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Energy and Buildings 76 (2014) 176184

Contents lists available at ScienceDirect

Energy and Buildings

j ourna l ho me pa g e: www.elsev ier .com/ locate /enbui ld

Model tenaccoun

AgnieszkDepartment of 4 31-

a r t i c l

Article history:Received 10 JuReceived in re28 November Accepted 21 F

Keywords:Intermittent hTransient heatControl volum

thodsed poff mos not es a m

heati capa

accut temicteduringtemp

1. Introduction

Reducinronmental asavings is incan work inperature asweekend reheating moa constant lmode, a buithermal com

Not all bent structurlight, mediuattention isair temperaature hot-wseason. In tcapacity sh

CorresponE-mail add

The paper is organised as follows. The model of unsteady state

http://dx.doi.o0378-7788/ g energy consumption in buildings is an important envi-nd economic issue. One of the methods leading to suchtermittent heating in which the central heating system

continuous heating mode at a constant set-point tem- well as in switch-off mode with a night time and/orduced set-point temperature [15]. During continuousde, thermal comfort indices PMV and PPD [6] are set atevel and no energy is saved. During switch-off heatingldings energy consumption is lowered, although insidefort is also decreased.

uildings are constructed similarly. They can have differ-es, i.e. with different thermal heat capacities: very light,m, heavy and very heavy elements [1]. In this paper the

focused on modelling the heat dynamics of the indoorture in a light building heated by a low surface temper-ater radiator in a moderate climate during the heatinghe building with a very light or light structure, radiatorould be taken into account.

ding author. Tel.: +48 12 632 09 48; fax: +48 12 628 20 48.ress: [email protected] (A. Lechowska).

heat exchange in buildings is introduced in Section 2. The calcula-tion data and test room is described in Section 3. The measurementand calculation results are presented in Section 4, and nally, con-clusions with discussion are given in Section 5.

2. Mathematical model

An energy balance of room internal air can be written as [79]:

VaacadTadt

= Qr + Qgn 6

j=1Qs Qwin (1)

where:

dTadt

= Ta2 Ta1t2 t1

(2)

Qwin = (UwinAwin + Hinf ) (Ta2 Tsol) (3)6

j=1Qs =

6j=1

AsjRsj + Rsj

(Ta2 Tsj2

)(4)

In Eq. (3) Hinf is constant value accounting for an external airinltration, dependent on several factors, including building height,window air inltration rate, and total length of gaps around the

rg/10.1016/j.enbuild.2014.02.0622014 Elsevier B.V. All rights reserved.of unsteady heat exchange for intermitt hot water radiator capacity

a Lechowska , Artur Guzik Environmental Engineering Cracow University of Technology ul. Warszawska Cracow 2

e i n f o

ly 2013vised form2013ebruary 2014

eating exchangee method

a b s t r a c t

Intermittent heating is one of the meheating system can work with reducoccupied. At the beginning of the cut-oto its thermal capacity. This capacity ivery light structures. This paper outlinlight wall structure with intermittentroom air capacity, hot water radiatoras well as air inltration. Reasonableachieved. With known air and radianpredicted mean vote (PMV) and predverify how thermal comfort changes dmeasured and calculated internal air t heating taking into

155, Poland

leading to savings in energy consumption. The intermittentwer or it can be completely cut off when the rooms are notde, the radiator remains warm for a specic period of time, duenegligible and should be considered for buildings with light orathematical model of unsteady heat exchange in rooms with

ng. The air heat balance of a given room takes into account thecity, heat transfer through walls, ceiling, oor and windowsracy between calculation and measurement results has beenperatures, air humidity and velocity, thermal comfort indices

percentage of dissatised (PPD) were evaluated in order to radiator cut-off mode. The satisfactory convergence betweeneratures has been achieved.

