Mathematical model of a multi-loop network of gas pipelines at various modes of current

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613

UDC 622.691.4.0 0 4

Ma th e m a tica l m o d e l o f a m u lti-lo o p n e tw o rk o f ga s p ip e lin e s a t va rio u s m o d e s o f c u rre n t

1 _{Ism atulla Q. Khujaev} 2 _{Norm ahm ad Ravshanov}

3 _{Orifjon Sh. Bozorov} 4 _{J am ol I. Khujaev}

1 _{Tashkent University of Inform ation Technologies}

Centre for the developm ent of software and hardware program com plexes, Uzbekistan 25, Durm on yuli str., 10 0 125, Tashkent

Dr. (Technical), leading research scientist E-m ail: i_ k_ hujayev@m ail.ru

Dr. (Technical)

E-m ail: ravshanzade-0 9@m ail.ru

3 _{Tashkent Institute of Textiles and Light Industry, Uzbekistan}

5, Shohjahon str., 10 0 10 0 , Tashkent PhD

E-m ail: b_ orif_ sh@m ail.ru

Research scientist

E-m ail: jam olhoja@m ail.ru

Abs tra c t. A m ethod of hydraulic calculation of a m ulti-loop network of gas pipelines based on Kirchhoff’s laws is offered. As com pleting relations, the form ula for the change of pressure on elem entary sites of the horizontal gas pipe, received on the basis of Leybenzon’s generalized form ula of resistance is used.

Ke yw o rd s: hydraulic calculations; Leybenzon’s generalized form ula of resistance; Kirchhoff’s laws; m ulti-loop pipeline network.

The wide use of pipeline networks in various branches of a national econom y and social sphere is due to their profitability and ecological cleanliness. Application of m ulti-loop structures at construction of these networks prom otes the increase of their reliability and reduction of working costs. The role of m athem atical m odeling and com puting experim ents in these processes is extrem ely im portant as they replace the natural experim ents dem anding greater resources in the form of energy, m aterials and specialists of various branches.

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614

second law should consider different degrees of unknowns. Meanwhile, in an electric circuit of a direct current the potential drops on length of a wire has linear connection with the value of electric current, as it served at the form ation of Kirchhoff’s second law (the first hypothesis). In this sense, laws of resistance serve as the fifth axiom of Euclidean geom etry and it opens new directions for research of sim ultaneous equations with different degrees of unknowns.

Taking into account the com pressibility of ideal gas due to Clapeyron equations or real gas with the super com pressibility coefficient of environm ent the analogue of Kirchhoff’s second law shows a linear dependence between difference of squares of pressures and square of the gas outflow at realization of the square law of resistance. Subsequently, the system of the equations becom es com plicated even m ore.

In the given work we will analyze Leybenzon’s generalized form ula of resistance [1]. The advantages of this form ula are the account of all five m odes of current, whereas in m any other approxim ated form ulas not all m odes of current are considered, and its applicability for incom pressible and com pressible environm ents in quasi-one dim ensional representation. And the account of all m odes of current during hydraulic calculations is im portant, as the pipeline network can work both with sm all loading, and with an overload.

In opinion of [1, 2] and som e other authors, at pipeline transportation of liquids (waters, oil and its derivatives) the m ost observed m odes of current are lam inar (where

Re

=

wD

/

ν

<220 0 ) and transitive (where 220 0≤Re≤40 0 0 ), and also a sm ooth m ode of a turbulent flow of a rough surface when the equivalent roughness of the m oistened surface of the pipeline has

Re

k

/

D

<

10

condition. For the gas m ixture currents, m ainly, turbulent m odes of current are observed, where Reynolds's criterion has greater values (Re>40 0 0 ). These are sm ooth (

Re

k

/

D

<

10

), m ixed (

10 ≤

Re

k

/

D

≤

158

) and developed (

Re

k

/

D

>

158

) m odes of a rough surface turbulent flow. At such classification of m odes of current the resistance factor is approxim ated by this unique form ula

(

k D

/

)

θ

Re

ϕ

λ ς

=

(1)

with piecewise constant values of param eters

ζ θ ϕ

, ,

.

