- Email: [email protected]
The dynamic thermal characteristics of lightweight building elements with a forced ventilated cavity and radiation barriers
Energy and Buildings 37 (2005) 972–981 www.elsevier.com/locate/enbuild
The dynamic thermal characteristics of lightweight building elements with a forced ventilated cavity and radiation barriers Bosˇtjan Cˇerne a,*, Sasˇo Medved b a
b
Trimo d.d, Prijateljeva 12, 8210 Trebnje, Slovenia University of Ljubljana, Faculty of Mechanical Engineering, Asˇkercˇeva 6, 1000 Ljubljana, Slovenia
Received 9 October 2004; received in revised form 15 December 2004; accepted 17 December 2004
Abstract This paper presents an analysis of the dynamic thermal characteristics of lightweight building elements (LBEs) with a forced ventilated cavity and radiation barriers. A two-dimensional numerical model was used for the analysis, which was verified with an analytical method and outdoor experiments. The influence of the temperature boundary conditions at the cavity inlet and along the ambient surface of the cavity on the dynamic thermal characteristics of the LBEs was analyzed separately. The influence of the LBE width, the cavity width, the specific air flow rate and the cavity surface emissivities was also demonstrated. Multi-parametric equations were derived for predicting the heat flux amplitude and the decrement factor along the length of the LBE. # 2005 Elsevier B.V. All rights reserved. Keywords: Lightweight building element; Ventilated cavity; Dynamic thermal characteristics; Two-dimensional heat transfer
1. Introduction Lightweight building elements (LBEs) are building constructions made of thermal insulation and two thin metal sheets. They are especially appropriate as envelope elements for buildings such as warehouses, commercial buildings, shopping centres, production plants as well as residential buildings. The steady thermal characteristics (U-value) of an LBE are good, while the dynamic thermal characteristics show some disadvantages in terms of ensuring indoor thermal comfort due to the low thermal capacity of an LBE compared with massive building constructions. Two of the most important dynamic thermal characteristics of a building construction are the decrement factor and the time lag. The decrement factor is the ratio between the dynamic thermal transmittance and the steady-state thermal transmittance, while the time lag is the period of time between the maximum ambient temperature and the maximum inner surface heat flux [1]. For LBEs the decrement factor is high, while the time lag is small compared to massive building constructions. Several * Corresponding author. Tel.: +386 1 4771316; fax: +386 1 2518567. E-mail address: [email protected] (B. Cˇerne). 0378-7788/$ – see front matter # 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.enbuild.2004.12.006
studies involving ventilated cavities, radiant barriers and phase-change materials were made to reduce these disadvantages of LBEs. Medina et al. [2] studied the influence of a ventilated attic on the heat flux through a lightweight ceiling construction in two small test houses. One of the houses had a radiant barrier installed in the attic. The authors reported that the heat flux through the lightweight ceiling did not decrease when the air flow rate through the ventilated attic was higher than 0.0013 m3/sm2 of ceiling construction. The radiant barrier was effective because the heat flux through the ceiling decreased by as much as 40% at the highest sol–air temperature. Medina [3] also analyzed the efficiency of a radiant barrier with different emissivities (due to dust particles). He reported a decrease in the ceiling heat flux from 40 to 15% when the emissivity of the radiant barrier increased from 0.05 to 0.3. He also analyzed the efficiency of a radiant barrier as a function of the steady-state thermal transmittance of a lightweight ceiling [4]. Winiarski and O’Neal [5] developed a model for calculating the heat flux through a lightweight ceiling, where they supposed the value of the decrement factor was equal to 1. Kairys and Karbauskaite [6] analyzed the transient heat flux through a
B. Cˇ erne, S. Medved / Energy and Buildings 37 (2005) 972–981
Nomenclature arg C d f g h K L P q q˜ _ q Q _ Q R t T T˜ _ T Dt U v w Z jj
argument of complex number (rad) specific heat capacity (J/kg K) width (mm) decrement factor gravitational acceleration (m/s2) heat transfer coefficient (W/m2K) correction coefficient length (m) period (s) heat flux (W/m2K) harmonic heat flux oscillation (W/m2) specific heat flux amplitude (W/m2K) heat flux (W/m2) heat flux amplitude (W/m2) thermal resistance (m2K/W) time (s) temperature (K) harmonic temperature oscillation (K) temperature amplitude (K) time lag (h) thermal transmittance (W/m2K) specific air flow (m3/sm2) wind speed (m/s) element of transfer matrix absolute value of complex number
Greek letters a absorptivity b phase (rad) e emissivity l conductivity (W/mK) r density (kg/m3) v frequency (rad/s) Subscripts a ambient an analytical av average c cavity d downward i inner in inlet ir longwave radiation max maximum nu numerical out outlet s shortwave radiation sin sine function sol sol–air u upward 1 m, 3 m, 5 m distances from cavity inlet
973
vertical lightweight wall with and without a naturally ventilated cavity for a selected extreme day. They showed a 30% decrease in the maximum heat flux on the inner surface of a wall with a naturally ventilated cavity compared to a wall with no cavity. Peippo et al. [7] analyzed the temperature swings in a lightweight house with phasechange material (PCM) impregnated into the plasterboard compared to a reference house without any PCM. The temperature swings decreased by as much as 2 8C when the PCM was used. Athienitis et al. [8] performed experiments on a lightweight test-room with a PCM gypsum board and on a gypsum board without PCM. They showed that the daily maximum room temperature was about 4 8C lower in the room with the PCM gypsum board. In these investigations the transient heat transfer in the lightweight building constructions was analyzed onedimensionally. Longwave radiation on the building construction was not measured. The sol–air temperature was calculated with the measured ambient temperature and shortwave radiation. The influence of the ratio between the ambient temperature amplitude and the sol–air temperature amplitude was not analyzed in detail. The one-dimensional analytical model for a transient flux calculation from the EN ISO 13786 standard [1] can only be used for a building construction with no cavity or with a closed cavity. For a transient heat flux calculation through a long lightweight building construction with a ventilated cavity a twodimensional model must be used. In this case the ratio between the temperature boundary conditions at the cavity inlet and the temperature boundary conditions along the ambient surface of the cavity must be analyzed. In our work, we also formed multi-parametric equations to predict the inner surface heat flux amplitude and the decrement factor.
2. Analytical method for calculating transient heat transfer in a building construction The dynamic thermal characteristics of a building construction can be calculated using the analytical method described in EN ISO 13786 [1]. This method is based on the work of Carslaw and Jaeger [9], who showed that the temperatures and the heat fluxes on both sides of the building construction can be calculated with linear equations when the temperatures vary sinusoidally. These equations can be written in matrix form: Z ii Z ia T˜ i T˜ a ¼ (1) Z ai Z aa q˜ i q˜ a When the ambient temperature amplitude is known and the inner temperature is constant the heat flux amplitude on the inner surface of the construction can be calculated from Eq. (1): _ qi
_
¼
Ta jZ ia j
(2)
974
ˇ erne, S. Medved / Energy and Buildings 37 (2005) 972–981 B. C
The amplitude characterizes the fluctuation around the average heat flux, which can be calculated from steady-state conditions. The decrement factor, f, is one of the two basic dynamic characteristics of building constructions according to the cited standard. It is the ratio of the dynamic thermal transmittance to the thermal transmittance under steadystate conditions, known as the U-value. This factor has values between 0 and 1, and can be calculated using the following equation: 1 (3) jZ ia jU The period of time between the ambient temperature amplitude and the inner surface heat flux amplitude can be calculated using the following equation: f ¼
P argðZ ia Þ (4) 2p In our work the analytical method will be used for two purposes: first, to verify the numerical model in the case of an LBE with a closed cavity; second, as a source of standardized boundary conditions in the two-dimensional numerical model and to calculate the dynamic thermal characteristics. The latter allows a direct comparison of oneand two-dimensional transient problems.
Dt ¼
Fig. 1. Geometrical model with boundary conditions.
