product characterization and processing of pharmaceutical particulate solidsppps/fig... ·...
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Fig_PC&P_3 Lecture Product Characterization and Processing of Pharmaceutical Particulate Solids, mechanics of cohesive particulate solids, Jürgen Tomas 13/05/2013
Figure 3.0 Prof. Dr.-Ing. habil. J. Tomas – chair for Mechanical Process Engineering 3. Introduction into mechanics of cohesive particulate solids 3.1 Microscopic particle bonds and resulting adhesion forces 3.2 Contact mechanics of adhesive particles 3.3 Particle interaction test methods 3.4 Micro-macro transition of contact forces and bulk stresses in a particle packing 3.5 Macroscopic biaxial stress state, powder flow criteria 3.6 Product characterization test equipment and shear testing techniques 3.7 Flow and consolidation functions of cohesive particulate solids 3.8 Compression functions and shear work 3.9 Consolidation functions for hopper design 3.10 Permeation and fluidisation behaviour 3.11 Data sheet of product properties …
Fig_PC&P_3 Lecture Product Characterization and Processing of Pharmaceutical Particulate Solids, mechanics of cohesive particulate solids, Jürgen Tomas 13/05/2013
Figure 3.1 Prof. Dr.-Ing. habil. J. Tomas – chair for Mechanical Process Engineering
Survey of constitutive functions, processing and handling problems of cohesive powders Property, problems
Physical prin-ciple
Physical assessment of product quality Particle size
d in µm Physical law Assessment charac-
teristic Value range
Evaluation
Large adhe-sion poten-tial1) FG
FH0
2s
20
sls,H
G
0H
dag2C
FF
⋅ρ⋅⋅π= 2
2
d)µm100(
WeightAdhesion
≈ 1 - 100
100 - 104 104 - 108
slightly adhesive adhesive
very adhesive
10 - 100 1 - 10
0.01 - 1 Large intensi-fication of adhesion2)
FN
FH(FN)FN
FH(FN)
pA
p
κ−κκ
=κ Contact consolidation
coefficient κ by flattening
0.1 – 0.3 0.3 – 0.77
> 0.77
soft very soft
extreme soft
< 10 < 1
< 0.1
Poor flowab-lity2)
σ1 σc
c
1cff
σσ
= Flow function ffc 2 - 4
1 - 2 < 1
cohesive very cohesive non-flowing
< 100 < 10 < 0.1
Large com-pressibility2)
σ1
∆h
n
0
st,M
0,b
b 1
σ
σ+=
ρρ
Compressibility in-
dex n 0.05 – 0.1
0.1 - 1 compressible
very compressible < 100 < 10
Small perme-ability3,4)
∆hW
∆hb
u b
Wf h
hku∆∆
⋅= Permeability
kf in m/s < 10-9
10-9 - 10-7 10-7 - 10-5
non-permeable very low
low
< 1 1 - 10
10 - 100
Poor fluidi-sation5,6)
( ))d(ufp P=∆ Pressure drop ∆p (Channelling)
Group C, non-fluidising
< 10
1) Rumpf, H.: Die Wissenschaft des Agglomerierens. Chem.-Ing.-Technik, 46 (1974) 1-11. 2) Tomas, J.: Product Design of Cohesive Powders - Mechanical Properties, Compression and Flow Behavior. Chem. Engng. & Techn., 27 (2004) 605-618. 3)Förster, W.: Bodenmechanik - Mechanische Eigenschaften der Lockergesteine, 4. Lehrbrief, Bergakademie Freiberg 1986. 4) Terzaghi, K., Peck, R. B., Mesri, G.: Soil mechanics in engineering practice, Wiley, New York 1996.
5) Geldart, D.: Types of Gas Fluidization, Powder Techn. 7 (1973) 285-292. 6) Molerus, O.: Fluid-Feststoff-Strömungen, Springer, Heidelberg 1982.
Fig_PC&P_3 Lecture Product Characterization and Processing of Pharmaceutical Particulate Solids, mechanics of cohesive particulate solids, Jürgen Tomas 13/05/2013
Figure 3.2 Prof. Dr.-Ing. habil. J. Tomas – chair for Mechanical Process Engineering
Fig_PC&P_3 Lecture Product Characterization and Processing of Pharmaceutical Particulate Solids, mechanics of cohesive particulate solids, Jürgen Tomas 13/05/2013
Figure 3.3 Prof. Dr.-Ing. habil. J. Tomas – chair for Mechanical Process Engineering
3.1 Microscopic particle bonds and resulting adhesion forces
Interactions between Atoms and Molecules
1. Interaction pair potential for various bonding types
ao,k a o,i ao,m a o,w ao,v
+ repulsion
atomic centre separation a
UB,V
0
UB,W
UB,m
UB,k
UB,i
VAN DER WAALS
hydrogen bridge bond
metallic
covalent
ionic
pair
pot
entia
l
U
Na+
Na+
Cl- Na+
Cl- Cl-
CC CC
C
Na+
Na+
Na+
Na+ Na+
e-
e-
e-
e-e-
HO H O
H
H..
+ - + -- + - +
+ - + -
typical crystal lattice
examplespacking
separationa0 in nm
bond energyUB in kBT*
diamondNaClNa - NaH ... OCH4
∗ θ = 25°C
aU=0 0n = 9n = 8n = 7n = 6n = 5n = 4n = 3
n = 2
n = 1
n = 0
h · ν
∆a expansion
T = 0
a0 equilibrium distance (F=0)atomic centre separation a, x
quan
tum
num
bers
UB bonding energy
h PLANCK constantν absorption frequency
Etot = En = (n + ) · h · ν (1)12
T ↑
w = exp (- ) (2) EnkBT
ψ ψ 2
x
n=2
n=1
n=0U(x) U(x)
δ2 ψ 1 δ2 ψδ x² vp
2 δ t2 - · = 0
vp2 =
h2 ν2
2m [Etot - U(x)](5) wave propagation velocity
(4) SCHRÖDINGER equation
ψ = ψmax.exp[ j(k.x - ω.t)] (3) complex wave function
2. Interaction pair potential U and thermally excited atom oscillations
3. Wave function (probability amplitude) ψ and residence frequency ψ ²
BOLTZMANNdistribution factor
energy of harmonic quantummechanical oscillator(parabolic potential curve)
0.1540.2760.3720.1760.400
290310 45 9 4
F 3.3
Fig_PC&P_3 Lecture Product Characterization and Processing of Pharmaceutical Particulate Solids, mechanics of cohesive particulate solids, Jürgen Tomas 13/05/2013
Figure 3.4 Prof. Dr.-Ing. habil. J. Tomas – chair for Mechanical Process Engineering
3. Influence of repulsive potential Ure on total interaction potential U a) ideal stiff sphere ( dA) b) compliant sphere c) compliant sphere (particle) and electric double-layer
4. Elasticity of an atomic pair at direct contact
0 0 0
0 0 0
Ure
dA dA dA
dA
UB
Ubar
a a
aacentre separation a a0
Ure ~ ( )dA a
m
m = ∞
Ure ~ ( )dA a
m
m = 9 - 12
,Ure,sphUre,el ~ exp( ) a
a37
m
∅
expansion ε = ∆a a0
MIE pair potential in a homogeneous lattice
U = - + (5)
β = Σ (7)≠i 1
α = Σ ±k1,i
pn1,i
(6)
p1,i = a1,i /a separation ratios of lattice neighbours i of contact 1 k1,i coordination numbers of packing
EM = (m - n) · n · = n · m ·α · A a0
n+3 UB
a03
(9)
atomic centre separation aaFmax
U F
tota
l pot
entia
l U
tota
l for
ce
F
F = - dUda
+ repulsion
- attraction
a0dA
U
Fmax
norm
al st
ress
σ
- pressure
- compression
εmax
+
+ tension
σmax = Fmax/A
dσdε = E
α · A β · B an am
k1,ipm
1,i≠i 1
Ure
UU
UB
UB
UB
(8) EM
ρscS,M
2= = n · m · UB,m
M
Interactions between Atoms and Molecules
centre separation a
tota
l pot
entia
l U
repu
lsio
n p
oten
tial
Ure
+ repulsion
- attraction
dσ = E · (1)
σ = = (2)
1 - εA packing density dA atomic hard sphere diameter a0 atomic packing diameter
atomic modulus of elasticity:
EM = - (3)
EM = - (4)
da a0
F 1 - εA FA εA d²A
1 - εA a0 dF εA dA
2 da a0
1 - εA a0 d²U εA dA
2 da² a0
F 3.4
Fig_PC&P_3 Lecture Product Characterization and Processing of Pharmaceutical Particulate Solids, mechanics of cohesive particulate solids, Jürgen Tomas 13/05/2013
Figure 3.5 Prof. Dr.-Ing. habil. J. Tomas – chair for Mechanical Process Engineering
Interactions of Polar Molecule Pair
Interaction pair potential due to MIE (1903) and e.g. the LENNARD-JONES potential:
UAa
Ban m= − + integer exponents n < m (1) U U
aa
aaB
U U= ⋅ ⋅
−
= =4 0
120
6
(2)
Pot. equilibrium separ.: aBAU
m n
=
−=
0
1
(3) equilibrium separation: am Bn AF
m n
=
−=
⋅⋅
0
1
(4)
Bond energy: Um n
mA
aBFn= −
−⋅
=0 (5) potential ratio:
UU
m nm
B
an aF=
=−
<0
1 (6)
Maximum attraction force: d Uda
dFda
2
2 0= − = : Fm nm
n Aa F
nmaxmax
= −−+
⋅⋅
+1 1 (7)
