Thermal Analysis of Vertical Ground Exchangers of Heat Pumps

Heat Transfer Engineering, 27(2):2–13, 2006Copyright C©© Taylor & Francis LLCISSN: 0145-7632 print / 1521-0537 onlineDOI: 10.1080/01457630500397088

Thermal Analysis of Vertical GroundExchangers of Heat Pumps

JAN SK�LADZIEN, MA�LGORZATA HANUSZKIEWICZ-DRAPA�LA,and ADAM FICInstitute of Thermal Technology, The Silesian University of Technology, Gliwice, Poland

The heat rate absorbed from the ground by a vertical ground exchanger of a heat pump unit is considered. The aim isto investigate the time variation of this energy rate for a set of parameters. The analyzed set of alternatives encompassesarrangements of the exchanger tubes, values of the temperature of the heat carrier, thermal parameters of the ground, periodicoperation of the compressor, and, when appropriate, different values of seepage velocity. To achieve this goal, the transienttemperature distributions in the soil surrounding the ground exchanger are evaluated. The calculations are carried out usingboth PATRAN-THERMAL, a commercial finite volume code, and FEMCONV, an in-home FEM package. Characteristicfeatures of the latter are discussed briefly along with some results of simulations for a ground exchanger with tube-in-tube(Field-type) elements. It is shown that in every case, the heat rate absorbed from the ground depends on the season andreaches the minimum value in the second part of winter. As expected, a strong influence of the arrangement of the exchangertubes and the motion of moisture is observed. It is shown that if the prices of the electric energy are variable during a day,it may be profitable to operate an HP unit compressor in a periodic regime. The approximate values of the heat pump unitcoefficient of performance, defined as the ratio of heat output of HP and compressor driving power, are evaluated. It ispointed out that this coefficient depends on the heat carrier temperature, and therefore this temperature may also be a subjectof optimization calculations.

INTRODUCTION

Recently, heating systemswith heat pumps (HP) supported bythe low-temperature energy of natural surroundings have beenapplied more frequently [1–4]. They are mainly used for theheating of decentralized units, such as small houses or hotels,and sometimes as a source for bigger heat consumers. The to-tal amount of heat pumps installed all over the world at thebeginning of the twenty-first century was estimated at 60 mil-lion [2]. At the end of the 1990s, more than 150,000 heat pumpswere installed in Europe [3],mainly in Sweden (50%),Germany,Austria, and Switzerland. There are about 200,000 HPs installedonly in Sweden today. The strong interest and development ofHPs can be also noticed in Poland.

Low-temperature energy for HPs can be transferred from at-mospheric air, water, or the ground. The energy of the ground is

The financial assistance of the National Committee for Fundamental Re-search, Poland, within grant 4T10B 039 23 is gratefully acknowledged.

Address correspondence to Professor Jan Sk�ladzien, Institute of ThermalTechnology, The Silesian University of Technology, Gliwice, Poland. E-mail:[email protected]

the most likely to be utilized in heat pump systems [4]. A low-heat source of such systems is usually accomplished by specialground heat exchangers (GHE). Energy from the ground is trans-ferred by a medium of a ground loop (e.g., refrigerating brineor a water solution of ethylene glycol), and it is then transportedto the evaporator of the vapor compressor HP (see Figure 1a).The pipes of the ground exchangers of the ground source heatpumps are arranged horizontally or vertically in the ground. InPolish conditions, vertical ground exchangers seem to be usedmore frequently. In this case, the tubes of the exchanger (of alength from about 10 to 100 m) are of an elongated U-type shapeor have a tube-in-tube form (i.e., the so-called Field tubes form;Figure 1b). The temperature of the medium of a ground loop isusually equal to about −4◦ to −10◦C.

The heat rate, which is transferred from the ground to a heatcarrier liquid of the ground loop, is essential in the consideredHP systems. This heat rate depends strongly on many param-eters, such as the distance and arrangement of pipes, thermalparameters of the moist ground, velocity of the moisture seep-age, and temperature of the heat carrier. On the other hand, theability of the ground to transfer energy to the system determines,for the given output of an HP, the desired level of the saturation

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J. SK�LADZIEN ET AL. 3

Figure 1 Scheme of the HP with GHE: a) a water heating system, b) a Fieldtube as the element of GHE.

temperature in an evaporator, which influences the coefficientof performance (COP) of HP. This coefficient, defined as therate of useful energy over consumption of driving energy of theHP compressor, plays a decisive role with respect to the eco-nomic efficiency of the HP system. Thus, the analysis of theheat rate transferred from the ground in the GHE for differentdesign parameters of the exchanger, different properties of moistground, as well as different kinds of work of HP systems is ofgreat importance. For this purpose, the time-dependent temper-ature distributions in the ground surrounding the pipes of theexchanger should be calculated. This kind of complex numer-ical analyses of the considered heat transfer processes are themain aim of the paper. They were performed for real Polish

Figure 2 Horizontal view of the considered arrangements of the GHE pipes, computational domains, and pipe numeration.

conditions. Some results of such calculations were presentedpreviously [5–7]; however, they were obtained assuming a sim-ple model of vertical exchangers. The analyses presented in thispaper are extended on the following problems:

• the influence of the temperature of a heat carrier and valuesof selected thermal parameters of the ground on the heat ratetransferred from the ground.

