Liuotuksen kinetiikka – sileiden pintojen karheus
Dissolution kinetics – the roughness of even surfaces
Tapio Salmi and Henrik Grénman
Outotec 10.2.2012
Outline
 Background of solid-liquid reactions
 New methodology for solid-liquid kinetic
modeling
 Description of rough particles
 General product layer model
 Particle size distribution
 Conclusions
Milestones from ÅA perspective
 Lectures in chemical reaction engineering at ÅA in 70’s: Ready
formulae were presented for ideal surfaces for gas solid reactions 
students did not understand anything
 At undergraduate library: Denbigh-Turner Chemical reactor theory
– the ideal concepts logically explained
 Organic liquid-phase reaction kinetics [ideal non-porous particles]
(Tirronen et al. 1998)
 Cellulose substitution [completely porous particles] (Valtakari et al.
2003)
 Zink leaching – old theory and experimental observations in conflict
(Heidi Markus (Bernas) et al. 2004)
 General theory of rough particles (Salmi et al. 2010)
 General theory for product layer model (Salmi et al. 2011)
 Particle size distribution (Grénman et al. 2011)
Solid-liquid reaction kinetics
• The aim is to develop a mathematical model for the dissolution
kinetics
Why modeling is useful?
 Modeling helps in effective process and equipment design
as well as control
 Empirical process development is slow in the long run
 The optimum is often not achieved through empirical
development, at least in a reasonable time frame
What influences the kinetics
A + B → AB → C (l)
• Reaction rate depends on
– Mass transfer
• External
• Internal (often neglected)
– Intrinsic kinetics (the “real”
chemical rates
A
AB
C
Practical influence of mass transfer
 External mass transfer resistance can be overcome by
agitation
 It is important to recognize what you actually are measuring
What influences the kinetics
 Reaction rate depends on
 Surface area of solid
 Morphological changes
 Reactive surface sites on solid
 Heterogeneous solids
 Possible phase transformations in solid phase
 Equilibrium considerations
 Complex chemistry in liquid phase
Traditional methodology
The conversion is followed by measuring the solid or liquid phase
12
Koncentration (gram/liter)
Concentration
10
8
50°C
6
80°C
4
2
0
0
2
4
6
Time
Tid (min)
8
10
Traditional hypothesis in modeling
solid-liquid reactions
Shrinking particle
Sphere
Cylinder
Slab
Shrinking core
Traditional kinetic modeling –
screening models from literature
• The kinetics depends on the
surface area (A) of the particles
kt  1  (1   )
1/ 3
dc solid

  kA particlescliquid
dt
• Because of the difficulties associated
with measuring the surface area online, the change is often expressed
with the help of the conversion
• Experimental test plots are used to
determine the reaction mechanism
nr
g()
f(cS)
Type of model
1
-ln(1-)
cS/c0S
First-order kinetics
2
(1-)-1/2 - 1
(cS/c0S)3/2
Three-halves-order kinetics
3
(1-)-1
(cS/c0S)2
Second-order kinetics
4
1 - (1-)1/2
(cS/c0S)1/2
One-half-order kinetics; two-dimensional
advance of the reaction interface
5
1 - (1-)1/3
(cS/c0S)2/3
Two-thirds-order kinetics; threedimensional advance of the reaction
interface
6
1 - (1-)2/3
(cS/c0S)1/3
One-thirds-order kinetics; film diffusion
7
[1 - (1-)1/3]2
(cS/c0S)2/3/(1 - (cS/c0S)1/3)
Jander; three-dimensional
8
1 - 2/3 - (1-)2/3
(cS/c0S)1/3/(1 - (cS/c0S)1/3)
Crank-Ginstling-Brounshtein, mass transfer
across a nonporous product layer
9
[1/(1-)1/3 – 1]2
(cS/c0S)5/3/(1 - (cS/c0S)1/3)
Zhuravlev-Lesokhin-Tempelman, diffusion,
concentration of penetrating species varies
with 
10
[1 - (1-)1/2]2
(cS/c0S)1/2/(1 - (cS/c0S)1/2)
Jander; cylindrical diffusion
11
1/(1-)1/3 - 1
(cS/c0S)4/3
Dickinson, Heal, transfer across the
contacting area
12
1-3(1-)2/3+2(1-)
(cS/c0S)1/3/(1 - (cS/c0S)1/3)
Shrinking core, product layer (different
form of Crank-Ginstling-Brounshtein)
Surface area of solid phase
25
• The change in the total surface
area of the solid depends
strongly on the morphology of
the particles
Cracking
2
Total surface area (m /L)
20
15
Mineral 2
Mineral 1
10
Cylinder
Sphere
5
Steadily
increasing
porosity
0
0
20
40
60
Conversion (%)
80
100
• Models based on ideal
geometries can be inadequate for
modeling non-ideal cases
• The particle morphology can be
implemented into the model
with the help of a shape factor
New methodology for general shapes
• The morphology can be flexibly implemented with the help of a shape
factor (a)
dcsolid

