G-Functions for multiple interacting pile heat exchangersFleur
Loveridge*, William PowrieFaculty of Engineering & Environment,
University of Southampton, UKarti cle i nfoArticle history:Received
31 May 2013Received in revised form31 October 2013Accepted 6
November 2013Available online 8 December 2013Keywords:Ground heat
exchangerPileGround energy systemGround source heat pump
systemabstractPile heat exchangers e where heat transfer pipes are
cast into the building piled foundations e offer
anopportunitytousegroundenergysystemswithout theadditional
constructioncostsrelatedtotheprovision of special purpose heat
exchangers. However, analysis methods for pile heat exchangers
arestill underdevelopment.
Inparticularthereisanabsenceofavailablemethodsandguidancefortheamountofthermal
interactionthatmayoccurbetweenadjacentpileheatexchangersandthecorre-sponding
reductionin available energythatthis willcause.This is of
particular importance as the lo-cationsof
foundationpilesarecontrolledbythestructural demandsof
thebuildingandcannotbeoptimised with respect to the thermal
analysis. This paper presents a method for deriving G-functions
forusewithmultiplepileheatexchangers.
Examplefunctionsillustratetheprimaryimportanceof pilespacing in
controlling available energy, followed by the number of piles
within any given arrangement.Signicantly it was found that the
internal thermal behaviour of a pile is not inuenced appreciably
byadjacent piles. 2013 Elsevier Ltd. All rights reserved.1.
IntroductionInstallations of ground energy systems e where ground
sourceheat pumps are connected to a series of ground heat
exchangers toobtainsustainableheatingandcoolingeareincreasing
atasig-nicant rate. Lund et al. [1] report that the energy provided
by suchsystems hasmorethandoubledbetween2005and2010. Asthemarket
has expanded, so has the range of types of heat
exchangerusedtoprovidetheheatsource. Traditionally,
eldsofboreholeheat exchangers have been used to provide a large
thermal capacityfor major commercial or municipal buildings.
However, wherebuildings require deep foundations it is becoming
more common tomake dual use of the piled foundations, equipping
them with heattransfer pipesso thatthey canactasheat
exchangersaswellasstructural components, e.g. Refs. [2e4]. This can
lead to cost savingson projects by removing the need to construct
special purpose heatexchangers.Design methods forelds of borehole
heat exchangers are welldeveloped. Numerous different commercial
software packages areavailable, some of which will be mentioned
below. Typically
thesemethodsarealsoappliedforthedesignofpileheatexchangers.However,
thereareimportantdifferencesbetweenboreholeandpileheat exchangers
whichmeanthat applicationof boreholemethodswill
underestimatetheenergyavailablefrompiles[5].Heat transfer problems
are typically controlled by their geometryand the thermal
properties of the materials. The thermal propertiesof the materials
for borehole and concrete pile construction are in asimilar range,
but their geometries arenot. Piles
haveamuchsmalleraspectratio(AR(aspectratio)
heatexchangerlength/diameter)thanboreholes,
whichaffectstheirlongtermthermalcharacteristics [6]. In addition,
the larger diameter of piles results ina different short term
behaviour with a greater proportion of
shorttermenergystoragehappeningwithintheheatexchangeritself,rather
than just in the ground [5].Previous work on pile heat exchanger
design has focused mainlyon determining the temperature changes
with time, usuallyexpressed as temperature response functions or
G-functions, for asingle pile [5,7,8]. However, all ground energy
systems using pileheat exchangers will comprise multiple heat
exchangers which willinteract. With the exception of the Duct
Storage Model [9] (whichwill be discussed below) no other specic
solution for interactingpileheat exchangers has beenpublished. This
paper builds onprevious work [5] and presents G-functions for
multiple pile heatexchangers in some simple congurations, together
with anapproach for determining G-functions for any arrangement of
piles.The impact on the available energy of different pile
arrangementsandspacings will bediscussed, alongwithstrategies for
max-imising the energy output.*Correspondingauthor. Facultyof
Engineering&Environment, UniversityofSouthampton, Higheld,
Southampton SO17 1BJ, UK. Tel.: 44 (0)2380592662.E-mail address:
[email protected] (F. Loveridge).Contents lists available
at ScienceDirectEnergyj ournal homepage: www. el sevi er. com/ l
ocat e/ energy0360-5442/$e see front matter 2013 Elsevier Ltd. All
rights
reserved.http://dx.doi.org/10.1016/j.energy.2013.11.014Energy 64
(2014) 747e7572. Background2.1. Design of single heat
exchangersG-Functions describe the change in temperature in the
groundaroundaheat exchangerwithtimeastheresult of anappliedthermal
load. Usually both time and temperature change are nor-malised.
