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[Jun Katagiri](https://orcid.org/0000-0002-6399-1951), [Masahiro Kusano](https://orcid.org/0000-0002-5061-0195), [Makoto Watanabe](https://orcid.org/0000-0002-5064-9583)

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[Statistical Variability of Image-Derived Observables in Rotating-Drum Tests of a Cohesive Metal Powder: Dynamic Angle of Repose Versus Contour-Length Statistics](https://mdr.nims.go.jp/datasets/dd01df27-d2ce-47b8-933b-47a857757a34)

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Statistical Variability of Image-Derived Observables in Rotating-Drum Tests of a Cohesive Metal Powder: Dynamic Angle of Repose Versus Contour-Length StatisticsAcademic Editor: Paul F. LuckhamReceived: 21 May 2026Revised: 3 July 2026Accepted: 16 July 2026Published: 21 July 2026Copyright: © 2026 by the authors.Licensee MDPI, Basel, Switzerland.This article is an open access articledistributed under the terms andconditions of the Creative CommonsAttribution (CC BY) license.ArticleStatistical Variability of Image-Derived Observables inRotating-Drum Tests of a Cohesive Metal Powder:Dynamic Angle of Repose Versus Contour-Length StatisticsJun Katagiri 1,* , Masahiro Kusano 2 and Makoto Watanabe 21 Degradation Diagnostics Research Team, Integrated Research Center for Resilient Infrastructure, NationalInstitute of Advanced Industrial Science and Technology, Higashi 1-1-1, Tsukuba 305-8567, Ibaraki, Japan2 Research Center for Structural Materials, National Institute for Materials Science, Sengen 1-2-1,Tsukuba 305-0047, Ibaraki, Japan* Correspondence: j-katagiri@aist.go.jpAbstractRotating-drum tests are widely employed for the characterization of powder flowabilitythrough image-derived free-surface observables. However, the statistical stability of theselected observable is rarely examined, although it directly determines the number ofimages required for reliable evaluation. In the present study, two observables obtainedfrom rotating-drum tests of a cohesive metal powder are compared: the dynamic angle ofrepose (AoR) and a contour-length-based index, R, derived from the extracted free-surfacecontour. Using identical experimental conditions and image sequences, the observablesare evaluated in terms of their mean values, coefficients of variation (CVs), and the samplesizes required to achieve a prescribed relative precision. The CV of AoR ranged from0.095 to 0.186, whereas that of R remained below 0.03. For a target relative error of 3%,the required numbers of cases were estimated to be 21, 84, and 189 for AoR at the 1σ, 2σ,and 3σ confidence levels, respectively, whereas those for R were 1, 1, and 2. These resultsdemonstrate that, even under identical apparatus and powder conditions, the statisticalburden of characterization depends strongly on observable choice. In the present study, thecontour-length-based index R is interpreted as a geometry-based descriptor of free-surfaceirregularity rather than as a direct measure of cohesion or adhesion. Because the analysis islimited to a single cohesive metal powder, the present findings should be interpreted as acase-specific demonstration within the current experimental framework rather than as ageneral law for all cohesive powders.Keywords: rotating drum; cohesive powder; dynamic angle of repose; image-derivedobservable; statistical variability1. IntroductionPowder flowability is important in a wide range of engineering processes, includinghandling, filling, mixing, granulation, and additive manufacturing. Among the avail-able experimental methods, rotating-drum tests have been widely used because theyallow continuous observation of free-surface dynamics with a relatively simple apparatusconfiguration [1–3].In practical applications, commercially available rotating-drum powder rheometerssuch as the GranuDrum and the Revolution Powder Analyzer (RPA) are also used, andPowders 2026, 5, 27 https://doi.org/10.3390/powders5030027https://crossmark.crossref.org/dialog?doi=10.3390/powders5030027&domain=pdf&date_stamp=2026-07-21https://creativecommons.org/licenses/by/4.0/https://creativecommons.org/licenses/by/4.0/https://www.mdpi.com/journal/powdershttps://www.mdpi.comhttps://orcid.org/0000-0002-6399-1951https://orcid.org/0000-0002-5061-0195https://orcid.org/0000-0002-5064-9583https://doi.org/10.3390/powders5030027Powders 2026, 5, 27 2 of 15image-derived angle-based and surface-shape-based observables have become establishedtools for powder characterization [4–7].In such instruments, the measured quantity is not limited to a single angle-based out-put. Depending on the analysis logic and the built-in processing workflow, rotating-drumcharacterization may provide observables related to average slope, interface irregularity,or other morphology-dependent features. Some descriptors are defined internally by theinstrument software, whereas others can be derived by custom image processing from thesame side-view images. This means that, even when the apparatus, powder, and operatingconditions are fixed, the outcome of characterization may still depend strongly on whichobservable is selected for analysis. From a measurement-design perspective, the choice ofobservable is therefore not merely a post-processing detail, but part of the experimentalprotocol itself.Among image-derived observables, the dynamic angle of repose (AoR) is the mostwidely used and serves as a practical descriptor of flowability, internal friction, andcohesion [8–12]. For cohesive or adhesive powders, however, the free surface often fluc-tuates irregularly owing to localized agglomeration, intermittent collapse, and