1. Bases of creation of the determined mathematical models of the cts elements
As it was told above, the determined or physical and chemical mathematical models reflect the theoretical regularities of processes proceeding in the CTS elements. In case of development of such models use the mass conservation laws and energies, laws of transfer of substance, energy and an impulse, regularity of kinetics of the proceeding chemical reactions, hydrodynamics of flows, etc. In connection with complexity of real engineering procedures, in case of development of their mathematical models usually enter a number of the assumptions simplifying the description of real process, and allowing to apply the block principle of creation of models according to which the mathematical description of an object is received in general as set of descriptions of the separate elementary processes proceeding in the considered object. We will cover the basics of creation of the determined mathematical models of processes on some examples.
Mixer module.
The module of the mixer is one of the simplest modules. According to an initial task, two flows of substance having expenses of G1 and G2 (mol/sec.), temperatures of T1 and T2 (a hail. J), structures of X1i and X2i (molar shares), warmth of Q1 and Q2 (W) move in the mixer from where there is one flow with G3 expense, T3 temperature, structure of X3i and warmth of Q3 (see Fig. 4.11).
Fig. 4.11. Scheme of the module of the mixer
The physical and chemical model of the mixer is intended for calculation of material and thermal balances of process of mixture of two flows of substance. There are modules for mixture of several flows of substance, but they are expanded modification of the mixer for mixture of two flows.
Usually in case of creation of the simplified determined model some assumptions are accepted. For the mixer, assumptions will be the following:
1. The structure of a flow in the device corresponds to the mode of ideal mixture;
Otherwise, the flow at the exit of the mixer will be not completely mixed, and, it will be in that case necessary or to complicate model taking into account hashing coefficient, or to complicate model taking into account hydrodynamics of flows in the device. It can be not justified on specific costs of time for development of model, and in case of incomplete accounting in model of all proceeding physical and chemical phenomena to lead to considerable mistakes.
2. Mixture process – adiabatic, isn't considered warmth of mixture;
Otherwise it is necessary to consider processes of a supply and withdrawal of warmth, and also the warmth of mixture which is marked out in case of mixture of substances (in particular cases, in thermal balance of the mixer it is required to consider warmth of mixture).
3. All flows have one phase condition;
Otherwise the model will need to be complicated considerably since it is necessary to use the mixer having 2 or 3 output flows (gas, liquid and firm) since one flow it is impossible to express at the same time various phase conditions, it will be necessary to consider phase balance in system firm - liquid-gas, and conditions of its establishment, and also thermal balance of processes of establishment of phase balance.
4. Pressure of entrance and output flows – identical;
In case of change of pressure there can be conditions leading to change of a phase condition.
In case of observance of all assumptions stated above we will consider the equations entering a basis of the mathematical description of model of the mixer.
The general equation of a material balance will register:
[4.2]
With use of the equation of material balance for substance it is possible to calculate structure of an output stream:
,
for i=1…k [4.3]
In case of creation of a material balance special attention is required to be paid on units of measure of expenses and structures. It is usually recommended to use a molar expense [mol/sec.] and structure [% mol.] or mass expense [kg/sec.] and structure [%масс.] or, in case of mixture of gas flows, a volume expense under normal thermodynamic conditions (0OC and 1 atm), i.e. [nm3/sec.], and volume structure [%об.].
It should be noted that when calculating the structure of a flow is usually used not as a percentage, and in shares (the amount = 1), and use of various units of measure for an expense and structure is inadmissible.
The general equation of thermal balance will register:
[4.4]
At unknown warmth of a stream, it can be calculated on the basis of material balance on the equation:
[4.5]
where SP – a specific isobaric thermal capacity of a stream (mix of substances) which pays off by the rule of additivity:
[4.6]
where SPi – an isobaric thermal capacity of i-go of a component of a stream which can be calculated on the equation:
[4.7]
coefficients of a which b, c and d for i-go of substance undertake from the reference book.
Temperature of an output stream pays off by method of iterations:
[4.8]
Divider module.
The module of a divider is one of the simplest modules. According to an initial task, the substance flow having an expense of G1 (mol/sec.), T1 temperature (a hail. J), structures of X1i (molar shares) and warmth of Q1 (W) moves in a divider from where there are two flows with expenses of G2 and G3, temperatures of T2 and T3, structures of X2i and X3i and warmth of Q2 and Q3 (see Fig. 4.12).
Fig. 4.12. Scheme of the module of a divider
The physical and chemical model of a divider is intended for calculation of material and thermal balances of process of division of one flow of substance into two flows. There are modules for division of a flow into bigger quantity of flows, but they are expanded modification of a divider on two flows.
For a divider, assumptions will be the following:
• The structure, temperature and pressure of output flows are equal to structure, temperature and pressure of an entrance flow;
• All flows have one phase condition.
Two methods of division of a flow are known. For the FIRST method it is required to know an expense of the first leaving flow, and for the SECOND – coefficient of division of a flow. Depending on type of the equipment connected with a divider, both methods can be applied, however the FIRST method has restrictions which are that absolute values, but not relative are used. For example, in the course of calculations, the expense of the entering flow will be less than the set expense of the first leaving flow, i.e. the second leaving flow will have a negative expense that is impossible. The SECOND method stable in calculations since relative values, however, depending on type of the equipment connected with a divider are used, use of the fixed coefficient of division can not correspond to real CTS.
For implementation of the FIRST method it is necessary to know: expense of G1 (mol/sec.), T1 temperature (hail. J), structure of X1i (molar shares) and warmth of Q1 (W), also an expense of the first leaving G2 flow.
The main equation of a material balance will register:
[4.9]
Proceeding from an assumption, the structure of output streams will be equal to structure of an entrance stream:
,
for i=1…k [4.10]
Warmth of the leaving flows can be calculated in proportion to expenses of the leaving flows (temperature and structure of the leaving flows are equal to entering) or are calculated based on a material balance on the equation:
[4.11]
where SP – a specific isobaric thermal capacity of a stream (mix of substances) which pays off by the rule of additivity:
[4.12]
where SPi – an isobaric thermal capacity of i-go of a component of a stream which can be calculated on the equation:
[4.13]
coefficients of a which b, c and d for i-go of substance undertake from the reference book.
For implementation of the SECOND method of calculation shall be known: expense of G1 (mol/sec.), T1 temperature (hail. J), the structure of X1i (molar shares) and warmth of Q1 (W), is also set coefficient of division of the entering Kf flow (according to designations in Fig. 4.3, Kf = G2/G1).
