Keywords 

COP; Efficiency; Entropy production; Lower and upperbounds;
Rankin cycle; Absorption cycle 

Introduction 

In the analysis of thermal power, cooling and heat pump cycles
it has been the general practice to use the ideal Carnot efficiency and
coefficient of performance (COP). However, due to the ideal nature of
Carnot cycle the resulting efficiency and COP relations are the highest
upper bound to the real values of efficiency and COP. The marvelously
simple and highly cited Carnot cycle and its related efficiency and
COP relations [1] were proposed at a time when principles of
thermodynamics were at their infancy. The genius Nicolas Léonard Sadi
Carnot who proposed his cycle in 1823 recognized the need to develop
his theory independent of the knowledge about properties of working
fluids, at a time of lack of any accurate thermodynamic property data
for such fluids. 

Presently, thanks to extensive research and development in
thermodynamics of irreversible processes [2,3] and our knowledge
about accurate thermodynamic properties of materials (see [4] and
many databooks published by IUPAC, JANAF, NIST, TRC, etc.). Since
the time of Sadi Carnot, we can now develop upper bounds to efficiency
and COP which are much closer to their real values than those of
Carnot cycle values. Also the methodology introduced in this report
has allowed us to generate lower bounds to efficiency and COP. 

In this report we demonstrate, through the application of the 2nd
law of thermodynamics for irreversible processes, it is possible to derive
both upper and lowerbounds to efficiency and COP of cycles. The
upper bounds derived and reported here are lower upper bounds than
the Carnot cycle values. Availability of both, lower and upper bounds
to efficiencies and COPs of cycles will allow us to acquire a better
understanding about the real performance of thermodynamic cycles. 

The Theory 

According to thermodynamics of flow processes for an open system
with incoming and outgoing flows the first law of thermodynamics can
be written in the following form [2,3,5]: 

(1) 

In this equation is the rate of energy accumulation in the
system, and are the rates of work and heat added to the system,
respectively. P is the pressure, v is the specific volume, e is the energy per unit mass and is the mass flow rate of incoming and outgoing
flows. 

The second law of thermodynamics for an open flow process takes
the following form [2,3,5]: 

(2) 

In this equation p_{s} is the rate of production of entropy in the
system, is the entropy accumulation rate in the system, s is the
entropy per unit mass of the incoming and outgoing flows to and from
the system, and T_{ej} is the temperature of the external heat source or
sink . In what follows we apply Eq.s (1) and (2) to develop the upper
and lower bounds to the efficiency of thermal power cycles and COP of
cooling / heat pump cycles. 

Rankine Thermal Power Cycle 

In Figure 1 we demonstrate a basic Rankine thermal power cycle as
it is wellknown: 

For a basic Rankine thermal power cycle, Figure 1, considering it
to be in the steady state and steady flow conditions, application of the
second law for the boiler produces the following inequality for 

(3) 

Application of the 2nd law for the condenser gives us 

(4) 

In the above two relations equality signs are for the reversible cases
and inequality signs are for the irreversible cases. As a result of the
above two inequalities we get, 

(5) 

Considering that the actual efficiency of the cycle is defined as 

(6) 

We conclude from inequality (5) the following upper bound for
efficiency 

(7) 

The upper bound of efficiency as it is shown by Inequality (7) is
lower than the Carnot efficiency, i.e. 

(8) 

This is because (s_{4}s_{3}) ≥ (s_{1}s_{2}) as it is shown in Figure 2 and
the fact that the Carnot efficiency depends only on the temperatures of
the heat source and heat sink and it is independent of the working fluid
characteristics. 

In order to derive the lower bound to the efficiency, we use the first
law for the turbine which gives us 

(9) 

From relations (3) and (9) we conclude 

(10) 

Finally we have the following upper and lower bounds to the actual
efficiency of the cycle: 

(11) 

The above inequalities can be used to calculate the upper and lower
bounds of the efficiency of a cycle. We should add, the efficiency of the
cycle according to the first law of thermodynamics is in the following
form: 

(12) 

Obviously 

, (13) 

and we may expect η_{1st Law} to be larger than η_{Actual}, but there is
no theoretical indication of the relative values of η_{1st Law} and the upper
bound to efficiency, i.e. 

The above inequalities can be used to calculate the upper and lower
bounds of the efficiency of a Rankine thermal power cycle. In what
follows we show two expels of applications of the above expressions of
efficiencies. 

Example 1: 

As the first example we consider the data of the cycle shown on
Figure 2 in which water is the working fluid with T_{H}=500°C=773K,
TC=100°C=373K, h_{1}=3460, h_{2}=1320, h_{3}=515, h_{4}=2650 [kJ/kg], and
s_{1}=7.35, s_{2}=3, s_{3}=1.35, s_{4}=7.4 [kJ/kg.K]. We calculate the following value
for the Carnot, upper bound, lower bound, and first lawefficiencies: 



According to the above calculations 24.1% ≤ η_{Actual} ≤ 32.9% while
η_{Carnot} =51.7% which is much higher than 32.9%, the upper bound of
efficiency of the Rankine thermal power cycle for the data of Figure
2. The efficiency based on the first law of thermodynamics η_{1st Law}
= 37.8% is still much lower than the Carnot efficiency and closer to
the actual efficiency of the cycle.It is worth mentioning that the actual
efficiency of Rankin thermal power cycles at best possible conditions
has not exceeded much above 40%. 

