Thermal inertia effect of reactive sources on one-dimensional discrete combustion wave propagation

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Thermal inertia eﬀect of reactive sources on

one-dimensional discrete combustion wave propagation

Daoguan Ning, Yuriy Shoshin

Department of Mechanical Engineering, Eindhoven University of Technology, the

Netherlands

Abstract

In the present work, the discrete ﬂame model [1] is augmented by intro-

ducing the thermal inertia of particles in the preheating zone. The eﬀect of

particle thermal inertia on ﬂame speed, propagation limits, and near-limits

dynamics of one-dimensional discrete combustion waves is studied using the

new model. It is found that, with the increase of particle thermal inertia, the

propagation velocity of the discrete ﬂame decreases due to a smaller heating

rate of the particles. Besides, particle thermal inertia extends the propaga-

tion limits compared to the prediction of the old model. Furthermore, it is

mathematically proven that the nonphysical branch of the solutions for the

discrete ﬂame speeds, found using the old discrete model, is a set of solu-

tions for the propagation limits of steady-state discrete ﬂames with particle

thermal inertia included. The ﬂame speed predicted using the new model is

also compared with that determined analytically using a continuum model

considering the thermal inertia of the condensed phase [2]. We ﬁnd that

the discrete ﬂame speeds predicted by the both models become closer to

Email address: d.ning@tue.nl (Daoguan Ning)

Preprint submitted to Combustion and Flame October 28, 2022

arXiv:2210.15046v1 [physics.flu-dyn] 26 Oct 2022

each other with increasing particle thermal inertia. Finally, the two models

converge regardless of the discrete nature of the heat sources when particle

thermal inertia is large enough so that can limit the ﬂame propagation. The

particle thermal inertia controlled ﬂames could be regarded as a new kind of

combustion regime.

Keywords: Discrete combustion wave, Thermal inertia, Heterogeneous

ﬂame speed, Propagation limits, Flame dynamics

Nomenclature

Aparticle surface area [m2]

Bparticle concentration [kg/m3]

cspeciﬁc heat [J/kg/K]

hheat transfer coeﬃcient [W/m2/K]

HHeaviside function

iindex of particle

lmean inter-particle distance [m]

Nnumber of particles

rparticle radius [m]

tDimensional time [s]

Vparticle volume [m3]

xdimensional coordinate vector

ydimensionless spacial coordinate

Greek symbols

αthermal diﬀusivity [m2/s]

γdimensionless particle thermal inertia, deﬁned Eq. (A.1)

∆ time interval

κsquared dimensionless ﬂame speed

λthermal conductivity [W/m/K]

θdimensionless temperature

ρdensity [kg/m3]

τDimensionless time

ξdeﬁned in Eq. (4), ξ=ρscsr2

p/(3λg,utc)

ωdimensionless heat source term

Subscripts

aadiabatic

ccombustion

dheat diﬀusion in the gas phase between particles

eheat exchange between gas phase and particles

ggas

ign ignition

pparticle

uunburned mixture at y→ −∞

1. Introduction

Flame propagation in heterogeneous media consisting of a continuum gas

phase and condensed reactive sources is an important phenomenon. It is

encountered in a variety of practical processes, such as burning of liquid

fuel spray and combustion of solid particle aerosols including coal, biomass,

and metals. Depending on the heterogeneous system, the size of the reac-

tive source varies from mircon-sized particles to millimeter-sized droplets.

For low-volatile condensed fuels, exothermic chemical reactions happen at or

close to the interfaces between localized reactive sources and a continuum

gaseous oxidizer, which leads to spatially discrete heat sources. In spite of

the discrete nature of the heat sources, theoretical models developed before

early 90s, as typical examples [3, 4, 5], commonly adopted the homogeneous

continuum approximation, in which the heat sources were modeled by a con-

tinuous function of spatial coordinates. These models fail to reveal impor-

tant phenomena related to the discreteness of the heat sources, in particular

independence of ﬂame speed on particle combustion time for fast burning

particles. On the contrary, continuous models, developed for both gaseous

and heterogeneous systems, predict the ﬂame speed to be proportional to

the square root of the reaction rate, which is inversely proportional to the

particle combustion time in the later case.

A novel theoretical model, elucidating the eﬀects of spatial discreteness

of reactive sources on the combustion wave propagation, was ﬁrst proposed

by Shoshin et. al [6] and further explored by Goroshin et. al [1, 7] and

Beck and Volpert [8]. The theory (i.e., discrete ﬂame model) idealizes re-

acting particles as point-like heat sources based on the fact that the mean

distance between the particles is around two orders of magnitude larger than

their sizes in fuel lean suspensions. Besides, the model introduces a step-

wise kinetic, i.e., the exothermic reactions are triggered when a ﬁxed igni-

tion temperature is reached. After ignition, the particles combust at a con-

stant reaction rate within a burn time. The stepwise kinetic is a simpliﬁed

physical representation of reactive particles burning in the external diﬀusion

regime [9]. The temperature ﬁeld of the continuum phase is then obtained

through superimposing heat diﬀusion waves, generated by every point-like

heat source, using the Green’s function and the exact spatial coordinate of

each heat source. Thus, the continuum approximation is eliminated. An

analytical solution for the ﬂame propagation velocity can be derived from

the model when mono-dispersed particles are regularly distributed in space.

For systems with randomly located point-like reactive particles, the mean

propagation speed and the front structure of a combustion wave can be ob-

tained by computer-assisted numerical simulations for systems with adiabatic

[10, 11, 12] or conductive boundaries [13].

The discrete ﬂame model provides a criterion to diﬀerentiate between

the discrete and continuum combustion waves. It points out that owing to

the spatially discrete nature of the heat sources, a heterogeneous ﬂame can

propagate in continuum or discrete regimes depending on the dimensionless

combustion time (also known as the discreteness parameter [1]):

τc=tc

=tcαu

l2,(1)

which is the ratio between the physical combustion time of an individual

reactive particle tc(also the inverse of chemical reaction rate) and the average

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摘要：

Thermalinertiaeectofreactivesourcesonone-dimensionaldiscretecombustionwavepropagationDaoguanNing,YuriyShoshinDepartmentofMechanicalEngineering,EindhovenUniversityofTechnology,theNetherlandsAbstractInthepresentwork,thediscreteamemodel[1]isaugmentedbyintro-ducingthethermalinertiaofparticlesintheprehe...

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Thermal inertia effect of reactive sources on one-dimensional discrete combustion wave propagation

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