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Marchetti, L., Hulme-Smith, C. (2021)Flowability of steel and tool steel powders: A comparison between testing methodsPowder Technology, 384: 402-413https://doi.org/10.1016/j.powtec.2021.01.074

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Powder Technology 384 (2021) 402–413

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Flowability of steel and tool steel powders: A comparison betweentesting methods

Lorenzo Marchetti ⁎, Christopher Hulme-SmithDepartment of Material Science and Engineering, KTH Royal Institute of Technology, Brinellvägen 23, 11324 Stockholm, Sweden

⁎ Corresponding author.E-mail addresses: lormar@kth.se (L. Marchetti), chrihs

https://doi.org/10.1016/j.powtec.2021.01.0740032-5910/© 2021 The Author(s). Published by Elsevier B

a b s t r a c t

a r t i c l e i n f o

Article history:Received 24 September 2020Received in revised form 29 January 2021Accepted 30 January 2021Available online 16 February 2021

Keywords:Powder flowabilitySteel powdersComparative flowability testingFlowability correlationPowder rheometer

The flow behaviour of a powder is critical to its performance in many industrial applications and manufacturingprocesses. Operations such as powder transfer, die filling and powder spreading all rely on powder flowability.Multiple testing methods can help in assessing flowability, but it is not always clear which may better representspecific flow conditions or how different metrics correlate. This study compares 8 different flowability testingmethods using 11 steel powders varying in chemistries and size fractions. Regression analysis was used to testthe relationship between each flowabilitymetric obtained. Somemetrics, such as the conditioned bulk density, re-late to many flowability indicators. Others, such as the basic flowability energy, show poor correlations to othervariables, likely describing different aspects of the powder flow behaviour.When twometrics show a strong cor-relation, as between conditioned bulk density and Hausner ratio, a numerical relationship is derived: CBD =− (5.65 ± 0.86)HR g cm−3.

© 2021 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).

1. Introduction

The flow of a powder is a complex phenomenon to understand ormodel. It is sensitive to properties of the powder itself and external fac-tors such as humidity [1,2], geometrical impediments [3], or appliedstresses [4]. Within the powder, many factors affect flow properties,such as the size distribution and shape of the particles, their surfaceroughness and oxidation state, the material density, and electrostaticand magnetic characteristics [5,6]. Powder handling also affects flowby altering the packing of particles and the ratio of solid to other phasesin the ensemble. This complexity has significant consequences formanyfields that rely on powder and its propensity to flow or remain cohesive,such as powder metallurgy, soil mechanics, civil engineering, the phar-maceutical industry, paint production and food processing. If the pow-ders are more cohesive than predicted, this could limit the materialsstream in a process or affect the final product quality [7]. Therefore, itis crucial to provide an accurate description of the flow of a powderfor a specific process. When the same powder is used in different appli-cations, it may be important to provide a description that is relevant fora general process [6].

The growing focus on metal powder-based additive manufacturingtechnologies increasingly highlights the need to deliver componentsthat comply with industry standards and operate under demandingconditions. The powder plays a key role in the final component quality[8–11]. For example, to produce low defects components in a powder-

@kth.se (C. Hulme-Smith).

.V. This is an open access article und

bed based additive manufacturing process, the metallic powder shouldbe spread in a thin, compact, and even layer [12–14]. Similarly, in pressand sinter and hot isostatic pressing a good powder flowability canavoid cavities and improve the die filling [15], while in cold spray depo-sition the flow rate of powder and gas can directly affect the coating de-position rate [16]. There are many viable techniques to measure flow ina powder, but no agreement as to which is the best testingmethod for agiven application [6,17–21]. Furthermore, there is no agreement onwhat flow behaviour is required for a powder to be considered suitablefor any specific manufacturing process.

The flow behaviour depends on the balance between forces thatdrive the motion of particles and those that prevent it. Forces promot-ing flow are gravity and applied external forces, while the ones oppos-ing it include frictional forces between the powder particles and theexternal surfaces, as well as forces between neighbouring particles.Such inter-particular forces include mechanical friction, the mechani-cal interlocking of particles, van deer Waals forces, capillary forces,electrostatic force, and magnetic force [22–24]. The magnitude ofthese resistive forces is determined by an array of properties arisingfrom the powder, such as powder particle size and particle size distri-bution, particle shape, material density, particles roughness, surfacechemistry, oxidation, and water content [18,25]. In addition, the ma-terial of the testing equipment, its surface conditions, geometry, andoperating conditions (e.g., if the test uses stationary or moving pow-der, stress state, temperature, gas flow) can also have a significant ef-fect on the flow behaviour. For example, when a stress is applied to apowder column of irregular particles, its density increases, locking theparticles in position. Hence, mechanical forces will propagate across

er the CC BY license (http://creativecommons.org/licenses/by/4.0/).

Table 1Overview of the powders used in the experimental trials.

Material classification Particle size distribution

Production process Alloy Designation D10 / μm D50 / μm D90 / μm

Water atomization 17–4 PH 1 7.4 19.9 44.6316L 2 8.2 20.7 42.3

Gas atomization 17–4 PH 3 5.7 10.7 15.44 7.5 17.0 37.25 59.8 75.1 95.6

316L 6 16.5 35.4 50.47 14.5 26.4 46.2

420 8 11.5 28.3 45.4Tool steel 1 9 25.0 37.8 57.0Tool steel 2 10 23.9 35.8 55.4Tool steel 3 11 18.4 30.9 55.7

1 There is a contradiction that a “rate” measures how much or how often somethinghappens per unit time, but the flow rate is the time required for a fixedmass (or volume)of powder to leave the funnel. More correctly, this is a specific (or volumetric) time, not arate, but the term flow rate is common.

L. Marchetti and C. Hulme-Smith Powder Technology 384 (2021) 402–413

particles more easily, strongly affecting the flow behaviour and over-coming the effect of the weaker cohesive forces. This web of proper-ties and mechanisms that compete and act concurrently makespowder flow a complex multivariate problem [17], where is not fullyclear exactly how properties of the powder itself affects flowability[17,19].

