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Calculation of the mass transfer coefficient for the combustion of a carbon particle
Combustion and Flame 157 (2010) 137–142
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Combustion and Flame j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / c o m b u s t fl a m e
Calculation of the mass transfer coefficient for the combustion of a carbon particle Fabrizio Scala * Istituto di Ricerche sulla Combustione – CNR, P.le Tecchio 80, 80125 Napoli, Italy
a r t i c l e
i n f o
Article history: Received 26 March 2009 Received in revised form 11 May 2009 Accepted 1 June 2009 Available online 21 June 2009 Keywords: Mass transfer coefficient Sherwood number Combustion Carbon particle Multi-component diffusion
a b s t r a c t In this paper we address the calculation of the mass transfer coefficient around a burning carbon particle in an atmosphere of O2, N2, CO2, CO, and H2O. The complete set of Stefan–Maxwell equations is analytically solved under the assumption of no homogeneous reaction in the boundary layer. An expression linking the oxygen concentration and the oxygen flux at the particle surface (as a function of the bulk gas composition) is derived which can be used to calculate the mass transfer coefficient. A very simple approximate explicit expression is also given for the mass transfer coefficient, that is shown to be valid in the low oxygen flux limit or when the primary combustion product is CO2. The results are given in terms of a correction factor to the equimolar counter-diffusion mass transfer coefficient, which is typically available in the literature for specific geometries and/or fluid-dynamic conditions. The significance of the correction factor and the accuracy of the different available expressions is illustrated for several cases of practical interest. Results show that under typical combustion conditions the use of the equimolar counter-diffusion mass transfer coefficient can lead to errors up to 10%. Larger errors are possible in oxygen-enriched conditions, while the error is generally low in oxy-combustion. Ó 2009 The Combustion Institute. Published by Elsevier Inc. All rights reserved.
1. Introduction In many practical situations combustion of carbon particles is significantly or even completely controlled by external (boundary layer) mass transfer of oxygen. While a huge amount of literature on combustion of carbon has appeared focusing on the rates and mechanisms of surface reactions and intraparticle diffusion, only very few papers have addressed the estimation of the mass transfer coefficient under conditions of practical interest. In fact, most of the model formulations to date use a simplified estimation of the mass transfer coefficient (e.g. for pulverized fuel and fluidized bed cases), without considering the complex interplay of diffusive fluxes of the different gaseous species around the particle. Hayhurst [1] showed that the implicit assumption of equimolar counter-diffusion of oxygen and carbon dioxide around a carbon particle (which is typically made for the calculation of the mass transfer coefficient) might not always be valid. For example, if CO is the only primary product of carbon combustion and it is further oxidized away from the carbon particle, assuming equimolar counter-diffusion (i.e. neglecting the Stefan flow) would lead to a 10–20% underestimation of the real mass transfer coefficient. However, in his analysis Hayhurst assumed that the problem could be schematized as a pseudo-binary system and did not take into account the presence of the other relevant
* Fax: +39 081 5936936. E-mail address: [email protected]
gaseous species. A more detailed analysis of the multi-component diffusion in the boundary layer around a burning carbon particle should consider that the different gaseous species are characterized by different diffusion coefficients one from the other. As a consequence the Stefan–Maxwell approach should be used to treat such a problem, while the use of the binary case approach (Fick’s law) in the multi-component case (e.g. see [2–6]) might introduce significant errors. The Stefan–Maxwell equations were applied by Mon and Amundson [7], Sundaresan and Amundson [8], Biggs and Agarwal [9], Wang and Bhatia [10], and Sadhukhan et al. [11] to model the multi-component diffusion in the boundary layer around a burning spherical carbon particle, with or without the presence of water vapor. Numerical solutions of the equations were reported by these authors, but no indication was given on the value of the mass transfer coefficient to be used for practical purposes. Recently, Förtsch et al. [12] derived an approximate analytical expression of the mass transfer coefficient for oxygen reacting with a small carbon particle in an atmosphere of O2, N2, CO2, CO, and H2O by solving the Stefan–Maxwell equations with some simplifying approximations and assuming no homogeneous reaction in the boundary layer. The authors showed that the correction with respect to the simple equimolar counter-diffusion case can be up to 17%, depending on the primary combustion product (CO or CO2), reaction rate, and gas-phase composition. In this work we extend the analysis by removing the above approximations (but keeping the no homogeneous reaction assumption) and analytically solving the complete set of Stefan–Maxwell equations.
0010-2180/$ - see front matter Ó 2009 The Combustion Institute. Published by Elsevier Inc. All rights reserved. doi:10.1016/j.combustflame.2009.06.002
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F. Scala / Combustion and Flame 157 (2010) 137–142
Using the mechanism factor f defined in Eq. (1) the molar fluxes of the different gaseous species at the particle surface can be linked one to the other by the following expressions:
2. Theory 2.1. Physical system We consider the pseudo-stationary multi-component diffusion of the five species O2, N2, CO2, CO, and H2O around a spherical carbon particle. At the surface of the particle oxygen is consumed while CO and/or CO2 are formed according to the overall stoichiometry [6,12]:
ð1 þ f Þ C þ O2 ! 2f CO þ ð1 f Þ CO2
N0N2 ¼ 0
ð4aÞ
N0CO2 ¼ ð1 f Þ N0O2
ð4bÞ
N0CO ¼ 2f N0O2
ð4cÞ
N0H2 O
ð4dÞ
¼0
ð1Þ
where f is a mechanism factor ð0 6 f 6 1Þ. For f ¼ 0 the only primary product of carbon consumption is CO2, while for f ¼ 1 only CO is formed. Any CO and/or CO2 formation by combustion and/or CO homogeneous oxidation within the internal pores of the carbon sphere as well as any possible carbon gasification by CO2 can be effectively lumped in Eq. (1), since we are only interested in the relative proportion (at the particle surface) of the incoming O2 molar flux with respect to the outgoing CO and CO2 molar fluxes. It is noted that also any possible carbon gasification by H2O can be treated in the same way: in fact hydrogen is practically instantaneously oxidized to water vapour by oxygen (at the particle surface or inside the pores) and the net result of the combination of steam gasification and hydrogen oxidation is the production of CO with parallel consumption of carbon and oxygen. In addition, for the same reasons, the model can be applied also in the case that an inert coherent ash layer forms around a shrinking carbon core. We will not make any assumption on the rate expression for carbon consumption in the particle, but the equations derived in the following section have general validity and can be applied to any system, provided oxygen can reach the particle surface (which is the case in most situations of practical interest). The simplifying assumption is here made that no homogeneous oxidation of CO occurs in the gas phase around the particle. This assumption is reasonable for small particles, like those used in pulverized combustion [7,8,10,11,13]. However, this assumption might be reasonable also for large particles burning in a fluidized bed at atmospheric pressure and temperature below 1000 °C, since the inert bed material is reported to provide a CO oxidation quenching effect in these conditions [14]. For simplicity, we will derive the equations for the perfectly stagnant case, but exactly the same results are obtained by considering a stagnant film around the particle. Further, it is worth to note that even if the derivation is carried out for a spherical particle, the results can be extended to any geometry [15,16]. On the other hand, one should keep in mind that the use of the stagnant film model is a simplifying approximation when the particle is in relative motion with respect to the gas phase and that more rigorous equations should be formulated using the more complex boundary layer theory (thus preventing any analytical solution).
