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ISSN 00405795, Theoretical Foundations of Chemical Engineering, 2012, Vol. 46, No. 5, pp. 437–445. © Pleiades Publishing, Ltd., 2012.
Original Russian Text © D.S. Dvoretskii, S.I. Dvoretskii, G.M. Ostrovskii, B.B. Polyakov, 2012, published in Teoreticheskie Osnovy Khimicheskoi Tekhnologii, 2012, Vol. 46,
No. 5, pp. 501–510.
A New Approach to the Optimal Design of Industrial
ChemicalEngineering Apparatuses
D. S. Dvoretskiia,*, S. I. Dvoretskiia, G. M. Ostrovskiib, and B. B. Polyakova
a
b
Tambov State Technical University, ul. Sovetskaya 106, Tambov, 392000 Russia
Karpov Institute of Physical Chemistry, per. Obukha 31/12, Moscow, 105064 Russia
* email: dvoretsky@tambov.ru
Received December 21, 2011
Abstract—An algorithm is described for the twostage problem of the optimal design of industrial chemical
engineering apparatuses based on a new approach, in which uncertainty in the mathematical description
coefficients and process parameters is taken into account in the formulation of a designing problem. A typical
feature of twostage optimal design problems is the possibility of adjusting the regime (control) variables of a
control system depending on the refinement (measurement) of uncertain parameters at the operating stage
of an industrial apparatus. An algorithm for solving the twostage problem of the optimal design of engineer
ing systems has been developed, and its efficiency is exemplified by industrial chemicalengineering appara
tuses; a turbulent tube reactor of fine organic synthesis, an adsorption oxygen concentrator, and a press mold
for the hightemperature synthesis of hardalloy materials.
DOI: 10.1134/S0040579512040112
INTRODUCTION
When designing industrial chemicalengineering
apparatuses, we always encounter uncertainties of two
kinds. Some of them, such as the parameters of raw
materials and the external temperature, may change
during their operation, remaining within a certain
range. It is impossible to specify their unique value in
principle. The other may be virtually constant for the
given industrial apparatus, but their values are known
within the accuracy of a certain range, for example,
some coefficients in kinetic and heat and mass transfer
equations. To take into account the uncertainties in
the mathematical description of an industrial appara
tus, it is sufficient to distinguish them in the depen
dences for the target function (optimality criterion)
F and the constraint functions gj of the optimal design
problem, assuming that F = F(d, z, ξ), gj = gj(d, z, ξ),
j = 1, …, m, where ξ is the vector of uncertain param
eters taking any values within a specified area Ξ, which
is usually considered to be rectangular as follows: Ξ =
{ξ: ξL ≤ ξ ≤ ξU}.
In this case, the solution of an optimal design prob
lem with respect to the criterion F = F(d, z, ξ) using
the constraint functions gj = gj(d, z, ξ), j = 1, …, m
proves to be uncertain and depends on the value that
has been taken by the vector ξ. The traditional way of
overcoming the given difficulty consists of the follow
ing. The vector of uncertain parameters is assigned a
certain nominal value ξ = ξNom, and a designing prob
lem is solved at nominal ξNom to obtain the vector of
structural parameters dNom for a specified type of
equipment implementation. Thus, using the available
knowledge on a designed object and intuition, so
called margin coefficients ki (ki > 1) are introduced,
and it is assumed during the design stage that di =
ki diNom, where di is the ith component of the vector d
and i = 1, …, n (reactor length and diameter, heat
transfer surface area in a heat exchanger, tray number
in a distillation column, etc.).
The disadvantages of this approach are obvious,
since it does not assure either the optimality of the
obtained solution or the fulfillment of all constraints
during the operation of the industrial apparatus. If the
margin coefficients are low, the constraints will be vio
lated and, if they are very high, the design will not be
economical.
The approach in which the uncertainty in the
mathematical description coefficients and process
parameters are taken into account in the formulation
of the optimal design problem is much more correct
and scientifically substantiated.
FORMULATION OF TWOSTAGE PROBLEM
OF OPTIMAL DESIGN OF CHEMICAL
ENGINEERING APPARATUSES
Traditionally, the problem of the technical imple
mentation of chemicalengineering processes is for
mulated as the nonlinear programming problem
437
min F (d, z, ξ)
(1)
d, z
y = Ψ(d, z, ξ);
(2)
g j (d, z, ξ) ≡ y j,giv − y j ≤ 0, j = 1,..., m,
(3)
438
DVORETSKII et al.
where F(⋅) is the optimal design criterion; y, d, z, and
ξ are the vectors of outlet, structural, and regime (con
trol) and uncertain variables of a designed object,
respectively; y = Ψ(d, z, ξ) is the operator of the math
ematical model of the chemicalengineering appara
tus; yj, giv is the limit admissible value of the jth outlet
variable of the chemicalengineering apparatus; and
gj(d, z, ξ) ≤ 0, j = 1, …, m are the constraint functions.
Let ξ belong to the area Ξ, i.e., ξ ∈ Ξ. Let us rewrite
problem (1)–(3) as
min u
(4)
d, z,u
F (d, z, ξ) ≤ u
(5)
g j (d, z(ξ), ξ) ≤ 0, j = 1,…, m.
(6)
At constant ξ, these two formulations are equivalent.
However, when uncertain parameters are taken into
consideration, the advantage of formulation (4)–(6) is
that the optimal design criterion F = F(d, z, ξ) of the
initial problem (1)–(3) is taken into account in the
same manner as the other constraints.
When an optimal design problem is formulated
under the uncertainty of initial information, it is nec
essary to specify the types of the target function (opti
mality criterion) and constraints. This is based on the
concept of two stages in the life cycle of an industrial
apparatus, i.e., the design stage and operating stage. At
the operating stage, the following cases are possible:
(1) All uncertain parameters can be precisely deter
mined at each time moment (either by direct measure
ment or by solving the inverse problem based on infor
mation obtained from measurements).
(2) The area of uncertain parameters at the operat
ing stage is the same as that at the design stage.
(3) At the operating stage, some of the parameters
ξi can be precisely determined, while others have the
same interval as those at the design stage.
(4) At the operating stage, all parameters ξi contain
uncertainty, but their uncertainty intervals are smaller
than the corresponding intervals at the design stage.
The constraints of an optimal design problem may
be strict (unconditional) and mild (probabilistic).
Strict constraints must not be violated under any con
dition. Mild constraints may be met with a specified
probability. Most of the real problems pertain to the
case when some constraints are strict and others are
mild. For example, the safety constraints of an indus
trial apparatus as classified as strict, and its productiv
ity and selectivity constraints may be categorized as
mild.
Let us consider the twostage problem of the opti
mal design of industrial chemicalengineering appara
tuses under interval uncertainty. A typical feature of
the twostage problems of the optimal design of indus
trial apparatuses is the possibility of adjusting the
regime (control) variables z at the stage of their opera
tion depending on the refinement of the vector of
uncertain parameters ξ; i.e., the control variables z are
multidimensional functions z = z(ξ).
Let there be a mathematical model of the steady
state operation of an industrial apparatus y = Ψ(d, z,
ξ), where y is the vector of the outlet variables of the
designed object; the constraints with indices j = 0, j ∈
J1 = {1, 2, …, m1} are mild (probabilistic); and the con
straints with indices j ∈ J2 = {m1 + 1, m1 + 2, …, m} are
strict (unconditional).
The twostage problem of the optimal design of
industrial apparatuses in statics is formulated as fol
lows: it is required to determine the vectors d* and vec
tor functions z* that provide an extremum of the target
function F(d, z, ξ) and the fulfillment of mild (proba
bilistic) and strict (unconditional) constraints inde
pendently of the change in the vector of uncertain
parameters ξ within a certain specified area Ξ. The
mathematical formulation of this problem can be writ
ten as follows:
F * = min u
(7)
Pr { g 0(d, z(ξ), ξ) = F (d, z(ξ), ξ) ≤ u} ≥ ρ0;
(8)
Pr {g j (d, z(ξ), ξ) ≤ 0} ≥ ρ j , j ∈ J 1;
(9)
χ1(d) = max min max g j (d, z, ξ) ≤ 0.
(10)
d,u,z(ξ)
ξ∈Ξ
z
j∈J 2
In problems (7)–(10), u is a scalar variable (ana
logue of structural variables); Pr{⋅} is the probability of
meeting the constraint {⋅}; g0 and gj are the constraint
functions; g0(d, z(ξ), ξ) = F(d, z(ξ), ξ) is the optimal
design criterion; gj(d, z, ξ) ≡ yj, giv – yj ≤ 0, j = 1, …, m is
the function of constraints; yj = Ψ(d, z, ξ), j = 1, …, m
and ρ0 and ρj are the specified probabilities of meeting
the constraints; and χ1(d) is the flexibility function of
the apparatus.
ALGORITHM OF TWOSTAGE
OPTIMIZATION OF CHEMICAL
ENGINEERING SYSTEMS
Let us introduce the notations
⎧g j (d, z, ξ) − u, j = 0;
g j (d, u, z, ξ) = ⎨
⎩g j (d, z, ξ), j ∈ J 1;
and the set S(k) = {ξi: i ∈ I (k)}, which accumulates the
points ξ (i ∈ I (k)) at which constraints (8)–(10) are
violated, and the sets S1(k) and S 2(k) will accumulate the
points at which strict and mild constraints are vio
lated, respectively. Moreover, we shall use the auxiliary
nonlinear programming problem (A) as follows:
F * = mini u
d,u,z
g j (d, u, z , ξ ) ≤ 0,
i
i
g j (d, z , ξ ) ≤ 0,
i
i
j ∈ J 1, i ∈ I (k);
(A)
j ∈ J 2, i ∈ I .
JOURNAL OF CTHEORETICAL FOUNDATIONS OF CHEMICAL ENGINEERING Vol. 46
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A NEW APPROACH TO THE OPTIMAL DESIGN
The solution of problem (A) consists of finding the
minimum of the scalar variable u under all constraints
of this problem within a specified set of points ξi, i ∈
I(k).
The algorithm consists of the following steps.
Step 1. At the first step, the numbers of iterations
and critical points are taken to be equal to k = 1 and
n = 0, respectively. The initial set S(k – 1) is selected
from the condition of the best approximation of
functions z(ξ). The initial estimates d(k – 1), u(k – 1),
and zi, (k – 1) are specified.
Step 2. Auxiliary problem A is solved, and d (k), u(k),
and z(k) are its solutions.
Step 3. Using the outer approximation algo
rithm [1],
(
χ1(d ) = max min max g j d , z, ξ
(k )
ξ∈Ξ
z
j∈J 2
(k )
)
χ1(d , ξ ) ≤ 0
(k)
(k)
(12)
is verified. If condition (12) is true, we pass to Step 4;
otherwise, we pass to Step 5.
Step 4. The set S1(k), of points at which constraints
(12) are violated is complemented by the following:
S1(k) = S1(k −1) ∪ ξ (k), where ξ (k) : χ1(d (k)) > 0;
}
Pr g j (d , z(ξ), ξ) ≤ 0 ≥ ρ j , j ∈ J 1.
(k)
j∈J 1
(
)
(14)
where J 1 = (0, 1, 2, …, m1). The solution of problem
(14) is denoted as ξ (k ) , and the set S 2(k) of points at
which mild constraints are violated is complemented
as follows:
S 2(k) = S 2(k −1) ∪ ξ (k); where χ 2(d (k)) > 0;
I 2(k) = I 2(k −1) ∪ (n + 1); n := n + 1.
i
∑ (ξ
−ξ j ) ,
i 2
j
j =1
i∈I
(k)
=
(k)
I1
∪ I 2 , nξ = dim ξ,
(k )
i
(l )
i
(i)
(iˆ)
r (ξ, ξ ) ⇒ iˆ = arg min
r (ξ, ξ ) ⇒ zˆ = z .