• the influence of the neighboring pipes on the heat rate trans-ferred from the ground, depending on the distance betweenpipes and the arrangement of the pipes.

• the influence of the moisture flow on the output heat rate fordifferent velocities and directions of the moisture flow into thesystem.

In practice, the soil is always moist. Moreover, sometimes theseepage of the moisture can occur in some layers of the ground.The analyses mentioned above involve, among others, the fol-lowing features of the heat transfer process in the consideredsystem (i.e., a ground-heat exchanger):

• the process is generally unsteady, even if the HP works con-tinuously with constant capacity,

• HP can work periodically within the day,• the phase change of water in the ground takes place,• convective heat transfer can occur in the considered domain

if the flow of moisture takes place in the ground surroundingthe GHE,

• the temperature field is often considered to be 3D,• the computational domain should approximate a semi-infinite

domain,• the considered ground can be inhomogeneous.

heat transfer engineering vol. 27 no. 2 2006

4 J. SK�LADZIEN ET AL.

Figure 3 Computational domain and the mesh used in variant B of GHE, shown in Figure 2b.

It should also be noted that the identification of some bound-ary conditions is usually difficult. These features of the heattransfer process in ground heat exchanger systems make the cal-culations of the temperature distributions and heat transfer raterather complex. Suitable calculations may be profitable, espe-cially because experimental investigations performed for GHEof the HP unit are rather expensive [8]. Additionally, such exper-iments are long-lasting, as they should extend over at least twoyears. The results of simulations may be helpful for designers ofHP systems with GHE and instrumental both in optimizing theparameters of the system and finding optimal operation regimes.It isworth noting that the operation cost of anHP systemdependsstrongly on the cost of consumed electric energy. This in turn is afunction of COP, the value of which changes during the heatingseason. As already mentioned, the paper describes the depen-dence of the average unitary heat rate taken from the ground onseveral variables. The analyzed set of parameters includes thearrangement of the GHE pipes, heat carrier temperature, heatconductivity and latent heat of the soil, periodical work of thecompressor, and the rate of the moisture seepage in the ground.This is illustrated by results of simulations. Additionally, somecharacteristic values of HP unit COP for a set of heat carriertemperatures are given.

NUMERICAL MODEL OF THE HEAT AND MOISTURETRANSPORT IN THE GROUND

To determine the heat rate transferred from the ground bythe GHE, the unsteady temperature distributions in the groundsurrounding the pipes of the exchanger within a heating season

Figure 4 Selected isotherms in the surroundings of pipes of the GHE as in Figure 2b: a) 115 days—beginning of January, b) 185 days—half of March.

have to be calculated. The modeled ground is assumed to bemoist, and the solid/liquid change of phase takes place in thesystem. The calculations have been accomplished mostly usinga 3D model, except in the case when the moisture seepage in theground is taken into account. The radiative heat transfer on thesurface of the ground (absorption of solar radiation, emission ofthe radiation into cosmic space) has been neglected. The influ-ence of this phenomenon on the considered heat rate transferredfrom the ground by GHE is negligible, especially in case thevertical heat exchanger is used [2].

For cases when the seepage flow is taken into account, a2D heat and moisture transport has been analyzed. Unsteadycoupled energy and groundwater transport in a single water-saturated horizontal layer of the ground is then considered. Theflow of groundwater is modeled as a fluid flow in a porousmedium (grains of the soil). The main simplifying assumptionsare in this case as follows:

• the considered horizontal layer of the ground is located be-tween two impermeable layers

• the vertical heat transfer and the vertical fluid flow are ne-glected

• the porous matrix and groundwater are in a local thermal equi-librium

• the porous matrix and the solid are rigid• the porous matrix and groundwater are incompressible• the velocity distribution of groundwater in each current timet is assumed to be steady, within a current subdomain.

The detailed description of the mathematical model of heattransfer phenomena is given in Appendix A. This model isadopted in the in-house FEMCONV code used in the work.



Figure 5 Heat rate per unit length versus time in following pipes of the GHE: a) variant B compared with variant A (repetitive pipe) and D (single pipe), b)variant C.

