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HUNGARIAN JOURNAL
OF INDUSTRIAL CHEMISTRY
VESZPRÉM
Vol. 31. pp. 13-21 (2003)
APPLICATION OF FOURIER TRANSFORMATION FOR WASTE
MINIMIZATION IN BATCH PLANTS. 2. PROCESS-UNITS ASSIGNMENT
N. G. VAKLIEVA-BANCHEVA*, E. G. SHOPOVA, B. B. IVANOV
Institute of Chemical Engineering – Bulgarian Academy of Sciences
“Acad. G. Bontchev” Street, Bl.103
1113 Sofia, Bulgaria
The problem for determining the minimum environmental impact for compatible products manufacturing in
multipurpose batch plants is considered in this study. It is based on the use of the Fourier transformation for
mathematical descriptions of the waste emissions from routine sources appearing into the horizon cyclically -
an approach which has been proposed in its first part [4]. The problem takes into accounts both the used
materials compositions and the constructed production routes. The formulated sets of constraints follow for
feasibility and compatibility of the chosen production routes and justify the accomplishment of the
production demands into the determined horizon. Global or Local Environmental Impact Assessments are
used as the objective function.
An example concerning simultaneous manufacturing in a dairy of two types of curds is used to illustrate
the considered problem. The aim is to determine the milkfat content in the skimmed milk used as a raw
material for both products, and plant units assignment for the respective processing tasks, at which the BOD
generated from the process is minimal for accomplishment of some production requirements in a given
horizon. Both the BOD generated due to the amount and composition of the processed milk and the one due
to inherent losses are taken into account in the formulated problem.
Keywords: Waste minimization, Fourier transformation, Multipurpose batch plants, BOD, Dairy processing
* To whom correspondence should be addressed: e-mail vaklieva@bas.bg
Introduction
Following contemporary trends to reduce the
environmental impact by developing systematic
methodologies for waste minimization in sources,
Pistikopoulos at al. have created the Minimum
Environmental Impact Methodology for continuous
and batch plants [1-3]. They have embedded the
Life Cycle Analysis principles within a design and
optimization framework and introduced appropriate
quantitative environmental impact assessments.
The latter are defined on the basis of the
introduced environmental impact indices, such as
CTAM, CTWM, SMD, GWI, POI, SODI, etc., and
determine the environmental impact of the whole
system or of a particular pollutant over the time
horizon.
Later, in the first part of the current study [4],
an approach describing the wasting from the batch
routine sources has been proposed. It is based on
the transformation of the periodic discontinuous
function of the waste mass rate of pollutants w in
the Fourier series into the time horizon and allows
for the relevant pollutant to be followed within the
horizon. The obtained continuous time function
presents a mathematical model of the waste
producing from the routine source, which appears
cyclically into the horizon during batch product
manufacturing. The model involves the general
characteristics of the batch product, such as batch
size, cycle time, and processing times as well as
accounts for the raw materials composition. It can
14
be propagated readily not only to all the routine
sources of pollutant - w at a current manufacture,
but also to those, appearing at the compatible
processing of a group and campaign of batch
products in the definite horizon. Its integration
over the time allows for the determination of the
produced waste - w from the relevant routine
sources. This permits the use of the environmental
impact assessments introduced in [1-3]. As an
illustration, the proposed approach has been applied
for production recipes analysis of examples from
dairy industry (curds processing). The aim was to
determine the raw material composition at which
the environmental impact, assessed by BOD, for
manufacturing of 1-kilogram target product was
minimal.
In order to reduce the waste from the
multipurpose batch plant, at accomplishing the
production demands for a group of compatible
products, both the raw materials compositions and
the process/units assignments must be taken into
account. They affect not only the production
characteristics, such as batch size and the number
of processed batches, but also the environmental
impact through the amount of the waste produced.
Usually, production and environmental
requirements are in conflict. The objective of this
study will be to find the best tradeoff between
them. It will be based on the application of the
approach for mathematical modeling of the
wasting from appearing cyclically routine sources
proposed in the first part. The aim will be reached
through simultaneously determining the used
materials compositions and the process/units
assignments at which the production demands will
be fulfilled into the given horizon with minimum
environmental impact.
