Energy 38 (2012) 255e263 Contents lists available at SciVerse ScienceDirect Energy journal homepage: www.elsevier.com/locate/energy New temperature response functions (G functions) for pile and borehole ground heat exchangers based on composite-medium line-source theory Min Li, Alvin C.K. Lai* Department of Civil and Architectural Engineering, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong, China a r t i c l e i n f o a b s t r a c t Article history: Received 19 September 2011 Received in revised form 1 December 2011 Accepted 3 December 2011 Available online 26 January 2012 This paper presents a new approach to modeling of heat transfer by ground heat exchangers, involving unsteady heat conduction in composite media together with complex geometry. Analytical solutions for continuous line and cylindrical-surface sources are developed based on Jaeger’s instantaneous linesource solution for composite media. New temperature response functions (G functions) are also presented for pile ground heat exchangers with spiral coils and for borehole ground heat exchangers with single or double U-shaped tubes. These temperature response functions can be used to analyze the impact of difference between properties of materials inside and outside boreholes or piles on the performance of ground heat exchangers. Theoretical results show that the difference in properties is an important factor affecting the temperature response of ground heat exchangers. Since the heat capacity of grout, inside boreholes, is fully considered by this line-source theory, the new G functions should be particularly suitable for predicting small-time dynamic behaviors of ground heat exchangers. Therefore, these G functions may be significant for energy analysis and in-situ thermal response tests of groundcoupled heat pump systems. Ó 2011 Elsevier Ltd. All rights reserved. Keywords: Ground heat exchanger Energy pile Temperature response function Ground-coupled heat pump system Composite medium 1. Introduction Ground-coupled heat pump systems (GCHPs) use the ground as a heat source and sink as they transfer heat through buried ground heat exchangers (GHEs). Compared to ambient air, the ground provides low temperature for cooling and high temperature for heating, and the temperature fluctuation is low. Therefore, GCHPs offer an environment-friendly and energy-efficient way of providing cooling and heating, as well as hot water. Because of its economic benefits, installation of GCHP systems in both residential and commercial buildings has become increasingly commonplace in many countries [1e7]. However, the promotion of GCHPs is impeded by relatively high initial cost, large land area requirement, and difficulties and uncertainty of computation and design of GHEs. Much research interest exists in resolving these practical problems. To improve the reliability of design and simulation of GCHEs, many researchers have investigated heat transfer processes associated with GHEs. A borehole GHE consists of one or two U-shaped tubes inserted into a vertical borehole and connected to a heat pump to form a closed loop (Fig. 1), and water with or without antifreeze is circulated in the closed loop. The space between the * Corresponding author. Tel.: þ852 3442 6299; fax: þ852 2788 7612. E-mail addresses: [email protected] (M. Li), [email protected] (A.C.K. Lai). 0360-5442/$ e see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.energy.2011.12.004 borehole wall and the U-tubes is filled with grout to enhance the heat transfer between soil and the circulated fluid. When analyzing the heat transfer processes, the borehole GHE is theoretically separated into two parts with the radius of the borehole. Typically, the radius of the borehole used for GHE is about 0.06 m, which is very small compared to its length, which is tens of meters. So, heat conduction outside a borehole may be modeled as a line of heat sources of infinite or finite length liberating heat to the surrounding soil [8,9]. This process can also be treated as unsteady heat conduction in an infinite region bounded internally by a circular cylinder [10,11]. In contrast, heat transfer inside boreholes is generally assumed to be in steady-state due to the much smaller dimensions and heat capacity. Several models with varying complexity were developed for this purpose; for example, the simplest one-dimensional model [12], two-dimensional