1、Saqib Javed is a graduate student at Chalmers University of Technology, Sweden. Johan Claesson is a professor at Chalmers University of Technology and Lund University of Technology, Sweden. New Analytical and Numerical Solutions for the Short-term Analysis of Vertical Ground Heat Exchangers Saqib Ja
2、ved, P.E. Johan Claesson, Ph.D. Student Member ASHRAE ABSTRACT This paper presents the background, development and the validation of new analytical and numerical solutions for the modeling of short-term response of borehole heat exchangers. The new analytical solution studies the boreholes heat tran
3、sfer and the related boundary conditions in the Laplace domain. A set of equations for the Laplace transforms for the boundary temperatures and heat-fluxes is obtained. These equations are represented by a thermal network. The use of the thermal network enables swift and precise evaluation of any th
4、ermal or physical setting of the borehole. Finally, very concise formulas of the inversion integrals are developed to obtain the time-dependent solutions. The new analytical solution considers the thermal capacities, the thermal resistances and the thermal properties of all the borehole elements and
5、 provides a complete solution to the radial heat transfer problem in vertical boreholes. The numerical solution uses a special coordinate transformation. The new solutions can either be used as autonomous models or easily be incorporated in any building energy simulation software. INTRODUCTION Long-
6、term response of a borehole represents the development of ground temperatures over time in response to the overall ground heat injections and extractions. On the other hand, short-term response of the borehole shows the variations in circulating fluid temperatures not associated to the long-term res
7、ponse of the ground. The short-term response of the borehole corresponds to time periods ranging from a few minutes to a number of days. Today many commercial buildings, like super markets and shopping centers, have simultaneous heating and cooling demands. Many other commercial and office buildings
8、 have a cooling demand during the day, even in cold climates, and a heating demand during the night. For such buildings, a significant amount of thermal energy is just pumped up and down the borehole system with heat transfer mainly occurring in the borehole. Similarly, the circulating fluid tempera
9、ture of a borehole system operating under peak load conditions depends largely on the internal heat transfer of the borehole. For these cases, the borehole exit fluid temperature depends on the short-term thermal response of the borehole. As operation and performance of a heat pump both depend on th
10、e fluid temperature from the borehole system, the thermal energy use and electrical demands of the heat pump and the overall ground source heat pump (GSHP) system are considerably affected by the short-term response of the borehole. Therefore, when optimizing the operation, control and performance o
11、f a GSHP system, the short term response of the borehole is quite important. The evaluation of thermal response tests (TRTs) and heat-flux build-up analysis of the borehole are also conducted using models based on the short-term response of the borehole. Various numerical, analytical and semi-analyt
12、ical solutions have been developed to model the short-term response of the borehole. Analytical solutions, because of their flexibility and superior computational time efficiencies, are of particular LV-11-C001 2011 ASHRAE 32011. American Society of Heating, Refrigerating and Air-Conditioning Engine
13、ers, Inc. (www.ashrae.org). Published in ASHRAE Transactions, Volume 117, Part 1. For personal use only. Additional reproduction, distribution, or transmission in either print or digital form is not permitted without ASHRAES prior written permission.interest; however, numerical solutions are also re
14、quired to obtain precise solutions and for parametric analysis. The first major contribution came from Yavuzturk (1999), who extended Eskilsons concept of non-dimensional temperature response functions (1987) to include the short-term analysis. As with Eskilsons approach, the work of Yavuzturk also
15、requires the response to be pre-computed for individual cases. Shonder and Beck (1999) and Austin (1998) also developed solutions which numerically solve the heat transfer in the borehole. However, both these solutions are aimed at evaluation of TRTs. Young (2004) modified the classical Buried Elect
16、ric Cable (BEC) solution (Carslaw and Jaeger, 1959) to develop his analytical Borehole Fluid Thermal Mass (BFTM) solution. The solution is based on an analogy between a buried electric cable and a vertical ground heat exchanger and needs to be optimized for individual cases. Lamarche and Beauchamp (
