Particle-scale modeling of asphaltene pyrolysis in thermal plasma

Fuel 175 (2016) 294–301

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Particle-scale modeling of asphaltene pyrolysis in thermal plasma Yan Cheng, Tianyang Li, Binhang Yan, Yi Cheng ⇑ Department of Chemical Engineering, Tsinghua University, Beijing 100084, PR China

a r t i c l e

i n f o

Article history: Received 23 October 2015 Received in revised form 10 February 2016 Accepted 17 February 2016 Available online 23 February 2016 Keywords: Asphaltene Thermal plasma Pyrolysis Particle-scale modeling Acetylene

a b s t r a c t Thermal plasma technique is proposed as a potential approach to pyrolyze asphaltenes to useful chemicals at ultrahigh temperature. Thermal plasma pyrolysis experiments show that acetylene is the major gaseous product with concentration up to 45 wt.%, together with some hydrogen and methane as well as the solids residue as by-products. To predict such a millisecond process, a particle-scale heat transfer model coupled with a modified competition kinetic model has been proposed, in which the heat exchange between the plasma gas and the particles as well as the heat transfer inside the particles are taken into account. Typical results indicate that the pyrolysis temperature would remarkably increase the heating rate. The heating rate would exceed 107 K/s at 2000 K in H2 thermal plasma and the devolatilization process would complete in 0.4 ms for 50 lm asphaltene particles, H2 thermal plasma provides better heating exchange efficiency between the plasma and particles than He, N2 or Ar. Linear relationship is found between the heating rate and the devolatilization time under bi-logarithm coordinates at a certain temperature, regardless of the plasma gas compositions and pulverized sub-millimeter particle size, which gives powerful guidance for process optimization of the asphaltene pyrolysis. Ó 2016 Elsevier Ltd. All rights reserved.

1. Introduction Various upgrading methods [1–3] have been developed for the utilization of heavy crude oil [1] owing to the limited light crude oil resources. Solvent deasphalting method [3] is one of the techniques that upgrades crude oil to higher quality feedstock by separating the asphaltene and metals from the light components in the crude oil. The deasphalted oil (DAO) is employed as a lighter raw material to be further converted to valuable products by hydrogenation [4] or catalytic cracking [5]. Consequently, a large quantity of deoiled asphaltenes (DOAs) are generated at the same time as by-products [6]. However, those DOAs can be hardly transformed to valuable chemicals using conventional methods in petrochemical industry. Nowadays, efficient utilization of asphaltenes becomes crucial to improve the profit of the entire solvent deasphalting technology. Generally, asphaltene contains more than 60 wt.% volatile matter, which makes it possible to acquire valuable products via thermochemical processes. Pyrolysis via thermal plasma has been developed as an alternative technology to convert various kinds of carbonaceous materials, including coal [7], tar [8] and paraffins [9]. Thermal plasma has the features of high enthalpy and high ⇑ Corresponding author. Tel.: +86 10 62794468; fax: +86 10 62772051. E-mail address: [email protected] (Y. Cheng). http://dx.doi.org/10.1016/j.fuel.2016.02.053 0016-2361/Ó 2016 Elsevier Ltd. All rights reserved.

temperature, and thus could realize efficient conversion of asphaltene to gas products and carbon black. Devolatilization is the first step of most thermochemical conversion processes, and is directly controlled by the heating conditions. Studying the devolatilization characteristics and particle heating history will benefit the understanding and optimization of the process. However, in thermal plasma pyrolysis, the heating rate reaches 104–107 K/s [10], which is significantly different from traditional thermochemical processes (e.g., 102–104 K/s in combustion and gasification processes [11]). Owing to the extreme heating conditions, direct measurement of asphaltene particle behavior in thermal plasma reactor is difficult. Therefore, it is necessary to propose a model approach to theoretically analyze the heating procedure as well as the devolatilization process in detail. In the present work, experiments of thermogravimetric analysis and thermal plasma pyrolysis were carried out to acquire the basic thermochemical properties of asphaltene firstly. Then a particlescale model, coupled with the heat transfer model and the asphaltene devolatilization kinetic model, was introduced to investigate the pyrolysis behaviors of asphaltene under thermal plasma conditions. The heat exchange between the thermal plasma and the particle and the heat transfer inside the particles were considered. The effects of atmosphere temperature, plasma gas compositions and the particle size were discussed in detail. Further comparison

Y. Cheng et al. / Fuel 175 (2016) 294–301

was made to investigate the relationship between the heating rate and the devolatilization time.

