Model of unsteady heat exchange for intermittent heating taking into account hot water radiator capacity

Energy and Buildings 76 (2014) 176–184

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Model of unsteady heat exchange for intermittent heating taking into account hot water radiator capacity Agnieszka Lechowska ∗ , Artur Guzik Department of Environmental Engineering Cracow University of Technology ul. Warszawska Cracow 24 31-155, Poland

a r t i c l e

i n f o

Article history: Received 10 July 2013 Received in revised form 28 November 2013 Accepted 21 February 2014 Keywords: Intermittent heating Transient heat exchange Control volume method

a b s t r a c t Intermittent heating is one of the methods leading to savings in energy consumption. The intermittent heating system can work with reduced power or it can be completely cut off when the rooms are not occupied. At the beginning of the cut-off mode, the radiator remains warm for a specific period of time, due to its thermal capacity. This capacity is not negligible and should be considered for buildings with light or very light structures. This paper outlines a mathematical model of unsteady heat exchange in rooms with light wall structure with intermittent heating. The air heat balance of a given room takes into account the room air capacity, hot water radiator capacity, heat transfer through walls, ceiling, floor and windows as well as air infiltration. Reasonable accuracy between calculation and measurement results has been achieved. With known air and radiant temperatures, air humidity and velocity, thermal comfort indices predicted mean vote (PMV) and predicted percentage of dissatisfied (PPD) were evaluated in order to verify how thermal comfort changes during radiator cut-off mode. The satisfactory convergence between measured and calculated internal air temperatures has been achieved. © 2014 Elsevier B.V. All rights reserved.

1. Introduction Reducing energy consumption in buildings is an important environmental and economic issue. One of the methods leading to such savings is intermittent heating in which the central heating system can work in continuous heating mode at a constant set-point temperature as well as in switch-off mode with a night time and/or weekend reduced set-point temperature [1–5]. During continuous heating mode, thermal comfort indices PMV and PPD [6] are set at a constant level and no energy is saved. During switch-off heating mode, a building’s energy consumption is lowered, although inside thermal comfort is also decreased. Not all buildings are constructed similarly. They can have different structures, i.e. with different thermal heat capacities: very light, light, medium, heavy and very heavy elements [1]. In this paper the attention is focused on modelling the heat dynamics of the indoor air temperature in a light building heated by a low surface temperature hot-water radiator in a moderate climate during the heating season. In the building with a very light or light structure, radiator capacity should be taken into account.

The paper is organised as follows. The model of unsteady state heat exchange in buildings is introduced in Section 2. The calculation data and test room is described in Section 3. The measurement and calculation results are presented in Section 4, and finally, conclusions with discussion are given in Section 5. 2. Mathematical model An energy balance of room internal air can be written as [7–9]:

http://dx.doi.org/10.1016/j.enbuild.2014.02.062 0378-7788/© 2014 Elsevier B.V. All rights reserved.

(1)

j=1

where: dTa Ta2 − Ta1 = t2 − t1 dt

(2)

Q˙ win = (Uwin Awin + Hinf ) (Ta2 − Tsol )

(3)

6  j=1

∗ Corresponding author. Tel.: +48 12 632 09 48; fax: +48 12 628 20 48. E-mail address: [email protected] (A. Lechowska).

 dTa Q˙ s − Q˙ win = Q˙ r + Q˙ gn − dt 6

Va a ca

Q˙ s =

6  j=1

Asj Rsj + R˛sj



Ta2 − Tsj2



(4)

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

A. Lechowska, A. Guzik / Energy and Buildings 76 (2014) 176–184

In Eq. (1) Q˙ gn is a heat flow rate from internal heat gains from: occupants, equipment and lighting. The internal heat gains distribution is put in the model as known values [12–14]. Assuming that the room heating system consists of the lowsurface-temperature horizontal hot-water radiator the supplied heat flux can be expressed as:

Nomenclature a A c C I m n N Q˙ t T U V

177

solar radiation absorptivity area, m2 specific heat capacity, J/(kg K) radiator constant global solar irradiance, W/m2 mass, kg radiator constant exponent number of measurements or calculated values heat flow rate, W time, s temperature, K overall heat transfer coefficient, W/(m2 K) volume, m3

