Outline. Numerical Heat Transfer. Review All Black Surfaces. Review View Factor, F i j or F ij. Review Gray Diffuse Opaque II

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1 umercao Heat raser ay 9 ad, 7 umercal Heat raser arry Caretto ecacal geerg 75 Heat raser ay 9 ad, 7 Outle Wat s umercal aalyss Cosderatos o coducto, covecto ad radato evew umercal aalyss bascs ervatve epressos, trucato error ad roudo error umercal solutos o te coducto equato oe space dmeso plct versus mplct algortms evew Vew actor, j or j j or j s te racto o radato, leavg surace, tat strkes surace j j j j Σ j j d vew actors rom carts or equatos asc compoet o radato ecage evew ll lack Suraces Heat traser rom surace reacg surace s σ Heat traser rom surace reacg surace s σ σ et eat traser rom surace s σ( egatve value dcates eat to surace or multple suraces j j j j ( gure - rom Çegel, Heat ad ass raser σ j evew Gray use Opaque Krco s law apples to te average: α at all temperatures or opaque suraces τ so α ρ or gray, dusve, opaque suraces te ρ α ee radosty, J b ρg emtted ad relected radato b J ( b J were 5 evew Gray use Opaque II J j j ( J J j j j j Combg two equatos or J J j J b J J J j j j j J j j j j Q & Solve system o smultaeous lear equatos or values o J lack or reradatg surace ( Q & as J b σ j b 6 75 Heat raser

2 umercao Heat raser ay 9 ad, 7 evew Crcut alogy evew ree-surace Crcut ook at smple eclosure wt oly two suraces pply crcut aalogy wt total resstace b b b b otal I Q &, Q & et, ca be oud rom crcut wt two parallel resstaces et, b b otal b b c c 7 8 evew adato cage wo possble surace codtos: ( kow temperature, ( kow Q & ( b J j ( J J j, K, j ( j J j J j b σ j, j j, j Solve ts set o ( j J j J j smultaeous j, j j, j equatos or values o 9 J evew adato cage II Oce all J values are kow we ca compute ukow values o ad Q & or kow b or kow Q & J ( J ( σ J b σ J umercal alyss eretal equatos gve aalytcal soluto at ay pot Usually ot possble to obta aalytcal solutos or real problems umercal aalyss provdes algebrac equatos or values at a set o pots rego arger sets gve better values o temperature ad temperature gradet requred or eat traser umercal alyss II Use deret metods te dereces use aylor seres to get potwse epressos or deretal equatos wt measure o error te elemet metods use tegral approac to get smlar results te volume metods are mdway betwee te two Ca use pyscal argumets to derve algebrac equatos (see tet 75 Heat raser

3 umercao Heat raser ay 9 ad, 7 te erece Grds Subdvde rego to dscrete pots Spacg betwee te pots may be uorm or o-uorm ample: grd or m ma wt odes umbered rom zero to Ital ode value, m al grd ode value, ma ode spacg betwee Δ ad - Uorm spacg, Δ ( m ma / odes gve spaces te erece Grds II o-uorm grd llustrated below ~ ~ Uorm grd below ~ ~ Ca ave two, tree or our-dmesoal grds or, y, z, ad tme otato jk (, y j, zk, t ervatve pressos Obta rom deretatg terpolato polyomals or rom aylor seres apulate equato to get epresso ( ( a or dervatve plus error term d d a ( a d! d a ( - a pply to a ad d! d a ( - a... d d d ( ( ( ( (...!! d d d d ( ( d ( d (... d!! d d 5 ervatve pressos II ook at te rst term te error Ca sow tat ts equals error locato dervatve s ukow d ( ( d ( d (... d!! d d d d d d ( ( d ( d! ξ or uorm Δ ξ ( ( d ( ( O(! d 6 d d O( s Order o te rror ( ( d ' O( or O( d otato O( meas trucato error s proportoal to rror proportoal to called t order Cagg step sze rom to cages t order error power o 7 rst ervatve pressos rst order orward rst order backward ' ' O( O( Secod order cetral Secod order orward Secod order backwards ' O( ' ' ( O ( O 8 75 Heat raser