Other normalisations are used, but this paper applies thefollowing
expressions, based on precedent from other studies, e.g.Refs.
[7,10]. Thetemperaturechange(DT) is normalisedbytheapplied thermal
load q (in W/m) and the thermal conductivity ofthe groundlg (in W/m
K) such that:F 2plgqDT (1)The elapsedtime (t) is normalisedbythe
groundthermaldiffusivity (ag) and the heat exchanger radius (rb):Fo
agtr2b(2)Eskilson[11] pioneeredtheG-functionconcept. Heassumedthat
groundheat exchangers act likea nitelineheat
source,operatingwithinauniformmediumwithaninitial temperatureequal
tothefar eldboundaryconditionsandalsothesurfaceboundary condition.
As a result of the constant temperature surfaceboundary condition
the temperature response (F) will reach steadystate at large values
of time as the heat output is matched by thatavailable. The time at
which this occurs and the equilibrium tem-perature change depends
on the aspect ratio of the heat exchanger.Eskilsons G-functions,
published for AR 1000, were developed bymeans of a combination of
analytical and numerical methods andform the basis of many
commercial software
packages.Anothercommonlyadoptedapproachisthecylindrical heatsource
[12,13]. This is identical to the line heat source in the
longerterm, but accounts for the nite radius of the heat exchanger
in theshort term. Thisapproachhasbeenappliedto
thedesignofpileheatexchangersdueto theirlargerdiameter,
butithasrecentlybeen shown [14] that the either the line heat
source approach orthe solid cylinder approach is more appropriate.
While the classiccylindrical heat source assumes that the heat
source is hollow, thesolid cylinder model [8] assumes that heat may
alsoow into thecentre of the heat exchanger. This makes it much
more suitable forsimulating pile heat exchangers.All the above
approaches take no account of the arrangement ofheat transfer
pipeswithinthepilecross-section. Onlytwoap-proaches overcome this
limitation. Li & Lai [7] published G-func-tions based on the
superposition of multiple line heat sources, witheachline source
representing a single pipe. This
canproducetheoreticallyaccurateG-functions, but has
thedisadvantageofrequiringacomplicatedcalculation
processforeachspecicpilediameter and pipe arrangement. This will be
a barrier to its adop-tion on a routine basis. Loveridge &
Powrie [5] presented upper
andlowerboundpileG-functionsbasedonnumerical analysisof arange of
commonly constructed pile geometries (Fig. 1). Thisremoves the need
to make new analytical solutions for every
ge-ometrybutthereisatrade-off intermsof asmall
reductioninaccuracy.TraditionallyG-functions,
describingthetemperaturechangewithintheground,
arecombinedwithasteadystateresistancewhich determines the
temperature change between the heattransfer uid and the ground.
However, recent work [15] has shownthat this approach is
conservative for pile heat exchangers and
thatgreaterenergyefciencycanbeobtainedbytakingatransientapproachtothetemperaturechangewithintheheatexchanger.ConsequentlyLoveridge&Powrie[5]alsoproposedconcreteG-functions
which described the proportion of the steady state pileresistance
which is appropriate for different values of normalisedtime,
Fo.2.2. Multiple heat exchangersIf ground heat exchangers are
positioned within a certainproximity of each other they will
interact thermally to an increasingdegreeastimepasses. If not
accountedfor indesign, thiswillreduce the sustainability of a
ground energy system over time. Allmultiple heat exchanger
G-functions are based on the principle ofspatial superposition.
Eskilson [11] developed G-functions for manydifferent arrangements
of borehole heat exchangers. Softwarepackages, such as EED (Earth
Energy Designer) [16] and GLHEPRO[17] are based on these, and
typically use interpolation to cater forboreholearrangements
different fromthosegiveninRef. [11].However, in all cases Eskilsons
published G-functions are limited tothe case where AR 1000 and are
therefore not suitable for
usewithpileheatexchangerswhichtypicallyhaveAR 200. Full details of
the model are given in Ref. [14]with summary characteristics
provided in Table
2.TocalculatetheG-functionfortwopilesinteractingthetem-perature
change at r rb at the edge of a single pile was added tothe
temperature change in the model at r B where B is the centreto
centre spacing between the two piles. As the pile arrangement
issymmetrical and both piles suffer the same degree of
interactionthen the G-function only needs to be calculated for one
pile. Thelength of both piles is then taken into account when
determiningthe applied heatux as q Q=2H. When presenting his
G-func-tions, EskilsonnormalisedtheheatexchangerspacingBbythelength
of the heat exchanger, H [11]. However, with piles we havefound it
more practical to normalise by the pile diameter 2rb, as aresult of
existing practice in the piling industry. First, pile spacingsTable
1Main characteristics of 3D pile G-function model, see also Ref.
