ОСНОВИ НА МОДЕЛИРАНЕТО И СИМУЛИРАНЕТО

ОСНОВИ НА МОДЕЛИРАНЕТО И СИМУЛИРАНЕТО
В ИНЖЕНЕРНАТА ХИМИЯ И ХИМИЧНАТА ТЕХНОЛОГИЯ ЛЕКТОР: проф. дтн Христо Бояджиев ВЪВЕДЕНИЕ (5) ТЕМА 1. МОДЕЛИ НА ЕЛЕМЕНТАРНИ ПРОЦЕСИ (26) 1.1. Механика на непрекъснатите среди (30) 1.2. Хидродинамични процеси (45) 1.3. Дифузионни процеси (68) 1.4. Топлинни процеси (74) 1.5. Химически процеси (79) 1.6. Адсорбционни процеси (91)

ТЕМА 2. МОДЕЛИ НА ИНЖЕНЕРНО-ХИМИЧНИ ПРОЦЕСИ (96)
2.1. Механизъм и математично описание (101) 2.2. Теоретични модели (122) 2.3. Моделни теории (133) 2.4. Критериални модели (137) 2.5. Аналогови модели (164) 2.6. Регресионни модели (171) 2.7. Дифузионни модели на колонни апарати (172) ТЕМА 3. ИДЕНТИФИЦИРАНЕ НА ПАРАМЕТРИ (182) 3.1. Права и обратна задача. Некоректност на обратни задачи (183) 3.2. Методи за решаване на некоректни идентификационни задачи (189) 3.3. Идентификация при многопараметрични модели (197) ТЕМА 4. СТАТИСТИЧЕСКИ АНАЛИЗ НА МОДЕЛИ (213) 4.1. Основни понятия в статистическия анализ (215) 4.2. Проверка на хипотези (229) 4.3. Значимост на оценки на параметри и адекватност на модели (233)

ТЕМА 5. КАЧЕСТВЕН АНАЛИЗ НА МОДЕЛИ (260)
5.1. Обобщен анализ (261) 5.2. Нестационарен процес (268) 5.3. Стационарен процес (269) ТЕМА 6. МАЩАБЕН ПРЕХОД (275) 6.1. Подобие и мащабен преход (278) 6.2. Мащабен ефект (290) 6.3. Дифузионен модел и мащабен преход (297) ТЕМА 7. ХИМИКО-ТЕХНОЛОГИЧНИ СИСТЕМИ (303) 7.1. Симулиране на ХТС (305) 7.2. Симулиране при регламентирани изходни променливи (331) 7.3. Модели на отделни блокове (345) 7.4. Оптимален синтез на ХТС (356) 7.5. Реновация на ХТС (373)

ТЕМА 8.ТЕОРЕТИЧНИ ПРОБЛЕМИ (391)
8.1. Some theoretical problems of chemical engineering (391) 8.2. Similarity and modeling (434) 8.3. Regression models (456) 8.4. On the modeling of an airlift photobioreactor (482) 8.5. Non-linear mass transfer from a solid spherical particle dissolving in a viscous fluid (496) 8.6. Diffusion models and scale-up of a spouted bed (511)

Количествено описание. Инженерна химия и химична технология.
ВЪВЕДЕНИЕ Количествено описание. Инженерна химия и химична технология. Основни проблеми.

Планиране на експеримента Изясняване на механизма Моделиране
Симулиране Количествено описание Хипотеза за механизма Алгоритъм Програма Оптимална реконструкция (реновация) Оптимално управление Оптимално проектиране Мащабен преход Математична структура Качествен анализ Изчислителен (математичен) експеримент Идентификация на параметри Статистически анализ на адекватност Регуляризация (обратна некоректна задача)

ТЕМА 1. МОДЕЛИ НА ЕЛЕМЕНТАРНИ ПРОЦЕСИ
1.1. Механика на непрекъснатите среди Скаларни и векторни полета. Тензор на напрежението. 1.2. Хидродинамични процеси Основни уравнения. Начални и гранични условия. Математично описание на конкретни процеси. Обобщени променливи. Основни параметри. 1.3. Дифузионни процеси Основни уравнения. Гранични условия. Скорост на дифузия. 1.4. Топлинни процеси Основни уравнения. Скорост на топлопренасяне. 1.5. Химически процеси Стехиометрични уравнения. Механизъм и маршрут. Кинетика на прости реакции. Кинетика на сложни реакции. 1.6. Адсорбционни процеси. Физична адсорбция. Химична адсорбция.

Литература: Л. Д. Ландау, Е.М. Лифшиц, Теоретическая физика. Том. VI. Гидродинамика, Москва, Наука, 1988. Г. Шлихтинг, Теория пограничного слоя, Наука, Москва, 1969 (H= Schlichting, Granzshicht – Theorie, Verlag G. Braun, Karlsruhe, 1965). Chr. Boyadjiev, V. Beschkov, Mass Transfer in Liquid Film Flows, Publishing House Bulg. Acad. Sci., Sofia, (Хр. Бояджиев, В. Бешков, Массоперенос в движущихся пленках жидкости, Мир, Москва, 1988). Р. Рид, Т. Шервуд, Свойства газов и жидкостей, ГОСТЕХИЗДАТ, Москва, (R.C. Reid, T.K. Sherwood, The properties of Gasses and Liquids, McGraw-Hill, New York, 1958.) Д. А. Франк-Каменецкий, Диффузия и теплопередача в химической кинетике, Наука, Москва, 1967.

ТЕМА 2. МОДЕЛИ НА ИНЖЕНЕРНО-ХИМИЧНИ ПРОЦЕСИ
2.1. Механизъм и математично описание Механизъм на физична абсорбция при съвместно движение на газ и течност. Математично описание. Обобщени променливи, характерни и характеристични мащаби. Безразмерни параметри и механизъм на процеса. Гранични условия и механизъм. Кинетика и механизъм. 2.2. Теоретични модели Модел на физична адсорбция в стичащ се филм. Модел на кинетиката в обобщени променливи. Общи свойства. 2.3. Моделни теории Модел на Хигби. Филмови модели. 2.4. Критериални модели Модел на абсорбция в колона с нареден пълнеж. Обобщени променливи. Обобщен индивидуален случай – подобие. Критериални уравнения. Анализ на измеренията. Математична структура на критериалните модели. Определяне на критериите за подобие. Определящи и определяеми комплекси. Обобщен анализ. Някои грешки при критериалните модели.

2.5. Аналогови модели Моделиране масообменни процеси в колонни апарати. Формална аналогия. Определяне на параметрите. 2.6. Регресионни модели Моделиране без хипотеза за механизма. Регресионни уравнения 2.7. Дифузионни модели на колонни апарати Масопренасяне с химична реакция. Междуфазно масопренасяне. Процеси в ерлифтни апарати. Моделиране на бавни процеси. Литература: В. В. Рамм, Абсорбция газов, Химия, Москва, 1976. Хр. Бояджиев, В. Бешков, Массоперенос в движущихся пленках жидкости, Мир, Москва, 1988). А. А. Гухман, Введение в теорию подобия, Высшая школа, Москва, 1973. А. А. Безденежных, Математические модели химических реакторов, Техника, Киев, 1970. Дж. Астарита, Масопередача с химической реакции, Химия, Ленинград, 1971.

2.7. ДИФУЗИОННИ МОДЕЛИ НА КОЛОННИ АПАРАТИ
Many mass transfer processes in column apparatuses may be described by the convection – diffusion equation with a volume reaction. These are gas absorption in column with (or without) packet bed, chemical reactors for homogeneous or heterogeneous reactions, air - lift reactors for biochemical or photochemical reactions. The convective transfer in column apparatuses is result of a laminar or turbulent (large-scale pulsation’s) flows. The diffusive transfer is molecular or turbulent (small-scale pulsation’s) The volume reaction is mass sours as a result of chemical reactions or interphase mass transfer. The scale - up theory shows that the scale effect in mathematical modeling is result of the radial nonunformity of the velocity distribution in the column.

4.1. DIFFUSION MODEL Let’s consider liquid motion in column apparatus with chemical reaction between two of the liquid components. If suppose for velocity and concentration distribution in the column: , the mathematical description has the form: . The radial nonuniformity of the velocity is the cause for the scale effect (decreasing of the process efficiency with increasing of the column diameter) in the column scale - up. That is why average velocity and concentration for the cross - section’s area must be used.

4.2. AVERAGE CONCENTRATION MODEL
Let’s the average values of the velocity and concentration for the cross - section’s area are: As result it is possible to present the velocity and concentration as: , where: .

The average concentration model may be obtained if put these expressions in the convective – diffusion equation, multiply by r and integrate over r in the interval [0,R]. As a result is obtained: where:

4.3. INTERPHASE MASS TRANSFER MODEL
In the cases of interphase mass transfer in gas - liquid or liquid - liquid systems, in the model equations must be to introduce convection - diffusion equations for the two phases and the chemical reaction rate must be replaced with interphase mass transfer rate: As a result the diffusion model for interphase mass transfer in the column apparatuses has the form: where Di and εi (i=1,2) are diffusivities and hold - up coefficients (ε1 +ε2=1). The boundary conditions of counter-current gas-liquid bubble column with column height l have the form: where are inlet average velocities in gas and liquid phases.

The average concentration model:
where and are velocity and concentration radial nonuniformities in the column.

ТЕМА 3. ИДЕНТИФИЦИРАНЕ НА ПАРАМЕТРИ
3.1. Права и обратна задача. Некоректност на обратни задачи. 3.2. Методи за решаване на некоректни идентификационни задачи. 3.3. Идентификация при многопараметрични модели. Литература: А. Н. Тихонов, В. Я. Арсенин, Методы решения некоректных задач, Наука, Москва, 1979. Chr. Boyadjiev, Some theoretical problems of chemical engineering, in: Proceedings of 11-th Workshop on Transport Phenomena in Two-phase flow, Sunny Beach Resort, Bulgaria, 1-30, September 1-5, 2006.

