Modelling convective heat transfer to non-spherical particles

Accepted Manuscript Modelling convective heat transfer to non-spherical particles

H. Gerhardter, R. Prieler, C. Schluckner, M. Knoll, C. Hochenauer, M. Mühlböck, P. Tomazic, H. Schroettner PII: DOI: Reference:

S0032-5910(18)30935-5 https://doi.org/10.1016/j.powtec.2018.11.031 PTEC 13866

To appear in:

Powder Technology

Received date: Revised date: Accepted date:

25 April 2018 27 September 2018 6 November 2018

Please cite this article as: H. Gerhardter, R. Prieler, C. Schluckner, M. Knoll, C. Hochenauer, M. Mühlböck, P. Tomazic, H. Schroettner , Modelling convective heat transfer to non-spherical particles. Ptec (2018), https://doi.org/10.1016/ j.powtec.2018.11.031

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ACCEPTED MANUSCRIPT Modelling convective heat transfer to non-spherical particles H. Gerhardter a,* [email protected], R. Prieler a [email protected], C. Schluckner a [email protected], [email protected],

Knolla

M. M.

Mühlböckb

[email protected],

C.

[email protected],

Hochenauer a P.

Tomazic b

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[email protected], H. Schroettner c,d [email protected] a

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Graz University of Technology, Institute of Thermal Engineering, Inffeldgasse 25/B, 8010 Graz, Austria

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M. Swarovski Gesellschaft m.b.H., Industriestraße 10, 3300 Amstetten, Austria Graz University of Technology, Institute for Electron Microscopy and Nanoanalysis (FELMI),

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c

Steyrergasse 17/3, 8010 Graz, Austria d

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Graz Centre for Electron Microscopy (ZFE), Steyrergasse 17/3, 8010 Graz, Austria

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*Corresponding author.

ACCEPTED MANUSCRIPT Abstract A powder is a composition of many unique particles in almost all cases. The grains have different shapes and sizes which influence drag and heat transfer, so in multiphase flows, each particle also interacts differently with the continuous phase. In this work, a practical approach to calculate the convective in-flight heating of differently shaped, non-spherical particles, is presented. A model from literature, which uses the same parameters for the calculation of

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particle drag coefficients and Nusselt numbers, was selected. The model parameters were

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determined by simply measuring the settling velocity of multiple particles in still air. A novel method for particle classification, which has already been proven to be essential for precise

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drag calculations of powders with differently shaped grains, was extended for heat transfer calculations. The approach was validated by numerical calculations and experimental data of a

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simple test rig where the particles were heated from room temperature in free fall through hot

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air. After a drop height of 1.7 m, the mean particle temperature was measured with a specially developed device. The particle temperature in the test rig increased in the order of 300 K. The commercial CFD-code Ansys Fluent 17.0 in combination with an Euler-Lagrangian approach for

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multiphase flow calculations was used for the simulations. The numerically and experimentally

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determined particle temperatures agreed very well in the case of spherical and as well for non-

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spherical particles. In comparison to the standard model of the CFD code, which only considers spherical particles, the error in calculating the mean particle temperature was reduced from -

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59 K to just 2 K. Moreover, it was shown that the standard deviation of the particle temperature distribution increases significantly when the different particle shapes are included in the calculation. It was concluded that the particle classification improves the calculation of maximum and minimum particle peak temperatures. The proposed approach is fully independent of the particle type and it can be used to determine the heat transfer characteristics of various different powders. Further, the customized model can be added to nearly every CFDcode on the market and improve the results considerably while computational cost remains low due to the numerically efficient Euler-Lagrangian approach.

ACCEPTED MANUSCRIPT Keywords:

Convective heat transfer, Multiphase flow, Non-spherical particles, Particle

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temperature measurement, Nusselt number calculations

ACCEPTED MANUSCRIPT 1

INTRODUCTION

Models to describe the transfer of heat and momentum between non-spherical particles and a continuous phase in multiphase flows were investigated by numerous researchers. Many different drag models are available in literature, a brief overview can be found in [1]. Most of these models use the particle sphericity or a similar shape factor to incorporate the particle

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shape. In recent years, the results of drag calculations were highly improved by more suitable descriptions of the particle shape [2–5] and the incorporation of the particle orientation. In

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contrast to this, significantly less researchers have published studies on the calculation of heat

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transfer coefficients or Nusselt numbers of non-spherical particles. The limit value for the particle Nusselt number of differently shaped particles was investigated by Wadewitz et al [6].

