Development and application of coupled discrete and continuum models in solid particles classfication

Development and Application of Coupled Discrete and
Continuum Models in Solid Particles Classication
(Doctoral dissertation)
Rotich Nicolus Kibet1
(B.Eng., M. Sc., PhD Chem. Eng)
Lappeenranta University of Technology
(Auditorium 2310)
09.06.2017
1
nicholas.rotich(at)gmail.com

CONTENTS
i Title Explanation
ii Industrial Signicance of Solids Classication
iii Problem Setting
a Global economic problem
b Local engineering problem
iv Problems with conventional approaches
v Hypotheses Novelty
vi Methodology
a Continuum models
b Discrete models
c Experimental verication
vii Results
viii Conclusion future focus
Rotich Nicolus Kibet2 (B.Eng., M. Sc., PhD Chem. Eng) (Lappeenranta University of Technology (Auditorium 2310))Development and Application of Coupled Discrete and Continuum Models in Solid09.06.2017 2 / 27

TITLE EXPLANATION
DISCRETE CONTINUUM

SOLID PARTICLES CLASSIFICATION
Aiming to distinguish unique particles by predetermined criteria e.g.
resilience, hardness, shape, size and density etc. Synonyms: Screening,
sieving, solid-solid separation or sorting.

INDUSTRIAL SIGNIFICANCE
Key application areas:

INDUSTRIAL SIGNIFICANCE ...
KEY APPLICATION AREAS:
Granular materials are abundant in our day-to-day life comprising of over
60% of domestic and industrial products.
i Mining: Gangue/Ore removal, grading of mineral product etc.
ii Pharmaceuticals: Sizing of raw materials and drugs
iii Agriculture: Mechanized planting, fertilizer production/application
iv Power generation: Grading of pulverized coal, forest residue etc.
v Waste management: Separation of e-waste upon crushing
vi Foods industry: Food grading, salts, sugar, cereals processing
vii Building Construction: Asphalt, sand, and gravel grading etc.

PROBLEM SETTING
GLOBAL PROBLEMS
(1) The US pharmaceutical industry reject about 5% of manufactured
products, due to problems related to solid-solid separation, each BATCH
costing $ 650,000 (INIMS, 2009)
(2) In Seed/fertilizer produce precise size classication is important in
maintaining ow over agricultural equipment e.g. Planting, fertilizer
application etc.
(3) In mining, allowing excess earth and rocks through grinding circuits
lowers the overall operating and recovery eciency.
(4) Waste processing (E-waste/municipal) is nowadays a priority sub sector
In these industries, passing of oversized or retaining undersized particles
constitutes a quality problem additional cost of re-manufacturing, e.g.
grinding (size reduction), and agglomeration (size enlargement).

PROBLEM SETTING..
LOCAL ENGINEERING DESIGN PROBLEM
i Granular materials presents a mix of properties e.g.,
take the shape of container like liquids, but unlike it,
ii It forms a heap when poured onto a at surface. Can ow
continuously or fall as single particles and therefore,
iii Granular materials in general are not fully understood8
iv Sorting in particularly is faced with problems ranging from those
aecting equipment design strategies, to those directly related
to the physicochemical properties of the solids. As such,
v today: there is no universal set of equations that describes
granular material ow - In general
8
Refer to pp 16 (dissertation)

EQUIPMENT MARKET CHALLENGE
CURRENT SITUATION
The most common design handbook is the guide developed by the
Vibrating Screens Manufacturers Association (VSMA) and distributed by
Association of Equipment Manufacturers (AEM):
i The industry is highly
commercialized/monopolized
ii Dicult to penetrate, yet
iii The design process need input
from the community (users,
academia, researchers etc.)
iv The handbook is largely based
on statistical/empirical data10
10
Can deviate by a factor of 3+, pp 23 (dissertation)

CONVENTIONAL APPROACH
FIRST ORDER RATE LAW
FORL Eq.(1) has been the basis for solid-solid separation since it's
inception. It is the simplest form of the population balance equation (PBE).
∂m(t, x)
∂t
= −km
∂m(t, x)
∂x
= −Km



k K are constants (1)
Problem with FORL: It does not take into account design information
such as screen size, dimensions, holed area, screening energy needed, and
transient motion (velocity) of material on the screen surface. It doesn't
link separation eciency to these factors.

