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Gas-transport Networks Modelling

on the Basis of Energetic Analogies Theory

Saukh S. Ye., Semagina E. P. National Academy of Sciences of Ukraine, G.Ye.Pukhov's Institute of Modelling in Power Engineering, saukh@svitonline.com

Abstract

Saukh S.Ye., Semagina E.P. Gas-transport Networks Modelling on the Basis of Energetic Analogies Theory. The problem of computer models creation is considered for energy systems, which allows studying heterogeneous physical events. A common theory of energy concepts is proposed for such models building. This common theory allows selecting valid base variables from the mathematical description of heterogeneous events and applying circuit theory methods to build computer models of topologically complex objects. The efficiency of our approach is illustrated by the problem solution for a gas-transport system model building.

Introduction

The creation of computational models of power supply systems includes many difficulties. A significant variety of the generating elements of the system, the heterogeneity of the observed physical phenomena and the complexity of their mathematical description are the main aspects of the problem .A general solution for this problem is possible with the help of the common theory which provides the harmonized development of mathematical models and computational algorithms.

Description of interrelated phenomena forcomponents with lumped parameters

The basic motivation for the application of circuit theory analogies for constructing models of energetic circuit components with heterogeneous physical phenomena is the possibility to use the laws of energy conservation.

Introducing vector groups of generalized sequential and parallel variables allows describing the interrelations between variables by means of analogues of resistances, conductivities, capacities and inductances. As a result, the basic principles and analysis methods for electric circuits appear to be usable

for the energetic circuit analysis.

Sequential Parallel

Figure.1. Method of measurement

All system variables are subdivided into two sets: sequential one, which may be measured by sequentially switched device, and parallel one, which may be measured by parallel switched device (Fig. 1).The main classification of variables was made on the basis of a generalized approach on a higher level of abstraction, i.e. the principle of energetic analogies, according to which: the concepts of generalized state variables (Table 1) and generalized action variables (Table 2) are introduced. The action variables determine the level of energy dissipation of the component and the state variables represent the integral characteristics of the action variables; the product of parallel and sequential action variables determines the level of energy dissipation in the system element; kinetic energy is defined as the integration of the sequential action variables with respect to the sequential state variables;potential energy is defined as the integration of the parallel action variables with respect to the parallel state variables.

Table 1. Parallel and sequential state variables

Type of interaction

Parallel variables

Sequential variables

Electricity

Charge q, к

Magnetic flow F, Wb

Mechanics

Displacementx , m

Impulse к, kgm/sec

 

Rotation angle 6,rad

Angular Moment М, kg m'/sec

Heat

Entropy S, J/K

Undefined

Hydraulics

Displacement    Xh ,

m

Impulse of moving liquid

KH = \p •     • dV

V    dt       kg m/sec

P- density, kg/m3; V - volume, m .

Table 2. Parallel and sequential action variables

Type of interaction

Parallel variables

Sequential variables

Electricity

dФ

u =-

Voltage      dt , V

dq і = —-

Current    dt , A

Mechanics

F = dK Force      dt , N

dx

Velocity     dt , m/sec

 

Momentum

N = dM

dt    N ■ m

Angular velocity dt , rad/sec

Heat

Temperature T, K

dS о =

Entropy flow       dt ,

Wt

K

Hydraulics

Force

Fh =J pdn = dKJ

s          dt n

S - surface of volume V,

m2

p - pressure, Pa .

Average  velocity of liquid motion in cross-section

dxH m

°H =— -

dt sec

The law of energy conservation in passive components of energetic

circuits with lumped parameters is represented as dWD + dWK + dWP = 0, where       the       summands       are       the       dissipation energy

dWD =(u,i)dt = (R i, i )dt =(u, R1 i) • dt,

re ,e

re,fm

re,rm

 

re h

rfm ,e

rfm,fm

rfm,rm

rfm,t

rfm h

rrm ,e

rrm,fm

rrm,rm

rrm,t

rrm h

rt ,e

rt,fm

rt ,rm

RRt t

rt h

rh ,e

rh,fm

rh,rm

rh t

rh h

the kinetic energy dWК = d(L i, i)/ 2 = (L i, di),


Le ,e

Le ,fm

Le ,rm

Le ,t

Le ,h

Lfm ,e

Lfm ,fm

Lfm ,rm

Lfm ,t

Lfm ,h

Lrm ,e

Lrm ,fm

Lrm ,rm

Lrm ,t

Lrm ,h

Lt ,e

Lt ,fm

Lt ,rm

Lt ,t

Lt ,h

Lh ,e

Lh ,fm

Lh ,rm

Lh ,t

Lh ,h

the potential energy dWP = d (С u, u)/2 = u, d u).

C

C

e ,fm

C

e ,rm

C

e ,t

C

e ,h

C

C

C

C

C

 

 

 

 

 

C

C

C

C

C

C

C

C

C

C

In these relations ((, f2) is the scalar product of the vectors fx and f2;

the   parallel   and   sequential   variables   are   organized  in   the vectors

t

u = u F N T FH\ and i = \i и a a vHf; R, L and C are the matrices

of the parameters with the elements. Nondiagonal elements of matrix parameters are obviously defined by interrelations of the physical phenomena.

The formulas for determination of power, kinetic and potential energies for different physical areas are shown in Table 3.

Table 3. Energetic analogies

Type of interaction

Power

(is defined by

action

variables)

Kinetic energy (is   defined by sequential variables)

Potential energy (is   defined by parallel variables)

Electricity

RE = u i

= Re i2 = GE u 2

ф

W/ = J i dф

0

= le t

Е 2

q

WEp = J u dq

0

Е 2

Mechanics

RFM = F U = RFM u2

= G    ^ F2

К

wFM = jU dK

0

= l

—  fm 2

x

WM =j F ■ dx

0

F 2

= с -_

 

RRM = N a

= Rrm = G N2

М

WRm = {« dM

0

=l aL

в

=J N  de

0

N 2

= C

Heat

RT = T ■a = RT-a2 = GT T2

Undefined

S

WTP =J T ■ dS

0 T 2

Hydraulics

= Rh '"I

= Gh fH

K г

=\vh dKH

0

= lh &

2

WH = JFh ■

0

F 2

2

Description   of   homogeneous   and heterogeneous phenomena in resistive type components

For the case of homogeneous phenomena the generalized law for components is defined as J = /■ E, where Y is the specific conductance of the flow J induced by the intensity E. In particular, for components of homogeneous systems this law is formulated as following laws:

Ohm's law, i.e. the relation between density JE of the electrical current and the potential gradient <p have the form:

J е =-ye ^ grad ((P)=ye ^ E е ,       ee =- grad ((P),   ye > 0;

Fourier's law, i.e. the relation between the density JT of the heat flow

and the temperature gradient T, has the form:

JT = -YT grad (T) = yt ET, ET = - grad (T), yt > 0;

Darcy's law, i.e. the relation between the density J H of the mixture component flow and the pressure gradient p has the form:

jh = -yh ^grad(p) =yh'eh ,     eh = -grad(p), yh > °.

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