1. Introduction
The diffusion of ground-source heat pumps, and in particular, of ground-coupled heat pumps (GCHPs), is rapidly increasing. Indeed, GCHPs are a very efficient technology for the climatization of buildings and the production of domestic hot water [
1]. The performance of these systems has been analyzed by experiments and by simulation tools [
2,
3,
4,
5,
6,
7]. GCHPs usually utilize buried vertical heat exchangers, named borehole heat exchangers (BHEs), that are mostly composed of a single or double polyethylene U-tube, installed in a drilled hole subsequently filled with a grouting material. A BHE has a length that commonly ranges from 50 to 200 m, and a diameter of about 15 cm. Improving the BHE-field design is an important way to enhance the efficiency of GCHP systems.
Several models for a BHE-field design have been proposed [
8,
9,
10,
11,
12]. Most methods adopt dimensionless factors of thermal-response, also named
g-functions. The
g-function of a borefield yields the time-dependent dimensionless mean temperature of the boundary surface of the borefield caused by a constant total power released by the BHEs. Different methods to obtain the
g-function of a borefield have been developed, either assuming that the heat flux released by the BHEs is uniform along the surface between BHEs and soil [
13,
14,
15,
16,
17], or assuming that the total heat flux released by the BHEs is constant in time and the boundary between the ground and the borefield has a uniform temperature [
18,
19,
20,
21]. More accurate thermal response factors have also been determined, by assuming time-constant total heat flux and same inlet temperature in all the BHEs, and considering the influence of the borehole thermal resistance on the borefield surface temperature [
22,
23].
The
g-functions or thermal response factors determined by the methods cited above are not precise in the short term, i.e., during the first one or two hours of operation, because the models employed do not consider accurately the thermal inertia of the BHE. Moreover, they yield the surface averaged time-dependent temperature of a borefield,
Tsm, but not the time-dependent mean temperature of the fluid,
Tfm. The latter is then determined by the following equation:
where
Rb is the BHE thermal resistance per unit length and
qlm is the mean linear thermal power exchanged between the fluid and the surrounding ground. Equation (1) is correct when the heat transfer in the borehole is quasi-steady, but is not precise in the short term and yields an unphysical jump of
Tfm when
qlm changes from zero to a given constant value.
In order to predict accurately the time evolution of Tfm during the first hours of operation, several researchers developed simulation models that are accurate even in the short term.
De Carli et al. [
24] and Zarrella et al. [
25] developed a Capacity Resistance Model (CaRM) of BHE suitable for the short-term analysis and employable also in the long term. Quaggiotto el al. [
26] applied the CaRM model proposed in [
25] for a numerical and experimental comparison between coaxial and double U-tube BHEs. Other resistance and capacity models were presented by Bauer et al. [
27] and by Pasquier and Marcotte [
28]. Ruiz-Calvo et al. [
29] proposed to separate the short-term and the long-term simulation and developed a short-term model based on that by Bauer et al. [
27]. Li and Lai [
30,
31] developed a 2D analytical BHE model where the tubes are schematized as infinite line sources that supply a uniform and constant linear heat flux. Zhang et al. [
32] presented a transient quasi-3D line source model that introduces the concept of transient borehole thermal resistance and gives a full-time-scale thermal response.
Beier and Smith [
33] proposed an analytical cylindrical BHE model, where the borehole is represented by a grout annulus with external radius equal to that of the borehole, and internal radius such that the grout annulus has a thermal resistance equal to that of the borehole.
Xu and Spitler [
34] developed a numerical model that approximates the real BHE structure with several concentric cylinders, that include a fluid layer, an equivalent convective-resistance layer, a tube layer, and a grout layer. Man et al. [
35] presented the analytical solution for a simple cylindrical BHE model, where the thermal properties of the BHE materials coincide with those of the ground and the thermal power is supplied by a generating cylindrical surface that represents the fluid. The solution is given both for the 1D scheme, that neglects the heat conduction along the BHE axis, and for the 2D axisymmetric scheme, that takes into account the finite length of the BHE.