ξ = min
(k )
(k )
i∈I
i∈I
Actually, the piecewise constant approximation of
the functions, z = z(ξ) is used in the described proce
dure.
At Step 6, the inequality χ2(d (k)) ≤ 0 means that
mild constraints are fulfilled with a probability of 1.
Therefore, if constraint (13) is not met, it is certain
that χ2(d (k)) > 0 and, consequently, we obtain the point
(13)
χ 2(d (k)) = max min max g j d (k), u (k), z, ξ ,
z
i
(k)
are verified.
At the given step, we have no functions z = z(ξ), and
know only their values at discrete points ξi, i ∈ I (k). For
this reason, we shall use these points to approximate
the functions z = z(ξ).
If condition (12) is true, and condition (13) is false,
we pass to Step 6.
If conditions (13) and (14) are true, the solution
d* = d (k) and z* = zi, (k) is found.
Step 6. Using the outer approximation algorithm
[1], we calculate
ξ∈Ξ
nξ
r (ξ, ξ ) =
ξ , at which mild constraints are violated.
In the case of using the additional variable u, we
scale search variables so as to make the ranges of their
variation nearly equal.
(k )
(k −1)
I 1 = I 1 ∪ (n + 1); n := n + 1.
Step 5. Mild (probabilistic) constraints
{
Step 7. We form the sets S (k) = S1(k) ∪ S 2(k) and
I(k) = I 1(k) ∪ I 2(k), and assume k to be as follows: = k + 1,
and pass to Step 2.
Let us give some explanation for this algorithm.
At Step 5, we perform multivariate interpolation
through the known discrete points ξi, zi, i ∈ I(k) using
the functions z = z(ξ). This may be done by means of
multivariate cubic splines or the rough approximation
procedure, which consists of the following. When
implementing a simulation model, we accept z(ξ),
which corresponds to each obtained random ξ equal to
zl(ξl), l ∈ I (k), which corresponds to the point ξi, which
is closest to the point ξ, i.e.,
(11)
is calculated. The solution of problem (11) is denoted
as ξ (k ) and the condition
439
EXAMPLES OF OPTIMAL DESIGN
OF INDUSTRIAL CHEMICAL
ENGINEERING APPARATUSES
The efficiency of the proposed algorithm will be
demonstrated with some examples of the optimal
design of a number of industrial chemicalengineering
apparatuses, including a turbulent tube reactor of fine
organic synthesis, a shortcycle adsorption unit, and a
press mold for the hightemperature synthesis of hard
alloy materials.
The mathematical model y = Ψ(d, z, ξ) of the stat
ics of a nonlinear process of fine organic synthesis,
namely, the diazotization of aromatic amines in a tur
bulent tube reactor allows us to calculate the following
variables y of the diazotization reactor outlet: the pro
ductivity Q; the concentrations c(out) = (cD, cNA, cχ, cσ)
of diazocompound, nitric acid, diazotars, and nitrous
gases; the flow rates of the liquid and solid phases of a
diazosolution suspension G (out) = (G1(out), G s(out)); and
the amounts of the solid phase of an amine Πη, diazo
tars Πξ, and nitrous gases Πσ in a diazosolution at the
outlet of the diazotization reactor [2], where d, z, and
THEORETICAL FOUNDATIONS OF CHEMICAL ENGINEERING Vol. 46
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DVORETSKII et al.
ξ are the vectors of structural regime (control) and
uncertain variables of the diazotization reactor.
Let us formulate the technical specifications for
designing a turbulent tube reactor for the diazotization
of aromatic amines with diffuser–confuser flow turbu
lization devices (see figure).
At a specified reactor productivity (with respect to
a diazocompound) of Q = 1000 t/yr, it is necessary to
provide the following outlet variables: the break
through of an aromatic amine Πη = G s(out) G s(0) × 100%,
as well as concentrations of diazotars of Πχ = cχ(out) ×
(out)
×
G1(out)) × 100% and nitrous gases Πσ = cNA
(0)
(out)
(out )
(0)
(0)
G1
(c N × G N ) × 100%, where [cA ]s and cχ are the
inlet concentrations of aromatic amine in the solid
phase and the outlet concentration of diazotars, and
( 0)
(0)
cN and G N are the inlet concentration and flow rate of
sodium nitrite, respectively) below their limit admissi
ˆ η = 0.25%, Π
ˆ χ = 0.9%, Π
ˆ σ = 0.5%, i.e.,
ble values Π
ˆ η, Π χ ≤ Π
ˆ χ , and Π σ ≤ Π
ˆ σ . These require
Πη ≤ Π
ments must be met in the interval of the uncertainty of
some parameters of the process and coefficients of the
mathematical model of diazotization, namely, the
inlet concentration of the solid phase of an amine
[cA(0)]s = 370.0 (±4%) mol/m3 and the kinetic coeffi
cient in the equation of dissolution of the solid phase
of an aromatic amine А = 5.4 × 105(±5%).
The optimal design problem consists of determin
ing the structural parameters d (tube reactor diameter
D and length L, number of diffuser–confuser devises
mdc and their mounting sites lj, j = 1,2, …), as well as
nd the regime (control) variables z (inlet temperature
of an aromatic amine suspension Т(0) and the distribu
tion of the sodium nitrite flow rate GN(i), i = 1,2, …, p)
along the reactor length) that ensures the minimum
reduced expenditures C(d, z, ξ) in the creation of a
reactor and its operability independently of the ran
dom variations in the vector of uncertain parameters ξ
in the area Ξ. Constraints may be specified in strict or
mild (probabilistic) forms. As a rule, strict constraints
include the specified quality requirements for
obtained products and the technical regulation
requirements for the production’s explosion, fire, and
environmental safety. Let us formulate the twostage
problem of the optimal design of a turbulent tube reac
tor for the diazotization of aromatic amines with
mixed constraints; it is necessary to determine the vec
tors d* and z* at which C attains a minimum, i.e.,
TC* = min u;
(15)
Pr { g 0(d, z(ξ), ξ) ≤ u} ≥ ρ0;
(16)
Pr { g1(d, z(ξ), ξ) ≤ 0} ≥ ρ1;
(17)
χ1(d) = max min max g j (d, z, ξ) ≤ 0.
(18)
d,u, z(ξ)
ξ∈Ξ
z
j∈J 2
In problem (15)–(18), u is a scalar variable (ana
logue of structural variables); Pr{⋅} is the probability of
meeting the constraint {⋅}; g0 and g1 are the mild con
straint functions; ρ0 and ρ1 are the specified probabil
ities of meeting mild constraints; g0(d, z(ξ), ξ) = C(d,
z(ξ), ξ) is the criterion of the optimal design of a diaz
otization reactor (reduced expenditures on the cre
ation of a reactor); g1(d, z(ξ), ξ) = Qgiv – Q(d, z(ξ), ξ),
Qgiv, and Q are the specified and current reactor pro
ductivities with respect to a diazocompound; χ1(d) is
the flexibility function of a diazotization reactor; the
constraints with indices j ∈ J2 = {2,3,4} are strict;
ˆ η − Π η(d, z(ξ), ξ), Π
ˆ η, and Π η are the
g 2(d, z(ξ), ξ) = Π
limit admissible and current amounts of the solid
phase of an unconverted amine in a diazocompound
ˆ χ − Π χ(d, z(ξ), ξ); Π
ˆ χ, and
solution; g3(d, z(ξ), ξ) = Π
Π χ are the limit admissible and current amounts of
diazotars in a diazocompound solution; g 4(d, z(ξ), ξ) =
ˆ σ − Π σ(d, z(ξ), ξ), and Π
ˆ σ and Π σ are the limit admis
Π
sible and current amounts of nitrous gases in a diazo
compound solution.
The results of solving the twostage problem of the
optimal design of an industrial turbulent diazotization
reactor at each iteration are listed in the table.
The problem of the optimal design (with respect to
the reduced expenditure criterion) of a shortcycle
adsorption unit for enriching air with oxygen is formu
lated as follows: it is necessary to determine the struc
tural (adsorbent type b ∈ B, adsorbent bed height H,
and adsorber diameter Dinner) and regime parameters
(pressures Pad and Pdes, cycle duration τc, and blow
back coefficient θ) at which minimum reduced expen
ditures C on the creation of the unit are attained for the
given adsorption unit of type a ∈ A at specified values
of the productivity Qgiv and the outlet oxygen concen
tration cOout2 . Some initial design data are uncertain,
e.g., the concentration of oxygen cOin2 in air delivered by
a compressor to an adsorber for enrichment may be
varied from 18 to 23 vol %, the limit adsorption capac
ity W0 of a zeolite adsorbent may range from 0.160 to
0.230 cm3/g, and the masstransfer coefficient β may
be changed from 1.2 to 1.8 × 10–5 1/s.
The problem is mathematically formulated as
I* =
min
u
a,b,H ,Dinner ,u,Pad,Pdes,τc,θ
for relations expressed as equations of mathematical
model of a nonsteadystate airoxygen enrichment
process [3] and the following constraints:
for the target designing function,
Pr { g 0(a, b, H , Dinner, Pad, Pdes, τ c, θ, ξ)
= TC(a, b, H , Dinner, Pad, Pdes, τ c, θ) ≤ u} ≥ ρ 0;
JOURNAL OF CTHEORETICAL FOUNDATIONS OF CHEMICAL ENGINEERING Vol. 46
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A NEW APPROACH TO THE OPTIMAL DESIGN
441
2
3
Sodium
nitrite
4
5
Cooling agent
Cooling agent
1
Sodium
nitrite
Amine
suspension
lch = (8–10)Dch
7
8
dlube
Dch
6
Diazocompound
αdif/2
αcon/2
Turbulent tube reactor with diffuser–confuser mixing chambers: (1) tube block, (2) bend; (3) sodium nitrite spraying nozzles,
(4) diffuserconfuser device, (5) heatexchange jacket, (6) diffuser, (7) straight section, (8) confuser, dtube is the reactor tube sec
tion diameter, Dch is the diameter of mixing chamber, lch is the length of a mixing chamber, αdif is the diffuser divergence angle,
and αcon is the confuser convergence angle.
for the unit productivity,
χ1(a, b, H , Dinner ) = max min
ξ∈Ξ Pad,Pdes,τc,θ
Pr { g1(a, b, H , Dinner , Pad, Pdes, τc, θ, ξ)
= (Qgiv − Q) ≤ 0} ≥ ρ1;
× max g j (a, b, H , Dinner, Pad, Pdes, τ c, θ, ξ) ≤ 0,
for the oxygen concentration and the overall unit
dimensions,
k p ≤ kˆp, H ≤ Hˆ , Dinner ≤ Dˆ,
j =2,3
THEORETICAL FOUNDATIONS OF CHEMICAL ENGINEERING Vol. 46
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2012
442
DVORETSKII et al.
Results of soling the problem of the optimal design of an industrial turbulent diazotization reactor
Iteration no., k
1
Structural
variables, d
D = 0.04 m;
L = 115 m;
m = 3;
l1 = 40 m;
l2 = 80 m.
Regime (control) variables,
z
T(0) = 296°C;
p = 3;
C, u
Flexibility
function, χ
Probability of meeting
mild constraints, Pr{}
⋅
2225 $
0.326
Pr{g0 ≤ u} = 92.1%
Pr{g1 ≤ 0} = 95%
2230 $
0.00075
Pr{g0 ≤ u} = 94.1%
Pr{g1 ≤ 0} = 97.7%
(1)
G N = 5.1 × 10–5 m3/s;
(2)
G N = 2.55 × 10–5 m3/s;
(3)
G N = 2.55 × 10–5 m3/s.
