Design, Simulation and Optimization of a Novel Transpired Tubular Solar Air Heater

Abstract

In this paper, a novel tubular solar air heater is introduced. In this air heater, the hot boundary layer is drawn into the absorber tube and can provide thermal energy at moderate temperatures. Several different cases were simulated and a correlation was proposed to predict the collector’s effectiveness as a function Rayleigh number and Reynolds number. An equation was derived to find the effectiveness of this collector. Finally, a real case was studied with non-uniform solar flux distribution, as well as radiation heat loss. Good agreement was found between the results and those derived by the proposed analytical method. For different suction values, the first-law and the second-law efficiencies were calculated. Based on the exergy analysis, exergy destruction in absorption is the dominant factor that is unavoidable in low-temperature collectors. It was shown that there is an optimum suction value at which the second-law efficiency is maximized. At the optimum point, temperature rise can reach 54 K, which is hardly possible with a flat plate collector. Based on the exergy analysis, the relation between tube wall temperature and air outlet temperature in their dimensionless forms at the optimum working condition was derived, and it was shown that effectiveness at the optimum working condition is around 0.5. This means that the air temperature rise shall be half of the temperature difference between collector wall and the ambient temperatures. A high outlet temperature besides the low cost of construction and maintenance are the main advantages of this air heater. With such a high temperature rise, this type of collector can increase the use of solar energy in domestic applications.

1. Introduction

1.1. Introduction

Solar energy is a clean and sustainable source of energy that must be included in our daily life. Besides the use of solar energy in industrial areas such as power generation, the use of this energy source in domestic and semi-industrial areas such as cooking, air heaters, and crop dryers should not be neglected. Several studies can be found concerning domestic solar collectors [1,2,3,4]. For example, Nemati and Javanamrdi [1] optimized domestic solar cylindrical–parabolic cookers using exergy analysis. They proposed a simple correlation to find the optimum dimensionless temperature for various geometries and solar radiations. Moghimi et al. [5,6] used the MOGA algorithm to optimize a Linear Fresnel Collector. They selected seven geometrical parameters for optimization. They found that the most sensitive parameters were the top insulation thickness and the cavity depth. Lingayat and Chandramohan [7] used corrugated absorber surfaces to increase the heat transfer rate between air and absorber. They used the proposed geometry in an indirect solar dryer for drying banana samples. They concluded that the average collector and dryer efficiencies were 64.5% and 55.3%, respectively. For this solar dryer, banana samples were dried from 3.5566 to 0.2604 kg/kg on a dry basis. A similar study was conducted by El-Sebaey et al. [8]. They used an indirect solar dryer to reduce the moisture content of banana slices. In their experimental studies, they compared chimney-type and fan-type collectors and found that chimney-type collector has a higher thermal efficiency. The mean thermal efficiency of the chimney-type was 14.5% in comparison to 12.76% for the fan-type solar dryer.

Among the domestic applications of solar energy, solar air heaters have a special place. Hot air can be used for space heating and crop drying [9,10,11]. Because a solar air heater is simple and cost-effective for construction, it is one of the main interests of researchers in this field [12]. There are different types of solar air heaters like unglazed transpired collectors (UTCs) [13,14,15], solar air heaters [16,17], and many other novel collectors [18,19]. Regardless of new developments in air heaters to increase thermal performance, they are mostly flat plate types. Flat plate collectors are very simple and easy to construct. However, the temperature rise, and consequently the outlet exergy in these collectors, are very low. Moreover, there is no special control available to adjust the outlet temperature in this type of collector.

In contrast, the temperature rise in concentrating collectors is high. So, it can be mixed with cold fresh air to achieve a large amount of air with the proper temperature. It means that there is no need to transfer a large amount of air from the collector to the destination, but just enough to mix hot air with ambient air at the destination. This also provides control over temperature. On the other hand, there are some types of reflectors, such as CPC, that do not require high-precision technology for construction, or sun tracking systems, which do not require tracking without losing too much performance. However, convection loss is another issue. A high temperature means a strong natural convection loss. To eliminate natural convection, it is common to use a glass tube around the absorber to intercept its direct contact with the ambient air. But the use of a glass tube has many problems, including the cost of construction and maintenance, the weight, and the possibility of breakage. In the following, a novel type of collector is proposed to utilize the advantages of high temperature, which also allows low-cost construction and maintenance. This type of collector can be used for indirect crop drying, and heating or pre-heating houses and large buildings.

1.2. Problem Description

Figure 1 shows a temperature contour plot for natural convection around a horizontal tube (simulated by author). Based on this figure, a narrow thermal boundary layer starts at the bottom of the tube and it gradually gets thicker toward the top of the tube. At the top of the tube, a thick layer of air is separated from the solid wall in the form of a plume. Since in a round tube, the shape of the tube wall changes from a vertical plane at both sides (right and left sides) to a flat horizontal plane at the top of the tube, the boundary layer is thick and wide at the top of the tube. So, it is a great source of high-temperature thermal energy or exergy. The solid black line in this figure shows the isothermal line of 380 K (80% of the temperature difference between wall and ambient). Based on the rulers in the figure, for the current case, the isothermal line is around 12 mm in width (from −6 mm to +6 mm).

This rising plume carries a considerable amount of high-temperature thermal energy that is lost. So, if a narrow slit is installed at the top of the tube, with proper suction the hot air accumulated on the top of the tube can be drawn in. Figure 2 clarifies this idea. In Figure 2b, the thermal plume vanishes and hot air is sucked in.

Since a tubular absorber is a proper shape for collectors with a concentration ratio of more than one, this Unglazed Transpired Tubular Absorber (UTTA) can be used in concentrating collectors like trough collectors, linear Fresnel collectors, and CPCs. High-temperature air can later be mixed with fresh cold air to achieve an adequate amount of warm air at a proper temperature. Figure 3 schematically shows the schematic of a UTTA.

