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]:
which shows the value of thermal energy gained at the slit to the maximum theoretically accessible thermal energy.
is the outlet air temperature at the slit (see
Figure 3). The two other contributing parameters are:
where
is the slit width and
is the air velocity at the slit entrance.
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, .
For a series of solutions, the variation in
with respect to
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
(over a vertical flat plate,
[
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
. 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
and
, 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
and
. 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
,
Using Equations (11), (15) and (17), the non-dimensional form of the above equation (Equation (18)) is:
is the heat transfer area per unit length of the tube. Finally, using Equation (10), the final form of effectiveness at high
will be
Based on the above explanation and through trial and error, the following correlation is proposed for
and
.
in which
In the above equation, for , the effectiveness approaches , 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
, Equation (21) reduces to:
For very low
, the form of the equation shows a very weak dependency on
, and for very high
, the equation approaches the analytical result (Equation (20)). The value of R
2 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
and
. Based on this figure, effectiveness decreases monotonically with increasing suction (
. Moreover,
is higher at higher
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 (
) shall be equal to energy losses. The energy balance for a UTTA is:
This means that the absorbed solar heat input (
is wasted by radiation and convection [
29].
In the above equation,
is the Stefan–Boltzmann constant and
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:
This can be presented as a function of dimensionless parameters. Starting from Equation (17), the air mass flow rate per unit length is:
Alternatively, using Equations (11) and (15),
Replacing the above equations as well as Equation (24) in Equation (26) yields:
Knowing that
and using Equation (28), the above equation is simplified to:
where
3.3. Exergy Analysis of a UTTA
The exergy balance for a UTTA can be written as:
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]:
In the above equation,
is the effective sun temperature [
31].
The exergy gain at the outlet of the collector (on the slit) is [
13]:
The exergy loss due to the reflection from the absorber is:
The exergy loss from the absorber due to the radiation and convection is:
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]:
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:
Replacing Equations (11), (27) and (32), the second-law efficiency will be:
Using Equation (29), the above equation is simplified to:
Alternatively, using Equation (31),
where
is the ratio of temperature to ambient temperature:
.
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
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
m and a thickness of 4 mm. The reflectivity of the parabolic reflector is
and the absorptivity of the absorber tube is
. All materials are gray. Geometrical parameters are summarized in
Table 2.
For a parabolic collector with an aperture plane normal to incident sun rays, the flux distribution over the tube is [
32]:
In the above equation,
W/m
2 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
W/m.
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 (
,
) are calculated using Equation (45).
To calculate the tube wall temperature, the procedure shown in
Figure 12 is used.
Finally, one should guess
and then calclulate
at
. Effectiveness is then calculated using Equation (21), and
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. Furthermore, the intermediate results are also presented in
Table 4 for more clarity.
Based on the results in
Table 3, the maximum difference in the value of
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.
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:
In this case,
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
, other forms are presented in
Figure 15.
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
. For a constant heat flux, absorber wall temperature, and consequently radiation loss, are constant. So,
is also constant and independent of
. However, the behaviors of
and
are basically different from those of the others.
Based on
Figure 15, by increasing the suction value and consequently
(See Equation (17)), the part of energy that is recovered increases, and as a result the convection loss as well as
decreases. However, the temperature of air outlet from the collector decreases, resulting in the increase in
or
. 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.,
or
kg/s.m, the heat flux was varied between 1000 W/m
2 and 2000 W/m
2.
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.