Pa ge 1 Pa ge 7 American Journal of Geospatial Technology (AJGT) Comparative Assessment of Comminution Energy Equations (Models) Using Some Selected Legumes, Tubers, Cereals and Sea Food Akpan Abasiama J.1, Olosunde William A.1*, Antia Orua O.1 Volume 2 Issue 1, Year 2022 ISSN: 2833-8006 (Online) DOI: https://doi.org/10.54536/ajgt.v2i1.1969 https://journals.e-palli.com/home/index.php/ajgt Article Information ABSTRACT Received: August 23, 2023 Accepted: September 27, 2023 Published: September 30, 2023 The quest to predict more precisely energy that is just enough to achieve closely any desired final particle size of food material subjected to comminution is necessary. This is to avoid energy wastage, obtain finer product size(s) that are uniform and can encourage easy mixing, dehydration/drying, etc. In this regard, some selected legumes (soybeans and beans), cereals (sorghum, millet, corn and wheat), sea food (crayfish) and tubers (cassava and yam) were subjected to comminution through the application of some major energy equations (mod- els); as they find various applications in food industries. Four energy equations (models) namely the Kick’s, Rittinger’s and Bond’s for size reduction and the OruaAntia’s minimum energy equation for mass-size reduction were employed. The constant in each equation (model) to be applied in grinding of selected food materials was determined and used in obtaining the required specific energy of comminution needed to accomplish desired final product size(s). The corresponding grinding time expected to achieve the desired final prod- uct were computed and used in operating the grinding machine. Results revealed that the OruaAntia’s minimum energy equation for mass-size reduction operation may be applied on the selected food materials to achieve very closely any desired final average particle diameter with a percentage deviation of 1.66% followed with Rittinger’s equation having average percentage deviation of 6.88%. Technical analysis re-affirm that Kick’s equation could only achieve coarse particles as the final particle size showed average percentage deviation of 38.40%, while Bond’s equation may be limited to prediction of coarse and intermediate particles; since the final particle size average percentage deviation was 16.19%. Besides using Rittinger’s energy equation to obtain fine particles, the use of Orua Antia minimum energy equation may possibly further achieve desired finer particle size(s). Keywords Comminution, Energy Equations, Food Materials, Grinding Machine Particle Size 1 Department of Agricultural and Food Engineering University of Uyo, Uyo, Nigeria * Corresponding author’s e-mail: williamolosunde@uniuyo.edu.ng INTRODUCTION The amount of energy used for grinding materials may depend on the materials physical characteristics, machine geometrical and kinematic parameters used for the grinding. Size reduction operation is very important in food sector as it enhances powder mixing, reduction in sedimentation rate among several things (Sushaut & Archana, 2013; Song., Mycong., Hyong., & Won., 2014; Dabbour., Bahnasawy., Ali., & El-Haddad., 2015; Mulla., Hajare., & Doijad., 2016). During size reduction, particles may experience elastic and inelastic deformation. Hence, requires appropriate energy to produce product that may be averagely uniform in size or categorized into distinct size ranges. Thus, accurate estimation of energy requirements necessary