In this case expenses of the flows leaving a divider can be calculated by formulas:
[4.14]
[4.15]
Further, the algorithm of calculation doesn't differ from an algorithm of the FIRST way of the calculation given above.
Heat exchanger module.
Unlike modules of the mixer and a divider, the module of the heat exchanger isn't so simple since at change of temperature of streams change of their phase state is possible, and, therefore, when calculating it is necessary to consider such changes. In this regard, for example, only for systems gas-gas, liquid-liquid and gas-liquid distinguish the following models of heat exchangers:
• The heat exchanger gas-gas or liquid-liquid without phase transitions (heaters and refrigerators);
• The heat exchanger gas-gas or liquid-liquid with phase transition (for system gas-gas it is called the condenser, and for system liquid-liquid – the evaporator, also exists more difficult option when heat from the condensed gas is used for liquid evaporation);
Besides, as his design exerts impact on process of calculation of the heat exchanger, for everyone stated above like heat exchangers distinguish the following models:
1. Counter flow ("cold" and "hot" agents meet halfway each other, i.e. a countercurrent);
2. Direct-flow ("cold" and "hot" agents go in parallel, i.e. a direct flow);
3. Crossflow (intermediate option between stated above);
4. One-pass or multiple-pass heat exchangers (in multiple-pass heat exchangers a part of pipes works in the countercurrent mode, and a part – in the direct flow mode, or in the multiple-pass crossflow heat exchanger liquid or gas on pipes can move on the course or against the stream course in interpipe space);
5. Options when one of agents (or both agents) moves due to natural convection which, on intensity of hashing of a stream due to natural convection, in turn share on horizontal and vertical;
6. Mixture heat exchangers ("cold" and "hot" agents directly contact with each other, for example, in the device to a nozzle).
And, at last, the heat exchangers differing on an operating mode on:
• periodic;
• continuous.
It should be noted that according to an initial task for the CTS module, it is required to calculate parameters of output streams at the known parameters of entrance streams and parameters of the CTS element (in case of the heat exchanger, at the known area of heat exchange and coefficient of a heat transfer) that corresponds to test calculation of the heat exchanger. However sometimes there is a need to calculate the sizes of the heat exchanger entering CTS. In that case it is necessary to use design calculation of the heat exchanger.
As an example we will consider test calculation of the heat exchanger heater for system gas-gas, liquid-liquid or gas-liquid taking into account the following assumptions:
• The one-pass shell-and-tube heat exchanger in the stationary mode;
• The heat transfer isn't followed by change of aggregate state;
• Thermalizes coefficients for "cold" and "hot" streams pays off at the reference temperatures of heat carriers;
• The scheme of the movement of streams – counter flow;
• Losses of warmth are absent.
The scheme of the heat exchanger is submitted in Fig. 4.13.
Fig. 4.13. Scheme of the heat exchanger
According to an initial task, for an entrance "hot" flow GGN expense, TGN temperature, structure of HGN and warmth of QGN is known, for an entrance "cold" flow – GHN expense, THN temperature, structure of HHN and warmth of QHN. Except flow parameters, for the heat exchanger thermalizes coefficients for "cold" and "hot" flows of X and G, and the area of a heat transfer of F are known.
Because phase transition doesn't happen, the material balance of the heat exchanger will register equalities of expenses and structures of output flows entrance:
[4.16]
[4.17]
According to thermal balance:
[4.18]
or
[4.19]
where, heat transfer coefficient:
[4.20]
where, CT and CT – thickness and heat conductivity of a wall
"driving force" of a heat transfer:
[4.21]
where tБ and tM – big and smaller differences of temperatures on entrances and heat exchanger exits taking into account the mutual course of streams, for example, for a countercurrent:
i.e. tБ = bigger of (TGN - TGC), (THK - THN) [4.22]
tM = smaller of (TGN - TGC), (THK - THN) [4.23]
As temperatures of "hot" and "cold" flows at the exit of the heat exchanger are unknown, it isn't possible to carry out simple calculation therefore, it is possible to recommend carrying out calculation by method of search or minimization according to the following algorithm:
Some initial value of temperature of a "cold" flow at the exit from the heat exchanger is set. Usually: THK = TXH+0,1;
At the known temperature of a "cold" flow at the exit, its structure and an expense, on a formula [4.11] taking into account formulas [4.12 and 4.13] its heat content of QHK is calculated;
Thermal load of the QTP heat exchanger and heat content of an output "hot" flow of QGK is calculated with use of a formula of thermal balance [4.18];
Its temperature (see [4.7]) and a driving force of a heat transfer is determined by the size of heat content of an output "hot" flow of QGK [4.21];
The difference of warmth of the heat transfer calculated in item 3 and on a formula of the main equation of a heat transfer [4.19] is determined;
If the difference determined in item 5 is less than set calculation accuracy, then calculation comes to an end. Otherwise temperature of the leaving "cold" flow increases by some value, and calculation repeats with item 2.
When using mathematical methods of minimization, it is necessary to add logical conditions on the analysis of crossing of lines of heating/chilling i.e. that temperature of a "cold" flow on an entrance and an exit of the heat exchanger wasn't higher than temperature of a "hot" flow to item 4.
In conclusion it should be noted that except the specified algorithm of testing calculation of the heat exchanger, there are other algorithms resulting in similar results.
From all variety of the heat exchange equipment, proceeding from entry conditions, the simplest option of the one-pass counter flow shell-and-tube heat exchanger heater without phase transitions and losses was considered above. In case of phase transitions calculation considerably becomes complicated since in this case it is required not only to consider warmth of phase transitions, their completeness and "break" of lines of heating/chilling in thermal balance, but also to recalculate a material balance taking into account phase balance and change of structure and mass of flows owing to phase transitions. Undoubtedly, this task is rather difficult and not universal since requires creation of algorithms of calculation for each case.
Except the integrated approach to calculation of the heat exchanger considered above, there is also a differential approach consisting in integration of the differential equations of material and thermal balance of the heat exchanger on the area of heat exchange taking into account all nuances. The main benefit of differential approach is lack of difficult accounting methods of all features of a heat transfer for all heat exchanger and its universality since various features of heat exchange are always considered in the differential equations for the elementary area of a heat transfer of dF. For this reason, this approach is used by the majority of software products for calculation of CTS about which speech will go in the final chapters.
