Exergetic Optimization of a Parabolic Trough Solar Collector

Abstract

In this chapter, detailed thermal and optical models for PTSCs were formed. The purpose of the model is to determine the collector losses in PTSCs and thus to obtain the collector efficiency. A parametric study was conducted to assess the effect of some key design and operating parameters on the performance of the PTSC. An exergy analysis for PTSC is conducted to obtain the exergetic efficiency of PTSCs and to find the exergy destruction of the PTSCs. In addition, an optimization study using Taguchi method was applied to find the design parameters that give the maximum exergetic efficiency of the PTSC. The results show that when the solar radiation and aperture width increase, the exergetic efficiency increases for any heat transfer fluid used. On the other hand, when the outer diameter and wind speed increase, the exergetic efficiency decreases. In addition, Taguchi results show that the exergetic efficiency gets its maximum value (50.19%).

1 Introduction

Parabolic trough solar collectors (PTSCs) are one of the most promising technologies among the different types of solar collectors (e.g., flat plate, evacuated tube, solar dishes) [1]. PTSCs are widely used in many engineering applications such as heating buildings, hybridization with geothermal power plants, and other types of energy systems. The first usage of PTSC in power plant was in 1984 in California [2]. Since then, its usage in power plants has increased significantly. The schematics of the PTSC and its components are shown in Fig. 48.1. As can be seen in this figure, the collector is made of a reflective material sheet which has a form of parabola. The receiver is a metal tube, which is surrounded by a glass cover. Glass cover reduces the heat losses. Collector performance is significantly affected by reflectivity, absorptivity, operating conditions of the heat transfer fluid, tracking mechanism, and so on [3].

Fig. 48.1

The schematics of PTSC and its layers

There are several studies on the modeling of PTSCs. Manikandan et al. [4] conducted a thermal model for PTSC and obtained the thermal performance of the PTSC. In addition, a parametric study was conducted using the following parameters : mass flow rate, concentration ratio, heat transfer fluid (HTF) , heat removal factor, and the solar radiation. The results of this study showed that when the inlet temperature of PTSC fluid increases, the efficiency of the collector decreases for various mass flow rates and different HTF. In addition, when collector temperature increases, radiation losses increase. Qu et al. [5] programed a single dimensional model by using EES and considered the effect of solar intensity, incident angle, collector dimensions, and fluid properties on the performance of the PTSC. The model is validated by the experimental data. After that, the model is used to improve PTSC design and to optimize system operation. Yılmaz and Söylemez [6] developed solar, optical, and thermal models by differential and non-algebraic correlations . Differential equations were solved simultaneously using EES. The developed model was compared to the experimental data. The results showed that the optical loss is the most effective parameter on the collector efficiency. From a theoretical viewpoint, the best suggestion to reduce the heat loss was given as vacuuming the annulus. However, when the vacuuming is used, maintenance is more difficult. As a result of this study, using less-conductive fluid to decrease heat losses through the annulus was recommended. The required pumping power increases when mass flow rate increases; hence, for more efficient PTSC, the optimal range of mass flow rate should be considered. Padilla et al. [7] developed a one-dimensional numerical heat transfer model of a PTSC. To analyze the receiver and glass envelope by using mass and energy balances, both of these components were divided into segments. Their numerical model compared well with experimental data. Kalogirou [8] proposed a thermal model of the PTSC and solved it using EES . This model was compared with the known performance of existing collectors, which were tested in Sandia National Laboratories. Finally, the model is used for installation of the collector at the Archimedes Solar Energy Laboratory of the Cyprus University of Technology. Conrado et al. [9] presented a review of thermal and mathematical models for components of PTSC. Abbas et al. [1] developed a model using Monte Carlo ray-trace code to compare PTSCs and linear Fresnel collectors in terms of energy effectiveness and flux intensity map at the receiver for different days and orientations. Their results showed that the maximum efficiency is obtained from PTSCs when the linear Fresnel collectors have multitube receivers. In addition, NS orientation ensures higher efficiency for PTSCs, but orientation type has not affected the linear Fresnel collectors’ efficiency.