models [13], and a quasi-three-dimensional model [14]. But the steadystate assumption may cause somewhat errors when modeling short-time temperature responses of borehole GHEs, in which the heat capacity of grout affects the heat transfer process. Modeling short-time responses of GHEs is significant and important for analysis of energy consumption of GCHPs, design of hybrid GSHPs, and in-situ thermal response tests of GHEs [15,16]. A great challenge, however, occurs when analyzing this short-time responses. The challenge of this problem is solving the unsteady 256 M. Li, A.C.K. Lai / Energy 38 (2012) 255e263 Fig. 1. Schematic layout of Borehole GHE with a single or double U-shaped tubes. heat conduction in composite solids with complex geometric arrangements. One way to solve this problem is numerical methods [17,18]. But numerical methods are computationally expensive and lack enough flexibility to model various forms of heat transfer tubes. Another way is to simplify the geometric arrangements, for example, by modeling U-shaped tubes as a pipe with an “equivalent” diameter. This assumption reduces the original problem to a problem of transient heat conduction in a hollow cylindrical composite region. This problem has been solved by several researchers using the generalized orthogonal expansion technique [19] or the method of Laplace transform [20,21]. Better ways to model the short-time response of borehole GHEs are still lacking. Another practical problem of GCHPs is that large land area is required for installing GHEs. To reduce this requirement, foundation piles of buildings may be used for installing heat transfer tubes, which are called “energy piles” or pile ground heat exchangers [4,22]. In energy piles, heat transfer tubes are arranged near the steel frame of foundation piles in various forms (Fig. 2). These forms usually have large heat transfer areas compared to single or double U-shaped tubes used in borehole GHEs, though they cause complications in analysis of relevant heat transfer processes. Since energy Fig. 2. Schematic layout of Energy piles with a spiral coils and a W-shaped tube. M. Li, A.C.K. Lai / Energy 38 (2012) 255e263 piles can reduce the requirement of land area, this technology has evoked increasing interest in research community [23e28]. A solid cylindrical-source and a ring-coil-source models have been proposed for pile GHEs with spiral coils [26,27], assuming that heat is released from the coil pipe wall and ignoring the difference between properties of soil and foundation piles. The ring-coil-source model treats the spiral coil as a series of separated rings liberating heat in an infinite or semi-infinite homogenous medium [26]. The solid cylindrical-source model models spiral coil as a cylindrical surface releasing heat in the ground [27]. Similarly, the traditional line-source theory has also been used to evaluate the performance of energy piles [28], ignoring the difference between thermal properties of piles and soil. So far, there are two knowledge gaps that need to be closed in the field of modeling heat transfer by GHEs. The first is that no reliable analytical solution is available for modeling the short-term unsteady behaviors of borehole GHEs with geometric complexity inside boreholes. The traditional heat source models ignore the difference between properties of soil and piles. Soil composition varies widely not only with locations but also from wet clay to sandy soil. So, heat conduction associated with an energy pile definitely occurs in composite media. Therefore, the second gap is that no analytical model can explicitly consider the difference between properties of soil and piles. This paper aims to fill these two gaps by introducing the line-source theory for composite media to GCHP applications. The advantages of this theory are that it can be used to analyze transient heat conduction inside and outside boreholes, and it includes the effect of the difference between properties of soil and piles/grout. Further, it is flexible enough to model many configurations of heat transfer tubes, including single and double U-shaped tubes, W-shaped tubes and spiral coils, etc. In this paper, we first introduce Jaeger’s solution for an instantaneous line-source in composite solids. Next, based on the Jaeger’s solution we develop the continuous line and cylindrical-surface source solutions. Then, these solutions are used to obtain temperature distributions around two borehole and two pile GHEs. After that, new temperature response functions for pile and borehole GHEs are presented, which is applicable not only for short-time scale but also medium time scale. Finally, a preliminary quantitative analysis of the influence of difference between properties of grout (or pile) and surrounding soil is presented. 