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Applied Thermal Engineering 31 (2011) 3066e3077

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Applied Thermal Engineering

journal homepage: www.elsevier .com/locate/apthermeng

Coupling of geothermal heat pumps with thermal solar collectors using doubleU-tube boreholes with two independent circuits

Parham Eslami-nejad, Michel Bernier*

École Polytechnique de Montréal, Département de génie mécanique, Case Postale 6079, succursale “centre-ville”, Montréal, Québec, Canada H3C 3A7

a r t i c l e i n f o

Article history:Received 2 March 2011Accepted 28 May 2011Available online 13 June 2011

Keywords:Heat pumpGeothermalBoreholeHeat transferSolar injectionDouble U-tubeSimulation

* Corresponding author. Tel.: þ1 514 340 4711x438E-mail address: michel.bernier@polymtl.ca (M. Be

1359-4311/$ e see front matter � 2011 Elsevier Ltd.doi:10.1016/j.applthermaleng.2011.05.040

a b s t r a c t

This study presents an analytical model to predict steady-state heat transfer in double U-tube boreholeswith two independent circuits operating with unequal mass flow rates and inlet temperatures. Themodel predicts the fluid temperature profiles in both circuits along the borehole depth. It accounts forfluid and pipe thermal resistance and thermal interaction among U-tube circuits. The proposed model isused to study a novel double U-tube borehole configuration with one circuit linked to a ground-sourceheat pump operating in heating mode and the other to thermal solar collectors. The performance of thisconfiguration is compared to a conventional ground-source heat pump system (without thermalrecharge of the borehole) and to a single-circuit solar assisted ground-source heat pump system. Allthree systems are simulated over a 20-year period for a residential-type single-borehole configuration.Results indicate that winter solar recharging, either for the proposed configuration or the solar assistedground-source heat pump system, reduces by 194 and 168% the amount of energy extracted from theground by the heat pump. It is also shown that, for a ground thermal conductivity of 1.5 W m�1 K�1, theborehole length can be reduced by up to 17.6%, and 33.1% when the proposed configuration or the solarassisted ground-source heat pump system are used. The impact on the annual heat pump energyconsumption is less dramatic with corresponding reductions of 3.5% and 6.5%.

� 2011 Elsevier Ltd. All rights reserved.

1. Introduction

Systems that link heat pumps to vertical borehole ground heatexchangers are proving to be energy-efficient systems to heat andcool buildings. However, the cost associated with the boreholeremains relatively high which handicaps the widespread applica-tion of this technology. Furthermore, when the heat pump operatesin heatingmode, it collects heat from the groundwhich reduces theground temperature near the borehole. In turn, the lower groundtemperature decreases the Coefficient of Performance (COP) of theheat pump. Thus, it might be advantageous to inject heat into theborehole to increase the ground temperature and heat pumpperformance. One possibleway is to inject solar energy by using thenovel borehole configuration presented schematically in Fig. 1. Itconsists of a 4 pipe (2 U-tubes) vertical borehole with two inde-pendent circuits. One U-tube is linked to a heat pump and the otherto a thermal solar collector. As shown in Fig. 1, these circuits arereferred to as the heat pump circuit (hp-circuit) and the heat sourcecircuit (hs-circuit), respectively. In effect, this configuration withtwo independent circuits acts as a heat exchanger between the heatsource and the heat pump. The system can operate in three

1; fax: 1 514 340 5917.rnier).

All rights reserved.

different modes: heat pump only; solar charging only, or simulta-neous heat pump and solar charging operation. This latter case isthe main subject of the present investigation. A newmodel for sucha configuration is elaborated in this paper to accurately assess itsperformance. The proposed model predicts steady-state heattransfer in double U-tube boreholes with two independent circuitsoperating with unequal mass flow rates and inlet temperatures.The model predicts the fluid temperature profiles in both circuitsalong the borehole depth. It accounts for fluid and pipe thermalresistance and thermal interaction among U-tube pipes.

1.1. Literature review

Several numerical and analytical models have been developedto simulate heat transfer in single U-tube boreholes. He et al. [1]developed a finite-volume based three-dimensional model tosimulate the dynamic response of the circulating fluid as well astransient heat transfer in and around boreholes. Young [2] evalu-ated analytically the short-time response of boreholes based on the“buried cable” solution given by Carslaw and Jaeger [3] whichaccounts for the grout and fluid thermal capacities. Lamarche andBeauchamp [4] approximated the short-time response of singleU-tube boreholes using an analytical solution to the unsteady one-

Fig. 1. Schematic representation of the system under study.

Fig. 2. Schematic representation of conventional solar assisted ground-source heatpump system.

P. Eslami-nejad, M. Bernier / Applied Thermal Engineering 31 (2011) 3066e3077 3067

dimensional heat conduction problem of concentric cylinders. Manet al. [5] developed an analytical model which accounts for the heatcapacity of boreholes by assuming a homogenous medium for thewhole calculation domain including the ground in the vicinity ofthe borehole. Remund [6] proposed a set of relationships fordifferent single U-tube borehole configurations to calculate thesteady-state borehole thermal resistances based on conductionshape factors obtained from empirical data. Equal fluid tempera-tures in both circuits are assumed in his approach. Marcotte andPasquier [7] proposed a so-called “p-linear” average temperatureusing a 3-D numerical simulation to evaluate the borehole thermalresistance from experimental data. A review paper by Lamarcheet al. [8] compared different existing approaches to calculateborehole thermal resistance including thermal short-circuitbetween pipes. They also performed an unsteady 3D numericalsimulation of a standard single U-tube borehole.

A few studies have also modeled conventional single-circuitdouble U-tube boreholes. Al-Khoury and Bonnier [9] applied thethree-dimensional finite element method to analyze transient heattransfer in a double U-tube parallel arrangement. Hellström [10]derived steady-state two-dimensional analytical solutions forborehole thermal resistances with an arbitrary number of pipes.Zeng et al. [11] used this work to establish an analytical quasi-three-dimensional model for single and double U-tube configuredeither in parallel or in series. Diao et al. [12] used this latterapproach to simulate heat transfer inside boreholes, includingthermal interactions among tubes, along with the finite-line sourcesolution to predict heat transfer in the ground. Wetter and Huber[13] modeled the transient behavior of a double U-tube boreholewith a single equivalent pipe diameter which does not allowmodeling of two independent circuits in one borehole. Even thoughthese last five studies examined double U-tubes none of theminvestigated the use of two independent circuits.

Despite the fact that ground-source heat pump (GSHP) systemsconstitute attractive options for heating and cooling applications, ithas been shown that in heating-dominated climates, the perfor-mance of conventional GSHP system decreases gradually over time.This is due to the fact that more heat is extracted from the groundthan the amount rejected during the cooling season. Thus, theground temperature in the vicinity of the borehole decreases overthe years with a resulting decrease in the inlet temperature to the

heat pumps which translates into a reduction in the coefficient ofperformance (COP). Bernier [14] indicated that for a typicalconstant heat extraction of 37.5 W/m, the borehole wall tempera-ture decreases by approximately 5 �C over a 24 h period. Trillat-Berdal et al. [15] also reported a reduction of 2 �C of the soiltemperature in the vicinity of double U-tube boreholes over twentyyears of heat pump operation.

One possibility to let the ground temperature recover from heatextraction is to use, alternatively, the ground and another source forthe heat pump. This idea was first introduced by Penrod and Pra-sanna [16,17] who proposed such a dual-source (solar collector andground) system. Yang et al. [18] recently recommended using solar-source heat pump (SSHP) and ground-source heat pump (GSHP)systems alternately for a period of 10e14 h hours a day to achievea 30e60% recovery-rate for the ground temperature in the vicinityof the borehole.

