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Performance evaluation of flat rectangular earth-to-air heat exchangers in harmonically fluctuating thermal environments
Applied Thermal Engineering 162 (2019) 114262
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Performance evaluation of flat rectangular earth-to-air heat exchangers in harmonically fluctuating thermal environments
T
⁎
Haibin Wei, Dong Yang
National Centre for International Research of Low-carbon and Green Buildings, Chongqing University, Chongqing 400045, China School of Civil Engineering, Chongqing University, Chongqing 400045, China
H I GH L IG H T S
heat exchanger (EAHE) with flat rectangular cross section is proposed. • Earth-to-air capacity exceeds that of circular-cross-section EAHE. • Heating/cooling temperature fluctuation amplitudes are lower, and phase shift is larger. • Air pipe wall temperature is more stable. • EAHE • The analytical model for such EAHE is proposed and validated.
A R T I C LE I N FO
A B S T R A C T
Keywords: Earth-to-air heat exchanger Shallow geothermal energy Renewable energy Flat rectangular cross section Cooling/heating potential Fluctuating thermal environment
An earth-to-air heat exchanger (EAHE) is an excellent passive technology that uses shallow geothermal energy directly to generate cooled or heated air for thermal regulation of buildings. Previous studies have focused on circular EAHEs, paying insufficient attention to the effect of cross-sectional shape. In this paper, we propose an EAHE having a flat rectangular cross section, develop a mathematical model to calculate its thermal performance in a harmonic thermal environment, and validate the theoretical results using a three-dimensional numerical model. Our results indicate that compared with a circular EAHE shape of the same cross-sectional area, the outlet air temperature fluctuation of the flat rectangular EAHE is 24.7% lesser and its phase shift is 22.67 days longer, and the soil's thermal disturbance range is shorter, resulting in a temperature wave in the soil around the buried pipe that decays faster and creates a more stable pipe wall temperature. The cooling/heating capacity of the flat rectangular EAHE is greater than that of the circular EAHE. The pressure drop and fan power consumption of the flat rectangular EAHE is greater than that of circular EAHE, and thus the coefficient of performance (COP) value of the flat rectangular EAHE is less than that of the circular EAHE.
1. Introduction To reduce building energy consumption and alleviate environmental pollution, many renewable energy sources have been introduced to building air conditioning systems in recent decades [1–3]. Geothermal energy, for example, is a natural energy source that is limitless, environmentally friendly, and sustainable. Owing to the high thermal inertia of soil, the soil temperature at a depth of 2–3 m is nearly constant and is approximately equal to the annual average outdoor air temperature [4,5]. One application of geothermal energy, earth-to-air heat exchangers (EAHEs), use the soil directly to exchange heat with the ventilation air and absorb or store the natural coolness or heat from the underground soil in summer or winter and then supply the cooled or
⁎
heated air into a building's interior spaces to regulate the indoor thermal environment [6–8]. A number of experimental and analytical studies have been conducted under various climatic conditions worldwide. For example, AlAjmi et al. [9] developed a theoretical model for evaluating EAHE thermal performance in the arid Saharan climate. They found that the EAHE could reduce cooling energy demand for a typical house by 30% in summer. Kumar et al. [10] proposed a numerical model to predict the performance and energy conservation potential of an earth–air tunnel system coupled with a non-air-conditioned building. The numerical model was in excellent agreement with experimental data for Mathura (India) and indicated that the cooling potential of an 80-m earth pipe can provide 19 kW of cooling energy, adequate to maintain an average
Corresponding author at: School of Civil Engineering, Chongqing University, Chongqing 400045, China. E-mail address: [email protected] (D. Yang).
https://doi.org/10.1016/j.applthermaleng.2019.114262 Received 12 June 2019; Received in revised form 6 August 2019; Accepted 13 August 2019 Available online 14 August 2019 1359-4311/ © 2019 Elsevier Ltd. All rights reserved.
Applied Thermal Engineering 162 (2019) 114262
H. Wei and D. Yang
Nomenclature A Ac a b C Ca dc F h i L Nu P Pr Re T t V x z
λa λs νa ρa φn ω
temperature fluctuation amplitude (K) cross-sectional area of pipe (m2) pipe width (m) pipe height (m) pipe circumference (m) specific heat of air (J/(kg·K)) equivalent diameter of non-circular pipe (m) combination of modified Bessel functions convective heat transfer coefficient at inner surface of pipe (W/(m2·K)) the imaginary unit pipe length (m) Nusselt number fluctuation period (s) Prandtl number Reynolds number temperature (°C) time (s) velocity of airflow in pipe (m3/s) distance from pipe surface (m) burial depth of pipe (m)
Diacritical marks — ~ ⌢
time-averaged term fluctuation term Laplace transformation
Subscripts n o s w z
EAHE outlet air outdoor air soil buried-pipe surface soil depth
Abbreviations CFD COP EAHE
Greek symbols
αs κn
temperature thermal conductivity of air (W/(m·K)) thermal conductivity of the soil (W/(m·K)) kinematic viscosity coefficient of air (m2/s) air density (kg/m3) phase shift of EAHE outlet air temperature (rad) fluctuation frequency (s−1)
thermal diffusivity of the soil (m2/s) normalized fluctuation amplitude of EAHE outlet air
computational fluid dynamics coefficient of performance earth-to-air heat exchanger
(a) Longitudinal view of EAHE pipe.
(b) Cross-sectional view of a flat rectangular EAHE pipe. Fig. 1. Schematic diagram of heat transfer between pipe and surrounding soil. 2
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During an annual cycle, the variation in soil temperature decreases exponentially with the increase in depth, and its cosine changes with time [24–26]:
indoor temperature of 27.65 °C. Kabashnikov et al. [11] proposed a mathematical model to calculate the EAHE outlet temperature and analyzed the effects of the EAHE engineering parameters. Rouag et al. [12] proposed a transient semi-analytical model to calculate the temperature distribution in the soil surrounding an EAHE pipe. They found that the soil radius is a function of the operation duration, soil thermal diffusivity, pipe diameter, and inlet air temperature. Yang et al. [13–16] proposed an analytical approach for evaluating EAHE thermal performance under periodic fluctuations in both ambient air temperature and soil temperature and noted that the effects of coupling EAHEs with building thermal mass could maintain a building's thermal environment. Furthermore, Yusof et al. [17] developed a laboratory EAHE simulator to analyze EAHE thermal performance. Cuny et al. [18] found that the difference in soil types has an indispensable effect on EAHE heat transfer. Elminshawy et al. [19] found that EAHE thermal performance is highly dependent on operating conditions and soil compaction levels. Mathur et al. [20] proposed a spiral-shaped EAHE system to reduce the space taken up by the EAHE; its coefficient of performance (COP) was comparable to that of a straight system. Rodrigues et al. [21] performed a numerical investigation on different EAHE geometrical configurations using Constructal Design to seek the optimal geometries and found that increasing the number of ducts (complexity of geometry) improved thermal performance by up to ~73% for cooling and ~115% for heating. Zukowski et al. [22] compared the thermal performance of tube and plate EAHEs and found that their winter heating capacities were 13.5 MWh and 16.35 MWh, respectively, and their cooling capacities were 10.3 MWh and 20.41 MWh, respectively. The above investigations found that many parameters influence the heat transfer between the flowing air and the earth around the buried pipe. However, the pipe cross-sectional shapes' influence on heat transfer is still unknown. To improve EAHE energy efficiency, this study proposes a flat rectangular EAHE (Fig. 1). Unlike those in previous studies, the cross section of this pipe is not cylindrical, and the soil's heat conduction around the pipe is not in a cylindrical coordinate system; therefore, an analytical model for evaluating the proposed EAHE's thermal performance is also proposed. In this study, to reflect the effects of periodically fluctuating outdoor air temperatures in an annual cycle, the harmonically fluctuating air temperature profiles rather than constant air temperature values are specified as the inlet boundary condition for both the analytical model and numerical simulations. The model's results are validated by numerical simulation. The differences in thermal performance between the flat rectangular EAHE and the conventional cylindrical EAHE are also investigated.