2
D = 0.04 m;
L = 120 m;
m = 3;
l1 = 42,5 m;
l2 = 82,5 m.
T(0) = 300°C;
p = 3;
(1)
G N = 6.3 × 10–5 m3/s;
(2)
G N = 1.95 × 10–5 m3/s;
(3)
G N = 1.95 × 10–5 m3/s.
3
D = 0.04 m;
L = 123 m;
m = 3;
l1 = 43m;
l2 = 84 m
T(0) = 300°C;
p = 3;
2232 $
–0.036
Pr{g0 ≤ u} = 100%
Pr{g1 ≤ 0} = 98.8%
(1)
G N = 6.1 × 10–5 m3/s;
(2)
G N = 2.05 × 10–5 m3/s;
(3)
G N = 2.05 × 10–5 m3/s.
where u is a scalar variable; Pr{⋅} is the probability of
meeting the constraint {⋅}; ρ0 and ρ1 are specified prob
abilities; g0(a, b, H, Dinner, Pad, Pdes, τc, θ, ξ) = C(a, b,
H, Dinner, Pad, Pdes, τc, θ) is the unit optimal design cri
terion (reduced expenditures); g1(a, b, H, Dinner, Pad,
Pdes, τc, θ, ξ) = (Qgiv –Q(a, b, H, Dinner, Pad, Pdes, τc,
θ)); χ1 is the flexibility function of the unit; Qgiv,
[cOout2 ]giv are the specified unit productivity and outlet
oxygen concentration, respectively; g2(a, b, H, Dinner,
Pad, Pdes, τc, θ, ξ) = ([cOout2 ]giv – cOout2 ); g3(a, b, H, Dinner,
ˆ − M , Mˆ , kˆp, Hˆ , D̂inner are the
Pad, Pdes, τc, θ, ξ) = M
limit admissible mass, pressure coefficient, and overall
dimensions of the adsorbers in the unit, respectively.
Let us exemplify the optimal design of a shortcycle
adsorption unit with the development of a portable
medical oxygen concentrator, the technical specifica
tion on the designing of which includes the following
characteristics to be attained: the concentrator pro
ductivity Qgiv= 0.05 × 10–3 m3/s, the outlet oxygen
concentration [cOout2 ]giv ≥ 90%; ρ0, ρ1 = 0.9, the limit
admissible adsorber mass M = 0.6 kg, adsorption/des
orption pressure ratio Pad/Pdes = k p = 3;, adsorbent bed
height H = 0.4 m, and adsorber diameter Dinner =
0.1 m, respectively.
The alternate variants of equipment implementa
tion included a column adsorber, a twoadsorber con
centrator without pressure equalization, a two
adsorber concentrator with pressure equalization, a
fouradsorber concentrator with pressure equaliza
tion, and a fiveadsorber concentrator with twostage
pressure equalization. For each case, we analyzed dif
ferent variants of airoxygen enrichment (pressure,
with vacuum desorption, vacuumpressure) and types
of adsorbent (grained and block, NaX and LiLSX).
In the course of optimal design, we selected the
twoadsorber variant of a portable medical oxygen
concentrator with vacuum desorption and determined
its optimal structural parameters H* = 0.22 m and
* = 0.035 m; regime variables Pad* = 1.5 × 105 Pa,
Dinner
* = 0.5 × 105 Pa, θ* = 2.5, τ*c = 1.6 s, and Qinit
* =
Pdes
–4
3
2.93 × 10 m /s; and engineering and economic
parameters TC* = 45250 rub, M* = 0.5 kg, and N* =
76 W.
Our practical recommendations on the designing
of medical oxygen concentrators with a productivity
below 0.08 × 10–3 m3/s imply the use of adsorbers with
dimensions 4 ≤ H/Dinner ≤ 6, the pressure variant with
vacuum desorption (kp = Pad/Pdes ≤ 3), and LiLSX
block zeolite adsorbents with deff ≤ 0.5 ×10–3 m. This
improves the energysaving characteristics of medical
oxygen concentrators by 20% on the average in com
parison with world analogues.
The traditional methods for calculating the
strength of the thermally loaded cylindrical shells of
apparatuses, press molds, etc. use the assumption of a
linear temperature profile in the wall of the calculated
equipment, which results in the unreasonably overes
JOURNAL OF CTHEORETICAL FOUNDATIONS OF CHEMICAL ENGINEERING Vol. 46
No. 5
2012
A NEW APPROACH TO THE OPTIMAL DESIGN
timated thickness and mass of process equipment
shells. The selfpropagating hightemperature synthe
sis of hardalloy materials by press molding combines
hightemperature and force loadings; high tempera
tures of ∼2000–3000°С are generated in a press mold,
and excessive pressures of ~200 MPa are attained in a
material in the course of press molding. High thermal
and force loads superimposed on one another within
different time intervals and the nonstationarity,
nonisothermicity, and qualitatively different level of
temperature gradients in the walls of process equip
ment shells require a detailed study.
To calculate the strength of the press mold, we used
a mathematical model that includes nonlinear heat
transfer and combustion front motion equations with
boundary conditions [4]. The model inlet variables are
the molding delay time td (the time from the end of
material combustion to the beginning of loading with
internal pressure) and the molding pressure P. In the
calculation of temperature fields, the model takes into
account the combustion rate Ucom and temperature
Tcom of the material of the specimen. The mathemati
cal model allows one to calculate the outlet variables;
the internal wall temperature T1w, the wall’s boundary
layer thickness δ1, and the equivalent stress σeq appear
ing in the wall under thermal and mechanical loads.
The value of δ1 is specified by the admissible wall
across temperature drop, at which the changes in the
material of a press mold are reversible and do not lead
to any loss in the mechanical properties of the wall
material.
The combustion rate Ucom and temperature Tcom of
a pressed material in the synthesis of a product were
considered to be uncertain parameters ξ. The uncer
tainty of information with respect to Ucom and Tcom is
caused by different factors that depend on the proper
ties of an initially prepared molding mixture (bulk
density, moisture content, etc.). The problem of the
strength calculation of a press mold for the selfprop
agating hightemperature synthesis of hardalloy
materials is formulated as follows. It is necessary to
determine the delay time td and the pressure P at which
the minimum thickness δ of the wall of a press mold is
attained, i.e.,
min δ,
δ,t d,P
for the relations expressed as the equation of the heat
transfer mathematical model [1] and the following
constraints:
for the temperature on the internal wall of a press
mold,
g 1(δ, t d, P, ξ) = max min (T1 (δ, t d, P, ξ) − T
w
ξ∈Ξ
t d ,P
lim
) ≤ 0,
443
for the boundary layer thickness of the wall of a
press mold,
g 2 (δ, t d, P, ξ) = max min (10δ1(δ, t d, P, ξ) − δ) ≤ 0,
ξ∈T
t d ,P
for the equivalent stress in the wall,
g 3 (t d, P, ξ) = max min (σ eq (t d, P, ξ) − [σ]) ≤ 0.
ξ∈T
t d ,P
As an example, we solved the problem of optimiz
ing the wall thickness of a press mold for the experi
mentally established ranges of varying the combustion
rate Ucom ∈ [5…25] mm/s and temperature Tcom ∈
[1950…2050]°C of a molding mixture. The abovefor
mulated problem was solved in three iterations. As a
result of solving the problem of optimization, we
determined an optimal press mold wall thickness δ* =
48.3 mm, delay time t d* = 4.3 s, molding pressure P* =
100 MPa, and χ0(δ*) = –0.00019. The thickness of a
press mold δ*= 42 mm, as well as t d* = 4.7 s and P* =
100 MPa, was previously calculated in our works at
nominal Ucom = 15 mm/s and Tcom = 2000°C.
A comparative analysis shows that a press mold
with a wall thickness of 48.3 mm will continue to func
tion during operation independently of the random
change in uncertain parameters ξ within specified
intervals. The scientifically substantiated margin coef
ficient for the wall thickness of a press mold of 15%
was obtained with consideration for the real tempera
ture profile in its wall.
CONCLUSIONS
The proposed efficient algorithm of solving the
twostage problem of the optimal design of engineer
ing systems represents a new scientifically substanti
ated approach, in which the uncertainty in mathemat
ical description coefficients and process parameters is
taken into consideration in the formulation of an opti
mal design problem itself. As a result of using this
approach, we solved the problem of the optimal design
of an industrial turbulent diazotization reactor, devel
oped practical recommendations on the design of
medical oxygen concentrators, and optimized the wall
thickness of a press mold for the hightemperature
synthesis of hardalloy materials.
NOTATIONS
A—set of equipment implementation variants of
industrial chemicalengineering apparatuses;
A—dimensionless kinetic coefficient in the equa
tion of dissolution of the solid phase of an aromatic
amine;
B—set of adsorbent types used in a shortcycle
adsorption unit;
b—adsorbent type;
C—reduced expenditures;
THEORETICAL FOUNDATIONS OF CHEMICAL ENGINEERING Vol. 46
No. 5
2012
444
DVORETSKII et al.
[c A(0)]s—concentration of an aromatic amine in the
solid phase at the inlet of a reactor, mol/m3;
cD—concentration of a diazocompound, mol/m3;
( 0)
c N —concentration of sodium nitrite at the inlet of
a reactor, mol/m3;
cNA—concentration of nitric acid, mol/m3;
cOout2 —concentration of oxygen at the outlet of a
shortcycle adsorption unit, vol %;
cOin2 —concentration of oxygen in the air delivered
to an adsorber for enrichment, vol %;
c(out)—concentration at the outlet of a reactor,
mol/m3;
cσ—concentration of nitrous gases, mol/m3;
cχ—concentration of diazotars, mol/m3;
D—diameter of a tube reactor, m;
Dch—diameter of a mixing chamber, m;
Dinner—diameter of an adsorber, m;
d—vector of structural parameters;
d*—vector d providing an extremum of the target
function F(d, z, ξ);
deq—equivalent diameter of the pore channels in
block zeolite adsorbents, m;
di—ith component of the vector d;
dN—nominal vector of structural parameters;
dtube—diameter of the tube section of a reactor, m;
F—target function (optimality criterion);
F**—function F(d, z, ξ) at an extremum point;
G1(out)—flow rate of the liquid phase of a diazosolu
tion suspension at the outlet of a reactor, m3/s;
0
G N( ) —flow rate of sodium nitrite at the inlet of a
reactor, m3/s;
G N(i)—flow rate of sodium nitrite at the inlet of the
ith reactor section, m3/s;
G(out)—flow rate of a diazosolution suspension at
the outlet of a reactor, m3/s;
(0)
G s —flow rate of the solid phase of a diazosolution
suspension at the inlet of reactor, m3/s;
G s(out)—flow rate of the solid phase of a diazosolu
tion suspension at the outlet of a reactor, m3/s;
g—constraint function;
H—adsorbent bed height, m;
I(k)—set of indices i for points ξ at which con
straints may be violated;
J1—set of indices j for mild constraints;
J2—set of indices j for strict constraints;
k—counter of algorithm iterations;
ki—dimensionless margin coefficient for the ith
component of the vector d;
kp—adsorption/desorption pressure ratio;
L—length of a tube reactor, m;
lj—mounting sites of diffuserconfuser devices, m;
lch—length of a mixing chamber, m;
m—number of constraints in a problem;
m1—number of mild constraints in a problem;
mdc—number of diffuserconfuser devices;
N—power of a portable medical oxygen concen
trator, W;
n—counter of indices i accumulated in the set I(k);
nξ—length of the vector ξ;
P—molding pressure, MPa;
Pad—adsorption pressure, MPa;
Pdes—desorption pressure, MPa;
p—number of sections supplied with sodium
nitrite along the length of a reactor;
Pr{⋅} —probability of meeting the constraint
{{⋅}, }, %;
Q—productivity, t/yr;
Qgiv—specified productivity of a shortcycle
adsorption unit, m3/s;
r—rough approximation function;
S
(k )
—set of the points, at which constraints are
violated;
Т(0)—temperature of an aromatic amine suspen
sion at the inlet of a reactor, K;
Tcom—combustion temperature of the material of a
specimen, °С;
T1w —temperature on the internal wall of a press
mold, °С;
tinit—molding delay time, s;
Ucom—combustion rate of the material of a speci
men, mm/s;
u—scalar variable;
W0—limit adsorption capacity of a zeolite adsor
bent, cm3/g;
y—vector of mathematical model outlet variables;
yj, giv—limit admissible value of the jth outlet vari
able of a chemicalengineering apparatus;
z—vector of control variables;
z*—vector z providing an extremum of the target
function F(d, z, ξ);
αcon—confuser convergence angle;
αdif—diffuser divergence angle;
β—masstransfer coefficient, 1/s;
δ—thickness of the wall of a press mold, m;
δ1—thickness of the boundary layer of the wall of a
press mold, m;
θ—blowback coefficient;
Ξ—variation area of uncertain parameters;
ξ—vector of uncertain parameters;
ξL—lower boundary of variation range of uncer
tain parameters;
ξU—upper boundary of variation range of uncer
tain parameters;
ξΝ—nominal vector of uncertain parameters;
Πη—amount of the solid phase of an amine in a
diazosolution at the outlet of a reactor, %;
JOURNAL OF CTHEORETICAL FOUNDATIONS OF CHEMICAL ENGINEERING Vol. 46
No. 5
2012
A NEW APPROACH TO THE OPTIMAL DESIGN
Πσ—amount of nitrous gases in a diazosolution at
reactor outlet, %;
Πχ—amount of diazotars in a diazosolution at
reactor outlet, %;
ρ0—specified probability of meeting the constraint
g0, %;
ρj—specified probability of meeting the constraint
gj, %;
[σ]—admissible stress, MPa;
σeq—equivalent stress, MPa;
τc—adsorption cycle duration, s;
χ(d)—flexibility function;
Ψ—operator of the mathematical model of a
chemicalengineering apparatus.