Reaction rate:
 kAparticles cliquid
Geometry
Shape factor x=
dt
Reaction rate:
Shape factor:
dc solid
x

 kc1particles
cliquid
dt
a
AP
R0
VP
1-x
(a)
1/a
Slab
1
1
0
Cylinder
2
½
1/2
Sphere
3
1/3
2/3
Rough,
porous particle
high value
0
1
Geometry
Shape factor
(a)
x=
1/a
1-x
Slab
1
1
0
Cylinder
2
½
1/2
Sphere
3
1/3
2/3
Rough,
porous particle
high value
0
1
Often kinetics is
closer to first order!
The roughness is
always there, σ=1
m2/g is not a perfect
sphere!
 Detailed considerations give a relation
between area
(A),
specific surface area (σ),
amount of solid
(n),
1 / a 11 / a
A


Mn
n
0
initial amount of solid(n0),
and molar mass
(M);
a=shape factor
New methodology
 The solid-liquid reaction mechanism should be considered
from chemical principles, exactly like in organic chemistry!
dcprod
dt
x
 kc1particle
f (cliquid )
Solid
contribution
Liquid
contribution
The dissolution of zink with ferric iron
ZnS(s) + Fe3+ ↔ I1
(I)
I1+ Fe3+ ↔ I2
(II)
I2 ↔ S(s) + 2 Fe2+ + Zn2+
(III)
________________________________________________
ZnS(s) + 2Fe3+ ↔ S(s) + 2 Fe2+ + Zn2+
The mechanism gave the following rate expression
k (cFeIII  cFeII cZnII / K )
r
D
2
2
The dissolution of zink with ferric iron
The reaction order is not 2/3 but clearly higher!
0.2
75°C
85°C
Fe3+ (mol/L)
0.15
95°C
0.1
0.05
0
0
25
50
75
100
125
150
Time (min)
Wrong reaction order in the kinetic model is the worst mistake!
General product layer model
General product layer model in a nutshell
S
(a  2) Dei (cLi b  cLi s )
s


R
(
c
ik k Li )  0
a2
R(1  (1  (a  2) / BiMi )(r / R) )(r / R) k 1

d 2 ci (a  1) dci
Dei ( 2 
)0
r
dr
dr
Ni  DeiCR1 a  k Li (cLi b  cLi *)
 (a  2) Dei (c Li  c Li )
b
Ni 
s
R(1  (1  (a  2) / BiMi )(r / R) a 2 )(r / R)
S
Ni A 

k 1
 ik Rk (cLi s )
S
dni

dt

dc j
 jM
dt

ik Rk A
k 1
x0 j
x
c0 j c j
1 x
r
dci  iM
x
1 x

c0 j c j r
dt
x0 j
r  f (cLiS )
Comparison of shrinking particle and
product layer model
Effect of shape factor
Particle size distribution
• If the particle size distribution deviates significantly from the Gaussian
distribution, erroneous conclusions can be drawn about the reaction mechanism
VC = standard deviation / mean particle size
Shrinking sphere
VC=1.5
VC=0
VC=0
VC=1.2
Implementing the particle size
distribution into modeling
Total surface area in reactor
m² / 100 ml
5
6M
4M
2M
4
3
2
1
0
0
20
40
60
80
% dissolved
•
Gibbsite is rough/porous and cracks during dissolution
• The surface area goes through a maximum, non-ideal behavior
100
Implementing the particle size
distribution into modeling
x
f ( x)  x
k SP 1
e 
 k (k SP )
 k 1 t
SP
(kSP )   t
e dt
0
Var( x)  kSP 2
E( x)  kSP
• The Gamma distribution is fitted to the fresh particle size distribution and
the distribution is divided into fractions
• The shape parameter (k) and the scale parameter (θ) are kept constant
Implementing the particle size
distribution into modeling
0.09
ct mt Vt


