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Solar Energy Vol. 68, No. 1, pp. 69–75, 2000 1999 Elsevier Science Ltd All rights reserved. Printed in Great Britain 0038-092X / 00 / $ - see front matter

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AN IRREVERSIBLE-THERMODYNAMIC MODEL FOR SOLAR-POWERED ABSORPTION COOLING SYSTEMS N. E. WIJEYSUNDERA† Department of Mechanical and Production Engineering, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260 Received 7 July 1998; revised version accepted 11 May 1999 Communicated by BYARD WOOD

Abstract—The ideal three-heat-reservoir cycle with constant internal irreversibilities and external heat transfer irreversibilities is used to model the absorption refrigeration machine of a solar operated absorption cooling system. Analytical expressions are obtained for the variation of the entropy transfer with storage tank temperature and the variation of the coefficient of performance (COP) with the cooling capacity of the plant. These expressions give the operating points for the maximum cooling capacity and the maximum COP. The results for ideal irreversible cycles are compared with those obtained by detailed simulation of the absorption cooling system. The effect of internal and external irreversibilities on the second-law efficiency of the plant is examined. The ideal cycles that include internal and external irreversibilities are found to give realistic limits and trends for the cooling capacity and the COP of solar powered absorption cooling systems. 1999 Elsevier Science Ltd. All rights reserved.

External heat transfer irreversibilities between the heat reservoirs and the working fluid were included in the analysis presented by Yan and Chen (1989). Wu (1993) considered the optimization of waste-heat driven absorption refrigeration systems. Bejan et al. (1995) studied the optimal allocation of heat exchanger inventory in absorption systems while Vargas et al. (1996) presented the optimization of solar-powered absorption systems. External heat transfer irreversibilities were included in the models used in the above studies. Entropy production in absorption refrigerators was compared with experimental data in the study by Chua et al. (1997). Wijeysundera (1996) found that the performance limits predicted by the threeheat-reservoir cycle including external heat transfer irreversibilities were much closer to the simulated performance of actual absorption cooling systems. In a recent paper Wijeysundera (1997) studied the thermodynamic performance of solar powered absorption systems using a model which included external heat transfer irreversibilities. The aim of the present work is to extend the three-heatreservoir model to include entropy generation in the working fluid due to internal irreversible processes of the cycle. The effect of these irreversibilities on the second-law efficiency is investigated. The predicted performance of the absorp-

1. INTRODUCTION

Performance and design data for solar-powered absorption cooling systems are usually obtained by detailed simulation of the plant as described by Duffie and Beckman (1991). This procedure entails the solution of a large set of equations and requires as input the properties of the working fluids. Gommed and Grossman (1990) presented the results of a detailed computer simulation of absorption heat pumps. A computer code, ABSIM (1995), which simulates absorption cooling systems in a modular manner is now available. However the inclusion of such a computer code in a solar system simulation would require considerable effort. It is therefore useful to develop simple models to predict the performance trends and limits of solar-powered absorption cooling systems. The ideal coefficient of performance (COP) of absorption cycles may be obtained by analyzing the corresponding reversible cycle, which consists of three heat reservoirs. To obtain more realistic limits for the COPs of absorption cycles, Chen and Yan (1989) studied three methods for decomposing the absorption cycle into a reversible power cycle and a reversible refrigeration cycle. †

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to the cooling system shown in Fig. 1 to obtain the variation of important performance parameters. Assuming the storage tank temperature to be uniform, the rate of useful energy transfer, Q u can be written as: Q u 5 A c FR (ta )Q s 2 A c FRUL (T s 2 T o ) 2 A tUt (T s 2 T o )

Fig. 1. Schematic diagram of a solar-operated absorption cooling system.

tion system at the maximum cooling capacity and maximum COP is compared with simulation data obtained using the ABSIM (1995) computer programme. 2. ANALYSIS