Due to the complexity of the powder flow, there is no single tech-nique to characterise flowability for a specific application [26] and stud-ies do not always agree on the best one to use for any specific case[18,21,27]. Furthermore,many commonmethods found in the literatureare not standardised, such as the rotating drum angle of repose, powderrheometry and spreading devices [18,21,28]. Others, such as tap densitytesting, static angle of repose and shear cell testing are standardised[29–32], but not included in the recommended standards whencharacterising the flowability for additive manufacturing metallic pow-ders [26]. ASTM F3049-14 suggests instead to characterise powders foradditive manufacturing with a Hall funnel, Carney funnel or Arnoldmeter [26]. However cohesive powders might not be suited for thesemethods as they B. Spierings, M. Voegtlin, T. Bauer, K.Wegener, Powderflowability characterisation methodology for powder-bed-based andgive non-quantitative results [21]. When ranking the flowability of sev-eral powders, it will not be possible to differentiate between the non-flowing ones, making a comparison difficult. Also, characterisingflowabilitywith differentmethodologiesmay lead to contrasting results[33,34]. To avoid this, it is often recommended that testing should beperformed under conditions that reproduce the specific case of interestas far as possible [17,26].

A holistic evaluation of the flow behaviour of a powder based onmultiple testing would solve the limitations of using a singleflowability test for powders intended for applications with differentparameters or where it is not clear which test is the most suited [6].However, a comprehensive model describing flowability has provenproblematic to obtain. This is due to the complexity the high numberof inter-dependent variables that must be accounted for in powderflow [10]. Some models are published in literature, mostly restrictedto specific flow conditions and without explicit physical variables[21,35]. Comparative testing can complement existing models andgive a more complete description of the problem [6,8,17,19]. Numer-ical modelling simulations such as the direct element method (DEM)are also a powerful tool to link interactions between particles totheir flow and the behaviour of the ensemble. In the case of finemetal-lic particles, such modelling techniques may have some limitations.This is partly due to increased computational demands of modellingmicrometre-scale particles and the need to consider several factors,such as the particle size distribution, shape, roughness, van deerWaals forces, liquid bridging and eventual electromagnetic forcesthat are known to affect the flow. It may be possible to simplify theforce models used in the simulations, but more research is requiredbefore such solutions may be applied with confidence and be realisticfor industrial applications [36,37].

In the present work we compare a range of flowability measuringtechniques available using a range of metal powder size fractions andchemistries. The results are used to evaluate the relationships betweenthe metrics from the different flowability measuring techniques usingstatistical analysis. By including a range of powder chemistries andsize distributions, it is intended that relationships should not be specificto individual alloys or powder production conditions. The commonflowability measurement techniques are assigned to categories, whereone representative method is selected from each. Every powder isthen measured using each selected technique and statistical analysis isperformed on pairs of metrics to derive any correlations. From theseevaluations, it may be understood which techniques are equivalent ofeach other and which techniques do not correlate. These are likely tobe measuring different responses in the powder to flow. Multiple tech-niques would be needed to describe all aspects of the powder flowbehaviour.

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2. Materials and method

2.1. Sample preparation

Flowability tests were performed on 11 steel powders with variouschemistries, atomization routes and particle size ranges (Table 1). Theparticle size distribution of each powder was measured using aCamsizer XT particle analyser with compressed air dispersion. Addi-tional data regarding particle shape and particle size distribution are in-cluded in the supplementary files (Supplementary Fig. 1 andSupplementary Table 1). Each powder batch was sampled with a rotat-ing sampler divider (spinning riffler) to obtain 5 individual specimens,prepared according to the relevant industrial standard (ASTM B215)[38]. Three specimens with a mass of 50 g were prepared to test theflow rate and tap density [39–41]. Two further specimens were pre-pared with a volume of 25 ml for testing in a powder rheometer[29,42] and a fixed height funnel angle of repose meter [43]. The speci-mens were dried for 30 min at 105 °C to remove moisture immediatelybefore testing. Adsorbed water layers and liquid bridging between par-ticles are known to deteriorate the flow properties of metallic powders,especially at higher relative humidity [44–46]. While it is reasonablethat during testing powders adsorbs some moisture from the environ-ment, it is unlikely that such uptakewithin the short time between dry-ing and testing will have a large impact on the results [47].

2.2. Flowability measurements

Flowability was evaluated with multiple measuring techniques,which can be divided into separate categories (Fig. 1). The differencewithin these techniques can be important, as a testing method affectsthe stress state, velocity of the powder particles, packing density andother factors in a powder. The ambient conditions and the history ofthe powder are also well known to influence its flow behaviour[42,48,49].

2.2.1. Flow rate measurementsWhen in a funnel, a powder may flow out due to gravity. A way to

quantify the phenomenon is the time required for a fixed mass (or vol-ume) of powder to flow entirely out of the funnel. This metric is com-monly referred to as the flow rate.1 A lower flow rate is an indicator ofa better flowability. When frictional forces are low, the powder flowsout of the funnel and the powder remaining in the funnel forms a sur-face that is approximately horizontal and gradually descends. Whenfrictional forces dominate, powders may form an arch near to the outletof the funnel, which leads to the presence of stable vertical cavities

Fig. 1. Flowability testingmethods overview. For every testingmethod class, some of the common testingmethods applicable are listed below. The highlightedmethods have been chosenfor this work [19,21,29,39,40,42,43,50–57].

L. Marchetti and C. Hulme-Smith Powder Technology 384 (2021) 402–413

(ratholes) or jamming, where the powder does not flow at all. These ef-fects are well studied in the literature and can be observed whenpowders are cohesive, when the ratio of the orifice diameter to the par-ticle diameter is too low or when the particle size distribution is toowide [58].

Such techniques have been common for a century and are currentlyrecommended for additive manufacturing [21], with several funnel ge-ometries described in the active standards. The best-known experimen-tal setup is the Hall flowmeter (ASTM B213–17) [39], whichworks wellfor non-cohesive powders. Cohesive powders may be tested using theCarney funnel (ASTM B964−16) [40], with a wider bottom opening,and the Gustavsson funnel [56], with a steeper conical surface. In thecurrent study, Hall, and Carney funnels from Qualtech Products Ltd.(Manchester, U. K.). were used.

