Simulation model of Heat Transfer through the Wall

Transcription

1 Proceeding of the International Conference on Alied Mathematic and Comutational Method Simulation model of Heat Tranfer through the Wall Jana Mižáková Stella Hrehová Aleander Hošovký Deartment of Mathematic Informatic and Cybernetic Faculty of Manufacturing Technologie Technical Univerity of Košice ith a eat in Prešov Preov Slovak Reublic janamizakova@tukek tellahrehova@tukek aleanderhoovky@tukek Abtract Thi aer deal ith decribing of mathematical model of heat tranfer through the all and imulation hich ere obtained by MATLAB Simulink Model i a art of comle model of heating ytem During our model deign reearch e olve artial differential euation ytem and roblem ith invere Lalace tranform occur becaue of function of real argument from image function of comle argument i not define Keyord heat tranfer through the all Lalace tranform alication of artial differential euation ytem I INTRODUCTION The goal of the utainable develoment in the field of heat roduction and uly can only be achieved by automatic roce control uing digital controller and other control device hich erform advanced control algorithm bae on the intelligent control method The claical verified aroach to the heating roce control i uing of outdoor temerature comenation hich rovide for the otimal temerature of uly ater to heating bodie according to outdoor temerature To achieve thi reuirement it i neceary to find a euilibrium beteen the ulied heat outut and heat loe i e to enure otimum temerature of heating ater Thi i realized by uch method to the ater temerature in the heating ytem a controlled by o-called euithermic curve [ ] For that reaon it i neceary to deign artial model of heating ytem including model of heat tranfer through the all model of heating body and euithermic curve Sub-model imlemented into the comle model ill be the bai for heating control and different controller can be imlemented and comared The model of heat tranfer dynamic through the all a deigned on the bae of mathematical decribing of the energy balance for the elementary layer of lane all and roblem of olving ytem of artial differential euation by Lalace tranform occured lane all ith thickne d at ditance y from the heated urface (Fig ) Let temerature of the heated all urface i θ temerature of the refrigerated all urface i θ and temerature of elementary layer i θ The heat flo ulied into heated all urface i and the heat flo taken aay from refrigerated all urface i The heat flo inut into unit urface of layer dy and the heat flo d outut from it B Mathematical decrition of energy balance According to [] the heat energy doen t originate either doen t diolve in conidering elementary layer of the all Then difference of inut heat and outut heat in the layer ha to be eual to the time variation of the energy in layer Let c i ecific heat caacity (ecific heat) and ρ i volume eight of the all material then: ( d ) ( c dy) ρ θ () Conidering that heat flo d i: d dy () II THE BASIS FOR MODEL DESIN A Heat tranfer dynamic throught the all For deign of the heat tranfer dynamic model through the all e conider a lane all here the all i conidered a continuum ith continuouly ditributed thermal reitance and caacity We chooe elementary layer ith thickne dy in the Fig Plane all 95

2 Proceeding of the International Conference on Alied Mathematic and Comutational Method If ecific heat caacity and volume eight of the all material are contant then: θ c ρ () According to Fourier la the heat flo i directly roortional to the temerature gradient θ λ (4) here λ i heat conductivity coefficient of the all material Partial differential euation () and (4) ith relevant initial and border condition comletely decribe non-tationary onedimenional heat flo It i coniderable for automatic control the dynamic deendence of the control deviation according to change of variable hich have effect on the deviation For that reaon e ere deendent variable by their value and their increment a: θ λ () To imlify comutation e ere each of deendence in non-dimenional form (relative change of variable i e change comared ith initial value of variable): θ θ () θ θ Then euation () and () ill be: c ρ ( θ θ ) θ () ( θ θ ) (4) θ λ According (9) it i oible to ere: θ θ θ (5) By ubtitution (5) to () and (4) e get: θ θ θ c ρ c ρ (6) and then θ θ d (5) λ d θ c ρ (6) λ θ θ λ (7) For initial teady-tate i valid: (8) θ θ θ θ θ λ λ λ (9) d d and then by ubtitution (8) to (6) and ubtraction (9) from (7) e get artial differential euation ytem of heat tranfer dynamic through the all: θ d (7) C Lalace tranform By the Lalace tranform of artial differential euation (6) and (7) and the other mathematical oeration it i oible to get a ytem of euation hich decribe deendence of non-dimenional variable for heat flo and temerature θ θ [4] We ubtitute d and T a thickne η and time τ for ecluion of contant: y η (8) d θ c ρ () t τ T T d ρ c (9) λ 96