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They used numerical methods for their research and concluded, that the lower limit of particle Nusselt number depends exponentially on the ratio of the Sauter diameter to the sieve

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diameter. A computational study on heat transfer to non-spherical particles was performed by Richter et al [7]. They used numerical methods to study the influence of particle orientation and

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angle of attack on convective heat transfer. Further, they proposed a formulation for the

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calculation of the particle Nusselt number in dependence of the particle sphericity and crosswise sphericity as the governing particle shape descriptors and reported a maximum error

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below 5% for a cuboidal particle at 𝑅𝑒𝑃 = 250. The crosswise sphericity was defined as the ratio between the particle projection area in flow direction and the cross-sectional area of the volume

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equivalent sphere [7]. Novel formulations for the drag coefficient and Nusselt numbers of spherical, cubic and ellipsoidal particles were proposed in a further work of Richter et al [8]. The influence of porosity on the drag coefficient and the particle Nusselt number was investigated by Wittig et al [9]. They performed numerical simulations on differently packed agglomerates of spherical bodies to model particles with different porosities and developed a formulation for the particle Nusselt number which incorporates the particle porosity. It was concluded that the influence of particle porosity sharply increases with the particle Reynolds number. Considering non-spherical particles has a major impact on the results in combustion modelling of biomass particles. Selecting a suitable drag law for cylindrical biomass particles instead of

ACCEPTED MANUSCRIPT using spherical drag correlations improves the calculation results of flame shapes and influences the gasification characteristics of the fuel particles [10–14]. Similar effects are expected by the authors of the present study if not only drag, but also heat transfer to the nonspherical particles is taken into account.

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MODEL PARAMETERS

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The main emphasis of this work was to determine a valid formulation for the convective heat transfer between non-spherical particles and a gaseous phase and to develop a simple and cost

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efficient procedure for the determination of model parameters. Crushed coal slag particles were

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selected as an appropriate test material. This material was of particular interest, as based on the heat transfer model evaluated in this work, an industrial furnace in-flame spheroidization of

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coal slag particles is to be designed. The proposed methods can be extended to various other kinds of solid particles.

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The basis was a model for the calculation of momentum and convective heat transfer proposed by Kishore and Gu [15]. They investigated heat and momentum transfer between a fluid and

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ellipsoidal particles. Based on numerical calculations, they obtained formulations for the

24∙𝐴𝑅0.49 𝑅𝑒𝑃

∙ (1.05 + 0.152 ∙ 𝑅𝑒𝑃 0.687 ∙ 𝐴𝑅 0.671 )

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𝑐𝐷 =

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calculation of the drag coefficient 𝑐𝐷 and Nusselt number 𝑁𝑢 , given by Eq. 2.1 and Eq. 2.2.

Eq. 2.2

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𝑁𝑢 = 2 ∙ 𝐴𝑅 0.3 + 𝑃𝑟 0.4 (0.4 ∙ 𝑅𝑒𝑃 0.5 𝐴𝑅 0.83 + 0.06 ∙ 𝑅𝑒𝑃 2/3 ∙ 𝐴𝑅 0.1 )

Eq. 2.1

Besides 𝑅𝑒𝑃 and 𝑃𝑟, the particle aspect ratio 𝐴𝑅 was the only model parameter. Kishore and Gu [15] defined the particle aspect ratio as the ratio between the equatorial and polar radii of the particle. It incorporated the influence of the particle shape on heat and momentum transfer. A side view of three randomly picked particles is displayed in Figure 2.1, the image was taken using environmental scanning electron microscopy (ESEM). The right particle had the largest volume but more importantly, the different particle shapes are visible in Figure 2.1. The left particle was very flat and almost flaky while the shape of the right particle was less oblate. It was concluded that a model, that is able to describe the interaction between non-spherical particles and a continuous phase in multiphase flow simulations, has to incorporate a range of

ACCEPTED MANUSCRIPT differently sized and shaped grains instead of just a single shape category with different sizes. Within this study, the particle shape was incorporated by their aspect ratio. The slag particles used for this study were sharp-edged and rough, so their shape was different from that of an ideal ellipsoid but still, the ellipsoid has been proposed as a reasonable simplification for a wide range of particle shapes [16]. Due to the fact that 𝐴𝑅 was not clearly defined for rough and sharp-edged particles, the model parameters were determined

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empirically.

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2.1 Previous work

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Values for 𝐴𝑅 were already evaluated in an earlier work [17] and hence, only a brief overview of the determination of valid model parameters will be given here. The main emphasis of previous

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research was to determine a valid drag formulation for the same particle type. One of the three evaluated models was the drag formulation given by Eq. 2.1. As proposed by Loth [18],

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geometric quantities were not used as an input parameter for drag calculations but instead, 𝐴𝑅 was treated as an empirically determined drag model parameter.

