Equations Flashcards
Van der waals force (adhesion force)
F(ad)=(A*Rp)/6x^2
A=Hamakar constant (7.2*10^-20)
Rp=radius of particle
x= Gap between particles
** When distance is small force is high**
Gravity force
F(gr)=4/3piRp^3*rho
Forces due to liquid bridges
F(s)=2piy*Dp
y= surface tension
Dp=Particle diameter
Breakage equations (3)
Ribbon tensile strength
tensile strength= 3FG/2wt^2
F= force
G= gap between beams
W=ribbon width
t=thickness
pressing force
Pmax=2Rf/WDF
Rf= pressing force
W=width of roll
D= roll diameter
f=force factor
Pmax = peak pressure
Hardness
H=Fmax/Ap(Hc)
Fmax = max force
Ap(Hc) = projected contact area
Hc indentor contact depeth at Hmax
Hmax=Hc+Hf
Hmax = total distance
Hf = distance after loading
Hc = sinking depth
Granule size distribution
Y=1-exp(-x/xo)^n
Gravitational force
Fg=(d^3pi/6)rho*g
d= diameter
rho=particle density
buoyancy force
Fb=(d^3pi/6)rho*g
rho=particel density
drag force
carmen kozey
pressure drop=180((1-e)^2/e^3)((viscU)/(d^2rho))
e=voidage
visc = visccity
U=velocity
min fluidisation velocity (baeyan and geldart correlation)
look at notes but used when particle size <100Um
min fluidsation velocity (Wen and Yu)
look at notes but use when particle size>100Um
archemedies number
Ar=(pf(pp-pf)gd^3)/visc*2
pp=particle density
pf=fluid density
visc= viscocity
Bed density
Pb=(1-e)pp
e=voidage
pp= particle density
moisture content
X=xw/1+xw
Xw=x/1-x
x=dry basis
xw = wet basis
relative humidty
RH=p/ps
P
p=PY/0.622+Y
y=absolute humidity
P=total pressure
p = vapour pressure
ps = saturation pressure
Convective heat flux
Q”=H(ts-Tinf)
h= proportanity constant
ts= surface temp
tinf= fluid temp
Constant drying rate
X0-X=(rhoFgCpg(Tgi-Tgo))/(Mslambda)t
rho= gas density
Fg= gas vol flowrate
Cpg=specific heat of gas
t= drying time
Ms=mass of solids
lambda= latent heat
constant drying time
T=(M/AR)(xf-xi)
M= mass of dry material
A= area
R= drying rate
Falling rate time (linear)
t=(m/a)(Xcr/Rcr)ln(Xin/Xfin)
xcr = critical moisture content
m= mass of dry material
a=area
Rcr = crticial drying rate
Falling rate time (parabolic)
t=(m/a)(Xcr/Rcr)ln(Xin/Xfin)+(m/a)(1/a)ln((a+bXfin)Xcr/((a+bXcr)Xfin)
xcr = critical moisture content
m= mass of dry material
a=area
Rcr = crticial drying rate
Falling rate time (Linear+ parabolic)
t=(m/a)(Xcr/Rcr)ln(Xin/Xfin)+(m/a)(1/a)ln((a+bXfin)Xcr/((a+bXcr)Xfin)
xcr = critical moisture content
m= mass of dry material
a=area
Rcr = crticial drying rate
dimensionless fragmentation number
Fa= applied stress/ agglomerate strength
rate of extent of dispersion increases with fragmentation number
Fa<1 then no dispersion
Fa>1 then agglomerates begin to erode
tensile strength for collision dispersion
tensile strength is when particles collide with each other or boundary
tensile strength = 2/3* rhoPDp(Vi/delta T)
rhoP= particle density
Dp=particle dimater
vi= impact velocity
delta T= impact time
dispersion efficency
defined as particle size distribution of dispersed particles and primary particles
look at equation in notes
imbibition rate
imbibition rate = (7Dpe^2y^lvcos())/(75udagg^2(1-e))
e=voidage
y^lv = surface tension between liquid and vapour
* = contact angle
u- fluid viscocity
Dagg= agglomerate diameter
stagnant conditions
- mass
- particle radius
- dissolution time
m=Kl(4piRp^2)*(Cs-Cb)
Rp= particle radius
Cs= saturation conc
Cb= bulk conc
assumptions
- conc at the surface is the saturation conc
Cs»Cn dissolution under sink conditions
Rp^2=Rp0^2= (2DlCs)/rhoP)*t
RP0 is radius when dissolution is finished
Td= (- rhoPRp0)/(2Dl*Cs)
non zero velocity
- slip velcocity for laminar flow
stagnant conditions give no dependence to flow field and implies rate is independent of stirrer speed
slip velcocity can be approximated for laminar flow as slip increase mass transfer coefficent increases
** look at laminar slip equation in notes**
volume fraction of particles that dissolve at time t
t*=1-(ri^3)/(ri0^3)
total fraction of powder dissolved
t=Vit
Vi= inital volume fraction
t*= volume fraction of particles dissolved at time t
weight fraction in saturated phase?
Fsat=(Ctot-Ccr)/ (Csat-Ccr)
weight fraction in crystalline phase phase?
Fcr=(Ctot-Csat)/(Ccr-Csat)
gibbs free energy
crtiical radius and crticial gibbs
delta G= 4pir^2y-4/3pir^3(deltaU/Vm)
r= crystal cluser radius
y= interfacail free energy
delta U=chemical potential difference per solute molecule in the fluid and in the solid
Vm= molecular volume
r=(2yVm)/(delta U)
delta G=(16piy^3Vm^2)/(3deltaU^2)
driving force from solution (crystalisation)
delta U= KT*ln(C/Cs)
driving force from melt (crystallisation)
delta U= delta Hm*deltaT/Tm
Tm= crystaliisation temp
rate of formation of 3D nuclei
J=Aexp(-delta G*/KbT)
limear growth rate (growth rate of a face in the direction normal to the face
v=dl/dt
growth is a kinetic phemomen driven by superaturation that is determined by thermodynamic data
mass deposition rate (crystal growth kinetics)
Kgm=3alpha/betarho*Kgl
kgl= linear growth rate
alpha and beta = volume and area shape factors
for spheres and cubes
Kgm=rho*v
v= mean lineat growth velocity
kgm and kgl are both temperature dependent and fit arrenhius
