Exergy analysis of heat transfer in buildings

Exergy analysis of heat transfer in buildings

4.1

4

Summary

Traditionally, the analysis of heat transfer through the envelope of buildings has been carried out by applying the First Law of Thermodynamics. However, this type of analysis has its limitations, and it has been shown that, by itself, it does not provide a total understanding of the processes of heat transfer and their consequences for the energy consumption of buildings. In this chapter, we will apply exergy analysis to the processes of heat transfer through the envelope of buildings. After a review of the heat exchange mechanisms that take place on the interior and exterior surfaces of a wall, these exchanges are analysed from an exergy point of view. For this, first, the energy balance and then the exergy balance is performed inside a wall, in which the heat transport mechanism is conduction, taking into account the steady state, but above all looking at the dynamic case. The main objective of the application of the exergy method is to provide a new point of view to thermal inertia and to determine what inertia a façade must have, in order to acquire its best thermal behaviour. For this purpose, a calculation method has been developed, based on the results obtained with a building energy simulation

Exergy Analysis and Thermoeconomics of Buildings. https://doi.org/10.1016/B978-0-12-817611-5.00004-7 Copyright © 2020 Elsevier Inc. All rights reserved.

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program, which allows for selecting from among the different constructive solutions the one that presents the best energy performance for each of the defined climatic zones in the Spanish Technical Building Code (TBC). Next, the destruction of exergy that occurs in the boundary layer due to convective heat transfer is evaluated. Afterwards, the exchanges of heat by long-wave radiation between the interior surfaces are analysed and, once these exchanges are known, the corresponding exergy balances are considered, which allows the destruction of exergy associated with the absorption and emission of radiation to be evaluated. The case of the exterior surfaces is also considered, in which there is absorption of short-wave solar radiation and the exergy balance in said surfaces is described, which in turn allows the exergy destruction that takes place in them to be quantified. We present a review of the different ways of analysing the behaviour of opaque envelopes and we propose a method, which we have called the detailed dynamic method, which allows us to calculate, in a more precise way, the exergy destruction in the building envelope. A new index is suggested to characterize exergy behaviour, which allows us to classify the walls according to the destroyed exergy. The chapter ends with a summary of the methods for calculating the energy demand of a building and, based on this demand, the two existing methods to calculate the corresponding exergy demand are shown. In short, in order to promote the improvement of energy efficiency of buildings, a methodology based on exergy analysis is given, which allows us to take advantage of everything it offers in terms of identification and quantification of irreversibilities, in order to be able to compare the constructive solutions for façades and roofs and select the most suitable one from the point of view of its energy behaviour.

4.2

Heat exchanges in a building

Both the characteristics of the envelope and those of the interior elements of a building influence the differences between the characteristics of the environment that is generated inside and the outside conditions. Phenomena of exchange of mass flows (air and humidity) and energy occur between these two interior and exterior environments that define the thermal and environmental behaviour of the building. It is appropriate to use the concept of a thermodynamic system when assessing buildings. Thus, we consider the building as a system consisting of a volume of air, which is limited by the exterior and interior envelopes, so that each of the architectural elements (facades, roofs, floors, etc.) cause a filtering of the external climate inward, which results in a global thermal response of the entire building. In addition to the exchanged airflows, the thermal state of the air in the building is the result of the different heat fluxes that take place within it. These heat fluxes are caused by exterior and interior solicitations, ASHRAE [1]. Among the most notable exterior solicitations are: • • • •

Solar radiation. Outside air temperature. Temperature of the surroundings. Sky temperature.

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Interior solicitations come from within the occupied space and include: • • • •

Occupants. Illumination. Heating and air conditioning equipment. Miscellaneous equipment.

The various architectural factors, such as the shape, orientation, and inclination of walls, the size and location of openings, as well as the characteristics of the surfaces, of the materials making up the envelope and those of the structure condition the behaviour of the building, all of which, in short, act as an intermediary with the external climate. The solicitations listed above give rise to the different heat fluxes that we describe below, for which we consider the surfaces of the envelope in contact with the outside air, the surfaces in contact with the interior air and the interior air of the building itself. Surfaces in contact with the outside air: • • • •

Absorption of short-wave radiation (from the sun). In semi-transparent enclosures (windows), part of that radiation is transmitted to the interior. Emission and absorption (exchange) of long-wave radiation between the surface, the sky and the surroundings. Convection with the outside ambient air. Conduction through the enclosure. This conduction is usually considered 1D, except in the case of thermal bridges.

Fig. 4.1A, shows a diagram of the different mechanisms of heat exchange that take place on the exterior surface of a building. Surfaces in contact with the indoor air: • • • •

Absorption of short-wave radiation from the sun (after redistribution) and that from internal sources (lighting). Emission and absorption of long-wave radiation between the surface and the internal elements and the other internal surfaces of the premises. Convection with the indoor air. Conduction through the wall.

Figure 4.1 Mechanisms of heat exchanges (A) on an exterior surface (B) on an interior surface.

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Fig. 4.1B shows a scheme of the mechanisms of heat exchange that take place on a surface in contact with the indoor air of a building. Indoor air of the building: • •

Convection with the interior surfaces and the various objects with which it is in contact. Convection with the surfaces of heating or air conditioning equipment.

Keep in mind that air is practically impervious to exchanges of radiation for distances that we may consider inside a building (tens of meters) so that it only exchanges heat by convection. Once we have presented the different mechanisms of heat exchange that appear in the envelope of buildings, we will show how to perform the exergy balances associated with those exchanges. To do this, we will start by performing energy balances and then compare them with those of exergy, both through the corresponding expressions and in numerical form, through several examples.

4.3

Heat conduction in a wall

4.3.1

Energy balance

We will start by analysing conduction through a wall like the one in Fig. 4.2, whose interior and exterior surface temperatures at a given moment are respectively Tis and Tes, with the situation in winter being Tis > Tes. As we have said, we will first look at the balance of energy in the wall and then we will consider the balance of exergy. The Law of the Conservation of Energy in a dynamic state for the wall in Fig. 4.2a, for any time interval, allows us to say that fEnergy that is storedg ¼ fEnergy that entersg  fEnergy that leavesg

Figure 4.2 (A) Balance of energy in the wall; (B) Balance of exergy in the wall.

(4.1)

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267

Referring to the previous balance per unit area of the wall and per unit of time, we have the equation duw ðtÞ ¼ q_is ðtÞ  q_es ðtÞ dt

(4.2)

with uw(t) being the internal energy of the wall in the instant t, q_is ðtÞ being the heat flux that is transmitted by conduction from the inner surface and q_es ðtÞ the heat flux that arrives by conduction to the outer surface, at that time t under consideration. For a homogeneous wall, where r is the density, c the specific heat and L the thickness, the equation can be written as dTðtÞ ¼ q_is ðtÞ  q_es ðtÞ dt

crL

(4.3)

and in the case of a wall composed of N homogeneous layers as N X

ci ri Li

i¼1

dTi ðtÞ ¼ q_is ðtÞ  q_es ðtÞ dt

(4.4)

The above equations refer to the general case of the dynamic state. If we look at the steady-state the term corresponding to the variation of stored energy is zero and the heat fluxes would not be a function of time. In reality, this situation can only be considered in the case of very thin walls or low-density materials, such as a sheet of glass or an insulation sheet, C¸engel [2]. For this steady-state, we have q_is ¼ q_es ¼

Tis  Tes Rw

(4.5)

where Rw is the thermal resistance of the wall, which in the case of a multi-layered wall, is Rwall ¼

N X Li l i¼1 i

(4.6)

with li being the thermal conductivity of layer i and Li its thickness.

4.3.2

Exergy balance

Performing an exergy balance per unit of time and per unit of area of the wall, we have the equation dbw ¼ b_q;is ðtÞ  b_q;es ðtÞ  d_ w dt

(4.7)

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where dbw/dt is the rate of change of the exergy stored in the wall, d_ w is the exergy destruction per unit of time that occurs in the wall due to the temperature gradient in the heat transfer by conduction and b_q;is ðtÞ and b_q;es ðtÞ are the rates of exergy associated with the heat fluxes q_is and q_es , respectively. Eq. (4.7) is telling us that of the exergy flow rate that enters the wall b_q;is ðtÞ, a part is destroyed as a consequence of the irreversibility due to the temperature gradient d_ w , another part is stored in the wall dbw/ dt and the rest leaves it, b_q;es ðtÞ. We can also write the above equation as dbw ¼ dt



   T0 ðtÞ T0 ðtÞ 1 q_ ðtÞ  1  q_ ðtÞ  d_ w ðtÞ Tis ðtÞ is Tes ðtÞ es

(4.8)

For an N-layers wall, the above equation becomes       N X dTi ðtÞ T0 ðtÞ T0 ðtÞ T0 ðtÞ ri ci Li 1 ¼ 1 q_ ðtÞ  1  q_ ðtÞ dt Ti ðtÞ Tis ðtÞ is Tes ðtÞ es i¼1 

N X

d_ i ðtÞ

i¼1

(4.9) where, according to the Gouy-Stodola equation N X

d_ i ðtÞ ¼ T0

i¼1

N X

s_g;i ðtÞ

(4.10)

i¼1

The change in the exergy stored in the wall over a period (for example, a day or a year) is obtained by adding the exergy change for each of the I intervals calculated, that is Dbw ¼

I X i¼1

I X   Tf ;i ri ci Li Tf ;i  Ti;i  T0 ri ci Li ln Ti;i i¼1

(4.11)

where Tf,i and Ti,i are the final and initial temperatures, respectively, of the layer i. As can be seen in the previous equations, the resolution of the balances requires knowing the internal temperatures of the different layers that form the wall. The calculation of these temperatures is in no way trivial and to be able to carry it out, either some simplification needs to be made, or a numerical resolution technique needs to be used. Finally, if we consider the steady-state, the term on the left of Eq. (4.9) is zero and everything is independent of time, so that the equation of the exergy balance in an N-layers wall becomes     N X T0 ðtÞ T0 ðtÞ 1 (4.12) d_ i q_is ðtÞ ¼ 1  q_es ðtÞ þ Tis ðtÞ Tes ðtÞ i¼1

Exergy analysis of heat transfer in buildings

4.3.3

269

Examples

Example E.4.1.

An exterior wall of a house is considered to be made up of a 10.2 cm layer of brick (lbrk ¼ 0.7W/m C) followed by a 3.8 cm layer of plaster (lplstr ¼ 0.48W/m C). If the interior surface temperature is 20 C, that of the exterior surface is 12 C and the ambient temperature is 10 C, determine (a) The rate of heat transfer per unit of wall area. (b) The flow exergy rate that is transferred by the interior and exterior surface of the wall. (c) The exergy destroyed in the wall per unit of time.

Solution (a) This is a steady-state so that the heat flux that is transferred by conduction in one layer of the wall is the same as in the other. Therefore Q_ Tis  Tes Tis  Tes 20  12 W ¼ ¼ ¼ 35:57 2 ¼ q_ ¼ Lbrk Lplstr 0:102 0:038 Rwall A m þ þ lbrk lplstr 0:7 0:48 (b) The exergy flows at the boundary surfaces associated with that heat flux are   T0 W 1 q_ ¼ 1:21 2 m Tis   T0 W 1 q_ ¼ 0:25 2 Tes m (c) Undertaking an exergy balance on the wall, we have     T0 T0 1 q_  1  q_ ¼ d_ w Tis Tes W d_w ¼ 0:96 2 m

Consider a 12 m2 façade consisting of a 11 cm thick layer of solid moulded brick with mortar joints, with a thermal resistance of 0.25 m2K/W and with an internal lime mortar render of 2 cm and thermal resistance of 0.03 m2K/W. On a winter day, the outside temperature is 0 C, with the indoor air temperature being 20 C. Using the values of the convection-radiation coefficient of the Spanish Building Code (BTC) for exterior and interior surfaces, determine:

Example E.4.2.

(a) The thermal conductivity of the brick and mortar. (b) The heat transfer rate and the external, internal and intermediate (between the two layers) surface temperatures. (c) The rate of exergy transfer on both outer and inner surfaces and through the interlayer.

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(d) The exergy destroyed in the façade per unit of time.

Solution (a) The thermal conductivity of the solid brick layer is Rbrk ¼

Lbrk lbrk

/

lbrk ¼

0:11 W ¼ 0:44 0:25 mK

Likewise, the thermal conductivity of the mortar is Rmor ¼

Lmor lmor

/ lmor ¼

0:02 W ¼ 0:67 0:03 mK

(b) According to the BTC for vertical walls and horizontal flows, the external thermal resistance is Res ¼ 0.04 m2K/W, while the interior thermal resistance is Ris ¼ 0.13 m2K/W. Therefore, the heat flux that is transferred through the facade is q_ ¼

Ti  T0 20 W ¼ 44:44 2 ¼ m Ris þ Rmor þ Rbrk þ Res 0:13 þ 0:03 þ 0:25 þ 0:04

_ ¼ 533 W Q_ ¼ qA To calculate the temperatures, we will use the following equations q_ ¼

Ti  Tis Ris

/

Tis ¼ 14:2 C

q_ ¼

Tes  T0 Res

/

Tes ¼ 1:8 C

q_ ¼

Ti  Tin Ris þ Rmor

/ Tin ¼ 20  44:44ð0:13 þ 0:03Þ ¼ 12:9 C

(c) The corresponding exergy flows are     T0 _ 273:1 1 Q¼ 1 533 ¼ 26:3 W Tis 287:3   T0 _ Q ¼ 3:5 W 1 Tes   T0 _ Q ¼ 24:0 W 1 Tin (d) The exergy destroyed in the façade per unit of time is Tis  Tes _ Q ¼ 22 W D_ ¼ T0 Tis Tes

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271

Example E.4.3. A renovation is carried out in the wall of Example E.4.2 by means of a direct cladding, consisting of 4 cm rock wool and 3 mm plasterboard, with a total thermal resistance of 1.08 m2 K/W. Answer the questions in Example E.4.2, now with the wall renovated and compare the results obtained with the previous figures.

Solution A new thermal resistance has been added so that the rate of heat transfer is now Ti  T0 20 ¼ Ris þ Rins þ Rmor þ Rbrk þ Res 0:13 þ 1:08 þ 0:03 þ 0:25 þ 0:04 W ¼ 13:07 2 m

q_ ¼

_ ¼ 157 W Q_ ¼ qA As we can see, the rate of heat transfer has been reduced by 71%. The interior surface temperature is q_ ¼

Ti  Tis Ris

/

Tis ¼ 18:3 C

while that of the exterior surface is q_ ¼

Tes  T0 Res

/ Tes ¼ 0:5 C

and the temperature of the intermediate surface is q_ ¼

Ti  Tin Ris þ Rins þ Rmor

/

Tin ¼ 3:8 C

The exergy destruction in the wall per unit time is Tis  Tes _ D_ ¼ T0 Q ¼ 27 ¼ 9 W Tis Tes The fact of having added the insulation layer reduces the exergy destruction by almost 60%.

4.4

Exergy and inertia of walls

For many years, improvements to the envelope of buildings have fundamentally meant lowering the transmittance values of the opaque parts and the transparent elements as much as possible. So-called low-energy buildings are generally based on reducing heat transfer through facades, roofs and windows, basically by increasing the thickness of insulation, Feng [3].

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This approach has even been used in some national regulations on energy efficiency in buildings, for example, in the first version of the BTC in Spain, Spanish Ministry of Housing [4] or its Italian equivalent DL n311/2006 [5], where, according to the climatic zone, limiting values are set for the transmittance of the walls, roof, walls in contact with the ground and windows. Currently, the trend in regulations is aimed at including dynamic effects, either through various dynamic characterization parameters, such as the Réglamentation Thermique [6] or The Building Regulation [7], or as in the latest version of the BTC, Spanish Ministry of Housing [8], by establishing limits on the demand of the building. Therefore, during the last few years, the idea that it is impossible to design energyefficient buildings using only an approach based on the thermal transmittance values of their envelopes has become generally accepted. As such it is essential to look at other aspects, such as their dynamic behaviour, or as it is colloquially called, thermal inertia.

4.4.1

The concept of thermal inertia

Thermal inertia can be defined as the ‘property of a material that expresses the degree of slowness with which its temperature reaches that of the environment’ Ng et al. 2011 [9]. However, the definition that probably best expresses the effects it causes in an enclosure is the ‘capacity of a material to store heat and to delay its transmission’, Ferrari [10]. The term inertia, often used by scientists and engineers, is an analogy with that used in mechanics to relate mass and velocity, where inertia, in that case, is that which limits the acceleration of the object. Similarly, thermal inertia can be interpreted as a measure of the ‘thermal mass’ and the speed with which the heat wave is transmitted through the material. For this reason, it is common to find references to inertia in which it is directly called thermal mass. From a scientific point of view, the diffusion of heat through a solid is a well-known phenomenon. This diffusion plays a double role: on the one hand, the thermal resistance (function of the insulation level) between the interior and the exterior reduces the transferred heat flux; on the other hand, the thermal inertia causes a shift between the maximum external temperature and the maximum instantaneous heat flux transmitted to the interior space. Both effects combined in an appropriate way can serve to reduce the energy consumption of the HVAC equipment. This effect that the inertia causes in the interior conditions (temperature) of a room and associated consumption of energy is something known and used for a long time, and of course before air conditioning equipment existed. Throughout history and the world, there are numerous examples such as castles, churches, wineries and even cave houses dug in the mountains, where the differences between indoor and outdoor environmental conditions can be perceived as soon as one enters, with these being obtained in a ‘natural’ way. These differences can be summarized, on the one hand, by a greater attenuation of the temperature oscillations in the interior in relation to the external oscillations and, on the other hand, in a delay of the instant in which

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273

the temperature peaks on the inside, compared to when that peak was produced on the outside. Unlike thermal insulation that can be characterized by thermal resistance, thermal inertia is not quantified by a single parameter. Over the years, groups of researchers have used different indicators to characterize it. Thus, coefficients such as thermal diffusivity, and thermal admittance are used, AENOR [11], as well as the offset, Stephan [12], the amortization factor, UNE EN-ISO 1786 [13], the effective heat capacity per unit area, Tsilingiris [14], the time constant, C¸engel and Ghajar [15], etc.

