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Modelling and experiments on the effect of air humidity on the flow properties of glass powders
Powder Technology 207 (2011) 437–443
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Powder Technology j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / p ow t e c
Modelling and experiments on the effect of air humidity on the flow properties of glass powders Giovanna Landi, Diego Barletta, Massimo Poletto ⁎ Dipartimento di Ingegneria Chimica e Alimentare, Università di Salerno, Via Ponte Don Melillo, I-84084 Fisciano (SA), Italy
a r t i c l e
i n f o
Article history: Received 12 May 2010 Received in revised form 22 November 2010 Accepted 26 November 2010 Available online 3 December 2010 Keywords: Cohesion Powder flow properties Humidity Capillary forces
a b s t r a c t The effect of air humidity on flow properties of two different cuts of fine glass beads was experimentally studied by means of shear tests. The moisture content of powder samples was conditioned by humid air at relative humidities between 13% and 98% in a fluidization column. In spite of the very low moisture contents in the powder (b 0.2%) obtained by this technique, a significant change in the powder cohesion was observed, while the effect on the angle of internal friction was limited. A model based on the Kelvin equation and the Laplace–Young equation was applied to describe the capillary condensation between touching asperities found on the surface of neighbouring particles and to estimate the relevant interparticle forces. This result was used to derive the powder tensile strength following the Rumpf approach and compare it with values derived from shear experiments. A single value of the unknown capillary bridge gap was chosen on the basis of the observed dependence of interparticle cohesive forces. This gap was able to provide quantitative agreement between model predictions and experimental results at relative humidity lower than 80%. Comparison between the total moisture content and the amount of water in liquid bridges indicated that water mainly condenses on rough surfaces of the particles and only a small portion of this condensed humidity contributes to change the powder flow properties by interparticle capillary forces. © 2010 Elsevier B.V. All rights reserved.
1. Introduction Powders often show flow problems due to cohesive properties that may compromise the reliability of many operations of industrial processes such as storage, handling, feeding and fluidization. In particular, problems are more troublesome when the powder treated is really cohesive, because it can form dead zones, steady channels and agglomerates. Powders flow properties are strongly dependent not only on intrinsic physical properties of powders such as particle shape, roughness and particle size distribution, but also on operating conditions such as system temperature, presence of a liquid phase or moisture content. The latter is one of the properties dependent on the environmental conditions which may more strongly affect the powder behavior. In fact, moisture can cause the formation of liquid bridges between particles giving rise to interparticle capillary forces. These forces are due to the combined effects of the surface tension acting on the liquid bridge interface and of the curvature of the bridge surface [1–3]. As a result, capillary forces cause an increase of powder cohesion and tensile strength and, thus, negatively affect the powder flowability [4–8]. Moreover, capillary forces modify also other powder bulk properties such as voidage, aggregative structure and permeability. These properties and the relative importance of capillary forces with respect to fluid dynamic forces seriously affect also the ⁎ Corresponding author. Tel.: +39 089 964 132; fax: +39 089 964 057. E-mail address: [email protected] (M. Poletto). 0032-5910/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.powtec.2010.11.033
fluidization behavior of fine powders [9–11]. On the other hand, the effect of capillary forces can be beneficial in some applications such as powder agglomeration [12,13]. Moisture content of powders can be determined by the capillary condensation of the surrounding atmosphere humidity [14]. Consequently, powder flow properties are influenced by the humidity of the storage environment [6,15–19]. In some of these experimental studies the liquid bridges formed by capillary condensation interact chemically with the particle surface giving rise to the so called “caking” phenomenon. The latter is not addressed in this paper, which is focused only on the contribution of capillary forces to cohesion. A large number of theoretical and experimental studies in the literature are devoted to the effect of capillary condensation on interparticle forces [3]. Results of the direct measurements of interparticle forces by atomic force microscopy show wide scatter mainly due to the complex effect of the particle size, shape and surface roughness and of the chemical nature of the surface [20]. Nevertheless, the theoretical model based on the combination of the Kelvin equation with the Laplace–Young equation is widely accepted [3]. In this framework a dramatic effect of particle surface roughness on capillary condensation and thus on capillary forces was highlighted [3,21]. In particular, capillary forces are substantially lower in presence of nanoscale roughness since liquid bridges form between asperities of smaller curvature radius. Moreover, the critical relative humidity, at which capillary condensation is first observed, increases as roughness on the nanoscale increases.
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A similar modelling approach accounting for the role of asperities was followed by Bocquet et al. [22] to successfully predict a bulk property as the angle of avalanche of glass beads as a function of air humidity. This paper aims at developing a model for the prediction of the flow properties of powders exposed to humid air, with the basic hypothesis that, in such conditions, capillary condensation is the main phenomenon which is responsible for the formation of liquid bridges. It has been pointed out in previously published works that humidity has a completely different effect on porous and non porous materials [9]. In porous materials most of the moisture content is found within the internal particle porosity and does not play a significant role in the determination of interparticle capillary forces due to liquid bridges. In order to approach the problem from a simpler perspective and to close the material balances on the condensed water more easily, in this work only non porous particles are considered. Fig. 1. Scheme and nomenclature in a capillary bridge formed between two particle asperities.