∼ −z Ts (t , z ) = T¯s + Ts = T¯s + Ao e =T¯s + As ei (ωt − φs, z )
ω 2α s
cos ⎛ωt − z ⎝ = T¯s + As′ eiωt
ω 2αs
⎞ ⎠ (2)
where Ts is the time-averaged soil temperature, z is the soil depth, Ao is the fluctuation amplitude of the ground surface temperature, and αs is the soil's thermal diffusivity. As = Ao e−z ω (2αs) , φs, z = z ω (2αs ) , and As′ = As e−iφs, z . When the soil depth reaches 2–3 m, the fluctuation amplitude of the soil temperature for an annual cycle is small and can be approximated as the annual average outdoor air temperature, Ts = To [4,5]. The heat transfer process between the buried pipe and the surrounding soil is complex. The soil temperature around the pipe is affected not only by the depth and time but also by the flowing-air temperature inside the pipe (Fig. 1). The temperature wave originating from the pipe penetrates the soil, changes the soil temperature around the pipe, and decays with increasing propagation distance. At an infinitely large distance from the pipe, the disturbance by the flowing-air temperature is negligible, and thus the soil temperature approaches the original undisturbed-soil temperature. Yang et al. [13] proposed an excess fluctuating temperature to deal with the difference between a certain prescribed fluctuating temperature and the undisturbed-soil fluctuating temperature. The soil's excess fluctuating temperature at ∼ ~ ~ distance x from the pipe is U (t , x ) = T (t , x ) − Ts , where Ts is the un~ ~ disturbed-soil fluctuating temperature; therefore, Un = Tn − Ts and ~ ~ Uo = To − Ts . 2.2. Soil temperature around EAHE We regard the heat conduction process of the soil in the height direction around the flat rectangular buried pipe as semi-infinite plate heat conduction. To simplify the heat transfer model, we assumed that the original temperature around the pipe is not affected by the ground environment temperature. Therefore, the equation governing the heat conduction can be established by the excess fluctuating temperature:
∂U (t , x ) ∂2U (t , x ) = αs ∂t ∂x 2
(3)
The corresponding boundary and initial conditions can be described as
2. Mathematical model for flat rectangular EAHE
− λs
∂U (t , x ) ∂x
= h (Un − U (t , x )) x=0
(4)
2.1. Basic considerations
U (t , x ) = 0
x=∞
(5)
In this study, the EAHE buried pipe is assumed to have a flat rectangular cross section whose width-to-height ratio is 10 or greater (see Fig. 1). Therefore, the soil heat conduction transfer around the pipe can be considered to occur mainly in the height direction, and that in the width direction can be neglected. To simplify the heat transfer model, only the sensible heat exchange is considered, not the latent heat change; soil thermal conductivity and capacity are assumed to be homogeneous and constant; and the thermal resistance of the pipe wall is neglected [21,23,34,35]. In [13–16], both the outdoor air temperature To and the EAHE outlet air temperature Tn are assumed to be harmonic fluctuations and are represented as sums of the time-averaged term and fluctuating term: ∼ T (t ) = T + T = T + Aei (ωt − φ) = T + A′eiωt (1) ~ where T is the time-averaged air temperature, T is the fluctuating air temperature, A is the temperature fluctuation amplitude, ω = 2π P is the fluctuation frequency, and P is the fluctuation period.
U (0, x ) = 0
t=0
(6)
where h is the convective heat transfer coefficient at the pipe's inner surface. The equivalent diameter is used as the characteristic scale to calculate both the convective heat transfer coefficient and the pressure drop [27]. The usual equivalent-diameter calculation formula for a noncircular pipe is
dc = 4Ac C
(7)
where Ac and C are the cross-sectional area and circumference of the pipe, respectively. The convective heat transfer coefficient at the pipe's inner surface is given by
h = Nu·λa dc
(8)
where λa is the thermal conductivity of air, and the Nusselt number of the pipe flow is given as [28,29] 3
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⎧ Nu = 8. 23 0.3 0.8 ⎨ ⎩ Nu = 0. 023Pr Re
2.3. Air temperature along EAHE
Re<2300 Re>2300
(9) The heat-balance equation between the inner surface of the EAHE pipe and the flowing air inside a rectangular pipe is
where Pr is the Prandtl number, and the Reynolds number of the pipe flow is
ρa Ca Ac
(10)
Re = V ·dc νa
where V is the air velocity inside the pipe and νa is the kinematic viscosity of air. According to the values of Re of airflow inside the pipe, there are two flow regimes, Re<2300 , where the airflow is laminar, and Re>2300 , where the airflow is turbulent. Thus, the heat transfer coefficient at the pipe surface is related to the values of Re. Applying the Laplace transformation, Eqs. (3)–(6) can be written as ⌢ ∂2U (s,
x)
∂x 2
−
s ⌢ U (s, x ) = 0 αs
∂U (s, x ) ∂x
~ ~ 1 ∂Tn hC ∂T ~ ~ (Tn − Tx = 0) =− n − V ∂t ρa Ca VAc ∂L
⌢
∂Un iω hC iω ~ +⎡ + (1 − G (iω, 0)) ⎤ Un + Ts = 0 ⎢V ⎥ ∂L ρ C VA Va c (21) a a ⎣ ⎦ ~ ~ With the boundary condition Un L = 0 = Uo = To − Ts , the solution of Eq. (21) is
⌢
(12)
x=0
⌢
U (s, ∞) = 0
∼ −⎡ iω + hC (1 − G) ⎤⎥ L Un = To e ⎣⎢ V ρa Ca VAc ⎦
(13)
x=∞
Combined with the initial condition given by Eq. (12), the general solution of Eq. (11) is expressed as a combination of modified Bessel functions ⌢
U (s, x ) = X1 e x
s as
+ X2 e−x
s as
+ ρ ChCVA (1 − G ) ⎤ L ⎡ hC (G − 1) e−⎡⎣ iω ⎤ V a a c ⎦ + hC (1 − G ) ∼ ⎥ Ts +⎢ ρa Ca Ac iω + hC (1 − G ) ⎢ ⎥ ⎣ ⎦
(14)
⌢
s as
An′ = Ao′ e
⌢
⌢
⌢
s as
(h + λs s as )
(23)
The normalized fluctuation amplitude and phase shift of the pipe air temperature with respect to the outdoor air temperature are
(16)
By applying the inverse Laplace transformation together with its convolution theorem [30], Eq. (16) can be expressed as
U (t , x ) = Un G (iω, x )
hC −⎡ iω + (1 − G ) ⎤ L ⎥ ⎢ ⎦ ⎣ V ρa Ca VAc
+ ρ ChCVA (1 − G ) ⎤ L ⎤ ⎡ hC (G − 1) e−⎡⎣ iω V a a c ⎦ + hC (1 − G ) ⎥ As′ +⎢ ρa Ca Ac iω + hC (1 − G ) ⎥ ⎢ ⎦ ⎣
(15)
Substituting Eq. (15) into Eq. (12), X2 = hUn (h + λs s as ). Then, Eq. (15) can be rewritten as
U (s, x ) = hUn e−x
(22)
The expression for the air temperature at the pipe outlet is then given by
where both X1 and X2 are determined by boundary conditions. By substituting Eq. (13) into Eq. (14), x → ∞, e−x s as → 0 , and x s as → ∞ e , then X1 = 0 . Eq. (14) is rearranged as
U (s, x ) = X2 e−x
(20)
Inserting Eq. (18) into Eq. (20), we obtain
(11)
= h (Un − U (s, x ))
(19)
The fluctuating component of Eq. (19) can be rearranged as
⌢
− λs
Tn T dL = −ρa Ca VAc n dL − h (Tn − Tx = 0 ) CdL t L
(17)
κn = An Ao = abs (An′ ) Ao
(24)
φn = (−1)·angle (An′ )
(25)
where abs and angle denote the complex modulus and phase angle, respectively.
where G (iω, x ) = he−x iω as (h + λs iω as ) . The complex form of the excess fluctuating temperature around the pipe is ~ ~ ~ T (t , x ) = Tn G (iω, x ) + (1 − G (iω, x )) Ts (18) ~ ~ When x → 0 , Tw = T (t , 0) is the fluctuating component of the surface temperature of the pipe.
2.4. Pressure drop and cooling/heating performance The fan pressure drop caused by the buried pipe is calculated as [31]
(a) Computational domain.
(b) Cross-sectional mesh.
Fig. 2. Computational domain of EAHE model. 4
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Δp = λ
L ρa V 2 dc 2
Q = Ca ρa Ac V (Tn − To) (26)
The EAHE's COP is determined by the amount of cooling/heating produced and the amount of power required to circulate air through the buried pipe:
where L is the length of the pipe, and the coefficient of friction λ is given by [31,32] 64
COP = |Q| Qfan
Re < 2000
⎧ λ = Re ⎪ 0 . 3164 λ = 0.25 Re ⎨ ⎪ λ = (1. 82logRe − 1. 64)−2 ⎩
2000⩽Re <
(29)
(30)
105
Re ⩾ 105
3. Comparison of theoretical model's results with numerical simulations
(27)
Then, the fan energy consumption of the flowing air inside the pipe is
Qfan =
3.1. Numerical simulation setup
Δpρa VAc η
For this comparison, commercial computational fluid dynamics (CFD) software, Ansys Fluent, was employed to simulate the airflows and the air–earth heat transfer processes. The soil computational domain was 60 m × 10 m × 8 m (Fig. 2). A flat rectangular EAHE pipe
(28)
where η is the fan efficiency, assumed to be 0.6 [33]. The EAHE's cooling or heating capacity is determined by
30.0 27.5
Temperature ( )
25.0 22.5
Initial stage
20.0 17.5 15.0 12.5 10.0 7.5 5.0 0
100
200
300
400
Time (day)
Inlet Numeric 20m Numeric 40m Theory 40m Theory 60m
500
600
700
Numeric 60m
Model 20m
(a) Transient profiles of EAHE outlet air temperatures. 30.0 27.5
Temperature ( )
25.0 22.5 Initial stage 20.0 17.5 15.0 12.5 10.0 7.5 5.0 0
100
200
300
400
Time (day)
Inlet Numeric 20m Numeric 40m Theory 40m Theory 60m
500
Numeric 60m
600
Theory 20m
(b) Transient profiles of EAHE buried-pipe wall temperatures. Fig. 3. Theoretical and numerical simulation results compared. 5
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results. Both the fluctuation amplitudes and phase shifts obtained from the proposed theoretical approach agree well with the numerical simulation results. Except in the initial stage, the maximum relative error between the theoretical results and the simulated results is 3%, indicating that the model is sufficiently accurate to analyze the system's thermal performance.
