AP m H THEORETICAL ANALYSIS FRESNEL LEN. I l l

Transcription

1 I l l AP m H FRESNEL LEN THEORETICAL ANALYSIS

2 31 CHAPTER 2 FRESNEL LENS : THEORETICAL ANALYSIS Solar thermal systems for intermediate temperature range (100 C C) essentially use solar concentrators. The Parabolic Trough and Fresnel Lens Concentrators are best suited for medium temperature applications. Parabolic Trough solar concentrators are mostly used for power generation using low pressure steam. Fresnel Lens concentrators are less developed and are mostly used for photovoltaic applications. The main objective of work presented here is to develop refractive type Fresnel Lens Solar Concentrator for generating low pressure steam. In the following sections theoretical considerations for Fresnel Lens Solar Concentrator are discussed in brief. 2.1 Principle of Fresnel Lens There are two types of Fresnel Lenses viz ' Refractive Lens' and 'Reftective Mirrors'. The Refractive Fresnel Lenses are mostly used in the photovoltaic applications where as Reflective Mirrors are used in the photothermal applications. Fresnel Lenses are more flexible as far as optical designs are considered. And can produce uniform flux density on the absorber. Figure 2.1 shows the schematic view of Refractive Fresnel Lens and Reflecting Mirror Fresnel Concentrator. a. Refractive Fresnel Lens

3 32 b. Reflective Fresnel Lens Figure 2.1: Schematics of refractive and reflecting Fresnel Lens The Fresnel Lenses are also classified as 'imaging' and 'non-imaging lenses. Nonimaging lenses does not produce image of the light source. Instead it is designed to concentrate radiation at a density as high as theoretically possible. Imaging lenses in contrast form images of the source on the absorber. Here we are concerned more with the nonimaging Fresnel Lens for generating process heat. The cross-sections of a Fresnel Lens are shown in the figure 2.2. There are two types of Fresnel Lens depending on the groove positioning. : either grooves in or grooves out designs. Each groove in a lens is equivalent to a prism extended along the length of the lens. If grooves are co-centric then a point image of the sun is formed at the focal point. The angle of prism (groove) is adjusted in such a way that the sun's ray after refraction by each prism is brought to focus at the focal line.

4 33 snnnn A^Al^i^^^J#l\ a. Grooves-in Fresnel Lens b. Grooves-out Fresnel Lens c. Point Focus Fresnel lens d. Line Focus Fresnel Lens Figure 2.2: Various types of Fresnel Lenses CHAPTER 2 : FRESNEL LENS: THEORETICAL ANALYSIS

5 34 focal point. Figure 2.3 shows the ray diagram which explains how the rays are focused at the Figure 2.3: Focusing property offresnel Lens schematically shown for refracting Fresnel Lens

6 35 The Fresnel Lens follows the same principle of geometrical optics used for other lenses. The imaging Fresnel Lens can be considered as three dimensional lens while nonimaging Fresnel Lenses are treated as two dimensional lenses. The focal length and the aperture are the two main parameters of the lens. The focal length of the plano-convex Fresnel Lens is given by usual lens formula i.e.: = V (2.0 Where, f is the focal length, i is the distance of image from the centre of the lens and o is the distance of object (sun here) from the lens. Since o = oo, f = i i.e. the image of the sun is formed as the focal point at a distance off. The f /number is a measure of the aperture of the lens and is a ratio of the focal length (f) to diameter of aperture of lens (for imaging) or width of the lens (for non-imaging lens), i.e. f f I number = 2/? for imaging lenses f - for non-imaging lenses (2.2) where, R is the distance of the extreme paraxial ray from the optical axis of the system. In our case, 2 R will be width of the Fesnel lens. Smaller f/number indicates larger aperture area. Fresnel lenses are mostly free from spherical aberrations since each groove/ prism is designed separately for focusing the rays. Since they are thin, absorption losses over the lens profile are small [1,2]. 2.2 Grooves -in Fresnel Lens The most important parameter in Fresnel Lens design is the prism angle a. Figure 2.4 shows simple Fresnel Lens with grooves facing inwards. With reference to figure 2.4, three equations to describe the lens are written as [1,3] n sin a = sin p (2.3) tan*y = (2.4) / p = a + co (2.5) Substituting value of P in above equation we get, CHAPTER 2 : FRESNEL LENS: THEORETICAL ANALYSIS

7 36 n sin a = sin (a + co) n sin a =sin a cos co + cos a sin co From above equation we get, sin ft; tanar = - n - cos co (2.6) (2.7) We have, R sm co- cos co f R cos co Hence, tan a - f n- cos co Since cos<2> = (2.8) tan a = R W* 2 +/ 2 )- / (2.9) R vnknnnnn^ Figure 2.4: Fresnel Lens with grooves-in

