ADSORPTIVE SOLAR POWERED ICE MAKER EXPERIMENTS .pdf

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Solar Energy Vol. 69, No. 3, pp. 249–263, 2000  2000 Elsevier Science Ltd S 0 0 3 8 – 0 9 2 X ( 0 0 ) 0 0 0 6 3 – 3 All rights reserved. Printed in Great Britain 0038-092X / 00 / $ - see front matter

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ADSORPTIVE SOLAR POWERED ICE MAKER: EXPERIMENTS AND MODEL A. BOUBAKRI* , †, J. J. GUILLEMINOT** and F. MEUNIER*** ˆ *ERBEM, Institut Universitaire de Technologie, Rue de la Raperie, 93290 Tremblay, France **CNRS-LIMSI, BP 133, 91403 Orsay Cedex, France ***CNAM-IFFI, 292, Rue St Martin, 75141 Paris Cedex, France Received 29 October 1998; revised version accepted 18 January 2000 Communicated by BYARD WOOD

Abstract—A solar adsorptive ice maker model is presented and experimentally validated. In order to determine the model parameters, the identification procedure is carried out employing an experimental data base obtained from tests carried out on two adsorptive solar-powered ice-makers using a methanol / carbon pair. The packaged component, the collector–condenser, represents the main new feature of these units. The prediction is compared with results of correlations. This allows a comparison of the collector–condenser behavior in the two units since the collector–condenser of one of these units is equipped with a radiation ` shield. The model is then used to study daily ice production sensitivity vis-a-vis critical physical parameters of the unit and to estimate the limits of the collector–condenser technology with flat plate collectors.  2000 Elsevier Science Ltd. All rights reserved.

moniac pair is also usable for ice production (Critoph and Gong, 1992). Those adsorptive systems are still limited, just as all solar systems are, by the cost of solar collectors and by the fluctuation of insolation. Their cost generally depends on the price of the reactor, which serves two functions: solar collector and chemical compressor. Different technologies were tested to reduce the manufacturing cost. Some of them were applied: a collector equipped with fins (Guilleminot and Meunier, 1981), a collector and condenser in one packaged component (Paye and Guyader) and a tubular collector field with an external parabolic circle concentrator (Niemann et al., 1997). At the same time, studies were carried out on the adsorbate conditioning and on the improvement of heat transfer inside the collector (Mauran et al., 1992; Guilleminot et al., 1993) considering the effect of these parameters on the dimensions of the reactor and hence on its cost. As far as the fluctuation of the insolation is concerned, a recent study (Critoph, 1996) proposed an adsorptive machine working with solar energy, biomass or combination of both. This study showed that the daily ice production (D.I.P.) could notably increase. In previous studies (Boubakri et al., 1992a,b), the performances of three commercial adsorptive solar-powered ice-makers using a methanol / carbon pair were presented. They were dimensioned to produce 5 kg m 22 of ice daily. Their character-

1. INTRODUCTION

Solar refrigeration is more and more recognized as a priority in developing countries (UNEP report, 1997). This is due to the needs for refrigeration, for food as well as for vaccine preservation, and to the fact that solar energy is generally available in these countries. Therefore, it is important to revisit solar refrigeration systems which have been studied recently. Different solar systems such as liquid absorption or solid gas systems were studied and tested with success (Chung et al., 1963; Chinnappa, 1974; Guilleminot and Meunier, 1981; Worse-Schmidt, 1983; Balat and Crozat, 1988). However, the industrial phase was not successful, due to the lack of market. This situation might change if solar refrigeration actually becomes a priority in developing countries as shown in the UNEP report. In the field of adsorptive systems, different types of solid–gas pairs were studied to build adapted cooling solar systems. At LIMSI, the zeolite–water pair was chosen (Guilleminot and Meunier, 1981) for refrigeration, and the active carbon–methanol pair for ice production (Boubakri et al., 1985). The active carbon–am†

Author to whom correspondence should be addressed. Tel.: 133-1-4151-1243; fax: 133-1-4861-3817; e-mail: [email protected] 249