A schematic diagram of a solar powered absorption cooling system is shown in Fig. 1. The useful energy collected is transferred to the hot liquid storage tank from which the generator of the absorption machine is supplied with input thermal energy. The heat extracted from the cooling load together with the energy input to the generator is rejected to the ambient through a cooling tower. The absorption machine can be idealized by a three-heat-reservoir cycle with finite heat conductances between the working fluid and the heat reservoirs as indicated in the thermal network diagram shown in Fig. 2. The entire system including the collector-storage unit is assumed to operate in a quasi-steady state. The first-law and the second-law can be applied

(1)

where Q s is the solar radiation intensity, A c and A t are collector and tank outside areas, UL and Ut are the collector and tank heat loss coefficients, FR is the efficiency factor, (ta ) is the transmittance–absorptance product and T s and T o are the tank and ambient temperatures, respectively. The energy transfer rate from the storage tank to the generator is given by Q u 5 Kh (T s 2 T s1 )

(2)

where Kh is an effective thermal conductance between the storage tank fluid and the working fluid in the generator of the absorption cycle, whose temperature is T s1 . The thermal conductance depends on the mass flow-rate of the storage tank fluid and the effectiveness of the generator heat exchanger. Combining Eqs. (1) and (2) the overall energy equation can be written as: Q sa 5 Kh (T s 2 T s1 ) 1 KL (T s 2 T o )

(3)

where Q sa 5 A c FR (ta )Q s and K L 5 A c FR U L 1 A t U t . The heat rejected to the ambient reservoir is given by: Q o 5 Ko (T o1 2 T o ) 5 Q u 1 Kc (T c 2 T c1 )

(4)

Kc and Ko are the effective conductances for the heat interactions between the cooling load reservoir at temperature T c and the ambient reservoir at temperature T o with the working fluid at temperatures T c1 and T o1 , respectively. For real absorption machines, the above conductances are functions of the effectiveness of the evaporator, the absorber and the condenser and the corresponding fluid flow-rates. The thermal network diagram shown in Fig. 2 can represent the energy transfers. The working fluid undergoes a cycle and therefore the net entropy change is zero. This may be expressed as: Fig. 2. Thermal network and energy-flow diagram.

Q o /T o1 2 Q c /T c1 2 Q u /T s1 5 DSgi

(5)

An irreversible-thermodynamic model for solar-powered absorption cooling systems

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where DSgi is the entropy increase in the working fluid due to adiabatic irreversible processes within the system. In real absorption machines, the internal entropy generation is a complex function of the various processes undergone by the working fluid in the cycle (Alefeld and Radermacher, 1995). In the present analysis the internal entropy generation rate is taken as a constant to be determined in a semi-empirical manner by matching the predictions of the model with actual performance data. It is convenient to express the above equations in terms of dimensionless variables. The following dimensionless variables are defined for this purpose: qs 5 A c FR (ta )Q s /T o KL ,

qc 5 Q c /T o Kc , Fig. 3. Variation of entropy transfer with storage tank temperature. Conditions: r 5 0.7, s 5 2.0, u 5 6.75, uc 5 0.926.

s 5 Ko /Kc , r 5 Kh /Kc ,

u 5 Kh /KL ,

w 5 r /(1 1 u),

d s i 5 DSgi /Kc , ui 5 T i /T o

(i 5 o, o1, s, s1, c, c1, su).

(6)

2.1. Entropy transfer equation When the absorption cycle operates under steady-state conditions, the net entropy transfer from the working fluid to the heat reservoirs is equal to the internal entropy generation in the working fluid as stated by Eq. (5). It is possible to express (5) in terms of the dimensionless variables of the system and the storage tank temperature. The dimensionless form of the Eqs. (3)–(5) is as follows: qs 5 u(us 2 us1 ) 1 (us 2 1)

(7)

uo1 5 1 1 qc /s 1 r(us 2 us1 ) /s

(8)

s /uo1 5 s 1 r 2 d s i 1 qc /(qc 2 uc ) 2 rus /us1 .