for size reduction equipment is economically essential in many aspect (Antia., Obahiagbon., Aluyor., & Ebunilo., 2013; Antia & Aluyor, 2018). In an effort to minimize cost coupled with achieving the desire size range of material, various equations (models) have been proposed and developed based on the differential energy (dE) required to produce a small change (dX) in size of a unit of a material x (Akinoso., Lawal., & Aremu., 2013; Ndukwu., Nwakuba., & Henry., 2016; Antia, 2021, pp. 672-673) as: dE/dX ∝ x(-n) (1) dE= -kx(-n) dX (2) Where, k and n are constant. n=1,2,3⁄2for kicks, Rittinger’s and Bond’s concept respectively. These expressions are as follows: Ek=Kk [ln x1/x2 ] (3) ER=KR [1/x2 -1/x1 ] (4) EB=KB [1/√(x2 )-1/√(x1 )] (5) Where,Ek,ER and EB are energy by Kick’s,Rittinger’s and Bond’s respectively and may be expressed in J or J⁄kg or kWh⁄kg or Ws⁄kg. Kk, KR and KB may be expressed in J or J⁄kg or kWh⁄kg,Jm or Jm⁄kg or (kWh.m)⁄kg,Jm(1⁄2) or Jm(1⁄2)⁄kg or kWh.m(1⁄2)⁄(kg) respectively as Kick’s, Rittinger’sand Bond’s constants while x1 and x2 are respectively initial and final particle sizes(diameter(s)) in m. A new dimension considered by Antia., Obahiagbon., Aluyor., & Ebunilo., 2014 in this regard was that energy (dE) is required to produce a small change (dA) in area of a unit material. This approach is because it is easier to measure effectively any material by mass (m) irrespective of it size and shape. Moreso, mass could be related to area (A) as: m=ρV=ρAT (6) Where, V= volume of the material ρ= density of the material T= thickness of the material Where, A= m/ρT (7) Pa ge 8 https://journals.e-palli.com/home/index.php/ajgt Am. J. Geo Spat. Technol. 2(1) 7-14, 2023 Hence, the concept was expressed as: dE/dA ∝ A(-n) (8) dE= -KA(-n) dA (9) Emin= Bm(1-n)/(1-n) (10) Where, n= power index E_min=Minimum energy required for mass reduction of the materialThe power index n was experimentally determined as n =1⁄2. The minimum energy for the mass reduction was obtained as: Emin=2Bm(1⁄2) (11) Where,B = mass index, m= mass of the material A combination of analytical and empirical approaches to relate size(diameter) of the material resulted in one of the expressions (models) for mass-size reduction operations (Antia, 2020) as: EA=K(A3 ) [1/(D2 (3⁄2) )-1/(D1 (3⁄2) )] (12) Where, K(A3 ) = Orua Antia Energy constant expressed in Jm(3/2) or Jm(3/2)/kg or KWh. m(3/2)/kg EA = Orua Antia minimum energy for mass-size reduction of material and may be expressed in J/kg.kWh/kg or Ws/kg D1= Initial diameter of the material in m. D2 = Final diameter of the material in m. K(A3)= (2Bρm (-½))/(Cf Mf) (0.2304)= (0.4608 Bρ(-½))/(Cf Mf) (13) Where, Cf= crushing efficiency Mf= Mechanical efficiency Besides the mentioned energy equations (models), various researchers have proposed other models during studies on energy consumption for grinding materials. Mani., Tabil., & Sokhansanj (2004) worked on energy consumption of switch grass, corn stover and wheat straw using hammer mill with screen size of 0.8, 1.6 and 3.2mm. The data obtained fitted a straight line and a second order polynomial respectively at 8 and 12% moisture content of the material as: E = k1+ k2 d (14) E = k1+ k2 d+k2 d 2 (15) Where, E = specific energy,(kWh⁄t) d is the hammer mill screen size (mm) k1, k2= constants. Bitras., Womac., Chevanan., Miu., Igathinathane., Sokhansanj et al. (2009) also worked on switch grass, corn stover and wheat straw