60 m × 1 m × 0.1 m was buried inside the computational domain of the soil at a depth of 4 m. The thermal inertia of the buried pipe is significantly smaller than that of the soil surrounding the pipe, and the thermal conductivity of the buried pipe is generally larger than that of the soil. The heat transfer resistance caused by the pipe wall is much lower than that of the soil. Thus, the effects of pipe thickness are neglected in the simulations. The side boundaries of the soil surface were at 10 m, which is ten times the width of the buried pipe; therefore, the temperature distribution was considered undisturbed at these boundaries. These side boundaries were set as adiabatic walls. The pipe's inner surface was set as fluid–solid coupled surfaces to account for the convective heat transfer at the wall surfaces. The pipe inlet was set as an inlet-velocity boundary condition, and the inlet velocity was a constant 1 m/s. The pipe outlet was treated as a pressure outlet. Meteorological data for Chongqing (a hot-summer/cold-winter region) indicate an annual average air temperature of 17.84 °C, with fluctuation amplitude 10.1 °C. The soil's thermal conductivity is λs = 1.1 W/(m·K) , and its thermal diffusivity is αs = 7. 1 × 10−7 m2/s [36]. The boundary conditions for both at the upper ground surface and the temperature variation at the pipe inlet were set via the user-defined functions. The temperature for the bottom surface of the computational domain, i.e., z = −8 m, is set to follow a transient profile, which is calculated using Eq. (2). Because [36] provides hourly data for outdoor air temperature, the time step in the simulations was set as 1 h (3600 s). The number of hexahedral grid cells was 7,518,021, and grid sensitivity tests were performed. According to Eq. (10), the Reynolds number is approximately 11,806 and the flow inside the pipe is fully turbulent. Thus, the standard k–ε model was employed to account for the turbulence effects [17,18,20,37,38]. The SIMPLE scheme was used for the pressure–velocity coupling. The calculations were considered converged when the residuals for mass, momentum, and energy between two consecutive iterations were under 10−3, 10−3, and 10−6, respectively.
4. Comparison of the effects of flat rectangular EAHE with those of circular EAHE In this section, we use theoretical analysis to compare the thermal performance of the flat rectangular and the circular EAHEs. The flat rectangular EAHE results are from the theoretical model (Section 2); the circular EAHE results are from the model of [13]. For Sections 4.1–4.4, the cross section of the flat rectangular pipe is 1 m × 0.1 m, and the radius of the circular pipe is 0.18 m, so that their cross-sectional areas are equal. For Sections 4.1–4.4, the input engineering parameters for the EAHE (burial depth, airflow velocity, length) are identical to those in Section 3.1, as are the climate parameters and the thermal properties of the soil. 4.1. Air temperature Fig. 4 shows EAHE air temperature variations for the flat rectangular and circular cross sections at various buried-pipe lengths. It is seen that the fluctuation amplitudes of the air temperature of the flat rectangular EAHE at 20, 40, and 60 m are lower than those of the circular EAHE and that the phase shifts of the air temperature of the flat rectangular EAHE at 20, 40, and 60 m are larger than those of the circular EAHE. The fluctuation amplitudes of the air temperature for the flat rectangular EAHE are 1.37 °C, 1.91 °C, and 2.49 °C lower than those of the circular EAHE at 20, 40, and 60 m, respectively, representing reductions of 13.7%, 24.7%, and 24.7%, respectively. In addition, the phase shifts for the flat rectangular EAHE were 4.64, 12.21, and 22.67 days larger than those of the circular EAHE at 20, 40, and 60 m, respectively.
3.2. Model validation Fig. 3 shows the transient profiles of the outlet air temperature and the buried-pipe wall temperature for various pipe lengths obtained from the numerical simulation and the proposed theoretical model. The theoretical air temperatures and buried-pipe wall temperatures at 20, 40, and 60 m are in close agreement with the numerical simulation
24
Rectangular 60m
Circular 60m
Rectangular 40m
Circular 40m
4.2. Buried-pipe wall temperature Fig. 5 shows the buried-pipe wall temperature variations for the flat
20
Temperature ( )
16 12 24 20 16 12 Inlet
30
Rectangular 20m
Circular 20m
24 18 12 6 0
100
200
300
400
500
600
Time (day) Fig. 4. Comparison of air temperatures for flat rectangular and circular EAHEs in annual cycle. 6
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24
Rectangular 60m
Circular 60m
Rectangular 40m
Circular 40m
21 18
Temperature ( )
15 24 21 18 15 12 Inlet
30
Rectangular 20m
Circular 20m
24 18 12 6 0
100
200
300
400
500
600
700
Time (day) Fig. 5. Comparison of buried-pipe wall temperatures for flat rectangular and circular EAHEs in annual cycle.
4.3. Cooling/heating capacity provision and COP
rectangular and circular EAHEs at various EAHE lengths. It can be seen that the fluctuation amplitudes of the flat rectangular EAHE wall temperature were 1.73 °C, 1.36 °C, and 1.02 °C lower than those of the circular EAHE at 20, 40, and 60 m, respectively. In contrast, the phase shifts of the flat rectangular EAHE wall temperature were 9.88, 18.02, and 26.15 days larger than those of the circular EAHE at 20, 40, and 60 m, respectively. This is because the thermal disturbance range in the soil for the flat rectangular pipe is smaller than that for the circular pipe, resulting in the temperature wave of the soil around the buried pipe decaying faster and creating a more stable pipe wall temperature. Therefore, the flat rectangular EAHE has a relatively stable wall temperature, which is beneficial for heat transfer performance.
70
Rectangular EAHE
The EAHE has length 60 m, airflow velocity 3 m/s, and burial depth 4 m. Fig. 6(a) shows the variations in outlet air temperature for the flat rectangular and circular EAHEs in annual cycle. The fluctuation amplitudes and phase shift of the flat rectangular EAHE outlet air temperature are lower and larger than those of the circular EAHE, respectively. This indicates that the thermal performance of the flat rectangular EAHE is better than the circular EAHE's at the same crosssectional area. Fig. 6(b) shows the variation in cooling/heating capacity (Eq. (29)) provided by the flat rectangular and circular EAHEs in annual cycle. It is observed that the cooling/heating of the flat rectangular EAHE is much stronger than that of the circular EAHE. Fig. 6(c) shows (c)
Circular EAHE
COP
56 42 28
0
Rectangular EAHE
Circular EAHE
(b)
(W)
1800 0
-1800
Temperature ( )
Cooling/heating capacity
14
Inlet
Outlet air temperature of rectangular EAHE
Outlet air temperature of circular EAHE
(a)
30 24 18 12 6 0
100
200
300
400
500
600
700
Time (day) Fig. 6. Comparison of cooling/heating capacity and COP for flat rectangular and circular EAHEs in annual cycle. 7
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the variation in the COP value (Eq. (30)) of the flat rectangular and circular EAHEs in annual cycle. It is seen that the COP values of the flat rectangular EAHE is less than that of the circular EAHE. This is because the pressure drop and fan power consumption of the flat rectangular EAHE is greater than that of the circular EAHE because of the former's narrow airflow space.
boundary conditions were the same as those given in Section 3.1. Fig. 7 shows the winter and summer temperature distributions for the circular and flat rectangular EAHEs at the cross-section at Y = 5 m. The results indicate that as the soil depth Z increases, the temperature waves of the soil layer gradually stabilize and the temperature waves do not reach the bottom surface boundary of the soil domain, which indicates that the computational domain of the soil is sufficiently large. Owing to the strong thermal inertia of the soil, the temperature distribution of the soil layer has obvious stratification by depth. Therefore, soil temperatures at the pipe depth are higher/lower than the outdoor air temperature in summer/winter, which provides an opportunity to utilize the temperature difference between the soil and the flowing air. In addition, the soil temperature is essentially constant at Z = −4 m. From Fig. 7, it is seen that the thermal disturbance range in the
4.4. Discussion of reasons for the differences between flat rectangular EAHE and circular EAHE To understand the thermal performance differences between the flat rectangular EAHE and the circular EAHE, we used the same numerical simulation method as described in Section 3.1 to study the temperature distributions for the two types of EAHE. The input parameters and
(a) Circular EAHE.
