Numerical simulation of cooling energy consumption ...

Applied Energy 88 (2011) 2871–2884

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Numerical simulation of cooling energy consumption in connection with thermostat operation mode and comfort requirements for the Athens buildings C. Tzivanidis ⇑, K.A. Antonopoulos, F. Gioti National Technical University of Athens, School of Mechanical Engineering, Thermal Department, Heroon Polytechniou 9, 157 73 Zografou, Athens, Greece

a r t i c l e

i n f o

Article history: Received 29 July 2010 Received in revised form 16 December 2010 Accepted 20 January 2011 Available online 26 February 2011 Keywords: Cooling energy saving Air-conditioning control Thermostat operation Thermal comfort Athens buildings

a b s t r a c t A model and a corresponding numerical procedure, based on the finite-difference method, have been developed for the prediction of buildings thermal behavior under the influence of all possible thermal loads and the ‘‘guidance’’ of cooling control system in conjunction with thermal comfort requirements. Using the developed procedure analyses have been conducted concerning the effects of thermostat operation mode and cooling power in terms of the time, on the total cooling energy consumption for the ideal space cooling, as well as for various usually encountered real cases, thus trying to find ways to reduce cooling energy consumption. The results lead to suggestions for energy savings up to 10%. Extensive comparisons between the ideal and various real cooling modes showed small differences in the 24-h cooling energy consumption. Because of the above finding, our detailed ideal cooling mode predictions gain considerable value and can be considered as a basis for comparison with real cases. They may also provide a good estimate of energy savings obtained if we decide to increase thermostat set point temperature. Therefore, as the extent of cooling energy saving is a priori known, one can decide if (and how much) it is worthy to increase thermostat set point temperature at the expense of thermal comfort. All results of the study, which refer to the Typical Athens Buildings during the typical Athens summer day, under the usual ranges of thermal loads, may be applicable to other regions with similar conditions. Ó 2011 Elsevier Ltd. All rights reserved.

1. Introduction Numerous methods have been devised and proposed for reduction of the energy consumed in building heating and cooling. Examples of these are summarized below. The largest group of energy saving methods appears to be the one suggesting interventions to existing buildings or proposing new construction ideas related or not to renewable energy sources in conjunction with passive or active systems. A characteristic example in this category is Ref. [1] concerning the monitored excellent energy performance of a rehabilitated passive office building in Tubingen/Germany. Other examples from the large number of studies of this group are Refs. [2–9], briefly discussed below. In Ref. [2] the performance of vegetated roofs is investigated. It is concluded that green roof outperforms the reference roof. In Ref. [3], it is shown that thin antireflective coatings located in ‘‘switchable or smart windows’’ neither influence thermal emissivity nor heat-transfer coefficient of the glazing but increase significantly the light transmittance in the transparent state. The simulation results of Ref. [4] revealed that mechanically ventilated double-skin façade systems can save energy from 21% to 26% in ⇑ Corresponding author. E-mail address: [email protected] (C. Tzivanidis). 0306-2619/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.apenergy.2011.01.050

summer and from 41% to 59% in winter as compared to conventional double-skin façades without thermal mass. The thermal problem of space cooling using ceiling embedded piping systems is studied by developing analytical models [5], transient twodimensional [6] and three-dimensional [7] numerical solutions, as well as experimentally [8]. An alternative to space cooling using ceiling embedded piping is presented in the experimental study of Ref. [9], in which a metal sheet incorporating cooling tubes is placed on the undersurface of the ceiling. All the above ceiling cooling studies lead to the conclusion that they offer both considerable energy savings and higher level of thermal comfort as compared to the conventional space cooling systems. Applications with new building materials of improved thermal or thermodynamic properties, for example phase change materials (PCM), insulating elements or improved glass with coatings may be included in the same group of energy saving methods. PCM energy saving applications seem to be promising as deduced from the latest studies. Characteristic examples from the large number of related studies are Refs. [10–12]. In Ref. [10] a numerical parametric analysis of space cooling systems based on night ceiling cooling with PCMembedded piping is conducted and shows the proper combinations of the numerous parameters involved for obtaining the lowest energy consumption in conjunction with the highest level of thermal comfort. In Ref. [11] experiments and numerical simulation

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C. Tzivanidis et al. / Applied Energy 88 (2011) 2871–2884

Nomenclature A AS a B b C c DT⁄ Eb Ed ðT i Þ E1,2 ed f, F GT G0T g HD h i k L m n Pd ðT i Þ

P1,2

pd

Qc Q c Qr QV Q1 Q2 q, Q

total floor area of a building (m2) fenestration area (m2) coefficient given by Eq. (27) (W) layers thickness of a multilayer building element (m) coefficient given by Eq. (28) (W/°C) number of air changes per hour (change/h) specific heat at constant pressure (J/kg K) thermostat throttling range (°C) part of the total daily cooling energy, which causes indoor air temperature to drop below the low limit of thermostat throttling range (J/day) daily cooling energy consumption as a function of the thermostat set point temperature T i (J/day) daily cooling energy saving obtained if T i is increased from T i1 to T i2 (J/day) daily cooling energy consumption per temperature degree (J/K-day) minimum and maximum values of fenestration percentage, respectively (%) solar radiation incident on outdoor walls or roof surface (W/m2) solar radiation incident on fenestration (W/m2) radiation heat-transfer factor (W/m2 K) hydraulic diameter (m) Heat-transfer coefficient (W/m2 K) number from 0 to 1 used to include infiltration into ventilation heat load thermal conductivity (W/mK) normalized radiative loss (K or °C) mass (kg) exponent in Eq. (9) percentage of daily cooling energy saving with reference to 24 °C as a function of the thermostat set point temperature T i (%/day) percentage (with reference to 24 °C) of daily cooling energy saving obtained if T i is increased from T i1 to T i2 (%/ day) percentage (with reference to 24 °C) of daily cooling energy saving obtained by 1 K temperature increase of thermostat set point (%/K-day) cooling power (W) specified (constant) value of cooling power (W) radiative load (W) heat load from ventilation and infiltration (W) heat from building envelope, partitions and equivalent furniture slabs (W) indoor load from various sources (W) minimum and maximum values of indoor load, respectively (W/m2)

showed that gypsum wallboards incorporated with different PCM content can be used to improve comfort, save energy and even reduce the weight of wallboards. The analytical and numerical results of Ref. [12] show that PCM improves considerably the thermal behavior of lightweight passive solar rooms. Another large class of space heating and cooling energy saving methods borrows ideas, approaches and techniques from scientific fields other than the classic thermal sciences. Such fields are related to control and optimization techniques, expert systems, fuzzy logic, neural networks, genetic algorithms, predictive control or auto tuning methods and generally information technology. For example, in Ref. [13] a model predictive control strategy is proposed for building temperature regulation using electrical convectors. It is concluded that significant consumption reductions can be

qc qi, qo R r T Ti T i To t td to Uf Um Vo v, V x

specific value of Q c (cooling power per unit flow area, W/m2) heat flows at an indoor and outdoor surface, respectively (W/m2) radiative load absorbed by a building element (W/m2) fraction of radiative load absorbed by a building element temperature (K or °C) indoor air temperature (K or °C) thermostat set point temperature (K or °C) equivalent outdoor temperature (K or °C) time (s) total daily operation time of a cooling unit (s or h/day) starting time (for initial conditions) (s) fenestration overall heat-transfer coefficient (W/m2 K) average overall heat-transfer coefficient of building envelope (W/m2 K) average outdoor windspeed (m/s) minimum and maximum values of ventilation, respectively (changes/h) cartesian coordinate (m)

