Mixing and Combustion in Dense Mixtures by William A. Sirignano and Derek Dunn-Rankin
At very high pressures and densities, what is different and what is similar about the processes of – Injection and Atomization, Phase Change, Molecular Transport, Turbulent mixing, Oxidation, and Soot Formation – and the Abilities for Modelling and Measurement.
What do we mean by “high pressure”? • High density is actually meant. • In particular, density is so high and molecular distances so small that molecular force fields reach beyond the distance to neighboring molecules. • Since molecular collisions become continuous rather than intermittent, the ideal gas law must be abandoned. A cubic equation of state is commonly used, e,g., van der Waals, Peng-Robinson, or Redlich-Kwong. • Certain mixing rules are needed in the state equation applied to multicomponent fluids to determine two “constants” which actually vary over space and time with composition. Two phases or a continuous fluid ? • It is widely and often wrongly assumed that a liquid injected into a surrounding gas at a subcritical liquid temperature but supercritical pressure for all components, may always be treated as a continuous fluid. • The critical pressure of a mixture can be several times greater than the critical values for the components. Thus, critical pressure becomes a variable over the domain as composition varies and many problems must be viewed as transcritical with both subcritical and supercritical subdomains. • Near the critical point, gas more easily dissolves in liquid and liquid more easily vaporizes; surface tension and energy of vaporization are reduced. Both phases must be considered as mixtures. • Whether we have a distinct liquid phase with a distinct gas phase or one continuous fluid without interfaces, transport of mass will be important because of composition gradients. Heat transport will occur because of inherent temperature gradients. Hydrocarbon – O2 mixtures Two-phase behavior at up to 7x“critical” pressures
Propane
Heptane Computed by Albert Jordà Juanós
Octane
Decane
Mixture of decane, O2, N2, CO2, and H2O at 4x the critical pressure of decane. Two phases clearly exist !!.
A. Jordà Juanós
Surface Tension for near-critical behavior Value depends on all components Calculation by Delplanque & Lengsfeld -- Heptane-Nitrogen Surface tension depends strongly on liquid temperature, more modestly on pressure and ambient gas temperature. Enthalpy of Vaporization
-- Latent heat of vaporization applies exactly to single species systems. Enthalpy of vaporization is needed for multispecies systems. -- Enthalpy of vaporization decreases with increasing temperature and increasing pressure. V. Yang & Co-workers A. Jordà Juanós A. Jordà Juanós
• Density versus composition in a two-component decane-O2 mixture at subcritical temperature and a pressure supercritical for each component. • A fuzzy zone in images may or may not have a discontinuous density. Breakup of Injected Fuel Stream (aka atomization) at High Reynolds and Weber numbers • The disintegration of the injected fuel stream determines the chamber mixture ratio distribution. • The process also determines length scales for the resulting fuel droplets, ligaments, or ‘blobs’ thereby affecting rate controlling processes such as vaporization and mixing. • Thus, without knowledge of these processes, we have no starting point for practical combustion analysis. • Except for final stages, the atomization process is weakly dependent on surface tension and viscosity. • Consequently, substantial similarities are found for homogeneous jets (e.g., air-into-air or water-into-water) and liquid jets into gas. Round jets with 3D instabilities of axisymmetric vortex rings
Liepmann & Gharib (1992), water-into-water
Marmottant & Villermaux (2004) Co-axial flow of liquid with air
Ring Braid Vorticity analysis shows common cause between the above side jets and the ligaments and lobes to the left. Liquid transient, round jet into still air Shinjo & Umemura (2010), NS-LS Re = 440 – 1470; We = 1270 – 14,100
Jarrahbashi & Sirignano (2014), NS-LS-PP
Re = 1600 ; We = 230000 ; ρ1 / ρ2 = 0.1
-- Cap forms on starting jet. -- Mass and size of cap increase. -- Instability causes shredding of cap. time -- Cones have azimuthal instability. Lobes form. -- Lobes stretch and tear under azimuthal instability. -- Ligaments form from torn rims. -- Capillary instabilities on ligaments. Density ratio affects 3D instability but a qualitative similarity exists over a wide range. There is no basis for assuming large qualitative distinction in cone, lobe, and ligament formations for supercritical injection at low Mach numbers.
Re = 1600 We = 230,000 -4 μ1 / μ2 = 9 x 10 t = 90 μs
ρ1 / ρ2 = 0.1 , 0.5 , 0.9 Turbulent Mixing • Molecular transport remains important even with turbulent eddies enhancing mixing. • Simple models developed first by Marble (1985), Karagozian & Marble (1986), Cetegen & Sirignano (1989, 1990) explain how molecular mixing and reaction occur in the strained flow within an eddy. • The temporal and spatial behavior within a two-dimensional viscous vortex placed at the interface between reactants is analyzed. Diffusion across the strained material lines is determined. • Mixing rates increase with both vortical strength (circulation) and diffusivities for heat and mass. • Results can be shown for the mixing of two nonreacting fluid species and for the mixing with reaction of two reactants and a product. • A mixedness parameter f(t) is used based on integrations over the volume eddy: f(t) =
Mixedness Parameter
Mixedness enhancement Δf due to eddy circulation and strain increases with time, vortex strength, and mass diffusivity.
D = υ ; R = Г/ υ Turbulent Mixing of O3 and NO with isothermal chemical reaction For reacting case, similar conclusions occur about dependence of mixedness on both vortex strength and molecular transport. Molecular Transport
Viscosity
Prandtl Number
Water – dramatic effects near the critical point diminishing at more extreme conditions Prandtl Number of Binary Mixtures of Noble Gases -- Complex behavior for mixtures, strongly nonlinear behavior. -- Variable mixture ratios over space and time gives highly non-uniform transport properties. -- Beware of simplifying assumptions. -- Sub-grid models become huge challenge.
Tournier & El-Genk, 2008 Challenges with Molecular Transport
• Transport properties of mixtures are complex. • More information about properties for mixtures of combustion reactants and products is needed. • Discontinuities with phase change are replaced by regions of steep gradients at supercritical conditions. • Viscosity can decrease with increasing density causing higher Reynolds numbers, smaller scales of turbulence, and greater demands on sub-grid modelling. Vaporization A cold fuel droplet might be sub-critical for its local gas mixture. --A discontinuity in composition and density exists at the interface. -- For a hot environment, critical temperature occurs at some distance. -- Eventually, after some heating and vaporization, the critical surface reaches the interface. Phase distinction ceases.
Challenges with Near-critical Vaporization -- Enthalpy of vaporization and surface tension still exist, can be important, and more difficult to calculate because they are mixture-dependent. -- Gases dissolve in liquid and diffuse through liquid, creating a multicomponent liquid-phase diffusion problem. -- A quasi-steady gas-phase assumption becomes unreasonable; magnitudes of transport properties are similar in both phases which now must be considered unsteady. -- The droplet heating and vaporization problem becomes much more difficult. Chemical Reactions
• While oxidation rates increase with density, transport rates decrease possibly modifying rate control in some situations. • At high densities, new pathways must be evaluated. • Can chemical kinetic theory be built around bi- molecular and tri-molecular collisions for a dense gas with continual rather than intermittent collisions? • Use fugacity, not partial pressure, in law of mass action for chemical equilibrium. • How can kinetic laws without fugacity variables predict chemical equilibrium? Summary of Key Knowledge
