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

2 Combustion Theory
Moreover, experiments at microgravity provide an opportunity to merge theories
and reality for a better understanding of the fundamental combustion
phenomena involved [153, 219, 222, 223].
Godsave [152] and Spalding [415] first developed a basic, spherically symmetric
model for an isolated single component droplet in a stagnant environment.
The model is well-known as “D² law” and predicts a linear relationship between
droplet surface area and time. Quasi-steadiness 7 is assumed and temperature
inside of the droplet is taken to be uniform and constant at the wetbulb
temperature. In addition, constant values are employed for the thermophysical
properties as well as a unity Lewis number (Le = 1). In order to include
the effect of liquid-phase heating, Law [231] modified the D² law. While
the gas phase is taken to be spherically symmetric and quasi-steady (as in the
D² law before), the droplet temperature is assumed to be spatially uniform but
temporally varying. Results of this extended model showed that droplet heating
is a significant source of unsteadiness and should be incorporated into
any realistic model of droplet vaporization and combustion [231, 350]. In this
context, Sirignano [401] illustrates that the assumption of a spatially uniform
temperature within the droplet requires an infinite thermal conductivity in
the associated liquid phase. Hence, this model is denoted as the “infiniteconductivity
model”. A further advancement of the D² law is the “conductionlimit
model” due to Law and Sirignano [236]. In this third model, molecular
diffusion exclusively controls the nonuniform temperature field in the liquid
phase, implying that liquid circulation within the droplet is negligible. The
droplet surface temperature is regarded as uniform. As summarized by Renksizbulut
et al. [350], this is reasonable for a nonconvective environment and
represents the slowest heat transfer limit. In fact, Law and Sirignano [236]
showed that transient heating dominates the first 10 to 20% of the droplet life
time. Additionally, Aggarwal et al. [9] conducted a comparative study on the
effect of the different liquid-phase models. It shows that the selection of the
particular heating model is substantial for the liquid phase temperature. The
infinite-conductivity and conduction-limit models are commonly considered
as two limiting cases bounding the possible range of real conditions [4, 350].
Still, droplet vaporization occurs in a convective environment in almost all
7 Despite the fact that a fuel droplet may not attain steady-state evaporation/vaporization during its lifetime, it
is often convenient to consider a quasi-steady gas phase. This means that the process can be described as if it
were in steady state at any instant of time. This assumption eliminates the need to deal with partial differential
equations and still provides a reasonable level of accuracy [244, 341, 443].
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