2. Theory Humidity can condense in powders even from unsaturated air due to reduction of the fugacity of water in the condensed phase in turn due to either soluble material present on particle surfaces or to the curvature of the surface of the condensed phase in small volumes delimited by solid particle surfaces, the so-called capillary condensation. In this study the main hypothesis is that condensation of air humidity is only due to the formation of curved air liquid interfaces in which the joint action of surface tension and interface curvature determines some vapor pressure reduction within the liquid. This is the case of the concave surfaces in capillary condensation, where the liquid vapor pressure P'0 is reduced with respect to that measured over a planar surface P0. The Kelvin equation relates the reduced vapor pressure with the total pressure difference between gas and liquid, ΔP: P0′ −ΔP = exp P0 ρl RT
P0′ P0
ð1Þ
ð2Þ
The total pressure difference between gas and liquid pressure is determined by the action of surface tension, σs, over the concave interface and is a function of the local average radius, rm ΔP =
2σ s rm
ð3Þ
The convention used here is that a concave surface has a positive radius and a liquid pressure smaller than the gas pressure. The opposite occurs for convex surfaces. The theory of capillary bridges, reported by Pierrat and Caram [5], corrects Eq. (3) for the saddle shaped surface of the bridge as represented in Fig. 1, accounting for the radius of the bridge smaller cross section, r2, bearing a convex curvature and of the radius of the bridge axial section r1, bearing a concave curvature. These geometrical considerations yield the Laplace–Young equation:
r1 = ra
a 1+ sec β−1 2ra
ð5Þ
a a 1+ tan β− 1 + sec β + 1 2ra 2ra
ð6Þ
and
where R is the ideal gas constant, T is the absolute temperature and ρl is the molar density of the liquid. At equilibrium between humid air and water in the capillary bridges, the relative humidity is given by: RH =
characterized by some roughness. Therefore, the particle asperities are responsible for particle–particle contacts, as depicted in the schematics reported in Fig. 1. If the capillary bridge is formed between the asperities of two particles, both radii appearing in Eq. (4) can be expressed as a function of the radius of the spherical asperity on which the bridge is located, ra, of the angle β as defined in Fig. 1, and of the gap a between the bridged asperities:
r2 = ra
By combining Eqs. (1) and (2) and (4) to (6) it is possible to evaluate the angle β and the bridge radii r1 and r2 given the values of the air relative humidity RH, of the asperity radius ra and of the distance a. All these values can be used to evaluate the liquid bridge volume, Vbridge:
Vbridge
3 h i 2π r13 cos β 2 2 = 2π r1 + ðr1 + r2 Þ r1 cos β− + 3 h π i 2 −2πr1 ðr1 + r2 Þ cos β sin β −β 2 −
ð7Þ
2πra 3ð2 + cos βÞð1− cos βÞ2
and the tensile force of the bridge, Fc, given by the sum of the effects of the surface tension and the pressure difference: 2
Fc = 2πr2 σ s + πr2 σ s
1 1 − r1 r2
ð8Þ
ð4Þ
In order to estimate the relationship between Fc and the powder tensile strength we make the following further assumptions:
Following the Rabinovich et al. [21] approach we do not consider particles as smooth spheres but we assume that particle surfaces are
1. a single bridge is responsible for each contact of the particle; 2. the particle coordination number depends on the powder porosity, ε, and is of the order of π/ε [12].
ΔP = σ s
1 1 − r1 r2
G. Landi et al. / Powder Technology 207 (2011) 437–443
Consequently, an estimate of the tensile strength as a function of the particle diameter, d, can be obtained by applying the equation proposed by Rumpf [12]: σt =
1−ε Fc ε d2
ð9Þ
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Table 1 Particle size distribution of the powders used in the experiments. Material
d10, μm⁎
d50, μm⁎
d90, μm⁎
dVM, μm†
Glass beads 50 μm Glass beads 95 μm
35.6 69.7
48.1 95.9
65.0 125.5
49.4 96.3
⁎ d10, d50, d90 are the 10th, 50th and 90th percentile of the cumulative particle size distribution by volume. † dVM is the volume moment mean diameter.
3. Experimental The experimental procedure includes two steps: the powder preconditioning in a fluidization column and the characterization of the powder flow properties in a Schulze shear tester. 3.1. Apparatus A sketch of the preconditioning apparatus is shown in Fig. 2. It is made of an air humidification section in which air at the desired humidity is obtained by mixing a stream of saturated air and a stream of dry air. The stream of saturated air is obtained by forcing the air flow through some water contained in a temperature controlled bubble column. A second bottle is used to disengage residual water drops from the humid air stream before it is mixed with the dry air stream. The air flow rates of the two streams are separately adjusted with two thermal mass flow controllers in order to be mixed at any fixed proportion required for a desired air relative humidity. The flow rate of the mixed air stream fed to a fluidization column is regulated by a third mass flow controller and the excess air flow coming from the humidification section is diverted to the atmosphere through a low permeability packed bed of fine particles which keeps the pressure of the gas entering the third mass flow meter always higher than the atmospheric and sufficiently high to feed the fluidization column. The fluidization column (50 mm ID) is made of Pyrex with a porous distributor about 5-mm thick, made of sintered glass particles with a diameter of about 100 μm. The water content of the humidified powders samples was measured with a thermo-gravimetric moisture analyzer (OHAUS MB 45). The shear tester used is a standard Schulze ring shear tester RST01.01 equipped with a small cell of 96 mL. A software application, developed in the LabVIEW environment (National Instrument), is used to acquire, visualize, and record the main data measured during the shear experiments.
Shape characteristics of the material under investigation were observed and photographed using a scanning electron microscope (SEM). The glass beads are perfectly spherical as it is shown in Fig. 3. In Fig. 4 a magnification of Fig. 3 is reported. The microphotograph shows on the surface of each particle several superficial corrugations and asperities probably due to the production process of the beads and to adhering impurities that persisted even after a washing procedure with acetone. 3.3. Procedures During the preconditioning procedure, a batch of about 200 g of ballotini was loaded in the fluidization column of the humidification set up. The humid air flow rate in the fluidization column was kept above the minimum fluidization velocity in order to allow a good and homogeneous humidity distribution in the powder bed. Few grams of the conditioned glass beads were tested into the thermo-gravimetric moisture analyzer to measure the water content of the powder sample while about 130 g were used in the shear experiments. The latter were performed according to the standard procedure for the Schulze shear tests to evaluate a single yield locus which, following the simplified Mohr–Coulomb approach, gave back the values of the angle of static internal friction, ϕ, and of the cohesion, C, of the sample tested under a certain consolidation load. Both values were used to calculate the unconfined yield strength, σc, and the tensile strength of the powder, σt, that according to the Mohr–Coulomb approach, are given by σc =
ð10Þ
−2C cosϕ 1 + sinϕ
ð11Þ
and
3.2. Materials In these experiments two different cuts of non porous glass beads having a particle density of 2500 kg/m3 were used. The particle size distribution (PSD) of the samples was performed by a laser diffraction method, using a Malvern Instruments Mastersizer Hydro 2000s. Table 1 reports the measured characteristic particle diameters of the cumulative volumetric particle size distribution. In this table, d10, d50, d90 are the 10th, 50th and 90th percentile of the cumulative particle size distribution and dVM is the volume moment mean diameter [23].
2C cosϕ 1− sinϕ
σt =
4. Results and discussion The water sorption isotherm of the glass beads as a function of the air relative humidity is reported in Fig. 5. Inspection of the plot reveals that the equilibrium moisture content increases with increasing air
Fig. 2. Sketch of the humidifying system.
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Fig. 3. SEM microphotograph of dry 50-μm glass beads magnified at 1.00 kX.
Fig. 6. Yield loci for 50-μm glass beads powder, at a consolidation stress of 1300 Pa and at different values of air relative humidity.