Fig. 2. Forces acting on a single particle in a powder pile. Fg is the gravitational force, Fa isthe sumof the frictional forces, Fc the sumof the cohesive forces, FN is the normal force. F isthe gravitational component promoting the flow. (a) With a steeper pile, F > Fa and theparticle will move down the inclined surface; (b) when the powder pile is flatter, F < Faso the particle slows down, and eventually stop.

2.2.2. Density ratiosThe ability of a powder to pack can be taken as an indicator for its

flowability, as particles must move past each other (i.e., flow mustoccur) to increase packingdensity [54]. In general, the packing of a pow-der is quantified through its bulk density (defined as themass of a sam-ple divided by the total occupied volume, inclusive of particles and airgaps). Santomaso et al. [54] make a distinction between measuringthe bulk density of a powder in a loose or a dense packing condition.For example, a loose random packing condition is achieved throughtest as aerated, poured, or apparent density [57,59–61]. Other tests pro-mote a denser random packing through different methods, allowingair to exit the bulk volume. The most well-known example is the tapor tapped density, which is achieved after the powder is gently shakenand allowed to settle to a constant volume [41]. In this study, 3 samplesof 50 g were tapped using an Autotap tap density tester fromQuantachrome (Boynton Beach, Florida, U. S. A) for twice as manytaps as was necessary for the volume to stop changing, in accordancewith ASTM B527-15.

When comparing different powders, the ratio between high and lowpacking densities, or Hausner Ratio, RH is commonly used. This is oftenexpressed as the ratio between the tappeddensity ρt is and the apparentdensity ρa. However, some authors replaced the apparent density withother alternative metrics, such as aerated density [35,62], poured den-sity [63], apparent density determined with the Scott volumeter [64].A high value of Hausner ratio indicates a high variation between thestarting loose packing density and the final dense packing density.This is usually due to weak forces, such as mechanical contact forcesor cohesive interparticle forces, which prevent the powder fromreaching an optimal loose packing density. The effect of these forces

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can be overcome by shaking the powder and allowing particles to rear-range through tapping. Similarly, in the first stages of the particle mo-tion, powders will locally vary in their densities when particles start tomove one past each other. In the current work, the apparent density isreplaced by the conditioned bulk density (CBD, ρc) [42],which is a similarquantity measured in the Freeman FT4 powder rheometer from Free-man Technology Ltd. (Tewkesbury, U. K.) (Eq. (1)). Conditioned bulkdensity is measured after the powder sample has been gently stirredby an upwards-moving blade and reduced to a fixed volume.

RH ¼ ρt=ρcð1Þ

2.2.3. Angle of reposeThe angle of repose is the angle of a pile of powder at which it stops

flowing. It is the complementary property to avalanche angle, which isthe angle at which a stationary powder will begin to flow. A well-flowing powder will not stop flowing when the powder pile has highangles. However, lowering the pile angle, the gravitational componentmoving the powder will diminish. At the point where the powderstops flowing, the gravitational component driving the flow is balancedby the attractive and frictional forces within the powder (Fig. 2). Previ-ous research has used the angle of repose to qualitatively classify theflowability powders [65, Table 1].This angle can be measured in severalalternative methodologies: by allowing a powder to fall through a

L. Marchetti and C. Hulme-Smith Powder Technology 384 (2021) 402–413

funnel onto a flat surface and form a pile (fixed funnel method), by fillinga hollow cone with powder and removing the cone, allowing the pow-der to settle into a pile (hollow cylinder method), or by causing powderto run along a planar surface tilted at some angle and finding the tiltangle at which the powder stops moving (tilting box method) [43].While this final method measures the avalanche angle, it is commonlyreferred to as a technique to measure angle of repose. In all thesecases the cone angle of the pile of powder is measured statically. Itshould be noted that there is evidence that the angle of repose dependsstrongly on the method chosen to measure it. For example, the proce-dure outlined in the standard ASTMC1444 (a variant of the fixed funnelmethod) gives a higher angle of repose than the cone lifting technique[65]. In the current study, a fixed funnel method was used, with a pro-tractor fitted to the funnel stand to measure the angle of the side ofthe pile directly. The funnel has a cone angle of 60°, an orifice diameterof 5mmand a distance between the orifice and the traywhere the angleforms of 80 mm (Qualtech Products Ltd., Manchester, U. K). When pos-sible, two samples of 25mlweremeasured three times for each powder,where everymeasurement consisted of three angles observations at an-gles from the same pile, as suggested in the instrument operatingman-ual. The angle of repose was taken as the average angle of the pile.Measuring the anglewith a protractormounted on the funnel assembly,it was possible to exclude the effect of possible distortions on the pile,such as the presence of a small flat cone tip, or a wide base, by takingthe angle in the remaining part of the pile edge. Other methods, whichcalculate the angle based only on the diameter and the height of thepile, could be more prone to these errors.

These methods are inexpensive and traditionally used to character-ise powder flow, but none of them are described in the current stan-dards for the characterisation of metallic powders [30–32,51]. Arelated technique, which is mostly automated and not standardised, isthe dynamic angle of repose or rotating powder analyser, in which pow-ders are placed in a drum and rotated for a pre-programmed speedcycle, during which pictures are taken and evaluated automatically.Powders rise the drum until some critical angle is reached (avalancheangle), at which it begins to flow. It then stops at some lower angle(rest angle). Based on these angles and the irregularity of the powderprofile, several parameters are automatically calculated and can beused to indicate flow behaviour.

Fig. 3. (a) Sketch of the FT4 vessel and impeller in the Stability and Variable flow rate,(b) sketch of the vessel and the flange for the Compressibility test.

2.2.4. Shear cellIn a shear cell, a powder sample is confined between two flanges,

which subject the powder to a pre-set cycle of normal and shearstresses. First, the powder sample is consolidated to a set pre-compaction normal stress (σ1c). It is then sheared at different compres-sive stresses below σ1c. Some powder particles move coherently withone moving flange, while some other adopt the velocity of the secondflange, which is usually fixed. There is a regionwhere particleswill tran-sition from following one flange to following the other. In this region,particles move past to each other under a shear stress. The shear stressnecessary to initiate the flow is recorded at different compressivestresses. The normal and shear stresses are then used in a Mohr's circleto plot the stress state at which flow first began in at each compressivestress, known as the locus of incipient flow. Several parameters can bederived from this diagram, such as cohesion (C), unconfined yieldstrength (UYS), major principal stress (MPS), angle of internal friction(AIF) and the flow function coefficient (ffc or FF) [42,66]. In general, agood flowability leads to lower shear values of these metrics, exceptfor the flow function coefficient, where a higher value should indicatea better flow behaviour transitioning from static to dynamic friction.In this work, a shear cell supplied with the FT4 powder rheometerwas used following ASTM D7891–15 [29]. Pre-compaction stresses of3 kPa and 9 kPa were chosen. To increase the data reliability, two sam-ples were tested for each powder. The pre-compacted volume of thetested sample was of 10 ml, while the vessel diameter 25 mm.