Example 2: 

We would like to search for the best working fluid which can be
used in a Rankine power cycle operating between temperatures of 40°F
(4.4°C) and 80°F (26.7°C). A real life example of this cycle is the Ocean
Thermal Energy Conversion (OTEC) system in which the hot source is
the surface ocean water and cold source is the water about 1000 meters
deep in the ocean [6]. We bound our search here to pure working fluids
even though mixtures can also be considered as possible candidates for
such application. The Carnot efficiency of the cycle is 7.41% which is
independent of the nature of working fluid.By considering the vapor
coming out of the boiler to be a saturated vapor at 80°F and application
of expressions for the upper bound, lower bound and 1st law efficiency
of the cycle, the following table is produced for eight different candidate
working fluids. 

Experimental data needed to produce this table were taken from Perry’s Handbook [7]. Because of the low temperature difference
between the heat source and heat sink all the efficiencies are rather
small. But it is clear that among all the fluids investigated 1, 3, Butadiene
will be a better working fluid from the thermodynamics point of view.
It is worth noting that by the mere use of the Carnot cycle efficiency
there is no way to compare capabilities of the working fluids (Table 1).
Comparison of efficiencies of various working fluids which may be used
in an OTEC Rankine power cycle operating between temperatures of
40°F (4.4°C) and 80°F (26.7°C). 

In what follows we consider two different cooling and heat pump
cycles. One is the Rankine cycle, and the other type is the absorption
cooling cycle [8,9]. 

Rankine Cooling and Heat Pump Cycle 

In Figure 3 we demonstrate a basic Rankine cooling and heat pump
cycle as it is wellknown: 

For the basic Rankine cooling and heat pump cycle, Figure 3,
considering that to be in the steady state and steady flow conditions,
application of the second law of thermodynamics for the evaporator
(refrigerator) and the condenser produces the following inequalities: 

, (13) 

and 

(14) 

In the above two relations equality signs are for the reversible cases and inequality signs are for the irreversible cases. Considering that the
coefficient of performance (COP) of the cycle is defined as 

(15) 

and knowing that from Relations (13) and (14) 

, (16) 

we get the following upper bound for the cycle COP 

(17) 

The upper bound of COP as shown by the right side of (17) is lower
than the Carnot COP, i.e. 

(18) 

This is because (s_{1}s_{2})≥ (s_{4}s_{3}) as it is shown in Figure 4 and
the fact that Carnot COP depends only on the temperature of the
heat source and heat sink and it is independent of the working fluid
characteristics. 

An example of stages of ammonia phase transitions going through
the basic Rankine cooling cycle of Figure 4. Stage numbers 1, 2, 3, 4 on
this figure correspond to the same stages shown in Figure 3. 

In order to derive a lower bound to the COP for this cycle we use
the first law for the compressor which gives 

, (19) 

we also know that 

(20) 

Now by dividing the left side of (20) by right side of (19) we get, 

(21) 

The right side of (21) provides us with the lower bound of the COP
of the cycle. 

Finally we have the following upper and lower bounds (UB, LB) to
the actual COP of the cycle: 

(22) 

The above inequalities can be used to calculate the upper and lower
bounds of the COP of a Rankine cooling cycle. However, the COPof the
cycle according to the first law of thermodynamics is in the following
form: 

(23) 

Example 

Inequalities Eq. (22) can be used to calculate the upper and lower
bounds of the COP of a Rankine cooling cycle. As an example for the
data of the cycle shown on Figure 4 in which T_{R}=20°C =253K, T_{C}=60°C
=333 K, h_{1}=1775, h_{2}=480, h_{3}=480, h_{4}=1410 [kJ/kg], and s_{1}=5850,
s_{2}=1950, s_{3}=2150, s_{4}=5850 [J/kg.K], we calculate the following value for
the Carnot, upper bound, lower bound, and first law efficiencies: 

LB = 2.558 ≤ COP_{Actual} ≤ UB = 2.581≤ COP_{Carnot} = 3.163 

and 



According to the above calculations 2.558 ≤ COP_{Actual} ≤ 2.581while
COP_{Carnot}=3.163 which is much higher than 0.2.581, the upper bound
of actual COP of the Rankine cooling cycle for the example of Figure 4. 

