Preparation of particle oxide for biomedical applications employing a safe thermo-chemical technique: An analytical study

Journal Pre-proof Preparation of particle oxide for biomedical applications employing a safe thermo-chemical technique: An analytical study Sadegh Sadeghi, Mehdi Bidabadi

PII:

S0255-2701(19)30419-2

DOI:

https://doi.org/10.1016/j.cep.2019.107680

Reference:

CEP 107680

To appear in:

Chemical Engineering and Processing - Process Intensification

Received Date:

10 April 2019

Revised Date:

25 August 2019

Accepted Date:

2 October 2019

Please cite this article as: { doi: https://doi.org/ This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. 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. © 2019 Published by Elsevier.

Preparation of particle oxide for biomedical applications employing a safe thermo-chemical technique: An analytical study Sadegh Sadeghi, Mehdi Bidabadi* School of Engineering, Mechanical Engineering Department, Iran University of Science and Technology, Narmak,

*

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Tehran, Iran. Corresponding author E-mail: [email protected]

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Highlights

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applications is proposed.

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 A thermo-chemical technique for preparation of particle oxide for biomedical

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 Preheat, reaction, melting, vaporization and oxidation processes are analyzed.  An asymptotic concept is employed for the analysis.

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 Mass fraction and temperature distributions of particles, particle oxide and oxidizer are obtained.

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 Path of the particles through the system is presented.

Abstract Due to the potential use of particle oxides in biomedical industry and tissue engineering, particularly for cancer diagnosis and therapy, promising preparation ways of the oxides are crucial. In this paper, an analytical approach is proposed to model the production of a particle oxide for biomedical applications using a thermochemical technique. For this purpose, a multi-zone

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structure including preheat, reaction, melting, vaporization and oxidizer zones is developed for the system. To detect the behavior of the flame, an asymptotic method is used. Accordingly, melting

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and vaporization processes are modelled by different Heaviside functions. Mass and energy

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conservation equations are presented for determining the amount of particle oxide and temperature distribution. To preserve the continuity and trace the positions of flame, melting and vaporization

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fronts, jump conditions are determined at the inner interfaces. Finally, variations of temperature and mass fraction of the particles and particle oxide with position and Lewis number are obtained.

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Based on the results, maximum non-dimensional temperature of the system corresponding to the vaporization front is found to be 0.56. Non-dimensional positions of flame front (at which particle

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oxide is produced), melting front and vaporization front are found to be -1.8, -1.78 and -1, respectively.

Keywords: Biomedical applications; Cancer diagnosis and therapy; Particle oxide; Non-premixed

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structure; Counter-flow design; Mathematical model.

Nomenclature 𝑎

1

Strain rate (s )

1

𝐵

Frequency factor (s )

𝐶

Mixture specific heat (kg.K)

𝐶𝑎

Specific heat capacity of oxidizing gas (kg.K)

𝐶𝑝

Specific heat capacity of particle (kg.K)

𝐷

Mass diffusion coefficient ( m2/s)

𝐷𝐹

Mass diffusion coefficient of particle ( m2/s)

𝐷𝑂

Mass diffusion coefficient of oxidizer (m2/s)

𝐷𝑇

Thermal diffusion coefficient of particle (m2/s)

D𝑇𝑂

Thermal diffusion coefficient of particle oxide

𝐸

Overall activation energy (kJ)

𝐻

Heaviside function

𝐿𝑒𝐹

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kJ

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𝐿𝑒𝑂

kJ

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𝐿𝑒

kJ

Lewis number Fuel Lewis number Oxidizer Lewis number kg

𝑚

Mixture molecular mass (mol)

𝑚𝑂

Oxidizer molecular mass (mol)

𝑛𝑝

Density number of particles per unit volume (m3)

kg

1

kJ

𝑄

Heat of reaction per unit mass of particle (kg)

𝑄𝑚𝑒𝑙𝑡

Latent heat of melting per unit mass of particle (kg)

𝑞𝑚𝑒𝑙𝑡

Normalized heat of melting

𝑄𝑣𝑎𝑝

Latent heat of vaporization per unit mass of particle (kg)

𝑞𝑣𝑎𝑝

Normalized heat of vaporization

𝑅

Universal constant of gases (mol.K)

𝑟

Particle radius (𝑚)

𝑇

Temperature (K)

𝑇𝑎

Activation temperature (K)

𝑇𝑓

Flame temperature (K)

𝑇𝑚𝑒𝑙𝑡

Melting temperature of particle oxide (K)

𝑢

of

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re

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m3 Pa

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𝑣𝑂

kJ

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𝑇𝑣𝑎𝑝

kJ

vaporization temperature of particle oxide (K) 𝑚

Velocity ( 𝑠 ) Oxidizer stoichiometric coefficient

𝑣𝑝𝑟𝑜𝑑𝑢𝑐𝑡

Product stoichiometric coefficient

𝑋

Horizontal direction (𝑚)

𝑥

Normalized form of position in 𝑋 direction

Flame front position

𝑥𝑚𝑒𝑙𝑡

Normalized position of melting front

xp

Position of the particles in X-direction

𝑥𝑣𝑎𝑝

Normalized position of vaporization front

𝑌𝑂

Oxidizer mass fraction

𝑦𝑂

Normalized mass fraction of oxidizer

𝑦𝑝

Position of particles in Y-direction

𝑌𝑠

Mass fraction of particle

𝑦𝑠

Normalized mass fraction of particle

𝑍𝑒

Zeldovich number

φ

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𝜃

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Greek symbols 𝛼

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𝑥𝑓

Initial mass fraction of oxidizer Arbitrary particle Normalized temperature

𝜃𝑓

Normalized flame temperature

𝜃𝑚𝑒𝑙𝑡

Normalized temperature of melting

𝜃𝑣𝑎𝑝

Normalized temperature of vaporization

𝜗𝑂

Moles of oxidizer combusting with one mole of fuel

𝜆

Thermal conductivity (m.s.K)