4 umercao Heat raser ay 9 ad, 7 Secod ervatves Secod-order, cetral-derece, secod dervatve '' ( O Oter epressos avalable, but ts s most commo 9 ' d ad or s at Secod order cetral ' '' O( O( ' s(. s(. s(. s(. s( '' (. (. ' s(. s(. (. s(. s(. s( '' (. s(. s(. (. s(. s(. s( '' (. oudo rror Possble dervatve epressos rom subtractg close dereces ample ( e : ( (e e - /( ad error at s (e e - /( - e (. Secod order error ( (. 9 rror vs. Step Sze og Plots Prevously sad tat log( log( log log( log( log( log( log( log( log( log( log( log( [ [ log( e slope o a le o a log(error vs. log(step sze plot s order o te error rror gure -. ect o Step Sze o rror...- d( e e e.- d log(error log(step sze log( log( log( ( Step Sze log( 5. (. log(error - log(step sze -5. umercal P Solutos ee a te-derece grd te depedet varables (, y, z, t Place grd pots o rego boudary wose values are oud rom boudary codtos or te problem t some grd locato covert deretal equato to a te derece equato Observe trucato error process eglect trucato error to get set o algebrac equatos to solve 75 Heat raser

5 umercao Heat raser ay 9 ad, 7 Usteady Heat raser pply derece ormulas derved or ordary dervatves to partal dervatves Use otato to cosder deret coordate drectos pply to duso equato α t Grds Δ ad t t ry te derece epressos below to get smple te-derece equato O( ad O[( Δ 5 Usteady Heat raser II Substtute te derece epressos to deretal equato α O[,( Δ Igore trucato error, solve or α α ( Obta temperature at ad t t terms o values at old tme step 6 plct (CS etod etod just derved s called eplct metod; ca solve oe equato at a tme ( ( ( α α t αδ does ot deped o oter values at te ew tme step ( 7 plct etod ample Pck α, Δ.5,,. α/(δ (./(.5.6 Pck tal ad boudares, or tme > ( pply ( ( [ (.6[.68[ 8 [ (.6[.68[ [ (.6[.68[ 8 epeat or subsequet tme steps 8 plct etod esults.6 plct etod esults t t t. t. t. t. t.5 t.6 t.7 t act rror t. t. t. t.5 t.6 t.7 t.8 t.9 t. t. t Heat raser 5

6 umercao Heat raser ay 9 ad, 7 plct esults. plct esults act rror t t t. t. t.6 t.6 t.8 t. t. t act rror t t t. t.8 t. t.6 t. t. t Wat Happeed? We are seeg eects o stablty erece equatos may ot coverge Ustable equatos grow wtout boud ay ave stable equatos tat produce correct results Codtoal stablty requres step sze less ta tat eeded or accuracy Goal o absolute stablty ot always possble scussos o stablty comple, ca sometmes use pyscal argumets Stablty o plct etod I te values o ad - are ed a crease sould crease ( ( umercal : I s greater ta.5, a crease wll cause a decrease We ca avod ts correct result by keepg α/(δ.5 s mposes a tme step lmt tat may be less ta te lmt requred or accuracy te soluto Crak-colso etod Seek more accurate tme dervatve Provdes mplct metod Value o depeds o oter ore work per step, but ca take loger tme steps wt ts metod pply to deretal equato at tme / O[( O[( α 5 Crak-colso quato See dervato o sldes 5-55 al result sow below [ ( ( System o equatos easly solved by specal applcato o Gauss elmato called omas algortm (sldes ( [ ( ( 6 75 Heat raser 6

