Cohesion number

In simulation of granular materials, scaling the particle size with regards to the other particles physical and mechanical properties is a challenging job. Especially in simulation of cohesive powders, lack of a robust criterion for tuning the level of the surface energy of the particles can waste enormous amount of time during the process of calibration. The Bond number {{Cite journal|last=Bond|first=W. N.|date=1935|title=The surface tension of a moving water sheet|url=http://stacks.iop.org/0959-5309/47/i=4/a=303|journal=Proceedings of the Physical Society|language=en|volume=47|issue=4|pages=549–558|doi=10.1088/0959-5309/47/4/303|issn=0959-5309|bibcode=1935PPS....47..549B|url-access=subscription}} has been used traditionally in this regards, where the significance of the adhesive force (pull-off force) is compared with the particles gravitational force (weight); nevertheless, the influence of the materials properties, particularly the particles stiffness, is not comprehensively observed in this number. The particles stiffness, which is not present in the Bond number, has a considerable impact on how particles respond to an applied force. If the forces in the Bond number are substituted with potential and cohesion energies, a new dimensionless number will be formed whereby the effect of the particles stiffness is also considered. This was firstly proposed by Behjani et al.{{Cite journal|last=Behjani|first=Mohammadreza Alizadeh|last2=Rahmanian|first2=Nejat|last3=Ghani|first3=Nur Fardina bt Abdul|last4=Hassanpour|first4=Ali|title=An investigation on process of seeded granulation in a continuous drum granulator using DEM|journal=Advanced Powder Technology|volume=28|issue=10|pages=2456–2464|doi=10.1016/j.apt.2017.02.011|year=2017|url=http://eprints.whiterose.ac.uk/113300/3/APT_Seeded_Granulation-accepted%20manuscript%20Feb%202017.pdf}} where they introduced a dimensionless number titled as the Cohesion number.

The Cohesion number is a dimensionless number which shows the ratio of the work required for detaching two arbitrary solid particles (work of cohesion) to their gravitational potential energy as expressed below,

$\backslash text\{Cohesion\; number\}=\backslash frac\{\backslash text\{work\; of\; cohesion\}\}\{\backslash text\{gravitational\; potential\; energy\}\}$

For example, in the JKR contact model {{Cite journal|last=Johnson|first=K. L.|last2=Kendall|first2=K.|last3=Roberts|first3=A. D.|date=1971-09-08|title=Surface energy and the contact of elastic solids|journal=Proc. R. Soc. Lond. A|language=en|volume=324|issue=1558|pages=301–313|doi=10.1098/rspa.1971.0141|issn=0080-4630|bibcode=1971RSPSA.324..301J|doi-access=free}} the work of cohesion is $7.09\backslash left\; (\; \backslash frac\{\backslash Gamma^5\{R^*\}^4\}\{\{E^*\}^2\}\; \backslash right\; )^\{\backslash frac\{1\}\{3\}\}$

{{Cite journal|last=Thornton|first=Colin|last2=Ning|first2=Zemin|title=A theoretical model for the stick/bounce behaviour of adhesive, elastic-plastic spheres|journal=Powder Technology|volume=99|issue=2|pages=154–162|doi=10.1016/s0032-5910(98)00099-0|year=1998}} by which the Cohesion number is derived as follows:

$$Coh=\backslash frac\{7.09\backslash left\; (\; \backslash frac\{\backslash Gamma^5\{R^*\}^4\}\{\{E^*\}^2\}\; \backslash right\; )^\{\backslash frac\{1\}\{3\}\}\}\{mgR^*\}$$

Mass can be shown in the form of density and volume and the constant number can be eliminated,

$$Coh=\backslash frac\{7.09\backslash left\; (\; \backslash frac\{\backslash Gamma^5\{R^*\}^4\}\{\{E^*\}^2\}\; \backslash right\; )^\{\backslash frac\{1\}\{3\}\}\}\{\backslash rho\; \{R^*\}^3\; gR^*\}$$

The final version of the Cohesion number is as following:

$$Coh=\backslash frac\{1\}\{\backslash rho\; g\}\backslash left\; (\; \backslash frac\{\backslash Gamma^5\}\{\{E^*\}^2\{R^*\}^8\}\; \backslash right\; )^\{\backslash frac\{1\}\{3\}\}$$

$\backslash rho$

is the particle density

$g$

is the gravity

$\backslash Gamma$

is the interfacial energy

$E^*$

is the equivalent Young’s modulus: $E^*=(\backslash frac\{1-\backslash nu\_\{1\}^2\}\{E\_\{1\}\}+\backslash frac\{1-\backslash nu\_\{2\}^2\}\{E\_\{2\}\})^\{-1\}$

$\backslash nu$

is the material Poisson's ratio

$R^*$

shows the equivalent radius: $R^*=(\backslash frac\{1\}\{R\_\{1\}\}+\backslash frac\{1\}\{R\_\{2\}\})^\{-1\}$

This number is dependent on the particles surface energy, particles size, particle density, gravity, and the Young’s modulus. It well justifies that the materials having lower stiffness become “stickier” if adhesive and it is a useful scaling method for the DEM simulations at which Young’s modulus is selected smaller than the real value in order to increase the computational speed.{{Cite journal|last=Alizadeh Behjani|first=Mohammadreza|last2=Hassanpour|first2=Ali|last3=Ghadiri|first3=Mojtaba|last4=Bayly|first4=Andrew|date=2017|title=Numerical Analysis of the Effect of Particle Shape and Adhesion on the Segregation of Powder Mixtures|journal=EPJ Web of Conferences|language=en|volume=140|pages=06024|doi=10.1051/epjconf/201714006024|issn=2100-014X|bibcode=2017EPJWC.14006024A|doi-access=free}} Recently, a rigorous analysis of the contact stiffness reduction for the adhesive contacts to speed up the DEM calculations shows the same fractional form.{{Cite journal|last=Hærvig|first=J.|last2=Kleinhans|first2=U.|last3=Wieland|first3=C.|last4=Spliethoff|first4=H.|last5=Jensen|first5=A.L.|last6=Sørensen|first6=K.|last7=Condra|first7=T.J.|title=On the adhesive JKR contact and rolling models for reduced particle stiffness discrete element simulations|journal=Powder Technology|volume=319|pages=472–482|doi=10.1016/j.powtec.2017.07.006|year=2017|url=http://vbn.aau.dk/da/publications/on-the-adhesive-jkr-contact-and-rolling-models-for-reduced-particle-stiffness-discrete-element-simulations(94635134-7b7b-4de2-bfd5-f77d2b4d515a).html}}

• Contact mechanics
• Surface tension

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Category:Dimensionless numbers of mechanics