Diffusional Effects in CO2 Gasification Experiments with Single

Energy & Fuels 2006, 20, 2211-2222

2211

Diffusional Effects in CO2 Gasification Experiments with Single Biomass Char Particles. 2. Theoretical Predictions A. Go´mez-Barea,* P. Ollero, and A. Villanueva Chemical and EnVironmental Engineering Department, Escuela Superior de Ingenieros (UniVersity of SeVille), Camino de los Descubrimientos s/n, 41092 SeVille, Spain ReceiVed NoVember 8, 2005. ReVised Manuscript ReceiVed April 10, 2006

We studied the diffusional effects that occur during the gasification of single char particles in a thermogravimetric analysis (TGA). In the first part of this study, which has been described in a companion paper, the diffusional effects in char particles were experimentally investigated at various particle sizes, CO2 partial pressures, and temperatures. In this second part, a kinetic particle model that takes into account the diffusion-reaction processes within a finite-size particle was formulated, based on a recently developed modeling approach. It was further extended to allow for intraparticle heat effects and heat- and mass-transfer phenomena that occurs in the external gas layer. The model satisfactorily explains the experiments reported in part I and highlights the importance of the diffusional effects at high temperatures and large particle sizessi.e., a large Thiele modulus. This theoretical treatment sheds light on the different physical aspects involved in a typical char gasification test. Intraparticle mass limitation was identified as the main factor responsible for the strong resistance found at large particle size and high temperature. External heat and mass transfer were also determined to have a relevant role. Although the situation in full-scale gasifiers is more complex than that in a TGA using a single oxidant, the methodology developed here is capable of capturing the major physicochemical processes with minor computational difficulties. Thus, it is a first step toward rigorously estimating the gasification rate of a single char particle with minor calculations.

Introduction The gasification of char is a key factor in the design and operation of gasifiers. The gasification rate of char is influenced by many process variables, such as particle size, char porosity, mineral content of the char, temperature and partial pressure of the gasifying reactant and products. Many of these variables have a complex impact on the process. As a result, several simplifications are used in practice to obtain a more tractable expression rate. If diffusional film and intraparticle mass- and heat-transfer processes in char particles are not rapid enough, the actual gasification rate differs from the intrinsic one evaluated under bulk-gas conditions. Under these conditions, the overall gasification rate of a single char particle is determined by combining the intrinsic chemical reaction rate with intraparticle and external diffusional rates. Therefore, the actual gasification rate may be strongly dependent on the particle size, effective properties of the char (if intraparticle resistance is limiting), and fluid-dynamic conditions (if film resistance is important). Numerous particle-kinetic studies have confirmed strong diffusional effects under common gasification conditions (see section 2 and references therein). Despite this observation, transport effects are usually disregarded when coal and biomass gasifiers are being modeled.1 A review of the existing literature given in ref 1 leads to the conclusion that this simplification, disregarding transport effects, is often used for the purpose of * To whom correspondence should be addressed. Tel.: +34 95 4487223. Fax: +34 95 4461775. E-mail address: [email protected]. (1) Go´mez-Barea, A. Modelling of diffusional effects during gasification of biomass char particles in fluidised-bed. Ph.D. Thesis, University of Seville, Seville, Spain, 2006.

