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Experimental investigation of laminar flame speed measurement for kerosene fuels: Jet A-1, surrogate fuel and its pure components Yi Wu, Vincent Modica, Xilong Yu, and Frederic Grisch Energy Fuels, Just Accepted Manuscript • DOI: 10.1021/acs.energyfuels.7b02731 • Publication Date (Web): 11 Jan 2018 Downloaded from http://pubs.acs.org on January 12, 2018
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Energy & Fuels
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Experimental investigation of laminar flame speed measurement for
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kerosene fuels: Jet A-1, surrogate fuel and its pure components
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Yi Wu1, 2*, Vincent Modica2, Xilong Yu1, Frédéric Grisch2
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State Key Laboratory of High Temperature Gas Dynamics, Institute of Mechanics, Chinese Academy of Sciences, Beijing 100190, China
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CORIA-UMR 6614, Normandie Université, CNRS, INSA et Université de Rouen Campus Universitaire du Madrillet, 76800 Saint-Etienne du Rouvray, France
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*Corresponding author: Dr. Yi WU Email:
[email protected] 18 19 20 21 22
State Key Laboratory of High Temperature Gas Dynamics, Institute of Mechanics, Chinese Academy of Sciences, Beijing 100190, China Phone: +86 132 6018 7692 Fax: +86 10 6256 1284
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CORIA-UMR 6614, Normandie Université, CNRS, INSA et Université de Rouen Campus Universitaire du Madrillet, 76800 Saint-Etienne du Rouvray, France Phone: (+33) 6 51 26 39 65 Fax: (+33) 6 51 26 39 65
Correspondence to:
[email protected] ACS Paragon Plus Environment
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Experimental investigation of laminar flame speed measurement for
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kerosene fuels: Jet A-1, surrogate fuel and its pure components
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Yi Wu1, 2*, Vincent Modica1, Xilong Yu1, Frédéric Grisch1
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1
State Key Laboratory of High Temperature Gas Dynamics, Institute of Mechanics, Chinese Academy of Sciences, Beijing 100190, China
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CORIA-UMR 6614, Normandie Université, CNRS, INSA et Université de Rouen Campus Universitaire du Madrillet, 76800 Saint-Etienne du Rouvray, France
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Abstract:
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The present work investigated the laminar flame speed measurement of kerosene-relevant fuel, including Jet A-1
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commercial kerosene, and surrogate kerosene fuel and its pure components (n-decane, n-propyl-benzene and
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propyl-cyclohexane), using a high-pressure Bunsen flame burner. The OH* chemiluminescence technique and
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the kerosene-PLIF technique were used for flame contours detection in order to calculate the laminar flame
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speed. The experiments were firstly conducted for n-decane/air flame at T = 400 K, φ = 0.6 − 1.3 and
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atmospheric pressure conditions in order to validate the whole experimental system and measurement
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methodology. The laminar flame speed of Jet A-1/air, surrogate/air and pure kerosene component (n-decane,
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n-propyl-benzene and propyl-cyclohexane) was then measured under large operating conditions, including
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temperature T = 400-473 K, pressure P = 0.1-1.0 MPa and equivalence ratio φ = 0.7 − 1.3. It was found that
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these three pure components of kerosene have very similar laminar flame speed. By comparing the experimental
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results of surrogate kerosene and Jet A-1 commercial kerosene, it was observed that the proposed surrogate
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kerosene, i.e., mixtures of 76.7wt%
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can appropriately reproduce the flame speed property of Jet A-1 commercial kerosene fuel. The experimental
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results were further compared with simulation results using a skeletal kerosene mechanism.
n-decane, 13.2wt% n-propyl-benzene and 10.1wt% propyl-cyclohexane,
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Keywords: laminar flame speed, kerosene, Jet A-1, surrogate fuel, n-decane, n-propyl-benzene and
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propyl-cyclohexane
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1. Introduction
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Understanding the fundamental combustion properties of commercial jet fuels is essential in the development of
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low-cost and efficient aircraft engines. For this purpose, as well as improve the performance of aeronautical
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propulsion systems, a comprehensive understanding of the chemical kinetic mechanisms of multicomponent
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commercial jet fuels is needed [1]. However, the detailed chemical mechanisms of commercial jet fuels, such as
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Jet A-1, consist of several hundreds of different species and about 1,000 elementary reactions. To perform
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mechanical CFD (Computational Fluid Dynamics) simulations using detailed kinetic mechanisms, it needs to get
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solutions of all transport equations for each species present in the detailed kinetic mechanism and calculate all
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the source terms for each species. This kind of simulation of an industrial complex geometry using
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aforementioned detailed kinetic mechanism remains elusive [2]. In practical investigations, an alternative
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approach is using pure hydrocarbon fuels or composition-simplified surrogate fuels to simulate the
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physicochemical characteristics of a practical kerosene fuels [3-8]. One of the evaluating indicator for the
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validation of surrogate kerosene is the similarity between the combustion properties and those of
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multicomponent practical fossil fuels [9]. Therefore, it is firstly of substantial interest to learn about the surrogate
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fuel combustion characteristics and associated chemical mechanism in order to more effectively exploit
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commercial jet fuels. Several methodologies have been used to quantify fundamental properties of surrogate jet
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fuels, such as ignition [10-15], laminar flame speed [7, 16-22], flame stretch and extinction limit [11, 23-27].
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Among these parameters, laminar flame speed is key parameter to understand the reactivity, diffusivity and
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exothermicity of fuel/air mixtures and validating both kinetic chemical mechanisms and turbulent models [28,
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29].
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The aforementioned necessities have motivated numerous previous works, which have investigated the laminar
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flame speed of composition-simplified surrogate kerosene fuel and their pure components [10, 19, 30-41].