405

2.2.5. RheometryPowder rheometry is becoming increasingly popular also for metal-

lic powders [6,8,67,68]. In addition to the shear cell unit, other testmethodologies are available to characterise the powder behaviourunder different conditions. The testing is mostly automated, while theinteraction with the operator very limited. To homogenise the sample,a pre-mixing (conditioning) is performed prior testing. Two standardprograms of the FT4 powder rheometer were used with a 25 ml sampleand a vessel diameter of 25 mm. Each test was performed on twosamples.

The first program was the stability and variable flow rate test, inwhich an impeller moves in a set path through the powder (Fig. 3a).For each sample, the test automatically runs 11 times: seven tests runat a fixed vertical blade tip speed and then four runs at increasing verti-cal blade speeds. The resistance met from the impeller is recorded andconverted into synthetic indices. The basic flowability energy (BFE) isthe energy spent from the blades moving downwards, where the pow-der is compressed and confined to a fixed volume by the blade and thewalls of the test vessel. The specific energy (SE) represents the energyspent when the blades move upwards, when the powder is free to oc-cupy more volume (unconfined flow) [42].

The second method performed was the compressibility test, wherethe powder is confined in a vessel and slowly subjected to uniaxial com-pression (Fig. 3b). At the same time, the variation in height is recorded.The compressibility (CPS) is then calculated as change in volume beforeand after the powder is compacted with a 15 kPa applied compressivestress [42].

3. Calculation

3.1. Parameters

In this study a broad overview of the relationships between severalflowability measurements is presented. For every flow test performed,one ormoremetric can be recorded to evaluate the behaviour of a pow-der (Table 2). Each of these metrics was then compared with all theother, as well as conditioned bulk density. To focus on the relationshipbetween testing methods, no distinction was made between differentpowders and all the eleven tests were taken as a dataset. The mean ofall repeats was reported for each powder. All results are supplied as asupplementary file in the online version of the article.

Table 2Samples, devices, test performed and flowability indicators considered.

Sample Testclass

Device Test Flowabilityindicator

Unit

50 g(3×)

Flowrate

Hall funnel Hall flow rate [31] Flow rate s (50 g)−1

Carneyfunnel

Carney flow rate [32] Flow rate s (150 g)−1

Packingdensity

Tap-meter, Tapped density [33], HausnerratioRheometer Conditioned bulk

density [34]25 ml(2×)

Other Rheometer Flowability [34] BFE mJSE mJ (g)−1

Other Rheometer Compressibility@ 15 kPa [34]

CPS (%)

Shearcell

Rheometerwith shearcell

Shear test@ 3 kPa [23]

Cohesion kPaUYS kPaMPS kPaFFAIF degrees

Shearcell

Rheometerwith shearcell

Shear test@ 9 kPa [23]

Cohesion kPaUYS kPaMPS kPaFFAIF degrees

Angle ofrepose

Fixedfunnelangle ofrepose

Fixed funnel angle ofrepose

Angle ofrepose

degrees

L. Marchetti and C. Hulme-Smith Powder Technology 384 (2021) 402–413

3.2. Linear correlation

As a first evaluation step, the correlation across each pair of param-eters (X and Y) was investigated using the programming language R.To analyse the data, the generic equation for a straight line was used(Eq. (2)). The intercept β10 and the slope coefficient β11 were deter-mined performing a linear regression with the ordinary least squaremethod, where the sum of the squares of the residual ϵi (Eq. (3),Fig. 4a) is minimized.

f 1 xið Þ ¼ β10 þ β11xi ð2Þ

yi ¼ f xið Þ þ ϵi ð3Þ

The solution of a system between Eqs. (2) and (3) leads to the deter-mination of β10 and β11 (Eqs. (4) and (5)).

Fig. 4.Graphical description of the correlation between theAngle of repose and theHausner ratifunction of angle of repose, f(xi), (red diamonds) and the actual Hausner ratio, yi, (black dotcoefficient of the gradient β11.

406

β11 ¼∑N

i¼1xi−xð Þ yi−yð Þ

∑N

i¼1xi−xð Þ2

¼ σxy

σ2x

ð4Þ

β10 ¼ y−β11x ð5Þ

β11 determines the steepness of the correlation between two vari-ables (Fig. 4b). However, this does not measure the dispersion of thedata. For this, the Pearson correlation coefficient, r, is used (Eq. (6)).For this reason, it can give an indication of the strength of the correlationand its direction. As the Pearson correlation coefficient is scaled for eachvariable, it oscillates between−1 (inverse perfect correlation) and +1(direct perfect correlation), while 0 implies no correlation.

r ¼∑N

i¼1xi−xð Þ yi−yð Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

∑N

i¼1xi−xð Þ2

s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi∑N

i¼1yi−yð Þ2

s ¼ σ xy

σ xσy¼ β11

σ x

σyð6Þ

3.3. Non-linear correlation

It is possible that pairs of metrics may be correlated in a non-linearway [35]. To assess the non-linear data trend between flowability pa-rameters, a multiple linear regression was performed. In addition to f1(xi), the data have been fitted to polynomial (Eqs. (7) and (8)) logarith-mic (Eq. (9)) ormixed (Eq. (10)) functions. The polynomial expressionsare a natural extension to the linear correlation. The logarithmic depen-dence was tested in the eventuality that a correlation could graduallydecrease in strength.

