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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 (k) No. 5 2012 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 No. 5 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 No. 5 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 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 Copyright of Theoretical Foundations of Chemical Engineering is the property of Springer Science & Business Media B.V. and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. 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 (k) No. 5 2012 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 No. 5 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 No. 5 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 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 Copyright of Theoretical Foundations of Chemical Engineering is the property of Springer Science & Business Media B.V. and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. 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.
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