*2.3.1 Mathematical model of the heating regime within the working chamber of a solar furnace*

The factors that influence the performances of the solar furnace used in industrial applications that impose the procurement of some large densities of the radiant power and the need of a geometric perfection of the concentrator are classified in three categories as follows:


The available potential power *P <sup>f</sup>* in sun's image from the focal plane of a solar furnace is given by the relation:

$$P\_f = \pi \bullet R\_d \bullet D\_a \bullet E\_0 \bullet f^2 \bullet \sin^2 \theta\_{\text{max}} \tag{1}$$

where:

*Rd* – directional reflection energetic factor of the parabolic mirror (including heliostats, if they exist);

*Da* – transmission energetic factor of the atmosphere where the furnace is installed;

*<sup>E</sup>*<sup>0</sup> <sup>¼</sup> 1353 W/m<sup>2</sup> – (constant);

*f* – focal distance;

*θmax* – the opening angle of the parabolic concentrator.

If a solid corp is disposed in the solar image, the available fraction of potential power effectively absorbed by the corps would be determined by the absorption factor and the form of this corps-receiver. As such, the maximum temperature that could be obtained into a solar furnace depends on the properties of the receiver disposed of in the focal zone of the furnace:

$$T\_{\max} = T\_s \bullet (R\_d \bullet D\_a)^{\natural\_4} \bullet (\sin \theta\_{\max})^{\natural\_2} \tag{2}$$

where *Ts* ¼ 5800 *K* – temperature at sun's surface.

*Assessment of Solar Energy Potential Limits within Solids on Heating-Melting Interval DOI: http://dx.doi.org/10.5772/intechopen.104847*

*2.3.2 The determination of the technological parameters involved in the heating process of a solar furnace*

The most important research is made on the behavior of metals and refractory materials at elevated temperatures, for purifying nonmetallic materials and for the achievement of some thermochemical synthesis.

One of the technological parameters is the temperature that is obtained by focusing on the sun's rays. The furnace uses the temperature in order to melt the metallic material in the crucible, without other complementary energy for the thermal process (**Figure 6**) [5, 18].

Steel or aluminum production needs very high quantities of energy. This is usually given by electric power, natural gases, or conventional fuels. A solar furnace uses the energy given by solar radiation.

We can see in the image how the sun's rays can be focused toward the crucible where the ore is. This is heated to a very high temperature until it melts.

Pollution is basically inexistent because solar energy is a pure form of energy. The melting materials with very high melting points are one of the main applications of solar furnaces.

The material melting on a portion whose area is approximately equal to the area of the sun's image can take place in case the exterior of solid material is exposed to very intense radiation from the focal zone of a solar furnace.

As heat enters the solid, the melted material quantity increases and forms a liquid cavity. Through such a process, it is possible to melt the material in the crucible; this happens because of the existence of a high-temperature gradient between melted material and the crucible's exterior.

In regular furnaces, the crucible is warmed from the exterior, and it has a high temperature continuously than the melted raw material. As a result, the crucible in such furnaces must be made of a material that is more refractory than the substance to be melted, as well as chemically inert toward the melted material.

As the melting point of the examined material rises above 2000°C, the difficulty of achieving these two conditions increases, as there are fewer options to avoid chemical reactions.

Solar furnaces overcome these significant constraints of conventional furnaces when melting materials have high refractivity. As a result, melting can take place in furnaces with a horizontal axis.

The furnace is rotated around its horizontal axis and has an inner diameter several times greater than the diameter of the solar image. When the rotation speed is modest, the melted material stays in the lower part of the furnace, and the turning aids in heat distribution uniformity.

The melted material is centrifuged, generating a cavity that prevents it from flowing out of the furnace at higher rotation rates. The furnace's external walls, which are typically composed of steel, can be water-cooled to maintain (if necessary) a high-temperature gradient through the walls.

When melting into a specific protective environment, a suitable gas current is passed, as shown in **Figure 7**. Quartz, zirconium dioxide, corundum, ceramic oxides, and materials like carbides, nitrides, and boron are among the materials that can be examined. Conventional melting processes have many drawbacks for these materials.

It is also possible to investigate the feasibility of employing solar furnaces for steel melting. Technically, the crucible can be readily made by inserting a refractory powder into the furnace's cavity and sintering or even melting it through the furnace's top that is exposed to solar radiation. After that, scrap iron is inserted, melted, and then molted in forms if necessary [6, 7].

The performance of such a solar furnace does not need to be exceptional because there is sufficient temperature of 2000–2500°C.

Other metals that are more expensive than steel, such as titanium, zirconium, and molybdenum, are expected to generate increased attention in the future.

In this instance, an inert protective atmosphere must be ensured, and the challenges and costs associated with this must be considered.

Impurities evaporation, zonal melting, fractioned crystallization, the separation of zirconium oxide from zirconium (zirconium silicate), and material investigation under thermal shock conditions are some of the other applications of sun furnaces.

**Figure 7.** *Details of melting installation of metallic alloys using solar energy.*

*Assessment of Solar Energy Potential Limits within Solids on Heating-Melting Interval DOI: http://dx.doi.org/10.5772/intechopen.104847*

#### **2.4 The analysis of melting/burning/purifying process within solar furnaces**

These objectives consist in developing some mathematical formalism, which allows a better capitalization of the advantages given by the evolution of melting/burning/ purifying within solar furnaces [14, 16].

Working out of this mathematical formalism was very useful for the documentary stage from Universidad de Las Palmas de Gran Canaria, Spain, as well as for the discussions on this theme with Prof. Agustin Santana Lopez.

### *2.4.1 Mathematical model of the melting/burning/purifying process within the working chamber of a solar furnace*

Thermal efficiency η<sup>t</sup> of an electrothermal installation based on solar energy is given by the ratio:

$$\eta\_t = \frac{Q\_u}{Q\_u + Q\_p + Q\_a} \tag{3}$$


The furnace will be in these working temperature classes that are considered as classification criteria in heating technology:


In order to define the calculus model of the furnace based on solar energy, it is necessary to define the following **input data**:

	- Heating time until reaching solidus temperature;
	- Heating temperature;
	- Overheating time for generalizing liquid state in the entire mass of the charge;
	- Overheating temperature;
	- Holding time at casting temperature necessary for eventual alloying in the liquid state;

#### **Figure 8.**

*Diagrams of possible functioning of the solar furnace: a) melting without holding a constant temperature; b) melting by holding at constant temperature (tc – A complete cycle; t<sup>î</sup> – Heating time; tm – Holding time at constant temperature; t0 – Loading – Unloading time).*


Heat consumption is calculated from functioning diagram of the furnace, that is, in fact, the variation diagram of load temperature–time function θp; taking into consideration the fact that this is a furnace with intermittent functioning, the diagrams presented in **Figure 8** are the possible ones.
