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Burning of single carbon particle approached by activation energy asymptotics

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Abstract

The method of activation energy asymptotics is used to treat the combustion of a single carbon particle in quiescent gas mixture with high temperature. Both heterogeneous reactions 2C+O2 → 2CO, C+CO2 → 2COand homogeneous reaction 2CO+O2 ⇌ 2CO2 are considered. It is shown that the burning of the particle principally is carried out during a diffusion-limited period. Four brief and complex periods through which the history of the particle evolves from a heat-up period to the diffusion-limited period are described. A comparison between results of activation energy asymptotics and exact numerical solutions is given. The agreement is considered satisfactory.

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Abbreviations

A=A(k) :

function of\(k, k(1 - \tilde c_p (1 - exp[ - k])^{ - 1} )\)

A cr :

A(k cr)

C 1 C2 :

preexponential factors for Reactions I and II

\(c_p , \tilde c_p \) :

specific heat capacity of the gas mixture at the particle surface and its nondimensional parameter

c s :

specific heat capacity of solid carbon

D=D(k) :

function of\(k, k(\exp [ - k]\tilde c_p (1 - exp[ - k])^{ - 1} + E)\)

D cr :

D(k cr)

E :

function ofk, Eq. (3.4)

\(h_s ,\tilde h_s \) :

enthalpy of solid carbon and its nondimensional quantityh s /c s T

\(h_p ,\tilde h_p \) :

enthalpy of gas mixture at the surface and its nondimensional quantityh

\(h_\infty ,\tilde h_\infty \) :

enthalpy of gas mixture at the ambient and its nondimensional quantityh /C s Th

k :

nondimensional rate of the mass loss of the carbon particlem p r p /η

k 1,k 2 :

parts ofk contributed by Reactions I and II, respectively

k cr :

critical value ofk with which the flame sheet detaches

k d :

maximum value ofk with the diffusion limit

m p :

total mass flux

P :

pressure

q r ,Q r :

radiative flux and its nondimensional quantity, Eq. (2.4)

r :

radial coordinate

r p :

radius of the particle

r p0 :

initial radius of the particle

R :

nondimensional radius of the particle, Eq. (2.4)

\(\hat R_1 ,\tilde R_1 ,\tilde R_2 \) :

scale stretch variables forR in the ignition period, postignition period and the period with flame sheet moving outward

s 1 :

transformed variable ofτ in the transient diffusion-limited period

s 2 :

transformed variable ofτ in the diffusion limited period

T p :

particle temperature

T i0 :

initial particle temperature

T :

gas temperature in the ambient

T a1 ,T a2 :

activation temperatures of Reactions I and II

T i1 T i2 :

ignition temperatures of Reactions I and II

t :

time

Y 1p ,Y 2p ,Y 3p ,Y 4p :

mass fractions of O2, CO2, CO, N2 at the particle surface

\(\tilde y_{1p} ,\tilde y_{2p} \) :

element mass fractions of oxygen and carbon at the particle surface

\(\tilde y_{1\infty } ,\tilde y_{2\infty } \) :

element mass fractions of oxygen and carbon in the ambient Greek

ε 1,ε 2 :

AEA expansion parameters for Reaction I and Reaction II, Eq. (3.7)

η :

viscosity of the ambient gas

λ 1,λ 2 :

parameters. Eq. (3.11a) and (3.21a)

μ ij :

number of grams of elementi in a gram of speciesj, j=1,2,3 denote O2, CO2, CO; i=1.2 denote element oxygen and element carbon

\(\hat \mu , \tilde \mu , \bar \mu , \mu '\) :

various combinations ofμ 11,μ 12/μ 22,μ 13/μ 23,μ 12- 13 μ 2223).(μ12μ2322)-μ 23

ρ s :

carbon particle density

φ r :

radiation related parameter, Eq. (3.5)

μ:

Stefan-Boltzmann constant

τ :

nondimensional time

τ1, τ2 :

ignition time for Reactions I and II

τ1 :

time of the flame sheet separation

\(\hat \tau ,\bar \tau ,\tilde \tau _2 \) :

scale stretch variables for τ in the ignition period, postignition period and the period with flame sheet moving outward

θ :

nondimensional temperature of the particle

θ 11,θ 12 :

ignition temperatures in terms ofθ for Reactions I and II, Eq. (3.8)

θ1 :

value ofθ at the moment of flame sheet separation

ξ :

nondimensional radial coordinate

ξ f :

flame sheet location in terms ofξ

Δ1234 :

species formation heats of O2, CO2, CO,N 2, respectively

\(\tilde \Delta _i (i = 1, 2, 3, 4)\) :

nondimensional quantities ofΔ 1,Δ 1/c 2,T

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Communicated by Tsai Shu-tang

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Ding-guo, X. Burning of single carbon particle approached by activation energy asymptotics. Appl Math Mech 8, 745–757 (1987). https://doi.org/10.1007/BF02017982

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  • DOI: https://doi.org/10.1007/BF02017982

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