2.2. Governing differential equations
Combination of Eqs. (3) and (4a)–(4d) gives
R2 0 N O2 r2 ¼0
NO2 ¼ NN2
ð5aÞ ð5bÞ
2
R ð1 f Þ N0O2 r2 R2 NCO ¼ 2 2f N0O2 r NH2 O ¼ 0
N CO2 ¼
ð5cÞ ð5dÞ ð5eÞ
For an ideal multi-component mixture, using the Stefan–Maxwell equations in radial coordinates, the molar fluxes of the gaseous species are linked to the mole fraction of species i ðyi Þ by
n yj Ni yi Nj dyi X ¼ dr cDij j¼1
ð6Þ
j–i
where Dij is the binary diffusion coefficient of species i in species j, and c is the total gas concentration. Following the common practice, the total gas concentration and the binary diffusion coefficients are calculated at the arithmetic mean of the bulk gas and particle temperature. It must be noted that Eq. (6) has been written with the underlying assumption that the thermal diffusion (Ludwig–Soret effect) contribution to the molar fluxes is negligible, which is reasonable for the system under consideration [17]. A close examination of the different binary diffusion coefficients shows that a number of these coefficients have values very close one to the other throughout the whole temperature range of interest in combustion processes [8,12]. In particular, it is
DO2 —N2 ¼ DO2 —CO ¼ DN2 —CO ¼ DCO2 —H2 O ¼ D DO2 —CO2 ¼ DN2 —CO2 ¼ DCO2 —CO ¼ uCO2 D
ð7aÞ ð7bÞ
DO2 —H2 O ¼ DN2 —H2 O ¼ DCO—H2 O ¼ uH2 O D
ð7cÞ
Using the relation by Fuller et al. [18] to estimate the diffusion coefficients, it can be shown that differences in the values of the coefficients for each set of equalities reported in Eqs. (7a)–(7c) are below ±2% in the whole temperature range 1000–2000 K. Values of uCO2 and uH2 O are approximately constant with temperature and can be estimated with the same relation to be
The pseudo-stationary molar balance in spherical coordinates for a gaseous species i with no homogeneous reaction is
uCO2 ¼ 0:79 and uH2 O ¼ 1:27:
d 2 r Ni ¼ 0 dr
Now it is useful to introduce the non-dimensional oxygen flux at the particle surface, defined as
ð2Þ
where Ni is the molar flux of species i in the radial direction. If we indicate with N 0i the molar flux of species i at the particle surface ðr ¼ RÞ, integration of Eq. (2) gives
/0O2 ¼
r 2 Ni ¼ R2 N0i
Using Eqs. (5a)–(5e), (7a)–(7c) and (8), Eq. (6) can be written for the five gaseous species as
ð3Þ
N0O2 R cD
ð8Þ
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F. Scala / Combustion and Flame 157 (2010) 137–142
"
dyO2 R f 1 þ 1 2f ¼ /0O2 2 yO2 r dr uCO2 !# 1 yH2 O 1
u H2 O "
! 1 yCO2
1
uCO2
"
!
y yb yO2 þ CO ¼ ybO2 þ CO exp /0O2 2f 2f
1 ð9aÞ
!#
dyN2 R f 1 þ 1 2f ð9bÞ ¼ /0O2 2 yN2 r dr uCO2 3 " ! uCO2 1 ð1 f Þ dyCO2 f 1f 0 R 5 ð9cÞ þ þy ¼ /O2 2 yCO2 r dr uCO2 uCO2 H2 O uCO2 " ! ! dyCO f 1 1 0 R þ 1 2f þ 2f þ yCO2 2f 1 ¼ /O2 2 yCO r dr uCO2 uCO2 !# 1 þyH2 O 2f 1 ð9dÞ
uH2 O
"
dyH2 O R 1 2f ¼ /0O2 2 yH2 O f 1 þ r dr uH2 O
y0O2 þ y0N2 þ y0CO2 þ y0CO þ y0H2 O ¼ 1 y0O2 ybO2 ybN2 þ ybCO þ 2f ybO2
¼
1 2f "
(
6 þ6 4f
ð9eÞ
ð10Þ
f 1
þ 1 2f
ð11Þ ð12Þ
Substitution of Eq. (12) into Eq. (9c) yields
8 ! < uCO2 1 ð1 f Þ dyCO2 f 1f 0 R b þ þ yH2 O ¼ /O2 2 yCO2 r : dr uCO2 uCO2 uCO2 " ! #) 1 2f R ð13Þ exp /0O2 f 1 þ uH2 O r which can be integrated with the boundary condition, Eq. (10), to give ! #) ( " 1f f R 1 exp /0O2 þ ybCO2 yCO2 ¼ f uCO2 r 3 2 ! # " ð1 f Þ 1 u 1 6 7 CO2 f R 7 þ ybH2 O 6 exp /0O2 4f 1 þ 12f þ f 5 uCO2 r u u H2 O
"
exp /0O2
H2 O
CO2
H2 O
" 1 exp /0O2
! # R r uCO2 " ! # 1 2f R ¼ ybH2 O exp /0O2 f 1 þ u H2 O r
(
u1
2 3 1 b y ð1 f Þ 1 O H u 2 7 6 b CO2 1 6y 1 f 7 þ 5 1 2f 4 CO2 f f 1 þ u12f þ u f
2.3. Solution procedure
yH2 O
uCO2
CO2
! #) " ! # 1 2f R f R 0 exp /O2 þ1f r uH2 O uCO2 r ð14Þ
Finally, if we divide both sides of Eq. (9d) by 2f and sum to Eq. (9a) we obtain
" !# d y R y f 1 yO2 þ CO þ 1 2f yO2 þ CO ¼ /0O2 2 dr r 2f 2f uCO2
ð15Þ
!#) þ 1 2f
3 " !#) ybH2 O ( 7 CO2 7 1 exp /0 f 1 þ 1 2f O2 uH2 O 1 þ u12f þ u f 5 1
uH2 O
(
"
!
f 1
1 exp /0O2
Boundary conditions for Eqs. (9a)–(9e) state that infinitely far from the particle surface the gaseous species mole fractions equal the bulk values:
yN2 ¼ ybN2 exp /0O2
ð17Þ
In this way we obtain the final relation
2
If we examine the set of equations (9a)–(9e), we immediately see that the two equations (9b) and (9e) can be directly integrated with the boundary conditions, Eq. (10), to give
ð16Þ
We now particularize Eqs. (11), (12), (14), and (16) at the particle surface r ¼ R; yi ¼ y0i and combine with the condition
!#
r ! 1 yi ¼ ybi for i ¼ O2 ; N2 ; CO2 ; CO; H2 O
! # f 1 R þ 1 2f r uCO2
f
CO2
!#) ð18Þ
uCO2
Eq. (18) links the oxygen mole fraction at the particle surface y0O2 with the non-dimensional oxygen flux at the particle surface /0O2 , paramet ric in the gas bulk composition ybi and in the mechanism factor f. Unfortunately, Eq. (18) cannot be inverted, so that it is not possible to obtain an explicit relation giving the non-dimensional oxygen flux as a function of the oxygen mole fraction at the particle surface. It can be noted that for /0O2 ! 0 the following simplification holds (for any constant K):
h i lim 1 exp /0O2 K ¼ /0O2 K
ð19Þ
/0O !0 2
Substitution of Eq. (19) for all the exponentials in Eq. (18) and rearrangement yields the approximate relation
y0O2 ybO2 " ¼ /0O2 ybN2 þ ybCO þ 2f þ
1f
uCO2
! ybO2 þ
1
u H2 O
ybH2 O þ
1
uCO2
# ybCO2 ð20Þ
As a final comment, examination of Eq. (18) shows that for the two cases f ¼ 0 and f ¼ 0:5 this equation cannot be used because the denominator of some of the terms on the RHS go to zero. For the first case ðf ¼ 0Þ if one uses a condition similar to Eq. (19):
lim½1 expðf KÞ ¼ f K
ð21Þ
f !0
Eq. (18) can be simplified in the following expression, only valid for f ¼ 0:
( " !#) 1 0 0 b b b yO2 yO2 ¼ yN2 þ yCO 1 exp /O2 1
uCO2
2
3 ( " !#) 1 1 yb 6 uH2 O uCO2 H2 O 7 /0O2 1 0 7 1 exp / þ6 1 þ O2 4 5 1 uH2 O uCO2 1 u H2 O
which can be integrated with the boundary conditions, Eq. (10), to give
ð22Þ
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F. Scala / Combustion and Flame 157 (2010) 137–142
For the second case ðf ¼ 0:5Þ, noting that the first and third exponentials on the RHS of Eq. (18) become equal, a simple rearrangement gives
2 6 b y0O2 ybO2 ¼ 6 42 þ yO2 þ 2 2 6 þ6 42
1
1
uH2 O
3 !# ybH2 O 7" /0O2 CO2 7 1 exp 5 2uCO2 1
u1 1
uH2 O
3 !# ybH2 O " 7 /0O2 7 1 exp 5 2 1 1
