A Systematic Approach to Determining the Kinetics of the Combustion of Biomass Char in a Fluidised Bed Reactor

Abstract

The aim of this work was to investigate the combustion of biochar in a fluidised bed and determine the intrinsic kinetic parameters for combustion: pre-exponential constant $A_i$ and activation energy $E_i$. When analysing the rates of reaction, Regimes I, II and III were demonstrated, with values for the activation energy of 155, 57 and 9 kJ/mol, respectively, when combustion was limited by different factors: intrinsic kinetics, intraparticle and external mass transport phenomena. These mass transport phenomena were decoupled from a set of ‘apparent’ kinetics incorporating effectiveness factors, which we used as a starting point in the determination of the intrinsic kinetic parameters. We also investigated a simple approach to model the evolution of the char structure over the course of oxidation using an empirical function, $f\left(X\right)$, fitted with an O(7) polynomial. We then reassessed the division into three combustion regimes by exploring the changes in $f\left(X\right)$ and the intraparticle effectiveness factor that occurred upon increasing the combustion temperature. Overall, we demonstrate that experiments in a fluidised bed can be used to determine biochar kinetics in a simplified but trustworthy way.

1. Introduction

Biomass is the most accessible and affordable fuel worldwide. The primary benefit of biomass compared to fossil-derived fuels is the potential to achieve net-negative CO₂ emission when utilised for bioenergy with carbon capture and storage (BECCS) [1]. Among the available technologies, fluidised bed combustors are particularly well fitted for the thermal conversion of biomass because of their operational flexibility. They offer fast heat transfer, resulting in a uniform temperature in the bed, while long residence times ensure the high conversion of solid fuels [2]. Combustion in fluidised beds has been already studied at length, with the majority of work focused on fossil fuels rather than biomass.

A key parameter in designing a combustor is information on the kinetics of combustion, especially useful when aiming for efficient operation at relatively low operating temperatures, e.g., in heating applications. To determine the kinetics of combustion, various methods have been presented in the literature. Conveniently, thermogravimetric analysers (TGA) inform us about the rates of mass loss and provide means for quick experimentation; thus, they are often employed in combustion research. However, the measured reactivity of chars in TGA differs from that in fluidised beds, arising from differences in gas delivery, heating rates and possible char restructuring [3]. Difficult to eliminate are the effects from various transport phenomena; thus, the results reported in the literature predominantly provide the apparent kinetics of char combustion without isolating the intrinsic reaction kinetics. To assess the true kinetics, careful experimental design is required to ensure purely kinetic limitations. For example, Morin et al. [4] investigated beech combustion at low temperatures (603 to 643 K), yet mass transfer remained important because of the large particles of char that were used (cylinders of 4 mm diameter and 9 mm length).

As recently reviewed by Kwong, reaching a set of trustworthy kinetic parameters for a very reactive fuel, such as biochar, is not a trivial task [1]. This paper aims to present a systematic approach to determining the intrinsic kinetics of the combustion of biomass char directly from fluidised beds.

To include the effects of different transport phenomena in a fluidised bed, the apparent rate of reaction—defined here as the observed rate of char consumption per unit mass of the fed char and thus in g s⁻¹—can be expressed as

$$r={\displaystyle \frac{dX}{dt}}={\eta }_{ip}{\eta }_e{\eta }_p{k}_i{P}_{{\mathrm{O}}_2}^{n}f\left(X\right)$$

(1)

where $X$ and $P_{{\mathrm{O}}_2}$ are the char conversion and partial pressure of O₂, respectively, while $f\left(X\right)$ is a function connecting the evolution of the pore structure to the char conversion. Then, ${\eta }_{ip}$, ${\eta }_e$, ${\eta }_p$ are effectiveness factors that account for interphase, external and intraparticle mass transfer phenomena, respectively. The interphase mass transfer refers to the transport of O₂ from gas bubbles to the particulate phase in the fluidised bed. The mass transfer of O₂ from the bulk of the particulate phase to the reactive surface of the char particle is referred to here as external mass transfer, whereas intraparticle mass transfer concerns the transport of O₂ from the external surface of the char particle through its pores to the site on the particle surface where combustion occurs.

The parameter $k_i$ in Equation (1) represents the reaction constant, describing the intrinsic kinetics of combustion—here, in bar⁻ⁿ s⁻¹. Here, the dependence of the apparent rate, $r$, on the partial pressure of O₂ is assumed in the form of a power law, where $n$ in Equation (1) is the reaction order with respect to oxygen—a similar approach to Morin et al. [4]. The evolution of the reactive surface of the char particle during combustion is represented by the function $f\left(X\right)$, which depends on the conversion of the particle, $X$.

