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Thermal radiative study of counterflow combustion of porous particles
Accepted Manuscript Title: Thermal Radiative Study of Counterflow Combustion of Porous Particles Authors: Hesam Moghadasi, Navid Malekian, Alireza Khoeini Poorfar, Mehdi Bidabadi PII: DOI: Reference:
S0255-2701(18)30450-1 https://doi.org/10.1016/j.cep.2018.10.018 CEP 7412
To appear in:
Chemical Engineering and Processing
Received date: Revised date: Accepted date:
12 April 2018 27 August 2018 28 October 2018
Please cite this article as: Moghadasi H, Malekian N, Khoeini Poorfar A, Bidabadi M, Thermal Radiative Study of Counterflow Combustion of Porous Particles, Chemical Engineering and Processing - Process Intensification (2018), https://doi.org/10.1016/j.cep.2018.10.018 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Thermal Radiative Study of Counterflow Combustion of Porous Particles
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Hesam Moghadasia, Navid Malekiana, Alireza Khoeini Poorfara, Mehdi Bidabadia
School of Mechanical Engineering, Department of Energy Conversion, Iran University of Science and
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Technology (IUST), Narmak, 16846-13114, Tehran, Iran.
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Graphical abstract
Corresponding author: School of Mechanical Engineering, Department of Energy Conversion, Iran University of Science and Technology, Email address: [email protected] Tel: +98 21 77 240 540; Fax: +98 77 240 488.
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Fig. Structure of diffusion counterflow combustion
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Highlights:
Non-premixed combustion of dust particles in a counterflow system is studied
Particles porosity and thermal radiation modeling are investigated
Effects of fuel Lewis number, particle porosity on flame temperature are studied
Effects of equivalence ratio and particles radius on flame temperature are studied
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Abstract
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The paper deals with modeling counterflow, non-premixed combustion of porous fuel particles (lycopodium) with the consideration of thermal radiation effects. Assuming that the streams of fuel particles and air as oxidizer, move towards the stagnation plane from the two opposing nozzles in a counterflow configuration. It is presumed that particles first vaporize in order to yield a gaseous fuel, methane, which then reacts with the oxidizer which is air. In this research, conservation equations with
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certain boundary conditions are solved using mathematical methods with the consideration of radiation heat transfer in different regions and compared to cases in which radiation heat transfer is not considered. Furthermore, flame temperature and mass fraction profiles are presented in terms of oxidizer and fuel
Lewis numbers. In addition, effects of particle porosity are investigated. As a result, with the increase of dust concentration and reduction of particles radius and porosity, it would lead to a rise in the flame
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temperature.
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Keywords: Thermal Radiative, Porous Particles, Counterflow Combustion, Non-Premixed, Lewis
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number.
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1. Introduction
Development in numerical and mathematical methods gives researchers the opportunity to study several
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types of problems in all fields of engineering with lower costs. Biomass gasification, combustion, chemical reactions are some of research works in different facets of engineering using these methods.
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Generating energy from the renewable resources is the one main focuses of the present research due to the universal awareness of reducing our dependency on fossil fuels. Also, in industries that produce
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combustible organic dust, a deep knowledge of the explosion hazards is essential. Process Intensification (PI) is a topic receiving considerable attention recently. Using the simple definition of Stankiewicz and Moulijn (2000), PI is ‘Any chemical engineering development that leads to a substantially smaller, cleaner, safer, and more energy efficient technology. Counterflow combustion mechanism would lead to
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smaller and more compact combustion chamber and propulsion systems. As one of important factors in aerospace industries are weight and space this would lead to reduction in costs [1, 2]. Due to the importance of the subject, in this paper an effort is made to improve the understanding of
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particle dust combustion that simultaneously involves the transfer of heat, mass, phase change and chemical reaction. Combustion of combustible fuel particles and oxidizer is used in many industries and safety fields. Hence, experimental studies and theory of combustion phenomena seem to be essential in
this field [3-5]. Combustion of particles is usually unstable and there are numerous interactions between particles. Therefore, combustion of particles is influenced by the physical and chemical properties of fuel,
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average particles size and shape and their distribution. Radiation heat transfer phenomena is usually an important process in the flame propagation mechanism. Furthermore, Radiation heat transfer is a
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dominant process in spreading of flame through a cloud of dust particles and leads to further release of
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heat energy. This increases the burning velocity and also the thickness of the reaction zone [6]. Proust [7] measured the laminar burning velocity and maximum flame temperature of combustion of
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particles-air mixture (organic particles including lycopodium-air and sulphur flour–air) in a planar
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configuration. Bidabadi et al [8] proposed an analytical model to investigate the effect of randomly distributed micro-organic dust particles on the flame propagation speed through the combustible mixture.
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Seshadri and Trevino [9] evaluated asymptotic structure of counterflow diffusion flames. In this research, maximum flame temperature with respect to the Lewis number variations of fuel and oxidizer for a
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specific gaseous fuel with a known chemical structure was provided. Also, quenching distance of the flame was determined. Linan [10] modeled the counterflow diffusion combustion by analyzing chemical
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reaction in flame zone and mixing quality of the fuel and oxidizer which were moving toward each other from two opposite jet. Li et al. [11] investigated counterflow methanol and heptane spray combustion both theoretically and experimentally and showed that by increasing strain rate, the distance of vaporization’s starting point and flame formation from the nozzle will be decreased.
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In the last decades, study of heat transfer mechanism by radiation in combustion of dust particles were performed. Proust et al. [12] presented a new experimental measurement of radiation heat transfer in dust flames (methane-air, methane-air seeded with inert particles, aluminum-air flames). The results focused on the amount of heat radiated by a propagating flame. The effects of radiation on flame acceleration and
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transition from deflagration to detonation (DTD) in hydrogen dust cloud were studied by Liberman et al. [13]. They evaluated the role of radiation heat transfer diffusion from the flame zone and the gaseous
products to the pre-heat zone and the unburned particles. Bidabadi et al. [14] investigated the effect of
radiation heat transfer on the flame propagation of micro organic dust cloud with variable Lewis number. Their results show that, by increasing the temperature of flame and burning velocity, the Lewis number
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will be decreased and equivalence ratio will be increased. A numerical investigation on self-absorption
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and soot radiation in laminar diffusion flame of the methane and air mixture were studied by Datta and
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should be considered at least in a thin layer.
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Saha [15]. They showed that the influence of thermal radiation in the combustion of methane and air
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Levin and Lutsenkoa [16] investigated a two-dimensional unsteady gas flow in a porous media with
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heterogeneous combustion. Using numerical methods, it was shown that, flows with vortex can be formed in combustion of solid fuels in porous media. De Souza [17] determined a one-dimensional model for
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counterflow combustion using the Riemann solution due to injection of air into a porous medium containing solid fuel. Assuming that the combustion front is modeled by a traveling wave, they obtained a
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family of Riemann solutions for the same initial and injection conditions. Hurevich et al. [18] analyzed the behavior of porous particles in a plasma flame, theoretically. The porous particles were supposed to be
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a conglomerate of fine precursors with a size of roughly 1μm. Gremyachkin [19] extended a model for combustion of porous carbon particles in oxygen. This model considers heat and mass transfer in both gas phase above particle’s surface and inside the porous particle. The conditions for the model are determined by solving the diffusion equation in the gas phase around the particle.
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In this study, an analytical model is presented for combustion of organic porous particles in a nonpremixed counterflow configuration. It is presumed that the flame structure consists of three zones including pre-heat, post-vaporization and post-flame zones. In contrast to the previous analytical studies, effects of radiation heat transfer and particles porosity are taken into account in this thermal model. Also,
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thermal radiation equations state the radiation exchanges between these zones. The model was considered to be one-dimensional and insulated against thermal heat loss. The carrying gas of dust particles is
assumed nitrogen. Fuel and oxidizer mass fractions along with temperature distribution are studied with respect to position. Moreover, effects of radiation, particles porosity, Lewis number, etc. on flame
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temperature are investigated.
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2. Governing Equations
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In this research, the counterflow configuration is considered in a way that organic particles enters the combustion domain from the direction −∞ and move towards the stagnation plane and the oxidizer flows
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away from the nozzle from the direction +∞. At first, fuel particles vaporize to produce a gaseous fuel
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with a specific chemical structure, Methane. Surface reactions are ignored in this study. Then, the gaseous
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fuel reacts with the oxidizer (air). The position of flame formation depends on the initial conditions which can occur in the left or right hand side of the stagnation plane. Changing the initial conditions will lead to
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a change in this position. The structure of diffusion counterflow combustion of organic particles is shown in Fig. 1 in a model with thin reaction and vaporization zones. Fuel particles in an asymptotic zone which
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is called vaporization front, suddenly vaporize and a gaseous fuel will be yielded. The gaseous fuel reacts with the oxidizer flow in an asymptotic zone which is called flame front. The flame front position can be formed on the sides of the stagnation plane which depends on the initial conditions of the problem. As seen in Fig. 1, the flame position is located at the left side of the stagnation plane which also can happen at the right side.
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Fig. 1. Structure of diffusion counterflow combustion
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2.1. Radiation heat transfer modelling
To analyze radiation heat transfer in a two-phase mixture, organic particles and air are considered as a gray medium. For a two-phase gray medium which consists of scattering, absorbing and emitting, the radiative heat transfer equation can be expressed as [20, 21]: 𝐾
= 𝐾𝑎 𝐼 + 𝐾𝑠 𝐼 − 𝐾𝑎 𝐼𝑏𝑃 − 4𝜋𝑠 ∫4𝜋 𝐼(𝛺)𝑃(𝜃,𝜑)𝑑𝛺
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(1)
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𝑑𝐼 𝑑𝑥
The right hand side terms of Eq. (1) are radiation intensity caused by absorption, scattering, emission and
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incoming scattering brought about by other particles, respectively. Also, 𝐾𝑎 , 𝐾𝑠 and I are absorption
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coefficient, scattering coefficient and radiation intensity, respectively. 𝑃(𝜃,𝜑) is scattering phasic
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function. Also, 𝐼𝑏𝑃 is the blackbody emissivity power which depends on local temperature of particles. In
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this model, the shape of particles is spherical and their size is uniform. The properties of a two-phase mixture including uniform sized particles can be expressed as [22]:
𝐾𝑎 =
2 𝜋𝑑𝑃 𝑛𝑠 𝑄𝑎𝑏𝑠 4
= 2𝜌
𝐾𝑠 =
2 𝜋𝑑𝑃 𝑛𝑠 𝑄𝑠𝑐𝑎 4
= 2𝜌
𝜑
𝑄𝑎𝑏𝑠
(2)
𝑄𝑠𝑐𝑎
(3)
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𝑃 𝑑𝑃
3
𝜑
𝑃 𝑑𝑃
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which 𝑑𝑃 and 𝜑 are particle diameter and mass density, respectively. Also, 𝑄𝑎𝑏𝑠 and 𝑄𝑠𝑐𝑎 are efficiency factors of scattering and absorption, respectively. If particles are diffusely reflecting spheres, one can write [22]: 𝑄𝑎𝑏𝑠 = 𝜖𝑃
,
𝑄𝑠𝑐𝑎 = 1 − 𝜖𝑃
(4)
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where 𝜖𝑃 is emissivity of solid particles. Eq. (1) has a differential-integral nature which is difficult to be solved using analytical approaches. So as to overcome this difficulty and simplify the problem, integral term of the Eq. (1) is neglected. In this part of study, the radiative heat transfer (𝑄𝑟𝑎𝑑 ) is analyzed. By
structure is obtained as: Pre-heat zone
𝑄𝑟𝑎𝑑 = 𝐾𝑎 𝐼𝑓 𝑒𝑥𝑝[𝐾𝑡 (𝑥 − 𝑥𝑓 )]
𝑅1 : − ∞ < 𝑥 ≤ 𝑥𝑣
Post-vaporization zone
𝑄𝑟𝑎𝑑 = 𝐾𝑎 𝐼𝑓 𝑒𝑥𝑝[𝐾𝑡 (𝑥 − 𝑥𝑓 )]
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Post-flame zone
(6)
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𝑅2 : 𝑥𝑣 ≤ 𝑥 ≤ 𝑥𝑓
(5)
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solving the general radiation heat transfer equation in every zone, radiative heat transfer in the flame
𝑅3 : 𝑥𝑓 ≤ 𝑥 < +∞
(7)
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𝑄𝑟𝑎𝑑 = −𝛺𝐾𝑎 𝐼𝑏𝑃
, 𝐼𝑏𝑃 =
𝜎𝑇𝑓4 𝜋
, 𝐼𝑓 =
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𝐾𝑡 = 𝐾𝑎 + 𝐾𝑠
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Where
𝐾𝑎 𝐼 𝐾𝑡 𝑏𝑃
𝐾
𝑠 , 𝛺 = 𝐾 +𝐾 𝑠
𝑎
(8)
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where 𝐼𝑓 is the induced radiation intensity from the flame front into pre-heat and vaporization zones. Also, σ and 𝑇𝑓 are Stefan-Boltzmann constant and flame temperature in the thin reaction zone,
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respectively.
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In the current research study, the structure of particles porosity is considered as shown in Fig. (2).
Fig. 2. Structure of particles porosity
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As stated previously, porosity of a material is a criteria to measure the empty spaces within that material. Porosity is defined as the ratio of void volume (𝑉𝑣𝑜𝑖𝑑 ) per total volume (𝑉𝑡𝑜𝑡𝑎𝑙 ) of the material, hence: 𝑉
𝜀 = 𝑉 𝑣𝑜𝑖𝑑
(9)
𝑡𝑜𝑡𝑎𝑙
𝑒=
𝑉𝑣𝑜𝑖𝑑 𝑉
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In which 𝜀 denotes porosity. In addition, apparent porosity is defined as:
(10)
Consequently, a general statement for porosity is written as follows: 𝑉
𝑒
𝑣𝑜𝑖𝑑 𝜀 = 𝑉+𝑉 = 1+𝑒
(11)
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𝑣𝑜𝑖𝑑
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Here, an important parameter is defined which is called porosity function: (12)
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𝑓(𝜀) = 1 − 𝜀
In combustion of organic particles, the vaporization rate defined as produced gaseous fuel mass per unit
considered as [23]: 𝑌
𝑣
(13)
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𝜔𝑣 = 𝜏𝑠 𝐻(𝑇 − 𝑇𝑣 )
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volume and time, is a controller parameter of combustion process. In this research, vaporization rate is
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Where 𝑌𝑠 mass fraction of solid fuel is, 𝜏𝑣 is characteristic time of vaporization, 𝑇 and 𝑇𝑣 are fuel and vaporization temperatures. Also, 𝐻 is Heaviside function which is defined as:
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0 𝐻(𝑇 − 𝑇𝑣 ) = { 1
𝑇 < 𝑇𝑣 𝑇 ≥ 𝑇𝑣
(14)
Another factor which controls the combustion process is Lewis number which is defined as the ratio of heat diffusion to mass diffusion, thus:
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𝐿𝑒𝑖 =
𝜆 𝜌𝐶𝐷𝑖 (15)
where 𝜆, 𝜌, 𝐶 and 𝐷𝑖 are conductivity, density, specific heat and mass diffusive coefficient of gaseous fuel
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or oxidizer, respectively. In this paper, Lewis number of oxidizer and fuel are considered to be non-unity and their variations on the flame are investigated. Chemical kinetic is assumed as a general one-stage reaction, therefore: 𝑣𝐹 [𝐹] + 𝑣𝑂 [𝑂] → 𝑣𝑃 [𝑃]
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(16)
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where [F], [O] and [P] denote fuel, oxidizer and products, respectively. Quantities of 𝑣𝐹 , 𝑣𝑂 and 𝑣𝑃
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express stoichiometric coefficient of fuel, oxidizer and products, respectively. Velocity field is considered
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as (𝑢,𝑣) = (−𝑎𝑋,𝑎𝑌) , where 𝑢 and 𝑣 are velocities in 𝑋 and 𝑌 directions. In this relation, a is known as velocity gradient at the stagnation point which also represents the strain rate. If the strain rate is assumed
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to have a small amount, the problem can be solved in one dimension and only in 𝑋 direction. Also, it is
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considered that oxidizer and fuel velocity flows are equal. The amounts of density, specific heat and other transient coefficient are assumed to be constant. Other assumptions considered in this work are as
The processes between the preheating and beginning of the lycopodium particles’ vaporization
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follows:
are neglected and is assumed that lycopodium dust particles vaporize first to yield a gaseous fuel,
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which is presumed to be methane.
