Sphericity

Schematic representation of the difference in particle shape. Two parameters are shown: sphericity (the higher the object, the higher the sphericity) and roundness (the right of the object, the higher the roundness).

Sphericality is a measure of how spherical (round) an object is. Defined by H. Wadell in 1935, ^[1] sphericity $\Psi$ ${\ displaystyle \ Psi}$ ${\ displaystyle \ Psi}$ particle is the ratio of the surface area of the sphere (the same volume as this particle) to the surface area of the particle:

\Psi ={\frac {\pi ^{\frac {1}{3}}(6V_{p})^{\frac {2}{3}}}{A_{p}}},

{\ displaystyle \ Psi = {\ frac {\ pi ^ {\ frac {1} {3}} (6V_ {p}) ^ {\ frac {2} {3}}} {A_ {p}}},}

{\ displaystyle \ Psi = {\ frac {\ pi ^ {\ frac {1} {3}} (6V_ {p}) ^ {\ frac {2} {3}}} {A_ {p}}},}

Where $V_{p}$ ${\ displaystyle V_ {p}}$ $V_ {p}$ equal to the volume of the particle and $A_{p}$ ${\ displaystyle A_ {p}}$ $A_ {p}$ equal to the surface area of the particle. The sphericity of a sphere is unity by definition, and due to isoperimetric inequality, the sphericity of any other body is less than unity.

Ellipsoidal objects

Sphericity $\Psi$ ${\ displaystyle \ Psi}$ $\Psi$ flattened spheroid is equal

\Psi ={\frac {\pi ^{\frac {1}{3}}(6V_{p})^{\frac {2}{3}}}{A_{p}}}={\frac {2{\sqrt[{3}]{ab^{2}}}}{a+{\frac {b^{2}}{\sqrt {a^{2}-b^{2}}}}\ln {\left({\frac {a+{\sqrt {a^{2}-b^{2}}}}{b}}\right)}}},

{\ displaystyle \ Psi = {\ frac {\ pi ^ {\ frac {1} {3}} (6V_ {p}) ^ {\ frac {2} {3}}} {A_ {p}}} = { \ frac {2 {\ sqrt [{3}] {ab ^ {2}}}} {a + {\ frac {b ^ {2}} {\ sqrt {a ^ {2} -b ^ {2}}} } \ ln {\ left ({\ frac {a + {\ sqrt {a ^ {2} -b ^ {2}}}} {b}} \ right)}}},}

\Psi ={\frac {\pi ^{\frac {1}{3}}(6V_{p})^{\frac {2}{3}}}{A_{p}}}={\frac {2{\sqrt[{3}]{ab^{2}}}}{a+{\frac {b^{2}}{\sqrt {a^{2}-b^{2}}}}\ln {\left({\frac {a+{\sqrt {a^{2}-b^{2}}}}{b}}\right)}}},

where a and b are equal to the major and minor semiaxes of the spheroid.

Formula Output

Hakon Waddell defined sphericity as the ratio of the surface area of a sphere equal to the volume of a given particle to the surface area of a given particle. Express the surface area of the sphere $A_{s}$ ${\ displaystyle A_ {s}}$ $A_s$ through the volume of a particle $V_{p}$ ${\ displaystyle V_ {p}}$ $V_{p}$ :

A_{s}^{3}=\left(4\pi r^{2}\right)^{3}=4^{3}\pi ^{3}r^{6}=4\pi \left(4^{2}\pi ^{2}r^{6}\right)=4\pi \cdot 3^{2}\left({\frac {4^{2}\pi ^{2}}{3^{2}}}r^{6}\right)=36\pi \left({\frac {4\pi }{3}}r^{3}\right)^{2}=36\,\pi V_{p}^{2}.

{\ displaystyle A_ {s} ^ {3} = \ left (4 \ pi r ^ {2} \ right) ^ {3} = 4 ^ {3} \ pi ^ {3} r ^ {6} = 4 \ pi \ left (4 ^ {2} \ pi ^ {2} r ^ {6} \ right) = 4 \ pi \ cdot 3 ^ {2} \ left ({\ frac {4 ^ {2} \ pi ^ { 2}} {3 ^ {2}}} r ^ {6} \ right) = 36 \ pi \ left ({\ frac {4 \ pi} {3}} r ^ {3} \ right) ^ {2} = 36 \, \ pi V_ {p} ^ {2}.}

A_{s}^{3}=\left(4\pi r^{2}\right)^{3}=4^{3}\pi ^{3}r^{6}=4\pi \left(4^{2}\pi ^{2}r^{6}\right)=4\pi \cdot 3^{2}\left({\frac {4^{2}\pi ^{2}}{3^{2}}}r^{6}\right)=36\pi \left({\frac {4\pi }{3}}r^{3}\right)^{2}=36\,\pi V_{p}^{2}.

