heating capacity

Energy 147 (2018) 587e602

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Coupling of earth-to-air heat exchangers and buoyancy for energyefficient ventilation of buildings considering dynamic thermal behavior and cooling/heating capacity Haibin Wei a, b, Dong Yang a, b, *, Yuanhao Guo a, b, Mengqian Chen a, b a b

Key Laboratory of the Three Gorges Reservoir Region's Eco-Environment, Ministry of Education, Chongqing University, Chongqing 400045, China Faculty of Urban Construction and Environmental Engineering, Chongqing University, Chongqing 400045, China

a r t i c l e i n f o

a b s t r a c t

Article history: Received 31 July 2017 Received in revised form 3 January 2018 Accepted 12 January 2018 Available online 4 February 2018

Energy-efficient technologies such as earth-to-air heat exchangers (EAHEs) and buoyancy-driven natural ventilation (BV) are employed for space conditioning. However, a fan is required in conventional EAHEs for air circulation, and BV normally serves as a passive cooling measure in temperate transitional seasons. This paper proposes the coupling of EAHEs and the buoyancy generated inside a building to achieve passive and autonomous ventilation without requiring any mechanical system. A model is developed to investigate the effects of the coupled system, with a primary focus on the dynamics of the airflow temperature, flow rate, and cooling/heating capacity provision. The model results are in good agreement with those of the computational fluid dynamics simulation. The model is applied in a hypothetical building located in a hot-summer/cold-winter region (Chongqing, China). The proposed coupled scheme is superior to the BV in hot and cold seasons. The indoor air temperature, ventilation flow rate, and cooling/heating capacity are found to fluctuate asynchronously. The cooling capacity is 56.3 kWh for the hottest day, and the heating capacity is 111.1 kWh for the coldest day. The maximum cooling or heating capacity is nearly achieved at the hottest or coldest times with the help of ventilation flow rate fluctuation. © 2018 Elsevier Ltd. All rights reserved.

Keywords: Energy-efficient ventilation Earth-to-air heat exchanger Buoyancy coupling Thermal dynamics Ventilation flow fluctuation Heat storage

1. Introduction To maintain a comfortable indoor thermal environment, active heating, ventilation, and air conditioning (HVAC) devices are usually employed in buildings. The energy consumed by HVAC devices accounts for approximately 30e50% of the total energy consumption of the building [1,2]. Electricity is the main source of energy consumed by HVAC, and its generation process consumes a significant amount of fossil fuels and causes environmental pollution [3e5]. To decrease the energy consumed by HVAC devices, many passive technologies have been developed in recent years, such as natural ventilation [6e12], solar energy [5,13], geothermal energy [5,14,15], and wind energy [16]. Earth-to-air heat exchangers (EAHEs), a passive measure, have received a significant amount of attention worldwide

* Corresponding author. Faculty of Urban Construction and Environmental Engineering, Chongqing University, Chongqing, 400045, China. E-mail address: [email protected] (D. Yang). https://doi.org/10.1016/j.energy.2018.01.067 0360-5442/© 2018 Elsevier Ltd. All rights reserved.

[5,14,15,17e26]. Studies have explained the heat transfer occurring in the pipes of an EAHE. Hollmuller [18] analyzed the temporal variation in the air temperature of a circular EAHE pipe in the case with a constant airflow rate and sinusoidal-type temperature input. Hollmuller and Lachal [19] provided climate-independent design guidelines for dampening the temperature fluctuation and analyzed the advantages and limitations. However, the yearly or daily variation in the soil temperature was not directly included in the analysis. Because the temperature of a shallow soil layer fluctuates in an annual climatic period [20], Yang et al. proposed an analytical approach to evaluate the performance of EAHEs subjected to a periodically fluctuating ambient air temperature and fluctuating soil temperature [21]. They incorporated the concept of an “excess fluctuating temperature” to consider the variation in the soil temperature around the pipes due to the interaction between the temperature wave transmitted from the ground surface and that emitted from the EAHE pipes. The model results indicate that an EAHE pipe could decrease the air temperature by 7
C and produce a cooling capacity of 3000 W on a summer day. Bansal et al. [22] developed a transient and implicit model based on

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Nomenclature A Ab An Aq As,z At A* Ca Cd CM Cs d D E F g h h1 h2 i Ke L M Nu P Pr q r Q Qd Re R Se SM T Tn(x) Tn TR Ts,z t

temperature fluctuation amplitude (K) opening area of the bottom inlet (m2) fluctuation amplitude of EAHE pipe outlet temperature (K) fluctuation amplitude of the ventilation flow rate (m3/s) fluctuation amplitude of soil temperature at the depth of z (K) opening area of the upper outlet (m2) effective opening vent area (m2) specific heat capacity of air (J/(kg$K)) discharge coefficient specific heat capacity of the internal thermal mass (J/ (kg$K)) specific heat capacity of the soil (J/(kg$K)) diameter of the EAHE pipe (m) dimensionless air exchange time for the interior space effective heat input (W) combination of modified Bessel functions acceleration due to gravity (m/s2) height difference between two openings (m) convective heat-transfer coefficient of the inner surface of the EAHE pipe (W/(m2$K)) convective heat-transfer coefficient of the surface of the internal thermal mass (W/(m2$K)) the imaginary unit effective heat transfer coefficient of the building surface (W/(m2$K)) length of the EAHE pipe (m) mass of the internal thermal mass (kg) Nusselt number fluctuation period (s) Prandtl number ventilation flow rate (m3/s) radius of the internal thermal mass (m) heating or cooling capacity (W) characteristic geometric parameter (m6/s2) Reynolds number radius of the EAHE pipe (m) area of the building surface (m2) area of the internal thermal mass surface (m2) temperature (K) EAHE pipe air temperature as a function of distance x (K) air temperature at the EAHE pipe outlet (K) inner surface temperature of the EAHE pipe (K) soil temperature at the depth of z (K) time (s)

computational fluid dynamics (CFD) to predict the heating capacity of EAHEs. They demonstrated that the increase in the temperature of a pipe system with a length of 23.42 m is in the range of 4.1e4.8
C for flow velocities in the range of 2e5 m/s on a winter day. Al-Aimi et al. [23] developed a model to predict the outlet air temperature and cooling potential of EAHEs for a hot and arid climate. They showed that EAHEs could yield a reduction in the peak cooling load of 1700 W and reduce the cooling energy demand by 30% over the summer season. Khabbaz et al. [24] performed investigations on EAHEs installed on a residential building in a hot

u Vi x

bulk flow velocity (m/s) volume of the building (m3) position along the length of the EAHE pipe (m)