Another possibility which can be used for ground temperaturerecovery is to inject heat into the boreholes. In this configurationa GSHP is combined with thermal solar collectors using an addi-tional heat exchanger as shown in Fig. 2. Although adding an addi-tional heat exchanger may add to the complexity of the system [19],several studies indicated that it is a viable option for heating-dominated climates. Yang et al. [20] reviewed various GSHPsystems assisted by supplemental heat rejecters or heat supplydevices such as solar collectors. They concluded that in general, theuse of a supplemental source such as solar energy can make GSHPsystems economically attractive. Wang et al. [21] performed anexperimental study of a GSHP system coupled with seasonal bore-hole storage. Solar energy is stored in the summer in 12 singleU-tube boreholes. They indicated that the operation of sucha system improves significantly the heat pump heating COP. Chias-son and Yavuzturk [22] performed a 20-year life-cycle cost analysisto evaluate the economics of ground heat pump systems coupled tothermal solar collectors for six different climates in the U.S. Theyconcluded that GSHP systems combined with solar collectors areeconomically viable for heating-dominated climates. Stojanovic andAkander [23] conducted a two-year performance test on a full-scalesolar assistedheat pump system for a residential building in a nordicclimatic. They indicated that despite unfavorable building condi-tions, the proposed system succeeded in fulfilling the heatingrequirements. Han et al. [24] investigated different operationalmodes of a solar assisted GSHP system in the presence of latentheat energy storage tanks in a heating-dominated climate. Theyindicated that using storage tanks increases system performance by12.3%. Kjellsson et al. [25] analyzed five alternatives to supply solar

Fig. 3. Borehole cross-section showing the 1e3, 2e4 configuration.

Fig. 4. Nomenclature used in Eq. (1).

P. Eslami-nejad, M. Bernier / Applied Thermal Engineering 31 (2011) 3066e30773068

energy to a ground-source heat pump (GSHP) system and comparedthem against a base case without solar collectors. They concludedthat the option with the highest electrical consumption savings isa hybrid systemwith solar heat injection into the borehole inwinterand solar domestic hot water production during the summer. Solarheat is first directed to the heat pump and then to the borehole.Double U-tube boreholes with two independent circuits were notconsidered. Based on a series of experiments, Georgiev et al. [26]indicated that only a small portion of solar energy injected intothe ground can be extracted after a month from the first injection.They injected solar energy to a shallow single U-tube borehole foraperiod of 700hand afterward they started to extract heatout of theborehole for 300 h. Only 32% of the total accumulated energy couldeventually be extracted due to the heat dissipation into the groundand heat losses to the surface.

In order to reduce the relative complexity of the system pre-sented in Fig. 2 and to make simultaneous heat injection and heatextraction possible, double U-tube boreholeswith two independentcircuits can be used (Fig. 1). Chapuis and Bernier [27,28] are at theorigin of two studies which attempted to model double U-tubeswith two independent circuits. In the first study, Chapuis and Ber-nier [27] used an external heat exchanger combined to the ductground heat storage (DST) model [29] to mimic simultaneouscharging and discharging in the ground heat exchanger. Later, theymodified the DST model to handle double U-tube boreholes withtwo independent circuits [28]. However, thermal interactionbetween the U-tubes in the boreholewas not taken into account norwas the axial temperature variation in the fluid in both legs. Kum-mert andBernier [30] proposed anewsystem for space conditioningand domestic hot water heating in a cold climate using a gas-firedabsorption heat pump coupled to a vertical geothermal boreholewith two independent circuits linked, respectively, to the evapo-rator and condenser of the heat pump. They considered threemodesof operations including heating only, cooling only, and simultaneousheating/cooling. The ground heat exchanger wasmodeled using theDST model but simultaneous flows in the two circuits could not beevaluated. Bernier andSalimShirazy [31] andEslamiNejadet al. [32]evaluated the impact of solar heat injection on borehole length andheat pump energy consumption. However, they approximated thedouble U-tube configurationwith a single U-tube based on a simpleapproach which assumed that heat collected/rejected into theground is the sumof the solar energy injected andheat pumpenergyextraction/rejection. Furthermore, both outlet fluid temperatureswere assumed equal. In a related work, Eslami Nejad and Bernier[33] developed a double U-tube borehole model with two inde-pendent circuits. However, the model is restricted to identicalworking fluid and equal mass flow rate in each circuit.

2. Model development

Fig. 3 presents a cross-section of a four-pipe borehole. The spacebetween the pipes and the boreholewall is assumed to befilledwitha solidmaterial (grout). There are three possible ways of connectingthese four pipes to form two independent circuits: As suggested byZeng et al. [11], each configuration can be identifiedwith a four-digitnotation. For example, in the 1e3, 2e4 configuration, the fluid fromthefirst circuit enters pipe#1goes to the bottomof the borehole andthen up pipe #3. Similarly, the fluid from the second circuit enterspipe #2 goes to the bottomof the borehole and then up pipe #4. Theother possible configurations are: 1e2, 3e4, 1e2, 4e3.

Zeng et al. [11] examined the performance of all three configura-tions for the single-circuit case with pipes arranged in series orparallel. They concluded that the 1e3, 2e4 configuration presents thelowest borehole thermal resistance. Based on this result itwas decidedto restrict the model development to the 1e3, 2e4 configuration.

As shown in Fig. 3, the model assumes that the pipes are placedsymmetrically in the borehole with identical center-to-centerdistance (2D) between two opposing pipes. Both circuits are inde-pendent and have different inlet temperatures T 0

fsT 00f and heatcapacities ð _mCÞ1�3sð _mCÞ2�4. Other modeling assumptionsinclude: (i) the heat capacities of the grout and pipe inside theborehole are neglected; (ii) the ground and the grout are homoge-neous and their thermal properties are constant; (iii) the boreholewall temperature (Tb in Fig. 4) is uniform over the borehole depth;(iv) heat conduction in the axial direction is neglected; (iv) thecombined fluid convective resistance, pipe wall thickness conduc-tion resistances are assumed to be equal in both circuits. Theseassumptions have been used in the past by a number of researchers(e.g. Zeng et al. [11] and Hellstrom [10]). More recently, Lamarcheet al. [8] compared borehole thermal resistance calculationmethods against an unsteady three-dimensional borehole model. Agood agreement for the axial fluid temperature distribution ofa single U-tube borehole was reported between the approachproposed by Zeng et al. [11], which uses the assumptionsmentionedabove, and the three-dimensional simulation results. Yang et al. [34]coupled a steady-state single U-tube borehole model, assuminga uniform Tb, to an unsteady one-dimensional ground heat transfermodel based on the cylindrical heat source approach. This two-region model was validated experimentally. Results indicated thatthe calculated fluid outlet temperatures are in very good agreementwith experimental data in the steady-state regime.