RESULTS OF CALCULATIONS AND DISCUSSION

Basic Data and Assumptions

The vertical ground heat exchanger considered consists ofvertical Field-type pipes of length equal to 10 m and an ex-ternal diameter equal to 0.052 m. As mentioned previously,the radiative heat transfer on the surface of the ground is ne-glected. The temperature of the heat carrier is assumed to beconstant along the pipe and equal to −8◦C in the basic alter-native, and there is a distance between the pipes equal to 2 m.The value of the heat carrier temperature may be assumed to beconstant if the flow rate of this agent is high enough. The heattransfer coefficient in the pipes of GHE is 100 W/(m2·K), and10 W/(m2·K) on the surface of the ground. The temperature ofatmospheric air is assumed typical for Central Europe conditions[5]. Thermal parameters of the ground in basic alternatives are asfollows:

density, 1700 kg/m3,specific heat capacity, 1200 J/(kg.K),

Figure 6 Heat rate per unit length of the GHE pipe arranged in a row with a distance: a) a = 4 m, b) a = 6 m.

thermal conductivity, 1.5 W/(m.K),latent heat, 30,000 J/kg,initial temperature of the ground, 10◦C.

The beginning of the heating season is assumed to be mid-September, and the heating season takes 230 days. The presentedresults of calculations relate to the second heating season. It wasshown [5] that the temperature distributions in the ground sur-rounding the pipes of the GHE are repetitive from the secondheating season. (In the first heating season, they differ from thenext ones.) As mentioned before, the direct results of the calcu-lations have a form of temperature distributions in the groundsurrounding the pipes. These time-dependent temperature distri-butions enable the determination of a heat rate transferred fromthe ground to the GHE.

No Seepage of Moisture, Continuous Performance of HP

The results presented in this subsection have been obtainedemploying commercial PATRAN-THERMAL code.



Figure 7 Heat rate per unit length of the GHE pipe for different values oftemperature of an intermediate medium.

Influence of the Arrangement of Pipes on the Heat Ratein the GHE

In order to investigate the influence of the arrangement ofGHE pipes, proper comparative calculations have been carriedout for various alternatives of the arrangements shown in Fig-ure 2. The computational domain is bounded in each case byadiabatic symmetry planes or planes located far enough frompipes to limit the semi-infinite domain. The assumed depth ofthe computational domain is always equal to 30 m, the distancebetween pipes a = 2 m, and x0 = y0 = 30 m, except the al-ternative A. In alternative A (Figure 2a), the GHE consists ofan infinite row of pipes. The computational domain contains arepetitive fragment limited by two parallel planes of symme-try perpendicular to a row of pipes: the first crosses the cen-ter of the pipe and the second is the plane of symmetry of theneighboring pipes. Thus, geometrical parameters are: x0 = 1m, y0 = 30 m. In alternative B (Figure 2b), the GHE consistsof a row of nine pipes; in alternative C (Figure 2c), 25 pipes ar-ranged in a square grid; and, in alternative D (Figure 2d), a singlepipe.

Thewhole computational domain and its part in the vicinity ofthe pipes, as well as the mesh used in alternative B, are presentedin Figure 3.

Table 1 An estimation of selected parameters of the HP system referred to one pipe of GHE forminimal values of heat flow rate (see Figure 7)

Temperature ofan intermediate

medium,T ◦inC

Evaporationtemperature,Tev, ◦C

Coefficient ofperformance,

εhp

Heat flowrate fromthe ground,Qg, W

HP drivingpower,Nc , W

Heat powerof HP,Qhp, W

Ratio,Qg/Qhp

0 −10 2.91 62 32 94 0.66−5 −15 2.72 144 84 228 0.63

−10 −20 2.56 225 144 369 0.61

Selected isotherms obtained for alternative B are shown as anexample in Figure 4. They relate to the time (measured from thebeginning of the second heating season) equal to 115 days (be-ginning of January) and 185 days (half of March). The shape ofthe isotherms is similar in the vicinity of the neighboring pipes.It is worth noting the visible influence of heat transfer along thesurface of the ground and the temperature of atmospheric airon the temperature field. Moreover, the flat horizontal shape ofisotherms far from the pipes confirms the correct assumption ofthe domain size.

