Conservative Averaging Method and its Application for One Heat Conduction Problem

Transcription

1 Poceedings of the 4th WSEAS Int. Conf. on HEAT TRANSFER THERMAL ENGINEERING and ENVIRONMENT Elounda Geece August - 6 (pp6-) Consevative Aveaging Method and its Application fo One Heat Conduction Poblem RAIMONDS VILMS ANDRIS BIKIS Institute of Mathematics and Compute Science nivesity of Latvia Raina bulv. 9. Riga LV459 LATVIA Abstact: - In this pape the desciption of the method of consevative aveaging fo patial diffeential equations in two-laye domain is given. As numeical example the non-linea heat conduction in electical wie with insulation is consideed. This poblem is solved numeically. Key-Wods: - patial diffeential equations discontinuous coefficients consevative aveaging non-classical bounday conditions heat conduction electical wie insulation tempeatue. Intoduction Consevative aveaging method was developed fo patial diffeential equations with discontinuous coefficients. Because of its publication in non- English language and had accessible papes [] [] we give hee a shot geneal conception of the method. Application of this method to heat conduction in wie with insulation is demonstated. Desciption of Consevative Aveaging Method fo Rectangula Two-laye Domain in Catesian coodinates Size of this pape allow us to descibe this method fo second type bounday condition (5) only.. Initial Poblem Let us loo at domain D whee n n ( x D R x R y ( y... y n ) R. Domain D consists of seveal sub domains (fo simplicity we can image (n ) dimensional paallelepiped see Fig.): D G G H G {( x x ( ) y D} G {( x x > y D} With notation x > we undestand that domain G is located to the ight fom domain G. Shaed bode-hype plane of domains G and G we denote as H: H G G {( x x y D}. Right bode of the domain G we denote as H : H {( x x y G }. We will use notation x and x fo hype planes H and H. y H G H G x Fig. Objective is to find function (x (continuous in domain G ) and function (x (continuous in domain G ) that fulfills following equations: a) diffeential equations: ( ) L F x () L( ) F( x () whee L and L ae diffeential opeatos; b) conjugation conditions: () x x ; (4) x c) bounday condition on bode H : ϕ ( ; (5) x

2 Poceedings of the 4th WSEAS Int. Conf. on HEAT TRANSFER THERMAL ENGINEERING and ENVIRONMENT Elounda Geece August - 6 (pp6-) d) bounday conditions on the est bodes that ae not necessay to concetize at this moment: l ( ) Ψ( x ( x D D \ H. (6) Hee continuous on D (see ()) function ( x is defined to be equal to function ( x in domain G and to be equal to ( x in domain G. We equie that all deivatives of equation () ae continuous in domain G and deivatives of equation () in domain G. Deivative x is continuous function in domain D except hype plane H whee it has the fist ind discontinuity and condition (4) is fulfilled. Note that value is compaatively small but still finite aveaging will be not made by passage to the limit. We assume that coefficient and coefficients of the diffeential opeato L ae not dependent on agument x but could be dependent on agument y. Besides opeato L is linea and it do not contain deivatives egading the agument x. Let us see two examples of opeatos L and L : ) opeatos fo D heat tansfe poblem: L ( ) c (7) L( ) c ; (8) ) opeatos fo steady-state D heat tansfe: L ( ) (9) y y z z L( ). () y y z z Such estimate is valid: max max x [ ] [ ] x It shows that values of the solution changes compaatively little in the diection of agument x if coefficient is significantly lage than o if is small (o both conditions tae place). That means that function ( x could be appoximately substituted by constant egading agument x with accuacy O. Othewise substitution by constant in the x diection could be inappopiate polynomial o exponential appoximation should be used.. Tansfomed poblem with non-classical bounday condition We will convet diffeential equation () to nonclassical bounday condition fo the equation (). It means that we will have othe one poblem instead of the poblem ()-(6). To mae diffeence between these two poblems cleae new solution of the equation () we denote as uxy ( ) instead of the function ( x. We ewite equation (): L( u) F( x ( x G () and intoduce the integal aveaged value function: u ( ( x dx. () Afte integating equation () ove the segment [ ] and taing into account conjugation condition (4) we get: L ( u ) f () whee x f ( F ( x dx. (4) Equation () is not a valid bounday condition yet because it has two unnown functions: ( and u (. We use bounday condition (5) to exclude function fom the equation (): L ( u ) ( ϕ f ). (5).. Appoximation with constant To move fowad we need to appoximate function ( x in espect to agument x. Accuacy of this depends on paticula tas. We stat with simplest one appoximation with constant. Taing into account expessions () and () we get: ( x u ( u(. (6) New bounday condition is found if we put this equality into equation (5): L ( u ) ( ϕ f x ). (7) So the tansfomed poblem consists of diffeential equation () non-classical bounday condition (7) on the bode H and oiginal bounday conditions (6) on the othe bodes. New poblem is defined in the sub domain G of the initial domain D. To be coect we need also to add bounday condition on the bode H :

