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Functional exergy efficiency at near-environmental temperatures.


This paper is concerned with the effectiveness of temperature levels on the exergy efficiency of thermal exergy transfer at near-environmental temperatures.

The notions of heat exchanger heat transfer effectiveness, functional exergy efficiency and exergy consumption are combined with the concept of warm and cool exergy, in order to obtain new insights that may be useful when specifying the operating temperatures of air-to-air sensible heat exchangers used at near-environmental temperatures.

It is possible to define exergy efficiencies in various ways, depending on the significance of various conditions such as sensitivity for changes in a system, applicability in practice, accuracy and accessibility (Sami, 2008; Semenyuk, 1990; Sorin and Brodyansky, 1992; Tsatsaronis, 1993 and 2002; Kotas, 2001).

Kanoglu, Dincer and Cengel (2008) discuss exergy efficiency in heat exchange involving phase change, as well as for other processes. Hepbasli (2008) presents a review on exergy analysis of renewable energy resources. Kanoglu, Dincer and Rosen (2007) present expressions for and examples of exergy analysis for power plants.

In the literature, links have been made between sustainability, exergy consumption and heat transfer at near-environmental temperatures, including warm and cool exergy (Shukuya, 1996; Shukuya and Hammache, 2002) and tepidology (Wall, 1990). Links have also been made between exergy resource efficiency and sustainability (Granovskii, Dincer and Rosen, 2008; Swaan Arons et. al., 2004; Connely and Koshland, 2001).

Semenyuk (1990) discusses heat exchanger exergetic efficiency as a function of a dimensionless temperature, and indicates domains of technically inexpedient heat exchanger operation. Similarly to the present study, his analysis shows that using hot thermal carriers (above environmental temperature) to heat cold thermal carriers (below environmental temperature) is irrational since this heating could be accomplished by using environmental air. However, his approach is computationally more complex, as it requires knowledge of heat exchanger inlet and outlet temperatures. Moreover, his study focuses on operating temperatures relatively far from the environmental temperature.

Wu et al. (2006) use the concept of heat transfer effectiveness (ASHRAE, 2000; Holman, 2002) to indicate the relative magnitude of the heat transfer, and perform detailed comparisons of exergy transfer effectiveness with heat transfer effectiveness, for parallel flow, counter-flow and cross-flow heat exchangers operating above and below the surrounding temperature. They analyze variations of exergy transfer effectiveness with number of transfer units (NTU), with the ratio of the heat capacity of cold fluid to that of hot fluid ([C.sub.c]/[C.sub.h]) and with finite pressure drops. They note that there is not an optimal combination of NTU and [C.sub.c]/[C.sub.h] for maximizing exergy transfer effectiveness. They do not elaborate on the effects of temperature variations.

Peng (2007) address the dependence of heat transfer effectiveness on the number of heat transfer units, the heat capacity ratio and the flow pattern. They present an expression for calculating the optimal value of the heat capacity ratio for achieving maximum exergy efficiency, and conclude that this optimum is different for heat exchangers operating above or below environmental temperature. They present results of exergy efficiency with different flow patterns for counterflow, cross flow and parallel flow heat exchangers, all operating below environmental temperature. Their study does not directly address the effect of temperature variations.

Hirs (2003) defines the thermodynamic performance of a heat exchanger as being determined by the input exergy of the stream that is cooled and the output exergy of the stream that is heated, and the difference between these two as being the exergy loss due to loss in quality of the transferred heat. He defines the thermodynamic efficiency as the ratio between output and input exergy, and omits exergy losses due to the flow resistance to enable a direct comparison between the thermodynamic efficiency and the heat exchanger heat transfer effectiveness. Hirs defines two ratios [R.sub.c] and [R.sub.h] of temperatures at the heat exchanger inlets and the environmental temperature ([T.sub.c,in]/[T.sub.e] and [T.sub.h,in]/[T.sub.e]). For a countercurrent heat exchanger with equal thermal capacities of hot and cold fluid, he plotted heat exchanger effectiveness against thermodynamic efficiency and found that at higher values of the temperature ratio, [R.sub.c] > 1, the thermodynamic efficiency is clearly above the effectiveness. At lower values of the temperature ratio, [R.sub.c] < 1, the thermodynamic efficiency is lower than the effectiveness, because the exergy below [T.sub.e] is completely lost. He notes that hot streams should in practice be cooled as much as possible by streams from the environment and not by means of a chiller, and states that the thermal effectiveness is no suitable yardstick for heat exchanger design and operation.

These conclusions from Hirs (2003) are in line with the results from the present study. This paper elaborates further by also considering the case of unbalanced heat capacity ratios, by presenting expressions for calculating exergy efficiency as a function of heat transfer effectiveness and temperatures, and by proposing a single number for designating temperature combinations.