Greek symbols as thermal diffusivity of the soil (m2/s) q dimensionless heat input of the building l dimensionless convective heat transfer number of the internal thermal mass l0 friction factor la thermal conductivity of air (W/(m$K)) ls thermal conductivity of the soil (W/(m$K)) lM thermal conductivity of the thermal mass (W/(m$K)) l0w dimensionless effective heat transfer number of the building envelopes m dynamic viscosity coefficient of air ((N$s)/m2) v kinematic viscosity coefficient of air (m2/s) ra air density (kg/m3) t dimensionless time constant for measuring the thermal storage capability of the internal thermal mass fi phase shift of the indoor air temperature with respect to the outdoor air temperature (rad) fM phase shift of the internal thermal mass temperature with respect to the outdoor air (rad) fn phase shift of the outlet air temperature of the EAHE with respect to the outdoor air temperature (rad) fq phase shift of the ventilation flow rate with respect to the outdoor air temperature (rad) phase shift of the soil temperature with respect to the fs;z outdoor air temperature (rad) u fluctuation frequency (s1) Superscripts e time-averaged term ~ fluctuation term Subscripts g ini i M n o s w y; d z t; b

ground surface initial condition indoor air internal thermal mass EAHE outlet air outdoor air soil external wall values of the annual or daily fluctuation periods depth top opening or bottom opening

semiarid climate. Their TRNSYS simulations show that the air temperature could be decreased by 19.5 and 18.3
C for one and three pipes, respectively, using EAHEs [24]. The airflow circulation is crucial for both enhancing the heat exchange in underground EAHE pipes and delivering the heating or cooling capacities produced by the EAHEs to the interiors of a building. Currently, mechanical driving systems are largely used to circulate the flow for EAHEs. However, the mechanical driving system used for air circulation consumes electricity, and the operation of electrical machinery could release excrescent heat to the

H. Wei et al. / Energy 147 (2018) 587e602

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air temperature by only a couple of hours, and the resulting decay in the amplitude of the temperature fluctuation is limited. This implies that the BV, even after combination with the building thermal mass, has a negligible regulation effect in an annual fluctuation cycle. Thus, BV may not be applicable in the seasons when the outside climate is not temperate. It is noted that the time lag of the pipe outlet air temperature of the EAHEs could reach up to tens of days and the air temperature fluctuation amplitude can be greatly attenuated by the underground thermal storage effects [18,21,27]. This indicates EAHEs have greater regulation potential than BV, especially in annual cycles. A new mode is proposed in the current paper, in which the EAHEs and the buoyancy force are combined for ventilating buildings and improving the indoor thermal environments. In the new mode, the buoyancy is responsible for air circulation through both the EAHE pipes and the building interior spaces, and the EAHEs are responsible for providing heating or cooling capacity for the buildings. The coupled system is denoted by EAHEBV and Fig. 1(b) shows its schematic. Unlike the traditional EAHEs that are driven by mechanical fans, the buoyancy is the only force for airflow circulation in the current coupled system, and thus both fresh air and a heating or cooling capacity is provided simultaneously without additional energy consumption. The objective of coupling BV and EAHEs is to combine both their benefits and also exert the regulation effects in hot and cold seasons. The time suitable for the use of passive conditioning is thus expected to be

circulated air. This may weaken the benefits of EAHEs with regards to both energy conversation and indoor thermal environment improvements. Some recent studies proposed the combination of EAHEs with other passive systems as a substitute for mechanical driving systems. For example, Yu et al. [25] connected EAHEs and solar chimneys to ventilate a building, where the solar chimney provided the driving force required to circulate the flow. Benhammou et al. [26] presented a design for EAHEs assisted by a wind tower. The interior heat generation rate and building height are both considerable in some buildings such as industrial workshops or theaters, whose characteristic is beneficial for producing buoyancy to drive natural flows passively. The advantages of natural buoyancy-driven ventilation (BV) have been recognized [2,6e8]. Some researchers also proposed the coupling of natural BV and the thermal mass inside the building to enhance regulation effects, as shown in Fig. 1(a). Herein, the buoyancy-driven ventilation scheme shown in Fig. 1(a) is denoted by BV. Yam et al. [9] concluded that the phase shift between the indoor and outdoor air temperatures in BV varies between 0 and 6 h and that of the ventilation flow rate lies between 6 and 12 h. Zhou et al. [10] incorporated the effects of external walls into their analysis and found that the time shift between the indoor and outdoor air temperatures could slightly exceed 6 h. Yang et al. [11] analyzed the inharmonic fluctuation behavior induced by the nonlinear coupling of the thermal mass and BV. It is noted that BV could delay the fluctuation in the indoor

Internal thermal mass

Ti Insulation

To

q

At TM

Internal thermal mass

Ab E

(a) BV: the naturally ventilated space under the effects of buoyancy-driven ventilation (BV) and the internal thermal mass of the building Internal thermal mass

q At

To

Ti Insulation

Ab

TM E

Internal thermal mass

Tn (b) EAHEBV: the naturally ventilated space under the coupled effects of the buoyancy and earth-to-air heat exchanger Fig. 1. Two natural ventilation schemes that employ the buoyancy for circulating the driving flow.

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extended. To evaluate the effects of EAHEBV, a mathematical model is necessary for taking into account the nonlinear coupling of buoyancy, the ventilation flow rate and the thermal storage effects induced by EAHEs. Although the previous studies, e.g., Ref. [27,28], have proposed a mathematical approach and developed a model to consider the coupling effects of building thermal mass and EAHEs, only traditional EAHEs were taken into account and thus the ventilation flow was considered to be constant. However, the nonlinear dynamic behaviors resulted from the newly proposed system, EAHEBV, are much more complicated than the previous ones. Because the buoyancy is determined by the temperature difference between indoor and outdoor environments, the fluctuation of indoor air temperature leads to the buoyancy fluctuation, and then the buoyancy-driven flow rate varies with time as well. As compared to the BV without the use of EAHEs, i.e., the system analyzed in Ref. [11], the fluctuation of ventilation flow rate in EAHEBV would remarkably enhance the nonlinear coupling between the EAHE thermal storage effects and the indoor air temperature. This is attributed to two reasons: First, the heating or cooling capacity supplied to the building interior environments is embedded in the airflow passing through the EAHEs. The heating or cooling capacity of EAHEs, as the key energy performance indicator, varies with time due to the fluctuations of both airflow temperature and flow rate. Second, the indoor air temperature fluctuation results in the variation of the flow rate across the EAHEs and then it affects the heat transfer process as well. Therefore, there are nonlinear interactions between the four main EAHEBV parameters: air temperature, ventilation flow rate, buoyancy and the thermal storage capability of EAHEs. The mathematical approach proposed in Refs. [27,28] should be further improved to deal with the nonlinear dynamic behaviors embedded in EAHEBV. Developing a model that couples heat transfer in both EAHEs and building interiors, air flow temperature and rate, buoyancy and the cooling/ heating capacity provision is another objective of the current study. We expect to develop an explicit model to describe the dynamic behavior of EAHEBV and provide an analytical method for evaluating the effects and energy performance of EAHEBV. In Section 2, a model is developed for EAHEBV, and the dynamic solutions of indoor air temperature, ventilation flow rate and heating/cooling capacity provision are derived. In Section 3, the analytical solutions are compared with a CFD simulation for validation. In Section 4, the effects of the proposed EAHEBV are compared with those of the conventional BV scheme for hotsummer/cold-winter regions. The effects of the two schemes in an annual cycle are investigated. The two schemes in a daily cycle are also analyzed for both summer and winter days. Finally, the conclusions are presented in Section 5. 2. Theoretical analysis 2.1. Characterization of temperature and ventilation flow rate fluctuations The indoor air temperatures are assumed to be uniform. Further, the Biot number of the internal thermal mass is assumed to be 2r small, i.e., Bi ¼ 2h l < < 0:1, leading to relatively uniform tempera-

fluctuations. Therefore, we only consider the cases in which the effective heat input inside the building is sufficient to produce a positive temperature difference between the indoor and outdoor environments and to assist the upward air flow during the entire fluctuation cycle. Therefore, both the indoor air temperature and ventilation flow rate follow continuously harmonic fluctuations. The effects of external winds are neglected. The outdoor air temperature To , the outlet air temperature of the EAHE Tn , the internal thermal mass temperature TM , the indoor air temperature Ti , and the ventilation flow rate q fluctuate with a main fluctuation frequency. The variables are characterized by the summations of time-averaged terms and fluctuating terms:

To ¼ T o þ T~ o ¼ T o þ Ao eiut

(1)

Tn ¼ T n þ T~ n ¼ T n þ An eiðutfn Þ ¼ T n þ A0n eiut

(2)

TM ¼ T M þ T~ M ¼ T M þ AM eiðutfM Þ ¼ T M þ A0M eiut

(3)

Ti ¼ T i þ T~ i ¼ T i þ Ai eiðutfi Þ ¼ T i þ A0i eiut

(4)

~ ¼ q þ Aq eiðutfq Þ ¼ q þ A0q eiut q¼qþq

(5)

where u ¼ 2p=P is the main fluctuation frequency, P is the fluctuation period, and P ¼ 3:1536  107 s in an annual cycle and P ¼ 8:64  104 s in a daily cycle. Ao , An , AM , Ai , and Aq are the fluctuation amplitudes, and A0n ¼ An eifn , A0M ¼ AM eifM , A0i ¼ Ai eifi , and A0q ¼ Aq eifq . T o , T n , T M , T i , and q are the time-averaged components. The thermo-physical parameters of the soil are assumed to be constant. The variation of soil temperature is a function of depth and time, which is given elsewhere [29,30] as

Ts;z ðtÞ ¼ T s;z þ As;ground ez ¼ T s;z þ As;z eiðutfs;z Þ

pffiffiffiffi ffi u 2as

rffiffiffiffiffiffiffiffi  u cos ut  z 2as (6)

¼ T s;z þ A0s;z eiut where T s;z is the time-averaged soil temperature at a depth of z, As;ground is the amplitude of the fluctuation of the ground surface temperature, and as is the soil thermal diffusivity. pffiffiffiffi ffi qffiffiffiffiffiffi u As;z ¼ As;ground ez 2as , fs;z ¼ z 2uas and A0s;z ¼ As;z eifs;z . Eq. (6) shows that the soil temperature is defined by both the external climate and thermal properties of the soil [20,31e33]. The thermal balance between the ambient environment and the soil is achieved in an annual cycle. Hence, the average soil temperature at all buried depths in an annual cycle T s;z;y is approximately equal to the annually averaged outdoor air temperature:

T s;z;y zT o;y

(7)

M

ture distribution inside the internal thermal mass. Unlike previous studies wherein the mechanically driven flow rate is assumed to be constant [21,27,28], it is assumed in this study that the buoyancydriven flow rate fluctuates. Then, both the temperature and ventilation flow rate fluctuations should be considered in the analysis. In addition, the mathematical approach employed here could only be applicable for the analysis of regular harmonic

where the subscript y represents the annual cycle. Hence, the average outlet air temperature of the EAHE in an annual cycle T n;y can be obtained as follows:

T n;y ¼ T o;y

(8)

H. Wei et al. / Energy 147 (2018) 587e602

2.2. Governing equations

ra Ca pR2

2.2.1. Buoyancy-driven flow rate The flow rate is given as [34,35].

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ffi  Ti  To   q ¼ Cd A sg nðTi  To Þ 2gh To  *

(9)

where Cd is the discharge coefficient, A* is the effective opening vent area.

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Cd A* ¼ ðCdt At ÞðCdb Ab Þ ðCdt At Þ2 þ ðCdb Ab Þ2

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A2t þ A2b A ¼ At Ab

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2ghðTi  To Þ=To zCd A* 2ghðTi  To Þ T o

where Tn ðxÞ, R, h1 , and TR are the pipe air temperature at the distance x, the radius, the convective heat-transfer coefficient of the inner surface, and the inner surface temperature of the EAHE pipe, respectively.

2.3. Solutions for indoor air temperature and ventilation flow rate

(11)

If an upward flow is ensured in the entire fluctuation cycle, the following ventilation flow rate can be obtained:

q ¼ Cd A*

vTn ðxÞ vTn ðxÞ dx þ ra Ca q dx þ 2pRh1 ½Tn ðxÞ  TR dx ¼ 0 vt vx (15)

(10)

where Cdt , Cdb are the discharge coefficients of the top and bottom openings, respectively, and At and Ab are the opening areas of the top and bottom openings, respectively. If the discharge coefficients are the same, i.e., Cd ¼ Cdt ¼ Cdb , the effective opening vent area can be obtained as follows: *

591

(12)

Eq. (9) is applicable to both a pure BV scheme and an EAHEBV scheme. However, the resistance effects due to the underground pipes are incorporated in the discharge coefficient. Hence, the discharge coefficient in the EAHEBV scheme is lower than that in the corresponding BV case, which will be further discussed in the subsequent sections.

The average indoor air temperature T i;y in an annual cycle is derived in Appendix A and given as

T i;y ¼

T o;y 2 q þ T o;y Qd y

(16)

where the yearly averaged ventilation flow rate qy is derived in Appendix A.2. For both annual and daily cycles, the fluctuating terms of the indoor air temperature and ventilation flow rate are derived in Appendix A.3 and are given as Fluctuating indoor air temperature 0

A0i

ifi

¼ Ai e

¼

A0n þ lw Ao þ

 Tn  Ti

0

i 1 þ lw þ Di þ llt þti 0

¼

A0q q

A0n þ lw Ao 

ðT n T i ÞAo

2ðT o =Qd Þq

i  1 þ l0w þ Di þ llt þti

2

ðT n T i Þ

(17)

2ðT o =Qd Þq

2

Fluctuating ventilation flow rate 2.2.2. Energy balance inside the building In the EAHEBV scheme, the inflow air passes through the EAHEs instead of directly being drawn from the outside environment. Hence, the inlet air temperature of the ventilated space is Tn . The energy balance equation for the air inside the building is expressed as

ra Ca Vi

vTi ¼ ra Ca qðTn  Ti Þ þ Ke Se ðTo  Ti Þ þ E  h2 SM ðTi  TM Þ vt (13)

The effective heat input E includes the constituents induced from the sensible heat input, the heat gain caused by solar radiation through the windows, the radiation heat transfer through the external wall, and the convective heat exchange due to air infiltration. The energy conservation equation for the internal thermal mass is written as

MCM

vTM þ h2 SM ðTM  Ti Þ ¼ 0 vt

 Aq eifq s A0i  Ao ¼ ¼ q q 2T o

A0q

It is seen from Eqs. (17-18) that, because of buoyancy-driven ventilation fluctuation, the expression for the indoor air temperature of EAHEBV is different from that of an EAHE scheme with constant flow rate (as given in Ref. [27]). The daily averaged parameters of a specific day are then given by: Daily averaged ventilation flow rate 2pty

qd ¼ qy þ A0q;y e 365 i

where M and CM are the mass and specific heat of the internal thermal mass, respectively.

(19)

Daily averaged indoor air temperature 2pty

T i;d ¼ T i;y þ A0i;y e 365 i

(20)

Daily averaged outlet air temperature of EAHE 2pty

(14)

(18)

T n;d ¼ T o;y þ A0n;y e 365 i

(21)

where ty is the day of the year since the start of an annually fluctuating cycle.

