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Improving the thermal performance of coaxial borehole heat exchangers

E. Zanchini*, S. Lazzari, A. PriaroneDipartimento di Ingegneria Energetica, Nucleare e del Controllo Ambientale, Alma Mater Studiorum - Universita di Bologna, Viale Risorgimento 2, I-40136, Bologna, Italy

a r t i c l e i n f o

Article history:Received 29 July 2009Received in revised form29 October 2009Accepted 29 October 2009Available online 19 November 2009

Keywords:Energy savingGround coupled heat pumpsBorehole heat exchangersFinite-element simulations

a b s t r a c t

Borehole heat exchangers are the fundamental component of ground coupled heat pumps, which arenow widely employed for energy saving in building heating and cooling. The improvement of thethermal efficiency of Coaxial Borehole Heat Exchangers (CBHEs) is pursued in this paper by investigatingthe effects of thermal short-circuiting and of flow rate, as well as of the constituent materials and of thegeometrical configuration of the CBHE cross section. The analysis is performed by means of finite-element simulations, implemented through the software package COMSOL Multiphysics. The real 2-Daxisymmetric unsteady heat transfer problem is modelled, for both winter and summer workingconditions, by considering CBHEs with a length of 100 m placed either in a high conductivity or in a lowconductivity ground. The results point out that the effects of flow rate and of thermal short-circuiting areboth important, and that the latter can be reduced considerably by employing a low conductivitymaterial, such as PPR80, for the inner tube. Finally, it is shown that the performance of the CBHE could beimproved, with respect to the commonly used geometry, by increasing the diameter of the inner tubewhile leaving the outer tube unchanged.

� 2009 Elsevier Ltd. All rights reserved.

1. Introduction

On account of the global climate change and of the reduction ofthe world oil and gas reserves, techniques for a rational use ofenergy resources are employed all over the world. In this frame-work, ground coupled heat pumps, based on borehole heatexchangers (BHEs), are widely adopted to reduce primary energyuse in building heating and cooling. Therefore, a broad researchactivity on the analysis and the improvement of the efficiency ofBHEs is being developed.

BHEs for ground coupled heat pumps may have three differentmain geometries: a single U-bent polyethylene tube, placed ina borehole which is then grouted; two U-bent polyethylene tubes,placed in a borehole which is then grouted; two coaxial circulartubes, typically an internal polyethylene tube and an externalstainless steel tube, either inserted in a borehole which is thengrouted or directly driven into the soil, without drilling. Forcoaxial heat exchangers directly driven into the soil, the lengthdoes not exceed 20 m and the external diameter is about 50 mm.In order to install longer and thicker coaxial heat exchangers,the traditional method based on drilling and grouting must beemployed.

The total length of the BHEs necessary for a plant is usuallydetermined by a method recommended by ASHRAE [1], which wasproposed by Ingersoll and Zobel [2] and adjusted by Kavanaugh [3].The final expressions for the total length require the knowledge ofthe effective thermal conductivity of the soil, kgd, of the effectivethermal diffusivity of the soil, agd, and of the borehole thermalresistance per unit length, Rb. In order to determine kgd, agd and Rb,a Thermal Response Test (TRT) is performed, according to theprocedure prescribed by ASHRAE [1].

The research on BHEs carried out in the last decade concernseither detailed investigations of the thermal behaviour of a singleBHE or studies of the long-term performance of a BHE field. Somerelevant references, for the second kind of studies, are as follows.

Signorelli et al. [4] simulate the long-term behaviour of an arrayof six double U-tube BHEs working only in heating mode. Theyconsider an operation period of 30 years and determine thetemperature decrease which occurs in the ground. Trillat-Berdalet al. [5] propose the coupling of thermal solar collectors with BHEs,to maintain a stable temperature in the ground when the heatingload is higher than the cooling load. Fan et al. [6,7] study numeri-cally the performance of a BHE field for a building in SouthernChina, where the annual cooling load is higher than the heatingload. With the aid of a cooling tower, they design the groundcoupled heat pump system in a way such that an almost completecompensation between the energy extracted and that supplied tothe ground during one year is obtained.

* Corresponding author. Tel.: þ39 51 2093295; fax: þ39 51 2093296.E-mail address: enzo.zanchini@unibo.it (E. Zanchini).

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0360-5442/$ – see front matter � 2009 Elsevier Ltd. All rights reserved.doi:10.1016/j.energy.2009.10.038

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Some relevant references, for the first kind of studies, are listedbelow.

In Ref. [8], a 2-D finite-volume numerical method for thesimulation of unsteady heat transfer between a single U-tube BHEand the soil is presented. The method is validated by comparisonwith analytical models. In Ref. [9], the authors report the experi-mental results of some TRTs as well as the results of numericalevaluations of the TRTs, performed by the method proposed inRef. [8]. The numerical evaluation allows the authors to determinethe thermal conductivities of both ground and grout. In Ref. [10],four methods for the evaluation of TRTs are applied to the samedata set and compared; in each case, the parameters kgd and Rb areevaluated. The methods are: two different applications of the lineheat source method, a cylindrical heat source analytical method[10,11], a 1-D finite-difference numerical method. The authorsconclude that, while the line heat source method and the numericalmethod are in fair agreement with each other, the cylindrical heatsource method overestimates the thermal conductivity of theground and does not seem reliable.

Roth et al. [12] compare two methods for the evaluation of a TRTperformed in Latin America: the analytical evaluation by the lineheat source method and a 1-D numerical evaluation. In thenumerical method, the borehole is sketched as the union of threedomains: an internal water cylinder, a thin cylindrical layer withoutthermal inertia which represents the convective thermal resistance,a cylindrical layer which represents the grout.

In Ref. [13], the authors study the heat transfer within a BHE andexclude the ground. They consider steady-state conditions andassume that the temperature of the external surface of the BHE is

uniform. They neglect the heat conduction in the vertical directionand, by other suitable assumptions, they evaluate analytically thedistribution of the fluid bulk temperature along the channels ofa BHE. They consider both single U-tube and double U-tube BHEs.By means of the obtained temperature distributions, they evaluatethe borehole thermal resistance as

Rb ¼Tm � Twall

ql(1)

where Tm is the mean water temperature, evaluated as the arith-metic mean of the outlet and inlet water temperatures, Twall is thetemperature of the external surface of the BHE, and ql is the powertransferred from the BHE to the surrounding ground per unitlength. The results point out that the thermal resistance of doubleU-tube BHEs is about 30% lower than that of single U-tube BHEs.