Fig. 4. SEM microphotograph of dry 50-μm glass beads magnified at 2.11 kX. Fig. 7. Experimental values of the powder cohesion for glass beads as a function of air relative humidity for different particle sizes and consolidation normal stresses: ●, 50μm beads at 700 Pa; ○, 50-μm beads at 1300 Pa; ▼, 95-μm beads at 700 Pa; ▽, 95-μm beads at 1300 Pa.
respectively. The effect of the air relative humidity on the angle of internal friction (Fig. 8) is less significant than the effect on the cohesion (Fig. 7), even at high humidity. This finding is in agreement with what was previously found by Pierrat et al. [6] on glass beads and other powders at higher moisture content. An accurate inspection of Fig. 7 reveals that cohesion decreases with the mean size of the glass beads as expected and displays a small
Fig. 5. Water sorption isotherm of 50-μm (●) and 95-μm (▲) glass beads at 20 °C.
relative humidity. The water content ranges between 0.02 up to 0.14 wt.%. For each conditioned sample at a given RH in the range 13–98%, two static yield loci were obtained corresponding to the normal consolidation stresses of about 1300 and 700 Pa. Fig. 6 reports as example the yield loci corresponding to a consolidation stress of 1300 Pa at different values of air relative humidity (20%, 50%, 70%, 98%) for 50-μm glass beads. The linear approximation, according to the Mohr–Coulomb failure model, fits well the experimental points of each yield locus for a single consolidation condition and, therefore, was considered for the description of the powder yield loci. According to this finding, yield loci are unequivocally identified through the cohesion, C (the line intercept), and the angle of internal friction, ϕ (the line slope). The values of C and ϕ are reported in Figs. 7 and 8,
Fig. 8. Experimental values of the powder angle of internal friction for glass beads as a function of air relative humidity for different particle sizes and consolidation normal stresses: ●, 50-μm beads at 700 Pa; ○, 50-μm beads at 1300 Pa; ▼, 95-μm beads at 700 Pa; ▽, 95-μm beads at 1300 Pa.
G. Landi et al. / Powder Technology 207 (2011) 437–443 Table 2 Mean value of the consolidation stress, σ1 and of the unconfined yield strength, σc , at 1300 and 700 Pa consolidation loads, 50-μm glass beads. Consolidation stress
1300 Pa
RH, %
σ1, Pa
σc, Pa
σ1, Pa
700 Pa σc, Pa
13 20 48 72 81 90 98
1199 ± 25 1244 ± 22 1253 ± 45 1278 ± 79 1294 ± 37 1321 ± 113 1444 ± 128
131 ± 36 148 ± 1 163 ± 19 241 ± 217 219 ± 50 386 ± 219 528 ± 237
671 ± 32 664 ± 24 679 ± 29 701 ± 46 732 ± 43 728 ± 64 780 ± 58
120 ± 25 92 ± 10 134 ± 16 164 ± 61 174 ± 46 236 ± 89 261 ± 52
dependence on the pre-consolidation stress. By examining Fig. 7, it is also possible to notice that the cohesion value for the 50-μm glass beads increases sharply when the 80% of relative humidity value is reached showing, correspondingly, larger variations between different samples of powders at the same humidity. Instead, the cohesion value for the 95-μm glass beads increases with a less steep slope, displaying also error bars of smaller amplitudes. Table 2 reports the mean value of the consolidation stress, σ1 and of the unconfined yield strength, σc , calculated according to Eq. (10) for all the experiments at 1300 and 700 Pa and at RH between 13% and 98% for the 50-μm glass beads. Each value reported for each condition is the average calculated over about eight different experiments. Inspection of Table 2 reveals an increase of the unconfined compressive strength with increasing air relative humidity for both consolidation conditions. This result might be due to an increase of both the angle of internal friction and the cohesion value or by the increase of one of two properties when relative humidity increases. The tensile strength was calculated by Eq. (11) from the experimental values of cohesion and the angle of internal friction measured for each of the two powders cuts conditioned at different air relative humidity. The resulting tensile strength values are reported in Fig. 9: the black full circles refer to 50-μm glass beads whereas the upside down full triangles refer to the 95-μm glass beads. The variation of σt with the relative humidity is similar to that of the cohesion. In fact, at the lowest air relative humidity tested, the average tensile strength is almost constant and the experimental values show a very small scatter, whereas, at high air relative humidity (RH N 70%), the experimental tensile strength of the 50-μm glass beads sharply increases displaying, however, a bigger variance as shown by the error bar amplitude.
Fig. 9. Values of the powder tensile strength for glass beads as a function of air relative humidity for different particle sizes and consolidation normal stresses. Experimental results: ●, 50-μm beads; ▼, 95-μm beads. Model results: , 50-μm beads and r a = 0.5 μm; , 95-μm beads and ra = 0.5 μm; ∙ ∙ , 95-μm beads and ra = 0.95 μm.
–––
—
– –
441
On modelling grounds, as it was described in the Theory section, the tensile strength is function of the cohesive force which, in turn, is a function of the bridge gap a, and of the asperity radius ra. As Fairbrother and Simons [24] demonstrated, it is found that the higher the gap value, the smaller the cohesive force between particles. In order to investigate how the gap value affects the cohesive force for the analyzed powder, a sensitivity analysis on the a value was performed. The analysis here reported is referred to the 50-μm glass beads with an asperity radius 0.01 times smaller than the mean particle diameter. This value of the asperity derives from the direct inspection of Fig. 4, where it appears that asperities are around 1/100 times smaller than the particle size. Fig. 10 shows the binary interparticle force calculated from the theory for a couple of 50-μm glass beads and an asperity radius ra = 0.5 μm, as a function of the air humidity and at different distances a of the bridged asperities. By inspecting Fig. 10 it is possible to see that for gap values higher than 0.01 μm, the cohesive force is nearly equal to zero for a large range of air relative humidity values and suddenly increases when relative humidity achieves a critical value. This can be understood by considering that bridge condensation at low relative humidity is possible only if low values of r1 are attained (see Eqs. (1) and (4)). Inspections of Fig. 1, instead, indicate that r1 cannot be smaller than a/ 2. This limit determines the corresponding minimum relative humidity for which water can condense in a capillary bridge given a certain gap. Fig. 10 also shows that the higher value of the binary interparticle force is reached for a small but non-zero separation distance, probably due to the possibility that, at these distances, smaller r2 values in the bridge lead to a decrease in the internal pressure of the capillary bridge producing an increase of the interparticle binary force. Observing Fig. 10, it appears that with a gap as small as 0.01 μm Fc shows a slight increase with relative humidity between 20% and 80%. Such increase is close to the observed trend of the experimental powder tensile strength as reported in Fig. 9. As a result, in the following analysis the separation distance between two particles was set to 0.01 μm for both cuts of glass beads. The experimental tensile strength was compared in Fig. 9 to the tensile strength values estimated according to the procedure described in the Theory section with the use of the a value just found. In order to proceed with the theoretical calculation, it is necessary to set up the asperity radius value. As it was underlined above for glass beads of 50 μm, the asperity radius was assumed about 0.01 times the particle diameter, because this value can be inferred from the visual observation of the glass beads made by SEM. For this powder, inspection of Fig. 9 reveals that, in spite of several assumptions, the comparison between theory and experiments shows a satisfactory agreement up to a RH of 80%. For glass beads
Fig. 10. Cohesive binary forces as a function of air relative humidity at different separation distances a: , 1 μm; ∙ ∙ ∙ ∙ ∙, 0.1 μm; , 0.01 μm; ∙ ∙ , 0.001 μm.