having respectively the initial mean sizes of 8.3, 7.1, and 8.3 mm. The feeding rate of 41.7 g/s and hammer mill with 3.2 mm screen were used. The data obtained fitted equation given as: Ee= KΔXg (16) Where, K is a function of the rotational speed, N [i.e. K=f(N)] Ee= specific energy (MJ/Mg) ΔXg= unit size reduction (mm) Adapa., Tabilo., & Schoenau (2011) obtained barley straw, canola straw, oat straw, and wheat straw for grinding using two hammer mills. The first had 30 mm screen while the second had 6.4, 3.2, and 1.6 mm screen. The energy equation proposed was given as: E=k1 S (-k2 ) (17) Where, E = specific energy, (kWh⁄t) s= hammer mill screen size (mm) k1, k2= constants. Miao., Grift., Hansen., & Ting (2011) used a knife mill with screens that ranges from 10mm to 1mm to grind some biomass at two moisture content levels of 15 and 7%. The energy consumption and particle size were related as: E=axb (18) Where, E= specific comminution energy, J/gx= comminution ratio(ratio of average size of initial biomass to average size of comminuted particles)a and b = regression constants. In another development, the Kick’s, Rittinger’s and Bond’s equations were used to grind hard wood and soft wood (Naimi., Sokhansanj., Bi., Lim., Womac., Lau et al., 2013). It was observed that the predicted specific comminution energy fitted best using Rittinger’s equation followed by Bond’s and Kick’s equations. Ladan., Flavien., Xiaotao., Jim., & Shahab (2016) used rotary knife mill and Lingocellulose biomass conditioned at 11.5% moisture content coupled with Kick’s, Rittinger’s and Bond’s equations. The experimental data generated fitted best the Rittinger’s followed by Bond’s and Kick’s energy equations for size reduction operations. In this study, the energy Equations (models) 3,4,5 and 12 were used to assess the most applicable energy of comminution for some selected cereals (corn, millet, wheat and sorghum), legumes (soya beans and beans), tubers (yam and cassava) and sea food (cray fish). MATERIALS AND METHODS Equipment, Sourcing and Preparation of Materials The equipment used includes: vernier calliper, sieves, digital weighing balance, polytene bags, sieve shaker, stop watch and grinder (Hammer-Attrition mill). The selected materials used were cereals (corn, millet, wheat and sorghum), legumes (soybeans and beans), tuber (cassava and yam) and sea food (cray fish,) obtained from Uyo local market, Akwa Ibom State, Nigeria. Each material was cleaned manually to remove any unwanted material on it; and then air dried at 105℃ till constant weight (bone dried mass) was achieved (Antia., Oboh., & Olosunde., 2019). Thereafter, each material was stored for use in this study. Evaluation of Energy Equations (models) Constants The grinding machine energy was first determined by reading the power of electric motor used from the machine description panel. A constant time for grinding was chosen to evaluate all the constants in the energy equations (Kick’s,Rittinger’s, Bond’s and OruaAntia’s). The energy of the grinding machine was given (MohdRozalli., Chin., & Yusof., 2015; Antia, 2021, pp. 681-692) as: Energy (J)= power(J⁄s)x time(s) (19) Therefore, the specific energy E1 * of the grinding machine powered by the electric motor may be given as: E1 *=([Energy (J)])/(mass of material @subjected to Pa ge 9 https://journals.e-palli.com/home/index.php/ajgt Am. J. Geo Spat. Technol. 