𝜌

Mixture density (m3 )

𝜌𝑎

Density of gas (m3 )

𝜌𝑝

Density of particle (m3 )

𝜏𝑚𝑒𝑙𝑡

Characteristic time of melting (𝑠)

𝜏𝑣𝑎𝑝

Characteristic time of vaporization (𝑠)

𝜔𝑚𝑒𝑙𝑡

Melting rate (m.s2 )

𝜔𝑆

Chemical reaction rate of particle (m.s2 )

𝜔𝑣𝑎𝑝

Vaporization rate (m.s2)

kJ

kg

kg

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kg

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kg

kg

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kg

1. Introduction

In the last couple of decades, particle oxide is introduced as one of the substances attractively used for cancer diagnosis and treatment, purification of waste water and air from hospitals, and food

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factories [1,2]. Particle oxide possesses a highly hydrophilic property, and is thermodynamically stable and harmless [1]. Furthermore, immunity to corrosion, biocompatibility, strength, low modulus and density, and capacity for joining with bone have made some particulate substances suitable for tissue engineering applications [1-6]. Numerous potential techniques, such as coprecipitation, hydrothermal and combustion, have been introduced for synthesizing the particles

oxides for biomedical applications [1]. Combustion of particle in air can be considered as a potential and common method for preparing particle oxide [7]. Generally, combustion can be occurred in premixed, non-premixed and partially-premixed modes amongst them non-premixed flames are safer and more controllable due to its structure [8]. Up to now, different efforts have been made to propose different techniques for providing particle

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oxide used for treatment of cancer and tissue engineering. Daoush [9] used co-precipitation technique to synthesize particle oxide for biomedical applications in presence of

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ethylenediaminetetraacetic acid and sodium hydroxide as a precipitating agent taking into account

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the particle shape and size. Mata-Pérez et al. [10] proposed a simple three-step process for preparing stable and uniform particle oxide at low temperatures using x-ray diffraction (XRD),

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Fourier transform infrared spectroscopy (FT-IR), transmission electron microscopy (TEM), and vibrating sample magnetometry (VSM). Andrade et al. [11] employed a direct reduction-

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precipitation method to synthesize size-controlled particle oxide and investigated the effects of aging time and temperature on the size and monodispersion characteristics of the produced oxide.

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Setyawan and Widiyastuti [12] utilized an electrochemical method carried out by passing an electric current through an anode and a cathode in an electrolyte solution in order to prepare a particle oxide. Dresco et al. [13] suggested an experimental technique composed of a two-stage microemulsion process and a seed precipitation polymerization for synthesizing particle oxide

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considering particle size. Koushika et al. [14] employed a thermal plasma method to prepare powder of particle oxide conducting a phase analysis and chemical composition of the particles. Rashid et al. [15] applied an in-situ precipitation method to synthesize particle oxide employing X-ray diffraction, energy-dispersive X-ray spectroscopy and Raman spectroscopy. Silva et al. [16] proposed a precipitation/photoreduction combined process to produce particle oxide for reducing

environmental damages in coal mining industry. Lei and Girshick [17] used a DC thermal plasma method to produce particle oxide considering aerosol sampling probe interfaced to a scanning mobility particle sizer for measuring the particle size distribution. Owens et al. [18] described different potential sol-gel methods for preparing particle oxide for biomedical applications. With respect to the literature review, former research studies have mostly investigated the co-

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precipitation, electrochemical method and sol-gel technique for preparing particle oxide for biomedical applications. Although, thermochemical technique one of the efficient ways of

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synthesizing the oxides [7], no effort has been devoted to modelling the safe combustion of

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particles for production of particle oxide. It should be noted that non-premixed combustion is safer and more controllable than premixed or partially-premixed combustion for producing particle

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oxide for biomedical applications [8].

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In this work, an analytical model is proposed for production of particle oxide by counter-flow nonpremixed combustion of the particles. For this purpose, a multi-zone structure consisting of preheat, reaction, melting, vaporization and oxidizer zones is mathematically investigated. In order

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to study the heat transfer behavior of the flame, an asymptotic concept is used. In addition, melting and vaporization processes are described by different Heaviside functions. Mass and energy conservation equations are derived in the zones to calculate the amount of particle oxide and

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temperature distribution. In order to enforce the continuity and obtain the locations of flame, melting and vaporization fronts, proper jump conditions are achieved. Eventually, variations of temperature and mass fraction of particle and particle oxide with position and Lewis number are presented.

2. Description of the system In this research, production of particle oxide by non-premixed combustion technique in counterflow configuration is mathematically modelled. Schematic of the counter-flow non-premixed combustion system is shown in Fig. 1. As can be seen in this figure, micron-sized particles (accompanied by an inert gas) and oxidizer are injected into the system by two different nozzles

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located at opposite sites. The particles gain some heat in the preheat zone until reaching the ignition temperature. In the reaction zone, preheated particles are combusted with oxidizer during a

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heterogeneous reaction resulting in production of particle oxide. It should be pointed out that

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Zeldovich number is assumed to be very large; therefore, thickness of the reaction zone will be very small. Initial activation energy for onset of the combustion is created by a spark. In the next

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stage, all amount of particle oxide is quickly transformer into liquid phase at the melting front. Subsequently, liquid particle oxide is rapidly vaporized at the vaporization front and converted

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into gaseous phase. As it is assumed that initial momentum of fuel and oxidizer streams are equal, a stagnation line exists in the middle of the distance between the oxidizer and fuel nozzles. In this

line.