(9)

The dimensionless temperatures uo1 and us1 can be eliminated between Eqs. (7)–(9) to obtain the net entropy transfer equation in the following form:

d s tr 5 d s i 5 1 1 s 1 w 1 uc /(qc 2 uc ) 1 a 4 /(us 2 l4 ) 2 a 5 /(us 2 l5 )

(10)

where a 4 5 s 2 u /r, a 5 5 w 2 u(1 1 qs ) /r,

l4 5 1 1 qs 1 (qc 1 s)u /r, l5 5 w(1 1 qs ) /r.

(11)

A plot of the variation of the entropy transfer, d s tr given by (10), with the storage tank tempera-

ture us , for fixed values of the cooling capacity qc , and the solar radiation level qs , is shown in Fig. 3. It is seen that point A, where d s tr is zero, corresponds to the internally reversible cycle. At point B the cycle has the maximum entropy transfer and therefore the maximum internal entropy generation in the working fluid. All physically meaningful operating points of the irreversible three-heat-reservoir cycle must fall between A and B. The variation of the cooling capacity with storage tank temperature for a given d s tr may also be deduced by referring to the curves shown in Fig. 3. A straight line drawn parallel to the us axis at a height equal to the given d s tr will in general intersect the curves at two points as shown in Fig. 3. The point of intersection that lies between A and B gives the physically meaningful value of the tank temperature at the cooling capacity represented by the curve. As the cooling capacity is increased the height of the constant qc curves decrease and the points of intersection between the straight line and the curves get closer to each other until they finally coincide. At this value of the cooling capacity, the line of constant d s tr will be tangential to the curve. If the cooling capacity is increased further, the resulting curves will not intersect the straight line as seen in Fig. 3. Therefore, the constant qc curve that touches the straight line of constant d s tr at its turning point B corresponds to the maximum cooling capacity that may be obtained when the entropy transfer is at the given value d s tr . The coefficient of performance of the absorp-

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tion system with irreversibilities is given by the expression, COP 5 Q c /Q u 5 Q c /Kh (T s 2 T s1 ).

(12)

This may be expressed in the dimensionless form COP 5 qc u /r(qs 2 us 1 1).

(19)

In a recent paper (Wijeysundera, 1997) expressions were obtained for entropy generation and the second-law efficiency of solar-powered absorption cooling systems which included only external heat transfer irreversibilities. These expressions are applicable to the present system that includes internal irreversibilities of the absorption cycle because time-dependent entropy changes occur only in the reservoirs. The entropy generation is given by

(15) Sg 5 Q sa (1 /T o 2 1 /T su ) 2 Q c (1 /T c 2 1 /T o )

where a2

a 3 5 s 2 /qc .

h2 5 (qc /qs )(u /r)(1 /uc 2 1)[usu /(usu 2 1)] (16)

2.3. Maximum cooling capacity and COP The general COP-cooling capacity characteristic shows that for given internal entropy generation and solar radiation level, the cooling capacity of the system has a maximum value. Calculations indicate that the maximum cooling capacity and its trends of variation predicted by the present irreversible model agree reasonably well with simulation data for real plants. It is therefore useful to obtain an expression for the maximum cooling capacity. The storage tank temperature at the maximum cooling capacity operating point is obtained by applying the condition ds tr / dus 50 to Eq. (10). This gives the storage tank temperature at maximum qc as:

usm 5 [u(s 1 qc ) 1 r(1 1 qs ) 1 s(1 1 qs )1 / 2 ] / [r 1 s(1 1 u) /(1 1 qs )1 / 2 ].

(20)

where T su is the temperature of the sun. The second-law efficiency h2 can be expressed in terms of the dimensionless variables as follows:

l2 5 1 1 s /qc ,

5 rl1 /(1 1 u),

(17)

On substituting for the tank temperature in Eq. (10), the maximum cooling capacity, qcm is obtained as the positive root of the following quadratic equation: 2 m q cm 1 ( m x 2 muc 1 uc 2 f )qcm 1 (fuc 1 xuc

2 xmuc ) 5 0

2

1 s] .

