PSI - Issue 36

Ye. Kryzhanivskyy et al. / Procedia Structural Integrity 36 (2022) 370–377 Ye. I. Kryzhanivskyy et al. / Structural Integrity Procedia 00 (2021) 000 – 000

374

5

0  L

0  L

0  L

   

  

2

L

L

2

P nx 

P nx 

nx P nx   

n

nx

n

nx

n   =      L 





L 







(8)

cos

cos

sin

sin

cos

;

dx

dx

P

P

dx

P

=

−

=

−

с

2

L

x

L

L x 

L

L L

L



x



0

0

L

0  L

c dP

P nx 

d

nx

  



(9)

,

cos

cos

;

dx

P

dx

=

=

t

L dt

L

dt

0

0  L

0  L

n

n

n

M

M

M

nx

nx

nx













(10)

i

i

i

(

) cos

(

) cos

cos .

x x −

dx

x x −

dx





=

=

i

i

F

L

F

L

F L

i

i

i

1

1

1

i

i

i

=

=

=

Then, based on (3), we obtain the equation:

2

2

dt d P

L nx

a

n





n

 =  

(11)

.

F P M c 1  = i

cos

  − 

c

i

i

c

L

i

Initial condition:

L 

nx



(12)

0 (0) 2 cos a P L =

0.

P

dx

=

c

L

Solution (11) has the form:

t

c M 2

  

  

n 2 2

æ

L nx





n

(13)

,

(

)

 = i 1 0

exp

cos

P

t

d





=

−

−

i

i

c

2

L

F

i

2

c 2

where

a . As a result of obtaining the initial from the integral, we obtain the following expression: æ =

n i   =    L t 1 0 0

  

c M 2

  

  

L 2 2 2 n

2

æ

L nx

L nx







( ) t  −

(14)

( , )

exp

cos

cos

.

dx

d

P x t

= −

−



i

i

F



i

The obtained expression is a solution of the problem and can be used to predict the change in pressure in the pipeline during the injection of carbon dioxide with a flow rate of M i at the corresponding points x i . 4. Results and discussion The obtained dependence allows modelling the injection of carbon dioxide from two or more sources with the possibility of simultaneous (parallel) selection of the product at several fixed points of the route. The problem is spatio-temporal, so the result can be obtained for any point in time. It is obvious that from the point of view of the set tasks there is a moment of time when in injection points pressure reaches the maximum acceptable level from the point of pipe strength. In this case, it is not allowed to continue the injection process. The pressure distribution along the length for this value of time characterizes the amount of carbon dioxide injected into the pipeline (Fig. 2). Graphical dependences of the pressure depression in the pipeline, obtained using the software package Schlumberger Olga, with a different number of injection points are presented in Figure 3.