4.4.2

Inertia and exergy

All the references, parameters and aspects related to inertia and commented on so far show that this is a well-known topic, at least from the energy point of view. Unfortunately, there is hardly any work that addresses the inertia of walls through exergy analysis, except for the doctoral thesis of I. Flores [16]. In this respect, Choi et al. [17] is also of great interest, in which a methodology for exergy analysis of heat transmission problems by conduction in dynamic states is shown. In this section, we will interpret the inertia from this exergy perspective, which will provide additional information that may be of interest when selecting the type of envelope. In order to simplify the analysis, the simple case of a homogeneous wall subjected to a 24-hour sinusoidal variation in external surface temperature will be considered, Tes, with the interior surface temperature Tis constant. As a consequence of this sinusoidal excitation, heat fluxes are established, periodic in time, both on the exterior surface q_es and the interior surface q_is , of different amplitudes and out of phase. As an example, we shall consider a homogeneous wall of 20 cm thickness, with a thermal conductivity l ¼ 1 W/mK and a heat capacity c ¼ 1.5 MJ/m3$K. The temperature Tes is characterized by the sun-air temperature, which includes the effects of solar radiation (to which we will refer in Section 4.9.1.3) together with the outside air temperature. The values used are Tis ¼ 20 C and constant while Tes is varying sinusoidally around an average temperature of 10 C and with an amplitude of 15 C during a period of P ¼ 24 h, according to the following expression   2pt p  Tes ðtÞ ¼ 10 þ 15 sin (4.13) P 2 This type of function is a reasonable approximation of what would be obtained with real climatological data, Asan [18]. The following Fig. 4.3 shows the profiles of the external and internal surface temperature for the case under consideration. The numerical values of the heat fluxes obtained corresponding to a full day are shown in Fig. 4.4. These heat fluxes are obtained from the application of the corresponding energy balance in the dynamic state, Eq. (4.2). If the graphs of the temperature profiles and heat fluxes are superimposed, we get Fig. 4.5. Taking into

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Figure 4.3 External surface temperature (left) and interior surface temperature (right) for the example analyzed.

Figure 4.4 Heat fluxes for the example analyzed.

account the respective signs of the temperature difference Tis  Tes, and the heat flux q_is , we can see that there are four different cases: •

•

Case I: q_is < 0 and Tis > Tes. This is the usual situation in winter. The heat flux is from inside to outside, as the interior temperature is higher than the exterior. The exergy flow is also from inside to outside, so that the exergy inside the room will decrease and, to maintain the indoor air temperature constant (constant exergy of indoor air), a contribution of exergy through a heating equipment will be necessary. A heat engine with its thermal energy sources in the indoor and outdoor air in such a case would work as a heat pump and would consume work. Case II: q_is < 0 and Tis < Tes. The heat flux is from inside to outside, although the temperature of the exterior surface is higher than that of the interior. The exergy flow is the opposite, so that exergy is being provided to the room, which allows it to maintain its constant

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275

Figure 4.5 Possible situations depending on the sign of heat flux qis and the temperature difference between the inner and outer surface.

•

•

temperature. This is, therefore, a favourable case, since air conditioning equipment is not necessary for maintaining comfortable conditions. On the other hand, the supposed heat engine to which we referred earlier would produce work. Case III: q_is > 0 and Tis < Tes. This is the usual situation in summer. The wall gives heat to the interior when the temperature outside is higher than inside. As a consequence of this heat flux there is an exergy flow towards the outside, so to keep the interior temperature constant an air conditionning equipment will be necessary. The heat engine would work like a refrigerating machine and would consume work. Case IV: q_is > 0 and Tis > Tes. The wall gives heat to the interior, although the temperature of the interior surface is greater than that of the exterior. Due to this heat transfer, there is an exergy flow towards the interior, so that in these conditions no air conditioning equipment is necessary to keep the indoor air temperature constant. In this case, the heat engine would extract heat from the heat source (the indoor environment) and give it to the cold source (external environment), generating work.

The exergy flows associated with the heat fluxes for each of the four cases, which keep these indoor temperature conditions constant with the given variation of the outside temperature, are shown schematically in Fig. 4.6. In cases I and IV, the temperature of the exterior surface is lower than the interior. However, due to the effect on the exergy value of the ambient air temperature, although in case I there is an exergy flow that leaves the interior surface, which will have to be compensated for, in case IV the wall provides exergy to the room to be

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air conditioned. Likewise, cases II and III correspond to summer, when the temperature of the exterior surface is higher than that of the interior. Although case III corresponds to the usual case in which there is an exergy flow that leaves the interior environment through the wall and will have to be compensated for to maintain the interior temperature, in case II there is a contribution of exergy from the wall to the room to be air-conditioned. Thus, in cases I and III, the exergy is transmitted from the indoor air to the wall. These situations force the energy system, whether heat pump or refrigeration machine, to replace the lost exergy and, therefore, are not desirable from the point of view of the building’s energy efficiency. On the other hand, in the other two situations, cases II and IV, it is the wall that returns part of the exergy that the indoor air had previously given to it. These situations are desirable from the point of view of energy efficiency, as they allow the constant comfortable interior temperature to be maintained which means, in short, the desired level of exergy of the indoor air without the need for an external contribution, which would be the work consumed by the corresponding equipment.

Figure 4.6 Energy and exergy flow in the four cases.

As time passes, the wall goes through the four previous situations, depending on its inertia and the climatic conditions. If a parametric study of b_q;is is carried out as a function of the thermal transmittance and the heat capacity of the wall, it can be determined, for certain climatic conditions, which wall implies the lowest contribution of exergy to the room to be air-conditioned in order to maintain certain comfort conditions. This study has been carried out by Flores [16] in his doctoral thesis. Taking the above into account, exergy can be a very useful parameter when designing the envelope of energy-efficient buildings. For this, walls will need to be chosen with a thermal transmittance and dynamic characteristics that minimize the

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277

net extracted exergy through the envelope, or in other words, minimize the exergy that needs to be added to the indoor air with the energy supply installation.

4.5

Transport of exergy by convection

Convection is present both on the internal and external surfaces of the envelope and is due to the difference in temperature between the surface of the envelope and the environment in which it is located. It is a phenomenon that occurs within the boundary layer and as we know from the texts on heat transfer, C¸engel [19], despite the complexity of convection, the heat flux is expressed by Newton’s Law of Cooling by introducing a convection coefficient. For the interior surface of a wall we have Q_ cv;i ¼ q_cv;i Ais ¼ Ais hcv;i ðTi  Tis Þ

(4.14)

and analogously for external surfaces. In general, this coefficient of convection depends on the configuration, airspeed, temperature difference between surface and air, and the thermophysical properties of the fluid, in this case, the air. However, given that the usual configurations in buildings are flat surfaces, and the temperature ranges are close enough that we not need to take into account any variation in properties, we can conclude that the main factors that govern the convection coefficient are: the direction and sense of the heat flux (horizontal, vertical, and if vertical, ascending or descending) and, above all, the airspeed. Therefore, there are important differences in their values for interior and exterior surfaces. Indeed, on the outside of buildings, the movement of air is mainly due to the wind. On the other hand, in the interior of buildings, the movement of the air is generally due to natural convection, which is generated by a difference in densities associated with a difference in temperatures close to the wall. This means that the internal convection coefficients hcv,i and external convection coefficients hcv,e will be very different, Ito and Kimura [20]. The differences in values, although not as pronounced, are maintained even in situations with forced ventilation systems, since for reasons of comfort, the interior airspeed is considerably lower than that of the wind outside. As a consequence of the non-slipping condition, the air in contact with the surface of a wall has a zero velocity. Therefore, the heat transfer between the surface of the wall and the air in contact is done by pure conduction, since that air layer is motionless. Then, that heat moves away by convection, as a result of air movement. For a study of convection, consult the aforementioned work of C¸engel or that of Kays and Crawford [21].

4.5.1

Energy balance

Despite these differences between interior and exterior surfaces, the way to mathematically analyse heat transfer is the same. As an example, we shall consider the following case of the boundary layer on the interior surface of a wall, see Fig. 4.7, with Ti and Tsi being the temperatures of the indoor air and the interior surface, respectively. If it is assumed that Ti > Tis, the indoor air in the vicinity of the interior surface will cool,

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Figure 4.7 Heat transfer by convection on the interior boundary layer of the wall.

then increasing its density and, consequently, will tend to descend activating the mechanism of convection. By taking into account the low inertia of the air, the analysis can be carried out as a steady-state without appreciable loss of precision, when the boundary layer is already configured. The energy balance in these conditions, per unit of wall area and per unit of time, is reduced to q_cv;i ¼ q_cd;is ¼ lai

vT j vx x¼0

(4.15)

where lai is the thermal conductivity of the air, T represents the temperature distribution in the air and vT/vxjx¼0 is the temperature gradient in the surface. As we said before, the rate of heat transfer by convection [W/m2] for an interior surface, per unit area, is q_cv;i ¼ hcv;i ðTi  Tis Þ

(4.16)

Referring now to the outer surface, where Tes is the surface temperature and hcv,e the convection coefficient, the rate of heat transferred by convection on the outer surface is q_cv;e ¼ hcv;e ðTes  T0 Þ

(4.17)

Table 4.1 shows the regular values that should be adopted for these convection coefficients. The experimental determination of the external convection coefficient is limited to establishing correlations with air velocity v, generally given by hcv,e ¼ aþbv, with a and b being both constant. The problem with these correlations is that, in the vicinity of the surface, the air velocity can be very different from the natural speed of the wind.

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Table 4.1 Internal and external convection coefficients according to the standard UNE-EN ISO 6946 (AENOR 2007).

4.5.2

Direction of heat flux

hcv,i [W/m2$K]

hcv,e [W/m2$K]

Horizontal

2.5

20

Vertical ascending

5

20

Vertical descending

0.7

20

Exergy balance

By performing the corresponding exergy balance in the boundary layer of the inner surface, see Fig. 4.7, we obtain the equation b_cv;i  b_cd;is ¼ d_ cv;i

(4.18)

Clearing the exergy destruction d_ cv;i in the previous expression, we finally have d_ cv;i ¼ T0 q_cv;i



1 1  Tis Ti

 ¼ T0 hcv;i

ðTi  Tis Þ2 Ti Tis

(4.19)

This expression allows us to quantify the rate of exergy destruction that occurs in the heat transfer by convection between the air and the internal surface of a wall. Obviously, for the case of convection in the boundary layer of an external surface, the expression is analogous, simply by substituting the variables hcv,i, Ti and Tis for the corresponding exterior surface variables hcv,e, T0 and Tes, respectively. In effect, if the exergy balance is performed on the outer surface, the resulting equation is b_cd;es ¼ b_cv;e þ d_ cv;e

(4.20)

obtaining the following expression for the rate of exergy destruction 2

ðTes  T0 Þ d_ cv;e ¼ T0 hcv;e Tes T0

4.5.3

(4.21)

Examples

Hot water flows through a pipe with an outer diameter of 50 mm and a length of 9 m, the temperature of the outer surface of the pipe being 42 C. If the

Example E.4.4.

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Exergy Analysis and Thermoeconomics of Buildings

outside air is at the temperature of 15 C and the coefficient of convection between the pipe and the air is 24 W/m2K, determine: (1) The rate of heat exchanged and the associated rate of exergy. (2) The rate of exergy destruction.

Solution (1) The rate of heat given to the air by the pipe is Q_ ¼ pDe LhðTs  T0 Þ ¼ 916 W The rate of exergy associated with that heat is   T0 _ Q ¼ 78:5 W 1 Ts (2) The flow of exergy calculated above is completely destroyed in the environment (external irreversibilities) so that the rate of exergy destruction is D_ ¼ 78:5 W

The façade of a house of dimensions 7  4 m has a thermal resistance of 4.5 m2K/W. The house is maintained at a temperature of 20 C on a day when the outdoor air temperature drops to 2 C, and the wind speed is 60 km/h. Without taking into account the heat transfer by radiation, determine:

Example E.4.5.

(1) The rate of heat lost through the wall. (2) The rate of exergy coming out of the wall. (3) The rate of exergy destroyed in the inner boundary layer, in the facade and in the outer boundary layer of the wall.

Solution (1) According to the ASHRAE Fundamentals the internal convection coefficient for horizontal convection is hcv,i ¼ 3.06 W/m2K. Therefore, the thermal resistance of the inner boundary layer is Rcv,i ¼ 1/hcv,i ¼ 0.327 m2K/W. On the other hand, for a wind speed of 60 km/h, the ASHRAE Fundamentals proposes a value of hcv,e ¼ 65.5 W/m2K, so the thermal resistance in the outer boundary layer is Rcv,e ¼ 1/hcv,e ¼ 0.015 m2K/W. In short, the rate of heat lost per unit area is q_ ¼

Ti  T0 W ¼ 3:72 2 Rcv;i þ Rfac þ Rcv;e m

and the total heat flux is Q_ ¼ qA _ ¼ 104 W

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281

(2) The flow of exergy associated with that heat flux coming from the indoor air is   T0 _ 1 Q ¼ 6:4 W Ti (3) This flow of exergy is destroyed in the inner boundary layer, in the facade and in the outer boundary layer. Calculating the temperature on the inner surface Tis ¼ Ti  Rcv;i q_ ¼ 291:8 K the exergy destroyed in the inner boundary layer is Ti  Tis ¼ 0:43 W D_ i ¼ T0 Q_ Ti Tis The temperature on the outer surface is Tes ¼ T0 þ Rcv;e q_ ¼ 275:2 K so the rate of exergy destroyed in the outer boundary layer is Tes  T0 ¼ 0:04 W D_ e ¼ T0 Q_ T0 Tes Obviously, the rate of exergy destroyed in the facade is D_ fac ¼ D_  D_ i  D_ e ¼ 5:9 W Practically, all the exergy destruction takes place inside the facade, due to the irreversibility of conduction. In fact Tis  Tes ¼ 5:9 W D_ fac ¼ T0 Q_ Tis Tes

4.6

Exchange of radiation exergy between surfaces

The transfer of heat by radiation represents a very important part of the energy exchanges that occur in buildings. In the case of radiant floor systems, approximately 50% of the heat is directly transmitted to the air by convection, and the other 50% arrives by convection after the mechanisms of conversion of radiant energy to heat of convection, Olesen [22]. The calculation of the exergy associated with this thermal radiation has traditionally been a very controversial topic and has led to much scientific discussion, Torio [23]. Proofs for this are the different approaches and definitions proposed by various authors as shown in Chapter 2.

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One of the main difficulties of radiation problems is the calculation of energy exchanged and, of course, the radiant exergy exchanged between different surfaces. In the ideal case in which the system is formed by black surfaces, the problem is relatively simple, because there is only emission and absorption, and no reflected component. On the other hand, when the system is formed by real surfaces, even when using the model of grey surfaces the problem is considerably complicated, as we need to calculate the reflections of the radiation emitted. In those situations with surfaces of high emissivity, close to unity, an approximate value can be obtained with an acceptable error if it is assumed that there is no reflection. In general, in spite of this greater complexity, the calculations of the energy flows are perfectly described in heat transfer books, such as C¸engel and Ghajar [15], Incropera and DeWitt [24], and Lienhard [25]. However, the calculation of the radiation exergy exchanged is not usually described in the heat transfer books and, moreover, is somewhat more complex. For each flow of radiation exergy that reaches an opaque surface, in addition to the absorbed and reflected components, the remaining part that is destroyed due to irreversibilities will need to be considered, and all of this in the multiple processes of emission, reflection, absorption and destruction that occur in the radiant exchange between surfaces.