The air flow rate through the ventilated cavity is constant during the whole period. The inlet velocity profile is uniform. The gravitational forces are taken into account when an LBE with a non-ventilated cavity is analyzed. The radiation, the absorption and the scattering of the air in the cavity are zero. The emissivities of the cavity surfaces are diffuse. The shortwave and longwave radiation are uniform across the ambient surface of the LBE. Adiabatic conditions are assumed at the edge of the LBE. 4. Verification of the numerical model 4.1. Analytical verification
3. Numerical model of an LBE with a ventilated and a non-ventilated cavity Transient heat transfer in an LBE with a ventilated cavity must be solved numerically using a CFD method. The control volume method was used in our case. For control volumes, differential equations of the conservation of mass, momentum and energy, known as the set of Navier–Stokes equations, must be solved. Together with Fourier’s law of heat conduction and the Stefan–Boltzmann radiation law, the pressure, velocity and temperature fields in the system can be calculated. The commercial CFD software PHOENICS was used to solve the set of equations using the set of boundary conditions. A model called IMMERSOL was used to calculate the longwave radiation within the cavity [10]. In all cases the numerical calculations were performed with a 10 min time step. The two-dimensional geometrical model with the boundary conditions used in the numerical analyses is shown in Fig. 1. Various boundary conditions were used in the numerical simulation and they are described in detail in the following sections. The influence of the initial conditions was neglected by using a 12 h pre-period calculation. The following assumptions were used in the numerical model: The heat transfer is transient with a 24 h period. The heat transfer is two-dimensional. The k–e model is used in the case of turbulent flow.
For the analytical verification of the numerical model an LBE with a non-ventilated cavity was used. The dimensions and the material properties are shown in Table 1. Two different cavity surface emissivities were analyzed: eci = eca = 0.9 and eci = eca = 0.1. The combined convection and radiation surface resistance Ri = 0.04 m2K/W and Ra = 0.13 on the ambient and the inner surface were analyzed [11]. According to [11] the thermal resistance of a closed cavity (Rc) with high emissivity cavity surfaces 0.12 and 0.26 m2K/W for the low emissivity cavity surfaces, were used. The transient ambient temperature was defined using a sine function with amplitude _ (T a ) 1 K around an average daily value of 25 8C with a maximum value at 12 h (noon) and a minimum at 24 h (midnight). The indoor air temperature was a constant 25 8C _ (T i = 0 K). No shortwave radiation was taken into account. The gravitational forces in the numerical simulations were calculated by considering the inclination of the LBE. The elements of the transfer matrix were calculated as
Table 1 Dimensions and material properties of the LBE r (kg/m3)
C (J/kg K)
Layer
Thickness (mm)
k (W/mK)
Metal sheet Thermal insulation Metal sheet Air Metal sheet
0.6 78.8 0.6 7 0.6
43 7800 460 0.04 120 840 43 7800 460 Rc,ec = 0.9 = 0.12; Rc,ec = 0.1 = 0.26 43 7800 460
B. Cˇ erne, S. Medved / Energy and Buildings 37 (2005) 972–981
975
Fig. 2. Temperatures and heat fluxes for 2 days with equal initial and boundary conditions for high (left) and low (right) cavity surface emissivities.
follows: Zii,ec = 09 = 1.31, Zia,ec = 09 = 2.29 (m2K/W), Zai,ec = 09 = 1.10 (W/m2K), Zaa,ec = 09 = 1.52, Zii,ec = 01 = 1.38, Zia,ec = 01 = 2.45 (m2K/W), Zai,ec = 01 = 1.11 (W/m2K), Zaa,ec = 01 = 1.53. Fig. 2 shows the temperature boundary conditions Ta and Ti and the ambient and inner surface heat fluxes calculated with analytical and numerical methods. It is clear that the numerically calculated heat flux amplitudes and the time lag are in good agreement with the analytically calculated values, with a maximum difference of 2%. This also indicates that the numerical time step and the control volume mesh are appropriate.
surface of the LBE. The influence of the ambient was neglected by an additional 5 cm of polystyrene thermal insulation, as shown in Figs. 3–5. The electrical resistance heating wire for the heating and the polyethylene pipes for
4.2. Verification of the numerical model with experiments A series of experiments was carried out to validate the numerical simulations for LBEs with closed and ventilated cavities. The layers and dimensions were the same as described in Section 4.1. The air flow rate through the ventilated cavity was ensured by forced ventilation. The air enters the cavity directly from the ambient. Some hydrodynamic devices were used in the collecting duct to provide a uniform air flow rate through the cavity. The indoor air temperature was regulated by the heating and cooling of an additional 2 cm airspace below the bottom
Fig. 3. Photo of the LBE where the experiments were performed. The dimensions of LBE were 6 m 6 m, the inclination was 358. The locations of temperature, wind velocity and heat flux sensors are shown schematically.
Fig. 4. Thermostated cavity below the LBE with resistance heating wire and cooling pipes. The opening for the installed heat flux meter was later thermally insulated. The heat flux meter was installed on the ventilated cavity bottom sheet, as shown in Fig. 5, in such a way that even small fluctuation of the thermostated air cavity temperature, Ti, were reduced.
Fig. 5. Cross-section of the tested LBE with heating and cooling elements in the insulated cavity below the LBE.
976
ˇ erne, S. Medved / Energy and Buildings 37 (2005) 972–981 B. C
Fig. 6. Measured temperatures, wind speed, shortwave and longwave radiation on the LBE surface.
the water cooling were mounted in the thermally insulated air cavity below the LBE (Figs. 4 and 5). All the temperatures were measured with calibrated Ni– CrNi thermocouples. Four temperature sensors were used at the inlet- and outlet-cavity cross-sections; they also indicated the uniformity of the air flow rate through the cavity of the LBE. The temperature gradients in the cavity were measured at three locations above the heat flux meters. Three heat flux meters (size: 120 mm 120 mm) were mounted 1, 3 and 5 m from the inlet opening. The meteorological parameters were measured with a calibrated pyranometer (for the shortwave radiation) and pyrgeometer (for the downward and upward longwave radiation), a ventilated and shielded temperature sensor, a wind speed and direction sensor and a relative humidity sensor. Both the pyranometer and the prygeometer were mounted directly on the test roof and therefore had the same inclination as the LBE. The air flow rate was measured using an orifice plate and a differential pressure gauge [12]. The measurement data were stored every 5 min using a data logger. Measurements with the non-ventilated and the ventilated cavities were performed from June to August 2003. Two different air flows were analyzed: 0.0029 and 0.0043 m3/ sm2. Fig. 6 shows the measured parameters for the three
selected days that were used for the experimental verification of the numerical model. The main difficulty when comparing the numerical model and the results of the experiments is the description of the real meteorological parameters with an appropriate periodic function. The following boundary conditions were used in the numerical calculations: The inlet air temperature was described with a sine function: _
T a;sin ¼ T a;av þ T a sin ðvt bÞ
(5)
In Eq. (5), the average ambient temperature is the mean value between the maximum and minimum temperatures, while the temperature amplitude is half the difference between the maximum and minimum temperature on the selected day. The sol–air temperature was used to represent the influence of the ambient temperature as well as the shortwave and longwave radiation on the ambient surface of the LBE: T sol ¼ T a þ
Qs aa;s þ Qir;d aa;ir Qir;u ea;ir ha
Fig. 7. Sol–air temperatures used in the numerical calculations, Tsol is calculated with Eq. (6) using measurement data.
