some theoretical problems in chemical engineering chr. bojadjiev bulgarian academy of sciences,...

B A S - I C h EL A B O R AT O R Y

P R O C E S SS Y ST E M S

E N G IN E E R IN GSOME THEORETICAL PROBLEMS IN CHEMICAL ENGINEERING

Chr. BojadjievBulgarian Academy of Sciences, Institute of Chemical Engineering,

“Acad. G.Bontchev” str., Bl.103, 1113 Sofia, Bulgaria, E-mail: [email protected]

CONTENTS INTRODUCTION

1. NON-LINEAR MASS TRANSFER1.1. Non-linear mass transfer kinetics1.2. Gas – liquid and liquid – liquid systems

2. HYDRODYNAMIC STABILITY2.1. Non-stationary absorption kinetics2.2. Non stationary evaporation kinetics

3. INCORRECT INVERSE PROBLEM3.1. Incorrectness of the inverse problem3.2. Regularization of the iterative method for parameter identification3.3. Inverse problem solution3.4. Incorrect inverse problem ”diagnostics”

CONCLUSIONS

INTRODUCTION

Many models of the industrial processes are based on the hypothesis for linearity of the heat and mass transfer mechanism. However, there are many cases of a big differences between the experimental data and the prediction of the linear theory of heat and mass transfer kinetics. A composition of the non-linear models of heat and mass transfer is one of the mane problems of the modern chemical engineering.

Theoretical analysis of the non-linear effects in the heat and mass transfer processes show that its order is about 10-30%. However in many cases the difference between experimental data and linear theory is bigger. This effects are result of the lost of stability of the systems, when small disturbances increase to the stable periodical structure with constant amplitude. This self – organizing dissipate structures are characterized with very big heat and mass transfer rates. In this cases the dissipation energy value is not very big, because the system is before the flow turbulization. The modelling of this processes with intensive heat and mass transfer is related with the theoretical analysis of hydrodynamic stability, which is another main problem in chemical engineering. Many models in chemical engineering contain parameters, which must be obtain on the bases on the experimental data. Very often a solution of this inverse problem for parameter identification lead to big mathematical difficulties, related with the inverse problem incorrectness. These are the cases when problem solution (parameters values) is sensible with respect to experimental data error of the objective function. This is another important problem in chemical engineering.



E N G IN E E R IN G

1. NON-LINEAR MASS TRANSFER

The main idea follows from the non-linearity of the convection-diffusion equation:

ρ(c)W(c) grad c = div[ρ(c)D(c) grad c] + kcn .

The velocity W is governed by the hydrodynamic equations. However, the principal non-linear phenomenon is due to the concentration effects on the velocity W(c), density (c), viscosity (c), diffusivity D(c) and on the chemical reaction rate kcn (for n 1).

The mathematical model allows the following principle characteristics of the linear mass transfer to be drawn: the interphase mass transfer rate does not depend on the mass transfer direction;the interphase mass transfer coefficient does not depend on the characteristic concentrations.

All deviations of the experimental data from the predictions of the non-linear mass transfer theory are results of the non-linear effects.



E N G IN E E R IN G The thermal effect of the chemical reactions could lead to the temperature non-uniformity on the interface and to consequent surface tension gradients. This calls for new boundary conditions taking into account the equality of the tangential components of the stress tensor on the interface:

xy

u

y

u,0y 2

21

1

One of the most interesting non-linear effects arises from the conditions imposed by the high concentration gradients. The latter induce secondary flows at the interface. The velocity of these flows is directed normally to the interface:

0y

i

i0

*i y

cMDv

This effect has been discussed in details for a large number of systems taken as examples and it has been termed “non-linear mass transfer effect”.



