Proceedings of the 7th World Congress on Mechanical, Chemical, and Material Engineering (MCM'21)
Prague, Czech Republic Virtual Conference – August, 2021
Paper No. HTFF 141
DOI: 10.11159/htff21.141

Influence of Surface Temperature Distribution on Reaction Rateof Solid Fuel Char Particles
Ewa Karchniwy1,2, Sławomir Sładek1, Adam Klimanek1, Andrzej Szlęk1,Agnieszka Korus1, Wojciech Adamczyk1

Department of Thermal Engineering, Silesian University of Technology
Konarskiego 22, 44-100 Gliwice, Poland{ewa.karchniwy; slawomir.sladek; adam.klimanek; andrzej.szlek; agnieszka.korus; wojciech.adamczyk}@polsl.pl2
Department of Energy and Process Engineering, Norwegian University of Science and Technology
Kolbjørn Hejes vei 1B, NO-7491 Trondheim, Norway
1

Abstract - In the paper the effect of surface temperature distribution on the reaction rate and conversion of char particles is investigated.
The distribution of temperature at reacting char particles was observed experimentally and was attributed to the non-uniformity ofchemical composition, porosity and shape of particles. Based on the measured temperature variation, the influence of the temperaturenon-uniformity on reaction rate at the particle surface was investigated analytically and numerically using a kinetic-diffusion model. It
was found that the temperature non-uniformity translates into reaction rate variation with a temperature dependent standard deviationthat has two extrema in the considered temperature range. However, the minimum lies in the region of high temperatures, where the bulk
diffusion resistance limits the reaction rate. The effect was also investigated during combustion of char particles in a CFD model of asimple combustion chamber. It was shown that the non-uniform temperature has small but noticeable influence on mean reaction rate
and particle conversion, leading to a larger distribution of conversion of solid fuel particles.
Keywords: reacting particle temperature, solid fuel combustion, char combustion, reaction rate, kinetic-diffusion model

1. Introduction

Coal is still, after oil, a second most important fuel in global primary energy supply. Its share comprises 38% inworldwide electricity generation [1] and 27% of World’s primary energy consumption in 2019, however the contribution
decreased to its lowest level since 2003 [2]. This is attributed to the continuous reduction in coal consumption in the OECD
countries in favor of growth of natural gas consumption and the use of renewables. The demand for coal is, however, still
increasing in many developing countries in Asia and Pacific regions [2]. Other solid fuels, whose share in the energy
consumption is foreseen to be growing in the future, are wastes and biomass. It is therefore important, due to environmental,
economic and energy efficiency reasons, to understand the processes involved in solid fuel conversion. Nowadays, much ofthe research and design of solid fuels combustion and gasification systems is carried out using mathematical modelling. The
modelling tools should be reliable, and their predictions should reflect the investigated processes with satisfactory accuracy.
In numerical modelling of combustion and gasification of pulverized solid fuels, it is usually assumed that the reacting
particles have a single temperature, uniform throughout their volume [3]. As a result, surface reactions occur at the same rateover the entire particle surface. Consequently, also the flow of products of surface chemical reactions, known as Stefan flow,is also symmetric, and there is no net force acting on the particles due to this transport phenomenon. Similarly, in most ofexperimental studies, only a single mean particle surface temperature is measured, even for quite large particles. Such an
approach was applied in the works [4]-[7], where valuable characteristics of O2/N2 and O2/CO2 combustion were provided,
however the temperature distributions at particles’ surfaces were not quantified. Küster et al. [8] applied Raman spectroscopy
and thermal imaging to measure both, the temperature, and species distribution on milimeter-sized particles’ surfaces ingasification conditions, and significant variation of surface temperature was observed in the measured data. The implicationsof this variation were not further studied. Recently Sładek et al. [9] measured surface temperatures of burning pulverizedcoal char (bituminous coal and lignite) particles ( 100 µm) in O2/N2 and O2/CO2 atmospheres using two-color pyrometry
and observed considerable temperature non-uniformity at various combustion temperatures, and a weak dependence on fueltype. It was found that the temperature variation is increasing almost linearly with increasing mean particle temperature. TheHTFF 141-1

temperature non-uniformity was attributed to inhomogeneous composition and structure of the reacting char particles.
The non-uniformity of Stefan flow induced by temperature distribution on particle surface was studied by Karchniwy etal. [10]. It was found that the acceleration of particle due to non-uniform Stefan flow is non-negligible and can influence
the reacting particles’ trajectories, however the effect is small.
In this study we focus on the influence of the particle temperature non-uniformity on reaction rate. First the influence
was examined analytically, and then numerically, by studying the reaction rate and conversion of pulverized solid fuelparticles in a simple combustion chamber.