Separation ratios: 111
0
0
1
0
1
< =
< =
⋅ +⋅ +
=
=
−
=
−aa
mn
aa
m mn n
F
U
m n F
U
m nmax ( )
( ) (8)
Strain: ε ε εUU
FF F
F
F
aa
aa=
=
==
=
= − < = < = −00
00
01 0 1
max
max
(9)
YOUNG modulus: ( )
Ea
d Uda
m n nA
an m
UaF a F
nB
FF
= − ⋅ = − ⋅ ⋅ = ⋅ ⋅−
= =+
==
1
0
2
203
03
0
( ) (10)
Pull-off strength: σ Z
nm n
E mnm
,max =+
⋅++
+−1
111
1
(11)
-20
-15
-10
-5
0
5
10
15
20
0,00 0,10 0,20 0,30 0,40 0,50
atomic centre separation a in nm
inte
ract
ion
pair
pot
entia
l U in
10
-21 J
-20
-15
-10
-5
0
5
10
15
20
pote
ntia
l for
ce F
in 1
0-11 N
repulsion potential Uab
repulsion force Fab
attraction force Fanattraction potential Uan
aF=0aU=0
+ repulsion
- attraction aFmax
bond energy UB
total force F
total potential U
Fig_PC&P_3 Lecture Product Characterization and Processing of Pharmaceutical Particulate Solids, mechanics of cohesive particulate solids, Jürgen Tomas 13/05/2013
Figure 3.6 Prof. Dr.-Ing. habil. J. Tomas – chair for Mechanical Process Engineering
1. Shear deformation of quasi-cubical molecular packing at uniform probability of point defects flow, dV = 0
a) initial state b) uniform or simple shear y0 /a3 position shift events3
c) sharp shear crack y0 /a2 position shift events2
2. Creation of opportunities for discrete position shift events
a) no opportunity for a position shift event b) position shift events by oscillating potential curves at thermal wave transfer
centre separation a
pote
ntia
l U
centre separation a
pote
ntia
l U
∆U1
3. Opportunities for a temperature dependent distribution of various potential wells
boundary curves
a
∆WPW
= τ
· a3 (b)
(a)∆Ua
U
y
x
y0
x0
D
F'E'
AB'
y0E'
AB
C CD
E F
D
A' B'
F'
x0
x0 dx.a y0
Ey0
dx
a
γ
ux, τ
ux, τ
a
∆U1
U∆U2
Rate Dependent Flow of Monodisperse Spherical Molecules
S2
0F
0FB
aB
caa2Tk
UexpTk3
⋅
⋅π⋅λ
⋅∆
⋅⋅π⋅
=η=
=
apparent shear viscosity:
S2
0F
0FB
2B
caTa2Tk
UexpTk3
⋅⋅∆⋅α
⋅π⋅λ
⋅∆
⋅⋅π⋅=η
=
=
apparent shear viscosityfor viscoplastic flow:
F 3.6
Fig_PC&P_3 Lecture Product Characterization and Processing of Pharmaceutical Particulate Solids, mechanics of cohesive particulate solids, Jürgen Tomas 13/05/2013
Figure 3.7 Prof. Dr.-Ing. habil. J. Tomas – chair for Mechanical Process Engineering
Type Structural Model Interaction Energy U(a) in J
chemical orcovalentbond
ion bondcharge - charge
hydrogenbridge bond
charge - dipole
dipole - dipole
charge - non-polar
dipole - non-polar
non-polar - non-polar
short range, a = 0.1 - 0.2 nm1) U 60 - 350 kB · T per bond Um 150 - 800 kJ/mol
≈≈
a = 74 pm
H - H H2
directed
a = 97pm 0.1 nm≈
H
HO H2O
Q1 Q2
a
Q1 = z1 · eQ2 = z2 · enon-directed
long range a > 0.1 nm Q1 · Q2
4π ε0 · a= µi COULOMB
O O
H H
H H
H H
H H
O
Oa 0.176 nm
0.1 nm
short range a 0.3 nm≈
Ca2-
fixed dipole
freely rotating
Q · p · cos θ 4π ε0 · a²
Q² p²
6 (4π ε0)² kBT · a4
-
-
fixed dipole
rotating dipole
permanentfixed dipoles
freely rotating
p1 φ p2
aθ2θ1
p1 p2
a
Q p
aθ
Q p
a
p2 p2 [ 2cosθ1cosθ2 - sin θ1 sin θ2 cos φ] 4π ε0 · a³
p2 p2
3 (4π ε0)² kBT · a6
1 2-
1 2- KEESOM
Q² · α2 (4π ε0)² · a4
-induced
α1 α2
a
induced
p² α (1 + 3 cos² θ)2 (4π ε0)² a6
-
p² α(4π ε0)² · a6
- DEBYE
3 α1α2 hν4 (4π ε0)² · a6
- LONDON dispersion
1)1 kB·T = 4.05 · 10-21 J for T = 293 KkB = 1.38 · 10-23 JK-1 BOLTZMANN constanth = 6.628 · 10-34 Js-1 PLANCK constantε0 = 8.854 · 10-12 AsV-1m-1 permittivity of vacuume = 1.602 · 10-19 As electronic charge
p = Q · a electric dipole moment (As · m)Q electric charge (As)α electric polarizability (A²s²m²J-1)ν electronic absorption (ionization) frequencyz ion valencyµi ion potential (chemical potential)
F = - bonding forcedUda
Q αa
p α
aθ
induced
p α
a
directed
Interaction Free Energies between Atoms, Ions andMolecules in Vacuum
ISRAELACH VILI, J.: Intermolekular & Surface Forces, Academic Press London 1992, p.28
F 3.7
Fig_PC&P_3 Lecture Product Characterization and Processing of Pharmaceutical Particulate Solids, mechanics of cohesive particulate solids, Jürgen Tomas 13/05/2013
Figure 3.8 Prof. Dr.-Ing. habil. J. Tomas – chair for Mechanical Process Engineering
Molecule - plate Sphere - plate Conductor Non-conductor
Interaction Forces and Potentials between Smooth and Stiff Model Bodies
Partner Dipol moment and dispersion Electrostatic (COULOMB) (VAN-DER-WAALS)
1 a 2Sphere - sphere Point charge sphere-sphere q = 2ε0εrUel/d
d 1
d2
a
d = 2d1d2
d1 + d2
EVdW =
FVdW = -
- CH · d 24 · a
CH · d 24 a²
a1 ρn,2
EVdW = -
FVdW = -
CH · d1
12 a
CH · d1
12 a²
d 1
aAS
Q1 = π d1 ε0 εr · EQ2 = AS ε0 εr · E
UVdW = -
FVdW = -
π ρn,2 A
6 a³π ρn,2 A
2 a4
EVdW = -
FVdW = -
CH · l · d 24 2 a
√¬
√¬ 3/2
Plate - plate Conductor Non-conductorQ = AS · ε0 εr · E
EVdW =
FVdW =
- CH · AS
12 π a²
- CH · AS
6 π a3
2 Crossed cylinders
a
d1
d²
EVdW =
FVdW =
- CH· d1d2
12 a√¬
- CH · d1d2
12 a²√¬
Molecule-molecule
√¬ CH · l · d
16 2 a√¬ 5/2
Eel= ε0 εr Uel d1 lnπ2
ad1
2
l
a
d1 d2
l
a
d Mll
Eel = · q1 q2 · a l · d2ε0 εr
Eel = · q1 q2 · a AS
2 ε0 εr
a
AS
FC = · q1 q2 l · d2ε0 εr
Eel = ε0 εr · Uel1a
AS2
2
FC = ε0 εr · Uel1a2
AS2
2FC = · q1 q2
AS
2 ε0 εr
Eel = · q1 q2 · a π d1
2 ε0 εr
2
FC = · q1 q2 π d1
2 ε0 εr
2
FC = ε0 εr Uel · π2
d1a
2
2
2 Parallel chain Cylinder - cylinder Conductor Non-conductor molecules Q = π d l · ε0 εr · E
UVdW = -
FVdW = -
Α a6
6 A a7
UVdW = -
FVdW = -
3 π A l 8 dM a5
15 π A l 8 dM a6
2
2
HAMAKERconstant = f(A):
CH = π2 ρn,1 ρn,2 A
d = 2d1d2
d1 + d2
q = Q/AS =
ρn = ρ·NA/M number densitye = 1.6·10-19 A·s electronic charge ε0= 8.854·10-12 A·s/(V·m) permittivity of vacuum
nQ·eAS
(1+2· )·E surface charge densityεr,s - 1εr,s + 2
ε0≈ E electric field strengthUel electrostatic potentialF = - dU/da potential (counter) forcez ion valencyεr permittivity of interstitial medium
ISRAELACH VILI, J.: Intermolekular & Surface Forces, Academic Press London 1992, p.177SCHUBERT, H.: Handbuch der Mechanischen Verfahrenstechnik, Whiley-VCH Weinheim 2003, S.217
z1 z2 e²4π ε0 a
z1 z2 e²4π ε0 a²
π q1 q2 · d1 d22 2
2 ε0 εr (d1 + d2 + 2a)Eel =
π q1 q2 · d1 d22 2
2 ε0 εr (d1 + d2 + 2a)2FC =
UC =
FC =
FC = ε0 εr · Uell·da2
12
2
Eel = ε0 εr · Uell·da
12
2
F 3.8
Fig_PC&P_3 Lecture Product Characterization and Processing of Pharmaceutical Particulate Solids, mechanics of cohesive particulate solids, Jürgen Tomas 13/05/2013
Figure 3.9 Prof. Dr.-Ing. habil. J. Tomas – chair for Mechanical Process Engineering
Fig_PC&P_3 Lecture Product Characterization and Processing of Pharmaceutical Particulate Solids, mechanics of cohesive particulate solids, Jürgen Tomas 13/05/2013
Figure 3.10 Prof. Dr.-Ing. habil. J. Tomas – chair for Mechanical Process Engineering
Adhesion Forces between Stiff Solid Particles
a) Smooth sphere - smooth plate b) Rough sphere - smooth plate
102
10-2
1
104
10-4
10-1 1 10 102 103
particle separation a in nm
adhe
sion
forc
e F
H0
in n
N α = 20°
liquid bridgerel. humidity 50%
non-conductor
van der Waals conductor
a0 = 0.4 nm
10-7
10-9
10-5
10-10
adhe
sion
forc
e F
H0
in N
10-2 10-1 1 10 102 103 104
10-8
10-6
10-4
10-3
particle size d in µm
h r = 0
nm1 n
m5 n
m10
nm
1 µm
100 nm
FH0,VdW = . 1 + CH hr d / hr
6 a02 2.(1 + hr / a0)2
a0 = 0.4 nm molecular force equilibrium separation σlg = 72 10-3 N/m surface tensionα = 20° bridge angleθ = 0° wetting angleCH = 19 10-20 J Hamaker constant acc. to Lifschitz
.