Test questions
1. Bases of creation of the determined mathematical models of the CTS elements
2. Describe modules of the mixer, a divider, the heat exchanger.
3. Give the equations of thermal balance, a material balance, expression for heat transfer coefficient.
Lecture No. 19 - 22. Bases of creation of statistical models of the CTS elements
Lecture purpose: studying of methods of creation of statistical models of the CTS elements.
Plan of a lecture:
1. Method of the smallest squares.
2. Active (factorial) and passive experiment.
3. Fractional factorial test
4. Orthogonal central composite plan.
5. Equation of regression of an object
6. Matrixes of planning of a two-factor experiment.
7. Check of adequacy of model.
8. Assessment of the importance of factors,
In case of development of statistical models of CTS modules the task of the detailed description of regularities of the processes happening in an object since its mathematical description is under construction in the form of regression dependences of output parameters of an object on entrance isn't set and represents the linear or nonlinear polynomial equations. Coefficients of the polynomial equations find by handling of the experimental data obtained on production or on the specified physical and chemical model of an object with use of a method of the smallest squares. Thus, approach to a technological object as to "a black box", i.e. without the processes which are taking place in the object allows to create model with the minimum costs for data collection and processing.
The essence of a method of the smallest squares is that through a number of experimental points carry out such dependence (Y=f (X1,X2, Xm)), which amount of squares of deviations from experimental values in case of the corresponding X1, X2 and Xm values – is minimum (see Fig. 4.14).
Fig. 4.14. Illustration of a method of the smallest squares
The type of dependence of Y=f (X1,X2, Xm) can be various. However usually dependence of Y=f (X1,X2, Xm) represents a polynomial:
Xm – is minimum (see Fig. 4.14).
[4.24]
where ai – polynomial coefficients;
Xi – changeable factors;
m – quantity of factors.
Coefficients of a polynomial of ai at which the sum of squares of differences of experimental (YIE) and settlement (YIR) values will be minimum (the equation 4.25) can be calculated with use of various mathematical methods (the solution of system of the linear equations, minimization, etc.).
[4.25]
It should be noted that the method of the smallest squares is rather widely used when processing experimental data since allows not only to determine parameters of the polynomial dependences which are describing work of an object, and not making physical sense but also to specify parameters of physical and chemical models.
For example, when calculating coefficient of a heat transfer (the equation 4.21) the coefficients of a thermalizes (X and G) depending on parameters of the movement of a stream of hot and cold heat carriers which can be calculated on criteria dependences which are kinds of physical and chemical models are used:
[4.26]
where the A, B, C and D parameters are in result of processing of experimental data by method of the smallest squares.
It is possible to give the equation of dependence of a constant of speed of chemical reaction of the steam conversion of monoxide of carbon for the iron chromic of catalysts in the range of temperatures 400-500OC which is the cornerstone of physical and chemical model of the reactor as other example:
[4.27]
where values of coefficients: "34000", "4,57" and "10,2" have also been found by data processing of an experiment on studying of kinetics of steam conversion of monoxide of carbon method of the smallest squares.
The polynomial equation of dependence of an isobaric thermal capacity on temperature (the equation 4.13) which is used in physical and chemical model can be one more example. Coefficients of this equation have been also received by processing of experimental data by method of the smallest squares.
Except specified, it is possible to give still a set of similar examples where concepts the statistical model and physical and chemical model have rather closely intertwined among themselves. However, it should be noted that the side between physical and chemical and statistical models is very thin since in fact, distinctions depend only on depth of consideration of the processes happening in a real object and completeness of their mathematical description.
Feature of physical and chemical models is that at their use process is considered at two levels: lower – the level of change of parameters of processes and properties of streams, and, top – the level describing features of processes.
For example, Arrhenius's equation is rather widely used as dependence of a constant of speed of various chemical reactions on temperature:
[4.28]
where the coefficients of k0 and EAKT (making physical sense) can be determined, for example, for process of steam conversion of methane, processing of experimental data by method of the smallest squares. For other process coefficients and EAKT will have other values, but the type of the equation will remain the same. Moreover, experimentally it is also theoretically proved that, for example, at decrease of the activity of the catalyst the size of energy of activation (EAKT) will remain invariable, and only the k0 parameter will change. Thus, knowing a general view of physical and chemical dependence and value of two constants, the possibility of the description of rather difficult dependence of a constant of speed on temperature in a wide interval of change of parameters of process appears.
In case of use of statistical models an object is considered as a unit without specification of the processes happening in him, i.e. in the form of "a black box" for which functions of transformation of input parameters during the week-end. are defined. These functions of transformation can have various appearance even for identical technological objects since don't make any physical sense. Moreover, if installation for which the statistical model has been made passes into an operating mode which parameters weren't used by drawing up model, then her statistical model has to be constructed anew since application of statistical model is limited to limits of a variation of input and output parameters within the data used by drawing up model (see Fig. 4.15), i.e. in limits for which her adequacy to a real object has been proved.
Fig. 4.15. Illustration of limits of applicability of statistical model
In Fig. 4.5. it is visible that dependence of output parameter (Y) on entrance (Xm) well describes experimental points only within a variation of input parameter from Xmin to Xmax, and output parameter from Ymin to Ymax. Outside a variation of parameters dependence can pass randomly. Thus, unlike physical and chemical model, the statistical model can't be extrapolated out of limits of a variation of parameters. For expansion of parameters of a variation it is necessary to collect additional data and to anew constitute model.
Usually carry the models received by data processing of an active (factorial) or passive experiment on a real object or by means of adequate model to statistical models. In case of active factorial test if linear dependences between output and entrance variables are observed, then plans of the first order are used: complete factorial test (CFT) and fractional factorial test (FFT). If dependences between output and entrance variables of active factorial test have obviously nonlinear nature, for receipt of the mathematical description of an object use composite plans of the second order, for example, the orthogonal central composite plan (OCCP). In case of data processing of a passive experiment receive the regression equation which complexity is determined depending on complexity of an object, quantity of basic data and required accuracy.
PFE in comparison with passive statistical methods of receipt of the mathematical description of model has an advantage in what allows to obtain a maximum of information on an object in case of the minimum quantity of experiences. However, PFE is generally applied to receipt of statistical model of an object on the basis of its physical and chemical model or if on installation there is an opportunity to plan change of the technological modes without prejudice to production. One more condition of use of PFE is importance and mutual independence of initial parameters.
In a general view, the equation of regression of an object can be provided by means of a polynomial:
[4.29]
where ai – polynomial coefficients;
Xi – changeable factors;
m – quantity of factors.