Absorption Cooling Cycle 

In Figure 5 we demonstrate a basic absorption cooling and heat
pump cycle as it is wellknown: 

The coefficient of performance (COP) of the absorption cooling
cycle, Figure 5, is defined as the ratio of cooling effect by the evaporator
and the heat input to the generator, 

(24) 

According to the first law of thermodynamics the following balance
equation holds for the whole cycle, 

(25) 

According to the second law of thermodynamics the following
inequality can be written for the cycle, 

(26) 

or 

(27) 

Assuming the work input to the liquid pump negligible as compared
to the other terms. Now by consideration of definition of COP_{Actual}, Eq.
(24), the above inequality can be rearranged to the following form 

(28) 

This upper bound to COP_{Actual} is the Carnot cycle COP. According to
the first law of thermodynamics for flow systems the following relations
hold between the heat and work transfer rates and the properties of the
working fluids in a steady state steady flow condition: 

(29) 

(30) 

(31) 

(32) 

(33) 

In the above equations is the mass flow rate of refrigerant passing
through the throttling valve (I) and is the mass flow rate of the
solution passing through the liquid pump.The following relation exist
between and 

(34) 

where x_{A} is the mass fraction of refrigerant in the strong liquid
phase coming out of absorber and x_{G} is for the liquid phase coming
out of the generator. In deriving Eq. (34) it is assumed the vapor coming
out of the generator is pure refrigerant. 

According to the second law of thermodynamics for open
systems, Inequality Eq. (2), the following relation also holds the heat
transfer rates and properties of working fluids in a steady state steady
flow absorption cooling cycle: 

(35) 

(36) 

(37) 

(38) 

In the above four relations equality signs are for the reversible cases
and inequality signs are for the irreversible cases. By joining Eq. (35)
and (37) we get, 

(39) 

Also by joining Eq. (36) and (38) we get, 

(40) 

Now by assuming negligible as compared with the other terms
in Eq. (25) we can write: 

(41) 

Then from relations Eq. (39)  (41) we conclude that 

(42) 

By dividing Eq. (42) by and consideration of Eq. (24) for the
Figure 5: Absorption cooling cycle. definition of COP we derive the following relation, 

(43) 

Where lower bound (LB) and upper bound (UB) of COP will have
the following expressions, 

(44) 

(45) 

Where, 

(46) 

Relations Eq. (44)(46) can be used to calculate the upper and lower
bounds of COP of the cycle knowing the working fluid properties.Also
with the understanding that the Carnot COP as given by the right side
of Eq. (28) is the upper bound of COP regardless of working fluid it is
always larger than UL as given by Eq. (45). In conclusion we can write 

(47) 

These inequalities can be used to calculate the lower bound and
upper bound of the actual COP of absorption cooling cycles. 

The following relations also exist for the isenthalpic expansion valve
(I) and (II) in the cycle: 

h_{2}=h_{3} and h_{7}=h_{8}. (48) 

By assuming the power input to the liquid pump
negligible the
following relation will also hold 

h_{2}=h_{5} (49) 

Based on the above equations the COP of the cycle based on the
first law of thermodynamics alone is defined by the following relation: 

(50) 

These inequalities can be used to calculate the lowerbound and
upper bound of the COP of absorption cooling cycles. 

Example 

Solar energy as the heat source can be utilized through the
absorption cooling cycle shown in Figure 5 for cooling (refrigeration
and air conditioning) purposes. The major questions in the design of
solar absorption cooling cycles are the choice of combined working
fluids (refrigerant and absorbent) and thermal energy storage system
for the cycle to continue working during evening and cloudy days. The
latter subject is out of the scope of this report and the reader is referred
to other literature [10,11]. 

The upper and lowerbond expressions for the COP absorption
cooling cycle as reported by Eq’s (44) and (45) are used in order to make
comparative studies of candidate working fluid combinations of the
cycle. We have reported the details of the methodsand results of various
calculations of the upper and lowerbounds of COP of absorption
cooling cycle in our earlier publications [8,9,12,13]. In general the
present approach has allowed us to compare various absorbentrefrigerant
combinations which would have been otherwise impossible
to do with the use of Carnot cycle COP calculation. 

Conclusions 

The inequalities reported in this paper can be used to calculate the
lower bounds and upper bounds of efficiencies of Rankine thermal
power cycles, COPs of Rankine cooling and heat pump cycles and
COPs of absorption cooling and heat pump cycle. There are several
advantages in using these inequalities over the Carnot upper bound
values for efficiency and COP: i. We are now able to calculate, both
upper and lower bounds of efficiency and COPs which are quite useful
for a more proper design of power and cooling cycles. ii. In the study of
specification of better working fluids for alternative power and cooling
cycles such upper and lower bounds will help to choose the optimum
kind of working fluid. iii. Overall, the inequalities presented in this
report are the thermodynamics criteria for the optimum design of
thermal power cycles and cooling and heat pump cycles from the point
of view of energy conservation and sustainability. 

Acknowledgements 

The author would like to thank Prof. A.L. Gomez and Mr. V. Patel for their
helpful comments. 

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