Greek symbols a absorption coefficient for incident solar radiation DN indoor air-surface temperature difference (K or °C) Dt time step (s) e emission coefficient q density (kg/m3) s fenestration transmission coefficient for solar radiation Subscripts a air amb ambient E equipment e element of building envelope f equivalent furniture slabs i indoor J total number of layers of a multilayer building element j layer of a multilayer building element L lighting l lower (i.e. indoor) surface of a ceiling o outdoor P people p partition S solar radiation v indoor surfaces of building envelope, partitions or equivalent furniture slabs

achieved by optimizing the transitions between inoccupation and occupation phases. In the same study it is identified that a main problem in air-conditioning systems is their slow dynamics, usually with time delays. Approaches to cope with this are related to fuzzy logic [14,15], neural networks [16] or genetic algorithms [17]. A living space comfort regulator is described in Ref. [14] using fuzzy logic. After the tests made, it is concluded that the proposed systems shows satisfactory performance. In Ref. [15] it is concluded that the proposed combined air-conditioning system operated under different ventilation strategies and controlled by an intelligent fuzzy logic controller can be considered as an efficient technology to achieve good thermal comfort, improved indoor air quality and energy conservation in the modern air-conditioning applications. In Ref. [16], general regression neural networks are


used to optimize air-conditioning setback scheduling in public buildings. The results show that the neural control-scheme is a powerful instrument for optimizing air-conditioning setback scheduling based on external temperature records. In Ref. [18], a knowledge-based automation approach is proposed to support air-conditioning operations, aimed at improving energy conservation and indoor air quality control. It is concluded that a proposed intelligent operation support system, consisting of expert systems, provides a real-time integrated operation planning method and can be used to assist or train operators to achieve better operation in air-conditioning systems. In many studies energy saving is based on the proper combination between thermal comfort requirements and thermostat operation. For example, in Ref. [19] building thermal behavior analysis is integrated with thermal comfort models for determining the appropriate cooling technologies with low energy consumption. It is concluded that systems using a chilled ceiling offer the best thermal comfort for occupants in subtropical climates. Experiments showed [20] that the extent of energy savings in a building is a function of its occupancy schedule during day and week and may reach 46% if a programmable thermostat is used. Considerable energy savings may be obtained if the recent findings on transient effects concerning the conventional steady-state thermal comfort theories are taken into account [21]. The analysis of Ref. [22] concludes that ‘‘thermostat management’’, based on larger sample sizes, uniform sampling designs and instruments, collection of engineering, social, behavioral and attitudinal data, multivariate analysis and longterm studies will produce considerable energy savings. Additional examples in this category of space heating and cooling energy saving efforts are the monthly optimization approach [23] and the thermal comfort compromise with adaptive opportunities [24]. The first study [23] utilizes thermal comfort chart where indoor air temperatures are selected inside the summer and winter comfort-zones, as a function of relative humidity, in a manner to provide the highest comfort level while maximizing energy savings. It is concluded that considerable energy savings are achieved by utilizing the optimized monthly-fixed thermostat setting scheme developed in the study. In the second study [24], which is based on a quantitative interview survey of 3094 respondents in Finland, it is concluded that the perceived control over room temperature is remarkably low in offices, while the higher thermal comfort levels in homes are supported by greater adaptive opportunities. In some studies the nature of thermal comfort is called in question and this facilitates expropriation from building occupants of their autonomy to control their own immediate environment. For example, in Ref. [25] it is concluded that there now exist deeply entrenched but restricted notions about the nature of comfort itself, and about how and by whom, acceptable environmental conditions should be created and maintained in buildings. Thus, control is transferred to automated and centralized energy saving systems. Energy savings are reported in Ref. [26] which follows mild tactics by providing inhabitants with information on energy consumptions. The conclusion is that energy-saving consciousness was raised and energy consumption was reduced by the energy-saving activities of the household members. In Ref. [27] it is shown that energy requirements for heating and cooling buildings depend strongly on the combination of local climate with the operation control of air-conditioning equipment. Considerable energy savings are reported for decreasing and increasing thermostat settings during winter and summer, respectively, in conjunction with natural ventilation under certain outdoor conditions. Cooling energy saving efforts of the present study fall, in principle, within the above class of studies [19–27], which seek the energy saving in the proper combination between thermal comfort requirements and thermostat operation. A portion of our efforts is based on the combination of thermal comfort requirements with

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our detailed predictions concerning the ideal space cooling mode. The latter is defined as the mode, in which the thermostat throttling range is equal to zero and the indoor air-temperature remains constant and equal to the thermostat set point temperature. The above mentioned predictions of the ideal as well as of various real space cooling modes have been obtained by use of a developed rigorous model and a corresponding finite-difference procedure, suitable for the analysis of buildings thermal behavior under the influence of all possible thermal loads and the ‘‘guidance’’ of cooling control system. The predictions obtained are used in the analysis of the effects of thermostat operation mode and cooling power function in terms of the time on the total cooling energy consumption, for various usually encountered cases. The results of the analysis lead to energy saving suggestions. The study refers to the Typical Athens Buildings (TAB) during the typical Athens summer day, under various values combinations of the parameters involved. The daily variation of ambient temperature and solar radiation used for the typical Athens summer day were obtained by statistical processing of hourly measurements over a period of 20 years [28–30]. The characteristics of the TAB and the ranges of indoor and outdoor heat loads were defined by ‘‘averaging’’ the data of a local research. The latter have been modified in order to be in agreement with the Hellenic Directive on the energy performance of buildings, which was published in the Hellenic Official Government Gazette Issue 407/9-4-2010, as provided for in Law 3661/2008. The Hellenic Directive adapted the previous legislation on the subject of buildings to the European Union Directive 91/2000 on energy performance of buildings. Most of the conclusions of the study may be used for regions with a climate similar to that of Athens. 2. Differential equations and solution procedure As existing codes for the energy analysis of buildings, for example [31,32], required extensive modifications for the purposes of our analysis, it was decided to develop a detailed finite-difference procedure adjusted exactly to the present needs. The procedure is based on a previous one suitable for the simulation of buildings thermal behavior, developed and tested in previous studies [5– 8,33]. For the presentation of our analysis concerning the relation among cooling energy consumption, thermostat operation and comfort indoor air temperature in Athens buildings, it is necessary to summarize below the basic characteristics of the method and point out the new elements that needed to be introduced. As there are no validation data for testing these new elements, the reader must be aware of the fact that some of the results of the study are subject to plausibility and accuracy of a few unchecked modeling practices followed. 2.1. Thermal behavior of building envelope The thermal behavior of the multilayer building envelope elements e (exterior walls, fenestration, ceiling and floor) is expressed by the transient one-dimensional heat conduction differential equation, as only the direction normal to the wall, fenestration and other extended surfaces present significant temperature variation, i.e.

qej cej @T ej ðt; xÞ=@t ¼ kej @ 2 T ej ðt; xÞ=@x2 ; xj 6 x 6 xj þ Bej ; j ¼ 1; 2 . . . ; J

ð1Þ

where Tej(t, x) is the temperature of any layer j of multilayer envelope element e at time t and depth x, measured from the outdoor surface; J is the number of layers each envelope element is composed of; qej, cej, kej and Bej are the density, thermal capacity, thermal conductivity and thickness of each layer j of multilayer element

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e, respectively; and xj, xj + Bej are the coordinates of the jth layer surfaces of element e. The boundary conditions for the building envelope may be written as:

qo;e ðtÞ ¼ ho ½T o ðtÞ T e1 ðt; xÞ; qi;e ðtÞ ¼ hi ½T eJ ðt; xÞ T i ðtÞ þ

X v

x¼0

ð2Þ

g e;v ½T eJ ðt;xÞ T v ðtÞ þ Re ðtÞ; x ¼ xJ þ BeJ ð3Þ

where Ti (t) and To(t) are the indoor and outdoor air temperatures, respectively; qi,e(t) and hi are the heat flow and the convection coefficient at the indoor surface of element e of building envelope, respectively, while qo,e(t) and ho stand for the corresponding quantities for the outdoor surface; Rv denotes summation over indoor surfaces v; and ge,v is the radiation heat-transfer factor between indoor surface of element e and any other indoor surface v of temperature Tv(t). The solar radiation GT(t) incident on outdoor walls or roof surface is taken into account by using the sol–air temperature concept [34]:

T o ðtÞ ¼ T amb ðtÞ þ aGT ðtÞ=ho L

ð4Þ

where Tamb(t) is the real ambient temperature, a is the absorption coefficient of the exterior surface for the incident solar radiation GT(t) and L expresses normalized radiative loss, as analyzed in [34]. The term Re (t) in Eq. (3) expresses the part of solar radiation transmitted through any opposite fenestration, and the parts of the radiative loads from lighting Qr,L(t), equipment Qr,E(t) and people Qr,P(t), respectively, which are absorbed by the indoor surface of any element e of the envelope, i.e.

