Optimization of the Brownie Pan

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1 Optimizatio of the Browie Pa Briaa Oshiro Patrick Larso February 4, 2013 Sujay Cauligi Abstract I this paper we address the effect that pa shape has o uiformity of heat distributio, as well as efficiecy of spacial utilizatio. Specifically, we address the family of super ellipses kow as Lamé Curves, aalyzig the midway shapes betwee squares ad circles. Regrettably, due to a ufortuate overlookig of a assumptio which tured out false, our aalysis is icomplete. So istead, we addressed some more creative aswers that focus o solutios to these issues, rather tha mere compromise. Cotets 1 Itroductio 2 2 Theory of Operatio Heat Distributio Optimizatio Spacial Optimizatio Methodology Numerical results of heat distributio optimizatio Numerical results for optimal spacial usage Simulatios 5 5 Coclusio Limitatios Results Other solutios Hilbert s Space Fillig Curve Wire mesh

2 A Proof that Lamé Curves approach a square 7 1 Itroductio A browie is a terrible thig to waste. It is a shame how so may corer pieces have lost the fight to remai soft ad moist simply because of a o-optimal pa desig. Of course, a cylidrical pa avoids these dagerous corers, but the space i the ove is o loger used efficietly, ad agai, a browie is a terrible thig to waste. It is with these issues i mid that we have developed a model to fid the optimal bakig solutio. We focus o a compromise betwee ice, evely-cooked browies, ad maximal browies per batch. I the followig sectios we outlie how we parameterize this compromise i terms of superellipses ad idividually examie the effects o uiformity of heat trasfer throughout the pa, as well as efficiecy of packig i the ove. We the examie these two fuctio simultaeously i order to maximize the beefits of both extremes. 2 Theory of Operatio We focus our aalysis o the family of superellipses commoly kow as Lamé Curves. These are curves of the form x + y = 1, (1 a b where is some iteger greater tha 2, a, b are real umbers, ad a = b. For our optimizatios, we hold the area of the pa fixed, regardless of pa shape. The area for a superellipse is give as follows: ( ( Γ A = 4ab Γ (, ( where Γ is Euler s gamma fuctio. So the costats a, b of our curve will be ( a = b = A ( Γ Γ ( (3 Note also that as approaches ifiity, the graph of the image teds to a square. A proof of this fact is give i the appedix. I the followig sectios we examie how uiformity of heat distributio as well as efficiecy of space usage is affected with chagig variable. 2

3 2.1 Heat Distributio Optimizatio To aalyze the heat distributio i various pas, we use a scheme similar to that of solvig the heat equatio i a cylider. Notice that for Lamé curves of degree greater tha two, although we kow that the boudary x + y = a is held at the temperature of the ove, parameterizig that boudary i a maer that is useful i solvig the heat equatio is difficult. To combat this, we impose a chage of coordiates, sedig x = r cos 2/ θ; y = r si 2/ θ. (4 Furthermore, we ca remove the absolute values from x ad y, by oly examiig x > 0 ad y > 0. Values i the other quadrats will be similar by symmetry. After this chage i coordiates, our Lamé curves will be mapped to a circle aroud the origi of radius a. Due to the symmetry of the image we ca assume that the temperature T at ay poit i the browie pa is idepedet of θ (NOTE: As it turs out, this is assumptio does ot hold. We proceed as i sectio 2.2 i Wilkiso, but istead cosiderig the pa whose edge is defied by x + y = a, 0 z Z, where a is a costat depedig o the predefied area ad the degree, as i the previous sectio. We examie the homogeeous equatio u(r, z, t = T (r, z, t T b, (5 where T b is the ove temperature, ad impose the boudary coditios u(a, z, t = u(r, 0, t = u(r, Z, t = 0. (6 We let T i be the iitial temperature of the browie batter, so u(a, z, 0 = T i T b. (7 We apply these coditios to the followig heat equatio: ( ( δu 1 δt = δ D 2 u = D r δu + δ2 u. (8 r δr δr δz 2 where D is a costat relatig to thermal coductivity ad other physical variables which we are holdig costat. The, we ca employ separatio of variables to fid the geeral solutio u(r, z, t = k=1 m=1 ( (2k 1πz A km si Z ( r J 0 a j 0m e λ kmdt ( ( 2 where λ km = (2k 1π ( Z + j0m 2, for k, m positive itegers, J a 0 defies some Bessel fuctio of the first kid, ad j 0m is the m th root of J 0. The applyig our iitial 3 (9