2014 Elsevier B.V. All rights reserved.

A. Lechowska, A. Guzik / Energy and Buildings 76 (2014) 176184 177

Nomenclature

a solar radiation absorptivityA c C I m n N Q t T U V

Greek let Qir

AbbreviaMAPE PMV PPD RMSE

Subscripta c calc e ew f gn h in iw meas n o r s sol w win

0 1 2

window. Exthe solair calculated a

Tsol = Te +a

Eq. (4) dboundary (the room ai

In Eq. (1) Qgn is a heat ow rate from internal heat gains from:occupants, equipment and lighting. The internal heat gains distri-bution is put in the model as known values [1214].

uming that the room heating system consists of the low--temperature horizontal hot-water radiator the supplied

ux can be expressed as:

Ur Ar

+ Ur Ar2mwcw(Tin1 Ta2) (6)

the overall radiator heat transfer coefcient is dened by:

n( )narea, m2

specic heat capacity, J/(kg K)radiator constantglobal solar irradiance, W/m2

mass, kgradiator constant exponentnumber of measurements or calculated valuesheat ow rate, Wtime, stemperature, K

Asssurfaceheat

Qr =1

whereoverall heat transfer coefcient, W/(m2 K)volume, m3

tersheat transfer coefcient, W/(m2 K)infrared radiation due to difference between theexternal air temperature and the sky temperature,W/m2

density, kg/m3

tionsmean absolute percentage errorpredicted mean votepredicted percentage od dissatisedroot mean square error, K

sinternal airceilingcalculated valueexternalexternal walloorinternal heat gainsend of heating season conditionsinlet, supply waterinternal wallmeasured valuenumber of time stepsdesign conditionsradiatorstructure, room opaque elementssolairwaterwindowheat convectionheat conductioninitialbeginning of time stepend of time step

ternal air temperature Tsol in Eq. (3), referred to astemperature, takes into account solar radiation and iss [10,11]:

I Qire

(5)

escribes the heat conduction and convection from aroom external and internal walls, oor and ceiling) tor.

Ur = C Ta1

In Eqs. (lated from:

Tin1 = Tin o

Once thtransfer coelated, both into (1), thon-mode isD + E + F

where:

D = Vaacat2 t1

E = Ur A1 + Ur2mw

F = UwinAw

Gj =Asj

Rsj +

Eq. (9) fance equatstep.

During rdecreasing.

(Vwwcw +

The soluinside air te

Tw1 = exp(K

where:

K = VwwcTin1Ta1

Ta1 TeTa1 Te o

Tr o2Ta1

1 (7)

6) and (7) radiator supply water temperature is calcu-

+ Tin h Tin oTe h Te o

(Te Te o) (8)

e supply water temperature and overall radiator heatfcient based on the previous time step data are calcu-values are inserted into Eq. (6). Inserting Eqs. (2) (6)e room air heat balance equation for radiator in the

given as:

+6

j=1Gj

Ta2 6

j=1GjTsj2 = D Ta1 + F Tsol + E Tin1 + Qgn

(9)

(10)

rArcw

(11)

in + Hinf (12)

Rsj(13)

or inside air during radiator on-mode with energy bal-ions comprise a set of equations solved at each time

adiator off-mode, water temperature is continuously Its energy balance equation can be expressed as:

mrcr)dTwdt

= rAr (Tw Ta) (14)

tion of Eq. (14), taking into account that both water andmperatures are time-dependent, is given by:

t2)

[Tw0 KB Ta1 KC Ta2 K

n1i=1

(Bi Ta1,i + Ci Ta2,i

)]

(15)

rAr

w + mrcr (16)

178 A. Lechowska, A. Guzik / Energy and Buildings 76 (2014) 176184

Fig. 1. Plan of analysed room.

B = 1K (t2 t1)

{[t2 exp (Kt2) t1 exp (Kt1)] [exp (Kt2)

exp( )}

C = 1K (t2 [t2 ex

Initial wator supplycut-off mod

Radiator

Qr = rAr (T

After substituting Eqs. (15) (19) into (1), the following heatbalance equation for radiator off-mode is obtained:

F +6

j

Gj + M + N

Ta2 6(GjTsj2)

P)

:

rAr

ArKC

ArKB

Fig. 2. I(Kt1)] t2 1K

(17)

t1)

{[exp (Kt2) exp (Kt1)]

(t1

1K

)p (Kt2) t1 exp (Kt1)]

}(18)

ater temperature Tw0 is assumed as an average of radi- and return temperatures from the last time step beforee.

heat ow rate during off-mode can be written as:

w1 Ta2) (19)

D +

= (D

where

M =

N = r

P = rnternal air temperature measured and calculated, solair temperature, radiator water te=1 j=1

Ta1 + FTsol + S W + Qgn (20)

(21)

exp (Kt2) (22)

exp (Kt2) (23)mperature measured and calculated 48 h period in February.