(6)
B. Cˇ erne, S. Medved / Energy and Buildings 37 (2005) 972–981
The ambient temperature and the shortwave radiation were measured. The downward longwave radiation was calculated from the measured net longwave radiation and the temperature of the pyrgeometer. The Stefan–Boltzmann law was used to calculate the upward longwave radiation from the LBE. For this calculation the average temperature of an ambient metal sheet was used. In Eq. (6) the absorptivity was 0.85 [13] and the emissivity was 0.9. The convective heat transfer coefficient, ha, does not take into account the radiative heat transfer, and is calculated using the equation [14]: ha ¼ 3:1 þ 4:1w
(7)
The daily wind speed average was used for the calculation of ha. For the purpose of the numerical calculation the sol–air temperature was approximated in two ways (Fig. 7). First, only a single sine function (Tsol,1) was used: _
T sol;1 ¼ T sol;av þ T sol sin ðvt bÞ
T sol;2
8 Qir;d;av aa;ir Qir;u;av ea;ir > > < T a;av þ ha _ ¼ þT a sin ðvt bÞ; night time > > : _ T sol;av þ T sol sin ðvt bÞ; day time
977
(9)
Fig. 8 shows a comparison of the measured and the calculated heat fluxes along the cavity length. The results show that the approximation of the thermal boundary conditions with two sine functions of the sol–air temperature is necessary for good agreement of the heat fluxes measured with experiments and calculated using numerical simulations. However, when the heat flux amplitude is of interest, a single sine sol–air temperature approximation provides good agreement between the measured and the calculated extreme daily heat flow. Therefore, the single sine approximation is appropriate. When the solar heat gains or the longwave radiant cooling is analyzed, an approximation of the sol–air temperature with two sine functions is necessary.
(8)
Second, the sol–air temperature was divided in two parts: the first part for night time and the second part for day time (between sunrise and sunset). Each part was approximated with a single sine function:
5. Multi-parametric analysis of the heat flux amplitude The presented and verified numerical model was used for a set of transient numerical calculations for different
Fig. 8. Measured and numerically calculated heat fluxes: top, with boundary condition Tsol,1 (Eq. (8)); bottom, with temperature boundary condition Tsol,2 (Eq. (9)).
ˇ erne, S. Medved / Energy and Buildings 37 (2005) 972–981 B. C
978
_
Table 2 Analyzed air velocities in the ventilated cavity Specific air flow (m3/sm2)
Cavity width (mm)
0.001
0.002
0.005
0.010
0.020
10 25 50 100
0.9 0.36 – –
1.8 0.72 0.36 –
4.5 1.8 0.9 0.45
– 3.6 1.8 0.9
– – 3.6 1.8
geometric, thermal and hydraulic initial and boundary conditions. The aim of this set of numerical calculations was to provide the necessary input data for a multi-parametric model of the inner surface heat flux amplitude for a wide range of influential parameters. These parameters were the thermal insulation thickness (78.8, 148.8 and 198.8 mm), the cavity surfaces’ emissivities (three different combinations of emissivities were used: eci = 0.9 and eca = 0.9; eci = 0.9 and eca = 0.1; eci = 0.1 and eca = 0.1). The range of specific air flow rates through the cavity used in the numerical calculations is presented in Table 2. The length of the LBE with the ventilated cavity was 9 m in all cases. The ambient and inner surface resistances were 0.04 and 0.13 m2K/W, respectively. The material properties used in the calculation are given in Table 1. The validity of the multi-parametric model must be ensured for any combination of both transient boundary conditions—at the inlet of the cavity and along the cavity surface. For this reason the transient boundary conditions must be normalised. Both normalised boundary conditions were in phase and varied sinusoidally. In all the numerical calculations the normalised air _ temperature amplitude was T a = 1 K, the normalised increase of the air temperature amplitude due to absorbed _ solar radiation was T s = 0 and 1 K. In the case when _ _ T a = 1 K and T s = 0 K the transient temperature boundary conditions on the inlet were the same as along the cavity surface—the influence of solar radiation was not taken into _ account. With these boundary conditions qL;i;a was
_
calculated. When T a = 1 K and T s = 1 K the combined _ _ boundary condition along the cavity surface is T a þ T s = 2 K—the normalised influence of solar radiation is taken _ into account. In such a way qL;i;s can be calculated. _ Fig. 9 shows the inner surface heat flux amplitudes qL;i;a _ and qL;i;s along the LBE with a ventilated cavity for an LBE width of 80 mm and a cavity width of 25 mm in the case of two different specific air flows. The influence of all three combinations of cavity surface emissivities is shown. From Fig. 9 it is clear that the most influential parameter when it comes to a decrease in the heat flux amplitude of an LBE with a ventilated cavity is the emissivity of the cavity _ surfaces. The air flow rate has the opposite influence on qL;i;a _ _ and qL;i;s : qL;i;a increases with an increase in the air flow _ rate, while qL;i;s decreases with an increase in the air flow. When the ventilated cavity has low emissivity surfaces, an increase in the air flow rate over 0.010 m3/sm2 contributes to _ a relatively small decrease of qL;i;s . The amplitude of the heat flux on the inner surface of an LBE with a ventilated cavity for different combinations of _ _ T a and T s can be calculated using the equation: _
__
__
QL;i ¼ T a qL;i;a þ T s qL;i;s _ Ts _
Ts ¼
(10)
can be calculated using the following equation: Qs;max aa;s ha
(11)
Altogether, 252 transient numerical simulations were made to provide sufficient data for a statistical analysis. The least-square error method was used to form two multiparametric equations for each component of the heat flux _ _ amplitude (qL;i;a and qL;i;s ) used in Eq. (10): _ qL;i;a
Da ¼ Aa LBa vCa dLBE dcEa
(12)
_ qL;i;s
Ds ¼ As LBs =ln dc vCs dLBE dcEs
(13)
The values of the constants used in Eqs. (12) and (13) as function of the emissivities of the cavity surfaces are given in
Fig. 9. Inner surface heat flux amplitude for an LBE with an 80 mm width and with 25 mm of ventilated cavity; specific air flow rates 0.002 m3/sm2 (left) and _ _ _ 0.010 m3/sm2 (right). (1) qL;i;a ; (2) qL;i;s ; (3) qLBE without cavity ; (A) eci = 0.9, eca = 0.9; (B) eci = 0.9, eca = 0.1; (C) eci = 0.1, eca = 0.1.
B. Cˇ erne, S. Medved / Energy and Buildings 37 (2005) 972–981 Table 3 _ _ Values of the constants for calculation of qL;i;a and qL;i;s ei
eo
Aa
Ba
Ca
Da
Ea
Table 4 Values of constants for calculation of KL,a and KL,s R2
0.9 0.9 101.1394 0.01008 0.00995 1.21654 0.01365 0.9924 0.9 0.1 126.3316 0.03369 0.04297 1.20232 0.06438 0.9895 0.1 0.1 136.1368 0.03939 0.05198 1.19968 0.07883 0.9876 ei
eo
0.9 0.9 0.9 0.1 0.1 0.1
As
Bs
Cs
Ds
Es
R2
4.15038 0.91473 0.24666 1.19512 0.25357 0.9699 0.28997 1.60445 0.50365 1.17216 0.24383 0.9578 0.15227 1.82506 0.59344 1.17234 0.16108 0.9461
Table 3, together with the R2 value of multi-parametric regression. Like with the heat flux amplitude changes along the length of an LBE with a ventilated cavity, the decrement factor changes too. A comparison between the decrement factor of an LBE with a ventilated cavity and an LBE without a cavity is of interest to us. Therefore, the influence of a ventilated cavity was analyzed with a correction coefficient. The correction coefficient at a selected distance from the cavity inlet with regard to the previously described normalised transient thermal boundary conditions can be calculated with the following equation: K L;a ¼ K L;s ¼
_ qL;i;a
U LBE f LBE _ qL;i;s
979
eI
eo
Fa
Ga
Ha
Ia
R2
0.9 0.9 0.1
0.9 0.1 0.1
1.054616 1.451800 1.592288
0.012020 0.038390 0.044620
0.011422 0.047308 0.056988
0.014970 0.071030 0.086190
0.9903 0.9296 0.9095
eI
eo
Fs
Gs
Hs
Is
R2
0.9 0.9 0.1
0.9 0.1 0.1
0.046632 0.003694 0.001984
0.937760 1.631020 1.843530
0.246490 0.501100 0.589540
0.254260 0.240790 0.154060
0.9489 0.9523 0.9407
such an element. Using a ventilated cavity the decrement factor of an LBE can be reduced by up to 10 times, even in the case of very long elements. Cavity surfaces with low emissivity contribute greatly to the reduction of the decrement factor. The decrease of KL,s is much higher than the decrease of KL,a; this is especially important because in _ _ nature T s can be much higher than T a . Like in Eq. (10), the decrement factor of an LBE with a ventilated cavity along the LBE length for different _ _ combinations of T a and T s can be calculated with the following equation: _
_
T a K L;a þ T s K L;s f LBE (16) _ _ T a þ Ts _ Like in the case of the heat flux amplitude (qL;i;a and _ qL;i;s ) the multi-parametric model of the correction coefficient KL,a and KL,s has been formed. These two coefficients can be calculated with the following equations: fL ¼
(14)
(15) U LBE f LBE In Eqs. (14) and (15) ULBE and f LBE are calculated for an LBE without a cavity. We found that using such a formulation, Ka and Ks become independent of the LBE’s width. Fig. 10 shows the correction coefficients Ka and Ks for an LBE with a 25 mm ventilated cavity and two specific air flow rates. It is clear that a two-dimensional approach is necessary for calculating the proper decrement factor of
K L;a ¼ F a LGa vH a dcI a
(17)
K L;s ¼ F s LGs =ln dc vH s dcI s
(18)
The values of the constants used in Eqs. (17) and (18) as a function of the emissivities of the cavity surfaces are given in Table 4, together with the R2 values of multi-parametric regression.