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istic lies in the solar collector and condenser assembly in a single component, i.e. the collector–condenser (CC). These units were tested for over 2 years in South Morocco (Agadir, 308 239 N, 98 349 WG), the climate of which is very moderate. The recorded yearly means of insolation and of diurnal and nocturnal ambient temperature are, respectively, 19,540 kJ m 22 , 248C and 158C. Consequently, performances of solar ice making units are quite good in this area. The D.I.P. exceeds 4 kg m 22 for over 60% of the year corresponding to an average net solar coefficient of performance (ratio of the net heat extracted to produce the ice to the total solar radiation incident on the collector during the considered day), COPs ¯ 12%. Until now, the estimation of the performance of these units was centered on prototype testing and on the analysis of the influence of meteorological parameters. In this work, an extension of a previous model (Guilleminot et al., 1987) is proposed in order to study the D.I.P. sensitivity with respect to various physical parameters of

units. This model is consequently used to estimate the limit of the CC technology with flat plate collectors. Identification and validation procedures are carried out to obtain the model parameters from experimental data. The model is then validated by comparison between predicted and measured D.I.P. 2. DESCRIPTION OF ADSORPTIVE SOLARPOWERED ICE-MAKERS

The modeled units are schematically shown in Fig. 1. These units are mainly composed of a single glassed collector–condenser connected by a flexible tube with an evaporator. • The collector–condenser (1 m 2 , 90 mm thick, 208 tilted) is made of two identical stainless steel shells. A grid holds 20 kg of activated carbon (A.C.) in the upper shell, which plays the role of solar collector. The outside face of the solar collector is covered by ‘Maxsorb’ selective surface. Inside the solar collector, internal fins improve heat transfer between the

Fig. 1. (a) Schematic of adsorptive solar powered ice makers; (b) cross section through collector–condenser.

Adsorptive solar powered ice maker: experiments and model

upper face and the A.C. The rear shell of the collector–condenser plays the role of aircooled condenser with external fins (7.5 m 2 ). The packaged component, CC, represents the main new feature of these units. • The evaporator (0.3 m 2 ) is made of steel, immersed in a highly insulated, 5.2 l, ice making tank. The two modeled units will be denoted as MS and MEAR. MS is the standard unit. MEAR also uses the activated AC35–CH 3 OH pair but its CC is equipped with a radiation shield. This device which is installed between the adsorbent and the condenser limits radiative transfer between the upper and rear shells of the CC and hence leads to an improvement of the solar caption efficiency of the collector. Fig. 2 shows this effect, since the solar collector temperature for the MEAR unit is higher than that of the MS unit. In fact, the difference between collector temperatures of the two units can reach 158C when the difference between condenser temperatures is around 28C. The temperatures of the main components of each unit and the meteorological parameters (total solar radiation incident on the collector and ambient temperature) were automatically measured and recorded by a computerized data acquisition system at 10-min intervals. Temperatures were measured by K-type thermocouples with the uncertainty equaling 0.5 K. Total solar radiation incident on the collector was measured by a Kipp-Zonen solarimeter with the uncertainty equaling 1.6%. The accuracy of D.I.P. measurement is 2%.

Fig. 2. Measured temperature of the collector–condenser during heating and cooling periods.

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These measured temperatures allow to validate the model whereas the meteorological parameters are used by the model to generate the solution and the D.I.P. to estimate the limits of the collector– condenser technology. 3. ADSORPTIVE CYCLE

3.1. Principle of solid adsorption Adsorption is the fixation of the gas molecules on the surface of a solid, by Van der Waals interaction. Adsorption is reversible, exothermic, it induces no changes of solid volume and does not involve any chemical reaction (Ruthven, 1984). In the case of the pure vapor, the thermodynamic equilibrium between the adsorbed and the gaseous phases is divariant and it can be described by an equation of state, correlating concentration of the adsorbed phase m (adsorbed mass / adsorbent mass), pressure P, and temperature T, so that f(m, T, P) 5 0.

3.2. Principle of adsorptive refrigeration cycle The adsorptive cycle is intermittent. The adsorber containing the adsorbent is alternatively connected with a condenser and with an evaporator. The cycle is the succession of two periods: (a) period of heating–desorption–condensation at high pressure, (b) period of cooling– adsorption–evaporation at low pressure. High and low pressures are the saturation pressures of the adsorbate at the temperature of the condenser and the evaporator, respectively. For a solar adsorptive cycle, the first period consists of regeneration of the adsorbent by solar energy when the adsorbate is condensed and the second period occurs during the night when the evaporation of the adsorbate and the adsorption take place. Fig. 3 shows an ideal cycle in the Clapeyron diagram (ln P vs. 21 /T ). This cycle includes four thermodynamic steps: 1. isosteric (constant concentration) heating 2. isobaric desorption 3. isosteric cooling 4. isobaric adsorption. The cycle begins at point A, where the adsorbent is at low temperature TA , and low pressure Pev 5Ps (T ev ). The adsorbent is heated, disconnected from any exchanger first: the system follows the isoster mA 5m(TA , Pev ). Pressure increases and when it reaches the value Pcd 5 Ps (T cd ), (point B), the adsorbent is connected with the condenser ideally at constant temperature T cd .

