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# Heat Transfer EML 4142

University of Central Florida

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This 33 page Class Notes was uploaded by Araceli Kohler on Thursday October 22, 2015. The Class Notes belongs to EML 4142 at University of Central Florida taught by Staff in Fall. Since its upload, it has received 50 views. For similar materials see /class/227605/eml-4142-university-of-central-florida in Engineering Mechanical at University of Central Florida.

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Date Created: 10/22/15

By Alain Kassab Mechanical Materials and Aerospace Engineering UCF EXCEL Applications of Calculus G UCF EXCEL Application of Series in Heat Transfer transient heat conduction I Part background and review of series Monday 14 April 2008 1 Taylor and Maclaurin series section 1210 2Fou erse es 0 Part II applications Monday 21 April 2008 1 Transient heat conduction 2 Finite Difference 8a UCF EXCEL Application of Series in Heat Transfer transient heat conduction ckq round and review of series 1 Taylor series section 1210 developed through the works of J Gregory L Euler B Taylor and C Maclaurin 18th Century named after B Taylor Can be used represent any function fx that is infinitely differentiable about a point X0 I called the expansion point and the series is fx fxo rum 1 0 Mme a nil f Where f first derivative of fx f second derivative of fx Taylor Series location expansion point evaluate the Series ff nth derivative of fx 0 The location X where the series is evaluated can be taken anywhere within the radius of convergence of the series in order to yield the correct value for fx 8i UCF EXCEL 1 James Gregory 16381675 Scottish mathematician and Brook Taylor 16851731 English C I39 M I 39 16981746 Mathematician Cambridge 0 m ac aurm 39 Scottish mathematician at the astronomer at the University University Published his discoveries University of Edinburgh Published 0f 31 AHdFEWS in 1557 PUbliSheS on series and What we can the Taylor series for trigonometric functions series expansions for Sing theorem that according to Joseph that now bear his name and 00500 300500 and arCSinOO Lows Lagrange was the main popularizes them in his Calculus that turn out t0 be Mactaurtn foundation of differential calculus book Treatise ofFIUXIons 1742 series for these functions in Methodus Incrementorum Directa etlnversa 1715 Note earlier Madhava of Sangamagrama 14th Century GED1425 an Indian mathematician and astronomer had V also discovered series for some trigonometric functions 1 Heat Transfer transient heat conduction w Leonhard Paul Euler 17071783 a pioneering Swiss mathematician and physicist who made seminal discoveries in all branches of pure and applied mathematics and Calculus He introduced much ofthe modern mathematical terminology and notation He made major contributions to solid and fluid mechanics optics and astronomy Euler is considered to be the preeminent mathematician ofthe 18th century and one ofthe greatest of all time He was blind for nearly half of his life and despite that affliction he was such a prolific scientific authorthat his discoveries are still being published PierreSimon Laplace said of Euler Read Euler read Euler he is the master us allquot w In your studies you ll encounter his contributions in nearly every subject and when referring to Euler s Formula you will wonder which one One of his most famous often called the Euler Formula is 6197 COS RP isi 50 w He discovered the Maclaurin series for ex 31 Z Biggie g n20 F EXCEL o Application of Series in Heat Transfer transient heat conduction O infinitely differentiable about a pointI X0 means the function can t be badly behaved at X0 0 examples of functions that are badly behaved and not differentiable at X0 5 and therefore cannot be expanded in a Taylor series about that point wow 8a UCF EXCEL 39 39 Application of Series in Heat Transfer transient heat conduction I The radius of convergence of the series can be found using the ratio test section 1261 0 Practically the radius of convergence is from the point x0 to the nearest point where fx is no longer well behaved m fx Inx expanded about Xo2 0 Taking the derivatives and evaluating at Xo2 Plot oflnxforxgt0 f2 1712 f r2 Taylor series converges n 1 2 Taylor series diverges finme lnli l finkg Wt unify Introducing into the Taylor series definition