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Development of a simulation analysis environment for ventilated slab systems
Applied Thermal Engineering 87 (2015) 66e78
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Applied Thermal Engineering journal homepage: www.elsevier.com/locate/apthermeng
Research paper
Development of a simulation analysis environment for ventilated slab systems Benjamin Park, Moncef Krarti* University of Colorado at Boulder, Building Systems Program, Civil, Environmental, and Architectural Engineering Department, UCB 428, University of Colorado, Boulder, CO 80309, USA
h i g h l i g h t s A new dynamic simulation environment has been developed for ventilated slabs. Thermal bridging effects for ventilated slabs have been evaluated. Impact of a wide range of design and operating parameters is determined.
a r t i c l e i n f o
a b s t r a c t
Article history: Received 18 February 2015 Accepted 20 April 2015 Available online 8 May 2015
In this paper, a new simulation environment is presented to evaluate the energy performance of ventilated slab systems in multi-floor buildings. The simulation environment combines a transient twodimensional finite difference solution of a ventilated slab system comprising slab-wall joints with thermal network model for indoor spaces and associated exterior walls. The developed simulation environment can assess the impact of thermal bridging effects on both heating and cooling building thermal loads. First, the predictions of the developed simulation environment are verified against those obtained from a detailed whole-building energy simulation tool when thermal bridging effects are neglected. Then, a series of parametric analyses are performed to determine the performance of ventilated slab systems under various design and operating conditions considering the thermal bridging effects. It is found that the energy performance of ventilated slab systems and thermal bridging impact depend on a wide range of factors including size of the slab, supply air inlet temperature, air mass flow rate, core diameter, core pitch, and depth of hollow cores. In particular, it is found that the thermal bridge affects significantly the energy performance of ventilated slab systems and can increase both heating and cooling energy consumptions by 17% and 11%, respectively. © 2015 Elsevier Ltd. All rights reserved.
Keywords: Air conditioning Hollow cores Thermal bridging Ventilated slab
1. Introduction In most countries, approximately 20e40% of total energy consumption is due to building heating and cooling operations [1]. The use of building thermal mass is one of the effective techniques to reduce the building energy consumption. Recently, active cooling/ heating systems integrated with building thermal mass, such as ventilated slab systems, have been considered to heat and cool residential and commercial buildings. Indeed, ventilated slab systems have been employed in northern Europe and Australia.
* Corresponding author. Tel.: þ1 303 492 3389; fax: þ1 303 492 7317. E-mail address: [email protected] (M. Krarti). http://dx.doi.org/10.1016/j.applthermaleng.2015.04.065 1359-4311/© 2015 Elsevier Ltd. All rights reserved.
Ventilated slab systems, utilize air channels within precast slabs to cool or heat thermal zones. The basic operation of ventilated slab systems is similar to hydronic radiant systems, but it uses air instead of water as the heat transfer fluid. The air is then supplied directly to condition various thermal spaces. Compared to conventional air systems, hollow core slab systems have the potential to significantly reduce energy use to heat and/or cool buildings. In particular, it is reported that ventilated slab systems can serve as passive thermal storage media, and can effectively reduce the daytime cooling demand for several types of buildings. During the summer season, outside air can be used to cool down the temperature of the slab thermal mass specially at nighttime. During the winter season, the slab thermal mass can store heat generated by internal sources (such lighting and
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equipment) and then release it to the conditioned zone during the on-peak period. Barnaby et al. simulated the thermal performance of a hollow core slab system for a commercial building using an energy analysis program and concluded that the system could provide an energy reduction between 13% and 30% for the peak cooling load for dry US climates [2]. However, the energy analysis evaluation neglected thermal bridging effects at the floor-wall joints. Using an implicit finite difference model for a ventilated slab system serving a single zone office in Montreal, Canada, Zmeureanu and Fazio found that the daily cooling load could be reduced by over 35% compared to conventional HAVC system [3]. Similarly, Russell and Surendran investigated the ventilated slab system performance using a two-dimensional finite difference model. Assuming that air at 16.9 C is continuously circulated through hollow cores for 14 h, they indicated that the cooling potential of the system increased by 335% compared to a traditional slab configuration with night ventilation [4]. Chae and Strand developed a heat balance model suitable for EnergyPlus for the ventilated slab using a modification of the conduction transfer function (CTF) formulation with heat sources/sinks. The authors have found that daily total demand savings of 23% and peak demand savings of 28% can be achieved by the ventilated slab system if outdoor air is circulated through the hollow cores during unoccupied hours [5]. Moreover, compared to all air HVAC systems, the benefits of ventilated slab systems include better thermal comfort. Better thermal comfort is associated to increased heat transfer by radiation. It is estimated that more than 50 percent of heat can be transferred from the controlled ventilated slab surface to other surfaces by radiation [6]. Indeed, Shaw et al. reported that ventilated slab systems can control slab surface temperatures and maintain comfortable environment for the occupants [7]. Corgnati and Kindinis investigated the thermal performance of active hollow core slabs under a Mediterranean climate. The authors concluded that the hollow core ventilated slab system reduces thermal cooling load and provides better thermal comfort to the occupants compared to the traditional ventilation system. However, no specific analysis was carried to estimate the level of thermal cooling load reduction and thermal comfort improvement compared to the traditional ventilation system [8]. Several studies have found that thermal bridging effects especially at the wall-floor joints can be significant and can increase building thermal loads [9e11]. In particular, Theodosiou and Papadopoulos reported that actual heating loads can be up to 30% higher when thermal bridging effects are considered [10]. Most of the simulation models developed for ventilated slab systems do not take account for thermal bridge effects which can be magnified due to the controlled slab temperatures. Indeed, higher slab surface temperature for hollow core ventilated slab system could increase heat losses through slab edges during the heating operation. Existing detailed whole-building simulation tools including, eQUEST and EnergyPlus which are the most widely used building energy analysis tools in the US, are not suitable for estimating of thermal bridging effects on the energy performance of ventilated slabs. Indeed, eQUEST has no capabilities to perform any 2D heat transfer analysis (i.e., analysis of thermal bridging in building envelope including joints between floors and walls). Moreover, while the current version of EnergyPlus has some capabilities to perform 2-D heat transfer analysis for slab floors, the boundary conditions along the joints between the floor and the walls have to be set as adiabatic. Thus, EnergyPlus is also not capable to evaluate the thermal bridging effects for the ventilated slab due to the wall-floor joints. In paper, the impact of thermal bridges on the performance of ventilated slab systems is estimated under various design and operating conditions using a simulation analysis environment for multi-floor buildings.
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2. Development of the simulation analysis environment A simulation environment that combines finite difference model and thermal network technique is developed to evaluate the performance of ventilated slab systems and assess the impact of thermal bridging effects associated with floor-wall joints. Specifically, 3R2C thermal network model is used to estimate heating and cooling loads for the thermal zones. A 2-dimensional FDM ventilated slab model is employed to accurately address the heat transfer through slab-wall joint. Fig. 1 illustrates the schematic of the simulation environment using both an RC thermal network model and a finite difference method (FDM) ventilated slab system model. 2.1. FDM ventilated slab model A control volume approach and pure implicit finite difference technique is used to model hollow core ventilated slab system by solving the two dimensional heat conduction equation within the building envelope components that include embedded heat source/ sink (i.e. floor and ceiling) [12]:
k
v2 T v2 T vT þ k 2 þ Q ¼ rcp 2 vt vx vy
(1)
The actual heat transfer between the building envelope elements (i.e. slab floor and walls) and the ventilated slab system depends on the fluid inlet temperature and mass flow rate. To model the ventilated slab system using Eq. (1), it is assumed that the air inlet temperature and the mass flow rate are known while the remaining parameters are to be calculated [13]. Specifically, air flows through hollow cores of the ventilated slab system are supplied to heat and cool the indoor space. A ventilated slab system can be thought of as a heat exchanger between air flowing through the hollow cores and the slab floor. The effectiveness-NTU heat exchanger method has been shown to be convenient to utilize when the air outlet temperature is not known [14]. Specifically, heat released or absorbed from the air flowing in the hollow cores can be estimated using Eq. (2):
_ p air Tair; in Tair; out Q ¼ mc
(2)
The maximum heat transfer potential between the air and the slab is estimated based on the temperature of heat source or sink, Tsrc:
_ p air Tair; in Tsrc qmax ¼ mc
(3)
The effectiveness of the heat exchanger, ε, is defined as the ratio of the actual energy transfer to the maximum amount of energy transfer. When one fluid is stationary for a heat exchanger, the effectiveness can be related to the number of transfer units (NTU) [15]:
ε≡
Q ¼ 1 eNTU Qmax
(4)
where NTU is defined by:
NTU≡
UA _ p air mc
(5)
The heat transfer coefficient, UA, is can be estimated using the convection heat transfer along the contact surface area of the cores:
UA ¼ hair ðpDLÞ
(6)
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Fig. 1. Schematic of the FDMRC simulation environment.