The heat rate per unit length of the pipe versus time, in thefollowing pipes of the GHEs of considered alternatives, is pre-sented in Figure 5. It should be interpreted as an average heatrate per unit length along the length of the pipe. The heat ratefor the single pipe (alternative D) constitutes the maximal valueof the heat rate, which can be transferred from the ground byone pipe. When the pipes of GHE are arranged in a row (Fig-ure 5a), the heat rate of the pipes located inside is in practiceabout 20% less than that obtained in alternative D for single pipe(comparing these values within the period of maximum demandof heat in winter). In the side pipes, the heat rate is about 10%higher than in internal ones. When the GHE consists of a largenumber of pipes (e.g., ten pipes or more) arranged in a row, itcan be analyzed using a simple model assumed in the alterna-tive A. The results obtained for alternative C (pipes of the GHEexchanger in a square grid) show that the arrangement of pipesin a row is better than in a square grid with a view to the GHEheat rate. The heat rate in the internal pipes of GHE in alterna-tive C can be even twice less than in internal pipes located in arow.

It is also worth noting that the heat rate attains a minimalvalue in the vertical GHE after about 180 days of heating seasonin March (i.e., after the period of maximal demand for heatingin winter). Afterward, the heat rate increases slightly as a resultof the temperature growth of atmospheric air.

Influence of a Distance between Pipes on the Heat Ratein the GHE

The heat rate per unit length of the pipe versus time in thesuccessive GHE pipes arranged in a row with a distance equalto 4 m and 6 m is presented in Figure 6. It can be compared withthe heat rate shown in Figure 5a obtained for a distance equal to2 m. The difference between the heat rate in the side pipe and



Figure 8 Heat rate per unit length of the GHE pipe for different values of (a) heat conductivity and (b) latent heat of the ground.

internal ones becomes smaller for higher values of the distancebetween pipes. The difference between the heat rate in pipes ofthe considered GHE and maximal value of the heat rate (for asingle pipe located in the ground—alternative D) also decreasesin this case. The so-called “heat stealing” by neighboring pipesof GHE for a distance equal to about 4–6 m has no essentialmeaning, and a further increase of the distance does not causeany advantages.

Influence of the Temperature of the Heat Carrier on the HeatRate in the GHE

The heat rate per unit length of the pipe versus the time fordifferent values of temperature of a heat carrier is presented inFigure 7. The results have been obtained using a simple modelof the GHE described in alternative A. As expected, a signifi-cant influence of the carrier temperature on the heat rate in theGHE can be observed. It is worth noting that the carrier tempera-ture also influences the temperature in the HP evaporator, whichhas a direct effect on COP and the economical efficiency of theHP. To estimate this influence, a simple calculation of COP εhpand ratio Qg/Qhp for different values of the temperature of aheat carrier has been carried out, as presented in Table 1. Quan-tity Qg here is the heat rate transferred from the ground in onepipe of the GHE, Qhp stands for the heat power of HP, and Nc

represents the driving power of the HP compressor, when HPuses GHE consisting of one pipe. A simple model of HP, witha typical refrigerating medium, has been assumed in these cal-culations. The condensing temperature has been assumed to beequal to 50◦C. The results show that a decrease of temperatureof the heat carrier involves the distinct increase of heat rate inthe GHE. It also increases both the demand of driving powerand heat output of HP. Concurrently, the COP becomes lower,and the ratio Qg/Qhp does not change significantly. For a highervalue of the temperature of the heat carrier, the GHE consistingof more pipes is required to obtain the same value of HP heatoutput. This involves increasing the investment costs, and oper-ation costs decrease in this case. Thus, the temperature of the

heat carrier is an important parameter that should be optimizedeconomically.

Influence of Thermal Parameters of the Ground on the HeatRate in the GHE

The heat rate per unit length of the GHE pipe versus time,for different values of the ground conductivity and latent heat,is presented in Figure 8. The results show a distinct influenceof the ground conductivity on the heat rate. The value of latentheat, depending on the amount of moisture in the ground, doesnot affect the heat rate as much. The heat capacity of the grounddoes not have an important influence in the considered analyses.Proper calculations have been carried out, but their results arenot presented in the paper.

Figure 9 Heat rate per unit length of the GHE pipe for the periodic andcontinuous performance of HP.



No Seepage of Moisture, and the Periodical Performanceof HP

The results presented in this subsection have been obtainedusing in-house FEMCONV code, which employs the FEM. Itwas assumed that the HP works eight hours per day. Within thenext 16hr periodof a day,while the exchanger is out of operation,some kind of regeneration of the ground surrounding the GHEpipes takes place, and the gradients of the temperature close tothe pipes become in this case distinctly higher when comparedto the case of the continuous work of HP. As a result, the heatrate transferred from the ground is then higher during the work-ing period of the system, which is shown in Figure 9. The heatrate per unit length of the GHE pipe in the exchanger workingcontinuously is about twice less than the average heat rate for an8 hr working period of the exchanger, which works periodically,and only about 50%higher than latter exchangerwhen averagingover 24 hrs. The considered periodical work of the heat pump

Figure 10 Selected isotherms in the surroundings of pipes in the half of February obtained in 2D calculations: a) no seepage, b) un = 0.5E-6 m/s, c) un =1.0E-6 m/s for perpendicular seepage flow, d) un = 0.5E-6 m/s, e) un = 1.0E-6 m/s in parallel seepage flow.

can therefore be profitable from an economical point of view.The regeneration of the ground also implies a strong depen-dence of the heat rate versus time during the working period ofHP.

