The conjugate equations in the heterogeneous systems heat and mass exchange tasks Текст научной статьи по специальности «Физика»

УДК 536:620

The Conjugate Equations in the Heterogeneous Systems Heat and Mass Exchange Tasks

Evgeniy P. Khagleev*

Siberian Federal University 79 Svobodny, Krasnoyarsk, 660041 Russia

Received 15.04.2013, received in revised form 22.06.2013, accepted 18.07.2013

Conjugate equations based on a conception of the uniform mathematic description of a naturally equal heating transfer process in all points of a heterogeneous system is proposed. The equations may be applied to describe the heat and mass exchange processes in the divided boundaries of the heterogeneous systems instead of conventional IV type boundaries conditions. The modeling results of the underground coal gasification heat and mass exchange is adduced in the new problem definition.

Keywords: heterogeneous system, divided boundaries, heat and mass exchange, type IV boundary conditions, conjugate equations, underground gasification, coal-bed reaction channel.

Introduction

The research of heat-mass exchange processes (HME) is very actual area today. The results of the researches is much used power engineering, metallurgy, chemicals, building and space exploration.

Mathematical modeling HME processes in heterogeneous systems is consider. In the systems separate phases have different physical properties and is in close contact. The systems describes by boundary conditions IV type. It’s writes as equal of temperature and heat transfer rate of contact phases. The equation accuracy is heating effects in boundaries (surfaces division)

ti = t2 = k; ^ ^-X2 ^ = Q&t, t (£;), (1)

on on

1, 2 - indexes of first and second boundary contacts phases; t1,t2,t? - temperature of first, second and third phases in the boundary, °C; X1,X2 - coefficient of heat conductivity first and second phases, W/(m-K); n - normal on division surface; 4 - point coordinate, which placed on the division surface, m; Q(£,, t, t(^) - heat effect on the division surface, W/m2.

HME tasks in heterogeneous systems described IV type boundary conditions refers to conjugate tasks HME.

© Siberian Federal University. All rights reserved

* Corresponding author E-mail address: sfu118@mail.ru

1. Conjugate heat-mass exchange tasks with static and mobile division boundary

The HME tasks divides to two class conjugate tasks in dependence of specific conditions on phases division boundary. The first class is task with static boundaries, the second is task with mobile bound aries.

The tnsks of first class includes tasks inside/outside convectional HME between liquid (gas) flow solid body. And besides it tncludes many tasks of heat-maos exchange between static bodies, e.g. solid - solid, so lid - lcquid and no mi xed liquids [1 -6].

In c ase oS static b odies with otatic division boundaries in formula (1) heat effect is null and IV type boundary condition becomes simple:

Heterogeneous sy stems HME tasks with phase and chemical transformations bringi ng to changing di-visions boundary put into the second class [7-17].

Stefsn task was an eaaly one of the second class. Historicalry just in Stefan task the IV type boundary condition was used for thefirst rime in 1889. It was described wet ground freeze process wrng phase transmisrions of water [h]. The boundary condhion witd origin (draina as hidden heat of phnse Otansmissions write down on mobile boundary:

tph - wbter phase transmission temperature (to ice and inversely), °C; qph; p - specific phase transfer hidden heat of dry ground J/kg and try ground dentity, kg/m3; d^/dx - phase tsansfer boundary movemrnr velocity, m/c.

It should be noted that ire coarse-dispersed grounds the water phase transmission temperature .s constant - tph =const because water is in fred. And water in fine-dispersed grounds is bound, so freezing occur in range of temperatures tph =var.

Tasks with state of mattes change s and relating boundaries movement would refer to Stefan tasks in what follows. In ahis case in the systems of bodies with the phrre transitions allows for heat emission in solidification zone by solution or melt phasing dcagram [8-12]:

Ccr, Xcr - volumetric heat capacity, J/(m3K), and heat conductivity in solidification zone as function of coordinates and temperature; S(t) - function of heat emission in solidification zone by state of body diagram. In the phase division boundary writes IV boundaries conditions as before (3).