The inert carrying gas of dust particle is assumed nitrogen which does not take part in the combustion reaction.
Heterogeneous and surface reactions are negligible.
Vaporization happens in a thin asymptotic front.
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Gas thermal conductivity (𝜆) is assumed constant in all temperatures.
The particles shape is spherical and their size is uniform. Also, volumetric porosity has been considered for particles. Effects of forces including thermophoresis, buoyancy and drag are neglected.
The influences of pressure gradient, heat loss and heat recirculation are ignored.
Lewis number of oxidizer and fuel are considered to be non-unity.
There is no temperature difference between the gas and particles.
Mass conservation of gaseous fuel = 𝐷𝐹
−
𝜔𝐹 𝜔 𝑓(𝜀) + 𝑣 𝑓(𝜀) 𝜌 𝜌
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𝑑 2 𝑌𝐹 𝑑𝑋 2
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𝑑𝑌𝐹 𝑑𝑋
(17)
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−𝑎𝑋
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where 𝐷𝐹 and 𝑌𝐹 are mass diffusivity of fuel and mass fraction of gaseous fuel, respectively. 𝜔𝐹 is the
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rate of chemical reaction which follows the Arrhenius rule and is in the first order relative to fuel and oxidizer. It is define as [10, 24]:
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𝐸 𝜔𝐹 = 𝐵𝜌2 𝑣𝐹 𝑣𝑂 ̅̅̅ 𝑌𝐹 ̅̅̅ 𝑌𝑂 𝑒𝑥𝑝 (− 𝑅𝑇)
(18)
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𝐵 is a frequency constant and ̅̅̅ 𝑌𝑂 and ̅̅̅ 𝑌𝐹 are defined as: 𝑚 ̅̅̅ 𝑌𝐹 = 𝑌𝐹 𝑣 𝑚
𝐹
𝑚 , ̅̅̅ 𝑌𝑂 = 𝑌𝑂 𝑣 𝑚 𝑂
𝑂
(19)
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𝐹
where 𝑌𝐹 , 𝑌𝑂 , 𝑚𝐹 , 𝑚𝑂 and 𝑚 are mass fraction of fuel and oxidizer, molecular weight of fuel and oxidizer and molecular weight of mixture, respectively.
Mass conservation of solid fuel
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If the solid particles diffusion is negligible and solid particles don’t have any reactions together, the mass conservation of solid fuel particles is written as: 𝑑𝑌
−𝑎𝑋 𝑑𝑋𝑠 = −𝑓(𝜀)𝜔𝑣
(20)
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where 𝑌𝑠 is mass fraction of solid particles and 𝜔𝑣 is the vaporization rate which has been defined in Eq. (13), previously.
Energy Conservation of Mixture 𝑑𝑇
𝑑2 𝑇
𝑄
𝑄𝑣 𝑄𝑟𝑎𝑑 𝑓(𝜀) + 𝜌𝐶 𝐶
(21)
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−𝑎𝑋 𝑑𝑋 = 𝐷𝑇 𝑑𝑋 2 + 𝜔𝐹 𝜌𝐶 𝑓(𝜀) − 𝜔𝑣
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where 𝑄 is the heat released per unit of consumed fuel mass, 𝑄𝑣 is the latent vaporization heat of particles 𝜆
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and 𝑄𝑟𝑎𝑑 is radiation heat transfer. Also, 𝐷𝑇 = 𝜌𝐶 is thermal diffusivity and 𝐶 is specific heat of mixture
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which is obtained from combination of gaseous phase specific heat 𝐶𝑎 and solid particles specific heat 𝐶𝑃
4𝜌𝑃 𝜋𝑟 3 𝐶𝑃 𝑛𝑃 3𝜌
(22)
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𝐶 = 𝐶𝑎 + 𝑓(𝜀)
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as below [24, 25]:
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where 𝜌𝑃 is the solid particle density and 𝑛𝑃 shows the number of particles per unit of volume. Thus: 4
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𝜌 = 𝜌𝑎 + 𝑓(𝜀) 3 𝜋𝑟𝑃3 𝑛𝑃 𝜌𝑃
A
(24)
(23)
Mass conservation of oxidizer −𝑎𝑋
𝑑𝑌𝑂 𝑑𝑋
= 𝐷𝑂
𝑑 2 𝑌𝑂 𝑑𝑋 2
−𝜐
𝜔𝐹 𝑓(𝜀) 𝜌
In the above equation 𝐷𝑂 , 𝑌𝑂 and 𝜐 are oxidizer mass diffusivity, oxidizer mass fraction and stoichiometric mass ratio of oxygen to fuel, respectively. 2.2. Dimensionless Governing equations
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For dimensionless form of equations, some variables are defined as below:
𝑥=
𝑋
𝐶(𝑇−𝑇∞ ) 𝑄𝑌𝐹 −∞
Г=
𝜆 √𝜌𝐶𝑎
𝑦𝐹 = 𝑌
𝑌𝐹
𝑌
𝐹 −∞
𝐾𝑎 𝐼𝑓 𝜌𝑎𝑄𝑌𝐹 −∞
𝑦𝑜 = 𝜐𝑌 𝑂
𝐹−∞
𝜆 𝜌𝐶𝑎
𝜓 = 𝐾𝑡 √
𝛬=
𝑦𝑠 = 𝑌
𝑌𝑠
𝐹 −∞
𝛺𝐾𝑎 𝐼𝑏𝑃 𝜌𝑎𝑄𝑌𝐹 −∞
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𝜃=
(25)
where 𝜃, 𝑦𝐹 , 𝑦𝑂 , 𝑦𝑠 and 𝑥 are dimensionless forms of temperature, mass fraction of fuel, mass fraction of oxidizer, mass fraction of solid particles and position, respectively. 𝑌𝐹−∞ is the mass fraction of fuel at the position −∞ where the fuel is coming from the fuel nozzle. Here, 𝑇∞ represents the temperature in the
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outlets of nozzles. By substituting dimensionless parameters into the conservation equations,
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dimensionless conservation equations will be achieved.
𝑣
(26)
(27)
Dimensionless equation of oxidizer mass conservation 1 𝑑 2 𝑦𝑂 2 𝑂 𝑑𝑥
+ 𝐿𝑒
𝑇
= 𝑓(𝜀)𝐷𝐶 𝑦𝐹 𝑦𝑜 𝑒𝑥𝑝 (− 𝑇𝑎)
(28)
Dimensionless equation of energy conservation in the 𝑅1 and 𝑅2 𝑑𝜃
𝑞
𝑇
+ 𝑥 𝑑𝑥 − 𝑓(𝜀) 𝑎𝜏 𝑦𝑠 𝐻(𝑇 − 𝑇𝑣 ) + Г𝑒𝑥𝑝[𝜓(𝑥 − 𝑥𝑓 )] = −𝑓(𝜀)𝐷𝐶 𝑦𝐹 𝑦𝑜 𝑒𝑥𝑝 (− 𝑇𝑎)
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𝑑2 𝜃 𝑑𝑥 2
𝑇 𝑇
− 𝑇𝑣 ) = 𝑓(𝜀)𝐷𝐶 𝑦𝐹 𝑦𝑜 𝑒𝑥𝑝 (− 𝑎)
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𝑦
= 𝑓(𝜀) 𝑎𝜏𝑠 𝐻(𝑇 − 𝑇𝑣 )
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𝑑𝑦𝑂 𝑑𝑥
𝑦𝑠 𝐻(𝑇 𝑎𝜏𝑣
Dimensionless equation of solid fuel mass conservation
𝑥
+ 𝑓(𝜀)
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𝑑𝑦𝑠 𝑑𝑥
𝑑𝑦𝐹 𝑑𝑥
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𝑥
+𝑥
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1 𝑑 2 𝑦𝐹 𝐿𝑒𝐹 𝑑𝑥 2
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Dimensionless equation of gaseous fuel mass conservation
𝑣
In which 𝑞 =
𝑄𝑣 𝑄
𝐸
and 𝑇𝑎 shows dimensionless activation energy which 𝑇𝑎 = 𝑅.
Dimensionless equation of energy conservation in 𝑅3 13
(29)
𝑑2 𝜃 𝑑𝑥 2
𝑑𝜃
𝑞
𝑇
+ 𝑥 𝑑𝑥 − 𝑓(𝜀) 𝑎𝜏 𝑦𝑠 𝐻(𝑇 − 𝑇𝑣 ) + Г𝑒𝑥𝑝[𝜓(𝑥 − 𝑥𝑓 )] = −𝑓(𝜀)𝐷𝐶 𝑦𝐹 𝑦𝑜 𝑒𝑥𝑝 (− 𝑇𝑎) 𝑣
(30 )
𝐷𝐶 = 𝜌𝐵 𝜐𝑂 𝑌𝐹 −∞ /𝑊𝐹 𝑎
SC RI PT
where 𝐷𝐶 the Damkohler number is defined as [9]: (31)
In which 𝜐𝑂 is the number of stoichiometric oxygen moles reacting with one mole of fuel and 𝑊𝐹 is the molecular weight of the fuel.
N
U
2.3. Boundary conditions
To solve equations, it is necessary to apply the boundary conditions. Here, 𝑥𝑣 shows the formation
M
A
position of the thin vaporaization zone and 𝑥𝑓 is the formation position of the flame. In this zone, reaction and vaporiztion terms are ignored relative to diffusion and convection terms. Vaporization and reaction
D
are limited in the thin positions of 𝑥𝑣 and 𝑥𝑓 , respecrtively. Vaporization of solid fuel particles is
TE
considered to happen at the particles vaporization temperature in a thin asymptptic zone, which means before the vaporization zone, all the fuel is in the solid phase and after that, no solid particles exist.
EP
Therefore:
𝑦𝑠 = 1
CC
−∞ < 𝑥 ≤ 𝑥𝑣 : 𝑥𝑣 ≤ 𝑥 < ∞:
𝑦𝑠 = 0
(32)
A
With respect to the previous description and the defined zones, the boundary conditions are stated as: 𝑦𝐹 = 0
,
𝑦𝑂 = 0
,
𝑦𝑠 = 1
, 𝜃=0
@ 𝑥 → −∞
(33)
𝑦𝐹 = 0
,
𝑦𝑂 = 𝛼
,
𝑦𝑠 = 0
, 𝜃=0
@ 𝑥 → +∞
(34)
14
1
𝑑𝑦
𝑦
𝑑𝜃
𝑞
𝑑𝑦
− 𝐿𝑒 [ 𝑑𝑥𝐹 ] = 𝑓(𝜀) 𝑎𝜏𝑠 𝑥𝑣 , [𝑑𝑥 ] = 𝑓(𝜀) 𝑎𝜏 𝑦𝑠 𝑥𝑣 , [ 𝑑𝑥𝑂 ] = 0 𝐹
𝑣
1 𝑑𝑦𝑜 𝑑𝜃 [ ] + [𝑑𝑥 ] 𝐿𝑒𝑜 𝑑𝑥
=
@ 𝑥 = 𝑥𝑣
𝑣
1 𝑑𝑦𝐹 𝑑𝜃 [ ] + [𝑑𝑥 ] 𝐿𝑒𝐹 𝑑𝑥
= 𝛬 𝑥𝑓 +
𝛤 𝜓
(35)
@ 𝑥 = 𝑥𝑓
(36)
𝑌𝑂−∞ 𝜐𝑌𝐹−∞
𝛼=
SC RI PT
In which 𝛼 is the initial fuel mass fraction and is expressed as [9]:
(37)
With respect to the asymptotic vaporization zone and neglecting the reaction and convection terms against the vaporization and diffusion terms, Eqs. (26), (28) and (29) are rewritten as: 𝑞
+ 𝑓(𝜀) 𝑎𝜏𝑠 𝐻(𝑇 − 𝑇𝑣 ) = 0
1 𝑑 2 𝑦𝑂 𝐿𝑒𝑂 𝑑𝑥 2
=0
N
𝑣
1 𝑑 2 𝑦𝐹 𝐿𝑒𝐹 𝑑𝑥 2
(38)
U
− 𝑓(𝜀) 𝑎𝜏 𝑦𝑠 𝐻(𝑇 − 𝑇𝑣 ) + Г exp[𝜓(𝑥 − 𝑥𝑣 )] = 0 𝑦
𝑣
(39)
M
A
𝑑2 𝜃 𝑑𝑥 2
(40)
1
𝑑𝑦
𝑦
TE
D
By integrating the above equations from 𝑥𝑣− to the 𝑥𝑣+ and considering 𝑎𝜏𝑣 = 1, one can obtain: − 𝐿𝑒 [ 𝑑𝑥𝐹 ] = 𝑓(𝜀) 𝑎𝜏𝑠 𝑥𝑣 , 𝑣
𝑑𝑦
𝑑𝜃
𝑞
[ 𝑑𝑥𝑂 ] = 0 , [𝑑𝑥 ] = 𝑓(𝜀) 𝑎𝜏 𝑦𝑠 𝑥𝑣 𝑣
(41)
EP
𝐹
The above equation is known as the jump condition. Also, for the flame position, jump condition is
CC
derived from the Eqs. (26), (28) and (29) with the same procedure for vaporization position and ignoring the convection and vaporization terms against the diffusion and reaction terms.