3.2. Методи за решаване на некоректни идентификационни задачи
A main problem of the modelling of the heat and mass transfer processes is the parameters identification in the mathematical description, based on experimental data. The inverse identification problem is often an incorrect (ill-posed), i.e. the solution is sensible with respect to the errors of the experimental data. Let us consider a numerical model: y = f (x,b) , where f is an objective function, expressed analytically, numerically or through an operator (algorithm); x = (x1,…,xm) is a vector of independent variables, b = (b1,…,bJ) - vector of parameters. The parameters of the model should be determined by means of N experimental values of the objective function ŷ = (ŷ1,…,ŷN) . This requires the introduction of a least square function: where yn = f (xn ,b) are the calculated values of the objective function of the model (1), while xn = (x1n,…,xmn) are the values of the independent variables from the different experimental conditions (regimes), n = 1,…,N. The parameters of the model can be determined upon the conditions imposed by the minimum of the function Q = (b1,…,bJ) with respect to the parameters b = (b1,…,bJ).

3.1 INCORRECTNESS OF THE INVERSE PROBLEM
Let us consider the one-parameter model: y = 1 - exp (-bx). The relation between the objective function and the parameter in the figure is typical for a number models of heat and mass transfer processes. y Δy Δy In the figure is seen that if b = 5, the inverse identification problem is correct (when 0 < x < 0.3), incorrect (if 0.31 < x < 0.65) and essentially incorrect (when 0.66 < x < 1). y0 Δy Δb1 b0 Δb2 Δb3 b

Essentially Incorrect problem
Let us consider the two-parameter model: y = 1 – b1 exp (-b2 x) , where and are exact parameter values. The parameter identification problem will be solved by the help of the “experimental” data, obtained by a generator of random numbers. In the figures, are seen the horizontals of the least square function in different interval of x when inverse problem is correct, incorrect and essentially incorrect. These results show that the least square method is correct when the differences between exact parameter values and the parameter values in the least square function minimum are very small. These differences are too large, when the inverse problem is incorrect. In the case, when inverse problem is essentially incorrect the least square function has not a minimum. Incorrect problem Essentially Incorrect problem Correct problem The results obtained show figures, that in the cases of incorrect inverse problems, the least square function minimization is not lead to solution of the inverse problem and for the problem solution must be use additional conditions.

3.2 REGULARIZATION OF THE ITERATIVE METHOD FOR PARAMETER IDENTIFICATION
Let the iteration procedure starts with an initial approximation b(0)=(b1(0),…,bJ(0)) . The values of bi = (b1i,…,bJi) , where i is iteration number, are result of the conditions imposed by the movement towards the anti - gradient of the function Q(b): bji = bj(i-1) – β(i-1) Rj(i-1) , j = 1,…,J , where Here βi is the iteration step and β0 = (arbitrary small step value). Each iteration step is successful if two conditions are satisfied: The first condition indicates that iterative solution (bi) approaches the solution at the minimum (b*), while the second condition in concerns the approach of the iterative solution (bi) towards the exact solution ( ). Obviously, it is due to the effect of the problem incorrectness The results obtained permit to create an algorithm for solution of the inverse identification problems.

3.3 INVERSE PROBLEM SOLUTION
The proposed algorithm was used for the correct problem solution (0 < x < 0.3) and the results are shown on the table. One and two-parameter model solutions Δŷ [%] b* i b1* b2* 5 4.9678 337 1.0025 5.0674 128 10 4.9351 339 4.9218 172 The parameters identification problem will be solved by minimization of the least square function, where xn = 0.01 n , n = 31,...,65 . The incorrect problem solution for the one-parameter model (b(0) = 6, γ = 0.5) and two-parameter model (b1(0) = 1.1, b2(0) = 6, γ = 0.05) are shown on the table. Incorrect problem solution Δŷ [%] b* i b1* b2* 5 5.0614 1213 1.1797 5.4666 642 10 5.1232 1217 1.3778 5.9106 416

Statistical analysis of the model adequacy (0.31 ≤ x ≤ 0.65)
The model adequacy is defined by the variance ratio F = S2 / Sε2, where S is model error variance, Sε – experimental data variance. The condition of the model adequacy is F ≤ FJ (α,ν,νε) , where FJ is tabulated value of the Fisher’s distribution (criteria). The statistical analysis of the model adequacy was tested for 0 ≤ x ≤ 0.30 and 0.31 ≤ x ≤ The results are presented on tables. Statistical analysis of the model adequacy (0 ≤ x ≤ 0.30) J Δŷ [%] b1* b2* γ Sε .10-2 S .10-2 F FJ 1 5  4.9678 0.9 1.7933 1.7071 0.9061 2.24 10 4.9351 3.5867 3.4139 0.9059 2 1.0025 5.0674 1.8354 1.0475 2.25 4.9218 3.4434 0.9217 Statistical analysis of the model adequacy (0.31 ≤ x ≤ 0.65) J Δŷ [%] b1* b2* γ Sε .10-2 S .10-2 F FJ 1 5  5.0614 0.5 2.6042 2.3588 0.8205 2.19 10 5.1232 5.2083 4.7328 0.8257 2 1.1797 5.4666 0.05 2.3656 0.8252 2.20 1.3778 5.9106 4.7349 0.8265

Statistical analysis of the model adequacy (0.66 ≤ x ≤ 1) .
The parameters identification problem when inverse problem is essentially incorrect if solved for 0.66 ≤ x ≤ 1 . One and two-parameter model solutions (0.66 ≤ x ≤ 1) . Δŷ [%] b* i b1* b2* 5 5.1828 2066 2.1720 6.1731 54 10 5.3816 2156 4.9003 7.4004 128 Statistical analysis of the model adequacy (0.66 ≤ x ≤ 1) . J Δŷ [%] b1* b2* γ Sε .10-2 S .10-2 F FJ 1 5  5.1828 5 2.7850 2.5988 0.8707 2.19 10 5.3816 5.5701 5.2482 0.8723 2 2.1720 6.1731 2.7851 2.6221 0.8855 2.20 4.9003 7.4004 0.8877

The solution of the model parameters identification problem by the help of the least square function minimization manifests a large difference between the exact and calculated (as a function minimum) parameter values i.e. the minimization of the least square function is not a solution of the parameter identification problem. This difference is not result of the experimental data size and can be explained with the inverse problem incorrectness, i.e. the parameter value sensibility with respect to the experimental data errors. An additional condition is introduced for the inverse problem regularization, which permits to use least square function minimization for a solution of the model parameter identification problem. A statistical analysis of the model adequacy is a criterion for the applicability of the presented iterative method for the model parameters identification.

3.3. Идентификация при многопараметрични модели
HIERARCHICAL APPROACH FOR PARAMETER IDENTIFICATION OF BIOENGINEERING MULTIPARAMETER MODELS INTRODUCTION The kinetic of many chemical, biochemicals, photochemical or catalytic reactions is very complex, i.e. the kinetic model consists of many equations. The number of the parameters of the separate equations is not great, the total number of parameters is great. Model parameters identification in these cases is very difficult to be done, because of the multiextremal least square function and because of the fact that some minima are “ravine” type.

PROBLEM FORMULATION Let’s consider the following multiparameter model: where t is the time, ci(t) and ki (i = 1,…,M) are objective functions (concentrations of the reagents) and parameters in the model, αi - number of parameters of the i-th equation. For parameter identification problem solution we will use experimental data for the objective functions : ci(e) = ci(e)(tn) , n = 1,…,N , where N is the number of experimental data measurements. The least square functions for the separate objective functions are: where ci(t) , i = 1,…,M are solutions of the model equations. The least square function of the parameter identification in the model is: The total parameters number: is very large and in many cases it is not possible to minimize the equation for the least square function.

The experimental data for the objective functions (concentrations) can be presented, using polynomial approximations: ci(e)(tn) = Pi(t) , i = 1,…,M , where Pi(t) are polynomials of 3 or 4 power. Let’s consider the first equation of the model, where ci ≡ Pi , 2 ≤ i ≤ M, i.e. The minimization of the least square function permits to obtain parameter values in the first equation in the model. This procedure is possible to repeat for all equations in the model (step by step). It is possible the obtained parameter values to be used as an initial approximation for the model parameters identification. For this purpose it is possible to use a hierarchical approach and to obtain the first approximation of the parameter identification problem solution. On this base it is possible to minimize Q , i.e. to obtain “exact” parameter values. The hierarchical approach for parameter identification of multiparameter models will be tested for fermentation systems modeling.

FERMENTATION SYSTEMS MODELING
Mathematical model of the fermentation kinetics of gluconic acid production by Gluconobacter oxydans consists of 4 equations for biomass, product (gluconic acid) and substrates (glucose and oxygen). The dependence of the specific growth (μ) rate from glucose and oxygen substrates was assumed to follow the Monod kinetic model which considers substrate limitation. The biomass growth can be described as where the specific growth rate μ is given by the Monod type model as The kinetics of gluconic acid formation was based on the Luedeking-Piret equation, originally developed for the fermentation of lactic acid. It is a model, which combines growth - and non-growth - associated contributions towards product formation, i.e. the growth dx/dt and instantaneous biomass concentration (x): , where k1 and k2 are Luedeking-Piret equation parameters for growth- and non-growth-associated product formation respectively.

The rate of glucose utilization is presented by mass balance equation, i.e. the glucose form cell material dx/dt, metabolic product dcGa /dt and cells activity k4 x : The dependence of biomass growth dx/dt, product formation dcGa /dt and cells activity x on the oxygen consumption rate is given by: . The initial conditions of the model equations are: The model equations will be solved in the time interval [0 ≤ t ≤ t0] , where the biomass concentration increases.

