xe-vi

entlel,nsfeicl teods o

convolved with any desired heat load to estimate uid temperatures at the GHE inlet for a wide variety of

lable iesultinng theute thangerre can

used a linear system of equations solved at each time step. Theirapproach requires the numerical evaluation, at each time step, of aninverse Laplace transform. Pasquier and Marcotte [20,21] proposeda quite different approach where they work simultaneously on alltime steps by perturbing iteratively an initial guess heat transfer

ngement betweenall with parallel

without providingmmodates parallel

lds: i. adapt thearotto [15] to thethe unit response

functions based on the uid temperature at the GHE inlet ratherthan based on the BHE average wall temperature, hence allowingsimulation of temperatures for parallel, series and mixed arrange-ments, and iii. test the proposed approach with a full 3D niteelement numerical model.

The paper is structured as follows. The methodological sectionpresents the idea introduced by Lamarche and Beauchamp [13] ofsplitting the response function in a historical and a contemporarypart. Using the FLS and the assumption of linear uid temperature

* Corresponding author. Tel.: 1 514 340 4711x4620; fax: 1 514 340 3970.

Contents lists availab

Renewable

els

Renewable Energy 68 (2014) 14e24E-mail address: denis.marcotte@polymtl.ca (D. Marcotte).apply than the Laplace approach when the heat transfer occurringat each BHE is already known. However, in a GHE, the interactionsbetween the BHEs imply that the heat transfer at each BHE varies intime according to its position in the network, hence, it has to bedetermined at each time step.

Cimmino et al. [4] and Lazzarotto [15] both used the Laplacetransform approach of Lamarche [11] to determine sequentially theheat transfer and the average wall temperature at the BHEs. They

perature which implicitly assume a parallel arrathe BHEs. The examples in Lazzarotto [15] arearrangement although Lazzarotto [15] mentions,the pertaining equations, that his approach accoand series arrangements.

The objectives of this paper are three-fosequential idea of Cimmino et al. [4] and Lazzeasier to apply FFT spectral approach, ii. developspectral methods using either Fast Fourier Transform (FFT) [16] orLaplace transform [11,13]. The FFT approach is faster and easier to

[9] and nd noticeable differences at long times. Moreover, theprevious methods focused on the determination of BHE wall tem-1. Introduction

Several analytical models are avaipredict the temperature response rground heat exchanger (GHE). Amosource (FLS) is often used to compalong thewall of a borehole heat exchcomputation of BHE wall temperatu0960-1481/$ e see front matter 2014 Elsevier Ltd.http://dx.doi.org/10.1016/j.renene.2014.01.023scenarios. 2014 Elsevier Ltd. All rights reserved.

n the literature [3,10] tog from operation of amodels, the nite linee average temperature(BHE) [5,7,14,17,18]. Fastbe done efciently by

distribution so as to meet imposed temperature signals on the BHEwall temperatures. Their algorithm converges in a few tens of it-erations in all examples tested, and is proven to be exact when thenumber of iterations reaches the number of time steps.

Neither Cimmino et al. [4], Lazzarotto [15] and Pasquier andMarcotte [20], compared their results to the temperatures obtainedwith a full 3D numerical model, although Cimmino et al. [4]compared their results to the numerical g-functions of EskilsonGround heat exchanger

Series arrangement nite element numerical model shows good agreement with the computed uid temperatures. The

proposed approach is computationally very efcient. The uid temperature unit response function can beUnit-response function for ground heat eor mixed borehole arrangement

D. Marcotte*, P. PasquierCivil, Geological and Mining Department, Polytechnique Montral, C.P. 6079 Succ. Centr

a r t i c l e i n f o

Article history:Received 22 July 2013Accepted 20 January 2014Available online 14 February 2014

Keywords:Finite line sourceParallel arrangement

a b s t r a c t

A novel approach is presExchanger (GHE) for paraleach time step the heat traat the GHE inlet for a speccompute the borehole walalong the pipes. The methsteps. The different system

journal homepage: www.All rights reserved.changer with parallel, series

lle, Montral H3C 3A7, Canada

ed that allows to predict uid temperatures entering a Ground Heatseries and mixed arrangements of boreholes. The method determines atr rates occurring at each borehole so as to reproduce the uid temperatureborehole arrangement. The analytical nite line source model is used tomperatures, whereas the uid temperatures are assumed to vary linearlyrequires to solve a linear system of equations at a small number of timef equations for each arrangement are determined. A comprehensive 3D

le at ScienceDirect

Energy

evier .com/locate/renene

variation along the pipes, the remarkably simple linear systems ofequations for different BHE arrangements (parallel, series ormixed)are then presented. The model is applied for a GHE with radialsymmetrical distribution of BHE, for parallel and mixed arrange-ments. The results are compared to those of a full 3D numericalmodel developed in COMSOL. Computational aspects are discussedto prove the practicality of the sequential FFT approach, even fornumerous BHE and long time series. Finally, a detailed numericalexample is provided in Appendix A.