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paper, locations of flame, melting and vaporization fronts are measured relative to the stagnation

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3. Mathematical model

In this section, the considered physical processes are mathematically modelled. For this purpose, dimensional and non-dimensional forms of mass and energy conservation equations, and corresponding boundary and jump conditions are derived. Velocity field in 𝑋 direction is obtained

by 𝑢 = −𝑎𝑋 in which 𝑎 represents the flow strain rate [19]. In this model, Arrhenius one-step reaction is considered [19]: 𝑣𝐹 [𝐹] + 𝑣𝑂 [𝑂] → 𝑣𝑝𝑟𝑜𝑑𝑢𝑐𝑡 [𝑃]

(1)

where [𝐹], [𝑂] and [𝑃] denote fuel, oxidizer and products, respectively. Also, 𝜐𝐹 , 𝜐𝑂 and 𝜐𝑃 are fuel, oxidizer and product stoichiometric coefficients, respectively.

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In order to analyze the heat transfer behavior of the system, following assumptions are exerted: Particles are uniformly distributed in the preheat zone.



The analysis is conducted for the very small values of Knudsen number, less than 10−3. In

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

Since large quantities of Zeldovich number, 𝑍𝑒 =

𝐸𝑄𝑌𝐹−∞ 𝑅𝐶𝑇𝑓2

, are considered, the thickness of

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

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this regard, a continuum model is considered for the system [19].

the flame zone will be very small [19].

For simplicity, density and specific heat are considered to be constant [19].



Ambient temperature is considered to be 300 K.

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

Reaction rate of the particles is obtained by the following correlation [20]: ωS = k 0 YS YO exp (−

Ea ) RTf

(2)

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In the above equation, YS , YO , Ea , R, Tf and k 0 are mass fraction of the particles, mass fraction of oxidizer, activation energy, universal gases constant, flame temperature and reaction coefficient, respectively.

Since melting process of particle oxide occurs rapidly, a Heaviside step function is applied:

ωmelt =

YS τmelt

(3)

H(T − Tmelt )

where τmelt and Tmelt represent constant characteristic time of melting and melting temperature of the particles, respectively. Similarly, to model the vaporization process of particle oxide, following Heaviside step function is used: Ym H(T − Tvap ) τvap

(4)

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ωvap =

where Ym , τvap and Tvap are mass fraction of liquid particle oxide, constant characteristic time of

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vaporization and vaporization temperature of particle oxide, respectively.

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Density and specific heat capacity of particles/oxidizer mixture are obtained by the following formulas [19]:

4𝜌𝑝 3 𝜋𝑟 𝐶𝑝 𝑛𝑝 3𝜌

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𝐶 = 𝐶𝑎 +

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4 𝜌 = 𝜌𝑎 + 𝜋𝑟 3 𝑛𝑝 𝜌𝑝 3

(5) (6)

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In the above equations, 𝜌𝑎 , 𝑟, 𝑛𝑝 and 𝜌𝑝 demonstrate the density of oxidizing gas, radius of the particles, average density number of the particles per unit volume and density of the particles, respectively. Also, 𝐶𝑎 and 𝐶𝑝 are specific heat capacity of oxidizing gas and the particles, respectively.

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Now, considering the aforementioned definitions, general forms of mass and energy conservation equations are derived in the following. Dimensional form of energy conservation equation is presented below:

𝒬vap dT d2 T 𝒬 𝒬melt −ax ( ) = DT ( 2 ) + ωS ( ) − ωmelt ( ) − ωvap ( ) dx dx ρc c c

(7)

where DT , 𝒬, 𝒬melt and 𝒬vap demonstrate the thermal diffusivity of the particles, heat of reaction, heat of melting and heat of vaporization, respectively. DT is determined by the following formula [19]: λ DT = ( ) ρc

(8)

Mass conservation equation for the particles is written as follows:

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dYS ωS −ax ( ) = − ( ) dx ρ

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where λ is thermal conductivity of the particles.

(9)

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In the above equation, the interaction between the particles is disregarded; thus, diffusion term in

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preheat zone is neglected.

Mass conservation equation for particle oxide in liquid phase is written as below:

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dYm d2 Ym −ax ( ) = Dm ( 2 ) + ωmelt dx dx

(10)

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Mass conservation equation for particle oxide in gaseous phase is presented as below: dYF d2 YF −ax ( ) = DF ( 2 ) + (ωvap ) dx dx

(11)

where YF represents the mass fraction of gaseous particle oxide.

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Mass conservation equation for oxidizer is written as follows: dYO d2 YO ωs −aX = DO − ϑ dX dX 2 ρ

(12)

where YO and DO represent the mass fraction and mass diffusivity coefficient of oxidizer, respectively.