For combustion mechanisms in different operating conditions and regimes, we provided an overview in our recent review paper [1]. Multiple mechanistic models were revised by Hurt and Calo for coal chars. McMurchie et al. from our group looked into the combustion of char from oak wood, assessing the order of reaction $n$ to infer about the global mechanism and the rate-determining elementary steps [5]. A good level of discussion on the Langmuir–Hinshelwood approach to explain combustion was also provided by Morin et al. [4]. The mechanism of combustion by itself is a substantial and complex topic. In contrast, our goal in this paper is to focus on the intricacies of determining global kinetic parameters of biochar combustion in fluidised beds, presenting a step-by-step approach to decoupling the effect of transport phenomena from the experimentally observed rates. This aims to provide guidance for the extraction of information about the intrinsic kinetics of combustion directly from fluidised bed experiments.

2. Materials and Methods

Particles of biochar, created from mixed sources of hardwood, were purchased from Soilfixer (UK). Granules of biochar (<8 mm diameter) were crushed and sieved to different ranges, e.g., 200–250, 250–300, 300–355, 355–422, 422–500 and 500–600 µm. The fuel was used in our previous study [6], where the results of the proximate and ultimate analysis, BET area and BJH pore size are presented. Silica sand (David Ball Group Ltd., Cambridge, UK, Fraction D) was sieved to 200–250 µm, washed in deionised water and dried for 48 h at 393 K before being used as the bed material in the fluidised bed reactor.

The combustion experiments were performed in a bubbling fluidised bed housed in a quartz reactor (i.d. 30 mm), with a porous disk (grade 1) located 110 mm above the gas inlet. The disk’s function was to distribute the gas and support the bed material. The quartz reactor was filled with ~30 g of silica sand and was heated electrically by a tubular furnace. The temperature was raised to the desired setpoint as measured by a K-type thermocouple, which was inserted into the reactor with the tip located ~1 cm above the porous disk. The outlet gas from the reactor was sampled continuously with a pump (1 L min⁻¹ at NTP) and passed through a drying tube of CaCl₂, and then directed to an ABB EL3020 analyser equipped with an NDIR cell to measure the concentrations of CO and CO₂. A lid was placed on top of the quartz tube to prevent air ingress to the reactor, as also described elsewhere [6,7].

Gases with different concentrations of O₂ (5, 11, 21 vol% O₂) in N₂ were used to fluidise the bed with $U/U_{mf}$~3, where the superficial, minimum fluidisation velocity, $U_{mf}$, was determined experimentally at each experimental temperature. To study char combustion, batches of biochar—3 to 40 mg—were dropped into the reactor at a temperature between 723 and 1123 K. From preliminary experiments, a minimum temperature of 673 K and mass of 3 mg were required to obtain CO and CO₂ profiles that were distinguishable from the background noise. The offset of combustion, i.e., $t=0$, was defined as the moment when the measured concentration of CO₂ increased above 0.01 vol%, while the endpoint, i.e., $t=t_{burnout}$, was achieved once the CO₂ reading decreased below 0.01 vol%.

A summary of the conditions used in the experimental campaigns to investigate different transport phenomena is presented in

Table 1. Experimental campaigns conducted to discern mass transfer and reaction kinetics.

Campaign	Temperature (K)	Mass of Char (mg)	Particle Size (µm)	Concentration of O2 (vol%)	Purpose
1	823, 873	6, 10, 14, 19, 21	200–250	5, 11, 21	assessment of the influence of interphase mass transfer on the apparent kinetics
2	773	6, 12	200–255, 250–300, 300–355, 355–422, 422–500, 500–600	21	influence of interparticle (external) mass transfer on the apparent kinetics
3	723, 748, 773, 798, 823, 848, 863, 873, 883, 898, 923, 948, 973, 1023, 1073, 1123	10 1	250–300	5, 21	identification of the combustion regimes, assessment of the intrinsic kinetics of combustion

¹ Doubled and halved masses were also used to study the influence of interphase mass transfer.

. Every experiment was repeated at least once, and only results with a carbon balance above 80% were analysed for kinetics. Consequently, for the experiments at 5 vol% O₂ and lower temperatures, only a reduced number of data were obtained, further contributing to the simplified assessment of Regime I for this O₂ concentration—see Section 3.3.