In addition Stefan tasks to second class of HME mobile boundary conjugation tasks refers tasks of chemical conversions with heat emissions or absorptions in the phase division boundary [13-16]. For example it’s burning and thermo chemical destruction of solid fuels wich blows high-enthalpy gas. IV type boundary condition writes for temperature fields and heat flows conjugation in phase division boundary in this case [13]:

(3)

(4)

g, s, w, e - gas flow, solid body, phase division boundary (wall) and external surface of boundary layer indexes; y - coordinate of orthogonal system; e - Stefan-Boltzmann constant, 5,7-10-8 W/(m2K4), and emissivity; q,, Rsi - heating effect and solid boby disappekrance mass velocity by ,-th heterogeneous chemical reaction in the wall, kg/(m2s).

Besides condition (5) components mass-conservation conditions writes in the phase division boundary. For gas the heat exchange describes energy equation subject to homogeneous reactions heat volume sources. For solid fuel the heat exchange describes thermal conductivity equation or if the solid fuel would have porosity and so diffusion-convective thermal conduction process would be have energy equation too.

When III type boundary conditions uses to describe nonstationary HME processes in heterogeneous systems the temperature fields of the modeling system divided to independend sections. The problem statement result to improbable and inconsistent solutions [3, 8, 13]. Instead of this case IV type boundsey conditions give able to consider ehe heterogeneous eystems as single whole by considaration thermal interaction between all paaticles.

However a question it appear - Why the transfer of substance, for example tnergy, impulse, amount ek the subslance, m the united syctem modeling by different mathematical ways? In the volume of bodies the heat iransfer describes ley energy equation and otherwise in boundary by equation of IV type boundary condition, when the physical laws nett change. May the energy or heat conductivity equations use as conjugation of volumes and boundaries of HME taska?

2. The uniform heat transfer mathematic description in phase volume and boundaries conception

The conception was to bring forward lay authors [17] and where was apply conjugation equation instead of IV type boundary condition. The conception based on the next reasoning.

Let’s look flat HME Insh between iolid body and washed liquid which have different temperature in initial time:. Let’s place elementary volume Ax, Ay thereby one half is oe solid body and another is on washed liquid (Fig. 1). Let’s call it conjugate elementary volume.

Thermophysic properties of solid body is the same by nature in conjunction volume 1’and in volume of body 1. As well washed liquid 2 and 2’ is. The properties have difference by quantitatively but not qualitatively.

On the other hand thermal transfer processes is unchangeable by their nature thermal conductivity, convection and radiation regardless of place in the considered body either in volume of phase or divided boundary.

Thereby substance thermophysic properties sameness and thermal transfer processes unchangeable in the volume and in boundary implicate their identical formalized description as energy equation independently to current point place either in heterogeneous system.

2.1. The conjunction equation in fixed boundary tasks

Let’s conclude differential heat transfer equation in fixed divided boundary conjunction voluentary unit (Fig. 1). Let’s make next assumptions in the case:

dx

Fig. 1. The conjugate elementary volume: 1 - the fluid flow(gas mixture); 2 - the solid body; 1’, 2’-fluid flow and the solid body in the conjugate elemenOaoy volume torresponOently

- bodies in the conjunction unit 1’ and 2’ is homogenous and isomorphous;

- p = cons’

- liqudd is inc ompre ssible ;

- liquid motion is stationary with velocity vector projection Ox, Oy-wx = const, wy = const;

- b odier inside hert origins specify as space roondinat-i o-d time runcti-ns;

- fiction io neglect.

The interactive bodies heat quantity changing in voluentary unit through time interval At may to

AQi - heat qudntitn coming in voluendary unit AV = AxAy byhea) conducdivity and convection, J;

AQv - hoat quantity emtiiidg in SV ai thf expense of inside heod origins, J.

Let’s define first c omponent of (6) by examine lower and upper sfmi-elements AV1t2 = AxAy/2 (Fig . 1). The first body valuen will denote by dsymbol and second - s.

In lower semi-elemena from liquiW through Ax face in the axis Oy directkin will have come a heat

qualtty. It will OQq = qj/’WoT fos f At. tAitc1 the heat quality which will rome (mot through opposite

define

AQ = AQht +AQv,

(6)

0ace wHl AQy+Ay /0 = qr+Ay/oAxAT.