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Typically, commercial kerosene, such as Jet A-1, involves around 50 - 65% paraffin, 10 - 20% aromatics and 20
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– 30 % naphthenic, which cause the elucidation of the main components of previously proposed surrogate
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kerosene [42]. In the early stages of a kerosene surrogate investigation, the composition of the surrogates
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generally comprised more than 10 kinds of hydrocarbons fuels [30, 35]. Investigations were then performed to
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reducing the number of compositions to less than 10, because it was commonplace to compose the surrogates in
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using some compositions in practical fuels, including low-concentration components, such as naphthalene and
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several hydrocarbons of the same category [36]. More recently, the composition of surrogate fuels has
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continually been simplified to culminate in one-, two- or three-component surrogate fuels [10,30-34]. A review 3
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of literature reveals that, the Aachen surrogate [5] and the UCSD surrogate [34] can well reproduce the property
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of auto ignition and extinction rate of commercial kerosene fuels. Laminar flame speeds measurement of the
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Aachen surrogate fuel have been performed with the spherically expanding flame method under conditions of P
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= 0.1 MPa and T = 473 K. Holley et al. [25] conducted measurements of the extinction strain rate and ignition
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temperature of the surrogate fuels with a counter flow burner under atmospheric pressure and an elevated
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preheating temperature conditions. In their work, they also compared single-component hydrocarbon with
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surrogate jet fuel of JP-8. More recently, Comandini et al. [7] measured laminar flame speeds of surrogate fuel
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composed by n-decane, n-butyl-benzene and n-propyl-benzene using the spherically expanding flame method
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under conditions of T = 403 K and P = 0.1 MPa. Laminar flame speeds of surrogate mixtures were compared
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with each pure component and discussed in detail [7]. Besides the aforementioned Aachen surrogate [5] and the
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surrogate referred in the work of Comandini et al. [7], few investigations of laminar flame speeds measurements
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for other surrogate fuels can be found in literature. Comparisons of laminar flame speeds between surrogate fuel
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and commercial kerosene fuels have been rarely performed. In the present work, surrogate kerosene consisting of
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n-decane (76.7wt%), n-propyl-benzene (13.2wt%) and propyl-cyclohexane (10.1wt%) are investigated for their
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laminar flame speeds using a high-pressure Bunsen flame burner [43, 44]. This composition of surrogate for
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kerosene was initially referred in the work of Dagaut et al. [8] and Luche et al. [6], who developed skeletal
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kinetic mechanism targeting to emulate the combustion properties of commercial kerosene fuels. However, to
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the best of our knowledge, no measurements of laminar flame speeds were performed and compared with
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commercial kerosene fuels for this three-component surrogate kerosene. In the present work, in order to evaluate
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the performance of this surrogate fuel, its laminar flame speed will be measured under large range of conditions,
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including variations of temperature, pressure and equivalence ratio, and compared with the laminar flame speed
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of commercial kerosene fuel (Jet A-1).
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In order to compare the flame speed of the surrogate kerosene proposed in the present work with commercial
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kerosene, laminar flame speeds measurements of Jet A-1 kerosene fuel/air mixtures under the same conditions as
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for the aforementioned surrogate kerosene have been performed. A review of the literature reveals that, even
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though commercial kerosene (such as Jet A-1) fuel has been used for decades in aeronautics, it has only been in
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recent years that its laminar flame speeds have been measured with reasonable accuracy, owing to the
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development of the measurement devices used [18, 19, 37-41]. For instance, Kumar et al. [37] measured laminar
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flame speeds of Jet A-1 using the counter flow flame method under atmospheric pressure conditions and
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temperatures of 400, 450 and 470 K. Hui et al. [19] also measured the laminar flame speed of Jet A-1 using the 4
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counter-flow method, in their work efforts were focused on measurements in higher pressure conditions of P =
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0.1 - 0.3 MPa, with preheated temperatures between 350 and 470 K and equivalence ratio from 0.7 to 1.3. In
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their work, the un-stretched laminar flame speed was determined by the linear extrapolation of the stretched
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laminar flame speeds. While both of the measurements [19, 37] were performed using the same counterflow
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flame burner, large data deviation was observed. Other laminar flame speed measurements of Jet A-1 were
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depicted in the investigation of Chong et al. [18]. The PIV optical technique has been applied on a jet wall
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stagnation flame to determine the laminar flame speed of Jet A-1. Regarding the laminar flame speed of Jet A-1
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measured under pressure conditions higher than 0.5 MPa, to the best of the authors’ knowledge, the only
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experimental data available are found in the work of Vukadinovic et al. [38]. In their study, the laminar flame
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speeds were measured at a pressure of up to 0.8 MPa using the spherically expanding flame method. Generally,
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as a summary of the above literature analysis, large data scattering (up to 15 cm/s) are observable between the
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different results of laminar flame speeds for Jet A-1 fuel [19, 37, 39-41]. Even though part of the discrepancies
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are due to the fact that the composition of Jet A-1 varies among different production regions, further
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investigation is still necessary to restrain the uncertainties introduced by methodologies, such as linear or
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nonlinear correlations, and by measurements devices. Moreover, measurements under high pressure and elevated
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temperature conditions in the propulsion system are still deficient, such that further investigations into accurate
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laminar flame speed measurements of kerosene fuels are still necessary.
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In the present work, laminar flame speeds of kerosene fuels, including Jet A-1 commercial kerosene, as well as
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Luche surrogate fuel and its pure components, i.e., n-decane, n-propyl-benzene and propyl-cyclohexane, are
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measured under large operating conditions, including variations in temperature, pressure and equivalence ratio
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based on a high-pressure Bunsen flame burner. The OH* chemiluminescence optical technique and the
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kerosene-PLIF technique were used for flame contours detection in order to calculate the laminar flame speed.
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OH* chemiluminescence is a convenient and widely applied combustion diagnostic technique; however, for
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Bunsen flames with large molecular weight fuels, flame tip opening phenomenon occurs under slight fuel-rich
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conditions (around φ > 1.15~1.2). Under these conditions, OH* chemiluminescence has difficulty in obtaining
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the whole flame contours due to local quenching at the cusp of the flame [45-48]. Kerosene-PLIF has the
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advantage of visualizing the contours of the unburned fronts of kerosene/mixtures, such as Jet A-1, which is free
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from the influence of local quenching located in the burned gas [49, 50]. Hence, in the present work,
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kerosene-PLIF is applied to validate the suitable flame contours definition for the OH* chemiluminescence
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technique when the flame-opening phenomenon conditions are present. 5
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Experimental results of n-decane/air mixtures using the OH* chemiluminescence technique will first be
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presented to validate the accuracy of the whole experimental system and the determination methodology. For
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further investigating the effect of the flame-opening phenomenon on the laminar flame speed measured by OH*
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chemiluminescence, comparison of the laminar flame speeds obtained by OH* chemiluminescence and the
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kerosene-PLIF technique for Jet A-1/air mixtures under the same operating conditions are performed.