Bandyopadhyay et al. [
36] found an analytical solution for a borehole model composed of a high-conductivity solid cylinder with heat generation, representing the fluid, and a grout layer. The solution was given in the Laplace transformed domain, and the authors employed a numerical inversion to determine the thermal response in the time domain. Javed and Claesson [
37] developed a complete analytical solution of the borehole model employed by Bandyopadhyay et al. [
36]. Claesson and Javed [
38] proposed an analytical method to determine the thermal response of a borehole, valid both in the short term and in the long term, by coupling the short-term model presented in Javed and Claesson [
37] to a long-term model based on the finite line-source solution.
Beier [
39] developed an analytical BHE model that approximates the U-tube as two half tubes, and considers both the fluid and the grout thermal capacity. The analytical solution is given in the Laplace transformed domain and is inverted by using the Stehfest algorithm. Lamarche [
40] developed an analytical cylindrical borehole model in which the borehole is composed of a solid cylinder subjected to heat generation, representing the fluid, surrounded by cylindrical layers representing the polyethylene pipes, and the grout. Naldi and Zanchini [
41] proposed a numerical BHE model (OMEC) composed of a homogeneous equivalent cylinder having an internal heat-generating surface and thermal properties suitable to reproduce both the thermal resistance and the heat capacity of the BHE. By means of that model, the authors determined the full-time-scale evolution of
Tfm for a borefield with BHEs having equal inlet temperatures.
Most of the short-term or full-time-scale simulation models cited above yield directly the time evolution of Tfm, for a borefield subjected to a time constant heat load, without employing Equation (1). However, they do not yield directly the time evolution of the outlet fluid temperature, Tout, that is needed for the simulation of the heat pump. Therefore, it is interesting to complete these models by providing relations between Tfm and Tout.
Beier and Spitler [
42] presented a method that allows determining the dimensionless factor
f, given by:
where
Tin is the inlet fluid temperature. Equation (2) and the energy balance equation:
where
is the thermal power supplied by the heat pump to the borehole fluid,
and
cp are the mass flow rate and specific heat capacity of the fluid, yield:
Through 3D finite element simulations and best fit of simulation results, Zanchini and Jahanbin [
43] determined simple correlations to evaluate the dimensionless coefficient
φ defined as:
where
is the volume flow rate,
is a reference value of
, namely 12 L per minute (L/min), and
Tave = (
Tin +
Tout)/2. The correlations reported in [
43] apply to double U-tube boreholes. Equation (5) and the balance Equation (3) yield:
In the present paper, new correlations to determine
φ are provided, for single U tube BHEs, through the best fit of the results of 3D numerical simulations. These correlations can be employed to obtain an accurate evaluation of the time evolution of
Tout by means of a BHE simulation code that yields the time evolution of
Tfm. An example of this use is reported, in the case of constant flow rate and constant power supplied to the ground, by evaluating the time evolution of
Tfm through the simple analytical BHE model proposed by Man et al. [
35], and that of
Tout through Equation (6). Then it is shown that our correlations for
φ can be applied to determine accurately the time evolution of
Tout from that of
Tfm even in the simulation of BHE fields subjected to a time dependent heat load. Finally, it is shown that the correlation for
φ valid for the quasi-stationary regime can be employed for an immediate calculation of the effective BHE thermal resistance.
2. Simulation Cases and Method
We considered nine BHE geometries, with BHE diameter
Db = 152 mm, shank spacing
d = 84, 94 and 104 mm, length
L = 50, 100 and 200 m, pipes with internal diameter
Dpi = 32.6 mm and external diameter
Dpe = 40 mm. Sketches of the BHE cross sections considered are illustrated in
Figure 1.
