The most efficient way of heating biological andtechnical objects from the point of view of electric ener�gy conservation is a local surface�distributed electric he�ating which may be implemented by composite electricradiators.

Multi�electrode composite electric radiators (MCE)are a complex system transforming electric energy intothermal one and supporting specific temperature on thesurface of electric radiator according to electro�, ther�malphysic parameters of MCE. In this connection theseparameters should be accurately calculated in respect tothe system of electrode arranged in electrocunductivecomposite material.

Rigorous solution of the problem may be obtainedonly as a result of calculation of stationary electric fieldformed by system of electrodes in quasi�homogeneousmedium. The calculated model of electric radiator is ac�cepted subject to the following boundary conditions:MCE may be considered possessing lumped parametersas periodic processes at frequency f=50 Hz are conside�red to be quasi�stationary; electrode surface along thelength may be accepted as equipotential owing to medi�um low specific conductivity; perimeter boundary of re�sistance material may be considered as impermeable forelectric field lines; taking into account the fact thatelectrode length coincides with resistive layer size in di�rection of electrode length, the electric radiator electricfield is accepted to be plane�parallel and the same in allz sections directed along electrode length. In this casecurrent density is the same in all points of the field be�ing z situated.

The problem was solved by the method of direct de�termination of electric field strength [1] in combinationwith the conformal mapping method. This method isbased on introduction of auxiliary function γ 0(х, у) sta�ting the value of angle formed by electric�field vector ofplane�parallel field in any point of the examined areawith one of axes of rectangular coordinate system. Fun�ction γ 0(х, у) is a harmonic one satisfying two�dimen�sional Laplace equation and boundary conditions of thefirst sort arranged subject to orthogonality of force andequipotential lines of the field on the areas where one ofthe conditions is specified:

Us=const or

Electric field of the systems of axis�symmetric MCEmay be described by the Laplace equation in plane me�ridian system equal in all meridian planes in cylindricalcoordinates. (R, ϕ, z). In the case l/R>1 in cylindricalcoordinate system the axis of which coincides with thecylinder axis, the electric field turns out to be plane�pa�rallel and the same for any z.

As a result of introduction of plane�parallel calcula�ted model and conformal mapping of the original planeof complex variable Z to the plane of new complex vari�able ζ keeping the required conformances of the points ofthe original and mapped planes the systems of nonlineartranscendental equations were obtained. The system wassolved numerically by modification of Newton discretemethod. The Jacobian matrix of partial derivative fun�ctions of the system was approximated by the first diffe�rences; in this case minimal step of argument of functionwas selected by criterion of significance regret of properdifference. Certain intervals entering the functions of theequation system were calculated at each integration byuse of quadrature formula of Newton�Kotess of theeighth order. Simultaneously with calculation of furthervalues of dimensionless mapping parameters the error oftheir determining is estimated at each integration.

Applying this method for the most abundant MCEsystems the calculation of electric conductivity was car�ried out:

1. between two pairs of coplanar electrodes arranged inconductor of rectangular cross section;

2. between three�electrode systems;

3. multi�electrode systems of low�temperature com�posite electric radiators;

4. two�electrode axis�symmetric system;

5. three�electrode axis�symmetric system;

6. multi�electrode axis�symmetric systems.

The calculated models and systems in the mappedplane are given in Table 1. The mapping parameters, ac�curate and approximate design formulas are given inTable 2.

The introduced plane�parallel calculated models al�lowed reflecting not only qualitative features of opera�tion of multi�electrode composite electric radiators butalso obtaining quantitative response.( ) 0.s

Un

∂ =∂

Mathematics and mechanics. Physics

141

UDC 621.316.8: 678.01:537.311

CALCULATION OF STATIONARY ELECTRIC FIELDS IN QUASI�HOMOGENEOUS MEDIA

OF MULTI�ELECTRODE COMPOSITE ELECTRIC RADIATORS

V.V. Evstigneev, T.M. Khalina, M.V. Khalin

I.I. Polzunov’s Altai state technical university

E�mail: temf@yandex.ru

Theoretical bases of calculation of stationary electric fields parameters in quasi�homogeneous media of multi�electrode composite elec�tric radiators on the basis of methods of direct determination of field intensity and conformal transformations have been given. The cal�culated models, exact and approximate formulas necessary for determining electric conductivity of various electric radiators are offered.Experimental acknowledgement of quasi�homogeneity of resistive layer by specific electrical conductivity is given.

