Computers and Geotechnics

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Comuters and Geotechnics 9 (3) Contents lists available at SciVerse ScienceDirect Comuters and Geotechnics journal homeage: www.elsevier.

and Environmental Engineering, KAIST, Daejeon 35-7, Reublic of Korea b Deartment of Mathematical Sciences, KAIST, Daejeon 35-7, Reublic of Korea article info abstract Article history: Received Aril

1 Comuters and Geotechnics 9 (3) Contents lists available at SciVerse ScienceDirect Comuters and Geotechnics journal homeage: Characteristics of an analytical solution for a siral coil tye ground heat exchanger SKhan Park a, Seung-Rae Lee a,, Hyunku Park a, Seok Yoon a, Jaywan Chung b a Deartment of Civil and Environmental Engineering, KAIST, Daejeon 35-7, Reublic of Korea b Deartment of Mathematical Sciences, KAIST, Daejeon 35-7, Reublic of Korea article info abstract Article history: Received Aril Received in revised form November Acceted November Available online 7 December Keywords: Siral coil tye GHE model Analytical model Thermal resonse test Numerical analysis This aer resents an efficient siral coil source model and its analytical solution, develoed to consider 3-dimensional shae effects and radial dimension effects of a siral coil tye ground heat exchanger (GHE) using Green s function method. To avoid singular integrals, the solution is exressed using an error function by which the comutational limitation is avoided. To analyze the characteristics of the analytical model, redicted results of the model were comared with test measurements by a thermal resonse test conducted in a model chamber, and numerical analysis results imlemented in ABAQUS/Standard. Ó Elsevier Ltd. All rights reserved.. Introduction Interest in renewable energy has gradually increased throughout the world. In articular, ground-couled heat um (GCHP) systems have been alied to obtain shallow-deth geothermal energy with high energy efficiency rate. The GCHP uses constant ground temerature to transfer heat during the summer and winter for cooling and heating buildings. Recently, coil tye heat exchangers are being used because the siral coil configuration has the advantage of more heat transfer area and better flow attern without air chocking in the ies comared to the serial of arallel U-tubes in the ile []. In addition, the siral coil system can reduce the comlexity of the ie connections and decrease to a certain extent the thermal short-circuit between suly and return ies []. The coil tye heat exchanger has more comlex geometry than the U-tube but engineers continue to use commercial design rograms based on one-dimensional analytical solutions without considering the comlex geometry. In engineering fields, desite limits stemming from simlifying assumtions, analytical solutions are used for their simlicity and convenience of comutation. Reresentative classical models for GHE are the line source model [3] and cylindrical source model []. A widely used model is the infinite line source model. The model, develoed and evaluated by Carslaw and Jaeger [3], alies the Kelvin heat source theory to ground heat exchangers. Ingersoll Corresonding author. Tel.: ; fax: address: srlee@kaist.ac.kr (S.-R. Lee). and Plass [5] rovided tabulated values of the evaluated integral solution art. Simle correlations have been also roosed to aroximate the integral by Bose []. Another widely used model is the infinite cylindrical source model. The model was mentioned by Carslaw and Jaeger [3], but later roosed by Ingersoll et al. []. The infinite cylindrical model was reviewed by Baudoin [7] who obtained imroved comutational seed by numerically inverting the Lalace transform. The finite line source model was develoed by Eskilson []. These classical models, however, because of limitations due to simlifying assumtions of the heat source, are not adequate for coil tye heat exchangers because the heat transfer coils are disosed in the roximity of the ile circumference []. The heat source in the line source model is located at the center of borehole. Thus, the line source model might estimate somewhat lower soil temerature rise comared to the others. On the other hand, the heat source in the cylindrical source model is located near the borehole wall with a cylindrical face. Then, the cylindrical source model might estimate somewhat higher soil temerature rise comared to the others. Lamarche and Beaucham [9] resented an imroved solution of the finite cylindrical source model. Bandos et al. [] roosed a new analytical finite line source model considering the effects of the geothermal gradient and temerature changes at the soil surface. Recently, Man et al. [] resented a new solid cylindrical source model evolved from the classical models. Also, the following models for a siral coil tye GHE have been develoed: a ring coil source model [], and a siral heat source model []. However, they do not rovide exact solutions for the siral coil source and -35X/$ - see front matter Ó Elsevier Ltd. All rights reserved. htt://dx.doi.org/./j.comgeo...

