1 DEVELOPMENT AND ANALYSIS OF TECHNOLOGICAL OBJECTS MATHEMATICAL MODELS AT SCALING-UP AND CREATING ТНЕ CONTROL SYSTEMS 1 Yu.V. Sharikov, 1 F.Yu. Sharikov,

1

DEVELOPMENT AND ANALYSIS OF TECHNOLOGICAL OBJECTS MATHEMATICAL MODELS AT SCALING-UP AND CREATING ТНЕ CONTROL SYSTEMS1Yu.V. Sharikov, 1F.Yu. Sharikov, 2V.V. Zhukov, 2I. Turunen1National Mineral Resources University (MINING UNIVERSITY), Saint-Petersburg, Russia2Lappeenranta University of Technologies (LUT, Lappeenranta, Finland)

A new approach is proposed to develop mathematical models of technological objects at the base of kinetic study of chemical reactions with the use of heat flow calorimetry and chemical analysis of the reaction mixtures. The results of modeling the processes of epoxy resins modification and gold thiosulphate leaching are given and syntheses of their proposed control systems are presented.

2

Heat flow calorimeter allows you to record the heat flow, the flow that occurs in its sample cell due to chemical reactions and phase changesChange in heat, fixed in the calorimeter heat flow can be described by the following system of equations:

PiRjdt

QqHwdQ

i

P

iij

R

jj

gen ,...,1;,...,1;11

(1.1)

R

j

N

iij

i wdtdc

1 1 (1.2)

Np

Nri

ni

ij

Nr

i

ni

ijij ckckw11

Где- total thermal effect of the chemical process, kJQgen

conversion of rate of the reaction of i-th componentwij

))/()exp(ln(,0

RTEkk jjj

))/()exp(ln(,0

RTEkk jjj

rate constant of the direct reaction

rate constant reverse reaction

Qqi

P

ii

1

-heat generation of phase transformation

3

As the observed response we measure either the total heat release rate when using DSC, or total heat release rate and change in mass of the sample over time using TG DSC,Integrated responses to stroke the whole process in general have a very large number of points, taking into account all the extreme points and points of inflection. These features make it the heat curve highly informative, and can, if necessary, extending the analysis of additional points in the most characteristic points to describe a rather complex and detailed physico-chemical mechanism of the process, taking into account the possible intermediate steps. The procedure is usually used to determine the kinetic parameters (pre-exponential factor, activation energy, reaction order and thermal effects), is the selection of the kinetic parameters from the condition of minimum deviation between the experimental data and the results of mathematical modeling of the process. As a measure of this mismatch is usually used sum of squared deviations between experimental and calculated data.The task of finding the kinetic parameters is reduced to a problem of minimizing the sum of squared deviations of R as a function of the kinetic parameters up.

4

)(exp(,

calc)(,

1 1

2

xx u p

K

k

S

s

RskskR

(1.4)

Mismatch function

up - the unknown kinetic parameters - pre-exponential factor, energyactivation reactions, orders for components, heat effects.As a method of extremum function mismatch usually use different versions of the method of nonlinear programming. The main problem of the problem of finding solutions of the kinetic parameters from experimental data is the problem of the use of an incomplete search of the state variables of the object for which the developed mathematical model. In particular the use of a calorimetric method kinetics we record the time a sufficient number of points to modify the generated heat or heat flux, but have not generally possible to measure the other state variables, in particular the concentrations of individual components. In fact, in this case we are talking about the possibility of fully parametric identification of the non fully observed object. In order to study the possibility of identifying such an object, we investigated this problem in the example of parametric identification of sequential reactions with thermal effects in the case where we can observe the different number of state variables

5

For a model experiment using the following reaction scheme:

k

H

k

HFBA

2

2

1

1

(1.5)

