Effect of heating time of adsorber-collector on the performance of a solar adsorption refrigerator

Abstract

This paper presents a numerical study of heat transfer inside the adsorber-collector of a solar adsorption refrigerator using the activated carbon AC35-methanol pair. The objective is to estimate the amount of the heat loss through the adsorber-collector, during the solar heating phase, and to determine the effect of heating time on the thermal efficiency of the system. The numerical results showed that the heating time is the most important factor affecting the amount of energy loss. It has shown that the shorter heating time corresponds to the higher efficiency of the adsorber-collector. In addition, a new optimal coefficient of performance, COP _optm, is proposed to determine the number of adsorbers to be added to a machine. This latter is considered for consuming an energy equivalent to that received by the adsorber-collector. These additional adsorbers use a heat transfer fluid, coming from the adsorber-collector, instead of direct heating by solar radiation. An application example is presented using experimental results obtained from the literature. It has shown that the number of the additional adsorbers can reach three adsorbers.

Background

The fossil energy is the foundation of the global economy; the consumption, the transport, and the delivery of this energy source are harmful to the human being, the animals, and the nature. We know that the combustion of oil rejects a thousand polluting substances in the atmosphere; one of the well-known direct impact is global warming.

The search for alternative energy sources brought together environmentalists and those who for other reasons do not want to be dependent on oil suppliers. Certainly, other sources of energy may be an alternative to oil, such as solar energy.

Solar power in itself does not present a problem. The problem of its use comes mostly from the investment cost which is relatively expensive compared to fossil energy because the profitability of certain types of solar systems is still weak. The improvements are not intended only for the materials of the solar installation but also for the performance of the system, which includes all energy losses caused by its components.

The ordinary refrigerating machines, even those that are operated by electricity, are indirect consumer devices of petroleum energy, so a system that must be replaced in the future by an equipment belonging to solar power system technology. In the refrigeration machines powered by the electricity, the compressor is the driving force of the thermodynamic cycle of the system, while in refrigeration machines powered by the sun, the phenomenon of desorption by solar heating can be an alternative to the compression phase. This phenomenon takes place in the adsorber-collector of the solar machine; the more the desorption is important, the more the cold production is high.

The estimation of coefficient of heat loss per unit of time and temperature within the adsorber-collector is already developed by Klein 1975 and discussed by Duffie and Beckman 2013. Jing and Exell 1994 used the coefficient of loss developed by Klein 1975 for simulation and sensitivity analysis of an intermittent solar-powered charcoal/methanol refrigerator. Anyanwu et al. 2001 have made a study of an adsorption refrigerator using the charcoal/methanol pair, where the numerical model takes into account loss coefficient. The results of coefficient of performance (COP) are given according to the physical and dimensional characteristics of adsorber-collector. The loss coefficient was also considered by Leite and Daguenet 2000. It was a numerical simulation for instant transfers of heat and mass in each element of the refrigerating adsorption machine during the typical average day of each month. Another use of the coefficient of Klein was in the work of Chekirou et al. 2014, who introduced a modeling of heat and mass transfer in the tubular adsorbent of a refrigerating adsorption machine.

The objective of our study is to analyze the effect of heat loss within the adsorber-collector on the performance of solar adsorption refrigeration machines using AC35-methanol as working pair and to determine the criteria affecting the increase of this amount of energy.

The amount of heat loss through the adsorber-collector was calculated using the coefficient of loss over the solar heating time and an interval of temperatures between the temperature of adsorption (temperature at start of heating) and temperature of generation (temperature at end of heating). The study is done using a numerical program (Fortran) and validated with experimental results obtained from the literature.

Methods

Basic components

The main components of solar adsorption refrigerator are shown in the Fig. 1: the adsorber-collector containing the adsorbent-adsorbate pair (activated carbon AC35-methanol), the air-cooled condenser, and the evaporator within the cold chamber (Rouag et al. 2016).

Fig. 1

Schematic representation of the solar adsorption refrigerator. a Process A-C. b Process C-A

A valve is set at inlet and outlet of adsorber-collector, to control the flow of the refrigerant from the adsorbent towards the condenser during the desorption phase, and from the evaporator towards the adsorbent during the adsorption phase.

