Application of Series in Heat Transfer - transient heat conduction

Transcription

1 Application of Series in Heat Transfer - transient heat conduction By Alain Kassab Mechanical, Materials and Aerospace Engineering UCF EXCEL Applications of Calculus

2 Part I background and review of series (Monday 4, April 28). Taylor and Maclaurin series (section 2.) 2. Fourier series Part II applications (Monday 2, April 28). Transient heat conduction 2. Finite Difference

3 Background and review of series. Taylor series (section 2.) - developed through the works of J. Gregory, L. Euler, B. Taylor and C. Maclaurin ~ 8 th Century - named after B. Taylor Can be used represent any function, f(), that is infinitely differentiable about a point, o, called the epansion point, and the series is where, first derivative of f() R. second derivative of f(). n-th derivative of f() o Taylor series epansion point location where we evaluate the series The location,, 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 f().

4 James Gregory (638-67) Scottish mathematician and astronomer at the University of St. Andrews in 667 publishes series epansions for sin(), cos(), acos() and arcsin() that turn out to be Maclaurin series for these functions. Brook Taylor (68-73) English Mathematician, Cambridge University. Published his discoveries on series and what we call the Taylor theorem that according to Joseph Louis Lagrange was the main foundation of differential calculus in Methodus Incrementorum Directa et Inversa (7). Colin Maclaurin ( ), Scottish mathematician at the University of Edinburgh. Published series for trigonometric functions that now bear his name and popularizes them in his Calculus book Treatise of Fluions (742). ote: earlier Mādhava of Sangamagrama (4 th Century), (3-42) an Indian mathematician and astronomer had also discovered series for some trigonometric functions

5 Leonhard Paul Euler (77-783) a pioneering Swiss mathematician and physicist who made seminal discoveries in all branches of pure and applied mathematics and Calculus. He introduced much of the 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 of the 8th century and one of the greatest of all time. He was blind for nearly half of his life and, despite that affliction, he was such a prolific scientific author that his discoveries are still being published. Pierre-Simon Laplace said of Euler Read Euler, read Euler, he is the master us all. 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: He discovered the Maclaurin series for e :

6 infinitely differentiable about a point, o, means the function can t be badly behaved at o eamples of functions that are badly behaved and not differentiable at o = and, therefore, cannot be epanded in a Taylor series about that point. ln( o ) Φ ( o)

7 The radius of convergence of the series can be found using the ratio test (section 2.6). Practically, the radius of convergence is from the point o to the nearest point where f() is no longer well behaved. Eample: f()= ln() epanded about o =2 Taking the derivatives and evaluating at o =2 Plot of ln() for > ln () X o = R=2 Taylor series converges Taylor series diverges Introducing into the Taylor series definition, we have the series for ln() about o =2.and since ln() is badly behaved (goes to - ) at =, the series has radius of convergence R=2 or can be taken anywhere <<4 Let s compute this series with MATHCAD

8 Epand ln() about o =2 f (, ) := ln( 2) Evaluate series at point c := 2. + n = ( ) n+ ( 2)n n2 n Take M-terms in the series M := := 2,.. M X=2. is inside radius of convergence Series converges umber of terms in series = Taylor series for ln() about o =2 fc (, ) take anti-log of result to check = e fc (, ) =.9.94 fc (, ) e fc (, ) e fcm (, ) = Observation: =2. close to epansion point and series converges quickly (takes very few terms)

9 Epand ln() about o =2 f (, ) := ln( 2) Evaluate series at point c := n = ( ) n+ ( 2)n n2 n Take M-terms in the series M := := 2,.. M X=3.9 is inside radius of convergence Series converges umber of terms in series = Taylor series for ln() about o =2 fc (, ) take anti-log of result to check = e fc (, ) = fc (, ) e fc (, ) e fcm (, ) = Observation: =3.9 far from epansion point and series converges slowly (takes many terms)

10 Epand ln() about o =2 f (, ) := ln( 2) Evaluate series at point c := 4. + n = ( ) n+ ( 2)n n2 n Take M-terms in the series M := := 2,.. M umber of terms in series = Taylor series for ln() about o =2 fc (, ) take anti-log of result to check = e fc (, ) = fc (, ) e fc (, ) X=4. is outside radius of convergence Series diverges e fcm (, ) =