surfaceundulations [8,9,13–15]. Under such conditions, observables should be evaluated not onlyin terms of physical meaning but also in terms of how stably they can be estimated.Even when an observable has a clear physical interpretation, large frame-to-framevariability implies that many images are required to estimate its mean reliably. Conversely,if another observable extracted from the same image sequence has substantially smallervariance, it may provide a more efficient and reproducible descriptor under the same appa-ratus and operating conditions. Therefore, in rotating-drum characterization, observableselection should account not only for physical interpretability but also for statistical stability,required sample size, and experimental cost.Despite its practical importance for measurement design, previous studies have mainlyfocused on mean AoR values, rotation-speed dependence, hysteresis, and qualitativefeatures of free-surface morphology [1,2,16,17]. In particular, for cohesive powders inrotating drums, irregular free surfaces and clustering have been discussed extensively,whereas direct comparisons of the statistical burden associated with different observablesremain limited [13,14,16,17]. A systematic discussion of coefficients of variation and thenumber of cases required for statistically reliable estimation is still uncommon.At the same time, the broader science of powder flowability characterization includeswell-established shear-test-based frameworks in which bulk quantities such as yield lo-cus, cohesion, internal friction, unconfined yield strength, and flow function are used tocharacterize powder behavior and its reproducibility [18,19]. Schulze’s round-robin studyon ring shear testers is particularly relevant because it treats not only mean shear stressesand yield loci, but also standard deviations, confidence intervals, reference ranges, andthe influence of ambient conditions on reproducibility [18]. Tomasetta et al. further illus-trate how annular shear-cell measurements can provide Coulomb-type descriptors suchas cohesion, internal friction angle, major principal stress, unconfined yield strength, andflow factor, and can detect changes in cohesive behavior caused by temperature-dependentinterparticle interactions or liquid-bridge formation [19]. In parallel, recent morphology-oriented studies have also highlighted that full-field or profile-shape information mayprovide complementary insight beyond single angle-based descriptors alone [20].The relevant literature can be grouped into several partially overlapping streams.One stream focuses on classical rotating-drum or related angle-of-repose measurements,where mean values, flow regimes, or hysteresis are the main outcomes. A second streamis associated with commercial powder-rheometry tools, in which built-in image-deriveddescriptors, including angle-based and morphology-related indices, are used for practi-https://doi.org/10.3390/powders5030027https://doi.org/10.3390/powders5030027Powders 2026, 5, 27 3 of 15cal powder characterization. A third stream addresses repeatability, reproducibility, ormeasurement sensitivity, often from the viewpoint of instrument performance rather thandirect comparison between observables. A fourth stream examines cohesive or partiallysaturated granular flow, where clustering, adhesion, and irregular free-surface morphologyare central issues. These studies are all relevant to the present work, but they are not usuallydiscussed within a common framework of statistical burden and observable selection.Table 1 summarizes representative studies relevant to rotating-drum characterization.In addition to classical angle-based measurements, the recent literature includes commercialimage-derived descriptors implemented in instruments such as the GranuDrum and theRevolution Powder Analyzer, standardization efforts, and studies addressing repeatabilityor measurement sensitivity. However, these aspects are not always discussed together. Asindicated in Table 1, previous studies have commonly emphasized mean AoR values, flowregimes, or instrument-defined descriptors. By contrast, direct comparisons of statisticalburden across different observables extracted from the same image sequence remain limited.This distinction is especially important for cohesive powders, in which clustering, adhesion,and intermittent collapse may increase the frame-to-frame variability of some observablesmuch more strongly than others.Table 1. Representative studies relevant to rotating-drum characterization, with emphasis on observ-ables, variability reporting, and morphology-related descriptors.Ref. System Method Observable Variability Morphology[1] Dry grains Drum AoR Ltd. No[16] Wet grains Drum AoR/flow Partial Qual.[4] Powders GranuDrum Drum indices No Yes[5] AM powder RPA AoR/fractal No Yes[7] Metal powders Drum rheom. R&R metrics R&R Built-in[6] Powders GranuDrum Flowability Standard Built-in[21] Powders Drum tools Sensitivity No Poly-fit[22] Metal powders Multi-test AoR/AIFE No No[13] Sat. grains Drum Flow regime No Qual.[15] Cohesive grains Drum Regime/size No Qual.[23] Cohesive grains Drum AoR/velocity No Qual.