2(1) 7-14, 2023 grinding, kg)=(Electric motor power (J⁄s)×@grinding Time (s))/(mass of material @subjected to grinding, kg) (20) The specific energyE1 * may be expressed in J⁄kg or kWh⁄kg or Ws⁄kg. The grinding time chosen and the electric motor power used were 120sec and 0.746kW respectively. The mass (500g) of each selected material to be used per experimental run was chosen and its average initial (feed) diameter was determined. The average final (product) diameter of each selected material after being subjected to grinding was determined using sieve analysis (Antia, 2021, pp. 700-710; Okoro, 2001, pp. 31-53). The energy constants of Equations 3,4,5 and 12 were determined using specific Energy E1 *, the sample average initial feed diameter of the feed and the average final product diameter. Estimation of Specific Energy of Comminution and Grinding Time Applicable to Each Energy Equation In assessment of the energy Equations 3,4,5 and 12 for size reduction operations;three desired final particle sizes were chosen as 300μm, 400μm, and 500μm. Sieve aperture size that correspond to the chosen (desired) final product was used to sieve out each of the selected materials after grinding.The specific comminution energyE2 *based on the four energy Equations 3,4,5 and 12 were computed for each of the selected samples by using each corresponding energy equation constant obtain in section 2(ii)coupled with the initial average diameter of the feed and each desired (chosen) average diameter of the final product.Each time expected to grind the particles to achieve each of the three (3) chosen sizes (diameters) 300μm, 400μm, and 500μm were calculated based on the Equation 21(MohdRozalli., Chin., & Yusof., 2015, Antia, 2021, pp. 681-692) as: tg=(E2 *×ms)/Pm (21) Where, E2 *= Expected specific comminution energy required in (kWh.Kg(-1)) ms=Chosen quantity of sample to be ground (Kg) Pm=power of the grinding machine (kW) tg=Expected time required for grinding (hr) The estimated time required to achieve each final product size chosen per selected material was used to run and grind the sample(s) accordingly. The final product of eachselected sample per grinding time was subjected to sieve analysis; in order to determine the most appropriate energy equation for application, to achieve the expected desired size (diameter) (d2 *) of the final product. Determination of Particle Diameter for Comparative Assessment of the Energy Equations Sieving tests were conducted to determine the initial and final average/mean particle size (dp), of each selected material subjected to comminution. Each test was conducted by placing a series of sieves with progressively smaller mesh sizes on top of each other and passing the sample through the stacked sieve “tower” using a sieve shaker for a given time.The following expressions were used for the analysis (Antia, 2021, pp. 700-710; Okoro, 2001, pp. 54-69): Rf = ᵶ⁄m (22) Where, ᵶ=weight of material on a given sieve m= mass of sample dpi= Si Rf (23) dp = ∑dpi (24) Where, i=1,2,..... n is the serial number of sieve aperturesincluding the final bottom material n sieved out Si=aperture size of the sieve used dpi=average particle size (diameter) on i sieve The values of the average/mean diameters obtained using each energy equation on each selected material were compared; to determine the equation that could be used to predict closely the comminution of each selected material. This may be achieved by computing the percentage deviation