—
–––
– –
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having a particle diameter of 95 μm a sensitivity analysis on the asperity radius was made, and in particular two cases where considered for the 95-μm beads. In the first case the asperity was taken the same absolute value of the 50-μm beads, that is 0.5 μm. In the second case the asperity to particle diameter ratio was considered constant for the two powders, which corresponds in the case of 95-μm beads to an asperity radius of 0.95 μm. Inspection of Fig. 9 indicates that the latter case produces the better agreement between experimental and theoretical results. At air relative humidity higher than 80% the trend of the model results differs significantly from the experiments; in particular, for the 50-μm glass beads, the measured tensile strength shows a considerable increase with the relative humidity that is not reproduced by the model. On the contrary the experimental results relative to the 95-μm glass beads do not show such a significant change of the tensile strength at high air relative humidity, and the agreement between the model and the experiments is better than in the other case. The trend inconsistency at high relative humidity for 50-μm glass beads supports the idea that, in spite of the fact that both the Kelvin and the Laplace–Young equations describe the formation of capillary bridges at low humidity quite well, for relative humidity larger than 80%, the hypothesis that capillary interparticle interactions rely on single liquid bridges anchored on particle asperities may not be correct. In order to investigate this point, the volume of the single liquid bridge obtained by Eq. (7), was compared with the amount of water condensed over each particle calculated from the experimental water content absorbed by the powder. Carrying out a mass balance on water, in fact, it is possible to calculate the water content of powders with the hypothesis that condensation occurs only in connecting bridges: Xwbridge ≈
π ρl Vbridge 2ε ρs πd3 = 6
ð12Þ
where ρs and ρl are the solid and liquid densities, respectively. These values of Xw are compared in Fig. 11 with those measured by thermogravimetry. It appears that the water content obtained from Eq. (12) is several order of magnitudes smaller than the experimental values. This suggests that interparticle capillary bridges between particles only represent a small portion of the total condensed water on a single sphere, thus implying that only a small amount of this water contributes to the change of the powder flow properties. The physical interpretation may be that for rough beads the condensed water may fill the free volume of the beads roughness locally. In fact, since the water vapor pressure decreases when the curvature raises, a
Fig. 11. Powder water content: ●, experimental values obtained by termogravimetry; ○, estimated values assuming condensation in liquid bridges only, Xw-bridge, Eq. (12); ▼, estimated values assuming condensation in diffused cavities, Xw-capill, Eq. (13).
small amount of water may condense in the small cavities of the bead. The water content condensed over each bead can be easily estimated by considering the water amount spread off the total bead area having the same thickness of the curvature radius of the liquid meniscus filling the cavity of the roughness. The corresponding water content can thus be obtained with the following equation: Xwcapill ≈
π ρl πd2 rm 3π ρl rm = 2ε ρs πd3 = 6 ε ρs d
ð13Þ
where d is the particle diameter and rm is the curvature radius formed by condensed water between the roughness and the sphere itself, which can be calculated from Eq. (3), combined with (1) and (2), assuming that the depth of the condensed pool of water is of the same order of rm. The water content evaluated from Eq. (13) and reported in Fig. 11 is one order of magnitude bigger than the condensed water volumes calculated from Eq. (12), though it is still smaller than the experimental water content. This means that the proposed interpretation can only partially explain the experimental values found for the condensed water. Probably, in order to completely justify the amount of water condensed in the powder samples some mechanisms other than capillary condensation should be included in the model. 5. Conclusions The experimental procedure developed to characterize the powder flow properties at bulk level has allowed to study powders shear behavior exposed to different air humidity. Tests performed on humidified powder in a Schulze shear tester show that powder cohesion increases with air relative humidity, while the powder angle of internal friction is not significantly affected by air humidity. The cohesion increase is dependent on the mean particle size and the consolidation stress at which experiments are performed. The model developed and based on the hypothesis of capillary condensation is able to determine the capillary forces and to predict the powder tensile strength corresponding to the estimates of the same parameter, derived from shear experiments at air relative humidity lower than 80%. In order to investigate the model consistency, the measured values of the condensed water were compared to those estimated by assuming that all the condensed water is found within the connecting capillary bridges. The values of the moisture content predicted with the latter assumption are significantly smaller than the experimental ones and this suggests that interparticle capillary bridges between particles only represent a small portion of the total condensed water on a single sphere. Nomenclature a gap between bridged asperities, m C powder cohesion, Pa d particle diameter, m d10 10th percentile of the cumulative particle size distribution, m d50 50th percentile of the cumulative particle size distribution, m d90 90th percentile of the cumulative particle size distribution, m dVM volume moment mean particle diameter, m Fc bridge tensile force, N P0 liquid vapor pressure over capillary bridge planar surface, Pa P'0 liquid vapor pressure over capillary bridge curve surface, Pa r1 radius of bridge axial section, m r2 radius of bridge smaller cross section, m ra spherical asperity radius, m rm local average curvature radius, m RH air relative humidity, kgwater/kgair R ideal gas constant, J/kg∙K T absolute temperature, K Vbridge liquid bridge volume, m3 Xw powders water content, kgliquid/kgsolid
G. Landi et al. / Powder Technology 207 (2011) 437–443
Xw-bridge water content condensed in bridges, kgliquid/kgsolid Xw-capill water content condensed over each bead, kgliquid/kgsolid
Greek letters β water filling angle in the bridge, deg ΔP total pressure difference between gas and liquid, Pa ε powder porosity, ρl liquid density, kg/m3 ρl liquid molar density, kg/m3 ρs solid density kg/m3 σc unconfined yield strength, Pa σs surface tension, N/m σt powder tensile strength, Pa ϕ angle of static internal friction, deg
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Modelling and experiments on the effect of air humidity on the flow properties of glass powders Giovanna Landi, Diego Barletta, Massimo Poletto ⁎ Dipartimento di Ingegneria Chimica e Alimentare, Università di Salerno, Via Ponte Don Melillo, I-84084 Fisciano (SA), Italy
a r t i c l e
i n f o
Article history: Received 12 May 2010 Received in revised form 22 November 2010 Accepted 26 November 2010 Available online 3 December 2010 Keywords: Cohesion Powder flow properties Humidity Capillary forces
a b s t r a c t The effect of air humidity on flow properties of two different cuts of fine glass beads was experimentally studied by means of shear tests. The moisture content of powder samples was conditioned by humid air at relative humidities between 13% and 98% in a fluidization column. In spite of the very low moisture contents in the powder (b 0.2%) obtained by this technique, a significant change in the powder cohesion was observed, while the effect on the angle of internal friction was limited. A model based on the Kelvin equation and the Laplace–Young equation was applied to describe the capillary condensation between touching asperities found on the surface of neighbouring particles and to estimate the relevant interparticle forces. This result was used to derive the powder tensile strength following the Rumpf approach and compare it with values derived from shear experiments. A single value of the unknown capillary bridge gap was chosen on the basis of the observed dependence of interparticle cohesive forces. This gap was able to provide quantitative agreement between model predictions and experimental results at relative humidity lower than 80%. Comparison between the total moisture content and the amount of water in liquid bridges indicated that water mainly condenses on rough surfaces of the particles and only a small portion of this condensed humidity contributes to change the powder flow properties by interparticle capillary forces. © 2010 Elsevier B.V. All rights reserved.