In a common concentrating collector, there is a glass tube around the absorber tube to reduce the natural convection heat loss. But, due to the different thermal expansion coefficients for the absorbent tube and the glass, and also due to the significant difference in temperature between these two tubes, breaking the glass cover is not unlikely to occur. In a UTTA, since the natural convection heat loss is reduced and controlled by proper suction, there is no reason to keep the glass tube cover around the absorber. Eliminating the glass tube significantly reduces construction and maintenance costs. On the other hand, since the temperature in the circumferential direction of the absorber tube varies considerably, thermal stress is one of the main concerns of designers [20,21,22]. However, the narrow slit at the top of the absorber tube circumferentially provides a free displacement for the tube, and thermal stress is released.

In the following, the method of simulation of a UTTA and effective non-dimensional parameters are presented. Nearly fifty different cases related to a wide range of non-dimensional parameters were simulated, and a correlation was proposed to predict the effectiveness of UTTA. Furthermore, the effects of radiation loss and non-uniformity in solar heat flux were studied, and it was shown that the proposed analytical method effectively predicts UTTA effectiveness.

2. Numerical Simulation

2.1. Governing Equations

For a narrow slit where the pressure drop of the flow passing through the gap is much higher than the pressure drop of the air flow along the pipe, the suction is uniform along the slit and the problem can be simplified to a two-dimensional case. For steady and laminar natural convection over the outer surface of a tube, the governing equations, as well as the boundary conditions, are presented in the following.

Continuity equation:

$$\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0$$

(1)

Momentum equations:

$$\rho \left(u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}\right)=-\frac{\partial P}{\partial x}+\mu \left(\frac{{\partial }^{2}u}{\partial x^2}+\frac{{\partial }^{2}u}{\partial y^2}\right)$$

(2)

$$\rho \left(u\frac{\partial v}{\partial x}+v\frac{\partial v}{\partial y}\right)=-\frac{\partial P}{\partial y}+\mu \left(\frac{{\partial }^{2}v}{\partial x^2}+\frac{{\partial }^{2}v}{\partial y^2}\right)-\rho g$$

(3)

Energy equation in the air zone:

$$\rho Cp\left(u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}\right)=k\left(\frac{{\partial }^{2}T}{\partial x^2}+\frac{{\partial }^{2}T}{\partial y^2}\right)$$

(4)

The energy equation in the solid zone:

$$k_{\mathrm{s}}\left(\frac{{\partial }^{2}T}{\partial x^2}+\frac{{\partial }^{2}T}{\partial y^2}\right)=0$$

(5)

Air properties including density were assumed as a function of temperature. The dependency of properties on temperature is shown in Appendix A.

The diameter of the air domain around the wall was considered to be ten times the outer diameter of the tube. It was large enough to avoid numerical errors due to the false impression of boundary conditions [23,24,25]. The boundary conditions at the outer surface of the tube were

$$u=v=0, T=T_{\mathrm{w}}$$

(6)

and at the far field they were:

$$P=P_{\mathrm{a}\mathrm{t}\mathrm{m}}, T=T_{\mathrm{\infty }}=300 \mathrm{K}$$

(7)

At the solid–fluid interface, there was no heat flux and temperature jump. Moreover, the no-slip condition was assumed. So:

$${\left.k_{\mathrm{s}}\frac{\partial T}{\partial n}\right|}_{\mathrm{S}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{d} \mathrm{s}\mathrm{i}\mathrm{d}\mathrm{e}}={\left.k\frac{\partial T}{\partial n}\right|}_{\mathrm{F}\mathrm{l}\mathrm{u}\mathrm{i}\mathrm{d} \mathrm{s}\mathrm{i}\mathrm{d}\mathrm{e}}, T_{\mathrm{S}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{d} \mathrm{s}\mathrm{i}\mathrm{d}\mathrm{e}}=T_{\mathrm{F}\mathrm{l}\mathrm{u}\mathrm{i}\mathrm{d} \mathrm{s}\mathrm{i}\mathrm{d}\mathrm{e}}, u=v=0$$

(8)

where $n$ is the normal direction. Because of the geometry and flow symmetry, only one-half of the domain was considered in the simulation. Finally, the air mass flow rate per unit length ($\dot{m}$) was specified over the hole on the tube.

2.2. Solution Methodology

A very fine quadrilateral structured grid was used to discretize the domain. To capture the thermal boundary layer, the distance of the first gird to the tube wall was set to 3.95 × 10⁻⁵ m. The total number of grids was 248,000 (See Figure 4). Equations were solved numerically, using Ansys-Fluent 2020. A couple algorithm was used for pressure–velocity coupling and the second-order upwind method was used for momentum and energy discretization. Residuals were set to 10⁻⁶ and 10⁻⁹ for continuity and energy equations, respectively, and the Pseudo Transient method was used to solve equations. For the grid study, natural convection over a horizontal tube with a 0.06 m diameter was simulated by the above settings. The results are presented in Figure 5. Based on this figure, the difference between ${Nu}_{\mathrm{c}}$ for the highest and lowest grid numbers is less than 0.17. Therefore, the selected number of the grid is satisfactory.

Near fifty different cases were simulated using the aforementioned settings, and their results are verified and studied in the next parts. The ranges of contributing parameters in the simulation are presented in

Table 1. Ranges of simulating parameters.

D	${\mathit{T}}_{\mathbf{w}}-{\mathit{T}}_{\mathit{\infty }}$	$\dot{\mathit{m}}$
(mm)	(k)	(kg/s·m)
[21, 140]	[50, 250]	[$2\times {10}^{-5}, 0.012]$

.