The paper is structured as follows: A
description of the problem is presented in next
part. A mathematical formulation of the waste
minimization problem is laid out in part 3. An
application of the proposed approach on an
example from the dairy industry – simultaneous
processing of two types of curds into a given plant
is presented in part 4, while the concluding
remarks are in part 5.
Problem description
Let us consider a multipurpose batch plant
comprising P units Pp ∈ of different types.
Each unit has a volume pV , Pp ∈ and could be
connected with others in the plant. The plant
provides an opportunity for a compatible
manufacturing of I different products, Ii ∈
within a given time horizon H . iQ [kg] is the
production demand for the respective products i .
Each product i comprises ,Li processing
tasks, iLl ∈ . A set of iN key components
iNn ∈ is used for product manufacturing. The
key components are introduced into the process by
the raw and/or other supporting materials. Only
one material source for any particular key
component n is allowed. The composition of the
key components could be changed within the
technologically defined boundaries
nn )imax(x,)imin(x . Physical, chemical or
biochemical transformation takes place in each
task until the target product is obtained. The tasks
processing times ilT are supposed to be constant.
The cycle time depends on the chosen operational
mode. For the overlapping case it is
{ }ili TmaxTC = , while for the non-overlapping -
∑=
l
ili TTC .
Each processing task could be performed in
one or more appropriate plant units. A variety of
process/unit assignments exists, which results in
multiple routes for product manufacturing. Binary
matrices lp)i(ID are introduced to identify the
appropriate plant units for products i as follows:
1)i(ID lp = , for units p suitable to process task
l , and 0)i(ID lp = , for the others.
The volumes of the involved plant units and
the size factors for the respective tasks determine
the batch size of a structured production route.
Usually, the size factors are taken to be constant.
But in fact they depend on the used materials
composition. The effect of the key components
composition on batch size will be taken into
account in the problem.
Manufacturing the products is accompanied by
producing different waste types. The standard limit
values www s,w,a µµµ , for air, water and soil are
15
given for the processed pollutants w , Ww ∈ . In
principle, each task iLl ∈ of the products i could
be a potential routine source of effluent with
pollutants w . The amount of the wastes produced
over the time horizon also depends on both the key
components composition and the respective batch
sizes.
The above description brings the waste
minimization problem to determining the
composition of raw materials, solvents and other
components used in the process and the respective
production routes for each product i so that the
demands iQ could be accomplished into the
horizon H at minimum environmental impact.
Mathematical formulation of the optimization
problem
Variables and constraints
Variables
A set of continuous variables n)i(x is
introduced for each product i to account for the
change of composition of the key components n
in the relevant material sources.
A set of binary variables p)i(ζ is used to
structure the different production routes for each
product i , as follows: 1)i( p =ζ if unit p is used
for product i manufacturing, and 0)i( p =ζ ,
otherwise.
Constraints
Structural constraints. The objective of this set
of constraints is to structure feasible and
compatible production routes for products i .
Manufacturing each product i is feasible, if at
least one of the plant units, suitable for tasks l , is
assigned in the structured production route:
∏ ∑
= =
∈∀≥
iL
1l
P
1p
plp Ii,i1)i()i(ID ζ
. (1)
The structured production routes for all I
products are compatible if there are not any
common shared units.
Pp,p1)i(
I
1i
p ∈∀≤∑
=
ζ . (2)
Production constraints. This group of constraints is
introduced to account for the demands iQ to be
accomplished within the time horizon H .
Taking into account that the composition of the
key components in the used materials affects both,
batch size and environmental impact, admissible
vectors )n,i(X of the independent variable values
n)i(x should be formed for each product i ,
{ }
iNn1
)i(x,...,)i(x,...,)i(x)n,i(X = .
Ii,iNn,n i ∈∀∈∀ , (3)
subject to:
IiiNnnixixix innn ∈∀∈∀≤≤ , , ,)max()()min( .