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In the first step, drop velocities of individual particles were measured in order to determine the

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aerodynamic properties of the slag particles, resulting in values between 2.04 m/s and 4.47 m/s. The high variation of the velocity measurements could not sufficiently be explained by

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incorporating the particle sizes between 400 µm and 850 µm. As proposed by Bagheri and

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Bonadonna [4], the particle shape had a large impact on the velocity of freely falling particles. A microscopic analysis of individual particles confirmed the strong deviations between the shapes of individual particles. To incorporate different shapes, four particle classes were created. Each of the four classes was characterized by its individual 𝐴𝑅, resulting in an individual calculated mean drop velocity. 𝐴𝑅 was determined by fitting calculated velocities to the measurements. Higher particle aspect ratios resulted in lower calculated drop velocities. The velocity measurements and calculated mean drop velocities of each particle class are displayed in Figure 2.2. The obtained values for the mass fraction 𝑥𝑖 and mean particle aspect ratio AR of each particle class are given in Table 2.1. Later, this approach and the corresponding drag

ACCEPTED MANUSCRIPT model parameters were validated by experimental work and numerical calculations of an air tunnel separator [17]. The sphericity 𝜓 is the ratio between the surface area of the equal-volume sphere and the actual particle surface area. An air tunnel separator, schematically displayed in Figure 2.3, was used to separate the particles according to their size and shape in order to validate the sphericity of each particle class listed in Table 2.1. After being injected at the top of the air

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tunnel separator, the non-spherical slag particles were deflected by an airstream. Light particles

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with low sphericity were deflected farther downstream, marked by a dotted line in Figure 2.3, while heavy particles with higher sphericity settled in the first few rows, marked by a dashed

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line. The particle sphericity in these sections was determined by dynamic image analysis [19], using a “Retsch Camsizer P4” particle analyzer. According to the numerical results in [17], 68%

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of the particles in the sample from row 1-4 corresponded to class 1 and 32% to class 2, the

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experimentally determined sphericity of this particle sample was determined at 𝜓 = 0.844 which was a very close fit to the generic values given in Table 2.1. The values corresponding to the

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other particle classes were validated in a similar way.

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2.2 Extension for convective heat transfer A literature research revealed no practical approach for precise heat transfer calculations on

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non-spherical particles dispersed in a continuous phase. A numerically efficient way to calculate

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the temperatures of individual particles of a powder was also not found. Since particle classification significantly increased the precision of drag computations [17], the same approach has also been extended to the computation of convective heat transfer. The model parameters given in Table 2.1 were retained within this work whereby the particle Nusselt numbers of non-spherical slag particles were calculated using Eq. 2.2 while Eq. 2.3, proposed by Ranz and Marshall [20], was used in for spherical particles.

𝑁𝑢 = 2 + 0.6 𝑅𝑒𝑃 1/2 𝑃𝑟 1/3

Eq. 2.3

Figure 2.4 shows the particle Nusselt numbers of all four classes as a function of the particle Reynolds number in comparison to a spherical particle. Kishore and Gu [15] found that 𝑐𝐷 and

𝑁𝑢 of an oblate ellipsoid were higher in comparison to a spherical particle and that both values

ACCEPTED MANUSCRIPT increase with 𝐴𝑅. As already stated in Table 2.1, 𝐴𝑅 increases from particle class 1 to particle class 4 and as displayed in Figure 2.4, the same is the case for 𝑁𝑢 .

𝛼=

𝑁𝑢∙𝜆𝐴𝑖𝑟

Eq. 2.4

𝑑 𝑒𝑞,𝑉

𝑄̇ = 𝐴𝑃 ∙ 𝛼 ∙ ∆𝑇

Eq. 2.5

The convective heat transfer coefficient was calculated via Eq. 2.4. For the calculation of the

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convective heat flow rate 𝑄̇ in form of Eq. 2.5, where ∆𝑇 is the temperature difference between the solid particles and the continuous phase, the actual surface area 𝐴𝑃 of the non-spherical

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particle was used. As listed in Table 2.1, the sphericity decreased from particle class 1 to 4

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while 𝐴𝑅 and therefore 𝑁𝑢 increased. Lower sphericities are equivalent to higher particle surface areas involved in heat transfer. Generally, the convective heat transfer is thus

EXPERIMENTAL SETUP

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3

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increased.

