HEAT AND MASS TRANSFER PROCESSES

HEAT AND MASS TRANSFER PROCESSES

by

Christo BOYADJIEV and Maria DOICHINOVA

Original scientific paper UDC: 536.24:519.876.5:66.011 BIBLID: 0354–9836, 8 (2004), 1, 95-105

Many sys tems with non-lin ear heat and mass trans fer pro cesses might be un sta ble at cer tain con di tions. Small dis tur bances might bring out them of their equi lib rium state, af ter which they achieve it self to a new sta ble state. The method de vel oped here con cerns a non-lin ear anal y sis of hy dro dy namic sta bil ity of the sys tems with in ten sive heat and mass trans fer. It al lows the de ter mi na tion of the ki netic en ergy dis tri bu tion be tween the main flow and the dis tur bance, when the equi lib rium value of the dis tur bance am pli tude is de ter mined.

Key words: non-linear stability, method of analysis, heat and mass transfer

Introduction

Many systems with non-linear heat and mass transfer processes might be unstable at certain conditions •1•. Small disturbances bring them out of their equilibrium state, after which they achieve itself to a new stable state. As a result the disturbances amplitude of self-organized dissipative structure is a constant.

The linear theory of stability •2• allows the determination of parameters of the stable condition, but it could not determine the amplitude of dissipative structures.

The theoritical analisis of self-organized dissipative structures is possible only in the approximation of non-linear theory of stability •3•, where the disturbances could be significant.

Mathematical model

As an example a gas (liquid) laminar boundary layer with intensive heat and mass transfer •1•, leading to a change of velocity distribution in the laminar boundary layer is considered below. For this particular case the Prandtl equations are:

~ ~

~ ~ ~

,

~ ~

u u

x v v

y

u y

u

x y

¶

¶

¶

¶ u ¶

¶

¶

¶

¶ n

+ = + ¶ =

2

2 0

x=0, u~=u; y=0, u~=a₀, n~=b₀; y®4, ~u=u

The last boundary condition for eqs. (1) is usually substituted by:

y u

y c

=0, = ₀

¶~

¶ (2)

and c₀ is determined in order to satisfy the last boundary condition of eqs. (1).

In eqs. (1) and (2) ~u and ~n are the components of the velocity, x and y are the co-ordinates, u is the kinematic viscosity, and a₀, b₀, and c₀ are the boundary conditions, which might conform (correspond) to various effects at the phase boundary (y = 0) such as motion of the second phase, occurrence of secondary flows as a result of the non-linear heat and mass transfer.

The existence of disturbances in the system (u¢, n¢, p¢) leads to their interaction with the main flow (~u, ~n), which creates a new main flow (u, n, p). The new flow (U, V, P) is non-stationary because of the non-stationary character of the disturbances:

U x y t( , , )=u x y t( , , )+ ¢u x y t( , , )

V x y t( , , )=n( , , )x y t + ¢v x y t( , , ) (3) P x y t( , , )= p x y t( , , )+ ¢p x y t( , , )

The new flow defined by eqs. (3) satisfies the full system of Navier-Stokes equations:

¶

¶

¶

¶

¶

¶ r

¶

¶ u ¶

¶

¶

¶ U

t U U

x

U y

P x

U x

U y

+ + = - + æ +

è çç

ö ø

÷÷

V 1 ²

2 2

2

¶

¶

¶

¶

¶

¶ r

¶

¶ u ¶

¶

¶

¶ V

t U V

x V V

y

P y

V x

V

+ + = - + æ + y

è çç

ö ø

÷÷

1 ²

2 2

2 (4)

¶

¶

¶

¶ U

x + Vy = 0

We can introduce eqs. (3) into eqs. (4) and eliminate the pressure using the differentiation and subsequent subtracting of the first two equations in eqs. (4). The following result is obtained:

¶y

¶

¶y

¶ n n ¶y

¶ u ¶ y

¶

¶ y t u u ¶

x y x y

+ + ¢ + + ¢ = æ +

è çç

ö ø

÷÷

( ) ( )

2 2

2 2

¶y

¶

¶y

¶ n n ¶y

¶ u ¶ y

¶

¶ y

¶

¢+ + ¢ ¢

+ + ¢ ¢

= ¢

+ ¢

æ è çç

t u u

x y x y

( ) ( )

2 2

2 2

ö ø

÷÷ (5)

¶

¶

¶ n

¶

¶

¶

¶ n

¶ u

x y

u

x y

+ = ¢

+ ¢

=

0, 0

(1)

where

y ¶

¶

¶

¶ y ¶

¶

¶

= - ¢ = ¢ ¶

- ¢

u y

v x

u y

v

and x (6)

Thus, eqs. (5) stresses on the segregation of the effects:

– the first equation renders the influence of disturbance over the main flow, – the second equation renders the influence of main flow over the disturbance.