The minimization of the least square function will be used for parameter identification. The problem will be solved in dimensionless form, where the characteristic scales are the maximal experimental values of the concentrations in the interval [0 ≤ t ≤ t0]: As a result the model equations have the form: where The initial conditions of the equations are:

EXPERIMENTAL DATA The parameter identification problem will be solved, using the real experimental data (S. Velizarov , V. Beschkov, Production of Free Gluconic Acid by Cells of Gluconobacter Oxydans, Biotechnology Letters, 16 (7), 715 – 720 , 1994). The concentrations of biomass, gluconic acid, glucose and oxygen as time functions will be presented in dimensionless forms The dimensionless experimental data for the concentrations permit to obtain their polynomial approximations and to calculate polynomial approximation error variances :

; It could be supposed that the differences between polynomial approximation error variances and experimental data error variances are negligible. That is why we could use polynomial approximation instead of concentration as time function.

INITIAL APPROXIMATION FOR THE MODEL PARAMETERS
The initial approximation of the model parameters could be obtained, by solving model equations, where the concentration time distribution has to be replaced with their polynomial approximations. For the biomass model parameters identification the following problem must to be solved: and the initial parameter values approximations and will be obtained after least square function minimization:

The next steps are the consecutive solutions of the problems for the gluconic acid productions, glucose and oxygen consumption: after the minimization of the least square functions: , , . The obtained initial approximations of the parameter values are shown in the table. In these equations the experimental data values are replaced by their polynomial approximations.

HIERARCHICAL APPROACH FOR MODEL PARAMETERS IDENTIFICATION
For the model parameters identification a hierarchical approach is used. The first step is to solve the equations for the biomass growth and gluconic acid production. and to minimize: The obtained parameter values are the first approximation in the identification problem solution. The next steps are consecutive solutions of the problems:

The last step is to solve the model equations, where the obtained parameter values for the oxygen consumption are replaced: and the biomass parameter values to be calculated by least square function minimization:

The obtained by hierarchical approach values of the parameters are the first approximation (see the table). The first parameter values approximation permit to obtain the exact parameter values. For this purpose model equations should be solved and the exact parameter values will be obtained using the least square function minimization (see the table): where N = 8 is the number of the experimental data. The first approximations of the parameter values are used as a zero approximations in the minimization procedure. A comparison of the calculated biomass, gluconic acid, glucose and oxygen concentrations with the experimental data is given in figures 1 – 4. The variances Sexp and Spol are obtained using experimental data and their polynomial approximations.

Table Parameter Evaluation Initial parameter values approximation
First parameter values approximation Exact parameter values

Fig. 1. Comparison of the calculated values and experimental data for biomass dimensionless concentration Fig. 2. Comparison of the calculated values and experimental data for gluconic acid dimensionless concentration Fig. 3. Comparison of the calculated values and experimental data for glucose dimensionless concentration Fig. 4. Comparison of the calculated values and experimental data for oxygen dimensionless concentration

CONCLUSIONS The proposed method solves parameter identification problem for multiparameter models, when the least square function is multiextremal. The using of polynomial approximations of the experimental data permit to obtain parameter values in the separate model equations. On this base is proposed a hierarchical approach for parameter identification. The method is tested on fermentation process modeling, resulting in the gluconic acid production. The negative value of the parameter K1 reveals that a part of gluconic acid becomes a substrate for the biomass.

ТЕМА 4. СТАТИСТИЧЕСКИ АНАЛИЗ НА МОДЕЛИ
4.1. Основни понятия в статистическия анализ Случайни събития. Случайни величини. Математично очакване. Дисперсия. Закон на Гаус. 4.2. Проверка на хипотези. Правила за проверка. Равенство на дисперсии. Еднородност на дисперсии. 4.3. Значимост на оценки на параметри и адекватност на модели. Значимост на оценка на параметър в регресионен анализ. Адекватност на регресионен модел. Пригодност на модели. Адекватност на теоретични модели и моделни теории. Литература: Н. Драйпер, Г. Смит, Прикладной регресионный анализ, Статистика, Москва, 1976.

ТЕМА 5. КАЧЕСТВЕН АНАЛИЗ НА МОДЕЛИ
5.1. Обобщен анализ Качествен анализ на масопренасяне с химична реакция. Обобщени променливи. Влияние на характерната скорост. Влияние на геометричните мащаби. 5.2. Нестационарен процес Кратковременни процеси. Продължителни процеси. 5.3. Стационарен процес Граничен слой. Влияние на химичната кинетика. Влияние на числото на Фурие. Литература: А. А. Гухман, Введение в теорию подобия, Высшая школа, Москва, 1973. А. А. Гухман, Итоги работ в области развития и применения методов обобщенного анализа, ИФЖ, 53, No. 5, , 1987.

ТЕМА 6. МАЩАБЕН ПРЕХОД 6.1. Подобие и мащабен преход Еднофазни системи. Двуфазни системи.Масопренасяне с химична реакция. Математичен модел и подобие. 6.2. Мащабен ефект Коефициент на мащабния ефект. Природа на мащабния ефект. Видове неравномерности. 6.3. Дифузионен модел и мащабен преход Еднофазен модел. Еднофазен модел с химична реакция. Двуфазен модел с междуфазно масопренасяне. Литература: Масштабной переход в химической технологии, Под ред. А. М. Розена. Химия, Москва, 1980. Chr. Boyadjiev, Some theoretical problems of chemical engineering, in: Proceedings of 11-th Workshop on Transport Phenomena in Two-phase flow, Sunny Beach Resort, Bulgaria, 1-30, September 1-5, 2006.

6.3. Дифузионен модел и мащабен преход

4.1. DIFFUSION MODEL Let’s consider liquid motion in column apparatus with chemical reaction between two of the liquid components. If suppose for velocity and concentration distribution in the column: , the mathematical description has the form: . The radial nonuniformity of the velocity is the cause for the scale effect (decreasing of the process efficiency with increasing of the column diameter) in the column scale - up. That is why average velocity and concentration for the cross - section’s area must be used.

4.2. AVERAGE CONCENTRATION MODEL

The average concentration model may be obtained if put these expressions in the convective – diffusion equation, multiply by r and integrate over r in the interval [0,R]. As a result is obtained: where:

4.3. INTERPHASE MASS TRANSFER MODEL
In the cases of interphase mass transfer in gas - liquid or liquid - liquid systems, in the model equations must be to introduce convection - diffusion equations for the two phases and the chemical reaction rate must be replaced with interphase mass transfer rate: As a result the diffusion model for interphase mass transfer in the column apparatuses has the form: where Di and εi (i=1,2) are diffusivities and hold - up coefficients (ε1 +ε2=1). The boundary conditions of counter-current gas-liquid bubble column with column height l have the form: where are inlet average velocities in gas and liquid phases.

The average concentration model:
where and are velocity and concentration radial nonuniformities in the column.

ТЕМА 7. ХИМИКО-ТЕХНОЛОГИЧНИ СИСТЕМИ
7.1. Симулиране на ХТС Модел на ХТС. Методи за симулиране. Последователно модулен (йерархичен) подход. Ациклични ХТС. Циклични ХТС. Независими контури. Разкъсващи множества. Оптимален ред. 7.2. Симулиране при регламентирани изходни променливи. Зона на въздействие. Абсолютно независимо въздействие. Независимо въздействие. Комбинирани зони. 7.3. Модели на отделни блокове Видове модели. Топлообмен. Сепариране. Химически процеси. 7.4. Оптимален синтез на ХТС Оптимизация. Оптимален синтез. Основни задачи. Методи на синтез. Оптимален синтез на системи за рекуперативен топлообмен (СРТ). 7.5. Реновация на ХТС Математично описание. Математични модели. Основни задачи. Реновация посредством оптимален синтез на ХТС. Реновация чрез въвеждане на високоинтензивни апарати. Реновация чрез въвеждане на високоефективни процеси. Литература: Ф. М. Островский, Ю. М. Волин, Методы оптимизации сложных химико-технологических схем, Химия, Москва, 1970. Ф. М. Островский, Ю. М. Волин, Моделирование сложных химико-технологических схем, Химия, Москва, 1975.

ТЕМА 8.ТЕОРЕТИЧНИ ПРОБЛЕМИ
8.1. Some theoretical problems of chemical engineering INTRODUCTION The main purpose of the chemical engineering and chemical technology is the quantitative description of the processes and systems for their optimal design, control or renovation. This problem is being solved by modeling and simulation methods. According to the association for the Advancement of Modeling and Simulation techniques in Enterprises (AMSE), the modeling concerns the schematic description of systems and devices, whereas the simulation is the use of the models to investigate and/or optimize the processes without experimenting on real systems. The modeling consists of three stages: construction of the mathematical description structures (on the base of a hypothesis for the process mechanisms), parameters identification in the model equations (using experimental data) and statistical analysis of the model adequacy.

The simulation creates (use) methods, algorithms and software and solves model equations, i.e. quantitative description is a result of mathematical (numerical) experiments. The models of the chemical engineering consists process kinetics equations while the chemical technology models are based on the mass and heat balance equations. The theoretical problems of chemical engineering are different on different stage of the modeling and simulation. The theoretical problems in the construction of mathematical description structures are related with the necessity of full correspondence between every physical effect in the process and the mathematical (differential) operator in the model equations. The parameter identification problems ensue from the incorrectness of the inverse problem and multiextremity of the least square function. The using of commercial software for differential equations solutions make easy the processes simulation except for the scale-up problem solutions.

y = h , u = ug , v = vg , pn = pσ + png , pτ = pτg , c = c*,
1. MASS TRANSFER IN FILM FLOW 1.1.Laminar films The mathematical model of mass transfer in laminar film flow over vertical solid interface contain the Navier - Stokes equations, continuity equation, film thickness equation and convection-diffusion equation in the approximation of the boundary layer theory: , , , , x = 0 , u = u0 , c = c0 ; y = 0 , u = v = 0 , ; y = h , u = ug , v = vg , pn = pσ + png , pτ = pτg , c = c*, where Pσ- capillary pressure, c*- equilibrium concentration. The mass transfer rate ( J ) is possible to express by the mass transfer coefficient ( k ) and average mass flux on the gas-liquid interphase: .

