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

4.6 Model Validation
utilized model accounts for gradients in the r -direction, transient effects of
droplet heating can be resolved correctly on an appropriate scale. The liquid
phase model resembles the so-called “conduction limit” or “diffusion limit”
model due to Law and Sirignano [236], which is recommended if the droplet
Reynolds number is negligible compared to unity [9, 236, 381, 402, 403]. Comparing
the thermal diffusivities α of the liquid and gas phase yields α l ≪ α g .
Thus, the transport processes in the gas phase happen much faster than in the
liquid phase, and the gas phase may be treated as quasi-steady.
Figures 4.5 and 4.6 clarify the temperature sensitivity of the utilized single
droplet model. In particular, Figure 4.5 shows the temperature distribution
inside a water droplet during the initial, transient heating with T l,0 = 283.15 K
and T ∞ = 473.15 K. At t = 1×10 −3 s, the droplet surface temperature T S is highest
due to the initialization of the computational domain. However, it decreases
immediately, before it starts to increase continuously, while accounting
for the heat flux from the gas phase into the droplet. In Figure 4.6, water
droplets are vaporized in an ambient gas environment at T ∞ = 473 K with different
initial temperatures of the liquid phase (283, 303, 323, 343, and 363 K).
The respective wet-bulb temperatures are reached after 3 to 10% of the droplet
lifetime, depending on the temperature difference between T l,0 and T ∞ . All
wet-bulb temperatures match with a relative humidity on the droplet surface
in the range of ϕ= 10%. Hence, the heat transfer inside the droplet as well as
heat and mass transfer over the gas-liquid interface show a correct and consistent
behavior. As far as the experiments on droplet array combustion (Chap. 3)
are concerned, the heating phase of the droplets is significant and cannot be
neglected. It must be included in the calculation of the pre-vaporization rate
Ψ (see also Tabs. D.1 through D.3).
4.6.1 Validation of Vaporization Mechanism Without Combustion
Since the calculated vaporization rate k is a further indicator of the accuracy of
the simulation results, it is evaluated for water and n-decane droplets against
the classical D² law as well as experimental measurements. Ansatz, assumptions,
and resulting equations are discussed here. In order to facilitate the
comparison of the simulation results with the simplified validation model, the
vaporization process is studied as an example for droplets in an atmosphere
of air.
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