1. Introduction Powders often show flow problems due to cohesive properties that may compromise the reliability of many operations of industrial processes such as storage, handling, feeding and fluidization. In particular, problems are more troublesome when the powder treated is really cohesive, because it can form dead zones, steady channels and agglomerates. Powders flow properties are strongly dependent not only on intrinsic physical properties of powders such as particle shape, roughness and particle size distribution, but also on operating conditions such as system temperature, presence of a liquid phase or moisture content. The latter is one of the properties dependent on the environmental conditions which may more strongly affect the powder behavior. In fact, moisture can cause the formation of liquid bridges between particles giving rise to interparticle capillary forces. These forces are due to the combined effects of the surface tension acting on the liquid bridge interface and of the curvature of the bridge surface [1–3]. As a result, capillary forces cause an increase of powder cohesion and tensile strength and, thus, negatively affect the powder flowability [4–8]. Moreover, capillary forces modify also other powder bulk properties such as voidage, aggregative structure and permeability. These properties and the relative importance of capillary forces with respect to fluid dynamic forces seriously affect also the ⁎ Corresponding author. Tel.: +39 089 964 132; fax: +39 089 964 057. E-mail address: [email protected] (M. Poletto). 0032-5910/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.powtec.2010.11.033
fluidization behavior of fine powders [9–11]. On the other hand, the effect of capillary forces can be beneficial in some applications such as powder agglomeration [12,13]. Moisture content of powders can be determined by the capillary condensation of the surrounding atmosphere humidity [14]. Consequently, powder flow properties are influenced by the humidity of the storage environment [6,15–19]. In some of these experimental studies the liquid bridges formed by capillary condensation interact chemically with the particle surface giving rise to the so called “caking” phenomenon. The latter is not addressed in this paper, which is focused only on the contribution of capillary forces to cohesion. A large number of theoretical and experimental studies in the literature are devoted to the effect of capillary condensation on interparticle forces [3]. Results of the direct measurements of interparticle forces by atomic force microscopy show wide scatter mainly due to the complex effect of the particle size, shape and surface roughness and of the chemical nature of the surface [20]. Nevertheless, the theoretical model based on the combination of the Kelvin equation with the Laplace–Young equation is widely accepted [3]. In this framework a dramatic effect of particle surface roughness on capillary condensation and thus on capillary forces was highlighted [3,21]. In particular, capillary forces are substantially lower in presence of nanoscale roughness since liquid bridges form between asperities of smaller curvature radius. Moreover, the critical relative humidity, at which capillary condensation is first observed, increases as roughness on the nanoscale increases.
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A similar modelling approach accounting for the role of asperities was followed by Bocquet et al. [22] to successfully predict a bulk property as the angle of avalanche of glass beads as a function of air humidity. This paper aims at developing a model for the prediction of the flow properties of powders exposed to humid air, with the basic hypothesis that, in such conditions, capillary condensation is the main phenomenon which is responsible for the formation of liquid bridges. It has been pointed out in previously published works that humidity has a completely different effect on porous and non porous materials [9]. In porous materials most of the moisture content is found within the internal particle porosity and does not play a significant role in the determination of interparticle capillary forces due to liquid bridges. In order to approach the problem from a simpler perspective and to close the material balances on the condensed water more easily, in this work only non porous particles are considered. Fig. 1. Scheme and nomenclature in a capillary bridge formed between two particle asperities.
