UNIVERSIDADE FEDERAL DE SANTA CATARINA
PROGRAMA DE PÓS-GRADUAÇÃO EM
ENGENHARIA MECÂNICA
Herlon de Faveri Linemburg
ZERO-DIMENSIONAL MODEL WITH A WIEBE
FUNCTION AND SHIFTING CHEMICAL
EQUILIBRIUM FOR SPARK IGNITED
COMBUSTION ENGINES
Florianópolis
2017
Herlon de Faveri Linemburg
ZERO-DIMENSIONAL MODEL WITH A WIEBE
FUNCTION AND SHIFTING CHEMICAL
EQUILIBRIUM FOR SPARK IGNITED
COMBUSTION ENGINES
Dissertação submetida ao Programa de Pós-Graduação em Engenharia Mecânica da Universidade Federal de Santa Catarina para a obtenção do Grau de Mestre em Engenharia Mecânica. Orientador:
Prof. Amir Antônio Martins de Oliveira Jr, Ph.D.
Coorientador:
Prof. Rafael de Camargo Catapan, Dr.Eng.
Florianópolis
2017
Herlon de Faveri Linemburg
ZERO-DIMENSIONAL MODEL WITH A WIEBE
FUNCTION AND SHIFTING CHEMICAL
EQUILIBRIUM FOR SPARK IGNITED
COMBUSTION ENGINES
Esta dissertação foi julgada adequada para a obtenção do título de Mestre em Engenharia Mecânica, especialidade Engenharia e Ciências Térmicas e aprovada em sua forma final pelo Programa de Pós-Graduação em Engenharia Mecânica.
Florianópolis, Outubro de 2017
Prof. Jonny Carlos da Silva, Dr. Eng.
Coordenador do Programa de Pós-Graduação em Eng. Mecânica
Banca examinadora:
Prof. Amir Antônio Martins de Oliveira Jr, Ph.D. Orientador
-UFSC
Prof. Rafael de Camargo Catapan, Dr. Eng. Coorientador
-UFSC
Prof. Leonel Rincón Cancino, Dr.Eng.- UFSC
Prof. Mario Eduardo Santos Martins, Ph.D. - UFSM
Ao meu avô, Alcinio de Faveri (in memorian) e à minha prima Sileir V. Soares (in memorian), estarão sempre em minhas lembranças.
AGRADECIMENTOS
Primeiramente, e em especial, agradeço aos meus pais, Mª Terez-inha de F. Linemburg e Heriberto Linemburg, pelo apoio e incentivo constantes durante toda minha vivência acadêmica e, sobretudo, pelos valores morais que me ensinaram durante todos esses anos.
Ao meu primo Leonel da Silva Soares pelo incessante e incondi-cional apoio durante todo período deste trabalho.
À toda minha família, em especial ao meu tio, Morivaldo de Faveri, pelos momentos de lazer e descontração que me mantiveram motivado.
Aos meus professores Amir A. M de Oliveira Jr. e Rafael de C. Catapan por todo aprendizado, incentivo e ajuda essencial na realização deste trabalho. Serei eternamente grato.
Não posso deixar de agradecer especialmente meus grandes ami-gos Bruno Backes, Cassiano Tecchio, Edemar Morsch Filho, Lucas F. Possamai e Rafael Goulart pela amizade verdadeira.
À todos meus colegas do LabCET pela convivência maravilhosa, amizade e apoio durante todo o período de pós-graduação. Agradeço em especial os colegas Camilla R. Medeiros, Renzo Figueroa, Daniel Bonin, Carlos A. M. Spohr, Nury A. N. Garzón, Amir R. De Toni Jr., Marcos V, Oro, Flavia B. Artuso, Jônatas Vicente, Thiago Rios e Raul A. Puentes. À UFSC e ao POSMEC pela oportunidade de cursar uma pós-graduação de excelência.
"Todos os esforços tendem a levar a mente humana cada vez mais próxima daquela inteligência, mas que ainda permanecerá sempre infinitamente inatingível" Pierre Simon LaPlace "A teoria do cientista sonhador venceu e vencerá sempre o imediatismo grosseiro do ambicioso sem ideal filosófico" Malba Tahan
RESUMO
A performance de um motor flex-fuel foi avaliada a partir de um modelo numérico zero-dimensional de uma zona para motores de combustão in-terna de ignição por centelha desenvolvido no presente trabalho. Neste modelo, o cálculo da energia liberada durante o processo de combustão foi implementado utilizando a função de Wiebe junto com a hipótese de equi-líbrio químico a cada instante de tempo durante o intervalo da combustão. Este método de cálculo difere do uso comum do poder calorífico inferior do combustível que superestima as temperaturas do processo. Além disso, o modelo compreende os fenômenos físicos de transferência de calor e vazões mássicas de admissão e exaustão, permitindo assim simulações de um ciclo completo do motor. As análises feitas foram baseadas em um motor CFR monocilíndrico operando em regime permanente a 900 rotações por minuto com carga fixa de 330 kP a de pressão média efetiva indicada líquida e abastecido com misturas de gasolina e etanol. Ambas funções de Wiebe simples e dupla para foram utilizadas para representar a fração mássica de combustível queimada. Parâmetros como pressão de pico, consumo específico de combustível e eficiência da conversão de combustível foram avaliados perante variações da concentração de etanol, razão de compressão e razão de equivalência. As concentrações de CO e CO2 ao fim do processo de combustão também foram avaliadas. A comparação com os resultados experimentais indicou uma redução na diferença de pressão de pico de 164.4 para 112 kP a no caso da Wiebe simples e de 17.00 para 10.42 no caso da Wiebe dupla. O aumento na quantidade de etanol resultou num aumento do consumo específico de combustível como também na eficiência de conversão de combustível. A concentração de CO2ao final da combustão aumentou com a quantidade de etanol e a de CO diminuiu. Com respeito ao aumento na razão de compressão, um menor consumo de combustível foi alcançado e também uma maior eficiência na conversão de combustível. A concentração de CO2 ao fim da combustão aumentou com a razão de compressão e a de CO diminuiu. No caso de razão de equivalência, para maiores valores desse parâmetro, um maior consumo de combustível foi observado e uma menor eficiência na conversão de combustível também. A concentração de CO2 ao final da combustão diminuiu e a de CO aumentou com o aumento da razão de equivalência. Além do mais, o comportamento do motor perante variações na concentração de etanol, razão de compressão e razão de equivalência se mostrou coerente com a literatura revisada. A nova abordagem sugerida utilizando equilíbrio químico apresentou-se apropriada para o cálculo da energia liberada durante o processo de combustão para as condições de operação testadas.
