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This work presents an algorithm able to simulate the heating of a solar collector throughout the day. The discussed collector is part of a solar adsorption refrigerator, and is used to regenerate the activated carbon contained inside a cylindrical recipient (absorber), which is located in the focal line of a parabolic trough concentrator. The developed algorithm takes into account all the transfer mechanisms when analyzing the heat transfers taking place between the collector’s components and the environment, as well as the transfer mechanisms towards the absorber’s interior. The temperature evolution for the collector’s elements is obtained, and the model is validated by comparing the experimentally measured surface temperature of the absorber with the one determined by the algorithm. The experimental data were gathered from similar collectors in two different scenarios: Santo Domingo (Dominican Republic) and Buenos Aires (Argentina). The model is satisfactorily validated with experimental data.

In the last decades, the research about adsorption refrigeration systems has risen since it can offer an alternative which is less harmful to the environment than the conventional refrigeration systems. Unlike the traditional vapor-compression systems, the adsorption refrigeration technology uses refrigerants with zero ozone depletion potential and zero global warming potential [

While the traditional refrigeration systems convert the consumed electricity into mechanical work in order to compress the working fluid, the functioning of an adsorption refrigeration system relies on the variable adsorbent capacity of a given refrigerant pair, which is caused by a temperature variation in the adsorbent bed, making that this kind of device can be easily powered by solar energy or waste heat.

When driven by solar energy, an adsorption refrigeration system can be built from affordable materials and its construction is relatively simple. Additional features such as silent operation and zero operative costs, make this kind of system worth implementing.

Many researchers are seeking to develop this type of refrigeration system, which could be used for everyday comfort as well as for higher priority purposes, such as vaccines and food storage, especially in those inhabited areas where the non-existent electricity distribution networks make difficult and expensive the utilization of the conventional refrigeration systems. Also regions with abundant solar energy resources could benefit from this king of technology. However, there are major drawbacks of the solar adsorption refrigeration (SAR) systems to be addressed: their low coefficient of performance compared with the traditional refrigeration systems, its discontinuous cold production, and the fact that the external operation conditions strongly affect the performance of the SAR cooling device; making difficult to predict its behavior.

A solar adsorption refrigeration system is based in the dynamics of the adsorbent-adsorptive refrigeration pair, which manifests as the capacity of the adsorbent material to adsorb and desorb the adsorptive fluid through the day.

A solar adsorption refrigeration device, as can be seen on

The refrigeration cycle of this device has been described by many researchers [

These devices are able to produce the cold intermittently, in a one-day cycle. For that reason is essential to produce a certain amount of ice during the night, in order to maintain a 0˚C temperature in the cold chamber or perhaps allow a minimal temperature change during the day.

Some researchers have performed a thermodynamic design of a solar adsorption refrigerator [

In order to attain a more precise prediction of the behavior of a solar adsorption refrigerator, it is fundamental to understand the processes taking place in the absorber, where the refrigerant pair interact. Some research have been done concerning the dynamic of sorption; Luo and Tounder [

In realistic conditions, the temperature on the surface of the absorber changes throughout the day, and this variation is influenced by multiple factors. Hence, in order to accurately determine the temperature evolution in the surface of the absorber, it is indispensable to develop a model that takes into account the absorber’s net incoming energy and the energy consumed by the desorption process.

The objective of this study is to develop a physical model able to simulate the thermal behavior of a solar collector, which takes in account the climatic factors of the place where it is located, such as the solar radiation and the environmental temperature. The solar collector model could be coupled to models of devices that are powered by solar energy such as heat pumps, water heaters and, as intended in this work, adsorption refrigerators.

In order to obtain the usable heat entering the absorber tube, it is required to determine the incident solar radiation directed towards the absorber, and it is necessary to calculate the absorber’s heat losses by performing a thermal analysis of the solar collecting system and its heat transfers with the absorber tube and with the environment.

The analyzed adsorption refrigeration device uses two parabolic trough collectors (PTCs), where each one is composed by a parabolic-shaped reflector in whose focal line is located the absorber. In this work, only half of it (one PTC) is analyzed, assuming that the other half would behave in the same way.