 X  1
c0 m0 V0
0.08
Frequency (counts/min)
0.07
0.06
0.05
ri ,t  ri ,0 a X
time
0.04
0.03
0.02
0.01
0
0
20
40
60
80
100
120
140
160
180
AP
a
R0
VP
Ari ,tP 
aVri ,tP
ri ,tP
Diameter (μm)
• A new radius is calculated for each fraction and each fraction is summed to
obtain the new surface area in the reactor
• The new surface area is implemented into to rate equation
The fit of the model and
sensitivity analysis
80
Concentration (g/L)
Concentration (g/L)
80
60
40
20
0
0
5
10
15
20
40
20
0
30
25
60
35
0
20
10
30
Time (min)
Time (min)
x 10 4
4.5
8000
1000
3.5
5000
900
3
800
2.5
4000
3000
700
2
600
1.5
2000
1
500
1000
0.5
400
0
2
0
3
4
5
6
7
8
shape factor
9
10
11
12
Ea (J/mol)
Obj. function
Obj. function
Obj. function
1100
4
7000
6000
40
0
0.1
0.2
0.3
k0 (1/(min m2))
0.4
0.5
300
0.8
0.9
1
1.1
1.2
1.3
x 105
Selection of the experimental system and equipment
Kinetic investigations
Structural investigations
Mass- and heat transfer studies
Ideas on the reaction mechanism including structural changes of the solid
Derivations (and simplification) of rate equations
Estimation of kinetic and mass transfer parameters
Model verification by numerical simulations and additional experiments
Conclusions
 Modeling is an important tool in developing new processes as
well as optimizing existing ones
 Solid-liquid reactions are in general more difficult to model
than homogeneous reactions
 Traditional modeling procedures have potholes, which can
severely influence the outcome
 Care should be taken in drawing the right conclusions about
the reaction mechanisms
Things to consider in modeling
 Some important factors:
1. Be sure about what you actually are measuring
2. Evaluate if the particle size distribution needs to be taken into
account (VC<0.3)
3. If the morphology is not ideal use a shape factor to describe
the change in surface area (surface area, density and
conversion measurements needed)
4. Use sensitivity analysis to see if your parameter values are well
defined
Some relevant publications

Salmi, Tapio; Grénman, Henrik; Waerna, Johan; Murzin, Dmitry Yu. Revisiting shrinking
particle and product layer models for fluid-solid reactions - From ideal surfaces to real
surfaces.Chemical Engineering and Processing 2011, 50(10), 1076-1084.

Salmi, Tapio; Grénman, Henrik; Bernas, Heidi; Wärnå, Johan; Murzin, Dmitry Yu.
Mechanistic Modelling of Kinetics and Mass Transfer for a Solid-liquid System: Leaching
of Zinc with Ferric Iron. Chemical Engineering Science 2010, 65(15), 4460-4471.

Grénman, Henrik; Salmi, Tapio; Murzin, Dmitry Yu.; Addai-Mensah, Jonas. The Dissolution
Kinetics of Gibbsite in Sodium Hydroxide at Ambient Pressure. Industrial &
Engineering Chemistry Research 2010, 49(6), 2600-2607.

Grénman, Henrik; Salmi, Tapio; Murzin, Dmitry Yu.; Addai-Mensah, Jonas. Dissolution of
Boehmite in Sodium Hydroxide at Ambient Pressure: Kinetics and Modelling.
Hydrometallurgy 2010, 102(1-4), 22-30.

Grénman, Henrik; Ingves, Malin; Wärnå, Johan; Corander, Jukka; Murzin, Dmitry Yu.; Salmi,
Tapio. Common potholes in modeling solid-liquid reactions – methods for avoiding them.
Chemical Engineering Science (2011), 66(20), 4459-4467.

Grénman, Henrik; Salmi, Tapio; Murzin, Dmitry Yu.. Solid-liquid reaction kinetics –
experimental aspects and model development. Rev Chem Eng 27 (2011): 53–77