LEARNING UNIT $1$ Particle TechnologyPresentation III EXAMPLE $142205000\mathrm{.}0205990\mathrm{.}040\mathrm{.}086990\mathrm{.}070\mathrm{.}218530\mathrm{.}120\mathrm{.}4410030\mathrm{.}190\mathrm{.}6812040\mathrm{.}320\mathrm{.}8814050\mathrm{.}5116760\mathrm{.}820571$ Size d $($m$)$ EXAMPLE$100$grams of a random mixture consists of two components A and B inproportions $60\%$ and $40\%$ by mass, respectively. The particles are spherical$(=1)$andAandBhaveparticledensities$500$kg$/$m$3$and$700$kg$/$m$3,$ vrespectively. The cumulative undersize mass distributions of the two components are shown in the table below:XA $[$FA $($x$)]$XB $[$FB $($x$)]$For component A$1.$ Determinethenumberofparticlesineachfractionandthetotalnumberof particles of A in$100$grams of the mixture$\mathrm{.}2.$ Calculate the following statistical mean diameters $($a$)$ Arithmetic; $($b$)$ Length; $($c$)$ Surface; $($d$)$ Volume; $($e$)$ Square root; $($f$)$ Cube root and $($g$)$ Harmonic$3.$ DrawaDifferentialCumulativeDistributionCurvedeterminethemedian SOLUTIONFirst, we estimate the number of particles per unit mass of A and B$,$ by converting the size distributions into differential frequency number distributions and using:mass of particles number of particles mass of one $=\backslash$times in each size range in size range particle $3$ davgi mi $=$Ni $\backslash$times v P xi $3$ davgi mi $=$ Ni $\backslash$times P $-$ forasphericalparticle $6$where is the particle density and dis the arithmetic mean of Pavg adjacent sieve sizes. $3512849$ kg $\backslash$times $\backslash$times $(0\mathrm{.}6\backslash$times $100$g$)$NAkg $1000$g $(100$g mix$)=210771$ N$=$ iN$=16($x $)$ i$3$ d P avg$6(0\mathrm{.}2)63(500)(1866\mathrm{.}5\backslash$times $10)$N $=117484$particles$/$kg $1$Ni$1174843134493097223695373345694087114191974604247797563512849210771$xi$0\mathrm{.}20\mathrm{.}30\mathrm{.}180\mathrm{.}130\mathrm{.}070\mathrm{.}050\mathrm{.}030\mathrm{.}020\mathrm{.}021$Mean size of range davg $($m$)1866\mathrm{.}51540\mathrm{.}51304\mathrm{.}51103\mathrm{.}5928776649549\mathrm{.}5461$Total $($particles per kg$)$Particles A in $100$g mixture x$1=10\mathrm{.}8=0\mathrm{.}2$ x$2=0\mathrm{.}80\mathrm{.}5=0\mathrm{.}3$ davg $=0\mathrm{.}5$x$(2057+1676)=1866\mathrm{.}5$SOLUTION cont...A particles: $.$t n o cm m m m m m$3$ m $4\mathrm{.}15\mathrm{.}92\mathrm{.}42\mathrm{.}73\mathrm{.}6\mathrm{.}87\mathrm{.}6683012534920117=1=1=1==$R $==$ DA DL Ds Dv DS RC DH D$121325285909109010101010210951099301481111111414481\backslash$times $2\backslash$times $\backslash$times $\backslash$times $\backslash$times $\backslash$times $\backslash$times $\backslash$times $8\backslash$times $82\backslash$times $621564620264480262102115100\mathrm{.}31\mathrm{.}0\mathrm{.}8\mathrm{.}1\mathrm{.}1\mathrm{.}8\mathrm{.}41\mathrm{.}5328\mathrm{.}5339\mathrm{.}3333353334$ N$492791746$Ni$84427356517912457$O$739948909$I : $113036330414647$T sU elcit $)5555$L r m$(\mathrm{.}6\mathrm{.}0\mathrm{.}4\mathrm{.}38695\mathrm{.}91$O ap $684503012746-$g $1111976454$vaS Ad N$/$d$6\mathrm{.}294$E$+012\mathrm{.}035$E$+022\mathrm{.}374$E$+023\mathrm{.}349$E$+023\mathrm{.}605$E$+025\mathrm{.}267$E$+026\mathrm{.}459$E$+028\mathrm{.}379$E$+021\mathrm{.}691$E$+034\mathrm{.}901$E$+03$ Nxd$41\mathrm{.}426$E$+181\mathrm{.}765$E$+188\mathrm{.}969$E$+175\mathrm{.}480$E$+172\mathrm{.}481$E$+171\mathrm{.}482$E$+177\mathrm{.}437$E$+164\mathrm{.}198$E$+163\mathrm{.}522$E$+165\mathrm{.}184$E$+18$ Nxd$37\mathrm{.}639$E$+141\mathrm{.}146$E$+156\mathrm{.}876$E$+144\mathrm{.}966$E$+142\mathrm{.}674$E$+141\mathrm{.}910$E$+141\mathrm{.}146$E$+147\mathrm{.}639$E$+137\mathrm{.}639$E$+133\mathrm{.}820$E$+15$ Nxd$24\mathrm{.}093$E$+117\mathrm{.}439$E$+115\mathrm{.}271$E$+114\mathrm{.}500$E$+112\mathrm{.}881$E$+112\mathrm{.}461$E$+111\mathrm{.}766$E$+111\mathrm{.}390$E$+111\mathrm{.}657$E$+113\mathrm{.}146$E$+12$ Nxd$2\mathrm{.}193$E$+084\mathrm{.}829$E$+084\mathrm{.}040$E$+084\mathrm{.}078$E$+083\mathrm{.}105$E$+083\mathrm{.}172$E$+082\mathrm{.}721$E$+082\mathrm{.}530$E$+083\mathrm{.}595$E$+083\mathrm{.}026$E$+09$ Ni$1174843134493097223695373345694087114191974604247797563512849$davg $-($m$)1866\mathrm{.}51540\mathrm{.}51304\mathrm{.}51103\mathrm{.}5928776649549\mathrm{.}5461$ SUM SOLUTION cont... F X$10\mathrm{.}80\mathrm{.}50\mathrm{.}320\mathrm{.}190\mathrm{.}120\mathrm{.}070\mathrm{.}040\mathrm{.}020$davg $-($m$)20571696140512041003853699599500422$ SOLUTION cont...Median $1405$m$10\mathrm{.}90\mathrm{.}80\mathrm{.}70\mathrm{.}60\mathrm{.}50\mathrm{.}40\mathrm{.}30\mathrm{.}20\mathrm{.}1002004006008001000120014001600180020002200$Differential Cumulative Distribution Curved in m