The Citical Angle and Pecent Eiciency o Paabolic Sola Cookes Aiel Chen Abstact: The paabola is commonly used as the cuve o sola cookes because o its ability to elect incoming light with an incoming angle o 90 degees to a single ocus point. Howeve, because o this vey speciic popety, the paabolic sola cooke must constantly be ealigned with the obiting sun o maximum eectiveness. Sola cookes concentate light to a lage suace, so it is slightly moe toleant o maginal eos o alignment. Howeve, to what point ae the elected ays no longe toleant, and stat to miss the taget? How http://www.blog.thesietch.og/wp content/uploads/2007/02/vaja 2006 1.JPG quickly do the est o the elected ays miss? Is thee an optimal ocal height o taget size that will always be moe toleant o eos in alignment? Using Geomety Expessions, a constaint based symbolic geomety system, in conjunction with Maple, a compute algeba system, these situations wee modeled and investigated. We ound that the sola concentato always has the lagest toleance o eo when the ocal height is a outh o the diamete, o level with the boundaies o the concentato. The pecentage o light elected to the taget is at 100% o a small ange o angles, and pomptly completely misses the taget at angles outside this ange. This miss, called the spill ate, is essential o compaing the paabola with othe cuves. Although thee may not be anothe cuve moe accuate than the paabola when peectly aligned with the light souce, thee may be anothe cuve that is moe accuate at wide angles. Intoduction: Sola cookes have become popula aound the wold because they heat and cook without consuming uels which ae not always eadily available. They can each highe tempeatues than the suounding atmosphee because sola cookes concentate a lage amount o light to a elatively smalle absobing suace. The aea that sunlight is collected om is called the apetue aea. Beoe how sola cookes concentate light is discussed, the law o election must ist be undestood. The diagam [1] below illustates how a single ay o light elects o a lat mio. The incoming ay, called the incident ay, hits the mio at the incidence point. At the incident point, a line pependicula to the mio, the nomal, acts as a line o symmety o the incident ay and the elected ay. The angle between the elected ay and the nomal, the angle o election, and the angle between the incident ay and the nomal, the angle o incidence, ae always identical. 1
Nomal Angle o Incidence Angle o Relection Incident Ray Relected Ray Relective Suace [1] Fo cuved suaces, the nomal is pependicula to the tangent at the incidence point shown in the diagam [2] below. Sola cookes come in all dieent shapes and sizes, but because o the paabola s ability to elect light onto a single ocus point, paabolic cookes ae commonly used. Howeve, this special popety is only useul when light has an incoming angle o 90 o. The elected ays completely miss the ocal point at any othe incoming angle. Because sola cookes ae concentating light to a suace (the pot), athe than a single ocus point, the sola cooke can still unction at angles slightly less than 90 o. In the model [3] ceated with Geomety Expessions, we examine the popeties o the elected light ays in a 2D view. The paabola is given the geneal equation o, whee is the ocal height. The ed line epesents the indeinite numbe o incoming ays, the oange lines epesent all the elected ays, Θ epesents the incoming angle o light, and the cicle epesents a pot with adius. [2] 2
8 6 4 2 X 2 =4 Y [3] This simulates eal wold paabolic sola concentatos such as the ones shown below. http://digitaljouno.wodpess.com/2008/03/14/sun ie and cooking/ http://www.ecowold.com/uels/sola themal powe.html Poblem Finding the Citical Angle: Because the position o the sun in the sky is constantly changing, we ae inteested in detemining the ocal height and adius that will collect the most elected ays with the most angle toleance. I a sola cooke has a lage angle toleance, it means that it is able to elect light onto the absobing suace at exteme o smalle incoming angles. Fo a geneic model [4] the apetue length o the sola cooke is held constant at 1 while the adius () and ocal height () ae intepeted as atios o the diamete. 3