Fig. 10. Correction coefficient for LBE with a 25 mm ventilated cavity; different cavity surface emissivities and specific air flow rates of 0.002 m3/sm2 (left) and 0.010 m3/sm2 (right); (A) eci = 0.9, eca = 0.9; (B) eci = 0.9, eca = 0.1; (C) eci = 0.1, eca = 0.1.
980
ˇ erne, S. Medved / Energy and Buildings 37 (2005) 972–981 B. C
Fig. 11. Time lag of an 80 mm thick LBE with a 25 mm ventilated cavity; air flow rates 0.002 m3/sm2 (left) and 0.010 m3/sm2 (right); (A) eci = 0.9, eca = 0.9; (B) eci = 0.9, eca = 0.1; (C) eci = 0.1, eca = 0.1; (3) LBE without cavity.
Fig. 12. Time lag of a 200 mm thick LBE with a 25 mm ventilated cavity and air flow rates of 0.002 m3/sm2 (left) and 0.010 m3/sm2 (right); (A) eci = 0.9, eca = 0.9; (B) eci = 0.9, eca = 0.1; (C) eci = 0.1, eca = 0.1; (3) LBE without cavity.
The time lag was also analyzed in this investigation. Figs. 11 and 12 show the numerically calculated time lag for an LBE with a ventilated cavity for two different LBE widths, three different combinations of cavity surface emissivities and two different air flow rates. The analytically calculated time lag for the same LBE without a cavity is also shown. It is clear that the ventilated cavity has a small influence on the time lag along the LBE length and small influence with regard to the LBE without a cavity. In the case of a higher air flow rate and high emissivity surfaces, the time lag is even a little bit smaller due to the higher convection heat transfer. Therefore, we can conclude that the influence of a ventilated cavity on the time lag can be neglected.
6. Conclusions In this work we analyzed the dynamic thermal characteristics of an LBE with a ventilated cavity. Presented
dynamic characteristics, decrement factor, f, and time lag, Dt, must be defined in such a way that different building constructions can be compared. The important difference between LBE with a ventilated cavity and other nonventilated constructions is in two-dimensional heat transfer and in two different thermal boundary conditions: at the cavity inlet and along the ambient cavity surface. Both were taken into consideration in our work. A two-dimensional numerical model based on the control volume method was made to calculate the dynamic heat transfer through the LBE with a forced ventilated cavity. It was verified with an analytical method and experiments. The influences of the ambient temperature variation and the solar radiation variation on the dynamic heat transfer were analyzed separately. For comparison purposes, the ambient temperature variation and the solar radiation variation were normalised. At the cavity inlet only normalised ambient temperature variation was taken into account, while along the cavity surface the normalised ambient temperature variation and the normalised solar radiation variation were
B. Cˇ erne, S. Medved / Energy and Buildings 37 (2005) 972–981
taken into account as sol–air temperature. The variation of ambient temperature at the cavity inlet is defined with one sine function. The sol–air temperature along the cavity surface can be defined with one or two sine functions. We found that for calculating dynamic thermal characteristics of an LBE one sine function is sufficient. A multi-parametric analysis was made for different parameters: LBE width and length, cavity surfaces emissivity, cavity widths and air flow rates through the cavity. Based on numerical simulations correction factors for normalised ambient temperature amplitude and normalised sol–air temperature amplitude were determinated in form of multi-parametric equations. The set of two multiparametric equations were determinated for prediction of the length dependent inner surface heat flux amplitude. We found that a ventilated cavity can greatly reduce the influence of solar radiation on the inner surface heat flux compared to an LBE without a ventilated cavity. In combination with a low emissivity cavity surfaces the decrement factor of an LBE can be reduced by up to 10 times. Increasing the air flow rate over 0.010 m3/sm2 _ contributes to a relatively small decrease of qL;i;s when the cavity surfaces have low emissivities. The ventilated cavity does not contribute to a greater shift of the cooling load to off-peak electricity consumption. Therefore, the time lag calculated for an LBE without a ventilated cavity can also be used for an LBE with a ventilated cavity.
References [1] European Standard EN ISO 13786, Thermal performance of building components—dynamic thermal characteristics—calculation methods, European Committee for Standardization, Brussels, Belgium, 1999.
981
[2] M.A. Medina, D.L. O’Neal, W.D. Turner, Effect of attic ventilation on the performance of radiant barriers, Journal of Solar Energy Engineering 114 (1992) 234–239. [3] M.A. Medina, Effects of shingle absorptivity, radiant barrier emissivity, attic ventilation flow rate and roof slope on the performance of radiant barriers, International Journal of Energy Research 24 (2000) 665–678. [4] M.A. Medina, On the performance of radiant barrier in combination with different attic insulation levels, Energy and Buildings 33 (1) (2000) 31–40. [5] D.W. Winiarski, D.L. O’Neal, A quasi-steady-state model of attic heat transfer with radiant barriers, Energy and Buildings 24 (3) (1996) 183–194. [6] L. Kairys, J. Karbauskaite, Solar radiation influence to the thermo physical parameters of thermal wave that penetrates through the lightweight buildings partitions, in: Proceedings of the Conference on Dynamic Analysis and Modelling Techniques for Energy in Buildings, Ispra, Italy, 2003, , pp. 105–113. [7] K. Peippo, P. Kauranen, P.D. Lund, A multicomponent PCM wall optimized for passive solar heating, Energy and Buildings 17 (4) (1991) 259–270. [8] A.K. Athienitis, C. Liu, D. Hawes, D. Banu, D. Feldman, Investigation of the thermal performance of a passive solar test-room with wall latent heat storage, Building and Environment 32 (5) (1997) 405–410. [9] H.S. Carslaw, J.C. Jaeger, Conduction of Heat in Solids, second ed. Oxford University Press, 1959. [10] PHOENICS 3.5.1, Parabolic, hyperbolic, elliptic numerical integration code system, CHAM, Wimbledon, England, 2003. [11] European Standard EN ISO 6946, Building components and building elements—thermal resistance and thermal transmittance—calculation method, European Committee for Standardization, Brussels, Belgium, 1996. [12] European Standard EN ISO 5167-1, Measurement of fluid flow by means of pressure differential devices inserted in circular cross-section conduits running full—Part 1: general principles and requirements, European Committee for Standardization, Brussels, Belgium, 1995. [13] S. Medved, C. Arkar, B. Cˇ erne, A large-panel unglazed roof-integrated liquid solar collector—energy and economic evaluation, Solar Energy 75 (6) (2003) 455–467. [14] Australian Standard AS 3634, Solar heating systems for swimming pools, 1989.