The work presented in this paper is based on exergy efficiency definitions from Woudstra (2002), who distinguishes two different types of exergy efficiency definitions: universal exergy efficiency and functional exergy efficiency. Kotas (2001), Tsatsaronis (1993), Lazzaretto and Tsatsaronis (2006) call them 'simple' and 'rational' exergy efficiencies, while Hepbasli (2008) refers to them as 'brute force' and 'functional' exergy efficiencies. The universal exergy efficiency considers gross inputs and gross outputs, and has relatively little sensitivity to changes in a system when the exergy loss within the system is small compared to the exergy of the incoming and outgoing flows. The universal exergy efficiency also makes no distinction regarding usability (exergy transferred inside the heat exchanger or discarded with the outgoing flows), or intended use in heating or cooling (Boelman and Sakulpipatsin, 2005). Functional exergy efficiency is more sensitive to changes in exergy loss within the system, and yields 'net' efficiency values since it excludes the exergy discarded with outgoing flows. In this study, the functional exergy efficiency is used as a measure for the results of the exergy analysis of a heat exchanger.


This item starts by defining exergy for the purposes of this work, in relation to a pre-defined reference environment, and by presenting the concept of warm and cool exergy. Then, it introduces the simplified model of sensible heat exchangers used in this work, thermal exergy profiles assumed for a counter-flow heat exchanger, the main relevant parameters as well as working definitions of heat transfer effectiveness and functional exergy efficiency.

Exergy and Environment

Exergy is always evaluated with respect to the intensive properties of the reference environment, and becomes zero when the system is in equilibrium with the reference environment. Many researchers have examined characteristics of the reference environment (Gaggioli and Petit, 1977; Wepfer et. al., 1979; Sussman, 1981; Ahrendts, 1980; Rodriguez, 1980; Rosen, 1986; Sakulpipatsin et. al., 2007b), which acts as an infinite system and is a sink and source for thermal energy and substances.

The exergy value of air in a building can be expressed as a function of temperature, pressure, and chemical composition. In this paper air pressure inside and outside the building is assumed constant and equal to 101.325 kPa, and dry air of standard chemical composition is considered.

For the purposes of this work, the thermal exergy Exth of an ideal gas stream is calculated as a function of mass flow rate m, isobaric heat capacity [c.sub.p], temperature T, and standard environmental air temperature [T.sub.e] (Moran and Shapiro, 1998; Sakulpipatsin et al, 2007b)

Exth = m x [[bar.c].sub.p]((T - [T.sub.e]) - [T.sub.e]ln(T/[T.sub.e])) (1)

where [c.sub.p] is an appropriate mean heat capacity over the temperature interval [T.sub.e] - T.

Warm and Cool Exergy

Shukuya (1996) named "warm exergy" (or "cool exergy") the exergy contained by air at a temperature higher (or lower) than its environment. The exergy value of indoor air (equation 1) is positive when its temperature [T.sub.i] is either higher or lower than the environmental temperature [T.sub.e], and zero if [T.sub.i] is equal to [T.sub.e]. For indoor air, the direction of the exergy flow is always from indoors to outdoors, although the direction of the energy flow changes depending on whether the indoor air temperature [T.sub.i] is higher or lower than the environmental air temperature [T.sub.e] (Shukuya 1996; Shukuya and Hammache, 2002).

Simplified Sensible Heat Exchangers

This study focuses on the effect of operating temperatures, and uses a simplified black box model of a counter-flow heat exchanger where thermal energy is simply assumed to be transferred from a hot to a cold air stream. Relevant parameters are heat transfer fluid (air) temperature levels, [T.sub.c,in], [T.sub.c,out], [T.sub.h,in] and [T.sub.h,out], environmental air temperature [T.sub.e], total heat capacities of both air streams [C.sub.c], [C.sub.h] and exchanger heat transfer effectiveness [epsilon]. The concept of exchanger heat transfer effectiveness [epsilon] (ASHRAE, 2000; Holman, 2002; Wu et al., 2006) is used as a lumped parameter to characterize heat exchanger thermal performance. Airflow and other effects are neglected, as outlined in Table 1. More detailed studies of heat transfer and fluid flow phenomena in heat exchangers are beyond the scope of this work and can be found in the literature, e.g. Bart (2002); Heun and Crawford (1994); Vaidya et. al. (1992). Sakulpipatsin et. al. (2007a) considered pressure drops and fan power requirements in an exergy analysis of residential heat recovery systems.

Table 1. Simplifying Assumptions for the Heat Exchange Model

* counter-flow heat exchange

* only sensible heat exchange considered

* heat exchange between 2 dry airflows

* temperatures and heat transfer coefficients constant and uniform

* constant temperature difference [T.sub.h] - [T.sub.c] throughout the heat exchanger

* mass flows for cold air: 1 kg*[s.sup.-1] ([C.sub.c] = [C.sub.h] and [C.sub.c] < [C.sub.h]) and 10 kg*[s.sup.-1] ([C.sub.c] > [C.sub.h])

* mass flow for hot air: 1 kg*[s.sup.-1] ([C.sub.c] = [C.sub.h] and [C.sub.c] > [C.sub.h]) and 10 kg*[s.sup.-1] ([C.sub.c] < [C.sub.h])

* isobaric heat capacity of dry air constant [c.sub.p, air]: 1.005 kJ*[kg.sup.-1]*[K.sup.-1]

* exchanger heat transfer effectiveness [epsilon] is a parameter, at 70% and 90%

* well insulated

* air flow effects neglected

* only thermal exergy considered

Figure 1 schematically illustrates airflow and air temperature profiles as considered in this study, assuming the total heat capacities of both airflows to be the same ([C.sub.h] = [C.sub.c]) for simplicity. The case of [C.sub.h] [not equal to] [C.sub.c] is discussed further on in the paper. Tables 1 and 2 list the main relevant parameters and boundary conditions.


