Energy and exergy analysis of parabolic trough collectors

After mesh generation in the tube geometry, the in-house computational code has been used to simulate the absorber tube as a receiver of radiations. The reached heat flux functions are defined and linked to main program with self-developed subroutine program. Also, the properties of Syltherm-800 are defined in code as polynomial functions. Therefore, the gravity must be activated.

The governing equations are discretized with finite volume method (FVM). Considering, the convective terms of momentum and energy are discretized with the second upwind order. The SIMPLE algorithm is applied to couple velocity and pressure together. The convergence criterion of all equations is considered less than 10^-6.

The results which are obtained by simulation of LS-2 absorber tube (Dp=50.8 mm), are used to compare and validate of numerical method. Tab. 2 indicates the comparison between numerical results and Dudley's data. $\Delta T \%$ was selected as comparison criterion and its definition is as below:

$\Delta T \%=\frac{T_{\exp }-T_{\text { num }}}{T_{\text { exp }}} \times 100 \%$    (20)

The useful heat gain by the parabolic trough collector exposed to solar radiation is defined as:

$Q_{\mathrm{u}}=\dot{m} c_{\mathrm{p}}\left(T_{\mathrm{out}}-T_{\mathrm{in}}\right)$    (21)

And the thermal efficiency of the collector is obtained by:

$\eta_{\mathrm{th}}=Q_{u} /_{A_{\mathrm{c}}} . I$    (22)

where $A_{\mathrm{c}}$ is the area of the collector and I is the solar radiation incident on the collector mirror. It can be seen from table 2, there is a logical agreement between numerical and experimental results.

7.1 Average Nu

After validation, the effect of horizontal wind speed and the orientation of collector on exergy efficiency are numerically studed. Also, the influence of glass cover on prevention of exergy destruction is investigated. After simulation of each case-studies the average Nuarround absorber tube is calculated. Fig.6shows the effect of collector orientation on average Nu variation.

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Figure 6. Variationdiagram of Nu around absorber tube of collector versus several orientations and wind speeds=5, 10, 15 m/sec

According to fig. 6, the maximum Nu is occurred versus orientation angle=90 deg. With decreasing of wind speed, the slope of diagrams is decreased and becomes horizontal. It means that, if the wind speed decreases, the average Nu will be constant and free convection will invigorate. Also, it is cancluded, when wind blows on the concave side of the parabolic mirror, the impressibility of average Nu from wind speed is most than other orientations.

7.2 Outlet temperature

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Figure 7. Diagram of outlet temperature versus several orientations and wind speeds=5, 10, 15 m/sec

According to eq. (21) the outlet temperature (T₀) is the most important in the calculation of collector thermal efficiency. Therefore, increasing of the outlet temperature leads to increasing of efficiency. Fig. 7 inhibits the impressibility of T₀ from collector orientation. Considering, when wind blows on the concave side of the parabolic mirror, the impressibility of T₀ from wind speed is most than other orientations. In contrast, when wind blows on the convex side of the parabolic mirror, the impressibility of T₀ from wind speed is least than other orientations. Also, if the wind blows on the convex side of the parabolic mirror, the impressibility of T₀ from the variation of orientation is most than other orientations. The mathematical statement of this sentence is shown in the slopes of diagrams. While, the distance between diagrams shows  the impressibility of T₀ from wind speeds.

7.3 Air path lines

Typical air path lines around the absorber tube for wind speeds= 5, 10, 15 m/sec are illustrated in fig. 8 for the angle of orientation=0 deg. These schematics exhibit that several recirculation regions are made around the receiver tube. Similar results are formed when angle of orientation is increased. This figure shows that, air flow field around the receiver tube is depended on the wind speed and strongly related to the collector orientation. Heat transfer from receiver is important factor in calculating of heat loss in receiver tube. If the wind speed is very high (higher than 5 m/sec), the effect of collector on absorber tube cannot beneglected and is important. This result is concluded from fig. 7 too. In low speeds (less than 2.5 m/sec) the free convection is more effective than forced convection around absorber tube.

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Figure 8. Distribution of air velocity around absorber tube versus wind speed

7.4 Exergy analysis

One of the governing equations on the solar collector is the usefull heat gain ($Q_{\mathrm{u}}$) for energy analysis. It is calculated by the working fluid property and mass flow rate as:

$Q_{\mathrm{u}}=\dot{m} c_{\mathrm{p}}\left(T_{\mathrm{out}}-T_{\mathrm{in}}\right)$    (21)

The Hottel-Whillier equation calculates theuseful heat gain for a solar collector system, considering the heat losses from the absorber tube to the ambient.Two different approaches are applied to calculate heat gain:“brute-force”, and “functional.” [15].

• A “brute-force” exergy efficiency is calculated by the ratio of the sum of all output exergy terms to the sum of all input exergy terms.
• A “functional” exergy efficiency specially for solar systems is calculated by as the ratio of the exergy expended to the expected output.