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Measurements are then extended to higher temperatures T = 400, 423 and 473 K for pure surrogate components,
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i.e., n-decane, n-propyl-benzene and propyl-cyclohexane under atmospheric pressure conditions. Finally, the
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laminar flame speed of surrogate kerosene, i.e., mixtures of n-decane (76.7 wt %), n-propyl-benzene (13.20 wt
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%) and propyl-cyclohexane (10.1 wt %), are measured under conditions of T = 400-473 K, P = 0.1-1.0 MPa
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and φ = 0.7 − 1.3, then compared with the experimental results of commercial kerosene (Jet A-1) measured
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under identical conditions. The laminar flame speed of surrogate fuel is also simulated with skeletal kerosene
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mechanism. Further analysis and discussions will be detailed.
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2. Experimental methodology
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2.1 High pressure burner and gas feeds
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In the present study, a high pressure Bunsen flame burner was developed to measure laminar flame speed of
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kerosene fuels. The detailed description of the experimental apparatus can be found in Wu et al.’s previously
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published work [43, 44]; as such, only a brief introduction is given here.
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Figure 1: Schematic of the experimental setup 6
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Energy & Fuels
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As illustrated in Figure 1, the whole experimental apparatus consists of a high-pressure Bunsen flame burner,
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liquid fuel vaporization with a gas feed system and an optical diagnostic set-up. With this system, a steady
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conical flame can be generated on the outlet of the contoured nozzle that has an outlet diameter of 7 mm. A
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concentric contoured nozzle with an outlet diameter of d2 = 7.6 mm, which surrounds the central nozzle, is used
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to produce a pilot flame in order to stabilize the flame under high-pressure operating conditions. The equivalence
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ratio and the flow rate of the pilot flame are regulated by two mass flow controllers. The whole burner is placed
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in an N2 (guard flow) ventilated high-pressure chamber. A liquid flow controller (Bronkhorst mini-CORIFLOW)
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is used to deliver liquid kerosene fuel to the Controlled Evaporator and Mixer (CEM, Bronkhorst), where the
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liquid fuel and N2 carrier gas are mixed and heated to a preset temperature. The CEM system allows to vaporize
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liquid kerosene fuel and two additional flow controller were used to deliver nitrogen and oxygen to mix at the
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outlet of CEM to reproduce the synthetic species’ composition of air and control the equivalence ratio of
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vaporized fuel/air mixtures. The whole chamber and all the gas feed lines are preheated with electrical wire
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heaters. Five thermocouples located in different positions of the high pressure chamber are used to measure and
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monitor the uniformity of the temperature throughout of the pressure chamber. The pressure in the chamber is
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measured with a piezoelectric transducer located at the outlet of the high pressure chamber. Two circulation
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heaters are used to heat the guard flow and the vaporized fuel/air mixture to avoid condensations of the fuel
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vapor. There are four UV quartz windows at the high pressure chamber allowing probing the flame with optical
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diagnostics.
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2.2 Optical diagnostic setup 2.2.1 OH* chemiluminescence
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One of the optical techniques used in the present work to detect the flame contour is the OH*
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chemiluminescence imaging technique. A thermoelectrically cooled, 16-bit intensified CCD camera (PI-MAX 3,
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Roper Scientific) was used to record the OH* chemiluminescence flame image. With an achromatic UV lens
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(CERCO)of f/2.8, f = 100 mm, and an interference band pass filter centered at 310 nm, the camera can visualize
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a 40 × 40 mm2 area of the flame with a 1024×1024 array so that the spatial resolution is about 40 µm per pixel.
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The exposure time to record the OH* chemiluminescence image is controlled by opening the intensifier gate 1
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µs. The acquisition repetition rate of the camera is set to 10 Hz.
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2.2.2 Kerosene-PLIF
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It is well known that the composition of kerosene commonly used in civil engines (i.e. Jet A1) consists of a wide
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range of hydrocarbon molecules. Among these, aromatic molecules are known to have UV and visible transitions 7
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exhibiting strong fluorescence. Therefore, kerosene fuel benefits from its broadband fluorescence emission
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between the 260-420 nm spectral domains [49, 50]. This fluorescence emission results from the excitation of
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aromatics (i.e., mono- and di-aromatics) allows visualizing the fresh gas region of the flame. Although aromatics
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may be excited on a wide range of wavelengths, for a flame to visualize only the unburned gas, the excitation
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wavelength of kerosene must be selected carefully to avoid overlaps with the resonance of an OH radical
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transition located in the burned gas. An observation of the fluorescence spectrum of the OH transition reveals
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that the Q1(5) and Q1(6) rotational lines are well separated, thus presenting the possibility to tune the excitation
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wavelength of aromatics to 282.85 nm in a spectral region in which only the aromatics fluorescence will be
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collected.
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Figure 2: Jet A-1/air flames under the condition of T = 400 K, P = 0.1 MPa for equivalence ratios φ = 0.8, 1.0
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and 1.3: (a) OH* chemiluminescence technique; (b) Kerosene-PLIF technique
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In the present work, the kerosene planar laser-induced fluorescence system consists of a cluster system, which
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includes an Nd:YAG laser (Quanta-ray Pro-190), a dye laser (Sirah PRSC-G-300) and a high-resolution ICCD
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camera (PI-MAX 3, Roper Scientific). A frequency-doubled, Q-switched Nd:YAG laser was used to pump a dye
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laser, which was then frequency-doubled to obtain wavelengths in the 280-290 nm spectral range. The dye laser
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was tuned to 282.85 nm to excite the aromatics in the Jet A-1 fuel. The laser energy was regulated to 5 mJ in
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order to keep the fluorescence of kerosene within the linear regime, thus maintaining the proportionality between
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the intensity of fluorescence and the concentration of kerosene fuel. The UV laser beam at the exit of the laser
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source was transported via optical mirrors. It was then transformed into one 50 mm collimated sheet using a set
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of converging and divergent cylindrical lenses and a spherical lens. The two cylindrical lenses formed a
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telescope, which sped up the beam into a collimated sheet to cross the target flame. The kerosene fluorescence
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image was then recorded by the ICCD camera (the same one used for the OH* chemiluminescence technique).