The dynamic thermal characteristics of lightweight building elements with a forced ventilated cavity and radiation barriers Bosˇtjan Cˇerne a,*, Sasˇo Medved b a
b
Trimo d.d, Prijateljeva 12, 8210 Trebnje, Slovenia University of Ljubljana, Faculty of Mechanical Engineering, Asˇkercˇeva 6, 1000 Ljubljana, Slovenia
Received 9 October 2004; received in revised form 15 December 2004; accepted 17 December 2004
Abstract This paper presents an analysis of the dynamic thermal characteristics of lightweight building elements (LBEs) with a forced ventilated cavity and radiation barriers. A two-dimensional numerical model was used for the analysis, which was verified with an analytical method and outdoor experiments. The influence of the temperature boundary conditions at the cavity inlet and along the ambient surface of the cavity on the dynamic thermal characteristics of the LBEs was analyzed separately. The influence of the LBE width, the cavity width, the specific air flow rate and the cavity surface emissivities was also demonstrated. Multi-parametric equations were derived for predicting the heat flux amplitude and the decrement factor along the length of the LBE. # 2005 Elsevier B.V. All rights reserved. Keywords: Lightweight building element; Ventilated cavity; Dynamic thermal characteristics; Two-dimensional heat transfer
1. Introduction Lightweight building elements (LBEs) are building constructions made of thermal insulation and two thin metal sheets. They are especially appropriate as envelope elements for buildings such as warehouses, commercial buildings, shopping centres, production plants as well as residential buildings. The steady thermal characteristics (U-value) of an LBE are good, while the dynamic thermal characteristics show some disadvantages in terms of ensuring indoor thermal comfort due to the low thermal capacity of an LBE compared with massive building constructions. Two of the most important dynamic thermal characteristics of a building construction are the decrement factor and the time lag. The decrement factor is the ratio between the dynamic thermal transmittance and the steady-state thermal transmittance, while the time lag is the period of time between the maximum ambient temperature and the maximum inner surface heat flux [1]. For LBEs the decrement factor is high, while the time lag is small compared to massive building constructions. Several * Corresponding author. Tel.: +386 1 4771316; fax: +386 1 2518567. E-mail address: [email protected] (B. Cˇerne). 0378-7788/$ – see front matter # 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.enbuild.2004.12.006
studies involving ventilated cavities, radiant barriers and phase-change materials were made to reduce these disadvantages of LBEs. Medina et al. [2] studied the influence of a ventilated attic on the heat flux through a lightweight ceiling construction in two small test houses. One of the houses had a radiant barrier installed in the attic. The authors reported that the heat flux through the lightweight ceiling did not decrease when the air flow rate through the ventilated attic was higher than 0.0013 m3/sm2 of ceiling construction. The radiant barrier was effective because the heat flux through the ceiling decreased by as much as 40% at the highest sol–air temperature. Medina [3] also analyzed the efficiency of a radiant barrier with different emissivities (due to dust particles). He reported a decrease in the ceiling heat flux from 40 to 15% when the emissivity of the radiant barrier increased from 0.05 to 0.3. He also analyzed the efficiency of a radiant barrier as a function of the steady-state thermal transmittance of a lightweight ceiling [4]. Winiarski and O’Neal [5] developed a model for calculating the heat flux through a lightweight ceiling, where they supposed the value of the decrement factor was equal to 1. Kairys and Karbauskaite [6] analyzed the transient heat flux through a
B. Cˇ erne, S. Medved / Energy and Buildings 37 (2005) 972–981
Nomenclature arg C d f g h K L P q q˜ _ q Q _ Q R t T T˜ _ T Dt U v w Z jj
argument of complex number (rad) specific heat capacity (J/kg K) width (mm) decrement factor gravitational acceleration (m/s2) heat transfer coefficient (W/m2K) correction coefficient length (m) period (s) heat flux (W/m2K) harmonic heat flux oscillation (W/m2) specific heat flux amplitude (W/m2K) heat flux (W/m2) heat flux amplitude (W/m2) thermal resistance (m2K/W) time (s) temperature (K) harmonic temperature oscillation (K) temperature amplitude (K) time lag (h) thermal transmittance (W/m2K) specific air flow (m3/sm2) wind speed (m/s) element of transfer matrix absolute value of complex number
Greek letters a absorptivity b phase (rad) e emissivity l conductivity (W/mK) r density (kg/m3) v frequency (rad/s) Subscripts a ambient an analytical av average c cavity d downward i inner in inlet ir longwave radiation max maximum nu numerical out outlet s shortwave radiation sin sine function sol sol–air u upward 1 m, 3 m, 5 m distances from cavity inlet
973
vertical lightweight wall with and without a naturally ventilated cavity for a selected extreme day. They showed a 30% decrease in the maximum heat flux on the inner surface of a wall with a naturally ventilated cavity compared to a wall with no cavity. Peippo et al. [7] analyzed the temperature swings in a lightweight house with phasechange material (PCM) impregnated into the plasterboard compared to a reference house without any PCM. The temperature swings decreased by as much as 2 8C when the PCM was used. Athienitis et al. [8] performed experiments on a lightweight test-room with a PCM gypsum board and on a gypsum board without PCM. They showed that the daily maximum room temperature was about 4 8C lower in the room with the PCM gypsum board. In these investigations the transient heat transfer in the lightweight building constructions was analyzed onedimensionally. Longwave radiation on the building construction was not measured. The sol–air temperature was calculated with the measured ambient temperature and shortwave radiation. The influence of the ratio between the ambient temperature amplitude and the sol–air temperature amplitude was not analyzed in detail. The one-dimensional analytical model for a transient flux calculation from the EN ISO 13786 standard [1] can only be used for a building construction with no cavity or with a closed cavity. For a transient heat flux calculation through a long lightweight building construction with a ventilated cavity a twodimensional model must be used. In this case the ratio between the temperature boundary conditions at the cavity inlet and the temperature boundary conditions along the ambient surface of the cavity must be analyzed. In our work, we also formed multi-parametric equations to predict the inner surface heat flux amplitude and the decrement factor.
2. Analytical method for calculating transient heat transfer in a building construction The dynamic thermal characteristics of a building construction can be calculated using the analytical method described in EN ISO 13786 [1]. This method is based on the work of Carslaw and Jaeger [9], who showed that the temperatures and the heat fluxes on both sides of the building construction can be calculated with linear equations when the temperatures vary sinusoidally. These equations can be written in matrix form: Z ii Z ia T˜ i T˜ a ¼ (1) Z ai Z aa q˜ i q˜ a When the ambient temperature amplitude is known and the inner temperature is constant the heat flux amplitude on the inner surface of the construction can be calculated from Eq. (1): _ qi
_
¼
Ta jZ ia j
(2)
974
ˇ erne, S. Medved / Energy and Buildings 37 (2005) 972–981 B. C
The amplitude characterizes the fluctuation around the average heat flux, which can be calculated from steady-state conditions. The decrement factor, f, is one of the two basic dynamic characteristics of building constructions according to the cited standard. It is the ratio of the dynamic thermal transmittance to the thermal transmittance under steadystate conditions, known as the U-value. This factor has values between 0 and 1, and can be calculated using the following equation: 1 (3) jZ ia jU The period of time between the ambient temperature amplitude and the inner surface heat flux amplitude can be calculated using the following equation: f ¼
P argðZ ia Þ (4) 2p In our work the analytical method will be used for two purposes: first, to verify the numerical model in the case of an LBE with a closed cavity; second, as a source of standardized boundary conditions in the two-dimensional numerical model and to calculate the dynamic thermal characteristics. The latter allows a direct comparison of oneand two-dimensional transient problems.
Dt ¼
Fig. 1. Geometrical model with boundary conditions.
