Optical concentration ratio of a parabolic trough collector with flat receiver and concentrator with surface irregularities

Abstract

In a parabolic trough collector (PTC), the geometric concentration ratio (C_g), which is the ratio of the aperture width of the concentrator and the width of the receiver, is generally used for evaluating the solar radiation collected by the flat receiver. This paper argues that this approach is valid only up to a certain width of receiver and that the optical concentration ratio (C_opt) should be used when the receiver exceeds that width. A model for evaluating C_opt has been developed in this work. It is based on tracing a large number of rays, sampled probabilistically using the Latin Hypercube Sampling (LHS) technique. The model also considers the incidence angle of individual rays, and concentrator surface irregularities, which are ignored in the analytical relationships for C_g. For a perfect parabolic surface where $100\leq C_{g}\leq 10$, the C_opt was found to be between ≈80% to 90% of C_g. Also, the concentration ratio was found to be significantly affected by surface irregularities. A useful graphical tool, a set of mathematical relationships and an online program are presented to help designers and researchers evaluate the C_opt by providing the design parameters.

Appendix

In this section, the width of the sun’s image (d_sun) is calculated. Consider that two rays, one from the centre of the sun and the other from the edge of the sun, δ from the centre, are incident at the edge of a truncated parabola (O), as shown in Fig. 10. The rays are reflected to the receiver and are collected at points A and B, which are d_sun∕2  apart. If the law of sines is applied to $\Updelta AOB$ as in Eq. 18, then:

$$\frac{d_{\mathbf{sun}}/\mathbf{2}}{\mathbf{\sin }\boldsymbol{\delta }}=\frac{\boldsymbol{r}}{\mathbf{\sin }\mathbf{\Phi }}$$

(18)

which uses:

$$d_{\mathbf{sun}}=\mathbf{2}\boldsymbol{r}\frac{\mathbf{\sin }\boldsymbol{\delta }}{\mathbf{\sin }\mathbf{\Phi }}$$

(19)

where r is the distance between the centre of the receiver and the edge of the parabola and Φ is the angle at point A.

The value of r can be calculated as the hypotenuse of the right angled $\Updelta BQO$, as shown in Eq. 20:

$$\boldsymbol{r}=\sqrt{\left(\boldsymbol{D}/\mathbf{2}\right)^{\mathbf{2}}+\left(\boldsymbol{f}-\boldsymbol{h}\right)^{\mathbf{2}}}$$

(20)

where h is the height of the truncated parabola, which can be calculated by substituting $y=h$ and $x=D/2$ in Eq. 3 and considering Eq. 8 while simplifying, yielding Eq. 21:

$$\boldsymbol{h}=\frac{\boldsymbol{D}\mathbf{\tan }\left(\frac{\boldsymbol{\psi }}{\mathbf{2}}\right)}{\mathbf{4}}$$

(21)

The value of Φ in Eq. 18 can be calculated by equating the sum of all angles of $\Updelta AOB$ to 180°, given by Eq. 22:

$$\mathbf{\Phi }=\mathbf{90}{^{\circ}}-\left(\boldsymbol{\psi }+\boldsymbol{\delta }\right)$$

(22)

For a typical case when $\psi =45{^{\circ}}$ [23], calculation of d_sun can be simplified to a linear relationship, given in Eq. 23:

$$\boldsymbol{d}_{\mathrm{sun}}=\mathbf{9.469}\times \mathbf{10}^{-\mathbf{3}}\boldsymbol{D}$$

(23)

Fig. 10

Schematic for calculating the width of the sun’s image at the receiver of a parabolic trough collector