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Fig. 3. Ideal adsorptive cycle in the Clapeyron diagram.

Then, isobaric desorption proceeds, the mass m of adsorbate decreases while adsorbent temperature increases, up to point C where the remaining adsorbed mass is m C 5 m(T C , Pcd ). Condensed liquid amount, Dm (kg / kg) 5 mA 2 m C , is transferred into the evaporator (e.g. by gravity). Further on, symmetrically on the path ABC, the adsorbent is cooled, first disconnected from the evaporator, from point C to point D, and then connected with the evaporator (ideally at constant temperature, T ev ) from point D back to point A. During this last period, the adsorbent adsorbs the quantity of vapor Dm from the evaporator. This evaporation induces heat extraction, which occurs only during this period: this is the reason why the refrigeration cycle is intermittent. The real cycle is different from the ideal cycle (Boubakri et al., 1985; Pons and Grenier, 1987), mainly because the adsorbent is continuously connected with exchangers (there are no valves). The exchanger temperature may also change and the system does not follow the isosteric path.

4. MODELING

The difference among the main developed models to simulate adsorptive cooling units (Guilleminot et al., 1987; Jing and Exell, 1994; Leite, 1997) generally lies on the differences in the simplifying assumptions, numerical resolution method, design and use of the modeled system. The mathematical modeling described in the present work is based on previous studies (Pons and Guilleminot, 1986; Guilleminot et al., 1987; Bentayeb et al., 1994). It is extended to simulate a

solar adsorptive ice-maker equipped with a CC tested in Morocco. The predictive feature of this model represents an additional interest when compared with the previously developed correlation (Boubakri et al., 1992b). Thus, the effects of physical parameters and the geometry of the machine on the D.I.P., or on the manufacturing cost, can be separately evaluated. The limits of the CC technology can also be estimated. Some parameters used in the present model have already been identified, like the heat transfer coefficient between the adsorbent and fins, physico-chemical parameters of activated carbon– methanol pair, and the heat losses coefficient for the ice-making tank (Boubakri et al., 1985, 1992a; Guilleminot et al., 1987). The model is mainly based on the following assumptions. • Pressure is uniform in the CC. • Convective heat transfer in the vapor phase inside the solar collector is neglected. • Resistance to mass diffusion through the interparticle voids and pores is neglected. • Equilibrium adsorption occurs. • Temperature is uniform in the direction parallel to the collector plate that leads to a bidimensional model. The model needs two input data, i.e. ambient temperature and total solar radiation incident on the collectors. The main mathematical equations of the model and flow charts (Figs. A.1 and A.2) showing the calculation sequence are reported in Appendix A. The details about the numerical resolution were previously reported (Guilleminot et al., 1987; Bentayeb et al., 1994).


The model is used: (a) first to identify convection heat transfer coefficients on the upper plate outside of the collector h av , and outside the condenser h ar , and the equivalent heat transfer coefficient h inside the condenser. By equivalent heat transfer coefficient, we mean the global heat transfer coefficient including convection, gas conduction and radiation between the rear adsorbent bed and the inner condenser plate. This coefficient h is also the adsorbent rear losses coefficient. These coefficients, which are required for the simulation of those special types of machines, control a major part of heat exchanges between the machines and the external environment. Therefore they have a direct influence on their efficiency; (b) then, to study the sensitivity of the D.I.P. ` vis-a-vis critical physical parameters of the unit and to estimate the limit of the collector–condenser technology. Specific experiments were carried out to extract h av , h ar , and h. These experiments consisted in transferring the quasi totality of the adsorbate into the evaporator via a succession of desorption– condensation phases. Then, the connecting valve located between CC and evaporator was closed. Subsequently, the temperature of this CC working without mass transfer was measured during the heating phase and during the cooling phase. Fig. 4 shows the evolution of the adsorbent temperature when functioning without mass transfer and when functioning normally. On the normal curve, the beginning of desorption (point 1), the end of desorption (point 2) and the beginning of ad-

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sorption (point 3), are clearly observed. On the curve without mass transfer, these particular points do not appear. This allows one to simplify the model: only the modeling of the CC (without mass transfer) is needed to identify the triplet (h av , h ar , h) which is independent of the mass transfer in the CC. In this case, Eq. (A.2) becomes:

r2 (T )c 2 (T )(≠T 2 / ≠t) 5 l2 = 2 T 2

(1)

where the residual adsorbent mass is neglected, Eq. (A.6) becomes: h(T al (t) 2 T p (t)) 5 2 l1 (≠T p / ≠n).