4.6.1

Radiation exergy exchange between two grey surfaces

Petela [26] developed a formulation to calculate the exergy exchange between two surfaces 1 and 2, grey, flat, parallel, infinitely long and facing each other. He considered there is a vacuum between the surfaces, so there are no exchanges for conduction and convection. The surfaces are isotherms, and their temperatures T1 and T2 are constant thanks to the action of thermal energy reservoirs that provide or withdraw the necessary heat. The emissivity, absorptivity and reflectivity of the surfaces are ε1 and ε2 , a1 and a2, and r1 and r2 respectively, with the surfaces A1 ¼ A2 ¼ A. The heat exchanged between both surfaces per unit of time and area q_12 is the fraction of the energy absorbed by 2 of the energy emitted by 1 minus the fraction absorbed by 1 of that emitted by 2 and therefore   (4.22) q_12 ¼ ε12 s T14  T24 where ε12 ¼

1 1 1 þ 1 ε1 ε2

(4.23)

Petela similarly deduced that the exergy of radiation exchanged between both surfaces is    4  b_r;12 ¼ ε12 s T14  T24  T0 T13  T23 (4.24) 3

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283

Once the exergy exchange is calculated, it is important to evaluate the exergy destruction on each surface due to irreversibilities in the emission and absorption. That destruction of exergy can be broken down into the sum of two parts. So for surface 1, we let d_ 1;1 be the term that represents the exergy destruction caused by the emission of the surface itself and the absorption of the fractions of that emission that, having been reflected by surface 2, are absorbed by 1. The second addition d_ 2;1 corresponds to the exergy destruction on surface 1 caused by the absorption of the radiation emitted by surface 2, either directly or through various reflections, so that d_ 1 ¼ d_ 1;1 þ d_ 2;1

(4.25)

and analogously for surface 2. The following expressions are derived in the referenced work of Petela [26].    ε1 ε2 b_r;b1 T0 ε 1 r2 d1;1 ¼ ε1 e_b;1 1  (4.26) 1  T1 1  r1 r2 1  r1 r 2 d_ 2;1 ¼

   ε1 T0 _ _ ε2 br;b2  ε2 eb;2 1  1  r1 r2 T1

(4.27)

where, as we saw in Chapter 2, e_b;1 , e_b;2 are the emission power, that is, the black radiation emitted per unit of time and area of surfaces 1 and 2 respectively and b_r;b1 , b_r;b2 are the exergy of that radiation. Analogous expressions are obtained for the exergy destruction in surface 2. However, when what we want to obtain is the total exergy destruction, which is the sum of the one that takes place on surfaces 1 and 2, there is a simpler alternative method than the previous one. This method consists of looking at the exergy balance of the whole process, so that, given that the temperatures remain constant, the global balance is     T0 T0 q_12 1  ¼ q_12 1  þ d_ 1 þ d_ 2 T1 T2

(4.28)

which gives the rate of total exergy destruction as d_ ¼ d_ 1 þ d_ 2 ¼ q_12 T0



1 1  T2 T1

 (4.29)

an expression which corresponds to the rate of exergy destruction related to the heat exchanged between two systems of temperatures T1 and T2, as we saw in Chapter 2.

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Exergy Analysis and Thermoeconomics of Buildings

4.6.2

Radiation exchange between the interior surfaces of a room

For those more frequent situations, where there are more than two grey surfaces exchanging radiation, the calculation becomes more complex. This is what happens in the exchange of long-wave radiation between the interior surfaces of a building: the radiation emitted by one surface propagates until it impinges on another, being then partially reflected and again re-reflected and so on, while also being partially absorbed in each contact with a surface. It is therefore impossible to try to follow the radiation as it passes through these complicated processes, unlike the case of two flat and parallel surfaces. Fortunately, this is not necessary, since many methods have been developed over the years to resolve the exchange of radiation between surfaces; among them are, for example, those proposed by Hottel and Sarofin [27], Sparrow and Cess [28], Gebhart [29] or Clark and Korybalski [30]. Although basically all of them are equivalent, the Gebhart method has been chosen due to its simplicity and adaptation to the tools used. The method is based on the so-called radiant exchange factor or Gebhart factor, Gij, which represents the fraction of energy being emitted by the surface i and absorbed by surface j. In this fraction, all possible ways of reaching the surface j are included, that is, the direct path, as well as those originating from the various reflections

Gij ¼

4.6.2.1

Q_ ij εi Si sTis4

(4.30)

Radiative energy exchange

Before addressing the radiative exergy exchanges, let us first consider the energy exchanges. A surface loses energy by emission and gains energy by absorbing the radiation emitted by other surfaces and its own emission, which has been reflected by the other surfaces. Depending on which of the two quantities is the highest, there will be a net gain or loss of energy. The rate of net energy transfer on a surface i of an enclosure composed of N surfaces can be calculated from the Gebhart factors as the net balance between the energy emitted and the energy absorbed, which is Q_ i ¼ Ai εi sTis4 

N X

Aj εj sGji Tjs4

(4.31)

j¼1

Taking into account the following geometric relationship εi Ai Gij ¼ εj Aj Gji

(4.32)

Exergy analysis of heat transfer in buildings

285

the net heat exchanged between two surfaces i and j can be expressed as  Q_ ij ¼ εi Ai Gij s Ti4  Tj4

(4.33)

For its part, considering the energy balance and using the vision factors, the Gebhart factors are calculated by solving the following system of equations Gij ¼ Fij εj þ

N X

Fik ð1  εk ÞGkj

i ¼ 1; :2.N

(4.34)

k¼1

In order to calculate the configuration factors Fij the rule of addition, the rule of superposition, the reciprocal relationship, the crossed string method, as well as the graphs and analytical expressions that can be found in the books of heat transfer are used. In the previous equation, the first summand of the member on the right represents the fraction of energy emitted by i, which is spread directly on j and is absorbed. The summation, on the other hand, represents the fraction of energy that reaches j after suffering at least one reflection. Thus, the emission of the surface i that reaches the surface k and is reflected will be Fikrk ¼ Fik(1  εk). Of that part, only the fraction Gkj is absorbed by N P the surface j. As with the vision factors, we can verify that Gij ¼ 1. Once the Gebj¼1

hart factors have been obtained, all the components of the radiation energy exchange between the surfaces of the room can be determined by applying Eq. (4.33).

4.6.2.2

Radiation exergy exchange

For its part, the exchange of exergy in an enclosure composed of diffuse-grey surfaces, is at least as complex as the case of energy exchange, since in addition to the components seen above, we must add the inevitable exergy destruction. Fortunately, if diffuse-gray surfaces are considered, the same coefficients of the Gebhart matrix can also be used to study exergy balances. Thus, the term Gij also represents the fraction of radiation exergy that is emitted by the surface i and is absorbed by the surface j, including all possible trajectories. In this way, from Eq. (4.33) and considering two surfaces i and j, the fraction of radiation exergy emitted by the surface i, which is absorbed by the surface j will be  s B_ r;i/j ¼ Gij B_ r;i ¼ Ai Gij εi 3Ti4 þ T04  4T0 Ti3 3 Analogously, the emission by j which is absorbed in i is s B_ r;j/i ¼ Gji B_ r;j ¼ Aj Gji εj 3Tj4 þ T04  4T0 Tj3 3

(4.35)

(4.36)

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Exergy Analysis and Thermoeconomics of Buildings

Therefore, the radiation exergy exchange per unit of time between both surfaces is  i sh  B_ r;ij ¼ Ai Gij εi 3 Ti4  Tj4  4T0 Ti3  Tj3 (4.37) 3 Finally, the net rate of exergy exchanged by the surface i is obtained from the previous expression by adding up over the total of the interior surfaces of the room, and therefore

B_ r;i ¼

N X

B_ r;ij ¼

j¼1

N X j¼1

Ai Gij εi

 i sh  4 3 Ti  Tj4  4T0 Ti3  Tj3 3

(4.38)

On the other hand, the exergy destruction that takes place on the surface i will be the sum of two contributions, as expressed in the following equation

D_ i ¼ D_ i;i þ

N X

D_ j;i

(4.39)

jsi

The first of the addends of the member on the right represents the exergy destruction caused by the emission of the surface i and the absorption on that surface of its own emission, which, after having been reflected by the other surfaces, is finally absorbed by the surface i. For its part, the second addend represents the exergy destruction caused by the absorption on surface i of the exergy emitted by the other surfaces, which either directly or after a series of reflections, ends up being absorbed by the surface i. For the case of three or more surfaces, a detailed approach, for the calculation of each of the addends of Eq. (4.25), such as the one developed by Petela, is unfeasible. For those situations, however, one can calculate the exergy destruction on each surface i from the corresponding exergy balance on the surface.

4.7

Energy and exergy balances on the interior surface of a façade

So far, we have separately considered the exergy associated with conduction, convection and long-wave radiation. Let us now refer to the interior surface of a façade, in which, as we saw in Section 4.2, the three mechanisms of heat exchange exist. Consider Fig. 4.8 which represents a wall and in which, as a dashed line, we have indicated the system, of infinitesimal thickness, on which we are going to consider the energy balance and later the exergy balance. We shall consider the case of summer so that the heat flux moves from outside to inside.

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287

Figure 4.8 Interior surface of a building envelope.

4.7.1

Energy balance

Carrying out an energy balance per unit of time in the surface under consideration, we have (

Heat transfer by conduction

)

( ¼

Exchange of

)

long wave radiation ) ( Heat exchange þ by convection

( þ

Absorption of

)

short wave radiation

(4.40) which, we show by means of the following mathematical equation Q_ cd ¼ Q_ lwr þ Q_ swr þ Q_ cv

(4.41)

where: • •

• •

Q_ cd : rate of heat transfer by conduction. Q_ lwr : rate of long-wave radiation exchanged (absorption e emission). This radiant exchange is, in turn, often broken down into two terms: one which takes place with the other interior surfaces of the enclosure that are at different temperatures while the second is the radiant exchange with internal components such as furniture, etc. Q_ swr : rate of absorption of redistributed short-wave radiation from the sun and internal sources, such as lighting. Q_ cv : rate of heat exchanged by convection with the indoor air.

The radiant exchange in interior surfaces of a building is of great complexity, due to the different nature of this radiation and the irregular behaviour of those surfaces. In order to simplify the calculations and since both convection and radiation flows are in parallel, a convection-radiation coefficient is used with which the heat exchange

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Exergy Analysis and Thermoeconomics of Buildings

with the air in the room is calculated, directly by convection and through exchanges by radiation with the other interior surfaces and which is exchanged, finally, with the air by convection, so that hcvr ¼ hcv þ hr

(4.42)

Regulations like the BTC give the limit values of the thermal transmittance that the interior surfaces must have. These values of the thermal transmittance are obtained using values of the normalized convection-radiation coefficients, which depend on whether the flow is horizontal or vertical, and in the latter case of whether it is ascending or descending, BTC [31].

4.7.2

Exergy balance

We refer now to the exergy balance. This balance establishes that 9 9 8 9 8 8 Exergy associated > > Short wave > > Long wave radiation > > > > > > = < = > < = > <  radiation exergy exergy exchanged with heat transfer ¼ > > > > > > > > > > ; : ; > : ; > : absorbed by the surface by conduction 9 9 8 8 Exergy associated > > Exergy destroyed > > > > > = < = > < in the þ with heat transfer þ > > > > > > > ; : ; > : surface by convection (4.43) Mathematically, the exergy balance is expressed according to the following equation     T0 T0 _ _ _ _ Qcd;i 1  (4.44) ¼ Br;i  Bswr;i þ Qcv;i 1  þ D_ i Ti Ti From this equation, the rate of exergy destruction on the surface is obtained. This destruction is due to the absorption of the long-wave radiation that comes from the interior surfaces, to the emission of the surface itself, to the redistributed short-wave absorption from the sun and lights and also includes the exergy destruction associated with convection in the boundary layer between the air and the surface. Naturally, as a prerequisite, it is necessary to have calculated the configuration factors, Gebhart factors and resolve the exchanges of both energy and exergy. Besides, in the interior of the wall, the exergy destruction associated with conduction takes place, which is calculated according to Eq. (4.9).

Exergy analysis of heat transfer in buildings

4.7.3

289

Examples

Example E.4.6.

The main façade of a rectangular industrial warehouse consists of a double-layer base wall. The outer layer is made of solid brick facing the outside with a polyurethane coating of 2 cm, and the inner layer is a double-hollow brick partition of 7.5 cm with internal mortar and plastering, see Table E.4.1. The interior surface of the facade is 16 m2 and the rest of the interior surfaces 90 m2. We will assume that the emissivity of the interior surface of the facade and the rest of the interior surfaces is 0.9.

Table E.4.1 Data for the main facade. Description of layer

Thickness

R(m2/WK)

1

/2 solid facing brick

10.5

0.25

Polyurethane insulation

2

0.72

Double hollow brick partition

7.5

0.15

Layer of mortar

1

0.008

Plastering

0.5

0.017

With a wind speed of 3.3 m/s, the temperature of the indoor air at 20 C, the ambient air at 2 C and assuming that the temperatures of the other internal partitions are at 16 C, determine: (a) The interior surface temperature of the façade. (b) The long-wave radiation exchanged by the inner surface of the facade with the other surfaces, and the corresponding radiation exergy exchanged. (c) The rate of exergy destruction due to convection in the inner boundary layer. (d) The rate of exergy destruction inside the facade due to conduction.

Solution (a) The thermal resistance of the facade, that is, the sum of the thermal resistances of the layers is 1.145 m2K/W. According to ASHRAE Fundamentals, the interior convection-radiation coefficient is 8.9 W/m2K, for a surface of emissivity ε ¼ 0.9. Therefore, the interior resistance is Ri ¼ 0.112 m2K/W. For a wind speed of 3.3 m/s the convection-radiation coefficient is 22.7 W/m2K, so the exterior resistance is Re ¼ 0.044 m2K/W. The rate of heat transfer through the façade is q_ ¼

Ti  T0 W P ¼ 13:8 2 Ri þ j Rj þ Re m

Q_ ¼ 221 W

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Exergy Analysis and Thermoeconomics of Buildings

so the temperature of the interior surface is Tis ¼ Ti  Ri q_ ¼ 18:4 C (b) Since all the other surfaces are at the same temperature of 16 C and have the same emissivity, the heat exchange by radiation is calculated as that which takes place between two surfaces, with is the interior surface of the facade and s2 the other interior surfaces. Therefore, the heat exchanged by radiation is Q_ r;iss2 ¼

    4 s Tis4  Ts2 5:67x108 291:54  2894 ¼ 197:9 W ¼ 1  εis 1 1  εs2 1  0:9 1 1  0:9 þ þ þ þ Ais Fis;s2 As2 εs2 16$0:9 16 90$0:9 Ais εis

The exergy of the exchanged radiation is " sTis4 B_ r;iss2 ¼

# " #     1 T0 4 4 T0 1 T0 4 4 T0 4 1þ  sTs2 1 þ   3 Tis 3 Tis 3 Ts2 3 Ts2 1  εis 1 1  εs2 þ þ Ais Fiss2 As2 εs2 Ais εis

¼ 10:3 W

(c) The coefficient of convection (without radiation) corresponding to the vertical inner surface and horizontal heat flux, according to ASHRAE Fundamentals, is hcv,i ¼ 3.06 W/m2K. The rate of heat transfer due to convection through that boundary layer is Q_ cv;i ¼ Ais hcv;i ðTi  Tis Þ ¼ 78:3 W and the exergy associated with that heat flux by convection B_ cv;i ¼

  T0 _ Qcv;i ¼ 4:4 W 1 Tis

The rate of exergy destruction in the convective boundary layer is Ti  Tis ¼ 0:4 W D_ cv;i ¼ T0 Q_ cv;i Ti Tis (d) To calculate the exergy destruction inside the facade due to conduction, we first determine the temperature of the outer surface Tes ¼ T0 þ Re q_ ¼ 2:6 C with the rate of exergy destruction being Tis  Tes ¼ 11:9 W D_ fac ¼ T0 Q_ Tis Tes

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291

Example E.4.7.

The end elements of a dwelling heating installation are cast iron radiators whose average surface temperature is 60 C, with the interior surface temperature of the walls being 20 C. The total heat given by the radiators is 12 kW, of which 65% is convection. Evaluate the error in the calculation of the transferred exergy, when using the expression corresponding to the exergy of convection, when the ambient temperature is 290 K. Solution Considering that all the heat given is by convection, the exergy of that heat is   T0 _ Q ¼ 1:55 kW 1 Ts

If we consider that the emissivity of the radiators and the surfaces of the walls is the same, we have that the exergy of the radiation exchanged is ! 4 Ts3  Tis3 _ Qr 1  T0 4 3 Ts  Tis4 By breaking down the heat flux into a convective part and a radiative part, the associated exergy is   T0 _ þ 1 0:65Q Ts

! 4 Ts3  Tis3 1  T0 4 0:35Q_ ¼ 1:05 kW 3 Ts  Tis4

Therefore, the error made in this approach is 47%.

4.8

Energy and exergy balances in the exterior surface of a façade

After analysing the mechanisms of heat exchange on the interior surface of a façade and performing an exergy balance on an interior surface, it remains to analyse the exchanges of energy and exergy on the exterior surface of the building envelope.

4.8.1

Energy exchanges

For greater clarity Fig. 4.9 shows the heat exchange mechanisms that act on the exterior surface. Taking as a reference the direction of the flows in the figure, that is to say, under conditions of winter in which there is a net heat flux from the interior to the exterior, the rate of energy balance in the exterior surface per unit area is

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Exergy Analysis and Thermoeconomics of Buildings

Figure 4.9 Energy exchanges on the exterior surface’.

q_cd;es ¼ q_cv;e  q_r;sun þ q_r;sky þ q_r;sur

(4.45)

where: • • • • •

q_cd;es is the rate of heat that is transmitted by conduction from the exterior surface of opaque envelopes and by a combination of mechanisms for semi-transparent envelopes. q_cv;e is the rate of heat exchanged by convection with the outside air. q_r;sun is the rate of short-wave radiation absorbed from the sun. q_r;sky is the rate of long-wave radiation exchanged with the sky, and finally q_r;sur is the rate of long-wave radiation exchanged with the surroundings, such as the terrain, other buildings, etc.

Developing the previous balance equation gives  4  q_cd;es ¼ hcv;e ðTes  T0 Þ  q_r;sun þ εes Fes;sky sTes  εsky sT04  4   sT04 þ εes Fes;sur sTes

(4.46)

where εes is the emissivity of the surface, εsky is the emissivity of the sky and Fes,sky and Fes,sur are the configuration factors of the surface/sky and the surface/surroundings respectively. Next, we will make a series of comments on each of the terms in this equation.