Intermittent cycle

Four thermodynamic processes form the intermittent ideal cycle of solar-powered adsorption refrigerator are shown in Fig. 2.

Fig. 2

Clapeyron diagram for adsorption refrigeration cycle

First, the adsorber-collector is heated by solar energy, which increases the pressure of the adsorbent-adsorbate pair up to the condensing pressure P _c necessary for the condenser (processes A–B). In state B of cycle, the valve at the inlet of condenser is open, and the continuing of the heating at constant pressure induces the desorption of refrigerant (adsorbate) from the adsorbent towards the air-cooled condenser to be condensed at condensing temperature T _c (processes B–C).

The desorbed mass during processes B–C is the difference between the maximal mass of adsorbate in state B and the minimal mass of adsorbate in state C.

The maximum temperature (generating temperature T _g) reached by the adsorbent will be in state C. At this value of temperature, the valve is closed and solar irradiance starts to decrease, which induces the drop of temperature and pressure of adsorber-collector (processes C–D).

When the pressure reaches the value P _e, which reigns in the evaporator, the valve is open, and the continuing of the cooling of adsorbent at constant pressure induces the evaporation of refrigerant at an evaporating temperature T _e from the evaporator towards the adsorbent by the adsorption phenomena (processes D–A). By this way, the evaporated refrigerant extracts heat and generates cold production within the cold chamber.

The adsorbed mass during processes D–A is the difference between the maximal mass of adsorbate in state A and the minimal mass of adsorbate in state D.

This cycle is said to be intermittent because the cold production starts only at sunset.

Adsorbed mass

Dubinin and Astakov proposed a state equation for mass adsorbed by microporous medium in equilibrium with polymodal distribution of pore size (Leite and Daguenet 2000):

$$ X\left( T, P\right)={W}_0{\rho}_l(T) exp\left\{- D{\left[ T. ln\left(\frac{P_s(T)}{P}\right)\right]}^n\right\} $$

(1)

Where W ₀ is the maximum volume adsorbed, ρ _l is the density of the adsorbate in the liquid state, D is the coefficient of affinity, T is the temperature of the adsorber-collector, P _s is the saturation pressure of the adsorbate, P is the equilibrium pressure of adsorbent-adsorbate pair, and n is the parameter of adjustment of the D–A equation.

The adsorbate mass concentration maximal X _max and minimal X _min are calculated by Eq. (1), respectively, at the adsorption temperature T _a and the generating temperature T _g as the following:

$$ X\left({T}_{\mathrm{a}},{P}_{\mathrm{e}}\right)= X\left({T}_{\mathrm{s}1},{P}_{\mathrm{c}}\right)={X}_{\max } $$

(2)

$$ X\left({T}_{\mathrm{g}},{P}_{\mathrm{c}}\right)= X\left({T}_{\mathrm{s}2},{P}_{\mathrm{e}}\right)={X}_{\min } $$

(3)

Where T _s1 and T _s2 are respectively the temperature at start of desorption and temperature at start of adsorption.

During the adsorption, the evaporating pressure is P _e = P _s (T _e), and for the desorption, the condensing pressure is P _c = P _s (T _c). T _e and T _c are respectively the evaporating temperature and the condensing temperature.

Thermal efficiency of the adsorber-collector

The total energy received by the adsorber-collector Q _ge can be divided in five parts:

$$ {Q}_{\mathrm{ge}}={Q}_1+{Q}_2+{Q}_3+{Q}_{\mathrm{des}}+{Q}_{\mathrm{L}} $$

(4)

The heats Q ₁, Q ₂, and Q ₃ are respectively energy supplied to heat the adsorbent, energy supplied to heat the tubes in metal, and energy supplied to heat the adsorbate mass. They can be calculated as the following (Chekirou et al. 2011):

$$ {Q}_1={m}_{\mathrm{d}}{\mathrm{Cp}}_{\mathrm{d}}\left({T}_{\mathrm{g}}-{T}_{\mathrm{a}}\right) $$

(5)

m _d is the adsorbent mass and Cp _d is the specific heat of adsorbent.

