Nucleation and Growth Rates 
and Number, Size, Area and Mass of crystals  
from Crystal Size Distribution 
Author's Data: Housam BINOUS
Department of Chemical Engineering
National Institute of Applied Sciences and Technology
Tunis, TUNISIA
Email: binoushousam@yahoo.com

Acknowledgement: The crystal size distribution data were obtained from Harrison at al., Bioseparations Science and Engineering, OUP, 2003 and N. S. Tavare, Industrial Crystallization, p. 180, Plenum Press, 1995.

Crystal size distribution data from Harrison at al.

L = {17.4, 21.9, 27.6, 34.8, 43.8, 55.2, 69.6, 87.7, 110.5, 139.2, 
            175.4, 221. } 10^-6;

ΔL = {4. , 5. , 6.4, 8. , 10.1, 12.7, 16. , 20.2, 25.4, 
            32. , 40.3, 50.8} 10^-6;

(Subscript[n, 3600]) = {180.9, 43.6, 24.5, 21.8, 13.8, 13.9, 6.4, 3.8, 0.7, 0.4, 
            0.06, 0} 10^10;

(Subscript[n, 4500]) = {176.3, 55.7, 32.1, 23.4, 17.6, 18.1, 9.3, 4.8, 1.3, 0.3, 
            0.12, 0.09} 10^10;

\(tbl = 
      Transpose[{Join[{"L"}, L\ 10^6], 
            Join[{"ΔL"}, ΔL\ 10^6], 
            Join[{\*""\!\(n\_3600\)""}, n\_3600\ 10^\(-10\)], 
            Join[{\*""\!\(n\_4500\)""}, n\_4500\ 10^\(-10\)]}] // 
        TableForm\)

Zeroth moments give number of crystals

(Subscript[μ, 0, 3600]) = (Subscript[n, 3600]) . ΔL

(Subscript[μ, 0, 4500]) = (Subscript[n, 4500]) . ΔL

First moments give size of crystals

(Subscript[μ, 1, 3600]) = (Subscript[n, 3600]) . (ΔL L)

(Subscript[μ, 1, 4500]) = (Subscript[n, 4500]) . (ΔL L)

Second moments gives area of crystals

(Subscript[μ, 2, 3600]) = (Subscript[n, 3600]) . (ΔL L L)

(Subscript[μ, 2, 4500]) = (Subscript[n, 4500]) . (ΔL L L)

Third moments gives mass of crystals

(Subscript[μ, 3, 3600]) = 
      (Subscript[n, 3600]) . (ΔL L L L)

(Subscript[μ, 3, 4500]) = 
      (Subscript[n, 4500]) . (ΔL L L L)

Nucleation and growth rates from crystal size distribution data expressed in no./(Kg s) and m/s; respectively.

B = ((Subscript[μ, 0, 4500]) - (Subscript[μ, 0, 3600]))/(4500 - 
            3600)

G = ((Subscript[μ, 1, 4500]) - (Subscript[μ, 1, 3600]))/(((Subscript[μ, 0, 4500]) + (Subscript[μ, 0, 3600]))/2 (4500 - 3600))

Size of crystals in μm

(Subscript[CS, 3600]) = (Subscript[μ, 1, 3600])/(Subscript[μ, 0, 3600]) 10^6

(Subscript[CS, 4500]) = (Subscript[μ, 1, 4500])/(Subscript[μ, 0, 4500]) 10^6

Area of crystals in (μm)^2

(Subscript[CA, 3600]) = Pi (Subscript[μ, 2, 3600])/(Subscript[μ, 0, 3600]) 10^12

(Subscript[CA, 4500]) = Pi (Subscript[μ, 2, 4500])/(Subscript[μ, 0, 4500]) 10^12

density of crystals

ρ = 2.5 10^3;

Mass of crystals in milligrams

(Subscript[CM, 3600]) = 
      Pi/6 ρ (Subscript[μ, 3, 3600])/(Subscript[μ, 0, 3600]) 10^6

(Subscript[CM, 4500]) = 
      Pi/6 ρ (Subscript[μ, 3, 4500])/(Subscript[μ, 0, 4500]) 10^6