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Applied Thermal Engineering 26 (2006) 1169–1175

Heat transfer analysis of ground heat exchangerswith inclined boreholes

Ping Cui a,*, Hongxing Yang b, Zhaohong Fang a

a Ground Source Heat Pump Research Center, Shandong University of Architecture and Engineering, Jinan, Chinab Department of Building Services Engineering, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, China

Received 18 August 2005; accepted 29 October 2005Available online 9 December 2005

Abstract

Consisting of closed-loop of pipes buried in boreholes, ground heat exchangers (GHEs) are devised for extraction or injection ofthermal energy from/into the ground. Evolved from the vertical borehole systems, the configuration of inclined boreholes is consid-ered in order to reduce the land plots required to install the GHEs in densely populated areas. A transient three-dimensional heatconduction model has been established and solved analytically to describe the temperature response in the ground caused by a singleinclined line source. Heat transfer in the GHEs with multiple boreholes is then studied by superimposition of the temperatureexcesses resulted from individual boreholes. On this basis, two kinds of representative temperature responses on the borehole wallare defined and discussed. The thermal interference between inclined boreholes is compared with that between vertical ones. Theanalyses can provide a basic and useful tool for the design and thermal simulation of the GHEs with inclined boreholes.� 2005 Elsevier Ltd. All rights reserved.

Keywords: Ground-coupled heat pump; Inclined borehole; Ground heat exchanger; Heat conduction

1. Introduction

Due to reduced energy consumption and mainte-nance costs, ground-coupled heat pump (GCHP) sys-tems, which use the ground as a heat source/sink, havebeen gaining increasing popularity for space condition-ing in buildings [1,2]. The efficiency of the GCHP sys-tems is inherently higher than that of air source heatpumps because the ground maintains a relatively stabletemperature throughout the year. The system is environ-ment-friendly, producing less CO2 emission than theconventional alternatives. The ground heat exchanger(GHE) is devised for extraction or injection of heatfrom/into the ground. These systems consist of a sealedloop of pipes, buried in the ground and connected to a

1359-4311/$ - see front matter � 2005 Elsevier Ltd. All rights reserved.doi:10.1016/j.applthermaleng.2005.10.034

* Corresponding author. Tel.: +852 2766 4611; fax: +852 2774 6146.E-mail addresses: cuip@sdai.edu.cn (P. Cui), behxyang@polyu.

edu.hk (H. Yang), fangzh@sdai.edu.cn (Z. Fang).

heat pump through which water/antifreeze is circulated.The GCHP systems require a certain plot of ground forinstalling the GHEs, which often becomes a significantrestriction against their applications in densely popu-lated cities and towns. The vertical GHE is the mostpopular design of GCHP systems currently employed,since it requires less ground area than the horizontaltrench systems. These boreholes should be separatedby certain distances to ensure long term operation ofthe system. Evolved from the vertical borehole systems,inclined boreholes are considered as a favorable alterna-tive to further reduce the land areas required for theGHEs. The inclined boreholes can alleviate the thermalinterference among them in the ground while occupyingless land area on the ground surface than the verticalGHEs.

Despite all the advantages of the GCHP systems,commercial growth of the technology has been hinderedby higher capital cost of the system, of which a significant

xt0

(x ,y ,0)

P (x,y,z)

o

ds

_r

0 0

-ql

virtual line sink

β

α

Nomenclature

a ground thermal diffusivity (m2 s�1)B space between boreholes (m)H borehole depth (m)k ground thermal conductivity (W m�1 K�1)l variable of the depth of borehole (m)P temperature at point P

ql heat flow per unit length of borehole (W m�1)rb borehole radius (m)t temperature (�C)s integral parameter in Eq. (1)t0 ground far field temperature (�C)x axial coordinate (m)y axial coordinate (m)z axial coordinate (m)

Greek symbols

a inclined angle of boreholeb direction angle of borehole

x angle in cross-section circleH dimensionless temperatures time (s)

Subscripts

c mean temperature of cross-section circlee temperature of borehole wall in ground heat

exchangersi base boreholej adjacent boreholeL mean temperature over borehole depth L

r representative temperaturex temperature at the angle x of a cross-section

1170 P. Cui et al. / Applied Thermal Engineering 26 (2006) 1169–1175

portion is attributed to the GHEs. Thus, it is crucial towork out appropriate and validated tools, by which thethermal behaviour of the GCHP systems can be assessedand then, optimised in technical and economical aspects.However, the thermal analysis on the GHE with inclinedboreholes is extremely difficult for engineering applica-tions, for it has to be treated as transient and three-dimensional. Few studies, therefore, have been carriedout on the GHE with inclined boreholes due to complex-ity of its heat transfer analysis, except some qualitativediscussions from a Swedish researcher [3] who did somenumerical simulation on the heat conduction of inclinedboreholes in a specific GHE. However, the numericalsolution of transient three-dimensional heat transfer istoo computationally intensive to be applied generallyin engineering designs.