f 2 xið Þ ¼ β20 þ β21xi þ β22x2i ð7Þ

f 3 xið Þ ¼ β30 þ β31xi þ β32x2i þ β33x

3i ð8Þ

f 4 xið Þ ¼ β40 þ β41 log xið Þ ð9Þ

f 5 xið Þ ¼ β50 þ β51xi þ β52 log xið Þ ð10Þ

Multiple linear regression is similar to the linear regression, wherelinear coefficients (βk0, βk1, βk2, βk3) are found minimizing the matrix

o. (a) shows the straight linefitted to the data, the predicted values of theHausner ratio as as). The vertical lines highlight the residuals, ϵi; (b) the uncertainty σ11 around the slope

L. Marchetti and C. Hulme-Smith Powder Technology 384 (2021) 402–413

of the square residuals [ϵ], obtained subtracting the fitted values [fk(x)]from the observed values [y]. The adjusted coefficient of determinationRadj2 was used to test the suitability of themodel. As R2, Radj2 explains how

much of the total variation from of the data from the average value isexpressed through themodel (Eq. (11)). If the data had a similar disper-sion to themodel, then R2 will tend to 1,meaning that themodel fits thedata well. Otherwise, if there are additional sources of variance in thedata, such as random scatter, R2 will fall to 0. In this case the modeldoes not explain any variance of the data. However, R2 increases withmany coefficients, leading eventually to overfitting. Radj2 adjusts for thenumber of coefficients in the model (k) relative to its data points (N)(Eq. (12)).

R2 ¼∑N

i¼1f k xið Þ−yð Þ2

∑N

i¼1yi−yð Þ2

¼ σ xy

σ xσy

� �2

ð11Þ

R2adj ¼ 1− 1−R2

� � N−1N−k−1

ð12Þ

3.4. Statistical significance

Although the linear coefficients βk (or the regression coefficient r)are obtainedminimizing the sum of residuals ϵi, there is always some fi-nite probability that the fit is simply due to random scatter in the data(i.e., there is not enough evidence to confirm the relationship betweenvariables). This probability is given from the p-value. When a p-valueis smaller than a pre-set significance level, there is enough evidence toconsider the relationship statistically significant. In this work the re-quired significance level for all the p-values was set at 0.05, thus mean-ing that there is 95% probability that the calculated fits are genuinetrends. The p-values were calculated with a Student's t-test, based ona Student t distribution. This is preferred in regression analysis when asmall data set with unknown variance is used [69,70].

Fig. 5. Linear relationships across flowability indicators: (a) correlation matrix, where the sizedescribes the sign of the Pearson coefficients (white for a positive coefficient, black for a negat

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4. Results and discussion

4.1. Linear correlation

Whilemost of the flowability parameters reveal a positive linear cor-relation, the conditioned bulk density (CBD) and the flow function coeffi-cient (FF) clearly show an inverse one (Fig. 5a).

Hall and Carney flow rate exhibits a strong correlation with severalvariables (e.g., positive with Hausner ratio and negative with compress-ibility, CPS), while showing aweak dependencewithmost others. How-ever, a powder with a good flowability would be expected to have alower flow rate, as well as a lower Hausner ratio and compressibility. Itis useful to note that, due to the cohesiveness of many of the powdersused in this study, few could give quantitative results in flow rate test-ing: jamming or ratholing were observed on many samples, in agree-ment with what was stated from Marnani et al. for fine and ultra-fineparticles [71] and Muñiz-Lerma et al. for fine metallic powders [73].The low number of flow rate data available prevents us from drawingconclusions on possible correlations (Figs. 2–4 in the Supplementaryfile).

Conversely, Hausner ratio, angle of repose (AOR) compressibility andspecific energy (SE) all show a strong positive correlation betweenthem (Fig. 5 and Figs. 8–11 in the Supplementary file). These variablesalso exhibit strong interactionswith other shear test parameters. For in-stance, Angle of repose and compressibility are strongly correlated withcohesion, unconfined yield strength (UYS), major principal stress (MPS)and flow function coefficient (FF), correlated with the conditioned bulkdensity and weakly correlated with the angle of internal friction (AIF).Conversely, Hausner ratio and Specific energy are weakly correlatedwith cohesion, unconfined yield strength and flow function coefficients,while strongly correlatedwith themajor principal stress, angle of internalfriction and conditioned bulk density (Fig. 5). Many interactions betweenshear test variables were similarly found in a study from Li et al. [73],where 11 powders were tested with different Freeman FT4 setups. Co-hesion was found positively correlated with unconfined yield strengthand major principal stress, while negatively correlated with the flowfunction coefficient, indicating that a more cohesive has a lowerflowability.

of the circles represents the modulus of the Pearson coefficients (from 0 to 1). The colourive one); (b) map of the most important correlations between metrics.

L. Marchetti and C. Hulme-Smith Powder Technology 384 (2021) 402–413

Basic flowability energy (BFE) shows weak between most of the var-iables (Fig. 5 and Fig. 5 in the Supplementary file) and only some corre-lation with the angle of internal friction and the flow function coefficients.It is worth noting that basic flowability energy is one of the few parame-ters to be poorly related to the conditioned bulk density (Fig. 6 in the Sup-plementary file). This disagrees from what was found by Li et al. [74],where the basic flowability energy clearly correlates with the condi-tioned bulk density. In this study the powders tested had a significantdifference in density. In our study instead, all powders used are ironbased steels, with a relatively similar size fraction and density, so thatthe effect of the density on the basic flowability energy goes unseen.Other parameters poorly related to the conditioned bulk density areshear cell metrics. Cohesion, unconfined yield strength and flow functioncoefficients show a strong correlation between each other, but a poorcorrelationwith the conditioned bulk density. Conversely, the angle of in-ternal friction is correlated with the conditioned bulk density. The Majorprincipal stress always showed a strong correlation with the conditionedbulk density (Fig. 5).

Fig. 5b further summarizes some of the correlations analysed. Pa-rameters showing ambiguous relationships with other variables, suchas Hall flow rate, Carney flow rate or the major principal stress (MPS) inrelationship to some shear cell parameters, were not included.

All the Pearson correlation coefficients obtained (Fig. 5a) were sub-sequently filtered according to their significance level. The matrix ofthe statistically significant correlation coefficients (with a confidenceof 95% or more) is shown in Fig. 6a. Despite the data filtering, 51 ofthese relationships proved to be statistically significant, while the re-maining 102 did not. For example, almost all the correlation coefficientsrelated to Hall and Carney flow rate are found to be not significant(Fig. 6a). The only exception is the correlation between Carney flowrate and the flow function coefficient with pre-compaction at 3 kPa. Al-though this proved to be statistically significant, the data suggests thatthe correlation is due to the lack of data available and not due to under-lying physical causes (Fig. 4 in the Supplementary info).