295

3. Mathematical model 3.1. Kinetic model

2. Experimental 2.1. Asphaltene sample The asphaltene particle sample, provided by Chinese University of Petroleum, was made by the spray granulation process [6,12]. The basic characterization analysis of the sample is shown in Table 1. The bulk density and porosity were measured by mercury porosimetry analysis (Micromeritics Corp., AutoPore IV 9500). The ultimate analysis was made via two elemental analyzers (PE Corp., PE-2400 II for oxygen test; EAI Corp., CE-440 for carbon and hydrogen test; Changsha Kaiyuan Instruments Co., Ltd., 5E-8S II for sulfur test). The proximate analysis was tested with a muffle furnace (CCRI, GW-300C) and a loft drier (Huamao Corp., DGF25012C) with the guidance of standard ASTM D7582-15. The size distribution of the sample particles was measured in a laser particle size analyzer (Malvern Corp., Mastersizer 2000).

The established kinetic model to describe the asphaltene pyrolysis is mainly based on a two-step model [13]. However, this model has the disadvantage that the kinetic parameters should be refitted according to different heating conditions [13], and thus cannot be used to predict devolatilization behavior under unknown conditions. Instead of the two-step model, a modified competition model has been developed in the present work. The competition model, proposed by Kobayashi et al. [14], assumed that raw material pyrolysis process was governed by several first order competition reactions, and has been proved capable in describing the devolatilization of both coal [14] and biomass [15]. The first order assumption has also been proved successful in asphaltene kinetics investigation by Trejo et al. [16] and Ancheyta and Speight [17]. The mass loss rate and the volatile generation rate could be expressed as

dm ¼  2.2. Thermal plasma experiments

X

ki mdt

ð1Þ

ci ki mdt

ð2Þ

i¼1;2;

A 10-kilowatt (maximum) lab-scale device was used to carry out thermal plasma pyrolysis experiments of asphaltene. The device includes a plasma torch, a mixing section, a reaction chamber, a quenching section as well as a separator [7]. An insulated gate bipolar transistor (IGBT) rectifier power with a maximum output of 10 kW was used as the power source. The plasma torch consists of a cerium–tungsten cathode and a copper anode insulated with a PTFE pad. The working gases of plasma torch are Ar and H2. Asphaltene particles are delivered with a screw feeder and conveyed with Ar. The feedstock is then mixed with the plasma jet in the mixing section. Devolatilization takes place in the stainlesssteel reaction chamber. To avoid secondary decomposition of product gas, additional Ar gas is fed into the quenching zone. Products then flow into the separation unit to separate the coke and carbon black from gas phase. Gas samples are collected to be further analyzed via an online mass spectrometer (AMETEK Corp., DCU 200) and two gas chromatography analyzers (Tianmei Corp., GC 7890II; Shimadzu Corp., GC 2014). Light gases (including small molecule hydrocarbons and inorganic species like CO and H2) are analyzed respectively: CH4, C2H2, C2H4 and C2H6 are detected by GChydrogen flame ionization detector (FID) with a capillary column (HP-plot Al2O3, Agilent Corp.), while H2, Ar, CO and CO2 are detected by GC-thermal conductivity detector (TCD) with a packed column (TDX-01, Lanzhou Chemical Engineering Research Institute).

Table 1 Basic characterization analysis of asphaltene samples. Unit

Value

Bulk density (293 K) Porosity Volume average diameter

g/cm3 % lm

1.37 41.2 55.6

Ultimate analysis (Md) Carbon Hydrogen Oxygen Nitrogen Sulfur

wt.%

Proximate analysis (Mar) Volatile Moisture Fixed carbon Ash

wt.%

dv ¼

X

i¼1;2;

ki ¼ Ai expfEi =RTg

ð3Þ

where m and v represent the remained weight and generated volatile matter during pyrolysis, being function of time t; c represents the volatile yield; k represents the reaction rate constant, and is described in the form of Arrhenius law; A and E represent the pre-exponential factor and activation energy. It is widely accepted that the weight loss of asphaltene is mainly attributed to two clusters of reactions: the devolatilization of small molecule groups (methyl, ethyl, carboxyl, etc.) and the cracking reactions [18]. So, the number of the competing reactions in the present model is set as two to simplify the problem. On the other hand, original samples undergoes continuous changes in molecular structures during the pyrolysis process, which means the reactions are much easier to take place at the beginning and harder at the end of the devolatilization. So, a modification on the activation energy is added to reflect the effect:

E ¼ f ðE0 ; DE; mÞ ¼ E0 þ DE  gð1  mÞ

ð4Þ

In Eq. (4), E is a function of initial value E0, increment DE and sample remaining mass fraction m; g represents the modification function using ‘1  m’ as a controlling variable, and is regressed from the thermogravimetric data. The parameters of the competition model can be fitted from thermogravimetric analysis at several different heating rates, and then further verified at extreme heating conditions. In thermogravimetric analysis, the standard error for TGA temperature reading is ±0.25 K and for weight reading is ±10 lg.