Thermal analysis of this kind of solar collectors has been made by some researchers [

The absorber of the parabolic trough collector analyzed in this work, unlike the ones mentioned above, is not bounded by a cylindrical concentric crystal; but by a thermally isolated trapezoidal enclosure as shown in

In this work we determine the energy flow towards the absorber throughout the day, by performing an analysis of the transfers of energy taking place between the elements of the collector, considering all the heat transfer mechanisms. The model includes an algorithm to obtain the solar irradiation as a function of time, which also depends on the geographic location of the prototype and the

day of the year. The model is coupled with the mathematical model of the absorber of a solar adsorption refrigerator developed by Echarri et al. [

The numerical model is based on an energy balance. In this balance the solar irradiation, the optical losses, as well as the thermal losses, are analyzed in order to obtain the net heat gained by the absorber. This net heat would be the responsible for temperature change on the surface of the absorber, and also for the desorption, which is a temperature-dependent process.

In order to determine the incident solar irradiation ( q ˙ i n c ), the Hottel’s model is used [

Once the irradiance is known, it is possible to estimate the solar radiation received by the upper and lower half of the absorber.

The upper half receives the solar irradiance (both direct and diffuse components), obtaining the expression (1) for the absorbed energy per unit length.

q ˙ i n c − u p p e r = 2 r q ˙ s o l (1)

The lower half receives the direct radiation ( q ˙ d i r ) which is redirected by the concentrator. This magnitude depends on the captation area of the concentrator ( A c ), as well as on the reflectivity (R) of the material the concentrator is built, obtaining the expression (2).

q ˙ i n c − l o w e r = A c R q ˙ d i r (2)

The final expression for the incident radiation is as seen on equation (3). The incident radiation is further conditioned by the transmissivity of the glass cover and the absorptivity of the absorber’s material. The product of these parameters ( L o p t ) is a factor representing the optical losses [

q ˙ i n c = ( q ˙ i n c − u p p e r + q ˙ i n c − l o w e r ) L o p t (3)

During the day, the solar irradiation is concentrated and directed towards the absorber, causing a temperature increment in its surface. Then, since the absorber is not in thermal equilibrium with the rest of the collector, it transfers energy by convection to the air trapped in the enclosure, which subsequently starts transferring energy to the walls of the enclosure.

Given the amount of energy gained for the walls of the enclosure by convection with the air, and also for the radiative transfer involving the other elements of the PTC, the walls raise their temperature. To the increase of the temperature of the walls, follows an energy transfer (radiative and convective) between the walls and the environment.

As seen on

q ˙ n e t = q ˙ i n c − q ˙ c o n v − a b s − q ˙ r a d − a b s (4)

In a similar way, the energy balance for each wall of the enclosure can be obtained as follows:

q ˙ s = q ˙ c o n v − w i + q ˙ r a d − w i − q ˙ c o n v − w e − q ˙ r a d − w e (5)

Assuming that the temperature around the circumference of the absorber is uniform, the convection heat transfer between the absorber and the air trapped inside the enclosure is determined by using Newton’s cooling law;

q ˙ c o n v − a b s = h A ( T a b s − T a i r ) (6)

where, T a b s is the temperature of the absorber, T a i r is the temperature of the air at an intermediate point between the absorber and the walls, A is the heat transfer area, and h is the convection heat transfer coefficient. Since the

cross-sectional area of the absorber cylinder is negligible compared to its transverse area, only the last one is considered for the heat exchanges calculations.

The convection heat transfer coefficient is evaluated through the dimensional analysis and similitude method. In this method, the conservation equations are expressed as a function of non-dimensional parameters and can be solved analytically or empirically. Once solved, the results can be applied to surfaces geometrically similar.

In this work, the convection heat transfer coefficient is determined as a function of the Nusselt number (Nu);

h = k N u L (7)

where, k is the thermal conductivity and L is the characteristic longitude of the given element.

