Global Journal of Researches in Engineering, J: General Engineering, Volume 22 Issue 1

and a fixed point are equal, where the fixed line is called the directrix and the fixed point F, the focus. The length FR equals the length RD. The line perpendicular to the directrix and passing through the focus F is called the axis of the parabola. The parabola intersects its axis at a point V called the vertex, which is exactly midway between the focus and the directrix [25]. Fig. 3.1: The Parabola If the origin is taken at the vertex V and the x- axis along the axis of the parabola, the equation of the parabola is given by: y2 = 4fx (1) y2 = 4f(x-f) (2) In polar coordinates, using the usual definition of r as the distance from the origin and θ the angle from the x-axis to r, we have for a parabola with its vertex at the origin and symmetrical about the x-axis; (3) Usually, in solar studies, it is more useful to define the parabolic curve with the origin at F and in terms of the angle ( ψ ) in polar coordinates with the origin at F. The angle ψ is measured from the line VF and the parabolic radius p, is the distance from the focus F to the curve. Shifting the origin to the focus F, [26&27] describe it as: (4) The parabolic shape is widely used as the reflecting surface for concentrating solar collectors because it has the property that, for any line parallel to the axis of the parabola, the angle p between it and the surface normal is equal to the angle between the normal and a line to the focal point. Since solar radiation arrives at the earth in essentially parallel rays and by Snell's law the angle of reflection equals the angle of incidence, all radiation parallel to the axis of the parabola will be reflected to a single point F, which is the focus and then the following is true: (5) ψ is the angle of reflection and P is distance RF. The general expressions given so far for the parabola define a curve infinite in extent. Solar concentrators use a truncated portion of this curve. The extent of this truncation is usually defined in terms of the rim angle ( ψ rim) or f/d which represents the ratio of the focal length (f) to diameter of dish (d). The scale (size) of the curve is then specified in terms of a linear dimension such as the aperture diameter d or the focal length f. This is readily apparent in fig 3.2 below which shows various finite parabola having a common focus and the same aperture diameter: Fig. 3.2: Segments of a parabola having a common focus F and the same aperture diameter. (6) In a like manner, the rim angle ( ψ rim ) may be found in terms of the parabola dimensions (7) Another property of the parabola that may be of use in understanding solar concentrator design is the arc lengths. This may be found for a particular parabola. From Equation (7) by integrating a differential segment of this curve and applying the limits x = h and y = d/2. L ocating The F ocal P oint Of The D ish In locating the focal point of the dish, two methods are often used: Manual construction which entails finding the focal point by placing the receiver on the approximate or assumption point till the right point of Analysis of Thermal and Optical Efficiency of Parabolic Concentrating System for Thermal Application lobal Journal of Researches in Engineering ( ) Volume XxXII Issue I Version I J G 53 Year 2022 © 2022 Global Journals VII. where x = h i.e. the dept while f is the focal length between the vertex and focus, when the origin is shifted to the focus F as is often done in optical studies, with the vertex to the left of the origin, the equation becomes;