Seepage of Moisture in the Ground

The results presented in this subsection have also been ob-tained using home code FEMCONV, which employs the FEMapproach presented in Appendix A. The calculations have beenperformed in a 2D domain. It is assumed that the seepage takesplace in a single homogeneous horizontal layer of the ground,permeable for the moisture. The layer is located deepenough for the influence of the heat transfer along the surfaceof the ground on the considered process to be neglected. Thus,only the horizontal heat and moisture transport is taken intoaccount.



Figure 11 Velocity vectors of moisture in the surroundings of pipes in the half of February obtained for boundary velocity un = 0.5 E-6 m/s: a) normal seepageflow, b) parallel seepage flow.

The GHE exchanger consists of 10 pipes arranged in a singlerow, and the distance is a = 2 m. In the considered cases,moisture flows into the domain perpendicularly or in parallel tothe row of pipes, with different realistic values of the boundaryseepage velocity (Darcy’s velocity) un . To compare, the casewith no seepage flow has been also calculated. The other data arethe same as in the basic alternative in the previously describedcalculations. The discretization of the domain has been madeusing four-node isoparametric elements.

Selected isotherms in the ground surrounding the GHE pipesobtained in the considered cases are shown in Figure 10, andvectors of the moisture velocity are shown in Figure 11. Theyrelate to about half of February. The heat rate transferred fromthe ground per unit length of the pipe versus time is presentedin Figure 12. The pipes are numbered from left to right.

In the cases with no seepage flow or with seepage flow per-pendicular to the row of pipes, the heat rate in the side pipes (1and 10) is distinctly higher than in the side-but-one pipes (2 and9). In the internal pipes, the heat rate is almost the same, slightlyless than in the latter ones. This is consistent with the results of3D calculations described previously. It changes in the case ofseepage flow parallel to the row of pipes. If the convective heattransfer has enough important meaning, as in cases considered,the highest values of the heat rate appear in the first pipe fromthe left (i.e., from the inflow of moisture) and become smallerin the successive pipes.

Seepage flow can distinctly enlarge the heat rate transferredin the GHE pipes, especially if the flow is normal to the rowof pipes. In this case, the heat rate for the inflow velocity un =0.5 E-6 m/s is about 50% higher during the winter (within theperiod 100–200 days of the heating season) than in the case ofno seepage flow, and for un = 1.0 E-6 m/s, it is about 100%higher. Besides, for such seepage flow, the heat rate becomesalmost constant in time very quickly. These advantages cannotbe observed if the seepage flow is parallel to the row of the pipes.The heat rate then becomes distinctly higher only in two or threelimiting pipes from the left side.

FINAL REMARKS

The present paper addresses the results of the numerical cal-culations of the temperature fields in the ground surrounding

the vertical ground heat exchangers of heat pumps and the heatrate transferred from the ground in these exchangers. Numericalresults have been obtained employing commercial PATRAN-THERMAL code and the in-house FEMCONV code describedin Appendix A.

Plots of unitary heat rate taken from the ground versus timeare given for several parameter sets. Specifically, different ar-rangements of GHE pipes, various heat carrier temperatures, theheat conductivity and latent heat of the ground, the periodicalwork of a compressor, and different rates of moisture seepage inthe soil are investigated. Some exemplary values of the HP unitCOP corresponding to different temperatures of the heat carrierare also given.

Detailed remarks resulting from the calculations have beenpresented in the previous sections. More general conclusions areas follows:

• The effect of the so-called “heat stealing” by neighboringpipes can be observed if the compact arrangement of GHEpipes is used. This effect can be distinctly decreased if thepipes are located in a row with a distance equal to about 4 to6 m. A further increase of the distance between the pipes doesnot magnify the heat rate of the GHE pipe.

• The periodic operation of HP can be profitable from an eco-nomical point of view. Only about 50% more pipes are re-quired in this case (HP works 8 hrs per day) to obtain the samedaily average value of the GHE heat rate when compared withthe HP working continuously. An additional advantage can beachieved if the HP considered works only during the periodof the low cost of electric energy.

• The seepage of moisture intensifies distinctly the heat transferin the ground surrounding theGHEpipes, especially if a rowofpipes is perpendicular to the direction of themoisture flow. Forrealistic values of the flow velocity, the heat rate can increaseeven twicewhen comparedwith the casewith no seepage flow.

