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

4.2 Basics for Numerical Modeling
Presuming the boundary is far enough away from the droplet and the main
reaction zone, the boundary conditions may also be set to constant ambient
conditions (Dirichlet formulations) according to Equations (4.19) through
(4.21):
T| r →∞ = T ∞ , (4.19)
p ∣ ∣
r →∞
= p ∞ , and (4.20)
Y m | r →∞ = Y m,∞ . (4.21)
4.2.5 Modeling of Gas-Liquid Interface
Conservation of mass, species, momentum, and energy is not only valid in
the gas and liquid phase but also at the interface S of the two phases, as
shown in Figure 4.1 [65]. The interface is presumed to be an infinitely thin
layer. Its time-dependent position is denoted by R(t ) and its velocity by u S,r
(Eq. (4.22)). The summation convention for the indices i and j , as introduced
in Chapter 4.2.1, is also pursued in the following paragraphs.
u S,r = ∂R
∂t
(4.22)
The change in mass of the liquid droplet l is
ṁ l = d ∫ ∫
( )
ρ l dV l =− ρ l ul,i − u S,i ni dS l
dt V l (t)
S l (t)
(
( )∣
= −4πR 2 ρ l ul,r − u ∣R=R(t) S,r = −4πR 2 ρ l u l,r − ∂R )∣
∣∣∣R=R(t)
(4.23)
∂t
and the change in mass of the gas phase g
ṁ g = d ∫ ∫
( )
ρ g dV g =− ρ g ug ,i − u S,i ni dS g
dt V g (t)
S g (t)
(
( )∣
= 4πR 2 ρ g ug ,r − u ∣R=R(t) S,r = 4πR 2 ρ g u g ,r − ∂R )∣
∣∣∣R=R(t)
.
∂t
(4.24)
The term u g ,r denotes the radial velocity in the gas phase g , and u l,r the velocity
in the liquid phase l. As mass conservation is valid despite vaporization of
the liquid droplet, the change of mass in the volume V (t ) has to be balanced
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