obtaining reasonable and tractable models, but it is seldom justified in the literature publications. An experimental study on kinetic tests performed with single wood matter from pressed-oil stone WPOS-char particles in thermogravimetric analysis (TGA) was described in Part 1 of this work.2 The results made it possible to elucidate the impact of particle size, CO2 partial pressure, and temperature on the observed gasification rate and calculate experimentally the diffusional effects that occur during the char reactivity tests. This article addresses the second objective of the work: to verify the experiments of Part 1 computationally. This was achieved via the development of a kinetic particle model capable of reproducing the most relevant aspects of the tests with a simple theoretical treatment. In addition, the method of solution presented here is, by itself, a second contribution of this work. In fact, the model developed makes it possible to solve any noncatalytic gas-solid reaction involving a single oxidant rapidly. Despite the method being computationally simple, it is rather general, because it allows for the incorporation of any nonlinear chemical reaction rate and also the changes in porous structure during conversion. In principle, the situation in a char particle in a gasifier is more complex than that simulated here. To simulate the gasification of a single char particle inside an atmospheric gasifier, one needs a model that is capable of simultaneously considering two heterogeneous char-gasification reactions (with CO2 and H2O in atmospheric gasifiers) and the water-gas shift reaction. The model developed in this work is applicable for the gasification of a char particle with a single oxidant. (2) Go´mez-Barea, A.; Ollero, P.; Ferna´ndez-Baco, C. Diffusional Effects in CO2 Gasification Experiments with Single Biomass Char Particles. 1. Experimental Investigation. Energy Fuels. 2006, 20, 2202-2210.

10.1021/ef0503663 CCC: $33.50 © 2006 American Chemical Society Published on Web 07/12/2006

2212 Energy & Fuels, Vol. 20, No. 5, 2006

Therefore, the model developed here should be considered as a first step toward a more-realistic estimate of the char gasification rate in a gasifier. For the prediction of real gas composition and char-consumption rate in an industrial gasifier, it is necessary to extend the model presented here to cover the aforementioned scenario. This treatment is currently under development, but it is beyond the scope of this work and will be presented in a future publication. 2. Literature on Modeling of Char Gasification in Single Particles Char gasification has been widely simulated and numerous models are available, either structural3-5 or volumetric.6 A structural model allows for explicitly including changes in the internal structure with conversion. An inherent problem of these models is the need for extensive experimental input data.7 In the volumetric approach, however, this problem is overcome by using experimental correlations and effective properties.6,7 Several volumetric models have analyzed CO2-char gasification with a computational approach.6,8-14 Some of them have analyzed the diffusional effects of single particles, predicting major limitations under some operating conditions. Bliek et al.8 presented a model under conditions in which intraparticle mass transfer was rate-controlling and intraparticle heat transfer was negligible. The main controlling parameters during the gasification of single particles were determined to be gasification temperature, particle size, and diffusive permeability of the particle. Chen et al.9 formulated a nonisothermal model for gasification of single char particles in an environment of a gas mixture commonly occurring in moving-bed reactors. They found significant intraparticle gradients of gas concentrations, temperature, and carbon-conversion profiles at the largest particle size and highest temperature tested. Bandyopadhyay et al.15,16 derived a modified version of the formulation developed by Turkdogan and Vinters14 to explain the intraparticle heat (3) Bathia, S. K.; Perlmutter, D. D. A random pore model for fluidsolid reactions: I. Isothermal, Kinetic control. AIChE J. 1980, 26, 379386. (4) Bathia, S. K.; Perlmutter, D. D. A random pore model for fluidsolid reactions: II. Diffusion and transport effects. AIChE J. 1981, 27, 247254. (5) Szekely, J.; Evans, J. W.; Sohn, H. Y. Gas-Solid Reactions; Academic Press: New York, 1976. (6) Groeneveld, M.; van Swaaij, W. Gasification of Char Particles with CO2 and H2O. Chem. Eng. Sci. 1980, 35, 307-313. (7) Standish, N.; Tanjung, A. F. Gasification of single wood charcoal particles in CO2. Fuel 1988, 67, 666-672. (8) Bliek, A.; Lont, J. C.; Van Swaajj, W. P. M. Gasification of coalderived chars in synthesis gas mixtures under intraparticle mass-transfercontrolled conditions. Chem. Eng. Sci. 1986, 41, 1895. (9) Chen, J. S.; Gunkel, W. W. Modeling and simulation of co-current moving bed gasification reactorsspart I. A nonisothermal particle model. Biomass 1987, 14, 51-72. (10) Dasappa, S.; Paul, P. J.; Mukunda, H. S.; Shrinivasa, U. The gasification of wood-char spheres in CO2-N2 mixtures: analysis and experiments. Chem. Eng. Sci. 1994, 49, 2, 223-232. (11) Hastaolu, M. A.; Karmann, M. G. Modelling of catalytic carbon gasification. Chem. Eng. Sci. 1987, 42, 1121-1130. (12) Haynes, H. W. An improved single particle char gasification model. AIChE J. 1981, 28, 517-521. (13) Srinivas, B.; Amundson, N. R. A Single-Particle Char Gasification Model. AIChE J. 1980, 26, 487-496. (14) Turkdogan, E. T.; Vinters, J. V. Effect of carbon monoxide on the rate of oxidation of charcoal, graphite and coke in carbon dioxide. Carbon 1970, 8, 39-53. (15) Bandyopadhyay, D.; Chakraborti, N.; Ghosh, A. Heat and mass transfer limitations in gasification of carbon by carbon dioxide. Steel Res. 1991, 62, 143-151. (16) Bandyopadhyay, D.; Chakraborti, N.; Ghosh, A. Re-evaluation of heat transfer effects in carbon gasification reaction. Steel Res. 1988, 59, 537-541.