Exergy analysis of heat transfer in buildings

4.8.1.1

293

Convection coefficient on the exterior surface

Unlike what happens inside the building, the exchange of energy by convection on the outside is affected by the presence of wind. As we stated in Section 4.5, the wind can significantly vary the value of the convection coefficient hcv,e, especially in those buildings that are very exposed, Brau [32]. There are various expressions in the literature for the calculation of the coefficient as a function of wind speed. Unfortunately, these types of expressions are generally not very useful. The fundamental reason is that, as we said in Section 4.5.1, weather stations are usually found at airports or areas that are quite exposed, so the wind speed data is not applicable to most buildings. For this reason, it is usual to work with normalized convection coefficient values, such as those seen previously in Table 4.1. Regardless of whether standardized or particularized values are used for specific wind conditions, the heat exchanged between the surface and the environment is given by Newton’s law of cooling, in an identical way to what happens on the inner surface.

4.8.1.2

Radiation exchange with the sky and surroundings

We are now looking at the exchange of long-wave radiation, which takes place between the exterior surface and the sky q_r;sky and between said surface and the surroundings q_r;sur . Covered by this term ‘surroundings’ are the floor and all those objects that the building ‘sees’, such as trees, other buildings, etc. The emission of radiation by the atmosphere is a consequence of the presence of participatory gases, H2O and CO2 fundamentally, and it is concentrated in the regions of the spectrum between 5 and 8 mm and around 13 mm. Although this emission is far from resembling that emitted by a black body, it is convenient and very usual in calculations to consider the atmosphere as an ideal black surface at a fictitious temperature, which emits the same amount of radiant energy as the atmosphere. This fictitious temperature is what is called the effective sky temperature Tsky. Its value depends on atmospheric conditions, fundamentally on the ambient temperature, the relative humidity, the degree of cloud cover and the ambient pressure. This dependence is usually brought together as the so-called emissivity of the sky εsky, so that the effective temperature of the sky can be calculated from the ambient temperature T0 through the 4 ¼ ε T 4 , so that the emission power of the sky is relationship Tsky sky 0 4 e_sky ¼ εsky sT04 ¼ sTsky

(4.47)

This effective sky temperature varies from 230 K for clear and cold sky conditions up to around 285 K for warm sky and with clouds. There are numerous proposed equations for determining the emissivity of the sky, some as simple as supposing a constant value equal to 0.74, while more complex ones are given depending on atmospheric

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Exergy Analysis and Thermoeconomics of Buildings

conditions. Several of the latter are collected in Gliah et al. [33]. The TRNSYS simulation software uses the proposal in Martin and Berdahl [34]. εsky ¼ ε0 þ 0:8ð1  ε0 ÞCcld

(4.48)

where Ccld is the cloud cover factor and ε0 is the emissivity corresponding to a clear sky. The latter can be obtained by the following mathematical expression [34] ε0 ¼ 0:711 þ 0:005Tsat þ

2 7:3x105 Tsat

þ 12x105 ðp0  psea Þ

  time þ 0:013 cos 2p 24 (4.49)

with Tsat being the saturation temperature for that temperature and humidity of the air, time the moment of the day expressed in hours, and p0 and psea are the pressures of the place in question and at sea level, respectively. Thus, the exchange of radiation [W/m2] between the exterior surface of emissivity εes and the sky is given by  4 4  Tsky q_r;hv ¼ εes $s$Fes;sky $ Tes

(4.50)

where Fes,sky represents the configuration factor between the surface and the sky. For its part, the radiation exchange between the surface and the surroundings is obtained through the equation  4  4  Tsur q_r;sur ¼ Aes εes $s$Fes;sur $ Tes

(4.51)

with, in this case, Fes,sur being the configuration factor between the surface and the surroundings. As the exterior surface is usually flat Fes,es ¼ 0, we can see that Fes,sur ¼ 1  Fes,sky. In simulation programs, the exchange of long-wave radiation with the sky and with the surroundings usually appears grouped in a single term, calculated using a fictitious temperature Tf,sky. This temperature is obtained from the temperatures of the sky and the surroundings, weighted according to the respective configuration factors, which is   Tf ;sky ¼ 1  Fes;sky Tsur þ Fes;sky Tsky (4.52) In this way, the heat exchanged by long-wave radiation with the sky and the surroundings, is  4  Tf4;sky q_r;skyþsur ¼ εes s Tes (4.53)

Exergy analysis of heat transfer in buildings

4.8.1.3

295

Equivalent temperature and sun-air temperature

Given these comments on the different mechanisms of energy exchange, we will present in summary the way in which we work with the balance equation. Expression (4.46) can be transformed, in a first step, into an equation of the type q_cd ¼ q_r;sun þ hcvr ðTeq  Tes Þ

(4.54)

with Teq being the equivalent temperature and hcvr a mixed transfer coefficient of convection and radiation. Since both convection and long-wave radiation heat flows are in parallel, a mixed transfer coefficient can be defined, hcvr ¼ hcv þ hr, with hr being a coefficient that satisfies  4  hr ðTes  T0 Þ ¼ εes s Tes  T04

(4.55)

In order to linearize the above expression we use the approximation 4  T 4 y4T 3 ðT  T Þ where T is the arithmetical mean of T and T . The relative Tes 0 m es 0 m es 0 error of this approach, in the usual ranges of temperatures is less than 0.1%, Alvarez [35]. Therefore, the convection-radiation coefficient is hcvr ¼ hcv þ 4sεes Tm3

(4.56)

By carrying out appropriate development, we find that this equivalent temperature is Teq

  εes sT04 Fes;sky εsky þ Fes;sur  1 ¼ T0 þ hcvr

(4.57)

The exchange by short-wave radiation q_r;sun on the exterior surface is produced as a result of the absorption by that surface of a fraction of the incident solar radiation that reaches it. It will be, therefore, a gain of energy for the wall and also, of great consequence in the balance. It can be calculated by the following expression q_r;sun ¼ aes GT

(4.58)

where aes represents the absorptivity for short-wave radiation of the exterior surface of the wall and GT is the solar irradiation [W/m2] that is incident on this surface. Its value will depend on the location (latitude), orientation of the wall and the day and time. As can be seen, direct and diffuse radiation has been considered jointly. According to what was said in Chapter 2, in exergy balances, both types of radiation must be

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considered separately, since the associated exergy is different. According to what we have been saying, the equation of energy balance in a given instant is q_cd;es ¼ aes GT þ hcvr ðTeq  Tes Þ

(4.59)

which can be expressed in the following way q_cd;es ¼ hcvr ðTsa  Tes Þ

(4.60)

where what is known as the sun-air temperature has been introduced Tsa ¼ Teq þ

aes GT hcvr

(4.61)

The heat exchange of a wall on its exterior surface is usually analysed by encompassing the mechanisms of convection and heat exchange by short-wave and long-wave radiation through this concept of sun-air temperature. In this way, all these exchanges can be expressed in a similar way to Newton’s equation, using the convection-radiation coefficient, the temperature of the exterior surface and this sun-air temperature. Given the different thermodynamic quality of the energy exchanged by these different mechanisms, it is evident that it is not possible to apply the concept of sun-air temperature for exergy analysis, as demonstrated by Flores [16] in his doctoral thesis.

4.8.2

Exergy balance

Taking into account the heat fluxes described above, looking at the exergy balance for the exterior surface we have 9 9 8 9 8 8 Exergy associated > > Solar radiation > > Exergy associated > > > > > > = < = > < = > < with the heat flux ¼ with the heat flux  exergyðshort waveÞ > > > > > > > > > > ; : ; > : ; > : absorbed of convection by conduction 9 9 8 8 Radiation exergy > Radiation exergy > > > > > > > = = < < exchanged with þ exchanged with þ > > > > > > > ; : ; > : the surroundings the sky ) ( Exergy destruction þ in the surface (4.62)

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The exergy destruction associated with radiation has two origins: on the one hand, due to the absorption of the short wave coming from the sun and, on the other hand, due to the absorption of the long wave coming from the sky and the surroundings and to emission from its own surface. The previous balance can be written b_cd;es ¼ b_cv;e  aes b_r;sun þ b_r;hv þ b_r;sur þ d_ es

(4.63)

where the exergy flows due to the heat transferred by conduction and by convection are obtained from the expressions seen in Sections 4.3.2 and 4.5.2, respectively, similar to what has been shown for the interior surface. For its part, the exergy of the absorbed solar radiation can be calculated from the equation ( )   1 T0 4 4 T0 _ aes br;sun ¼ aes GT 1 þ  3 Tsun 3 Tsun

(4.64)

where aes is the absorptivity of the surface for short-wave radiation. According to what was stated in Chapter 2, to make the analysis more accurate, we take into account the components of direct and diffuse irradiation, so that the exergy of the solar radiation absorbed will be " b_r;sun ¼ aes GD

"   #   # 4 T0 1 T0 4 4 T0 1 T0 4 1 sin a þ aes Gd 1  þ þ 3 TD 3 TD 3 Td 3 Td (4.65)

The sky emits radiation at a temperature of Tsky which, as we have seen, is significantly lower than the ambient temperature. The exergy of radiation emitted per unit area and time, according to Eq. (2.83), is 2 1 T0 4 4 sTsky 1þ 3 Tsky

!4

3 4 T0 5  3 Tsky

(4.66)

The Earth’s surface receives this radiation exergy, which is a cold exergy, following the terminology of Shukuya [36]. Shukuya obtained values for this exergy, depending on the ambient temperature and the relative humidity of the air. Expressing the above equation in terms of the emissivity of the sky we have   1 3 3 þ εsky  ε4sky sT04 3 4

(4.67)

Finally, the exergy flow due to the exchange of long-wave radiation between the exterior surface and the environment (considered together with the sky and

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surroundings through the fictitious temperature Tf,sky) can be roughly evaluated by the expression 



2



3  T3 Tes f ;sky

3

4 4 5  Tf4;sky 41  T0  b_r;hv þ b_r;sur ¼ εes s Tes 3 4  T4 Tes f ;sky

(4.68)

Now, the heat that leaves the exterior surface of the envelope is finally exchanged with the environment by convection and long-wave radiation until reaching the ambient temperature. Therefore, the exergy flows that leave the exterior surface by convection and radiation are finally destroyed in the environment, meaning it is a lost exergy so that the previous terms of Eq. (4.63) can be grouped together, finally resulting in the equation b_cd;es þ aes b_r;sun ¼ I_e

(4.69)

In winter, the heat flow and exergy flow have the same direction, while in summer the heat flow is inward and exergy flow, on the contrary, outward. However, in both cases, the conclusion is the same, and all the exergy associated with the heat flow on the exterior surface is finally destroyed.

4.8.3

Examples

Example E.4.8. Let there be a facade formed by a layer of solid moulded brick with mortar joints, 11 cm thick and of thermal resistance 0.25 m2K/W and with an internal lime mortar plastering of 2 cm and thermal resistance 0.03 m2K/W. On a summer day when the outdoor air temperature is 28 C, and the indoor temperature is 24 C, the overall irradiation is 750 W/m2, with the absorptivity of the exterior surface for solar radiation being 0.45 and its emissivity 0.9. The temperature of the exterior surface is 40 C and that of the surrounding area is 32 C. Using the values of the convectionradiation coefficient of the Spanish BTC for exterior and interior surfaces, determine the following per m2 of facade:

(a) (b) (c) (d)

The heat flux by conduction and sun-air temperature. The interior surface temperature and intermediate temperature between the two layers. The exergy flows in the exterior surface and through the interlayer. The rate of exergy destroyed in the interior of the facade.

Solution (a) Carrying out an energy balance on the exterior surface of the facade, in accordance with Eq. (4.59), we have q_cd;es ¼ aes GT þ hcvr;e ðTeq  Tes Þ z aes GT þ hcvr;e ðT0  Tes Þ

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We use the value of the convection-radiation coefficient of the BTC, according to which, the value to be taken for an exterior vertical surface is hcvr,e ¼ 25 W/m2K. As we will use it later in the exergy calculations, from this data we now calculate the pure convection coefficient. Taking into account that hcvr,e ¼ hcv,e þ hr,e, and that hr;e ¼ 4sεes Tm3 where Tm¼(Tes þ Tsur)/2, we have hr;e ¼ 4sεes Tm4 ¼ 4$5:67$108 $0:9

  313 þ 305 3 W ¼6 2 2 m K

so the convection coefficient is hcv,e ¼ 19 W/m2K. Returning to the energy balance equation, the heat flux by conduction gives q_cd;es ¼ q_cd ¼ 37:5

W m2

The sun-air temperature is Tsa ¼ Teq þ

aes GT aes GT z T0 þ ¼ 41:5 C hcvr;e hcvr;e

(b) The heat of conduction can be worked out using this calculated temperature, since according to Eq. (4.60) we have q_cd;es ¼ q_cd ¼ 25ð41:5  40Þ ¼ 37:5

W m2

As according to the BTC, the interior surface resistance in a vertical enclosure is Ris ¼ 0.13 m2/W, that is, hcvr,is ¼ 7.69 W/m2, the inner surface temperature is q_cd ¼ hcvr;i ðTis  Ti Þ / Tis ¼ 28:9 C while the temperature of the intermediate layer is Tin ¼ Tis þ q_cd Rin ¼ 30:0 C (c) The exergy associated with the heat flow exchanged by convection is     T0 T0 W 1 q_cv;e ¼ 1  hcv;e ðTes  T0 Þ ¼ 8:7 2 m Tes Tes The exergy flow associated with the long-wave radiation exchanged with the sky and surroundings is calculated from the fictitious temperature Tf,sky. Being a vertical facade, we can consider that Fes,sky ¼ 0 and therefore Tf,sky ¼ Tsur ¼ 32 C. Hence, the heat exchanged by long-wave radiation is   4 4 q_r;ðhvþsurÞ ¼ εes s Tes ¼ 48:2 W  Tsur

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 3   3  _br;ðskyþsurÞ ¼ q_r;ðskyþsurÞ 1  4T0 Tes  Tsur  ¼ 1:3 W 4  T4 3 Tes m2 sur The exergy of the solar radiation that is absorbed by the exterior surface is ( )  4 4 T0 W _br;sun ¼ aes GT 1 þ 1 T0  ¼ 314 2 3 Tsun 3 Tsun m We have considered that Tsun ¼ 5780 K and the irradiation has not been broken down into its direct and diffuse components. Finally, the exergy flow of conduction on the exterior surface is   T0 W 1 q_ ¼ 1:4 2 m Tes cd;es while the exergy flow associated with conduction in the interlayer is   T0 W q_ ¼ 0:24 W 2 1 m Tin cd (d) The rate of exergy destroyed on the interior of the façade per unit of surface is Tes  Tis W ¼ 1:3 2 d_ cd ¼ T0 q_cd m Tes Tis Example E.4.9.

To calculate the temperature of the sky in cloudless skies, we can use Swinbank’s equation, which states that Tsky ¼ 0:0552T01:5 . Determine: (a) The temperature of a hot black surface that emits the same exergy of radiation as the sky, on a day when T0 ¼ 300 K. (b) And if the ambient temperature is T0 ¼ 250 K.

Solution (a) We calculate the temperature of the sky Tsky ¼ 0:0552T01:5 ¼ 287 K If the exergy of radiation emitted by the sky is the same as that emitted by a black surface of temperature T, then 4 3 3T 4 þ T04  4T0 T 3 ¼ 3Tsky þ T04  4T0 Tsky

and hence T ¼ 310 K

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301

Therefore, the ‘cold’ exergy emitted by the sky is the same as that emitted by a hot black surface at a temperature of 310 K. (b) The temperature of the sky is Tsky ¼ 0:0552T01:5 ¼ 218 K 3T 4  4: 250$T 3 ¼ 3: 2184  4: 250$2183

/ T ¼ 280 K

We see that the lower the ambient temperature, the lower is the temperature of the hot surface that emits the same radiation exergy as the sky. Example E.4.10.

The roof of an industrial building consists of a rough concrete slab with dimensions of 10  25 m and 16 cm of thickness. On a winter night, the wind speed is 40 km/h, the ambient air temperature is 2 C, and the temperature of the sky is 23 C. The exterior surface of the roof is at 0 C, while the temperature of the indoor air and internal parititons is 10  C. Determine: (1) The heat transfer through the roof. (2) The flow of exergy associated with that heat flux. (3) The rate of exergy destruction on the exterior surface of the roof.