0.4-1 2, 1 16 0.2 1 2, 1 16-0.4-0.2 0.2 0.4 [4] In this paticula situation [4], all the elected light ays ae hitting the pot. But at what angle Θ do the light ays stat missing the pot? Geomety Expessions calculates the citical incoming angle that light stats missing the pot as. This citical angle is the widest incoming angle o a given and and an apetue length o 1 that concentates all the elected ays to the pot with adius. 0.4 0.2-0.6-0.4-0.2 0.2 0.4 0.6 actan 1+32 2 +256 4-256 2 2 16-0.2 [5] To uthe undestand the meaning o this omula, the angle omula was gaphed using Maple. The gaph o the omula and a coss section at whee =0.12 is shown: 4
[6] [7] These gaphs [6],[7] show the citical angle o a given and whee the elected ays stat to miss the pot. Because ou goal is to detemine the combination that will have the lagest angle toleance, the smalle the citical angle, the moe angle toleant the sola cooke. As visible om the gaph [6], the sola cooke becomes moe angle toleant as the pot size () inceases. This is expected because as inceases, the elected ays have a lage absobing aea to hit. Howeve, having a lage adius means that thee will be a smalle concentation atio (apetue aea to absobing aea atio). Fom obseving the gaph, it becomes appaent that thee is always a ocal height that will have the minimal citical angle. To solve o the minimum, the angle omula was dieentiated and solved o in Maple: > actan((((1+(()^(2)*32)+(()^(4)*256)+(()^(2)*()^(2)*(-256))))^(1/2)*()^((-1))*()^((- 1))*1/16)); > di(%,); actan 1+ 32 2 + 256 4 256 2 2 16 > simpliy(%); > solve(%,); 64 + 1024 3 512 2 32 1 + 32 2 + 256 4 256 2 2 1 + 1 + 32 2 + 256 4 256 2 2 256 2 2 1 + 32 2 + 256 4 256 2 2 16 2 16 ( 16 2 1 ) ( 1 + 16 2 ) 1 + 32 2 + 256 4 256 2 2 5
1, 4 Fom the calculations, the optimal ocal height is always when the ocal height () is a outh o the length o the diamete, egadless o the taget size. This is also when the ocal height is level with the boundaies o the sola cooke. -1 4 Poblem: Spill Rates Peviously, we investigated the citical angle that elects all ays within a diamete atio o 1 to a pot with adius and ocal height. Ate detemining the bounday that light stats spilling ove, it led to the question o how quickly the est o the light misses the pot as the incoming angle deceases. 0.4 A 0.2 0.6 z -0.4-0.2 0 X 2-4 Y =0 (0,0) t 0.2 0.4 0.6 [8] Using the above diagam [8] in conjunction with Maple, the pecentage o light in the sola cooke with diamete atio o 1 that hits the pot o a given pot adius and incoming angle was expessed in the piecewise unction [9] below. Results ae pesented with the optical ocal height. [9] The omula is gaphed in 3D below [10]. The diagam [11] at the ight is coss sections o the 3d gaph at speciic values o (gay: 0.25, geen: 0.20, blue: 0.15, black: 0.10, gay: 0.05). 6
[10] [11] When the adius o the pot is smalle, the pecent eiciency deceases at a aste ate. When the adius is lage, the pecent eiciency deceases at a slowe ate. Now that we know how quickly light spills out o the pot in a paabolic sola cooke, the next step is to ind a cuve that may not be at peect as the paabola as incoming angles aound 90 degees, but can hold moe light in the pot at smalle incoming angle than the paabola because light spills out elatively quickly o a paabolic sola cooke. Conclusion: The sola concentato is always most angle toleant when the ocal height is one outh o the the apetue length egadless o the taget size. When the sola cooke is positioned this way, it will be able to concentate all light to the taget at the smallest incoming o widest angle possible o a given diamete and taget. Thee isn t, howeve, an optimal taget size. The lage the taget, the moe angle toleant the sola concentato is. Howeve, the lage the taget, the smalle the concentation atio is. The tade o between angle toleance and sola concentation depends on the taget size. To impove one, the othe needs to be saciiced. It is impossible to maximize both. The esults om gaphing the eiciency o the concentato at angles beyond its citical angle shows how the eiciency deceases vey dastically with smalle tagets, while lage tagets ae eicient at smalle incoming angles. The gaph s applicability will be vital when compaing the paabola with othe cuves. Although thee may not be anothe cuve as peect as the paabola when peectly aligned, thee may be cuves that will have bette eiciency at smalle incoming and wide angles. This mateial is based upon wok suppoted by the National Science Foundation unde Gant No. 0750028. Any opinions, indings, and conclusions o ecommendations expessed in this mateial ae those o the autho(s) and do not necessaily elect the views o the National Science Foundation. 7