Ollero and VillanueVa

effects that they observed experimentally. Dasappa et al.10 developed a rigorous nonisothermal computational model to explain Ergun’s experiments,17 finding that the observed gasification rate using particle sizes of 5 mm. Nevertheless, the model is too simple and does not make it possible to discern the mechanism responsible for the limitation or take into account the effect on the magnitude of these limitations as the reaction progresses. Moreover, the gas kinetics is first-order. Generally, simple modeling approaches are mostly applicable to isothermal conditions and usually limited for one (or both) of the following reasons: (i) they are only applicable to firstorder kinetics, with respect to gas or solid reactant, and/or (ii) they do not explicitly make allowance for structural changes with reaction. Rafsanjani et al.20 applied a new simple mathematical method for solving gas-solid noncatalytic reactions to predict char activation processes. In their model, they included a term to take into account the variation in the activation energy as the reaction proceeds, but they assumed isothermal conditions and first-order kinetics for the gas reactant. Go´mez-Barea and Ollero23 recently extended the spectrum of applications for existing simplified methods, using an approximate methodology that can accommodate both any general kinetics and any explicitly given intrinsic behavior of the solid structure variation with reaction. Their methodology has been shown to be potentially capable of providing major computational savings while exhibiting very close agreement with the exact (numerical) solution. In this study, the model described by Go´mez-Barea and Ollero23 has been expanded to incorporate external mass- and heat-transfer effects and intraparticle heat effects, both in a very simple manner. Also, a correction for the nonequimolarity of the Boudouard gasification reaction was applied. The model developed and validated in this work has the essential advantage of requiring minor (17) Ergun S. Kinetics of the reaction of carbon dioxide with carbon. Phys. Chem. 1956, 60, 480-485. (18) Ramachandran, P. A. Analytical prediction of conversion-time behaviour of gas-solid noncatalytic reaction. Chem. Eng. Sci. 1983, 38, 1385-1390. (19) Brem, G.; Brouwers, J. J. H. Analytical solutions for nonlinear conversion of a porous solid particle in a gas. Chem. Eng. Sci. 1990, 45, 1905-1924. (20) Rafsanjani, H.; Jamshidi, E.; Rostam-Abadi, M. A new mathematical solution for predicting char activation reactions. Carbon 2002, 40 (8), 11671171. (21) Doraiswamy, L. K.; Sharma, M. M. Heterogeneous Reactions. Analysis, Examples and Reactor Design, Vol. 1; Wiley: New York, 1984. (22) Hawley, M. C.; Boyd, M.; Anderson, C.; DeVera, A. Gasification of wood char and effects of intraparticle transport. Fuel 1983, 62, 213216. (23) Go´mez-Barea, A.; Ollero, P. An approximate method for solving gas-solid noncatalytic reactions. Chem. Eng. Sci. 2006, 61, 3725-3735.

