geothermal helical heat exchanger · geothermal helical heat exchanger xavier moch1,2, marc...

GEOTHERMAL HELICAL HEAT EXCHANGER

Xavier MOCH1,2, Marc PALOMARES2, Fabrice CLAUDON3, Bernard SOUYRI1 and Benoıt STUTZ1

1LOCIE - CNRS UMR 5271, Le Bourget-du-Lac, France2RYB-Terra, Le Bourget-du-Lac, France3CEA/INES, Le Bourget-du-Lac, France

ABSTRACTThis paper presents helical heat exchangers forgeothermal use. The differences with morewidespread geothermal exchangers are given, and acomplete thermal model for the underground such asfor the exchanger is developed, including the localfreezing of water contained in the underground whentemperatures are lower than 0

�C. This model is com-

pared with published data and with experimental re-sults.

In a second part, the model is applied to the heatingand the cooling of buildings. The choice was made toconsider that the thermal energy needed was directlyextracted from the underground, so that results do notdepend on the model of heat pump.

INTRODUCTIONHelical heat exchangers belong to geothermal ex-changers which are said to be compact. Contraryto vertical exchangers – or borehole heat exchangers(BHE) –, they are buried in the first dozen of metersunder the ground surface, in order to reduce the instal-lation costs. Several geometries and ways of buryingthese exchangers exist. In this paper, we consider anexchanger “Terra-Spiral”, which height is H = 2.4 m,radius R = 0.5 m, and pitch p = 0.08 m. The upperpart of the exchanger is buried at around 1 meter depth:zup

= �1 m, with a vertical Z-axis going upwards.The ground surface serves as the reference z = 0.

According to Philippe (2010), the vertical heat ex-changers are widespread over the world. Geothermalhorizontal exchangers are quite common in France:the installation costs is reduced since they are buriedat around 1 meter depth on large surfaces (typicallybetween once and twice the area to be heated/cooled).That is the reason why many thermal models exist forthese exchangers (Philippe (2010)).

Figure 1: A house connected with four spiral heat ex-changers (courtesy of RYB-Terra)

Helical heat exchangers constitute an alternative to

these exchangers: compared to horizontal exchang-ers, they require few place on the ground; and com-pared to BHE, the installation costs are significantlyreduced. Few thermal models exist: at our knowl-edge, the main work was done for energy storage inarid zones (Doughty et al. (1991), Rabin et al. (1991),Rabin and Korin (1996)). These studies concern ob-jects of 6 m height which upper part is buried at 4 mdepth. The encountered problematic is not exactly thesame as our, since our goal is to use these exchangersin order to heat and to cool buildings in western Eu-rope, where the climate and even the underground isdifferent.

In this paper, we adapt an existing model, presentedby Rabin and Korin (1996), in order to make it usablewith our objectives. This model is 2D axisymmetric,with the vertical axis of axisymmetry being the axis ofthe helical heat exchanger.

THERMAL MODEL OF THE UNDERGROUNDTemperatures not disturbed by the exchangersThe first step is to model the underground. Thegeothermal gradient does not need to be taken intoaccount since the depths are quite low. We consideras upper boundary condition the temperature of theambient air. We denote h

tot

(W.m�2.K�1) the heattransfer coefficient between the air and the groundsurface. Moreover, we assume constant values ofthermal conductivity � and heat capacity ⇢c

p

of theground, considered as homogeneous and semi-infinite.Heat conduction is considered to be the only trans-fer of thermal energy occurring in the underground.These hypothesis are widespread in geothermal stud-ies (Doughty et al. (1991), Rabin et al. (1991), Neb-bali and Makhlouf (2007), Rabin and Korin (1996),Deveughele and Vercamer (1983)): Rabin and Korin(1996) insist on the difficulty to take into account allphenomena, and Doughty et al. (1991) show an ex-ample of thermal parameters depending on temper-ature and humidity: the results are not significantlyimproved while the calculation is more complex andtime-consuming.

Let consider the air temperature variation:

Text

(t) = Tmoy

� ˇTamp,atm

cos

�!(t� t

c

)

�(1)

The mathematical solution for the undisturbed temper-ature in the underground ˇT (z, t) is given as the sum of

Proceedings of BS2013: 13th Conference of International Building Performance Simulation Association, Chambéry, France, August 26-28

- 2948 -

four sinusoids with the same pulse !:

ˇT (z, t) = Tmoy

�ˇTamp,atm

1 + ↵4exp(z

r!