Assuming that carbon combusts only to CO and CO₂, the mass of the carbon burnt ($m_C$) during an experiment and the conversion at a given time, $X\left(t\right)$, could be calculated from the molar flowrates of the produced CO and CO₂, ${\dot{n}}_{CO,out}$ and ${\dot{n}}_{C{O}_2,out}$:

$$m_C=M_C\underset{t=0}{\overset{t_{burnout}}{{\displaystyle \int }}}{\dot{n}}_{\mathrm{CO},out}+{\dot{n}}_{{\mathrm{CO}}_2,out} dt$$

(2)

$$X\left(t\right)={\displaystyle \frac{M_C{{\displaystyle \int }}_{t=0}^t{\dot{n}}_{\mathrm{CO},out}+{\dot{n}}_{{\mathrm{CO}}_2,out} dt}{m_{\mathrm{C}}}}$$

(3)

where $M_C$ is the molecular mass of carbon (12.01 g mol⁻¹) and $t$ is an arbitrary duration of time, aligned with the time of char combustion and the time of measurement of CO and CO₂ traces ($t\le t_{burnout})$.

The instantaneous rate of reaction, as inferred by the rate of consumption of carbon for a unit mass of biochar, $dX/dt$ in s⁻¹, is determined as

$$r={\displaystyle \frac{dX}{dt}}={\displaystyle \frac{M_C\left({\dot{n}}_{\mathrm{CO},out}+{\dot{n}}_{{\mathrm{CO}}_2,out}\right)}{m_{\mathrm{C}}}}$$

(4)

3. Results

3.1. Influence of Interphase Mass Transfer (Campaign 1)

Fluidised beds can be considered as systems of two interacting phases—(1) a particulate phase, with solid particles fluidising in a portion of the provided gas, and (2) a gas phase, where the rest of the provided gas travels through the particulate phase as bubbles. Solid particles of char combust in the particulate phase, consuming, in the first instance, oxygen from the interstitial gas in this phase. The depletion of O₂ in the particulate phase drives interphase diffusion. If the mass transfer between the two phases is slower than the rate of reaction in the particulate phase, the transport of gaseous O₂ from bubbles to the mixture of particles and interstitial gas (particulate phase) limits the rate of combustion. Consequently, the char particles in the particulate phase are exposed to a different oxygen concentration than the O₂ concentration at the entrance to the reactor. To determine the intrinsic kinetics, first, the observed rate should be decoupled from the interphase mass transfer. To demonstrate the significance of the interphase mass transfer, the mass loads of biochar used in the experiments were varied from 6 to 20 mg, and the results of the instantaneous rate of combustion in 11 vol% O₂ at 823 K are presented in Figure 1a.

No change in the instantaneous rate of combustion was found when varying the mass of biochar, $m_{biochar}$, indicating that the measured rate was independent of the introduced amount of carbon. Thus, we conclude that, as seen in Figure 1a, combustion was not limited by interphase mass transfer. In addition to the experiments carried out at 11 vol% O₂ and 823 K, the instantaneous rate of reaction was also measured for 5 vol% and 873 K, comparing the values at a fixed conversion point of X = 0.1. As presented in Figure 1b, the rate changed very little with the input mass—the apparent trend of some decrease in rate is within the experimental error as seen when comparing the results at 5vol% O₂, 873 K with those at 21vol% O₂, 823 K. Consequently, for the conditions shown in Figure 1, i.e., temperatures of 823–873 K, pO₂ and $m_{biochar}$, the apparent rate of combustion in Equation (1) can be simplified by taking ${\eta }_{\mathrm{ip}}$ = 1.

Learning from Campaign 1, experiments that aimed to determine the kinetics of combustion were performed using 10 mg of biochar (see Campaign 3 in Table 1), noting also the rest of the conditions (temperatures, pO₂) which helped to mitigate the effect of the interphase mass transfer of O₂.

3.2. Influence of External Mass Transfer (Campaign 2)

External mass transfer, the transport of O₂ from the bulk concentration in the particulate phase to the reactive surface of the char particle, limits the rate of combustion when large char particles are used, especially severely at higher temperatures, when the intrinsic kinetics are rapid. Thus, first, combustion experiments at varied particle sizes were conducted. The results of the instantaneous rates of reaction at 773 K, fluidised by air, with varied sizes of biochar particles, d_p, are presented in Figure 2. As seen in Figure 1a, the maximum rates of combustion of the studied biochar align with the early stage of combustion at conversion $X$ below 0.2. Here, instantaneous rates at X = 0.1 are used for the analysis, representing the fastest stages of combustion but also the initial rates when the char diameter is still close to the nominal particle size. When the particle burns in a regime limited by O₂ delivery to the external surface, its diameter decreases with conversion, as in a shrinking-core situation. Thus, taking $X$ for the largest particle size in the burnout process corresponds to the “worst-case scenario”, i.e., a situation most severely affected by external mass transport.