Diffusional Effects in CO2 Gasification. 2

Energy & Fuels, Vol. 20, No. 5, 2006 2213

computations while still reproducing the major physicochemical processes that may affect the observed reaction rate. 3. Theoretical Modeling 3.1. Volumetric Local Reaction Rate. Using the same nomenclature as that given in Part 1,2 the reactivity at any degree of char conversion can be expressed as

R(X) ) f(X)R50

(1)

where f(X), the so-called structural profile, takes into account the changes in the number of active sites during conversion. If we use nth-order kinetics and Arrhenius kinetics for the kinetic constant, the reactivity expression at 50% char conversion can be written as

( )

n K ) A exp R50 ) KpCO 2

E RgT

(2)

If the kinetic experiments are conducted with very fine char that is well-exposed to the reacting gas, both the structural profile and the reference reactivity (R50) can be considered intrinsic. In that case, a volumetric reaction rate at any point in a particle can be defined as follows23

-r (mol/m3/s) ) r(c,T)F(X)

(3)

introduce structural profiles merely as a way to calculate the reactivity at conversions other than that taken as the reference and usually include situations where diffusional effects are appreciable. As a result, they are global or overall structural profiles and not intrinsic. If this is the case, they should not be directly used in the intrinsic local reaction rate, such as that defined in eq 3. This point was recently discussed in detail in refs 1, 23, and 24. Normally, for the experimental determination of the function F(X), it is necessary to assume certain functional forms. In other words, this function must be determined by fitting one or various parameters that best reproduce the experimental variation of reactivity with conversion. Many researchers make the fit using an arbitrary polynomial of X. Nevertheless, in principle, nothing is “banned” a priori; there are some forms that have been derived on the basis of physicochemical arguments that describe the evolution of the solid matter. These provide an “ideal” initial guess for the determination of the function F(X). Examples of this are the random pore model,3 Simons model,25 etc., in the kinetically controlled regime. These models have free parameters, and, therefore, the actual values of the parameters can be evaluated for a given char based on the experimental data by applying the least-squares procedure. However, other models such as the grain model5 or the uniform model26 do not have free parameters in the kinetic regime. For instance, nth-order kinetics, with respect to a gas reactant, and a uniform model, with respect to solid consumption, gives

where r(c,T) and F(X) are defined as

-r ) kcn(1 - X)

R50Fc0 r(c,T) ) Mc

(4a)

F(X) ) f(X)(1 - X)

(4b)

and

The r(c,T) function is the part of the reaction rate that is dependent on gas composition and temperature, whereas F(X) is the function that treats the change in accessible reacting surface at any conversion level. This can be interpreted as the ratio of the reacting surface available at a given conversion to the surface available at the reference conversion levelsi.e., F(X) ) 1 at Xp ) 0.5. The use of eq 3 presumes that the largest length scale that is characteristic of the solid structure is much smaller than the characteristic length that is associated with concentration and temperature gradients. The usefulness of eq 3 is realized if it is further postulated that the particle is composed of small particles or grains where diffusional effects are absent. The latter is a reasonable hypothesis, because these grains have very small particle sizes. At this local scale, intrinsic kinetics should be applicable. Note that the complex problem of surface pore enlargement and development in each of the grains is not explicitly considered, but it is implicitly taken into consideration by the F(X) function. Moreover, the actual size and/or shape of the microparticles or grains are not relevant in our treatment. We only assume that they are small enough. This assumption ensures that diffusional effects are not present within the grains and, thus, intrinsic reactivity can be applied at this local level. In practice, the function F(X) is obtained by performing kinetic experiments, which should be conducted in a kinetic regime. Otherwise, the resulting function F(X) would not be intrinsic. In TGA experiments, for instance, it is a common practice to report this information in the form of an experimental structural profile. However, most of these experimental studies