E N G IN E E R IN G

1.1 NON-LINEAR MASS TRANSFER KINETICS

The kinetics of the non-linear mass transfer in the approximations of the boundary layer theory will be discussed on the basis of the solution of the equations of hydrodynamics and convection-diffusion, with boundary conditions that take into consideration the influence of the mass transfer on the hydrodynamics. In a rectangular co-ordinate system, where y = 0 corresponds to the interphase surface gas (liquid) – solid, the mathematical description of the non-linear mass transfer has the form:

,0y

v

x

u,

y

u

y

uv

x

uu

2

2

;y

cD

y

cv

x

cu

2

2

**0

0000 cc,y

cMDv,0u,0y;cc,uu,;ycc,uu,0x

where a potential flow, with a velocity u0 along a plate, and a concentration (c0) of the transferred substance are assumed. As a result of the rapid establishment of thermodynamic equilibrium, the concentration c* is always constant on the solid surface. The normal component of the velocity at the interpface is a consequence of intensive interphase mass transfer. The mass transfer rate for a plate of length L could be determined from the average mass flux:

,IdxL

1)cc(MkJ

L

00

*

where k is the mass transfer coefficient and I can be expressed from as follows: 5

ε = 1 ε = 2 ε = 10 ε = 20

θ -ψ'N(0) -ψ'(0) -ψ'N(0) -ψ'(0) -ψ'N(0) -ψ'(0) -ψ'N(0) -ψ'(0)

0.00 0.664 0.6640 0.535 0.535 0.314 0.305 0.250 0.246

+0.03 0.650 0.6500 0.515 0.516 0.270 0.265 0.190 0.199

-0.03 0.679 0.6790 0.553 0.555 0.384 0.365 0.406 0.363

+0.05 0.641 0.6410 0.503 0.504 0.248 0.250 0.166 0.205

-0.05 0.689 0.6890 0.572 0.570 0.459 0.415 - 0.479

+0.10 0.620 0.6200 0.475 0.478 0.207 0.250 - 0.355

-0.10 0.716 0.7162 0.616 0.611 - 0.581 - 0.903

+0.20 0.581 0.5840 0.429 0.442 0.160 0.418 - 1.229

-0.20 0.779 0.7760 0.736 0.707 - 1.080 - 2.325

+0.30 0.548 0.5550 0.393 0.425 - 0.808 - 2.868

-0.30 0.855 0.8430 0.936 0.822 - 1.800 - 4.512

where ′(0) is the dimensionless diffusion flux, which was obtained by asymptotic and numerical methods. Comparison of the results of the asymptotic theory ′(0) with the results of the numerical experiment ′N(0).

.D

LuPe),0(Pe

D

kLSh 05.0

*0

*

.y

cMDI

0y*0

*

B A S - I C h E

L A B O R AT O R YP R O C E S SS Y ST E M S

E N G IN E E R IN G

Similarity variables permit to obtain Sherwood number:



E N G IN E E R IN G

The obtained results show that the direction of the intensive mass transfer significantly influences the mass transfer kinetics and this can not be predicted in the approximations of the linear theory ( = 0). When the mass transfer is directed from the volume towards the phase boundary ( < 0), the increasing of the concentration gradient in the diffusion boundary layer (c* - c0) leads to an increase in the diffusion mass transfer. In the cases when the mass transfer is directed from the phase boundary towards the volume ( > 0), the increasing of the concentration gradient leads to a decrease in the diffusion mass transfer.

The above mentioned two effects (the Marangoni effect and the effect of the non-linear mass transfer) can manifest themselves separately as well as in combination in gas – liquid and liquid – liquid systems. Co-current gas and liquid flows in the laminar boundary layer along the flat phase surface will be considered. One of the gas components is absorbed by the liquid and reacts with a component in the liquid phase. The chemical reaction rate is of first order. The thermal effect of the chemical reaction creates a temperature gradient, i. e. the mass transfer together with a heat transfer can be observed. Under these conditions the mathematical model takes the following form:

1.2 GAS – LIQUID AND LIQUID – LIQUID SISTEMS



E N G IN E E R IN G ,0

;

2

2

y

v

x

u

y

u

y

uv

x

uu jjj

jj

jj

j

,)1(

2

2

jj

jj

jj

j kcjy

cD

y

cv

x

cu

;kcc

q)1j(

y

ta

y

tv

x

tu j

pjj2

j2

jj

jj

j

j=1,2 ,

where the indexes 1 and 2 are referred to the gas and the liquid respectively. The boundary conditions determine the potential two-phase flows far from the phase boundary. Thermodynamic equilibrium and continuity of velocity and momentum, mass and heat fluxes can be detected on the phase boundary.