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The intensifier gate width was set to 1 µs and the framing rate of the acquisition of fluorescence images was 10
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Hz. The camera was equipped with the same optical lens with OH* chemiluminescence, while only the optical
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filter was changed to 300-550 nm. As shown in Figure 2, images of Jet A-1/air flames using OH*
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chemiluminescence technique and kerosene-PLIF technique are presented.
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2.3 Fuel properties
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The chemicals used in the different mixtures were n-decane (C10H22, Sigma–Aldrich, 99+ %), n-propyl-benzene
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(C9H12, Aldrich, 99+ %), and n-propyl-cyclohexane (C9H18, Aldrich, 99+ %). The Luche surrogate fuel for
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emulating the commercial jet fuel is a mixture of these three components by n-decane (76.7wt %),
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n-propyl-benzene (13.2wt %) and propyl-cyclohexane (10.1wt %). The average chemical formula is C9.74H20.05
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with a molar mass equal to 136.93 g/mol. Meanwhile, the target commercial kerosene fuel used in the present
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work was analyzed by gas chromatography and its chemical composition is shown in Table 1. The chemical
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composition of this fuel consists mainly of paraffin, naphthenic and alkyl-benzenes. In addition, there is no
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hydrocarbon poly-aromatic hydrocarbon heavier than naphthalene. The average chemical formulation of Jet A-1
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is C11.16H20.82 with a molar mass equals to 154 g/mol. Table 2 summarizes the properties of studied fuels in the
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present work.
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Species
Proportion (wt%)
Paraffin
46.5
Non-condensed naphthene
22.3
Condensed naphthene
13.2
Alkylbenzenes
13.2
Indanes and tetralines
2.8
Naphthalenes
1.8
Acenaphthenes and diphenyls
0.1
Benzothiophenes
0.1
Saturated
82
Monoaromatic
16
Diaromatic
1.9
Sulfur
0.1
Table 1: Chemical composition of Jet A-1 in the present study
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Fuel
Formula
Boiling point (K)
Density (kg/m3)
n-decane
C10H22
447.3
724.0
n-propyl benzene
C9H12
432.4
825.7
propyl cyclohexane
C9H18
429.9
793.0
Luche surrogate
C9.74H20.05
432~447
747.0
Jet A-1
C11.16H20.82
423~533
815.9
Table 2: Fuel property of n-decane, n-propylene, propyl cyclohexane, Luche surrogate and Jet A-1 fuel.
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2.4 Extraction of flame speed
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The Bunsen flame method is a convenient and widely used approach for laminar flame speed measurements
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because of its advantages of simplicity and a well-defined flame structure. Laminar flame speed is defined as the
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velocity at which a planar flame front travels relative to the unburned gas in the direction of the flame surface
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[51]. For a conical flame, with the hypothesis that the flame speed is identical all over the entire surface area of
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the flame, the laminar flame speed can be calculated, based on the mass conservation between the outlet nozzle
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and the flame front [44, 52-56]. As shown in Figure 3, the average flame speed in the transverse plane is
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expressed by:
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= → = /( )
Eq. 1
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Where is the total mass flow rate of the fuel/air gaseous mixture, and ρu is the density of fresh gas. This
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method requires the knowledge of the total area of the flame surface A, which is determined in the present
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investigation by analyzing the OH* chemiluminescence image and the kerosene-PLIF image (for Jet A-1 fuel) of
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the flames. Kerosene-PLIF is applied to validate the OH* chemiluminescence image-processing algorithm for
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the cases when the flame tip-opening phenomenon is present. For large molecule weight fuel, such as kerosene,
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the flame tip-opening phenomenon appears when the equivalence ratio is higher than around φ > (1.15~1.2).
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The tip opening of the laminar Bunsen flame involves local combustion extinction, which occurs at the top part
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of the flame where a strong negative flame stretch presents [45,46]. When the OH* chemiluminescence
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technique is applied, as illustrated in Figure 2 (for the case φ = 1.3), the tip opening phenomenon greatly
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increases the uncertainties of flame contours determination of the top part of the flame. Fortunately,
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Kerosene-PLIF, as applied in the current work, has the advantage of directly imaging the unburned fuel/air
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mixture contour, which is free from the influence of the tip-opening phenomenon. The image-processing
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procedures are detailed in the following sections.
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Energy & Fuels
2.4.1 OH* chemiluminescence
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In order to obtain the two-dimensional flame front of the reaction zone of the OH* chemiluminescence, the
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recorded image is transformed by Abel inversion using a MATLAB program [57]. By taking the inside flame
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front of image after Abel inversion, the surface A can be calculated. The inside boundary is used because its
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laminar flame speed is defined by the velocity relative to the unburned premixed gas in the direction of the flame
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surface. As the inside boundary is more attracted to fresh gas, it is more appropriate to use it as the flame
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boundary in order to calculate flame speed.
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A specific high-frequency filtering function was used to smooth the resulting curve plotted along the red line in
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Figure 3 and obtain the final flame front. For equivalence ratio fuel-lean conditions when the flame tip-opening
283
phenomenon has not occurred, the flame contours can be accurately detected using the above-described
284
image-processing method. However, for cases where the flame-opening phenomenon is present, due to the local
285
quenching of the OH* signal, the flame contours are difficult to detect at the flame tip part. In such cases, the
286
flame area that is used for flame speed determination is obtained by extending the linear peripheral contour to
287
the tip of the flame with the application of a high-order interpolation method, as illustrated in Figure 3. The
288
accuracy of the OH* chemiluminescence image-processing method will be further evaluated by the
289
kerosene-PLIF technique (Part 3.1).