we have the series for Inx about Xo2 x J n 7 1 I 7 2V and since Inx is badly behaved goes to 71 71 2k2r 1 2n l at x0 the series has radius of conver ence mm 7 7 INN 2 R2 orx can be taken an here 0ltXlt4 n 1 722 Let s compute this series with MATHCAD 8a UCF EXCEL Application of Series in Heat Transfer transient heat con uction N n Expand Inx about x 2 ax N m2 Z Hwy n n 1 quot392 Evaluate series at point X25 is inside radius of convergence Series converges Take Mterms in the series M 12quot M Number of Taylor series take antii terms In for nx about of result e xcM 24999999559 fxcN 09431471806 09118971806 09171055139 09161289514 09163242639 09162835738 09162922931 09162903857 09162908096 09162907142 I Observation 2 4 N x25 close to expansion point and series converges guicklv takes very few terms Application of Series in Heat Transfer transient heat co Expand lnx about x o2 N 2 n fxNln2 Z fowlxi n 1 quot392quot Evaluate series at point Kc 39 Take Mterms in the series M 100 M 12quot M Number of Taylor series terms in for lnx about series x 2 take antilog of result to check efxc M X39 is inside radius of convergence Series converges 38998880882 efxc N W 3293 4383 3575 4174 3692 408 3755 4028 3794 3995 3819 I 40 I Observation x39 from expansion point and series converges slowly takes many terms 100 Application of Series in Heat Transfer transient heat co Expand nx about x o2 Evaluate series at point Take Mterms in the series Number of Taylor series terms in for nx about RMN n 1 xc 41 M 150 N12M m take antilog of result to check N 2n fxNln2 Z HWIX n2 quot X41 is outside radius of convergence Series diverges efXC39M 00241986898 efxc N W W 4844 W 4614 W 4512 3751 4457 I I 50 100 Ga Usz EXCEL 150 Application of Series in Heat Transfer transient heat conduction I As we just saw in practice when we compute using series we can never add an infinite number of terms so we truncate the series after Nterms stop after adding Nterms and from the Taylor remainder theorem we have an approximation of the function fx as i izc w 20quot f U m mgr 1 Mthis expression 3 n n 1 for R X Is called the 310 E z temp we compute the term we tmncate Lagrange for m or the and add to approximate and 3901 0111 our remainder There are the function f 1 Computing Off 17 is two other alternative the tnulcatlon eu or TE forms Integral form of the remainder and Taylor P yn mia39 Tn X Remainder R quotX Taylors Inequality While we do not know g exactly we do know at least that its location is somewhere xolt f lt X More importantly although we do not know the truncation error TE exactly we know at least that it is proportional to XXoquot1 helps explain rate of convergence How many terms do we take in the series we take enough terms such that adding more terms will not make a significant change in the value of the sum we compute to approximate fx For converging alternating series can rely on the Alternating Series Estimation Theorem section 125 Application of Series in Heat Transfer transient heat conduction I The Maclaurin series is a Taylor series expanded about the particular expansion point X0 0 that is x f0 f 0w f 0 20 xquot 0 What the big deal discovered independently and the advantage is that if a function behaves well everywhere on the real axis and can be expanded about any point then the special point x0 gives the easiest form of the series to differentiated and to compute V Example computing 11 with the Maclaurin series for tan3910 1 mm 2 f i wo 1 U r2 film 72 7 7 2 1 41 12 l fll00 39 my 1quot warm 7 Iquot f WU 24 7 4 i 24w x 1 3 7f 12110971 71 m 14 8a UCF EXCEL 39 39 Application of Series in Heat Transfer transient heat conduction 0 so that the Maclaurin series for tan7X also called Gregory s series is m although the series a DC 2n1 1 V x 1 n I converges on the tan wm gZ 1 interval 1Sx 1 1 from our previous I and for the specific case that x 1 and tan71 1T4 then experience we expect that since we are FT 1 r 1 1 CC 1quot interested to evaluate Ztan l1 E39 n l2n1 theseriesatX1 slow convergence 0 or a way to compute Tr called the Leibniz formula for Tr is 74 39 2n1 0 Let s compute with MATHCAD 8a UCF EXCEL 39 39 Application of Series in Heat Transfer transient heat co d ction N Leibnizformula MN 4 Z 0 1N39 2 667 3 467 W 3 34 2 976 Z 61gt n 1 Take Miterm 5 M5 500 Aka M p1M 31435886596 1 31415926536 500 Application of Series in Heat Transfer transient heat conduction OK that s