Variation in temperature and exergy metrics with the outlet pressure for a throttling process of air.

## Abstract

The various definitions of the coefficient of exergy efficiency (CEE), which have been proposed in the past for the thermodynamic evaluation of compression and expansion devices, operating below and across ambient temperature as well as under vacuum conditions, are examined. The shortcomings of those coefficients are illustrated. An expression for the CEE based on the concept of transiting exergy is presented. This concept permits the quantitative and non-ambiguous definition of two thermodynamic metrics: exergy produced and exergy consumed. The development of these CEEs in the cases of an expansion valve, a cryo-expander, a vortex tube, an adiabatic compressor and a monophasic ejector operating below or across ambient temperature is presented. Computation methods for the transiting exergy are outlined. The analysis based on the above metrics, combined with the traditional analysis of exergy losses, allows pinpointing the most important factors affecting the thermodynamic performance of sub-ambient compression and expansion.

### Keywords

- exergy efficiency
- expansion
- compression
- sub-ambient
- across ambient

## 1. Introduction

Cooling is part of twenty-first century life. Air conditioning, food conservation, industries such as steel, chemicals, and plastics depend on cooling. By mid-century people will use more energy for cooling than heating [1]. Almost all cold is produced by vapor-compression refrigeration and requires large amounts of electricity for its production. And since electricity is still overwhelmingly produced by burning fossil fuels, the rise in cold production will inevitably increase both fuels consumption and power plant emissions. A climate-change irony is that cooling makes the planet hotter. Besides the development of new cooling devices using renewable energy, an important way to reduce refrigeration power consumption is through the energy efficiency improvement of vapor-compression cycles and their associated elementary processes. The processes of compression and expansion play a central role in air-conditioning, refrigeration and cryogenics. An important question still remains: How to define the efficiency of these processes by taking into account the constraints of the first and second laws of thermodynamics? The answer will be discussed in this paper.

The introduction of exergy, the thermodynamic function that takes into account the quality as well as the quantity of energy, has paved the way for a unified approach to the concept of efficiency, a subject pioneered by Grassmann [2]. Serious difficulties concerning the practical application of this concept to sub-ambient systems, however, retarded the acceptance of exergy analysis by the air-conditioning and refrigeration engineering profession. One can mention, in particular, the difficulty of formulating a coefficient of exergy efficiency (CEE) for elementary processes such as compression and expansion. The coefficient should evaluate the exergy losses, quantify the extent to which the technical purpose of an elementary process is achieved, as well as quantify the exergy consumption within the process. Finally, a uniquely determined value (not several) should be assigned to the coefficient. This paper examines some important points pertinent to these issues and presents a definition of the CEE for the thermodynamic evaluation of expansion and compression devices operating below and across ambient conditions. The definition is based on the concept of transiting exergy, introduced by Brodyansky et al. [3], that allows non-ambiguous computation of two metrics: exergy produced and exergy consumed.

## 2. Basic concepts of exergy analysis of sub-ambient systems

The maximum amount of work obtained from a given form of energy or a material stream, using the environment as the reference state, is called exergy [4, 5]. Three different types of exergy are important for thermodynamic analysis of the sub-ambient processes: exergy of heat flow, exergy of work (equivalent to work) and thermo-mechanical exergy, also known as physical exergy by some authors [4, 6]. Chemical exergy [7], important for some refrigeration systems based on the mixing of streams of different composition, is not considered in the present paper.

### 2.1. Exergy of heat flow at the sub-ambient conditions

The exergy of heat flow

where Θ = 1 − T_{0}/T is the Carnot factor determined by the temperature T of heat flow, and the ambient temperature T_{0}. Contrary to conditions above ambient, Θ is negative for sub-ambient temperatures. However, according to Eq. (1), _{0}. This means that the exergy of a heat flow at T < T_{0} is looked upon as a product of the refrigeration system rather than as feed. The exergy transfer of a RR characterizes the rate of transformation of power

### 2.2. Thermo-mechanical exergy

The thermo-mechanical exergy equals the maximum amount of work obtainable when the stream of substance is brought from its initial state to the environmental state, defined by pressure P_{0} and temperature T_{0}, by physical processes involving only thermal interaction with the environment [3, 4]. The specific thermo-mechanical exergy e_{P,T} is calculated according to:

The value of e_{P,T} may be divided by two components: thermal exergy e_{T} due to the temperature difference between T and T_{0}, and mechanical exergy e_{P} due to the pressure difference between P and P_{0}. It is important to emphasize that this division is not unique, because e_{T} depends on pressure conditions and e_{P} in its turn depends on temperature conditions. As a result, the division has no fundamental meaning and leads, as will be illustrated further, to ambiguities for the exergy efficiency definition. By conventional agreement [4], e_{T} and e_{P} are defined as:

The contribution of e_{T} and e_{P} to the value of e_{P,T} can be clearly visualized on the exergy-enthalpy diagram presented in Figure 2. For instance, the thermal exergy for point 1 is illustrated as the segment (e_{T})_{1} defined by the intersections of two isotherms T_{1} and T_{0} with the isobar P_{1}. The mechanical exergy for point 1 is illustrated as the segment (e_{P})_{1} defined by the intersections of two isobars P_{1} and P_{0} with the isotherm T_{0}. Whatever the temperature conditions are (T < T_{0} or T > T_{0}), the thermal exergy is always positive [3], as clearly presented on the e-h diagram. In this sense the e_{T} behavior is similar to that of the exergy of heat flow, that is always positive, as has been discussed. Meanwhile, e_{P} is only positive for conditions P > P_{0} (see for example point 1), but it is negative for P < P_{0}, as illustrated by point 2 in Figure 2.

### 2.3. Exergy efficiency of processes operating below and across the ambient temperature

The exergy balance around any process under steady state conditions and without external irreversibilities (the case considered in this paper) may be written as [4]:

Here

The input-output efficiency η_{in-out}, first proposed by Grassmann [2], is computed according to Eq. (6):

The shortcomings of this definition, particularly for its application to sub-ambient problems, are well documented [3, 4, 5, 6, 7, 8, 9]. The main one is the fact that often η_{in-out} does not evaluate the degree to which the technical purpose of a process is realized; the subject will be illustrated in Section 3. The products-fuel efficiency η_{pr-f} proposed by Tsatsaronis [10] and Bejan et al. [6] in the context of expansion and compression processes is computed as:

Under the terms “products” and “fuel” the authors meant either the differences in exergies of the streams at the inlet and outlet of a process, or the exergies of streams themselves. For example, while evaluating the efficiency of an adiabatic compression operating above ambient conditions, the “fuel” is the supplied work, and the “product” is the increment of thermo-mechanical exergy. The problem with this approach is that it is possible to obtain different values of η_{pr-f} of the sub-ambient expansion and compression processes due to the fact that different things can be understood under the notions “products” and “fuel”. It should be also mentioned that some authors used a different terminology to express the numerator and denominator of Eq. (7). For example, Kotas [4] used “desired output” vs. “necessary input”; Szargut et al. [5] used “exergy of useful products” vs. “feeding exergy”.

Brodyansky et al. [3] proposed a definition of efficiency based on the subtraction of the exergy that has not undergone transformation within an analyzed process. The latter was named “transiting exergy”,

where Δ_{tr} to assess the rate of exergy transfer of the mechanical exergy component to the thermal exergy component for these processes will be based on these metrics.

## 3. Transiting thermo-mechanical exergy and its link to the thermal and mechanical components

Following the equations proposed by Brodyansky et al. [3], the specific transiting thermo-mechanical exergy e_{tr} of an analyzed system is defined as the minimum exergy value that can be assigned to a material stream, considering the pressure P and temperature T at the inlet and outlet, as well as the ambient temperature T_{0}. With this definition, there are three possible combinations of P_{in}, T_{in}, P_{out}, T_{out} and T_{0} that determine the value of e_{tr}:

Inspection of these equations shows that for all three cases e_{tr} is determined by using the lowest pressure _{0} for the case of processes operating across ambient temperature. In order to understand the physical meaning of the transiting exergy, let us analyze the throttling process of a real gas taking place under these three different temperature conditions.