8 37 From equation (2.9) we can derive prism angle of each step assuming certain focal length and f / number. 2.3 Grooves-out Fresnel Lens The cross section of Fresnel Lens having grooves out is shown in the figure (2.5). Figure 2.5: Fresnel Lens with grooves-out A set of four equations can be derived from figure (2.5), viz. sin a = n sin p - a = y- p n sin y = sin co R tan co- f (2.10) (2.11) (2.12) (2.13) CHAPTER 2 : FRESNEL LENS: THEORETICAL ANALYSIS

9 38 The paraxial rays in case of Grooves out fresnel lens are refracted twice. From above equation we get, sin a = n sin (y+ a) sin a = n (sin y cos a + cos y sin a) nsiny... tan or = (2.14) 1 - n cos y Since, n sin y = sin co and n 2 (l-cos 2 y) = sin 2 oo we get, n - cos y = (n 2 -sin 2 oo) 1/2 (2.15) Substituting above values in equation (2.14) we get, tanor = (2.16) l-(n 2 -sin 2 6» 1/2 with sine; = (2.17) (R 2 +f 2 r 2 The above two equations enables us to design the Fresnel Lens with grooves out [1,3,7]. 2.4 Arched Fresnel Lenses Fresnel lenses can be shaped into dome-form (three dimensional) or arched form (two dimensional). In these shapes, prisms are arranged along a semi circular surface around the focal point. The focal length is always kept constant with reference to prism location along the circle. A shaped Fresnel Lens has following advantages 1. the shaped Fresnel Lens has more mechanical strength and stability. 2. the shaped Fresnel Lens reduces focal aberrations. 3. there is no sagging of fresnel Lens when mounted on the frame. 4. thickness of lens could be small without loosing mechanical strength. However these lenses are difficult to manufacture due to complicated processes and requirement of higher accuracy. Similarly, there will be more reflection losses due to curved surface. To find the angle of prism for shaped Fresnel lens, consider a prism of right hand side as shown in the figure

10 39 Figure 2.6: Arched Fresnel lens The focal length f» d where d is the path of ray in the lens. From figure (2.6), following two equations can be derived. sin a = n sin(a-) (2.18) n sin (P + y) = sin (p + co) (2.19) Similarly, a = co From equations (2.18) and (2.19) sin co = n sin (co - y) r sin (O^ fy-^=arcsin (2.20) V n ) f sin co^ y= <y-arcsin (2.21) V n J sin(/? + co) (2.22) n = sm(p+y) n +1 _ sin(/? + co) + sin(/? + y) n -1 sin(/7 + CO) - sin(/7 + y).. x + y x-y sin x + sin y = 2sin -cos x+y. x-y sinx-siny = 2cos -sin (2.23) (2.24) (2.25) CHAPTER 2 : FRESNEL LENS: THEORETICAL ANALYSIS

11 40 n + \ ^7 2 sin 2 cos 2/3 + co+y cos 2/3 + co+y" sin co-y co-y (2.26) i - tan fi + co+y \ cot co-y P = arctan n + \ tan 'co-y^ co-y (2.27) (2.28) The above equation enables us to find the prism inclination angle P for each step. Similarly the deviation 5 of beam is given by, sin<y 5 = a + (3 = arctan f sm. co ncos arcsin 1 v. \ n ) J ) Using above equations, one can design Fresnel lens for given focal length and aperture [1]. 2.5 Concentration Ratio for Fresnel Lens Imaging and Non-imaging Fresnel Concentrators have following components; 1. Fresnel Lens to concentrate solar radiation on an absorber 2. Absorber for converting solar radiation into heat. 3. Enclosure for absorber. 4. Heat Removal System (2.29) Figure (1.7) shows the schematic view of imaging (a) and (b) non-imaging Fresnel Lens Solar Concentrator. Solar Concentrators are characterized by Geometrical Concentration Ratio C and Optical Concentration Ratio n c, which are given by, Th U^Q-4 C = ^L 5 7 and n (2.30) (2.31)