P. Eslami-nejad, M. Bernier / Applied Thermal Engineering 31 (2011) 3066e3077 3069

2.1. Heat flow balance equations

The difference between the borehole wall temperature and thefluid temperatures in each circuit is the result of net heat flows perunit length, q1, q2, q3, and q4 in and out of the four pipes. Based onthe approach presented by other authors [10,11] and using thenomenclature presented in Fig. 4, the following heat flow balancesare obtained:

Tf1ðzÞ � Tb ¼ R11q1 þ R12q2 þ R13q3 þ R14q4Tf2ðzÞ � Tb ¼ R21q1 þ R22q2 þ R23q3 þ R24q4Tf3ðzÞ � Tb ¼ R31q1 þ R32q2 þ R33q3 þ R34q4Tf4ðzÞ � Tb ¼ R41q1 þ R42q2 þ R43q3 þ R44q4

(1)

In Eq. (1), TfiðzÞði ¼ 1;2;3;4Þ represents the fluid temperatureat a certain borehole depth z, Riiði ¼ 1;2;3;4Þ is the thermalresistance between the fluid in pipe i and the borehole wall, andRijði; j ¼ 1;2;3;4Þ is the thermal resistance between pipes i and j.Since the pipes are assumed to be positioned symmetrically in theborehole, Rij ¼ Rji;Rii ¼ Rjj;R24 ¼ R13; andR23 ¼ R14 ¼ R34 ¼ R12.Thus, only three thermal resistancesR11; R12; R13, need to be eval-uated. Hellström [10] presented a technique to evaluate Rii and Rijbased on the line source solutions for each pipe which are thensuperimposed. This leads to:

R11 ¼ 12pkb

"ln�rbrp

�� kb � kkb þ k

ln

r2b � D2

r2b

!#þ Rpipe

R12 ¼ 12pkb

"ln�

rbffiffiffi2

pD

�� kb � k2ðkb þ kÞ ln

r4b þ D4

r4b

!#

R13 ¼ 12pkb

"ln� rb2D

�� kb � kkb þ k

ln

r2b þ D2

r2b

!# (2)

Where k and kb are the ground and grout thermal conductivities,respectively, rp is the outer radius of the pipe, Rpipe, which isassumed constant over the borehole depth, is the combined fluidconvective resistance, pipe wall thickness conduction resistance,and contact resistance associated with gaps between the pipes andthe grout. It is defined as follow:

Rpipe ¼ 12phiri;p

þ ln�rp=ri;p

�2pkp

þ Rair (3)

where ri,p is the inner pipe radius, kp is the pipe thermal conduc-tivity, and hi is fluid convective heat transfer coefficient. The secondterm on the right hand side is the conductive thermal resistance ofthe pipe. The third term, Rair, is a contact resistance at the grout/pipe interface. This last resistance was set to zero in this work.However, it could easily be included in the value of Rpipe if required.The reader is referred to the work of Philippacopoulos and Berndt[35] for a discussion on the effect of air-filled gaps at the grout/pipeinterface. Finally, the value of hi is assumed to be the same in bothcircuits and constant along the depth of the borehole.

The fluid temperature varies along the borehole depth so eachheat flow per unit length in Eq. (1) can be replaced with the firstorder derivative of the corresponding fluid temperature as a func-tion of zmultiplied by the corresponding fluid thermal capacity, i.e.the product of fluid mass flow rate, _m, and specific heat, C. Forconfiguration 1e3, 2e4 the net heat flows per unit length, q1, q2, q3,and q4 are then:

q1 ¼ �� _mC�1e3dTf1ðzÞ=dz; q2 ¼ �� _mC

�2e4dTf2ðzÞ=dz;

q3 ¼ þ� _mC�1e3dTf3ðzÞ=dz; q4 ¼ þ� _mC

�2e4dTf4ðzÞ=dz (4)

The z-coordinate direction is defined as downward (from theground surface) and, as indicated in Fig. 4, an outward heat flow isconsidered positive. Thus, a negative sign in Eq. (4) indicates thatthe fluid is flowing in the downward direction. Conversely, a posi-tive signs implies an upward flow. It is to be noted that the modelcan handle different thermal capacities in both circuits, i.e.ð _mCÞ1�3sð _mCÞ2�4.

Eq. (1) can be rearranged in terms of the net heat flows in eachpipe and afterward by substituting Eq. (4) into Eq. (1), a set ofcoupled linear differential equations are obtained:

�� _mC�1�3dTf1ðzÞ=dz ¼ Tf1�Tb

RD1þTf1�Tf2

RD12þTf1�Tf3

RD13þTf1�Tf4

RD12

�� _mC�2�4dTf2ðzÞ=dz ¼ Tf2�Tf1

RD12þTf2�Tb

RD1þTf2�Tf3

RD12þTf2�Tf4

RD13�_mC�1�3dTf3ðzÞ=dz ¼ Tf3�Tf1

RD13þTf3�Tf2

RD12þTf3�Tb

RD1þTf3�Tf4

RD12�_mC�2�4dTf4ðzÞ=dz ¼ Tf4�Tf1

RD12þTf4�Tf2

RD13þTf4�Tf3

RD12þTf4�Tb

RD1(5)

where

RD1 ¼ R11 þ R13 þ 2R12RD12 ¼

�R211 þ R213 þ 2R11R13 � 4R212

�.R12

RD13 ¼hðR11 � R13Þ

�R211 þ R213 þ 2R11R13 � 4R212

�i.�R213 þ R11R13 � 2R212

�

2.2. Dimensionless governing equations and boundary conditions

Eq. (6) are non-dimensionalized using the following dimen-sionless variables:

qi¼1;2;3;4 ¼h�

TfiðzÞ � T 0f�þ�TfiðzÞ � T 00f

�i.�T 0f � T 00

f

�; T 0fsT 00f

qb ¼h�

Tb � T 0f�þ�Tb � T 00f

�i.�T 0f � T 00

f

�R*1 ¼ �

_mC�1�3R

D1=H ; R*12 ¼ �

_mC�1�3R

D12=H ;

R*13 ¼ �_mC�1�3R

D13=H and Z ¼ z=H

(6)

where, as indicated in Fig. 3, T 0f and T 00

f are the inlet temperatures tothe 1e3 and 2e4 circuits, respectively. The dimensionless thermalresistances, R*1, R*12,R

*13 are all defined based on the thermal

capacitance of circuit 1e3, ð _mCÞ1�3. Then, the heat flow balanceequations are simplified as follows:

�dq1=dZ ¼ aq1 þ bq2 þ cq3 þ bq4 þ d�dq2=dZ ¼ aðbq1 þ aq2 þ bq3 þ cq4 þ dÞdq3=dZ ¼ cq1 þ bq2 þ aq3 þ bq4 þ ddq4=dZ ¼ aðbq1 þ cq2 þ bq3 þ aq4 þ dÞ

(7)

where

a ¼ 1=R*1 þ 2=R*12 þ 1=R*13; b ¼ �1=R*12; c ¼ �1=R*13;

d ¼ �qb=R*1

and

a ¼ �_mC�1�3=

�_mC�2�4

Fig. 5. Temperature difference between the inlet and outlet of both circuits asa function of a ( _mhp is kept constant).

P. Eslami-nejad, M. Bernier / Applied Thermal Engineering 31 (2011) 3066e30773070

Considering that T 0f ¼ Tf1ð0Þ and T 00f ¼ Tf2ð0Þ, the dimension-less boundary conditions are:

q1ð0Þ ¼ 1; q2ð0Þ ¼ �1; q1ð1Þ ¼ q3ð1Þ; q2ð1Þ ¼ q4ð1Þ (8)

Finally, the dimensional and dimensionless outlet temperatures ofboth circuits will be denoted by Tf3(0), Tf4(0), q3ð0Þ;q4ð0Þ,respectively.

2.3. Laplace transforms

Using Laplace transforms the dimensionless heat flow balanceequations are transformed to yield a set of linear algebraic equa-tions as follows:

ðaþ pÞq1 þ bq2 þ cq3 þ bq4 ¼ 1� d=pabq1 þ ðaaþ pÞq2 þ abq3 þ acq4 ¼ �1� ad=pcq1 þ bq2 þ ða� pÞq3 þ bq4 ¼ �q3ð0Þ � d=pabq1 þ acq2 þ abq3 þ ðaa� pÞq4 ¼ �q4ð0Þ � ad=p

(9)

where qðpÞi ¼RN0 e�pZqiðzÞdZ.

