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 Q [4] is defined as: \_

$$
\dot{\mathbf{E}}^{\dot{Q}} = \dot{\mathbf{Q}} \cdot \boldsymbol{\Theta} \tag{1}
$$

where Θ = 1 � T0/T is the Carnot factor determined by the temperature T of heat flow, and the ambient temperature T0. Contrary to conditions above ambient, Θ is negative for sub-ambient temperatures. However, according to Eq. (1), E\_ <sup>Q</sup> is positive due to the fact that heat is removed from a cooled object, and thus Q has a negative sign in Eq. (1). The energy and exergy balances \_ of a reversible refrigerator (RR) are presented in Figure 1. One can notice that the directions of energy and exergy flows are opposite below T0. This means that the exergy of a heat flow at T<T0 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 W to exergy of heat flow \_ Q\_ (exergy of produced cold). Given that the system presented in Figure 1 is reversible, the minimum power W\_ min necessary to maintain a cooling rate Q equals \_ E\_ Q. Obviously it is not the case for a real (non-reversible) refrigerator, where E\_ <sup>Q</sup> is lower than W by the value of \_ exergy losses D. \_

#### 2.2. Thermo-mechanical exergy

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

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

thermodynamics? The answer will be discussed in this paper.

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

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

The exergy of heat flow Q [4] is defined as: \_

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

where Θ = 1 � T0/T is the Carnot factor determined by the temperature T of heat flow, and the ambient temperature T0. Contrary to conditions above ambient, Θ is negative for sub-ambient

<sup>E</sup>\_ <sup>Q</sup> <sup>¼</sup> <sup>Q</sup>\_ � <sup>Θ</sup> (1)

consumed.

62 Energy Systems and Environment

present paper.

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 P0 and temperature T0, by physical processes involving only thermal interaction with the environment [3, 4]. The specific thermo-mechanical exergy eP,T is calculated according to:

$$\mathbf{e}\_{\mathbf{P},\mathbf{T}} = \left[\mathbf{h}(\mathbf{P},\mathbf{T}) - \mathbf{h}(\mathbf{P}\_0, \mathbf{T}\_0)\right] - \mathbf{T}\_0 \cdot \left[\mathbf{s}(\mathbf{P}, \mathbf{T}) - \mathbf{s}(\mathbf{P}\_0, \mathbf{T}\_0)\right] \tag{2}$$

The value of eP,T may be divided by two components: thermal exergy eT due to the temperature difference between T and T0, and mechanical exergy eP due to the pressure difference between P and P0. It is important to emphasize that this division is not unique, because eT depends on pressure conditions and eP 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], eT and eP are defined as:

$$\mathbf{e}\_{\Gamma} = \left[ \mathbf{h}(\mathbf{P}, \mathbf{T}) - \mathbf{h}(\mathbf{P}, \mathbf{T}\_0) \right] - \mathbf{T}\_0 \cdot \left[ \mathbf{s}(\mathbf{P}, \mathbf{T}) - \mathbf{s}(\mathbf{P}, \mathbf{T}\_0) \right] \tag{3}$$

$$\mathbf{e}\_{\mathbf{P}} = \left[ \mathbf{h}(\mathbf{P}, \mathbf{T}\_0) - \mathbf{h}(\mathbf{P}\_0, \mathbf{T}\_0) \right] - \mathbf{T}\_0 \cdot \left[ \mathbf{s}(\mathbf{P}, \mathbf{T}\_0) - \mathbf{s}(\mathbf{P}\_0, \mathbf{T}\_0) \right] \tag{4}$$

Figure 1. Energy (a) and exergy (b) balances of a reversible refrigerator (RR).

The contribution of eT and eP to the value of eP,T can be clearly visualized on the exergyenthalpy diagram presented in Figure 2. For instance, the thermal exergy for point 1 is illustrated as the segment (eT)1 defined by the intersections of two isotherms T1 and T0 with the isobar P1. The mechanical exergy for point 1 is illustrated as the segment (eP)1 defined by the intersections of two isobars P1 and P0 with the isotherm T0. Whatever the temperature conditions are (T < T0 or T > T0), the thermal exergy is always positive [3], as clearly presented on the e-h diagram. In this sense the eT behavior is similar to that of the exergy of heat flow, that is always positive, as has been discussed. Meanwhile, eP is only positive for conditions P>P0 (see for example point 1), but it is negative for P < P0, 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]:

$$
\dot{\mathbf{E}}\_{\text{in}} = \dot{\mathbf{E}}\_{\text{out}} + \dot{\mathbf{D}}\_{\text{int}} \tag{5}
$$

Here E\_ in and E\_ out are the inlet and outlet exergy flows; D\_ int is the rate of internal exergy losses. There is abundant scientific literature [3–9] on the subject of the exergy performance criteria definition based on Eq. (5). However, there are only a few definitions of this criteria applied to the processes of expansion and compression operating below and across ambient temperature.