.
qmax = 160 10-19 As/µm2 surface charge densityU = 0.5 V contact potential
CH,sls = ( CH,ss - CH,ll )2 Hamaker constant particle-water-particle
adhe
sion
forc
e F
H0
in N
particle size d in µm
10-5
10-6
10-7
10-8
10-9
non-
cond
ucto
r
liquid
bridg
e
van d
er W
aals,
dry
cond
uctor
van d
er W
aals,
wet
wei
ght o
f sph
ere
10-1 1 10 102 103
αd2
a0
hr
a0
d2
1 10 102 103
roughness height 2.hr in nm
10-5
10-6
10-7
10-8
10-9
10 µm
1 µm
adhe
sion
forc
e F
H0
in N
liquid bridge, d = 10 µm 10 µm, α = 2,5 °
d = 100 µm van der
Waals
conductor, d = 10 µ
m
non-conductor, d = 10 µm
acc. to H. Schubert (1979):
Models according to Rumpf et al. (1974):
F 3.10
Fig_PC&P_3 Lecture Product Characterization and Processing of Pharmaceutical Particulate Solids, mechanics of cohesive particulate solids, Jürgen Tomas 13/05/2013
Figure 3.11 Prof. Dr.-Ing. habil. J. Tomas – chair for Mechanical Process Engineering
Moisture Bonding in a Particle Packing
2. Liquid bridge at direct contact (a = aF=0) of two equal-sized spheres
a) Pendular state (liquid bridges)
b) Funicular state (bridges + filled pores)
c) Capillary state (filled pores)
for a real packing:
for cubic packing of monodisperse spheres:
Fs
FH
α
d/2
R1
R'2
h
R2 Fs
d/2
σlg
σlg
1. Bond types
F 3.11
Fig_PC&P_3 Lecture Product Characterization and Processing of Pharmaceutical Particulate Solids, mechanics of cohesive particulate solids, Jürgen Tomas 13/05/2013
Figure 3.12 Prof. Dr.-Ing. habil. J. Tomas – chair for Mechanical Process Engineering
wat
er c
onte
nt X
W
XWK
Desorption
Adsorption
capillarycondensation
multimolecular layersmonolayer
relative partial pressure ϕϕK 1
Sorption isotherme of a capillary-porous powder
dewatering
moisten
satu
ratio
n
capi
llary
con
dens
atio
n
adso
rptio
n
pKe
XWC water content XWXWS
capi
llary
pre
ssur
e p
K
Capillary pressure hysteresis of a particle packing
Moisture Bonding in a Particle PackingF 3.12
Fig_PC&P_3 Lecture Product Characterization and Processing of Pharmaceutical Particulate Solids, mechanics of cohesive particulate solids, Jürgen Tomas 13/05/2013
Figure 3.13 Prof. Dr.-Ing. habil. J. Tomas – chair for Mechanical Process Engineering
Fig_PC&P_3 Lecture Product Characterization and Processing of Pharmaceutical Particulate Solids, mechanics of cohesive particulate solids, Jürgen Tomas 13/05/2013
Figure 3.14 Prof. Dr.-Ing. habil. J. Tomas – chair for Mechanical Process Engineering
Fig_PC&P_3 Lecture Product Characterization and Processing of Pharmaceutical Particulate Solids, mechanics of cohesive particulate solids, Jürgen Tomas 13/05/2013
Figure 3.15 Prof. Dr.-Ing. habil. J. Tomas – chair for Mechanical Process Engineering
Fig_PC&P_3 Lecture Product Characterization and Processing of Pharmaceutical Particulate Solids, mechanics of cohesive particulate solids, Jürgen Tomas 13/05/2013
Figure 3.16 Prof. Dr.-Ing. habil. J. Tomas – chair for Mechanical Process Engineering
Liquid Bridge Bonds
Range of adsorption layers:
(9)
−⋅
⋅π⋅σ⋅
εε−
=σ⋅ 0m,W
W0A,Z
0
00 a2
aXX
d
a1
or:
( )( )
75,0
Wl
S
i75,0
ilgc X
sin1
sin18,88
d
⋅
ρρ
⋅ϕ⋅⋅ε
ϕ⋅σ⋅ε−⋅=σ
−
(10)
Range of liquid bridges:
(11)( ) ( )
( ) Wl
S
i
ilgc X
sin1d
sin2125,8⋅
ρρ
⋅ϕ−⋅⋅ε⋅ε
ϕ⋅σ⋅ε−⋅ε−⋅=σ
2
1F 3.17
Fig_PC&P_3 Lecture Product Characterization and Processing of Pharmaceutical Particulate Solids, mechanics of cohesive particulate solids, Jürgen Tomas 13/05/2013
Figure 3.17 Prof. Dr.-Ing. habil. J. Tomas – chair for Mechanical Process Engineering
( ) ( ) ( )[ ]63WEWOSs,Dt,c ttexp1XXY1 −−⋅−⋅⋅ε−⋅σ=σ
( )1tk
tkM
XM1W
W
WW
WASDst,c +⋅
⋅⋅ϑ⋅
⋅⋅ε−⋅σ=σ
t,0HNtt,H FFF +⋅κ=
s
Zst 5
t2η⋅
⋅σ⋅=κ
stit
tt,i tan/tan2
tantanϕϕ⋅κ+
ϕ=ϕ
s
sgZst,0H 5
td4F
η⋅⋅⋅σ⋅σ⋅π⋅
=
Solid Bridge Bonds
2
(13)
C hem ical reaction:
Electrostatic attraction forces
Bonds by interlocking
Magnetic attraction forces
Crystallization:
(12)
Sinter bridges: (14)
(17)
(15)
(16)
F 3.18
Fig_PC&P_3 Lecture Product Characterization and Processing of Pharmaceutical Particulate Solids, mechanics of cohesive particulate solids, Jürgen Tomas 13/05/2013
Figure 3.18 Prof. Dr.-Ing. habil. J. Tomas – chair for Mechanical Process Engineering
σct(t) Atot
σDsfAsf(t)
(1)
(2)
(3)
dt)t(dA
dt)t(dA sf
Dsfct
tot ⋅σ=σ⋅
dtV)t(dV
dt)t(d
tot
sfDsf
ct ⋅σ=σ
dtdtm
)t(dm)1()t(Lt
0 s
sf
sf
sDsfct ∫⋅
ρρ
⋅ε−⋅σ=σ
Stress Transmission at Time Consolidation (Caking)F 3.19
Fig_PC&P_3 Lecture Product Characterization and Processing of Pharmaceutical Particulate Solids, mechanics of cohesive particulate solids, Jürgen Tomas 13/05/2013
Figure 3.19 Prof. Dr.-Ing. habil. J. Tomas – chair for Mechanical Process Engineering
Micro Process Kinetics of Solid Bridge Formation by Crystal-lization
Crystallization of superaturated solution at particle contacts (YS mass re-
lated solubility):
dtdX
Ymm
mdt(t)dm
mdt(t)dm W
SW
W
s
sf
s
sf ⋅=⋅⋅
=⋅
Kinetics of water mass transfer (evaporation) from supersaturated solution
to environment:
( )WEWspWW XXAK
dtdX
−⋅⋅−=
Integration with initial condition for t = 0: XW = XW0
if ( )W63
spW Xft1AK ≠=⋅
Uniaxial compressive strength:
( ) ( )
−−⋅−⋅⋅−⋅=
63WEW0SDsct t
texp1XXYε1σσ
Flow function:
( )
−−⋅−⋅⋅
⋅=
63WEW0SDs
1ct
ttexp1XXYσ
σ2ff
Minimum hopper outlet width to avoid bridging:
( ) ( )
−−⋅
⋅−⋅⋅⋅+
=63s
WEW0SDstmin, t
texp1gρ
XXYσ1mb
Fig_PC&P_3 Lecture Product Characterization and Processing of Pharmaceutical Particulate Solids, mechanics of cohesive particulate solids, Jürgen Tomas 13/05/2013
Figure 3.20 Prof. Dr.-Ing. habil. J. Tomas – chair for Mechanical Process Engineering
Micro Process Kinetics of Solid Bridge Formation by Hydra-tion Reactions
E.g.: cement X + water
HXOHX 2W ↔⋅+ϑ
Results in hydrated (hardened) cement paste
( ) ( )dt
tdmM
Mdt
tdm WH
WW
HXHX ⋅⋅
=ϑ
( )nHWR
WH wkdt
d−= 1,
For reaction order of n = 2:
Uniaxial compressive strength:
( )tk
tkM
XM
WR
WR
WW
WHXDHXct ⋅+
⋅⋅
⋅⋅
⋅−=,
,
11
ϑεσσ
Minimum hopper outlet width to avoid bridging:
( )tk
tkMg
XMmb
WR
WR
WWs
WAHXDHXt ⋅+
⋅⋅
⋅⋅⋅⋅⋅⋅+
=,
,min, 1
1ϑρ
σ
Fig_PC&P_3 Lecture Product Characterization and Processing of Pharmaceutical Particulate Solids, mechanics of cohesive particulate solids, Jürgen Tomas 13/05/2013
Figure 3.21 Prof. Dr.-Ing. habil. J. Tomas – chair for Mechanical Process Engineering
Micro Process Kinetics of Solid Bridge Formation by Viscous Contact Flow, Fusion and Sintering
Rumpf u.a. (1976): Diameter of sinter neck:
⋅
π+
σ⋅⋅
η⋅⋅
=
2Nsg
s
22
dF1
d2
5t8
dd
( ) ( ) ( ) Ntt,HOHt FttFtF ⋅κ+=
dt1
54
s
sgZs
0
0t0 ⋅η
⋅σ⋅σ⋅
εε−
⋅π⋅
=σ
s
Zst 5
t2η⋅
⋅σ⋅=κ
Superposition of adhesion forces:
( ) ( )tFFtF t,HvdWaals,Htot,H +=
Angle of internal friction:
st
it
it,i
tantan1
tantan
ϕϕ⋅κ+
ϕ=ϕ
Uniaxial compressive strength:
( ))sin1()sin1(
sin)sin1(2)sin1()sin1(
tantan2
itst
t,0stit1
itst
itstct ϕ−⋅ϕ+
σ⋅ϕ⋅ϕ+⋅+σ⋅
ϕ−⋅ϕ+ϕ−ϕ⋅
=σ
Minimum hopper outlet width to avoid bridging:
( ) ( )( ) ( )[ ]1ff2sinsinsinsin1g
sin)sin1(2sin1m2b
itstitstb
t,0stitWtmin, −⋅⋅ϕ−ϕ−ϕ⋅ϕ−⋅⋅ρ
σ⋅ϕ⋅ϕ+⋅Θ+ϕ⋅+⋅=
FN
FN
d2
d