Palavras-chave: Motor de combustão interna ; etanol; simulação;
MODELO ZERO-DIMENSIONAL COM UMA
FUNÇÃO DE WIEBE E EQUILÍBRIO QUÍMICO
PARA MOTORES DE COMBUSTÃO DE IGNIÇÃO
POR CENTELHA
Introdução: As mudanças climáticas globais provenientes de causas
antropogênicas ocorrem devido ao acúmulo de CO2na atmosfera origi-nado do consumo de derivados do petróleo. Em 2015, o setor de transportes e industrial representaram, respectivamente, 32.2% e 32.5% da energia total consumida no Brasil. O consumo de derivados do petróleo nesses setores representa respectivamente, 56.7% e 9.8% [1]. Devido ao expressivo consumo de derivados do petróleo, o setor de transportes e industrial emitiram para a atmosfera 204x109 kg e 171x109 kg de CO
2, respecti-vamente [2]. De 2003 a 2016, a substituição dos derivados de petróleo por biomassa, como o etanol, evitaram a emissão de quase 352x109 kg de CO2 no Brasil. Esse padrão pode ser ampliado para outros modais de transportes e para outros biocombustíveis, desde que critérios de sustentabilidade sejam satisfeitos. Motores de combustão interna (MCI) compreendem um papel vital nesta discussão já são amplamente utilizados nestes setores. Esses dispositivos são conhecidos pelas baixas eficiências e substanciais emissões de poluentes o que ocasiona prejuízos ambientais e desvantagens econômicas devido ao alto gasto energético. Para que se amenize esses problemas, o uso de biocombustíveis, como o etanol, vem sendo adotado como alternativa aos derivados do petróleo [3]. Tais mudanças de combustível requerem o estudo de métodos para testar, analisar e otimizar MCI de maneira eficiente. Essas tarefas são usual-mente realizadas por modelos numéricos. Tais modelos são amplausual-mente empregados no projeto e análise de MCI devido à facilidade de uso e baixo custo se comparados às medições em testes de bancada. Diversos parâmetros podem ser alterados para simulação sem que haja a necessi-dade de se modificar o motor real, como por exemplo diferentes misturas de combustível, características geométricas, rotação, atraso de ignição, carga, ângulos de abertura e fechamento de válvulas e etc [4]. Uma vasta diversidade de modelos computacionais para MCI são conhecidos da liter-atura variando, por exemplo, de 0D até 3D, e de uma zona até múltiplas zonas. A escolha do tipo de modelo depende do detalhamento requerido na análise. Neste trabalho um modelo 0D de uma zona é elaborado para simulações do ciclo do motor. São englobados pelo modelo os processos físicos de admissão, exaustão, combustão e transferência de calor visando, ao final da simulação, a determinação de parâmetros de performance para uma condição de operação específica. A metodologia utilizada para o cálculo da energia liberada durante a combustão utiliza o equilíbrio
químico acoplado com uma função de Wiebe. Esta abordagem permite diversas analises com diferentes misturas de combustíveis e razões de equivalência.
Objetivo:O objetivo principal desse trabalho é o desenvolvimento de
um modelo numérico 0D de uma zona orientado à avaliação geral da performance indicada de um motor de ignição por centelha utilizando misturas de etanol e gasolina.
Metodologia:O modelo 0D de uma zona foi desenvolvido em três seções,
geométrica, física e química. A seção geométrica leva em conta as medidas dos componentes internos ao motor e suas relações trigonométricas, a física envolve o cálculo das vazões mássicas de admissão e exaustão, transferência de calor e a conservação de energia no volume de controle. A seção química abrange os cálculos para determinação do mapeamento das espécies químicas durante o processo de combustão de modo que a energia liberada na queima do combustível seja determinada. As simulações feitas foram baseadas num caso da literatura visto em Yelianna [4], no qual um motor monocilíndrico abastecido com misturas de etanol e gasolina opera em regime permanente a 900 RPM com uma carga constante de 330 kP a de pressão média efetiva indicada líquida. Uma comparação da precisão entre uma função de Wiebe simples e dupla é feita. No caso da Wiebe simples cinco diferentes composições de combustível foram utilizadas, E0 (ou gasolina pura), E20 (mistura com 20% de etanol), E40, E60 e E84. Já para a Wiebe dupla apenas E84 foi relatado. As simulações realizadas aqui visam a reprodução das curas de pressão obtidas em Yelianna [4] como também a análise do comportamento do motor quando submetido a variações na razão de compressão e de equivalência.
Resultados: Mesmo com uma relativa falta de dados para todos os
fenômenos físicos ao longo de um ciclo completo as simulações foram realizadas e com o ajuste da carga baseando-se no formato das curvas de pressão a condição operacional de [4] foi reproduzida com sucesso. Utilizando esta nova abordagem para o cálculo da energia liberada na queima do combustível houve uma minimização dos erros na pressão de pico. Fazendo-se uma comparação entre as curvas experimentais e simuladas, nos casos utilizando Wiebe simples o erro máximo na curva de pressão foi de 5.8% (equivalente a 112.00 kP a) e no caso da Wiebe dupla, 6.6% (equivalente a 100.00 kP a). Os perfis de outros parâmetros
importantes, como temperatura, calor transferido, vazões mássicas estão de acordo com a literatura revisada. O comportamento do motor sob diferentes concentrações de etanol no combustível foi testado, quanto maior a presença de etanol maior se tornam a pressão de pico, consumo específico de combustível e eficiência de conversão de combustível. Uma combustão mais completa também foi observada nestes casos onde se verificou maiores frações de CO2 e menores de CO ao fim do processo de combustão. Variações na razão de compressão e equivalência foram impostas ao motor para uma análise qualitativa, já que a mudança desses parâmetros muda o comportamento da combustão, requerendo assim uma nova função de Wiebe. Como esperado maiores raz es de compressão alcançaram um consumo específico mais baixo e uma melhor eficiência na conversão de combustível para todos os combustíveis testados. Além do mais, baseando-se nas frações de CO e CO2uma combustão mais completa também foi observada. No caso da razão de equivalência diversos fatores influenciam na escolha de um valor ótimo, então para a dada condição de operação e amplitude de variação deste parâmetro, um maior consumo acompanhado de uma menor eficiência de conversão foram observados com o aumento deste. Para altas razões de equivalência também observou-se uma combustão mais incompleta de acordo com as frações de CO e CO2 ao final da queima. A abordagem sugerida para o cálculo da energia liberada pela queima do combustível baseada numa função de Wiebe acoplada com equilíbrio químico se mostrou apropriada para reproduzir o comportamento do motor na condição de operação estudada.
ZERO-DIMENSIONAL MODEL WITH A WIEBE
FUNCTION AND SHIFTING CHEMICAL
EQUILIBRIUM FOR SPARK IGNITED
COMBUSTION ENGINES
ABSTRACT
The performance of a flex-fuel engine was evaluated by a zero-dimensional single-zone model for spark ignited internal combustion engines devel-oped in the present work. In this model, the combustion energy release calculation was implemented utilizing a Wiebe function with the chemical equilibrium hypothesis in each time step during the combustion interval. This method differs from the traditional use of the fuel lower heating value, which overestimates the temperature of the process. Besides that, the model comprehends the physical phenomena of heat transfer and intake and exhaust mass flows, allowing complete cycle simulations to be performed. The analyses were based on a CFR single cylinder engine operating in steady state condition at 900 RPM with a constant load of 330 kP a of net indicated mean effective pressure (nimep) fueled with gasoline-ethanol blends. Both single and double Wiebe functions were used to represent the burned mass fraction. Parameters such as peak pressure, specific fuel consumption and fuel conversion efficiency were evaluated for variations of ethanol concentration, compression ratio and equivalence ratio. The concentrations of CO and CO2 at the end of the combustion were also examined. Comparison to measurements indicated peak pressure difference reduction from 164.40 to 112.00 kP a in the single Wiebe case and from 17.00 to 10.42 kP a in the double Wiebe case. The increase in the ethanol concentration resulted in higher fuel consumption but also a higher fuel conversion efficiency. The CO2 concentration at the end of combustion increased with the ethanol content and the CO decreased. Regarding the increase in the compression ratio, a lower fuel consumption was achieved and also, a higher fuel conversion efficiency occurred. The CO2concentration at the end of the combustion increased with the compression ratio and the CO decreased. In the case of the equivalence ratio, for higher values of the parameter, a higher fuel con-sumption was observed and a lower fuel conversion efficiency too. The CO2 concentration at the end of the combustion decreased and the CO increased with the increase in the equivalence ratio. The behavior of the engine for ethanol content, compression ratio and equivalence ratio variations was coherent with the reviewed literature. The new suggested approach using chemical equilibrium came out to be appropriate to the combustion analyses for the tested operational conditions.