A
𝑑2 𝜃 𝑇𝑎 + 𝑄̂𝑟𝑎𝑑 = −𝑓(𝜀)𝐷𝐶 𝑦𝐹 𝑦𝑜 𝑒𝑥𝑝 (− ) 2 𝑑𝑥 𝑇 (42) 1 𝑑 2 𝑦𝐹 𝐿𝑒𝐹 𝑑𝑥 2
𝑇
= 𝑓(𝜀)𝐷𝐶 𝑦𝐹 𝑦𝑜 𝑒𝑥𝑝 (− 𝑇𝑎)
(43)
15
1 𝑑 2 𝑦𝑜 𝐿𝑒𝑜 𝑑𝑥 2
𝑇
= 𝑓(𝜀)𝐷𝐶 𝑦𝐹 𝑦𝑜 𝑒𝑥𝑝 (− 𝑇𝑎)
(44)
(45)
𝑑2 𝜃 𝑑𝑥 2
+ 𝐿𝑒
1 𝑑 2 𝑦𝐹 2 𝐹 𝑑𝑥
+ 𝑄̂𝑟𝑎𝑑 = 0
(46)
𝑑2 𝜃 𝑑𝑥 2
+ 𝐿𝑒
1 𝑑 2 𝑦𝑜 2 𝑜 𝑑𝑥
+ 𝑄̂𝑟𝑎𝑑 = 0
SC RI PT
By summing Eqs. (42), (43) and also, Eqs. (42), (44), one can write:
By integrating Eqs. (45) and (46) from 𝑥𝑓− to 𝑥𝑓+ and assuming 𝑎𝜏𝑣 = 1, it becomes: (47)
1 𝑑𝑦𝐹 𝑑𝜃 [ ] + [𝑑𝑥 ] 𝐿𝑒𝐹 𝑑𝑥
= 𝛬 𝑥𝑓 +
𝛤 𝜓
(48)
1 𝑑𝑦𝑜 𝑑𝜃 [ ] + [𝑑𝑥 ] 𝐿𝑒𝑜 𝑑𝑥
= 𝛬 𝑥𝑓 +
𝛤 𝜓
U
Using Eqs. (47) and (48), the equations can be rewritten in a new format. It should be noted that in the
N
three zones of 𝑅1 , 𝑅2 and 𝑅3 , vaporization and reaction terms are neglected in comparison with
A
convection and diffusion terms. According to these assumptions, Eq. (29) is expressed as:
In 𝑅3 zone: 𝑑2 𝜃 𝑑𝑥 2
D
𝑑𝜃
+ 𝑥 𝑑𝑥 − 𝛬 = 0
CC
(50)
𝑑𝜃
+ 𝑥 𝑑𝑥 + Г 𝑒𝑥𝑝[𝜓(𝑥 − 𝑥𝑓 )] = 0
TE
𝑑2 𝜃 𝑑𝑥 2
EP
(49)
M
In 𝑅1 and 𝑅2 zones:
A
Also, by neglecting reaction and vaporization terms in the triple zones, Eqs. (26) and (28) can be written as:
(51)
1 𝑑 2 𝑦𝐹 𝐿𝑒𝐹 𝑑𝑥 2
+𝑥
𝑑𝑦𝐹 𝑑𝑥
=0
(52)
1 𝑑 2 𝑦𝑂 𝐿𝑒𝑂 𝑑𝑥 2
+𝑥
𝑑𝑦𝑂 𝑑𝑥
=0
16
2.4. Solving equations Governing equations are solved using an analytical method. According to the boundary conditions, constant coefficients in each zone are calculated for differential equations. Using Eqs. (49) to (52) and
SC RI PT
applying related boundary conditions, conservation equations of gaseous fuel mass fraction, oxidizer mass fraction and energy are solved. 2.4.1.
Temperature Distribution In 𝑹𝟏
In this zone, fuel particles are released from fuel nozzle outlet to reach to the vaporization position.
N
U
Boundary conditions governing on the equations are written as:
(53)
A
𝑥 = −∞ → 𝜃 = 0 , 𝑥 = 𝑥𝑣 → 𝜃 = 𝜃𝑣
By using reduced-order method and changing the variables, one can write:
(55)
𝑑𝜂 𝑑𝑥
𝑑𝜃 𝑑𝑥
,
𝑑𝜂 𝑑𝑥
=
𝑑2𝜃 𝑑𝑥 2
M
𝜂=
D
(54)
TE
+ 𝑥𝜂 + Г𝑒𝑥𝑝[𝜓(𝑥 − 𝑥𝑓 )] = 0
EP
Solving Eqs. (54) and (55) leads to:
1 2
𝑥2
𝜓2
𝜓+𝑥 )) √2
𝜂 = 𝑒 − 2 (2𝐶2 − Г√2𝜋𝑒 − 2 𝑒𝑟𝑓𝑖 (
CC
(56)
A
For gaining temperature relations, Eq. (56) should be integrated. The function of 𝑒𝑟𝑓𝑖(𝑥) is defined as:
𝜓+𝑥 𝑒𝑟𝑓𝑖 ( ) √2
(57)
=
𝜓 𝑒𝑟𝑓𝑖 ( ) + √2
𝑒
𝜓2 2
2 √ 𝑥 𝜋
+
𝜓2
(3+6𝜓2 +𝜓4 )𝑒 2 60√2𝜋
𝜓2
𝜓2
𝜓𝑒 2 √2𝜋
𝑥 5 + 𝑂(𝑥)6
Replacing Eq. (57) to Eq. (56) and integrating will lead to:
17
2
𝑥 +
(1+𝜓2 )𝑒 2 3√2𝜋
𝜓2
3
𝑥 +
𝜓(3+𝜓2 )𝑒 2 12√2𝜋
𝑥4 +
𝑥2
𝑥2
𝜃 = 𝐶1 + 𝑒 − 2 [1.303 + 0.002𝑥 + 0.226𝑥 2 + 0.0003𝑥 3 + 0.021𝑥 4 + 𝑒 2 (−0.006 +
(58)
𝑥
1.253 𝐶2 )𝑒𝑟𝑓 ( 2)] √
By applying boundary conditions, constant coefficients of Eq. (58) are obtained as follows: In 𝑹𝟐
SC RI PT
By substituting Boundary dimensionless temperature distribution is derived as: 𝑥2
(59)
𝑥2
𝜃 = 𝐶1 + 𝑒 − 2 [1.303 + 0.002𝑥 + 0.226𝑥 2 + 0.0003𝑥 3 + 0.021𝑥 4 + 𝑒 2 (−0.006 + 𝑥 √2
1.253 𝐶2 )𝑒𝑟𝑓 ( )]
N
In 𝑹𝟑
A
U
By applying boundary conditions, constant coefficients of Eq. (49) are obtained as follows:
M
This zone shows the presence of oxidizer in the environment and there is no fuel. The governing temperature equation is stated as:
TE
D
Solving Eq. (50) will lead to: 𝜋
𝑥
1
3
𝜃 = √ 2 𝑐1 𝑒𝑟𝑓 ( 2) + 2 𝛬𝑥 2 2 𝐹2 [(1,1); (2 ,2) ; (−
(60)
√
𝑥2 )] + 𝑐2 2
EP
In which 𝑃 𝐹𝑄 function is known as Hypergeometric function. Boundary condition governing on this zone
CC
is defined as:
𝑥 = 𝑥𝑓 → 𝜃 = 𝜃𝑓 , 𝑥 = +∞ → 𝜃 = 0
(61)
A
By applying boundary condition of Eq. (61) into Eq. (60), constant coefficients and dimensionless temperature distribution will be completed and expressed as: 2.4.2.
Oxidizer Mass fraction distribution In 𝑹𝟏
18
In this zone, there is no oxidizer in the environment and mass fraction of the oxidizer is equal to zero, hence: 𝑦𝑂 = 0
SC RI PT
(62)
In 𝑹𝟐
Obtaining mass fraction of the oxidizer in this zone in which only oxidizer exists, will be done by solving
2
√𝐿𝑒
) + 𝑐2
𝑜
M
𝑜
𝑥
A
𝜋
𝑦𝑜 = 𝑐1 √2𝐿𝑒 𝑒𝑟𝑓 (
(63)
N
U
Eq. (52) as follows:
The boundary conditions governing on this zone exist in the two position of oxidizer nozzle outlet and
D
flame position, therefore: 𝑥 = 𝑥𝑓 → 𝑦𝑜 = 0 ,𝑥 = +∞ → 𝑦𝑜 = 𝛼
TE
(64)
𝛼=
𝑌𝑂−∞ 𝜐𝑌𝐹−∞
CC
(65)
EP
As mentioned, α is the initial oxidizer mass fraction and expressed as:
By using boundary conditions in Eq. (65), constant coefficients and oxidizer mass fraction
A
distribution can be gained as:
𝑒𝑟𝑓(
(66)
𝑦𝑜 = 𝛼
𝑥𝑓 𝑥 ) −𝑒𝑟𝑓( ) 2 2 √𝐿𝑒 √𝐿𝑒 𝑜 𝑜
1−𝑒𝑟𝑓(
𝑥𝑓 ) 2 √𝐿𝑒 𝑜
2.4.3 Gaseous Fuel Mass fraction distribution 19
In 𝑹𝟏
Gaseous Fuel mass fraction distribution is achieved solving Eq. (51) as:
𝐹
𝑥
) + 𝑐2
2
√𝐿𝑒
𝐹
SC RI PT
𝜋
𝑦𝐹 = 𝑐1 √2𝐿𝑒 𝑒𝑟𝑓 (
(67)
The boundary conditions governing on this zone exist in the two position of fuel nozzle outlet and vaporization position, therefore: 𝑥 = −∞ → 𝑦𝐹 = 0 ,𝑥 = 𝑥𝑣 → 𝑦𝐹 = 𝑦𝐹𝑣
(68)
U
By using boundary conditions into Eq. (67), constant coefficients and gaseous fuel mass fraction
𝑥𝑣 )+1 2 √𝐿𝑒 𝐹
In 𝑹𝟐
EP
TE
D
𝑒𝑟𝑓(
A
𝑦𝐹 = 𝑦𝐹𝑣
(69)
𝑥 )+1 2 √𝐿𝑒 𝐹
M
𝑒𝑟𝑓(
N
distribution can be gained as:
Gaseous Fuel mass fraction distribution in this region is achieved as previous regions. The
CC
boundary conditions governing on this zone exist in vaporization position and flame position, therefore, by using boundary conditions in Eq. (69), constant coefficients and gaseous fuel mass
A
fraction distribution can be gained as: 𝑥 = 𝑥𝑣 → 𝑦𝐹 = 𝑦𝐹𝑣 , 𝑥 = 𝑥𝑓 → 𝑦𝐹 = 0
(70)
(71)
20
𝑒𝑟𝑓(
𝑦𝐹 = 𝑦𝐹𝑣 𝑒𝑟𝑓(
𝑥𝑓 𝑥 )−𝑒𝑟𝑓( 2 ) 2 √𝐿𝑒 √𝐿𝑒 𝐹 𝐹 𝑥𝑓 𝑥𝑣 )−𝑒𝑟𝑓( 2 ) 2 √𝐿𝑒 √𝐿𝑒 𝐹 𝐹
In 𝑹𝟑
SC RI PT
In the post-flame zone, the amount of fuel is equal to zero, thus: 𝑦𝐹 = 0
(72)
To determine the temperature, fuel and oxidizer mass fraction distributions, it is necessary to obtain all parameters of 𝑦𝐹𝑣 , 𝑥𝑣 , 𝑥𝑓 𝑎𝑛𝑑 𝜃𝑓 . Thus, four boundary conditions should be applied using jump
U
conditions at vaporization and flame positions to solve the governing equations simultaneously.
N
Substituting the aforesaid parameters in the temperature, fuel and oxidizer mass fraction distributions in
A
various zones will complete the conservation equations. Therefore, various temperature and mass
M
fractions are acquired. It should be said that to solve these equations, numerical methods should be used.
Results and discussion
TE
3.
D
Hence, the methods of Ref. [26] have been used for non-linear system of equations.
Lycopodium fuel particles are considered and the value of quantities utilized in conservation equations
EP
will be obtained using Table 1. It is necessary to note that these particles are of organic type that release combustible gases while receiving heat. In Table 1, lycopodium properties are presented.
A
CC
Table 1. Constant properties of equations
The released gas from lycopodium is considered to be methane [27, 28]. The combustion reaction is denoted as: (73)
𝐶𝐻4 + 2 𝑂2 + 7.52 𝑁2 → 𝐶𝑂2 + 2 𝐻2 𝑂 + 7.52 𝑁2
21
For reaction of fuel and air which methane gas is made from vaporization of the fuel particles, the equivalence ratio is obtained as [28]: 𝑌
𝜙𝑢 = 17.18 1−𝑌𝐹−∞
𝐹−∞
SC RI PT
(74)
where 𝑌𝐹−∞ is defined as: (75)
𝑌𝐹−∞ =
4 3
𝑓(𝜀) 𝜋𝑟𝑃3 𝑛𝑃 𝜌𝑃 𝜌
Fig. 3 shows a comparison between gaseous fuel mass fraction versus position for different particle
porosities at mass concentration of 100 𝑔𝑟/𝑚3 . Results are presented for the two cases of applying and
U
not considering radiation heat transfer. As can be observed, gaseous fuel mass fraction rises until it
N
reaches a maximum point at the vaporization position and then, slowly decreases until reaches zero at the
A
flame formation position due to consumption and reaction in the flame zone. In addition, considering
M
thermal radiation, increases the amount of generated gaseous fuel in pre-heat and post-vaporization zones and moves both vaporization and flame positions to the oxidizer side. Moreover, the more particle
D
porosity increases, more reduction in fuel mass fraction can be observed. In other words, rising porosity is
EP
TE
equivalent to decreasing of available fuel to be burnt with oxidizer.
Fig. 3. Fuel mass fraction variations against position for various particle porosities.
CC
Fig. 4 demonstrates a comparison between the oxidizer mass fraction in terms of the non-dimensional position for various particle porosities at mass concentration of 100 𝑔𝑟/𝑚3. As shown, by getting closer
A
to the flame formation position, mass fraction of oxidizer gradually decreases until reaches the flame position and then it achieves the value of zero. Furthermore, applying thermal radiation decreases the amount of initial oxidizer mass fraction in the post-flame zone. Also, increasing of the particle porosity decreases the reaction rate with oxidizer which means less oxidizer takes part in the reaction process. Hence, the curves related to the higher porosities are positioned at lower levels.
22
Fig. 4. Oxidizer mass fraction variations against position for various particle porosities.
Fig. 5 depicts temperature profile of the domain with respect to the non-dimensional position for different values of particle porosities. The temperatures gradually increase to reach the flame temperature.
SC RI PT
Observations reveal that increasing particle porosity leads to reduction in temperature distribution as the amount of available fuel would be reduced and therefore the amount of heat released will be decrease.
Also, the temperature profile moves toward the right side as particle porosity rises and the flame will be
Fig. 5. Temperature profile of the domain for various particle porosities.
U
formed farther form the fuel nozzle.
N
Figs. 6 and 7 respectively illustrate flame temperature variations with respect to the fuel and oxidizer
A
Lewis numbers for different mass concentrations and porosities of particles. As mentioned previously,
M
Lewis number is defined as the ratio of heat diffusion to the mass diffusion. In Fig. 6, increasing of the
D
fuel Lewis number is equivalent to the reduction in fuel mass fraction, thus this decrease will cause the flame temperature to reduce as the amount of heat released will be decreased. Moreover, increasing mass
TE
concentration from 67 𝒈𝒓/𝒎𝟑 to 83 𝒈𝒓/𝒎𝟑 and decreasing porosity from 6% to 0% will cause a growth
EP
in flame temperature. By increasing fuel mass concentration and/or decreasing porosity, the amount of available fuel will be increased and then, by reacting more fuel, flame temperature increases as well. The
CC
vaporization temperature used in this research was obtained from [6]. It should be noted that oxidizer Lewis number is assumed to be unit and the initial equivalence ratio is equal to 𝝋 = 𝟏.𝟒. In Fig. 7, flame
A
temperature is plotted against oxidizer Lewis number in which fuel Lewis number is assumed to be unit. Increasing of the oxidizer Lewis number equals to the reduction in oxidizer mass fraction which causes flame temperature to decrease. As exhibited in Fig. 7, with a rise in mass concentration and/or a decline in particle porosity, flame temperature will get a higher value.
23
Fig. 6. Effect of fuel Lewis number, fuel mass concentration and particle porosity on flame temperature.
Fig. 7. Effect of oxidizer Lewis number, fuel mass concentration and particle porosity on flame temperature.
In Fig. 8, flame temperature variations versus equivalence ratio are shown for two different particles
SC RI PT
radiuses of 30 and 60 microns. Influence of particle porosity is investigated as well. As demonstrated, increasing equivalence ratio leads to a growth in flame temperature. The reason is that when equivalence ratio gets higher values, the amount of available fuel will increase and it causes the temperature to
increase. Also, by increasing fuel particles radius, flame temperature will decrease. An increase in
particles size will lead to decrease in the particles surface-to-volume contact ratio which is the reason of
U
increasing of the required energy for particles vaporization and therefore flame temperature will be
A
N
reduced. As porosity increases, for a constant equivalence ratio and a radius, flame temperature reduces.