Re ðtÞ ¼

r s G0T ðtÞAS =Ae

s

þ r P Q r;P ðtÞ=Ae

þ r L Q r;L ðtÞ=Ae þ rE Q r;E ðtÞ=Ae ð5Þ

where G0T ðtÞ (in W/m2) is the incident to the fenestration solar radiation; AS and Ae denote the areas of the fenestration and the opposite envelope element, respectively; s is the fenestration transmission coefficient for solar radiation; and coefficients rS, rL, rE and rP are the fractions of transmitted solar radiation and radiative loads from lighting, equipment and people, respectively, absorbed by the indoor surface of the envelope element e. The remaining fractions of radiative loads (1 rS), (1 rL), (1 rE) and (1 rP) are directly or after reflection absorbed by the indoor air, as implied by Eq. (14), given later. The surface orientation is taken into account in the values of GT(t) and G0T ðtÞ by using the model described in detail in Ref. [30]. This model provides the value of the total solar radiation incident on surfaces of any orientation in terms of the values of direct and diffuse solar radiation incident on the horizontal plane. The latter values have been calculated in Ref. [30] for the Athens area, by a statistical processing of local hourly measurements over a period of 20 years. Boundary conditions for outdoor and indoor fenestration surfaces are similar to those expressed by Eqs. (2) and (3), respectively, with the differences that To (t) is replaced by Tamb (t) and term Re (t) is omitted. The percentage of solar radiation absorbed by fenestration is taken into account as a source term in the corresponding transient heat conduction differential equation (1). On the lower surface of floors directly in contact with the ground, or over an underground non-ventilated basement, adiabatic boundary conditions are imposed [34]. If the floor lower surface is in contact with the ambient, boundary condition expressed by Eq. (2) is used. On the upper floor surface boundary condition expressed by Eq. (3) is applied. Convection heat-transfer coefficients at the indoor, hi, and outdoor, ho, wall and fenestration surfaces, contained in Eqs. (2) and (3), are calculated using the following correlations [35–37]:

hi ¼ ½½1:5ðDT=3Þ1=4 6 þ ð1:23DT 1=3 Þ6 1=6 ; ho ¼ 4:1V o þ 5:8;

in W=m2 K

in W=m2 K

ð6Þ ð7Þ

where DT (in K) is the indoor air-surface temperature difference, which varies with time, and Vo (in m/s) is the average outdoor windspeed. The values given by Eq. (6) have been slightly increased, in order to be in agreement with ASHRAE [34] suggestions. The 24-h mean value of ho calculated from Eq. (7) is in agreement with ASHRAE [34]. Convection heat-transfer coefficient hl at the indoor (i.e. lower) surface of the ceiling has been calculated from equation [38]:

hl ¼ 2:175ðDTÞ0:308 =ðHDÞ0:076 ;

in W=m2 K

ð8Þ

where DN (in K) is the time dependent indoor air-surface temperature difference and HD (in m) is the ceiling ‘‘hydraulic diameter’’. The same values of convection coefficient are used for the upper floor surface. For the outdoor ceiling surface, Eq. (7) is used. The heat-transfer coefficients for radiative heat exchange among indoor surfaces ge,v, in Eq. (3), are expressed by the following correlation [39]:

g e;v ¼ 8eðDTÞn ;

in W=m2 K

ð9Þ

where e is the emission coefficient taken equal to 0.7 for usual indoor finishing layer surfaces; DT (in K) is the indoor air-surface temperature difference, which varies with time; and the value of the exponent n ranges from 0.06 to 0.1, depending on the indoor surface temperatures. In our calculations it has been taken n = 0.1. 2.2. Thermal behavior of indoor walls and ceilings (partitions) The temperature distribution Tpj (t, x) within any indoor multilayer partition p, composed of j = 1, 2, . . . , J layers, each one of thickness Bpj, is calculated from the transient one-dimensional heat conduction Eq. (1), in which subscript e is replaced by p. Boundary conditions at the two sides of any partition p may be expressed as

qp1 ðtÞ ¼ hi ½T p1 ðt; 0Þ T i ðtÞ þ

X v

g p;v ½T p1 ðt; 0Þ T v ðtÞ þ Rp1 ðtÞ ð10Þ

qpJ ðtÞ ¼ hi ½T pJ ðt;xJ þ BpJ T i ðtÞ þ

X v

g p;v ½T pJ ðt;xJ þ BpJ Þ T v ðtÞ þ RpJ ðtÞ ð11Þ

where qp1(t) and qpJ(t) are the heat flows at the two sides (first and last layers j = 1 and j = J) of partition p at time t, respectively; Tp1(t, 0) and TpJ(t, xJ + BpJ) are the temperatures of the corresponding sides of the partition; Ti(t) stands for the indoor air temperature and hi denotes the convection heat-transfer coefficient at partition surfaces, calculated from Eq. (6); gp,v is the radiation heat-transfer factor between surfaces of partition p and any other indoor surface v of temperature Tv(t); and Rp1(t), RpJ(t) express the parts of solar radiation, transmitted through any opposite fenestration, and the parts of the radiative loads from lighting, equipment and people, respectively, which are absorbed by the two partition sides 1 and J. Terms Rp1(t) and RpJ(t) are calculated from Eq. (5), in which Ae is replaced by the area Ap of partition side, which absorbs the radiation. 2.3. Thermal behavior of furniture An accurate calculation of indoor furnishings heat transfer behavior requires solution of the transient three-dimensional heat conduction differential equation. Such a solution would be difficult and time consuming mainly because of the complex geometry of furnishings boundaries. In the present study, as well as in some of

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the previous ones [6–8] a much more convenient (but less accurate) approach was followed, in which furniture is simulated by equivalent multilayer slabs, composed of the usual furnishings materials (wood, plastics, glass, textile mater, metal, etc.), with approximately equal mean thermal capacity, mean specific thermal conductivity and exterior surface area. The thermal behavior of the equivalent furniture slabs is predicted by using the same equations as those for the indoor partitions, with subscript p replaced by f. 2.4. Thermal energy balance of indoor air The indoor air thermal energy balance may be expressed by differential equation:

ma ca @T i ðtÞ=@t ¼ Q 1 ðtÞ þ Q 2 ðtÞ Q c ðtÞ

ð12Þ

where Ti(t), ma and ca denote the temperature at time t, which is a single value for the whole building, the mass and the thermal capacity of indoor air, respectively; Q1(t) stands for the sum of heat flows from envelope indoor surface, partitions and equivalent furnishings slabs; Q2(t) denotes the total indoor load from various sources; and Qc (t) is the cooling power provided by the air-conditioning system. Loads Q1(t) and Q2(t) may be expressed as:

Q 1 ðtÞ ¼

X e

X X qi;e ðtÞAe þ ½qp1 ðtÞ þ qpJ ðtÞAp þ ½qf 1 ðtÞ þ qfJ ðtÞAf p

f

ð13Þ

ð14Þ

where summation Re refers to the e (=1, 2, . . .) elements of building envelope, with corresponding indoor heat-transfer surfaces Ae and heat flows qi,e(t); summations Rp and Rf refer to the p(=1, 2, . . .) indoor partitions and to the f(=1, 2, . . .) equivalent furnishings slabs, respectively, with corresponding heat-transfer surfaces Ap and Af and heat flows at either sides (qp1, qpJ) and (qf1, qfJ); QV(t) is the load resulting from ventilation and infiltration; and (1 rL)QL(t), (1 rE)QE(t), (1 rP)QP(t) and ð1 r s ÞsAS G0T ðtÞ represent the parts of radiative loads from lighting, equipment and people, and the part of transmitted solar radiation through fenestration, respectively, which are directly or after reflection absorbed by the indoor air. The heat load resulting from ventilation and infiltration Qv (t), contained in Eq. (14), is calculated as:

Q v ðtÞ ¼ ð1 þ iÞC½cao T o ðtÞ ca T i ðtÞma =3600

T i ðt þ DtÞ ¼ T i ðtÞ þ ½Q 1 ðtÞ þ Q 2 ðtÞ Q c ðtÞDt=ðma ca Þ

ð16Þ

The rate of heat removal from the space Qc(t), contained in the above equation, may be specified according to the following different ways, depending on the operation mode of thermostat and cooling unit, as described in the following section. Solution is repeated for consecutive days with identical outdoor conditions until convergence to the periodic steady-state, in which solution repeats itself every 24-h. 3. Operation mode of thermostat and cooling unit 3.1. Ideal operation: constant indoor air-temperature T i ðtÞ ¼ T i In this case the thermostat throttling range DT⁄ is equal to zero and the indoor air-temperature remains constant and equal to the thermostat set point temperature T i , i.e.