Lewis number which is described as the ratio of thermal diffusivity to mass diffusivity is formulated as follows [19,21]: λ

(13)

Le = ρcD

In order to non-dimensionalize the governing equations, following dimensionless parameters are applied:

), θ =

𝓆melt =

𝒬melt , 𝒬

𝓆vap =

𝒬vap 𝒬

yS = (Y

,ω ̃ melt =

YS S−∞

), yF = (Y

yS H(T aτmelt

YF S−∞

), yO =

− Tmelt ), ω ̃ vap =

ym =

Y (Y m ), S−∞

(14)

ym H(T aτvap

− Tvap ), ω ̃S =

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k0 ϑO YS −∞ T ys yo exp (− Ta) 𝜌a f

Y (ϑY O ), S−∞

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λ

√ρca

c(T−T ) ( 𝒬Y ∞ ), S−∞

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𝒳=(

x

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where 𝒳, θ, yS , yF , yO , ym , 𝓆melt , 𝓆vap , ω ̃ melt , ω ̃ vap and ω ̃ S represent the non-dimensional forms of position, temperature, mass fraction of particles, mass fraction of gaseous particle oxide, mass

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fraction of oxidizer, mass fraction of molten particle oxide, heat of melting, heat of vaporization, melting rate, vaporization rate and reaction rate, respectively. Also, YS−∞ is defined as the initial

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mass fraction of the particles injected by the fuel nozzle. By substituting the non-dimensional parameters into the governing equations (Eqs. 7-12), the equations are rewritten as below:

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H(T − Tvap ) d2 θ dθ yS H(T − Tmelt ) yS Ta +𝒳 − 𝓆melt + 𝓆vap = −𝒟c yS yO exp (− ) 2 d𝒳 d𝒳 a τmelt a τvap T

𝒳

dyS Ta = 𝒟c yS yO exp (− ) d𝒳 T

1 d2 ym dym yS H(T − Tmelt ) +𝒳 − =0 2 Le d𝒳 d𝒳 a τmelt

(15) (16) (17)

1 d2 yF dyF ym H(T − Tvap ) + 𝒳 − =0 Le d𝒳 2 d𝒳 a τvap

(18)

1 d2 yO dyO Ta +𝒳 = 𝒟c yS yO exp (− ) 2 Le d𝒳 d𝒳 T

(19)

In order to solve the non-dimensionalized governing equations, following regions are considered: ℛ1 : − ∞ < 𝒳 ≤ 𝒳f

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ℛ2 : 𝒳f < 𝒳 ≤ 𝒳melt

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ℛ3 : 𝒳melt < 𝒳 ≤ 𝒳vap

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ℛ4 : 𝒳vap < 𝒳 < +∞

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Necessary boundary conditions are presented in Table 1 considering the interfaces shown in Fig. 1. In order to preserve the continuity, proper jump conditions are derived at the flame, melting and

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vaporization fronts. At the flame front, convection term and rates of melting and vaporization are neglected compared with the reaction rate and diffusion terms. At the melting front, convection

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term and rates of reaction and vaporization are disregarded compared with the melting rate and diffusion terms. At the vaporization front, convection term and rates of reaction and melting are neglected compared with the vaporization rate and diffusion terms. By following similar procedure

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provided in Refs. [19] and [22], jump conditions at 𝒳f , 𝒳melt and 𝒳vap are obtained as follows:

x+ f

x+ f x− f

x+ f

1 dyF ( ) | Le d𝒳

=−

x− f

=−

x− f

{

1 dym ) | Le d𝒳

| d𝒳 x− x+ f

dθ

| d𝒳 x− f

x+ f

(

dθ

f

x+ f

1 dyO ( ) | Le d𝒳

x+ f

=−

x− f

x+ f

dθ

| d𝒳 x− f

+ dy xmelt

1

(Le) d𝒳F |

x− melt

+ dyO xmelt | d𝒳 x−

𝓆

melt

|

d𝒳 x− melt x+ vap

=

x− vap

yS

= aτ

melt

ym H(T − Tvap )xvap aτvap

dyO d𝒳 x+ vap

(22)

x+ vap

|

=0

x− vap

𝓆𝑣𝑎𝑝 y H(T − Tvap )xvap aτvap m

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dθ

H(T − Tmelt )xmelt

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+ dym xmelt

1 dyF ( ) | Le d𝒳

@ 𝒳 = 𝒳vap

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= − aτ𝑚𝑒𝑙𝑡 yS H(T − Tmelt )xmelt

melt

1

=0

melt

+ dθ xmelt | d𝒳 x−

( ) { Le

(21)

=0

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@ 𝒳 = 𝒳melt

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x− f

@ 𝒳 = 𝒳f

(20)

1 dyO =( ) | Le d𝒳

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1 dyF ( ) | Le d𝒳

| d𝒳 x−

vap

=−

x+ vap

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{

1 dym ( ) | Le d𝒳

=0

x− vap

It should be noted that solution of the equations are implemented in Matlab software. Temperature distribution in each of the considered zones is presented as below: Zone ℛ1 :

θ=

θp √2 [1 + erf ( 2 𝒳f )]

[1 + erf (

(23)

√2 𝒳)] 2

Zone ℛ2 :

√2 √2 erf ( 𝒳melt ) − erf ( 𝒳f ) 2 2

√2 erf ( 2 𝒳) (θf − θmelt )

−

erf (

√2 √2 𝒳 − erf ( 𝒳f ) 2 melt ) 2

√2 √2 erf ( 2 𝒳melt ) − erf ( 2 𝒳vap )

−

√2 erf ( 2 𝒳) (θvap − θmelt )

√2 √2 erf ( 2 𝒳melt ) − erf ( 2 𝒳vap )

√2 𝒳 2 vap )]

√2 𝒳)] 2

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[1 − erf (

[1 − erf (

(26)

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θvap

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Zone ℛ4 :

θ=

(25)

-p

√2 √2 θvap erf ( 2 𝒳melt ) − erf ( 2 𝒳vap ) θmelt

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Zone ℛ3 :

θ=

(24)

of

θ=

√2 √2 θf erf ( 2 𝒳melt ) − erf ( 2 𝒳f ) θmelt

Mass fraction of molten particle oxide in each of the considered zones is obtained as below: Zone ℛ1 :

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ym = 0

(27)

Zone ℛ2 :

ym =

√2Le √2Le ymM (erf ( 2 𝒳) − erf ( 2 𝒳f )) √2 Le √2Le erf ( 2 𝒳melt ) − erf ( 2 𝒳f )

