A transient model for the energy analysis of indoor spaces

Applied Energy 87 (2010) 3084–3091

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A transient model for the energy analysis of indoor spaces K.A. Antonopoulos *, F. Gioti, C. Tzivanidis School of Mechanical Engineering – Thermal Department, National Technical University of Athens, 9, Heroon Polytechniou Str., Zografou, 157 73 Athens, Greece

a r t i c l e

i n f o

Article history: Received 5 November 2009 Received in revised form 19 February 2010 Accepted 8 April 2010

Keywords: Building Indoor space Surface thermal capacitance Heat-loss coefficient Indoor space models Indoor thermal pulse

a b s t r a c t Using a finite-difference procedure, the dynamic energy response of indoor spaces under the influence of indoor energy pulses is analyzed. The method of analysis is simple and explicit and is based on the indoor surface thermal capacitance and heat-loss coefficient Cs and Ls respectively. It is demonstrated that these parameters characterize fully any specified indoor space, as far as its energy behaviour is concerned. Their values are calculated for an extended variety of indoor spaces, i.e. for various floor areas, floor dimensions ratios, indoor surface materials of envelope, partitions and furnishings, fenestration and indoor partitions areas. The range of validity of the present method of analysis is also defined and the corresponding deviations are quantified with reference to rigorous finite-difference solutions. The provided values of indoor space characteristics Cs and Ls may be used in a wide range of technological building applications, including comparisons and classifications of indoor spaces, design and selection of construction materials and furnishing as well as the investigation of effects from electric equipment, windows or doors opening, short-time ventilations, brief stay of visitors, etc. Ó 2010 Elsevier Ltd. All rights reserved.

1. Introduction Various kinds and components of thermal capacitance related to a building have been introduced and analyzed in previous publications [1–6]. Briefly: (a) The apparent thermal capacitance Ca of a building [1] results by adding distributed heat capacities of building elements into a lumped value and does not represent the real ability of a building to store heat, because the storage ability of construction elements is obviously different when these are distributed in the building or considered together forming a unified volume. Therefore, the only use of Ca may be that it provides a gross way for evaluating, characterizing, classifying or comparing buildings with respect to their ability to store heat. (b) The real or effective thermal capacitance Ceff of a building [1], which differs considerably from the apparent thermal capacitance Ca, quantifies the ability of a building to store thermal energy and is useful in dynamic thermal performance calculations, as for example when the ‘‘effective thermal capacitance model” [1,2] is employed. A method for the calculation of Ceff has been presented in [1].

* Corresponding author. E-mail address: [email protected] (K.A. Antonopoulos). 0306-2619/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.apenergy.2010.04.002

(c) The above mentioned effective thermal capacitance Ceff may be analyzed [3,4] into components Cenv and Cind, which quantify the ability of the building envelope and indoor mass (i.e. partitions and furnishings), respectively, to store thermal energy. The heat capacitances Cenv and Cind for any specified building may be calculated as described in [3], where the practical usefulness of this analysis is also discussed. (d) The above defined envelope thermal capacitance Cenv may be further analyzed into components corresponding to discrete sections of the envelope [5], i.e. the roof, a whole wall of a specific orientation, to envelope parts of different compositions (i.e. brickwork, concrete parts, etc.), or even to layers of the exterior multilayer walls. This analysis quantifies the thermal contribution of every element of the envelope and may improve its thermal behaviour if the related conclusions are taken into consideration in the design of buildings [5]. A method for performing the analysis of Cenv to its components is given in [5]. (e) The ‘‘indoor surface thermal capacitance”, Cs, characterizes the thermal inertia of an indoor space, which is supposed to consist of the indoor air, the indoor (thin) surface layers of the construction elements enclosing the space and the (thin) surface layers of indoor partitions and furnishings. Thus, Cs expresses the heat stored within indoor air and surface layers of walls and furnishings, per degree of mean temperature difference between indoor air and building envelope [6]. The indoor surface thermal capacitance Cs,