2. Theory Humidity can condense in powders even from unsaturated air due to reduction of the fugacity of water in the condensed phase in turn due to either soluble material present on particle surfaces or to the curvature of the surface of the condensed phase in small volumes delimited by solid particle surfaces, the so-called capillary condensation. In this study the main hypothesis is that condensation of air humidity is only due to the formation of curved air liquid interfaces in which the joint action of surface tension and interface curvature determines some vapor pressure reduction within the liquid. This is the case of the concave surfaces in capillary condensation, where the liquid vapor pressure P'0 is reduced with respect to that measured over a planar surface P0. The Kelvin equation relates the reduced vapor pressure with the total pressure difference between gas and liquid, ΔP: P0′ −ΔP = exp P0 ρl RT
P0′ P0
ð1Þ
ð2Þ
The total pressure difference between gas and liquid pressure is determined by the action of surface tension, σs, over the concave interface and is a function of the local average radius, rm ΔP =
2σ s rm
ð3Þ
The convention used here is that a concave surface has a positive radius and a liquid pressure smaller than the gas pressure. The opposite occurs for convex surfaces. The theory of capillary bridges, reported by Pierrat and Caram [5], corrects Eq. (3) for the saddle shaped surface of the bridge as represented in Fig. 1, accounting for the radius of the bridge smaller cross section, r2, bearing a convex curvature and of the radius of the bridge axial section r1, bearing a concave curvature. These geometrical considerations yield the Laplace–Young equation:
r1 = ra
a 1+ sec β−1 2ra
ð5Þ
a a 1+ tan β− 1 + sec β + 1 2ra 2ra
ð6Þ
and
where R is the ideal gas constant, T is the absolute temperature and ρl is the molar density of the liquid. At equilibrium between humid air and water in the capillary bridges, the relative humidity is given by: RH =
characterized by some roughness. Therefore, the particle asperities are responsible for particle–particle contacts, as depicted in the schematics reported in Fig. 1. If the capillary bridge is formed between the asperities of two particles, both radii appearing in Eq. (4) can be expressed as a function of the radius of the spherical asperity on which the bridge is located, ra, of the angle β as defined in Fig. 1, and of the gap a between the bridged asperities:
r2 = ra
By combining Eqs. (1) and (2) and (4) to (6) it is possible to evaluate the angle β and the bridge radii r1 and r2 given the values of the air relative humidity RH, of the asperity radius ra and of the distance a. All these values can be used to evaluate the liquid bridge volume, Vbridge:
Vbridge
3 h i 2π r13 cos β 2 2 = 2π r1 + ðr1 + r2 Þ r1 cos β− + 3 h π i 2 −2πr1 ðr1 + r2 Þ cos β sin β −β 2 −
ð7Þ
2πra 3ð2 + cos βÞð1− cos βÞ2
and the tensile force of the bridge, Fc, given by the sum of the effects of the surface tension and the pressure difference: 2
Fc = 2πr2 σ s + πr2 σ s
1 1 − r1 r2
ð8Þ
ð4Þ
In order to estimate the relationship between Fc and the powder tensile strength we make the following further assumptions:
Following the Rabinovich et al. [21] approach we do not consider particles as smooth spheres but we assume that particle surfaces are
1. a single bridge is responsible for each contact of the particle; 2. the particle coordination number depends on the powder porosity, ε, and is of the order of π/ε [12].
ΔP = σ s
1 1 − r1 r2
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Consequently, an estimate of the tensile strength as a function of the particle diameter, d, can be obtained by applying the equation proposed by Rumpf [12]: σt =
1−ε Fc ε d2
ð9Þ
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Table 1 Particle size distribution of the powders used in the experiments. Material
d10, μm⁎
d50, μm⁎
d90, μm⁎
dVM, μm†
Glass beads 50 μm Glass beads 95 μm
35.6 69.7
48.1 95.9
65.0 125.5
49.4 96.3
⁎ d10, d50, d90 are the 10th, 50th and 90th percentile of the cumulative particle size distribution by volume. † dVM is the volume moment mean diameter.
3. Experimental The experimental procedure includes two steps: the powder preconditioning in a fluidization column and the characterization of the powder flow properties in a Schulze shear tester. 3.1. Apparatus A sketch of the preconditioning apparatus is shown in Fig. 2. It is made of an air humidification section in which air at the desired humidity is obtained by mixing a stream of saturated air and a stream of dry air. The stream of saturated air is obtained by forcing the air flow through some water contained in a temperature controlled bubble column. A second bottle is used to disengage residual water drops from the humid air stream before it is mixed with the dry air stream. The air flow rates of the two streams are separately adjusted with two thermal mass flow controllers in order to be mixed at any fixed proportion required for a desired air relative humidity. The flow rate of the mixed air stream fed to a fluidization column is regulated by a third mass flow controller and the excess air flow coming from the humidification section is diverted to the atmosphere through a low permeability packed bed of fine particles which keeps the pressure of the gas entering the third mass flow meter always higher than the atmospheric and sufficiently high to feed the fluidization column. The fluidization column (50 mm ID) is made of Pyrex with a porous distributor about 5-mm thick, made of sintered glass particles with a diameter of about 100 μm. The water content of the humidified powders samples was measured with a thermo-gravimetric moisture analyzer (OHAUS MB 45). The shear tester used is a standard Schulze ring shear tester RST01.01 equipped with a small cell of 96 mL. A software application, developed in the LabVIEW environment (National Instrument), is used to acquire, visualize, and record the main data measured during the shear experiments.
Shape characteristics of the material under investigation were observed and photographed using a scanning electron microscope (SEM). The glass beads are perfectly spherical as it is shown in Fig. 3. In Fig. 4 a magnification of Fig. 3 is reported. The microphotograph shows on the surface of each particle several superficial corrugations and asperities probably due to the production process of the beads and to adhering impurities that persisted even after a washing procedure with acetone. 3.3. Procedures During the preconditioning procedure, a batch of about 200 g of ballotini was loaded in the fluidization column of the humidification set up. The humid air flow rate in the fluidization column was kept above the minimum fluidization velocity in order to allow a good and homogeneous humidity distribution in the powder bed. Few grams of the conditioned glass beads were tested into the thermo-gravimetric moisture analyzer to measure the water content of the powder sample while about 130 g were used in the shear experiments. The latter were performed according to the standard procedure for the Schulze shear tests to evaluate a single yield locus which, following the simplified Mohr–Coulomb approach, gave back the values of the angle of static internal friction, ϕ, and of the cohesion, C, of the sample tested under a certain consolidation load. Both values were used to calculate the unconfined yield strength, σc, and the tensile strength of the powder, σt, that according to the Mohr–Coulomb approach, are given by σc =
ð10Þ
−2C cosϕ 1 + sinϕ
ð11Þ
and
3.2. Materials In these experiments two different cuts of non porous glass beads having a particle density of 2500 kg/m3 were used. The particle size distribution (PSD) of the samples was performed by a laser diffraction method, using a Malvern Instruments Mastersizer Hydro 2000s. Table 1 reports the measured characteristic particle diameters of the cumulative volumetric particle size distribution. In this table, d10, d50, d90 are the 10th, 50th and 90th percentile of the cumulative particle size distribution and dVM is the volume moment mean diameter [23].
2C cosϕ 1− sinϕ
σt =
4. Results and discussion The water sorption isotherm of the glass beads as a function of the air relative humidity is reported in Fig. 5. Inspection of the plot reveals that the equilibrium moisture content increases with increasing air
Fig. 2. Sketch of the humidifying system.
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Fig. 3. SEM microphotograph of dry 50-μm glass beads magnified at 1.00 kX.
Fig. 6. Yield loci for 50-μm glass beads powder, at a consolidation stress of 1300 Pa and at different values of air relative humidity.