Zone ℛ3 :

(28)

ymM (erf ( ym =

√2Le √2Le 𝒳) − erf ( 𝒳vap )) 2 2

(29)

√2 Le √2Le erf ( 2 𝒳melt ) − erf ( 2 𝒳vap )

Zone ℛ4 : ym = 0

of

(30)

Mass fraction of gaseous particle oxide in each of the considered zones is presented in the

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following:

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Zone ℛ1 : yF = 0

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Zone ℛ2 :

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yF = 0

Zone ℛ3 :

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yF =

√2Le √2Le yFV (erf ( 2 𝒳melt ) − erf ( 2 𝒳)) erf (

yF = yFV

Mass fraction of oxidizer in each of the considered zones is calculated in the following: Zone ℛ1 :

(32)

(33)

√2 Le √2Le 𝒳melt ) − erf ( 𝒳vap ) 2 2

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Zone ℛ4 :

(31)

(34)

yO = 0

(35)

Zone ℛ2 : yO

(36) 1 xf 2 2 1⁄Le )

1 1 −12 x + −2 ( Le LeO e

(

x2 1⁄Le ) + x√2π

Le erf (

(

1 xf 2 − 2 (e 2 1⁄Le )

1 1 −12 x + −2 ( melt Le Le e

x2melt 1⁄Le ) +

xf x − erf √2 √2 ( Le) ( Le)

2π xmelt √ Le erf (

(

(

Zone ℛ4 :

𝒳vap 𝒳melt − erf √2 √2 ( Le ) ( Le )

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erf

α(erf (

𝒳vap

))

(37)

erf

𝒳vap 𝒳melt − erf √2 √2 ( Le ) ( Le )

x x ) − erf ( )) + yO𝑉 (1 − erf ( )) √2 √2 √2 𝒳vap −1 + erf ( ) √2

)

(38)

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yO =

−

𝒳melt yOM erf − yOV erf √2 √2 ( Le ) ( Le )

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yO =

𝒳vap

xf

re

x (yOM − yOV ) erf √2 ( Le)

)) )

xf x − erf melt √2 √2 ( Le) ( Le )

-p

Zone ℛ3 :

xf yOM

of

=

−

ro

2 (e

In order to trace the path of particles over the system, it is assumed that the particles are uniformly distributed before they enter the domain, and there is no collision between the particles at the entrance. It is worth noting that the particles are carried by an inert gas with the same velocity. In X-direction, the particles are affected by drag and thermophoretic forces, while in Y-direction,

drag and buoyancy forces are exerted on the particles. From a Lagrangian point of view, the curves of motion of each particle are achieved by integrating the following equations: Ux =

dxp dUx , mp = FDx + FT dt dt

(39)

Uy =

dyp dUy , mp = FDy + FB dt dt

(40)

of

where drag, thermophoretic and buoyancy forces can be determined by Eqs. (41), (42) and (43), respectively [19,20]:

Cs k g ∇T k p + 2k g ρg Tu

-p

FT = −6πμ2 dp

ro

FD = −3πμdp U

re

1 FB = − πd3p (ρp − ρg )g 6

obtained in X and Y directions:

dU dt

(42)

(43)

dU

with U dx , following equations are

lP

By using the aforementioned formulas and replacing

(41)

Cs k g dU ∇T = −3πμdp U − 6πμ2 dp dx k p + 2k g ρg Tu

(44)

mp . U

dU 1 = −3πμdp U − πd3p (ρp − ρg )g dy 6

(45)

ur na

mp . U

Similar to the computations reported in Ref. [20], following solutions are obtained for Ux and Uy :

Jo

3πμdp 2 ) (x) mp −1− 2 −6πμ dp Cs kg ∇T mp (kp +2kg ) ρg Tu

(

Ux = −

(46)

(−

−6πμ2 dp Cs k g ∇T e ) 1 + Lambert − mp (k p + 2k g ) ρg Tu −6πμ2 dp Cs k g ∇T mp (k p + 2k g ) ρg Tu {

3πμdp − m p

[

]}

3πμdp 2 ) (y) mp −1− 3 −πμdp g(ρp −ρg ) 6mp

(47)

(−

−πμd3p g(ρp − ρg ) e ( ) 1 + Lambert − 6mp −πμd3p g(ρp − ρg ) 6mp 3πμdp − m p

[

]}

of

{

Uy = −

ro

4. Results and discussion

In this section, results of the current study obtained for temperature and mass fraction distributions

-p

of the particles, particle oxide and oxidizer are presented. Complete reaction of the particle with

φ + (O2 + 3.76N2 ) →

φO2 + 3.76N2

re

air is written in the following:

(48)

lP

where φ is an arbitrary particle whose oxide chemical formula is φO2 .

Z=

4 3

ur na

Mass particle concentration is defined as follows [19]:

In this study, initial value of mass particle concentration is considered to be 0.67 𝑘𝑔

𝑘𝑔 𝑚3

(less than 1

). For this reason, the flame front is initially located on the left-hand side of the stagnation line

Jo

𝑚3

(49)

𝜋𝑟 3 𝑛𝑝 𝜌𝑝

(See Fig. 1). In order to follow the temperature and mass fraction distributions, properties of the considered particle, particle oxide and oxidizer, and values of input parameters are listed in Table 2. Fig. 2 shows the variation of non-dimensional temperature distribution of the system in terms of position (distance from the stagnation line). From left to right, it can be observed from Fig. 2 that

the particles heat up in absence of oxidizer and their temperature continuously increases until reaching the ignition temperature. In the very thin reaction zone, the particles are combusted by the oxidizer, and temperature of the particles strikingly rises. In this zone, particle oxide is produced. At the melting and vaporization fronts, all amount of the particle oxide is completely melted and vaporized, respectively. According to Fig. 2, maximum non-dimensional temperature corresponding to the vaporization zone is found to be 0.56. Finally, due to the heat exchange

of

between particle oxide and oxidizer stream, temperature of the products is consistently reduced to

ro

the ambient temperature. It should be noted that non-dimensional values of flame front and

-p

vaporization front positions are found to be -1.81 and -0.96, respectively.