2. The infinite line-source theory in composite solids Kelvin’s theory of heat sources and sinks has been proved most useful in solving the problem of heat conduction [10]. Starting from fundamental solutions of instantaneous point sources, one can obtain solutions for continuous sources or line, plane, and volume sources by integrating the fundamental solutions with respect to time or appropriate space variables. The fundamental solution used in this work has been developed by Jaeger [29], which is the infinite instantaneous line-source solution in composite cylindrical media. 2.1. Instantaneous line-source in composite media In a composite medium, heat conduction due to an instantaneous line-source of infinite length is solved by the following idea [29]. Consider an infinite composite solid, expressed in cylindrical coordinates, region r < rb of which is of one medium and region r > rb of another. There is an infinite line-source of strength q0I (J/m) instantaneously releasing heat into the composite solid at zero time. The line-source is parallel to the height direction, z-axis, and is located through a point (r0, q0 ). The key step to resolve this problem is application of the addition theorem for the modified Bessel function of the second kind of 257 zeros order. This theorem makes this problem solvable in cylindrical coordinates by the Laplace transform method. The final result for the case of r0 < rb is [29]: T1 ðt;r; qÞ ¼ þN X q0I 0 cos n q q 2r1 c1 p n¼N ZþN 0 T2 ðt; r; qÞ ¼ J ðurÞJ ður 0 Þ40 g 0 j0 f 0 n n exp a1 u2 t udu (1a) 402 þ j02 þN q0I a1 X 0 cos n q q 2 p rb n ¼ N ZþN 0 J ður0 Þj0 J ðaurÞ 40 Y ðaurÞ n n n exp a1 u2 t du(1b) 402 þ j02 where 40 ¼ k1 akJn ðrb uÞJn0 ðarb uÞ Jn0 ðrb uÞJn ðarb uÞ j0 ¼ k1 akJn ðrb uÞYn0 ðarb uÞ Jn0 ðrb uÞYn ðarb uÞ (2a) (2b) f 0 ¼ k1 akYn ðrb uÞJn0 ðarb uÞ Yn0 ðrb uÞJn ðarb uÞ (2c) g0 ¼ k1 akYn ðrb uÞYn0 ðarb uÞ Yn0 ðrb uÞYn ðarb uÞ (2d) In these expressions, Jn and Yn denote the Bessel functions of the first kind and the second kind of order n; u is the integral variable (1/m); subscripts 1 and 2 denote regions r < rb and r > rb; k, r, and c are thermal conductivity, density and specific heat, respectively, of the media; pffiffiffiffiffiffiffiffiffiffiffiffiffiand a and k are dimensionless variables k ¼ k2/k1, a ¼ a1 =a2 , where a1 and a2 are thermal diffusivities of the composite solid. This solution provides a good basis for developing solutions for other heat sources. 2.2. Continuous line-source in composite media If a line-source in a composite medium liberates heat continuously at a constant rate ql per unit time and per unit length (W/m), the temperature response can be easily obtained by integrating the instantaneous line-source with respect to time t0 : þN X ql 0 cos n q q 2r1 c1 p n ¼ N Z t ZþN exp a1 u2 ðt t 0 Þ Jn ðurÞJn ður 0 Þ 40 g 0 j0 f 0 Tl;1 ðt; r; qÞ ¼ 0 402 þ j02 0 ududt 0 (3a) þN ql a1 X 0 cosn q q 2 p rb n¼N Z t ZþN exp a1 u2 ðtt 0 Þ Jn ður0 Þ j0 Jn ðaurÞ 40 Yn ðaurÞ Tl;2 ðt;r; qÞ¼ 0 402 þ j02 0 dudt 0 (3b) Performing the integration with respect to t0 , we reduce Eqs. (3a) and (3b) to Ql;1 ðFo; R; qÞ ¼ þN X n ¼ N 0 cos n q q ZþNh i 1 exp y2 Fo 0 Jn ðyRÞJn ðyR0 Þð4g jf Þ dy y 42 þ j2 (4a) 258 M. Li, A.C.K. Lai / Energy 38 (2012) 255e263 Ql;2 ðFo; R; qÞ ¼ 2 þN X p n ¼ N 0 cos n q q ZþNh 1 exp y Fo 2 i Qs;1 ðFo; R; R0 Þ ¼ 0 0 Jn ðyR0 Þ½jJn ðayRÞ 4Yn ðayRÞ dy y2 42 þ j2 ZþNh (4b) 0 Qs;2 ðFo; R; R Þ ¼ Here, we use dimensionless variables Q(Fo, R, q) ¼ 2pk1T/ql, Fo ¼ a1 t=rb2 , R ¼ r/rb, R0 ¼ r0 /rb, y ¼ urb; and definitions (2a)e(2d) become: 2 p i J ðyRÞJ ðyR0 Þð4g jf Þ 0 0 1 exp y2 Fo dy y 42 þ j2 ZþNh 1 exp y Fo 2 i (9a) 0 J0 ðyR0 Þ½jJ0 ðayRÞ 4Y0 ðayRÞ dy y2 42 þ j2 (9b) 4 ¼ akJn ðyÞJn0 ðayÞ Jn0 ðyÞJn ðayÞ (5a) In Eqs. (8,9), values of n in defining Eqs. (5a)e(5d) are equal to 0. j ¼ akJn ðyÞYn0 ðayÞ Jn0 ðyÞYn ðayÞ (5b) 3. Application to borehole and pile GHEs f ¼ akYn ðyÞJn0 ðayÞ Yn0 ðyÞJn ðayÞ (5c) g ¼ akYn ðyÞYn0 ðayÞ Yn0 ðyÞYn ðayÞ (5d) The operation of GCHPs spans