uH2 O uCO2 1
uH2 O
From a practical point of view the important quantity to be used for modeling purposes is the apparent mass transfer coefficient app k , defined by
¼k
c y0O2 ybO2
ð24Þ
which includes both the Stefan flow and the diffusive contributions to oxygen mass transfer from the bulk phase to the particle surface (note that the oxygen flux in Eq. (24) is negative, since it is directed towards the particle surface). An apparent Sherwood number can be defined as app
app
Sh
¼
2k R D
ð25Þ
where the particle diameter is used as the characteristic length of the system. For the system under consideration (sphere in stagnant gas) under the assumption of either equimolar counter-diffusion or of dilute oxygen diffusion ðyO2 ! 0Þ – both assumptions implying zero Stefan flow – and of equal diffusion coefficients for all species, the well known result Sh ¼ 2 can be easily calculated. Taking this as app a reference value for the Sherwood number, Sh can be expressed as app
Sh
¼ h Sh
ð26Þ
where h is a correction factor accounting for the deviation of the Sherwood number from the simple equimolar counter-diffusion case. Combining Eqs. (24)–(26) and setting Sh ¼ 2, we obtain
h¼
app app N0O2 R /0O2 Sh k R ¼ ¼ ¼ D Sh y0O2 ybO2 cD y0O2 ybO2
!(
"
ð27Þ
It is here underlined that even if Eq. (27) has been derived for a specific case (spherical particle and stagnant gas), it has a general validity [12]. This illustrates the advantage of using the correction factor h rather than the Sherwood number in the following discussion. In fact, if the equimolar counter-diffusion value of the Sherwood number is known for a specific geometry and/or fluid-dynamic field around the particle (e.g. the well-known Frössling [19] – Ranz and Marshall [20] correlation for a sphere in convective flow), then the apparent Sherwood number can be easily calculated from Eq. (26) once the correction factor h is known. Substitution of Eq. (18) into Eq. (27) gives the value of the correction factor as a function of the non-dimensional flux of oxygen at the particle surface, parametric in the gas bulk composition and the mechanism factor:
!#)
f 1
/0O2
1 exp þ 1 2f 1 2f uCO2 3 ( " !#) 1 1 yb 6 uH2 O uCO2 H2 O 7 1 2f 0 7 þ6 4 f 1 þ 12f þ f 5 1 exp /O2 f 1 þ u H2 O u u
2
H2 O
CO2
2 3 ð1 f Þ 1 u 1 ybH2 O 6 7 CO 1 2 6yb 1 f 7 þ CO2 4 5 12f f 1 2f f f 1þ þ uH2 O
2.4. Mass transfer coefficient
app
ybN2 þ ybCO þ 2f ybO2
( ð23Þ
N0O2
( h ¼ /0O2
"
1 exp /0O2
uCO2
!#))1
f
ð28Þ
uCO2
For the cases f ¼ 0 and f ¼ 0:5, a similar substitution of Eqs. (22) and (23) into Eq. (27) yields
h¼
/0O2
(
ybN2
þ
2
ybCO
( " !#) 1 0 1 exp /O2 1
uCO2
91 3 > " !#) 1 u 1 ybH2 O ( 0 > u 6 H2 O 7 CO2 /O2 = 1 7 1 exp /0 1 þ þ6 O2 4 5 1 uH2 O uCO2 > > uH2 O 1 ; ð29Þ
h ¼ /0O2
8 2 > > < 6 b 6 42 þ yO2 > > :
3 " !# 1 1 b uH2 O uCO2 yH2 O 7 /0O2 7 1 exp þ2 5 1 2uCO2 1 u H2 O
2
91 3 " !#> 1 1 b > y 0 6 uH2 O uCO2 H2 O 7 /O2 = 7 1 exp þ6 42 5 1 > 2 1 > u ;
ð30Þ
H2 O
If a rate expression is given expressing carbon consumption at the particle surface by chemical reaction (possibly including the effect of intraparticle diffusion and internal burning), this can be put in the form of an equation linking the oxygen flux and the oxygen concentration at the particle surface, parametric in the particle temperature [1,12]. This expression can then be combined with Eq. (18) to find the oxygen flux, which can be used to calculate the correction factor via Eq. (28). In the limit /0O2 ! 0 the approximate relation, Eq. (20), holds, which combined with Eq. (27) gives
" h ¼ ybN2 þ ybCO þ 2f þ
1f
uCO2
! ybO2 þ
1
u H2 O
ybH2 O þ
1
uCO2
#1 ybCO2
ð31Þ
which is independent of both oxygen flux and oxygen mole fraction at the particle surface. It is noted that in this case the oxygen flux is linear in the oxygen concentration and so the external diffusion resistance can be easily combined with the other resistances (kinetic and intraparticle) to yield an explicit expression for the oxygen consumption rate (or equivalently the carbon consumption rate). 3. Analysis of the mass transfer correction factor The typical assumption that is implicitly formulated in most models is that the equimolar counter-diffusion mass transfer coefficient can be used to estimate the external mass transfer resis-
141
F. Scala / Combustion and Flame 157 (2010) 137–142 0.99
tance. This corresponds to setting a value h ¼ 1. Hayhurst [1] showed that in the case that CO is the only primary product of carbon combustion and it is further oxidized away from the carbon particle (corresponding to the case f ¼ 1), for a binary system the correction factor can be calculated to be
!
ln h¼ 1 þ ybO2 y0O2 ybO2
/0 h O2 i ¼ 1 þ ybO2 1 exp /0O2
h¼
ybN2 þ ybCO þ 2f þ "
1f
uCO2
!
ybO2 þ
1
uH2 O
!#91 f 1 > > 1 exp /0O2 þ 1 2f > > = uCO2 ! > > f 1 > > þ 1 2f /0O2 ;
ybH2 O þ
1
uCO2
0.96 0.95
ð32Þ
In a more detailed analysis, Förtsch et al. [12] considered the multicomponent diffusion problem around a burning carbon particle using the Stefan–Maxwell approach. However, the treatment was simplified by considering the CO2 and H2O concentrations approximately constant in Eq. (9a). With this assumption Eq. (9a) could be directly integrated to give an approximate solution to the oxygen mole fraction profile around the particle. Using this profile and Eq. (27) an approximate correction factor was calculated, which can be written after some manipulations as
("
0.97
Correction factor θ, -
1 þ y0O2
1
f=1
0.98
0.94
f = 0.5
0.98 0.97 0.96 0.95 0.94
f=0
0.98 0.97
#
Complete expression, eqs (28-30) Approximate expression, eq (31) Expression by Fortsch et al. [12], eq (33) Expression by Hayhurst [1], eq (32)
0.96
ybCO2
0.95 0.94 -0.040
-0.035
ð33Þ
uCO2
It is interesting to note that in the limit /0O2 ! 0 Eq. (33) simplifies into Eq. (31). In the following we will compare Eqs. (28)–(33) for several cases of practical interest to show the significance of the correction factor and the accuracy of the different expressions.
-0.025
-0.020
-0.015
-0.010
-0.005
0.000
Fig. 1. Calculated values of the correction factor to the mass transfer coefficient for the air combustion base case conditions as a function of the non-dimensional oxygen flux at the particle surface, for three values of the mechanism factor f.