3.2. Heat transfer model inside a single particle 88.0 8.4 1.6 1.6 0 66.7 0 32.8 0.5

In the literature, it is usually assumed to be isothermal inside the particles. However, there is obvious temperature gradient inside the particles in thermal plasma pyrolysis, which would consequently affect the devolatilization process [10,19]. Therefore, it is important to employ a particle-scale model in order to investigate the heating procedure and the devolatilization process of asphaltene particles during thermal plasma pyrolysis. Some basic assumptions are made according to the process characteristics:

296

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(1) The porous asphaltene particle is assumed to be an ideal sphere, with one-dimensional distribution of physical properties along radius. (2) The volatiles in the pore and the solid phase are assumed to be in local thermal equilibrium inside the particle, and the convective heat transfer inside the particle caused by volatile evolution is ignored. (3) The effect of the volatiles on the surrounding atmosphere is ignored, considering that the sample mass is small compared with the purge flow. A particle-scale coal devolatilization model was proposed in our previous work [10], and the particle-scale heat transfer model was proved to be capable to describe particle heating procedures in thermal plasma. So, the heat transfer-model is coupled with the asphaltene pyrolysis kinetic model in this work. The model parameters are modified specially for asphaltene properties, which are listed in Table1. The radiation coefficients (ep = 0.8), the specific heat capacity (c = 1.2 kJ kg1 K1) and thermal conductivity of asphaltene solid phase (k = 0.2 W m2 K1) are also updated according to the feedstock.

Fig. 1. TG experimental data and calculated results from modified competition model.

Table 3 Working conditions and experimental results of thermal plasma pyrolysis experiments.

4. Results and discussion 4.1. Asphaltene devolatilization kinetic model

case-1

The parameters of asphaltene decomposition kinetic model were firstly fitted using TG experimental data. The kinetic parameters are shown in Table 2. The correlation equation for activation energy was also regressed, and the function g in Eq. (4) was chosen as the inverse function of gamma distribution function Ga (0.15,0.5). Gamma distribution is a statistical function used in mathematical analysis. Fig. 1 shows the TG experimental data and results calculated from modified competition model. The calculated results agree well with the TG data from 300 K to 1300 K at different heating rates. TG experiments provide relatively slow heating rates (e.g., 0.1– 2 K/s), which ensures steady and homogeneous heating processes. However, under thermal plasma pyrolysis conditions, the heating rate is much faster (e.g., 105–107 K/s), and the pyrolysis temperature is higher than that of TG analysis. It is assumed that the modified competition kinetic model can be used at higher temperature range, which is proved reasonable in the following discussion. 4.2. Model validation with thermal plasma pyrolysis experiments Asphaltene pyrolysis experiments were carried out in the labscale thermal plasma device. Table 3 gives the experimental results under different working conditions. The pyrolysis temperature was estimated based on the gas components using thermodynamic analysis. The estimation method for the temperature via thermodynamic analysis can be referred to the work of Wu et al. [20,21]. Table 3 shows that the gas yield from asphaltene pyrolysis is 57.9% in Ar/H2 thermal plasma, and the acetylene yield is up to 45.2%. The Ar/H2 thermal plasma shows better pyrolysis capability than pure Ar plasma. The pyrolysis temperature in the lab-scale device is estimated to be 1520–1625 K under different conditions, which is much higher than that of TG experiments. The residence

Table 2 Kinetic parameters of modified competition model. Cluster

c (%)

A (s1)

E (kJ/mol)

DE (kJ/mol)

1st 2nd

63.2 64.1

40.5 8.1  1014

62.7 235.1

70 120

Plasma torch Current Power Cathod gas Ar H2 Feeding Ar Asphaltene Reactor size Diameter Length Yield C2H2 CH4 H2 Light gas Residue