For validation purposes, slag particles were dropped through hot air and their mean

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temperature was measured before and after the experiment. The setup is displayed in Figure

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3.1. The test bench consisted of welded stainless steel tubes with an outer diameter of 60 mm and a wall thickness of 2 mm. The lower section was equipped with 80 mm thick thermal

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insulation with a thermal conductivity of 0.25 W/(m∙K). Four thermocouples in the positions

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“TC1” to “TC4” were installed in order to monitor the fluid temperature on the pipe axis. The thermocouples type K were 1 mm in diameter and in tolerance class 1 according to [21]. Hot air was provided by two heat guns. The particles were placed in a conical reservoir with a 5mm bore at the bottom. As in an hourglass, the particles were fed to the test bench evenly and without any oscillations. The particle mass flow was independent of the head of material in the reservoir [22]. The bed in the reservoir additionally provided a seal against ambient air. By measuring the air temperatures 𝑇1 and 𝑇2 directly after the nozzles and examining the electrical mean input 𝑃1 and 𝑃2 of each heat gun, the air mass flow rates 𝑚̇1 and 𝑚̇2 were calculated via Eq. 3.1. Here, 𝑇0 is the ambient temperature. Values for the temperature dependent specific heat capacity of air 𝑐𝑝,𝐴𝑖𝑟 (𝑇) were taken from [23].

ACCEPTED MANUSCRIPT 𝑚̇1,2 = 𝑐

𝑃1,2

Eq. 3.1

𝑝,𝐴𝑖𝑟(𝑇 )∙(𝑇1,2 −𝑇0 )

The measured data of both heat guns and the resulting mass flow rates and air temperatures are given in Table 3.1. The steady state condition in three different working points was investigated. No particles were injected in the first working point, further referred to as “WP1”. The air temperature measurements were compared to numerical results in order to validate the

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material data for the thermal insulation and the boundary conditions used for numerical work given in section 4.1 and Table 3.1. In the second working point, further referred to as “WP2”,

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spherical slag beads were used as a test material while in the third working point “WP3”, the

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non-spherical slag particles were injected. Drag and heat transfer on spherical particles were already investigated extensively [20,24] so that data collected in WP2 was used for further

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validation of the multiphase model. The boundary conditions of the particle inlet are given in Table 3.1. The mean diameter and spread parameter given in Table 3.1 are the parameters of a

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Rosin-Rammler distribution [25,26]. The cumulative particle size distributions and the corresponding Rosin-Rammler data fittings are displayed in Figure 3.2. The sieve curves of the

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two particle types differed significantly, the spherical particles were much finer. However, both

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powders were produced with raw material from the same batch, so the slag particles had the same composition and, subsequently, also the same material properties.

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3.1 Particle temperature measurement

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The particles escaped from the test rig at the air outlet, displayed in Figure 3.1. The particle mean temperature was monitored at this position. Based on a concept proposed by Wu et al [27], a very simple but effective measurement device was designed. As displayed in Figure 3.3, the particles escaped the test rig in co-flow with the hot air. The air exited through a 5 mm wide radial gap into the environment and the particles were collected in a stainless steel funnel. A thermocouple type K with a sheath diameter of 1 mm was installed on the thermally insulated bottom of the funnel. The small thermocouple had a very fast response and only a low thermal mass. Temperature measurements were performed according to the procedure proposed by Wu et al [28]. The transient temperature signal of the thermocouple installed within the funnel across a period of 300 s is displayed in Figure 3.4. The temperature reading increased while

ACCEPTED MANUSCRIPT filling the chamber, covering the thermocouple with particles. This part of the cycle is marked by the points 1 and 2 in Figure 3.4. After the peak temperature was reached at point 2, a temperature drop in the range of 1 K - 3 K due to heat loss indicated a full chamber. The chamber was emptied quickly which resulted in the temperature drop between the points 2 and 3 in Figure 3.4 and the device was put back in position to begin another cycle. The measuring device was simply emptied by hand, the exact positioning was ensured by a jig. The repeated

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filling and emptying of the device compensated for its thermal mass, bringing it close to thermal

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equilibrium with the particles. The procedure was repeated until 15 peak temperature readings differed less than +/-4 K from each other. The highest value of the 15 measured peak

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temperatures was considered as the mean particle temperature, marked by point 4 in Figure 3.4. This point was considered a valid measurement because the heat loss of the measuring

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device and its thermal mass tend to result in too low a particle temperature.

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3.2 Slag properties

The temperature dependent material properties of air as the continuous phase were already

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investigated extensively, the values used within this work are listed in [23]. In contrast to this,

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the properties of slags depend on their chemical composition [29,30] which is affected by the composition of fired coal [31]. An overview of models for the calculation of slag properties as a

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function of their chemical composition is available in [32]. To avoid deviations due to insufficient

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material data, measurements of the slag density and its specific heat capacity were performed at different temperatures. Based on the work of Aineto et al [29], differential thermal analysis with sapphire as the reference material was used to determine the specific heat capacity. The particle density at ambient temperature was determined with a helium pycnometer, the Archimedean double bob method [33] was used for measurements at 1473 K and 1673 K. Polynomial and linear data fittings were used to incorporate the measurements in numerical calculations. The coefficient of determination of both curves was above 0.99, measurements and data fittings for the density and specific heat capacity of boiler slag are displayed in Figure 3.5.