0j21010jj tt,0c,cc,uu,0x

01101101 tt,cc,uu,y

022202 tt,0c,uu,y

21 0v,y

cMDv,

xy

u

y

u,uu,0y 2

1

*10

11

2121

y

cD

y

cD,cc 2

21

*10

*1

121

*1

*10

*1 Mc

2, 1j,y

ttvc

y

t,tt 2

2111p11

121

;

;

;

,

,,

.



E N G IN E E R IN G

At high enough values of c0 a large concentration gradient directed normally to the interface (∂c1/∂y)y=0 , can be observed, which induces a

secondary flow with the rate ν1. The tangential concentration and temperature gradients along the phase boundary create surface tension gradient:

x

t

tx

c

cx2

2

2

2

which induces a tangential secondary flow, which rate is proportional to ∂σ/∂x. Later the use of substance, which is not surface active, i. e. ∂σ/∂c2 ≈ 0, will be examined. The mass transfer rate (Jt) and the heat transfer rate (Jt) can be determined from the local mass (Ic) and heat (It) fluxes after taking the average of these fluxes along a length (L) of the interface:

dxIL

1ckJ

L

0c0cc

0y

1

*10

*11

c y

cMDI

dxIL

1tkJ

L

0t0tt 0y111p1

0y

11t )tv(c

y

tI

The solution of the problem allows the determination of Sherwood and Nusselt numbers:

)JScJ(ReMSh 21311 )JJ(PrJReNu 411331 ,

where J1 and J2 are the main parts of the dimensionless mass and heat fluxes:

,

, ,

, ,

.



E N G IN E E R IN GdX

Y

CJ

L

0 0Y1

11

1

dXY

TJ

L

0 0Y1

13

1

, .

The results obtained by the solving of the problem are shown in the tables, where θ3 and θ4 are parameters of the non-linear mass transfer effect and the Marangoni effect:

1*10

03 Sc

Mc

,

luu

t

t 20

2

220

0

24

,

The comparative analysis of the non-linear mass transfer effect and the Marangoni effect in gas-liquid and liquid-liquid systems show, that the Marangoni effect does not affect on the heat and mass transfer kinetics, because in real systems the parameter θ4 is very small. Influence of the non-linear mass transfer effect and Marangoni effect on the heat and mass transfer kinetics in gas-liquid systems.

Gas-liquid1=0.1 2=0.145

No 3 4 J1 J2 J3 J4

1 0 0 0.5671 0.09721 0.01855 -0.013372 0.2 0 0.6129 0.01155 0.02143 -0.015543 -0.2 0 0.5274 0.08542 0.01623 -0.011624 0 10-4 0.5671 0.09721 0.01855 -0.013385 0 10-3 0.5671 0.09721 0.01855 -0.013376 0 10-2 0.5670 0.09718 0.01857 -0.013397 0 10-1 0.5658 0.09696 0.01879 -0.013648 0 1 0.5658 0.09696 0.01879 -0.013649 0 5 0.5660 0.09696 0.01854 -0.01345



E N G IN E E R IN G

Influence of the non-linear mass transfer effect and Marangoni effect on the heat and mass transfer kinetics in liquid-liquid systems.

Liquid-liquid

1=0.9 2=3 (u2(X,Y2)=1)

No 31 32 4 J1 J2 J3 J4

1 0 0 0 21.1000 4.8778 0.3320 -0.0524

2 4.10-4 -8.10-4 0 22.5419 5.7854 0.4288 -0.0628

3 0 0 2.10-4 21.1000 4.8778 0.3320 -0.0524

4 0 0 1.10-3 21.0999 4.8778 0.3320 -0.0524

5 0 0 1.10-2 21.0990 4.8774 0.3320 -0.0524

6 0 0 1.10-1 21.0899 4.8736 0.3319 -0.0524

7 0 0 5 20.5698 4.6527 0.3291 -0.0513

Influence of the non-linear mass transfer effect and Marangoni effect on the heat and mass transfer kinetics in liquid-liquid systems when the second liquid is immobile.