17、2007) presented an exact solution assuming two legs of the U-tube as a hollow equivalent-diameter cylinder. The solution solves the heat transfer problem assuming a steady heat-flux condition across the hollow cylinder boundary. However, it ignores the thermal capacity of the fluid present in the U-
18、tube. Bandyopadhyay et al. (2008) also presented an exact solution for the case of boreholes backfilled with the borehole cuttings. This solution has limited practical application as most of the boreholes are backfilled with a material quite different from the borehole cuttings. Gu and ONeal (1995)
19、developed an analytical short-term response solution assuming a cylindrical source in an infinite composite region. The solution solves the borehole transient heat transfer problem using the generalized orthogonal expansion technique which requires calculation of multiple eigenvalues. Beier and Smit
20、h (2003) also developed a semi-analytical solution, which first solves the borehole heat transfer problem in the Laplace domain and then uses a numerical inversion to obtain the time domain solution. Bandyopadhyay et al. (2008) have also reported a similar solution. Javed et al. (2010) studied the e
21、xisting short-term solutions in detail and noted the need of an analytical solution which should consider the thermal capacities, the thermal resistances and the thermal properties of all the borehole elements and which could be easily incorporated in building energy simulation software to optimize
22、the operation and control of GSHP systems. NEW ANALYTICAL SOLUTION A new analytical solution has been developed to model the short-term response of the borehole (Claesson, 2010). The new solution studies the heat transfer and the related boundary conditions in the Laplace domain. The solution assume
23、s radial heat transfer in the borehole. To meet this requirement, the U-tube in the borehole is approximated as a single equivalent-diameter pipe. The fluid temperatures entering and leaving the U-tube are represented using a single average value. The resulting problem is shown in Figure 1. The heat
24、 flux qinjis injected to the circulating fluid with temperature Tf (). The fluid has a thermal capacity of Cp. The pipe thermal resistance is Rp, and the pipe outer boundary temperature is Tp(). The heat flux qp() flows through the pipe wall to the grout. The thermal conductivity and the thermal dif
25、fusivity of the grout are gand ag, respectively. The heat flux qb() flows across the borehole boundary to the surrounding ground (soil). The borehole boundary temperature is Tb(). The thermal conductivity and the thermal diffusivity of the ground (soil) are sand as, respectively. Figure 1 Geometry,
26、temperatures, heat fluxes and thermal properties of the borehole. rrpg , agGrouts , asGround (Soil)FluidrbTb()Tp()Rpqp()qb()Tf ()CpqinjT(r,)4 ASHRAE TransactionsThe temperature T(r, ) satisfies the radial heat conduction equation in the grout and the ground (soil) regions. 1null(null)nullnullnullnul
27、l=nullnullnullnullnullnull+1nullnullnullnullnull,null(null) =nullnullnull,nullnullnullnull. (1) The radial heat flux in the grout and the soil regions is: null(null,null) = 2nullnullnull(null)nullnullnullnull,null(null)=nullnullnull,nullnullnullnull. (2) The heat flux at the grout-soil interface is
28、continuous and hence the boundary condition from Equation 2 at r=rbis: nullnullnullnullnullnullnullnullnullnullnullnullnull=nullnullnullnullnullnullnullnullnullnullnullnullnull. (3) The boundary condition at the pipe-grout interface is: nullnull(null) nullnullnullnull,nullnull=nullnullnullnullnullnu
29、ll,nullnull. (4) The heat balance of the fluid in the pipe with the injected heat qinjis: nullnullnullnull=nullnullnullnullnullnullnull+nullnullnullnull,nullnull, null0.(5) The initial temperatures in the pipe, the grout and the ground (soil) are all taken as zero. nullnull(0) =0, null(null,0) =0, n
30、ullnullnull. (6) The mathematical problem defined above (i.e. Equations 1 to 6) is solved using Laplace transforms of temperatures and heat fluxes at null=nullnull, nullnullnullnullnull, null=nullnulland nullnullnull. For null=nullnull, two relation between the Laplace transforms of temperature and
31、heat flux are readily obtained by taking Laplace transforms of Equations 4 and 5. nullnullnull(null) nullnullnull(null) =nullnullnullnullnull(null), null = nullnull. (7) nullnullnullnullnull=nullnullnullnullnullnullnull(null) 0null+nullnullnull(null), null = nullnull. (8) Here, s is the complex-valu
32、ed variable in the Laplace domain. For the annular grout region defined by nullnullnullnullnull, the Laplace transform of the radial heat equation is considered. General solutions of the resulting equation involving Bessel functions are then obtained together with the Laplace transform of the radial
33、 heat flux. From this, two equations between the Laplace transforms of the boundary temperatures and boundary fluxes are obtained. The full derivation of the formulas below is presented in Claesson (2010). nullnullnull(null) =nullnullnull(null) nullnullnullnull(null) 0null+nullnullnull(null) nullnul