Keywords: Internal combustion engine; ethanol; simulation; chemical
LIST OF FIGURES
Figure 1 – Burned mass fraction as a function of the crank angle calculated using a single Wiebe function for a=2; 6 and 9 (b = 1.08, θCS= 350◦, ∆θ = 33.1◦and f = 2.0). 37
Figure 2 – Burned mass fraction as a function of the crank angle calculated using a single Wiebe function for f = 0.5; 1 and 2 (b = 1.08, θCS= 350◦, ∆θ = 33.1◦and a = 6.0). 38
Figure 3 – Variation of BMF as a function of the crank angle calculated using a single Wiebe function for a=2; 6 and 9 (b = 1.08, θCS= 350◦, ∆θ = 33.1◦and f = 2.0). 39
Figure 4 – Variation of BMF as a function of the crank angle calculated using a single Wiebe function for f = 0.5; 1 and 2 (b = 1.08, θCS= 350◦, ∆θ = 33.1◦and a = 6.0). 40
Figure 5 – Engine geometrical parameters. . . 46 Figure 6 – Schematics of valve lifts as a function of the crank angle. 48 Figure 7 – Combustion chamber of the engine representing the
control volume where the conservation equations will actuate. . . 49 Figure 8 – Schematics representing the gas flow through a nozzle
analog to the engine valve ports restriction. . . 50 Figure 9 – Heat exchanger surfaces represented by the cylinder
metallic media. . . 56 Figure 10 – Equivalent thermal resistances circuit. . . 57 Figure 11 – Equilibrium process scheme. . . 59 Figure 12 – States change during the evolution of the combustion
process in one time step. . . 62 Figure 13 – Schematic of ICE P-V diagram emphasizing the gross
indicated work (WG) and the pump work (WP). . . . 63
Figure 14 – Convergence of the variables mg, pg and Tg along the
engine cycles during simulation (E84, double-Wiebe, rc = 8, speed = 900 RPM, SA = 6◦ BTDC, nimep =
330 kP a, φ = 1). . . 68 Figure 15 – Blocks diagram of algorithm organization. . . 69 Figure 16 – Intake ( ˙mi) and exhaust ( ˙me) mass flow rates as a
function of the crank angle (E84, rc = 8, speed = 900
Figure 17 – Mean flow velocities of the intake (vi) and exhaust (ve)
valve passages as a function of the crank angle (E84, rc = 8, speed = 900 RPM, SA = 6◦ BTDC, nimep =
330 kP a, φ = 1). . . 74 Figure 18 – Mach number of the mass flow of the inlet (Mi) and
exhaust (Me) valve passages as a function of the crank
angle (E84, rc= 8, speed = 900 RPM, SA = 6◦BTDC,
nimep= 330 kP a, φ = 1). . . 75 Figure 19 – Mass of gas in the cylinder as a function of the crank
angle (E84, rc= 8, speed = 900 RPM, SA = 6◦BTDC,
nimep= 330 kP a, φ = 1). . . 76 Figure 20 – Combustion energy release as a function of the crank
angle (E84, rc= 8, speed = 900 RPM, SA = 6◦BTDC,
nimep= 330 kP a, φ = 1). . . 77 Figure 21 – Pressure in the combustion chamber as a function of
the crank angle (E84, rc = 8, speed = 900 RPM, SA
= 6◦ BTDC, nimep = 330 kP a, φ = 1). . . 78 Figure 22 – Temperature in the combustion chamber as a function
of the crank angle (E84, rc = 8, speed = 900 RPM,
SA = 6◦ BTDC, nimep = 330 kP a, φ = 1). . . 79 Figure 23 – Cylinder heat transfer rate as a function of the crank
angle (E84, rc= 8, speed = 900 RPM, SA = 6◦BTDC,
nimep= 330 kP a, φ = 1). . . 80 Figure 24 – Schematic of ICE P-V diagram (E84, rc = 8, speed =
900 RPM, SA = 6◦ BTDC, nimep = 330 kP a, φ = 1). 81 Figure 25 – Pressure traces comparison between the experimental
data and the LHV and shifting equilibrium approach (E84, rc = 8, speed = 900 RPM, SA = 6◦ BTDC,
nimep= 330 kP a, φ = 1). . . 82 Figure 26 – Comparison of the BMF variation predicted by the
double-Wiebe function and the experiment [4] (E84, rc = 8, speed = 900 RPM, SA = 6◦ BTDC, nimep =
330 kP a, φ = 1). . . 83 Figure 27 – Comparison of the measured and predicted pressure
traces for the combustion period for E0, E20, E40 and E60 blends (rc = 8, speed = 900 RPM, SA = 10◦
BTDC, nimep = 330 kP a, φ = 1). . . 85 Figure 28 – Experimental and single-Wiebe results for the peak
pressure inside the cylinder for E84 (rc = 8, speed =
Figure 29 – Comparison of the BMF variation between the experi-ment data and the single-Wiebe function adjusted (E0, rc = 8, speed = 900 RPM, SA = 10◦ BTDC, nimep
= 330 kP a, φ = 1). . . 87 Figure 30 – Specific fuel consumption as a function of the
compres-sion ratio (speed = 900 RPM, SA = 10◦BTDC, nimep = 330 kP a, φ = 1). . . 88 Figure 31 – Volumetric efficiency as a function of the compression
ratio (speed = 900 RPM, SA = 10◦ BTDC, nimep = 330 kP a, φ = 1). . . 89 Figure 32 – Fuel conversion efficiency as a function of the
compres-sion ratio (speed = 900 RPM, SA = 10◦BTDC, nimep = 330 kP a, φ = 1). . . 90 Figure 33 – Ratio between FCE and standard Otto cycle efficiency
(speed = 900 RPM, SA = 10◦ BTDC, nimep = 330
kP a, φ = 1). . . 91 Figure 34 – CO molar fraction at the end of the combustion phase
as a function of the compression ratio (speed = 900 RPM, SA = 10◦BTDC, nimep = 330 kP a, φ = 1). . 92 Figure 35 – CO2molar fraction at the end of the combustion phase
as a function of the compression ratio (speed = 900 RPM, SA = 10◦BTDC, nimep = 330 kP a, φ = 1). . 93 Figure 36 – O2 molar fraction at the end of the combustion phase
as a function of the compression ratio (speed = 900 RPM, SA = 10◦BTDC, nimep = 330 kP a, φ = 1). . 94 Figure 37 – H2molar fraction at the end of the combustion phase
as a function of the compression ratio (speed = 900 RPM, SA = 10◦BTDC, nimep = 330 kP a, φ = 1). . 95 Figure 38 – Specific fuel consumption as a function of the
equiv-alence ratio (rc = 8, speed = 900 RPM, SA = 10◦
BTDC, nimep = 330 kP a). . . 96 Figure 39 – Specific fuel consumption as a function of the
equiv-alence ratio (rc = 8, speed = 900 RPM, SA = 10◦
BTDC, nimep = 330 kP a). . . 97 Figure 40 – Fuel conversion efficiency as a function of the
equiv-alence ratio (rc = 8, speed = 900 RPM, SA = 10◦
BTDC, nimep = 330 kP a). . . 98 Figure 41 – CO molar fraction at the end of the combustion phase
as a function of the equivalence ratio (rc = 8, speed =
Figure 42 – CO2molar fraction at the end of the combustion phase as a function of the equivalence ratio (rc = 8, speed =
LIST OF TABLES
Table 1 – CFR engine data. . . 41 Table 2 – Single-Wiebe function parameters for each fuel blend. . 41 Table 3 – Double Wiebe function parameters for E84. . . 42 Table 4 – Relevant literature works. . . 43 Table 5 – Engine dimensions and fixed operation parameters. . . 71 Table 6 – Constants for the physical model. . . 72 Table 7 – Indicated performance parameters for double-Wiebe
E84 case. . . 81 Table 8 – Properties of each ethanol-gasoline blends . . . 84 Table 9 – Parameters comparison between predicted and
measure-ments. . . 84 Table 10 – Chemical species utilized in the chemical equilibrium
NOMENCLATURE
Acronyms
ABDC After bottom dead center ATDC After top dead center BBDC Before bottom dead center BC Before combustion
BDC Bottom dead center BMF Burned mass fraction BTDC Before top dead center CE Combustion end