2. Analysis and Results

The analysis is based on experimental results presented by Sładek et al. [9] and Karchniwy et al. [10], where it was
found that the standard deviation of temperature at the particle surface 𝜎𝜎𝑇𝑇 is a function of mean particle temperature 𝑇𝑇𝑝𝑝 .
The function weakly depends on fuel type (bituminous and lignite coal chars were tested) and the resulting fit to theexperimental data is [10]where 𝑇𝑇𝑝𝑝 should be provided in K.

𝜎𝜎𝑇𝑇𝑝𝑝 = 0.187𝑇𝑇𝑝𝑝 − 110.0

2.1. Influence on the Reaction Rate
In this section the influence of the temperature variation on the reaction rate during solid fuel particle combustion
is presented. The analysis assumes that the solid fuel particle char is burning according to the kinetic-diffusion model,
so that the reaction rate 𝛩𝛩 is given by𝜋𝜋𝑑𝑑𝑝𝑝2 𝑝𝑝𝑜𝑜𝑜𝑜1𝛩𝛩 =𝑚𝑚𝑝𝑝0 1⁄𝑅𝑅𝑑𝑑 + 1⁄𝑅𝑅𝑘𝑘

where 𝑚𝑚𝑝𝑝0 is the initial particle mass, 𝑑𝑑𝑝𝑝 is the particle diameter, 𝑝𝑝𝑜𝑜𝑜𝑜 is the partial pressure of oxidizer, 𝑅𝑅𝑘𝑘 is thechemical reaction rate and 𝑅𝑅𝑑𝑑 is the reaction rate due to diffusion defined as

𝑅𝑅𝑑𝑑 =𝐶𝐶3⁄4
��𝑇𝑇𝑝𝑝 + 𝑇𝑇�⁄2�𝑑𝑑𝑝𝑝

(1)

(2)

(3)

where 𝐶𝐶 is a reaction-dependent constant 𝑇𝑇𝑝𝑝 is the particle temperature and 𝑇𝑇 is the temperature of the bulk gas.
The kinetic rate is expressed as

𝑅𝑅𝑘𝑘 = 𝐴𝐴exp�−𝐸𝐸 ⁄𝑅𝑅 𝑇𝑇𝑝𝑝 �(4)where 𝐴𝐴 is the pre-exponential factor, 𝐸𝐸 is the activation energy, and 𝑅𝑅 is the universal gas constant. The kinetic
rate 𝑅𝑅𝑘𝑘 is the apparent rate, thus the intrinsic reactivity and pore diffusion are already accounted for in parameters 𝐴𝐴 and𝐸𝐸. In order to examine the influence of the particle temperature variation during combustion on the reaction rate thevariance formula is applied, thus the standard deviation of the reaction rate can be expressed as2

𝑑𝑑𝑑𝑑𝜎𝜎𝛩𝛩 = ��𝜎𝜎 �
𝑑𝑑𝑇𝑇𝑝𝑝 𝑇𝑇𝑝𝑝where the derivative of the reaction rate with respect to particle temperature isHTFF 141-2(5)𝐸𝐸 exp�𝐸𝐸/𝑅𝑅𝑇𝑇𝑝𝑝 �3𝑑𝑑𝑝𝑝+27/4𝐴𝐴𝐴𝐴𝑇𝑇𝑝𝑝
8𝐶𝐶�(𝑇𝑇 + 𝑇𝑇)/2�𝑑𝑑Θ 𝜋𝜋𝑑𝑑𝑝𝑝2 𝑝𝑝𝑜𝑜𝑜𝑜𝑝𝑝=2𝑑𝑑𝑇𝑇𝑝𝑝
𝑚𝑚𝑝𝑝0 exp�𝐸𝐸/𝑅𝑅𝑇𝑇 �𝑑𝑑𝑝𝑝
𝑝𝑝�+3/4 �𝐴𝐴𝐶𝐶�(𝑇𝑇 + 𝑇𝑇)/2�(6)𝑝𝑝

In Fig. 1 the standard deviation 𝜎𝜎𝛩𝛩 and the derivative 𝑑𝑑𝑑𝑑⁄𝑑𝑑𝑑𝑑𝑝𝑝 are shown in a function of temperature. Also, the resulting
deviation in the reaction rate is presented. In the calculations the standard deviation of particle temperature, as given by Eq.
(1), was used and the results correspond to combustion in air with 𝐶𝐶= 5.0e-12 sK-3/4, 𝐴𝐴 = 0.002 s/m, 𝐸𝐸 = 7.9e7 J/kmol,𝑑𝑑𝑝𝑝 = 5e-4 m, particle density 𝜌𝜌𝑝𝑝 = 800 kg/m3 and 𝑇𝑇𝑝𝑝 = 𝑇𝑇.