However the influences in both equations have non-linear character.

Form of the disturbances

The particular solutions of eqs. (5) in the form of “normal” disturbances, i. e.

periodic disturbances which amplitude depends exponentially of time will be commented further:

u x y t( , , ) exp(= -wt u) ₀( , )x y

n w ¶

( , , )x y t exp( t) ¶u x dy

= - -

ò

⁰

(7)

¢ = - + +

u x y t( , , ) exp( wt)[n₀( , )x y u x y₁( , )sinnx n₁( , )cosx y nx]

¢ = - - +æ -

è

ç ö

ø

÷ +

n w ¶ n

¶

¶

¶ n ¶ n

( , , )x y t exp( t) sin ¶

x

u

x n nx

0 1

1 1

x +nu nx dy æ

è

ç ö

ø é ÷

ëê

ù ûú

ò

^{1 cos}

The substitution of the eqs. (7) into eqs. (5) allows the determination of a stable dissipative structure at w = 0. This partial solution depends on the eigenvalues, values of the wave number n = 2p/l, where l is the wave length of disturbance.

As a result we have two equations for u₀, v₀, u₁, and n₁, corresponding to the eqs.

(5), where cos² nx = 1 – sin² nx. From these equations it is possible to obtain a set of equations, if we put their apperiodical parts and all parts containing sin nx, cos nx, sin² nx, and sin nx cos nx to be equal to zero.

Set of equations

From apperiodical parts we can obtain directly two equations in order to determine u₀ and n₀:

(u ) u

x y

u

x dy u

x dy

x d

0 0

2 0

3 0 3

0 0

+ æ +

è çç

ö ø

÷÷- +

ò ò

n ¶

¶ ¶

¶

¶

¶

¶

¶n

¶ y u

x

u

ò

y æ

è

ç ö

ø

־ +

è çç

ö ø

÷÷=

¶

¶

¶

¶

2 0 2

2 0 2

= æ + +

è çç

ö ø

÷÷

ò

u ¶

¶

¶

¶ ¶

¶

¶

4 0 4

3 0 2

3 0

2 3

u x

dy u

x y

u y

(8a)

(u ) v x y

v

x dy u

x dy v

0 0 x

2

0 3

0 3

0 2

0

+ æ + 2

è çç

ö ø

÷÷- +

ò

n ¶

¶ ¶

¶

¶

¶

¶

¶

¶

¶ n

¶

2 0

y2

æ è çç

ö ø

÷÷=

ò

= æ + +

è çç

ö ø

÷÷-

ò

u ¶

¶

¶

¶ ¶

¶

¶

4 0 4

3 0 2

3 0

2 3

v

x dy v

x y

v y

-v v + + + - -

x y n u

y

v

x n u

x n v

x n u

1 2

1 1 3

1 3

2 1 2

2 1 3

3 3 1

¶

¶ ¶

¶

¶

¶

¶

¶

¶

¶

¶ æ

è çç

ö ø

÷÷ é

ë ê ê

ù û ú ú

ò

^dy +

+æ + + -

è çç

ö ø

÷÷ æ + è

ç ö

¶

¶

¶

¶

¶

¶

¶

¶

2 1 2

2 1 2

1 2

1

1

2 1

v x

v

y n u

x n v v

x nu ø

ò

÷^{dy (8b)}

The periodical part in the equation for u' contains terms with sin nx and cos nx (after the summing up separately the parts containing sin x and cos x and equalization of these sums in separate to zero):

u u

x y

u

x dy u

x

u

1 y

2 0

3 0 3

2 0 2

2 0 2

¶

¶ ¶

¶

¶

¶

¶

¶

¶ æ +

è çç

ö ø

÷÷+æ + è

çç

ò

ö^ø÷÷

ò

^{æèçnn1- ¶¶ux1öø÷dy=0}

(9)