2a)⇥

h↵0

cos

�!(t� t

c

) + z

r!

2a

�

+ ↵ cos

�!(t� t

c

) + z

r!

2a+

⇡

4

�

+ ↵2cos

�!(t� t

c

) + z

r!

2a+

⇡

2

�

+ ↵3cos

�!(t� t

c

) + z

r!

2a+

3⇡

4

�i

where a =

�

⇢c

p

is the thermal diffusivity of the under-

ground and ↵ =

�p

�⇢c

p

!

h

tot

.

The term exp(zp

!

2a ) reflects the fact that the ampli-tude of high-frequency air temperature variation de-creases significantly with the depth. If the period ofthe sinusoid is equal to 1 day (it is colder at dawnthan in the late afternoon) and if the diffusivity isa = 10

�6 m2/s, values of z �0.4 m implyexp(z

p!

2a ) < 0.1; values of z �0.8 m implyexp(z

p!

2a ) < 0.01. That is the reason why a meantemperature over the day is sufficient to describe thetemperatures at the depths of the exchangers.

With this assumption, 1-order atmospheric tempera-ture model is sufficient. From now on, ! is set tothe annual pulse. An order of magnitude is ↵ =p

�⇢c

p

!

h

tot

' 0.06, ranging from 0 to 0.2, dependingon the type of underground and of its upper “insula-tion”. Except for special cases with really specific con-ditions and high accuracy of measurement, it is rea-sonable to consider that the undisturbed temperatureof the ground can be written as:

ˇT (z, t) = Tmoy

�Tamp

exp(z

r!

2a) cos

�!(t�t

c

)+z

r!

2a

�

(2)with T

amp

' ˇTamp,atm

the amplitude of the tempera-ture on the ground surface.

EQUATION 2 was already used by Doughty et al.(1991) in this form. Indeed, the author consider thatthe temperature of the ground surface is the same asthe atmospheric temperature (what we actually equallydid when considering ↵ = 0); and since their geother-mal heat exchanger is buried deep enough in the un-derground, they consider that this temperature can notbe disturbed by its use.

Boundary condition expressed with heat fluxEQUATION 2 suffices to give the initial temperaturecondition in the underground. The derivative with re-spect to z — multiplied by (��) — indicates the cor-

responding heat flux:

e'(z, t) = ��@ eT@z

=

8>><

>>:

Ep!T

amp

exp

�r !

2az�

⇥ cos�!(t� t

c

) +

r!

2az +

⇡

4

�

(3)

with E =

p�⇢c

p

the thermal effusivity of the under-ground.

Moreover, if the temperature of the ground surface dif-fers from the undisturbed one, an additional flux mustappear on this place, which value is h

tot

�T (r, 0, t) �

eT (0, t)�. h

tot

takes into account the natural convectionsuch as the (linearized) radiation. According to Rabinet al. (1991), we can consider that a reasonable valuewith a low wind condition is h

tot

= 15 W.m�2.K�1.

SummaryThe underground is described with a 2D axisymmetricway. It is modeled as a cylinder, with a height go-ing from z1 < 0 to 0 and a radius ranging from 0 tor1. z1 and r1 are chosen “far enough” from the ex-changer – depending on the simulation done –, so thatthe temperatures at these limits remain undisturbed bythe exchanger. Typical values are z1 = �10 m andr1 = 7 m. The conditions applied to the undergroundare:1. Initial condition at t = t

ini

T (z, tini

) = Tmoy

� Tamp

exp(z

r!

2a)

⇥ cos

�!(t

ini

� tc

) + z

r!

2a

� (4)

2. Boundary conditions:

• axial symmetry at r = 0;• adiabatic condition at r = r1;• heat flux condition at z = z1

'(z1, t) = Ep!T

amp

exp

�r !

2az1

�

⇥ cos

�!(t� t

c

) +

r!