Liquid-liquid1=1 2=1 (u2(X,Y2)=10-4)

No 31 32 4 J1 J2 J3 J4

1 0 0 0 16.9333 3.3960 0.3041 -0.0460

2 4.10-4 -8.10-4 0 18.3164 4.0715 0.3967 -0.0551

3 0 0 2.10-4 16.9333 3.3960 0.3041 -0.0460

4 0 0 1.10-3 16.9331 3.3959 0.3042 -0.0460

5 0 0 1.10-2 16.9314 3.3952 0.3041 -0.0596

6 0 0 1.10-1 16.9145 3.3885 0.3040 -0.0592

7 0 0 1 16.7421 3.3201 0.3026 -0.0456

8 0 0 5 15.8955 2.9669 0.2968 -0.0437

The obtained results show that the Marangoni effect is negligible in two-phase systems with movable phase boundary and absence of surface active agents. The deviations from the linear mass transfer theory have to be explained by the non-linear mass transfer effect in conditions of the large concentration gradients. However, in many cases the deviations from the linear theory are significantly greater than those predicted by the non-linear mass transfer theory. This may be attributed to the loss of hydrodynamic stability as a result of secondary flows induced by the large concentration gradients.



E N G IN E E R IN G

2. HYDRODYNAMIC STABILITY

Theoretical studies of the influence of the suction (injection) from (to) the boundary layer, as a result of the secondary flow, show that it leads to a significant change in the flow stability. The linear stability analysis consider a non-stationary flow (U, V, P, C), obtained as a combination of a basic stationary flow (u, v, c) and two-dimensional periodic disturbances (u1, v1, p1, c1) with small amplitudes ( << 1) :

U(x,y,t) = u(x,y) + u1(x,y,t) , V(x,y,t) = v(x,y) + v1(x,y,t) ,

P(x,y,t) = p1(x,y,t) , C(x,y,t) = c(x,y) + c1(x,y,t).

The non-stationary flow thus obtained satisfies the full system of Navier - Stokes equations. After linearizing about small disturbances we have the following problem:

,y

u

x

u

x

p1

y

uv

x

uu

y

uv

x

uu

t

u2

12

2

12

111

111

,y

v

x

v

y

p1

y

vv

x

vu

y

vv

x

vu

t

v2

12

2

12

111

111

;0y

v

x

u 11

;pp,0v,0u,0x 0111

;pp,0v,0u,0y 0111 .0v,0u,y 11



E N G IN E E R IN G

The differentiation on y and x of the first two equations provides the opportunity to exclude the pressure p1. The stability of the basic flow will be examined considering periodic disturbances of the form:

),tx(iexp)y(Fu1 ),tx(iexp)y(Fiv1 where F(y) is the amplitude of an one-dimensional disturbance (regarding y); and / are its wave number and phase velocity respectively:

.i,2

ir

In the expressions is the wave length, r - the circle frequency, i - the increment factor. Obviously, the condition for stability of the flow is i < 0. In the case of i > 0 the basic flow is unstable (the amplitude grows with time). Linear stability analysis use the solution of Orr - Sommerfeld type equations for the amplitude of the disturbances (in similarity variables):

fACf,0)(,0)(,0)0(,0)0( 2

,ff2

Aff

2

1ff

2

1AA2

ReA

i 242

where ,0fff2 ;b

4)0(f,0)0(f,a)0(f

2

.u

72.1Re,iCCu

,A 0ir

0

C



E N G IN E E R IN G

Values of the critical Reynolds number Recr corresponding to the wave velocities Cr wave number A and Cr min , Amin obtained.

Recr A Cr Amin Cr min

1 -0.30 1619 0.259 0.3281 0.301 0.3310-0.20 1014 0.285 0.3587 0.322 0.3599-0.10 689 0.290 0.3816 0.340 0.38480.0 501 0.305 0.4035 0.359 0.40670.10 386 0.309 0.4196 0.373 0.42430.20 310 0.320 0.4351 0.387 0.43960.30 258 0.331 0.4488 0.398 0.4526

10 -0.05 555 0.300 0.3960 0.351 0.39900.0 501 0.305 0.4035 0.359 0.40670.05 476 0.305 0.4062 0.360 0.40970.10 459 0.305 0.4085 0.361 0.41240.20 437 0.310 0.4123 0.367 0.4155

20 -0.05 558 0.305 0.3959 0.351 0.3978-0.03 528 0.305 0.4010 0.354 0.40370.0 501 0.305 0.4035 0.359 0.40670.03 488 0.305 0.4064 0.362 0.4099

It could be seen from the table, that the intensive interphase mass transfer directed toward the phase boundary ( < 0) (the effect of "suction") stabilises the flow. In the case of intensive interphase mass transfer directed from the phase boundary toward the volume ( >0) (the effect of "injection") a destabilization of the flow is observed.