CFR Cooperative Fuel Research CI Compression ignited CS Combustion start
ICE Internal combustion engine
LabCET Laboratório de combustão e engenharia de sistemas térmicos LHV Lower heating value
LSM Least Square Method
ODE Ordinary Differential Equation RPM Rotations per minute
SA Spark advance
SFC Specific fuel consumption SI Spark ignited
TDC Top dead center
UFSC Universidade Federal de Santa Catarina
Greek alphabet
α Crank radius [m]
β Double-Wiebe weighting factor [–]
∆θ0−90% Angle interval to burn 90% of the fuel [°] ∆hvap Latent heat of vaporization [J/kg]
∆θ Combustion duration [°]
˙θ Engine rotational speed [°/s]
ηf Fuel conversion efficiency [–]
ηv Volumetric efficiency [–]
ηOtto Otto cycle efficiency [–]
γ Ratio of the specific heats [–]
µ Gas viscosity [Pa · s]
φ Equivalence ratio [–]
ρ Density [kg/m3]
θ Crank angle [°]
Roman alphabet
˙m Mass flow rate [kg/s]
˙Q Lost heat [W]
˙Srf Combustion generation therm [W]
˙
W Flow power [W]
ReB Reynolds number [–]
Sp Mean piston speed [m/s]
A Internal cylinder total area [m2]
a Combustion duration related Wiebe parameter [–]
aa Annand constant [–]
Ae Exhaust valve port area [m2]
Ai Inlet valve port area [m2]
Av Valve flow-through area [m2]
Av Valve port area [m2]
B Bore [m]
b Wiebe function correction factor [–]
ba Annand constant [–]
cp Specific heat at constant pressure [J/(kg K)]
cv Specific heat at constant volume [J/(kg K)]
D Valve diameter [m]
EV C Exhaust valve closing angle [°]
f Wiebe shape factor [–]
h Specific enthalpy [J/kg]
h∗ Convective heat transfer coefficient [W/m2K]
IV C Inlet valve closing angle [°]
IV O Inlet valve opening angle [°]
J Generic time dependent variable [–]
k Thermal conductivity [W/(m K)]
L Stroke [m]
l Wall thickness [m]
le Exhaust valve lift [m]
li Inlet valve lift [m]
M Mach number [–]
m Mass [kg]
M EL Maximum exhaust valve lift [m]
M IL Maximum inlet valve lift [m]
N Engine rotational speed [s−1]
nimep Net indicated mean effective pressure [Pa]
N uB Nusselt number [–]
p Pressure [Pa]
p∗ Critical pressure [Pa]
R Universal gas constant [J/(kg K)]
rc Compression ratio [–]
Req Equivalent thermal resistance [m2K/watt]
s Rod length [m]
ss Sound speed [m/s]
T Temperature [K]
u Gas internal energy [J/kg]
V Volume [m3]
v Speed [m/s]
X Vector of species mass fractions [–]
xb Burned mass fraction [–]
z Horizontal coordinate [m]
FCE Fuel conversion efficiency [–]
sfc Specific fuel consumption [kg/J]
Subscripts
0 Initial or reference state accu Accumulated air Air b Burned c Clearance ch Cylinder head cw Cylinder Walls d Displacement e Exhaust f Fuel G Gross g Gas i Inlet lub Lubricant mix Mixture
n Previous time step n+ 1 Current time step N I Net indicated P Pump p Piston prev Previous prod Products reac Reactants w Water
CONTENTS
1 INTRODUCTION . . . . 31 1.1 Motivation . . . 31 1.2 Objectives . . . 32 1.2.1 Overall objective . . . 32 1.2.2 Specific objectives . . . 32 1.3 Structure of the thesis . . . 322 LITERATURE REVIEW . . . . 35 2.1 Introduction . . . 35 2.2 Combustion models . . . 35 2.3 Wiebe function . . . 36 2.3.1 Historical perspective . . . 36 2.3.2 Wiebe equation . . . 36 2.4 Relevant literature works . . . 40
3 ENGINE MODEL . . . . 45
3.1 Introduction . . . 45 3.2 Geometrical Model . . . 46 3.3 Physical Model . . . 48 3.3.1 Conservation of Mass . . . 49 3.3.2 Inlet and Exhaust Mass Flows . . . 50 3.3.3 Energy Conservation . . . 53 3.3.4 Cylinder Heat Transfer . . . 55 3.4 Chemical Model . . . 58 3.4.1 Chemical Equilibrium . . . 58 3.4.2 Species Mass Balance . . . 59 3.4.3 Combustion Energy Release . . . 61 3.5 Performance Parameters . . . 63
4 SOLUTION METHOD . . . . 65
4.1 Introduction . . . 65 4.2 Method of Solution . . . 65 4.3 Algorithm . . . 69
5 RESULTS AND DISCUSSION . . . . 71
5.1 Introduction . . . 71 5.2 Double-Wiebe Baseline Case . . . 72 5.2.1 Engine operation at baseline . . . 72 5.2.2 Comparison of predictions and measurements . . . 81 5.3 Single-Wiebe Results for E0, E20, E40, E60 and E84 . . . . 83 5.4 Parametric Analysis . . . 87
5.4.1 Effect of the Compression Ratio . . . 87 5.4.2 Effect of Equivalence Ratio . . . 95
6 CONCLUSIONS . . . 101
6.1 Concluding Remarks . . . 101 6.2 Suggestions for future works . . . 102
REFERENCES . . . 105 Appendix A Chemical species utilized in the chemical
31
1 INTRODUCTION
1.1 MOTIVATION
The global climate change from anthropogenic causes occurs due to the CO2 accumulation in the atmosphere originated from consump-tion of fossil fuels, mostly coal and oil derived products. In 2015, the transportation and industrial sector represented 32.2% and 32.5% of the total energy consumption in Brazil, respectively. The consumption of oil products in these sectors represents respectively 56.7% and 9.8% [1]. Due to the massive consumption of oil products, the transportation and indus-trial sector emitted 204x109kg and 171x109kg of CO2to the atmosphere, respectively [2].
From 2003 to 2016, the substitution of oil products by biofuels, such as ethanol, avoided the emission of almost 352x109 kg of CO
2 in Brazil [3]. This pattern can be extended to other modes of transportation and other biofuels, if sustainability criteria are fulfilled.
Internal combustion engines (ICE’s) play a vital role in this scenario since they are largely used. ICE’s are limited by Carnot cycle efficiencies and may emit harmful gases and particulates. Given their large importance in today’s economies, it is vital to explore ways to test, analyze and optimize ICE’s efficiency.
Numerical models are widely employed in the design and analysis of ICE due to the low costs compared to measurements in test benches. A great variety of parameters can be screened using numerical simulation without the need to change the real engine. Parameters to be evaluated include fuels and their blends, geometrical characteristics and different operational conditions as spark advance (SA), engine rotational speed, mechanical load, valve phasing, etc [4].
There is a wide range of engine models varying, e.g., from zero-dimensional (0D) to three-zero-dimensional (3D), and from single-zone to multi-zone to represent the reaction space. The choice of a given model depends on the detail required from the analysis.
Here, a 0D, single-zone model is developed to perform the time simulation of the engine cycle aiming at predicting performance parame-ters for specific operational conditions. The combustion energy release method utilizes chemical equilibrium coupled with a Wiebe function. Such approach enables several analyzes with different fuel blends and equivalence ratios (φ).
32
1.2 OBJECTIVES 1.2.1 Overall objective
This work aims at developing a Zero-Dimensional, single-zone numerical model aiming at evaluating the overall indicated performance of a spark ignited engine fueled by gasoline-ethanol mixtures.