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Nomenclature A a,b c Ca Ceff Cenv Cind Cs Csr E e fp fr F Fe Fp g H h i k Ls Lsr q Q Qi r

area (m2) floor dimensions (m) specific heat (J/kg K) apparent thermal capacitance (J/K) effective thermal capacitance (J/K) envelope thermal capacitance (J/K) indoor mass thermal capacitance (J/K) indoor surface thermal capacitance for r = 1 (J/K) indoor surface thermal capacitance for r < 1 (J/K) construction element of building envelope error (°C) indoor partitions factor factor for floor dimensions ratio floor area (m2) indoor area of building envelope indoor partitions area radiation heat-transfer factor (W/m2 K) height of an indoor space (m) convection heat-transfer coefficient (W/m2 K) factor taking the values 1 or 2 for floors covered or not by a carpet, respectively thermal conductivity (W/mK) indoor space heat-loss coefficient for r = 1 (W/K) indoor space heat-loss coefficient for r < 1 (W/K) heat flow (W/m2) thermal load (W) indoor thermal load (W) dimensions ratio of a floor (r < 1)

when used in the related indoor surface thermal capacitance model [6] quantifies the response of indoor spaces to positive or negative thermal ‘‘pulses”, i.e. to indoor thermal loads of limited duration (i.e. up to 2:00 h). These will be noted in the following as (Qi, tl), where Qi (in W) is the power of the pulse and tl its duration (loading time). The above mentioned model falls within the class of ‘‘indoor building space models”, which by contrast with the ‘‘building envelope models”, are suitable for predicting the dynamic thermal response of an indoor space to an indoor heat pulse, but they are inefficient to predict the effect of lasting indoor or outdoor thermal loads. For the latter loads ‘‘envelope models” should be used. Examples of ‘‘indoor space models” are the effusivity [7], the thermal time constant [8] and the admittance models [9,10] while models based on numerical techniques (i.e. finite differences, finite elements, etc.) [1–5,11–14] belong to the category of ‘‘envelope models”, because solution covers the whole mass of the building and takes into account outdoor and indoor boundary conditions. Other examples of ‘‘envelope models” are the transfer functions [15] the thermal network simulations [16], procedures based on measurements [17] or derived from actual building performance data [18,19]. Related computer codes, such as the well known TRNSYS [20], and ESP-r [21] fall in the above category. Although a great number of studies has been published recently on buildings energy analysis using ‘‘envelope models”, for example Refs. [22–26], no recent studies have been found on ‘‘indoor space models”. The explanation for this disproportion may lie on the potentiality of the general and rigorous ‘‘envelope models”, which may provide accurate solutions for indoor space problems, but at the expense of simplicity, convenience and promptitude. The applied value and main characteristic of the present indoor space model lies on its simplicity and its easy and quick application. Our model can be applied even without the use of a computer.

S Sr t t1 tl tr T Tenv Ti Til Tio v x

q

indoor area of space envelope for r = 1 (m2) indoor area of space envelope for r < 1 (m2) time (s) time during which error remains lower than e loading time (s) restoration time (s) temperature (K or °C) envelope temperature (°C) indoor temperature (°C) indoor temperature at the end of loading period (°C) initial indoor temperature (°C) volume (m3) cartesian coordinate density (kg/m3)

Subscripts a air A surface e element of building envelope f furniture i indoor j wall layer J number of wall layers Max maximum o outdoor or initial value p partition s thermal loads from various sources t total

As will be demonstrated later, an indoor space is fully characterized by the indoor surface thermal capacitance Cs, defined earlier, and the ‘‘indoor space heat-loss coefficient” Ls, which represents the rate of indoor space heat-loss per degree of mean temperature difference between indoor air and building envelope. When a heat pulse Qi is given to the indoor space during the loading period tl, the indoor space energy balance during loading and restoration periods yields the differential equations:

C s dTiðtÞ=dt ¼ Q i  LS ½T i ðtÞ  T env ; C s dT i ðtÞ=dt ¼ Ls ½T i ðtÞ  T env ;