1.00 0.95
Correction factor θ, -
3.1. Air combustion conditions We set as base case conditions the following bulk gas composition: ybO2 ¼ 0:04; ybN2 ¼ 0:75; ybCO2 ¼ 0:13; ybCO ¼ 0; ybH2 O ¼ 0:08. Fig. 1 reports the correction factor calculated by Eqs. (28)–(33) for the base case conditions as a function of the non-dimensional oxygen flux at the particle surface, for the three values of the mechanism factor f ¼ 0, 0.5 and 1. In the figure the left limit value the of the non-dimensional oxygen flux, /0O2 ’ 0:386, represents condition for complete external mass transfer control y0O2 ¼ 0 . For the base case condition the maximum error in neglecting the correction factor is slightly above 5%. The results at f ¼ 0:5 are somewhat intermediate between those at f ¼ 0 and f ¼ 1. The approximate correction factor, Eq. (31), is very accurate for the case f ¼ 0 and obviously when the non-dimensional oxygen flux tends to zero, but fails to predict the correct values in the other cases. The Hayhurst’s expression (to be used only for f ¼ 1), Eq. (32), clearly always overestimates the real correction factor. The expression of Förtsch et al., Eq. (33), is very accurate for f ¼ 1, but shows a reasonably good accuracy also for f ¼ 0:5 and 1. Fig. 2 reports the correction factor as a function of the bulk oxygen concentration for the limiting cases f ¼ 0 and 1, under the assumption of complete external mass transfer control y0O2 ¼ 0 . In the calculations the sum ybO2 þ ybCO2 þ ybH2 O =2 has been kept equal to 0.21 and ybCO ¼ 0. The error in neglecting the correction factor increases with the bulk oxygen concentration reaching a value slightly lower than 10% for f ¼ 1 and ybO2 ¼ 0:20. Larger errors would be predicted for oxygen-enriched conditions. It can be noted that the Hayhurst’s expression, Eq. (32), increases its accuracy as the oxygen concentration increases (because CO2 and H2O concentrations decrease correspondingly), while the expression of Förtsch et al., Eq. (33), decreases its accuracy (for f ¼ 0) at higher O2 concentrations.
-0.030
Non-dimensional oxygen flux φ0O , -
0.90 0.85
f=1 0.80 0.95 0.90
Complete expression, eqs (28-29) Approximate expression, eq (31) Expression by Fortsch et al. [12], eq (33) Expression by Hayhurst [1], eq (32)
0.85
f=0 0.80 0.00
0.05
0.10
0.15
0.20
Oxygen mole fraction yO , Fig. 2. Calculated values of the correction factor to the mass transfer coefficient for air combustion conditions as a function of the bulk oxygen concentration, under the assumption of complete external mass transfer control (for the limiting cases f ¼ 0 and 1).
3.2. Oxy-combustion conditions For this case, we set as base concentrations the following values: ybO2 ¼ 0:04; ybN2 ¼ 0; ybCO2 ¼ 0:88; ybCO ¼ 0; ybH2 O ¼ 0:08. In oxycombustion conditions the reference value of the Sherwood number is calculated using the diffusion coefficient of oxygen in carbon dioxide ðDO2 CO2 ¼ uCO2 DÞ rather than that of oxygen in nitrogen. As a consequence, the correction factor calculated with Eqs. (28)– (31) and (33) should be divided by uCO2 . For the same reason, to obtain the correct correction factor using Eq. (32) the quantity /0O2 both on the numerator and on the denominator in the term on the RHS should be divided by uCO2 .
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F. Scala / Combustion and Flame 157 (2010) 137–142 1.04
the real correction factor. The expression of Förtsch et al., Eq. (33), is very accurate for f ¼ 0:5, but shows a reasonably good accuracy also for f ¼ 0 and 1. Fig. 4 reports the correction factor in oxy-combustion conditions as a function of the bulk oxygen concentration for the limiting cases f ¼ 0 and 1, under the assumption of complete external mass transfer control. In the calculations it has been set: ybN2 ¼ 0 and ybCO ¼ 0. In this case, the error in neglecting the correction factor decreases with the bulk oxygen concentration. The maximum error is slightly above 3% for ybO2 ! 0. It can be noted that The Hayhurst’s expression, Eq. (32), fails to predict the correct correction factor values in all conditions, while the expression of Förtsch et al., Eq. (33), decreases its accuracy at higher O2 concentrations.
f=1 1.02 1.00
Correction factor θ, -
0.98 0.96
f = 0.5 1.02 1.00 0.98 0.96
f=0 1.02
4. Conclusions and recommendations
1.00
Complete expression, eqs (28-30) Approximate expression, eq (31) Expression by Fortsch et al. [12], eq (33) Expression by Hayhurst [1], eq (32)
0.98 0.96 -0.040
-0.035
-0.030
-0.025
-0.020
-0.015
-0.010
-0.005
0.000
Non-dimensional oxygen flux φ0O , Fig. 3. Calculated values of the correction factor to the mass transfer coefficient for the oxy-combustion base case conditions as a function of the non-dimensional oxygen flux at the particle surface, for three values of the mechanism factor f.
1.05
Correction factor θ, -
1.00 0.95 0.90
f=1 0.85 1.00 0.95
Complete expression, eqs (28-29) Approximate expression, eq (31) Expression by Fortsch et al. [12], eq (33) Expression by Hayhurst [1], eq (32)
0.90
f=0 0.85 0.00
0.05
0.10
0.15
0.20
Oxygen mole fraction yO , Fig. 4. Calculated values of the correction factor to the mass transfer coefficient for oxy-combustion conditions as a function of the bulk oxygen concentration, under the assumption of complete external mass transfer control (for the limiting cases f ¼ 0 and 1).
Fig. 3 reports the correction factor for the oxy-combustion base case conditions as a function of the non-dimensional oxygen flux at the particle surface, for the three values of the mechanism factor f ¼ 0, 0.5 and 1. In this case, the left limit value of the non-dimensional oxygen flux, representing the condition for complete external mass transfer control, is /0O2 ’ 0:325. The maximum error in neglecting the correction factor is below 3%. Again, the results at f ¼ 0:5 are intermediate between those at f ¼ 0 and f ¼ 1. Also in this case, the approximate correction factor, Eq. (31), is very accurate for the case f ¼ 0 and when the non-dimensional oxygen flux tends to zero, but fails to predict the correct values in the other cases. The Hayhurst’s expression, Eq. (32), always underestimates
A complete analytical solution to the set of Stefan–Maxwell equations is given in this paper under the assumption of negligible homogeneous reaction in the boundary layer. An expression linking the oxygen concentration and the oxygen flux at the particle surface (as a function of the bulk gas composition) is derived which can be used to calculate the mass transfer coefficient. A very simple approximate explicit expression is also given for the mass transfer coefficient, which is relatively accurate when the primary combustion product is mostly CO2 or in the low oxygen flux limit. Modeling results show that under typical combustion conditions the use of the equimolar counter-diffusion mass transfer coefficient can lead to errors up to 10%. Larger errors are possible in oxygen-enriched conditions, while the error is generally low in oxy-combustion. The simple binary case correlation given by Hayhurst [1] for the case of CO being the primary combustion product is shown to be less accurate in most of the conditions investigated. The approximate equation given by Förtsch et al. [12], instead, is found to be reasonably accurate in all conditions of practical interest. As a final consideration, it is worth noting that the total accuracy of the model to predict the mass transfer rates around a burning particle is obviously dependent on how accurately the different species’ diffusion coefficients can be estimated. Both experimentally measured values and estimates obtained with available correlations are typically reported to have a 10–20% reliability. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]
A.N. Hayhurst, Combust. Flame 121 (2000) 679–688. H.S. Caram, N.R. Amundson, Ind. Eng. Chem. Fund. 16 (1977) 171–181. P.A. Libby, T.R. Blake, Combust. Flame 36 (1979) 139–169. P.A. Libby, T.R. Blake, Combust. Flame 41 (1981) 123–147. A. Makino, C.K. Law, Proc. Combust. Inst. 21 (1986) 183–191. R.H. Essenhigh, Energy Fuels 5 (1991) 41–46. E. Mon, N.R. Amundson, Ind. Eng. Chem. Fund. 17 (1978) 313–321. S. Sundaresan, N.R. Amundson, Ind. Eng. Chem. Fund. 19 (1980) 351–357. M.J. Biggs, P.K. Agarwal, Chem. Eng. Sci. 52 (1997) 941–952. F.Y. Wang, S.K. Bhatia, Chem. Eng. Sci. 56 (2001) 3683–3697. A.K. Sadhukhan, P. Gupta, R.K. Saha, Int. J. Chem. Kinet. 40 (2008) 569–582. D. Förtsch, U. Schnell, K.R.G. Hein, R.H. Essenhigh, Combust. Flame 126 (2001) 1662–1668. R.E. Mitchell, R.J. Kee, P. Glarborg, Proc. Combust. Inst. 23 (1990) 1169–1176. A.N. Hayhurst, R.F. Tucker, Combust. Flame 79 (1990) 175–189. W.R. Paterson, Chem. Eng. Sci. 55 (2000) 3567–3570. R. Krishna, Int. Comm. Heat Mass Transfer 28 (2001) 39–48. D.E. Rosner, Transport Processes in Chemically Reacting Flow Systems, Dover, New York, 2000. E.N. Fuller, P.D. Schettler, J.C. Giddings, J. Ind. Eng. Chem. 58 (1966) 18–27. N. Frössling, Gerl. Beitr. Geophys. 52 (1938) 170–216. W.E. Ranz, W.R. Marshall Jr., Chem. Eng. Prog. 48 (1952) 141–146; W.E. Ranz, W.R. Marshall Jr., Chem. Eng. Prog. 48 (1952) 173–180.