The influence of fuel Lewis number on the position of flame front for several particle diameters is

re

plotted in Fig. 3. As can be observed from Fig. 3, flame front shifts toward the fuel nozzle with a rise in the value of fuel Lewis number. Based on the definition of Lewis number (Eq. 13), larger

lP

fuel Lewis number implies that less amount of fuel particles are diffused into the reaction zone resulting in motion of flame front toward the preheat zone. With regard to Fig. 3, it can be

ur na

concluded that enhancing the particle diameter leads to motion of flame front toward the fuel nozzle. In other words, as the particle diameter decreases, the amount of accessible fuel supplied for the flame reduces; therefore, flame front moves toward the fuel nozzle. It is notable that the influence of particle diameter at lower fuel Lewis numbers is more significant than that at larger

Jo

fuel Lewis numbers. For the investigated diameters, when fuel Lewis number varies from 0.1 to 1.8, magnitude of flame front position ranges from about -0.7 to -1.7. The effect of oxidizer Lewis number on the position of flame front for different particle diameters is plotted in Fig. 4. It can be seen in Fig. 4 that flame front moves toward the oxidizer nozzle with increasing the oxidizer Lewis number. This can readily be justified by the definition of Lewis

number presented in Eq. 13. Higher oxidizer Lewis number implies that less amount of oxidizer mass is diffused into the reaction zone leading to a reduction in the distance between the flame front and oxidizer nozzle. According to Fig. 4, it can also be implied that increasing the particle diameter leads to a shift in the position of flame front toward the oxidizer nozzle. As particle diameter decreases, the amount of accessible fuel reached the reaction zone reduces so the flame front shifts toward the fuel nozzle. It should be pointed out that the effect of particle diameter at

of

lower oxidizer Lewis numbers is more intense than that at higher oxidizer Lewis numbers. For the

-p

flame front position varies between about -1.8145 to -1.8128.

ro

considered particle diameters, when oxidizer Lewis number ranges from 0.1 to 1.8, the value of

Fig. 5 indicates the variation of mass fraction of the particles with position for different particle

re

diameters over the combustion system. With respect to Fig. 5, diameter of the particles has insignificant effect on the flame front position at which particle oxide is rapidly produced. It can

lP

be implied from this figure that with an increase in diameter of the particles, the flame front travels slightly toward the oxidizer zone. When the particles are combusted and transformed into liquid

ur na

particle oxide, mass fraction of the particles is found to be zero at the position of about -1.8. Fig. 6 shows the mass fraction of the particle oxide in liquid phase through a quick melting process. It can be observed from Fig. 6 that maximum amount of liquid particle oxide is produced at the

Jo

melting front. According to this figure, position of melting front is found to be -1.78. In the vaporization and oxidizer zones, there is no liquid particle oxide so mass fraction of the molten particle oxide is found to be zero. It is worth mentioning that the vaporization front position at which molten particle oxide is transformed into gaseous phase is found to be about -1. Fig. 7 depicts the mass fraction of gaseous particle oxide produced over a rapid vaporization process with position for different particle diameters. As can be seen in Fig. 7, all amount of the

molten particle oxide is vaporized at the position of about -1 which corresponds to the vaporization front. Subsequently, the vaporized particle oxide exits the system by rejecting heat to the oxidizer stream. It can also be concluded that the position of vaporization front is slightly affected by particles diameter. The larger the size of the particles, the lower the distance between the vaporization front and oxidizer nozzle.

of

Fig. 8 illustrates the variation of oxidizer mass fraction with position. As described in Fig. 1, oxidizer stream enters the system from +∞. From right to left, mass fraction of oxidizer decreases

ro

to zero at which all amount of oxidizer is consumed for formation of the flame. It is notable that

-p

no oxidizer exists in preheat zone; hence, mass fraction of oxidizer will be zero throughout the preheat zone.

re

In order to assess the reliability of the current analytical study, obtained results are compared to

lP

the results reported by Shafirovich et al. [25]. Fig. 9 compares the results obtained for mole fractions of the particles and particle oxide as a function of air initial mole fraction under the same conditions (adiabatic condition at atmospheric pressure when relative increment of diameter of

ur na

particle oxide is equal to 1.17 μm). As can be seen in Fig. 9, there is a reasonable compatibility between the results of the present study and the data measured by Shafirovich et al. [25]. Fig. 10(a) depicts the path of the particles and the carrier inert gas considering different particle

Jo

diameters. As previously expressed, the particle enters the preheat zone with the same velocity as the inert gas stream from the fuel nozzle. As can be seen in this figure, an increment in the particle diameter results in a greater deviation from the X-direction. Fig. 10(b) presents the influence of thermophoretic force on the path of the particles. In this figure, particle diameter is considered to be 50 μm. Regarding Fig. 10(b), existence of thermophoretic force term in Eq. (4) results in less deviation from the X-direction. The reason is that the mixture molecules closer to the stagnation

line have greater temperatures leading to higher kinetic energy and further collisions between the particles.

5. Conclusions In this study, an analytical model was suggested for preparing particle oxide for biomedical

of

applications by counter-flow non-premixed combustion technique. For this purpose, a multi-zone

ro

structure consisting of preheat, reaction, melting, vaporization and oxidizer zones was examined. In order to investigate the combustion behavior of the flame, an asymptotic approach was used.