Literature survey conducted shows that there is very limited study on the detailed optical and thermal models of PTSC and its optimization. In the first part of this study, detailed thermal and optical models are first developed; then an exergy analysis is conducted; and finally parametric studies were conducted. The effect of some key design and operating parameters, which are the outer diameter of receiver, the aperture width, the wind speed, the type of heat transfer fluid, and the solar radiation, on the performance of the PTSC is assessed. The performance parameters are the outlet temperature of receiver and the exergetic efficiency of the PTSC. In the second part of this paper, an optimization study was carried out using Taguchi method to find the optimum design parameters that maximize the exergetic efficiency of the PTSC.

2 Modeling

In this section, modeling equations and approach for PTSCs are presented. The operating principles of PTSCs are as follows. When the solar rays reach the parabolic mirror, these rays are focused onto the receiver tube. The concentrated radiation heats the heat transfer fluid; in this way, the solar radiation transforms to useful energy. When the solar beams reach to the receiver from the mirror, optical and thermal losses are observed. These optical and thermal losses are calculated as discussed in the following subsections.

2.1 Optical Model

The aim of the optical model is to find the optical efficiency of the PTSC. Therefore, the heat absorbed by the receiver should be first found by using Eq. (48.1) [8].

$$ S={G}_{\mathrm{b}}\cdotp {\eta}_{\mathrm{r}}\kern0.75em $$

(48.1)

The optical efficiency for PTSC is given by Eq. (48.2).

$$ {\eta}_{\mathrm{r}}=\rho \cdotp \gamma \cdotp \tau \cdotp \alpha \cdotp \left(\left(1-{A}_{\mathrm{f}}\cdotp \tan \left(\varTheta \right)\right)\cdotp \cos \left(\varTheta \right)\right) $$

(48.2)

where 𝐴_𝑓 is geometric factor and is given in Eq. (48.3).

$$ {A}_{\mathrm{f}}=\frac{A_{\mathrm{loss}}}{A_{\mathrm{a}}} $$

(48.3)

The geometric factor is caused due to the shaded area of the PTSC. Total shaded area can be obtained from Eq. (48.4).

$$ {\mathrm{A}}_{\mathrm{loss}}=\frac{2}{3}{W}_{\mathrm{a}}\cdotp {h}_{\mathrm{p}}+f\cdotp {W}_{\mathrm{a}}\cdotp \left[1+\frac{{W_{\mathrm{a}}}^2}{48\cdotp {f}^2}\right] $$

(48.4)

2.2 Thermal Model

Thermal losses should be first found to find the useful energy of PTSC. The loss coefficient, U_L, is used to estimate the thermal losses from the receiver. This loss coefficient is given in Eq. (48.5) [10].

$$ {U}_{\mathrm{L}}={h}_{\mathrm{w}}+{h}_{\mathrm{r}}+{h}_{\mathrm{c}} $$

(48.5)

where h_r, radiation loss, is given in Eq. (48.6).

$$ {h}_{\mathrm{r}}=4\upsigma \boldsymbol{\varepsilon} {\mathrm{T}}_{\mathrm{r}}^3 $$

(48.6)

To find the wind loss coefficient, Nusselt correlations can be used and as shown below.

For 0.1 < Re < 1000,

$$ \mathrm{Nu}=\frac{h_{\mathrm{w}}}{D_{\mathrm{g}}\cdotp k}=0.4+0.54{\left(\operatorname{Re}\right)}^{0.52} $$

(48.7)

For 1000 < Re < 50,000,

$$ \mathrm{Nu}=\frac{h_{\mathrm{w}}}{D_{\mathrm{g}}\cdotp k}=0.3{\left(\operatorname{Re}\right)}^{0.6} $$

(48.8)

The convection losses from the receiver to air are negligible , when the space between the glass cover and the receiver is evacuated.