### 3.1. Adiabatic throttling process

The case of the throttling process operating above T_{0} is presented on an e-h diagram (see Figure 3a). According to Eq. (9a), the value of e_{tr} is:

This value coincides with the value e_{2} as illustrated in Figure 3a. The specific exergy losses (d) are also presented on the diagram. Following Eq. (8), the values Δ

where

As a result, the efficiency η_{tr} = 0, meaning that the exergy consumed is completely lost during the process and there is no produced exergy. It should be mentioned that the input-output efficiency calculated according to Eq. (6) has a negative value in this particular case and has no physical meaning. This is due to the fact that _{tr}.

Now, let us analyze the case of throttling at sub-ambient conditions presented in Figure 3b. According to Eq. (9b):

Thus Δ

The term (Δe_{T})_{Pout} in Eq. (14) is the increase of the specific thermal exergy due to an isobaric temperature drop under **sub-ambient** conditions at constant pressure P_{out}. The term (∇e_{P})_{Tin} in Eq. (15) is the decrease of the specific mechanical exergy due to an isothermal pressure drop at constant temperature T_{in}. Finally, the case presented in Figure 3c illustrates a throttling process started above ambient and ended at sub-ambient conditions. According to Eq. (9c):

The values Δ

Again, the input-output efficiency is not suitable for the evaluation of the processes presented in Figure 3b and c, given that the outlet exergy e_{2} = e(P_{out}, T_{out}) is negative. Another difficulty is linked to the application of the products-fuel efficiency for these cases. The exergy transfer of the throttling process at sub-ambient conditions consists in the partial transformation of mechanical exergy (“fuel”) into thermal exergy (“product”). The problem stems from the fact that there are multiple possibilities to define “fuel” and “product” in this case; as a result multiple values of η_{pr-f} may be formulated, leading to the ambiguity in the products-fuel efficiency application. Indeed, the different increments of thermal exergy may be considered as a “product” for the case in Figure 3b, for example, the increase in thermal exergy following the isobar P_{1} or the isobar P_{2}. In the same way, different decrements of mechanical exergy may be considered as a “fuel” in the same figure, for example, the decrease of mechanical exergy following the isotherms T_{1} or T_{2}.

Contrary to “products-fuel”, the transiting exergy approach does not attempt to individually compute the thermal and mechanical exergy component variations. It relies, rather, on the unaffected part of the thermo-mechanical exergy entering and leaving the system.

As illustrated in Figure 3a and c, the transiting exergy may be considered as the introduction of a new reference state to evaluate exergy consumed and produced. Instead of the reference point e = 0 (the intersection of the isobar P_{0} and the isotherm T_{0}), the new reference point is presented by e_{tr}: the intersection of the isobar P_{2} and the isotherm T_{2} for the case 3a; of the isobar P_{2} and the isotherm T_{1} for the case 3b; and of the isobar P_{2} and the isotherm T_{0} for the case 3c. Finally, the transiting exergy approach provides the foundation for the non-ambiguous definition of the terms Δ_{tr}.

**Example 1**

The initial parameters of air at the inlet of a throttling valve are: _{1} = 3 MPa, T_{1} = 140 K. The ambient temperature T_{0} = 283 K. Calculate the variation of _{tr} as a function of the outlet pressure P_{2} in the range 0.1–1 MPa.

**Solution**

The outlet temperature of the air is calculated by using the software Engineering Equation Solver (EES) [11]. Given that the expansion of air takes place at sub-ambient conditions, Eqs. (13)–(15) are used to evaluate _{tr} go along with the decreasing outlet pressure P_{2}. Less obvious is that the exergy produced Δ_{2}, reflecting the production of a more important cooling effect. It can also be noticed that the transiting exergy _{2}.

**Example 2**

The expansion valve of a refrigeration mechanical vapor compression cycle is supplied with the subcooled working fluid R152a at the rate _{1} = 615.1 kPa. The fluid is expanded to a pressure of P_{2} = 142.9 kPa. The ambient temperature T_{0} = 278 K. Calculate the variation of _{tr} as a function of the subcooling ΔT_{subC} in the range 275–281 K.