(4)
The sizes of batches that are determined by the
plant units assigned to the processing task of
product i are:
,
))n,i(X(s
)i(.)i(ID.V
))i(),n,i(X(B
l
p
plpp
l
∑
=
ζ
ζ iNn,n ∈∀
Ii,iLl,l i ∈∀∈∀ , . (5)
The size factors l))n,i(X(s in equations (5)
are functions of the key components composition
and can be calculated from the mass balances,
according to the production recipes:
))n,i(X(Y
)i(v
))n,i(X(s lj
j
l
∑
∈= ,
{ } Ii,i,Ll,l,Llllj ii ∈∀∈∀∈∈ , iNn,n ∈∀ (6)
where:
)i(v is the volume of the input flow j in the
processing task l of product i
)),(( niXY is the yield of the target product i
presented as a function of the key components
compositions.
The batch size for each product i is limited by
the processing task with the minimum batch size:
{ } ,))i(),n,i(X(Bmin))i(),n,i(X(B lζζ = iNn,n ∈∀ ,
Ii,iLl,l i ∈∀∈∀ , . (7)
16
The number of batches being carried out in
order to manufacture the planned amounts iQ for
the products i is:
=
))i(),n,i(X(B
Q
))i(),n,i(X(NB i
ζ
ζ ,
iNn,n ∈∀ Ii,i ∈∀ . (8)
Finally, the time required to manufacture the
demand iQ for the products i :
iTC)).i(),n,i(X(NB))i(),n,i(X( ζζ =Θ , iNn,n ∈∀ ,
Ii,i ∈∀ , (9)
must be within the time horizon H :
H))i(),n,i(X( ≤ζΘ , iNn,n ∈∀ , Ii,i ∈∀ . (10)
Mathematical models of the wasting from batch
routine source accounting for the composition of
the key components and production routes
The mathematical description of the waste - w
produced from the cycle batch routine source is
analogous to that proposed in [4]. The Fourier
transformation is used. However, here it must
account for the structured particular production
routes done through the respective batch sizes, (see
equation (5)), and for the key components
composition done through both the batch size and
the mass wl))n,i(X(m of the pollutant - w ,
generated in the task l for processing of 1 kg.
target product:
( ) ( )( ) ( ) ( )[ ] ( )[
( ) ( )( ) ( ) ( )[ ] ( )]]
+−+
++−+
=
∑
tkTkTskTkTsk
tkTkTskTkTsk
Tk
TC
niXminiXB
tiniXF
ilililil
k
ilililil
il
i
wl
wl
ϕϕϕϕϕ
ϕϕϕϕϕ
ϕζζ
cossincos1cossin
sinsinsincos1cos
1
2
1
.
)),(()).(),,((2
)),(),,((
TCt0 ≤≤
Ll,l,Ww,w,Nn,n ∈∀∈∀∈∀ ,
Ii,i ∈∀ , (11)
where ϕ
π
=
TC
2
.
The mass wl))n,i(X(m of the pollutant - w ,
for products i , can be determined from the
pollutants mass balance of the production recipes
as it is proposed in [5, 6]:
,)i(C.)i(MO))n,i(X(R)i(C.)i(MI
))n,i(X(Y
1
))n,i(X(m
lj l'j
wjjwlwjjwl
−+= ∑ ∑
∈ ∈
Ii,i,Ll,l,Ww,w,Nn,n ii ∈∀∈∀∈∀∈∀ (12)
where:
)i(MI and )i(MO note the amount of
materials input into processing task l of product i
by the flows j and output by the flows;
)i(C is the composition of pollutant )i(C w is the relevant flow;
))n,i(X(R is the waste produced in the task l .
