On the Formation of Nitrogen Oxides During the Combustion of ...

4 Numerical Modeling and Simulation
4.2.4 Boundary Conditions
Boundary conditions have to be applied in order to solve the coupled governing
equations. In the case of a spherical setup with the droplet center in r = 0,
boundary conditions are necessary in the center and at the radial coordinate
r → ∞ or a finite radius r = R ∞ in the gas phase (Fig. 4.1).
For r = 0, in the liquid phase, boundary conditions follow from the requirements
that the functions of the conservation equations should be spherically
symmetric and twice continuously differentiable:
∂T
∂r
∣ = 0,
r=0
∂p
∂r
∣ = 0,
r=0
∂Y m
∂r
∣ = 0, and u r | r=0 = 0.
r=0
The zero gradient of the temperature T is necessary to avoid a heat sink or
source in r = 0. The gradient of the species mass fraction Y m is zero to conserve
species, and the radial velocity u r vanishes because of the conservation
of mass [297, 298].
At the outer boundary of the gas phase (r → ∞), boundary conditions must
be given for temperature, pressure, and the mass fractions of all species. However,
no condition is allowed for the radial velocity u r here because the convective
flow is directed outwards according to the calculation. Thus, the velocity
u r | r →∞ is a result of the conservation of mass. Furthermore, if an adiabatic,
closed system is studied, Neumann formulations must be used for temperature
T and mass fraction Y m at the outer boundary with:
∂T
∂r
∣ = 0
r →∞
and
∂Y m
∂r
∣ = 0.
r →∞
R
S
r
l
g
R ∞
Figure 4.1: Schematic of One-Dimensional Droplet Model with Interface. The interface S
links liquid droplet l and gaseous atmosphere g [297, 298].
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