For each geometry, we considered three grout thermal conductivities, 1.0, 1.6 and 2.3 W/(mK), two flow rates, = 12 and 24 L/min, and ground thermal conductivity kg = 1.8 W/(mK) (typical value). The following values were adopted for the thermal properties nearly uninfluential on φ: polyethylene thermal conductivity and volumetric heat capacity kp = 0.4 W/(mK) and (ρ c)p = 1.824 MJ/(m3K); grout and ground volumetric heat capacities (ρ c)gt = 1.600 MJ/(m3K) and (ρ c)g = 2.500 MJ/(m3K). We examined the cooling operation, with inlet fluid temperature 32 °C. Thus, 54 finite element simulations were performed to determine the correlations for φ. Additional simulations were performed to validate the simulation code, as well as to check the validity of the correlations for other values of Db and of kg and for other working conditions.
Water has been considered as working fluid. The water thermal properties at
Tin have been taken from NIST [
44]: density
ρw = 995.03 kg/m
3, dynamic viscosity
μw = 0.76456 mPa s, specific heat capacity
cpw = 4179.5 J/(kg K), thermal conductivity
kw = 0.61869 W/(mK). The water velocity was considered vertical and uniform. A heat flux per unit area given by the product of the convection coefficient and the temperature difference between fluid and solid surface was applied at the fluid-solid interface. The Reynolds, Nusselt and Prandtl numbers, and the heat transfer coefficient
h are given in
Table 1. The Nusselt number was calculated through the Churchill correlation with uniform wall heat flux [
45].
The ground surrounding the borehole has been represented as a cylinder coaxial with the borehole, having radius 10 m and 10 m longer than the BHE. The initial temperature of the ground and of the BHE has been set equal to the undisturbed ground temperature,
Tg. The latter has been supposed equal to 14 °C for
z = 10 m, with geothermal gradient 0.03 °C/m for
z > 10 m. The following distribution of
Tg(
z) has been assumed for
z < 10 m:
An adiabatic boundary condition has been imposed at the lateral and bottom ground surfaces and at the top of the borehole. The boundary condition Tg(0) = 24 °C has been applied at the horizontal ground surface.
Numerical simulations for a working period of 100 h have been carried out by a 3D finite element model, through COMSOL Multiphysics. The working period selected is more than sufficient to reach a quasi-stationary heat transfer regime in the BHE and to obtain abundant data of φ in this regime.
The reduced vertical coordinate
has been introduced to shorten the computational domain along
z [
46]. Consequently, a reduced thermal conductivity along
z,
, has been employed for every material; in addition, a reduced vertical water velocity
has been assumed. A more detailed description of the method can be found in Refs. [
43,
46]. The reduction coefficient
c has been taken equal to 5 for BHEs with length 50 m, equal to 10 for BHEs with length 100 m, and equal to 20 for BHEs with length 200 m.
In COMSOL Multiphysics, the time steps are non-uniform and are optimized by the software so that the solution matches the accuracy parameters imposed by the user. We have selected absolute tolerance 0.0001 and relative tolerance 0.001, instead of the default values 0.001 and 0.01, respectively.
For each geometry, the computational domain has been meshed with unstructured tetrahedral elements. Due to the rescaling coefficient, the selected mesh, illustrated in
Figure 2, is independent of the BHE length, and has 1,369,572 elements for
d = 84 mm, 1,444,394 elements for
d = 94 mm, and 1,510,647 elements for
d = 104 mm.
In order to ensure that the results are mesh independent, simulations have been carried out by employing meshes with 1,198,027, 1,311,663 and 1,444,394 tetrahedral elements, for L = 100 m, d = 94 mm, kgt = 1.6 W/(mK), kg = 1.8 W/(mK), and = 12 L/min. The values of Tave − Tfm determined at τ = 2 h, 20 h and 100 h from the operation start have been compared, and the maximum percent deviation from the values obtained with the third mesh, employed in the final simulations, has been 0.127%.