tens of years but no analytical model is available for all time scales. In this section, we apply the line-source theory for composite media to analyze heat transfer processes of GHEs and examine its performance in different time scales. Solution (4) gives the temperature distribution in a composite medium, where region r < rb is of one medium and r > rb of another. This solution can be applied to GHE situations as the grout inside borehole GHEs is different from soil or rocks outside boreholes and the material of foundation piles is also different from the surrounding soil or rocks. It should be noted that the common heat source models ignore these differences in properties. 2.3. Continuous cylindrical-surface source in composite media In energy piles with spiral coils (Fig. 2), one may model the GHE as a cylindrical-surface heat source [27]. A cylindrical-surface source of radius r0 (r0 < rb) can be imagined as many line-sources of strength qldq0 /2p are distributed around the circle of radius r0, which is also true for the case of composite media. Therefore, solutions for a continuous cylindrical surface source in composite media can be obtained by summing or integrating the distributed line-source solutions with regard to the azimuth q0 : Ts;i ¼ 1 2p Z2p 0 Tl;i ðt; r; qÞdq ; i ¼ 1; 2 (6) 0 Here, subscript s denotes surface source. Because we have the following relation for integer n: Z2p 0 0 cos n q q dq ¼ 2p 0 n ¼ 0 ns0 (7) 0 the infinite series in Eq. (4) only leave the term with n ¼ 0, and Eq. (6) finally reduces to q Ts;1 ðt; rÞ ¼ l 2pk1 ZþNh i 1 exp a1 u2 t 0 J0 ðurÞJ0 ður0 Þ 40 g0 j0 f 0 du u 402 þ j02 q Ts;2 ðt; rÞ ¼ 2 l p rb (8a) ZþNh i 1 exp a1 u2 t 3.1. Scale analysis All analytical models for GHEs make some assumptions relevant to time scales of heat transfer by GHEs and, therefore, it is necessary to analyze these time scales before evaluating the performance of the new method. To the best of our knowledge, these time scales have never been explicitly analyzed before. Using the heat conduction equation, the time scale, when unsteady effect is still dominant, can be estimated by: twðDrÞ2 =a1 (10) where t denotes the estimated time scale, Dr is the chosen space range and a1 denotes thermal diffusivity of the medium. Although the above scale relation produces only the order-of-magnitude estimates for the time scale of interest, these results are thought to provide exact results within a factor of order one [30]. Therefore, the transient effect may be ignored when the time period is equal to or larger than (5e10) (Dr)2/a1. Now we examine three time scales relevant to three space ranges: radius of the heat transfer pipe, radii of boreholes or piles, and one half of length of GHEs. The order-of-magnitude of these three space ranges are about 0.013 m for inner radii of pipes, 0.06 m for borehole radii or 0.3 m for radii of piles, and 20e50 m for one half of length of boreholes. Based on one half of length of boreholes, the scale analysis can give time scales used to determine whether the ground surface affects the heat transfer processes. The order-ofmagnitude of thermal diffusivities is 106 m2/s in GSHP applications. According to these data, the time scale when the effect of heat capacity inside a borehole is dominant is t w 1 h. This time scale is equal to that of energy simulation and variation of cooling loads. For a foundation pile of radius 0.3 m, the corresponding time equals 25 h. These scales imply that the transient responses inside boreholes or piles are important for hourly energy analysis of GSHPs. One half of length of the borehole gives a time scale of at least 10 years, implying that the impact of ground surface is negligible during the first several years. In the space of heat transfer tubes (Ushaped or spiral tubes), expression (10) gives an estimation of time, 3 min. These time periods are useful when evaluating the performance of the following new models. 