The air convection heat transfer coefficient, hair, can be determined using the DittuseBoelter correlation. Using Eq. (4), the heat generation per volume of air flowing in the hollow cores can be expressed in W/m3 as shown by Eq. (7):
εQmax Q ≡ pD2 4 L
(7)
A non-uniform geometric discretization scheme is applied in order to reduce the computational efforts as well as the memory requirements to obtain an accurate numerical solution. In particular, the discretization grid is very fine near the surface boundaries where thermal interactions are strong between two different materials and/or near the heat sources. The grid is gradually expanded in the area where relatively smaller temperature changes are
Fig. 2. Geometric mesh discretization used for the two-dimensional FDM slab model.
expected. Fig. 2 illustrates the selected mesh configuration used for the two-dimensional building envelope model. The finest distance between nodes in the model is 0.008 cm. A sensitivity analysis is carried out to assess the accuracy in predicting slab inside/outside surface temperatures, wall inside/ outside surface temperatures, and zone mean air temperature as functions of the grid size (using a detailed grid with 40,625 nodes as a reference). Fig. 3 depicts the results of the assessment analysis to determine the optimal grid size (expressed in terms of the number of nodes) on the prediction accuracy (expressed in terms of RMSE between the predictions and the reference case results) and on the computational effort (expressed in terms of CPU of processing time). Based on the results of Fig. 1, an optimal grid size of about 11,000 allows minimizing both the CPU requirement and the RMSE value. A comprehensive convective algorithm developed by Walton [16] is adopted in the FDM model for interior convective coefficient for any surface related to the hollow core ventilated slab system. Specifically, the convective coefficients along the surfaces depend on the direction of heat flow and the buoyancy as shown by Eqs. (8) and (9), respectively.
Fig. 3. Impact of the discretization nodes on the prediction accuracy and computational effort.
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1
h¼
9:482jDTj3 7:283 jcos Sj
(8)
dT2 T1 T2 ¼ þ hi AðTzone T2 Þ þ QLWX þ QSW dt R
(11)
where,
1
1:810jDTj3 h¼ 1:382 jcos Sj
C2
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(9)
In the developed simulation environment, algorithms have been included to control the zone indoor air temperature to be within pre-defined set point temperatures and throttling ranges. In particular, variable temperature control strategies are considered to modulate the operation of the ventilated slab systems. Specifically, the ventilated slab system can vary the supply air temperature based on user input of minimum and maximum supply air temperatures. The supply air temperature is varied linearly on an hourly basis, meanwhile the air mass flow rate remains constant [5]. In the analysis presented in this paper, the indoor mean air temperature is used as a thermal load indicator to control the supply air temperature. However, other thermal comfort indicators such as radiant mean temperature can be used to control the supply air temperature. Fig. 6 illustrates variable supply air temperature control strategies used for the ventilated slab system model.
R ¼ l=kA C ¼ rcp lA=2 h ¼ convective coefficient QLWR ¼ long-wave radiation on the outside surface QSOLAR ¼ solar incident on the outside surface QLWX ¼ long-wave radiation exchange between surrounding surfaces QSW ¼ short-wave radiation absorbed by the inside surface T ¼ temperature
Interior convective heat flux along interior surfaces is a function of the surface temperature and the air-layer temperature directly in contact with the surfaces. Various convective heat transfer coefficient correlations have been developed and reported in the literature. For the simulation environment, a correlation to estimate heat transfer coefficient for natural convection along vertical walls is adopted [6,13]:
2.2. RC thermal network model
h ¼ 1:31jDTj3
Thermal networks have been widely used to simulate the thermal performance of the building envelope. In particular, 3R2C models have been successfully applied to simulate the building envelopes for transient building load predictions [17e21]. For instance, Dewson et al. have demonstrated that a 3R2C model is capable of capturing accurately thermal behavior of a test solar cell [19]. Moreover, Fux et al. have shown that a 3R2C thermal network model can accurately predict the room air temperature compared to the measured average room air temperature for a lodging building in the Swiss Alps [21]. Fig. 4 depicts a 3R2C thermal network of exterior wall connecting the indoor space to the outdoor environment. All resistances and capacitances are assumed to be time invariant. The heat balance on the outside wall surfaces can be expressed using Eq. (10). The incident solar radiation includes both direct and diffuse incident solar radiation absorbed by the outside surface. A radiation exchange between the surface, the ambient air, the sky, and the ground is included as a formulation of long-wave radiation flux on the surface. The convective term and conduction term are modeled in RC thermal network using the classical formulation. The heat balance involving the inside surfaces can be written as indicated by Eq. (11) with relevant heat transfer components [13]:
Two surfaces at different temperatures exchange heat energy by thermal radiation. The FDMRC simulation environment uses a gray interchange model to account for thermal radiation heat exchange between interior surfaces. In particular, the radiosity concept developed by Hottel and Sarofim is used by the simulation environment [22]. The net radiative heat transfer at a surface can be determined by Eq. (13).
C1
dT1 T2 T1 ¼ þ ho AðTout T1 Þ þ QLWR þ QSOLAR dt R
1
QLWX;i ¼
Ai εi 4 sTi Ji 1 εi
(12)
(13)
where the radiosity, J, is the sum of the gray body radiation of temperature T, and the incident radiation, H, as expressed by Eq. (14):
J ¼ εsT 4 þ ð1 εÞH
(14)
The incident radiation, H, is normally unknown. If a certain surface i is hit by radiation from another surface j, the radiation heat energy incident on surface i can be described as Eq. (15). Where Fji is the view factor from surface j to i [23].
PN (10)
Hi ¼
j¼1 Fji Aj Ji
Ai
(15)
2.3. Window thermal model
Fig. 4. RC thermal network model of the exterior wall.
The developed simulation environment (FDMRC) utilizes the simplified window model based U-factor and SHGC values to determine the properties of the window glazing. Arasteh et al. outlined a procedure to determine window properties such as glass-to-glass resistance, thickness, thermal conductivity, transmittance, and glazing reflectance by using only U-factor and SHGC values [24]. Eq. (16) expresses the heat balance for single-glazing window. Few assumptions are made in deriving the window heat balance equations. It is assumed that the glass is thin enough so that heat storage within the glazing is neglected and that the short
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wave radiation absorbed is equally distributed to the two faces of the window glazing.
thermal network model, and the simplified window model in order to estimate heating and cooling thermal loads of conditioned
4 Eo εwin εwin Twin;o þ kwin Twin;i Twin;o þ ho To Twin;o þ Swin;o ¼ 0 4 þ kwin Twin;i Twin;o þ hi Ti Twin;i þ Swin;i ¼ 0 Ei εwin εwin Twin;i
The DOE-2 convection model is adopted for the FDMRC to calculate exterior convective coefficient. The DOE-2 convection model combines the MoWiTT and BLAST detailed convection algorithms [13]. The convection coefficient for very smooth surfaces is calculated by Eq. (17):
hc;glass ¼
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i2 h h2n þ aV bz
(16)
spaces. Fig. 5 illustrates the heat transfer mechanisms for various surfaces using the developed FDMRC simulation environment of zones equipped with hollow core ventilated slab systems. In particular, Eq. (21) expresses the indoor air heat balance and includes convective heat transfer from inside surfaces of building envelope and internal loads [6]:
(17)
hn is calculated using Eqs. (8) and (9) and the constants a and b are given in Table 1. For rough surfaces, the convection coefficient is modified according to the Eq. (18):
hc ¼ hn þ Rf hc;glass hn
qconv þ qCE þ qIV þ qair; sys ¼ 0
(21)
The air is assumed to be an ideal gas. Moreover, supply air is assumed fully mixed with the thermal zone indoor air. Heat
(18)
Apart from window heat balance, it is also necessary to consider the interior surfaces that absorb solar radiation that has been transmitted through the fenestration. The developed simulation environment uses a simplified interior solar distribution model that assumes that all transmitted direct radiation is incident on the floor and absorbed as a function of the floor solar absorptance. The reflected portion is assumed to be diffuse and is uniformly absorbed by all surfaces. All transmitted diffuse radiation is uniformly absorbed by all of the zone surfaces including floor surface. If the total transmitted diffuse radiation and the reflected direct radiation (from the floor) are divided by the total interior surface area of the zone and distributed uniformly, then for all surfaces except for the floor. Eqs. (19) and (20) provide expressions for the absorbed solar radiation for interior surfaces and floor, respectively.
P QSW;in;j;q ¼
P q_ TSHG; diffuse þ 1 afloor q_ TSHG; direct PN j¼1 Aj (19) P q_ TSHG;diffuse þ 1 afloor q_ TSHG; direct PN j¼1 Aj P afloor q_ TSHG; direct
P QSW;in;floor;q ¼ þ
Afloor (20)
2.4. Indoor air heat balance The Indoor air heat balance model is an important feature of the developed simulation environment to link FDM slab model, RC Table 1 MoWiTT coefficients. (Source: Refs. [13] and [14]). Wind direction (Units)
Ct (W/m2 K4/3)
a (W/m2 K(m/s)b)
b (e)
Windward Leeward
0.84 0.84
3.26 3.55
0.89 0.617
Fig. 5. Heat balance calculation procedure used by the developed simulation environment.