The following parameters are allowed to vary: environmental air temperature [T.sub.e]; hot and cold air temperature at the heat exchanger inlets [T.sub.c,in] and [T.sub.h,in]; and exchanger heat transfer effectiveness [epsilon]. Air temperatures at the heat exchanger outlets [T.sub.h,out] and [T.sub.c,out] are calculated by assuming [Q.sub.h,out] = [Q.sub.c,in] (well insulated heat exchanger). The total heat capacities of the air streams are taken as [C.sub.h] = [C.sub.c], [C.sub.c] = [C.sub.h]/10 and [C.sub.c] = [C.sub.h] x 10.

The air is assumed to be dry and to have a constant chemical composition, and pressure drops in the heat exchange process are neglected. Therefore the exergy values of the cold and hot air at the inlet and outlet points are equal to their thermal contributions.

Thermal Exergy Loss

Figure 2 schematically shows an example of thermal exergy profiles in a well-insulated counter-flow heat exchanger. Operating temperatures of hot air [T.sub.h] and cold air [T.sub.c] are all above environmental air temperature [T.sub.e]. Exergy values can be calculated from equation 1.


As a result of thermal energy and exergy transfer from hot to cold air, the thermal exergy of the cold air increases by [DELTA][Ex.sub.c] while that of the hot air decreases by [DELTA][Ex.sub.h]. It can be seen from the figure that [DELTA][Ex.sub.c] < [DELTA][Ex.sub.h], the difference being [DELTA][Ex.sub.loss] or the exergy loss resulting from heat transfer.

These exergy values depend not only on the operating temperatures [T.sub.h] and [T.sub.c], but also on how near or far these temperatures are from the environmental temperature [T.sub.e].

Dimensionless Temperature

In order to relate environmental air temperature [T.sub.e] to heat exchanger operating temperatures [T.sub.c,in] and [T.sub.h,in], a dimensionless temperature T' is defined (Boelman and Sakulpipatsin, 2004, 2005, 2008, Sakulpipatsin et. al., 2007c).

T' = [[[T.sub.h,in] - [T.sub.e]]/[[T.sub.h,in] - [T.sub.c,in]]] (2)

In the numerator of equation (2), [T.sub.h,in] - [T.sub.e], provides a measure of the potential of the hot air to heat environmental air. This potential can be expressed as thermal exergy [Ex.sub.h,in] (calculated from equation 1) or simply as the temperature difference [Th.sub.h,in] - [T.sub.e] as in equation 2. In the denominator, [T.sub.h,in] - [T.sub.c,in] gives a measure of the potential of the hot air to heat the cold air going into the heat exchanger. This can be expressed by [T.sub.h,in] - [T.sub.c,in] or by [Ex.sub.h,in] - [Ex.sub.c,in]. The dimensionless temperature compares the potential of the hot air relative to the environment ([T.sub.h,in] - [T.sub.e]) and to the cold air ([T.sub.h,in] - [T.sub.c,in]). When the ratio of these two temperature differences (given by T') is relatively close to 1, the heat exchange process is considered to take place at near-environmental conditions. When the magnitude of T' increases (e.g. T' > 4 or T' < -4), then we can consider [T.sub.h,in] - [T.sub.e] [much greater than] [T.sub.h,in] - [T.sub.c,in] and the heat exchange process can be regarded as relatively far from environmental conditions.

Figure 3 illustrates how the dimensionless temperature T' can express different temperature combinations of [T.sub.c,in] and [T.sub.h,in], for a given environmental air temperature [T.sub.e] = 10[degrees]C.


For T' = 1, [T.sub.c,in] is equal to [T.sub.e]. In practice, this could correspond e.g. to a heat exchanger taking up environmental air at the cold heat exchanger inlet in order to (pre) heat it for use in balanced ventilation systems. For T' > 1, [T.sub.c,in] is above [T.sub.e]. This could be the case for ventilation air being pre-heated (e.g. by a sun room or in buried air ducts) above [T.sub.e] before reaching the heat exchanger inlet.

For 0 < T' < 1, [T.sub.e] is between [T.sub.c,in] and [T.sub.h,in]. This could be the case of heat exchange between air from a freezer and from an office, but is unlikely to occur in space heating applications. For T' < 0, [T.sub.e] is above [T.sub.c,in] and [T.sub.h,in]. This could be the case upon heat exchange between two cold air streams and is also unlikely to be the case in space heating applications.