(b) Flat rectangular EAHE. Fig. 7. Temperature distribution at Y = 5 m. 8
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acceptable. Fig. 8 also shows that the range of disturbed soil for the flat rectangular pipe is much shorter than that for the circular pipe in the length direction. As the pipe length increases, the air temperature in the flat rectangular pipe is higher/lower than that of the same-length circular pipe in winter/summer. Fig. 9 shows the temperature distributions for the circular and flat rectangular EAHEs at the cross section at X = 10 m. It indicates that the range of disturbed soil around the flat rectangular pipe is much smaller than that for the circular pipe, which has a major effect on the design of EAHE pipe spacing for multi-tube arrangements. Therefore, the flat rectangular EAHE can improve the temperature wave of the soil around the buried pipe, providing a faster decay rate and creating a more stable
surrounding soil for the flat rectangular pipe is less than that of the circular pipe in both the depth direction and the length direction. As the pipe length increases, the air temperature in the flat rectangular pipe is higher/lower than that of the same-length circular pipe in winter/ summer. Fig. 8 shows the winter and summer temperature distributions for the circular and flat rectangular EAHEs at a burial depth of Z = −4 m. It shows that the temperature waves emitted from the EAHE do not reach the side boundaries of the soil domain. In the width direction, the thermal disturbance range in the soil temperature for the rectangular pipe is much smaller than that for the circular pipe, which demonstrates that neglecting the heat transfer process in the width direction is
(a) Circular EAHE.
(b) Flat rectangular EAHE. Fig. 8. Temperature distribution at Z = −4 m. 9
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(a) Circular EAHE.
(b) Flat rectangular EAHE.
Fig. 9. Temperature distribution at X = 10 m.
pipe wall temperature.
is larger.
4.5. The key factors influencing the differences between the effects of flat rectangular EAHE and those of circular EAHE
4.5.3. Effect of airflow velocity in pipe For this section, the EAHE has length 100 m and burial depth 3 m. Fig. 12 shows that as the airflow velocity increases, the phase shifts of the flat rectangular and circular EAHEs initially increase to the highest point and then decrease. The normalized fluctuation amplitudes of both EAHE types first decrease and then increase with the increase in airflow velocity, which indicates that there is no advantage in increasing the airflow velocity for improving the heat transfer performance of the rectangular or the circular EAHE. When the airflow velocities in the EAHE pipes are the same, the normalized fluctuation amplitude of the flat rectangular EAHE is smaller than that of the circular EAHE, and its phase shift is larger.
In this section, the effects of the main engineering parameters on the outlet air temperature of the flat rectangular and circular EAHEs are investigated. In Sections 4.5.1–4.5.3, the cross section of the flat rectangular pipe is 1 m × 0.1 m and the radius of the circular pipe is 0.18 m, so that their cross-sectional areas are equal. In Section 4.5.4, although the cross-sectional areas change, the rectangular EAHE and the circular EAHE cross-sectional areas remain equal. 4.5.1. Effect of pipe length For this section, the EAHE has burial depth 3 m and airflow velocity 1.5 m/s. Fig. 10 shows that the normalized fluctuation amplitudes of both EAHE types decrease with the pipe length, decreasing faster at the EAHE entry portion and then slowing down. In contrast, the phase shift of the outlet air temperature of the flat rectangular and circular EAHEs increases with the length, increasing faster at the EAHE inlet section and then increasing less and less. This is mainly because the temperature difference between the air and the surrounding soil at the EAHE inlet is large and the heat flux is high. 4.5.2. Effect of pipe's burial depth For this section, the EAHE has length 100 m and airflow velocity 1.5 m/s. Fig. 11 indicates that the phase shifts of both EAHE types initially increase to the highest point and then decrease with an increase in burial depth. The normalized fluctuation amplitudes of the flat rectangular and circular EAHEs decrease with the increase in EAHE burial depth and then stabilize. The phase shifts differ between the flat rectangular and circular EAHEs at the highest point, and the normalized fluctuation amplitudes also differ when the stable value is reached. However, the phase shift of the flat rectangular EAHE reaches a much larger maximum value than the circular EAHE. When the burial depths are the same, the normalized fluctuation amplitude of the flat rectangular EAHE is smaller than that of the circular EAHE, and its phase shift
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Fig. 10. Effect of pipe length. 10
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4.5.4. Effect of pipe's cross-sectional area For this section, the EAHE has length 100 m, burial depth 3 m, and airflow velocity 1.5 m/s. Fig. 13 shows that the phase shifts of the flat rectangular and circular EAHEs decrease with an increase in the crosssectional area of the pipe and that the normalized fluctuation
Applied Thermal Engineering 162 (2019) 114262
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indicate the following:
• The outlet air temperature fluctuation of the flat rectangular EAHE Phase shift
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H. Wei and D. Yang
• •
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Phase shift of circular EAHE Phase shift of rectangular EAHE
is 24.7% lesser and its phase shift is 22.67 days longer than that of circular EAHE with the same cross-sectional area. The soil's thermal disturbance range of the flat rectangular EAHE is shorter than that of circular EAHE, and resulting in a temperature wave in the soil around the buried pipe that decays faster and creates a more stable pipe wall temperature. Although the cooling/heating capacity of the flat rectangular EAHE is greater than that of the circular EAHE, the pressure drop and fan power consumption of the flat rectangular EAHE are greater than that of the circular EAHE as a result of the former's narrow airflow space, Therefore, the COP values of the flat rectangular EAHE is less than that of the circular EAHE.
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Acknowledgments
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This work proves that the flat rectangular cross-section of EAHE can markedly reduce the outlet air temperature fluctuation, extend the phase shift and improve the cooling/heating capacity of EAHE although the flat rectangular EAHE has a lower COP, which implies that the pipe cross-sectional shape is also an important influencing factor on the performance of EAHE. It is recommended that the flat rectangular pipes use the high-strength stainless steel, cast iron or precast concrete material, or make use of the reinforcement method inside the pipe to avoid deformation.
Phase shift
Normalized fluctuation amplitude
Fig. 11. Effect of burial depth.
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The authors acknowledge support from the National Natural Science Foundation of China (NSFC) under Grant No. 51578087, the Projects Nos. 2017M622967 and 2018T110945 funded by the China Postdoctoral Science Foundation, 2019CDXYCH0027, 2018CDXYCH0015 and CQU2018CDHB1B06 supported by the Fundamental Research Funds for the Central Universities.
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Appendix A. Supplementary material
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Supplementary data to this article can be found online at https:// doi.org/10.1016/j.applthermaleng.2019.114262.
Phase shift
Normalized fluctuation amplitude
Fig. 12. Effect of airflow velocity.
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Across-sectional area (m 2)
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Fig. 13. Effect of cross-sectional area.
amplitudes increase with the increase in the cross-sectional area. When the cross-sectional areas are equal, the normalized fluctuation amplitude of the flat rectangular EAHE is smaller than that of the circular EAHE. However, its phase shift is larger. 5. Conclusions This paper develops a mathematical model for evaluating the thermal performance of flat rectangular earth-to-air heat exchangers in harmonically fluctuating thermal environment. Unlike the conventional EAHE, the flat rectangular buried pipe has a width-to-height ratio of 10 or greater. In addition, the model was validated by three-dimensional CFD numerical simulations. Then, studies comparing the flat rectangular and conventional circular EAHEs were conducted. Our results 11
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[12] A. Rouag, A. Benchabane, C.E. Mehdid, Thermal design of Earth-to-Air Heat Exchanger. Part I a new transient semi-analytical model for determining soil temperature, J. Clean. Prod. 182 (2018) 538–544. [13] D. Yang, Y.H. Guo, J.P. Zhang, Evaluation of the thermal performance of an earthto-air heat exchanger (EAHE) in a harmonic thermal environment, Energy Convers. Manage. 109 (2016) 184–194. [14] D. Yang, J.P. Zhang, Analysis and experiments on the periodically fluctuating air temperature in a building with earth-air tube ventilation, Build. Environ. 85 (2015) 29–39. [15] D. Yang, J.P. Zhang, Theoretical assessment of the combined effects of building thermal mass and earth-air-tube ventilation on the indoor thermal environment, Energy Build. 81 (2014) 182–199. [16] H.B. Wei, D. Yang, Y.H. Guo, M.Q. Chen, Coupling of earth-to-air heat exchangers and buoyancy for energy-efficient ventilation of buildings considering dynamic thermal behavior and cooling/heating capacity, Energy 147 (2018) 587–602. [17] T.M. Yusof, H. Ibrahim, W.H. Azmi, M.R.M. Rejab, Thermal analysis of earth-to-air heat exchanger using laboratory simulator, Appl. Therm. Eng. 134 (2018) 130–140. [18] M. Cuny, J. Lin, M. Siroux, V. Magnenet, C. Fond, Influence of coating soil types on the energy of earth-air heat exchanger, Energy Build. 158 (2018) 1000–1012. [19] N.A.S. Elminshawy, F.R. Siddiqui, Q.U. Farooq, M.F. Addas, Experimental investigation on the performance of earth-air pipe heat exchanger for different soil compaction levels, Appl. Therm. Eng. 124 (2017) 1319–1327. [20] A. Mathur, S. Priyam, G.D. Mathur, J. Agrawal, Mathur, Comparative study of straight and spiral earth air tunnel heat exchanger system operated in cooling and heating modes, Renew. Energy 108 (2017) 474–487. [21] M.K. Rodrigues, R.D. Brum, J. Vaz, L.A.O. Rocha, E.D. dos Santos, L.A. Isoldi, Numerical investigation about the improvement of the thermal potential of an Earth-Air Heat Exchanger (EAHE) employing the constructal design method, Renew. Energy 80 (2015) 538–551. [22] M. Zukowski, J. Topolanska, Comparison of thermal performance between tube and plate ground air heat exchangers, Renew. Energy 115 (2018) 697–710. [23] V. Bansal, R. Misra, G.D. Agrawal, J. Mathur, Performance analysis of earth-pipe air heat exchanger for summer cooling, Energy Build. 42 (2010) 645–648. [24] F. Niu, Y. Yu, D. Yu, H. Li, Heat and mass transfer performance analysis and cooling capacity prediction of earth to air heat exchanger, Appl. Energy 137 (2015) 211–221. [25] H.B. Derbel, O. Kanoun, Investigation of the ground thermal potential in Tunisia
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Contents lists available at ScienceDirect
Applied Thermal Engineering journal homepage: www.elsevier.com/locate/apthermeng
Performance evaluation of flat rectangular earth-to-air heat exchangers in harmonically fluctuating thermal environments
T
⁎
Haibin Wei, Dong Yang
National Centre for International Research of Low-carbon and Green Buildings, Chongqing University, Chongqing 400045, China School of Civil Engineering, Chongqing University, Chongqing 400045, China
H I GH L IG H T S
heat exchanger (EAHE) with flat rectangular cross section is proposed. • Earth-to-air capacity exceeds that of circular-cross-section EAHE. • Heating/cooling temperature fluctuation amplitudes are lower, and phase shift is larger. • Air pipe wall temperature is more stable. • EAHE • The analytical model for such EAHE is proposed and validated.