SUBSCRIPTS AND SUPERSCRIPTS
* solution to the problem;
0—at the inlet of a reactor;
A—aromatic amine;
ad—adsorption;
c—cycle;
ch—mixing chamber;
com—combustion;
con—confuser;
D—diazocompound;
d—delay;
dc—diffuserconfuser device;
des—desorption;
dif—diffuser;
eq—equivalent;
giv—specified value;
i, j—vector’s component indices;
inner, internal;
k—number of an algorithm iteration;
L—lover boundary;
l—liquid phase;
445
l—index of vector components corresponding to
approximation points;
lim—critical;
N—sodium nitrite;
NA—nitric acid;
Nom—nominal;
O2—oxygen;
out—outlet;
p—pressure;
s—soli phase; tube, tube section of a reactor;
U—upper boundary;
w—wall;
η—amine “breakthrough”;
σ—nitrous gases;
χ—diazotars.
REFERENCES
1. Ostrovskii, G.M. and Volin, Yu.M., Tekhnicheskie
sistemy v usloviyakh neopredelennosti: Analiz gibkosti i
optimizatsiya (Technical Systems under Uncertainty:
Flexibility Analysis and Optimization), Moscow:
BINOMi, 2008.
2. Dvoretskii, D.S., Dvoretskii, S.I., and Peshkova, E.V.,
Computer Simulation of Turbulent Reactors for Fine
Organic Synthesis under Uncertainty, Izv. Vyssh.
Uchebn. Zaved., Khim. Khim. Tekhnol., 2007, vol. 50,
no. 8, p. 70.
3. Akulinin, E.I., Dvoretskii, D.S., Dvoretskii, S.I., and
Ermakov, A.A., Mathematical Simulation of Air
Enrichment with Oxygen in a ShortCycle Adsorption
Device, Vestn. Tomsk. Gos. Tekh. Univ., 2009, vol. 15,
no. 2, p. 341.
4. Stel’makh, L.S., Stolin, A.M., and Dvoretskii, D.S.,
Nonisothermal Method for Calculating the Mold
Equipment of an Apparatus for Compacting the Hot
Products of SelfPropagating HighTemperature Syn
thesis, Theor. Found. Chem. Eng., 2010, vol. 44, no. 2,
p. 192.
THEORETICAL FOUNDATIONS OF CHEMICAL ENGINEERING Vol. 46
No. 5
2012
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Literature critique #1
Date due: April 27, 2015
By Abdulrahman Batarfi
Dvoretskii, D. S., Dvoretskii, S. I., Ostrovskii, G. M., & Polyakov, B. B. (2012). A new approach
to the optimal design of industrial chemical-engineering apparatuses. Theoretical Foundations of
Chemical Engineering, 46(5), 437-445.
In developing a new approach to the optimal design of industrial chemical-engineering
apparatuses, an algorithm has been described. It stems from a two stage problem in the design of
the apparatus and addresses the uncertainty in the mathematical description coefficients and
processes parameters are taken into account for the formulation of the designing problem. This is
advantageous because of the possibility of adjusting the control variables depending on the
measurement of uncertain parameters at the operating stage of an industrial apparatus.
The efficiency of the algorithm is exemplified in apparatuses such as the turbulent tube reactor of
fine organic synthesis and the adsorption oxygen concentrator. Some of the uncertainties
encountered in design include change in the parameters of raw materials. the external
temperatures are Also within certain during operation making it impossible to specify the unique
value.
Optimality criterion takes into account the uncertainties in the mathematical description as it is
sufficient to distinguish them in the dependencies for the target function. The disadvantages are
such that there is no assurity of either the optimality of the obtained solution or the fulfillment of
all constraints, during the operation of the industrial apparatus.
While formulating an optimal design problem, the specific types of target functions and
constrains were based on the concept of two stages in the life cycle of an industrial apparatus.
The constraints may be strict or mild where the former may not be violated under any condition.
I was particularly impressed by the fact that the proposed algorithm was tested with the optimal
design of a number of industrial chemical engineering apparatuses. the optimality criterion and
the possible constraints were also addressed. These being taken into account, the demonstrated
results were impressive e.g. the problem of optimal design of an industrial turbulent diazotization
reactor was solved and practical recommendations on the design of medical oxygen
concentrators were developed among other things.
IEGR 204 – INTRODUCTION TO IE AND COMPUTERS
MiniProject #1 (Slides Due no later than Monday,
5/11/15 via Bb by 11:59pm)
Using Microsoft PowerPoint (PP), complete a presentation about the
sub-area of interest in Industrial Engineering (IE) which you have
started your research on with the earlier assignments (Research
Assignments #1 and #2, and Lab #2). You must also find a 2nd
refereed research article on the same topic area, and a 3rd source of
your choice (webpage, magazine article, textbook, podcast, etc.).
Thus, a total of three sources are mandatory for this project (and
anything short of this will result in a deduction on your project
grade).
The PP slides must meet the following criteria:
8 slides minimum ( including the mandatory 1st slide for the
title page and a last slide entitled “References” for listing your
full APA-style bibliographic citations); both of these were
started during your Lab #2 assignment
The remaining slides in between must meet the following
specifications:
o Your 2nd slide must be entitled “Outline”
o The 3rd slide entitled “Background”, 4th slide entitled
“Research Problem”, 5th and 6th slides should be entitled
“Math Equations Used”, 7th slide entitled “Conclusions”,
and 8th entitled “References” as stated earlier
o
o
o
o
Keep in mind that on the “Math Equations Used”
slide, you must explain 2 or more equations used
within the articles in a brief statement or phrase
If there are no equations in the 1st article, then use
whatever is math related from the article…a
segment of a computer program described, a
mathematical algorithm shown, etc.
Be sure that the 2nd article contains mathematical
equations if the 1st article does not
If neither of the refereed articles contain
mathematical equations, your project will receive a
deduction
Remember, you must use your literature critique
assignment (from your 1st approved article) and a 2nd
refereed article for this project, as well as 3rd source
The presentation title should reflect the area of IE
which your project focuses on and not the title of
any of the articles (e.g., Robotics in IE, Energy
Systems in IE, Human Factors in IE, etc.).
Also, be sure to include your name, class, and due
date of project on the title slide
Remember, you can ‘pull’ some information from your
literature critique assignment into the “Background” or
“Research Problem” slides of the project
You can also use a non-refereed article for your 3rd source
to gather information for your “Background” slide
You must have a minimum of 3 sources/references as
stated earlier, you can gather more if they are related.
However, do not list any sources that are not used in this
project. In other words, having 5 sources will not gain
o
o
o
o
o
o
you any extra points on the project, but could only
enhance your project if utilized properly.
Thus, a minimum of 3 sources (2 refereed research
journal articles, and 1 additional source) must be
cited in the project on the “References” slide
You must include two slides with some IE mathematical
formulations and explain briefly how/why the author(s)
were using the math
You can include a figure(s) in your slides from your IE
research sources as long as you properly cite the
reference source (i.e., you must give the author(s) credit
for the information which you are utilizing!) – remember
to use the APA-format in-text citation with Author’s last
name and date in parenthesis at the end of the caption!
You must make good use of animation on slides and
between slides during transitions (if you need help on
this, ask/see the instructor!)
Again, you must use the APA journal style for your
bibliographic citations on the “References” slide as shown
on my Research webpages (i.e., the same format used on
your Literature Critique assignment – if done correctly)
If the citation(s) is/are incorrect, you will receive
deductions on the project
DO NOT use paragraphs of information on the slides and
please use a serif type font (such as Times New Roman,
Times Roman, etc.) and no size less than 14 pt font on
each slide
Use last week’s lab (from 4/29/15) which your submitted as part of
your PowerPoint presentation slides for this project (i.e., don’t
waste the work you have already done). While working on the
project (or if you complete the work before the due date), if you
have any additional questions, be sure to ask the instructor and not
just a classmate. Do not ask me questions such as (1) “Does this
look right?” or (2) “Is this OK?”…please only ask legitimate questions
which deal with uncertainty about a specific issue/item.
This is a major part of your grade, and if your friend gives you the
wrong information you will not be able to blame them! DO NOT
ASSUME anything if you are unsure about something.
Lastly, submit your PowerPoint presentation slides on-time along
with the 2 additional sources via Bb to receive full credit. If the Bb
submission time stamp is beyond 11:59pm on the due date, you will
receive a 20% deduction per day in points for lateness as on other
assignments. Take this very seriously! If no file is ever submitted
via Bb, you will NOT receive a project grade nor make it up at a
later time.
The timely submission will be part of the overall MiniProject grade.
The remaining portions of the grade will be made up from meeting
the specifications listed above. A project grading table for the PP
slides will be shown at a future date in class or posted via Bb. This
table will detail the full project requirements and allows for
everyone to know ‘up front’ what is necessary to achieve the
maximum number of points.
Do your best!
Be sure to submit your MiniProject Slides by the assigned due date and time.
It may be submitted ahead of time as well!
FINAL NOTE: Again, any assignment that has an Bb submission time stamp beyond the
assigned due time will be reduced (i.e., the grade assigned will have a deduction for
lateness of 20% off per 24 hour period). NO EXCEPTIONS!
If you have any questions, do one of the following: (1) please ask them in class, (2) ask via
email (Richard.Pitts@morgan.edu), or (3) come by during office hours to see me.
Created on May 2, 2015 by Dr. Richard Pitts, Jr.
Last Updated on May 3, 2015 by Dr. Richard Pitts, Jr.