0  t  tl

ð1Þ

tl < t  tr

ð2Þ

respectively, where Ti(t) is the air temperature of the indoor space (supposed uniform) at time t, and Tenv is the mean envelope temperature. Restoration period expressed by Eq. (2), begins at time tl, i.e. at the moment at which load Qi is interrupted, and ends at time tr defining the moment at which the room temperature takes again its initial value Ti(0), i.e. Ti(tr) = Ti(0). Integration of Eqs. (1) and (2) with initial conditions Ti(0) = Tio and Ti(tl) = Til, respectively, gives:

T i ðtÞ ¼ T env þ ½T io  T env  Q i =Ls  expðLs t=C s Þ þ Q i =Ls ; 0  t  tl T i ðtÞ ¼ T env þ ½T il  T env  exp½ðLs ðt  tl Þ=C s ;

ð3Þ tl < t  tr

ð4Þ

Eqs. (3) and (4), provide the indoor space temperature Ti(t) during the ‘‘loading period” (0 6 t < tl) and the ‘‘restoration period” (tl < t 6 tr) provided that the indoor space characteristics Cs and Ls are known. The accuracy of this approach has been successfully tested against experimental and theoretical data in [6]. The purpose of the present article is:

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(a) To provide the required values of the characteristics Cs and Ls for various indoor space cases of practical interest. (b) To quantify the range of validity of the present analysis, i.e. to define the allowed range of values of the independent parameters Qi, tl, Tenv and Tio and the related level of the accuracy obtained. The value Til is not independent, as it is calculated from Eq. (3), i.e. Til = Ti(tl). Once tasks (a) and (b) are obtained, Eqs. (3) and (4) can be used in practice in a wide range of building applications, as discussed in Section 4. 2. Calculation of indoor space characteristics Cs and Ls

2.1. The finite-difference procedure As TRNSYS [20], or other similar codes, required extensive modifications for the calculation of indoor space characteristics Cs and Ls, it was decided to develop a detailed finite-difference procedure adjusted exactly to the present needs. The procedure is based on a previous one developed for the simulation of building thermal behaviour [11] and solves the following set of differential equations, which describe the one-dimensional transient heat conduction within exterior and interior elements of buildings (including furnishings suitably modeled) and the indoor thermal energy balance:

qEj cEj ½@T Ej ðt; xÞ=@t ¼ kEj ½@ 2 T Ej ðt; xÞ=@x2 ; xj 6 x 6 xjþ1 ;

j ¼ 1; 2; . . . ; J;

qE1 ðtÞ ¼ kE1 ½@T E1 ðt; 0Þ=@x ¼ ho ½T o ðtÞ  T E1 ðt; 0Þ; qEJ ðtÞ ¼ kEJ ½@T EJ ðt; xJþ1 Þ=@x ¼ hi ½T EJ ðt; xJþ1 Þ  T i ðtÞ þ

X

ð5Þ ð6Þ

g E;A ½T EJ ðt; xJþ1 Þ  T A ðtÞ;

ð7Þ

j ¼ 1; 2; . . . ; J;

ð8Þ

A

T Ej ð0; xÞ ¼ fEj ðxÞ;

xj 6 x 6 xjþ1 ;

qa v a ca ½@T i ðtÞ=@t ¼

X

qEJ ðtÞAE þ Q t ðtÞ;

ð9Þ

E

Q t ðtÞ ¼

X X  ½qp1 ðtÞ þ qpJ ðtÞAp þ qf 1 ðtÞ þ qfJ ðtÞ Af þ Q s ðtÞ; p

ð10Þ

f

where TEj (t, x) is the temperature at time t and location x within any layer j of construction element E (i.e. wall or roof or part of them, fenestration, etc.) of the building envelope, composed of J parallel layers. The remaining symbols are explained in detail in the list of symbols. The set of the above differential equations is closed by the corresponding transient heat-conduction equations for the interior partitions (subscript p) and ‘‘equivalent furniture slabs” (subscript f). Solution of the outlined set of differential equations is obtained by a usual finite-difference technique [27]. The results include the indoor air temperature variation with time Ti (t), as well as the temperature distribution within the elements of the building envelope TEj (t, x), the interior partitions Tpj (t, x) and the equivalent furniture slabs Tf (t, x). The code developed was tested successfully against TRNSYS code [20] and measurements [14], as follows: Transient thermal behaviour calculations were carried out by using TRNSYS [20] and our code for the same buildings and under the same conditions. Comparison of the results showed that the indoor air temperature differences, between the two sets of predictions, in all cases examined, were less than 0.5 °C. Comparisons