-p

Melting and vaporization processes were described by two different Heaviside step functions.

re

Mass and energy conservation equations were derived in the zones to obtain the mass fraction and temperature distributions. To enforce the continuity throughout the system and calculate the

lP

locations of the flame, melting and vaporization fronts, jump conditions were derived. Eventually, variations of temperature and mass fraction of the particles and particles oxide with position and

following: 

ur na

Lewis number were illustrated. Main conclusions of this analytical study are summarized in the

Maximum non-dimensionalized temperature of the system which corresponds to the vaporization front is found to be 0.56. Non-dimensional temperature of the flame is found to be 0.48.



Non-dimensional values of positions of flame, melting and vaporization fronts are found

Jo



to be -1.8, -1.78 and -1, respectively. 

Diameter of the particles has insignificant influence on the positions of melting and vaporization fronts.



Effect of particle size on the flame front position is insignificant when oxidizer Lewis number is greater than 0.9. For the investigated particle diameters, when oxidizer Lewis number ranges from 0.1 to 1.8, value of flame front position varies between about -1.8145 to -1.8128. Flame front travels toward the fuel nozzle with increasing the fuel Lewis number.



With enhancing the oxidizer Lewis number, flame front shifts toward the oxidizer nozzle.

ro

of



References

-p

[1] A Akbarzadeh, M Samiei, S Davaran, Magnetic nanoparticles: preparation, physical properties,

re

and applications in biomedicine, Nanoscale Research Letters 7.1 (2012): 144. [2] S Laurent, D Forge, M Port, A Roch, C Robic, L Vander Elst, R N Muller, Magnetic iron oxide

lP

nanoparticles: synthesis, stabilization, vectorization, physicochemical characterizations, and biological applications, Chemical Reviews 108.6 (2008): 2064-2110.

ur na

[3] J B Park, Biomaterials science and engineering, Springer Science & Business Media, 2012. [4] J A Hubbell, Biomaterials in tissue engineering, Bio/technology 13.6 (1995): 565. [5] C Xu, S Sun, Monodisperse magnetic nanoparticles for biomedical applications, Polymer International 56.7 (2007): 821-826.

Jo

[6] A Ito, M Shinkai, H Honda, T Kobayashi, Medical application of functionalized magnetic nanoparticles, Journal of Bioscience and Bioengineering 100.1 (2005): 1-11. [7] D Chen, Y Meng, Synthesis of magnetic oxide nanoparticles for biomedical applications, Global Journal of Nanomedicine 2.3 (2017): 555588.

[8] T J Poinsot, D P Veynante; Combustion, Encyclopedia of Computational Mechanics Second Edition (2018): 1-30. [9] W M Daoush, Co-precipitation and magnetic properties of magnetite nanoparticles for potential biomedical applications, Journal of Nanomedicine Research 5.1 (2017): e6. [10] F Mata-Pérez, J R Martínez, A L Guerrero, G Ortega-Zarzosa, New Way to Produce Magnetite Nanoparticles at Low Temperature, Advanced Chemical Engineering Research 4.1

of

(2015): 48-55.

ro

[11] Â L Andrade, M A Valente, J M Ferreira, J D Fabris, Preparation of size-controlled nanoparticles of magnetite, Journal of Magnetism and Magnetic Materials 324.10 (2012): 1753-

-p

1757.

re

[12] H Setyawan, W Widiyastuti, Progress in the Preparation of Magnetite Nanoparticles through the Electrochemical Method, KONA Powder and Particle Journal (2019): 2019011.

lP

[13] P A Dresco, V S Zaitsev, R J Gambino, B Chu, Preparation and properties of magnetite and polymer magnetite nanoparticles, Langmuir 15.6 (1999): 1945-1951.

ur na

[14] E M Koushika, G Shanmugavelayutham, P Saravanan, C Balasubramanian, Rapid synthesis of nano-magnetite by thermal plasma route and its magnetic properties, Materials and Manufacturing Processes 33.15 (2018): 1701-1707. [15] H Rashid, M A Mansoor, B Haider, R Nasir, S B Abd Hamid, A Abdulrahman, Synthesis and

Jo

characterization of magnetite nano particles with high selectivity using in-situ precipitation method, Separation Science and Technology (2019): 1-9. [16] R de Almeida Silva, C D Castro, E M Vigânico, C O Petter, I A Schneider, Selective precipitation/UV production of magnetite particles obtained from the iron recovered from acid mine drainage, Minerals Engineering 29 (2012): 22-27.

[17] P Lei, S L Girshick, Thermal Plasma Synthesis of Superparamagnetic Iron Oxide Nanoparticles for Biomedical Applications, Plasma Chemistry and Plasma Processing (2012); DOI: 10.1007/s11090-012-9364-1. [18] G I Owens, R K Singh, F Foroutan, M Alqaysi, C M Han, C Mahapatra, H W Kim, J C Knowles, Sol–gel based materials for biomedical applications, Progress in Materials Science 77 (2016): 1-79.

of

[19] M Bidabadi, M Ramezanpour, A K Poorfar, E Monteiro, A Rouboa, Mathematical modeling

ro

of a Non-Premixed organic Dust flame in a Counterflow configuration, Energy & Fuels 30.11

-p

(2016): 9772-9782.

[20] S A Madani, M Bidabadi, N M Aftah, A Afzalabadi, Semi-analytical modeling of non-

re

premixed counterflow combustion of metal dust, Journal of Thermal Analysis and Calorimetry

lP

137.2 (2019): 501-511.