ACCEPTED MANUSCRIPT Even though the convective heat transfer to the particles was investigated, particle radiation was considered in the numerical calculations. A value of 0.83 for the emissivity of boiler slag was reported [34]. In order to clarify the low influence of the thermal radiation on the heating of the slag particles, a case study was conducted where particle radiation was neglected. The impact on the results was low, further details can be found in section 0.

NUMERICAL MODEL

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To validate the model for convective heat transfer onto the particles, experimental and

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numerical data was evaluated. The commercial CFD-code Ansys Fluent 17.0 was used for

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numerical simulations of the test rig. Here, the determination of any modelling errors and other

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deviations which might influence further work was crucial.

4.1 Numerical grid and boundary conditions

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Three different numerical grids were created in order to ensure a mesh-independent solution. The upper section of each three-dimensional mesh is displayed in Figure 4.1. In case of mesh

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2, tetrahedral elements were used to generate a numerical grid of the upper section of the test

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bench, including the particle and air inlets. The thermally insulated part of the test bench, as well as the thermal insulation itself, were meshed with wedge elements. Mesh 2 was converted

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to polyhedral in order to create mesh 1. Mesh 3 was a hexahedral mesh with significantly higher

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resolution than the other two grids. The mesh metrics are listed in Table 4.1. The mesh independence was checked in operation without particles, as well as with non-spherical particles. Here, the applicability of the numerical model on coarse grids was evaluated. When simulating industrial plants, it is not always possible to produce a high quality hexahedral mesh, which is generally preferable. Being able to use tetrahedrons or polyhedrons and still obtain a solution with sufficient quality offers a reduction in the number of elements and thus reduces the computing time. Both air inlets were modelled as a “mass flow inlet”, the nominal mass flow rates and air temperatures are given in Table 3.1. A turbulent intensity of 1% and the pipe diameter as the hydraulic diameter were set as the turbulent boundary conditions.

ACCEPTED MANUSCRIPT A convective heat transfer coefficient of 5 W∙m-2K-1 and a free stream temperature of 298 K was applied to the outer walls in order to incorporate heat loss due to convection. Especially in the upper section of the test bench, where no thermal insulation was installed, a significant amount of heat was lost due to radiation. For the calculation of radiative losses, the emissivity of the blank stainless steel walls was set to 0.5 [35]. The thermal insulation was covered by sheet metal and a layer of clear, heat resistant silicone lacquer, the surface emissivity was set to 0.7

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[35]. The external radiation temperature was set to 298 K.

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4.2 Modelling of fluid flow and turbulence

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A pressure-based, coupled solver [36] was used to solve the Reynolds Avereged Navies Stokes equations (RANS) in a numerically efficient manner. The Flow Courant Number was set to 20.

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The Reynolds Stress Model [37] was used for turbulence modelling. The laminar sublayer near the walls was not resolved by mesh 1 and 2, a standard wall function was applied to model

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turbulence in the near-wall region [36]. The dimensionless wall distance of the first cell layer was set to 𝑦 + ≈ 30. In the calculations based on mesh 3, the dimensionless wall distance was

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significantly lower, so the option "enhanced wall treatment" was chosen for modelling the near-

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wall region. The “Least Squares Cell Based”-method was used for the calculation of gradients, the “Body Force Weighted”-scheme for the spatial discretization of pressure. A “Second Order

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Upwind”-scheme was used for the spatial discretization of all the other variables.

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4.3 Radiation modelling Even though the convective heat transfer characteristics of the particles were investigated within this paper, thermal radiation was considered. The discrete ordinates (DO) radiation model [38,39] was used to solve the radiative transfer equation (RTE). A spatial discretization of 4x4 was used for every octant, so the RTE was solved in 128 directions.