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Combustion and Flame j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / c o m b u s t fl a m e
Calculation of the mass transfer coefficient for the combustion of a carbon particle Fabrizio Scala * Istituto di Ricerche sulla Combustione – CNR, P.le Tecchio 80, 80125 Napoli, Italy
a r t i c l e
i n f o
Article history: Received 26 March 2009 Received in revised form 11 May 2009 Accepted 1 June 2009 Available online 21 June 2009 Keywords: Mass transfer coefficient Sherwood number Combustion Carbon particle Multi-component diffusion
a b s t r a c t In this paper we address the calculation of the mass transfer coefficient around a burning carbon particle in an atmosphere of O2, N2, CO2, CO, and H2O. The complete set of Stefan–Maxwell equations is analytically solved under the assumption of no homogeneous reaction in the boundary layer. An expression linking the oxygen concentration and the oxygen flux at the particle surface (as a function of the bulk gas composition) is derived which can be used to calculate the mass transfer coefficient. A very simple approximate explicit expression is also given for the mass transfer coefficient, that is shown to be valid in the low oxygen flux limit or when the primary combustion product is CO2. The results are given in terms of a correction factor to the equimolar counter-diffusion mass transfer coefficient, which is typically available in the literature for specific geometries and/or fluid-dynamic conditions. The significance of the correction factor and the accuracy of the different available expressions is illustrated for several cases of practical interest. Results show that under typical combustion conditions the use of the equimolar counter-diffusion mass transfer coefficient can lead to errors up to 10%. Larger errors are possible in oxygen-enriched conditions, while the error is generally low in oxy-combustion. Ó 2009 The Combustion Institute. Published by Elsevier Inc. All rights reserved.
1. Introduction In many practical situations combustion of carbon particles is significantly or even completely controlled by external (boundary layer) mass transfer of oxygen. While a huge amount of literature on combustion of carbon has appeared focusing on the rates and mechanisms of surface reactions and intraparticle diffusion, only very few papers have addressed the estimation of the mass transfer coefficient under conditions of practical interest. In fact, most of the model formulations to date use a simplified estimation of the mass transfer coefficient (e.g. for pulverized fuel and fluidized bed cases), without considering the complex interplay of diffusive fluxes of the different gaseous species around the particle. Hayhurst [1] showed that the implicit assumption of equimolar counter-diffusion of oxygen and carbon dioxide around a carbon particle (which is typically made for the calculation of the mass transfer coefficient) might not always be valid. For example, if CO is the only primary product of carbon combustion and it is further oxidized away from the carbon particle, assuming equimolar counter-diffusion (i.e. neglecting the Stefan flow) would lead to a 10–20% underestimation of the real mass transfer coefficient. However, in his analysis Hayhurst assumed that the problem could be schematized as a pseudo-binary system and did not take into account the presence of the other relevant
* Fax: +39 081 5936936. E-mail address: [email protected]
gaseous species. A more detailed analysis of the multi-component diffusion in the boundary layer around a burning carbon particle should consider that the different gaseous species are characterized by different diffusion coefficients one from the other. As a consequence the Stefan–Maxwell approach should be used to treat such a problem, while the use of the binary case approach (Fick’s law) in the multi-component case (e.g. see [2–6]) might introduce significant errors. The Stefan–Maxwell equations were applied by Mon and Amundson [7], Sundaresan and Amundson [8], Biggs and Agarwal [9], Wang and Bhatia [10], and Sadhukhan et al. [11] to model the multi-component diffusion in the boundary layer around a burning spherical carbon particle, with or without the presence of water vapor. Numerical solutions of the equations were reported by these authors, but no indication was given on the value of the mass transfer coefficient to be used for practical purposes. Recently, Förtsch et al. [12] derived an approximate analytical expression of the mass transfer coefficient for oxygen reacting with a small carbon particle in an atmosphere of O2, N2, CO2, CO, and H2O by solving the Stefan–Maxwell equations with some simplifying approximations and assuming no homogeneous reaction in the boundary layer. The authors showed that the correction with respect to the simple equimolar counter-diffusion case can be up to 17%, depending on the primary combustion product (CO or CO2), reaction rate, and gas-phase composition. In this work we extend the analysis by removing the above approximations (but keeping the no homogeneous reaction assumption) and analytically solving the complete set of Stefan–Maxwell equations.
0010-2180/$ - see front matter Ó 2009 The Combustion Institute. Published by Elsevier Inc. All rights reserved. doi:10.1016/j.combustflame.2009.06.002
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F. Scala / Combustion and Flame 157 (2010) 137–142
Using the mechanism factor f defined in Eq. (1) the molar fluxes of the different gaseous species at the particle surface can be linked one to the other by the following expressions:
2. Theory 2.1. Physical system We consider the pseudo-stationary multi-component diffusion of the five species O2, N2, CO2, CO, and H2O around a spherical carbon particle. At the surface of the particle oxygen is consumed while CO and/or CO2 are formed according to the overall stoichiometry [6,12]:
ð1 þ f Þ C þ O2 ! 2f CO þ ð1 f Þ CO2
N0N2 ¼ 0
ð4aÞ
N0CO2 ¼ ð1 f Þ N0O2
ð4bÞ
N0CO ¼ 2f N0O2
ð4cÞ
N0H2 O
ð4dÞ
¼0
ð1Þ
where f is a mechanism factor ð0 6 f 6 1Þ. For f ¼ 0 the only primary product of carbon consumption is CO2, while for f ¼ 1 only CO is formed. Any CO and/or CO2 formation by combustion and/or CO homogeneous oxidation within the internal pores of the carbon sphere as well as any possible carbon gasification by CO2 can be effectively lumped in Eq. (1), since we are only interested in the relative proportion (at the particle surface) of the incoming O2 molar flux with respect to the outgoing CO and CO2 molar fluxes. It is noted that also any possible carbon gasification by H2O can be treated in the same way: in fact hydrogen is practically instantaneously oxidized to water vapour by oxygen (at the particle surface or inside the pores) and the net result of the combination of steam gasification and hydrogen oxidation is the production of CO with parallel consumption of carbon and oxygen. In addition, for the same reasons, the model can be applied also in the case that an inert coherent ash layer forms around a shrinking carbon core. We will not make any assumption on the rate expression for carbon consumption in the particle, but the equations derived in the following section have general validity and can be applied to any system, provided oxygen can reach the particle surface (which is the case in most situations of practical interest). The simplifying assumption is here made that no homogeneous oxidation of CO occurs in the gas phase around the particle. This assumption is reasonable for small particles, like those used in pulverized combustion [7,8,10,11,13]. However, this assumption might be reasonable also for large particles burning in a fluidized bed at atmospheric pressure and temperature below 1000 °C, since the inert bed material is reported to provide a CO oxidation quenching effect in these conditions [14]. For simplicity, we will derive the equations for the perfectly stagnant case, but exactly the same results are obtained by considering a stagnant film around the particle. Further, it is worth to note that even if the derivation is carried out for a spherical particle, the results can be extended to any geometry [15,16]. On the other hand, one should keep in mind that the use of the stagnant film model is a simplifying approximation when the particle is in relative motion with respect to the gas phase and that more rigorous equations should be formulated using the more complex boundary layer theory (thus preventing any analytical solution).