(5)

where k is the kinetic constant, expressed in units of (mol/m3)1-n/ s. Some examples of functions F(X) for kinetic models under the kinetic regime can be found in ref 23. To summarize, for the estimation of diffusional effects, it is possible to use eq 3 with experimentally determined F(X) and r(c,T) functions. These are dependent mainly on the fuel nature and the conditions in which the char is generated si.e., the heating rate and the pyrolysis temperatures. 3.2. Modeling of Reaction-Diffusion Process Inside a Char Particle. We modeled the process of a reaction with diffusion within an isothermal particle on the basis of a recently developed model.23 This model makes it possible to incorporate a nonlinear chemical reaction rate, expressed in dimensionless form by R(C), and the changes in porous structure during conversion by the specification of F(X). It also allows for changes in the effective diffusivity with reaction through the input of g(X). For a given time τ and particle position z, we can obtain the dimensionless concentration profiles within the particle with this method by solving the following twodimensional set of equations:23

[

C(z) ) C* + (1 - C*) exp -

]

λ(1 - z2) 1 - zh(z) 21 + 2/λ

X(z) ) Θ-1(τR(C(z)))

(6)

(7)

Table 1 displays the values of the parameters involved in eqs 6 and 7 for spherical geometry and nth-order kinetics. For this (24) Gomez-Barea, A.; Ollero, P.; Arjona, R. Reaction-diffusion model of TGA gasification experiments for estimating diffusional effects. Fuel 2005, 84, 1695-1704. (25) Simons, G. A. The unified coal-char reaction. Fuel 1980, 59, 143. (26) Adanez, J.; DeDiego, R. F. Int. Chem. Eng. 1993, 33, 656.



Table 1. Go´ mez-Barea and Ollero Model for nth-Order Kinetics and Spherical Particlesa Equations/Parameters

()

dp φs2 ) 2 M2(X)

) φs

h(z) ) λ)

X

0

dX F(X)

12 n 5 n+1

(

)

1 - exp(-λz) 1 - exp(-λ)

( )

M η 3 GOT

ηGOT ) [M*2 + exp(-aM*2)]-1/2

R(C) )

2r dp

r(c,T) r(cs,Ts)

(10)

k(T) n C k(Ts)

(11)

( ) kcsn cB0

The ratio between the two kinetic constants can be expressed as21

c cs

k(T) 1-C ) exp B(X)γs k(Ts) 1 + B(X)(1 - C)

C) X)1-

[

( ) cB cB0

Method of Solution Given τ; for i ) 1 to N, solve f C(zi) and X(zi) (N ) 5-10, valid for most situations)

R(C) ) Cn exp[ζ(X)(1 - C)]

Input (1) reaction rate: R(C) and F(X) (2) diffusivity: De0 and g(X) Output C(z,τ) and X(z,τ) f Xp(τ), ηiiso(τ), ηR(τ) The nomenclature is the same as that given in ref 23.

case, the value of C* in eq 6 is null. Table 1 includes further details on how to solve the problem with this method. As discussed in ref 23, the model applies to isothermal particles when external mass and heat transfer are not ratelimiting (large Biot heat and mass numbers). These assumptions are probably violated in TGA experiments such as that described in this study. Therefore, the model must be expanded to include intraparticle heat effects, as well as external heat- and masstransfer effects. The extension of the model is developed below. 3.3. Modeling Intraparticle Heat Effects. In principle, the temperature field within the particle can be readily estimated by incorporating the following expression into the method given in section 3.2:

T′ ) 1 + B(X)(1 - C)

]

(12)

where γs is the classical Arrhenius number, which is defined as γs ) E/(RTs). If B(X)(1 - C) is assumed to be small enough, the so-called two-parameter model27 is obtained and the dimensionless reaction rate, R(C), can be expressed as

R(C) ) Cn

a

(9c)

For nth-order kinetics, this leads to

Dimensionless Variables

τ)t

(9b)

(- ∆Hr)De0cs ke0Ts

R(C) )

2

z)