0 J0 ður 0 Þ j0 J0 ðaurÞ 40 Y0 ðaurÞ du u2 402 þ j02 or in dimensionless form 3.2. Temperature distributions of borehole and pile GHEs (8b) Fig. 3 shows the layout of line and surface sources for modeling heat transfer by borehole and pile GHEs, where line and surface M. Li, A.C.K. Lai / Energy 38 (2012) 255e263 sources are there, in place of tubes. Points A and B in Fig. 3 correspond to those in Figs. 1 and 2. These two points are used for developing temperature response functions. It is important to analyze the temperature distribution around a GHE. Fig. 4 shows some contours of temperatures near energy piles and boreholes, corresponding to the arrangements of Fig. 3. These temperature fields are obtained by Eq. (4) or (9) with prescribed positions of heat sources. Each line-source is assumed to be of the same strength. Obviously, temperature fields around boreholes (Fig. 4a and b) are symmetric because of symmetrical arrangements of U-tubes or line-sources. The temperature around the energy pile with a spirally bound tube is a one-dimensional distribution in a cylindrical coordinate system (Fig. 4c) because we adopted the solid cylindrical surface source model. Fig. 4d shows an example of an energy pile with a W-tube, showing the line-source theory has the flexibility for dealing with any arbitrary arrangement of heat transfer pipes. As shown below, when developing temperature response functions of GHEs we are only interested in average temperature at walls of U-shaped pipes or heat transfer coils. These temperature contours clearly indicate that temperatures at the pipe walls vary, and temperatures at positions A and B (as labeled in Fig. 3) should be the maximum and minimum of temperatures at pipe walls for these symmetrical arrangements. Therefore, arithmetic mean values of temperatures at A and B can approximate average temperature of pipe walls with good accuracy. Based upon this observation, we derive response functions of GHEs as follows. 3.3. Temperature response functions When deriving the line-source solutions we assumed the strength of line or surface sources to be constant. In fact, loads of GCHPs vary continuously due to varied cooling or heating loads of buildings. In such situations, solutions can be obtained by the Duhamel’s theorem [10,31], or the principle of superposition, a method that uses solutions to cases of constant load ql. Now, we assume that the load of GHEs is a function of time ql(t). According to Duhamel’s theorem, the temperature, T1, say, reads T1 ðr; q; tÞ ¼ T1;0 þ 259 Zt ql ðsÞ 0 vGðr; q; t sÞ ds vt (11) where T1,0 is the initial temperature of the ground, and G, called G function in GSHP literature, is the temperature response function. In this context, physical significance of the G function represents temperature response in the composite media due to a unit-step change in GHEs loads ql. Since heating and cooling loads of buildings are commonly expressed in step-wise constant values (on hourly, daily or other bases), Eq. (11) further reduces to [31] T1 ðr; q; tÞ ¼ T1;0 þ N1 X Dql;j Gðr; q; t jDtÞ (12) j¼0 The G function is generally evaluated at borehole walls r ¼ rb, representing the average temperature response at a bore wall. Thus, the temperature of circulating fluid Tf of GHE loops is obtained by Tf ¼ Tb þ ql Re;b (13) where Re,b is the effective fluid-to-ground thermal resistance, and Tb is the average temperature at a borehole wall obtained by substituting coordinates of the borehole wall into Eq. (12). Thermal resistance Re,b is derived traditionally by assuming that heat conduction inside a borehole is steady-state. 3.3.1. New response functions for GHEs with a U-tube Line-source models for composite solids can do without the steady-state assumption for Re,b and model the U-shaped pipe inside a borehole as two line-sources in a composite medium. In terms of definition of the G function, Eq. (11), the G function for single U-tube should be the temperature response of two line heat sources of strength 1/2 at (R0 , 0) and (R0 , p). More importantly, by using the new solutions, the G function can be evaluated at the Upipe wall, and not at the borehole wall, representing the average temperature response at the pipe wall. For simplicity, average value of temperature responses at points A and B at the wall of U-shaped tube is used to evaluate the G function (based on the observation in Fig. 4a): Fig. 3. Schematic layout of heat sources for pile and borehole GHEs. 260 M. Li, A.C.K. Lai / Energy 38 (2012) 255e263 Fig. 4. (a) Borehole with single U-pipe. (b) Borehole with double U-pipes. (c) Pile with a spiral pipe. (d). Pile with a W-tube. G Fo; Rp ; R0 ¼ 1 2pk1 þN X m ¼ N ZþNh i 1 exp y2 Fo 0 ðyR0 Þð4g J2m ðyRA Þ þ J2m ðyRB Þ J2m jf Þ dy (14) 2 y 4 2 þ j2 In Eq. (14), RA and RB are dimensionless radial coordinates of points A and B; values of n in definitions