Let suppose that q^yf+'>Ay/2 function is continued on Ay/2 interval. The Taylor series of the function is

Let’s take two first sie ries term s. In the lower semi-element will collect a lie at quantity by axis Oy heat flux; for a At time. The heat quantity is

= Q-aQtqyn = . (8)

In much the same way a lower semi-element heat quantity by axis Ox heat flux for a At time and a upper semi-element he at quantity by heat flux of axis Ox, Oy for the same time is

AgO^-M^Ar^Ao;

dx 2 ’ (9)

AgM^^A^Ax; A^ ^-O^Ax^

r hy 2! y — 2

As a -esult of applying of (8), (9- in the lower semi-element will be

fa-a) f

OQ-) = AQif)+OQ ( ) = -

And in upper semi-element will be

dx dy

Ax—At = -div a (f) Ox—Ox

oq^ = —da ad Ox—tLox.

A total heat quantity accumulated in the conjuncOion elementary volume AV may define

as

oqq = AQrp +AQhi> = ~^div a(f) adLv a(t))OxOa/O'1 ■ (10)

Let’sdenote a tnside =eyt origjins^ powers in first atd second semi-elements as qv/H qvs. Consequently second eomponent o. (6) is

hQi = ^ . (11)

In the isobayic procese tAe heat ■'^llicl:l brou=ht to elemenCary ^o^rnn- ix tpent to inarease enthalpy AH in tOe tolume completely. In that way the (6) would change to AQt = AH. In view of dh = cpdt is to incompressible liquid, we can write

AH = AH f + A//s=^ pfdh + cp ps dtj Ax Ay At , (12)

Let’s set (9) - (11) to (6)

cpf p f dr+°psPs dr=-(div q( f)+div q(s))+qvf+qvs■ (13)

The Ox, Oy axis projection of liquid and solid bo dy heat flux may prese nt as

- 489 -

if) p dt 7 (fd . dt 7

qx = a dX + pf wx/; qy ~xf dy5+pf W/;

Js) =_x, -d=- ■ >) = -A, —

AS ^.5 Hy AS ~ '

dx dy

Let’s put (14-) into (13)

dt dt ( d (. 3t'\ d f. df '' ( dt dt ,

c# p/ ^p+ Pd\ww dX+Wf df J +

dydx(Xsfdl'^qf^s^S-;))Hh(7^+q''s’

(14)

(15)

cpa cas - the specific heat capacityf Jt(kg-K); p) pd - the depsity, kgAmf cpfi cpsL pf, p, lcP js - tli.<s fhermal conductivity \^/Im^]<L) for liquid and so)id phases accordingly.

Let’s transfer the convective derivative in the left part equation (I5)

I dt de dt | dt

C’tp 'lfte"XSt' "•> J + C's|1sft"

(i..f^P-V-( = t P-( "-('

dx\ 3 dx) dy ^ 3 dy

4^1+^ *

SSx ^ dx) dy\sdy

(151)

■ qvJ + qvs

The energy equation (15) is different from the basic energy equation which written separately for the volume of thebody d and the volumeod the body 2(Figs 1). The rquation (I5( tncludes Ihe

heat transfer oS both interactive b odie s among themselver in tde boundary division. Let’s name the equation (id) as interactive bodies energy conjunction equation or more simple as conjunction equation.

Thus eonjunction equation C15), (15’) allow to model tde natvrally equal proce(ses of heaf rsansfer in heterogfneour rystem by the saructuratfy equal enurgy equahion ai im tict]Li;^:nt^^ of bodies vsintheti divided boundaries without tde ieregulas ineluskun in the foum of IV type boundary co ndition.

Let’s i ntroduce tlie sjiLj^naitiic:»nfosihe tubatantial derivftive of liquidflux as ^ = if -1- w^ewr

dr St dx dy

. Taking in(o c ons id^ rait)on Itliat thermalphy ric properties of the both bodies is constan), tlie conjunction

equation may rewrite as

Dt dt 2 2

cfP fdx+ Csps 3x=iLV t eisV t + + qvs’ (16)

V2 =dj+—j is Laplace operator.

dx dy

In contrast to considered case (Fig. 1) an occurrence is possible when two bodies moves relative to each over. "The examples is liquid - gas or two immiscible dropping liquids. In thecase a additional term appears. The term describes the convectional heat transfer in boundary area in tht second body which early considered as immovable. Thus the both derivative in the left side of (15) equation is eubstantial.