2az1 +

⇡

4

�

(5)

• heat flux condition at z = 0

'(0, t) = Ep!T

amp

cos

�!(t� t

c

) +

⇡

4

�

+ htot

�T (0, t)� eT (0, t)

�

(6)

Freezing of the undergroundThe underground contains water, which will freeze iftemperatures goes below 0

�C. We model this freez-

ing considering the following hypothesis for numeri-cal reasons:


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1. Gas contained in the underground (air) can flowout during the freezing, when the water expanses,so that there is no variation of volume of the un-derground;

2. Water is liquid over 0�C and solid below �1

�C.

A smooth Heaviside function gives the ratio ⌘ ofliquid water between these temperatures.

The local thermal conductivity of the undergroundsuch as its local volumetric thermal capacity ⇢c

p

areaffected by this transition. We assume that these prop-erties depend only and directly on the local amount ofsolid/liquid water. Denoting the water content of theunderground and 1 � ⌘ the amount of ice in the totalwater (⌘ varies between 0 and 1), we assume that thethermal conductivity of the underground is given by

�mix

= (1�)�mat

+⇣(1�⌘)�

ice

+⌘�liq.wat

⌘(7)

The same applies for volumetric thermal capacity:�⇢c

p

�mix

= (1� )�⇢c

p

�mat

+ ⇣(1� ⌘)

�⇢c

p

�ice

+ ⌘�⇢c

p

�liq.wat

⌘

(8)

Moreover, latent heat is released during this change ofstate. Mathematically, the heat capacity of the under-ground seems to be affected by this change of state.The heat equation is

�⇢c

p

�mix

@T

@t= r

✓�mix

rT

◆� @⌘

@t

�⇢L

�liq.wat

that is:

r✓�mix

rT

◆=

✓�⇢c

p

�mix

+

d⌘

dT�⇢L

�liq.wat

◆@T

@t

Thus, the equivalent heat capacity of the undergroundcan be defined with:

�⇢c

p

�mix,eq

=

�⇢c

p

�mix

+

d⌘

dT�⇢L

�liq.wat

(9)

One can keep the classical aspect for the heat equation:

r✓�mix

rT

◆=

�⇢c

p

�mix,eq

@T

@t

MODEL OF THE HEAT EXCHANGERForewordsJunctions between the heat pump and the extremitiesof the exchangers are not taken into account. It is gen-erally assumed that the heat transfer fluid flows fromthe upper to the lower bound of the spiral coil. In otherwords, the outlet to the heat pump is downside and theinlet upside.

The thermal resistance between the underground andthe heat transfer fluid is calculated through the geom-etry of the wall, the thermal properties of the materialconstituting the exchanger, the thermal conductivity ofthe heat transfer fluid, and the Nusselt number. Thetotal thermal resistance of the object modeled has tobe the same as the thermal resistance of the real ob-ject, i.e. the sum of the thermal resistance of the pipe,R

th,wall

, and of the flow, Rth,flow

:

Rth,wall

=

ln

�r

e

r

i

�

2⇡Ltot

�wall

(10)

Rth,flow

=

1

⇡NuLtot

�f

(11)

with �wall

the thermal conductivity of the wall, ri

(re-spectively, r

e

) the internal (respectively, external) ra-dius of the pipe, �

f

the thermal conductivity of theheat transfer fluid and L

tot

the helical length of theexchanger.

Moreover, there exists a thermal resistance of externalcontact R

th,ext

, due to the potential development of anair layer between the pipe and the underground, inher-ent to freezing and thawing cycles during extraction.Considering the other thermal resistances, we assumethat this resistance can be neglected.

An order of magnitude of the volumetric flow rate inan exchanger is 4 L/min, that is q

v

' 67 10

�6 m3/s.The heat transfer fluid is mono-propylene glycol, sothat the kinematic viscosity should remain between2 10

�6 and 10 10

�6 m2/s, with high viscosity whenthe fluid is cold and concentrated. Through a crosssection 2r

i

= 20.4 10�3 m, the Reynolds number isRe < 2000: the flow in the exchanger is assumed tobe laminar, as in Rabin et al. (1991).

We use Nu = 4.36: it matches a constant flux condi-tion. Indeed, this hypothesis is realistic: assuming asteady state for the heat pump, the temperature of theheat transfer fluid along the depth is approximately astraight line.

Annular cylindrical conduitThis model appears in Doughty et al. (1991), Rabinet al. (1991) and in Rabin and Korin (1996) but is notdetailed. This is the goal of this part.