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strategies used for the ventilated slab systems and implemented in the simulation environment. 3. Model verification
Fig. 6. Variable temperature control scheme for ventilated slab system.
transfer rate from supply air to the zone air is considered as heat source term for thermal zone as Eq. (22).
qair; sys ¼ m_ air $cp; air Tair; supply Tzone
(22)
In the developed simulation environment, algorithms have been included to control the zone indoor air temperature to be within pre-defined set point temperatures and throttling ranges. In particular, variable temperature control strategies are considered to modulate the operation of the ventilated slab systems. Specifically, the ventilated slab system can vary the air flow rate based on user input of minimum and maximum supply air temperatures. The supply air temperature is varied linearly on an hourly basis, meanwhile the air mass flow rate remains constant [5]. In the analysis presented in this paper, the indoor mean air temperature is used as a thermal load indicator to control the supply air temperature. However, other thermal comfort indicators such as radiant mean temperature can be used to control the supply air temperature. Fig. 6 illustrates variable supply air temperature control
a
Predictions of mean air temperature and radiant energy consumption obtained from EnergyPlus are employed in order to verify results obtained from the developed simulation environment (FDMRC). Fig. 7 shows a 2-story building section with two thermal zones considered in the verification analysis using both EnergyPlus and the FDMRC simulation environment. The verification analysis is conducted using weather data for Golden, CO obtained from TMY3 weather file for select days representing winter and summer conditions. The zone model includes the ventilated slab system within a concrete slab. The length of the ventilated slab is 8 m. The thickness of the exterior walls is 0.15 m. A diameter of hollow core is assumed to be 0.10 m. The total area of the hollow core ventilated slab is 64 square meter. Heating and cooling set point temperature is assumed to be 20 C and 25 C, respectively, for 24 h per day. All outside surfaces of building envelope except for lower zone floor are exposed to outdoor air. The floor of the lower zone is assumed to be in contact with the ground set with a constant temperature (18 C). Table 2 provides a summary of the building model features and the control parameters used for the ventilated slab system. Using the developed simulation environment, the heating energy consumption and cooling energy for ventilated slab system are calculated for each time step using Eq. (23):
Eheating ¼ m_ h; air cp;h Th;air;in Th;air;out
(23)
Ecooling ¼ m_ c;air cp;c Tc; air; out Tc;air; in As part of the verification analysis, the predictions of mean air temperature and ventilated slab system energy consumption for the lower zone (zone 1) and the upper zone (zone 2) obtained from the one-dimensional FDMRC model during heating season (January 15th to 18th) and summer season (August 17th to 20th) are verified
b
Fig. 7. A section view of the building slab and exterior walls for two thermal zones used for the verification analysis: (a) ventilated heating slab and (b) ventilated cooling slab.
Table 2 Basic features of building envelope and the ventilated slab system used in the whole-building energy analysis. Slab Exterior wall Roof Fenestration
U-value U-value U-value U-value SHGC Window-to-wall ratio
0.219 W/m2-K 0.493 W/m2-K 0.221 W/m2-K 2.96 W/m2-K 0.385 34.6% (East and West)
Internal load
Occupancy
Number of people: 2 (each zone) Schedule:
Lighting
Power density: 10.4 W/m2 Schedule:
Infiltration Core diameter Throttling range Set point
0.015 m3/s
Ventilated slab system
Air inlet temperature
Air mass flow rate
Heating Cooling Min. heating Max. heating Min. cooling Max. cooling Heating Cooling
0.10 m ±1 C 20 C 25 C 25 C 30 C 10 C 15 C 0.1 kg/s 0.05 kg/s
Fig. 8. Comparison of mean air temperatures and heating energy use for both zones 1 and 2 obtained from EnergyPlus and developed 1-dimensional FDMRC during the period from January 15th to 18th.
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Fig. 9. Comparison of mean air temperatures and cooling energy use for both zones 1 and 2 obtained from EnergyPlus and developed 1-dimensional FDMRC during the period from August 17th to 20th.
against results obtained from EnergyPlus. It should be noted that the one-dimensional FDM solution is obtained by setting adiabatic boundary conditions at the slab and wall joint in the model of Fig. 4. Fig. 8 shows a comparative analysis of mean air temperature and heating energy consumption of ventilated slab system for each zone obtained from one-dimensional FDMRC and EnergyPlus. As shown in Fig. 8, the mean air temperature of the upper zone swings more than that of the lower zone because the ceiling in the upper zone is exposed to outdoor conditions, whereas the lower zone is in contact with the ground medium set at a constant temperature. Moreover, the heating thermal load of the upper zone is greater than that of the lower zone. By comparing the mean air temperatures and heating energy consumption during the heating season, the results of both simulation tools showed good agreement with
similar patterns under the same operational and climatic conditions. Similarly, time variations of outdoor temperature, zone mean air temperatures, and cooling energy usage for each zone predicted by the two simulation tools during three summer days are depicted in Fig. 9. The predictions of mean air temperature and cooling energy consumption obtained from the 1-dimensional FDMRC agreed well with the results of EnergyPlus. 4. Parametric analysis A series of parametric analyses is carried out to determine the effect of a wide range of design and operating parameters for ventilated slabs. Specifically, the impact of inlet air temperature, air mass flow rate, depth of hollow core, and contact area of hollow core is determined on the overall performance of ventilated slab system to maintain desired temperature for thermal zones. In order to account for the thermal bridging effects, the FDMRC-2D simulation environment is used for parametric analyses during heating season (January 15th to 18th) and summer season (August 17th to 20th) in Golden, CO. Fig. 10 shows a cross-section view of the ventilated slab medium with the hollow cores. As defined in Fig. 10, the average heat transfer rates along the top and bottom surfaces of the slab (Qup, Qdown) are obtained and compared for various parameters. For all parametric analyses, zone mean air temperature is assumed to be fixed at 20 C and 25 C as heating set point temperature and cooling set point temperature, respectively. 4.1. Effect of inlet air temperature
Fig. 10. Cross-section view of ventilated slab and wall joint and illustration of variables used for parametric analyses.
The inlet air temperature is varied with a constant mass flow rate of 0.6 kg/s, and a hollow core diameter of 0.1 m located in
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Fig. 11. Average heat transfer rates along the slab surfaces with no insulation as functions of inlet supply air temperature.
Fig. 12. Average heat transfer rates along the slab surfaces with no insulation as functions of air mass flow rate.
0.25 m slab construction. Fig. 11 illustrates the average heat transfer rate through the slab surface as a function of inlet air temperature. It is observed that the change of supply air temperature dramatically influences the slab surface temperatures and average heat transfer rates along the slab surfaces. During heating mode, the increase in supply air temperature increases slab surface temperatures, and the larger temperature differences between zone air and slab surfaces result in an increase of the average heat transfer rates along the slab surfaces. Whereas, as the inlet air temperature
increases, the average heat transfer rates on the slab surfaces rapidly decrease during cooling mode. 4.2. Effect of air mass flow rate The mass flow rate of inlet hot air is varied from 0.1 kg/s to 2.0 kg/s with a constant supply air temperature of 27.5 C and 12.5 C for the heating mode and the cooling mode, respectively. Fig. 12 shows the effect of air mass flow rate on average heat
Fig. 13. Average heat transfer rates along the slab surfaces with no insulation as functions of the depth of hollow cores.
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Fig. 14. Average heat transfer rates along the slab surfaces with no insulation as functions of hollow cores contact area embedded in fixed concrete slab thickness for two core diameters.
transfer rate along the slab surface. An increase of air mass flow rate increases the average slab surface heat transfer rate for both heating mode and cooling mode. It should be noted, however, that when the air mass flow rate is larger than 1.0 kg/s, but the impact gradually decreases. 4.3. Effect of core depth The depth of hollow core is varied by changing the slab thickness from 0.20 to 0.50 m. Thus, the depth of hollow cores,
defined by the distance from center of heat source to slab surface, is varied from 0.1 to 0.25 m. The supply temperature is maintained at 27.5 C and 12.5 C, for heating season and cooling season, respectively. A diameter of hollow core is assumed to be 0.1 m, and inlet air mass flow rate of 0.6 kg/s. As shown in Fig. 13, an increase in core hollow depth proportionally reduces average heat transfer rates along the slab surfaces. As the core depth increases, lower increases of the slab surface temperatures in heating mode and decreases of the slab surface temperatures in cooling mode can be achieved due to the insulating effect of the
Fig. 15. Comparison of mean air temperatures and heating energy use for both zones 1 and 2 obtained from the developed simulation environment with both 1-dimensional and 2dimensional solutions during the period from January 15th to 17th in Golden, CO.
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concrete layer between the cores and the indoor space. As a result, the differences between zone air temperatures and slab surface temperatures are reduced, and the average heat transfer rates are reduced. 4.4. Effect of core pitch In this analysis, the core pitch is varied by changing the number of hollow cores in the slab construction. The core pitch is large with the small number of cores in the slab. As the number of cores changes, the contact area, number of cores times circumference of cores, also varies. Diameter of 0.05 and 0.10 m for a hollow core is considered. The heat generation rate of 1.8 kW and 1.2 kW for heating and cooling operation, respectively, is applied. Fig. 14 combines all the results of the parametric analyses when both core diameters and pitches are varied and shows the average heat transfer rates along the top and bottom slab surface as functions of core contact area (number of core circumference of core) for both heating and cooling modes. As shown in Fig. 14, when the contact area of core increases, the average heat transfer rates along the slab surfaces increases but with diminishing return until reaching asymptotic values. The smaller hollow cores need a larger number of cores to have the same contact area. When the contact area of cores is the same for all the diameters, the average heat transfer rates for the smaller cores are slightly greater than those for the larger cores as shown in Fig. 14. Indeed, the higher density of small cores results in higher heat transfer along the surfaces even though they are located deeper in the concrete layer compared to the larger cores.
5. Impact of thermal bridging on the performance In this section, the effect of thermal bridging caused by the floor slab-wall joint is investigated by estimating the total heating and cooling energy consumption of the ventilated slab system operated with variable temperature control strategy. To estimate the impact of thermal bridging effects, predictions of the developed simulation environment, combining the RC network model with the 2dimensional FDM numerical solution (FDMRC-2D), are compared to those obtained from FDMRC-1D simulation model. Specifically, the FDMRC-1D simulation model, which agrees well with EnergyPlus, does not account for thermal bridging and has no heat losses through the slab edges. In the other hand, the FDMRC-2D accounts for the thermal bridging between the floor slab-wall joint and considers heat transfer through the slab edges. Figs. 15 and 16 show the time variations of zone mean air temperature and energy consumption predicted by the developed simulation environment using both the 1-dimensional numerical solution and 2-dimensional numerical solution during respectively, heating and cooling seasons. Based on the results, the time-variations of mean air temperatures computed by FDMRC-1D and FDMRC-2D are almost identical because the variable temperature control strategy is maintaining the same desired indoor temperature with each thermal zone. However, there is a substantial difference in heating energy use as well as cooling energy use predicted using the two numerical solutions. As summarized in Table 3, the un-insulated ventilated slab system without accounting for the thermal bridging effects (i.e.,
Fig. 16. Comparison of mean air temperatures and cooling energy use for both zones 1 and 2 obtained from the developed simulation environment with both 1-dimensional and 2dimensional solutions during the period from August 17th to 19th in Golden, CO.