with the measurements presented in [14] also showed temperature differences less than 0.5 °C, apart from a decreasing disagreement less than 2.0 °C, observed at the initial part of the transient state, owing to inevitable differences in the initial conditions between experiment and numerical solution. 2.2. Calculation procedure of Cs, Ls The purpose is to find, for a specified indoor space, the pair (Cs, Ls) which provides sufficiently accurate values of the indoor air temperature Ti(t), when used in Eqs. (3) and (4), for 0 6 t 6 tl and tl < t 6 tr, respectively, for any value of the parameters Qi, tl, Tenv, Tio. This is obtained by ‘‘tuning” Eqs. (3) and (4) to follow the results Ti(t) obtained from the rigorous and accurate finite-difference solution described earlier. The restrictions imposed for the developed procedure include that (a) the effect of a heat pulse (Qi, tl) on the space envelope is restricted to a thin indoor layer corresponding to a small percentage of the space envelope width (i.e. less than 1%) for error in predicted indoor air temperature less than 1 °C. Detailed values for these quantities will be given later, (b) restriction (a) implies that there is no influence of the outdoor conditions. The developed procedure includes the following steps for a specified indoor space: 1. Select values of the parameters Qi, tl, Tenv, Tio. 2. Run the detailed finite-difference code which gives the (Ti(t), t) pairs with a specified time step (i.e. 1–60 s, depending on the case) during loading (0 6 t < tl) and restoration (tl < t 6 tr). 3. Select the first pair (Ti(t), t) of the loading period and, for various arbitrary Ls, calculate Cs from Eq. (3). Plot curve Cs = f(Ls) using the calculated values, as shown in the example of Fig. 1. 4. Repeat the above for the first pair of restoration period, using Eq. (4). The result is a new curve Cs = f(Ls), as shown in Fig. 1. 5. The intersection of the two curves in Fig. 1 gives the correct pair (Cs, Ls) because it satisfies both loading and restoration periods. 6. Repeat steps 3–5 for the following pairs (Ti(t), t) during loading and restoration periods. For each of the above pairs a new pair (Cs, Ls) is calculated. These pairs, which are very close to each other, are averaged to give the (Cs, Ls) pair corresponding to the values of the parameters selected in step 1. 7. Repeat steps 2–6 systematically for a great number of values combinations for parameters Qi, tl, Tenv, Tio. The result is a set of pairs (Cs, Ls), which are very close to each other. These are averaged to give the final pair (Cs, Ls).

Fig. 1. The intersection of the Cs = f(Ls) curves corresponding to loading and restoration of an indoor space, defines the values of Cs and Ls of this space.

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3. Results 3.1. Calculation of (Cs, Ls) for various usual cases Following the procedure outlined above, the values of (Cs, Ls) have been calculated for spaces with envelopes having indoor surface layers composed of (a) wood, (b) concrete and (c) usual finishing layer, which is the most often encountered material. Figs. 2 and 3 show the calculated values of Cs and Ls, respectively, for the above cases (a)–(c), in terms of the floor area of the indoor space. For simplicity, fenestration, indoor partitions and furniture have not been taken into account in the above figures. Also, the dimensions ratio r of the floor is equal to 1 (i.e. floor of a square shape). The influence of all the above characteristics will be introduced later. The examined materials of the indoor layers show the expected trends, i.e. the wooden layer has the highest values of Cs and the lowest of Ls, while the differences between concrete and finishing layer are small. Linear and 2nd order polynomials were tested for expressing analytically the calculated values of Cs and Ls in terms of the floor area F, with corresponding errors less than 6% and 3%, respectively. The errors introduced in the final results (i.e. Ti(t) given by Eqs. (3) and (4)) were very small, i.e. less than 0.5% and 0.3% for the linear and the 2nd order fitting, respectively. Therefore, the linear representation was selected because of its simplicity, i.e. For wooden indoor surface layer:

C s ¼ 2721F þ 89; 602

ð11Þ

Ls ¼ 2:3352F þ 304:45

ð12Þ

For concrete indoor surface layer:

C s ¼ 2667F þ 19; 665

ð13Þ

Ls ¼ 3:3598F þ 533:59

ð14Þ

Fig. 3. Calculated indoor space heat-loss coefficient Ls in terms of floor area F for spaces with envelope indoor surface layer composed of different materials.

Fig. 4. Calculated indoor surface thermal capacitance Cs in terms of floor area F for spaces with envelope indoor surface layer composed of the usual finishing material and for various percentages of fenestration area.

For indoor surface layer made by usual finishing material:

C s ¼ 3163F  51; 330

ð15Þ

Ls ¼ 4:5974F þ 306:66

ð16Þ

where F is the floor area of the indoor space. In Eqs. (11)–(16), Cs, Ls and F are in J/K, W/K and m2, respectively. The effect of fenestration is illustrated in Figs. 4 and 5, which show the calculated values of Cs and Ls, respectively, in terms of the floor area for 5% and 15% fenestration area. These percentages, which haven been taken on the basis of the floor area F, represent the usual lowest and highest values of fenestration area, so that all

Fig. 5. Calculated indoor space heat-loss coefficient Ls in terms of floor area F for spaces with envelope indoor surface layer composed of the usual finishing material and for various percentages of fenestration area.

other cases can be covered by interpolating to these values. The above results correspond to spaces with indoor surface composed of usual finishing material, as most commonly is encountered in practice. The results show the expected trend, i.e. for increasing fenestration area, Cs decreases while Ls increases. Linear fitting was selected for expressing the calculated values of Cs and LS in terms of the floor area F, with the same error level as in the case of Eqs. (11)–(16). The derived linear expressions are:

C s ¼ 3150F  57; 431

ð17Þ

Ls ¼ 5:0707F þ 666:54

ð18Þ

for 5% fenestration area, and

Fig. 2. Calculated indoor surface thermal capacitance Cs in terms of floor area F for spaces with envelope indoor surface layer composed of different materials.

C s ¼ 2948F  59; 758

ð19Þ

Ls ¼ 4:8392F þ 816:38

ð20Þ

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for 15% fenestration area, where F is the floor area of the indoor space. In Eqs. (17)–(20), Cs, Ls and F are in J/K, W/K and m2, respectively. The values of Cs and Ls for spaces with indoor partitions have been calculated using the procedure outlined in Section 2. It has been found that the total (i.e. of both sides) area of indoor partitions has a practically liner effect on both Cs and Ls. Thus, an excellent approximation of Cs and Ls can be obtained by multiplying Cs and Ls given by Eqs. (11)–(20) by factor

5 C < T env < 40 C

ð24Þ

f p ¼ 1 þ F p =F e

5 C < T io < 40 C

ð25Þ

ð21Þ

For the above purpose, Eqs. (3) and (4) have been applied using systematically all values of practical interest of the parameters Qi, tl, Tenv and Tio. The results in each case were compared with the corresponding results of the rigorous finite-difference procedure outlined in Section 2.1 and the corresponding errors were calculated. The first conclusion of this extensive investigation is that for all values of practical interest of the parameters Tenv and Tio, i.e.

where Fp is the area of both sides of partitions and Fe the indoor area of the building envelope. Eq. (21) implies that, as expected, for increasing surface of indoor partitions, Cs increases. The same happens with Ls, because this characteristic expresses heat losses of the indoor space towards the mass of the walls and not towards the ambient. In all the above cases, a square floor area has been supposed, i.e. the ratio r of floor dimensions was r = 1. The values of Cs and Ls for spaces with non-square floors (r < 1) have been calculated using the procedure of Section 2. It has been found that the indoor area of the space envelope has a practically linear effect on both Cs and Ls, i.e.