Lamarche and Beauchamp [14] propose a new analyticalmethod for modelling BHEs, which can be applied also for shorttime transient responses, and compare the analytical results withsimulations obtained by means of COMSOL Multiphysics. Theborehole is sketched as a water cylinder surrounded by a ring ofgrout, buried into the soil. Two thermal conditions are consid-ered: uniform and constant heat flux at the internal surface ofthe grout; uniform water temperature and convective heat fluxbetween water and grout. The paper refers to single U-tubeBHEs; analytical results are in fair agreement with numericalsimulations.

Marcotte and Pasquier [15] perform 3-D numerical simulationswith COMSOL Multiphysics, with reference to a single U-tube BHE

Nomenclature

a rescaling coefficientA1 cross section area of the inner tube (m2)c specific heat capacity (J kg�1 K�1)gg geothermal gradient (K m�1)H Heaviside unit step functionh1 convection coefficient at the inner tube wall (W m�2 K�1)k thermal conductivity (W m�1 K�1)bK thermal conductivity matrix (W m�1 K�1)l reference length (m)L length of the borehole heat exchanger (m)LD length of the computational domain (m)p1 internal wall perimeter (m)_Q thermal power exchanged between the CBHE and the

surrounding soil (W)ql power per unit length (W m�1)r radial coordinate (m)rD radius of the computational domain (m)r1 internal radius of the inner tube (m)r2 external radius of the inner tube (m)r3 internal radius of the outer tube (m)r4 external radius of the outer tube (m)r5 external radius of the grout layer (m)Rb borehole thermal resistance per unit length (Km W�1)Re Reynolds numberT temperature (K)Tb water bulk temperature (K)Tgd undisturbed ground temperature (K)Tin water inlet temperature (K)Tm ¼ (Tinþ Tout)/2, arithmetic mean water temperature (K)Tout water outlet temperature (K)Ts ground surface temperature (K)

Twall CBHE temperature at the external surface (K)U velocity vector (m s�1)U water velocity in the axial direction (m s�1)Um mean water velocity in the axial direction (m s�1)_V water volume flow rate (m3 s�1)z axial coordinate (m)

Greek symbolsa thermal diffusivity (m2 s�1)g Eulero constantm dynamic viscosity (Pa s)n kinematic viscosity (m2 s�1)r density (kg m�3)s time (s)

Subscriptsa AISI 304 steeleff effectivegd groundgd1 ground 1gd2 ground 2gt groutp at constant pressurepe polyethylene PE100ppr polypropylene PPR80sum summert turbulentw waterwin winter0 initial, for s¼ 0

Superscriptsw rescaled quantity

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which absorbs a constant thermal power from the ground. Theyobtain the time evolution of the fluid bulk temperature distributionalong the tubes. The results show that the true mean fluidtemperature does not coincide with the arithmetic mean fluidtemperature, Tm, and that if Tm is employed instead of the truemean fluid temperature to evaluate Rb by Eq. (1), then Rb is over-estimated. The authors then propose a new expression of the meanfluid temperature for the evaluation of TRTs.

Most of the studies on BHEs concern U-tube BHEs, which,indeed, are the most commonly used. Even for this geometry, theeffect of thermal short-circuiting is seldom considered. In thequoted references, it is taken into account only in Refs. [13,15] and,even there, is not evidenced by comparison with ideal conditions inwhich the effect is prevented by adiabatic partitions. On the otherhand, to account for thermal short-circuiting, a 4–6% increase in theeffective thermal resistance of ground for a daily heat pulse isrecommended by ASHRAE [1], in the use of the expressions forsizing U-tube BHEs piped in parallel, so that the effect seems to berather important. In addition, also coaxial BHEs are employed forground coupled heat pumps, although less extensively. To ourknowledge, data on the effects of thermal short-circuiting forcoaxial ground heat exchangers are available in the literature only

for small size heat exchangers, driven into the soil without drillingand having a length of no more that 20 m [16]. For these heatexchangers, the effect of thermal short-circuiting is not relevant,because, on account of the short length, the difference between theoutlet and inlet water temperatures is very small. However, for longcoaxial BHEs the effect of thermal short-circuiting could beimportant, and techniques to reduce it could be recommendable.

In this paper, the improvement of the thermal efficiency ofCBHEs is pursued by investigating the effects of thermal short-circuiting and of flow rate, as well as of the constituent materialsand of the geometrical configuration of the CBHE cross section.

These effects are studied by finite-element simulations per-formed through the software package COMSOL Multiphysics 3.4.Reference is made to 100 m long coaxial BHEs with an external AISI304 stainless steel tube, having an outer diameter of 88.9 mm anda thickness of 2 mm. A 35 mm thick cylindrical grout layer is placedaround the external tube.

The effect of thermal short-circuiting is studied with reference toa coaxial borehole heat exchanger, denoted by CBHE1, having aninner tube with an external diameter of 50 mm and a thickness of

Table 2Geometrical parameters, properties of solid materials and working conditions.

Symbol Value Quantity

Geometrical data of CBHE1r1 20.4 Internal radius of inner tube [mm]r2 25 External radius of inner tube [mm]r3 42.45 Internal radius of outer tube [mm]r4 44.45 External radius of outer tube [mm]r5 79.45 External radius of grout [mm]L 100 Length [m]

Geometrical data of CBHE2r1 25.7 Internal radius of inner tube [mm]r2 31.5 External radius of inner tube [mm]r3 42.45 Internal radius of outer tube [mm]r4 44.45 External radius of outer tube [mm]r5 79.45 External radius of grout [mm]L 100 Length [m]

Thermal properties of AISI304 steelca 500 Specific heat capacity [J kg�1 K�1]ka 15 Thermal conductivity [W m�1 K�1]ra 7910 Density [kg m�3]

Thermal properties of high density polyethylene PE100cpe 1900 Specific heat capacity [J kg�1 K�1]kpe 0.38 Thermal conductivity [W m�1 K�1]rpe 958 Density [kg m�3]

Thermal properties of high molecular weight random copolymer polypropylenePPR80

cppr 2000 Specific heat capacity [J kg�1 K�1]kppr 0.24 Thermal conductivity [W m�1 K�1]rppr 900 Density [kg m�3]

