Extinction of burning particles due to unstable combustion modes

Fuel 81 (2002) 391±396

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Extinction of burning particles due to unstable combustion modes q Bernhard Peters* AVL List GmbH, Abt: CT, Hans-List-Platz 1, A-8020 Graz, Austria Received 12 July 2001; accepted 5 October 2001; available online 13 November 2001

Abstract The objective of this paper is to derive analytically the conditions under which burning particles extinguish. Experiments indicate that the extinction may occur with a rapid drop in particle temperature, thus causing an insuf®cient burn-out. Both qualitative and quantitative description of this phenomenon is presented in this study. Based on Semenov's theory of thermal explosion, global heat release and loss balance determines the critical particle size, at which a rapid change of particle temperature occurs while the particle is burning. Under more general conditions, particles experience a temperature and oxidizer distribution, which is described by the conservation equations of mass and energy. By applying a linear stability analysis to these equations, regions are identi®ed, for which the combustion mode becomes unstable, e.g. for which the hot solution changes rapidly to the cold solution and, thus leads to the extinction of the particle. The analysis shows that the regions of stable and unstable combustion are separated by critical heat and mass transfer conditions. These results enable engineers to assess the size of particles due to incomplete combustion, to apply appropriate ®lters or to change conditions favourable to further burn-out. q 2001 Elsevier Science Ltd. All rights reserved. Keywords: Solid fuel; Combustion; Extinction; Instability

1. Introduction Insuf®cient burn-out, e.g. extinction, of particles is a commonly encountered phenomenon in solid fuel combustion, such as pulverised fuel ¯ames. In the past, the incomplete burn-out was not considered to be signi®cant as it caused only marginal losses in the overall thermal ef®ciency. However, recently the problem of insuf®cient burn-out was brought up again, because the products of incomplete combustion (PIC) are considered precursors of unacceptable pollution formation [1] and lead to inferior ash quality. In the past the phenomenon of insuf®cient burn-out often was attributed to thermal annealing [2], sintering, and the effect of ash layers [3]. Other investigations based on numerical models suggest that a change in the combustion mode of the particles is likely to cause incomplete burn-out. Ubhayakar [4] used a one-dimensional model to describe the quasi-steady-state burning of a spherical particle and to determine the extinction limits in quiescent oxidising gas. His analysis yielded burning and extinction regimes dependent on the ambient temperature, the concentration of oxidiser, radiation and reaction conditions. Srinivas and Amundson [5] used a * Tel.: 149-7247-823491; fax: 149-7247-824837. E-mail addresses: [email protected] (B. Peters). q Published ®rst on the web via Fuel®rst.comÐhttp://www.fuel®rst.com

numerical model based on an unsteady mass and energy balance in conjunction with a probabilistic relation to predict the evolution of the internal surface area for a porous spherical particle and found that a critical particle size, ambient temperature, ambient oxygen concentration, and boundary layer thickness exist, above which a particle ignites. Recently, Hurt et al. [6] observed an immediate drop of particle temperature during the combustion of a fuel particle exposed to a hot environment of 1600 K. They injected spherical particles of sizes between R ˆ 0:55 and 0.60 mm into a heated ¯ow reactor. Due to conversion, the particle shrank and after having reached a critical size a rapid temperature drop occurred. As already pointed out by Wicke et al. [7] and Essenhigh et al. [8], a particle experiences different regimes of combustion, depending on the ratio of internal mass transport to reaction time scale which is referred to as the Thiele modulus. A rapid change of temperatures accompanies the shift from zone 2 (reacting core) to zone 1 (shrinking core) behaviour. This behaviour was con®rmed by investigations of Essenhigh et al. [9,10]. This temperature predictions obtained from an unsteady mass and energy balance for a spherical particle were in good agreement with the experiments of Hurt et al. [6]. Due to the non-linear behaviour of the reaction rate expression, several steady states for the combustion of a particle are possible, among which are the so-called hot