2.2. Governing differential equations
Combination of Eqs. (3) and (4a)–(4d) gives
R2 0 N O2 r2 ¼0
NO2 ¼ NN2
ð5aÞ ð5bÞ
2
R ð1 f Þ N0O2 r2 R2 NCO ¼ 2 2f N0O2 r NH2 O ¼ 0
N CO2 ¼
ð5cÞ ð5dÞ ð5eÞ
For an ideal multi-component mixture, using the Stefan–Maxwell equations in radial coordinates, the molar fluxes of the gaseous species are linked to the mole fraction of species i ðyi Þ by
n yj Ni yi Nj dyi X ¼ dr cDij j¼1
ð6Þ
j–i
where Dij is the binary diffusion coefficient of species i in species j, and c is the total gas concentration. Following the common practice, the total gas concentration and the binary diffusion coefficients are calculated at the arithmetic mean of the bulk gas and particle temperature. It must be noted that Eq. (6) has been written with the underlying assumption that the thermal diffusion (Ludwig–Soret effect) contribution to the molar fluxes is negligible, which is reasonable for the system under consideration [17]. A close examination of the different binary diffusion coefficients shows that a number of these coefficients have values very close one to the other throughout the whole temperature range of interest in combustion processes [8,12]. In particular, it is
DO2 —N2 ¼ DO2 —CO ¼ DN2 —CO ¼ DCO2 —H2 O ¼ D DO2 —CO2 ¼ DN2 —CO2 ¼ DCO2 —CO ¼ uCO2 D
ð7aÞ ð7bÞ
DO2 —H2 O ¼ DN2 —H2 O ¼ DCO—H2 O ¼ uH2 O D
ð7cÞ
Using the relation by Fuller et al. [18] to estimate the diffusion coefficients, it can be shown that differences in the values of the coefficients for each set of equalities reported in Eqs. (7a)–(7c) are below ±2% in the whole temperature range 1000–2000 K. Values of uCO2 and uH2 O are approximately constant with temperature and can be estimated with the same relation to be
The pseudo-stationary molar balance in spherical coordinates for a gaseous species i with no homogeneous reaction is
uCO2 ¼ 0:79 and uH2 O ¼ 1:27:
d 2 r Ni ¼ 0 dr
Now it is useful to introduce the non-dimensional oxygen flux at the particle surface, defined as
ð2Þ
where Ni is the molar flux of species i in the radial direction. If we indicate with N 0i the molar flux of species i at the particle surface ðr ¼ RÞ, integration of Eq. (2) gives
/0O2 ¼
r 2 Ni ¼ R2 N0i
Using Eqs. (5a)–(5e), (7a)–(7c) and (8), Eq. (6) can be written for the five gaseous species as
ð3Þ
N0O2 R cD
ð8Þ
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F. Scala / Combustion and Flame 157 (2010) 137–142
"
dyO2 R f 1 þ 1 2f ¼ /0O2 2 yO2 r dr uCO2 !# 1 yH2 O 1
u H2 O "
! 1 yCO2
1
uCO2
"
!
y yb yO2 þ CO ¼ ybO2 þ CO exp /0O2 2f 2f
1 ð9aÞ
!#
dyN2 R f 1 þ 1 2f ð9bÞ ¼ /0O2 2 yN2 r dr uCO2 3 " ! uCO2 1 ð1 f Þ dyCO2 f 1f 0 R 5 ð9cÞ þ þy ¼ /O2 2 yCO2 r dr uCO2 uCO2 H2 O uCO2 " ! ! dyCO f 1 1 0 R þ 1 2f þ 2f þ yCO2 2f 1 ¼ /O2 2 yCO r dr uCO2 uCO2 !# 1 þyH2 O 2f 1 ð9dÞ
uH2 O
"
dyH2 O R 1 2f ¼ /0O2 2 yH2 O f 1 þ r dr uH2 O
y0O2 þ y0N2 þ y0CO2 þ y0CO þ y0H2 O ¼ 1 y0O2 ybO2 ybN2 þ ybCO þ 2f ybO2
¼
1 2f "
(
6 þ6 4f
ð9eÞ
ð10Þ
f 1
þ 1 2f
ð11Þ ð12Þ
Substitution of Eq. (12) into Eq. (9c) yields
8 ! < uCO2 1 ð1 f Þ dyCO2 f 1f 0 R b þ þ yH2 O ¼ /O2 2 yCO2 r : dr uCO2 uCO2 uCO2 " ! #) 1 2f R ð13Þ exp /0O2 f 1 þ uH2 O r which can be integrated with the boundary condition, Eq. (10), to give ! #) ( " 1f f R 1 exp /0O2 þ ybCO2 yCO2 ¼ f uCO2 r 3 2 ! # " ð1 f Þ 1 u 1 6 7 CO2 f R 7 þ ybH2 O 6 exp /0O2 4f 1 þ 12f þ f 5 uCO2 r u u H2 O
"
exp /0O2
H2 O
CO2
H2 O
" 1 exp /0O2
! # R r uCO2 " ! # 1 2f R ¼ ybH2 O exp /0O2 f 1 þ u H2 O r
(
u1
2 3 1 b y ð1 f Þ 1 O H u 2 7 6 b CO2 1 6y 1 f 7 þ 5 1 2f 4 CO2 f f 1 þ u12f þ u f
2.3. Solution procedure
yH2 O
uCO2
CO2
! #) " ! # 1 2f R f R 0 exp /O2 þ1f r uH2 O uCO2 r ð14Þ
Finally, if we divide both sides of Eq. (9d) by 2f and sum to Eq. (9a) we obtain
" !# d y R y f 1 yO2 þ CO þ 1 2f yO2 þ CO ¼ /0O2 2 dr r 2f 2f uCO2
ð15Þ
!#) þ 1 2f
3 " !#) ybH2 O ( 7 CO2 7 1 exp /0 f 1 þ 1 2f O2 uH2 O 1 þ u12f þ u f 5 1
uH2 O
(
"
!