A kW

case-2

case-3

70 1.68

70 1.33

70 1.309

L/min

8.93 0.97

10.19 0

10.19 0

L/min g/min

1.01 0.4

1.01 0.34

1.01 0.46

cm

1 5

1 5

1 5

45.2 1.5 3.1 57.9 32.1

22.7 1.4 2.9 35.6 64.4

16.9 2 2.1 27.1 72.9

92.6 2.8 0.0 3.5 1.1

92.6 4.2 0.00 2.4 0.8

91.7 5.4 0.0 2.2 0.7

wt.%

Residue composition C wt.% H O N Ash

Temperature from thermodynamic analysis T K 1625.00

1565.00

1520.00

time of asphaltene particle, which is estimated from the reactor size and the gas properties, varies from 3.4 to 3.6 ms under different operating conditions. Since the kinetic parameters were acquired under TG heating rates, we first compared the simulated results with plasma pyrolysis experimental results to verify the kinetics feasibility under rapid heating conditions. Fig. 2 compares the simulated results using the particle-scale model with the experimental results. It can be seen that the simulated results match the experimental data well. The results prove that the particle-scale model can be employed to simulate the pyrolysis performance under thermal plasma heating conditions, and the kinetic model could give credible prediction up to 1625 K. Element analysis of the residue in case-1 was carried out. The residue consists of carbon (95.1 wt.

Y. Cheng et al. / Fuel 175 (2016) 294–301

Fig. 2. Comparison of simulation results with thermal plasma pyrolysis experiments.

%), hydrogen (3.8 wt.%) and oxygen (1.1 wt.%). As the residue mainly consists of solid carbon, it can hardly release a large amount of volatiles any more, which supports the assumption that the kinetic model could also provide convincing predictions at a slightly higher temperature. 4.3. Effect of reaction temperature Comparing case-2 and case-3, the reaction temperature significantly affects the volatile yield of asphaltene pyrolysis. Fig. 3 give the average temperatures and devolatilization processes at differ-

297

ent temperatures varying from 1300 K to 2000 K in both Ar thermal plasma and H2 thermal plasma. The average temperature refers to the volume average temperature inside the particle, and the particle diameter is set as 50 lm in diameter. The average temperature rises faster with the increase of the reaction temperature. In H2 thermal plasma, the final temperature could be reached within 1.5 ms for all cases, while the heating rates are much lower in Ar thermal plasma. The devolatilization time decreases with the increase of the reaction temperature. In H2 thermal plasma, the devolatilization time is 1.5 ms at 1300 K, while the devolatilization time is 0.4 ms at 2000 K. In contrast, for Ar thermal plasma, the asphaltene just begins to devolatilize at 4 ms at 1300 K. The devolatilization time at 1600 K in Ar thermal plasma is about 5 ms, which is longer than the residence times of case-2 and case-3. This explains the low volatile yield of the experimental results. To sum up, the pyrolysis temperature would significantly affect the heating rate, and consequently affect the devolatilization process.

4.4. Effect of gas composition of thermal plasma It can be justified from the comparison between case-1 and case2 that the gas yield increases remarkably in Ar/H2 thermal plasma. One reason is that H2 would increase the plasma voltage and thus increase the input power. The other reason may lie in that H2 would enlarge the gas-solid heat convection coefficient, and then strengthen the heat exchange between the plasma and the asphaltene particles. Fig. 4 gives the simulation results of different Ar/H2 mole ratio at 2000 K. With the increase of H2 mole fraction in plasma, the heating rate becomes faster, so does the devolatilization

Fig. 3. Effects of atmosphere temperature on the particle average temperature and devolatilization in (a, b) Ar and (c, d) H2 thermal plasma (particle diameter: 50 lm).

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Fig. 4. Effect of Ar/H2 ratio in thermal plasma on the pyrolysis process (particle diameter: 50 lm; atmosphere temperature: 2000 K).

