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A simplified mathematical study of thermochemical preparation of particle oxide under counterflow configuration for use in biomedical applications

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Abstract

This study mathematically presents a counterflow non-premixed thermochemical technique for preparing a particle oxide used for cancer diagnosis and treatment. For this purpose, preheating, reaction, melting, and oxidation processes were simulated considering an asymptotic concept. Mass and energy conservation equations in dimensional and non-dimensional forms were solved using MATLAB®. To preserve the continuity in the system and calculate the locations of melting and flame fronts, promising jump conditions were derived. In this research, variations in flame temperature, flame front location and mass fractions of the particle, particle oxide and oxidizer, with position, Lewis number and initial temperature of the particles were investigated. The simulation results were compared with those obtained from an earlier experimental study under the same conditions. Regarding the comparison, an appropriate compatibility was observed between the results. Based on the simulation results, flame temperature was found to be about 1310 K. Positions of flame and melting fronts were found to be − 1.8 mm and − 1.78 mm, respectively.

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Abbreviations

a :

Strain rate \(\left( {\frac{1}{a}} \right)\)

C :

Mixture specific heat capacity (kJ kg−1 K−1)

C a :

Heat capacity of the gas (kJ kg−1 K−1)

C p :

Heat capacity of the particle (kJ kg−1 K−1)

D C :

Damkohler number

D F :

Diffusion coefficient of particle (m2 s−1)

D m :

Diffusion coefficient of particle oxide in liquid phase (m2 s−1)

D O :

Diffusion coefficient of oxidizer (m2 s−1)

D T :

Thermal diffusion coefficient (m2 s−1)

E :

Overall activation energy (kJ)

erf(x):

Error function

H :

Heaviside function

Le:

Lewis number

m :

Mixture molecular mass (kg mol−1)

m F :

Fuel molecular mass (kg mol−1)

m O :

Oxidizer molecular mass (kg mol−1)

n p :

Local number density of particles (number of particles per unit volume)

\({\mathcal{Q}}\) :

Heat of reaction (kJ kg−1)

Q melt :

Latent heat of melting (kJ kg−1)

\({\fancyscript{q}}_{\rm melt}\) :

Ratio of latent heat of melting to the heat released from reaction

R :

The universal gas constant (m3 Pa mol−1 K−1)

r p :

Particle radius (µm)

T :

Temperature (K)

T a :

Activation temperature (K)

T ad :

Adiabatic temperature (K)

T f :

Flame temperature (K)

T ig :

Ignition temperature of particles (K)

T melt :

Melting temperature of particle oxide (K)

T ∞ :

Ambient temperature (K)

\({\mathcal{W}}_{\text{F}}\) :

Molecular weight of the particle (kg kmol−1)

x :

Dimensional length (m)

x f :

Dimensional flame sheet position (m)

X f :

Non-dimensional flame sheet position

x melt :

Onset position of melting in dimensional form (m)

X melt :

Onset position of melting in non-dimensional form

Y m :

Mass fraction of particle oxide in liquid phase

Y O :

Mass fraction of the oxidizer

Y S :

Mass fraction of the particle

Y S−∞ :

Mass fraction of the particle at the distance − ∞

y m :

Dimensionless form of the mass fraction of particle oxide in liquid phase

y O :

Dimensionless form of the mass fraction of oxidizer

y S :

Dimensionless form of the mass fraction of particle

ω melt :

Melting rate of particle oxide (kg m−1 s−2)

\(\widetilde{\omega }_{\text{melt}}\) :

Dimensionless form of melting

ω S :

Reaction rate (kg m−1 s−2)

\(\widetilde{\omega }_{\text{S}}\) :

Dimensionless form of the reaction rate

τ melt :

Constant characteristic time of melting

λ :

Thermal conductivity (kJ m−1 s−1 K)

ρ :

Density of the mixture (kg m−3)

ρ a :

Density of the gas (kg m−3)

ρ p :

Density of the particle (kg m−3)

θ :

Dimensionless form of the temperature

θ f :

Dimensionless form of the flame temperature

θ melt :

Dimensionless form of the melting temperature

v O :

Oxidizer stoichiometric coefficient

v p :

Product stoichiometric coefficient

v S :

Fuel stoichiometric coefficient

a:

Gas

f:

Flame

melt:

Melting

P:

Product

S:

Particle

V:

Velocity

∞:

Ambient condition

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Authors and Affiliations

  1. School of Engineering, Mechanical Engineering Department, Iran University of Science and Technology, Narmak, Tehran, Iran

    Amir Tabaei, Sadegh Sadeghi, Saman Hosseinzadeh & Mehdi Bidabadi

  2. IT Innovation Center, General Motors, Warren, MI, 48092, USA

    Qingang Xiong

  3. School of Engineering, University of Glasgow, Glasgow, UK

    Nader Karimi

Authors
  1. Amir Tabaei
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  2. Sadegh Sadeghi
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  4. Mehdi Bidabadi
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  5. Qingang Xiong
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Correspondence to Qingang Xiong.

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Tabaei, A., Sadeghi, S., Hosseinzadeh, S. et al. A simplified mathematical study of thermochemical preparation of particle oxide under counterflow configuration for use in biomedical applications. J Therm Anal Calorim 139, 2769–2779 (2020). https://doi.org/10.1007/s10973-019-08917-y

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