0016-2361/02/$ - see front matter q 2001 Elsevier Science Ltd. All rights reserved. PII: S 0016-236 1(01)00183-1

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B. Peters / Fuel 81 (2002) 391±396

Here V, A, Hm, Ea, Rg, a , e , s , T, Tw and T1 denote the volume and surface of a particle, the reaction enthalpy, activation energy, gas constant, heat transfer coef®cient, emissivity, Boltzmann constant, particle, wall and ambient temperature, respectively. The kinetics of the process is described by a global, unsteady energy balance of the following form:

Nomenclature A surface cp speci®c heat capacity cr reactive species D diffusion coef®cient Ea activation energy Hm reaction enthalpy k pre-exponential factor Qloss heat loss Qrel heat release r independent variable R radius Rg gas constant t time T temperature T1 ambient temperature Tw wall temperature V volume Greek symbols a heat transfer coef®cient b mass transfer coef®cient r density l conductivity s Boltzmann constant Dimensionless numbers Nu Nusselt number Sh Sherwood number Th Thiele modulus

rc p V

…3†

with r and cp being the density and the speci®c heat of the particle. For a qualitative description of the process the burn-up of initial material and its effect on the reaction rate will be neglected. The condition of stationarity requires equality of heat loss and release Qrel ˆ Qloss and determines the values of the steady-state temperatures of the particle. For a slab …n ˆ 1†; a cylinder …n ˆ 2† and a sphere …n ˆ 3†; the equality yields the following equation:  an an  4 Hm k e2Ea =Rg T ˆ …T 2 T1 † 1 …4† es T 2 Tw4 R R

and cold solutions. In the course of combustion, a steady state may be rendered unstable which is followed by a change of the combustion behaviour. This paper provides complementary explanations based on a linear stability analysis of this phenomenon. A global balance between heat release and loss of a particle already reveals the possible steady states, whereas in a more rigid approach additional temperature and concentration distributions inside the particles are taken into account. 1.1. Theory of thermal extinction Although Semenov [11±14] developed his theory for the thermal explosion of gases in vessels, its basics are also applicable to the burning and extinction of particles. If a combustible particle is introduced into a hot oxidising environment, the rate of heat release, assuming a zero-order Arrhenius reaction without any mass transfer limitations similar to a well-stirred reactor, is described as follows: Qrel ˆ VHm k e2Ea =Rg T

dT ˆ Qrel 2 Qloss dt

…1†

and the rate of heat loss due to heat transfer to the gaseous phase and radiation to walls is   Qloss ˆ aA…T 2 T1 † 1 esA T 4 2 Tw4 …2†

where R denotes a representative dimension of the geometry. Fig. 1 shows a qualitative plot of a heat release curve and different heat losses as a function of temperature. Depending on the heat transfer conditions, several values for the steady-state temperature are possible. While the curves for the heat loss rise steadily with the temperature, the curve for heat generation displays the common S-shape of the Arrhenius law with the point of in¯ection at the temperature TI ˆ Ea =2Rg : For an ambient temperature Ta . TI ; only one steady-state temperature exists at the point of intersection of the heat release and the heat loss (#1) curves. If the slope of the heat loss curve increases, e.g. due to a shrinking particle size, the point intersection moves downward on the hear release curve until the temperature Ta is reached. However, in the case of Ta , TI ; several modes of combustion are possible. For a very steep heat loss line (#4), one intersection point D with the heat release curve occurs, indicating an equilibrium between loss and generation. Thus, a stable steady-state temperature prevails at low reaction rates. For certain gradients of the heat loss curve (#3), three points …D; E; F† of intersection with the heat release curve exist. The intersection point E at the temperature TE corresponds to an unstable regime. For an initial particle temperature below TE the particle cools down to temperature TD, while for an initial temperature above TE the particle burns at a high reaction rate with the temperature TF. Similar conditions apply to the intersection point B, where the heat loss line (#2) is a tangent to the heat release curve. During the combustion of a particle, the slope of the heat loss line (#2) increases due to a shrinking size, so that the intersection point B represents the condition of an unstable regime. Therefore, the tangent to the heat release curve determines