Eqs. (5a)e(5d) equal 2m. function is also evaluated at points A and B, because of the symmetry: G Fo; Rp ; R0 ¼ þN þN Z h i X 1 1 exp y2 Fo 2pk1 l ¼ N 0 J ðyRA Þ þ J4l ðyRB Þ J4l ðyR0 Þð4g jf Þ dy 4l 2 y 42 þ j2 (15) In Eq. (15), values of n in definitions Eqs. (5a)e(5d) equal 4l. 3.3.2. New response functions for GHEs with double U-tubes Similarly, the response function G for a double U-tube can be obtained by assuming four line-sources of strength 1/4, located at positions (R0 , 0), (R0 , p), (R0 , p/2) and (R0 , 3p/2), and the G 3.3.3. New response functions for energy piles with spiral tubes In this case, the temperature field is one-dimensional, as shown in Fig. 4c. The average value of temperatures at points A and B can M. Li, A.C.K. Lai / Energy 38 (2012) 255e263 also approximate the average temperature response at tube walls with enough accuracy. Therefore, the G function for energy piles with spirally arranged tubes is G Fo; Rp ; R0 ¼ 1 2pk1 ZþNh i 1 exp y2 Fo 0 J ðyRA Þ þ J0 ðyRB Þ J0 ðyR0 Þð4g jf Þ dy 0 2 y 42 þ j2 (16) In Eq. (16), values of n in definitions Eqs. (5a)e(5d) equal 0. The three new G functions provide a new way of modeling the heat transfer by pile or borehole GHEs. The G functions can be used to determine the temperature of the circulating fluid by the following expression: Tf Tp ¼ ql Rp (17) Here, Tp denotes the average temperature at pipe walls obtained by substituting the new G functions into Eq. (12). Thermal resistance of pipe Rp is evaluated by Rp ¼ 1 ro ln 2pkp ri 261 used for predicting the behaviors of GHEs during short-time periods, except the very short periods (15e30 min) and that the new method should be more reasonable for short-time responses analysis than traditional ones that ignore all heat capacities inside a borehole (including those of U-shaped pipes and grout). The composite-medium G functions are derived from the theory of line-sources of infinite length, so they suffer from the common limit of infinite line-sources, i.e. they ignore the influence of the ground surface and are unable to predict long-term behaviors of GHEs in some situations. If cooling and heating loads balance each other, the ground surface has no influence on long-term behaviors of GHEs, and our method can model both short-term and long-term heat transfer by GHEs. Contrarily, if there is an imbalance between cooling and heating loads, the redundant heat due to unequal loads is accumulated in the ground, and the influence of the ground surface becomes significant after several years’ operations. The scale analysis suggests that the span of this time is ten years. As the time approaches theoretical infinity, the temperature field reaches a steady-state, in which redundant heat is transferred to ambient air through the ground surface. The models of infinite line-sources, however, ignore the ground surface, resulting in the calculated (18) where kp, ro, ri denote thermal conductivity, outer radius and inner radius of pipes, respectively. This modeling approach is different from traditional ones in several aspects, and its performance is further explored in the following subsection. 3.3.4. Analysis of G functions Before analyzing performance of the new G functions, the new line-source model was validated against the conventional infinite line-source solution for a homogenous medium. Table 1 indicates that when a medium is homogenous, the composite-medium linesource solution indeed approaches to that predicted by the conventional model, which has been widely used in design and simulation of GCHPs and has been partly validated by real engineering practice. An important aspect of heat transfer models for GHEs is applicability in different time scales. First of all, heat capacity of grout or concrete piles can be fully considered by the new models, replacing heat-transfer tubes by composite-medium heat sources. This replacement assumes that the convective heat transfer inside the tubes is replaced by heat conduction. So, the new line-source models is unsuitable for very small times when transient effect is dominant within the region of heat-transfer tubes. Heat transfer tubes have much small geometric dimensions and heat capacity compared to grout and boreholes. The scale analysis indicates that the unsteady effect within tubes is significant only for a few minutes and can be ignored when time scale is equal to or larger than 15e30 min. Therefore, we infer that the new method can be Table 1 Comparison of infinite line-source solutions with a ¼ 1 and k ¼ 1. An infinite linesource is parallel to the z-axis, through original point, and liberates heat at the rate of ql per unit time per unit length. Fourier number, Fo 1 10 100 1000 10,000 Dimensionless temperature 4pk1T/ql, at r ¼ rb Common line-source solution Line-source solution Eq. (4a) 1.04428 3.13651 5.41675 7.71708 10.0194 1.04422 3.13644 5.41668 7.71702 10.0194 Fig. 5. The variation of the dimensionless G functions of borehole GHEs with Fourier number and a (a) or k (b). 