B. Park, M. Krarti / Applied Thermal Engineering 87 (2015) 66e78 Table 3 Summary of energy consumption by the ventilated slab system and estimation of thermal bridging effects for heating season (January 15th to 17th) and cooling season (August 17th to 19th) for a 8-m wide slab in Golden, CO.
FDMRC-1D FDMRC-2D Absolute Diff. % Diff.
Heating energy use [MJ]
Cooling energy use [MJ]
Zone 1
Zone 2
Total
Zone 1
Zone 2
Total
450 540 90 20%
889 991 101 11%
1339 1531 191 14%
567 605 38 7%
311 361 50 16%
878 966 89 10%
estimated with FDMRC-1D) consumes 1339 MJ of heating source energy, meanwhile the same ventilated slab system but with consideration of the thermal bridging effects (i.e., estimated with FDMRC-2D) consumes 1531 MJ of heating energy in the heating season, that is, a 14% difference. Moreover, the uninsulated ventilated slab system without accounting for the thermal bridging effects (i.e., computed using FDMRC-1D) consumes 878 MJ of cooling source energy, meanwhile the same ventilated slab system but with consideration of the thermal bridging effects (i.e., computed using FDMRC-2D) consumes 966 MJ of cooling energy in the cooling season, that is a 10% difference. Thus, the FDMRC-1D neglects the additional heating and cooling thermal loads caused by thermal bridging effects at the slab-wall joint. In order to assess the impact of adding insulation on the thermal bridging effects for the ventilated slab system, energy consumption obtained from the 2-dimensional solution (FDMRC-2D) with slab edge insulation placement with various R-values are compared to energy consumption of the radiant system predicted by the 1dimensional numerical solution (FDMRC-1D). Fig. 17 illustrates the energy performance of the ventilated slab system as a function of R-value of slab edge insulation during both winter and summer seasons for various slab widths. As the R-value of the slab edge insulation increases, energy consumption obtained from FDMRC2D becomes closer to that predicted by FDMRC-1D. Furthermore, Fig. 17 shows that the relative percentage of energy losses due to thermal bridges is significantly increased as the width of the slab is decreased. Without any insulation, the FDMRC-2D model predicts that the ventilated slab system consumes up to 17% more heating energy and 11% more cooling energy than that predicted by the FDMRC-1D model for an 8-m wide slab during both heating and cooling modes due to thermal bridge effects. As the insulation is added, the thermal bridging effects are significantly reduced especially for smaller slabs. In particular, the energy impact
77
associated with thermal bridging effects can be approximately halved during the winter and the cooling seasons with the addition of R-5 h ft2- F/Btu thermal insulation to the wall-slab joint.
6. Summary and conclusions In this paper, the simulation environment that combines a twodimensional numerical model for ventilated slab system with a RC thermal network for exterior walls is developed and validated against existing model implemented in a whole building simulation tool, EnergyPlus. Several parametric analyses are performed to determine the performance of ventilated slab systems under various design and operating conditions. In particular, the parametric analyses include the effect of supply air inlet temperature, air mass flow rate, depth of embedded hollow cores, and contact area of hollow cores. It is found that hollow core inlet air temperature greatly influences the performance of ventilated slab system. The average heat transfer rate on the upper surface of slab increases proportionally to the increase in supply air inlet temperature when heating is required. As the inlet air temperature increases, the average heat transfer rate on the slab surface rapidly decreases for cooling mode. As the air mass flow rate increases, slab heat transfer rate is increased, but the impact gradually reduces. When the air mass flow rate is larger than 1.0 kg/s, the total slab heat transfer rate is found to be not significantly affected by the mass flow rate in the heating and cooling mode. As the depth of embedded hollow core increases, the total slab heat transfer rates along the slab surfaces are found to be proportionally reduced. The effect of the depth of hollow core follows a diminishing return pattern with a gradually reduced impact. The contact area of heat source/sink has less impact on the heat transfer rates along the slab surfaces. As the contact area of core increases, the average heat transfer rates along the slab surfaces increases gradually until reaching asymptotic values. When the contact area is the same, the average heat transfer rate for the smaller core is slightly greater than when the core is large. The impact of thermal bridge effects on the performance of ventilated slab system is explored by comparing FDMRC-1D to FDMRC-2D. It is found that the thermal bridge affects the energy performance of ventilated slab. As the heat losses through slab edge increases due to thermal bridge effects, a 16-m long ventilated slab consumes 14% more heating energy and 10% more cooling energy. These effects depend on several factors include the size of the slab floor. The relative percentage of energy losses due to thermal
Fig. 17. Impact of R-value of insulation at slab-wall joint on ventilated slab system energy consumption with various slab widths in Golden, CO during (a) heating season and (b) cooling season.
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B. Park, M. Krarti / Applied Thermal Engineering 87 (2015) 66e78
bridges is significantly increases as the width of the slab is reduced. It is estimated that thermal bridges increases heating and cooling energy consumption an 8 m slab by 17% and 11%, respectively. Therefore, it is very important to carefully address the thermal bridging effects in order to accurately evaluate the energy performance of ventilated slab systems especially for small buildings.
Si Tsrc t to ts Dd Dp
absorbed short-wave and long-wave radiation from internal load on the ith face the source location temperature [ C] time [s] outdoor air temperature [ C] surface temperature [ C] core depth core pitch
Nomenclature
a
ε
r
cp cp,h cp,c D Eheating Ecooling Ei Eo Et Fi,j hair ho hr K L Kair m_ h;air m_ c;air n Nu Pr Q Qmax qCE 00 qconv;o 00 qconv;i qconv qIV 00 qki 00 qko 00 qLWR 00 qLWS 00 qLWX 00
qasol 00 qsol 00
qSW qsys Re
absorptance of solar radiation hemispherical emittance of surface density [kg/m3] heat capacity [J/kg C] specific heat of hot air [J/kg- C] specific heat of cold air [J/kg- C] diameter of the hollow core [m] radiant heating energy consumption [J] radiant cooling energy consumption [J] inside surface long-wave radiation outside surface long-wave radiation the total solar radiation incident on surface [W/m2] view factor (ScriptF) convective coefficient of the air [W/m2-K] coefficient of heat transfer by both radiation and convection at outer surface radiative conductance [W/m2-K] thermal conductivity [W/m-K] the total length of the hollow core [m] thermal conductivity of the fluid [W/m-K] hot air mass flow rate [kg/s] chilled air mass flow rate [kg/s] number of the hollow core Nusselt number Prandtl number generated heating or extracted cooling rate [W] the maximum heat transfer [W] convective parts of internal loads convective exchange flux with outside air convective heat flux to zone air convective heat transfer rate sensible load caused by infiltration conductive flux through the interior surface conductive flux into wall net long-wave radiation flux long-wave radiation flux from equipment in zone net long-wave radiant flux exchange between zone surfaces absorbed direct and diffuse solar radiation flux transmitted solar radiative flux absorbed at interior surfaces net short-wave radiation flux to surface from lights heat transfer to/from HVAC system Reynolds number
References rez-Lombard, J. Ortiz, C. Pout, A review on buildings energy consumption [1] L. Pe information, Energy Build. 40 (3) (2008) 394e398. [2] C.D. Barnaby, D.H. Nall, E.D. Dean, Structural mass cooling in a commercial building using hollow core concrete plank, in: Proceedings of the National Solar Conference, Amherst, MA, 1980. [3] R. Zmeureanu, P. Fazio, Thermal performance of a hollow core concrete floor system for passive cooling, Build. Environ. 23 (3) (1988) 243e252. [4] M.B. Russell, P.N. Surendran, Influence of active heat sinks on fabric thermal storage in building mass, Appl. Energy 70 (2001) 17e33. [5] Y.T. Chae, R.K. Strand, Modeling ventilated slab systems using a hollow core slab: implementation in a whole building energy simulation program, Energy & Build. 57 (2013) 165e175, http://dx.doi.org/10.1016/j.enbuild.2012.10.036. [6] ASHRAE, Handbook of Fundamentals, American Society of Heating, Refrigerating and Air-Conditioning Engineers, Atlanta, GA, 2005. [7] M.R. Shaw, K. Treadaway, S. Willis, Effective use of building mass, Renew. Energy 5 (1994) 1028e1038. [8] S.P. Corgnati, A. Kindinis, Thermal mass activation by hollow core slab coupled with night ventilation to reduce summer cooling loads, Build. Environ. 42 (2007) 3285e3297, http://dx.doi.org/10.1016/j.buildenv.2006.08.018. que , F. Ollivier, J.J. Roux, Effect of 2D modelling of thermal bridges on the [9] F. De energy performance of buildings: numerical application on the Matisse apartment, Energy Build. 33 (6) (2001) 583e587. [10] T.G. Theodosiou, A.M. Papadopoulos, The impact of thermal bridges on the energy demand of buildings with double brick wall constructions, Energy Build. 40 (11) (2008) 2083e2089. [11] G. Evola, G. Margani, L. Marletta, Energy and cost evaluation of thermal bridge correction in Mediterranean climate, Energy Build. 43 (9) (2011) 2385e2393. [12] S.V. Pantankar, Numerical Heat Transfer and Fluid Flow, Hemisphere, New York, 1980. [13] EnergyPlus User's Guide, Engineering Reference e The Reference to EnergyPlus Calculations, DOE, 2010. [14] F. Kreith, R. Manglik, M. Bohn, Principles of Heat Transfer. Cengage Learning, 2010. [15] F.P. Incropera, D.P. DeWitt, T.L. Bergman, A.S. Lavine, Fundamentals of heat and mass transfer, in: F.P. Incropera, F.P.F.O.H.A.M.T. Incropera (Eds.), Water, vol. 6th, John Wiley & Sons, 2007, p. 997, http://dx.doi.org/10.1016/ j.applthermaleng.2011.03.022. [16] G.N. Walton, Thermal Analysis Research Program Reference Manual, NBSSIR 83-2655, National Bureau of Standards, 1983. [17] J.E. Braun, N. Chaturvedi, An inverse gray-box model for transient building load prediction, HVAC&R Res. 8 (1) (2002) 73e99. [18] J.E. Seem, Modeling of Heat Transfer in Buildings, Wisconsin Univ., Madison, USA, 1987. [19] T. Dewson, B. Day, A.D. Irving, Least squares parameter estimation of a reduced order thermal model of an experimental building, Build. Environ. 28 (2) (1993) 127e137. [20] S. Wang, X. Xu, Parameter estimation of internal thermal mass of building dynamic models using genetic algorithm, Energy Convers. Manage. 47 (13) (2006) 1927e1941. [21] S.F. Fux, M.J. Benz, L. Guzzella, Comparing Control-Oriented Thermal Models for a Passive Solar House (Submitted to CISBAT Conference), 2011. [22] H.C. Hottel, A.F. Sarofim, Radiative Transfer, McGraw-Hill, New York, 1967. [23] Yunus A. Cengel, Afshin J. Ghajar, Fundamentals of Thermal-fluid Sciences, McGraw Hill, 2001. [24] D. Arasteh, J.C. Kohler, B. Griffith, Modeling Windows in EnergyPlus with Simple Performance Indices, Lawrence Berkeley National Laboratory, 2009.