of actual final average diameter (d2 *) with predicted (chosen) final average diameter (d2) of the selected materials subjected to comminution.The equations may be given as: % deviation (Ep )=[(∑d2 *- d2 )/NT ]×100 (25) Where, NT=d2 n d2= desired (chosen) final average particle size (diameter), m n = number(s) of selected material(s) used in testing the energy equations d2 * = actual final average particle size (diameter) following comminution,m average % deviation (Epa)=( ∑Ep )/NTS (26) Where,NTS=the total number of size(s)(diameter(s)) chosen and used in testing the energy equations(s). RESULTS AND DISCUSSION Each constant required as applicable to Kick’s, Rittinger’s, Bond’s and OruaAntia’s energy Equations 3, 4, 5 and 12 respectively for each of the selected sea food (crayfish), cereals (sorghum. millet, corn and wheat), legumes (soybeans and beans) and tubers (cassava and yam) wasobtained and are presented in Table 1. From Table 1, it is observed that at the chosen grinding time 120sec (2min), the grinding machine with power of 0.746kWcoincidently produce final average product size of 600μm for each 500g of the selected materials. This may imply that at commencement of grinding, each equation accommodate this size within two (2) minutes range of grinding time; and beyond which each energy equation may likely start to exhibit its peculiar capability. In assessing each of the energy equations, the specific energy of comminution E2 * applicable to each of the energy equation considered was calculated based on the initial average feed diameter(s) and chosen final average product size(s) (300μm, 400μm, and 500μm) coupled with their respective determined constant in Table 1. The obtained values of E2 * using Equations 3, 4, 5 and 12 are presented in Tables 2 to 4. From Tables 2 to 4, each grinding time per selected material was obtained from Equation 21. Moreso, the desired (chosen) average particle size(s) d2 and actual average particle size(s) d2 *obtained after subjecting the Pa ge 10 https://journals.e-palli.com/home/index.php/ajgt Am. J. Geo Spat. Technol. 2(1) 7-14, 2023 Table 2: Specific comminution energy E2 *,predicted grinding timetg, desired (300μm) and actual final product(s) size(s) Se le ct ed m at er ia ls (s am pl es ) In iti al fe ed di am et er (d 1) (m ) D es ire d (c ho se n) Fi na l p ro du ct di am et er (d 2) (m ) E2 * Specific Energy Requirement (kWh⁄kg) tg Predicted Crushing Time (min:sec) Actual Final product Diameter (d2 *) (× 10(-4) m) K ic k' s R itt in ge r’s B on d' s O ru aA nt ia 's K ic k' s R itt in ge r’s B on d' s O ru aA nt ia 's K ic k' s R itt in ge r’s B on d' s O ru aA nt ia 's Se a fo od C ra yfi sh 0. 02 96 7 0. 00 03 0 0. 05 85 70 0. 10 04 93 0. 07 36 75 0. 14 09 29 02 :2 2 04 :0 2 02 :5 8 05 :4 0 5. 46 3. 13 3. 45 3. 03 C er ea ls So rg hu m 0. 00 31 7 0. 00 03 0 0. 07 04 43 0. 11 10 78 0. 08 61 98 0. 14 88 27 02 :5 0 04 :2 8 03 :2 8 05 :5 9 4. 96 3. 54 4. 17 3. 01 M ill et 0. 00 19 5 0. 00 03 0 0. 07 90 06 0. 12 16 03 0. 09 60 24 0. 15 94 16 03 :1 0 04 :5 3 03 :5 2 06 :2 5 4. 24 3. 08 3. 64 3. 01 C or n 0. 02 01 7 0. 00 03 0 0. 05 95 41 0. 10 09 92 0. 07 46 27 0. 14 11 36 02 :2 3 04 :0 4 03 :0 0 05 :4 1 5. 27 3. 10 3. 53 3. 06 W he at 0. 00 18 6 0. 00 03 0 0. 06 19 99 0. 10 26 51 0. 07 70 30 0. 14 20 30 02 :2 9 04 :0 8 03 :0 6 05 :4 3 5. 19 3. 27 3. 75 3. 09 Le gu m es So yb ea n 0. 00 51 6 0. 00 03 0 0. 06 57 58 0. 10 60 15 0. 08 09 98 0. 