10.07.2015   SonSolar

Example of a receiver surface for a paraboloidal dish

Fig. 3: Paraboloidal dish with surfaces with uniform illumination in front and behind the focal point of the dish, for different concentration ratios. The surfaces are not paraboloidal even though they are close to the paraboloidal shape.

A particular example of the application of the technique is the provision of a uniformly illuminated receiver surface for a paraboloidal solar dish concentrator. In Fig. 3, successive surfaces of different illumination levels in front and behind the focal point may be produced according to the concentration factor desired. Fig. 4 shows a particular case in the range of 500 suns and the distribution of concentration over the receiver surface; variation from the mean is less than 1%. The variation can be even lower if the tolerance of the optimisation process is set lower.

Fig. 4: Close-up of the receiver surface for uniform illumination for a concentration ratio in the range of 500 suns in front of the focal point as shown in Fig. 3, together with the distribution of concentration over the receiver surface along the x-axis of the receiver.

Fig. 5: A paraboloidal dish concentrator with three shapes for a receiver. The outer shape is a parabola, the middle shape the calculated shape as shown in Fig. 3 and the inner shape is a sphere.

Direct solar radiation falling on the reflector in Fig 5 has a given density of ESun. For an ideal system, it can be assumed that immediately after reflection the density of the radiation is still equal to ESun when it intercepts the area A^ish, which is the adjacent

area normal to the direction of the reflected radiation subtended by the area element of the reflector as shown. In the region of the receiver, the same quantity of radiation intercepts the area AReceiver, which is the area normal to the reflected radiation subtended by the area of the receiver illuminated by the beam element. AR, eceiver is smaller than AD, ish due to the concentration property of the reflector. The relation of the radiation flux density on these areas depends on dDish, the distance of the location on the reflector from the focal point. The distance dDish depends on a and ^Receiver, the distance of the corresponding location on the receiver from the focal point, according to the relation:

where C-1eceiver is the radiation flux concentration factor and Eleceiver the density of radiation °ver A^r.

Fig. 5 shows three possible receiver surfaces, a hemispherical surface, a paraboloidal surface and a surface in between these two extremes. For the paraboloidal receiver the relation of dDish and dRecever in Eq. 2 is constant for all a.

But since the area AR, eceiver is not tangential to the paraboloidal this equation does not describe the radiation flux concentration on the paraboloidal receiver. Therefore the illumination on a paraboloidal receiver is not constant. If /3(a) is defined as the angle between the normal at a specific point on the dish given by a and the normal of Aoish, Y(a) is the angle between the normal on the receiver and the normal of AR, eceiver. Therefore the concentration on the receiver is:

d 2

CRecever (a) = cos(Y(a)) (Eq. 3)

Receiver

dLk (a) = (a)

dReceiver (a) AReceiver (a)

(Eq. 2)

1

Receiver

(a)

E

= C1

Receiver

(a)

Sun

E

It can be seen that d1D)ih /dReceiver = const. for a paraboloid while cos(/(a)) is decreasing for the outer regions, while cos(/(a)) = const. for a hemisphere while d2sh /dReceiver is increasing for the outer regions. The desired surface must be therefore somewhere between a paraboloid and a hemisphere, which is the third surface in Fig. 3. In a future paper, we will show how Eq. 3 can be used to derive the function of the surface for even illumination by solving a corresponding differential equation.