In PFE all factors vary at two levels: upper (it is designated: +1) and lower (it is designated:-1). When carrying out experiments various combinations of factors at the chosen levels are implemented. For accounting of mutual influence of two factors use their pair works. In this case, the quantity of series of experiences can be counted on a formula:
N=2m [4.30]
The example of a matrix of planning of a two-factor experiment is provided in Table 4.2.
Table 4.2
Matrix of planning of a two-factor experiment
-
№ series of experiment
Factors
Х1*Х2
Х1
Х2
1
+1
+1
+1
2
+1
-1
-1
3
-1
+1
-1
4
-1
-1
+1
When handling results of PFE coefficients of the regression equation, adequacy dispersion, dispersion of an average, criterion of Fischer by whom adequacy of the regression equation, etc. is determined are determined. Upon termination of data processing the conclusion about adequacy of model is drawn. If the model is inadequate, for example, change basic data, change a type of the regression equation, and carry out handling anew.
However, methods of data processing of an active experiment aren't always applicable for creation of models of CTS modules based on production data since in the conditions of the real industrial plant it is rather difficult to observe the required intervals of a variation of parameters set in the plan. For this reason, the greatest distribution for creation of statistical models of CTS modules was gained by methods of handling of production data by method of a passive experiment.
For receipt of statistical model on the basis of data processing of a passive experiment, data collection is made from the operating installation and present them in the form provided on Table 4.3.
Table 4.3.
Form of representation of basic data
-
№ п/п (1…N)
Factors (1…К)
Response Y
Х1
Х2
…
ХК
1
X11
X21
…
XK1
Y1
2
X12
X22
…
XK2
Y2
3
X13
X23
…
XK3
Y3
…
…
…
…
…
…
N
X1N
X2N
…
XKN
YN
It was specified above that the statistical model "works" only within a variation of parameters therefore collection of basic data needs to be made for all the set operating modes of installation in the widest limits of their variation. Usually, the plan of change of parameters is developed by means of the methods accepted in PFE since these methods allow to obtain the maximum information on an object with the minimum quantity of changes of parameters.
In case of data collection it is necessary to consider the principle: the more it is collected data – the better. Special attention should be paid on an installation operating mode since it is possible to constitute statistical model of an element of technological installation only for the stationary modes of its work. However, really, after change of any parameters, the stationary operating mode of installation is reached only through certain time. Moreover, if installation has no ACS, then process of achievement of a stationary operating mode by it can be slowed down due to transition processes. For this reason, before data collection it is necessary to reduce as much as possible influence of a management system of the surveyed installation element on change of the modes of its work, for example, as transfer of an element of installation to manual control. If transfer to manual control is impossible, then in this case it is necessary to determine time of stabilization of an operating mode of installation upon termination of change of parameters of the technological mode and to begin data collection only after this time.
Because the main operating mode of any technological installation is the dynamic mode, i.e. installation constantly is in process of transition of one condition in another (the question only consists in the speed of this transition), in case of collection of the current technological parameters it is necessary to determine by sizes of an expense, temperature, structure, etc. approximate time of transition of installation to the stationary (pseudo-stationary) mode after change of any parameter of its work, and, to begin data collection only after this time.
According to Tab. 4.3, factors entrance variables (XKN), and a response – the output parameter (YN) are called. For example, concentration of substance at the exit from the reactor, and factors – temperature, pressure, initial concentration of reagents, stay time, etc. can be a response. In the presence of several output parameters constitute several initial tables and work out several regression equations.
The equation describing function of a response (Y) usually is presented in the form a Taylor’s series for multidimensional function. This equation is called the regression equation:
[4.31]
where YP – a calculated value of function of a response;
XK – values of parameters at which calculated function of a response.
Due to the high complexity of the regression equation, processing of experimental data is begun with use of simpler equation including only linear members. Further, at unsatisfactory result, pass to more difficult equations including square, cross, cubic and more difficult members. However at the choice of a type of the equation it is necessary to consider that with increase in complexity of the regression equation the probability that as a result of calculations it will be possible to receive smooth dependence even within a variation of parameters therefore are usually limited to a small number of members of the equation 4.30 decreases. It is connected with the fact that the adequate dependence having a set of local minima and maxima within a variation of parameters isn't suitable for the purposes of optimization and the analysis (see Fig. 4.16) since this dependence contradicts the physical nature of a real object (in the nature, with rare exception, all properties of objects have smooth dependences).
Fig. 4.16. Illustration of an inadmissible type of statistical model
In case the number of the parameters which are really influencing work of an object is high, then it is necessary to consider physical and chemical essence of an object and to carry out his engineering analysis. By results of the analysis of an object, depending on the physical and chemical dependences, and on the nature of this influence which are the cornerstone of an object, it is necessary to carry out a combination (association) of the influencing parameters to the minimum quantity of factors and to reconstruct the table of factors.
For example, the regression equation including linear, cross and square members for two factors will register:
[4.32]
This equation of regression will be linearly concerning coefficients of regression (bJ) in case to simplify the equation, i.e. to make replacement of factors:
[4.33]
Moreover, if to enter a factor h0=1, then in a general view, the equation of regression will take a form:
[4.34]
Thus, since 4.34 factors enter the equation linearly and independently, under value of a factor of XJ also more difficult expressions, than linear, cross or sedate members can be hidden. Thus, the equation can have any real kind, and the expression hidden behind a factor of XJ can be chosen not incidentally, and on the basis of theoretical reasons taking into account that the received dependence had the most smooth appearance.
Calculation of coefficients of the regression equation of bJ is performed by method of the smallest squares which essence is described above.
After calculation of coefficients of regression pass to the statistical analysis of this equation which includes the following stages:
• adequacy of model (ability to authentically describe function of a response) is estimated;
• the importance of the factors entering the regression equation is estimated.
Check of adequacy of the regression equation is carried out by means of Fischer's (FP) criterion on a condition:
[4.35]
where
[4.36]
where 
- dispersion of an average;
[4.37]
- adequacy dispersion.
[4.38]
where N – quantity of experimental points;
m – number of coefficients of regression in the equation, including the free member
Thus, than more there will be a data array of the basic data collected in Tab. 4.3 (size N) and the type of the regression equation will be simpler (size m), that the probability that the received equation of regression will be adequate will be higher. It should be noted that according to the principles of statistics, attempts of receiving the difficult equation of regression on a small amount of experimental data, for example, calculation of the equation of the line for one point or parabolas – on two are inadmissible, i.e. the condition always has to be met: N>m.