As piecewise approxim ation form ulas it is possible to use Stocks’s form ulas (

ζ

=64,

θ

=0,

ϕ

= −1) for a lam inar current m ode, Zaychenko’s form ulas (

ζ

=

0.0025,

θ

=

0,

ϕ

=

1/ 3

) for transitive m ode, Blasius’s form ulas (

ζ

=

0 .

3164

,

0, 1/ 4

θ

=

ϕ

= − ) for a sm ooth m ode of roughness flow, Leybenzon’s form ulas (

ζ

=

10

−0.627

,

θ

=

0 .

127 ,

ϕ

= −0.123) for m ixed m ode and Shifrinson’s form ulas (

ζ

=0.11,

θ

=0.25,

ϕ

=0) for developed m ode.

Param eters of a stationary isotherm al m ode of gas m ovem ent in a linear site of a gas pipeline with the length of l and diam eter of D can be found from the solution of this quasi-one dim ensional equation system [1]:

2

0,

,

2 dp

w dx

M

wF

p

Z RT

D

λ

_ρ

ρ

+

=

(2)

here, M – loss of m ass, T - absolute tem perature, R - gas constant, Z – super com pressibility coefficient of gas and

F

=

π

D

2 /

4

- the cross-sectional area of gas pipe are considered to be constant.

Let the site be characterized by the m ass loss of M (in kg/ s) and entrance pressure of

p

_H (in

Pa). H aving represented Reynolds's criterion

ð

D

MZRT

D

M

wD

4

1 Re

ν

π

ν

ρπ

ν

=

(3)

615

through static pressure, from system (2) we can work out the equation of 2 2

dp

b M

dx

ϕ ϕ + +

= −

, which

has a solution of

2 2 2

K H

p

+ϕ

=

p

+ϕ −

bM

+ϕ

l

_, ₍₃₎

where

(

)

1 3 2

2 5

(2

)2

k

ZRT

b

D

ϕ

ϕ θ

ϕ ϕ θ ϕ

ϕ

ζ

ν π

+ + + + +

+

=

_.

In practice, calculation of gas outflow is m ade com m ercially, i.e. led to standard conditions

st

T

=293.15 К and

p

_st=10 1325 Pa. In the form ulas resulted above, according to relation of

(

)

/

st st

M

=

Q p

RT

, it is possible to get the com m ercial gas outflow

Q

. In our m athem atical m odel of a m ulti-loop gas pipelines we are based on the m aterials given in [3] those are used for a heat supply system .

Calculation of a m ulti-loop network of a heat supply, according to [3], is m ade on the basis of set of the following three groups of the equations:

,

0,

.

Ax

=

Q

By

=

y

+

H

=

SXx

(4)

H ere, colum n vectors of

x

=

( ,

x

₁

x

₂

, ...,

x

_n

) ,

T

( ,

1 2

, ...,

) ,

T

т

y

=

y

1 2

(

,

, ...,

m

)

T

Q

=

Q

and

H

=

(

H

₁

,

H

₂

, ...,

H

_m

)

T represent outflow and pressure differences in arcs (branches and chords), intensity of selection and pressures on tops. The last from these vectors considers the change of leveling heights of the pipeline axis on a site, and the

pressure, produced by a supercharger if the junction represents a pum p. Matrices

A

and

B

represent a full m atrix of incidences and a m atrix of loops. In last group of the equations diagonal m atrices of resistance and absolute values of outflow am ount are used:

1

2

0 ... 0

... ... ... ...

0 0 ...

_n

s

S

s













=













, 1 2

0 ... 0

... ... ... ...

0 0 ...

_n

x

X

x













= 















.

In the literature these groups of the equations can be nam ed as representations of Kirchhoff’s laws and relations com pleting them . Solution uniqueness of the equations set (4) for incom pressible liquids is provided by rem oving one equation, corresponding to m ain junction - to one with known pressure from the first group of the equation, and replacem ent of last two groups

by unique group of

By

+

BH

=

BSXx

. Lobachev-Cross, Vyhandu and som e other m ethods are developed for the solution of the system of linear and nonlinear equations like these.