4 ad boudary values we solve for A km, ad arrive at the solutio ( T (r, z, t = T b + 8(T i T b si (2k 1πz Z J 0 ( r j a 0me λ kmdt. (10 π (2k 1 j 0m J 1 (j 0m k=1 m=1 Ivertig our chage of coordiates, we fid that T (x, y, z, t = T b + 8(T i T b π k=1 m=1 si ( (2k 1πz Z (2k 1 J 0 ( x +y j a 0m e λ kmdt. (11 j 0m J 1 (j 0m Usig this solutio, we ca plot how variable affects temperature at various poits ad times throughout the browies. 2.2 Spacial Optimizatio Uder the assumptio that the ove has dimesios L, W where L is the legth ad W is the width, we map the rectagle ito a square usig L W W L Thus both W, L map to 1 ad we are ow dealig with a uit square. Each Lamè curve ca be superimposed o top of a square with the same width ad height. Usig the equatio for the area of a Lamè curve, we ca write ( ( Γ A = 4a 2 Γ (, ( where a = 1s 2 ad s is the side legth of the superimposed square. Defie c as ( ( Γ c = 4 Γ (. ( Aalyzig extra area, w, of the superimposed square, we arrive at the fuctio. w = 1 4 s2 (4 c. (14 Usig this equatio, we fid that w > w +1 thus we kow that, per superimposed square, Lamè curves with higher degree will have less extra area per square ad thus are more space efficiet idividually. I optimizig the space efficiecy, we will eed square packig. As of ow, there is o algorithm for this, thus we will eed to proceed o a case-by-case basis. We will operate uder the assumptio that a ove tray caot hold more tha 50 pas. 4

5 3 Methodology It is apparet that the most uiform heat distributio will be from usig cylidrical pas, whereas the best use of space will be from square pas. Our goal i this sectio is to aalyze just how quickly we lose efficiecy whe strayig from these extremes. I order to do this, we assig to each a efficiecy ratig for uiform heat distributio, H, ad a efficiecy ratig for spacial utilizatio, S. We set H 2, (the circle to be 1, ad H (the square, to be 0. Similarly, we set S 2 = 0, ad S = 1. We ca the cosider these efficiecy ratigs simultaeously ad decide o the browie pa that captures the beefits of squares ad circles most effectively. 3.1 Numerical results of heat distributio optimizatio To assig H, we take measuremets of temperature from the heat equatio derived i sectio 2.1 at poits a small, fixed distace from the boudary. This will be costat for = 2. We call this costat α 2. For all other, we measure the least squares distace of the temperatures from the heat equatio ad the horizotal lie at α 2. Let ɛ be this least squares measuremet. The we ca map each to a poit i [0, 1] as follows: H = 1 ɛ ɛ. ( Numerical results for optimal spacial usage Usig square packig data, we ca fid the side legths of the pa whe give the umber, m, of pas to fit i the ove. I the data, we foud the optimal side legth for packig m uit squares iside a square was give as size t, thus the side legth 1 t is the maximum side legth to fit m pas iside the uit square. 4 Simulatios As for simulatios of the optimizatio, we have oe. But we did make browies ad cocluded that the corers of browies are ideed crispier. Furthermore, we oticed that as we icrease cookig time, the etire browie gets hard ad relatively more uiform regardless of the shape of the pa. 5