(9a)

where T′ is the dimensionless surface temperature; βi and B(X) are the classical and modified Prater numbers, respectively; and p(X) is a certain function of conversion (the last item is discussed in the Appendix). We can see that the modified Prater number is conversion-dependent. However, this feature does not add any supplementary difficulty, because, as explained in refs 20 and 23, the application of the quantize method allows us to take X as a parameter when integrating the concentration within the particle. Now, given eq 8, the R(C) function must be modified in the non-isothermal case as follows:

xn +2 1

M 3

a)1-

F(X) g(X)

∫

Θ(X) ) M* )

βi )

2G(X)

G(X) )

T Ts

B(X) ) βi p(X)

kcAsn-1 De0

2

T′ )

(8)

in which the following dimensionless parameters and variables are used:

(13)

where ζ(X) ) γsB(X). The nonisothermal model can now be approximately solved, using eq 13 instead of its isothermal counterpart, R(C) ) Cn. For the region of significant diffusion limitation and for small values of ζ(X), typically ζ(X) < 1, the relative error introduced by assuming isothermal operation for nth-order kinetics and any geometry may be estimated from the following expression:27

δt(X) ) 1 -

ζ(X) 2(n + 2)

(14)

For an endothermal reaction, the nonisothermal effectiveness factor can be calculated by means of the following approximate expression:27

ηi(τ) ) ηiso i (τ)δt(Xp(τ))

(15)

in which ηiso i (τ) is the intraparticle effectiveness factor, which is calculated according to the model in section 3.2. 3.4. Modeling External Mass and Heat Transfer. Under pseudo-steady-state conditions, the equations that govern the nonisothermal mass- and heat-transfer problem for general nthorder kinetics are (27) Satterfield, C. N. Heterogeneous Catalysis in Industrial Practice; McGraw-Hill: New York, 1991.


Energy & Fuels, Vol. 20, No. 5, 2006 2215

kG (c - cs) ) kacsn Le 0

(16)

h (-R)(-∆HR) ) (T0 - Ts) Le

(17)

(-R) )

The apparent kinetic constant (ka) lumps the intraparticle diffusion-reaction problem, that is, ka ) ηik(Ts), where ηi is the intraparticle effectiveness factor, defined as the ratio of the observed total reaction rate to the total reaction rate when the concentration of the reactant is equal to that at the surface. Mathematically, this is expressed as

(-R)

ηi )

k(Ts)cns

(18)

Similarly, the global effectiveness factor (ηG) is the ratio of the observed reaction rate and the reaction rate when the concentration of the reactant is equal to that in the bulk gas:

(-R)

ηG )

k(T0)cn0

(19)

The external effectiveness factor (ηe) is defined as the ratio between the intrinsic reaction rates evaluated at the surface and in bulk-gas conditions: n

k(Ts)cs

ηe )

n

)

k(T0)c0

(-R) ηik(T0)c0n

(20)

Equations 19 and 20 readily give

ηG ) ηeηi

(21)

Detailed derivation of the relations presented previously can be found in ref 28. For the isothermal case, an approximate explicit solution for ηe can be derived as29-31

ηe )

1 [{(1 - n)Da ηi}

1/n

+ 1]n + nDaηi

(for 0 < n < 1) (22)

where Da is the Damko¨hler number, which represents the ratio of the maximum diffusional rate (cs ≈ c0) to the diffusioncontrolled reaction rate (cs ≈ 0), which is

Da )

k(T0) n-1 c kG/Le 0

(23)

By considering the definition given for the Damko¨hler number Da in eq 23, we can transform eqs 16 and 17 into the following dimensionless equation set:32 (28) Go´mez-Barea, A.; Ollero, P.; Leckner, B. Mass transport effects during measurements of gas-solid reaction kinetics in a fluidised bed. Accepted for publication in Chem. Eng. Sci., 2006. (29) Fo¨rtsch, D.; Schnell, U.; Hein, K. R. G. The effect of boundary layer diffusion on the overall rate of heterogeneous reactions. Chem. Eng. Sci. 2001, 56, 4439-4443. (30) Frank-Kamenetskii, D. A. Diffusion and Heat Exchange in Chemical Kinetics; Princeton University Press: Princeton, NJ, 1955. (31) Froment, G. F.; Bischoff, K. B. Chemical Reactor Analysis and Design, 2nd Edition; Wiley: New York, 1990. (32) Carberry, J. J.; Kulkarni, A. A. The nonisothermal catalytic effectiveness factor for monolith supported catalysts. J. Catal. 1973, 31, 41-50.