3 Poceedings of the 4th WSEAS Int. Conf. on HEAT TRANSFER THERMAL ENGINEERING and ENVIRONMENT Elounda Geece August - 6 (pp6-) l ( u) ψ ( y H (8) whee opeato l is defined in domain D G and is equal to opeato l hee; ψ is analogue of aveaged value integal () fo the function Ψ. Ψ in its tun is G pat of the function Ψ %. If ϕ is not identical to zeo then bounday condition (5) and assumption of independency of x ae tue only fo That means that appoximation of. agument x of function by constant is applicable when values of coefficient is elevantly lage... Linea Appoximation If we apply linea appoximation to the function ( x y ) in diection of x axis x u ( xu ( ) (9) ( y and use conjugation condition () and bounday condition (5) we get x ( x u( ϕ (. () Now we apply integal () to the expession () and put the esult into equation (5) to get bounday condition on the bode H: L ( u ) ϕ x () whee ϕ ϕ f ( ) L ϕ. () Afte finding solution uxy ( ) we can use fomula () to estoe function ( x appoximately... Appoximation with polynomial of the second ode We assume such epesentation fo the function : x x ( x u( u( u (. () Conjugation condition (4) then gives: u( (4) Fom the condition (5) and integal () we get: ( u u ) ϕ (5) u u u. (6) x u Afte witing down u and u in tems of u we can tansfom equation (4) in the following fom: ϕ ( u u ). (7) x Now we need only to expess u fom this equation and put it into (5) to get bounday condition on the bode H: L u ϕ (8) ϕ ϕ f L ϕ. (9) whee ( ) 6 Fo the calculating puposes it could be useful to leave bounday condition in the fom of system. Then it would contain equations (7) and (5). If we put fomula (7) into (5) we could use following fom fo the second equation of the system as well: ( u ) ( u u ) ( ϕ f ) L x. () We can also use fomulas (7) and () togethe as the system of equation on the bounday x...4 Exponential Appoximation We could assume anothe epesentation fo function : qx ( x u( ( e ) u( () qx ( e ) u ( whee paamete q > is abitay (positive) constant. It could be feely chosen taing into account physical and geometical popeties of paticula poblem. Simplest way is to tae q. We get bounday condition on the bode H using simila mathematical modification as in pevious case: L u C ϕ () x x ϕ whee ϕ ϕ f C L. () Constants C C depend on choice of paamete q: q e ( q ) q C ( q) (4) q q e ( ) q q e qe C ( q). (5) q q ( e ) Again we can use system on the bounday x. Fist equation of the system is (5) and the second is C ( u u ) ϕ (6) x C C o L ( u ) ( u u ) x C (7) C ϕ f C