The loss coefficient can be estimated using Eq. (48.9).

$$ {U}_{\mathrm{L}}={\left[\frac{A_{\mathrm{r}}}{\left({h}_{\mathrm{w}}+{h}_{\mathrm{r},\mathrm{c}-\mathrm{a}}\right){A}_{\mathrm{c}}}+\frac{1}{h_{\mathrm{r},\mathrm{r}-\mathrm{c}}}\right]}^{-1} $$

(48.9)

where h_{r, c − a} is the radiation coefficient from cover to ambient which can be obtained from Eq. (48.10).

$$ {h}_{\mathrm{r},\mathrm{c}-\mathrm{a}}={\varepsilon}_{\mathrm{g}}\cdotp \sigma \cdotp \left({T}_{\mathrm{g}}+{T}_{\mathrm{a}}\right)\cdotp \left({T}_{\mathrm{g}}^2+{T}_{\mathrm{a}}^2\right) $$

(48.10)

where h_{r, r − c} is the radiation coefficient from the glass cover to the receiver which can be estimated by using Eq. (48.11).

$$ {h}_{\mathrm{r},\mathrm{r}-\mathrm{c}}=\frac{\sigma \left({T}_{\mathrm{r}}+{T}_{\mathrm{c}}\right)\left({T}_{\mathrm{r}}^2+{T}_{\mathrm{c}}^2\right)}{\frac{1}{\varepsilon_{\mathrm{r}}}+\frac{A_{\mathrm{r}}}{A_{\mathrm{c}}}\left(\frac{1}{\varepsilon_{\mathrm{c}}}-1\right)} $$

(48.11)

T_c is required to solve the previous equations . When the radiation absorbed by the cover is ignored, Tc can be estimated by using Eq. (48.12).

$$ {T}_{\mathrm{c}}=\frac{A_{\mathrm{r}}{h}_{\mathrm{r},\mathrm{r}-\mathrm{c}}{T}_{\mathrm{r}}+{A}_{\mathrm{g}}\left({h}_{\mathrm{r}.\mathrm{c}-\mathrm{a}}+{h}_{\mathrm{w}}\right){T}_{\mathrm{a}}}{A_{\mathrm{r}}{h}_{\mathrm{r},\mathrm{r}-\mathrm{c}}+{A}_{\mathrm{g}}\left({h}_{\mathrm{r}.\mathrm{c}-\mathrm{a}}+{h}_{\mathrm{w}}\right)} $$

(48.12)

The result obtained from this equation must be iterated until the accepted value is very close to it. The overall heat transfer coefficient, U_o, must be obtained to find collector efficiency factor and useful energy after this iteration by using Eq. (48.13).

$$ {U}_{\mathrm{o}}={\left(\frac{1}{U_{\mathrm{L}}}+\frac{D_{\mathrm{o}}}{h_{\mathrm{fi}}\cdotp {D}_{\mathrm{i}}}+\frac{D_{\mathrm{o}}\cdotp \ln \left({D}_{\mathrm{o}}/{D}_{\mathrm{i}}\right)}{2\cdotp k}\right)}^{-1} $$

(48.13)

where h_fi is convective heat transfer coefficient inside the receiver which can be obtained from Eq. (48.14).

$$ \mathrm{Nu}=\frac{h_{\mathrm{fi}}{D}_{\mathrm{i}}}{k} $$

(48.14)

Nu given in Eq. (48.14) can be obtained from Table 48.1.