M
Fig. 8. Effect of equivalence ratio, particles radius and porosity on flame temperature.
D
The obtained results are compared with the methane-air non-premixed counterflow combustion .Due to
TE
the lack of relevant experimental/numerical data on the non-premixed counterflow dust flames. The comparison shows that the same trend is obtained which reinforces the qualitative validation of the
EP
developed model. In Fig. 9, comparison between the ratio of critical quenching strain rate with Ref. [9] for different values of fuel and oxidizer Lewis numbers is illustrated. Results of the presented model
CC
indicate good agreement with those of Ref. [9]. According to the flame zone analysis section of the Ref. [9], the ratio of the critical strain rate in non-unity Lewis numbers of oxidizer and fuel (𝑎) to the critical
A
strain rate in 𝐿𝑒𝑂 = 𝐿𝑒𝐹 = 1 (𝑎0 ) is presented as:
(76)
𝑎
(𝑎 ) 0
𝐶𝑟
=
6 2 𝐿𝑒𝑂 𝐿𝑒𝐹 𝑇𝑓 𝜂 𝑇 ( ) 𝑒𝑥𝑝 [𝑇𝑎0 (1 2 0 𝐹𝑂𝑓 𝑇𝑓 𝑑𝐸 𝑓
𝑇𝑓0
2
− 𝑇 ) + (𝑥𝑓2 − (𝑥𝑓0 ) )]
24
𝑓
𝑎
This figure implies that at 𝐿𝑒𝑂 = 1, (𝑎 ) 0
𝑎
value of (𝑎 ) 0
𝐶𝑟
𝐶𝑟
decreases by a rise in the fuel Lewis number. For instance, the
declines from 7.6 to 0.4 when the magnitude of fuel Lewis number grows from 0.2 to 1.4 𝑎
at 𝑇𝑎 = 30000. Furthermore, the increase of 𝑇𝑎 is related with a rise in (𝑎 ) . By reducing value of fuel 0
𝐶𝑟
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Lewis number, the magnitude of 𝑥𝑓 (flame position) increases, which significantly rises the exponential 𝑇
term of Eq. (76). This increase is superior to the reduction in magnitude of the term ( 𝑓0) 𝑇𝑓
𝑎
to an increase in the ratio (𝑎 ) . 𝐶𝑟
𝜂2 𝐿𝑒𝐹, 𝑑𝐸
leading
U
0
6
Fig. 9. Effect of fuel Lewis number on critical rate of quenching strain ratio for 𝐿𝑒𝑂 = 1 and 𝑇𝑎 = 30000.
A
N
Conclusion
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In the present study, combustion of organic particles in a counterflow configuration of porous particles with the consideration of the effects of radiation heat transfer was studied. Assuming particles to vaporize
D
first to produce a specific chemical gas, gaseous fuel mass fraction and oxidizer mass fraction along with
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the energy equation in which the thermal radiation term was added, were written using specific boundary conditions and solved by analytical methods. It was found that by increasing Lewis numbers of fuel and
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oxidizer, flame temperature decreases and increasing Lewis number of oxidizer moves flame position toward the oxidizer nozzle. Results reveal that increasing of particle porosity decreases the reaction rate
CC
with oxidizer which means less oxidizer takes part in the reaction process. The increase of the particle porosity leads to reduction in temperature distribution. In addition, temperature profile moves toward the
A
right side as particle porosity rises. Finally, by increasing fuel mass concentration and/or decreasing the porosity, the amount of available fuel will increase and by reacting more fuel, flame temperature increases as well and with a rise in mass concentration and/or a decline in particle porosity, flame temperature will get a higher value. In dust flame concept, there is a new parameter, representative of organic particle's vaporization which should be embedded in the governing equations unlike the gas flame 25
structure. The concentration of gaseous mixture and dust particles are become related and coupled to each other in the governing equations. Furthermore, the heat capacity involved in the governing equations is a
A
CC
EP
TE
D
M
A
N
U
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function of gas and particle heat capacities.
26
Nomenclature
𝑎
Quenching strain rate
𝐵
Frequency constant
𝐶𝑎
Gaseous phase specific heat (𝑘𝑔.𝐾)
𝐶𝑃
Solid particle specific heat (𝑘𝑔.𝐾)
𝐷𝐶
Damkohler number
𝐷𝐹
Mass diffusivity coefficient of gaseous fuel (
𝐷𝑂
Mass diffusivity coefficient of oxidizer (
𝐷0𝐸
Critical Damkohler number
𝐷𝑇
Thermal diffusivity coefficient (
𝐸
Activation energy (𝑘𝐽)
𝑒
Apparent porosity
𝑒𝑟𝑓𝑖(𝑥)
Error function
𝑓(𝜀)
Porosity function
𝑘𝐽
𝑚2 ) 𝑠
N
TE
D
M
A
𝑚2 ) 𝑠
EP
Heaviside function
CC
𝐼𝑏𝑝
𝑊
Radiation intensity (𝑚2 ) 𝑊 ) 𝑚2
Blackbody emissivity power (
𝐼𝑓
Induced radiation intensity
𝐾𝑎
Absorption coefficient ( )
𝐾𝑠
Scattering coefficient (𝑚)
𝐿𝑒
Lewis number
𝑚
Mixture molecular mass (
A
U
𝑚2 ) 𝑠
𝐻 𝐼
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𝑘𝐽
1 𝑚
1
𝑘𝑔 ) 𝑚𝑜𝑙
27
Fuel molecular mass (
𝑚𝑂
Oxidizer molecular mass (𝑚𝑜𝑙)
𝑛𝑃
Number of particle per unit volume
𝑃(𝜃,𝛷)
Scattering phasic function
𝑄
Reaction heat per unit of fuel mass ( )
𝑄𝑎𝑏𝑠
Efficiency factors of absorption
𝑄𝑠𝑐𝑎
Efficiency factors of scattering
𝑄𝑟𝑎𝑑
Thermal radiation (𝑘𝑔)
𝑞
Dimensionless heat
𝑅
Universal constant of gases (𝑚𝑜𝑙.𝐾)
𝑇
Fuel temperature (𝐾)
𝑇𝑎
Dimensionless activation energy
𝑇𝑓
Flame temperature (𝐾)
𝑇𝑣
Particle temperature of vaporization (𝐾)
𝑉𝑣𝑜𝑖𝑑
TE
𝑘𝑔 ) 𝑚𝑜𝑙
𝑚𝑓
D
M
A
N
𝑚3 𝑃𝑎
Void volume
Total volume
Molecular weight of fuel
𝑥
Dimensionless length
𝑥𝑓
Flame front position
𝑥𝑣
Vaporization front position
A 𝑌𝐹
Gaseous fuel mass fraction
𝑌𝑂
Oxidizer mass fraction
28
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𝑘𝐽
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𝑊𝐹
𝑘𝐽 𝑘𝑔
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𝑉𝑡𝑜𝑡𝑎𝑙
𝑘𝑔
Solid particles mass fraction
𝑦𝐹
Dimensionless fuel mass fraction
𝑦𝑜
Dimensionless oxidizer mass fraction
𝑦𝑠
Dimensionless solid particles mass fraction of
𝑍𝑒
Zeldovich number
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𝑌𝑠
Initial mass fraction of oxidizer
𝜀
Porosity
𝜖𝑃
Emissivity of solid particles
𝜂
Defined in eq. (54)
𝜃
Dimensionless temperature
𝜆
Thermal conductivity of fuel or oxidizer (𝑚.𝑠.𝐾)
𝜈𝐹
Fuel stoichiometric coefficient
𝜈𝑂
Oxidizer stoichiometric coefficient
𝜈𝑃
N A
M
D
CC
𝜌𝑎
Products stoichiometric coefficient 𝑘𝑔 ) 𝑚3
Density (
EP
𝜌
U
𝛼
TE
Greek symbols
𝑘𝑔 ) 𝑚3
Gaseous phase density (
𝑘𝑔
Solid particle density(𝑚3 )
𝜎
Stefan-Boltzmann constant
A
𝜌𝑃
𝜏𝑣
Vaporization time
𝛤, 𝜓 and Ʌ
Dimensionless radiation parameters
𝛺
Defined in eq. (8)
29
𝑘𝐽
𝑘𝑔 ) 𝑚.𝑠2
𝜔𝑣
Rate of particle vaporization (
𝜔𝐹
Rate of chemical reaction (𝑚𝑠2 )
A
CC
EP
TE
D
M
A
N
U
SC RI PT
𝑘𝑔
30
References 1.
M. Bidabadi , M. Harati , Q. Xiong, E. Yaghoubi , M. Doranehgard , P. Aghajannezhad, Volatization & combustion of biomass particles in random media: Mathematical modeling and
Intensification, 126 (2018):232-238. 2.
A.I. Stankiewicz, J.A. Moulijn, Process intensification: transforming chemical engineering, Chemical engineering progress 96, no. 1 (2000): 22-34.
3.
R.K. Eckhoff, Dust Explosion in the Process Industries, Second ed., Butterworth Heinemann,
U
Oxford, (1997). 4.
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analyze the effect of Lewis number, Chemical Engineering and Processing - Process
H. Moghadasi, A. Rahbari, M. Bidabadi, A. Khoeini Poorfar, V. Farhangmehr, Mathematical
N
Investigation of Premixed Lycopodium Dust Flame in Small Furnaces, Journal of Energy
A. Afzalabadi, A. Khoeini Poorfar, M. Bidabadi, H. Moghadasi, S. Hochgreb, A. Rahbari, C.
M
5.
A
Resources Technology, (2018), DOI: https://doi.org/10.1115/1.4041106.
D
Dubois. Study on hybrid combustion of aero-suspensions of boron-aluminum powders in a
TE
quiescent reaction medium, Journal of Loss Prevention in the Process Industries 49 (2017): 645651.
H. Hanai, H. Kobayashi, T. Niioka, A numerical study of pulsating flame propagation in mixtures
EP
6.
of gas and particles, Proceedings of the Combustion Institute, 28(1) (2000):815-822. C. Proust, Flame propagation and combustion in some dust-air mixtures, Journal of Loss
CC
7.
8.
M. Bidabadi, Q. Xiong, M. Harati, E. Yaghoubi, M.H. Doranehgard, A. Rahbari, Study on the
A
Prevention in the Process, 19(1) (2006):89-100.
combustion of micro organic dust particles in random media with considering effect of thermal resistance and temperature difference between gas and particles. Chemical Engineering and Processing - Process Intensification 126 (2018): 239-247.
31
9.
K. Seshadri, C. Trevino, The Influence of the Lewis Numbers of the Reactants on the Asymptotic Structure of Counterflow and Stagnant Diffusion Flames. Combustion Science and Technology, 64(4-6) (1989): 243-261.
10.
A. Linan, The asymptotic structure of counterflow diffusion flames for large activation energies,
11.
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Acta Astronautica, 1(7-8) (1974):1007-1039.
S.C. Li, P.A. Libby, F.A. Williams, Experimental and theoretical studies of counterflow spray diffusion flames, Symposium (International) on Combustion, 24(1) (1992):1503-1512.
12.
C. Proust, R.B, Moussa, M. Guessasma, K. Saleh, J. Fortin, Thermal radiation in dust flame propagation, Journal of Loss Prevention in the Process Industries, 49(2017):896-904.
M.A. Liberman, M.F. Ivanov, A.D. Kiverin, Effects of thermal radiation heat transfer on flame
U
13.
N
acceleration and transition to detonation in particle-cloud hydrogen flames, Effects of thermal
14.
M
hydrogen flames,38 (2015):176-186.
A
radiation heat transfer on flame acceleration and transition to detonation in particle-cloud
M. Bidabadi, S. Montazerinejad, S.A. Fanaee, The influence of radiation on the flame
D
propagation through micro organic dust particles with non-unity Lewis number, Journal of the
15.
TE
Energy Institute, 87(4) (2014):354-366. A. Datta, A. Saha, Contributions of self-absorption and soot on radiation heat transfer in a
EP
laminar methane—air diffusion flame. Proceedings of the Institution of Mechanical Engineers, Part A: Journal of Power and Energy, 221(7) (2007):955-970. V.A. Levin, N.A. Lutsenko, Two-dimensional gas flows under heterogeneous combustion of
CC
16.
solid porous media. Doklady Physics, 62(9) (2017): 425-429.
A
17.
18.
A.J. Souza, Counterflow combustion in a porous medium, Hyperbolic Problems: Theory, Numerics, and Applications, (2008):1005-1012 V. Hurevich, I. Smurov, L. Pawlowski, Theoretical study of the powder behavior of porous particles in a flame during plasma spraying, Surface and Coatings Technology,151 (1) (2002):370-376. 32
19.
V.M. Gremyachkina, D. Förtsch, U. Schnell, K.R.G Hein, A model of the combustion of a porous carbon particle in oxygen, Combustion and Flame, 130 (3) (2002):161-170.
20.
K.S. Adzerikho, E.F. Nogotov, V.P. Trofimov, Radiative heat transfer in two-phase media. 1st ed. CRC, Boca Raton (1993). J.R. Howell, M.P. Menguc, R. Siegel, Thermal radiation heat transfer. Fifth ed., CRC Press, Boca
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21.
Raton, London, New York, (2010). 22.
M.F. Modest, Radiative Heat Transfer, 3rd ed., Academic Press, New York, (2013).
23.
A. Haghiri, M. Bidabadi, Sciences, and undefined, Modeling of laminar flame propagation through organic dust cloud with thermal radiation effect ,International Journal of Thermal
F.E. Fendell, Ignition and extinction in combustion of initially unmixed reactants, Journal of
N
24.
U
Sciences, 49(8) (2010): 1446-1456.
C. Proust, Experimental determination of the maximum flame temperatures and of the laminar
M
25.
A
Fluid Mechanics, 21(1965):281-303.
burning velocities for some combustible dust-air mixtures, International Colloquium on Dust
D
Explosions, Varsovie, Poland, (1993):161-184, 1993. R.L. Burden, J.D. Faires, Numerical analysis, 9th ed., Brooks/Cole, Pacific Grove, USA, (2011).
27.
Rockwell SR, Rangwala AS. Modeling of dust air flames. Fire Safety Journal. 2013 Jul 1; 59:22-
Adzerikho KS, Nogotov EF, Trofimov VP. Radiative heat transfer in two-phase media. CRC
CC
28.
EP
9.
TE
26.
Press; 1992 Nov 10.
A
29. Seshadri, K., A. L. Berlad, and V. Tangirala. "The structure of premixed particle-cloud
30.
flames." Combustion and flame 89, no. 3-4 (1992): 333-342. Rockwell, Scott R., and Ali S. Rangwala. "Modeling of dust air flames." Fire Safety Journal 59 (2013): 22-29.
33
Caption of figures
Structure of diffusion counterflow combustion.
Figure 2
Structure of particles porosity.
Figure 3
Fuel mass fraction variations against position for various particle porosities.
Figure 4
Oxidizer mass fraction variations against position for various particle porosities.
Figure 5
Temperature profile of the domain for various particle porosities.
Figure 6
Effect of fuel Lewis number, fuel mass concentration and particle porosity on flame
N
U
SC RI PT
Figure 1
Effect of oxidizer Lewis number, fuel mass concentration and particle porosity on flame
M
Figure 7
A
temperature.
D
temperature.
Effect of equivalence ratio, particles radius and porosity on flame temperature.
Figure 9 30000.