The air flow rate through the ventilated cavity is constant during the whole period. The inlet velocity profile is uniform. The gravitational forces are taken into account when an LBE with a non-ventilated cavity is analyzed. The radiation, the absorption and the scattering of the air in the cavity are zero. The emissivities of the cavity surfaces are diffuse. The shortwave and longwave radiation are uniform across the ambient surface of the LBE. Adiabatic conditions are assumed at the edge of the LBE. 4. Verification of the numerical model 4.1. Analytical verification
3. Numerical model of an LBE with a ventilated and a non-ventilated cavity Transient heat transfer in an LBE with a ventilated cavity must be solved numerically using a CFD method. The control volume method was used in our case. For control volumes, differential equations of the conservation of mass, momentum and energy, known as the set of Navier–Stokes equations, must be solved. Together with Fourier’s law of heat conduction and the Stefan–Boltzmann radiation law, the pressure, velocity and temperature fields in the system can be calculated. The commercial CFD software PHOENICS was used to solve the set of equations using the set of boundary conditions. A model called IMMERSOL was used to calculate the longwave radiation within the cavity [10]. In all cases the numerical calculations were performed with a 10 min time step. The two-dimensional geometrical model with the boundary conditions used in the numerical analyses is shown in Fig. 1. Various boundary conditions were used in the numerical simulation and they are described in detail in the following sections. The influence of the initial conditions was neglected by using a 12 h pre-period calculation. The following assumptions were used in the numerical model: The heat transfer is transient with a 24 h period. The heat transfer is two-dimensional. The k–e model is used in the case of turbulent flow.
For the analytical verification of the numerical model an LBE with a non-ventilated cavity was used. The dimensions and the material properties are shown in Table 1. Two different cavity surface emissivities were analyzed: eci = eca = 0.9 and eci = eca = 0.1. The combined convection and radiation surface resistance Ri = 0.04 m2K/W and Ra = 0.13 on the ambient and the inner surface were analyzed [11]. According to [11] the thermal resistance of a closed cavity (Rc) with high emissivity cavity surfaces 0.12 and 0.26 m2K/W for the low emissivity cavity surfaces, were used. The transient ambient temperature was defined using a sine function with amplitude _ (T a ) 1 K around an average daily value of 25 8C with a maximum value at 12 h (noon) and a minimum at 24 h (midnight). The indoor air temperature was a constant 25 8C _ (T i = 0 K). No shortwave radiation was taken into account. The gravitational forces in the numerical simulations were calculated by considering the inclination of the LBE. The elements of the transfer matrix were calculated as
Table 1 Dimensions and material properties of the LBE r (kg/m3)
C (J/kg K)
Layer
Thickness (mm)
k (W/mK)
Metal sheet Thermal insulation Metal sheet Air Metal sheet
0.6 78.8 0.6 7 0.6
43 7800 460 0.04 120 840 43 7800 460 Rc,ec = 0.9 = 0.12; Rc,ec = 0.1 = 0.26 43 7800 460
B. Cˇ erne, S. Medved / Energy and Buildings 37 (2005) 972–981
975
Fig. 2. Temperatures and heat fluxes for 2 days with equal initial and boundary conditions for high (left) and low (right) cavity surface emissivities.
follows: Zii,ec = 09 = 1.31, Zia,ec = 09 = 2.29 (m2K/W), Zai,ec = 09 = 1.10 (W/m2K), Zaa,ec = 09 = 1.52, Zii,ec = 01 = 1.38, Zia,ec = 01 = 2.45 (m2K/W), Zai,ec = 01 = 1.11 (W/m2K), Zaa,ec = 01 = 1.53. Fig. 2 shows the temperature boundary conditions Ta and Ti and the ambient and inner surface heat fluxes calculated with analytical and numerical methods. It is clear that the numerically calculated heat flux amplitudes and the time lag are in good agreement with the analytically calculated values, with a maximum difference of 2%. This also indicates that the numerical time step and the control volume mesh are appropriate.
surface of the LBE. The influence of the ambient was neglected by an additional 5 cm of polystyrene thermal insulation, as shown in Figs. 3–5. The electrical resistance heating wire for the heating and the polyethylene pipes for
4.2. Verification of the numerical model with experiments A series of experiments was carried out to validate the numerical simulations for LBEs with closed and ventilated cavities. The layers and dimensions were the same as described in Section 4.1. The air flow rate through the ventilated cavity was ensured by forced ventilation. The air enters the cavity directly from the ambient. Some hydrodynamic devices were used in the collecting duct to provide a uniform air flow rate through the cavity. The indoor air temperature was regulated by the heating and cooling of an additional 2 cm airspace below the bottom
Fig. 3. Photo of the LBE where the experiments were performed. The dimensions of LBE were 6 m 6 m, the inclination was 358. The locations of temperature, wind velocity and heat flux sensors are shown schematically.
Fig. 4. Thermostated cavity below the LBE with resistance heating wire and cooling pipes. The opening for the installed heat flux meter was later thermally insulated. The heat flux meter was installed on the ventilated cavity bottom sheet, as shown in Fig. 5, in such a way that even small fluctuation of the thermostated air cavity temperature, Ti, were reduced.
Fig. 5. Cross-section of the tested LBE with heating and cooling elements in the insulated cavity below the LBE.
976
ˇ erne, S. Medved / Energy and Buildings 37 (2005) 972–981 B. C
Fig. 6. Measured temperatures, wind speed, shortwave and longwave radiation on the LBE surface.
the water cooling were mounted in the thermally insulated air cavity below the LBE (Figs. 4 and 5). All the temperatures were measured with calibrated Ni– CrNi thermocouples. Four temperature sensors were used at the inlet- and outlet-cavity cross-sections; they also indicated the uniformity of the air flow rate through the cavity of the LBE. The temperature gradients in the cavity were measured at three locations above the heat flux meters. Three heat flux meters (size: 120 mm 120 mm) were mounted 1, 3 and 5 m from the inlet opening. The meteorological parameters were measured with a calibrated pyranometer (for the shortwave radiation) and pyrgeometer (for the downward and upward longwave radiation), a ventilated and shielded temperature sensor, a wind speed and direction sensor and a relative humidity sensor. Both the pyranometer and the prygeometer were mounted directly on the test roof and therefore had the same inclination as the LBE. The air flow rate was measured using an orifice plate and a differential pressure gauge [12]. The measurement data were stored every 5 min using a data logger. Measurements with the non-ventilated and the ventilated cavities were performed from June to August 2003. Two different air flows were analyzed: 0.0029 and 0.0043 m3/ sm2. Fig. 6 shows the measured parameters for the three
selected days that were used for the experimental verification of the numerical model. The main difficulty when comparing the numerical model and the results of the experiments is the description of the real meteorological parameters with an appropriate periodic function. The following boundary conditions were used in the numerical calculations: The inlet air temperature was described with a sine function: _
T a;sin ¼ T a;av þ T a sin ðvt bÞ
(5)
In Eq. (5), the average ambient temperature is the mean value between the maximum and minimum temperatures, while the temperature amplitude is half the difference between the maximum and minimum temperature on the selected day. The sol–air temperature was used to represent the influence of the ambient temperature as well as the shortwave and longwave radiation on the ambient surface of the LBE: T sol ¼ T a þ
Qs aa;s þ Qir;d aa;ir Qir;u ea;ir ha
Fig. 7. Sol–air temperatures used in the numerical calculations, Tsol is calculated with Eq. (6) using measurement data.