4 Poceedings of the 4th WSEAS Int. Conf. on HEAT TRANSFER THERMAL ENGINEERING and ENVIRONMENT Elounda Geece August - 6 (pp6-) Consevative Aveaging Method in Cylindical Coodinates. Initial Poblem Let us tae domain D G G H with points 4 D R R y ( ϕ z t) R. G { y ( ( ) ) y D} G { ( ) y D} H G G { y D} H { y G }. Agument epesent adius in this case. Similaly as in pevious section we denote domain (suface) H as and domain H as. We have a) diffeential equations fo domains G and G: L ( ) F (8) L( ) F( ; (9) whee L is linea diffeential opeato that is not dependent on agument ; b) conjugation conditions: (4) ; (4) c) bounday condition at % : % ϕ ( ) y. (4) Condition in such fom is used when domain is a ing o coed cylinde. Bounday condition is homogeneous fo : ; (4) d) bounday conditions on othe bodes: l ( ) Ψ( D D \ H (44) whee function is defined similaly as in pevious section.. Tansfomed poblem Let us apply consevative aveaging method to the subdomain G that is located close to the cente of the domain D. Solution of tansfomed poblem we denote as u(. It is defined in the domain G. We ewite equation (4) with new notation: L( u) F(. (45) Again we intoduce integal aveaged value function that is diffeent fom equation () because of cylindical coodinates: u ( d. (46) Fo simplicity we will conside only the case % in futhe tansfomations. We integate equation (8) ove the segment [ ] and tae into account conjugation condition (4): L ( u) f( (47) whee f ( F d. (48) We will use equality (47) to obtain non-classical bounday condition fo the equation (45) on the bode... Appoximation with constant Let us assume that function is constant ove agument. Taing into account conjugation condition (4) we find bounday condition on the bode : L ( u ) f (. (49) Appoximation with linea function leads to the same equation because of bounday condition (4) at the point... Appoximation with polynomial We assume such epesentation fo the function : u( u ( u(. (5) Conjugation conditions (4) (4) and bounday condition (4) allow us to find unnown functions u u u. This allows to ewite the expession fo the tempeatue as follows: u ( u u ). (5) Hee u( u. (5) 4 The equality (47) gives the fist bounday equation: 8 ( u u ) L ( u ) f. (5) Deivation of function (5) by agument and putting the esult into conjugation condition (4)

5 Poceedings of the 4th WSEAS Int. Conf. on HEAT TRANSFER THERMAL ENGINEERING and ENVIRONMENT Elounda Geece August - 6 (pp6-) gives the second equation on the bounday : 4 ( u u ). (54) The new fomulation of the poblem consists of diffeential equation (45) bounday conditions (5) (54) on the bode and oiginal bounday conditions on the othe bodes of the domain G. Afte solving this poblem we can find appoximation fo the function ( t) fom (5)... Exponential Appoximation We assume following epesentation fo function : u( ( e ) u ( (55) ( e ) u( We can obtain unnown functions u u u and wite down function afte using bounday condition (4) and both conjugation conditions: u sinh() (56) ( cosh( ) cosh() ) Integal aveaged value function in this case is given by expession u ( u (57) α e e 4e α. (58) This is the fist equation on the bode. We find the second one by using equality (47): α ( u u ) L ( u ) f. (59) Afte finding solution we can appoximately econstuct values in the domain G by using fomula (56). Function could be expessed also in tems of u that is moe convenient fo calculation puposes: u ( ) u u whee constant ( ) ( 5) ( ) β (6) e whee β cosh cosh() ( ). (6) e 4e 5 4 Computation of Heat Tansfe in Cylindical Wie with Insulation Calculation of heat tansfe and heat emission in the electical wies is a vital issue in many industies. Electic cicuity is used in cas houses consume electonics etc. Damage and accidents ae possible if insulation melts because of heightened tempeatue in the wies. Themal popeties of conducto and insulation significantly diffe that maes calculation of citical tempeatues moe difficult. Besides wies fequently ae combined in bundles. In this section a single ound wie is consideed. Since length of the wie is much lage than diamete D model is used as in [] and [4]. 4. Statement of the Poblem We have two-laye domain that consists of conducto and insulation. With we denote adius of metallic wie with adius of the wie with insulation. We have a) heat conduction equation fo conducto ( ) : c F t) (6) and homogeneous non-linea heat conduction equation fo the insulation ( ) : c ; (6) b) conjugation conditions at : (64) ; (65) c) bounday conditions: : ; (66) : h( θ ) (67) d) initial conditions: t : θ θ. (68) Hee c c h coefficients which coespond to heat capacity heat conductivity and heat exchange; θ tempeatue of envionment. 4. Tansfomed poblem We use consevative aveaging method to exclude wie fom definition domain we inspect heat tansfe only in the insulation. Calculation of tempeatue in the wie was discussed also in pape [] whee authos used finite volume method fo both egions conducto (wie) and insulation. The new solution tansfomed by consevative aveaging method we denote as u( t). This statement of poblem consists of diffeential equation (6) (eplacing by u) bounday condition (67) initial condition (68) and bounday condition on the bode that we obtain fom