DT ¼ 0;

T i ðtÞ ¼ T i ðt þ DtÞ ¼ T i

ð17Þ

Therefore, Eq. (16) gives:

Q c ðtÞ ¼ Q 1 ðtÞ þ Q 2 ðtÞ

ð18Þ

i.e. the cooling power Qc(t) varies with time. 3.2. Constant or zero cooling power with specified thermostat set point temperature and throttling range

Q 2 ðtÞ ¼ Q V ðtÞ þ ð1 r L ÞQ L ðtÞ þ ð1 r E ÞQ E ðtÞ þ ð1 r P ÞQ P ðtÞ þ sð1 r S ÞAS G0T ðtÞ

door air-temperature Ti(t + Dt) at any time level t + Dt is calculated from the discretized form of differential equation (12), i.e.

ð15Þ

where C is the number of air changes per hour, cao is the outdoor air thermal capacity, and i, which takes the values from 0 to 1, is used for increasing Qv(t) value, so as to include infiltration load. All other symbols, contained in the above equation, have been defined earlier. The cooling energy consumption accounts for the outdoor air load as a result of mixing between circulating cool air and fresh warmer outdoor air. The mass percentage of fresh air used depends mainly on the number of air changes per hour. For example, for two changes per hour, 12 °C inlet cool air temperature and 26 °C indoor air temperature, the above percentage is 10%. Its load is included in the analysis through term Qv(t) contained in Eq. (14). 2.5. Finite-difference solution procedure Initial conditions (for t = to) are first prescribed for all temperature fields Tej(to, x), Tpj(to, x), Tfj(to, x) and for the indoor air-temperature Ti(to). Then, the temperature fields Tej(to + Dt, x), Tpj(to + Dt, x) and Tfj(to + Dt, x) at the next time level t = to + Dt are calculated by solving the transient one-dimensional heat conduction differential equation for the above variables, given in the previous sections, by employing a usual implicit finite-difference procedure [40]. The in-

A constant value Q c is assigned to Qc (t), which may be taken approximately equal to the 24-h mean value of the space load Qc (t) according to Eq. (18). Using this value, the indoor air-temperature Ti(t + Dt) is calculated from Eq. (16) and:

If

T i ðtÞ P T i þ DT =2;

then Q c ðtÞ ¼ Q c

ð19Þ

If

T i ðtÞ P T i DT =2;

then Q c ðtÞ ¼ 0

ð20Þ

If

T i DT =2 < T i ðtÞ < T i þ DT =2 and T i ðt DtÞ T i ðtÞ > 0;

If


ð21Þ

T i DT =2 < T i ðtÞ < T i þ DT =2 and T i ðt DtÞ T i ðtÞ < 0;

then Q c ðtÞ ¼ 0

ð22Þ

Eqs. (21) and (22) imply that if the indoor air-temperature decreases or increases with time within the thermostat throttling range DT⁄, the cooling power is taken equal to Q c or 0, respectively. 3.3. Linear variation of cooling power in terms of the indoor airtemperature within thermostat throttling range The cooling power Qc(t) is expressed as a linear function of the indoor air-temperature Ti(t) within the thermostat throttling range DT⁄, while for Ti(t) above or below of this range, Qc(t) is taken constant or zero, respectively, i.e.


If

T i ðtÞ P T i þ DT =2;

If

T i DT =2 < T i ðtÞ < T i þ DT =2;

then Q c ðtÞ

¼ a þ bT i ðtÞ If

T i ðtÞ 6 T i DT =2;

ð23Þ

then Q c ðtÞ ¼ 0

ð24Þ ð25Þ

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The value of Qc(t) resulting from the above equations is used in Eq. (16) for the calculation of Ti(t + Dt). Coefficients a and b, which may be considered as characteristics of the cooling unit performance, are calculated by applying the linear relation between Qc (t) and Ti(t) at the ends of the thermostat throttling range, i.e.

Q c ¼ a þ bðT i þ DT =2Þ

ð26Þ

0 ¼ a þ bðT i DT =2Þ

ð27Þ

Solution of the above set of equations for a and b gives:

a ¼ Q c =2 Q c T i =DT

ð28Þ

b ¼ Q c =DT

ð29Þ

The above approach is used by ASHRAE [41] in the transfer function technique for the calculation of heat extraction rate and space temperature. 3.4. Microprocessor-based operation In this case a microprocessor-based control system regulates cooling unit operation. Sensors signals from indoor and outdoor environment are processed by the microprocessor, which executes a software program and provides an output signal concerning the values of the cooling power Q c and parameters T i and DT⁄ in terms of the time. Microprocessor-based operation is not examined in the present study. It is mentioned just to inform the reader of all possible cases encountered in practice. 3.5. Comments The daily cooling energy consumption Ed is calculated as:

Ed ¼

X

Q c ðtÞDt

ð30Þ

Dt

where RDt denotes summation over all consecutive time steps Dt along 24-h. In the case of the ideal operation mode (Section 3.1) it is Qc(t) > 0 for all time steps, as implied by Eq. (18), while in all other operation modes there are time steps for which it is Qc (t) = 0, as stated in Eqs. (20), (22), and (25). Relation between cooling power Qc(t) and ventilation and infiltration load Qv(t) differs for each operation mode. In the case of the ideal operation mode (Section 3.1) Qc (t) and Qv (t) are related through Q2(t), as implied by Eqs. (14) and (18). For the operation modes of Sections 3.2 and 3.3, Qc(t) and Qv(t) are related through the indoor and outdoor air temperatures Ti (t) and To (t), respectively, as implied by Eqs. (19)–(25) for Qc(t) and Eq. (15) for Qv(t). The daily cooling energy consumption Ed is related to Qv (t) through Qc(t) as stated in Eq. (30). 4. Parameters of the analysis The effect of the thermostat set point temperature T i and throttling range DT⁄ on the cooling energy consumption will be examined for the ‘‘Typical Athens Buildings’’ (TAB), defined below. 4.1. Typical Athens Buildings (TAB) Our analysis refers to a typical multi-storey, detached building of square shape with floor directly in contact with the ground. The number of storeys are related to the total floor area A of the building, as follows: For 100 m2 6 A 6 300 m2, 2 storeys. For 300 m2 < A 6 600 m2, 3 storeys.

For 600 m2 < A 6 800 m2, 4 storeys. For 800 m2 < A 6 1000 m2, 5 storeys. and the total area of indoor walls is equal to the outdoor area of the exterior walls. The indoor air-temperature Ti(t) is taken as a single value for the whole building, as already mentioned in Section 2.4, otherwise our results and conclusions would lose their generality as they would depend on the design of rooms and storeys. Therefore, for the sake of generality, indoor partitions and furnishings are considered to be uniformly distributed within the building, operating only as a thermal mass. Under these conditions a schematic representation of the TAB is not required, as only the amount and thermal properties of indoor mass (and not the design of indoor partitions) affects the uniform indoor temperature Ti(t). The total area of indoor walls, as well as the number of storeys, in terms of the TAB total floor area, were defined by ‘‘averaging’’ the results of our research concerning the characteristics of the Athens buildings. The above results were suitably modified in order to be consistent with the related Directive (Official Government Gazette Issue 407/9-42010). Our intention to examine the effects of the total floor area A up to the value A = 1000 m2 made us consider multi-storey TAB, because: (a) one-storey buildings of 1000 m2 floor area are rare in Athens and (b) such buildings are not consistent with the above mentioned Hellenic Directive, as implied in Section 4.3 (iv). The exterior walls are constructed from five layers in the following order: 0.02 m exterior finishing layer, 0.09 m brickwork, 0.04 m insulation, 0.09 m brickwork and 0.02 m indoor finishing layer. The roof is composed of 6 layers in the following order: 0.07 m exterior gravel concrete, 0.01 m waterproof layer, 0.06 m insulation, 0.07 m concrete mixture, 0.14 m reinforced concrete and 0.015 m indoor finishing layer. The indoor walls are made of single bricks with 0.02 m finishing layers on both sides, while indoor ceilings or floors (i.e. between storeys) are constructed from 0.10 m upper floor tiles with cement mixture sub-layers, 0.05 m insulation layer, 0.14 m reinforced concrete slab and 0.015 m lower finishing layer. Of similar composition is the floor directly in contact with the ground (waterproof layer instead of the lower finishing layer). The four sides of the building are oriented towards the four main orientations. The exterior envelope surface is light-colored. Because of the uniform indoor air-temperature Ti(t) in the entire building, thermal insulation within indoor partitions (walls, ceilings, floors) is not required in our simulation. However, insulation is prescribed in indoor ceilings or floors of the TAB, as it is applied in practice to all new multi-storey buildings in Athens, which are supplied with autonomous storey-heating and cooling, paid by the inhabitant of the storey. 4.2. Ambient temperature and solar radiation The Athens daily variations of ambient temperature and solar radiation are used, obtained by statistical processing of hourly measurements over a period of 20 years, as described in Refs. [28–30]. 4.3. Ranges of parameters values for the TAB and corresponding heat loads Although the parameters involved in the present analysis are numerous, our preliminary calculations showed that their number can be considerably reduced because some of them: (i) Have very small effect on the final results. (ii) Should be fixed to constant or nearly constant values for physical, technical or construction reason.


