Application of the green element method to chemical engineering problems تطبيق طريقة قرين لحل مسائل الهندسة الكيميائية

Application of the Green Element Method to Chemical Engineering Problems

M. E. E. Abashar

Chemical Engineering Department, College of Engineering, King Saud University, P.O. Box 800, Riyadh 11421, Saudi Arabia

E-mail address:

(Received 08 March, 2003; accepted for publication 19 October, 2003)

Abstract. The novel Green element method (GEM), which applies the singular boundary integral theory (a Fredholm integral equation of the second kind) on each element at a time, is herein implemented on chemical engineering problems. The method overcomes some of the limitations of classical boundary element approach and uses the finite element methodology to achieve optimum inter nodal connectivity. The global coefficient matrix is banded and numerical difficulties from a densely populated matrix are eliminated. Two numerical examples are used to demonstrate the capabilities of the method. The results are compared with orthogonal collocation method, finite element method and experimental data. It has been shown that the Green element method is very reliable and efficient.

Keywords: Finite element method, green element method, orthogonal collocation method, diffusion, reaction

Introduction

Numerical methods of an advanced nature such as orthogonal collocation and finite element methods have become increasingly popular in chemical engineering discipline [1-4]. However, very little is known about the Green element method (GEM) in chemical engineering literature [5-7]. These numerical schemes differ essentially in the way they address the problems. The role of understanding engineering fundamentals is of primary importance because the overall accuracy of these schemes is very closely related to the accuracy with which the problems are formulated.

For decades, the boundary element method (BEM) is known to be as an efficient and accurate computational method for differential equations [8,9]. The attractive feature of the method is that the numerical solution proceeds along the boundary of the computational domain, thereby effectively reducing the dimension of the region. The

47

application of the methods are in general directed to elliptic boundary value problems which are transformed into integral equations that essentially have to be implemented on the boundary. However, it has been shown in the mid 1980s that the method fails to retain its strictly boundary nature for a large class of problems [10]. Moreover, in many cases to retain the boundary-only character of the method too many nodes on the boundary are needed and the solution at one node must directly involve every node on the boundary in which the computational efficiency is sacrificed.

Recently the Green element method (GEM) is proposed which has the computational efficiency and versatility of the finite element method (FEM) and the second order accuracy of the boundary element method (BEM) [10]. The method is based essentially on the Fredholm singular integral theory of the boundary element method, but implements the theory in each element in the same fashion of the finite element method. By this combination, many limitations of the boundary element method are eliminated and making the method more generally reliable. The Green element method has shown great success in many branches of science and engineering e.g. civil and petroleum engineering [10,11].

The purpose of this work is to show the potential application of the Green element method to solve chemical engineering problems. Examples of stiff ordinary differential equations of the boundary value type and coupled partial differential equations are included showing how this method can be applied. We felt that the superiority of the Green element method can be better judged by comparing the results of the method with other numerical techniques such as the orthogonal collocation and finite element methods. The method is also validated by experimental data.

Boundary Value Problems

We consider here diffusion and reaction in a porous catalyst pellet. The first order reaction of A→ B is to be carried isothermally. The conservation of mass gives

2

2 2

d Y a dY

=- φ Y+ x dx dx

(1)

B.C:

o o

x=0 dY =j =0 dx

x=1 Y=1 (2)

where Y is the dimensionless concentration, x is dimensionless coordinate, φis the Thiele modulus,ϕis the gradient (dY

dx ), and a=0,1,2 for slab, cylindrical and spherical

geometry respectively.

Green Element Method (GEM)

The Green element formulation is based on the Fredholm singular integral theory which employs the free space Green’s function of the term with the highest derivative. A complementary differential equation to that of equation (1) is given by

2 2 i

d G (x x ), x

dx = δ − − ∞ ≤ ≤ ∞ (3)

where G is the free space Green’s function, (δ −x x_i)is Dirac delta function (spike function) and x is commonly referred to as the field point and xi as the source point. The Dirac delta function can be expressed mathematically as

i i

i

, x x

(x x )

0, x x

∞ =

δ − =  ≠ (4)

Some properties of the Dirac delat function are its symmetry about the mean and a unit area under the distribution [10].