**Solution**

A vapor compression cycle is presented on a Ts-diagram in Figure 4. The subcooling process is represented by the line 3f-3. Given that the expansion of R152a takes place across ambient temperature, Eqs. (16)–(18) are used to evaluate

The transiting exergy does not change with the subcooling, because it is the function of constant parameters T_{0} and P_{2}, meanwhile the exergy produced increases and exergy consumed decreases. The new result is that η_{tr} is rising with the subcooling. It should be mentioned that increasing the amount of subcooling is well documented as a way to increase the COP (coefficient of performance) of vapor compression cycles [4]. Thus, the rise in η_{tr} of an expansion device guarantees the COP improvement of the overall cycle, a conclusion that may lead to practical recommendations for optimization of refrigeration cycles.

P_{2}(MPa) | T_{2}(K) | ∇ (kW) | Δ (kW) | (kW) | (kW) | η_{tr}(%) |
---|---|---|---|---|---|---|

1.0 | 118.6 | 101.8 | 30.8 | 71.0 | 245.9 | 30.3 |

0.9 | 117.3 | 110.8 | 32.4 | 78.4 | 236.9 | 29.2 |

0.7 | 114.7 | 132.0 | 35.6 | 96.4 | 215.7 | 26.9 |

0.5 | 111.9 | 160.1 | 38.8 | 121.3 | 187.6 | 24.2 |

0.3 | 109.1 | 202.3 | 42.1 | 160.2 | 145.4 | 20.8 |

0.1 | 106.0 | 292.3 | 45.5 | 246.8 | 55.5 | 15.6 |

ΔT_{subC}(K) | ∇ (kW) | Δ (kW) | (kW) | (kW) | η_{tr}(%) |
---|---|---|---|---|---|

275 | 4.083 | 3.208 | 0.875 | 1.744 | 78.6 |

276 | 4.066 | 3.230 | 0.836 | 1.744 | 79.4 |

278 | 4.034 | 3.274 | 0.760 | 1.744 | 81.2 |

279 | 4.020 | 3.295 | 0.725 | 1.744 | 82.0 |

281 | 3.994 | 3.339 | 0.655 | 1.744 | 83.6 |

### 3.2. Expansion in low temperature systems with work production and heat transfer

The primary purpose of expansion processes in the sub-ambient region is the production of cooling effect. The power that may be produced can be considered as a useful by-product. This type of expansion takes place in cryo-expanders. There are two types of these devices: adiabatic and non-adiabatic gas expansion machines. The energy and exergy balances around a non-adiabatic expander are presented in Figure 5. It should be emphasized that the directions of heat flow _{0}. As a result

The process of gas expansion in a non-adiabatic cryo-expander is presented on an e-h diagram (see Figure 6). Similar to the case of adiabatic throttling (Figure 3b), the transiting exergy in the gas flow is defined according to Eq. (9b). As a result, the exergy efficiency is calculated as:

In the case of an adiabatic cryo-expander η_{tr} is calculated according to Eq. (19), but with the term _{in} equals to zero.

**Example 3**

An adiabatic turbine (η_{T} = 0.80) is supplied with air at the rate _{1} = 6 MPa, T_{1} = 320 K. The ambient temperature T_{0} = 283 K. Calculate the variation of _{tr} as a function of the outlet pressure P_{2} in the range 0.1–3 MPa.

**Solution**

Given that the expansion of air takes place across ambient temperature, Eqs. (16)–(18) are used to evaluate _{tr} is calculated according to Eq. (19), but with the term _{in} equals to zero. The results are shown in Table 3. It is illustrated that _{2} reduction, and as a result Δ_{tr} to decrease. The negative value of

P_{2}(MPa) | T_{2}(K) | ∇ (kW) | Δ (kW) | (kW) | (kW) | (kW) | η_{tr}(kW) |
---|---|---|---|---|---|---|---|