In a similar relation, exergy efficiency is defined as the ratio of output exergy to input exergy, as follow:

$\eta_{\mathrm{ex}}=\frac{\dot{E}_{\mathrm{out}}}{\dot{E}_{\mathrm{in}}}=1-\frac{\dot{E}_{\mathrm{dest}}}{\dot{E}_{\mathrm{in}}}$    (22)

In other definition, Exergy is determined as the maximum amount of work which can be produced by a system or flow of mass or energy as it regards to equilibrium with a reference point at surrounding[16]. Consider that, this method is used in this paper. The main form of exergy equilibrium equation is as:

$\dot{E}_{\mathrm{in}}+\dot{E}_{\mathrm{s}}+\dot{E}_{\mathrm{out}}+\dot{E}_{1}+\dot{E}_{\mathrm{dest}}=0$    (23)

where, $\dot{E}_{\mathrm{in}}, \dot{E}_{\mathrm{s}}, \dot{E}_{\mathrm{out}}, \dot{E}_{1}$ and $\dot{E}_{\mathrm{dest}}$ are the inlet, stored, outlet, loss and destroyed exergy rate, respectively. The inlet exergy rate is sum of the inlet exergy rate with fluid flow and the absorbed solar radiation exergy rate. The inlet exergy rate with fluid flow is obtained by: [16, 17]

$\dot{E}_{\mathrm{in}, \mathrm{f}}=\dot{m} c_{\mathrm{p}}\left(T_{\mathrm{in}}-T_{\mathrm{a}}-T_{\mathrm{a}} \ln \left(\frac{T_{\mathrm{in}}}{T_{\mathrm{a}}}\right)\right)+\frac{\dot{m} \Delta P_{\mathrm{in}}}{\rho}$    (24)

Also, according to the Petela theorem, the absorbed solar radiation exergy rate is calculated by: [18]

$\dot{E}_{\mathrm{in}, \mathrm{Q}}=\eta_{o} I_{\mathrm{T}} A_{\mathrm{P}}\left[1-\frac{4}{3}\left(\frac{T_{\mathrm{a}}}{T_{\mathrm{Sun}}}\right)+\frac{1}{3}\left(\frac{T_{\mathrm{a}}}{T_{\mathrm{Sun}}}\right)^{4}\right]$    (25)

The term in the bracket is named the Petela efficiency ($\eta_{\mathrm{p}}$). However, this equation contravenes the second law of thermodynamics in some systems [19]. If the sun is assumed as an infinite thermal source, the corrected form of eq. (25) will be: [19, 20]

$\dot{E}_{\mathrm{in}, \mathrm{Q}}=\eta_{\mathrm{o}} I_{\mathrm{T}} A_{\mathrm{P}}\left(1-\frac{T_{\mathrm{a}}}{T_{\mathrm{Sun}}}\right)$    (26)

where $T_{\mathrm{Sun}}$ is the apparent sun temperature and equals 75% of blackbody temperature of sun [21].

The summations of Eqs. (24) and (26) will result in total inlet exergy rate of the solar collector. The stored exergy rate is negligible at steady-state conditions. Also, the outlet exegy rate of outlet fluid flow in the solar collector is:

$\dot{E}_{\text { out }, \mathrm{f}}=-\dot{m} c_{\mathrm{p}}\left(T_{\text { out }}-T_{\mathrm{a}}-T_{\mathrm{a}} \ln \left(\frac{T_{\text { out }}}{T_{a}}\right)\right)+\frac{\dot{m} \Delta P_{\text { out }}}{\rho}$    (27)

In eqs. (24) and (27), $\Delta P_{\mathrm{in}}$ and $\Delta P_{\text { out }}$ are the pressure differences of the working fluid with the surrounding atentrance and exit of the receiver tube. The loss exergy rate consists of two terms: the first is created by heat loss rate from the absorber tube to the environment because of wind blast. It is calculated by: [22]

$\dot{E}_{1, w}=-U_{1} A_{c}\left(T_{P}-T_{a}\right)\left(1-\frac{T_{\mathrm{a}}}{T_{\mathrm{r}}}\right)$    (28)

Thesecond is created by the temperature difference between the receiver tube and the sun. In other words, the radiation loss makes these term of exergy loss: [22]

$\dot{E}_{1, \Delta \mathrm{T}_{\mathrm{S}}}=-\eta_{\mathrm{o}} I_{\mathrm{T}} A_{\mathrm{c}} T_{\mathrm{a}}\left(\frac{1}{T_{r}}-\frac{1}{T_{\mathrm{Sun}}}\right)$    (29)

where $\eta_{0}$ is optical efficiency of collector and calculated by $\eta_{0}=\tau \alpha$. The destroyed exergy rate consists of two terms: The first term is caused by the pressure drop in the receiver tube duct: [23, 24]

$\dot{E}_{\mathrm{dest}, \Delta \mathrm{p}}=-\frac{\dot{m} \Delta P}{\rho} \frac{T_{\mathrm{a}} \ln \frac{T_{\mathrm{out}}}{T_{\mathrm{a}}}}{\left(T_{\mathrm{out}}-T_{\mathrm{in}}\right)}$    (30)

And the second term is concluded by the temperature difference between the receiver tube surface and the working fluid which is obtained by: [23, 24]