[ (

ηe ) (1 - Daηiηe)n exp -γ0

1 -1 1 + βeDa ηiηe

)]

(24)

T′s ) 1 + βeDa ηiηe

(25)

Cs ) 1 - Da ηiηe

(26)

where the βe parameter is defined as

βe )

(

)

(-∆HR)c0

(1 + ξ)FcpT0

Le-2/3

(27)

To derive eqs 24 and 25, the heat- and mass-transport analogy, jD ) jh, was invoked, where jD and jh are the Chilton-Colburn j-factors for mass and heat transfer, respectively. However, the coefficient h in eq 17 should take into account the radiant contribution of the heat transfer, i.e. h)hrd+hcv, as computationally verified in ref 24. Thus, the coefficient ξ in eq 27 is an approximate correction that is made to include the radiation effects. This factor is defined as the ratio between the “radiative” and convective film coefficients:

ξ)

hrd hcv

(28)

where the radiative film coefficient is defined according to the approximation

hrd ) 4cσT03

(29)

This approximation is assessed in Figure 1, where ξ is displayed as a function of T0, taking ∆T/T0 as a parameter. The solid lines are the “exact” solution resulting from calculating the radiative flux by its formal expression, which is

ξexact )

cσ(T04 - Ts4) hcv(T0 - Ts)

(30)

whereas the asterisk-marked curve is calculated by considering the approximation for hrd given by eq 29. The value of ∆T (the temperature drop between the bulk gas and the char surface) is estimated to be 20-40 °C for drawing the figure. This hypothesis is verified in the section on results. As seen in Figure 1, the approximation is very good for the entire range of bulkgas temperatures. Moreover, a linear relation can be derived between ξ and T0 as follows:

ξ ) 0.66 + 0.0016(T0 - 1073)

(31)

This approximation avoids making the correction factor dependent on the surface temperature and allows for the direct estimation of the radiative effects under different bulk-gas conditions. With regard to char emissivity, experimental measurements of carbonaceous ash have shown values as low as 0.4, as well as a marked variation with temperature.33 However, the absorptivity of a carbonaceous porous surface has been shown to be very high and, consequently, its emissivity should be close to 1. In the course of an actual test, the surface of a char particle is made up of ash and carbonaceous material and the proportion between both materials changes as the reaction (33) Elliot, M. A. Chemistry of Coal Utilization (Second supplementary volume); Wiley: New York, 1981.



noted, we assume the same structural profile for the full range of operating conditions. In the present case, eq 36 turns out to be almost a straight line, and we can therefore choose an easier polynomial. This point is discussed further in the Results and Discussion section. 3.6. Model Outputs. We used the model developed previously for the computation of particle conversion as well as the internal and external effectiveness factors and, therefore, the overall effectiveness factor. The particle conversion is calculated with

Xp(τ) ) 3

∫01Xz2 dz

(37)

The intraparticle effectiveness, previously defined by eq 18, is computed according to: Figure 1. Radiant approximation.

progresses. Thus, the uncertainty about this parameter is very high. In the absence of further information, we used a value of 0.5. 3.5. Model Inputs. 3.5.1. Rate Law. We use the char reactivity at 50% conversion that was determined in Part 1 for the experiments that have been conducted at 0.060 mm (eq 8 in Part 1).2

[

(

R50 (g/g/s) ) 1993 exp -

14260 pCO20.4 T

)]

(32)

This expression can be transformed in terms of the model’s needs as

r(c,T) ((mol/m3)/s) ) kcn

(33)

with a rate constant given by

k ((mol/m3)1-n/s) )

[c

B0(RgT)

n

(

× 1993 exp -

14260 T

)] (34a)

and

n ) 0.4

(34b)