Fig_PC&P_3 Lecture Product Characterization and Processing of Pharmaceutical Particulate Solids, mechanics of cohesive particulate solids, Jürgen Tomas 13/05/2013
Figure 3.22 Prof. Dr.-Ing. habil. J. Tomas – chair for Mechanical Process Engineering
3.2 Contact mechanics of adhesive particles
Particle Contact Deformation in Normal Direction without Adhesion
rrK,el << 1
rK,plr << 1
material data: E* effective modulus of elasticity, pf micro-yield strength, ηΚ contact viscosity
hK,el
FN
FN
rrK,el
FN
hK,pl
FN
rK,pl
runloadingyield
ing
loading
WD = ∫ FR (hK) dhK
particle centre approach hK
cont
act n
orm
al fo
rce
FN
3π pf E*hK,f = ( )2r
2
kN = dFNdhK
elastic plastic andviscoplastic
force
response FR =π · r · pf · hK,pl
π · r · ηK · hK,vis·
13 E* ·√d · hK,el
3
stiffness kN = π · r · pf12 E* ·√ d · hK,el
deformation
work WD = 215 E*·√d · hK,el
5 · r · pf ·(hK,pl - hK,f)
π2
2 2
π2 · r · ηK · hK,vis · t
2·
F 3.23
Fig_PC&P_3 Lecture Product Characterization and Processing of Pharmaceutical Particulate Solids, mechanics of cohesive particulate solids, Jürgen Tomas 13/05/2013
Figure 3.23 Prof. Dr.-Ing. habil. J. Tomas – chair for Mechanical Process Engineering
Adhesive particle contacts:
Fig_PC&P_3 Lecture Product Characterization and Processing of Pharmaceutical Particulate Solids, mechanics of cohesive particulate solids, Jürgen Tomas 13/05/2013
Figure 3.24 Prof. Dr.-Ing. habil. J. Tomas – chair for Mechanical Process Engineering
3.3 Particle interaction test methods
Testing the Adhesion Force between Particle and Surface
H. Masuda and K. Gotoh, Adhesive Force of a Single Particle, pp.141, in K. Gotoh, M. Masuda, K. Higashitani, Powder Technology Handbook, Marcel Dekker, New York 1997
FHFN FH
c) Vibration method d) Impact separation method
e) Hydrodynamic method
a) Spring balance method b) Centrifugal method
u
Pressing Detachment
FC
F 3.51
Fig_PC&P_3 Lecture Product Characterization and Processing of Pharmaceutical Particulate Solids, mechanics of cohesive particulate solids, Jürgen Tomas 13/05/2013
Figure 3.25 Prof. Dr.-Ing. habil. J. Tomas – chair for Mechanical Process Engineering
Fig_PC&P_3 Lecture Product Characterization and Processing of Pharmaceutical Particulate Solids, mechanics of cohesive particulate solids, Jürgen Tomas 13/05/2013
Figure 3.26 Prof. Dr.-Ing. habil. J. Tomas – chair for Mechanical Process Engineering
Fig_PC&P_3 Lecture Product Characterization and Processing of Pharmaceutical Particulate Solids, mechanics of cohesive particulate solids, Jürgen Tomas 13/05/2013
Figure 3.27 Prof. Dr.-Ing. habil. J. Tomas – chair for Mechanical Process Engineering 3.4 Micro-macro transition of contact forces and bulk stresses in a particle packing
Fig_PC&P_3 Lecture Product Characterization and Processing of Pharmaceutical Particulate Solids, mechanics of cohesive particulate solids, Jürgen Tomas 13/05/2013
Figure 3.28 Prof. Dr.-Ing. habil. J. Tomas – chair for Mechanical Process Engineering
Fig_PC&P_3 Lecture Product Characterization and Processing of Pharmaceutical Particulate Solids, mechanics of cohesive particulate solids, Jürgen Tomas 13/05/2013
Figure 3.29 Prof. Dr.-Ing. habil. J. Tomas – chair for Mechanical Process Engineering
Fig_PC&P_3 Lecture Product Characterization and Processing of Pharmaceutical Particulate Solids, mechanics of cohesive particulate solids, Jürgen Tomas 13/05/2013
Figure 3.30 Prof. Dr.-Ing. habil. J. Tomas – chair for Mechanical Process Engineering
From the Coulomb friction limit of adhesive particle contact wit h additional normal force FNV due to pre-consolidation
, (5)
by splitting in radius and centre components (index R, M)
(6)
after elimination of the angle α of contact normal
(7)
by FR → σR, FHR → σVR, FM → σM, FHM → σVM a non-linear equation of the yield locus follows:
(8)
( ) ( )[ ]α⋅⋅κ++⋅κ++⋅κ+⋅ϕ=α⋅ 2cosFFFFF1tan2sinF HRRHMM0HiR
( )( )
ϕ⋅⋅κ−ϕ⋅⋅κ−+⋅κ+⋅κ+⋅ϕ= iHRi
22HR
22MHM0HiR sinFcosFFFF1sinF
( ) ( ) ( )
ϕϕ−ϕ
−ϕ⋅ϕ
ϕ−ϕ−
σ
σ+σ−σ+σ⋅ϕϕ
⋅σ⋅ϕ=σst
ist
i2
st2
ist2
2
VR
MVM0VMi
st
VRiR cossin
tancossintan
tan
sin
( )[ ] ( ) ( )[ ]NVN0HiNVNHNiH,C,T FFF1FFFFF ⋅κ++⋅κ+⋅µ=++⋅µ=
From the Coulomb friction limit of adhesive particle contact wit h additional normal force FNV due to pre-consolidation
, (5)
by splitting in radius and centre components (index R, M)
(6)
after elimination of the angle α of contact normal
(7)
by FR → σR, FHR → σVR, FM → σM, FHM → σVM a non-linear equation of the yield locus follows:
(8)
( ) ( )[ ]α⋅⋅κ++⋅κ++⋅κ+⋅ϕ=α⋅ 2cosFFFFF1tan2sinF HRRHMM0HiR
( )( )
ϕ⋅⋅κ−ϕ⋅⋅κ−+⋅κ+⋅κ+⋅ϕ= iHRi
22HR
22MHM0HiR sinFcosFFFF1sinF
( ) ( ) ( )
ϕϕ−ϕ
−ϕ⋅ϕ
ϕ−ϕ−
σ
σ+σ−σ+σ⋅ϕϕ
⋅σ⋅ϕ=σst
ist
i2
st2
ist2
2
VR
MVM0VMi
st
VRiR cossin
tancossintan
tan
sin
( )[ ] ( ) ( )[ ]NVN0HiNVNHNiH,C,T FFF1FFFFF ⋅κ++⋅κ+⋅µ=++⋅µ=
Mikro-Makrotransition, Force and Stress Transmissionin a Sheared Particle Packing , cont.acc. to TOMAS
Usually, the stationary flow is the stressing pre-history (σVR = σR,st, σVM = σM,st):(9)
and the yield locus results in:
(10)
By a Taylor series expansion at transition to stationary flow
(11)
suitable linear equations of the yield locus follow as σR(σM) or τ(σ) functions:
(12)
(13)
)(sin 0st,Mstst,R σ+σ⋅ϕ=σ
( ) ( ) ( )
ϕ−ϕ⋅ϕ−ϕ−ϕ−
σ+σσ−σ
ϕϕ
−ϕϕ
σ+σ⋅ϕ=σ istiist2
2
0st,M
Mst,M
st
i
i
st0st,MiR sintansin
tantan1
tantansin
( ) ( )st,MMM
Rst,MMRR
st,MMdd
σ−σ⋅σσ
+σ=σσ=σσ=σ
( )
σ−
ϕσ
+σ⋅ϕ=σ+σ⋅ϕ=σ st,Mi
st,RMiZMiR sin
sinsin
( )
σ−
ϕσ
+σ⋅ϕ=σ+σ⋅ϕ=τ st,Mi
st,RiZi sin
tantan
Usually, the stationary flow is the stressing pre-history (σVR = σR,st, σVM = σM,st):(9)
and the yield locus results in:
(10)
By a Taylor series expansion at transition to stationary flow
(11)
suitable linear equations of the yield locus follow as σR(σM) or τ(σ) functions:
(12)
(13)
)(sin 0st,Mstst,R σ+σ⋅ϕ=σ
( ) ( ) ( )
ϕ−ϕ⋅ϕ−ϕ−ϕ−
σ+σσ−σ
ϕϕ
−ϕϕ
σ+σ⋅ϕ=σ istiist2
2
0st,M
Mst,M
st
i
i
st0st,MiR sintansin
tantan1
tantansin
( ) ( )st,MMM
Rst,MMRR
st,MMdd
σ−σ⋅σσ
+σ=σσ=σσ=σ
( )
σ−
ϕσ
+σ⋅ϕ=σ+σ⋅ϕ=σ st,Mi
st,RMiZMiR sin
sinsin
( )
σ−
ϕσ
+σ⋅ϕ=σ+σ⋅ϕ=τ st,Mi
st,RiZi sin
tantan
F 3.57
Fig_PC&P_3 Lecture Product Characterization and Processing of Pharmaceutical Particulate Solids, mechanics of cohesive particulate solids, Jürgen Tomas 13/05/2013
Figure 3.31 Prof. Dr.-Ing. habil. J. Tomas – chair for Mechanical Process Engineering
3.5 Macroscopic biaxial stress state, powder flow criteria
Fig_PC&P_3 Lecture Product Characterization and Processing of Pharmaceutical Particulate Solids, mechanics of cohesive particulate solids, Jürgen Tomas 13/05/2013
Figure 3.32 Prof. Dr.-Ing. habil. J. Tomas – chair for Mechanical Process Engineering
Constitutive Models for Elastic, Plastic and Viscous Material Behaviour
strain (displacement) ε = ∆l / l0
norm
al st
ress
σ
linear elasticσE
ε =non-linear elastic
anelasticelastic hysteresis
plastic hardening
shear rate (gradient) or strain rate γ = dux/dy
shea
r st
ress
τ
τ0
perfect plastic
linear viscous
shear-thickeningn > 1 (dilatant)
n = 1 linear viscoplastic
shear-thinningn < 1 (pseudoplastic)
τ = ηp. γ n + τ0
.