On the basis of our previous studies on heat transferof GHEs with vertical boreholes, a model has beenestablished and solved analytically to describe the tem-perature response in the ground caused by a singleinclined line source. Heat transfer in the GHE with mul-tiple boreholes can then be studied by superimpositionof the temperature excesses resulted from individualboreholes. The main objective of this paper is to providea practical algorithm for engineers to design or analyzethe GCHPs with inclined boreholes.

z

H

y

s

ds

r+

ql line source

α

Fig. 1. The geometry of a finite line source in a semi-infinite medium.

2. Heat transfer analysis of an inclined line source

of finite length

In order to develop the theoretical model of inclinedGHEs, a basic and simple case is to study a single

inclined borehole and introduce other complicationsstep by step. In a similar way to the vertical boreholeanalysis [4–7], the inclined borehole buried in the groundcan be approximated as an inclined line source of finitelength in a semi-infinite medium. In the model, theground is regarded as a homogeneous semi-infinite med-ium; and its thermophysical properties do not changewith temperature; the boundary of the medium, i.e.the ground surface, keeps a constant temperature allthe time t0 as its initial one throughout the periodconcerned.

A diagram of the physical model for a single inclinedline source is illustrated in Fig. 1. The coordinate of thetop of the line source at the ground surface is (x0, y0); itslength is H; the inclined angle of the line source with thez-axis is denoted by a; the direction of the inclination isb; and the heating rate per length of the line source is ql.In order to solve this problem, a virtual line-sink with

P. Cui et al. / Applied Thermal Engineering 26 (2006) 1169–1175 1171

the same length H but a negative heating rate �ql is seton symmetry to the boundary as shown in Fig. 1. Thus,the temperature rise at time s in a random pointP(x,y,z) in the semi-infinite medium can be deduced [8]:

tðx; y; z; sÞ � t0

¼ ql

4kp

Z H

0

erfc rþ2ffiffiffiasp

� �rþ

�erfc r�

2ffiffiffiasp

� �r�

8<:

9=;ds; ð1Þ

where

rþ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðx� x0 � s sin a cos bÞ2 þ ðy � y0 � s sin a sin bÞ2 þ ðz� s cos aÞ2

q;

r� ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðx� x0 � s sin a cos bÞ2 þ ðy � y0 � s sin a sin bÞ2 þ ðzþ s cos aÞ2

q.

Introducing the following dimensionless variables:

X ¼ xH; Y ¼ y

H; Z ¼ z

H; X 0 ¼

x0

H; Y 0 ¼

y0

H;

Fo ¼ as

H 2; Hp ¼

ðtðx; y; z; sÞ � t0Þ4kpql

;

the dimensionless temperature excess caused by a singleinclined line source can be expressed as a function of thefollowing dimensionless variables:

Hp ¼ f ðX ; Y ; Z;X 0; Y 0; a; b; FoÞ. ð2ÞOnce the temperature response to a single inclined

line source is determined, it can be superimposed inspace to obtain the temperature response to multipleinclined line sources. For a GHE with n boreholes, ofwhich the parameters of the ith borehole is distinguishedas (x0i,y0i,ai,bi,Hi), the dimensionless temperature riseat the point P(x,y,z) can be obtained according to thelinear superposition principle:

H ¼Xn

i¼1

f ðX ; Y ; Z;X 0i; Y 0i; ai; bi; FoÞ. ð3Þ

r�1þ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðL sin aþ Rb cos a cos x� S sin aÞ2 þ ðRb sin xÞ2 þ ðL cos a� Rb sin a cos x� S cos aÞ2

q;

r�1� ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðL sin aþ Rb cos a cos x� S sin aÞ2 þ ðRb sin xÞ2 þ ðL cos a� Rb sin a cos xþ S cos aÞ2

q;

Rb ¼ rb=H ; L ¼ l=H ; S ¼ s=H .

3. Temperature response on the inclined borehole wall

In design and simulation of the GHEs in GCHPsystems, a characteristic temperature response on theborehole wall is usually required [9]. However, thetemperature responses on the inclined borehole wallperimeter of any cross-section perpendicular to its axisare unequal and vary with the borehole depth becausethe heat transfer of the inclined line source is three-dimensional.