Geometry

For usual geometries, this model leads to an increaseof the exchange area between the exchanger and theunderground. The cylinder has the same (mean) ra-dius and the same height as the exchanger, so that theunderground volume in the middle of the exchangercorrespond to the real one.The flow section is chosen so that that the volume ofthe heat transfer fluid is kept. This is the only influenceof the pitch.


- 2950 -

The thickness of the wall is not very important, as longas it is small when compared to the radius of the cylin-der. The value of the thermal conductivity of the cylin-der walls is such that the total thermal resistance be-tween the heat transfer fluid and the underground isequal to the one of the helical heat exchanger.

Heat transfer fluid flow

Since the model is 2D axisymmetric, the flow cannothave an orthoradial component. Its velocity is verti-cal. It corresponds to the vertical component of thevelocity of the heat transfer fluid inside the real helicalexchanger. Since the height is kept, the transit time iskept.

Moreover, since the fluid volume is kept, the volumet-ric flow rate is kept too.

The boundary conditions are quite easy to express. Ateach time step, two parameters are needed:

• the geothermal power P to be extracted from theunderground;

• the outlet temperature Tout

(computed at thelower side of the cylinder).

Both parameters enable the calculation of the inlettemperature T

in

to be applied on the upper side of thecylinder:

Tin

= Tout

� P�⇢c

p

�f

qv

(12)

with P the geothermal power, positive when the ex-changer extracts energy from the underground (heat-ing mode for the building), q

v

the volumetric flow ofthe heat transfer fluid, and

�⇢c

p

�f

its volumetric heatcapacity.

VALIDATION OF THE MODEL

Comparison with published results

At first, we use the same conditions as Rabin and Ko-rin (1996). The test consist in storing heat in the un-derground by injecting hot water at 70�C for 150 days,before recovering it by injecting water at 20�C. Thissimulation is done using two undergrounds with differ-ent thermal properties (case 1 and case 2). The maindifference between both models is the use of a ther-mal resistance between the heat transfer fluid and theunderground: moreover, this resistance is quite low sothat results should have the same orders of magnitude.Figure 2 shows the temperatures at the outlet for bothmodels. The coherence is really good.

20

25

30

35

40

45

50

55

60

65

70

0 15 30 45 60 75 90 105 120 135 150 165 180

Out

lett

empe

ratu

re(�

C)

Days since beginning

Thermal load and unload of the undergroundCase 1 (reference)Case 2 (reference)

Case 1Case 2

Figure 2: Outlet temperatures for two cases (refer-ences given by Rabin and Korin (1996))

Comparison for the freezing modelNumerical approach

To prove the interest of taking into account the freez-ing of the underground water, three simulations aredone. At the beginning, the underground temperatureis homogeneous (10�C). A constant cooling power,high enough to make the underground freeze, is con-tinuously extracted during 3 days. Then the heat ex-traction is stopped, but the fluid circulation still worksduring 11 additional days (relaxation time).

Three soils are simulated, each of them containing adifferent volumetric ratio of water (0%, 20%, 50%).These soils are chosen so that the thermal parametersare the same when the whole water content is liquid.Figure 3 shows how the outlet temperature evolves(the inlet temperature is nearly 3.5 K lower than theoutlet temperature during the heat extraction).

-10

-5

0

5

10

0 2 4 6 8 10 12 14

Tem

pera

ture

(�C

)

Days since beginning

Outlet temperature, depending on water content

0%20%50%

Figure 3: Influence of underground freezing on outlettemperatures

It can be noted that the outlet temperatures after 3 daysstrongly depend on this freezing phenomenon. Thisphenomenon also occurs in standard condition and hasto be taken into account when modeling the under-ground: it would not suffice to consider only the ther-mal parameters of the underground at the beginning,when the entire water content is liquid.


- 2951 -

Experimental approach

An experimental platform was created some kilome-ters away from Chambery (Savoy, French Alps). He-lical heat exchangers were connected to a heat pump.The exchangers are H = 2.4 m high and R = 0.5 mlarge. The platform is a dozen meters away from ariver, the Leysse, and the underground is mainly com-posed of silt and sand. Water was found at a depthof 3.5 m or 4 m, just below the lowest part of the ex-changer. As a consequence, it can be assumed thatthere is a high water content in the underground. Ac-cording to van Genuchten (1980) and Wosten et al.(1999), we first estimate the volumetric water contentto = 40%.