2.4. Cooling/heating potential of the EAHEBV scheme 2.2.3. Thermal balance in an underground pipe The thermal balance equation for the air passing through the EAHE is given by Ref. [21].

The variation of heating or cooling capacity is determined by both the fluctuations of air temperature and ventilation flow rate:

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H. Wei et al. / Energy 147 (2018) 587e602

Q ¼ Ca ra qðTn  To Þ



 ¼ Ca ra q þ A0q eiut T n  T o þ A0n eiut  Ao eiut

Table 1 Thermo-physical parameters employed in the simulations.

(22)

Material Soil properties

Note that Tn , To , and q are fluctuating asynchronously. Hence, the fluctuation in the heating/cooling capacity results in a different phase shift with respect to the air temperature and ventilation flow rate.

3. Comparisons between the theoretical results and numerical simulations for validation

The real annual fluctuation period is P ¼ 365  24  3600 s. However, such a long simulation time requires a very large amount of computational resources. To decrease the computational cost to an affordable level, the fluctuation period in the numerical simulation is set to be significantly shorter than the real annual fluctuation period, and the soil thermal diffusivity as ¼ ls =ðrs Cs Þ is amplified accordingly. A similar methodology was introduced elsewhere [21], and it has successfully been applied to simulate the dynamic thermal behavior of EAHEs. However, in Ref. 21, the rate of pipe flow is set to be constant, whereas the buoyancy-driven flow rate in the current simulation case must be fluctuating. Furthermore, a similarity criterion for preserving the heat storage capability of the soil for such a short fluctuation cycle was not proposed by Ref. 21. In the current case, the Fourier number of the soil in the simulation, Fos ¼ as P=z2 , is set to be equal to that of the real soil, which leads to the following scaling law:

(23)

where Cas is the ratio of the soil thermal diffusivity set in the numerical simulation to the real value of the soil thermal diffusivity, Cz is the ratio of the buried depth set in the numerical model to the buried depth corresponding to a real annual fluctuation period, and CP is the ratio of the fluctuation period set in the numerical simulation to the real annual fluctuation period. The fluctuation period is set to 1000 s in the current case. The variations in the outdoor air temperature and ground surface temperature follow a harmonic profile with a fluctuation period of 1000 s, a fluctuation amplitude of 7 K, and a time-averaged value of 293 K:



To ¼ Ts;ground ¼ 293 þ 7:0 cos 6:28  103 t  p

Thermal mass properties

Specific heat (J/(kg$K)) Thermal conductivity (W/(m$K)) Densitya (kg/m3) Thermal diffusivity (m2/s)

Air properties

Specific heat (J/(kg$K)) Thermal conductivity (W/(m$K)) Densitya (kg/m3) Viscosity (kg/(m s)) Specific heat (J/(kg$K)) Thermal conductivity (W/(m$K) Density (kg/m3)

3.1. Artificially reducing the fluctuating period in a numerical simulation by modifying the thermal properties of the soil

. Cas ¼ Cz2 CP

Buried pipe depth (m) Thermal diffusivity (m2/s)

(24)

where t is the number of seconds since the start of the simulation. Although the buried depth in the numerical model is fixed, the effects of different buried depths corresponding to a real annual fluctuation period could be simulated by varying the specified thermal diffusivity of the soil in the numerical simulation. In the simulation, the fluctuation period is set to 1000 s; thus, CP ¼ 1=31536. The buried depth of the EAHE in the numerical model is 4 m. On the basis of Eq. (23), to reflect the effects of an EAHE buried at 2 m in a real annual period, i.e., Cz ¼ 2, the thermal diffusivity of the soil in the simulation should be 126,144 times the real value, i.e., 5:57  102 m2/s, where the original value is 4:4  107 m2/s. Since thermal diffusivity as ¼ ls =ðrs Cs Þ includes the density, thermal conductivity and specific heat, the soil thermal diffusivity in the simulation can be artificially specified by changing any of these three parameters. In the presented case, the density is

4 5:57  102 1170 0.93 0.014 2:32  103 8710 202.4 10 1:79  105 1006 0.0242 1.225

a The soil and thermal mass densities were artificially specified in the numerical simulation.

adjusted to meet the similarity criterion. To obtain the change in the thermal diffusivity of the soil, the density of the soil is decreased by 126,144 times. However, the real values of the specific heat and thermal conductivity of the soil are used. Table 1 lists the thermal properties of the soil specified in the simulation.

3.2. Numerical simulation setup The commercial CFD software ANSYS FLUENT was employed for the simulation. The dimensions of the building are 2 m (length)  2 m (width)  4 m (height). The upper outlet of the building is circular, the diameter of which is 0.5 m. The lower inlet of the building is connected to an EAHE pipe. The diameter of the lower inlet is 0.5 m. The difference in the height between the center of the air inlet and that of the outlet is 3 m. A 4000 W heat source was located at the center of the building floor. Six cylinders, each with a diameter of 0.2 m and a length of 2 m, were placed inside the building to serve as the internal thermal mass. Table 1 lists the thermal properties of the internal thermal mass. To adapt to the reduced fluctuation period and incorporate the heat storage effects of the internal thermal mass of the building, the thermal diffusivity of the internal thermal mass was also increased in the simulation. Hence, the density of the thermal mass is specified to be remarkably lower than that of the thermal mass of a real building such as that made of concrete or wood. The external walls of the building were set to be adiabatic. A circular EAHE pipe with a length of 40 m and a diameter of 0.5 m was buried inside the computational domain of the soil. The buried depth of the EAHE in the numerical model was 4 m. Because the ventilation flow was driven by the buoyancy, the computational domain adjacent to the air outlet of the building and that adjacent to the inlet of the EAHE were extended, as shown in Fig. 2. The variation in the air temperature in these extended domains corresponds to the variation in the outdoor air temperature, which follows Eq. (24), and is specified via a user-defined function (UDF). The ambient pressure is 101,325 Pa, and the boundaries of the extended computational domain are defined as pressure boundaries. The internal thermal mass surface and the inner surface of the EAHE are set to be fluidesolid coupled surfaces. The total height and width of the computational domain of the soil are 14 and 20 m, respectively. The bottom surface temperature of the soil domain is set to follow a transient profile, which is calculated using Eq. (6). The side boundaries of the computational domain of the soil are set to be adiabatic. The initial condition of the soil temperature is given as

H. Wei et al. / Energy 147 (2018) 587e602

20m

593

Ground surface temperature Eq. (24) 40m 2m 2m

Internal thermal mass 4m

4m EAHE pipe

Heat input

10m

Outdoor environment temperature Eq. (24)

Outdoor environment temperature Eq. (24)

Bottom surface temperature Eq. (6)

Fig. 2. Computational domain in the numerical simulations.

The time step is set as 0.5 s, and the standard keε model is employed to model the turbulence. The number of hexahedral grid cells used in the simulation is 6,484,798, and grid sensitivity tests were performed. We carried out transient simulations to study the variations in the air temperatures and ventilation flow rate in five fluctuation periods. The simulation results shown in Fig. 3 demonstrate that the temperature waves emitted from the EAHE cannot reach the side boundaries of the soil domain, indicating that the computational domain of the soil is sufficiently large. The temporal profiles of both Tn and the ventilation flow rate q were monitored at the pipe outlet of the EAHE, and the temporal profile of the indoor air temperature was monitored at the upper outlet vent of the building. 3.3. Theoretical and numerical simulation results

1000 s

1500 s

Fig. 3. Temperature distributions at the cross section of the computational domain of the soil (x ¼ 20 m).