Objective function
The Local and Global Environmental Impact
Assessments introduced in [1-3] are used for the
objective function definition. The need for the
purpose relevant environmental impact indices
such as STAM, WTAM, SDM etc. can be presented
by means of the mathematical models (11)
introduced above firstly as time dependent
functions. Thus, they will account for the used
materials composition and process-unit
assignment:
∑=
l
wl
w
w tiniXFa
tiniXCTAM )),(),,((
1
)),(),,(( ζ
µ
ζ ,
∑=
l
wl
w
w tiniXFw
tiniXWTAM )),(),,((
1
)),(),,(( ζ
µ
ζ ,
∑=
l
wl
w
w tiniXFс
tiniXSMD )),(),,((
1
)),(),,(( ζ
µ
ζ ,
Ww,w ∈∀ , ,Lll,Nn,n ∈∀∈∀
Ii,i ∈∀ , TCt0 ≤≤ . (13)
The environmental impact indices are obtained
after the integration of the equations (13) over the
17
cycle time duration and multiplying by the number
of batches (equations (8)), that can be performed to
produce the planned amounts iQ for products i :
∑ ∫=
=
Θ
l
TC
wl
w
iniX
w
dttiniXF
a
iniXNB
iniXCTAM
0
))(),,((
)),(),,((
1
)).(),,((
))(),,((
ζ
µ
ζ
ζ
ζ
∑ ∫=
=
Θ
l
TC
wl
w
iniX
w
dttiniXF
w
iniXNB
iniXWTAM
0
))(),,((
)),(),,((1
1
))(),,((
))(),,((
ζ
µ
ζ
ζ
ζ
∑ ∫=
=
Θ
l
TC
wl
w
iniX
w
dttiniXF
s
iniXNB
iniXSDM
0
))(),,((
)),(),,((
1
))(),,((
))(),,((
ζ
µ
ζ
ζ
ζ
,
Ww,w ∈∀ , ,Lll,Nn,n ∈∀∈∀ Ii,i ∈∀ ,
TCt0 ≤≤ . (14)
It follows, the relevant Local Environmental
Impact Assessments with regard to the particular
pollutant - w and respectively Global
Environmental Impact Assessment, for all
produced pollutants are obtained as:
=
=
Θ
ΘΘ
Θ
))(),,((
))(),,(())(),,((
))(),,((
))(),,((
))(),,(())(),,((
))(),,((
iniX
w
iniX
w
iniX
w
iniX
w
iniXSDM
iniXCTWMiniXCTAM
iniXEI
ζ
ζζ
ζ
ζ
ζζ
ζ
Ww,w ∈∀ , Ii,i ∈∀ , TCt0 ≤≤ , (15)
∑
Θ
Θ
=
=
w
iniX
w
iniX
iniXEI
iniXGEI
))(),,((
))(),,((
))(),,((
))(),,((
ζ
ζ
ζ
ζ
.
Ww,w ∈∀ , Ii,i ∈∀ , TCt0 ≤≤ . (16)
Depending on the particular problem, the Local
Environmental Impact Assessments or the Global
one could be used as the objective functions in the
problems for minimization of the environmental
impact of the multipurpose batch plants at the
simultaneous production of a group of products.
))i(),n,i(X(
w
)i(),n,i(X
))i(),n,i(X(EIMIN ζ
ζ
ζ Θ , (17)
))i(),n,i(X(
)i(),n,i(X
))i(),n,i(X(GEIMIN
ζ
ζ
ζ
Θ
. (18)
As a result, the optimal composition of the
used raw and other supporting materials and
necessary process-units assignment for the
production tasks will be obtained.
The non-linear–objective functions (17) or
(18), equations (14), (15) (and (16) - at the
objective function (18)), the mathematical models
of the wasting from routine batch sources (11) and
eqs. (12), and as well as constraints (1)-(3) and (5)-
(10) represent the problem for environmental
impact minimization on the process/unit level. The
problem comprises the sets of the two types of
independent variables - continuos )i(x , forming
the vectors - )n,i(X equ. (3), the key components
compositions in raw materials, and binary - )i(ζ
structuring the production routes for products i ,
and constraints in the form of equalities and non-
equalities. Its result is the mixed integer non-linear
programming (MINLP) problem.
Environmental impact minimization in curds
processing
Wastewater is a common factor in the dairy
industry that generates a considerable treatment
cost. The biological oxygen demand (BOD)
measures the effluent strength of the wastewater in
terms of the amount of dissolved oxygen utilized
by microorganisms to oxide the organic
components. The BOD load depends not only on
the composition and amount of processed whole
and/or skimmed milk, but also on the inherent
losses due to spilled whey, milk coagulated and
glued on the unit’s walls, products and by-products
lost etc. Since the latter could not be avoided, it is
accepted to regulate them to the inherent levels,
which are accounted by BOD “produced” in the
relevant processes.
The example under consideration concerns
simultaneous manufacturing in a dairy of two types
of curds ( 2I = ), one with a low fat content -
0.3%, called product A; and the other with a high
fat content- 1%, - product B. The aim is to
determine the milkfat content in the skimmed milk
used as a raw material for both products, and plant
units assignment for the respective processing
tasks, in which the BOD generated from the
process is minimal for the accomplishment of the
posed production requirements in a given horizon.