For this experiment, the heat pump worked continu-ously during several days, until temperatures below0

�C could be reached. Temperatures were measured

with 4-wired Pt100 and the geothermal power was cal-culated thanks to the flow rate. This power was thenused as a working condition for the model.

Moreover, the values of atmospheric temperaturesover a “mean” year were used to obtain the valuesof T

moy

, Tamp

and tc

. We measured variation ofnatural temperatures in the underground in order toestimate its average diffusivity using EQUATION 2:a ' 1.1 106 m2/s.

Besides, the thermal effusivity of the undergroundwas estimated experimentally. This lead to � =

2.6 W.m�2.K�1 and ⇢cp

= 2.3 106 J.m�3.K�1:this values were used in the simulations. The valueof thermal conductivity is slightly high accordingto Verein Deutscher Ingenieure (2000), but is reason-able.

15

10

5

Outlet (simulation)Axis (simulation)

Observation (simulation)

Tem

pera

ture

s(�

C)

Hours since beginning

Outlet (experimental)Axis (experimental)

Observation (experimental)

0

-55004003002001000

Freezing of the underground

Figure 4: Comparison between simulation and exper-imental work

Figure 4 compares the experimental and simulatedtemperatures at following points: “outlet” of the heattransfer fluid, “axis” (at middle-height), “observation”at 1 m from the axis (at middle-height).

The temperatures on the axis keep a value of 0�C dur-ing a long time (more than 200 hours). This can bebound with the release of latent heat in the under-ground. This proves the necessity of taking into ac-count the effect of freezing.

We probably overestimated the water content, sincethe simulated temperatures on the axis remain at 0�Cfor a higher time in the simulation than in reality. Ata distance of 1 m of the axis, the model overestimatesthe temperatures: this effect can be tied to a bad esti-mation of the thermal properties for the underground.Conversely, the model underestimates the outlet tem-peratures when the water freezes. This may be a con-sequence of the way we modeled the freezing of wa-ter, between 0

�C and �1

�C: indeed, experimental and

simulated outlet temperatures converge again when allthe water is solid (after 400 hours). We could notcheck this hypothesis by reducing this interval, be-cause of a lack of computer memory.

The model is not perfect and should be improved;moreover, the thermal parameters are not preciselyknown. Nevertheless, a good adequateness can befound with experimental results.

COUPLING WITH A LOW-ENERGY HOUSEThe thermal energy need of a low-energy house wassimulated besides. It handles on daily-mean val-ues required for heating and cooling a 120 m2 com-pact building located in Savoy (France). The under-ground is supposed to have a thermal conductivity� = 1.7 W.m�1.K�1, a volumetric thermal capac-ity ⇢c

p

= 2.5 106 J.m�3.K�1, and a volumetric wa-ter content = 20%. These values are representativefor many types of undergrounds which would not bedried. 3 helical heat exchangers are connected in par-allel: we assume that they do not to have any thermaleffect on each other.

-10

0

10

20

30

40

-100 0 100 200 300 400

Tem

pera

ture

s(�

C)

Day of the year

Results over 1 year

OutletAxis

Observation at 1 m

Figure 5: Coupling with a low-energy building over 1year

In the following, the energy required for the buildingis assumed to be extracted from the underground on acontinuous way. Different temperatures evolutions are


- 2952 -

represented on Figure 5: the outlet temperature, thetemperature on the axis at nearly mid-height of the ex-changer, and the temperature at 1 m of the axis at thesame depth.The temperatures of the ground, and in particular thespatial extension of the freezing zone, appear on Fig-ure 6 at the day when the ice has the maximal exten-sion. The solid line represents the isotherm �1

�C: in-

side it, water is frozen. The dashed line represents theisotherm 0

�C: outside, the water remained liquid.

-6

-4

-2

0

2

4

6

8

10

Dep

th(m

)

Radius (m)

Temperatures in the underground (�C) and limits of freezing

-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

0 0.5 1 1.5 2 2.5 3 3.5 4

Figure 6: Temperatures in the underground and maxi-mal volume of ice

Inside the exchanger, the underground is completelyfrozen. Conversely, ice propagates slowly outside.