The simplest form of conjunction equation (16) would to be in case of an immovable bodies interaction with an immovable divi°ed boundary ant0 the absence (if volume thermal sources:

In this foom the conjuncOion equaOion (17) is the functionoi analogue of IV type boundary concdition (2). There is Iwo difference between ehey. Firstly the he at trannfer in divided boundary and

Secondary the nonatetionariOy (if the temperature field individing boundary environment models by conjunction equation itself. Due to this the conjunction equation (17) consistently blends with the maOhematical model of nonstationary heaO exchange in heterogeneous system. In case oj the type IV boundary conditfons (2) the nonntationonity oO the temponature field don’t denote explicitly as well as in (3) and (5). The nonstationariiy express indirectly Ohoough nonstationary heat equation which writes foo the bodies inside points (4-). In other woods: the IV type boundary conditions in nonstationary conjugate HME Oasks brings the induced heat inertia in the divided boundary environment. In reality nonsrarionany conjugaOn HME processno in heterogeneoua eystems proceeds in continual iemperature changes conditions in spatial nnd temporal as wnll in the inside bodies as in the divided boundary environment.

2.2. The conjugate equation in the floating boundary tasks

If the gas mixture stream flows around the solid body and at that the heterogeneous chemical reactions of mixture components and the solid body takes place (Fig. 2) then the terms qhr is appear in equation (15). Exactly the terms qhr take account the thermal effects of the chemical reactions. The material dimension of Ihe solid body will be obsermed with a displacement the phases divided boundary as n result of the; chemical re action.

The conjugate equation will complicate if the solid body is porosity and is permeable for gases. The porous solid fuels is thermal decomposed under heating. The decomposed process lead to the volatile matters vaporization. Under a overpressure the volatile matters filteos through the porous to the phase divides sueface (Fig. 2). Thereby ihe convectional heao transferq^ created. Besides the thermal decomposition process is endathermic reaction witii heet absorption - qvs. And so if ahe HME occurs in the presences of high temperatuees then in equation (a5) need to input the origins heat terms. The terms would to take iato acc ouat the radiation oft the inUerphase surface - qRs and the uadiatio n of the gas - qRf. Then in base ot the reasoning equal 2.1 item the co njugate equaUio n in vector form may write as:

+ nhr + nks + nvf + nvs + nRs + nRf,

D, Ci - diffusion coefficient, m2/s and mas s concentration of i-th c omponent of gas mixture.

Let’s give some examples of uspng the conjugate equution (18) which take place upon mathematical modeling of thie HME procesa in the feterogeneous systems with ihe floating divided boundary.

- 491 -

(17)

the heat transfer in inside points oo bodies describes equally in formalizod form- In rhis case they describes by heat equation. Whereas in (2) instead of ohe heat equation uses this boundary conditions.

(18)

r 2, 3 4 5

VVVVVV^nPj XX

,] * \

—1 / ) ^

J I

Fig. 2. The firing elaboration processes scheme of the coal-bed reaction channel: 1 - the flow of gas mixture; 2 - the peripheral layer of the coal-bed; 3 - the near-wall layer of the coal-bed which prone to the thermal decom-posit ion; 4 - the volatile matters filtration in the coal-bed pores; ; r0, rrt - the radiuses of the initial channel, the mobile divided boundary of the solid body-gas and the coal Ihermal decomposition mobile boundary correspondently; l - the reaction channellength

3. The conjugate equation in the underground coal gasificationHME task

In the [16] Kreinin I0. V. and Shiftin E. I. produced a mathematical model wich quite complete described physicaC and chemical effects oO the stable phase undergeound coal gasification (UCG) in the gas generator (GG). But strictly speaking no one of ihe pracere etages is stabie. Since a moment of a fireplace creating to a reaction eanal forming cnd further GG expktitalion their material composition, a gas mixture componenta mass concenteatieos and tempeoaeure fields of the coal layer and the gas flow changes in fime of the essence.