14 44 25 02 :4 3 04 :1 6 03 :1 6 05 :4 9 5. 10 3. 64 4. 17 3. 07 Be an s 0. 00 63 0 0. 00 03 0 0. 05 88 26 0. 10 06 15 0. 07 39 76 0. 14 09 77 02 :2 2 04 :0 3 02 :5 8 05 :4 0 5. 27 3. 34 3. 66 2. 97 Tu be rs C as sa va 0. 00 14 0 0. 00 03 0 0. 09 36 24 0. 14 11 43 0. 11 31 62 0. 18 11 13 03 :4 6 05 :4 1 04 :3 3 07 :1 7 5. 68 3. 23 3. 58 3. 14 Table 1: Energy equations constants obtained for selected materials with respect to grinding machine power (0.746kW) and time (120sec) used. Se le ct ed m at er ia ls (s am pl es ) (d 1) In iti al fe ed di am et er (m ) (d 2) Fi na l pr od uc t di am et er (m ) (E 1* )S pe ci fic en er gy re qu ire m en t (J ⁄k g or k W h⁄ kg ) Energy equation constants K ic k’ s, K k (J ⁄k g) (1 0(- 2) ) R itt in ge r’s , K R (J m ⁄k g) (1 0(- 5) ) B on d’ s, K B (J m (1 ⁄2 ) ⁄k g) (1 0(- 3) ) O ru aA nt ia ’s , K k ( Jm (3 ⁄2 ) ⁄k g) (1 0(- 7) ) Sea food Crayfish 0.02967 0.00060 0.049733 1.274932 3.045596 1.418761 7.330361 Cereals Sorghum 0.00317 0.00060 0.049733 2.987782 3.680654 2.156346 7.965170 Millet 0.00195 0.00060 0.049733 4.223176 4.312190 2.737464 8.816342 Corn 0.02017 0.00060 0.049733 1.414946 3.075503 1.472139 7.346981 Wheat 0.00186 0.00060 0.049733 1.769614 3.175078 1.614205 7.418804 Legumes Soybean 0.00516 0.00060 0.049733 2.311909 3.376890 1.848848 7.611337 Beans 0.00630 0.00060 0.049733 1.311845 3.052907 1.433589 7.334147 Tubers Cassava 0.00140 0.00060 0.049733 6.332049 5.484559 3.750929 10.560293 Yam 0.00196 0.00060 0.049733 5.283426 4.892742 3.245050 9.664068 Pa ge 11 https://journals.e-palli.com/home/index.php/ajgt Am. J. Geo Spat. Technol. 2(1) 7-14, 2023 Ya m 0. 00 19 6 0. 00 03 0 0. 08 63 55 0. 13 12 79 0. 10 46 08 0. 16 99 63 03 :2 8 05 :1 6 04 :1 3 06 :5 0 4. 76 3. 23 3. 44 3. 10 Table 3: Specific comminution energy E2 *,predicted grinding timetg, desired (400μm) and actual final product(s) size(s) Se le ct ed m at er ia ls (s am pl es ) In iti al fe ed di am et er (d 1) (m ) D es ire d (c ho se n) Fi na l p ro du ct di am et er (d 2) (m ) E2 * Specific Energy Requirement (kWh⁄kg) tg Predicted Crushing Time (min:sec) Actual Final product Diameter (d2 *) (× 10(-4) m) K ic k' s R itt in ge r’s B on d' s O ru aA nt ia 's K ic k' s R itt in ge r’s B on d' s O ru aA nt ia 's K ic k' s R itt in ge r’s B on d' s O ru aA nt ia 's Se a fo od C ra yfi sh 0. 02 96 7 0. 00 04 0 0. 05 49 03 0. 07 51 13 0. 06 27 01 0. 09 14 86 02 :1 3 03 :0 1 02 :3 1 03 :4 1 4. 61 4. 18 4. 41 4. 14 C er ea l So rg hu m 0. 00 31 7 0. 00 04 0 0. 06 18 48 0. 08 04 05 0. 06 95 18 0. 09 51 02 02 :2 9 03 :1 4 02 :4 8 03 :1 3 5. 14 4. 27 4. 91 4. 03 M ill et 0. 00 19 5 0. 00 04 0 0. 06 68 57 0. 08 56 68 0. 07 48 50 0. 09 99 50 02 :4 1 03 :2 7 03 :0 0 04 :0 1 4. 82 4. 25 4. 59 4. 07 C or n 0. 02 01 7 0. 00 04 0 0. 05 54 70 0. 07 53 63 0. 06 32 40 0. 09 15 81 02 :1 4 03 :0 2 02 :3 2 03 :4 0 5. 53 4. 18 4. 53 4. 06 W he at 0. 00 18 6 0. 00 04 0 0. 05 69 08 0. 07 61 92 0. 06 45 44 0. 09 19 90 02 :1 7 03 :0 4 02 :3 6 03 :4 2 5. 81 4. 46 4. 70 4. 15 Le gu m es So yb ea n 0. 00 51 6 0. 00 04 0 0. 05 91 07 0. 07 78 74 0. 06 66 97 0. 09 30 86 02 :2 2 03 :0 8 02 :4 1 03 :4 4 5. 43 4. 25 5. 04 4. 06 Be an s 0. 00 63 0 0. 00 04 0 0. 05 50 52 0. 07 51 74 0. 06 28 87 0. 09 15 08 02 :1 3 03 :0 1 02 :3 2 03 :4 1 5. 53 4. 26 4. 44 4. 16 Tu be rs C as sa va 0. 00 14 0 0. 00 04 0 0. 07 54 08 0. 09 54 38 0. 08 41 49 0. 10 98 83 03 :0 2 03 :5 0 03 :2 3 04 :2 5 4. 92 4. 39 4. 41 4. 02 Ya m 0. 00 19 6 0. 00 04 0 0. 