Figure 7 represents the outlet temperatures over 10years, when the exchangers are only used to cope withthe heating needs. It shows that the natural load of theunderground suffices to get similar temperatures overyears.

-5

0

5

10

15

-2 0 2 4 6 8 10

Out

lett

empe

ratu

re(�

C)

Year

Evolution over 10 years

Figure 7: Simulation over 10 years

EFFECT OF CYCLINGLet define the medium temperature of the heat transferfluid as the mean value between inlet and outlet tem-peratures. Figure 8 shows this temperature for threecases:1. one with a steady operating mode with constant

power (labelled Steady).

2. a second one with an operating mode “15 min-utes on / 45 minutes off” each hour: the circulatorworks during both periods (labelled On).

3. a third one with an operating mode “15 minuteson / 45 minutes off” each hour: the circulator doesnot work during the second period (labelled Off ).

The powers are chosen such that the energy extractedover an hour is the same for the three cases.

5

6

7

8

9

10

0 2 4 6 8 10 12Te

mpe

ratu

re(�

C)

Hours since beginning

Average temperature of heat transfer fluid

SteadyOnOff

Figure 8: Evolution of the mean heat transfer fluidtemperature over 12 hours

Figure 8 shows the inconvenience of short cycling: thetemperatures of the heat transfer fluid are lower whenextracting geothermal power. Two effects play a majorrole in this phenomenon:

• the energy is extracted next to the exchanger(there is not enough time for the underground tohomogenize temperatures on a further distance);

• the gap between the temperature of heat trans-fer fluid and the temperature of the undergroundoutside the exchanger is increased, due to thethermal resistance R

th

presented above.

Therefore, better coefficient of performance (COP)can be reached with long cycles. Moreover, the twocases “cycling” show no major differences on out-let temperatures when geothermal heat is extracted.There is apparently no use in making the circulatorwork when no heat is required.

INFLUENCE OF SEVERAL PARAMETERSThe results presented in this section are done with thethermal needs of another house. It would correspondto a 120 m2 house built at the same place as the pre-vious one, with thermal norms corresponding to year2005. For the same reasons as previously, the heatpump is not modeled and the energy needed for thebuilding is supposed to correspond to the energy ex-tracted from the underground. For the reference case,six exchangers are connected in parallel and are usedto heat and to cool the building.

Results over 1 yearFigure 9 shows the outlet temperatures, for the case ofreference (heating and cooling) and for the case “heat-ing only”. Once again, the natural thermal load during


- 2953 -

summer is good, whatever the use of the geothermalexchangers.

-10

-5

0

5

10

15

20

25

-100 -50 0 50 100 150 200 250 300

Tem

pera

ture

(�C

)

Day of the year

Outlet temperature

Heating onlyReference

Figure 9: Outlet temperature over 1 year

Influence of the pitch

On Figure 10, we made the pitch grow from 0.08 m(reference case) to 0.2 m, keeping values for radius Rand height H constant. The outlet temperature vari-ations increase with the increase of the pitch, mainlybecause of the greater thermal resistance R

th

(the totallength is reduced).

-10

-5

0

5

10

15

20

25

-100 -50 0 50 100 150 200 250 300

Tem

pera

ture

(�C

)

Day of the year

Outlet temperature

Pitch: 0.2 mReference

Figure 10: Outlet temperature, making the pitch vary

Influence of height and radius

On figure 11, the number of exchangers was reduced toN = 5, and the total flow rate “at the heat pump” waskept constant. On a first model, the height was mul-tiplied by 1.2; on a second one, the radius was multi-plied by this same value. It appears that the tempera-tures are nearly the same for the three cases. As a firstapproximation, the thermal resistance of the whole in-stallation is the same as in the reference case, and theoutlet temperatures evolve on a similar way when theproduct N ⇥H ⇥R is kept constant.

-10

-5

0

5

10

15

20

25

-100 -50 0 50 100 150 200 250 300

Tem

pera

ture

(�C

)

Day of the year

Outlet temperature

H multiplied by 1.2R multiplied by 1.2

Reference

Figure 11: Outlet temperature, making geometry vary

Influence of the groundOn figure 12, the ground is supposed to be wet sand(� = 2.4 W.m�1.K�1). The temperatures gain in sta-bility (a greater volume around the exchanger is used).