(2)

The initial and boundary conditions remain unchanged. For a given triplet (h av , h ar , h), the numerical resolution of the system generates, at every interval, a particular solution S(h av , h ar , h) for the temperature values, respectively, for the collector and the condenser. The criterion of error F is evaluated by the mean square deviation between the measured temperatures T and the calculated ones S, i.e.:

SSO O(S 2 T ) D / 2ND 2

N

1/2

2

F(h av , h, h ar ) 5

ij

ij

j 51 i 51

(3) where j 5 1 is the index of the adsorbent, j 5 2 is the index of the condenser and N being the number of recorded measured temperatures. The minimization of F(h av , h ar , h) allows the identification of the triplet (h 0av , h 0 , h 0ar ) for which the prediction most closely approaches the experiment. 5. RESULTS AND DISCUSSION

Fig. 4. Effect of adsorption and desorption on heating and cooling of the collector–condenser.

The triplet h 0av , h 0 , h 0ar for which the difference between the prediction and the experiment has a minimum value leads to the curves in Figs. 5 and 6, respectively, for MEAR and MS units. Each figure represents a comparison between the calculated and the measured temperatures during the cycle in the collector and in the condenser. Table 1 gives the optimum identified triplet values (h 0av , h 0 , h 0ar ) and the associated minimal mean square deviation F0 for each unit. The results in Table 1 show a difference between the heat transfer coefficient h 0 in the two units. The smaller h 0 value obtained with the MEAR unit is due to the limitation of the heat losses between the adsorbent and the condenser

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improves solar caption efficiency, but hinders the cooling of the adsorbent (reduced rear loss). Both effects almost cancel each other. Finally, a lower evaporation temperature was observed with the MEAR unit, which resulted in a more effective ice making process. This constitutes another interesting effect of the radiation shield. The above results make it possible to estimate the physical parameters of the collector–condenser as described below.

5.1. Radiation heat transfer coefficient

Fig. 5. Comparison of calculated and measured average temperatures in the MEAR collector–condenser.

Assuming radiation shield efficiency to be 100%, the radiation heat transfer coefficient h r between the rear adsorbent bed and the inner condenser plate would be equal to the difference between the heat transfer coefficient h 0 of the two units i.e.: h r 5 Dh 0 5 2.5 2 1 W m 22 5 1.5 W m 22 .

(4)

5.2. Emissivity of the inner condenser plate. As explained in Appendix B, the emissivity of the inner condenser plate ´ is known as soon as h r is 22 known. We obtain ´ 5 0.2 for h r 5 1.5 W m 21 K . This value corresponds to a polished stainless steel emissivity, as it is the case of the studied inner condenser plate. This means that this estimation seems correct.

5.3. Convective heat transfer coefficient in the condenser Fig. 6. Comparison of calculated and measured average temperatures in the MS collector–condenser.

by the radiation shield. Therefore, the collector caption efficiency is improved, as shown by Fig. 3. But, this improvement has only a slight effect on the D.I.P. as shown by the experiment (Boubakri et al., 1992a). This can be roughly explained as follows: the limitation of heat transfer between the adsorbent and the condenser

Table 1. Optimum identified triplet values for MEAR and MS h 0av (W m 22 K 1 ) MEAR MS

3 3

h0 (W m

22

1 2.5

21

K )

h 0ar (W m 22 K 21 )

F0 (8C)

2 2.8

1.85 1.62

We consider the condenser of MEAR, where the heat transfer occurs essentially by convection and conduction of the methanol vapor. The conductivity l of the methanol vapor is available in the literature (Liney, 1988): l 5 0.02 W m 21 K 21 . Accordingly, the convection coefficient h f in the condenser can be obtained easily: h f 5 h 0 2 lf /e 5 1 2 0.02 / 0.04 W m 22 K 21 5 0.5 W m 22 K 21

(5)

where e is the condenser cavity thickness (m). This result is comparable to that obtained by a correlation proposed by Hollands presented in Appendix B. The convection coefficient between the collector upper plate and the glass cover was also estimated with the help of Hollands correlation. Hence, we obtained: h av 5 3.3 W m 22 K 21 .