07 11 56 0. 09 05 06 0. 07 95 07 0. 10 47 79 02 :5 2 03 :3 8 03 :1 2 04 :1 3 5. 05 4. 46 4. 49 4. 09 selected material to specific energy (E2 *) obtain from each energy equation were compared. The comparative assessment revealed that the new concept energy equation termed as OruaAntia energy equation had the actual and predicted (chosen) product size(s) to be reasonably close for all the selected samples. However, this new concept energy equation capability was followed by Rittingers, bond’s and then Kick’s energy equations. This is evident on the average percentage deviation of 1.66, 6.88, 16.19and 38.40% respectively computed in this study and presented in Table 5. Table 5 suggest that the new concept/approach of accomplishing any desired final product up to finer particle sizes may likely to be more reliable using OruaAntia’s energy equation followed by Rittinger’s equation. The percentage deviations obtain also confirm that using Kick’s equation may only achieve coarse particle while coarse and intermediate particle sizes are limited to Bond’s equation.Therefore, obtaining the desired coarse, intermediate and fine particles may likely be achieved Pa ge 12 https://journals.e-palli.com/home/index.php/ajgt Am. J. Geo Spat. Technol. 2(1) 7-14, 2023 Table 4: Specific comminution energy E2 *,predicted grinding timetg, desired (500μm) and actual final product(s) size(s) Se le ct ed m at er ia ls (s am pl es ) In iti al fe ed di am et er (d 1) (m ) D es ire d (c ho se n) Fi na l p ro du ct di am et er (d 2) (m ) E2 * Specific Energy Requirement (kWh⁄kg) tg Predicted Crushing Time (min:sec) Actual Final product Diameter (d2 *) (× 10(-4) m) K ic k' s R itt in ge r’s B on d' s O ru aA nt ia 's K ic k' s R itt in ge r’s B on d' s O ru aA nt ia 's K ic k' s R itt in ge r’s B on d' s O ru aA nt ia 's Se a fo od C ra yfi sh 0. 02 96 7 0. 00 05 0 0. 05 20 58 0. 05 98 85 0. 05 52 12 0. 06 54 21 02 :0 5 02 :2 5 02 :1 3 02 :3 8 5. 88 5. 11 5. 48 5. 08 C er ea ls So rg hu m 0. 00 31 7 0. 00 05 0 0. 05 51 81 0. 06 20 02 0. 05 81 36 0. 06 67 80 02 :1 3 02 :2 9 02 :2 0 02 :4 1 5. 70 5. 13 5. 32 5. 01 M ill et 0. 00 19 5 0. 00 05 0 0. 05 74 33 0. 06 41 07 0. 06 04 00 0. 06 86 01 02 :1 9 02 :3 5 02 :2 6 02 :4 6 5. 59 5. 25 5. 37 5. 06 C or n 0. 02 01 7 0. 00 05 0 0. 05 23 13 0. 05 99 85 0. 05 54 70 0. 06 54 57 02 :0 6 02 :2 5 02 :1 4 02 :3 8 5. 89 5. 11 5. 60 5. 08 W he at 0. 00 18 6 0. 00 05 0 0. 05 29 60 0. 06 03 17 0. 05 60 23 0. 06 56 11 02 :0 8 02 :2 6 02 :1 5 02 :3 8 5. 90 5. 30 5. 64 5. 01 Le gu m es So yb ea n 0. 00 51 6 0. 00 05 0 0. 05 39 48 0. 06 09 90 0. 05 69 37 0. 06 60 23 02 :1 0 02 :2 7 02 :1 7 02 :4 0 5. 58 5. 21 5. 32 5. 10 Be an s 0. 00 63 0 0. 00 05 0 0. 05 21 25 0. 05 99 10 0. 05 53 19 0. 06 54 29 02 :0 6 02 :2 5 02 :1 3 02 :3 8 5. 89 5. 23 5. 56 5. 03 Tu be rs C as sa va 0. 00 14 0 0. 00 05 0 0. 06 12 78 0. 06 80 15 0. 06 43 49 0. 07 23 34 02 :2 8 02 :4 4 02 :3 5 02 :5 5 5. 62 5. 18 5. 51 5. 04 Ya m 0. 00 19 6 0. 00 05 0 0. 05 93 66 0. 06 60 42 0. 06 23 78 0. 07 04 16 02 :2 3 02 :4 0 02 :3 1 02 :5 0 5. 69 5. 12 5. 51 5. 