1.2.2 Specific objectives
To accomplish the main objective described above, the following specific objectives are proposed:
• Develop a combustion energy release model based on Wiebe function and chemical equilibrium;
• Implement the calculation of engine performance parameters; • Evaluate the engine performance for different fuel compositions; • Compare the results of the simulations with measurements from the
literature obtained for a CFR engine at constant speed and mean indicated pressure;
• Perform a parametric analysis of the effects of compression ratio and equivalence ratio on engine performance.
1.3 STRUCTURE OF THE THESIS
This manuscript is divided in six chapters. Apart from this introduction, the content of the five following chapters is described below: • Chapter 2: Literature review. This chapter presents a description of the existing engine computational models and their applicability. Also, the Wiebe function and the relevant literature are discussed. • Chapter 3: Engine Model. This chapter is dedicated to the de-scription of the engine model focusing the geometrical, physical and chemical aspects of the operation of the engine.
• Chapter 4: Computational procedure. This chapter is devoted to the solution method, summarizing the system of equations, the integration method used and the algorithm.
• Chapter 5: Results and discussion. This chapter focuses on the presentation and analysis results for the different conditions aiming at validating the model by comparing the predictions with the literature. Followed by a discussion of the engine behavior.
33 • Chapter 6: Conclusion. This chapter lists the main conclusions
35
2 LITERATURE REVIEW
2.1 INTRODUCTION
This chapter covers a literature review on the existing types of engine computational models as well as their characteristics and appli-cability. Furthermore, the Wiebe function is comprehensively discussed regarding its parameters, behavior and utilization to model combustion heat release in SI engines.
2.2 COMBUSTION MODELS
The computational modelling of ICE’s has become a very effective tool for performance evaluation and optimization, as well as for providing detailed information on flow, reaction and heat transfer phenomena [5,6]. Engine models can be generally classified in zero-dimensional (0D) or thermodynamic models, and the more complex multi-dimensional (MD) models, based on computational fluid dynamics (CFD). While 0D models are based on integral forms of the equations for the conservation of mass and energy having only the time as independent variable, MD models are based on spatially resolved governing equations [7].
The models for combustion in spark ignited engines also follow this general framework. Usually, 0D models rely either on empirical mass burn rate models, such as the formulations based on Wiebe functions, or in turbulent front propagation models, where the turbulent flame velocity is obtained from empirical correlations, or in MD models, in which each zone is assumed as a constant mass reactor and combustion follows a defined chemical kinetics mechanism. MD models attempt for more detailed modelling of the reaction region, either using continuous reaction models, or front propagation models [6,8].
Single-zone models consider the thermodynamic state and prop-erties as being uniform throughout the cylinder and those are represented by spatial averaged values. On the other hand, multi-zone models di-vide the combustion chamber in a number of independent constant mass zones whose volumes, composition and temperature change as combustion proceeds [9]. While multi-zone models are more accurate in depicting the effect of chemical kinetics in the engine performance, single-zone models have shown effectiveness and robustness to investigate the com-bustion process and they are often used when there is a need for fast and preliminary analysis of the engine performance [4,10].
36
2.3 WIEBE FUNCTION 2.3.1 Historical perspective
According to the review by Ghojel [11], during the early 1950s researchers in the Union of Socialist Soviet Republics (USSR) were trying to come up with predictive combustion models that would allow more realistic engine cycle simulations. They were convinced that the state of engine technology at the time required a different computational approach to engine design.
In the mid-1950s the studies about chemical kinetics and chain chemical reactions were well established and known, also due to the fact that Semenov and Hinshelwood won the Nobel prize for chemistry in 1956 for their work in chemical reaction kinetics. With this subject emphasized, engine researchers started wondering whether better models could be constructed to characterize engine combustion processes and thus contribute towards the design of better engines.
Ivan Ivanovitch Wiebe was one of the first to attempt linking chain chemical reactions with the fuel reaction rate in ICE’s and his approach was based on the premise that a simple one-step rate equation will not be adequate to describe complex reacting systems such as those occurring in an internal combustion engine. He reasoned that the details of chemical kinetics of all the reactions could be bypassed and a general macroscopic reaction rate expression could be developed based on the concept of chain reactions.
2.3.2 Wiebe equation
The Wiebe function is a semi-empirical expression which repre-sents the fuel burned mass fraction (BMF) as a function of the crank angle (θ) during combustion in an ICE. The BMF is defined as the ratio between the burned and unburned fuel mass inside the chamber. The Wiebe function is characterized by the "S" shape and is adjustable to fit several engine behaviors.
The most common form is the single-Wiebe function expressed as, xb= b " 1 − exp −a θ − θ∆θCS f+1!# (2.1) where xbis the BMF, a is a parameter related to the combustion duration,
θis the current crank angle in degrees, θ0is the combustion start angle, ∆θ is the duration of the combustion in degrees and f is the shape factor. The original single-Wiebe function does not have the parameter b. It is
37 employed by some authors since the natural behavior of the exponential curve tends asymptotically to the value 1 but never reaches it, so this factor is used to correct the range of the function achieving a more complete combustion [4].
Figures 1 and 2 presents the BMF profile during the combustion phase of an ICE, where it is perceptible the "S" shape of the curve. These curves were generated using b = 1.08, θCS = 350◦, ∆θ = 33.1◦ and
variations of the parameters a and f. In figure 1 the dependency of the parameters a and ∆θ is shown. For higher values of a the combustion occurs more abruptly. At the final angles of the combustion the value of xb
varies significantly among the three curves. Figure 2 shows the deformation in the "S" shape due to the variation of f. The larger differences in the xb are observed in the first angles of the combustion.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 345 350 355 360 365 370 375 380 385 a=2 a=6 a=9 B u rn ed M a ss F ra ct io n ( B M F ), xb
Crank Angle, θ[degrees]
Figure 1 – Burned mass fraction as a function of the crank angle calculated using a single Wiebe function for a=2; 6 and 9 (b = 1.08, θCS = 350◦, ∆θ = 33.1◦
38 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 345 350 355 360 365 370 375 380 385 f=2 f=1 f=0.5 B u rn ed M a ss F ra ct io n ( B M F ), xb
Crank Angle, θ[degrees]
Figure 2 – Burned mass fraction as a function of the crank angle calculated using a single Wiebe function for f = 0.5; 1 and 2 (b = 1.08, θCS = 350◦,
∆θ = 33.1◦and a = 6.0).
For the combustion energy release calculation, the derivative of the Wiebe function represents the variation of the BMF with the time and also dictates the rate of the fuel consumption. Figures 3 and 4 shows the shape of the BMF variation for the previous single-Wiebe function. It is observed the same previous pattern, the higher difference in dxb/dt
occurs in the final angles for variations in a, and in the early angles for variations in f.
39 0 500 1000 1500 2000 2500 3000 345 350 355 360 365 370 375 380 385 a=2 a=6 a=9 V a ri a ti o n o f B M F , d xb /d t [1 /s ]
Crank Angle, θ[degrees]
Figure 3 – Variation of BMF as a function of the crank angle calculated using a single Wiebe function for a=2; 6 and 9 (b = 1.08, θCS = 350◦, ∆θ = 33.1◦
40 0 500 1000 1500 2000 2500 3000 345 350 355 360 365 370 375 380 385 f=2 f=1 f=0.5 V a ri a ti o n o f B M F , d xb /d t [1 /s ]
Crank Angle, θ[degrees]
Figure 4 – Variation of BMF as a function of the crank angle calculated using a single Wiebe function for f = 0.5; 1 and 2 (b = 1.08, θCS = 350◦, ∆θ = 33.1◦
and a = 6.0).
A combination of diverse Wiebe functions usually is adopted to represent more accurately the BMF profile. For example, in compression ignited (CI) engines the double-Wiebe function is widely used to replicate the two phase combustion. The first Wiebe models the premixed phase and the second one models the mixing-controlled phase. An usual form of the double-Wiebe function can be expressed as,
xb= β " 1 − exp −a1 θ − θ∆θ 1 1 f1+1!# +(1−β) " 1 − exp −a2 θ − θ∆θ 2 2 f2+1!# (2.2) where β is the weighting factor for the two distinctive combustion phases. 2.4 RELEVANT LITERATURE WORKS
The main reference of this work are the analyzes presented in Yelianna [4] due to the abundance of available data for simulation. In that work, a Cooperative Fuel Reseach (CFR) engine operating at 900 RPM and a load of 330 kP a of net indicated mean effective pressure (nimep) was used in the studies. Table 1 presents the main characteristics of the engine [4].