(6)


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6. SENSITIVITY OF IDENTIFICATION MODEL VERSUS HAV , H AND HAR

The error function F can be written near the minimum error F0 as: F 5 F0 1 (≠F / ≠h av ) hh ar dh av 1 (≠F / ≠h) h av h ar dh 1 (≠F / ≠h ar ) h av h dh ar .

(7)

` The sensitivity of F vis-a-vis h av , h and h ar is shown in Figs. 7 and 8 for MEAR and MS, respectively. In both figures, we note that the model is more sensitive to h av than to h or to h ar . Quantitatively, the variation of h av less than 5% increases F0 more than 10%, whereas h or h ar must vary more than 10% to vary F0 with the same value. Fig. 7. Sensitivity of identification model versus h av , h ar and h in the case of MEAR.

7. VALIDITY OF THE MODEL

This value is also comparable to the identified value obtained in this study. It can be concluded that the physical parameters identified by the model are correct and can be therefore used to simulate the units. Finally, note that the radiation heat transfer coefficient exceeds notably (three times) the convection heat transfer coefficient in the condenser. Therefore, it is worth supplying the units with a radiation shield. However, this device would improve the D.I.P. only if it was associated with an active nocturnal cooling device.

The D.I.P. is simulated with the model presented above, using the identified triplet h 0av , h 0 , 0 h ar and previously known parameters. The comparison between calculated and measured D.I.P. for MEAR and MS is represented in Figs. 9 and 10. The model describes the experiment fairly well. It has to be noted that this comparison has only included days with a definite background, considering that the model takes into account the effect of the preceding day, just as this type of machine actually does. In addition, the days for which the measured and calculated ice production are equal to 5.2 kg have also been excluded from com-

Fig. 8. Sensitivity of identification model versus h av , h ar and h in the case of MS.

Fig. 9. Comparison of calculated and measured daily ice production in the case of MEAR.

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Fig. 10. Comparison of calculated and measured daily ice production in the case of MS.

parison, since this value corresponds to the upper size of the ice making tank and does not mean that it is the actual production (the ice production could eventually have been higher).

Fig. 12. MS daily ice production evaluation without ice making tank capacity as a limiting factor.

as big (10.4 l) and we simulate only the days where a measured ice mass was equal to 5.2 kg. The result of this simulation is given in Figs. 11 and 12 for MEAR and MS, respectively. It shows that each of the two machines can produce daily an amount of ice higher than 5.2 kg and that it can even exceed 7 kg some days.

8. PREDICTION OF THE D.I.P.

Considering the appropriate results of the model, it becomes possible to estimate the D.I.P. which could be produced by those machines if they had not been limited by the ice making tank capacity. For this purpose, we assume that the units are equipped with an ice making tank twice

9. ICE MAKING TANK CAPACITY

As shown in Figs. 11 and 12, the ice making tank capacity could reach 7 l. Nevertheless, enlarging the ice making tank means increasing its heat capacity. Consequently, this modification affects the moderately sunny days since the amount of ice is reduced (until 50%) as displayed by simulation. Therefore we can conclude that the present capacity of the ice making tank is correctly estimated and constitutes a good compromise even if some ice production is missed for the most sunny days. 10. SENSITIVITY OF THE D.I.P. VERSUS SOME IMPORTANT PHYSICAL PARAMETERS OF THE UNITS

Fig. 11. MEAR daily ice production evaluation without ice making tank capacity as a limiting factor.

In order to study this sensitivity, we consider for a given day the mass of ice produced by MS unit supposed to be equipped with an enlarged ice making tank (10 l). The characteristics of the simulated day (4 April) are: nocturnal temperature equal to 148C, diurnal temperature equal to 218C and total solar radiation incident on the collector equal to 24,450 kJ m 22 . The following presumed physical parameters of the unit are considered as


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Fig. 13. Sensitivity of the daily ice production to variation of h av .