the error of the calculated values of indoor air temperature Ti(t), remains lower than e = 1 °C. This was an expected conclusion because the present analysis concerns indoor spaces without the influence of outdoor conditions. The remaining conclusions of the study, for maximum error less than 1 °C, are presented in a compact form in Figs. 6–8, which correspond to indoor space floor areas F = 60 m2, F = 120 m2 and F = 250 m2, respectively. The three curves in each of these figures correspond to loading times tl = 0.5 h, tl = 1.0 h and tl = 1.8 h, thus implying that tl should remain lower than 1.8 h. The allowed range of values of heat load Qi for each tl, is shown on the horizontal axes of the figures. For example, for F = 250 m2 (Fig. 8) the allowed ranges of Qi are

f r BC sr =C s ¼ Lsr =Ls ¼ Sr =S ¼ ½iF þ 2ða þ bÞH=ðiF þ 4F 0:5 HÞ

Q i < 13; 000 W for t l ¼ 0:5 h

ð26Þ

Q i < 11; 000 W for t l ¼ 1:0 h

ð27Þ

Q i < 4000 W for t l ¼ 1:8 h

ð28Þ

ð22Þ

where Cs and Ls are the indoor surface thermal capacitance and the indoor space heat-loss coefficient for the square floor space (i.e. for r = a/b = 1); Csr and Lsr are the corresponding quantities for spaces with non-square floor (i.e. for r = a/b < 1); S and Sr represent the indoor area of the space envelope for r = 1 and r < 1, respectively; H stands for the height of the indoor space; and a and b are the floor dimensions. Factor i is taken equal to 2 if the area F of both the floor and the ceiling are thermally active. If the floor is covered by a carpet or any other similar material, i is taken equal to 1, since it has been found that the thermal response of a covered floor to heat pulses is very small. Therefore, it is suggested that a covered floor should not be included in the indoor area of the space envelope. Solution of the set of equations F = ab and r = a/b with respect to a and b gives a = r0.5 F0.5 and b = F0.5/r0.5. Substitution of the above expressions of a and b into Eq. (22), and taking the usual value of storey height H = 3 m yields

f r ¼ ½iF

0:5

þ 6ðr0:5 þ 1=r 0:5 Þ=ðiF

0:5

þ 12Þ

The vertical axes of Figs. 6–8 show the period of time t1 for which error remains lower than 1 °C. For example, for F = 60 m2 (Fig. 6), for heat load Qi = 1000 W and loading time tl = 0.5 h, the period of time for which the error remains less than 1 °C, is

ð23Þ

Therefore, Csr and Lsr can be obtained by multiplying Cs and Ls of Eqs. (11)–(20) by the floor shape factor fr, given by Eq. (23). Our tests confirmed that this is an excellent approximation for the usual values of r, i.e. for 0.4 6 r 6 1. It can be easily seen from Eq. (23) that the indoor spaces with a square floor (r = 1) have the lowest values of Cs and Ls as they have the smallest envelope surface. Tests carried out using the numerical procedure of Section 2 confirmed that factors fp and fr can be used simultaneously in Eqs. (11)–(20) to simulate the combined effect of both indoor partitions and non-square floor (r < 1).

Fig. 6. Time t1, during which error remains lower than 1 °C, in terms of the heat load Qi for an indoor space of floor area F = 60 m2 and various loading times tl.

3.2. Validity range of the present analysis method The purpose of the present section is to calculate the allowed range of values of the independent operational parameters, i.e. – The heat load (pulse) offered to the indoor space, Qi. – The loading time, tl. – The mean envelope and initial indoor temperatures Tenv and Tio, respectively.

Fig. 7. Time t1, during which error remains lower than 1 °C, in terms of the heat load Qi for an indoor space of floor area F = 120 m2 and various loading times tl.

K.A. Antonopoulos et al. / Applied Energy 87 (2010) 3084–3091

Fig. 10. As in Fig. 7 for error less than 2 °C.