[18] Limestone powder Ring shear Yield locus/SD Yes No[19] Powders Annular shear C/ϕ/ fc/ f f c Rep. NoAoR, angle of repose; R&R, repeatability and reproducibility; Qual., qualitative; Ltd., limited; Sat., partially satu-rated; AIFE, avalanche angle/basic flowability evaluation; SD, standard deviation; Rep., repeated measurements;C, cohesion; ϕ, angle of internal friction; fc, unconfined yield strength; f f c, flow factor.The studies summarized above also indicate that the present work lies at the intersec-tion of two broader streams of research. One is the established literature on bulk flowabilitycharacterization, where shear-test-based descriptors such as cohesion, internal friction,yield locus, unconfined yield strength, and flow function are used to quantify powderbehavior and its reproducibility [18,19]. In that literature, the material response is obtainedunder prescribed consolidation and shear histories, and the scatter of the measured stressesis itself an important part of method validation [18]. The other stream is the growingrecognition that morphology- or profile-based descriptors can provide complementaryinformation beyond single angle-based metrics, especially when free-surface geometryfluctuates irregularly [20]. These broader contexts motivate the present comparison, whilethe scope of this study remains limited to two observables extracted from the same rotating-drum image sequence.The scope of the present study should therefore be defined precisely. The objectiveis neither to replace existing instrument-defined descriptors nor to argue that a singlehttps://doi.org/10.3390/powders5030027https://doi.org/10.3390/powders5030027Powders 2026, 5, 27 4 of 15morphology-related quantity should supersede angle-based characterization in general.Rather, the present work examines whether different observables extracted from the sameimage sequence impose materially different statistical burdens on experimental evalu-ation. A direct comparison within a single apparatus, a single powder system, and asingle image-analysis framework is advantageous in this respect, because it isolates theinfluence of observable definition from variations in hardware, operating protocol, andmaterial class. The study is thus positioned as an examination of measurement design inrotating-drum characterization, with particular emphasis on repeatability, required samplesize, and the experimental cost of obtaining statistically reliable representative values.Within this framework, the contour-length-based index R is treated as a geometry-basedobservable that reflects the tortuosity or morphology-related irregularity of the extractedfree-surface contour. It is not introduced here as a direct measure of cohesion, powder-walladhesion, or any other single bulk material property. Accordingly, the present results areintended as a methodological case study for the powder examined here, not as a universalcharacterization rule applicable to all cohesive powders.In this study, we use the GranuDrum rotating-drum analyzer to examine a cohe-sive Hastelloy X metal powder. From the same image sets, we extract two observables:the dynamic angle of repose, as a representative angle-based descriptor, and a contour-length-based index R, derived from the extracted free-surface contour. We compare theseobservables from a statistical perspective by evaluating not only their mean values butalso their standard deviations, coefficients of variation (CVs), and the number of casesrequired to estimate the mean at a prescribed relative precision. The purpose of this studyis not to propose a new universal powder metric but rather to demonstrate how observablechoice governs statistical stability and measurement burden in image-based rotating-drumcharacterization. Accordingly, the physical role of R is discussed in terms of the free-surfacegeometry that it reflects, rather than in terms of a claimed one-to-one correspondence witha specific material parameter.2. Materials and Methods2.1. MaterialThe powder used in this study was a gas-atomized Hastelloy X metal powder manu-factured by Högänas Högänas AB, Högänas, Skåne Län, Sweden. Hastelloy X is a Ni-basedsuperalloy with high corrosion resistance and good high-temperature strength, and it iscommonly supplied for additive manufacturing applications. The particle-size distributionof the powder was characterized by a mean particle size of 29 µm, with D10 = 19 µmand D90 = 46 µm. A nominal particle density of 8.22 × 103 kg m−3 was used as a mate-rial descriptor of the Hastelloy X alloy. In contrast, independent measurements of bulkdensity, cohesion, and powder–wall interaction parameters were not part of the presentrotating-drum campaign. Accordingly, the present study does not assign separate quanti-tative values to those properties, and the interpretation is restricted to the image-derivedobservables extracted from the rotating-drum tests. The powder was stored in a laboratoryenvironment at 20–25 ◦C without active humidity control.The powder adhered weakly to the inner wall of the storage container. During therotating-drum tests, the free surface did not remain smooth or straight but instead showedirregular temporal variations accompanied by the formation and disappearance of cohesiveclusters. In this study, such behavior was regarded as being consistent with cohesivepowder flow.https://doi.org/10.3390/powders5030027https://doi.org/10.3390/powders5030027Powders 2026, 5, 27 5 of 152.2. Rotating-Drum Apparatus and ProtocolExperiments were conducted using a commercially available GranuDrum GranutoolsSRL, Awans, Belgium. The drum is a horizontally mounted cylindrical cell with an innerdiameter of 84 mm. The powder was loaded to approximately half of the drum volume,corresponding to a filling fraction of about 0.5.The rotation speed ω was set to 2, 4, 6, 8, and 10 rpm. For each experimental series, anincreasing–decreasing protocol was adopted. The drum speed was first increased stepwisefrom 2 to 10 rpm and then decreased stepwise from 10 to 2 rpm. Representative side-view images for the experiments are shown in Figure 1, together with typical examplesof local