Fig. 4. SEM microphotograph of dry 50-μm glass beads magnified at 2.11 kX. Fig. 7. Experimental values of the powder cohesion for glass beads as a function of air relative humidity for different particle sizes and consolidation normal stresses: ●, 50μm beads at 700 Pa; ○, 50-μm beads at 1300 Pa; ▼, 95-μm beads at 700 Pa; ▽, 95-μm beads at 1300 Pa.
respectively. The effect of the air relative humidity on the angle of internal friction (Fig. 8) is less significant than the effect on the cohesion (Fig. 7), even at high humidity. This finding is in agreement with what was previously found by Pierrat et al. [6] on glass beads and other powders at higher moisture content. An accurate inspection of Fig. 7 reveals that cohesion decreases with the mean size of the glass beads as expected and displays a small
Fig. 5. Water sorption isotherm of 50-μm (●) and 95-μm (▲) glass beads at 20 °C.
relative humidity. The water content ranges between 0.02 up to 0.14 wt.%. For each conditioned sample at a given RH in the range 13–98%, two static yield loci were obtained corresponding to the normal consolidation stresses of about 1300 and 700 Pa. Fig. 6 reports as example the yield loci corresponding to a consolidation stress of 1300 Pa at different values of air relative humidity (20%, 50%, 70%, 98%) for 50-μm glass beads. The linear approximation, according to the Mohr–Coulomb failure model, fits well the experimental points of each yield locus for a single consolidation condition and, therefore, was considered for the description of the powder yield loci. According to this finding, yield loci are unequivocally identified through the cohesion, C (the line intercept), and the angle of internal friction, ϕ (the line slope). The values of C and ϕ are reported in Figs. 7 and 8,
Fig. 8. Experimental values of the powder angle of internal friction for glass beads as a function of air relative humidity for different particle sizes and consolidation normal stresses: ●, 50-μm beads at 700 Pa; ○, 50-μm beads at 1300 Pa; ▼, 95-μm beads at 700 Pa; ▽, 95-μm beads at 1300 Pa.
G. Landi et al. / Powder Technology 207 (2011) 437–443 Table 2 Mean value of the consolidation stress, σ1 and of the unconfined yield strength, σc , at 1300 and 700 Pa consolidation loads, 50-μm glass beads. Consolidation stress
1300 Pa
RH, %
σ1, Pa
σc, Pa
σ1, Pa
700 Pa σc, Pa
13 20 48 72 81 90 98
1199 ± 25 1244 ± 22 1253 ± 45 1278 ± 79 1294 ± 37 1321 ± 113 1444 ± 128
131 ± 36 148 ± 1 163 ± 19 241 ± 217 219 ± 50 386 ± 219 528 ± 237
671 ± 32 664 ± 24 679 ± 29 701 ± 46 732 ± 43 728 ± 64 780 ± 58
120 ± 25 92 ± 10 134 ± 16 164 ± 61 174 ± 46 236 ± 89 261 ± 52
dependence on the pre-consolidation stress. By examining Fig. 7, it is also possible to notice that the cohesion value for the 50-μm glass beads increases sharply when the 80% of relative humidity value is reached showing, correspondingly, larger variations between different samples of powders at the same humidity. Instead, the cohesion value for the 95-μm glass beads increases with a less steep slope, displaying also error bars of smaller amplitudes. Table 2 reports the mean value of the consolidation stress, σ1 and of the unconfined yield strength, σc , calculated according to Eq. (10) for all the experiments at 1300 and 700 Pa and at RH between 13% and 98% for the 50-μm glass beads. Each value reported for each condition is the average calculated over about eight different experiments. Inspection of Table 2 reveals an increase of the unconfined compressive strength with increasing air relative humidity for both consolidation conditions. This result might be due to an increase of both the angle of internal friction and the cohesion value or by the increase of one of two properties when relative humidity increases. The tensile strength was calculated by Eq. (11) from the experimental values of cohesion and the angle of internal friction measured for each of the two powders cuts conditioned at different air relative humidity. The resulting tensile strength values are reported in Fig. 9: the black full circles refer to 50-μm glass beads whereas the upside down full triangles refer to the 95-μm glass beads. The variation of σt with the relative humidity is similar to that of the cohesion. In fact, at the lowest air relative humidity tested, the average tensile strength is almost constant and the experimental values show a very small scatter, whereas, at high air relative humidity (RH N 70%), the experimental tensile strength of the 50-μm glass beads sharply increases displaying, however, a bigger variance as shown by the error bar amplitude.
Fig. 9. Values of the powder tensile strength for glass beads as a function of air relative humidity for different particle sizes and consolidation normal stresses. Experimental results: ●, 50-μm beads; ▼, 95-μm beads. Model results: , 50-μm beads and r a = 0.5 μm; , 95-μm beads and ra = 0.5 μm; ∙ ∙ , 95-μm beads and ra = 0.95 μm.
–––
—
– –
441
On modelling grounds, as it was described in the Theory section, the tensile strength is function of the cohesive force which, in turn, is a function of the bridge gap a, and of the asperity radius ra. As Fairbrother and Simons [24] demonstrated, it is found that the higher the gap value, the smaller the cohesive force between particles. In order to investigate how the gap value affects the cohesive force for the analyzed powder, a sensitivity analysis on the a value was performed. The analysis here reported is referred to the 50-μm glass beads with an asperity radius 0.01 times smaller than the mean particle diameter. This value of the asperity derives from the direct inspection of Fig. 4, where it appears that asperities are around 1/100 times smaller than the particle size. Fig. 10 shows the binary interparticle force calculated from the theory for a couple of 50-μm glass beads and an asperity radius ra = 0.5 μm, as a function of the air humidity and at different distances a of the bridged asperities. By inspecting Fig. 10 it is possible to see that for gap values higher than 0.01 μm, the cohesive force is nearly equal to zero for a large range of air relative humidity values and suddenly increases when relative humidity achieves a critical value. This can be understood by considering that bridge condensation at low relative humidity is possible only if low values of r1 are attained (see Eqs. (1) and (4)). Inspections of Fig. 1, instead, indicate that r1 cannot be smaller than a/ 2. This limit determines the corresponding minimum relative humidity for which water can condense in a capillary bridge given a certain gap. Fig. 10 also shows that the higher value of the binary interparticle force is reached for a small but non-zero separation distance, probably due to the possibility that, at these distances, smaller r2 values in the bridge lead to a decrease in the internal pressure of the capillary bridge producing an increase of the interparticle binary force. Observing Fig. 10, it appears that with a gap as small as 0.01 μm Fc shows a slight increase with relative humidity between 20% and 80%. Such increase is close to the observed trend of the experimental powder tensile strength as reported in Fig. 9. As a result, in the following analysis the separation distance between two particles was set to 0.01 μm for both cuts of glass beads. The experimental tensile strength was compared in Fig. 9 to the tensile strength values estimated according to the procedure described in the Theory section with the use of the a value just found. In order to proceed with the theoretical calculation, it is necessary to set up the asperity radius value. As it was underlined above for glass beads of 50 μm, the asperity radius was assumed about 0.01 times the particle diameter, because this value can be inferred from the visual observation of the glass beads made by SEM. For this powder, inspection of Fig. 9 reveals that, in spite of several assumptions, the comparison between theory and experiments shows a satisfactory agreement up to a RH of 80%. For glass beads
Fig. 10. Cohesive binary forces as a function of air relative humidity at different separation distances a: , 1 μm; ∙ ∙ ∙ ∙ ∙, 0.1 μm; , 0.01 μm; ∙ ∙ , 0.001 μm.
