## The mathematical formula

The solar radiation daily variation corresponding to the typical clear days characterized by a sunshine fraction a > 0,9 and a nebulosity index Kd < 0,2 [18], The variation of temperature, pressure and the total are obtained by establishing a mass and a thermal balance of the volume elements of the porous medium discretised on equal thickness and to evaluate the equations of heat and mass transfer in each slices separately.

In each slice, the transfer of heat is obtained by applying the first principle of thermodynamic for an open system by taking into account the fractions corresponding to the adsorbed fluid, the gas and the solid:

d(^U) + ^ qs hs — ^ qe he = Ф + E (2)

Su = Vc [(1 — s)psUs + (є — a)pgUg + a paUa] (3)

The combined of equations (3) and (4), enable us to obtain the general equation of heat and mass transfer in a layer, equation (5), these equations are written in the case of :

— cylindrical elements:

rn 4 4 , idT P d[(s-a)pg]

TOC o "1-5" h z [(1 -S)pscs + (є — a)pgcg +apaca] —

dt pg St

d(apa) P d(apa) d2T 1 d)T

-AHads (T, P) =2e ( +

dt pa dt dr2 r dr

These equations in the porous medium are completed by the initial and boundary conditions:

— Initial condition:

— T(r,0) = Ta ( r = 0,…, R ) (7)

— T(i, j,0) = Ta ( i = 1,., n ) and ( j = 1,., m ) (8)

Ta is the ambient temperature before the sunrise All the reactor is a constant temperature.

— Boundary conditions:

The boundary conditions to the center of the porous medium is a condition of symmetry;

(ffr) _ 0 , (ffr) _ (ffT) _ 0 (9)

(~&r ) r=°_0 (Ж) x=i, у — ~оу) x, y=i — 0 w

The thermal balance of the metallic wall is given by the following equations;

— Cylindrical tube without fins

pac VacCac — TvUacPsDe —UlDe(Tac—Ta) — hinDi(Tac — T) (10)

— Cylindrical tube with fins

Caofitc Vc— = Tv Oca Ps (De + 2 ШУ — U (-Dev + 2 Qi)(Tac-Ta ) — k 7t D, (Tac~T)

TOC o "1-5" h z dt 2

In this equation we take account the efficiency of the fins into consideration [19]

Q =tanh (m £) (19) and m = VUl/2acs (11)

m t

I is the wide of the fin

— Rectangular tube

Qj m

pac Vac Cac— = Tvaac Psu — Ul Sr (Tac ~ Ta) ~ 4 ^ hi AY (Tac ~ Tnj) —

» ‘ (12)

2 hi AY (Tac — Tnl) — 4 £ hi AX (Tac — Tim) — 2 hi AX (Tac — T 1m)

i

The obtained equations from a system of non linear differential equations that are solved by the implicit finite difference method [20].

The efficiency of the machine is characterized by the thermal coefficient of performance; COP and a solar performance coefficient COPs, deduced from the characteristic points of the obtained cycle using the following relations;

Tmax

Qc ^ mi Cpi dT + Qdes

Index i relates to ammonia, the activated carbon and the metal tube.

Qdes is the quantity of energy necessary to the desorption of the quantity Am [20];

5. Results:

The numerical simulation of the modelled reactor, under ambient temperature and solar radiation recorded in Tetouan, enable to describe aspects of heat and mass transfer inside the porous medium. The results gives the characteristic parameters of the functionning machine.

The numerical results obtained under real conditions of ambient temperature and solar irradiation relative to typical clear days of each season, allow the evaluation of the considered reactors performances from the cycle characterising points. The adsorption temperature is equal to the ambient one, the evaporation temperature is zero and the condensation temperature corresponds to the ambient temperature related with the beginning of ammonia desorption inside the condenser.

Figure 3 shows the variation of the thermal performance coefficient COP versus the normal and finned tubes diameter for the studied typical days. We observe that for each case there is a maximum value corresponding to a given tube diameter representing the optimum values.

Hence, under the applied functioning conditions the optimum COP value (diameter) are variable and depends strongly on solar radiation and on the ambient temperature. The same remarks are observed for the variation of the daily cycled mass versus the diameter figure 4-a, considering a collector of a 1 m2 of surface composed a number of equal tubes. The total cycled mass corresponds to the sum of the desorbed quantities by each cylindrical tube. We note that the optimum values are higher for the rectangular reactors compared to the cylindrical ones figure 4-b corresponding to the amount of the activated carbon used and thus to the offered volume to the reacting mixture.

The high values of the COP in April and October can be explained considering the fact that ammonia adsorption takes place before the sun rise in a uniform temperature porous medium, equal to the ambient temperature but less than that in July. So, the adsorption is very important, the choice of typical clear days characterised with high solar radiation allows to heat to the maximum values the reactors and thus the COP is a function of the considering temperature and that the maximum heating of the absorbent permits an important heat adsorption.