Fig. 1: The effect of temperature on derivative of reaction rate 𝑑𝑑 𝛩𝛩⁄𝑑𝑑𝑑𝑑𝑝𝑝 , standard deviations of the reaction rate 𝜎𝜎𝛩𝛩 and of particle
temperature 𝜎𝜎𝑇𝑇𝑝𝑝 (left) and on the reaction rate 𝛩𝛩 (right).

As can be seen the standard deviation of the reaction rate is a nonmonotonic function and has two extrema in the selected
temperature range. The second extremum is strictly linked with the increasing standard deviation of the particle temperature,
however, it occurs at very high temperatures. The maximum obtained at around 1500 K indicates the highest influence ofthe variation of particle surface temperature variation at that temperature. A reduction of this influence at higher temperatureslies in the nature of the process, as at higher temperatures the diffusion of the reactant to the surface controls the overall
reaction rate and thus the influence of temperature variation is limited. This effect can be also observed in Fig. 2, where thedeviation from the mean is also shown. Due to this deviation in the reaction rate, the distribution of fuel conversion can beaffected. This effect is examined in the next section.
2.2. Influence on Fuel Conversion – CFD ModelIn the next step the influence of the temperature variation on reaction rate and fuel conversion in a combustion chamber
was examined. For that purpose, a CFD model of a simple combustion chamber was developed using ANSYS Fluent code.
Geometry of the domain is presented in Fig. 2.

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Fig. 2: Geometry of the model.

Simulations were performed for various co-flowing gas temperatures and oxygen concentrations in the oxidizerstream. In the CFD model energy conservation equation is computed for each particle. However, since the model
assumes uniform composition and structure of the particles, the resulting temperature is treated as mean temperature 𝑇𝑇𝑝𝑝 .
To incorporate the variation of temperature observed in the experiments, it was accounted for that the surface
temperature of each particle can vary according to a normal distribution centered at 𝑇𝑇𝑝𝑝 with the standard deviation given
by Eq. (1). This modified temperature was then used in the rate expression (Eq. (2)). It was found that considerable
dispersion of carbon conversion 𝐶𝐶𝐶𝐶 occurred when the temperature non-uniformity at the particle surface was includedin the reaction rate computations. In Fig. 3 sample results of such analyses are presented for combustion in air attemperature 𝑇𝑇 = 1273 K. In left panel of Fig. 3 the dispersion of carbon conversion is due to the spread of particles in
the non-uniform flow, where 𝜇𝜇𝐶𝐶𝐶𝐶 is the mean and 𝜎𝜎𝐶𝐶𝐶𝐶 is the standard deviation of the carbon conversion. In the rightpart of the figure, the distribution is increased due to surface temperature non-uniformity. It can be seen, that the standard
deviation is much larger, which will also induce larger distribution of other flow variables in real combustion chambers.

HTFF 141-4

Fig. 3: Carbon conversion 𝐶𝐶𝐶𝐶 for uniform (left) and non-uniform (right) temperature distribution at particle surface.

3. Conclusion

In the paper the effect of temperature distribution at reacting particle surfaces on the reaction rate and carbon conversion
during solid fuel char combustion was investigated. The influence was examined by analysis of the variance in reaction rateinduced by variance in temperature, using the variance formula and applying the kinetic-diffusion model for reaction rate. It
was shown that the standard deviation of the reaction rate is a non-linear function of the mean particle surface temperature,despite the fact, that the deviation in the surface temperature is given by a linear expression. This translates into nonnegligible and temperature dependent distributions of the reaction rate. The influence of the surface temperature distribution
on carbon conversion was then studied by CFD modelling of char particles combustion for a simple combustion chambergeometry. It was observed that a broader spatial dispersion of carbon conversion can be expected when the temperature
nonuniformity is considered. The effect is not strong but noticeable. It is also expected that this broader dispersion willtranslate into redistribution of other flow variables in the combustion chamber.

Acknowledgements

The research has received funding from the Opus 13 Research Programme operated by the Polish National Science
Center in the frame of Project Contract No UMO-2017/25/B/ST8/00957. This support is gratefully acknowledged.

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