How To Calculate Sphericity

When comparing theoretical models of how things work to real-world applications, physicists often approximate the geometry of objects using simpler objects. This could be using thin cylinders to approximate the shape of an airplane or a thin, massless line to approximate the string of a pendulum.

Sphericity gives you one way of approximating how close objects are to sphere. You can, for example, calculate the sphericity as an approximation the Earth's shape which is, in fact, not a perfect sphere.

When finding sphericity for a single particle or object, you can define sphericity as the ratio of surface area of a sphere that has the same volume as the particle or object to the surface area of the particle itself. This is not to be confused with Mauchly's Test of Sphericity, a statistical technique to test assumptions within data.

Put into mathematical terms, the sphericity given by ​Ψ​ ("psi") is:

\(\Psi=\frac{\pi^{1/3}(6V_p)^{2/3}}{A_p}\)

for the volume of the particle or object ​_V_p​ and surface area of the particle or object ​A_p_​. You can see why this is the case through a few mathematical steps to derive this formula.

First, you find another way of expressing the surface area of a particle.

1. ​_A_s = 4πr_​² : Start with the formula for the surface area of a sphere in terms of its radius ​r​.
2. ​_( 4πr²​ ​)_​^{​3 ​}: Cube it by taking it to the power of 3.
3. ​_4³π_​³​r​⁶ : Distribute the exponent 3 throughout the formula. 
4. ​_4π(_​4²π²​r​⁶): Factor out the ​4π​ by placing it outside using parentheses. 
5. ​_4π x 3² (_​​_4²π_​²​r​⁶ ​/​​_3²)​ : Factor out ​3²._​
6. ​36​​_π (_​​4π​​r​³​_ /3​​)_​² : Factor out the exponent of 2 from the parentheses to get the volume of a sphere.
7. ​_36πV_p²​ : Replace the content in the parentheses with the volume of a sphere for a particle.
8. ​A_s = (36V_p²)^1/3_​ : Then, you can take the cube root of this result so that you are back to the surface area.
9. 36^1/3​π​^1/3​V​_p^2/3 : Distribute the exponent of 1/3 throughout the content in the parentheses. 
10. ​π​^1/3(6​V​_p)^2/3 : Factor out the ​π​^1/3 from the result of step 9. This gives you a method of expressing surface area. 

Then, from this result of a way of expressing surface area, you can rewrite the ratio of the surface area of a particle to the volume of a particle with

\(\frac{A_s}{A_p}=\frac{\pi^{1/3}(6V_p)^{2/3}}{A_p}\)

which is defined as ​Ψ​. Because it's defined as a ratio, the maximum sphericity an object can have is one, which corresponds to a perfect sphere.

You can use different values for changing the volume of different objects to observe how sphericity is more dependent upon certain dimensions or measurements when compared to others. For example, when measuring sphericity of particles, elongating particles in one direction is much more likely to increase sphericity than changing the roundness of certain parts of it.

Using the equation for sphericity, you can determine the sphericity of a cylinder. You should first figure out the volume of the cylinder.. Then, calculate the radius of a sphere that would have this volume. Find the surface area of this sphere with this radius, and then divide it by the surface area of the cylinder.

If you have a cylinder with a diameter of 1 m and height of 3 m, you can calculate its volume as the product of the area of the base and height. This would be

\(V=Ah=2\pi r^2 3 = 2.36\text{ m}^3\)

Because the volume of a sphere is ​_V = 4πr³/3_​, you can calculate the radius of this volume as

\(r=\bigg(\frac{3V\pi}{4}\bigg)^{1/3}\)

For a sphere with this volume, it would have a radius r = ​_(2.36 m³ x (3/4​​π)​​)^1/3 = .83 m._​

The surface area of a sphere with this radius would be ​_A = 4πr²​ or 4​πr²_​ or 8.56 m³. The cylinder has a surface area of 11.00 m² given by ​A = 2(πr​²​) + 2πr x h​, which is the sum of the areas of the circular bases and the area of the curved surface of the cylinder. This gives a sphericity ​Ψ​ of .78 from the division of the sphere's surface area with the cylinder's surface area.

You can expedite this step-by-step process involving volume and surface area of a cylinder alongside volume and surface are of a sphere using computational methods that can calculate these variables one-by-one much more quickly than a human can. Performing computer-based simulations using these calculations are just one application of sphericity.

Sphericity originated in geology. Because particles tend to take irregular shapes that have volumes that are difficult to determine, geologist Hakon Wadell created a more applicable definition that uses the ratio of the nominal diameter of the particle, the diameter of a sphere with the same volume as a grain, to the diameter of the sphere that would encompass it.

Through this, he created the concept of sphericity that could be used alongside other measurements like roundness in evaluating the properties of physical particles.

Aside from determining how close theoretical calculations are to real-world examples, sphericity has a variety of other uses. Geologists determine the sphericity of sedimentary particles to figure out how close they are to spheres. From there, they can calculate other quantities such as the forces between particles or perform simulations of particles in different environments.

These computer-based simulations let geologists design experiments and study features of the earth such as the movement and arrangements of fluids between sedimentary rocks.

Geologists can use sphericity to study the aerodynamics of volcanic particles. Three-dimensional laser scanning and scanning electron microscope technologies have directly measured the sphericity of volcanic particles. Researchers can compare these results to other methods of measuring sphericity such as the working sphericity. This is the sphericity of a tetradecahedron, a polyhedron with 14 faces, from the flatness and elongation ratios of the volcanic particles.

Other methods of measuring sphericity include approximating the circularity of a particle's projection onto a two-dimensional surface. These different measurements can give researchers more accurate methods of studying the physical properties of these particles when released from volcanoes.

The applications to other fields are worth noting as well. Computer-based methods, in particular, can examine other features of the sedimentary material such as porosity, connectivity and roundness alongside sphericity to evaluate the physical properties of objects such as the degree of osteoporosis of human bones. It also lets scientists and engineers determine how useful biomaterials may be for implants.

Scientists studying nanoparticles can measure the size and sphericity of silicon nanocrystals in finding out how they can be used in optoelectronic materials and silicon-based light emitters. These can later be put to use in various technologies like bioimaging and drug delivery.