3.00 | 271.8 | 57.9 | 0.2 | 45.6 | 12.1 | 274.2 | 79.1 |

2.55 | 263.8 | 68.3 | 0.7 | 53.0 | 14.6 | 263.9 | 78.7 |

2.50 | 255.1 | 80.1 | 1.5 | 61.2 | 17.4 | 252.0 | 78.2 |

1.85 | 245.4 | 94.1 | 2.9 | 70.3 | 21.0 | 238.0 | 77.7 |

1.50 | 234.3 | 111.1 | 4.9 | 80.7 | 25.5 | 221.1 | 77.1 |

1.15 | 221.2 | 132.6 | 8.2 | 93.1 | 31.3 | 199.6 | 76.4 |

0.50 | 204.9 | 162.0 | 13.6 | 108.4 | 40.0 | 170.1 | 75.3 |

0.45 | 182.4 | 208.9 | 24.1 | 129.7 | 55.1 | 123.2 | 73.6 |

0.10 | 138.9 | 333.2 | 57.7 | 171.6 | 103.9 | −1.05 | 68.8 |

### 3.3. Expansion in a vortex tube

Figure 7a illustrates a counter flow vortex tube [12]. High pressure gas enters the tube through a tangential nozzle (point 1). Colder low-pressure gas leaves via an orifice near the centerline adjacent to the plane of the nozzle (point 2), and warmer low-pressure gas leaves near the periphery at the end of the tube opposite to the nozzle (point 3). The vortex tube requires no work or heat interaction with the surroundings to operate. The cold mass fraction is μ; the hot gas mass fraction is (1 − μ). The exergy balance around the vortex tube is:

The expansion processes taking place within a vortex tube are presented on an e-h diagram (Figure 7b). The cold stream expands across T_{0}, the hot expands at T > T_{0}. By applying Eqs. (9a) and (9c) the transiting exergies may be determined for each mass stream, cold (1–2) and hot (1–3).

As a result, the exergy produced and consumed within the cold and hot streams are:

Thus Δ_{1} to T_{0}, and the decrease of mechanical exergy because of pressure drop from P_{1} to P_{3}. ∇_{tr}.

**Example 4**

An adiabatic vortex tube is supplied with air as ideal gas at the rate _{1} = 0.8 MPa, T_{1} = 308 K. The air expands at the cold end to pressure P_{2} = 0.1 MPa and at the hot end to the pressure P_{3} = 0.15 MPa. The ambient temperature T_{0} = 298 K. Calculate the variation of _{tr} as a function of the cold mass fraction μ in the range 0.2–0.9.

**Solution**

The results are shown in Table 4. It is illustrated that _{2} = P_{0} and T_{2} = T_{0}. As a result, for the cold stream Δ_{tr} increases, despite the rise in the exergy losses with the increasing cold mass fraction. This can be explained by the fact that the rise in exergy produced in the hot stream surpasses the increase in exergy losses. The exergy efficiency of the vortex tube is relatively low.

μ (–) | T_{2}(K) | T_{3}(K) | ∇ (kW) | Δ (kW) | (kW) | ∇ (kW) | Δ (kW) | (kW) | (kW) | η_{tr}(%) |
---|---|---|---|---|---|---|---|---|---|---|

0.35 | 268.0 | 329.5 | 61.9 | 0.6 | 0.0 | 93.1 | 0.9 | 21.9 | 153.5 | 0.97 |

0.60 | 273.0 | 360.4 | 106.2 | 0.7 | 0.0 | 57.3 | 2.3 | 13.5 | 160.5 | 1.83 |

0.75 | 278.0 | 397.6 | 132.7 | 0.5 | 0.0 | 35.8 | 3.4 | 8.4 | 164.6 | 2.32 |

### 3.4. Compression across ambient temperature

In most refrigeration plants and heat pumps compression starts at T < T_{0} and ends at T > T_{0}. The process is presented on an e-h diagram (see Figure 8). According to Eq. (9c), transiting exergy is:

The produced and consumed exergies are:

Δ_{2} and the rise in temperature from T_{0} to T_{2}. ∇_{1} to T_{0} under conditions of constant pressure P_{1}, plus the consumed power _{tr}.

**Example 5**

An adiabatic compressor of a refrigeration plant is supplied with the working fluid R152a at the rate _{1} = 142.9 kPa and T_{1} = 263 K (superheated). The fluid is compressed to a pressure of P_{2} = 615.1 kPa. The ambient temperature T_{0} = 298 K. Calculate the variation of _{tr} as a function of the isentropic efficiency η_{C} in the range 0.75–0.90.