ISSN 00405795, Theoretical Foundations of Chemical Engineering, 2012, Vol. 46, No. 5, pp. 437–445. © Pleiades Publishing, Ltd., 2012.
Original Russian Text © D.S. Dvoretskii, S.I. Dvoretskii, G.M. Ostrovskii, B.B. Polyakov, 2012, published in Teoreticheskie Osnovy Khimicheskoi Tekhnologii, 2012, Vol. 46,
No. 5, pp. 501–510.
A New Approach to the Optimal Design of Industrial
ChemicalEngineering Apparatuses
D. S. Dvoretskiia,*, S. I. Dvoretskiia, G. M. Ostrovskiib, and B. B. Polyakova
a
b
Tambov State Technical University, ul. Sovetskaya 106, Tambov, 392000 Russia
Karpov Institute of Physical Chemistry, per. Obukha 31/12, Moscow, 105064 Russia
* email: dvoretsky@tambov.ru
Received December 21, 2011
Abstract—An algorithm is described for the twostage problem of the optimal design of industrial chemical
engineering apparatuses based on a new approach, in which uncertainty in the mathematical description
coefficients and process parameters is taken into account in the formulation of a designing problem. A typical
feature of twostage optimal design problems is the possibility of adjusting the regime (control) variables of a
control system depending on the refinement (measurement) of uncertain parameters at the operating stage
of an industrial apparatus. An algorithm for solving the twostage problem of the optimal design of engineer
ing systems has been developed, and its efficiency is exemplified by industrial chemicalengineering appara
tuses; a turbulent tube reactor of fine organic synthesis, an adsorption oxygen concentrator, and a press mold
for the hightemperature synthesis of hardalloy materials.
DOI: 10.1134/S0040579512040112
INTRODUCTION
When designing industrial chemicalengineering
apparatuses, we always encounter uncertainties of two
kinds. Some of them, such as the parameters of raw
materials and the external temperature, may change
during their operation, remaining within a certain
range. It is impossible to specify their unique value in
principle. The other may be virtually constant for the
given industrial apparatus, but their values are known
within the accuracy of a certain range, for example,
some coefficients in kinetic and heat and mass transfer
equations. To take into account the uncertainties in
the mathematical description of an industrial appara
tus, it is sufficient to distinguish them in the depen
dences for the target function (optimality criterion)
F and the constraint functions gj of the optimal design
problem, assuming that F = F(d, z, ξ), gj = gj(d, z, ξ),
j = 1, …, m, where ξ is the vector of uncertain param
eters taking any values within a specified area Ξ, which
is usually considered to be rectangular as follows: Ξ =
{ξ: ξL ≤ ξ ≤ ξU}.
In this case, the solution of an optimal design prob
lem with respect to the criterion F = F(d, z, ξ) using
the constraint functions gj = gj(d, z, ξ), j = 1, …, m
proves to be uncertain and depends on the value that
has been taken by the vector ξ. The traditional way of
overcoming the given difficulty consists of the follow
ing. The vector of uncertain parameters is assigned a
certain nominal value ξ = ξNom, and a designing prob
lem is solved at nominal ξNom to obtain the vector of
structural parameters dNom for a specified type of
equipment implementation. Thus, using the available
knowledge on a designed object and intuition, so
called margin coefficients ki (ki > 1) are introduced,
and it is assumed during the design stage that di =
ki diNom, where di is the ith component of the vector d
and i = 1, …, n (reactor length and diameter, heat
transfer surface area in a heat exchanger, tray number
in a distillation column, etc.).
The disadvantages of this approach are obvious,
since it does not assure either the optimality of the
obtained solution or the fulfillment of all constraints
during the operation of the industrial apparatus. If the
margin coefficients are low, the constraints will be vio
lated and, if they are very high, the design will not be
economical.
The approach in which the uncertainty in the
mathematical description coefficients and process
parameters are taken into account in the formulation
of the optimal design problem is much more correct
and scientifically substantiated.
FORMULATION OF TWOSTAGE PROBLEM
OF OPTIMAL DESIGN OF CHEMICAL
ENGINEERING APPARATUSES
Traditionally, the problem of the technical imple
mentation of chemicalengineering processes is for
mulated as the nonlinear programming problem
437
min F (d, z, ξ)
(1)
d, z
y = Ψ(d, z, ξ);
(2)
g j (d, z, ξ) ≡ y j,giv − y j ≤ 0, j = 1,..., m,
(3)
438
DVORETSKII et al.
where F(⋅) is the optimal design criterion; y, d, z, and
ξ are the vectors of outlet, structural, and regime (con
trol) and uncertain variables of a designed object,
respectively; y = Ψ(d, z, ξ) is the operator of the math
ematical model of the chemicalengineering appara
tus; yj, giv is the limit admissible value of the jth outlet
variable of the chemicalengineering apparatus; and
gj(d, z, ξ) ≤ 0, j = 1, …, m are the constraint functions.
Let ξ belong to the area Ξ, i.e., ξ ∈ Ξ. Let us rewrite
problem (1)–(3) as
min u
(4)
d, z,u
F (d, z, ξ) ≤ u
(5)
g j (d, z(ξ), ξ) ≤ 0, j = 1,…, m.
(6)
At constant ξ, these two formulations are equivalent.
However, when uncertain parameters are taken into
consideration, the advantage of formulation (4)–(6) is
that the optimal design criterion F = F(d, z, ξ) of the
initial problem (1)–(3) is taken into account in the
same manner as the other constraints.
When an optimal design problem is formulated
under the uncertainty of initial information, it is nec
essary to specify the types of the target function (opti
mality criterion) and constraints. This is based on the
concept of two stages in the life cycle of an industrial
apparatus, i.e., the design stage and operating stage. At
the operating stage, the following cases are possible:
(1) All uncertain parameters can be precisely deter
mined at each time moment (either by direct measure
ment or by solving the inverse problem based on infor
mation obtained from measurements).
(2) The area of uncertain parameters at the operat
ing stage is the same as that at the design stage.
(3) At the operating stage, some of the parameters
ξi can be precisely determined, while others have the
same interval as those at the design stage.
(4) At the operating stage, all parameters ξi contain
uncertainty, but their uncertainty intervals are smaller
than the corresponding intervals at the design stage.
The constraints of an optimal design problem may
be strict (unconditional) and mild (probabilistic).
Strict constraints must not be violated under any con
dition. Mild constraints may be met with a specified
probability. Most of the real problems pertain to the
case when some constraints are strict and others are
mild. For example, the safety constraints of an indus
trial apparatus as classified as strict, and its productiv
ity and selectivity constraints may be categorized as
mild.
Let us consider the twostage problem of the opti
mal design of industrial chemicalengineering appara
tuses under interval uncertainty. A typical feature of
the twostage problems of the optimal design of indus
trial apparatuses is the possibility of adjusting the
regime (control) variables z at the stage of their opera
tion depending on the refinement of the vector of
uncertain parameters ξ; i.e., the control variables z are
multidimensional functions z = z(ξ).
Let there be a mathematical model of the steady
state operation of an industrial apparatus y = Ψ(d, z,
ξ), where y is the vector of the outlet variables of the
designed object; the constraints with indices j = 0, j ∈
J1 = {1, 2, …, m1} are mild (probabilistic); and the con
straints with indices j ∈ J2 = {m1 + 1, m1 + 2, …, m} are
strict (unconditional).
The twostage problem of the optimal design of
industrial apparatuses in statics is formulated as fol
lows: it is required to determine the vectors d* and vec
tor functions z* that provide an extremum of the target
function F(d, z, ξ) and the fulfillment of mild (proba
bilistic) and strict (unconditional) constraints inde
pendently of the change in the vector of uncertain
parameters ξ within a certain specified area Ξ. The
mathematical formulation of this problem can be writ
ten as follows:
F * = min u
(7)
Pr { g 0(d, z(ξ), ξ) = F (d, z(ξ), ξ) ≤ u} ≥ ρ0;
(8)
Pr {g j (d, z(ξ), ξ) ≤ 0} ≥ ρ j , j ∈ J 1;
(9)
χ1(d) = max min max g j (d, z, ξ) ≤ 0.
(10)
d,u,z(ξ)
ξ∈Ξ
z
j∈J 2
In problems (7)–(10), u is a scalar variable (ana
logue of structural variables); Pr{⋅} is the probability of
meeting the constraint {⋅}; g0 and gj are the constraint
functions; g0(d, z(ξ), ξ) = F(d, z(ξ), ξ) is the optimal
design criterion; gj(d, z, ξ) ≡ yj, giv – yj ≤ 0, j = 1, …, m is
the function of constraints; yj = Ψ(d, z, ξ), j = 1, …, m
and ρ0 and ρj are the specified probabilities of meeting
the constraints; and χ1(d) is the flexibility function of
the apparatus.
ALGORITHM OF TWOSTAGE
OPTIMIZATION OF CHEMICAL
ENGINEERING SYSTEMS
Let us introduce the notations
⎧g j (d, z, ξ) − u, j = 0;
g j (d, u, z, ξ) = ⎨
⎩g j (d, z, ξ), j ∈ J 1;
and the set S(k) = {ξi: i ∈ I (k)}, which accumulates the
points ξ (i ∈ I (k)) at which constraints (8)–(10) are
violated, and the sets S1(k) and S 2(k) will accumulate the
points at which strict and mild constraints are vio
lated, respectively. Moreover, we shall use the auxiliary
nonlinear programming problem (A) as follows:
F * = mini u
d,u,z
g j (d, u, z , ξ ) ≤ 0,
i
i
g j (d, z , ξ ) ≤ 0,
i
i
j ∈ J 1, i ∈ I (k);
(A)
j ∈ J 2, i ∈ I .
JOURNAL OF CTHEORETICAL FOUNDATIONS OF CHEMICAL ENGINEERING Vol. 46
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A NEW APPROACH TO THE OPTIMAL DESIGN
The solution of problem (A) consists of finding the
minimum of the scalar variable u under all constraints
of this problem within a specified set of points ξi, i ∈
I(k).
The algorithm consists of the following steps.
Step 1. At the first step, the numbers of iterations
and critical points are taken to be equal to k = 1 and
n = 0, respectively. The initial set S(k – 1) is selected
from the condition of the best approximation of
functions z(ξ). The initial estimates d(k – 1), u(k – 1),
and zi, (k – 1) are specified.
Step 2. Auxiliary problem A is solved, and d (k), u(k),
and z(k) are its solutions.
Step 3. Using the outer approximation algo
rithm [1],
(
χ1(d ) = max min max g j d , z, ξ
(k )
ξ∈Ξ
z
j∈J 2
(k )
)
χ1(d , ξ ) ≤ 0
(k)
(k)
(12)
is verified. If condition (12) is true, we pass to Step 4;
otherwise, we pass to Step 5.
Step 4. The set S1(k), of points at which constraints
(12) are violated is complemented by the following:
S1(k) = S1(k −1) ∪ ξ (k), where ξ (k) : χ1(d (k)) > 0;
}
Pr g j (d , z(ξ), ξ) ≤ 0 ≥ ρ j , j ∈ J 1.
(k)
j∈J 1
(
)
(14)
where J 1 = (0, 1, 2, …, m1). The solution of problem
(14) is denoted as ξ (k ) , and the set S 2(k) of points at
which mild constraints are violated is complemented
as follows:
S 2(k) = S 2(k −1) ∪ ξ (k); where χ 2(d (k)) > 0;
I 2(k) = I 2(k −1) ∪ (n + 1); n := n + 1.
i
∑ (ξ
−ξ j ) ,
i 2
j
j =1
i∈I
(k)
=
(k)
I1
∪ I 2 , nξ = dim ξ,
(k )
i
(l )
i
(i)
(iˆ)
r (ξ, ξ ) ⇒ iˆ = arg min
r (ξ, ξ ) ⇒ zˆ = z .