2 S. Park et al. / Comuters and Geotechnics 9 (3) 9 Nomenclature c secific heat caacity (J kg K ) h coil deth (m) L borehole length (m) N number of coil turns r o siral coil radius (m) r b borehole radius (m) q l heat rate er length of borehole (W m ) t time (s) T temerature (K) t, u integral variable x, y, z u integral variable vector in x, y, z Cartesian coordinates Greek letters a thermal diffusivity (m s ) k thermal conductivity (W m K ) q density (kg m 3 ) h variation of temerature (K) x wave number (m ) cause a comutational roblem because of the singularity in evaluating =ðt t 3= and need double integrations. Further research is thus necessary in order to develo a mathematically more efficient analytical solution for the coil tye heat exchanger. This aer describes the develoment of an efficient siral coil source analytical model. Its analytical solution was develoed to consider 3-dimensional shae effects and radial dimension effects of a siral coil tye GHE using Green s function method. Because of the limitation in comutation and the comlicated formula, the siral coil source analytical model was transformed by changing the variable into an error function to imrove and simlify the comutation for engineering alications. In addition, this aer validated the analytical model solution through comarison of the model s rediction results with test measurements by a thermal resonse test conducted in a model chamber and numerical analysis results imlemented in ABAQUS/Standard. The characteristics of the analytical model were also discussed.. Thermal resonse test Thermal resonse tests (constant flux tests) were carried out in the laboratory of Korea Electric Power Research Institute []. A schematic diagram of the thermal resonse test set-u is shown in Fig.. The setu includes a mocku steel box, a heater, a um, a flow meter, a water tank and temerature sensors (Fig. ). The size of the steel box, in which soils can be comacted to a certain density and GHEs are installed, is 5 m m m(l W H). The model chamber was filled with a reresentative Korean sand named Joomoonjin sand in a dry condition. Based on the sand-raining method, nearly homogeneous sand was reared in the chamber. Table resents the hysical and thermal roerties of the sand. Table shows the quantitative analysis of Joomoonjin sand. Various coil tye HDPE ies (ie diameter = 5 mm, sacing of coils = 5, 5, 5 cm) with coil diameter of 5 cm and coil length of m were installed horizontally in the soil. Two temerature sensors were installed at the inlet and outlet of the GHE ie. RTD (Resistance Temerature Detector) sensors were also attached at two locations (5 cm and 35 cm from the center and. m from the to, Fig. ). Thermal resonse tests were conducted during 5 h for which the temerature variation was less than C by a numerical analysis revealing that the test results might not be influenced by the size of box. Amount of heat flux injected to soil was calculated by the measured flow rate _m and water temerature at inlet T in and outlet T out. Table 3 shows the calculation results. 3. Numerical analysis The thermal resonse tests [] were numerically simulated based on a transient conductive heat transfer analysis scheme imlemented in ABAQUS/Standard [3]. In the simulations, a siral coil configuration of heat sources was considered, and accordingly, the coil heat exchanger was simly modeled as a zero-volume heat source emitting a constant heat flow. Fig. shows the 3-dimensional finite element model, where -noded three dimensional brick elements (DC3D) and -noded one dimensional link elements for diffusive heat transfer analysis (DCD) were used to model soil and siral coil, resectively. Based on the mesh convergence tests, single coil itch was determined to be modeled by twelve line elements for efficient calculation. Body heat flow from the line elements was considered as a thermal load. Hence, the amount of heat injection from each of line element is identical to / of q l, where q l is the heat injection from single coil itch. In the analyses, boundary conditions of constant temerature and zero heat flow were rescribed at the exterior of the finite element model.. analytical model Because of the comlex geometry of the siral coil tye GHE, an efficient siral coil source model and its analytical solution was develoed. In the model, the 3-dimensional shae effects and radial dimension effects of a siral coil tye GHE were considered using Green s function. A siral coil tye GHE is buried in a ile, as shown in Fig. 3. For the siral coil tye GHE, the coil radius is r o, coil deth is h, and number of coil turns is N, resectively. For modeling the heat transfer from the buried siral coil tye GHE, the following assumtions were made: () The ground is assumed to be a homogeneous infinite medium. The thermal roerties of the ground do not change with the variation of temerature. () The siral coil heat source is assumed to take a siral coil form without a vertical outlet ie. The siral coil heat source is buried in the ground from the ground surface to a certain deth, h. Mass, heat caacity, and thickness of the heat source are neglected. The heating rate er length of ile, q l, is constant from the starting instant, t =. (3) The medium has a uniform initial temerature, t. () The ground surface, the boundary of the medium, z =, maintains a constant temerature. The transient 3-D conduction roblem based on the governing equation of the transient heat conduction along with the given boundary and initial conditions in the infinite medium can then be formulated in the Cartesian adh þ Qðu; for z > Qðu; t q l dðu sðz qc for z h ð hðu; for z > hðu; t for z