34、lnullnull(null) nullnullnull(null)null, (9) nullnullnull(null) =nullnullnull(null) (nullnullnull(null) 0) +nullnullnull(null) nullnullnullnull(null) nullnullnull(null)null. (10) Here nullnullnull(null) may be interpreted as a transmittive conductance for the problem in Laplace domain, whereas nullnu
35、llnull(null) and nullnullnull(null) are absorptive conductances. The values of these conductances (and their inverse, the resistances) become: 2011 ASHRAE 5nullnullnull(null) =1nullnullnull(null)=2nullnullnullnullnullnullnullnullnullnullnull(nullnull) nullnullnullnullnullnullnullnull(nullnull), (11)
36、 nullnullnull(null) =1nullnullnull(null)=nullnullnullnullnullnullnullnullnullnullnull(nullnull) +nullnullnullnullnullnullnullnull(nullnull)null1nullnullnull(null), (12) nullnullnull(null) =1nullnullnull(null)=nullnullnullnullnull(nullnull) nullnullnullnullnullnull+nullnull(nullnull) nullnullnullnull
37、nullnullnull 1nullnullnull(null),(13) nullnull=nullnullnullnullnullnull ,nullnull=nullnullnullnullnullnull . (14) There is a corresponding relation between the Laplace transforms of the temperature and flux at null=nullnullto account for the region outside the borehole, which is derived from the Lap
38、lace transforms for the soil region. nullnullnull(null) = nullnullnull(null) (nullnullnull(null) 0), null = nullnull. (15) Here nullnullnull(null) is the ground thermal conductance and its value is: nullnullnull(null) =1nullnullnull(null)= 2nullnullnullnullnullnullnull(nullnull)nullnull(nullnull), n
39、ullnull=nullnullnullnullnullnull . (16) The heat transfer problem, shown in Figure 1, can now be represented by means of the thermal network shown in Figure 2. This network for the equivalent-diameter pipe, the circulating fluid, the borehole annulus region and the infinite ground outside the boreho
40、le is drawn using Equations 7 and 8 for the pipe region, Equations 9 and 10 for the annular region and Equation 15 for the soil region. The network involves a sequence of composite resistances. Figure 2 The thermal network for a borehole in the Laplace domain. The Laplace transform for the fluid tem
41、perature is readily obtained from the thermal network. Starting from the right in Figure 2, the conductances nullnullnull(null) and nullnullnull(null) lie in parallel and are added. The inverse of this composite conductance is added to the resistance nullnullnull(null). This composite resistance lie
42、s in parallel with nullnullnull(null) = 1 nullnullnull(null) and their inverses are added. This composite resistance lies in series with the resistance of the pipe wall nullnull. The total composite resistance from nullnulland rightwards lies in parallel with the thermal conductance nullnullnull. Th
43、e Laplace transform of the fluid temperature becomes: nullnullnullnullnull1nullnullnullnullnullnull(null) nullnullnull(null) nullnullnull(null) nullnullnull(null) nullnullnull(null) nullnullnullnullnull(null) nullnullnull(null) nullnullnull(null) nullnullnull(null) 0 0 0 0 6 ASHRAE Transactionsnulln
44、ullnull(null) =nullnullnullnullnull1nullnullnull+1nullnull+1nullnullnull(null) +1nullnullnull(null) +1nullnullnull(null) +nullnullnull(null).(17) In the type of problems considered here, the inversion formula to get nullnull(null) from nullnullnull(null) is: nullnull(null) =2nullnull1nullnullnullnul
45、l nullnullnullnullnull(null)nullnull. nullnull(18) The function L(u) in the above equation is given by: null(null) =Imnullnullnullnullnull(null)nullnull, : nullnullnull=nullnull+null 0, nullnullnullnull= null null, 0 null . (19) Here, 0is an arbitrary time constant, and Im. denotes the imaginary par
46、t. The first factor in the integral depends on the dimensionless time /0, and it is independent of the particular Laplace transform nullnullnull(null). The second factor, the function null(null), is independent of and represents the particular Laplace transform for the considered case. The inversion
47、 integral in Equation 18 is obtained by considering a closed loop in the complex s-plane. The original integral along a vertical line is replaced by an integral along the negative real axis. The following conditions have to be fulfilled: nullnull(0) =0,nullnullnullnullnull0, null.(20) The Laplace tr
48、ansform for the fluid temperature is given by Equation 17. When taken for s on the negative real axis, we get: null(null) =Imnullnullnullnullnullnullnullnullnullnull+1nullnull+1nullnullnull(null) +1nullnullnull(null) +1nullnullnull(null) +nullnullnull(null).(21) The final values of the conductances
49、(and their inverse, the resistances), when taken on the negative real axis are expressed using ordinary Bessel functions: nullnullnull(null) =1nullnullnull(null)=2nullnullnullnullnullnullnullnull(nullnullnull) nullnullnull(nullnullnull)nullnull(nullnullnull) nullnullnull(nullnullnull), (22) nullnullnull(null) = 1nullnullnull(null)=4nullnullnullnullnullnullnullnullnullnullnull(nullnullnull) nullnullnullnullnullnullnull