• The temperature of a heat carrier also significantly influencesthe heat rate transferred from the ground by the GHE. Theheat rate for the temperature equal to −8◦C is about twice ashigh as for the value equal to −2◦C. On the other hand, theincrease of this temperature involves the distinct increase ofthe coefficient of performance (COP). Thus, the carrier tem-perature is an important parameter that should be optimizedeconomically.



Figure 12 Heat rate per unit length of the GHE pipe versus time obtained in 2D calculations: a) no seepage, b) un = 0.5E-6 m/s, c) un = 1.0E-6 m/s forperpendicular seepage flow, d) un = 0.5E-6 m/s, e) un = 1.0E-6 m/s in parallel seepage flow.

• With careful estimating, the geothermal energy is nomore thanabout 1 to 2% of the total energy transferred in the GHE fromthe ground [6]. Thus, the solar energy stored in the ground isof crucial importance for the HP systems.

NOMENCLATURE

a distance between pipes of the GHE, ml volumetric latent heat of ground, J/m3

Nc driving power of HP compressor, WNi shape function associated with node iQg heat rate transferred from ground, WQhp heat output of HP, W

Greek Symbols

εhp coefficient of performance of heat pump systemφ velocity potential of moisture seepage, m2/s



Subscripts

a apparentf frozen subdomaing groundhp heat pumpin heat carriern normal componentph phase changeun unfrozen subdomainv volumetricw water

REFERENCES

[1] Brodowicz, K., and Dyakowski, T., Heat Pumps, PWN, Warsaw,1990 (in Polish).

[2] Zalewski, W., Compressor, Sorption and Thermoelectric HeatPumps, Theoretical Basis, Calculation Examples, IPPU MASTA,Gdansk, 2001 (in Polish).

[3] Kjellsson, E., Hellstrom, G., Tepe, R., and Ronnelid, M., Com-bination of Solar Heat and Ground Source Heat Pump forSmall Buildings, Proc. 9th Int. Conf. on Thermal EnergyStorage, FUTURESTOCK 2003, Warsaw, Poland, pp. 479–484,2003.

[4] Sanner, B., Current Status of Ground Source Heat Pumps inEurope, Proc. 9th Int. Conf. on Thermal Energy Storage, FU-TURESTOCK 2003, Warsaw, Poland, pp. 695–703, 2003.

[5] Fic, A., Hanuszkiewicz-Drapa�la, M., Piatek, R., and Sk�ladzien J.,Heat Transfer Analysis in Ground Heat Exchanger Systems ofHeat Pumps, Proc. European Congress on Computational Meth-ods in Applied Sciences and Engineering, ECCOMAS 2000, paperCFD 201, Barcelona, Spain, 2000.

[6] Sk�ladzien, J., Hanuszkiewicz-Drapa�la, M., and Fic, A., Groundas a Solar Energy Storage for Heat Pump Systems, Proc. 9thInt. Conf. on Thermal Energy Storage, FUTURESTOCK 2003,Warsaw, Poland, pp. 467–472, 2003.

[7] Fic,A., Hanuszkiewicz-Drapa�la,M., and Sk�ladzien, J., Heat PumpSystems Utilizing Solar Energy Accumulated in Ground, ZeszytyNaukowe Politechniki Slaskiej, Energetyka z. 139, InzynieriaSrodowiska z. 48, pp. 277–288, Gliwice, Poland, 2003 (in Pol-ish).

[8] Czekalski, D., Experimental Investigations of Vertical GroundExchanger Thermal Performance, Proc. Heat Transfer and Re-newable Sources of Energy 2004, Szczecin, Poland, pp. 341–346,2004.

[9] Eagleson, P. S., Dynamic Hydrology, McGraw-Hill Book Com-pany, New York, 1970.

[10] Nield,D.A., andBejan,A.,Convection inPorousMedia, Springer-Verlag, New York, 1998.

[11] Teng, Y. M., and Akin, J. E., An Effective Capacity Approach toStefan Problems Using Simple Isoparametric Elements, Int. Com.in Heat Mass Transfer, vol. 21, pp. 179–188, 1994.

[12] Zienkiewicz, O. C., and Taylor, R. L., The Finite Element Method,4th ed. McGraw-Hill Book Company, London, 1989.

[13] Fic, A., Numerical Analysis of Artificial Ground Freezing withSeepage, Archives of Thermodynamics.