Mathematical model for the conditions of the heat flow calorimeter has the form

T(t)T

;)exp(ln)exp(ln

;)exp(ln

;)exp(ln)exp(ln

;)exp(ln

2022

1011

202

202

101

101

cEkHcEkH

cEkdc

cEkcEkdc

cEkdc

BA

BF

BAB

AA

TRTRdt

dQ

TRdt

TRTRdt

TRdt

(1.6

With the reduced system of kinetic equations by their numerical solutions were generated initial data used as the experimental data for modeling problems of parametric identification incompletely observed objects

6

The solution of the model equations was performed for the following values of the kinetic parameters of the model shown in Table 1Table 1.1 Kinetic parameters for modeling the search procedure of the constants from the experimental data

lnk01=25, 1/min; E1=100 kJ/mol H1=30 kJ/mol;

lnk02=8, 1/min E2=40 kJ/mol H2 =10 kJ/mol

The initial conditions for the solutions of the model equations werethe following values :Table 1.2: Initial conditions.

cA(0) 10 kmol/m3

cB(0) 0 kmol/m3

cF(0) 0 kmol/m3

Temperature was chosen as a linear temperature increase from 400 to 500 K for 100 minutes.Solution of the problem was carried out in a complex software environment ReactOp [1] ..In Figures 1.1 and 1.2 present the results of the solutions used as experimental data

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Time, min100500

kmol

/m3

10

5

0

kJ/m3

400

300

200

100

0

Time, min100500

kmol

/m3/

min

0.4

0.2

0

-0.2

-0.4

kJ/m3/m

in

10

8

6

4

2

0

Figure 1.1 Integral kinetic curves of the simulation problem.

Figure 1.2 Differential curves modeling consecutive exothermic reactions

8

First, we use a function for generating error all state variables - cA (t), cB (t), cF (t), Heat (t) and Heat '(t).Search constants in accordance with the algorithm of the gradient search method is as follows:

u

Ruu

p

M

m

m

p

m

p

1

The following figure shows the results using all state variables of the initial starting point of the following:

Kinetic parameter Reaction 1 Reaction 2lnk0,j 18 3

Ej 80 15

Hj 20 12

R0 752058

Table 1.3 The starting point for finding the kinetic parameters.

9

10

11

12

Then there were used only variables measured in the calorimeter heat flow, i.e. magnitude Heat (t).The misalignment between the starting point was equal to the value 751482,.Once you start your search at least was found in 30 steps. The ultimate magnitude of the error was of the 3,7610 -20.

Use of the heat flux and concentration of the substances for forming mismatch function leads to successful finding a minimum point, through 19 steps was achieved at a

minimum Rmin = 8.7610-23.Using only the concentration curves can quite accurately determine the kinetic parameters, but it is not determined by the value of the thermal effects on the results of the research can conclude the following:Then there were used only variables measured in the calorimeter heat flow, i.e. magnitude Heat (t).The misalignment between the starting point was equal to the value 751482,.Once you start your search at least was found in 30 steps. The ultimate magnitude of

the error was of the 3,76 10 -20.Use of the heat flux and concentration of the substances for forming rassoglasovaniakzhe function leads to successful finding a minimum point, through 19

steps was achieved at a minimum Rmin = 8.76 10-23.Using only the concentration curves can quite accurately determine the kinetic parameters, but it is not determined by the value of the thermal effects on the results of the research can conclude the following:

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Statistical Summary===================Model fits experimental dataSum of squares: Residual = 2.723E-25 degree of freedom = 96 Regression = 605.8 degree of freedom = 6Percentage point = 0.05Fisher (calc) = 2.225E27Fisher (tab) = 2.195degrees of freedom = 6, 96Student (tab) = 1.985degree of freedom = 96Sigma ^ 2 = 2.723E-025Sigma = 5.219E-013Parameters of the model================================================== ======================== Parameter Parameter Standard Confidence Interval Parameter name estimate deviation min max units-------------------------------------------------- ------------------------ ln (Ko) {1} +25 +9.409 E-13 +25 +25 [min] E {1} +100 +3.221 E-12 +100 +100 kJ / mol (-H) {1} +30 +1.754 E-12 +30 +30 kJ / kmol ln (Ko) {2} +8 +2.242 E-12 +8 +8 [min] E {2} +40 +8.311 E-12 +40 +40 kJ / mol (-H) {2} +10 +1.925 E-12 +10 +10 kJ / kmol