262 M. Li, A.C.K. Lai / Energy 38 (2012) 255e263 temperatures increasing continuously and never reaching steadystates, as shown in Figs. 5 and 6. Although infinite line-source models cannot predict the long-term performance of GHEs, in the case of loads imbalance, the scale analysis demonstrates that these models still can predict the performance of GHEs in medium times, for example, over several years, depending on the length of GHEs. Another outstanding characteristic of the new models is that they can explicitly analyze the influence of the difference between thermal properties of soil and grout (or foundation piles). The influences of the differences in heat capacity on G functions are summarized in Fig. 5 for boreholes with U-pipes and in Fig. 6 for energy piles with spiral coils. In these figures, the differences between G functions along horizontal axes denote phase differences in temperature responses, and those along vertical axes are amplitude differences. According to the definition of the variables, k ¼ k2/k1 and a ¼ ða1 =a2 Þ1=2 , fixing k at 1 while increasing a can be understood as increasing heat capacity of soil, and fixing a at 1 while increasing k can be understood as increasing thermal conductivity or heat capacity of soil. Variations of G function with these two situations are shown in Figs. 5 and 6. These diagrams illustrate that the temperature magnitude decreases with increasing heat capacity of soil. Inferring from the figures, it can also be seen that the temperature response delays with heat capability of soil. The decrease and time-delay become significant for large values of time as the influence of soil outside boreholes or piles becomes more involved as time increases. In our cases, the impact of difference in properties is insignificant and may be ignored when Fo < 0.3, but when Fo > 0.3, the differences become significant and influence the temperature responses greatly. This observation suggests that ignoring differences in thermal properties, associated with GHEs, may induce considerable errors, including phase errors and amplitude errors. 4. Conclusions This paper presents a novel approach to modeling heat conduction by pile and borehole GHEs which uses the line-source solution to heat conduction in composite media. New solutions for continuous line and cylindrical surface sources are derived based on Jaeger’s instantaneous line-source solution. Several important conclusions are drawn, as follows: 1) Based on the new line-source solutions, this work develops new temperature response functions (G functions) for borehole GHEs with U-tubes and for energy piles with spiral coils, which provide a new way of designing GHEs and analyzing energy consumption of GCHPs. 2) The new G functions can be used to model the performance of GHEs in different time scales from an hour to several years. This characteristic is remarkable for annual energy analysis of GCHPs. 3) The property difference between the media inside and outside boreholes or piles is an important factor that should be considered when analyzing heat transfer by GHEs. Ignoring this property difference may induce considerable phase and amplitude errors. References Fig. 6. The variation of the dimensionless G functions of pile GHEs with Fourier number and a (a) or k (b). [1] Florides GA, Pouloupatis PD, Kalogirou S, Messaritis V, Panayides I, Zomeni Z, et al. The geothermal characteristics of the ground and the potential of using ground coupled heat pumps in Cyprus. Energy 2011;36:5027e36. [2] Yang W, Zhou J, Xu W, Zhang GQ. Current status of ground-source heat pumps in China. Energy Policy 2010;38:323e32. [3] Yang HX, Cui P, Fang ZH. Vertical-borehole ground-coupled heat pumps: a review of models and systems. Appl Energy 2010;87:16e27. [4] Moel M, Bach PM, Bouazza