Before starting the study, the method was validated by simulating the natural convection around a horizontal tube. The results were compared for a wide range of $Ra$, against two experimental correlations, i.e., Churchill and Chu [26] (Equation (9)), and Morgan [27] (Equation (10)), with a maximum uncertainty of ±5%, as reported by [27]. The comparison in Figure 6 shows good agreement between the numerical and experimental data and shows the validity of the CFD approach. The average relative error with Equation (9) is 0.9% and with Equation (10) is 3.4%.

$${Nu}_{\mathrm{c}}={\left[0.6+\frac{0.387{Ra}^{1/6}}{{\left[1+{\left(0.559/Pr\right)}^{9/16}\right]}^{8/27}}\right]}^2\hspace{1em}{10}^{-5}<Ra<{10}^{12}$$

(9)

$${Nu}_{\mathrm{c}}=0.48{Ra}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}\hspace{1em}{10}^4<Ra<{10}^7$$

(10)

where

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

(11)

in which the convective heat transfer coefficient, $h_{\mathrm{c}}$, is:

$$h_{\mathrm{c}}=\frac{Q}{A\left(T_{\mathrm{w}}-T_{\mathrm{\infty }}\right)}$$

(12)

and

$$Ra=\frac{g\beta d^3\left(T_{\mathrm{w}}-T_{\mathrm{\infty }}\right)}{{\nu }^2}$$

(13)

All contributing parameters were calculated at the film temperature:

$$T_{\mathrm{f}\mathrm{i}\mathrm{l}\mathrm{m}}=\frac{T_{\mathrm{w}}+T_{\mathrm{\infty }}}{2}$$

(14)

3. Data Processing and Discussion

3.1. Data Analysis

To analyze the results, it is necessary to introduce the governing dimensionless parameters. The main parameter, i.e., effectiveness is [13]:

$$\epsilon =\frac{T_{\mathrm{s}\mathrm{l}}-T_{\mathrm{\infty }}}{T_{\mathrm{w}}-T_{\mathrm{\infty }}}$$

(15)

which shows the value of thermal energy gained at the slit to the maximum theoretically accessible thermal energy. $T_{\mathrm{s}\mathrm{l}}$ is the outlet air temperature at the slit (see Figure 3). The two other contributing parameters are:

$$Ra=\frac{g\beta d^3\left(T_{\mathrm{w}}-T_{\mathrm{\infty }}\right)}{{\nu }^2}$$

(16)

$$Re=\frac{\rho v_{\mathrm{s}\mathrm{l}}w}{\mu }=\frac{\dot{m}}{\mu }$$

(17)

where $w$ is the slit width and $v_{\mathrm{s}\mathrm{l}}$ is the air velocity at the slit entrance. $\dot{m}=\rho v_{\mathrm{s}\mathrm{l}}w$ is the air mass flow rate per unit length.

Note: the thermo-physical condition required for Equation (16) is film temperature and for Equation (17) is the temperature of air at the entrance to the slit, $T_{\mathrm{s}\mathrm{l}}$.

For a series of solutions, the variation in $\epsilon$ with respect to $Re$ is shown in Figure 7. As is expected, the effectiveness decreases with increasing suction value. Generally, three zones can be assumed in this curve, i.e., low suction, high suction, and moderate suction, and three data points are selected for each zone, to be discussed in detail.

Figure 8 shows the temperature contour plot of three selected points (column (a)) as well as the velocity vector plot around the slit (column (b)). For point P1 associated with a low suction, a narrow layer in the vicinity of the wall is sucked in. So, the suction layer is located entirely beneath the boundary layer thickness.

It is known that in natural convection, the thermal boundary layer thickness is proportional to ${Ra}^N$ (over a vertical flat plate, $N=1/4$ [28]). Since for near-to-zero suction, only a thin wall-adjacent layer of gas is sucked in, the effectiveness has a weak dependency on thermal boundary layer thickness, and consequently on $Ra$. By reducing the suction value to zero, almost all of the thermal energy is wasted by natural convection, and nothing is recovered.

For moderate suction (point P2), part of the thermal boundary layer is drawn into the slit. So, some part of thermal energy is wasted and the rest is recovered. The momentum of the flow in natural convection and forced convection is proportional to $Ra$ and $Re$, respectively. So, the amount of flow that is trapped and sucked in involves a competition between the momentum of natural convection flow and forced convection flow. Consequently, the effectiveness is expected to depend on the ratio of $Ra$ and $Re$. From one point onwards, the entire boundary layer as well as fresh air is drawn in (point P3). So, almost nothing is wasted. But because of the fresh air, the temperature of the outlet from the slit and also effectiveness decrease. Under this condition, in the limiting case, i.e., high $Re$,

$$A.h_{\mathrm{c}}\left(T_{\mathrm{w}}-T_{\mathrm{\infty }}\right)=\dot{m}Cp\left(T_{\mathrm{s}\mathrm{l}}-T_{\mathrm{\infty }}\right)$$

(18)

Using Equations (11), (15) and (17), the non-dimensional form of the above equation (Equation (18)) is:

$$\epsilon =\frac{N{u}_{\mathrm{c}}}{Re.Pr}\frac{A}{D}$$

(19)

$A=\pi D-w$ is the heat transfer area per unit length of the tube. Finally, using Equation (10), the final form of effectiveness at high $Re$ will be

$$\epsilon =\frac{0.48R{a}^{1/4}}{Re.Pr}(\pi -\frac{w}{D})$$

(20)

Based on the above explanation and through trial and error, the following correlation is proposed for $Re\le 800$ and $1.8\times {10}^4\le Ra\le 7.8\times {10}^6$.

$$\epsilon =\frac{a+R{e}^b}{k{Ra}^f{\left(1+Re\right)}^n+\frac{Pr}{0.48\left(\pi -\frac{w}{D}\right)}{\left(\frac{Re}{R{a}^{\frac{0.25}{1+b}}}\right)}^{1+b}}$$

(21)

in which

$$a=13.1, k=11.1$$

$$b=0.57, n=0.14$$

$$f=0.02$$

In the above equation, for $Re=0$, the effectiveness approaches $a/k{Ra}^f$, which is slightly less than unity. At a very low suction, the velocity of the rising plume in the middle of the slit is dominant over the suction velocity, and consequently, some amount of the sucked air returns back to the ambient.