The solution to equation (3) is given by the free-space Green’s function as follows:

( )

i i

G(x, x ) 1 x x k

=2 − + (5)

and k is an arbitrary constant and is usually taken to be the length of the longest element in the problem domain (k=max [l^{( )e}]).

The derivative of G with respect to x is given by

[ ]

i *

i i i

dG(x, x ) 1

G (x, x ) H(x x ) H(x x)

dx = =2 − − − (6)

Where H is the Heaviside function and is defined as ( ) 1,

0,

i i

i

x x H x x

x x

 >

− =  < (7)

To obtain the integral representation of equation (1), the Green’s second identity is introduced. For any two functions Y and G, which should be at least twice differentiable with respect to the spatial variable x, the Green’s identity states:

L L L

o o o

x x

x 2 2

2 2

x x x

d G d Y dG dY

Y G dx Y G

dx dx

dx d x

 

− = −

 

 

∫

⁽⁸⁾

where xo and xL

Substitution of equations (1) and (3) into the Green’s identity of equation (8) gives are the starting and terminal points of the computational domain, respectively. This identity can be easily proved by integration by parts.

L L L L

o o o o

x x x x

2 i

x x x x

dG dY a dY

Y G Y (x x ) dx G Y dx 0

dx dx x dx

 

− − δ − + − + φ  =

 

∫ ∫

⁽⁹⁾

Equation (9) is the integral representation of equation (1). This equation is a degenerate form of the Fredholm integral equation of the second kind. Then the problem domain is discretized into M elements with a typical element (e) denoted by the interval [x₂^e-x₁^e], where x₁^e and x₂^e represent the x coordinates of node 1 and 2 respectively as shown in Fig. 1.

Using equations (5)-(7) and the properties of the Dirac delta function, equation (9) can be expressed as a summation of the integral representation in each element as follows

( ) ( )

( )

( e ) 2

( e ) 1

(e) (e)

M (e) (e) (e) (e) (e) (e) (e) 1 i (e)

i 2 i i 2 2 (e) (e) 1

e 1 i 1

(e) (e) (e) (e) (e) (e)

2 i 2 1 i 1

x

2 (e)

i x

H(x x )

2 Y H(x x ) H(x x ) Y Y

H(x x )

x x k x x k

Y a x x k dx 0, i 1, 2 x

=

 − −

 

− λ + − − −  − −  −

− + ϕ + − + ϕ

 

+ φ − ϕ − + = =

∑

∫

⁽¹⁰⁾

l

^(e)

1 2

x

₁^(e)

x

2

(e)

Y

₁^(e)

Y

₂^(e)

ϕ

₁^(e)

ϕ

₂^(e)

Fig. 1. Definition sketch for a 1D element.

where λ takes into account the location of the source point. According to the properties of Dirac delta function λ=1 when xi is within the domain [x₁^{( )e} ,x^{( )}₂^e ] and λ=0.5 when xi

( ) 1

xe

is at the end points and x^{( )}₂^e . It is necessary to use the Heaviside function so that equation (10) can be evaluated

(e) (e)

(e) (e) (e) (e) i 2

2 i i 2 (e) (e)

i 2

1, x x

H(x x ) H(x x )

0, x x

 <

− − − =  = (11)

(e) (e)

(e) (e) (e) (e) i 1

1 i i 1 (e) (e)

i 1

1, x x

H(x x ) H(x x )

0, x x

− >

− − − =  = (12)

Equation (10) can be evaluated at x_i^{( )e} =x₁^{( )e} and x_i^{( )e} =x^{( )}₂^e to give two equations

( )

^{( e )2}

( )

( e ) 1 M x

(e) (e) (e) (e) (e) 2 (e)

1 2 1 2 1

e 1 x

Y Y k k Y a x x k d 0x

= x

 

− + + ϕ − + ϕ + φ − ϕ − + =

 

∑

^

∫

⁽¹³⁾

( )

^{( e )2}

( )

( e ) 1 M x

(e) (e) (e) (e) (e) 2 (e)

1 2 1 2 2

e 1 x

Y Y k k Y a x x k dx 0

x

=

 

− + + ϕ − ϕ + φ − ϕ − + =

∑

^

∫

⁽¹⁴⁾

where l^{( )e} is length of the given element as shown in Fig. 1.