290 291 292 293 294
Figure 3: Determination of the flame surface area: (a) initial OH* chemiluminescence image and (b) left part of the Abel-inverted image (the red line is the initial flame contours determined by taking the inside front of the Abel-inverted image, while the dashed line is the flame contours used in flame speed calculations made following the smoothing function)
295
After obtaining the flame contours, flame area A can be calculated by pivoting flame front profile () along the
296
burner axis using the following equation:
297
= 2 () 1 ! | # ()| $ %
Eq. 2
298
Where & and ' are the boundary limits of the flame contours, and () is the flame contour profile function
299
obtained by image processing. Assuming the axisymmetric condition of the flame, the final flame speed is the 11
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300
averaged value of the flame surface area, obtained from each half of the recorded images. It should be noted that
301
the flame speed measured using the aforementioned equation is an averaged value of the entire flame surface,
302
which is different from the unstretched flame speed, as this conical flame is affected by its strain and curvature
303
effects. To avoid the effect of the flame, the stretch and the flame tip-opening phenomenon, for each case, the
304
measured flame was controlled in relation to the maximum height allowed by the current experimental
305
configuration in order to alleviate the tip of the flame. Fortunately, the difference between both values was found
306
to tiny and negligible, which is consistent with the results in [44, 52, 56].
307
2.4.2 Kerosene-PLIF
308
Compared to the processing procedures of OH* chemiluminescence imaging, the data processing of
309
kerosene-PLIF is largely simplified. The outer edge of the fresh gases is defined as the position at which the
310
fluorescence of the aromatics molecules disappears (Figure 4). This location corresponds to the frontier
311
delimiting a chemical transformation of these organic molecules through the action of chemical reactions.
312
Typically, these organic molecules disappear before the OH radical starts to be optically detected. Once the
313
flame front is obtained, the laminar flame speed is determined using the same calculation (Eq. 2) as the OH*
314
chemiluminescence technique. 18000 16000 14000 12000 10000
Intensity
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Page 12 of 29
8000 6000
B'
A'
4000 2000 0 0
200
400
A
600
800
1000
B
315 316 317
Figure 4: Illustration of the image processing of kerosene-PLIF techniques
2.5 Measurements uncertainties
318
For all the experimental results presented in this work, 30 single-shot images are systematically recorded and the
319
resulting laminar flame speed is determined by the data processing of averaged images, as determined from the 12
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320
set of single-shot images. The measurement uncertainties are mainly associated with the gas/liquid flow delivery
321
system (U)* ) and the uncertainty of the calculated flame area image (U)* ), as determined by the camera
322
resolution. The uncertainty of the total flow rate comes from the mass flow controller uncertainty (0.5% of
323
reading + 0.1% full scale), which is estimated to be ~2%, while uncertainty derives from the camera’s spatial
324
resolution, which is estimated to be ~3%. The overall uncertainty is calculated to be within ~4% (from the
325
$ relation+,- ! ,.$ ) for all the laminar flame speeds recorded under atmospheric pressure conditions. For
326
high-pressure measurements P > 0.5 MPa, the measurement uncertainty can be amplified due to flame stability.
327
Indeed, the fluctuation of the position of the flame displays an artificial thickening of the flame front during the
328
time integration of the signal on the camera. This effect modifies the position of the flame contours (2~3 pixels),
329
giving a maximum uncertainty of ~ 7% in the worst situation.
330
3. Experimental results and discussion
331
Experimental results of n-decane/air mixtures using the OH* chemiluminescence technique will first be
332
presented in order to validate the accuracy of the aforementioned image-processing methodology and the whole
333
experimental system. To further investigate the effect of the flame-opening phenomenon on the laminar flame
334
speed measured by OH* chemiluminescence, a comparison of the laminar flame speeds obtained by OH*
335
chemiluminescence and the kerosene-PLIF technique for Jet A-1/air mixtures under the same operating
336
conditions is performed. The laminar flame speed of pure surrogate components (n-propyl-benzene and
337
propyl-cyclohexane) is then measured under atmospheric pressure conditions at different temperatures, that is,
338
400, 423 and 473 K. The laminar flame speed of the kerosene surrogate (mixtures of n-decane, n-propyl-benzene
339
and propyl-cyclohexane) and the Jet A-1/air mixture are then subsequently measured at T = 400, 423 and 473 K.
340
High-pressure measurements from 0.1-1.0 MPa are performed with an equivalence ratio ranging from 0.7 to 0.8.
341
For high-pressure measurements, the preheated temperature is fixed at T = 423 K, with only the equivalence
342
ratio for the fuel-lean side performed. These temperature and equivalence ratio conditions are selected in order to
343
assure a stabilization of the flame at the nozzle outlet during the experiments and to prevent fuel pyrolysis for
344
temperatures higher than 423 K, especially under elevated pressure conditions. This is essential to guarantee
345
measurement accuracy and make sure that no chemical reaction modifies the fuel composition during the
346
transport of the fuel/air mixture inside the combustion chamber. Experiments are then compared with data found
347
in the literature and the simulation results using the Luche skeletal kerosene mechanism [6].