41 Table 1 – CFR engine data.
Compression Ratio 4.5 - 17.5
Bore (cm) 8.26
Stroke (cm) 11.43
Displacement Volume (cm3) 611.2
Intake Valve Opening (IV O) 10◦ATDC
Intake Valve Closing (IV C) 34◦ABDC
Exhaust Valve Opening (EV O) 40◦BBDC
Exhaust Valve Closing (EV C) 15◦ATDC
Maximum Speed (RP M) 900
Equivalence Ratio 0.4 - 1.4
Gasoline - Ethanol blends E0, E20, E40, E60, E84 Numerical data for the pressure inside the combustion chamber from a dual zone model was compared with the experimental results. The Wiebe function parameters estimation was made using the least square method (LSM) based on the experimental burned mass fraction (BMF) trace.
In the first analysis, simulations using a single Wiebe function were performed with a spark advance (SA) of 10◦ BTDC for all five fuel blends. Table 2 shows the Wiebe function parameters adjusted for each fuel blend [4].
Table 2 – Single-Wiebe function parameters for each fuel blend.
Parameter E0 E20 E40 E60 E84
θCS 350◦ 350◦ 350◦ 350◦ 350◦
a 2.30 2.30 2.30 2.30 2.30
f 2.43 2.57 2.59 2.57 2.46
∆θ0−90% 33.10◦ 31.77◦ 30.91◦ 30.88◦ 31.09◦
b 1.08 1.10 1.08 1.09 1.10
In the second analyze, a double-Wiebe function was employed with a SA of 6◦BTDC. Only numerical data for the E84 composition was provided for this study. Table 3 shows the parameters the double-Wiebe case [4].
42
Table 3 – Double Wiebe function parameters for E84.
θCS 354.00◦ θCE 388.50◦ a 1.00 f1 3.57 f2 1.23 β 0.75 ∆θ1 24.30◦ ∆θ2 16.24◦
Results showed best fit to the BMF profile for the double-Wiebe case, which minimized the error on the model results.
Various literature works were consulted to the development of this engine model and they are discussed below:
• Stockar et al. [12] studied wave propagation dynamics in a single-cylinder ICE’s air path system using lumped-parameter model (LPM) employing an ordinary differential equation (ODE) model. A comparison between experimental and simulated data is presented for the intake and exhaust pressure and mass flows. The results showed agreement of the model in capturing the behavior of this physical phenomena and the presented mass flow curves served as a confirmation for the obtained data simulated here.
• Lounici et al. [13] and Sanli et al. [14] investigated the heat transfer phenomena between gas and in-cylinder walls in a ICE model .A very complete review of the known correlations for estimating the gas convective heat transfer coefficient was presented. The magnitude and behavior of heat released and convective coefficient along the combustion phase were essential to verify the consistency of the outputted data from this work simulations.
• Vanzela et al. [15] assessed blends of ethanol and biodiesel for use in SI ICE’s. Physicochemical properties such as heating value, viscosity, flash point and density were studied. A brief review of the ethanol discussing its properties and effects when employed as a fuel for ICE’s is presented. From their work it is possible to extract information to compare the engine behavior obtained by this work simulations for the various fuel blends.
• Various authors - [16–24] - evaluated fuels blends and their influence on performance parameters such as specific fuel consumption (SFC),
43 fuel conversion efficiency (FCE), emissions of CO, CO2, HC and NOx, variations of compression ratio (rc) and equivalence ratio (φ).
Those parameters analysis were essential to confirm the numerical data from this work.
A summary of the revised literature works is presented in table 4.
Table 4 – Relevant literature works.
Author Methodology Combustionmodel parametersAnalyzed Results Stockar 1 cyl. Single-zone pressure and inlet, exhaust
et al.[12] SI ICE 1D valve flows mass flows
Lounici 1cyl. Single-zone heat transfer convective
et al.[13] Natural gasSI ICE 0D phenomena heat loss ( ˙Q)coef. (hg),
Sanli 1 cyl. SI ICE Experimental heat transfer convective
et al.[14] Gasoline phenomena heat loss ( ˙Q)coef. (hg),
Vanzela mixing eth. Experimental heating value, parameters
et al.[15] + biodiesel viscosity, density, behavior
flash point
Fang [16] 1 cyl. SI ICEeth.+gasol. Experimental CO emissionsSFC and ↑↑eth. ↑ SFCeth. ↓ CO
Fauzan Literature - SFC and ↑eth. ↓ CO
et al.[17] diverse fuelsreview CO emissions ↑eth. ↑ SFC Wechsatol 4 cyl. SI ICE
Experimental SFC and FCE Low load - RPM
et al.[18] eth. + gasol. ↑eth. ↑ SFC
Almeida 4 cyl. SI ICE Experimental SFC, CO, CO2, ↑eth. ↓ CO
et al.[19] eth., H2 + gasol. peak pressureHC, NOx ↑eth. ↑ SFC Erol 4 cyl. SI ICE
Experimental SFC, FCE, CO, ↑eth. ↓ CO
44
Thakur Literature - SFC(rc), FCE(rc), ↑ rc↓SFC
et al.[21] review SFC(φ) and ↑ rc↑FCE
eth. + gasol. flash point ↑ φ ↑SFC
Kumar and 3 cyl. SI ICE
Experimental torque, SFC ↑eth. ↑ FCE Trehan [22] eth.+gasol. and FCE ↑eth. ↑ torque Bharadwaz 1 cyl. SI ICE
Experimental SFC, CO, ↑ rc↓CO
et al.[23] meth.+gasol. HC, NOx
Yuqiang 1 cyl. SI ICE Experimental Pressure, BMF, ↑ φ ↑CO
et al.[24] fuels + gasol SFC(φ), FCE(φ)CO(φ) ↑ φ ↑SFC
↑ φ ↓FCE
Yelianna [4] 1cyl. SI ICE
Experimental Pressure, BMF, pressure eth. + gasol. 1D dual-zoneand Wiebe functioncomparison Wiebe analysisbehavior By studying the revised literature it is difficult to find works that study all the phenomena occurring in the complete engine cycle. Usually they focus on an isolated segment excluding or poorly detailing the other cycle periods. The numerical model developed here encompass the entire cycle simulation. Also, the combustion energy release calculation seen in those works employs the lower heating value approach, different from the present study which perform this task utilizing chemical equilibrium coupled with a Wiebe function.
45
3 ENGINE MODEL
3.1 INTRODUCTION
This chapter presents the governing equations employed in the engine model. In this case, a SI ICE model is developed with the thermo-dynamic properties uniform along the combustion chamber (single-zone) and assumed as time-dependent only (0D). The model is subdivided in three major parts:
• Geometrical Model • Physical Model • Chemical Model
The overall hypothesis of the model are:
1. Valve lifts are considered as a parabolic function of the crank angle; 2. No crevice volume is assumed;
3. No leakage occurs at the cylinder-wall gap (there is no blow-by); 4. The gas mixture is assumed ideal;
5. The specific heats are considered constant; 6. The mass flow rates follow an isentropic process; 7. The valve ports are treated as flow restrictions;
8. The air-fuel mixture is homogeneous and perfectly mixed;
9. There is no variation of kinetic and potential energies of the gas mixture;
10. No other forces of work other than flow work are considered; 11. Cylinder head and piston top are flat surfaces.
46
3.2 GEOMETRICAL MODEL
The geometrical model describes internal components geometry and their relations. Important parameters such as bore, stroke, rod length and crank radius are used to provide the relations for combustion chamber internal volume, superficial area and mean piston speed. Also, the opening area of each valve is determined based on the valve lifts. All of those parameters are functions of the crank angle which relates to the time variable through the engine rotational speed as follows,
˙θ = ∂θ
∂t, (3.1)
where ˙θ is the engine rotational speed in degrees per second (◦/s). It is considered constant for steady state simulation.