Using matrix algebra and indicia notation, equations (13) and (14) can be written as

M

(e) (e) (e) (e) (e)

ij j ij j i

e 1

R Y L F 0, i, j 1, 2

=

+ ϕ + = =

∑

(15)

where

(e) i j 1

ij

1 1

R ( 1)

1 1

− + −

 

= − = − (16)

(e) (e)

ij (e)

k ( k)

L ( k) k

 − + 

=  + − 



 (17)

( )

( e ) 2

( e ) 1 x

(e) 2 (e)

1 1

x

F Y a x x k dx

x

 

=

∫

φ − ϕ − + ⁽¹⁸⁾

( )

( e ) 2

( e ) 1 x

(e) 2 (e)

2 2

x

F Y a x x k dx

x

 

=

∫

φ − ϕ − + ⁽¹⁹⁾

to evaluate the line integral (F₁^{( )e} ,F₂^{( )e} ) over a typical element, it is necessary to prescribe a distribution of Y and ϕfor each element. We use the simplest interpolation functions to approximate these quantities

(e) (e) (e) (e)

1 1 2 2

Y(x)= Ω Y + Ω Y (20)

(e) (e) (e) (e)

1 1 2 2

ϕ(x)= Ω ϕ + Ω ϕ (21)

the element interpolating functions Ω₁^{( )e} and Ω^{( )}₂^e are given by

(e) (e)

1 ( ) 1 , 2

Ω ζ = − ζ Ω = ζ (22)

where ζ is a local co-ordinate given by

(e) (e)

(x x1 ) / , 0 1

ζ = −  ≤ ζ ≤ (23)

Substitution of equations (20)-(23) into equations (18) and (19) gives

( ) ( )

(e) (e) (e) (e) (e) (e) (e) (e) (e) (e) (e) (e) (e)

1 11 1 12 2 11 1 12 2 1j j 1j j

F = T Y +T Y + S ϕ +S ϕ =T Y +S ϕ , j 1, 2= (24)

( ) ( )

(e) (e) (e) (e) (e) (e) (e) (e) (e) (e) (e) (e) (e)

2 21 1 22 2 21 1 22 2 2 j j 2 j j

F = T Y +T Y + S ϕ +S ϕ =T Y +S ϕ , j 1, 2= (25) then equation (15) may be written as

M

(e) (e) (e) (e) (e) (e)

ij ij j ij ij j

e 1

(R T )Y (L S ) 0, i, j 1, 2

=

+ + + ϕ = =

∑

(26)

in which the element matrices T_ij^{( )e} and S_ij^{( )e} are given by

( ) ( )

1 1

(e) 2 (e) (e) (e) (e) 2 (e) (e) (e)

1j j 2 j j

0 0

T = φ^

∫

Ω ^ ζ +k d ,ζ T = φ^

∫

Ω ^ (1− ζ +) k d ,ζ j 1, 2= ⁽²⁷⁾

( )

( ) ( )

( )

(e) (e) (e) (e)

1 1

j j

(e) (e) (e) (e)

1j (e) (e) 2 j (e) (e)

0 1 0 1

k (1 ) k

S a d , S a d , j 1, 2

x x

Ω ζ + Ω − ζ +

= − ζ = − ζ =

ζ + ζ +

∫

^

∫

^

 

  (28)

The formulation of equation (26) is called mixed formulation because it gives the dependent variable (Y) and the gradient (ϕ) at each node. Equation (26) gives a global coefficient matrix with a half bandwidth of 2 and a row dimension of 6. Such formulation eliminates the numerical difficulties arising from a densely populated matrix.

Orthogonal Collocation Method (OCM)

The orthogonal collocation method is a weighted residual method. The characteristic of the collocation method is that it expresses the derivatives at each collocation point as a linear combination of the values at all collocation points. The application of the orthogonal collocation method to equation (1) gives a set of algebraic equations

N 1 N 1 2

j ji i ij i i

i 1 i 1

(a 1)

u B Y A Y Y 0

2 4

+ +

= =

+ φ

+ − =

∑ ∑

(29)

where x= uand the matrices A and B are calculated by subroutines given by Villadsen and Michelsen [3].