This set of equations can be solved, using Gaussian eliminationfor example, to obtain q1; q2; q3; and q4 as a function ofa; b; c; d; a; q3ð0Þ; q4ð0Þ; and p. Then, the inverse Laplace trans-forms of q1; q2; q3; and q4 are evaluated to obtain dimensionlesstemperature distributions. Boundary conditions are then applied toevaluate the dimensionless outlet temperature, q3ð0Þ and q4ð0Þ.The resulting dimensionless temperature distributions, qi¼1;2;3;4ðZÞ,are presented in the appendix. These results were validated againstthe equations developed by Zeng et al. [11] for the simpler case ofequal mass flow rates in a parallel single-circuit arrangement. It canbe shown that the resulting temperature distributions are identicalfor both approaches indicating that the proposed model has beencorrectly implemented.

3. Applications

In this section, the proposed model for double U-tube boreholeswith two independent circuits is used in two applications. The firstapplication evaluates the effect of unequal mass flow rates on heattransfer between each circuit for constant (but different) inlettemperatures in both circuits. In the second application, hourlysimulations over a twenty-year period are performed to quantifythe impact of solar recharging a borehole on heat pump energyconsumption, borehole length, and net ground heat extraction.Results are compared with the solar assisted GSHP system pre-sented in Fig. 2 and typical a GSHP systemwithout solar recharging.Typical geothermal borehole characteristics are used and are pre-sented in Table 1. The specific heats of both fluids are set to thesame value (4.00 kJ kg�1 K�1) but the mass flow rates are different.

3.1. Constant inlet conditions

For this first application, the borehole length (H) is set to 100 mand T 0f ¼ Tf1ð0Þ ¼ 25oC; T 00

f ¼ Tf2ð0Þ ¼ �5oC; Tb ¼ 10 �C leadingto a dimensionless borehole wall temperature qb (Eq. (6)) equal tozero. Themass flow rate of the hp-circuit is equal to _mref and is keptconstant. The thermal capacity ratio, a, is varied by changing themass flow rate in the hs-circuit. The flow regime is assumed to be

Table 1Borehole characteristics.

rb (cm) Rop (cm) D (cm) k (W m�1 K�1) _mref (kg s�1) kb (W m�1 K�1)

7.5 1.67 3.06 1.5 0.44 1

turbulent over the range of variation of a which lead to a value ofRpipe equal to 0.1 [m K W�1] for all four pipes.

Fig. 5 shows the temperature difference between the inlet andoutlet of each circuit as a function of a, the thermal capacity ratio.The left axis presents the results in dimensionless form and thedimensional values are given on the right axis. It is shown that thehs-circuit experiences large temperature difference changes whilethe corresponding hp-circuit changes are not as large. This is due tothe fact that the flow rate increases in the hs-circuit while it isconstant in the hp-circuit. Thus, the hs-circuit sees its inlet-outlettemperature difference decrease with an increase in the value ofa. As will be shown shortly this does not imply that the hs-circuitrejects less heat when a is increased. The observed temperaturedifference increase in the hp-circuit with increasing values of a isdue to the fact that the average temperature in the hs-circuit islarger leading to higher heat transfer from the hs-circuit to the hp-circuit. The black dot represents the case where a ¼ 1 with equalthermal capacities in both circuits. For this case, the resultingtemperature difference is the same in both circuits and since themass flow rates and specific heats are equal, the amount of energyextracted from the hp-circuit and rejected by the hs-circuit is thesame and there is no heat exchange with the neighboring ground.

Fig. 6 presents the fluid temperatures (Fig. 6a) as well as localand cumulative heat exchanges (Fig. 6b and c) as a function of thenon-dimensional depth for ten equally distant pipe segments andfour different thermal capacity ratios. As shown in Fig. 6a, the fluidtemperature evolution for a ¼ 0.8, 1, and 1.2 exhibit the samebehavior with a quasi linear profile in both the upward anddownward legs of both circuits. The exit temperature from thehp-circuit is approximately the same for these three cases(approximately þ1.5 �C) resulting in cumulative heat exchanges of11.48, 11.68, and 11.82 kW, respectively (Fig. 6c). The heat exchangeis not split equally between the downward and upward legs of thehp-circuit. For example, for a ¼ 1, the amount of heat exchanged inthe downward and upward legs are 6.40 kW and 5.28 kW, respec-tively. This is clearly seen in the local heat exchanges (Fig. 6b) wherethe local heat exchange in the first segment of the downward leg is0.70 kW while it is 0.49 kW in the last segment of the upward leg.The hs-circuit for a¼ 0.8, 1, and 1.2 shows a similar behavior exceptthat there are larger differences among these cases simply due tothe fact that ð _mCÞhs is varied while ð _mCÞhp is kept constant.

It is worth noting that the cumulative amounts of heatexchanged in the hs-circuit are 11.18, 11.68, 12.03 kW for a ¼ 0.8, 1,and 1.2, respectively. Comparing these values with the ones pre-sented above for the hp-circuit, it can be observed that there is a net

Fig. 6. Temperature profile and local and cumulative heat exchange along the boreholedepth for different flow rate ratios.

Fig. 7. Building heating load during the heating season.

P. Eslami-nejad, M. Bernier / Applied Thermal Engineering 31 (2011) 3066e3077 3071

amount of heat exchanged between both circuits and the ground.For a¼ 0.8, 0.3 kW is transferred to the borehole from the adjoiningground. For a ¼ 1.2, the heat transfer is in the opposite directionand the borehole transfers 0.21 kW to the ground. For a¼ 1.0, thereis no net heat exchanged between the borehole and the ground.

The differences are more significant, when a¼ 0.25. As shown inFig. 6a, the temperature in the hs-circuit drops from 25 �C to13.80 �C in the downward leg and from 13.80 �C to 8.59 �C in theupward leg. This translates into relatively small local heatexchanges in the upward leg as shown in Fig. 6b. Overall, the hs-circuit rejects 7.22 kW in the borehole. This relatively poorperformance has repercussions on the hp-circuit which collects9.88 kW, less than the other three cases. The net heat exchangedwith the ground is 2.66 kW.

3.2. Thermal recharging over a heating season

In this section, simulations are carried out over twenty years toexamine the impact of thermal recharging of a single-boreholeresidential system. The hourly heating load of this building is pre-sented in Fig. 7. It corresponds to a well-insulated building locatedin Montréal, Canada. As shown in Fig. 7, the building experiencesa peak space heating load of 5.2 kW. The annual space heatingrequirement is 11945 kW h over the heating season (mid-September to mid-May). During the summer, the building heatingload is zero and the cooling load is negligible. Thus, the heat pumpdoes not operate during that period.

This building is heated with a single-capacity GSHP. The heatingcapacity and compressor power requirements of the GSHP aregiven in Fig. 8 as a function of the inlet temperature to the heatpump, i.e. the outlet temperature from the hp-circuit of the bore-hole. These characteristics are based on a commercially available 3-ton (10.5 kW) water-to-water GSHP with a mass flow rate on theevaporator side, _mref , equal to 0.44 kg/s. As shown in Fig. 8, theoperation of this heat pump is not recommended when the inlettemperature is below �6 �C. System simulations are performedusing a 6 min time step with the GSHP cycling on and off to meetthe building load.

Three alternatives to provide space heating for this building arecompared; cases 1, 2, and 3 (Fig. 9). In all three cases, the double U-tube boreholewith characteristics presented in Table 1 is used. Case1 represents a conventional GSHP system with a standard paralleldouble U-tube borehole. In this case, the mass flow rate of the fluidcirculating in each U-tube of the borehole is equal to _mref =2. Thecirculating pump (I) is used to circulate the working fluid with thetotal mass flow rate of _mref through the heat pump evaporator andborehole as shown in Fig. 9a.