Fig. 8. Time t1, during which error remains lower than 1 °C, in terms of the heat load Qi for an indoor space of floor area F = 250 m2 and various loading times tl.

t1 = 8.5 h, starting from the beginning of the loading period. The time t1 may be shorter or longer than the restoration time tr. For t > t1, the error exceeds gradually the imposed limit of 1 °C. The unusual shape and trends of the curves in Figs. 6–8 is justified by the fact that these curves do not describe a physical phenomenon but only they summarize the behaviour of our model by defining in a compact and concise form (a) the ranges and combinations of parameters Qi, tl and F, for which the error of the model remains lower than 1 °C, and (b) the time t1 during which the model can be used with error less than 1 °C. An attempt to explain the features of the curves in Figs. 6–8 should be based on the nature of the suggested model, which is an ‘‘indoor space model”, i.e. the effect of indoor thermal load Qi, in conjunction with its duration tl, should be restricted to a thin indoor layer of the space envelope: The thinner the layer the more accurate the model. Therefore, (a) at a fixed loading time tl, for increasing Qi, indoor layer thickness increases thus making the duration of model validity t1 shorter, (b) for increasing loading time tl, the slope of t1 versus Qi curves increases for the same reason as above, (c) for the higher values of indoor thermal load (Qi > 3000 W), thermal mass and resistance of the influenced indoor layer become considerable and their combination seems to generate, for increasing Qi, a short and practically constant validity time t1. Figs. 9–11 show the corresponding results for error less than e = 2 °C. The allowed ranges of values of the parameters, as well as time t1, are now considerably increased, as expected. Successful comparisons of the proposed method with other analogous procedures have been made, concerning accuracy, calculation time and flexibility. An example is given in Fig. 12, which compares the indoor temperature variation in terms of the time with the corresponding predictions of a rigorous finite-difference solution. These results refer to a room of 60 m2 floor area with 20 °C initial indoor and envelope temperatures, which is heated for 0.5 h by a load of 1000 W. The maximum error observed is 0.36 °C, i.e. about 1.5%.

Fig. 9. As in Fig. 6 for error less than 2 °C.

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Fig. 11. As in Fig. 8 for error less than 2 °C.

Fig. 12. Results of the proposed analysis compared to the corresponding predictions of a rigorous finite-difference solution, under the conditions described in the text.

4. Applications The case studies of the present section refer to an indoor environment of square floor with area F = 60 m2 and indoor envelope layer made by usual finishing material. The initial mean envelope and indoor air temperatures are Tenv = Tio = 20 °C. Fig. 13 shows the air temperature of the above indoor space without partitions and furnishings, in terms of the time, calculated by using our model for the following usually encountered cases: (a) Doors open for tl = 5 min and the heat exchange between outdoor and indoor air gives a net thermal load of Qi = 2.5 kW to the indoor space. Curves 1 and 2 in Fig. 13 shows the predicted indoor air temperature variation for the cases of 5% and 15% fenestration, respectively. The effect of the different fenestration percentage is a 0.2 °C increase of indoor temperature and a small delay in temperature restoration to 20 °C in the case of less fenestration, as expected. The model predicts that the indoor air temperature returns to its initial value of 20 °C after 0.3 h = 18 min, measured from doors opening, or after 13 min measured from doors closing.

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Fig. 13. Indoor air temperature in terms of the time for six case-studies, calculated by using the model developed, under the conditions mentioned in the text.

Fig. 14. Indoor air temperature in terms of the time for nine case-studies, calculated by using the model developed, under the conditions mentioned in the text.