-10

-5

0

5

10

15

20

25

-100 -50 0 50 100 150 200 250 300

Tem

pera

ture

(�C

)

Day of the year

Outlet temperature

Wet sandReference

Figure 12: Outlet temperature, making ground vary

CONCLUSIONThis paper presented a reliable model of geothermalheat exchangers. This model may be used to definethe geometry of exchangers and to size an installation.Besides, it pointed out the fact that temperatures wereabout the same over years (no thermal unload of theunderground can be measured).

We showed that the influence of the thermal resistanceexisting between the heat transfer fluid and the under-ground had to be taken in account, in particular whendefining the pitch. Moreover, the influence of the cy-cling was explained: if possible, short cycles are to beavoided.

A limit of this model is the freezing of water, which isconsidered to occur at constant volume. Moreover, nomechanical effect is taken into account: for example,the freezing/thawing cycles may create an air layer be-tween the exchanger and the underground, increasingthe thermal resistance over years.


- 2954 -

NOMENCLATURELatin lettersa Thermal diffusivity (m2.s�1)cp

Specific heat (J.kg�1.K�1)E Thermal effusivity (J.s�1/2.m�2.K�1)

htot

Total heat transfer at theground surface (W.m�2.K�1)

L Specific latent heat (J.kg�1)Ltot

Helical length of the exchanger (m)N Number of helical heat exchangers (-)Nu Nusselt number (-)P Geothermal power (W)qv

volumetric flow rate (m3.s�1)r Radius (m)re

Exterior radius of the pipe (m)ri

Interior radius of the pipe (m)R

th

Thermal resistance (K.W�1)t Date (s)tc

Coldest date (s)T Temperature (K)Tmoy

Mean temperature over the year (K)

Tamp

Amplitude of temperature on theground surface (K)

ˇTamp,atm

Amplitude of temperature of theatmosphere (K)

z Depth (m)

Greek letters↵ Auxiliary coefficient (-)⌘ Ratio of water at liquid state (-) volumetric ratio of water in the underground (-)� Thermal conductivity (W.m�1.K�1)⇢ Mass density (kg.m�3)' Heat flux density (W.m�2)! Annual pulse (rad.s�1)

Subscriptsext External sidef Heat transfer fluidflow Flowice Solid waterin Inletliq.wat Liquid waterlow Lower partmat “Dry matrix” of the undergroundmix Multi statesout Outletup Upper partwall Wall of the exchanger

Accentuatione Undisturbed by the exchanger

REFERENCESDeveughele, M. and Vercamer, P. 1983. Exploitation

thermique du proche sous-sol par pompe a chaleur.In Bulletin of the international association of engi-neering geology n. 28.

Doughty, C., Nir, A., and Tsang, C.-F. 1991. Seasonalthermal energy storage in unsatured soils: Model

development and field validation. University of Cal-ifornia.

Nebbali, R. and Makhlouf, S. 2007. Determinationde la distribution du champ de temperatures dans lesol, par un modele semi-analytique. conditions auxlimites pour les besoins de simulation d’une serre deculture. Revue des energies renouvelables CER’07Oujda, pages 255–258.

Philippe, M. 2010. Developpement et vali-dation experimentale de modeles d’echangeursgeothermiques horizontaux et verticaux pour lechauffage de batiments residentiels. PhD thesis,Ecole des Mines de Paris.

Rabin, Y. and Korin, E. 1996. Thermal analysis ofa helical heat exchanger for ground thermal energystorage in arid zones. Int. J. Heat Mass Transfer,39(5):1051–1065.

Rabin, Y., Korin, E., and Sher, E. 1991. A simplifiedmodel for helical heat exchanger for long-term en-ergy storage in soil. In Springer-Verlag, editor, De-sign and operation of heat exchangers. Eurotherm.

van Genuchten, M. T. 1980. A closed-form equationfor predicting the hydraulic conductivity of unsatu-rated soils. Soil Science Society of America Journal,44:892–898.

Verein Deutscher Ingenieure 2000. VDI 4640.Technical report, VDI-Gesellschaft EnergietechnikFachausschuss Regenerative Energien.

Wosten, J., Lilly, A., Nemes, A., and Bas, C. L. 1999.Development and use of a database of hydraulicproperties of european soils. Geoderma.


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