Effect of fuel Lewis number on critical rate of quenching strain ratio for 𝐿𝑒𝑂 = 1 and 𝑇𝑎 =
A
CC
EP
TE
Figure 8
34
SC RI PT U N A
A
CC
EP
TE
D
M
Fig. 1. Structure of diffusion counterflow combustion
Fig. 2. Structure of particles porosity
35
SC RI PT U N A M
A
CC
EP
TE
D
Fig. 3. Fuel mass fraction variations against position for various particle porosities.
36
SC RI PT U N A M
A
CC
EP
TE
D
Fig. 4. Oxidizer mass fraction variations against position for various particle porosities.
37
SC RI PT U N
A
CC
EP
TE
D
M
A
Fig. 5. Temperature profile of fuel and oxidizer for various particle porosities.
Fig. 6. Effect of fuel Lewis number, fuel mass concentration and particle porosity on flame temperature. 38
SC RI PT U N
A
CC
EP
TE
D
M
A
Fig. 7. Effect of oxidizer Lewis number, fuel mass concentration and particle porosity on flame temperature.
Fig. 8. Effect of equivalence ratio, particles radius and porosity on flame temperature. 39
SC RI PT U N A M D
A
CC
EP
TE
Fig. 9. Effect of fuel Lewis number on critical rate of quenching strain ratio for 𝐿𝑒𝑂 = 1 and 𝑇𝑎 = 30000.
40
Caption of tables
Constant properties of equations
SC RI PT
Table 1
Table 1. Constant properties of equations
values
Reference
1000
𝐾𝑔 𝑚3
𝜌𝑎
1.164
𝐾𝑔 𝑚3
5.677688
𝐶𝑎
1.00416
M
D
TE
[30]
𝐾𝐽 𝑘𝑔.𝐾
[30]
𝐾𝐽 𝑘𝑔
[30]
64895.4
1.46538 × 10−4
A
CC
EP
𝜆
𝐾𝐽 𝑘𝑔.𝐾
[29]
A
𝐶𝑃
𝑄
[29]
N
𝜌𝑃
U
parameters
41
𝐾𝐽 𝑚.𝑠.𝐾
[25]
S0255-2701(18)30450-1 https://doi.org/10.1016/j.cep.2018.10.018 CEP 7412
To appear in:
Chemical Engineering and Processing
Received date: Revised date: Accepted date:
12 April 2018 27 August 2018 28 October 2018
Please cite this article as: Moghadasi H, Malekian N, Khoeini Poorfar A, Bidabadi M, Thermal Radiative Study of Counterflow Combustion of Porous Particles, Chemical Engineering and Processing - Process Intensification (2018), https://doi.org/10.1016/j.cep.2018.10.018 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Thermal Radiative Study of Counterflow Combustion of Porous Particles
a
SC RI PT
Hesam Moghadasia, Navid Malekiana, Alireza Khoeini Poorfara, Mehdi Bidabadia
School of Mechanical Engineering, Department of Energy Conversion, Iran University of Science and
U
Technology (IUST), Narmak, 16846-13114, Tehran, Iran.
A
CC
EP
TE
D
M
A
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Graphical abstract
Corresponding author: School of Mechanical Engineering, Department of Energy Conversion, Iran University of Science and Technology, Email address: [email protected] Tel: +98 21 77 240 540; Fax: +98 77 240 488.
1
SC RI PT U N A M D
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Fig. Structure of diffusion counterflow combustion
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Highlights:
Non-premixed combustion of dust particles in a counterflow system is studied
Particles porosity and thermal radiation modeling are investigated
Effects of fuel Lewis number, particle porosity on flame temperature are studied
Effects of equivalence ratio and particles radius on flame temperature are studied
A
CC
Abstract
2
The paper deals with modeling counterflow, non-premixed combustion of porous fuel particles (lycopodium) with the consideration of thermal radiation effects. Assuming that the streams of fuel particles and air as oxidizer, move towards the stagnation plane from the two opposing nozzles in a counterflow configuration. It is presumed that particles first vaporize in order to yield a gaseous fuel, methane, which then reacts with the oxidizer which is air. In this research, conservation equations with
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certain boundary conditions are solved using mathematical methods with the consideration of radiation heat transfer in different regions and compared to cases in which radiation heat transfer is not considered. Furthermore, flame temperature and mass fraction profiles are presented in terms of oxidizer and fuel
Lewis numbers. In addition, effects of particle porosity are investigated. As a result, with the increase of dust concentration and reduction of particles radius and porosity, it would lead to a rise in the flame
U
temperature.
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Keywords: Thermal Radiative, Porous Particles, Counterflow Combustion, Non-Premixed, Lewis
M
A
number.
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D
1. Introduction
Development in numerical and mathematical methods gives researchers the opportunity to study several
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types of problems in all fields of engineering with lower costs. Biomass gasification, combustion, chemical reactions are some of research works in different facets of engineering using these methods.
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Generating energy from the renewable resources is the one main focuses of the present research due to the universal awareness of reducing our dependency on fossil fuels. Also, in industries that produce
A
combustible organic dust, a deep knowledge of the explosion hazards is essential. Process Intensification (PI) is a topic receiving considerable attention recently. Using the simple definition of Stankiewicz and Moulijn (2000), PI is ‘Any chemical engineering development that leads to a substantially smaller, cleaner, safer, and more energy efficient technology. Counterflow combustion mechanism would lead to
3
smaller and more compact combustion chamber and propulsion systems. As one of important factors in aerospace industries are weight and space this would lead to reduction in costs [1, 2]. Due to the importance of the subject, in this paper an effort is made to improve the understanding of
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particle dust combustion that simultaneously involves the transfer of heat, mass, phase change and chemical reaction. Combustion of combustible fuel particles and oxidizer is used in many industries and safety fields. Hence, experimental studies and theory of combustion phenomena seem to be essential in
this field [3-5]. Combustion of particles is usually unstable and there are numerous interactions between particles. Therefore, combustion of particles is influenced by the physical and chemical properties of fuel,
U
average particles size and shape and their distribution. Radiation heat transfer phenomena is usually an important process in the flame propagation mechanism. Furthermore, Radiation heat transfer is a
N
dominant process in spreading of flame through a cloud of dust particles and leads to further release of
M
A
heat energy. This increases the burning velocity and also the thickness of the reaction zone [6]. Proust [7] measured the laminar burning velocity and maximum flame temperature of combustion of
D
particles-air mixture (organic particles including lycopodium-air and sulphur flour–air) in a planar
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configuration. Bidabadi et al [8] proposed an analytical model to investigate the effect of randomly distributed micro-organic dust particles on the flame propagation speed through the combustible mixture.
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Seshadri and Trevino [9] evaluated asymptotic structure of counterflow diffusion flames. In this research, maximum flame temperature with respect to the Lewis number variations of fuel and oxidizer for a
CC
specific gaseous fuel with a known chemical structure was provided. Also, quenching distance of the flame was determined. Linan [10] modeled the counterflow diffusion combustion by analyzing chemical
A
reaction in flame zone and mixing quality of the fuel and oxidizer which were moving toward each other from two opposite jet. Li et al. [11] investigated counterflow methanol and heptane spray combustion both theoretically and experimentally and showed that by increasing strain rate, the distance of vaporization’s starting point and flame formation from the nozzle will be decreased.
4
In the last decades, study of heat transfer mechanism by radiation in combustion of dust particles were performed. Proust et al. [12] presented a new experimental measurement of radiation heat transfer in dust flames (methane-air, methane-air seeded with inert particles, aluminum-air flames). The results focused on the amount of heat radiated by a propagating flame. The effects of radiation on flame acceleration and
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transition from deflagration to detonation (DTD) in hydrogen dust cloud were studied by Liberman et al. [13]. They evaluated the role of radiation heat transfer diffusion from the flame zone and the gaseous
products to the pre-heat zone and the unburned particles. Bidabadi et al. [14] investigated the effect of
radiation heat transfer on the flame propagation of micro organic dust cloud with variable Lewis number. Their results show that, by increasing the temperature of flame and burning velocity, the Lewis number
U
will be decreased and equivalence ratio will be increased. A numerical investigation on self-absorption
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and soot radiation in laminar diffusion flame of the methane and air mixture were studied by Datta and
M
should be considered at least in a thin layer.
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Saha [15]. They showed that the influence of thermal radiation in the combustion of methane and air
D
Levin and Lutsenkoa [16] investigated a two-dimensional unsteady gas flow in a porous media with
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heterogeneous combustion. Using numerical methods, it was shown that, flows with vortex can be formed in combustion of solid fuels in porous media. De Souza [17] determined a one-dimensional model for
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counterflow combustion using the Riemann solution due to injection of air into a porous medium containing solid fuel. Assuming that the combustion front is modeled by a traveling wave, they obtained a
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family of Riemann solutions for the same initial and injection conditions. Hurevich et al. [18] analyzed the behavior of porous particles in a plasma flame, theoretically. The porous particles were supposed to be
A
a conglomerate of fine precursors with a size of roughly 1μm. Gremyachkin [19] extended a model for combustion of porous carbon particles in oxygen. This model considers heat and mass transfer in both gas phase above particle’s surface and inside the porous particle. The conditions for the model are determined by solving the diffusion equation in the gas phase around the particle.
5
In this study, an analytical model is presented for combustion of organic porous particles in a nonpremixed counterflow configuration. It is presumed that the flame structure consists of three zones including pre-heat, post-vaporization and post-flame zones. In contrast to the previous analytical studies, effects of radiation heat transfer and particles porosity are taken into account in this thermal model. Also,
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thermal radiation equations state the radiation exchanges between these zones. The model was considered to be one-dimensional and insulated against thermal heat loss. The carrying gas of dust particles is
assumed nitrogen. Fuel and oxidizer mass fractions along with temperature distribution are studied with respect to position. Moreover, effects of radiation, particles porosity, Lewis number, etc. on flame
N
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temperature are investigated.
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2. Governing Equations
M
In this research, the counterflow configuration is considered in a way that organic particles enters the combustion domain from the direction −∞ and move towards the stagnation plane and the oxidizer flows
D
away from the nozzle from the direction +∞. At first, fuel particles vaporize to produce a gaseous fuel
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with a specific chemical structure, Methane. Surface reactions are ignored in this study. Then, the gaseous
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fuel reacts with the oxidizer (air). The position of flame formation depends on the initial conditions which can occur in the left or right hand side of the stagnation plane. Changing the initial conditions will lead to
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a change in this position. The structure of diffusion counterflow combustion of organic particles is shown in Fig. 1 in a model with thin reaction and vaporization zones. Fuel particles in an asymptotic zone which
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is called vaporization front, suddenly vaporize and a gaseous fuel will be yielded. The gaseous fuel reacts with the oxidizer flow in an asymptotic zone which is called flame front. The flame front position can be formed on the sides of the stagnation plane which depends on the initial conditions of the problem. As seen in Fig. 1, the flame position is located at the left side of the stagnation plane which also can happen at the right side.
6
Fig. 1. Structure of diffusion counterflow combustion
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2.1. Radiation heat transfer modelling
To analyze radiation heat transfer in a two-phase mixture, organic particles and air are considered as a gray medium. For a two-phase gray medium which consists of scattering, absorbing and emitting, the radiative heat transfer equation can be expressed as [20, 21]: 𝐾
= 𝐾𝑎 𝐼 + 𝐾𝑠 𝐼 − 𝐾𝑎 𝐼𝑏𝑃 − 4𝜋𝑠 ∫4𝜋 𝐼(𝛺)𝑃(𝜃,𝜑)𝑑𝛺
U
(1)
N
𝑑𝐼 𝑑𝑥
The right hand side terms of Eq. (1) are radiation intensity caused by absorption, scattering, emission and
A
incoming scattering brought about by other particles, respectively. Also, 𝐾𝑎 , 𝐾𝑠 and I are absorption
M
coefficient, scattering coefficient and radiation intensity, respectively. 𝑃(𝜃,𝜑) is scattering phasic
D
function. Also, 𝐼𝑏𝑃 is the blackbody emissivity power which depends on local temperature of particles. In
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this model, the shape of particles is spherical and their size is uniform. The properties of a two-phase mixture including uniform sized particles can be expressed as [22]:
𝐾𝑎 =
2 𝜋𝑑𝑃 𝑛𝑠 𝑄𝑎𝑏𝑠 4
= 2𝜌
𝐾𝑠 =
2 𝜋𝑑𝑃 𝑛𝑠 𝑄𝑠𝑐𝑎 4
= 2𝜌
𝜑
𝑄𝑎𝑏𝑠
(2)
𝑄𝑠𝑐𝑎
(3)
EP
3
CC
𝑃 𝑑𝑃
3
𝜑
𝑃 𝑑𝑃
A
which 𝑑𝑃 and 𝜑 are particle diameter and mass density, respectively. Also, 𝑄𝑎𝑏𝑠 and 𝑄𝑠𝑐𝑎 are efficiency factors of scattering and absorption, respectively. If particles are diffusely reflecting spheres, one can write [22]: 𝑄𝑎𝑏𝑠 = 𝜖𝑃
,
𝑄𝑠𝑐𝑎 = 1 − 𝜖𝑃
(4)
7
where 𝜖𝑃 is emissivity of solid particles. Eq. (1) has a differential-integral nature which is difficult to be solved using analytical approaches. So as to overcome this difficulty and simplify the problem, integral term of the Eq. (1) is neglected. In this part of study, the radiative heat transfer (𝑄𝑟𝑎𝑑 ) is analyzed. By
structure is obtained as: Pre-heat zone
𝑄𝑟𝑎𝑑 = 𝐾𝑎 𝐼𝑓 𝑒𝑥𝑝[𝐾𝑡 (𝑥 − 𝑥𝑓 )]
𝑅1 : − ∞ < 𝑥 ≤ 𝑥𝑣
Post-vaporization zone
𝑄𝑟𝑎𝑑 = 𝐾𝑎 𝐼𝑓 𝑒𝑥𝑝[𝐾𝑡 (𝑥 − 𝑥𝑓 )]
N
Post-flame zone
(6)
A
𝑅2 : 𝑥𝑣 ≤ 𝑥 ≤ 𝑥𝑓
(5)
U
SC RI PT
solving the general radiation heat transfer equation in every zone, radiative heat transfer in the flame
𝑅3 : 𝑥𝑓 ≤ 𝑥 < +∞
(7)
M
𝑄𝑟𝑎𝑑 = −𝛺𝐾𝑎 𝐼𝑏𝑃
, 𝐼𝑏𝑃 =
𝜎𝑇𝑓4 𝜋
, 𝐼𝑓 =
TE
𝐾𝑡 = 𝐾𝑎 + 𝐾𝑠
D
Where
𝐾𝑎 𝐼 𝐾𝑡 𝑏𝑃
𝐾
𝑠 , 𝛺 = 𝐾 +𝐾 𝑠
𝑎
(8)
EP
where 𝐼𝑓 is the induced radiation intensity from the flame front into pre-heat and vaporization zones. Also, σ and 𝑇𝑓 are Stefan-Boltzmann constant and flame temperature in the thin reaction zone,
CC
respectively.
A
In the current research study, the structure of particles porosity is considered as shown in Fig. (2).
