Question

A porous medium is constructed of ping-pong balls (of the same size, inner radius of Ri and outer radius of Ro) using cubic packing.

(a) Estimate the total and effective porosities of the pack. Make reasonable assumptions that you deem necessary to solve the problem.

(b) If the pore spaces of the cubic pack are filled with solid spherical grains of quartz that fit exactly in the pore spaces (poor sorting), calculate the effective porosity of the new pack. Quantify the effect of this poor sorting on the effective porosity of the cubic pack?

Answer 1

Answer(a)

Porosity(n) is given by (total volume-volume of solid)/total volume.

n=(V-V_s)/V.....................................................(where V=total volume, V_s=volume of solids).

Assuming a cube made out of ping pong balls(3*3*3)=27 balls..............................(assume any cube,result will be same).

,each of radius(R_o).

Now,

Volume of solids(V_s)=27*(4/3)*pi*(R_o)³

and Total volume(V)=(3*2R_o)³ =27*8*(R_o)³..............................................................(each sideof cube will be equal to 3*2R_o).

Porosity(n)=(V-V_s)/V ={(27*8*(R_o)³-27*(4/3)*pi*(R_o)³}/27*8*(R_o)³

from the above expression taking the term 27(R_o)³ out ,being common

n=27(R_o)³*(8-(4/3)*pi)/27*8*(R_o)³

n=(8-(4/3)*pi)/8

n=0.476 or 47.6%.

This value of porosity (0.476) is a constant for cubic arrangement of soil particles have perfectly spherical shape irrespective of the size of particles or cube .

Answer(b)

Since the value of porosity is constant for any cube formed out of perfectly spherical particles irrespective of the size of spherical particles i.e. 0.476 or 47.6%

so, after the filling of the voids of the current cubic packet with even smaller spherical quartz spheres the porosity will further decrease down as follows

n=0.476*0.476.....................................(i.e.47.6% of the previous value of porosity).

so, n=0.2265 or 22.65%

now the porosity is reduced to 22.65%.