• It is high density, not high pressure! • Molecular transport is always important in mixing and combustion problems, whether turbulent or laminar, continuous phase or two phase. • Molecular transport is generally slower, more controlling at high density. • Two-phases can exist at pressures well above the critical pressures of any component in a mixture. • Substantial mixing occurs across phase interface; thus, fuzzy, blurry images need not imply a continuous phase. • Length scales for composition variation and distribution of mixture ratio are vital characteristics for composition; thus, whether two-phase or continuous phase, “atomization” is important. • For the first stages of atomization, vorticity dynamics is controlling while viscosity and surface tension are of little importance; thus, there is little difference amongst homogeneous jets (e.g., water-into-water,), liquid jets into gas, and continuous-phase high-density jets. • Vaporization is a more complex problem at high densities and still is potentially rate-controlling. Knowledge Gaps for High-density Flows
• Atomization in the near-critical two-phase domain. • “Atomization” in the continuous fluid domain. • Sub-grid (small length-scale) mixing models. • Trans-critical vaporization models. • Chemical pathways. • Thermophysical and transport properties for MIXTURES at high density. Thank You (g/mol) (K) (bar) (cm3/mol)
푊 푇푐 푃푐 푣푐 휔 Ω푎 Ω푏
n/a H2 2.015 33.18 13.00 64.9 -0.216 n/a
n/a O2 31.998 154.6 50.4 73.4 0.025 n/a
0.0911 CO 2 44.009 304.1 73.8 93.9 0.239 0.4470 n/a H2O 18.015 647.3 221.2 57.1 0.344 n/a
0.0870 N2 28.013 126.2 33.9 89.8 0.039 0.4290
0.0867 CH4 16.043 190.4 46.0 99.2 0.011 0.4278
0.0889 C3H8 44.096 369.8 42.5 203 0.153 0.4380
0.0952 C7H16 100.202 540.3 27.4 432 0.349 0.4680
0.0968 C8H18 114.228 568.8 24.9 492 0.398 0.4760
n/a C10H22 142.282 617.8 21.1 624 0.484 n/a
] [1 Since hydrogen is a quantum gas, classical critical constants must be used [38]: Tco=43.6 K ; Pco=20.2 atm ; vco=51.5 cm3/mol • Critical point - distinction between liquid and vapor phases disappears; characterized by Tcr, Pcr and Vcr (or Definitions ρcr), unique for each pure substance • Near-critical point - narrow region around the critical point, where thermophysical properties of a pure fluid exhibit rapid variations • Pseudocritical line - consists of pseudocritical points • Pseudocritical point - pressure above the critical pressure and at a temperature Tpc of the maximum value of the specific heat at this pressure • Supercritical fluid - pressures and temperatures are higher than Pcr and Tcr (often, the term supercritical is used for both supercritical fluid and compressed fluid) • Compressed fluid – pressures above the critical pressure, but at One component, temperatures below Tcr e.g., water • Superheated fluid – pressures below Pcr, but temperatures above Tcr • Dramatic property variations occur only in the near-critical point region and only for pure fluid All of these definitions are complicated for multi-component fluids Piolo and Mokry, 2011 Cubic Equation of State Redlich-Kwong; Peng-Robinson; van der Waals
• = . ; = ; = 푹풖푻 풂 풂� 풃� ퟎ ퟓ ퟓ 풗−풃 푻 풗 풗+풃 ퟐ 푹풖푻 • 푷 −+ 푨 푹풖푻ퟐ = 푩 • ퟑ= ퟐ ퟐ = 풁 − 풁 푨 − 푩 − 푩 풁 − 푨� ퟎ 푵 푵 / 푵 풊=ퟏ 풋=ퟏ 풊 풋 풊� 풊=ퟏ 풊 풊 풂 =∑ =∑ 퓧 퓧 풂 ; 푩 ∑= 풙 풃 • ퟐ ퟓ ퟐ 휴풂풊푹풖푻풄풊 휴풃풊푹풖푻풄풊 풊풊 풊 풊 풂 풂 푷풄풊 풃 푷풄풊 /
• = ퟐ ퟓ ퟐ 휴풂풊+휴풂풋 푹풖푻풄풊� 풊� 풂 ퟐ푷풄풊�
Relations for Phase Equilibrium
• = , = , = 퓵 퓰 퓵 퓰 퓵 퓰 • 푻 푻 = 푷 푷 흁 흁 ⋄ 풇풊 풇풊 풊 풊 풖 ⋄ • 흁 −=흁 , 푹 푻 풍풍 풇풊 = 풂�풂 풍푷 ,� →풍ퟎ 푷 =풊 ⟶ ퟏ 퓵 퓰 퓵 퓵 퓰 퓰 풇풊 • 풇풊 풇풊 = 퓧풊 흓풊 퓧풊 흓풊 푷풊 흓풊 ∞ 흏� , , 푹풖푻 풖 풊 푽 풊 풖 • 푹 푻 풍풍=흓 ∫ 흏� 푻 푽 �풋 − 푽 𝒅 − 푹 푻 풍풍 풁
풊 + 풍풍 흓 푵 풃풊 푨 ퟐ ∑풋=ퟏ 퓧풋풂풊� 풃풊 푩 풃 풁 − ퟏ − 풍풍 풁 − 푩 − 푩 풂 − 풃 풍풍 ퟏ 풁 • , = 퓰 퓵 = 퓰 풉�풊 −풉�풊 흏 흓풊 ퟐ 퓵 횫푯� 풗 풊 푾풊 푹풖푻 흏� 풍풍 흓풊 Density of Gas and Liquid Temperature, composition, and density vary with distance from interface on both sides. (Calculations by Delplanque & Sirignano.)
As pressure increases, more gas can dissolve in the liquid. Consequently, liquid density can decrease with increasing pressure. (Calculations by V. Yang and team.) Back-up
A. Jordà Juanós
Delplanque & Sirignano Conductivity for Single-component Fluid
Tournier & El-Genk, 2008 Liquid Mixture Density Surface Tension –Heptane, Oxygen Deplanque & Lengsferd Dodecane-O2
V. Yang & Co-workers Results from Cetegen & Sirignano Discontinuities in Density