### 2.3 Concentration Ratio

The light concentration process is typically characterized by the **concentration ratio** (C). By physical meaning, the concentration ratio is the factor by which the incident energy flux (I_{o}) is optically enhanced on the receiving surface (I_{r}) - see Figure 2.4. So, confining the available energy coming through a chosen aperture to a smaller area on the receiver, we should be able to increase the flux.

${C}_{geo}=\frac{\text{area of the aperture}}{\text{area of the receiver}}=\frac{{A}_{a}}{{A}_{r}}$ |
(2.2) |

In the above equation, C_{geo} is called the **geometric concentration ratio**. It is easy to use, as the areas of the devices are known, although it is adequate only when the radiation flux is uniform over the aperture and over the receiver. Also, please note that for some imaging concentrators, the area of the available receiver surface can be different from the area of the image produced by the concentrator on the receiver. So, if the image does not cover the entire surface of the receiver, we need to use the image area to estimate the concentration ratio.

The concentration ratio can also be represented by the energy flux ratio at the aperture and at the receiver. In this case, it is termed **optical concentration ratio** C_{opt} (or flux concentration ratio) and can be directly applied to thermal calculations.

${C}_{opt}=\frac{\text{Average flux over the receiver}}{\text{Flux over the aperture (insolation)}}=\frac{\frac{1}{{A}_{r}}{\displaystyle \int {I}_{r}D{A}_{r}}}{{I}_{O}}$ |
(2.3) |

In case the ambient energy flux over the aperture (insolation) and over the receiver (irradiance) is uniform, the geometric and optical concentration ratios are equal (C_{geo} = C_{opt}).

The concentration ratios are important metrics used to characterize and rank optical concentrators. Next, we will look at several examples of concentrator designs and see what values of concentration ratios they can provide.

There is a theoretical limit to solar concentration. For circular concentrators - 45,000, and for linear concentrators - 212, based on the geometrical considerations; however, these limits may be unreachable by real systems because of non-idealities and losses. If you are interested in the analytical estimation of the concentration limits, refer to Duffie and Beckman's (2013) book (p.325) for more details.

In general sunlight, concentration systems are roughly classified into: low concentration range (C<10), medium concentration range (10<C<100), and high concentration range (C>100). However, only some of the systems provide uniform concentrated light flux (e.g., V-troughs or pyramidal plane reflectors) and can be characterized by a single concentration ratio. Many systems with curved reflecting surfaces (e.g., conical, parabolic, spherical) create a distribution of flux density over the receiver and would rather be characterized by a variable C over the receiver width. In that case, a **local concentration ratio** (C* _{l}*) is the main parameter to characterize the performance of the ideal concentrator:

${C}_{l}=\frac{\text{Local intensity}}{\text{Incident intensity}}=\frac{I\left(y\right)}{{I}_{ap}}$ |
(2.4) |

where I(y) is determined for any local position y from the center of the produced image, and I_{ap} is the intensity of the incident radiation at the aperture.

In many typical cases of imaging concentrators, the reflectance of the surface (ρ), i.e., the fraction of light radiation reflected from the surface compared to the total incident radiation, is also taken into account. Then the local intensity of the concentrated light, I(y), can be described as follows:

$I\left(y\right)={I}_{ap}\rho {C}_{l}$ |
(2.5) |

Further, in this lesson, we will study some examples that use this equation to estimate energy distribution within a concentrated image on the receiver. It would be better to have a specific type of concentrator to apply these concepts. Please read through the following text to enforce your understanding of the concentration ratios.

### Reading Assignment

**Book chapter**: Duffie, J.A. Beckman, W., *Solar Engineering of Thermal Processes*, Chapter 7: introduction through Section 7.2. pp. 322-327.

The above-referenced sections of the book in part repeat some of the material given here, but may give you more extensive commentary on the basics and probably provide deeper insight how concentration ratio is influenced by other parameters of the system.

After you have completed the above reading assignment, please answer a few self-check questions below.