Heat Exchange at Near-Environmental Temperatures

For the purposes of this paper, environmental temperatures are based on typical ranges applicable to HVAC systems in heating applications (Table 2).

Table 2. Temperature Domains

* 253.15 K [less than or equal to] [T.sub.e] [less than or equal to] 293.15 K

10K [less than or equal to] [T.sub.h,in] * [T.sub.c,in] [less than or equal to] 130K

* 0 [less than or equal to] T' [less than or equal to] 5

* 293.15 K [less than or equal to] [T.sub.h,in] [less than or equal to] 373.15 K

* 283.15 K [less than or equal to] [T.sub.c,in] [less than or equal to] 363.15 K

Heat exchanger operating temperatures are taken over a somewhat broader range than in usual HVAC applications, in order to enable the temperature combinations required to obtain dimensionless temperatures T' covering the relatively broad range of 0 [less than or equal to] T' [less than or equal to] 5.

A number of heat exchanger inlet air temperature combinations are defined, within the ranges given in Table 2. Air temperatures at the heat exchanger outlets are then determined, as simple functions of inlet air temperatures, exchanger heat transfer effectiveness and heat capacity ratios.

The equivalent temperatures (Cengel and Robert, 2001) of the cold air [T.sub.eq,c] and the hot air [T.sub.eq,h] are calculated from the air temperatures at the heat exchanger inlets [] and outlets [T.sub.out].

[T.sub.eq] = [[[T.sub.out] - []]/[ln([T.sub.out]/[])]] (3)

Heat Transfer Effectiveness

The heat transfer effectiveness [epsilon] is often used by HVAC designers as a lumped parameter to characterize the thermal performance of a heat exchanger. It can be simply defined in terms of the inlet and outlet air temperatures and the total heat capacities of the streams (ASHRAE, 2000), as in equation 4.

[epsilon] = [Q/[Q.sub.max]] = [[[C.sub.c]([T.sub.c,out] - [T.sub.c,in])]/[[C.sub.min]([T.sub.h,in] - [T.sub.c,in])]] = [[[C.sub.h]([T.sub.h,in] - [T.sub.h,out])]/[[C.sub.min]([T.sub.h,in] - [T.sub.c,in])]] (4)

[epsilon] = [Q/[Q.sub.max]] = [[[C.sub.c]([T.sub.c,out] - [T.sub.c,in])]/[[C.sub.min]([T.sub.h,in] - [T.sub.c,in])]] = [[[C.sub.h]([T.sub.h,in] - [T.sub.h,out])]/[[C.sub.min]([T.sub.h,in] - [T.sub.c,in])]] (4)

In the simplified model and operating conditions for this paper, mass flow rates, temperatures and heat transfer coefficients are assumed uniform and constant through the heat exchange process. The heat exchanger to be studied is used for heating the cold air from [T.sub.c,in] to [T.sub.c,out].

Air temperatures at the outlets [T.sub.c,out] and [T.sub.h,out] can be obtained from equation (3) for a well insulated heat exchanger ([Q.sub.h,out] = [Q.sub.c,in]). For unbalanced heat capacity ratios, the outlet temperatures are given in equations (5) to (8).

For [C.sub.c] < [C.sub.h] we obtain

[T.sub.c,out] = [T.sub.c,in] + [epsilon]([C.sub.min]/[C.sub.c])([T.sub.h,in] - [T.sub.c,in]) (5)

[T.sub.h,out] = [T.sub.h,in] - ([C.sub.c]/[C.sub.h])([T.sub.c,out] - [T.sub.c,in]) (6)

For [C.sub.c] > [C.sub.h] we have

[T.sub.h,out] = [T.sub.h,in] - [epsilon]([C.sub.min]/[C.sub.h])([T.sub.h,in] - [T.sub.c,in]) (7)

[T.sub.c,out] = [T.sub.c,in] + ([C.sub.h]/[C.sub.c])([T.sub.h,in] - [T.sub.h,out]) (8)

In the specific case of balanced heat capacity ratios ([C.sub.c] = [C.sub.h]), equations 5 and 7 become

[T.sub.c,out] = [T.sub.c,in] + [epsilon]([T.sub.h,in] - [T.sub.c,in]) (9)

[T.sub.h,out] = [T.sub.h,in] - [epsilon]([T.sub.h,in] - [T.sub.c,in]) (10)

Functional Exergy Efficiency

The functional exergy efficiency [[eta].sub.f] of the heat exchanger can be defined as a ratio of all product outputs [SIGMA][Ex.sub.product] to all source inputs [SIGMA][Ex.sub.source], as shown in equation 11 (Woudstra, 2002).

[[eta].sub.f] = [[[SIGMA][Ex.sub.product]]/[[SIGMA][Ex.sub.source]]] (11)

When the goal is to increase the thermal exergy of the cold air by exergy transfer from the hot air, the cold air thermal exergy increase [DELTA][Ex.sub.c] is taken as the net product output, and the absolute value of the hot air thermal exergy decrease |[DELTA][Ex.sub.h]| is considered as the net source input.