Angle of repose, specific energy, compressibility and Hausner ratio con-firmed most of their mutual strong positive correlation (Fig. 6a,Figs. 8–11 in the Supplementary file). The correlation between theangle of repose and Hausner ratio is often found in the literature

Fig. 6. Significant linear relationships acrossflowability indicators. a) Correlationmatrix showinof being representative). The size of the circles represents themodulus of the Pearson coefficienitive coefficient, black for a negative one). b) Scheme of the most important and statistically si

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[35,54], while there is less information available for the other variables.These variables still exhibit some correlation with shear cell parame-ters: while all except for compressibility are correlated with the majorprincipal stress, only the angle of repose and compressibility are correlatedwith the unconfined yield strength. Finally, only compressibility is corre-lated with both cohesion and the flow function coefficients. The correla-tions can be summarised in the map shown in Fig. 6b, where onlyvariables correlating both to shear variables with pre-compaction at3 kPa and 9 kPa are included.

Across the shear cell results, the strong correlation between cohe-sion, the ultimate yield strength and the flow function coefficient can beconfirmed. The uncertain correlation the major principal stress withother variables from shear cell testing is also confirmed, as well as itsstrong correlation with other variables (Hausner ratio, specific energy,angle of repose) and the conditioned bulk density (Fig. 6a). Finally, theConditioned bulk density shows a strong negative correlation with thespecific energy, Hausner ratio (Fig. 7 in the Supplementary file), Angleof repose, major principal stress, while none with compressibility.

The diagram in Fig. 6b shows only the significant relationships pre-viously reported in Fig. 5b. Parameters showing ambiguity, such asHall flow rate, Carney flow rate or major principal stress in relationshipto some shear cell parameters (cohesion, ultimate yield strength, flowfunction coefficient, angle of internal friction) were not included.

4.2. Non-linear correlation

Along with the linear correlations, Fig. 7 shows the statistically sig-nificant non-linear relationships obtained through the multiple linearregression analysis (where all the individual coefficients were foundto be significant). Quadratic (Eq. (7)), cubic (Eq. (8)), logarithmic(Eq. (9)) and mixed (Eq. (10)) relationships were all evaluated, butonly quadratic or cubic were found to be significant in any case. Allthe non-linear relationships found are like the ones reported in Figs. 9and 12 in the Supplementary file. While it is clear there is a statisticallysignificant variation from linearity, it is also difficult to explain the datatrend shown.

In some cases, non-linear relationships were found to fit the dataslightly better than some linear ones (for example the flow function

g only the Pearson correlation coefficientswith p a p-value<0.05 (orwith a 95%probabilityts (from 0 to 1). The colour describes the sign of the Pearson coefficients (white for a pos-gnificant (p<0.05) correlations.

Fig. 7. Correlation matrix summarizing the significant relationships between flowabilityindicators. The chart includes both linear and non-linear relationships. All thecorrelations and correlation coefficients shown have p < 0.05.

Fig. 8. Scheme of the correlations statistically significant (p < 0.05) and representative(Radj2 > 0.7).

L. Marchetti and C. Hulme-Smith Powder Technology 384 (2021) 402–413

coefficient with pre-compaction at 9 kPa and compressibility), with thelatter having already a significance level above the pre-set 95% value.For this reason, considering a non-linear relationship between thesevariables could lead to overfitting.

The correlation between Carney flow rate and flow function coefficientwith pre-compaction at 3 kPa was instead excluded from Fig. 7 as thereare few data available from the Carney flow rate trials (Fig. 4 in the Sup-plementary file). Taking this into account, the total number of statisti-cally significant relationships increased from 51 to 59. The angle ofrepose was confirmed to correlate linearly to the Hausner ratio, as alsoobserved from Geldart et al. [35] and Santomaso et al. [54].

4.3. Significant and representative quantitative relationships betweenmetrics

Although the relationships from in Fig. 7 are considered statisticallysignificant, this does not explain howwell the coefficientsβk fit the data.

Table 3Statistically significant (p < 0.05) and representative (Radj

2 > 0.7) correlations acrossflowability indicators.

Variables Function β1 ± σβ1 Radj2

[β2] [± σβ2]

MPS9 = ƒ(HR) Linear 10.1 ± 1.2 0.86CBD = ƒ(HR) Linear −5.65 ± 0.86 0.81SE = ƒ(AOR) Linear 0.083 ± 0.01 0.78MPS3 = ƒ(AOR) Linear 0.033 ± 0.006 0.74MPS9 = ƒ(SE) Linear 0.337 ± 0.16 0.74AIF9 = ƒ(SE) Polynomial,2 −34.9 ± 7.2 0.80

[6.52] [± 1.3]FF3 = ƒ(CPS) Linear −0.895 ± 0.16 0.74UYS3 = ƒ(C3) Linear 2.7 ± 0.1 0.98FF3 = ƒ(C3) Linear −14 ± 3 0.74UYS9 = ƒ(UYS3) Linear 1.3 ± 0.3 0.70UYS9 = ƒ(C9) Linear 2.5 ± 0.2 0.96FF9 = ƒ(UYS9) Linear −6.8 ± 1 0.82FF9 = ƒ(C9) Linear −20 ± 3 0.81CBD = ƒ(MPS9) Linear −0.502 ± 0.096 0.72

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In this study, only the correlations explaining the data to a good degreeshould be considered. Table 4 in theAppendix reports all the statisticallysignificant regression coefficients βk along with their confidence inter-val and the coefficient of determination R2 for the model considered,which can be used to evaluate the goodness of fit for each correlation.

Table 3 instead reports only the correlations coefficients having aRadj2 > 0.7 (meaning that 70% of the variation observed in the data can

explained through the proposed model). A visual explanation of theseresults is shown in in Fig. 8. Out of the 59 statistically significant rela-tionships, only 15 are shown to fit well the data. Many of the correla-tions previously found between specific energy, Hausner ratio, angle ofrepose and compressibility are not found to be explanatory. However,these metrics are still found to be important, relating either to shearcell parameters or to conditioned bulk density.