11 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 -terms (stop after adding -terms) and, from the Taylor remainder theorem, we have an approimation of the function f() as Taylor polynomial: T n () Remainder: R n () ote: this epression for R n () is called the Lagrange form of the remainder. There are two other alternative forms: integral form of the remainder and Taylor s inequality. While we do not know ξ eactly, we do know at least that its location is somewhere o < ξ < More importantly, although we do not know the truncation error (TE) eactly, we know at least that it is proportional to (- o ) n+ [helps eplain 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 approimate f(). For converging alternating series can rely on the Alternating Series Estimation Theorem (section 2.)

12 The Maclaurin series is a Taylor series epanded about the particular epansion point o =, that is What the big deal? discovered independently and the advantage is that if a function behaves well everywhere on the real ais (and can be epanded about any point) then the special point = gives the easiest form of the series to differentiated and to compute. Eample: computing π with the Maclaurin series for tan - (θ) y θ π tan( θ = ) = 4 π tan () = 4

13 so that the Maclaurin series for tan - (), also called Gregory s series, is ote: although the series converges on the interval,, from our previous and for the specific case that = and tan - () = π/4, then eperience we epect, that since we are interested to evaluate the series at =, slow convergence. or a way to compute π (called the Leibniz formula for π) is Let s compute with MATHCAD

14 ( ) Leibniz formula: pi( ) := 4 Take M-terms M := 2n + n = := 2,.. M = pi( ) = pi( M) = π = pi( )

15 OK, that s great, but how do we utilize such series in practice? In part II, we will consider practical eamples of the application of Taylor and Maclaurin series to the determination of the 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 442 Heat Transfer) in which we seek the temperature as a function of time and space, T(,t), involves what is called the error function, erf(), which is defined by the integral Quenching of a metal bar This integral has no closed form solution, that is we cannot find a formula, a method, or a combination of tricks from calculus to integrate and find an eplicit epression for the integral. How are we to then compute the temperature? 2 We can utilize the Maclaurin series for into the integral and integrate term by term (section 2.9). e

16 The Maclaurin series for is which converges on - < < +. Utilizing this result we can epress as a series that converges also on - < < + and therefore can be integrated term by term and will yield a series that will converge for any value of. e 2

17 Integrating term by term So that the error function can be written as the series: You will also encounter this function in statistics (STA 332) as it is also called the probability function, and, in statistics, you will likely be told to use look-up 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 of using MATHCAD

18 Error function ( ) Erf o, Erf(, ) o :=.2 M := 2 ( ) n 2n+ := π n! ( 2n + ) n = := 2,.. M n :=, 2.. M nn :=, 3.. M ( ) ( ) = := ( ) Erf( o, ) erf o = Erf o, M Error( ) erf o = Erf o, ( ) = Error( ) = Error( )

19 Error function Erf( o, ).. Erf(, ) := o :=. M := 2 ( ) n 2n+ π n! ( 2n + ) n = := 2,.. M n :=, 2.. M nn :=, 3.. M ( ) ( ) = := ( ) Erf( o, ) erf o = Erf o, M Error( ) erf o = Erf o, ( ) = Error( ) = Error( )

20 Error function Erf( o, ) 2 Erf(, ) o := 2. M := 2 ( ) n 2n+ := π n! ( 2n + ) n = := 2,.. M n :=, 2.. M nn :=, 3.. M ( ) ( ) = := ( ) Erf( o, ) erf o = Erf o, M Error( ) erf o Erf( o, M) = ( ) = Erf o, = Error( ) = Error( ) 2 3 4

21 The error function of : definition and series erf(.) =.2 erf( 2) =.99 erf( 2.) =.8 erf( )

22 2. Fourier series - due to the work of Jean-Baptiste Fourier (768-83) and named after him - L. Euler and Jacob Bernoulli also had made early discoveries summing certain sine and cosine series for specific functions. A function f() can be represented on [-L,L] as a sum of sines and cosines, the Fourier series where the coefficients in the series are given by: provide the amplitude for every trigonometric wave: applications in acoustics, vibrations, heat transfer, signal processing, imaging,.