E N G IN E E R IN G

No. 3 4 Recr Amax Cr max

1. 0.0 0.0 800 0.357 0.4503

2. 0.2 0.0 1411 0.329 0.4187

3. -0.2 0.0 512 0.382 0.4763

4. 0.0 10-4 800 0.357 0.4503

5. 0.0 10-3 800 0.357 0.4503

6. 0.0 10-2 800 0.357 0.4503

7. 0.0 10-1 799 0.356 0.4505

8. 0.0 1.0 799 0.356 0.4505

The results obtained show that Marangoni effect increase with increasing of the characteristic velocity of the liquid phase. That is why in the next part will be considered gas absorption and liquid evaporation in immovable gas-liquid layers. In these conditions is possible to create self – organizing dissipate structures.

Comparison analysis of the influence of the non-linear mass transfer effect and Marangoni effect on the hydrodynamic stability is made in the case of gas absorption with chemical reaction in liquid phase. The results are sown on the table. The increasing of the Marangoni effect parameter (4) in very large interval does not influence the critical Reynolds number.

Values of the critical Reynolds numbers Recr , the corresponding wave velocities Cr, the wave numbers A and Cr min , Amin (Da=10, 1=0.1, 2=0.145, 5=18.3, 6=0.034).



E N G IN E E R IN G

2.1 NON STATIONARY ABSORBTION KINETICS

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 (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. If the gas is with low solubility its concentration in the gas phase is constant.

The mass transfer rate could be obtained and the relationships for the Sherwood number and for the amount of the absorbed desorbed substance are:

Let consider gas absorption of low soluble gas in a vertical tube with a radius , in which an immovable liquid contacts an immovable pure (concentrated) gas. The gas is absorbed in the liquid, and the process is accompanied with a thermal effect. As a result several effects in the

200

0*t

0 0z00

m/kgDt

c2QtD

c2dtz

cDQ

Dt

1

tgD

klSh

0

,,



E N G IN E E R IN G

The experimental data for absorption of CO2 and Ar in H2O and C2H5OH show that the rate of the absorption is significantly great then the one that can be determined from expression for Q. This fact indicates that this non-stationary process (analogous to the Bernard problem) is unstable regarding small periodical disturbances. Their increasing may lead to new periodical flow (process) with a constant amplitude which will

change the mechanism and kinetics of heat and mass transfer. For this aim the linear stability analysis will be used. A process represented as a superposition of the basic process and small disturbances in the velocity (ν′z), pressure (p′), concentration (c′) , and temperature (θ′) will be considered:

,cc,pp,vv '''zz '

0 '

'

0

'

00

00

c,

c1,const

'

,

.

,

This new process should satisfy the Oberbeck-Boussinesq equations. After linearization according to the small disturbances, the perturbations ν′z, c′ and θ′

may be expressed through Fourier series of eigenfunctions, where and n are eigenvalues:

)ncos()r,z,t(c)texp(c)ncos()r,z,t(v)texp(v0n

n'

0nn

'z

,

)ncos()r,z,t()texp(0n

n'

.

,



E N G IN E E R IN G

After solution of the problem the final expressions for the velocity and the concentration are determined as follows:

1nnnz ncosf

l

D1

l

z

t

Dv

, 0*

01n

nn* cc,ncosf

l

z

Dt2

zerfcc

,

0

D

gl

00 fwhere and

The process rate can be determined as follows:

Shl

DckcJ

*

0

00 Dt

1

gt2

D

klSh

0*4

5

04

32

1

0* cctDgDt

2cQ

, ,

,

where Q is the quality (amount) of the adsorbed desorbed substance through a unit area per a time t0 [sec].The eigenvalue = av is determined by the least squares method applied to the experimental data.