[5].Pile geometry As Fig. 1, but with AR 15, 25, 331=3, 50Model
extent To a distance of 25 m radially and below the pile;quarter
pile modelled for computational efciencyHeat transfer By conduction
only; pipes anduid not modelledas these reach a steady state
rapidlyInitial conditions Constant temperature (T 0 C) applied
throughoutBoundary conditions Constant heatux applied at pipe outer
radiusConstant temperature (T 0 C) applied at fareld boundaries and
ground surfaceInsulated boundaries on the planes of
symmetryElements 8 Node linear heat transfer brick elementsMesh
sizes 5 mm at pipes; 10 mm at pile edge; 2.5 m atfareld; 0.5 m
along pile lengthMaterial properties Refer to Fig. 1Validity
Fo200Software used ABAQUS 6.10-2Fig. 2. Schematic of
axis-symmetricnite line source model.Table 2Main characteristics of
axis-symmetricnite line source model, see also Ref. [14].Pile
geometry As Fig. 2, with rb 0.6 m and AR 15, 25, 331=3, 50Model
extent To a radial distance of 150 m and depth of 200 mHeat
transfer By conduction only; pile not modelled as appliedin long
term onlyInitial conditions Constant temperature (T 0 C) applied
throughoutBoundary conditions Constant heatux applied at pile
radius over thefull length of the pile, insulated below the pile
toe.Constant temperature (T 0 C) applied at fareld boundaries and
ground surfaceElements 4-Node linear axisymmetric heat
transferquadrilateral elementsMesh sizes 10 mm at pile edge; 5 m at
fareld; 1 m alongpile lengthMaterial properties As Fig. 1Validity
Fo > 200Software used ABAQUS 6.10-2F. Loveridge, W. Powrie /
Energy 64 (2014) 747e757
750areoftenspeciedasacentretocentrespacingintermsof amultipleof
thediameter. Secondly, pilelengthsareoftendeter-mined by geological
features, such as the presence of a hard stra-tum or the need to
avoid penetrating into an underlying aquifer. Inaddition, while
Eskilson only published G-functions for AR 1000,it is important to
consider a range of smaller ARs for piles. Values of15, 25,
331=3and50havebeenusedonthebasisofasurveyofconstructed geometries
[6].3.1.2. ResultsTwo-pileG-functionsfortheextremecasesof AR
15andAR 50 are plotted in Fig. 3 for different B/2rb values from
1.2
(theclosestpilesaretypicallyspaced)to20(wheretheinuenceofadjacent
piles diminishes). The inuence of the pile spacing is clear,with
much greater temperature changes at steady state for thosepiles at
closest spacing. Table 3 gives an indication of the
maximuminteractionexperiencedbythepiles,
measuredasapercentageincreaseinFinthelongterm. This
clearlyshowsthegreaterinteraction for higher AR piles, but also the
important inuence ofthe spacing between the piles. For B/2rb 20,
increase inF valuesare always less than 15%. However, for B/2rb1.2,
Fincreases by upto76%. Theseincreases inFmeanthat
therearediminishingreturnsavailablefrommultiplepilesastheirspacingdecreases.This
means that for the closest spacing, less than 60% of the energythat
could be obtained from a single pile is available from each ofthe
pair of piles. Therefore in total 1.2q is available compared
with2qfor twoisolatedpiles. This means
signicantlydiminishingreturns as piles get closer together.Fig. 3
and Table 3 also show that the higher aspect ratios havegreater
values ofF as a steady state develops, and also a greaterdegree of
interaction as seen by the bigger increase inF and
cor-respondingreductioninequivalentenergyforthetwopilecase.Piles
with a lower aspect ratio interact to a lesser extent due to
theearlier andmoresignicant inuenceof thesurfaceboundarycondition.