[[eta].sub.f] = [[[DELTA][Ex.sub.c]]/|[DELTA][Ex.sub.h]]]| = [[[Ex.sub.c,out] - [Ex.sub.c,in]]/[[Ex.sub.h,in] - [Ex.sub.h,out]]] (12)

By assuming that the heat exchanger is well insulated and that airflow effects are neglected (Table 1), thermal energy is assumed to be completely transferred from the hot air to the cold air ([Q.sub.h,out] = [Q.sub.c,in]). Equation 12 can be rewritten as a function of the temperatures ([T.sub.c,in], [T.sub.h,in], [T.sub.c,out], and [T.sub.e]) and the total heat capacities of the air streams ([C.sub.h] and [C.sub.c]), as shown in equation 13.

[[eta].sub.f] = [[[C.sub.c]([T.sub.c,out] - [T.sub.c,in]) - [C.sub.c][T.sub.e]ln([T.sub.c,out]/[T.sub.c,in])]/[[C.sub.c]([T.sub.c,out] - [T.sub.c,in]) + [C.sub.h][T.sub.e]ln(1 - [[C.sub.c]([T.sub.c,out] - [T.sub.c,in])]/[[C.sub.h][T.sub.h,in]])]] (13)

A similar expression, as a function of heat capacity ratio ([C.sub.h]/[C.sub.c]), can be written as:

[[eta].sub.f] = [[([T.sub.c,out] - [T.sub.c,in]) - [T.sub.e]ln([T.sub.c,out]/[T.sub.c,in])]/[([T.sub.c,out] - [T.sub.c,in]) + [[C.sub.h]/[C.sub.c]][T.sub.e]ln(1 - [[T.sub.c,out] - [T.sub.c,in]]/[[[C.sub.h]/[C.sub.c]][T.sub.h,in]])]] (14)

The functional exergy efficiency [[eta].sub.f]can also be calculated as a function of dimensionless temperature T' and exchanger heat transfer effectiveness [epsilon] (ASHRAE, 2000; Holman, 2002; Wu et al., 2006), without the need to first determine outlet temperatures.

For [C.sub.c] < [C.sub.h] we can state that

[[eta].sub.f] = [[[epsilon]([[T.sub.h,in] - [T.sub.e]]/[T']) - [T.sub.e]ln(1 + [epsilon][[T.sub.h,in] - [T.sub.e]]/[T'[T.sub.c,in]])]/[[epsilon]([[T.sub.h,in] - [T.sub.e]]/[T']) + [[C.sub.h]/[C.sub.c]][T.sub.e]ln(1 - [epsilon][[T.sub.h,in] - [T.sub.e]]/[T'[T.sub.h,in][[C.sub.h]/[C.sub.c]]])]] (15)

For [C.sub.c] > [C.sub.h] we have

[[eta].sub.f] = [[[epsilon]([[T.sub.h,in] - [T.sub.e]]/[T']) - [[C.sub.c]/[C.sub.h]][T.sub.e]ln(1 + [epsilon][[T.sub.h,in] - [T.sub.e]]/[T'[T.sub.c,in][[C.sub.c]/[C.sub.h]]])]/[[epsilon]([[T.sub.h,in] - [T.sub.e]]/[T']) + [T.sub.e]ln(1 - [epsilon][[T.sub.h,in] - [T.sub.e]]/[T'[T.sub.h,in]])]] (16)

For balanced heat capacity ratios ([C.sub.h] = [C.sub.c]), equations 15 and 16 simplify to

[[eta].sub.f] = [[[epsilon]([[T.sub.h,in] - [T.sub.e]]/[T']) - [T.sub.e]ln(1 + [epsilon][[T.sub.h,in] - [T.sub.e]]/[T'[T.sub.c,in]])]/[[epsilon]([[T.sub.h,in] - [T.sub.e]]/[T']) + [T.sub.e]ln(1 - [epsilon][[T.sub.h,in] - [T.sub.e]]/[T'[T.sub.h,in]])]] (17)


Previous papers (Boelman, 2008; Boelman and Sakulpipatsin, 2005) discussed the temperature sensitivity of functional exergy efficiency regarding heat exchange at near-environmental temperatures, for balanced heat capacity ratios. Hirs (2003) related thermodynamic efficiency to thermal effectiveness for heat exchangers, for balanced heat capacity ratios, and noted the importance of selecting appropriate temperature combinations to avoid exergy losses below environmental temperature.

The present paper discusses how temperature combinations (expressed as a dimensionless temperature T') can affect the functional exergy efficiency [[eta].sub.f] of a simplified heat exchanger used for heating applications, and presents guidelines for selecting temperature combinations likely to be more effective from the viewpoint of thermal exergy.