image noise as described in the next subsection. The five values above representdistinct nominal rotation-speed setpoints. Because each setpoint was visited on both theincreasing and decreasing branches, each experimental series contained ten speed–branchconditions. The protocol was a stepwise setpoint protocol rather than a continuous ramp-rate experiment; images acquired during transitions between setpoints were not usedfor the analysis. Instead, the side-view images used for analysis were those recordedafter the flow had reached a steady state at each prescribed speed. The 2–10 rpm rangewas selected as a low-speed protocol that maintained a stable side-view free surface forthe tested cohesive Hastelloy X powder while providing multiple regularly spaced flowconditions. The purpose of this speed selection was to compare the statistical stability ofthe two image-derived observables under identical operating conditions, rather than tomap all possible flow regimes over the full operating range of the apparatus.Figure 1. Representative side-view images obtained during the increasing and decreasing speedsequences in the rotating-drum tests. The upper row shows the increasing branch (2, 4, 6, 8, and10 rpm), and the lower row shows the decreasing branch (2, 4, 6, 8, and 10 rpm). Dotted boxes indicatetypical examples of local image noise caused by wall-adhered particles or isolated powder fragments.At each speed, side-view images of the free surface were recorded after the flowhad reached a steady state. For each condition, 20 images were analyzed. With threeindependent series, two speed branches, five rotation speeds, and 20 images per condition,the total number of analyzed images was 600.In this study, apparatus and imaging conditions were kept as constant as possible.This design enabled a direct comparison of how differences in observable definition appearas differences in statistical variability.https://doi.org/10.3390/powders5030027https://doi.org/10.3390/powders5030027Powders 2026, 5, 27 6 of 15The main powder properties and experimental conditions are summarized in Table 2.This summary distinguishes between nominal descriptors available for the tested alloy andquantities that would require separate dedicated rheological measurements.Table 2. Powder properties and experimental conditions used in the rotating-drum tests.Item ValuePowder Gas-atomized Hastelloy XMean particle size 29 µmD10 19 µmD90 46 µmNominal particle density ρp 8.22 × 103 kg m−3Image-derived observables AoR and RIndependent bulk-density measurement Not performed in the present campaignIndependent cohesion/powder–wallinteraction Not measured in the present campaignStorage temperature 20–25 ◦CHumidity control Not controlledDrum inner diameter D 84 mmFilling fraction ∼0.5Rotation speeds ω 2, 4, 6, 8, and 10 rpmNumber of nominal speed setpoints 5Speed increment 2 rpmProtocol Increasing–decreasingSpeed–branch conditions per series 10Number of series 3Images per condition 20Images during speed transitions Not used for analysisTotal number of images 6002.3. Image Processing and Definition of ObservablesThe acquired images were first converted to grayscale and then binarized to extractthe powder region. Otsu’s method [24] was used to determine the threshold automatically.To avoid discontinuities and strong curvature near the wall boundaries, a centralsquare analysis window of side length 0.6D was used for free-surface analysis because theimage width of the circular drum region corresponds to the drum diameter D. Figure 2illustrates the analysis window, the fitted angle-of-repose line, the contour-length-basedindex (R), and typical examples of local image noise considered in the extraction procedure.The free-surface contour was extracted by scanning this analysis window column bycolumn. For each column, contiguous powder regions were identified, and the uppermostpixel of each contiguous region was treated as a free-surface candidate. A continuous free-surface profile was then constructed by selecting candidate points sequentially accordingto vertical continuity with the neighboring column. This procedure reduced the risk ofmisidentifying isolated adhered particles and detached powder clusters as part of thefree surface.The dynamic angle of repose, θ, was defined as the inclination angle of the best-fitstraight line to the extracted free-surface points. Because the image coordinate system ispositive downward, the sign of the vertical coordinate was inverted before linear fitting.The contour-length-based index R was defined from the same free-surface point set.The polyline length of the extracted free surface, Lsurface, was calculated and normalized bythe straight-line distance between the two endpoints of the contour, Lchord:R =LsurfaceLchord.https://doi.org/10.3390/powders5030027https://doi.org/10.3390/powders5030027Powders 2026, 5, 27 7 of 15For a nearly straight free surface, R ≈ 1. As the free surface becomes more tortuous,Lsurface increases and R becomes larger. In this study, R is therefore interpreted as ageometry-based descriptor of the irregularity or tortuosity of the extracted free surface.Because it is derived solely from the contour shape, R should not be interpreted as a directmeasure of cohesion, adhesion, or other single material properties. Instead, it provides acomplementary observable that is sensitive to the morphology of the free surface under thepresent rotating-drum conditions.Figure 2. Definition of the image-based observables and examples of local image noise. The upperpanel illustrates a central square analysis window of side length 0.6D, where D denotes the drumdiameter and