18
Curds manufacture is a typical cyclic batch
process. For it, processing standard milk skimmed
into the boundaries %05.0)imin(x = and
%4.1)imax(x = is used. The key component for
both products is the milkfat content Ii),i(x ∈∀ .
The composition of standard whole milk is given
in the first part of the current study [4]. A detailed
report of the curds processing is proposed in [4],
too. The description of the processing tasks is
presented in Table 1. The composition of the target
products and values of the respective recovery
factors are presented in Table 2.
Table 1 Processing tasks description. CY1(x)* is the yield of
curds by-product and x is milkfat content in skimmed milk
Production
tasks
Task
Duration
Input/Output Fractions
Task 1
pasteurization
30 min. In. Skim-milk
Out. Pasteurized
Skim-milk
1
1
Task 2
acidification
240 min. In. Skim-milk
In. Culture
Out. Curds by
product
Out. Whey
0.88
0.12
CY1(x)*
1-CY1(x)
Task 3
draining
30 min. In. Curds by-product
Out. Curds target
product
Out. Drained Whey
1
0.9
0.1
Table 2 The product compositions and values of the recovery
factors
Composition of the curds
target products
Values
of the recovery factors
Moisture% FC% CC% SC% RS RC RF
A 80 0.3 11.3 20 1.724 0.96 0.075
B 81.58 1.009 12.28 18.42 1.386 0.96 0.231
The Van Slyke equation is used for yield
calculation [7]:
[ ]
%SC
RS.))i(x%(MC.RC)i(x.RF
))i(x(CY
+
= , Ii ∈∀ (19)
where:
))i(x%(MC is the casein content in the
standardized skimmilk;
RS,RF,RC are the recovery factors for casein,
milkfat and other solids different from them;
%SC is the solids content in the target
products.
Since both products involve identical processing
tasks, identical relationships determined on the
base of the information given in Table 1 are used
for relating them to size factors
l))i(x(s :
=
1.1
))i(x(CY
1
))i(x(CY
88.0
))i(x(s
Ii,i ∈∀ . (20)
The manufacture of products A and B is
carried out simultaneously in a plant comprising
the apparatuses listed in Table 3. The suitable plant
units for the processing tasks of both products are
identified by the introduction of the following
binary matrices:
Table 3 Plant data
Type Pasteurizator Vat reactor Drainer
No 1 2 3 4 5 6 7 8 9 10 11
[m3] 300 250 150 100 300 400 250 80 60 60 100
2,1 '
1 1 1 1 0 0 0 0 0 0 0
0 0 0 0 1 1 1 0 0 0 0
0 0 0 0 0 0 0 1 1 1 1
)( =
= iiID . (21)
The cycle overlapping operational mode is
applied for the products manufacturing, where
[h] 4TmaxTC li == .
Two types of production demands named Q-I
and Q-II are considered for performing in two
different horizons H-I and H-II, see Table 4:
Table 4 Production demands
Products Q-I
[kg]
Q-II
[kg]
H-I
[h]
Q-I
[kg]
Q-II
[kg]I
H-II
[h]
A 5500 7000 5500 7000
B 6000 7000
360
6000 7000
400
Each cessing task from the curds production
generates the BOD due to:
I) The amount and composition of the used
skimmed milk. The BOD load of 1 kilogram
skimmed standard whole milk is determined as
follows:
))(%(.69.0
))(%(.031.1)(.89.0))((
ixML
ixMPixixBODM
+
++= , (22)
where:
))i(x%(MP and ))i(x%(ML are casein and
lactose contents presented as a function of the
milkfat content; and
19
II) Associated to the tasks inherent losses [8, 9]:
Task 1 – pasteurization. The pollutant
processed is due to the milk coagulated and glued
on the pasteurizer’s walls. The “generated” BOD
depends on the amount of the pasteurized milk.