v u

x y

u

x dy u

x

u

1 y

2 0

3 0 3

2 0 2

2 0 2

¶

¶ ¶

¶

¶

¶

¶

¶

¶ æ +

è çç

ö ø

÷÷-æ + è

çç

ò

öø

÷÷ æ + è

ç ö

ø

÷ =

ò

^{nu1 ¶n¶x1 dy 0}

By analogy, from the terms with sin nx and cos nx in the equation for n¢ we have two equations for determination of u₁ and v₁:

u u

x y n v

y

u

x n v

x n u

x n v

0 2

1 1

3 1 3

2 1 2

2 1 3

3 3 1

¶

¶ ¶

¶

¶

¶

¶

¶

¶

¶

- + æ - - ¶ +

è çç

ö ø

÷÷ é

ë ê ê

ù û ú ú

ò

^dy -

- æ + - -

è çç

ö ø

÷÷=

ò

^{¶¶ux0 dy ¶¶yu ¶¶xu n ¶¶vx n u}

2 1 2

2 1 2

1 2

2 1

= é - - +

ë êê

u ¶

¶ ¶

¶

¶ ¶

¶

2 4 2 ¶

3 1 2

2

1 2 1

u

x y n v

x y n u

y

+ æ - - + +

è çç

ö ø

¶ ÷

¶

¶

¶

¶

¶

¶

¶

4 1 4

3 1 3

2 2

1 2

3 1 4

4 6 4 1

u

x n v

x n u

x n v

x n u

÷ + ù

û úú

ò

^{dy ¶¶yu} -

3 1 3

- æ +

è çç

ö ø

÷÷- æ -

ò

è

u v

x y

v

x dy u

x y n v

1 y

2

0 3

0

3 0

2

1 1

¶

¶ ¶

¶

¶ n ¶

¶ ¶

¶

¶ çç

ö ø

÷÷-

- æ - - +

è çç

ö ø

÷÷ +

v

ò

u

x n v

x n u

x n v dy

0 3

1 3

2 1 2

2 1 3

3 3 1

¶

¶

¶

¶

¶

¶

+æ + - - è

çç

ö ø

÷÷ +

¶

¶

¶

¶

¶

¶

¶

¶

¶

¶

2 1 2

2 1 2

1 2

1

0 2

2 0

u x

u

y n v

x n u v

x dy v

x

v y

u

x nv dy

2 2

0 2

1

+ 1

æ è çç

ö ø

÷÷ æ -

èç ö

ø÷

ò

ò

^{¶¶ ¶¶ (10a)}

u v

x y n u

y

v

x n u

x n v

x n u

0 2

1 1 3

1 3

2 1 2

2 1 3

3 3 1

¶

¶ ¶

¶

¶

¶

¶

¶

¶

¶

+ + æ - - ¶ -

è çç

ö ø

÷÷ é

ë ê ê

ù û ú ú

ò

^dy -

- æ + + -

è çç

ö ø

÷÷=

ò

^{¶¶ux0 dy ¶¶xv ¶¶yv n ¶¶ux n v}

2 1 2

2 1 2

1 2

2 1

= é + - +

ë êê

u ¶

¶ ¶

¶

¶ ¶

¶

2 4 2 ¶

3 1 2

2

1 2 1

v

x y n u

x y n v

y

+ æ + - - +

è çç

ö ø

¶ ÷

¶

¶

¶

¶

¶

¶

¶

4 1 4

3 1 3

2 2

1 2

3 1 4

4 6 4 1

v

x n u

x n v

x n u

x n v

÷ + ù

û ú ú

ò

^{dy ¶¶yv} -

3 1 3

- æ +

è çç

ö ø

÷÷- æ +

ò

è

v v

x y

v

x dx v

x y n u

1 y

2 0

3 0

3 0

2

1 1

¶

¶ ¶

¶

¶ n ¶

¶ ¶

¶

¶ çç

ö ø

÷÷-

- æ + - -

è çç

ö ø

÷÷ +

ò

v v

x n u

x n v

x n u dy

0 3

1 3

2 1 2

2 1 3

3 3 1

¶

¶

¶

¶

¶

¶

+æ + + -

è çç

ö ø

÷÷ +

¶

ò

¶

¶

¶

¶

¶

¶

¶

2 1 2

2 1 2

1 2

1

2 0

v x

v

y n u

x n v v

x dy

+æ +

è çç

ö ø

÷÷ æ + è

ç ö

ø

ò

÷

¶

¶

¶

¶

¶

¶

2 0 2

2 0 2

1 1

v x

v y

v

x nu dy (10b)