4. Validation of the 3D Simulation Code
The validation of the 3D simulation code has been performed by comparison between the time-dependent values of
Tfm evaluated through this code and those calculated analytically through the BHE model proposed by Man et al. [
35]. The BHE selected for the comparison has
L = 100 m,
d = 94 mm,
kgt = 1.6 W/(mK), (
ρ c)
gt = 1.600 MJ/(m
3K), volume flow rate 18 L/min, is placed in a ground with
kg = 1.8 W/(mK) (
ρ c)
g = 2.500 MJ/(m
3K),
Tg = 14 °C, and receives a time constant linear power of 60 W/m, i.e., to a total thermal power
= 6000 W. The initial temperature coincides with
Tg(
z) in the whole domain. The properties of water are evaluated at 20 °C and are [
44]
ρw = 998.21 kg/m
3,
μw = 1.0016 mPa s,
cpw = 4184.1 J/(kg K),
kw = 0.59846 W/(mK). The convection coefficient, calculated by the Churchill correlation [
45], is
h = 1800.5 W/(m
2K).
In the 3D simulation code, the condition of constant thermal power
= 6000 W has been implemented by imposing the constant value of
Tin −
Tout given by Equation (3), namely 4.789 °C. The mesh is that illustrated in
Figure 2.
In the 2D axisymmetric BHE model proposed by Man et al. [
35], the BHE is represented by a cylindrical surface with radius
r0 that releases a constant and uniform power per unit area corresponding to the total thermal power received by the BHE. The generating surface represents the fluid, and the borehole has the same properties as the ground. The analytical solution for the temperature field is determined by the Green’s function method.
The mean temperature of the surface with radius
r0, that will be denoted by
Tfm, is given by [
35]:
where
is the thermal diffusivity of the ground, equal to 0.72 × 10
−6 m
2/s in the case considered.
The value of
r0 to be employed in the model has been determined by imposing that the thermal resistance of the cylindrical layer between
r0 and the BHE radius is equal to the BHE thermal resistance. The latter has been determined through a stationary 2D finite element simulation of a borehole cross section that includes a ground layer having radius 2 m and an isothermal external surface. The temperature difference between the fluid and the external ground surface has been set equal to 25 °C for one pipe and to 20 °C for the other pipe. The convection coefficient is
h = 1800.5 W/(m
2K). We adopted a very fine mesh, composed of 139,008 triangular elements. A particular of the mesh employed is illustrated in
Figure 5. The result is
Rb = 0.09863 mK/W and, as a consequence,
r0 = 2.491 cm.
The diagrams of
Tfm versus the decimal logarithm of time in hours obtained by the 3D finite element simulation and by the numerical integration of Equation (17) are compared in
Figure 6, in the time range between 10
−2 h and 10
2 h. The mean square deviation between the results of the finite element simulation and those obtained by the model by Man et al. [
35] is 0.26 °C. The small discrepancies that occur from 10
−2 to 10
−1 h are very probably due to the non-perfect accuracy of the analytical model, that does not consider the exact total value and distribution of the BHE heat capacity. On the contrary, those occurring from 10 to 10
2 h are probably due to numerical errors in the 3D simulation, that increase with time.
5. Validity of the Correlations for Other BHE Diameters, Thermal Conductivities of the Ground, Working Conditions
Although the correlations for
φ reported in
Section 3 were obtained by assuming
Db = 152 mm,
kg = 1.8 W/(mK), and summer operation with constant
Tin, they hold also for other BHE diameters, ground thermal conductivities, and working conditions.
The applicability of the values of
φ∞ reported in
Table 2 to other borehole diameters is shown in
Figure 7, that refers to BHEs with
L = 100 m,
d = 84 mm,
kgt = 1.0 W/(mK),
kg = 1.8 W/(mK),
= 12 L/min, and BHE diameters 134 mm and 170 mm. Clearly, the value of
Db has no effect on
φ if
kgt =
kg. Therefore, the value
kgt = 1.0 W/(mK) has been selected, to analyze a critical condition, with a high ratio
kg/
kgt. Higher values of this ratio should be avoided. The diagrams of
Tave −
Tfm versus
obtained by the 3D simulations with these BHE diameters are compared with the line obtained by employing the value of
φ∞ reported in
Table 2, namely
φ∞ = 0.07. The comparison shows the applicability of the correlation to other BHE diameters.