Table 48.1 Correlations to calculate Nusselt number for the heat transfer fluid in the receiver [11]

The collector efficiency is required to calculate the heat removal factor , which can be estimated using Eq. (48.15).

$$ {F}^{\prime }=\frac{1/{U}_{\mathrm{L}}}{\frac{1}{U_{\mathrm{L}}}+\frac{D_{\mathrm{o}}}{h_{\mathrm{fi}}\cdotp {D}_{\mathrm{i}}}+\left(\frac{D_{\mathrm{o}}}{2\cdotp k}\ln \left(\frac{D_{\mathrm{o}}}{D_{\mathrm{i}}}\right)\right)} $$

(48.15)

The heat removal factor is calculated from Eq. (48.16)

$$ {F}_{\mathrm{R}}=\frac{c_{\mathrm{p}}\cdotp \dot{m}}{A_{\mathrm{r}}\cdotp {U}_{\mathrm{l}}}\cdotp \left(1-\exp \left(-\frac{U_{\mathrm{L}}\cdotp {A}_{\mathrm{r}}\cdotp {F}^{\prime }}{c_{\mathrm{p}}\cdotp \dot{m}}\right)\right) $$

(48.16)

Finally, the useful energy is estimated by using Eq. (48.17).

$$ {Q}_{\mathrm{u}}={F}_{\mathrm{R}}\cdotp \left[S\cdotp {A}_{\mathrm{a}}-{A}_{\mathrm{r}}\cdotp {U}_{\mathrm{L}}\cdotp \left({T}_{\mathrm{i}}-{T}_{\mathrm{a}}\right)\right] $$

(48.17)

T_o can be obtained from energy balance around the PTSC as follows.

$$ {T}_{\mathrm{o}}={T}_{\mathrm{i}}+\frac{Q_{\mathrm{u}}}{c_{\mathrm{p}}\cdotp \dot{m}} $$

(48.18)

The collector efficiency is calculated by Eq. (48.19).

$$ \eta ={F}_{\mathrm{R}}\cdotp \left[{\eta}_{\mathrm{r}}-{U}_{\mathrm{L}}\cdotp \left(\frac{T_{\mathrm{i}}-{T}_{\mathrm{a}}}{G_{\mathrm{B}}\cdotp C}\right)\right] $$

(48.19)

In Eq. (48.19), C is the concentration ratio and can be calculated using Eq. (48.20).

$$ C=\frac{A_{\mathrm{a}}}{A_{\mathrm{r}}} $$

(48.20)

2.3 Exergy Analysis of PTSC

Exergy analysis is required to calculate the flow exergy change of the thermal oil and the exergetic efficiency of PTSC. Total exergy gained by PTSC can be obtained from Eq. (48.21)

$$ {\dot{\mathrm{Ex}}}_{\mathrm{s}\mathrm{olar}}={A}_{\mathrm{a}}\cdotp N\cdotp S\cdotp \left[1+\frac{1}{3}{\left(\frac{T_{\mathrm{a}}}{T_{\mathrm{s}\mathrm{un}}}\right)}^4-\frac{4}{3}\left(\frac{T_{\mathrm{a}}}{T_{\mathrm{s}}}\right)\right] $$

(48.21)

Flow exergy change is given by Eq. (48.22):

$$ {\dot{\mathrm{Ex}}}_{\mathrm{flow}}=\dot{m}\cdotp \left({h}_{\mathrm{out}}-{h}_{\mathrm{in}}\right)-{T}_{\mathrm{a}}\cdotp \dot{m\cdotp}\left({s}_{\mathrm{out}}-{s}_{\mathrm{in}}\right) $$

(48.22)

Exergetic efficiency can be calculated by using Eq. (48.23):

$$ {\eta}_{\mathrm{exergetic}}=\frac{{\dot{\mathrm{Ex}}}_{\mathrm{flow}}}{{\dot{\mathrm{Ex}}}_{\mathrm{solar}}} $$

(48.23)

3 Results and Discussion

The optical, thermal, and exergetic models of PTSC are solved for various geometric and environmental parameters by using different heat transfer fluids. The maximum and minimum values of these parameters are shown in Table 48.2.

Table 48.2 Value range for parameters

The fluid selection is one of the key factors affecting the performance of the PTSC. In general, a suitable heat transfer fluid should have the following properties in order to have favorable thermodynamic characteristics and low environmental impact: high working temperature range, high thermal conductivity, low viscosity, and non-corrosiveness. For this purpose, five different heat transfer fluids as PTSC fluid are chosen, which are Therminol VP1, Therminol XP, Therminol 59, Syltherm 800, and Dowtherm A.