Assessment of the importance of the factors used at the description of function of a response is carried out by means of Student's (tP) criterion on a condition:
[4.39]
where
[4.40]
where bJ – regression coefficient at the estimated factor;
– mean square deviation of coefficient of regression.
Usually the size of criterion of Student is in limits 2-4 therefore if the mean square deviation of coefficient of regression is more than 25-50% of size of the coefficient of regression (on the module), then this coefficient is considered insignificant and can be excluded from the regression equation.
After an exception of all insignificant members, the equation of regression takes a new form. Therefore, on the following step it will be necessary to estimate coefficients of regression again and again to check their importance. This cycle of operations is made until the adequate regression equation which all factors are significant is received.
It should be noted that when modeling CTS it is allowed to use only adequate mathematical models of processes in all range of change of input parameters regardless of that, the model is physical and chemical or statistical.
As an example of drawing up statistical model, we will consider the choice of parameters for definition of the dependences, which are the cornerstone of system of parametrical monitoring of emissions of the power copper using natural gas.
During the work of a copper on gas fuel, the measured technological parameters defining a copper operating mode are:
• pressure of fuel gas on torches (PGAS)
• air pressure on torches (PAIR)
• atmospheric pressure (PATM)
• temperature of combustion gases after the water economizer (VEK.) in parallel flues (TVEK-1 … 4)
• extents of opening of gates of forcing of air heaters of fans (% .zasl.vozdukha1 … 2)
• temperature of hot air after air heaters (tGOR.VOZDUKHA-1 … 2)
• extent of opening of gates of smoke exhausters (% gate DS1 … 2)
As direct use of these factors, their squares and pair works doesn't allow to receive adequate dependences, these factors were combined taking into account physical sense, in the following complexes:
[4.41]
[4.42]
[4.43]
[4.44]
[4.45]
[4.46]
The combination of technological parameters in complexes of this look can be explained with the following reasons:
• the copper torch, in fact, represents the narrowing device, therefore, amount of the fuel and air given on burning will depend on temperature of a stream, pressure before a torch and atmospheric pressure. Thus, the X1 and X2 complexes have included the parameters allowing to consider a deviation of parameters of streams from normal thermodynamic conditions (on the atmospheric pressure and temperature of the stream given to a torch);
• as formation of nitrogen oxides and недожог fuels depend on efficiency of the copper which is defined generally by temperature of combustion gases enter the X3 complex – the average temperature of combustion gases;
• formation of nitrogen oxides and недожог fuels is also influenced by the type of a torch of the burning fuel depending (with difficulty) on hydraulics of a path of combustion gases on which extent of opening of gates the air heater of the fans and smoke exhausters united in the X4, X5 and X6 complexes exerts impact (this type of complexes has been received after several unsuccessful attempts to consider influence of gates on content in combustion gases of pollutants).
It should be noted that association of factors in the specified complexes was made on a basis the empiricist, characteristic of the concrete power technological unit. For other coppers, even same, or for this copper after capital repairs of a fire chamber, a path of combustion gases or replacement of torches, the type of complexes (especially X4, X5 and X6) can be another. Besides, as hypotheses, by drawing up statistical model also other parameters of work of an object, which at data processing can be «eliminated» as insignificant, can be used.
Association of the specified factors in complexes has allowed to receive adequate dependences of concentration of NOX in combustion gases after the smoke exhauster:
,
ppm [4.47]
concentration
CO in combustion gases after the smoke exhauster:
,
ppm [4.48]
coefficient of excess of air after the smoke exhauster:
[4.49]
Values of coefficients of these dependences are presented in Table 4.4.
Table 4.4.
Values of coefficients of mathematical models for calculation of composition of combustion gases during the work of a copper on gas fuel
Calculation |
Number of coefficient in the equation |
||||
0 |
1 |
2 |
3 |
4 |
|
NOX |
1,792726102 |
-3,709418102 |
1,388268 |
7,19007610-2 |
-1,189317101 |
CO |
1,310595103 |
-3,560766102 |
4,162166102 |
-1,80793710-1 |
-1,730371103 |
|
6,932277 |
-1,26930110-1 |
7,28184610-4 |
- |
- |
Results of processing of experimental data, errors of calculations and statistical parameters of models are presented in Table 4.5 and on schedules of Fig. 4.17 and Fig. 4.18.
Table 4.5.
The key parameters of statistical models received as a result of processing of experimental data by method of the smallest squares.
Calculation |
Average absolute error |
Coefficient of multiple correlation |
Fisher’s criterion (calculated). |
NOX |
2,0 ppm |
95,94% |
10,89 |
CO |
4,8 ppm |
80,65% |
2,95 |
|
0,02 |
97,11% |
16,36 |
The tabular criterion of Fischer for the corresponding degrees of freedom has made 2,45, i.e. the received statistical dependences are adequate.
Fig. 4.17. Results of processing of experimental data: concentration of NOX and CO in combustion gases after the smoke exhauster
Fig. 4.18. Results of handling of experimental data: coefficient of excess of air after the smoke exhauster
Test questions
1. State an essence of a method of the smallest squares.
2. To what purposes are applied an active (factorial) and passive experiment.
3. Properties of the orthogonal central composite plan.
4. Creation of a matrix of planning of a two-factor experiment.
5. Check of adequacy of model.
6. Assessment of the importance of factors,
THEME 5. The MAIN SOFTWARE PRODUCTS FOR CALCULATION of CTS.
Lectures 23- 27. The overview of modern software products for calculation of CTS
Lecture purpose: studying of modern software products for calculation of CTS
Plan of a lecture:
1. A task of calculation of CTS by means of modern software products.
2. Program covers: off-line and on-line
3. Firms – software developers.
It is well known that recently special attention in the industry began to be paid on the engineering analysis and optimization of production processes. However, because of high integration of chemical engineering procedures their analysis and optimization are very difficult and steadily require application of computer facilities. Lack of the corresponding software, along with restriction of cost of works and time, necessary for performance of works, can lead to the analysis and optimization only parts of the existing technology or to consideration of smaller quantity of options of technical solutions. Besides, for more complete study of operating modes of technology and management in scales of the plant in certain cases there is a need of modeling of chemical and technological systems for dynamic conditions.