Thermal properties of ground(rc)gd 2.0 Heat capacity per unit volume [MJ m�3 K�1]kgd1 1.4 Thermal conductivity of GD1 [W m�1 K�1]kgd2 2.8 Thermal conductivity of GD2 [W m�1 K�1]

Thermal properties of groutcgt 1600 Specific heat capacity [J kg�1 K�1]kgt1 1 Thermal conductivity of GT1 [W m�1 K�1]kgt2 2 Thermal conductivity of GT2 [W m�1 K�1]rgt 1000 Density [kg m�3]

Working conditionsTin,win 4 Water inlet temperature, winter [�C]Tin,sum 28 Water inlet temperature, summer [�C]Ts,win 4 Ground surface temperature, winter [�C]Ts,sum 30 Ground surface temperature, summer [�C]Tgd 14 Undisturbed ground temperature [�C]_V1 12 Water flow rate 1 [dm3 min�1]_V2 18 Water flow rate 2 [dm3 min�1]_V3 24 Water flow rate 3 [dm3 min�1]

r1T0 (0)

T0 (z)

rD

weak form boundary condition

LD

ground

r5

grout

PE100 (PPR80)

water

AISI 304

flow direction

r4r3r2r

h1

L

0

z

Fig. 1. Sketch of the computational domain.

Table 1Overall view of all the cases investigated.

Thermal short-circuiting

Water flowrate

Cross sectiongeometry

Type of CBHE CBHE1 CBHE1 CBHE1, CBHE2Material of the

inner tubePE100, PPR80,Adiabatic

PE100 PPR80

Grout GT1, GT2 GT2 GT2Ground GD1, GD2 GD1, GD2 GD1, GD2Water flow rate _V2

_V1;_V2;

_V3_V2

Working season Winter, Summer Winter, Summer Winter, Summer

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4.6 mm, working with a water flow rate _V2 ¼ 18 dm3 min�1. Threedifferent materials for the inner tube are considered: High DensityPolyethylene PE100, Polypropylene Random Copolymer PPR80, anideal material with a vanishing conductivity. Calculations areperformed by considering two different grouts and two differentgrounds. The first grout, denoted by GT1, has a thermal conductivitykgt1 ¼ 1 Wm�1 K�1; the second one, denoted by GT2, has a thermalconductivity kgt2 ¼ 2 Wm�1 K�1. The first ground, denoted by GD1,has a thermal conductivity kgd1 ¼ 1:4 Wm�1 K�1; the second one,denoted by GD2, has a thermal conductivity kgd2 ¼ 2:8 Wm�1 K�1.Both winter and summer working conditions are examined.

The effect of water flow rate on the performance of CBHEs isanalyzed by considering CBHE1, with inner tube in PE100 andsealed by GT2, working with three different water flow rates:_V1 ¼ 12 dm3 min�1; _V2 ¼ 18 dm3 min�1; _V3 ¼ 24 dm3 min�1.Both GD1 and GD2, as well as both winter and summer workingconditions are considered. Also the effect of water flow rate on theloss of performance due to thermal short-circuiting is studied, inthe same conditions.

Finally, a coaxial borehole heat exchanger denoted by CBHE2 isconsidered, with the same external tube size and material as

CBHE1, but with a larger cross section of the inner tube anda smaller cross section of the outer annular passage. The inner tube,made of PPR80, has an external diameter of 63 mm and a thicknessof 5.8 mm. CBHE2 is analyzed to evaluate the increase in perfor-mance obtainable by reducing the water velocity in the inner tubeand increasing it in the outer annular passage. In this analysis, thegrout properties and the water flow rate are fixed (GT2 and _V2),while both GD1 and GD2 as well as both winter and summerworking conditions are examined.

An overall view of all the cases investigated is given in Table 1.

2. Numerical model

The CBHEs modelled have length L¼ 100 m. The real 3-Dunsteady conditions of the heat transfer problem are considered.However, on account of the axial symmetry of both the geometryand the initial and boundary conditions, a 2-D axisymmetric modelwith a cylindrical coordinate system (r, z) is employed, as sketchedin Fig. 1. The CBHE and a portion of the surrounding soil areincluded in the computational domain; the latter is a cylinder,coaxial with the CBHE, with radius rD¼ 4 m and length LD¼ 104 m.Thus, also a 4 m thick ground layer under the CBHE is modelled.

The geometrical parameters of CBHE1 and of CBHE2, the prop-erties of the solid materials and the working conditions are listed inTable 2. The thermal and rheological properties of water, evaluatedat the mean water temperature for each case examined, arereported in Table 3. The values of the mean water velocity and ofthe Reynolds number, both in the inner tube and in the outerannular passage, as well as the values of the convection coefficientat the internal wall of the inner tube are reported in Table 4.

The inlet water temperature is assumed equal to 4 �C for winteroperation and to 28 �C for summer operation. Both for CBHE1 andfor CBHE2, the water flow in the inner tube is always turbulent,even with the lowest flow rate. On the other hand, both for CBHE1and for CBHE2 the water flow in the outer annular passage is alwayslaminar for _V ¼ _V1, laminar in winter and turbulent in summer for_V ¼ _V2, always turbulent for _V ¼ _V3. In all the simulations, the

Table 3Thermal and rheological properties of water, for the working conditions examined.

Symbol Value Quantity

Water flow rate _V1, winter conditions, water mean temperature Tm¼ 6 �Ccpw 4202.8 Specific heat capacity at constant pressure [J kg�1 K�1]kw 0.57247 Thermal conductivity [W m�1 K�1]rw 999.94 Density [kg m�3]mw 1.4714 Dynamic viscosity [mPa s]

Water flow rate _V2, winter conditions, water mean temperature Tm¼ 5.3 �Ccpw 4204.3 Specific heat capacity at constant pressure [J kg�1 K�1]kw 0.57114 Thermal conductivity [W m�1 K�1]rw 999.96 Density [kg m�3]mw 1.5039 Dynamic viscosity [mPa s]

Water flow rate _V3, winter conditions, water mean temperature Tm¼ 5 �Ccpw 4205.0 Specific heat capacity at constant pressure [J kg�1 K�1]kw 0.57057 Thermal conductivity [W m�1 K�1]rw 999.97 Density [kg m�3]mw 1.5181 Dynamic viscosity [mPa s]

Water flow rate _V1 , summer conditions, water mean temperature Tm¼ 25.2 �Ccpw 4181.2 Specific heat capacity at constant pressure [J kg�1 K�1]kw 0.60754 Thermal conductivity [W m�1 K�1]rw 997.00 Density [kg m�3]mw 0.88604 Dynamic viscosity [mPa s]

Water flow rate _V2, summer conditions, water mean temperature Tm¼ 26.2 �Ccpw 4180.9 Specific heat capacity at constant pressure [J kg�1 K�1]kw 0.60923 Thermal conductivity [W m�1 K�1]rw 996.73 Density [kg m�3]mw 0.86629 Dynamic viscosity [mPa s]

Water flow rate _V3, summer conditions, water mean temperature Tm¼ 26.8 �Ccpw 4180.7 Specific heat capacity at constant pressure [J kg�1 K�1]kw 0.61024 Thermal conductivity [W m�1 K�1]rw 996.57 Density [kg m�3]mw 0.85477 Dynamic viscosity [mPa s]

Table 4Values of the mean water velocity, of the Reynolds number and of the convection coefficient h1, for the CBHEs and the working conditions examined.