This restricts both the overall temperature change thatcan occur,
and also the distance to which temperatures within theground are
inuenced by the heat exchanger. This suggests that lowaspect ratio
piles have the potential to be more efcient due to
i)reducedtemperaturechangesinthegroundandii)reducedin-teractions
between adjacent heat exchangers.Fig. 4 plotsF vs Fo for different
aspect ratios, showing that
theupperandlowerboundcasesexperienceidenticalinteractioninthe long
term for a given aspect ratio. However, in the short termthere is a
small increase in interaction for the upper bound case.This is to
be expected given the greater proximity of the pipes to thepile
edge. Overall Fig. 4 shows that the interactions are similar
atsmall values of time regardless of the pile spacing and pile
geom-etry. This is also to be expected as the zone of steep
temperaturegradientshasyetto reachfarbeyondthepile. However,
astimeincreases larger aspect ratio piles interact more and for
longer.Eskilson reportedthatthereisnointerference betweenbore-hole
heat exchangers if B > H and that any interactions are small
aslong as B > H/2 [11]. For the case of pile heat exchangers,
the lattercriterion is equivalent to B/2rb> AR. Reference to
Fig. 4 shows thatthese guidelines remain true for pile heat
exchangers and that inthis case small interactions are
approximately equivalent to a 5%increase in F. However, for pile
heat exchanger systems, where thepile layout and spacing are
governed by the structural andgeotechnical designofthe building and
foundations,it would beexceedinglyunlikelyfor
thepilespacingtobe15mor more.Therefore in most cases some
interactions will occur.Fig. 3. Pile G-functions for two piles
interacting at different B/2rb values: a) AR 15 lower bound; b) AR
15 upper bound; c) AR 50 lower bound; d) AR 50 upper bound. Ineach
case the curves are, from top to bottom, B/2rb 1.2, 1.5, 2, 3, 5,
10, 20 and N respectively.Table 3Steady state increase in F output
for a pair of piles for different spacings and aspectratios
(AR).B/2rbAR 15 AR 33 AR 25 AR 50Increase inF Increase inF Increase
inF Increase inF20 3% 6% 8% 12%10 9% 15% 19% 25%5 23% 30% 34% 40%3
37% 44% 48% 52%2 49% 56% 59% 63%1.5 59% 65% 67% 70%1.2 67% 71% 74%
76%F. Loveridge, W. Powrie / Energy 64 (2014) 747e757 7513.2.
Multiple pile G-functionsUsing Figs. 3 and 4 and Table 3 it is
possible to interpolate
theinuenceofpilesatanyspacingandhencecompiletheaveragemultiple pile
G-function for any arrangement of piles. For
brevity,twoexamplesareincludedbelow; threepilesinaline(Fig.
5),which may be an arrangement beneath a building column and 9piles
in a grid (Fig. 6), which gives an idea of the degree of inuenceof
many pile heat exchangers. In reality most pile arrangements
areirregularandrelatedtothestructural layoutof thebuilding,
inparticular under the positions of the columns that transfer most
ofthe vertical load to the foundations. A column may be supported
bytwo or three closely spaced piles, with many columns being
presentat larger spacings. However, the following method may be
used forany arrangement.It has been assumed, for the purpose of the
examples, that thepipes are installed near the edge of the pile and
therefore an upperbound pile G-function has been used. An aspect
ratio of 50 has beenchosenbecausethisprovidesanupperestimateto
thelikelyin-teractions for pile heat exchangers. Curvetting was
then appliedto Fig. 4d so that the percentage increase inF for any
pile spacingwould be determined. For three piles in a line (Fig. 5)
the G-functionwill then be equal to the average response of the
three piles, one ofwhich will suffer interactions fromtwo of its
neighbours at spacingB and two of which experience interactions
from two piles at B and2B spacing respectively. The same procedure
can be applied for thenine pile arrangement (Fig. 6), except here
there are 4 corner piles,4 centre edge piles and one central pile,
all of which have differentcombinations of interactions which must
be averaged.Figs. 5and6showthesignicant impact that
multiplepileheat exchangers can have when interacting. For a
singlepile(B innity) with AR 50, the long term steady state
nor-malisedtemperatureresponse(F)isapproximately3.6. Fortwopiles at
B/2rb 1.2 the response increases to 6.3, for three piles andnine
piles at the same spacing it is 8.6 and 22.2 respectively. Theseare
large increases and will result in a corresponding decrease
intheenergythat canbeexchangedperlinearmetreof thepile.Table 4
summarises the increase in F for the closest spacing. TheseFig. 4.