3 Proceeding of the International Conference on Alied Mathematic and Comutational Method Let ytem of artial differential euation (6) and (7) i ytem of artial differential euation of variable η τ: According to invere Lalace tranform () comle argument e get: η τ θ () coh inh θ θ () η Uing Lalace tranform defined ith comle argument : ( η ) ( η ) e τ d τ () e find image of function of real argument τ ith initial θ η (ee ubtitution (9)) and e get: condition ( ) d θ( ) dη dθ dη Secondly e ue Lalace tranform defined a ( ) ( η ) e η d η () of real argument η and comle argument initial : condition θ ( ) θ ( ) ( ) ( ) θ θ θ Lalace tranform image of artial differential euation ytem i : θ θ θ inh θ θ coh Let denote and θ θ And imultaneouly due to the fact that the unknon i heat flo taken aay from refrigerated all urface and temerature of the heated all urface θ e ere a θ We get: θ (4) coh tgh tgh (5) coh θ θ If e denote: (6) coh tgh (7) tgh (8) Euation (4) and (5) ill be: [ 5]: (9) θ () θ θ Net roblem i to find function hich correond to (9) and () uing invere Lalace tranform ith comle argument Vie of the fact that tranfer function (6) - (8) are not a image defined e have to ue Taylor erie: inh! 5! 7!

4 Proceeding of the International Conference on Alied Mathematic and Comutational Method coh! 4 4! 6 6! In dimenionle form for internal urface area of the euation: and e get tranfer function: 4 coh κθ (4) here include eternal condition on heat flo α change and κ d λ For eternal urface area ill be valid euation inh 6 4 ( 6 coh 4 ) 4 4 inh coh ( 4 )5 Uing ubtitution according (9) T 6 4 κθ (5) here include eternal condition on heat flo α change and κ d λ Finally imulation model of heat tranfer dynamic through the all a created baed on the block diagram in fig and in Matlab Simulink Problem of tranfer function () () () ecifying a to ytem (9) - () a table If e ued more term of Taylor erie ytem a untable Tranfer function in the form () - () have root negative and therefore i table [7 8 9] Block diagram in fig ill be etended uing (4) (5) to enure imact of heat flo ulied into heated all urface and and imact of eternal condition 4 4 T T () 4 ( 6T T ) 4 T T () T T () ( 4 T T )5 Fig Block diagram baed on the () - () here roertie d T ρ c i contant deended on all λ III APPLICATION OF SOLUTION OF PARTIAL DIFFERENTIAL EQUATIONS FOR SIMULATION MODEL Relation ereed by ytem of artial differential euation (9) and () can be ho a a block diagram in fig [ 7]: Euation (6) - (7) or () - () till need to be ulemented by the euation of heat tranfer on both ide of the all urface Fig Block diagram ith 98