(6)
B. Cˇ erne, S. Medved / Energy and Buildings 37 (2005) 972–981
The ambient temperature and the shortwave radiation were measured. The downward longwave radiation was calculated from the measured net longwave radiation and the temperature of the pyrgeometer. The Stefan–Boltzmann law was used to calculate the upward longwave radiation from the LBE. For this calculation the average temperature of an ambient metal sheet was used. In Eq. (6) the absorptivity was 0.85 [13] and the emissivity was 0.9. The convective heat transfer coefficient, ha, does not take into account the radiative heat transfer, and is calculated using the equation [14]: ha ¼ 3:1 þ 4:1w
(7)
The daily wind speed average was used for the calculation of ha. For the purpose of the numerical calculation the sol–air temperature was approximated in two ways (Fig. 7). First, only a single sine function (Tsol,1) was used: _
T sol;1 ¼ T sol;av þ T sol sin ðvt bÞ
T sol;2
8 Qir;d;av aa;ir Qir;u;av ea;ir > > < T a;av þ ha _ ¼ þT a sin ðvt bÞ; night time > > : _ T sol;av þ T sol sin ðvt bÞ; day time
977
(9)
Fig. 8 shows a comparison of the measured and the calculated heat fluxes along the cavity length. The results show that the approximation of the thermal boundary conditions with two sine functions of the sol–air temperature is necessary for good agreement of the heat fluxes measured with experiments and calculated using numerical simulations. However, when the heat flux amplitude is of interest, a single sine sol–air temperature approximation provides good agreement between the measured and the calculated extreme daily heat flow. Therefore, the single sine approximation is appropriate. When the solar heat gains or the longwave radiant cooling is analyzed, an approximation of the sol–air temperature with two sine functions is necessary.
(8)
Second, the sol–air temperature was divided in two parts: the first part for night time and the second part for day time (between sunrise and sunset). Each part was approximated with a single sine function:
5. Multi-parametric analysis of the heat flux amplitude The presented and verified numerical model was used for a set of transient numerical calculations for different
Fig. 8. Measured and numerically calculated heat fluxes: top, with boundary condition Tsol,1 (Eq. (8)); bottom, with temperature boundary condition Tsol,2 (Eq. (9)).
ˇ erne, S. Medved / Energy and Buildings 37 (2005) 972–981 B. C
978
_
Table 2 Analyzed air velocities in the ventilated cavity Specific air flow (m3/sm2)
Cavity width (mm)
0.001
0.002
0.005
0.010
0.020
10 25 50 100
0.9 0.36 – –
1.8 0.72 0.36 –
4.5 1.8 0.9 0.45
– 3.6 1.8 0.9
– – 3.6 1.8
geometric, thermal and hydraulic initial and boundary conditions. The aim of this set of numerical calculations was to provide the necessary input data for a multi-parametric model of the inner surface heat flux amplitude for a wide range of influential parameters. These parameters were the thermal insulation thickness (78.8, 148.8 and 198.8 mm), the cavity surfaces’ emissivities (three different combinations of emissivities were used: eci = 0.9 and eca = 0.9; eci = 0.9 and eca = 0.1; eci = 0.1 and eca = 0.1). The range of specific air flow rates through the cavity used in the numerical calculations is presented in Table 2. The length of the LBE with the ventilated cavity was 9 m in all cases. The ambient and inner surface resistances were 0.04 and 0.13 m2K/W, respectively. The material properties used in the calculation are given in Table 1. The validity of the multi-parametric model must be ensured for any combination of both transient boundary conditions—at the inlet of the cavity and along the cavity surface. For this reason the transient boundary conditions must be normalised. Both normalised boundary conditions were in phase and varied sinusoidally. In all the numerical calculations the normalised air _ temperature amplitude was T a = 1 K, the normalised increase of the air temperature amplitude due to absorbed _ solar radiation was T s = 0 and 1 K. In the case when _ _ T a = 1 K and T s = 0 K the transient temperature boundary conditions on the inlet were the same as along the cavity surface—the influence of solar radiation was not taken into _ account. With these boundary conditions qL;i;a was
_
calculated. When T a = 1 K and T s = 1 K the combined _ _ boundary condition along the cavity surface is T a þ T s = 2 K—the normalised influence of solar radiation is taken _ into account. In such a way qL;i;s can be calculated. _ Fig. 9 shows the inner surface heat flux amplitudes qL;i;a _ and qL;i;s along the LBE with a ventilated cavity for an LBE width of 80 mm and a cavity width of 25 mm in the case of two different specific air flows. The influence of all three combinations of cavity surface emissivities is shown. From Fig. 9 it is clear that the most influential parameter when it comes to a decrease in the heat flux amplitude of an LBE with a ventilated cavity is the emissivity of the cavity _ surfaces. The air flow rate has the opposite influence on qL;i;a _ _ and qL;i;s : qL;i;a increases with an increase in the air flow _ rate, while qL;i;s decreases with an increase in the air flow. When the ventilated cavity has low emissivity surfaces, an increase in the air flow rate over 0.010 m3/sm2 contributes to _ a relatively small decrease of qL;i;s . The amplitude of the heat flux on the inner surface of an LBE with a ventilated cavity for different combinations of _ _ T a and T s can be calculated using the equation: _
__
__
QL;i ¼ T a qL;i;a þ T s qL;i;s _ Ts _
Ts ¼
(10)
can be calculated using the following equation: Qs;max aa;s ha
(11)
Altogether, 252 transient numerical simulations were made to provide sufficient data for a statistical analysis. The least-square error method was used to form two multiparametric equations for each component of the heat flux _ _ amplitude (qL;i;a and qL;i;s ) used in Eq. (10): _ qL;i;a
Da ¼ Aa LBa vCa dLBE dcEa
(12)
_ qL;i;s
Ds ¼ As LBs =ln dc vCs dLBE dcEs
(13)
The values of the constants used in Eqs. (12) and (13) as function of the emissivities of the cavity surfaces are given in
Fig. 9. Inner surface heat flux amplitude for an LBE with an 80 mm width and with 25 mm of ventilated cavity; specific air flow rates 0.002 m3/sm2 (left) and _ _ _ 0.010 m3/sm2 (right). (1) qL;i;a ; (2) qL;i;s ; (3) qLBE without cavity ; (A) eci = 0.9, eca = 0.9; (B) eci = 0.9, eca = 0.1; (C) eci = 0.1, eca = 0.1.
B. Cˇ erne, S. Medved / Energy and Buildings 37 (2005) 972–981 Table 3 _ _ Values of the constants for calculation of qL;i;a and qL;i;s ei
eo
Aa
Ba
Ca
Da
Ea
Table 4 Values of constants for calculation of KL,a and KL,s R2
0.9 0.9 101.1394 0.01008 0.00995 1.21654 0.01365 0.9924 0.9 0.1 126.3316 0.03369 0.04297 1.20232 0.06438 0.9895 0.1 0.1 136.1368 0.03939 0.05198 1.19968 0.07883 0.9876 ei
eo
0.9 0.9 0.9 0.1 0.1 0.1
As
Bs
Cs
Ds
Es
R2
4.15038 0.91473 0.24666 1.19512 0.25357 0.9699 0.28997 1.60445 0.50365 1.17216 0.24383 0.9578 0.15227 1.82506 0.59344 1.17234 0.16108 0.9461
Table 3, together with the R2 value of multi-parametric regression. Like with the heat flux amplitude changes along the length of an LBE with a ventilated cavity, the decrement factor changes too. A comparison between the decrement factor of an LBE with a ventilated cavity and an LBE without a cavity is of interest to us. Therefore, the influence of a ventilated cavity was analyzed with a correction coefficient. The correction coefficient at a selected distance from the cavity inlet with regard to the previously described normalised transient thermal boundary conditions can be calculated with the following equation: K L;a ¼ K L;s ¼
_ qL;i;a
U LBE f LBE _ qL;i;s
979
eI
eo
Fa
Ga
Ha
Ia
R2
0.9 0.9 0.1
0.9 0.1 0.1
1.054616 1.451800 1.592288
0.012020 0.038390 0.044620
0.011422 0.047308 0.056988
0.014970 0.071030 0.086190
0.9903 0.9296 0.9095
eI
eo
Fs
Gs
Hs
Is
R2
0.9 0.9 0.1
0.9 0.1 0.1
0.046632 0.003694 0.001984
0.937760 1.631020 1.843530
0.246490 0.501100 0.589540
0.254260 0.240790 0.154060
0.9489 0.9523 0.9407
such an element. Using a ventilated cavity the decrement factor of an LBE can be reduced by up to 10 times, even in the case of very long elements. Cavity surfaces with low emissivity contribute greatly to the reduction of the decrement factor. The decrease of KL,s is much higher than the decrease of KL,a; this is especially important because in _ _ nature T s can be much higher than T a . Like in Eq. (10), the decrement factor of an LBE with a ventilated cavity along the LBE length for different _ _ combinations of T a and T s can be calculated with the following equation: _
_
T a K L;a þ T s K L;s f LBE (16) _ _ T a þ Ts _ Like in the case of the heat flux amplitude (qL;i;a and _ qL;i;s ) the multi-parametric model of the correction coefficient KL,a and KL,s has been formed. These two coefficients can be calculated with the following equations: fL ¼
(14)
(15) U LBE f LBE In Eqs. (14) and (15) ULBE and f LBE are calculated for an LBE without a cavity. We found that using such a formulation, Ka and Ks become independent of the LBE’s width. Fig. 10 shows the correction coefficients Ka and Ks for an LBE with a 25 mm ventilated cavity and two specific air flow rates. It is clear that a two-dimensional approach is necessary for calculating the proper decrement factor of
K L;a ¼ F a LGa vH a dcI a
(17)
K L;s ¼ F s LGs =ln dc vH s dcI s
(18)
The values of the constants used in Eqs. (17) and (18) as a function of the emissivities of the cavity surfaces are given in Table 4, together with the R2 values of multi-parametric regression.