Fig. 2. Structure of particles porosity
8
As stated previously, porosity of a material is a criteria to measure the empty spaces within that material. Porosity is defined as the ratio of void volume (𝑉𝑣𝑜𝑖𝑑 ) per total volume (𝑉𝑡𝑜𝑡𝑎𝑙 ) of the material, hence: 𝑉
𝜀 = 𝑉 𝑣𝑜𝑖𝑑
(9)
𝑡𝑜𝑡𝑎𝑙
𝑒=
𝑉𝑣𝑜𝑖𝑑 𝑉
SC RI PT
In which 𝜀 denotes porosity. In addition, apparent porosity is defined as:
(10)
Consequently, a general statement for porosity is written as follows: 𝑉
𝑒
𝑣𝑜𝑖𝑑 𝜀 = 𝑉+𝑉 = 1+𝑒
(11)
U
𝑣𝑜𝑖𝑑
A
N
Here, an important parameter is defined which is called porosity function: (12)
M
𝑓(𝜀) = 1 − 𝜀
In combustion of organic particles, the vaporization rate defined as produced gaseous fuel mass per unit
considered as [23]: 𝑌
𝑣
(13)
EP
𝜔𝑣 = 𝜏𝑠 𝐻(𝑇 − 𝑇𝑣 )
TE
D
volume and time, is a controller parameter of combustion process. In this research, vaporization rate is
CC
Where 𝑌𝑠 mass fraction of solid fuel is, 𝜏𝑣 is characteristic time of vaporization, 𝑇 and 𝑇𝑣 are fuel and vaporization temperatures. Also, 𝐻 is Heaviside function which is defined as:
A
0 𝐻(𝑇 − 𝑇𝑣 ) = { 1
𝑇 < 𝑇𝑣 𝑇 ≥ 𝑇𝑣
(14)
Another factor which controls the combustion process is Lewis number which is defined as the ratio of heat diffusion to mass diffusion, thus:
9
𝐿𝑒𝑖 =
𝜆 𝜌𝐶𝐷𝑖 (15)
where 𝜆, 𝜌, 𝐶 and 𝐷𝑖 are conductivity, density, specific heat and mass diffusive coefficient of gaseous fuel
SC RI PT
or oxidizer, respectively. In this paper, Lewis number of oxidizer and fuel are considered to be non-unity and their variations on the flame are investigated. Chemical kinetic is assumed as a general one-stage reaction, therefore: 𝑣𝐹 [𝐹] + 𝑣𝑂 [𝑂] → 𝑣𝑃 [𝑃]
U
(16)
N
where [F], [O] and [P] denote fuel, oxidizer and products, respectively. Quantities of 𝑣𝐹 , 𝑣𝑂 and 𝑣𝑃
A
express stoichiometric coefficient of fuel, oxidizer and products, respectively. Velocity field is considered
M
as (𝑢,𝑣) = (−𝑎𝑋,𝑎𝑌) , where 𝑢 and 𝑣 are velocities in 𝑋 and 𝑌 directions. In this relation, a is known as velocity gradient at the stagnation point which also represents the strain rate. If the strain rate is assumed
D
to have a small amount, the problem can be solved in one dimension and only in 𝑋 direction. Also, it is
TE
considered that oxidizer and fuel velocity flows are equal. The amounts of density, specific heat and other transient coefficient are assumed to be constant. Other assumptions considered in this work are as
The processes between the preheating and beginning of the lycopodium particles’ vaporization
CC
EP
follows:
are neglected and is assumed that lycopodium dust particles vaporize first to yield a gaseous fuel,
A
which is presumed to be methane.
The inert carrying gas of dust particle is assumed nitrogen which does not take part in the combustion reaction.
Heterogeneous and surface reactions are negligible.
Vaporization happens in a thin asymptotic front.
10
Gas thermal conductivity (𝜆) is assumed constant in all temperatures.
The particles shape is spherical and their size is uniform. Also, volumetric porosity has been considered for particles. Effects of forces including thermophoresis, buoyancy and drag are neglected.
The influences of pressure gradient, heat loss and heat recirculation are ignored.
Lewis number of oxidizer and fuel are considered to be non-unity.
There is no temperature difference between the gas and particles.
Mass conservation of gaseous fuel = 𝐷𝐹
−
𝜔𝐹 𝜔 𝑓(𝜀) + 𝑣 𝑓(𝜀) 𝜌 𝜌
U
𝑑 2 𝑌𝐹 𝑑𝑋 2
N
𝑑𝑌𝐹 𝑑𝑋
(17)
M
A
−𝑎𝑋
SC RI PT
D
where 𝐷𝐹 and 𝑌𝐹 are mass diffusivity of fuel and mass fraction of gaseous fuel, respectively. 𝜔𝐹 is the
TE
rate of chemical reaction which follows the Arrhenius rule and is in the first order relative to fuel and oxidizer. It is define as [10, 24]:
EP
𝐸 𝜔𝐹 = 𝐵𝜌2 𝑣𝐹 𝑣𝑂 ̅̅̅ 𝑌𝐹 ̅̅̅ 𝑌𝑂 𝑒𝑥𝑝 (− 𝑅𝑇)
(18)
CC
𝐵 is a frequency constant and ̅̅̅ 𝑌𝑂 and ̅̅̅ 𝑌𝐹 are defined as: 𝑚 ̅̅̅ 𝑌𝐹 = 𝑌𝐹 𝑣 𝑚
𝐹
𝑚 , ̅̅̅ 𝑌𝑂 = 𝑌𝑂 𝑣 𝑚 𝑂
𝑂
(19)
A
𝐹
where 𝑌𝐹 , 𝑌𝑂 , 𝑚𝐹 , 𝑚𝑂 and 𝑚 are mass fraction of fuel and oxidizer, molecular weight of fuel and oxidizer and molecular weight of mixture, respectively.
Mass conservation of solid fuel
11
If the solid particles diffusion is negligible and solid particles don’t have any reactions together, the mass conservation of solid fuel particles is written as: 𝑑𝑌
−𝑎𝑋 𝑑𝑋𝑠 = −𝑓(𝜀)𝜔𝑣
(20)
SC RI PT
where 𝑌𝑠 is mass fraction of solid particles and 𝜔𝑣 is the vaporization rate which has been defined in Eq. (13), previously.
Energy Conservation of Mixture 𝑑𝑇
𝑑2 𝑇
𝑄
𝑄𝑣 𝑄𝑟𝑎𝑑 𝑓(𝜀) + 𝜌𝐶 𝐶
(21)
U
−𝑎𝑋 𝑑𝑋 = 𝐷𝑇 𝑑𝑋 2 + 𝜔𝐹 𝜌𝐶 𝑓(𝜀) − 𝜔𝑣
N
where 𝑄 is the heat released per unit of consumed fuel mass, 𝑄𝑣 is the latent vaporization heat of particles 𝜆
A
and 𝑄𝑟𝑎𝑑 is radiation heat transfer. Also, 𝐷𝑇 = 𝜌𝐶 is thermal diffusivity and 𝐶 is specific heat of mixture
M
which is obtained from combination of gaseous phase specific heat 𝐶𝑎 and solid particles specific heat 𝐶𝑃
4𝜌𝑃 𝜋𝑟 3 𝐶𝑃 𝑛𝑃 3𝜌
(22)
TE
𝐶 = 𝐶𝑎 + 𝑓(𝜀)
D
as below [24, 25]:
EP
where 𝜌𝑃 is the solid particle density and 𝑛𝑃 shows the number of particles per unit of volume. Thus: 4
CC
𝜌 = 𝜌𝑎 + 𝑓(𝜀) 3 𝜋𝑟𝑃3 𝑛𝑃 𝜌𝑃
A
(24)
(23)
Mass conservation of oxidizer −𝑎𝑋
𝑑𝑌𝑂 𝑑𝑋
= 𝐷𝑂
𝑑 2 𝑌𝑂 𝑑𝑋 2
−𝜐
𝜔𝐹 𝑓(𝜀) 𝜌
In the above equation 𝐷𝑂 , 𝑌𝑂 and 𝜐 are oxidizer mass diffusivity, oxidizer mass fraction and stoichiometric mass ratio of oxygen to fuel, respectively. 2.2. Dimensionless Governing equations
12
For dimensionless form of equations, some variables are defined as below:
𝑥=
𝑋
𝐶(𝑇−𝑇∞ ) 𝑄𝑌𝐹 −∞
Г=
𝜆 √𝜌𝐶𝑎
𝑦𝐹 = 𝑌
𝑌𝐹
𝑌
𝐹 −∞
𝐾𝑎 𝐼𝑓 𝜌𝑎𝑄𝑌𝐹 −∞
𝑦𝑜 = 𝜐𝑌 𝑂
𝐹−∞
𝜆 𝜌𝐶𝑎
𝜓 = 𝐾𝑡 √
𝛬=
𝑦𝑠 = 𝑌
𝑌𝑠
𝐹 −∞
𝛺𝐾𝑎 𝐼𝑏𝑃 𝜌𝑎𝑄𝑌𝐹 −∞
SC RI PT
𝜃=
(25)
where 𝜃, 𝑦𝐹 , 𝑦𝑂 , 𝑦𝑠 and 𝑥 are dimensionless forms of temperature, mass fraction of fuel, mass fraction of oxidizer, mass fraction of solid particles and position, respectively. 𝑌𝐹−∞ is the mass fraction of fuel at the position −∞ where the fuel is coming from the fuel nozzle. Here, 𝑇∞ represents the temperature in the
U
outlets of nozzles. By substituting dimensionless parameters into the conservation equations,
N
dimensionless conservation equations will be achieved.
𝑣
(26)
(27)
Dimensionless equation of oxidizer mass conservation 1 𝑑 2 𝑦𝑂 2 𝑂 𝑑𝑥
+ 𝐿𝑒
𝑇
= 𝑓(𝜀)𝐷𝐶 𝑦𝐹 𝑦𝑜 𝑒𝑥𝑝 (− 𝑇𝑎)
(28)
Dimensionless equation of energy conservation in the 𝑅1 and 𝑅2 𝑑𝜃
𝑞
𝑇
+ 𝑥 𝑑𝑥 − 𝑓(𝜀) 𝑎𝜏 𝑦𝑠 𝐻(𝑇 − 𝑇𝑣 ) + Г𝑒𝑥𝑝[𝜓(𝑥 − 𝑥𝑓 )] = −𝑓(𝜀)𝐷𝐶 𝑦𝐹 𝑦𝑜 𝑒𝑥𝑝 (− 𝑇𝑎)
A
𝑑2 𝜃 𝑑𝑥 2
𝑇 𝑇
− 𝑇𝑣 ) = 𝑓(𝜀)𝐷𝐶 𝑦𝐹 𝑦𝑜 𝑒𝑥𝑝 (− 𝑎)
D
𝑦
= 𝑓(𝜀) 𝑎𝜏𝑠 𝐻(𝑇 − 𝑇𝑣 )
CC
𝑑𝑦𝑂 𝑑𝑥
𝑦𝑠 𝐻(𝑇 𝑎𝜏𝑣
Dimensionless equation of solid fuel mass conservation
𝑥
+ 𝑓(𝜀)
TE
𝑑𝑦𝑠 𝑑𝑥
𝑑𝑦𝐹 𝑑𝑥
EP
𝑥
+𝑥
M
1 𝑑 2 𝑦𝐹 𝐿𝑒𝐹 𝑑𝑥 2
A
Dimensionless equation of gaseous fuel mass conservation
𝑣
In which 𝑞 =
𝑄𝑣 𝑄
𝐸
and 𝑇𝑎 shows dimensionless activation energy which 𝑇𝑎 = 𝑅.
Dimensionless equation of energy conservation in 𝑅3 13
(29)
𝑑2 𝜃 𝑑𝑥 2
𝑑𝜃
𝑞
𝑇
+ 𝑥 𝑑𝑥 − 𝑓(𝜀) 𝑎𝜏 𝑦𝑠 𝐻(𝑇 − 𝑇𝑣 ) + Г𝑒𝑥𝑝[𝜓(𝑥 − 𝑥𝑓 )] = −𝑓(𝜀)𝐷𝐶 𝑦𝐹 𝑦𝑜 𝑒𝑥𝑝 (− 𝑇𝑎) 𝑣
(30 )
𝐷𝐶 = 𝜌𝐵 𝜐𝑂 𝑌𝐹 −∞ /𝑊𝐹 𝑎
SC RI PT
where 𝐷𝐶 the Damkohler number is defined as [9]: (31)
In which 𝜐𝑂 is the number of stoichiometric oxygen moles reacting with one mole of fuel and 𝑊𝐹 is the molecular weight of the fuel.
N
U
2.3. Boundary conditions
To solve equations, it is necessary to apply the boundary conditions. Here, 𝑥𝑣 shows the formation
M
A
position of the thin vaporaization zone and 𝑥𝑓 is the formation position of the flame. In this zone, reaction and vaporiztion terms are ignored relative to diffusion and convection terms. Vaporization and reaction
D
are limited in the thin positions of 𝑥𝑣 and 𝑥𝑓 , respecrtively. Vaporization of solid fuel particles is
TE
considered to happen at the particles vaporization temperature in a thin asymptptic zone, which means before the vaporization zone, all the fuel is in the solid phase and after that, no solid particles exist.
EP
Therefore:
𝑦𝑠 = 1
CC
−∞ < 𝑥 ≤ 𝑥𝑣 : 𝑥𝑣 ≤ 𝑥 < ∞:
𝑦𝑠 = 0
(32)
A
With respect to the previous description and the defined zones, the boundary conditions are stated as: 𝑦𝐹 = 0
,
𝑦𝑂 = 0
,
𝑦𝑠 = 1
, 𝜃=0
@ 𝑥 → −∞
(33)
𝑦𝐹 = 0
,
𝑦𝑂 = 𝛼
,
𝑦𝑠 = 0
, 𝜃=0
@ 𝑥 → +∞
(34)
14
1
𝑑𝑦
𝑦
𝑑𝜃
𝑞
𝑑𝑦
− 𝐿𝑒 [ 𝑑𝑥𝐹 ] = 𝑓(𝜀) 𝑎𝜏𝑠 𝑥𝑣 , [𝑑𝑥 ] = 𝑓(𝜀) 𝑎𝜏 𝑦𝑠 𝑥𝑣 , [ 𝑑𝑥𝑂 ] = 0 𝐹
𝑣
1 𝑑𝑦𝑜 𝑑𝜃 [ ] + [𝑑𝑥 ] 𝐿𝑒𝑜 𝑑𝑥
=
@ 𝑥 = 𝑥𝑣
𝑣
1 𝑑𝑦𝐹 𝑑𝜃 [ ] + [𝑑𝑥 ] 𝐿𝑒𝐹 𝑑𝑥
= 𝛬 𝑥𝑓 +
𝛤 𝜓
(35)
@ 𝑥 = 𝑥𝑓
(36)
𝑌𝑂−∞ 𝜐𝑌𝐹−∞
𝛼=
SC RI PT
In which 𝛼 is the initial fuel mass fraction and is expressed as [9]:
(37)
With respect to the asymptotic vaporization zone and neglecting the reaction and convection terms against the vaporization and diffusion terms, Eqs. (26), (28) and (29) are rewritten as: 𝑞
+ 𝑓(𝜀) 𝑎𝜏𝑠 𝐻(𝑇 − 𝑇𝑣 ) = 0
1 𝑑 2 𝑦𝑂 𝐿𝑒𝑂 𝑑𝑥 2
=0
N
𝑣
1 𝑑 2 𝑦𝐹 𝐿𝑒𝐹 𝑑𝑥 2
(38)
U
− 𝑓(𝜀) 𝑎𝜏 𝑦𝑠 𝐻(𝑇 − 𝑇𝑣 ) + Г exp[𝜓(𝑥 − 𝑥𝑣 )] = 0 𝑦
𝑣
(39)
M
A
𝑑2 𝜃 𝑑𝑥 2
(40)
1
𝑑𝑦
𝑦
TE
D
By integrating the above equations from 𝑥𝑣− to the 𝑥𝑣+ and considering 𝑎𝜏𝑣 = 1, one can obtain: − 𝐿𝑒 [ 𝑑𝑥𝐹 ] = 𝑓(𝜀) 𝑎𝜏𝑠 𝑥𝑣 , 𝑣
𝑑𝑦
𝑑𝜃
𝑞
[ 𝑑𝑥𝑂 ] = 0 , [𝑑𝑥 ] = 𝑓(𝜀) 𝑎𝜏 𝑦𝑠 𝑥𝑣 𝑣
(41)
EP
𝐹
The above equation is known as the jump condition. Also, for the flame position, jump condition is
CC
derived from the Eqs. (26), (28) and (29) with the same procedure for vaporization position and ignoring the convection and vaporization terms against the diffusion and reaction terms.