corresponds to the image width of the circular drum region, together with the extractedfree-surface contour, the fitted line for the dynamic angle of repose (AoR), and the contour-length-based index R. The lower panel shows typical local image noise, such as wall-adhered particles andisolated powder fragments.The images may contain local noise caused by powder particles adhering to the wallor by isolated powder fragments (see Figures 1 and 2). Such noise may artificially extendthe extracted free-surface contour and can result in overestimation of the contour lengthand instability in slope estimation. In the present study, only the uppermost pixel of eachcontiguous powder region in a given column was regarded as a free-surface candidate, andthe final candidate was selected according to continuity with neighboring columns. Asillustrated in Figure 2, this procedure mitigated the influence of such noise on the extractedfree surface. Although the present extraction logic reduces the impact of image noise,some sensitivity to image-processing parameters is nevertheless expected. A systematicassessment of this sensitivity is left for future work.2.4. Statistical AnalysisFor each condition (series, branch, and rotation speed), the instantaneous values ofθ and R were obtained from individual images. From these values, the mean, standarddeviation, and coefficient of variation (CV) were calculated. The CVs were defined asCVθ =σθθ̄, CVR =σRR̄,https://doi.org/10.3390/powders5030027https://doi.org/10.3390/powders5030027Powders 2026, 5, 27 8 of 15where θ̄ and R̄ are the mean values of θ and R, and σθ and σR are their correspondingstandard deviations.To quantify the sampling burden, the minimum number of cases required to es-timate a mean value within a prescribed relative error ε was evaluated using thenormal approximation:n ≥(z CVε)2,where z is the standard normal quantile corresponding to the confidence level. In this study,ε = 0.03 was used, and the required number of cases was evaluated for 68.3% (1σ), 95.4%(2σ), and 99.7% (3σ) confidence levels.2.5. Temporal Dependence AssessmentBecause the image-by-image measurements were obtained from continuous rotating-drum sequences, the possible temporal dependence between successive observationswas examined additionally. For each condition (series, branch, and rotation speed), lag-k sample autocorrelation coefficients were calculated separately for the dynamic angleof repose (θ) and the contour-length-based index (R) using the image order within theanalyzed sequence:ρk =∑N−kt=1 (xt − x̄)(xt+k − x̄)∑Nt=1(xt − x̄)2,where xt denotes the observable at image index t, x̄ is its sample mean, and N is the numberof analyzed images for the condition. In the present study, N = 20 for each condition. Thepurpose of this supplementary analysis was not to establish an exact stochastic model forthe free-surface dynamics, but to assess whether strong serial dependence was present toan extent that would substantially affect the interpretation of the sample-size estimates.Because the image acquisition interval was not available from the outsourced experimentalprocedure, the autocorrelation analysis was used as an empirical diagnostic rather thanas a strict proof of independence. Accordingly, the required numbers of cases reported inthis study should be interpreted primarily as comparative indicators of relative statisticalburden within the present experimental framework.3. Results3.1. Representative Images and Extracted Free SurfacesWithin the analysis window shown in Figure 2, the free surface was tracked continu-ously from left to right, and both AoR and R were computed from the same set of extractedfree-surface points. The differences between AoR and R, as discussed below, can thereforebe attributed to differences in the definition of the observable rather than to variations inimage input or apparatus conditions.3.2. Mean Values of AoR and RFigure 3 shows the mean values of AoR and R as functions of rotation speed ω. Theupper panel shows the mean AoR, whereas the lower panel shows the mean contour-length-based index R. The mean AoR, θ̄, ranged approximately from 33 to 39◦, with anoverall mean of 35.4◦. The mean contour-length-based index, R̄, ranged from 1.028 to 1.056,with an overall mean of 1.041. Neither observable showed a strong systematic trend overthe tested speed range, although R remained confined to a narrower range than AoR.https://doi.org/10.3390/powders5030027https://doi.org/10.3390/powders5030027Powders 2026, 5, 27 9 of 15Figure 3. Mean values of the two image-derived observables as functions of rotation speed ω. Theupper panel shows the mean dynamic angle of repose, θ̄, and the lower panel shows the mean contour-length-based index, R̄. Black and red symbols denote the increasing (speed-up) and decreasing(speed-down) branches, respectively. Error bars indicate the standard deviations calculated from theimage-by-image measurements.3.3. Comparison of Variability: AoR Versus RFigure 4 compares the coefficients of variation of AoR and R on the same axis. The CVof AoR ranged from 0.095 to 0.186, with an overall mean of 0.137. In contrast, the CV ofR ranged from 0.0069 to 0.027, with an overall mean of 0.014. Thus, even though the twoobservables were extracted from the same image sets, the relative variability of AoR wasapproximately one order of magnitude larger than that of R.AoR is the slope of a fitted straight line and is therefore sensitive to local surfaceundulations and small collapse events. By contrast, R represents the total length of the freesurface and effectively integrates local fluctuations over the entire contour. For