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi    p p cos  z Ts;z;ini ¼ 293 þ 7 exp  z p 1000as 1000as (25)

Fig. 4 shows the temperature distributions in the entire computational domain at 1000 s and 1500 s, which correspond to the coldest and hottest times of the fluctuation period, respectively. The numerical results indicate that the soil temperature at the buried depth of the EAHE could be higher than the outside air temperature for some instances, e.g., that at 1000 s, thereby heating the air. The soil temperature at the buried depth of the EAHE is lower than the outside air temperature at 1500 s, thereby cooling the air. Moreover, there were differences in both the phase shift and the amplitude of fluctuation for the indoor air temperature and internal thermal mass temperature, as shown in Fig. 5. This discrepancy was due to the thermal storage effects of the thermal mass. However, the time-averaged temperature of internal thermal mass is 2.13
C lower than that of indoor air. The reason is that the air temperatures are not uniform inside the building. The temperatures of the air surrounding the internal thermal mass are generally lower than those of the air near the outlet vent of the building, as also shown in Fig. 4 (a). Fig. 6 shows the pressure distributions at 1000 and 1500 s, respectively. The static pressures are positive in the upper portions of the building; however, the static pressures in the EAHE pipe are negative, thus drawing the air from the outside environment. The buoyancy generated inside the building led to such a pressure distribution. Table 2 summarizes the amplitudes of fluctuation and the phase shifts obtained from the analytical model and numerical simulation results. Fig. 7 shows the transient profiles of the outlet air temperature of the EAHE, the indoor air temperature, and the ventilation flow rate obtained from the numerical simulation and proposed theoretical model. In the model, the discharge coefficient includes the effects of the resistance due to the EAHE pipe and the building vents, which is derived in Appendix B. The convective

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H. Wei et al. / Energy 147 (2018) 587e602

(a) 1000 s

(b) 1500 s Fig. 4. Temperature distributions in the computational domain obtained from the numerical simulations.

48

Temperature (ºC)

44 40 36 32 28 24 20

0

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

Time (s)

Simulation TM

Simulation Ti

Fig. 5. Differences between the indoor air temperature and internal thermal mass temperature.

heat-transfer coefficients of the inner surface of the EAHE pipe and the internal thermal mass surface are determined in Appendix C. Fig. 7 shows that the theoretical results are in very good agreement with the numerical simulation results with respect to the transient temperature profiles and the transient buoyancy-driven flow rates. 4. Discussion of the improvement in the indoor thermal environment and the cooling/heating potential induced by the BV and EAHEBV schemes 4.1. Case descriptions Two hypothetical cases were employed to compare the effects of the proposed strategy (EAHEBV) and those of the BV in this section. The EAHEBV and BV systems do not require additional energy to maintain operation. The EAHEBV results presented in this section are obtained from the theoretical model proposed in Section 2. The results of the BV are obtained from the method given elsewhere

[11]. When comparing the performance of the BV system to that of the EAHEBV system, the dimensions of the building and the magnitude of the internal thermal mass are identical, except that there are underground pipes in the EAHEBV system. The model for handling the nonlinear coupling between the thermal mass and the buoyancy-driven natural ventilation in the BV is the same as that in the EAHEBV system. The only discrepancy between them is that the flow resistance and heat storage resulting from underground pipes are not considered in the BV. The climate parameters of Chongqing (a hot-summer/cold-winter region) are employed as the input climate parameters, which are listed in Table 3 [36]. Tables 3 and 4 list the thermal properties of the soil and the internal thermal mass, respectively. In this section, real values are employed for the soil and thermal mass properties. The inner dimensions of the hypothetical building are 10 m (length)  6 m (width)  5 m (height), and the heat input given to the building is 7500 W. In the EAHEBV system, there are six circular lower vents, each of which is connected to an underground pipe

H. Wei et al. / Energy 147 (2018) 587e602

595

(a) 1000 s

(b) 1500 s Fig. 6. Pressure distributions obtained from the numerical simulations.

Table 2 Main results obtained from the simulation and analytical model. An (K) Simulation result Model result Relative error

5.69 5.76 1.2%

fn (rad) 0.48 0.45 6.3%

Ai (K)

fi (rad)

Aq (m3/s)

fq (rad)

5.33 5.4 1.3%

0.79 0.7 11.4%

0.017 0.016 5.9%

2.54 2.26 11%

with a diameter of 0.5 m, a length of 60 m, and a buried depth of 4 m. There is only one upper vent, the area of which is 1.18 m2. In the BV, one lower vent and one upper vent are employed, each with an area of 1.18 m2. The lower vent of the BV is directly connected to the outdoor environment, serving as the air inlet. In the EAHEBV and BV systems, the height difference between the center of the upper vent and that of the lower vent is 4 m. The flow discharge coefficients corresponding to the two schemes are different, which are determined using the method introduced in Appendix B. The flow discharge coefficient of the EAHEBV scheme is lower than that of the BV scheme. The heat-transfer coefficient of the external walls is set as 0.51 W/(m2$K) in both schemes [27,28], and the effective heat-transfer area of the external walls is 200 m2. The internal thermal mass employed in the two schemes is equal.

4.2. Comparisons and discussion 4.2.1. Comparisons for an annually fluctuating cycle Fig. 8(a) shows the variations in the outlet air temperature of the EAHE and the indoor air temperature in an annual cycle. The figure shows that in the EAHEBV, the fluctuation in the indoor air

temperature is delayed by approximately ten days. Furthermore, the amplitude of the fluctuation in the indoor air temperature is 31.7% smaller than that of the outdoor air temperature. This indicates that the improvements in the indoor thermal environment induced by the EAHEBV system are apparent. Moreover, in an annual cycle, the phase shift of the indoor air temperature is lower than that of the outlet air temperature of the EAHE; however, the amplitude of the fluctuation in the indoor air temperature is greater than that of the outlet air temperature of the EAHE. This is largely because the thermal mass contained in the building could reduce the positive effects of the EAHEs in an annual cycle. In the BV, the phase shift and the amplitude of the fluctuation in the indoor air temperature are close to those of the outdoor air temperature, indicating that the positive effects of the BV in an annual cycle are very limited. Fig. 8(b) shows the variation in the ventilation flow rate in an annual cycle. In the BV, the amplitude of the fluctuation in the ventilation flow rate is negligible. This is because the effects of the internal thermal mass are negligible in an annual cycle. However, in the EAHEBV system, the ventilation flow rate fluctuates significantly, the phase shift of which is considerably greater than that of the indoor air temperature. The phase shift of the ventilation flow rate induced by the EAHEBV system approaches p. In addition, because of the increased flow resistance due to the EAHEs, the annually averaged ventilation flow rate of the EAHEBV system is lower than that of the BV. Fig. 8(c) shows the temporal profiles of the heating or cooling capacities provided by the EAHEBV system in an annual cycle, which is calculated by Eq. (22). In this case, the maximum heating

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H. Wei et al. / Energy 147 (2018) 587e602

28

Temperature (ºC)

26 24 22 20 18 16 14 12 0

500

1000

Simulation To

1500

2000

2500

Time (s)

3000

3500

4500

5000

Theory Tn

Simulation Tn

Theory To

4000

(a) Transient profiles of the outlet air of the EAHE and the outdoor air temperatures. 48

Temperature (ºC)

44 40 36 32 28 24 20 16 12 0

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

Time (s)

Simulation To

Theory To

Simulation Ti

Theory Ti

(b) Transient profiles of the indoor and outdoor air temperatures.