Thus, in the result of the simulation procedure of finding the kinetic parameters by results of calorimetric studies in calorimeter heat flux was established that for reliable parametric identification shall be used as the initial data for forming the function of the mismatch, or the number of released heat - H(t), or heat flow H’(t) and one of the concentrations of the reactants ci(t). It should be considered at the organization of kinetic experiment.

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2.Matematicheskoe modeling modification process epoxy resins 2.1 Investigation of modification kinetics

Polymeric compositions based on epoxy resins have an extremely wide application in various industries. Procedure modification enables obtaining various indexes functionality, viscosity, composition of the active amine or hydroxyl groups to their compatibility in the final composite formulations. Modified epoxy oligomer with dihydric alcohols allows to obtain a new, unique set of properties and on this basis to synthesize a wide range of polymer compositions. These include anti-corrosion coatings for pipes, concrete structures, bridges, bonding and repair pastes for "cold fusion" of metals, etc.

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A powerful tool for intensification and optimization of relevant process steps is the method of mathematical modeling. Its application requires a kinetic studies and the development of mathematical models of the processes of modification. Evaluation of safe operation of the reactor site, and the problem of possible rational use of the reaction heat also require corresponding detailed kinetic studies. For the development of a kinetic model of the process of modification of epoxy oligomers butanediol we performed kinetic studies of the process of using heat flow calorimetry C-80 of French company «Setaram». For this study were selected two commercially important brand epoxy oligomers - "ED-20" and "OKSILIN-6", and as a modifying agent - a dihydric alcohol, butanediol-1, 4. The catalyst used was a solution of NaOH in known concentration butanediol-1, 4. safe operation of the reactor unit and the problem of the possible rational use of the reaction heat also require corresponding detailed kinetic studies.

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For epoxy oligomers "ED-20" and "OKSILIN-6" series of kinetic experiments conducted by modifying them in a wide range of experimental conditions (the molar ratio of the reactants, "epoxy group-alcohol" (1:20 - 1:1), catalyst concentration ( 0.1-0.4% wt.), temperature (linear heating in the range of 35-195 C with heating rates of 0.5 and 0,2 C / min and isothermal modes 110,120,130,150 C). Kinetic curves of heat release rate for the given experimental conditions , to determine the magnitude of the thermal effect of the reaction, and the reaction products for further conducted an independent analysis on the degree of conversion of the epoxy groups and viscosity measurements are made.

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The proposed kinetic model involves three steps:1. Equilibrium (reversible) formation step alkoxide ion-butanediol-1, 4;2. Irreversible step reacting the alkoxide ion of the epoxy group of the oligomer. 3. Equilibrium (reversible) step proton exchange between the charged moiety and butane diol.

ORHOOHMODIFOMODIFOMODIFORHOEPO

OHORHONa

}{OH-R-HO}){3

}{}){2

OH-R-HO1)NaOH2

The model has the following form: It has been observed over a mixture of epoxy resins with butanediol, epoxy oligomers have limited solubility in the butanediol, the solubility is dependent on temperature and the ratio of butanediol: oligomer.Given this observation to describe the kinetics of modification was designed biphasic model considering reactions of 1-3 butanediol solution and the dissolution rate of the epoxy oligomer butanediol.The model has the following form:

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)(

)()8

)7

/)(6)6

)5

)4

)3

/)/()2

)/()1

dTr

0

)3/2(

,

)3/1(

,0

EPO

TTFkFk

TTFkFkdV

JdCCVVkJ

dVH

dV

dV

rJVCdC

rVCdC

crjjcc

jrjj

jjEPOEPO

m

kEPOsEPOtEPOmm

EPOEPO

EPO

r

EPOmrEPO

jrj

j

cv

cvdtcv

Hdt

w

dtdt

dH

wdt

wdt

wwdt

wdt

Where Cj-concentration of all components of (1) in the reaction zone except for the concentration of epoxy groups kmol/m3

SEPO the concentration of epoxy groups in the reaction zone kmol/m3

Vr-volume of the reaction zone, m3

VEPO-volume epoxy resin in the reactor, m3

Jm-flow mole epoxy groups in the reaction zone kmol/m3

w-flow rate to the reaction zone m3/secdk ,0-volume droplets dissolving phase at the initial time, mdEPO-molar density of epoxy groups, kmol/m3.

Tr-reactor temperature, K; Tj-jacket temperature; Tc-coil temperature

19

Schematic model of two-phase system and the transition of the two immiscible liquids in a homogeneous system. Process is described by equation 6) of mathematical nodel

Initial state Intermediate state final state

20

The mathematical model of the modification process was used to determine the parameters of the mathematical model. Figure 1 shows a comparison of the calculated and experimental values of the heat generation rate. From the figures it is clear that the model parameter values to those found quite satisfactory, and the experimental data can be used to search for optimum conditions different brands of epoxy modified resins in

reactors of various sizes.

Figure-. 3.1. Comparison of experimental and calculated data modeling processmodified epoxy resins.a) an epoxy resin ED-20b) for chlorine-containing resin "Oksilin-6"

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3.2. Optimal control of the modifications Find a profile of the temperature T (t), which would provide maximum conversion x (tk) subject to the limitations on the temperature:

TTTTtj

jkFx

maxmin

max,)()(

Где x(tk)=F(Tj)max – objective

W j = 1 n-jacket temperature, J = (n +1) 2n-coil temperature for.Where, n-number of sections into which the specified process timeTmin, p, Tmax, p, Tmin, cTmax, c, minimum and maximum temperature for the jacket and the coil, T max - the maximum allowable temperature of the reaction mixture.the following limitations::

2n1j для,

n21)n(j для;

n1j для;

Tmax

max,min,

max,min,

TTTTTTT

cjc

JjJ

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Figure-3.2. Optimum modofitsirovaniya epoxy resin ED-20.a) The optimal temperature profileb) changes in the concentrations at the optimum the temperature profile

a) b)

23

Figure-3.3 Optimal modification using temperature jackets and coil as a control action. a) the optimal profile of the controls. b) Change in concentration in the optimal

a) b)

24

PID №1

PID №2

Contoller

Control system of modification process of epoxy resins

distillate

Cool water input

Cool water output

steam

25

Analyse of kinetic of the thiosulfate leaching

Reactions at leaching:

Reduction of the Cu(II).

(1)

(2)

(3)

Reaction (1) may be described by shrinking sphere (ss) model

Instead of reaction (1) may be used reaction (1’), taking into account sorption ofReagents on surface ores or concentrate: (1’)

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Integral form of the shriking sphere model:

After differentiation it is possible o obtain differential rate equation as follow:

ccxxk NHCuOSr

bk

dt

dxss )(33

4332)1()1(

3/23/2 (5)

In order to use equation (5) for description leaching rate it is necessary to divide value kss, obtained on eq.(4) on used at finding kss

(4)

cc NHCuOS )( and 4332

27

Temperature dependency of kss

On base of experimenta data from literature it is ossible to obtain activation energy ofLeaching proces.