this reason,R tends to show smaller relative variability.https://doi.org/10.3390/powders5030027https://doi.org/10.3390/powders5030027Powders 2026, 5, 27 10 of 15Figure 4. Direct comparison of the coefficients of variation (CVs) of the dynamic angle of repose (AoR)and the contour-length-based index R as functions of rotation speed ω. Black and red symbols denotethe increasing (speed-up) and decreasing (speed-down) branches, respectively. Square markerscorrespond to AoR, and circular markers correspond to R.3.4. Required Number of CasesAs suggested by the large difference in CV shown in Figure 4, the sampling burdendiffered markedly between the two observables. Table 3 summarizes the required numbersof cases for a target relative error of 3% at three confidence levels. The difference in CVdirectly appears as a difference in the number of cases required for reliable mean estimation.For AoR, the required numbers of cases were 21, 84, and 189 for 1σ, 2σ, and 3σ, respectively.For R, the corresponding numbers were 1, 1, and 2.Table 3. Required numbers of cases for estimating the mean values of AoR and R within a relativeerror of 3% at three confidence levels.Confidence Level z AoR R68.3% (1σ) 1 21 195.4% (2σ) 2 84 199.7% (3σ) 3 189 2These results indicate that, even under identical apparatus, powder, and operatingconditions, the experimental burden required to obtain a statistically reliable representativevalue depends strongly on which observable is selected.3.5. Autocorrelation AnalysisFigure 5 summarizes the lag-1 autocorrelation coefficients of θ and R for each rota-tion speed and branch. For both observables, the lag-1 autocorrelation coefficients weregenerally close to zero and showed no systematic positive trend across the tested speeds.Averaged over all 30 conditions, the lag-1 autocorrelation coefficient was −0.093 for θ and−0.116 for R.https://doi.org/10.3390/powders5030027https://doi.org/10.3390/powders5030027Powders 2026, 5, 27 11 of 15Figure 5. Lag-1 autocorrelation coefficients of the dynamic angle of repose (AoR) and the contour-length-based index R as functions of rotation speed ω. The upper panel shows AoR and the lowerpanel shows R. Black and red symbols denote the increasing (speed-up) and decreasing (speed-down)branches, respectively. Each point represents the mean over the three repeated series, and the errorbars denote the standard deviation. The horizontal dashed line indicates ρ1 = 0, and the dotted linesindicate the approximate white-noise reference bounds, ±1.96/√20.Using the approximate white-noise reference bound ±1.96/√20, only 4 of 30 condi-tions for θ and 4 of 30 conditions for R exceeded this range in absolute value, and all of thesecases corresponded to negative rather than positive lag-1 autocorrelation. These resultssuggest that, for the present image sets, the observed contrast in statistical burden betweenθ and R is unlikely to be explained primarily by strong positive frame-to-frame correlation.Accordingly, the required numbers of cases reported in Table 3 remain useful as approxi-mate comparative indicators of relative sampling burden. At the same time, because themeasurements were extracted from continuous image sequences and the image acquisitioninterval was not available from the outsourced experimental procedure, the sample-sizeestimates should not be interpreted as universally transferable exact requirements.4. Discussion4.1. Why Do AoR and R Impose Different Statistical Burdens?The difference in statistical stability between AoR and R can be understood directlyfrom their definitions. AoR is the inclination of a fitted straight line and therefore respondsdirectly to local changes in the free-surface geometry. In cohesive powder flow, localizedhttps://doi.org/10.3390/powders5030027https://doi.org/10.3390/powders5030027Powders 2026, 5, 27 12 of 15cluster formation and intermittent collapse can perturb the fitted slope significantly, leadingto relatively large frame-to-frame fluctuations in AoR.By contrast, R is based on the total contour length normalized by the chord length.Local roughness events do contribute to Lsurface, but their effects are distributed over thewhole contour. As a result, R is less sensitive to local surface events and tends to remainstatistically more stable. At the same time, this statistical stability should not be takento mean that R is universally more physically informative than AoR. Rather, the twoobservables emphasize different aspects of the same free surface: AoR is more directlyrelated to the mean inclination, whereas R reflects contour irregularity distributed over theanalyzed interface.4.2. Relationship to Established Powder Flowability Characterization and Implications forImage-Based ObservablesThe connection with established powder flowability characterization is important,but it should be interpreted carefully. In ring shear or annular shear tests, the powderis subjected to prescribed normal stresses and shear histories, and the resulting yieldloci are used to derive bulk material descriptors such as cohesion, internal friction angle,unconfined yield strength, and flow function [18,19]. Schulze’s round-robin test alsoshows that reproducibility is evaluated through the scatter of measured shear stressesand that ambient conditions such as humidity can contribute to the observed standarddeviation [18]. Tomasetta et al. show that annular shear-cell measurements can detectchanges in cohesive behavior when interparticle interactions change, for example throughtemperature-dependent liquid-bridge formation [19].An important point is that this study does not claim to introduce a fundamentallynew