Ventilation flow rate (m3/s)

0.18 0.16 0.14 0.12 0.10 0.08 0.06 0.04 0.02 0.00

0

500

1000

1500

2000

Theory q

Simulation q

2500

3000

3500

4000

4500

5000

Time (s)

(c) Transient profiles of the buoyancy-driven flow rates. Fig. 7. Comparisons of the theoretical and numerical simulation results.

Table 3 Climatic parameters for model inputs (a representative climate of Chongqing, China).

Annual cycle July 30 January 8

ls (W/(m$K))

as (m2/s)

T o (
C)

Ao (
C)

1.1

7.1  107

17.84 31.3 8.7

10.1 4.5 2.3

Table 4 Thermo-physical parameters of the internal thermal mass. Material

Density (kg/m3)

Weight (kg)

Specific heat (J/(kg$K))

Area (m2)

Wood

300

3515

2500

2000

capacity is achieved approximately on the coldest days, and the maximum cooling capacity is achieved approximately on the

40

0.8

35

0.7

Ventilation flow rate (m3/s)

Temperature ( )

H. Wei et al. / Energy 147 (2018) 587e602

30 25 20 15 10

Cooling by EAHEBV

5

Heating by EAHEBV

0

To

0.6 0.5 0.4 0.3 0.2 0.1 0.0

0 50

100

150

EAHEBV Tn

200

250

300

Time (day)

EAHEBV Ti

350

0

50

100

150

200

Time (day)

EAHEBV

BV Ti

(a) Temperatures Heating or cooling capacity (W)

597

250

300

350

BV

(b) Ventilation flow rate

6000 4000 2000 0 -2000 -4000 -6000

0

50

100

150

200

250

300

350

Time (day)

(c) Heating or cooling capacity induced by the EAHEBV scheme Fig. 8. Comparisons of the effects of the EAHEBV and BV schemes for an annual cycle.

hottest days, indicating that the phase shift of the heating or cooling capacity with respect to the outdoor air temperature approaches p. This phenomenon indicates that the fluctuation in the ventilation flow rate is beneficial for improving the cooling/heating potential of the EAHEs. 4.2.2. Comparisons for daily fluctuating cycles Fig. 9(a) and (b) show the variations in the air temperatures and ventilation flow rate in a summer daily cycle, i.e., on July 30. The daily averaged indoor air temperature of the EAHEBV scheme is 1.66
C lower than that of the BV scheme. Furthermore, the maximum indoor air temperature of the EAHEBV scheme is 3.35
C lower than that of the BV scheme, and the amplitude of the fluctuation in the indoor air temperature of the EAHEBV scheme is 1.69
C lower than that of the BV scheme. In EAHEBV scheme, the phase shift of the indoor air temperature is larger than that of the outlet air temperature of the EAHE, and the amplitude of the fluctuation in the indoor air temperature is smaller than that of the outlet air temperature of the EAHE. This phenomenon indicates that in a daily cycle, the thermal mass of the building helped the EAHEs in enhancing the regulation effects. Fig. 9(b) shows that the daily averaged ventilation flow rate of the EAHEBV scheme is lower than that of the BV scheme. This could be because of two reasons. First, the daily averaged indoor air temperature of the EAHEBV scheme is lower than that of the BV scheme on this day, thus resulting in a smaller buoyancy due to the EAHEBV scheme. Second, because of the resistance due to the EAHEs, the flow discharge coefficient in the EAHEBV scheme is less

than that of the corresponding BV scheme. In addition, the phase shift and the amplitude of the fluctuation in the ventilation flow rate in the EAHEBV scheme are greater than those of the BV scheme. Fig. 9(c) shows the variation in the cooling capacity of the EAHEBV scheme on this summer day. The phase shift of the cooling capacity is slightly greater than that of the ventilation flow rate. The maximum cooling capacity is achieved at approximately noon. The EAHEBV scheme yields a total daily cooling capacity of approximately 56.3 kWh. Fig. 10(a) and (b) show the variations in the air temperatures and ventilation flow rate in a winter daily cycle, i.e., on January 8. The daily averaged indoor air temperature of the EAHEBV scheme is 4.01
C higher than that of the BV scheme. The minimum indoor air temperature of the EAHEBV scheme increases by 5.15
C, and the amplitude of the fluctuation in the indoor air temperature of the EAHEBV scheme decreases by 1.14
C. Although the daily averaged indoor air temperature of the EAHEBV scheme is higher than that of the BV scheme, Fig. 10(b) shows that the daily averaged value and the amplitude of the fluctuation in the ventilation flow rate of the two ventilation schemes are similar. This is because the flow resistance in the EAHEBV scheme is greater than that of the BV scheme. Fig. 10(c) shows the variation in the heating capacity of the EAHEBV scheme on this winter day. The phase shift of the heating capacity approaches p, indicating that the maximum heating capacity is achieved at night. The EAHEBV scheme yields a total daily heating capacity of approximately 111.1 kWh.

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H. Wei et al. / Energy 147 (2018) 587e602

0.9

Ventilation flow rate (m3/s)

Temperature (ºC)

40

35

30

25

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0

20 0

To

2

4

6

8

10

12 14 16 Time (hour)

EAHEBV Tn

18

20

22

24

0

2

BV Ti

EAHEBV Ti

4

6

8

10

EAHEBV

(a) Temperatures

12

14

Time (hour)

16

18

20

22

24

BV

(b) Ventilation flow rate

Cooling capacity (W)

0

-1000

-2000

-3000

-4000

-5000 0

2

4

6

8

10

12

14

16

18

20

22

24

Time (hour)

(c) Cooling capacity induced by the EAHEBV scheme Fig. 9. Comparisons of the effects of the EAHEBV and BV schemes for the daily cycle of a typical summer day.