2. Methodology

An analytical model (e.g. FLS [5,14]) enables to compute themean temperature at the borehole wall. Under a steady state hy-pothesis the mean uid temperature is obtained as:

Tf t q=HRb Tbt (1)

obtained by spatial superposition. Summing the contribution from

and the transfer function f is the analytical model response for therstm time steps under a unit heat load at each BHE evaluated for adistance rij; hij(mDt) is obtained as the mth element of the

From

HP

To HP

Tin

Tout

a)

From

HP

To HP

Tin Tout

b)

From

HP

To

Tinc)

y (m

)

Coord. x (m)

Coo

rd. y

(m)

1 2 3

456

7 8 9b)

D. Marcotte, P. Pasquier / Renewable Energy 68 (2014) 14e24 153 m

Coord. x (m)

Coo

rd.

1 2 3

456each BHE, one has:

Tbit T0 Xnj1

DTj/it (3)

where DTj/i(t) is the temperature perturbation at BHE i caused byheat emanating from BHE j.

The heat loads are assumed to be a step function with time stepDt. The unit response function has to be calculated at times mDt,m 1.nt. It is convenient to split the temperature perturbations in

7 8 9a)where q is the heat transferred by the BHE, H is the borehole length,Rb is the BHE equivalent thermal resistance and Tb is the averagetemperature at the BHE wall. Assuming, for simplicity, a linearvariation of the uid temperature along the pipes, one has:

Tint Tf t q

2 _mCp; Toutt Tf t

q2 _mCp

(2)

where _m is the uid mass ow rate and Cp is the specic heat of theuid.

2.1. Borehole interactions

In a network of n BHEs, the wall temperature of BHE i can beFig. 2. Location of boreholes and arrtwo terms, the historical part hij(mDt) due to heat transfer from 0 to(m 1)Dt and the present time step contribution qj(mDt)fDt(rij):

DTj/it hijt qjtfDtrij

(4)

where fDt(rij) is the unit transfer function computed with theanalytical model for one time step Dt, and rij is the distance be-tween boreholes i and j. For i j, one has rii rb. Note that the totalheat transferred by the n BHEs is qt Pnj1 qjt. The historicalpart hij(mDt) can be easily computed by a discrete convolutionproduct:

hijmDt ~qj*f

mDt (5)

where, at time t mDt, the step increment vector eqj is the vector:eqj hqj;1; qj;2 qj;1;.; qj;m1 qj;m2;qj;m1i (6)

HPTout

Fig. 1. Different possible BHE arrangements: a) parallel; b) series; c) mixed.angements a) parallel, b) series.

convolution vector eqj*f. The convolution can be computed ef-ciently using FFT after padding vectors eqj and fwithm 1 zeros toaccount for the non-periodicity of functions as described in Mar-cotte and Pasquier [16], and Pasquier and Marcotte [20]. Note that

one convolution per BHE pair has to be computed. Moreover, thehistorical contribution has to be computed at each time step.However, Section 3.2 will show that the computations need to berealized at only a very small subset of the time steps, thereforemaking the approach practical.

Adopting a matrix notation and dropping the time reference,one has simultaneously:

T h Gqt (7)

where T is the vector with total temperature perturbation at each ofthe n BHEs, h is the vector containing the n temperature pertur-bations due to the historical part, G is the n n matrix whoseelement (i,j) is the temperature perturbation caused, at BHE i, byone unit of heat emanating from BHE j for the current time step, qtis the n 1 vector of heat ux emanating from the BHEs for time

Table 1Parameters.

Variable Value, Units

ks 2.5 W(m C)1

cs 2 106 J(m3 C)1Rb 0.12 m C W1

T0 0 C_m 0.3 kg/sCp 4180 J kg1 K1

rb 0.08 mH 30 m

a) b)

c)

D. Marcotte, P. Pasquier / Renewable Energy 68 (2014) 14e2416Fig. 3. Short-term heat transfer ratios (in %) for a) parallel and b) series; c) ~T in for the two arrangements.

ewaa)

D. Marcotte, P. Pasquier / RenmDt. Note that G is a function of the time step, not of the time,therefore it needs to be computed only once.