A R T I C LE I N FO
A B S T R A C T
Keywords: Earth-to-air heat exchanger Shallow geothermal energy Renewable energy Flat rectangular cross section Cooling/heating potential Fluctuating thermal environment
An earth-to-air heat exchanger (EAHE) is an excellent passive technology that uses shallow geothermal energy directly to generate cooled or heated air for thermal regulation of buildings. Previous studies have focused on circular EAHEs, paying insufficient attention to the effect of cross-sectional shape. In this paper, we propose an EAHE having a flat rectangular cross section, develop a mathematical model to calculate its thermal performance in a harmonic thermal environment, and validate the theoretical results using a three-dimensional numerical model. Our results indicate that compared with a circular EAHE shape of the same cross-sectional area, the outlet air temperature fluctuation of the flat rectangular EAHE is 24.7% lesser and its phase shift is 22.67 days longer, and the soil's thermal disturbance range is shorter, resulting in a temperature wave in the soil around the buried pipe that decays faster and creates a more stable pipe wall temperature. The cooling/heating capacity of the flat rectangular EAHE is greater than that of the circular EAHE. The pressure drop and fan power consumption of the flat rectangular EAHE is greater than that of circular EAHE, and thus the coefficient of performance (COP) value of the flat rectangular EAHE is less than that of the circular EAHE.
1. Introduction To reduce building energy consumption and alleviate environmental pollution, many renewable energy sources have been introduced to building air conditioning systems in recent decades [1–3]. Geothermal energy, for example, is a natural energy source that is limitless, environmentally friendly, and sustainable. Owing to the high thermal inertia of soil, the soil temperature at a depth of 2–3 m is nearly constant and is approximately equal to the annual average outdoor air temperature [4,5]. One application of geothermal energy, earth-to-air heat exchangers (EAHEs), use the soil directly to exchange heat with the ventilation air and absorb or store the natural coolness or heat from the underground soil in summer or winter and then supply the cooled or
⁎
heated air into a building's interior spaces to regulate the indoor thermal environment [6–8]. A number of experimental and analytical studies have been conducted under various climatic conditions worldwide. For example, AlAjmi et al. [9] developed a theoretical model for evaluating EAHE thermal performance in the arid Saharan climate. They found that the EAHE could reduce cooling energy demand for a typical house by 30% in summer. Kumar et al. [10] proposed a numerical model to predict the performance and energy conservation potential of an earth–air tunnel system coupled with a non-air-conditioned building. The numerical model was in excellent agreement with experimental data for Mathura (India) and indicated that the cooling potential of an 80-m earth pipe can provide 19 kW of cooling energy, adequate to maintain an average
Corresponding author at: School of Civil Engineering, Chongqing University, Chongqing 400045, China. E-mail address: [email protected] (D. Yang).
https://doi.org/10.1016/j.applthermaleng.2019.114262 Received 12 June 2019; Received in revised form 6 August 2019; Accepted 13 August 2019 Available online 14 August 2019 1359-4311/ © 2019 Elsevier Ltd. All rights reserved.
Applied Thermal Engineering 162 (2019) 114262
H. Wei and D. Yang
Nomenclature A Ac a b C Ca dc F h i L Nu P Pr Re T t V x z
λa λs νa ρa φn ω
temperature fluctuation amplitude (K) cross-sectional area of pipe (m2) pipe width (m) pipe height (m) pipe circumference (m) specific heat of air (J/(kg·K)) equivalent diameter of non-circular pipe (m) combination of modified Bessel functions convective heat transfer coefficient at inner surface of pipe (W/(m2·K)) the imaginary unit pipe length (m) Nusselt number fluctuation period (s) Prandtl number Reynolds number temperature (°C) time (s) velocity of airflow in pipe (m3/s) distance from pipe surface (m) burial depth of pipe (m)
Diacritical marks — ~ ⌢
time-averaged term fluctuation term Laplace transformation
Subscripts n o s w z
EAHE outlet air outdoor air soil buried-pipe surface soil depth
Abbreviations CFD COP EAHE
Greek symbols
αs κn
temperature thermal conductivity of air (W/(m·K)) thermal conductivity of the soil (W/(m·K)) kinematic viscosity coefficient of air (m2/s) air density (kg/m3) phase shift of EAHE outlet air temperature (rad) fluctuation frequency (s−1)
thermal diffusivity of the soil (m2/s) normalized fluctuation amplitude of EAHE outlet air
computational fluid dynamics coefficient of performance earth-to-air heat exchanger
(a) Longitudinal view of EAHE pipe.
(b) Cross-sectional view of a flat rectangular EAHE pipe. Fig. 1. Schematic diagram of heat transfer between pipe and surrounding soil. 2
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H. Wei and D. Yang
During an annual cycle, the variation in soil temperature decreases exponentially with the increase in depth, and its cosine changes with time [24–26]:
indoor temperature of 27.65 °C. Kabashnikov et al. [11] proposed a mathematical model to calculate the EAHE outlet temperature and analyzed the effects of the EAHE engineering parameters. Rouag et al. [12] proposed a transient semi-analytical model to calculate the temperature distribution in the soil surrounding an EAHE pipe. They found that the soil radius is a function of the operation duration, soil thermal diffusivity, pipe diameter, and inlet air temperature. Yang et al. [13–16] proposed an analytical approach for evaluating EAHE thermal performance under periodic fluctuations in both ambient air temperature and soil temperature and noted that the effects of coupling EAHEs with building thermal mass could maintain a building's thermal environment. Furthermore, Yusof et al. [17] developed a laboratory EAHE simulator to analyze EAHE thermal performance. Cuny et al. [18] found that the difference in soil types has an indispensable effect on EAHE heat transfer. Elminshawy et al. [19] found that EAHE thermal performance is highly dependent on operating conditions and soil compaction levels. Mathur et al. [20] proposed a spiral-shaped EAHE system to reduce the space taken up by the EAHE; its coefficient of performance (COP) was comparable to that of a straight system. Rodrigues et al. [21] performed a numerical investigation on different EAHE geometrical configurations using Constructal Design to seek the optimal geometries and found that increasing the number of ducts (complexity of geometry) improved thermal performance by up to ~73% for cooling and ~115% for heating. Zukowski et al. [22] compared the thermal performance of tube and plate EAHEs and found that their winter heating capacities were 13.5 MWh and 16.35 MWh, respectively, and their cooling capacities were 10.3 MWh and 20.41 MWh, respectively. The above investigations found that many parameters influence the heat transfer between the flowing air and the earth around the buried pipe. However, the pipe cross-sectional shapes' influence on heat transfer is still unknown. To improve EAHE energy efficiency, this study proposes a flat rectangular EAHE (Fig. 1). Unlike those in previous studies, the cross section of this pipe is not cylindrical, and the soil's heat conduction around the pipe is not in a cylindrical coordinate system; therefore, an analytical model for evaluating the proposed EAHE's thermal performance is also proposed. In this study, to reflect the effects of periodically fluctuating outdoor air temperatures in an annual cycle, the harmonically fluctuating air temperature profiles rather than constant air temperature values are specified as the inlet boundary condition for both the analytical model and numerical simulations. The model's results are validated by numerical simulation. The differences in thermal performance between the flat rectangular EAHE and the conventional cylindrical EAHE are also investigated.