—
–––
– –
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having a particle diameter of 95 μm a sensitivity analysis on the asperity radius was made, and in particular two cases where considered for the 95-μm beads. In the first case the asperity was taken the same absolute value of the 50-μm beads, that is 0.5 μm. In the second case the asperity to particle diameter ratio was considered constant for the two powders, which corresponds in the case of 95-μm beads to an asperity radius of 0.95 μm. Inspection of Fig. 9 indicates that the latter case produces the better agreement between experimental and theoretical results. At air relative humidity higher than 80% the trend of the model results differs significantly from the experiments; in particular, for the 50-μm glass beads, the measured tensile strength shows a considerable increase with the relative humidity that is not reproduced by the model. On the contrary the experimental results relative to the 95-μm glass beads do not show such a significant change of the tensile strength at high air relative humidity, and the agreement between the model and the experiments is better than in the other case. The trend inconsistency at high relative humidity for 50-μm glass beads supports the idea that, in spite of the fact that both the Kelvin and the Laplace–Young equations describe the formation of capillary bridges at low humidity quite well, for relative humidity larger than 80%, the hypothesis that capillary interparticle interactions rely on single liquid bridges anchored on particle asperities may not be correct. In order to investigate this point, the volume of the single liquid bridge obtained by Eq. (7), was compared with the amount of water condensed over each particle calculated from the experimental water content absorbed by the powder. Carrying out a mass balance on water, in fact, it is possible to calculate the water content of powders with the hypothesis that condensation occurs only in connecting bridges: Xwbridge ≈
π ρl Vbridge 2ε ρs πd3 = 6
ð12Þ
where ρs and ρl are the solid and liquid densities, respectively. These values of Xw are compared in Fig. 11 with those measured by thermogravimetry. It appears that the water content obtained from Eq. (12) is several order of magnitudes smaller than the experimental values. This suggests that interparticle capillary bridges between particles only represent a small portion of the total condensed water on a single sphere, thus implying that only a small amount of this water contributes to the change of the powder flow properties. The physical interpretation may be that for rough beads the condensed water may fill the free volume of the beads roughness locally. In fact, since the water vapor pressure decreases when the curvature raises, a
Fig. 11. Powder water content: ●, experimental values obtained by termogravimetry; ○, estimated values assuming condensation in liquid bridges only, Xw-bridge, Eq. (12); ▼, estimated values assuming condensation in diffused cavities, Xw-capill, Eq. (13).
small amount of water may condense in the small cavities of the bead. The water content condensed over each bead can be easily estimated by considering the water amount spread off the total bead area having the same thickness of the curvature radius of the liquid meniscus filling the cavity of the roughness. The corresponding water content can thus be obtained with the following equation: Xwcapill ≈
π ρl πd2 rm 3π ρl rm = 2ε ρs πd3 = 6 ε ρs d
ð13Þ
where d is the particle diameter and rm is the curvature radius formed by condensed water between the roughness and the sphere itself, which can be calculated from Eq. (3), combined with (1) and (2), assuming that the depth of the condensed pool of water is of the same order of rm. The water content evaluated from Eq. (13) and reported in Fig. 11 is one order of magnitude bigger than the condensed water volumes calculated from Eq. (12), though it is still smaller than the experimental water content. This means that the proposed interpretation can only partially explain the experimental values found for the condensed water. Probably, in order to completely justify the amount of water condensed in the powder samples some mechanisms other than capillary condensation should be included in the model. 5. Conclusions The experimental procedure developed to characterize the powder flow properties at bulk level has allowed to study powders shear behavior exposed to different air humidity. Tests performed on humidified powder in a Schulze shear tester show that powder cohesion increases with air relative humidity, while the powder angle of internal friction is not significantly affected by air humidity. The cohesion increase is dependent on the mean particle size and the consolidation stress at which experiments are performed. The model developed and based on the hypothesis of capillary condensation is able to determine the capillary forces and to predict the powder tensile strength corresponding to the estimates of the same parameter, derived from shear experiments at air relative humidity lower than 80%. In order to investigate the model consistency, the measured values of the condensed water were compared to those estimated by assuming that all the condensed water is found within the connecting capillary bridges. The values of the moisture content predicted with the latter assumption are significantly smaller than the experimental ones and this suggests that interparticle capillary bridges between particles only represent a small portion of the total condensed water on a single sphere. Nomenclature a gap between bridged asperities, m C powder cohesion, Pa d particle diameter, m d10 10th percentile of the cumulative particle size distribution, m d50 50th percentile of the cumulative particle size distribution, m d90 90th percentile of the cumulative particle size distribution, m dVM volume moment mean particle diameter, m Fc bridge tensile force, N P0 liquid vapor pressure over capillary bridge planar surface, Pa P'0 liquid vapor pressure over capillary bridge curve surface, Pa r1 radius of bridge axial section, m r2 radius of bridge smaller cross section, m ra spherical asperity radius, m rm local average curvature radius, m RH air relative humidity, kgwater/kgair R ideal gas constant, J/kg∙K T absolute temperature, K Vbridge liquid bridge volume, m3 Xw powders water content, kgliquid/kgsolid
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Xw-bridge water content condensed in bridges, kgliquid/kgsolid Xw-capill water content condensed over each bead, kgliquid/kgsolid
Greek letters β water filling angle in the bridge, deg ΔP total pressure difference between gas and liquid, Pa ε powder porosity, ρl liquid density, kg/m3 ρl liquid molar density, kg/m3 ρs solid density kg/m3 σc unconfined yield strength, Pa σs surface tension, N/m σt powder tensile strength, Pa ϕ angle of static internal friction, deg
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