For a single particle, the rate of external mass transfer is proportional to the external surface of the char particle, i.e., $\propto d_p^2$. As the mass of the particle is proportional to $d_p^3$, the rate of combustion per unit mass would be inversely proportional to $d_p$, if the process were solely limited by external mass transfer. However, as shown in Figure 2, the experimental rate of combustion was found to be independent of the particle diameter in the investigated region, indicating strongly that the effect of the external mass transfer had little influence on the rate of combustion.

To further investigate the importance of external mass transfer, we estimated the range of mass transfer rates expected in the presented experiments. The rate of mass transfer of O₂ (molar flowrate) to a single particle of char is $4\pi d_p{k}_g\left(c_B-c_S\right)$, where $c_B$ is the bulk concentration of O₂ and $c_S$ is the concentration of O₂ at the external surface of the char. Assuming a fast reaction and slow O₂ delivery, the surface concentration, $c_S,$ can be taken as 0, i.e., O₂ becoming fully depleted. The external mass transfer coefficient, $k_g$, connects to the Sherwood number, $Sh=k_g{d}_p/{\mathcal{D}}_G$, where ${\mathcal{D}}_G$ is the molecular diffusivity of O₂ in N₂. The Sherwood number, $Sh$, can be estimated from a suitable correlation—for a bubbling fluidised bed, this is provided by Scala [8]:

$$Sh=2{\epsilon }_{mf}+0.70{\left(R{e}_{mf}/{\epsilon }_{mf}\right)}^{1/2}S{c}^{1/3}$$

(5)

where ${\epsilon }_{mf}$ is the voidage of the particulate phase at the onset of fluidisation—here, taken as 0.51, as estimated by fitting the Carman–Kozeny equation to the experimental results. Then, $R{e}_{mf}$ is the Reynolds number based on $U_{mf}$, and $Sc$ is the Schmidt number. The resulting rate of external mass transfer (per mass of biochar) to particles of char differing in size is presented in Figure 2.

As expected, the theoretical rate of mass transfer decreases with the particle size. In the considered conditions, i.e., pO₂ = 0.21 bar and 773 K, the calculated rate of external mass transfer is at least an order of magnitude larger than the rates observed experimentally. Thus, we conclude that the external mass transport in our experiments was not a limiting factor. Consequently, the particle size of the fraction 250–300 μm was deemed suitable for further work in Campaign 3, where we aimed to study the intrinsic kinetics of combustion, as shown in Table 1. Deciding on a particle size and experimental conditions in which the effects of external mass transfer are mitigated enables the setting of ${\eta }_e=1$ in Equation (1).

3.3. Influence of Combustion Temperature (Campaign 3)

The combustion reaction of biochar could proceed under three possible reaction regimes depending on the temperature: Regime I, where combustion is limited by the intrinsic kinetics; Regime II, where combustion is influenced by both the kinetics and mass transfer; and Regime III, where only the mass transfer limits the combustion. Therefore, the intrinsic kinetics can only be extracted directly when the combustion is in Regime I. Alternatively, the assessment necessitates the decoupling of the kinetics from transport phenomena.

To identify the extent of the three regimes, batches of particles of biochar ($d_p$ = 250–300 µm) were combusted, varying the combustion temperature. Results taken as logarithms of the instantaneous rate of reaction (per mass of biochar added) at conversion of 0.1 are plotted in Figure 3.

Three straight lines, each with different gradients because of the difference in the apparent activation energy, are depicted in Figure 3a,b. The decision about which points to classify under each regime was made by maximising the sum of the R² (correlation coefficient) values for the three lines.

As shown in Figure 3a, the transition temperature between Regimes I and II is ~773 K and that between Regimes II and III is ~883 K during combustion in 21 vol% O₂. In Regime II, the gradient is nearly half the gradient of Regime I, as expected from the theoretical findings in the literature [9]. The gradient of Regime III is close to zero.