In the next section, the inuence of the connections betweenBHEs is examined. More specically, the pure parallel arrangementand the pure series arrangement and a mix of parallel/seriesarrangement are considered, as depicted in Fig. 1. As in Lazzarotto[15], it is assumed that the uid coming from the heat pumps isthorough-fully mixed when entering the GHE, so a unique Tin valueapplies. Similarly, the uid leaving the GHE is mixed beforeentering the heat pumps, so a unique Tout applies.

2.2. Pure parallel arrangement

The n BHEs are fed simultaneously by a common uid source.Therefore the Tin temperatures at the BHEs are all equal at eachtime step. Moreover the sum of the heat ux over the n BHEs is set

c)

Fig. 4. Long-term heat transfer ratios (in %) for a) parab)

ble Energy 68 (2014) 14e24 17to one unit of heat for the entire boreeld. The latter condition iswritten in vector form as:

1Tqt 1ct (8)

where 1 is a n 1 vector of ones.The n 1 unknowns at each time step are the n heat transfer

coefcients (one per BHE) and the Tin T0 value. Solution of thelinear system of equations described by Equation (9) gives, for agiven time step, qt and Tin since T0 is known.

G D 11T 0

qt

Tin T0h

1

(9)

with D diagRbi=Hi 1=2 _miCp; i 1:::n, where diag representsa diagonal matrix with the n terms on the diagonal. The _mi are the

llel and b) series; c) ~T in for the two arrangements.

Tin;1 Tin Tout;n Q_mCp

(12)

The n 1 unknowns at each time step are the n heat transfercoefcients of qt and Tin. All other uid and wall temperatures canbe obtained from these. They are obtained by solving the linearsystem of equations:

6

4

2

0

2

4

6

8

A BProfile (Fig. 14)

N

Plan

e of

sym

met

ry

Profile section (Fig. 11)

Inflow OutflowExterior

Outflow InflowCenter

Coo

rd. y

(m)

a)

Fig. 5. Sensitivity of the ~T in to the time step; 9 boreholes as in Fig. 2 with parallelarrangement; the four curves indicated in the legend are almost perfectly superposed,

D. Marcotte, P. Pasquier / Renewable Energy 68 (2014) 14e2418mass ow rate entering BHE i. They are assumed known at all timesteps.

2.3. Pure series arrangement

The n BHEs form an ordered sequence going from BHE 1 to BHEn. The uid temperature at the outlet of a BHE is considered to bethe uid temperature at the inlet of the next BHE in the chain(hence, uid heat losses occurring during the transfer between theBHEs are neglected). Moreover, we have:

_mi _mci; i 1.n (10)

and

T T ci; i 1.n 1 (11)

therefore indistinguishable.out;i in;i1

Fig. 6. ~T in computed at varying time steps from 1 h to 200 y; 9 boreholes as in Fig. 2with parallel arrangement.8 6 4 2 0 2 4 6 8

8

Coord. x (m)

Fig. 7. BHE locations, planes of symmetry, and proles used in Figs. 11 and 14. The twoseries arrangements are illustrated: 1 e Exterior: Inow from the eight exteriorboreholes and outow from the eight center boreholes, and 2 e Center: Inow fromthe eight center boreholes and outow from the eight external boreholes.G D L 1

1T 0

qt

Tin T0h

1

(13)

Fig. 8. Geometry and mesh in the horizontal plane.

Table 2Thermal properties and geometry of the reference model.

Description Parameters, Units Values

Fluid thermal conductivity kf, W/(m K) 0.55a

Pipe thermal conductivity kp, W/(m K) 0.40Grout thermal conductivity kg, W/(m K) 1.45Soil thermal conductivity ks, W/(m K) 2.5Fluid volumetric heat capacity rfcf, J/(m3 K) 4 180 000Pipe volumetric heat capacity rpcp, J/(m3 K) 2 000 000Grout volumetric heat capacity rgcg, J/(m3 K) 2 000 000Soil volumetric heat capacity rscs, J/(m3 K) 2 000 000Borehole radius rb, m 0.08Inner radius of the pipe ri, m 0.017Outer radius of the pipe ro, m 0.022Half pipe spacing D, m 0.04Depth of BHE summit D, m 2.0BHE length H, m 30.0Mass ow rate per BHE _m, kg/s 0.3

a To represent convection in the pipes, a conductivity 500 times higher is used in

D. Marcotte, P. Pasquier / Renewable Energy 68 (2014) 14e24 19where L is a lower triangular matrix with zeros on the diagonal andLij 1= _mCp; cj < i; i 1:::n.