roughness metric. Nor does it claim that R is a direct surrogate for cohesion, adhesion,internal friction, unconfined yield strength, or flow factor. The practical relevance of R inthe present work lies in the fact that it captures morphology-related irregularity of the freesurface while requiring a much smaller sampling burden under otherwise identical imagingconditions. For example, the GranuDrum itself provides an automated roughness-relatedindex, and previous studies have already recognized the importance of free-surface lengthand contour geometry in rotating-drum characterization [3,5,6,23]. At the same time, thebroader powder-characterization literature includes established shear-test-based descrip-tors such as yield locus, cohesion, internal friction angle, unconfined yield strength, andflow function, as well as recent arguments that morphology-based information may com-plement angle-based metrics [18–20]. The present rotating-drum observables are thereforebest regarded as image-derived descriptors that are complementary to conventional bulkflowability measurements, rather than as substitutes for them. However, although previousstudies have reported that cohesion can increase variability in angle-based measurements,systematic discussions of statistical accuracy, such as comparisons of CV and requiredsample size across different observables, remain limited [13,14,16,17].The key contribution of the present study is to show that, even within a single ex-periment, the reliability of measured values depends strongly on observable choice. Asa consequence, the number of cases required to obtain a statistically reliable representa-tive value can change by more than an order of magnitude. In this sense, the presentwork should be viewed not as a proposal of a replacement for AoR, but as an experimen-tal demonstration that observable choice governs measurement burden in image-basedrotating-drum characterization. The statistical stability of an observable and its physicalrelevance should therefore be discussed separately: the former determines how efficientlya representative value can be estimated, whereas the latter depends on which feature ofthe free-surface dynamics the observable is designed to reflect. Because only one cohesivemetal powder was examined, the magnitude of the difference observed here between AoRhttps://doi.org/10.3390/powders5030027https://doi.org/10.3390/powders5030027Powders 2026, 5, 27 13 of 15and R should not be assumed to be universally transferable across powder classes, particlemorphologies, or cohesion levels.A further implication of the present results concerns DEM parameter calibration.The angle of repose has long been used as a convenient macroscopic target for DEMcalibration because of its experimental simplicity. However, recent inverse-analysis workhas shown that calibration based solely on the angle of repose can become ill-conditioned,and that observational variability in AoR may strongly amplify the uncertainty of calibratedparameters, particularly those associated with rolling resistance [25]. From this perspective,the present result is relevant: the contour-length-based index R exhibited substantiallysmaller CV and a much lower sampling burden than AoR under the same experimentalconditions. This suggests that low-variability observables such as R, when combinedwith AoR, may provide a more stable experimental basis for future multi-objective DEMcalibration. At present, however, this study does not perform such calibration explicitly,and the extent to which R improves parameter identifiability should be examined infuture work. The additional autocorrelation analysis showed no systematic positive serialdependence in either observable sequence. Therefore, the large contrast in required samplesize between AoR and R is unlikely to be explained solely by temporal persistence inthe image acquisition order, although the reported sample-size values should still beinterpreted as approximate comparative indicators within the present framework.4.3. LimitationsThis study has several limitations that should be acknowledged. First, only onecohesive metal powder was examined, and generalization to non-cohesive powders orpowders with varying degrees of cohesion requires further investigation. Accordingly,the present study should be interpreted as a methodological case study rather than asa cross-material validation. In addition, the tested rotation-speed range was limited to2–10 rpm; therefore, the conclusions should not be extrapolated to higher-speed regimes orto other operating protocols without additional validation. Second, the analysis was basedon two-dimensional side-view images and did not address possible three-dimensionalfree-surface effects. Third, the contour-length-based index R is a deliberately simple geo-metrical quantity, and comparison with other roughness metrics or multiscale descriptorsremains a subject for future work. Likewise, its relationship to independent measure-ments of cohesion, adhesion, or bulk flowability has not been established in the presentstudy and should be examined separately in future work. Because independent ring-shear,annular-shear, wall-friction, compression, or other powder-rheometry measurements werenot performed in the present campaign, no quantitative correlation can be established herebetween R and conventional flowability descriptors such as yield locus, cohesion, internalfriction angle, unconfined yield strength, flow function, compressibility, or stress-relaxationparameters. Cross-material validation, including powders with different morphologiesand cohesion levels and comparison with independent