Earlier, process of modeling of engineering procedures and systems required application of programming languages and therefore it was used only by the specialists who are freely understanding chemical technology, modeling and programming. Rapid development of powerful personal computers and intuitive graphical interfaces of the user allowed to create the specialized program covers which are automating difficult calculations and visually displaying results of calculation. Now in the world there is a small choice of the competitive program covers allowing to calculate material and thermal balances of the technological systems taking into account physical and chemical regularities and intended for modeling of stationary, dynamic and periodic chemical and technological systems. When using these program covers the user doesn't have any more need to well know programming languages as process of creation of model of production consists in use of the screen interface by means of which on the computer screen in a convenient type the technological scheme is constituted. Further the software itself determines the optimum sequence of calculation of CTS, interacts with databases on processes and substances, starts process of the solution of a task and removes results in a user-friendly type.
Until the end of the 90th years such software in Russia wasn't applied that was possibly connected with the fact that along with rather high price of acquisition of the license on its use and rather high costs for training of skilled users it belonged to the category of "high technologies" on which there was a restriction for sale to Russia for political reasons until recently. Now these restrictions don't exist, however unlike countries of Western Europe and America, in Russia of wide use of the similar software didn't occur. Possibly, it is connected with level of training of an engineering personnel as successful application of similar program covers requires availability at the entity of the highly skilled specialists technologists having the corresponding theoretical preparation and a work experience with similar software products.
There are two families of program covers: off-line and on-line. The on-line family of covers is connected with PILES devices in real time. In case of its functioning information from systems, sensors and controllers in real time gathers, further this information is archived and provided to operators, technologists and managers in a form, convenient for them. These data are also transferred to the operational database from where get the software for handling. In the presence of feedback (system of "intellectual" regulation of work as production), on the basis of the acquired information the corresponding modules calculate optimum values of managing parameters, transfer them to external devices and monitor process reaction, consistently optimizing a production operating mode. However for work of on-line of products it is necessary to have the working system of the distributed management (industrial control system: sensors, controllers, computer-controlled equipment) and adequate models of basic processes and system (shop or plant) in general. In a general view, the scheme of work on-line of the software (without feedback) is provided in Fig. 5.1.
Fig. 5.1. Scheme of work on-line of the software.
The optimizer of a ratio of H2/N2 in ammonia production delivered on some new productions of ammonia, allowing to support the set H2/N2 ratio in the nitrogen hydrogenic mix given to a synthesis cycle can be an example of on-line of the software.
The off-line family of the covers (which aren't interacting directly with engineering procedure) is used in technical departments of the companies. They allow to design new production, help to eliminate bottlenecks in a technological chain, model separate installations or all plant, allow to model reconstruction of the operating installations for assessment of opportunities of transition from the existing technology to perspective. For the purpose of optimization of production or the analysis of the existing problems and emergencies, these systems help to estimate economic aspects of production, to plan resources, products and the working schedule, etc. The scheme of work off-line of the software is provided in Fig. 5.2.
Fig. 5.2. Scheme of work off-line of the software: (a) – on production; (b) – when designing.
It should be noted that these covers are universal, i.e. can be applied to various productions at the same time. Besides, depending on requirements to calculations, there are program covers as for modeling of CTS in the stationary (set) operating modes (steady state), and in the dynamic (dynamic) modes. The broadest application was found by the software products for modeling of CTS in stationary operating modes realizing an iterative method of calculation. The program covers intended for modeling of dynamic operating modes of production are applied rather seldom and generally to development of systems of management (development of the system of management of transition processes).
Now in the world there is rather small choice of competitive program covers for the specified modeling of CTS in stationary and dynamic operating modes. The basic principles of functioning of covers are single and rather well described in domestic and foreign literature, and also in this abstract of lectures.
Rapid development of computer modeling of CTS began in 1958, and was connected with transfer of calculation of material and thermal balances of CTS to the computer. Calculations were carried out by means of the modeling Flexible Flowsheet program. Further, within the 60-70th years, there was a rapid development of both the concept of computer modeling of CTS, and the software products realizing this concept. Except Flexible Flowsheet programs were abroad created: Cheops, Macsim, Chess, Flowtran, Process, etc. Software products gained the most rapid development with the advent of personal computers. A number of programs for modeling of CTS was created also in the USSR: RSS and ROSS (NIFHI), ASTERS and BASTR (GIAP), NEFTEKHIM (VNIPINEFT), CAMXTC (NIUIF), SYNSYS-78 (MHTI), etc. However, from the beginning of a transition period, the majority of domestic software products stopped the existence therefore now domestic software products are practically not used. Prime vendors of the software for modeling of CTS are the American companies now: AspenTech Inc. (http://www.aspentech.com), Honeywell (http://www.honeywell.com) and Simulation Science Inc. (http://www.simsci.com). Some time ago suppliers of the software was more, however in the world continuously there are processes of enlargement of one companies due to purchase of others that leads to reduction of their quantity.
As it was told above, all software products for calculation of CTS are based on unified theoretical bases of synthesis, the analysis, calculation and optimization. Possibly, these can explain the single functional structure of the specified covers provided on Rice 5.3.
Fig. 5.3. The functional structure of software products for simulation of CTS
According to the functional structure of software products, a basis of shell program the functional kernel of system which directly makes the analysis of structure of CTS and calculations of the material and heat balances is, exchanges data with databases, makes an input/output, etc.
Necessary part of the software are the databases on pure substances (viscosity, density, a heat capacity, heat conduction, etc.) and thermodynamic rules of their mixing, and also the database on elementary processes (the specified models of reactors, adders, dividers, columns of rectification, heat exchangers, etc.) completed with firm manufacturer. As all substances and all processes existing in the nature can't be put in databases, software products usually have an opportunity to expand databases on substances and on processes, creating temporal databases of the user. Thus, in case of unified bases and one functional structure shell programs differ from each other only in different set of databases on substances and on processes, quality of the interface and opportunities of a kernel of system. In turn these differences affect the price of a software product and conditions of its acquisition.