Fig. 3. The deviation of measured and calculated temperatures both of internal air and radiator water 48 h period in February.

S = rArTw0 exp (Kt2) (24)

W = rArK exp (Kt2)n1i=1

(Bi Ta1,i + Ci Ta2,i

)(25)

Using aiequation, throom durintions can buser-denesimulation

3. Experim

The matvalidated bment data c

The room had a single external wall, three internal walls andboth internal ceiling and oor. This is schematically presentedin Fig. 1. The window was orientated into north. The room wasequipped with a single low surface temperature, horizontal hot-

radiaed dinto l as teed. F-owle 1 lts an

meprob, prcy r energy balance Eq. (9) or Eq. (20) as the constrainte mathematical model of transient heat transfer in a

g radiator on- and off-mode can be solved. These equa-e relatively easily incorporated, via external library ord constraints, into existing commercial heat transferpackages.

ent

hematical model presented in the previous section wasy comparing the calculation results with the measure-ollected in an actual room.

water occupitaken as welrecordand air

Tabelemen

Thesors: 0.15 K)accuraFig. 4. Difference between measured and calculated values of internal air tor, working in on- and off-mode. The room was noturing measurements, so no internal heat gains wereaccount. Both the external and internal temperatures,mperatures of the adjacent rooms were measured andurthermore, radiant temperatures, relative humidity,

speed were also measured.ists values of input data of room parameters, its opaqued the radiator.asurement system included the following sen-es for air temperature (Pt100 with accuracy obe for radiant temperature measurement (with0.15 K), probe for measurement of surface temperaturetemperatures 48 h period in February.


Fig. 5. Internal air temperature measured and calculated, solair temperature, radiator water temperature measured and calculated 24 h period in April.

(Pt100 with accuracy 0.5 K), thermohygrometric probe (withrelative air humidity accuracy 3%) and hot wire anemometer(with air speed accuracy 0.05 m/s). Measurements were takenevery 3 min

As a numthe Controinternal walayers, and centre. Theaccount theand/or, in cacore and borounding aias adjacent ments), wh

4. Results

This section presents selected calculation and measurement. Fig.sultsd off

3 preraturand K andte detivel

4 pr valu and logged automatically on a connected PC [1518].erical method used for digitizing mathematical model

l Volume Method (CVM) was selected. External andlls, as well as the ceiling and oor were divided intoeach layer represented by its core located at the mass

energy balance equation for each core takes intormal conductivity between the core and its neighboursse of boundary core, thermal conductivity between theundary and convection between the boundary and sur-r. The model assumed external air temperature as wellrooms temperatures to be known (taken from measure-ile cores and internal air temperatures to be unknown.

resultstion reon- an

Fig.tempeimum to 2.7 absolurespec

Fig.culatedFig. 6. The deviation of measured and calculated temperatures both of internal 2 presents a comparison of measurement and simula- for a selected 48-h period in February, containing two- modes.sents the deviations between measured and calculatedes of both internal air and the radiator water. The max-average absolute deviations of internal air are equal

0.5 K respectively, while the maximum and averageviations of radiator water are equal to 4.2 K and 0.7 Ky.esents the difference between measurements and cal-es of internal air temperatures. air and radiator water 24 h period in April.


Fig. 7. Difference between measured and calculated values of internal air temperatures 24 h period in April.

The medium absolute percentage error was calculated using thefollowing formula:

MAPE = 1N

NTmeas Tcalc 100 (26)while the ro

RMSE =

The med

the root metively. The m

the root mean square error were equal to 2.5% and 1.0 K respec-tively.

Fig. 5 presents another measurement of 24 h period with radia-tor in off- and on-mode during a sunny day in April. It demonstrates

onab.

devias weerag

K reons o

diffel air

simThe tagei=1Tmeas

ot mean square error from:

1N

Ni=1

(Tmeas Tcalc)2 (27)

ium absolute percentage error of air temperature andan square error were equal to 3.1% and 0.6 K respec-edium absolute percentage error of radiator water and

a reasresults

Thenal air and avand 0.3deviati

Theinterna

Thement. percenFig. 8. Predicted mean vote (PMV) and predicted percentage of dissatisle agreement between measurement and calculation

ation of measured and calculated temperatures of inter-ll as radiator water is presented in Fig. 6. The maximume absolute deviations of internal air are equal to 2.4 Kspectively, while the maximum and average absolutef radiator water are equal to 7.5 K and 1.1 K respectively.rence between measurements and calculated values of

temperatures are presented in Fig. 7.ulation results and measurements are in good agree-calculated, using Eqs. (26) and (27), medium absolute

error of air temperature and the root mean square errored (PPD) 48 h period in February.