2.4. Mixed arrangements

We consider a parallel arrangement for the m BHE that are theheads of as many clusters of BHEs arranged in series (a cluster canhave a single BHE). The clusters are of size ni, i 1.m. Combiningthe results of the two previous sections, one gets:

Fig. 9. Geometry and mesh for the whole model.

Fig. 10. Illustration of the boundary conG D L 1

1T 0

qt

Tin T0h

1

(14)

where the D and Gmatrices are dened as before, and the matrix Lhas the following structure:

L L1 0 / 00 L2 0 // / / /0 / 0 Lm

2664

3775 (15)

and where each Li corresponds to a head BHE. It is a lower trian-gular matrix of size ni ni with zeros on the diagonal and value1= _miCp for all other entries in the lower part of the matrix ( _mi is theuid mass ow entering head BHE i).

There is one linear system of Equations (9) and (13), or (14) pertime step to be solved. However, each system is of small size (n 1)and moreover the left hand matrix is time invariant, so it needs tobe inverted only once. Appendix A presents a small numericalexample to illustrate the computations involved.

2.5. GHE dimensionless unit response function

the xy plane.The Tin(mDt) function evaluated in Equations (9), (13) and (14),represents the time temperature evolution (in C) of the uidentering the GHE due to the constant application of 1 W of heat

ditions used at the base of a BHE.

rangements described in Fig. 2 are computed with parameters

maximal relative difference between any pair of curves being only

4. Comparison with a numerical model

The analytical model is compared to a full 3D numerical modelbuilt in COMSOL Multiphysics [6]. The case of radially and sym-metrically located BHEs depicted in Fig. 7 is used with parallel andseries feeding strategies. Two different strategies are considered forthe series case: feeding by the central BHE of each branch orfeeding by the external BHE of each branch.

4.1. Description of the numerical model in COMSOL

To provide a reference set of uid temperature, the modeldeveloped by Marcotte and Pasquier [16] for a single BHE in theComsolMultiphysics [6] environment has been extended to include12 complete BHEs. The 12 boreholes lie on the same side of any ofthe symmetry planes found at azimuth 22.5, 67.5, 112.5 and157.5. The modeled geometry includes the heat carrier uidcirculating in the supply and return pipes, the pipe material itselfand the grout for each BHE, as well as the geological materiallocated below, above and around the BHEs. The model howeveromits the horizontal return loop located at the base of every pair ofpipes, and the horizontal pipe segment transporting the heat car-rier uid between the BHEs. The geometry around a single BHE isshown in Fig. 8 while the complete geometry is shown in Fig. 9.

The state equation behind the model is given by

rCpvTvt

rCpu,VT V,kVT (17)

ewable Energy 68 (2014) 14e240.11%. The drawback to using a large time step is to provide the rstvalue at a larger time. This suggests, as in Cimmino et al. [4] todiminish the total number of time steps by adopting increasingtime steps and merging the results. As an example, the solution canbe computed at every hour for one day, then at every day for onemonth, then at every month for one year, then at every year for 10years, then every 10 years for a few centuries (if required). Thisstrategy ensures the number of time steps remains alwaystractable.

Fig. 6 shows the result of such strategy for the previous casewith 9 boreholes. It was computed in 1.5 s on a standard laptop(Intel coreI3 at 2.66 GHz, Windows 7; average time over 100given in Table 1. The nine H 30 m long BHEs are located on a 3 mregular grid. Fig. 3 shows the heat transfer coefcients for each BHEand the ~T in obtained with the parallel and the series arrangementsin the short-term, i.e. within the rst 20 days (ts 2.537y, ln(t/ts) 3.83, where ts is the Eskilsons characteristic time, ts H2ks/(9Cs)). For the parallel arrangement, due to the symmetry, only 3different heat transfer proportion curves appear. For the seriesarrangement, each BHE shows a different heat transfer proportionordered according to the BHE sequence in the series. The ~T in issignicantly lower for the parallel case than for the series case,indicating a larger heat transfer to the ground for this arrangementcompared to the series arrangement. Note that the uid mass owper BHE is the same for both arrangements, implying that 9 timesmore uid goes to the HP in the parallel case.