For the two limiting cases, i.e., very low and very high $Re$, Equation (21) reduces to:

$$\underset{\mathit{Re}\to 0}{\lim }\epsilon =\frac{a}{k{Ra}^f}$$

(22)

$$\underset{\mathit{Re}\to \mathrm{\infty }}{\lim }\epsilon =\frac{0.48R{a}^{1/4}}{Re.Pr}(\pi -\frac{w}{D})$$

(23)

For very low $Re$, the form of the equation shows a very weak dependency on $Ra$, and for very high $Re$, the equation approaches the analytical result (Equation (20)). The value of R² for Equation (21) is 0.998 and the RMSE is 1.1%, which shows the accuracy of the proposed correlation. The comparison of Equation (21) with numerical results is presented in the form of a scatter plot in Figure 9.

Figure 10 shows the variation in effectiveness for various $Re$ and $Ra$. Based on this figure, effectiveness decreases monotonically with increasing suction ($Re)$. Moreover, $\epsilon$ is higher at higher $Ra.$

3.2. Energy Analysis of a UTTA

There are two main sources of energy loss, i.e., natural convection and radiation. So, in the absence of other sources, the total absorbed inlet energy ($\alpha .Q_{\mathrm{i}\mathrm{n}}$) shall be equal to energy losses. The energy balance for a UTTA is:

$$\alpha .Q_{\mathrm{i}\mathrm{n}}=\underset{Q_{\mathrm{r}}}{\underbrace{A.h_{\mathrm{c}}\left(T_{\mathrm{w}}-T_{\mathrm{\infty }}\right)}}+\underset{Q_{\mathrm{c}}}{\underbrace{A.h_{\mathrm{r}}\left(T_{\mathrm{w}}-T_{\mathrm{\infty }}\right)}}$$

(24)

This means that the absorbed solar heat input ($\alpha .Q_{\mathrm{i}\mathrm{n}})$ is wasted by radiation and convection [29].

$$h_{\mathrm{r}}=\sigma .ϵ.\left(T_{\mathrm{w}}+T_{\mathrm{\infty }}\right)\left(T_{\mathrm{w}}^2+T_{\mathrm{\infty }}^2\right)$$

(25)

In the above equation, $\sigma$ is the Stefan–Boltzmann constant and $ϵ$ is tube emissivity, which is equal to its absorptivity as per Kirchhoff’s law of thermal radiation [29].

The first-law efficiency can be defined as:

$${\eta }_{\mathrm{I}}=\frac{\dot{m}Cp\left(T_{\mathrm{s}\mathrm{l}}-T_{\mathrm{\infty }}\right)}{Q_{\mathrm{i}\mathrm{n}}}$$

(26)

This can be presented as a function of dimensionless parameters. Starting from Equation (17), the air mass flow rate per unit length is:

$$\dot{m}=Re.\mu$$

(27)

Alternatively, using Equations (11) and (15),

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

(28)

$$T_{\mathrm{s}\mathrm{l}}-T_{\mathrm{\infty }}=\epsilon \left(T_{\mathrm{w}}-T_{\mathrm{\infty }}\right)$$

(29)

Replacing the above equations as well as Equation (24) in Equation (26) yields:

$${\eta }_{\mathrm{I}}=\frac{Re.\mu .Cp.\epsilon .\alpha }{A\frac{{Nu}_{\mathrm{c}}k}{D}\left(1+\frac{h_{\mathrm{r}}}{h_{\mathrm{c}}}\right)}$$

(30)

Knowing that $Pr=\mu .Cp/k$ and using Equation (28), the above equation is simplified to:

$${\eta }_{\mathrm{I}}=\frac{Re.Pr}{{Nu}_{\mathrm{c}}}\frac{D}{A}\frac{\epsilon .\alpha }{1+{Nu}_{\mathrm{r}}/{Nu}_{\mathrm{c}}}$$

(31)

where

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

(32)

3.3. Exergy Analysis of a UTTA

The exergy balance for a UTTA can be written as:

$${Ex}_{\mathrm{i}\mathrm{n}}={Ex}_{\mathrm{g}\mathrm{a}\mathrm{i}\mathrm{n}}+{Ex}_{\mathrm{r}\mathrm{e}\mathrm{f}\mathrm{l}}+{Ex}_{\mathrm{l}\mathrm{o}\mathrm{s}\mathrm{s}}+{Ex}_{\mathrm{d}\mathrm{e}\mathrm{s}\mathrm{t}}$$

(33)

Equation (33) shows that the incoming exergy from the sun is partly recovered via the suction, a part is reflected and not absorbed, a part is lost because of the heat loss, and finally, the rest is degraded due to heat transfer with a finite temperature difference.

The inlet exergy from the sun is obtained by multiplying the inlet energy from the sun (Equation (24)) by the maximum efficiency [30]:

$${Ex}_{\mathrm{i}\mathrm{n}}=Q_{\mathrm{i}\mathrm{n}}\left[1-\frac{T_{\mathrm{\infty }}}{T_{\mathrm{s}\mathrm{u}\mathrm{n}}}\right]=\frac{A\left(h_{\mathrm{c}}+h_{\mathrm{r}}\right)\left(T_{\mathrm{w}}-T_{\mathrm{\infty }}\right)}{\alpha }\left[1-\frac{T_{\mathrm{\infty }}}{T_{\mathrm{s}\mathrm{u}\mathrm{n}}}\right]$$

(34)

In the above equation, $T_{\mathrm{s}\mathrm{u}\mathrm{n}}~4500 \mathrm{K}$ is the effective sun temperature [31].