3.1. Temperature response on the cross-section circle

of the borehole

In order to simplify the problem, set the coordinate ofthe top of the line source as x0 = y0 = 0 and b = 0 (i.e.the line source lies in the plane xoz) as the direction ofinclination has no effect on the temperature rise for asingle borehole. The cross-section at the borehole depthl is a tilted circle, whose coordinates can be convenientlyexpressed by the following parameters:

x ¼ l sin aþ rb cos a cos x;

y ¼ rb sin x;

z ¼ l cos a� rb sin a cos x;

8>><>>:

ð4Þ

where rb denotes the radius of the borehole, x means theangle between the radius passing the concerned point onthe circle and the specific radius at the plane xoz

(0 6 x < 2p). Substituting Eq. (4) into Eq. (1), the tem-perature response at the concerned point in the circlecan be derived. Accordingly, the dimensionless expres-sion is given as follows:

HxðL;Rb;Fo;a;xÞ¼Z 1

0

erfcr�1þ

2ffiffiffiffiffiFop

� �

r�1þ�

erfcr�1�

2ffiffiffiffiffiFop

� �

r�1�

0BB@

1CCAdS;

ð5Þ

where

Here S is an integral variable, i.e. the relative distancebetween a point on the wall and the top of the borehole,and L means the relative depth of the concerned cross-section circle.

According to the symmetry of the temperature fieldon the plane xoz, the dimensionless integral mean tem-perature along the circle can be presented as follows:

HcðL;Rb; Fo; aÞ ¼ 1

p

Z p

0

Hx dx. ð6Þ

Fig. 2. Temperature profiles along the borehole depth with differentrelative radii.

1172 P. Cui et al. / Applied Thermal Engineering 26 (2006) 1169–1175

For fixed parameters of Rb, a and Fo, the boreholewall mean temperature varies with its depth, which iscomputed and plotted in Fig. 2. The variation of thetemperature on the inclined borehole wall is almost sim-ilar to that on the vertical borehole (see Ref. [9]), whichillustrates that drilling the inclined borehole is unneces-sary for a GHE with a single borehole.

3.2. Representative temperature of the cross-section circle

For a specific cross-section circle of the borehole, thedimensionless temperature at any point on the circle isonly a function of x. The radius of the borehole is minorcompared with its depth, which means the thermal influ-ence of the boundary condition (ground surface) on thecircle may be negligible. Hence, the temperatures varia-tion along the circle is substantially little. Take the mid-dle cross-section (i.e. l = H/2) as an example,computation shows that the temperature (named asHir) of the point on the circle with x = p/2, i.e. its coor-dinate is (H sina/2, rb, Hcosa/2), can be recommendedto the representative temperature instead of its meantemperature (Hc). Table 1 lists Hir and Hc for differentconditions.

From Table 1, it is noted that the representative tem-peratures are almost equal to the mean temperatureswith different Fo and a for a borehole with Rb = 0.001,

Table 1Comparisons of the representative and mean temperatures(Rb = 0.001)

Fo a = 10� a = 30�

Hir Hc Hir Hc

0.01 10.002 10.002 10.022 10.0220.1 12.048 12.048 12.012 12.0121 12.662 12.662 12.550 12.550

and the relative errors between the two temperaturesare less than 0.001%. Therefore, the representative tem-perature Hir, which is easier to be computed, can be usedto replace the mean temperature of the cross-section cir-cle on the borehole wall in engineering applications.

3.3. Representative temperature of the borehole wall

There are usually two characteristic borehole walltemperatures in practice. One is the integrated meantemperature along the borehole depth:

HiðRb; Fo; aÞ ¼Z 1

0

Hc dL. ð7Þ

The integrated temperature is regarded as a morereasonable characteristic temperature. However, it istoo complicated and, therefore, inconvenient to beemployed directly in engineering applications. A morecommon scheme is to use the temperature at the middleof the borehole wall as its representative temperature.Thus, the representative temperature Hir of the middlesection circle can be employed instead of the integralmean temperature Hi over the borehole depth. BothHir and Hi are the function of a, Rb, Fo, so that the rel-ative error, D ¼ Hir=Hi � 1, is related with these threevariables. The variations of the relative errors with a,Rb, Fo are illustrated in Figs. 3–5, respectively, wherethe data concerned can be used for engineeringcalculations.