γ τη=
.dy τ
dux
.
perfect plastic
σ
l0∆dd0
∆llateral expansion εq
= -∆d /d0
Poisson ratio ν = -εq/ε
εpl
Flow Functions
yield stress
elastic
Uniaxial Stress-Strain-Curves
σ
ε
.ε1
.ε2>
viscoelastic
F 3.68
Fig_PC&P_3 Lecture Product Characterization and Processing of Pharmaceutical Particulate Solids, mechanics of cohesive particulate solids, Jürgen Tomas 13/05/2013
Figure 3.33 Prof. Dr.-Ing. habil. J. Tomas – chair for Mechanical Process Engineering
Fig_PC&P_3 Lecture Product Characterization and Processing of Pharmaceutical Particulate Solids, mechanics of cohesive particulate solids, Jürgen Tomas 13/05/2013
Figure 3.34 Prof. Dr.-Ing. habil. J. Tomas – chair for Mechanical Process Engineering
Fig_PC&P_3 Lecture Product Characterization and Processing of Pharmaceutical Particulate Solids, mechanics of cohesive particulate solids, Jürgen Tomas 13/05/2013
Figure 3.35 Prof. Dr.-Ing. habil. J. Tomas – chair for Mechanical Process Engineering
(1) shear and dilatancy dV > 0
cohesionτc
ϕi
0 normal stress σ
shea
r st
ress
τ
τc
yield locus
angle of internal friction
σc σc
uniaxial pressure
ϕi
0normal stress σ
shea
r st
ress
τ
yield locus
−σZ1−σZ
τc
σc
σZ1
σZ1
uniaxial tension
σZσZ
σZ σZ
isostatictension
τσ
∆h→
angle ofdilatancy ν (+)
ϕi
0normal stress σ
shea
r st
ress
τ
yield locus
σc
τc
τ c
Biaxial Stress States of Sheared Particle Packing F 3.73
Fig_PC&P_3 Lecture Product Characterization and Processing of Pharmaceutical Particulate Solids, mechanics of cohesive particulate solids, Jürgen Tomas 13/05/2013
Figure 3.36 Prof. Dr.-Ing. habil. J. Tomas – chair for Mechanical Process Engineering
σ0 σ0
σ0σ0
no deformation
isostatic tensilestrength
σiso
σisoσiso
σiso
isostatic pressure,compression dV < 0
(3) shear and compression dV < 0
(2) stationary shear dV = 0
τσ
∆h→
ν (-)angle ofdilatancy
Biaxial Stress States of Sheared Particle Packing
0normal stress σ
shea
r st
ress
τyieldlocus
−σ0 σ1σ2
ϕst
stationaryyield locus
σM,st
σR,st
stationary angle of internal friction
σ στ
ϕi
0normal stress σ
shea
r st
ess τ
yield locus
−σZσ1σ2 σM,st
σR,st
σiso
ϕi
consolidationlocusϕi ϕi
F 3.74
Fig_PC&P_3 Lecture Product Characterization and Processing of Pharmaceutical Particulate Solids, mechanics of cohesive particulate solids, Jürgen Tomas 13/05/2013
Figure 3.37 Prof. Dr.-Ing. habil. J. Tomas – chair for Mechanical Process Engineering
shea
r st
ress
τ
normals stress σ
Yield Locus
−σ00
ϕi
σM,st
Stationary Yield Locus
End point
ϕst
ϕi angle of internal friction,ϕst stationary angle of internal friction,σ0 isostatic tensile strength of unconsolidated packing; andσM,st centre stress for steady-state flow
shea
r st
ress
τ
σ1normal stress σ
τc
Yield Locus
σVRσVM
ϕi
0 σc−σZσiso
Stationary Yield Locus
−σZ1 σ2
Consolidation Locus
σM,st
σR,st
σ1 major principal stress,σ2 minor principal stress,σc uniaxial compressive strength,σZ1 uniaxial tensile strength,σZ isostatic tensile strength,σiso isostatic pressure;
a) The three flow parameters
b) Stress states
c) Stress states at Mohr circle of steady-state flow:
shea
r st
ress
τ
normal stress σ
Yield Locus
−σ0
0
ϕi
σM,st
End point
ϕst
σ1σst
ϕst σR,st
τst
Stationary Yield Locus:
τst = cosϕst.σR,st
σst = σM,st - sinϕst.σR,st
σR,st = sinϕst.(σM,st + σ0)
Tangential point:
Yield Locus:τ = tanϕi
.(σ + σΖ)−σZ
Stationary Yield Locus
Biaxial Stress States of Sheared Particle Packing F 3.75
Fig_PC&P_3 Lecture Product Characterization and Processing of Pharmaceutical Particulate Solids, mechanics of cohesive particulate solids, Jürgen Tomas 13/05/2013
Figure 3.38 Prof. Dr.-Ing. habil. J. Tomas – chair for Mechanical Process Engineering
σz(1) uniaxial tensile strengthσz(3) isostatic tensile strengthτc cohesionϕst stationary angle of internal frictionϕi angle of internal friction
ϕi = 0τ
τ = f ( γ ).
c) a wet-mass viscoplastic powder without Coulomb friction
σ
A preshear pointE end pointγ shear rate gradientρb bulk densityσ1 major principal stressσc uniaxial compressive strength
Yield Loci and Powder Flow Parameters for:
ϕi = ϕstϕi
Eτ
σ
a) a dry, cohesion-less or free flowing particulate solid
τ
ρb = const. EA
τc
−σZ(3) -σZ(1) σc σ1σ
ϕst
ϕi
b) a general case of moist or fine cohesive powder
.
F 3.76
Fig_PC&P_3 Lecture Product Characterization and Processing of Pharmaceutical Particulate Solids, mechanics of cohesive particulate solids, Jürgen Tomas 13/05/2013
Figure 3.39 Prof. Dr.-Ing. habil. J. Tomas – chair for Mechanical Process Engineering
3.6 Product characterization test equipment and shear testing techniques
Fig_PC&P_3 Lecture Product Characterization and Processing of Pharmaceutical Particulate Solids, mechanics of cohesive particulate solids, Jürgen Tomas 13/05/2013
Figure 3.40 Prof. Dr.-Ing. habil. J. Tomas – chair for Mechanical Process Engineering
Fig_PC&P_3 Lecture Product Characterization and Processing of Pharmaceutical Particulate Solids, mechanics of cohesive particulate solids, Jürgen Tomas 13/05/2013
Figure 3.41 Prof. Dr.-Ing. habil. J. Tomas – chair for Mechanical Process Engineering
normal stress or centre stress
direction
shea
r stre
ss or
radi
us st
ress
direct shear test triaxial test
F 3.79
Fig_PC&P_3 Lecture Product Characterization and Processing of Pharmaceutical Particulate Solids, mechanics of cohesive particulate solids, Jürgen Tomas 13/05/2013
Figure 3.42 Prof. Dr.-Ing. habil. J. Tomas – chair for Mechanical Process Engineering
load lift
twister
shear cell
yoke
powder box
control unit
FN
FS
svS = 1 - 5 mm/min
Translational Shear Cell
σ < 80 kPaτ < 70 kPa
shear cell with load yoke
F 3.80
Fig_PC&P_3 Lecture Product Characterization and Processing of Pharmaceutical Particulate Solids, mechanics of cohesive particulate solids, Jürgen Tomas 13/05/2013
Figure 3.43 Prof. Dr.-Ing. habil. J. Tomas – chair for Mechanical Process Engineering
Fig_PC&P_3 Lecture Product Characterization and Processing of Pharmaceutical Particulate Solids, mechanics of cohesive particulate solids, Jürgen Tomas 13/05/2013
Figure 3.44 Prof. Dr.-Ing. habil. J. Tomas – chair for Mechanical Process Engineering
Bulk Shear Modulus - Centre Stress Diagramof Load/Unload for Limestone Powder
0.0 5 10 15 20 25 30centre stress for steady-state flow σM,st in kPa
bulk
shea
r m
odul
us G
b in
kN
/m2
200
250
150
100
50
0.0
350
300
Gb,0
physical modelfit factor = 3.10-3
load
unload
G = 60 kN/mm²el
astic
wav
e pr
opag
atio
n ve
loci
ty c
S,b i
n m
/s
20
25
15
10
5
0.0
35
30
cS,b
( ) 0F
T
0Szb
TddF
d1
h/sddG
→→τδ
⋅⋅ε
ε−=
τ= (1)
( ) 3/1
Z23/1
22,1
V,N0Hb *E
61*Gr*E2
FF)1(3*G1G
σ⋅⋅
εε−
⋅=
⋅⋅⋅κ+⋅κ+⋅
⋅⋅ε
ε−= (2)
3/1
0
2
0
0
3/1
22,1
0H
0
00,b *E
61*Gr*E2
F3*G1G
σ⋅⋅
ε
ε−⋅=
⋅⋅⋅
⋅⋅ε
ε−= (3)
F 3.82
Fig_PC&P_3 Lecture Product Characterization and Processing of Pharmaceutical Particulate Solids, mechanics of cohesive particulate solids, Jürgen Tomas 13/05/2013
Figure 3.45 Prof. Dr.-Ing. habil. J. Tomas – chair for Mechanical Process Engineering
displacement s
shea
r fo
rce
FS
σpre
normal stress σ = FN / A
Incipient Yield and Steady-State Flow
preshearplastic yielding dV=0
instantaneousyield locus
steady-state flow
σ<σpreσpre
FN
FS
s
preshear FN
FS
s
shear
incipientyielding
0
σpre
shear dV>0
σ
shea
r st
ress
τ =
FS /
A
F 3.83
Fig_PC&P_3 Lecture Product Characterization and Processing of Pharmaceutical Particulate Solids, mechanics of cohesive particulate solids, Jürgen Tomas 13/05/2013