The BOD load of 1-kilogram pasteurized milk is:
milkdpasteurizekg
Okg
10.5.1BOD 23P
−= . (23)
Task 2 – acidification. The pollution results
entirely from spilled whey. The inherent leaks are
WL%=1.6% from the processed whey mass. The
BOD load of 1-kilogram acid whey is:
wheyacidkg
Okg
10.32BOD 23W
−= . (24)
Task 3 – draining. The polluting is due to:
i) Draining and discharging of the whey
remained in the curds. The BOD load of 1-
kilogram acid whey is the same as in task
2; and
ii) Inherent losses of target product gluing on
the drainer’s wall. They depend on the
curds fat content (FC%) -
%FC.0017.0%CL = , The BOD load of
1-kilogram curds depends on the yield and
the BOD of used skimmilk:
cheesekg
Okg
))i(x(BOD)).i(x(CY))i(x(BOD 2MC = . (25)
The mass
wl))n,i(X(m of the pollutant - w , for
both products i , is determined from the pollutants
mass balance by using the data presented in Table
1 and equations (23)-(25):
i,l,w,
%CL.100
9.0
1.0
%WL.
))i(x(CY
9.0
1.0
1))i(x(CY1
0
00
))i(x(CY
88.0
))i(x(m l,w ∀∀∀
+−
=
.
(26)
The above data and the obtained relations,
equations (20) and (26), are completely sufficient
to formulate the posed optimization problem,
according to equations (1)-(12).
As an optimization criterion the “generated”
Global BOD in the simultaneous manufacturing of
the products A and B is used. It is obtained on the
base of the Local BODs of the pollutants due to the
milk processed, whey spilled and curds lost, from
the respective processing tasks – pasteurization,
acidification and draining:
∑ ∫
=
Θ
=
=
3
1 0
))(),((
)),(())(()).(),((
))(),((
l
TC
wlw
iix
w
dttixFixBODiixn
iixBOD
ζ
ζ
ζ
,
Ii,i ∈∀ , 3,2,1w = . (27)
where:
=
))i(x(BOD
BOD
BOD
))i(x(BOD
C
W
P
w
.
Taking into account equation (27) the objective
function - Global Biological Oxygen Demand -
GBOD, presented as a function of milkfat content -
)i(x for products A and B and structured
production routes for them by means of vectors
)i(ζ is:
∑∑
= =
=
3
1w
2
1i
))i(),i(x(
w
)i(),i(x
))i(),i(x(BODGBODMIN
ζ
ζ
ζ
Θ
.(28)
The problem formulated above in the
optimization criteria (28) is the environmental
impact model for the simultaneous manufacturing
of products A and B in the dairy. A– an outcome
of its solution the values of the independent
variables - milkfat in the skimmilk used for both
products and the assigned plant units for the
respective tasks are obtained in a way to minimize
the GBOD for the plant. These data are listed in the
Table 5.
Table 5 Optimal values of GBOD and the values of the
variables at which they are obtained
Pro-
duction
demands
H,
[h]
GBOD ,
kg O2
Pro-
ducts
x(i),
%
B(i),
[kg]
NB(i) Units
assigned
Q-I H-I 146.943 A 0.633 61.798 89 1, 7, 11
B 1.071 68.966 87 2, 3, 4, 5, 8
Q-II H-I 178.096 A 0.93 112.904 62 2, 3, 5, 7, 8, 9
B 1.131 78.683 89 1, 6, 10, 11
Q-I H-II 146.943 A 0.633 61.798 89 1, 7, 11
B 1.071 68.966 87 2, 3, 4, 5, 8
Q-II H-II 178.058 A 1.055 70.707 99 2, 5, 9, 11
B 1.079 104.482 67 1, 4, 6, 7, 8, 10
The batch sizes and number of batches subject
to processing into the horizons H-I and H-II are
shown in the same table. In Table 5 it could be
seen that for the production demands Q-I in both
horizons, equal values of GBOD are obtained,
while for the demands Q-II, which are more
strained, the optimal values of GBOD are diverse
and are reached at different values of the
independent variables. In real practice, the
20
different milkfat content effects on the raw
material consumption are what could be taken into
account. But in the case considered here it is not
accounted. For illustration this affect is shown in
Table 6.