The transition between different regimes was also observed in experiments carried out in 5 vol% O₂, as shown in Figure 3b. Here, we made a deliberate choice to limit Regime I to the same temperature range as for 21 vol% O₂. This then necessitated the description of Regime I with two temperature points only, as we did not have further results for temperatures between 750 and 798 K. Consequently, our analysis is a showcase and only a first approximation of Regime I at 5 vol% O₂, as the fitted line—needed for parameter estimations—has no associated standard error.

The transition between Regimes II and III, judged solely on the changes in the gradient, occurred at a similar temperature for both the 5 vol% and 21 vol% O₂ experiments. This is again an approximated result. With a lower concentration of O₂, the driving force for external mass transfer is lower; hence, the reaction would have been more readily limited by external mass transfer in the 5 vol% case, with the transition to Regime III occurring at a lower temperature—not clearly observed in Figure 3b.

The kinetic constant of a chemical reaction can be expressed as a function of the temperature, following the Arrhenius equation, i.e., $k_i=A_i\exp \left(-{\displaystyle \frac{E_i}{RT}}\right)$. Substituting $k_i$ in Equation (1) leads to the dependency of the rate of combustion on the temperature. After taking a log on Equation (1) and asserting from the previous results that ${\eta }_e={\eta }_{ip}=1$ (i.e., negligible external and interphase mass transfer), Equation (1) now becomes

$$\ln \left({\displaystyle \frac{dX}{dt}}\right)=-{\displaystyle \frac{E_i}{R}}{\displaystyle \frac{1}{T}}+\ln \left({\eta }_p{A}_i{P}_{O_2}^{n}f\left(X\right)\right)$$

(6)

The gradient and the y-intercept of the lines of best fit in Figure 3a,b provide information on the apparent activation energy, $E_i$, and pre-exponential factor, $A_i$, from the transformation of Equation (1). From Equation (6), the activation energies at different reaction regimes are determined, as presented in

Table 2. Extracted activation energies and apparent pre-exponential factors for different regimes of combustion in different concentrations of O₂, according to Equation (6). The values in parentheses give the estimated parameters within the standard error, assessed from the experimental results. Regime I describes combustion limited by intrinsic kinetics, Regime II corresponds to limitations from intraparticle mass transfer, and Regime III from interparticle mass transfer.

Parameter	Concentration of O2 (vol%)	Regime
		I	II	III
Temperature range (K)	21	<798	798–873	>883
	5	<798	798–883	>898
$E_i$ (kJ mol−1)	21	155 (126, 183)	57.0 (49.0, 65.0)	9.3 (4.2, 14.3)
	5	142 (-) *	57.0 (55.8, 58.1)	29.2 (24.6, 33.8)
$A_i{\eta }_{\mathrm{p}}{P}_{O_2}^{n}f\left(X\right)$ (s−1)	21	4.7 × 108 (2.56 × 106, 1.61 × 1010)	1.26 × 102 (4.02 × 101, 3.97 × 102)	1.78 × 10−1 (9.24 × 10−2, 3.44 × 10−1)
	5	3.11 × 107 (-) *	5.36 × 101 (4.53 × 101, 6.34 × 101)	1.46 × 100 (8.09 × 10−1, 2.64 × 100)

* Assessed based on only two temperature points; thus, the linear fit gives R² = 1. See the text for a discussion.

. Under a kinetically controlled regime, the activation energy for 21 vol% O₂ is slightly higher than in 5 vol%, but both remain within the standard error of the results at 21 vol% O₂. Worth noticing is that the obtained values of E_i are similar to that for coal chars given by Smith, 179.4 kJ mol⁻¹, and often used by the combustion community, also when studying biochar combustion [1].

As mentioned, the values of $E_i$ can be calculated directly from the slopes of the linear fits in Figure 3. In contrast, the ln term in Equation (6), which denotes the intercepts, demonstrates that the pre-exponential factors, $A_i$, depend on ${\eta }_p$ (effectiveness factor for intraparticle mass transfer). As we discuss in Section 4.2, ${\eta }_p$ contains information about the reaction kinetics; thus, the ln term is rich in information that is highly coupled, including the coupling between $A_i$ and $E_i.$ Additionally, because $E_{i }$ is already present in the first term of Equation (6), ($-E_i/RT)$, the two terms (and, in the linear fits, the intercept and slope) are also coupled. Finally, the structure function $f\left(X\right)$ is also coupled with intra-particle mass transfer, ${\eta }_p$, a relation that is indirectly explored in Section 4.1.