(b) Operation of a 1600 W fan heater within the space considered for tl = 30 min. Curves 3 and 4 in Fig. 13 show the calculated indoor air temperature variation for the cases of 5% and 15% fenestration, respectively. The predicted indoor air temperature difference, as well as the slight delay in temperature restoration, owing to the different fenestration percentage, is clearly shown. The above temperature difference is slightly higher than that in case (a), although heat input is lower, because of the longer loading time in case (b) (tl = 30 min versus tl = 5 min). Return of the indoor air temperature to its initial value of 20 °C occurs after 0.75 h = 45 min, measured from the start of heater’s operation, or after 15 min from the end of its operation. Curves 3 and 4 in Fig. 13 show that a practically constant indoor air temperature is established after 10 min from the start of heaters operation. This behaviour can be considered as a simulation of the thermal equilibrium between the heat offered by the heater and the heat absorbed by the walls. (c) Persons enter and stay within the space for tl = 20 min with convected sensible heat load Qi = 1000 W. Curves 5 and 6 in Fig. 13 show the calculated indoor air temperature variation for the cases of 5% and 15% fenestration, respectively. The predicted indoor air temperature difference and the slight delay in temperature restoration, owing to the different fenestration percentages, are clearly shown, as in the previous applications (a) and (b). The above temperature difference is clearly lower than that in application (b), owing to the lower values of both the thermal load Qi and its duration tl. Return of the indoor air temperature to its initial value of 20 °C occurs after 0.55 h = 33 min, measured from persons entry into the space, or after 13 min from their departure. (d) For examining the behaviour of the model with regard to indoor walls, calculations have been made for the same space, as above, with 5% fenestration and 46 m2 indoor walls. Fig. 14 shows the calculated indoor air temperature in terms of the time for nine applications of fan heaters operation, i.e. for thermal loads Qi = 2.4 kW, 1.6 kW and 0.8 kW combined with loading times tl = 10 min, 20 min and 30 min.

indoor air temperature is higher, i.e. Ti,max = 21.65 °C and the restoration occurs quicker, i.e. after 15 min from the end of heaters operation. These trends are due to the surface thermal mass of indoor walls existing in the case of curve 6. Therefore, our model reproduces the expected differences in the behaviour of indoor spaces with and without indoor walls.

The most interesting of the curves in Fig. 14 is the one under number six, which corresponds to Qi = 1.6 kW and tl = 30 min. This curve can be compared to curve 3 of Fig. 13, which corresponds to the same values of Qi, tl and fenestration percentage (5%), as well as to the same indoor space, which does not contain indoor walls. In the case of curve 6 (Fig. 14) the maximum indoor air temperature is Ti,max = 21.25 °C and the restoration to its initial value of 20 °C occurs after 18 min from the end of fan heaters operation. In the case of curve 3 (Fig. 13, without indoor walls), the maximum

5. Conclusions The transient thermal response of indoor spaces to indoor heat pulses may be easily predicted with satisfactory accuracy by using the calculated values of characteristics Cs and Ls, in conjunction with Eqs. (3) and (4). According to the present analysis a specified indoor space is fully characterized by the indoor surface thermal capacitance and heat-loss coefficient Cs and Ls, respectively. Their values have been calculated for the indoor spaces usually encountered in practice, taking into account their basic characteristics, as for example the floor area F, its dimensions ratio r, the indoor surface materials of space envelope and partitions, the fenestration and indoor partitions surfaces. The present study also quantifies the allowed range of values of the basic parameters of the problem, as for example are the offered indoor thermal load, its duration and the initial temperatures of building envelope and indoor air. The error of the analysis is also quantified in all cases, with reference to rigorous finite-difference solutions. The proposed method of analysis can be used in practice in a wide range of applications. Related examples are the operation of electric equipment (i.e. in coffee or meal preparation, computer operation, photocopying, etc.), during a ventilation of a short duration (i.e. during doors or windows opening), when visitors remain indoor for a short period, when a small electric heater is switched on for a short period of time, etc. Further applications may be found in the comparison and classification of indoor spaces, as well as in the design and selection of construction materials in order to obtain a specified level of thermal inertia of indoor spaces. Also, use of the method may be made in passive applications as, for example, in regular or night ventilation for cooling. References [1] Antonopoulos KA, Koronaki EP. Apparent and effective thermal capacitance of buildings. Energy – Int. J. 1998;23:183–92. [2] K.A. Antonopoulos, E.P. Koronaki, On the heat capacity of Greek buildings. In: Proceedings of the 1st international conference on energy and the environment, Limassol, Cyprus, vol. 2, 1997, pp. 463–70. [3] Antonopoulos KA, Koronaki EP. Envelope and indoor thermal capacitance of buildings. Appl Therm Eng 1999;19:743–56.

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