(iii) Exhibit expected or even obvious behavior. (iv) Moreover, the Directive on the energy performance of buildings, which was published in the Official Government Gazette Issue 407/9-4-2010, as provided for in Greek Law 3661/2008, adapted the previous legislation on the subject of buildings to the European Union Directive 91/2000 on the energy performance of buildings. The new legislation prescribes limits to the values of the numerous building parameters. For example, the value of the average overall heat-transfer coefficient of buildings envelope, Um, should not exceed a maximum value, which depends on the climatic characteristics (climatic zones A,B,C,D) of building location and the ratio of the outdoor envelope surface to the volume of the building. Therefore, there is no discretion to change considerably the value of the average overall heattransfer coefficient Um. In our analysis Um variation in each TAB case is due only to variations of fenestration area, as mentioned below in paragraph (b). According to the above remarks, the most important parameters, which exhibit considerable practical interest and the ranges of their values, usually encountered in the TAB, are: (a) The size of TAB, expressed by the total floor area A. The range examined is from 100 m2 to 1000 m2. (b) The total load through fenestration, expressed by the total fenestration area of the TAB. The range examined is from 5% to 20% of the exterior walls area. The overall heat-transfer coefficient of fenestration Uf is taken equal to 2.8 W/m2 K for all cases apart from TAB of small floor areas (i.e. 100 m2 6 A 6 150 m2) with high fenestration percentages (i.e. from 15% to 20%). In these cases Uf is taken equal to 2.2 W/m2 K so that the resulting values of the average overall heat-transfer coefficient Um remains lower than the maximum value allowed by the Directive mentioned earlier in (iv). Indicative values showing the changes of Um, resulting by the variation of fenestration percentage, are given below for the limiting TAB cases: For A = 100 m2 and fenestration percentages from 5% to 20%, Um varies from 0.695 W/m2 K to 0.850 W/m2 K, respectively, while for A = 1000 m2 and the same range of fenestration percentages, Um varies from 0.706 W/m2 K to 0.966 W/m2 K, respectively. Fenestration area is equally divided among the four building sides. No shading devices are considered. (c) The ventilation and infiltration thermal load, expressed by the indoor air changes per hour. The range examined is from 2 to 6 changes per hour. The data collected from the Athens buildings research, mentioned in the Introduction, showed that an approximate value of infiltration load may be expressed as a 5% increase of the ventilation load, i.e. it was taken i = 0.05 in Eq. (15). (d) The indoor load owing to lighting, equipment and people is taken from 20 to 50 W per m2 of floor area.

4.5. Finite-difference solution parameters With the aid of the flow chart of Fig. 1, a brief description of the numerical algorithm is given below: (i) Calculate temperature fields in the envelope, indoor partitions and equivalent furniture slabs at time level t + Dt, using the corresponding known temperature fields and the known indoor air temperature at the previous time level t. (ii) Using the above results, calculate heat flows and all other quantities required for the calculation of the indoor air temperature at time level t + Dt from Eq. (16). (iii) If time level has not reached the 24-h period, then increase time by Dt and repeat steps (i)–(iii), otherwise go to step (iv). (iv) Repeat steps (i)–(iii) for the next 24-h period and if temperature fields of the last two 24-h periods differ less than 0.01 °C at all grid nodes, periodic steady-state solution has been reached. Otherwise proceed to the next 24-h period by repeating steps (i)–(iv). Different storeys are not handled separately, finite-difference equations are not coupled between the storeys and heat transfer does not take place between them, for the reasons explained in Section 4.1. The finite-difference equations sets are solved noniteratively by the tri-diagonal matrix algorithm. The number of cycles needed to reach the periodic steady-state solution depends on the initial conditions. A good initial guess is to assign a tempera-

All calculations have been carried out for thermostat set point temperatures ranging from 24 °C to 28 °C, as this is the range used during summer in the Athens area. 4.4. Values of various coefficients for convective and radiative heat transfer Coefficients hi, ho, hi,l and ge,v = gp,v = gf,v are calculated from Eqs. (6)–(9), while the values 0.85 and 0.44 have been assigned to coefficient s and a, respectively [34]. Because of the periodic steadystate solution (see Sections 2.5 and 4.5) the values of rS, rL, rE and rP have negligible effect on the final results.

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Fig. 1. Concise flow chart of the developed numerical algorithm.

2878


ture value equal to the average of the indoor and the 24-h mean outdoor temperature to all fields. In this case six 24-h cycles are needed. Finite-difference procedure was applied in all cases examined with time and space grid finess Dt = 60 s and Dx = 0.002 m, respectively. For the insulation it was taken Dx = 0.001 m. Convergence criterion was fixed to 0.01 °C. Grid independence study was carried out using 40%, 70%, 100% and 130% of the above mentioned grid spacing. The latter two cases gave maximum temperature differences of 0.01 °C, which were considered sufficiently small. Therefore, the 100% case was selected. The final results for all cases examined are independent of initial conditions because they correspond to the periodic steady-state, i.e. the solution repeats itself every 24-h, as mentioned earlier. With the time and space grid finess, convergence criterion and initial conditions given above, the computation time needed is one hour, using a PC with Intel processor i7@, 2.67 GHz, 32-bit, with Windows 7, Fortran Lahey 95 and 6 GB ram memory.

binations of the ends of parameters ranges. Small and capital symbols are used for the lower and upper ends, respectively, i.e.:

5. Results and discussion

Pd ðT i Þ ¼ 100½Ed ð24Þ Ed ðT i Þ=Ed ð24Þ

5.1. Ideal operation: constant indoor air-temperature T i ðtÞ ¼

T i

Fenestration:

from f ¼ 5% to F ¼ 20%

ð31Þ

Ventilation:

from

v ¼ 2changes=h to V ¼ 6 changes=h

ð32Þ

Indoor load:

from q ¼ 20 W=m2 to Q ¼ 50 W=m2

ð33Þ

Each line is characterized by the combination of the above ends of parameters ranges. For example, (F, v, q) denotes the combination F = 20%, v = 2 changes/h, q = 20 W/m2. Fig. 2b and Fig. 3b correspond to the 100 m2 and 1000 m2 TAB and show the percentage of daily cooling energy saving P d ðT i Þ with reference to 24 °C, in terms of the thermostat set point temperature T i , for each one of the parameters values combinations of Fig. 2a and Fig. 3a, respectively, calculated as

ð34Þ

In this case the thermostat throttling range DT⁄ is equal to zero and the indoor air-temperature Ti(t) remains constant and equal to the thermostat set point temperature T i , according to Eq. (17). Fig. 2a and Fig. 3a show the predicted daily cooling energy consumption, Ed ðT i Þ, in terms of the thermostat set point temperature T i for the 100 m2 and 1000 m2 TAB, respectively, and for all com-

Fig. 4 shows the predicted daily cooling energy consumption Ed ðT i Þ and the corresponding percentage of cooling energy saving with reference to 24 °C, P d ðT i Þ, in terms of the thermostat set point temperature T i for the 100 m2, 500 m2 and 1000 m2 TAB, with all other parameters fixed at the middle values of their ranges, given by Eqs. (31)–(33). Solid lines and the left side axis correspond to the energy consumption, while dash-lines and the right side axis refer to the percentage of energy saving. Fig. 4 complements Figs.

Fig. 2. Predicted (a) daily cooling energy consumption Ed and (b) percentage of daily cooling energy saving Pd with reference to 24 °C, in terms of the thermostat set point temperature T i for the 100 m2 TAB and for all combinations of the ends of parameters ranges, for the ideal operation.

Fig. 3. As in Fig. 2, but for the 1000 m2 TAB.