The exergy gain at the outlet of the collector (on the slit) is [13]:

$${Ex}_{\mathrm{g}\mathrm{a}\mathrm{i}\mathrm{n}}=\dot{m}Cp\left[\left(T_{\mathrm{s}\mathrm{l}}-T_{\mathrm{\infty }}\right)-T_{\mathrm{\infty }}\ln \left(\frac{T_{\mathrm{s}\mathrm{l}}}{T_{\mathrm{\infty }}}\right)\right]$$

(35)

The exergy loss due to the reflection from the absorber is:

$${Ex}_{\mathrm{r}\mathrm{e}\mathrm{f}\mathrm{l}}=Q_{\mathrm{i}\mathrm{n}}(1-\alpha )\left[1-\frac{T_{\mathrm{\infty }}}{T_{\mathrm{s}\mathrm{u}\mathrm{n}}}\right]$$

(36)

The exergy loss from the absorber due to the radiation and convection is:

$${Ex}_{\mathrm{l}\mathrm{o}\mathrm{s}\mathrm{s}}=\underset{{Ex}_{\mathrm{l}\mathrm{o}\mathrm{s}\mathrm{s},\mathrm{r}}}{\underbrace{Q_{\mathrm{r}}\left[1-\frac{T_{\mathrm{\infty }}}{T_{\mathrm{w}}}\right]}}+\underset{{Ex}_{\mathrm{l}\mathrm{o}\mathrm{s}\mathrm{s},\mathrm{c}}}{\underbrace{\left[Q_{\mathrm{c}}-\dot{m}Cp\left(T_{\mathrm{s}\mathrm{l}}-T_{\mathrm{\infty }}\right)\right]\left[1-\frac{T_{\mathrm{\infty }}}{T_{\mathrm{w}}}\right]}}$$

(37)

The first term is the exergy loss due to radiation heat loss, while the second term is exergy loss due to convection heat loss.

The exergy destruction is [13]:

$${Ex}_{\mathrm{d}\mathrm{e}\mathrm{s}\mathrm{t}}=\underset{{Ex}_{\mathrm{d}\mathrm{e}\mathrm{s}\mathrm{t},\mathrm{c}}}{\underbrace{\dot{m}Cp{T}_{\mathrm{\infty }}\left[\ln \left(\frac{T_{\mathrm{s}\mathrm{l}}}{T_{\mathrm{\infty }}}\right)-\frac{T_{\mathrm{s}\mathrm{l}}-T_{\mathrm{\infty }}}{T_{\mathrm{w}}}\right]}}+\underset{{Ex}_{\mathrm{d}\mathrm{e}\mathrm{s}\mathrm{t},\mathrm{a}\mathrm{b}\mathrm{s}}}{\underbrace{\alpha .Q_{\mathrm{i}\mathrm{n}}\left[\frac{T_{\mathrm{\infty }}}{T_{\mathrm{w}}}-\frac{T_{\mathrm{\infty }}}{T_{\mathrm{s}\mathrm{u}\mathrm{n}}}\right]}}$$

(38)

where the first term is destruction due to convection heat transfer with a finite temperature difference and the second term is exergy destruction due to absorption from sun temperature to wall temperature.

The second-law efficiency is defined as the ratio of gained exergy to inlet exergy from the sun [1]. So:

$$\begin{gathered} {\eta }_{\mathrm{I}\mathrm{I}}=\frac{{Ex}_{\mathrm{g}\mathrm{a}\mathrm{i}\mathrm{n}}}{{Ex}_{\mathrm{i}\mathrm{n}}} \\ =\left[\frac{\dot{m}Cp.\alpha }{A\left(h_{\mathrm{c}}+h_{\mathrm{r}}\right)\left(T_{\mathrm{w}}-T_{\mathrm{\infty }}\right)}\right]\left[\frac{\left(T_{\mathrm{s}\mathrm{l}}-T_{\mathrm{\infty }}\right)-T_{\mathrm{\infty }}\ln \left(\frac{T_{\mathrm{s}\mathrm{l}}}{T_{\mathrm{\infty }}}\right)}{1-\frac{T_{\mathrm{\infty }}}{T_{\mathrm{s}\mathrm{u}\mathrm{n}}}}\right] \end{gathered}$$

(39)

Replacing Equations (11), (27) and (32), the second-law efficiency will be:

$${\eta }_{\mathrm{I}\mathrm{I}}=\left[\frac{Re.Pr.\alpha }{{Nu}_{\mathrm{c}}(1+{Nu}_{\mathrm{r}}/{Nu}_{\mathrm{c}})}\frac{D}{A}\right]\frac{\left(T_{\mathrm{s}\mathrm{l}}-T_{\infty }\right)}{\left(T_{\mathrm{w}}-T_{\mathrm{\infty }}\right)} \left[\frac{1-\frac{T_{\infty }}{\left(T_{\mathrm{s}\mathrm{l}}-T_{\infty }\right)}\ln \left(\frac{T_{\mathrm{s}\mathrm{l}}}{T_{\infty }}\right)}{1-\frac{T_{\mathrm{\infty }}}{T_{\mathrm{s}\mathrm{u}\mathrm{n}}}}\right]$$

(40)