Fig. 9. Predicted mean vote (PMV) and predicted percentage of dissatised (PPD) 24 h period in April.

were equal to 1.9% and 0.5 K respectively. The medium absolutepercentagewere equal

Moreovelated usingcomfort indin Table 2 [6

Measureand velocitculating of tthermal comdecreased aroom air tem

about 5 h was required to raise internal air temperature from 13 CC. Thrmal

in Fiok a

19.5 C as ae mt duadiatating

of seing t error of radiator water and the root mean square error to 3.3% and 1.8 K respectively.r, the microclimate indices were additionally calcu-

InfoGAP program. The input parameters of thermalices PMV and PPD for sedentary ofce activity are given].d air and radiant temperatures, internal air humidityy were, along with data, listed in Table 2, used for cal-hermal comfort indices. As seen in Fig. 8, in 48 h periodfort index PMV during radiator off mode continuouslynd reaches after about 20 h the value of 2 when theperature drops to just 13 C. During radiator on-mode,

to 18

Thesented

It to15 to sideredthat thried ouOnce rthe heperiodreheatFig. 10. Internal air temperature calculated, solair temperature, PMVe values of PMV index then grew from about 2 to 0.7. comfort indices PMV and PPD for 24 h period are pre-g. 9.bout 5 h to increase the internal air temperature from

and thus reach the PMV value of 0.5, widely con-cceptable level. It should be mentioned here, however,easurements and the experiment itself were not car-ring a typical working pattern of ofce working hours.or was in on-mode and the air temperature increased,

system was immediately cut off again, without a usualveral hours when the radiator is in on-mode and thus

he room structure. calculated 24 h period in January.


Fig. 11. Internal air temperature calculated, solair temperature, PMV

Fig. 12. Internal air temperature calculated, solair temperature, PMV

Table 1Input parameters.

Room length 4.50 mRoom width 2,83 mRoom height 2.65 mWindow area Awin 7.42 m2

External wall area Aew 0.85 m2

Internal walls area Aiw1 + Aiw2 + Aiw3 29.75 m2

Ceiling and oor area Ac = Af 12.74 m2

External wall Reinforced concrete 0.140 mInternal walls Gypsum cardboard plate 0.020 m

Mineral wool 0.050 mGypsum cardboard plate 0.020 m

Floor/Ceiling Linoleum 0.002 mConcrete 0.080 mSteel plate 0.010 mGypsum cardboard plate 0.020 m

Radiator dimensions (height/length) 0.3 m/1.8 mRadiator mass 29.34 kgRadiator water volume 0.00612 m3

In orderther simulato 6 PM wefrom 7 PM (on-mode frirradiance fin January, gains werecan be easiair tempera

Table 2Input paramet

Metabolic raEffective meClothing ins calculated 24 h period in February.

calculated 24 h period in March.

to check the required period of radiator on-mode fur-tions were performed. The working hours from 9 AMre assumed as well as that the radiator was in off-modewhen there were no occupants in the ofce) and again inom 5 AM. The external air temperatures and global solarrom meteorological data for Poland for a 24-h periodsFebruary and March were assumed. No internal heat

accounted for, although once measured/known theyly accounted for (see Eq. (1)). The calculated internaltures and PMV values are shown in Figs. 10 to 12.

ers for thermal comfort indices calculations.

te M 1.2 met = 70 W/m2

chanical power W 0 W/m2

ulation Icl 1.01 clo = 0.157 m2 K/W


It can be seen in Figs. 10 12, that the required time, to reachthe acceptable PMV = 0.5, is about 4 h after 10 h off mode. Therequired time does not depend on the external conditions due tothe fact that radiator temperature is adjusted to the external con-ditions.

