Building a Dynamic Two Dimensional Heat Transfer Model part #1

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1 Buildig a Dyamic Two Dimesioal Heat Trasfer Model part #1 - Tis is te first alf of a tutorial wic sows ow to build a basic dyamic eat coductio model of a square plate. Te same priciple could be used to model differet sape 2D objects. - Tis presetatio explais ow to partitio te plate ito elemetary square sectios eac avig four eigborig elemets (rigt, frot, left, back) ad eac elemet beig caracterized by tree parameters: te eat capacitace, te eat coductace (betwee eigborig elemets) ad te eat coductace wit te iet eviromet. - Te sectio also explais ow te eat trasfer occurs betwee te elemets by followig te two rules of eat trasfer: te storage ad te trasport equatio. Te fial umerical temperature calculatio formula is derived. < by eorge Lugu 1

2 Partitioig te body ad creatig a equivalet elemet scematic: - We divide te plate i equal squares. Eac square as a eat (or termal) capacitace, a eat (termal) coductace it ad te Top Neigbor Ambiet coductace betwee eac square (eleme ad te iet eviromet is. - Eac elemet iside te plate as four immediate eigbors, te rigt, te frot, te left ad te back eigbor. Left Neigbor Elemet Rigt Neigbor - I te calculatios we give eac elemet te same treatmet. We will see tat we ca use te same formulas eve o te edges ad i te corers were a elemet as oly tree or two eigbors Bottom Neigbor by applyig some tricks (add fake eigbors by matrix paddig). - Eac elemet cages eat wit its immediate eigbors as well wit te iet eviromet. T iitial_ T frot ( T - We use a electrical scematic because it is te easiest S iitial to draw ad read. it - Notice tat eac elemet is modeled as a capacitor to groud storig eat (represeted by temperature) istead of electric carge. Te eat coductaces wit it T rigt ( te eigbors are bi-color to represet te fact tat tey are sared betwee te eigbors. T left ( ( T it - Te switc S iitial is beig opeed at te begiig of te simulatio. At tat poit te eat capacitace is already set to temperature T iitial_. it ND temp T back ( < 2

3 Applyig te storage ad te trasport equatios: (storage equatio) dt (trasport equatio) ( T 1 2) - Let s first write te trasport equatio for eac of te five coductaces separately: total_ rigt( it Trigt( frot( it frot( left it left bottom( it bottom( ( ( - Now let s write te storage equatio for elemet : - rate of eat trasfer from te rigt eigbor - rate of eat trasfer from te frot eigbor - rate of eat trasfer from te left eigbor -rate of eat trasfer from te bottom eigbor - rate of eat trasfer from te iet rigt( frot( left back iet _ dt - After some maipulatio te above formula becomes: it 4 dt rigt( frot left back < 3

4 < 4 Let s write te previous equatio i a umerical form: From te previous page we ave: it 4 dt rigt( frot left back By usig te followig otatios: t m, t m 1, te above equatio becomes: it 4 ( m1) rigt( frot left back - We are tryig to obtai T ( ad if we solve te equatio above, T ( will be depedet o T rigt (, T frot (, T left ( ad T back (. At teir tur eac of te four eigborig temperatures are calculated fuctio of T (. Terefore i te form above te T ( system of equatios for all te compoet elemets leads to a mess of circular refereces. Solutio: - Istead of usig a back estimate for te differetial of temperature fuctio we ca use a forward estimate: dt T( ( m 1) dt T( m 1) ( (back estimate) (forward estimate) Usig te forward estimate te mai temperature equatio becomes: it 4 ( m1) rigt( frot left back

5 Solvig for T (m+1): 4 it T ( m 1) T rigt frot left back -Above we ave te eeded equatio for fidig te future temperature fuctio of curret temperature of te elemet itself ad te curret temperature values of te ear eigbor elemets. - Te equatio ca be writte i a more suggestive way as: T _ future T _ curret it rigt_ curret frot_ curret left _ curret back_ curret 4 _ curret _ curret _ curret - Te equatio above refers to ay two cosecutive momets i time (future & curre separated by oe time step. I order to be aliged wit our previous models we will rewrite te equatios as follows (I used istead of previous to save space o te slide). T _ curret T _ it rigt_ frot_ left _ back_ 4 _ - We ca see te similarity to te 1D equatio: T _ curret_1d T _ it rigt_ left _ 2 - I 3D, a elemet would ave 6 eigbors istead of just four so te equatio would become: T T < 5 curret_ 2D _ it rigt_ frot_ left _ back_ top_ bottom_ 6 _

6 Excel implemetatio of te 2D coductio eat-trasfer model: -Ope a ew workbook ad save it as 2D_Heat_Trasfer_Tutorial - Isert te labels as see i te sapsoot to te rigt - Usig te otrol Toolbox create two buttos ad ame tem oductiotoambiet ad Time_Step - Set te rage of te first butto (rigt click te Properties meu ad cage Mi ad Max ) to [0,100] ad cage te rage of te secod butto to [1,100] - After creatig te buttos double click tem wile i desig mode ad te VBA editor will come up. Write te followig code i te editor ad save. - After you fiis make sure you exit te Desig Mode before testig te fuctioality of te buttos. Private Sub oductiotoambiet_age() [B11] = oductiotoambiet.value / 100 Ed Sub Private Sub Time_Step_age() [B15] = Time_Step.Value / 500 Ed Sub to be cotiued < 6

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