12 41 on the absorber surface. The energy gain from the system depends on the radiation received by the absorber. Therefore the optical concentration ratio n c is defined as, c = Flux density on the receiver (absorber) (2.32) Flux density received at the aperture <p x IS x If the concentrator is a perfect concentrator in terms of optical losses, i.e. r\= 1, the optical concentration ratio and the geometric concentration ratio are identical: rj c = C. Since non-imaging Fresnel Concentrator is assumed to be two dimensional, The geometrical concentration ratio = R/W (2.34) Where R is the width of aperture and W is the width of absorber. The maximum linear concentration for two- dimensional concentrator is, Q Dm ax=-rv C ) sm 0 S For three-dimensional concentrator, concentration happens along both y axis and x axis and the third axis of the coordinate system yields a maximum theoretical concentration ratio in air of, Qzw = h~ (2.35.2) sin 0 S The refractive index of the materials involved is often n =1 for air or vacuum. The maximum concentration of solar image on the absorber of a two dimensional concentrator is, C 2Dmax = * 212 (2.35.3) sin# v For three dimensional concentrators, C = _J * 45 ooo (2.35.4) ^ 3D max. 2 n ~ H -'> U ««For two-dimensional concentrators, the maximum temperature of an absorber under the sun, once the real concentration ratio C, is known can be given by, r* 1M,=7,,J^ (2.36) V max Where, T s is the surface temperature of the sun. It is assumed that there are no heat losses from the absorber [1]. 2.6 Flux Density on the Absorber The distribution of flux on the absorber surface is important for designing the receiver. The flux density distribution in the focal plane depends on the optical v ' CHAPTER - 2: FRESNEL LENS: THEORETICAL ANALYSIS

13 42 concentration ratio of the lens, the size of the solar disc, brightness distribution and spectral distribution of incident solar radiation. The focal plane for non-imaging Fresnel Lens is defined as a plane parallel to aperture plane and passing through the focal line. By following ray tracing method we can get flux distribution of incident solar radiation focused on focal plane. Grilikhes [4] has given a highly mathematical model for determining flux distribution in a image formed by concentrator. However, to apply such simulation process for nonimaging Fresnel Lens is very difficult due to their shapes. The flux density on the absorber of non-imaging Fresnel Lens can be determined by considering the shape and size of image of sun formed by the lens on the focal plane. The formation of image of sun by using ray tracing method is shown in the figure (2.7). pooaood Ax Wo Image formed by central element C30 Z$30GI~1I~^) Image formed by last prism w R Figure 2.7: Formation of image by Fresnel Lens

14 Let us consider rays falling on the aperture plane along the normal direction. Consider the image formed by ray (1) shown in figure (2.8) falling on the aperture plane. The shape of the image formed by this ray will be circular. The size of the image will be given by, W,=2ftan0 s (2.37) Similarly the images formed by the rays (2) and (3) will be circular one having same size but A shifted towards left or right by a distance. The effective width of image will be given by, Wo = A x + 2f tan 0 S (2.38) Now consider the prism located at the end of the Fresnel Lens i.e. near the edge of the Fresnel Lens. The image of the sun formed by the ray (1)' will be at the centre of the image i.e. focal point. The width of the image will be given by, Wr = 2psin0.tan0 s (2.39) The shape of the image will be elliptical with major axis WV and minor axis 2ptan0 s. The images formed by rays (2)' and (3)'will have same shape but shifted to right / left side A of central image by distance -. Therefore, the width of theimage will be, W R = A X + 2 p sin 0. tan 0 S (2.40) The flux at the centre of the image will be maximum. There will be hot spot at the centre having maximum flux over the distance (A x + 2f tan 0 S ). The flux density will reduce as we go away from the centre and will be zero beyond image of the sun. Therefore flux distribution of nonimaging Fresnel Lens will have form as shown in the figure (2.9). The curve A shows the ideal flux distribution for nonimaging Fresnel Lens. It is impossible to make Fresnel Lens without any error. Deviation from the ideal system due to optical aberrations will produce an image of larger size. Therefore, in reality, flux distribution will be as shown in the curves B or C depending on perfectness of lens. We have tried to measure the flux distribution by measuring stagnation temperature (Chapter 4) from which we designed the absorber. 43

15 44 Theoretical / ^ Q " J) 50-0 \ y Practical The intercept factor y is defined as Figure 2.9: Flux distribution across the focal line \I(x).dx \I(x)dx (2.41) Where I(x)is flux density at x and AB is the width of the absorber. If we know the flux distribution, we can optimize the intercept factor for maximum collection of solar radiation [5,6,8]. 2.7 Tracking of Fresnel Lens Concentrator Tracking of the sun is essential for solar concentrators for collecting maximum energy from the sun. Ideally the optical axis of the concentrator should be along the direction of the sun. Therefore two axis tracking system will allow the system to orient optical axis along the direction of sun rays. However such tracking system will be very costly. There are other tracking systems which keep minimum angle between sun's rays and

