Laws of size reduction

Laws of size reduction

• The estimation of the power requirement of crushing and grinding operations are very important in the design and selection of size reduction equipment
• But it is not possible to estimate accurately the power requirement of crushing and grinding to affect the size reduction of a given material.
• There are a number of  empirical laws to find power requirement  crushing and grinding operations

Rittinger's law

It states that the work required for the crushing operation is directly proportional to the newly generated surface area.

• Mathematically

• where

                        P= power required by the machine

                        m = feed rate to machine

                        K_r = Rittinger's constant

Kick's law

• Kick's law related the energy to the sizes of the feed particles and the product particles

It states that the work required for crushing a given material is proportional to the logarithm of the ratio between the initial and final diameters

• where

                      K_K = Kick's constant

                      D and d = initial and final diameters

• Kick's law is more accurate than Rittinger's law for the course where the amount of surface produced is considerably less.

Bond's law

• Bond has proposed a law intermediate between Rittinger's and kick's law for estimating the power required for crushing and grinding operations.

Bond postulated that work required to form particles of size D_p from the very large feed is proportional to the square root of the surface-to-volume ratio of the product, S_p/V_p.

• By relation S_p/V_p = 6/ɸsD_p, from which it follows that

• where

                     D_p = the particle size

                     Kb = Bond's constant that depends on the type of machine and the         material being crushed

Hence Bond's law also states that the total work useful in breakage is inversely proportional to the square root of the diameter of the product particles.

• from above equation, a work index W_i is defined as the gross energy required in kilowatt-hours per ton of feed material, required to reduce a very large feed to such a size that 80 % of the product passes a 100-µm screen.
• This definition leads to a relation between K_b and W_i. If D_p is in millimeters, P in kilowatts, and ṁ in tons per hour

• If 80% of the feed passes a mesh size of D_pa millimeters and 80% of the product passes through a mesh of D_pb millimeters, then from above two equation we get