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

4.5 Simulation of Single Droplets
energy equation (Eq. (4.56)) is the only equation that has to be solved here.
The governing equations as applied can be summarized for the liquid phase
as follows:
Conservation of mass:
u l,r = 0, with ∂u l,r
∂r
= 0 and u l,r
∣
∣r=0
= 0 (4.54)
Conservation of momentum:
∂p l
∂r = 0 (4.55)
Conservation of energy:
∂T
ρ l c p,l
∂t − ∂p l
∂t =−∂ ˙q l,r
∂r
− 2 ˙q l,r
r , with u l,r = 0 (4.56)
Heat flux:
˙q l,r =−λ l
∂T
∂r
(4.57)
Coupling at Gas-Liquid Interface
Gas and liquid phase are modeled with one set of governing equations each, as
stated directly above. The governing equations are one-dimensional and account
for spherically symmetric droplets under microgravity conditions. The
coupling of the two phases is achieved by an additional set of equations, as
introduced in Chapter 4.2.5. This set of equations takes into account the conservation
laws at the interface, and they finally read as follows:
Conservation of mass:
( ) ∂R
∣
ρg − ρ l ∂t
∣ = ρ g u ∣R=R(t) g ,r (4.58)
R=R(t)
Conservation of vaporizing species m (liquid-phase):
(
Y g ,m 1+ ∆u )∣
∣∣∣∣R=R(t)
g ,m,r
= 1 (4.59)
u g ,r − ∂R
∂t
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