CBHE1 CBHE2

_V1_V2

_V3_V2

Tube Annulus Tube Annulus Tube Annulus Tube Annulus

W [m s�1] 0.153 0.054 0.229 0.081 0.306 0.108 0.145 0.118Rewin 4242 1283 6225 1883 8222 2487 4941 1717Resum 7023 2124 10772 3258 14554 4402 8550 2972h1,win [W m�2 K�1] 594 – 996 – 1251 – 630 –h1,sum [W m�2 K�1] 921 – 1284 – 1632 – 847 –

0.025 0.030 0.035 0.0400.00

0.02

0.04

0.06

0.08

0.10

0.12U (m s-1)

r (m)

(a)

(b)

Fig. 2. Velocity distributions in the outer annular duct, for CBHE1, with_V ¼ 18 dm3 min�1: the laminar velocity profile (a), for winter operation, is comparedwith the turbulent velocity profile (b), for summer operation.

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fluid inlet is placed at the top of the external annular passage, whilethe fluid outlet is placed at the top of the inner circular tube. Indeed,this flow direction is more efficient than the opposite one [16].

In the inner tube, the convection coefficient is not stronglydependent on the thermal boundary condition. On the contrary, inthe outer annular passage the convection coefficients between fluidand walls are strongly dependent on the ratio between the wallheat fluxes [17], which is unknown, non-uniform along z and timedependent. For this reason, the forced convection in the externalannular passage is simulated, while the internal forced convectionis not simulated. In the inner tube, the distribution of the waterbulk temperature Tb along the flow direction z and the heat transferbetween fluid and wall are determined by means of the so-calledweak form boundary condition available in COMSOL Multiphysics.In this method, the following energy balance equation for the fluidis coupled with the convective heat transfer boundary condition atr¼ r1 (internal wall of the inner tube):

A1rwcpwvTb

vs¼ � _Vrwcpw

vTb

vzþ h1p1ðTr1 � TbÞ (2)

In Eq. (2): A1 ¼ pr21 is the inner tube cross section area, s is time,

p1 ¼ 2pr1 is the internal wall perimeter, Tr1 is the internal walltemperature, h1 is the convection heat transfer coefficient. Thevalues to be assigned to the internal Nusselt number, and thereforeto h1, are determined by means of the Churchill correlation [18].

In the annular passage, the forced convection flow is consideredas fully developed, because the duct length is about three thousandtimes the duct mean diameter. For laminar flow, the velocity profileis given by [19]

UðrÞ ¼ Um

2

8><>:"�

r2

r3

�2

�1

#ln

rr3�"�

rr3

�2

�1

#ln

r2

r3

9>=>;1�

�r2

r3

�2

þ"

1þ�

r2

r3

�2#

lnr2

r3

(3)

where Um ¼ _V=½pðr23 � r2

2Þ� is the mean fluid velocity in the duct,which is negative, because the flow direction is downward. For

turbulent flow, the velocity profiles is determined by employing themethod proposed in Ref. [20]. As an example, the laminar velocityprofile for winter working conditions and the turbulent velocityprofile for summer conditions, for CBHE1 and _V ¼ _V2, are illus-trated in Fig. 2.

For laminar flow in the outer annular passage, the differentialequations to be solved are

rwcpw

�vTvsþ ðU$VÞT

�¼ kwV2T (4)

with U ¼ ð0;UÞ and U¼U(r) given by Eq. (3), in the annular duct;

rcvTvs¼ kV2T ; (5)

with the thermal properties of the specific material present in thesubdomain, for all the other subdomains. For turbulent flow in theannular passage, Eq. (4) must be replaced by

rwcpw

�vTvsþ ðU$VÞT

�¼ kweff V2T (6)

where kweff is the effective thermal conductivity [17] of water, forturbulent flow. The latter is given by

kweff ¼ kw þ rwcpwnwt

Prwt(7)

where nwt is the turbulent kinematic viscosity and Prwt is theturbulent Prandtl number, both evaluated for the water flow. Thedistribution of nwt in the annular passage is determined byemploying the method suggested in Ref. [21], which is animprovement of that proposed by Reichardt [22]. The turbulentPrandtl number is evaluated by the equation

Prwt ¼ 0:85þ 0:015Prw

(8)

recommended in Ref. [23].As initial condition (s¼ 0), a depth-dependent temperature

distribution in the soil T0(z) is considered, which accounts both forthe steep temperature gradient near the ground surface and for thegeothermal gradient. The following expression of T0(z) is assumed,

T0ðzÞ ¼ Tgd þ�

Ts � Tgd

�ez=l þ Hð�10� zÞ$gg$ð�10� zÞ (9)

where Tgd is the undisturbed ground temperature, l is a referencelength, assumed equal to 1 m, H is the Heaviside unit step function,gg ¼ 0:03 �C m�1 is the geothermal gradient and �104 m � z � 0.The same initial condition is adopted for the CBHE, which isconsidered in thermal equilibrium with the ground. For s> 0, theinlet water temperature Tin is assumed as prescribed and constant,as well as the surface ground temperature Ts. The external verticaland bottom surfaces of the modelled ground are considered as keptat the initial temperature T0(z).

Table 5Energy collected in selected time intervals by CBHE1 with internal tube in PE100,grout GT2, placed in GD2, working with flow rate _V2, during winter operation,calculated by employing 4 different meshes.