As shown in the figures, the representative tempera-ture Hir is higher than the mean temperature. The max-imum relative error between them may be as high as10.6% in the studied ranges (a 6 30�, Rb 6 0.005), whichis still acceptable in most practices as in the similar situ-ations recommended for the vertical borehole GHEs.Employing the representative temperature on the middlesection as the characteristic temperature of the wholeborehole can greatly simplify the process of the calcula-

Fig. 3. Relative error between Hir and Hi vs. Fo.

Fig. 4. Relative error between Hir and Hi vs. Rb.

Fig. 5. Relative error between Hir and Hi vs. a.

x

z

y

0 (x ,y ,0)o 0 i 0 i

i j

i j

(x ,y ,0)0 j0 j

borehole i borehole j

l

β

α α

β

Fig. 6. A schematic of two inclined boreholes.

P. Cui et al. / Applied Thermal Engineering 26 (2006) 1169–1175 1173

tion in engineering practice. Of course, more reasonableintegrated mean temperature defined in Eq. (7) may alsobe used with sophisticated computer software.

4. Thermal interference among inclined boreholes

In engineering practices the radius of a borehole (typ-ically from 0.05 m to 0.1 m) is much smaller than thespace between adjacent boreholes, which is usually above3 m. Thus, the temperature response on the concernedborehole wall caused by its adjacent boreholes can beapproximately treated as the response on its axis sinceits radius can be ignored. Take two boreholes (referredto as i and j) as an example, and suppose the borehole i

is the one concerned, and j is its adjacent one as shownin Fig. 6. The coordinates of the point at any axial dis-tance l of the ith borehole may be presented as

x ¼ x0i þ l sin ai cos bi;

y ¼ y0i þ l sin ai sin bi;

z ¼ l cos ai.

8><>: ð8Þ

Substituting Eq. (8) into Eq. (1), the temperature re-sponse at any axial distance l of the ith borehole causedby the adjacent jth one can be derived accordingly.Therefore, the dimensionless expression can be rear-ranged as a function of the following variables:

Hij;L ¼ fLðX 0i; Y 0i; L; ai; bi;X 0j; Y 0j; aj; bj; FoÞ. ð9Þ

A representative temperature rise on the boreholewall caused by its adjacent one is also required to char-acterize the thermal interference between them. As thesame with a single inclined borehole, there are two char-acteristic temperatures. One is the mean temperaturerise, which can be obtained by integrating along thedepth of the concerned borehole:

Hij ¼Z 1

0

Hij;L dL. ð10Þ

The other is a representative temperature response atthe middle of the concerned borehole wall. The processesof the two methods, however, are both a little compli-cated for practical applications of GHEs with multipleinclined boreholes. Again, while it is possible to computesuch temperature rises according to Eq. (1) or Eq. (10)with appropriate computer software, an approximateapproach is recommended to calculate the thermal inter-ference between adjacent inclined boreholes on the basisof computations and comparison, i.e. the thermal inter-ference between two inclined boreholes may be estimatedas that of two relevant vertical boreholes disposed at adistance between the middle points of the inclined bore-holes. Thus the temperature rise on the ith boreholecaused by the jth one can be obtained by the two sup-posed vertical boreholes, which is recommended as anew representative temperature (Hij,r). Involving a two-dimensional process, the expression of Hij,r is much sim-pler, and can be found in references [5,6].

Now take an example of the two inclined boreholeswith the following parameters: X0i = 0, Y0i = Y0j = 0,ai = aj = 20�, and the value of X0j is obviously equalto the relative space B* between them. Fig. 7 presentsthe comparisons of the representative and the meantemperatures vs. Fo in different inclining directions. It

Fig. 7. Comparisons of Hij and Hij,r.

1174 P. Cui et al. / Applied Thermal Engineering 26 (2006) 1169–1175

shows that the thermal influence between boreholesincreases with Fo and gradually approaches constantwhen Fo is large enough. Though the maximum relativeerror resulted from the approximated approach reachesnearly 17% when Fo = 1.5 for B* = 0.05 and a = 20�, thetemperature rise caused by the heat source in adjacentboreholes is much smaller than that caused by the heatsource in the concerned borehole itself due to the signif-icant difference. Furthermore, the representative temper-ature is higher than the mean temperature, which canresult in a conservative design for the GHEs. Therefore,the approximate method can be acceptable for engineer-ing applications.

Fig. 8. An inclined GHE with different spaces in rectangle.

Fig. 9. Comparisons of He between different GHEs (4 · 5).