3.5.2. Structural Profile. We realized that, strictly, the structural profile is dependent on the reaction temperature and the gas composition.34 However, for the sake of simplicity, this model uses an average structural profile derived from the first set of experiments, i.e., without diffusional limitations.34 The structural profile used is

f(X) ) 66X5 - 140X4 + 110X3 - 37X2 + 6.3X - 0.09 (35) For this 5th-order polynomial, no analytical solution can be found for Θ(X). To avoid numerical integration in every run, we used the best-fit (in the sense of the least-squares) 5th-order polynomial of the numerical integration of Θ(X). The result is the following polynomial:

Θ(X) ) - 18.9X5 + 53X4 - 51X3 + 18.9X2 + 0.61X 0.06 (36) which can be used for all the computations, because, as already (34) Ollero, P.; Serrera, A.; Arjona, R.; Alcantarilla, S. The CO2 gasification kinetics of olive waste. Biomass Bioenergy 2003, 24, 151161.

∫01 [R(C)F(X)z2] dz ηi(τ) ) ∫01 [F(X)z2] dz

(38)

The second effectiveness factor is defined as the ratio of the actual conversion, Xp, at any instant to the conversion that would be observed if there were no intraparticle gradients:

ηR(τ) )

Xkp(τ) Xint p (τ)

(39)

where the superscripts “k” and “int” represent experiments performed with diffusional effects and without (“intrinsic”), respectively. The first effectiveness factor, which is defined as ηi, is the classical definition, as used by Wen,35 and is merely the direct translation from catalytic systems. The effectiveness factor ηR, which was originally introduced by Ramachandran,18 is especially useful for directly discerning diffusional effects in experimental Xp vs τ curves. Finally, the global effectiveness factor is computed according to eq 21. 3.7. Method of Solution and Numerical Treatment. Several steps must be taken to determine a solution: (1) Assume an internal effectiveness factor, ηi. (2) With ηi given in step (1) and from bulk-gas data, calculate the particle effectiveness factor external effectiveness factor ηe, using eq 24, and, thus, Cs and T′s, with eqs 26 and 25, respectively. A first guess for the solution of eq 24 is that calculated according to the isothermal case (eq 22). (3) Given Cs and T′s from step (2), solve the intraparticle nonisothermal problem (eqs 6 and 7) to calculate the intraparticle effectiveness factor, ηi. Use Table 1 to solve this problem. The reaction rate must be expressed in the form given in eq 3, with the nonisothermal kinetics presented in eq 13. The remaining inputs are defined in section 3.5. In a simpler approach, it is possible to estimate the isothermal effectiveness factor and make the correction given in eq 15. Both procedures give similar results for the experiments simulated in this work. However, the first method is more general and, because it is very simple to implement, it is recommended for other situations. (4) Check to determine whether the difference between the intraparticle effectiveness factor ηi value assumed in step (1) and that calculated in step (3) is small enough. If so, go to step (5). If not, go to step (2) and repeat the procedure in steps (2)(4) with a new value of ηi. The selection of a new candidate can be implemented with a Newton-Raphson algorithm. (35) Wen, C. Y. Noncatalyitic heterogeneous solid fluid reactions. Ind. Eng. Chem. 1968, 60, 34-54.


Figure 2. Experimental (symbols, +) and theoretical predictions (solid line) of intrinsic Xp vs t curves, for the case where xCO2 ) 0.20.

(5) Generate outputs (particle conversion and effectiveness factors). The system described by eqs 6 and 7 is solved by dividing the radial coordinate (z) into N + 1 points [zi ) (i - 1)∆z, where i ) 1, ..., N + 1 and ∆z ) 1/(N - 1)]. The solution (Ci, Xi) is determined by solving (N + 1) systems of two nonlinear equations, which is done by applying the Newton-Raphson method. A value of N ) 5 was chosen for all the calculations in this work. Normally no more than three iterations were necessary to achieve an absolute error of

Diffusional Effects in CO2 Gasification Experiments with Single

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