Solution (1) We shall take an emissivity of 0.94 for the rough concrete. For a wind speed of 40 km/h, the net convection coefficient according to Burberry for a horizontal surface and upward flow is hcv,e ¼ 50.3 W/m2K. According to these values, the heat flux exchanged by the exterior surface of the roof is  4 4  Tsky ¼ 47; 128 W Q_ es ¼ Q_ cv;e þ Q_ r;e ¼ Ahcv;e ðTes  T0 Þ þ Aεes s Tes It is the heat flux lost from the surface of the roof (2) The flow of exergy associated with that heat flux is   T0 _ 1 Qcv;e þ Tes

!   3 3 4 Tes  Tsky _ 271 25; 165 Q 1  T0 4 ¼ 1  r;e 4 3 Tes  Tsky 273   4 2733  2503 þ 1  271 21; 963 2734  2504 3 ¼ 584 W

The heat flow associated with convection involves an exergy flow that leaves the surface. With regards to radiation, although the emission of radiation is greater than the absorption of radiation from the sky, due to the low temperature of the sky, the exergy

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associated with the radiation emitted by the sky and absorbed on the surface is greater than the exergy emitted by the surface itself and is associated with heat by convection; hence the negative sign. In effect, the quality factor for the radiation emitted by the

 4

surface is 1 þ 1 3 T04 Tes  4 3ðT0 =Tes Þ ¼ 4:4$104 while for the radiation from the . .  . 4  4 3ðT0 =Tsky Þ ¼ 0:021. sky it is 1 þ 1 3 T04 Tsky (3) All the exergy of the heat flux that reaches the exterior surface by conduction is finally destroyed in the surface itself, in the boundary layer and external environment, so that we have D_ ¼

  T0 _ Qes ¼ 345 W 1 Tes

A surface of 1.3 m2 that is at a temperature of 24 C, with an ambient temperature of 17 C, has an absorptivity of 0.9 for solar radiation and an emissivity of 0.6. It is observed that when the direct and diffuse components of solar radiation are 380 and 470 W/m2, respectively, with direct radiation having an incidence angle of 30 degrees, the surface temperature is 320 K. If the sky temperature is 280 K, determine:

Example E.4.11.

(a) The net heat transfer by radiation to the surface at that moment. (b) The exergy associated with the radiation exchanged.

Solution (a) The absorbed solar radiation is Q_ r;sun ¼ AaðGD cos q þ Gd Þ ¼ 1:3$ 0:9ð380 cos 30 þ 470Þ ¼ 618:5 W

while the heat exchanged by radiation with the sky is  4 4 Q_ r;hv ¼ Aεs Tsky  Tes ¼ 192 W so that the net heat transfer by radiation is  4  Ts4 ¼ 426:5 W Q_ r ¼ AaðGD cos q þ Gd Þ þ Aεs Tsky The exergy associated with the absorbed solar radiation is " B_ r;sun ¼ aGD

"   #   # 4 T0 1 T0 4 4 T0 1 T0 4 þ þ 1 cos q þ aGd 1  3 TD 3 TD 3 Td 3 Td

¼ 335 W

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303

while the exergy of the radiation exchanged with the sky is  3 2 3 Ts3  Tsky 4 5 ¼ 185 W B_ r;hv ¼ Q_ r;hv 41  T0  3 4 Ts4  Tsky Although the emitted radiation to the sky is greater than the absorbed radiation from the sky, due to the exergy of the radiation from the sky being greater than that of the emitted radiation the exergy of the exchanged radiation is positive.

4.9

Exergy exchanged by a building through an opaque envelope

In the previous sections, we obtained expressions to calculate the exergy associated with heat fluxes of conduction, convection and thermal radiation. Likewise, we have highlighted the exergy destruction that takes place in these three mechanisms of exergy transport, undertaking the corresponding exergy balances on the interior and exterior surfaces of a building. The exergy flows are very sensitive to the variations experienced by the RE when the temperatures of these flows and those of the RE do not differ much from each other, as is the case of buildings. This fact must be borne in mind when introducing simplifications in the evaluation of the exergy behaviour of the envelope. There are different methods in the application of exergy analysis to opaque enclosures, with a variable degree of complexity and detail. Below, we present a summary of the different methods, from lower to higher complexity and precision.

4.9.1

Steady-state method

It is the simplest method. The heat transmitted through an envelope is calculated from its thermal transmittance U, by means of the expression Q_ ¼ UAðTi  T0 Þ

(4.70)

where Ti is the indoor air temperature and A the surface of the envelope, C¸engel and Ghajar [15]. For the sizing up of air conditioning equipment, we usually assume that indoor air temperature Ti and exterior temperature T0 are constants, with their values being those corresponding to the design conditions (generally based on the hottest or coldest temperature of the year for that location). In some cases, especially when sizing up equipment whose performance or COP is greatly influenced by the outside temperature, the analysis is usually carried out considering the corresponding average outside temperature for each month, Angelotti and Caputo [37].

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Exergy Analysis and Thermoeconomics of Buildings

Once the heat transmitted by the opaque envelope has been determined, the exergy associated with this heat is calculated as the sum of the heats transmitted through the different surfaces of the envelope, in accordance with the expression B_ Q ¼

 1

 en T0 X Q_ Ti j¼1 j

(4.71)

Even if a more detailed analysis were carried out, including considering the 8760 hourly values of outside temperature of the climatic year of the locality, it would still be a steady-state method, since the calculation of the energy losses through the envelope using the thermal transmittances does not take into account the thermal inertia of the walls. This method is, therefore, the first approximation and is not valid for a calculation of energy demands or exergy with a certain precision and it is basically for two reasons: • •

Performing a steady-state analysis does not permit taking into account the effect of the inertia of the envelope since by definition it cancels the term for energy variation (exergy) of the envelope in the corresponding balance. As discussed above, exergy flows are more sensitive to changes in the RE when the properties of the system are closer to those of the RE. This circumstance leads to the fact that steadystate analysis, common in studies of power plants or industrial facilities, are not valid for the exergy analysis of buildings in general and of facades or roofs in particular.

4.9.2

Quasi-steady method

It is an intermediate method between the steady-state and the dynamic method. In this method, the energy flows are calculated dynamically, while the exergy flows are evaluated by a steady-state approach during each time step of the simulation, that is, avoiding the possible storage phenomena. Together with the steady-state method, it is one of the methods proposed and used by legislation and regulations for the calculation of energy demand in buildings in many European countries, AENOR [38]. Being a dynamic method in the calculation of heat fluxes, spatial and temporal discretization of the problem is required. Thus, the envelope is represented by a set of nodes j equi-spaced at a distance Dx, forming an RC circuit. The resistance between two consecutive nodes is equal to the thermal resistance to the existing conduction between both. On the other hand, each node concentrates the heat capacity corresponding to the volume element associated with said node. If it is an interior node this element will have a thickness Dx, whereas if it is an exterior (surface) node, it will have a thickness Dx=2.

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305

Figure 4.10 Discretization of the envelope.

Fig. 4.10 represents a wall through a series of nodes connected to each other by a series of thermal resistances with there being in each node a capacitor representing the thermal capacity of the associated layer. Following the symbology used by Shukuya [36] the internal nodes are represented by circles in black, while the nodes representing the interior and exterior surfaces appear as white circles. These nodes are in turn connected to the interior and exterior air and the surfaces with which they exchange radiation, which are represented by white squares. In this way, the differential equation of heat transfer by unidimensional conduction in the non-steady state without heat sources or sinks is   l v2 T vT (4.72) ¼ rc vx2 vt which becomes the equation in finite differences   Tj;nþ1  Tj;n l Tj1;nþ1  2Tj;nþ1 þ Tjþ1;nþ1 ¼ 2 9$c Dx Dt

(4.73)

whose solution in the node j in that moment of time n is Tj,n. When expressing the spatial derivative in finite differences, there are, as we know, two options. Since the nodal temperatures, in general, vary during each time interval, the temperatures in the previous time interval or in the new time interval can be used, as in Eq. (4.73). The first option constitutes what is known as the explicit method and the second is the so-called implicit method.

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In the explicit method, the set of equations for each node is independent of each other, so the numerical resolution is very simple. As a disadvantage, the method imposes limits on the maximum allowed values of distances between nodes ðDxÞ and time interval ðDtÞ, in order to be stable (adimensional Fourier number < 0.5). On the other hand, the implicit method supposes a greater computational effort, since it requires resolving all the nodal temperatures simultaneously for each instance, but it is intrinsically stable regardless of the number of nodes and the time interval Dt chosen, C¸engel and Ghajar [15]. Considering that in many cases the behaviour of the envelope will be simulated for long periods of time, even a whole year, the implicit method for resolution is recommended, as it does not need extremely small time steps. Eq. (4.73), properly reordered, gives for a generic node j the following  kTj1;nþ1 þ ð1 þ 2kÞTj;nþ1  kTjþ1;nþ1 ¼ Tj;n

(4.74)

where k is the adimensional Fourier number, that is k¼

Dt l Dx2 rc

(4.75)

If the wall is divided into M intervals of width Dx, there will exist Mþ1 nodes, see Fig. 4.10. Of all of them, in the extreme nodes (nodes 1 and Mþ1) the boundary conditions are applied and their temperatures are known at all times, as they will have been previously obtained in TRNSYS. Therefore, Eq. (4.73) only needs to be applied in the instant n þ 1 to the M  1 internal nodes. Assuming that all the temperatures in the instant n are known (they have been calculated previously or are initial conditions) and that also temperatures T1,nþ1 and TMþ1,nþ1 are known (they are boundary conditions), the system of M  1 equations with M  1 unknowns is 8 ð1 þ 2kÞT2;nþ1  kT3;nþ1 ¼ T2;n þ kT1;nþ1 > > > > > > > kT2;nþ1 þ ð1 þ 2kÞT3;nþ1  kT4;nþ1 ¼ T3;n > > > > > > < kT3;nþ1 þ ð1 þ 2kÞT4;nþ1  kT5;nþ1 ¼ T4;n (4.76) > > / > > > > > > kTM2;nþ1 þ ð1 þ 2kÞTM1;nþ1  kTM;nþ1 ¼ TM1;n > > > > > : kTM1;nþ1 þ ð1 þ 2kÞTM;nþ1 ¼ TMþ1;n

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307

and in matrix form

(4.77) This system written compactly is A$T nþ1 ¼ T n

(4.78)

where the matrix of coefficients A depends solely on the properties of the wall (l, r, c), of the nodal distance Dx and the chosen time step Dt. Once these parameters are set, the matrix is constant throughout the simulation, so the vector of new temperatures Tnþ1 can be easily obtained for any time as T nþ1 ¼ A1 T n

(4.79)

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Once the energy balance is done, we will know the heat flux in successive instants and, of course, the surface temperatures Tis(tk) and Tes(tk). Performing an exergy balance for the total envelope in the instant tk we have _ kÞ Dbðtk Þ ¼ b_q;is ðtk Þ  b_q;es ðtk Þ  dðt

(4.80)

In this method called quasi-steady-state, the term for the variation of the exergy _ k Þ. accumulated in the envelope Dbðtk Þ is grouped with the exergy destruction dðt This then is a term, that following Annex 49 [39] is called exergy consumed, b_cons . Therefore _ k Þ ¼ b_q;is ðtk Þ  b_q;es ðtk Þ b_cons ðtk Þ ¼ Dbðtk Þ þ dðt

(4.81)

Consequently, the exergy consumed in the interval tk is     _bcons ðtk Þ ¼ q_is ðtk Þ 1  T0 ðtk Þ  q_es ðtk Þ 1  T0 ðtk Þ Tis ðtk Þ Tes ðtk Þ

(4.82)

This type of approach is based on a valid approximation only in those systems that do not have a significant storage capacity. However, in many cases, it is not an appropriate method. Taking advantage of the matrix expression of the problem and in order to automate both the calculation of the inverse of the matrix A as well as the resolution of the system of equations for any situation, a code can be implemented in MATLAB. The composition of the envelope (number of layers and thickness, thermal conductivity, density and specific heat of each layer), ambient temperature and interior and exterior surface temperatures are used as input data. These last two are precisely the boundary conditions at the extreme nodes (1 and M þ 1) that allow the resolution of the system of Eq. (4.78). The values of these extreme nodes for each instant of time are obtained by linear interpolation from the time values previously calculated with TRNSYS. For an easier treatment of the results, taking into account that the time step chosen may be, for example, 1 min, the energy and exergy flow values obtained at each instant are subsequently accumulated in hourly values. Finally, it is worth mentioning that, for avoiding possible errors due to the initialization of the internal temperatures in the wall, the calculations can be made with the data corresponding to two full years, taking the values of the second year, as results. The selection of such a large past period is not really necessary, since the values used as initialization of the problem affect, in the worst case, the first 240 h. Proposing a previous period of 1 year is for the simplicity of programming, with the additional computational effort not being excessive. For a laptop with an Intel Core Duo processor at 2.53 GHz and 4 GB of RAM, the simulation time in MATLAB for each case is a few minutes.

Exergy analysis of heat transfer in buildings

4.9.3

309

Simplified dynamic method

Once the energy flows are calculated by the method described in the previous Section 4.10.2, the simplified dynamic method separately considers the exergy stored in the enclosure and the exergy destructions, Torio and Schmidt [39]. Therefore, the consumed exergy includes only the inevitable irreversibilities associated with the temperature difference necessary for heat transfer. The exergy stored in the time interval between tk and tk-1 is " # N X Tj ðtk Þ Dbðtk Þ ¼ rj cj lj Tj ðtk Þ  Tj ðtk1 Þ  T0 ln (4.83) Tj ðtk1 Þ j¼1 From the exergy balance, Eq. (4.9) and taking into account the previous expression, we have that the rate of exergy destroyed in the envelope in the time interval tk is     _ k Þ ¼ q_is ðtk Þ 1  T0 ðtk Þ  q_es ðtk Þ 1  T0 ðtk Þ dðt Tis ðtk Þ Tes ðtk Þ " # (4.84) N X Tj ðtk Þ rj cj lj Tj ðtk Þ  Tj ðtk1 Þ  T0 ðtk Þln  Tj ðtk1 Þ j¼1 As can be observed from the previous expressions, for the evaluation of the stored exergy and the destroyed exergy it is necessary to know the interior temperatures of the wall, at least in each layer (subscript j) and each moment. This necessity represents a problem in many cases. In fact, most of the energy simulation programs for buildings enable knowledge of the energy stored in the wall through the application of the corresponding energy balance. On the other hand, it is not possible to directly obtain the internal temperatures of the wall since it is very common to use the method of transfer functions (CTF), developed by Stephenson and Mitalas [40] for the calculation of the transient heat transfer through it. This is the case for software as popular as EnergyPlus [41] or TRNSYS [42], Klein [43]. An approximate way to solve this problem, proposed by the working group of Annex 49 [39] and Torio [44], has been to replace the real wall with an equivalent homogeneous wall and to approximate its average temperature at every moment Tm(tk) to the average value between the interior and exterior surface temperature, Tis(tk) and Tes(tk), at said moment, that is, Tm(tk) ¼ (Tis(tk)þTes(tk))/2.

4.9.4

Detailed dynamic method

The problem presented by the approximation of the simplified dynamic method described in the preceding paragraphs is that it does not provide the values of internal temperatures of the wall, or at best, approximates them to a linear variation between the two surface temperatures. When calculating the average temperature of the wall in a

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Figure 4.11 (A) Profile of possible temperatures not considered by the simplified dynamic method. (B) Values of the actual average temperature, Tm,dd (black) and the average temperature according to the simplified dynamic method, Tm,sd (blue [grey in print version]).

linear way, this approach can avoid possible maximums and/or minimums of temperature that can occur inside the envelope at certain times of the day. These unconsidered maximums or minimums give rise to an inversion in the direction of the heat fluxes with respect to those obtained with the approximation, see Fig. 4.11a. It could even happen that the actual average temperature of the wall Tm,dd would be greater than the highest of the surface temperatures, Tis or Tes, a situation which is impossible according to the simplified dynamic method hypothesis, see Fig. 4.11b. As for the calculation of exergy balances, it is required to know the accurate internal temperatures of the wall; in the Flores thesis [16], a methodology was proposed that allows them to be calculated, both in homogeneous walls and in those formed by different layers of materials. The calculations are made with MATLAB using a code based on the finite difference method. This method allows the equation of the exergy balance to be solved more precisely, Eq. (4.84), and constitutes what we call the detailed dynamic method. It involves calculating the interior temperatures of the wall, using the surface temperatures obtained with dynamic simulation software, such as TRNSYS, as boundary conditions. The method consists of approximating the differential equation of heat transfer by conduction in the dynamic state to a finite difference equation, obtained from the truncation of the Taylor series, as explained in Section 4.10.2. As Flores explains in his doctoral thesis, this method has been compared with the different existing methods, for a series of cases with different climates, inertias and transmittances. Although in some cases there are hardly any differences, it is in the range of inertias commonly used in construction where the greatest discrepancies in the results are found. For this reason, and given the low computational cost involved, it is advisable to use the detailed dynamic method for the exergy analysis of envelopes.