ξ = min
(k )
(k )
i∈I
i∈I
Actually, the piecewise constant approximation of
the functions, z = z(ξ) is used in the described proce
dure.
At Step 6, the inequality χ2(d (k)) ≤ 0 means that
mild constraints are fulfilled with a probability of 1.
Therefore, if constraint (13) is not met, it is certain
that χ2(d (k)) > 0 and, consequently, we obtain the point
(13)
χ 2(d (k)) = max min max g j d (k), u (k), z, ξ ,
z
i
(k)
are verified.
At the given step, we have no functions z = z(ξ), and
know only their values at discrete points ξi, i ∈ I (k). For
this reason, we shall use these points to approximate
the functions z = z(ξ).
If condition (12) is true, and condition (13) is false,
we pass to Step 6.
If conditions (13) and (14) are true, the solution
d* = d (k) and z* = zi, (k) is found.
Step 6. Using the outer approximation algorithm
[1], we calculate
ξ∈Ξ
nξ
r (ξ, ξ ) =
ξ , at which mild constraints are violated.
In the case of using the additional variable u, we
scale search variables so as to make the ranges of their
variation nearly equal.
(k )
(k −1)
I 1 = I 1 ∪ (n + 1); n := n + 1.
Step 5. Mild (probabilistic) constraints
{
Step 7. We form the sets S (k) = S1(k) ∪ S 2(k) and
I(k) = I 1(k) ∪ I 2(k), and assume k to be as follows: = k + 1,
and pass to Step 2.
Let us give some explanation for this algorithm.
At Step 5, we perform multivariate interpolation
through the known discrete points ξi, zi, i ∈ I(k) using
the functions z = z(ξ). This may be done by means of
multivariate cubic splines or the rough approximation
procedure, which consists of the following. When
implementing a simulation model, we accept z(ξ),
which corresponds to each obtained random ξ equal to
zl(ξl), l ∈ I (k), which corresponds to the point ξi, which
is closest to the point ξ, i.e.,
(11)
is calculated. The solution of problem (11) is denoted
as ξ (k ) and the condition
439
EXAMPLES OF OPTIMAL DESIGN
OF INDUSTRIAL CHEMICAL
ENGINEERING APPARATUSES
The efficiency of the proposed algorithm will be
demonstrated with some examples of the optimal
design of a number of industrial chemicalengineering
apparatuses, including a turbulent tube reactor of fine
organic synthesis, a shortcycle adsorption unit, and a
press mold for the hightemperature synthesis of hard
alloy materials.
The mathematical model y = Ψ(d, z, ξ) of the stat
ics of a nonlinear process of fine organic synthesis,
namely, the diazotization of aromatic amines in a tur
bulent tube reactor allows us to calculate the following
variables y of the diazotization reactor outlet: the pro
ductivity Q; the concentrations c(out) = (cD, cNA, cχ, cσ)
of diazocompound, nitric acid, diazotars, and nitrous
gases; the flow rates of the liquid and solid phases of a
diazosolution suspension G (out) = (G1(out), G s(out)); and
the amounts of the solid phase of an amine Πη, diazo
tars Πξ, and nitrous gases Πσ in a diazosolution at the
outlet of the diazotization reactor [2], where d, z, and
THEORETICAL FOUNDATIONS OF CHEMICAL ENGINEERING Vol. 46
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2012
440
DVORETSKII et al.
ξ are the vectors of structural regime (control) and
uncertain variables of the diazotization reactor.
Let us formulate the technical specifications for
designing a turbulent tube reactor for the diazotization
of aromatic amines with diffuser–confuser flow turbu
lization devices (see figure).
At a specified reactor productivity (with respect to
a diazocompound) of Q = 1000 t/yr, it is necessary to
provide the following outlet variables: the break
through of an aromatic amine Πη = G s(out) G s(0) × 100%,
as well as concentrations of diazotars of Πχ = cχ(out) ×
(out)
×
G1(out)) × 100% and nitrous gases Πσ = cNA
(0)
(out)
(out )
(0)
(0)
G1
(c N × G N ) × 100%, where [cA ]s and cχ are the
inlet concentrations of aromatic amine in the solid
phase and the outlet concentration of diazotars, and
( 0)
(0)
cN and G N are the inlet concentration and flow rate of
sodium nitrite, respectively) below their limit admissi
ˆ η = 0.25%, Π
ˆ χ = 0.9%, Π
ˆ σ = 0.5%, i.e.,
ble values Π
ˆ η, Π χ ≤ Π
ˆ χ , and Π σ ≤ Π
ˆ σ . These require
Πη ≤ Π
ments must be met in the interval of the uncertainty of
some parameters of the process and coefficients of the
mathematical model of diazotization, namely, the
inlet concentration of the solid phase of an amine
[cA(0)]s = 370.0 (±4%) mol/m3 and the kinetic coeffi
cient in the equation of dissolution of the solid phase
of an aromatic amine А = 5.4 × 105(±5%).
The optimal design problem consists of determin
ing the structural parameters d (tube reactor diameter
D and length L, number of diffuser–confuser devises
mdc and their mounting sites lj, j = 1,2, …), as well as
nd the regime (control) variables z (inlet temperature
of an aromatic amine suspension Т(0) and the distribu
tion of the sodium nitrite flow rate GN(i), i = 1,2, …, p)
along the reactor length) that ensures the minimum
reduced expenditures C(d, z, ξ) in the creation of a
reactor and its operability independently of the ran
dom variations in the vector of uncertain parameters ξ
in the area Ξ. Constraints may be specified in strict or
mild (probabilistic) forms. As a rule, strict constraints
include the specified quality requirements for
obtained products and the technical regulation
requirements for the production’s explosion, fire, and
environmental safety. Let us formulate the twostage
problem of the optimal design of a turbulent tube reac
tor for the diazotization of aromatic amines with
mixed constraints; it is necessary to determine the vec
tors d* and z* at which C attains a minimum, i.e.,
TC* = min u;
(15)
Pr { g 0(d, z(ξ), ξ) ≤ u} ≥ ρ0;
(16)
Pr { g1(d, z(ξ), ξ) ≤ 0} ≥ ρ1;
(17)
χ1(d) = max min max g j (d, z, ξ) ≤ 0.
(18)
d,u, z(ξ)
ξ∈Ξ
z
j∈J 2
In problem (15)–(18), u is a scalar variable (ana
logue of structural variables); Pr{⋅} is the probability of
meeting the constraint {⋅}; g0 and g1 are the mild con
straint functions; ρ0 and ρ1 are the specified probabil
ities of meeting mild constraints; g0(d, z(ξ), ξ) = C(d,
z(ξ), ξ) is the criterion of the optimal design of a diaz
otization reactor (reduced expenditures on the cre
ation of a reactor); g1(d, z(ξ), ξ) = Qgiv – Q(d, z(ξ), ξ),
Qgiv, and Q are the specified and current reactor pro
ductivities with respect to a diazocompound; χ1(d) is
the flexibility function of a diazotization reactor; the
constraints with indices j ∈ J2 = {2,3,4} are strict;
ˆ η − Π η(d, z(ξ), ξ), Π
ˆ η, and Π η are the
g 2(d, z(ξ), ξ) = Π
limit admissible and current amounts of the solid
phase of an unconverted amine in a diazocompound
ˆ χ − Π χ(d, z(ξ), ξ); Π
ˆ χ, and
solution; g3(d, z(ξ), ξ) = Π
Π χ are the limit admissible and current amounts of
diazotars in a diazocompound solution; g 4(d, z(ξ), ξ) =
ˆ σ − Π σ(d, z(ξ), ξ), and Π
ˆ σ and Π σ are the limit admis
Π
sible and current amounts of nitrous gases in a diazo
compound solution.
The results of solving the twostage problem of the
optimal design of an industrial turbulent diazotization
reactor at each iteration are listed in the table.
The problem of the optimal design (with respect to
the reduced expenditure criterion) of a shortcycle
adsorption unit for enriching air with oxygen is formu
lated as follows: it is necessary to determine the struc
tural (adsorbent type b ∈ B, adsorbent bed height H,
and adsorber diameter Dinner) and regime parameters
(pressures Pad and Pdes, cycle duration τc, and blow
back coefficient θ) at which minimum reduced expen
ditures C on the creation of the unit are attained for the
given adsorption unit of type a ∈ A at specified values
of the productivity Qgiv and the outlet oxygen concen
tration cOout2 . Some initial design data are uncertain,
e.g., the concentration of oxygen cOin2 in air delivered by
a compressor to an adsorber for enrichment may be
varied from 18 to 23 vol %, the limit adsorption capac
ity W0 of a zeolite adsorbent may range from 0.160 to
0.230 cm3/g, and the masstransfer coefficient β may
be changed from 1.2 to 1.8 × 10–5 1/s.
The problem is mathematically formulated as
I* =
min
u
a,b,H ,Dinner ,u,Pad,Pdes,τc,θ
for relations expressed as equations of mathematical
model of a nonsteadystate airoxygen enrichment
process [3] and the following constraints:
for the target designing function,
Pr { g 0(a, b, H , Dinner, Pad, Pdes, τ c, θ, ξ)
= TC(a, b, H , Dinner, Pad, Pdes, τ c, θ) ≤ u} ≥ ρ 0;
JOURNAL OF CTHEORETICAL FOUNDATIONS OF CHEMICAL ENGINEERING Vol. 46
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2012
A NEW APPROACH TO THE OPTIMAL DESIGN
441
2
3
Sodium
nitrite
4
5
Cooling agent
Cooling agent
1
Sodium
nitrite
Amine
suspension
lch = (8–10)Dch
7
8
dlube
Dch
6
Diazocompound
αdif/2
αcon/2
Turbulent tube reactor with diffuser–confuser mixing chambers: (1) tube block, (2) bend; (3) sodium nitrite spraying nozzles,
(4) diffuserconfuser device, (5) heatexchange jacket, (6) diffuser, (7) straight section, (8) confuser, dtube is the reactor tube sec
tion diameter, Dch is the diameter of mixing chamber, lch is the length of a mixing chamber, αdif is the diffuser divergence angle,
and αcon is the confuser convergence angle.
for the unit productivity,
χ1(a, b, H , Dinner ) = max min
ξ∈Ξ Pad,Pdes,τc,θ
Pr { g1(a, b, H , Dinner , Pad, Pdes, τc, θ, ξ)
= (Qgiv − Q) ≤ 0} ≥ ρ1;
× max g j (a, b, H , Dinner, Pad, Pdes, τ c, θ, ξ) ≤ 0,
for the oxygen concentration and the overall unit
dimensions,
k p ≤ kˆp, H ≤ Hˆ , Dinner ≤ Dˆ,
j =2,3
THEORETICAL FOUNDATIONS OF CHEMICAL ENGINEERING Vol. 46
No. 5
2012
442
DVORETSKII et al.
Results of soling the problem of the optimal design of an industrial turbulent diazotization reactor
Iteration no., k
1
Structural
variables, d
D = 0.04 m;
L = 115 m;
m = 3;
l1 = 40 m;
l2 = 80 m.
Regime (control) variables,
z
T(0) = 296°C;
p = 3;
C, u
Flexibility
function, χ
Probability of meeting
mild constraints, Pr{}
⋅
2225 $
0.326
Pr{g0 ≤ u} = 92.1%
Pr{g1 ≤ 0} = 95%
2230 $
0.00075
Pr{g0 ≤ u} = 94.1%
Pr{g1 ≤ 0} = 97.7%
(1)
G N = 5.1 × 10–5 m3/s;
(2)
G N = 2.55 × 10–5 m3/s;
(3)
G N = 2.55 × 10–5 m3/s.