A mathematical model of a nonstationary HME reacttoe channel d= 200 mm oxidative and reduction domains crafting process in tine well GG was build in lire article [17] in the simplest problem definition. In the case the coal-bed is entire solid body and the only heterogeneous redox reaction C + 02=C02; C + C02=2C0 takes place in a reaftton channel wall in tUe oxidative and ceduction domains:

Dt dt . „ 2 r, c , ( Et 'l dCt

c fp+dT+c* p*dT +v 2*v k± itkokexp[- -rt j-dt’ (19)

qk, kok, Ek - thermal effect, JAnol, preexponential efficienr, m/s, activation energy, ../mol, k-th heteroganeouk reaction k = 1, 2,1-02, 2 - C02; R - the gas constant, J/(moMC); Ck - k-th component molar concentration, mol/m3.

In the problem definition the last: term of equation (19) define a two heterogeneous reactions heat effect intensity with O2, CO2 components. The fitst occurs with calorification in oxidative domain and the second occurs with the thermal absorption in reduction domain of the reaction channel. And at the same time the wall of the channel burns down under heterogeneous reaction exposure. A reaction channel cavity enlarges by degrees. Primarily the cylindrical cavity is transformed to spindle-shaped.

The task is solved by the numerical implicit finite difference sweep method. This allowed to build the optimal algorithm by a time step variation in the condition of an essential temporal discontinuity.

For examplemost intensive temperature field forming in the channei and the coal-bed occurs for first 24 hours. The time sOep toro accepted as one hour for the time span. After first 24 hours the time step was increased to 24 hours by degreas.

A mathemaeocal modei offered inthe artielee [18] is mere sophisticated then [17] one. The model takes inta account an endothermic procnrs erf the cork thermal decomposition in the coal-bed body. Coal is structural chmnged by the the rmal decompomitio n wich bring tea coke, pited and a volaOile matte rs generation. The volatile matters filters through the coal-bed clnftt and poune in the neaction channel (Fig. 2). In this connecOitn in tke probeem definition new floating boundary appears. This is a coal thermal decomposition boundaay. With all thir the coal-bed et considered as a double layer system wich consists from a peripheral layer rnd a near-wnll layet(F-g. 2C. The heat transfer in the peripheral layer is described bythe heat equation ae before. The coal thermolysis occurs tn rhe neae-wall layet and -his heat transfer there is described by theeneegy equation wtth a consideration of a convective heat transfer by the filtering volatile matters flow in the coal-bed porous space and a coal thermolysis heat origin:

* dt =t d(* , dt ^ 15(4 aA 5

— PiC— — + cp3c3P3vT- = t— — PiCi ^ I +-t— — PiV^ \ + tthPi -.rr- (20)

i=1 or dr dx \j=1 dx) r dr Ti=1 dr) or

9,- - the /'-tp c:o.elI component in [he sltare umis: [ -, 2 - pitch, 3 - the volatilm matters, 4 . a mineral share; X, - the /-th coal component tkermal conductivity; q,r - heat of the coal thermae decomposition, J/kg; p1 - a density of the comhuatible share of coal ii solid phase; v - radial component of a velocity vector of a filtration the volatile1matters in the coal-bed, m/s.

In the cuannel wall an energy conjugatioa equation is

^7 r Dt+ -0 dt + dt d l ,7 C / dt^

- rjcpjpj— + -—P pi - + 93^— = 1= - - r -

"j jjcj 7 1 l^iri - 1 ^3K'3r3y ~t

j=1 dx i=1 dx dr dx

d

t------

dr

( 7 dt A d ( o dt A

dr

d

dx

dt

— r,-C ,-rw— t---------— Pi/p — +-------------— Pi/prw—

■^ - j w r.. Z+ ^-i i C, .. ^^i i C r..

Vx=l )

1 d

d=1 ^

( 0 dt A

rw dr

di=1 dw.