Q˙ r =

Ur Ar 1+

(Tin1 − Ta2 )

Ur Ar ˙ w cw 2m

(6)

where the overall radiator heat transfer coefficient is defined by: Ur = C Ta1 n

Greek letters ˛ heat transfer coefficient, W/(m2 K) infrared radiation due to difference between the Qir external air temperature and the sky temperature, W/m2  density, kg/m3 Abbreviations MAPE mean absolute percentage error predicted mean vote PMV PPD predicted percentage od dissatisfied root mean square error, K RMSE

T

in1

−

Ta1

(7)

In Eqs. (6) and (7) radiator supply water temperature is calculated from: Tin1 = Tin

o

+

Tin Te

− Tin o (Te − Te o ) h − Te o h

(8)

Once the supply water temperature and overall radiator heat transfer coefficient based on the previous time step data are calculated, both values are inserted into Eq. (6). Inserting Eqs. (2) ÷ (6) into (1), the room air heat balance equation for radiator in the on-mode is given as:

⎛ ⎝D + E + F +

6 

⎞ Gj ⎠ Ta2 −

j=1

Subscripts a internal air ceiling c calc calculated value external e ew external wall floor f gn internal heat gains end of heating season conditions h in inlet, supply water internal wall iw meas measured value number of time steps n o design conditions radiator r s structure, room opaque elements sol–air sol w water window win ˛ heat convection  heat conduction 0 initial beginning of time step 1 2 end of time step

n

Ta1 − Te Tr o −1 Ta1 − Te o 2Ta1

6 

Gj Tsj2 = D Ta1 + F Tsol + E Tin1 + Q˙ gn

j=1

(9) where: D=

E=

Va a ca t2 − t1

(10)

Ur Ar 1+

(11)

Ur Ar ˙ w cw 2m

F = Uwin Awin + Hinf Gj =

(12)

Asj

(13)

Rsj + R˛sj

Eq. (9) for inside air during radiator on-mode with energy balance equations comprise a set of equations solved at each time step. During radiator off-mode, water temperature is continuously decreasing. Its energy balance equation can be expressed as: (Vw w cw + mr cr )

dTw = −˛r Ar (Tw − Ta ) dt

(14)

The solution of Eq. (14), taking into account that both water and inside air temperatures are time-dependent, is given by: window. External air temperature Tsol in Eq. (3), referred to as the sol–air temperature, takes into account solar radiation and is calculated as [10,11]: Tsol = Te +

a I − Qir ˛e

 Tw1 = exp(Kt2 )

Tw0 − KB Ta1 − KC Ta2 − K

Bi Ta1,i + Ci Ta2,i





i=1

(15)

(5)

Eq. (4) describes the heat conduction and convection from a boundary (room external and internal walls, floor and ceiling) to the room air.

n−1  

where: K=

−˛r Ar Vw w cw + mr cr

(16)

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Fig. 1. Plan of analysed room.

B=

1 K (t2 − t1 )



− exp (−Kt1 )] t2 −

C =

1 K (t2 − t1 )

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

[t2 exp (−Kt2 ) − t1 exp (−Kt1 )] − [exp (−Kt2 ) 1 K

 (17)





[exp (−Kt2 ) − exp (−Kt1 )] t1 −

− [t2 exp (−Kt2 ) − t1 exp (−Kt1 )]



1 K

⎝D + F +



6 

⎞ Gj + M + N ⎠ Ta2 −

j=1

6  

Gj Tsj2



j=1

= (D − P) Ta1 + FTsol + S − W + Q˙ gn (18)

Initial water temperature Tw0 is assumed as an average of radiator supply and return temperatures from the last time step before cut-off mode. Radiator heat flow rate during off-mode can be written as: Q˙ r = ˛r Ar (Tw1 − Ta2 )

⎛

(20)

where: M = ˛r Ar

(21)

N = ˛r Ar KC exp (Kt2 )

(22)

P = ˛r Ar KB exp (Kt2 )

(23)

(19)

Fig. 2. Internal air temperature measured and calculated, sol–air temperature, radiator water temperature measured and calculated – 48 h period in February.