Fig. 15. Sensitivity of the daily ice production to variation of h be A be .

reference values: h av 53 W m 2 K 21 , h ar 52.8 W m 22 K 21 , h be A be 5 0.3 W K 21 and h eve 540 W m 22 K 21 . Figs. 13–16 show the sensitivity of the calculated D.I.P. versus h av , h ar , h be A be and h eve , respectively. It is first noticed that the current physical parameters corresponding to points represented on the curves by (^) are quite well located with regard to the optimum of the curve. This means that the units are quite well designed. The variation of the D.I.P. with the various values of h av and h ar is shown in Figs. 13 and 14, respectively. Note that the D.I.P. increases with increasing h ar

and decreases slightly beyond optimum with increasing h av . This means that the nocturnal cooling of the CC takes place essentially in the rear part of the CC during the adsorption phase, while increasing h av only enhances day time heat loss during the desorption. On the other hand, the D.I.P. is more sensitive to the variation of h ar than to the variation of h av and the maximum D.I.P. for the simulated day (8 kg) can only be reached with h ar corresponding to a forced convection type (h ar ¯20 W m 22 K 21 ). Beyond these values, a decrease in the D.I.P. takes place. Sensitivity of D.I.P. to variation of h be A be

Fig. 14. Sensitivity of the daily ice production to variation of h ar .

Fig. 16. Sensitivity of the daily ice production to variation of h eve .

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coefficient, which represents the ice making tank insulation, is given in Fig. 15. It appears that even for an extreme insulation, only an 8% gain in ice production is obtained. An additional cost to improve evaporator and ice making tank insulation is therefore unwarranted. Moreover, Fig. 16 shows that beyond h eve 540 W m 22 K 21 , which is the presumed value of the machine, the D.I.P. varies slightly with the increase of the global heat exchange coefficient between methanol and water. An improvement of 140% of h eve with regard to the reference state induces only a 10% increase in ice production. Above h eve 5100 W m 22 K 21 , the ice production is still limited to 7 kg. If h ar A cd coefficient, which controls the heat transfer at the rear part of CC, is not adapted, the cooling of the condenser is poor and the cycled mass remains limited.

11. LIMIT OF THE COLLECTOR– CONDENSER TECHNOLOGY

Finally, an important question arises: what would the D.I.P. be, if the MS physical parameters had the best values suggested in Figs. 13–16, i.e. in the case where MS is highly optimized? If the coefficients were h av 51.5 W m 22 K 21 , h ar 5 20 W m 22 K 21 , h eve 5100 W m 22 K 21 and h be A be 5 0.1 W K 21 , ice making tank capacity5 12 l, and under very favorable meteorological conditions, MS might produce 11.5 kg of ice per m 2 of collector (Table 2). This value determines the limit of the collector–condenser technology with flat plate collectors for the Agadir area. It corresponds to: • a coefficient of performance (ratio of the cooling load to the heat input): COP533.9%;

Table 2. Sensitivity of the limit of the collector–condenser ` technology with flat plate collectors vis-a-vis some climates simulated through total solar radiation incident on the collector during the day (En ), diurnal ambient temperature (T d ) and nocturnal ambient temperature (T n ) En (kJ m 22 )

Td (8C)

Tn (8C)

D.I.P. (kg m 22 )

COPs (%)

COP (%)

29,000 29,000 23,240 28,050 17,000 28,500 25,000 24,320 21,170 19,540

20 35 20.5 26.5 30 22 33 18 18 24

10 25 10.5 15.5 20 12 21 10 7 15

11.5 5 9.5 8.7 2.4 10.4 5.6 10.6 9.5 5.79

19 15 19.9 17.4 15.4 18.15 16.3 20.6 20.3 18.5

33.9 29.5 35.4 31.6 31.1 33.0 30.9 36.1 36.0 34.7

• a solar coefficient of performance (ratio of the cooling load to the total solar radiation incident on the collector): COPs519%. However, as shown in Table 2, this limit is very sensitive to the climate, mainly to the influence of the ambient temperature (nocturnal and diurnal) on the performance of the activated carbon–methanol pair (Boubakri et al., 1992b). 12. CONCLUSION

With the help of a model based on experimental measurements obtained from tests on adsorptive solar-powered ice makers, the convective heat transfer coefficients on the front and the rear part of the collector–condenser and the global heat transfer coefficient inside the condenser are identified (Table 1). These coefficients are first analyzed, and thus the effect of a radiation shield which equips one of the machines is evaluated. This component leads to a 40% reduction of adsorbent rear losses. In practice, it leads to two effects: (a) a better heating of the collector, (b) a poorer cooling. This last effect limits the improvement of D.I.P., as observed during the experiments. These coefficients have then been used to develop a global model in order to simulate the operating of the adsorptive solar-powered ice makers using collector–condenser technology. Conclusive results are obtained, especially in the calculation of the D.I.P., regarded as a validation criterion. Thus, the validated model allows us firstly to study the D.I.P. sensitivity to variation of the main physical parameters of the machines. The major limiting factor appears to be h ar . Except for this parameter, the machines are found to be quite well designed. The model then permits to estimate the limits of ice production by means of adsorptive collector–condenser technology: if the machines were working in the best conditions of physical and meteorological parameters, D.I.P. could reach 11.5 kg per m 2 of collector, corresponding to a COPs5 19% (Table 2). This represents the limit of collector–condenser technology with flat plate collectors. APPENDIX A