Contents lists available at ScienceDirect
Applied Thermal Engineering journal homepage: www.elsevier.com/locate/apthermeng
Research paper
Development of a simulation analysis environment for ventilated slab systems Benjamin Park, Moncef Krarti* University of Colorado at Boulder, Building Systems Program, Civil, Environmental, and Architectural Engineering Department, UCB 428, University of Colorado, Boulder, CO 80309, USA
h i g h l i g h t s A new dynamic simulation environment has been developed for ventilated slabs. Thermal bridging effects for ventilated slabs have been evaluated. Impact of a wide range of design and operating parameters is determined.
a r t i c l e i n f o
a b s t r a c t
Article history: Received 18 February 2015 Accepted 20 April 2015 Available online 8 May 2015
In this paper, a new simulation environment is presented to evaluate the energy performance of ventilated slab systems in multi-floor buildings. The simulation environment combines a transient twodimensional finite difference solution of a ventilated slab system comprising slab-wall joints with thermal network model for indoor spaces and associated exterior walls. The developed simulation environment can assess the impact of thermal bridging effects on both heating and cooling building thermal loads. First, the predictions of the developed simulation environment are verified against those obtained from a detailed whole-building energy simulation tool when thermal bridging effects are neglected. Then, a series of parametric analyses are performed to determine the performance of ventilated slab systems under various design and operating conditions considering the thermal bridging effects. It is found that the energy performance of ventilated slab systems and thermal bridging impact depend on a wide range of factors including size of the slab, supply air inlet temperature, air mass flow rate, core diameter, core pitch, and depth of hollow cores. In particular, it is found that the thermal bridge affects significantly the energy performance of ventilated slab systems and can increase both heating and cooling energy consumptions by 17% and 11%, respectively. © 2015 Elsevier Ltd. All rights reserved.
Keywords: Air conditioning Hollow cores Thermal bridging Ventilated slab
1. Introduction In most countries, approximately 20e40% of total energy consumption is due to building heating and cooling operations [1]. The use of building thermal mass is one of the effective techniques to reduce the building energy consumption. Recently, active cooling/ heating systems integrated with building thermal mass, such as ventilated slab systems, have been considered to heat and cool residential and commercial buildings. Indeed, ventilated slab systems have been employed in northern Europe and Australia.
* Corresponding author. Tel.: þ1 303 492 3389; fax: þ1 303 492 7317. E-mail address: [email protected] (M. Krarti). http://dx.doi.org/10.1016/j.applthermaleng.2015.04.065 1359-4311/© 2015 Elsevier Ltd. All rights reserved.
Ventilated slab systems, utilize air channels within precast slabs to cool or heat thermal zones. The basic operation of ventilated slab systems is similar to hydronic radiant systems, but it uses air instead of water as the heat transfer fluid. The air is then supplied directly to condition various thermal spaces. Compared to conventional air systems, hollow core slab systems have the potential to significantly reduce energy use to heat and/or cool buildings. In particular, it is reported that ventilated slab systems can serve as passive thermal storage media, and can effectively reduce the daytime cooling demand for several types of buildings. During the summer season, outside air can be used to cool down the temperature of the slab thermal mass specially at nighttime. During the winter season, the slab thermal mass can store heat generated by internal sources (such lighting and
B. Park, M. Krarti / Applied Thermal Engineering 87 (2015) 66e78
equipment) and then release it to the conditioned zone during the on-peak period. Barnaby et al. simulated the thermal performance of a hollow core slab system for a commercial building using an energy analysis program and concluded that the system could provide an energy reduction between 13% and 30% for the peak cooling load for dry US climates [2]. However, the energy analysis evaluation neglected thermal bridging effects at the floor-wall joints. Using an implicit finite difference model for a ventilated slab system serving a single zone office in Montreal, Canada, Zmeureanu and Fazio found that the daily cooling load could be reduced by over 35% compared to conventional HAVC system [3]. Similarly, Russell and Surendran investigated the ventilated slab system performance using a two-dimensional finite difference model. Assuming that air at 16.9 C is continuously circulated through hollow cores for 14 h, they indicated that the cooling potential of the system increased by 335% compared to a traditional slab configuration with night ventilation [4]. Chae and Strand developed a heat balance model suitable for EnergyPlus for the ventilated slab using a modification of the conduction transfer function (CTF) formulation with heat sources/sinks. The authors have found that daily total demand savings of 23% and peak demand savings of 28% can be achieved by the ventilated slab system if outdoor air is circulated through the hollow cores during unoccupied hours [5]. Moreover, compared to all air HVAC systems, the benefits of ventilated slab systems include better thermal comfort. Better thermal comfort is associated to increased heat transfer by radiation. It is estimated that more than 50 percent of heat can be transferred from the controlled ventilated slab surface to other surfaces by radiation [6]. Indeed, Shaw et al. reported that ventilated slab systems can control slab surface temperatures and maintain comfortable environment for the occupants [7]. Corgnati and Kindinis investigated the thermal performance of active hollow core slabs under a Mediterranean climate. The authors concluded that the hollow core ventilated slab system reduces thermal cooling load and provides better thermal comfort to the occupants compared to the traditional ventilation system. However, no specific analysis was carried to estimate the level of thermal cooling load reduction and thermal comfort improvement compared to the traditional ventilation system [8]. Several studies have found that thermal bridging effects especially at the wall-floor joints can be significant and can increase building thermal loads [9e11]. In particular, Theodosiou and Papadopoulos reported that actual heating loads can be up to 30% higher when thermal bridging effects are considered [10]. Most of the simulation models developed for ventilated slab systems do not take account for thermal bridge effects which can be magnified due to the controlled slab temperatures. Indeed, higher slab surface temperature for hollow core ventilated slab system could increase heat losses through slab edges during the heating operation. Existing detailed whole-building simulation tools including, eQUEST and EnergyPlus which are the most widely used building energy analysis tools in the US, are not suitable for estimating of thermal bridging effects on the energy performance of ventilated slabs. Indeed, eQUEST has no capabilities to perform any 2D heat transfer analysis (i.e., analysis of thermal bridging in building envelope including joints between floors and walls). Moreover, while the current version of EnergyPlus has some capabilities to perform 2-D heat transfer analysis for slab floors, the boundary conditions along the joints between the floor and the walls have to be set as adiabatic. Thus, EnergyPlus is also not capable to evaluate the thermal bridging effects for the ventilated slab due to the wall-floor joints. In paper, the impact of thermal bridges on the performance of ventilated slab systems is estimated under various design and operating conditions using a simulation analysis environment for multi-floor buildings.