Using Equation (29), the above equation is simplified to:

$${\eta }_{\mathrm{I}\mathrm{I}}=\left[\frac{Re.Pr.\epsilon .\alpha }{{Nu}_{\mathrm{c}}(1+{Nu}_{\mathrm{r}}/{Nu}_{\mathrm{c}})}\frac{D}{A}\right]\left[\frac{1-\frac{\ln \left(1+\epsilon \left({\theta }_{\mathrm{w}}-1\right)\right)}{\epsilon \left({\theta }_{\mathrm{w}}-1\right)}}{1-\frac{1}{{\theta }_{\mathrm{s}\mathrm{u}\mathrm{n}}}}\right]$$

(41)

Alternatively, using Equation (31),

$${\eta }_{\mathrm{I}\mathrm{I}}={\eta }_{\mathrm{I}}\left[\frac{1-\frac{\ln \left(1+\epsilon \left({\theta }_{\mathrm{w}}-1\right)\right)}{\epsilon \left({\theta }_{\mathrm{w}}-1\right)}}{1-\frac{1}{{\theta }_{\mathrm{s}\mathrm{u}\mathrm{n}}}}\right]$$

(42)

where $\theta$ is the ratio of temperature to ambient temperature: $\theta =T/T_{\mathrm{\infty }}$.

3.4. Analysis of a Real Case

In a real case, instead of temperature, the heat flux over the absorber tube is known, and is not uniform. Furthermore, besides natural convection, radiation heat loss shall be considered. In a real case, a parabolic trough collector with a focal length $f=0.1$ m and a width of 0.4 m is considered. The absorber tube in this collector is made of aluminum with an outer diameter of $D=0.06$ m and a thickness of 4 mm. The reflectivity of the parabolic reflector is $\rho =0.95$ and the absorptivity of the absorber tube is $\alpha =0.9$. All materials are gray. Geometrical parameters are summarized in

Table 2. Geometrical parameters of the trough collector.

Parameter	Value
Reflector width	0.4	M
Reflector focal length	0.1	m
Tube outer diameter	0.06	m
Tube thickness	4	mm
Slit width	8	mm
${\varphi }_{\mathrm{r}\mathrm{i}\mathrm{m}}$	$\pi /2$	rad
${\varphi }_{\mathrm{s}\mathrm{h}\mathrm{d}}$	0.298	rad
A	0.18 *	m2/m

* Area of one meter of absorber tube excluding slit.

.

For a parabolic collector with an aperture plane normal to incident sun rays, the flux distribution over the tube is [32]:

$$q=\left\{\begin{array}{cc}0& \left|\varphi \right|\le {\theta }_{\mathrm{s}\mathrm{h}\mathrm{d}}\\ \frac{4f.\rho .I}{D\left(1+\cos \left(\varphi \right)\right)}& {\varphi }_{\mathrm{s}\mathrm{h}\mathrm{d}}<\left|\varphi \right|\le {\varphi }_{\mathrm{r}\mathrm{i}\mathrm{m}},{\varphi }_{\mathrm{r}\mathrm{i}\mathrm{m}}\le \frac{\pi }{2}\\ 0& {\mathsf{\theta }}_{\mathrm{r}\mathrm{i}\mathrm{m}}<\left|\varphi \right|\le \frac{\pi }{2}\\ -I.\cos \left(\varphi \right)& \frac{\pi }{2}<\left|\varphi \right|\le \pi ,{\varphi }_{\mathrm{r}\mathrm{i}\mathrm{m}}\le \frac{\pi }{2}\end{array}\right.$$

(43)

In the above equation, $I=1000$ W/m² is the solar intensity. The other parameters used are illustrated in Figure 11. For one meter of the tube, the absorbed solar radiation (per Equation (44)) is $\alpha .Q_{\mathrm{i}\mathrm{n}}=333.34$ W/m.

$$\alpha .Q_{in}={\int }_A\alpha .q.dA$$

(44)

For the above collector, the absorber tube was modeled based on the previous procedure. To study the collector analytically, radiation and natural convection heat loss shall be calculated first. Radiation and natural convection heat loss ($Q_{\mathrm{r}}$, $Q_{\mathrm{c}}$) are calculated using Equation (45).

$$\left[\begin{array}{c}{Q}_r\\ Q_c\end{array}\right]=\left[\begin{array}{c}{h}_r\\ h_c\end{array}\right]A\left(T_w-T_{\infty }\right)$$

(45)

To calculate the tube wall temperature, the procedure shown in Figure 12 is used.

Finally, one should guess $T_{\mathrm{s}\mathrm{l}}$ and then calclulate $Re$ at $T_{\mathrm{s}\mathrm{l}}$. Effectiveness is then calculated using Equation (21), and $T_{\mathrm{s}\mathrm{l}}$ is calculated using Equation (15). This loop shall be repeated until the difference between the calculated and the guessed values approaches zero. The results are presented in

Table 3. Calculation of $T_{\mathrm{s}\mathrm{l}}$ using analytical and numerical methods.

$\dot{\mathit{m}}$	$\mathit{R}\mathit{e}$	$\mathit{\epsilon }$	${\mathit{T}}_{\mathbf{s}\mathbf{l}}$ (Analytical)	${\mathit{T}}_{\mathbf{s}\mathbf{l}}$ (Numerical)
(kg/s·m)	(---)	(---)	(K)	(K)
0.0001	8.95	0.801	385.9	385.8
0.0005	45.64	0.706	375.8	374.0
0.001	93.93	0.573	361.4	361.7
0.002	196.47	0.374	340.1	339.0
0.003	302.47	0.264	328.3	326.1
0.004	409.62	0.200	321.5	319.4

. Furthermore, the intermediate results are also presented in

Table 4. Intermediate results in the calculation of $T_{\mathrm{s}\mathrm{l}}$ using the analytical method.

$\beta =0.003$ (1/K)	$h_{\mathrm{r}}=7.98$ (W/m2 K)
$Ra=1.17\times {10}^6$	$Q_{\mathrm{c}}=154.6$ (W/m)
$Nu=15.77$	$Q_{\mathrm{r}}=178.8$ (W/m)
$h_{\mathrm{c}}=9.23$ (W/m2 K)	$T_{\mathrm{w}}=407.3$ (K)

for more clarity.

