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

4 Numerical Modeling and Simulation
In addition to the assumptions made for the single droplet model, the following
simplifications are introduced to achieve an analytical solution with the
D² law: The temperature in the droplet is homogeneous and equal to the temperature
at the droplet surface T S . The problem is quasi-steady, i.e. partial
derivatives in time ∂/∂t vanish. Profiles of temperature and mass fraction are
assumed to be in steady state at each time t and droplet radius R(t ). This implies
that the droplet radius changes slowly compared to the transport of heat
and mass [298, 443].
The transport of mass can be written for a water droplet as
ṁ H2 O
A
= ρ dY H2 O
g D H2 O,air + ṁ
dr A Y H 2 O, (4.64)
where ṁ H2 O/A is the mass flux of water in kg m −2 s −1 , ṁ/A the total mass flux,
D H2 O,air the diffusion coefficient of water in air, ρ g the density of the humid
air, and Y H2 O the mass fraction of water. The area A reflects the droplet surface
with A = 4πR 2 . The mass flow of vaporizing water ṁ H2 O is equal to the total
mass flow ṁ, as air is insoluble in water in a first estimate. The mass flow
ṁ is constant in space, as a quasi-steady state assumption was postulated.
The mass flow value is generally defined as positive, if water vaporizes and the
droplet shrinks.
Heat transfer is described by:
(
d
r 2 ṁ )
dr A c pT − d dr
(
r 2 λ dT )
= 0. (4.65)
dr
As the vaporization mass flow ṁ H2 O in Equation (4.65) is linear in R(t ) and
accordingly in D, the mass fraction Y H2 O,S and droplet temperature T S at the
droplet surface (on the gas side) are independent of time t for this simplified
validation model. Thus, the droplet diameter is
D 2 = D 2 0 − k t , (4.66)
where D 0 is the initial droplet diameter at t = 0 and k the vaporization rate
[244, 443, 461]. Equation (4.66) is known as the D² law. It is used only within
this section (Chap. 4.6) for validation purposes but not for any further scientific
studies. Equation (4.67) further clarifies the definition of k:
k =− d (D 2 )
dt
=−2ḊD = 2ṁ H 2 O
= const. (4.67)
ρ l πR(t )
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