Figure 5 shows those geometrical parameters in the one-cylinder scheme, T DC BDC L B li, le s α θ ˙θ Vc Vd
47 where Vc is the clearance volume, L is the stroke, B is the bore, Vdis the
displacement volume, s is the rod length and α is the crank radius. T DC and BDC stands for the top and bottom dead center while li and le are
the inlet and exhaust valve lifts respectively.
The displacement volume (Vd) is by definition the swept volume
when the piston moves between T DC and BDC while the clearance volume (Vc) is the remaining volume when the piston is in the T DC
position. Usually the Vc is provided by engine manufacturer but it can
also be determined through the following relation, Vc=
Vd
rc−1
, (3.2)
where rc is the engine compression ratio.
The total combustion chamber volume and inside superficial area are then defined, respectively [5], as
V = Vch1 + 1/2(rc−1)[s/α + 1 − cos θ − ((s/α)2−(sin θ)2)1/2] i (3.3) and A= Ach+ Ap+ πBL 2 [s/α + 1 − cos θ − ((s/α)2−(sin θ)2)1/2], (3.4) where Ach and Ap are the cylinder head and piston area respectively,
whose are considered to be flat surfaces. Then,
Ach = Ap = πB2/4. (3.5)
Another important parameter that will be necessary is the mean piston speed, which is defined as,
Sp= 2LN, (3.6)
where N is the engine rotational speed in revolutions per second. To calculate the valves flow-through area, it was considered that the cam lobes have a parabolic shape. Then valves lift can be determined using the opening and closing angles, as shown in figure 6.
48 IV O EV O IV C EV C M IL M EL Valve lift
θ
Figure 6 – Schematics of valve lifts as a function of the crank angle. where IV O and EV O are the inlet and exhaust valve opening angle, IV C and EV C are the inlet and exhaust closing angles while MIL and M ELare the maximum inlet and exhaust valve lifts.
Then, the flow-through area may be determined by,
Ai= πDili (3.7)
and
Ae= πDele, (3.8)
where Di and De are the inlet and exhaust valve external diameters,
respectively.
3.3 PHYSICAL MODEL
The physical model describes the thermodynamic state of the gas in combustion chamber. It accounts for inlet and exhaust mass flows as well as the mass and energy conservation in the control volume. Parameters like pressure, temperature, and mass of the gas in the combustion chamber are determined as a function of the time. Figure 7 presents a rendering of the control volume, the mass flux and the heat transfer rate.
49
− ˙
Q
˙m
pm
g,p
g,T
g,V
gp
e˙m
eT
ep
i˙m
iT
iFigure 7 – Combustion chamber of the engine representing the control volume where the conservation equations will actuate.
In this figure p, T , m and V are the pressure, temperature, mass and volume respectively. The gas mass flow through the valve ports is ˙mi,e
and ˙Q stands for the heat transfer between the gas and chamber walls. The subscripts g, i, e and p correspond to gas, inlet, exhaust and piston respectively.
3.3.1 Conservation of Mass
The mass conservation equation applied to a fixed control volume around the gas in the combustion chamber can be written as,
∂mg
∂t = ˙mi− ˙me+ ˙mp, (3.9) where the first term represents the mass variation inside the control volume while the ˙m’s are the mass flow rates entering and leaving the control volume respectively. The ˙mp accounts for the mass flow rate
caused by the piston motion.
In this equation, control volume has a volume Vg momentarily
constant (∆t → 0). Then, the transient term may be written as, Vg∂ρg
50
where ρg is the gas density.
The mass flow produced by the piston motion is related to the instantaneous variation of the gas volume through,
˙mp= ρgvpAp= ρgAp
∂yp
∂t = −ρg ∂Vg
∂t , (3.11)
where vp, yp are the piston velocity and vertical position
respec-tively. Substituting this relation to equation 3.10 it is obtained Vg ∂ρg ∂t = ˙mi− ˙me− ρg ∂Vg ∂t , Vg ∂ρg ∂t + ρg ∂Vg ∂t = ˙mi− ˙me, (3.12) which can be written as,
∂mg
∂t = ˙mi− ˙me. (3.13) 3.3.2 Inlet and Exhaust Mass Flows
Considering the engine valve ports as a flow restriction, the mass flow can be calculated assuming an isentropic flow corrected by a flow coefficient [5]. Figure 8 is a rendering of the one-dimensional flow of an ideal gas along a flow restriction.
z ˙m p0 T0 h0 p T h throat
Figure 8 – Schematics representing the gas flow through a nozzle analog to the engine valve ports restriction.
51 The ideal steady flow one-dimensional energy equation is given as ∂ ∂z(h 2+ v2/2)(ρvA v) = 0, (3.14) in which, h2+ v2/2 = constant, (3.15) h2+ v2/2 = h0 (3.16) where h stands for enthalpy.
Assuming an ideal gas behavior and constant specific heat cp,
equation 3.15 may be written as,
cpT+ v2/2 = cpT0. (3.17)
Rearranging the terms it is obtained, 1 +2cv2
pT =
T0
T . (3.18)
Using the following relation for the constant pressure specific heat, cp =
γR
γ −1, (3.19)
where γ is the specific heats ratio of the gas and R is the universal gas constant, equation 3.18 becomes,
T0 T = 1 + (γ − 1) 2 v2 γRT. (3.20)
Defining the Mach number M as M = v
ss
, (3.21)
where ssis the sound speed, ss=
√
γRT [5], equation 3.20 may be written as,
T0
T = 1 +
(γ − 1)
2 M2. (3.22)
Using the isentropic relation, p0 p = (T 0) T γ−γ1 , (3.23)
52
an equation for the pressure is obtained as, p0 p = 1 + (γ − 1)2 M2 γ−γ1 . (3.24)
A critical condition occurs when the flow reaches the sound speed at the restriction (M = 1). This condition is known as choked flow. For M = 1 equation 3.24 is rewritten as,
p∗ p0 = 2 γ+ 1 γ−γ1 . (3.25)
Equation 3.25 provides a pressure criteria, which is used to determine the flow behavior as follows:
• p > p∗ the flow is subsonic;
• p < p∗ the flow is sonic at the restriction (choked flow).
Using the ideal gas law and the relations for p and T , the mass flow rate through the restriction may be written as [5],
˙m = Avγp0M √ γRT0 1 + (γ − 1)2 M2 −(γ+1)2(γ−1) , (3.26) or, ˙m = Avγp0 √ γRT0 p p0 1γ" 2 γ −1 1 − p p0 γ−γ1!# 1 2 . (3.27) At the critical condition (M = 1),
˙m =r γ RT0Avp0 2 γ+ 1 2(γ−1)γ+1 . (3.28)
When applying those equations for the cylinder presented in figure 7, variables p, p0 and T0 are replaced by pg, pi, pe, Tg, Ti or
Te depending on the flow direction, which is dictated by the pressure
difference between the valve ports and the interior of the combustion chamber.
Here, the pressure and temperature in the valve ports were considered to be constants for the calculations, except for the exhaust port temperature which is considered to be an isentropic expansion of the gas.
53 It is notable that there is no discharge coefficient in those mass flow rate equations. That is because the lack of data for the intake and exhaust valves required a load loss factor multiplying the intake mass flow rate. This factor is adjusted so the engine achieves the desired load of the study.
The set of equations presented in this section provides the mass flow rates at each time step based on the current thermodynamic state of the combustion chamber gas.