A. Lechowska, A. Guzik / Energy and Buildings 76 (2014) 176–184

179

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

S = ˛r Ar Tw0 exp (Kt2 ) W = ˛r Ar K exp (Kt2 )

(24)

n−1  

Bi Ta1,i + Ci Ta2,i



(25)

i=1

Using air energy balance Eq. (9) or Eq. (20) as the constraint equation, the mathematical model of transient heat transfer in a room during radiator on- and off-mode can be solved. These equations can be relatively easily incorporated, via external library or user-defined constraints, into existing commercial heat transfer simulation packages. 3. Experiment The mathematical model presented in the previous section was validated by comparing the calculation results with the measurement data collected in an actual room.

The room had a single external wall, three internal walls and both internal ceiling and floor. This is schematically presented in Fig. 1. The window was orientated into north. The room was equipped with a single low surface temperature, horizontal hotwater radiator, working in on- and off-mode. The room was not occupied during measurements, so no internal heat gains were taken into account. Both the external and internal temperatures, as well as temperatures of the adjacent rooms were measured and recorded. Furthermore, radiant temperatures, relative humidity, and air-flow speed were also measured. Table 1 lists values of input data of room parameters, its opaque elements and the radiator. The measurement system included the following sensors: probes for air temperature (Pt100 with accuracy ± 0.15 K), probe for radiant temperature measurement (with accuracy ± 0.15 K), probe for measurement of surface temperature

Fig. 4. Difference between measured and calculated values of internal air temperatures – 48 h period in February.

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

(Pt100 with accuracy ± 0.5 K), thermohygrometric probe (with relative air humidity accuracy ± 3%) and hot wire anemometer (with air speed accuracy ± 0.05 m/s). Measurements were taken every 3 min and logged automatically on a connected PC [15–18]. As a numerical method used for digitizing mathematical model the Control Volume Method (CVM) was selected. External and internal walls, as well as the ceiling and floor were divided into layers, and each layer represented by its core located at the mass centre. The energy balance equation for each core takes into account thermal conductivity between the core and its neighbours and/or, in case of boundary core, thermal conductivity between the core and boundary and convection between the boundary and surrounding air. The model assumed external air temperature as well as adjacent rooms temperatures to be known (taken from measurements), while cores and internal air temperatures to be unknown.

4. Results This section presents selected calculation and measurement results. Fig. 2 presents a comparison of measurement and simulation results for a selected 48-h period in February, containing two on- and off- modes. Fig. 3 presents the deviations between measured and calculated temperatures of both internal air and the radiator water. The maximum and average absolute deviations of internal air are equal to 2.7 K and 0.5 K respectively, while the maximum and average absolute deviations of radiator water are equal to 4.2 K and 0.7 K respectively. Fig. 4 presents the difference between measurements and calculated values of internal air temperatures.

Fig. 6. The deviation of measured and calculated temperatures both of internal air and radiator water – 24 h period in April.

A. Lechowska, A. Guzik / Energy and Buildings 76 (2014) 176–184

181

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 the following formula: 1 MAPE = N

  N   Tmeas − Tcalc  i=1

Tmeas

· 100

(26)

while the root mean square error from:

  N 1  RMSE =  (Tmeas − Tcalc )2 N

(27)

i=1

The medium absolute percentage error of air temperature and the root mean square error were equal to 3.1% and 0.6 K respectively. The medium absolute percentage error of radiator water and

the root mean square error were equal to 2.5% and 1.0 K respectively. Fig. 5 presents another measurement of 24 h period with radiator in off- and on-mode during a sunny day in April. It demonstrates a reasonable agreement between measurement and calculation results. The deviation of measured and calculated temperatures of internal air as well as radiator water is presented in Fig. 6. The maximum and average absolute deviations of internal air are equal to 2.4 K and 0.3 K respectively, while the maximum and average absolute deviations of radiator water are equal to 7.5 K and 1.1 K respectively. The difference between measurements and calculated values of internal air temperatures are presented in Fig. 7. The simulation results and measurements are in good agreement. The calculated, using Eqs. (26) and (27), medium absolute percentage error of air temperature and the root mean square error

Fig. 8. Predicted mean vote (PMV) and predicted percentage of dissatisfied (PPD) – 48 h period in February.

