## 2 Materials and Methods  
### Modeling and Simulating of the Solar Collector  
#### 2.1.1 Modeling  
The modeling of a direct absorption solar collector, such as the one shown in Figure 1 was carried out as in [39; 40]. In the figure, the direction of the fluid flow, the incident solar radiation and the proportional relationship between the dimensions of the solar collector are indicated. For the development of this model, the following assumptions were made:  
**Assumption 1**. : _The flow is assumed to be in a steady state, incompressible, laminar and fully developed in the collector._  
**Assumption 2**. : _Top, bottom and lateral thermal losses are determined as recommended in [41]._  
**Assumption 3**. : _The side walls and the bottom of the collector are considered adiabatic._  
**Assumption 4**. : _Nanofluid properties change with temperature, even with the base fluid._  
The useful heat in a direct absorption collector is calculated as:  
\[\dot{\mathrm{Q}}_{\mathrm{u}}=\mathrm{A}_{\mathrm{c}}\mathrm{F}_{\mathrm{R}}[ \mathrm{I}-\mathrm{U}_{\mathrm{L}}(\mathrm{T}_{\mathrm{i}}-\mathrm{T}_{ \mathrm{a}})] \tag{1}\]  
where \(\mathrm{F}_{\mathrm{R}}\) is evaluated from:  
\[\mathrm{F}_{\mathrm{R}}=\frac{\dot{\mathrm{m}}\mathrm{C}\mathrm{P}_{\mathrm{nf }}(\mathrm{T}_{\mathrm{o}}-\mathrm{T}_{\mathrm{i}})}{\mathrm{A}_{\mathrm{c}}[ \mathrm{I}-\mathrm{U}_{\mathrm{L}}(\mathrm{T}_{\mathrm{i}}-\mathrm{T}_{ \mathrm{a}})]} \tag{2}\]  
In this equation, \(\mathrm{F}_{\mathrm{R}}\) is the removal factor in the direct absorption collector, \(\mathrm{C}\mathrm{P}_{\mathrm{nf}}\) is the specific heat of the nanofluid, \(\mathrm{U}_{\mathrm{L}}\) defines the heat losses, \(\mathrm{I}\) is the solar radiation, \(\mathrm{A}_{\mathrm{c}}\) is the collector area, \(\mathrm{m}\) is the mass flow rate of the fluid, while \(\mathrm{T}_{\mathrm{o}}\), \(\mathrm{T}_{\mathrm{i}}\) and \(\mathrm{T}_{\mathrm{a}}\) are the outlet, inlet and ambient temperatures, respectively. The heat transfer coefficient, \(\mathrm{h}\), is the combination of radiative and convective heat transfer and is determined experimentally from [7]:  
\[\mathrm{U}_{\mathrm{L}}=\mathrm{h}=\ \frac{\dot{\mathrm{Q}}_{\mathrm{sol}}}{ \mathrm{T}_{\mathrm{s}}-\mathrm{T}_{\mathrm{a}}} \tag{3}\]  
where \(\mathrm{T}_{\mathrm{s}}\) is the stagnation temperature, equal to 65 taken from [7]. The steady-state stagnation temperature is measured when the flow of nanofluids with various concentrations in the collector is performed at zero flow rate. To test the collector system, the overall heat transfer coefficient can be varied from 19.2 to 24.3 W/m\({}^{2}\) K.  
Figure 1: Diagram of the direct absorption solar collector. (**a**) Isometric, (**b**) side view. The efficiency of a direct absorption collector is defined by [41]. In this equation, FR is the removal factor in the direct absorption collector, Cpnf is the specific heat of the nanofluid, UL defines the heat losses, I is the solar radiation, Ac is the collector area, m is the mass flow rate of the fluid, while To, Ti and Ta are the outlet, inlet and a  
\[\eta_{\rm i}=F_{\rm R}(\tau\alpha)-F_{\rm R}U_{\rm L}\frac{T_{\rm i}-T_{\rm a} }{I} \tag{4}\]  
where \(\tau\) is the transmittance of the glass and \(\alpha\) is the absorptance of the nanofluid. Since the collector has a reflective background, \(\alpha\) is calculated by [7]:  
\[\alpha=1-\exp(-2(\rm Abs)H) \tag{5}\]  
In Equation (5), Abs is the mean of the absorption coefficient of the working fluid and H is the height of the direct absorption solar collector in cm. The nanoparticles are normally metallic or metal oxide in nature, while the base fluids are usually water, oils and ethylene glycol. The general objective of nanofluid development is to improve the properties of the base fluid with the minimum addition of nanoparticles, normally with a volume fraction (f\({}_{\rm v}\)) of less than 1% of the base fluid.