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348
3.1 OH* chemiluminescence methodology validation
349
Preliminary measurements using OH* chemiluminescence are performed for n-decane/air mixtures under
350
condition φ = 0.7 − 1.3, T = 400 K and atmospheric pressure conditions to validate the whole experimental
351
setup and measurement methodology. Laminar flame speeds calculated in the present work are compared with
352
literature found experimental results. A review of the literature reveals that n-decane is one of the main
353
components in kerosene fuels, while its laminar flame speed has been extensively explored. Figure 5 illustrates
354
the comparison between the experimental results in the present work and the experimental data found from the
355
literature. The simulated flame speeds using detailed kinetic mechanism JetSurF 2.0 are plotted at the same
356
figure as well. JetSurf 2.0 is a detailed mechanism for kerosene fuels, which consists of 348 species and 2,162
357
reactions, as established by Wang et al. [58]. Except for the results presented by Hui et al. [19] and those
358
reported by Kumar et al. [37], which shows significant difference from the other measurements, there is a
359
general agreement between our laminar flame speeds and those in the literature, albeit with a better agreement
360
between our data and the JetSurF 2.0 simulation results. Our experimental results show good agreement with the
361
laminar flame speed measured using spherical flames as well [7, 59, 60]. However, a tiny difference between our
362
experimental results and the numerical predictions is observed at condition of equivalence ratio φ = 1.3. 70 60 50
SL (cm/s)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
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40 30
Jet surf 2.0 Hui et al. 2013 Singh et al. 2011 Kumar et al. 2007 Comandini et al. 2015 Kim et al. 2015 Munzar et al. 2013 Present work
20 10 0 0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
Equivalence ratio
363 364 365 366
Figure 5: Comparison between our experimental results and the data from the literature of n-decane/air
367
Figure 6 shows experimental results for higher temperature conditions of T = 423 K and 473 K. As expected,
368
the laminar flame speed increased with the preheating temperature of the mixture. The conditions where the
369
flame tip-opening phenomenon was present are also labeled in the figure. In order to further clarify the influence
370
of the flame tip-opening phenomenon on the determination of the flame speed using the OH*
371
chemiluminescence
mixtures: φ = 0.6-1.5, T = 400K and P = 0.1 MPa
method,
experiments are
performed with both the
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kerosene-PLIF and OH*
Page 15 of 29
372
chemiluminescence techniques for Jet A-1/air mixtures at T = 400 K, P = 0.1 MPa and φ = 0.6 − 1.3. As
373
depicted in Figure 7, laminar flame speeds derived from the kerosene-PLIF technique show general agreement
374
with those determined with OH* chemiluminescence, with a maximum difference of 3 cm/s on the fuel-lean side.
375
There is a better agreement between the results of the OH* chemiluminescence technique and the kerosene-PLIF
376
technique under approaching stoichiometric conditions. For the fuel-rich side, where the flame tip-opening
377
phenomenon is present, the difference between these two techniques is less than 2 cm/s. This illustrates that the
378
OH* chemiluminescence image-processing method proposed in the present work is able to determine the flame
379
speeds with reasonable accuracy when the flame tip-opening phenomenon is present. The experimental results
380
presented in the following sections were obtained using OH* chemiluminescence technique considering the
381
simplicity of this methodology.
382 90 85
n-decane T = 400 K n-decane T = 423 K n-decane T = 473 K
80 75 70
SL (cm/s)
65 60 55 50 45 Flame tip opening
40 35 30 25 20 0.4
0.5
0.6
0.7
0.8
0.9
383 384 385
1.0
1.1
1.2
1.3
1.4
1.5
ϕ
Figure 6: Laminar flame speed of n-decane /air mixture at T = 400, 423 and 473 K, P = 0.1 MPa and φ = 0.65-1.3 60 55 50
SL (cm/s)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
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45 40
OH* Chemiluminescence Kereosene-PLIF
35 30 0.5
386 387 388 389
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.5
Equivalence ratio
Figure 7: Comparison between laminar flame speeds determined from OH* chemiluminescence and Kerosene-PLIF of Jet A-1/air mixture, T = 400 K, P = 0.1 MPa, φ = 0.65-1.3
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390
3.2 Laminar flame speed of neat surrogate components 3.2.1 N-propyl-benzene
391 392
N-propyl-benzene is a large molecular weight fuel that can be largely found in kerosene fuels. However, its
393
laminar flame speed has not been widely studied. The only literature results can be found is the measurements
394
recently performed with the stagnation flame [11] and the heat flux methods [61]. Figure 8 plotted the
395
experimental results obtained in the present work and those presented in [11] [61]. It shows that our
396
experimental results are in accordance with those reported in the work of Mehl et al. [61] for all the equivalence
397
ratio conditions investigated. The maximal difference observed from this comparison is around 5 cm/s, which is
398
a value that remains comparable with our measurement uncertainty. Meanwhile, our experimental results are
399
shows lower flame speeds compared with the results obtained by Hui et al. [11]. As shown in Figure 9, in order
400
to investigate the effect of preheating temperature on laminar flame speeds, experiments were performed with
401
higher temperature T = 423 K and 473 K. As expected, the experiments results show that the laminar flame
402
speed increases with higher temperature conditions. 70 65 60 55 50
SL (cm/s)
45 40 35 30 25 20
Hui et al. 2012, T=400 K Mehl et al. 2015, T = 403 K Present work
15 10 5 0 0.4
0.6
0.8
1.0
403 404
1.2
1.4
1.6
ϕ
Figure 8: Laminar flame speed of n-propyl-benzene at T = 400 K, φ = 0.6-1.3 and P = 0.1 MPa [11] [61] propylbenzene T=400K propylbenzene T=423K propylbenzene T=473K
80
SL (cm/s)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Page 16 of 29
60
Flame tip opening
40
20 0.4
0.6
0.8
1.0
405 406 407
1.2
1.4
1.6
ϕ
Figure 9: Laminar flame speed of n-propyl-benzene at T = 400 K, 423 K and 473 K, φ = 0.6-1.3 and P = 0.1 MPa 16
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3.2.2 Propyl-cyclohexane
408
Propyl-cyclohexane is one of the main cycloalkanes that are usually present in kerosene fuels. Different from
410
n-alkanes, branched alkanes or aromatics, laminar flame speeds measurements of propyl-cyclohexane over a
411
large range conditions of preheating temperature and equivalence can be rarely found in literature. In the present
412
work, experimental measurements were firstly performed at T = 400 K, P = 0.1 MPa and φ = 0.65-1.3, then
413
compared with the data found in the literature. As shown in
414
Figure 10, at the equivalence ratio lean side our experimental results are in accordance both with the results
415
presented in the work of Comandini et al. [7] (nonlinear extrapolation at a zero stretch rate) and with the results
416
presented in the work of Dubois et al. [62] (linear extrapolation at a zero stretch rate). However, discrepancy was
417
observed at fuel rich side that our experimental results give higher laminar flame speed compared with literature
418
found results. This discrepancy could come from the influence of the tip-opening phenomenon, when active for
419
these operating conditions of φ = 1.15-1.3. Experiments were also performed for higher temperature conditions
420
of T = 423 K and 473 K, as plotted in Figure 11.
