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-(μ 13 μ 22/μ23).(μ12μ23/μ22)-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
- Δ1,Δ2,Δ3,Δ4 :
-
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