As well as the majority of software products for simulation of CTS, Design-II for Windows it is intended for the specified simulation of stationary chemical and petrochemical processes, including rectification, cooling, movement of liquid and gas on pipelines and other processes of oil processing, gas processing, productions of ammonia, methanol and hydrogen. The database on substances contains data on 886 substances and 50 thermodynamic methods of their mixing, tables of properties of saturated and superheated steam, parameters of models of vapor-liquid equilibrium for compounds etc. The database on processes contains 63 models of processes, including their varieties and modifications. Along with opportunities to make simulation and optimization of difficult chemical and technological systems this shell program allows to make at the same time project calculation of parameters of some technology equipment and has the opportunities which are absent at other shell programs:
• the specified modeling of system of pipelines (horizontal, vertical, inclined) for two-phase systems taking into account a heat transfer;
• calculation of parameters of various mixes of the amines allowing to model columned devices (with mixes of amines) taking into account mass transfer kinetics;
• the specified calculation of rectifying columns with determination of their diameter;
• calculation of design data of heat exchangers and separators;
• detailed setup of the modes of calculation of each module by means of keywords and by means of implementation of programs of the user in algorithmic language of FORTRAN;
• handlings of experimental data and calculation of missing properties of substances for the existing properties and structure of substances with simultaneous creation of the file of the database of the user;
• possible extension of databases on substances and processes;
• creations of CTS of the unrestricted sizes by means of "a stitching of sheets";
• access for users to the database on properties of net substances including: the molecular weight, structure, critical properties, pressure of saturated steam, warmth of steam formation, a thermal capacity of ideal gas, viscosity of gas and liquid, heat conductivity of gas and liquid, specific amount, a superficial tension, etc. as to the reference book;
• export of results of calculation to MS Excel.
The screen type in case of the solution of a task on modeling of a cycle of synthesis of ammonia is provided in Fig. 5.4.
The graphical interface of a program cover isn't really convenient, however also the price of its license which for the beginning of 2003 constitutes US$3895 a year in case of the annual conclusion of the agreement, or US$6995 upon purchase of the termless license (a comment is rather low: other suppliers don't sell the termless license). Except one-time payment it is necessary to pay US$1295 a year for support of the software product. Thus, without being "favourite" in the software market for technological calculations, but, having effectively working kernel and expanded opportunities, Design-II for Windows allows to achieve result for smaller money and is purchased by small engineering firms.
Fig. 5.4. The copy of the screen in case of the solution of a task in Design-II for Windows
If to exclude influence on the price of the license of a variable marketing component, then the cost of the license is usually determined by convenience of use of a program cover (the graphical interface), amount of databases on substances and processes, and also performance of a kernel. Besides, undoubted impact on the price is exerted by trademark of firm. However, it should be noted that the cost of the license of the most expensive software products different from the cost of the license Design-II for Windows for orders isn't comparable with the cost of the capital expenditure for reconstruction of chemical plant estimated by millions of dollars. For example, the cost of one cubic meter of the catalyst loaded into the reactor by tens of cubic meters usually constitutes US$5000030000-US$. Thus, even in case of standard standard rates, the profit got as a result of reconstruction will be many times more, than the cost of the most expensive tool for calculations. However in case of the solution of the small engineering tasks having rather low profitability, the price of the software will be determining since payback of less expensive, and even less productive software, can be reached quicker, than more productive and more expensive.
It should be noted that when carrying out any calculations correct basic data since the software for modeling of CTS accepts the basic data entered by the user as the truth are necessary. Therefore, when using wrong basic data results of calculations can be far from reality. By use reason in calculations of wrong basic data it can be connected with the fact that production work parameters (temperature, pressure, an expense and structure) are measured by the instrumentations (I) with some error. Depending on flow parameter, the size of an error can reach 5-10%. Besides, KIP can fail, showing plausible, but incorrect values.
From the specified parameters of the flows necessary for calculation of material and thermal balances, temperature and pressure can be measured rather precisely and cheap, and the structure of a flow can be checked in laboratory of the entity on the model laboratory equipment. Thus, the main problems on production arise with measurement of expenses of flows since the most widely used method of measurement of an expense by means of pressure difference on a diaphragm has rather big errors. Therefore, sizes of expenses of the flows received from PILES can't be used for job evaluation of production without additional handling. Use of the devices based on other physical principles and allowing to measure an expense with high precision will lead to rather high costs for their acquisition and servicing, and doesn't exclude mistakes.
For preprocessing of these expenses of the flows received from flowmeters there is a software allowing to make preprocessing of the measured expenses with subsequent their coordination. Along with coordination of expenses of material flows the software determines the sensors measuring an expense with a margin error, exceeding set. Because the technological scheme may contain as continuous processes (the reactor, columns, mixers etc.) and periodic processes (reservoirs, reservoirs, etc.) coordination of data on amount of substance is made per day taking into account structure of movement of material flows and errors of sensors. DATACON can be examples of this software (SimSci Inc.), SIGMAFINE (OSI Software Inc.) and some other. The principle of work of the software is provided in Fig. 5.5.
Fig. 5.5. The diagram of functioning of the software in coordination of primary information
According to the diagram, data on process are transferred to the operational database "uncoordinated data" through industrial control system and/or entered by a technological staff (in the absence of industrial control system or if industrial control system doesn't envelop all installation), and further are transferred to the software in coordination of data. Matched data register in the operational database "matched data" and can be used as for input in the program of the specified calculation of CTS, and for the analysis of a correctness of indications of PILES or for other purposes.
Example of submission of the technological installation diagram AT-6 in a software product of SIGMAFINE and results of coordination are provided in Fig. 5.6.
Fig. 5.6. Results of coordination of data for AT-6 by means of SigmaFine
Information on operability of sensors of an expense of flows and levels in reservoirs can be received, comparing the approved and uncoordinated data (in Fig. 5.6. near the sensor – a circle with figure – the approved value of a mass daily issue of a flow is under uncoordinated). Apparently from Fig. 5.6, the approved data differ from uncoordinated in spite of the fact that the KIP is issued by the indications which aren't exceeding them an error. Thus, the real data obtained from installation PILES originally shall be approved, and only after that can be used in the specified calculations.
Any mathematical model of an object or CTS is only its analog within the accepted assumptions and restrictions. When carrying out technological calculations with use of the specialized software for calculation of CTS issued by well-known companies, or carrying out calculations by means of the calculator or own programs, it is necessary to pay special attention on adequacy to the used model. If adequacy of model or its proximity to a real object isn't proved, then results of calculation can't be used to do any technological conclusions and recommendations on their basis. There are several methods of assessment of adequacy of model, however their essence comes down to comparison of settlement and experimental data by means of statistical criteria based on which draw a conclusion about adequacy or inadequacy of model.
As an example we will consider system of parametrical monitoring of the copper producing dry saturated steam which technological scheme is provided in Fig. 5.7.
Fig. 5.7. The traffic pattern of material flows of the copper producing dry saturated steam.