A
𝑑2 𝜃 𝑇𝑎 + 𝑄̂𝑟𝑎𝑑 = −𝑓(𝜀)𝐷𝐶 𝑦𝐹 𝑦𝑜 𝑒𝑥𝑝 (− ) 2 𝑑𝑥 𝑇 (42) 1 𝑑 2 𝑦𝐹 𝐿𝑒𝐹 𝑑𝑥 2
𝑇
= 𝑓(𝜀)𝐷𝐶 𝑦𝐹 𝑦𝑜 𝑒𝑥𝑝 (− 𝑇𝑎)
(43)
15
1 𝑑 2 𝑦𝑜 𝐿𝑒𝑜 𝑑𝑥 2
𝑇
= 𝑓(𝜀)𝐷𝐶 𝑦𝐹 𝑦𝑜 𝑒𝑥𝑝 (− 𝑇𝑎)
(44)
(45)
𝑑2 𝜃 𝑑𝑥 2
+ 𝐿𝑒
1 𝑑 2 𝑦𝐹 2 𝐹 𝑑𝑥
+ 𝑄̂𝑟𝑎𝑑 = 0
(46)
𝑑2 𝜃 𝑑𝑥 2
+ 𝐿𝑒
1 𝑑 2 𝑦𝑜 2 𝑜 𝑑𝑥
+ 𝑄̂𝑟𝑎𝑑 = 0
SC RI PT
By summing Eqs. (42), (43) and also, Eqs. (42), (44), one can write:
By integrating Eqs. (45) and (46) from 𝑥𝑓− to 𝑥𝑓+ and assuming 𝑎𝜏𝑣 = 1, it becomes: (47)
1 𝑑𝑦𝐹 𝑑𝜃 [ ] + [𝑑𝑥 ] 𝐿𝑒𝐹 𝑑𝑥
= 𝛬 𝑥𝑓 +
𝛤 𝜓
(48)
1 𝑑𝑦𝑜 𝑑𝜃 [ ] + [𝑑𝑥 ] 𝐿𝑒𝑜 𝑑𝑥
= 𝛬 𝑥𝑓 +
𝛤 𝜓
U
Using Eqs. (47) and (48), the equations can be rewritten in a new format. It should be noted that in the
N
three zones of 𝑅1 , 𝑅2 and 𝑅3 , vaporization and reaction terms are neglected in comparison with
A
convection and diffusion terms. According to these assumptions, Eq. (29) is expressed as:
In 𝑅3 zone: 𝑑2 𝜃 𝑑𝑥 2
D
𝑑𝜃
+ 𝑥 𝑑𝑥 − 𝛬 = 0
CC
(50)
𝑑𝜃
+ 𝑥 𝑑𝑥 + Г 𝑒𝑥𝑝[𝜓(𝑥 − 𝑥𝑓 )] = 0
TE
𝑑2 𝜃 𝑑𝑥 2
EP
(49)
M
In 𝑅1 and 𝑅2 zones:
A
Also, by neglecting reaction and vaporization terms in the triple zones, Eqs. (26) and (28) can be written as:
(51)
1 𝑑 2 𝑦𝐹 𝐿𝑒𝐹 𝑑𝑥 2
+𝑥
𝑑𝑦𝐹 𝑑𝑥
=0
(52)
1 𝑑 2 𝑦𝑂 𝐿𝑒𝑂 𝑑𝑥 2
+𝑥
𝑑𝑦𝑂 𝑑𝑥
=0
16
2.4. Solving equations Governing equations are solved using an analytical method. According to the boundary conditions, constant coefficients in each zone are calculated for differential equations. Using Eqs. (49) to (52) and
SC RI PT
applying related boundary conditions, conservation equations of gaseous fuel mass fraction, oxidizer mass fraction and energy are solved. 2.4.1.
Temperature Distribution In 𝑹𝟏
In this zone, fuel particles are released from fuel nozzle outlet to reach to the vaporization position.
N
U
Boundary conditions governing on the equations are written as:
(53)
A
𝑥 = −∞ → 𝜃 = 0 , 𝑥 = 𝑥𝑣 → 𝜃 = 𝜃𝑣
By using reduced-order method and changing the variables, one can write:
(55)
𝑑𝜂 𝑑𝑥
𝑑𝜃 𝑑𝑥
,
𝑑𝜂 𝑑𝑥
=
𝑑2𝜃 𝑑𝑥 2
M
𝜂=
D
(54)
TE
+ 𝑥𝜂 + Г𝑒𝑥𝑝[𝜓(𝑥 − 𝑥𝑓 )] = 0
EP
Solving Eqs. (54) and (55) leads to:
1 2
𝑥2
𝜓2
𝜓+𝑥 )) √2
𝜂 = 𝑒 − 2 (2𝐶2 − Г√2𝜋𝑒 − 2 𝑒𝑟𝑓𝑖 (
CC
(56)
A
For gaining temperature relations, Eq. (56) should be integrated. The function of 𝑒𝑟𝑓𝑖(𝑥) is defined as:
𝜓+𝑥 𝑒𝑟𝑓𝑖 ( ) √2
(57)
=
𝜓 𝑒𝑟𝑓𝑖 ( ) + √2
𝑒
𝜓2 2
2 √ 𝑥 𝜋
+
𝜓2
(3+6𝜓2 +𝜓4 )𝑒 2 60√2𝜋
𝜓2
𝜓2
𝜓𝑒 2 √2𝜋
𝑥 5 + 𝑂(𝑥)6
Replacing Eq. (57) to Eq. (56) and integrating will lead to:
17
2
𝑥 +
(1+𝜓2 )𝑒 2 3√2𝜋
𝜓2
3
𝑥 +
𝜓(3+𝜓2 )𝑒 2 12√2𝜋
𝑥4 +
𝑥2
𝑥2
𝜃 = 𝐶1 + 𝑒 − 2 [1.303 + 0.002𝑥 + 0.226𝑥 2 + 0.0003𝑥 3 + 0.021𝑥 4 + 𝑒 2 (−0.006 +
(58)
𝑥
1.253 𝐶2 )𝑒𝑟𝑓 ( 2)] √
By applying boundary conditions, constant coefficients of Eq. (58) are obtained as follows: In 𝑹𝟐
SC RI PT
By substituting Boundary dimensionless temperature distribution is derived as: 𝑥2
(59)
𝑥2
𝜃 = 𝐶1 + 𝑒 − 2 [1.303 + 0.002𝑥 + 0.226𝑥 2 + 0.0003𝑥 3 + 0.021𝑥 4 + 𝑒 2 (−0.006 + 𝑥 √2
1.253 𝐶2 )𝑒𝑟𝑓 ( )]
N
In 𝑹𝟑
A
U
By applying boundary conditions, constant coefficients of Eq. (49) are obtained as follows:
M
This zone shows the presence of oxidizer in the environment and there is no fuel. The governing temperature equation is stated as:
TE
D
Solving Eq. (50) will lead to: 𝜋
𝑥
1
3
𝜃 = √ 2 𝑐1 𝑒𝑟𝑓 ( 2) + 2 𝛬𝑥 2 2 𝐹2 [(1,1); (2 ,2) ; (−
(60)
√
𝑥2 )] + 𝑐2 2
EP
In which 𝑃 𝐹𝑄 function is known as Hypergeometric function. Boundary condition governing on this zone
CC
is defined as:
𝑥 = 𝑥𝑓 → 𝜃 = 𝜃𝑓 , 𝑥 = +∞ → 𝜃 = 0
(61)
A
By applying boundary condition of Eq. (61) into Eq. (60), constant coefficients and dimensionless temperature distribution will be completed and expressed as: 2.4.2.
Oxidizer Mass fraction distribution In 𝑹𝟏
18
In this zone, there is no oxidizer in the environment and mass fraction of the oxidizer is equal to zero, hence: 𝑦𝑂 = 0
SC RI PT
(62)
In 𝑹𝟐
Obtaining mass fraction of the oxidizer in this zone in which only oxidizer exists, will be done by solving
2
√𝐿𝑒
) + 𝑐2
𝑜
M
𝑜
𝑥
A
𝜋
𝑦𝑜 = 𝑐1 √2𝐿𝑒 𝑒𝑟𝑓 (
(63)
N
U
Eq. (52) as follows:
The boundary conditions governing on this zone exist in the two position of oxidizer nozzle outlet and
D
flame position, therefore: 𝑥 = 𝑥𝑓 → 𝑦𝑜 = 0 ,𝑥 = +∞ → 𝑦𝑜 = 𝛼
TE
(64)
𝛼=
𝑌𝑂−∞ 𝜐𝑌𝐹−∞
CC
(65)
EP
As mentioned, α is the initial oxidizer mass fraction and expressed as:
By using boundary conditions in Eq. (65), constant coefficients and oxidizer mass fraction
A
distribution can be gained as:
𝑒𝑟𝑓(
(66)
𝑦𝑜 = 𝛼
𝑥𝑓 𝑥 ) −𝑒𝑟𝑓( ) 2 2 √𝐿𝑒 √𝐿𝑒 𝑜 𝑜
1−𝑒𝑟𝑓(
𝑥𝑓 ) 2 √𝐿𝑒 𝑜
2.4.3 Gaseous Fuel Mass fraction distribution 19
In 𝑹𝟏
Gaseous Fuel mass fraction distribution is achieved solving Eq. (51) as:
𝐹
𝑥
) + 𝑐2
2
√𝐿𝑒
𝐹
SC RI PT
𝜋
𝑦𝐹 = 𝑐1 √2𝐿𝑒 𝑒𝑟𝑓 (
(67)
The boundary conditions governing on this zone exist in the two position of fuel nozzle outlet and vaporization position, therefore: 𝑥 = −∞ → 𝑦𝐹 = 0 ,𝑥 = 𝑥𝑣 → 𝑦𝐹 = 𝑦𝐹𝑣
(68)
U
By using boundary conditions into Eq. (67), constant coefficients and gaseous fuel mass fraction
𝑥𝑣 )+1 2 √𝐿𝑒 𝐹
In 𝑹𝟐
EP
TE
D
𝑒𝑟𝑓(
A
𝑦𝐹 = 𝑦𝐹𝑣
(69)
𝑥 )+1 2 √𝐿𝑒 𝐹
M
𝑒𝑟𝑓(
N
distribution can be gained as:
Gaseous Fuel mass fraction distribution in this region is achieved as previous regions. The
CC
boundary conditions governing on this zone exist in vaporization position and flame position, therefore, by using boundary conditions in Eq. (69), constant coefficients and gaseous fuel mass
A
fraction distribution can be gained as: 𝑥 = 𝑥𝑣 → 𝑦𝐹 = 𝑦𝐹𝑣 , 𝑥 = 𝑥𝑓 → 𝑦𝐹 = 0
(70)
(71)
20
𝑒𝑟𝑓(
𝑦𝐹 = 𝑦𝐹𝑣 𝑒𝑟𝑓(
𝑥𝑓 𝑥 )−𝑒𝑟𝑓( 2 ) 2 √𝐿𝑒 √𝐿𝑒 𝐹 𝐹 𝑥𝑓 𝑥𝑣 )−𝑒𝑟𝑓( 2 ) 2 √𝐿𝑒 √𝐿𝑒 𝐹 𝐹
In 𝑹𝟑
SC RI PT
In the post-flame zone, the amount of fuel is equal to zero, thus: 𝑦𝐹 = 0
(72)
To determine the temperature, fuel and oxidizer mass fraction distributions, it is necessary to obtain all parameters of 𝑦𝐹𝑣 , 𝑥𝑣 , 𝑥𝑓 𝑎𝑛𝑑 𝜃𝑓 . Thus, four boundary conditions should be applied using jump
U
conditions at vaporization and flame positions to solve the governing equations simultaneously.
N
Substituting the aforesaid parameters in the temperature, fuel and oxidizer mass fraction distributions in
A
various zones will complete the conservation equations. Therefore, various temperature and mass
M
fractions are acquired. It should be said that to solve these equations, numerical methods should be used.
Results and discussion
TE
3.
D
Hence, the methods of Ref. [26] have been used for non-linear system of equations.
Lycopodium fuel particles are considered and the value of quantities utilized in conservation equations
EP
will be obtained using Table 1. It is necessary to note that these particles are of organic type that release combustible gases while receiving heat. In Table 1, lycopodium properties are presented.
A
CC
Table 1. Constant properties of equations
The released gas from lycopodium is considered to be methane [27, 28]. The combustion reaction is denoted as: (73)
𝐶𝐻4 + 2 𝑂2 + 7.52 𝑁2 → 𝐶𝑂2 + 2 𝐻2 𝑂 + 7.52 𝑁2
21
For reaction of fuel and air which methane gas is made from vaporization of the fuel particles, the equivalence ratio is obtained as [28]: 𝑌
𝜙𝑢 = 17.18 1−𝑌𝐹−∞
𝐹−∞
SC RI PT
(74)
where 𝑌𝐹−∞ is defined as: (75)
𝑌𝐹−∞ =
4 3
𝑓(𝜀) 𝜋𝑟𝑃3 𝑛𝑃 𝜌𝑃 𝜌
Fig. 3 shows a comparison between gaseous fuel mass fraction versus position for different particle
porosities at mass concentration of 100 𝑔𝑟/𝑚3 . Results are presented for the two cases of applying and
U
not considering radiation heat transfer. As can be observed, gaseous fuel mass fraction rises until it
N
reaches a maximum point at the vaporization position and then, slowly decreases until reaches zero at the
A
flame formation position due to consumption and reaction in the flame zone. In addition, considering
M
thermal radiation, increases the amount of generated gaseous fuel in pre-heat and post-vaporization zones and moves both vaporization and flame positions to the oxidizer side. Moreover, the more particle
D
porosity increases, more reduction in fuel mass fraction can be observed. In other words, rising porosity is
EP
TE
equivalent to decreasing of available fuel to be burnt with oxidizer.
Fig. 3. Fuel mass fraction variations against position for various particle porosities.
CC
Fig. 4 demonstrates a comparison between the oxidizer mass fraction in terms of the non-dimensional position for various particle porosities at mass concentration of 100 𝑔𝑟/𝑚3. As shown, by getting closer
A
to the flame formation position, mass fraction of oxidizer gradually decreases until reaches the flame position and then it achieves the value of zero. Furthermore, applying thermal radiation decreases the amount of initial oxidizer mass fraction in the post-flame zone. Also, increasing of the particle porosity decreases the reaction rate with oxidizer which means less oxidizer takes part in the reaction process. Hence, the curves related to the higher porosities are positioned at lower levels.
22
Fig. 4. Oxidizer mass fraction variations against position for various particle porosities.
Fig. 5 depicts temperature profile of the domain with respect to the non-dimensional position for different values of particle porosities. The temperatures gradually increase to reach the flame temperature.
SC RI PT
Observations reveal that increasing particle porosity leads to reduction in temperature distribution as the amount of available fuel would be reduced and therefore the amount of heat released will be decrease.
Also, the temperature profile moves toward the right side as particle porosity rises and the flame will be
Fig. 5. Temperature profile of the domain for various particle porosities.