PROBLEM WITH FORL
Rosin-Ramler: Screen function
Eq.(2) was developed by a French
mathematician Fréchet in 1927 and
rst applied by Rosin-Ramler in
1933 in PSD.
η = 1 − e
−0.693
Dp
D50
5.9
(2)

END RESULT ...

HYPOTHESIS
Breakdown of k and K into:
1. Screen and material surface
characteristics, µk/µ/ν
2. Surface area (width, W /HAF)
3. Material travel speed, ¯v
4. Material PSD (Dp/D50)
5. Mechanical/electrical energy
6. Surface inclination, θ
7. Material mass ow, ˙m

HYPOTHESES ...
Hypothesis statement: the rate of particle classication is proportional to
the rate of change in momentum and the screening open area, but
inversely proportional to the particle size
Mathematically:
∂m(t, x, ...)
∂t
= −τ
a
Dp
m (x, t)
∂v (x, t)
∂t
where
a = Eective surface/HAF
Dp = Mean particle diameter
v = Bulk ow mean velocity
(3)

METHODOLOGY
CONTINUUM MODEL
The continuum model is based on uid mechanics approximation of the
generalized transport equations.
ρ
∂v
∂t
+ v. v = − p − µ 2
.u +
−→
F
∂ρ
∂t
= − · (ρu)
(4)

QUALITATIVE DESCRIPTION
Assumptions: Granular ow is
described as turbulent, unsteady,
Newtonian uid ow, with nominal
to high densities, low viscosities, and
subsonic velocities see Eq.(5). M is
the Mach number, v ow velocity,
and vs speed of sound.
M =
v
vs
0.011 (5)

QUALITATIVE DESCRIPTION ...
Assumptions: It is
incompressible within the continuum,
but compressible around the bulk
ow regime. The volume of the
continuum changes with space, and
thus the density varies only
macroscopically, but remains nearly
constant microscopically.

SIMPLIFIED CONTINUUM MODEL
ρ
∂v
∂t
+ u
∂ρ
∂t
= − p + µ 2
u +
−→
F
ρ
∂v
∂t
+ u
∂ρ
∂t
= ¨¨¨B0
− p +¨¨
¨¨B0
µ 2
u +
−→
F
ρ
∂u
∂t
+ u
∂ρ
∂t
=
−→
F
(6)
With units of
−→
F being force per unit volume, and thus:
m
∂v
∂t
+ u
∂m
∂t
= F (7)

METHODOLOGY ...
DISCRETE MODEL
The discrete models is based on one by one particle analysis and later
generalized to cover the whole ow:

SIMULTANEOUS SOLUTION
The simultaneous solutions between the rate equation (hypothetical) and
Newton's second law for variable mass yields Eq.(8)22
:
¯v = − ln(ε)/β
β = τ
a
Dp
η = 1 − e−β¯v
G = sinθ − µk cosθ (1 + ln(1 − η))
(8)
22
β¯v of 2π gives η of 99.8 %

GRAPHICAL SOLUTIONS

GRAPHICAL SOLUTIONS ...

EXPERIMENTAL VERIFICATIONS
Experiments were conducted with mono-disperse glass beads of diameters
0.75, 1, 2, 3 mm (mean Dp 1.7 mm) at various inclination angles (5 -
20◦), frequencies (f= 0 - 20 Hz, amplitudes ∼ 2 ± 0.5 mm ). The holed
are fraction was constantly kept at 0.4031:

EXPERIMENTAL VERIFICATIONS ...
Refer to clearer image(s)27
27
Publication I pp 341

RESULTS
Eciency measurement was based on two methods: The net and gross
eciency Eq.(9).
ηn 70%
ηg 93%
(9)

CONCLUSION FUTURE FOCUS
i The study successfully tested and veried the hypothesis
ii New parameters have been introduced to the FORL that controls the
system rate eciency
iii A methodology for optimizing (above parameters) was developed
iv The study linked the operating energy to the eciency of solid
particles classication
v Future studies could focus on putting all these factors in a standalone
computer program for easy click-calculate during design phase.

All models are wrong, but some are useful- George Box
THANK YOU

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