+tthPi—r-+t3k(3exp\~j~~ 1dp3PpP±qkkokd<rk

(21)

dx T RT) dr T RT) dr

q3, ko3, E3, - thermal effect, J/mol, preexponential efficient and energy of a volatile matters burning activation; TIV, -a volatilematters molecular mass, kg/mol.

In comparison of ( 17) the equation (19) has terms which allows coal decompo sition heat in the upper semielemend of a conjugate elementary volume and volatile matters burning heat in the lower semielement. It it fifth and sixth terms in right side of the equation.

In the [18] problem definition a radiation components of the energy transmission qRs, qRf did’t consideo by supporition thnt the process proceeds in well walls self-irradiation condition. And also a coal properties changing supposes quasistationary on account of slowec coal-bed heating. In every new time step thermalphysics properties values recalculates by empidic formulas. For example an equivaleni c oal themral conductiviOy factor rs calculated by Roussel formula [19]:

m273 + (10 /1 )7 - m273)

K = 10-57,---------------- s)-----27T------------------------------------------1 • (22)

eq 0 m273 - m + (10 / 1s;(( - m273 + m)

m

10,1 s - veritable coal thedmal c:on^uc:‘ti's,i^rr and gas thermal conductivity factors, W/(m-K); m - coal porosity. A temperature dependence of effective heat capacity factor of Irsha-Borodino deposit on the materials [19] is approximated at:

Kinetrccoal deseripttonis supposed in actordance wtth data offering in [20].

A ptich share which generated by the coal decomposition defines from continuify equation on the analogy of [B]:

A volatile meitees share defines under the aisumption (if the mineral share is constant 94 = const:

The mathematical medel besides the temperature fields conjf gate equdtion includes concentration fields conjugate equations of the gas components whichsimuftaneourly praeents in the sotid body and the gas mixfuie of channel. The concenteation fields conjugate equations is similar to temperature fields conjugate equation fy they structnre and so don’t; bring here.

A computing experiment of an HME of firing wel; credting process with account od coal thermal decomposhion had reahzed fon the sama conditions ar bn [17]. In the Itrown coal-bed from the Irsha-Borodino deposit had bored a well with the diameter 200 mm and the length 100 m. In a distance 41 m from the well entry had created a fireplace by 4 m length and simultaneous oxygen-containing blowing supply.

The results of the experiment has showed the mathematical model of an HME of firing well creating process in coal-bed with using the energy conjugate equations (21) and the diffusion conjugate equation truly adjust with physical representation of the processes nature.

For example the blowing velocity value influences deeply on the reaction channel wall temperature status and on the expansion velocity of the reaction domain front along the well axis. A essential channel wall convectional cooling on the scope of the combustion zone 41- 45 m is observed by changing the middle blowing velocity w0 from 0,1 to 0,5 m/s (Fig. 3). With middle blowing velocity w0=0,06 m/s to twentieth hour from the combustion zone creating the wall temperature is higher then the initial temperature (t0=1000°C, dotted line 1 Fig. 3) everywhere and achieves 1220°C. The blowing velocity increasing brings to enlargement of the setback temperature domain (t < t0). And when the blowing velocity reaches w0=0,5 m/s the wall temperature becomes lower than 1000°C along the whole length of the combustion zone (Fig. 3). A beginning section of the combustion domain is cooled to the utmost. If the blowing velocity w0 is 0,1 and 0,3 m/s the temperature of the beginning section of the combustion domain will fall from t0=1000°C to 750 and 305°C correspondently. And if the w0 is 0,5 m/s the temperature will become lower ignition point t < tig=200°C (Fig. 3).