Based on the results in Table 3, the maximum difference in the value of $T_{\mathrm{s}\mathrm{l}}$ between analytical and numerical methods is 2.2 K, which shows the accuracy of the analytical method. Figure 13 shows the temperature rise relative to ambient temperature, as well as energy gain. Based on this figure, for the current selected mass flow rates, the temperature rise varies between 85.8 K and 19.4 K, while energy gain varies between 8.8 W/m and 88.6 W/m. At a suction flow rate of around 0.002 kg/s.m, the useful energy reaches its saturated value. At this stage, the thermal boundary layer is completely sucked in, and after that, fresh air is drawn in in addition to it. Therefore, for a higher flow rate, the energy gain does not change significantly, and the rise in temperature is also reduced.

For the above case, the variations in effectiveness, as well as first- and second-law efficiencies, are plotted in Figure 14. Based on this figure, we can see that the second-law efficiency is much smaller (one order of magnitude) than the first-law efficiency. This conclusion was observed before in other research [1]. Furthermore, there is an optimum point at which the value of second-law efficiency is at its maximum. The optimum values are presented in

Table 5. Optimum values.

$\mathit{R}\mathit{e}$	$\mathit{\epsilon }$	${\mathit{\eta }}_{\mathbf{I}}$	${\mathit{\eta }}_{\mathbf{I}\mathbf{I}}$	${\mathit{T}}_{\mathbf{s}\mathbf{l}}$ (K)
120.4	0.51	0.376	0.0325	354.2

.

Based on Table 5, at the optimum working point, the outlet temperature is 354.2 K. This shows that the ambient temperature has risen more than 54 K. This considerable temperature rise shows the superiority of this air heater in comparison to the common flat-type air heaters. It must be emphasized here that the elimination of the glass cover over the tube absorber reduces the maintenance and the capital cost in comparison to concentrating collectors.

To clarify the existence of an optimal working point as well as the contribution of each exergy loss or destruction mechanisms, the dimensionless forms of exergy losses and destruction are presented:

$$B_{\mathrm{r}\mathrm{e}\mathrm{f}\mathrm{l}}=\frac{{Ex}_{\mathrm{r}\mathrm{e}\mathrm{f}\mathrm{l}}}{{Ex}_{\mathrm{i}\mathrm{n}}}=(1-\alpha )$$

(46)

$$B_{\mathrm{l}\mathrm{o}\mathrm{s}\mathrm{s},\mathrm{r}}=\frac{{Ex}_{\mathrm{l}\mathrm{o}\mathrm{s}\mathrm{s},\mathrm{r}}}{{Ex}_{\mathrm{i}\mathrm{n}}}$$

(47)

$$B_{\mathrm{l}\mathrm{o}\mathrm{s}\mathrm{s},\mathrm{c}}=\frac{{Ex}_{\mathrm{l}\mathrm{o}\mathrm{s}\mathrm{s},\mathrm{c}}}{{Ex}_{\mathrm{i}\mathrm{n}}}$$

(48)

$$B_{\mathrm{d}\mathrm{e}\mathrm{s}\mathrm{t},\mathrm{c}}=\frac{{Ex}_{\mathrm{d}\mathrm{e}\mathrm{s}\mathrm{t},\mathrm{c}}}{{Ex}_{\mathrm{i}\mathrm{n}}}$$

(49)

$$B_{\mathrm{d}\mathrm{e}\mathrm{s}\mathrm{t},\mathrm{a}\mathrm{b}\mathrm{s}}=\frac{{Ex}_{\mathrm{d}\mathrm{e}\mathrm{s}\mathrm{t},\mathrm{a}\mathrm{b}\mathrm{s}}}{{Ex}_{\mathrm{i}\mathrm{n}}}=\alpha .\left[\frac{1}{{\theta }_{\mathrm{w}}}-\frac{1}{{\theta }_{\mathrm{s}\mathrm{u}\mathrm{n}}}\right]/\left[1-\frac{1}{{\theta }_{\mathrm{s}\mathrm{u}\mathrm{n}}}\right]$$

(50)

In this case, $B_{\mathrm{d}\mathrm{e}\mathrm{s}\mathrm{t}, \mathrm{a}\mathrm{b}\mathrm{s}}=64.7\%$ is the main source of exergy destruction. However, this is inevitable in low-temperature solar applications, because there is a large gap between the sun’s temperature and that of the solar absorber. This form of exergy destruction is the main reason for low second-law efficiency. Apart from the $B_{\mathrm{d}\mathrm{e}\mathrm{s}\mathrm{t}, \mathrm{a}\mathrm{b}\mathrm{s}}$, other forms are presented in Figure 15. $B_{\mathrm{l}\mathrm{o}\mathrm{s}\mathrm{s},\mathrm{r}\mathrm{e}\mathrm{f}\mathrm{l}}$ is the part of the incoming exergy that is not absorbed and reflected. Since it depends only on absorptivity, it is constant and does not vary with $Re$. For a constant heat flux, absorber wall temperature, and consequently radiation loss, are constant. So, $B_{\mathrm{l}\mathrm{o}\mathrm{s}\mathrm{s}, \mathrm{a}\mathrm{b}\mathrm{s}}$ is also constant and independent of $Re$. However, the behaviors of $B_{\mathrm{l}\mathrm{o}\mathrm{s}\mathrm{s},\mathrm{c}}$ and $B_{\mathrm{d}\mathrm{e}\mathrm{s}\mathrm{t}, \mathrm{c}}$ are basically different from those of the others.