Bulletin of the Томsк Pоlytеchnic University. 2007. V. 311. № 2

142

Table 1. The calculated models of multi�electrode composite electric radiators and system in the mapping plane

№№ Calculated model System in the mapped plane

1

2

3

4

5

6

Mathematics and mechanics. Physics

143

Table 2. Formulas for determining parameters of conformal mapping and dimensionless electric conductivities for multi�electrodesystems given in Table 1

№№ Mapping parameters Accurate and approximate calculated expressions

1

where K0 and K’0 are the complete elliptic integrals of the first sortwith modules k0 and k'0

where I1–I8 are the hyperelliptic integrals determined numerically accor�ding to [2]

2

I1–I16 are the hyperelliptic integrals determined numerically according to [3]

3

where K'(k) and K(k) are the complete elliptic integrals with modules k

and √⎯k=

⎯1–k2

⎯[4]

4

5

6

1 14 0 0 0 0 0 0

4 4

201 40 0 0 0 0

4 0 5

sn( , ), sn( , ), sn( , ),

sn( , ), , , 1 ,

a az d da K k K k K kl a l a l

Ka ad hK k k k ka l l K a

ζ = = =

′′= = = = −

1 20 0 0 0

6 6

3 40 0 0 0

6 6

5 70 0

6 6 0

sn( , ), sn( , ),

2sn( , ), sn( , ),

2 2 1sn( , ), .

a ad d aK k K ka l a la ad a n d a nK k K ka l a l

a àd a nK ka l à k

+= =

+ + + += =

+ += =

( )

2

13 01 01 01 3 4 01 01

3

201 01

3

211 2 2 3 3 41 1 1 1

01 1 4 91 1 1 1

21

sn( , ), / , sn( , );2 2

2sn( , ),2

14 ,1 2 2 2 2

where .

n n

n

hl

a bZa K k k a a K kl a la l aK ka l

q q q qk qq q q q

q eπ

ζ

+⋅ ⋅ ⋅

−

= = =

−=

⎡ ⎤+ + + + += ⎢ ⎥

+ + + + +⎢ ⎥⎣ ⎦

=

( )

( )

2

21 2 1 1 0 0

2 2 0 010 0

1 20 0

20 0 0 0

211 2 2 30

0 4

ln , , sn , ,

, ( , ), ,( , )

where 1 , ( , ) is the delta amplitudinis,

2 1, 4 ,1 2 2 2ln

n n

n

ZZ Z Z Z d l a K klkab sn K kl adn K kl

ak k dn K kl

a q q qa k ql R q q qRr r

q e

ζ

α αω ζ

α α

⋅ +⋅ ⋅

⎛ ⎞= = − + = ⎜ ⎟⎝ ⎠

′= + = =

′ ′= −

⎡ ⎤+ + + += = ⎢ ⎥+ + + +⎣ ⎦

=2

0 02, , , , are the design values ln

of electric radiator

R k aRr

ρ πρ− =

2

0

1 1 4 10 0 0

5 2 5

0 01 2 1

4 5 1 00 0

ln /0

; ( , , );

, ,( , , )

4 .R r

à à à hk dn K kà à à hk àà à h

à à h h Rdn K kh

k lπ

π

−

′ ′= = =

′= = =

′ ′

=

23

0

3 3 01 0 0 0 1 4

0

1 12

40 0 0 0

3 3

2ln /1 23 0

3 50 0

3

3 2 1

4 4 4

32 1

4 4 4

sn( , ), where / ; ;

; ;( , ) ( , )

; 4 4 ;( , )

11

1

I I

hR rl

Z h Ka K k k a al l Ka aay hdn K k dn K kh h

aa k e eh hdn K kh

a a aa a a dnuk dn uaa aa a a

π π

ζ

ξ ξ

ξ− −

′= = =

= = =′ ′

= = = =−′ ′

⎛ ⎞⎛ ⎞− −⎜ ⎟⎜ ⎟ −⎝ ⎠⎝ ⎠= =−⎛ ⎞⎛ ⎞− −⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠

00

0 0 00 0

0 3 0 00 0

;

2 .ln ln

kk

a a au K KR h R RR Rr r

π′= = ≈

17 5 6 8

2

1 43

2

( ),l l

II I I IG I IU I I II

γ

− + −= =

Δ −

1 2 32 2 2 2

7 01 02 8 01 02 9

2 21 10 13 16 2 01 02 11 14 17

2 23 01 02 12 15 18

,( )

where , ( )( ),

( ),

lG D D DI c c I c c I

D I I I D c c I I ID c c I I I

γ− +

=− + +

− − + − + − +

− − +

3

2 21 22 21 3

'( )4 ,( )

1- ,1-

G K kK k

a aka a

γ=

=

0 010

1 2 0

00

20

( , )( ) , ,( ) ( , )2where ,

ln

.2 1 ln

l

l

à sn u kàK kG k kK k à à dn u kàu K

r RR R rR rGR r R rr ar

γ

γπ

ππ

′= = ⋅ =′ ′

=

−≈⎛ ⎞− −+⎜ ⎟⎝ ⎠

( )( )

4 1

5 20

2 4 1 0

5 2 2

0 00

0 0 00 2

0 0

(1 )(1 )14 ; ,

' (1 )( )

2 4( )where , .4( )ln 1 ln 2 ( )

l

l

à àK k à à dnuG k kà a à dnu kK k

à a àa R ru K GR R r a rR r shr R R rr

γ

ππ

π

− −−= = =

−− −

−= ≈

−⎛ ⎞−⎜ ⎟−⎝ ⎠

10 0

2

10 0 0

20 0

( )8 ;( )

8ln( / ) 8ln / )1 ln sh 2

8( ) 8( )1 ln sh .2 ( )

l

l

K kG K k

R r R r uG

R r R r a rr R R rr

γ

π π

ππ π

−

−

= ′

⎡ ⎤≈ − ≈⎢ ⎥⎣ ⎦

− −⎡ ⎤≈ −⎢ ⎥−⎣ ⎦

Quasi�homogeneity of the medium by the condit�ions of specific electric conductivity was confirmed bythe direct researches of composite material (CM) ofMCE electroconductive layer by the methods of elec�tron scanning, transmission diffraction and optical mic�roscopy.

Metallographic examination on proper scale levelsshowed that there are isotropic and anisotropic particleswith the size from 0,4 to 100 mkm in CM. Isotropic par�ticles have the form of globe and polyhedron. Smallerparticles are combined in separate groups. Sizes and den�sity of propagation of anisotropic and isotropic particlesas well as small particle groups indicate the fact that par�ticles visible on scale level supported by metallographicmicroscopes are not basic elements of regular structure ofelectrocunductive filler implementing the mechanism ofMCE electrocunductivity. Besides, on this level it is pos�sible to estimate the consistency of CM productiontechnology by the methods of plane geometry.

To study electroconductive structure technologiesand methods allowing studying CM with higher incre�ase and resolution are required. The technique of scan�ning electronic microscopy is such method. The inves�tigations were carried out at microscope Tesla BS 301 inthe mode of reflected and secondary electrons. The sur�face of fresh cleavage obtained by sample destruction atliquid nitrogen temperature was irradiated. Structureradiated in wide range of scale of magnification from 50to 20000 times with maximum resolution 10 nm. Partic�le sizes were determined by the method of randomlythrown secant studying sample micrographs obtained inscanning electronic microscope and digitized by perso�nal computer.

The investigations were carried out at the samples ofbutyl rubber (BR) with filler in the form of industrialsorts of carbon black (CB) П�234, П�324 of four diffe�rent concentrations (45, 58, 75 and 145 m.f. per 100m.f. of polymer). Studying cleavage surface structure bythe technique of scanning electronic microscopy at inc�reases from 500 to 5000 times the high density of sepa�rate particles is observed. Identification of these partic�les is possible only when applying complex methods of

optical, scanning and transmission microscopy at onesample [6, 7]. But on the basis of the fact that at the gi�ven increases the average sizes of separate particles am�ount to units of micrometers and relying on the data ofinvestigations of the original state of various ingredientsone can state that visible particles are not separate par�ticles of CB. Taking into account the fact that CB par�ticles may be coagulated when obtaining CM one canstate with considerable degree of confidence that visibleparticles are the largest agglomerates of CB or separateparticles of other ingredients of CM of proper size.