67
2. Development of the simulation analysis environment A simulation environment that combines finite difference model and thermal network technique is developed to evaluate the performance of ventilated slab systems and assess the impact of thermal bridging effects associated with floor-wall joints. Specifically, 3R2C thermal network model is used to estimate heating and cooling loads for the thermal zones. A 2-dimensional FDM ventilated slab model is employed to accurately address the heat transfer through slab-wall joint. Fig. 1 illustrates the schematic of the simulation environment using both an RC thermal network model and a finite difference method (FDM) ventilated slab system model. 2.1. FDM ventilated slab model A control volume approach and pure implicit finite difference technique is used to model hollow core ventilated slab system by solving the two dimensional heat conduction equation within the building envelope components that include embedded heat source/ sink (i.e. floor and ceiling) [12]:
k
v2 T v2 T vT þ k 2 þ Q ¼ rcp 2 vt vx vy
(1)
The actual heat transfer between the building envelope elements (i.e. slab floor and walls) and the ventilated slab system depends on the fluid inlet temperature and mass flow rate. To model the ventilated slab system using Eq. (1), it is assumed that the air inlet temperature and the mass flow rate are known while the remaining parameters are to be calculated [13]. Specifically, air flows through hollow cores of the ventilated slab system are supplied to heat and cool the indoor space. A ventilated slab system can be thought of as a heat exchanger between air flowing through the hollow cores and the slab floor. The effectiveness-NTU heat exchanger method has been shown to be convenient to utilize when the air outlet temperature is not known [14]. Specifically, heat released or absorbed from the air flowing in the hollow cores can be estimated using Eq. (2):
_ p air Tair; in Tair; out Q ¼ mc
(2)
The maximum heat transfer potential between the air and the slab is estimated based on the temperature of heat source or sink, Tsrc:
_ p air Tair; in Tsrc qmax ¼ mc
(3)
The effectiveness of the heat exchanger, ε, is defined as the ratio of the actual energy transfer to the maximum amount of energy transfer. When one fluid is stationary for a heat exchanger, the effectiveness can be related to the number of transfer units (NTU) [15]:
ε≡
Q ¼ 1 eNTU Qmax
(4)
where NTU is defined by:
NTU≡
UA _ p air mc
(5)
The heat transfer coefficient, UA, is can be estimated using the convection heat transfer along the contact surface area of the cores:
UA ¼ hair ðpDLÞ
(6)
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B. Park, M. Krarti / Applied Thermal Engineering 87 (2015) 66e78
Fig. 1. Schematic of the FDMRC simulation environment.
The air convection heat transfer coefficient, hair, can be determined using the DittuseBoelter correlation. Using Eq. (4), the heat generation per volume of air flowing in the hollow cores can be expressed in W/m3 as shown by Eq. (7):
εQmax Q ≡ pD2 4 L
(7)
A non-uniform geometric discretization scheme is applied in order to reduce the computational efforts as well as the memory requirements to obtain an accurate numerical solution. In particular, the discretization grid is very fine near the surface boundaries where thermal interactions are strong between two different materials and/or near the heat sources. The grid is gradually expanded in the area where relatively smaller temperature changes are
Fig. 2. Geometric mesh discretization used for the two-dimensional FDM slab model.
expected. Fig. 2 illustrates the selected mesh configuration used for the two-dimensional building envelope model. The finest distance between nodes in the model is 0.008 cm. A sensitivity analysis is carried out to assess the accuracy in predicting slab inside/outside surface temperatures, wall inside/ outside surface temperatures, and zone mean air temperature as functions of the grid size (using a detailed grid with 40,625 nodes as a reference). Fig. 3 depicts the results of the assessment analysis to determine the optimal grid size (expressed in terms of the number of nodes) on the prediction accuracy (expressed in terms of RMSE between the predictions and the reference case results) and on the computational effort (expressed in terms of CPU of processing time). Based on the results of Fig. 1, an optimal grid size of about 11,000 allows minimizing both the CPU requirement and the RMSE value. A comprehensive convective algorithm developed by Walton [16] is adopted in the FDM model for interior convective coefficient for any surface related to the hollow core ventilated slab system. Specifically, the convective coefficients along the surfaces depend on the direction of heat flow and the buoyancy as shown by Eqs. (8) and (9), respectively.
Fig. 3. Impact of the discretization nodes on the prediction accuracy and computational effort.
B. Park, M. Krarti / Applied Thermal Engineering 87 (2015) 66e78
1
h¼
9:482jDTj3 7:283 jcos Sj
(8)
dT2 T1 T2 ¼ þ hi AðTzone T2 Þ þ QLWX þ QSW dt R
(11)
where,
1
1:810jDTj3 h¼ 1:382 jcos Sj
C2
69
(9)
In the developed simulation environment, algorithms have been included to control the zone indoor air temperature to be within pre-defined set point temperatures and throttling ranges. In particular, variable temperature control strategies are considered to modulate the operation of the ventilated slab systems. Specifically, the ventilated slab system can vary the supply air temperature based on user input of minimum and maximum supply air temperatures. The supply air temperature is varied linearly on an hourly basis, meanwhile the air mass flow rate remains constant [5]. In the analysis presented in this paper, the indoor mean air temperature is used as a thermal load indicator to control the supply air temperature. However, other thermal comfort indicators such as radiant mean temperature can be used to control the supply air temperature. Fig. 6 illustrates variable supply air temperature control strategies used for the ventilated slab system model.
R ¼ l=kA C ¼ rcp lA=2 h ¼ convective coefficient QLWR ¼ long-wave radiation on the outside surface QSOLAR ¼ solar incident on the outside surface QLWX ¼ long-wave radiation exchange between surrounding surfaces QSW ¼ short-wave radiation absorbed by the inside surface T ¼ temperature
Interior convective heat flux along interior surfaces is a function of the surface temperature and the air-layer temperature directly in contact with the surfaces. Various convective heat transfer coefficient correlations have been developed and reported in the literature. For the simulation environment, a correlation to estimate heat transfer coefficient for natural convection along vertical walls is adopted [6,13]:
2.2. RC thermal network model
h ¼ 1:31jDTj3
Thermal networks have been widely used to simulate the thermal performance of the building envelope. In particular, 3R2C models have been successfully applied to simulate the building envelopes for transient building load predictions [17e21]. For instance, Dewson et al. have demonstrated that a 3R2C model is capable of capturing accurately thermal behavior of a test solar cell [19]. Moreover, Fux et al. have shown that a 3R2C thermal network model can accurately predict the room air temperature compared to the measured average room air temperature for a lodging building in the Swiss Alps [21]. Fig. 4 depicts a 3R2C thermal network of exterior wall connecting the indoor space to the outdoor environment. All resistances and capacitances are assumed to be time invariant. The heat balance on the outside wall surfaces can be expressed using Eq. (10). The incident solar radiation includes both direct and diffuse incident solar radiation absorbed by the outside surface. A radiation exchange between the surface, the ambient air, the sky, and the ground is included as a formulation of long-wave radiation flux on the surface. The convective term and conduction term are modeled in RC thermal network using the classical formulation. The heat balance involving the inside surfaces can be written as indicated by Eq. (11) with relevant heat transfer components [13]:
Two surfaces at different temperatures exchange heat energy by thermal radiation. The FDMRC simulation environment uses a gray interchange model to account for thermal radiation heat exchange between interior surfaces. In particular, the radiosity concept developed by Hottel and Sarofim is used by the simulation environment [22]. The net radiative heat transfer at a surface can be determined by Eq. (13).
C1
dT1 T2 T1 ¼ þ ho AðTout T1 Þ þ QLWR þ QSOLAR dt R
1
QLWX;i ¼
Ai εi 4 sTi Ji 1 εi
(12)
(13)
where the radiosity, J, is the sum of the gray body radiation of temperature T, and the incident radiation, H, as expressed by Eq. (14):
J ¼ εsT 4 þ ð1 εÞH
(14)
The incident radiation, H, is normally unknown. If a certain surface i is hit by radiation from another surface j, the radiation heat energy incident on surface i can be described as Eq. (15). Where Fji is the view factor from surface j to i [23].
PN (10)
Hi ¼
j¼1 Fji Aj Ji
Ai
(15)
2.3. Window thermal model
Fig. 4. RC thermal network model of the exterior wall.
The developed simulation environment (FDMRC) utilizes the simplified window model based U-factor and SHGC values to determine the properties of the window glazing. Arasteh et al. outlined a procedure to determine window properties such as glass-to-glass resistance, thickness, thermal conductivity, transmittance, and glazing reflectance by using only U-factor and SHGC values [24]. Eq. (16) expresses the heat balance for single-glazing window. Few assumptions are made in deriving the window heat balance equations. It is assumed that the glass is thin enough so that heat storage within the glazing is neglected and that the short
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wave radiation absorbed is equally distributed to the two faces of the window glazing.
thermal network model, and the simplified window model in order to estimate heating and cooling thermal loads of conditioned
4 Eo εwin εwin Twin;o þ kwin Twin;i Twin;o þ ho To Twin;o þ Swin;o ¼ 0 4 þ kwin Twin;i Twin;o þ hi Ti Twin;i þ Swin;i ¼ 0 Ei εwin εwin Twin;i
The DOE-2 convection model is adopted for the FDMRC to calculate exterior convective coefficient. The DOE-2 convection model combines the MoWiTT and BLAST detailed convection algorithms [13]. The convection coefficient for very smooth surfaces is calculated by Eq. (17):
hc;glass ¼
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i2 h h2n þ aV bz
(16)
spaces. Fig. 5 illustrates the heat transfer mechanisms for various surfaces using the developed FDMRC simulation environment of zones equipped with hollow core ventilated slab systems. In particular, Eq. (21) expresses the indoor air heat balance and includes convective heat transfer from inside surfaces of building envelope and internal loads [6]:
(17)
hn is calculated using Eqs. (8) and (9) and the constants a and b are given in Table 1. For rough surfaces, the convection coefficient is modified according to the Eq. (18):
hc ¼ hn þ Rf hc;glass hn
qconv þ qCE þ qIV þ qair; sys ¼ 0
(21)
The air is assumed to be an ideal gas. Moreover, supply air is assumed fully mixed with the thermal zone indoor air. Heat
(18)
Apart from window heat balance, it is also necessary to consider the interior surfaces that absorb solar radiation that has been transmitted through the fenestration. The developed simulation environment uses a simplified interior solar distribution model that assumes that all transmitted direct radiation is incident on the floor and absorbed as a function of the floor solar absorptance. The reflected portion is assumed to be diffuse and is uniformly absorbed by all surfaces. All transmitted diffuse radiation is uniformly absorbed by all of the zone surfaces including floor surface. If the total transmitted diffuse radiation and the reflected direct radiation (from the floor) are divided by the total interior surface area of the zone and distributed uniformly, then for all surfaces except for the floor. Eqs. (19) and (20) provide expressions for the absorbed solar radiation for interior surfaces and floor, respectively.