7 umercao Heat raser ay 9 ad, 7 Crak colso esults esults or α,, Δ.,.5, α/(δ t t t.5 t. t.5 t. t.5 t Crak colso esults II t.5 t. t.5 t.5 t.55 t.6 t.65 t.7 t.75 t.8 t Crak colso esults III ully Implct etod act rror t.9 t.95 t. t.5 t. t.5 t. t.5 t.5 t scretze duso equato at t O( ad α ( α O[( Δ rdagoal system o equatos lmost same work as C ad o spurous oscllatos, but less accuracy O[(, 9 ully Implct esults ully Implct esults II Same as C results: α,, Δ.,.5, α/(δ t t t.5 t. t.5 t. t.5 t t.5 t. t.5 t.5 t.55 t.6 t.65 t.7 t.75 t.8 t Heat raser 7

8 umercao Heat raser ay 9 ad, act rror ully Implct esults III t.9 t.95 t. t.5 t. t.5 t. t.5 t.5 t cardso/eaprog Use two tme step cetral dereces O[( α α esult s eplct wt secod order accuracy tme O[( Δ ( ( α However result s ustable or ay ad caot be used uort rakel odcato o cardso metod to provde stablty eplace secod dervatve by average at tme steps ad - Itroduces aoter O[( error O[( α α O[( O[( Δ ( α O( Δ,(, uort rakel earrage ad troduce α/(δ αδ ( ( t ( ( ( esult s eplct or values at tme plct start requred to get rst set o values at tme rror plct Soluto o Coducto quato.- rror Crak-colso Soluto o Coducto quato.-.- S emperature rro ermal usvty < < ( (, (,t (,t d me S emperature rro ermal usvty < < ( (, (,t (,t d me α/(δ α/(δ 75 Heat raser 8

9 umercao Heat raser ay 9 ad, 7 rror ully Implct Soluto o Coducto quato.- rror uort rakel Soluto o Coducto quato..- S emperature rro.... ermal usvty < < ( (, (,t (,t d me.... α/(δ S emperature rro α/(δ ermal usvty < < ( (, (,t (,t d me S emerature rror S emperature rror by ecuto me ecuto me (secods Crak-colso plct ully Implct uort-rakel ervato etals Sldes 5 to 55 cover dervato o Crak-colso metod Sow ow usg mdpot o terval or derece equato gves trucato error tat s O[(Δ, ( Sldes 56 to 58 dscuss omas algortm Provdes ecet umercal solutos or smultaeous equatos o ollowg orm ( 5 Space ervatve at t / ake average o space dervatve at tme steps ad Sow average s secod order accurate ' '' '''... 6 ' '' '''... 6 ''' ''''... '' ''''... O( 8 5 Usg Space ervatve at t / pply average to space dervatve O[( Substtute to usteady equato α α [( Δ,( Δ O t Itroduce α/(δ ad rearrage ( ( [ 5 75 Heat raser 9

10 umercao Heat raser ay 9 ad, 7 75 Heat raser 55 Crak-colso quatos ( ( ( ( ( ( O ewrte equatos matr orm to sow trdagoal structure (boudary values ad speced [ ( ( 56 omas lgortm Geeral ormat or trdagoal equatos C C C C O 57 omas lgortm II O Gauss elmato upper tragular orm 58 omas lgortm III orward computatos Ital: C / / or, -: C ack substtute: Get last value rst

Introduction to Numerical Differentiation and Interpolation March 10, !=1 1!=1 2!=2 3!=6 4!=24 5!= 120

Introduction to Numerical Differentiation and Interpolation March 10, !=1 1!=1 2!=2 3!=6 4!=24 5!= 120 Itroducto to Numercal Deretato ad Iterpolato Marc, Itroducto to Numercal Deretato ad Iterpolato Larr Caretto Mecacal Egeerg 9 Numercal Aalss o Egeerg stems Marc, Itroducto Iterpolato s te use o a dscrete