For natural convection on horizontal cylinders, the correlation used to determine the Nusselt number, is the one proposed by Churchill and Chu [

N u = { 0.6 + 0.378 [ R a ( 1 + ( 0.559 / P r ) 9 / 16 ) 16 / 9 ] 1 / 6 } 2 (8)

The Rayleigh number used in expression (8) is given by expression (9) [

R a = g β ( T a b s − T a i r ) D 3 ν 2 P r (9)

The convection of the internal faces and the enclosed air is as shown on Equation (10), while the convection of the external faces and the environment (convection is assumed natural) is shown on Equation (11).

q ˙ c o n v − w i = h i A ( T a i r − T w − i n ) (10)

q ˙ c o n v − w e = h e A ( T w − e x − T a ) (11)

where T w − i n is the temperature of the internal surface of the collector wall/glass cover, T w − e x is temperature of the external surface of the collector wall/glass cover, T a is the environmental temperature, and h i and h e are the convection heat transfer coefficients in each respective case.

In order to calculate the convection heat transfer coefficient between each of the cover surfaces with the surrounding air, the dimensional analysis method is used. This way, h i and h e are determined similarly as in Equation (7).

For the cases of convective heat transfer involving the lateral and inferior faces of the enclosure, the empirical correlation found for the Nusselt number is a follows [

N u = { 0.825 + 0.387 [ R a ( 1 + ( 0.492 / P r ) 9 / 16 ) 16 / 9 ] 1 / 6 } 2 (12)

The Rayleigh number in Equation (12) is calculated as in Equation (13) for the cases where h i is being evaluated, and as in Equation (14) when h e is being determined.

R a = g β ( T a i r − T w − i n ) L 3 ν 2 P r (13)

R a = g β ( T w − e x − T a ) L 3 ν 2 P r (14)

For the convective heat transfer involving the glass cover, where the internal face is assumed to be colder than the enclosed air and the external face is assumed to be at higher temperature than the environment, the expression for the Nusselt number is given by [

N u = 0.23 ( G r a P r ) 1 / 4 (15)

where the Grashof number (Gra) is obtained as follows:

G r a = g β ( Δ T ) L 3 ν 2 (16)

The used value for Pr in expressions (8), (9) and (12)-(15) was obtained as the mean value of the Prandtl number for the maximum and minimum expected temperature of the enclosed air (when internal convection was analyzed) and as the as the mean value of Prandtl number for T w − e x and T a (when external convection was examined).

The radiation emitted by the different elements of the PTC is calculated through:

q ˙ r a d − e m = ε σ T 4 A (17)

where ε is the emissivity of the material, σ is the Stefan-Boltzmann constant, T is the absolute temperature and A is the radiating area.

When analyzing the radiation received by the absorber and the inward faces of the enclosure, the expression (18) is used:

q ˙ r a d − r e c = ∑ j ε i j F i j σ T j 4 A j (18)

where i denotes the absorbing surface, j the radiating surface and ε i j refers to the effective emissivity for two surfaces with different emissivity.

The F i j in expression (18) stands for the view factor between the i^{th} and j^{th} surfaces. These factors are determined by the Crossed Strings Method [

The net heat lost by the absorber per unit length can be obtained as following:

q ˙ r a d − a b s = ε a b s σ T a b s 4 A a b s − ∑ j = 1 3 ε j − a b s F j − a b s σ T j 4 A j (19)

Similarly, the net heat gained by the i^{th} wall of the enclosure can be calculated as (being the absorber the 4th surface):

q ˙ r a d − w i = ∑ j = 1 4 ε i j F i j σ T j 4 A j − ε i σ T i 4 A i (20)

The outward surfaces of the collector interact with the environment, so the net radiative flow rate is given by:

q ˙ r a d − w e = ε σ ( T 4 − T a 4 ) A (21)

Since the convective and radiative heat flows have been determined, the sensible heat is obtained. This heat causes a variation in the temperature of the enclosure. The new temperatures of the walls and glass cover of the collector, are calculated from:

q ˙ s = m c Δ T w Δ t (22)

where m is the mass of the wall and c represents the specific heat of the material.