Based on Figure 15, by increasing the suction value and consequently $Re$ (See Equation (17)), the part of energy that is recovered increases, and as a result the convection loss as well as $B_{\mathrm{l}\mathrm{o}\mathrm{s}\mathrm{s},\mathrm{c}}$ decreases. However, the temperature of air outlet from the collector decreases, resulting in the increase in $B_{\mathrm{d}\mathrm{e}\mathrm{s}\mathrm{t}, \mathrm{c}}$ or ${Ex}_{\mathrm{g}\mathrm{a}\mathrm{i}\mathrm{n}}$. A summation of these two factors is also shown in Figure 15. As expected, there is a minimum value for the summation of these two factors. In the next section, the optimum value will be presented.

The last parameter that should be studied is the solar heat flux. For the above collector, at the optimum working point, i.e., $Re=120.4$ or $\dot{m}=0.0013$ kg/s.m, the heat flux was varied between 1000 W/m² and 2000 W/m². Figure 16 shows the variation in temperature rise, with respect to heat flux. Based on this figure, as the received solar heat flux increases, the temperature rise at the slit increases almost linearly. Consequently, the thermal effectiveness of the collector increases also. Furthermore, as is expected, the second-law efficiency increases too. This is because, at a constant air mass flow rate, by increasing the air temperature at the slit, the exergy gain increases according to Equation (34). The variations in the effectiveness as well as the second-law efficiency with respect to heat flux are shown in Figure 17.

The variation in the second-law efficiency can also be observed in Figure 17. Unlike the second-law efficiency, the first-law efficiency is reduced with the increase in the solar heat flux. The main reason is that at higher solar heat flux and thus higher tube wall temperature, the radiation heat loss is higher, and as a result, the efficiency decreases.

4. UTTA Optimization

For a known geometry, the main challenge is determining the optimum working condition, i.e., optimum effectiveness. According to Figure 15, it is known that there is a suction value at which the second-law efficiency is maximized. Unfortunately, there is no analytical solution for the optimum working condition. Therefore, the optimization was performed numerically for 100 different cases for $d$ = 0.02 to 0.1 m and $Q_{\mathrm{i}\mathrm{n}}$ = 50 to 2000 W/m. A subroutine was written in MATLAB to calculate the second law efficiency for a wide range of Re and the specified different working conditions, and for each Re range, the maximum value was derived using the adaptive Newton–Raphson method. The optimum working condition can be approximated by the following curve-fitted equation with R² = 99.94% and RMSE = 0.0052.

$${\theta }_{\mathrm{s}\mathrm{l}}-1=0.4898{\left({\theta }_{\mathrm{w}}-1\right)}^{0.9665}$$

(51)

The interesting result of Equation (51) is that the effectiveness at the optimum working condition is around 0.5, with a weak dependency on ${\theta }_{\mathrm{w}}$.

Equation (51) is an applicable equation that can be helpful in the design of a UTTA. For example, for a tube with a diameter of 0.03 m and a slit width of 0.008 m, it is known that the absorbed heat is 400 W/m and the ambient temperature is 300 K. Under this condition, by following the procedure shown in Figure 12, ${\theta }_{\mathrm{w}}=1.64$. It just remains to find the best working condition. Based on Equation (15), to reach the optimum working condition, ${\theta }_{\mathrm{s}\mathrm{l}}$ shall be 1.32. This means that the outlet air temperature shall be 396 K. As a result, the collector’s effectiveness is 0.5. Using Equations (21) and (27), the fluid capacity of the collector is $\dot{m}$ = 0.00193.

5. Summary

In this paper, a novel tubular solar air heater was introduced. The thickness of the thermal boundary layer in natural convection is thick atop a horizontal tube. So, this offers a great source of energy that can be sucked into the tube and used. Furthermore, in this type of collector:

-

The glass tube is eliminated, which results in a considerable reduction in construction and maintenance costs;

-

Hot air can be mixed with fresh air to derive an adequate amount of warm air at the desired temperature. So, the air outlet temperature is under the control;

-

The mixing can be done at the destination, and therefore, smaller ducts are required to transfer hot air;

-

The temperature rise in this collector is relatively high. A higher temperature for thermal energy means a higher level of availability and exergy.

Several different cases were simulated, and based on the results, the suction of the boundary layer can be categorized into three zones:

-

Low suction—Only a narrow layer of gas adjacent to the wall is drawn in and the effectiveness has a weak dependency on $Ra$. The effectiveness approaches very slowly to a value less than unity when the suction value reduces toward zero. Consequently, most of the thermal energy is wasted and nothing is recovered;

-

Moderate suction—A part of the thermal boundary layer is sucked. So, only a part of thermal energy is recovered and the rest is wasted. The effectiveness depends on the ratio of $Ra$ and $Re$;

-

High suction—Fresh air in addition to the boundary layer is sucked. So, almost nothing is wasted. From this point on, the amount of total energy is fixed, and by increasing the air flow rate, temperature rise proportionally decreases.

A correlation was proposed to calculate effectiveness as a function of the Reynolds number and Rayleigh number. In the end, the first-law and second-law efficiencies for the case under consideration were calculated for different suction values. At the optimum point where the second-law efficiency is maximized, temperature rise can reach 54 K, which is much higher than the working zone of a flat plate collector. Based on the exergy analysis, exergy destruction in absorption is the dominant factor. This mode of destruction is unavoidable in low-temperature collectors, and is the main source of low efficiency.

Finally, a correlation was proposed to find the optimum working condition. It was shown that in the optimum working condition, the effectiveness is around 0.5.

This new type of solar collector can produce air at higher temperatures than flat plate collectors. Therefore, it can extend the use of solar energy in daily life. Since the performance of different solar collectors as well as the proposed solar collector in this research is affected by wind, it is suggested to investigate the effects of wind on the performance of such collectors in future research.