Numerical Experimentation

The numerical experimentation is carried out for a spherical geometry (a=2). The cases of weak (φ=1) and strong (φ=100) diffusion limitations are considered. Table 1 summarizes the numerical experimentation for the case when the concentration of the reactant does not drop appreciably within the pores (φ=1). For the Green element method (GEM), a 3-element uniform spatial discretization of the computational domain is employed i.e. k=^{( )e} =1/ 3and the collocation method with three collocation points is used (N+1=3).

The exact solution is given by

sinh( x) Y x sinh ( )

= φ

φ (30)

Consequently, the gradient is

2

dY 1 ( x) cosh( x) sinh( x)

dx sinh( ) x

φ φ − φ

 

ϕ = = φ   (31)

and the effectiveness factor is

3 1 1

tanh( )

 

η =φ φ −φ (32)

Table 1. GEM, OCM and Exact solutions for φ =1

Green Element Method (GEM) 3 Elements, φ =1

Orthogonal Collocation Method (OCM)

N+1=3, φ =1

Exact solution φ =1

Node x Y ϕ η N+1 x Y η x Y ϕ η

1 0 0.8508 0.0000

0.9391

1 0 0.8509

0.9391

0 0.8509 0.0000

0.9391

2 1/3 0.8668 0.0906 2 1/3 0.8668 1/3 0.8668 0.0906

3 2/3 0.9154 0.1944 3 2/3 0.9154 2/3 0.9154 0.1944

4 1 1.0000 0.3095 4 1 1.000 1 1.000 0.3095

It is clear that identical results are obtained with the two methods and the results agree accurately with the exact solution. However, the Green element method provides additional information about the gradient.

The numerical experimentation is carried further for large value of φ i.e. when the reaction rate is high and the concentration profiles inside the catalyst pellet drop rapidly.

For the Green element method (GEM) a 5-element uniform spatial discretization of the computational domain is employed i.e. k=^{( )e} =1/ 5and 7-interior collocation points (N+1=8) are used for collocation method. Table 2 summarizes the numerical experimentation for this case. It can be seen that the Green element method predicts satisfactorily the effectiveness factor with 4.04% error.

Table 2. GEM, OCM and Exact solutions for φ =100

Green Element Method (GEM) 5 Elements, φ =100

Orthogonal Collocation Method (OGM) N+1=8, φ =100

Exact solution φ =100

Node x Y η %

Error

x Y η %

Error

x Y η

1 0.0 1.9764E-10

0.0309 4.04

0.0 -0.2888

0.0340 14.48

0.0 7.4402E-42

0.0297

2 0.2 5.1268E-10 0.2 1.5934E-2 0.2 9.0243E-35

3 0.4 4.2241E-11 0.4 -2.3828E-2 0.4 2.1891E-26

4 0.6 7.2930E-11 0.6 3.7260E-2 0.6 7.0806E-18

5 0.8 2.7052E-9 0.8 -1.8254E-2 0.8 2.5764E-9

6 1.0 1.000 1.0 1.000 1.0 1.000

Partial Differential Equations

Here the Green element method is formulated for unsteady state coupled partial differential equations that describe one-dimensional transport of the biochemical oxygen demand (BOD) and dissolved oxygen (DO) in a biological reactor. The differential mass balances involve the contribution of the axial mass transfer, axial diffusion, mass transfer relative to the equilibrium (re-aeration), contribution of the external sources at certain point along the axial direction of the reactor (point loads), consumption (decay) terms and accumulation terms. The governing equations are given by [13]:

2 NP

j Aj

A A A

A A j

2

A j 1 j

C 1 C C Q C

V k C (x x )

D x t A

x =

 

∂ ∂ ∂

=  + + − δ − 

∂ ∂

∂ 

∑

 ⁽³³⁾

2 NP

j Bj

B B B *

B A B B B j

2

B j 1 j

C 1 C C Q C

V k C K (C C ) (x x )

D x t A

x =

 

∂ ∂ ∂

=  + + − − − δ − 

∂ ∂

∂ 

∑



(34) where C_A is the BOD concentration, C_B the DO concentration, C^*_B the saturation DO concentration, DA the BOD dispersion coefficient, DB the DO dispersion coefficient, V the velocity, kA the BOD decay rate, kB the BOD deoxygenation rate, KB the re- aeration rate, Qj the volumetric flow rate of load at point j, CAj the BOD concentration at point j , CBj the DO concentration at point j, Aj the cross-sectional area at point j and Np

is number of point loads.

