MATHEMATICAL MODELING
OF PROCESSES
IN FOOD INDUSTRIES
(Modellistica matematica
dei processi
dell’industria alimentare)
Prof. Michele MICCIO
Prof. Gianpiero PATARO
a.a. 2013-14
Course Syllabus
General
Classification of
models
Modeling
2
Course Syllabus
Mathematical Modeling
General
Classification of
Mathematical
Models
Examples of
mathematical models
of interest for Food
Engineering
3
Course Syllabus
Mathematical Modeling
Solution of
Mathematical
Models
Numerical Methods
for parabolic PDEs
•Finite differences
•Finite elements
•Collocation Methods
4
Course Syllabus
Optimization
Mathematical
modeling and
optimization
• Intro to Optimization
• General definitions
Linear Programming (LP)
• theory
• examples
5
Course Syllabus
Optimization
Mathematical
modeling and
optimization
Linear Programming (LP)
•The graphical method
•The simplex algoritm
6
Objectives of the Course
To provide the basic knowledge, methods
and software instruments in order to:
1)discriminate the complexity of the systems
of process engineering and meet the
possibilities of abstract representation;
2)classify the models, particularly the ones
related to processes;
3)handle the Matlab® software to solve some
simple models and understand the results;
4)numerically solve the parabolic partial
differential equations;
5)understand and solve Linear Programming
problems.
7
Recommended readings
and
Study materials
Textbook
Himmelblau D.M. e Bischoff K.B., “Process
Analysis and Simulation”, John Wiley & Sons Inc.,
1967 (Collocazione: 660.281 HIM 1)
Consultation book
Snieder R., “A Guided Tour of Mathematical
Methods For the Physical Sciences”, 2nd Edition,
Cambridge University Press, ISBN-13:
9780521834926, ISBN-10: 0521834929, 2004
(Collocazione: 515 SNI - Inv: 19569 Ing. Bcode:
00118373 )
Other
• Teaching aids provided by the lecturer
• Lecture slides, course handouts and past examination
texts can be retrieved from the web:
http://comet.eng.unipr.it/~miccio
8
Assessment method
software-based test
½ score
oral colloquium
½ score
9
WHAT WILL STUDENTS
LEARN?
WHAT CAN STUDENTS
DO FOR THESIS?
10
Salami Simulation
and
Optimal Operation
of a ripening chamber
11
Example
Stazione Sperimentale per l’Industria
delle Conserve Alimentari (Parma/Angri)
Example
Stazione Sperimentale per l’Industria
delle Conserve Alimentari (Parma/Angri)
progetto di
Ricerca e Sviluppo
“Safemeat” (PON01_1409)
TITLE
Process and product innovations
aimed at increasing food safety
and at diversifying pork-based
products (SAFEMEAT)
OR 2.7
Simulazione, ottimizzazione e
controllo automatico delle celle di
stagionatura nella preparazione dei
prodotti carnei stagionati innovativi
(DICA-UniSA, SSICA, Dodaro SpA)
14
Optimization study
for salami ripening
OPTIMIZATION MODEL
Aim: Drying salamis to a given final moisture
content in the minimum process time.
Initial salami data: All known.
Objective Function:
C = c1*Wbatch*t + c2*Wbatch /t
min!
where:
C [€] is the overall cost for the production of
an industrial salami batch
c1 [€/(kg day)] and c2 [(€ day)/kg] are cost
coefficients
15
NOTE: C(t) is a non-linear function!
Optimization study
for salami ripening
OPTIMIZATION MODEL
Total cost:
optimum conditions at the intersection of
two mechanisms
16
Optimization study
for salami ripening
Independent variable (decision variable):
t [days] is the overall industrial ripening time
for a salami batch
t is a function of the main Operating
Variables of the ripening chamber through a
suitable math model of salami drying
N
t   ti
i 1
where ti is the process time in the i-th phase
or step in industrial ripening.
17
Optimization study
for salami ripening
Operating Variables:
Air Velocity ( vair )
Air Temperature ( Tair )
Relative Humidity of Air ( RHair )
No. of phases or process steps ( N ) in industrial
ripening
NOTE: As an initial trial, N may fixed, e.g., N=2
Constraints
Air Velocity:
natural convection →
forced convection
Air Temperature:
→
vair = 0
vair,low < vair < vair,up
Tair,low < Tair < Tair,up
Relative Humidity of Air: RHair,low < RHair < RHair,up
18
Fluidized Systems.
Application of
Fluidization
19
Typical fluidized bed systems
Gas
Gas and entrained solids
Solids Feed
Disengaging Space
(may also contain a
Freeboard
Dust
Separator
cyclone separator)
Fluidized Bed
Bed depth
Dust
Gas in
Solids Discharge
Windbox or plenum chamber
Gas distributor or constriction plate
20
Video
of a lab-scale fluidized bed
videos_katia_ing_02.wmv

http://www.fluidizacao.com.br/ing/home.php
21
Animation
of a Fluidized (bubbling) bed
borb_med.swf

http://www.fluidizacao.com.br/ing/home.php
22
Animation
of Liquid-Solid Fluidization
FBRMov.avi
23
Gas-Fluidized bed:
“bubbling” bed phenomenology
24
Fluidized bed dryer
of “bubbling” type
25
Fluidized bed dryer:
an example (1)
26
Fluidized bed dryer:
an example (2)
27
Fluidized bed features:
Liquid-like behavior
 Kunii and Levenspiel, Fluidization Engineering (1991)
28
Fluidized bed features:
Liquid-like behavior
from Galdo’s project work (2008)
29
Fluidized-bed systems
• When a fluid flows upward through a bed of
solids, beyond a certain fluid velocity
(minimum fluidization velocity) the solids
become suspended. The suspended solids:



have many of the properties of a fluid,
seek their own level (“bed height”),
assume the shape of the containing vessel.
• Bed height typically varies between 0.3m and
15m.
• Particle sizes vary between 1 mm and 6 cm.
Very small particles can agglomerate. Particle
sizes between 10 mm and 150 mm typically
result in the best fluidization and the least
formation of large bubbles. Addition of finer
size particles to a bed with coarse particles
usually improves fluidization.
• Superficial gas velocity = Q/S (based on cross
sectional area of empty bed) typically ranges
from 0.15 m/s to 6 m/s.
30
Fluidized bed uses
• Fluidized beds are generally used for gas-solid
contacting in process industry (chemical, food,
petroleum, power production, etc.). Typical
uses include:
 Chemical reactions:
• Catalytic reactions (e.g., hydrocarbon cracking)
• Non-catalytic reactions (both homogeneous and
heterogeneous)

 biomass gasification
Physical contacting:
• Heat transfer: to and from fluidized bed; between
gases and solids; temperature control; between
points in bed.
• Solids mixing.
• Gas mixing.
• Drying (solids or gases).
• Size enlargement or reduction.
• Classification (removal of fines from gas or fines
from solids).
• Adsorption-desorption.
• Heat treatment.
31
• Coating.
Gas-solid Fluidization:
basic calculations
interactive webpage
http://asp.dica.unisa.it/MCS/miccio/flui
dizzazione/fluidizzazione.asp
32
ERGUN sw
for fluidized bed design
installed in the PC lab No. 134
http://www.utc.fr/ergun/
33
EXTRA
34
PARABOLIC PDE
SOLVER
A project by
Ugo AVAGLIANO and Caterina SOMMA