Cases 2 and 3 involve solar recharging during the heating season(mid-September tomid-May). In Case 2 (Fig. 9b) thenewlyproposedborehole configuration, is used. It consists of a solar collector andaheat pump, connected to thehs-circuit andhp-circuit, respectively.Two separate pumps, I and II, are used to circulatefluids in these twoloops. When the heat pump is not working or/and solar energy isunavailable, pump I or/and II are turned off. The mass flow ratecirculating in the hp-circuit is equal to _mref . The solar collector isa standard single-glazed flat plate collector whose efficiency can bedescribedbya secondorder curve relating the efficiency to (Tm� Ta)/G .The intercept of this curve is 0.78, andfirst and second order slopecoefficients are 3.20Wm�2 K�1 and 0.015Wm�2 K�2, respectively.

Fig. 8. Heat pump capacity and corresponding compressor power requirement asa function of the inlet temperature.

Fig. 9. Schematic representation of three cases.

P. Eslami-nejad, M. Bernier / Applied Thermal Engineering 31 (2011) 3066e30773072

The collector area is set at 10m2 and themassflowrate circulating inthe hs-circuit is equal to 0.11 kg/s resulting in a thermal capacityratio, a, of 0.25.

Case 3 (Fig. 9c) shows a more conventional solar assisted GSHPsystem. It consists of solar and heat pump loops which are linkedusing a heat exchanger. Themass flow rate of the fluid circulating inthe solar loop is equal to 0.11 kg/s and themass flow rate of the fluidcirculating in each U-tube of the borehole is equal to _mref =2. Theefficiency of the heat exchanger is assumed equal to 70%. In thiscase, the borehole is modeled using the analytical model of Zeng[11] for the parallel 1e3, 2e4 configuration.

As shown in Fig. 9c, two pumps circulate the fluids through thetwo loops. When solar energy is not available, pump (II) stopsworking and a three-way valve (II) directs the fluid to the boreholebypassing the heat exchanger. Pump (I) is turned off when solarenergy is unavailable and the heat pump is off. When the heatpump is off and solar energy is available both circulating pumps areworking and the three-way valve (I) directs the fluid away from theheat pump loop while valve (II) lets the flow go through the heatexchanger. The solar collector is the same as the one used in Case 2.

In all three cases, transient ground heat transfer from theborehole wall to the far-field is evaluated using the multiple loadaggregation algorithm (MLAA) developed by Bernier et al. [36]. Thisground model is coupled to a borehole model. In Cases 1 and 3, theborehole model provided by Zeng [11] is used while Case 2 uses themodel proposed here. An iterative solution is required at each timestep in order to match the heat transfer rate at the borehole wallgiven by both the borehole and ground models. This results ina variation of the borehole wall temperature at each time step.Finally, for this case, the ground thermal conductivity and diffu-sivity are 1.5 W m�1 K�1 and 0.06 m2$day�1, respectively and theundisturbed far-field ground temperature is set at 10 �C.

Simulation results, including cumulative heat pump energyconsumption (Whp), extracted energy from the ground (qb), andinjected solar energy to the borehole (qsolar) for the first and the lastyear and the average over 20 years, are presented in Table 2 for allthree cases.

Table 2 has two separate sections referred to as “equalWhp” and“equal H”. The results in the left portion, for equalWhp, are obtainedby varying the borehole length so that the total heat pump energyconsumptions over the 20-year period are equal (within less than0.5%) for all three cases without allowing Tinhp to fall below �4 �C(thus 2 �C above the recommended limit). The resulting boreholelengths are 142,117, and 95 m for Cases 1, 2, and 3, respectively. Theright portion of Table 2, for equal H, represents the results obtained

by setting an equal length of 142 m in Cases 2 and 3. These tworesults can be readily compared to the results for Case 1, also for142 m, presented in the left portion of Table 2.

As shown for the reference case (Case 1), the heat pump extracts,on average over 20 years, 8.62 MW h/year from the ground (qb) anduses3.32MWh/year for thecompressor (Whp) toprovide therequiredbuilding load of 11.94 MW h/year (qbuild). Heat extraction from theground inCase 1 decreases slightly over the years (from8.67MWh inthefirst year to 8.60MWh in the last year). In turn, heat pumpenergyconsumption increases by approximately 2% from the first year to the20thyear (3.27e3.34MWh)due to the ground temperature decreaseafter 20 years of heat extraction from the ground.

In Case 2, an averageof 5.77MWh/year of solar energy is injectedinto the borehole, thereby reducing the energy extracted from the

Table 2Simulation results for the first and the last years and the average over 20 years.

Case

Equal Whp Equal H

1st year 20th year 20 y average 1st year 20th year 20 y average

1 2 3 1 2 3 1 2 3 2 3 2 3 2 3

H (m) 142 117 95 142 117 95 142 117 95 142 142 142

qbuild (MW h) 11.94 11.94 11.94 11.94 11.94 11.94

Whp (MW h) 3.27 3.32 3.31 3.34 3.33 3.32 3.32 3.32 3.32 3.21 3.11 3.22 3.12 3.21 3.11qb (MW h) 8.67 2.84 3.05 8.60 2.83 3.03 8.62 2.85 3.04 2.93 3.22 2.92 3.21 2.93 3.22qsolar (MW h) e 5.78 5.58 e 5.78 5.59 e 5.77 5.58 5.80 5.61 5.80 5.61 5.80 5.61

Fig. 10. Heat pump inlet temperature (i.e. borehole outlet temperature) for all threecases for the first and last year of simulation.

P. Eslami-nejad, M. Bernier / Applied Thermal Engineering 31 (2011) 3066e3077 3073

ground by 67% compared to case 1 (8.62 MW h/year in Case 1 to2.85 MW h/year in Case 2). As explained earlier, a good heatexchange between the two borehole circuits reduces significantlythe energy required from the ground for the heat pump operation.When using the configuration of Case 2, the borehole length can bereduced to 117 m (a 17.6% reduction compared to Case 1) whilekeeping the same average heat pump energy consumption of3.32 MW h/year over 20 years. This reduction is essentially due tosolar energy injected and stored into the ground over the heatingseason which increases ground temperature in the vicinity of theborehole. Due to the solar heat injection into the borehole, the heatpumpenergyconsumption remains almost constantover 20years ofoperation asWhp varies from3.32MWh to 3.33MWhover 20 years.

In Case 3, the GSHP system requires a shorter borehole, 95 m (a33.1% reduction compared to Case 1), since the solar heat is trans-ferred directly to the heat pump during heat pump operation. Asshown in the left portion of Table 2, more heat is extracted from theground for Case 3when compared to Case 2 (3.04MWh/year for Case3 and 2.85 MW h/year for Case 2) since less solar energy is injectedinto the ground due to the system configuration (5.58 MW h/year inCase 3 compared to 5.77 MW h/year in Case 2). As was the case forCase 2, the heat pump energy consumption remains almost constantover the 20-year simulation due to solar heat injection.

Even though the average heat pump energy consumption over20 years is equal for the three cases, Case 1 consumes the leastamount of energy during the first year due to the longer boreholeand thus higher inlet temperature to the heat pump. However, heatpump energy consumption for Case 1 increases gradually over theyears and it surpasses the corresponding value for Cases 2 and 3after 20 years. This is due to the relatively large amount of heatextraction which induces a ground temperature reduction anda corresponding reduction in the inlet heat pump temperature. Incontrast, the heat pump energy consumption remains constant forCases 2 and 3 because of solar energy injection.

In the right portion of Table 2 the results for Cases 2 and 3 withthe same length as Case 1 in the left portion, i.e. 142 m, are pre-sented to compareWhp, qb, and qsolar for the three cases. For Case 2,the heat pump receives an average of 5.80 MW h/year and2.93 MW h/year from the solar and the ground, respectively, anduses 3.21 MW h/year for the compressor to provide the requiredbuilding load of 11.94MWh/year. When compared to Case 2, Case 3receives about 3% less energy from solar, extracts 10% more energyfrom the ground and consumes 3% less energy for the heat pump.The amount of energy extracted from the ground in Case 1 is,respectively, 194% and 168% higher than in Cases 2 and 3. The heatpump energy consumption improves only marginally over Cases 2and 3 by 3.5% and 6.5%, respectively. Consequently, Case 3, with theborehole length of 142 m, has the lowest heat pump energyconsumption among all three cases. As will now be shown, this isdue to higher inlet fluid temperature to the heat pump.