**Solution**

The results are shown in Table 5. The transiting exergy does not change with the isentropic efficiency η_{C}, because is a function of constant parameters T_{0} and P_{1}. The exergy consumed does not change either. The produced exergy decreases. This drop in Δ_{tr} increases.

η_{C}(–) | (kW) | ∇ (kW) | Δ (kW) | (kW) | (kW) | η_{tr}(kW) |
---|---|---|---|---|---|---|

0.75 | 66.0 | 2.3 | 53.3 | 15.0 | 12.6 | 78.0 |

0.80 | 61.9 | 2.3 | 52.9 | 11.3 | 12.6 | 82.4 |

0.85 | 58.3 | 2.3 | 52.6 | 8.0 | 12.6 | 86.8 |

0.90 | 55.0 | 2.3 | 52.3 | 5.0 | 12.6 | 91.3 |

### 3.5. Compression and expansion in a one phase ejector

A combination of the processes of vapor expansion and compression takes place within a one-phase ejector presented in Figure 9a. A primary (pr) stream at high pressure P_{1} and temperature T_{1} expands and entrains a secondary stream (s) at low pressure P_{2} and temperature T_{2} < T_{0}. The ratio _{2} < P_{3} < P_{1} and T_{2} < T_{3} < T_{1} leaves the ejector. The exergy balance around the ejector is:

The processes of expansion of the primary stream and compression of the secondary stream are presented on an e-h diagram (Figure 9b). The secondary stream is compressed across T_{0}, meaning that Eq. (9c) is applied to calculate (e_{tr})_{s}. As a result, the transiting exergy for secondary and primary streams are:

This means that the exergies produced and consumed may be computed as:

Δ_{2} to P_{3} and the rise in temperature from T_{0} to T_{3}. The exergy consumed ∇_{2} to T_{0} (the partial cold destruction). The exergy consumed within the primary stream ∇_{tr}. The detailed analysis of efficiencies for different parts of an ejector is given in [13].

**Example 6**

An ejector of a refrigeration plant is supplied with the working fluid R141b. The parameters of the secondary stream are: P_{2} = 22.3 kPa, T_{2} = 268 K. The pressure of the mixed stream is P_{3} = 91 kPa. The ambient temperature T_{0} = 289 K. Calculate the variation of (_{pr}, (_{s}, Δ_{tr} as a function of the entrainment ratio ω =

**Solution**

The calculation results are shown in Table 6. The transiting exergy in the secondary flow is negative because the parameters P_{2} and T_{0} define the state of the flow under vacuum conditions. The exergy produced and exergy consumed increase with the entrainment factor. The increase in Δ_{tr} increases.

ω (–) | ∇ (kW) | ∇ (kW) | Δ (kW) | (kW) | (kW) | (kW) | (kW) | η_{tr}(%) | P_{1}(kPa) | T_{1}(K) |
---|---|---|---|---|---|---|---|---|---|---|

0.15 | 9.99 | 0.017 | 1.189 | 8.818 | 4.286 | −0.546 | 3.740 | 11.9 | 1000 | 418 |

0.17 | 10.07 | 0.019 | 1.334 | 8.755 | 4.205 | −0.619 | 3.586 | 13.2 | 1000 | 418 |

0.20 | 10.19 | 0.023 | 1.547 | 8.666 | 4.090 | −0.729 | 3.361 | 15.2 | 1000 | 418 |

0.23 | 10.29 | 0.026 | 1.754 | 8.562 | 3.983 | −0.838 | 3.145 | 17.0 | 1000 | 418 |

0.25 | 10.36 | 0.028 | 1.890 | 8.498 | 3.915 | −0.911 | 3.004 | 18.2 | 1000 | 418 |