$$ {Q}_2={m}_{\mathrm{t}}{\mathrm{Cp}}_{\mathrm{t}}\left({T}_{\mathrm{g}}-{T}_{\mathrm{a}}\right) $$

(6)

m _t and Cp _t are respectively the mass and specific heat of metallic tubes containing the adsorbent.

$$ {Q}_3={m}_{\mathrm{d}}\left({X}_{\max }{\displaystyle {\int}_{T_{\mathrm{a}}}^{T_{\mathrm{S}1}}{\mathrm{Cp}}_{\mathrm{l}}(T) dT}+{\displaystyle {\int}_{T_{\mathrm{S}1}}^{T_{\mathrm{g}}} X(T){\mathrm{Cp}}_{\mathrm{l}}(T) dT}\right) $$

(7)

Cp _l is the specific heat of adsorbate in liquid state.

The heat necessary for desorption process Q _des is given by (Chekirou 2008):

$$ {Q}_{\mathrm{d}\mathrm{es}}={m}_{\mathrm{d}} n D{\displaystyle {\int}_{T_{\mathrm{S}1}}^{T_{\mathrm{g}}} X(T){T}^{\mathrm{n}}{\left( ln\frac{P_{\mathrm{s}}(T)}{P_{\mathrm{c}}}\right)}^{n-1}\frac{{q_{\mathrm{s}\mathrm{t}}}^2(T)}{r{ T}^2} dT} $$

(8)

Isosteric heat q _st is calculated as the function of the pressure and temperature:

$$ {q}_{\mathrm{s}\mathrm{t}}= L\left({T}_{\mathrm{c}}\right)+ rTln\left(\frac{P_{\mathrm{s}}(T)}{P_{\mathrm{c}}}\right)+\left[\frac{\upalpha rT}{nD}\right]{\left[ Tln\left(\frac{P_{\mathrm{s}}(T)}{P_{\mathrm{c}}}\right)\right]}^{\left(1- n\right)} $$

(9)

L is the latent heat of the adsorbate, r is the particular gas constant of the adsorbate, and α is the thermal expansion coefficient of the adsorbate.

According to Anyanwu et al. 2001, the heat loss from the lateral surfaces of adsorber-collector is assumed to be negligible.

The global loss coefficient is given by Duffie and Beckman 2013 as follows:

$$ {U}_{\mathrm{L}}={U}_{\mathrm{t}}+{U}_{\mathrm{b}} $$

(10)

U _t is the loss coefficient from the top of the adsorber-collector, and U _b is the loss coefficient from the bottom of the adsorber-collector.

where:

$$ {U}_{\mathrm{t}}\left({T}_{\mathrm{p}},{T}_{\mathrm{am}}\right)={\left[\frac{n_{\mathrm{g}}}{\frac{c_0}{T_{\mathrm{p}}}{\left(\frac{T_{\mathrm{p}}-{T}_{\mathrm{am}}}{n_{\mathrm{g}+\mathrm{f}}}\right)}^{\mathrm{e}}}+\frac{1}{h_{\mathrm{w}}}\right]}^{-1}+\left[\frac{\upsigma \left({T}_{\mathrm{p}}+{T}_{\mathrm{am}}\right)\left({T}_{\mathrm{p}}^2+{T}_{\mathrm{am}}^2\right)}{{\left({E}_{\mathrm{p}}+0.00591{n}_{\mathrm{g}}{h}_{\mathrm{w}}\right)}^{-1}+\frac{2{n}_{\mathrm{g}}+ f-1+0.133{E}_{\mathrm{p}}}{E_{\mathrm{g}}}-{n}_{\mathrm{g}}}\right] $$

(11)

with:

$$ e=0.43\left(1-\frac{100}{T_p}\right) $$

$$ f=\left(1+0.089{h}_w-0.1166{h}_w{E}_p\right)\left(1+0.07866{n}_g\right) $$

$$ {c}_0=520\left(1-0.000051{\boldsymbol{\varOmega}}^2\right) $$

$$ 0{}^{\circ}<\boldsymbol{\varOmega} <70{}^{\circ},\ \mathrm{and}\ \mathrm{f}\mathrm{o}\mathrm{r}\ 70{}^{\circ}<\boldsymbol{\varOmega} <90{}^{\circ}\ \mathrm{we}\ \mathrm{take}\ \boldsymbol{\varOmega} =70{}^{\circ} $$

$$ {h}_w=2.8+3.0{W}_V $$

T _p is the wall temperature of the top of adsorber-collector, T _am is the ambient temperature, n _g is the number of glass cover of adsorber-collector, E _p is the emissivity of the top wall of the adsorber-collector, E _g is the emissivity of the glass cover of the adsorber-collector, Ω is the adsorber-collector inclination, and W _v is the wind velocity.