Percentage increase inF for different normalised pile spacings and
aspect ratios (AR): a) AR 15; b) AR 25; c) AR 33; d) AR 50. Solid
lines for upper bound cases;dashed lines for lower bound
cases.Fig.5. G-Functionfor three upperbound pilesinaline, withAR
50. From top tobottom B/2rb 1.2; 1.5; 2; 3; 5; 10; 20; N.Fig. 6.
G-Function for nine upper bound piles arranged on a grid, with AR
50. Fromtop to bottom B/2rb 1.2; 1.5; 2; 3; 5; 10; 20; N.F.
Loveridge, W. Powrie / Energy 64 (2014) 747e757 752increases mean
that at steady state with nine piles in a grid thateach pile is
only delivering 16% of the energy of an individual iso-latedpile.
However, itisrareforsomanypilestobespacedsoclosely and in reality
combination arrangements as described aboveare more common. In
addition, this analysis has assumed that all ofthe nine piles are
equipped as heat exchangers. The results showthat while each
additional pile used as a heat exchanger does in-crease the overall
quantity of energy available there are diminish-ing returns. When
the energy required foruid circulation is alsotaken into account it
may be more economical to equip only someof these piles with heat
transfer pipes.4. Multiple concrete G-functions: internal pile
temperatureresponseThemultiplepileG-functionspresentedinSection3describethe
average temperature change with time in the ground aroundthe heat
exchangers. However, it is important to determinewhether the nature
of the temperature changes within the pile
isalsoaffectedbytheinteractionof theheat exchangers.
Toourknowledgethishasnotbeenascertainedbefore. Previouslytheauthors
have presented concrete G-functions to describe the tran-sient
behaviour of the pile concrete [5]. This is an improvement onthe
previous approach of assuming the pile concrete is at steadystate.
Therst concern for multiple piles is whether the change
intemperatureeld for two or more piles will inuence the
steadystatevalueof thepilethermal resistance. Secondly,
weneedtounderstand whether the shape of the concrete
G-function,expressedasaproportionof
thesteadystateresistanceRcwithtime, changes when there are more
than one pile in proximity.To answer these questions, the numerical
model used in Ref. [5]was extended to represent two piles, taking
advantage of a line ofsymmetry between the piles to minimise the
computational effortrequired (Fig. 7). Full details of the model
set up and validation aregiven in Refs. [5,14], with updated
geometry and a summary of theconditions analysed provided in Table
5. Only the upper and lowerbound pile concrete conditions, as
identied in Ref. [5] and given inTable 6, were analysed. These
include upper and lower bounds forthe two cases of pipes installed
centrally and those closer to the pileedge. As the lower bound
piles are 300 mm in diameter with twopipesinstalled,
itisnecessarytocomparetwocases. Case#1iswhere the twopipes are
alignedperpendicular tothe line ofsymmetry in the model (as shown
in Fig. 7) and Case #2 is wherethe pipes are aligned parallel to
the line of symmetry. It should alsobe noted that lower and upper
bound cases used for the concrete G-function are different from the
lower and upper bound cases usedfor thepileG-functions as
theyrelatetothebehaviour of theconcrete not of the ground.4.1.
Steady state resistance for two interacting pilesAt any given time
the thermal resistance of the concrete part ofthe pile is
calculated as follows:Rc TpTbq(8)where Tp and Tb are the integral
mean values of the temperature atthe pipes and the pile edge
boundaries respectively, andq is thetotal heatux (in W/m) applied
to the all the pipe boundaries. Thesteadystateresistanceis
theasymptoticvalueRccalculatedatlargervaluesof time,
typicallywhenFoapproaches10. Table6presents the steady state values
of Rc for single and pairs of inter-acting piles. In most cases the
presence of an additional pile has nonoticeableimpact
onthesteadystateresistance, despitesomechanges to the heatow paths
that the additional pile must cause.The exception to this is the
lower bound case with pipes near to theedge. Here the steady state
resistance now depends on the relativearrangement of the two pipes
in the adjacent piles. For Case #1 theresistance is increased by
2%, while for Case #2 it is decreased by2%. Thesedifferencesarenot
signicant inthecontext of
con-structiontolerancesforthepositioningof thepipeswithinthepiles.
In addition, the construction process will give limited
controlovertheorientationofanypairofpipes,
andthereforethenalorientation will be almost random. Therefore,
overall, it is consid-ered appropriate to continue using the steady
state resistance of asingle pile regardless of the number and
spacings of adjacent piles.4.2. Concrete G-functions for two
interacting pilesThe second question that needs to be answered is
whether theshape of the concrete G-function curve, i.e. how the
transient valueof thermal resistance for the pile changes with
time, is affected as aresult of piles interacting. Fig. 8a and b
plots concrete G-functionsfor upper and lower bound cases for the
scenario of pipes placedcentrallyandneartheedgerespectively.