, , . , i.e. the mass transfer rate (k) increase if the film length decrease. This result is a base for the intensification of the gas absorption in column apparatuses with packing bed, using packing elements with small size (small liquid film length l). The effect of the chemical reaction in the liquid phase on the mass transfer rate was solved too .

Fig.2 Comparison between theoretical
and experimental results for Sherwood number Fig.1 Comparison between theoretical and experimental results for the film thickness

1.2. Effect of gas flow An addition of the gas phase equations in permit to solve the iterphase mass transfer problems. , , , , , , where Q is flow rate, ai and ni (i = 0,1,2) are shown in the table and .

i ai ni 3.011 0.348 1 1.608 2 3.052 1.283 The functions f0(η) and f1(η1) are solutions of the problem: ; , where f0 is the Blasius function. The solution of Eqs. permits to obtain initial conditions: and the problem can be solved as a Cauhy problem. The main results of these investigations are the expressions for velocity distributions and mass transfer rate in the two phases and the new universal functions f1, φ0 , φ1 and φ2 .

1.3. Effect of surfactants Many experimental investigations show the influence of the surface active agents (SAA) on the hydrodynamics and mass transfer.The theoretical analysis of SAA effect uses the same equations, where in the boundary conditions must add the surface tension gradient as a result of SAA: . The problem was solved for the cases of adsorption or diffusion mechanism of the SAA transfer. The theoretical and experimental results show that SAA effects is localized at the two ends of the film flow and mass transfer rate decrease. SAA effect is absent from the middle part of the long films and the agreement between theoretical and experimental results for the film thickness is very well.

Fig.3. Theoretical and experimental results of the SAA influence on the film thickness
Fig.4. Theoretical and experimental results for the wave number

1.4. Effect of capillary waves
Many experimental data show that for Re > 6 the flow pattern of the liquid film is wavy. In the cases of fully absence of SAA the waves are possible for long films at Re > 2. The theoretical and experimental investigations of the flow stability, velocity distribution and mass transfer rate permit to obtain wave number (n), phase velocity (αr), wave amplitude (A). Fig.5 Theoretical and experimental results for the wave amplitude Fig.6 Theoretical and experimental results for the phase velocity

On the Fig.7 is seen the difference between the streamlines and liquid particles trajectories, i.e. an absence of convective mass transfer from the liquid volume to the gas-liquid interface. The main theoretical result is absence of surface renovation caused by the waves (see Fig. 7) and the increase of the mass transfer rate is result of the new velocity distribution only. Fig.7. Streamlines (continuous lines) and liquid particle trajectories lines (broken lines) Fig.8. Theoretical and experimental results for velocity distribution in co-current gas-liquid flow

2. MASS TRANSFER IN BOUNDARY LAYERS
The main part of the mass and heat transfer processes are realized in the boundary layers near the face interface in the gas-liquid, liquid-liquid or gas (liquid)-solid systems. These layers are very thin which permit to use simplified model equations in boundary layer approximation. 2.1. Boundary layer approximation In two phase flow the boundary layer approximation of the hydrodynamic equations and convective-diffusion equation has the form: , ; with boundary conditions: ; ; .

The theoretical analysis of the full system of Navier - Stokes equations and convection-diffusion equation show that the boundary layer approximation is valid for big values of the Reynolds and Peclet numbers . In these conditions the face interphase y = 0 is flat and all deviations are result of the loss of stability. The shape of the face interphase is not possible to be obtained in the boundary layer approximation. The model in dimensionless form has 6 parameters: Rej , Pej , θj , j = 1,2 , . The dimensionless numbers play the role of similarity criteria. The comparison of the θ2 with Rej ( j = 1,2 ) show that they are incompatible, i. e. the physical modeling is impossible in two - phase flow systems. The using of the asymptotic method permits to obtain the velocity distributions in gas and liquid phases which are in good agreement with the experimental data (Fig. 8).

2.2. Interphase mass transfer in gas-liquid and liquid-liquid systems.
The theoretical and experimental investigations of the interphase mass transfer in two phase gas-liquid flow were made in boundary layer approximation. If similarity variables are used: , j = 1,2 , , the problem has the form ; , , , , where .

The using of the asymptotic method permit to obtain the solutions of by the help of special functions of the Schmidt numbers Scj (j = 1,2) . On the Figs. (9,10) are shown the comparison between theoretical result and experimental data (dimensionless concentrations). After the solution is possible to obtain Sherwood numbers: . Fig. 9. Theoretical and experimental results for dimensionless mass flux (ψ’1(0)) for well soluble gas (ammonia desorption) Fig.10. Theoretical and experimental results for dimensionless mass flux (ψ’2(0)) for slightly soluble gas (carbon dioxide desorption)

2.3. Counter current flow The theoretical analysis of counter current flows shows that it is a non-classical problem of mathematical physics (parabolic boundary value problem with changing direction of the time). It was shown that this non classical problem can be described as consisting of several classical problems. The obtained theoretical results are in a good agreement with the experimental data . The obtained velocity and concentration in the boundary layers permit to calculate Sherwood number and energy dissipation (E) in the boundary layers. Its ratio A = Sh / E is the energy efficiency of the mass transfer. The obtained results show that co-current flow regime is more efficient than counter-current one.

Fig.11. Theoretical and experimental results of velocity distribution
Fig.12. Theoretical and experimental results of interface velocity

3. NON-LINEAR MASS TRANSFER
The non - linear mass transfer is a result of the non-linearity of the convection - diffusion equation: ρ(c) w(c) grad c = div [ρ(c) D(c) grad c] ± kcn In the linear cases ρ, w, D do not depend on concentration, n = 1 and the linear mass transfer theory lead to next principle characteristics: the mass transfer rate does not depend on the mass transfer direction; the mass transfer coefficient does not depend on the concentration. All deviations from these conditions manifest an existence of the non-linear effects which are quantitative and qualitative . The qualitative non-linear effects are result of concentration dependence of the density, viscosity and diffusivity. The quantitative non-linear effects are result of secondary flows arised out of: big concentration gradient; surface tension gradient (Marangoni effect); vertical density gradient (natural convection); pressure gradient (Stephan flow). Practically all deviations from the linear mass transfer theory was explained as a Marangoni effects but the theoretical analysis show that the main non-linear mass transfer effect is a result of the big concentration gradients.

x = 0 , u = u0 , c = c0 ; y = 0 , u=0 , y → ∞ , u = u0 , c = c0 .
3.1. Gas (liquid) – solid systems In the cases of systems with intensive interphase mass transfer the big concentration gradient induces secondary flow at the face interphase. The velocity of this flow is directed normally to the face interphase and it is determined from the concentration gradient: . As a result the velocity distribution is a function of the concentration i.e. the convection-diffusion equation is non-linear. The mathematical model of non-linear mass transfer near solid interphase y = 0 has the form: , , ; ; x = 0 , u = u0 , c = c0 ; y = 0 , u=0 , y → ∞ , u = u0 , c = c0 .

The using of the similarity variables lead to a new form of the model:
, , , where θ is the non-linear mass transfer parameter : . For the Sherwood number was obtained: . The solution of this problem was obtained using asymptotic method and special function of Shmidt number. On the Table are shown comparison results of the asymptotic theory and numerical experiments. On the Table is seen that a change of the mass transfer direction lead to the change of the mass transfer rate. If the mass transfer is directed from the volume to the interface (θ < 0) the mass transfer rate increase as a result of the non-linear effect. In the opposite direction the mass transfer rate decrease.

θ ε = 1 ε = 2 ε = 3 ε = 4 -ψ’N (0) -ψ’(0) Table
Comparison of the results of the asymptotic theory ψ’(0) with the results of the numerical experiment ψ’N (0) θ ε = 1 ε = 2 ε = 3 ε = 4 -ψ’N (0) -ψ’(0) 0.00 0.664 0.535 0.314 0.305 0.250 0.246 +0.03 0.650 0.515 0.516 0.270 0.265 0.190 0.199 -0.03 0.679 0.553 0.555 0.384 0.365 0.406 0.363 +0.05 0.641 0.503 0.504 0.248 0.166 0.205 -0.05 0.689 0.572 0.570 0.459 0.415 - 0.479 +0.10 0.620 0.475 0.478 0.207 0.355 -0.10 0.716 0.7162 0.616 0.611 0.581 0.903 +0.20 0.584 0.429 0.442 0.160 0.418 1.229 -0.20 0.779 0.776 0.736 0.707 1.080 2.325 +0.30 0.548 0.393 0.425 0.808 2.868 -0.30 0.855 0.843 0.936 0.822 1.800 4.512

3.2. Two phase systems The similar results were obtained for gas-liquid and liquid-liquid systems. The comparison between this effect and Marangoni effect show [that Marangoni effect is negligible in comparison of the big concentration gradient effect. As a result of this non-linear effect the absorption rate of highly soluble gases is grater then its desorption rate. 3.3. Another problems The non-linear mass transfer was investigated in the cases of liquid film flows and counter-current flows. The new velocity distribution as a result of the big concentration gradient influences on similar way simultaneous heat transfer or multicomponent mass transfer.

4. HYDRODYNAMIC STABILITY
The theoretical analysis shows that non-linear mass transfer effect is about 10-20% but in many cases the deviation of the experimental data from the linear mass transfer theory may be grater then two times. This difference is possible to explain with the loss of stability because non-linear mass transfer induce secondary flow which is equivalent to a suction (injection) from (to) the boundary layer and leads to a significant change of the flow stability. 4.1. Linear stability analysis The linear stability analysis permits to obtain the critical Reynolds number (Recr) for the transition between laminar and turbulent flow regimes. The solution of Orr-Sommerfeld equation (as an eigen values problem ) in the case of non-linear mass transfer show that flow stability (Recr) increase when the intensive mass transfer is directed from the volume toward the phase boundary (θ < 0) (the effect of “suction”). In the opposite case (θ > 0) the intensive interphase mass transfer is directed from the phase boundary toward the volume (the effect of “injection”) and the flow stability (Recr) decrease. The comparison of the influence of the non-linear mass transfer and Marangoni effect on the hydrodynamic stability in the boundary layers show that the influence of the Marangoni effect is negligible.