APPENDIX A

Temperature Distribution

Taking into account the assumptions given here, the temper-ature field is described by:

• the governing equation:

(ρc)∂T

∂t+ (ρc)w�u · ∇T = ∇k∇T + q ′′′, �r ∈ � (A1)

where T (�r , t) denotes the temperature at time t and point �rwithin the domain �,(ρc) is the volumetric heat capacity ofthe moist ground, (ρc)w is the volumetric heat capacity ofwater, �u stands for the Darcy’s velocity of the seepage flow, krepresents the thermal conductivity, and q ′′′ is the volumetricheat source equal to 0 in the ground. Darcy’s velocity of theseepage flow �u is defined as a volumetric flow rate of waterper unit area, and it is equal to zero in the frozen subdomain.Eq. (A1) is valid if temperature of moisture flowing throughpores in the ground is equal to the temperature of the groundgrains. This assumption is reasonable for small velocities ofmoisture.

• standard boundary conditions (Dirichlet’s, or Robin’s in thecase considered) along the external surface � of the domain�

• initial condition:

T (�r , 0) = T0(�r ) (A2)

• Stefan’s boundary condition along the phase change interface�ph :

k f∂T f∂nph

− kun∂Tun∂nph

= −lu ph, �r ∈ �ph (A3)

and

T f = Tun = Tph, �r ∈ �ph (A4)

where l is the volumetric latent heat of the moist ground, u phdenotes the normal component of the velocity of the movinginterface, nph represents the unit normal to the interface, andsubscripts f , un, and ph relate to the frozen domain, unfrozensubdomain, and interface, respectively.

Velocity Distribution of Moisture

The solution of boundary problems (A1)–(A4) requires thedetermination of the distribution of the groundwater velocity �u.If the freezing of moisture in the ground takes place around thepipes of GHE, the flow of moisture exists only in the unfrozenpart of the considered domain �, say �un . Obviously, the subdo-main �un changes its shape during freezing, making the velocitydistribution ofmoisture to become generally unsteady.However,the changes of the unfrozen subdomain �un are in fact very slow.



Therefore, the velocity distribution of moisture at the consideredtime t is assumed to be steady within the actual subdomain �un .

According to the above assumptions, the velocity distributionof moisture �u is governed by Darcy’s law [9]:

�u = −∇φ (A5)

where the velocity potential φ is described by the followingequation [9, 10]:

∇2φ = 0, �r ∈ �un (A6)

Darcy’s law (A5) allows the formulation of the boundaryconditions for Eq. (A6) along the external surface �un of thesubdomain �un . For this purpose, it is convenient to rewriteEq. (5) to the form:

�nun · ∇φ + un = 0, �r ∈ �un (A7)

where �nun denotes the external unit normal to the surface �un(including interface �ph) and un represents the normal compo-nent of the moisture velocity to the considered surface. Alongthe plane of symmetry and over the interface, as well as on theexternal surface of the freezing tubes, the velocity un is equalto zero. It is worth noting that Eq. (A7) has a general form ofNeuman’s boundary condition.

Solution of the Problem

The apparent heat capacity method—spatial averaging tech-nique [11]—is used in the FEMCONV code due to the existenceof a moving boundary. Employing the apparent heat capacitymodel allows one to take into account the latent heat on a fixedgrid and avoid tracking the interface. The apparent heat capac-ity (ρc)a at the quadrature point is calculated using the so-calledspatial averaging technique. For a considered 2D plane geome-try, the apparent heat capacity is determined using the Cominiformula [11]:

(ρc)a =∂T

∂x

∂hv

∂x+ ∂T

∂y

∂hv

∂y(∂T

∂x

)2

+(

∂T

∂y

)2 (A8)

where volumetric enthalpy hv has to be a continuous function oftemperature. In the case of isothermal phase change, an abruptbut finite jump of enthalpy is approximated linearly within avery small temperature interval �Tph around the phase changetemperature Tph .

The derivatives in Eq. (A8) are calculated using a typicalFEM interpolation, i.e.,

∂T

∂x=

LW∑l=1

Tl∂Ni∂x

(A9)

where subscript l denotes that quantity is referred to the node land LW is a number of nodes in the element considered.

The spatial discretization of Eq. (A1) by FEM leads to thesystem of ordinary differential equations [12]:

CdT

dτ= KT + F (A10)

where T is the vector of the nodal temperature, F denotes theload vector, and C and K stand for matrices of heat capacity andheat conductivity, respectively.

Equation (A10) is integrated in time using the first-orderfinite-difference approximation [12]. Components of matricesC, K, and F are defined by a well-known sum of integrals overindividual elements. To obtain the symmetric matrix K, the con-vective term in Eq. (A1) is treated as the heat source term inFEM. It leads to the following element components Fei of thematrix F referred to element e:

Fei =∫�e

N iq′′′dV (A11)

where Ni is the shape function associated with node i , the ap-parent heat source q ′′′ is defined as:

q ′′′ = −(ρc)w

(ux

∂T

∂x+ uy

∂T

∂y

)(A12)

within the unfrozen subdomain (T > Tph), and

q ′′′ = 0 (A13)

within the frozen subdomain (T ≤ Tph), where ux and uy arethe components of the moisture velocity in the x and y direction,respectively.