After obtaining this new central temperature, and assuming a linear temperature profile (as seen on

q ˙ i n = q ˙ c o n v − w i + q ˙ r a d − w i = − k A T w − T w − i n e / 2 (23)

q ˙ o u t = q ˙ c o n v − w e + q ˙ r a d − w e = − k A T w − e x − T w e / 2 (24)

The temperature of the enclosed air also varies. Assuming that the convective heat being transferred from the absorber to the air is divided towards the inner faces of the enclosure’s walls according to the thermal resistance of each one, the new temperature of the enclosed air can be obtained from Equation (25).

T a i r = ∑ h j A j T j − i n ∑ h j A j (25)

The algorithm is based on the physical model obtained from the heat transfer equations described above. In

In the “parameters and initial values” section is where the environmental temperature is specified and the temperatures of the PTC parts are initialized. In this section are also introduced the parameters for the particular prototype/case being evaluated: the dimensions of the elements of the PTC, view factors, properties of the materials (absorptivity of the absorber tube, reflectivity of the concentrator, transmissivity of the glass cover, thermal conductivity of the PTC elements, captation area of the concentrator, emissivity and specific heat of the PTC elements, Pr and kinematic viscosity of the air) as well as the day of the year of the desired analysis, the coordinates where the given prototype is located and

its height above sea level.

In section “incident radiation”, the total solar radiation directed towards the absorber is determined, after the incoming irradiation is calculated according to the geographic location and the day of the year.

In the “heat transfers” section, the energy fluxes between each pair of elements is analyzed by determining the convection heat transfer coefficients and the net radiative transfer. The sensible heat is calculated, as well as the new temperatures of the walls of the enclosure for the next temporal step.

Since all the heat fluxes are known, the absorbed heat is calculated and given as a boundary condition to the absorber model developed by Echarri et al. [

This process continues along the whole day, making possible to determine the temperature evolution of each one of the elements composing the PTC.

As previously stated, in order to validate the thermal model, it is coupled to the numerical model of an absorber of a solar adsorption refrigerator [

The measurements were carried out with thermocouples (type J) with a working range of (−290˚C to 1,190˚C) with an error of +/−1˚C. The measurements were automatically recorded with the EL-USB-TC Thermocouple Data Logger, manufactured by Lascarc Ekectrinic Inc.

In

prototypes at different locations and different dates. In both cases, simulation results show good agreement with the experimental results.

Since the solar irradiance model employed [

Also, if the PTC is not completely horizontal, the temperature on both absorbers can present a slight difference between them, since the asymmetry is not contemplated in the model and air stratification may occur. In

In

In

of a SAR using the same refrigerant pair, at different places (Santo Domingo: 69.96, 18.5 and Buenos Aires: 58, −34.5) during their respective summer solstice and supposing a constant 25˚C environmental temperature. From this can be seen how the mere fact of the location of a SAR prototype can affect its behavior.

A detailed numerical model based on energy balance of a PTC has been developed. The proposed model included a detailed convective heat transfer analysis in which different empirical correlations have been selected according to the conditions under study and a thermal radiative heat transfer analysis based on the crossed-string method.

The thermal model is validated with experimental results available after the construction of prototypes, showing a quite good agreement. Results obtained matched experimental ones, although some discrepancies are observed due to unaccounted cloudiness and air stratification effects during the prototype operation.

It is shown that the present model is capable of estimating reasonably well the heat losses and the temperature in the PTC. According to the results obtained, it can be concluded that the current numerical model is suitable for predicting the thermal behavior of the PTC under different operating conditions.

The PTC model developed in this paper can be coupled to absorbers used for different purposes; it can be paired to absorbers with a heat transfer fluid inside, and can be paired (as in this case) to a regenerator of a solar adsorption refrigerator. The model can predict accurately the net heat directed towards the absorber of a solar adsorption refrigerator based on; the structure and material properties of the system, the date and the geographic location of the prototype.

This work has been financially supported by the Ministerio de Educación Superior, Ciencia y Tecnología of Dominican Republic through projects INNOVACION 2012-2013-2E1-28 and 2015-2E4-091.

Mateo, M., Echarri, R. and Samsón, I. (2017) Thermal Analysis and Experimental Validation of Parabolic Trough Collector for Solar Adsorption Re- frigerator. Energy and Power Engineering, 9, 687-702. https://doi.org/10.4236/epe.2017.911044