Fig. 4 shows the heat transfer ratios for each BHE and the ~T intemperatures obtained with the parallel and the series arrange-ments for the long-term, i.e. up to 10 years. The parallel and seriescurves have similar behavior except that the ~T in is lower for theparallel case. After approximately 2.5 years (ln(t/ts) z 1), a quasi-steady state is reached for the heat transfer coefcients in bothscenarios and for all BHEs. Note that in the series arrangement, theBHE heat transfer depends of its location (corner, sides or center)and its order in the sequence. Hence, corner BHEs transfer moreheat than side BHEs and side BHEs transfer more heat than thecentral BHE. Among the corner BHEs, the heat transfer proportionfollows the BHE sequence and similarly for the side BHEs.

3.2. Sensitivity to the time step

Fig. 5 shows the ~T in obtained for the parallel arrangement ofFig. 2 with different choice of time steps Dt. Whatever, the time stepadopted, the results are almost identical at the same time, theload for the entire GHE. It is convenient to dene the dimensionlessunit response function ~T in as:

~T in Tin T02pksnH (16)This function is similar to the g-function of Eskilson [9], except

that it represents the uid temperature at the GHE inlet instead ofthe average BHE wall temperature. Contrary to the g-function ofEskilson, it already integrates the effects of BHE arrangement(parallel, series or mixed) and can be used directly to predict uidtemperatures for any heat load scenarios.

3. Results

3.1. A simple example

The time varying heat transfer coefcients for the two ar-

D. Marcotte, P. Pasquier / Ren20simulations).

Fig. 11. Temperature distribution in a vertical section including six BHEs, after 180 daysof 1 W heat injection for the whole GHE.

where r is the material density, Cp is the specic heat capacity, k isthe thermal conductivity and where u is a velocity vector. The latteris used only to represent the vertical uid advection within thepipes and is equal to zero everywhere else in the domain. In everysupply pipe, uworths _m=rfpr2i while a value of _m=rfpr2i is used inevery return pipe.

The boundary conditions used include a zero constant temper-ature condition along the surface (z 0 m) and the base (z 67 m)of the domain. To represent a constant temperature boundarylocated at innity, innite boundary conditions constrained at T 0are used for the lateral boundary located at r 20 m. To reduce themodels size, a symmetry boundary condition is used to split in twoequal parts the problem. To create small inner boundaries corre-sponding to the inlet and outlet uid at the extremities of eachpipe, small voids are created in themodel as depicted in Fig.10. Thisfeature allows to assign an outow boundary condition along theoutlet boundary of each pipe. To model the U-loop at the base of aBHE, a numerical integration is performed at the outlet of eachsupply pipe. The correspondingmean temperature is then used as atime-dependent prescribed boundary condition along the inletboundary of the return pipe, leading to a coupled nonlinear prob-lem. The same technique is used to emulate BHE in series andensure the same uid temperature between the outlet of each

return pipe and the inlet of the supply pipe located downstream.Finally, for BHE connected in series, a prescribed temperatureboundary condition given by Tint Toutt qt=Cp _m is usedto link the temperature leaving the last BHE (Tout) to the inlettemperature of the rst BHE (Tin). If the BHEs are connected inparallel, the same condition is used to link the supply and returnpipes of each BHE.

The sizes and the thermal properties of the materials present inthemodel are summarized in Table 2. Care has been taken to ensurethat the equivalent borehole resistance of the numerical modelcorresponds to the Rb value integrated in matrix D. In fact, whenused in conjunctionwith the Multipole Method [2], the parameterspresented in Table 2 correspond to a Rb value of 0.121 mK/W. Notethat turbulent ow conditions within the pipes are represented byassigning to the uid a much higher thermal conductivity in thehorizontal direction than in the vertical direction.

The volume where the BHEs are embedded is divided into 60layers of triangular prisms while an unstructured mesh is usedabove and below the BHEs. Ultimately, the domain is discretizedin 2 334 674 linear nite elements, which corresponds to aproblem of 1 006 175 degrees of freedom. This meshing was thenused to compute a solution at each time step from an initialground temperature of T(0) 0. A cross-section showing a

a) b)

c

D. Marcotte, P. Pasquier / Renewable Energy 68 (2014) 14e24 21Fig. 12. Heat transfer ratios (%) obtained by the analytical and numerical model for parallel (aequals 100/8 as there are 8 branches in the GHE.)) and series arrangements fed by (b) BHE1 (center) or (c) BHE3 (external). The sum of %

temperature solution across two of the eight branches is illus-trated in Fig. 11.