∼ −z Ts (t , z ) = T¯s + Ts = T¯s + Ao e =T¯s + As ei (ωt − φs, z )
ω 2α s
cos ⎛ωt − z ⎝ = T¯s + As′ eiωt
ω 2αs
⎞ ⎠ (2)
where Ts is the time-averaged soil temperature, z is the soil depth, Ao is the fluctuation amplitude of the ground surface temperature, and αs is the soil's thermal diffusivity. As = Ao e−z ω (2αs) , φs, z = z ω (2αs ) , and As′ = As e−iφs, z . When the soil depth reaches 2–3 m, the fluctuation amplitude of the soil temperature for an annual cycle is small and can be approximated as the annual average outdoor air temperature, Ts = To [4,5]. The heat transfer process between the buried pipe and the surrounding soil is complex. The soil temperature around the pipe is affected not only by the depth and time but also by the flowing-air temperature inside the pipe (Fig. 1). The temperature wave originating from the pipe penetrates the soil, changes the soil temperature around the pipe, and decays with increasing propagation distance. At an infinitely large distance from the pipe, the disturbance by the flowing-air temperature is negligible, and thus the soil temperature approaches the original undisturbed-soil temperature. Yang et al. [13] proposed an excess fluctuating temperature to deal with the difference between a certain prescribed fluctuating temperature and the undisturbed-soil fluctuating temperature. The soil's excess fluctuating temperature at ∼ ~ ~ distance x from the pipe is U (t , x ) = T (t , x ) − Ts , where Ts is the un~ ~ disturbed-soil fluctuating temperature; therefore, Un = Tn − Ts and ~ ~ Uo = To − Ts . 2.2. Soil temperature around EAHE We regard the heat conduction process of the soil in the height direction around the flat rectangular buried pipe as semi-infinite plate heat conduction. To simplify the heat transfer model, we assumed that the original temperature around the pipe is not affected by the ground environment temperature. Therefore, the equation governing the heat conduction can be established by the excess fluctuating temperature:
∂U (t , x ) ∂2U (t , x ) = αs ∂t ∂x 2
(3)
The corresponding boundary and initial conditions can be described as
2. Mathematical model for flat rectangular EAHE
− λs
∂U (t , x ) ∂x
= h (Un − U (t , x )) x=0
(4)
2.1. Basic considerations
U (t , x ) = 0
x=∞
(5)
In this study, the EAHE buried pipe is assumed to have a flat rectangular cross section whose width-to-height ratio is 10 or greater (see Fig. 1). Therefore, the soil heat conduction transfer around the pipe can be considered to occur mainly in the height direction, and that in the width direction can be neglected. To simplify the heat transfer model, only the sensible heat exchange is considered, not the latent heat change; soil thermal conductivity and capacity are assumed to be homogeneous and constant; and the thermal resistance of the pipe wall is neglected [21,23,34,35]. In [13–16], both the outdoor air temperature To and the EAHE outlet air temperature Tn are assumed to be harmonic fluctuations and are represented as sums of the time-averaged term and fluctuating term: ∼ T (t ) = T + T = T + Aei (ωt − φ) = T + A′eiωt (1) ~ where T is the time-averaged air temperature, T is the fluctuating air temperature, A is the temperature fluctuation amplitude, ω = 2π P is the fluctuation frequency, and P is the fluctuation period.
U (0, x ) = 0
t=0
(6)
where h is the convective heat transfer coefficient at the pipe's inner surface. The equivalent diameter is used as the characteristic scale to calculate both the convective heat transfer coefficient and the pressure drop [27]. The usual equivalent-diameter calculation formula for a noncircular pipe is
dc = 4Ac C
(7)
where Ac and C are the cross-sectional area and circumference of the pipe, respectively. The convective heat transfer coefficient at the pipe's inner surface is given by
h = Nu·λa dc
(8)
where λa is the thermal conductivity of air, and the Nusselt number of the pipe flow is given as [28,29] 3
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⎧ Nu = 8. 23 0.3 0.8 ⎨ ⎩ Nu = 0. 023Pr Re
2.3. Air temperature along EAHE
Re<2300 Re>2300
(9) The heat-balance equation between the inner surface of the EAHE pipe and the flowing air inside a rectangular pipe is
where Pr is the Prandtl number, and the Reynolds number of the pipe flow is
ρa Ca Ac
(10)
Re = V ·dc νa
where V is the air velocity inside the pipe and νa is the kinematic viscosity of air. According to the values of Re of airflow inside the pipe, there are two flow regimes, Re<2300 , where the airflow is laminar, and Re>2300 , where the airflow is turbulent. Thus, the heat transfer coefficient at the pipe surface is related to the values of Re. Applying the Laplace transformation, Eqs. (3)–(6) can be written as ⌢ ∂2U (s,
x)
∂x 2
−
s ⌢ U (s, x ) = 0 αs
∂U (s, x ) ∂x
~ ~ 1 ∂Tn hC ∂T ~ ~ (Tn − Tx = 0) =− n − V ∂t ρa Ca VAc ∂L
⌢
∂Un iω hC iω ~ +⎡ + (1 − G (iω, 0)) ⎤ Un + Ts = 0 ⎢V ⎥ ∂L ρ C VA Va c (21) a a ⎣ ⎦ ~ ~ With the boundary condition Un L = 0 = Uo = To − Ts , the solution of Eq. (21) is
⌢
(12)
x=0
⌢
U (s, ∞) = 0
∼ −⎡ iω + hC (1 − G) ⎤⎥ L Un = To e ⎣⎢ V ρa Ca VAc ⎦
(13)
x=∞
Combined with the initial condition given by Eq. (12), the general solution of Eq. (11) is expressed as a combination of modified Bessel functions ⌢
U (s, x ) = X1 e x
s as
+ X2 e−x
s as
+ ρ ChCVA (1 − G ) ⎤ L ⎡ hC (G − 1) e−⎡⎣ iω ⎤ V a a c ⎦ + hC (1 − G ) ∼ ⎥ Ts +⎢ ρa Ca Ac iω + hC (1 − G ) ⎢ ⎥ ⎣ ⎦
(14)
⌢
s as
An′ = Ao′ e
⌢
⌢
⌢
s as
(h + λs s as )
(23)
The normalized fluctuation amplitude and phase shift of the pipe air temperature with respect to the outdoor air temperature are
(16)
By applying the inverse Laplace transformation together with its convolution theorem [30], Eq. (16) can be expressed as
U (t , x ) = Un G (iω, x )
hC −⎡ iω + (1 − G ) ⎤ L ⎥ ⎢ ⎦ ⎣ V ρa Ca VAc
+ ρ ChCVA (1 − G ) ⎤ L ⎤ ⎡ hC (G − 1) e−⎡⎣ iω V a a c ⎦ + hC (1 − G ) ⎥ As′ +⎢ ρa Ca Ac iω + hC (1 − G ) ⎥ ⎢ ⎦ ⎣
(15)
Substituting Eq. (15) into Eq. (12), X2 = hUn (h + λs s as ). Then, Eq. (15) can be rewritten as
U (s, x ) = hUn e−x
(22)
The expression for the air temperature at the pipe outlet is then given by
where both X1 and X2 are determined by boundary conditions. By substituting Eq. (13) into Eq. (14), x → ∞, e−x s as → 0 , and x s as → ∞ e , then X1 = 0 . Eq. (14) is rearranged as
U (s, x ) = X2 e−x
(20)
Inserting Eq. (18) into Eq. (20), we obtain
(11)
= h (Un − U (s, x ))
(19)
The fluctuating component of Eq. (19) can be rearranged as
⌢
− λs
Tn T dL = −ρa Ca VAc n dL − h (Tn − Tx = 0 ) CdL t L
(17)
κn = An Ao = abs (An′ ) Ao
(24)
φn = (−1)·angle (An′ )
(25)
where abs and angle denote the complex modulus and phase angle, respectively.
where G (iω, x ) = he−x iω as (h + λs iω as ) . The complex form of the excess fluctuating temperature around the pipe is ~ ~ ~ T (t , x ) = Tn G (iω, x ) + (1 − G (iω, x )) Ts (18) ~ ~ When x → 0 , Tw = T (t , 0) is the fluctuating component of the surface temperature of the pipe.
2.4. Pressure drop and cooling/heating performance The fan pressure drop caused by the buried pipe is calculated as [31]
(a) Computational domain.
(b) Cross-sectional mesh.