[21] M Bidabadi, M Harati, Q Xiong, E Yaghoubi, M H Doranehgard, P Aghajannezhad, Volatization & combustion of biomass particles in random media: Mathematical modeling and

ur na

analyze the effect of Lewis number, Chemical Engineering and Processing-Process Intensification 126 (2018): 232-238.

[22] K Seshadri, C Trevino, The influence of the Lewis numbers of the reactants on the asymptotic

Jo

structure of counterflow and stagnant diffusion flames, Combustion Science and Technology 64.46 (1989): 243-261. [23]

American

Elements.

The

Advanced

Materials

https://www.americanelements.com/titanium-particles-7440-32-6.

Manufacturer,

(2019).

[24]

AZO

Network.

AZO

Materials,

(2019).

https://www.azom.com/properties.aspx?ArticleID=1179. [25] E Shafirovich, S K Teoh, A Varma, Combustion of levitated titanium particles in air,

Jo

ur na

lP

re

-p

ro

of

Combustion and Flame 152.1-2 (2008): 262-271.

lP

re

-p

ro

of

Figures

Jo

applications

ur na

Fig. 1 Schematic of the proposed counter-flow non-premixed system for preparation of particle oxide for biomedical

0.6

0.5

0.4

0.3

of

0.2

0.1

ro

Non-dimensional temperature

θmax = θvap = 0.56

0 -2

-1

0

1

2

3

-p

-3

Position

0

0.2

-0.1

-0.7 -0.9 -1.1 -1.3 -1.5

0.8

1

1.2

ur na

-0.5

0.6

1.4

1.6

1.8

Particle Diameter= 60 Particle Diameter= 40 Particle Diameter= 20

Jo

Flame front position

-0.3

0.4

lP

re

Fig. 2 Variation of non-dimensional temperature distribution in terms of position

-1.7 -1.9

Fuel Lewis number

Fig. 3 The effect of fuel Lewis number on the flame front position for several different particle diameters

-1.813 -1.8132

-1.8136 -1.8138 -1.814 Particle Diameter=60

-1.8142

of

Flame front position

-1.8134

Particle Diameter=40 -1.8144 -1.8146 0

0.3

0.6

0.9

1.2

ro

Particle Diameter=20 1.5

1.8

-p

Oxidizer Lewis number

lP

re

Fig. 4 The effect of oxidizer Lewis number on the flame front position for several particle diameters

1

0.6

0.4

Reaction Zone

Jo

0.2

ur na

ys

0.8

0

-3

-2

-1

0

1

Position

Fig. 5 Variation of mass fraction of particles with position for different particle diameters

2

3

1

0.8

0.4

of

ymelt

0.6

0 -3

-2

-1

0

1

2

3

-p

Position

ro

0.2

lP

re

Fig. 6 Variation of mass fraction of molten particle oxide with position

yF

0.8

0.6

Vaporization Zone

Jo

0.4

ur na

1

0.2

0 -3

-2

-1

0

1

Position Fig. 7 Variation of mass fraction of gaseous particle oxide with position

2

3

0.3 0.25 0.2 0.15 0.1 0.05

of

Oxidizer mass fraction

0.35

0 -3

-2

-1

0

1

2

0.9

Particle (Reported by Shafirovich et al. [25])

lP

0.8

Particle (Present study)

0.7

Particle oxide (Reported by Shafirovich et al. [25])

0.6

Particle oxide (Present study)

ur na

Mole fraction

-p

re

Fig. 8 Variation of oxidizer mass fraction with position

ro

Position

3

0.5 0.4 0.3 0.2

Jo

0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Air initial mole fraction

Fig. 9 Validation of results obtained for the particle, and particle oxide mole fractions as a function of air initial mole fraction

Gas 50 µm 20 µm 10 µm 5.4 µm

Y-direction

0.2

0.15

0.1

0 -1.8

-1.3

-0.8

ro

of

0.05

-0.3

-p

X-direction

0.2

lP

re

Fig. 10(a) Path of the particles and the inert gas considering different particle diameters

0.25

With Thermophoresis

Without Thermophoresis

0.1

ur na

0.15

Jo

Y-direction

0.2

0.05

0 -2

-1.5

-1

-0.5

0

0.5

X-direction Fig. 10(b) Path of the particles in presence and absence of thermophoretic force for particle diameter of 50 μm

Tables

Table 1 Boundary conditions considered for the combustion system

Description

𝓧 = −∞

𝓧 = 𝓧𝐟

𝓧 = 𝓧𝐦𝐞𝐥𝐭

𝓧 = 𝓧𝐯𝐚𝐩

𝓧 = +∞

θvap

0

θ

Non-dimensional temperature

0

θf

θmelt

yS

Mass fraction of the particles

1

0

0

ym

Mass fraction of liquid particle oxide

0

0

ymM

yF

Mass fraction of gaseous particle oxide

0

0

yO

Mass fraction oxidizer

0

0

lP ur na Jo

0

0

0

0

yFV

0

yOM

yOV

α

-p

ro

0

re

of

of

Variable

Table 2 Properties of the particle, particle oxide, oxidizer, and values of input parameters [23,24]

𝒬 ρp ρa a cp ca ϑ 𝒬melt 𝒬vap d

1.1 20 0.125

kg/m3 cm/s cal⁄g. K

1.004 2 14.15 425

J⁄kg. K − kJ⁄mol kJ⁄mol

1 20 11.8 1941 3839

− μm W⁄m. K K K

300 21.9

K W⁄m. K

re

ur na

lP

λ

Jo

kJ/kg kg/m3

-p

D𝑇𝑂 Tmelt Tvap T∞

Unit

ro

YS−∞

Value 6540 4540

of

Parameter