06 Table 5: % Deviation of Actual Final average diameter d2 * with predicted (chosen) final average diameter d2 of selected materials subjected to comminution Diameter of selected materials (m) C ra yfi sh (1 0(- 4) ) So rg hu m (1 0(- 4) ) M ill et (1 0(- 4) ) C or n (1 0(- 4) ) W he at (1 0(- 4) ) So yb ea n (1 0(- 4) ) B ea ns (1 0(- 4) ) C as sa va (1 0(- 4) ) Ya m (1 0(- 4) ) ∑ d 2* -d 2 (1 0(- 4) ) E p K ic k’s e qu at io n d2 * 5.46 4.96 4.24 5.27 5.19 5.10 5.27 5.68 4.76 d2 3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00 d2 *-d2 2.46 1.96 1.24 2.27 2.19 2.10 2.27 2.68 1.76 18.93 70.11 d2 * 4.61 5.14 4.82 5.53 5.81 5.43 5.53 4.92 5.05 d2 4.00 4.00 4.00 4.00 4.00 4.00 4.00 4.00 4.00 Pa ge 13 https://journals.e-palli.com/home/index.php/ajgt Am. J. Geo Spat. Technol. 2(1) 7-14, 2023 d2 *-d2 0.61 1.14 0.82 1.53 1.81 1.43 1.53 0.92 1.05 10.84 30.11 d2 * 5.88 5.70 5.59 5.89 5.90 5.58 5.89 5.62 5.69 d2 5.00 5.00 5.00 5.00 5.00 5.00 5.00 5.00 5.00 d2 *-d2 0.88 0.70 0.59 0.89 0.90 0.58 0.89 0.62 0.69 6.74 14.99 Epa 38.40 Ri tti ng er ’s eq ua tio n d2 * 3.13 3.54 3.08 3.10 3.27 3.64 3.34 3.23 3.23 d2 3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00 d2 *-d2 0.13 0.54 0.08 0.10 0.27 0.64 0.34 0.23 0.23 2.56 9.49 d2 * 4.18 4.27 4.25 4.18 4.46 4.25 4.26 4.39 4.46 d2 4.00 4.00 4.00 4.00 4.00 4.00 4.00 4.00 4.00 d2 *-d2 0.18 0.27 0.25 0.18 0.46 0.25 0.26 0.39 0.46 2.70 7.50 d2 * 5.11 5.13 5.25 5.11 5.30 5.21 5.23 5.18 5.12 d2 5.00 5.00 5.00 5.00 5.00 5.00 5.00 5.00 5.00 d2 *-d2 0.11 0.13 0.25 0.11 0.30 0.21 0.23 0.18 0.12 1.64 3.64 Epa 6.88 Bo nd ’s eq ua tio n d2 * 3.45 4.17 3.64 3.53 3.75 4.17 3.66 3.58 3.44 d2 3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00 d2 *-d2 0.45 1.17 0.64 0.53 0.75 1.17 0.66 0.58 0.44 6.39 23.67 d2 * 4.41 4.91 4.59 4.53 4.70 5.04 4.44 4.41 4.49 d2 4.00 4.00 4.00 4.00 4.00 4.00 4.00 4.00 4.00 d2 *-d2 0.41 0.91 0.59 0.53 0.70 1.04 0.44 0.41 0.49 5.52 15.33 d2 * 5.48 5.32 5.37 5.60 5.64 5.32 5.56 5.51 5.51 d2 5.00 5.00 5.00 5.00 5.00 5.00 5.00 5.00 5.00 d2 *-d2 0.48 0.32 0.37 0.60 0.64 0.32 0.56 0.51 0.51 4.31 9.58 Epa 16.19 O ru aA nt ia ’s eq ua tio n d2 * 3.03 3.01 3.01 3.06 3.09 3.07 2.97 3.14 3.10 d2 3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00 d2 *-d2 0.03 0.01 0.01 0.06 0.09 0.07 -0.03 0.14 0.10 0.48 1.78 d2 * 4.14 4.03 4.07 4.06 4.15 4.06 4.16 4.02 4.09 d2 4.00 4.00 4.00 4.00 4.00 4.00 4.00 4.00 4.00 d2 *-d2 0.14 0.03 0.07 0.06 0.15 0.06 0.16 0.02 0.09 0.78 2.17 d2 * 5.08 5.01 5.06 5.08 5.01 5.10 5.03 5.04 5.06 d2 5.00 5.00 5.00 5.00 5.00 5.00 5.00 5.00 5.00 d2 *-d2 0.08 0.01 0.06 0.08 0.01 0.10 0.03 0.04 0.06 0.47 1.04 Epa 1.66 using Orua Antia’s and Rittinger’s equations but much finer particles may possibly be further achieved using the Orua Antia’s energy equation. CONCLUSION Predicting and achieving closely the desired final particle size following comminution of an initial particle size could be achieved more closely using the OruaAntia’s energy equation for mass-size reduction operation followed by Rittinger’s energy equation. The computed average percentage deviation 38.40%, 16.19%, 6.88% and 1.66% of the desired particle size(s) with the actual particle size(s) of material subjected to comminution re- affirm the fact that kick’s equation may be limited only to coarse particle size reduction; while Bond’s energy equation may be for both coarse and intermediate particle size prediction. Moreso, the likely prediction of coarse, intermediate and fine particles could be achieved using Rittinger’s and OruaAntia’s energy equations. However, further size reduction to finer particle sizes may be better achieved using the OruaAntia’s minimum energy equation which is a new concept/approach for mass-size reduction operations of a given material. 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