B. Peters / Fuel 81 (2002) 391±396

393

linear stability analysis in conjunction with the collocation method [15]. The one-dimensional unsteady energy equation for the temperature distribution T of a spherical particle in dimensionless coordinates, taking conduction and a ®rst order Arrhenius reaction into account, is written as follows:   rcp R2 2T 1 2 2 2T kHm cr R2 2Ea =Rg T ˆ 2 r e …6† 1 2r l 2t l r 2r

Fig. 1. Heat release and loss curves.

the radius of the particle at which instability occurs. The gas temperature of Tg ˆ 1600 K in the experiments of Hurt et al. [6] is below the temperature of the point of in¯ection TI ˆ 10 820 K; so that instability may occur. At a temperature of T ˆ 1712 K the heat loss curve is a tangent onto the heat release curve, so that this condition suf®ces to determine the particle radius for which the combustion becomes unstable. In conjunction with the equality between heat loss and release the critical particle radius is estimated as follows:   3a…T 2 T1 † 1 3es T 4 2 Tw4 Rcrit ˆ …5† Hm k e2Ea =Rg T Eq. 5 yields a value of R ˆ 47 mm, at which instability occurs and the burn-out time extends signi®cantly due to a temperature drop. Essenhigh et al. [8] predicted the temperature and conversion rate of a particle under the same conditions and obtained a critical radius of R < 44 mm, which agrees well with the present value.

2. Extinction under a non-uniform temperature distribution Within the previous analysis, uniform temperature and concentration pro®les were assumed for a particle, which lead to a global energy balance. However, a gradient of the temperature always exists in order to allow for a heat transfer, which holds also for small particles. This applies certainly to larger particles, for which the inertia effects of heat transport enhance a non-uniformity of the temperature distribution. The conservation equation of energy describes the evolution of temperature in space and time for a particle, whereas the pro®le of reactive species is assumed to be uniform e.g. no mass transfer limitations. This assumption will be neglected in the analysis in Section 3. For the identi®cation of instabilities the energy equation is subject to a

where r , cp, l and R denote density, heat capacity, heat conductivity and the particle radius, respectively. The reactive source term contains k, Hm, cr, Ea and Rg as the pre-exponential factor, reaction enthalpy, concentration of a reactive specie, activation energy and the gas constant, respectively. The thermal boundary conditions representing convective and radiative heat transfer are employed in a dimensionless form as follows: rˆ0:

2T ˆ0 2r

rˆ1:

2

 2T esR  4 ˆ Nu…T 2 T1 † 1 T 2 Tw4 2r l

…7† …8†

where e , s , l , Tw, T1 and Nu are the emissivity, the Boltzmann constant, the conductivity, the wall temperature, the ambient gas temperature and the Nusselt number Nu ˆ aR=l: In the present analysis the orthogonal collocation method [15] is employed to discretise the conservation equation. The orthogonal collocation method has the advantage of distributing discontinuous masses along the domain of discretisation as compared to the ®nite volume and ®nite element method. This results in a functional evaluation at discrete points whereas the other methods require the evaluation of integrals. With respect to the non-linear reaction term, the collocation method, therefore, signi®cantly facilitates the mathematical analysis. In order to discretise the radiative boundary condition, the temperature dependence is linearised around T0 in the following form:     T 4 2 Tw4 ˆ T03 1 T02 Tw 1 T0 Tw2 1 Tw3 …T 2 Tw † ˆ gR …T 2 Tw †