23 Jean-Baptiste Joseph Fourier (768-83) who lived in the days of apoleon whom he accompanied in his 798 campaigns to Egypt and served as governor of Lower Egypt under French rule. He later became Prefect of the Department of Isere in South-Western France where he carried out the famous eperimental 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 epressed in terms of summations of trigonometric functions that now bear his name: Fourier series. Actually, some of the greatest mathematicians and physicists of his time were not convinced of this and made his life a bit miserable. He published these results and the foundations of heat transfer in solids in his famous 822 treatise Théorie Analytique de la Chaleure (the Analytical Theory of Heat) which got him elected as a member of the French ational 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 ormale and Ecole Polytechnique Fourier is also credited with the discovery in 824 that gases in the atmosphere might increase the surface temperature of the Earth: the greenhouse effect.

24 The Bernoullis were a family of Swiss traders and scholars. The founder of the family, icolaus Bernoulli, immigrated to Basel from Flanders in the 6 th Century. The Bernoulli family has produced many notable mathematicians, scientists, philosophers and artists, and in particular several of the most well-known mathematician the 8th century: the brothers Jacob and Johann Bernoulli and Daniel Bernoulli. icolaus Bernoulli (623-78) Jacob Bernoulli (64 7; also Jacques): Bernoulli numbers and icolaus Bernoulli (662 76) separation of variables icloaus I Bernoulli (687 79) Johann Bernoulli ( ): developed L Hopital s rule & sold it to him icolaus II Bernoulli (69 726) Daniel Bernoulli (7 782): Bernoulli Equation Johann Bernoulli II (7 79) Johann Bernoulli II (744 87) Daniel II Bernoulli (7 834) Jacob Bernoulli II (79 789) Johann s son Daniel The brothers: Jacob and Johann 2 p + ρ V + gh = constant 2

25 How does this series work? Lets eamine the terms. the mean value: 2. amplitude of cosine waves: w_cos (, n) := cos n π L also called the discrete Fourier cosine transform w_cos(, ) w_cos(, 2) w_cos(, 3)..

26 3. amplitude of sine waves: w_sin(, n) := sin n π L also called the discrete Fourier sine transform w_sin(, ) w_sin(, 2) w_sin(, 3).. J.B Fourier, 822 treatise Théorie Analytique de la Chaleure J.P. Dirichlet (8-89) defines the Dirichlet conditions for f() to be represented by a Fourier series as well as the behavior of the Fourier series of f():. f() must be a piece-wise regular real-valued function defined on some interval, say [-L,L] or [,L] 2. f() has only a finite number of discontinuities and etrema in that interval 3 then Fourier series of this function converges to f() where f() is continuous and to the arithmetic mean of the limit to the left and limit to the right of the function f() at a point where f() is discontinuous.

27 EXAMPLE: f() = Є [,L] f() := Calculate the Fourier series coefficients: f ( ) L 2 a := f () d a = mean value of f() L L on [-L,L] L L a n π f ( ) cos n L L := d b f ( ) sin n π n L L := d L L Let s compute this series with MATHCAD

28 n = a = n b = n a n -.64 b n All the a n s are zero!! actually, we already know this.why? 2 n and f()= Integral of odd function (cos even, odd and cos() is odd) between symmetric limits=!

29 . F(, ) := a + n = a cos n π n L b sin n π n L + As we add more terms series converges F (, ) F (, ) F (, ) F(, ma)..... terms terms terms ma terms ma:= f()=. what s going on here? F(, ma).. what s going on here?...

30 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 J.P. (Dirichlet) was right!!! M := 3 := M L, M L +... ML 3 2 mean value of the limit from the left and the limit from the right F (, ) F(, ma) this wiggling is called the Gibb s effect terms ma terms f()=

31 M := 6 := M L, M L +... ML F (, ) F(, ma) terms ma terms f()=

32 Step function. Fourier series can handle even discontinuous functions f (). ma:= F(, ) := a + n = a cos n π n L + b sin n π n L F(, ma) F (, ) F (, ) terms F (, ) F(, ma). f ( )... terms terms terms ma terms f() modern version: I. Daubechie s wavelets handles jumps without wiggles

33 Conclusions we studied two series:. 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