Fig. 10. Correction coefficient for LBE with a 25 mm ventilated cavity; different cavity surface emissivities and specific air flow rates of 0.002 m3/sm2 (left) and 0.010 m3/sm2 (right); (A) eci = 0.9, eca = 0.9; (B) eci = 0.9, eca = 0.1; (C) eci = 0.1, eca = 0.1.
980
ˇ erne, S. Medved / Energy and Buildings 37 (2005) 972–981 B. C
Fig. 11. Time lag of an 80 mm thick LBE with a 25 mm ventilated cavity; air flow rates 0.002 m3/sm2 (left) and 0.010 m3/sm2 (right); (A) eci = 0.9, eca = 0.9; (B) eci = 0.9, eca = 0.1; (C) eci = 0.1, eca = 0.1; (3) LBE without cavity.
Fig. 12. Time lag of a 200 mm thick LBE with a 25 mm ventilated cavity and air flow rates of 0.002 m3/sm2 (left) and 0.010 m3/sm2 (right); (A) eci = 0.9, eca = 0.9; (B) eci = 0.9, eca = 0.1; (C) eci = 0.1, eca = 0.1; (3) LBE without cavity.
The time lag was also analyzed in this investigation. Figs. 11 and 12 show the numerically calculated time lag for an LBE with a ventilated cavity for two different LBE widths, three different combinations of cavity surface emissivities and two different air flow rates. The analytically calculated time lag for the same LBE without a cavity is also shown. It is clear that the ventilated cavity has a small influence on the time lag along the LBE length and small influence with regard to the LBE without a cavity. In the case of a higher air flow rate and high emissivity surfaces, the time lag is even a little bit smaller due to the higher convection heat transfer. Therefore, we can conclude that the influence of a ventilated cavity on the time lag can be neglected.
6. Conclusions In this work we analyzed the dynamic thermal characteristics of an LBE with a ventilated cavity. Presented
dynamic characteristics, decrement factor, f, and time lag, Dt, must be defined in such a way that different building constructions can be compared. The important difference between LBE with a ventilated cavity and other nonventilated constructions is in two-dimensional heat transfer and in two different thermal boundary conditions: at the cavity inlet and along the ambient cavity surface. Both were taken into consideration in our work. A two-dimensional numerical model based on the control volume method was made to calculate the dynamic heat transfer through the LBE with a forced ventilated cavity. It was verified with an analytical method and experiments. The influences of the ambient temperature variation and the solar radiation variation on the dynamic heat transfer were analyzed separately. For comparison purposes, the ambient temperature variation and the solar radiation variation were normalised. At the cavity inlet only normalised ambient temperature variation was taken into account, while along the cavity surface the normalised ambient temperature variation and the normalised solar radiation variation were
B. Cˇ erne, S. Medved / Energy and Buildings 37 (2005) 972–981
taken into account as sol–air temperature. The variation of ambient temperature at the cavity inlet is defined with one sine function. The sol–air temperature along the cavity surface can be defined with one or two sine functions. We found that for calculating dynamic thermal characteristics of an LBE one sine function is sufficient. A multi-parametric analysis was made for different parameters: LBE width and length, cavity surfaces emissivity, cavity widths and air flow rates through the cavity. Based on numerical simulations correction factors for normalised ambient temperature amplitude and normalised sol–air temperature amplitude were determinated in form of multi-parametric equations. The set of two multiparametric equations were determinated for prediction of the length dependent inner surface heat flux amplitude. We found that a ventilated cavity can greatly reduce the influence of solar radiation on the inner surface heat flux compared to an LBE without a ventilated cavity. In combination with a low emissivity cavity surfaces the decrement factor of an LBE can be reduced by up to 10 times. Increasing the air flow rate over 0.010 m3/sm2 _ contributes to a relatively small decrease of qL;i;s when the cavity surfaces have low emissivities. The ventilated cavity does not contribute to a greater shift of the cooling load to off-peak electricity consumption. Therefore, the time lag calculated for an LBE without a ventilated cavity can also be used for an LBE with a ventilated cavity.
References [1] European Standard EN ISO 13786, Thermal performance of building components—dynamic thermal characteristics—calculation methods, European Committee for Standardization, Brussels, Belgium, 1999.
981
[2] M.A. Medina, D.L. O’Neal, W.D. Turner, Effect of attic ventilation on the performance of radiant barriers, Journal of Solar Energy Engineering 114 (1992) 234–239. [3] M.A. Medina, Effects of shingle absorptivity, radiant barrier emissivity, attic ventilation flow rate and roof slope on the performance of radiant barriers, International Journal of Energy Research 24 (2000) 665–678. [4] M.A. Medina, On the performance of radiant barrier in combination with different attic insulation levels, Energy and Buildings 33 (1) (2000) 31–40. [5] D.W. Winiarski, D.L. O’Neal, A quasi-steady-state model of attic heat transfer with radiant barriers, Energy and Buildings 24 (3) (1996) 183–194. [6] L. Kairys, J. Karbauskaite, Solar radiation influence to the thermo physical parameters of thermal wave that penetrates through the lightweight buildings partitions, in: Proceedings of the Conference on Dynamic Analysis and Modelling Techniques for Energy in Buildings, Ispra, Italy, 2003, , pp. 105–113. [7] K. Peippo, P. Kauranen, P.D. Lund, A multicomponent PCM wall optimized for passive solar heating, Energy and Buildings 17 (4) (1991) 259–270. [8] A.K. Athienitis, C. Liu, D. Hawes, D. Banu, D. Feldman, Investigation of the thermal performance of a passive solar test-room with wall latent heat storage, Building and Environment 32 (5) (1997) 405–410. [9] H.S. Carslaw, J.C. Jaeger, Conduction of Heat in Solids, second ed. Oxford University Press, 1959. [10] PHOENICS 3.5.1, Parabolic, hyperbolic, elliptic numerical integration code system, CHAM, Wimbledon, England, 2003. [11] European Standard EN ISO 6946, Building components and building elements—thermal resistance and thermal transmittance—calculation method, European Committee for Standardization, Brussels, Belgium, 1996. [12] European Standard EN ISO 5167-1, Measurement of fluid flow by means of pressure differential devices inserted in circular cross-section conduits running full—Part 1: general principles and requirements, European Committee for Standardization, Brussels, Belgium, 1995. [13] S. Medved, C. Arkar, B. Cˇ erne, A large-panel unglazed roof-integrated liquid solar collector—energy and economic evaluation, Solar Energy 75 (6) (2003) 455–467. [14] Australian Standard AS 3634, Solar heating systems for swimming pools, 1989.