For adaptation of the presented m aterial to com pressible gas it is necessary to consider the following features.

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616

ends of a site, but a difference of 2+

ϕ

-th powers (exponents) of pressure at the ends. Accordingly at drawing up of a colum n vector it is necessary to consider the given fact.

Thirdly, in a diagonal m atrix of

S

it is necessary to take into account

b

i

,

l

i and the scale

m ultiplier, and m atrix X should be replaced by 1

1

1 2

1

0 ...

0

0 ...

...

0

0 ...

_n

x

X

x

ϕ

+













= 



















.

This developed set of the equations represents the m athem atical m odel of a m ulti-loop network of gas pipelines at various m odes of current. The possibility of form ation of different m odes of current on different parts of a network is not excluded. In such cases, the values of

param eters

ζ θ ϕ

, ,

are chosen according to param eters of current on each part.

If the developed m ode of a flow of turbulent stream through roughness (i.e. at square law of resistance) is established, for the solution of the equations system it is possible to use known solution m ethods (e.g. Lobachev-Cross and others) of current distribution problem s. In this case, in the equations, concerning the case of an incom pressible liquid, changes occur at only pressure differences which are replaced with differences of the squares of pressure.

For the lam inar m ode of current the offered system of the equations becom es sim pler up to a level of linear equations system . The only difference with the case of an electric circuit of a galvanic current will be the additional account of intensities of outflows at the junctions of a gas pipelines network. As com puting experim ent is down on the track, we have to accept this fact as the proof of reliability of the offered m odel.

Re fe re n ce s :

1. Трубопроводный транспорт нефти и газа / Под общ. ред. В.А.Юфина. М.: Недра,

1978. 407 с.

2. Ионин А.А. Газоснабжение: Учеб. для ВУЗов. М.: Стройиздат, 1989. 439 с.

3. Сеннова Е.В., Сидлер В.Г. Математическое моделирование и оптимизация

развивающихся теплоснабжающих систем. Новосибирск: Наука, 1987. 222 с.

4. Садуллаев Р., Вагапов И.Х., Зайниев Н.З., Хужаев И.К., Хуррамова Р.И. Расчёт

магистрального газопровода с учётом рельефа местности // Газовая промышленность. 2003. №8. С. 58-59.

УДК 622.691.4.0 0 4

Математическая модель многоконтурной сети газопроводов при различных режимах течения

1 Исматулла Кушаевич Хужаев 2 Нормахмад Равшанов 3 Орифжон Шодиевич Бозоров 4 Жамол Исматуллаевич Хужаев

1 Центр разработки программных продуктов и аппаратно_-программных комплексов

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617

10 0 125 г. Ташкент, ул. Дурмон йули 25

Доктор технических наук, ведущий научный сотрудник

E-m ail: i_ k_ hujayev@m ail.ru

2 Центр разработки программных продуктов и аппаратно_-программных комплексов

при Ташкентском университете информационных технологий, Узбекистан 100125 г. Ташкент, ул. Дурмон йули 25

Доктор технических наук

E-m ail: ravshanzade-0 9@m ail.ru

3 Ташкентский институт текстиля и легкой промышленности

Узбекистан, 100100 г. Ташкент, ул. Шохжахон 5 Кандидат физико-математических наук

E-m ail: b_ orif_ sh@m ail.ru

4 Центр разработки программных продуктов иаппаратно_-программных комплексов

при Ташкентском университете информационных технологий, Узбекистан 100125 г. Ташкент, ул. Дурмон йули 25

Научный сотрудник

E-m ail: jam olhoja@m ail.ru

Аннотация. В рамках законов Кихгофа предложен способ гидравлического расчета многоконтурной сети газопроводов. В качестве замыкающих соотношений использована формула для изменения давления на элементарных участках горизонтального газопровода, полученная на основе обобщенной формулы сопротивления Лейбензона.