Chekirou et al. 2014 took U _b as the constant.

To estimate the amount of total heat loss towards the air ambient during processes A–C (Fig. 2), the following equation is used:

$$ {Q}_{\mathrm{L}}= S.\left({U}_{\mathrm{t}}+{U}_{\mathrm{b}}\right).\left( T{p}_{\max }- T{p}_{\min}\right).{t}_{\max } $$

(12)

Using Eq. (12), we cover the full time of heating process t _max on a range of wall adsorber-collector temperatures from Tp _min (temperature of adsorber-collector wall at the start of heating) to Tp _max (temperature of adsorber-collector wall at the end of heating).

The temperatures Tp _max and Tp _min given in Eq. (12) can be substituted respectively by the temperatures T _g and T _a, so:

$$ {Q}_{\mathrm{L}}= S.\left({U}_{\mathrm{t}}+{U}_{\mathrm{b}}\right).\left({T}_{\mathrm{g}}-{T}_{\mathrm{a}}\right).{t}_{\max } $$

(13)

And for Eq. (11), the temperature ambient T _am is considered to be equal to T _a.

It is known that the refrigerator adsorber-collector needs a useful energy Q _u and the remainder is a lost energy Q _L; therefore,

$$ {Q}_{\mathrm{u}}={Q}_{\mathrm{ge}}-{Q}_{\mathrm{L}} $$

(14)

While the recovery heat Q _r from the adsorber-collector is a part or all the lost energy Q _L, so Q _r ≤ Q _L, it varies from 0 to Q _L, then we can write:

$$ {Q}_{\mathrm{r}}={Q}_{\mathrm{L}}\left(1-\frac{t_{\mathrm{h}}}{t_{\max }}\right) $$

(15)

Equation (15) means that Q _r = 0 for a heating time t _h = t _max and Q _r = Q _L for heating time t _h = 0.

On the other hand, the thermal efficiency of the solar-powered adsorber-collector can be expressed as follows:

$$ \upeta =\frac{Q_{\mathrm{u}}}{Q_{\mathrm{ge}}}=\frac{Q_{\mathrm{ge}}-{Q}_{\mathrm{L}}}{Q_{\mathrm{ge}}}=1-\frac{Q_{\mathrm{L}}}{Q_{\mathrm{ge}}} $$

(16)

Coefficient of performance

The coefficient of performance of an adsorptive cooling system is equal to the heat extracted in the evaporator per the heat received in the adsorber-collector (Douss and Meunier 1988):

$$ \mathrm{COP}=\frac{Q_{\mathrm{ev}}}{Q_{\mathrm{ge}}} $$

(17)

Where:

$$ {\mathrm{Q}}_{\mathrm{e}\mathrm{v}}={m}_{\mathrm{d}}\left({X}_{\max }-{X}_{\min}\right)\left[ L\left({T}_{\mathrm{e}}\right)-{\displaystyle {\int}_{T_{\mathrm{e}}}^{T_{\mathrm{c}}} C{p}_{\mathrm{l}}(T) dT}\right] $$

(18)

The first term of Eq. (18) represents the heat absorbed by the evaporation of the refrigerant (adsorbate) at the evaporation temperature T _e. The second term represents the sensible heat necessary to bring the condensate of its condensing temperature T _c to evaporating temperature T _e (Chekirou et al. 2011).

Results and discussion

Tables 1 and 2 give all the relevant data of numerical computation.

Table 1 Data used in the present numerical study

Table 2 Values of Dubinin-Astakhov equation parameters

The pressure of saturation P _s, the latent heat L, density ρ _l, and the mass specific heat of adsorbate (methanol) in the liquid state Cp _l are estimated by interpolation of the data given in Table 3.