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Fig. 4. Predicted daily cooling energy consumption Ed (solid lines, left side axis) and percentage of daily cooling energy saving Pd with reference to 24 °C (dash-lines, right side axis), in terms of the thermostat set point temperature T i , for the 100 m2, 500 m2 and 1000 m2 TAB, with all other parameters fixed at the middle values of their ranges, for the ideal operation.

Fig. 5. Predicted daily cooling energy consumption per temperature degree ed in the range from 24 °C to 28 °C, in terms of the TAB floor area A, for all combinations of the ends of parameters ranges and for the ideal operation.

2 and 3 so that a better view of the cooling energy consumed and the percentage of saving, in terms of the selected indoor air temperature, is gained. The way of presentation of Ed(T⁄) and Pd(T⁄) in terms of T⁄ in the above Figs. 2–4 is related to our attempt to show the above functions for all combinations of the ends of parameters ranges. As, for clarity reasons, this information could not be presented in one figure only, it was decided to divide the results into Figs. 2 and 3 for the lowest and highest values of A-range, i.e. A = 100 m2 and A = 1000 m2, respectively. Moreover, in order to provide a more complete picture, a third figure (Fig. 4) has been added to show Ed(T⁄) and Pd(T⁄) functions for all parameters fixed at the middle of their ranges. Our results for the ends of A-range (i.e. A = 100 m2 and A = 1000 m2) have also been included in the same figure in order to show that the rate of drop of Ed and Pd with T⁄ increases with A. With reference to Figs. 2 and 3, the following comments may be made:

pd ¼ ½Pð 28Þ Pd ð24Þ=4 ¼ ½Ps ð28Þ 0=4 ¼ Pd ð28Þ=4

Ed ðT i Þ

T i

(a) The variation of the daily cooling energy in terms of is practically linear and the same happens with function Pd ðT i Þ, as implied by Eq. (34). (b) Among heat loads, ventilation rate exerts the strongest influence on the slope of Ed ¼ f ðT i Þ linear function. This influence is clearly shown in Fig. 2a and Fig. 3a, where lines may be separated into two groups, i.e. the solid ones with the small slope, which correspond to the small value of ventilation rate (v = 2 changes/h), and the dash-lines with the high slope, which correspond to the high value of ventilation rate (V = 6 changes/h). T i

on the Because of the predicted practically linear effect of daily cooling energy Ed, the latter may conveniently be expressed as the daily cooling energy per temperature degree, ed, i.e.

ed ¼ ½Ed ð24Þ Ed ð28Þ=4

ð35Þ

Fig. 5 shows the introduced ed in terms of the TAB floor area A (instead of T i ). It is interesting that the eight lines in Fig. 2a and Fig. 3a become four in Fig. 5, because lines with small (q = 20 W/ m2) and high (Q = 50 W/m2) indoor load coincide, as the latter is expressed per unit of floor area, just as ep is plotted in terms of the floor area. The daily cooling energy per temperature degree, ed, was introduced on the basis of the linearity found. In the same way a per-

centage of daily energy saving per temperature degree, pd, may also be introduced as

ð36Þ

and the percentage P1,2 and energy savings E1,2, obtained by increasing T i from T i1 to T i2 , become

P1;2 ¼ ðT i2 T i1 Þpd

ð37Þ

E1;2 ¼ ðT i2 T i1 Þed

ð38Þ

Alternatively, P1,2 and E1,2 can be calculated as:

P1;2 ¼ Pd ðT i2 Þ Pd ðT i1 Þ

ð39Þ

E1;2 ¼ Ed ðT i1 Þ Ed ðT i2 Þ

ð40Þ

A relation between the above quantities P1,2 and E1,2 may be derived by applying Eq. (34) for T i ¼ T i1 and T i ¼ T i2 , subtracting the resulting equations and then using Eqs. (39) and (40). The resulting relation between P1,2 and E1,2 is

E1;2 ¼ P 1;2 Eð24Þ=100

ð41Þ 2

Example: With reference to Fig. 3, for the case of an 1000 m TAB with heat loads (f, V, Q), i.e. f = 5%, V = 6 changes/h and Q = 50 W/ m2, Eqs. (36) and (35) give

Pd ¼ 32:55=4 ¼ 8:14%=K

ð42Þ

ed ¼ ð7:30 4:93Þ=4 ¼ 0:59 GJ=K

ð43Þ

Using the above values, the percentage P1,2 and the cooling energy saving E1,2, obtained by increasing T i from T i1 ¼ 25 C to T i2 ¼ 26:5 C, may be calculated from Eqs. (37) and (38), respectively, i.e.

P1;2 ¼ ð26:5 25Þ8:14 ¼ 12:21%

ð44Þ

E1;2 ¼ ð26:5 25Þ0:59 ¼ 0:89 GJ

ð45Þ

Alternatively, P1,2 and E1,2 may be calculated from Eqs. (39) and (40), respectively, i.e.

P1;2 ¼ Pd ð26:5Þ Pd ð25Þ ¼ 20:35 8:14 ¼ 12:21%

ð46Þ

E1;2 ¼ Ed ð25Þ Ed ð26:5Þ ¼ 6:71 5:82 ¼ 0:89 GJ

ð47Þ

The same value of E1,2 results by applying Eq. (41), i.e.

E1;2 ¼ 12:21 7:30=100 ¼ 0:89 GJ

ð48Þ

Therefore, in the case examined, the daily energy saved is 0.89 GJ, i.e. energy saving of 12.21% is obtained.

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5.2. Constant or zero cooling power with specified thermostat set point temperature and throttling range In this case, the thermostat set point temperature T i is specified and a non-zero constant value is assigned to the thermostat throttling range DT⁄. During the typical 24-h, cooling power is either constant, i.e. Q c ðtÞ ¼ Q c –0 or zero, according to Eqs. (19)–(22). The upper part of Fig. 6 shows the predicted indoor air-temperature Ti(t) in terms of the time for the typical 24-h. The triangular symbols in the lower part of the figure depict the top and bottom edges of the above Ti(t) variation for each time t. The diagonal dashline crossing the figure gives on the right side axis the total time during which indoor temperature Ti(t) remains below the low limit of thermostat throttling range. This time should be as short as possible, as it makes cooling energy consumption higher without improving thermal comfort. Fig. 6 corresponds to thermostat set point temperature and throttling range T i ¼ 27 C and DT⁄ = 2 °C, specific cooling power qc ¼ Q c =A ¼ 120 W=m2 , while all other parameters are fixed at the middle value of their ranges defined in Section 4.3. An interesting case is shown in Fig. 7, which corresponds to T i ¼ 24 C and DT⁄ = 2 °C, while all other parameters are fixed again at the middle value of their ranges. Here, the indoor air temperature from 11:00 h to 17:30 h remains within thermostat throttling range, as heat load is increased by the solar radiation during the above period.

Fig. 7. As in Fig. 6, but for T i = 24 °C and qc = 100 W/m2.

Predictions have been obtained for a great number of parameters values combinations within the parameters ranges defined in Section 4.3. The predictions showed that the specific cooling power qc (i.e. the cooling power per unit floor area, qc ¼ Q c =A) and the thermostat set point temperature T i and throttling range DT⁄ present, in all cases, the same trends of influence on the daily consumed cooling energy Ed and the operation time of the cooling unit. On the basis of the above finding, only the prediction corresponding to parameters values fixed at the middle of their ranges, defined in Section 4.3, are presented below in Figs. 8–10.

Fig. 6. Predicted indoor air-temperature Ti in terms of the time (upper part of the figure, left side axis) and top and bottom edges of the same variation (lower part of the figure, left side axis). Also, predicted total time during which indoor temperature remains below thermostat throttling range (dash-line, right side axis). Thermostat parameters: T i = 27 °C, DT⁄ = 2 °C. Non-ideal operation mode.

Fig. 8. Predicted daily cooling energy Eb, which causes indoor air-temperature to drop below the low limit of thermostat throttling range DT⁄, in terms of DT⁄, for various values of T i and two values of qc . Non-ideal operation mode.


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(c) Eb increases with specific cooling power qc , as the higher values of the latter give rise to indoor temperatures Ti(t) lower than the low end of DT⁄. (d) According to the above remarks (a)–(c), low values of the unwanted cooling power Eb are obtained by assigning low values to qc and T i and a high value to DT⁄, within the ranges allowed by the required level of thermal comfort. Fig. 9 shows the predicted total daily operation time of the cooling unit td in terms of the thermostat throttling range DT⁄ for various values of the thermostat set point temperature T i and two values of the specific cooling power, qc . With reference to the above Fig. 9, the following comments may be made:

Fig. 9. Predicted daily operation time td, in terms of the thermostat throttling range DT⁄, for various values of T i and two values of qc . Non-ideal operation mode.