6 Poceedings of the 4th WSEAS Int. Conf. on HEAT TRANSFER THERMAL ENGINEERING and ENVIRONMENT Elounda Geece August - 6 (pp6-) pevious section fomula (49) if we use appoximation by constant fomulas (5)(54) fo polynomial appoximation and fomulas (57)(59) fo exponential one. 4. Numeical solution Since it is not possible to obtain analytical solution fo the stated poblem because of non-lineaity we solve it in numeical way. We constucted standad diffeence scheme with second ode appoximation egading to and fist ode appoximation fo t. As an example we tae wie of coppe with polyvinylchloide (PVC) insulation. Coefficients in this case ae following: a) heat conductivity coefficient and (W/m K) 4. ; b) specific heat capacity coefficient c c (W/g K) c ( T ) 8. 7T c ( T ) 9.T.74T ; c) heat geneation is achieved by electic cuent; only diect cuent and ohm esistance is consideed: I env f ( T ) ( α ( θ )) T A whee I electic cuent A coss sectional aea of the metallic conducto α coppe esistance α α α ( θ )) α.9 /( 8 α α.75 env / specific esistance of coppe at efeence tempeatue C; d) heat convection coefficient h(t) was calculated using given compute suboutine fo lamina unfoced convection to ai of hoizontal cylindical suface. Range of coefficient s values ae if tempeatue ange is C and tempeatue of envionment θ C. All mentioned coefficients ae valid in the tempeatue ange of ou inteest. Tempeatue is given in centigade degees. t t Fig. Fig. We can obseve tempeatue distibution afte it become stable in Fig. if we use following values fo paametes:.7 m. m θ C I A. Gaphics of solutions matches even if we use diffeent appoximations fo function. Diffeence stats only by fifth significant digit (see Table). Constant appox Polynomial appox Exponential appox Note that consevative aveaging method does not mae solution linea as it is visually obseved in Fig.. As an example we can tae oute adius times lage:.6 - (Fig. ). We also numeically veified that we could choose such coefficients fo stated poblem that choice of appoximation of by polynomial o exponential instead of constant is significant. But such paametes ae useless fom pactical point of view. 5 Conclusions Consevative aveaging method can be applied to stationay and non-stationay poblems. Constant appoximation is sufficient in many pactical heat tansfe poblems fo the one mateial with good heat conductivity. Acnowledgements: Reseach was suppoted by Euopean Social Fund and Council of Sciences of Latvia (gant 5.55). Refeences: [] Buiis A. Aufgabenstellung und Loesung eine Klasse von Poblemen de mathematischen Physi mit nichtlassischen Zusatzbedingungen. Rostoc. Math. Kolloq pp (In Geman) [] Buiis A. The change of fomulation of poblems of mathematical physics with discontinuous coefficients in compound domains. The Electonic Modelling 986 Vol.8 No.6 pp (In Russian) [] Ilgevicius A. Liess H.-D. Calculation of the heat tansfe in cylindical wies and electical fuses by implicit finite volume method. Mathematical Modelling and Analysis Vol.8 No. pp [4] Buiis A. Liess H.-D. Vilums R. Consevative Aveaging Method fo Calculation of Heat Tansfe in Cylindical Wie with Insulation. Mathematical Modelling and Analysis. Abstacts of the th Intenational Confeence MMA 5 5 p. 48.