Table 3 Thermophysical properties of methanol along the saturation line (Bejan and Kraus 2003)

For each reference of validation, i.e., Douss and Meunier 1988 and Lemmini and Errougani 2007, the mass m _d of the adsorbent (activated carbon AC35) has been taken the same as that considered by the authors. Lemmini and Errougani 2007 have given the value of the adsorbent mass, which is equal to 14.5 kg, and for the machine studied by Douss and Meunier 1988, it is estimated by 12.5 kg.

The same for the mass m _t of metallic tubes containing the adsorbent, which are estimated to be equal to 45.68 and 26.024 kg for Douss and Meunier 1988 and Lemmini and Errougani 2007 respectively.

The maximum time t _max of heating processes A–C (Fig. 2) is considered for all day, so it is an average of 12 h.

Validation of numerical model

First, we have validated our numerical model by the results given by Douss and Meunier 1988. The authors have studied an experimental unit of an adsorption refrigeration machine that operates by electrical heating instead of the solar heating; therefore, heat loss Q _L cannot be estimated by Eq. (13), which is destined to solar adsorption adsorber-collectors. Douss and Meunier 1988 have given this amount of heat; it is equal to 365 kJ. For the temperatures’ data, we used the data of working fluid (WF) by Douss and Meunier 1988.

The compared results of COP are shown in Figs. 3, 4, and 5.

Fig. 3

COP versus the evaporating temperature variation (T _a = 20 °C, T _c = 14.5 °C, and T _g = 80 °C)

Fig. 4

COP versus the generating temperature variation (T _a = 20 °C, T _c = 14.5 °C, and T _e = −5 °C)

Fig. 5

COP versus the adsorbing temperature variation (T _g = 80 °C, T _c = 14.5 °C, and T _e = −5 °C)

The average relative errors between the results compared in the Figs. 3, 4, and 5 are respectively 4.77, 5.22, and 9.96%.

The validation with the work of Douss and Meunier 1988 was made for the four calculated heats, i.e., Q ₁, Q ₂, Q ₃, and Q _des, while Q _L is given by the author. Our numerical model takes into account the prediction of this quantity (Q _L), that is why we are going to carry out a second validation with Lemmini and Errougani 2007 to validate all the total energy Q _ge, ensuring by this way the validation of Eq. (13).

Lemmini and Errougani 2007 studied an experimental prototype of a solar adsorption refrigeration machine, where the recorded results are given in Table 4.

Table 4 Numerical results of COP variation compared with those calculated during the tested days by Lemmini and Errougani 2007

We have calculated the cooling production quantities Q _ev obtained by Lemmini and Errougani 2007 for eleven tested days by using Eq. (17), where COP and Q _ge are known. After that, we have predicted the operating temperatures of the studied machine.

The advantage of the developed numerical program is the prediction of operating temperatures T _a, T _c, and T _g for given cold production Q _ev and evaporating temperature T _e. The calculated temperatures will be used to estimate the amount of heat Q _ge.

The numerical results of COP variation using Eq. (17) compared with those obtained during the tested days by Lemmini and Errougani 2007 are shown in Table 4 and Fig. 6. The comparison shows an average relative error of 5.03%.

Fig. 6

Comparison of the numerical COPs of the present study with those obtained by Lemmini and Errougani 2007

Thermal efficiency and performance coefficient

The obtained results for the amounts of heats estimated, the thermal efficiency of the adsorber-collector, and the coefficient of performance of solar adsorption refrigerator in this subsection are based on the data considered for validation with the results of Lemmini and Errougani 2007.

As shown in Fig. 7, for one adsorber-collector, 25% of total received heat during t _max is useful and the rest must be recovered. The thermal efficiency takes two ends, maximum and minimum, which are 100% for th = 0 and 25% for t _h = t _max. This can be deduced from Eq. (16).

Fig. 7

Energy ratios and efficiency of adsorber-collector as function of the ratio of heating time and maximum heating time (T _a = T _c = 25 °C, T _e = 0 °C, and T _g = 90 °C)

The center of this curve takes a parabolic form, where it must coincide with the recovered heat quantity Q _r(t _h)/Q _ge(t _max) = 50% and the quantity of heat received Q _ge(t _h)/Q _ge(t _max) = 50% at t _h/t _max = 0.33.