The applicability of the values of
φ∞ reported in
Table 2 to other ground thermal conductivities is shown in
Figure 8, that refers to BHEs with
L = 100 m,
d = 94 mm,
kgt = 1.6 W/(mK),
= 12 L/min, ground thermal conductivities 1.4 and 2.2 W/(mK). The diagrams of
Tave −
Tfm versus
obtained by the 3D simulations with
kg = 1.4 and 2.2 W/(mK) are compared with the line obtained by employing the value of
φ∞ listed in
Table 2, namely
φ∞ = 0.0847. The results reveal that the correlation can be applied to other values of the ground conductivity.
The applicability of the values of
φ∞ reported in
Table 2 to other working conditions is shown in
Figure 9, that refers to thermal response tests (TRTs) performed on a borehole with
L = 100 m,
d = 94 mm,
kgt = 1.6 W/(mK),
kg = 1.8 W/(mK). The working conditions are
= 12 L/min and
ql = 50 W/m for the first TRT,
= 18 L/min and
ql = 60 W/m for the second TRT,
= 24 L/min and
ql = 80 W/m for the third TRT. In a TRT,
Tin −
Tout is a constant and
Tave −
Tfm becomes constant after 2 h. Therefore, only one value of
Tave −
Tfm as a function of
is obtained for each TRT, in the quasi-stationary regime. In
Figure 9, the values of
Tave −
Tfm as a function of
obtained for the TRTs described above are compared with the line obtained by employing the value of
φ∞ reported in
Table 2. The points that represent the results for the TRTs lay on the correlation line.
7. Calculation of Rb,eff Through φ∞
In this section we show that, in quasi-stationary working conditions, the effective BHE thermal resistance, Rb,eff, can be easily calculated by means of the dimensionless coefficient φ∞. Then, we compare the values of Rb,eff evaluated through our general correlation for φ∞, given by Equation (11), with those determined by 3D finite element computations and by Hellström’s analytical solution.
One can rewrite Equation (23) as:
In quasi-stationary working conditions, the energy balance of the fluid yields:
By substituting Equation (26) in Equation (25), one gets:
In quasi-stationary conditions, Equation (5) yields:
By substituting Equation (28) in Equation (27), one obtains:
Equations (29) and (11) allow a very simple calculation of the difference between
Rb,eff and
Rb, for single U-tube BHEs. Equation (29) can be employed also in combination with Equation (27) of Ref. [
43], for double U-tube BHEs. The BHE thermal resistance
Rb can be calculated either by employing a suitable approximate expression, or, more accurately, through a 2D numerical simulation of a borehole cross section.
A comparison between the values of
Rb,eff obtained by Equations (29) and (11), denoted by
Rb,eff,φ, those evaluated through 3D numerical computations, denoted by
Rb,eff,3D, and those obtained by Equations (18) and (19), denoted by
Rb,eff,H, is illustrated in
Table 4, for single U-tube BHEs having
L = 100 m or 200 m,
Db = 152 mm,
kg = 1.8 W/(mK),
= 12 L/min and inlet temperature
Tin = 32 °C. The convection coefficient is
h = 1462.2 W/(m
2K), as indicated in
Table 1. The values of
Ra and of
Rb to be employed in Equations (18) and (19) have been determined by performing stationary 2D numerical simulations of a borehole cross section with the method described in
Section 4 and evaluating
Ra with Equation (20).
The table shows that the values of Rb,eff evaluated through our correlation for φ∞ are in fair agreement both with those computed directly by 3D numerical computations and with those yielded by Hellström’s analytical method. The highest percent deviation from Rb,eff,3D is −1.46% for Rb,eff,φ and −1.81% for Rb,eff,H; the mean square deviation from Rb,eff,3D is 0.00093 mK/W for Rb,eff,φ and 0.00119 mK/W for Rb,eff,H.
The values of
Rb,eff,φ reported in
Table 4 have been determined by calculating
φ∞ through Equation (11). If one employs the values of
φ∞ reported in
Table 2, the highest percent deviation from
Rb,eff,3D reduces to −1.05% and the mean square deviation reduces to 0.00086 mK/W.