The modified shape function Ni in Eq. (A11), used to avoidnumerical oscillations (so-called “wiggles”), is defined as fol-lows [12]:

Ni = Ni + βd

2

ux∂Ni∂x + uy

∂Ni∂y

|�u| (A14)

where d denotes the length of element e in the streamline direc-tion (the length of the segment of the line inside element e alongthe velocity direction and crossing the center of element) and β

is the upwind parameter.The velocity potential φ, which is assumed steady in each

time step, is also calculated by FEM. The method yields thefollowing equation system for a current time:

G� = R (A15)

wherematrixesG andR are defined according to FEMprocedure[12].

In the analyzed case, the matrix R has non-zero elements onlyfor nodes located along the boundaries with a non-homogeneousboundary condition, that is, if un �= 0 in Eq. (A7).

The components of the groundwater velocity in quadraturepoints are calculated applying Darcy’s law and a FEM



interpolation, i.e.,

ux =LW∑i=1

φi ·∂Ni∂x

(A16)

for the x component and a similar formula for the y componentof the velocity.

Themaindifficultywhile solvingEq. (A15) relates to changesof the computational subdomain �un in time. The employedapproach [13] avoids remeshing and simplifies the assembly ofelement contributions of the matrices G and R.

At the beginning of the freezing process, the velocity poten-tial φ is calculated when the whole domain � is unfrozen. Thematrix R and the element components Ge of matrix G are stored.In the next time steps, the calculations are performed using a pri-mary mesh. It should be noted that the computational domainis defined by the unfrozen elements of the primary mesh andthe interface of the unfrozen parts of the elements runs throughthem. The primary matrix R has at the beginning the same formas the stored one. In the FEM procedure, a new matrix G is builtthrough assembling its element components Ge. However, thisprocess runs only over unfrozen elements (these components arethe same as the stored ones) and unfrozen sub-elements. In or-der to retain the local nodal connections, the original interface isapproximated by a temporary relocation of frozen nodes of par-tially frozen elements [13]. This is performed to obtain the samenodal connections of sub-elements as connections of the originalelements. In this way, the structure of the new matrix G is thesame as the structure of the primary one. Next, a constant valueof φ (arbitrarily selected) is assumed for frozen nodes belongingto fully frozen elements by proper modification of matrixes Gand R, and, finally, the trivial equations are removed from thesystem. The obtained nodal values of the velocity potential areused to calculate the groundwater velocity components throughEq. (A16) and the convective term inEq. (A1).However, the con-vective term in partially unfrozen elements is neglected becausethe moisture flow is parallel to isotherms near the interface.

Jan Sk�ladzien is a professor and deputy man-ager for research of the Institute of Thermal Tech-nology at the Silesian University of Technology,Faculty of Power and Environmental Engineer-ing, Gliwice, Poland. He is the head of the Di-vision of Heat Transfer, Nuclear Energy and Re-newable Energy Sources. His research interestsinclude heat exchanger theory, heat transfer withphase change, thermal phenomena in nuclear re-actor containments, and the theory ofMHD-steam

systems and power plants. He has published a number of articles in all of thesefields. He is a member of Committee of Thermodynamic and Combustion at thePolish Academy of Science, and a member of Main Board of Polish NucleonicSociety.

Ma�lgorzata Hanuszkiewicz-Drapa�la is an as-sistant professor in the Institute of Thermal Tech-nology at the Silesian University of Technology,Faculty of Power and Environmental Engineer-ing, Gliwice, Poland. She defended her Ph.D. the-sis in 1996. Her research interests include prob-lems of heat transfer and thermodynamics, es-pecially concerning heat exchangers and heatpumps, and numerical modeling of thermal pro-cesses. She is the co-author of proceedings of con-

ferences and articles connected with these fields.

Adam Fic is an assistant professor in Institute ofThermal Technology, Faculty of Power and En-vironmental Engineering of the Silesian Univer-sity of Technology, Gliwice, Poland, where hereceived his M.Sc. and Ph.D. in mechanical engi-neering. His research interests are in heat transfer,mathematical modeling and their applications toprocesses with change of phase (artificial groundfreezing, ground exchangers of heat pumps,Czochralski process, fire refining of copper, pro-

cesses in containments of nuclear reactors during the loss of coolant accidents),and inverse heat transfer problems. He has published a number of papers in thesefields. His teaching experience relates to thermodynamics, heat transfer, nuclearengineering, and numerical methods.


Documents

Thermal Analysis of Vertical Ground Exchangers of Heat Pumps