E N G IN E E R IN G

In the relationships for Q the first term depends on the rate of the non-stationary diffusion in the stagnant liquid. The second terms occur due the loss of a stability of the process provoked by small disturbances of the concentration of absorbed gas at the liquid surface. These periodic disturbances with small amplitude grows continuously up to the establishing of a new stable state, i.e. self-organizing dissipative structure (process).

There are a lot of experimental data concerning the non-stationary absorption of low-soluble gases. Some data are summarized in table. Experimental data used:

No Process System T,oC D.109 m2/s .106 m2/s c*,kg/m3 .104 av.104

1 Absorption 100% CO2 - H2O 23 1.88 0.97 1.60 3.70 4.20

2 Absorption 30% CO2(70%N2) - H2O 23 1.88 0.97 0.471 2.73 2.68

3 Absorption 7.1% CO2(92.9 %N2) - H2O 23 1.88 0.97 0.112 1.90 1.20

4 Absorption 100 % Ar - H2O 20 0.47 1.00 0.0599 1.63 1.79

5 Absorption 100 % Ar - H2O 10 0.35 1.30 0.0719 1.70 2.12

6 Absorption 100 % Ar - C2H5OH 20 0.49 1.52 0.427 2.66 3.29

7 Desorption (CO2 / H2O) - N2 20 1.88 0.97 0 0 -0.303 Obviously the values of γav , depend on the interphase concentration c*. The suitable

4

1*4 c10.29.3 correlation developed is .



E N G IN E E R IN G

The relationship expressing the amount of the absorbed desorbed substance is:

4

5

04

32

1

4

54

2

1

0 tDg

c10.29.3Dt

c2Q

,

where under desorption c* = 0.

The figures show a part of the experimental data (the labels) concerning the system summarized in the table. They correlate well (lines 1) with the values of γav developed by

the least squares method and the expression for Q (lines 2).

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 that leads to γ = 0, i.e. c0

* = 0 (the disturbances attenuate).

Comparison of Q for =aν (line 1) and (39) for = 3,29.10-4c*1/4 (line 2) with experimental

data (labels) under absorption of 100% CO2 in water (c*=1.60 kg/m3) at 23°C.



E N G IN E E R IN G

Comparison of Q for =aν (line 1) and (39) for =0 (line 2) with experimental data

(labels) under desorption of CO2 from a saturated water solution (c*=0 kg/m3) in N2 at 20°C.



E N G IN E E R IN G

2.2 NON STATIONARY EVAPORATION KINETICSB A S - I C h EL A B O R AT O R Y


E N G IN E E R IN G

The 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 be investigated.

The mechanism of the non-stationary evaporation may be considered as a non-stationary diffusion complicated with additional effects of a temperature gradient at the liquid surface (as a result of the thermal effect of the evaporation phenomenon) and a convection (secondary Stefan flow) as well as a natural convection.

The difference between the evaporation rate and the rate of the non-stationary diffusion indicates that a convective contribution exists. The evaporation of a liquid in an inert gas is a results of a phase transition liquid-vapours, so there is a volumetric effect of a heterogeneous reaction at the interface that creates the Stephan flow. If the process occurs in a thermostatic conditions it is limited by both the diffusive and the convective transports in the gas phase.

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, indicate the existence of an additional convective transport, that could be provoked by of a loss of stability of the system. Thus, the small disturbances grow up to the establishment of stable amplitudes and the dissipative structures formed have greater rate from the transport processes.



E N G IN E E R IN G

As a result it is possible to find the amount of the evaporated liquid:

,exp1

10

0

222

1

t

n

n dtDt

ln

t

dtDt

ln

ttAJdtQ

t

n

nt 00

0

22

10

exp1

11

where is amplitude of the velocity disturbances, and are related with distance of the velocity and concentration changes of the disturbances and A is determined from:

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 weighter 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). The process is unstable when the vapours are lighter then gas. In these conditions a natural convection appeared as a result of the instability. Thus the evaporation rate ( = 1.70) is essentially increased which is 2.7 times larger than the diffusion rate.

.D

c

c2A

0

0



E N G IN E E R IN G

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.