5 Proceeding of the International Conference on Alied Mathematic and Comutational Method Fig 4 Simulation model heat tranfer dynamic through the all Fig 5 Simulation reult: internal temerature θ eternal temerature θ temerature of the heated all urface θ temerature of the refrigerated all urface θ ith inut condition arameter c ρ d IV DISCUSION AND RESULTS Net e can reent ome reult of imulation For the imulation model in Matlab (ee Fig 4) ere ued the folloing real arameter value meaured on tyical all of building: all thickne d 5 m here all conit of internal later ith thickne 5 m eternal later ith thickne 5 m internal iolation 5 m and eternal iolation 5 m and finally thickne of brick 9 m Volume of all material ρ 4 kg/m and ecific heat caacity c 84 J/(kgK) Thermal conductivity i λ W/(mK) and alo conit of thermal conductivitie of individual all layer Outdoor temerature i imulated a ine ave ith freuency π/864 rad/ and amlitude 5 C Internal reueted temerature i contant C Fig 5 cature day i e 78 ec Behavior of temerature correond to imulated ine ave but internal all layer i refrigerated by about degree of Celiu In Fig 6 i imilar ituation but all conit of ood layer ithout iolation layer A e can ee of heat loe aeared Refrigerated all urface ha higher temerature due to higher heat tranfer through the all And on the other hand heated all urface ha loer temerature a reueted internal temerature i 99

6 Proceeding of the International Conference on Alied Mathematic and Comutational Method Fig 6 Simulation reult: internal temerature θ eternal temerature θ temerature of the heated all urface θ temerature of the refrigerated all urface θ ith inut condition arameter c ρ d CONCLUSION To ummarize the firt art of aer deal ith olving of artial differential euation ytem here roblem of olution invere Lalace tranform aear becaue tranfer function (6) - (8) are not a image defined To olve thi roblem e have to find Taylor erie of image function a a reult of firt invere Lalace tranform Thee reult are in the ued in econd art of aer here ere ued a tranfer function for imulation model Net roblem a to find uch number of term of Taylor erie to ytem a table There i decribed creation of model by Matlab Simulink and there are reented obtained imulation reult Baic model in Fig had to be etended by for eamle tandardization of inut temerature variable or unit converion from Celiu to Kelvin cale etc The deigned model of heat tranfer dynamic through the all together ith model of heating body and ith model of euithermic curve ill be imlemented into control ytem ill be teted and ill be comared ith real heating ytem in future ork ACKNOWLEDMENT The reearch ork i uorted by the Project of the Structural Fund of the EU Oerational Programme Reearch and Develoment Meaure Tranfer of knoledge and technology from reearch and develoment into ractice Title of the roject: Reearch and develoment of intelligent control ytem for bioma baed heat roduction and uly ITMS code: 6 REFERENCES [] F Hruška "Regulation and control of environment arameter according to thermal comfort" In: Zborník referátov z medzinárodnej konferencie I STU Bratilava 6-67 [] J Piteľ Mižák J "Otimization of heat uly control baed on the euithermic regulation" In: Princiia Cybernetica Liberec Czech reublic 8 9 etember [CD-ROM] Technical univerity of Liberec 66-7 [] U Čermák V Paterka and J Závorka "Dynamic of controlled ytem in thermal energy and chemitry" Academia Praha [4] D Mamrilla J Seman A Vagaká "On the olution of the firt order nonlinear differential euation" in: Journal of the Alied Mathematic Statitic and Informatic (JAMSI) Vol no [5] J Piteľ "Mathematical model of heat tranfer through the all for imulation of heat control roce" Acta Metallurgica Slovaca 7 roč no / [6] J Piteľ "Model of the Heated Sace for Uing in Heating Proce Control" In: Sbornik trudov I Meždunarodnaja naučnaja konferencija Matematičikije metody v technike i technologijach Voronež Ruian [7] A Vagaká "Mathematical modeling of technological rocee uing method of multidimenional tatitic" in: Moderní matematické metody v inženýrtví VŠB - TU Otrava ISBN [8] A Vagaká "Statitical uality otimalization of milled urface" in: Proce Control 8 Univerity of Pardubice Pardubice 8 C54a--C54a-5 ISBN [9] S Hrehová A Vagaká "Alication of fuzzy rincile in evaluating uality of manufacturing" in: WSEAS Tranaction on Poer Sytem Vol 7 no 5-59 ISSN [] C Oald V Plaček B Šulc A Hošovký "Tranfer Iue of Control Otimizing Combution from Small-cale to Medium-cale Biomafired Boiler" In: PPPS : 8th Poer Plant & Poer Sytem Control Touloue -5