From the terms with sin² nx and sin nx cos nx we have:

u u

x y n v y

u

x n v

x n u

x n v

1 2

1 1 3

1 3

2 1 2

2 1 3

3 3 1

¶

¶ ¶

¶

¶

¶

¶

¶

¶

¶

- + æ - - ¶ +

è çç

ö ø

÷÷ é

ë ê ê

ù û ú ú

ò

^dy -

-æ + - -

è çç

ö ø

÷÷ æ - è

ç ö

¶

¶

¶

¶

¶

¶

¶

¶

2 1 2

2 1 2

1 2

1 1

2 1

u x

u

y n v

x n u u

x nv ø

÷ -

ò

^dy

-v v + + + - -

x y n u

y

v x

n u

dx

n v

x n u

1 2

1 1 3

1 3

2 1 2

2 1 3

3 3 1

¶

¶ ¶

¶

¶

¶

¶

¶ ¶

¶ æ

è çç

ö ø

÷÷ é

ë ê ê

ù û ú ú

ò

^dy +

+æ + + -

è çç

ö ø

÷÷ æ + è

ç ö

¶

¶

¶

¶

¶

¶

¶

¶

2 1 2

2 1 2

1 2

1 1

2 1

v x

v y

n u

x n v v

x nu ø

÷ =

ò

^{dy 0 (11a)}

u v

x y n u

y

v x

n u

x

n v

x n u

1 2

1 1

3 1 3

2 1 2

2 1 3

3 3 1

¶

¶ ¶

¶

¶

¶

¶

¶

¶

¶

+ + æ + - ¶ -

è çç

ö ø

÷÷ é

ë ê ê

ù û ú ú

ò

^dy +

+v u - + - - +

x y n v

y

u x

n v

x

n u

x n v

1 2

1 1

3 1 3

2 1 2

2 1 3

3 3 1

¶

¶ ¶

¶

¶

¶

¶

¶

¶

¶

¶ æ

è çç

ö ø

÷÷ é

ë ê ê

ù û ú ú

ò

^dy -

-æ + + -

è çç

ö ø

÷÷ æ -

èç ö

¶

¶

¶

¶

¶

¶

¶

¶

2 1 2

2 1 2

1 2

1 1

2 1

v x

v

y n u

x n v u

x nv ø÷ -

ò

^dy

-æ + - -

è çç

ö ø

÷÷ æ + è

ç ö

¶

¶

¶

¶

¶

¶

¶

¶

2 1 2

2 1 2

1 2

1 1

2 1

u x

u

y n v

x n u v

x nu ø

÷ =

ò

^{dy 0 (11b)}

Here must be marked that the right hand side of eq. (8b) is different if we put sin² nx = 1 – cos² nx. On the other hand from the eq. (11a) can be seen that for these two cases the right hand sides are equal.

Similarity variables

The solution of eqs. (1) can be expressed in similarity variables:

~ ( ), ~ ( ),

u uF v u

x F F y u

= ¢ x = u ¢ - = x

x x

4 u (12)

Introducing of variables (12) into eqs. (1) leads to the following:

2 0

0

¢¢¢ + ¢¢ =

= = ¢ = ¢¢ =

F FF

F a F b F c

x , , ,

(13)

where a, b, and c are determined in advance taking into account the movement of the second phase, the secondary flows, etc. (see, for example, •1•).

Those similarity variables can be also introduced into eqs. (8)-(11):

u uf v u u u f v u u

0 = 0¢( ),x 0 = bj0¢( ),x 1 = b ¢( ),x 1 = bj x¢( ), b= u1 (14) where u and u₁are characteristic velosities of the main flow and disturbances and b is the dimensionless amplitude of the disturbances.

The problem will be solved in approximation of the laminar boundary layer theory, i. e. in zero approximation of the small parameter g ²:

g ² = 0 (15)

where

g d

d u

u

2 2

=æ 1 2

èç ö

ø÷ = ^- = =

x

x u

Re ^/ , , Re ux (16)

In this approximations (g ² = 0) A₀, b, and Re are parameters:

A n x

A A x

= d= ^- = =

l g

l

2 1 2 2

0 0

p p

Re ^/ , (17)

because the dependence of parameters on x is very weak.