At increases up to 500 times inclusive only particleswith the size from 3...4 mkm to 20...30 mkm are distinc�tly singled out; such particles are coarse grain of suchingredients of CM as: phenol�formaldehyde resin, he�xole and barite that is conditioned by imperfection oftechnological production process. However, it should benoted that such particle concentration in the examinedmaterial is negligibly low.

At increases 1000 and 2000 times a great number ofseparate particles which are rather homogeneous in sizemay be seen; they should be referred to the largest CBagglomerates which participate in formation of electro�cunductive structure of he examined material. Typicalmicrographs of BR vulcanizates with CB of differentsorts of concentration are given in Fig. 1.

Absolute majority of particles in Fig. 1 may be divi�ded into two groups by morphological characters. Par�ticles with correct surface faceting and sharp image offission�fragment form as a rule are referred to the firstgroup and particles with round shape the images ofwhich are amorphous and have no clearly defined boun�daries are referred to the second one. Average distances,diameter and surface density of particles are determinedfor each group. To estimate homogeneity of CB propa�gation in rubber matrix the dispersion and root�mean�square deviation of distance of particles of propergroups from each other were calculated (Table 3).

In addition, to estimate homogeneity, the obtainedmicrographs (Fig. 1) were divided into four fragments.Then average distances and surface density of determi�ned types of particles were determined in each fragment

Bulletin of the Томsк Pоlytеchnic University. 2007. V. 311. № 2

144

a bFig. 1. Micrograph of vulcanizate on the basis of butyl rubber BR�1675 with content of CB of the type: a) П�324 58 m.f. and b) П�234

45 m.f. per 100 m.f. of polymer at increase of 200 times

(Fig. 4). The obtained data indicate statistically disorde�red arrangement of particles that allows considering theproposed technology of producing CM to be rather suc�cessful supporting generation of the system with randomdistribution of dispersed filler particles.

Table 3. Particle size, their surface density, distance betweenparticles, dispersion and root�mean�square deviationby the results of scanning microscopy

Table 4. Average distance between particles and their surfacedensity corresponding to different fragments (F) ofmicrographs

At higher increase the structure of agglomerates ofcarbon black in butyl rubber matrix may be estimated(Fig. 2).

To identify the particles observed at increase of 2000times which form electrocunductive structure of CMthe material was studied by transmission diffractionmicroscopy by coal replica technique with extraction.

Objects of investigation in the form of samples ofelectroconductive layer of MCE for transmission mic�roscopy were prepared in the following way: samplefragment was frozen in liquid nitrogen and damaged.The extracted finest grain was arranged on a slide andplaced in exhaust cart where thin coal layer was sprayed.The formed film was segregated from glass by gelatinwater solution and placed on special copper meshes.Copper meshes with extracted replicas were placed inthe column of electronic microscope which was used forstudying phase composition, morphology, granulometryand impurity defect structure. Method of planimetrywas used for determining average size of particles [6].Micro electron�diffraction patterns were indicated bystandard method.

In spite of the fact that the obtained coal replica isthe replica with extraction and does not bear informati�on on morphology of macroobject surface it confirmsidentification of particles forming electroconductivestructure of CM as the main quantity of particles extrac�ted to the replica were the CB particles with hexagonalcrystal lattice that is indicated by typical structure ofmicro electron�diffraction patterns.

Typical electron microscope image of CB particlewith hexagonal crystal lattice is given in Fig. 3, a; andmicro electron�diffraction pattern of CB particle is gi�ven in Fig. 3, b.

Inhomogeneity of CB distribution determined bymicrographs with increase of 20000 times and calcula�ted by the level of grey of different areas amounted to≈7 %. The degree of dispersion determined at macrole�vel by standard method of Li�Dagmor, as well as by themethod of comparison with sample micrographs in therange from 93,0 to 97,5 %.