The evolution of the fluid temperature into the borehole, Tinhp,for the first and the last years of simulation are given in Fig. 10. As

shown in this figure, the average fluid temperature to the heatpump, Tinhp, for Case 1 with the borehole length of 142 m is equal to1.3 �C for the first year and it decreases to 0.4 �C for the 20th yeardue to a relatively large amount of heat extraction. With the sameborehole length, the average of Tinhp for the first year for Cases 2 and3 is slightly higher than for Case 1, 2.2 �C and 3.6 �C, respectively.The average value of Tinhp drops only by 0.1 �C over 20 years ofoperation because of solar heat injection. As shown in Fig. 10, thepeaks for Case 3 are slightly higher as solar heat is transferreddirectly to the heat pump when it is operating.

Fig. 11 provides finer details over a 24 h period (January 17th,20th year) when solar energy is available for almost 8 h with a peakof about 6 kW. The top two figures present Tb and Tinhp for all threecases. The bottom three figures show the instantaneous net heattransfer from the ground for all three cases, solar energy injectionfor Cases 2 and 3 and the building load. The top two figures showthat solar injection has a significant impact on Tinhp and Tb. Forexample, at the peak of solar injection, Tinhp reaches 10.6 �C and6.0 �C and Tb reaches 7.3 �C and 7.7 �C for Cases 3 and 2, respectivelywhile Tinhp is �0.9 �C and Tb is 2.1 �C in Case 1. As shown in thesecond figure from the top, Tinhp is higher for Case 3 than for Case 2when solar heat is available. As mentioned earlier, this is due to thefact that solar heat is transferred directly to the heat pump in Case

Fig. 11. Comparison between three cases over 24 h period in winter.

Table 3Simulation results for ground thermal conductivity of 1.5 and 3.0 W m�1 K�1

Casek ¼ 1.5 W m�1 K�1 k ¼ 3.0 W m�1 K�1

1 2 3 1 2 3

H (m) 142 103

qbuild (MW h) 11.94 11.94

Whp (MW h) 3.32 3.21 3.11 3.32 3.29 3.16qb (MW h) 8.62 2.93 3.22 8.62 2.94 3.24qsolar (MW h) e 5.80 5.61 e 5.71 5.54

P. Eslami-nejad, M. Bernier / Applied Thermal Engineering 31 (2011) 3066e30773074

3. As shown in the first figure from the top, Tb in Case 2 is slightlyhigher than in Case 3 due to the fact that more solar energy isinjected into the ground.When solar injection stops, at around 16 h,Tinhp and Tb are higher for Cases 2 and 3 than for Case 1 indicatingthat previous solar heat injection into the ground, is still present inthe vicinity of the borehole and it contributes to the warming of theground and the observed higher values of Tinhp and Tb.

It is also worth examining the values of qb. Positive and negativevalues of qb represent, respectively, heat extraction from the groundand heat injection into the ground. As shown in the bottom threefigures, the values of qb are cyclic indicating the on-off nature of theheat pump operation. Usually, a value of qb ¼ 0 indicates that theheat pump is not operating. There are some rare cases, includingone presented below, where solar heat injection is exactly equal tothe amount of heat extraction. The heat pump does not cycle at thesame frequency in all three cases even though the building load isthe same. Since Tinhp is higher for Cases 2 and 3, the heat pumpcapacity is higher and the operating time of the heat pump isreduced when compared to Case 1. As shown in the bottom twofigures, when solar energy is available (from 8 to 16 h) and the heatpump is not operating, qb is negative and equal to qsolar. When theheat pump is operating, qb is simply equal to qsolar minus the heatextraction from the heat pump. For example, for Case 3 at 12 h, qsolaris equal to 6 kW and the heat extraction is also 6 kW (i.e. the heatpump capacity is 7.76 kW and the compressor power is 1.76 kW fora Tinhp of 10.5 �C) resulting in a value of qb ¼ 0.

Finally, it is interesting to evaluate the changes on H, qb,Whp, andqsolar when the ground thermal conductivity is doubled from 1.5 to

3Wm�1K�1 (withotherparameters remaining the same). The resultsof this analysis arepresented inTable3 for a 20-yearaveragingperiod.Also included in this Table are results obtained earlier fork ¼ 1.5 W m�1 K�1. As shown in Table 3, the borehole length can bereduced from 142 m to 103 m (27% reduction) with the same heatpump consumption for Case 1 when k increases from 1.5 to3Wm�1K�1.Highgroundthermalconductivities reduce the requiredborehole length;however theyhavesomewhatofadetrimentaleffecton solar heat injection. For example, the results for Case 2 in Table 3,indicates that the heat pump energy consumption is 3.29MWh/yearon average. This represents amodest decrease (1%) from 3.32MWh/year when compared to Case 1. The corresponding decrease fork ¼ 1.5 W m�1 K�1 is 3.5% (from 3.32 to 3.21 MW h/year). This indi-cates that despite the fact that solar energy is injected over a shorterlength, good heat diffusion distributes solar heat injection away fromtheborehole sothat itdoesnot contribute toan increase in thegroundtemperature in the vicinity of the borehole.

4. Conclusion

This study presents an analytical model to predict steady-stateheat transfer in double U-tube boreholes with two independentcircuits operating with unequal mass flow rates and inlet temper-atures. The model predicts the fluid temperature profiles in bothcircuits along the borehole depth. It accounts for fluid and pipethermal resistance and thermal interaction among U-tube circuits.

The proposedmodel is used for twoapplicationswhich could notbe previously examined. Both applications can be representedschematically by Fig. 1 where one circuit is linked to a heat pumpoperating in heating mode and the other to a heat source. The firstapplication evaluates the effect of unequal mass flow rates on heattransfer betweeneach circuit for constant inlet temperatures inbothcircuits. Results show that the flow rate variation of one circuit hasa small effect on heat transfer and temperature profile of the othercircuit due to the thermal interaction among pipes in the borehole.

In the second application, simulations over 20 years are per-formed to examine the impact of thermal recharging of a single-borehole residential system. A conventional solar assisted heatpump system (Case 3) and the novel system proposed in this study(Case 2) are compared against a reference case (ground-source heatpump system without thermal recharging, Case 1). The resultsindicate that despite a relatively large amount of solar energyinjected into the system in Cases 2 and 3, the annual heat pumpenergy consumption is not reduced significantly. For example, foran average ground thermal conductivity of 1.5 W m�1 K�1 andidentical borehole length in all three cases, Cases 2 and 3 consume,respectively, 3.5% and 6.5% less energy than Case 1. When the heatpump energy consumption is the same in all three cases, Boreholelength reductions of 17.6% and 33.1% for Cases 2 and 3 is reported.Based on these results, it can be concluded that solar rechargingalternatives do not improve the annual heat pump energyconsumption of single boreholes. However they might contributeto reduce installation costs as they lead to shorter boreholes.