The effect of the solar radiation on the exergetic efficiency is shown in Fig. 48.2. It is clear from Fig. 48.2 that the exergetic efficiency increases with an increase in the solar irradiation for various heat transfer fluids. These efficiency differences of the heat transfer fluids are due to their specific heat value at a given working temperature. When the solar radiation increases, the useful energy also increases. This increase causes a higher outlet temperature of PTSC. On the other hand, the maximum exergetic efficiency is provided by using Dowtherm A for various solar radiations as shown in the Fig. 48.2.

Fig. 48.2

Exergetic efficiency vs. solar radiation for various heat transfer fluids

The exergetic efficiency of the PTSC for different wind speed is plotted in Fig. 48.3. When the wind speed increases, convective heat transfer coefficient and heat removal factor decrease. As a result, useful energy and output temperature also decrease. Figure 48.3 shows that the exergetic efficiency decreases as the useful energy increases . For the same wind speed, the highest exergetic efficiency is obtained by using Dowtherm A or Therminol 59.

Fig. 48.3

Exergetic efficiency vs. wind speed for various heat transfer fluids

The unshaded area of the PTSC increases when the aperture width of the parabola increases. When the unshaded area increases, the useful energy gain and the flow exergy increase. Consequently, the exergetic efficiency increases as the aperture width increases as shown in the Fig. 48.4. On the other hand, Dowtherm A yields the highest exergetic efficiency for the same aperture width.

Fig. 48.4

The effect of aperture width on the exergetic efficiency of the PTSC for different heat transfer fluids

The effect of the outer diameter of the receiver on the exergetic efficiency is shown in Fig. 48.5. When the outer diameter of the receiver increases, the outlet temperature, heat removal factor, and useful energy decrease. As a result, when the outer diameter of the receiver increases, the exergetic efficiency of PTSC increases. Additionally, Therminol VP1 ensures the highest exergetic efficiency for the selected receiver diameters.

Fig. 48.5

The effect of outer diameter of receiver on the exergetic efficiency for different heat transfer fluids

The outlet temperature of the heat transfer fluids, which are used in this study, is shown in Fig. 48.6. These fluids are used because of their compatibility for wide temperature range. The heat transfer fluids must remain in the liquid phase to achieve the correct results for the exergetic efficiency [13]. According to Fig. 48.6, when the Syltherm 800 is used as the PTSC fluid, it is found that the outlet temperature of receiver has the highest value (547.8 K) among the other type of fluids. This trend is mainly due to the fact that Syltherm 800 has favorable thermophysical properties (e.g., specific heat and enthalpy).

Fig. 48.6

Outlet temperatures for various heat transfer fluids

4 Optimization by Using Taguchi Method

The exergetic efficiency of the PTSC is calculated by using different values of the mentioned parameters. In this paper, optimization study is conducted to calculate the maximum exergetic efficiency for chosen conditions. The optimization of this system is done with Minitab 17 software by using Taguchi method. Taguchi method is an experimental optimization method and developed by Dr. Genichi Taguchi [12, 13]. In this study, standard L9 orthogonal array for a three-level design is selected. Four different design parameters (factors) were chosen for the Taguchi method as follows: the type of PTSC fluids, receiver diameter, aperture width, and wind speed. Three different values (levels) for each factor that represent a low, medium, and high value in a given range were taken. The Taguchi method gives us nine different types of parameter sets to be used in simulations. These simulations are then run in EES software to find the desired performance parameters (exergetic efficiency). The design parameters of this optimization are shown in Table 48.3.

Table 48.3 The design parameters of Taguchi method

In this study, heat transfer fluids, outer diameter of the receiver, aperture width of the parabola, and wind speed are used as design parameters (factors). The exergetic efficiency is calculated for these factors.