16 45 optical axis of the concentrator. These systems are simple and economical. A linear Fresnel Lens can be tracked by using four different ways. These are shown in the figure (2.10). A linear fresnel Lens Concentrator can be oriented with its line of focus either along East-West direction or along North-South direction. The four different ways of tracking are: 1. Fresnel Lens is mounted on a stand which can be rotated along two axis (figure a). the sun is tracked by rotating Fresnel Lens about these two axis to align optical axis along the sun rays. 2. The Fresnel lens is mounted along East-west direction and tracked by rotating about the line of focus (figure 2.10-b). 3. The Fresnel Lens is mounted along North-South direction and inclined to a certain angle so as to minimize angle between sun's rays and optical axis and tracked above the line of focus (figure 2.10-c). 4. The Fresnel Lens is mounted along North-South direction with focal axis parallel to the horizontal direction (figure 2.10-d). The lens is rotated about the line of focus for tracking the sun. Apperturs Plane A1/A2: Axis of Rotation 2.10 (a)

17 46 w A1 Axis of Rotation 2.10 (b) 2.10 (c) A 1 : Axis of Rotation

18 47 z 2.10 (d) Figure 2.10 : Different methods of Tracking the Sun for Linear Fresnel Lens We have used the fourth choice for tracking the sun. The focal line is set along the North-South direction and kept horizontal. The collector is rotated about a horizontal North- South axis and sun is tracked continuously so that the solar beam makes a minimum angle of incidence with the aperture plane at all the times. The absorber tube is mounted along the line of focus and is stationary. The slope (3 is the angle made by the plane surface with the horizontal. It can vary from 0 to 180. It can be shown as, cos 9 = sin cp ( sin 8 cos p + cos 5 cos y cos to sin P) + cos cp ( cos 5 cos co cos (3- sin 8 cos y sin P) + cos 8 sin y sin co sin P (2.42) In order to find the condition to be satisfied for 8 to be minimum, we differentiate the right hand side of the resulting equation with respect to P and equate it to zero. Thus, we get, tan (<p - P) = [ tan 8 / cos co] for y = 0 (2.43) and tan(cp +P) = [tan 8/cosco] fory= 180 (2.44) Equations (2.43) and (2.44) can be used for finding the slope of the aperture plane. Equation (2.43) corresponding to y = 0 is used if the magnitude of the solar azimuth angle y s is less than 90 while equation (2.44) corresponding to y = 180 is used if the magnitude of the solar azimuth angle is greater than 90. The expression for the corresponding

19 48 minimum angle of incidence is obtained by combining equations (2.43), (2.44), and (2.42) [6]. For both the cases we obtain, cos 9 = ( 1 - cos 2 5 sin 2 co) 1/2 (2.45) 2.8 Design criteria for Fresnel Lens Concentrator While designing Fresnel Lens Concentrator, following parameters are considered: Operating Temperature Requirement Fresnel Lens Concentrators are used for low and medium temperature applications. Once the operating temperature requirement has been finalized, concentration ratio for the system can be decided Solar Resource Availability Nonimaging Fresnel Lens can only concentrate the direct fraction of solar radiation. Solar resource assessments are necessary to evaluate the suitability of locations for the installation of solar concentrating collector and to estimate the potential for energy conversion. Hence it is important to design the system according to a minimum requirement of solar radiation for the total energy need of the system. This will determine the size and quality of the system to be built Lens Design According to the requirement, reflecting or refracting type lens is to be selected and accordingly design of lens, material for fabrication of lens, fabrication techniques or procurement of ready lens are to decided Absorber Design Proper absorber design is important to reduce convective and radiative heat losses. Since the shape of absorber is not flat, heat pipes or fluid operation pipes have to be installed Tracking Arrangement Concentrators of medium concentration ratio have to be equipped with tracking around at least one axis. Design and fabrication of Fresnel Lens Concentrator System is discussed in detail in Chapter 3. CHAPTER 2 : FRESNEL LENS: THEORETICAL ANALYSIS

20 : References 1. R. Leutz, A. Suzuki, "Nonimaging Fresnel Lenses, Design and Performance of Solar Concentrators", Published by Springer series in optical sciences, " Fresnel Technologies", lenses, E.V. Tver'yanovich, " Profiles of solar engineering Fresnel Lenses", Applied Solar Energy vol. 19, pages (1984). 4. V.A. Grilikhes, " Transfer and distribution of radiant energy in concentration systems", Photovoltaic conversion of concentrated sunlight, Wiley Publication (1997). 5. H.P.Garg, J.Prakash, "Solar energy Fundamentals and applications", Tata McGraw-Hill Publishing Company Limited, New Delhi.(1997). 6. Sukhatme S.P., ' Solar Energy, Principles of Thermal Collection and Storage'. Tata McGraw-Hill Publishing Company Limited, New Delhi.(1996). 7. B.N Patil,. 'Design, Fabrication and Testing of a Cylindrical Fresnel Lens', M.Phil Project Report submitted to University of Pune, J.A. Duffie, W.A. Beckman, " Solar engineering of thermal process", John Wiley, New York, 1991.