2.4. Entropy generation and second-law efficiency of the plant

a 1 (1 / COP)2 1 [a 1 ( l2 2 l1 ) 1 a 2 2 a 3 ](1 / COP)

a 1 5 s 1 qc /(qc 2 uc ) 1 r /(1 1 u) 2 d s i ,

1/2

(14)

The variation of the COP of the absorption system with the cooling capacity is an important characteristic of the plant. The dimensionless temperatures uo1 , us1 and us can be eliminated between Eqs. (7–9) and (14) to obtain the following quadratic equation for the variation of (1 / COP) with cooling capacity,

l1 5 r(1 1 qs ) /(1 1 u)qc ,

5 [w(1 1 qs )

The COP at the maximum cooling capacity is computed by substituting in Eq. (14) for us and qc from Eqs. (17) and (18), respectively.

2.2. COP-cooling capacity characteristic

1 (a 2 l2 1 a 3 l1 2 a 1 l1 l2 ) 5 0

m 5 1 1 s 1 w 2 d s i , x 5 s 1 w(1 1 qs ), f

(13)

Therefore

us 5 1 1 qs 2 (qc u /r)(1 / COP).

where

(18)

(21)

where usu is the dimensionless temperature of the sun. In the next section the predictions of the above model will be compared with data obtained by detailed simulation of real absorption systems. 3. RESULTS AND DISCUSSION

Fig. 4 shows the variation of the COP of the irreversible three-heat reservoir cycle, with the cooling capacity for a given value of qs and several values of the internal irreversibility. For a given qc , the COP has two values resulting from the two positive roots of Eq. (15). The continuous section of the curves gives the physically meaningful values of the COP. In the section shown by the broken lines, a large portion of the heat from the high temperature reservoir flows directly to the heat-sink reservoir, thus resulting in the low COP for the cycle. When d s i is zero, the COP has a finite value at zero qc which is the Carnot COP of the cycle. As qc increases, the COP decreases until at maximum qc the COP has the lowest value. For systems with internal irreversibilities (d s i ), the COP is zero when qc is zero. As qc increases, the COP reaches a maximum (point C) and then decreases

An irreversible-thermodynamic model for solar-powered absorption cooling systems

73

Fig. 4. Variation of COP and second-law efficiency with cooling capacity for different values of entropy production. Conditions: r 5 0.7, s 5 2.0, u 5 6.75, uc 5 0.926.

to a lower value at the maximum qc. This variation is similar to that for real absorption cooling systems (Gommed and Grossman, 1990). The second-law efficiency given by (21) does not depend directly on the internal entropy generation as seen from Fig. 4. However, the second-law efficiency at the maximum cooling capacity (point D) decreases as the internal entropy generation increases. Fig. 5 shows the variation of COP and h2 with cooling capacity for different solar radiation levels at a given entropy generation. As expected the maximum cooling capacity increases with increasing solar radiation level. At given cooling

capacity h2 decreases with increasing solar radiation due to the increased heat transfer to the ambient heat reservoir which results from the higher storage tank temperature reached. A series of calculations were done to investigate how well the expressions derived in the preceding section predict the performance of real solar-powered absorption cooling systems. The ABSIM (1995) computer programme was used to simulate the performance of the absorption system under different operating conditions. The conductances and other system parameters are similar to those used by Gommed and Grossman (1990). The data obtained from the simulation of single-

Fig. 5. Variation of COP and second-law efficiency with cooling capacity for different values of solar radiation. Conditions: r 5 0.7, s 5 2.0, u 5 6.75, uc 5 0.926.