A, Singh RM, Sun JLO. Technological advances and applications of geothermal energy pile foundations and their feasibility in Australia. Renew Sust Energy Rev 2010;14:2683e96. [5] Blum P, Campillo G, Koelbel T. Techno-economic and spatial analysis of vertical ground source heat pump systems in Germany. Energy 2011;36: 3002e11. [6] Bakirci K, Ozyurt O, Comakli K, Comakli O. Energy analysis of a solar-ground source heat pump system with vertical closed-loop for heating applications. Energy 2011;36:3224e32. [7] Yu X, Wang RZ, Zhai XQ. Year round experimental study on a constant temperature and humidity air-conditioning system driven by ground source heat pump. Energy 2011;36:1309e18. [8] Ingersoll LR, Plass HJ. Theory of the ground pipe source for the heat pump. ASHRAE Trans 1948;54:339e48. [9] Zeng HY, Diao NR, Fang ZH. A finite line-source model for boreholes in geothermal heat exchangers. Heat Transf Asian Res 2002;31:558e67. [10] Carslaw HS, Jaeger JC. Conduction of heat in solids. 2nd ed. Oxford: Claremore Press; 1959. [11] Deerman JD, Kavanaugh SP. Simulation of vertical U-tube ground-coupled heat pump systems using the cylindrical heat source solution. ASHRAE Trans 1991;97:287e95. [12] Gu Y, O’Neal DL. Development of an equivalent diameter expression for vertical U-Tubes used in ground-coupled heat pumps. ASHRAE Trans 1998; 104:347e55. [13] Hellstrom G. Ground heat storage: thermal analyses of duct storage systems. Lund: Department of Mathematical Physics University of Lund; 1991. M. Li, A.C.K. Lai / Energy 38 (2012) 255e263 [14] Zeng HY, Diao NR, Fang ZH. Heat transfer analysis of boreholes in vertical ground heat exchangers. Int J Heat Mass Transf 2003;46:4467e81. [15] Yavuzturk C, Spitler JD. A short time step response factor model for vertical ground loop heat exchangers. ASHRAE Trans 1999;105:475e85. [16] Li M, Lai ACK. Parameter estimation of in-situ thermal response tests for borehole ground heat exchangers. Int J Heat Mass Transf (Accepted). [17] Bozzoli F, Pagliarni G, Rainieri S, Schiavi L. Estimation of soil and grout thermal properties through a TSPEP (two-step parameter estimation procedure) applied to TRT (thermal response test) data. Energy 2011;36:839e46. [18] Kim SK, Bae GO, Lee KK, Song Y. Field-scale evaluation of the design of borehole heat exchangers for the use of shallow geothermal energy. Energy 2010;35:491e500. [19] Gu Y, O’Neal DL. An analytical solution to transient heat conduction in a composite region with a cylindrical heat source. J Sol Energy Eng TransASME 1995;117:242e8. [20] Lamarche L, Beauchamp B. New solutions for the short-time analysis of geothermal vertical boreholes. Int J Heat Mass Transf 2007;50:1408e19. [21] Bandyopadhyay G, Gosnold W, Mann M. Analytical and semi-analytical solutions for short-time transient response of ground heat exchangers. Energy Build 2008;40:1816e24. [22] Hamada Y, Saitoh H, Nakamura M, Kubota H, Ochifuji K. Field performance of an energy pile system for space heating. Energy Build 2007;39:517e24. 263 [23] Gao J, Zhang X, Liu J, Li K, Yang J. Thermal performance and ground temperature of vertical pile-foundation heat exchangers: a case study. Appl Therm Eng 2008;28:2295e304. [24] Wood CJ, Liu H, Riffat SB. An investigation of the heat pump performance and ground temperature of a piled foundation heat exchanger system for a residential building. Energy 2010;35:4932e40. [25] Jalaluddin, Miyara A, Tsubaki K, Inoue S, Yoshida K. Experimental study of several types of ground heat exchanger using a steel pile foundation. Renew Energy 2011;36:764e71. [26] Cui P, Li X, Man Y, Fang ZH. Heat transfer analysis of pile geothermal heat exchangers with spiral coils. Appl Energy 2011;88:4113e9. [27] Man Y, Yang HX, Diao NR, Liu JH, Fang ZH. A new model and analytical solutions for borehole and pile ground heat exchangers. Int J Heat Mass Transf 2010;53:2593e601. [28] Bozis D, Papakostas K, Kyriakis N. On the evaluation of design parameters effects on the heat transfer efficiency of energy piles. Energy Build 2011;43: 1020e9. [29] Jaeger JC. Some problems involving line sources in conduction of heat. The London, Edinburgh and Dublin Philos Mag J Sci 1944;242:169e79. [30] Bejan A. Convection heat transfer. 3rd ed. Hoboken: John Wiley & Sons, Inc; 2004. [31] Ozisik MN. Heat conduction. 2nd ed. New York: John Wiley & Sons, Inc.; 1993.
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