f 1
1 exp /0O2
Boundary conditions for Eqs. (9a)–(9e) state that infinitely far from the particle surface the gaseous species mole fractions equal the bulk values:
yN2 ¼ ybN2 exp /0O2
ð17Þ
In this way we obtain the final relation
2
If we examine the set of equations (9a)–(9e), we immediately see that the two equations (9b) and (9e) can be directly integrated with the boundary conditions, Eq. (10), to give
ð16Þ
We now particularize Eqs. (11), (12), (14), and (16) at the particle surface r ¼ R; yi ¼ y0i and combine with the condition
!#
r ! 1 yi ¼ ybi for i ¼ O2 ; N2 ; CO2 ; CO; H2 O
! # f 1 R þ 1 2f r uCO2
f
CO2
!#) ð18Þ
uCO2
Eq. (18) links the oxygen mole fraction at the particle surface y0O2 with the non-dimensional oxygen flux at the particle surface /0O2 , paramet ric in the gas bulk composition ybi and in the mechanism factor f. Unfortunately, Eq. (18) cannot be inverted, so that it is not possible to obtain an explicit relation giving the non-dimensional oxygen flux as a function of the oxygen mole fraction at the particle surface. It can be noted that for /0O2 ! 0 the following simplification holds (for any constant K):
h i lim 1 exp /0O2 K ¼ /0O2 K
ð19Þ
/0O !0 2
Substitution of Eq. (19) for all the exponentials in Eq. (18) and rearrangement yields the approximate relation
y0O2 ybO2 " ¼ /0O2 ybN2 þ ybCO þ 2f þ
1f
uCO2
! ybO2 þ
1
u H2 O
ybH2 O þ
1
uCO2
# ybCO2 ð20Þ
As a final comment, examination of Eq. (18) shows that for the two cases f ¼ 0 and f ¼ 0:5 this equation cannot be used because the denominator of some of the terms on the RHS go to zero. For the first case ðf ¼ 0Þ if one uses a condition similar to Eq. (19):
lim½1 expðf KÞ ¼ f K
ð21Þ
f !0
Eq. (18) can be simplified in the following expression, only valid for f ¼ 0:
( " !#) 1 0 0 b b b yO2 yO2 ¼ yN2 þ yCO 1 exp /O2 1
uCO2
2
3 ( " !#) 1 1 yb 6 uH2 O uCO2 H2 O 7 /0O2 1 0 7 1 exp / þ6 1 þ O2 4 5 1 uH2 O uCO2 1 u H2 O
which can be integrated with the boundary conditions, Eq. (10), to give
ð22Þ
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F. Scala / Combustion and Flame 157 (2010) 137–142
For the second case ðf ¼ 0:5Þ, noting that the first and third exponentials on the RHS of Eq. (18) become equal, a simple rearrangement gives
2 6 b y0O2 ybO2 ¼ 6 42 þ yO2 þ 2 2 6 þ6 42
1
1
uH2 O
3 !# ybH2 O 7" /0O2 CO2 7 1 exp 5 2uCO2 1
u1 1
uH2 O
3 !# ybH2 O " 7 /0O2 7 1 exp 5 2 1 1
uH2 O uCO2 1
uH2 O
From a practical point of view the important quantity to be used for modeling purposes is the apparent mass transfer coefficient app k , defined by
¼k
c y0O2 ybO2
ð24Þ
which includes both the Stefan flow and the diffusive contributions to oxygen mass transfer from the bulk phase to the particle surface (note that the oxygen flux in Eq. (24) is negative, since it is directed towards the particle surface). An apparent Sherwood number can be defined as app
app
Sh
¼
2k R D
ð25Þ
where the particle diameter is used as the characteristic length of the system. For the system under consideration (sphere in stagnant gas) under the assumption of either equimolar counter-diffusion or of dilute oxygen diffusion ðyO2 ! 0Þ – both assumptions implying zero Stefan flow – and of equal diffusion coefficients for all species, the well known result Sh ¼ 2 can be easily calculated. Taking this as app a reference value for the Sherwood number, Sh can be expressed as app
Sh
¼ h Sh
ð26Þ
where h is a correction factor accounting for the deviation of the Sherwood number from the simple equimolar counter-diffusion case. Combining Eqs. (24)–(26) and setting Sh ¼ 2, we obtain
h¼
app app N0O2 R /0O2 Sh k R ¼ ¼ ¼ D Sh y0O2 ybO2 cD y0O2 ybO2
!(
"
ð27Þ
It is here underlined that even if Eq. (27) has been derived for a specific case (spherical particle and stagnant gas), it has a general validity [12]. This illustrates the advantage of using the correction factor h rather than the Sherwood number in the following discussion. In fact, if the equimolar counter-diffusion value of the Sherwood number is known for a specific geometry and/or fluid-dynamic field around the particle (e.g. the well-known Frössling [19] – Ranz and Marshall [20] correlation for a sphere in convective flow), then the apparent Sherwood number can be easily calculated from Eq. (26) once the correction factor h is known. Substitution of Eq. (18) into Eq. (27) gives the value of the correction factor as a function of the non-dimensional flux of oxygen at the particle surface, parametric in the gas bulk composition and the mechanism factor:
!#)
f 1
/0O2
1 exp þ 1 2f 1 2f uCO2 3 ( " !#) 1 1 yb 6 uH2 O uCO2 H2 O 7 1 2f 0 7 þ6 4 f 1 þ 12f þ f 5 1 exp /O2 f 1 þ u H2 O u u
2
H2 O
CO2
2 3 ð1 f Þ 1 u 1 ybH2 O 6 7 CO 1 2 6yb 1 f 7 þ CO2 4 5 12f f 1 2f f f 1þ þ uH2 O
2.4. Mass transfer coefficient
app
ybN2 þ ybCO þ 2f ybO2
( ð23Þ
N0O2
( h ¼ /0O2
"
1 exp /0O2
uCO2
!#))1
f
ð28Þ
uCO2
For the cases f ¼ 0 and f ¼ 0:5, a similar substitution of Eqs. (22) and (23) into Eq. (27) yields
h¼
/0O2
(
ybN2
þ
2
ybCO
( " !#) 1 0 1 exp /O2 1
uCO2
91 3 > " !#) 1 u 1 ybH2 O ( 0 > u 6 H2 O 7 CO2 /O2 = 1 7 1 exp /0 1 þ þ6 O2 4 5 1 uH2 O uCO2 > > uH2 O 1 ; ð29Þ
h ¼ /0O2
8 2 > > < 6 b 6 42 þ yO2 > > :
3 " !# 1 1 b uH2 O uCO2 yH2 O 7 /0O2 7 1 exp þ2 5 1 2uCO2 1 u H2 O
2
91 3 " !#> 1 1 b > y 0 6 uH2 O uCO2 H2 O 7 /O2 = 7 1 exp þ6 42 5 1 > 2 1 > u ;
ð30Þ
H2 O
If a rate expression is given expressing carbon consumption at the particle surface by chemical reaction (possibly including the effect of intraparticle diffusion and internal burning), this can be put in the form of an equation linking the oxygen flux and the oxygen concentration at the particle surface, parametric in the particle temperature [1,12]. This expression can then be combined with Eq. (18) to find the oxygen flux, which can be used to calculate the correction factor via Eq. (28). In the limit /0O2 ! 0 the approximate relation, Eq. (20), holds, which combined with Eq. (27) gives
" h ¼ ybN2 þ ybCO þ 2f þ
1f
uCO2
! ybO2 þ
1
u H2 O
ybH2 O þ
1
uCO2
#1 ybCO2
ð31Þ
which is independent of both oxygen flux and oxygen mole fraction at the particle surface. It is noted that in this case the oxygen flux is linear in the oxygen concentration and so the external diffusion resistance can be easily combined with the other resistances (kinetic and intraparticle) to yield an explicit expression for the oxygen consumption rate (or equivalently the carbon consumption rate). 3. Analysis of the mass transfer correction factor The typical assumption that is implicitly formulated in most models is that the equimolar counter-diffusion mass transfer coefficient can be used to estimate the external mass transfer resis-
141
F. Scala / Combustion and Flame 157 (2010) 137–142 0.99
tance. This corresponds to setting a value h ¼ 1. Hayhurst [1] showed that in the case that CO is the only primary product of carbon combustion and it is further oxidized away from the carbon particle (corresponding to the case f ¼ 1), for a binary system the correction factor can be calculated to be
!
ln h¼ 1 þ ybO2 y0O2 ybO2
/0 h O2 i ¼ 1 þ ybO2 1 exp /0O2
h¼
ybN2 þ ybCO þ 2f þ "
1f
uCO2
!
ybO2 þ
1
uH2 O
!#91 f 1 > > 1 exp /0O2 þ 1 2f > > = uCO2 ! > > f 1 > > þ 1 2f /0O2 ;
ybH2 O þ
1
uCO2
0.96 0.95
ð32Þ
In a more detailed analysis, Förtsch et al. [12] considered the multicomponent diffusion problem around a burning carbon particle using the Stefan–Maxwell approach. However, the treatment was simplified by considering the CO2 and H2O concentrations approximately constant in Eq. (9a). With this assumption Eq. (9a) could be directly integrated to give an approximate solution to the oxygen mole fraction profile around the particle. Using this profile and Eq. (27) an approximate correction factor was calculated, which can be written after some manipulations as
("
0.97
Correction factor θ, -
1 þ y0O2
1
f=1
0.98
0.94
f = 0.5
0.98 0.97 0.96 0.95 0.94
f=0
0.98 0.97
#
Complete expression, eqs (28-30) Approximate expression, eq (31) Expression by Fortsch et al. [12], eq (33) Expression by Hayhurst [1], eq (32)
0.96
ybCO2
0.95 0.94 -0.040
-0.035
ð33Þ
uCO2
It is interesting to note that in the limit /0O2 ! 0 Eq. (33) simplifies into Eq. (31). In the following we will compare Eqs. (28)–(33) for several cases of practical interest to show the significance of the correction factor and the accuracy of the different expressions.