2
D = 0.04 m;
L = 120 m;
m = 3;
l1 = 42,5 m;
l2 = 82,5 m.
T(0) = 300°C;
p = 3;
(1)
G N = 6.3 × 10–5 m3/s;
(2)
G N = 1.95 × 10–5 m3/s;
(3)
G N = 1.95 × 10–5 m3/s.
3
D = 0.04 m;
L = 123 m;
m = 3;
l1 = 43m;
l2 = 84 m
T(0) = 300°C;
p = 3;
2232 $
–0.036
Pr{g0 ≤ u} = 100%
Pr{g1 ≤ 0} = 98.8%
(1)
G N = 6.1 × 10–5 m3/s;
(2)
G N = 2.05 × 10–5 m3/s;
(3)
G N = 2.05 × 10–5 m3/s.
where u is a scalar variable; Pr{⋅} is the probability of
meeting the constraint {⋅}; ρ0 and ρ1 are specified prob
abilities; g0(a, b, H, Dinner, Pad, Pdes, τc, θ, ξ) = C(a, b,
H, Dinner, Pad, Pdes, τc, θ) is the unit optimal design cri
terion (reduced expenditures); g1(a, b, H, Dinner, Pad,
Pdes, τc, θ, ξ) = (Qgiv –Q(a, b, H, Dinner, Pad, Pdes, τc,
θ)); χ1 is the flexibility function of the unit; Qgiv,
[cOout2 ]giv are the specified unit productivity and outlet
oxygen concentration, respectively; g2(a, b, H, Dinner,
Pad, Pdes, τc, θ, ξ) = ([cOout2 ]giv – cOout2 ); g3(a, b, H, Dinner,
ˆ − M , Mˆ , kˆp, Hˆ , D̂inner are the
Pad, Pdes, τc, θ, ξ) = M
limit admissible mass, pressure coefficient, and overall
dimensions of the adsorbers in the unit, respectively.
Let us exemplify the optimal design of a shortcycle
adsorption unit with the development of a portable
medical oxygen concentrator, the technical specifica
tion on the designing of which includes the following
characteristics to be attained: the concentrator pro
ductivity Qgiv= 0.05 × 10–3 m3/s, the outlet oxygen
concentration [cOout2 ]giv ≥ 90%; ρ0, ρ1 = 0.9, the limit
admissible adsorber mass M = 0.6 kg, adsorption/des
orption pressure ratio Pad/Pdes = k p = 3;, adsorbent bed
height H = 0.4 m, and adsorber diameter Dinner =
0.1 m, respectively.
The alternate variants of equipment implementa
tion included a column adsorber, a twoadsorber con
centrator without pressure equalization, a two
adsorber concentrator with pressure equalization, a
fouradsorber concentrator with pressure equaliza
tion, and a fiveadsorber concentrator with twostage
pressure equalization. For each case, we analyzed dif
ferent variants of airoxygen enrichment (pressure,
with vacuum desorption, vacuumpressure) and types
of adsorbent (grained and block, NaX and LiLSX).
In the course of optimal design, we selected the
twoadsorber variant of a portable medical oxygen
concentrator with vacuum desorption and determined
its optimal structural parameters H* = 0.22 m and
* = 0.035 m; regime variables Pad* = 1.5 × 105 Pa,
Dinner
* = 0.5 × 105 Pa, θ* = 2.5, τ*c = 1.6 s, and Qinit
* =
Pdes
–4
3
2.93 × 10 m /s; and engineering and economic
parameters TC* = 45250 rub, M* = 0.5 kg, and N* =
76 W.
Our practical recommendations on the designing
of medical oxygen concentrators with a productivity
below 0.08 × 10–3 m3/s imply the use of adsorbers with
dimensions 4 ≤ H/Dinner ≤ 6, the pressure variant with
vacuum desorption (kp = Pad/Pdes ≤ 3), and LiLSX
block zeolite adsorbents with deff ≤ 0.5 ×10–3 m. This
improves the energysaving characteristics of medical
oxygen concentrators by 20% on the average in com
parison with world analogues.
The traditional methods for calculating the
strength of the thermally loaded cylindrical shells of
apparatuses, press molds, etc. use the assumption of a
linear temperature profile in the wall of the calculated
equipment, which results in the unreasonably overes
JOURNAL OF CTHEORETICAL FOUNDATIONS OF CHEMICAL ENGINEERING Vol. 46
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2012
A NEW APPROACH TO THE OPTIMAL DESIGN
timated thickness and mass of process equipment
shells. The selfpropagating hightemperature synthe
sis of hardalloy materials by press molding combines
hightemperature and force loadings; high tempera
tures of ∼2000–3000°С are generated in a press mold,
and excessive pressures of ~200 MPa are attained in a
material in the course of press molding. High thermal
and force loads superimposed on one another within
different time intervals and the nonstationarity,
nonisothermicity, and qualitatively different level of
temperature gradients in the walls of process equip
ment shells require a detailed study.
To calculate the strength of the press mold, we used
a mathematical model that includes nonlinear heat
transfer and combustion front motion equations with
boundary conditions [4]. The model inlet variables are
the molding delay time td (the time from the end of
material combustion to the beginning of loading with
internal pressure) and the molding pressure P. In the
calculation of temperature fields, the model takes into
account the combustion rate Ucom and temperature
Tcom of the material of the specimen. The mathemati
cal model allows one to calculate the outlet variables;
the internal wall temperature T1w, the wall’s boundary
layer thickness δ1, and the equivalent stress σeq appear
ing in the wall under thermal and mechanical loads.
The value of δ1 is specified by the admissible wall
across temperature drop, at which the changes in the
material of a press mold are reversible and do not lead
to any loss in the mechanical properties of the wall
material.
The combustion rate Ucom and temperature Tcom of
a pressed material in the synthesis of a product were
considered to be uncertain parameters ξ. The uncer
tainty of information with respect to Ucom and Tcom is
caused by different factors that depend on the proper
ties of an initially prepared molding mixture (bulk
density, moisture content, etc.). The problem of the
strength calculation of a press mold for the selfprop
agating hightemperature synthesis of hardalloy
materials is formulated as follows. It is necessary to
determine the delay time td and the pressure P at which
the minimum thickness δ of the wall of a press mold is
attained, i.e.,
min δ,
δ,t d,P
for the relations expressed as the equation of the heat
transfer mathematical model [1] and the following
constraints:
for the temperature on the internal wall of a press
mold,
g 1(δ, t d, P, ξ) = max min (T1 (δ, t d, P, ξ) − T
w
ξ∈Ξ
t d ,P
lim
) ≤ 0,
443
for the boundary layer thickness of the wall of a
press mold,
g 2 (δ, t d, P, ξ) = max min (10δ1(δ, t d, P, ξ) − δ) ≤ 0,
ξ∈T
t d ,P
for the equivalent stress in the wall,
g 3 (t d, P, ξ) = max min (σ eq (t d, P, ξ) − [σ]) ≤ 0.
ξ∈T
t d ,P
As an example, we solved the problem of optimiz
ing the wall thickness of a press mold for the experi
mentally established ranges of varying the combustion
rate Ucom ∈ [5…25] mm/s and temperature Tcom ∈
[1950…2050]°C of a molding mixture. The abovefor
mulated problem was solved in three iterations. As a
result of solving the problem of optimization, we
determined an optimal press mold wall thickness δ* =
48.3 mm, delay time t d* = 4.3 s, molding pressure P* =
100 MPa, and χ0(δ*) = –0.00019. The thickness of a
press mold δ*= 42 mm, as well as t d* = 4.7 s and P* =
100 MPa, was previously calculated in our works at
nominal Ucom = 15 mm/s and Tcom = 2000°C.
A comparative analysis shows that a press mold
with a wall thickness of 48.3 mm will continue to func
tion during operation independently of the random
change in uncertain parameters ξ within specified
intervals. The scientifically substantiated margin coef
ficient for the wall thickness of a press mold of 15%
was obtained with consideration for the real tempera
ture profile in its wall.
CONCLUSIONS
The proposed efficient algorithm of solving the
twostage problem of the optimal design of engineer
ing systems represents a new scientifically substanti
ated approach, in which the uncertainty in mathemat
ical description coefficients and process parameters is
taken into consideration in the formulation of an opti
mal design problem itself. As a result of using this
approach, we solved the problem of the optimal design
of an industrial turbulent diazotization reactor, devel
oped practical recommendations on the design of
medical oxygen concentrators, and optimized the wall
thickness of a press mold for the hightemperature
synthesis of hardalloy materials.
NOTATIONS
A—set of equipment implementation variants of
industrial chemicalengineering apparatuses;
A—dimensionless kinetic coefficient in the equa
tion of dissolution of the solid phase of an aromatic
amine;
B—set of adsorbent types used in a shortcycle
adsorption unit;
b—adsorbent type;
C—reduced expenditures;
THEORETICAL FOUNDATIONS OF CHEMICAL ENGINEERING Vol. 46
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DVORETSKII et al.
[c A(0)]s—concentration of an aromatic amine in the
solid phase at the inlet of a reactor, mol/m3;
cD—concentration of a diazocompound, mol/m3;
( 0)
c N —concentration of sodium nitrite at the inlet of
a reactor, mol/m3;
cNA—concentration of nitric acid, mol/m3;
cOout2 —concentration of oxygen at the outlet of a
shortcycle adsorption unit, vol %;
cOin2 —concentration of oxygen in the air delivered
to an adsorber for enrichment, vol %;
c(out)—concentration at the outlet of a reactor,
mol/m3;
cσ—concentration of nitrous gases, mol/m3;
cχ—concentration of diazotars, mol/m3;
D—diameter of a tube reactor, m;
Dch—diameter of a mixing chamber, m;
Dinner—diameter of an adsorber, m;
d—vector of structural parameters;
d*—vector d providing an extremum of the target
function F(d, z, ξ);
deq—equivalent diameter of the pore channels in
block zeolite adsorbents, m;
di—ith component of the vector d;
dN—nominal vector of structural parameters;
dtube—diameter of the tube section of a reactor, m;
F—target function (optimality criterion);
F**—function F(d, z, ξ) at an extremum point;
G1(out)—flow rate of the liquid phase of a diazosolu
tion suspension at the outlet of a reactor, m3/s;
0
G N( ) —flow rate of sodium nitrite at the inlet of a
reactor, m3/s;
G N(i)—flow rate of sodium nitrite at the inlet of the
ith reactor section, m3/s;
G(out)—flow rate of a diazosolution suspension at
the outlet of a reactor, m3/s;
(0)
G s —flow rate of the solid phase of a diazosolution
suspension at the inlet of reactor, m3/s;
G s(out)—flow rate of the solid phase of a diazosolu
tion suspension at the outlet of a reactor, m3/s;
g—constraint function;
H—adsorbent bed height, m;
I(k)—set of indices i for points ξ at which con
straints may be violated;
J1—set of indices j for mild constraints;
J2—set of indices j for strict constraints;
k—counter of algorithm iterations;
ki—dimensionless margin coefficient for the ith
component of the vector d;
kp—adsorption/desorption pressure ratio;
L—length of a tube reactor, m;
lj—mounting sites of diffuserconfuser devices, m;
lch—length of a mixing chamber, m;
m—number of constraints in a problem;
m1—number of mild constraints in a problem;
mdc—number of diffuserconfuser devices;
N—power of a portable medical oxygen concen
trator, W;
n—counter of indices i accumulated in the set I(k);
nξ—length of the vector ξ;
P—molding pressure, MPa;
Pad—adsorption pressure, MPa;
Pdes—desorption pressure, MPa;
p—number of sections supplied with sodium
nitrite along the length of a reactor;
Pr{⋅} —probability of meeting the constraint
{{⋅}, }, %;
Q—productivity, t/yr;
Qgiv—specified productivity of a shortcycle
adsorption unit, m3/s;
r—rough approximation function;
S
(k )
—set of the points, at which constraints are
violated;
Т(0)—temperature of an aromatic amine suspen
sion at the inlet of a reactor, K;
Tcom—combustion temperature of the material of a
specimen, °С;
T1w —temperature on the internal wall of a press
mold, °С;
tinit—molding delay time, s;
Ucom—combustion rate of the material of a speci
men, mm/s;
u—scalar variable;
W0—limit adsorption capacity of a zeolite adsor
bent, cm3/g;
y—vector of mathematical model outlet variables;
yj, giv—limit admissible value of the jth outlet vari
able of a chemicalengineering apparatus;
z—vector of control variables;
z*—vector z providing an extremum of the target
function F(d, z, ξ);
αcon—confuser convergence angle;
αdif—diffuser divergence angle;
β—masstransfer coefficient, 1/s;
δ—thickness of the wall of a press mold, m;
δ1—thickness of the boundary layer of the wall of a
press mold, m;
θ—blowback coefficient;
Ξ—variation area of uncertain parameters;
ξ—vector of uncertain parameters;
ξL—lower boundary of variation range of uncer
tain parameters;
ξU—upper boundary of variation range of uncer
tain parameters;
ξΝ—nominal vector of uncertain parameters;
Πη—amount of the solid phase of an amine in a
diazosolution at the outlet of a reactor, %;
JOURNAL OF CTHEORETICAL FOUNDATIONS OF CHEMICAL ENGINEERING Vol. 46
No. 5
2012
A NEW APPROACH TO THE OPTIMAL DESIGN
Πσ—amount of nitrous gases in a diazosolution at
reactor outlet, %;
Πχ—amount of diazotars in a diazosolution at
reactor outlet, %;
ρ0—specified probability of meeting the constraint
g0, %;
ρj—specified probability of meeting the constraint
gj, %;
[σ]—admissible stress, MPa;
σeq—equivalent stress, MPa;
τc—adsorption cycle duration, s;
χ(d)—flexibility function;
Ψ—operator of the mathematical model of a
chemicalengineering apparatus.