A.1. EQUATIONS OF THE SYSTEM ( Figs. A.1 and A.2) A.1.1. Collector–condenser. The energy balance for the steel is given by:


259

Fig. A.1. Flow chart of thermal parameters identification.

r1 c 1 (≠T 1 / ≠t) 5 l1 = 2 T 1

(A.1)

where r1 , c 1 , T 1 , l1 represent, respectively, the 23 21 21 specific mass (kg m ), specific heat (J kg K ), the average temperature (K), the heat conductivity (W m 21 K 21 ) of the metal; t represents the time (s).

The heat balance for the adsorbent layer is given by:

r2 (T )fc 2 (T 2 ) 1 m(P, T 2 )c 3 (T 2 )g(≠T 2 / ≠t) 5 l2 = 2 T 2 1 I(x, y)

(A.2)

where r2 , c 2 , T 2 , l2 represent, respectively, the spe-

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Fig. A.2. Flow chart of the ice production process.

cific mass (kg m 23 ), specific heat (J kg 21 K 21 ), the average temperature (K), the heat conductivity (W m 21 K 21 ) of the anhydrous adsorbent; P, c 3 , represent the pressure (Pa) in the collector and the specific heat (J kg 21 K 21 ) of the adsorbate.

I(x, y) 5 DH(≠m / ≠t)

(A.3)

represents the heat source per volume unit (W m 23 ) where x, y are the x-axis perpendicular to the upper plate and the y-axis perpendicular to


the fin, respectively; DH is the latent heat of adsorption (J kg 21 ) obtained from the following Clapeyron equation: DH 5 RT 22s≠ln P/ ≠T 2d m

(A.4)

where R is the ideal gas constant of the adsorbate 21 21 (J kg K ); m is the mass of adsorbate in the adsorbent (kg kg 21 ) obtained from the following Dubinin equation (Bering et al., 1972):

A.1.3. Ice making tank. The ice making tank may contain either only water (Mi 5 M4 ), or only ice (Mi 5 M5 ) or a mixture of water and ice. Therefore, the energy balance equation depends on the ice concentration:

fCbe 1 Mi cig(dT be / dt) 5 d Ls (dM5 / dt) 1 h eve A eve (T ev (t) 2 T be (t)) 1 h be A be (T am (t) 2 T be (t))

m 5 W0 r3 (T 2 ) exp(2(E 21 RT 2 ln(Ps (T 2 ) /P))n) (A.5) where r3 , Ps are, respectively, the specific mass 23 (kg m ) and the saturation pressure of the adsorbate (Pa). The parameters (E, W0 , n) and the numerical value of DH were determined in a previous study (Boubakri et al., 1985). The heat conservation on the inner condenser plate where the condensation occurs can be obtained by:

SEE

V

m(T 2 (x, y), P)dxdy

DY

5 2 l1 (≠T p / ≠n)

dt (A.6)

where h is the global heat transfer coefficient including convection, gas conduction and radiation between the adsorbent bed and the inner condenser plate (W m 22 K 21 ); Lc is the latent heat of condensation (J kg 21 ); dcd is a coefficient equal to 1 during condensation phase and equal to 0 otherwise; T al is the temperature of the rear adsorbent bed (K); T p is the temperature of the inner condenser plate (K). A.1.2. Evaporator. The evaporator is immersed in the ice making tank. The energy balance equation is

fCev 1 M3 (t)c 3g(dT ev / dt) 5 dev Lv (dM3 / dt) 2 h eve A eve (T ev (t) 2 T be (t))

(A.8)

where Cbe is the heat capacity of the ice making 21 tank (J K ); c i is the specific heat of component 21 i (J kg K 21 ); A be is the heat transfer surface area between the ice making tank and ambient 2 (m ); h be is the ice making tank heat loss coefficient (W m 22 K 21 ); d is a coefficient equal to 1 during freezing phase and equal to 0 otherwise; T am is the ambient temperature (K). A.2. INITIAL CONDITIONS The initial conditions in the reactor are those found at the end of the night-time. They depend on the history of the previous cycle:

h(T al (t) 2 T p (t)) 1 dcd Lc r2 d

261

(A.7)

where Cev is the heat capacity of the evaporator (J K 21 ); M3 (t) is the methanol mass contained in the evaporator (kg); T ev is the average temperature of the evaporator (K); T be is the average temperature of the ice making tank (K); dev is a coefficient equal to 1 during evaporation phase and equal to 0 otherwise; Lv is the latent heat of evaporation (J kg 21 ); A eve is the heat transfer surface area between adsorbate and water (m 2 ); h eve is the global heat transfer coefficient between adsorbate and water (W m 22 K 21 ).