4.4. Physical explanation

Many of the powders tested in this work showed a cohesive behav-iour during the flow rate trials. When the powders obstructed the Halland Carney funnels, no flow rate could be recorded. Table 4 comparesthe flow rates and the powder size in relationship to the funnel outletdiameter, showing how for powders of similar chemistry and produc-tion technique, the finer fractions have a greater tendency to jamthe funnels (sample 3 and 4, 7 and 11), while the coarsest powder

Table 4Measured Hall and Carney flow rates and ratio D50 (Doutlet)−1 for each sample.

Powder Hall funnel flow test Carney funnel flow test

D50 (Doutlet)−1 Flow rate / s D50 (Doutlet)−1 Flow rate / s

1 8.0 Jamming 4.0 Jamming2 8.3 Rathole 4.1 Rathole3 4.3 Jamming 2.1 Jamming4 6.8 Jamming 3.4 Jamming5 30 21 15 4.76 11 11 5.7 2.87 11 Jamming 5.3 Jamming8 11 12 5.7 2.49 15 12 7.6 4.410 14 11 7.2 2.611 12 11 6.2 Jamming

L. Marchetti and C. Hulme-Smith Powder Technology 384 (2021) 402–413

(sample 5) is always able to flow through the funnels (table). This is op-posite from what found from Janda et al. [74], where bulk solids jamwhen their particle size is too coarse in relationship to the funnel orifice.While in this case the interlocking mechanisms, affecting coarser parti-cles, can explain the jamming, in our work the fractions that are finer inrelationship to the funnel diameter aremore prone to the jammingphe-nomenon due to their higher cohesive forces. This can be seen veryclearly with sample 11, where the same powder was able to flowthrough theHall funnel, but not through the Carney funnel. Our findingsagree with other results in the literature [75]. With less data available,any outlier can have a significant impact on any data trend observedfor the flow rate trials. This can explain why flow rate shows little corre-lation with other variables (Figs. 10–12 in the Supplementary file).

Basic flowability energy measures flowability dynamically in pow-ders confined under a compressive load. These conditions are uniqueamongst the techniques used in the current study and it explains whybasic flowability energy shows no correlations with other parameters(Figs. 5–6 and Figs. 5–6 in the Supplementary file). This suggests thatpowders subjected to a dynamic flow and confined under the bladestress behave very differently than other cases and highlights the needto test flowability under conditions that are representative of the pro-cess for which the powders are intended.

In our study the conditioned bulk densitywas found to be clearly con-nected with multiple flowability metrics (Figs. 6,8 and Fig. 7 in the Sup-plementary file), but a weak correlation between conditioned bulkdensity and basic flowability energy (Fig. 6 in the Supplementary file).The correlation between flowability and powder density in generallyknown in the literature: Santomaso et al. brings evidence of a correla-tion between the density ratios and the angle of repose [54], while Ab-dullah and Geldart find an important contribution of the particle shapedistribution to both bulk density and flowability for fluid-cracking cata-lyst and fire-retardant filler powders [76]. Hou and Sun note that higherparticle density corresponds to a better flow behaviour for pharmaceu-tical powders tested with a powder shear cell [77]. The reason to thiscould be that in many flowability testing techniques, only mild forcesare applied to the powder, limiting the influence of mechanisms suchas mechanical friction, or the propagation of forces through particleinterlocking. These mechanisms are more pronounced when strongerforces are applied to the powder andwith an increased particle packing.For this reason, in flow tests with cohesive powders, weaker forces be-tween neighbouring particles may predominate. Similarly, theseweaker forces are important when determining the conditioned bulkdensity of these powders. Conversely, when a stress is applied to thepowder, the mechanical friction and interlocking are more important,and the flow is therefore determined by other powder properties. Thiscould explain why the basic flowability energy does not correlated withthe conditioned bulk density (Fig. 6 in the Supplementary file).

Specific energy, Hausner ratio, compressibility, and angle of repose(Fig. 6 and Figs. 7–11 in the Supplementary file) seem to be strongly re-lated and are all correlated to the conditioned bulk density, suggestingthat the effect of weak cohesive forces is limiting in these flowabilitytests. However, there is variation between these flowability indicators.From the current data, specific energy and Hausner ratio are more corre-lated to each other than to other metrics (Fig. 8 in the Supplementaryfile) andwith themajor principal stress (Table 3 and Fig. 8). The correla-tion observed between Hausner ratio and angle of repose agrees withwhat was found from Riley and Mann for glass particles of differentshapes [78]. Similarly, Geldart et al. found a clear correlation betweenthe angle of repose and the Hausner ratio for different non-metallicpowders [35]. The reason may be that Hausner ratio considers bothweak cohesive forces and the rearrangement of particles, as powdersare shaken in the tapped density measurement. It is reasonable thatboth these phenomena are also affecting the specific energy and majorprincipal stress measurements, where the powder resistance to an ap-plied stress is recorded under conditions of low stress. Similarly, duringthe angle of repose measurement powders are forced to fall from a

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funnel and come to rest in a pile: the cohesive and frictional forces actto oppose a motion that is imposed on the powder, which are partlyallowed to rearrange (Fig. 10 in the Supplementary file). Conversely,in the compressibility test, powders are slowly compressed up to a pre-set stress, overcoming weak cohesive forces. However, the pre-set lowlevel of stress and the single actionmovement prevent powders to rear-range with a higher packing fraction. For this reason, this test does notcorrelate well with other metrics, such asHausner ratio, for all the pow-dersmeasured (Fig. 9 in the Supplementaryfile). Instead, compressibilitycorrelates with cohesion and ultimate yield strength, where the powderresistance to flow is driven by weak applied forces and in static condi-tions (Table 3 and Fig. 8). In these tests, themechanism of particle rear-rangement is inhibited, allowing results to be compared to thecompressibility test. Zegzulka et al. found that flow function coefficient,angle of internal friction and angle of repose vary all similarly across dif-ferent metallic powders, and the particle shape has a prominent effecton their flowability. This result is somewhat different from what wefound in our study, but it highlights the existence of a similar trend be-tween the angle of repose and some rheometry variable [79].