The variation of the maximum temperature at the center of the porous medium is a decreasing function of its width, it is had has the thermal conductivity of the porous medium and the thermal capacity of the whole of the reactor figure 5. We notice that the finned tubes improves the thermal exchanges between the metallic walls and the porous medium, consequently the maximum temperature attained is greater allowing an important desorption for the finned reactor with regard the same diameter normal tube, figure 5-a. Figure 6 shows the evolution of temperature at the center of the porous medium versus the time for the three optimum widths reactors. For the cylindrical tube, we compare the temperature variation inside the tubes with a similarly diameter for the finned and normal reactor. The studied cycle begin the morning where all the reactor is at the ambient temperature and finish at midnight marking the start of a new cycle relatively to the temperature recorded at LT.

The rectangular reactor heating duration is higher, owing to the important volume of the fixed bed containing the mixture, than the cylindrical cases.

Figure 7 shows the pressure evolution inside the reactors versus time, causes by the temperature variation. The temperature elevation during the heating phase of the closed reactors causes an increases in the gas pressure until it becomes just larger than the condensation pressure which corresponds to the saturation pressure at the temperature condensation, then the desorption of ammonia into the condenser starts at a constant pressure and the heating of the fixed bed continues until the temperature reaching the maximum value. The reactors are closed and both temperature and pressure decrease.

At the pressure value of 4,2 bar the reactors are opened and the adsorption phenomena of ammonia vapour start with a cooling product quantity.

These evolution of temperature and pressure is represented in a Clapeyron diagrams, corresponding to the variation of Log P versus the temperature figure 8, and shows the daily thermodynamic cycle characterised by two isosters and two isobars representing four phases relatively to the heating or the cooling of the reactors.

In figure 9, we show the daily evolution of ammonia total mass, both adsorbed and gaseous, inside two cylindrical tubes having the same diameter in the two optimal cases. At the beginning the temperature is the same inside both of the tubes implying that their respective ammonia masses are also the same. During the heating of the closed reactor, condensation pressure inside the finned reactor is reached before the tube without fins, causing the opening of valve V1 and hence ammonia desorption. This desorption is important considering the temperature elevation and the values of 2,42 kg and 1,55 kg are collected for the unit area respectively for the normal and finned reactor.

Inside a 1m2 surface captor, 1 m long and 1 m wide, equals to the multiplication of the number of tubes by their external diameter. The total desorbed mass represents the sum of all the desorbed amounts in each tube. The non desorbed mass is the total fluid mass inside the reactor which the variation during a cycle is showed in figure 10.

We gives in table I the values of the computed amounts and those of the parameters under which the reactors functioning for the typical clear days of July, of which can be compared the three reactors. The obtained optimal geometry of each reactor presents an evaluation of the parameters that characterises the functioning conditions, the efficiency of the machine and the computed both provided and useful energy.

6. Conclusion

In this work, the aims is to present a model and an optimisation of solar adsorption cooling machine using ammonia / activated carbon couples, that allows a design according to the real functioning condition. The prediction of the performance of the solar refrigerator require the knowledge of various parameters, which characterise the daily thermodynamic cycle. The optimisation is based on heat and mass transfer in the porous medium consider the collected mass, the thermal and solar performance coefficient, allow to give an idea of the transitory evolution of temperature, pressure and ammonia concentration inside the reactors. The efficiency of each reactors are enhanced and the preferential adsorber depends on the desired role to generate (the useful cooling quantity).

A presentation of temperature and adsorption ammonia quantity inside the reactor that develop solar radiation is carried out in this paper. Thus, the simulation has been performed using some assumption will be applied to an experimental test.

Table II. Comparison of the operating parameters and results of each reactors

 Height optimum (diameter) (cm) rectangular 8 without fins 7.29 with fins 6.94 Tads (K) 297,2 297,2 297,2 Tcond (K) 299,37 299,05 298,79 Tmax (K) 360,3 356,7 369,4 Pcond (bars) 10,41 10,31 10,23 Mass used AC (kg) 40 27,134 13,244 Desorbed mass (kg) 5,8 2,42 1,55 Total desorbed mass fraction (kg/kg) 0.145 0.089 0.117 Time of beginning condensation (LT) 10 h 24 min 9 h 48 min 9 h 18 min Time of end condensation (LT) 16 h 12 min 14 h 36 min 14 h 30 min Quantity of cooling product at the evaporator (kj ) 4443,8 2759,7 1777,2 Quantity of heat the reactor (kj) 10371,2 7538,8 4293,9 Thermal COP 0,43 0,366 0,414 Solar COP 0,17 0,105 0,068

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Figure 4. Total daily condensed mass versus the tube dimensionless -—- January April July -0- October

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