…9†

so that the boundary condition at r ˆ 1 writes as follows: 2

2T ˆ Nu…T 2 T1 † 1 Nu 0 …T 2 Tw † 2r

…10†

Due to a similar appearance between the convective and the radiation boundary condition, the term  esR  3 T 0 1 T02 Tw 1 T0 Tw2 1 Tw3 l is labeled Nu 0 : In the present context the collocation method of ®rst order for spherical geometries is applied, which leads to analytical expressions and represents the behaviour suf®ciently

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B. Peters / Fuel 81 (2002) 391±396

accurately. For higher accuracy, a high-order collocation formula may be applied. Hence, the energy equation is discretised in space in the following form:

rcp R2 2T1 kHm cr R2 2Ea =Rg T1 ˆ 210:5…T1 2 T2 † 1 e l 2t l

…11†

Application of the collocation method to the boundary conditions eliminates the unknown temperature T2 and yields the following equation:

rcp R2 2T 10:5…Nu 1 Nu 0 † 10:5…NuT1 1 Nu 0 Tw † ˆ2 T1 0 Nu 1 Nu 1 3:5 Nu 1 Nu 0 1 3:5 l 2t 1

kHm cr R2 2Ea =Rg T e l

(12)

For the stability analysis the reaction source term may be linearised around the steady state solution T0 to obtain:

rcp R2 2T 10:5…Nu 1 Nu 0 † 10:5…NuT1 1 Nu 0 Tw † ˆ2 T 1 Nu 1 Nu 0 1 3:5 Nu 1 Nu 0 1 3:5 l 2t ! kHm cr R2 2Ea =Rg T0 e2Ea =Rg T0 Ea …T 2 T0 † 1 e 1 l Rg T02 …13† A linear perturbation equation for the temperature T ˆ T0 1 eu with u ˆ A eSt is chosen for insertion into Eq. (13). It yields a condition for S , under which a stable and unstable region is identi®ed depending on the radius R of the particle. Under the condition of S ˆ 0 the thermal instability occurs and it is deduced from the following equation:   e Rs  3 10:5 aR=l 1 T 0 1 T02 Tw 1 T0 Tw2 1 Tw3 l Sˆ2
esR 3 aR=l 1 T 0 1 T02 Tw 1 T0 Tw2 1 Tw3 1 3:5 l 1

kHm cr R2 e2Ea =Rg T0 Ea ˆ0 l Rg T02

(14)

Eq. (14) represents a third-order polynomial in R and has three roots, of which only one is positive and is expressed as follows: Rcrit ˆ 1

1700 K; where the heat loss curve is a tangent to the heat release curve and assuming Tw ˆ 1500 K; the abovementioned analysis yields a value of R ˆ 36:4 mm, which is in fair agreement with the analysis of Section 1.

27l 4…a 1 seg R †

3. Extinction under non-uniform temperature and concentration distributions For heterogeneous combustion, in general a distribution for one or several reactive species has to be considered additionally to the temperature pro®le. The conservation of energy is therefore accompanied by the conservation of mass including a reaction source term. This results in a coupled set of differential equations, to which the same procedure to analyse stability is applied. Assuming a ®rst order reaction for the combustion of a porous and spherical particle, the energy equation for the solid material neglecting the contribution of the gaseous pore space and the mass conservation equation for the gaseous phase of the pores are written as follows: Energy:   rcp R2 2T 1 2 2 2T kHm cr R2 2Ea =Rg T ˆ 2 r e 1 2r l 2t l r 2r

…16†

Mass:   R2 2cr 1 2 2 2cr kc R2 r ˆ 2 2 r e2Ea =Rg T D 2t 2r D r 2r

…17†

where cr and D denote the concentration of a reactive species and the diffusion coef®cient, respectively. The following boundary conditions are applied for heat and mass transfer: Heat transfer: rˆ0:

2T ˆ0 2r

rˆ1:

2

…18†

 2T esR  4 ˆ Nu…T 2 T1 † 1 T 2 Tw4 …19† 2r l

Mass transfer:

q 49c2r Ea2 Hm2 k2 l2 r2 1 168cr e2Ea =Rg T0 Ea Hm kRg T02 lr…a 1 seg R †2

rˆ0:

2cr ˆ0 2r

rˆ1:

2

4cr Ea Hm kr…a 1 seg R †

…15† For values of R , Rcrit S becomes negative, which indicates a stable solution at low reactions rates, e.g. extinction. This result agrees well with the ®ndings of Essenhigh et al. [9], according to which, after the size of the particle is reduced to a critical radius, the temperature distribution changes from the hot branch to the cold branch and, thus signi®cantly extends the time for burn-out. By setting T 0 ˆ

2cr ˆ Sh…cr 2 cr;1 † 2r

…20†

…21†

where cr,1 and Sh are the ambient concentration and the Sherwood number Sh ˆ bR=D: By applying the orthogonal collocation method to Eqs. (16) and (17) in conjunction with the boundary conditions and linearising the reaction source term around the steady state solution T0 and cr,0 leads to the following system

B. Peters / Fuel 81 (2002) 391±396

of differential equations:

rcp R2 2T 10:5…Nu 1 Nu 0 † 10:5…NuT1 1 Nu 0 Tw † ˆ2 T1 0 Nu 1 Nu 1 3:5 Nu 1 Nu 0 1 3:5 l 2t  kHm R2  cr;0 e2Ea =Rg T0 1 gT …T 2 T0 † 1 gc …cr 2 cr;0 † l …22†

1

R2 2cr 10:5Sh 10:5cr;1 Sh c 1 ˆ2 Sh 1 3:5 r Sh 1 3:5 D 2t  kR2  cr;0 e2Ea =Rg T0 1 gT …T 2 T0 † 1 gc …cr 2 cr;0 † (23) D

2

Here gT and gc are the partial derivatives of temperature and concentration of the source term due to linearisation. The perturbed temperature and concentration equations T ˆ T0 1 eA eSt and c ˆ c0 1 eB eSt inserted into the system yield a system of linear equations for the unknown variables A and B: 0

r c R2 10:5…Nu 1 Nu 0 † kH m R2 gT B p S1 2 0 B l Nu 1 Nu 1 3:5 l B B 2 @ kR gT D

395

Taking uncertainties of the measurement into account, the predicted values of the radii deviate by less than 15%. A simple energy balance with a uniform representative temperature of the particle yields already good estimation. The two remaining approaches predict values, which differ to a small extent only, so that in the present case the simpler method considering only a temperature distribution suf®ces. However, the principal applicability of the abovementioned methods to predict extinction of spherical particles is shown in the present contribution. For engineering applications the previous results as a relationship between a critical particle radius and heat/ mass transfer conditions can be arranged in a graph to determine critical conditions for a particle. This relationship with the mass transfer coef®cient as a parameter is shown for the Illinois coal as a representative example in Fig. 2. In Fig. 2 the parameter of a constant mass transfer coef®cient distinguishes into a region of extinction above the curve and a region of combustion below the curve. Thus, for given Nusselt and Sherwood numbers the behaviour,

kH R2 gc 2 m D

1

C A! C C ˆ C 2 2 R 10:5Sh kR gc A B 1 S1 Sh 1 3:5 D D

The system has a non-trivial solution for a vanishing determinant, which leads to the following equation for S : gT Hm k 10:5DSh cp R2 …Sh 1 3:5† ! 10:5l…Nu 1 Nu 0 † 1 S rcp R2 …Nu 1 Nu 0 1 3:5†

0 0

! (24)

whether extinction or combustion takes place, is easily determinable.