Inbothcasesthesolidlines represent the G-function for a single pile
and the dashed linesTable 4Steady state increase in Fenergy output
for different pile arrangements compared toa single pile (AR 50).No
piles Spacing Increase inFSingle pile N/A N/A2 Piles B/2rb 1.2 76%3
Piles in a line B/2rb 1.2 141%9 Piles in a grid B/2rb 1.2 521%Fig.
7. Schematic of numerical model for 2Danalysis of
concreteresistance andtemperature response (showing Case #1, not to
scale). Refer also to Table 5.Table 5Main characteristics of 2D
concrete G-function model, see also Refs. [5,14].Pile geometry
Upper and lower bound pile geometries as per Table 6Model extent To
a distance of 25 m from the model centre (Fig. 7)Heat transfer By
conduction only; pipes anduid not modelledas these reach a steady
state rapidlyInitial conditions Constant temperature (T 0 C)
applied throughoutBoundary conditions Constant heatux applied at
pipe outer radiusConstant temperature (T 0 C) applied at model
edgeInsulated boundary at plane of symmetryElements TriangularMesh
sizes 2 mm at pipes; 10 mm at pile edge; 0.5 m model edgeMaterial
properties Refer to Table 6Validity Fo10Software used COMSOL 4.3F.
Loveridge, W. Powrie / Energy 64 (2014) 747e757 753represent those
for two piles interacting. There is some differencebetween these
cases, but typically the discrepancies are within afewpercent. The
exception is the lower bound case with pipes
neartheedgewherethedifferencebetweenthesingleandtwopilescenarios is
initially 20%, rapidly falling to around 2% by Fo 0.1.Given that
the lower bound represents smaller diameter piles andthat Fo 0.1 is
equivalent to less than 1 h for a 300 mm diameterpile,
thisdifferenceisminorinthecontext ofstandard1htimestepping used in
design. Therefore, while it is possible to producenew concrete
G-functions for the interacting cases in Fig. 8, it
willbesufcientinmostscenariostousethecurvesforsinglepilespresented
(with curvet data) in Ref. [5].5. Impact on energy
storageToillustratetheimpactofinteractionsbetweenmultiplepileheat
exchangers on the thermal energy that can be extracted fromthe
ground, three cases have been compared using the multiple
pileG-functions presented in Section 3:A single pile (B/2rb innity
in Fig. 3d),Two piles installed at a spacing of B/2rb 1.2 (Fig.
3d),Nine piles installed on a grid pattern with B/2rb 3 (Fig. 6).In
allcasesAR 50hasbeen assumed asthis representstheworst case
interactions. The piles have been selected to be 450 mmdiameter and
22.5 m long with a concrete thermal conductivity oflc 1 W/mK. The
surrounding ground is assumed to have thermalpropertieslg 2 W/mK
andag 1 106m2/s. The pile concretethermal resistance Rc is 0.075
mK/W and the pipe resistance Rp is0.025mK/Wfor four pipes
placednear the edge of the pile.Consequently an upper bound pile G
function for AR 50 is used
todescribethetemperaturechangeintheground, whilealowerbound
concrete G-function for pipes placed near the edge is used
todetermine the temperature change within the pile. The
followingequation is used to calculate the mean temperature change
in theheat transferuid:DTfqRpqRcGcq2plgGg(9)where Gcis the concrete
G-function and Ggis the pile G-function. Asthe applied thermal load
q (in W/m) is not constant, but changeswitheachtimestep,
superpositionmust beusedwitheachG-function so that the change in
temperature can be calculated:DTn Xi ni 1qi2plghG
FonFoi1
GFonFoii(10)where n is the point in normalised time in which the
superpositionis evaluated and G is the G-function (whichever one is
being used
inaparticularcase)calculatedatthevalueofFoprescribedintheequation.