Non - linear mass transfer as a result of the big concentration gradients creates conditions for self – organizing dissipative structures, where heat and mass transfer processes are very intensive.

In these conditions there are not small turbulent pulsations in the flows, i.e. the amount of the energy dissipation is very small.

It is important to note, that the parameters of the dissipative structure (as a result of instability) are equal ( = 0.332, = 1.7, = 2.4) for different liquid-gas systems (H2O/N2, H2O/Ar, CH3OH /Ar). The figures show

comparison of the values of Q and experimental data from the systems H2O/N2 and H2O/He.

Evaporation in system H2O-N2

0 1 0 2 0 3 0 4 0 5 0

t

0

0 .0 0 1

0 .0 0 2

0 .0 0 3

0 .0 0 4Q

0 1 0 2 0 3 0 4 0 5 0t

0

0 .0 0 1

0 .0 0 2

0 .0 0 3

0 .0 0 4

0 .0 0 5Q

Evaporation in system H2O-He



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Buy online with a credit card in the Elsevier Science & Technology Bookstore:

By C.B. Boyadjiev , Institute of Chemical Engineering, Bulgarian Academy of Sciences, Acad. G. Bontchev Str. Bl. 103, 1113 Sofia, BulgariaV.N. Babak , Institute of Problems of Chemical Physics, Russian Academy of Sciences, Chernogloovka, 142432 Moscow, Russia

http://books.elsevier.com/?isbn=0444504281



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3. INCORRECT INVERSE PROBLEM

identification of 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:

The main problem of the modelling of the heat and mass transfer processes is the build-up mathematical structure, describing the processes based on the hypothesis (knowledge) concerning to their physical mechanisms. Moreover, the procedure needs of the parameters

,yyQN

1n

2nn

b

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).



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3.1 INCORRECTNESS OF THE INVERSE PROBLEM

The relation between the objective function and the parameter in the figure is typical for a number models of heat and mass transfer processes.

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 figure is shown a dependence of the objective function from the model parameter at a constant value of the independent variable x = x0 .

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Objective function y for different values of the model parameter b at x = x0 = const.

Δy

Δy

Δy

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Δy

Δy

Δy

Δb1 Δb2 Δb3

y0

b0

y

b



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and large objective function values. For the small objective function values the error b1

is small and the inverse identification problem is correct. If the objective function values is large the error b2 is large and the inverse problem is incorrect (ill-posed). In the case

of very large objective function values b3 is very large and the inverse identification

problem is essentially incorrect. The results in the figure show, that inverse method incorrectness is not result of the error size and the cause is the parameter sensitivity with respect to the experimental errors of the objective function. 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:

The figure 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.

Let y is an experimental error of the objective function. In the figure is seen, that the error of the parameter identification is different for small

1b1 5b2

nnnnnn yAyyAy 2.09.0ˆ,1.095.0ˆ 21 Here, An are random numbers at the interval [0,1], and yn is obtained from the model

for x=0.01n (n = 1,…,100). Obviously, the maximum relative errors of the “experimental” data (Δŷ) are 5% and 10%, which are normally distributed. The values of yn, ŷn

(1) and

ŷn(2) are shown in figure.



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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1.2

1.4

Mathematical model and “experimental” data:[] - ŷn

(1) - values of with a maximal “experimental” error of 5%;

[] - ŷn(2) - values of with a maximal “experimental” error of 10%;

[—] - y = 1 – exp (-5x) . In the figure is seen that 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.

ŷn(1)

ŷn(2)

y

x



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In the figures, are seen the horizontals of the least square function in the cases of ±5% relative experimental data error and different interval of , 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.

The horizontals of the least square function Q (n = 1÷30; Δŷ [%] = ±5);

[] – b = [1 ; 5];

The horizontals of the least square function Q (n = 31÷65; Δŷ [%] = ±5);

[] – b = [1 ; 5];

b1 b1

b2

b2



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The horizontals of the least square function Q (n = 66÷100; Δŷ [%] = ±5); [] – b = [1;5];

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.

b2

b1

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

.J,...,1j,b

Q

b

QR

21J

1j

2

1ij1ij1ij

where

Here βi is the iteration step and β0 = 10 -2 (arbitrary small step value).

