.51), with initial and boundary conditions is
solved numerically at every time and spatial location within the droplet using second
order explicit finite difference method, given as
Y j+1
i
∆t
− Y j
i
[
r
2
= D i+1 (Y j
i+1 − Y j
i ) − r2 i−1(Y j
i − Y j
i−1 )]
12 , (3.1)
ri 2∆r2 where r i+1 = r i + ∆r, r i−1 = r i − ∆r, and i and j are the spatial location within the
droplet and time step indices, respectively. Equation (3.1) can be simplified to yield
the following equation
Y j+1
i
= Y j
i
+ D 12∆t [
r
2
ri 2 i+1 (Y j
∆r2 i+1 − Y j
i ) − r2 i−1(Y j
i − Y j
i−1 )] . (3.2)
The initial condition to compute Eq. (3.2) is provided as a Dirichlet condition, i.e.,
Y = Y i0 at every location inside the droplet at t = 0. A Neumann boundary condition
is applied at the center of the droplet, i.e., ∂Y i /∂r = 0 at r = 0, which implies the
radial symmetry within the droplet. A Robin boundary condition is employed at the
droplet surface, and it is given by Eq. (2.52).
The energy Eq. (2.59) is an ordinary differential equation, solved using Runge-Kutta
4 th order method. The droplet is discretized into equal distant grid points at any given
time. As the droplet size decreases with time thereby the grid size changes because
grid points are fixed, thus a moving grid problem is solved, and grid independency of
the numerical method is tested using different grid sizes with the number of grid points
varying from 10 to 100. The value of 50 grid nodes is found to perform well.
The numerical stability of the method is tested using various time steps following
the Courant-Friedrichs-Lewy (CFL) condition [188]. The CFL condition defines the
limiting criteria for the numerical grid size when the time step and fluid velocity are
known, and it is defined as
C = u∆t
∆x ≤ C max, (3.3)
where C is the dimensionless number known as Courant number, u is the velocity, ∆t
and ∆x are the time step and grid size, respectively. C max is the maximum possible
Courant number to get a stable numerical solution, and it is generally taken as any
positive value lower than or equal to 0.5 [188]. The step-by-step procedure of Abramzon
and Sirignano [62] is applied to calculate the mass evaporation rate given by Eq. (2.55).
Numerical simulations of pure water, mannitol dissolved in water droplet evaporation