SUBSCRIPTS AND SUPERSCRIPTS
* solution to the problem;
0—at the inlet of a reactor;
A—aromatic amine;
ad—adsorption;
c—cycle;
ch—mixing chamber;
com—combustion;
con—confuser;
D—diazocompound;
d—delay;
dc—diffuserconfuser device;
des—desorption;
dif—diffuser;
eq—equivalent;
giv—specified value;
i, j—vector’s component indices;
inner, internal;
k—number of an algorithm iteration;
L—lover boundary;
l—liquid phase;
445
l—index of vector components corresponding to
approximation points;
lim—critical;
N—sodium nitrite;
NA—nitric acid;
Nom—nominal;
O2—oxygen;
out—outlet;
p—pressure;
s—soli phase; tube, tube section of a reactor;
U—upper boundary;
w—wall;
η—amine “breakthrough”;
σ—nitrous gases;
χ—diazotars.
REFERENCES
1. Ostrovskii, G.M. and Volin, Yu.M., Tekhnicheskie
sistemy v usloviyakh neopredelennosti: Analiz gibkosti i
optimizatsiya (Technical Systems under Uncertainty:
Flexibility Analysis and Optimization), Moscow:
BINOMi, 2008.
2. Dvoretskii, D.S., Dvoretskii, S.I., and Peshkova, E.V.,
Computer Simulation of Turbulent Reactors for Fine
Organic Synthesis under Uncertainty, Izv. Vyssh.
Uchebn. Zaved., Khim. Khim. Tekhnol., 2007, vol. 50,
no. 8, p. 70.
3. Akulinin, E.I., Dvoretskii, D.S., Dvoretskii, S.I., and
Ermakov, A.A., Mathematical Simulation of Air
Enrichment with Oxygen in a ShortCycle Adsorption
Device, Vestn. Tomsk. Gos. Tekh. Univ., 2009, vol. 15,
no. 2, p. 341.
4. Stel’makh, L.S., Stolin, A.M., and Dvoretskii, D.S.,
Nonisothermal Method for Calculating the Mold
Equipment of an Apparatus for Compacting the Hot
Products of SelfPropagating HighTemperature Syn
thesis, Theor. Found. Chem. Eng., 2010, vol. 44, no. 2,
p. 192.
THEORETICAL FOUNDATIONS OF CHEMICAL ENGINEERING Vol. 46
No. 5
2012
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Literature critique #1
Date due: April 27, 2015
By Abdulrahman Batarfi
Dvoretskii, D. S., Dvoretskii, S. I., Ostrovskii, G. M., & Polyakov, B. B. (2012). A new approach
to the optimal design of industrial chemical-engineering apparatuses. Theoretical Foundations of
Chemical Engineering, 46(5), 437-445.
In developing a new approach to the optimal design of industrial chemical-engineering
apparatuses, an algorithm has been described. It stems from a two stage problem in the design of
the apparatus and addresses the uncertainty in the mathematical description coefficients and
processes parameters are taken into account for the formulation of the designing problem. This is
advantageous because of the possibility of adjusting the control variables depending on the
measurement of uncertain parameters at the operating stage of an industrial apparatus.
The efficiency of the algorithm is exemplified in apparatuses such as the turbulent tube reactor of
fine organic synthesis and the adsorption oxygen concentrator. Some of the uncertainties
encountered in design include change in the parameters of raw materials. the external
temperatures are Also within certain during operation making it impossible to specify the unique
value.
Optimality criterion takes into account the uncertainties in the mathematical description as it is
sufficient to distinguish them in the dependencies for the target function. The disadvantages are
such that there is no assurity of either the optimality of the obtained solution or the fulfillment of
all constraints, during the operation of the industrial apparatus.
While formulating an optimal design problem, the specific types of target functions and
constrains were based on the concept of two stages in the life cycle of an industrial apparatus.
The constraints may be strict or mild where the former may not be violated under any condition.
I was particularly impressed by the fact that the proposed algorithm was tested with the optimal
design of a number of industrial chemical engineering apparatuses. the optimality criterion and
the possible constraints were also addressed. These being taken into account, the demonstrated
results were impressive e.g. the problem of optimal design of an industrial turbulent diazotization
reactor was solved and practical recommendations on the design of medical oxygen
concentrators were developed among other things.
IEGR 204 – INTRODUCTION TO IE AND COMPUTERS
MiniProject #1 (Slides Due no later than Monday,
5/11/15 via Bb by 11:59pm)
Using Microsoft PowerPoint (PP), complete a presentation about the
sub-area of interest in Industrial Engineering (IE) which you have
started your research on with the earlier assignments (Research
Assignments #1 and #2, and Lab #2). You must also find a 2nd
refereed research article on the same topic area, and a 3rd source of
your choice (webpage, magazine article, textbook, podcast, etc.).
Thus, a total of three sources are mandatory for this project (and
anything short of this will result in a deduction on your project
grade).
The PP slides must meet the following criteria:
8 slides minimum ( including the mandatory 1st slide for the
title page and a last slide entitled “References” for listing your
full APA-style bibliographic citations); both of these were
started during your Lab #2 assignment
The remaining slides in between must meet the following
specifications:
o Your 2nd slide must be entitled “Outline”
o The 3rd slide entitled “Background”, 4th slide entitled
“Research Problem”, 5th and 6th slides should be entitled
“Math Equations Used”, 7th slide entitled “Conclusions”,
and 8th entitled “References” as stated earlier
o
o
o
o
Keep in mind that on the “Math Equations Used”
slide, you must explain 2 or more equations used
within the articles in a brief statement or phrase
If there are no equations in the 1st article, then use
whatever is math related from the article…a
segment of a computer program described, a
mathematical algorithm shown, etc.
Be sure that the 2nd article contains mathematical
equations if the 1st article does not
If neither of the refereed articles contain
mathematical equations, your project will receive a
deduction
Remember, you must use your literature critique
assignment (from your 1st approved article) and a 2nd
refereed article for this project, as well as 3rd source
The presentation title should reflect the area of IE
which your project focuses on and not the title of
any of the articles (e.g., Robotics in IE, Energy
Systems in IE, Human Factors in IE, etc.).
Also, be sure to include your name, class, and due
date of project on the title slide
Remember, you can ‘pull’ some information from your
literature critique assignment into the “Background” or
“Research Problem” slides of the project
You can also use a non-refereed article for your 3rd source
to gather information for your “Background” slide
You must have a minimum of 3 sources/references as
stated earlier, you can gather more if they are related.
However, do not list any sources that are not used in this
project. In other words, having 5 sources will not gain
o
o
o
o
o
o
you any extra points on the project, but could only
enhance your project if utilized properly.
Thus, a minimum of 3 sources (2 refereed research
journal articles, and 1 additional source) must be
cited in the project on the “References” slide
You must include two slides with some IE mathematical
formulations and explain briefly how/why the author(s)
were using the math
You can include a figure(s) in your slides from your IE
research sources as long as you properly cite the
reference source (i.e., you must give the author(s) credit
for the information which you are utilizing!) – remember
to use the APA-format in-text citation with Author’s last
name and date in parenthesis at the end of the caption!
You must make good use of animation on slides and
between slides during transitions (if you need help on
this, ask/see the instructor!)
Again, you must use the APA journal style for your
bibliographic citations on the “References” slide as shown
on my Research webpages (i.e., the same format used on
your Literature Critique assignment – if done correctly)
If the citation(s) is/are incorrect, you will receive
deductions on the project
DO NOT use paragraphs of information on the slides and
please use a serif type font (such as Times New Roman,
Times Roman, etc.) and no size less than 14 pt font on
each slide
Use last week’s lab (from 4/29/15) which your submitted as part of
your PowerPoint presentation slides for this project (i.e., don’t
waste the work you have already done). While working on the
project (or if you complete the work before the due date), if you
have any additional questions, be sure to ask the instructor and not
just a classmate. Do not ask me questions such as (1) “Does this
look right?” or (2) “Is this OK?”…please only ask legitimate questions
which deal with uncertainty about a specific issue/item.
This is a major part of your grade, and if your friend gives you the
wrong information you will not be able to blame them! DO NOT
ASSUME anything if you are unsure about something.
Lastly, submit your PowerPoint presentation slides on-time along
with the 2 additional sources via Bb to receive full credit. If the Bb
submission time stamp is beyond 11:59pm on the due date, you will
receive a 20% deduction per day in points for lateness as on other
assignments. Take this very seriously! If no file is ever submitted
via Bb, you will NOT receive a project grade nor make it up at a
later time.
The timely submission will be part of the overall MiniProject grade.
The remaining portions of the grade will be made up from meeting
the specifications listed above. A project grading table for the PP
slides will be shown at a future date in class or posted via Bb. This
table will detail the full project requirements and allows for
everyone to know ‘up front’ what is necessary to achieve the
maximum number of points.
Do your best!
Be sure to submit your MiniProject Slides by the assigned due date and time.
It may be submitted ahead of time as well!
FINAL NOTE: Again, any assignment that has an Bb submission time stamp beyond the
assigned due time will be reduced (i.e., the grade assigned will have a deduction for
lateness of 20% off per 24 hour period). NO EXCEPTIONS!
If you have any questions, do one of the following: (1) please ask them in class, (2) ask via
email (Richard.Pitts@morgan.edu), or (3) come by during office hours to see me.
Created on May 2, 2015 by Dr. Richard Pitts, Jr.
Last Updated on May 3, 2015 by Dr. Richard Pitts, Jr.