## 4. Environmental life cycle analysis and exergy efficiency of cooling systems

Life Cycle Analysis (LCA) is an important tool to analyze environmental problems associated with the production, use, and disposal of products or systems [14]. For every product produced within a system the total inflow and outflow of energy and materials are evaluated. The environmental burdens are associated by quantifying the energy and materials used, as well as the wastes released into the environment. The impact of these uses and releases on the environment is assessed. The multidimensional approach of LCA causes some problems when different substances need to be compared and general agreement is required. This problem may be avoided if exergy is used as a common quantity as proposed by Life Cycle Exergy Analysis [15]. The crucial idea behind this method is the distinction between renewable and non-renewable resources. In order to illustrate the method, let us consider three defined time periods within the life cycle of an ejector refrigeration system driven by solar energy [16]. At first, exergy is required during the construction stage to build the plant and put it into operation. During this period the spent exergy is stored in materials, such as metals, glass etc. For the second period, maintenance required for the system’s operation takes place. Exergy necessary for this maintenance is evaluated. The third period is the clean-up stage, including the plant demolition and the recycling of materials. Exergy used for the clean-up is assessed. The exergy used for the construction, maintenance, and clean-up is assumed to originate from non-renewable resources and is named indirect exergy, _{tr} of an ejector, as presented in Section 3.5, and its subsequent maximization, may lead to the construction and operation of more sustainable solar driven refrigeration plants.

## 5. Conclusion

The common feature of expansion processes operating below or across ambient temperature is the partial transformation of the mechanical exergy component into the thermal exergy component. Sub-ambient compression processes are characterized by the transformation of work into the mechanical exergy component and the partial destruction of the thermal exergy component below T_{0}. In order to evaluate the efficiency of these transformations the calculations of the variation in mechanical and thermal exergy components are required. These calculations may be done in many different ways, for example the variation in e_{P} depends on the chosen temperature conditions, while the variation in e_{T} depends on the chosen pressure conditions. This multiplicity in the exergy variation evaluation leads to ambiguity in the exergy efficiency definition. The approach based on the exclusion of the “transiting flow” from thermo-mechanical inlet and outlet exergies of an analyzed process overcomes this difficulty. This improvement is possible because the transiting exergy is uniquely defined by a specific combination of the process intensive parameters, namely the inlet and outlet pressures and temperatures, as well as T_{0}. The transiting exergy approach allows non-ambiguous evaluation of two thermodynamic metrics: exergy produced and exergy consumed. Their ratio represents the exergy efficiency; the difference between exergy consumed and exergy produced equals the exergy losses within the process. The phenomenological significance of the transiting exergy and the way in which it can be computed for processes below and across T_{0} has been illustrated for the cases of an expansion valve, a cryo-expander, a vortex tube, an adiabatic compressor, and a monophasic ejector. The input-output exergy efficiency is not an appropriate criterion for evaluation of these processes.

## Acknowledgments

This project is a part of the Collaborative Research and Development (CRD) Grants Program at “Université de Sherbrooke”. The authors acknowledge the support of the Natural Sciences and Engineering Research Council of Canada, Hydro Québec, Rio Tinto Alcan and CanmetENERGY Research Center of Natural Resources Canada (RDCPJ451917-13).

## Nomenclature

D ̇ | Destroyed exergy, (kW) |

d | Specific exergy losses, (kJ/kg) |

e | Specific exergy, (kJ/kg) |

E ̇ | Exergy, (kW) |

h | Specific enthalpy, (kJ/kg) |

H ̇ | Enthalpy, (kW) |

ṁ | (Total) Mass flowrate, (kg/s) |

P | Pressure, (MPa, kPa) |

Q ̇ | Heat rate, (kW) |

s | Specific entropy, (kJ/kg K) |

T | Temperature, (K, °C) |

W ̇ | Mechanical power, (kW) |

η | Efficiency, (%) |

∇ | Consumption |

Δ | Production |

μ | Cold mass fraction, μ = ṁC/ṁ |

ω | Entrainment ratio |

0 | Ambient state |

1, 2, 3… | States in a process |

C | Cold, Compressor |

f | Fuel |

H | Hot |

in | Inlet |

ind | Indirect |

int | Internal |

max | Maximal |

min | Minimal |

out | Outlet |

pr | Primary, Product |

S,s | Secondary |

subC | Subcooling |

tr | Transiting |