We start by obtaining the integral representation of equations (33) and (34) using the Green element formulation. Since the same steps applied to equation (33) and (34), we shall present only the integral representation of equation (33). Applying the same procedure for equation (1), the integral representation of equation (33) within a generic element is given by

( e ) ( e )

( e )

2 2

2

( e ) ( e ) ( e )

1 1 1

x x

x

A A

A A i A A j

x x x A

dC C

dG G

C G C (x x ) u (x, t) k C f dx 0

dx dx D t

  ∂ 

− −

∫

 δ − −  ϕ + ∂ + −  = (35) where

NP j Aj A

j j

j 1 j

C Q C

(x, t) , f (x x )

x ₌ A

ϕ =∂ = δ −

∂

∑

(36)

Substituting equations (3),(5) and (6) into equations (35) gives

(e) (e) * (e) (e) (e) (e) * (e) (e) (e) (e)

i A i 2 i A 2 1 i A 1

A (e) (e) (e) (e) (e) (e) (e) (e)

2 i 2 1 i 1

(e)

(e) (e) (e) A (e)

i A A j

C (x , t) G (x , x )C (x , t) G (x , x )C (x , t) D

G(x , x ) (x , t) G(x , x ) (x , t)

G(x , x ) u (x , t) C k C f dx 0, i

t

−λ + − 

 −

− ϕ + ϕ

 

 

 ∂ 

ϕ + + − = =

 ∂ 

 

( e ) 2

( e ) 1 x

x

∫

1, 2 ⁽³⁷⁾

In order to evaluate the line integral over an element, the dependent variables are interpolated in line with a typical finite element procedures as follows:

(e) (e) (e) (e)

A 1 A1 2 A 2

C (x, t)= Ω ( )C (t)ζ + Ω ( )Cζ (t) (38)

(e) (e) (e) (e)

1 1 2 2

(x, t) ( ) (t) ( ) (t)

ϕ = Ω ζ ϕ + Ω ζ ϕ (39)

The time derivative in equation (37) can be handled by finite difference approximation as follows

m

Aj Aj m Aj m Aj,m 1 Aj,m

t t t

dC C (t t) C (t ) C C

, 0 1

dt t t

+

= +α∆

+ ∆ − −

≈ = ≤ α ≤

∆ ∆ (40)

where m+1 and m represents the current and previous time respectively and Δt=tm+1-tm

is the time step. Solution of the integral equation on the elements of the problem domain results in a set of equations for each element in which there are as many equations as there are nodal unknowns. This set of equations is then assembled to give a set of global equations in which the unknown are the nodal values.

Substitution of equations (38)-(40) into equation (37) gives a system of discrete equations for the two-level time scheme as follows

( )

( )

M (e)

(e) (e) ij (e) (e) (e) (e)

A ij A ij Aj,m 1 A ij ij j,m 1

e 1

(e)

(e) (e) ij (e) (e) (e) (e)

A ij A ij Aj,m A ij ij j,m

(e) (e) (e)

j,m 1 j,m ij

D R k T T C D L u T

t

(1 ) D R k T T C (1 ) D L uT

t

F (1 )F T 0, i

+ +

=

+

 

 

α + + + α + ϕ +

 ∆   

 

 

 

 

− α + + + − α + ϕ +

 ∆   

 

 

α + − α  =

 

∑

, j 1, 2, 0= ≤ α ≤1 (41)

where

1 1 (e) (e)

(e) (e) (e) (e) (e) (e)

11 22

0 0

(3 )

T (1 )(1 ) d T 1 ( 1) d

6

  +

=^

∫

− ζ + ζ ζ =^ =^

∫

ζ + ^ ζ −  ζ =^{  (42)}

1 1 (e) (e)

(e) (e) (e) (e) (e) (e)

12 21

0 0

(3 2 )

T (1 ) d T (1 ) 1 ( 1) d

6

  +

=^

∫

ζ + ζ ζ =^ =^

∫

− ζ  +^ ζ −  ζ =^{  (43)}

the integral values of the point source are given by

( ) ( )