Time Energy [MJ], for CBHE1, GD2, GT2, PE100, _V2, WINTER

Mesh 1 (21396elements)

Mesh 2 (31072elements)

Mesh 3 (41193elements)

Mesh 4 (51588elements)

300 s 0.0382 0.0388 0.0392 0.0394900 s 1.044 1.060 1.063 1.0653600 s 13.72 13.86 13.87 13.856 h 107.9 108.3 108.5 108.424 h 402.7 404.0 404.5 404.75 days 1751 1755 1757 1757

Table 6Energy collected by CBHE1 in GD1, during winter operation, and effects of thermal short-circuiting: %PE100¼ percent efficiency loss with PE100; %PPR80¼ percent efficiencyloss with PPR80.

Time Energy [MJ]: CBHE1 – _V2 – GD1 – WINTER

GT1 GT2

PE100 PPR80 Adiabatic %PE100 %PPR80 PE100 PPR80 Adiabatic %PE100 %PPR80

300 s 0.00618 0.00618 0.00619 0.03 0.02 0.0329 0.0329 0.0329 0.07 0.05900 s 0.415 0.418 0.428 3.06 2.47 0.902 0.905 0.939 3.88 3.573600 s 8.66 8.82 9.76 11.29 9.65 12.0 12.2 13.5 11.57 9.466 h 76.0 77.1 80.6 5.73 4.37 91.7 93.2 98.1 6.46 4.9424 h 278 281 290 4.09 3.09 323 327 339 4.58 3.435 days 1160 1170 1200 3.38 2.58 1310 1320 1360 3.50 2.62

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Clearly, the continuity conditions are imposed at the interfacesbetween different materials, while the outflow pressure conditionhas been imposed at the fluid outlet.

Since each CBHE has a length of 100 m and an external diameterof 88.9 mm, a rescaling of the vertical coordinate z is adopted, toobtain a more compact computational domain. Thus, a rescaledvertical coordinate ~z and a thermal conductivity matrix bK aredefined, as follows:

~z ¼ za; bK ¼ � k 0

0 ~k

�; ~k ¼ k

a2 (10)

where a is a dimensionless rescaling coefficient, which is assumedequal to 10. As a consequence, Eq. (2) is rewritten as

A1rwcpwvTb

vs¼ �~_Vrwcpw

vTb

v~zþ h1p1ðTr1 � TbÞ (11)

where ~_V ¼ _V=a. The velocity distribution given by Eq. (3) isreplaced by the rescaled distribution ~UðrÞ ¼ UðrÞ=a, obtained byintroducing the rescaled mean velocity, ~Um ¼ Um=a. Eqs. (4) and(5) are rewritten as

rwcpw

�vTvsþ ~U

vTv~z

�¼ kw

v2Tvr2 þ

1r

vTvr

!þ ~kw

v2T

v~z2 (12)

rcvTvs¼ k

v2Tvr2þ 1

rvTvr

!þ ~k

v2T

v~z2(13)

Eq. (6) is rewritten as

rwcpw

�vTvsþ ~U

vTv~z

�¼ kweff

v2Tvr2 þ

1r

vTvr

!þ ~kweff

v2T

v~z2 (14)

where, ~kweff ¼ kweff =a2. Eq. (9) is replaced by:

T0

�~z�¼ Tgd þ

�Ts � Tgd

�ea~z=l þ H

��10

a� ~z�

$gg$

��10

a� ~z�(15)

A period of five days of operation, with given values of the waterinlet temperature and of the water flow rate, is considered for eachCBHE. Clearly, the temperature changes are steeper during the firstminutes and become much slower as time goes by. Therefore thefollowing non-uniform time steps are adopted in computations: 1 sfor 0 < s < 60 s; 5 s for 60 s < s < 600 s; 60 s for 600 s < s < 3600 s;600 s for 3600 s < s < 43200 s; 7200 s for 43;200 s < s < 432;000 s.The thermal power _Q exchanged between the CBHE and thesurrounding soil, as a function of time, is determined as

_Q ¼ �a

0BB@2pr5

Z0

�~L

kgdvTvr

����r¼r5

d~zþ 2p

Zr5

0

~kgdvTv~z

����~z¼~L

rdr

1CCA (16)

where ~L ¼ L=a is the rescaled depth of the computational domainand the spatial derivatives of temperature are evaluated in theground. The sign þ is employed in the study of winter operation,where _Q is considered as positive if collected by the CBHE; thesign � is employed in the study of summer operation, where _Q isconsidered as positive if transmitted to the ground.

The solution convergence is granted by setting, for the solver,a relative tolerance equal to 10�6 and checking the completeconvergence of all solutions.

In order to grant the results to be independent of the domainextension, most simulations are repeated by replacing theboundary condition T¼ T0(z), at the external vertical and bottomsurfaces, with the boundary condition of adiabatic surfaces. Thecheck has always given the same results as those obtained with thefirst boundary condition.

In order to ensure the grid independence of results, fourdifferent mapped meshes are tested: Mesh 1, with about 21,000elements; Mesh 2, with about 31,000 elements; Mesh 3, with about41,000 elements, Mesh 4, with about 51,000 elements. A compar-ison of the results obtained with the different meshes is illustratedin Table 5, where the values of the energy collected by CBHE1, withinternal tube in PE100 and grouted with GT2, placed in GD2,working with flow rate _V2, during winter operation, are reportedfor some time intervals after the start up. The results show that thediscrepancies between Mesh 3 and Mesh 4 are very small.

Table 7Energy collected by CBHE1 in GD2, during winter operation, and effects of thermal short-circuiting: %PE100¼ percent efficiency loss with PE100; %PPR80¼ percent efficiencyloss with PPR80.

Time Energy [MJ]: CBHE1 – _V2 – GD2 – WINTER

GT1 GT2

PE100 PPR80 Adiabatic %PE100 %PPR80 PE100 PPR80 Adiabatic %PE100 %PPR80

300 s 0.00719 0.00719 0.00719 0.07 0.06 0.0392 0.0392 0.0392 0.08 0.06900 s 0.474 0.478 0.490 3.16 2.44 1.06 1.07 1.10 3.51 2.843600 s 9.77 10.0 11.1 11.74 9.37 13.9 14.2 15.6 11.23 9.126 h 87.6 89.0 93.4 6.21 4.71 108 110 117 7.20 5.5324 h 337 342 355 4.91 3.68 404 410 429 5.75 4.415 days 1500 1520 1570 4.30 3.27 1760 1780 1850 4.82 3.66

Table 8Energy removed by CBHE1 in GD1, during summer operation, and effects of thermal short-circuiting: %PE100¼ percent efficiency loss with PE100; %PPR80¼ percent efficiencyloss with PPR80.