P. Eslami-nejad, M. Bernier / Applied Thermal Engineering 31 (2011) 3066e3077 3075

Appendix

The dimensionless fluid temperature profiles as a function ofborehole depth are derived in this appendix for the configuration1e3, 2e4 with different thermal capacitances.

qi¼1;2;3;4ðzÞ ¼ Gi1þGi2þG0

i2q3ð0ÞþG00i2q4ð0Þ

g2�h2

coshðgzÞþ

Gi3þG0

i3q3ð0ÞþG00i3q4ð0Þ�

g2�h2�g

sinhðgzÞ

þGi4þG0

i4q3ð0ÞþG00i4q4ð0Þ

g2�h2

coshðhzÞþ

Gi5þG0

i5q3ð0ÞþG00i5q4ð0Þ�

g2�h2�h

sinhðhzÞ

Gi1ji¼1;2;3;4 ¼ qb

G12 ¼ g2þh2qbþdða�cÞ�a2�a2�c2

��bða�cÞðaþ1Þ;G0

12 ¼ 0;G0012 ¼ bða�cÞð1�aÞ

G13 ¼ a2ða�cÞh�2bdþdðaþcÞ�2b2þaðaþcÞ

iþabða�cÞ2þg2½b�a�d�

G013 ¼ a2c

�a2�c2

��2a2b2ða�cÞ�cg2;G00

13 ¼ abða�cÞ2�bg2

G14 ¼ �hh2þg2qbþdða�cÞ�a2

�a2�c2

��bða�cÞðaþ1Þ

i;G0

14 ¼ 0;G0014 ¼ �G00

12

G15 ¼ �na2ða�cÞ

h�2bdþdðaþcÞ�2b2þaðaþcÞ

iþabða�cÞ2þh2½b�a�d�

oG015 ¼ �

ha2c�a2�c2

��2a2b2ða�cÞ�ch2

i;G00

15 ¼ �habða�cÞ2�bh2

iG22 ¼ �g2þh2qbþda2ða�cÞþa2�c2þbaða�cÞðaþ1Þ;G0

22 ¼ �aG0022;G

0022 ¼ 0

G23 ¼ aða�cÞh�2bdþdðaþcÞþ2b2�aðaþcÞ

i�a2bða�cÞ2þg2a½�bþa�d�

G023 ¼ aG00

13;G0023 ¼ ac

�a2�c2

��2ab2ða�cÞ�cag2

G24 ¼ �h�h2þg2qbþda2ða�cÞþa2�c2þbaða�cÞðaþ1Þ

i;G0

24 ¼ aG0012;G

0022 ¼ 0

G25 ¼ �naða�cÞ

h�2bdþdðaþcÞþ2b2�aðaþcÞ

i�a2bða�cÞ2þh2a½�bþa�d�

oG025 ¼ aG00

15;G0025 ¼ �

hac�a2�c2

��2ab2ða�cÞ�cah2

iG32 ¼ h2qbþdða�cÞþbða�cÞða�1Þ;G0

32 ¼ g2�a2�a2�c2

�;G00

32 ¼ bða�cÞð1þaÞ

G33 ¼ a2ða�cÞh2bd�dðaþcÞþ2b2�cðaþcÞ

iþabða�cÞ2þg2½�bþcþd�

G033 ¼ �a2a

�a2�c2

�þ2a2b2ða�cÞþag2;G00

33 ¼ abða�cÞ2þbg2

G34 ¼ �hg2qbþdða�cÞþbða�cÞða�1Þ

i;G0

42 ¼ h2�a2�a2�c2

�;G00

42 ¼ bða�cÞð1þaÞ

G35 ¼ �na2ða�cÞ

h2bd�dðaþcÞþ2b2�cðaþcÞ

iþabða�cÞ2þh2½�bþcþd�

oG035 ¼ �

h�a2a

�a2�c2

�þ2a2b2ða�cÞþah2

iG0035 ¼ �

habða�cÞ2þbh2

iG42 ¼ h2qbþda2ða�cÞþbaða�cÞða�1Þ;G0

42 ¼ aG0032;G

0042 ¼ g2�a2þc2

G43 ¼ aða�cÞh2bd�dðaþcÞ�2b2þcðaþcÞ

i�a2bða�cÞ2þg2a½b�cþd�

G043 ¼ aG00

33;G0033 ¼ �aa

�a2�c2

�þ2ab2ða�cÞþaag2

G44 ¼ �hg2qbþda2ða�cÞþbaða�cÞða�1Þ

i;G0

44 ¼ �aG0032;G

0044 ¼ �h2þa2�c2

G45 ¼ �naða�cÞ

h2bd�dðaþcÞ�2b2þcðaþcÞ

i�a2bða�cÞ2þh2a½b�cþd�

oG045 ¼ aG00

35;G0045 ¼ �

h�aa

�a2�c2

�þ2ab2ða�cÞþaah2

i(A.1)

P. Eslami-nejad, M. Bernier / Applied Thermal Engineering 31 (2011) 3066e30773076

Configuration 1e3, 2e4

Where

g2 ¼��

a2�c2��1þa2

�þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�a2�c2

�2�a2�1

�2þ16b2a2ða�cÞ2q �

2

h2 ¼��

a2�c2��1þa2

�� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�a2�c2

�2�a2�1

�2þ16b2a2ða�cÞ2q �

2

a ¼ 1R*1

þ 2R*12

þ 1R*13

;b ¼ � 1R*12

;c ¼ � 1R*13

;d ¼ �qbR*1

a ¼ � _mC�1�3=

�_mC�2�4

q3ð0Þq4ð0Þ

¼A11 A12A21 A22

�1C1C2

q3ð0Þ ¼A22C1

A11A22�A12A21� A12C2A11A22�A12A21

q4ð0Þ ¼ � A21C1A11A22�A12A21

� A11C2A11A22�A12A21

A11 ¼ �G012�G0

32

�coshðgÞþ�G0

13�G033

�sinhðgÞþ�G0

14�G034

�coshðhÞþ�G0

15�G035

�sinhðhÞ;

A12 ¼ �G0012�G00

32

�coshðgÞþ�G00

13�G0033

�sinhðgÞþ�G00

14�G0034

�coshðhÞþ�G00

15�G0035

�sinhðhÞ;

A21 ¼ �G022�G0

42

�coshðgÞþ�G0

23�G043

�sinhðgÞþ�G0

24�G044

�coshðhÞþ�G0

25�G045

�sinhðhÞ;

A22 ¼ �G0022�G00

42

�coshðgÞþ�G00

23�G0043

�sinhðgÞþ�G00

24�G0044

�coshðhÞþ�G00

25�G0045

�sinhðhÞ;

C1 ¼ G31�G11þðG32�G12ÞcoshðgÞþðG33�G13ÞsinhðgÞþðG34�G14ÞcoshðhÞþðG35�G15ÞsinhðhÞ;C2 ¼ G41�G21þðG42�G22ÞcoshðgÞþðG43�G23ÞsinhðgÞþðG44�G24ÞcoshðhÞþðG45�G25ÞsinhðhÞ;

Nomenclature

a, b, c, d dimensionless parameters, defined in Eq. (7)C fluid specific heat (J kg�1 K�1)D half of the shank spacing between U-tube (m)hi fluid convective heat transfer coefficient on the inside

surface of the pipes (W m�2 K�1)H active borehole depth (m)k ground thermal conductivity (W m�1 K�1)kb grout thermal conductivity (W m�1 K�1)_m mass flow rate of the circulating fluid (kg s�1)p Laplace transform operatorq heat flow per unit length of pipe (W m�1)rb borehole radius (m)rp pipe outer radius (m)R thermal resistance, defined in Eq. (1) (m K W�1)RD thermal resistance, defined in Eq. (5) (m K W�1)R* dimensionless thermal resistanceRpipe combined thermal resistance of the fluid and pipe wall

(m K W�1)Gij; G0

ij;G00ij dimensionless parameters, defined in equations (A.1)

Tb borehole wall temperature (�C)Tf fluid temperature (�C)Tm solar collector mean fluid temperatureTa ambient temperatureG solar radiationT 0f ; T

00f inlet fluid temperatures (�C)

z axial coordinate along the borehole depth (m)Z dimensionless z-coordinate

Greek symbolsa ratio of thermal capacities (defined in Eq. (8))g;h dimensionless parameters in equations (A.1)q dimensionless fluid temperatureq Laplace transform of q

Subscripts1,2,3,4 pipe sequence in the borehole1e3 circuit in 1e3,2e4 configuration2e4 circuit in 1e3,2e4 configurationinhp heat pump inletouths heat source outlet

References

[1] M. He, S. Rees, L. Shao, Simulation of a domestic ground source heat pumpsystem using a transient numerical borehole heat exchanger model, in: Proc.11th Intern. IBPSA Conf. (2009) 607e614.