4.2. Self – organizing dissipative structures
In many cases for Re < Recr the system is not stable and small disturbances increase to the stable state (subcritical biffurcation) with constant amplitude. In these conditions the convective transfer is very intensive. These self-organizing dissipative structures are very useful because the process intensification is not result of the flow turbulization (big energy dissipation). This effect can see in the cases of gas absorption or liquid evaporation. The experimental data for absorption of CO2 and Ar in H2O and C2H5OH show that the absorption rate is significant great than the one that can be determined from the linear theory. The gas is absorbed in the liquid, and the process is accompanied with a thermal effect. As a result several effects in the liquid may occur in the form of secondary flows due to the big concentration gradients on the phase boundary (non-linear mass transfer), a density gradient in the volume (natural convection) and a surface tension gradient at the interphase (Marangoni effect). The theoretical analysis of the Oberbeck - Boussinesq equations in gas and liquid phases and three non-linear effects (natural convection, non-linear mass transfer and Marangoni effect) shows that the temperature is practically constant and Marangoni effect is negligible.

The linear stability analysis of this problem permits to obtain the amount (Q) of the absorbed (desorbed) substance: , where under desorption c* = 0 . Fig. 13 Comparison of the theoretical and experimental data for Q under absorption of 100% CO2 in water . Fig. 14 Comparison of the theoretical results for Q with experimental data under desorption of CO2 from a saturated water solution in N2 . The results explain the differences exhibited by one and the same system under absorption and desorption respectively. Under a desorption process the process is stable because c* = 0 .

A similar problem was solved in the cases of non-stationary evaporation of a liquid with a moderate partial pressure (water, methanol, ethanol and i-propanol) at 20° C in an inert gas (nitrogen, argon and helium) was investigated. The convective mass transfer upon non-stationary evaporation from a stagnant liquid into a stagnant gas above it ( within a large initial time interval) could be attributed to the Stephan flow and the natural convection. The problem was solved by the similar way. As a result it is possible to find the amount of the evaporated liquid: where  is amplitude of the velocity disturbances,  and  are related with the distance of the disturbance changes in velocity and concentration. The values of ,  and  are obtained on the bases of experimental data for different gas-liquid systems. In the cases when the vapours of the liquid are weightier than the inert gas (H2O/He, C2H5OH/Ar, i-C3H7OH/Ar) the process is stable ( =  =  = 0) and the rate of the evaporation could be determined from the non-stationary diffusion rate (it has not conditions for naturale convection).

Comparative analysis of the non-linear mass transfer and Marangoni effect in gas – liquid and liquid – liquid systems shows that Marangoni effect in negligible in the cases of an absence of surface active agents and a direct heating of the liquid surface. Fig. 15 Theoretical and experimental data for evaporation in system H2O – N2 Fig. 16 Theoretical and experimental data for evaporation in system H2O – He

5. PARAMETERS IDENTIFICATION
A main problem of the modelling of the heat and mass transfer processes is the parameters identification in the mathematical description, based on experimental data. The inverse identification problem is often an incorrect (ill-posed) , i.e. the solution is sensible with respect to the errors of the experimental data. Let us consider a numerical model: y = f (x,b) , where f is an objective function, expressed analytically, numerically or through an operator (algorithm); x = (x1,…,xm) is a vector of independent variables, b = (b1,…,bJ) - vector of parameters. The parameters of the model should be determined by means of N experimental values of the objective function ŷ = (ŷ1,…,ŷN) . This requires the introduction of a least square function: where yn = f (xn,b) are the calculated values of the objective function of the model, while xn = (x1n,…,xmn) are the values of the independent variables from the different experimental conditions (regimes), n = 1,…,N . The parameters of the model can be determined upon the conditions imposed by the minimum of the function Q = (b1,…,bJ) with respect to the parameters b = (b1,…,bJ).

5.1. Incorrectness of the inverse problem
Let us consider the one-parameter model: y = 1 - exp (-bx) . The relation between the objective function and the parameter in Fig. 17 is typical for number models of heat and mass transfer processes. In the Fig. 17 is seen that the inverse identification problem is correct (when 0 < x < 0.3), incorrect (if 0.31 < x < 0.65) and essentially incorrect (when 0.66 < x < 1). Let us consider the two-parameter model: y = 1 – b1 exp (-b2 x) ,where and are exact parameter values. The parameter identification problem will be solved by the help of the “experimental” data, obtained by a generator of random numbers. In the figures, are seen the horizontals of the least square function in different interval of x when inverse problem is correct, incorrect and essentially incorrect. These results show that the least square method is correct when the differences between exact parameter values and the parameter values in the least square function minimum are very small Fig. 18. These differences are too large, when the inverse problem is incorrect Fig.19. In the case, when inverse problem is essentially incorrect the least square function has not a minimum Fig. 20. The results obtained show that in the cases of incorrect inverse problems, the least square function minimization is not lead to the solution of the inverse problem and for the problem solution must be use additional conditions.

Fig. 17 Incorectness of the inverse problem
Fig. 18 Horrizontals of the least square function - correct problem Fig. 17 Incorectness of the inverse problem Fig. 19 Horrizontals of the seast square function – incorrect problem Fig. 20 Horrizontals of the seast square function – essentially incorrect problem

bji = bj(i–1) – β(i–1) Rj(i–1), j = 1,…,J ,
5.2. Regularization of the iterative method for parameter identification Let the iteration procedure starts with an initial approximation b(0)=(b1(0),…,bJ(0)) . The values of bi = (b1i,…,bJi) , where i is iteration number, are result of the conditions imposed by the movement towards the anti - gradient of the function Q(b): bji = bj(i–1) – β(i–1) Rj(i–1), j = 1,…,J , Here βi is the iteration step and β0 = (arbitrary small step value). Each iteration step is successful if two conditions are satisfied: The first condition indicates that iterative solution (bi) approaches the solution at the minimum (b*), while the second condition in concerns the approach of the iterative solution (bi) towards the exact solution ( ). Obviously, it is due to the effect of the problem incorrectness. The results obtained permit to create an algorithm for solution of the inverse identification problems.

5.3. Inverse problem solution
The proposed algorithm was used for the correct problem solution (0 < x < 0.3) and the results are shown on the Table 2. Table 2 One and two-parameter model solutions Δŷ [%] b* i b1* b2* 5 4.9678 337 1.0025 5.0674 128 10 4.9351 339 4.9218 172 The incorrect problem solution for the one-parameter model (b(0) = 6, γ = 0.5) and two-parameter model (b1(0) = 1.1, b2(0) = 6, γ = 0.05) are shown on the Table 3. Table 3 Incorrect problem solution Δŷ [%] b* i b1* b2* 5 5.0614 1213 1.1797 5.4666 642 10 5.1232 1217 1.3778 5.9106 416 The parameters identification problem when inverse problem is essentially incorrect if solved for 0.66 ≤ x ≤ 1 (see Table 4) . Table 4 Essential incorrect problem solutions Δŷ [%] b* i b1* b2* 5 5.1828 2066 2.1720 6.1731 54 10 5.3816 2156 4.9003 7.4004 128

5.4. Parameter identification in complicated multiparameters models
In many cases the process model consist 3-5 equations and parameters. The identification of these parameters is a difficult task because usually the used least square functions are of “ravine” type and with multiple extreme. A hierarchical approach is possible to use for the solving of this problem if the parameter identification is made for separated equations step by step. In every equation the unknown functions are replaced by the polynomial approximations of the experimental data. In the cases of fermentation systems the comparison of the model with the experimental data is shown on the Fig

and experimental data for glucose dimensionless concentration
Fig. 21. Comparison of the calculated values and experimental data for biomass dimensionless concentration Fig.22. Comparison of the calculated values and experimental data for gluconic acid dimensionless concentration Fig. 23 Comparison of the calculated values and experimental data for glucose dimensionless concentration Fig. 24. Comparison of the calculated values and experimental data for oxygen dimensionless concentration

6. DIFFUSION MODELS AND SCALE-UP

6.1. Diffusion model of column apparatuses
Let’s consider liquid motion in column apparatus with chemical reaction between two of the liquid components. If suppose for velocity and concentration distribution in the column: the mathematical description has the form: . The radial non uniformity of the velocity is the cause for the scale effect (decreasing of the process efficiency with increasing of the column diameter) in the column scale - up. That is why velocity average and concentration for the cross - section’s area must be used.

6.2. Average concentration models and scale-up

α using experimental data.
The average concentration model may be obtained if put these expressions in the convection – diffusion equation, multiply by r and integrate over r in the interval [0,R] . As a result is obtained: ; where: Practically and g = 0, As a result the scale-up problem is to obtain α using experimental data.

7. PROCESS SYSTEM ENGINEERING
A general problem of the chemical technology is modeling and simulation processes of Chemical Engineering Systems (CES). Their object is a quality assessment of the production systems for its optimal design, control and renovation. CES comprises a combination of mutual effecting processes in the chemical productions. Concerning to that, the CES model obviously represents an aggregation of particular process models as equations of the relations (interactions) between them are included as well. In this sense, the problem of CES model design is solved when the particular process models have been designed and only the equations of relations have to be added to it, as it is no complicated task. In this manner, CES quality assessment as a problem is related formerly to the CES process simulation.