Time Energy [MJ]: CBHE1 – _V2 – GD1 – SUMMER

GT1 GT2

PE100 PPR80 Adiabatic %PE100 %PPR80 PE100 PPR80 Adiabatic %PE100 %PPR80

300 s 0.0171 0.0172 0.0177 3.52 2.80 0.0748 0.0748 0.0771 2.99 2.92900 s 0.622 0.635 0.672 7.44 5.56 1.37 1.41 1.49 8.53 5.563600 s 11.2 11.8 13.6 17.67 13.21 16.2 17.0 19.5 16.67 12.756 h 95.9 98.8 106 9.68 6.94 118 122 132 10.56 7.5324 h 344 351 372 7.58 5.59 405 414 437 7.53 5.295 days 1410 1430 1490 5.29 3.74 1600 1630 1700 5.62 3.92

E. Zanchini et al. / Energy 35 (2010) 657–666662

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Therefore, Mesh 3 is adopted for all the computations reported inthe following.

3. Effect of thermal short-circuiting

In order to analyze the effect of thermal short-circuiting,CBHE1 is considered, working with a water flow rate_V2 ¼ 18 dm3 min�1, with three different materials as constitu-ents of the inner tube: High Density Polyethylene PE100, Poly-propylene Random Copolymer PPR80, an ideal material witha vanishing conductivity. Both GT1 and GT2, both GD1 and GD2,as well as both winter and summer working conditions areconsidered. Then, also the effect of water flow rate on the loss ofefficiency due to thermal short-circuiting is investigated, forCBHE1 with inner tube made of PE100 and sealed by GT2, forboth grounds and seasons. For each simulation, the energyexchanged between the BHE and the ground is evaluated, duringselected time intervals from the start up. The energy exchanged isconsidered as positive if collected, during winter, and if suppliedto the ground during summer.

The results for the energy exchanged, with the flow rare _V2, arereported in Tables 6–9. The results show that, for long CBHEs, theeffects of thermal short-circuiting are considerable. The percentloss of efficiency, with respect to the ideal condition of adiabaticinner tube, reaches its maximum for a time interval of one hour,then decreases but remains important. It is higher for GT2 and forGD2, because a high conductivity grout and, even more, a highconductivity ground, enhance the energy exchange and, asa consequence, the temperature difference between water outletand inlet. If the internal tube is in PE100, for a time interval of 5days and for GT2 and GD2, the percent loss of efficiency due tothermal short-circuiting is 4.82% during winter operation and8.02% during summer operation. The higher effect of thermal short-circuiting during summer operation is due to these reasons:

(a) following ASHRAE recommendations [1], the differencebetween water inlet temperature and undisturbed groundtemperature has been assumed larger in summer (14 �C) thanin winter (10 �C);

(b) with the water flow rate considered, _V2, the flow in theexternal annular duct is laminar in winter and turbulent insummer.

Conditions (a) and (b) enhance, during summer, the differencebetween the water inlet and outlet temperatures.

If an internal tube in PPR80 is employed, the effect of thermalshort-circuiting is reduced considerably. With reference to a timeinterval of 5 days, the reduction obtained is: 24–25% for winteroperation, about 30% for summer operation in GD1, about 40% forsummer operation in GD2. Therefore, an inner tube in PPR80 isrecommended.

The effect of water flow rate on the loss of efficiency due tothermal short-circuiting is studied with reference to CBHE1, grou-ted by GT2, operating with the flow rates _V1, _V2 and _V3; severalcases are examined: two materials for the inner tube (PE100 andPPR80), both grounds (GD1 and GD2), winter and summer workingconditions. The results for the inner tube in PE100 are illustrated inTables 10 and 11 with reference to winter and summer operation,respectively.

Table 11 shows that, for time intervals longer that 1 h, thepercent loss of efficiency due thermal short-circuiting decreases asthe water flow rate increases. This is reasonable since a higher flowrate implies a lower difference between water inlet and outlettemperatures. Table 10 shows that, unexpectedly, during winter thepercent loss of efficiency due to thermal short-circuiting increasesif the water flow rate increases from _V2 to _V3. This behaviour is dueto the fact that, during winter, _V3 determines a transition toturbulent flow in the outer annular passage.

Although the percent loss of efficiency due to thermal short-circuiting depends on several parameters, it is reasonable toassume that, for a given material of the inner tube, this lossdepends mainly on the magnitude of the difference between waterinlet and outlet temperatures, DT ¼ jTin � Tout j. In Fig. 3, thepercent efficiency loss due to thermal short-circuiting is plottedversus DT , for CBHE1, with reference to time intervals between 6 hand 5 days. Clearly, some data scatter is present, because grout andground properties, water mass flow rate and seasonal workingconditions affect the percent efficiency loss due to thermal short-

Table 9Energy removed by CBHE1 in GD2, during summer operation, and effects of thermal short-circuiting: %PE100¼ percent efficiency loss with PE100; %PPR80¼ percent efficiencyloss with PPR80.

Time Energy [MJ]: CBHE 1 – _V2 – GD2 – SUMMER

GT1 GT2

PE100 PPR80 Adiabatic %PE100 %PPR80 PE100 PPR80 Adiabatic %PE100 %PPR80

300 s 0.0197 0.0199 0.0205 3.53 2.81 0.0886 0.0887 0.0914 3.01 2.92900 s 0.711 0.726 0.768 7.45 5.57 1.61 1.66 1.76 8.55 5.563600 s 12.8 13.4 15.5 17.60 13.15 18.9 19.8 22.6 16.43 12.536 h 112 116 125 10.35 7.37 143 148 163 12.14 8.6424 h 425 439 464 8.56 5.43 520 542 580 10.19 6.535 days 1870 1920 2000 6.80 4.04 2210 2290 2410 8.02 4.83

Table 10Percent efficiency loss due to thermal short-circuiting, for CBHE1 with inner tube inPE100 and grouted by GT2, with water flow rates _V1, _V2 and _V3: winter operation.