Fig. 2. Computational domain of EAHE model. 4
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Δp = λ
L ρa V 2 dc 2
Q = Ca ρa Ac V (Tn − To) (26)
The EAHE's COP is determined by the amount of cooling/heating produced and the amount of power required to circulate air through the buried pipe:
where L is the length of the pipe, and the coefficient of friction λ is given by [31,32] 64
COP = |Q| Qfan
Re < 2000
⎧ λ = Re ⎪ 0 . 3164 λ = 0.25 Re ⎨ ⎪ λ = (1. 82logRe − 1. 64)−2 ⎩
2000⩽Re <
(29)
(30)
105
Re ⩾ 105
3. Comparison of theoretical model's results with numerical simulations
(27)
Then, the fan energy consumption of the flowing air inside the pipe is
Qfan =
3.1. Numerical simulation setup
Δpρa VAc η
For this comparison, commercial computational fluid dynamics (CFD) software, Ansys Fluent, was employed to simulate the airflows and the air–earth heat transfer processes. The soil computational domain was 60 m × 10 m × 8 m (Fig. 2). A flat rectangular EAHE pipe
(28)
where η is the fan efficiency, assumed to be 0.6 [33]. The EAHE's cooling or heating capacity is determined by
30.0 27.5
Temperature ( )
25.0 22.5
Initial stage
20.0 17.5 15.0 12.5 10.0 7.5 5.0 0
100
200
300
400
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Inlet Numeric 20m Numeric 40m Theory 40m Theory 60m
500
600
700
Numeric 60m
Model 20m
(a) Transient profiles of EAHE outlet air temperatures. 30.0 27.5
Temperature ( )
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100
200
300
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Inlet Numeric 20m Numeric 40m Theory 40m Theory 60m
500
Numeric 60m
600
Theory 20m
(b) Transient profiles of EAHE buried-pipe wall temperatures. Fig. 3. Theoretical and numerical simulation results compared. 5
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results. Both the fluctuation amplitudes and phase shifts obtained from the proposed theoretical approach agree well with the numerical simulation results. Except in the initial stage, the maximum relative error between the theoretical results and the simulated results is 3%, indicating that the model is sufficiently accurate to analyze the system's thermal performance.
60 m × 1 m × 0.1 m was buried inside the computational domain of the soil at a depth of 4 m. The thermal inertia of the buried pipe is significantly smaller than that of the soil surrounding the pipe, and the thermal conductivity of the buried pipe is generally larger than that of the soil. The heat transfer resistance caused by the pipe wall is much lower than that of the soil. Thus, the effects of pipe thickness are neglected in the simulations. The side boundaries of the soil surface were at 10 m, which is ten times the width of the buried pipe; therefore, the temperature distribution was considered undisturbed at these boundaries. These side boundaries were set as adiabatic walls. The pipe's inner surface was set as fluid–solid coupled surfaces to account for the convective heat transfer at the wall surfaces. The pipe inlet was set as an inlet-velocity boundary condition, and the inlet velocity was a constant 1 m/s. The pipe outlet was treated as a pressure outlet. Meteorological data for Chongqing (a hot-summer/cold-winter region) indicate an annual average air temperature of 17.84 °C, with fluctuation amplitude 10.1 °C. The soil's thermal conductivity is λs = 1.1 W/(m·K) , and its thermal diffusivity is αs = 7. 1 × 10−7 m2/s [36]. The boundary conditions for both at the upper ground surface and the temperature variation at the pipe inlet were set via the user-defined functions. The temperature for the bottom surface of the computational domain, i.e., z = −8 m, is set to follow a transient profile, which is calculated using Eq. (2). Because [36] provides hourly data for outdoor air temperature, the time step in the simulations was set as 1 h (3600 s). The number of hexahedral grid cells was 7,518,021, and grid sensitivity tests were performed. According to Eq. (10), the Reynolds number is approximately 11,806 and the flow inside the pipe is fully turbulent. Thus, the standard k–ε model was employed to account for the turbulence effects [17,18,20,37,38]. The SIMPLE scheme was used for the pressure–velocity coupling. The calculations were considered converged when the residuals for mass, momentum, and energy between two consecutive iterations were under 10−3, 10−3, and 10−6, respectively.
4. Comparison of the effects of flat rectangular EAHE with those of circular EAHE In this section, we use theoretical analysis to compare the thermal performance of the flat rectangular and the circular EAHEs. The flat rectangular EAHE results are from the theoretical model (Section 2); the circular EAHE results are from the model of [13]. For Sections 4.1–4.4, the cross section of the flat rectangular pipe is 1 m × 0.1 m, and the radius of the circular pipe is 0.18 m, so that their cross-sectional areas are equal. For Sections 4.1–4.4, the input engineering parameters for the EAHE (burial depth, airflow velocity, length) are identical to those in Section 3.1, as are the climate parameters and the thermal properties of the soil. 4.1. Air temperature Fig. 4 shows EAHE air temperature variations for the flat rectangular and circular cross sections at various buried-pipe lengths. It is seen that the fluctuation amplitudes of the air temperature of the flat rectangular EAHE at 20, 40, and 60 m are lower than those of the circular EAHE and that the phase shifts of the air temperature of the flat rectangular EAHE at 20, 40, and 60 m are larger than those of the circular EAHE. The fluctuation amplitudes of the air temperature for the flat rectangular EAHE are 1.37 °C, 1.91 °C, and 2.49 °C lower than those of the circular EAHE at 20, 40, and 60 m, respectively, representing reductions of 13.7%, 24.7%, and 24.7%, respectively. In addition, the phase shifts for the flat rectangular EAHE were 4.64, 12.21, and 22.67 days larger than those of the circular EAHE at 20, 40, and 60 m, respectively.
3.2. Model validation Fig. 3 shows the transient profiles of the outlet air temperature and the buried-pipe wall temperature for various pipe lengths obtained from the numerical simulation and the proposed theoretical model. The theoretical air temperatures and buried-pipe wall temperatures at 20, 40, and 60 m are in close agreement with the numerical simulation
24
Rectangular 60m
Circular 60m
Rectangular 40m
Circular 40m
4.2. Buried-pipe wall temperature Fig. 5 shows the buried-pipe wall temperature variations for the flat
20
Temperature ( )
16 12 24 20 16 12 Inlet
30
Rectangular 20m
Circular 20m
24 18 12 6 0
100
200
300
400
500
600
Time (day) Fig. 4. Comparison of air temperatures for flat rectangular and circular EAHEs in annual cycle. 6
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24
Rectangular 60m
Circular 60m
Rectangular 40m
Circular 40m
21 18
Temperature ( )
15 24 21 18 15 12 Inlet
30
Rectangular 20m
Circular 20m
24 18 12 6 0
100
200
300
400
500
600
700
Time (day) Fig. 5. Comparison of buried-pipe wall temperatures for flat rectangular and circular EAHEs in annual cycle.
4.3. Cooling/heating capacity provision and COP
rectangular and circular EAHEs at various EAHE lengths. It can be seen that the fluctuation amplitudes of the flat rectangular EAHE wall temperature were 1.73 °C, 1.36 °C, and 1.02 °C lower than those of the circular EAHE at 20, 40, and 60 m, respectively. In contrast, the phase shifts of the flat rectangular EAHE wall temperature were 9.88, 18.02, and 26.15 days larger than those of the circular EAHE at 20, 40, and 60 m, respectively. This is because the thermal disturbance range in the soil for the flat rectangular pipe is smaller than that for the circular pipe, resulting in the temperature wave of the soil around the buried pipe decaying faster and creating a more stable pipe wall temperature. Therefore, the flat rectangular EAHE has a relatively stable wall temperature, which is beneficial for heat transfer performance.
70
Rectangular EAHE
The EAHE has length 60 m, airflow velocity 3 m/s, and burial depth 4 m. Fig. 6(a) shows the variations in outlet air temperature for the flat rectangular and circular EAHEs in annual cycle. The fluctuation amplitudes and phase shift of the flat rectangular EAHE outlet air temperature are lower and larger than those of the circular EAHE, respectively. This indicates that the thermal performance of the flat rectangular EAHE is better than the circular EAHE's at the same crosssectional area. Fig. 6(b) shows the variation in cooling/heating capacity (Eq. (29)) provided by the flat rectangular and circular EAHEs in annual cycle. It is observed that the cooling/heating of the flat rectangular EAHE is much stronger than that of the circular EAHE. Fig. 6(c) shows (c)
Circular EAHE
COP
56 42 28
0
Rectangular EAHE
Circular EAHE
(b)
(W)
1800 0
-1800
Temperature ( )
Cooling/heating capacity
14
Inlet
Outlet air temperature of rectangular EAHE
Outlet air temperature of circular EAHE
(a)
30 24 18 12 6 0
100
200
300
400
500
600
700
Time (day) Fig. 6. Comparison of cooling/heating capacity and COP for flat rectangular and circular EAHEs in annual cycle. 7
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the variation in the COP value (Eq. (30)) of the flat rectangular and circular EAHEs in annual cycle. It is seen that the COP values of the flat rectangular EAHE is less than that of the circular EAHE. This is because the pressure drop and fan power consumption of the flat rectangular EAHE is greater than that of the circular EAHE because of the former's narrow airflow space.
boundary conditions were the same as those given in Section 3.1. Fig. 7 shows the winter and summer temperature distributions for the circular and flat rectangular EAHEs at the cross-section at Y = 5 m. The results indicate that as the soil depth Z increases, the temperature waves of the soil layer gradually stabilize and the temperature waves do not reach the bottom surface boundary of the soil domain, which indicates that the computational domain of the soil is sufficiently large. Owing to the strong thermal inertia of the soil, the temperature distribution of the soil layer has obvious stratification by depth. Therefore, soil temperatures at the pipe depth are higher/lower than the outdoor air temperature in summer/winter, which provides an opportunity to utilize the temperature difference between the soil and the flowing air. In addition, the soil temperature is essentially constant at Z = −4 m. From Fig. 7, it is seen that the thermal disturbance range in the
4.4. Discussion of reasons for the differences between flat rectangular EAHE and circular EAHE To understand the thermal performance differences between the flat rectangular EAHE and the circular EAHE, we used the same numerical simulation method as described in Section 3.1 to study the temperature distributions for the two types of EAHE. The input parameters and
(a) Circular EAHE.