Equation (10) has been coded in the software Matlab toallow
calculation of the sum in hourly timesteps for the period ofone
year.For the examples being consideredq is obtainedfromthebuilding
thermal load prole shown in Fig. 9. The prole is given fora typical
year and has been developed from a numerical simulationof a modern
multi-use development in the South East of England,scaled down to
an appropriate level for a single pile. The total de-mand is 1.72
MWh heating and 1.76 MWh cooling per pile. How-ever, despite the
overall energy demand being close to balanced,the peak power values
for cooling are much higher than those forTable 6Steady state
thermal resistance (Rc) for individual and pairs of interacting
piles.Lower bound300 mm diameterpile, 2 pipeslc 1;lg 2Upper
bound1200 mm diameterpile, 4 central pipesor 8 near edgelc 2;lg
1Central pipes; Single pile 0.267 m K/W 0.231 m K/WTwo piles#1
0.267 m K/W 0.231 m K/WTwo piles#2 0.267 m K/W N/APipes near edge;
Single pile 0.183 m K/W 0.029 m K/WTwo piles#1 0.187 m K/W 0.029 m
K/WTwo piles#2 0.180 m K/W N/AFor piles with centrally placed pipes
the concrete cover is c 105 mm for 2rb 300mm and c 555 m for
2rb1200 mm. For piles with pipes placed near the edge theconcrete
cover is c 50 mm for 2rb 300 mm and c 75 m for 2rb 1200 mmFig. 8.
Concrete G-functions for individual and pairs of interacting piles:
a) pipes placed centrally; b) pipes placed near the pile edge.
Solid lines are for individual piles; short dashedlines are two
piles Case #1; long dashed lines are two piles Case #2.Fig. 9.
Example thermal loads for one year commencing in January (insets
show dailycycle detail). Note: positive thermal loadsare
heatinjectionto the ground (buildingcooling); negative thermal
loads are heat extraction fromthe ground (building heating).F.
Loveridge, W. Powrie / Energy 64 (2014) 747e757 754heating,
beingcomprisedofshorterdurationbutgreatermagni-tude power peaks. It
will be seen that this has an impact on theresulting temperature
changes.5.1. ResultsThe calculated mean uid temperatures (Tf) are
shown in Fig. 10.The actual inlet and outlet temperatures to the
heat pump wouldactually cover a wider range than this (as per
Equations (6) and (7)).However, thesevalueshavenot
beencalculatedastheywoulddepend on the mechanical design of the
pipework for any partic-ular system,
whichwouldneedtobeoptimisedtomaintainasensibletemperaturedifferences
across theheat pump. Fig. 10shows the calculated temperature
changes for the rst year,which range from 8.3 C to 15.9 C for
single pile to 11.1 Cto 20.1 C for two piles at B/2rb 1.2 and 11.0
C to 21.8 C fornine piles at B/2rb 3. It is interesting to note
that initially the twopile scenario has a greater temperature
change that the nine pilescenario. This is due to the closer
spacing and the interaction effectscommencing at a shorter time. In
the longer term the temperaturechange for the nine pile arrangement
is much greater, especially inthe summer when the rate of change of
thermal load is greatest.
Itisexpectedthatthedifferencesbetweenthethreecaseswouldincrease
with time.Assumingatypical initial groundtemperatureof 12
Cit isreasonable to apply limits on the meanuid temperature
changeof 10 C and 20 C (i.e. absolute limits of 2 C and 32 C).
Thenthe average energyavailable per pile canbe
calculatedusingEquation (9) assuming thatDTf remains within the
range of theselimits. As in Ref. [5], for simplicity, the shape of
the thermal loadproles in Fig. 9 has not been changed, but the
values of appliedthermal load have been adjusted pro-rata to give
an indication ofthe impact the interactions are having on the
thermal capacity ofthe piles. In reality the system would be
adjusted to cover as muchbaseloadas
possiblewithsomeextremepeakloads beingbesupplied by an auxiliary
system. The assessment would also coverthe full design life of the
structure rather than just the rst year as isshown here.Fig.
11shows therelationshipbetweenthe average energyavailable per pile
and the imposed temperature limits for the oneyearperiodanalysed.
Basedonthe 10
Ctemperaturechangelimittheavailableaverage heating
energyperpileis2069kWh,1546kWhand1564kWhforonepile,
twopileandninepilesrespectively. Interestinglytheninepilecaseis
actuallyslightlygreater than the 2 pile case. This is because the
nine piles are atFig. 10. Calculated meanuid temperatures for
different numbers and arrangements of piles.Fig. 11. Relationship
between allowableuid temperature change and available energy for
example pile arrangements: a) heating; b) cooling.F. Loveridge, W.
Powrie / Energy 64 (2014) 747e757 755greater spacing and the
analysis starts in the winter with heating.
Iftheanalyseswerecarriedout overalongertimeperiod, therewould
undoubtedly be less energy available for the nine pile casethan for
the 2 pile case. In terms of cooling, using the 20 C limit,the
available average energy for each pile is 2219 kWh, 1746 kWhand
1611 kWh for one pile, two pile and nine piles respectively.