5. Conclusions An energy-efficient strategy is proposed for ventilating a building by combining EAHEs and BV. The EAHEs are responsible for promoting the heat exchange between the ventilation inflow and the underground soil or rocks. Unlike the common characteristics of EAHEs observed in previous studies, the buoyancy is the only driving force for circulating the flow across the EAHE pipes and the interior of the building in the proposed coupled system. The underground thermal storage is used to regulate the indoor thermal environment autonomously and passively. Thus, the provisions of the ventilation air and the heating and cooling capacities could be achieved simultaneously through a zero-energy manner. Another contribution of the paper is a theoretical model is developed to consider the nonlinear coupling between the EAHE and BV. The dynamic behaviors of indoor air temperature, ventilation flow rate and heating/cooling capacity are described explicitly. This is the first time a quantitative model concerning the coupled effects of the EAHE and BV is given. The model is then compared with a three-dimensional CFD simulation model for

validation. Good agreement between them is obtained, thereby validating the theoretical model. Thereafter, the regulation effects of the proposed coupled scheme and the pure BV scheme without EAHEs are compared for a hypothetical building located in a hot-summer/cold-winter region. In an annual cycle, the proposed coupled scheme remarkably decreases the indoor air temperature on summer days and increases the indoor air temperature on winter days, exhibiting clear superiority to the pure BV scheme. This means that the time suitable for natural ventilation could be extended to even hot and cold seasons using the proposed coupled system, thereby enhancing the positive effects and extending the applicability of natural ventilation. The cooling/heating capacity provision fluctuates in both annual and daily cycles, and it does not synchronize with the variations in both the air temperature and ventilation flow rate. In addition, the maximum cooling and heating capacities of the coupled system are asymmetric. The coupled system yields a cooling capacity of approximately 56.3 kWh on the hottest day and a heating capacity of 111.1 kWh on the coldest day. The time shift between the fluctuation in the cooling/heating capacity and the fluctuation in the

H. Wei et al. / Energy 147 (2018) 587e602

599

0.9 0.8

Ventilation flow rate (m3/s)

Temperature (ºC)

20

15

10

5

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0

0 0

To

2

4

6

8

EAHEBV Tn

10

12 14 16 Time (hour)

18

20

22

24

0

2

BV Ti

EAHEBV Ti

4

6

8

10

EAHEBV

(a) Temperatures

12

14

Time (hour)

16

18

20

22

24

BV

(b) Ventilation flow rate

Heating capacity (W)

7000 6000 5000 4000 3000 2000 1000 0 0

2

4

6

8

10

12

14

16

18

20

22

24

Time (hour)

(c) Heating capacity induced by the EAHEBV scheme Fig. 10. Comparisons of the effects of the EAHEBV and BV schemes for the daily cycle of a typical winter day.

outdoor air temperature is approximately half of the fluctuation period. This phenomenon is quite interesting because the maximum cooling capacity is achieved on the hottest days of a year or during the hottest hours of a day, and the maximum heating capacity is achieved on the coldest days of a year or during the coldest hours of a day. The phase shift of ventilation flow rate fluctuation helps to achieve the above characteristic. These characteristics are very beneficial for reducing the energy demand in terms of the conditioning of the thermal environment of a building. The newly proposed mode could be applied in most of the building types, and it is advantageous to the heat processing plants and theaters. In such buildings, the heat input is considerable and the height is large, and thus the strength of the buoyancy force is large enough to drive the airflows.

Appendix A. Derivation of solutions A.1. Decomposition and normalization of the coupled equations Eqs. (9)e(15) are converted into the time-averaged and fluctuating equations given by Time-averaged equations

To 2 q ¼ Ti  To Qd 



(A1)







ra Ca q T n  T i þ Ke Se T o  T i þ E ¼ h2 SM T i  T M



(A2)

Acknowledgments The authors acknowledge support from the National Natural Science Foundation of China (NSFC) under Grant No. 51578087.

TM  Ti ¼ 0

(A3)

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H. Wei et al. / Energy 147 (2018) 587e602

ra Ca q



vT n ðxÞ þ 2pRh1 T n ðxÞ  T R ¼ 0 vx

(A4)

where Qd ¼ 2ghCd 2 A*2 is a characteristic geometric parameter.

The annually averaged ventilation flow rate qy is obtained by solving the above cubic equation:

qy ¼

Fluctuating equations

 To

~þq ~2 ¼ T~ i  T~ o 2qq Qd

(A5)

h   i





ra Ca q~ T n  T i þ q~ T~ n  T~ i þ q T~ n  T~ i þ Ke Se T~ o  T~ i ¼ ra Ca Vi



vT~ i þ h2 SM T~ i  T~ M vt

 (A6)

~

ra Ca pR

þ r a Ca vt h i þ 2pRh1 T~ n ðxÞ  T~ R

vT ðxÞ vT~ ðxÞ vT~ n ðxÞ ~ n ~ n þq þq q vx vx vx

¼0

qy ¼

b 

(A7)

b þ

 pffiffiffi

A cos Y34 þ sin Y34 3a

Y1 ¼ Ab þ 3a

Tn  Ti To

0

þ lw

To  Ti To

þq¼0

 Y4 ¼ arccos

K¼

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi B2  4AC

(A18)

B

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi B2  4AC

(A19)

2 2Ab  3aB pffiffiffiffiffiffi 2 A3

 (A20)

  B 2Ab  3aB pffiffiffiffiffiffi < 1 As0; 1 < A 2 A3

(A21)

The annually averaged indoor air temperature is then calculated from (A1) and expressed by Eq. (16).

A.3. Derivation of fluctuating components The fluctuating component of the inner surface temperature of the EAHE pipe was derived elsewhere [21] as

Fluctuating dimensionless equations

T~ R ¼ T~ n ðxÞ$Fðiu; RÞ þ T~ s;z ½1  Fðiu; RÞ

~þq ~2 T~ i  T~ o 2qq ¼ Qd To

(A11) where Fðiu; RÞ ¼





 ~ ~ q q 0 þ 1 T~ n  T~ i þ lw T~ o  T~ i Tn  Ti þ q q

 ~ ~ ¼ l T i  T M þ DiA0i eiðutÞ

Bþ

(A10)

where s ¼ Qd =q2 .





tiA0M eiut þ l T~ M  T~ i ¼ 0

(A12) (A13)

where the dimensionless parameters t ¼ MCM u=ðra Ca qÞ, l ¼ h2 SM =ðra Ca qÞ, q ¼ E=ðra Ca T o qÞ, D ¼ Vi u=q, and 0

lw ¼ Ke Se =ðra Ca qÞ have been proposed in Refs. [9,11].

h1 K0

h1 K 0

pffiffiffi  iu as R

pffiffiffi 

þls

iu a R

s

pffiffiffi . pffiffiffi iu iu

as K1

as R

 2 

To 2qA0q eiðutÞ þ A0q eiðutÞ ¼ A0i eiðutÞ  Ao eiðutÞ Qd h







(A23) 



ra Ca A0q eiðutÞ T n  T i þ A0q eiðutÞ A0n eiðutÞ  A0i eiðutÞ þ q A0n eiðutÞ i

 þ Ke Se Ao eiðutÞ  A0i eiðutÞ

¼ ra Ca Vi iuA0i eiðutÞ þ MCM iuA0M eiðutÞ

If an upward bulk flow is ensured, the combination of Eqs. (A1)e(A3) leads to the following.

Ke Se 2 EQd q  ¼0 ra Ca y ra Ca T o;y

(A22)

In this paper, by substituting T~ o , T~ i , T~ n , T~ M , and the fluctuating ~, into Eqs. (A5)e(A8), the following intermediate equaflow rate, q tions can be obtained:

 A0i eiðutÞ

A.2. Derivation of time-averaged components

q3y þ

(A17)

2

Y2 ¼ Ab þ 3a

(A9)

when D < 0

a

The dimensionless forms of Eqs. (A1)e(A8) are rewritten as.

q2 1 T  To ¼ ¼ i Qd s To

(A16)

where D ¼ B2  4A$C, A ¼ b2  3ac, B ¼ bc  9ad, C ¼ c2  3bd,  a ¼ 1, b ¼ rKeCSae , c ¼ 0, and d ¼ EQd = ra Ca T o;y .



Time-averaged dimensionless equations

when D > 0

3a

!