The essence of work of parametrical system of monitoring is that emission of pollutants in the atmosphere continuously is calculated based on a consumption of combustion gases, certain of material and thermal balances, and concentration of the pollutants calculated on adequate model of a copper. The example of handling of experimental data in case of creation of model is provided in Fig. 5.8.
Fig. 5.8. An example of handling of experimental data in case of creation of mathematical model of emissions of nitrogen oxide (NO).
Apparently from the drawing, both mathematical models (lines) rather well describe the corresponding sets of experimental values (point). The average error of models constitutes 2,52% and 2,76%. Adequacy of models was checked by criterion of Fischer, which in both cases exceeds tabular value, therefore, both models are adequate, and can be used for calculation of mass emission of nitrogen oxide in the environment.
If adequacy of mathematical model isn't proved, i.e. the mathematical model doesn't correspond to engineering procedure, to use this model inadmissibly. Thus, check of adequacy of model for all planned intervals of change of parameters is the necessary procedure before use of any mathematical model for the purposes of optimization.
Test questions
1. What problems of calculation of CTS by means of modern software products are solved.
2. Features of program covers of off-line and on-line.
3. Firms – software developers.
THEME 6. OPTIMIZATION of CTS
Lecture No. 28 - 30. CTS optimization methods
Lecture purpose: studying of optimization methods of CTS
Plan of a lecture:
1. The tasks of optimization of CTS this for the decision.
2. Optimality criterions of calculation of CTS.
3. Analytical and numerical methods of finding of an optimum.
Optimization, for example, according to/12/, are called the purposeful activities consisting in receipt of the best results under the corresponding conditions.
Historically, the problem of optimization arose with technical progress and emergence of the competition, i.e. producers of identical goods began to look for the production conditions allowing to issue the same goods with the minimum expenses. Searches of optimal solutions led to creation of special mathematical methods and in the 18th century the mathematical foundation of optimization (calculus of variations, numerical methods and B'day) were laid. However, to the second half of the 20th century optimization methods in many fields of science and technology were applied very seldom as practical use of mathematical optimization methods required huge computing work, which without COMPUTER was extremely difficult to be realized, and in some cases – it is impossible. Especially great difficulties arose in case of the solution of tasks of a process optimization in chemical technology because of a large number of parameters and their difficult interrelation among themselves.
In case of problem definition of optimization it is necessary:
1. Availability of the purpose of optimization. At the same time the formulation of each task of optimization shall require extreme value of only one size, i.e. at the same time two and more criteria of optimization since usually the extremum (a minimum or a maximum) of one criterion doesn't correspond to an extremum of another shan't be attributed to system.
Typical example of the wrong problem definition of optimization: "To receive the maximum performance in case of the minimum cost value". The mistake is that the task of search of an optimum of 2 sizes contradicting each other in essence is set.
The correct problem definition shall be following:
a) to receive the maximum performance in case of the set cost value;
b) to receive the minimum cost value with the set productivity;
In the first case criterion of optimization – performance, and in the second – cost value.
2. Availability of resources of optimization as which understand a possibility of the choice of values of some parameters of the optimized object. An object shall possess certain degrees of freedom - corrective actions.
3. A possibility of quantitative assessment of the optimized size as only in this case it is possible to compare effects of the choice of these or those corrective actions.
4. Usually optimized size is connected with profitability of work of the considered object (the device, the shop, the plant), therefore, the optimized option of work of an object shall be estimated by some quantitative measure – an optimality criterion.
In conclusion, it should be noted that it is accepted to differentiate tasks of static optimization for the processes proceeding in the set modes and tasks of dynamic optimization. In the first case issues of creation and implementation of optimum model of process, and in the second – tasks of creation and implementation of system of optimum control of process are resolved in case of the unsteady modes of operation.
Optimality criterion.
Optimality criterion is called quantitative assessment of the optimized quality of an object. Otherwise, the optimality criterion is the main sign on which judge that, the technological system how well functions, this process works, etc., and also, the optimization task is how well solved.
The optimality criterion is one of system exits, and, the following requirements are imposed to it:
• The optimality criterion shall be expressed quantitatively;
• The optimality criterion shall be the only thing;
• The size of an optimality criterion shall change monotonously (without gaps and jumps);
• The optimality criterion shall reflect the most essential parties of process;
• It is desirable that the optimality criterion made a clear physical sense and easily was calculated.
Based on the chosen optimality criterion the criterion function representing dependence of an optimality criterion on the parameters influencing its value is constituted. The type of an optimality criterion or criterion function is determined by a specific objective of optimization. Thus, the task of optimization comes down to finding of an extremum of criterion function.
The most general statement of an optimum task is expression of an optimality criterion in the form of an economic evaluation (performance, product cost, profit, profitability). However in private tasks of optimization when an object is a part of engineering procedure, it isn't always possible or it isn't always reasonable to allocate a direct economic indicator which completely would characterize overall performance of the considered object. Technical characteristics, indirectly estimating profitability of operation of the aggregate (contact time, a product yield, extent of transformation, temperature) can serve in such cases as an optimality criterion. For example, the optimum temperature profile, cycle duration - "reaction - regeneration" and etc. is established. However, anyway the optimality criterion has the economic nature.
Differentiate simple and difficult criteria of optimization. The optimality criterion is called simple if it is required to determine an extremum of criterion function without task of conditions for any other sizes. Such criteria are usually used in case of the solution of private tasks of optimization (for example, determination of the maximum concentration of a target product, optimum time of stay of reactionary mix in the device, etc.).
operation modes.
The optimality criterion is called difficult if it is necessary to establish an extremum of criterion function under some conditions which are imposed on some other sizes and restrictions. Thus, the procedure of the solution of a task of optimization surely includes, in addition to the choice of managing parameters, also establishment of restrictions for these parameters. Restrictions can be imposed both on technological, and on economic purposes. Differentiate the following main restrictions:
• By quantity and quality of raw materials and products (composition of raw materials, product quality, performance, etc.);
• Under the terms of technology (the device sizes, stay time, temperature of ignition and restructuring of the catalyst, etc.);
• On economic purposes;
• On labor and environmental protection;
Thus, it is necessary for the solution of a task of optimization:
1. to constitute mathematical model of an object of optimization;
2. to choose an optimality criterion and to constitute criterion function;
3. to set possible restrictions which shall be imposed on variables;
4. to choose an optimization method which will allow to find extreme values of required sizes.