At this time where Q _u(t _h)/Q _ge(t _max) = Q _L(t _h)/Q _ge(t _max) = 25% and using Eq. (16), the value of thermal efficiency η is 50%. It is the moment where the three quantities must take the same value which equals to 50% and corresponds to t _h/t _max = 0.33.

The shorter heating times correspond to higher efficiencies of the adsorber-collector. It can allow to recover up to 75% of total heat received by the adsorber-collector during t _max with zero heat loss. This means that improving design of the adsorber-collector is required to ensure the useful amount of solar energy necessary for system functioning during a very short time. So, it is recommended to use a heat transfer fluid instead of direct heating by solar radiation. Thus, we can isolate the upper side of the adsorber-collector that represents the large hole of the heat, and we command the heating time by the temperature and the flow rate of the heat transfer fluid. The fluid will flow in the tubes of a heat exchanger set inside the adsorber-collector. Even if the prior heating of heat transfer fluid by solar energy takes time, we can use a system of energy storage in the form of heat. Thus, we ensure the heating of the adsorber-collector even at night, and the cycle of the machine becomes continuous instead of intermittent cycle with a very good performance.

With a recovery energy system, the number of additional adsorbers can be added to the machine with a total received energy Q _ge by only one additional adsorber is Num _re, so we can write:

• One additional adsorber needs →Q _u (*)

• Number of additional adsorbers (Num _re) to be added need →Q _r (**)

From (*), (**), Eqs. (14) and (15):

$$ {\mathrm{Num}}_{\mathrm{r}\mathrm{e}}=\frac{Q_{\mathrm{r}}}{Q_{\mathrm{u}}}=\frac{Q_{\mathrm{L}}\left(1-\frac{t_{\mathrm{h}}}{t_{\max }}\right)}{Q_{\mathrm{ge}-}{Q}_{\mathrm{L}}} $$

(19)

As shown in Fig. 8, for a heating time t _h less than 4 h (t _max is assumed to be equal to 12 h), we can recover a useful energy for functioning of two to three additional adsorbers.

Fig. 8

Number of additional adsorbers and ratio of the recovery energy and the useful energy, as a function of the ratio of heating time and maximum heating time (T _a = T _c = 25 °C, T _e = 0 °C, and T _g = 90 °C)

The recovery of energy loss Q _L leads us to propose a new optimal COP, it may be given by:

$$ {\mathrm{COP}}_{\mathrm{optm}}=\left({\mathrm{Num}}_{\mathrm{re}}+1\right).\mathrm{COP} $$

(20)

By Eq. (20), we can see that a high optimal coefficient of performance COP _optm corresponds to a heating of shorter time t _h . If t _h  = t _max, no energy recovery (by Eq. (19) Qr = 0⇒Num _re = 0) and COP optm would take the value of the normal COP.

As shown in Fig. 9, with an energy recovery system from one adsorber-collector, the COP _optm value can reach up to four times the COP in the normal case.

Fig. 9

Ratio of the new optimal COP _optm and the COP as a function of ratio of heating time and maximum heating time (T _a = T _c = 25 °C, T _e = 0 °C, and T _g = 90 °C)

Conclusions

In the present paper, a numerical model was developed to predict the different amounts of heat received by the adsorber-collector of a solar-powered adsorption refrigeration machine using activated carbon AC35/methanol as a working pair.

Four amounts of heat are useful for the operation of system, while the fifth amount is an energy loss. The estimate of this lost energy showed that the time of heating of the adsorber-collector is a very important factor affecting its thermal efficiency.

The shorter heating time allows to cut off the continuation of heat loss into the external environment and gives a good performance.

If we use a system heated by solar energy, the reduction of the heating time is somewhat difficult because it is related to the radiation intensity, which is as function of the time. So, it is useful to use a heat transfer fluid heated by the sun with storage system using the water for example instead the direct heating by solar radiation.