T 2 (x, y, t 5 0) ¯ T am P(t 5 0) ¯ Pev m 0 (x, y, t 5 0) ¯ m(T am , Pev ).

(A.9)

A.3. ENERGY BOUNDARY CONDITIONS The temperature of the upper collector plate is given by the energy balance including emissivity (´1 ) and absorptivity (a ) of the upper plate collector, solar radiation, convection (and radiation) heat loss: 4 aF (t) 1 s´1 (T am (t) 2 T 14 (t)) 1 h av (T am (t) 2 T 1 (t))

5 2 l1 (≠T 1 / ≠n)

(A.10)

where F is the global solar radiation flux incident on the collector plate (W m 22 ). At the interface S between the upper plate, fins of the collector and the adsorbent, there is a contact conductance ho so that the energy balance is: 2 l1 (≠T 1 / ≠n) 5 hS (T 1 (t) 2 T 2 (t)) 5 2 l2 (≠T 2 / ≠n)

(A.11)

where T 1 and T 2 are the temperatures of each body at the interface. At the interface between the rear adsorbent bed and the condenser: h(T al (t) 2 T p (t)) 5 2 l2 (≠T al / ≠n).

(A.12)

262

A. Boubakri et al.

At the rear condenser plate, external heat exchange is simulated through h ar : h ar (T am (t) 2 T p (t)) 5 2 l1 (≠T p / ≠n).

(A.13)

A.4. MASS BOUNDARY CONDITION For an isosteric evolution, the methanol mass variation in the reactor is equal to 0: dm / dt 5 0.

(A.14)

The total mass variation is calculated from the saturated pressure and the temperature using Eq. (A.5).

collector and the horizontal plane (0 , c , 708). This correlation gives the following. (a) Convection coefficient between the upper plate of the collector and the glass, h av 5 Nu L l /L 5 3.3 W m 22 K 21

(B.5)

where l is the thermal conductivity W m 21 K 21 . (b) Convection coefficient in the condenser h f 5 0.5 W m 22 K 21 .

(B.6)

Note that all properties of fluids are evaluated at their average temperature (70 and 408C, respectively) and C 5208.

APPENDIX B

B.1. EMISSIVITY OF THE INNER SIDE OF THE CONDENSER The emissivity may be obtained from the radiation heat transfer coefficient equation. The radiation heat transfer coefficient h r between the adsorbent bed and the inner condenser plate (parallel planes) can be expressed as: 4 h r 5 (s (T al 2 T p4 )(T al 2 T p )21 ) 21 21 3 (´ 21 al 1 A al A p (´ p 2 1))

(B.1)

where A al is the rear adsorbent bed surface area (m 2 ); A p is the inner condenser plate surface area (m 2 ); ´al and ´p are, respectively, the rear adsorbent bed and the inner condenser plate emissivity. With assumptions: A al and A p are surfaces that form an enclosure (end effects are negligible): A al 5 A p ; active carbon is assimilated to a blackbody i.e. ´al 5 1, Eq. (B.1) becomes: 4 h r 5 ´p s (T al 2 T 4p )(T al 2 T p )21

(B.2)

and ´p is given by:

´p 5 h r (s (T 4al 2 T 4p ))21 (T al 2 T p ) 5 0.2.

(B.3)

B.2. CONVECTION COEFFICIENT The convection coefficient in inclined rectangular (length L, wide w, high H ) cavities can be obtained from a correlation proposed by Hollands et al. (1976) for aspect ratios H /L $ 12 and tilt angles less than the critical value C * 5 708: Nu L 5 1 1 1.44(1 2 1708(Ra L cosc )21 ) 3 (1 2 1708(sin1.8c )1.6 (Ra L cosc )21 ) 1 ((Ra L cosc / 5830)1 / 3 2 1)

(B.4)

where Nu L is the Nusselt number; Ra is the Rayleigh number; C is the tilt angle between

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