Calculated

parameter

CB П�234, concentra�

tion 45 m.f. per

100 m.f. of polymer

CB П�324, concentrta�

ion 58 m.f. per

100 m.f. of polymer

Ф 1 Ф 2 Ф 3 Ф 4 Ф 1 Ф 2 Ф 3 Ф 4

Average distance

Lср, mkm10,286 10,381 10,346 8,746 10,138 10,524 11,500 9,399

Particle surface

density, mm–29452 9279 9343 13072 9730 9029 7561 11318

Par

ticl

eg

rou

p The calculated pa�

rameter

CB П�234 with con�

centration 45 m.f.

per 100 m.f. of po�

lymer

CB П�324 with con�

centration 58 m.f.

per 100 m.f. of po�

lymer

1

Average diameter

of particle dср, mkm2,158 0,943

Average distance

Lср, mkm10,299 8,890

Particle surface

density, mm–2 9426 12651

Dispersion 21,498 12,628

Root�mean�square

deviation4,637 3,554

2

Average diameter

of particle dср, mkm1,411 1,388

Average distance

Lср, mkm9,224 9,963

Particle surface

density, mm–2 11753 10074

Dispersion 15,692 28,714

Root�mean�square

deviation3,961 5,359

Mathematics and mechanics. Physics

145

a b

Fig. 2. Micrograph of vulcanizate on the basis of butyl rubber BR�1675 with content of CB of the type: a) П�324 45 m.f. and b) П�23458 m.f. per 100 m.f. of polymer at increase of 20000 times

Fig. 3. Electron microscope image (a) and micro electron�dif�fraction pattern of CB particle with hexagonal crystal lat�tice (b)

The obtained indices of homogeneity and dispersiondegree confirm definitely quasi�homogeneity of theexamined medium by conditions of specific electrocon�ductivity.

Thus, parameters of stationary electric field (Tab�le 2, 3) were determined by specific structural dimen�sions of multi�electrode electric radiators by the para�

meters of conformal transformation and selection of ac�curate or approximate formulas. The given expressionsshow that in spite of methodological identity the calcu�lations are individual for each type of systems.

The obtained formulas determine, in particular, thatdependence of electric conductivity of axis�symmetricelectric radiator on electrode width is of logarithmiccharacter, increase of ratio of inner radius of electro�cunductive layer to outer one results in decreasing valu�es of dimensionless conductivity at constant magnitudesof ratios of electrode width to the difference of the abo�ve mentioned radii; the increase of the last ratio at con�stant ratios of radii of electroconductive layer increaseselectric radiator conductivity.

The developed models of complex CE systems inconjunction with the obtained complex of accurate andapproximate expressions of parameters of stationary elec�tric fields in quasi�homogeneous media are theoreticalbasis of engineering design procedure of constructive pa�rameters of wide range of composite electric radiators.

0,1

Bulletin of the Томsк Pоlytеchnic University. 2007. V. 311. № 2

146

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1. Iossel Yu.Ya., Kochanov E.S., Strunskii M.G. Capacity calculation.2d issue. – Leningrad: Energoizdat, 1981. – 288 p.

2. Khalina T.M. Calculation of electrical conductivity between two pa�irs of co�planar electrodes disposed in conductor of rectangular sec�tion // Аzdгbаусаn Elmldr Akademiyasi H.M. Abdullayev adina Fi�zika Institutu. Fizika. – BAKl: ELM NcSRIYYATI. – 1999. –Cild 5. – № 2. – Р. 14–22.

3. Khalina T.M. Calculation of electrical conductance between theelectrode system in a composite electric heater // Electrical Techno�logy Russia. – 2003. – № 4. – Р. 43–57.

4. Khalina T.M. Theoretical analysis and calculation of electric con�ductivity of multi�electrode low�temperature composite electric ra�diators // Electrotekhnika. – 2001. – № 8. – P. 57–62.

5. Evstigneev V.V., Khalina T.M. Stationary electric field parameterscalculation in quasi�homogeneous medium of the low temperaturemulti�electrode composite electric heater complex system // ThirdIntern. Conf. Technical and Physical Problems in Power Engende�ring (TPE�2006), Ankara, Turkey, May 29–31, 2006. – P. 505–511.

6. Chernyavskii K.S. Stereology in metal science. – Moscow: Metal�lurgiya, 1977. – 280 p.

7. Practical Methods in Electron Microscopy / Ed. A.M. Glauert. –North�Holland Publishing Company. Amsterdam, 1972.

Received on 14.11.2006