Dimensionless Temperature T' and Functional Exergy Efficiency [[eta].sub.f] for Heating Applications

Figure 4 schematically illustrates the relationship between inlet air temperatures [T.sub.h,in], [T.sub.c,in] and environmental air temperatures [T.sub.e]. This relationship is expressed in terms of the dimensionless temperature T' defined in equation (2), for temperature combinations corresponding to 0 < T' < 1. The exergy efficiency values in the figure are based on the assumptions that: environmental air temperature [T.sub.e] is constant at 10[degrees]C; air temperature difference at the heat exchanger inlets ([T.sub.h,in] * [T.sub.c,in]) is constant at 15[degrees]C; [T.sub.h,in] is between -50[degrees]C and 100[degrees]C; [T.sub.c,in] is between -65[degrees]C and 85[degrees]C. Table 3 presents this relationship in more detail and for a broader temperature range, based on the same set of assumptions. For the sake of simplicity, balanced heat capacity rates are assumed in this item.



The range of dimensionless temperature T' [less than or equal to] 0.3.5 corresponds to heating below environmental temperature [T.sub.e]. When T' = 0.35, the equivalent temperature of the hot air and the outlet temperature of the cold air are about the same as the environmental temperature ([T.sub.eq,h] = [T.sub.e] and [T.sub.c,out] [approximately equal to] [T.sub.e]). The exergy change undergone by the hot air is zero ([DELTA][Ex.sub.h] = 0). This is because the warm exergy lost by the hot air above [T.sub.e] (as it cools down from [T.sub.h,in] to [T.sub.e]) has the same magnitude as the cool exergy gained below [T.sub.e] (as the cold air further cools down from [T.sub.e] to [T.sub.h,out]). For the cold air, the exergy change is negative ([DELTA][Ex.sub.c] < 0), because it loses cool exergy as it is warmed up from [T.sub.c,in] to [T.sub.e]. From [DELTA][Ex.sub.h] = 0 and [DELTA][Ex.sub.c] < 0, we obtain [[eta].sub.f][right arrow] -[infinity] from the definition in equation 12. Here the functional exergy efficiency clearly indicates that it is highly ineffective to heat air below ambient temperature when the desired product is air at ambient temperature.

When T' = 0, the inlet temperature of the hot air is the same as the environmental temperature ([T.sub.h,in] = [T.sub.e]). Air on the hot side gains cool exergy as it cools down from [T.sub.h,in] = [T.sub.e] to [T.sub.h,out] < [T.sub.e], and air on the cold side loses cool exergy as it warms up from [T.sub.c,in] [much less than] [T.sub.e] to [T.sub.c,in] < [T.sub.e]. This loss of cool exergy on the cold side results in negative [[eta].sub.f] values when heating occurs below [T.sub.e]. These [[eta].sub.f] values are even smaller than minus one, because the exergy change on the cold side is bigger than the exergy change on the hot side of the heat exchanger: |[DELTA][Ex.sub.c]| > |[DELTA][Ex.sub.h]|. From an exergy viewpoint this heat exchange could have been effective if the desired product had been cooled air, although another definition would have been needed for [[eta].sub.f]. Heating below [T.sub.e] is considered out of scope and will not be discussed further in this paper.

Dimensionless temperatures T' between ca. 0.35 and 0.65 correspond to heating across the environmental temperature [T.sub.e]. For example when T' = 0.5 the hot air enters the heat exchanger at [T.sub.h,in] > [T.sub.e] and exits at [T.sub.h,in] < [T.sub.e]. Conversely, the cold air enters at [T.sub.c,in] < [T.sub.e] and exits at [T.sub.c,in] > [T.sub.e]. On the cold air side, the loss in cool exergy is bigger than the gain in warm exergy, which results in [DELTA][Ex.sub.c] < 0 and thus in [[eta].sub.f] < 0. These negative functional exergy efficiencies clearly show that it is ineffective to exchange heat across the environmental temperature.

The range of T' [greater than or equal to] 0.7 corresponds to heating above [T.sub.e]. For T' = 0.7 we have [T.sub.h,out] = [T.sub.e] and [T.sub.eq,c] [approximately equal to] [T.sub.e]. Although the cold air loses cool exergy as it warms up from [T.sub.c,in] to [T.sub.e], this loss is smaller than the warm exergy gains it undergoes as it warms up from [T.sub.e] to [T.sub.c,out]. However, the functional exergy efficiency is still low, at [[eta].sub.f] = 0.14. For T' =1 we have [T.sub.c,in] = [T.sub.e], so we can say that heat transfer takes place above [T.sub.e]. Since there is no more heating and cooling across environmental temperature, the functional exergy efficiency increases to [[eta].sub.f] = 0.55. As T' increases and heat exchange takes place relatively further from environmental temperature, the functional exergy efficiency also increases. The smaller the magnitude of [T.sub.h,in] - [T.sub.c,in] relative to [T.sub.h,in] - [T.sub.e] (and hence the bigger T'), the smaller the influence of [T.sub.e] on the relative magnitudes of [DELTA][Ex.sub.c] and [DELTA][Ex.sub.h]. This in turn results in [DELTA][Ex.sub.c] [approximately equal to] [DELTA][Ex.sub.h] and hence in [[eta].sub.f][right arrow] 1 for this simplified counterflow heat exchanger model with matched heat capacity rates ([C.sub.c] = [C.sub.h]).