7.1. CES simulation CES model. The CES model composes of mathematical structures, which get a relation between input regime variables (x), output regime variables (y), design variables (a), characterizing apparatus construction and the variables (b), characterizing the state of equipment. For a block with number , they could be represented by following vectors: Block i model represents a system of equations, which get the relation of all of variables: The aggregation of all blocks models (i = 1,…, I , where i is the common block numbers in CES) forms CES model if equations of the relations are added to it, as follows: namely, input regime variable yn0 in s-th block is output regime variable xm0 of i-th block.

F (X,Y,A,B) = 0 The CES model variables are:
where A and B are common aggregation forms (vectors) of the design variables and the variables, characterizing the state of equipment. In this sense, CES model is obtained analogically, as a system equations type of: F (X,Y,A,B) = 0 CES simulation as an object of control processes represents the creation of methods, algorithms and programming systems so that the output regime variables to be determined: Y = Φ (X,A,B) . An extending application of sequential modules approach is that the CES process simulation leads to consecutive simulation of CES blocks (modules). The problem under consideration consists of determination of the sequence order. For this purpose a universal method is proposed. Recently, this problem is solved with different commercial software products. CES simulation as an object of the process design carries out methods, algorithms and commercial software to determine the design variables, as the others variables values have been assigned: A = Φ2 (X,A,B) . The problem under consideration consists of process simulation at the controlled output regime variables. It was be solved using affection zones at block.

7.2. Optimization Determination of CES optimization depends on a predefined object, i.e. an objective function. where is structure variables vector, expressing the presence (absence) of technological flows between particular blocks (units) in CES. In the cases of optimal control The optimal design uses optimization. The last problem could be extended to such one of the optimal CES structure synthesis. An important problem is optimal synthesis of systems for heat recuperation, where the proposed combinatorial heuristic method solves heat integration problem too. The established methods for CES modeling, simulation and optimization allow of a method for renovation to be proposed, i.e. the system turns into a new state, which is more effective (economically) compared to the previous one. Another important class of optimization problems is an optimization problem of multiple products CES, when batch processes system must be optimally realized (in any sense) at the universal units system. A general problem is related to the creation of an optimal scheduling of the units.

CONCLUSIONS The importance of theoretical investigations in the science is formulated more straightforward from Max Planck (see Motto), i.e. theory motivates formulation of the experiment and explains the obtained results. Though, the theory in chemical engineering solves not definitely scientific problems but it has to solve applied science problems too. For this purpose it needs of a quality assessment of the processes for their control, design, renovation. Modeling and simulation use different theoretical approaches at the different quality assessment stages. The variety of practical problems emerged requires for a consequential development of these theoretical methods.

8.2. Similarity and modeling
INTRODUCTION Modeling and simulation – An united approach for quantitative description of the processes and systems and its optimal design, control or renovation. Modeling - the creation of a model on the bases of hypothesis of physical mechanism, mathematical structure, model parameter identification using experimental data and analysis of the model adequacy. Simulation – the numerical (mathematical) experiment for quantitative description of the process on the bases of an algorithm and computer program for solution of the model equations.

SIMILARITY 1.1. Mechanism and mathematical description
The model equations are a full correspondence between different physical effect in the process and the mathematical operators in the mathematical description. Model of the physical absorption in falling film Convection–diffusion equation - mass balance between convective longitudinal and cross transfer and diffusive transfer : . For the vertical film with constant thickness h , , .

0 ≤ x ≤ l , h – δ ≤ y ≤ h , 0 ≤ u ≤ u0 , c0 ≤ c ≤ c*,
1.2. Generalized variables Every variable have a characteristic scale (very often the maximal value increase in a characteristic interval): 0 ≤ x ≤ l , h – δ ≤ y ≤ h , 0 ≤ u ≤ u0 , c0 ≤ c ≤ c*, where u0 and δ are interface velocity and thickness of the diffusion boundary layer: . The characteristic scales permit to obtain the generalized (dimensionless) variables X, Y, C : , ; , where .

Fo < 1 (10 -1) , C = C0 + FoC1 + Fo2C2 + … ;
1.3. Qualitative analysis Fo ~ 1 , C = C ( X,Y ) ; Fo < 1 (10 -1) , C = C0 + FoC1 + Fo2C2 + … ; Fo << 1 (10 -2) , C = C0. 1.4. Generalized individual case (process) and similarity ; , . Generalized case – Fo have all possible values Generalized individual case (GIC) - Fo have a concrete value All processes in the general individual case are similar, i.e. every process is a model of the all others.

Fo – similarity criteria; Pe, Sh is not similarity criterion
The equality of the dimensionless parameters – in the GIC is similarity criterion Mass transfer kinetics , , Fo – similarity criteria; Pe, Sh is not similarity criterion Sh Pe-½=ψ(Fo)

2. MODELING 2.1. Similarity and physical modeling Let’s consider a real process (1) and its physical model (2). From the similarity condition follows: . If J2 is an experimental value of the mass transfer rate for the model, the mass transfer coefficient value is . i.e. As a result the mass transfer rate for the real process is , where .

Rei(0) (i=1,2), α0, β0, i.e. 2.2. Incompatible criteria
Let’s consider two phase flow in the boundary layer approximation with boundary conditions at the interphase (y=0 is flat face interphase). If ui(0) and δi (i = 1,2) are the scales of the flow the dimensionless equations are The parameters of the general individual case are: Rei(0) (i=1,2), α0, β0, i.e. It is seen that it is not possible to be satisfied these two conditions in the scale - up, i.e. the similarity criteria Rei (i =1,2) and β are incompatible.

Let’s consider mass transfer with chemical reaction:
where u0, c0 and l are the scales of the process. The dimensionless form of the equation is where The parameters of the general individual case are Pe0 and Bi0 , i. e. the similarity conditions are It is seen that the similarity criteria Pe and Bi are incompatible, i. e. the generalized individual case consist one process only and it is not possibility to obtain process for the physical model. The dimensionless model shows that similarity criteria express the balance between two physical effects. For example: Pe – convective and diffusive transfer, Bi – convective transfer and chemical reaction , Bi/Pe – chemical reaction and diffusive transfer.

2.3. Mathematical modeling
The criteria incompatibility leads to impossibility to be used physical modeling for quantitative description of many processes. It is a main cause for the mathematical modeling development in the chemical engineering. If the rate of absorption with chemical reaction is for the Sherwood number is obtained If we have experimental data for the processes rate J for different values of the Pe and Bi numbers we will obtain an experimental dependence between Sherwood number and Peclet and Bio numbers: The introduction of a mathematical structure for the function ψ and an obtaining of the parameters in this structure (using experimental data) is equivalent to the mathematical modeling of the process, i.e. the equations with the similarity criteria are mathematical models of the process.

2.4. Model equation structure
The model equation must be a function which approximation the experimental data. This function must be invariant with respect to the similarity transformations, i.e. do not change the type of this function when the model variables are replaced with the real process variables. Let’s consider model equations f(x1,…,xm)=0 . The function f is invariant with respect to the similarity transformations if it is homogenous function f(k1x1,…,kmxm)=0 , i.e. f(k1x1,…,kmxm)=φ(k1,…,km) f(x1,…,xm)=0 . If it use short record The problem is to obtain a function with this property. A differentiation of with respect to the first parameter k1 lead to

As a result If i.e. As a result and A repetition of this procedure for x2,…,xm lead to i.e. f is homogenous function if represents a power complex and as a result is invariant with respect to the similarity (metric) transformations. The result obtained explains the mathematical structure of the criteria models.

2.5. Scale-up and scale effect
The scale-up on the base of the physical modeling. It is impossible if the similarity criteria are incompatibles. The scale effect is the problem of the mathematical modeling because the increase of the apparatuses size lead to the decrease of the process efficiency. This effect is result of the radial non-uniformity of the velocity in the column apparatuses which is none predicted in the model equations. The solution of the scale-up problem has two steps. The first is to decrease the radial non-uniformity of the velocity distribution by means of design modification of the column ends. The second is to give an account of the non-uniformity in the model equations. The scale-up of the column apparatuses uses very often diffusion type models. The behavior in these cases is very complicated but the processes include convective transport, diffusion transport and volume reaction, i.e. a convective – diffusion equation with volume reaction may be used as a mathematical structure of the model. The convective transfer is result of the laminar flow or large scale turbulent pulsations. The diffusion transport is molecular turbulent as a result of the small scale turbulent pulsations. The volume sources are result of the chemical, biochemical reactions or interphase mass transfer.

Let’s consider gas absorption with chemical reaction in the liquid phase:
where k is interphase mass transfer coefficient, χ – Henry’s number, k0 – chemical reaction rate constant. The velocities and concentration may be presented as where are average velocities and concentrations and radial non -uniformities of the velocities and concentrations:

As a result the average concentration model has the form:
where In many cases , i.e. and The parameters Di, εi, k, k0, χ are related with the process and αi, βi, γi – with the apparatuses, i.e. with the scale-up.

2.6. Parameter identification and incorrectness of the inverse problem.
In the chemical engineering are used models of the continuum mechanics where the parameter must be obtained on the bases of experimental data. Let us consider a numerical model : y = f (x , b) , where f is an objective function, expressed analytically, numerically or through an operator (algorithm); x = (x1,…,xm) is a vector of independent variables, b = (b1,…,bj) - vector of parameters. The parameters of the model (1) should be determined by means of N experimental values of the objective function This requires the introduction of a least square function: where yn = f ( xn , b ) are the calculated values of the objective function of the model (1), while xn = ( x1n , … , xmn ) are the values of the independent variables from the different experimental conditions (regimes), n = (1,…,N) . The parameters of the model (1) can be determined upon the conditions imposed by the minimum of the function Q = (b1,…,bj) with respect to the parameters b = (b1,…,bj) .