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

were equal to 1.9% and 0.5 K respectively. The medium absolute percentage error of radiator water and the root mean square error were equal to 3.3% and 1.8 K respectively. Moreover, the microclimate indices were additionally calculated using InfoGAP program. The input parameters of thermal comfort indices PMV and PPD for sedentary office activity are given in Table 2 [6]. Measured air and radiant temperatures, internal air humidity and velocity were, along with data, listed in Table 2, used for calculating of thermal comfort indices. As seen in Fig. 8, in 48 h period thermal comfort index PMV during radiator off mode continuously decreased and reaches after about 20 h the value of −2 when the room air temperature drops to just 13 ◦ C. During radiator on-mode,

about 5 h was required to raise internal air temperature from 13 ◦ C to 18 ◦ C. The values of PMV index then grew from about −2 to −0.7. Thermal comfort indices PMV and PPD for 24 h period are presented in Fig. 9. It took about 5 h to increase the internal air temperature from 15 to 19.5 C and thus reach the PMV value of −0.5, widely considered as acceptable level. It should be mentioned here, however, that the measurements and the experiment itself were not carried out during a typical working pattern of office working hours. Once radiator was in on-mode and the air temperature increased, the heating system was immediately cut off again, without a usual period of several hours when the radiator is in on-mode and thus reheating the room structure.

Fig. 10. Internal air temperature calculated, sol–air temperature, PMV calculated – 24 h period in January.

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183

Fig. 11. Internal air temperature calculated, sol–air temperature, PMV calculated – 24 h period in February.

Fig. 12. Internal air temperature calculated, sol–air temperature, PMV calculated – 24 h period in March.

Table 1 Input parameters. Room length Room width Room height Window area Awin External wall area Aew Internal walls area Aiw1 + Aiw2 + Aiw3 Ceiling and floor area Ac = Af External wall Internal walls

Floor/Ceiling

Radiator dimensions (height/length) Radiator mass Radiator water volume

Reinforced concrete Gypsum cardboard plate Mineral wool Gypsum cardboard plate Linoleum Concrete Steel plate Gypsum cardboard plate

4.50 m 2,83 m 2.65 m 7.42 m2 0.85 m2 29.75 m2 12.74 m2 0.140 m 0.020 m 0.050 m 0.020 m 0.002 m 0.080 m 0.010 m 0.020 m 0.3 m/1.8 m 29.34 kg 0.00612 m3

In order to check the required period of radiator on-mode further simulations were performed. The working hours from 9 AM to 6 PM were assumed as well as that the radiator was in off-mode from 7 PM (when there were no occupants in the office) and again in on-mode from 5 AM. The external air temperatures and global solar irradiance from meteorological data for Poland for a 24-h periods in January, February and March were assumed. No internal heat gains were accounted for, although once measured/known they can be easily accounted for (see Eq. (1)). The calculated internal air temperatures and PMV values are shown in Figs. 10 to 12.

Table 2 Input parameters for thermal comfort indices calculations. Metabolic rate M Effective mechanical power W Clothing insulation Icl

1.2 met = 70 W/m2 0 W/m2 1.01 clo = 0.157 m2 K/W

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It can be seen in Figs. 10 ÷ 12, that the required time, to reach the acceptable PMV = − 0.5, is about 4 h after 10 h off mode. The required time does not depend on the external conditions due to the fact that radiator temperature is adjusted to the external conditions. 5. Conclusions In this paper, a mathematical model describing the heat dynamics of a room heated by a hot-water radiator is presented. The model is relatively simple and can be implemented, in any commercial or in-house simulation software, as a set of room temperature constraint equations. The model was validated by comparing the calculations results with the measurement data collected in an existing room. The results clearly demonstrate that the model provides quite a satisfactory description of heat dynamics of the considered room, as well as that of radiator power during heating system switch-off mode. The calculation and measurement data for both radiator water and internal air are in reasonable agreement. The root mean square errors 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 it is necessary to switch the radiator to on-mode about 4–5 h prior to the working time in order to achieve the acceptable levels of PMV.

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