[2] T.R. Young, Development, verification, and design analysis of the boreholefluid thermal mass model for approximating short-term borehole thermalresponse. Master of Science Thesis, Oklahoma State University, United States,(2004).

[3] H.S. Carslaw, J.C. Jaeger, Conduction of Heat in Solids, second ed. Oxford Press,U.K, 1959.

[4] L. Lamarche, B. Beauchamp, New solutions for the short-time analysis ofgeothermal vertical boreholes, Int. J. Heat Mass Transf. 50 (7 & 8) (2007)1408e1419.

[5] Y. Man, H. Yang, N. Diao, J. Liu, Z. Fang, A new model and analytical solutionsfor borehole and pile ground heat exchangers, Int. J. Heat Mass Transf 53 (13 &14) (2010) 2593e2601.

[6] C.P. Remund, Borehole thermal resistance: laboratory and field studies,ASHRAE Trans. 105 (1) (1999) 439e445.

[7] D. Marcotte, P. Pasquier, On the estimation of thermal resistance in boreholethermal conductivity test, Renew. Energy 33 (11) (2008) 2407e2415.

[8] L. Lamarche, S. Kajl, B. Beauchamp, A review of methods to evaluate boreholethermal resistances in geothermal heat-pump systems, Geothermics 39 (1)(2010) 187e200.

[9] R. Al-Khoury, P.G. Bonnier, Efficient finite element formulation for geothermalheating systems. Part II: transient, Int. J. Numer. Methods Eng. 67 (5) (2006)725e745.

[10] G. Hellström, Ground heat storage, thermal analysis of duct storage system.Doctorial Thesis, Department of Mathematical Physics, University of Lund,Sweden, (1991).

[11] H. Zeng, N. Diao, Z. Fang, Heat transfer analysis of boreholes in vertical groundheat exchangers, Int. J. Heat Mass Transf 46 (23) (2003) 4467e4481.

[12] N.R. Diao, H.Y. Zeng, Z.H. Fang, Improvement in modeling of heat transfer invertical ground heat exchangers, Int. J. Heat Vent. Air-Cond. Refrig. Res. 10 (4)(2004) 459e470.

[13] M. Wetter, A. Huber, TRNSYS Type 451: Vertical Borehole Heat Exchanger. 3.1Ed (1997).

P. Eslami-nejad, M. Bernier / Applied Thermal Engineering 31 (2011) 3066e3077 3077

[14] M. Bernier, A review of the cylindrical heat source method for the design andanalysis of vertical ground-coupled heat pump systems, in: 4th Intern. Conf.Heat Pumps Cold Climates, (2000) (14 pages).

[15] V. Trillat-Berdal, B. Souyri, G. Achard, Coupling of geothermal heat pumpswith thermal solar collectors, Appl. Therm. Eng. 27 (10) (2007) 1750e1755.

[16] E.B. Penrod, D.V. Prasanna, Design of flat-plate collector for solar earth heatpump, Solar Energy 6 (1) (1962) 9e22.

[17] E.B. Penrod, D.V. Prasanna, Procedure for designing solar-earth heat pumps.Heat, Piping Air Cond 41 (6) (1969) 97e100.

[18] W.B. Yang, M.H. Shi, H. Dong, Numerical simulation of the performance ofa solar-earth source heat pump system, Appl. Therm. Eng. 26 (17 & 18) (2006)2367e2376.

[19] B. Sibbitt, T. Onno, D. McClenahan, J. Thornton, A. Brunger, J. Kokko, B. Wong,The Drake Landing solar Community Project e Early results, in: Proc. 2nd Can.Sol. Build. Conf. M2-1-3 (2007) 11 pages.

[20] H. Yang, P. Cui, Z. Fang, Vertical- borehole ground-coupled heat pumps:a review of models and systems, J. Appl. Energy 87 (1) (2010) 16e27.

[21] X. Wang, M. Zheng, W. Zhang, S. Zhang, T. Yang, Experimental study of a solar-assisted ground-coupled heat pump system with solar seasonal thermalstorage in severe cold areas, Energy Build 42 (11) (2010) 2104e2110.

[22] A.D. Chiasson, C. Yavuzturk, Assessment of the viability of hybrid geothermalheat pump systems with solar thermal collectors, ASHRAE Trans. 109 (2)(2003) 487e500.

[23] B. Stojanovic, J. Akander, Build-up and long-term performance test of a full-scale solar-assisted heat pump system for residential heating in Nordicclimatic conditions, Appl. Therm. Eng. 30 (2 & 3) (2010) 188e195.

[24] Z. Han, M. Zheng, F. Kong, F. Wang, Z. Li, T. Bai, Numerical simulation of solarassisted ground-source heat pump heating system with latent heat energystorage in severely cold area, Appl. Therm. Eng. 28 (11 & 12) (2008)1427e1436.

[25] E. Kjellsson, G. Hellström, B. Perers, Optimization of systems with thecombination of ground-source heat pump and solar collectors in dwellings, J.Energy 35 (6) (2010) 2667e2673.

[26] A. Georgiev, A. Busso, P. Roth, Shallow borehole heat exchanger: response testand charging-discharging test with solar collectors, Renew. Energy 31 (7)(2006) 971e985.

[27] S. Chapuis, M. Bernier, Étude préliminaire sur le stockage solaire saisonnierpar puits géothermique, Proc. 3rd Can. Sol. Build. Conf. (2008) 14e23.

[28] S. Chapuis, M. Bernier, Seasonal storage of solar energy in borehole heatexchangers, in: Proc. 11th Intern. IBPSA Conf. (2009), pp. 599e606.

[29] G. Hellström, Duct Ground Heat Storage Model. Manual for Computer Code.Department of Mathematical Physics, University of Lund, Sweden, 1989.

[30] M. Kummert, M. Bernier, Analysis of a combined photovoltaic-geothermalgas-fired absorption heat pump system in a Canadian climate, J. Buil. Perform.Simul 1 (4) (2008) 245e256.

[31] M. Bernier, A. Salim Shirazi, Solar heat injection into boreholes: a preliminaryanalysis, in: Proc. 2nd Can. Sol. Build. Conf. T1-1-1 (2007) 8 p.

[32] P. Eslami Nejad, A. Langlois, S. Chapuis, M. Bernier, W. Faraj, Solar heatinjection into boreholes, in: Proc. 4th Can. Sol. Buil. Conf. (2009) pp. 237e246.

[33] P. Eslami-nejad, M. Bernier, Heat transfer in double U-tube boreholes withtwo independent circuits, ASME J Heat Trans. 103 (8) (2011) 12.

[34] W. Yang, M. Shi, G. Liu, Z. Chen, A two-region simulation model of vertical U-tube ground heat exchanger and its experimental verification, Appl. Energy 86(10) (2009) 2005e2012.

[35] A.J. Philippacopoulos, M.L. Berndt, Influence of debonding in ground heatexchangers used with geothermal heat pumps, Geothermics 30 (5) (2001)527e545.

[36] M. Bernier, R. Labib, P. Pinel, R. Paillot, A multiple load aggregation algorithm forannual hourly simulations of GCHP systems, Int. J. Heat. Vent. Air-Cond. Refrig.Res. 10 (4) (2004) 471e487.