I,05 + 0,0014t, 0 < t < 200;

II,68, 200 < t < 560;

c0t)# - 1,6566 - 0,004°t - 569°), 560 < t < 900;

(23)

0,28,

t > 900°C.

qe -1 - (qn + 9i + 94) ■

%0L1 1200 1000 800 600 WO 200 0

W 4/ t>2 4J 44 45 46 47 4<? 49 50

Fig. 3. The reaction channel wall temperature status on 20 th hour of the firing elaboration: 1 - the initial wall temperature; 2 - the wall temperature when blowing velocity w0= 0,06 m/s; 3 - the wall temperature when blowing velocity w0= 0,10 m/s and with taking into account the volatile matters filtration in the pores of the near-wall layer; 3’ - the same without account of the volatile matters filtration; 4 - the wall temperature when blowing velocity w0= 0,30 m/s; 5 - the wall temperature when blowing velocity w0= 0,50 m/s; 6 - the coal ignition temperature; 7 - the coal thermal decomposition temperature

From the other hand blowing velocity increasing brings to increasing combustion domain front velocity along the stream. The wall temperature is distinctly increased after twenty hours of firing elaboration with w0=0,06 m/s. But the combustion domain length lr had stays the same 4 m (from 41to 45 m). When wcp becomes 0,3 m/s the combustion domain length becomes lr=8 m and the front of ignition moves along to stream from 45 m point to 49 m (Fig. 3). The front of ignition moving the utmost is when the w0=0,5 m/s and the front achieve 49,5 m point. But on the velocity the first combustion point fades as noted above.

Besides the blast velocity the reaction channel forming is influenced by oxygen concentration on the blast. The oxygen-enriched blast C0=0,3-0,4 supply more intense coke burning on the channel wall and increasing the wall temperature (Fig. 3). This provides coal-bed heating and more deep combustion front moving along stream in compare an ordinary air blowing.

A preliminary blast heating 20 to 300°C before blowing it to well don’t influence to reaction channel forming because the blowing has relatively low thermal capacity. The capacity is less than coal-bed thermal capacity in three times. When blowing air has passed a distance from well entry to fireplace (it’s 40 m in the case) it would cooled by heat transfer with well walls. So there is no difference between lower and high temperature blowing.

The feature of the mathematical model [18] is to take into account the processes of the coal thermal decomposition in the bed and the volatile matters filtering (20) passes in the opposite direction to the coal-bed heating direction. As a consequence of this the coal-bed heating in the model is more slowly than in the model [17]. For example in the case w0=0,1 m/s and taking into account the coal

thermolysis and the volatile matters filtering the wall temperature in the combustion domain 3 is less in three-five times than in the model which don’t consider the process 3’ (Fig. 3).

Conclusions

1. The energy (thermal conductivity) conjugate equation was received. It bases on the conception of the uniform mathematic description of a naturally equal heating transfer process in all points of a heterogeneous system.

2. The energy conjugate equation on the divided boundary describes processes of the both interacting bodies. The bodies may be in different states of matter and be mobile relative to each other.

3. In the non-stationary HME processes the IV type boundary conditions obviously reflect only the process temporal homogeneity but the thermal inertia in points of the divided boundaries they includes only by artificially (unnaturally). In the other hand the conjugate equations equally describes the energy transfer essence in all points of heterogeneous system. And besides the conjugate equations reproduces temporal changeability of temperature and the thermal flows in an explicit form.

4. For appropriate describes of HME processes in heterogeneous systems isn’t enough to use only temperature fields conjugate equations. Because besides heat exchange processes in the systems there is mass exchange processes with interpenetration. Therefore there is need to add the components concentration fields conjugate equations in the equation set.

5. The conception of the uniform mathematic description of HME process in the heterogeneous systems may apply to the underground coal gasification task. The numerical experiment results is well conform with physical representation of the reaction channel firing elaboration processes in the coal-bed.

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Уравнения сопряжения в задачах тепломассообмена гетерогенных систем

Е.П. Хаглеев

Сибирский федеральный университет Россия 660041, Красноярск, пр. Свободный, 79

Предложены уравнения сопряжения, основанные на концепции единообразного формализованного представления одинаковых по своей природе механизмов переноса теплоты во всех точках гетерогенной системы. Уравнения сопряжения могут быть применены при описании процессов тепломассообмена на границах раздела фаз гетерогенных систем вместо традиционных граничных условий четвертого рода. Приведены результаты моделирования задачи тепломассообмена при подземной газификации угля в постановке с использованием уравнений сопряжения.

Ключевые слова: гетерогенная система, границыраздела, тепломассообмен, граничные условия IVрода (ГУ IVрода), уравнения сопряжения, подземная газификация угля, реакционный канал угольного пласта.