Consequently,

A_{s}=\left(36\,\pi V_{p}^{2}\right)^{\frac {1}{3}}=36^{\frac {1}{3}}\pi ^{\frac {1}{3}}V_{p}^{\frac {2}{3}}=6^{\frac {2}{3}}\pi ^{\frac {1}{3}}V_{p}^{\frac {2}{3}}=\pi ^{\frac {1}{3}}\left(6V_{p}\right)^{\frac {2}{3}}.

{\ displaystyle A_ {s} = \ left (36 \, \ pi V_ {p} ^ {2} \ right) ^ {\ frac {1} {3}} = 36 ^ {\ frac {1} {3} } \ pi ^ {\ frac {1} {3}} V_ {p} ^ {\ frac {2} {3}} = 6 ^ {\ frac {2} {3}} \ pi ^ {\ frac {1 } {3}} V_ {p} ^ {\ frac {2} {3}} = \ pi ^ {\ frac {1} {3}} \ left (6V_ {p} \ right) ^ {\ frac {2 } {3}}.}

A_{s}=\left(36\,\pi V_{p}^{2}\right)^{\frac {1}{3}}=36^{\frac {1}{3}}\pi ^{\frac {1}{3}}V_{p}^{\frac {2}{3}}=6^{\frac {2}{3}}\pi ^{\frac {1}{3}}V_{p}^{\frac {2}{3}}=\pi ^{\frac {1}{3}}\left(6V_{p}\right)^{\frac {2}{3}}.

Then the expression for sphericity $\Psi$ ${\ displaystyle \ Psi}$ $\Psi$ has the form

\Psi ={\frac {A_{s}}{A_{p}}}={\frac {\pi ^{\frac {1}{3}}\left(6V_{p}\right)^{\frac {2}{3}}}{A_{p}}}.

{\ displaystyle \ Psi = {\ frac {A_ {s}} {A_ {p}}} = {\ frac {\ pi ^ {\ frac {1} {3}} \ left (6V_ {p} \ right) ^ {\ frac {2} {3}}} {A_ {p}}}.