S. Park et al. / Comuters and Geotechnics 9 (3) Fig.

Table Physical and thermal roerties of Joomoonjin sand. m m Parameters Values Unit Unit dry density, c 397 kn m 3 Void ratio, e.

b Sand at material roerties of engineeringtoolbox.com (5).

$Biotite Small quantity a XRD (X-ray diffraction) test by Korea Institute of Geoscience and Mineral Resources. Fig.$

3 S. Park et al. / Comuters and Geotechnics 9 (3) Fig.. Diagram for thermal resonse test (uer) and thermal resonse test equiment (lower). Table Physical and thermal roerties of Joomoonjin sand. m m Parameters Values Unit Unit dry density, c 397 kn m 3 Void ratio, e.9 Thermal conductivity, k a. W m Secific heat caacity, c b J kg K 5 m y a Measured using heat flow meter in the laboratory. b Sand at material roerties of engineeringtoolbox.com (5). z m Line elements for siral coil x Table Quantitative analysis of Joomoonjin sand. a.5 m Materials Comonent ratio (%) Quartz 9. Microcline 5. Albite 5. Biotite Small quantity a XRD (X-ray diffraction) test by Korea Institute of Geoscience and Mineral Resources. Fig.. Numerical model for simulation of TRT. y x Table 3 Total heat flux in thermal resonse tests. Sacing (cm) Total heat flux (W) h N = Number of coil turns where the wave number is x N=h, the heat source is sðz ðr cosðxz; r sinðxz; z, and dðu sðz is the Dirac d- function. z r Fig. 3. Schematic view of siral coil source.

4 S. Park et al. / Comuters and Geotechnics 9 (3) (a) Sacing = 5cm, measured oint=5cm (c) Sacing = 5cm, measured oint=5cm (b) Sacing = 5cm, measured oint=35cm (d) Sacing = 5cm, measured oint=35cm (e) Sacing = 5cm, measured oint=5cm (f) Sacing = 5cm, measured oint=35cm Fig.. Comarisons of the measured soil temeratures and redictions. The siral coil heat source Q(u,t) can be formulated according to Green s function method. The detailed Green s function method is given in Aendix A. Then, for modeling a semi-infinite interval z >, a virtual siral coil heat source with the same heat rate q l and negative deth h is assumed on a symmetric sace to the boundary. An analytical solution of the siral coil source roblem can then be derived as Eq. (). The detailed rocedure of derivation is given in Aendix B. hðu; t q Z l ~Gðu; t; x r cosðxz ; y qc r sinðxz ; z ; t dz dt q l ða 3= qc ðt t 3= e Fðx;y;z aðt t e ðz z aðt t e ðzþz aðt t dz dt ð