(A8)

(A15)

pffiffiffiffiffiffi pffiffiffiffiffiffi 3 Y1 þ 3 Y2



 vT~ MCM M þ h2 SM T~ M  T~ i ¼ 0 vt 2 vT n ðxÞ

qy ¼

b þ K when D ¼ 0 a

(A14)

(A24)

 MCM iuA0M eiðutÞ þ h2 SM A0M eiðutÞ  A0i eiðutÞ ¼ 0

(A25)

H. Wei et al. / Energy 147 (2018) 587e602



vA0n ðxÞ iðutÞ vA0 ðxÞ e þ A0q eiðutÞ n eiðutÞ vx vx h i 0 iðutÞ 0 iðutÞ ½1  Fðiu; RÞ ¼ 0  As;z e þ2pRh1 An ðxÞe

ra Ca pR2 iuA0n ðxÞeiðutÞ þ ra Ca q



(A26)   A0i  Ao ¼ 2qA0q T o Qd

(A27)

h   i  ra Ca A0q T n  T i þ q A0n  A0i þ Ke Se Ao  A0i

(A28)

¼ ra Ca Vi A0i iu þ MCM A0M iu

MCM iuA0M þ h2 SM A0M ¼ h2 SM A0i

 A0s;z



(A29)

vA0 ðxÞ þ 2pRh1 ½1  Fðiu; RÞ A0n ðxÞ þ ra Ca q n vx

¼0 (A30) h S A0

where A0M ¼ MC 2iuMþhi S . M 2 M Eqs. (A27)e(A30) are converted into the following dimensionless forms:

A0q q

¼

 s A0i  Ao

(A31)

2T o

A0q    lti 0 A T n  T i þ A0n  A0i þ l0w Ao  A0i ¼ DiA0i þ q l þ ti i

(A32)

 2  vA0n ðxÞ pR ui 2pRh1 ð1  FÞ 0 2pRh1 ð1  FÞ 0 þ þ An ðxÞ  As;z ¼ 0 vx q ra Ca q ra Ca q (A33)

¼

2pRh1 ð1  FÞA0s;z

þ

Ao 

2pRh1 ð1  FÞ þ ra Ca pR2 iu   g 0 g 0 As;z þ Ao  As;z egzi ¼ g þ zi g þ zi

2pRh1 ð1  FÞA0s;z

2

a

a

represent the geometrical size and the heat transfer capability of the EAHE, respectively. The following equation is derived using Eq. (A32):

" ¼

A0n

A0q  þ T n  T i þ l0w Ao q

In the theoretical analysis, the flow discharge coefficient of the entire flow route is correlated with the total flow resistance.

(A36)

In Eq. (A36), Cd and xtotal are the global flow discharge coefficient and global resistance coefficient, respectively. 0 The frictional factor of the EAHE pipe l is derived as [26,37].

8 64 > > > Ren < 2000 > > Re n > > < l0 ¼ 0:3164 2000  Ren < 105 > > Re0:25 > n > > > > : ð1:82 logðRen Þ  1:64Þ2 Ren  105

(A37)

where Ren ¼ pn4qd is the Reynolds number of the EAHE pipe flow, q is a the time-averaged flow rate across the pipe, na is the kinematic viscosity coefficient of air, and d is the inner diameter of the pipe. If N EAHE pipes are connected in parallel, the global discharge coefficient of the EAHEBV is given as

N 8 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi Ren <2000

 > 64 l 2 > > þx þ N þ1 xoutlet > > Ren d inlet > > > > > > N > < sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2000Ren <105

 0:3164 l Cd ¼ þ xinlet þ N 2 þ1 xoutlet > > 0:25 d > Ren > > > > > > > N > > Ren 105 : rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

 2 l 2 ð1:82logðRen Þ1:64Þ þx þ N þ1 xoutlet d inlet (A38)

#, Di þ

! 

2pRh1 ð1  FÞ þ ra Ca pR2 iu

1 ð1FÞ where the dimensionless parameters z ¼ pRqxu and g ¼ 2pRxh rCq

A0i

Appendix B. Flow discharge coefficient

where the local resistance coefficient of the inlet xinlet is approximately 0.5, and that of the outlet xoutlet is usually considered to be

A0n ðxÞ is then obtained from Eq. (A33):

A0n ðxÞ

Eqs. (17) and (18).

qffiffiffiffiffiffiffiffiffiffi Cd ¼ 1 xtotal

Eqs. (A23)e(A26) are rearranged as

ra Ca pR2 A0n ðxÞiu

601

lti þ l0w þ 1 l þ ti



(A35) By substituting Eqs. (A31)e(A34) into Eq. (A35), the final solutions of the fluctuating components are obtained, as expressed in

e

2pRh1 ð1FÞþra Ca pR2 iu

ra Ca q

x

(A34)

0.5. In Section 3.3, a numerical simulation is conducted prior to the theoretical analysis. The time-averaged flow rate obtained from the numerical simulation q ¼ 0:154m3 =s is then substituted into Eqs. (A37) and (A38) to calculate the global flow discharge coefficient. The value of the global flow discharge coefficient is 0.54, which is then considered an input parameter for the theoretical model. In Section 4.2, both the ventilation flow rate and the global discharge coefficient of the EAHEBV scheme are unknown beforehand, and they are interactive. The ventilation flow rate and the global discharge coefficient of the EAHEBV scheme are obtained by iteratively solving the combined equations, Eqs. (A14) and

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H. Wei et al. / Energy 147 (2018) 587e602

(A36)e(A38). In addition, the global discharge coefficient of the BV is directly considered to be 0.6 ± 0.1 [11,38]. Appendix C. Heat transfer coefficient Although the fluctuating flow rate leads to a variation in the convective heat-transfer coefficient, we only consider the value corresponding to the time-averaged flow rate. The Nusselt number of the pipe flow is given as [39].

8 Nu ¼ 4:36 Re < 2300 n n > < ðf =8ÞðRen  1000ÞPr

 > : Nun ¼ 1 þ 12:7ðf =8Þ1=2 Pr 2=3  1

Ren > 2300

(A39)

where f is the friction factor, f ¼ ð1:82 lnRen  1:64Þ2 , and Pr is the Prandtl number, e.g., 0.703 for air. The convective heat-transfer coefficient of the inner surface of the EAHE can then be obtained as follows:

h1 ¼ Nun

la

(A40)

d

By substituting un ¼ 0.76 m/s into Eqs. (A39) and (A40), the convective heat-transfer coefficient of the inner surface of the EAHE can be obtained as

h1 ¼ Nun

la d

.

 ¼ 3:13W m2 $K

(A41)

For the surfaces of the internal thermal mass of a building, Churchill and Bernstein provided the Nusselt number for the flow across a circular cylinder [40]:

" 5=8 #4=5  Pr 1=3 Rei Nui ¼ 0:3 þ h i1=4 1 þ 282000 1 þ ð0:4=PrÞ2=3 1=2

0:62Rei

(A42) ir where Rei ¼ 2u na . The spatially averaged flow rate across the cross section of the building is ui ¼ 0.039 m/s. By substituting ui ¼ 0.039 m/s into Eq. (A42), the convective heat-transfer coefficient of the surface of the internal thermal mass can be obtained as

h2 ¼ Nui

la 2r

¼ 1:4W

.

 m2 $K

(A43)

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