SL (cm/s)
409
90 85 80 75 70 65 60 55 50 45 40 35 30 25 20 15 10 5 0
Dubois et al. 2009, T=403K Ji et al. 2011, T=403K Comandini et al. 2015, T=403K
0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9
ϕ
421 422
Figure 10: Laminar flame speed of propyl-cyclohexane at T = 400 K, P = 0.1 MPa, φ = 0.65-1.3 [7] [62] 90 85
T=473K T=423K T=400K
80 75 70 65
SL (cm/s)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Energy & Fuels
60 55 50 45
Flame tip opening
40 35 30 25 0.3
0.4
0.5
0.6 0.7 0.8
0.9
423 424 425
1.0
1.1
1.2 1.3 1.4 1.5
1.6
ϕ
Figure 11: Laminar flame speed of propyl-cyclohexane/air mixture at T = 400, 423 and 473 K, P = 0.1 MPa and φ = 0.65-1.3 17
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426
3.2.3 Comparison between the pure components
427
Figure 12 plots the laminar flame speeds of the three component in Luche surrogate fuel at condition of T = 400
428
K, P = 0.1 MPa and / = 0.65 − 1.3. It shows that for all equivalence ratio conditions investigated in the present
429
work, the three pure components have quite similar laminar flame speed values. However, a more careful
430
observation of the results reveals that n-decane has slightly higher flame speed values (about a maximum of 2.5
431
cm/s at fuel lean side) than n-propyl-benzene. This tiny difference can be explained by the different molecular
432
structure between n-decane and n-propyl-benzene. As discussed in the work of Comandini et al. [7],
433
n-propyl-benzene is an aromatic molecule who has a ring chemical structure which potentially have an influence
434
on the combustion reaction. Indeed, a sensitivity analysis explained that the flame speed is affected by the ring
435
structure, in particular, by the reactions involving radical intermediates, such as phenoxy, benzyl and the phenyl
436
radicals. Moreover, the work of Mehl et al. [61] also mentioned that the increase of benzyl radicals in the
437
pre-heating zone restrains flame propagation and can decrease flame speed. The above analysis indicates that the
438
laminar flame speed of propyl-cyclohexane is higher than that of n-propyl-benzene. 70 65 60 55 50
SL (cm/s)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Page 18 of 29
45 40
Propylbenzene Propylcyclohexane n-decane
35 30 25 20 0.5
0.6
0.7
0.8
0.9
1.0
439 440 441
1.1
1.2
1.3
1.4
1.5
1.6
ϕ
Figure 12: Laminar flame speeds of n-decane, n-propyl-benzene, and n-propyl-cyclohexane at T=400 K, P=0.1 MPa and / = 0.65 − 1.3
442 443
As n-decane and propyl-cyclohexane both belong to the alkane category, they can be directly compared. Our
444
experiments results show that laminar flame speed of n-decane (n-alkane) is almost identical but uniformly
445
higher than those of propyl-cyclohexane (mono-alkylated cyclohexane). This observation is with agreement of
446
the work of Ji et al. [63] and Wu et al. [64]. In their study, it shows that event though laminar flame speed of
447
n-decane is influenced by the small intermediates, the decomposition of linear structure of n-alkane (such as
448
n-decane) is faster than the decomposition of the ring structure exist in mono-alkylated cyclohexane (such as
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Energy & Fuels
449
propyl cyclohexane). Therefore, these experimental results shed light on the fact that laminar flame speeds are
450
influenced by both the small intermediates involved in the combustion reactions and the initial step in the fuel
451
decomposition. [7]. For lean equivalence ratio conditions, above mentioned tendency is well observed in Figure
452
12: n-decane > propyl-cyclohexane > n-propyl-benzene. However, discrepancies among these measurements are
453
limited (about 2-3 cm/s), meaning that it is difficult to estimate this tendency with accuracy. According to our
454
measurement results, laminar flame speed of these three components is basically the same at fuel-lean conditions.
455
For fuel-rich mixtures, the propyl-cyclohexane obviously shows higher laminar flame speeds than those
456
observed for n-decane, which contradicts the results found in the literature. These discrepancies could be caused
457
by the aforementioned flame tip-opening phenomenon, which makes the flame contour determination more
458
difficult. Consequently, the measurement uncertainties increase, making the elucidation of the evolution of the
459
different laminar flame speeds more complex. For greater clarity, error bars of the propyl-cyclohexane
460
measurements in fuel-rich conditions have been added in Figure 12, representing a global uncertainty of around
461
4.5%.
462 463
3.3 Laminar flame speed of surrogate kerosene
464
In this part, laminar flame speeds of the Luche surrogate fuel consisting of n-decane (76.7 wt%),
465
n-propyl-benzene (13.2 wt%) and propyl-cyclohexane (10.1wt%), are investigated in a large rang of working
466
conditions, including variation of temperature (400-473 K), pressure (0.1-1.0 MPa) and equivalence ratio
467
(/ = 0.6 − 1.3). The measurements are compared with simulation results using the Luche kinetic mechanism
468
and compared to experimental results of neat component of surrogate Luche. Experimental data of Luche
469
surrogate is firstly plotted with neat component of surrogate in Figure 13. It depicts that the difference between
470
laminar flame speed of Luche surrogate and its three component is tiny, meanwhile its laminar flame speed is
471
more approaching to n-decane.