5 S. Park et al. / Comuters and Geotechnics 9 (3) with Fðx; y; z x þ y þ r xr cosðxz yr sinðxz The analytical solution of Eq. () cause a singularity roblem in calculating /(t t ) 3/ time integral, which makes the integration difficult. Thus, the analytical solution is modified for comutational imrovement. Let A be a ositive constant. Then, by changing the variable n A=ðaðt t : rffiffiffiffiffiffi ðt t 3= e A Z a aðt t dt A ffiffiffiffi e n dn A at rffiffiffiffiffiffi r ffiffiffiffiffiffiffiffi! a A erfc A at where erfc denotes the comlementary error function: erfcðx ffiffiffi Z x e n dn Using this formula and after changing the order of integrations: q l hðu; t ða 3= qc ðt t 3= e ðzþz aðt t dz dt q l erfcða ðu; z = at ffiffiffiffiffi k A ðu; z erfcða þðu; z = at ffiffiffiffiffi dz A þ ðu; z with ffiffiffiffi A ðu; z Fðx; y; z þðzz 5. Verification and comarison e Fðx;y;z aðt t ðe ðz z aðt t Fig. shows a comarison of the test results, numerical analysis redictions, and analytical solution redictions. The results show reasonable agreements between the test results, numerical analysis redictions, and the analytical solution of siral coil source model. Relatively close agreements are shown in all the tests as shown in Fig.. However, the analytical solution of the line source model estimates a lower soil temerature rise comared to the others. On the contrary, the analytical solution of the cylindrical source model estimates a greater soil temerature rise comared to the others. ð3 ð ð5 ð ð7 The magnitude of the difference between the various models becomes changed by the injected heat flux. The magnitude of the difference between the line source model and the siral coil source model was over C, also. The magnitude of the difference between the cylindrical source model and the siral coil source model was over C even if the tests duration was short term.. Discussion.. 3-D effect of siral coil source Fig. 5 shows 3-D characteristics of the siral coil source model. The results rovide a comarison of the temerature distributions between the x side and x side (where Z = z/r o, H = kh/q l ). Because of the siral form along the deth, Fig. 5 shows an exactly oosite result (exactly -hase difference) at the different location on the same deth level. Similarly, the differences in the results show /- hase and 3/-hase differences at the y side and y side. 3-D effect of heat transfer in the ile can be well reresented by the siral coil analytical model. The thermal behavior was caused by siral coil source along the coil itch. Therefore, the siral coil analytical model can calculate the temerature variation along borehole wall and in the ground with resect to a deth more accurately than other analytical models as shown in Fig Secial case of the analytical solution When a siral coil becomes a line, which means the radius of coil r o is zero, the solution of the siral coil source model reduces to the finite line source model [] because F(x,y,z )=x + y = r in cylindrical coordinates, as described in Eq. (). hðu; t q Z h erfcð x þ y þðz z = ffiffiffiffiffi at l k x þ y þðz z erfcð x þ y þðzþz = ffiffiffiffiffi at dz x þ y þðzþz q Z h erfcð r þðz z = ffiffiffiffiffi at l k r þðz z erfcð r þðzþz = ffiffiffiffiffi at dz ð r þðzþz 5 Dimensionless deth, Z x=cos or π x=cos π..... Dimensionless temerature rise, Θ Deth (m) 3 5 Measured oint=35cm Measured oint=5cm r o =5.cm r o =.5cm r o =.cm Fig. 5. Comarison of temerature distributions between x side and x side (where Z = z/r o, H = kh/q l ). Fig.. Comarison of temerature distributions by the siral coil analytical model with various coil radius (sacing = 5 cm, measured oints = 5 cm, 35 cm).