Table 6 Raw material consumption for processing demand Q-
II in horizons H-I and H-II
Pro-
duction
demand
Pro-
ducts
Amount
,
[kg]
x(i),
%
Skimmed
milk
con-
sumption
in H-I [kg]
x(i),
%
Skimmed
milk
con-
sumption
in H-II [kg]
Q-II, A 7000 0.93 2.818.104 1.055 2.812.104
B 7000 1.131 3.034.104 1.069 3.046.104
Finally, the Local BOD values for the
processed pollutants are listed in Table 7 to
illustrate the weight of the inherent losses into the
“generated” GBOD. They constitute approximately
28.7% ofOD generated in accomplishing
production demands Q-I and Q-II.
Table 7 Values of Local BODs for the pollutants from
the tasks for demands Q-I, and Q- II accomplished in
horizons H-I and H-II
Products
A and B, kg
Routine
source
BOD [kg O2]
Pasteurized milk Whey Curds
Q-I, H-I and H-II
xÀ=0.633 %
xÂ=1.071 %
Task 1 63.822
Task 2 18.213
Task 3 40.889 24.019
Q-II, H-I [h]
xÀ =0.93 %
x =1.131 %
Task 1 77.251
Task 2 21.998
Task 3 49.788 29.059
Q-II, H-II [h]
xÀ =1.055 %
x =1.079 %
Task 1 77.326
Task 2 22.028
Task 3 49.783 28.925
Conclusions
The general problem for determining the minimum
environmental impact for compatible products
manufacturing in multipurpose batch plants is
considered in this study. It presents an evolving of
the process/units assignment level, proposed in the
first part [4] approach, based on the Fourier
transformation use, for mathematical descriptions
of the waste emissions from routine sources
appearing into the horizon cyclically. The problem
takes into accounts both the used materials
compositions and the constructed production
routes, which are set as independent variables. The
formulated sets of constraints follow for feasibility
and compatibility of the chosen production routes
and justify the accomplishment of the production
demands into the determined horizon. Global or
Local Environmental Impact Assessments are used
as the objective function, and for their definition
the mathematical descriptions of pollutants from
cyclic batch routine sources are used.
An example from the dairy industry is
considered to illustrate the possibilities of the
proposed system oriented approach for modeling
the environmental impact of the multipurpose
batch plants. Simultaneous processing of two types
of curds - low fat and high fat is regarded. The
biological oxygen demand is used to assess the
dairy environmental impact. Both the BOD
generated due to the amount and composition of
the processed milk and the one due to inherent
losses, are taken into account in the formulated
problem.
The most appropriate milkfat content of the
skimmed milk used in products manufacturing and
respective process-units assignments (production
routes) is determined to result in minimum values
of the “processed” Global BOD from the entire
plant. The considerable contribution of inherent
losses into the GBOD is illustrated too.
Acknowledgement
This study is conducted with the financial support
of the Bulgarian National Science Fund under the
Contract X-1108/01.
21
SYMBOLS
B – batch sizes [kg];
C – waste mass concentration [kg/kg]
%,CC – casein content in the curds;
CL% – inherent loss of curds;
%FC – fat content in the curds;
H – time horizon [h] ;
H-I, H-II – names of horizons used in the example;
I – number of products;
ID – units/tasks identification matrix
L – number of production tasks;
%MC – casein content in the skim-milk;
MI – amount of material input to the task [kg];
%ML –lactose content in the skim-milk;
MO – amount of material output from the task [kg];
%MP –protein content in the skim-milk;
N – number of key components in used materials
NB – number of batches;
P – number of plant units
Q – production demand [kg];
Q-I, Q-II – names of production demands used in
the example;
R – waste processed in the task [kg];
RC – recovery factor for casein;
RF – recovery factor for milkfat;
RS – solids recovery factor;
%SC – solids content in curds;
T – processing time [h];
TC – cycle time [h];
Ts – starting time of task with regard to the cycle
beginning;
V – apparatus volume [m3];
WL% – inherent loss of whey;
X – vector of the key components;
m – mass of waste processed from the production
task per 1 kilogram target product [kg/kg];
s – size factor [m3/kg];
t – time [h];
x –particular key component composition in used
materials continuos variables, (milkfat content
for the example)
SUBSCRIPTS
i – product; j
j, j’ – input output flows of the task;
k – a series order;
l – production task;
n – key component;
p – plant unit;
w – waste
GREEK LETTERS
µ – the standard limit value for the pollutant;
ν – volume of input to the task flow [m3];
ζ – binary variables presenting the process-units
assignments
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