Each iteration step is successful if two conditions are satisfied:

,0b

fR

b

fRy2,f2QQ

J

1j1ij

j

N

1n

J

1j1ij

j1in1in1ii1i

bx

.,...,1,02 11111

22

1 JjRRbbbbbb ijijijijijjijij 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.

the anti - gradient of the function Q(b): bji = bj(i-1) – β(i-1) Rj(i-1) , j = 1,…,J ,

b*bb



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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

* i

5 4.9678 337 1.0025 5.0674 128

10 4.9351 339 0.99401 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

* i

5 5.0614 1213 1.1797 5.4666 642

10 5.1232 1217 1.3778 5.9106 416



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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 for0 ≤ x ≤ 0.30 and 0.31 ≤ x ≤ 0.65 . The results are presented on tables.

Statistical analysis of the model adequacy (0 ≤ x ≤ 0.30)

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 4.9678 0.9 1.7933 1.7071 0.9061 2.24

1 10 4.9351 0.9 3.5867 3.4139 0.9059 2.24

2 5 1.0025 5.0674 0.9 1.7933 1.8354 1.0475 2.25

2 10 0.99401 4.9218 0.9 3.5867 3.4434 0.9217 2.25

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

1 10 5.1232 0.5 5.2083 4.7328 0.8257 2.19

2 5 1.1797 5.4666 0.05 2.6042 2.3656 0.8252 2.20

2 10 1.3778 5.9106 0.05 5.2083 4.7349 0.8265 2.20



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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) .

Statistical analysis of the model adequacy (0.66 ≤ x ≤ 1) .

Δŷ [%] b* i b1* b2

* i

5 5.1828 2066 2.1720 6.1731 54

10 5.3816 2156 4.9003 7.4004 128

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

1 10 5.3816 5 5.5701 5.2482 0.8723 2.19

2 5 2.1720 6.1731 5 2.7851 2.6221 0.8855 2.20

2 10 4.9003 7.4004 5 5.5701 5.2482 0.8877 2.20



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3.4 INCORRECT INVERSE PROBLEM “DIAGNOSTICS”

In all these cases the difference between correct and incorrect inverse identification problem is based on the distance between exact solution point and least square function minimum point. In practice however the exact parameter values are unknown and a criterion for the inverse problem “diagnostics” will be very useful.

On the tables are shown the solutions of correct and incorrect inverse problems on the bases of different experimental data sets. It is seen that a criterion of the inverse problem incorrectness is the large difference between solutions which are obtained on the bases of different experimental data sets.

Solutions of correct and incorrect problems using different “experimental” data sets:

,

b1(0) = 1.1 , b2

(0) = 6

Different “experimental” data b1* b1

* γ i

1 1.0025 5.0674 0.9 128

2 1.0115 5.1706 0.9 120

3 1.0068 5.1881 0.9 179

1 1.1564 5.2675 0.05 798

2 0.5789 3.7056 0.05 1803

3 1.1723 5.2624 0.05 776

0 ≤

x ≤

0.3

0.31

≤ x

≤ 0

.65



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The proposed iterative method and algorithm for model parameters identification in the cases when inverse problem is incorrect shows that a large difference between parameter values, obtained on the bases of different experimental data sets, is a criterion for inverse problem incorrectness.

Solutions of essentially incorrect problem and general case,using different “experimental” data sets :

b1(0) = 1.1 , b2

(0) = 6

Different “experimental” data b1* b1

* γ i

1 4.5933 6.7246 5 680

2 0.1161 2.3417 5 390

3 2.7943 5.9219 5 133

1 1.0106 5.1716 2 66

2 1.0100 5.1963 2 70

3 1.0134 5.1913 2 76

0.66

≤ x

≤ 1

0 ≤

x ≤

1



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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.

5. CONCLUSIONS

The considered theoretical method permits to solve many fundamental problems of chemical engineering. Theoretical analysis of the non-linear heat and mass transfer processes is a base of the intensive industrial processes modeling. The method of the hydrodynamic stability theory allows to obtain conditions for existing of self-organizing dissipative structures with very big heat and mass transfer rates. Numerical method for model parameter identification solves the inverse problems when there are incorrect or essentially incorrect.

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

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