Furthermore, the performed study allows to define a new optimal coefficient of performance (COP _optm) more generalized. It takes the value of normal coefficient of performance (COP) for zero recovery of energy loss, and greater value for more energy recovery, keeping the same total energy necessary for the functioning of the system.

The usefulness of this optimal coefficient of performance is to determine the number of additional adsorbers that can be added to the machine for an equivalent consumption to energy received by the adsorber-collector.

The cold production increases with the number of additional adsorbers. Even though the days when solar radiation is low, the decrease in cold production will be better than that without an energy recovery system.

Nomenclature

• COP Coefficient of performance

• COP _optm Optimal coefficient of performance

• Cp _d Specific heat of adsorbent, J kg⁻¹ K⁻¹

• Cp _l Specific heat of adsorbate in liquid state, J kg⁻¹ K⁻¹

• Cp _t Specific heat of tubes in metal containing the adsorbent, J kg⁻¹ K⁻¹

• D Coefficient of affinity

• E _g Emissivity of glass cover of adsorber-collector

• E _P Emissivity of top wall of adsorber-collector

• L Latent heat of adsorbate, J kg⁻¹

• m Adsorbate mass, kg

• m _d Adsorbent mass, kg

• m _t Mass of metallic tubes containing the adsorbent, kg

• n Parameter of adjustment of D–A equation

• n _g Number of glass cover of adsorber-collector

• Num _re Number of additional adsorbers

• r Particular gas constant of adsorbate, J kg⁻¹ K⁻¹

• P Equilibrium pressure of adsorbent-adsorbate pair, Pa

• P _c Condensing pressure of adsorbate, Pa

• P _e Evaporating pressure of adsorbate, Pa

• P _s Saturation pressure of adsorbate, Pa

• Q _des Heat necessary for desorption process, J

• Q _ev Heat load in evaporator (cooling production), J

• Q _ge Total energy received by adsorber-collector, J

• q _st Isosteric heat, J kg⁻¹

• Q ₁ Energy supplied to heat the adsorbent, J

• Q ₂ Energy supplied to heat the tubes in metal containing the adsorbent, J

• Q ₃ Energy supplied to heat the adsorbate mass, J

• Q _L Heat loss by adsorber-collector during the time of heating (processes A–C of Fig. 2), J

• Q _r Recovered heat from adsorber-collector, J

• Q _u Useful energy for functioning of system, J

• S Adsorber-collector surface, m²

• T Temperature, K

• T _a Adsorption temperature (temperature at end of adsorption), K

• T _am Ambient temperature, K

• T _c Condensing temperature of adsorbate, K

• T _e Evaporating temperature of adsorbate, K

• T _g Generating temperature (temperature at end of desorption), K

• T _p Wall temperature of the top of adsorber-collector, K

• Tp _maxMaximal temperature of heated walls of adsorber-collector (including upper and bottom parts), K

• Tp _minMinimal temperature of heated walls of adsorber-collector (including upper and bottom parts), K

• t _h Heating time of adsorbent, s

• t _maxMaximum time of heating processes A–C (Fig. 2), s

• U _b Loss coefficient from the bottom of the adsorber-collector, W K⁻¹

• U _L Global loss coefficient from the adsorber-collector surfaces, W K⁻¹

• U _t Loss coefficient from the top of the adsorber-collector, W K⁻¹

• T _s1 Temperature at start of desorption, K

• T _s2 Temperature at start of adsorption, K

• W _v Wind velocity, m s⁻¹

• W ₀ Maximum adsorbed volume, m³ (of adsorbate) kg⁻¹ (of adsorbent)

• X _max Maximal adsorbate mass concentration in 1 kg of adsorbent, kg kg⁻¹

• X _min Maximal adsorbate mass concentration in 1 kg of adsorbent, kg kg⁻¹

^{Greek symbols}

• α Thermal expansion coefficient of adsorbate, K⁻¹

• ΔT _am Variation of ambient temperature, K

• ΔT _p Variation of upper wall temperature of adsorber-collector, K

• η Thermal efficiency of adsorber-collector

• $\varOmega$ Adsorber-collector inclination, °

• ρ _l Density of the adsorbate in the liquid state, kg m⁻³

• σ Stefan-Boltzmann constant, W m⁻² K⁻⁴