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Fig. 6. Variation of maximum cooling capacity with solar radiation. Conditions: r 5 0.7, s 5 2.0, u 5 6.75, uc 5 0.926.

stage and two-stage absorption machines were compared with predictions of the present irreversible model. It is interesting to study the variation of the COP at maximum qc because the second-law efficiency of the plant is a maximum at this point (Fig. 4). The variation of the maximum cooling capacity with the solar radiation level qs for different values of d s i is shown in Fig. 6. The graphs show a nearly linear variation which is a trend obtained in previous studies. The simulated results for single-stage and double-stage systems are in good agreement with the predictions of the model. By adjusting the value of internal entropy generation, d s i , the agreement can be improved compared with the internally reversible cycle. Fig. 7 shows the variation of the COP at maximum qc with the solar radiation level qs for different values of d s i . The general trend is well predicted by the model. The agreement is better for two-stage systems. However, the magnitude of the internal irreversibilities has to be adjusted to obtain better agreement between the simulated

results and the predictions of the model. It is seen that the internal entropy generation, d s i provides an additional degree of freedom to match the predictions of the model to simulation data. A comparison of the cooling capacity and the COP at the maximum COP operating point with simulation results showed poor agreement, which could only be improved marginally by adjusting the internal entropy generation. It can be concluded that the analytical expressions obtained in the present study for the maximum cooling capacity and the COP at the maximum cooling capacity can be used for design studies of solar operated absorption cooling systems. The present model yields analytical expressions for the important system parameters like the cooling capacity and the COP. These expressions can be used in preliminary design studies. Moreover, by selecting suitable values for the internal entropy generation the agreement between the predictions of the model and actual performance data can be improved. This procedure may be used as a convenient means to represent performance data in simulation studies where the solar-powered absorption cooling unit is a sub-component of a larger system. 4. CONCLUSION

A three-heat-reservoir model with external irreversibilities and constant internal irreversibilities was used to analyze the performance of solar-powered absorption cooling systems. The inclusion of the internal entropy production in the model makes it possible to predict a COP-cooling capacity characteristic, which is similar to that of real absorption systems. This model predicts the existence of a maximum COP in addition to the maximum cooling capacity. The maximum cooling capacity and the COP at the maximum cooling capacity show trends that agree well with results obtained by detailed simulation. The agreement between the predictions of the model and the simulation data can be improved by adjusting the magnitude of the internal irreversibilities. REFERENCES

Fig. 7. Variation of COP at maximum cooling capacity with solar radiation. Conditions: r 5 0.7, s 5 2.0, u 5 6.75, uc 5 0.926.

ABSIM User’ s Guide and Reference (version 1.2). Alefeld G. and Radermacher R. (1995). Heat Conversion Systems, CRC Press, Ann Arbor. Bejan A., Vargas J. V. C. and Sokolov M. (1995) Optimal allocation of a heat-exchanger inventory in heat driven refrigerators. Int. J. Heat Mass Transfer 38(16), 2997. Chen J. and Yan Z. (1989) Equivalent combined systems for three-heat-source heat pumps. J. Chem. Phys. 90(9), 4951– 4955.

An irreversible-thermodynamic model for solar-powered absorption cooling systems Chua H. T., Gordon G. M., Ng K. C. and Han Q. (1997) Entropy production analysis and experimental confirmation of absorption systems. Int. J. Refrigeration 20(3), 179. Duffie J. A. and Beckman W. A. (1991). Solar Engineering of Thermal Processes, 2nd ed, pp. 599–601, Wiley Interscience, New York. Gommed K. and Grossman G. (1990) Performance analysis of staged absorption heat pumps: water-lithium bromide systems. ASHRAE Trans. 93(2), 2389. Vargas J. V. C., Sokolov M. and Bejan A. (1996) Thermodynamic optimization of solar-driven refrigerators. Trans. ASME J. Solar Energy Eng. 118, 130.

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Wijeysundera N. E. (1996) Performance limits of absorption cycles with external heat transfer irreversibilities. Appl. Thermal Eng. 16(2), 175. Wijeysundera N. E. (1997) Thermodynamic performance of solar-powered ideal absorption cycles. Solar Energy 61(5), 313. Wu C. (1993) Cooling capacity optimization of a waste heat absorption refrigeration cycle. Heat Recovery Systems CHP 13(2), 161. Yan Z. and Chen J. (1989) An optimal endoreversible threeheat-source refrigerator. J. Appl. Phys. 65(1), 1.

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