5. Temperature response on the borehole wall in GHEs

with multiple inclined boreholes

U-tubes in the multiple boreholes of GHEs are gener-ally joined in parallel configurations. The temperatureresponses on each borehole caused by its adjacent bore-holes are obviously different, which mainly depends onthe spacing and geometric disposal of the boreholes.Hence, a base borehole needs to be found out, whichhas the highest temperature rise or the worst heat trans-fer condition among all the boreholes, as the benchmarktemperature rise on the borehole wall in a GHE.

For each borehole, its temperature response on theborehole wall to heating of the GHE consists of twoparts: the primary temperature rise due to the line source(U-tube) in the borehole itself and the second onecaused by the rest boreholes in the GHE. Thus, the tem-perature response on the base borehole wall in a GHEwith n boreholes can be expressed as follows:

He ¼ maxðHiÞ ¼ max Hir þXn

j¼1j 6¼i

Hij;r

0BB@

1CCA; ð11Þ

where the function means the maximum of the temper-ature responses among the n boreholes.

A typical configuration of the GHEs with inclinedboreholes is the rectangle pattern in practical applica-tions. Fig. 8 depicts a layout of 20 boreholes on theground surface in four rows with five boreholes each(4 · 5). The six inner boreholes are vertical, with relativespace B�1 ¼ 0:1 and the peripheral boreholes are inclinedoutward with the equal tilting angle and the relativespace B�2 of 0.05. The temperature responses of such aGHE with different tilting angles are plotted in Fig. 9.The curve for a = 0�, i.e. the vertical GHE, is also plot-ted in Fig. 9 for comparison purpose. The temperaturerises for a = 10� and a = 20� are obviously 16.3% and27.3% lower, respectively, than that in the verticalGHE case when Fo = 5.0.

Consider a GHE in a (2 · 5) rectangle pattern in twodifferent cases of vertical and inclined boreholes withidentical spacing. The tilting angle of each borehole is20� outward in the case of inclined boreholes. The dimen-sionless temperature responses of the cases are presentedin Fig. 10 with different relative borehole space B*. Boththe temperature responses of the inclined and the verticalGHEs increase with decrease in the space between bore-holes. However, the thermal influence of the boreholes in

Fig. 10. Comparisons of He between different GHEs (2 · 5).

P. Cui et al. / Applied Thermal Engineering 26 (2006) 1169–1175 1175

the inclined GHE is much less than that in the verticalone, especially in the case of B* = 0.03, where theinclined GHE (2 · 5) with a = 20� may gain 35% reduc-tion in the temperature rise compared to that in the caseof vertical boreholes.

The curves in Figs. 9 and 10 show that the thermalinterference between boreholes can be subdued consid-erably by either expanding the planar spacing amongboreholes or deviating the boreholes away from eachother along their depth. Hence, drilling inclined bore-holes can be a favorable alternative to minimize the tem-perature rise of the GHEs in case of limited ground areato install the GHEs.

6. Conclusions

This paper presents an extensive analysis of the heattransfer of the inclined GHEs from a basic case of a sin-gle inclined borehole to practical cases of the GHEs withmultiple boreholes of mixed vertical and inclined ones.For a single inclined borehole, the model of an inclinedline source with finite length in a semi-infinite medium isdeveloped to describe the heat conduction process inGHEs, especially for long-term operation. The represen-tative temperature of a specific point on the middlecross-section circle of the borehole wall is recommendedfor the design of GHEs instead of using its integral meantemperature along the cross-section circle.

For multiple inclined boreholes, an expression is pre-sented to determine the thermal interference betweenadjacent boreholes. Meanwhile, an approximate methodis proposed to calculate the thermal interferencebetween two inclined boreholes where two vertical bore-

holes are supposed to substitute for the inclined ones.The representative temperature obtained from theapproximate method is recommended for engineeringdesigns.

Comparisons between the GHEs of typical rectangu-lar patterns with inclined or vertical boreholes show thatthe temperature rise on the borehole wall of the inclinedGHE can be 10–35% lower than that of the verticalGHE for long-term performance in commonly encoun-tered conditions in engineering practice. Thus, inclusionof inclined boreholes in the GHE configuration canimprove its thermal performance especially for theGCHP systems with imbalanced annual loads and lim-ited land allowance to install the GHE. The benefit fromdrilling inclined boreholes is evident in such situations.

Acknowledgements

The authors wish to thank the financial supportsfrom the Natural Science Foundation of China (ProjectNo. 50476040) and the Hong Kong Research GrantsCouncil (RGC) (Project No. 530204).

References

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