In the stability and variable flow rate test, specific energy is mostly de-termined by interactions due to attractive forces between particles, suchas electrostatic forces or polar attractions due to the presence of water.Since these forces generally weak, the inertia of the powder due to thework done by the impeller strongly affects the results. For this reason,specific energy is normalized by the sample mass (hence the use of theword “specific” in the name of the metric). However, the current datastill shown a strong and statistically significant correlation betweenthe specific energy and the conditioned bulk density. While normalizingthe force recorded from the impeller for the sample mass removes theeffect of different powder densities, it is also true that the conditionedbulk density, as well as the specific energy, can be an indicator of the at-tractive forces between particles in the powder (Fig. 6).

4.5. Testing strategy

Flow ratemetrics proved unsuitable to evaluate flowability with oursamples. For this reason, in disagreement with current standards [26],thesemetrics are not recommended tomeasure flowability for cohesivesteel powders.

Many singlemetrics proved to offermeaningful results, however notalways in agreement. For example, when evaluating flowability,Hausner ratio, specific energy, and angle of repose showed an agreement.However, a funnel flow device to measure angle of repose like the oneused in this study is not indicated with cohesive powders as the resultsreading heavily relies on the operator as the funnel could get blockedwith cohesive powders. Compressibility behaves similarly butunderestimating the effect due to particle rearrangement compared toHausner ratio and specific energy. Conversely, basic flowability energygives unique results, strongly enhancing the effect of frictional mecha-nisms and particle interlocking. Flowability results that, similarly toour case, are not in agreement were found also from Mellin et al.when comparing the Basic flowability energy and the specific energywith the Hall and Gustavsson flow rates [80]. These differences suggestthat when powder flowability is needed to be evaluated comprehen-sively, a multiple tests strategy is needed, in agreement with similarconsiderations made from Leturia et al. [6] and Prescott [17]. For exam-ple, basic flowability energy, compressibility and one metric betweenHausner ratio andspecific energy can measure the flow behaviour in dif-ferent conditions. Instead, when the flow characterisation is needed fora specific application, it is recommended to select a flow test that canreplicate adequately the powder condition. However, Ghadiri et al. [7]observe that this could not always be possible, as when evaluating thespreading of powders for additive manufacturing.

Shear cell testing records accurately the powder stress state and canoffers several metrics to evaluate flowability. The major principal stresswas found correlated to other metrics (Hausner ratio, specific energy,

L. Marchetti and C. Hulme-Smith Powder Technology 384 (2021) 402–413

and angle of repose), while cohesion and ultimate yield strength werefound linked to the compressibility (Table 3 and Fig. 8). This meansthat it is possible to reliably express all the shear cell metrics withother metrics (and vice-versa) for the powders in our possession. How-ever, as it was experienced that all shear cell metrics are sensitive to in-terferences, including a correct sample preparation, these values can bereliably obtained from other metrics (Hausner ratio, specific energy,angle of repose, compressibility), as shown in Table 3 and Fig. 8.

5. Conclusions

• A testing strategy using multiple techniques is necessary whencharacterising the flow behaviour of a powder in a comprehensiveway: different metrics depend on different properties of the powderand, so, give different results. Basic flowability energy, compressibilityand at least one of Hausner ratio and specific energy revealed to repre-sent different aspects of theflowbehaviour of cohesive steel powders.

• Hall and Carney flow rate methods were found not suitable forassessing the flowability for cohesive steel powders. The funnelmethod is not suitable for the cohesive powders because the jammingdoes not allow to record a flow rate, useful to compare different pow-ders. For this reason, it is not possible to determine any significant andrepresentative correlations between either Hall or Carney flow rateand any other flowability metric.

• The conditioned bulk density (CBD) showed a strong correlation withmostmetrics. Many of these correlationswere found to be statisticallysignificant, while 2 (CBD with Hausner ratio (HR) and major principalstress with pre-compaction at 9 kPa (MPS9), Eq. (13) and (14) werealso found to be representative of the data.

CBD ¼ − 5:65� 0:9ð Þ∙HR R2adj ¼ 0:81

��� ð13Þ

CBD ¼ − 0:502� 0:22ð Þ∙MPS9 R2adj ¼ 0:72

��� ð14Þ

• The basic flowability energy (BFE) showed a weak correlation withmost flowability metrics, consistent with the fact that powders wereconfined and compressed during testing. It is reasonable that thisstress state heavily promoted mechanical interlocking and inter-particle mechanical friction.

• Specific energy (SE), Hausner ratio (HR), compressibility (CPS) andangle of repose (AOR) were found to be inter-correlated. Specific en-ergy,Hausner ratio and angle of reposewere found to behave very sim-ilarly and the results from Hausner ratio and angle of repose, beingstatistically significant and representative of the data, can be reliablyconverted between them (Eq. (15)). Compressibility was found tolead to some different results, probably because the test conditionslead to underestimate the effect of particle rearrangement comparedto other techniques.

SE ¼ 0:083� 0:03ð Þ∙AOR R2adj ¼ 0:78

��� ð15Þ

• Other metrics were found strongly correlated. For example, themajorprincipal stress (MPS) was found to be more correlated to Hausnerratio, specific energy, and angle of repose. Conversely, cohesion and ulti-mate yield strengthwere found to bemore strongly correlated to com-pressibility and angle of repose (Table 3). This means that is possible toreliably express all the shear cellmetricswith othermetrics (and vice-versa) for the powders in our possession, making the shear cell testflexible in representing different flow behaviours.

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6. Future work

In this work, we focused on the comparison between severalflowability testing methods. However, this will be completed whenwe will establish a procedure to characterise the spreadability of steelpowders. In addition, the influence of some relevant powder propertieson flowability should be determined. For example, the effects of the par-ticle size distribution, morphology, and environmental moisture. Fi-nally, is could be useful to expand this work to a wider range ofpowder chemistries, such as titanium or aluminium alloys and non-metal powders.

Declaration of Competing Interest

The authors declare that they have no known competing financialinterests or personal relationships that could have appeared to influ-ence the work reported in this paper.

Acknowledgments

This work wasmade possible by financial support from the SwedishGovernmental Agency for Innovation Systems (Vinnova), project num-ber 2019-01087.

Appendix A. Supplementary data

Supplementary data to this article can be found online at https://doi.org/10.1016/j.powtec.2021.01.074.

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