shear-cell or powder-rheometermeasurements, remains an important subject for future work. Finally, because the im-age acquisition interval was not available from the outsourced experimental procedure,temporal dependence could only be assessed indirectly from the observable sequencesthemselves. Although the additional autocorrelation analysis did not indicate systematicpositive serial dependence, the sample-size estimates reported here should still be regardedas approximate comparative indicators rather than exact universal requirements.5. ConclusionsIn this study, a commercially available GranuDrum was used to evaluate the statisticalstability of two image-derived observables for a cohesive Hastelloy X metal powder: thehttps://doi.org/10.3390/powders5030027https://doi.org/10.3390/powders5030027Powders 2026, 5, 27 14 of 15dynamic angle of repose (AoR) and a contour-length-based index R. The main findings canbe summarized as follows.1. An improved free-surface extraction procedure enabled stable tracking of the freesurface within the central square analysis window of side length 0.6D.2. The coefficient of variation of AoR ranged from 0.095 to 0.186, with an average of0.137, whereas the coefficient of variation of R ranged from 0.0069 to 0.027, with anaverage of 0.014. Thus, R showed approximately one order of magnitude smallerrelative variability than AoR.3. For a target relative error of 3%, the required numbers of cases for AoR were 21, 84,and 189 at the 1σ, 2σ, and 3σ confidence levels, respectively, whereas those for R were1, 1, and 2. The additional autocorrelation analysis showed no systematic positivelag-1 serial dependence in either observable sequence, supporting the use of thesevalues as approximate comparative indicators of relative sampling burden.4. These results demonstrate that, in rotating-drum image analysis, observable choicestrongly affects statistical stability and measurement efficiency. Within the presentframework, R should be interpreted as a geometry-based descriptor of free-surfaceirregularity, not as a direct measure of cohesion or adhesion.Overall, these results indicate that image-based characterization of cohesive powderflow should consider not only the physical meaning of an observable but also its statisticalstability and sampling burden. For the cohesive Hastelloy X powder examined here, thispoint is established as a case-specific experimental result within the present rotating-drumimage-analysis framework.Author Contributions: Conceptualization, J.K.; methodology, J.K.; software, J.K.; validation,J.K.; formal analysis, J.K., M.K. and M.W.; investigation, J.K.; resources, J.K.; data curation, J.K.;writing—original draft preparation, J.K. and M.K.; writing—review and editing, J.K., M.K. and M.W.;visualization, J.K. All authors have read and agreed to the published version of the manuscript.Funding: This research received no external funding.Data Availability Statement: The processed data and analysis scripts supporting the findings of thisstudy are available from the corresponding author upon reasonable request. The raw image data arenot publicly available at this stage because they have not yet been prepared for public deposition.Conflicts of Interest: The authors declare no conflicts of interest.AbbreviationsThe following abbreviations are used in this manuscript:AM Additive manufacturingAoR Angle of reposeCV Coefficient of variationDEM Discrete element methodQual. QualitativeR&R Repeatability and reproducibilityRPA Revolution Powder AnalyzerReferences1. Liu, X.Y.; Specht, E.; Mellmann, J. Experimental study of the lower and upper angles of repose of granular materials in rotatingdrums. Powder Technol. 2005, 154, 125–131. [CrossRef]2. Liu, X.Y.; Zhang, Y.Y. 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MDPI and/or the editor(s) disclaim responsibility for any injury topeople or property resulting from any ideas, methods, instructions or products referred to in the content.https://doi.org/10.3390/powders5030027http://dx.doi.org/10.1016/j.measurement.2022.111548http://dx.doi.org/10.1016/j.addma.2018.09.023http://dx.doi.org/10.1016/j.powtec.2024.119810http://dx.doi.org/10.1080/00018730500167855http://dx.doi.org/10.1080/00018730600626065http://dx.doi.org/10.1073/pnas.2107965118http://dx.doi.org/10.1016/j.powtec.2021.01.010http://dx.doi.org/10.1016/j.powtec.2008.04.089http://dx.doi.org/10.1063/5.0166241http://dx.doi.org/10.1103/physreve.64.051301http://dx.doi.org/10.1017/jfm.2025.26http://dx.doi.org/10.1016/j.powtec.2011.09.010http://dx.doi.org/10.1103/lygy-vjljhttp://www.ncbi.nlm.nih.gov/pubmed/40826592http://dx.doi.org/10.1016/j.apt.2010.10.015http://dx.doi.org/10.1016/j.apt.2012.11.007http://dx.doi.org/10.48550/arXiv.2605.09371http://dx.doi.org/10.1016/j.powtec.2024.120231http://dx.doi.org/10.1038/s41598-020-77974-3http://www.ncbi.nlm.nih.gov/pubmed/33273528http://dx.doi.org/10.1103/PhysRevE.79.011305http://www.ncbi.nlm.nih.gov/pubmed/19257028http://dx.doi.org/10.1109/TSMC.1979.4310076http://dx.doi.org/10.1016/j.cpms.2026.03.008https://doi.org/10.3390/powders5030027 Introduction Materials and Methods Material Rotating-Drum Apparatus and Protocol Image Processing and Definition of Observables Statistical Analysis Temporal Dependence Assessment Results Representative Images and Extracted Free Surfaces Mean Values of AoR and R  Comparison of Variability: AoR Versus R  Required Number of Cases Autocorrelation Analysis Discussion Why Do AoR and R  Impose Different Statistical Burdens? Relationship to Established Powder Flowability Characterization and Implications for Image-Based Observables Limitations Conclusions References