4.2. Results of comparison test

The heat transfer ratios and the unit response functions ob-tained by the analytical and the numerical models for the paralleland the two series arrangements are presented in Figs. 12 and 13.

The numerical model shows similar results to the analyticalmodel regarding the values and the time evolution of the heattransfer occurring at the different BHEs. The numerical values ob-tained for ~T in are slightly smaller than the analytical model values.The steady 3.5% difference for ~T in is deemed acceptable and can beattributed to the difference of assumptions inherent to the twoapproaches. Hence, the analytical model assumes that the heattransfer is distributed evenly along the BHE length and that theuid temperature varies linearly along the pipes. These assump-tions are notmade in the numerical model. Moreover, the analyticalmodel assumes instantaneous effects of heat transferred to theuid, whereas the numerical model incorporates the time delaydue to advection. Nevertheless, the analytical model is much easier

distribution in the ground is evidently different in both cases, as

One of the interesting ndings is that, for the mixed arrange-ment in the radial case, the ~T in is insensitive to the location of thestart of the series, center or exterior. This somewhat surprisingresult was conrmed by the 3D numerical model and can haveimportant consequences for heat storage applications.

The FLS model is of course a simplication of the heat transferprocess occurring in a GHE. It was shown to lie within 3.5% of the3D numerical solution obtained with a comprehensive niteelement GHEmodel developed in COMSOL. However, for very shorttimes (less than 1 h) the FLS cannot represent the complex heattransfer occurring within the BHE radius. But at short times theinteractions between BHEs are non-perceptible (as a consequence,matrix G becomes diagonal in Equations (9), (13) and (14)). Thissuggests to use alternative models to the FLS for the short times.One approach is to use the thermal resistance-capacitance models(TRCM) [1,8,19]. With these models, Tin can be computed directly atshort times andmergedwith the FLS Tin obtained at longer times. Asingle TRCMmodel would be required for the parallel arrangementwhereas the TRCM can easily be coupled in series for the series casedue to the lack of interactions occurring between BHEs at thesetimes. This opens the possibility to generate realistic uid tem-perature unit response function covering time intervals from mi-nutes to centuries.

One limitation of the proposed approach is the assumption,

D. Marcotte, P. Pasquier / Renewa22~veried in Fig. 14. This observation is valid for both the analyticaland the numerical model.

5. Discussion

The analytical model described enables to compute, withsimple algebraic manipulations, the heat transfer rate and theTin temperature at the end of each time step for any arrange-ment of the BHEs, parallel, series or mixed, and any locations ofthe BHEs. The model relies on the FLS, and the assumption oflinear variation of uid temperature along the pipes. Into manipulate and constitutes a good and exible alternativeapproximation to the heavy numerical model.

One notes that the parallel ~T in presents only slightly inferiorvalues compared to the mixed ~T in. This reects that the series isshort along each branch with only 3 BHEs. More important, and toour knowledge this was not previously noticed in the literature, thetwo series arrangements, i.e. fed by the central BHE or by theexternal BHE, provide identical ~T in, despite the fact that the heatFig. 13. T in obtained by the analytical and numerical models for parallel and seriesarrangements.principle, any model allowing computation of the transferfunction f in Equation (4) can be used (e.g. the inclined FLS Cuiet al. [7], Marcotte and Pasquier [17], Lamarche [12]). The al-gorithm involves only convolutions and the computed responsesare exact in the sense that the FLS does reproduce exactly the Tintemperature with the time varying heat transfer qt computed ateach BHE under the same assumption for the linear variationalong the pipes of uid temperature. As for the CPU aspect, thediscrete convolutions can be computed efciently by FFT as inMarcotte and Pasquier [16]. More importantly, only a smallsubset of the time steps need to be evaluated to estimate thewhole ~T in function as shown in Fig. 6, therefore reducingconsiderably the CPU effort required. The ~T in function is theninterpolated to the desired time step for the convolution withthe heat load function, allowing to predict the uid temperaturefor any scenario of heat load.