472
0.6 - 1.3 conditions are compared and plotted in Figure 14, which shows that the model developed by Luche is
473
effective at predicting laminar flame speeds. For all equivalence ratio investigated, simulation results using
474
Luche mechanism give slightly lower values compared with experimental results. Meanwhile, it should be noted
475
that this difference is quite close to measurement uncertainties. A comparison of the different temperature
476
conditions also illustrated that the Luche kinetic mechanism effectively predicts the effect of preheating
477
temperature on laminar flame speed that higher temperature increases the laminar flame speed. However, the
Simulation results of Luche surrogate at T= 400 - 473 K, P = 0.1 MPa and φ =
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478
mechanism underestimates flame speed of Luche surrogate about 2-3 cm/s, compared with our measurements for
479
all temperature conditions investigated. 65 60 55
SL (cm/s)
50 45 40
Propylbenzene Propylcyclohexane n-decane LUCHE experimental results
35 30 25 0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
ϕ
480 481 482
Figure 13: Laminar flame speeds of n-decane, n-propyl-benzene, propyl-cyclohexane and the LUCHE surrogate at condition of T=400 K, P=0.1 MPa and. / = 0.65 − 1.3
80
60
SL (cm/s)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Page 20 of 29
40 LUCHE mechanism T = 400 K LUCHE mechanism T = 423 K LUCHE mechanism T = 473 K present work measurements T = 400 K present work measurements T = 423 K present work measurements T = 473 K
20
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
ϕ
483 484 485 486
Figure 14: Comparisons between experimental results and simulation results of Luche surrogate fuel with
487
In order to evaluate the capability of Luche kinetic mechanism under high pressure conditions, laminar flame
488
speed measurements of Luche surrogate fuel are performed at T = 423 K, φ = 0.7 and 0.8, and under pressure
489
conditions from 0.1 to 0.8 MPa. In Figure 15, experimental results of Luche surrogate fuel under high pressure
490
conditions are compared with simulation results using Luche skeletal mechanism in a logarithmic graphic. The
491
same as observed at atmospheric pressure conditions, the Luche mechanism gives lower flame speed values
492
compared with our measurements for both equivalence ratios. Meanwhile, Luche mechanism can well reproduce
493
the effect of pressure on laminar flame speed that the higher pressure gives lower flame speed values for
temperature variations (T = 400, 423, 473 K, P = 0.1 MPa and / = 0.6 − 1.3)
20
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494
condition of at φ = 0.8, while a discrepancy in the dependence of pressure is observed for conditions of
495
equivalence ratio φ = 0.7. With the increments of pressure, the discrepancy between experiments and simulations
496
tends to increase. However, aforementioned discrepancy must be put into perspective because the maximum
497
difference between the simulation and the experimental results does not exceed 5 cm/s, which is a value
498
comparable to the uncertainty of our measurements under high pressure conditions. This indicates that the Luche
499
skeletal mechanism performs better under higher rather than lower pressure conditions.
500 60 55 50 45 40
SL (cm/s)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Energy & Fuels
35 30 25 20 Experimental results ϕ = 0.8 Experimental results ϕ = 0.7 LUCHE mechanism ϕ = 0.8 LUCHE mechanism ϕ = 0.7
15 10 5 0 0.1
501 502 503 504 505
0.2
0.3
0.4
0.5
0.6 0.7 0.8 0.9 1
Pressure (MPa)
Figure 15: Comparison of laminar flame speeds of the Luche surrogate between experimental results and simulation results at condition of T= 423 K, φ = 0.7 and 0.8, P = 0.1 – 0.8 MPa
3.4 Laminar flame speed of Jet A-1
506
In this part, laminar flame speed measurements of commercial kerosene fuel Jet A-1 are conducted over a large
507
range of working conditions. Measurements are also performed at high temperature and elevated pressure to
508
investigate temperature and pressure dependency on laminar flame speed of kerosene Jet A-1. An empirical
509
correlation of laminar flame speed in taking account of temperature and pressure dependency is then proposed.
510
As illustrated in Figure 16, the general tendency of the effect of the preheating temperature on the laminar flame
511
speed is in agreement with theoretical predictions, i.e., the laminar flame speed increases in line with the
512
preheating temperature. According to the power law correlation theory of temperature dependency =
513
0 1 4
514
flame speed at the reference condition of temperature and pressure ( 0 ) and multiplied by correction factors
515
displaying the temperature and pressure dependencies. As shown in Figure 17, laminar flame speeds are plotted
516
as a function of the preheating temperature using log-log scales, while straight lines for all the equivalence ratios
517
tested are observed.
2 5
23
[44], the laminar flame speed under higher temperature conditions can be expressed by a laminar
21
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85 80 75 70 65
SL (cm/s)
60 55 50 45 40 35
T = 400K T = 423K T = 445K T = 473K
30 25 20 15 0.4
0.5
0.6
0.7
0.8
519 520 521
0.9
1.0
1.1
1.2
1.3
1.4
1.5
ϕ
518
Figure 16: Laminar flame speed of Jet A-1/air with temperature variation – T = 400 K, 423 K, 445 K and 473 K with φ = 0.6-1.3, P = 0.1MPa
522 1.9
log(SL)
1.8
1.7
1.6
0.7 0.9 1.1
0.8 1.0 1.2
1.5 1.0
1.1
1.2
log (T/T0)
523 524 525
Figure 17: Log-log plot of the laminar flame speed of Jet A-1/air under atmospheric pressure and different preheating temperatures (the symbols are the experiments and the lines are linear fits) 60 55 50 45 40
SL (cm/s)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Page 22 of 29
35 30 25 20 Jet A-1 ϕ = 0.7 Jet A-1 ϕ = 0.8
15 10 5 0 0.1
526 527 528 529
0.2
0.3
0.4
0.5
0.6 0.7 0.8 0.9
Pressure (MPa)
Figure 18: Laminar flame speed of Jet A-1/air mixture at T = 423 K, P = 0.1 – 0.8 MPa and φ = 0.7 and 0.8 (the points are our measurements and dash lines are the linear fits)
22
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Energy & Fuels
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With the experimental results obtained in the present work, an empirical temperature dependency correlation was
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proposed by using the power law equation S7 = S7 0 1 4 , where S7 0 is defined as the laminar flame speed at
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T = 400 K and P = 0.1 MPa. The effect of the equivalence ratio is then taken into account by using the extended
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formulation proposed by Metghalchi and Keck [65] :
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S0 (φ) = S73,9:; ! S73,; (φ − 1) ! S7,$ (φ − 1)$ ! S7,< (φ − 1)