Fig. 5. Effects of surrounding gas composition on the pyrolysis process (particle diameter: 50 lm; atmosphere temperature: 2000 K).

rate. These results prove that H2 thermal plasma would strengthen the heat transfer. This is one of the reasons that H2 thermal plasma is usually used in industrial applications. This phenomenon provides the possibility to control the devolatilization time by means of adjusting the Ar/H2 ratio to match the residence time in the designed reactor. Fig. 5 compares the volatile yields and average temperatures of different gas compositions in the plasma, including H2, He, N2 and Ar. It is assumed that these gas species only affect the heat exchange, and do not affect the devolatilization kinetics owing to non-oxidizing environment. It can be concluded from Fig. 5(b) that H2 thermal plasma has the best heat transfer effect and the fastest heating rate, followed by He plasma, while N2 and Ar are apparently worse, which mainly owes to the differences of heat convection coefficient. Meanwhile, H2 and He showed similar trend, which indicated that H2 and He have similar heat transfer coefficients. The heat exchange effect directly causes the difference in devolatilization time, as shown in Fig. 5(a). Fig. 6 shows the temperature distribution inside the particle in different pyrolysis atmosphere. The outer layer of the particle is heated up first, followed by the inner part. The particle temperature gradient inside particle is smaller for N2 and Ar thermal plasma. For example, the maximum temperature difference inside the particle for Ar thermal plasma is 123 K. On the contrary, the particle surface is heated up rapidly for H2 and He thermal plasma, while the maximum temperature difference inside the particle

reaches 809 K for H2 thermal plasma. This is due to the interplay of heating rate and inner heat transfer resistance inside particle. 4.5. Effect of particle size The diameter of the asphaltene particle is another important factor that affects the heat transfer. Fig. 7 gives the devolatilization processes of different particle sizes varying from 25 lm to 200 lm, and Fig. 8 gives the temperature distributions inside the particles. The devolatilization time increases significantly with the particle size. It can be concluded from Fig. 7(a) and (b) that the heating rate in Ar thermal plasma is slower than that of H2 thermal plasma. In other words, H2 thermal plasma could be used for a wider range of particle sizes efficiently. From the viewpoint of temperature gradient inside the particle, the temperature difference inside the particle is small in Ar thermal plasma. That is to say, the local temperature at different radius is similar, which is shown in Fig. 8. So the devolatilization process along the radius begins and ends at a close time, which explains the large slope of the curve in Fig. 8(a). For H2 thermal plasma, the heating rate is faster and the temperature gradient is larger even for 25 lm particles. The particle surface is heated up rapidly, and thus starts to react before the inner part does, shown in Fig. 8, which gives explanation for the quick beginning of devolatilization even for 200 lm particles in Fig. 7(b). However, with the increase of particle size, the heat transfer resistance inside the particle would play a more important

Y. Cheng et al. / Fuel 175 (2016) 294–301

299

Fig. 6. Effects of surrounding gas composition on the temperature distribution inside the particle (particle diameter: 50 lm; atmosphere temperature: 2000 K).

Fig. 7. Effects of particle size on the pyrolysis process under (a) Ar and (b) H2 thermal plasma.

role. So it takes time to heat up the inner part of the particles, which results in the increasing slope in Fig. 7(b). 4.6. Relationship between the heating rate and the devolatilization time In the above discussion, there are many factors that affect the heating rate of the particle, and thus affect the devolatilization time. However, for the pyrolysis at a certain final temperature (e.g., 2000 K), there is a strong correlation between the heating rate and the devolatilization time at different heating conditions. Fig. 9

gives the bi-logarithmic profile of the heating rate and the devolatilization time. Here, the heating rate (a) is defined as the final temperature divided by the time needed for the average temperature to reach the atmosphere temperature; the devolatilization time (s) is defined as the time needed to reach 99% of the final volatile yield. In Fig. 9, there is an obvious linear relationship between the heating rate and the devolatilization time under bi-logarithm coordinates. The relationship is independent of atmosphere components and sub-millimeter particle size, which gives powerful guidance for process control of the asphaltene pyrolysis.

300

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Fig. 8. Effects of particle size on the temperature distribution inside the particle.

Fig. 9. Correlation between the heating rate and devolatilization time.

were further verified with the experimental results of thermal plasma analysis, and showed the capacity to be used in extreme heating conditions. Simulations were carried out to investigate the influences of atmosphere temperature, plasma composition and particle size on heating rate and devolatilization time of asphaltene particle. The results showed that, for 50 lm asphaltene particles, the heating rate would exceed 107 K/s at 2000 K in H2 thermal plasma, and the devolatilization would complete in 0.4 ms. The heating rate has a positive correlation with the plasma temperature. The increase of H2 in Ar/H2 thermal plasma would remarkably increase the heating rate and decrease the devolatilization time. From the viewpoint of properties of heating exchange between the plasma phase and particles, H2 thermal plasma is better than He, followed by N2 and Ar. The devolatilization time also increases significantly with the particle size. To summarize, there is a linear relationship between the heating rate and the devolatilization time under bi-logarithm coordinates at a certain final temperature, regardless of the plasma components and sub-millimeter particle size, which gives power guidance for process control of the asphaltene pyrolysis.