Dimensionless Temperature and Effective Exergy Use

This sub-item starts with a brief discussion of the sensitivity of functional exergy efficiency [[eta].sub.f] of the heat exchanger model to dimensionless temperature T' and exchanger heat transfer effectiveness [epsilon] for balanced and unbalanced heat capacity ratios. Then [[eta].sub.f] and [epsilon] are compared for various T', for balanced and unbalanced heat capacity ratios, and the effectiveness of different temperature combinations is discussed from an exergy viewpoint.

Sensitivity of Functional Exergy Efficiency to Dimensionless Temperature and Heat Capacity Ratio

Previous papers (Boelman, 2008; Boelman and Sakulpipatsin, 2005) have discussed the sensitivity of functional exergy efficiency to temperature and heat exchanger heat transfer effectiveness, for operation at near-environmental temperatures with balanced heat capacity ratios ([C.sub.h] = [C.sub.c]).

The case of [C.sub.h] = [C.sub.c] is compared below with [C.sub.h] [not equal to] [C.sub.c], for [C.sub.c] = [C.sub.h]/10 and [C.sub.c] = [C.sub.h] x 10. These heat capacity ratios are arbitrarily defined, for the sake of illustration. Equation 14 is used to calculate [[eta].sub.f] and equations 5 to 10 are used to obtain [T.sub.h,out] and [T.sub.c,out], assuming a well insulated heat exchanger.

Figure 5 shows a plot of functional exergy efficiency [[eta].sub.f] as a function of dimensionless temperature T' and heat capacity ratio ([C.sub.c] = [C.sub.h]/10 and [C.sub.c] = [C.sub.h] x 10), for [epsilon] =90%. [[eta].sub.f] values are highest for [C.sub.h] = [C.sub.c], due to the similar temperature differences on the hot and cold sides ([T.sub.c,out] - [T.sub.c,in] = [T.sub.h,in] - [T.sub.h,out]). [[eta].sub.f] values decrease when [C.sub.h] [not equal to] [C.sub.c], the decrease being bigger for [C.sub.c] > [C.sub.h] than for [C.sub.c] < [C.sub.h], as could be expected from the bigger temperature increase of the cold stream (product) compared to the temperature decrease of the hot stream (source). As discussed in a previous paper (Boelman, 2008), the temperature sensitivity of [[eta].sub.f] shown in Figure 5 generally applies to temperature combinations typically found in HVAC applications.


Functional Exergy Efficiency and Heat Transfer Effectiveness for Different Dimensionless Temperatures and Heat Capacity Ratios

Three plots of functional exergy efficiency [[eta].sub.f] versus heat exchanger effectiveness [epsilon] are presented below, for balanced and unbalanced heat capacity ratios.

Figure 6 shows a plot of [[eta].sub.f] versus [epsilon] for balanced heat capacity ratios and heat exchange above environmental temperature [T.sub.e]. Dimensionless temperatures T' [less than or equal to] 1 correspond to [T.sub.c,in] [less than or equal to] [T.sub.e], and the resulting [[eta].sub.f] < [epsilon] indicates that from an exergy viewpoint it is not effective to heat air below [T.sub.e] when the desired product is air above [T.sub.e]. Instead, it could be more effective to heat air at [T.sub.e]. For T' [greater than or equal to] 1.5, the functional exergy efficiency [[eta].sub.f] is higher than the heat exchanger effectiveness [epsilon]. In these temperature ranges, the hot and cold sides of the heat exchanger operate relatively far above environmental temperature, and the functional exergy efficiency is relatively high for a well insulated heat exchanger. The ideal situation for balanced heat capacity ratios would correspond to [epsilon] = 1, leading to [DELTA][Ex.sub.c] = [DELTA][Ex.sub.h] and hence to [[eta].sub.f] = 1.


The plot in figure 7 shows similar curves of [[eta].sub.f] versus [epsilon] for unbalanced heat capacity ratios, [C.sub.c] = [C.sub.h]/10. The heat exchanger effectiveness is taken as a lumped parameter. The functional exergy efficiency clearly shows less sensitivity to [epsilon], particularly for the two upper and two lower values of T' considered. Also, the lines do not converge towards [[eta].sub.f] = 1 as they do for balanced heat capacity rates.


Figure 8 shows another similar plot [[eta].sub.f] of versus [epsilon], for [C.sub.c] = [C.sub.h] x 10. Functional exergy efficiencies are clearly lower, particularly for smaller values of T'. For T' < 1, functional exergy efficiencies are negative and do not appear in the plot. These low exergy efficiencies reflect the ineffectiveness of heating a cold stream when [C.sub.c] > [C.sub.h], in particular when heat exchange across [T.sub.e] is involved.