Fig. 10. Predicted daily consumption of cooling energy Ed, in terms of the thermostat throttling range DT⁄, for various values of T i and two values of qc . Non-ideal operation mode.

When the indoor air-temperature Ti(t) is below the low limit of the thermostat throttling range ðT i DT =2Þ, the cooling power Qc (t) is taken equal to zero, according to Eq. (20). However, the nonzero value of Qc(t) given to the space when the indoor temperature was just above the low limit of the thermostat throttling range ðT i DT =2Þ may cause a drop of the indoor temperature below ðT i DT =2Þ, as shown in Fig. 6. Fig. 8 shows the predicted part Eb of the total 24-h cooling energy, which caused the indoor temperature to drop below ðT i DT =2Þ, as a function of DT⁄ for various values of the thermostat set point temperature T i and two values of the specific cooling power, i.e. qc ¼ 100 W=m2 and qc ¼ 200 W=m2 . Our interest in cooling energy Eb is due to the fact that it makes the total cooling energy higher, without improving thermal comfort. With reference to the above Fig. 8, the following remarks may be made: (a) Eb decreases with decreasing T i for all values of DT⁄, because the low end of DT⁄ drops with decreasing T i , so that the available specific cooling power qc is not enough for lowering indoor air-temperature Ti(t) below this low end. (b) For increasing DT⁄ both Eb and the effect of T i on Eb decrease because qc is not enough for lowering Ti(t) below the low end of DT⁄, which drops with increasing DT⁄.

(a) td increases for decreasing T i because, for obtaining lower indoor temperatures with the same specific cooling power, longer operation time of the cooling unit is required, as expected. (b) td increases for decreasing qc , as a long operation time of the cooling unit is required for obtaining a specified level of thermal comfort, when its specific cooling power is low. (c) For the lower values of qc , td increases with increasing DT⁄, while the opposite happens for the higher ones. The explanation is as follows: (i) For low qc : For increasing DT⁄ with constant T i , the low end of DT⁄ drops and the cooling unit should operate for a longer time to reach the low end of DT⁄. Therefore, td increases with DT⁄. (ii) For high qc : For increasing DT⁄ the low end of DT⁄ drops as above, but now, because of the high qc , the rate of temperature drop is higher. Therefore, indoor air-temperature Ti(t) drops below the low end of DT⁄ when DT⁄ is small. For increasing DT⁄, Ti (t) gradually leaves off dropping below the low end of DT⁄ and this makes cooling unit operation time shorter. Therefore, td decreases for increasing DT⁄. Fig. 10 provides the predicted daily consumption of cooling energy Ed, as a function of the thermostat throttling range DT⁄ for various values of the thermostat set point temperature T i and two values of the specific cooling power qc . The following remarks may be made with reference to the above figure: (a) The expected increase of Ed with decreasing T i is clearly shown. (b) For the lower values of qc , Ed increases with increasing DT⁄, while the opposite happens for the higher ones. The explanation is the same as in (c) for Fig. 9. (c) Because of the above behavior of function Ed = f(DT⁄), the pairs of the Ed versus DT⁄ lines for the low and high values of qc and for T i =24, 25, 26, 27 and 28 °C intersect at points P24, P25, P26, P27 and P28, respectively. At the lower values of DT⁄, i.e. towards the left side of points P, Ed is higher for the high values of qc , while the opposite happens towards the right side of points P. The above behavior is better pronounced in Fig. 11, where points P lie at about the middle values of DT⁄ range. The latter figure differs from Fig. 10 in the total floor area value (A = 200 m2 instead of 500 m2) and in the specific cooling power values (qc = 50 W/m2 and 100 W/m2 instead of 100 W/m2 and 200 W/m2). (d) The above finding suggests that energy saving may be obtained by selecting the appropriate values combination of DT⁄, T i and qc . For example, with reference to Fig. 11, for T i from 25 °C to 28 °C and a low value of qc , a value lower than 1.0 °C should be assigned to DT⁄, while for a high value of qc , DT⁄ should be higher than 1.0 °C. For T i = 24 °C, DT⁄

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Fig. 11. As in Fig. 10, but for different values of A and qc . The intersections of lines for different qc are better shown.

should be lower or higher than 1.2 °C for a low or high value of qc , respectively. In the above cases, the cooling energy saving may reach 0.04 GJ/day, i.e. savings up to 5% or to 8% may be obtained for T i = 24 °C or T i = 28 °C, respectively. Similarly, with reference to Fig. 10, cooling energy savings up to 9% may be obtained.

Fig. 12. As in Fig. 4, but for thermostat throttling range DT⁄ = 2 °C and linear variation of cooling power Qc in terms of the indoor air-temperature Ti within DT⁄, with maximum specific cooling power qc = 100 W/m2.

range DT⁄ and to the specific cooling power qc ¼ Q c =A, respectively, while all other parameters were fixed at the middle values of their ranges, given by Eqs. (31)–(33).

Extensive tests have been made for the present non-ideal operation, extracts of which have been presented in Figs. 6–11, above. The tests correspond to thermostat set point temperatures T i from 24 °C to 28 °C and throttling range DT⁄ from 0.1 °C to 2 °C, with numerous values combinations of the TAB parameters and related heat loads defined in Section 4.3. Comparison of the predictions obtained with the corresponding ones for the ideal operation (Section 5.1, Figs. 2–5) showed differences of up to 5% in the 24-h cooling energy consumption Ed ðT i Þ and of up to 20% in the 24-h energy saving percentage P d ðT i Þ when T i is increased from 24 °C to 28 °C. As in the above range, T i was found to have a practically linear effect on P d ðT i Þ, the up to 20% total energy saving gives a difference of up to 5% saving per °C, between the ideal (i.e. DT⁄ = 0) and real (i.e. DT⁄ from 0.1 °C to 2 °C) cases. Because of the above relatively small differences between the ideal and the real operation modes, Figs. 2–5 provide a good estimate of the energy savings obtained during the typical summer day in Athens, if we decide to increase thermostat set point temperature T i by any specified amount in the range from 24 °C to 28 °C, starting from any temperature within the above range. 5.3. Linear variation of cooling power in terms of the indoor airtemperature within thermostat throttling range In this case, the thermostat set point temperature T i and the throttling range DT⁄ are specified while the cooling power Qc(t) is defined as a linear function of the indoor air-temperature Ti (t) within the thermostat throttling range DT⁄, according to Eqs. (23)–(29). For Ti(t) above or below throttling range, cooling power takes the maximum Q c ðtÞ ¼ Q c and the minimum Qc(t) = 0 values of the linear variation, as implied by Eqs. (23) and (25), respectively. An example of our predictions for the present case is given in Fig. 12, which shows the daily cooling energy consumption Ed ðT i Þ (solid lines and left side axis) and the corresponding percentage of cooling energy saving with reference to 24 °C P d ðT i Þ (dash-lines and right side axis) in terms of the thermostat set point temperature T i for the A = 100 m2, 500 m2 and 1000 m2 TAB. The values of 2 °C and 100 W/m2 were assigned to the thermostat throttling

Fig. 13. Comparison of the daily cooling energy Ed ðT i Þ in terms of the thermostat set point temperature T i for the three operation modes of Sections 5.1–5.3.