3.3.3 Energy Conservation
The first law of thermodynamics applied to a control volume, neglecting the kinetic and potential energy variations, is
∂
∂t(mu) − ( ˙mu)in+ ( ˙mu)out= ˙Q + ˙W , (3.29) where u is the gas specific internal energy, ˙Q is the heat transfer rate and
˙
W is the flow power, which is defined by, ˙ W = ˙mp ρ in − ˙mp ρ out . (3.30)
Using h = u + p/ρ and neglecting contribution of the viscous stresses, equation 3.29 becomes,
∂
∂t(mu) − ( ˙mh)in+ ( ˙mh)out= ˙Q. (3.31) The transient term applied to the cylinder gas may be written as,
∂
∂t(mgug) = ∂
∂t(mghg− Vgpg). (3.32) Recalling that the control volume is assumed to have a constant value of Vg momentarily (∆t → 0), the transient term becomes,
∂ ∂t(mgug) = mg ∂hg ∂t + hg ∂mg ∂t − Vg ∂pg ∂t . (3.33) Substituting this relation in equation 3.31 it is obtained
mg ∂hg ∂t = ˙Q + ( ˙mh)in−( ˙mh)out− hg ∂mg ∂t + Vg ∂pg ∂t , (3.34) where the left-hand side term is the mixture enthalpy variation with time, the first term in the right-hand side is the heat transfer rate to the control
54
surface, the second and third terms are the enthalpy flows through the control volume, the fourth is the enthalpy variation due to mass variation and the last is the fluid compression power.
Using the mass conservation, equation 3.34 becomes, mg ∂hg ∂t = ˙Q + [( ˙mh)in− hg( ˙m)in]− −[( ˙mh)out− hg( ˙m)out] + Vg ∂pg ∂t . (3.35) Applying to the cylinder and considering uniform flows for both inlet and exhaust ports,
mg∂hg ∂t = ˙Q+[ ˙mi(hi−hg)+ ˙mp(hg−hg)]− ˙me(hg−he)+Vg ∂pg ∂t . (3.36) or, mg ∂hg ∂t = ˙Q + ˙mi(hi− hg) − ˙me(hg− he) + Vg ∂pg ∂t (3.37)
For an ideal gas with constant specific heats, equation 3.37 becomes, mgcp ∂Tg ∂t = ˙Q + ˙micp(Ti− Tg) − ˙mecp(Tg− Te) + Vg ∂pg ∂t . (3.38) It is important to recall that during the combustion process, this equation gains one more term due to fuel chemical energy release and equation 3.38 may be written as,
∂Tg ∂t = 1 mgcp ˙Q + ˙mi cp(Ti− Tg)− −˙mecp(Tg− Te) + Vg ∂pg ∂t + ˙Srf , (3.39)
where the combustion energy generation term ( ˙Srf) will be deduced later.
The ideal gas law is used as the closing equation,
pgVg= mgRTg. (3.40)
The time variation of pressure becomes ∂pg ∂t = ∂ ∂t mgRTg Vg = RTg Vg ∂mg ∂t + mgR Vg ∂Tg ∂t − mgRTg Vg2 ∂Vg ∂t , (3.41)
55 or, ∂pg ∂t = pg mg ∂mg ∂t + pg Tg ∂Tg ∂t − pg Vg ∂Vg ∂t . (3.42)
From the set of equations presented in this section the thermo-dynamic state of the gas inside the combustion chamber (mg, pg and Tg)
can be calculated.
3.3.4 Cylinder Heat Transfer
During combustion, the burned gas temperature increases greatly and heat transfer becomes significant. When heat is lost to the chamber walls, the combustion gas temperature and pressure decrease, reducing the work per cycle and consequently the specific power and efficiency [5]. Figure 9 presents a rendering of the surfaces and heat transfer rates for this engine model. Only conduction and convection are considered due to the fact that the radiation is considered negligible for SI engines since it comprehends only 3-4% of the total heat transfer [13].
56
Lubricant
Water Cylinder Head Wall
Cylinder Wall Piston
Qhead Qcyl Qpiston
Figure 9 – Heat exchanger surfaces represented by the cylinder metallic media. The heat transfer surfaces are:
• Piston (assumed as a flat surface); • Cylinder walls;
• Cylinder head (assumed as a flat surface).
For the cylinder head and walls, heat is transfered from the gas to the coolant water and for the piston head, it is transfered from the gas to the lubricant. Figure 10 shows an equivalent thermal resistance circuit representing each one of those heat transfer rates.
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Metallic Wall
Inside Gas Water/Lub
Tg Tw= Tlub Qhead: Qcyl: Qpiston: 1 h∗gAch R1= 1 h∗gAcw R4= 1 h∗gAp R7= lch kchAch R2= lcw kcwAcw R5= lp kpAp R8= 1 h∗wAch R3= 1 h∗wAcw R6= 1 h∗ lubAp R9=
Figure 10 – Equivalent thermal resistances circuit.
where l is the metallic wall thickness, h∗ is the convective heat transfer coefficient and k is the thermal conductivity of the respective wall. The subscripts cw, ch, w, lub stand for cylinder walls, cylinder head, water and lubricant respectively. All surfaces are assumed to be made of cast iron, then kch= kcw= kp. For steady state conditions, Tlub is considered
to be equal to Tw [5]. To simplify the calculations, the cylinder wall
thermal resistance R5 is considered for a plane wall.
With the thermal resistances defined, the heat transfer rate are calculated as,
˙Q = (Tg− Tw)
Req , (3.43)
where Req is the equivalent thermal resistance of the circuit, which is
calculated by, Req = 1 R1+ R2+ R3 + 1 R4+ R5+ R6+ 1 R7+ R8+ R9 −1 . (3.44) The heat transfers coefficients between the gas and chamber walls are quite complex to evaluate since they are non-uniform and unsteady. Lounici et al.[13] proposes several correlations to perform this analysis and provide a spatially-averaged value for the cylinder.
A natural convection assumption was adopted by the pioneer models. They described for the first time the influence of engine parame-ters such as gas temperature, pressure and rotational speed on the heat
58
transfer [13]. These correlations became obsolete since they were found not accurate [25].
Several authors adopted a forced convection assumption since it is more realistic because the motion of mechanical moving parts cause fluid flow inside the chamber. Usually those correlations were developed using measurements in diesel engine ( [13]), but Annand [26] proposed a correlation for both CI and SI engines.
For SI engines, Annand’s correlation may be written as, N uB = hgB kg = aa ReB ba , (3.45) where ReB= ρgSpB µ , (3.46)
and kg and µ are the gas thermal conductivity and viscosity respectively
and aa and ba are adjustable parameters of the correlation, e.g., 0.35 ≤
aa≤0.8 and b = 0.7 for normal combustion [5]. Sp is the average piston
speed, which provides an order of magnitude of the main axial flow within the combustion chamber.
3.4 CHEMICAL MODEL
This section provides the procedure to calculate the chemical composition and the combustion energy release rate ( ˙Srf). The model
assumes shifting equilibrium to calculate the combustion energy release. 3.4.1 Chemical Equilibrium
The chemical equilibrium provides the current burned gas com-position for a given condition of pressure and temperature. This task is performed using the library routines of the Stanford software STANJAN, which accepts the input in CHEMKIN format. STANJAN algorithm works minimizing the mixture free-energy to find the equilibrium state of the ideal gas mixture [27].
Here, the chemical equilibrium calculation involves 75 chemical species obtained from the GRI mechanism, as shown in table 10. The main species are iC8H18 (iso-octane as a surrogate for gasoline), CH3CH2OH (ethanol), O2, CO2, H2O, H2, CO, CH4 and N2 [5]. The combustion at each crank angle increment is considered to occur adiabatically and at constant pressure, where the products temperature is the final tempera-ture achievable by the initial reactants mixtempera-ture. Hence, the combustion during a time interval ∆t follows the adiabatic flame temperature for the mixture, given the initial composition, temperature and pressure [28].