0 1 2 3 4 5 6 7 8

5

6

7

8

9

10

11

12

13

A B

Position along profile (m)

Gro

und

Tem

p. (o

C)

CenterExteriorParallel

Fig. 14. Ground temperature perturbation (T(x) T0) along prole AeB of Fig. 7, for aconstant heat injection of 15 W/m, at day 180. The ground temperature is the averagetemperature integrated along a vertical line parallel to the BHE.

ble Energy 68 (2014) 14e24intrinsic to the FLS model, of spatially constant ground thermal

ewaparameters. Situations where the ground thermal parameters aredeemed to vary in space would have to be dealt with by otherapproaches such as TRCM or numerical models.

6. Conclusion

The different BHE arrangements, parallel, in series or mixed, canbe represented in a unied way by a linear system of equations thatcan be solved at each time step. This enables to compute the heattransfer specic to each BHE and the temperature unit responsefunctions ~T in proper to the arrangement and the BHE network.Only a few time steps need to be considered, which allows efcientcomputation of the uid unit response function ~T in. The latter canthen be convolved in the spectral domain with any desired heatload to provide uid or ground temperature solutions.

Acknowledgments

This research was nanced by the National Science and Engi-neering Research Council of Canada. Helpful comments by twoanonymous reviewers are acknowledged.

Nomenclaturecs ground volumetric heat capacity (J/(m3 K))Cp uid specic heat capacity (J kg1 K1)D diagonal matrix for effects due to the BHE resistance and

the uid temperature linear variation along the pipes (K/W)

fDT unit temperature response at one time stepG matrix n n of temperature perturbation caused, at BHE i

by one unit of heat transfer at BHE j (K/W)hij historical part of the temperature perturbation caused by

BHE j at BHE i (K)h vector of temperature perturbations at the n BHEs due to

the historical part (K)H borehole length (m)k thermal conductivity (W/(m K))ks ground thermal conductivity (W/(m K))L lower triangular matrix for effects due to series

arrangement (K/W)_m mass ow rate (kg/s)n number of BHEsr ground density (g/cm3)q(t) heat transfer for the whole boreeld at time t

qt Pnj1 qjt (W)qj(t) heat transfer by BHE j at time t (W)eqj vector of m heat transfer increments at BHE j (W)qt vector of the n BHE heat transfer coefcients at time t (W)rb borehole radius (m)rij distance between BHEs i and j (m)Rb BHE effective thermal resistance (m K/W)t timets Eskilsons characteristic time, ts H2Cs=9ks (s)Tf uid temperature (K)Tin uid temperature at the entrance of the boreeld (K)~T in dimensionless uid temperature response ()Tout uid temperature at the outlet of the boreeld (K)T0 undisturbed ground temperature (K)T vector of temperatures at the n BHEs (K)u uid velocity vector (m/s)x, y spatial coordinates (m)1 vector of size n of ones

Subscripts

D. Marcotte, P. Pasquier / Renf Fluidi, j Borehole indexin, out inlet, outlet of the boreeldm time step indexs groundt time (s)Dt time step (s)

Appendix A

Three 30 m long BHEs are located along a line at x 0 m,x 2 m and x 5 m. The parameters used for the computationsare given in Table 1. Computations are illustrated for a 1000 h timestep at times 1000 h and 2000 h, for the three different arrange-ments: parallel, series and mixed. In the mixed cases, BHE1 andBHE2 form one branch, BHE3 is connected in parallel with theBHE1e2 branch.

The G matrix is time invariant and invariant for the differentarrangements. It is computed using the FLS model evaluated at therst time step (here 1000 h) for a 1 W heat injection per BHE (i.e. 1/30 W/m as H 30 m), for the distances:24 r11 r12 r13r21 r22 r23r31 r32 r33

35

240:08 2 52 0:08 3

5 3 0:08

35m (18)

One nds:

G 24 7:6489 1:1413 0:116711:1413 7:6489 0:557670:11671 0:55767 7:6489

35 103K=W (19)

D is diagonal with the constant value 0.12/30 1/(2 0.3 4180) 4.3987 103 (K/W) as each BHE has the same resistanceand the same mass ow rate for the 3 arrangements. The timeinvariant L matrices are for the series arrangement:

Lseries 24 0 0 00:79745 0 00:79745 0:79745 0

35 103K=W (20)

and for the mixed arrangement:

Lmixed 24 0 0 00:79745 0 0

0 0 0

35 103K=W (21)

A1. Solution for the rst time step (t 1000 h)

At the rst time step, there is no history to account for. Thereforeh 0. Applying Equations (9), (13) and (14) with the above matrixdenitions, one nds the solutions:

Parallel Series Mixedq11000 0:33283 0:35625 0:34153q21000 0:31874 0:31699 0:30215q31000 0:34842 0:32676 0:35631Tin T0 0:0044143 0:0046919 0:0045011

and ~T in1000 6:2405; 6:6330; 6:3633.