Let us consider the one-parameter model : y = 1 – exp(–bx) ,
The determination of b faces many troubles due to the incorrectness of the problem. They are a result of the sensibility of the solution with respect to the experimental errors associated with the determination of . They can be avoided by applications of regularization methods that make the problem conditionally correct. Let us consider the one-parameter model : y = 1 – exp(–bx) , where y is an objective function, x is an independent variable and b is an parameter. In the fig. is shown a dependence of the objective function from the model parameter at a constant value of the independent variable. The relation between the objective function and the parameter in the fig. is typical for number models of heat and mass transfer processes. The fig. permits to obtain objective function y0 for a given parameter value b0, i.e. this is the direct problem solution. The inverse problem is an obtaining of the parameter value b0 if the experimental value of the objective function y0 is known. Objective function y for different values of the model parameter b at x = x0 = const.

y = 1 – b1 – exp(–b2x) Let us consider the two -parameter model:
where and are exact parameter values. The horizontals of the least square function Q – correct inverse problem The horizontals of the least square function Q – incorrect inverse problem The horizontals of the least square function Q – essentialy incorrect problem

Let the iteration procedure starts with an initial approximation b(0) = (b1(0) , … , bJ(0)). The values of bi = (b1i , … , bJi), where i is iteration number, are result of the conditions imposed by the movement towards the anti-gradient of the function Q (b) : Each iteration step is successful if two conditions are satisfied:

2.7. Some specific problems
Similarity with out mathematical description - method of the dimensions The dimension of the left part of the equation of the physical processes must be equal to the dimension of the right part. Let’s consider the model equation z = f (x1 ,…, xm ; y1 ,… , yr) , where x1 , … , xm are primary quantities (length, mass time, …) and y1 , … , yr – secondary quantities (velocity, concentration, …). The secondary quantities may be expressed as a power complex The physical equation is invariant to the similarity transformations, i.e. it is possible to present as a power complex: The dimensions equality conditions lead to , … , i.e. n equations with n + r variables.

To obtain dimensionless numbers must to put concrete values of r variables and to obtain another values, solving the linear system. These dimensionless numbers are not similarity criteria because: are not ratio between two physical effects; are not uniquely obtained; an error in the selection of the quantities x1,…, xm , y1,…,yr is fatal; do not present an opportunity for to make the difference between similarity criteria and dimension numbers. Criteria models and autocorrelations On the figure is seen correlation between Reynolds and Nuselt numbers in the case of heat transfer in fluidized bed:

The same experimental data are used for the correlation between Veber and Stanton numbers
The same experimental data for u and h are shown on the next picture without correlation because they are obtained with the generator of random numbers. The autocorrelation is result of the common parameter (n) in Re and Nu, or ρu in St and We.

CONCLUSION The comparison analysis of the similarity and modeling shown: the similarity and modeling are related in the cases of physical modeling for a quantitative description of the general individual cases; the mathematical modeling permit a quantitative description of the general cases; the mathematical technique of modeling is very simple but the similarity has very great sense. The using of the modeling without an understanding (realization) of the similarity may to lead to reduce the similarity theory into a similarity of theory (Prof. P. G. Romankov). In all cases the mathematical modeling must to finish with statistical analysis of the model adequacy, i.e. equality of the variances of model error and experimental error.

8.3. Regression models INTRODUCTION
An iterative method for model parameter identification has been proposed for solution of the correct, incorrect and essentially incorrect problems. The method has been tested by two-parameter model: where and are exact parameter values. For different values of x, the inverse problem is correct (0<xn<0.3) , incorrect (0.3<xn<0.6) and essentially incorrect (0.6<xn<1) .

REGRESSION MODELS where y is an objective function,
bi (i=0,…,I)- model parameters, fi(x) (i=0,…,I) - linear or non-linear function, x=xm(k=0,…,M) - independent variables (regime parameters).

LINEAR REGRESSION MODELS WITH STRONG NON-LINEAR FUNCTION
Let consider the model where are exact parameter values. The least square method for parameter identification use the matrix A of the normal set of equations. For the model was obtained:

correct inverse problem incorrect inverse problem
If the normal set of equations has a form (for two parameters model): and the scale of the matrix A is: where The conditions of correctness are: correct inverse problem incorrect inverse problem essentially incorrect problem.

0<xn<0.6 b2 b1 x

0.6<xn<0.8; 0.8<xn<1; 1<xn<1.5

0.6<xn<0.8; 0.6<xn<0.8; 0.8<xn<1; 0.8<xn<1; b2

1<xn<1.5; 1<xn<1.5; b2 b1 x

1.5<xn<2 b2 b1 x

THREE PARAMETER MODEL where exact parameters values are

STATISTICAL ANALYSIS OF THE PARAMETER SIGNIFICANCE AND MODEL ADEQUACY

CONCRETE REGRESSION MODELS

MODEL SUITABILITY The degree of freedom number of Q, Q0 and Q1 are:

INCORRECTNESS CRITERION

EXACTNESS INCREASING OF THE IDENTIFICATION PROBLEM SOLUTION

CONCLUSION The results obtained show that an iterative method for parameter identification is applicable for the regression models too. The statistical analyses of the model adequacy or suitability are as a criterion for the correctness of the parameter identification results. The scaling of the variables and parameters or decreasing of the initial iterative step value, lead to decreasing of the model error variance.

8.4. On the modeling of an airlift photobioreactor
INTRODUCTION Photobioprocesses include dissolution of an active gas component (CO2 , O2 ) in liquid and its reaction with a photoactive material (cells) . These two processes may take place in one environment (mixed bioreactors ; bubble columns) or in different environments (air lift photobioreactor) . The comparison of these systems shows apparent advantages in the use of airlift photo –bioreactors , because the possibility of manipulation of the light - darkness history of the photosynthetic cells . The hydrodynamic behaviour of the gas and the liquid in airlift reactors is very complicated , but in all cases the process includes convective transport, diffusion transport and volume reactions . That is why convection - diffusion equation with volume reaction may be use as a mathematical structure of the model .

, J = J (x1, r, t) , J1 = J1 (x1, r, t) . MATHEMATHICAL MODEL
Let’s consider an airlift reactor with a horizontal cross-section’s area F0 for the riser zone and F1 for the downcomer zone. The length of the working zones is l. The gas flow rate is Q0 and the liquid flow rate (water) - Q1. The gas and liquid hold - up in the riser are ε and (l – ε) . The concentrations of the active gas component (CO2) in the gas phase is c(x,r,t) and in the liquid phase – c0 (x, r, t) for the riser and c1 (x1 ,r ,t) – for the downcomer, where x1 = l - x. The concentration of the photoactive substance in the downcomer is c2 (x1,r,t) and in the riser - c3 (x1 ,r ,t). The average velocities in gas and liquid phases are: The interphase mass transfer rate in the riser is: . The photoreaction rates in the downcomer and the riser are taken, respectively, as: , J = J (x1, r, t) , J1 = J1 (x1, r, t) . Let consider cylindrical surface with radius R0 and length 1 m, which is regularly illuminated with a photon flux density J0. The photon flux densities over the cylindrical surfaces with radiuses r  R0 is: .

The increasing of the photon flux density between r and r -  r is:
Ro r J The increasing of the photon flux density between r and r -  r is: . The volume between the cylindrical surfaces with radiuses r and r - r is: , and the decreasing of the photon flux density as a result of the light absorption in this volume is: , The difference between photon flux densities for r and r - r is: . As a result where J(R0) = J0. The solution is: .

The equations for the distribution of the active gas component in the gas and liquid phases in the riser are: , ε = const . The equations for the distribution of the active gas component in the gas and liquid phases in the downcomer are: x1 = l – x . Photochemical reaction may take place in riser too, and the equation for the cell concentration is: .

The initial conditions will be formulated for the case , of thermodynamic equilibrium between gas and liquid phases , i. e. a full liquid saturation with the active gas component and the process starts with the starting of the illumination : where c(0) and c2(0) are initial concentrations of the active gas component in the gas phase and the photoactive substance in the liquid phase. The boundary conditions are equalities of the concentrations and mass fluxes at the two ends of the working zones - x = 0 (x1 = l) and x = l (x1 = 0). The boundary conditions for c (x, r, t) and c0 (x, r, t) are:

The boundary conditions for c1 (x1, r, t) , c2 (x1, r, t) and c3 (x, r, t) are:
The radial non - uniformity of the velocity in the column is the cause for the scale effect (decreasing of the process efficiency with increasing of the column diameter) in the column scale-up. In the specific case of photo-reactions, an additional factor is the local variations of light availability. Here an average velocity and concentration in any cross – section’s area is used. This approach has a sensible advantage in the collection of experimental data for the parameter identification because measurement the average concentrations is very simple in comparison with local concentration measurements.

AVERAGE CONCENTRATION MODELS
The average velocities and concentration in the cross-section’s area in the riser are: As a result is possible to present velocities and concentration distribution: where If introduce the average velocity and concentration in the equation of the gas phase in riser, multiply by r and integrate over r in the interval [0 , r0] the average concentration model has the form:

must be obtained from the continuity equation if multiply by r2 and integrate over r in the interval [0 , r0] : As a result is obtained: with boundary condition:

The parameters in the model are of two types - specific model parameters (D , k ,  , ) and scale model parameters (A , B , G) . The last ones (scale parameters) are functions of the column radius r0 . They are a result of the radial non-uniformity of the velocity and the concentration and show the influence of the scale - up on the equations of the model . The parameter  may be obtained beforehand from thermodynamic measurements. From the model follows that the average radial velocity component influences the transfer process in the cases , i. e. when the gas hold - up in not constant over the column height. As a result for many cases of practical interest and the radial velocity component will not taken account in this case. The hold - up  can be obtained using: where l and l0 are liquid level in the riser without gas motion. The values of the parameters D , k , A , B , G must be obtained using experimental data for measured on a laboratory column. In the cases of scale - up A, B and G must be specified only (because they are functions of the column radius and radial non-uniformity of the velocity and concentration) , using a column with real diameter.