( e )

2 P P

( e ) 1

x N N

j Aj j Aj

(e) (e) (e) (e) (e)

1 j 1 j 1

j 1 j 1

A j A j

x

Q C Q C

1 1

F (x x ) x x d x x x

D = A D = A

=

∫ ∑

δ − − +^ =

∑

− +^{ (44)}

( ) ( )

( e )

2 P P

( e ) 1

x N N

j Aj j Aj

(e) (e) (e) (e) (e)

2 j 2 2 j

j 1 j 1

A j A j

x

Q C Q C

1 1

F (x x ) x x d x x x

D = A D = A

=

∫ ∑

δ − − +^ =

∑

− +^{ (45)}

In order to compare the numerical results obtained herein with those in the literature, the following experimental values are given by Brebbia and Skerget [14]:

Length of the biological reactor is 4.4 m, C_A(0, )t =C_A( , 0)x =8.817mg l/ , (0, ) ( , 0) 7.56 /

B B

C t =C x = mg l, DA=DB= 37.16 cm²/s, V=0.08 m/s, kA=kB

* 8.0 /

CB = mg l

=2.0/day, , ∆x=0.1 m and ∆ =t 0.05days.

The comparison of the simulation results obtained by the Green element method (GEM), finite element method (FEM) and experimental data after 10 days is shown in Table 3.

Table 3. Comparison of experimental data and numerical results

BOD (mg/l) DO (mg/l)

x EXP. [11] GEM % Error FEM [11] % Error EXP.[11] GEM % Error FEM [11] % Error

1.1 7.12 7.33 2.95 7.55 6.04 6.27 6.39 1.91 6.64 5.90

2.2 6.83 6.89 0.88 7.10 3.95 4.86 4.91 1.03 4.98 2.47

3.3 4.64 4.73 1.94 4.61 -0.65 3.85 3.89 1.04 3.86 0.26

4.4 3.91 4.00 2.30 3.71 -5.12 3.77 3.86 2.39 3.58 -5.04

Table 3 shows good agreement between the experimental and the Green element results.

It can be seen in this example that the results of the Green element method compare favorably with the finite element method.

Conclusions

In this paper the Green element method (GEM) has been successfully applied to chemical engineering systems. This has been done by making the computation for two important chemical engineering systems. The treatment of these examples illustrates the superiority of the Green element method and shows that the method is very efficient and reliable. The good predictive features of the Green element model may be due to the fact

that it combines the features of the boundary element and the finite element methods.

Moreover, the slender and sparse nature of the global coefficient matrix of the GEM saves considerable computation time and storage when it is inverted, compared to the short and robust matrix of the other numerical methods. It seems that the potential application of the method outlined herein in chemical engineering domain is very promising.

References

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ﺔﻴﺋﺎﻴﻤﻴﻜﻟا ﺔﺳﺪﻨﻬﻟا ﻞﺋﺎﺴﻣ ﻞﺤﻟ ﻦﻳﺮﻗ ﺔﻘﻳﺮﻃ ﻖﻴﺒﻄﺗ ﺮﺸﺑأ ﻦﻴﻣﻷا ﺮﻴﺸﺒﻟا ﺪﻤﺤﻣ

ﺔﻴﺋﺎﻴﻤﻴﻜﻟا ﺔﺳﺪﻨﳍا ﻢﺴﻗ -

ﺔﺳﺪﻨﳍا ﺔﻴﻠﻛ -

دﻮﻌﺳ ﻚﻠﳌا ﺔﻌﻣﺎﺟ

ص . ب . ٨٠٠ - ضﺎﻳﺮﻟا ١١٤٢١ - ﺔﻳدﻮﻌﺴﻟا ﺔﻴﺑﺮﻌﻟا ﺔﻜﻠﻤﳌا

ﻲﻓ ﻡﻠﺗﺳﺍ ) ۰۸

۰۳ / ۲۰۰۳ / ﻲﻓ ﺭﺷﻧﻠﻟ ﻝﺑﻗﻭ،ﻡ ۱۹

۱۰ / ۲۰۰۳ / ﻡ (

ﺙﺣﺑﻟﺍ ﺹﺧﻠﻣ ﻁ ﺕﻣﺩﺧﺗﺳﺍ ﺩﻗ .