In similarity variables (14) b is small parameter and problem will be solved in zero approximation of the small parameter b², b² = 0.

The introduction of similarity variables (14) and approximation (15) in eqs. (8a) and (8b) leads to the equations for f₀ and j₀ :

2f₀¢¢¢+(f₀+bj₀)f₀¢¢ =0 (18a) 2j¢¢¢ +₀ (f₀+bj₀)j₀¢¢ =0 (18b) The introduction of similarity variables (14) and approximation (15) in eqs. (9a) and (9b) leads to the conditions:

2 2

2 0

0

0

¢¢¢+ ¢ 0

- ¢¢ =

f f

f A f

j

(19)

2 2

2 0

0

0

¢¢¢+ ¢ 0

- ¢¢ =

f jA f f j

From eqs. (19) and the eq. (18a) are obtained the conditions:

f f

f A A f

0 0

0 0

2 2

2

+ = ¢ 2

- = ¢

bj +

j

j

j (20)

The introduction of eq. (20) in eqs. (18) leads to:

2 2

2 0

0

0

¢¢¢+ ¢ 0

- ¢¢ =

f f

f A f

j

(21)

2 2

2 0

0

0

¢¢¢ + ¢ 0

+ ¢¢ =

j j

j j

A f

The introduction of similarity variables (14) and approximation (15) in eqs. (10a) and (10b) leads to the equations for f and j:

2f^IV + ¢ ¢¢ +f f₀ f f₀ ¢¢¢ +2A f_{0 0}¢ ¢¢ +j b(f¢ ¢¢ + ¢¢ ¢ + ¢¢¢j₀ f j₀ f j₀+ fj₀¢¢¢ +) 2bA₀(j j₀¢ ¢¢ -jj¢¢¢ =₀) 0 2j^IV + ¢ ¢¢ +f₀j f₀j¢¢¢ -2A₀f f₀¢ ¢¢ +b j j( ¢ ¢¢ + ¢¢ ¢ + ¢¢¢₀ j j₀ j j₀+jj )₀¢¢¢ - (22)

-2bA₀(j¢ ¢¢ -₀f fj¢¢¢ =₀) 0

With the approximations (b² = 0, g² = 0) used, the eqs. (11) are eliminated.

The boundary conditions are obtained using two considerations:

(1) boundary conditions for F and f₀ are equal, and

(2) the dependences of the disturbances from the main flow are identical in the volume and at the interface.

From eq. (13) follows:

F( )0 =a, F¢( )0 =b, F¢¢( )0 =c

¢¢¢ = - ¢¢

= -

F F F ac

( ) ( ) ( )

0 0 0

2 2

(23)

F_IV bc a c

( )0

2 4

2

= - +

F_V c abc a c

( )0 2

3

4 8

2 3

= - + -

where F^IV(0) and F^V(0) were obtained after double differentiation of eq. (13).

From eq. (18a) follows:

j⁰ b b

0 0

2 0

= - ¢¢¢

¢¢- f f

f (24)

By double differentiation of eq. (24) and introduction of x = 0 in j₀, ¢j₀, and ¢¢j₀ and using eq. (24) allows the determination of boundary conditions in f₀ and j₀:

f₀( )0 =a, f₀¢( )0 =b, f₀¢¢ =c

(25) j₀( )0 =0, j¢₀( )0 =0, j₀¢¢ =0

From the conditions (20) is directly obtained:

2f¢ =(f₀ +bj₀)(f-2A₀j)

(26) 2j¢ =(f₀+bj₀)(j+2A f₀ )

If we note a₁ = f(0) and a₂ = j(0), we could determine from eqs. (26) boundary conditions for eq. (22) after differentiation of eqs. (26) and use of eq. (25):

f f a

A

( )0 , ( )0 ( )

2 2

1 1 0 2

=a ¢ = a - a

¢¢ = æ + -

è çç

ö ø

÷÷- +

f b a

A a A a b

( )0 ( )

2¹ 2 2

2 0 2 2

0 2

a 2

a (27a)

¢¢¢ = æ + - + -

èç ö

ø÷ - +

f c ab

abA a a A A c

( )0 (

2 3

2 6 1

4 3

1

0

2 3 3

0 2

2 0

a a 3 4

4

3 3

0

ab+ a -a A2)

j( )0 a , j( )0 (a a )

2 2

2 2 0 1

= ¢ =a +

A

¢¢ = + + æ + -

èç ö

ø÷

j a a

( )0 ( )

2 1

2 2

0 1 2 2 2 2

02

A a b b a a A (27b)

¢¢¢ = æ + - + -

èç ö

ø÷ + +

j a

( )0 a 2

3

2 6 1

4 3 3

2

02 3 3

02

1 0

c ab

abA a a A A æc ab+ a -a A

èç ö

ø÷ 3

4

3 3

03

where a₁ and a₂ are eigenvalues of the problem.