great but how do we utilize such series in practice In part II we will consider practical examples ofthe application of Taylor and Maclaurin series to the determination ofthe temperature of a hot metal bar being quenched rapidly cooled in water bath and the applications of Taylor series in computational methods in heat transfer and fluid flow finite difference methods The solution to certain heat conduction problems EML 4142 Heat Transfer in which we seek the temperature as a function oftime and space 0 W b m TXt involves what is called the error function x aer a erfx which is defined by the integral T 1chc Quenching of a metal bar This integral has no closed form solution that is we cannot find a formula a method or a combination oftricks from calculus to integrate and find an explicit expression for the integral How are we to then compute the temperature We can utilize the Maclaurin series for 2 into the integral and integrate term by term section 129 8a UCF EXCEL Application of Series in Heat Transfer transient heat conduction C The Maclaurin series for air Is 2 3 4 2 7 r i i x e ilir i347 2 which converges on lt x lt 2 as a series that converges also on lt x lt Utilizing this result we can express 2 and therefore can be integrated term by term and will yield a series that will converge for any value ofx 8a UCF EXCEL Application of Series in Heat Transfer transient heat conduction I Integrating term by term 2 z 1 er r 1 e dz V a 2 z 4 5C 1nI2nl nl2n1 0 gt You will also encounter this function in statistics STA 3032 as it is also called the probability function and in statistics you will likely be told to use lookup tables to evaluate this function After this discussion you will know another way than to use a table Let s compute the error function series for various values ofx using MATHCAD 8a UCF EXCEL Application of Series in Heat Transfer transient heat co d ction ion X2nl N 2 Error function E V r x m 5 2 m Zn 1 n n 1z af x n ZZZ7U258921U479 x n ZZZ7U258921U478 Er Emma erfxu 7 Er XDN I 22271 an N z n 2227 9 83710 3914 Application of Series in Heat Transfer transient heat co d ction ion sz N Z Errorfunctlon E z Km 5 2 m 2M 1 n I 11 erfxu U5661U5146475311 ErEMU9661168918U9457 ErmrN erfxu 7 Er XUN 54 7031 014 70 05 001 7521510 132110 7300710 621110 4 UCF EXCEL Application of Series in Heat Transfer transient heat co d ction 70an m 2m 1 N Errorfunction mix m f 2 n n I E z s 5n 3512 M erfxu n 999593U47982555 Ediwa n 999593U47982557 EYE 1 Emma erf g 7 mam mix Em UCF EXCEL Application of Series in Heat Transfer transient heat conduction The errorfunction ofx definition and series erf05 052 erf2 0995 erf25 1 DC r i iljanlJ 0 n2nl n er x e adx 8a UCF EXCEL Application of Series in Heat Transfer transient heat conduction 2 Fourier series due to the work of JeanBaptiste Fourier 17681830 and named after him L Euler and Jacob Bernoulli also had made early discoveries summing certain sine and cosine series for specific functions 0 A function fX can be represented on LL as a sum of sines and cosines the Fourier series 10 l ian605r 711 n1 where the coefficients in the series are given by 1 L a0illfwdz an coszdz bn sinmdx provide the amplitude for every trigonometric wave applications in acoustics vibrations heat transfer signal processing imaging 8a UCF EXCEL 39 39 Application of Series in Heat Transfer transient heat conduction JeanBaptiste Joseph Fourier 17681830 who lived in the days of Napoleon whom he accompanied in his 1798 campaigns to Egypt and served as governor of Lower Egypt under French rule He later became Prefect ofthe Department of lsere in SouthWestern France where he carried out the famous experimental and analytical research that laid the foundations of heat transfer It was a great accomplishment by Fourier to show that any function with some reasonable degree to continuity could be expressed in terms of summations oftrigonometric functions that now bear his name Fourier series Actually some ofthe greatest mathematicians and physicists of his time were not convinced ofthis and made his life a bit miserable He published these results and the foundations of heat transfer in solids in his famous 1822 treatise The orie Analytique de la Chaleure the Analytical Theory of Heat which got him elected as a member ofthe French National Academy of Science with special mention for lack of rigor are the French tough or what of which he was later