4.4 Multiphase modelling The discrete phase model (DPM) was used to incorporate the interaction between the fluid and the particles in the numerical simulations. Per default, this model has already been implemented in Ansys Fluent 17.0 [36]. The main advantage of the model is that the particles do not need to

ACCEPTED MANUSCRIPT be resolved by the mesh, but are considered to be discrete mass points, thus reducing computation times. A two-way coupling was used to calculate the transfer of heat and momentum between the fluid and the particles. 𝑑𝑢 ⃗𝑝 𝑑𝑡 𝜏𝑟 =

=

𝑢 ⃗ −𝑢 ⃗𝑝

+

𝜏𝑟

2 𝜌𝑃 𝑑𝑒𝑞,𝑉

18𝜇 𝐴𝑖𝑟

∙

𝑔 ∙ (𝜌𝑃 − 𝜌𝐴𝑖𝑟 )

Eq. 4.1

𝜌𝑃 24

Eq. 4.2

𝑐𝐷 ∙ 𝑅𝑒𝑃

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Particle motion was calculated in a Lagrangian reference frame by Eq. 4.1 [36] the particle

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relaxation time 𝜏𝑟 is given by Eq. 4.2. Drag calculations were performed with Eq. 2.1, the model

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was customized according to [17]. In case of spherical particles, the drag law of Morsi and Alexander [24] was applied. The effects of turbulence on particle motion were incorporated by a

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stochastic tracking approach [40]. In Ansys Fluent 17.0, this model is referred to as "Discrete Random Walk" [36]. d𝑇𝑃 dt

= 𝛼𝐴𝑃 ( 𝑇𝐹𝑙𝑢𝑖𝑑 − 𝑇𝑃 ) + 𝜀𝑃 𝐴𝑃 𝜎(𝑇4𝑅𝑎𝑑 − 𝑇4𝑃 )

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𝑚𝑃 𝑐𝑝,𝑃

Eq. 4.3

The heating of the inert slag particles was calculated by Eq. 4.3. The first term on the right hand

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side of the equation represents the convective heat transfer. The heat transfer coefficient 𝛼 was

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calculated as outlined in section 2.2 for non-spherical particles and according to Ranz and

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Marshall [20] in the case of spherical particles. Radiative heat transfer to the particle was taken into account by the second term on the right hand side of Eq. 4.3.

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The particle size distribution was divided into 15 discrete diameter classes. Every DPM iteration, the tracks of n=101400 particles were calculated. Each one of the calculated tracks represents a share 𝑚̇ 𝑖 on the total particle mass flow rate 𝑚̇ 𝑃 . The magnitude of 𝑚̇ 𝑖 depends on the diameter of the tracked particle and the proportion of the diameter class in the particle size distribution. At the air outlet, the mean particle temperature was measured and compared with the mass flow-weighted mean particle temperature 𝑇𝑃 from numerical simulations. It was calculated by Eq. 4.4 where 𝑇𝑖 is the temperature of each individual particle. 𝑛

𝑇𝑃 = ∑ 𝑖=1

𝑚̇ 𝑖 𝑚̇ 𝑃

∙ 𝑇𝑖

Eq. 4.4

ACCEPTED MANUSCRIPT For the validity of the multiphase model used, there are two limits: First, the model is only valid if particle-particle interactions are negligible. This assumption is valid for a particle volume fraction of less than 0.1% [41]. Secondly, Eq. 4.3 is only valid for thermally thin particles. A particle is considered to be thermally thin if the condition given by Eq. 4.5 is satisfied [42]. The temperature and composition of the slag dependent thermal conductivity was set to 𝜆𝑃 = 1 𝑊/(𝑚 ∙ 𝐾) [43]. < 0.1

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𝐴𝑃 ∙ 𝜆𝑃

5

Eq. 4.5

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𝛼 ∙ 𝑉𝑃

RESULTS AND DISCUSSION

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𝐵𝑖 =

Temperature data of all three working points listed in section 3 was logged throughout the

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experiment, the data log is displayed in Figure 5.1.

Data from WP1 was used to validate the numerical model for single-phase flow, assuring that

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the wall heat losses and material properties of the thermal insulation were correctly implemented in the calculations. In WP2, spherical slag particles were injected in order to

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validate the multiphase model and evaluate the material properties of the slag particles. Finally,

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in WP3, the heating characteristics of non-spherical slag particles were investigated in application of the validated numerical model for multiphase flow.

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In the first step, the test rig was operated without particle injection in steady state in WP1. The

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temperature readings of the thermocouples TC3 and TC4, displayed in Figure 3.1, were compared to the numerically obtained values. The measured air temperatures and the corresponding limiting deviations according to [21] are compared with the numerical results in Table 5.1. In general, the measured fluid temperatures were close to the numerical results. The maximum deviation of 4 K between the results based on mesh 1 and 3 the measurements in position TC4 appeared acceptable and the numerical model for the single-phase flow was considered to be valid. The calculated fluid temperature profiles in WP1 are displayed in Figure 5.2. The fluid temperature was monitored throughout WP2 and WP3 to ensure a steady state during the experiment. Measured and calculated fluid temperatures in WP2 and WP3 were in very

ACCEPTED MANUSCRIPT good agreement with the numerical results, the maximum deviation was still at 4K in position TC4. In WP2, the mean temperature of spherical slag particles was investigated in order to validate the multiphase model, ensure correct material properties and boundary conditions. Calculations were performed on the basis of mesh 2. The measured mean particle temperature was 578 K, the numerically calculated value was 577 K when radiative particle heating was considered. Neglecting the radiative heat transfer to the particles resulted in a calculated

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particle temperature of 575 K. The small difference of 2 K between the numerically obtained

the particles were heated by convection almost exclusively.