Exergy analysis of heat transfer in buildings

4.10

311

Indicator of exergy behaviour of a wall

A drawback that characterizes the exergy behaviour of systems is that there is a great variety of coefficients or indicators that reflect the efficiency of their behaviour, as we have already seen in Chapter 1. This drawback arises precisely because of the versatility of exergy analysis, as it can be applied to different disciplines and areas of life. This fact has led to a certain lack of standardization, and the choice of one or the other indicator is often open to individual interpretation, Marmolejo-Correa and Gundersen [45]. Despite this great diversity, there is no an exergy index that can be properly applied to the characterization of facades or roofs. Indeed, the most common indicators such as the universal exergy efficiency, used by authors such as Boelman and Sakulpipatsin [46], Cornelissen and Hirs [47], Torio et al. [23], or the functional efficiency used by authors, such as, Kotas [48] and Tsatsaronis [49], are originally intended for application in industrial processes or power generation facilities. Even when applied to the building sector, as in Favrat et al. [50] or Gonçalves et al. [51] they are not used to evaluate the behaviour of envelopes. The only parameter related a priori to the exergy behaviour of the building envelope found in the bibliography is that proposed by Tronchin and Fabbri [52]. But in the calculation of this parameter, the authors do not take into account the dynamic behaviour of the envelope and do not distinguish between the mechanisms of convection and radiation. This fact and its lack of physical meaning have resulted in it not being used. However, it is necessary to define a parameter that characterizes the exergy behaviour of building envelopes. In order for this parameter to meet the required needs, we must consider the behaviour of the envelope as a dynamic system and, in addition, for a greater ease of application, it would be of value if this parameter did not involve tedious operations, was easy to interpret and did not involve a radical change with respect to what already exists. In his doctoral thesis, Flores [16] defined five different possible parameters. Once the results were analysed for the different walls, climates, etc. he proposed the one that is of most interest from a theoretical and practical point of view. As a starting point, taking the expression proposed by the ISO 9869-1 standard [53], for the ‘in situ’ determination of the thermal resistance of a wall, he defines a dynamic exergy transmittance according to the expression PN ex Udyn

_

j¼1 bq;is;j

¼ PN

j¼1 DTj

! T0;j j¼1 q_is;j 1  Tis;j  PN  j¼1 Ti;j  T0;j

PN ¼

(4.85)

Among the advantages of this proposed parameter are: •

The possibility of being measured ‘in situ’: this parameter can be measured with the same equipment with which the value of the thermal resistance of a wall is determined according to the ISO 9869-1 standard. By means of three temperature probes (air outside, indoor air and

312

•

• • •

Exergy Analysis and Thermoeconomics of Buildings

interior surface) and a heat flux-meter on the interior surface all the terms that appear in Eq. (4.85) can be determined. The difference with respect to the measurement of the thermal transmittance is that, in this case, it is necessary to measure for a whole year, or at least for the heating (or cooling) season, due to the dependence of exergy on the climatic conditions. The other limitations for its experimental determination are the same as in the case of thermal transmittance. It is an alternative to testing: the value of this index can be obtained by simulation of the building. As the variables used (heat flow and temperatures) are the usual output variables in energy simulation programs, it does not require important modifications or the development of complicated codes. Similarity to thermal transmittance: being a similar concept and formula to the thermal transmittance, its use by technicians, or even by administrations, as a control parameter in regulations, would be easy to introduce. ex Ease of interpretation: when comparing two envelopes, the one with the lowest value of Udyn would be the best from an exergy point of view. It considers the dynamic state of the wall: when calculated from the values of heat fluxes and temperatures, it implicitly takes into account both its resistance and its thermal capacity.

A priori someone could raise as a possible drawback that the dynamic exergy transmittance of the same facade gives different values for different climates. This circumstance is unavoidable and inherent in any parameter in which exergy operates, given its dependence on the ambient temperature. However, this drawback could be avoided, if the regulations establish limiting values depending on the locality, in a similar way to what was done in the BTC of 2006 with thermal transmittance, or with energy consumption and demand in the latest version of 2013. The latter proposal could at the same time serve to limit, albeit partially, a building’s energy demand. In fact, once internal temperature levels are set, for example, depending on the outside temperature as stated in the EN 15,026 AENOR 2007 standard [54], the denominator of Eq. (4.85) is constant for a given location. On the other hand, the level of insulation that is currently required and considerations of comfort for the user means that the value of the interior surface temperature moves by very limited values and is close to the indoor air temperature. In this way, establishing a limiting value for ex for a locality, is implicitly limiting the energy demand (exergy) of a building due Udyn to losses through the envelope.

4.10.1

Examples

Example E.4.12.

We want to know the effect of the retrofitting of a façade by an SATE (External Thermal Insulation System). For this, the façade was tested, before and after its renovation, in order to calculate the steady-state and dynamic thermal transmittances, as well as the dynamic exergy transmittance before and after placing the SATE. It consists of a vertical double-layer façade made up, from outside to inside, by 2 cm of mortar plastering, perforated solid brick of 11.5 cm thickness with continuous horizontal and vertical mortar joints, 1 cm of mortar plastering, an air chamber 5 cm wide, single hollow brick 4 cm thick with continuous horizontal and vertical mortar joints, covered with 2 cm of thick plaster and 1 mm thin layer of gypsum plaster.

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313

Solution The façade was tested in a PASLINK test chamber following the procedures described in the document Van Dick, H.A.L. and Van Der Linden, G.P. PASLINK Calibration and component test procedures, TNO, Delf, 1995 developed by the European network PASLINK EEIG. The dimensions of the sample were 2.7  2.7 m (surface area of the sample 7.29 m2) and it was built on an insulating pre-frame, as can be seen in Fig. E.4.1.

Figure E.4.1 Sample of the enclosure and its pre-frame prepared for the test.

Next, a sample of the façade renovated by the SATE was prepared and its dynamic behaviour was tested to calculate its dynamic transmittance and dynamic exergy transmittance. The SATE system consists of rock wool panels 5 cm thick placed on a metal substructure, a 5 cm air chamber, a water barrier film which is permeable to vapour and plates fixed on the outside of the structure. Once the plates were fixed and dried, the reinforcement and levelling plaster was applied with a mortar base incorporating a fibreglass mesh and a stone base plus a finish with acrylic plaster. Fig. E.4.2 shows the façade with the SATE already placed in the test cell.

Figure E.4.2 Sample of the façade with SATE placed in the test chamber.

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Exergy Analysis and Thermoeconomics of Buildings

Figure E.4.3 Effect of SATE on the lag and attenuation of heat wave.

In addition to the instrumentation required by the PASLINK test itself, to characterize the sample, twenty-one PT100 sensors were placed in the different layers of the sample, three Type T thermocouples, five heat flux sensors for contrast and three relative humidity sensors. Fig. E.4.3 represents the difference of temperature between the interior and exterior environment, as well as the heat flux through the façade per unit area, for a day of January in the city of Vitoria-Gasteiz, which is chosen as representative of climatic area D1. The effect of SATE is clearly seen, both in the lag of the heat wave and in its damping. The objective of the tests carried out was to have the necessary data to be able to construct a model of the initial wall and the retrofitted one, which would allow us to know the thermal behaviour for any climatic condition. In this way, the heat gains and losses through the walls can be quantified for any climate and the value of renovation with the SATE can, therefore, be evaluated; in short, the thermal model allows us to understand the thermal behaviour of the façade whatever be the climatic conditions. The thermal model constructed was of the distributed parameter type, that is, an RC model. The heat fluxes on the interior and exterior surfaces of the single hollow brick and solid perforated brick were used as objective functions. The identification system used was based on Monte Carlo and the downhill method, with the idea of finding the vector of resistance and thermal capacities that minimizes the error between the objective function and the equivalent function obtained through the model. The resolution of the equations for the calculation of the resistances and thermal capacities as a function of time was done by the application of the LORD 3.21 software. Both the tools and the calculation procedure were developed by the PASLINK network. In Fig. E.4.4 the simplified scheme of the RC model is shown, both for the non-renovated façade and for the one renovated with the SATE.

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315

Figure E.4.4 RC model of the façade.

Once the models were obtained, the thermal transmittance of both walls was calculated, according to the equation U¼

1 Ris þ Rm þ Res

where Ris and Res are the normalized surface resistances, that is, the values indicated by the TBC in the DB-HE-1 Document and Rm is the sum of the thermal resistances of the different layers of the wall. In a dynamic system, the effect of the capacity of the wall to accumulate heat and the interaction with solar radiation implies that the thermal load associated with the façade differs from that calculated by the use of thermal transmittance. One way of evaluating the effect of the said capacity is using a coefficient that represents the average behaviour, throughout the day, of the thermal gain with respect to the temperature differences between interior and exterior environment. We call this coefficient dynamic transmittance, Udyn and it is obtained by the following expression PN

j¼1 q_is;j

Udyn ¼ PN  j¼1

Ti;j  T0;j



The dynamic exergy transmittance is a sophistication of the previous expression since, instead of considering the heat flux in the inner surface of the envelope, the exergy associated with that heat flux is taken into account. As we have seen, it is calculated from Eq. (4.85).

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In Table E.4.2 the values of the thermal gains and losses are shown per m2 of façade throughout the year for the city of Vitoria-Gasteiz, climatic zone D1, as well as the monthly values of the dynamic thermal transmittance Udyn for the wall before and after being renovated. Fig. E.4.5 shows the values of thermal transmittance U (in steady state) and dynamic transmittance for the tested wall, before and after the renovation, for each month of the year. One can clearly see, that unlike the steady-state U, the dynamic transmittance Udyn varies over the months, increasing significantly in the summer months for the base case. In Fig. E.4.5 the effect of the renovation with the SATE is also recognized, since both the steady-state U as well as the Udyn decrease in a significant way.

Figure E.4.5 Values of U and Udyn of the two façades in Vitoria.

Lastly, in Table E.4.3 and Fig. E.4.6 the values obtained for the dynamic exergy transmittance are shown. Comparing the values with those of the dynamic transmittance we see that the behaviour is qualitatively similar, but naturally, its values are much lower, since now it is the exergy values that appear in the numerator, and these are much smaller than those of the energy.

4.11

Exergy and thermal comfort

The purpose of air conditioning and ventilation is the attainment of thermal comfort conditions; therefore, the definition of a suitable comfort standard is the basis of its design. Given the importance that the method of exergy analysis can have in its application to buildings, it is very important to understand the exergy balance of the human body, in order to appreciate how the heating and cooling demands can be supplied with the highest efficiency, guaranteeing comfortable conditions at all times. We present first a brief summary of the thermal comfort standards, in order to then give an introduction to the application of exergy in this context.

BASE Thermal losses Udyn(kWh/m2 month) 2

Thermal gains (kWh/m month) 2

U Dynamic (W/m K)

Jan

Feb

Mar

Apr

May

Jun

Jul

Aug

Sep

Oct

Nov

Dec

Total

11.22

9.08

7.97

7.22

5.54

4.07

2.99

2.77

3.1

4.61

8.13

11.05

77.75

0.38

0.43

0.74

0.68

0.82

1.02

1.97

2.46

2.45

1.6

0.28

0.26

13.09

1.03

0.97

0.91

0.95

0.95

0.95

1.09

1.19

1.11

0.97

0.94

1.03

4.37

3.59

3.26

2.25

2.25

1.58

1.02

0.83

0.9

1.69

3.21

4.3

29.25

0

0

0

0

0

0.01

0.1

0.07

0.13

0.01

0

0

0.32

0.38

0.36

0.33

0.33

0.33

0.29

0.29

0.23

0.22

0.26

0.35

0.3

Exergy analysis of heat transfer in buildings

Table E.4.2 Gains, losses and dynamic transmittances of the two walls for Vitoria.

BASE D STATE Thermal losses Udyn(kWh/m2 month) 2

Thermal gains (kWh/m month) 2

U Dynamic (W/m K)

317

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Exergy Analysis and Thermoeconomics of Buildings

Figure E.4.6 Dynamic exergy transmittance of the two façades for Vitoria-Gasteiz.

4.11.1

Thermal comfort standards

Currently, the most important thermal comfort standards are the UNE-EN-ISO 7730 (European) [55] and ASHRAE-55 (American) [56] standards. Both standards are based on the assumption that human beings are thermo-regulated machines, which must maintain body temperature while exchanging mass and energy with the environment. Through these balances, we can analyse the influence of the physical parameters of the environment, such as temperature, relative humidity, airspeed and average radiant temperature in relation to the thermal insulation of a person (CLO) and the metabolic activity of an individual (MET). First of all, carrying out a water balance and applying the First Law of Thermodynamics, we obtain a result that determines the load/discharge of energy that the human body experiences in relation to its own thermoregulation mechanisms and which are activated in different modes according to whether the environment is cold, hot or temperate. This energy balance creates a scale of sensations, which later, according to a purely statistical criterion, determines the degrees of dissatisfaction. There are also local discomfort criteria in the cited standards, which qualify the thermal sensation of people and that must be taken into account when designing HVAC systems. These criteria of local discomfort determine the dissatisfaction created by exchanges of heat localized in parts of the human body, which activate the body’s defences, regardless of whether the overall balance is of comfort. This may include possible uncomfortable air currents, asymmetries of radiant temperature with vertical gradients of temperature, situations producing cold feet, etc. In relation to these criteria, there are certain differences between the European and American standards, though not concerning conceptual issues, but that basically the American standard sets ‘comfortable’ as being slightly colder than the European standard. On the other hand, both standards introduce the possibility that human beings have certain inertia in their thermal sensations so that they can assume situations to be comfortable that are not actually in the comfort zone during a period of time.

Exergy analysis of heat transfer in buildings

Table E.4.3 Values of the dynamic exergy transmittance of the two façades for Vitoria-Gasteiz BASE

Jan

Feb

Mar

Apr

May

Jun

Jul

Aug

Sep

Oct

Nov

Dec

Uex dynamic (mW/m2K)

46.0

40.1

25.1

29.2

22.7

16.1

18.5

20.1

18.9

23.1

38.7

42.6

16.9

14.8

9.0

10.1

7.8

4.9

4.9

3.9

3.7

6.2

14.4

16.1

BASE D SATE Uex dynamic (mW/m2K)

319

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Exergy Analysis and Thermoeconomics of Buildings

This circumstance is lightly treated in both standards, and it can be seen that there is a deficit of research on this aspect. The ASHRAE 55 standard even differentiates between cyclic thermal modifications and deviations and ramps, setting temporary acceptance limits for cyclical situations. This standard mentions the concept of adaptive comfort and presents it as an optional method to determine comfort in buildings without air conditioning, in which people have freedom in clothing and have access to the opening of windows and natural means of ventilation. This concept also appears in the European standard UNE-EN 15521, in its annex A, and is defined as informative. This criterion considers variable comfort values depending on the external conditions. The adaptive approach does not yet figure in the aspects that determine the design of buildings and their concepts relative to comfort. However, it is a valuable contribution, which comes from exhaustive statistical studies developed by different authors, among which we should mention Humphreys and Nicol [57]. In short, there are three criteria for thermal comfort, which we could call: static, transient and adaptive. The European standard UNE-EN-ISO 7730 is developed based on Fanger’s postulates [58], through the PMV (Predicted Mean Vote) and PPD (Predicted Percentage of Dissatisfied) indices. The PMV index is applied to humans exposed to constant environmental conditions during a long period in which an invariable metabolic rate is maintained. For its evaluation, the energy conservation equation is used. The resolution of the equation, which requires certain iterative processes, leads to a PMV value that depends on the following parameters PMV ¼ f ðT0 ; pv ; Tmr ; vra ; MET; CLOÞ

(4.86)

where T0 represents the dry bulb temperature of the environment, pv the vapour pressure of the air, Tmr the mean radiant temperature, vra is the residual velocity of the air, MET the individual’s metabolic rate, and CLO an index that represents the insulation of the clothing. The value of PMV obtained from the above equation from the physical parameters mentioned can take values on a scale ranging from ‘e3’ very cold to ‘þ3’ very hot. The range of intermediate values expresses the sensation of comfort, which can be related to a more intuitive interpretation value called PPD (Percentage of Persons Dissatisfied) by means of an equation. The ASHRAE 55 standard uses the ET-DISC (ET Effective Temperature, DISC Discomfort) model. The DISC value represents the relative thermoregulatory stress necessary to reach a state of thermal equilibrium and uses a scale (Cold/Warm/Hot) the same as in the European case. The effective temperature ET is the temperature of an environment with 50% relative humidity in which a person experiences the same amount of losses as in the situation under analysis. The American model determines the heat and vapour flow between the interior of the human organism, the skin and through clothing by using a model based on two concentric cylinders (one represents the skin and the other the clothing) and the use of a two-node calculation module. The model allows us to obtain solutions in time

Exergy analysis of heat transfer in buildings

321

from an initial moment until stabilization is reached. The output values of this model are used to obtain the ET-DISC values, in addition to other indices, such as SET (Standard Effective Temperature) which represents the thermal stress experienced by the inner cylinder referring to a standard person in a standard environment.

4.11.2 Thermal model of the human body and energy balance Different authors have come up with different models to represent the thermodynamic behaviour of the human body. In the ECBS Annex 49 [39] the human body is considered to be made up of two subsystems: the core and the envelope. The core is a subsystem whose temperature remains constant at approximately 37 C, independently of the variations of temperature and humidity of the ambient air, whereas on the contrary, the envelope is a subsystem highly dependent on those variations. Between both subsystems, there is a variable blood flow dependent on the internal and external conditions of the body, see Fig. 4.12. Other authors like Ferreira and Yanagihara [59] have modelled the human body as a set of 15 cylinders that represent the head, neck, trunk, arms, forearms, hands, thighs, legs and feet. Each cylinder contains a set of tissues, such as skin, fat, muscles, etc. and are interrelated to each other through the bloodstream. Whichever model is used, the energy balance in each of the subsystems that make up the human body is resolved. For this it is necessary to specify first the thermophysical properties of each subsystem, that is, the values of density, specific heat and thermal conductivity. One of the terms of the energy balance is the metabolism M, which is the set of chemical reactions of oxidation that release energy and maintain the processes of life. There are different models like that of Harris and Benedict [60] that correlate metabolic activity with body mass, age and height for each sex. Another term in the energy balance is the heat transferred to the environment by convection and radiation. With hcv being the convection coefficient, hr the linear radiation coefficient of the subsystem under consideration, A the exterior surface of the

Figure 4.12 Thermal model of the human body.