5.4. Comparison of operation modes Fig. 13 compares the daily cooling energy Ed ðT i Þ in terms of the thermostat set point temperature T i for the three operation modes discussed above in the corresponding Sections 5.1–5.3. In all operation modes the values of the parameters are fixed at the middle values of their ranges. The figure shows that the lowest values of the daily cooling energy consumption Ed (T⁄) are obtained in the case of the linear variation of Qc(t). This happens because, in the ideal operation (DT⁄ = 0) the indoor air-temperature Ti (t) remains constant and equal to the set temperature T i , while in the case of linear variation of Qc(t), Ti(t) varies within the thermostat throttling range DT⁄ taking for longer periods values T i ðtÞ > T i than values T i ðtÞ < T i . The same explanation applies for the lower values of Ed ðT i Þ obtained in the linear operation mode, as compared to the constant or zero qc operation mode. Tests made for various values of the parameters involved showed that in the case of linear variation of Qc(t), the daily cooling energy consumption Ed ðT i Þ is up to 10% lower than the corresponding energy consumptions of the ideal and the constant or zero qc operation modes. Comparison between the ideal and the constant or zero qc operation modes (Fig. 13) shows that lower Ed ðT i Þ values are obtained in the ideal operation mode. The explanation is the same as in the previous cases. Tests made for various values of the parameters involved showed that qc is the parameter defining the difference between Ed ðT i Þ values in the two operation modes. The maximum energy savings in the constant or zero qc operation mode are obtained for the minimum qc , i.e. for the value below which comfort conditions cannot be achieved. A suggested value for qc is the 24-h mean value of the specific total space load Qc(t)/A, as already mentioned in Section 3.2. 6. Conclusions The developed model and the corresponding numerical procedure is one of the contributions of the present study, since no other similar finite-difference approaches combining buildings heattransfer and thermostat and cooling unit operation mode with thermal comfort requirements, have been found in the literature. The procedure was applied to the Typical Athens Buildings (TAB) during the typical Athens summer day, under various combinations of the parameters involved. Our local research for obtaining a concise classification of the Athens buildings and a definition of the ranges of indoor and outdoor loads may be considered as another contribution of the present study. However, the central contribution is related with the analysis of the effects of thermostat operation mode and cooling power function on the total cooling energy consumption for various usually encountered cases, in conjunction with thermal comfort requirements. The results of the analysis lead to energy saving suggestions and conclusions, the main of which are summarized below: 6.1. Ideal operation mode (constant indoor air-temperature T i ðtÞ ¼ T i and thermostat throttling range DT⁄ = 0) Extensive comparisons of the predictions obtained for the ideal operation with the corresponding ones for various real cases (i.e. for DT⁄ – 0 and variable indoor air-temperature Ti(t)) showed differences, in the 24-h cooling energy consumption Ed ðT i Þ, of up to 5% in most cases, and of up to 10% only in a few ones. Because of the above relatively small differences, our detailed predictions for the ideal operation mode (Figs. 2–5) gain considerable value: Although they can only approximately be reproduced in practice, they may be used as a basis for comparisons with real operation modes and provide estimates of expected energy savings. For

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example, our detailed ideal operation mode predictions provide a good estimate of the energy savings obtained if we decide to increase thermostat set point temperature T i by any specified amount in the range from 24 °C to 28 °C, starting from any temperature within the above range. Therefore, since the extent of cooling energy saving is a priori known, one can decide if (and how much) it is worthy to increase thermostat set point temperature T i at the expense of thermal comfort. Main conclusions referring to the ideal operation, which are approximately applicable to many real cases, are summarized below: (a) The variation of the daily cooling energy Ed ðT i Þ in terms of T i is practically linear in the range from T i = 24 °C to T i = 28 °C and the same happens with Pd ðT i Þ. (b) Among heat loads, ventilation rate exerts the strongest influence on the slope of Ed ¼ f ðT i Þ linear function. Thus, all Ed ¼ f ðT i Þ lines corresponding to the same ventilation rate have the same slope regardless of all other thermal load values. (c) If Ed ¼ f ðT i Þ is expressed as ed = f(A), i.e. in the form of daily energy consumption per temperature degree in terms of the TAB floor area A, the ed = f(A) lines coincide for any value of indoor load (expressed in W/m2) if all other thermal loads are the same, as shown in the example of Fig. 5. (d) The predicted limiting daily cooling energy values at thermostat set point temperature T i = 24 °C, may be considered characteristic and useful in practice: For A = 100, 500 and 1000 m2, Ed(24) = 0.418, 1.723 and 3.271 GJ/day for the lowest and Ed(24) = 1.070, 4.295 and 8.089 GJ/day for the highest thermal loads, respectively. (e) Characteristic and useful in practice are also the predicted limiting values of the daily cooling energy per temperature degree: For A = 100, 500 and 1000 m2, ed (24) = 29.5, 125.8 and 240.8 MJ/K-day for the lowest and ed (24) = 69.5, 312.5 and 608.3 MJ/K-day for the highest thermal loads, respectively. 6.2. Constant or zero cooling power with specified thermostat set point temperature and throttling range In this operation mode (Sections 3.2 and 5.2) the predictions showed that: (a) In most cases the indoor air-temperature Ti(t) falls below the low limit of the specified thermostat throttling range DT⁄. The daily cooling energy Eb, which causes Ti(t) to drop below the low limit of DT⁄ should be kept to a minimum, as it increases the daily total cooling energy Ed, without improving thermal comfort. It has been found that Eb decreases with increasing thermostat throttling range DT⁄ and decreasing specific cooling power qc ð¼ Q c =AÞ and thermostat set point temperature T i . (b) The total 24-h operation time of the cooling unit td increases with decreasing specific cooling power qc and thermostat set point temperature T i . For each value of T i there is a value qco of the specific cooling power, for which td is independent of DT⁄. For qc > qco or qc < qco , td decreases or increases with increasing DT⁄, respectively. (c) For each value of the thermostat set point temperature T i there is a value qco of the specific cooling power, for which the 24-h cooling energy consumption Ed is independent of thermostat throttling range DT⁄. For qc > qco or qc < qco , Ed decreases or increases with increasing DT⁄, respectively (Fig. 11). It has been found that energy savings of up to 10% may be obtained because of the above behavior of func-

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tion Ed = f(DT⁄). For example, for qc ¼ qc1 > qco and qc ¼ qc2 < qco the corresponding lines Ed1 = f(DT⁄) and Ed2 = f(DT⁄) intersect at a point P (DT P , EdP) (Fig. 11). For qc ¼ qc1 > qco the daily cooling energy consumption is lower for DT > DT P , while for qc ¼ qc2 < qco it is lower for DT < DT P . 6.3. Linear variation of cooling power in terms of the indoor airtemperature within thermostat throttling range DT⁄ In this case (Sections 3.3 and 5.3), the daily cooling energy consumption Ed was found to be up to 10% lower than the corresponding energy consumptions of the ideal and the constant or zero cooling power modes. 6.4. Other operation modes The model and corresponding numerical procedure developed may be used to simulate, analyze and test any operation mode before this is applied in practice, in order to select, in each case, the operation mode providing the highest energy saving in conjunction with an acceptable thermal comfort level. References [1] Eicker U. Cooling strategies, summer comfort and energy performance of a rehabilitated passive standard office building. Appl Energy 2010;87:2031–9. [2] Fioretti R, Palla A, Lanza LG, Principi P. Green roof energy and water related performance in the Mediterranean climate. Build Environ 2010;45:1890–904. [3] Jonsson A, Roos A. Visual and energy performance of switchable windows with antireflection coatings. Solar Energy 2010;84:1370–5. [4] Fallahi A, Haghighat F, Elsadi H. Energy performance assessment of doubleskin facade with thermal mass. Energy Build 2010;42:1499–509. [5] Antonopoulos KA. Analytical and numerical heat transfer in cooling panels. Int J Heat Mass Trans 1992;35:2777–82. [6] Antonopoulos KA, Democritou F. Periodic steady-state heat transfer in cooling panels. Int J Heat Fluid Flow 1993;14:94–100. [7] Antonopoulos KA, Tzivanidis C. Numerical solution of unsteady threedimensional heat transfer during space cooling using ceiling-embedded piping. Energy 1997;22:59–67. [8] Antonopoulos KA, Vrachopoulos M, Tzivanidis C. Experimental and theoretical studies of space cooling using ceiling-embedded piping. Appl Therm Eng 1997;17:351–67. [9] Antonopoulos KA, Vrachopoulos M, Tzivanidis C. Experimental evaluation of energy savings in air-conditioning using metal ceiling panels. Appl Therm Eng 1998;18:1129–38. [10] Tzivanidis C, Antonopoulos KA, Kravvaritis ED. Parametric analysis of space cooling systems based on night ceiling cooling with PCM-embedded piping. Int J Energy Res in press; Wiley on line library doi: 10.1002/er.1777. [11] Borreguero AM, Sanchez ML, Valverde JL, Carmona M, Rodriguez JF. Thermal testing and numerical simulation of gypsum wallboards incorporated with different PCMs content. Appl Energy 2011;88:930–7. [12] Xiao W, Wang X, Zhang Y. Analytical optimization of interior PCM for energy storage in a lightweight passive solar room. Appl Energy 2009;86:2013–8. [13] Morosan PD, Bourdais R, Dumur D, Buisson J. Building temperature regulation using a distributed model predictive control. Energy Build 2010;42:1445–52.

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