U
formed farther form the fuel nozzle.
N
Figs. 6 and 7 respectively illustrate flame temperature variations with respect to the fuel and oxidizer
A
Lewis numbers for different mass concentrations and porosities of particles. As mentioned previously,
M
Lewis number is defined as the ratio of heat diffusion to the mass diffusion. In Fig. 6, increasing of the
D
fuel Lewis number is equivalent to the reduction in fuel mass fraction, thus this decrease will cause the flame temperature to reduce as the amount of heat released will be decreased. Moreover, increasing mass
TE
concentration from 67 𝒈𝒓/𝒎𝟑 to 83 𝒈𝒓/𝒎𝟑 and decreasing porosity from 6% to 0% will cause a growth
EP
in flame temperature. By increasing fuel mass concentration and/or decreasing porosity, the amount of available fuel will be increased and then, by reacting more fuel, flame temperature increases as well. The
CC
vaporization temperature used in this research was obtained from [6]. It should be noted that oxidizer Lewis number is assumed to be unit and the initial equivalence ratio is equal to 𝝋 = 𝟏.𝟒. In Fig. 7, flame
A
temperature is plotted against oxidizer Lewis number in which fuel Lewis number is assumed to be unit. Increasing of the oxidizer Lewis number equals to the reduction in oxidizer mass fraction which causes flame temperature to decrease. As exhibited in Fig. 7, with a rise in mass concentration and/or a decline in particle porosity, flame temperature will get a higher value.
23
Fig. 6. Effect of fuel Lewis number, fuel mass concentration and particle porosity on flame temperature.
Fig. 7. Effect of oxidizer Lewis number, fuel mass concentration and particle porosity on flame temperature.
In Fig. 8, flame temperature variations versus equivalence ratio are shown for two different particles
SC RI PT
radiuses of 30 and 60 microns. Influence of particle porosity is investigated as well. As demonstrated, increasing equivalence ratio leads to a growth in flame temperature. The reason is that when equivalence ratio gets higher values, the amount of available fuel will increase and it causes the temperature to
increase. Also, by increasing fuel particles radius, flame temperature will decrease. An increase in
particles size will lead to decrease in the particles surface-to-volume contact ratio which is the reason of
U
increasing of the required energy for particles vaporization and therefore flame temperature will be
A
N
reduced. As porosity increases, for a constant equivalence ratio and a radius, flame temperature reduces.
M
Fig. 8. Effect of equivalence ratio, particles radius and porosity on flame temperature.
D
The obtained results are compared with the methane-air non-premixed counterflow combustion .Due to
TE
the lack of relevant experimental/numerical data on the non-premixed counterflow dust flames. The comparison shows that the same trend is obtained which reinforces the qualitative validation of the
EP
developed model. In Fig. 9, comparison between the ratio of critical quenching strain rate with Ref. [9] for different values of fuel and oxidizer Lewis numbers is illustrated. Results of the presented model
CC
indicate good agreement with those of Ref. [9]. According to the flame zone analysis section of the Ref. [9], the ratio of the critical strain rate in non-unity Lewis numbers of oxidizer and fuel (𝑎) to the critical
A
strain rate in 𝐿𝑒𝑂 = 𝐿𝑒𝐹 = 1 (𝑎0 ) is presented as:
(76)
𝑎
(𝑎 ) 0
𝐶𝑟
=
6 2 𝐿𝑒𝑂 𝐿𝑒𝐹 𝑇𝑓 𝜂 𝑇 ( ) 𝑒𝑥𝑝 [𝑇𝑎0 (1 2 0 𝐹𝑂𝑓 𝑇𝑓 𝑑𝐸 𝑓
𝑇𝑓0
2
− 𝑇 ) + (𝑥𝑓2 − (𝑥𝑓0 ) )]
24
𝑓
𝑎
This figure implies that at 𝐿𝑒𝑂 = 1, (𝑎 ) 0
𝑎
value of (𝑎 ) 0
𝐶𝑟
𝐶𝑟
decreases by a rise in the fuel Lewis number. For instance, the
declines from 7.6 to 0.4 when the magnitude of fuel Lewis number grows from 0.2 to 1.4 𝑎
at 𝑇𝑎 = 30000. Furthermore, the increase of 𝑇𝑎 is related with a rise in (𝑎 ) . By reducing value of fuel 0
𝐶𝑟
SC RI PT
Lewis number, the magnitude of 𝑥𝑓 (flame position) increases, which significantly rises the exponential 𝑇
term of Eq. (76). This increase is superior to the reduction in magnitude of the term ( 𝑓0) 𝑇𝑓
𝑎
to an increase in the ratio (𝑎 ) . 𝐶𝑟
𝜂2 𝐿𝑒𝐹, 𝑑𝐸
leading
U
0
6
Fig. 9. Effect of fuel Lewis number on critical rate of quenching strain ratio for 𝐿𝑒𝑂 = 1 and 𝑇𝑎 = 30000.
A
N
Conclusion
M
In the present study, combustion of organic particles in a counterflow configuration of porous particles with the consideration of the effects of radiation heat transfer was studied. Assuming particles to vaporize
D
first to produce a specific chemical gas, gaseous fuel mass fraction and oxidizer mass fraction along with
TE
the energy equation in which the thermal radiation term was added, were written using specific boundary conditions and solved by analytical methods. It was found that by increasing Lewis numbers of fuel and
EP
oxidizer, flame temperature decreases and increasing Lewis number of oxidizer moves flame position toward the oxidizer nozzle. Results reveal that increasing of particle porosity decreases the reaction rate
CC
with oxidizer which means less oxidizer takes part in the reaction process. The increase of the particle porosity leads to reduction in temperature distribution. In addition, temperature profile moves toward the
A
right side as particle porosity rises. Finally, by increasing fuel mass concentration and/or decreasing the porosity, the amount of available fuel will increase and by reacting more fuel, flame temperature increases as well and with a rise in mass concentration and/or a decline in particle porosity, flame temperature will get a higher value. In dust flame concept, there is a new parameter, representative of organic particle's vaporization which should be embedded in the governing equations unlike the gas flame 25
structure. The concentration of gaseous mixture and dust particles are become related and coupled to each other in the governing equations. Furthermore, the heat capacity involved in the governing equations is a
A
CC
EP
TE
D
M
A
N
U
SC RI PT
function of gas and particle heat capacities.
26
Nomenclature
𝑎
Quenching strain rate
𝐵
Frequency constant
𝐶𝑎
Gaseous phase specific heat (𝑘𝑔.𝐾)
𝐶𝑃
Solid particle specific heat (𝑘𝑔.𝐾)
𝐷𝐶
Damkohler number
𝐷𝐹
Mass diffusivity coefficient of gaseous fuel (
𝐷𝑂
Mass diffusivity coefficient of oxidizer (
𝐷0𝐸
Critical Damkohler number
𝐷𝑇
Thermal diffusivity coefficient (
𝐸
Activation energy (𝑘𝐽)
𝑒
Apparent porosity
𝑒𝑟𝑓𝑖(𝑥)
Error function
𝑓(𝜀)
Porosity function
𝑘𝐽
𝑚2 ) 𝑠
N
TE
D
M
A
𝑚2 ) 𝑠
EP
Heaviside function
CC
𝐼𝑏𝑝
𝑊
Radiation intensity (𝑚2 ) 𝑊 ) 𝑚2
Blackbody emissivity power (
𝐼𝑓
Induced radiation intensity
𝐾𝑎
Absorption coefficient ( )
𝐾𝑠
Scattering coefficient (𝑚)
𝐿𝑒
Lewis number
𝑚
Mixture molecular mass (
A
U
𝑚2 ) 𝑠
𝐻 𝐼
SC RI PT
𝑘𝐽
1 𝑚
1
𝑘𝑔 ) 𝑚𝑜𝑙
27
Fuel molecular mass (
𝑚𝑂
Oxidizer molecular mass (𝑚𝑜𝑙)
𝑛𝑃
Number of particle per unit volume
𝑃(𝜃,𝛷)
Scattering phasic function
𝑄
Reaction heat per unit of fuel mass ( )
𝑄𝑎𝑏𝑠
Efficiency factors of absorption
𝑄𝑠𝑐𝑎
Efficiency factors of scattering
𝑄𝑟𝑎𝑑
Thermal radiation (𝑘𝑔)
𝑞
Dimensionless heat
𝑅
Universal constant of gases (𝑚𝑜𝑙.𝐾)
𝑇
Fuel temperature (𝐾)
𝑇𝑎
Dimensionless activation energy
𝑇𝑓
Flame temperature (𝐾)
𝑇𝑣
Particle temperature of vaporization (𝐾)
𝑉𝑣𝑜𝑖𝑑
TE
𝑘𝑔 ) 𝑚𝑜𝑙
𝑚𝑓
D
M
A
N
𝑚3 𝑃𝑎
Void volume
Total volume
Molecular weight of fuel
𝑥
Dimensionless length
𝑥𝑓
Flame front position
𝑥𝑣
Vaporization front position
A 𝑌𝐹
Gaseous fuel mass fraction
𝑌𝑂
Oxidizer mass fraction
28
SC RI PT
U
𝑘𝐽
CC
𝑊𝐹
𝑘𝐽 𝑘𝑔
EP
𝑉𝑡𝑜𝑡𝑎𝑙
𝑘𝑔
Solid particles mass fraction
𝑦𝐹
Dimensionless fuel mass fraction
𝑦𝑜
Dimensionless oxidizer mass fraction
𝑦𝑠
Dimensionless solid particles mass fraction of
𝑍𝑒
Zeldovich number
SC RI PT
𝑌𝑠
Initial mass fraction of oxidizer
𝜀
Porosity
𝜖𝑃
Emissivity of solid particles
𝜂
Defined in eq. (54)
𝜃
Dimensionless temperature
𝜆
Thermal conductivity of fuel or oxidizer (𝑚.𝑠.𝐾)
𝜈𝐹
Fuel stoichiometric coefficient
𝜈𝑂
Oxidizer stoichiometric coefficient
𝜈𝑃
N A
M
D
CC
𝜌𝑎
Products stoichiometric coefficient 𝑘𝑔 ) 𝑚3
Density (
EP
𝜌
U
𝛼
TE
Greek symbols
𝑘𝑔 ) 𝑚3
Gaseous phase density (
𝑘𝑔
Solid particle density(𝑚3 )
𝜎
Stefan-Boltzmann constant
A
𝜌𝑃
𝜏𝑣
Vaporization time
𝛤, 𝜓 and Ʌ
Dimensionless radiation parameters
𝛺
Defined in eq. (8)
29
𝑘𝐽
𝑘𝑔 ) 𝑚.𝑠2
𝜔𝑣
Rate of particle vaporization (
𝜔𝐹
Rate of chemical reaction (𝑚𝑠2 )
A
CC
EP
TE
D
M
A
N
U
SC RI PT
𝑘𝑔
30
References 1.
M. Bidabadi , M. Harati , Q. Xiong, E. Yaghoubi , M. Doranehgard , P. Aghajannezhad, Volatization & combustion of biomass particles in random media: Mathematical modeling and
Intensification, 126 (2018):232-238. 2.
A.I. Stankiewicz, J.A. Moulijn, Process intensification: transforming chemical engineering, Chemical engineering progress 96, no. 1 (2000): 22-34.
3.
R.K. Eckhoff, Dust Explosion in the Process Industries, Second ed., Butterworth Heinemann,
U
Oxford, (1997). 4.
SC RI PT
analyze the effect of Lewis number, Chemical Engineering and Processing - Process
H. Moghadasi, A. Rahbari, M. Bidabadi, A. Khoeini Poorfar, V. Farhangmehr, Mathematical
N
Investigation of Premixed Lycopodium Dust Flame in Small Furnaces, Journal of Energy
A. Afzalabadi, A. Khoeini Poorfar, M. Bidabadi, H. Moghadasi, S. Hochgreb, A. Rahbari, C.
M
5.
A
Resources Technology, (2018), DOI: https://doi.org/10.1115/1.4041106.
D
Dubois. Study on hybrid combustion of aero-suspensions of boron-aluminum powders in a
TE
quiescent reaction medium, Journal of Loss Prevention in the Process Industries 49 (2017): 645651.
H. Hanai, H. Kobayashi, T. Niioka, A numerical study of pulsating flame propagation in mixtures
EP
6.
of gas and particles, Proceedings of the Combustion Institute, 28(1) (2000):815-822. C. Proust, Flame propagation and combustion in some dust-air mixtures, Journal of Loss
CC
7.
8.
M. Bidabadi, Q. Xiong, M. Harati, E. Yaghoubi, M.H. Doranehgard, A. Rahbari, Study on the
A
Prevention in the Process, 19(1) (2006):89-100.
combustion of micro organic dust particles in random media with considering effect of thermal resistance and temperature difference between gas and particles. Chemical Engineering and Processing - Process Intensification 126 (2018): 239-247.
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33
Caption of figures
Structure of diffusion counterflow combustion.
Figure 2
Structure of particles porosity.
Figure 3
Fuel mass fraction variations against position for various particle porosities.
Figure 4
Oxidizer mass fraction variations against position for various particle porosities.
Figure 5
Temperature profile of the domain for various particle porosities.
Figure 6
Effect of fuel Lewis number, fuel mass concentration and particle porosity on flame
N
U
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Figure 1
Effect of oxidizer Lewis number, fuel mass concentration and particle porosity on flame
M
Figure 7
A
temperature.
D
temperature.
Effect of equivalence ratio, particles radius and porosity on flame temperature.
Figure 9 30000.
Effect of fuel Lewis number on critical rate of quenching strain ratio for 𝐿𝑒𝑂 = 1 and 𝑇𝑎 =
A
CC
EP
TE
Figure 8
34
SC RI PT U N A
A
CC
EP
TE
D
M
Fig. 1. Structure of diffusion counterflow combustion
Fig. 2. Structure of particles porosity
35
SC RI PT U N A M
A
CC
EP
TE
D
Fig. 3. Fuel mass fraction variations against position for various particle porosities.
36
SC RI PT U N A M
A
CC
EP
TE
D
Fig. 4. Oxidizer mass fraction variations against position for various particle porosities.
37
SC RI PT U N
A
CC
EP
TE
D
M
A
Fig. 5. Temperature profile of fuel and oxidizer for various particle porosities.
Fig. 6. Effect of fuel Lewis number, fuel mass concentration and particle porosity on flame temperature. 38
SC RI PT U N
A
CC
EP
TE
D
M
A
Fig. 7. Effect of oxidizer Lewis number, fuel mass concentration and particle porosity on flame temperature.
Fig. 8. Effect of equivalence ratio, particles radius and porosity on flame temperature. 39
SC RI PT U N A M D
A
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Fig. 9. Effect of fuel Lewis number on critical rate of quenching strain ratio for 𝐿𝑒𝑂 = 1 and 𝑇𝑎 = 30000.
40
Caption of tables
Constant properties of equations
SC RI PT
Table 1
Table 1. Constant properties of equations
values
Reference
1000
𝐾𝑔 𝑚3
𝜌𝑎
1.164
𝐾𝑔 𝑚3
5.677688
𝐶𝑎
1.00416
M
D
TE
[30]
𝐾𝐽 𝑘𝑔.𝐾
[30]
𝐾𝐽 𝑘𝑔
[30]
64895.4
1.46538 × 10−4
A
CC
EP
𝜆
𝐾𝐽 𝑘𝑔.𝐾
[29]
A
𝐶𝑃
𝑄
[29]
N
𝜌𝑃
U
parameters
41
𝐾𝐽 𝑚.𝑠.𝐾
[25]