Figure 3.46 Prof. Dr.-Ing. habil. J. Tomas – chair for Mechanical Process Engineering
displacement s
shea
r fo
rce
FS
shea
r st
ress
τ =
FS /
A
σ2 σcσpre σ1
normal stress σ = FN / A
ϕiτc
Incipient Consolidation and Yield, Steady-State Flow
preshearplastic yielding dV=0
ϕst
Stationary Yield Locus
ϕi
Yield Locus
Consolidation Locus
σΖ
end point
σiso σE
τE
σR,st=(σ1-σ2)/2
σM,st=(σ1+σ2)/2σ0
σ<σpre
FN
FS
s
preshear FN
FS
s
shear
shear dV>0incipientyielding
σ1 consolidation stressσc uniaxial compressive strengthϕi, ϕst angles of internal friction
steady-state flow
σpre
F 3.91
Fig_PC&P_3 Lecture Product Characterization and Processing of Pharmaceutical Particulate Solids, mechanics of cohesive particulate solids, Jürgen Tomas 13/05/2013
Figure 3.47 Prof. Dr.-Ing. habil. J. Tomas – chair for Mechanical Process Engineering
σ2 σc σ1
normal stress σ
ϕi
ϕst
σ0 σM,st=(σ1+σ2)/2
Stationary Yield Locus:
steady-state flowσR,st = sinϕst .(σΜ,st + σ0)
σΖ σiso
ϕi
Instantaneous Yield Locus:
incipient yielding
σR = sinϕi .(σΜ − σM,st) + σR,st
σR,st=(σ1-σ2)/2
τc
Consolidation Locus:σR = sinϕi .(- σΜ + σM,st) + σR,st
incipient consolidation
Constitutive Functions for Incipient Consolidation, Yield and Steady-State Flow sh
ear
stre
ss τ
0 σ
τ = tanϕi . σ - σM,st + [ ]σR,st
sinϕi
F 3.92
Fig_PC&P_3 Lecture Product Characterization and Processing of Pharmaceutical Particulate Solids, mechanics of cohesive particulate solids, Jürgen Tomas 13/05/2013
Figure 3.48 Prof. Dr.-Ing. habil. J. Tomas – chair for Mechanical Process Engineering
( )( ) ( )ist
ist1 sin1sin1
sinsin2aϕ−⋅ϕ+
ϕ−ϕ⋅=
( )( ) ( ) 0
ist
sti0,c sin1sin1
sinsin12σ⋅
ϕ−⋅ϕ+ϕ⋅ϕ+⋅
=σ
Yield Locus 1
YL 2
YL 3
YL 4
αϕst= arc sin (tanα)
σR,st= sin ϕst· (σM,st + σ0)
a) Stationary Yield Locus
radi
us st
ress
σR
,st=(
σ 1- σ
2)/2
0σ0 centre stress σM,st= (σ1 + σ2)/2
major principal stress during consolidation σ1
unia
xial
com
pres
sive
stre
ngth
σc
b) Consolidation function
ffc = 1
YL 2
YL 3
YL 4
YL 1
σ1= σc,st
σc,st
σc,0
0
F 3.93
σR,st = ast. σM,st + σR,0
σc = a1. σ1 + σc,0
ϕst= arcsin(tanα) = arcsin(ast)
σ0 = σR,0 / sin ϕst
( )( )
ϕ−⋅+
ϕ⋅+ϕ−⋅=ϕ
i1
ii1st sin1a2
sin2sin1aarcsin
(1 + sin ϕst) · (1 - sinϕi)2 · (1 + sin ϕi) · sin ϕst
σ0 = .σc,0
Experimental Determination of Cohesive Steady-State Flow Param eters of a Cohesive Powder
Fig_PC&P_3 Lecture Product Characterization and Processing of Pharmaceutical Particulate Solids, mechanics of cohesive particulate solids, Jürgen Tomas 13/05/2013
Figure 3.49 Prof. Dr.-Ing. habil. J. Tomas – chair for Mechanical Process Engineering
displacement s
shea
r fo
rce
FS
shea
r st
ress
τ
= F S
/ A
σc
σ1
σctnormal stress σ = FN / A
ϕit
ϕi
τc
Instantaneous, Stationary, Time Yield Locus and Wall Yield Locus
preshear
ϕst
t >> 0
−σ0
ϕW
steady-state flow
end point
σM,st
incipientyieldingshear
σ<σpre σ>σpreσpre σpre
stationaryyield locus
−σZ
FN
FS
s
preshearFN
timeconsolidationt >> 0 FN
FS
s
shear
FN
FS
s
wall shear
time yield locus
wall yield locusyield locus
F 3.98
time t (or displacement s = vS.t)
FS
FN
Fig_PC&P_3 Lecture Product Characterization and Processing of Pharmaceutical Particulate Solids, mechanics of cohesive particulate solids, Jürgen Tomas 13/05/2013
Figure 3.50 Prof. Dr.-Ing. habil. J. Tomas – chair for Mechanical Process Engineering
3.7 Flow and consolidation functions of cohesive particulate solids
Time Yield Loci for TiO2, Storage Time t = 24 h dS = 200 nm, Moisture XW = 0.4 %, Temperature θ = 20°C
shea
r st
ress
τ in
kPa
25
20
15
10
5
-5 0 5 10 15 20 25 30 35 40 normal stress σ in kPa
SYL
TYL 4
TYL 3
TYL 2
TYL 1
0
shea
r st
ress
τ in
kPa
15
10
5
-5 0 5 10 15 20 normal stress σ in kPa
SYL
YL 4
YL 3
YL 2
YL 10
−σ0
Instantaneous Yield Loci for TiO2
dS = 200 nm, Moisture XW = 0.4 %
F 3.99
Fig_PC&P_3 Lecture Product Characterization and Processing of Pharmaceutical Particulate Solids, mechanics of cohesive particulate solids, Jürgen Tomas 13/05/2013
Figure 3.51 Prof. Dr.-Ing. habil. J. Tomas – chair for Mechanical Process Engineering
Yield Loci and Stationary Yield Locus for TiO2
d50 = 0.61µm, XW = 0.4 %
YL 4
YL 3
CL 1
CL 2
Consolidation LocusCL 3
YL 1
-5.0 0.0 5.0 10.0 15.0 20.0 25.0−σZ -σ0
σisocentre stress σM in kPa
5.0
10.0
0.0
15.0
radi
us st
ress
σR
in k
Pa
CL 4σR,st(σM,st) σc(σ1)σR,st(σM,st) SYL
YL 2
TSC vS = 2 mm/min
-300 0.0 500 1000centre stress σM in kPa
radi
us st
ress
σR
in k
Pa
0.0
500
1000
YL 5
FO 3 CL 3
CL 2
CL 1
YL 2
YL 1
VL 4
CL 5σR,st(σM,st) SYLσR,st(σM,st) σc(σ1)
YL 4
Yield Loci and Stationary Yield Locus for TiO2
d50 = 0.61µm, XW = 27 - 30 %
σiso
PSC vS = 49 mm/min
−σZ -σ0
F 3.100
Fig_PC&P_3 Lecture Product Characterization and Processing of Pharmaceutical Particulate Solids, mechanics of cohesive particulate solids, Jürgen Tomas 13/05/2013
Figure 3.52 Prof. Dr.-Ing. habil. J. Tomas – chair for Mechanical Process Engineering
( )( )
( )( ) i
i22
i
i22
i
ic
sintan11
tan1tan11
tan11
2sin1ff
ϕ−ϕ⋅κ++
ϕ⋅κ+ϕ⋅κ++
ϕ⋅κ++
⋅ϕ−
=
Flowability assessment and contact consolidation coefficient κ(ϕi = 30°)flow
function ffc
κ-values ϕst in deg evaluation examples
100 - 10 0,01006 – 0,107 30,3 – 33 free flowing dry fine sand4 - 10 0,107 – 0,3 33 – 37 easy flowing moist fine sand2 - 4 0,3 – 0,77 37 – 46 cohesive dry powder1 - 2 0,77 - ∞ 46 - 90 very cohesive moist powder< 1 ∞ - non flowing,
hardened (ffct)moist powder
hydrated cement
0.0 5.0 10.0 15.0 20.0 25.0 30.0 35.0 40.0 consolidation stress σ1 in kPa
unia
xial
com
pres
sive
str
engt
h σ
c in
kPa
20.0
15.0
10.0
5.0
0.0ffc = σ1/σc 10≥
free flowing
4 < ffc < 10easy flowing
1 < ffc < 2very cohesive
σc(σ1) t = 0 σct(σ1) t = 24 h
25.0
-5.0
σ1 σc
ffc < 1hardenednon flowing
2 < ffc < 4cohesive
F 3.101Consolidation Function of Titaniaparticle size dS= 200 nm, moisture Xw= 0.4%, temperature = 20 °C
Fig_PC&P_3 Lecture Product Characterization and Processing of Pharmaceutical Particulate Solids, mechanics of cohesive particulate solids, Jürgen Tomas 13/05/2013
Figure 3.53 Prof. Dr.-Ing. habil. J. Tomas – chair for Mechanical Process Engineering
Comparison of Shear Test and Compressive Strength Testfor Time Consolidation (Caking) of Potassium Cloride K 60
0 0.1 0.2 0.3 0.4
unia
xial
com
pres
sive
stre
ngth
σ c
t in
kPa
100
80
90
70
60
50
40
30
20
10
0
moisture difference XWO - XWE in %
t = 4 . . . 6 hσ1 = 10 . . . 12 kPa
=
confidence interval
Θ 60 °C
σ
σct
τ
σ
Shear test
σ2
α
σ1 τ
Compr. strength test
σ1 = σct
σRσR = σΜ
σΜ
τσ
α
F 3.102
Fig_PC&P_3 Lecture Product Characterization and Processing of Pharmaceutical Particulate Solids, mechanics of cohesive particulate solids, Jürgen Tomas 13/05/2013
Figure 3.54 Prof. Dr.-Ing. habil. J. Tomas – chair for Mechanical Process Engineering
- 5 0.0 5 10 15 20 25 30 35 40 consolidation stress σ1 in kPa
unia
xial
com
pres
sive
str
engt
h σ
c in
kPa
20
15
10
5
0.0 ffc = σ1/σc 10≥ free flowing
4 < ffc < 10easy flowing
1 < ffc < 2very cohesive
25
σ1 σc
ffc < 1 non flowing
2 < ffc < 4cohesive
( )( )
( )( ) i
i22
i
i22
i
ic
sintan11
tan1tan11
tan11
2sin1ff
ϕ−ϕ⋅κ++
ϕ⋅κ+ϕ⋅κ++
ϕ⋅κ++
⋅ϕ−
=
Consolidation Function of Titania Powder
20
30
10
0.0- 4.0 0.0 5.0 10.0 15.0 20.0 25.0 normal force FN in nN
adhe
sion
forc
e F
H in
nN
40
κ = 0.77 for ffc = 2, ϕi = 30°
FH = f(FN, κ, d, E)
FH = (1 + κ) FH0 + κ FN.