S 2 1 gc k 2

4. Summary

1

110:25DlSh…Nu 1 Nu 0 † rc p R4 …Sh 1 3:5†…Nu 1 Nu 0 1 3:5†

1

10:5lkgc …Nu 1 Nu 0 † 10:5DgT Hm kSh 2 ˆ 0 (25) 2 0 rcp R …Nu 1 Nu 1 3:5† cp R2 …Sh 1 3:5†

The stability of the combustion mode for spherical particles in particular was investigated. Based on the classical theory of thermal explosion by Semenov the conditions for a shift from the hot to the cold solution of a reacting particle were derived both qualitatively and quantitatively. Under burning conditions, a particle experiences temperature and concentration pro®les which are determined by the combustion

Eq. (25) yields two values S 1,2 to determine the region of instability by setting S 1;2 ˆ 0: One of the two solutions does not have real roots and, therefore, is excluded for further analysis. The second solution furnishes a value of R ˆ 38:6 mm which agrees satisfactory with the results obtained from the previous analysis. For a better comparison the critical radii obtained by the different approaches are listed in the following table: Method

Critical radius (mm)

Experimental Energy balance Non-uniform temperature distribution Non-uniform temperature/species distribution

44 47 36.4 38.6

Fig. 2. Combustion and extinction regimes.

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B. Peters / Fuel 81 (2002) 391±396

of the particle and its heat/mass transfer rates. A system of conservation equations is applied to describe the evolution of temperature and concentration in time and space for a spherical particle assuming a ®rst-order reaction. A linear stability analysis in conjunction with the collocation method yields conditions dependent on the particle size, the heat and mass transfer conditions to determine regions of unstable combustion modes. In particular, for particle sizes smaller than a critical value, the combustion mode changes to slow reaction rates causing extinction of the particle. These conditions are usually encountered for particles, as the size shrinks during the combustion and eventually reaches a critical value, at which extinction occurs. This result enables engineers either to assess the size of particulate emissions due to sooting combustion or to design combustion chambers with suf®ciently high temperatures and residence times to allow for a suf®cient burn-out. References [1] Stieglitz L, Vogg H. Formation and decomposition of polychlorodibenzodioxins and- furans in municipal waste incineration. KfK 1988;4379:1±16. [2] Essenhigh RH. Fundamentals of coal combustion. Chemistry of coal utilization: second supplementary volume. New York: Wiley, 1981. p. 1153±312.

[3] Hurt RH, Sun JK, Lunden M. A kinetic model of carbon burnout in pulverized coal combustion. Combust Flame 1998;113:181±97. [4] Ubhayakar SK. Burning characteristics of a spherical particle reacting with ambient oxidizing gas at its surface. Combust Flame 1976;26:23±34. [5] Srinivas B, Amundson NR. Intraparticle effects in char combustion III transient studies. Can J Chem Engng 1981;59:728±38. [6] Hurt RH, Davis KA. 25th Symposium (International) on Combustion, 1994. [7] Wicke E, Rossberg M. Z Elektrochem 1953;57:641±5. [8] Essenhigh RH, Klimesh HE, FoÈrtsch D. Combustion characteristics of carbon: dependence of the zone I±zone II transition temperature (Tc) on particle radius. Energy Fuels 1999;13:826±31. [9] FoÈrtsch D, Essenhigh RH, Klimesh HE, Schnell U, Hein KRG. Modeling of time-dependent behavior of particle temperature and burnout during combustion of pulverized char particles. Joint Meeting of the US Sections of The Combustion Institute, 1999. p. 1±4. [10] Essenhigh RH, Klimesh HE, FoÈrtsch D. Combustion characteristics of carbon: in¯uence of the zone I±zone II transition on burn-out in pulverized coal ¯ames. Energy Fuels 1999. [11] Zel'dovich B, Semenov N. J Exp Theor Phys USSR 1940;10:1116. [12] Semenov NN. Z Physik 1928;48:571. [13] Kuo KK. Principles of combustion. New York: Wiley, 1986. [14] Zeldovich YaB, Barenblatt GI, Librovich VB, Makhviladze GM. The mathematical theory of combustion and explosions. Acad Sci USSR 1985. [15] Finalyson BA. The method of weighted residuals and variational principles. New York: Academic Press, 1972.