A2. Second time step (t 2000 h)

The response at t 2000 h due to the heat injected contin-uously at each BHE during the rst 1000 h has to be computedusing the heat transfer rate obtained in the previous solution. As

ble Energy 68 (2014) 14e24 23an example, for the parallel case, the historical contribution of

BHE2 to the BHE1 wall temperature is obtained by evaluating~qjf 0:31874; 0:318740:0011413; 0:001703 and retain-ing the mth element, here: 0.00017905. The process is repeatedfor each pair of BHEs. One nds the interactions matrix:

Finally, vector h is computed by summing along a given row thecontributions found in the different columns. Hence,h [0.00048425, 0.00055751, 0.0004584]T. The vectors at time2000 h for the different arrangements are computed similarly:

Parallel Series Mixed0:00048425 0:00049336 0:00048261

h 0:00055751 0:00055953 0:00055509

References

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inclined boreholes. Appl Therm Eng 2006;26:1169e75.[8] De Carli M, Tonon M, Zarrella A, Zecchin R. A computational capacity resis-

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[10] Ingersoll L, Zobel O, Ingersoll A. Heat conduction with engineering, geologicaland other applications. New York: McGraw-Hill; 1954.

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Injector j

1 2 3

Receiver i 1 0.0002194 0.00017905 8.5797e-0052 0.00018697 0.00021011 0.000160433 8.1957e-005 0.00014677 0.00022968

D. Marcotte, P. Pasquier / Renewable Energy 68 (2014) 14e24240:0004584 0:00044908 0:0004581

Finally the different solutions are obtained as:

Parallel Series Mixedq12000 0:33448 0:35711 0:34324q22000 0:31353 0:31153 0:29695q32000 0:35198 0:33136 0:35982Tin T0 0:0049129 0:0051898 0:0049987

fromwhich ~T in2000 6:9454; 7:337; 7:0667 is computed. Theprocess is repeated sequentially up to the last time. Oncecompleted, the ~T in signal can be convolved with any heat loadsignal so as to compute the uid temperature over time. Recall that~T in is a normalized (dimensionless) uid temperature corre-sponding to the constant injection of 1 W/m. Therefore, the in-cremental heat load signal to convolve with ~T in must be expressedin W/m.tems. ASHRAE Trans 2007;113(1). DA-07-050.[14] Lamarche L, Beauchamp B. A new contribution to the nite line-source model

for geothermal boreholes. Energy Build 2007;37:188e98.[15] Lazzarotto A. A network-based methodology for the simulation of borehole

heat storage systems. Renew Energy 2014;62:265e75.[16] Marcotte D, Pasquier P. Fast uid and ground temperature computation for

geothermal ground-loop heat exchanger systems. Geothermics 2008;37:651e65.

[17] Marcotte D, Pasquier P. The effect of borehole inclination on uid and groundtemperature for GLHE systems. Geothermics 2009;38:392e8.

[18] Marcotte D, Pasquier P, Sheriff F, Bernier M. The importance of axial effects forborehole design of geothermal heat-pump systems. Renew Energy 2008;35:763e70.

[19] Pasquier P, Marcotte D. Short-term simulation of ground heat exchanger withan improved TRCM. Renew Energy 2012;46:92e9.

[20] Pasquier P, Marcotte D. Efcient computation of heat ux signals to ensure thereproduction of prescribed temperatures at several interacting heat sources.Appl Therm Eng 2013;59:515e26.

[21] Pasquier P, Marcotte D, Kummert M, Bernier M. Simulation of ground-coupledheat pump systems using a spectral approach. In: Proceedings of BS2013:13th conference of international building performance simulation association,Chambry, France; 2013. pp. 2691e8.

Unit-response function for ground heat exchanger with parallel, series or mixed borehole arrangement1 Introduction2 Methodology2.1 Borehole interactions2.2 Pure parallel arrangement2.3 Pure series arrangement2.4 Mixed arrangements2.5 GHE dimensionless unit response function

3 Results3.1 A simple example3.2 Sensitivity to the time step

4 Comparison with a numerical model4.1 Description of the numerical model in COMSOL4.2 Results of comparison test

5 Discussion6 ConclusionAcknowledgmentsNomenclatureAppendix AA1 Solution for the first time step (t = 1000 h)A2 Second time step (t = 2000 h)

References