Time %PE100 – CBHE1 – GT2 – WINTER

GD1 GD2

_V1_V2

_V3_V1

_V2_V3

300 s 0.07 0.07 4.83 0.08 0.08 4.85900 s 4.62 3.88 12.86 4.65 3.51 12.883600 s 18.04 11.57 13.48 17.85 11.23 13.516 h 12.09 6.46 6.97 13.18 7.20 8.0624 h 9.21 4.58 4.67 10.97 5.75 6.185 days 7.26 3.50 3.57 9.42 4.82 5.00

Table 11Percent efficiency loss due to thermal short-circuiting, for CBHE1 with inner tube inPE100 and grouted by GT2, with water flow rates _V1, _V2 and _V3: summer operation.

Time %PE100 – CBHE1 – GT2 – SUMMER

GD1 GD2

_V1_V2

_V3_V1

_V2_V3

300 s �1.27 2.99 3.69 �1.22 3.01 3.70900 s 3.39 8.53 8.36 3.46 8.55 8.373600 s 17.92 16.67 13.55 17.93 16.43 13.626 h 11.58 10.56 7.13 12.38 12.14 7.9524 h 9.37 7.53 4.92 10.75 10.19 5.375 days 7.07 5.62 3.58 9.00 8.02 4.20

E. Zanchini et al. / Energy 35 (2010) 657–666 663

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circuiting, for each inner tube material and value of DT . Never-theless, linear best fits of the data for inner tube in PE100 and forinner tube in PPR80 allow approximate evaluations of the percentloss of efficiency due to thermal short-circuiting, for long CBHEs.The plots illustrate also the gain of efficiency obtainable byreplacing the traditional PE100 internal tube with a PPR80 tube.

The equations of the linear regression lines reported in Fig. 3,which hold in the range 2.0 �C<DT< 6.0 �C, are

eff loss% ¼ 2:8302 DT � 2:7245 (17)

eff loss% ¼ 2:2206 DT � 2:3307 (18)

where Eq. (17) refers to inner tube in PE100 and Eq. (18) refers toinner tube in PPR80. The standard deviation of data from thebest-fit curve is 1.34% for PE100 and 1.1% for PPR80.

As an example, Fig. 4 illustrates the loss of performance due tothermal short-circuiting, versus time, for CBHE1 grouted with GT2,placed in GD2, with water flow rate _V2, during summer operation:the power released by the CBHE is maximum for the ideal case ofadiabatic wall of the inner tube, lower for inner tube in PPR80,lower again for inner tube in PE100.

4. Effect of water flow rate

The effect of water flow rate on the performance of coaxial BHEsis analyzed with reference to CBHE1, inner tube in PE100, groutGT2, both grounds (GD1 and GD2), water flow rates _V1, _V2 and _V3.The results are illustrated in Tables 12 and 13, with reference towinter and summer working conditions, respectively.

In these Tables, the column loss% _V1 points out the percentenergy loss which occurs if the standard flow rate _V2 is replaced by_V1 while the column gain% _V3 points out the percent energy gainwhich occurs if the standard flow rate _V2 is replaced by _V3. Table 12shows that, especially if the CBHE is placed in GD2, the energy gainsobtainable by increasing the water flow rate from _V1 to _V2 and from_V2 to _V3 are both relevant. They are very high for short timeintervals, but remain remarkable even for a period of 5 days. Therelevant energy gain obtainable by replacing _V2 with _V3 duringwinter operation is due mainly to the fact that the increased waterflow rate produces a transition in the outer annular passage fromlaminar to turbulent flow. The effect of flow rate on the perfor-mance of CBHEs during winter operation is illustrated in Fig. 5,where the power collected by CBHE1, grouted with GT2 and placedin GD2, with internal tube in PE100, is plotted versus time.

Table 13 shows that, during summer, the energy gain obtainableby increasing the water flow rate from _V1 to _V2 is higher than inwinter, while the gain obtainable by replacing _V2 with _V3 is lower.In fact, during summer operation, the water flow rate in the outerannular passage is turbulent even with _V2.

Clearly, the optimization of the water flow rate must take intoaccount not only the thermal energy gain, but also the increasein electric energy consumption by the pumps. This combinedoptimization problem will not be studied in this paper.

5. Increase in thermal performance obtainable by CBHE2

In CBHE1, whose geometry is employed at present in Italy, themean water velocity in the inner circular tube is much higher thanin the outer annular passage (see Table 4). This circumstance seemscontrary to efficiency optimization. Indeed, without increasing thetotal pressure drop, one could lower the water velocity in the inner

2

4

6

8

10

12

14

2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0

Effic

ie

nc

y lo

ss

%

PE100PPR80PE100PPR80

Δ [°C]T

Fig. 3. Plots of the percent efficiency loss due to thermal short-circuiting versus DT, forCBHE1, for time intervals between 6 h and 5 days after start up.

0

1000

2000

3000

4000

5000

6000

7000

8000

0.0E+00 5.0E+04 1.0E+05 1.5E+05 2.0E+05 2.5E+05 3.0E+05 3.5E+05 4.0E+05

Q[W

]

τ [s]

PE100

PPR80adiabatic

.

Fig. 4. Power released by CBHE1 grouted with GT2, placed in GD2, with water flowrate _V2 during summer operation: internal tube made of PE100, PPR80, adiabatic.

Table 12Energy collected by CBHE1, with inner tube in PE100 and grouted by GT2, with water flow rates _V1, _V2 and _V3, during winter operation.

Time Energy [MJ]: CBHE1 – GT2 – PE100 - WINTER

GD1 GD2

_V1_V2

_V3 Loss% _V1 Gain% _V3_V1

_V2_V3 Loss% _V1 Gain% _V3

300 s 0.0220 0.0329 0.0864 33.03 162.58 0.0262 0.0392 0.103 33.06 162.26900 s 0.606 0.902 1.71 32.83 89.27 0.712 1.06 2.01 33.04 88.703600 s 8.63 12.0 18.2 27.81 52.12 9.92 13.9 21.3 28.49 53.336 h 76.6 91.7 117 16.49 27.99 88.5 108 145 18.44 33.3824 h 283 323 387 12.34 19.68 344 404 510 14.93 26.155 days 1180 1310 1500 9.44 14.44 1540 1760 2120 12.48 20.95

E. Zanchini et al. / Energy 35 (2010) 657–666664

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tube and increase it in the outer annular passage. The reduction ofthe water velocity in the inner tube would lower the internalconvection coefficient h1 and therefore the thermal short-circuitingeffect; the increase of the water velocity in the annular passagewould enhance the convection coefficient between water and theouter AISI tube.