The sphericity of some objects

Title	Volume	Surface area	Sphericity
Platonic solids
Tetrahedron	${\frac {\sqrt {2}}{12}}\,s^{3}$ ${\ displaystyle {\ frac {\ sqrt {2}} {12}} \, s ^ {3}}$	${\sqrt {3}}\,s^{2}$ ${\ displaystyle {\ sqrt {3}} \, s ^ {2}}$	$\left({\frac {\pi }{6{\sqrt {3}}}}\right)^{\frac {1}{3}}\approx 0.671$ ${\ displaystyle \ left ({\ frac {\ pi} {6 {\ sqrt {3}}}} \ right) ^ {\ frac {1} {3}} \ approx 0.671}$
Cube (hexahedron)	$\,s^{3}$ ${\ displaystyle \, s ^ {3}}$	$6\,s^{2}$ ${\ displaystyle 6 \, s ^ {2}}$	$\left({\frac {\pi }{6}}\right)^{\frac {1}{3}}\approx 0.806$ ${\ displaystyle \ left ({\ frac {\ pi} {6}} \ right) ^ {\ frac {1} {3}} \ approx 0.806}$
Octahedron	${\frac {1}{3}}{\sqrt {2}}\,s^{3}$ ${\ displaystyle {\ frac {1} {3}} {\ sqrt {2}} \, s ^ {3}}$	$2{\sqrt {3}}\,s^{2}$ ${\ displaystyle 2 {\ sqrt {3}} \, s ^ {2}}$	$\left({\frac {\pi }{3{\sqrt {3}}}}\right)^{\frac {1}{3}}\approx 0.846$ ${\ displaystyle \ left ({\ frac {\ pi} {3 {\ sqrt {3}}}} \ right) ^ {\ frac {1} {3}} \ approx 0.846}$
Dodecahedron	${\frac {1}{4}}\left(15+7{\sqrt {5}}\right)\,s^{3}$ ${\ displaystyle {\ frac {1} {4}} \ left (15 + 7 {\ sqrt {5}} \ right) \, s ^ {3}}$	$3{\sqrt {25+10{\sqrt {5}}}}\,s^{2}$ ${\ displaystyle 3 {\ sqrt {25 + 10 {\ sqrt {5}}}} \, s ^ {2}}$	$\left({\frac {\left(15+7{\sqrt {5}}\right)^{2}\pi }{12\left(25+10{\sqrt {5}}\right)^{\frac {3}{2}}}}\right)^{\frac {1}{3}}\approx 0.910$ ${\ displaystyle \ left ({\ frac {\ left (15 + 7 {\ sqrt {5}} \ right) ^ {2} \ pi} {12 \ left (25 + 10 {\ sqrt {5}} \ right ) ^ {\ frac {3} {2}}}} \ right) ^ {\ frac {1} {3}} \ approx 0.910}$
Icosahedron	${\frac {5}{12}}\left(3+{\sqrt {5}}\right)\,s^{3}$ ${\ displaystyle {\ frac {5} {12}} \ left (3 + {\ sqrt {5}} \ right) \, s ^ {3}}$	$5{\sqrt {3}}\,s^{2}$ ${\ displaystyle 5 {\ sqrt {3}} \, s ^ {2}}$	$\left({\frac {\left(3+{\sqrt {5}}\right)^{2}\pi }{60{\sqrt {3}}}}\right)^{\frac {1}{3}}\approx 0.939$ ${\ displaystyle \ left ({\ frac {\ left (3 + {\ sqrt {5}} \ right) ^ {2} \ pi} {60 {\ sqrt {3}}}} right) ^ {\ frac {1} {3}} \ approx 0.939}$
Axially symmetric bodies
Cone $(h=2{\sqrt {2}}r)$ ${\ displaystyle (h = 2 {\ sqrt {2}} r)}$	${\frac {1}{3}}\pi \,r^{2}h$ ${\ displaystyle {\ frac {1} {3}} \ pi \, r ^ {2} h}$ $={\frac {2{\sqrt {2}}}{3}}\pi \,r^{3}$ ${\ displaystyle = {\ frac {2 {\ sqrt {2}}} {3}} \ pi \, r ^ {3}}$	$\pi \,r(r+{\sqrt {r^{2}+h^{2}}})$ ${\ displaystyle \ pi \, r (r + {\ sqrt {r ^ {2} + h ^ {2}}})}$ $=4\pi \,r^{2}$ ${\ displaystyle = 4 \ pi \, r ^ {2}}$	$\left({\frac {1}{2}}\right)^{\frac {1}{3}}\approx 0.794$ ${\ displaystyle \ left ({\ frac {1} {2}} \ right) ^ {\ frac {1} {3}} \ approx 0.794}$
Hemisphere	${\frac {2}{3}}\pi \,r^{3}$ ${\ displaystyle {\ frac {2} {3}} \ pi \, r ^ {3}}$	$3\pi \,r^{2}$ ${\ displaystyle 3 \ pi \, r ^ {2}}$	$\left({\frac {16}{27}}\right)^{\frac {1}{3}}\approx 0.840$ ${\ displaystyle \ left ({\ frac {16} {27}} \ right) ^ {\ frac {1} {3}} \ approx 0.840}$
Cylinder $(h=2\,r)$ ${\ displaystyle (h = 2 \, r)}$	$\pi r^{2}h=2\pi \,r^{3}$ ${\ displaystyle \ pi r ^ {2} h = 2 \ pi \, r ^ {3}}$	$2\pi r(r+h)=6\pi \,r^{2}$ ${\ displaystyle 2 \ pi r (r + h) = 6 \ pi \, r ^ {2}}$	$\left({\frac {2}{3}}\right)^{\frac {1}{3}}\approx 0.874$ ${\ displaystyle \ left ({\ frac {2} {3}} \ right) ^ {\ frac {1} {3}} \ approx 0.874}$
Thor $(R=r)$ ${\ displaystyle (R = r)}$	$2\pi ^{2}Rr^{2}=2\pi ^{2}\,r^{3}$ ${\ displaystyle 2 \ pi ^ {2} Rr ^ {2} = 2 \ pi ^ {2} \, r ^ {3}}$	$4\pi ^{2}Rr=4\pi ^{2}\,r^{2}$ ${\ displaystyle 4 \ pi ^ {2} Rr = 4 \ pi ^ {2} \, r ^ {2}}$	$\left({\frac {9}{4\pi }}\right)^{\frac {1}{3}}\approx 0.894$ ${\ displaystyle \ left ({\ frac {9} {4 \ pi}} \ right) ^ {\ frac {1} {3}} \ approx 0.894}$
Sphere	${\frac {4}{3}}\pi r^{3}$ ${\ displaystyle {\ frac {4} {3}} \ pi r ^ {3}}$	$4\pi \,r^{2}$ ${\ displaystyle 4 \ pi \, r ^ {2}}$	$1\,$ ${\ displaystyle 1 \,}$

Notes

↑ Wadell, Hakon. Volume, Shape and Roundness of Quartz Particles (English) // Journal of Geology : journal. - 1935. - Vol. 43 , no. 3 . - P. 250-280 . - DOI : 10.1086 / 624298 .

[1] Wadell, Hakon. Volume, Shape and Roundness of Quartz Particles (English) // Journal of Geology : journal. - 1935. - Vol. 43 , no. 3 . - P. 250-280 . - DOI : 10.1086 / 624298 .

Sphericity

Ellipsoidal objects

Formula Output

The sphericity of some objects

See also

Notes

More articles: