APPENDIX B CALCULATION OF MEAN DIAMETER OF …

1 APPENDIX B: CALCULATION OF MEAN DIAMETER OF particles 141. APPENDIX B. CALCULATION OF MEAN DIAMETER OF particles . Sample CALCULATION : Two methods are used to calculate the Sauter mean DIAMETER of particle, there are: Sauter mean DIAMETER based on weight fraction and Sauter mean DIAMETER based on number of particles . Method 1: Sauter Mean DIAMETER , d32, based on Weight Fraction Table : Thickness fractions of 100% accepts with weigh fraction, average width, average length, and average thickness. Weight Fraction Average Average Average (xi) Width (mm) Length (mm) Thickness (mm). 6 mm thickness 4 mm thickness 2 mm thickness To determine the Sauter mean DIAMETER , d32, of wood chips (parallelepipedal shape), the specific surface area Sv is used: As Sv = ( ). Vp As 2 (w t + w l + t l ). Sv = = ( ). Vp w t l where As is mean surface area of the particles , Vp is the mean volume of the particles , w is width, l is length, and t is thickness.

2 For 6 mm thickness, the specific surface area Sv is 2 ( + + ). Sv = ( ). 6 mm APPENDIX B: CALCULATION OF MEAN DIAMETER OF particles 142. mm 2. Sv = ( ). 6 mm mm 3. For non-spherical particles , dp can be written as (Comiti and Renaud, 1989): 6. dp = ( ). Sv 6. dp = ( ). mm 2. mm 3. d p = ( ). Using the same procedures as above, the dp for 4 mm and 2mm thickness can be determined. The results are tabulated in Table Table : Results of mean surface area, particle volume, specific surface area, and equivalent DIAMETER of thickness fraction found in accepts. Weight Fraction (xi) As (mm2) V (mm3) Sv (mm2/ mm3) dp (mm). 6 mm thickness 1659 3328 4 mm thickness 1087 1541 2 mm thickness 777 798 d 32 = For a wide range of size distribution, the Sauter mean DIAMETER of particle can be written as follows (Coulson et al., 1978): 1. d 32 = ( ). xi i d p ,i where xi is the weight fraction of particles of size dp,i.

3 1. d 32 = ( ).. +. 8 .5. + .. APPENDIX B: CALCULATION OF MEAN DIAMETER OF particles 143. and the result is d 32 = ( ). Table : Average equivalent particle DIAMETER of 100% pins. Weight Fraction (xi) As (mm2) V (mm3) Sv (mm2/ mm3) dp (mm). 100% pins 1 248 156 d 32 = Table : Average equivalent particle DIAMETER of 75% accepts and 25% pins. Weight Fraction (xi) dp (mm). accepts pins d 32 = Table : Average equivalent particle DIAMETER of accepts and pins. Weight Fraction (xi) dp (mm). accepts pins d 32 = mm Method 2: Sauter Mean DIAMETER , d32, based on Number of particles Consider a unit mass of particles consisting of n particles of size d, constituting a weight fraction (Coulson et al., 1978). Then 3. x = nkd p ( ). or 1 x n = ( ). k d 3p where is the density of the particles k is a constant whose value depends on the shape of the particle n is number of particles APPENDIX B: CALCULATION OF MEAN DIAMETER OF particles 144.

4 X is weight fraction dp is the DIAMETER of particle, Equation For particle size distribution, the Sauter mean DIAMETER of the particle can be determined by d 32 =. [(n d )S ] f ,i p ,i i ( ). (n S ) f ,i i =. (n 3. f ,i k i d p ,i ) ( ). (n ). d 32 2. f ,i k i d p ,i =. (n 3. f ,i d p ,i ) ( ). (n ). d 32 2. f ,i d p ,i where S i = k i d i 2 , ki being a constant whose value depends on particle shape. nf,i is the number of particles in fraction d32 is the Sauter mean DIAMETER Table : Summarize calculated values of different size fraction in 100% accepts. Size dp (mm) dp2 (mm2) dp3 (mm3) Weight Number of Number of fraction particles particles in x n fraction, nf 6 mm thick 144 1728 10-6 4 mm thick 614 10-4 2 mm thick 238 10-3 Therefore, d32 of 100% accepts can be calculated using Equation with Table values, =. (n 3. f ,i d p ,i ) = 297 = 6 .80 mm ( ). (n ) 43 .7.

5 D 32 2. f ,i d p ,i APPENDIX B: CALCULATION OF MEAN DIAMETER OF particles 145. Table : Summarize calculated values of different size fraction in accepts + pins. Size dp (mm) dp2 (mm2) dp3 (mm3) Weight Number of Number of fraction particles particles in x n fraction, nf Pins 54 10-3 Accepts 314 10-3 Therefore, d32 of accepts + pins can be calculated using Equation with Table values, d 32 =. (n 3. f ,i d p ,i ) = 196 = 6 .18 mm ( ). (n f ,i d 2. p ,i ) 31 .7. Table : Summarize calculated values of different size fraction in 75% accepts + 25% pins. Size dp (mm) dp2 (mm2) dp3 (mm3) Weight Number of Number of fraction particles particles in x n fraction, nf Pins 54 10-3 Accepts 314 10-3 Therefore, d32 of 75% accepts + 255 pins can be calculated using Equation with Table values, d 32 =. (n 3. f ,i d p ,i ) = 143 = 5 .67 mm ( ). (n f ,i d 2. p ,i ) 25 .2. APPENDIX B: CALCULATION OF MEAN DIAMETER OF particles 146.

6 N u m b e r F r a c t io n Particle DIAMETER , d32, (mm). Figure : Measured number fraction vs. particle DIAMETER (d32) of 100% accepts. APPENDIX B: CALCULATION OF MEAN DIAMETER OF particles 147. N u m b e r F r a c tio n 4 5 6 7 8 9 10 11 12. Particle DIAMETER , d32 (mm). Figure : Calculated number fraction vs. particle DIAMETER (d32) of 100% accepts. The Sauter mean DIAMETER of particle is mm. APPENDIX B: CALCULATION OF MEAN DIAMETER OF particles 148. N u m b e r F r a c tio n 2 3 4 5 6 7 8 9 10 11 12. DIAMETER of Particle (mm). Figure : Number fraction vs. particle DIAMETER (d32) of accepts + pins.. The Sauter mean DIAMETER of particle is mm. APPENDIX B: CALCULATION OF MEAN DIAMETER OF particles 149. N u m b e r F r a c tio n 2 3 4 5 6 7 8 9 10 11 12. DIAMETER of Particle (mm). Figure : Number fraction vs. particle DIAMETER (d32) of 75% accepts + 25% pins. The Sauter mean DIAMETER of particle is mm.

7 APPENDIX B: CALCULATION OF MEAN DIAMETER OF particles 150. N u m b e r F r a c tio n 0. 2 3 4 5 6 7 8 9 10 11 12. Particle DIAMETER , d32 (mm). Figure : Number fraction vs. particle DIAMETER (d32) of 100% pins. The Sauter mean DIAMETER of particle is mm.