4. Discussion

4.1. The Conversion Function, $f\left(X\right)$

To determine the value of $A_i$ in Equation (1), the effect of the conversion function, $f\left(X\right)$, has to be decoupled. In Section 3.3, the rate of combustion was taken at the conversion of 0.1, often close to the maximum rate in a given experiment. Often, a small but arbitrarily selected value of $X$ is used to identify the initial rate of combustion. The goal is to find a rate corresponding to a situation where only a small amount of char has combusted so that the porous structure of the char can be assumed relatively unchanged, so as to then approximate the value of the conversion function as $f\left(X=0\right)=1$. Rates at very low values of X are avoided to mitigate the influence of oxygen adsorbed on the surface of the char, reacting with the fuel upon introduction to the hot fluidised bed [10]. Additionally, the experimental noise is often more pronounced at the start and end of the experiment; thus, unless a good resolution in the detection of the combustion products is ensured, the starting sections of experiments should be avoided.

Taking $X$ = 0.1 in Section 3.3 to establish the combustion regimes was an arbitrary choice. However, from Figure 1a, in a typical experiment, an increase in the rates was observed with X up to a value of ~0.15, followed by a decrease in the rate until complete burnout. The rise in the rate during the initial stage might be attributed to an increase in the char surface area from the gradual improvements in the access to micropores [11], whereas the drop in the rate after the maximum is because of the decrease in the net surface area caused by the enlargement and coalescence of pores [12].

To account for these effects in our work, an $f\left(X\right)$ function was fitted empirically to the experiment at which the particle combusted in Regime I, i.e., in the absence of an intra-particle gradient in the gas concentration. A similar approach was adopted in the literature earlier [13,14,15] when discussing the gasification of coal chars in CO₂. The conversion function, $f\left(X\right)$, was determined empirically as the ratio of the instantaneous rate at conversion $X$ to the initial rate, taken here at $X=0.01:$

$$f\left(X\right)=r\left(X\right)/r\left(X=0.01\right)$$

(7)

Equation (7) can be understood as a “normalised rate”. Examples of such normalised rates from experiments at temperatures between 723 and 923 K in an oxygen concentration of 21 vol% are given in Figure 4a. The normalised rates describe the structural evolution in char particles only if experiments are controlled by the reaction kinetics [13]. Changes in shape indicate a change in the regime [16]. Such changes are expected when oxygen delivery to the reactive sites in the char particle becomes too slow. Then, the effect from the opening of pores and from the evolution of the internal porosity becomes obscured by the diffusional effects, indicating the transition from Regime I (internal control) to Regime II (intraparticle control).

In Figure 4a, the normalise trends show no significant variation in trends between 723 and 773 K, but, with even higher temperatures, the maximum values in the normalised curves shift towards higher values of X. We can, thus, indirectly confirm the temperature points for the transition from Regimes I to II, demonstrating a simple approach to assessing the controlling phenomena in combustion. The temperature range for Regime I in Figure 4a agrees with that in Figure 3a.

To represent $f\left(X\right)$, we performed a multi-order regression in Excel (LINEST function), fitting the polynomials to the normalised rate from an experiment at 723 K, and the results are presented in Figure 4a. Clearly, the higher the order of the model, the more accurate the representation and fit of the $f\left(X\right)$ curve to the shape of the normalised rate. Still, all three models reach high correlation coefficients, R², with, the seventh-order polynomial giving the best overall fit. Taking the experimental noise into account, all three models are deemed satisfactory. The full regression equations for $f\left(X\right)$ are given in the Appendix A.

4.2. Effect of Intraparticle Mass Transfer

Thus far, our simplistic assessment of the effect of intraparticle mass transfer is embodied by the assumption of ${\eta }_p$ inherently being a unity, corresponding to Regime I, where combustion is limited by the intrinsic kinetics. However, this assumption is worth revisiting, as ${\eta }_p=1$ might not be valid, especially close to the transition temperature from Regimes I to II. To decouple the effect of the intraparticle mass transfer, the effectiveness factor, ${\eta }_p$, is calculated from the Thiele modulus, $\varphi$, a dimensionless number based on the ratio of the maximum rate of reaction under a purely kinetic limitation to the rate of diffusional transport through the porous structure of the char particle [17]:

$$\varphi =d_p\sqrt{{\displaystyle \frac{\left(n+1\right)k_i}{2{\mathcal{D}}_{p,E}}}{\displaystyle \frac{{\rho }_C}{m_{r,C}}}RT P_{{\mathrm{O}}_2}^{n-1}}$$