The model for the liquid phase in the riser is :
where A0 , B0 and G0 are obtained in the same way as A, B and G. The concrete expressions of A, B and G are not relevant because those values must be obtained, using experimental data in any case. In the equations of the downcomer must put the average velocity , concentrations and photon flux density:

In the same manner the problems has the form
and A1, A2, B1, B2, G1, G2 are obtained in a similar way as A, B, G, but taking into account that the limits of the integrals are [r0 , R0]. may be obtained after integration of the photon flux density model : .

may be obtained after integration:
The equation for and A3, B3 and G3 are obtained in a similar way. may be obtained after integration: For many cases of practical interest and the number of parameters of the model decreases, i.e. G = G0 = G1 = G2 = G3 = 0 . The photo-chemical reaction rates equation shown in are acceptable when J and c1 are very small.

A more general form of the photo-chemical reaction rate equation is (written here for the downcomer): Another possible form for these equations could be: where the kinetic parameters k0 , γ, γ1 , γ2 must be obtained wsing experimental data. Applying the last expressions in the model equations, the photo-chemical reaction rate equations could be:

HIERARCHICAL APPROACH
The obtained equations are the mathematical model of an airlift photobioreactor. The model parameters are different types: beforehand known (c(0) , c2(0) , R0 , J0 , r0); beforehand obtained (ε , χ , α , β , k0 , γ , γ1 , γ2); obtained without photo-bioreaction (k , D , D0 , A , A0 , B , B0 , G , G0); obtained with photo-bioreaction (D1 , D2 , D3), because diffusion of the gas and photoactive substance is result of the photobioreaction; obtained in the modelling and specified in the scale - up (A, A0 , A1, A2, A3, B, B0, B1, B2, B3, G , G0, G2, G3, M1, M3, P1, P2), because they are functions of the column radius and radial non-uniformity of the velocity concentration. The parameters c(0), c2(0), J0 , ε, χ, α, β, k0 , γ, γ1 , γ2 , k, D, D0 , D1 , D2 , D3 are related with the process (gas absorption with photobioreaction in liquid phase), but the parameters R0 , r0 , A, A0 , A1 , A2 , A3 , B, B0 , B1 , B2 , B3 , G , G0 , G2 , G3 , M1, M3 , P1 , P2 are related with the apparatus (column radius and radial non-uniformity of the velocities and concentrations). The equations allow to obtain (k, D, D0 , A, B, A0 , B0) without photo-bioreaction if it can be assumed that CONCLUSIONS The results obtained show a possibility of formulation airlift photobioreactor models using average velocities and concentrations. These models have two type parameters, related to the process and to the apparatus (scale - up). This approach permit to solve the scale - up problem concerning the radial nonuniformity of the velocity and concentration, using radius dependent parameters. The model parameter identification on the bases of average concentration experimental data is much simpler than considering the local concentration measurements.

8.5. Non-linear mass transfer from a solid spherical particle
dissolving in a viscous fluid Introduction Let’s consider mass transfer near a solid spherical particle dissolving in a viscous fluid. In the cases of a big concentration gradient the big mass flux induces secondary flow at the solid interface. As a result the convection-diffusion equation become non-linear.

Problem formulation Mass flux at the interface has diffusive and convective components

Dimensionless variables and governing parameters
Introduce scaled dimensionless variables according to The formulated above problem in new variables takes the form

Here characteristic time is chosen arbitrary so that 1sek<t0<10²sek.
For the length scale we chose the thickness of the concentration boundary layer at time t0, l=(Dt0)½, the choice of velocity scale, , reflects the fact that the secondary flow results solely by interfacial mass flux (concentration gradient). The values of the governing parameters of the problem, typically, are as follows The rate of mass transfer is characterized by a mass transfer coefficient k:

Problem solution The problem formulated above contains two small parameters We are going to construct several terms of the expansion of the solution in β assuming δ=0. The latter assumption reflects that our time scale is larger than the viscous time, so that the velocity field is fully established at any moment, depending on time parametrically through the boundary condition that links velocity and concentration, which is substantially unsteady at the chosen time scale. To this approximation the velocity problem is not evolutionary one, hence, the initial conditions for the velocity can not be satisfied and should be replaced by condition at infinity.

Problem solution At δ=0 the problem under consideration takes the form

Problem solution We look for the solution in the form
The problem for C0(R,T) is as follows The solution of this problem is of the form:

Problem solution The problem for U0 can be formulated as:
The solution is found explicitly as

Problem solution The first order expansion term for concentration C1(R,T) is to be determined by solving the problem

Problem solution The introduction of a new variable V(R,T) reduces to the standard problem for the non-homogeneous heat equation

Problem solution As soon as C0(R,T) and C1(R,T) are found, mass flux from the particle at any moment of time can be calculated from, while an amount of the matter dissolved by the moment t=t0 is given by

Results and discussion
The main result of the present work is the determination of , which can be approximated by the function J0(α) = α α2 – α , or J(α) =1/α J0(α) = α α – /α The results of computations are given in Table and in Fig.

Scaled amount of the matter dissolved by the moment as a function of parameter α. α J0(α) 1 161.72 2 3 4 5 6 7 8 9 10 11 -13542

Conclusions The non-linear mass transfer from a neutrally buoyant solid particle suspended in an unbounded viscous fluid is studied. The particle medium is soluble in the fluid or it contains a soluble admixture. Initially the continuous medium is quiescent and contains no admixture. An inter-phase mass transfer takes place that results in extremely high concentration gradients at the initial period of the processes indicating highly intensive mass transfer, which induces a secondary flow in the vicinity of the interface that, in turn, results in a convective mass transfer. The latter makes a substantial contribution to the net mass flux. Parametric analysis of the problem is carried out that revealed a range of small parameters for typical physical conditions of the process under consideration. The problem is solved analytically making use of regular expansion in small parameter. ACKNOWLEDGEMENT This study was performed in the framework of the joint research project "Non-linear stability analysis in dispersed systems" between Institute of Chemical Engineering, BAS and the Faculty of Chemical Engineering, Technion.

8.6. Diffusion models and scale-up of a spouted bed
Introduction Many technological processes, as a catalytic chemical reactions, burning, heating, drying, etc, are realized in column apparatuses with spouted bed. The hydrodynamic behavior of the spouted bed like to the motion in counter – current flows. A cylindrical surface, where velocity is equal to zero, is the border between the rising and the downcomer flows. In this case the spouted bed reactors are similar to the airlift reactors, where the solid phase plays the role of the liquid phase. On this base is possible to use the methods for modeling of the airlift apparatuses for spouted bed modeling.

Hydrodynamic model Let’s consider an spouted bed column with a horizontal cross - sections area f, for the riser zone and (F - f ) for the downcomer zone, where F is the horizontal cross - sections area of the column. The average velocities of the solid particle in the riser and downcomer are and , where is the average sedimentation velocity of the solid particles. Let’s assume that velocities of the solid phase and are known beforehand as a result of experimental or computing methods. From the mass balance of the solid phase is obtained: , . Let’s assume that gas phase in the riser away the solid phase while in the downcomer solid flow carry away the gas phase and the average gas velocities in the riser and downcomer are and . The gas flow rate in the riser zone Q is result of the inlet gas flow Q0 and circulation gas flow Q1 : , . . As a result was obtained: If accept a cylindrical form of the riser and downcomer zone, the radiuses may be obtained from:

Convection - diffusion model
Let’s consider spouted bed column with catalytic chemical reaction on the solid interface. The axial and radial gas velocity components in the riser are u and w: where l is spouted bad height. A component of the gas phase react on the catalyst particles interface and its concentration has radial and axial non - uniformity: The convection – diffusions equation for this case is: where the velocity components satisfy the continuity equation: The convective transfer is result of a laminar or turbulent (large – scale pulsations) flow, the diffusive transfer is molecular or turbulent (small – scale pulsations) and the order of the catalytic chemical reaction is 1. In the downcomer the convection – diffusion equation has the form . where z1 = 1 – z , u1 = u1 (r , z) , w1 = w1 (r , z) , c1 = c1 (r , z) are velocity components and concentration in the downcomer.

The boundary conditions are obtained on the bases of the assumption for an ideal mixing in the two ends of the column: where c0 is initial gas concentration, are average gas concentration: , . The radial non-uniformity of the velocity is the cause for the scale effect (decreasing of the process efficiency with increasing of the column diameter). That is why an average velocity and concentration for the horizontal cross-section’s area are used.

Average function values
For the average velocity and concentration in the riser zone may be obtained : If use a property of the average functions, the velocity and concentration may to present as a: The same results for the average velocity and concentration is obtained in the downcomer zone: , , .

Average concentration model
The average concentration model may be obtained if put average velocities and concentrations in the convection diffusion equations, multiply by r and integrate the equations over r in the intervals [0 , r0] and [r0 , R].

In many cases the concentration of the solid phase is constant, i. e
In many cases the concentration of the solid phase is constant, i.e and i.e. γ = γ1 = 0. A similar models may be obtained for burning or heat and mass transfer processes in spouted bed (heating or drying of the solid phase), but must be used two - phase models.

Hierarchical approach
The presented problems are mathematical model of a spouted bed catalytic reactor. The model parameters are different types: – beforehand known – F, R, Q0 , c0; – beforehand obtained ; – obtained with chemical reaction - D, D1 , α, γ, α1 , γ1; – specified in the scale-up - α, γ, α1 , γ1 . As a result an hierarchical approach in mathematical modeling (model parameters identification) is possible to be used. The parameters D, D1 , α, γ, α1 , γ1 , may be obtained using experimental data for and after solution of the inverse identification problems. In many cases D and D1 are small parameters and inverse problem is incorrect (ill-posed). Conclusion The results obtained show a possibility to present the spouted bed catalytic reactor model as a airlift reactor model. A diffusion model for the spouted bed catalytic reactor is used. The introducing of average velocity and concentration permit to solve the scale-up problem. The proposed model allow an hierarchical approach to be used for the model parameter identification.

ОСНОВИ НА МОДЕЛИРАНЕТО И СИМУЛИРАНЕТО

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