(b) Flat rectangular EAHE. Fig. 7. Temperature distribution at Y = 5 m. 8
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acceptable. Fig. 8 also shows that the range of disturbed soil for the flat rectangular pipe is much shorter than that for the circular pipe in the length direction. As the pipe length increases, the air temperature in the flat rectangular pipe is higher/lower than that of the same-length circular pipe in winter/summer. Fig. 9 shows the temperature distributions for the circular and flat rectangular EAHEs at the cross section at X = 10 m. It indicates that the range of disturbed soil around the flat rectangular pipe is much smaller than that for the circular pipe, which has a major effect on the design of EAHE pipe spacing for multi-tube arrangements. Therefore, the flat rectangular EAHE can improve the temperature wave of the soil around the buried pipe, providing a faster decay rate and creating a more stable
surrounding soil for the flat rectangular pipe is less than that of the circular pipe in both the depth direction and the length direction. As the pipe length increases, the air temperature in the flat rectangular pipe is higher/lower than that of the same-length circular pipe in winter/ summer. Fig. 8 shows the winter and summer temperature distributions for the circular and flat rectangular EAHEs at a burial depth of Z = −4 m. It shows that the temperature waves emitted from the EAHE do not reach the side boundaries of the soil domain. In the width direction, the thermal disturbance range in the soil temperature for the rectangular pipe is much smaller than that for the circular pipe, which demonstrates that neglecting the heat transfer process in the width direction is
(a) Circular EAHE.
(b) Flat rectangular EAHE. Fig. 8. Temperature distribution at Z = −4 m. 9
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(a) Circular EAHE.
(b) Flat rectangular EAHE.
Fig. 9. Temperature distribution at X = 10 m.
pipe wall temperature.
is larger.
4.5. The key factors influencing the differences between the effects of flat rectangular EAHE and those of circular EAHE
4.5.3. Effect of airflow velocity in pipe For this section, the EAHE has length 100 m and burial depth 3 m. Fig. 12 shows that as the airflow velocity increases, the phase shifts of the flat rectangular and circular EAHEs initially increase to the highest point and then decrease. The normalized fluctuation amplitudes of both EAHE types first decrease and then increase with the increase in airflow velocity, which indicates that there is no advantage in increasing the airflow velocity for improving the heat transfer performance of the rectangular or the circular EAHE. When the airflow velocities in the EAHE pipes are the same, the normalized fluctuation amplitude of the flat rectangular EAHE is smaller than that of the circular EAHE, and its phase shift is larger.
In this section, the effects of the main engineering parameters on the outlet air temperature of the flat rectangular and circular EAHEs are investigated. In Sections 4.5.1–4.5.3, the cross section of the flat rectangular pipe is 1 m × 0.1 m and the radius of the circular pipe is 0.18 m, so that their cross-sectional areas are equal. In Section 4.5.4, although the cross-sectional areas change, the rectangular EAHE and the circular EAHE cross-sectional areas remain equal. 4.5.1. Effect of pipe length For this section, the EAHE has burial depth 3 m and airflow velocity 1.5 m/s. Fig. 10 shows that the normalized fluctuation amplitudes of both EAHE types decrease with the pipe length, decreasing faster at the EAHE entry portion and then slowing down. In contrast, the phase shift of the outlet air temperature of the flat rectangular and circular EAHEs increases with the length, increasing faster at the EAHE inlet section and then increasing less and less. This is mainly because the temperature difference between the air and the surrounding soil at the EAHE inlet is large and the heat flux is high. 4.5.2. Effect of pipe's burial depth For this section, the EAHE has length 100 m and airflow velocity 1.5 m/s. Fig. 11 indicates that the phase shifts of both EAHE types initially increase to the highest point and then decrease with an increase in burial depth. The normalized fluctuation amplitudes of the flat rectangular and circular EAHEs decrease with the increase in EAHE burial depth and then stabilize. The phase shifts differ between the flat rectangular and circular EAHEs at the highest point, and the normalized fluctuation amplitudes also differ when the stable value is reached. However, the phase shift of the flat rectangular EAHE reaches a much larger maximum value than the circular EAHE. When the burial depths are the same, the normalized fluctuation amplitude of the flat rectangular EAHE is smaller than that of the circular EAHE, and its phase shift
1.0
1.2
0.9
1.0
0.8 0.8
0.7 0.6
0.6
0.5
0.4
0.4 0.2
0.3 0.2
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20
40
60
80
100
120
Length (m)
Normalized fluctuation amplitude of circular EAHE Normalized fluctuation amplitude of rectangularEAHE
140
160
180
0.0 200
Phase shift of circular EAHE Phase shift of rectangular EAHE
Fig. 10. Effect of pipe length. 10
Phase shift
Normalized fluctuation amplitude
4.5.4. Effect of pipe's cross-sectional area For this section, the EAHE has length 100 m, burial depth 3 m, and airflow velocity 1.5 m/s. Fig. 13 shows that the phase shifts of the flat rectangular and circular EAHEs decrease with an increase in the crosssectional area of the pipe and that the normalized fluctuation
Applied Thermal Engineering 162 (2019) 114262
1.0
2.0
0.9
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indicate the following:
• The outlet air temperature fluctuation of the flat rectangular EAHE Phase shift
Normalized fluctuation amplitude
H. Wei and D. Yang
• •
0.0
Phase shift of circular EAHE Phase shift of rectangular EAHE
is 24.7% lesser and its phase shift is 22.67 days longer than that of circular EAHE with the same cross-sectional area. The soil's thermal disturbance range of the flat rectangular EAHE is shorter than that of circular EAHE, and resulting in a temperature wave in the soil around the buried pipe that decays faster and creates a more stable pipe wall temperature. Although the cooling/heating capacity of the flat rectangular EAHE is greater than that of the circular EAHE, the pressure drop and fan power consumption of the flat rectangular EAHE are greater than that of the circular EAHE as a result of the former's narrow airflow space, Therefore, the COP values of the flat rectangular EAHE is less than that of the circular EAHE.
1.0
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Acknowledgments
0.2
0.3 0.2
This work proves that the flat rectangular cross-section of EAHE can markedly reduce the outlet air temperature fluctuation, extend the phase shift and improve the cooling/heating capacity of EAHE although the flat rectangular EAHE has a lower COP, which implies that the pipe cross-sectional shape is also an important influencing factor on the performance of EAHE. It is recommended that the flat rectangular pipes use the high-strength stainless steel, cast iron or precast concrete material, or make use of the reinforcement method inside the pipe to avoid deformation.
Phase shift
Normalized fluctuation amplitude
Fig. 11. Effect of burial depth.
0
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The authors acknowledge support from the National Natural Science Foundation of China (NSFC) under Grant No. 51578087, the Projects Nos. 2017M622967 and 2018T110945 funded by the China Postdoctoral Science Foundation, 2019CDXYCH0027, 2018CDXYCH0015 and CQU2018CDHB1B06 supported by the Fundamental Research Funds for the Central Universities.
0.0
Phase shift of circular EAHE Phase shift of rectangular EAHE
Normalized fluctuation amplitude of circular EAHE Normalized fluctuation amplitude of rectangularEAHE
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1.2
Appendix A. Supplementary material
0.9 1.0
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Supplementary data to this article can be found online at https:// doi.org/10.1016/j.applthermaleng.2019.114262.
Phase shift
Normalized fluctuation amplitude
Fig. 12. Effect of airflow velocity.
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Fig. 13. Effect of cross-sectional area.
amplitudes increase with the increase in the cross-sectional area. When the cross-sectional areas are equal, the normalized fluctuation amplitude of the flat rectangular EAHE is smaller than that of the circular EAHE. However, its phase shift is larger. 5. Conclusions This paper develops a mathematical model for evaluating the thermal performance of flat rectangular earth-to-air heat exchangers in harmonically fluctuating thermal environment. Unlike the conventional EAHE, the flat rectangular buried pipe has a width-to-height ratio of 10 or greater. In addition, the model was validated by three-dimensional CFD numerical simulations. Then, studies comparing the flat rectangular and conventional circular EAHEs were conducted. Our results 11
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