6 5 Coclusio 5.1 Limitatios Our most obvious limitatio is the restricted geometry of the pa. We ca see that Lamé curves are o-optimal whe the square root of the area does ot divide both the width ad the legth of the pa. To provide a more complete picture for arbitrary ove sizes, geeral superellipses (as well as other shapes should be cosidered. I this study we assumed the the diffusio rate remais costat throughout the bakig process, whereas previous studies have suggested otherwise (Wilkiso 2008, Olszewski Extesios of this study could take ito accout this o-liearity. Fially, we assume that the boudary values of the pa i the ove are costat (at ove temperature, igorig ay temperature fluctuatio or ay iteral heat geeratio. These limitatio are all due to the difficulty faced i solvig o-liear partial differetial equatios o arbitrary geometries. 5.2 Results Ufortuately, we were ot able to compute ay results (see abstract. 6 Other solutios Due to the failure of our previous aalysis, we ow preset two examples of more creative aswers: 6.1 Hilbert s Space Fillig Curve Although less suited for the real world (ad thus more difficult to market, we tur to fractals for a solutio. We oticed that the browies i square pas were burt as a result of the large curvature i the corers. We hypothesize that uiformity of heat distributio is related to chage i curvature. With this i mid, istead of miimizig this chage ad dealig with space utilizatio as a separate issue, we thought of maximizig the chage i curvature ad as a result the aalysis becomes trivial. Because of Hilbert s Curve s space fillig ature, a square of area A with this curve cut out will have A browie area. Thus, we choose a pa whose side legths are aw, al, such that A = a 2 W L. The we ca achieve maximal space utilizatio. Furthermore, because of the fractal patters, the distace betwee corers goes to zero, ad effectively every poit withi the pa becomes a corer. Thus, the heat distributio is also uiform, although the browie will ed up uiformly burt. Thus, this space fillig fractal is apparetly the most efficiet desig, as it maximizes both variables. 6

7 6.2 Wire mesh This solutio is actually implemetable. We propose that we should ot worry about ay of this silly compromise. Uiformity of heat distributio ad space utilizatio, represeted as percetages, are both bouded values ad so have maximums. So why ot maximize both? Agai it is apparet that the pa shape from the previous sectio is optimal i spacial utilizatio, while o optimal i uiformity of heat distributio. So we propose a solutio that maitais shape while fixig this heat issue. We propose fixig a wire mesh to the iside of the browie pa. The diffusio from metal to metal is much larger tha the diffusio of heat ito the browie. We must the desig this mesh to provide iteral heat geeratio i a maer which couteracts the hotter corers, creatig a uiform distributio of heat throughout the browie. A Proof that Lamé Curves approach a square The graph of a Lamé Curve approaches a square as. The equatio for a Lamé Curve is x + y = 1, (16 a a where is some positive iteger ad a is a real umber. By raisig both sides to the power 1/ ad pullig out the costat a, we get the equatio ( x + y 1/ = a. (17 If we let x be the vector x = (x, y, the equatio (17 ca be iterpreted as the set of all poits such that the -orm of x is equal to a : x = a. (18 We shall take the defiitio of the ifiity-orm of x to be the maximum of its compoets, that is, x = max {x, y}. (19 We will ow show that the limit of the -orm as approaches ifiity is i fact the ifiity-orm. Without loss of geerality, take x to be x. The value of x is clearly less tha or equal to that of x + y for all positive itegers. If we raise both sides of this iequality to the power 1/, we fid that x x for all. Thus, we fid that x lim x. (20 7

8 Agai, without loss of geerality, assume x to be x. Pullig x out of the equatio for the -orm, we have ( x 1 + y 1/. (21 x for all positive itegers. Sice x is the ifiity-orm of x, y must be less tha or equal to x, ad so y/x must be less tha or equal to 1. Thus, we have ( x = x 1 + y 1/ x 2 1/ = x x 2 1/. (22 The limit of 2 1/ as approaches ifiity is simply 1, so the limit of equatio (22 as approaches ifiity is lim x x. (23 Lookig at equatios (20 ad (23 together, we fid that lim x = x, (24 ad thus equatio (18, as approaches ifiity, represets the set of all poits (x, y i R 2 such that the maximum of x ad y is equal to a ; i other words, a square cetered o the origi with edges parallel to the axes located a uits from the origi. Refereces: 1. Wilkiso, Numerical Exploratios of Cake Bakig Usig the Noliear Heat Equatio 2. Olszewski, From Bakig a Cake to Solvig the Diffusio Equatio. America Joural of Physics, Jue C. H. Hamilto, A. Rau-Chapli, Compact Hilbert idices: Space-fillig curves for domais with uequal side legths, Iformatio Processig Letters 105 (5 ( www4.csu.edu/~ipse/ma580/lec_25ja.pdf 8

The picture in figure 1.1 helps us to see that the area represents the distance traveled. Figure 1: Area represents distance travelled

The picture in figure 1.1 helps us to see that the area represents the distance traveled. Figure 1: Area represents distance travelled 1 Lecture : Area Area ad distace traveled Approximatig area by rectagles Summatio The area uder a parabola 1.1 Area ad distace Suppose we have the followig iformatio about the velocity of a particle, how