P QSW;in;j;q ¼
P q_ TSHG; diffuse þ 1 afloor q_ TSHG; direct PN j¼1 Aj (19) P q_ TSHG;diffuse þ 1 afloor q_ TSHG; direct PN j¼1 Aj P afloor q_ TSHG; direct
P QSW;in;floor;q ¼ þ
Afloor (20)
2.4. Indoor air heat balance The Indoor air heat balance model is an important feature of the developed simulation environment to link FDM slab model, RC Table 1 MoWiTT coefficients. (Source: Refs. [13] and [14]). Wind direction (Units)
Ct (W/m2 K4/3)
a (W/m2 K(m/s)b)
b (e)
Windward Leeward
0.84 0.84
3.26 3.55
0.89 0.617
Fig. 5. Heat balance calculation procedure used by the developed simulation environment.
B. Park, M. Krarti / Applied Thermal Engineering 87 (2015) 66e78
71
strategies used for the ventilated slab systems and implemented in the simulation environment. 3. Model verification
Fig. 6. Variable temperature control scheme for ventilated slab system.
transfer rate from supply air to the zone air is considered as heat source term for thermal zone as Eq. (22).
qair; sys ¼ m_ air $cp; air Tair; supply Tzone
(22)
In the developed simulation environment, algorithms have been included to control the zone indoor air temperature to be within pre-defined set point temperatures and throttling ranges. In particular, variable temperature control strategies are considered to modulate the operation of the ventilated slab systems. Specifically, the ventilated slab system can vary the air flow rate based on user input of minimum and maximum supply air temperatures. The supply air temperature is varied linearly on an hourly basis, meanwhile the air mass flow rate remains constant [5]. In the analysis presented in this paper, the indoor mean air temperature is used as a thermal load indicator to control the supply air temperature. However, other thermal comfort indicators such as radiant mean temperature can be used to control the supply air temperature. Fig. 6 illustrates variable supply air temperature control
a
Predictions of mean air temperature and radiant energy consumption obtained from EnergyPlus are employed in order to verify results obtained from the developed simulation environment (FDMRC). Fig. 7 shows a 2-story building section with two thermal zones considered in the verification analysis using both EnergyPlus and the FDMRC simulation environment. The verification analysis is conducted using weather data for Golden, CO obtained from TMY3 weather file for select days representing winter and summer conditions. The zone model includes the ventilated slab system within a concrete slab. The length of the ventilated slab is 8 m. The thickness of the exterior walls is 0.15 m. A diameter of hollow core is assumed to be 0.10 m. The total area of the hollow core ventilated slab is 64 square meter. Heating and cooling set point temperature is assumed to be 20 C and 25 C, respectively, for 24 h per day. All outside surfaces of building envelope except for lower zone floor are exposed to outdoor air. The floor of the lower zone is assumed to be in contact with the ground set with a constant temperature (18 C). Table 2 provides a summary of the building model features and the control parameters used for the ventilated slab system. Using the developed simulation environment, the heating energy consumption and cooling energy for ventilated slab system are calculated for each time step using Eq. (23):
Eheating ¼ m_ h; air cp;h Th;air;in Th;air;out
(23)
Ecooling ¼ m_ c;air cp;c Tc; air; out Tc;air; in As part of the verification analysis, the predictions of mean air temperature and ventilated slab system energy consumption for the lower zone (zone 1) and the upper zone (zone 2) obtained from the one-dimensional FDMRC model during heating season (January 15th to 18th) and summer season (August 17th to 20th) are verified
b
Fig. 7. A section view of the building slab and exterior walls for two thermal zones used for the verification analysis: (a) ventilated heating slab and (b) ventilated cooling slab.
Table 2 Basic features of building envelope and the ventilated slab system used in the whole-building energy analysis. Slab Exterior wall Roof Fenestration
U-value U-value U-value U-value SHGC Window-to-wall ratio
0.219 W/m2-K 0.493 W/m2-K 0.221 W/m2-K 2.96 W/m2-K 0.385 34.6% (East and West)
Internal load
Occupancy
Number of people: 2 (each zone) Schedule:
Lighting
Power density: 10.4 W/m2 Schedule:
Infiltration Core diameter Throttling range Set point
0.015 m3/s
Ventilated slab system
Air inlet temperature
Air mass flow rate
Heating Cooling Min. heating Max. heating Min. cooling Max. cooling Heating Cooling
0.10 m ±1 C 20 C 25 C 25 C 30 C 10 C 15 C 0.1 kg/s 0.05 kg/s
Fig. 8. Comparison of mean air temperatures and heating energy use for both zones 1 and 2 obtained from EnergyPlus and developed 1-dimensional FDMRC during the period from January 15th to 18th.
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Fig. 9. Comparison of mean air temperatures and cooling energy use for both zones 1 and 2 obtained from EnergyPlus and developed 1-dimensional FDMRC during the period from August 17th to 20th.
against results obtained from EnergyPlus. It should be noted that the one-dimensional FDM solution is obtained by setting adiabatic boundary conditions at the slab and wall joint in the model of Fig. 4. Fig. 8 shows a comparative analysis of mean air temperature and heating energy consumption of ventilated slab system for each zone obtained from one-dimensional FDMRC and EnergyPlus. As shown in Fig. 8, the mean air temperature of the upper zone swings more than that of the lower zone because the ceiling in the upper zone is exposed to outdoor conditions, whereas the lower zone is in contact with the ground medium set at a constant temperature. Moreover, the heating thermal load of the upper zone is greater than that of the lower zone. By comparing the mean air temperatures and heating energy consumption during the heating season, the results of both simulation tools showed good agreement with
similar patterns under the same operational and climatic conditions. Similarly, time variations of outdoor temperature, zone mean air temperatures, and cooling energy usage for each zone predicted by the two simulation tools during three summer days are depicted in Fig. 9. The predictions of mean air temperature and cooling energy consumption obtained from the 1-dimensional FDMRC agreed well with the results of EnergyPlus. 4. Parametric analysis A series of parametric analyses is carried out to determine the effect of a wide range of design and operating parameters for ventilated slabs. Specifically, the impact of inlet air temperature, air mass flow rate, depth of hollow core, and contact area of hollow core is determined on the overall performance of ventilated slab system to maintain desired temperature for thermal zones. In order to account for the thermal bridging effects, the FDMRC-2D simulation environment is used for parametric analyses during heating season (January 15th to 18th) and summer season (August 17th to 20th) in Golden, CO. Fig. 10 shows a cross-section view of the ventilated slab medium with the hollow cores. As defined in Fig. 10, the average heat transfer rates along the top and bottom surfaces of the slab (Qup, Qdown) are obtained and compared for various parameters. For all parametric analyses, zone mean air temperature is assumed to be fixed at 20 C and 25 C as heating set point temperature and cooling set point temperature, respectively. 4.1. Effect of inlet air temperature
Fig. 10. Cross-section view of ventilated slab and wall joint and illustration of variables used for parametric analyses.
The inlet air temperature is varied with a constant mass flow rate of 0.6 kg/s, and a hollow core diameter of 0.1 m located in
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Fig. 11. Average heat transfer rates along the slab surfaces with no insulation as functions of inlet supply air temperature.
Fig. 12. Average heat transfer rates along the slab surfaces with no insulation as functions of air mass flow rate.
0.25 m slab construction. Fig. 11 illustrates the average heat transfer rate through the slab surface as a function of inlet air temperature. It is observed that the change of supply air temperature dramatically influences the slab surface temperatures and average heat transfer rates along the slab surfaces. During heating mode, the increase in supply air temperature increases slab surface temperatures, and the larger temperature differences between zone air and slab surfaces result in an increase of the average heat transfer rates along the slab surfaces. Whereas, as the inlet air temperature
increases, the average heat transfer rates on the slab surfaces rapidly decrease during cooling mode. 4.2. Effect of air mass flow rate The mass flow rate of inlet hot air is varied from 0.1 kg/s to 2.0 kg/s with a constant supply air temperature of 27.5 C and 12.5 C for the heating mode and the cooling mode, respectively. Fig. 12 shows the effect of air mass flow rate on average heat
Fig. 13. Average heat transfer rates along the slab surfaces with no insulation as functions of the depth of hollow cores.
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Fig. 14. Average heat transfer rates along the slab surfaces with no insulation as functions of hollow cores contact area embedded in fixed concrete slab thickness for two core diameters.
transfer rate along the slab surface. An increase of air mass flow rate increases the average slab surface heat transfer rate for both heating mode and cooling mode. It should be noted, however, that when the air mass flow rate is larger than 1.0 kg/s, but the impact gradually decreases. 4.3. Effect of core depth The depth of hollow core is varied by changing the slab thickness from 0.20 to 0.50 m. Thus, the depth of hollow cores,
defined by the distance from center of heat source to slab surface, is varied from 0.1 to 0.25 m. The supply temperature is maintained at 27.5 C and 12.5 C, for heating season and cooling season, respectively. A diameter of hollow core is assumed to be 0.1 m, and inlet air mass flow rate of 0.6 kg/s. As shown in Fig. 13, an increase in core hollow depth proportionally reduces average heat transfer rates along the slab surfaces. As the core depth increases, lower increases of the slab surface temperatures in heating mode and decreases of the slab surface temperatures in cooling mode can be achieved due to the insulating effect of the
Fig. 15. Comparison of mean air temperatures and heating energy use for both zones 1 and 2 obtained from the developed simulation environment with both 1-dimensional and 2dimensional solutions during the period from January 15th to 17th in Golden, CO.