Temperature dependency of kss

y = -8262.3x + 18.668R2 = 0.9194

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

0.00305 0.0031 0.00315 0.0032 0.00325 0.0033 0.00335 0.0034

1/T

ln(k

ss) Ряд1

Линейный (Ряд1)

E=68.69 kJ/molln(kss)0=18.69Lnk=25.598

Y=-8262.8x+18.688R2=0.9194

60_+10 kJ/ mol from literature: Minerals Engineering, Vol. 13, No. 10 1 l, pp. 1071-1081, 2000. © 2000 Published by Elsevier Science Ltd

28

Kinetic of reduction of Cu2+

Figure 2 shows a plot of the UV-Vis absorbance verses time for the conditions used in the leaching experiment.It is clear that the absorbance decreases with time, which is indicative of a decrease in copper(II). concentration. After a time of 13800 s (3.8 hrs), it can be seen that 50 % of the copper(II) has reacted withthiosulfate. From absorbance data, the rate of reaction can be calculated,

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Experimental data have been treated with aid of ReactOp and kinetic parameters have been speciefied. On the Fig.4 it is shown comparison experimental data with simulation results

Time, min3e32e31e30

abs-

Cu,

dim

less

1

0.5

0

Name Value Stepln(Ko), [sec] 41.06E=112 kJ/mol 1n(Cu(NH3)4), dimless 1 1n(S2O3), dimless1 1

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For obtaining of activation energy have been used experimental date of reduction ratesat different temperatures

Fig.6Tempeature dependency of reduction rate. Ряд 25 С, Ряд2-30 С, Ряд3-35 С, Ряд 4-40 С

y = 0.0216x2 + 0.0822x

R2 = 0.9943

y = 0.0123x2 + 0.0259x

R2 = 0.9958

y = 0.0073x2 + 0.0027x

R2 = 0.9977

y = 0.0045x2 - 0.0039x

R2 = 0.9947

-0.5

0

0.5

1

1.5

2

2.5

3

-2 0 2 4 6 8 10 12

Cu(II) conc

Re

du

cti

on

ra

te

Ряд1

Ряд2

Ряд3

Ряд4

Полиномиальный (Ряд4)




31

Activation energy 112 kJ/molFrom literature 110 kJ/kmol

Temp-dependency of reduction rate

y = -14501x + 39.357R2 = 0.9991c=5

E=120

y = -13308x + 36.077R2 = 0.9984 c=7

E=110

y = -12601x + 34.235R2 = 0.998 c=9

E=105

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

00.00318 0.0032 0.00322 0.00324 0.00326 0.00328 0.0033 0.00332 0.00334 0.00336 0.00338

1/T

lnR

r

Ряд1

Ряд2

Ряд3




32

Conversion of reaction of (2) may be made with using correlations on Fig.2

33

For taking into account temperature dependency it is necessary to take into account side reaction, for example

Formed cupric sulfide and sulfur may cause passivation of gold surface and to hinder gold leaching

34

Results of estimation proposed leaching model wirh using experimental data from Literature - Hydrometallurgy 39 (1995) 265-276 at temperature 25C

Time, min200150100500

B, k

mol

/m3

0.8

0.6

0.4

0.2

0

35

Results of the statistical analyze of estimation procedure:

Model fits experimental data

Sum of squares: Residual = 0.001652 degree of freedom = 5 Regression = 1.104 degree of freedom = 2Percentage point = 0.05Fisher (calc) = 668.2Fisher (tab) = 5.786degrees of freedom = 2, 5Student (tab) = 2.571degree of freedom = 5

Sigma^2 = 1.652E-003Sigma = 4.065E-002Parameters of the model E1=68.89; E2=112: n1=0.67========================================================================= Parameter Parameter Standard Confidence Interval Parameter name estimate deviation min max units ------------------------------------------------------------------------- ln(Ko){1} +15.1 +0.1238 +14.78 +15.41 [min] ln(Ko){2} +41.94 +0.1616 +41.52 +42.35 [min]

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Simulation leaching process at temperature 40C

Time, min200150100500

B, k

mol

/m3

0.8

0.6

0.4

0.2

0

Simulation results confirm experimental data about effect increasing temperature on leaching process

37

Modeling of thiosulphate gold leaching. Batch process.