Among them, three exergy efficiency definitions may be distinguished: input-output efficiency,

Exergy Flows Inside Expansion and Compression Devices Operating below and across Ambient Temperature

The input-output efficiency ηin-out, first proposed by Grassmann [2], is computed according to

<sup>η</sup>in-out <sup>¼</sup> <sup>E</sup>\_

The shortcomings of this definition, particularly for its application to sub-ambient problems, are well documented [3–9]. The main one is the fact that often ηin-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 ηpr-f proposed by Tsatsaronis [10] and Bejan et al. [6] in

<sup>η</sup>pr-f <sup>¼</sup> Exergy of Products

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 thermomechanical exergy. The problem with this approach is that it is possible to obtain different values of ηpr-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";

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

> out � <sup>E</sup>\_ tr

where ΔE and \_ ∇E are the exergy produced and exergy consumed in the process. It is clear that \_ the difference between the denominator and the numerator in Eq. (8) equals the exergy losses

for the uniquely determined thermodynamic metrics ΔE and \_ ∇E in the case of thermo- \_ mechanical exergy transformation for sub-ambient processes. The evaluation of ηtr to assess the rate of exergy transfer of the mechanical exergy component to the thermal exergy compo-

<sup>¼</sup> <sup>Δ</sup>E\_

tr, and the exergy efficiency is defined as:

E\_ in � <sup>E</sup>\_ tr

<sup>η</sup>tr <sup>¼</sup> <sup>E</sup>\_

within the process. It will be illustrated that the unambiguous definition of E\_

nent for these processes will be based on these metrics.

Exergy of Fuel (7)

http://dx.doi.org/10.5772/intechopen.74041

<sup>∇</sup>E\_ (8)

tr paves the way

out E\_ in

(6)

65

products-fuel efficiency, and efficiency that accounts for the transiting exergy.

the context of expansion and compression processes is computed as:

Szargut et al. [5] used "exergy of useful products" vs. "feeding exergy".

Eq. (6):

"transiting exergy", E\_

Figure 2. Thermal eT and mechanical eP exergy components on an exergy-enthalpy diagram.

Among them, three exergy efficiency definitions may be distinguished: input-output efficiency, products-fuel efficiency, and efficiency that accounts for the transiting exergy.

The contribution of eT and eP to the value of eP,T can be clearly visualized on the exergyenthalpy diagram presented in Figure 2. For instance, the thermal exergy for point 1 is illustrated as the segment (eT)1 defined by the intersections of two isotherms T1 and T0 with the isobar P1. The mechanical exergy for point 1 is illustrated as the segment (eP)1 defined by the intersections of two isobars P1 and P0 with the isotherm T0. Whatever the temperature conditions are (T < T0 or T > T0), the thermal exergy is always positive [3], as clearly presented on the e-h diagram. In this sense the eT behavior is similar to that of the exergy of heat flow, that is always positive, as has been discussed. Meanwhile, eP is only positive for conditions P>P0 (see for example point 1), but it is negative for P < P0, as illustrated by point 2 in Figure 2.

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

irreversibilities (the case considered in this paper) may be written as [4]:

Figure 2. Thermal eT and mechanical eP exergy components on an exergy-enthalpy diagram.

Here E\_

in and E\_

64 Energy Systems and Environment

E\_ in <sup>¼</sup> <sup>E</sup>\_

The exergy balance around any process under steady state conditions and without external

There is abundant scientific literature [3–9] on the subject of the exergy performance criteria definition based on Eq. (5). However, there are only a few definitions of this criteria applied to the processes of expansion and compression operating below and across ambient temperature.

out are the inlet and outlet exergy flows; D\_ int is the rate of internal exergy losses.

out <sup>þ</sup> <sup>D</sup>\_ int (5)

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

$$
\eta\_{\text{in-out}} = \frac{\dot{E}\_{\text{out}}}{\dot{E}\_{\text{in}}} \tag{6}
$$

The shortcomings of this definition, particularly for its application to sub-ambient problems, are well documented [3–9]. The main one is the fact that often ηin-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 ηpr-f proposed by Tsatsaronis [10] and Bejan et al. [6] in the context of expansion and compression processes is computed as:

$$
\eta\_{\text{pr}\text{-}\text{}\text{}} = \frac{\text{Exergy of Product}}{\text{Exergy of Fuel}} \tag{7}
$$

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 thermomechanical exergy. The problem with this approach is that it is possible to obtain different values of ηpr-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", E\_ tr, and the exergy efficiency is defined as:

$$
\eta\_{\rm tr} = \frac{\dot{\rm E}\_{\rm out} - \dot{\rm E}\_{\rm tr}}{\dot{\rm E}\_{\rm in} - \dot{\rm E}\_{\rm tr}} = \frac{\Delta \dot{\rm E}}{\nabla \dot{\rm E}} \tag{8}
$$

where ΔE and \_ ∇E are the exergy produced and exergy consumed in the process. It is clear that \_ the difference between the denominator and the numerator in Eq. (8) equals the exergy losses within the process. It will be illustrated that the unambiguous definition of E\_ tr paves the way for the uniquely determined thermodynamic metrics ΔE and \_ ∇E in the case of thermo- \_ mechanical exergy transformation for sub-ambient processes. The evaluation of ηtr 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.