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Exergy Analysis and Thermoeconomics of Buildings

subsystem, Tskin the surface temperature of the skin and fclo the relationship between the surface covered by clothing and the naked surface, so that for a naked person fclo ¼ 1 (so then the surface temperature of the clothing is that of the person’s skin), we have Q_ ¼ Q_ cv þ Q_ r ¼ A

Tskin  Top 1 Rclo þ fclo ðhcv þ hr Þ

(4.87)

In this equation Top is the operating temperature, that is, a unique temperature that the air and the interior surfaces should have so that an average person exchanges by convection and radiation the same amount of energy that is exchanged in the real situation. This operating temperature is calculated according to the expression Top ¼

hr T r þ hcv Ti hr þ hcv

(4.88)

with T r the average radiant temperature of the surfaces of the enclosure, that is, the unique and uniform temperature of the surfaces with which the heat transfer by radiation with a person located inside is the same as that produced with the real surface 

temperatures. In practice, as hr z hcv the arithmetic mean Top ¼ T r þ Ti 2 is normally used as the operating temperature. The enthalpy flow associated with evaporation from the skin can be calculated by the following expression ps;skin  f0 :ps;0 H_ ev ¼ AW hv ¼ m_ v hv 1 Rev;clo þ fclo hev

(4.89)

where W is the moisture of the skin, which varies between 1 when the skin is completely covered with sweat to 0.06 when there is only vapour diffusion, hev is the equivalent evaporation coefficient, Rev,clo it is the resistance to vapour diffusion of the clothes, ps,skin is the saturation vapour pressure at the temperature of the skin, pv,0 is the saturation pressure in the environment, f0 is the relative humidity of the environment and m_ v , hv are the mass flow rate and the specific enthalpy, respectively, of the vapour generated. Another term in the energy balance is associated with breathing. With m_ res being the mass flow rate of dry breathing air, Tex, uex the temperature and absolute humidity of the expired air and hv,ex the specific enthalpy associated with that expired air, the sensitive and latent losses associated with respiration are   H_ res ¼ m_ res cp;a ðTex  T0 Þ þ m_ res uex hv;ex  u0 hv;0

(4.90) 0

where the mass flow of breathed air is directly linked to the metabolism M .

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323

The variation of the internal energy of the subsystem under consideration is a sum of the metabolism level and the variation due to the change in environmental conditions, that is dU dU ¼ M_ þ jDT dt dt

(4.91)

With W_ being the mechanical power generated by the person, so that for the base metabolism W_ ¼ 0. According to the First Law the following equation must be fulfilled, expressing the internal energy variation of the body over time due to transient environmental conditions   dU j ¼ M_  Q_ cv þ Q_ r þ H_ ev þ H_ res þ W_ dt DT

(4.92)

4.11.3 Exergy balance in the human body A complete study of the energy behaviour of the human body requires the use not only of the First Law, but also of the Second, in order to evaluate the quality of the energy conversion processes that take place in the different organs and systems. Over the years, different models of the exergy behaviour of the human body have been developed, and we highlight the work of Shukuya [61], Iwamatsu and Asada [62], and Mady et al. [63]. Since 2013 there have been numerous works published on the exergy behaviour of the human body, Caliskan [64], Mady et al. [65] among others. In a way similar to what we said for internal energy, the rate of exergy change of the human body is due, on the one hand, to metabolism and, on the other, to the effect of the change of environmental conditions, so that with A being the exergy of the human body, we have dA dA ¼ B_ M þ jDT dt dt

(4.93)

Batato et al. [66] showed that the change of energy due to metabolism and the exergy change are practically identical, so that we can use the approximation M_ z B_ M . The exergy associated with the heat flux exchanged by convection and radiation is B_ Q ¼ 0



0

  T0  _ Qcv þ Q_ r 1 Tskin 00

(4.94)

00

With h , s and h , s the specific enthalpy and entropy of saturated water and saturated vapour respectively at skin temperature, the flow of exergy associated with vapourization on the skin is represented by the following equation B_ ev ¼ m_ v ½ðh00  h0  T0 ðs00  s0 ÞÞ þ m_ v Rv T0 ln

ps;skin pv;0

(4.95)

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Exergy Analysis and Thermoeconomics of Buildings

an expression in which the physical and chemical exergy of the vapour generated on the surface of the skin is taken into account. The exergy flows exchanged by respiration are associated with the inspired and expired air. Obviously, we assume ideal gas behaviour, and for convenience we separate the part of the exergy associated with dry air from that of water vapour, so that we have B_ res ¼ B_ a þ B_ v

(4.96)

where, according to Eq. (3.12) and Eq. (3.92) we have " B_ a ¼ m_ res ð1  uex Þ cp;a



Tex Tex  T0  T0 ln T0



# pa;ex þ Ra T0 ln pa;0

(4.97)

1 pa;ex þ m_ res ð1  uex ÞRa T0 ln 1 þ uex pa;0 " B_ v ¼ m_ res uex

#   Tex pv;ex þ Rv T0 ln cp;v Tex  T0  T0 ln T0 pv;0

þ m_ res uex Rv T0

uex pv;ex ln 1 þ uex pv;0

(4.98)

From the exergy balance in the human body we get that the rate of exergy destruction is     dA _ _ (4.99) D ¼ BM   B_ Q þ B_ ev þ B_ res  W_ dt The exergetic performance of the human body as an energy converting system is expressed by f¼1

D_ jdA=dtj

(4.100)

When the exergy balance is performed in each of the subsystems that are usually considered in the modelling of the human body, it is necessary to consider the exergy associated with arterial and venous blood flows exchanged by each subsystem. By knowing the specific heat of the blood, the corresponding temperatures and mass flows, rates, these exergy flows are calculated by applying Eq. (3.44). In short, calling B_ ar , B_ ven the flow of exergy associated with the flow of arterial and venous blood respectively, through the balance of exergy in the subsystem s we have that    dAs s s s s s D_ ¼ B_ M   B_ Q þ B_ ev þ B_ res  W_ þ ðB_ ar þ B_ ven Þin  ðB_ ar þ B_ ven Þout dt (4.101)

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325

Using this and other similar mathematical models of exergy balance in the human body, we come to some very interesting conclusions. Thus, Ala-Juusela [67] showed that, in the heating period, the minimum exergy consumption of the human body can be achieved at higher average radiant temperatures and lower indoor air temperatures. Therefore, in winter conditions, there is a set of indoor air temperatures between 18 and 20 C and average radiant temperatures between 23 and 25 C for which the consumption of exergy of the human body is minimal so that these conditions of minimum exergy consumption correspond to maximum thermal comfort conditions. These conditions can be achieved with radiant systems (systems of low temperature of heating and high temperature of refrigeration) that provide the energy required to satisfy the demand at a temperature close to the ambient temperature. Likewise, it has become clear that the trunk and the head are the parts of the body in which the greatest exergy destruction takes place and, consequently, they have the lowest efficiencies.

4.12

Energy and exergy demand of a building

To finish this chapter, we will address the calculation of the exergy demand of a building. To do this, we will first recall the essential ideas about the calculation of energy demand, since this is the basis for calculating the exergy demand.

4.12.1 Calculation of energy demand The energy demand for premises is the amount of energy required throughout the year to maintain the premises in the conditions of comfort required by the users. We will review first the distinction between what is gain and what is load, and we will make a brief summary of the existing methods for calculating the demand. It is evident that to calculate the energy (exergy) demand it is necessary to previously define the limits of the system on which the corresponding balance is to be made. This boundary surface corresponds to the interior surfaces of walls, floors and roofs. For a more detailed study, the reader should refer to the abundant literature on this subject, for example in O’Callaghan [68] or Calener [69].

4.12.1.1 Gains (losses) of heat Heat gains (losses) are understood to be the heat fluxes entering (leaving) the control volume defined by the established physical limits. These gains may be of external origin, such as •

•

Gain (loss) of heat by conduction through walls, ceilings, etc. in contact with outside air. Heat gains (losses) through external opaque envelopes are due not only to the temperature difference between the surface of the exterior wall and the ambient air but also due to the solar radiation absorbed. The variations of outside temperature and solar irradiation and the inertia of the walls make the problem dynamic, and the heat transfer equation must be solved for a non-steady system. Gain of incoming solar radiation through windows and skylights.

326

•

•

•

Exergy Analysis and Thermoeconomics of Buildings

The transmission of heat through semi-transparent media is due, on the one hand, to conduction and, on the other, to the incident solar irradiation. The resulting heat flux can be determined from a global energy balance, although for practical reasons both aspects are treated separately and the principle of superposition is applied, Stout and Billings [70] defining what is known as the solar factor. Gain (loss) of heat due to ventilation. Ventilation is the voluntary entry of outside air in order to maintain the quality of indoor air. The resulting gain (loss) of heat is obtained by an energy balance made on the control volume under consideration. Gain (loss) of heat due to infiltration. Unlike ventilation, infiltration is the involuntary input of outside air due to cracks and holes in the building envelope. The flow of infiltration is therefore unknown, but there are approximate experimental methods for its quantification, such as the blower door method, Odriozola [71]. In addition to these gains, there are others, whose common characteristic is that the source of heat is inside the building. Among gains of internal origin is Heat gain due to lighting, occupation and diverse equipment. The instantaneous gain due to lighting is expressed as a function of the installed power, the utilization factor and a characteristic coefficient for each light. The gains due to occupation are due to the exothermic transformations that take place in the human body and depend on the individual, degree of activity, clothing and environmental conditions. The gain is calculated based on the number of occupants and the occupation profile. As for the gain due to equipment such as computers, kitchens, etc. it is calculated analogously to lighting, ASHRAE Fundamentals [72]. Once the equations corresponding to each transfer mechanism have been established to calculate the gains (losses), the next step is to convert those gains into loads and ultimately obtain the thermal demand of the building. Therefore, the thermal load of a space is calculated in two stages: first, instantaneous heat losses (or gains) are calculated, that is to say, the heat fluxes that come out, named as losses, or that come into the defined volume, and in a second stage, from those gains the thermal loads are calculated; finally the demand is worked out.

4.12.1.2 Thermal load and energy demand The heat fluxes of the gains have in general two components: a convective part and a radiant one. The convective part directly affects the indoor air of the zone under consideration, while the radiant part is first absorbed by the surfaces that delimit the zone, to later pass by convection to the air. So, the thermal load corresponding to a zone is understood to be the amount of heat that must be supplied (heating) or extracted (cooling) to maintain the temperature and humidity of the air of the said space constant and equal to a previously fixed value. Fig. 4.13 schematically shows the difference between gain and load. Moisture exchanges contribute significantly to the energy exchanges of buildings, mainly due to the associated phenomena of evaporation and condensation. This is what

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327

Figure 4.13 Difference between gain and load.

is called the latent thermal load. By calculating the instantaneous loads, the demand comes from integrating these values over time, which normally will be a year, divided into a heating season and a cooling season. Thus, the integral of thermal loads over time, usually 1 year, is called the thermal demand of the building or the zone under consideration. Next, we will look at some basic ideas on how to calculate that energy demand in any building. According to ISO 13790 (2008), there are two types of methods to calculate this demand: • •

Quasi-steady methods, which calculate the energy balances over a sufficiently long period of time (1 month or the whole season), so that the dynamic effects are taken into account through gain/loss factors that are determined empirically. Dynamic methods, which calculate the energy balances in short periods (typically 1 hour) and which take into account the energy stored and released by the mass of the building.

Numerous national codes are based on the first type of methods. However, due to the software available (TRNSYS, EnergyPlus, etc.) and the possibilities of modern computers, dynamic methods are increasingly used. In turn, dynamic methods can be classified into two groups: direct and indirect methods. •

•

The direct methods form and solve the equations all at once so that in principle the system of equations does not have any restriction as to its character and all the requests are applied simultaneously. It is the most detailed but requires large memory capacity and long execution time. The indirect methods are based on the principle of superposition of the solicitations and the application of laws of convolution, Sanchez [73]. Recall that, in functional analysis, convolution is a mathematical operator that transforms two functions of the same variable (in our case time) into a third, which represents the integral of the product of both, having displaced one of them.

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4.12.1.3 Indirect method for calculating energy demand We are going to take a summary of the calculation of energy demand using this second type of methods, on which energy rating software such as CALENER and building simulation software such as TRNSYS and others are based. For a detailed review, consult AICIA [74] or Sala [75]. The indirect method is very interesting because of the reduced calculation effort required and because it provides results broken down by components. Basically, it consists of the determination of the response that would cause a unitary excitation of each one of the present solicitations and the subsequent obtaining of the global response, as superposition of the individual responses. This requires that the equations be linear and the coefficients that appear in them must be constant. Once the instantaneous heat gains (losses) that we have discussed above are calculated, the gains are converted into loads by the response factors, which are transfer functions that calculate the response of the zone under consideration to a unit impulse of heat gain (loss). Each of the components that constitute the gain (loss) has its own response factors, depending on the radiant and convective part and the thermal capacity of the walls and furnishings. Thus, the response factors are different for each room and each component of the thermal load. Since the gains are calculated discretely, for example, from hour to hour, the Z transform method is usually used to calculate the response factors. For the five types of gains to which we have referred above, there are four sets of response factors: conductive response factors, for solar gain, for lighting and for occupation and items of equipment. In this way, the thermal load at the time of calculation is expressed as a function of the load in the preceding hours and the heat gain at the current time and the preceding hours. Carslaw and Jaeger [76], using the Laplace transform of the temperature and radiation excitations, laid the foundations of the response factor method, Mitalas [77] that of the Z transfer functions, Stephenson and Mitalas [78] and Hittle [79] that of the frequency response. These three methods are basically similar, with the objective of determining the coefficients of the transfer function that relate the excitations on the two surfaces of the wall with the resulting heat flux. The difference between the methods lies in the type of excitation used: a triangular impulse the first, a unit slope the second and a unit sine wave the third, of variable frequency and phase shift. An alternative method to solve the non-steady state in multi-layered walls is to solve the resulting differential equations by numerical methods, the most used being the finite difference method, Harnett and Cho [80]. The fundamental problem of these methods is the choice of the numerical scheme and the discretization parameters, mesh size and time interval so that the desired precision is achieved with the minimum calculation effort. The calculation of energy demand is based on the resolution of a system of equations, which comes from applying the energy balances on the exterior surfaces of the building, on the interior surfaces and in the air of each zone. The unknowns are the surface temperatures, which must not be forgotten are variable in time, so that once

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329

these are calculated, instantaneous heat fluxes or any temporary integration thereof are obtained. From the energy balance in the exterior surfaces, we have an equation similar to Eq. (4.60) q_cd;es ¼ hcvr ðTsa  Tes Þ

(4.102)

and from the response factors q_cd;es ¼ a0 Tes ðtÞ  bo Tis ðtÞ þ P

(4.103)

and for interior surfaces q_cd;is ¼ b0 Tes ðtÞ  c0 Tis ðtÞ þ Q

(4.104)

where a0, b0, c0 are coefficients of the transfer function and P and Q are associated with temperatures and heat fluxes in previous instants. From Eqs. (4.102) and (4.103) we get Tes(t) and this expression substitutes Tes(t) in Eq. (4.104), thus obtaining qcd,is(t) as a function of Tis(t). In the equations of energy balance on the inner surfaces, we replace q_cd;is ðtÞ by the previous expression, as a function of Tis(t). For a system of N surfaces, we have a system of N equations, with (N þ 2) unknowns, which are the interior surface temperatures Tis, the indoor air temperature Ti and the thermal power supplied or extracted by the air conditioning equipment Q_ dem . We can write an additional equation, which is the sensitive energy balance in the air of the room. Considering a thermal zone, at a given moment that equation will be rcV

N dTi X ¼ Ai hcv;i ðTi  T0 Þ þ rcV_ e ðT0  Ti Þ þ Q_ IS þ Q_ dem dt i¼1

(4.105)

where • • • •

N P

Ai hcv;i ðTi  T0 Þ is the rate of heat exchanged with the interior surfaces.

i¼1

rcV_ e ðT0  Ti Þ is the mass flow rate of outside air that enters the premises (ventilation þ infiltrations). Q_ IS are the convective contributions of internal sources (occupants, lighting and equipment). Q_ dem is the thermal power supplied (or extracted) by the conditioning system.

In short, we have a system of (N þ 1) equations, with (N þ 2) unknowns. There are three ways to solve this problem: (1) Consider a situation in which the air temperature does not vary and therefore dTi/dt ¼ 0. The system of N algebraic equations can be solved and the values of Tis(t) are obtained. Once these are known, we can calculate Q_ dem for each time step Dt. (2) Consider that Q_ dem ¼ 0, so that temperature Ti evolves freely. We will have to solve a system of (N þ 1) equations (N algebraic and one differential) with (N þ 1) unknowns. (3) The most complete solution is to add the equations that characterize the behaviour of the conditioning equipment.

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Actually, the heat extracted (added) by the air conditioning system is different from the thermal load, because the internal temperature does not remain constant. This is due to some of the following reasons: • • •

Intentional stops of heating equipment (nights, weekends, holiday periods). Unintended stops (because the comfort conditions have already been reached). The systems do not maintain a temperature but a temperature range (thermostatic band).

4.12.2

Calculation of exergy demand

4.12.2.1 Preliminary comments As we have said, the energy demand is the amount of energy required throughout the year to maintain the premises in the conditions of comfort requested by the users. Similarly, the demand for exergy is the amount of exergy required to maintain the premises in the conditions of comfort requested by the users; it is, therefore, the exergy content of the energy demand. According to the exergy concept discussed in Chapter 2, we could also say that the demand for exergy is the minimum useful work required to satisfy the demand for energy. We know that the energy that is supplied to satisfy the demand must be of the minimum quality required, as otherwise, exergy destruction will take place. For example, this is what will happen when, in order to maintain air at a temperature of 21 C, we use a heating system at 80 C. When more energy is supplied (extracted) than necessary in a room, overheating (sub-cooling) occurs; similarly, the input in excess of exergy causes the destruction of exergy. Therefore, in the ideal situation, the minimum exergy should be supplied; that is, the minimum required to satisfy the conditions of comfort. Any excess exergy that is supplied to the premises will lead to exergy destruction. Once the energy demand is calculated, there are two methods to calculate the exergy demand, as described below. In principle, we could think that the demand for exergy is obtained through the application of exergy balance in the CV that constitutes the space to be conditioned, Eq. (2.53). However, in this equation, two unknowns appear: the demand for exergy and the exergy destroyed. Therefore, the calculation of exergy demand requires starting from the values obtained in the calculation of the energy demand.