(8)

where ${\mathcal{D}}_{p,E}$ is the effective diffusion coefficient in the pore, defined as the product of the molecular diffusion coefficient with the porosity of the char, $\mathcal{D}\epsilon$. Then, $k_i$ is the intrinsic rate constant to represent the chemical reaction. For a spherical particle, the effectiveness factor for intraparticle mass transfer, ${\eta }_p$, can be calculated as

$${\eta }_p={\displaystyle \frac{f_c}{\varphi }}\left[{\displaystyle \frac{1}{\tanh 3\varphi }}-{\displaystyle \frac{1}{3\varphi }}\right]$$

(9)

where $f_c$ is the correction factor to account for the effect of the reaction order being different from 1, and it is defined as [17]

$$f_c={\left(1+{\displaystyle \frac{1/\sqrt{2}}{2{\varphi }^2+\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2{\varphi }^2$}\right.}}\right)}^{0.5{\left(1-n\right)}^2}$$

(10)

The effectiveness factor, as described in Equation (9), is an implicit function of the intrinsic kinetics; hence, a numerical or iterative approach is needed to decouple the effect of intraparticle diffusion. When evaluating ${\eta }_p$ and $k_i$, the activation energy $E_i$ is taken to be constant, using the values from Table 2 obtained for Regime I; hence, only the pre-exponential factor $A_i$ in $k_i$ is iterated. However, another unknown that adds uncertainty is the order of the reaction with respect to oxygen, $n$. From the literature, the values of $n$ usually range from 0.6 to 1.0 [18]. Here, we take $n$ = 0.8 and represent the connected uncertainty as bounds for the resulting value of ${\eta }_p$, as presented in Figure 5.

The results in Figure 5 consider particles of char (250–300 μm) combusting in 21 vol% O₂, with ${\eta }_p$ describing the influence of intraparticle diffusion at conversion of X = 0.1, i.e., close to the maximum observed rate for experiments in the low-temperature range. Our intention in taking X = 0.1 is to assess the section of the experiment most prone to being limited by intraparticle mass transfer (worst-case scenario), pinpointing the transition between Regimes I and II. In experiments at higher temperatures, the maximum rates shift beyond $X$ of 0.1, but this is less important as, at these conditions, the limitations arise from the external mass transfer, transitioning to or within Regime III. Finally, when calculating ${\eta }_p$ from the intercepts in the Arhenius plots, we also account for the structure evolution function, f(X), taken as the O(7) polynomial from Figure 4.

As shown in Figure 5, ${\eta }_p$ decreases with the combustion temperature because the intrinsic rate of reaction increases more rapidly than the rate of intraparticle mass transport; hence, the latter phenomenon becomes rate-limiting. A non-uniform profile of O₂ within the particles of char is formed at a higher temperature; a lower value of ${\eta }_p$, therefore, is expected. As previously discussed in Section 4.1, the polynomial fitted $f\left(X\right)$ is normalised at $X=0.01$; hence, at $X=0.1$, the value of $f\left(X\right)$ is greater than 1, and, here, f(0.1) = 1.27.

4.3. Intrinsic Kinetics of Combustion

With the effect of the intraparticle mass transfer quantified, the intrinsic kinetics of the combustion are evaluated at Regime I using the proposed O(7) model for $f\left(X\right)$ and the kinetic parameters (pre-exponential factor and activation energy) are presented in

Table 3. Intrinsic kinetic parameters (activation energy and pre-exponential factors) from Regime I at different concentrations of O₂, using different $f\left(X\right)$.

	$\mathit{f}\left(\mathit{X}\right)$ Modelled as O(7) Polynomial; f(0.1) = 1.27
	${\mathit{P}}_{O_2}=\mathbf{0.21}$ bar	${\mathit{P}}_{O_2}=\mathbf{0.05}$ bar
$E_i$ (kJ mol−1)	154.6	141.8
$A_i$ (bar−n s−1)	(2.18 ± 0.06) × 109	(4.33 ± 0.23) × 108

.