Determination of β

The parameter b might be determined from the condition that the kinetic energy E of the main flow is distributed between the energy of new flow E₀ and the energy of disturbance E₁:

E = E0 + E1 (28)

If we assume that in all three cases kinetic energies are proportional to the velocity square summed in the area of the boundary layer (s):

E u dxdy E u dxdy E u dxdy

s s

s

~ ~ , ,

( ) ( )

( ) 2

0 2

1

=òò =òò ¢2

òò (29)

we obtain (w = 0):

E~ u dydx~² u² F ²d

0 6 0

0

= ò ¢

ò

ò dl x

d l

E₀ u dydx₀² u² f₀²d

0 6 0

0

~òò = dlò ¢ x

d l

(30) E_{1 0} u₁ nx ₂ nx ²dydx

0 0

~ (n sin n cos )

d l

+ + =

ò ò

= æò ¢ + ò ¢ + ò ¢

èç ö

ø÷

u^{2 2} ₀²d f ²d ²d

0 6 0

6 0

6 1

2

1 2

dlb j x x j x

The introduction of the expressions (30) in eq. (28) allows the determination of b:

b

x x

j x x j x

0

= 6

¢ -ò ¢ ò

ò ¢ + ¢ + ò ¢

F d f d

d f d d

2

02 0 6 0

6

02 0

6 2 2

0

16

2

1 ò 2

(31)

Method of solution

The eqs. (21) and (22) with boundary conditions (25) and (27) present an eigenvalues problem where a₁, a₂, b, and A₀ are eigenvalues. They may be determined from condition for minimum of the least square function Q:

Q(a1, a2, b, A0) = (bi–1 – bi)² (32) where i = 0, 1, 2, ... is the iteration number in the minimum search procedure of the function Q. The problem is reduced to the combined (joint) solving of eq. (21) and (22) with the corresponding boundary conditions (25) and (27). The first step in the solving is to assign initial values of a₁, a₂, b, and A₀, determine b and then search for which values of a₁, a₂, and A₀ the function Q has a minimum.

Conclusions

The proposed method for non-linear analysis of hydrodynamic stability in systems with intensive heat and mass transfer use the kinetic energy distribution between the main flow and the disturbances for determination of the equilibrium amplitude value of the disturbances.

Acknowledgment

This work was completed with the financial support of the National Science Fund, Ministry of Education and Science, Republic of Bulgaria under Contract MM-1102/2001.

Nomenclature

a0, b0, c0 – boundary conditions

n – wave number

~u – x component of the velocity, [m/s]

~n – y component of the velocity, [m/s•

x , y – coordinates, •m•

Greek letters

b – amplitude of the disturbance, •–•

g – small parameter

l – wave-length of disturbance u – kinematic viscosity, •m²/s•

References

[1] Boyadjiev, Chr., Babak, V. N., Non-Linear Mass Transfer and Hydrodynamic Stability, Elsevier, Amsterdam, 2000

[2] Betchkov, R., Crimminable, W. O., Stability of Parallel Flows, Academic Press, New York, 1967

•3• Joseph, D. D., Stability of Floid Motion, Springer Verlag, New York, 1976

•4• Gershuni, G. Z., Zhuhovitski, E. M., Convective Stability in Compressible Liquids (in Russian), Nauka, Moskow, 1972

Authors' addresses:

Chr. Boyadjiev, M. Doichinova

Institute of Chemical Engineering, Bulgarian Academy of Sciences Acad. G. Bontchev str., Bl. 103, 1113 Sofia, Bulgaria

Corresponding author (Chr. Boyadjiev):

E-mail: chboyadj@bas.bg

Paper submited: January 11, 2004 Paper revised: March 25, 2004 Paper accepted: March 26, 2004