5. Conclusion Thermal plasma pyrolysis has been proposed as an alternative thermochemical technology to convert asphaltene to gas products and carbon black. Thermal plasma pyrolysis experiments were carried out to validate the feasibility. Results showed that valuable gas products (including acetylene, hydrogen, methane, etc.) could be obtained via thermal plasma pyrolysis. Based on the nature of pyrolysis process in thermal plasma, a particle-scale model was established in the present work. The particle-scale model used a modified competition model for asphaltene pyrolysis kinetics, and took into consideration the heat exchange between the particle and the plasma as well as the heat transfer inside the particle. The kinetic parameters were fitted from thermogravimetric analysis. The kinetic model and the particle-scale heat transfer model

Acknowledgements Financial supports from the National Basic Research Program of China (973 Program no. 2012CB720301), the National Science and Technology Key Supporting Project (2013BAF08B04), PetroChina Innovation Foundation (2013D-5006-0508) are acknowledged.

Appendix A. Modeling of the heat transfer inside a single particle In the present work, the temperature gradient inside a particle was considered to simulate the heating process of asphaltene particle. With the assumptions and simplifications in the paper, the

Y. Cheng et al. / Fuel 175 (2016) 294–301

governing equation of heat transfer inside a particle could be expressed as:

ðqp cp Þ

  @T p ðr; tÞ 1 @ @T p ðr; tÞ ¼ 2 kp r 2  Dr H  kðr; tÞ @t r @r @r

ðA:1Þ

where

qp cp ¼ uqv cp;v þ ð1  uÞqs cp;s kp ¼ ukv þ ð1  uÞks

ðA:2Þ ðA:3Þ

Tp(r,t) and k(r,t) represent the local temperature (K) of the particle and its pyrolysis rate (kg/m3 s), which are functions of radial position r and time t; q represents the density; c (=1.2 kJ kg1 K1) and k (=0.2 W m2 K1) represent the specific heat capacity and thermal conductivity of solid phase, and the effect of temperature on these two properties is ignored; u stands for the porosity of the particle. DrH is the heat of pyrolysis reactions (J/kg), and is calculated from the DTA results. The subscripts p, v and s represent particle, volatile, and solid phase, respectively. The initial condition is set as following:

T p ðr; 0Þ ¼ T p;0 ;

u ¼ u0 ð0 6 r 6 RÞ

ðA:4Þ

where R is the particle radius; the subscript 0 represents the initial condition. The boundary conditions are given as following:

(

4pR2 kp @T p @r

@T p @r

¼ 4pR2 hðT g  T w Þh þ r0 ð4pR2 Þðeg T 4g  ep T 4w Þ r ¼ R

¼0

r¼0 ðA:5Þ

The subscript w represents the property at the surface of the particle, while g for the atmosphere gas; r0 is the Stefan–Boltzmann constant; eg and ep (=0.8) are the radiation coefficients of the gas and asphaltene particle; h is the gas-particle heat transfer coefficient, and h is a factor related to the effect of internal volatiles release on heat transfer. The gas-particle heat transfer coefficient is calculated from the Nusselt number Nu, which is estimated as a function of operating conditions and material properties [21]:

h ¼ Nukg =dp "

#0:5  0:42   qg lg 0:52 2 0:8 Pr w Nu ¼ 2 1 þ 0:63Reg  Prg C Prg qw lw Re ¼ qdp jus j=l Pr ¼ lcP =k

C¼

1  ðHw =Hg Þ1:14 1  ðHw =Hg Þ2

ðA:6Þ ðA:7Þ ðA:8Þ ðA:9Þ ðA:10Þ

In Eqs. (A.6)–(A.8), dp refers to the diameter of the particle. Re and Pr are Reynolds number and Prandtl number, respectively. These two factors are updated as a function of the gas properties in every time step. H and l are the specific enthalpy and viscosity of the surrounding gas, respectively. usl is the gas-particle slip velocity, which is set to be constant in this work. The subscripts w and g denotes the particle surface and surrounding gas, respectively. The factor h, reported by Spalding [22], represents the heat transfer resistance due to the fast release of volatiles:

B eB  1   cp;m dmv B¼ 2pdp km dt

h¼

301

ðA:11Þ ðA:12Þ

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