Chemical equations used in the model:1. Aus+Cu(NH3)4+5S2O3Au(S2O3)2+Cu(S2O3)3+4NH3

2. Cu(NH3)4+5S2O3Cu(S2O3)3+4NH3+S4O6

3. Au(S2O3)2+2NH3↔Au(NH3)2+S2O3

4. AB5. Auf+Cu(NH3)4+5S2O3Au(S2O3)2+Cu(S2O3)3+4NH3

s – “slow” gold, f – “fast” gold

reaction Ln(K0), min E, kJ/mol

1 11,65 58

2 42,53 112

3 20 40

4 10 40

5 14,07 58

Parameters were obtained by estimation of 8 laboratory experiments carried out by Matti Lampinen and Vladimir Zhukov

Kinetic parameters for other model were obtained by estimation of 6 laboratory experiments carried out by Dmirty Gradov and Nikolay Karastelev

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2. Modeling of thiosulphate gold leaching. Batch process.

Results: plots of chemical components during the leaching process

Time, min200150100500

Au

* 1e

+4, k

mol

/m3

0,5

0,4

0,3

0,2

0,1

Time, min200150100500

Auf

* 1

e+4,

km

ol/m

3

0,4

0,3

0,2

0,1

0

Time, min200150100500

B, k

mol

/m3

80

60

40

20

0

Time, min200150100500

z, d

imle

ss

1

0,95

0,9

0,85

0,8

Time, min200150100500

S2O

3, k

mol

/m3

0,16

0,155

0,15

0,145

39


Time, min200150100500

B, k

mol

/m3

100

50

0

Exp8

Con

vers

ion,

%Results: plots of simulation curve and experimental points

Time, min

40


Results: plots of simulation curve and experimental points

Time, min200150100500

B, k

mol

/m3

100

80

60

40

20

0

Run2

Con

vers

ion,

%

Time, min

41

2. Modeling of thiosulphate gold leaching. CSTR.

Time, min1,5e31e35000

kmol

/m3

100

80

60

40

20

0

Model is based on kinetic parameters taken from batch reactor model.

42

2. Modeling of thiosulphate gold leaching. CSTR.


B, k

mol

/m3

100

80

60

40

20

0


Au

* 1e

+5, k

mol

/m3

1,5

1

0,5

0


Auf

* 1

e+5,

km

ol/m

3

0,4

0,3

0,2

0,1

0


S2O

3, k

mol

/m3

0,15

0,1

5e-2

0


S4O

6, k

mol

/m3

1e-2

8e-3

6e-3

4e-3

2e-3

0

43

2. Modeling of thiosulphate gold leaching. Cascade of reactors.

• Model is based on the CSTR model with kinetic parameters taken from batch reactor model.

44

2. Modeling of thiosulphate gold leaching. Cascade of reactors.

Number of reactors

Volume of each reactor, m3

Total volume of cascade, m3 Max conversion, %

1 2 3 4 5

1 1 1 93,16

2 0,5 1 91,33 98,15

3 0,33 1 89,63 97,76 99,37

4 0,25 1 88,03 97,27 99,27 99,77

5 0,2 1 86,51 96,73 99,12 99,74 99,92

Volumetric Flow Rate, m3/min 0,0054 Residence time, min 185

45

Conclusion.These examples show the wide range of

applications of the method of development of kinetic models complex multistep schemes. Especially

effective is the use of integrated approach, provided analytical determination of the composition of the reacting mixture in the most characteristic points corresponding to the change in the mechanism of

the process. Creating a detailed mathematical model of the process allows use mathematical

optimization techniques to determine the optimum operation of the process.

1 DEVELOPMENT AND ANALYSIS OF TECHNOLOGICAL OBJECTS MATHEMATICAL MODELS AT SCALING-UP AND CREATING ТНЕ CONTROL SYSTEMS 1 Yu.V. Sharikov, 1 F.Yu. Sharikov,

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