6 S. Park et al. / Comuters and Geotechnics 9 (3) 3.3. Temerature variation with various coil radius Fig. resents temerature distributions by the siral coil analytical model with various coil radius. Under the same heat rate condition with coil radius of,.5 and 5 cm, the results were calculated by the siral coil analytical model for coil sacing 5 cm at measured oints of 5 and 35 cm. The results show that temerature rise becomes higher when coil radius becomes larger. 7. Conclusions In this study, an efficient siral coil source model and its analytical solution were develoed. Characteristics of the solution were analyzed and the analytical solution rediction results were comared with numerical analysis results imlemented in ABAQUS/ Standard and test measurements by a thermal resonse test. According to the research results, the following conclusion can be deduced. () The results of the analytical solution of the siral coil source model are in good agreement with the overall behavior. The line source model redicts underestimated results and the cylindrical source model redicts overestimated results for the siral coil GHE. The roosed analytical model seems to be an adequate and efficient rediction tool for the siral coil GHE. () The siral coil source model roerly resents 3-D effects caused by the siral form. When the radius of coil r o becomes zero, the solution of the siral coil source model reduces to Eskilson s finite line source model. According to the thermal resonse tests, the results were comared for short term erformance. It is necessary to conduct thermal erformance tests in order to verify the long term thermal behavior. Moreover, it is necessary to combine this efficient siral coil source model with other line source models in order to consider the vertical outlet ie either in order to more exactly evaluate the heat exchange rate. Acknowledgements This research was suorted by a grant from the National Research Foundation of Korea funded by the Ministry of Education, Science, Technology (No. -39) and the Construction Technology Innovation Project ( Technology Innovation E) under the Korea Institute of Construction and Transortation Technology Evaluation and Planning funded by the Ministry of Land, Transort, and Maritime Affair. Aendix A. Infinite sace Green s function The Green s function G(u,t;u,t ) reresents the resonse due to a adg þ dðu u dðt t ða In three dimensions: Gðu; t; u ; t j ðaðt t 3= e ju u =aðt t ða Using Green s function, the heat equation with sources can adh þ Qðu; t; hðu; h ðu ða3 Therefore, the solution is: Z hðu; t Gðu; t; u ; t Qðu ; t du dt R Z 3 þ Gðu; t; u ; h ðu du R 3 ða Aendix B. Derivation of the analytical solution for siral coil tye ground heat exchanger Let r o, N and h be the coil radius, number of coil turns and coil deth, resectively. The siral coil heat source can then be assumed to be: Qðu; t q l dðu sðz; for z h ðb qc where the wavenumber is x N=h, and the heat source is sðz ðr cosðxz; r sinðxz; z. Also the length of the siral coil is: js ðzjdz qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x r þ dz h x r þ The heat conduction roblem adh þ Qðu; for z > hðu; for z > hðu; t for z ðb ðb3 Because of the semi-infinite interval z >, the Green s function should be relaced by: ~Gðu; t; u ; t Gðu; t; u ; t Gðu; t; u ; t where u is a reflection in the z-axis: ðb u ðx ; y ; z for u ðx ; y ; z ðb5 The solution turns out to be: hðu; t Z Z Z By definition of the heat source: hðu; t q l qc with Z r sinðxz ; z ; t dz dt q l ða 3= qc ~Gðu; t; u ; t Qðu ; t dx dy dz dt ~Gðu; t; x r cosðxz ; y ðt t 3= e Fðx;y;z aðt t e ðz z aðt t e ðzþz aðt t dz dt Fðx; y; z x þ y þ r o xr cosðxz yr sinðxz References ðb ðb7 ðb [] Cui P, Li X, Man Y, Fang Z. Heat transfer analysis of ile geothermal heat exchangers with siral coils. Al Energy ;:3 9. [] Man Y, Yang H, Diao N, Cji P, Liu L, Fang Z. Develoment of siral heat source model for novel ile ground heat exchangers. HVAC R Res ;7(): 75. [3] Carslaw H, Jaeger J. Conduction of heat in solids. Oxford; 97. [] Ingersoll L, Zobel O, Ingersoll A. Heat conduction with engineering geological and other alications. New York: McGraw-Hill; 95. [5] Ingersoll L, Plass H. Theory of the ground ie heat source for the heat um. Piing Air Cond 9;:9. [] Bose J. Design and installations standards. Stillwater: International Ground Source Heat Pum Association; 99.

7 S. Park et al. / Comuters and Geotechnics 9 (3) [7] Baudoin A. Stockage intersaisonnier de chaleur dans le sol ar batterie d echangeurs baionnette verticaux: modele de redimensionnement. Ph.D. Thesis. France: Universite de Reims; 9. [] Eskilson. Thermal analysis of heat extraction boreholes. Ph.D. thesis. Sweden: University of Lund; 97. [9] Lamarche L, Beaucham B. A new contribution to the finite line-source model for geothermal boreholes. Energy Build 7;39: 9. [] Bandos T, Montero A, Fernandez E, Santander J, Isidro J, Perez J, et al. Finite line-source model for borehole heat exchangers: effect of vertical temerature variations. Geothermics 9;3:3 7. [] Man Y, Yang H, Diao N, Liu J, Fang Z. A new model and analytical solutions for borehole and ile ground heat exchangers. Int J Heat Mass Transf ;53:593. [] Yoon S, Park SK, Park HK, Go GH, Lee SR. Evaluation of heat transfer characteristics in double-layered and single-layered soils. Korea Soc Geotherm Energy Engrs ;7():3 5 [in Korean]. [3] ABAQUS Inc. ABAQUS user s manual Ver..5, Rhode Island;.

An equivalent diameter in calculating borehole thermal resistance for spiral coil type GHE

The 01 World Congress on Advances in Civil, Environmental, and Materials esearch (ACEM 1) Seoul, Korea, August 6-30, 01 An equivalent diameter in calculating borehole thermal resistance for siral coil