For the somewhat higher ranges of dimensionless temperature ( T' > 3), the functional exergy efficiency shows relatively little sensitivity to heat exchanger effectiveness [epsilon]. This is because heat transfer takes place relatively far from environmental temperature, whereby the relative proximity of either [T.sub.h,eq] or [T.sub.c,eq] to [T.sub.e] has little effect on the exergy changes undergone by the hot and cold air streams ([DELTA][Ex.sub.h] and [DELTA][Ex.sub.c]).


This paper discusses effective and ineffective temperature combinations for thermal exergy transfer at near-environmental temperatures, from the perspective of functional exergy efficiency, based on a simple air-to-air sensible heat exchanger model for heating purposes. The analysis combines functional exergy efficiency with exchanger heat transfer effectiveness, exergy consumption and warm/cool exergy, and proposes a dimensionless temperature T' to identify and analyze patterns in the sensitivity of the functional exergy efficiency to temperature combinations.

Depending on how the heat exchanger temperatures relate to the environmental temperature, different values of functional exergy efficiency can be obtained for the same exchanger heat transfer effectiveness. This is because the exchanger heat transfer effectiveness provides information on the relationship between the operating temperatures, but does not indicate whether these temperatures are near or far from the environment level. The functional exergy efficiency, on the other hand, does include information on the environmental temperature.

The notions of heat transfer effectiveness, functional exergy efficiency and exergy consumption can be combined with the concept of warm and cool exergy, in order to select effective temperature combinations in the preliminary design of systems involving heat exchange very near to environmental temperatures. For the examples considered in this paper, exergy efficient operation requires temperature combinations whereby the equivalent temperatures of hot and cold air are above environmental temperature [T.sub.e]. Although the hot air loses warm exergy, the heat exchange is exergy efficient because there is sufficient gain of warm exergy by the cold air. Heat exchanger operation is not recommended in the range of equivalent temperature of cold air below [T.sub.e], because heating this cold air implies losing its cool exergy. This cool exergy could be better used for cooling another flow of matter, at or below [T.sub.e]. Also, the warm exergy of the hot air could be better used to heat another flow of matter at or above [T.sub.e]. The functional exergy efficiency shows that it is exergy inefficient to use air above environmental temperature to heat air below environmental temperature, even with high exchanger heat transfer effectiveness.

This insight can be useful when designing a heat exchange system, for example when deciding between higher exchanger heat transfer effectiveness or pre-heating of outside air (e.g. by using a sunspace or the underground). In practice, such a decision would also have to consider the additional pressure drop usually associated with higher exchanger heat transfer effectiveness versus the possibility of using passive means to pre-heat environmental air. The analysis proposed in this paper can also be extended to other applications (e.g. in the food processing industry) involving heat exchange very near to environmental temperature.


Financial support for this research was provided by the Faculty of Architecture at the Delft University of Technology, and is gratefully acknowledged.


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A = area, [m.sup.2]

[c.sub.p]= isobaric heat capacity, [m.sup.2]*[s.sup.-2]*[K.sup.-1] (J*[kg.sup.-1]*[K.sup.-1])

C = heat capacity, [m.sup.2]*kg*[s.sup.-2]*[K.sup.-1] (J*[K.sup.-1])

Ex = exergy, [m.sup.2]*kg*[s.sup.-2] (J)

Ex = exergy per second, [m.sup.2]*kg*[s.sup.-3] (J*[s.sup.-1])

[DELTA]Ex = exergy difference, [m.sup.2]*kg*[s.sup.-2] (J)

m = mass flow rate, kg*[s.sup.-1]

Q = thermal energy, [m.sup.2]*kg*[s.sup.-2] (J)

Q = thermal energy per second, [m.sup.2]kg*[s.sup.-3] (J*[s.sup.-1])

T = air temperature, K ([degrees]C with notation)

T' = temperature in heating mode, dimensionless

T" = temperature in cooling mode, dimensionless

Greek Letters

[epsilon] = exchanger heat transfer effectiveness

[eta] = exergy efficiency


0 = reference environment

air = air

c = cold air

ch = chemical

e = reference environment state; dead state

eq = equivalent

f = functional

h = hot air

i = indoor; room

in = inlet

loss = loss

me = mechanical

min = minimum

out = outlet

product = product

source = source

th = thermal

E.C. Boelman, PhD

P. Sakulpipatsin, PhD

H.J. van der Kooi, PhD

L.C.M. Itard, PhD


E.C. Boelman and P. Sakulpipatsin are members of the faculty of architecture in the building technology section, H.J. van der Kooi is an assistant professor in the field of applied thermodynamics and chemical engineering, L.C.M. Itard is a researcher in the field of sustainable buildings and HVAC equipment and leads the group sustainable and healthy building of the Research Institute OTB, P.G. Luscuere is a professor of building services in the Climate Design Group in the faculty of architecture at Delft University of Technology, Delft, the Netherlands.
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Author:Boelman, E.C.; Sakulpipatsin, P.; van der Kooi, H.J.; Itard, L.C.M.; Luscuere, P.G.
Publication:ASHRAE Transactions
Article Type:Report
Geographic Code:4EUNE
Date:Jul 1, 2009
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