Therangeofvaluesissimilartotheheatingcase, butasthecoolingseason
comes later in the analysis period there is a greater differ-ence
between the two pile and nine pile cases. Despite the
similaramounts of energy extracted in heating and cooling the
tempera-ture changes are bigger in cooling due to the higher peak
loads. Thisillustrates the importance of understanding both the
monthly andtotal energy demands of a systems and the shorter term
variationsin demand which will result in the peak loads.Intotal
theaverageenergyavailableperpileinthethreear-rangements is 4288
kWh, 3292 kWh and 3175 kWh for the singlepile, twopileandnine
pilescasesrespectively. Thisrepresentsa23% drop in available energy
from one to two piles and a further26%dropfortheninepilecase.
Itisinterestingthatthesetwogures are of the same order, despite a
much larger number of pilesin the nine pile case. This reects the
importance of pile spacingand the much reduced interactions when
the pile spacing is openedfromB/2rb1.2 to B/2rb3. The equivalent
energy for the nine
pilearrangementis51%comparedwith23%forthatarrangementatsteady
state. This indicates that there is the potential for the
ef-ciencyof thesystemtodeclinefurtheroverthelifetimeof thebuilding.
For the 450 mm diameter piles analysed one year repre-sentsFo 623.
Steadystateconditions(underconstantthermalload) are not reached
until in excess of Fo 10,000, or around 16years.6. DiscussionGround
heat exchangers installed in multiple piles will
interactadverselyintermsofenergy availableperpile,
generallybylessthan5%aslongasB/2rb>AR. However,
giventhecostofcon-structing deep foundations, priority will always
be given to opti-mising pile layouts with respect to their
structural function, that ofsupporting the overlying building. This
means that it is not possibletoadjust thepositions of pileheat
exchangers toincreasethespacing andhence maximise the energy
output. However, anassessment of the potential for interactions
(increased
temperaturechangeorreduceenergyoutput)betweenadjacentpilesshouldstill
be carried out.The example analyses presented have shown how the
effect ofadjacent piles can greatly reduce the energy output from
each in-dividual pile. Overall the total energy obtained from
multiple pilesis always greater than from a single pile, but as the
number of
pilesisincreasedandthespacingreducedtheenergyreturnperpiledecreases.
Insomeextremecasesitmaybemoreeconomicaltoequip only some piles in a
foundation layout with heat exchangepipes. While the pipes
themselves are of relatively lowcost, there isadditional programme
time for installing the pipes and
additionalrunningcostsforalongerpipecircuit.
Thesewouldneedtobeweighed against the energy gains from the
interacting piles.The degree of interactions between adjacent pile
heat ex-changers will depend on a number of factors.The spacing
ofthepiles is important, but so is the number of piles. The spacing
willhavethebiggest impact onthetimetakenfor interactions tobecome
signicant, while the number of piles in the arrangement(in
combination with their spacing) will impact the long term en-ergy
obtainable. Generally smaller aspect ratio piles will interact toa
lesser extent than larger aspect ratio piles, which means that
theformercanbesuccessfullyimplementedat closerspacings. Thenature
of the thermal load is always important for the performanceof
ground heat exchangers and this case is no exception. Short
termvariationsinloadwillreduceinteractionscomparedwithasus-tained
base load.7. ConclusionsThis paper presents a method for
determining new G-functionsforuseinthethermal analysisof
multipleinteractingpileheatexchangers. ExamplepileG-functions,
whichdescribethetem-perature change in the soil around a pile with
time, are presentedfor anumber of examplescongurations. Thekey
conclusionsofthis study are:If multiple adjacent piles are used as
heat exchangers then therewill be adverse thermal interactions
betweenthe heat ex-changers. These interactions will become more
signicant overthe lifetime of the energy system, but will also be
dependent onthe nature of the thermal load. For example
highlyuctuatingthermal demands will reduce the potential for
interactions be-tween heat exchangers.Heat exchangers with smaller
aspect ratios are affected less bythermal interactions.
Thusinteractionsbetweenpileswill
belessthanthosebetweentraditionalboreholeheat exchangersinstalled
at the same spacing, potentially leading to more energyefcient
systems.However,as pilespacing is usually governedby the
overlyingstructure, piles are generally more closely spaced than
typicalborehole arrangements.Consequently, itisnot
alwaysadvantageoustoequipallpileswithin a pile group with heat
transfer pipes.The degree of interactions to be expected for a
given scheme canbe calculated using the methods for multiple pile
G-functionsdescribed in this paper.Signicantly, it was found that
the changes to the temperatureeldwithinconcretepilesarenotsufcient
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