The change of S/N ratios for each factor is shown in Fig. 48.7. The higher ratio corresponding to a level for each factor is taken to find the set of the parameters that gives the optimum performance. According to Fig. 48.7, Therminol VP1 as the heat transfer fluid, 0.04 m as the outer diameter of the receiver, 10 m as the aperture width, and 1 m/s as the wind speed are chosen to maximize the exergetic efficiency. After these selections, the obtained values are recalculated with EES. When chosen values are used in the EES code, the exergetic efficiency is found as 0.5019.

Fig. 48.7

S/N plot for the Taguchi method for maximized exergetic efficiency

5 Conclusions

In this chapter, the optical, thermal, and exergetic analyses of PTSC were carried out, and the equations were solved using EES. Parametric studies were conducted to calculate the effect of some key parameters on the outlet temperature of PTSC and the exergetic efficiency of PTSC. Then, the optimization of the PTSC using Taguchi method was done to maximize the exergetic efficiency of the PTSC. The main conclusions from this study are shown below.

• The parametric studies show that the exergetic efficiency increases when the solar radiation and aperture width increase for all the heat transfer fluids which are used in this study. In addition, when the outer diameter and wind speed increase, the exergetic efficiency decreases.

• Taguchi analysis shows that the effect of aperture width on the exergetic efficiency is the most dominant parameter, and Therminol VP1 is the most optimal heat transfer fluid for various working temperature.

• The maximum exergetic efficiency was obtained by using Dowtherm A as 38.51%.

• The parametric studies show that Syltherm 800 yields the lowest exergetic efficiency for different conditions (e.g., solar radiation, wind speed, or aperture width).

A _a :

Total aperture area (m²)

A _f :

Geometric factor (m²)

A _loss :

Total loss in aperture area (m²)

A _r :

Receiver area (m²)

D _g :

Glass cover diameter (m)

D _i :

Inside diameter of the receiver (m)

D _o :

Outside diameter of the receiver (m)

${\dot{\mathrm{Ex}}}_{\mathrm{flow}}$ :

Flow exergy rate (kW)

${\dot{\mathrm{Ex}}}_{\mathrm{solar}}$ :

Exergy rate from the sun (kW)

𝑓 :

Focal distance of the parabola, friction factor (m)

F _R :

Heat removal factor

G _b :

Solar radiation (W/m²)

h :

Specific enthalpy (kJ/kg)

h _c :

Conduction loss from the tubes (W/m²K)

h _fi :

Convective heat transfer coefficient inside the receiver (W/m²K)

h _p :

Parabola height (m)

h _r :

Radiation loss (W/m²K)

h _{r, c − a} :

Radiation coefficient from glass cover to ambient (W/m²K)

h _{r, r − c} :

Radiation coefficient from receiver to the glass cover (W/m²K)

h _w :

Wind loss coefficient (W/m²K)

k :

Thermal conductivity of fluid (W/mK)

N :

Number of PTSC

Nu :

Nusselt number

Pr :

Prandtl number

Re :

Reynolds number

𝑆 :

The heat absorbed by the receiver (W/m²)

s :

Specific entropy (kJ/kgK)

T _a :

Ambient temperature (K)

T _c :

Glass cover temperature (K)

T _sun :

The temperature of the sun (K)

T _i :

Inlet temperature of the heat transfer fluid (K)

U _o :

Overall heat transfer coefficient (W/m²K)

W _a :

Aperture of parabola (m)

a:

Ambient, aperture

i:

Inner, inlet

o:

Outer

r:

Radiation

α :

Absorptance of the receiver

γ :

Intercept factor

η _exergetic :

Exergetic efficiency of PTSC

η _r :

Optical efficiency

ε _g :

Emissivity of the glass cover

ρ :

Reflectance of the mirror

σ :

Stefan–Boltzmann constant (W/m²·K⁴)

τ :

Transmittance of the glass cover

θ :

Incidence angle