ﺔﻳﻠﻣﺎﻛﺗﻟﺍ ﻁﻳﺣﻣﻟﺍ ﺔﻳﺭﻅﻧ ﻲﻠﻋ ﺩﻣﺗﻌﺗ ﻲﺗﻟﺍﻭ ﺓﺯﻳﻣﺗﻣﻟﺍ ﻥﻳﺭﻗ ﺔﻘﻳﺭ

ﺔﻳﺩﺭﻔﻟﺍ ﺔﻳﻧﺎﺛﻟﺍ ﺔﺟﺭﺩﻟﺍ ﻥﻣ ﺔﻳﻠﻣﺎﻛﺗﻟﺍ ﻡﻟﻭﻫﺩﺭﻓ ﺔﻟﺩﺎﻌﻣ )

ﺔﻳﺋﺎﻳﻣﻳﻛﻟﺍ ﺔﺳﺩﻧﻬﻟﺍ ﻝﺋﺎﺳﻣ ﻝﺣﻟ ( ﻥﻣ .

ﺔﻣﻳﺩﻘﻟﺍ ﺭﺻﺎﻧﻌﻟﺍ ﻁﻳﺣﻣ ﺔﻘﻳﺭﻁ ﻲﻓ ﺓﺩﻭﺟﻭﻣﻟﺍ ﺕﺎﺑﻭﻌﺻﻟﺍ ﻲﻠﻋ ﺏﻠﻐﺗﺗ ﺎﻬﻧﺃ ﺔﻘﻳﺭﻁﻟﺍ ﻩﺫﻫ ﺹﺋﺎﺻﺧ ﺭﺻﺎﻧﻌﻟﺍ ﺔﻘﻳﺭﻁ ﺞﻬﻧ ﻡﺩﺧﺗﺳﺗ ﻙﻟﺫﻛﻭ ﺩﻘﻌﻟﺍ ﺩﻧﻋ ﺞﺋﺎﺗﻧ ﻝﺿﻓﺍ ﻲﻁﻌﺗﻟ ﺓﺩﻭﺩﺣﻣﻟﺍ

ﺔﻣﺎﻌﻟﺍ ﺔﻓﻭﻔﺻﻣﻟﺍ .

ﻑﻭﻔﺻﻟﺍ ﺕﺍﺫ ﺔﻓﻭﻔﺻﻣﻟﺍ ﻥﻣ ﺔﺟﺗﺎﻧﻟﺍ ﻝﻛﺎﺷﻣﻟﺍ ﺏﻧﺟﻳ ﺎﻣﻣ ﻑﻭﻔﺻﻟﺍ ﺓﺩﻭﺩﺣﻣ ﺔﻘﻳﺭﻁﻟﺍ ﻩﺫﻫ ﻥﻣ ﺓﺭﻳﺛﻛﻟﺍ ﺔﻘﻳﺭﻁﻟﺍ ﻩﺫﻫ ﺕﺍﺭﺩﻘﻣ ﻲﻠﻋ ﺓﺭﻛﻓ ءﺎﻁﻋﻻ ﻥﻳﻟﺎﺛﻣ ﺩﺭﻭﺃ . ﺔﻘﻳﺭﻁﻟﺍ ﻩﺫﻫ ﺞﺋﺎﺗﻧ ﺕﻧﺭﻗ ﺩﻘﻟﻭ .

ﻭﺩﺣﻣﻟﺍ ﺭﺻﺎﻧﻌﻟﺍ ﺔﻘﻳﺭﻁ ﻭ ﻥﺷﻛﻭﻛﺎﻛ ﻱﺭﺧﺍ ﻕﺭﻁﺑ ﺔﻳﻠﻣﻌﻣﻟﺍ ﺞﺋﺎﺗﻧﻟﺍ ﻙﻟﺫﻛﻭ ﺓﺩ

ﻩﺫﻫ ﻥﺍ ﺩﺟﻭ ﺩﻘﻟﻭ .

ﺓﺯﻳﻣﺗﻣ ﻭ ﺔﻳﻟﺎﻋ ﺓءﺎﻔﻛ ﺕﺍﺫ ﺔﻘﻳﺭﻁﻟﺍ

.