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values with and without the consideration of radiative particle heating proved the statement, that

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Finally, the heating characteristics of non-spherical particles were evaluated in WP3. As listed in Table 5.2, a calculation with the assumption of spherical particles was performed and compared

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to the model for non-spherical particles as described in section 2. The presented numerical

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results were obtained on the basis of mesh 2, the results based on the other grids were in the range +/- 2K which clearly indicated a mesh independent solution. The Biot number and the particle volume concentration were also checked, the results are displayed in Figure 5.3. Right

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after the particle inlet, where the particle volume fraction exceeds 0.01, the particles were

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dispersed in the flow field. The condition lined out in section 4.4, according to which the volume

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concentration of the particles should not exceed 0.001, is thus met almost in the entire test bench. The Biot numbers of the particles also proved to be sufficiently low. Thus, the multiphase

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model used was considered to be valid and the calculated particle temperatures were further evaluated, the results were listed in Table 5.2. The assumption of spherical particles led to deviation of -59 K between the experimental and numerical result, it was reduced to just 2 K by incorporating non-spherical particles of all four classes. The difference between the maximum and minimum calculated particle temperature was 𝑇𝑀𝑎𝑥 − 𝑇𝑀𝑖𝑛 of 191 K. The coolest particle was in class 1 and had a temperature of 473 K, the hottest particle with a temperature of 664 K occurred in class 4. The mean temperatures of particle class 1 and 2 were below the measured mean particle temperature, those of class 3 and 4 were higher than the measured value. This indicated, that the mean particle temperature could be calculated by just considering a single particle type. The aspect ratio and sphericity of

ACCEPTED MANUSCRIPT this particle type are between the values of class 2 and 3, thus representing a mean value of all powder grains. Particle classification becomes important if extremal values need to be calculated. The standard deviation of the particle temperature distribution, as well as the difference 𝑇𝑀𝑎𝑥 − 𝑇𝑀𝑖𝑛 between the highest and lowest particle temperature were significantly higher when all four particle classes were considered. The positive effect of considering aspheric particles with regard to heat and momentum transfer has thus been proven beyond

CONCLUSION AND OUTLOOK

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6

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any doubt.

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The maximum deviation between numerically and experimentally determined mean particle temperatures was -2 K. Due to the close fit between numerical and experimental data it was

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concluded, that the proposed model for the convective heating of inert particles was fully validated. Considering non-spherical particles as spherical resulted in a -59 K error, so aspheric

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particles must necessarily be treated as such in numerical calculations. The basis for the calculations were drag model parameters which were determined experimentally. Using the

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same particle shape descriptors for the calculations of drag coefficients and particle Nusselt

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numbers offers the advantage that the determination of model parameters according to the

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proposed procedure in [17] is very efficient. This way, not only the transfer of momentum, but also convective heat transfer can be incorporated by numerically efficient calculations using and

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Euler-Lagrangian approach. The proposal of Loth [18], that the drag model parameters should be determined and validated by experiments, must be extended to correlations for convective heat transfer. The importance of particle classification for particle temperature calculations was proven. In gasification or combustion processes it is crucial, that every fuel particle reaches a certain temperature level, otherwise the fuel remains partially unburnt. Reliable calculations of the minimum or maximum particle temperature are only possible, if the different particle shapes are considered by the numerical model. The proposed procedure in [17] is fully independent of the particle type, it can be easily adapted for other particle shapes, for example cylinders in case of biomass particles as proposed by Yin et al [10,11] or other fuel particles. A major impact of the correct incorporation of the particle shape on the calculation of the gasification

ACCEPTED MANUSCRIPT characteristics of fuel particles, flame shapes and flame temperatures and also on pollutant prediction is expected and should be investigated further.

ACKNOWLEDGEMENT This work was financially supported by the Austrian Research Promotion Agency (FFG) (Project

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861148, eCall 9336196), which is gratefully acknowledged by the authors.