Particle Adhesion Forces of Titania Particles
FN
FN FH(FN)
FH(FN)
F 3.103
Fig_PC&P_3 Lecture Product Characterization and Processing of Pharmaceutical Particulate Solids, mechanics of cohesive particulate solids, Jürgen Tomas 13/05/2013
Figure 3.55 Prof. Dr.-Ing. habil. J. Tomas – chair for Mechanical Process Engineering
Consolidation Functions of Titaniaparticle size dS = 200 nm, moisture Xw= 0.4%, temperature = 20 °C
unia
xial
com
pres
sive
and
tens
ile s
tren
gth
σc,
σ Z1 i
n kP
a
20.0
15.0
10.0
5.0
0.0ffc = σ1/σc 10≥
free flowing
4 < ffc < 10easy flowing
1 < ffc < 2very cohesive
25.0
- 5.0
σ1σc
ffc < 1 hardened non flowing
2 < ffc < 4cohesive
- 5.0
- 10.0
30.0σc(σ1) t = 0 σct(σ1) t = 24 hσZ1(σ1) t = 0 σZ1t(σ1) t = 24 h
σZ1
0.0 5.0 10.0 15.0 20.0 25.0 30.0 35.0 40.0
consolidation stress σ1 in kPa
F 3.104
Fig_PC&P_3 Lecture Product Characterization and Processing of Pharmaceutical Particulate Solids, mechanics of cohesive particulate solids, Jürgen Tomas 13/05/2013
Figure 3.56 Prof. Dr.-Ing. habil. J. Tomas – chair for Mechanical Process Engineering
dry cohesionlessmajor principal stress at consolidation
dry cohesive
moist cohesive incompressible
moist cohesive
moist cohesiveporous compressible
Typical Consolidation Functions of Particulate Solids
dry fibrous
time consolidation, hardened
unia
xial
com
pres
sive
stre
ngth
F 3.105
Fig_PC&P_3 Lecture Product Characterization and Processing of Pharmaceutical Particulate Solids, mechanics of cohesive particulate solids, Jürgen Tomas 13/05/2013
Figure 3.57 Prof. Dr.-Ing. habil. J. Tomas – chair for Mechanical Process Engineering 3.8 Compression functions and shear work
Bul
k de
nsity
ρb
ρb,0
ρb = ρb,0·(1 + )σM,stσ0
n
Centre stress of steady-state flow σM,stIsostatictensile strength -σ0
0
n = 0 incompressible
0 < n < 1 compressible
n = 1 Compressibility index of ideal gas
Compressibility index of cohesive powders for small (1 < σ < 50 kPa) and medium pressures (50 < σ < 1000 kPa)
Index n Evaluation Examples Flowabiliy 0 – 0.01 incompressible gravel free flowing
0.01 – 0.05 low compressibility fine sand 0.05 - 0.1 compressible dry powder cohesive
0.1 - 1 very compressible moist powder very cohesive
Adiabatic gas compression:
pV1
dpdV
adκ=− (1)
Isentropic powder compression:
∫∫σρ
ρ σ+σσ
⋅=ρρ st,Mb
0,b 0 0st,M
st,M
b
b dnd
(2)
Pow
der
pres
sure
σ
displace-ment ∆h
σ∆h
a) elastic
b) elastic-plastic
loading
Yunloading
elastic recovery
plastic compression
WV= σ(h) d(h/h0)
1) Uniaxial powder compression 2) Isentropic compression function
Uniaxial Compression of Cohesive Powder F 3.108
Fig_PC&P_3 Lecture Product Characterization and Processing of Pharmaceutical Particulate Solids, mechanics of cohesive particulate solids, Jürgen Tomas 13/05/2013
Figure 3.58 Prof. Dr.-Ing. habil. J. Tomas – chair for Mechanical Process Engineering
bu
lk d
ensit
y ρ
b
ρb,0
ρb = ρb,0 · (1 + )σM,stσ0
n
centre stress during consolidationor steady-state flow σM,st
isostatictensile strength -σ0
0
n = 0 incompressible
0 < n < 1 compressible
n = 1 ideal gas compressibility index
Isentropic Powder Compression
Compressibility index of powders, semi-empirical estimation index n evaluation examples flowability 0 – 0.01 incompressible gravel
0.01 – 0.05 low compressibility fine sand free flowing
0.05 - 0.1 compressible dry powder cohesive 0.1 - 1 very compressible moist powder very cohesive
Adiabatic gas compression:
pV1
dpdV
adκ=−
(1)
Isentropic powder compression:
∫∫σρ
ρ σ+σσ
⋅=ρρ st,Mb
0,b 0 0st,M
st,M
b
b dnd
(2)
F 3.109
Fig_PC&P_3 Lecture Product Characterization and Processing of Pharmaceutical Particulate Solids, mechanics of cohesive particulate solids, Jürgen Tomas 13/05/2013
Figure 3.59 Prof. Dr.-Ing. habil. J. Tomas – chair for Mechanical Process Engineering
bulk
den
sity
ρb
ρb,0
average pressure pisostatictensilestrength -σ0
0
0 < n < 1
Powder Compression
ρb = ρb,0 · (1 + )pσ0
n
com
pres
sion
rate
dρ b
/dp
ρb
p + σ0= n· dρb
dp
Wm,b = . . (1 + ) - 1pσ0
1-n n1 - n
σ0ρb,0
spec
ific
com
pres
sion
wor
k W
m,b
ρb,0
σ0n·
F 3.110
Fig_PC&P_3 Lecture Product Characterization and Processing of Pharmaceutical Particulate Solids, mechanics of cohesive particulate solids, Jürgen Tomas 13/05/2013
Figure 3.60 Prof. Dr.-Ing. habil. J. Tomas – chair for Mechanical Process Engineering
average pressure at steady-state flow σM,stisostatictensile strength -σ0
0
0 < n < 1
Compression and Preshear Work
Wm,b = . . (1 + ) - 1σM,st
σ0
1-n n1 - n
σ0ρb,0
spec
ific
com
pres
sion
and
pres
hear
wor
k W
m,b, W
m,b
,pre
displacement s
shea
r fo
rce
FS
preshear
spre
τpre, YL3
Wb, pre = ∫ FS(s) ds
τpre, YL2
τpre, YL1
FN
FS
s
Wm,b,pre= (1 + )σM,st
σ0
1-n. cosϕi
.sinϕst.spre
.σ0
hSz.ρb,0
. cosϕi.sinϕst
spre.σ0
hSz.ρb,0
F 3.111
Fig_PC&P_3 Lecture Product Characterization and Processing of Pharmaceutical Particulate Solids, mechanics of cohesive particulate solids, Jürgen Tomas 13/05/2013
Figure 3.61 Prof. Dr.-Ing. habil. J. Tomas – chair for Mechanical Process Engineering
- 5 0.0 5 10 15 20centre stress for steady-state flow σΜ,st in kPa
bulk
den
sity
ρb
in k
g/m
3
800
1000
600
400
200
0.0
ρb,0
ρb(σM,st)
−σ0
com
pres
sion
rat
e d
ρ b/d
σ M,st
in g
/J
40
50
30
20
10
0.0
dρb/dσM,st
Compression Function and Compression Rate of Titania, dS = 200 nm
- 2 0.0 5 10 15 20
centre stress for steady-state flow σΜ,st in kPa
mas
s rel
ated
pre
shea
r an
dco
mpr
essio
n w
ork
Wm
,b,p
re, W
m,b
,com
in J
/kg
4.0
5.0
3.0
2.0
1.0
0.0
Wm,b,prespre = 3 mm
Wm,b,com
Specific Preshear and Compression Work of Titania, dS = 200 nm
−σ0
v = 2 m/skinetic energy
spec
ific
pow
er c
onsu
mpt
ion
Pm
,b,p
re in
mW
/kg
40
50
30
20
10
0.0
Pm,b,pre
distortion γshea
r st
ress
τ
∫ γτ prepred
centre stress σM,st bulk
den
sity
ρb
∫ σρ
σ
)(d
st,Mb
st,M
F 3.112
Fig_PC&P_3 Lecture Product Characterization and Processing of Pharmaceutical Particulate Solids, mechanics of cohesive particulate solids, Jürgen Tomas 13/05/2013
Figure 3.62 Prof. Dr.-Ing. habil. J. Tomas – chair for Mechanical Process Engineering
Qualitative Comparisons of:
a) particle contact deformation b) particle adhesion
c) powder yield loci d) consolidation functions
Compliant and Stiff Particle Contact and Powder Behaviour
displace-ment hK
0
compliant
stiff
-FH0
forc
e F N
f) compression function
normal force FN0
compliant
stiff
adhe
sion
forc
e F H
consolidation stress σ1
0
compliantcohesive
stiff, free flowing
unia
xial
com
pres
sive
/te
nsile
stre
ngth
σc,
σ Z1
consolidation stress σ10
compliantcompressible
stiff, incompressible
bulk
den
sity
ρb
ρ b,0
−σ0
normal stress σ0
cohesive
free flowing
shea
r st
ress
τ
−σ0
SYL
SYLYLYL
e) powder constitutive models
average pressure σΜ0
cohesive
radi
us st
ress
σR
−σ0
SYLSYL
YL
YL
free flowing
CLCL
σiso
F 3.113
Fig_PC&P_3 Lecture Product Characterization and Processing of Pharmaceutical Particulate Solids, mechanics of cohesive particulate solids, Jürgen Tomas 13/05/2013
Figure 3.63 Prof. Dr.-Ing. habil. J. Tomas – chair for Mechanical Process Engineering
3.9 Consolidation functions for hopper design
Fig_PC&P_3 Lecture Product Characterization and Processing of Pharmaceutical Particulate Solids, mechanics of cohesive particulate solids, Jürgen Tomas 13/05/2013
Figure 3.64 Prof. Dr.-Ing. habil. J. Tomas – chair for Mechanical Process Engineering
Fig_PC&P_3 Lecture Product Characterization and Processing of Pharmaceutical Particulate Solids, mechanics of cohesive particulate solids, Jürgen Tomas 13/05/2013
Figure 3.65 Prof. Dr.-Ing. habil. J. Tomas – chair for Mechanical Process Engineering
3.10 Permeation and fluidisation behaviour
Fig_PC&P_3 Lecture Product Characterization and Processing of Pharmaceutical Particulate Solids, mechanics of cohesive particulate solids, Jürgen Tomas 13/05/2013
Figure 3.66 Prof. Dr.-Ing. habil. J. Tomas – chair for Mechanical Process Engineering
Fluid Flow through Particle Beds – Permeation Models
a) Fixed particle bed Valid for Model of No Author Equation
Lam
inar
flow
-aro
und
Re
< 0.