-0.025
-0.020
-0.015
-0.010
-0.005
0.000
Fig. 1. Calculated values of the correction factor to the mass transfer coefficient for the air combustion base case conditions as a function of the non-dimensional oxygen flux at the particle surface, for three values of the mechanism factor f.
1.00 0.95
Correction factor θ, -
3.1. Air combustion conditions We set as base case conditions the following bulk gas composition: ybO2 ¼ 0:04; ybN2 ¼ 0:75; ybCO2 ¼ 0:13; ybCO ¼ 0; ybH2 O ¼ 0:08. Fig. 1 reports the correction factor calculated by Eqs. (28)–(33) for the base case conditions as a function of the non-dimensional oxygen flux at the particle surface, for the three values of the mechanism factor f ¼ 0, 0.5 and 1. In the figure the left limit value the of the non-dimensional oxygen flux, /0O2 ’ 0:386, represents condition for complete external mass transfer control y0O2 ¼ 0 . For the base case condition the maximum error in neglecting the correction factor is slightly above 5%. The results at f ¼ 0:5 are somewhat intermediate between those at f ¼ 0 and f ¼ 1. The approximate correction factor, Eq. (31), is very accurate for the case f ¼ 0 and obviously when the non-dimensional oxygen flux tends to zero, but fails to predict the correct values in the other cases. The Hayhurst’s expression (to be used only for f ¼ 1), Eq. (32), clearly always overestimates the real correction factor. The expression of Förtsch et al., Eq. (33), is very accurate for f ¼ 1, but shows a reasonably good accuracy also for f ¼ 0:5 and 1. Fig. 2 reports the correction factor as a function of the bulk oxygen concentration for the limiting cases f ¼ 0 and 1, under the assumption of complete external mass transfer control y0O2 ¼ 0 . In the calculations the sum ybO2 þ ybCO2 þ ybH2 O =2 has been kept equal to 0.21 and ybCO ¼ 0. The error in neglecting the correction factor increases with the bulk oxygen concentration reaching a value slightly lower than 10% for f ¼ 1 and ybO2 ¼ 0:20. Larger errors would be predicted for oxygen-enriched conditions. It can be noted that the Hayhurst’s expression, Eq. (32), increases its accuracy as the oxygen concentration increases (because CO2 and H2O concentrations decrease correspondingly), while the expression of Förtsch et al., Eq. (33), decreases its accuracy (for f ¼ 0) at higher O2 concentrations.
-0.030
Non-dimensional oxygen flux φ0O , -
0.90 0.85
f=1 0.80 0.95 0.90
Complete expression, eqs (28-29) Approximate expression, eq (31) Expression by Fortsch et al. [12], eq (33) Expression by Hayhurst [1], eq (32)
0.85
f=0 0.80 0.00
0.05
0.10
0.15
0.20
Oxygen mole fraction yO , Fig. 2. Calculated values of the correction factor to the mass transfer coefficient for air combustion conditions as a function of the bulk oxygen concentration, under the assumption of complete external mass transfer control (for the limiting cases f ¼ 0 and 1).
3.2. Oxy-combustion conditions For this case, we set as base concentrations the following values: ybO2 ¼ 0:04; ybN2 ¼ 0; ybCO2 ¼ 0:88; ybCO ¼ 0; ybH2 O ¼ 0:08. In oxycombustion conditions the reference value of the Sherwood number is calculated using the diffusion coefficient of oxygen in carbon dioxide ðDO2 CO2 ¼ uCO2 DÞ rather than that of oxygen in nitrogen. As a consequence, the correction factor calculated with Eqs. (28)– (31) and (33) should be divided by uCO2 . For the same reason, to obtain the correct correction factor using Eq. (32) the quantity /0O2 both on the numerator and on the denominator in the term on the RHS should be divided by uCO2 .
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F. Scala / Combustion and Flame 157 (2010) 137–142 1.04
the real correction factor. The expression of Förtsch et al., Eq. (33), is very accurate for f ¼ 0:5, but shows a reasonably good accuracy also for f ¼ 0 and 1. Fig. 4 reports the correction factor in oxy-combustion conditions as a function of the bulk oxygen concentration for the limiting cases f ¼ 0 and 1, under the assumption of complete external mass transfer control. In the calculations it has been set: ybN2 ¼ 0 and ybCO ¼ 0. In this case, the error in neglecting the correction factor decreases with the bulk oxygen concentration. The maximum error is slightly above 3% for ybO2 ! 0. It can be noted that The Hayhurst’s expression, Eq. (32), fails to predict the correct correction factor values in all conditions, while the expression of Förtsch et al., Eq. (33), decreases its accuracy at higher O2 concentrations.
f=1 1.02 1.00
Correction factor θ, -
0.98 0.96
f = 0.5 1.02 1.00 0.98 0.96
f=0 1.02
4. Conclusions and recommendations
1.00
Complete expression, eqs (28-30) Approximate expression, eq (31) Expression by Fortsch et al. [12], eq (33) Expression by Hayhurst [1], eq (32)
0.98 0.96 -0.040
-0.035
-0.030
-0.025
-0.020
-0.015
-0.010
-0.005
0.000
Non-dimensional oxygen flux φ0O , Fig. 3. Calculated values of the correction factor to the mass transfer coefficient for the oxy-combustion base case conditions as a function of the non-dimensional oxygen flux at the particle surface, for three values of the mechanism factor f.
1.05
Correction factor θ, -
1.00 0.95 0.90
f=1 0.85 1.00 0.95
Complete expression, eqs (28-29) Approximate expression, eq (31) Expression by Fortsch et al. [12], eq (33) Expression by Hayhurst [1], eq (32)
0.90
f=0 0.85 0.00
0.05
0.10
0.15
0.20
Oxygen mole fraction yO , Fig. 4. Calculated values of the correction factor to the mass transfer coefficient for oxy-combustion conditions as a function of the bulk oxygen concentration, under the assumption of complete external mass transfer control (for the limiting cases f ¼ 0 and 1).
Fig. 3 reports the correction factor for the oxy-combustion base case conditions as a function of the non-dimensional oxygen flux at the particle surface, for the three values of the mechanism factor f ¼ 0, 0.5 and 1. In this case, the left limit value of the non-dimensional oxygen flux, representing the condition for complete external mass transfer control, is /0O2 ’ 0:325. The maximum error in neglecting the correction factor is below 3%. Again, the results at f ¼ 0:5 are intermediate between those at f ¼ 0 and f ¼ 1. Also in this case, the approximate correction factor, Eq. (31), is very accurate for the case f ¼ 0 and when the non-dimensional oxygen flux tends to zero, but fails to predict the correct values in the other cases. The Hayhurst’s expression, Eq. (32), always underestimates
A complete analytical solution to the set of Stefan–Maxwell equations is given in this paper under the assumption of negligible homogeneous reaction in the boundary layer. An expression linking the oxygen concentration and the oxygen flux at the particle surface (as a function of the bulk gas composition) is derived which can be used to calculate the mass transfer coefficient. A very simple approximate explicit expression is also given for the mass transfer coefficient, which is relatively accurate when the primary combustion product is mostly CO2 or in the low oxygen flux limit. Modeling results show that under typical combustion conditions the use of the equimolar counter-diffusion mass transfer coefficient can lead to errors up to 10%. Larger errors are possible in oxygen-enriched conditions, while the error is generally low in oxy-combustion. The simple binary case correlation given by Hayhurst [1] for the case of CO being the primary combustion product is shown to be less accurate in most of the conditions investigated. The approximate equation given by Förtsch et al. [12], instead, is found to be reasonably accurate in all conditions of practical interest. As a final consideration, it is worth noting that the total accuracy of the model to predict the mass transfer rates around a burning particle is obviously dependent on how accurately the different species’ diffusion coefficients can be estimated. Both experimentally measured values and estimates obtained with available correlations are typically reported to have a 10–20% reliability. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]
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