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

373

4

2

2 P x d +





(1)

0;

=



- mass conservation equation:

n

( ) 1 

1 i P M x x t F  =  −   ( i i 



(2)

),

+

i

2

x



c

where P ( x , t ), ρω ( x , t ) – pressure and mass flow as a function of time t and linear coordinate x ; d – the diameter of the pipeline; c – the speed of sound in the gas; M i – mass flow rate of pumping or extraction of gas into the pipeline at a point with a coordinate x i ; δ ( x - x i ) – Dirac delta function; λ – hydraulic resistance coefficient; F – an internal area of pipeline cross-section. By differentiating equations (1) and (2) we can come to the equation:

n

2

2 P a P a M x x c t F x  =   = + −    2 2 ( i

(3)

),

i

1

i

a 2  = – linearization coefficient.

where

2

d

To solve the problems, we form the initial and boundary conditions. We will assume that before the technological process of injection of carbon dioxide into the pipeline, all methane is displaced from it, as a result of which the pressure in the pipeline is atmospheric:

( , 0)

. a

(4)

P x

P =

In the process of injection of carbon dioxide, the initial and final sections of the pipeline are closed, so the mass flow in all sections: (0, ) 0 = M t ; ( , ) 0 = M L t . (5)

where L – pipeline length. Given that,

M F / =  , from equation (1) we will get:

P

P





0;

0.

=

=

(6)

x

x





0

x

x L =

=

Thus, to implement equation (3), we obtain a homogeneous boundary value problem with initial conditions (4). The solution is obtained based on the use of integral transformations, in particular, we use the cosine-Fourier transformation:

L

2 ( , ) cos

nx



с P P x t = 

.

dx

(7)

L

L

0

L nx  cos and integrate in the range from x = 0 to x = L, we will receive:

Multiply equation (3) by