4.12.2.2 Simplified method This method was proposed by Schmidt [81]. In the case of radiators or fan heaters, part of the energy demand is supplied by the terminal element of the installation as convection heat and another part is radiation heat, this proportion depending on the type of emitting element. Although the quality factor of one heat and another is different,

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331

the energy demand corresponds finally to the convection heat that is exchanged by the air of the premises. In short, the demand for exergy is calculated by the expression   To _ B_ dem ¼ 1  Qdem (4.106) Ti In Section 2.9 of Chapter 2, we explained the meaning of this expression for the different possible values of Ti with respect to T0 and both for heating and cooling. As the average temperature of the interior surfaces of the premises is generally different from that of the indoor air, the operating temperature can also be used Top, ISO 7726 [82] being then the exergy demand   T0 _ _ Qdem (4.107) Bdem ¼ 1  Top This expression does not take into account the fact that part of the demand is due to the need to heat (cool) the ventilation air and that its exergy content does not correspond to the previous expression. In addition, neither the component associated with the pressure nor the chemical component of the exergy has been taken into account. If we consider an air conditioning process in which there is humidification or dehumidification of the air, or if we carry out a study in which indoor air quality arises, then the chemical component is important and should be taken into account. In this regard, refer to Section 3.6.4 of Chapter 3.

4.12.2.3 Detailed method The detailed method, developed by Annex 49 [83] differs from the simplified one, in that it does separate the exergy demand associated with the ventilation air from the rest of the demand. Nevertheless, as the simplified method does not take into account the chemical exergy, and also it does not consider the small difference between the exergy of the convection heat and that of the radiation exchanged between surfaces with small temperature differences. Referring to the case of heating, as we have seen before, the total demand reflects the losses (by transmission through the walls taking into account the inertia, ventilation and infiltration) minus the gains (solar and internal). As discussed in Chapter 2, the quality factor of internal energy at temperature T is less than the quality factor associated with heat at that temperature T. Therefore, to determine the demand for exergy, it will be necessary to evaluate first what part of that demand is needed to heat (cool) the ventilation air, contributing the rest in the form of heat to the operating temperature of the room. Ultimately, to calculate the exergy demand, it is necessary to separate the demand into two components: (1) we determine the exergy needed to condition the ventilation

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air from the outside and mix it with the air of the premises. The exergy variation of the air between the interior and exterior conditions is the minimum exergy that must be provided to condition that air. (2) The rest of the exergy demand, if any, must be supplied as heat at the operating temperature Top. In accordance with the principles outlined above and following the methodology developed by the ECBCS Annex 49, we will present the general equations for the calculation of the exergy demand of premises. These equations can be programmed and coupled to energy simulation software, as has been done with TRNSYS. As we have said, the calculation of exergy demand by the detailed method requires prior knowledge of the energy demand. Once the energy balance is made, the total energy demand Q_ dem is compared first with ventilation losses Q_ vent . If these are less than the total demand, the ventilation air must be heated up to the temperature inside the premises, which implies a minimum contribution of exergy, which can be calculated with the expression  _ _ Bvent ¼ Qvent 1 

  Top T0 ln Top  T0 T0

(4.108)

with Q_ vent being the heat that must be provided to heat the air to the temperature of the premises, which is Q_ vent ¼ m_ vent cp ðTop  T0 Þ

(4.109)

The difference between the total demand and this heat Q_ vent must be provided as heat to the premises, at the temperature Top, so that the complementary exergy to be provided is B_ Q ¼



  T0  _ Qdem  Q_ vent 1 Top

(4.110)

In the case that the total demand is less than the losses by ventilation, the air does not need to be heated up to the temperature Top and no additional heat is required, as this has been given by the internal and solar gains. The temperature at which the air will have to be heated will be DTvent ¼

Q_ dem ðTop  T0 Þ Q_ vent

(4.111)

In short, the set of equations needed to calculate the exergy demand by the detailed method is summarized in the following set of equations   Q_ vent;d ¼ min Q_ dem ; Q_ vent

(4.112)

Exergy analysis of heat transfer in buildings

DTvent ¼

Q_ vent;d ðTop  T0 Þ Q_ vent

Tvent ¼ T0 þ DTvent  _ _ Bvent ¼ Qvent 1 

  T0 Tvent ln Tvent  T0 T0

Q_ ¼ Q_ dem  Q_ vent;d B_ Q ¼



 T0 _ Q 1 Top

333

(4.113) (4.114) (4.115) (4.116) (4.117)

In the case of refrigeration in which T0 > Top all natural energy flows represent unwanted gains, so that Q_ dem > Q_ ven is always going to be fulfilled. Therefore, the ventilation air will always have to be cooled to the temperature Top. In the case of refrigeration in which T0 < Top the need for cooling (energy output) does not represent an exergy demand, but rather is an undesired assignment of exergy. This exergy is given to the building by internal gains and could be somehow collected and used as heat at the temperature Top. In the final report of the ECBCS Annex 49 a comparison of the two calculation methods, simplified and detailed, is made in the case of an office. Different situations are compared, in one case maintaining constants T0 and the energy demand and modifying the level of insulation and air exchanges and in another, maintaining the characteristics of the room but varying the solar irradiation and the outside temperature. In all cases, it was found that the exergy demand is around 10% of the energy demand (obviously depending on T0 and Top). By comparing both methods with each other, the results obtained are quite different, with the demand for exergy calculated by the detailed method being smaller than by the simplified one. This difference becomes larger when the ventilation flow is greater, and Top comes closer to T0. Undoubtedly, the detailed method is more precise, so it should be used when more accurate information is needed, for example, when it comes to optimizing the building’s air conditioning and ventilation systems. Annex 49 recommends using the simplified method, at a preliminary stage, when it comes to analysing the energy supply chain of a building.

4.12.3 Examples Determine the quality factor of the exergy associated with a flow of water at temperature T and the heat quality factor at that temperature, for the temperature values T in Table E.4.4, with T0 ¼ 290 K

Example E.4.13.

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Table E.4.4 T values for T0 ¼ 290 K. T(8c) 10 0 10 20 30 40 50 60

Solution The quality factor of energy is the relationship between exergy and energy. In the case of a water mass flow rate m_ the associated exergy is given by Eq. (3.44), so its quality factor is ð2.1Þ FB ¼ 1 

T0 T ln T  T0 T 0

For its part, the quality factor of heat is as we well know ð2.2Þ FQ ¼ 1 

T0 T

Using these expressions and giving values to T we obtain the following Table E.4.5. In order to interpret the results more clearly, we present the absolute values in the table. Table E.4.5 Quality factors. T(8c)

FB

FQ

10

0.95

0.1

0

0.03

0.06

10

0.01

0.02

20

0.01

0.05

30

0.02

0.04

40

0.04

0.07

50

0.05

0.1

60

0.07

0.13

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335

We can appreciate that for any value of the temperature, whether it is above or below T0 the inequality FB < FQ is met. Both quality factors only coincide when T ¼ T0, since then its value is zero. Example E.4.14.

Let there be a house in which the losses by heat transfer through the envelope are 5 kW, the losses by ventilation and infiltrations are 3 kW, while the internal gains are 1.2 kW. The temperature of the outside air is 10 C and that of the interior of the house is 20 C. Determine the heating demand and the heating exergy demand. Solution The heating demand is the difference between the gains and losses, so that Qdem ¼ 5 þ 3  1; 2 ¼ 6:8 kW Next, we compare this demand with the losses by ventilation and infiltration, which are 3 kW, evidently less. Therefore, the entire airflow needs to be heated up to the indoor temperature, with the contribution of exergy    T0 Ti B_ vent ¼ 1  ln ¼ 0:05 kW Ti  T0 T0 The difference between the total demand, 6.8 kW and the one due to ventilation, 3 kW, is the heat that must be contributed to the premises, and therefore, the exergy that needs to be provided is    T0  _ Qdem  Q_ vent ¼ 0:13 kW 1 Ti In short, the total demand for exergy is B_ dem ¼ 0:05 þ 0:13 ¼ 0:18 kW so the quality factor of the energy contributed is 2.6%.

Example E.4.15.

In commercial premises, the heat losses through the envelope are 35 kW, the losses by ventilation are 25 kW and by infiltrations are 5 kW, with internal gains of 45 kW. The temperature of the outside air is 3 C and that of the interior of the room is 21 C. Determine the energy and exergy demands. Solution The heating demand for the premises is Q_ dem ¼ 35 þ 25 þ 5  45 ¼ 20 kW Comparing this demand with ventilation losses, we have that Q_ dem < Q_ vent , so that the ventilation airflow will only need to be heated up to a temperature of DTvent ¼

Q_ dem ðTi  T0 Þ ¼ 14:4 K Q_ vent

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Exergy Analysis and Thermoeconomics of Buildings

Tvent ¼ T0 þ DTvent ¼ 289:4 K The exergy needed to heat the ventilation air is    T0 Tvent _ _ Bvent ¼ Qvent 1  ln ¼ 2:28 kW Tvent  T0 T0 The rest of the demand is covered by the internal gains of the commercial premises. The quality factor of the energy contributed to the air of the premises is 9.1%. Example E.4.16.

In an office, the heat inputs through the envelope are 19 kW, those due to infiltrations are 7.6 kW, with internal heat gains of 8.4 kW. The air conditioning of the office is done through a centralized air conditioning system. The outside air temperature is 32 C, with the office temperature at 24 C and the ambient pressure 1 bar. Determine: (a) The airflow rate for the office conditioning. (b) The rate of exergy demand and the quality factor of the energy provided.

Solution (a) The demand for the refrigeration is Q_ dem ¼ 19 þ 7:6 þ 8:4 ¼ 34 kW To calculate the mass airflow rate, we take into account that _ p ðT0  Ti Þ ¼ 34 / mc

m_ ¼ 4:22

kg s

so the air conditioning volume flow rate is _ m_ ¼ V9

3

Ra T0 8:314$305 m ¼ 3:71 / V_ ¼ m_ ¼ 4:22 28:8$100 p0 s

(b) The rate of exergy demand is  B_ dem ¼ Q_ dem 1 

  T0 Ti ln ¼ 0:45 kW Ti  T0 T0

Cooling this airflow of 3.71 m3/s from the ambient temperature to 24  C involves extracting an amount of energy that, per second, is 34 kJ and this means that exergy of 0.45 kW is provided. The minimum electrical power consumed by the refrigerating compression machine to cool that air is 0.45 kW; actually, it is much greater, due to the irreversibilities and consequent exergy destruction. In short, the quality factor of the energy used is 1.3%.

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337

Example E.4.17.

Compare the heating exergy demand obtained for a dwelling using the simplified method and the detailed method. Fig. E.4.7 shows a section of the house, consisting of an unheated basement, a ground floor and a first floor, with a total useful area of 280 m2.

Figure E.4.7 Section of the single-family house.

Solution A simulation was carried out with TRNSYS v17 to obtain the heating demand, hour by hour, having used the meteorological data of the city of Bilbao. The monthly cumulative values are presented in Fig. E.4.8. The annual heating demand is 16,187 kWh, the month of maximum demand being January, with a value of 3,132 kWh. As for the exergy, the annual demand is 482 kWh, with the demand in January being 106 kWh. As we can see, the heating energy quality factor is very low, around 3% for the annual demand and slightly higher if we refer to the demand for January.

Figure E.4.8 Energy and exergy demand for heating.

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Exergy Analysis and Thermoeconomics of Buildings

Finally, the demand for heating exergy has been determined by both the simplified and detailed methods of the ECBS Annex 49. The monthly values are shown in Fig. E.4.9 and, as we can see, the values that come from the detailed method are lower than those from the simplified method. By looking at the annual demand, the value is 448 kWh for the detailed method, compared to 563 kWh for the simplified one, which is 23% higher. The exergy demand calculated by the detailed method is always lower than that obtained by the simplified method. The difference is greater when the indoor temperature is closer to the ambient temperature and the ventilation flow rate is higher. In this example, the difference is relatively important, so it is preferable to use the detailed method.

Figure E.4.9 Demand for heating exergy, by the detailed method and the simplified method.

Subscripts 0 s i, e i, f w cv, cd r, lwr, swr sky sun, sur v, a ex dem vent

Environmental state Surface Interior and exterior Initial and final Wall Convection and conduction Radiation, long-wave radiation and short-wave radiation Sky Sun and surroundings Water vapour and dry air Expiration Demand Ventilation

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339

Symbols r V h s T t m_ c Teq Tf Tm Top L A Dx N l R, U ex Udyn, Udyn U Q W d_ 4 u f εi ai ri e_i Gij

Density Volume Specific enthalpy Specific entropy Temperature Time Mass flow rate Specific heat Equivalent temperature Fictitious temperature Mean temperature Operative temperature Thickness Surface Distance Number of layers in a wall; number of surfaces in a room Thermal conductivity Thermal resistance and thermal transmittance Dynamic thermal transmittance and dynamic exergy transmittance Internal energy Heat Work Rate of exergy destruction per unit of area Exergy efficiency Absolute humidity Relative humidity Emissivity of surface i Absorptivity of surface i Reflectivity of surface i Emission power of surface i Gebhart factor of radiant heat exchange between surfaces i and j

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[5] DL n.311/2006, Corrective Provision and Integration to the Legislative Decree 19 Agosto 2005, No. 192, Implementing Directive 2002/91/CE, on the Energy Performance of Buildings (In Italian), 2007. [6] Thermal Regulation, 2012 (in French), http://www.rt-batiment.fr/. [7] The Building Regulations, 2010, 2015 (Internet), http://www.legislation.gov.uk/uksi/ 2010/2214/contents/made. [8] Ministry of Housing, Order FOM/1635/2013, 10 September, Updating Technical Building Code. Basic Document DB-HE: Energy Saving, 17 March, 2013 (in Spanish), approved by Royal Decree 314/2006 (in Spanish), Madrid. [9] S. Ng, K. Low, N. Tioh, Newspaper sandwiched aerated lightweight concrete wall panelsdthermal inertia, transient thermal behavior and surface temperature prediction, Energy and Buildings 43 (7) (2011) 1636e1645. [10] S. Ferrari, Building envelope and heat capacity: Re-discovering the thermal mass for winter energy saving, in: Proceedings of the 2nd PALENC Conference and 28th AIVC Conference on Building Low Energy Cooling and Advanced Ventilation Technologies in the 21st Century, Crete, Greece, 27e29 September, 2007. [11] UNE-EN ISO 13786 (AENOR 2011), 2011. [12] E. Stéphan, R. Cantin, A. Caucheteux, S. Tasca-Guernouti, P. Michel, Experimental assessment of thermal inertia in insulated and non-insulated old limestone buildings, Building and Environment 80 (2014) 241e248. [13] UNE-EN ISO 13786, Thermal Features of the Products and Components of Buildings. Dynamic Thermal Characteristics, Calculation methods, 2011 (in Spanish). [14] P.T. Tsilingiris, Parametric space distribution effects of wall heat capacity and thermal resistance on the dynamic thermal behavior of walls and structures, Energy and Buildings 38 (10) (2006) 1200e1211. [15] Y.A. C¸engel, A.J. Ghajar, Heat and Mass Tranfer: Fundamentals and Applications, fourth ed., McGraw-Hill, New York, 2011. [16] I. Flores, The Method of Exergy Analysis in Buildings. Its Application in the Characterization of the Dynamic Behavior of the Opaque Envelop (In Spanish) (Doctoral Thesis), University of the Basque Country, Bilbao, 2016. [17] W. Choi, R. Ooka, M. Shukuya, Exergy analysis for unsteady-state heat conduction, International Journal of Heat and Mass Transfer 116 (2018) 1124e1142. [18] H. Asan, Investigation of wall’s optimum insulation position from maximum time lag and minimum decrement factor point of view, Energy and Buildings 32 (2) (2000) 197e203. [19] Y.A. C¸engel, Heat Transfer (In Spanish), second ed., McGraw-Hill, Mexico, 2004. [20] N. Ito, K.I. Kimura, Convection Heat Transfer at the Exterior Surface of Buildings Exposed to Natural Wind, J.S:A.E. Transactions, 1968. [21] W.M. Kays, M.E. Crawford, Convective Heat and Mass Transfer, third ed., McGraw-Hill, New York, 1993. [22] B.W. Olesen, Radiant floor heating in theory and practice, ASHRAE Journal 44 (7) (2002) 19e26. [23] H. Torio, A. Angelotti, D. Schmidt, Exergy analysis of renewable energy based climatisation systems for buildings: a critical view, Energy and Buildings 41 (3) (2009) 248e271. [24] F. Incropera, D. DeWitt, Fundamentals of Heat and Mass Transfer, sixth ed., John Wiley & Sons, New York, USA, 2007. [25] J.H. Lienhard, A Heat Transfer Textbook, third ed., Phlogiston Press, Cambridge, MA, USA, 2006.

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