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ACCEPTED MANUSCRIPT Table 2.1 𝑥𝑖 15.91% 34.09% 34.09% 15.91%

1.365 1.69 2.295 2.875

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𝜓 [−] 0.85 0.8 0.75 0.7

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1 2 3 4

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Class Class Class Class

AR [−]

ACCEPTED MANUSCRIPT Table 3.1

Particle inlet

Unit 10-3∙kg/s K µm -

Electrical mean input Temperature Ambient Temperature Air mass flow rate

W K K 10-3 kg/s

Non-spherical 3.4981 298 700 6 Air inlet L 2675 825 298 4.6

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Air inlet

Mass flow rate Inlet temperature Mean diameter Spread parameter

Spherical 2.3853 298 520 6 Air inlet R 2628 825 298 4.51

ACCEPTED MANUSCRIPT Table 4.1 Elements

Aspect ratio Mean 1.43 2 1.21

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181562 320970 573973

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Mesh 1 Mesh 2 Mesh 3

Orthogonal skewness Mean Max. 0.06 0.78 0.15 0.73 0.04 0.67

Max. 12.5 16.3 9.4

ACCEPTED MANUSCRIPT Table5.1 Position

Measurement [K]

759 ±1.9

TC4

731 ±1.8

Mesh

758 760 759 735 734 735

1 2 3 1 2 3

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TC3

Numerical result [K]

ACCEPTED MANUSCRIPT Table 5.2

𝑇𝑀𝑖𝑛

𝑇𝑀𝑎𝑥

𝑇𝑀𝑒𝑎𝑛

𝑇𝑀𝑎𝑥 − 𝑇𝑀𝑖𝑛

𝑠𝑇

458

580

510

122

28.42

Deviation from measurement [K] -59

473

664

567

191

36.66

2

473 492 515 539

609 627 648 664

533 554 580 602

136 135 133 125

30.16 30.11 28.89 27.27

-36 -15 11 33

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Spherical Non-spherical 4 Classes Class 1 Class 2 Class 3 Class 4

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Particle Temperature [K] Considered particle type

ACCEPTED MANUSCRIPT Nomenclature Particle surface area, 𝐴𝑃 = 𝑑𝑒𝑞,𝑉2 ∙ 𝜋/(4 ∙ 𝜓)

𝐴𝑅

Particle aspect ratio

𝐵𝑖

Biot number

𝑐𝐷

Drag coefficient

𝑐𝑝

Isobaric specific heat capacity

𝑑𝑒𝑞 ,𝑉

Equal – volume particle diameter

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𝐴𝑃

Gravity

𝑚̇

Mass flow rate

𝑁𝑢

Nusselt number

𝑃𝑟

Prandtl number

𝑄̇

Heat flow rate

𝑄3

Volume –based, cumulative particle size distribution

𝑅𝑒𝑃

Particle Reynolds number, 𝑅𝑒𝑃 = 𝑑𝑒𝑞,𝑉 ∙ 𝑢𝑟𝑒𝑙 ∙ 𝜌𝐴𝑖𝑟 ⁄𝜇 𝐴𝑖𝑟

𝑡

Time

𝑇

Temperature

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𝑢𝑟𝑒𝑙

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Standard deviation

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𝑠𝑇

𝑢

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𝑔

Velocity Relative velocity, 𝑢𝑟𝑒𝑙 = |𝑢 ⃗𝑝−𝑢 ⃗|

𝑥𝑖

Particle mass fraction

𝑦+

Dimensionless wall distance

Greek symbols 𝜌

Density

𝛼

Heat transfer coefficient

𝜀

Emissivity

𝜆

Thermal conductivity

ACCEPTED MANUSCRIPT 𝜇

Dynamic viscosity

𝜓

Sphericity

𝜌

Density

𝜏𝑟

Particle relaxation time

𝜎

Stefan – Boltzmann constant

Minimal value

𝑀𝑎𝑥

Maximal value

𝑀𝑒𝑎𝑛

Mean value

𝑃

Particle parameter

𝑅𝑎𝑑 0

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Radiation

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𝑀𝑖𝑛

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Fluid parameter

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𝐴𝑖𝑟

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Subscripts

ACCEPTED MANUSCRIPT Highlights 

A model to calculate convective heat transfer to non-spherical particles is proposed



The particle shape is included in numerically efficient Euler-Lagrange calculations



A practical approach to determine the model parameters is shown



A novel method for particle classification is used



Precise and efficient numerical calculations of particle temperature are enabled

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Graphical abstract

Figure 1

Figure 2

Figure 3

Figure 4

Figure 5

Figure 6

Figure 7

Figure 8

Figure 9

Figure 10

Figure 11

Figure 12

Figure 13