to be the Permanent Secretary Mathematician physicist and historian Ecole Normale and Ecole Polytechnique Fourier is also credited with the discovery in 1824 that gases in the atmosphere might increase the Ga UCF EXCEL surface temperature ofthe Earth the greenhouse effect Application of Series in Heat Transfer transient heat conduction The Bernoullis were a family of Swiss traders and scholars The founder ofthe family Nicolaus Bernoulli immigrated to Basel from Flanders in the 16th Century The Bernoulli family has produced many notable mathematicians scientists philosophers and artists and in particular several of the most wellknown mathematician the 18th century the brothers Jacob and Johann Bernoulli and Daniel Bernoulli Nicolaus Bernoulli 16231708 Jacob Bernoulli 1654 1705 also Jacques Bernoulli numbers and Nicolaus Bernoulli 1662 1716 separation of variables Nicloaus Bernoulli 1687 1759 Johann Bernoulli 1667 1748 developed L Hopital s rule amp sold it to him Nicolaus Bernoulli 1695 1726 Daniel Bernoulli 1700 1782 Bernoulli Equation Johann Bernoulli II 1710 1790 Johann Bernoulli II 1744 1807 Daniel H Bernoulli 1751 1834 Jacob Bernoulli II 1759 1789 The brothers Jacob and Johan Johann s son Daniel G UCF EXCEL Application of Series in Heat Transfer transient heat conduction 0 How does this series work Lets examine the terms at a0 ianwsx ibnSMCILTx quot1 11 1 the mean value L aoLfmd1 1 Z TL 2 amplitude of cosine waves aquot fLfz C05 f1 d a80 called m discrete Fourier M cosine transform Wicosxn cos wicosx 1 wicosx 2 Ecosx3 l 05 1 X8 UCF EXCEL I W Application of Series in Heat Transfer transient heat conduction 1 L 7 3 amplitude of sine waves bu 7 fr 51 Idr also caHEd the L L L discrete FourIer nn j sine transform Wisinxnsm Tx 39 4M 439 1 wisinx 1 7 0 I JB Fourier 1822 treatise The orie Analytique de la Chaleure 0 JP Dirichlet 18051859 defines the Dirichlet conditions for fX to be represented by a Fourier series as well as the behavior of the Fourier series of fX fX must be a piecewise regular realvalued function defined on some interval say LL or 0L fX has only a finite number of discontinuities and extrema in that interval then Fourier series ofthis function converges to fX where x is continuous and to the arithmetic mean ofthe limit to the left and limit to the right ofthe function fX at a point where x is discontinuous 8n UCF EXCEL 1 um Pam 5mm Lejavm D39L39l L er Application of Series in Heat Transfer transient heat conduction EXAMPLE fx x x e 0L 0 Calculate the Fourier series coefficients BU 2 x dx mean vaiue of W L L n 7 L L L 3 1J fxcosn nxjdx n L L 7L 0 Let s compute this series with MATHCAD 8a UCF EXCEL Application of Series in Heat Transfer transient heat conduction 0 All the an s are zero actually we already know thiswhy an cosnf r d1 and fXX O Integral of odd function cos even x odd and xcosx is odd between symmetric imits0 8a UCF EXCEL Application of Series in Heat Transfer transient heat conduction 0 As we add more terms series converges I I I5 NmaX 15C whars going what s going X 0 hem 15 on here UCF EXCEL Application of Series in Heat Transfer transient heat conduction 0 Since we represent a function with sum of periodic functions result is periodic and at endpoints converges to mean value from limit to the left and limit to the right ol JP Dirichlet was right H M 3 K M39L7M39L 001 ML mean value of the limit from the left and the limit from the right Fx15 FxNmax this wiggling is called the Gibb s effect 8a UCF EXCEL Application of Series in Heat Transfer transient heat conduction M5 6 3g iML7ML 001 ML 55 5 45 35 3 25 15 Fx15 1 05 FX7NmaXl27897692436821013 05 Ga UCF EXCEL Application of Series in Heat Transfer transient heat conduction O Fourier series can handle even Step function 1 discontinuous functions 1 N gxm a Z ancosnxjbnsmni 1 modern version Daubechie s wavelets handles jumps without wiggles httpwwwpacmprincetonedulingridl Ga UCF EXCEL Application of Series in Heat Transfer transient heat conduction Conclusions we studied two series 1 Taylor series very powerful but can t model discontinuous functions applied to evaluation of error function the probability integral 2 Fourier series very general and can model to some degree discontinuous functions Coming attractions applications to heat transfer and finite difference methods in heat and fluid flow 8a UCF EXCEL

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