For this reason, a modified coaxial BHE is here proposed, calledCBHE2. In the latter, as shown in Table 2, a commercial PPR80internal tube with higher internal and external diameters isemployed, while the external AISI tube is unchanged. The values ofthe water mean velocity and of the Reynolds number are reportedin Table 4.

The values of the energy collected by CBHE2 in selected timeintervals after start up, during winter operation with water flowrate _V2 and GT2, are reported in Table 14 and compared with thosecollected by CBHE1, with inner tube in PPR80, in the same condi-tions. The corresponding results for summer operation are reportedin Table 15.

As is shown by Table 14, during winter operation the percentenergy gain provided by CBHE2, with respect to CBHE1, is consid-erable, especially for time periods up to one hour after start up; thepercent gain remains interesting also for longer time intervals, inparticular for GD2, where it is about 5% for a time interval of 5 days.

On the other hand, as is shown by Table 15, during summer theenergy gain obtainable by CBHE2 is much lower and is considerableonly for short time intervals. The reason is as follows. With thewater flow rate _V2, considered here, the water flow in the outerannular duct is turbulent during summer. In CBHE2, the meanvelocity in the annular duct is higher, but the mean tube diameter islower, so that the Reynolds number is slightly lower and, inturbulent regime, also the Nusselt number is lower. The decrease of

Table 13Energy released by CBHE1, with inner tube in PE100 and grouted by GT2, with water flow rates _V1, _V2 and _V3, during summer operation.

Time Energy [MJ]: CBHE1 – GT2 – PE100 – SUMMER

GD1 GD2

_V1_V2

_V3 Loss% _V1 Gain% _V3_V1

_V2_V3 Loss% _V1 Gain% _V3

300 s 0.0246 0.0748 0.105 67.07 40.31 0.0292 0.0886 0.124 67.06 40.34900 s 0.592 1.37 1.89 56.68 38.04 0.695 1.61 2.22 56.71 38.063600 s 9.12 16.2 20.3 43.85 25.24 10.5 18.9 23.7 44.39 25.456 h 85.0 118 133 28.18 12.79 98.1 143 165 31.32 15.5724 h 317 405 440 21.65 8.67 383 520 587 26.34 12.765 days 1330 1600 1700 17.00 6.25 1720 2210 2440 22.39 10.15

0

1000

2000

3000

4000

5000

6000

7000

0.0E+00 5.0E+04 1.0E+05 1.5E+05 2.0E+05 2.5E+05 3.0E+05 3.5E+05 4.0E+05

Q[W

]

τ [s]

V1

V2

V3

.

1

2

3

V

V

V

Fig. 5. Power collected by CBHE1 grouted with GT2, placed in GD2, with internal tubein PE100, during winter operation, with reference to the water flow rates _V1, _V2, _V3.

Table 14Energy collected by CBHE2, grouted by GT2, with water flow rate _V2, compared withthe energy collected by CBHE1, with inner tube in PPR80, in the same conditions:winter operation.

Time Energy [MJ]: PPR80 – GT2 – _V2 – WINTER

GD1 GD2

CBHE1 CBHE2 Gain % CBHE1 CBHE2 Gain %

300 s 0.0329 0.0602 45.32 0.0392 0.0715 45.23900 s 0.905 1.31 30.73 1.07 1.53 30.233600 s 12.2 14.4 14.86 14.2 16.6 14.696 h 93.2 100 6.90 110 120 7.7324 h 327 344 4.83 410 437 6.115 days 1320 1370 3.62 1780 1870 5.07

Table 15Energy collected by CBHE2, grouted by GT2, with water flow rate _V2, compared withthe energy collected by CBHE1, with inner tube in PPR80, in the same conditions:summer operation.

Time Energy [MJ]: PPR80 – GT2 – _V2 – SUMMER

GD1 GD2

CBHE1 CBHE2 Gain % CBHE1 CBHE2 Gain %

300 s 0.0748 0.102 26.86 0.0887 0.121 26.84900 s 1.41 1.72 18.07 1.66 2.02 18.053600 s 17.0 18.3 7.32 19.8 21.3 6.966 h 122 124 1.54 148 151 1.6824 h 414 417 0.74 542 546 0.875 days 1630 1640 0.46 2290 2300 0.56

0

1000

2000

3000

4000

5000

6000

0.0E+00 5.0E+04 1.0E+05 1.5E+05 2.0E+05 2.5E+05 3.0E+05 3.5E+05 4.0E+05

Q[W

]

τ [s]

CBHE1

CBHE2

.

Fig. 6. Powers collected by CBHE1 and by CBHE2, in the case GT2, GD2, PPR80, _V2,during winter operation.

E. Zanchini et al. / Energy 35 (2010) 657–666 665

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the Nusselt number damps the increase of the convection coeffi-cient due to a lower mean diameter.

The advantage offered by CBHE2 during winter operation isillustrated in Fig. 6, where the power collected by CBHE1 is plottedversus time and compared with that collected by CBHE2, in the caseGT2, GD2, PPR80, _V2. The figure shows that CBHE2 is more efficientthan CBHE1 at any time instant after start up.

6. Conclusions

The thermal performance of 100 m long coaxial borehole heatexchangers has been analyzed by finite-element simulations,carried out by means of the software package COMSOL Multi-physics 3.4 (�Comsol, Inc.). The real 2-D axisymmetric unsteadyheat conduction and convection problem has been considered, bothfor winter and for summer working conditions. The effects ofthermal short-circuiting and of water flow rate have been investi-gated, as well as the increase in performance obtainable by a slightmodification of the CBHE cross section.

The results point out that the effect of thermal short-circuitingon the performance of long coaxial borehole heat exchangers(CBHEs) is relevant. The percent loss of efficiency is more importantfor higher conductivity ground and grout, and it is higher duringsummer than during winter. The reduction of the thermal short-circuiting effect obtainable by employing a low conductivitymaterial for the wall of the inner tube, such as PPR80, is consid-erable; therefore, this technique is recommended.

Also the effect of water flow rate on the performance of longCBHEs is relevant. For CBHEs with an external diameter of about9 cm, a water flow rate of 12 dm3 min�1 is too low to ensure a goodperformance. The energy gain obtainable by enhancing the waterflow rate from 18 to 24 dm3 min�1 can be relevant especially duringwinter operation, where the flow rate enhancement produces theonset of turbulence in the outer annular duct.

Finally, it has been shown that a significant gain of efficiency canbe obtained, during winter operation, by increasing the cross sectionof the internal tube and decreasing that of the outer annular passage.

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