The activation energy values $E_i$ found here, as shown in Table 3, are in the range of 140–155 kJ mol⁻¹. For the combustion of lignocellulosic char, Di Blasi [19] summarised that the $E_i$ is in the range between 76 and 229 kJ mol⁻¹, where the activation energy in this work falls. In more recent work, Recari et al. [20] extracted the kinetics of combustion of spruce chars at 723 K in a 5 vol% O₂ environment using a TGA, assuming a shrinking-core model, and found that the activation energy was ~97 kJ mol⁻¹, lower than the value in Table 3. In a fluidised bed, assuming a volumetric model, i.e., setting $f\left(X\right)=1-X$, Morin et al. [4] found that the pre-exponential factor and the activation energy were $\left(8.0\pm 1.3\right)\times {10}^8$ bar⁻ⁿ s⁻¹ and 144.3 kJ mol⁻¹, respectively—similar to the values presented here, regardless of the differences in the fuel type. Similar to the work here, but using a TGA instead of a fluidised bed reactor, Peterson and Brown [21] studied the combustion of biomass char in a wide range of temperatures to establish the reaction regime. In Regime I, the pre-exponential factor and activation energy from the combustion of hardwood were found to be in the range of 3.12 $\times {10}^6$ to $1.08\times {10}^8$ s⁻¹ and 109 to 129 kJ mol⁻¹, respectively, which are also consistent with the values reported in Table 3 in this work.

4.4. Effect of Heat Transfer

When a char particle is combusted in a bed fluidised by air, the temperature of the char particle will increase above the bed temperature. The temperature of the char particle studied here can be estimated by a heat balance at steady state, assuming that (a) the temperature within the char particle is uniform (the validity of this assumption was demonstrated in our previous work [22]); (b) the solid matrix and the gas within the char pores are always in thermal equilibrium; (c) the heat evolved by combustion to the bed temperature is negligible due to the small mass of char, as measured by the thermocouple in the reactor; (d) the contribution of transpiration to the overall heat transfer is small; and (e) CO₂ is the only product from the combustion of char.

As the enthalpy of forming CO₂, $\mathsf{\Delta }{H}_{C{O}_2}$, is greater than the enthalpy of forming CO, $\mathsf{\Delta }{H}_{CO}$, the assumptions above would result in the maximum temperature deviation between the bed $T_b$ and particle temperature $T_p$. The energy balance can be written as

$$-r^{\prime }\mathsf{\Delta }{H}_{C{O}_2}=\pi d_p^{2}h\left(T_p-T_b\right)+\pi d_p^2ϵ\sigma \left(T_p^4-T_b^4\right)$$

(11)

where $r’$ is the rate of combustion for a single particle of char; $ϵ$ is the radiative emissivity of char, taken as 0.9 [23]; and $\sigma$ is the Stefan–Boltzmann constant. The heat transfer coefficient, $h$, is calculated from the Nusselt number, $Nu=h{d}_p/k_{air}$, where $k_{air}$ is the thermal conductivity of air. As the particle sizes of the bed materials and char are approximately the same, the $Nu$ number was calculated using the correlation by Baskakov et al. [24]:

$$Nu={\displaystyle \frac{2}{1-{\left(1-{\epsilon }_{mf}\right)}^{1/3}}}+0.117A{r}^{0.39}P{r}^{0.33}$$

(12)

where $Ar$ and $Pr$ are the Archimedes number and Prandtl number, respectively. For combustion at a bed temperature of 773 K, using the rate of combustion at the conversion of 0.1, the particle temperature was estimated to be ~774 K, only a degree greater than the bed temperature. Therefore, the temperature of the bed particle and char particle could be assumed as the same temperature, and the effect of heat transfer could be neglected.

5. Conclusions

In summary, this study demonstrates a step-by-step approach towards the decoupling of the mass transfer phenomena from the apparent kinetics in order to find the intrinsic kinetic parameters of biochar combustion. The obtained activation energy for combustion in air, E_i = 154 ± 19 kJ mol⁻¹, is similar to the values commonly quoted for coal chars. The value of $E_i$ drops quickly when moving to Regime II (internal transfer limitation), with $E_i$ of 57 ± 8 kJ/mol, and then to Regime III (external transfer limitations), with $E_i$ as low as 9 ± 5 kJ/mol. The obtained intrinsic pre-exponential constant A_i depends on the reaction model and thus is more difficult to compare. Here, we show that A_i is a strong function of the intraparticle effectiveness factor and f(X) and is also strongly coupled to $E_i$. As expected, the combustion of biochar is fast and thus the intrapacticle mass transfer becomes quickly prominent—in our study, already at 798 K. As we propose here, a useful approach when investigating the kinetics of biochar combustion in fluidised beds is to first determine the temperature range for the three combustion regimes, I, II and III, in order to minimise the effects of mass transfer deliberately.