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concrete layer between the cores and the indoor space. As a result, the differences between zone air temperatures and slab surface temperatures are reduced, and the average heat transfer rates are reduced. 4.4. Effect of core pitch In this analysis, the core pitch is varied by changing the number of hollow cores in the slab construction. The core pitch is large with the small number of cores in the slab. As the number of cores changes, the contact area, number of cores times circumference of cores, also varies. Diameter of 0.05 and 0.10 m for a hollow core is considered. The heat generation rate of 1.8 kW and 1.2 kW for heating and cooling operation, respectively, is applied. Fig. 14 combines all the results of the parametric analyses when both core diameters and pitches are varied and shows the average heat transfer rates along the top and bottom slab surface as functions of core contact area (number of core circumference of core) for both heating and cooling modes. As shown in Fig. 14, when the contact area of core increases, the average heat transfer rates along the slab surfaces increases but with diminishing return until reaching asymptotic values. The smaller hollow cores need a larger number of cores to have the same contact area. When the contact area of cores is the same for all the diameters, the average heat transfer rates for the smaller cores are slightly greater than those for the larger cores as shown in Fig. 14. Indeed, the higher density of small cores results in higher heat transfer along the surfaces even though they are located deeper in the concrete layer compared to the larger cores.
5. Impact of thermal bridging on the performance In this section, the effect of thermal bridging caused by the floor slab-wall joint is investigated by estimating the total heating and cooling energy consumption of the ventilated slab system operated with variable temperature control strategy. To estimate the impact of thermal bridging effects, predictions of the developed simulation environment, combining the RC network model with the 2dimensional FDM numerical solution (FDMRC-2D), are compared to those obtained from FDMRC-1D simulation model. Specifically, the FDMRC-1D simulation model, which agrees well with EnergyPlus, does not account for thermal bridging and has no heat losses through the slab edges. In the other hand, the FDMRC-2D accounts for the thermal bridging between the floor slab-wall joint and considers heat transfer through the slab edges. Figs. 15 and 16 show the time variations of zone mean air temperature and energy consumption predicted by the developed simulation environment using both the 1-dimensional numerical solution and 2-dimensional numerical solution during respectively, heating and cooling seasons. Based on the results, the time-variations of mean air temperatures computed by FDMRC-1D and FDMRC-2D are almost identical because the variable temperature control strategy is maintaining the same desired indoor temperature with each thermal zone. However, there is a substantial difference in heating energy use as well as cooling energy use predicted using the two numerical solutions. As summarized in Table 3, the un-insulated ventilated slab system without accounting for the thermal bridging effects (i.e.,
Fig. 16. Comparison of mean air temperatures and cooling energy use for both zones 1 and 2 obtained from the developed simulation environment with both 1-dimensional and 2dimensional solutions during the period from August 17th to 19th in Golden, CO.
B. Park, M. Krarti / Applied Thermal Engineering 87 (2015) 66e78 Table 3 Summary of energy consumption by the ventilated slab system and estimation of thermal bridging effects for heating season (January 15th to 17th) and cooling season (August 17th to 19th) for a 8-m wide slab in Golden, CO.
FDMRC-1D FDMRC-2D Absolute Diff. % Diff.
Heating energy use [MJ]
Cooling energy use [MJ]
Zone 1
Zone 2
Total
Zone 1
Zone 2
Total
450 540 90 20%
889 991 101 11%
1339 1531 191 14%
567 605 38 7%
311 361 50 16%
878 966 89 10%
estimated with FDMRC-1D) consumes 1339 MJ of heating source energy, meanwhile the same ventilated slab system but with consideration of the thermal bridging effects (i.e., estimated with FDMRC-2D) consumes 1531 MJ of heating energy in the heating season, that is, a 14% difference. Moreover, the uninsulated ventilated slab system without accounting for the thermal bridging effects (i.e., computed using FDMRC-1D) consumes 878 MJ of cooling source energy, meanwhile the same ventilated slab system but with consideration of the thermal bridging effects (i.e., computed using FDMRC-2D) consumes 966 MJ of cooling energy in the cooling season, that is a 10% difference. Thus, the FDMRC-1D neglects the additional heating and cooling thermal loads caused by thermal bridging effects at the slab-wall joint. In order to assess the impact of adding insulation on the thermal bridging effects for the ventilated slab system, energy consumption obtained from the 2-dimensional solution (FDMRC-2D) with slab edge insulation placement with various R-values are compared to energy consumption of the radiant system predicted by the 1dimensional numerical solution (FDMRC-1D). Fig. 17 illustrates the energy performance of the ventilated slab system as a function of R-value of slab edge insulation during both winter and summer seasons for various slab widths. As the R-value of the slab edge insulation increases, energy consumption obtained from FDMRC2D becomes closer to that predicted by FDMRC-1D. Furthermore, Fig. 17 shows that the relative percentage of energy losses due to thermal bridges is significantly increased as the width of the slab is decreased. Without any insulation, the FDMRC-2D model predicts that the ventilated slab system consumes up to 17% more heating energy and 11% more cooling energy than that predicted by the FDMRC-1D model for an 8-m wide slab during both heating and cooling modes due to thermal bridge effects. As the insulation is added, the thermal bridging effects are significantly reduced especially for smaller slabs. In particular, the energy impact
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associated with thermal bridging effects can be approximately halved during the winter and the cooling seasons with the addition of R-5 h ft2- F/Btu thermal insulation to the wall-slab joint.
6. Summary and conclusions In this paper, the simulation environment that combines a twodimensional numerical model for ventilated slab system with a RC thermal network for exterior walls is developed and validated against existing model implemented in a whole building simulation tool, EnergyPlus. Several parametric analyses are performed to determine the performance of ventilated slab systems under various design and operating conditions. In particular, the parametric analyses include the effect of supply air inlet temperature, air mass flow rate, depth of embedded hollow cores, and contact area of hollow cores. It is found that hollow core inlet air temperature greatly influences the performance of ventilated slab system. The average heat transfer rate on the upper surface of slab increases proportionally to the increase in supply air inlet temperature when heating is required. As the inlet air temperature increases, the average heat transfer rate on the slab surface rapidly decreases for cooling mode. As the air mass flow rate increases, slab heat transfer rate is increased, but the impact gradually reduces. When the air mass flow rate is larger than 1.0 kg/s, the total slab heat transfer rate is found to be not significantly affected by the mass flow rate in the heating and cooling mode. As the depth of embedded hollow core increases, the total slab heat transfer rates along the slab surfaces are found to be proportionally reduced. The effect of the depth of hollow core follows a diminishing return pattern with a gradually reduced impact. The contact area of heat source/sink has less impact on the heat transfer rates along the slab surfaces. As the contact area of core increases, the average heat transfer rates along the slab surfaces increases gradually until reaching asymptotic values. When the contact area is the same, the average heat transfer rate for the smaller core is slightly greater than when the core is large. The impact of thermal bridge effects on the performance of ventilated slab system is explored by comparing FDMRC-1D to FDMRC-2D. It is found that the thermal bridge affects the energy performance of ventilated slab. As the heat losses through slab edge increases due to thermal bridge effects, a 16-m long ventilated slab consumes 14% more heating energy and 10% more cooling energy. These effects depend on several factors include the size of the slab floor. The relative percentage of energy losses due to thermal
Fig. 17. Impact of R-value of insulation at slab-wall joint on ventilated slab system energy consumption with various slab widths in Golden, CO during (a) heating season and (b) cooling season.
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bridges is significantly increases as the width of the slab is reduced. It is estimated that thermal bridges increases heating and cooling energy consumption an 8 m slab by 17% and 11%, respectively. Therefore, it is very important to carefully address the thermal bridging effects in order to accurately evaluate the energy performance of ventilated slab systems especially for small buildings.
Si Tsrc t to ts Dd Dp
absorbed short-wave and long-wave radiation from internal load on the ith face the source location temperature [ C] time [s] outdoor air temperature [ C] surface temperature [ C] core depth core pitch
Nomenclature
a
ε
r
cp cp,h cp,c D Eheating Ecooling Ei Eo Et Fi,j hair ho hr K L Kair m_ h;air m_ c;air n Nu Pr Q Qmax qCE 00 qconv;o 00 qconv;i qconv qIV 00 qki 00 qko 00 qLWR 00 qLWS 00 qLWX 00
qasol 00 qsol 00
qSW qsys Re
absorptance of solar radiation hemispherical emittance of surface density [kg/m3] heat capacity [J/kg C] specific heat of hot air [J/kg- C] specific heat of cold air [J/kg- C] diameter of the hollow core [m] radiant heating energy consumption [J] radiant cooling energy consumption [J] inside surface long-wave radiation outside surface long-wave radiation the total solar radiation incident on surface [W/m2] view factor (ScriptF) convective coefficient of the air [W/m2-K] coefficient of heat transfer by both radiation and convection at outer surface radiative conductance [W/m2-K] thermal conductivity [W/m-K] the total length of the hollow core [m] thermal conductivity of the fluid [W/m-K] hot air mass flow rate [kg/s] chilled air mass flow rate [kg/s] number of the hollow core Nusselt number Prandtl number generated heating or extracted cooling rate [W] the maximum heat transfer [W] convective parts of internal loads convective exchange flux with outside air convective heat flux to zone air convective heat transfer rate sensible load caused by infiltration conductive flux through the interior surface conductive flux into wall net long-wave radiation flux long-wave radiation flux from equipment in zone net long-wave radiant flux exchange between zone surfaces absorbed direct and diffuse solar radiation flux transmitted solar radiative flux absorbed at interior surfaces net short-wave radiation flux to surface from lights heat transfer to/from HVAC system Reynolds number
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