5. Conclusions

In this paper, a mathematical model describing the heat dynam-ics of a room heated by a hot-water radiator is presented. Themodel is relatively simple and can be implemented, in any commer-cial or in-house simulation software, as a set of room temperatureconstraint equations. The model was validated by comparing thecalculations results with the measurement data collected in anexisting room. The results clearly demonstrate that the modelprovides quite a satisfactory description of heat dynamics of theconsidered room, as well as that of radiator power during heatingsystem switch-off mode.

The calculation and measurement data for both radiator waterand internal air are in reasonable agreement. The root mean squareerrors for internal air were equal to 0.6 K and 0.5 K respectively,while for radiator water were equal to 1.0 K and 1.8 K.

Simulations also indicate that in buildings with light structure itis necessary to switch the radiator to on-mode about 45 h prior tothe working time in order to achieve the acceptable levels of PMV.

References

[1] Standard ISO 13790: 2008, Energy performance of buildingsCalculation ofenergy use for space heating and cooling.

[2] C. Buratti, E. Moretti, E. Belloni, F. Cotana, Unsteady simulation of energyperformance and thermal comfort in non-residential buildings, Building andEnvironment, Elsevier 59 (2013) 482491.

[3] J.E. Frederick, S.K. De, Radiative exchange across a window and linksto indoor energy demand, Energy and Buildings, Elsevier 51 (2012)2128.

[4] M. Tabarki, S.B. Mabrouk, The Coupling in Transient Regime between the Mod-elings of Thermal and Mass Transfers Inside a Heated Room and its Radiator,Heat and Mass Transfer, 48, Springer-Verlag, 2012, pp. 18891901.

[5] P.T. Tsilingiris, Wall heat loss from intermittently conditioned spaces thedynamic inuence of structural and operational parameters, Energy and Build-ings, Elsevier 38 (2006) 10221031.

[6] Standard ISO 7730: 2005, Ergonomics of the thermal environment Analyticaldetermination and interpretation of thermal comfort using calculation of thePMV and PPD indices and local thermal comfort criteria.

[7] W.M. Rohsenow, J.P. Hartnett, Y.I. Cho, Handbook of Heat Transfer, McGraw-Hill, New York, 1998.

[8] J.P. Holman, Heat Transfer, McGraw-Hill Book Company, New York, 1981.[9] K.J. Moss, Heat and Mass Transfer in Buildings, Taylor & Francis, New York,

2007.[10] S. Danov, J. Carbonell, J. Cipriano, J. Marti-Herrero, Approaches to evalu-

ate building energy performance from daily consumption data consideringdynamic and solar gain effects, Energy and Buildings, Elsevier 57 (2013)110118.

[11] K.J. Kontoleon, Dynamic thermal circuit modelling with distribution of internalsolar radiation on varying fac ade orientations, Energy and Buildings, Elsevier47 (2012) 139150.

[12] A. Kashif, S. Ploix, J. Dugdale, X.H.B. Le, Simulating the dynamics of occupantbehaviour for power management in residential buildings, Energy and Build-ings, Elsevier 56 (2013) 8593.

[13] V. Tabak, B. de Vries, Methods for the prediction of intermediate activities byofce occupants, Building and Environment, Elsevier 45 (2010) 13661372.

[14] U. Wilke, F. Haldi, J.-L. Scartezzini, D. Robinson, A bottom-up stochastic modelto predict building occupants time-dependent activities, Building and Envi-ronment, Elsevier 60 (2013) 254264.

[15] E. Kuchen, M.N. Fisch, Spot Monitoring: Thermal comfort evaluation in 25ofce buildings in winter, Building and Environment, Elsevier 44 (2009)839847.

[16] S. Karjalainen, Thermal comfort and use of thermostats in Finnish homes andofces, Building and Environment, Elsevier 44 (2009) 12371245.

[17] K. Fabbri, Thermal comfort evaluation in kindergarten: PMV and PPD mea-surement through datalogger and questionnaire, Building and Environment,Elsevier 68 (2013) 202214.

[18] V. De Giuli, R. Zecchin, L. Salmaso, L. Corain, M. De Carli, Measured and per-ceived indoor environmental quality: Padua Hospital case study, Building andEnvironment, Elsevier 59 (2013) 211226.

Model of unsteady heat exchange for intermittent heating taking into account hot water radiator capacity1 Introduction2 Mathematical model3 Experiment4 Results5 ConclusionsReferences

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