$\dot{E}_{\mathrm{dest}, \Delta \mathrm{T}_{\mathrm{f}}}=-\dot{m} c_{\mathrm{p}}\left(\ln \left(\frac{T_{\mathrm{out}}}{T_{\mathrm{a}}}\right)-\frac{T_{\mathrm{out}}-T_{\mathrm{in}}}{T_{\mathrm{r}}}\right)$    (31)

Therefore, the eq. (23) is discretized to more terms as:

$\dot{E}_{\mathrm{in}, \mathrm{f}}+\dot{E}_{\mathrm{in}, \mathrm{Q}}+\dot{E}_{\mathrm{s}}+\dot{E}_{\mathrm{out}, \mathrm{f}}+\dot{E}_{1, \mathrm{w}}+\dot{E}_{1, \Delta \mathrm{T}_{\mathrm{s}}}+\dot{E}_{\mathrm{dest}, \Delta \mathrm{p}}+\dot{E}_{\mathrm{dest}, \Delta \mathrm{T}_{\mathrm{f}}}=0$    (32)

Exergy loss   Exergy destruction

If the absorbed solar radiation exergy rate ($\dot{E}_{\mathrm{in}, \mathrm{Q}}$) supplants to right hand and two hands of resulted equation divided to $\dot{E}_{\mathrm{in}, \mathrm{Q}}$ then:

$\frac{\dot{E}_{\mathrm{in}, \mathrm{f}}+\dot{E}_{\mathrm{out}, \mathrm{f}}+\dot{E}_{1, \mathrm{w}}+\dot{E}_{1, \Delta \mathrm{T}_{\mathrm{S}}}+\dot{E}_{\mathrm{dest}, \Delta \mathrm{p}}+\dot{E}_{\mathrm{dest}, \Delta \mathrm{T}_{\mathrm{f}}}}{-\dot{E}_{\mathrm{in}, \mathrm{Q}}}=1$                (33)

The exergy efficiency has been defined as the increase of fluid flow exergy to the radiation source in solar collectors. In other words:

$\eta_{\mathrm{ex}}=\frac{\dot{E}_{\mathrm{in}, \mathrm{f}}+\dot{E}_{\mathrm{out}, \mathrm{f}}}{-\dot{E}_{\mathrm{in}, \mathrm{Q}}}$                (34)

Subtitlingeqs.(24), (26) and (27) into Eq. (34), the exergy efficiency of the solar collector is calculated by: [19, 22, 25]

$\eta_{\mathrm{ex}}=\frac{\dot{m}\left[c_{\mathrm{P}}\left(T_{\mathrm{out}}-T_{\mathrm{in}}-T_{\mathrm{a}} \ln \left(\frac{T_{\mathrm{out}}}{T_{\mathrm{a}}}\right)\right)-\frac{\Delta P}{\rho}\right.}{I_{T} A_{\mathrm{c}}\left(1-\frac{T_{\mathrm{a}}}{T_{\mathrm{sun}}}\right)}=1-\left\{\left(1-\eta_{\mathrm{o}}\right)+\right.$

$\frac{\dot{m} \Delta P}{\rho I_{\mathrm{T}} A_{\mathrm{C}}\left(1-\frac{T_{\mathrm{a}}}{T_{\text { sun }}}\right)} \frac{T_{\mathrm{a}} \ln \left(\frac{T_{\text { out }}}{T_{\mathrm{a}}}\right)}{\left(T_{\text { out }}-T_{\text { in }}\right)}+\frac{\eta_{0} T_{\mathrm{a}}}{1-\frac{T_{\mathrm{a}}}{T_{\text { Sun }}}}\left(\frac{1}{T_{r}}-\frac{1}{T_{\text {sun}}}\right)+\frac{U_{1}\left(T_{\mathrm{r}}-T_{\mathrm{a}}\right)}{l_{\mathrm{T}}\left(1-\frac{T_{\mathrm{a}}}{T_{\text { sun }}}\right)}(1-$

$\frac{T_{\mathrm{a}}}{T_{\mathrm{r}}} )+\frac{\dot{m c}_{\mathrm{p}} T_{\mathrm{a}}}{I_{\mathrm{T}} A_{\mathrm{C}}} \frac{\left(\ln \left(\frac{T_{\mathrm{out}}}{T_{\mathrm{in}}}\right)-\frac{\left(T_{\mathrm{out}}-T_{\mathrm{in}}\right)}{T_{r}}\right)}{\left(1-\frac{T_{\mathrm{a}}}{T_{\mathrm{sun}}}\right)} \}$                (35)

The amounts of exergy have been calculated and presented in tabs. 3, 4 and 5 for the wind speeds 5, 10 and 15 m/sec respectively. Consider that, the amounts of exergy loss, exergy destruction and exergy efficiency have been obtained separately.

Table 3. Amounts of exergy for several orientations and wind speed= 5 m/sec

Table 4. Amounts of exergy for several orientations and wind speed= 10 m/sec

Table 5. Amounts of exergy for several orientations and wind speed= 15 m/sec

Table 6. Amounts of exergy for several orientations and wind speeds using avacuated tube