@TechReport{PereiraFach:2021:ShFlFo,
author = "Pereira, Lucas Soares and Fachini Filho, Fernando",
title = "Shvab-Zel’dovich and Flamelet formulations applied on quasi-steady
droplet combustion with soot formation and radiative heat
transfer",
institution = "Instituto Nacional de Pesquisas Espaciais",
year = "2021",
type = "RPQ",
address = "S{\~a}o Jos{\'e} dos Campos",
note = "{Bolsa PIBIC/INPE/CNPq.}",
keywords = "droplet combustion, Shvab-Zel’dovich, soot formation,
combust{\~a}o de got{\'{\i}}cula, forma{\c{c}}{\~a}o de
fuligem.",
abstract = "The present work, that started in August 2019, studies
theoretically the quasi-steady combustion of an isolated droplet
with the formation of soot. For this, an analysis was made on the
conservation equations to determine the characteristic spatial and
temporal scales of the problem, which were used for the
nondimensionalization of those equation. The problem has spherical
symmetry, which allows for a one-dimensional analysis of the
problem. The quasi-steady combustion regime is justified by the
fact that the thermal inertia of the gas phase close to the
droplet is much less than that of the liquid phase, so the
environment adapts thermically much faster than the droplet. The
boiling temperature is considered for the whole droplet, i.e., all
heat transferred to the droplet is used for the phase change
(vaporization). It was admitted that the chemical process occurs
at the Burke-Schumann limit, thus the reaction rate is infinitely
fast which leads to infinitely thin flame. To solve the system of
governing differential equations, the Shvab-Zeldovich formulation
was used, which eliminates the dependence of the chemical reaction
term, which is non-linear. Therefore, the mass fraction of species
and the temperature field are described by the mixture fraction,
Z, and excess enthalpy, H, equations. The boundary conditions were
imposed at the surface of the droplet and in a region far from it.
To describe the formation of soot in the problem, a simplified
mathematical model was adopted. The resulting system of second
order differential equations allow to be integrated analytically
once, and the final system of first order differential equations
is integrated numerically. The transport properties were
considered as constant, the that allowed us to find analytical
solutions for the system of differential equations resulting. The
next step will be to implement a computational code to solve the
system of nonlinear algebraic equations. RESUMO: O presente
trabalho, que foi iniciado em Agosto de 2019, estuda teoricamente
a combust{\~a}o no regime quase-estacion{\'a}rio de uma gota
isolada com a forma{\c{c}}{\~a}o de fuligem. Para isso, foi
feita uma an{\'a}lise das equa{\c{c}}{\~o}es de
conserva{\c{c}}{\~a}o para determinar as escalas espaciais e
temporais caracter{\'{\i}}sticas do problema, as quais foram
utilizadas para a adimensionaliza{\c{c}}{\~a}o dessas
equa{\c{c}}{\~o}es. O problema tem simetria esf{\'e}rica, o que
permite uma an{\'a}lise unidimensional. O regime de
combust{\~a}o quase-estacion{\'a}rio {\'e} justificado pelo
fato de que a in{\'e}rcia t{\'e}rmica da fase gasosa
pr{\'o}xima {\`a} gota {\'e} muito menor que a da fase
l{\'{\i}}quida, de modo que o ambiente se adapta muito mais
rapidamente que a gota. A temperatura de ebuli{\c{c}}{\~a}o foi
considerada para toda a gota, isto {\'e}, todo o calor
transferido para ela {\'e} usado para a mudan{\c{c}}a de fase
(vaporiza{\c{c}}{\~a}o). Foi admitido que o processo
qu{\'{\i}}mico ocorre no limite de Burke-Schumann, portanto a
taxa de rea{\c{c}}{\~a}o {\'e} infinitamente r{\'a}pida, o que
leva a chamas infinitamente finas. Para resolver o sistema de
equa{\c{c}}{\~o}es diferenciais governantes, foi utilizada a
formula{\c{c}}{\~a}o de Shvab-Zeldovich, que elimina a
depend{\^e}ncia do termo de rea{\c{c}}{\~a}o qu{\'{\i}}mica,
que n{\~a}o {\'e} linear. Portanto, a fra{\c{c}}{\~a}o de
massa das esp{\'e}cies e o campo de temperatura s{\~a}o
descritos pelas equa{\c{c}}{\~o}es da fra{\c{c}}{\~a}o da
mistura, Z, e de excesso de entalpia, H. As condi{\c{c}}{\~o}es
de contorno foram impostas na superf{\'{\i}}cie da gota e em uma
regi{\~a}o distante da mesma. Para descrever a
forma{\c{c}}{\~a}o de fuligem no problema, foi adotado um modelo
matem{\'a}tico simplificado. O sistema resultante de
equa{\c{c}}{\~o}es diferenciais de segunda ordem permite uma
integra{\c{c}}{\~a}o anal{\'{\i}}tica, e o sistema final de
equa{\c{c}}{\~o}es diferenciais de primeira ordem {\'e}
integrado numericamente. As propriedades de transporte foram
consideradas como constantes, o que permitiu encontrar
solu{\c{c}}{\~o}es anal{\'{\i}}ticas para o sistema de
equa{\c{c}}{\~o}es diferenciais resultante. O pr{\'o}ximo passo
ser{\'a} implementar um c{\'o}digo computacional para resolver o
sistema de equa{\c{c}}{\~o}es alg{\'e}bricas n{\~a}o
lineares.",
affiliation = "{Universidade Federal de Itajub{\'a} (UNIFEI)} and {Instituto
Nacional de Pesquisas Espaciais (INPE)}",
language = "en",
pages = "36",
ibi = "8JMKD3MGP3W34T/458SMQL",
url = "http://urlib.net/ibi/8JMKD3MGP3W34T/458SMQL",
targetfile = "Relatorio_Final_PIBIC_2020_2021_Lucas_Soares_Pereira.pdf",
urlaccessdate = "04 jun. 2024"
}