5 tr
ansi
tion
Re
= 0.
5 - 1
000
Tur
bule
nt fl
ow-a
roun
d
Re
> 10
00
Poro
sity
ε
Pore
syst
em
Ran
dom
pac
king
mon
odis
pers
e sp
here
s
Part
icle
shap
e
Remarks
1 Darcy uKukh
p
B⋅η⋅=⋅=
∆, K = Darcy constant
+ - - - + - - - Homoge-neous pore system
2 Carman, Kozeny
( )CK
2V,S3
2
B
KuA1h
p⋅⋅η⋅⋅
εε−
=∆
, KCK = Car-
man Kozeny constant, KCK = 5 for spheres of equal size with small deviation
+ - -
≈ 0.
4
+ - + - parallel bended, cylindrical pores
3 Gupte 5.5
B
K2
f Re6.5
hd
up
ε⋅=⋅
⋅ρ∆
+ - - + - + + - Dimension
analysis 4 Molerus,
Pahl, Rumpf
( ) 55.4B
K2
f Re6.514
hd
up
ε⋅⋅ε−⋅=⋅
⋅ρ∆
+ - -
0.35
– 0
.7 - + + - Basing on
results of Gupte
Fig_PC&P_3 Lecture Product Characterization and Processing of Pharmaceutical Particulate Solids, mechanics of cohesive particulate solids, Jürgen Tomas 13/05/2013
Figure 3.67 Prof. Dr.-Ing. habil. J. Tomas – chair for Mechanical Process Engineering
5 Pärnt ( )
3
2
2ST
2R
2F
hB
1d
uReh
pε
ε−⋅
⋅ψ⋅ψ⋅η
⋅⋅ξ=∆
, ε−
=1ReReh
Reh = hydraulic Reynolds number ψF = shape factor, ψR = roundness factor ξ = fluid drag coefficient
+ - - + - + - + Experiments with fine disperse particle beds
6 Burke, Plummer 3
B
K2
f
175.1hd
up
εε−
⋅=⋅⋅ρ
∆ - - + + - - - - Experiments
7 Ergun ( )3
B
K2
f
1hd
up
εε−
⋅λ=⋅⋅ρ
∆, 75.1
Re1150 +
ε−⋅=λ
+ + + + - - - - Experiments
8 Molerus
1.095.0
5.1
95.0
95.0
2
B
Re891.0
ad4.0
ad12.01
Re4
ad
21
ad692.01
Re24Eu
⋅
++
⋅+⋅+
+
⋅+⋅+⋅=
Euler number of fixed bed:
ε−ε
⋅ρ∆⋅
=1h
du3p4Eu
2
B2
fB
Re < 104
0.1
- 1
+ + + + η
ρ⋅⋅= fduRe
3
3
95.0 195.01
ad
ε−−ε−
=
Fig_PC&P_3 Lecture Product Characterization and Processing of Pharmaceutical Particulate Solids, mechanics of cohesive particulate solids, Jürgen Tomas 13/05/2013
Figure 3.68 Prof. Dr.-Ing. habil. J. Tomas – chair for Mechanical Process Engineering
b) Fluidized bed Valid for Model of No Author Equation
Lam
inar
flow
-aro
und
Re
< 0.
5 T
rans
ition
R
e =
0.5
- 100
0 T
urbu
lent
flow
-aro
und
R
e >
1000
Poro
sity
ε
Pore
syst
em
Ran
dom
pac
king
mon
odis
pers
e sp
here
s
Part
icle
shap
e
Remarks
9 Beranek, Rose, Winter-stein
( ) ( ) g1h
pfsL
B⋅ρ−ρ⋅ε−=
∆
εL = Porosity at point of bed expansion
- - + + + + + - ∆p = const. for complete range of flui-dized bed
10 Molerus
1.0
5.1
2
W
Re907.0
ad4.0
ad07.01
Re4
ad
21
ad341.01
Re24Eu
⋅++
⋅+⋅+
+
⋅+⋅+⋅=
Euler number of fluidized bed:
2f
fsW )/u(
gd34Eu
ε⋅
⋅ρ
ρ−ρ⋅= with W
W c1
Eulim=
→ε
Re < 104
0.5
- 1
+ + + + η
ρ⋅⋅= fduRe
3
3
9.0 19.01
ad
ε−−
ε−=
Fig_PC&P_3 Lecture Product Characterization and Processing of Pharmaceutical Particulate Solids, mechanics of cohesive particulate solids, Jürgen Tomas 13/05/2013
Figure 3.69 Prof. Dr.-Ing. habil. J. Tomas – chair for Mechanical Process Engineering
Fig_PC&P_3 Lecture Product Characterization and Processing of Pharmaceutical Particulate Solids, mechanics of cohesive particulate solids, Jürgen Tomas 13/05/2013
Figure 3.70 Prof. Dr.-Ing. habil. J. Tomas – chair for Mechanical Process Engineering Sample No........................ Locus/Time.......................
3.11 Data Sheet of Product Properties
Author ......................................... Phone/Fax..................... ................
Apparatus.................................. Object No. .......... Pos. No........
Material..................................... Serial No......Flow Sheet No.......
Date ..............
Sheet No ........
Next Sheet No.............
0 = not, 1 = small, 2 = large 7. Processing Behaviour 13. Physicochemical Data No Characteristic Grade 1 colour sensitive No Characteristic Unit Value 1. Flow Behaviour 2 dyeing 1 solid density kg/m3 1 free flowing (> 10) 3 moisture sensitive 2 internal porosity % 2 easily flowing (10) 4 freezing 3 bulk density kg/m3 3 cohesive (4) 5 heat sensitive 4 shaking density kg/m3 4 very cohesive (2) 8. Particle Size Distribution 5 tap density kg/m3 5 hardening (< 1) Nr. size fractions ....m % 6 angle of repose deg 6 sticking, greasy 1 7 angle of wall fric. deg 7 thixotrop 2 8 BET- surface m2/g 2. Fluidization Behaviour 3 9 Blaine-surface m2/g 1 homogenous (A) 4 10 surface diam. dST µm 2 bubbling (B) 5 11 median size d50 µm 3 pistoning (D) 6 12 upper pa. size d95 mm 4 channeling (C) 7 13 low. part. size d5 µm 5 flooding 8 14 hardness (Mohs) - 6 hygroscopic 9 15 Young modulus kN/mm2 3. Compressibility 10 16 Poisson ratio - 1 compressible 11 17 tensile strength N/mm2 2 pressure-erecting 12 18 compres. strength N/mm2 3 water excreting 13 Sum Σ 19 melting point °C 4 pressure sensitive 9. Lump and Particle Shape 20 melting enthalpy kJ/kg 5 impact sensitive 1 spherical 21 spec. heat capac. kJ/kgK 6 disintegrating 2 cubic 22 therm. conductiv. W/mK 4. Abrasion Behaviour 3 needle-like 23 expansion coeff. 10-6K-1 1 hard 4 fibrous 24 net calor. value MJ/kg 2 abrasion sensitive 5 thread-like 25 inflammation tem °C 3 abrasive 6 lamellar 26 spontan. ignition °C 4 corrosive 7 sharp-edged 27 min.ignition.ener. mJ 5. Endanger Behaviour 10. Chem. Constituents dry b. 28 low. explosion li. g/kg 1 electrically chargeable 1 29 upper expl. limit g/kg 2 dusty 2 30 therm. decompos. °C 3 inflammable 3 31 soluble constitu. g/kg 4 explosive 4 32 water solubility g/l 5 out gassing 5 33 solution enthalpy kJ/kg 6 odour intensive 6 34 sp..el. resistance Ωm 7 deteriorating 7 14. Storage and Process Conditions 8 biologically active 8 1 storage capacity m3 9 cancerogen 9 Sum Σ 2 feed mass fl. rate t/h 10 toxic 11. Chem. Aggressive Constit. 3 discharge m.fl.r. t/h 11 chemically reactive 1 4 solid content g/kg 12 2 5 residence time h 13 3 6 average moisture %
Fig_PC&P_3 Lecture Product Characterization and Processing of Pharmaceutical Particulate Solids, mechanics of cohesive particulate solids, Jürgen Tomas 13/05/2013
Figure 3.71 Prof. Dr.-Ing. habil. J. Tomas – chair for Mechanical Process Engineering 6. Health Risks 12. Dangerous Constituents 7 min/max moisture % 1 1 8 average temp. °C 2 2 9 min/max temp. °C Sources: 10 pressure kPa p.t.o. for supplementations